lllllllllllll||||l|lllll||||||||llllllWilllllllllllllll

31293 01563

This is to certify that the

dissertation entitled

APPLICATION OF E—PULSE AND CEPSTRAL ANALYSIS
TO TARGET DETECTION AND DISCRIMINATION

presented by

Glen Stuart WalIinga

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Electrical Eng.

 

Marloraarofessor

Date I I 3

MSU is an Affirmative Action/Equal Opportunity Institution 0- 12771

 

 

 

 

 

 

 

 

 

LIBRARY

Michigan State
University

 

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DATE DUE DATE DUE DATE DUE
DEC 11 2005
9913“6

MSU|sAn.“" _" . ' '1 'ch ‘ ',inetitution

czblmWedueun 3-D- 1

 

 

APPLICATION OF E-PULSE AND CEPSTRAL ANALYSIS TO RADAR TARGET
DETECTION AND DISCRIMINATION

By

Glen Stuart Wallinga

A DISSERTATION
Submitted to ~
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1997

 

 

Copyright by
GLEN STUART WALLINGA
1997

 

ABSTRACT

APPLICATION OF E-PULSE AND CEPSTRAL ANALYSIS TO RADAR TARGET
DETECTION AND DISCRIMINATION

By

Glen Stuart Wallinga

This thesis addresses several topics related to the use of ultra-wideband radar for
target detection and discrimination. A new method to determine the scattered field from
an infinite length, perfectly-conducting, periodic sea-surface has been formulated. This
method is based on a periodic surface-current representation. The motivation for doing
this work is to create a computationally efficient method for determining the scattered
field from periodic surfaces.

An enhanced detection algorithm for radar-target detection in a sea-clutter
environment has been formulated using the E-pulse method. The theory behind this new
method is discussed, several static test cases presented, and a dynamic test case presented
showing the functionality of the detection algorithm for a target moving over an evolving
sea-surface. The effect of different target types on the detection algorithm has been
tested. Also, the effect of multipath on the detection algorithm has been investigated.
Finally, the new method has been compared to a simple detection algorithm based on
clutter reduction using coherent signal processing.

A target discrimination scheme using only the magnitude of its spectral response
has been devised based upon real cepstral analysis. Basic cepstral analysis techniques and

the minimum-phase condition are discussed. A discrimination scheme based upon

 

 

 

Lie E-p‘
credo
spree:
tic-3m
Ma:
‘2'!”3."

it. ”A 'J

mink-DI‘

the E-pulse method was used. A library of E-pulse waveforms was generated from the
time-domain scattered return of each anticipated target type. The time-domain
representation of an unknown target was generated using the minimum-phase
reconstruction method. The target discrimination algorithm was used to identify an
associated geometry in the target library file. Test cases included: a) thin-wire scattering

geometries using a theoretical scattering program, and b) actual anechoic chamber

measurements of small-scale aircraft and missiles.

 

ACKNOWLEDGMENTS

Many people deserve acknowledgement for their participation in helping me
complete this work. My foremost appreciation must go to my advisor Dr. E]. Rothwell.
Thank you for your time, patience, and guidance over the past several years. I could have
never completed this thesis without your help. A special thanks to Dr. K.M Chen who
generously invited me to work in an outstanding lab during my studies. I am also
indebted to Dr. D.P. Nyquist for his guidance and teaching. To Dr. Byron Drachman,
thank you for participating on my guidance committee and for your assistance.

I also wish to thank Dr. William Peake, whose enthusiasm for electromagnetics
motivated me to pursue this area of study.

Many people in this lab have helped me during my study here. I am especially
grateful to Adam Norman for his generous help and guidance. Also, a special thank you

to Mike Havrilla and David Infante for their company and advice.

 

 

 

 

 

Maple]

Chime:

TABLE OF CONTENTS

 

TABLE OF CONTENTS ........................................ vi
LIST OF TABLES ............................................. viii
LIST OF FIGURES ............................................ ix
Chapter 1 ................................................... 1
Introduction ............................................ 1
Chapter 2 ................................................... 6
Scattering from a Periodic Surface Using a Periodic Current Function . . . . 6
2. 1 Introduction ....................................... 6
2.2 Theory ........................................... 7
2.2.1 Scattered field - simple expansion ................... 7
2.2.2 Scattered field - higher order expansion ............... 11
2.2.3 Electric Field Integral Equation (EFIE) solution ......... 16
2.2.4 Electric Field Scattering Solution in the Far Field ........ 18
2.3 Discussion ........................................ 21
2.3.1 Comparison with other methods .................... 21
2.3.2 Large surface ................................. 25
2.4 Computational Considerations ........................... 26
2.4 Conclusions ....................................... 27
Chapter 3 ................................................... 47
Review of Target Detection and Scattering Techniques .............. 47
3.1 Introduction ....................................... 47
3.2 Review of Target Detection Using the E-pulse Method ......... 48
3.3 E-pulse Target Detection Using Band-limited Signals .......... 51

3.4 Review of Theoretical Scattering Methods for Finite-Length,
Perfectly-Conducting Sea Surfaces ....................... 52
3.5 Conclusions ....................................... 60
Chapter 4 ................................................... 72
Enhanced Target Detection in a Sea Clutter Environment ............. 72
4. 1 Introduction ....................................... 72

vi

 

 

4.2 Theory ........................................... 74
4.3 Computational Considerations ........................... 77
4.4 Stationary Surface Demonstration ........................ 80
4.5 Simulated Sea-Surface Demonstration ..................... 83
4.6 Target Detection and Window Size ....................... 87
4.7 Application of CRTW Techniques to Different Target
Geometries ........................................ 91
4.8 Multipath Effect on Target Detection ...................... 94
4.9 Coherent Processing Clutter Reduction .................... 103
4. 10 Conclusions ...................................... 1 06
Chapter 5 .................................................. 175
Cepstral Analysis and Radar Target Response .................... 175
5. 1 Introduction ...................................... 1 75
5.2 Cepstral Analysis - Theory ............................ 178
5.3 Late-Time Analysis ................................. 186
5.4 Early-Time Analysis ................................ 202
5.5 Separation of Early and Late Time ...................... 211
5.6 Conclusions ...................................... 237
Chapter 6 .................................................. 238
Application of Cepstral Analysis to Radar Target Discrimination Using E-
Pulse Cancellation .................................. 238
6. 1 Introduction ...................................... 23 8
6.2 Theoretical Background .............................. 240
6.3 Target Discrimination Algorithm ........................ 243
6.4 Numerical Results .................................. 246
6.5 Conclusions ...................................... 254
Chapter 7 .................................................. 281
Conclusions ........................................... 28 1
7.1 Summary ........................................ 281
7.2 TOpics For Further Study ............................. 289
Appendix A ................................................ 292
Scattering Measurements .................................. 292
A. 1 Introduction ...................................... 292
A2 Frequency-Domain Synthesis and Scattering Measurements in the
EM lab ......................................... 293
A3 Calibration Procedures ............................... 295
A4 Windowing Functions ............................... 300
LIST OF REFERENCES ....................................... 321
vii

 

 

 

 

 

Table 2.1
Table 3.1

Table 3.2

Table 5.1

Table 6.1
Table 6.2

 

LIST OF TABLES

Matn'x fill time comparison for Stoke’s scattering problem. ......

Spatial decomposition iterative scheme efficiency results for

scattering from 255 segment sinusoid surface ................

Spatial decomposition iterative scheme efficiency results for

scattering from 500 segment sinusoid surface ................

Natural frequencies obtained from the E-pulse/least squares
extraction algorithm for composite sinusoidal input signals have

zero-phase and non-zero phase components. ................

E-pulse identifiers and descriptions. ......................
E-pulse discrimination ratio (dB) for each library E-pulse as a
fianction of input waveform response. ....................

viii

 

46

61

62

201

259

275

 

 

 

 

Figure 2.1
Figure 2.2
Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

Figure 2.8
Figure 2.9

Figure 2.10

Figure 2.11

Figure 2.12

Figure 2.13

Figure 2.14

 

 

LIST OF FIGURES

Scattering geometry for periodic surfaces. .................. 29
Infinite, conducting sinusoidal surface scattering geometry. ...... 29
Induced surface current on one period of infinite, conducting
sinusoidal surface for TE excitation at 2.95 GHz. Angle of
incidence and scattering 85° from vertical axis. ............... 30
Induced surface current on one period of infinite, conducting
sinusoidal surface for TE excitation at 5 GHz. Angle of incidence

and scattering 85° from vertical. ......................... 31
Induced surface current on one period of infinite, conducting
sinusoidal surface for TB excitation at 9 GHz. Incident and
scattering angles 85° from vertical. ....................... 32
Magnitude of backscattered electric field from 9-period sinusoidal
surface as a function of frequency for TB excitation. Finite length
scattering model used. ( L = .09m, h = .0254m, (1)0 = 30° ) ....... 33
Magnitude of bacscattered electric field from 9 period sinusoidal
surface as a function of frequency for TB excitation. Periodic
current model used. ( L = .09m, h = .0254, (1)0 = 30° ) .......... 34
1/8 cosine taper window for the frequency band .8 - 12.98 GHz. . . 35
Short input pulse, synthesized by inverse fourier transformng a 1/ 8

cosine tapered, uniform spectral response in the frequency band of

.8 - 12.98 GHz. .................................... 36
Periodic and non-periodic transient backscattered electric fields
created by a short pulse from a finite sinusoidal surface for TE
excitation. Incident and scattering angles are 30 degrees. ( L =

.09m, h = .0254m ) .................................. 37
Stokes wave representation showing only one period. .......... 38
Magnitude of backscattered electric field for 11 period Stoke’s
surface as function of frequency for TB excitation. Finite length
scattering model used. (L = .1778m, h = .O496m, (1)0 = 30°) ..... 39
Magnitude of backscattered electric field for 11 period Stoke’s
surface as a function of frequency for TB excitation. Periodic
current scattering model used. ( L = .1778m, h = .0496m, (p0 =

30° ) ............................................ 40
Periodic and non-periodic transient backscattered fields created by

a short pulse from a finite Stoke’s surface for TB excitation.

ix

 

 

 

 

'-:_ 3T).

v

533'

V'ITJ

1.9-"

P"-I.J

 

 

 

Figure 2.15

Figure 2.16
Figure 2.17
Figure 2.18

Figure 3.1
Figure 3.2

Figure 3.3
Figure 3.4
Figure 3.5

Figure 3.6
Figure 3.7
Figure 3.8

Figure 3.9
Figure 4.1
Figure 4.2
Figure 4.3
Figure 4.4
Figure 4.5
Figure 4.6
Figure 4.7
Figure 4.8
Figure 4.9
Figure 4.10
Figure 4.11

Figure 4.12
Figure 4.13
Figure 4.14

Figure 4.15
Figure 4.16

(L = .1778m, h = .0254m, (to = 30°) .......... 41
Magnitude of backscattered electric field from 11 periods of a
conducting sinusoidal surface for TB excitation in the frequency
domain. Periodic current model used. ( L = 1.0m, h = .25m, (1)0 =

30° ) ............................................ 42
1/8 cosine taper window from .5 - 3.0 GHz. ................. 43
Synthesized pulse constructed from 1/8 cosine taper. ........... 44

Transient backscattered electric field from incident pulse for 11
period conducting sinusoidal surface for TB excitation. ( L = 1.0m,

h = .25m, (1)0 = 45° ) ................................. 45
Simple sinusoid and double sinusoid surface geometry. ......... 63
Calculated scattered field from (a) single sinusoid surface and (b)

double sinusoid surface. ............................... 64
Spectral domain of incident waveform. .................... 65
Time domain representation of incident pulse. ............... 66
Band-limited response for (a) single sinusoid surface and (b)

double sinusoid surface. ............................... 67
Transient response for missile above (a) single sinusoid surface and

(b) double sinusoid surface. ............................ 68
Constructed CRTWs for (a) single sinusoid surface and (b) double

sinusoid surface. .................................... 69
Convolution of CRTW with missile response immersed in clutter

for (a) single sinusoid surface and (b) double sinusoid surface. . . . . 70
TB scatter field geometry. ............................. 71
CRTW flowchart process. ............................ 108
Genetic algorithm data structures. ....................... 109
Genetic algorithm block diagram. ....................... 110
Simple PEC scaled ocean surfaces. ...................... lll
Scattered field from the PEC Stokes surface. ............... 112
Scattered field from double sinusoid surface. ............... 113
Scattered field from Phoenix missile model. ................ 1 l4
Constructed CRTWs for measured surfaces. ................ 115
Convolution energy ratio for Stokes surface ................. 116
Convolution energy ratio for double sinusoid surface. ......... l 17
Constructed CRTW for scattered field from Stokes surface. CRTW

designed to eliminate the surface clutter only. ............... 118
Convolution of CRTW with surface return only. CRTW designed

to eliminate the surface clutter only. ..................... 119
Convolution ratio of CRTW with clutter and target return. CRTW

designed to eliminate the surface clutter only. ............... 119
Convolution energy ratio for Stokes surface. CRTW designed to

eliminate the surface clutter only. ....................... 120
Neumann spatial frequency spectrum. .................... 121

Covariance distribution generated using Neumann frequency

 

 

 

 

 

Figure 4.17

Figure 4.18

Figure 4.19
Figure 4.20
Figure 4.21
Figure 4.22
Figure 4.23
1

Figure 4.24
Figure 4.25

Figure 4.26
Figure 4.27
Figure 4.28
Figure 4.29
Figure 4.30
Figure 4.31
Figure 4.32
Figure 4.33
Figure 4.34

Figure 4.35

Figure 4.36
Figure 4.37

Figure 4.38
Figure 4.39

Figure 4.40

Spectrum with 20 knot winds. ..........................
Absolute value of typical band-limited sea clutter return from
stochastically-generated sea surface. Sea height is not to scale.
Profiles for an evolving stochastically-generated sea surface, and
computed energy ratios for Phoenix missile/clutter combination.
Arrow indicates spatial position of the missile. ..............
Surface response constructed by adding the missile response to the
clutter response.
Convolution energy ratio for an energy width A = .5 nsec.
Convolution energy ratio for energy window width A = .8 nsec. . .
Convolution energy ratio as a function of window Width and
time.
Two sea surfaces generated from the wind driven Kinsman model
at two different times.
CRTW construction for initial sea surface. .................
Convolution energy ratio (dB) detection diagram for a realistic sea
surface for a window width of 4 nsec. ....................
Convolution energy ratio for Kinsman sea surface as a function of
window size and time. ...............................
Missile models used in CRTW study. ....................
Missile scattering in the time domain. ....................
CRTW corresponding to missile type A. ..................
CRTW corresponding to missile type B.
CRTW corresponding to missile type C.
Convolution energy ratio response for each missile using a CRTW
designed for missile A.
Convolution energy ratio for each missile using a CRTW designed
for missile type B. ..................................
Convolution energy ratio for each missile type using a CRTW
designed for missile type C. ...........................
Scattering geometry for TM-polarization showing (a) Stoke’s
surface, and (b) relative position of cylinder with respect to Stoke’s
surface ..........................................
Transient scattered return from a Stoke’s surface only and from a
cylinder above the Stoke’s surface. TCR = —5.31 dB. .........
Difference between the transient scattered return from a Stoke’s
surface with cylinder and Stoke’s surface without cylinder. .....
Convolution energy ratio for Stoke’s surface and cylinder with no
multipath effect, and Stoke’s surface and cylinder with multipath
effect. TCR = - 5.31 dB. .............................
Convolution energy ratio for Stoke’s surface and cylinder with no
multipath effect, and Stoke’s surface and cylinder with the
multipath effect. ...................................
Convolution energy ratio for Stoke’s surface and cylinder with no

...........................................

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

OOOOOOOOOOOOOOOOOO

oooooooooooooooooo

xi

 

122

123

124

130

138

139

140

141

142

143

144

 

 

multipath effect, and Stoke’s surface and cylinder with the

 

 

 

 

and Stoke’s surface excited by a TB incident plane wave. Missile
is above trough of wave and incidence angle is 10 degrees from the

xii

 

multipath effect. TCR = -.22 dB. ....................... 145
Figure 4.41 Convolution energy ratio for Stoke’s surface and cylinder with
multipath effect as a function of different surface heights for a
fixed cylinder position. .............................. 146
Figure 4.42 Transient scattered return from a Stoke’s surface only and from a
cylinder above the Stoke’s surface. TCR = -5.31 dB. ......... 147
Figure 4.43 Difference between the transient scattered return from a Stoke’s
surface with cylinder and Stoke’s surface without cylinder. ..... 148
Figure 4.44 Transient scattered return from a Stoke’s surface only and from a
cylinder above the Stoke’s surface. TCR = -5.31 dB. ......... 149
i Figure 4.45 Difference between the transient scattered return from a Stoke’s
1 surface with cylinder and Stoke’s surface without cylinder. ..... 150
‘ Figure 4.46 Convolution energy ratio for Stoke’s surface and cylinder with
multipath effect as a function of cylinder height. ............. 151
Figure 4.47 Convolution energy ratio for Stoke’s surface and cylinder with no
multipath effect as a function of cylinder height. ............. 152
Figure 4.48 Transient scattered return from a Stoke’s surface only and from a
cylinder above the Stoke’s surface. TCR = -5.31 dB. ......... 153
Figure 4.49 Difference between the transient scattered return from a Stoke’s
surface with cylinder and Stoke’s surface without cylinder. ..... 154
Figure 4.50 Convolution energy ratio for Stoke’s surface and cylinder with
multipath effect as a function of target position. ............. 155
Figure 4.51 Convolution energy ratio for Stoke’s surface and cylinder with no
multipath effect interaction as a function of target position. ..... 156
Figure 4.52 Multipath missile/sea-surface scattering geometry. ............ 157
Figure 4.53 Normalized backscatter transient-response from 10 cm long
phoenix missile excited by a TB incident plane wave. Incidence
angle is 10 degrees from the horizontal axis. ............... 158
Figure 4.54 Normalized backscatter transient-response from a Stoke’s surface
excited by a TB incident plane wave. Incidence angle is 10
degrees from the horizontal axis. ........................ 159
Figure 4.55 Normalized backscatter transient-response from a double-sinusoid
surface excited by a TE incident plane wave. Incidence angle is 10
degrees from the horizontal axis. ........................ 160
Figure 4.56 CRTW corresponding to measured Stoke’s surface. ........... 161
Figure 4.57 Convolution energy-ratio as a function of phoenix missile position
with respect to Stoke’s surface. TCR = -23.2 dB. ........... 162
Figure 4.58 CRTW corresponding to measured double sinusoid surface. ..... 163
Figure 4.59 Convolution energy-ratio as a function of phoenix missile position
with respect to double-sinusoid surface. TCR = -4.75 dB. ..... 164
Figure 4.60 Normalized backscatter transient-response from a phoenix missile

 

Figure 4.61

Figure 4.62

Figure 4.63

Figure 4.64

Figure 4.65

Figure 4.66

Figure 4.67

Figure 4.68

Figure 4.69

Figure 5.1

Figure 5.2
Figure 5.3
Figure 5.4
Figure 5.5

Figure 5.6

horizontal axis. ...................................
Normalized backscatter transient-response from a phoenix missile
and double-sinusoid surface excited by a TB incident plane wave.
Missile is above trough of wave and incidence angle is 10 degrees
from the horizontal. ................................
Convolution energy-ratio for phoenix missile located above a
trough in Stoke’s surface. Ratio was calculated from composite
measured return and hybrid-composite measured return. TCR = -
23.2 dB. ........................................
Convolution energy-ratio for phoenix missile located above a
trough in double-sinusoid surface. Ratio was calculated from
composite measured return and hybrid-composite measured return.
TCR = -4.75 dB. ..................................
Convolution energy-ratio for phoenix missile located above a crest
in Stoke’s surface. Ratio was calculated from composite measured
return and hybrid-composite measured return. TCR = -23.2 dB.
Convolution energy-ratio for phoenix missile located above a crest
in double-sinusoid surface. Ratio was calculated from composite
measured return and hybrid-composite measured return. TCR = -
4.75 dB. ........................................
Amplitude of scattered return from an evolving sea surface. This
case corresponds to a time step AT = .25 seconds between
successive returns. ..................................
Target detection of simple missile simulation using coherent
detection algorithm. Time step between pulse returns is .25
seconds, with averaging over three successive pulse returns.
Amplitude of scattered return from an evolving sea surface. This
case corresponds to a time step of AT = .05 seconds between
successive pulse returns ...............................
Target detection of simple missile simulation using coherent
detection algorithm. Time step between pulse returns is .05
seconds, with averaging over three successive pulse returns. .....
Non-minimum phase systems showing (a) input sequence; (b) 2-
plane pole-zero plot; (c) 20 logIO | X(ej°)/Xm,,x(e’°’) I; (d) arg [
X(e“°) ] .........................................
Minimum phase systems showing (a) input sequence; (b) z-plane
pole-zero plot; (c) 20 log,O | X(ej°)/Xmax(e"°) I; (d) arg [ X(ej‘°) ] . .
Block diagram showing conversion process from real to complex
spectrum for a causal, stable input signal ...................
Block diagram showing construction of minimum phase signal
from the magnitude of the spectral response. ...............
Single mode decay sequence representing a minimum—phase signal
and its cepstral reconstruction. .........................
Zero locations with respect to the unit circle for a single mode

xiii

 

169

171

172

173

174

183

1 84

185

185

194

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

true
,
rare

:sgre

1:111?

" ft]
1.5 ,

mm
dim

minimum phase decay sequence. ........................ 194
5,7 Multi-mode decay sequence representing a minimum-phase signal
and its cepstral reconstruction. ......................... 195
5,8 Zeros with respect to the unit circle for a multi-mode minimum
phase decay sequence. ...............................
: 5.9

195
Single mode damped sinusoid signal illustrating a non-minimum
phase signal and its cepstral reconstruction. ................ 196
'e 5.10 Zeros with respect to the unit circle for a damped sinusoid non—

minimum phase sequence. ............................ 196
re 5.11 A second example of a single mode damped sinusoid signal
illustrating a non-minimum phase signal and its cepstral
reconstruction.

.................................... 197
ure 5.12 Zeros with respect to unit circle for the second example of a
damped sinusoid non-minimum phase sequence. ............. 197
gure 5.13 Single mode signal illustrating a maximum-phase signal and its
cepstral reconstruction ................................ 198
igure 5.14 Zeros with respect to the unit circle for a maximum-phase signal. .
‘igure 5.15

198
Composite 3-mode damped sinusoid signal and its minimum—phase
reconstruction.

.................................... 199
Figure 5.16 Zeros with respect to the unit circle for a composite damped

sinusoid signal. ....................................
Figure 5.17

199
A second example of a 3—mode composite damped sinusoid signal
and its minimum-phase reconstruction ..................... 200
Figure 5.18 Zeros with respect to the unit circle for the second example of a 3-
mode composite damped sinusoid ........................
Figure 5.19

200
Simulated early-time and its cepstral reconstruction using a 4-point
4—pu1se minimum-phase sequence. .......................

207
Figure 5.20 Zeros with respect to unit circle for simulated early-time 4-point 4-
pulse minimum-phase sequence. ........................
Figure 5.21

207
Simulated early-time and its cepstrum reconstruction using a 16-
point 4-pulse minimum-phase sequence ....................

208
Figure 5.22 Zeros with respect to unit circle for simulated early—time 16~point

4-pulse minimum-phase sequence. ....................... 208
Figure 5.23 Simulated early-time and its cepstral reconstruction using a 32-
point S-pulse non minimum-phase sequence. ............... 209
Figure 5.24 Zeros with respect to unit circle for simulated early-time 32-point
5-pulse non minimum-phase sequence ..................... 209
Figure 5.25 Simulated early-time and its cepstrum reconstruction using a
second 32-point S-pulse non minimum—phase sequence. ........ 210
Figure 5.26 Zeros with respect to unit circle for simulated early-time second
32-point 5-pulse non minimum-phase sequence. ............. 210
Figure 5.27 Thin wire aircraft model. ............................. 217
Figure 5.28 Planar view of B-58 scaled aircraft. ...................... 218
Figure 5.29 Frequency response magnitude for scattering from a simple wire

xiv

 

aircraft. E—field polarization perpendicular to fuselage. ........
5.30 Transient

2 1 9

scattered field for Wire—frame aircraft using
magnitude/phase transform and magnitude only transform. ...... 220

5.31 Late-time frequency filter response for scattering from a wire frame

aircraft. E-field polarization perpendicular to fuselage. ........

221
:5.32 Transient scattered field for late-time response for wire-frame
aircraft.
e 5.33

......................................... 222
Early-time frequency filter response for scattering from a wire
frame aircraft. E-field polarization perpendicular to fuselage .....

223
re 5.34 Transient scattered field using early-time frequency filtered
response for a wire-frame aircraft. .......................
ire 5.35

224
Frequency response magnitude for scattering from a wire frame
aircraft. E-field polarization 45 degrees with respect to fuselage. . 225
ure 5.36 Transient scattered field from Wire frame aircraft using
magnitude/phase transform and magnitude only transform. ...... 226
gure 5.37 Late-time frequency filter response for scattering from a wire frame
aircraft. E—field polarization 45 degrees with respect to fuselage.

. 227
igure 5.38 Transient scattered field for late-time response from wire frame
aircraft.

228
Figure 5.39 Early—time frequency filter response for scattering from a wire
frame aircraft.

E-field polarization 45 degrees with respect to
fuselage .......................................... 229
Figure 5.40 Transient scattered field using filtered early—time frequency
response from wire frame aircraft ........................
Figure 5.41

230
Measured frequency response magnitude for scattering from B-58
aircraft model.

Figure 5.42 Transient scattered field from B—58 aircraft. ................ 232
Figure 5.43 Late-time filtered frequency response for scattering from B-58
aircraft model.

....................................

233
Figure 5.44 Transient scattered field for late-time response from B-58
aircraft.

Figure 5.45

234
Early-time frequency filter response for scattering from B-58
aircraft.

.........................................

235
Figure 5.46 Transient scattered field using early-time filtered frequency
response for scattering from 13-58 aircraft. ................. 236
Figure 6.1 Frequency response magnitude for gaussian input pulse. ....... 256
Figure 6.2 Normalized gaussian incident wave pulse. .................
Figure 6.3

256
Noise-free backscattering response of a broadside 10.0 cm wire
with Na = 1000.
Figure 6.4

.................................. 257
Noise-free backscattering response of a 10.0 cm wire with l/a =
1000. Incident electric field 45 degrees from axis of wire. ..... 258
Figure 6.5 EDR values for broadside response of 10 cm wire with l/a =
1000. ........................................... 260
Figure 6.6 EDR values computed from response of 10 cm wire with l/a =

XV

 

Figure 6.7

Figure 6.8

Figure 6.9

Figure 6.10
Figure 6.11
Figure 6.12
Figure 6.13
Figure 6.14
Figure 6.15
Figure 6.16
Figure 6.17
Figure 6.18
Figure 6.19
Figure 6.20
Figure 6.21
Figure 6.22
Figure 6.23
Figure 6.24
Figure A.1

Figure A.2
Figure A.3

Figure A.4
Figure A.5

Figure A.6

Figure A.7

1000 oriented 45° from broadside. .......................
EDR values for broadside response of 9 cm wire with l/a =
1000. ...........................................
Complex frequency locations for 10.0 cm wire scattering. ......
Planar View of scaled models used for target measurements ......
Frequency domain scattered-field response of B-58 aircraft model
measured at 45 degree incident angle. ....................
Transient response of B—58 aircraft model. Dashed line indicates
the minimum-phase reconstructed response. ................
Frequency domain scattered-field response of F-14 aircraft model
measured at 45 degree incidence.
Transient response of F-14 aircraft model. Dashed line indicates
the minimum-phase reconstructed response. ................
Frequency domain scattered-field response of F-18 aircraft model
measured at 45 degree incident angle. ....................
Transient response of F-18 aircraft model. Dashed line indicates
the minimum phase reconstructed response. ................
Frequency domain scattered-field response of missile #1 for a
broadside measurement. ..............................
Transient response of missile #1. Dashed line indicates the
minimum phase reconstructed response. ...................
Frequency domain scattered-field response of missile #2 for a
broadside measurement. ..............................
Transient response of missile #2. Dashed line indicates the
minimum phase reconstructed response. ...................
Complex frequency locations for late-time response of B-58
Complex frequency locations of late-time response for F-14.
Complex frequency location for late-time response using F -18. . . .
Complex frequency for late—time reSponse of missile #1. .......
Complex frequency locations for late-time response of missile
#2 ..............................................
Anechoic chamber using a frequency—domain measurement system
at Michigan State University. ..........................
Measurement system block diagram for scattering measurement
analysis. Taken from Ross, [6]. ........................
Measured frequency response of 14 inch diameter metal calibration
sphere. ..........................................
Spectral response of 14 inch diameter calibration sphere after time
domain gating of chamber wall reflections ..................
Transfer function for the frequency-domain system using the 14
inch diameter sphere as a system calibrator. ................
Measured frequency response of a 3 inch diameter metallic
sphere. ..........................................
Spectral response of a 3 inch diameter sphere after time-domain

OOOOOOOOOOOOOOOOOOOOOOO

xvi

 

261

265

266

268
269
270
271
272
273
274
276
277
278
279
280
305
306
307
308
309

310

 

 

Figure A.8

Figure A.9

Figure A.10
Figure A.11
Figure A.12
Figure A.13
Figure A.14
Figure A.15

Figure A.16

gating of the chamber wall reflections .....................
Comparison between theory and experiment for a 3 inch diameter
metallic sphere. The measured data has had the system transfer
function removed. ..................................
Cosine taper weighting curves as a function of the window shape
parameter 1'. ......................................
Cosine taper weighting functions transformed to the time domain.
Original window bandwidth from 1.0 to 5.0 GHz. ............
Cosine taper weighting curves as a function of the window shape
parameter r. ......................................
Cosine taper weighting functions transformed to the time domain.
Original window bandwidth from 2.0 to 17.0 GHz. ...........
Gaussian modulated cosine weighting curves as a function of the
window shape parameter T. Center frequency fC = 3.0 GHz. . . . .
Gaussian modulated cosine weighting functions transformed to the
time domain. Original window centered at fC = 3.0 GHz. ......
Gaussian modulated cosine weighting as a function of window
shape parameter T. Center frequency fc = 0.0 GHz. ..........
Gaussian modulated cosine weighting functions transformed to the
time domain. Original windows centered at fc = 0.0 GHz. ......

xvii

 

311

312

313

314

316

317

319

320

 

 

Chapter 1

Introduction

Detecting the presence of small targets in a nonstationary clutter background is a
fundamental problem in radar detection and tracking scenarios. The detection of both low
altitude and low cross section antiship missiles is of prime concern to the navy, and early
detection becomes critical to a ship’s survival by successfully tracking and engaging the
missile. A radar detection system that provides clutter suppression and fine range
resolution offers a significant advantage over systems which lack these capabilities.

Interest in ultra-wideband (UWB) radar systems arises in their potential use for
target identification and for low altitude, low radar cross section target detection over
water [1]. Compared with conventional continuous wave (CW) radars, UWB radars are
characterized by very large relative bandwidth and fine range resolution [l]-[2]. Another
major advantage offered by UWB radar technology is its clutter suppression capability,
therefore making it useful for detecting low—flying missiles and aircraft in sea clutter
environments [1]-[3].

Within the electromagnetics laboratory (EM Lab) at Michigan State University
(MSU), a great deal of effort has been devoted to problems in the area of target detection
and discrimination using UWB radar returns. One proposed discrimination scheme, called
the "Extinction—pulse" or "E-pulse" technique, has been applied to a large number of
problems. Early deve10pment in this area can be found in the work of Baum [4] and,

Rothwell [5]. Several other authors have contributed to further research in this area [6] -

 

 

 

 

18]. .1 major portion of this them \Hl

themes using the E-pulse method.

This then trill enter a it Me ran
rtetdetettion and discrimination. In tl‘
tt‘pretious uteri“ done by prettuw \1.‘

-. ' .... . '. tt,‘ . .‘
at: nethestattered new from an 111111

1

1.. . . - f ‘ '- .. . . l- o I
Printer-innate 313.: seen tummutee

v.73.»

« ' ’ . - o
...-.. fl‘l'tfllllllt‘il. The .notix anon l
“ . v c ‘ 0
3721), ' 0 u
our menu: to: detemttnine the
5

. ' ‘
- 7...‘ v .n . . .

‘ I . . .P‘, - .
4.1‘KI AA: L313“ "

:..e scattered return

ct

Jenner are; later used- tnr 185111113 dirt
in enhanced algorithm for radar
tr: tonnulatetl usine the l’.-plll\t‘

:Llfiru . a 3‘ .
‘ “ lillon. tlttlltllllllgllt‘llt‘ SCiillt‘l'lllL'

itintrtuttt and a starttnu pom! 1‘t

Chapterl develops a new an;

irritant: ‘~
nt. This approach. rooted in

someof ' ‘
the difficulties encountered \t'hc

thron'f
. 0 ‘
tdetection enhancement mint
ttquiredt ~

0 solve the detection proble1

Ltmputational time is

[8]. A major portion of this thesis will be devoted to new detection and identification
schemes using the E-pulse method.

This thesis will cover a wide range of topics involving the use of UWB radar for
target detection and discrimination. In this respect, the present work will be an extension
of previous work done by previous MSU researchers. In chapter 2, a new method to
determine the scattered field from an infinite-length, periodic, perfectly electric conducting
(PEC) sea-surface has been formulated. This method is based on a periodic surface
current representation. The motivation for doing this work is to create a computationally
efficient method for determining the scattered field for periodic surfaces. This is
important in that the scattered return from a sea-like surface can be theoretically
computed and later used for testing different detection algorithms.

An enhanced algorithm for radar target detection in a sea clutter environment has
been formulated using the E-pulse scheme. Chapter 3 discusses basic E-pulse
formulation, electromagnetic scattering calculations, and target detection. This chapter
serves as a review and a starting point for the work on the enhanced detection algorithm.

Chapter 4 develops a new approach used to detect targets in a sea clutter
environment. This approach, rooted in the basic E-pulse detection scheme, overcomes
some of the difficulties encountered when using the E-pulse method. After discussing the
theory for detection enhancement, some time will be devoted to the numerical methods
required to solve the detection problem. One particular area requiring considerable
computational time is E-pulse construction. The construction of an optimal E-pulse

involves a global minimization scheme. A convenient approach is to use a genetic

 

 

 

 

 

 

 

algorithm. Implementation of the genet

To Verify the next enhanced deter
listastationar} sea-surface it 111 be te
roamed. In the latter case. the so;
striated as a function of time. The
strict: interaction will also ne 10365111231
incoherent processing clutter reductit

One area that :s common to a

mean: ‘3. one term or another. \los1

"l

more treqeene} domain and trans

etzmij.‘ require some careful plt‘mltm

.4: v
NC tllSClm

ti. data sets. This is especial

the"; ' ‘- '
-,n and 10 noise ratio obtained trot

inst~ ‘
t trequent} data contain both inaenit
Cohen '1 ~' ‘ '

t braintd itith a minimum of etlit

tthtrha ~ '
nd. under eenain conditions the

It still .- '
ma) be possmle to obtain a t‘
115m" '
mutations '
eheme us
.1an the lup
g . ttls
lOplc a
. I'l '
dseteral chapters in this thesi

ttitt the
. method of cepstral analvsis

Art or '
e
mew of cepstral analvt
method - i
an '
attempt Wlll be made to r
cco

algorithm. Implementation of the genetic algorithm will be discussed in detail.

To verify the new enhanced detection technique, several approaches will be taken.
First, a stationary sea-surface will be tested. Next, a simulated dynamic sea surface will
be examined. In the latter case, the scattered field from a changing sea surface will be
computed as a function of time. The effect on target detection from target and sea
surface interaction will also be investigated. Finally, this new technique will be compared
to a coherent processing clutter reduction algorithm.

One area that is common to all aspects of this research is the use of signal
processing in one form or another. Most analyses will numerically generate a set of radar
data in the frequency domain and transform this information to the time domain. This
effort may require some careful planning in order to avoid problems normally associated
with discrete data sets. This is especially true when requiring a transient response with
a high signal to noise ratio obtained from data measured in the frequency domain. Since
most frequency data contain both magnitude and phase information, the transient response
can be obtained with a minimum of effort by using an inverse Fourier transform. On the
other hand, under certain conditions the phase information might be absent. In this case
it still may be possible to obtain a transient response that can be used in a target
discrimination scheme using the E-pulse method. A great deal of time is spent on this
topic, and several chapters in this thesis will be devoted to UWB signal reconstruction
using the method of cepstral analysis.

An overview of cepstral analysis will be presented in chapter 5. Using this

method an attempt will be made to reconstruct a radar target transient response from the

 

 

 

magnitude of its frequency-domain spt
mount topics of minimum phase cont
.tergj; relation will proxide some phi
tritium phase condition. \lznimuzn p
the components of a target's transient

"senten- illustrating axe o. cepstr.

"earn; v '0- ‘,3 '9‘ v‘""'. ‘3 v‘ .t 3‘
~..i-.1.uTLH‘ 9.13.)» lsx0.;.‘.;d\tll‘tl “11:1

, ~ o y o
‘3 m\...\ n.- .. 0‘". ‘ h 7
T5. .-..t~...a..o.. o....no .tnst“

lino-rhin‘. . W' I“
._.\._.;1...:-.‘.t‘f‘.. iii-39:3? h "1‘” pf‘e‘st’ili ‘1

tit-pulse method :0: and the ltlt'.t\ p

lizrtn' or E-puise itatet‘orrns awn

‘\.~
8‘.

sacred. The E-pulse uaxet‘orm it

Emu ‘ ‘ V ‘
x ttnstmtted Using both the irequent

‘.
Rm:
that

.1 l ) r3 ' ‘
dtmam representation or an unkn

{til ‘ ‘ '
tinitnittion method. The tareet dist

in" ’
\lflltd geometry in a target libran'

gtomet' ~ ~'
net Using a theoretical scatter
measure
ments of small-scale aircraft an
ue - '
to the extensu'e use of seattt

tender ‘
(ted to this topic. Topics COVCT

, s

magnitude of its frequency-domain spectrum. Theoretical discussion will include the
important topics of minimum phase conditions and minimum phase energy relations. The
energy relation will provide some physical insight into signal characteristics and the
minimum phase condition. Minimum phase reconstructed signals for the early and late-
tirne components of a target’s transient signal will be discussed. Many examples will be
presented illustrating the use of cepstral reconstruction. Finally, the separation of the
early and late-time signal in both the frequency and time domain will be discussed using
the minimum phase reconstruction algorithm.

The motivation behind cepstral reconstruction is the application to target
discrimination. Chapter 6 will present a simple automated discrimination algorithm using
the E-pulse method [9] and the ideas presented in chapter 5. In this detection scheme,
a library of E-pulse waveforms associated with different target geometries will be
constructed. The E-pulse waveform will be generated from time-domain data that has
been constructed using both the frequency-domain magnitude and phase information. The
time-domain representation of an unknown target will be generated using the cepstral
reconstruction method. The target discrimination algorithm will attempt to identify an
associated geometry in a target library file. Test cases include: a) thin-wire scattering
geometries using a theoretical scattering program, and b) actual anechoic chamber
measurements of small-scale aircraft and missiles.

Due to the extensive use of scattering measurements, a section in the appendix has
been devoted to this topic. Topics covered will include measurement systems, procedures,

and windowing functions. Measurement systems will examine the physical setup and

 

 

dtrnption of measurements made in
:rotedttre section trill reiieu the \IL‘

Finally. the most common \1 indou me I

 

 

description of measurements made in the anechoic chamber at the EM Lab. The
procedure section will review the steps necessary to obtain reliable measurements.

Finally, the most common windowing functions used in this thesis will be discussed.

 

 

Scattering from a PeriodicI

2.1 Introduction
A erect deai of effort has

JiJ-‘t .’ .33'.‘ “‘fYA'rf'E-fi‘ Tf'tm ‘0‘]!5
hli.\10mdgn\ll\p st\\é «i. 9.3 ... l‘\\ohoo

‘ l t v -
'93'113 rt ' JY‘I‘ ': '.1931 ~' 'h z a"! -. ‘
..it.t.\llil \t.\0unt\.Lb ..‘ ...’\ . t. \ \.\ t:

- I ' ~
mywnaft '3‘"! '31 ~y~ Q 0h a \ v“ " i'
I , . .
must.) Ritual“. Jun 3:; LUNJ‘uw Dr”
I A O Q

“U. *1". \"FV).J"\\ I” . 1 t
. ‘- . t: I . " “I ‘0‘" .
3“"Luai.i\)\d.\fi Ha> Ste.:thRt'tCtl

‘3': i ‘-~ -
noceis. This "“113“

t\.\\ h

ch ..as :nclu

i-H-t
_‘\-r
A.»\.

anr‘f' px \ o s t \ .-
M...s 1.. an anechoic chamber

M. roughness in one dimension. it

to: sanous as a function of \ but

Rivals. - . ‘ '
ritual stud} or scatterine from the

El3s ‘ ‘
tit l ‘ ‘
or held integral equation thltlL’

tonur ' . .. '
P attonall} ttpensn'e for a Uener

surface, '
Hooeur. llllS leads to problem

tan be Si '
mPlified by putting some con~

xtens

it is to
propose another method

Ed .
. ‘ Ur

 

Chapter 2

Scattering from a Periodic Surface Using 3 Periodic Current
Function

2.1 Introduction

A great deal of effort has been devoted to the numerical solution of
electromagnetic scattering from ocean-like surfaces [10]-[11]. One of the problems
frequently encountered is the physical constraints imposed by the amount of computer
memory required and the computer processing speed available. In the MSU EM Lab, a
great deal of research has been devoted to the study of scattering from various sea surface
wave models. This research has included both theoretical scattering and experimental
measurements in an anechoic chamber. All the wave models used are constrained to
surface roughness in one dimension. For example, a simple sinusoid surface has a height
2 that various as a function of x but the wave height is invariant in the y direction.
Theoretical study of scattering from these surfaces involves the numerical solution of an
electric field integral equation using the method of moments. Most often this is
computationally expensive for a general surface and forces the use of a finite extent
surface. However, this leads to problems associated with edge effects. Often the problem
can be simplified by putting some constraints on the surface.

For periodic infinite surfaces, the scattered fields can be determined in a more
efficient way. Norman has done extensive research in this area [12]. The purpose of this
chapter is to propose another method whereby the current and scattered fields can be

calculated for an infinite periodic surface. The approach described in this chapter is

6

 

 

 

 

 

similar to a periodic-surface moment in

This chapter is diiided in the 11
item will be coined. Set era? sectit
tonputer scattering examples it 11‘; be ct
nil proud-e Lott comparisons to other
apes of surfaces. Smce researchers It

atlas trom rather sing; surfaces t 1} me.

are is detoted to axe: sertace~ ustnc

:g‘lr' 6. ‘ -I,‘ 0."

rated withing on ...e adtantaees an
H

Theon'

,, .
....l btaflflt‘d field - Simple expan-

‘ .31 _
Tim. .1 shotts a plane it toe. l

tenure sariace 0T11’3VClL‘Iltlel l). The

it’fl‘ ‘ - ‘
tool the innate and the annlc hem

tiienbt; . -- ‘ ‘
, o. .\ sortaee current K t It

Time
more. due to the periodic nature

modeled with the

lollou'ing expression

there Kit) 2 l

 

similar to a periodic-surface moment method described by Chen and West [13].

This chapter is divided in the following manner. First, a detailed discussion of
theory will be covered. Several sections will be devoted to this. Next, some simple
computer scattering examples will be computed using this new method. These examples
will provide both comparisons to other methods, and illustrate scattering from various
types of surfaces. Since researchers in the EM lab have typically calculated scattered
fields from rather small surfaces ( typical wavelength of surface is about 4 inches ), some
time is devoted to larger surfaces using the new method. A final discussion will also be

included focusing on the advantages and shortcomings of this new method.

2.2 Theory
2.2.1 Scattered field - simple expansion

Figure 2.1 shows a plane wave, having prOpagation constant k, impinging on a 2-d
periodic surface of wavelength D. The polarization of the electric field is parallel to the
crests of the surface and the angle between the horizon and the propagation vector E is

given by (1),. A surface current I? (x) = 2K(x) will be induced by the electric field.

Furthermore, due to the periodic nature of the surface, a periodic surface current can be

modeled with the following expression

K(x) = )3 Kotx—nD)e”’"” (2")
where "H.
K( ) -25 x 32 2 2
K006) = { x , 2 2 ( . )
0 elsewhere
[3” = nkcosth0 (2'3)
7

 

 

 

 

lhc scattered field from the induced CL

1|

.‘ k . I :
[.1101 = — 471 H” ’ Hi

: ' ‘ " I. ,‘l ‘9
there H- represent a HJRMI .UNin

“n ' I u' ' 1 l .
"‘ ' ' m ‘ nvn'fi an.

‘113'3 I 1 .3 J}; :3r' ‘ . , _ \

Lillhihnllal Lily \lbihc\.lo OS 5:. a He

"J

, . . . . . .
.. n~ ‘g ‘1?! \J O 0%,) "1 ‘13:."a 1 .-
l-l'lm‘h .mpi‘wztu O: a!» “Mam... \.

.v | ‘ ' I .
~.}'gfi} .A ‘11, ‘1‘"‘t""‘ ‘. Z - ”I ‘ "
li~ (: [5‘ )U?>Bis¥o1\‘éo 5‘ \ " ) . LC
3 ‘ .

.
. (,. . l
r . ’
urn. — V kn ( H
.1 “‘_ ’
I..." w\‘ n '-
..ae LML --.L. o. the x uterine

éezt.~t;~.,--:..::.-.- . ‘
. ma yet.) are Ll>Cittl l’L‘laIlttm

ft 11
Ll u
lsiag th

eaboie relations. the scattered

it in periodic surface

E:(.t.y) r hi

there

Kite" ~ 1‘ 1M
»-)-Ze ”H05

Ton '
umerically compute the he

~c lCS alt?

ltialiOn
that ca
ll be m -
ore easrly
~ compui

 

 

The scattered field from the induced current is given by [14]

 

Ezs(x,y) = if [Koch Ht§”(kt/(x—x’>2 +(f(x)-f(x’))2 ) L(x’)atx’ (14)

where H52) represent a Hankel function of the second kind. The surface height and
differential line element length are given by f(x) and L(x)dx respectively. n is the
intrinsic impedance of the medium. Using the periodic surface current given by (2.1) and

making the substitution u = x’- nD yields the following form for the scattered field

D/2 a,
Ezs(x,y) = #3 f E K0(u)ejB"DHéz)(ktf(x—u—nD)2+(f(x)—f(u+nD))2 )L(u+nD)du (2.5)
-D/2 n=-°°

 

The periodic nature of the scattering surface can be used to simplify (2.5). The

periodicity yields two useful relations

f(u+nD) = f(u) (2.6)
L(u +nD) = L(u)

Using the above relations, the scattered field can be written in terms of a kernel K(x,x’)

for the periodic surface D

Ezs(x,y) = 12%] K0(x’)K(x,x’)L(x/)dx/ (2.7)

where 2

 

K(x,x/) = Z” ejp"dHéZ)(k (x—x/—nD)2 + (f(x) —f(x/))2 ) (2.8)

To numerically compute the kernel the convergence of the series given by (2.8)

can be accelerated. One way of doing this is given by Kummer’s method [15]. In this

method like terms from the series are added and subtracted in hopes of yielding a new

relation that can be more easily computed. If (2.8) is expanded and like terms are added

8

 

 

 

hid <ubtracted the follow ing relation 15

__.—-—

————'—'————-

Kttxt = H,” [wt ‘1 r they

- E e°°H k.”

pot-3:21:01}! I

Lit

View I

H

isotherms for A, and A. must

\‘ol! 3.");

l _ ‘ " ‘ _ "
.o...t-.. the text: men to .4. Am
5 fi 0 I

 

T‘f‘fjvn 0v 5ny V'V\\ \nn?‘ [hm ll 1
-~\.ihll Eh. nil“.\ JiLun.\lat.\. l k 132
& ~

'1‘ “rim; 9
\ I. li\nu

statoltzag the t‘ollou me

5’:er l-‘I
\

to ..r large \dlUU ot n can 51mph

o. . ‘
themtort. tor large argument» the l

‘

Hft

)

hut th
eHankel function argument im
forth

eexponential term in t7 1”) Th
G. h . ‘

the l . .
UNIS Wllhlll the braces ()f i.)
obtained

 

 

 

and subtracted the following relation is formed

 

K(x,x’) = Hdz)(k\/(7“x/)2 + tftx) ~f(x’))2 )

 

+ z: { ejnkDCOS¢oHéz)(k\/(nD_(x_x/))2+(f(x) _f(x/))2 ) _ An‘(x’x/)
n=l
(2.9)

 

+ e‘jnkDCOS¢oHéz)(k\/(nD+(x_x/))2+(f(x) _f(x/))2 ) _ An+(x,x/) }

+

ZAQOM’) + ZAJOCJ’)
n=1

n=l

Next, the forms for An- and A; must be determined. If the series given by (2.9) is to
converge the terms given by A". and A; should. approach the value of the Hankel
function for large arguments. The Hankel function enclosed within the braces of (2.9)

has arguments involving the following term

 

Si = nD\J1 + (x-x/)2 + 20?- [)2 i 2——_(x_l§/) (2-10)
("13) n

which for large values of n can simply be written as

s" = nD :l: (x —x’) (2'11)
Furthermore, for large arguments the Hankel function can be written as
(2) _ 21' 6'” (2.12)
Ho (Z) ’ ‘1: ——

u;

Next, the Hankel function argument involving the term given in (2.11) can be substituted
for the exponential term in (2. 12). The denominator argument can be replaced by knD.

If the terms within the braces of (2.9) are compared then the following relation is

obtained

 

 

 

 

. , E
A,lx..tl \_‘ ‘

Some simplifying )ields

llith the form of A. chosen. the last i\

S 't.t'.,r

edoihso'tuting till! into the show 1

S't.t'..t ) * . :1,

\-t.

”“1116 01 mm in the summation :1

Wm) r \T

mom can be written in terms of th

(Dim v

v

(V .
Omlanngllltt) and (2.]9) allows th

nllle
rchIransc
endem giv
. en by an 1'

integral as [16]

 

 

. . -jknD
A:(x,x’) = [[21 e"""D°°S“’° —e eeW-X’) (2.13)
knD

Some simplifying yields

A: = /i eejkoc- 22’) (20) (2-14)
irkD fl
where
Zd : e-jkD(l a cosdJO) (2.15)

With the form of An chosen, the last two sums in (2.9) can be evaluated. Letting

S*(x,x’) = i A:(x,x’) (2-16)
n=1

and substituting (2.14) into the above relation yields

n- 1
S2t(x,x/) : [Te e¥]k(x x’) Z0 2 (ZO) (2.17)

 

A change of index in the summation gives

520,,” : lg; e2jktx- x) Zo 2: :2: (2.18)

This sum can be written in terms of the "Lerch transcendent", which is given by [16]

 

w

<I>(z,s,v) = Z (v+n)‘s z (2.19)
n=0
Comparing (2.18) and (2.19) allows the sum to be expressed as
3*(x x/) = [.21. e*1k(x‘x') Zoi <I>(ZJ,1,1) (2‘20)
’ 1tkD 2

The Lerch transcendent, given by an infinite summation, can be wrttten in terms of an

integral as [16]

10

 

 

 

 

 

emf.

Due to the SanUilnl} or‘ the integrand
numerically. lo eialuaze thzs :ntegrat

lollooirrg relation

(infill =

...i~
l

liesecotro integral 1: l:.::l 5 men

¢i:..:ll I ‘1‘“
1'1

p. ' ‘ ‘
.rtioiahtige or the atom c relation 1~

tuteth ' '
. eintegral argument 15 propttm

iii
..... Scattered field - higher order

I . .
he toniergence rate of (1‘)) t

irieo‘ - '
lcontergenee higher order terr

trim ‘ ‘
. Ptotic form ol the Hankel t‘unetio

iteloe '

Pdsrmple relations given by
lithe I
. rot

der terms necessary to incre'

lorAz '
Int '
i he preceding sections cont

Willi I ‘
lhlS SCCllOll i3 l0 Show a
(level

 

-1
t 2

 

(D(Z,?:',1) : dl‘ (2.21)

e'-z

ORB

L
«76

Due to the singularity of the integrand for t = 0, the above integral is difficult to solve
numerically. To evaluate this integral, like terms are added and subtracted to form the

following relation

 

 

1 =
(I) (Z, 391)

to

_£_ (”e—”1 ‘11 (2.23)
l

The advantage of the above relation is that the integral in this form converges much better

since the integral argument is proportional to the square root of t for small arguments.

2.2.2 Scattered field - higher order expansion

The convergence rate of (2.9) is determined by the forms of Ant. To increase the
rate of convergence higher order terms in both the Hankel function argument and the
asymptotic form of the Hankel function itself must be considered. The preceding section

developed simple relations given by (2.11) and (2.12). This section will deve10p the

higher order terms necessary to increase the rate of convergence. The terms developed

. . -l/2
for A: in the preceding sections contains a summation index n dependence of n . The

goal of this section is to show a development that has an index dependence for terms up

11

 

 

 

in" hithough the derelvpmcm 1‘ 1“
henteot‘conteraence for the series I

The argument to the Hankel tur

there

I: y .' ...v . r - ‘ q ‘
eta-inane re Mae .eo’. .n t-.-~' 10

5=nDtl - fit

Vine" .. "
turning tor n ma seeping 1mm t‘

12'3L‘lmPIOIiC form for the Hankel it

w
H l‘l . :1 1 ( 1 . j
i ~ ~ 8:

iii :'
hz ls.(2.28ibecomes

., l

Hém: ;} [in 1...]
Tilt g2: 81:“?

Otl’aluale
[he abOV ‘
e relation
. approf
ihile .
ms t. I
lllVOihlng anCTSC power
. S C
llpontnl' [m
l a c
3i [3 can be apprOle I
(

 

 

to n". Although the development is lengthy, the extra terms derived will greatly increase
the rate of convergence for the series in (2.9).
The argument to the Hankel function given in (2.9) is proportional to

S = nDi/l + u (234)

where

u = (x—x/)2 + (f'f/)2 + 2(x—x/) (2.25)

(nD)2 _ nD

 

Expanding the square root in (2.24) to include higher order terms gives

s=nD(1+lu-iu2+iu3__5_u4+,,.) (2.26)
2 8 16 128 -

Substituting for u and keeping terms only up to (nD)'3 yields

nD i (x-x/) + w ; (X‘xl)(f’fl)2 + (x—x/)2(f’f/)2 _ 1m (2.27)

5 z a
2110 2(nD)2 2 (nD)3 8 trim3

The asymptotic form for the Hankel function is given by [17]

f"_"

H02(z) = \j 2

I

1+j- 9 —'2251+_ll_0_2_5_i+,..] (2°28)

e'jz ____ J______._
82 12322 307223 9830424

 

3 lg.
NI

(

with z = ks , (2.28) becomes

Hrs) = new” _1_.2_1___2__1_ _,-_2_23_1__1_ . 112.222; . .] (2.29)
0 wk 51/2 8k 53/2 123/(2 55/2 3072 kasm 98304 k4 59/2

To evaluate the above relation, appropriate terms must be found for the exponential and

the terms involving inverse powers of 3. Using the relation given for s in (2.27) the

exponential term can be approximated as

12

 

 

 

there
”1“: , II'I Hf
a = Q ' #-
ZiiD ZtnDi‘

.lsimpieetpanshyn t '...'. erminemal

Shoe ti contains terms up to tnDi

thouldbe used. To see this. a substit
:tnsiptoihDi athereas .: term i
Hlfl‘ iv‘ v. V i ‘ I
.rte. ..t order .0 ethane the compo
tnssaouiu he :neiuded in the mains!

nonstop can he tiritten .13

lot ,
hack the problem of finding rel:

llll
ltan be expanded to give

his S

l
s‘ (MESH ‘1‘)”

§/*

 

 

e —jks z e -jknDe :jk(x-x’)e -jka (2.30)

where

a = M ,-. (x‘x/W‘f/V + UPI/>207]? _ 1% (2.31)
2nD 2(nD)2 2 (nD)3 8 (nD)3

A simple expansion of the exponential term containing the inverse powers of 11 yields

-jka ~ . k2 2 .k3 3 232
e ~1—Jka—7or+1—a (.)

Since or contains terms up to (nD)'3 , no higher order terms than those shown in (2.32)
should be used. To see this, a substitution for or into the second term of (2.32) yields
terms up to (nD)'3 whereas a term proportional to a4 will yield terms with (nD)“.
Hence, in order to expand the complex exponential to include additional terms, more
terms should be included in the evaluation of or. With terms up to (nD)‘3, the exponential

expansion can be written as

_' _- --. _’ 1 1. [2
jksz jknD .1L(xx)1_ -k _
e e e { —nD(21 (ff))

 

+ 1 (igjktx-x’itf—f’r — ékztf—f’r)

(nDiz (2.33)

+ ( Ly (-%jk(x-x’)2(f‘f/)2 + fijktf-f’f‘
n

.ngtx-x’itf—f’)‘ + zigjk3U‘f/)6) }

 

To attack the problem of finding relations for the inverse powers of s, the inverse of

(2.24) can be expanded to give

 

l = _1_(1 + u)-1/2 (2.34)
s nD
Thus
1_ 1 + -l/4: 1 __l_ +_5_2——1§—u3+)
ST? — (nD)”2(1 u) morn“ 4“ 32“ 128

13

 

 

ST i’nDtS:

1 1 ’1’.” . u) u _
S 17!le-

i, : —1-:_li' 14' u
5- ”10‘ '

lo keep only terms up to inDl -: 0nd.)
eedtobeused. A quick rent“ nil
proportional to nD and tnDr . and e ’
air}. The Hankel function in (2.29)
am iith the interse potter tenrrs.

nportential tie. the unit} temri 1)) th
order to hare the n'iI requirement. A]

l e' ‘
hietore. the potter terms can he re)l

l l
\ : \%, 7_ (
SUI ("DWZ

1 : J .
53/3 ("DPT
i a -1 ,
55/2 (nDWE
L . L
Sm MD)”2

llilti' -
pllcallon
0f the c m
0 PM ex
pon

r ilional
lei-ms n
0t needed ’
in the fin

 

 

 

1_ 1 (1+u)'3’4= 1 3u+§u2—77u3

 

 

 

 

 

__ _ ( — _ —

53/2 (220)3/2 (nD)3/2 4 32 128 + )

1 1 _ 1 5 45 195
__ = (1 +u) 5’4: 1-—u +—u2 ———u3 +... 2.35
35/2 (n D)” (n 1))5/2 ( 4 32 128 ) ( )
a- 2 .1 ”flu—1 renew—firm...)

s7/2 (nD)"2 (n 1))7/2 4 32 128

"2 only a limited number of terms in the above relations

To keep only terms up to (nD)
need to be used. A quick review of (2.25) and (2.33) shows that 11 has terms inversely
proportional to nD and (nD)2 , and e ’ij has terms inversely proportional to unity, n, n2,
and n3. The Hankel function in (2.29) is formed by multiplying the complex exponential
term with the inverse power terms. By multiplying the first term in the complex

"’2 term the highest power of u must be u3 in

exponential (i.e. the unity term) by the s
order to have the 11"”2 requirement. Also the highest power of u for s'3/2 term must be uz.

Therefore, the power terms can be rewritten as

5u2__L5_u3

 

 

1
__ .. 1 — — )
S1/2 (n D)1/2 ( 4 32 128
_1__ :1 1 (1 —3— + 31— u 2)
3/2 3/2 4 32
S (’1 D) (2.36)
1 1

 

 

12

Multiplication of the complex exponential term and the power terms will introduce

. . -9/2
additional terms not needed in the final expressron. In this case, terms such as (nD) ,

14

 

 

 

 

(nD)“. will of course be dropped
At this point a relation exists it

(233i. and the potter terms i2..‘~h l. \‘g

tilt terms up to n -i are retain-ed .-\\
—.— 'HD .~;_

H.-'t,t..t l. i - -. ‘e i e I

 

’1
g 31
LR
(D

§:~ = zit-I l 11"." f}

is: : 6l.t‘~.t' l: ‘ 3M .f l:

9 :t-Itf
81"

. 45
g..: = t-fll’fl') '160('

A k
2 601-1th i‘ 2 13C

: 96i3(.e.x )(f f l‘

hi 2
ththeaboie result the asymptotic it?

bill) Us '
. mg the relation for 7... uit

A’: T ti
n 5““ )ttxi ‘e
“we (20)]
f1

 

(nD)"“z, will of course be dropped.
At this point a relation exists for the Hankel function (2.29), the exponential term
(2.33), and the power terms (2.36). Next, (2.33) and (2.36) are substituted into (2.29) and

only terms up to n‘”2 are retained. Avoiding the pages of algebra, the final result is

H62) i ~ A -'knD $jk(x-x/) _1_ _1_ 1

(x,x) ~ irkDe J e { 1/251/2 20 €3/2 3/2

(2.37)
+ l l 1 1 }
161)2 5’2 5/2 2561)3 7’2 m
where

E1/2 = 1

€3/2 : +(X— ‘X/) —jk(f f/+)2 4k

55,, = 6(x- x) — 3(f- -’+f)2 - 3—k’tx— x’) + 121'k(x--x’)(f-f)2

-1 _ 2k2(f—f/)4 (2.38)
8k2
45 / 2 _ _1__51 2

€7/2=+k—(x x)+J (x x) 80(x-x)3 ' (f’lf)

+ 6ij(f—f’)4 i 120(x -x’)(f—f’)2 — 304jk(x—x’)2(f—f’)2
. 225 . 16 3 / 6
1- 96k2 x—x’ — ’)4 -1——- +J—k (f—f)
( )(f f 1223 3
With the above result the asymptotic forms for A ,f can be derived using the relation given
by (2.9). Using the relation for Z0 given by (2.15), the higher order forms for A: are

l 1
”D " " (2.39)

lg_1_}

+ 25603 7’2 n"2

15

 

 

 

 

 

' e esp
the evaluation of t2.l6l usrng th

:e‘w I Zr: ‘i 5374
Six-I) IlTiD

A . 1 ‘5 if
there the integral terms for the Len

@t:.33-

d>t:.7/3.l

2.23 Electric Field Integral Equat

The scattered field written in t

hl'llll. To detemrine the surface e

applied at the surface of the perfectly
E,‘(x,)2) ~ Ez'txq

ihtre Egg)”

represents the ineiden‘

D/Z

f K,(x/) Ktxac’
~D/2

Titsurface current Kntx‘) can be exp

Ko(x’)

 

The evaluation of (2.16) using the expanded form for A11 is given by

S(x,x’)* = ,l— k‘Derike x’;>z { £1,2<I>(ZJ,1/2,1) + iEMMZJJ/ZJ)

 

 

 

 

 

1 (2.40)
a 1 a
+ fingered/2, 1) + 25603E7,2<I>(Zo,7/2,1)}
where the integral forms for the Lerch transcendent are
°° t1/2
<I>(z,3/2,1)- ._ .2— f: (2.41)
(a? o -
°° 3/2
<1>(z,5/2,1) = i— f t dt (2.42)
3 fl 0 e t ‘Z
°° 5/2
<I>(z,7/2,1) = 8 f t dt (2.43)
15 1t 0 et‘z

2.2.3 Electric Field Integral Equation (EFIE) solution

The scattered field written in terms of the kernel for the period surface is given

by (2.7). To determine the surface current the following boundary condition must be
applied at the surface of the perfectly conducting surface

Ezs(x:}’) + EZ’tLy) = O x,y 6 surface (2.44)

where Ezi(x,y) represents the incident electric field. Combining (2.44) with (2.7) yields

D/2 .
f K0(x/) K(x,x’) L(x’)dx’ = 274: E; (x , f(x)) (2.45)
“D/Z

The surface current K0(x’) can be expanded as a finite series of terms taking the form

N
Kotx’) = E 0.. K..<x’> (2'46)

n=1

16

 

 

 

 

there Kit] represents the current ba
discrete points x. in the internal from
north tiritten as
\ D I
Zn, i K't.X)Kl1le)L
"‘ h:
Theabore equation can be mitten in
5
re A =
, '4 In:

there

lithe 'urterral between -D2 and D 2 l

Lipulse basis function can be defin

herein
erm (m = n) can be approxitr

i” S“flight l'
the with width w
.
m given

 

where Kn(x’) represents the current basis fimction. Next, point-matching is applied at N
discrete points xm in the interval from -D/2 to D/2. The integral equation in (2.45) can
now be written as
D/2
4

N
2 an I Kn(x/) K(xm,x/) L(x/)dx/ = k— Ezi(xm,f(xm)) m = 1,...,N (2°47)
T]

"=1 -D/2

The above equation can be written in matrix form as

N
ZanAmn=bm m=1,...,N (2.48)
n=1
where
D/2
Am” = f Kn(x’)K(xm,x’)L(x/)dx’ (2.49)
—D/2
bm = i5: (xm,f(x,,,)) (2.50)
kn

If the interval between -D/2 and D/2 is divided into N segments of length A and center

X", a pulse basis function can be defined as

A
sxsxn+—

1 x —
2 (2.51)

n

A
Kn(x’) = 2
0 elsewhere

With the pulse-basis function defined. in (2.51) the matrix term given by (2.49) becomes

(2.52)

The self-term (m = n) can be approximated by assuming that the segment between points

is a straight line with width Wm given by

w = W + [mm-i3) -f<xm+%>12

 

(2.53)

17

 

 

 

 

The kernel can be nritten in terms of

A :

M

The Hankel function can be e\alu2
argumentsas[18]
Hflul =1 -}

there t: 1.781. Evaluation of the ir

Fornontliagonal elements. the expre

rectangular rule integration as

All”! T

there the expression for the kernel 1

kitten.

) n u
2.4 Electric Fteld Scattering Solu
A solution to the matrix equal
filtil '
tcrentsan. These values can be u

Fits '
Leonnder the scattered field due

itli' ..
nttely periodic structure For th'
. 18‘

 

The kernel can be written in terms of the Hankel function. Hence, (2.52) becomes
wm/Z

A = f Hf’(k|x’|)dx’ (2.54)

mm
—wm/2
The Hankel function can be evaluated using the asymptotic expression for small

arguments as [18]

Hf’(u) z 1 filing) for “<1 (2.55)
112
where y = 1.781. Evaluation of the integral in (2.54) becomes
Am = wm[1 -j3(1n(_’ilwm) — 1)] (2.56)
Ti: 4

For non-diagonal elements, the expression given by (2.52) can be approximated using

rectangular rule integration as

Am = K(xm,xn) L(xn) A (2.57)

where the expression for the kernel must be determined as discuSsed in the previous

section.

2.2.4 Electric Field Scattering Solution in the Far Field

A solution to the matrix equation (2.48) yields values for the unknown current

coefficients an. These values can be used to obtain an expression for the scattered field.

First, consider the scattered field due to the surface current from a single period of the

infinitely periodic structure. For this case the scattered field is given by
D/2

Eire) = "lg f K.<x’> HER/ctr) — 6%)thth
-D/2

(2.58)

where the field and source points are

18

 

 

 

 

ltttheabore relation p and u represent
TheHartltel function argument can be
and 13.60! as

,
kfi-fi ”9"“:
p

ltrtlie far-field p >> p'. and (2.6] t ca
etptttding. The simplification becom

k 5 - t3 e k ‘I
hit. tor large arguments the Hanlo
relation

42)
H

0

Substituting (2.62) into (2.63) and t

argument yields

(2) t
Hoikp‘fi)

\

S . .
thstrtuttng the relations given by l7

is

E35) :: ‘n & e’lltD

8"I (1‘ 1
v9 "

The I I ( b d

 

f) = pcosa f + psina )7 (2.59)

5/ = x/e +f(x’)y (2.60)

In the above relation p and 0t represent the distance and scattering angle to the field point.
The Hankel fianction argument can be written in terms of the parameters given in (2.59)

and (2.60) as

2 /2 _1
klp’ — ("i/l = kp(1 - 3(x’cosoc +f(x’)sina) + x_+_f_2(x_) )2 (2.61)
P 9

For the far-field p >> p’, and (2.61) can be simplified by dropping the quadratic term and
expanding. The simplification becomes

klb’ - fi’l z k (p — (x’cosa +f(x/)sina)) (2.62)

Next, for large arguments the Hankel function can be expanded with the following
relation

H(2)(u) 2 “Z e—ju (2.63)
Substituting (2.62) into (2.63) and using only the p dependence for the amplitude
argument yields

_, _, 2 . 67k" +'k(x/c so: +f(x/) ina) 2.64
H52)(klp-p/l)e —-]--——6’ ° 5 ( )

kirf‘;

Substituting the relations given by (2.46), (2.51), and (2.64), the scattered field in (2.58)
is

. _- N / .

Ezs(p’) z _n 1]: 32 2 an f ejk(x cosoc +f(x/)sma) L(x/)dx/ (2.65)
\J8tr fl ":1 x -3
" 2

The relation given. by (2.65) provides an expression for the scattered field from a single

19

 

 

 

period of the infinitely periodic sur
additional periods of the surface. In t
uttering period and M periods are or

total field can be mitten as

 

hatesimplitication of the aboxe rela

Eftpt . -n\Lk LI”
8: \P

.\
l
t Z a
n _ l
The “ ~ ‘
tint summation tenn given bv I
toittibution ' '
tthtch can b '
e put in a m
(
M

l
X e} chosu ‘ 005%”
I=-M :

esumm '
atton on the right hand side
k

With H)
(2.69) the array factor in (2 68
- , l

 

 

period of the infinitely periodic surface. Next, consider the contribution from 2M
additional periods of the surface. In this case M periods appear to the left of the central
scattering period and M periods are on the right. With the surface period given by D, the

total field can be written as

 

é (2.66)

>1
+

3
ll
)_.

l>

Some simplification of the above relation yields the following expression

5 k e77” 1=M jlkD(cosa + cosd>)

E (5) = —n. ’— e a
Z 8n ‘5 ,ZM
N %

 

(2.67)

The first summation term given by the preceding form simply yields an array factor

contribution which can be put in a more useful form

M 2M
2 ejkD(cosa + cosdpa)! : e —jkMD(oosa + cosdta) E ejkD(cosu + cos¢0)l (2'68)
l=-M [=0

The summation on the right hand side of (2.68) is just a geometric series of the form

k _ k+l
2,1 = 1 r (2.69)
[=0 1 * r

With (2.69) the array factor in (2.68) becomes

20

 

 

H ”Drama-0059:” - 1‘
Xe ' e

l=-M

Simplifying (3.70) yields

M ,
X ejkDicosc ‘ 15%! ,

i V

This. the scattered field is given by

 

5- —'- 1‘:
film = ~71 i e __
'tp
\
l XL
It I
33 Discussion

33.1 Comparison with other meth
In the previous sections. the

Scattering from a conducting periodic

Purpose of those sections was to develi

site would allow the determination (

m y . .

ttalrdtty of the developed theory

Prettously developed by A No
. nnan l

lealmemf
6““8 0f pla
ne '

 

'kD(cosa cos¢ )1 'kMD( 4) ) 1 ej kD(2M+1)(cosa+°°S¢°)
+ - c + c —
e] O = e I 03a OS 0 (2.70)

1 _ ejkD(cosa +cosd>o)

Simplifying (2.70) yields

sin[k70(2M+ 1) (0030: +cosd>0)]

 

 

 

f: ejkD(cosa + coscbo)! : (2.71)
[=_M . kD
sm[7(cosa +cos<bo)]
Thus, the scattered field is given by
_. sin[k—D(2M+l)(cosd) +cosa)]
s _. jk e Jkp 2 0
Ez(p) 2'. -TI __
8r: \/— . kD
P sml—(cos¢o+cosoc)]
2 (2.72)
N x“ %
an ejk[x/cosa +f(x/)sina] L(JC/) dx/
t; I. }
n 2

2.3 Discussion
2.3.1 Comparison with other methods

In the previous sections, the theoretical aspects of TE incident plane wave
scattering from a conducting periodic surface was considered for the infinite case. The
purpose of those sections was to develop a method which is computationally efficient and
hence would allow the determination of the scattered fields from larger surfaces. To test
the validity of the developed theory several comparisons will be made with a model
previously developed by A. Norman [12]. Norman has developed a rigorous and general

treatment for the scattering of plane waves from a perfectly electric conducting (PEC)

21

 

 

 

 

ptrlodic surface. An integral‘ol’cm“
to as the electric field integral equztttt
izduced on the PEC surface are calcttl
functiontPGF t heme]. With the sort"
is scattered fields once again maktn
The goal of using \orman's
pelodic structure a: different frequent
E} the new model set era! ewerrmcn
Incas periodic surfaces. The calctila
:5 teen done extenstteh tn the pa
Shi‘ite task requiring the calcttlattt
radowing the data. and then perform
burier transtbmit to determine the Ir:

retittet ‘ ‘
hese sltps but the calculatiot

should ‘. ~.
tequm tar less computer prot
tittitesur‘ '
late a comparison between tl
dl‘littht
, on the us‘f
.L ulness of th '
.. e pertc
Nom
. ta ~ ~
n has used a geomett

“sure 2 2

"'- AS
can be seen. this i:

periodic surface. An integral-operator-based analysis has been employed and is referred
to as the electric field integral equation method of moments (EFIE-MOM). The currents
induced on the PEC surface are calculated as solutions to an EFIE with a periodic Green’s
function (PGF) kernel. With the surface currents known, it is then possible to determine
the scattered fields once again making use of the PGF.

The goal of using Norman’s model was to compare the surface current over a
periodic structure at different frequencies. In addition to analyzing the currents generated
by the new model several experiments were done to calculate the scattered fields from
various periodic surfaces. The calculation of scattered fields from a finite periodic surface
has been done extensively in the past. Solutions to these fields is a computationally
intensive task requiring the calculation of the scattered field at every frequency point,
windowing the data, and then performing an inverse discrete Fourier transform (via a fast
Fourier transform) to determine the transient scattered field. The new method would also
require these steps but the calculation of the current and field at each frequency point
should require far less computer processing time. Although one case is based upon a
finite surface a comparison between the transient scattered fields should shed a great deal
of light on the usefulness of the periodic current model.

Norman has used a geometry model from a sinusoidal surface defined in
Figure 2.2. As can be seen, this geometry is slightly different from that shown in
Figure 2.1. Major difference include the angle of incidence and the sinusoidal surface
placement with respect to the origin. The results of this study will use Norman’s

definition (forcing the author to make small changes in his program). Several frequency

22

 

 

 

 

points were chosen for the current ci
field “11h frequency points at 2.95. 5.‘
at 85 degrees lnear Wingl' The i]
mm: polarization). The surface
height of .013“ meters. The induc
conducting surface are shown in Figu
hagirtan' pans of the calculated cut
heart be seen in each of these figure
merits calculated using the new metl
Sinceh'onnan's method has been exit
talidates the new method

The spectral domain scatterec
L‘(llhthted for several periodic surfacr
tone approximation using two differ“
madUPON the solution of an EFIE t
model using the periodic current gener

the ' ‘
penodtc current only over a finite

hthi
srespect we can see that the pct

of '
this type of model is to multiply tl

factor.
The spectral domain scattered l

h= .09
meters, h = .0254 meters) g

 

 

 

points were chosen for the current comparison. The author selected to use an incident
field with frequency points at 2.95, 5.00, and. 9.00 GHz., while the angle of incidence was
at 85 degrees (near grazing). The incident field polarization was parallel to the wave
crests (TE polarization). The surface has a period length (L) of .1016 meters and wave
height of .0127 meters. The induced surface currents on one period of the infinite
conducting surface are shown in Figure 2.3 -Figure 2.5. Each figure shows the real and
imaginary parts of the calculated current as a function of normalized surface position.
As can be seen in each of these figures, there is excellent agreement between the surface
currents calculated using the new method and those calculated by Norman’s EFIE-MOM.
Since Norman’s method has been extensively researched and used , the good comparison
validates the new method.

The spectral domain scattered fields and their associated transient fields were
calculated for several periodic surfaces. The scattered field was calculated with the far
zone approximation using two different methods: (1) a 2-d finite length scattering model
based upon the solution of an EFIE using the method of moments, and (2) a scattering
model using the periodic current generated from an infinite surface. The latter model uses
the periodic current only over a finite number of periods to generate the scattered field.
In this respect we can see that the periodic current model is a hybrid model. The effect
of this type of model is to multiply the scattered field from a single period by an array
factor.

The spectral domain scattered fields from a 9-period conducting sinusoidal surface

(L = .09 meters, h = .0254 meters) generated by a TB plane wave with incidence and

23

 

 

 

 

 

gamingmgle of30degrees fromt

nefarzone as a ftmction of frequc
stepsize. The spectral amplitude of
length and infinite period current
figure 2.7.

To find the transient sutured
hya short pulse. the spectral result:
ton'ne taper function (see Appendix
urine taper window conesponding t:
irfigrrre 2.8. A short pulse is synth
thiswhidow. The synthesized incidct

The time-domain scattered fit
Modsare shown in Figure 2.10.
muse for both cases agree quite w
Mttailing edge of the surface. Th
wirent model is an exact solution wit
model is not an exact solution.

A second scattering surface,
The period of this surface is 17.78

Profile to fourth order [19] can be ex

[ex

 

 

 

scattering angle of 30 degrees from the horizon (see Figure 2.1) have been calculated in
the far zone as a function of frequency over the bandwidth .8 - 12.98 GHZ at .01 GHz
step size. The spectral amplitude of the backscattered field is computed using the finite
length and infinite period current models. The results are shown in Figure 2.6 and
Figure 2.7.

To find the transient scattered field from the conducting sinusoidal surface created
by a short pulse, the spectral results of the scattered field are windowed using a 1/8
cosine taper function (see Appendix A) and then inverse Fourier transformed. The 1/8
cosine taper window corresponding to the spectral band from .8 to 12.98 GHz is shown
in Figure 2.8. A short pulse is synthesized by applying an inverse Fourier transform to

this window. The synthesized incident pulse is shown in Figure 2.9.

 

The time-domain scattered fields created by the short pulse for both scattering
methods are shown in Figure 2.10. It is observed in this figure that the backscattered
response for both cases agree quite well. Exceptions to this agreement occur at the leading
and trailing edge of the surface. This is to be expected since the finite (non-periodic)
current model is an exact solution with edge discontinuities, whereas the periodic current

model is not an exact solution.
A second scattering surface, known as a Stoke’s surface, is shown in. Figure 2.1 1.
The period of this surface is 17.78 cm and wave height is 4.96 cm. A Stoke’s wave

profile to fourth order [19] can be expressed mathematically by

kx = fiksin(6) + 6 (2.73)
2 3 3 3
y = -Bcos(6) + 3k ([3 -—8-B )cos(2kx) (2,74)
24

 

 

 

where y is the vertical displacement

tisdieparameterofthe parametric

ample cycloid can be generated by

 

[2.74landsctting k = B = l. The wa
(.173) and (2.74) for the simple cycle
giten height and wavelength.

The frequency-domain scatten
figure 2.12. The scattered field wa
scattering angles of 30 degrees. The
0H2 step size. The solution method 1
nperiodicity in the current). Figun
period Stoke‘s nave illuminated unt
tolerated by windowing the magnitu
hpetnavefonn and then applying a
transient fields using both the peri
larger tiew of the scattering between
hthe scattering solution technique.

solutions, except at the front and bac

13.2 Large surface

The spectral domain scatte

Sinusoidal surface generated by a TE

 

 

 

where y is the vertical displacement of the surface and x is the horizontal displacement.
9 is the parameter of the parametric curve, while k and [3 are wave shaping variables. A
simple cycloid can be generated by dropping the second term on the right hand side of
(2.74) and setting k = B = 1. The wave profile shown in Figure 2.11 was generated using
(2.73) and (2.74) for the simple cycloid case and then scaling in the x-y direction for the
given height and wavelength.

The frequency-domain scattered field for the 11 period Stoke’s wave is shown in
Figure 2.12. The scattered field was generated using a TE wave having incident and
scattering angles of 30 degrees. The frequency range was from .8 to 12.86 GHz at .01
GHz step size. The solution method used an EFIE using only a finite length surface (i.e.
no periodicity in the current). Figure 2.13 shows the periodic current solution for a 11
period Stoke’s wave illuminated under the same conditions. The transient fields were
generated by windowing the magnitude of the frequency-domain data with a 1/8 cosine
taper waveform and then applying an inverse Fourier transform. Figure 2.14 shows the
transient fields using both the periodic and non-periodic solutions. The inset shows a
larger view of the scattering between the crests. As can be seen, even with a difference
in the scattering solution techniques there is remarkable agreement between the two

solutions, except at the front and back of the wave.

2.3.2 Large surface
The spectral domain scattered field from 11 periods of a large conducting

sinusoidal surface generated by a TB plane wave with an incidence and scattering angle

25

 

 

 

 

 

of 30degrees is calculated using the

ofthesurfaceare 1.0 and .25 m res
mahequency step size of .005
fieldisshoun in Figure 2.l5.

To find the time-domain scan
isfourier transformed after being w
Terrier transforming a uniform specn
hmweighted with a l 8 cosine tap
cosine taper nidow and the synthe
meted field created by the short pt
ntliedara in Figure 2.15 and then I
shown in Figure 2J8. The free
semblance to those generated fror
Figure 2.7. Similarly the transient so

the transient scattered field from t

1-4 Computational Considerati
The amount of computational
Currentmodel algorithm. As indicate

Stoke’s wave was generated using a

tan was generated over a 20 hou

 

 

of 30 degrees is calculated using the periodic current model. The wavelength and height
of the surface are 1.0 and .25 m respectively. The frequency band is from .5 to 3.0 GHz
with a frequency step size of .005 GHz. The spectral amplitude of the total scattered
field is shown in Figure 2.15. .

To find the time-domain scattered fields due to a short pulse, the frequency data
is Fourier transformed after being windowed. A short pulse is synthesized by inverse
Fourier transforming a uniform spectral response over a bandwidth .5 - 3.0 GHz that has
been weighted with a 1/8 cosine taper window. Figure 2.16 and Figure 2.17 show the
cosine taper widow and the synthesized short pulse respectively. The time-domain
scattered field created by the short pulse can be obtained by applying the same window
to the data in Figure 2.15 and then taking the inverse transform. The transient field is
shown in Figure 2.18. The frequency-domain scattered field shows remarkable
resemblance to those generated from a smaller sinusoidal surface in Figure 2.6 and
Figure 2.7. Similarly the transient scattered field shown in Figure 2.18 can be compared

to the transient scattered field from the smaller sinusoidal surface in Figure 2.10.

2.4 Computational Considerations

The amount of computational effort can be greatly reduced using the new periodic
current model algorithm. As indicated in Figure 2.12 the scattered field from a 11 period
Stoke’s wave was generated using an EFIE computational solution. The scattered field

data was generated over a 20 hour period using a Pentium 100 computer with 8

26

 

 

 

 

 

megabytes of memory. In contrast.

generated in approximately I hour
spwd indicates the usefulness of thi
One area of interest is the n
g'ven by (2.39). Table 2.l shows I
number of approximation terms. Th
ouupured for the Stoke's surface
computational speed improves as m
going from 3 to 4 terms is not highly
h‘adding more terms. but it must be
processing time. At some point the p‘
rupture the speed of the overall com
mount of processing time required
For 1 and 2 term approximations. t
solution The addition of the third te

trons are significant.

M Conclusions
In this chapter a new radar sc
new method was compared to sever

ltthnique. Several cases were run

method agreed quite well with previ

 

megabytes of memory. In contrast, the scattered field data represented in Figure 2.13 was
generated in approximately 1 hour using the same computer. This significant increase in
speed indicates the usefulness of this algorithm.

One area of interest is the number of terms to be used for the approximations
given by (2.39). Table 2.1 shows the matrix fill time associated with using different
number of approximation terms. The data corresponds to a subset of frequency points
computed for the Stoke’s surface as shown in Figure 2.13. In most cases the
computational speed improves as more terms are added, although the improvement in
going from 3 to 4 terms is not highly significant. Some improvements may be expected
by adding more terms, but it must be kept in mind that more terms will also require more
processing time. At some point the processing time to compute those extra terms will not
improve the speed of the overall computation. One significant feature of Table 2.1 is the
amount of processing time required to compute the matrix fill for the 10.0 GHz point.
For 1 and 2 term approximations, there was some difficulty in obtaining a converging
solution. The addition of the third term overcame this problem. In this respect the extra

terms are significant.

2.4 Conclusions

In this chapter a new radar scattering method has been developed and tested. This
new method was compared to several established methods in order to validate this new
technique. Several cases were run and findings indicated that the results of the new

method agreed quite well with previous methods. The utility of this new method is that

27

 

an!

 

the scattering problem from an infini
herenltsohtained can then be 0st

algorithms.

 

the scattering problem from an infinite surface can be solved in a more efficient manner.
The results obtained can then be used in conjunction with testing different target-detection

algorithms.

 

28

Figure 2.1 Scattering geometr} t}

Fi
lure 2.2 Infinite. conducting si

 

 

\/

 

Figure 2.1 Scattering geometry for periodic surfaces.

 

z=-hcos(2nx/L)

WAT t/\ /\ /\ .\
\/ UV \7

l—L—vl

 

 

Figure 2.2 Infinite, conducting sinusoidal surface scattering geometry.

29

 

 

.9: "
v..e' ~
A-
.4 _
.\
\
A _
'1
v
_ ‘4-
- ~$~v-.."
.
pr"
..
”A...
l”.- ..“'
h .

(Imaginary Dart)
r)
f)
3
u
1

Current
I
O
C)
O
C)
g
l

 

. o 0.26
Norrr.

ligur
e2.
3 Induced surface curr
c
surface for TE cxcitat
from vertical axis

0.0004

0.0002

0.0000

)

—0.0002

—0.0004

—0.0006

Current (Real Port

—0.0008

JllllllllllllllllllllllllllllllllllllIlllllllllllllllllllllllllll

Real Port of Current Comparing Solutions
Given by EFIE-PGF and Periodic Current Methods
(f = 2.956Hz, Ls=.1016m, h =.012m)

—— EFIE—PCP
— — Periodic Current

 

 

—0.0010

ii iii III II rrr or ill rrr rrr III II [II II III II] II it: iii—T1

0.00 0.20 0.40 0.60 0.80 1.00

0.0006

0.0004

0.0002

ry Port)

0.0000

—0.0002

—0.0004

Current (lmogino

—0.0006

ittirtrlirtritrrtlriiriritrlrirrrrrtrltititiitilrirtrtrliliitrtriirl

Normalized Surface Coordinate

Imaginary Port of Current Comparing Solutions
Given by EFIE—PGF and Periodic Current Methods
f = 2.956Hz, Ls=.1016m, h =.012m)

 

—— EFIE->PGF
— — Periodic Current

 

_0.0008 IlllllllllIIIIIIIITIIIIIUri—ITITIIIIIIIIIIIIIIIIII]

 

0.00

0.20 0.40 0.60 0.80 1.00
Normalized Surface Coordinate

Figure 2.3 Induced surface current on one period of infinite, conducting sinusoidal
surface for TB excitation at 2.95 GHz. Angle of incrdence and scattering

85° from vertical axis.

30

 

 

 

 

to" V
-a
I .'
a.
" n- - - ‘
"I' ‘ _
-
-
c
..
.
-p ' _
b
v o - _
4
‘\
a
A -
.— <
_ .
fl ‘
u
- -
.....
.. g
...... .
._ a
.—
¢ -
,, .,
6
..
.i
* -
v ..
-
- I o, -
—_ - .—
_ -
' -
s -
5 _.
- a
a
_ .-
v
-
.....
— - -
-
.
q
-
- -
. - "Mu-
‘
r
\
'v-A—
"C
,
_‘-,.
. _
-
-
-
..
AAA ‘
MW4 -
A
~
..
s.
a ..
\4 -
a
m '
>‘ -‘ «nah
L v '-
n ”Vvv
\.
N
v.
I
k ‘\
‘
V-” Dina
0 W4

Current

‘00008

.J...L.J.LL.1.AL1‘£J.LILJA1‘11Ll1‘11tall]I
,

 

 

Real Part of Current Comparing Solutions
Given by EFIE—PGF and Periodic Current Methods

 

 

 

 

0-0008 3 (f = 5.00GHz, Ls=.1016m, h =.012m)
0.0004 5
A E
t :
0 :
CL 0.0000 :
6 :
0) :
‘5 :
*5 —0.0004 5 i.
9 :
L _
3 -
o :
-0-0008 g — EFIE—PGF
; — — Periodic Current
-0.0012 lllllllllllllllllllllllilllll[Ilrlllllllllllllm
0.00 0.20 . 0.40 0.60 0.80 1.00
Normalized Surface Coordinate
Imaginary Part of Current Comparing Solutions
Given by EFIE-PGF and Period Current Methods
0.0008 -_ (f = 5.00GHz, Ls=.1016m, h =.012m)
0.0004 5
"E E
O .1
a :
6 0.0000 -j
C _
.E s
01 I
O :
_E :
v-0.0004 j
*5 E
g I — EFIE—PGF
3 E - — Periodic Current
0 —0.0008 :
—0.0012 3
0.00 0.20 0.40 0.60 0.80 1.00

Normalized Surface Coordinate

Figure 2.4 Induced surface current on one period of infinite, conducting sinusoidal
surface for TB excitation at 5 GHz. Angle of ineldence and scattering 85°

from vertical.

31

 

.9:
1. .C’
_ x :

ll]
“.1
”NHL...

..

1“.

J I \
_ -~: L‘m“
, , N
.-
we;
_ . ‘ 't :
. .335 i
’\ : 'J
I j 1
RC 1 “
a 1 /
> A 2
_ -ccox : /
C 1
c 4
e V
0 I
5 :
V~0.0005 j
.. 3
C -4
J
E’ .1
L
3. 3
U~0 00‘01
4

~00015 .
0.00 0.2l
Norm

Figurels Induced surface currc
surface for TE

cxcitati
vertical.

 

 

 

Real Part of Current Comparing Solutions
Given by EFIE—PGF and Periodic Current Methods

 

 

 

 

 

0.0010 3 f = 9.000Hz, Ls=.1016m, h =.012m)
0.0005 -j
‘E E
O :
fl —0.0000 :
E :
Q) .-
9; :
*5 —0.0005 —j
(1) _
t :
3 _
O I
‘0-0010 3 — EFIE—PGF
; - — Periodic Current
_0.0015 ll[llllll[lllllllll[lllllllll]lflllllll|llIll[FT—l7
0.00 0.20 . 0.40 0.60 0.80 1.00
Normalized Surface Coordinate
Imaginary Part of Current Comparing Solutions
Given by EFIE—PGF and Periodic Current Methods
@0010 '_‘ (f = 9.00GHz, Ls=.1016m, h =.012m)
0.0005 5
78 E
O _
a. :
33—00000 —j
C _
.E -
01 Z
O :
E :
5-00005 -_~
*5 E
8 :
L _
3 :
o ‘0'0010 : — EFIE—PGF
2 -— — Periodic Current
-0.0015 awn-1W
0.00 0.20 0.40 0.60 0.80 1.00

Normalized Surface Coordinate

Figure 2.5 Induced surface current on one period of infinite, conducting sinusoidal
surface for TB excitation at 9 GHz. Incident and scatterlng angles 85° from

vertical.

32

 

0,40 ‘

0.30 {

Mogmi'tude

0.20 f

 

 

Relotive

_O
O
I

 

 

Fl 11
gre 2.6 Magnitude of backsca

as a function of freq
model used. I L = ()6

 

 

0.40

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

osof
(D _
U _
3 _
t, _
C I
83 _
2020: I
m :
.2 I
-H -I
53 _
(D : f
Dfi0.10:
0.00 llllllllIITFIIIIIIIIIIIIIIIII[lllllllllj
0.00 4.00 8.00 12.00 16.00

Frequency (GHz)

Figure 2.6 Magnitude of backscattered electric field from 9-period sinusoidal surface
as a function of frequency for TB excitation. Finite length scattering

model used. ( L = .09m, h = .0254m, (1)0 = 30° )

33

 

Magnitude

Relative
3:)

Q

l\)

C)
I

U“!
11“

 

 

 

 

 

 

0.30

0.25

.O
I\)
o

 

 

 

 

Relative Magnitude
O
a

 

 

 

 

 

 

 

 

 

 

 

 

llllllllllllllllllllllllIlllllllllllllllllllllllllllllllllll

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.10 I

0.05 M fl l 0 If M

0.00 [[llllll I[[lllllIIIIIIIIIIIIIIIIIIj
0.00 4.00 8.00 12.00 16.00

Frequency (GHZ)

Figure 2.7 Magnitude of bacscattered electric field from 9 period sinusoidal surface
as a function of frequency for TB excitation. Periodic current model used.

( L = .09m, h = .0254, <1), = 30°)

34

 

O
4

O
n/..
0

0.00

4.00

0.00

8 cosme taper wine

Figure 2.8 1 /

—|4JI1I-J|—IJIqu—J.4JI4I—|«JJI<JIHIHI«11IWJI<I1IWAJI<
O 0 O
2 0. 60 6. .

i O O O
®UDfiC®0§ 0>3020K

4|

 

 

1.20

 

.0
00
0

Relative Magnitude
p o
4> a)
O O

 

 

lllllllllllllllllIIlllllllllIIIlllllllIllllllllIIlllllllllI

 

 

0.20
0.00 I I I I I I I I I I I I I I I I I I I I I I I I I Fl I I I I I I I I I I m
0.00 4.00 8.00 12.00 16.00

Frequency (GHz)

Figure 2.8 1/8 cosine taper window for the frequency band .8 - 12.98 GHz.

35

 

0.80

.1...1,Jw1 _l_z L..L_L. J __L . L_L_1_.L_I

Relative Arriplil‘ude
Q
4:.
0

1111111111111111

/

~0.00

 

~0.4o 3
MW

Flu

gre 2.9 IShort input pulse. svn
apered. ‘ "
GHz. uniform spe

 

 

1.20

 

0.80

0.40

111111111I111111111|1|1111111I

Relative Amplitude

 

 

-0.00 ..

 

 

 

 

 

 

lllllllll

 

_Oo4O llllIlllllllIlllllllllllllIIIIIIIIIIIIIIIIIIIllllI

0.00 1.00 2.00 3.00 4.00 5.00
Time (nsec)

 

Figure 2.9 Short input pulse, synthesized by inverse fourier transformng a 1/8 cosine
tapered, uniform spectral response in the frequency band of .8 - 12.98

GHz.

36

 

l
- I
g : 'II I
3 3 III . h
i :.:: aim“
3 3 Al} l I
E II I

r:l‘1liv(.

{\'

 

2.00 not
Tirr

FlS'Ire 2.10 Periodic and non-pen
ashon pulse from a
and scattering angles

 

1.00 .
3 Periodic Current Model
3 - - Non—Periodic Current Model

 

    

0.50 f

gnfiude

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.00 : _
0505 I
0) E i
s 5 "'
5% orm j 4 0M in It
on 3 i
o ..
2 :
a) I
.2 —0.50 "_‘ l
4.; _
2 : '
G) j I
0: I I
—t00§ .
3 Periodic Current Model
5 — - Non—Periodic Current Model
—1.SOuIIIIIIII[[lIIIIIIIl[IllIIIIIIIIIIIIIIIIIIIIIIIIII]
0.00 2.00 4.00 6.00 8.00 10.00

Tune (nsec)

Figure 2.10 Periodic and non-periodic transient backscattered electric fields created by
a short pulse from a finite sinusoidal surface for TB excitation. Incident

and scattering angles are 30 degrees. ( L = .09m, h = .0254m )

37

 

(rm)

9
C)
(24

l“‘lllllllll

l 1

Height

.C

O

I\)
ll‘llllll

Wave

3‘:

C)

_.
[l

 

0.00 3

0.10 —0.05

W

ave reprCSCI

Figure 2.11 Stokes w

 

0.05

0.04

(m)

.0
o
(.4

Wave Height
.0
8

0.01

0.00

Figure 2.11

 

 

F L = 17.78 cm
E h = 4.96 cm
I l l l l l I l l [jTl l I I l l l I l l l I l I I I l [ I l I l l l l [Fl—I
-0.10 —0.05 0.00 0.05 0.10

Wave Position (m)

Stokes wave representation showing only one period.

38

 

1 1 1 1 J

a
O
I

Magnitude

Relative
Q
()1
o

 

 

 

 

4.0C

Fl
31102.12 Magnitude of backse

function of f
requenc
used. (L = .I778m I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

L50—

% :

:3r00:

:5

C I

O?

O _

2 I

m _

.2 ‘

+, _

52050-1

a) _

O: _
-I 0 I I' WI“

0.00 llllllllIfFlllllFillllllFFlll'Illlll—I—
0.00 4.00 8.00 12.00

Frequency (GHz)

Figure 2.12 Magnitude of backscattered electric field for 11 period Stoke’s surface as
function of frequency for TE excitation. Finite length scattering model

used. (L = .1778m, h = .O496m, 0, = 30°)

39

 

 

 

 

 

 

I
I
1.50 :
0 i
‘c a
‘3 —.
011.00 :
2 I
0 :
.2
‘9’ .
® I
8050 ~
0.003
000 4.0C
Filure 2.13 Magnitude of backse
a function of frequc

model used. ( L =

gnhude

 

0.50

Relative Ma

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.00 “m“ l W I VI I I I I I I I I I I—Ijfi—
0.00 4.00 8.00 12.00
Frequency (GHZ)

Figul'e 2.13 Magnitude of backscattered electric field for 11 period Stoke’s surface. as
a function of frequency for TE excitation. Perlodic current scattering

model used. ( L = .1778m, h = .0496m, (1)0 = 30° )

4o

 

 

 

 

 

   
   

 

 

 

 

 

 

_
.
-
-

-C.

-
—
‘
-
-

~ ~ ‘

‘ A _

\

‘ ‘

-

- <

- ~

_ -

‘ ‘

~ ..

~—

- -

:- —

~ q
-

‘

‘. .

‘ -.

On l.

.5

\ w
..

.-

‘ <

.‘
...

u

- -

k V

L
-

5

fl -4

\v A“ ‘

7 ‘».CL -.

4..
.
‘
.
..
q
.1
I I
..
i
4
4
|
l h"
1
4 0‘ A
4

pulse from a finite S

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

1.20 3
0.60 -
1.20 : §
: '71 ‘
_ E .
- _ < 2
: I .0 0.00 : 1|
_ o :
_ .E
_ B
a) 0.60 — E 1
"O I Z-oeo:
3 _ :
t :
E. - — 11 period finite
E : --—- 11 period Infinite
_ llllll I IIIIIIIIIIIIIIIIIII I 111111111
_ 6.00 6.50 7.00 7 50
< 0 ()0 _ l ‘ ' I Time (nsec)
U ' -
(D I
J! - l
6 :
E :
O .a
I ——- 11 period finite
: ~-—-- 11 period infinite
“1.20 l I I l I I I I r | I l I I I l l l l I I I I I I I I I I I
0.00 5.00 10.00 15.00

Time (nsec)

Figure 2.14 Periodic and non-periodic transient backscattered fields created by a short
pulse from a finite Stoke’s surface for TB excitation.
( L = .1778m, h = .0254m, rho = 30° )

41

 

l.00 j

_C)
00
O

_O

O)

O
lllllllJlLJ|llll

111011

Relative Amplitude
O
h)
o

S:
l'\)
O

 

 

 

_C>
O
O

 

'o111111111111111111111100II

O

0 0.50 1

Fl

Ellie 2.15 Magnitude of backsc
Sinusoidal surface f(
current model used.

 

1.00

0.80

.0
a)
0

Relative Amplitude
'55

0.20

 

 

 

 

llllllllllllllllllIllllllllIIlllllllllIlllllllllI

 

 

  
  

 

 

o 00 . Juli 1.- . .
' 0.00””"0.'50 1.00 1.50 2.00 2.50 5.00
Frequency (GHZ)

 

Figure 2.15 Magnitude of backscattered electric field from 1] periods of a. conducting
sinusoidal surface for TB excitation in the frequency domain. Periodic

current model used. ( L = 1.0m, h = .25m, 4’0 = 300 )

42

 

i 1.20 :
1.00 5
0.80 —f

0.60

Relative Amplitude
o
34;
O

LlLIlllllll

0.20

0.00 E
0.00

 

1.00

II
0032.16 US cosine taper wint

 

 

1.20

 

1.00

 

.0
oo
o

 

Q
4:.
O

 

0.20

Relative Amplitude
O
(D
O
lllllllllIlllllllllIllllllllLIlllllllllIllllllllllllllllllll

 

 

0.00 lllllllll[lllllllll|lllIIIIIIIIIIIIIIIII

0 1.00 2.00 5.00 4.00
Frequency (GHz)

.0
0

Figure 2.16 1/8 cosine taper window from .5 - 3.0 GHz.

43

 

Relative Amplitude
l

~0.50 3
0.00

 

Ii
Eure2.17 Synthesized pulse cc

 

1.00

 

 

 

 

 

 

 

 

<0 _
'0 0.50 —
3 .u
,+_J -
a _
E I
< _
o) _
._>_ ‘
+, _
g -—0.00 e- a
0) _
m ..
_O.50 Tl—VRT—FTlijTTFIIIII'T—[TlllllTllRl
0.00 10.00 20.00 30.00

Time (nsec)

Figure 2.17 Synthesized pulse constructed from 1/8 cosine taper.

44

 

 

 

 

 

Relative Amplitude

_.
a
a

Q
(J1
C)

 

r. ,
C j: A’—
. .

 

0.50

C)

C)

C)
MJILIIJIIIIIIIJIAIIIIIllIllIlllllII

 

 

 

 

1.00

 

0.50

 

 

 

 

 

 

 

 

 

 

0.00

 

 

Relative Amplitude

 

—0.50

lllllllllIJJlLlllllIlllllllllIllllLllIII

 

 

 

 

 

 

 

 

 

 

 

 

_1.00 TIIII T—rll lrlrIFI ITI I—lrl ll TII lelllTll—r]

0.00 20.00 40.00 60.0 80. 100.00
Time (nsec)

 

Figure 2.18 Transient backscattered electric field from incident pulse for 11 period
conducting sinusoidal surface for TB excitation. ( L = 1.0m, h = .25m, (1)0
= 45° )

45

 

Table 2.] Matrix fill time comi

 

 

 

 

 

 

 

Fill Ti
Freq. 101121 1
i .
10 10.6:
10 9.73
‘0 9.4:
40 9.81
x"
‘0 10.49
\\
(‘0 10.22
\\
70 10.38
\\
8'0 10.55
\\
9'0 10.88
\_\
10'0 71.67
\\
”'0 11.64
\\
12'0 11.97
\%

 

 

 

 

Table 2.1 Matrix fill time comparison for Stoke’s scattering problem.
Fill Time (secs) for Different Number of Terms
Freq. (GHz) 1 2 3 4

_____________________J_____

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.0 10.65 3.57 2.03 1.76
2.0 9.78 3.13 1.05 1.26
3.0 9.43 3.02 1.70 1.32
4.0 9.83 3.02 1.70 1.36
5.0 10.49 3.02 1.75 1.43
6.0 10.22 3.13 1.82 1.43
7.0 10.38 3.18 1.85 1.43
8.0 10.55 3.35 1.87 1.48
9.0 10.88 3.40 1.92 1.48
10.0 71.67 33.89 1.93 1.48
11.0 11.64 3.40 1.92 1.53
12.0 11.97 3.68 1.98 1.59
46

 

 

 

 

 

Reiiew of Target DE

31 Introduction

Considerable elTon by resea
discrimination of radar targets u:
II].[9I.[20]-[23. This chapter \1
implemented at MSU. Particular at
inasea clutter environment using
roiijunction with the E-pulse tech
illnttate and validate the E-pulsc 1'
lay the foundation for the material
“IIIbe used for a new target detect:

11101111100000 with cepstral analvs

of the numerical methods to detenr
tonducting. 2-dimensional. finite-1C
Section 3.2 of this chaptel
method for radar target detection I
examples for a band-limited signal
section will present the theoretical
1111111 on, and scattered field, frr

It '
“11h. Difficulties such as comp
u

 

Chapter 3

Review of Target Detection and Scattering Techniques

3.1 Introduction

Considerable effort by researchers at MSU has been devoted to the detection and
discrimination of radar targets using the extinction pulse (E-pulse) technique [3]-
[4],[9],[20]-[22]. This chapter will review the E-pulse discrimination scheme as
implemented at MSU. Particular attention will be placed on the detection of radar targets
in a sea clutter environment using a stepped-frequency ultra—wideband (UWB) radar in
conjunction with the E-pulse technique. Several new examples will be presented to
illustrate and validate the E-pulse detection technique. The purpose of this review is to
lay the foundation for the material in chapters 4 and 6. In chapter 4 the E—pulse method
will be used for a new target detection algorithm. Chapter 6 will use the E-pulse method
in conjunction with cepstral analysis for target identification. In section 3.4 a review
of the numerical methods to determine the electromagnetic scattered fields for perfectly-
conducting, 2-dimensional, finite-length surfaces will also be reviewed.

Section 3.2 of this chapter reviews the theoretical foundation of the E-pulse
method for radar target detection in a sea clutter environment. In section 3.3 several
examples for a band-limited signal will be used to illustrate this technique. The final
section will present the theoretical methods that will be used for calculating the induced
current on, and scattered field, from a perfectly conducting sea-like surface of finite

length. Difficulties such as computer memory limitations and computational speed will

47

 

 

 

 

be discussed

3} Reiiew of Target Detectio

Detection of objects locate
corniderable detail at MSL'. 06100
the clutter signal from the ocean s
Interest in LII'B radar for targe-
background clutter and enhance the
thigh-resolution. time-domain rad
the alluring the clutter signal 10
1011511018 system. the period
increasing the probability of target

For a surface profile that i
Witd field response from the
measurement is made at time I am

1110 the scattered field will be av:

si
gnal from the surface profile in

damped sinusoids [4]

N
r(t) = Ea ea"
II
n=l

Where a
an () en h
n d n repres i l e i

S : '
1 0 +
1 10, represents the com]

1111110 I
Ximately periodic nature of t

be discussed.

3.2 Review of Target Detection Using the E-pulse Method
Detection of objects located above a disturbed sea surface has been studied in
considerable detail at MSU. Detection, using conventional radar, becomes difficult when

the clutter signal from the ocean surface becomes large compared to the target return.

 

Interest in UWB radar for target detection arises from the ability to reduce the
background clutter and enhance the overall resolution capabilities of the radar [23]. With
a high-resolution, time—domain radar, the dominant scattering events can be separated,

thus allowing the clutter signal to be reduced and the target signal to be extracted. By

 

 

using a UWB system, the periodic nature of the sea clutter return can be reduced,

increasing the probability of target detection.

 

For a surface profile that is approximately periodic in nature the time-domain
scattered field response from the surface is also approximately periodic. If a radar
measurement is made at time T and the two-way transit time across the range bin is TR
then the scattered field will be available in the time range '1: <t< “t? +TR. The clutter
signal from the surface profile in this time range bin can be modeled as a sum of N

damped sinusoids [4]

N
r(t) = Zane°”tcos(t0nt+¢n) t<t<t +TR (3-1)
n=1

where an and (1)” represent the amplitude and phase of the nth scattering mode,
s" = on + j 0)” represents the complex frequency of the nth mode. By using the

approximately periodic nature of this waveform, a clutter reducing transmit waveform

48

—+ -

 

 

101011 [3] can be created which
will reduce the background clutter
A CRTW eltl is a wavefon

sea surface clutter return 1111. resul
1 -7,

CI!) = e111 -rtri : f I
Application ofthe Laplace transfon

the CRTW

CI

there Elsi and R15) are the Lapl
condition for creating the C RTW [.

E15 :5") = E(-‘

there 31 is the complex conjugate
Numerical construction of t]

l ‘ .
111111 a set of K basrs functions as

w .
1050111)} is an appropriate set

0.5 '
)11110 the convolution integral

Wei ' '
thing function {W } gives
m .

 

 

 

(CRTW) [3] can be created which, when either transmitted or used in post-processing,
will reduce the background clutter yet preserve the target signal.

A CRTW e(t) is a waveform of finite duration TE that, when convolved with the
sea surface clutter return r(t), results in a null response given by

T+TR

60‘) = e(t)*r(t) = f r(t’)e(t—t’)dt’ = 0 T+TE<I<T+TR (3.2)

1'

Application of the Laplace transform to (3.2) leads to the following condition for creating

the CRTW

C(s) = E(s)R(s) = o (3.3)

where 13(5) and R(s) are the Laplace spectra of e(t) and r(t) respectively. Hence a

condition for creating the CRTW [24] is

E(s=sn)=E(s=sn*)=O n=1,2,...N (3.4)

where s; is the complex conjugate of sn.
Numerical construction of the CRTW is performed by expanding the waveform

e(t) in a set of K basis functions as [20]

K
e(t) = Z akgkU) (3-5)
k=l

where {gk(t)} is an appropriate set of basis functions. Substituting the expansion from
(3.5) into the convolution integral of (3.2) and taking inner products with a set of

weighting function {Wm} gives

49

 

 

 

 

TE 3 ' T‘

K
[up I [ g.(t’)r(t

H M , .1:

Using (3.6). a solution for a‘ for al
heformd by choosing M = 2N an
inhomogeneous matrix equation will
Itrix to be singular. Furthermo
simplified by using rectangular puls

Different values for T, resu
choice of TE can have a significant
resonant frequencies [4]. A suitable

enorper point between the original

situation. the following must be mi.

5 = |r(t)-

The extracted frequencies 5,, = 6n

amplitudes in (3.6) and then using t

equation of the form

 

 

If t+7k

[VS akf f gk(t') r(t—t')Wm(t) dt dt’ = o m = 1,2,...,M (3.6)
k=1 :’=o 1: +15
Using (3.6), a solution for ock for almost any choice of TE ("forced" E-pulse solution) can
be found by choosing M = 2N and K = 2N+1. For this condition (3.6) becomes an
inhomogeneous matrix equation with solution for any choice of TE that does not cause the
matrix to be singular. Furthermore, the evaluation of the integral in (3.6) can be
simplified by using rectangular pulse basis functions and weighting impulse functions.
Different values for TE result in significantly different CRTW waveforms. The
choice of TE can have a significant effect on the constructed E-pulse and the extracted
resonant frequencies [4]. A suitable choice of TE is one that yields the minimum squared
error per point between the original data r(t) and a reconstructed waveform f(t) . In this

situation, the following must be minimized over the sample interval

6 = “r(t) —r‘(t)ll = Elm.) —f(t,-)]2 (3.7)
where
N . A
f(t) = Zdneo"tcos(d>nt+¢n) (3'8)
n=1

The extracted frequencies sn = 6n + jcbn can be found by solving for the E-pulse basis
amplitudes in (3.6) and then using the relation given by (3.4). This leads to a polynomial
equation of the form

2N+1

2 ocka = 0 (3-9)
k=l

50

 

 

wheel = e"A andA is the
Once the natural frequen
applied to (3.7) and (3.8) to yiel
waveform. Different values of T,
tillE can be varied to construct th

process. the E-pulse amplitudes.

obtained.

33 E-pulse Target Detection

The use of the C RTW tee]
a sea surface clutter environme
implementation [22]. The followii
forthe case of band-limited signai

The theoretical pulse resp
mace models with surface profile
for an incident wave whose electrii
polarization). The incident and so
the horizon. The moment method
lnthe frequency range 0 to 14 GH
band limiting the scattered field 51:
SeeAppendix A) shown in Figure

the band 9-14 GHZ, with a 3 dl

 

 

 

where Z = e 'SA and A is the basis function width.

Once the natural frequencies are determined, a least-squares fitting routine is
applied to (3.7) and (3.8) to yield the amplitude and phase terms for the reconstructed
waveform. Different values of TE will of course lead to values of a that differ. The value
of TE can be varied to construct the waveform that yields the smallest value of 8. In this
process, the E-pulse amplitudes, frequencies, and best fit reconstructed waveform are

obtained.

3.3 E-pulse Target Detection Using Band-limited Signals

The use of the CRTW technique to detect the presence of a target embedded in
a sea surface clutter environment has previously been reported using a baseband
implementation [22]. The following examples will illustrate the validity of the technique
for the case of band-limited signals.

The theoretical pulse responses of two finite-length, perfectly-conducting, sea
surface models with surface profiles shown in Figure 3.1 were computed (see section 3.4)
for an incident wave whose electric field was polarized parallel to the surface crests (TE-
polarization). The incident and scattering angles were both 12.5 degrees with respect to
the horizon. The moment method was used to compute the frequency-domain response
in. the frequency range 0 to 14 GHz. A stepped, ultra-wideband signal was simulated by
band limiting the scattered field spectra with a GMC window (f(2 = 11 GHz, T = .5 nsec,
see Appendix A) shown in Figure 3.3. This window limited the frequency response to

the band 9-14 GHz, with a 3 dB bandwidth of 1.2 GHz (about 11% of the center

51

 

r www

 

 

frequency of 11 GHZ). The time
Figure 3.4. is obtained by applyin
frequency spectrum. The band
solace are shown in Figure 3.5.
allow comparison with measuren
match the dimensions of actual 5:
After the time-domain sip
response of a five inch long missil
to produce a prechosen target-to-
target signal strength to maximum
leading edge of the finite length
missile and clutter signals for the
to discern the presence of any targr
least-squares minimization fitting
surfaceare shown in Figure 3.7.
combinations are shown in F igurt

after application of the CRTW co

discerned from the clutter backgrt

3'4 Review of Theoretical :
Conducting Sea Surfaces

This section reviews elec

conducting surfaces. The numeric

 

 

 

frequency of 11 GHz). The time-domain representation of the incident pulse, shown in
Figure 3.4, is obtained by applying an inverse fast Fourier transform (IFFT) to the tapered
frequency spectrum. The band limited spectra for the sinusoid and double sinusoid
surface are shown in Figure 3.5. Also, the size of the sea surface models was chosen to
allow comparison with measurements taken within an anechoic chamber, and do not
match the dimensions of actual sea surfaces.

After the time-domain signals for the surfaces were computed, the measured
response of a five inch long missile target was added. The missile amplitude was scaled
to produce a prechosen target-to-clutter ratio (TCR) defined as the ratio of maximum
target signal strength to maximum clutter signal strength (excluding the response from the
leading edge of the finite length surface). Figure 3.6 shows the superposition of the
missile and clutter signals for the two surfaces. From this figure it is obviously difficult
to discern the presence of any target from the band-limited background clutter. Using the
least-squares minimization fitting routine (section 3.2), the CRTWs created for each
surface are shown in Figure 3.7. The convolution of the CRTWs with the clutter/target
combinations are shown in Figure 3.8. The presence of the target is clearly enhanced
after application of the CRTW convolution. For each surface, the missile can easily be

discerned from the clutter background.

3.4 Review of Theoretical Scattering Methods for Finite-Length, Perfectly-
Conducting Sea Surfaces

This section reviews electromagnetic scattering from finite-length, perfectly-

conducting surfaces. The numerical solution of an, electric field integral equation using

52

 

 

 

he method of moments will be fo
surface toughness in one dimensi
Figure 3.9 shows a plane
polarization of the electric field i
and the angle between the x-axis
inning electric field E; imprcs
tunent Kim on the conducting

electric field E: given by [‘4]

s _ It
E. (p) = -7"
there 116:) represents a Hankel fu
and source points respectively. an
The propagation constant and in
respectively. The differential line
integration for the surface are Ll

terms of their coordinates as

'Ol

where p represents the distance
scattering angle measured from the

the differential line element lengtl

 

the method of moments will be formulated for the case of scattering from surfaces having
surface roughness in one dimension.

Figure 3.9 shows a plane wave impinging on a perfectly conducting surface. The
polarization of the electric field is parallel to the crests of the surface (TE-polarization)
and the angle between the x-axis and the propagation vector is given by (1)0. The time-
varying electric field E,i impressed upon the surface generates a z-directed induced
current Kz(x) on the conducting surface. These currents in turn generate a scattered

electric field E: given by [14]

L2
E5(5) : _anfKZ(x/) H52)(kl5 _ §/|)L(x’)dx/ (3.10)

Z
Ll

 

where H52) represents a Hankel function of the second kind, p and 6’ represent the field
and source points respectively, and K(x) represents the induced surface current density.
The propagation constant and impedance of the medium are symbolized by k and 11
respectively. The differential line element length is given by L(x)dx and the limits of
integration for the surface are LI and L2. The field and source points can be written in
terms of their coordinates as

5 = pcosa )2 + psinoc )7 (3-11)

fi’ = x02 + f(x’)y‘ (3“)

where p represents the distance from the origin to the field point, or represents the
scattering angle measured from the x-axis, and f(x) represents the surface height. Finally,

the differential line element length can be written as

53

 

L(r/

To determine the surfac
applied at the surface of the perf

EXXJ) . E,

Combining (3.14) with (3.10) yie

L.

{tom Hf'm

L.

The surface current density can b

K1

there K,(x‘) represents the curre
Ndiscrete points Xm in the interv

now be written as

M:

L,
a” f K,(x') H;
Ll

The above equation can be writte

N
ZanA
71:]

where

1.1
A,” = f K,(x')H,§2’(

Lt

 

L(xr)dxt : 1 +fl(x)2 dx/ (3.13)

To determine the surface current, the following boundary condition must be

applied at the surface of the perfectly conducting surface

Ezs(xa)’) + Ezi(xa)’) = 0 x,y 6 surface (3-14)

Combining (3.14) with (3.10) yields the following integral equation

L

f K206)1’1’.§2)(k\/(x-JC’)2 + (rm—r(t)? ) L(x’)dx’
L

l

 

(3.15)
= i Ez‘(x,f(x)) Lrsstz
kn

The surface current density can be expanded as a finite series of terms taking the form

N
Kz(x') = 2 an Kn(x’) (3.16)

n = 1
where K,(x’) represents the current basis functions. Next, point-matching is applied at
N discrete points xm in the interval from L, to L2. The integral equation in (3.15) can

now be written as

 

N L2
2 an I Kn(x’) H52)(k¢(7nl-x’)2 + (f(xm) -f(x'))2) L(x’)dx’

”=‘ Ln (3.17)
= i Ez‘txmjtxm» m =1,...,N
kn
The above equation can be written in matrix form as
N
ZanAmn=bm m=19'°-9N (3.18)
n=1.

where
L2
A... = f 19.06)11'.§2’(I<((—x..-x’)2 + (f(xm)-f(x’))2 )th'>dx' (3.19)
L

l

 

54

 

 

 

 

 

lithe interval between L, and L

apulse basis function can be de

K,(x') =

With the pulse-basis function de

ulb

A": f Hflk

ml],

The self-term (m = n) can be appr

isa straight line having width w"

_.

The relation in (3.22) for the self

A

film

For small arguments the hankel f

Hf’ru) = 1
wherey= 1.781. Evaluation of ‘
yields

mm

Farnon~diagonal elements, the ex

 

 

bm = ire," (xm, f(xm)) (3.20)
kn

If the interval between LI and L2 is divided into N segments of length A and center x",

a pulse basis function can be defined as

l x -

n

_A_ A
Kn(x’) = 2 " 2 (3.21)

0 elsewhere

With the pulse-basis function defined in (3.21) the matrix term given by (3.19) becomes

 

A
” 3
Am= f Hizlkfin-x'r+<f(x,,,)—f(x')>2)th')dx' (3.22)
A
V?

The self-term (m = n) can be approximated by assuming that the segment between points

is a straight line having width wm given by

 

w = ([AZ + [f(x ——“—) -f(x +3) 12 (3.23)
m m 2 m 2
The relation in (3.22) for the self term becomes
wm/Z
A... = f H;Z>(k(x/() dx’ (3°24)
—wm/2

For small arguments the hankel function becomes [18]

Héz)(u) z 1 —j-%ln(-Y—2£) for “<1 (315)
where y = 1.781. Evaluation of the integral in (3.24) with the relation given by (3.25)
yields

A = w [1—j3(1n(flwm)—1)] (3.26)
at: 4

mm m

For non-diagonal elements, the expression given by (3.22) can be approximated by simple

55

 

 

 

 

 

rectangular rule integration as

All] : ’1‘?ka (

Evaluation of the matrix e
basis function amplitudes given
the pulse basis function amplitu
etaluated using (3.16) and (3.10
field approximation should be c

can be expanded as

for large arguments the Hankel f

Substituting (3.28) into (3.29) 2
argument yields

Hf’tkta - 6'!)

Substituting the relations given b

(3.10) becomes

 

 

rectangular rule integration as

 

An... = 1‘1’.§2)(kt/(JC,,,-x,.)2 + (f(xm)-f(xn))2 )L(xn)A (3.27)

Evaluation of the matrix elements in (3.26) and (3.27) is needed to solve the pulse
basis function amplitudes given by the solution of the matrix equation in (3.18). Once
the pulse basis function amplitudes are determined, the scattered electric field can be
evaluated using (3.16) and (3.10). However, to simplify the evaluation of (3.10) a far
field approximation should be considered. In this case, the Hankel function argument

can be expanded as

klp’ - 6’l z k(p — (x’cosoc +f(x') sinor)) (3-28)

For large arguments the Hankel function can be expanded with the following relation

113%) s P e’j“ (3.29)

Substituting (3.28) into (3.29) and using only the p dependence for the amplitude

argument yields

. —'k
H§2)(kif5 — 5'1) .. l f!— 67,—. warmed maps...) (3.30)
TB
p

Substituting the relations given by (3.16),(3.21), and (3.30) the scattered field given by

(3.10) becomes

 

. “'k - I , .
128(6) z ‘7] ‘élg- e I P 2 an 1‘ e}k(xcosa +f(x)sma) L(X’)dx’ (3.31)
\i 113

 

 

 

 

 

 

 

which for rectangular rule inte

.- 7
E.(p>=-n"—
1!

The determination of the
sstem of equations in many u
solution of large linear systems
computers. A typical linear syste
requires at least 2 Mbptes of real
complex arithmetic and 4 Mbyte
Doubling the number of equation
leading to memory resource prob]
louse some sort of virtual memorfi
Very attractive due to the large
memory and that found on the ha
problem using the spatial decomp:

If the process of calculat
necessary to store the entire mat:
into a number of subsections 0
scattering object separated from i
adjacent subzones carry tanger

electromagnetic field boundary cc

one side of the interface must be

 

 

 

which for rectangular rule integration becomes

5 jk e‘jk" N jk(x cosa +f(x )sina) 332
E(f>’)e-n — Zane n " L(xn)A (.)

Z 87: fl n=1

 

The determination of the surface current involves finding a solution to a large
system of equations in many unknowns. A typical problem often encountered in the
solution of large linear systems is the memory constraint imposed by most desktop
computers. A typical linear system composed of 512 equations in 512 unknown variables
requires at least 2 Mbytes of real memory to store the matrix values for single precision
complex arithmetic and 4 Mbytes of memory for double precision complex arithmetic.
Doubling the number of equations and unknowns requires four times as much memory,
leading to memory resource problems for small systems. One solution to the problem is
to use some sort of virtual memory management scheme. However, this alternative is not
very attractive due to the large amount of time required to swap data between real
memory and that found on the hard. disk. Another approach, taken here, is to solve the
problem using the spatial decomposition technique (SDT) [25].

If the process of calculating the matrix elements is fairly fast, it may not be
necessary to store the entire matrix. In the SDT method the scattering object is divided
into a number of subsections or subzones. Each subzone is considered a distinct
scattering object separated from its nearest neighbor by a virtual surface. Furthermore,
adjacent subzones carry tangential electric and magnetic virtual currents. The
electromagnetic field boundary conditions requires that the tangential virtual currents on

one side Of the interface must be equal to, but opposite from the other side. An integral

57

 

 

 

 

 

 

equation solution. using the m

approximate solution to the pro
the subzones. and then the solu
some stopping criteria is satisfie
Consider again the elec
into(3.14). A compact represe

aid of the operator 2?,” [25] as

there

sf=j
C

andC represents the contour of 11
the surface is divided into N sub:

be written as [25]

. N
EX (3) - X
where

JJ
S1” = f
C"
I
H916. Kzn(x ) represents the surf
lhe contour of integration over th

lellshand side of (3.35) the total e)

excitation and the additional tet

 

 

 

equation solution, using the method of moments, is applied to each subzone. An
approximate solution to the problem is determined by sequentially scanning through all
the subzones, and then the solution is refined through successive approximations until
some stopping criteria is satisfied.

Consider again the electric field integral equation formed by substituting (3.10)
into (3.14). A compact representation of this integral equation can be written with the

aid of the operator $131” [25] as

$91” [Kz(x’)] = Egg-5) (3.33)
where

281” = fkaéZ)(k(5-6’))L(x’)dx’ (3.34)
C

and C represents the contour of integration over the limits of the conducting surface. If
the surface is divided into N subzones, the electric field integral for the ith subzone can

be written as [25]

N
Eggs) — Z gcfj [Irma/)1 = gaff [Knot/)1 (3.35)
n=1
where
at; =f€1Ht§2)(k(6-t3"))L(x’)dx/ (3.36)
C

I!

Here, K,n(x/) represents the surface current density over the nth subzone, Cn represents
the contour of integration over the nth subzone, and (3 is located on subzone i. On the
left-hand side of (3.35) the total excitation on spatial subzone i is given by the plane—wave

excitation and the additional terms due to the other subzones. To implement the

 

 

 

 

 

 

algorithm. the value of i starts at

method of moments. The algo
subsection is calculated. This p
cmrent for each subsection on th
has been completed an approxi
Successive sweeps across the s
stopping criteria is satisfied.

The advantage of the SD
can be solved on a computer sy
anelecnically large scattering su
man-ix is of size NP x N,. Howev
intoN subsections of NL points. i
it‘ll. Hence. the memory limit:
surface into enough subsections.

To illustrate the use of t
sinusoid surface was computed.
wave and scattered field calcula
POlarization of the incident field v
The surface geometry consists of
peak wave height of .0254 meter

sections to determine the number

current. All computations were [

 

 

 

algorithm, the value of i starts at 1 and the current only on subzone 1 is calculated. by the
method of moments. The algorithm then shifts to subzone 2 where the current on this
subsection is calculated. This process sequentially steps through each subzone until the
current for each subsection on the entire surface has been calculated. Once the full sweep
has been completed an approximate solution for the surface current has been obtained.
Successive sweeps across the surface lead to a convergent iterative process until some
stopping criteria is satisfied.

The advantage of the SDT method is that the current on electrically large objects
can be solved on a computer system with a modest memory capacity. For example, if
an electrically large scattering surface is broken into Np discrete points, then the system
matrix is of size Np x Np. However, by using the SDT method the surface can be broken
into N subsections of NL points, where NL = Np/N. Now the system matrix is of size NL
x NL. Hence, the memory limitation problem can be managed nicely by breaking the
surface into enough subsections.

To illustrate the use of the above technique the scattered field from a simple
sinusoid surface was computed. The surface consists of 255 segments with the incident
wave and scattered field calculated at 30 degrees with respect to the horizon. The
polarization of the incident field was parallel to the crests of the surface (TE polarization).
The surface geometry consists of 11 periods, a period length of .1016 1n, and a peak to
peak wave height of .0254 meters. The surface was divided into a different number of
sections to determine the number of iterations and computational time to solve for the

current. All computations were performed on a Pentium-100 system (24 Mbytes). The

59

 

 

 

 

 

iteration stopping criteria for all

shows the computational result.
example illustrate the effectivcn
live segments does not lead to l
theralculations. However. div
longer computation times. This
ditidcd into 15 segments.

A second example consi.
owning the same test. Table
frequencies. The results in this

umber of surface segments use

airfare segments but not enough

35 Conclusions

This chapter reviewed tw
thesis. First, the E-pulse techni
illustrating the application of the
ellirulment. Next, the numeri
scattering from a finite-length. pe

The use of the spatial decomPO’

“868 were described. The 51"

comlltltet systems with a limited

 

 

iteration stopping criteria for all work was set at a relative tolerance of 10“. Table 3.1
shows the computational results for three different frequencies. The results of this
example illustrate the effectiveness of this technique. Dividing the surface into three or
five segments does not lead to longer computation time and requires less memory to do
the calculations. However, dividing the surface into too many segments can lead to
longer computation times. This can be seen for the case where the surface has been
divided into 15 segments.

A second example consists of dividing the same surface into 500 segments and
running the same test. Table 3.2 shows the computational results for three different
frequencies. The results in this table suggest that this technique is very effective for the
number of surface segments used. The computation time varies for different numbers of

surface segments but not enough to suggest any large increases in computation time.

3.5 Conclusions

This chapter reviewed two important topics which will be used throughout this
thesis. First, the E-pulse technique was presented. Several examples were presented
illustrating the application of the E-pulse method to the detection of target in a sea-clutter
environment. Next, the numerical solution to the electric field integral equation for
scattering from a finite-length, perfectly-conducting, 2-dimensional surface was reviewed.
The use of the spatial decomposition technique was presented and several sample test
cases were described. The spatial decomposition technique is extremely useful for

computer systems with a limited amount of memory.

60

 

 

 

     
    

Table 3.1 Spatial
from 255

 

* ‘ ‘ u l
double precrsron complex ant

 

 

 

 

 

 

 

 

 

 

Table 3.1 Spatial decomposition iterative scheme efficiency results for scattering
from 255 segment sinusoid surface
Ns/Np num. of iters. comp. time matrix storage
(sec) (kbytes)*
f = 3.0 GHz

1/255 1 15 1016

3/85 12 21 112

5/51 12 15 41

15/ 17 35 36 5

 

 

 

f= 3.1GHz

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1/255 1 15 1016
3/85 11 19 112
5/51 13 17 41

1 15/17 29 31 5

f= 3.2 GHZ

1/255 1 15 1016
3/85 ll 19 112
5/51 l4 19 41
15/17 34 35 5

 

* double precision complex arithmetic (16 byte arithmetic)

 

 

61

 

 

 

 

 

 

 

Table 3.2 Spatial decompos
from 500 segment

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1/500
2/250
4/125 1
5/100 1
10/50 ;

 

 

 

 

 

 

 

" double precision complex arit

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 3.2 Spatial decomposition iterative scheme efficiency results for scattering
from 500 segment sinusoid surface
Ns/Np mum. of iters. comp. time matrix storage
(sec) (kbytes)*
f = 3.0 GHz
1/500 1 107 3906
2/250 5 81 977
4/ 125 15 103 244
5/ 100 16 94 156
10/50 23 104 39
i f = 3.1 GHz
1/500 1 108 3906
2/250 4 65 977
4/ 125 16 110 244
5/100 15 88 156
I 10/50 24 108 39
f = 3.2 GHz
1/500 1 107 3906
2/250 5 80 977
4/125 15 103 244
5/ 100 15 88 156
10/50 24 109 39

 

 

 

 

 

* double precision complex arithmetic (16 byte arithmetic)

 

 

 

 

 

 

 

0.03 -
0.02 -
0.01 4

Height (m)

 

0.03 -
0.02 ‘
0.01 —

Height (m)

 

0.00 -

0.00 -t
6

C.6 —-O.4

Figure 3.1 Simple sinusoid a

 

 

 

 

 

E 0.03

v

4..» 0.02 —

.C

C) 0.01 a

$0.00 rlrlflllllfilrtlltrrirtr
I —0.6 —0.4 —0.2 0.0 0.2 0.4 0.6

Wave Position (m)

 

 

 

 

E 0.03

V

+.- 0.02 -—

.C

C) 0.01 —

a) 0.00 rrrlrrrij—FI—vflrrfrlrrrrnfi
I —O.6 —0.4 -—0.2 0.0 0.2 0.4 0.6

Wave Position (m)

Figure 3.1 Simple sinusoid and double sinusoid surface geometry.

63

 

 

 

 

< m a .
1W ) a a

lbw «in

u

0 0 w s

0. :.::—2.2.2:... . 22:. . .fn ad

0 0 O 1.1
s w. w w. m. m. m m. m. w. w m. m. m w m
0 0 o 0 0. o 0 o O o 0 0 O 0 low. M
A0>_uo_omv oUBEOOE Ao>52mmv onstcoofi C.m

Figure 3.2

 

 

 

 

 

 

 

Figure 3.2

 

 

 

 

 

 

 

 

 

0.50 :
i (o) Sinuoidol Surface
A 3
(D :
.2 0.40 —
+1 :1
g :
a) I
D: E
V0.30 -:
‘D E
'O _
:5 :
it” 2
C 0.20 ‘:
0‘ :
O :
2 1
0.10 i
0.00 BOT"lTlrrIIIIIWIIIIITYIIIIIIIIIIlllTlI—Fl]
0.04.00 8.00 12 00 16 00
Frequency (GHz)
0.60 :
0.50 3 (b) Double Sinuoid
A 3
<0 :
,2 0.40 :
+J _
g :
a) Z
0: ;
V0.30 —_
(D E
U ..
:3 :
It” I
C 0.20 '3
U” :
U :
E :
0.10 E
0.00ddldllflfillTlfIflllrlrllel—llerl—llelrl—j

 

 

4.00 8.00 1200 to 00
Frequency (GHz)

.0
0

Calculated scattered field from (a) single sinusoid surface and (b) double
sinusoid surface.

64

 

 

 

b .

Fi

_.J .quljiq i3. aj #1 «lqldlqlqlu- .alql— 110% 301th

.t- A0>_#C_o/mv 003..:C.m00_\<

411'

 

 

1.20

 

 

rmoé
A E
(D :
.2 0.80 :
+1 _
£3 :
<1) :
Ct: :
V0.60 E
a) :
“O :
3 :
.4: _
c 0.40 E
U) _
o :
2 E
0.20 E
0.00 - I I I I I I I I I I I I I I I I I I I I I I I I I l I l Ij
0.00 5.00 10.00 15.00

Frequency (GHZ)

Figure 3.3 Spectral domain of incident waveform.

65

 

4.114-4214. -qtafll_tr—t..-~..4-41.114!«la. _ _ q _ a a q a a q q _ . q 4 u . _ q q q q H
5.
O
‘ .

1.001

NU
r3
AHU

0

ob 0 O 0
0 0
AtU 4|
mv>mu..0\0wu

mo?3.w.lnuCL/\

 

Time domain rcpt

Figure 3.4

 

 

1.00

0.50

Irrtrttrrlrrrrrrrrrl
E
'3

 

 

 

 

 

 

 

 

Relative Amplitude
O
O
O
l

 

—O.50

 

 

 

 

 

 

 

 

rrrrrrrrrlrrrrrrrrr
;
K

t

—1oOO lIIIIIIIIIIIIIIIIII|IIFIIIIIIIIIlIlIIII—]

0.00 0.50 1.00 1.50 2.00
Time (nsec)

 

Figure 3.4 Time domain representation of incident pulse.

66

 

 

“the 3,5

A .—
4 J
4
‘
r.
" A x
,4 v'
.-
‘
..
.1
_.
F a
v
F n
> —.
V04
.—
g-v ..
.—
v -.
..
.-
v
A/
.-
_—
..
.-
a.
U
“ -
v
q .n
_4
H .-
.—
A A
,—
5 bid
H
w
a
V -
~.
_.
..
..
..
-.
\ AA
m
h 5»

)
’J
)

t
(
(

F
(\J
(I)

Q
r\)
C)

0.15

Magnitude (Relative)
Q
S

0.05

lrrirnrirrlrrnnrrrr11rrrr1111111rLLJJJJhirJJLLLLJJJrrritr.ritl

 

0.00

p
o
of

Band-limited resr
smusoid surface.

 

 

0.02

 

 

 

 

 

 

 

 

: (a) Sinuoidal Surface
A 7
Q) —
.2 0.01 -
44 ..
g _
G) ..
D: _
V _
a) _
.0 _
3 -
0t) —.
C 0.01 -
U) _
o _
2 _
0.00 IIIIIIIIIIIIII'IIII'IIIIIIIII IIIIIII
0.00 4.00 8.00 12.00 T5700
Frequency (GHz)
0.30 '2
0.25 —f
E (b) Double Sinuoid Surface
A E
(D :
.2 0.20 :
4.1 ..
_O :
a) 2
D: 5
V0.15 :
Q) E
"O _
3 :
.t’ :
C 0.10 ‘_'
U“ :
O :
2 :
0.05 E
0.00-IIIIIIIII IrIrIIIII IIIIIrTIIII'IIIIIIII
0.00 4.00 8.00 12.00 16.00

Frequency (GHZ)

Figure 3.5 Band-limited response for (a) single sinusoid surface and (b) double
sinusoid surface.

67

 

 

Figure 3-0

If"

.aa

a Pr

-.

4.:
,.
v
.4
r.
v
t
v
A/

v
n.

u
-v-‘
a
W
a
c—v
.—
A
u.
>—

A '-
h_ a
< V»

a
v
A
u
a «A
-V\r

0.00

~0.50

Amplitude (Relative)

OO

ttrtttrrrlrttrittrtl

 

rtrtrtrrrlrrtrrtttt

 

 

 

 

 

LLJLIJLIJAII 111111 11L1_1_1_1_1__1___1__L__L_1__1_11 1,1,1 11.]!

 

 

8?

Figure 3.6

(a) Simple Sinusoid Surface
TCR = -2.2 dB

.0
U!
a

:ll
‘lllfl.

9
o
o

‘l l‘.

u‘l
til

plitude (Relative)

—O.50

Am

 

_1.00IIIIIIllIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIImj—Ian ‘l

0.00 2.00 4:00 6.00 8.00 10.00
Time (nsec)

1‘00 (b) Double Sinusoidal Surface
TCR = —15.5 dB

.0
or
a

(Relative)

Amplitude

—O.50 ‘

_1.00 I I I I I I I I I I I I I I I I I I I I I I I I T I I I I I I I I I I I I—I—I
0.00 l0 4.00 6.00 8.00
Time (nsec)

Transient response for missile above (a) single sinusoid surface and (b)
double sinusoid surface.

68

_ p. 0 NH.
R
rk
.0
AV
_—~‘_«.‘_‘__._4u_<—~._.—4-__q ~_..4.~_.«.—_q_—-n .w —._-‘uwnqqq~<~uq-~dquqaianqrfina44|fllfiqr m m
u 3 C 3 .. .1. .. : c n- 0 0 0 00 W».
re re A. .. .. .. a. A.. Pr. r» O 0 rl e
. . . . . . . . Fw C . . O flu. I C
.L o . . . C T. we. 7.... 1 O «I 2 3 4 M 0m
_ _ _ _ . 2 0 l
AL>:C_LZV .;::_:?S_2 A©>ZGC®WC 0U3£C®U§ C W
al.
3
e
r
u.
:.we
F

 

2.00

 

 

 

 

1.00

0.00

111 111 111 111 111 111 J]! 111 111 III

 

—l.00

 

 

 

Magnitude (Relative)

 

 

 

—2.00

(a) Simple Sinusoidal Surface

11 11 It 11 It 11 11 11 ll

 

 

_3.00 IIIIIIIIIIIIIIIIIII‘IIIFIIIIIIIIFIIIIIIIIIIlllIIIII

0.00 0.05 0.10 0.15 0.20 0.25
Time (nsec)

3.00

 

2.00

 

 

 

 

1.00

0.00

 

—1.00

 

 

-2.00

Magnitude (Relative)

 

 

 

-3.00
(b) Double Sinusoidal Surface

trrrtrrrrlrttrrr1111111111111 lllllllllIlllllllllIlllllllllIlllllllllI

[I lit rrr ill I]

— I I I I I I I I II I I I I I I I I I I I I I I I I I I I I I I I I
4“0.00 0.05 0.1'0 0.15 0.20 0.25
Time (nsec)

 

Figure 3.7 Constructed CRTWs for (a) single sinusoid surface and (b) double sinusoid
surface.

69

 

 

 

,- an”

4.4U

t )
c )
t )

(lx’crlcrtivv)

A AAA

u.uu.a

A AA.

M()(]rlilll<lt‘

gov».

“ «AA

....4;

i)[rttrr1rrrlrtttrrrttgrtrttttttliriitrrrtl
() ’

" "K‘IA

v.\_.-L, 'U

 

0.0000

(F)
L)

O

C)

U1

11111141111111111111.1«11LL__1.1.11
m—
_—

Amplitude (Relative)

~0.0005

 

141111111111

.00
O
O

‘0001

 

Convolution of C
Single sinusoid su

 

 

 

 

0.002 :
E (b) Simple Sinusoidal Surface
_ with target
0001 —'
A -
Q) —.
> .—
L: - ,
O I 1‘
E - ”3‘ l
a: : ‘11 llt"
V 0.000 : , l” l
a) - l .1 - .
-O _ I fly
3 - ‘. l
.4: I l .
C _ I
0‘ :
g —0.001 :
—0.002 IIIIIIIIIIIIII7IIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
0.00 2.00 4.00 6.00 8.00 10.00
Time (nsec)
0.0010 :
I (b) Double Sinusoidal Surface
: with target
0.0005 f
d —
a) _
> _
‘4: _
—O ._
c0 2
I _
v 0.0000 :
(D I
"O _
3 ..
:4: _
E I
E _ :
< 0.0005 _
—0.0010‘IIIIIIIII IIIIIIFIIIIIIIIIIIIIIIIIIIIIII
0,00 2,60 4.00 6.00 8.00

Time (nsec)

Figure 3.8 Convolution of CRTW with missile response. immersed in clutter for (a)
single sinusoid surface and (b) double sinusord surface.

70

 

   

Figure 3.9

TE scatter field 0

h.
\

 

7‘1
j
59'

 

 

 

Figure 3.9 TB scatter field geometry.

71

 

 

 

 

 

 

Enhanced Target

4.! Introduction

A basic problem faced
skimming missile immersed in
nideband (UWB) radar system
use of an UWB system becom
is small compared to the
(characteristic of UWB radar) .
transmit waveform (C RTW) ca
enhance the target response [22

A new technique, basec
allows detection of low signal I:
difficulties in using the classic:
eradicate merely the sea-clutte
attenuated, resulting in a poor
problem is not to eradicate the
energy ratio.

The content of this cha

presents the theory and algorith

lichnique. The basic ideas behi

 

 

 

Chapter 4

Enhanced Target Detection in a Sea Clutter Environment

4.1 Introduction

A basic problem faced by on-board ship radar systems is the detection of a sea-
skimming missile irrnnersed in background clutter from the sea-surface. Interest in ultra-
wideband (UWB) radar systems arise from their potential use for target detection. The
use of an UWB system becomes more important when the signal returned by the target
is small compared to the background clutter. Using the increased bandwidth
(characteristic of UWB radar) and the periodic nature of sea swell, a clutter reducing
transmit waveform (CRTW) can be created which will eradicate the clutter return and
enhance the target response [22].

A new technique, based upon E-pulse concepts [24], has been devised which
allows detection of low signal targets in a sea-clutter environment. One of the inherent
difficulties in using the classical E-pulse method is that when an attempt is made to
eradicate merely the sea-clutter return, both the sea-clutter and target response are
attenuated, resulting in a poor target to clutter ratio (TCR). A new approach to the
problem is not to eradicate the clutter altogether but to maximize the target to clutter
energy ratio.

The content of this chapter will be divided into several sections. Section 2

presents the theory and algorithm deve10pment for target detection using the new CRTW

technique. The basic ideas behind the CRTW development will be presented as well as

72

 

 

 

 

 

 

qualitative arguments supporti

diagram summarizing the basi

The new CRTW techni
the drawbacks of this techniqu
function of many parameters an
optimization routine. Sectio
optimization problem.

Several examples will
technique in sections 4 and 5.
thighly conducting sea-surface
from a small missile model. A
whee-target retum data set. 1
short that this new technique vs
changing sea-surface is extreme
considers a more realistic mos
simulation of an evolving set
ntunen'cally calculated. Using at
initial sea-surface. A simulatior
the evolving sea-surface was
simulation show the effects of

need for periodic updates to an

Target detection using t

 

 

qualitative arguments supporting the usefulness of this new technique. Finally, a block
diagram summarizing the basic algorithm for target detection will be presented.
The new CRTW technique will be shown to be quite effective. However, one of
the drawbacks of this technique is in the construction of the CRTW. The CRTW is a
function of many parameters and therefore an optimal solution requires the use of a global
optimization routine. Section 3 will discuss a genetic algorithm solution to the
optimization problem.
Several examples will be presented showing the usefulness of this new CRTW
technique in sections 4 and 5. The first example uses the measured clutter return from
a highly conducting sea-surface model, in conjunction with the measured scattered return

from a small missile model. A CRTW is constructed and applied to a combined sea-

 

surface/target return data set. This first example, presented in section 4, is designed to
show that this new technique works for a simple static situation. Since the effect of a
changing sea-surface is extremely important, a second example presented in section 5
considers a more realistic model that can evolve over time. In this case, a time-
simulation of an evolving sea-surface was created and the scattered fields were
numerically calculated. Using a measured missile model, a CRTW was calculated for the
initial sea-surface. A simulation was then performed in which a missile traveling over
the evolving sea-surface was detected using the CRTW technique. Results of that
simulation show the effects of an evolving sea-surface on the CRTW technique and the
need for periodic updates to an initial CRTW.

Target detection using the new CRTW technique involves the computation of

73

 

fi

convolution energy ratios. The
nindow size. At certain point:
151 target enters the radar syst
the integral over which the can
determine the effect of window
nep of the CRTW algorithm.

CRTW construction is b
isoptimized for a specific targe
CRTW be effective in detectin;
inill address this topic.

The effect of multipath
detection problem will be discu

Finally. section 9 will

processing clutter reduction tecl

4.2 Theory

Consider a UWB radar
where a target is anticipated. I
finite portion of the sea-surface
time range r < t < r+W, where
pulse concepts, can be constrt

modeled as a series of complex

   

 

 

convolution energy ratios. The convolution energy ratio is a function of time and time-
window size. At certain points in time, the energy ratio will take on maximum values
as a target enters the radar system’s range bin. The window size defines the domain of
the integral over which the convolution energy is calculated. The goal of section 6 is to
determine the effect of window size on the convolution energy ratio during the detection
step of the CRTW algorithm.

CRTW construction is based on an anticipated or expected target; i.e., the CRTW
is optimized for a specific target. This, however, raises an important question: will the
CRTW be effective in detecting other targets of similar size or configuration? Section
7 will address this topic.

The effect of multipath and target/sea-surface electromagnetic coupling on the
detection problem will be discussed in section 8.

Finally, section 9 will compare the new CRTW technique to the coherent

processing clutter reduction technique as discussed by Iverson [26].

4.2 Theory

Consider a UWB radar system illuminating a finite portion of the sea surface
where a target is anticipated. If the two-way transit time of the radar signal across the
finite portion of the sea-surface is W, then the transient scattered field is available in the
time range 1: < t < t+W, where I is the time of measurement. A CRTW, based on E-
Pulse concepts, can be constructed if the clutter return from the sea-surface can be

modeled as a series of complex exponentials

74

 

 

 

e(t) =

where A, and Q, are comp
Furthermore. the CRTW e(t). li

when convolved with the sea-c

r(t) = e(t) -r(t) =

Hence. only a small signal will
seaelutter.

One of the problems at
signal embedded in the clutter r4
roclutter ratio is not really imp

arch that the following energy r

r-A/Z
f Ietx)

r-A
50.7,”) = #37

1M!

lnthis case, the energy ratio is r
is the time response of an antic

target response within the time

suPlotting the use of (4.3) can

 

 

 

 

 

 

N
C(t) = Z AneQn’ 1:< 2‘ <1,- +W (4.1)

n=—N
where A,1 and Qn are complex parameters appearing in complex-conjugate pairs.
Furthermore, the CRTW e(t), like the E-pulse, is a waveform of finite duration TE which
when convolved with the sea-clutter signal yields the null result

r+W

r(t) = e(l)*C(t) = f e(t/)e(t—t’)dt’ = 0 r+TE< t <17+W (4.2)

17

Hence, only a small signal will be returned if the CRTW is radiated in the presence of

sea-clutter.
One of the problems arising in the construction of the CRTW is that a target
signal embedded in the clutter return is also reduced, often to such a point that the target-

to-clutter ratio is not really improved. An alternative to (4.2) is to construct a CRTW

such that the following energy ratio is maximized

 

t-t-A/2
f {e(x)*[c(r+x)+T(t’+x)]1de
_ A A
/ _ r A/2 __ __ .
e(t,1:,t) — “M2 TE+ 2 < t <W 2 (4 3)

f {e(x) *c(1: +x) lzdx

t—A/2

In this case, the energy ratio is computed in a window of length A centered at time t. T
is the time response of an anticipated target and t’ is a parameter which time shifts the
target response within the time range bin of the clutter signal. A qualitative argument

SUpporting the use of (4.3) can be made by observing that the term in the denominator

75

 

 

 

   
  
  
 
  
 
  
 
   

should be quite small as give
terms: the convolution of the
CRTW with the time-shihed ta
CRTW/clutter convolution to
the net result is that the en
To envision the detectio
atsome initial time to a measu
hpre-recorded response T(t) o
clutter waveform. Next. a CR
in(-l.3). In this case the ener
represents the position of the e
Careful observation of (4.3) a1
window width A. The optimal
maximum energy and optimal t
that rm = tm’,
Once e(t) has been det

Surface return at some later tim

rvA/Z

f rec

E0) = ————-““”

is computed. If no target has

essentially stationary, the enert

 

 

 

 

 

 

should be quite small as give by (4.2). On the other hand, the numerator contains two
terms: the convolution of the CRTW with the clutter return and the convolution of the
CRTW with the time-shifted target response. Once again, consider the term involving the
CRTW/clutter convolution to be small, but hopefully the second convolution will not be.
The net result is that the energy ratio may be significant for the correct choice of e(t).
To envision the detection process, the ensuing procedure must be followed. First,
at some initial time To a measurement is made of the sea-clutter waveform cO(t) = C(T0+t).
A pre-recorded response T(t) of the anticipated target is then added to the measured sea-
clutter waveform. Next, a CRTW is constructed to maximize the energy ratio a(t,to,t’)
in (4.3). In this case the energy ratio is a function of the parameters t and t’ where t
represents the position of the energy window and t’ corresponds to the target time shift.
Careful observation of (4.3) also shows that the energy ratio is a function of the time
window width A. The optimal positions of tm and tm’ represent the window position for
maximum energy and optimal target position for detection. In most cases it is expected
that tm = tm’.
Once e(t) has been determined, detection can progress by measuring the sea-

surface return at some later time t > 130. At this time an energy ratio given by

t+A/2
{e(x) *c(r +x)}2dx

"' z “A” T A< <W-~A 4.4
e(t) mm 5+2 1? 2 ( )

] {e(x) *c(t0+x)}2dx

t—A/2
is computed. If no target has entered the observation bin and the sea surface remains

essentially stationary, the energy ratio will be unity for all t. Given that the denominator

76

 

 

 

 

 

term in (4.4) remains small. th

when a target enters the range
to the target position and sho
position corresponding to tn.

It is important to consi
>r,r the sea-surface will be di
computed using (4.4) will slov
detect a target in the range bit

retompute e(t). Figure 4.1 sho

43 Computational Considr
The construction of a
matimizing the ratio given in l

basis functions with amplitudes

where

8,,(1) =

The energy ratio given by (4.3)

amplitudes (1,, (b) E~pulse durat

the window duration A and nu

 

 

 

 

 

 

term in (4.4) remains small, the value of E(t) should be significantly greater than unity
when a target enters the range bin. The value of E(t) should be large for t corresponding
to the target position and should reach a maximum value when the target reaches the
position corresponding to tm.

It is important to consider the effect of an evolving sea-surface on EU). For t
> To, the sea-surface will be different than that used to compute e(t) and the energy ratio
computed using (4.4) will slowly change. As EU) rises above unity, the ability to
detect a target in the range bin will degrade. It is therefore necessary to periodically

recompute e(t). Figure 4.1 shows a flowchart for the detection process.

4.3 Computational Considerations
The construction of a CRTW for the new target detection scheme involves

maximizing the ratio given in (4.3). The CRTW, modeled. as a sum of N rectangular

basis functions with amplitudes oek, can be expressed as

N
300 = 2 “kg/(00 (4'5)
k=1
where i T T
1, —NE(k—1) < x < Xi
3.3/(0‘) = i (4.6)
0, elsewhere

 

The energy ratio given by (4.3) is dependent on the following: (a) E-pulse basis function
amplitudes ak, (b) E-pulse duration. TE, and (c) target response time shift t’. In addition,

the window duration A and number of rectangular basis functions N will effect the value

77

 

 

 

 

of a calculated in (4.3). Ho

respect to these variables. but
The process of numc
problem and choosing an appr
First the amount of compute

important. Second. but proba
aglobal maximum. This. h0“'t
isafunction of many variable
etident that it may be quite d
possible altemative is to design :
byapplping a global search ro
gadient search algorithm to do:
algorithm has extreme difficult}
often gets trapped in local minir
tortof initial solution guess to 1
'seed" or initial guess for the g
A global optimization 5c

M One scheme often used

problems based on the ideas of

 

Population of N individuals kn

Population can be represented b

Lbits. In analogy to genetics, 6

 

 

 

of 8 calculated in (4.3). However, the author did not choose to Optimize (4.3) with
respect to these variables, but regarded them as fixed during the optimization process.

The process of numerically optimizing (4.3) is a computationally expensive
problem and choosing an appropriate solution technique requires several considerations.
First, the amount of computer time required to find the solution can be extremely
important. Second, but probably more important, is the convergence of the solution to
a global maximum. This, however, can be computationally slow for an expression that
is a function of many variables and contains many local extreme values. It is quite
evident that it may be quite difficult to get both speed. and global optimization. A
possible alternative is to design some hybrid method whereby the problem may be solved
by applying a global search routine to target a global maximum and then applying a
gradient search algorithm to determine a better solution. By itself, a gradient search
algorithm has extreme difficulty finding a "best" solution since this type of algorithm
often gets trapped in local minima. In addition, these types of algorithms require some
sort of initial solution guess to get started. A hybrid method can be used to produce a
"seed" or initial guess for the gradient search routine.

A global optimization scheme should be used to find those values which maximize
(4.3). One scheme often used is the genetic algorithm (GA)[27]. This method solves
problems based on the ideas of heredity and evolution of the fittest. A GA maintains a
population of N individuals known as chromosomes. Each chromosome within the
pOpulation can be represented by a data structure containing M bit strings each of length

L bits. In analogy to genetics, each bit string is referred to as a gene. Furthermore, the

78

 

 

 

 

bit string representation in t
parameter. A problem contai
single chromosome. Figure
structures used in this analys
potential solution to the optim.
with each chromosome. a new
discards those with weak trai
process called crossover. in at
population. Each new generatit
and mutation. Successive ge
hopefully afier a number of
settings.

The GA is initiated by
be done for the entire populat
Next each chromosome is eva
done by decoding the binary gt

xj represent a design paramete‘

bit string of length L called gJ

where g“ is the 1th bit of g

function, such as (4.3), is ev

 

 

bit string representation in each gene is an encoding of the actual problem design
parameter. A problem containing M design variables can be encoded as ML bits on a
single chromosome. Figure 4.2 shows the relation between different genetic data
structures used in this analysis. Each chromosome from the population represents a
potential solution to the optimization problem. By calculating a fitness value associated
with each chromosome, a new population can be formed which selects fit individuals and
discards those with weak traits. Also, traits from fit individuals can be mixed in a
process called crossover. In addition, bit mutations can occur at random throughout the
pOpulation. Each new generation is formed through the processes of selection, crossover,
and mutation. Successive generations will tend to create more fit chromosomes and
hOpefully after a number of generations the genes will contain the optimized variable
settings.

The GA is initiated by filling each gene with random bit information. This must
be done for the entire population of N chromosomes ( where N is an even number ).
Next, each chromosome is evaluated to give some measure of its fitness value. This is
done by decoding the binary gene strings into actual design parameters. Let the variable

max
I

xj represent a design parameter defined on [ijm,Xj The variable xj is coded as a

bit string of length L called g- and recovered through

. X?” -— X .mi" L
: mm 1 J ' 1‘1 4.7
xj X] + ————-———————2L - 1 12:; 81,1 2 ( )

where ng is the lth bit of gene gj. Using the decoded. gene values, a user defined

function, such as (4.3), is evaluated. After all the individuals in the population are

 

 

(”it

 

fi

evaluated. the most fit chrome

calculating an expected allocar

where F, represents the fitness
chromosome are copied to the
the integer pan of the enclosed
areclrosen at random with pro
Following the selection
traits from the population pool
of chromosomes from the chro
defined probability Pc by swap;
data chain. This point. known
Here N, is the chromosome
probability Pm as bit string swa
mutation, the new population
repeated for a predetermined 1

genetic algorithm.

4-4 Stationary Surface De
To demonstrate some 0

fields from two highly cond

consisting of aluminum foil a

   

 

evaluated, the most fit chromosomes are selected. A simple selection scheme involves

calculating an expected allocation value defined as

ek = N114]: 17,)“1 (4-8)

where Fk represents the fitness value of the kth chromosome. Next, [ek] values of the kth
chromosome are copied to the new breeding population. Here, square brackets indicate
the integer part of the enclosed value. The remaining elements of the breeding population
are chosen at random with probability ek - [ek].

Following the selection process, a new population is created by mixing inherited
traits from the population pool. This is done by randomly selecting or mating N/2 pairs
of chromosomes from the chromosome pool. The chromosome pairs are bred with user
defined probability PC by swapping bit string information at a point along the chromosome
data chain. This point, known as the crossover point, is a random integer from 1 to NC-l.
Here NC is the chromosome length. Finally, a random chosen bit is mutated with
probability Pm as bit string swapping is taking place. Following selection, crossover, and
mutation, the new population is ready for its next generation. The above steps are
repeated for a predetermined number of times. Figure 4.3 shows a flowchart for the

genetic algorithm.

4.4 Stationary Surface Demonstration
To demonstrate some of the ideas presented in the preceding section, the scattered
fields from two highly conducting sea—like surfaces were measured. The surfaces,

consisting of aluminum foil adhered to styrofoam [12], have the cross—sectional shape

80

 

 

 

fi

shown in Figure 4.4. The St
measured within an anechoic c
points. The frequency-domain t
shapeparamcter t = 8 (see A
usingan IFFF to give the clutte
toallow measurements in the a
The first surface. knowt
and is a simple model used to s
isdiscussed in some detail in
cycloid of 10 periods. Each p
.ltt96 111. Using an electric fi
incidence angle of 10“ from the
infigure 4.5. As can be seen
main crests of the Stoke‘s wav
double sinusoid and was measr
surface is characterized by
lit) = .025(1-cos35.4x) +
shown in Figure 4.6. Once a
the sea-wave crests. A 10 cm
expected target. The scattere

measurement the electric field

angle of incidence was again

   

shown in Figure 4.4. The scattered fields from the two-dimensional surfaces were
measured within an anechoic chamber in the band 1 to 17 GHz using 1601 frequency
points. The frequency-domain data was then windowed using a cosine taper function with
shape parameter 1: = 8 (see Appendix A) and then transformed into the time domain
using an IFFT to give the clutter signal co(t). The dimensions of the surface were chosen
to allow measurements in the anechoic chamber.

The first surface, known as a Stoke’s wave [19], is characterized by steep slopes
and is a simple model used to simulate periodic ocean waves. The Stoke’s representation
is discussed in some detail in section 2.3.]. Figure 4.4 shows the case of a simple
cycloid of 10 periods. Each period for this surface is .1778 m and the wave height is
.0496 m. Using an electric field parallel to the wave crest (TE polarization), and an
incidence angle of 100 from the horizon, the scattered field for the Stoke’s wave is shown
in Figure 4.5. As can be seen, the scattered field is dominated by reflections from the
main crests of the Stoke’s wave. The second surface, also consisting of 10 periods, is a
double sinusoid and was measured under the same conditions as the Stoke’s wave. This
surface is characterized by two-scale roughness. The wave profile is give by
y(x) = .025(1 —cos 35.4x) + .06sin 177x (m). The scattered field for this surface is
shown in Figure 4.6. Once again the scattered fields are dominated by reflection from
the sea—wave crests. A 10 cm long scale-model Phoenix missile model was used as the
expected target. The scattered field from this target is shown in Figure 4.7. In this
measurement the electric field was perpendicular to the long axis of the missile and the

angle of incidence was again 10° with respect to the long axis of the missile.

 

 

 

 

fi

Using the target and c
Figure 4.8 were constructed
construcrion of the CRTW (tr
algorithm. To simulate the det
20% of the clutter maximum v
response. For the Stoke‘s wav
two locations: I = 4.9 nsec ant
raspouse was added at location
ratio response given by (4.4) v
Figure 4.9 and Figure 4.10 fort
included in these figures is the
(4.4). Figure 4.9 shows that v
reaches 22 dB and the target i:
=8nsec has a lower value E =
the target. The energy ratio c
sintilar patterns. At t = 11 nser
detection. In contrast, at t =

location for a target to be d

 

Window have a value of EU)
signal used in the detection so

As discussed in section

frequently reduces the target si

   

 

Using the target and clutter responses scaled to unity, the CRTWs shown in
Figure 4.8 were constructed by maximizing the ratio given in (4.3). The actual
construction of the CRTW (maximization of (4.3)) was implemented using a genetic
algorithm. To simulate the detection response, the missile scattered field was scaled by
20% of the clutter maximum value (TCR = -14 dB) and added to the sea-clutter surface
response. For the Stoke’s wave the target response was added to the clutter response at
two locations: t = 4.9 nsec and t= 7.8 nsec. For the double sinusoid surface the target
response was added at locations t = 5.6 nsec and t = 11.0 nsec. The convolution energy
ratio response given by (4.4) was computed for each surface. The results are shown in
Figure 4.9 and Figure 4.10 for the Stoke’s and double sinusoid waves, respectively. Also
included in these figures is the summed target and clutter response given by c(t+x) in
(4.4). Figure 4.9 shows that when the target is located at t = 5 nsec the energy ratioE
reaches 22 dB and the target is easily detected. On the other hand a target located at t
= 8 nsec has a lower value E = 3 dB indicating that this is not the best location to detect
the target. The energy ratio corresponding to the double sinusoid (Figure 4.10) shows
similar patterns. At t = l l nsec the ratio is 9 dB indicating a large jump and hence target
detection. In contrast, at t = 6 nsec the value of E is much smaller and not the best
location for a target to be detected. In both figures, points outside the convolution
window have a value of E(t) equal to unity (0 dB). This follows since the summed
Signal used in the detection scenario is identical to that used to create e(t).

As discussed in section 4.2, a CRTW designed to reduce only the clutter response

frequently reduces the target signal to such an extent that the target to clutter ratio in the

82

 

 

 

{coding convolution response
example of the Stoke’s wave g
eliminate only the sea-clutter
clutter only response and th
Figure 4.12 and Figure 4.l3
reduction in the sea-clutter res
inboth the sea-clutter and tar
clutter and target response I
Computation of the convolutio
compared to Figure 4.9. then s
using only the sea-clutter retu
magttitude of the convolution er
peaks in Figure 4.14 are signifit

indicate that this CRTW has Ii

needed to calculated the wavefr

45 Simulated Sea-Surface
A more realistic sea-su
evolving sea surface profile y(

We!) =

0

Where 6(a) is a phase shift ra

 

 

resulting convolution response is not improved. This effect can be seen using the
example of the Stoke’s wave given above. Figure 4.11 shows the CRTW constructed to
eliminate only the sea-clutter return. The convolution of this waveform with the sea-
clutter only response and the combined target/sea-clutter response are shown in
Figure 4.12 and Figure 4.13 respectively. Figure 4.12 clearly shows a significant
reduction in the sea-clutter response; however, Figure 4.13 shows a significant reduction
in both the sea—clutter and target response. In this case, the reduction in both the sea-
clutter and target response leads to a very poor target to clutter detection ratio.
Computation of the convolution energy (4.4) is shown in Figure 4.14. If Figure 4.14 is
compared to Figure 4.9, then several problems associated with constructing the CRTW
using only the sea-clutter retum can be seen. First, their is a significant reduction in the
magnitude of the convolution energy ratio. In addition, the number of and position of the
peaks in Figure 4.14 are significantly different than those in Figure 4.9. These findings
indicate that this CRTW has limited usefulness even though a great deal of effort was

needed to calculated the waveform.

4.5 Simulated Sea-Surface Demonstration

A more realistic sea-surface profile has been proposed by Kinsman [19]. An
evolving sea surface profile y(x,t) can be computed using the stochastic model

00

y(x,t) = fcosi—g—Z—x — or + d>(o)]i/[A(o)]2do (43)
g

0

where (I) ( 0) is a phase shift randomly distributed between 0 and 27:. Here the Neumann

83

 

 

 

 

spool frequency spectrum is

oth is the wind speed in
de = 3.05 mi’sec‘. A typi
Figure 415. A numerical
calculated the covariance fune

limction [19] may be mitten a

H(r}.r

In this case. an ensemble of

position. It is important to note
interval. This follows from the
320 knot wind. Figure 4.16 Si

interval T. For T = 0 the cova

Integrating the spectrum over

wave field, i.e.

Comparing (4.12) and (4.13) it

 

 

spatial frequency spectrum is used

[A(o)]2 = C; 0’66'282°_2U—2 (4-10)

where U is the wind speed in m/sec, g = 9.81 m/see2 is the acceleration due to gravity,
and C = 3.05 mz/secs. A typical spectrum generated using 20 knot winds is shown in
Figure 4.15. A numerical measure of the sea-surface evolution can be obtained by

calculated the covariance function at a fixed position on the surface. The covariance

function [19] may be written as

H(tj,tk) = 'ZifrAmnzcosro (z, —tj)]do (4.11)
0

In this case, an ensemble of functions {y(t)} is observed at times tj and tk at a fixed
position. It is important to note that the covariance is only a function of the observation
interval. This follows from the time-invariant statistics or stationarity of the process. For

a 20 knot wind, Figure 4.16 shows the covariance function in terms of the observation

interval T. For T = 0 the covariance can be written as

_ =1
H(tj,tk—tj) 2

f [A(o)]2do (4.12)
0

Integrating the spectrum over all frequencies gives a measure of the total energy in the

wave field, i.e.

°° 4.13
E = f[A(o)]2do ( )
0
Comparing (4.12) and (4.13) it is seen that
1
E = ~5H(tj,tk=tj) (4.14)

84

 

 

 

 

nuts ease the covariance for
through (4.14). Since the eneri
isdirectly related to the wind
inligure 4.16 coincides with

function is an indicator of sea

ofthe sea as a function of tim

 

thecovariance value decreases
seen that after about 2 secon
returns to its original value. a]
zero. With the covariance inf
scattered field must be remeas
beupdated more ofien than on
least once or twice a second.

A typical surface profile
shown in Figure 4.17 (sea heigl
profile was computed using a
to chapter 3, section 4). The
from the horizon. Due to com
afactor of 50 (from lOOOm tot

using l000 segments. A total

1.5 GHz.

To determine the effe

 

 

In this case the covariance for a step interval of 0 is directly related to the wave energy
through (4.14). Since the energy can be calculated from (4.13), it is seen that the energy
is directly related to the wind speed through (4.10). The point corresponding to T = O
in Figure 4.16 coincides with twice the energy given by (4.14). Since the covariance
function is an indicator of sea-surface evolution, Figure 4.16 illustrates the progression
of the sea as a function of time-separation T at a given point. As shown in this figure,
the covariance value decreases as a function of time separation. From this figure it is
seen that after about 2 seconds the covariance has dropped to about zero and never
returns to its original value, although it does slowly creep up and then returns back to
zero. With the covariance information it is possible to get some idea of how often the
scattered field must be remeasured. From Figure 4.16 the measurement must certainly
be updated more often than once every two seconds. A better update rate would be at
least once or twice a second.

A typical surface profile and scattered field generated. by the Kinsman model is
shown in Figure 4. I 7 (sea height is not to scale). The scattered field for a PEC with this
profile was computed using a 2-d Green’s function and moment method solution (refer
to chapter 3, section 4). The polarization is TE and the incidence angle is 10 degrees
from the horizon. Due to computational constraints the sea-surface was scaled down by
a factor of 50 (from lOOOm total length to 20 meter total length) and the field solved for

using 1000 segments. A total of 200 frequency points were computed in the band .5 ~
1.5 GHZ.

To determine the effects of an evolving sea-surface on the CRTW detection

85

 

 

 

technique, the following seen
generated using (4.9) at inte
scattered field was calculated
Phoenix missile model was
computed from the surface pro
scaled to a TCR of -l4dB and
the position of the missile wit
that the missile was flying at
evolving sea surface profile 3
Using the summed res
was computed for each time s
where it is assumed that the
simulation. At t = O the missil
has not changed, and therefore
dB). At t = .25 see the surface
this case the energy ratio is no
surface continues to evolve th
t= .75 sec the target enters th

the target. Since the baseline

been detected with a margin 0

 

thesimulation, the effect of be

htt= 2.0 see the sea surface

 

 

 

 

 

 

technique, the following scenario was devised. A series of sea-wave profiles were
generated using (4.9) at intervals of .25 seconds. Each surface was scaled and the
scattered field was calculated numerically as described above. The response from the
Phoenix missile model was scaled to match the clutter response and a CRTW was
computed from the surface profile at r = 0 sec. Next, the missile response was amplitude
scaled to a T CR of -l4dB and added to the evolving sea surface response. In this case
the position of the missile with respect to the sea surface was determined by assuming
that the missile was flying at 600 knots. The left hand side of Figure 4.18 shows the
evolving sea surface profile and the missile position (indicated by the small arrow).
Using the summed response and the initial clutter response, the energy ratio EU)
was computed for each time step. This is shown on the right hand side of Figure 4.18
where it is assumed that the CRTW computed at t = 0 does not change during the
simulation. At t = O the missile has not yet entered the range bin and the clutter signal
has not changed, and therefore the energy ratio computed from (4.4) must be unity (0
dB). At t = .25 see the surface has evolved but no target has entered the range bin. In
this case the energy ratio is no longer unity but has reached a value of 3 dB. As the sea
surface continues to evolve the baseline energy ratio (max value) continues to rise. At
t = .75 see the target enters the range bin and EU) reaches a max value of 20 dB near
the target. Since the baseline value of EU) is about 5 dB at t = .75 see, a target has
been detected with a margin of about 15 dB above the baseline level. Continuing with
the simulation, the effect of both the moving target and evolving sea surface can be seen.

At t = 2.0 see the sea surface has evolved to the point where only a 10 dB margin exists

86

 

 

 

 

 

between the baseline clutter

4.6 Target Detection and

The detection of any ta
involves computing convoluti
ratio involves integrating ove
the time-window interval in
convolution energy ratio. At t
the scattered return) the com
indicating the presence of a tar
of window size on the convoli
algorithm.

The study of the windr
investigation in sections 4 and
Stoke’s surface was measurer
backscattered field was measu
target. A CRTW was then c
composite return was cons
measurement from the Stoke’

algorithm was then applied I

calculate the convolution ener

 

only on the effects of the win

 

 

 

 

 

 

 

between the baseline clutter ratio and the target ratio.

4.6 Target Detection and Window Size

The detection of any target using the clutter reducing transmit waveform procedure
involves computing convolution energy ratios. The calculation of the convolution energy
ratio involves integrating over a time-window interval. Both the placement and size of
the time-window interval in the radar’s system range bin will affect the value of the
convolution energy ratio. At certain points in time (corresponding to different points of
the scattered return) the convolution energy ratio will possibly take on large values
indicating the presence of a target. The purpose of this section is to determine the effects
of window size on the convolution energy ratio during the detection phase of the CRTW
algorithm.

The study of the window-size effect on the convolution energy ratio parallels the
investigation in sections 4 and 5. First, the backscattered field from a highly conducting
Stoke’s surface was measured inside the anechoic chamber at MSU. In addition, the
backscattered field was measured from a small missile model to simulate an anticipated
target. A CRTW was then constructed from these two measurements. A synthesized
composite return was constructed by scaling the missile return and adding it to the
measurement from the Stoke’s sea surface model. The detection section of the CRTW
algorithm was then applied to this composite signal using a variable window width to
calculate the convolution energy integral. In this procedure the investigation concentrates

only on the effects of the window width since the simulated sea surface does not change

87

 

 

 

 

 

nthedctection step. The sec
field from an evolving sea 3
calculation The fust time ste
step was used to generate a
procedure. Using a variable
detection for non-static clutte
A simple Stoke's surf
wave model is a 2-d electric CI
mi The measured return fro
usingan HP network analyzer
inthe measurement was from
incident radiation was paralle
inclined 10 degrees from the
The frequency-domain return
ttansfonned into the time dor

return are shown in Figure 4.‘

A simple Phoenix mis

 

anticipated target. This mod
the physical similarity to an

surface was made on this mi
Figure 4.7. Using the norm

was constructed using (4.3).

 

 

 

 

in the detection step. The second example involves calculating the theoretically scattered
field from an evolving sea surface. Two time steps were used in the scattered field
calculation. The first time step was used in the construction of a CRTW. The next time
step was used to generate a composite clutter/target return to be used in the detection
procedure. Using a variable window size in the detection step the effect upon target
detection for non-static clutter background could be studied.

A simple Stoke’s surface model has been described in section 4. The Stoke’s
wave model is a 2-d electric conductor of wavelength 17.78 cm and wave height of 5.08
cm. The measured return from the surface was initially done in the frequency domain
using an HP network analyzer and amplifier (see Appendix A). The frequency range used
in the measurement was from 1-17 GHz using 1601 data points. The electric field of the
incident radiation was parallel to the wave crests and the direction of propagation was
inclined 10 degrees from the horizontal with 0 degrees representing grazing incidence.
The frequency-domain return was then windowed (using a cosine taper waveform) and
transformed into the time domain using an IFFT. The surface and transient scattered
return are shown in Figure 4.4 and Figure 4.5 respectively.

A simple Phoenix missile model approximately 10 cm in length was used for the
anticipated target. This model was chosen due to its availability from a model kit and
the physical similarity to an Exocet-type missile. A measurement similar to the Stoke’s
surface was made on this missile. The time—domain return from this model is shown in
Figure 4.7. Using the normalized returns from the sea surface and the target a CRTW

was constructed using (4.3). This constructed CRTW is shown in Figure 4.8. The author

 

 

 

 

elected to realize the CRTW
selection was the ease of use

values. it must be kept in mi
the use.

As a detection simulat
maximum value (TC R = -1
figure 4.l9 shows both the
expanded view gives a better 1
16mm.

Next. the convolution e
fora window width A = .5 ns
removed from the target the cr
target the energy ratio is abo
of a target. In this example tl
to calculate the CRTW. To 5
.8 nsec was chosen and the cor
the effect of widening the e
about 6 dB at the peak (see

A series of different

ratio with (4.4). Figure 4.22

seen from this figure, the pea

 

values of the energy window

 

 

 

 

 

elected to realize the CRTW by using a genetic algorithm. The reason behind this
selection was the ease of use since no initial guessing is required other than a range of
values. It must be kept in mind that this algorithm is not an efficient method for real-
tirne use.

As a detection simulation, the missile response was scaled to 20% of the clutter
maximum value (TCR = -14 dB) and added to the clutter signal at t = 6 nsec.
Figure 4.19 shows both the original clutter response and the summed response. The
expanded view gives a better idea of the effect of adding the target return to the surface
return.

Next, the convolution energy was calculated using (4.4). The convolution energy
for a window width A = .5 nsec is shown in Figure 4.20. As can be seen, for positions
removed from the target the convolution energy ratio is unity (0 dB). For points near the
target the energy ratio is about 19 dB. This large jump in value indicates the presence
of a target. In this example the energy window was chosen to be the same as that used
to calculate the CRTW. To see the effect of a different energy window a value of A =
.8 nsec was chosen and the convolution energy ratio recomputed using (4.4). In this case,
the effect of widening the energy window is to lower the convolution energy ratio to
about 6 dB at the peak (see Figure 4.21).

A series of different values of A were used to compute the convolution energy
ratio with (4.4). Figure 4.22 shows a three dimensional graph of the results. As can be
seen from this figure, the peak value of the convolution energy ratio decreases for larger

values of the energy window. Also, the single peak spreads out for small values of the

 

 

 

 

 

energy window t eil- values 1‘

From the results of the
window size should be used
analysis is the problem assoc
problem. two sea-like surfaces
Kinsman. Two surfaces. gr
figureili The first surface
senate is generated at 3 subs
figure 4.33 is not drawn to .\
the height is 3.3 meters.
computed using l-d (irccn's
section 4). Due to computing
5‘1:in the frequency range u
field from the initial sca surf
healed aIII-impriatcly). a CRT
5W" Figure 4.24.

Next. the missile mod
addEd ‘0 the Clutter return fr:

finally. target detection was :

h . .
eresult IS shown in Figure

sea '
surface has raised the max

is - ‘
detected With a margin of

 

 

energy window ( e.g. values less than .5 nsec).

From the results of the preceding paragraphs it might be suggested that a small
window size should be used. An important parameter missing from the preceding
analysis is the problem associated with an evolving sea surface. To investigate this
problem, two sea-like surfaces were computed from the wind driven model proposed by
Kinsman. Two surfaces, generated with wind speed of 20 knots, are shown in
Figure 4.23. The first surface was generated at an initial time of t = 0 sec. The second
surface is generated at a subsequent time step t = 1.25 sec. Notice that the vertical in
Figure 4.23 is not drawn to scale, the total length of the wave is 1000 meters, and the
wave height is 3.3 meters. The scattered fields for each surface were numerically
computed using 2-d Green’s function and the method of moments (refer to chapter 3,
section 4). Due to computing constraints the surfaces were scaled down by a factor of
50, and the frequency range was set at .5 - 1.5 GHz with 200 points. Using the scattered
field from the initial sea surface and the target return from the Phoenix missile model
(scaled appropriately), a CRTW was computed using (4.3). The CRTW for this case is
shown Figure 4.24.

Next, the missile model response was scaled in amplitude to TCR = ~14 dB and
added to the clutter return from the surface (at t = 115 nsec) in the second time step.
Finally, target detection was attempted by computing E(t) with a window A = 4 nsec.
The result is shown in Figure 4.25. This figure clearly shows a target, but the evolved
sea surface has raised the maximal baseline value of e(t) to about 10 dB. Thus, the target

is detected with a margin of about 10 dB. Using different values for the window size,

90

 

 

the convolution energy ratio
figure436. The significant ft
loner the target values for E(
is also lon-ered. Therefore. so
to false detection. If a sma

CRTW more often.

4.7 Application of C RTV

A clutter reducing tra
rerunr front a surface (clutter
expected target can be meas
produced trill maximize the
tttrlace return for a specific I

question raised is. cart the C

This situation can be S
and
the retum from a stationa
return
and the scattering frt
algorithm can

then be used w

a .
rid the different targets
The scattered return f
the '
prevrous sections. The

Fitu
, re 4.5 respectively. Sew

the convolution energy ratio €(t) can be recomputed. The results are shown in
Figure 4.26. The significant feature in this figure is that a wide window will broaden and

lower the target values for 5(1). On the other hand the maximal baseline value of E( t)

 

is also lowered. Therefore, some care should be taken for a small window size in regards
to false detection. If a small Window size is used it is also advisable to update the

CRTW more often.

4.7 Application of CRTW Techniques to Different Target Geometries

A clutter reducing transmit waveform can be constructed using the measured
return from a surface (clutter producer) and the return from an anticipated target. The
expected target can be measured in the lab or calculated theoretically. The CRTW
produced will maximize the ratio of target to clutter return at some point along the
surface return for a specific target. Since a particular target is anticipated an important
question raised is, can the CRTW be effective in detecting other targets?

This situation can be studied be measuring the return from several different targets

 

and the return from a stationary surface. A CRTW can be constructed using the surface
return and the scattering from one of the measured targets. The CRTW detection
algorithm can then be used with the returns from a combination of the stationary surface

and the different targets.

 

The scattered return from a Stoke’s surface representation has been discussed in
the previous sections. The surface and transient return are shown in Figure 4.4 and

Figure 4.5 respectively. Several highly conductive missile models were constructed and

91

 

 

 

 

 

measured in the anechoic Chat
emit-n in Figure 4.3"”. The
respectitely. This notation n:
rrtissile which has an appearan
generic cruise missiles of the
tune-domain returns from the
The CRTW for each r.
tonesponding to the return 1]
figurel.3l. As cart be seen
tussile retum it was Ctmgtnlc

A detection scheme us

\‘\

etteetn‘e for detecting that I

measured. the detection scltet

not Work as well. lf target id

some adt‘antage in hatinU '

\ t
\

u ' s '
ceometnes. In this case dill

‘

1 . ‘

u' .
q rte different depending on
To determine the sens

taro - '
.et retums in the time dorr

Iime- '
domain retum of the St(

ter '
sron of the targets shown

 

 

 

measured in the anechoic chamber at MSU. The missile models used for this study are
shown in Figure 4.27. The missiles are labeled as A, B, and C from top to bottom
respectively. This notation will be used throughout this section. Missile A is a Phoenix
missile which has an appearance similar to that of an Exocet. Missiles B and C represent
generic cruise missiles of the same geometry but slightly different size. The normalized
time-domain returns from these models are shown in Figure 4.28.

The CRTW for each missile was created using a genetic algorithm. The CRTW
corresponding to the return for the surface and each target are shown in Figure 4.29 -
Figure 4.31. As can be seen from these figures each CRTW is unique to the particular
missile return it was constructed from.

A detection scheme using a CRTW designed for a particular target should be quite
effective for detecting that target. On the other hand, if a different target return is
measured, the detection scheme (using a CRTW designed for the original missile) may
not work as well. If target identification is not particularly important, then there may be
some advantage in having a CRTW scheme that is not sensitive to different target
geometries. In this case differences may be characterized by missile size and control—

surface geometry. This could be important if the return from a missile is not static but
quite different depending on radar to missile geometry.

To determine the sensitivity of the CRTW technique, different combined surface~

target returns in the time domain were generated. This was essentially done by taking the

time-domain return. of the Stoke’s surface shown in Figure 4.5 and adding a time~shifted

version of the targets shown in Figure 4.28. In addition to time shifting, the magnitude

9.2

 

 

 

 

 

 

oi the normalized scattered tar
atombined return that was gt“
was located between peaks 5 a
between the retum from the st
C were also added to the Stol
ans stored in different files.
Applying the detectior
combined target surface returr
shows the cont oltrtion energs
surface. lfno missile were pr

.3

sartax. Figtrre 4.32 clearl}

(

int.)

motion. .\lore significant i

CRT“ was designed for mis

' , 1 “
it Figure 4.32 except Figure

Fio ‘ '~ -
ture 4.34 is based upon a (

plots points to the lack of s

r~ . .
easons tor this lack of sensit

produ ' -
crng surface return. l3

sh
ould be close to zero. then

ache '
me 18 somewhat insensit

identific '
atron tt'

. ts expecu
det ‘ '
ectron wrll still work but

 

of the normalized scattered target return was reduced by 90 percent. Figure 4.19 shows
a combined return that was generated using target A. As shown in the inset, the target
was located between peaks 5 and 6. An expanded view of the inset shows the difference
between the return from the surface only, and from the surface and target. Target B and
C were also added to the Stoke’s surface at the same position as target A, but the data
was stored in different files.

Applying the detection algorithm for a CRTW designed for missile A for each
combined target/surface return results in the response shown in Figure 4.32. This figure
shows the convolution energy ratio expressed in dB as a function of position along the
surface. If no missile were present, the response should be flat at 0 dB with a stationary
surface. Figure 4.32 clearly shows a large jump in the response indicating target
detection. More significant is that missiles B and C are also detected ( remember, the
CRTW was designed for missile A). Figure 4.33 and Figure 4.34 show similar results
as Figure 4.32 except Figure 4.33 is based upon a CRTW designed for missile B and
Figure 4.34 is based upon a CRTW designed for missile C. The significance of all three
plots points to the lack of sensitivity to missile size or geometry. One of the main
reasons for this lack of sensitivity is that the CRTW is constructed for a particular clutter
producing surface return. For this matched surface return the denominator in (4.4)
should be close to zero, therefore making the response quite large. Since the detection
scheme is somewhat insensitive to target type, this method should not be used for target

identification. It is expected that for an even smaller measured bandwidth, target

detection will still work but identification is out of the question.

 

 

 

 

 

 

 

4,3 Multipath Effect on '1
Precious sections in 1'
rmsient scattered electric fiel
ainidual transient fields frt
amplitude between the target .
easily be controlled. One prt'
eiectromagnetic coupling inter
bower er. is the absence of mu
surface. i.e. the rnultipat
s‘attered field from the target
fanned by adding the two int
The impact which the

is intestigated in this sectio
theoretical scattered field fror
different surface heights

{ill

talculatrons were of the train

th - .- '
esurfacbtarget pair with tlt

sca .
ttered fields from a surfac

The -
scattered field from sever

sc ‘ '
aled rnrssrle model located

aputoaches a CRTW was calt

surface. Using this CRTW

 

 

4.8 Multipath Effect on Target Detection
Previous sections in this chapter have considered several examples where the
transient scattered electric field from the target/surface pair was formed by adding the
individual transient fields from the target and the surface. By scaling the relative
amplitude between the target and surface return, the TCR in the composite return could
easily be controlled. One problem associated with this technique is that it neglects the
electromagnetic coupling interaction between the surface and the target. More important
however, is the absence of multiple excitation due to reflection of the incident wave from
the surface, i.e. the multipath effect. If the multipath effect is significant, then the
scattered field from the target/surface pair may be quite different from the transient field
formed by adding the two individual fields.
The impact which the multipath effect has on the new CRTW detection scheme
is investigated in this section. Several different approaches were taken. First, the
theoretical scattered field from a cylinder/Stoke’s surface pair was calculated for several
different surface heights and cylinder positions with respect to the surface. The
calculations were of the transient fields from the target and surface separately and then
the surface/target pair with the multipath effect. A second approach was to measure the
scattered fields from a surface/target pair in the anechoic chamber at the MSU EM Lab.
The scattered field from several different surfaces was measured with and without a small
scaled missile model located at different positions with respect to the surface. In both
approaches a CRTW was calculated from the separate transient fields from the target and

surface. Using this CRTW, the convolution energy ratio was calculated from the return

 

 

 

 

 

 

 

 

 

of the surface target PM ’1
composite return was formed
the scattered field was fonne
cases. a comparison can be r
interaction between the surfat
Figure 4.35 depicts tht
located above a Stokes surfat
field parallel to the surface er
consists ot‘a Stokes was c ha
a and indit'idual wax elength
of diameter t and the bottom
street to the surface is shot
center peak on the Stokes st
measured with respect to the

A lar-lrcld approxima

am
ethod of moments solutio

talid r ‘
only for a closed surfact

direc '
tron for 320 frequenes' t‘

transi ‘ .
ent field was determine

to the '
trrne domain using art

plO‘e’ed “1 ‘

res ' .
Pectttely (see Appendix
I

 

 

 

of the surface/target pair. This surface/target pair return includes two cases. First, a
composite return was formed by adding the separate target and surface return. Second,
the scattered field was formed by including the multipath effect. By considering both
cases, a comparison can be made showing the effect of multipath and electromagnetic
interaction between the surface and target.

Figure 4.35 depicts the 2-dimensional scattering geometry for a cylindrical target
located above a Stoke’s surface. The incident plane wave is polarized with its magnetic
field parallel to the surface crests (TM). The enclosed surface shown in Figure 4.35 (a),
consists of a Stoke’s wave having N periods (where N is an odd number), surface height
h, and individual wavelength L. The sides of the enclosed surface consist of half-circles
of diameter t and the bottom is a flat surface. The position of the scattering target with
respect to the surface is shown in Figure 4.35 (b) with the y—axis coincident with the
center peak on the Stoke’s surface. The incident angle (1),, shown in Figure 4.35 (a), is
measured with respect to the x-axis.

A far-field approximation to the scattered field was theoretically calculated using
a method of moments solution to the magnetic—field integeral equation (MFIE) which is
valid only for a closed surface [18]. The calculated field was computed in the backscatter
direction for 320 frequency points starting at 1.0 GHz at a step size of .025 GHz. The
transient field was determined by windowing the frequency data and then transforming
to the time domain using an IF FT. For all trials a Gaussian modulated cosine window

was employed with frequency and shaping parameters of f0 = 5.0 GHz, and T = .25 nsec

respectively (see Appendix A for window description). The surface wavelength L for

 

 

 

 

 

each trial was set to .1016 m
of the side half-circles was SC
pp m, The Value for the
reasonable Value for the TCR
the scattering calculations the
each. the sides and cylinder w
seamenng angle was set to O.

Seyeral scattering cal
surface fora ll\€d cylinder p
be position ofthe cylinder w
:ielol computed.

Figure 4.36 shows the
clone and from the surface
coupling interaction. For thi:
and has horizontally displao
Fiau

r-‘ . .
. 64.36.1llc two returns it

about 4.4 nsec). After this

re '
tum. Figure 4.37 shows tl‘

onlvr
. Etum. The response of

the multipath effect.

To detennine the effe

‘he t '
‘ S

 

 

 

each trial was set to .1016 m, the number of periods N was set to 11, and the diameter
of the side half-circles was set to .0254 m. Also, the radius of the cylinder R was set to
.005 m. The value for the variable R was determined by trial and error to get a
reasonable value for the TCR (about -3 to -5 dB) for a surface height h of .0254 m. For
the scattering calculations the top and bottom of the surface were divided into 200 points
each, the sides and cylinder were divided into 16 and 32 points respectively. Finally, the
scattering angle was set to (la = 20° for all trials.

Several scattering calculations were performed by varying the height h of the
surface for a fixed cylinder position. After that, the height of the surface was fixed and
the position of the cylinder was varied both as function of height yC and range x, and the
field computed.

Figure 4.36 shows the normalized transient scattered field from the Stoke’s surface
alone and from the surface/target pair with the multipath effect and electromagnetic
coupling interaction. For this case, the cylinder was located .0254 m above the surface,
and was horizontally displaced .0508 m to the right of the central peak. As shown in
Figure 4.36, the two returns overlap exactly before the incident wave strikes the target (at
about 4.4 nsec). After this event, the composite return differs from the surface-only
return. Figure 4.37 shows the difference between the composite return and the surface
only return. The response of the cylinder is clearly shown as well as the later effect from
the multipath effect.

To determine the effects that the multipath effect has on the detection algorithm

the transient response of t'h surface only and the cylinder only were used to construct a

 

 

 

 

 

turn: using the “fumed
energy ratio was calculated 11‘
sliotrs the convolution energ."
r'ere calculated usinéI ”'4’ ll
“.3, generated by calculating
response haying no electrom.
only transient response and ll
astrong conyolution energy
iconi'olution timer. The origi
benreen the cylinder and the
:‘oliou is from the cylinder tr
.lnotlier path is from the sur
Point. Also. multiple scatter
crest before the scattered tic
multipath scattering events o.

Figure 4.39 and Figu
SlOltes surfaces haying heig
lhe Size and position of the 0
defined convolution energy
multipath effect. For. the sr

tn '
crgy ratio peaks at nearly

cr' -
tinder does not change but

CRTW. Using the calculated CRTW and the surface plus target response, the convolution
energy ratio was calculated for the detection phase of the CRTW algorithm. Figure 4.38
shows the convolution energy ratio as a function of time. The curves shown in this figure
were calculated using (4.4) for an energy window width of .25 nsec. The dashed cuwe
was generated by calculating the convolution energy ratio for the target plus surface
response having no electromagnetic interaction. This was done by adding the surface
only transient response and the cylinder only transient response. Both curves indicated
a strong convolution energy ratio peak of about 14 dB occurring at about 4.6 nsec
(convolution time). The origin of the secondary peaks are due to the multiple reflections
between the cylinder and the surface crests. One simple path the scattered field can
follow is from the cylinder to the surface crest and then back to the observation point.
Another path is from the surface crest to the cylinder and then back to the observation
point. Also, multiple scattering events can occur between the cylinder and the surface
crest before the scattered field goes back to the observation point. In each case the
multipath scattering events occur after the initial scattering from the cylinder.

Figure 4.39 and Figure 4.40 Show the convolution energy ratio calculations for
Stoke’s surfaces having heights of .0127 m and .00635 m, respectively. In both cases,
the size and position of the cylinder have not changed. Both figures clearly show a well
defined convolution energy ratio peak and the secondary peaks associated with the
multipath effect. For the small surface height shown in Figure 4.40, the convolution
energy ratio peaks at nearly 20 dB. This result can be expected since the size of the

Cylinder does not change but the surface height has decreased by a factor of 4 (compared

 

 

 

 

toh= .0354 ml. HOWCVCT- ‘
smaller peak than in Fit—’11rc '1-
of3 has compared to h = .0
surprising since the TC R diff
3.63 dB in Figure 4.39).
Another imponant fac
peaks of the cylinder lie or
oierlapping peaks lie. surfat
ratio will be smaller. The Ct
the three different surface he
The effect upon the
ttlt‘estigated. This was done
“1%“ hfight )1. Three differ
.lll3 m. In each case. the
icoordinate of the cy

hnder‘

scattered fields from the surf;

effect fora target height of
Position and some of the inte
two curves in Figure 4.42. i
Flgltre 4.36 and Figure 4.37.

are ' '
shown in Figure 4.44 and

tar c '
gtand interaction is clea

 

to h = .0254 111). However, the convolution energy ratio shown in Figure 4.39 shows a
smaller peak than in Figure 4.38 even though the surface height has decreased by a factor
of 2 (as compared to h = .0254 m ). Although this is a bit unexpected, it is not too
surprising since the TCR difference is not that great ( from -5.31 dB in Figure 4.38 to -
3.63 dB in Figure 4.39).

Another important factor to consider for these different cases is where the transient
peaks of the cylinder lie with respect to the peaks from the scattering surface. For
overlapping peaks (i.e. surface peaks overlapping target peaks) the convolution energy
ratio will be smaller. The convolution energy ratio for the multipath scattered field for
the three different surface heights are combined in Figure 4.41.

The effect upon the detection process of different target heights was also
investigated. This was done by fixing the Stoke’s surface parameters and varying the
target height yc. Three different target heights were considered: .0635 m, .0889 m, and
.1143 m . In each case, the height of the Stoke’s wave was set to h = .0254 m and the
x-coordinate of the cylinder was fixed at xC = .0508 m. Figure 4.42 shows the transient
scattered fields from the surface alone and from the surface plus target with the multipath
effect for a target height of yC = .0889 m. The dashed curve clearly shows the target
position and some of the interaction effects. Figure 4.43 shows the different between the
two curves in Figure 4.42. Similar figures for a target height of .0635 m are shown in
Figure 4.36 and Figure 4.37. Also, corresponding figures for a target height of .1 143 m.
are shown in Figure 4.44 and. Figure 4.45. Using the difference figures, the effect of the

target and interaction is clearly seen. For an evolving surface it would be much more

 

 

 

 

 

 

difficult to distinguish the tar
After creating a CRT)
transient response. the com 0
target surface pairs. Figure
target heights. Once again.
Associated with the main pet
effect. The change in the ct
function of the electromagnt
target and surface. In Figun
figure the peak Values are \ e
is caused by the electromag
cont‘olution energy Value cal
The position of the 1;

height of the surface and tare

t ‘~ . ‘
he scattered return from If

sur‘ ' ’
face target pair with the r

set at .0254 m and .0635 m 1

St "
ole 3 peak by setting x

—
_

curt ' '
es tn Figure 4.48. For c

a l ‘ '
argct posmon between tht

IS clearly lined up with surfa

lot
he surface peak cause
. s a

difficult to distinguish the target by just taking the difference between the two curves.

After creating a CRTW from the surface only transient response and cylinder only
transient response, the convolution energy ratio was calculated for each of the composite
target/surface pairs. Figure 4.46 shows the convolution energy ratios for the different
target heights. Once again, a well defined peak indicates the presence of a target.
Associated with the main peaks are the secondary peaks corresponding to the multipath
effect. The change in the convolution energy value for the different target heights is a
function of the electromagnetic interaction and the relative peak location between the
target and surface. In Figure 4.47 the interaction effect has not been included. In this
figure the peak values are very close in value. Therefore, the effect seen in Figure 4.46
is caused by the electromagnetic interaction distorting the signal enough to effect the
convolution energy value calculation.

The position of the target can also be varied along the surface. In this case, the
height of the surface and target are fixed and the value of x0 is varied. Figure 4.48 shows
the scattered return from the Stoke’s surface alone and from. the combined. Stoke’s
surface/target pair with the multipath terms. The surface height and target height were
set at .0254 m and .0635 m respectively. The target was placed directly over the central
Stoke’s peak by setting xC = 0.0 m. Figure 4.49 shows the difference between the two
curves in Figure 4.48. For comparison, Figure 4.36 and Figure 4.37 were calculated for
a target position between the surface peaks at .0508 m. In Figure 4.48 the target return
is clearly lined up with surface peak return. In addition, the close proximity of the target

to the surface peak causes a greater interaction between the surface and the target. To

 

 

 

 

 

 

 

see the effect upon detection
was determined for the co
figure 4.50 shows the results
contolution energy ratio is m
tl‘téCOIlVOlUIlOH energy ratio 1
term is neglected. Herc. flu
difficult to detect the target.

The scattered electric
neasured in the anechoic cl
placed aboye these surface a
the relatiye geomcm m“
scattering measurements. 'l‘l
field is parallel to the watt
Plt

oenn missile model. appr

abote the crests of the surf;

those ‘ ~
or for the measurements

surf - ' '
ace peak (posrtton l ) and

lposuton 2). Figure 4.4 sl

Secti . '
on 4.4 discusses the pat

 

see the effect upon detection, 3 CRTW was calculated, and the convolution energy ratio
was determined for the composite return shown in Figure 4.36 and Figure 4.48.
Figure 4.50 shows the results. For a target located extremely close to a surface peak the
convolution energy ratio is much lower as shown by the dashed curve. Figure 4.51 shows
the convolution energy ratio calculation for the case when the electromagnetic interaction
term is neglected. Here, the position of the target over the peak makes it much more
difficult to detect the target.

The scattered electric fields from various scaled sea-surface models have been
measured in the anechoic chamber at MSU. A Phoenix missile model has also been
placed above these surface and the resultant electric field measured. Figure 4.52 shows
the relative geometry between the missile and sea-surface model using during the
scattering measurements. The incident electric field was polarized such that the electric
field is parallel to the wave crests at an incident angle of 100 from the horizon. A
Phoenix missile model, approximately 10 cm in length, was position at a height of 12"
above the crests of the surface. Two missile positions with respect to the peaks were
chosen for the measurements. In the first case the missile was placed directly above the
surface peak (position 1) and in the second case the missile was placed between the peaks
(position 2). Figure 4.4 shows the surface profiles used during the measurements.
Section 4.4 discusses the parameters required to generate these surfaces.

The time-domain scattered field from the Phoenix missile model is shown in
Figure 4.53. This response was synthesized from a frequency domain measurement in

the anechoic chamber at MSU. The frequency domain response was measured in the

 

 

 

 

 

frequency band from 1 0112

data was windowed and the
taper function with a shape
the data

The construction of a
target and the initial sea-5t
Figure 4.55 show the transi
suface for an incident angl
identical to that of the Phoe
Figure 4.55. a CRTW was cc
surface.

The position of the 1
peaks of the scattering surfa
the transient scattered return
of .09, time shifted, and add
=~23.2 dB). Next, the ma
CRTW (see Figure 4.56), wa
time shift. Figure 4.57 sho
‘ nearlya 17 to 18 dB differe
sinusoid surface the magnitu

added to the double sinusoi

values of the convolution en

 

 

frequency band from 1 GHz to 17 GHz at a step size of .01 GHz. The frequency domain
data was windowed and then transformed to the time domain using an IFFT. A cosine
taper function with a shape parameter of ‘C = 8 (see Appendix A) was used to window all
the data.

The construction of a CRTW requires the transient response from the anticipated
target and the initial sea—surface return having no target present. Figure 4.54 and
Figure 4.55 show the transient scattered return from the Stoke’s and double sinusoid
surface for an incident angle of 10°. The measurement process and windowing were
identical to that of the Phoenix missile model. Using the data shown in Figure 4.53 -
Figure 4.55, a CRTW was constructed for the Stoke’s surface and for the double sinusoid
surface.

The position of the target’s transient return peaks with respect to the transient
peaks of the scattering surface is extremely important for target detection. To see this,
the transient scattered return from the Phoenix missile was amplitude scaled by a factor
of .09, time shifted, and added to the transient response from the Stoke’s surface (TCR
= -23.2 dB). Next, the maximum convolution energy ratio, using the Stoke’s wave
CRTW (see Figure 4.56), was calculated for the composite return as a function of missile
time shift. Figure 4.57 shows the results for these calculations. In this figure, there is
nearly a 17 to 18 dB difference in the convolution energy detection ratio. For the double
sinusoid surface the magnitude of the Phoenix missile was scaled by .05, time shifted, and
added to the double sinusoid return (TCR = -4.75). Figure 4.59 shows the maximum

values of the convolution energy ratio using the double sinusoid CRTW (see Figure 4.58)

 

 

 

 

 

as a function of missile tim

detection ratio.

In the preceding di
were included. The measure
Figure 4.60 and Figure 4.61
includes the surface only ret
for the missile in position
electromagnetic interaction)
and each missile position.
Stoke's surface. Similarly. I
The peak convolution energy
14dBrespectively. The sec
and surface. For position 1,
surface and are shown in Fig
the double sinusoid surface 1
This is most prevalent in Fir
the target itself. However, it
enough to detect the targe
probably not be the case.

The effect due to t

theoretically calculated sca

between the surface and the

 

 

 

as a function of missile time shift. In this case there is nearly a 25 dB difference in the
detection ratio.

In the preceding discussion no multipath or electromagnetic interaction effects
were included. The measured transient return from the surface/missile pairs is shown in
Figure 4.60 and Figure 4.61 for the Stoke’s and double sinusoid surface. Each figure
includes the surface only return and the return from the composite surface/missile return
for the missile in position 1 or position 2. With the composite return (including
electromagnetic interaction) the convolution energy ratio was calculated for each surface
and each missile position. Figure 4.62 shows this ratio for missile position 2 on the
Stoke’s surface. Similarly, Figure 4.63 shows the results for the double sinusoid surface.
The peak convolution energy ratios for the Stoke’s and double sinusoid surface are 10 and
14 dB respectively. The secondary peaks correspond to the coupling between the target
and surface. For position 1, the convolution energy ratios were also calculated for each
surface and are shown in Figure 4.64 and Figure 4.65. For both the Stoke’s surface and
the double sinusoid surface the convolution energy ratio has been considerably reduced.
This is most prevalent in Figure 4.64 where the effects of the interaction are as great as
the target itself. However, in Figure 4.65 the convolution. energy ratio may still be large
enough to detect the target, although for a rapidly evolving sea surface this would
probably not be the case.

The effect due to the target-surface coupling have been illustrated using both
theoretically calculated scattering data and measured data. Although the interaction

between the surface and the target does effect target detection, the relative position of the

 

 

 

 

 

target with respect to the s

the target scatter return did
was able to find the target.

in the presence of the mulip

4.9 Coherent Processinr
A clutter suppressior
clutter has been discussed bj
Nsamples each are collected
clutter and target return. A

the cross-range average fron

§i(k) = siU

Alittle reflection shows why
clutter from all the M puls
subtracted from the sample
any clutter present in the si
improve for increasing valu
clutter suppressed signal sh

achieved. The chief proble

is that the clutter environm

 

 

 

 

target with respect to the surface wave is probably more important. For all cases where
the target scatter return did not coincide with a surface crest return the detection scheme

was able to find the target. The detection algorithm was able to detect the target even

in the presence of the mulipath effects.

4.9 Coherent Processing Clutter Reduction

A clutter suppression technique useful for moving targets on a static background
clutter has been discussed by Iverson [26]. In this technique, a set of M pulse returns of
N samples each are collected. The ith pulse return Sr( k) is composed of both background
clutter and target return. A clutter suppressed signal §r(k) can be formed by subtracting

the cross-range average from each pulse return, i.e.

M
§i(k) = Si(k) — 112 Zsjuc) k=1...N, i=1...M (4.15)

i=1

A little reflection shows why this formula should work. For a given sample point k, the
clutter from all the M pulse returns is simply averaged. This average value is then
subtracted from the sample point k for each pulse return. The net effect is to suppress
any clutter present in the signal. For a static background clutter, clutter suppression will
improve for increasing values of M. Following the clutter suppression step, the resulting
clutter suppressed signal should be aligned and added so that coherent integration can be
achieved. The chief problem associated with clutter suppression in the sea environment

is that the clutter environment is far from static. For a slowly evolving sea surface

103

 

 

 

 

Iverson’s method may wor

time interval. On the other

pulse returns must be samp
An alternative form

moving window of discrete ‘

can be written as

where i, is the pulse responSv
the cross-range average valL
issubtracted from the pulse
changing clutter retum. it i
probably fluctuate about the
important point since subtrar
the edge of the window will
of the window may be an c:
then poor performance can b
change appreciably over the
To demonstrate the
theoretically calculated fro

stochastic model given by

evolving sea surface was cal

 

 

Iverson’s method may work by using a moderate number of pulse returns M in a finite
time interval. On the other hand, if the sea surface is evolving at a faster rate, then the
pulse returns must be sampled at a higher rate.

An alternative form for (4.15) can be used for evolving background clutter. If a
moving window of discrete width L is applied to a subset of the pulse returns, then (4.15)
can be written as

. (L-l)
lr+_

ask) = ark) - % Z sjtk) k = 1 N (4.16)
M

J=z,- 2

where ir is the pulse response index corresponding to the center of the window. In (4.16)
the cross-range average value is computed over the range of the window and this value
is subtracted from the pulse return corresponding to the center of the window. For a
changing clutter return, it is expected that the pulse returns within the window will
probably fluctuate about the pulse response at the center of the window. This is an
important point since subtracting the cross-range average from a pulse retum located at
the edge of the window will not lead to good clutter suppression results. Also, the size
of the window may be an extremely important factor. If the window size is too large,
then poor performance can be expected since the sea surface (and the clutter return) can
change appreciably over the window’s time space.

To demonstrate the use of Iverson’s modified technique, the scattered field was
theoretically calculated from an evolving sea surface which was generated from a
stochastic model given by Kinsman. As in section 4.5, the scattered. field from the

evolving sea surface was calculated over a 2.0 sec interval every .25 seconds. Figure 4.66

104

 

 

 

 

shows the scattered return f
figure represents time-dom
time; successive lines repr
bottom The sensor (radar
shown on the graph. A tin
calculate the cross range as
second scene. shown in Fig
across the signal space. A
distinguish between clutter
too large for the rate of sea
To remedy the abos
asmaller time step could
Figure 4.68 shows the sc:
generated in the previous 5
returns were sampled over
Once again a pulse return it
calculate the cross-range th
process using the smaller tir
the background clutter. Si:
algorithm is very useful if

compared to the new CR'l

implement for real-time apj

 

 

shows the scattered return from the clutter/target combination. Each horizontal line in the
figure represents time-domain samples from the scatter response over some interval of
time; successive lines represent a series of scatter returns with the first return on the
bottom. The sensor (radar) is located on the left and successive missile locations are
shown on the graph. A time window using three successive measurements was used to
calculate the cross range average. After application of the clutter suppression process a
second scene, shown in Figure 4.67, was generated to show the movement of the target
across the signal space. As can be seen from the second figure it is very difficult to
distinguish between clutter and missile return. In this case the time step is most likely
too large for the rate of sea surface evolution.

To remedy the above problem a second simulation was also performed to see if
a smaller time step could lead to an improvement in the clutter suppression problem.
Figure 4.68 shows the scatter return and missile position for the same sea-surface
generated in the previous simulation using the Kinsman model. In this case 10 pulse
returns were sampled over a .45 sec duration with the time interval AT = .05 seconds.
Once again a pulse return window containing three successive pulse returns was used to
calculate the cross-range average. Figure 4.69 shows the results of the clutter suppression
process using the smaller time step. In this case the target is clearly distinguishable from
the background clutter. Since the calculation given by (4.16) is extremely simple, this
algorithm is very useful if the time step between successive pulse returns is small. As

compared to the new CRTW technique this method. is extremely simple and easy to

implement for real-time applications.

 

 

 

 

4.10 Conclusions

This chapter has pr
pulse method. The motiv
difficulties associated with 1
important topics were prese:
scheme was described. Near
The author has found this a]
algorithm several examples
static sea-surface demonst:
dynamic sea surface was n
surface environment and th
surfacet'target coupling on t1
generated test cases and me
energy window size on the
see if the detection schem
missile models were measu
models was used in conjun
CRTW and to test the deter
the detection algorithm is q
The final topic covered i
technique. This method v

update rate is sufficiently 1

 

 

4.10 Conclusions

This chapter has presented a new target detection technique based upon the E-
pulse method. The motivation for doing this chapter was to overcome some of the
difficulties associated with the conventional use of the E-pulse detection method. Several
important topics were presented in this chapter. First, the theory behind the new detection
scheme was described. Next, the genetic algorithm was presented and described in detail.
The author has found this algorithm to be robust but quite slow. To test the new CRTW
algorithm several examples were presented showing the effectiveness of this method. A
static sea-surface demonstration was presented showing proof of concept. Next, a
dynamic sea surface was modeled in order to show the effect of a more realistic sea-
surface environment and the need to regularly update the CRTW. The effect from sea—
surface/target coupling on the detection algorithm was also studied. Several theoretically
generated test cases and measured results were presented. The effect of the convolution
energy window size on the detection phase of the CRTW algorithm was presented. To
see if the detection scheme was tolerant to different target geometries several scaled
missile models were measured in the anechoic chamber. The scattered return from these
models was used in conjunction with the return from a sea-surface model to generate a
CRTW and to test the detection phase of the CRTW algorithm. This study showed that
the detection algorithm is quite tolerant to some variation in the target’s scattered return.
The final topic covered in this chapter is the coherent processing clutter reduction
technique. This method was tested and found to work remarkably well provided the

update rate is sufficiently fast. In addition, this technique can easily be applied to real-

106

 

 

 

time processing with very

 

 

time processing with very little code modification.

 

107

 

 

 

SYSTEM BASEU
irrrALtzA’tOrr fl SEA-CLU
CAPTUI

Figure 4.1 CRTW flow

 

 

l

 

   

Yes

 

 

   
 

 
    
 

 

 

 

SYTTTEhd p i TmASElJNEE lE-PULSE _ IE-PULSE i TARGE:\\\
INITIALIZATION #1 SEA-CLUTTEH consmucmu SEéApgflggER cohbé‘tfihm 1 passem
i CAPTURE i
NO

     
 

BASUN;\\\\

UPDATE?

IYes

Figure 4.1 CRTW flowchart process.

108

 

 

 

 

Population of N chr

X1:
X2:
Xi: 1
XN:
N =
M =
L =
*aHQenes.

Figur
”'2 Generic a]m

Population of N chromosomes:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

X1: 1
X2: 1
Xi: 1121 iii 1M1
jth gene* (L bit length)
11121 1L1
individual bit: 9 _'
J.
j = gene number
| = bit position
XN: ' ” TT' _ _- z

 

 

 

 

 

N = population size
M = number of genes (design parameters)
L = number of encoding bits per gene

* all genes of same length

Figure 4.2 Genetic algorithm data structures.

109

 

 

 

Figure 4.3 Genetic alg<

 

 

Select:
Number of Generations
Population Size
Mutation Rate
Crossover Rate

J

Initialize
Population

 

 

 

 

 

 

 

 

Decode
Gene
Strings

 

r

Population
Function
Evaluation

I

Select
Mates

1

Cross
Genetic
Information

 

 

 

 

 

 

 

 

Mutate 1
Genetic

le

L/Mfi/

 

 

 

Figure 4.3 Genetic algorithm block diagram.

110

 

0.00 ‘

___.d4q«.__.q

4
O

_
6
o. .
O O

AEV 29m:

2
0

AU.

id

®>O>>

0.00

 

 

O
a q a 4 a 4 ~ _ a aJ _ q _ flu.

0
5 3 4| 4|
0 O O O

_
Arte Eofil ®>o>>

 

Simple PEC

Figure 4.4

 

0.06

(m)

.0
O
4:

 

 

 

 

 

 

 

 

 

0.02

 

iiiiliiiiliiiil

Wove Height

 

 

0.00 lllillllllllllllll lllllllllllllllll1fi
0.00 0.530 1.60 1.50 2.00

Surface Position (m)

.0
O
on

 

 

 

 

 

lllllillllll

 

 

 

 

 

 

   

.0
o

 

Wove Height (m)
C
8

 

—0.01IIIIIIIIll[IIIIIIIIIIIIIIIIIIIITIlllllllI

0.00 0.50 1.00. . 1.50 2.00
Surface Posrtion (m)

Figure 4.4 Simple PEC scaled ocean surfaces.

111

 

 

 

W
00
Scattered m

 

 

:11i

 

—«‘-«dfiqq-1<q-~«—<uquqnq—_~——----—~o
O O O O 00
0 5. flu. 5. nU. 5
11 O O O .l 4"
_ z N
©033QE< ®>Zozmm us
uh
A

 

 

 

1.00

 

 

 

 

 

 

 

 

 

 

 

 

 

 

050-:
Q) .—
‘O :
3 _
.4: _
E :
E _
< 0.00 :
(D :
.2 I
44 _
g _
<1) :
Df—oso:

E Stokes
—1.00 llIllllTrlllllllllifllllllll
0.00 5.00 10.00 F5700

Time (nsec)

Figure 4.5 Scattered field from the PEC Stokes surface.

112

 

 

 

LT???—
O
Scattered fi<

 

 

 

AU.
0 00
nu.

—_fiJduq-uudquu~—————-—--~—-—---qfi~
O
5
O
_ .

1
Figure 4.6

O O
flu. : V. O.
O O

4"

®UJEQE< ®>30zmm

 

 

1.00

 

 

 

 

 

 

050—:
a) .—
U :
3 _
:2: _
EL :
< 0.00 : ,
(D :
.2 I
44 .—
2 _
<1) :
0: —0.50 :
3 Double Sinusoid
—1.0o_lllllllllIIIIIIITTIIIIIIIII
0.00 5.00 10.00 115.00

Time (nsec)

Figure 4.6 Scattered field from double sinusoid surface.

113

 

Scattered fir

.1.
0
0.

 

—4d«—_—d_—dfifi~1q-—_q——1d1—~qq‘———~_——~——————————~—
O O 00
m. w. m 5. 0. 5 7
4i 0 O 0 all 4' Au
_ _ z e
003.:QE< 020.90% m.
w“
A

 

 

1.00

 

 

 

 

 

 

0.50-E
cv E
‘0 _
3 :
E 0.00 _ A Ma
Q_ ._
< E
(J) I
.>—O.50-_'
L _
2 :
<D :
Q: _
—1.00—:
_1.50 lllllllTTlllllr1lll|lllllllll]lllllllllllllllillllllllllllfl

 

 

0.00 1.00 2.00 3.00 4.00 5.00 6.00
Time (nsec)

Figure 4.7 Scattered field from Phoenix missile model.

114

 

 

File, |
e
O O m
ji4lidl|dill4|l<i4i ialJliq inlaid-11.1 jig-I i nu. — A a _ q a u q 1 u — u u q u u ~ ~ — q u 0 n
I O O t
.6 O O O S
now 0. 0.. nouu 0. 0. m0.“
. C l. O .l n .
_ _ . z
®U3fi.QE< ®>LO7®E
. .
4..
e
r
u
g
I]
‘

MiVC<Z__QCL< C>_,..C_OZ

 

 

Stokes Wave

 

 

 

 

 

 

 

 

 

 

 

 

1.00—
0) —4
‘O -.
3 _
,4: _
E. _
E _.
(D _
> —l
1: ..
2 _
(D _
D: - Q
‘1.00 IIITIITTIIIjT—TlllllllllllIII—[llllIl
0.00 0.10 0.20 0.30
Time (nsec)
1.00 — Double Sinusoid Wave
(D a
”O —t
3 _.
:0: .u
E. _
E f.
<—0.00_
(D _
.2 ‘
44 _
2 —
(D _
m i
_. lllllllllllllllllllllllllllll
10%.00 0.05 0.10 0.15

Time (nsec)

Figure 4.8 Constructed CRTWs for measured surfaces.

115

 

30.00

20.00

 

10.00

Rotio (dB)

0.00

Convolution

*1000

1111111111111111111ll1111111lllllllJlllJiLilLilllli

~20.00
0.00

 

Fi
gure 4.9 Convolutior

 

 

30.00

 

 

 

 

 

20.00 —:
A _
m :
‘0 ..
V :
O I
“4: 10.00 ':
O :
D: _
C I
.9 0 00 E g Convolution
+5 . _ Ratio
9 E Scattered
C 2 red
O I T1
0 —10.00 ':

: T1 — target between wave peaks
I T2 — target over wave peak
_20.00 FIITFIIIIIllIlllrlrlllllllllfl
0.00 5.00 10.00 15.00

Time (nsec)

Figure 4.9 Convolution energy ratio for Stokes surface.

116

 

a W

F‘
'gure4.10 Convolutior

 

- a — q u — u — q u u q — — u H a H 4 q d d u 0 u ~ ~ — u is fl H q
0 O

0 mud W moo 0. 5. 0.

_ . ‘
C033‘0>C00

00. .
9 6
AmUV 030W. \Amemqu

 

 

 

 

 

 

 

 

 

    

 

 

 

9.00 —-
“8 6.50 3
.9 : Convolution Ratio
+2 _
D? 4.00 — M
>‘ _
U) _
s— _
“g 1.50 —
LIJ -
C - L,
.9 -
E —i .00 :
. -—1
> _
g - Scattered Response
0 —3.50 -

- T1 = target between wave peaks
: T2 = target over wave peak
—6oOO I I I I I I I I I I I I I fr I I I I [j I I I I I I I
0.00 4.00 8.00 12.00

Time (nsec)

Figure 4.10 Convolution energy ratio for double sinusoid surface.

117

 

b
O

 

0.50
(D
0
3
:1 0.00
O.
E
<1
0
3 —0.50
:6
Q)
tr

l
b
O

 

l
£11

00
b1111111111lit111111111111114111111111iriliiiiiiiimJ
j

FIgure 4.11 Constructed

designed to

 

 

 

1.00

 

 

 

0.50 —:
<1) 3
U _
3 I
7E 0.00 :
O- :
E :
< :
“3) —0.50 f
4.) _
2 :
<1) :
D: :
—1.00 f
_1.50 IIIfiTIIIIIIIIIIIIITIIIIIIIIlI—l
0.00 0.40 0.80 1.20

Time (nsec)

Figure 4.11 Constructed CRTW for scattered field from Stokes surface. CRTW
designed to eliminate the surface clutter only.

118

 

 

 

 

I.

F A“
'J.J‘
p
V
«7-
V
-w
_’
V
f‘
‘-
,—
)—
5 f‘ (‘1‘ _
< .4 \Jv
f.
‘1
.J
r-w
V
f;‘ -
\e
A _
g
_h _" ."i‘ __
U-U‘
..
q
‘A «A
”U.Jé “‘
A.
‘I
at

Figure 4.12 C OllVOiLlIlOl
eliminate 11'

0.01

Relative Amplitude
O
O
O

*0.01

rilririliliiirrriril11Aiii111_L_1__;_;ii1_1_1L_J

 

0.02

 

 

 

 

 

 

 

 

 

0.01 f
a) —
U 2
3 _
:5 _
E I
E _
< 0.00 :
a.) —.
.2 I
4.; _
_o _
(D :
0: -0.01 —_

_O-02 I I I I I I I I I I I I I I I I I I I I I I I I I I I I F1
.00 5.00 10.00 15.00

Time (nsec)

Figure 4.12 Convolution of CRTW with surface return only. CRTW designed to
eliminate the surface clutter only.

0.02

0.01

 

0.00

 

 

 

Relative Amplitude

 

 

111lllllIlllllllllIlllllllllIlllllllllI

 

 

_0a020‘00 I I I I I I I 5I.OIOI I I I I I I I1IO.|06 I I I I I I '1'5‘100
Time (nsec)
Figure 4.13 Convolution ratio of CRTW with clutter and target return.

designed to eliminate the surface clutter only.

CRTW

119

 

10.00

5.00

 

0.00

Ratio (dB)

-5.00

Convolution

~10.00

 

‘1 5.00

'0liriirirriiritrrirrilii1111111111111111111111111111

O

0

F. .
Igure 4.14 C0nv0lutior

the surface

 

 

10.00

 

 

 

 

 

5.00 —:
/'\ ._
m :
"o _
v :
O I Convolution Ratio
:5 0.00 :
O :
D: :
C I Scattered Field
0 I
143 —5.00 :
3 _
T5 '_' T1
> :
C _
0 :
O —10.00 :
3 T1 — target between wave peaks
2 T2 — target over wave peak
—15.00 IIIIIIIIIIIIIIIIIIIFIIIIIIIT—“l
0.00 5.00 10.00 1500

Time (nsec)

Figure 4.14 Convolution energy ratio for Stokes surface. CRTW designed to eliminate
the surface clutter only.

120

 

 

 

i\>
C)

'Olljllll1llillllllllLilllllllllil111111![llllllLlLlIllllllLlJi

O

'0
o

0.80

0.60

0.40

[A(o)]2 do (m2 sec)

0.20

 

0.00

O

‘11
0'5
=
-7
(D
A
—
or
Z
('9
S
:5
m
:3
:3
fl

 

1.20

1.00

0.80

0.60

0.40

[A(o)]2do (m2 sec)

 

0.20

lllllllllllllllllllllllllllllilllllllllllllllllllIlllllllllI

 

 

IIIIIIIIIIIIIIIIIIFIIIIIIIIIIIIIIIIII—I_l
0'000. 0 1.00 2.00 3.00 4.00

o (sec'1)

0

Figure 4.15 Neumann spatial frequency spectrum.

121

 

 

 

0.40

0.30

0.20

0.10

0.00

l
Q
Q

HCyC'tJD’Y<tk>>

-020
~0.30

~0.40
0.00

Figure 4.16 Covariance

20 knot wi

0.40

0.30

0.20

0.10

0.00

|
.0
o

H(y(ti),y(tk))

I
.0
N
o

—O.30

lllllllIlllllllllIlJlllllIIIlllllllllllllllllllIlllllllllIlllllllllIlllllllllI

 

_Oo4O IIIIIIIII IIIIIIIIIIIIIIIII1
0.00 5.00 10.00 1

T (sec)

.00 20.00

()1-

Figure 4.16 Covariance distribution generated using Neumann frequency spectrum wrth
20 knot winds.

122

 

 

 

Magnitude

Seo Height

 

\

Figure 4.17 Absolute
stochastica

 

 

 

Magndude

   

Scattered Response — Time

 

 

Sea Height

Sea Surface — Position

 

 

Figure 4.17 Absolute value of typical band-limited sea clutter return from.
stochastically-generated sea surface. Sea height is not to scale.

123

 

 

-
-
._ \ " K
-.
. p f
-.
'. ‘x
z — V ‘ ‘ I F
g.
—
x ‘ x
/-
‘ —,\ ‘ \ .Jv' ’ q
, .
qf-NN‘

, \

..
V1 1 ’\
\J/‘NJ" \"JVJ "Pf",

Distance Along 39

Figure 4.18 Profiles for

energy rat
spatial pos

 

 

.25 sec l0 —

 

W
.50 sec 10 -

WWW ongWN/CWW

_1O .1

20 a
_. .75 sec -
W 10 _
0 —1
1.0 sec

‘
W 10
O

20 -

_. 1.25sec -
WWW 10:
0 _

 

 

-10-i
20
_. 1.5OSec
FNMA/“WM 1O ‘-
0 .1
-101
20
1.75 sec

 

_, 2.0 sec 10 _
WWW/WWW 0 14
-10 1
. Distance Along Sea Surface ‘ 1 Radar Return Time ‘
0-0 1000 m 0.0 6.67 usec

stically-generated sea surface, and computed

Figure 4.18 Profiles for an evolving stocha
le/clutter combination. Arrow indicates

energy ratios for Phoenix missi
spatial position of the missile.

124

 

'0
o

0.50

 

0110

Relative Amplitude

I
Q
()1
CD

IlllllllllIllllJllllIlllllllllIllllllllLJ

 

l

O
.010
4:.
00%

.37
cra
=
'1
(p
A
—
\O
C/)
C
:1.
a:
O
(D
H
(D

resPOnse.

 

elotive Amplitude

 

 

 

 

 

 

1.00 :
E 'o.o’ 5.0gime (nseggoo Too
050{§
<1) _
‘0 I
3 _
,4: _
—Q_ I
E _
(D I
.2 I
+J _
_O_ _
3:) I —— Clutter Only
—O.50 : ----- Clutter and Target(A)
—1.00 - IIIIIIIIITIIIIIITTI—[IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII—I—I
5.418 5.68 5.88 6.08 6.28 6.48

Time (nsec)

Figure 4.19 Surface response constructed by adding the missile response to the clutter
response.

125

20.00

15.00

10.00

Energy Ratio (dB)

Convolution
ICJ'l
o
o

iriiriiiiiiiiiiiiiiiliiiiiiliiliriiliiill

 

O

b
N0
O
O

F'
igure 4.20 Convolutit

20.00

)

15.00

 

 

Convolution Energy Ratio (dB
.3
8

_
-
-
-
-
—
—
-
—
—

.—
—
-
—
—
-
—
—
—:
_

—
—
-
—
_
_
—
I—
.-
—

—
-
c-
_
-
—
.1
.—
_

 

 

 

 

5.00
0.00 llTllllll[lllllfilllllflT‘Y’FllllllllllffllllllllT1_|
2.00 4.00 6.00 8.00 10.00 12.00

Time (nsec)

Figure 4.20 Convolution energy ratio for an energy width A = .5 nsec.

126

 

m
0
V
O m
10a«__~q__q___-qq_.__qq~.qqu__fi~+~__~_.~_~__~u.~__AU. C
0 0 0 0 0 02
0. 0. 0. 0. 0. 0. N.
0 DO 0 4 2 0 4"
i e
AmUV OEOK \ADL0CU C0.L.3.0>COU hula
mu
‘

 

 

 

10.00

8.00

6.00

4.00

2.00

 

Convolution Energy Ratio (dB)

 

llllllllllllllllllilllllllllIlllllllllllllllllllI

 

 

III TIIIIIIII lilll‘ffilllTTFFllilllillllllm
0002.00” 2.00 6.00 8.00 10.00 12.00

Time (nsec)

Figure 4.21 Convolution energy ratio for energy window width A = .8 nsec.

127

-32

24

7.

((J/ 3’)
6.

 

R0 “'0
5

 

 

 

 

 

.e: 5
o
.0
’1;

F‘
Igure 4.22 C onvolutii

 

 

.32

24

7,6

 

Rat/o (d5)

 

 

 

 

 

 

 

 

‘C‘L
\
\
\
\

 

X

 

m9

 

Figure 4.22 Convolution energy ratio as a function of window width and time.

128

 

Figure 4.23 Tu 0 sea 51
different il

b
O
iiiLiiLJJ

.0

(J1

O
1111

l

llllIlllllll

r

l 1

Relative Amplitude
O
O
O

~O.50

lllll

 

t = 0 secs.

W

t = 1 .25 secs.

 

Figure 4.23 Two sea surfaces generated from the wind driven Kinsman model at two
different times.

 

1.00

 

 

0.50

 

0.00

 

Relative Amplitude

l
.0
or
o

 

 

 

llllLllllIllIllllllIllllllllllllllLllllI

 

 

 

 

IIIIIIII IIIIIIIII]
40% """'o'.o'.o' 0.60 1.20

Time (nsec)

 

 

O
0

Figure 4.24 CRT W construction for initial sea surface.

129

 

15.0

10.0

Ratio (dB)

 

5.0

0.0

Convolution

IllllIllllIllllIIlllILlllIlllJ_I

‘10'0 FTTTT‘T‘i‘r-rj
30.00 50

 

Figure 4.25 C onvolutii

fOF a wine:

 

 

20.0

15.0

1 0.0

 

 

5.0

 

0.0

Convolution Ratio (dB)

lllIllllIllllIlllllllllIllllI

 

-1O’0 IIIIIIIITI—ITIIIIIIIIIIIIIIIIIIIIIIII I IIIIIIIIIIIF‘I

30.00 50.00 70.00 90.00 110.00 130.00
Time (nsec)

Figure 4.25 Convolution energy ratio (dB) detection diagram for a realistic sea surface
for a window width of 4 nsec.

130

 

 

 

 

5 101520
|g

ROUO(dB>

 

 

Figure 4.26 C OITVOIUIll

size and ti

 
           

 

  
  

  
   

   
 
  

  
    
  
 

   
            
 
 

   
  

 

     
    

 
  
 
  

  

  
  

   
     

    
   
 
 

o 9.
"o :s
s‘o
s‘o

c ‘0‘

  
 
    

‘

 

  

531 ii"
Ed in Ii,
‘0 f—
.9 v- 1% I III“
4.4
o W in r H I . :‘biée’l’ r//////
0: {‘2‘}; I‘./ ”" :\- “ III“: "0‘ '10"! II/l,\

9:: ll‘ii‘ég ‘ “ 1kg”); ’ '0 \‘0 III

9’, W‘. “ O I 0 g‘oq“; ‘2‘“. 3 ‘3. z’l’q’lll,\

 

‘\ . \
“ ‘ “\ \\“ “\
‘ 3 M e ‘0 Q ‘. \ ““‘
\\\‘\‘ “““\\\ 3“ ‘6'. \ ‘ ‘

 

 

Figure 4 26 Convolution energy ratio for Kinsman sea surface as a function of window
size and time.

131

 

 

 

< ' (A)

 

 

f
@- (B)

 

 

_. (o)

 

 

 

 

 

Figure 4.27 Missile models used in CRTW study.

132

 

 

W
0.00 0.
“We 4.23

Missile Sc

 

 

(A)

 

 

 

(B)

 

 

(C)

 

 

 

 

IIIIIIIII[[IlIIIIIIIIIIIIIIIIIIIIIIIIIIII

0.00 0.50 . 1.00 1.50 2.00
Time (nsec)

Figure 4.28 Missile scattering in the time domain.

133

0.
0 00

0.

._1

—id_~—J_~d~d—q_qd-«_—~——-——~—_—~_—_—~——~—
0 0 0
0 5. 0. 5

0 0 0.0

4|

Figure 4.29 CRTW cr

0035.0CL0 0>3020W2

 

 

1.00

 

 

 

050—:
G) _
.0 _
3 I
it” _
E :
E _
(D I
.2 I
-+_’ —
2 _
(D I
01*050:
_1.00 rlIIIIIIIIIIIIIIIIIITIIIIIIII—l
0.00 0.04 0.08 0.12

Time (nsec)

Figure 4.29 CRTW corresponding to missile type A.

134

 

r 2 WW

Figure 4,30 CRTW u

 

__ u . a _ _ . q . 3 q _ _ a . q _ a . _ . ~ 3 _ _ . _ a _ _ _ _ . _ _ _ _ _ .a
0
0 0 0 0 0
0. 5. 0. 5. 0.
.i 0 0 mo .zi
003wZQfC0 0>.L.0_0m
‘

 

 

 

1.00

 

0.50

 

 

 

0.00

 

 

Relative amplitude

—0.50

 

 

lllllllllilllllllllilllllllllIlllllllllI

 

IIIIT—l

_ IIIIIIIII IIIIIIIIIIIIIIIIIIII
10(000 0.05 0.10 0.16

Time (nsec)

Figure 4.30 CRTW corresponding to missile type B.

135

_l|\\_i W0
0.
0 00

0 5. 0.

0 4|.

_ 2

__d———_——_—-d—uu—~—q——-——__———~—~—_q——~—
0 0
0 5 n0.

0 0

AI

003wZQfC0 0>3020K

 

 

 

 

 

1.00

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.50 f
G) —.
U _
3 :
.4: _
E :
E _ .
O 0.00 t f
(D :
.2 I
.44 —
g _
(D : —/__i
01 —0.50 : A _/_i
—1.00 - I I I I I I I I I I I I I I I I I I I I I I I I I I I I
0.00 0.05 0.10 0‘15

Time (nsec)

Figure 4.31 CRTW corresponding to missile type C.

136

20.00

1111111111J

15.00

10.00

Energy Ratio ((318)

5.00

 

0.00
2.

Oirrrirrrirlrrrirriirli11141111

0

Figure 4.32 Convolut

dCSlgned

 

 

 

20.00

15.00

 

Energy Ratio (dB)
8
8

 

—
-
-
.—
-
—1
-
.-
-
—
—
_
—
_
—
_
—
-
—
-
— I
-|
—
_
-
—
—
_
—
—
_
cu
-
—
—
—
—I
_
-

 

 

 

' — Missile A

-------- Missile B

---- Missile C
5.00 '
l
’l
I
I

0.00 IIIIIIIIFIIIIIIrITErIIIIIIIIIIerIIIIIIIIIIIIIIrI
2.00 4.00 6.00 8.00 10.00 12.00

Time (nsec)

Figure 4.32 Convolution energy ratio response for each missile using a CRTW
designed for missile A.

137

20.00

15.00

 

10.00

Energy Ratio (dB)

5.00

lllllllLlIlllllllllLl111111111111111111]

 

Q

O

O
I

2.00

Figure 4.33 Convolut
missile t}

 

 

20.00

 

 

 

 

15.00 5 .
/'\ " II
no I 1
"O _
\./ _
O .—
113 I i
O 10 00 — '
D: I i -——— Missile A
- ' -------- issile B
5% I I ---- Missile C
L -1
0) _
C _ I
I._L.I -
5.00 : I
0.00uIIIIIIIIIIlIIIIIMIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIm
2.00 4.00 6.00 8.00 10.00 12.00

Time (nsec)

Figure 4.33 Convolution energy ratio for each missile using a CRTW designed for
missile type B.

138

 

8.00

rgY R'atiO (dB)
Ene

 

0

Figure 4.34 Convolut

missile tj

 

 

16.00

 

 

 

 

 

 

12.00 —'
/"\ T
m I .
U _ 1
v ..
O I
'-g I
8.00 -
D: I I —— Missile A
>\ _ l -------- Missile B
U) I ---- Missile C
L —
0) _
C _
I_LI _
4.00 —_-
: 1
0.00 _IrIrlIIIIIIIIIII-I'IJIIIIjTIIIIIIIIIIIIIIIIIIIIIllfl
2.00 4.00 6.00 8.00 10.00 12.00

Time (nsec)

Figure 4.34 Convolution energy ratio for each missile type using a CRTW designed for
missile type C.

139

 

 

 

N periods

r141
1‘

 

 

Figure 4.35 Scatterin
(b) relati

 

4 '4. f x1"! 1‘

~<
XL
1

N periods

 

 

 

 

 

 

 

 

 

 

 

r TM-polarization showing (a) Stoke’s surface, and

Figure 4.35 Scattering geometry fo
respect to Stoke’s surface

(b) relative position of cylinder with

140

 

 

 

 

X7

0.50

IlllllllllllllLllllli

1

0.00

Amplitude (Relative)

“0.50

 

Irririiririiiiirrrii

l

b

.0 O
O
0

Figure 4.36 Transien

above th

 

 

N = 11, h = .0254m, L = .1016m, <I>i = 20°

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.00 : X0 = .0508m, Yc = .0635m, R = .005”.
0.50 -‘ .
/'\ ‘ I
(D _
> I
1+3 _
2 - i
<1) I i: .
D: - i I ‘ 1 1
V 0.00 :M N ’
(D ‘ ii:i '1 ' I .
‘0 j I
3 _ t I
:1: _
E. :
: —— Surface only
‘ ----- Surface and Target
—1oOO IITIIIII IrIIIIIIIrrIIIIIIIIIIIIIIIIIIIIIIIIIllm
0.00 2.00 4.00 6.00 8.00 10.00

Time (nsec)

Figure 4.36 Transient scattered return from a Stoke’s surface only and from a cylinder
above the Stoke’s surface. TCR = -5.31 dB.

141

 

 

 

 

0.50

3

I N =

2 X0
0.30j
010—:

Amplitude (Relative)
.c'2
S

I
Q
(A
C)

 

0.50
0.

oIiiiiiii11111111111111111

0

Figure 4.37 Differeni
with cyli

 

0.50

 

 

 

 

 

 

 

 

 

E N = 11, h = .0254m, L =.1016m, <1>1= 20°
: Xc = .0508m, Yc = .0635m, R = .005m

030—:
A :
(I) Z
.2 :
+1 _
2 0.10 j
G) _.
0: :

V :mew‘
g E
3-0.10j
.4: :
E :
E :
< :
—0.30:

—o IIIIIIIII IIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIm
O 5(0.00 2.00 4.00 6.00 8.00 10.00

Time (nsec)

Figure 4.37 Difference between the transient scattered return from a Stoke’s surface
with cylinder and Stoke’s surface without cyllnder.

142

 

20.00

 

 

1

3 N

: Xc
15.00 :
A .
m :
0 .
V _
0 :
310.00 :
0 -
0: :
C :
O :
'5 5.00 ‘
3 :
O :
> _
C _
O :

0 0.005 n

~5001r~r~r~

2.00

Figure 4.38 Convolu

multipat
TCR = -

 

 

 

 

20.00

 

 

 

 

 

 

E N = 11, h = .0254m, L = .1016m, <l>i = 20°
2 Xc = .0508m, Yc = .0635m, R = .005m
15.00 '_‘_
A _
m : '1
‘0 — I
V - I
T |
o 3 '
.4: 10.00 j
0 :
05 :
C I
O I
113 5.00 ‘_‘
_:5_ _
o 2
> _
C I
O :
Q 0.00 ':
E ----- No interaction
E Interaction
_5.00 IIIIIIIIIIIIIIIIIIIIIIIIIIIII—I
2.00 4.00 6.00 8.00

Time (nsec)

Figure 4.38 Convolution energy ratio for Stoke’s surface and cylinder with no
multipath effect, and Stoke’s surface and cylinder with multipath effect.

TCR = - 5.31 dB.

143

 

20.00

11111111111

15.00

 

- 10.00

Ratio (dB)

.CJ'l
O
O

Convolution

Q
O
O

 

~s.00 1a..
2. 0

O1111111111111111111111111111111111111111

Figure 4.39 Convolu

multipat
Effect,

 

 

 

 

20.00

 
  

 

 

 

 

 

 

 

E N = 11, h = .0127m, L = .1016m, <l>i = 20°
-_- XC = .0508m, Y0 = .0635m, R = .005m
15.00 :

/'\ _

m :

U _ I

O 2

5 10.00 :

U I

CK :

C 3 I

O I l

113 5.00 - I

3 I :

6 : .

> _ I

C 2 I

O I l

O 0.00 '3 A A I‘""
E ----- No interaction
— interaction

—5.00 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I—I
2.00 4.00 6.00 8.00

Time (nsec)

Figure 4.39 Convolution energy ratio for Stoke’s surface and cylinder with no
multipath effect, and Stoke’s surface and cylinder with the multipath

effect.

144

 

25.00

20.00

15.00

Ratio (dB)

10.00

5.00

Convolution

0.00

lillIllllIllllIllllIllllIlll_1__I

I
p1
O
O

I

H

. 0

 

{\D
C

Figure 4.40 C onvolu

multipar
effect. ’

 

25.00

 

 

 

 

 

‘ N = 11, h = 0.00635m, L = .1016m, <l>i = 20°
: X0 = .0508m, Y0 = .0635m, R = .005m
20.00 -
A T'
m _
‘0 _
V1500 -
.9 I
4.1 l
g I :1
10.00 — ; i
C ‘ i I
O ’ I i
a: - I l
3 ‘ ,’ l
6 5.00 n l I
> T I, l
C ~ r I
8 : 1
0.00 — -- "" " T" T T TT
: ----- No interaction
_ Interaction
—5.00 iiiiiiIlllll||"'|""""'1
2.00 ' 4.00 6.00 8.00

Time (nsec)

Stoke’s surface and cylinder with no

Fi ur 4.40 1 ton ener ratio for
g e Convo U] gy cc and cylinder with the multipath

multipath effect, and Stoke’s surfa
effect. TCR = —.22 dB.

145

 

 

15.00 ‘

 

10.00 ‘

ratio (dB)

5.00 ‘

Convolution

 

1
~5.00 ‘TTT‘F
2.00

Figure 4.41 Convolu
effect a:
position.

 

 

 

 

Xc=.0508m, Yc=.0635m, R=.005m

 

 

 

   

 

 

20.00 : N=11, L=.1016m, ¢i = 20°
‘ 11
15.00 -— 55
A ‘ II
in _ I;
U _ l
V _ 1
.9 10.00 - ,1.
-I-J _ '|
O ,i
L - I I
C - i 'i
o ' i 1".
:3 5.00 _ I 1“,
2 'T I 1‘
O ..
>
C _
O ..
O 0.00 —
Z — h=.0254m, TCR= I—5.31 dB
_ --—- h=.0127m, TCR= —-3.63 dB
------- h=.00636m, TCR= -.22 dB
—5.00 I I r I I I I 1 I I I I I I I I I I I I I l
2.00 3.00 4.00 5.00 6.00 700 8 00

Time (nsec)

Figure 4.41 Convolution energy ratio for Stoke’s surface and cylinder with multipath
effect as a function of different surface heights for a fixed cylinder

position.

146

 

 

0.50

 

IiiiiLiiiiliiiiLiiiiJ

1

0.00

Amplitude (Relative)

-0.50

 

Iiiiiriiiiliiiiiiiii

I

C)

.O o
o
0

Figure 4.42 Transier
above 0'

 

All

 

 

N = 11, h = .0254m, L = .1016m, ¢i = 20°

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1-00 : Xc = .0506m, Yc = .0889m, R = .005m
0.50 1
A _
(D _
> I
1+3 _
2 _
(D I
E — N I N I
0.00 : ' l ’
(D '_' II I, A ii I
TO _ i
3 _
:1: _
TE; :
{—0.50:
_ I
Z — Surface only
‘ ----- Surface and Target
_1.00 IIIIIIIITIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII‘I_I
4.00 6.00 8.00 10.00

0.00 2.00
Time (nsec)

Figure 4.42 Transient scattered return from a Stoke’s surface only and from a cylinder
above the Stoke’s surface. TCR = -5.31 dB.

147

 

 

 

 

Arnplitude (Relative)

 

 

 

 

0.75

 

 

 

 

 

 

 

 

 

: N = 11, h = .0254m, L = .1016m, ¢i = 20°
_ Xc = .0508m, Yc = .0889m, R = .005m
0.50 —_
a 3
.2 0.25 -
g _
2 _
(D -
m _
v 0.00 -*I
<0 I
U _
3 —
3—0 25—
C. . _
E _
< _.
—0.50—
— -IIIIIIIII IIIIIIrII IIIIIIIIIIIIIIIIIIIIIIIIIIIm
0750.00 2.00 4.00 6.00 8.00 10.00

 

Time (nsec)

Figure 4.43 Difference between the transient scattered return from a Stoke’s surface
with cylinder and Stoke’s surface without cylinder.

148

 

 

q
-4

—l

c-

d

_

-

-1

d
050—

l

d

d

.1

fl

‘

-1

-

‘

c1

—.

0.00

Amplitude (Relative)

 

 

 

 

 

N = 11 h = .0254m, L = .1016m, <I>i = 20°

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1'00 : X0 = .0’508m, Yc = .1143m, R = .005m

0.50 —'
A - i
(D T i
.2 Z
+J _
_O _.
(D I
G: _ '1
V 0.00 : N N ’
CD " ii
‘0 j i r
3 _
5r: _
E? :
<1: —O.50 :

3 Surface only
‘ ----- Surface and Target
—1.00 IIIIlIIIIIIIIIIIIllIllIrIIIIIIIIIIIIIIIIlllllllm
0.00 2.00 4.00 6.00 8.00 10.00

Time (nsec)

Figure 4.44 Transient scattered return from a Stoke’s surface only and from a cylinder
above the Stoke’s surface. TCR = -5.31 dB.

149

 

; 0.50 1
‘ Xc

0.25 -

 

Amplitude (Relative)
.c'>
o
o
1 l

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0’50 — N = 11, h = .0254m, |_ = .1016m, q); = 20°
‘ Xc = .0508m, Yc = .1143m, R = .005m
0.25-
/'\ _
(D
.2 _
4.2
2 ._
(D _
D:
V—OOO Mil N W
(D _
‘0 ..
:5
.4: _
B. _
E_ _
<1: 0.25
—050 Illlllll lllllllll|lllllllll|llllllllllllllllllll
0.00 2.60 4.00 6.00 8.00 10.00

Time (nsec)

Figure 4.45 Difference between the transient scattered return from a Stoke’s surface
with cylinder and Stoke’s surface without cyllnder.

150

 

25.00
l

 

 

: N

a X<
20.00 -
A I
% 15.00 —
V "l
.9 I
E l0.00 :
l1 _
C I
,9 5.00 —
:3 _
O I

E 0.00 ~ _
o _
0 I
43.00 I

2.00

 

 

 

25.00

 

 

 
   

  

 

   

 

        

 

 

 

: N = 11, h = .0254m, L = .1016m
_ X0 = .0508m, R = .005m, ¢i = 20°
20.00 :-
_ '5";
ES 3 '
3 1500 : bl
.9 f l:
*5 10.00 :
Oi ..
c 2
.9 5.00 —
4g _
2 I l
e - «l
C 0.00 : = ":u'
8 : ‘a. .5 'v’
“5'00 : Yc = .0635 m
— -------- Yc = .0889 m
; —-—— Y0 = .1143 m
"' llllllllllllllllll|llllllllll
10002,00 4.60 6.00 8.00

Time (nsec)

Figure 4.46 Convolution energy ratio for Stoke’s surface and cylinder with multipath

effect as a function of cylinder height.

151

 

l
l
15.00 a
l0.00 ‘
5.00 ~

0.00 ~ __

Convolution Ratio (dB)

 

~5.oo ~+fi
2.00

Figure 4.47 Convoll
multipa

 

 

 

 

 

 

 

 

 

 

 

 

 

N = 11, h = .0254m, L = .1016m
15.00 - Xc = .0508m, R = .005m, <l>i = 20°
$10.00 -
U _
\_../
O _
.4:
O ..
D: 5.00 —
C _
O
1: _.
2 _
O
E I L
0 OOO “ — 3:7
0 _.
‘ Y0 = .0635 m
‘ --------- Yc = .0889 m
_ ————— Yc = .1143 m
_ I I I I I I I I I I I I I I I I I I I I I I I
5002,00 4.60 6.00 8.00

Figure 4.47 Convolution energy ratio for Stoke’s surface and cylinder with no

multipath effect as a function of cylinder height.

152

 

_q_d___q_-____._._a~_..a_..__a_._.__~_
0
.l.

O
Ru.
0

mac 5
o. mu.
A®>:O_®Kv ®U3§QE<

 

m0
AU.

00

0

.

Figure 4.48 Transie

abovet

ii

 

 

 

, .0 54m, L = .1016m, (M = 20°

1 h =
1.00 Xe = 0.0m, Yc = .0635m, R = .005m

 

0.50

 

 

 

 

 

 

 

 

 

 

=‘ll WW

‘E’

 

 

 

 

 

 

 

 

 

 

 

 

 

Amplitude (Relative)

O
O
O
IllllllllllllllllllllllllllllIllllllllLI
.2
?
S

 

—O.50

 

 

 

 

 

 

 

 

 

 

 

 

Surface only
----- Surface and Target

—1.00 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I
0.00 2.00 4.00 6.00 8.00 10.00
Time (nsec)

 

 

 

Figure 4.48 Transient scattered return from a Stoke’s surface only and from a cylinder
above the Stoke’s surface. TCR = -5.31 dB.

153

 

 

0.25 -

Amplitude (Relative)
.c'>
o
o
l I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0'50 ‘ N = 11, h = .0 54m, L = .1016m, 121 = 20°
- Xe = 0.0m, Yc = .0635m, R = .005m
0.25 —
f0? ‘
.Z -
4.2
2 _.
(I) -
CK
(D _
U _
3
:‘I-_’ _
_Q_ a
E_ _
< 0.25
— Illlllll llllIIlll IIIIIIIII|IIIIIlllllllllllllll
05%.00 2.00 4.00 6.00 8.00 10.00

Time (nsec)

Figure 4.49 Difference between the transient scattered return from a Stoke’s surface
with cylinder and Stoke’s surface without cylinder.

154

 

l
' 20.00 1

15.00

 

l0.00

5.00

0.00

Convolution Ratio (dB)

~5.00

liiiiliiiiliiiiliiiiliiiiliii

I
Q
O
O

 

N
o
a

Figure 4.50 Convoli
effect a

 

 

20.00

 

 

 

 

   

 

 

 

' N =11, h = .02 4m, L = .1016m
: Yc = .0635m, R = .005m, ¢i = 20
15.00 -
A '-
m _
‘0 _
V 10.00 -
O I
1*: _
D? _
5.00 —
C _
O ‘ 'l
1": _ l
g - '1
O 0.00 — " " I
> ‘ I
C — : ,4
8 Z :l
__ _ 1:
5'00 - ii: ----- Xc = 0.0 m
— \ Xc = .0508 m
— IIIIIIIIIIlillllllllllllllirl
10002,00 4.60 6.00 8.00

Time (nsec)

Figure 4.50 Convolution energy ratio for Stoke’s surface and cylinder with multipath
effect as a function of target position.

155

 

20.00 1
15.00
l0.00

5.00

0.00

Convolution Ratio (dB)

l
911
O
O

111111111]111111111111111111

I
9
CD
CD

I

 

fi‘

2.00

Figure 4.51 C 0W0]

multipa

 

20.00

 

 

 

 

 

 

 

 

- N =11, h = .02 4m, L = .1016m
: Yc = .0635m, R = .005m, (bi = 20°
15.00 n
u
A _
m -‘ W
'0 u
v 10 OO -
0 Z
'43 _
a? _
5.00 - 1
c - r.
.9 - l
+a ‘ ,n
3 — I:
6 0.00 _ —'-“'"’| V’Vf
> ‘ I N
C - I ‘1
8 : l 1'
_ _ I',’l"
5'00 .. ‘ ----- Xe = 0.0 m
— Xc = .0508 m
_ I'I'l'lllIIIIIIIIIIIIIIIIIIII
10002.00 4.00 6.00 8.00

Figure 4.51

Convolution energy ratio for
multipath effect interactlon as a

156

Time (nsec)

Stoke’s surface and cylinder with no
function of target position.

 

 

 

H

l N
k

 

positi<
‘

_/\

Figure 4.52 Multip

 

 

 

 

E
N=1o°
i2

position 1 position 2

 

 

position 1 = over crest
position 2 = over trough

Figure 4.52 Multipath missile/sea-surface scattering geometry.

157

 

 

b
O
J_I

 

Q
()1
O

0.25

0.00

i
Q
(\J
U“:

 

Normalized Amplitude

l
9
()1
CD

I
F3
\1
U1

 

IllliillllIllllIllllIllllIllllIllllI111

 

l
b
o

 

9
O
C

Figure 4.53 Norma

missile
degree:

 

 

 

1.00

 

 

 

 

 

 

 

 

 

 

 

: .a
- E
0.75f 91=10 o
T A
i H
(D 0.50 - A
U _ k ‘
3 I
.‘L’ _
E :
<5 ..
.0 0.00 :wli
G) .—
E I
6—025:
E :
5 ..
Z—O.50:
—0.75-'_~'
— _ lllllllIIIIIIIII|IIIIIIII||
10%.00' ' 1.00 200 3.00

Figure 4.53 Normalized backsc
missile excited by

Time (nsecj

atter transient-response from 10 cm long phoenix
a TE incident plane wave. Incidence angle is 10

degrees from the horizontal axis.

158

 

.\.I

 

 

 

Q
UT
0

0.00

lllIlllllllllIlllllllllI

CL

 

NomwaHzed Anuflflude

l
FD
an
CD

 

lrirrrriiilriiiii

l
a:
(3‘33

i‘l‘r
o

0

Figure 4.54 Norma

byal
hofizo

 

1.00 -

 

 

 

 

 

 

 

 

 

 

 

 

 

@0505
U _
3 _
.t’ I
5— :
E _
< I
.0000:
(D _
.51 I
5 :
E _
L I
§—0.50:
_IOOIIirrrrrrllllllllrlllllIlrr11—1
0.00 5.00 10.00 15.00

Time (nsec)

Figure 4.54 Normalized backscatter transient-response from a Stoke’s surface excited
by a TB incident plane wave. Incidence angle is 10 degrees from the

horizontal axis.

159

 

 

 

i
u 0.501
F :
3 -4
r: _
71 :
E :
< —
D 0.00 :fi
(1) -
.N 3
F :
E _
L :
§-0.5o —_
~i00 3m
0.00

Figure 4‘55 Norma

exonec
the 110'

 

1.00

 

 

 

 

 

 

 

 

 

 

o 0.503

"O ..

3 I

."L’ _

E :

E :

< ._

TO 0.00:

(I) _.

.E I

‘5 j

E .4

’5 I
— ‘ lllllllllllllllll||llllllll|
“0%.00' 5.00 10.00 1500

Time (nsec)

-response from a double-sinusoid surface

Figure 4.55 Normalized backscatter transient _
ence angle 18 10 degrees from

excited by a TE incident plane wave. Incid
the horizontal axis.

160

 

 

 

_ L

R
0.00

 

EJ- _ u _ _ _ — _ _ — _ ~ _ — _ ~ _ _ _ u _ — . _ _ _ — u _ _ _ _ _ _ _ _—

O O O O O

O. 5. O. 5. 0.
oDJEQE< 020.90%

Figure 4.56 CRT“

 

 

1.00

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

050—:

(D _ .

e ; f

3 _

7': _ .

E : F.

g 0.00-E

(D :

.2 I

+J .—

2 _

<1) :

01—050:
—1.00qIFIIIIIIIIIIIIIIIII‘IIIIIIIIIIIjIIlllllfl

0.00 0.04 0.08 0.12 0.16

Time (nsec)

Figure 4.56 CRTW corresponding to measured Stoke’s surface.

161

 

 

will. .3.
VP»
new

0 me

filaquu__a_~_a_-___q_____________._._..~.~nfl/_U. r

m 5 O 5 flu. 4"
4| III e

Amzov 0:01. c0530>c00 on.
mull.
‘

 

 

 

20.0

.01
o

10.0

 

 

 

 

.U‘
o

 

Convolution ratio (dB)

 

 

 

 

 

 

 

llllllllIlllllIlllIlllllllllllllllllllI

 

l

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I
0'02 00 ' ' ' ' '400 6.00 8.00 10.00 12.00
Time (nsec)

 

 

 

Figure 4.57 Convolution energy-ratio as a function of phoenix missile position with
respect to Stoke’s surface. TCR = -23.2 dB.

162

   

1.00

0
av.
O

O
flu.
flu

003,:QCL< o>506m

~O.50

______..___._____.______
OO
0

Le

all

L

Figure 4-58 CRT“

 

 

 

1.00 -

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.505
G) _
U I
3 _
7': -
E :
E 0.00-f
0) : m
.2 :
+J .—
2 _
(D I
01—050:
—100 lllIIIIIIIIIIIIIIIIII'IIIIIIIIIIII'I'Ij—I
0.00 0.05 0.10 0.15 0.20

Time (nsec)

Figure 4.58 CRTW corresponding to measured double sinusoid surface.

163

I\)
91
CD

 

 

i

20.0 5
o I
m .
o :
V _
.9150 5
+J -4
O ..
L _
C I
O -.
510.0 1
3 :
0 :
> ..
C I
O _
0 5.0 f
0.0 l

3.00

Figure 459 C 0W0

reSpool

 

 

 

 

 

 

 

 

 

 

 

 

 

 

20.0
A
00
U
V
.9150
_I_J
O
L
C
O
3100
3
O
>
C
O
O 5.0
000 IFIIIIIIIIIIIIIIIIrIIIIIIIIIIIII IIIIII
3.00 5.00 7.00 9.00 11.00 13000

Time (nsec)

Figure 4.59 Convolution energy-ratio as a function of phoenix missile position with
respect to double-sinusoid surface. TCR = -4.75 dB.

164

C.)
N
U!

Arriplit udc‘:

0.00

Normalized

l
Q
I\.)
(J7

I11111111111111111]!III]1441_L1..I_.I_J1_JI11.1.J_.1.J

 

I
Q
(J?
O

iTTt‘ru-w

.50

\I

Figure 4.60 N 011113
Stoke‘s
trOUgh

 

 

1.00 :

0.50

 

 

 

 

   

 

 

 

 

 

: E
- :0.003
a) 025-3 E i
"U - e
3 -' Z
.t’ I
a :
E -
< I
T) 0.00:
(I) ._
.N :
5 :
E :
5 - No target
22‘4125': ---- Target over crest
: -------- Target over though
—OSO —lIIIIIIIIIllllllIlIIliliillllllllllllllIIIlllllll]
7.50 7.75 8.00 8.25 8.50 8.75

Time (nsec)

scatter transient-response from a phoenix missile and
TE incident plane wave. Missile is above
10 degrees from the horizontal axis.

Figure 4.60 Normalized back
Stoke’s surface excited by a
trough of wave and incidence angle is

165

 

r.)
N
C‘)

 

]

jg 6.103 i
V -
3 _
”F. I
K .
>5 1
< :

f‘.
D 0.00 - r'
0 ~ /
N _ \
,2 i
V
L I
0 _
Zrflm~

‘0'20 "i‘fi‘rn-Tn

_\l
\l
UT
m

Figure 4.61 Nonna

double
abOVC
hOflzo

 

 

 

 

 

 

 

 

 

 

0.20 -" g 0.50
:
: i D 000
_ 1‘ g
' 1‘ E
: l I £4.50
'0 - /
.3 I " i:
Z ‘ I i ii i 0'60 5'00_ IIIIIII 15.106 IIIIIII 1‘5'00
D. : l I ,‘l i A V‘ Time (nsec)
E - I I A ':\I :1 \‘ l I
<1: : i I‘ \ / ,. / I“; I l ‘ ‘1‘ III
0.00 “ I i X‘ ’ \ F - I
‘O - F' \ \ I J’ \ \p’i’ ti 1'
(D - \l \/ l | I I I 1
N " I ‘ ‘1 I I t
Z "‘ I I,I I I ‘I \I
O - ' 1 1 V
‘ l 1'
E ~ , n
L - l '1'
O “ ‘1‘
Z—ono— I t
Z l
_ 1, No target
1 '1 ---— Target over crest
: “1 -------- Target over trough
— l
- |
—O.20_IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
7./5 8.00 8.25 8.50 8.75 9.00

Tune (nsec)

Figure 4.61 Normalized backscatter transient-response from a phoenix missile and
double-sinusoid surface excited by a TB incident plane wave. Missile is
above trough of wave and incidence angle is 10 degrees from the

horizontal.

166

 

 

 

“ W AVubenennu
“ HO mmw
—dn——_~4—q—————q——-u_———q—q—_———~qqn—q——~—~_—qq—~—_-—~_O. C81“
0 flu 0 0 0 0. flu. 02 MM
miu. oo. 6. 4. Z 0 JA 41 M.
AmUV 03-01. C0.53.0>C00 Wis
F
4

 

 

10.0

 

8.0

.03
o

------ No coupling/multipath
—— With coupling/multlpath

 

.t‘
o

 

I"
o

LllllllIlllllllIlllllllllllllllIlllllllIlllllllIlllllllI

 

0.0

Convolution ratio (d B)

 

 

IIIIIIIIIIIIIIIIIIIIIIII

1 1 1 1 I I
_4.02 1 1 1 1 141.661 600 800 1000 1200
Time (r1890)

y-ratio for phoenix missile located above a trough in

. ' ner -
Figur e 4'62 COHVOlunon e g om compos1te measured return and

Stoke’s surface. Ratio was calculated fr

hybrid-composite measured return. TCR = -23.2 dB.

167

 

Q
Q

Convolution ratio (dB)
0 o o Q

Q
Q

111111111111111111111111lrirriirliiriiriliiLiirrliiiiililiiiiiriJ

i
no
0

 

I\)
O
C

Figure 4.63 C onvt

dOUbh
fetUm

 

 

14.0

 

I"
o

 

.0
o

------ No coupling/multipath
—-— With coupling/multlpath

.00
o

.42
o

2.0

Convolution ratio (d B)

 

 

 

 

03
O
1111111l111111111111111|1111111|1111111l1111111|1111111|1111111I

 

1 1| 11 11 11 1| 11 11

T" 11 11 1 11 11 11 11 11 11 1| 11 111
2'92. 4.00 6.00 8.00 10.00 1200
Time (nsec)

 

y-ratio for phoenix missile located above a trough in

face. Ratio was calculated from composite measured
= -4.75 dB.

Figure 4.63 Convolution energ

double-sinusoid sur
return and hybrid-composite measured return. TCR

168

 

 

[\D
0
CJ

b

ratio (dB)

l

_..L 0C3
. .
liiiiLirlri11111l111111L1111111

l
(\3
O

Convolution

 

[\J
O
O

Figure 4.64 Com

Stokt
hybn

 

 

2.0

 

 

 

 

 

 

 

 

: ------ No coupling/multipath
- —-— With coupling/multipath
A 1.0 i
m _
“o _
\./ A
o I
113 _
E .4
0.0 - _
C _
O _
.4: _
3. Z
O _
E _
O —1.0 —
Q :
—o _IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
202.00 4.00 6.00 8.00 10.00 12.00

Time (nsec)

ratio for phoenix missile located above a crest in
ated from composite measured retum and
= -23.2 dB.

Figure 4.64 Convolution energy—
Stoke’s surface. Ratio was calcul

hybrid-composite measured return. TCR

169

 

 

 

. x 1
_ wo mum
ad—d~__~_q__—~—_——___—___———~_————~_q~—_—_~—~—~—~—A-—~_#d__-—m (dr
0 0 0. flu. 0. 0. 0. 0. nfl/Lu ”m.
Ru. r0. Ar 3 2 1 flu .ul _ M.
AmUV 0.501. COEJC0>COU We
4

 

 

 

6.0

   

 

 

 

5.0 —f
/\ 40-;
[I] 3 ----- No coupling/multipath
U - -— With coupling/multipath
V :
O 3.0 E
'43 :
E :

2.0 E
C I
.9 E
‘S 1x3€
6 E
> :
g 0.0 _:
o s

-—Lo§
i i
—2.0-l[lllerT—lllllllllllIll]l:lllll|llllllllll—T—I
2.00 4.00 6.00 8.00 10.00 12.00

Thne (nsec)

Figure 4.65 Convolution energy—ratio for phoenix missile located above a crest in
double-sinusoid surface. Ratio was calculated from composue measured

return and hybrid-composite measured return. TCR = -4.75 dB.

170

 

COI'I‘E

O
7

WWWWWWW W W W W.

WOC®FC®L3WU®§

.0
Figure 4.66 Amp

meUC.

 

 

 

 

 

m. . liIllllIl

. lllll 0.
m
.. M I 0.
Im

8
iii!!! 0.

illlllilll
fix?

I)?

WWW

_

 

 

 

_ nU
0000000

ooooooooooooooo

algOI
over

Figure 4.67 Tan:

X®UC_ L®wC®O >>OUC._>>

 

 

 

 

CD
I

i if:
i is:
if e

 

O I I I
60.0 70.0

 

I ' I ' I I I ' I ' I I I
80.0 90.0 100.0 110.0 120.0 130.0 140.0
Time (nsec)

n using coherent detection

Figure 4.67 Target detection of simple missile simulatio
.25 seconds, with averaging

algorithm. Time step between pulse returns is
over three successive pulse returns.

172

 

 

 

 

I‘CI II II

COITC

1010 0
Figure 4.68 Amp

.WWWWWWWWW.

I
0
X®UC_ wC®CL®L3mU®

 

 

 

 

é 8“wa
73 /”\/\J\/\/\VA\/\4f¥~/\¢AVF\/Ah/\eF-y\~f‘v’—~AV/\/\JAL/\/~v~/VAV/\
.E
E; 6-/\rv~flfVNf\»~»vxrvoJ\/\~yrv~wv\flvN/v«»va\
(I)
g WWW/W
3 4- J
U) I
O I
(I)
O I | I I ' 7
110.0 120.0 130.0 140.0

Tune (nsec)

om an evolving sea surface. This case

Figure 4.68 Amplitude of scattered return fr
= .05 seconds between successive pulse

corresponds to a time step of AT
returns.

173

 

 

 

ihdex

Wihdow Cehter

 

O‘\

110.0

FigUre 4'69 Tame

algori
O‘ICI. .

 

 

 

10-

 

 

 

Ihdex
CI)
1
1

 

 

 

 

 

 

 

 

Wmdow Center
_;>.
I

i 1

 

 

 

r

r F —1
110.0 120.0 130.0 140.0

Time (nsec)

Figure 4.69 Target detection of simple missile simulation using coherent detection
algorithm. Time step between pulse returns is .05 seconds, with averaging
over three successive pulse returns.

174

 

Cept

5,1 lntroductior
A challengir
and or identification
in the ability of rad:
enhanced both the q
airbome threats is b
with that target. 01
target‘s complex t‘re
transient EM field
illuminated by an
modelled as having

that the total scatter

A simple model for

Where f"(t) is the r

In. The late-time c

 

Chapter 5
Cepstral Analysis and Radar Target Response

5.1 Introduction

A challenging problem encountered in radar system technology is the detection
and/or identification of airborne targets in a highly cluttered environment. Advancement
in the ability of radar systems to transmit and analyze signals of very short duration has
enhanced both the quality and quantity of target information [1]. Recognition of potential
airborne threats is based upon a system’s ability to analyze specific signatures associated
with that target. One technique used to identify targets is based on the uniqueness of a
target’s complex frequencies in the late-time transient scattered signal [4]. Consider the
transient EM field scattered by a perfectly conducting, finite size radar target when
illuminated by an incident EM pulse. The scattered return from the target can be
modelled as having an early-time component fe (t) and. a late-time component fl(t) , such

that the total scattered field is given by

ft!) = f.(t) + f,(t) (5.1)

A simple model for the early-time component is [42]

N
fett) = Z f.(t-T..> (5'2)
n=l

where fn(t) is the pulse response of the n‘h scattering center located. at temporal position

Tn. The late-time component can be written as

M
fl(t) = Z ameomtcos(wmt +d>m) t > TL (5.3)
m=l

175

 

 

 

 

 

 

where Is” = om +

late time. The sc
However, the targe
target. thereby pro
Inherent in
both magnitude an
spectral magnitude
transient signal can
under certain condi
a signal‘s spectru
through a Hilbert
frequency spectrum
more restrictive am
both the poles ant
minimum phase cor
through a Hilbert
response of the sp
needed if these tra
An altemat

cepstrum approach

transient signal usi

transform (FFT), t

 

 

where {Sm = om + j mm} are the target natural frequencies and TL is the beginning of
late time. The scattered return from the early-time signal is aspect angle dependent.
However, the target’s natural frequencies are aspect angle independent and unique to the
target, thereby providing a target identifier.

Inherent in the generation of these complex frequencies is the measurement of
both magnitude and phase from the target scattered return. In general, a knowledge of
spectral magnitude does not allow the calculation of the phase and vice-versa. Hence, the
transient signal cannot be recovered with only spectral magnitude or phase. Nonetheless,
under certain conditions a relationship does exist between the real and imaginary parts of
a signal’s spectrum. For a real, causal signal, the real and imaginary parts are related
through a Hilbert transform integral [28],[29]. Typically, the magnitude of a signal’s
frequency spectrum is measured, rather than the real or imaginary part, and a somewhat
more restrictive approach must be used. If a finite length sampled signal is causal, and
both the poles and zeros of its z-transform lie inside the z-plane’s unit circle (the
minimum phase condition) then the phase and logarithm of the magnitude can be related
through a Hilbert transform. This relation is certainly useful if only the magnitude
response of the spectrum is available. On the other hand, a certain amount of caution is
needed if these transforms are blindly applied. to any signal.

An alternate approach for calculating the minimum, phase signal is to use the
cepstrum approach [28]-[29]. In this approach an attempt is made to recover the original
transient signal using only the spectral magnitude. This method employs the fast Fourier

transform (F F T), thus requiring considerably shorter computation time than the Hilbert

 

 

 

 

 

integral approach.

diverse fields as sp
unique application
an ultra-wideband
ultra-wideband rad
scattered signal.
Two questi
to the above signal
I. How must
What is the
2. Under wha
IF ,(w)| or
F (to) = .9

early and la

This chapte
many examples w
instances, cepstral
signal. Hopefully,
determine if the c

scattered return of

sections. In sectio

 

 

 

integral approach. The use of this technique has found specific applications in such
diverse fields as speech and image processing, seismology, and acoustics [30]-[32]. A
unique application of this method is to reconstruct a target’s transient scattered field from
an ultra-wideband radar signal. However, in order to apply this new methodology to an
ultra-wideband radar signal, some fundamental questions must be answered about the
scattered signal.
Two questions concerning the characteristic and application of cepstral analysis
to the above signal models should be examined:
1. How must fn(t) and sn behave so that fe(t) and. fl(t) are minimum phase signals?
What is the significance of these waveforms being minimum phase?
2. Under what circumstances can fe (t) and fl(t) be obtained fi‘om |Fe(<o) I,
1F,(00)I or lF(o>)|? HereFe(oo) = Flfe(t)},FI(w) = 9{fl(t) } , and
F ( (o) = .9{f(t) } . That is, is it possible to use cepstral analysis to separate the

early and late-time portions of a radar signal?

This chapter will attempt to answer these questions as rigorously as possible, and
many examples will be presented for clarification. It will be shown that in many
instances, cepstral analysis will work remarkably well even for a non-minimum phase
signal. Hopefully, a set of guidelines or rules of thumb can be developed which will help
determine if the cepstral technique can be used for reconstruction of the time—domain
scattered return of a radar target. The body of this chapter will be divided into several

sections. In section 2 background material is presented covering the necessary theory to

177

 

 

 

 

 

understand the has

in section three. N
discusses the use
radar signal. The
some further areas

identification in ch

5.2 Cepstral A
Two simpl
both figures the po
magnitude and pha
The important featt
for both sequences,
sequence for a give
using only the mag
in turn affect the p‘
To derive a

to see if some rel

response. A real, 0

odd non-causal seq

 

 

 

understand the basic ideas of cepstral analysis. Late-time characterization is presented
in section three. Next, the early-time response is discussed in section four. Section five
discusses the use of cepstral analysis to separate the early and late-time portions of a
radar signal. The final section summarizes the major points in this chapter, and suggests
some further areas of study. The ideas presented in this chapter will be applied to target

identification in chapter 6.

5.2 Cepstral Analysis - Theory

Two simple, real causal sequences are shown in Figure 5.0 and Figure 5.1. In
both figures the poles and zeros associated with the z-transform are shown as well as the
magnitude and phase responses obtained from the discrete time Fourier transform (DTF T).
The important feature to notice is that the modulus of the Fourier transform is the same
for both sequences, but the phase is not. These simple examples illustrate that the input
sequence for a given magnitude response is not unique. To construct the input sequence
using only the magnitude response, certain restrictions must be applied on the input which
in turn affect the placement of the poles and zeros in the z-plane.

To derive a relationship between the magnitude and phase, it is quite instructive
to see if some relationship exists between the real and. imaginary part of the spectral

response. A real, causal sequence x[n] can be written in terms of the sum of an even and

odd non-causal sequence as

x[n] = xe[n] + x0[n] (5.4)

 

 

 

 

 

 

A little reflection

even part as

where the window

The real part of t

The result of the

where z = e”.

calculated by taki
used to determine
real and imagina
process. The rest

the system poles

fora stable causa

 

 

 

 

xe[n] = %[x[n] + x[—n]]

(5.5)

x0[n] = %[x[n] - x[—n]]

A little reflection shows that the causal sequence can be written entirely in terms of the

even part as

x[n] = xe[n]°w[n] (5.6)

where the window function is just

2, n>0
w[n] = 1, n=0 (5-7)
0, n<0

The real part of the Fourier transform can be obtained by applying the DTFT to xc[n].

The result of the transform is

oo

2: xelnlz'"

n=-w

XR(e’“’)

II

CD

2 xe[n](coswn — jsinwn) (5-8)

n=~w

00

Z xe[n]coso>n

n=-0°

where z = ej “’. Hence, if the real part of the spectral response is known, xc[n] can be
calculated by taking the inverse Fourier transform. Once this is determined, xc[n] can be
used to determine x[n] from (5.6) and (5.7). Finally, a forward DTFT will yield both the
real and imaginary part of the spectrum. Figure 5.3 shows a block diagram of the
process. The restrictions placed on x[n] is that it be causal and stable. In this case all
the system poles must be located within the unit circle, since the region of convergence

for a stable causal sequence lies outside the unit circle.

179

 

 

 

 

 

 

 

 

Taking the natural

Consider (5.10) to
restriction that the
in the minimum p
with X(ej‘°) = 0,

stable sequence th
must lie with the u
is properly windov
corresponds to the
i[n] can he get
transform will yie
exponential can 0
One final inverse

diagram of the prt
sampled sequence

is implemented

periodicity in the

 

 

 

A similar argument can be used to derive a relation between the magnitude and
phase of the transform. Consider the complex spectrum

Xtejs) = IX(e"‘°) rem-we“) (5.9)

Taking the natural log of (5.9) yields

item) = 10g1X(e"°) t + jargmm) (51“)

Consider (5.10) to be the transform of a causal stable sequence fin] with the additional
restriction that the zeros also lie within the unit circle. This additional restriction results
in the minimum phase condition. Since the real part of X(ej“’) = log |X(ej“’)| diverges
with X (ej “) = 0, no zeros must exist within the region of convergence. For a causal,
stable sequence the region of convergence lies outside the unit circle; therefore, all zeros
must lie with the unit circle. fin] can also be written in terms of an even sequence that
is properly windowed as in the previous argument. In this case, the even sequenceJEe[n]
corresponds to the inverse transform of the log of the spectrum’s modulus. Using fie [n] ,
fin] can be generated by using the window function given in (5.7). A forward
transform will yield the complex spectrum Jae”) = log X (6’ w). Finally, the complex
exponential can be applied to log X (6”) to yield the original input spectrumX(ej°’).
One final inverse transform will yield the original sequence. Figure 5.4 shows a block
diagram of the process. In this diagram, the transforms have been written in terms of a
sampled sequence in the time and. frequency domain; hence, the discrete Fourier transform
is implemented with the FFT. As is well known, sampling in one domain will force a

periodicity in the other domain with a period determined by the sampling rate. By taking

180

 

 

 

 

 

the inverse FFT 0

and aliased. To co

must be used

One note 0
of the sampling ra
not be an exact d
frequency domain

The seque

following equatior

where IX[k] |is s
w[n] defined in (5

following relation

An accura
requires the minir

must lie within t

However, if the

 

 

 

the inverse FFT of the Spectrum’s modulus, the cepstrum (see below) becomes periodic
and aliased. To compute fin], a new window function defined by the following equation
must be used
2, 1 s n < N/2
w[n] = 1, n=O,N/2 (5-11)
0, N/2 < n s N —1
One note of caution is that aliasing will occur, which is a function of the number
of the sampling rate in the frequency domain. In this case the final sequence fin] will
not be an exact duplicate of the input sequence x[n]. A higher sampling rate in the
frequency domain will yield a better approximation to the input sequence.
The sequence c[n] is known as the real cepstrum and can be obtained with the

following equation

c[n] = FFT"(log |er] |) (5.12)

Where |X [k] I is simply the magnitude of the input signal spectrum. Using the window
w[n] defined in (5.1 l), the minimum phase sequence fin] can be computed using the
following relation

fin] = FFT—l(eFFT(w[n]c[n])) (5.13)

An accurate reconstruction of a signal using the cepstral analysis technique
requires the minimum phase restriction: all poles and zeros of the signal’s z-transform
must lie within the unit circle. Certainly, most signals will not be minimum phase.

However, if the system can be modeled. as minimum phase, the cepstral analysis

181

 

 

 

 

 

technique should

energy concentrat

sequence is define

By Parseval‘s the
total energy conte
near the beginnin

spectral magnitud

A sequence can b1

near the origin ( r

 

 

technique should be able to reconstruct the original signal using only the magnitude of
the spectral response.

One important property of minimum phase sequences concerns the location of
energy concentration [29],[33]. The energy from m+1 samples of a finite length

sequence is defined by the following relation

m

E(m) = Z |x[n]|2 (5.14)

n=0

By Parseval’s theorem, two sequences having the save spectral magnitude have the same
total energy content. However, for a minimum phase signal the energy is concentrated
near the beginning of the sequence. So, for signals x[n] and erninl having the same

spectral magnitude, the following relation holds

i lx[n]|2 -<- Z I’flninlnll2 for all m (5.15)
”=0 n=0

A sequence can be considered minimum phase-like if most of its energy is concentrated

near the origin ( n = 0 ).

182

 

 

mmghov

a
0
~

mcUDfiCOUCC 004

n/Z

NC

Figure 5.1

Z€l

 

 

 

 

z—plone

20— x[n]

O

O
N—
U—
C)
4*

 

-10-

 

“ZO—i

 

 

(a) (b)

 

 

 

 

 

 

 

 

 

 

 

 

 

180 —- I I I r r I I F;
120- q

A

m

'0

/'\

a) C»

D a)

3 "o

44 v

‘E a) 0

3 a

E e

O) —60—

O

__r

—120--
—180 r
0 1r/2 1r 3111/2 Zn 0 1r/ 2 1t arr/2 2"
(C) (d)

nput sequence; (b) z-plane pole-

Non-minimum phase systems showing (a) i .
(d) are I XW‘”) ]

Figure 5.1 . .
zero plot; (c) 20 logo I X(elw)/Xmax(ejm) I;

183

 

::— llt‘l

/
/

. //
i I
l -’

’\ _ i ‘

m ‘ 3 ‘1 (I

u-a .‘ ‘

I} ‘ I

\1‘
|

Log rrragnitudc:

‘15..

 

 

 

 

flak

Figure 52
pk

 

 

 

_' —" -=---—-i—....'.___ _- _~

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

z—plone
20— x[n]
104»
O n A A
t) i i ' v #3; V v
l 0
._10-1
_20_a
(a) (b)
0‘ 180—
1204
as _5-
U
V 604
/'\
‘1’ 0‘1
'0 a)
3 '0
.t’ v 0
a -10- a) n ,A
O a
E a
C» -60— a
3
-15-
—120~
—20 —180
0 n/z 1r 311/2 2n 0 n/2 1! Sir/2 2“
(c) ((1)

Figure 5.2 Minimum phase systems showing (a) input sequence; (b) z-plane pole-zero
plot; (C) 20 logic I Xte’w)/Xm..(e"”) l; (d) are [ X(€ ) ]

184

 

 

 

X(ei°’ )-

Tigure 5.3 810

for
x[n] 2.
-> FFT .

A.

Figure 5.4 311

1112

 

 

 

 

Xelnl x [n]

Xlgejw )‘f DTFT-1 —> DTFT _>X(ejoo)

l

w[n]

 

 

 

 

 

Figure 5.3 Block diagram showing conversion process from real to complex spectrum
for a causal, stable input signal.

 

x[n] 1“ X[k] ]X[k]] ———log]x[k]i _1 ctnt
—»i FFTH ] ] I—> Log »—> FFT —->?—>IA\

Win]

Block diagram showing construction of minimum phase signal from the
magnitude of the spectral response.

 

 

 

 

 

Figure 5.4

185

 

 

 

 

 

 

5.3 Late—Time
The late-tin

as given by (5.3 ).

time Variable t can

Tas

The corresponding

there

In order for the n
altasrng)‘ the pole
Z‘Plane‘s unit cit-c

iS just

F01 any fillii€~Ier

However. the zen

on the zeros are 1

c .
a" be Written as

 

 

5.3 Late-Time Analysis

The late-time signal can be modelled using a sum of damped sinusoid components
as given by (5.3). Consider a time-sampled sequence whose sampling period is T. The

time variable t can then be written in terms of a sequence index n and sampling period

T as
t = nT (5.16)
The corresponding time-sampled sequence is then
M
fllnl = Z Cman',I cos(an + (pm) (5.17)
where "1:1
am — e°1~T (5.18)
Qm = me (5.19)

In order for the minimum phase reconstruction to match the original signal (neglecting
aliasing), the poles and zeros of the sampled signal’s z-transform must lie within the
z-plane’s unit circle. The z-transform for an N-point sampled sequence given by (5.17)
is just

Nel

F,(z) = Zfitnlz‘" (5-20)

n=0
For any finite-length sequence, the above polynomial contains N-l poles at the origin.
However, the zeros of the above polynomial are not easy to find. More properly, bounds

on the zeros are difficult to establish. By expanding the z-transform polynomial (5.20)

can be written as

186

 

 

 

 

For/1101 : 0, the
roots.
Consider tl

relation

For 0< am<l (C

(5.33) are related

For an N‘h degree

where (10 < a] < a:
lie within the z-pI
by (5.22) is min
sequence if the 3;

To illustra
constructed with
frequency-domair
The appending 0

any Pl'Oblems as

 

F,(z) = 71—1 [f,[0]zN‘1 +f,[1]zN‘2 + +f,[N—2]z +fl[N—1] ] (5.21)
Z

For f,[0] at O, the above polynomial, whose coefficients are given by (5.17), has N-l

I‘OOIS.

Consider the simple case of a purely damped sequence given by the following

relation

M
f,[n] = Z cm(am)" (5.22)
m=1

For 0 < am< 1 (corresponding to —oo< om< 0 ), successive values of the sequence in

(5.22) are related with the following inequality

fl[n+1] <f,[n] n = 0,1, N—2 (5.23)
For an Nlh degree polynomial of the form
P(Z) = aNZN + aN_]zN’l + + alz + a0 (5.24)

where (10 < a] < a2 < < a N, the Kakeya-Enestrom theorem [34] states that all zeros must
lie within the z-plane’s unit circle. Using this theorem, a simple damped sequence given
by (5.22) is minimum phase and a cepstral reconstruction should match the original
sequence if the sampling rate is high enough to avoid any aliasing effects.

To illustrate the above ideas, a simple 32-point ,single-mode (M=1) sequence was
constructed with damping constant a1 = .9, and amplitude coefficient cl = 1.0 . The
frequency-domain response was formed by taking a 512-point FFT of the input signal.
The appending of zeros to the input sequence in forming the frequency response limits

any problems associated with aliasing. Using only the magnitude of the frequency

187

 

 

 

 

 

 

response. a minim
original sequence
reconstructed 5qu
original 312-point E
zeros lie within 1h
origin due to the f
will Show the pol:
A second 1
Figure 5.7. In this
parameters a1 = .7
=5. This combi
point transform a
original sequenCI
reconstmcted mir.
since all the zero:
inside the unit cir
For the ca:
extremely well
Unfortunately, 1h(

asum of damped

\

l
. 111 the follor
eigenvalues of th

137] A

' II calculat

 

 

response, a minimum phase reconstructed signal was formed [35]. Figure 5.5 shows the
original sequence and its cepstral (minimum phase) reconstruction. As expected, the
reconstructed sequence matches the original sequence. Taking the z-transform of the
original 32-point sequence and plotting the zeros1 in the z-plane gives the result that all
zeros lie within the unit circle (see Figure 5.6). Note that all the poles are located at the
origin due to the finite length of the signal sequence. None of the following z-plane plots
will show the pole locations.

A second example of a decaying sequence consisting of two modes is shown in
Figure 5.7. In this case a 32-point sequence was formed from one mode having sequence
parameters aI = .7 and c] = 1.0 and the other sequence having parameters a2 = .9 and c2
= .5. This combined sequence was transformed to the frequency domain using a 512-
point transform and the subsequent magnitude response was used to reconstruct the
original sequence. As Figure 5.7 clearly shows, the original sequence and the
reconstructed minimum phase sequence match nicely. This again should be expected,
since all the zeros corresponding to the z~transform of the original sequence are located
inside the unit circle as shown in Figure 5.8.

For the case of a purely damped signal the cepstral reconstruction technique works
extremely well if aliasing problems caused. by low sampling rates are avoided.
Unfortunately, the late-time model proposed consists of a sum. of damped sinusoids. For

a sum of damped sinusoids of arbitrary parameters it is difficult to put bounds on zero

' lIn the following z-plane plots, zero locations were determined by computing the
eigenvalues of the companion matrix associated with the polynomial equation [34],[36]-
[37]. All calculations were performed in MATLAB.

188

 

 

 

 

locations in the Z-‘
sampled damped S?
64-point sequence
Figure 5.9 and I
respectively. Figu
are inside the uni“
sequence. Howey
point transform)
reconstruction a a
reconstruction.

As another
adarnping param
slowly. Figure
reconstruction. 0
sequence. The 2.
indicating an orig

Both of tl‘

condition is not

srgnal. One 0

concentration of
sinusoid more e1

phase~like appro

 

 

locations in the z-plane. In fact, it is quite easy to show with a simple example that a
sampled damped sinusoid sequence may not be minimum phase. Consider a single mode,
64-point sequence with parameters given by c1 = 1.0, a1 = .95, Q, = 1.0, d), = 80.0".
Figure 5.9 and Figure 5.10 show the sequence and the zeros of the z-transform
respectively. Figure 5.10 clearly shows a zero well outside the unit circle (the other zeros
are inside the unit circle). In this case the original sequence is not a minimum phase

sequence. However, if the magnitude of the frequency response is obtained (using a 512—

 

point transform) and used to construct the minimum phase sequence using cepstral
reconstruction a nice match results between the original sequence and the minimum phase
reconstruction.

As another example, consider the same signal as in the previous example but with

 

a damping parameter of al = .99. In this example the signal sequence damps out more
slowly. Figure 5.11 shows this new sequence as well as its minimum phase
reconstruction. Once again, the reconstructed sequence is a good rendition of the original
sequence. The z-plane zero plot (Figure 5.12) shows a single zero outside the unit circle
indicating an original sequence that is not minimum phase.

Both of the previous examples show that strict observance of the minimum phase
condition is not always necessary to reconstruct a sequence that matches the original
Signal. One of the key characteristics of the minimum phase sequence is the
concentration of the signal’s energy at the beginning of the sequence. For the damped
sinusoid more energy is located early in the life of the signal, indicating a minimum

phase-like approximation. For a non-damped sinusoid (i.e. the damping parameter > 1)

 

 

asimple example \
Consider a 64-poiI
3111111qu Figu
zero locations rest:
the original seque
maximum phase).
signal sequence. a
[after obtaining th
In this case the m
sequence. A datr
correct sequence
For radar returns
tXpected.

The next c
modes. The first
150.0. The sec
The third mode
and t. = 0.0.
HOWever, the aut
a 128-point sequ

generate the mag

minimum Phase

 

 

 

 

a simple example will show that the cepstral reconstruction behavior does not match well.
Consider a 64-point sequence using only a single mode with cl = 1.0, al = 1.2, Q] = 1.0,
and d3] = 0.0. Figure 5.13 and Figure 5.14 show the original sequence and the z-transform
zero locations respectively. Figure 5.14 clearly shows the non-minimum phase nature of
the original sequence with all the zeros outside the unit circle (commonly denoted as
maximum phase). For this sequence , the concentration of energy occurs late in the
signal sequence, a feature that is clearly not minimum phase. The cepstral reconstruction
(after obtaining the magnitude response with a 1024-point FFT) is shown in Figure 5.13.
In this case the minimum phase sequence does not come close to matching the original
sequence. A damping coefficient with values greater than unity is a real problem if the
correct sequence is to be reconstructed from the magnitude of the frequency response.
For radar returns in the late-time, a damping ratio greater than unity should not be
expected.

The next example shows a. composite damped sinusoidal signal consisting of three

modes. The first mode consists of parameter values cl = 1.0, aI = .987, (2' = .1645, and

d), = 0.0. The second mode has parameters c2 = 1.0, a2 .9935, (22 = .366, and (1)2 = 0.0.
The third mode consists of a signal having parameters c3 = 1.0, a3 = .978, S23 = .4945,
and (1)3 = 0.0. The natural frequencies were obtained from actual measurements.
However, the author chose the magnitude and phase coefficients. Using the above data
a 128-point sequence was constructed. An FFT consisting of 2048 points was used to

generate the magnitude response. Figure 5.15 shows the original sequence as well as the

minimum phase reconstruction. In. this case the match between the two sequences is

 

 

 

 

 

 

 

extremely close. /
example nearl)l al
barely outside 0‘“
In the rum
ot‘non-zero phase
consider a compOS
consists of paramt
mode has parame‘
consists of a sign
[sing this data a '
has used to gener
as well as the mi
sequences is mucl
shift between the
of the zeros for It
zeros outside the
ambiguity in the
waveform was it.
target identificati

The capa

specrfrc signature

based on the un

 

extremely close. A plot of the zeros for the z—transform is shown in Figure 5.16. In this
example nearly all the zeros are inside the unit circle. However, due to a few zeros
barely outside of the unit circle, the original sequence is not minimum phase.

In the previous example, all phase terms were set to a value of zero. The effect
of non-zero phase terms can be seen by considering a similar example. Once again,
consider a composite damped sinusoidal signal consisting of three modes. The first mode
consists of parameter values cI = 1.0, al = .987, Q, = .1645, and d)l = 30.0. The second
mode has parameters c2 = 1.0, a2 = .9935, $22 = .366, and (1)2 = 45.0. The third mode
consists of a signal having parameters c3 = 1.0, a3 = .978, Q3 = .4945, and (1)3 = 90.0.
Using this data a 128-point sequence was constructed. An FFT consisting of 2048 points
was used to generate the magnitude response. Figure 5.17 shows the original sequence
as well as the minimum phase reconstruction. In this case the match between the two
sequences is much worse than in the previous example. This plot not only shows a phase
shift between the two waveforms, but also a difference in the waveform shapes. A plot
of the zeros for the z-transform is shown in Figure 5.18. In this example there are more
zeros outside the unit circle than in the previous case. This example illustrates the
ambiguity in the phase terms for cepstral reconstruction. Even though the original
waveform was not duplicated, cepstral reconstruction may still be a powerful tool for
target identification.

The capability to identify a specific target depends on the ability to recover
Specific signatures associated with that target. One technique used to identify a target is

based on the uniqueness of a target’s complex (natural) frequencies in the late-time

191

 

 

 

 

 

 

transient scattered
minimum-phase t
sequence. HOW“
match those in the
technique for targl
frequencies in a si
The extracted. nar
an E-pulse least-s
two examples can
The second case
reconstruction. T
is the 3~mode. n
Table 5.] shows t
the input signals i
closely match the
does not match II
in Table 5.1 shou
very Closely. T
technique will

I

identification alg

A late-tin

phase Cepstral tr

 

 

transient scattered signal. If a waveform is composed of a sum of damped sinusoids the
minimum-phase technique may or may not be able to reconstruct the exact input
sequence. However, if the natural frequencies contained in the reconstructed waveform
match those in the original sequence then it should be possible to use the minimum-phase
technique for target identification. A simple example will be used to show if the natural

frequencies in a signal remain unchanged in the minimum phase reconstructed waveform.

 

The extracted, natural frequencies for the previous two examples were calculated using
an E-pulse/Ieast-squares fitting algorithm (see chapter 3). Four cases from the previous
two examples can be considered. The first case is the 3-mode, zero—phase, input signal.
The second case is the 3-mode, zero-phase, composite signal obtained by cepstral
reconstruction. The third case is the 3-mode, non-zero phase, input signal. The final case

is the 3-mode, non-zero phase, composite signal obtained by cepstral reconstruction.

 

Table 5.1 shows the extracted frequencies for each case. The frequencies used to generate
the input signals is case 1 and 3 are also shown. In every case, the extracted frequencies
closely match the original values. As shown in Figure 5.17 the reconstructed waveform
does not match the original sequence for the non-zero phase case. However, the results
in Table 5.1 show that the extracted frequencies for case 4 match the original frequencies
very closely. This is a very nice result, indicating that the cepstral reconstruction
technique will probably work very well in conjunction with the E-pulse target
identification algorithm.

A late-time signal modeled using (5.3) can be reconstructed using the minimum

phase cepstral technique even though the original sequence is not minimum phase.

 

 

 

 

However. the late
minimum phase 5
should be used to
frequency domain
measurements tak

cepstral algorithrr

 

 

However, the late-time signal has an energy concentration nature similar to that of a
minimum phase signal. It also should be kept in mind that a small sampling interval
should be used to avoid aliasing. In the above examples, a small sampling interval in the
frequency domain was used which lead to very good cepstral reconstruction. However,
measurements taken at or near the Nyquist rate will not lead to good results using the

cepstral algorithms.

193

 

 

 

 

Amnli’rrrrip

Figure 5.5 Sir
ce]

Figure 5-6 Zf
pl

 

1.20

   

 

 

1.00 £9
0. 369 NM” Ori inal Se uence
80 E CCCIJOCegstral Rgconstruction
0) :
U :
3 - G)
1:. 0.60 a
O— :
E : (9
< E
0.40 5 ®
3 ®
3 ®
3 (9
0.20 3 ®
I ®©
: ®@
(9690000 0
000 Fri WW I It I I F] I r l 0‘ 00900099999
0.00 0.50 1.00 1.50

Figure 5.5

Time (nsec)

cepstral reconstruction.

z—plane

 

2.00

Single mode decay sequence representing a minimum-phase signal and its

 

Figure 5.6

 

 

 

194

 

 

Zero locations with respect to the unit circle for a single mode minimum
phase decay sequence.

 

 

 

Figure 5.7

Figllre 5.8

Amnlitrrde

Mr
ce]

Zr

 

2.00

 

 

1.50 Ea
: "Ht Original Sequence
_ CCCCOCepstral Reconstruction
Q) _
‘0 ~ (9
3 I
:1 1.00 —
Q- _
- (9
E :
<1: -
- G)
I 6)
0.50 : ®
I @®
I (9
_ ®@
_ ®
_ 00000000000
0.00 rfirrft r—FlrrrulrrrlllrlrW
0.00 0.50 1.00 1.50 2.00

Time (nsec)

Figure 5.7 Multi-mode decay sequence representing a minimum-phase signal and its
cepstral reconstruction.

 

z—plone

 

 

 

 

Figure 5.8 Zeros with respect to the unit circle for a multi-mode minimum phase

decay sequence.
195

 

Amplitude

Figure 5.9 Sir

sig

Figure 5.10 ZE
ph

 

 

1.00

 

 

 

 

 

 

 

: Original Sequence
- --- Cepstral Reconstruction
0.50 f
a) I
'O _
3 I
E 0.00 -
Q- I
E :
<1: -
—o.50 f:
-l
l
:I
—1.00 rlIllTllI'llIlllllllllllllllllllllilli‘l—l
0.00 1.00 2.00 3.00 4.00

Time (nsec)

Figure 5.9 Single mode damped sinusoid signal illustrating a non-minimum phase
signal and its cepstral reconstruction.

 

z—plone

 

 

 

Figure 5.10 Zeros with respect to the unit circle for a damped sinusoid non—minimum

phase sequence.
196

 

 

 

Amolitude

Figure 5.11 A
no

“Elite 512 Ze

SIT

1.00

0.50

rrlrrrrrrtrrl

Amplitude
.0
8

—O.50

llllllllllllllllllllllIll]

 

-~-

 

 

 

 

 

 

 

 

‘o———_—_——_—_——-—-—-————--——
4

 

 

 

Original Sequence
- — - Cepstral Reconstruction

—
_-~

 

 

 

_——
_————-——-————
—
‘-——__—_—--
_v
—

 

 

~4-

 

 

-1.00 r
0.00

IIITIIIFTITIIlllllIllllTIlll[IIIIIIITII

1.00

2.00 3.00 4.00

Time (nsec)

Figure 5.11 A second example of a single mode damped sinusoid signal illustrating a
non-minimum phase signal and its cepstral reconstruction.

z—plane

f"\
\J

 

Figure 5.12 Zeros with respect to unit circle for the second example of a damped
sinusoid non-minimum phase sequence.

197

 

 

 

 

Amplitude

Figure 5.13 Si

1'0

FlgUFe 514 Z

 

 

 

1.0E-l-005 —

 

 

 

 

 

:l
:l
I . .
:r Ongrnal Sequence
_t --- Cepstral Reconstruction
qt
_ r
5.0E+OO4 — l
_ I
~ I
_ 1 ~
‘ I It
(I) ‘ I r I
“O : I| I i
3 r r ’
+2 ‘ I t t f X /\ As __ A A
_ _ I V
O. _ I l \ ,
E _ I t \_,
_ I I
<1: _ I l
.. I r
_ l r
_ I I
_ ll
—5.0E+004- lI
‘1.OE+005 lllllllllllllIllllllllIllllll|
0.00 1.00 2 00 3.00

Time (nsec).

Figure 5.13 Single mode signal illustrating a maximum-phase signal and its cepstral
reconstruction.

 

 

° 0
0 o
0 o
0 o
0 o
C o
o o
o o
o o
o o

o

 

 

Figure 5 14 Zeros with respect to the unit circle for a maximum-phase srgnal.

198

 

Amnli'l’l Ir‘lto

Figure 5.15 C

TC

Figure 5.16

 

 

3.00 -

 

2.00 ‘ Original Sequence
_ --- Cepstral Reconstruction

1.00 -

0.00 - r

 

Amplitude

 

-1.00-

—2.00 -

 

—3.00 IIIIIIITTIIIIIllilllllllllIIIIIIIITIITII

0.00 2.00 4.00 6.00 8.00
Time (nsec)

Figure 5.15 Composite 3-mode damped sinusoid signal and its minimum-phase
reconstruction.

z—plane

 

 

 

 

Figure 5.16 Zeros with respect to the unit circle for a composite damped sinusoid

signal.
199

_____A

 

 

Figure 5.17

“We 5.18

Amnlii‘t trim

A
in

C0

3.00 —

 

 

 

 

 

 

 

' 0" IS
2'00 _ l ---- ngsrtlfal lsgg::sct:uction
_r|
_l
I
_|
I
<1) 1.00—I
“O I
3 "‘I
t -'.
EL .1
I
< 0.00"I
“l
—1.00-
—2.00 IIIITIIIIITIIIIIIIIIIIIIIIITT[IIITIITI
0.00 2.00 4.00 6.00 8.00

Time (nsec)

Figure 5.17 A second example of a 3-mode composite damped sinusoid signal and its
minimum-phase reconstruction.

 

z—plane

/
\s

Figure 5.18 Zeros with respect to the unit circle for the second example of a 3-mode
composite damped sinusoid.

200

 

 

 

 

Table 5.1 NE
alg
ze‘

 

Case Numbe

 

Original \'alt

K

 

 

Table 5.1 Natural frequencies obtained from the E-pulse/least squares extraction
algorithm for composite sinusoidal input signals have zero-phase and non-
zero phase components.

 

 

 

 

 

 

 

 

! Case Number Complex Complex Complex
Frequency #1 Frequency #2 Frequency #3
1 .9870 + .1645j .9935 + .3660j .9780 + .4945j
2 .9870 + .1645j .9935 + .3660j .9780 + .4945j
3 .9870 + .l645j .9935 + .3660j .9780 + .4940j
4 .9872 + .1660j .9921 + .3675j .9758 + .4920j
Original Values .9870 + .1645 j £235 + .3660 j .9780 + .4945 j

 

 

 

 

 

 

201

 

 

 

 

.4 Early-Tit

'JI

A simple
section. A samp

33

where n is the 5:
center at tempor
difficult to deter
addition. a deter
guarantee that t'
fanning the z-t:
However. the 2
correspond to thi
energy concentrz
to create a comp-
phase.

The disc
Iept‘esertted as a

Where

 

 

 

5.4 Early-Time Analysis

A simple model for the early-time component was introduced in the introductory
section. A sampled, early-time signal sequence for M scattering centers can be written
as M
fem] : £1 L(n'mi) (5.25)

i:

where n is the sample index and mi represents the index corresponding to ith scattering
center at temporal position Ti. Clearly, for a general pulse response it will be very
difficult to determine whether (5.25) corresponds to a minimum phase sequence. In
addition, a determination that each pulse response in (5.25) is minimum phase is no
guarantee that the composite signal is minimum phase. This can easily be seen by
forming the z-transform polynomial for each pulse response and finding the zeros.
However, the zeros of the composite z-transform polynomial will not necessarily
correspond to the zeros of the individual components. Since a minimum phase signal has
energy concentration characteristics described in the previous two sections, it is quite easy
to create a composite early-time signal having energy characteristics that are not minimum
phase.

The discussion to follow assumes that the individual pulse responses can be

represented as a discrete pulse of amplitude ai over one sample interval. This response

can be written as

(5.26)

.- (5.27)

 

 

 

 

The z-transform

which is miniml
0n the or

composite respo

which is general
the first pulse is

parameters a2 =

This polynomia
conclusion shor
match to the or

An imp

phase. COIlSIdt

and

 

 

E _____,_.. _.- '..

The z-transform of (5.26) is given as
ai
Fi(z) = — (5.28)

’"i

z
which is minimum phase.

On the other hand, if all the pulse responses are considered, the z-transform of the

composite response is

M a.
F (z) = Z —n:- (5-29)

i=1zi

 

which is generally not minimum phase. Consider a two pulse sequence (M = 2) in which
the first pulse is characterized by parameters a1 = 1.0, ml = 2 and the second pulse has

parameters a2 = 2, and m2 = 3. The z-transform of this sequence is just

m) = i, (2 + z) (5.30)
Z

This polynomial clearly has a root outside the unit circle at z = -2. At this point, no

 

conclusion should be made that a cepstral reconstruction will fail to reproduce a close
match to the original sequence; however, some caution should be considered.

An important question to ask is whether a sequence does exist that is minimum
phase. Consider the case where

(5.31)

a >a2>a >...>a

l 3

and

m1 < m2 < m3 < < mM_1 < mM (5.32)

203

 

 

 

 

 

 

 

If the conditions
that the zeros ol
sequence shown
satisfies the m
magnitude respt
of the original
matches the orig
the z-transfonn
The pre\
using a Io—poin
zeros correspon
points in each
wasefonns. OI
match between
(1024-point FF'
correct placemc

What it;

general. for a s
reconstruction i
energy concent
be expected.

S‘mPle Signals.

 

 

 

 

If the conditions in (5.31) and (5.32) are met, then the Kakeya—Enestrom theorem states
that the zeros of (5.29) must all lie within the unit circle. Consider the simple 4—point
sequence shown in Figure 5.19. As can be seen, this sequence is decaying and easily
satisfies the minimum phase requirements. Using a 128-point FFT, the spectral
magnitude response was generated and “used to form the minimum—phase reconstruction
of the original pulse sequence. As shown in Figure 5.19, the reconstructed sequence
matches the original sequence quite well. Figure 5.20 shows the zeros corresponding to
the z-transform of the original sequence; in this case all zeros lie inside the unit circle.

The previous example has all the sample pulses equally spaced. Another example
using a 16-point sequence with unequally spaced pulses is shown in Figure 5.21. The
zeros corresponding to the z-transform of this sequence are shown in Figure 5.22. The
points in each signal have been connected to visually enhance the separation of the
waveforms. Once again the input sequence in minimum phase; hence, there is a good
match between the original sequence and the cepstral reconstruction shown in Figure 5.21
(1024-point FFT used to generate the frequency response). This example illustrates the
correct placement of pulse locations for a minimum phase signal.

What happens when the sequence of pulses is not a decaying sequence? In
general, for a sequence of pulses that does not behave in a decaying manner, a cepstral
reconstruction will not yield the original sequence. However, if the original sequence has
energy concentration characteristics similar to a minimum phase signal, better results can

be expected. A few simple examples will illustrate the problems posed by some very

simple signals.

 

 

 

 

 

ngeS.
thssequence.a
expected that th
rqnoducethe or
monunucnon i
figme 524. Tl
numnnnn phas
reterse die ntptr
reconstruction c

A final
Figure 5.25. 1
aircraft \\llll tis
propagatknris r
mroneivnrg.th
low amplitude
findage. h um
good represent
reconstructed s
locations of th
example, the u
VFW disappoin

without phase

 

Figure 5.23 shows a series of five pulses embedded in a 32-point input signal. In
this sequence, a much larger pulse is located late in the input waveform. Therefore, it is
expected that the signal is non-minimum phase and cepstral reconstruction will fail to
reproduce the original sequence. This result is indeed the case, as shown by the cepstral
reconstruction in Figure 5.23 and the location of zeros outside the unit circle in
Figure 5.24. The location of the zeros indicate that this signal has characteristics of a
maximum phase signal [29]. In this case, the cepstral reconstruction algorithm will
reverse the input waveform. This simple example illustrates just how badly the cepstral
reconstruction can be for some time sequences.

A final example of a simple 32-point sequence using five pulses is shown in
Figure 5.25. This sequence is very similar to the measured early-time responses of
aircraft with two engines mounted on each wing (see Figure 5.44). If the direction of
propagation is perpendicular to the fuselage, the first two pulses represent the two engines
on one wing, the middle pulse corresponds to the return from the fuselage, and finally the
low amplitude pulses represent returns from the engines that are shadowed by the
fuselage. It would quite useful if the cepstral reconstruction would be able to provide a
good representation of the original signal. However, as Figure 5.25 shows, the
reconstructed signal does not match the original sequence. Figure 5.26 shows the zero
locations of the z-transform clearly indicating a non—minimum phase signal. In this
example, the use of cepstral reconstruction to reproduce the original waveform leads to

very disappointing results. Clearly the input signal in this example can not be reproduced

without phase information.

 

 

 

 

 

 

 

The anal
difficult to recon
should not be us
ofthe signal is c
target identificat
transient signal
beginning of ea

presence of sort

 

The analysis presented above indicates that the early-time component is very
difficult to reconstruct using only the magnitude of the spectral response. This technique
should not be used for any type of imaging algorithm where correct spatial representation
of the signal is critical. However, the use of the early-time response could be useful for
target identification. One benefit of the cepstrum technique is that the largest point of the
transient signal is usually place first (at the origin). This might be used to find the

beginning of early time which has been notoriously difficult to find, especially in the

presence of some noise.

206

 

 

 

 

 

 

Figure 5.19 f
r

Figul‘e 5.20

 

. =.-__-..___.‘_~.. _.——‘ ~.-'.-_‘-_Z...,—-‘-

5.00

 

4.00 Ea
: ***** Original Sequence
2 00000 Cepstral Reconstruction
o) 3.00 E 9
‘O _
3 —r
.t' 2
Ti 3
(E: 2.00 E t
1.00 E a
1
0.00 —
0.00 0.05 0.10 0.15 0.20

Time (nsec)

Figure 5.19 Simulated early-time and its cepstral reconstruction using a 4-point 4-pulse
minimum-phase sequence.

z—plane

0

 

Figure 5.20 Zeros with respect to unit circle for simulated early—time 4-point 4-pulse
minimum-phase sequence.

207

 

 

 

 

Figure 5.21 f
I

 

 

 

5.00

 

 

 

 

3
4.00 E
: Original Sequence
I --- Cepstral Reconstruction
<1) 3.00 5
.0 ..
3 _
7‘: I
E 2
E 2.00 E
1.00 E
0.00 A Tlllllll IIIIIIrl‘hrr‘I’ITl‘I’I‘IIIIIIT‘I‘VII’III
0.00 0.2'0 0.40 0.60 0.80

Time (nsec)

Figure 5.21 Simulated early—time and its cepstrum reconstruction using a 16-point 4-
pulse minimum-phase sequence.

z—plane

 

 

 

Figure 5 22 Zeros with respect to unit circle for simulated early-time 16-point 4-pulse

minimum-phase sequence.
208

_____A

Figure 5.23 f
l

FlgUre 5.24

 

 

10.00

 

 

 

 

 

 

 

E;
_r
-I
:l
8.00 :i
:I
~' Original Sequence
:: --- Cepstral Reconstruction
:l
-I
(I) 6.00 jr
‘0 _I
3 -I
it: 2*
Q. I:
:I
<Et 400 ii I
'_‘ I :1
I: ‘ I"
a: ,.
: I I“ i“
2.00 ': I I ‘I l I
_ I H I l I l
- I II I i' I
: I II I‘ I ‘i I
I
3i I‘I I i lI i
.. I \I l g ‘1
' AL \I I J l . A 4- J
0.00 fifijTfiTfirIfirlfifITIY—rrr1—fil T1 l’l‘]
0.00 0.50 1.00 1.50

Time (nsec)

Figure 5.23 Simulated early-time and its cepstral reconstruction using a 32-point 5-
pulse non minimum-phase sequence.

z—plane

3
O 0
F
\X
C

 

 

Figure 5.24 Zeros with respect to unit circle for simulated early-time 32-point 5—pulse
non minimum-phase sequence.

209

—;

I Figure 5.25

Figure 5.26

E;—

 

12.00

 

 

 

 

 

 

“I
“I
-I
-t
_|
__l
r
8.00 —I
—r . .
OrrgInal Sequence
(D ‘j --- Cepstral Reconstruction
g “I
.4: :l
E. _.
I
E ‘I
‘l
< —l
4.00 -'| .
-I I H‘
_ I
-l ,l .r
| ,I t I
' I I‘ ' ‘
_ I I I I I
. . r '.
I I
I A ..
0.00 ’TlF’FRWflfY—Jfil—‘a‘Y—fi—II—Flfifill‘rlllmrlll
00 V 0.50 1.00 1.50 2.00

Time (nsec)

Figure 5.25 Simulated early-time and its cepstrum reconstruction using a second 32-
point 5-pulse non minimum-phase sequence.

z—plane

 

 

 

 

Figure 5.26 Zeros with respect to unit circle for simulated early—time second 32-point
5-pulse non minimum-phase sequence.

210

—;

 

 

very good reali

reconstruction

component du

apoor recons
begging to be
composite ma
More importa

phase reconstn
phase signal l‘.
early-time con
component is I
Therefore, a n
the original si
not necessarilj
time compon
reconstructed
Signal.

To se

 

 

5.5 Separation of Early and Late Time

The previous sections discussed some of the characteristics of early-time and late-
time cepstral reconstruction if only the magnitude of the frequency response is present.
As demonstrated, the minimum phase reconstruction obtained from IF 1(a)) | yields a
very good realization of the late-time component. On the other hand, the minimum phase
reconstruction obtained from lFe(w) | generally yields poor results for the early-time
component due to the extremely non-minimum phase nature of the signal. In both cases
a poor reconstruction will occur if aliasing is not taken into account. One question
begging to be answered is the following: Will a cepstral reconstruction using the
composite magnitude spectrum |F(w) | yield a good realization of the actual signal?
More importantly, will the early and late-time components separate using the minimum
phase reconstruction? One way to look at this problem is to remember that a minimum
phase signal has its energy concentrated earlier in the time sequence. Normally, the
early—time component contains a higher concentration of energy whereas the late-time
component is a decaying signal whose energy is concentrated later in the time sequence.
Therefore, a minimum phase reconstruction. should yield a fairly good representation of
the original signal. On the other hand, reconstruction of the early-time component will
not necessarily be correct. A possible rule of thumb to use is that the reconstructed early-
time component will occur before the late-time component. Furthermore, the
reconstructed late~time component should be a fair representation of the actual late—time
signal.

To separate the early and late-time components from the composite spectral

211

 

 

 

 

 

 

magnitude I F

representation
time componen
spectral respon
late-time freq
reconstruction
and late-time
spectrum -- i
components.
spectral energ
occurs even w
The following
using the comp
frequency can
simple, multipl
was a simple ‘
Two different
fuselage (but ]
45 degrees w
measurement
Figure

aircraft for an

magnitude |F(o>) l, two options are available. First, simply obtain the time—domain
representation using a minimum phase reconstruction and then separate the early and late-
time components with an appropriate window. A second option is to filter the composite
spectral response IF( (0) [ into a separate early-time frequency component |Fe( 0)) | and
late-time frequency component |F [(00) l. Once this is done, a minimum phase
reconstruction can be applied to each spectrum to yield an approximate fit to the early
and late-time components. This technique will require some a-priori knowledge of the
spectrum -- i.e., which part of the spectrum contributes to the early and late-time
components. One of the real problems occurs when there is a great deal of overlap in the
spectral energy for the early and late-time components. Of course, this problem also
occurs even when using both magnitude and phase to reconstruct the transient response.
The following analysis will present several examples showing cepstral reconstruction
using the composite spectrum and then filtering the spectrum into early—time and late-time
frequency components. The spectrum for the first two examples was created from a
simple, multiple thin—wire scattering routine. For these two examples, the scattering target
was a simple thin-wire aircraft made up of fuselage, wings, and tail (see Figure 5.27).
Two different polarizations were used: in one case the E-field was perpendicular to the
fuselage (but parallel to the tail and wings), in the second case the E-field was oriented
45 degrees with respect to the fuselage. The third example consists of a scattering
measurement from a highly conducting scaled B—58 aircraft model.

Figure 5.29 shows the spectral magnitude of the scattered field from a thin-wire

aircraft for an electric field polarized normal to the fuselage but parallel to the wings and.

 

 

 

 

 

 

parameters fc =

domain repres
phase recons
good match be
A low-
capture late-ti
the filtered out
output of the
wings). Figu
both the magni
be seen, there
phase reconstr
Arr atte
frequency spec
spectral data a:
the eliminatio
eliminated too
representation

made between

and the magni

 

 

tail. The figure contains both the raw spectral response and a spectrum formed by
windowing the raw data with a Gaussian modulated cosine (GMC) window having
parameters fC = 0 GHz and, T = .06 nsec (see Appendix A). Figure 5 .30 shows the time-
domain representation of this windowed spectrum computed using both the magnitude-
phase reconstruction and the magnitude only reconstruction. In this example there is a
good match between the late-time components using the minimum phase reconstruction.

A low-frequency filter was applied to the raw spectral data in an attempt to
capture late-time frequency information. Figure 5.31 shows the raw frequency data and.
the filtered output. The filter consists of a GMC window (fc = 0 GHz, T = .3 nsec). The
output of the filter consists of one large late-time resonant mode (from the aircraft’s
wings ). Figure 5.32 shows the time-domain representation of the filter’s response using
both the magnitude—phase reconstruction and the magnitude only reconstruction. As can
be seen, there is a good match between the late-time components using the minimum
phase reconstruction.

An attempt was made to filter out the early-time component from the composite
frequency spectrum. Using a 1/8 cosine taper window function (see Appendix A) the raw
spectral data and filter output are shown in Figure 5.33. The resulting filter output shows
the elimination of the low frequency resonant made. However, this filter may have
eliminated too much frequency data from the high end of the spectrum. The time~domain
representation of the filter output is shown in Figure 5.34. Once again a comparison is
made between the time domain obtained using both magnitude and phase reconstruction

and the magnitude only reconstruction. The match between the two representations is not

 

 

 

 

 

    

reconstruction
early—time si
time compone
well with the
presence of m

Figure
time-domain t
window with t'.
between the rr
that some ear
selection.

The ear
are shown in i
uses a 1/8 co

cepstral recon

 

 

 

perfect, but the main peaks in each case line up. Some improvement might be expected
depending on the filter chosen, but in most cases a match is quite unlikely.

The computation of the scattering from the simple wire frame aircraft was also
repeated for an electric field polarized 45 degrees with respect to the aircraft’s fuselage.
In this case, a new resonant mode due to the aircraft’s fuselage appears in the frequency

spectrum. Figure 5.35 shows the raw data frequency spectrum and the windowed

 

spectrum computed using a GMC window (fc = 0 GHz, T = .06 nsec). The time-domain
reconstruction for the windowed spectrum is shown in Figure 5.36. As can be seen more
early-time signal exists for this polarization. For the cepstral reconstruction, the early-
time component as a whole is in the proper position but the components do not match
well with the magnitude-phase reconstruction. In addition, the late time is affected by the
presence of more early-time signal.

Figure 5.37 and Figure 5.38 shows the late-time frequency filter output and the

 

time-domain transform outputs respectively. The low frequency filter used a GMC
window with the same parameters as in the previous case. There is quite good agreement
between the magnitude-phase reconstruction and the cepstral reconstruction. Also note
that some early-time components exist in the time-domain signal due to the filter
selection.

The early-time frequency filter response and time-domain transform representations
are shown in Figure 5.39 and Figure 5.40 respectively. The early—time frequency filter
uses a V8 cosine taper window exactly like the previous example. Once again the

cepstral reconstruction does not match the original sequence obtained from both the

 

 

 

magnitude and
all the late-tim
The fin
B-58 aircraft
side of the air
following figu
presents some
data down to
window to the
should be take
use of the log
the values to
Figure 5.42 8
frequency resp
again a good I
Figure
reconstruction
T = .4 nsec).
magnitude-13h;
filter results.
a 1/8 cosine t:

Surprisingly g

 

 

 

magnitude and phase frequency components. Also the filter did not adequately remove
all the late-time components.

The final example is a measured frequency response for scattering from a. scaled
B-58 aircraft model. For this particular case the incident electric field is incident on the
side of the aircrafi and polarized 45 degrees with respect to the roll-axis (see insets in
following figures). The raw frequency data was measured from .5 to 5.5 GHz. This
presents some problem for the cepstral reconstruction routines which require frequency
data down to 0 GHz. A usable spectrum was obtained by applying a 1/8 cosine taper
window to the raw frequency spectrum and zero filling the low frequency. Although, care
should be taken in zero filling frequency data for cepstral reconstruction ( remember the
use of the logarithm function ) the cepstral routines do check for this condition and set
the values to a small non-zero value. Figure 5.41 shows the frequency data and
Figure 5.42 shows the time-domain representation using both the magnitude—phase
frequency response and the magnitude only response. The plots in Figure 5 .42 show once
again a good comparison for the late-time periods and a fairly good early—time match.

Figure 5.43 and Figure 5.44 show the low~pass filter and time-domain
reconstruction respectively. The low-pass filter used was a GMC window (fC = 0 GHz,
T = .4 nsec). The time-domain cepstral, reconstruction shows a good match with the
magnitude~phase reconstruction. Finally, Figure 5.45 and Figure 5.46 show the early-time
filter results. Figure 5.45 shows the filter output which was obtain by windowing with
a 1/8 cosine taper window. The minimum phase reconstruction shown in Figure 5.46 is

surprisingly good. Although this does not occur in general, under certain conditions the

215

 

 

 

 

early-time sea
In each
representation
time compone
non-minimum
quite crude bu
using only th
difficult to we

of constructin

 

 

 

early-time scattered response is nearly minimum phase.

In each example, the late-time component using cepstral reconstmction is a good
representation of the actual signal using the composite frequency response. The early-
time component does not keep its temporal order under a cepstral transform due to the
non-minimum phase character of the early—time signal. The filters presented above are
quite crude but they do show that a late-time component can be filtered and reconstructed
using only the magnitude of the spectrum. The early-time component is much more
difficult to work with -- both due to the non-minimum phase character and the difficulty

of constructing a good early-time frequency filter.

216

 

 

 

 

 

0.90

 

Figure 5.27

 

 

 

 

 

 

1 .875

0.906 3.203
0.984

 

 

 

 

wire radius = .018
All dimensions in inches

 

Figure 5.27 Thin wire aircraft model.

217

 

 

Figure 5.28

 

 

 

 

 

 

 

 

Figure 5.28 Planar view of B-58 scaled aircraft.

218

 

Figure 5.29

___________ _________

6 5

O. O. . .

O O O O
®UDfiC®O§ ®>5

_
4
O
O_®m¢

 

 

 

 

 

 

 

 

 

 

 

0.06 -;
0.05 -: Row Spectrum
5 ---- Windowed Spectrum
g i
:5 0.04 E _.
,4: — E
C I
goose r l
g 3 :i
+’ :
g 0.02 : ,‘l
(D : 1‘
n: — I ‘
: i \
.a \ r l
0.01 E \J' .
: I"
2 \
" \
I \V,\\/,\\ ,_
ODO- llllllllIllllllllllll/lllll‘i-l-l—l—ll—l—lll|lllllIllll
0.00 5.60 10.'oo 15.00 20.00 25.00
Freq (GHZ)

Figure 5.29 Frequency response magnitude for scattering from a simple wire aircraft.
E-field polarization perpendicular to fuselage.

219

 

l.5E+0(

1.0E+Ol

5.0E+O

0.0E+O

-5.0E+C

Relative Amplitude

-l.OE+(

~1.5E+(

Figure 5.30

 

 

———-——-—._- .-a_._.'.-1..‘;.(_

1.5E+008

l —- Mog 8c Phase Recon.

LOE+008
‘ R —--- Mog Recon.

 

5.0E+OO7

 

 

0.0E+OOO

 

 

 

—5.0E+OO7

 

Relative Amplitude

 

 

—1.0E+008

 

lllJLlllllllllllllllllllllllllllllllLllillllllllllllllllllll
\
/

 

llllllllllllllllllIllllllI

-l.5E+OO(8).O(r)rrlrlrrzidéllllIla-idiom: 6.00 8.00 1000
Time (nsec)

Figure 5 30 Transient scattered field for wire-frame aircraft using magnitude/phase
transform and magnitude only transform.

220

 

j-——_._—____—_ ______ _ ......... _ ......... _ .................. o
6 5 4 3 2 1| 00 M“
O. O. 0. O. O 0. 0 5
O O O O O O O e
®UDULC®O§ ®>.LO_®K We

H

 

 

 

 

 

 

 

 

 

 

 

0.05 :
0-05 E Row Spectrum
3 ---- Windowed Spectrum
a 2
3 0.04 E _.
.t’ — E
r: E L
U) : l
g 0 03 E c |
<1) 3 i
> : H
Li: 3 l'
2 0.02 -_ ii
(D : ll
0.”. : i'
_ ,I
I r'
0.01 E ,‘l
: l
: \
_ \
OoOO llllllrllIllllllllllllllllllllllllllIIIIIIIIIIII
0.00 5.60 to.'oo 15.00 20.00 25.00
Freq (GHZ)

Figure 5.31 Late-time frequency filter response for scattering from a wire frame
aircraft. E-field polarization perpendicular to fuselage.

221

 

30E+Oi

20E+Oi

LOE+OI

00E+0

 

-LOE+C

Reijve Amnphtude

-20E+C

~30E+C

~4.0E+t

Figure 5.32

 

 

 

3.0E+OO7

2.0E+007
—-— Mog & Phase Recon.

‘ ---- Mog Recon.

1.0E+OO7

___———
.’

0.0E+000

 

 

—1.0E+OO7

 

~——__

 

C

_
-_—
——_————_
—

—2.0E+OO7

iflflofive Anufldude

 

 

—-I-—
—-——
C

-3.0E+OO7

 

llllllllllllllllllllllLlllllllllllllllllllllllllllllLllllllllIllllllll

 

ll|llllll Ill llllTll]

‘4'OEJ’OOSDO"'"3:65””'4.'do”””é.'do 8.00 10. 0 12.00
Tune (nsec)

Figure 5.32 Transient scattered field for late-time response for wire-frame aircraft.

222

 

0.00

_________.

3

0
0.01 -
Figure 5.33

O m

.0

 

 

 

0.06

 

 

 

 

 

 

 

    

 

 

0.05 “I Row Spectrum
5 ---- Windowed Spectrum
g E
3 0.04 E _.
7': '_' E
C ..
o. _:_ L I
20 0 03 —_‘_ r —'
<1) 3
> :
'43" :
2 0.02 :
<1) :
01 :
5 ll“
0.01 -3 ,' x
Z I l
_ I r
I I I
Z A /’/ /‘\\’~
l1/ITTTIII Illlrrrll llllllIlllllllll‘l‘lllllllllli—“I
0000.00 5.60 10.00 15.00 20.00 25.00

Freq (GHZ)

Figure 5.33 Early-time frequency filter response for scattering from a wire frame
aircraft. E-field polarization perpendicular to fuselage.

223

 

 

 

 

LOE+0(

50E+0'

00E+0

-50E+(

Hekjhve Amnphtude

—tDE+i

~LSE+

Figure 5.34

 

 

 

O
m
+
O
O
00
l

 

 

 

 

 

 

 

 

 

 

5.0r-:+007{
(D E
'0 _
3 I
_t_’ 0.0E+OOO _
Q— :
E :
<3 :
Q) I
> —5.or-:+007:
J3 _
2 2
CD I
35 :
409008: Mag & Phase Recon.
2 ———- Mag Recon.
3
—1.5E+008— llilllllllllllllllllllllll|llll|llll|
0.50' ' ' ' ' 'ii66” 1.50 2.00 2.50 3.00

Thne (nsec)

Figure 5.34 Transient scattered field using early-time frequency filtered response for

a wire-frame aircraft.

224

 

nu.
®UJECQOE 02040.0

0.01 ‘

mm

0.00

Figure 5.35

 

0.04

0.03

 

Relative Magnitude
8
l\)

 

 

.0
o

 

 

 

 

 

Raw Spectrum
---- Windowed Spectrum

 

 

 

 

IiIIIIIIIIIIIIIIIIIIIIIIIIIIIlIIIIIIIIII

 

 

I u .
ii \/\\
\ ’\ "I\—\
III llllllll\l \IIi/IIII/IIIIIIITfi—I—lllIIIIIII]
O'000 06m” 5.60 10.00 15.00 20.0 25.00
Freq (GHZ)

Figure 5.35 Frequency

response magnitude for scattering from a wire frame aircraft.

E-field polarization 45 degrees with respect to fuselage.

225

 

 

LOE+Ol

50E+0

00E+C

Hekjhve Amnphtude

—50E+l

~10E+

Figure 5.36
;

 

__._¥.._ M_._.._. ._ _ _..... ... . '34,-

1.0E+008 —

—— Mag & Phase Recon.
-—-- Mag Recon.

5.0E+OO7

 

 

0.0E+OOO

Remdwe Anufldude

—5.0E+OO7

 

 

IIIIIIIIIliILIIrIIIJIImIIILJJIJJIIIIIII

 

lllllllll'lllllllll

— I I I I I I I I I I
1.0E+00E3).OO 5.60 10.00 15.00
Tune (H860)

 

Figure 5 36 Transient scattered field from wire frame aircraft using magnitude/phase
transform and magnitude only transform.

226

 

0.04 f

0.03 '

2
flu.
0

003.:[002 ®>E.O.®

 

nu.
0
Wm

0.00

Figure 5.37

 

 

 

0.04

 

 

 

 

 

 

 

 

 

 

 

 

 

: Raw Spectrum
2 - - - - Windowed Spectrum
0.031
(D _
U _
3 _.
i"- I
r: _
8‘ :
2002-:
(D 3k
> _I
'5 I
g _
(D I
050.01: I
l
:l
... ll
_ II H
_. I
.I ‘4‘
000 .1 V\TIII llllll|l||llllllll||lllllllll|IIllIlllll
' 0.06"" 5.60 10.00 15.00 20.00 25.00

Freq (GHz)

ency filter response for scatterlng from a W1re frame

F‘ . L t -t' e fre u
Igure 5 37 a e 1m q degrees with respect to fuselage.

aircraft. E-field polarization 45

227

 

LSE+0

LOE+C

50E+C

00E+C

—5.0E+I

~10E+

Heijve Amhphtude

~15E+

~20E+

Figure 5.38

 

 

 

 

_A______.h._’ ~ -__.. ._ _.___ _.

1.5E+OO7

1.0E+007
—— Mag 85 Phase Recon.

- - - - Mag Recon.

5.0E+006

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

I

‘-

0.0E+OOO

 
 
  

 

 

 

 

—5.0E+006

 

—1.0E+007

 

Relative AmplItude

-‘| .5E+007

IIIIIIIIlILLIIIIIIlIIIIIIILIlIIIIII

 

llllllllllllllllllllli]

_2’OE+OO6.00W"'5i60"””1'0.b'0”"15.00 20.00 25.00
Time (nsec)

 

Figure 533 Transient scattered field for late-time response from wire frame aircraft.

228

 

0.04 ‘

0.03 '

2
O

flu.
®UD£CDOE 02010.0

0.
0

K

0.00

Figure 5.39

 

 

 

0.04

 

Raw Spectrum
---- Windowed Spectrum

0.03

 

 

 

Relative Magnitude
3
NJ

 

 

 

 

 

 

III Lll III III II III III III III III III III III II

 

 

 

l I
0.01 U :1 n
H II /
"I \
ii n ; /\\ "\nx ,-
II\\ ‘V’i’, \Jl I l J \\
I i1 "‘\__ I
0900 IJrFIIIIIIIlllllllllllIlllllllllllllllllllllllllll
0.00 5.00 10.00 15.00 20.00 25.00

Freq (GHZ)

lter response for scattering from a wire frame

. _ ' 0 fi
Figure 5'39 Early time frequen y degrees with respect to fuselage.

aircraft. E—field polarization 45

229

 

1.0E+C

5.0E+(

0.0E+(

HethIve Amplitude

-5.0E+

-i .OE+

Figure 5.40

 

 

1.0E+008 -

 

 

 

 

 

 

 

 

 

 

 

 

 

5.0E+007 I:
Q) .—
U I
3 _
t’ ..
B. :
E _
CD
> 1
43 _
2 ‘i
(D I
I —5.0£+007—
I —— Mag & Phase Recon.
: ---- Mag Recon.
_1°OE+008—jllllllrl|IIIIIIITIIIIIIITIIllllllllllI—l
0.00 0.50 1.00 1.50 2.00

Time (nsec)

Figure 5.40 Transient scattered field using filtered early-time frequency response from
wire frame aircraft.

230

 

0.02

0.02

flu.
0

®UDfiC®O§ @3590

l
flu.
0

m

0.00

Figure 5.41

 

 

 

 

 

0.02

 

 

0.02

 

 

 

 

.0
o

 

Relative Magnitude
O
9.
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

 

 

I I I I I I I I I I I I I I I I I I I I I I I I I I r I I I I I I I I I I I I
0'000.00 2.60 4.00 6.00 8.00
Freq (GHZ)

Figure 5 4] Measured frequency response magnitude for scattering from B-58 aircraft

model.

231

 

5.0E+'

3.0E+'

l.OE+'

-l .OE+

RelotIve Amplitude

~3.0E+

~5.0E+

Figure 5.42

 

 

 

5.0E+OO7

 

 

 

 

 

 

 

 

3 ———- Mag & Phase Recon.
3°OE+OO7i ---- Mag Recon.
0) E
'O s
3 :
+_: 1.0E+OO7:
0- :
< 2
3 —1.0E+OO7-:
I; _
2 :
(D I
35 :
—3.0E+OO7—:
—5.0E+OO7‘ llilllllllllllllllllilil
0.00‘ ' ' ' ' ' 5.00 10.00 1500

Time (nsec)

Figure 5.42 Transient scattered field. from B-58 aircraft.

232

 

0.02

0.02

1

O

0.
@UDfiCGOE @2090

AI.

0.
0

K

0.00

Figure 5.43

 

 

0.02

 

 

 

 

 

 

 

 

 

 

: Raw Spectrum
I ---- Windowed Spectrum
0.02 ':
(D _
"Q ..
3 —
if I
C - I
a“ : I
2 0.01 - i:
: 'I
0) : ll:
.2 _ ..
+2 _ I |
O _ I l
6 _ i '4
D: 0.01 -: lI L[II
_. l l
_ 1 \II
: 'i i
_ \
I K
0.00 IIIIIIIII -I.I‘T“i~l—l—lll[lTlllllll[lIllll[IT—1
0.00 2.60 4.00 6.00 8.00

Freq (GHZ)

Figure 5.43 Late-time filtered frequency response for scattering from B-58 aircraft

model.

233

4.0E+'

2.0E+

p
O
m
+

-2.0E+

Relative Amphtude

~4.0E-i

~6.0E-I

Figllre 5,44

 

 

 

 

 

 

 

4.0E+OO7 :
: i
_ lI Mag & Phase Recon.
2'OE+OO7 : i i: ----- Mag Recon.

_ II

c» 3

U _

3 :

E 0.0E+OOO -_- :5 s

Q— :

E :

‘< :

0)) —2.0I—:+007-§

+3 _

2 I

(D Z

I :

—4.0£+007—:
—6.0E+OO7 IIIIIIIII[IIIIIIIIIIIIIIIIITIIITIIIIlllflIIllllm
0.00 5.00 10.00 15.00 20.00 25.00

Tune (nsec)

Figure 5.44 Transient scattered field for late-time response from B-58 aircraft.

234

0.02

0.02

0.
0

®UJfiC©O§ ®>.LO.®

nu.
0
K

0.00

Figure 5.45

 

 

0.02

 

 

 

 

 

 

 

 

: Raw Spectrum
: ---- Windowed Spectrum
0.02 :
(D _
P :
5:5 - e _,
C I ,,
on _
g 0.01 —_- k ,
CD : , //
> — \,
'43 I I,
O _ I I
00:) : ' \I‘I‘
" l
OOI : i ll"
.. 1 I
: Iclia r’; Illa
: l\ \i/‘4 ‘\
.. /I \\
_ I‘ \A
0.00 III‘IJIIIII WIIIIITIIIIIIITIIIIIIIIIIIIrq
0.00 2.66 4.00 6.00 8.00
Freq (GHZ)

Figure 5.45 Early—time frequency filter response for scattering from B-58 aircraft.

235

 

3.0E+

2.0E+

l.0E+

5:)
O
FFT
+

-l .OE+

~2.0E+

HeiotIve AmplItude

~4.0E+

“am 5.46

 

 

 

$5.1..- . -

3.0E+007

2.0E+OO7

1.0E+007

0.0E+OOO

LLIIllllllIIJIIIJJIIJIIIILIIIILLLI

 

L

 

—-l .OE+OO7

 

 

~2.0E+OO7

RelatIve AmplItude

 

-— Mag & Phase Recon.

—3.0E+007 -—-- Mag Recon.

IIIIILIIIlIIIIIIIIIILIJILIIIIILLIIIJ

 

 

~4.0E+OO7 IflTITIIIIIITIFIIWITII—TITII—FITII—FIT—I Frl1IIIIrIITI|TTIIITII—I_]

0.00 2.00 4.00. 6.00 8.00 10.00 12.00
TIme (nsec)

 

Figure 5.46 Transient scattered field using early—time filtered frequency response for
scattering from B-58 aircraft.

236

 

 

56 ConcI
Some
cepstral reCOr
1. Mi

includ

recon:
respor
3. E:
filtere
4. Tl
obtaiI

specu

Since
frequency s1
detection alg
should not b
possibly be
components.

features dire

 

 

5.6 Conclusions

Some important results have been demonstrated for radar target responses and

cepstral reconstruction:
1. Minimum phase conditions 2% be guaranteed for any scattered radar signal
including the late-time component.
2. Although the minimum phase condition is rarely met exactly, cepstral
reconstruction can be used to obtain the late-time component of the target
response.
3. Early and late time can be separated using the composite spectrum and the
filtered components.
4. The early-time signal obtained by cepstral reconstruction will not match that

obtained from using both the magnitude and phase components of the frequency

 

spectrum

 

Since the late-time component can be obtained from only the magnitude of the
frequency spectrum , this method should work in conjunction with E-pulse target
detection algorithms. Early-time reconstruction performs poorly and the cepstral method
should not be used in conjunction with any imaging techniques. Two future issues could
possibly be worked on : First, better filtering algorithms to separate late and early-time
components. Second, some consideration should be given to the analysis of the late-time

features directly in the cepstral domain.

237

 

 

Applicat

6.1 Intro
ln at
specific targc
honian anal
unknoxrn tar
mnedonunn
addfimntm
Inahods of
obtained. A
orconurflled
toanunknOI
the time-don
has bth a
\I'avefonns.
Chen
E‘Pulse disc
Whhh,Ivher
the annihilat

the convolut

 

Chapter 6

Application of Cepstral Analysis to Radar Target Discrimination
Using E-Pulse Cancellation
6.1 Introduction

In a typical radar target discrimination problem, an attempt is made to identify a
Specific target from a set of target features or signatures. These features can be obtained
from an analysis of the measured time-domain response. If the spectral waveform of the
unknown target is measured, an inverse Fourier transform can be applied to obtain the
time-domain representation. However, if only the modulus of the spectrum is available,
a different approach must be used to reconstruct the time-domain waveform. Using the
methods of cepstral analysis, an approximation to the time-domain waveform can be
obtained. A set of target discriminants (signatures) obtained fi‘om scattering calculations
or controlled measurements must be obtained prior to applying the discrimination scheme
to an unknown target. For this study, a set of discriminant features were obtained from

the time~domain waveform using the Fourier transformed spectrum. A target data base

 

was built and later applied to the cepstrally reconstructed unknown radar-target
waveforms.

Chen, et a], [20]~[21], [38] have developed a target identification method using the
E-pulse discrimination scheme. In this method an E-pulse wavefonn is constructed
which, when convolved with the late-time pulse response of a matched target, results in
the annihilation of the natural resonant modes excited by radar illumination. However,

the convolution of the same E-pulse with a target having a different resonant structure

238

_‘—

 

 

does not resr
can be cons
response.
automated ta
on a Visual ii
target discrii
An u
to determine
domain repn
conjunction
information
be used to g
ideas of cep
there have bi
rtESponse fro
C6PSii'al thee
in this cliapi
A rct
PFObably not
However, a 1
detection Sci

 

 

 

does not result in a null convolution response. Hence, a data base of E-pulse waveforms
can be constructed which, when convolved with a matched target, results in the null
response. In order to effectively use the phenomena of E-pulse cancellation, an
automated target discrimination scheme has been devised [9] which does not need to rely
on a Visual inspection of the convolved responses. The use of the ideas in the automated
target discrimination scheme will be discussed and used in this chapter.

An unknown target waveform can be interrogated by E-pulse library waveforms
to determine if any of the target files match the unknown target. In this scheme a time-
domain representation waveform is used. A spectral response of the unknown target in
conjunction with an inverse Fourier transform yields the transient response. If only
information on the modulus of the spectral response is known, a different method must

be used to get the time-domain waveform. The approach taken in this study is to use the

 

ideas of cepstral analysis to reconstruct a transient response approximation. Although
there have been many applications of cepstral analysis, the reconstruction of the late-time
response from short pulse radars has not been investigated. A useful investigation into
cepstral theory can be found in Openheim and Schaefer [29]. Much of the investigation
in this chapter is based upon the work done by these authors.

A reconstructed waveform formed from a cepstral reconstruction algorithm will
probably not match a waveform generated using both magnitude and phase information.
However, a perfect match between the two waveforms may not be necessary for the target
detection scheme to work. If the target’s natural resonant modes exist within the late-time

Portion of the minimum-phase reconstructed signal, then the E-pulse discrimination

 

 

 

 

 

scheme shou
late-time res
many cases.
the use of t
addition. the
€V€Il though
in mind that
for cepstral
llO\\'€\'€r. sii
reconstructei

Thei
0i E-pulse a
target discrii
presented sli.
The perforn

del‘il’ed scatt

6'2 Then

The s

can be Char;

can be mode

 

fiv . ‘_ __. , - . - y. - - ‘. “
n A -—. —....-._... _ . . ...F .. . . -
. I
I

scheme should be able to identify the target. Chapter 5 has shown that the reconstructed
late-time response, using cepstral analysis, approximates the true late—time response in
many cases. Since the natural resonance modes are derived from this late-time reSponse,
the use of target discrimination using cepstral reconstruction should be effective. In
addition, the reconstructed natural frequencies often match the true natural fi‘equencies
even though the reconstructed waveform does not exactly match the original signal. Keep
in mind that the scattered retum does not pass the minimum-phase condition necessary
for cepstral reconstruction. This even applies to the late-time return. Practically,
however, since the late-time return is decaying, it is minimum-phase like and the
reconstructed signal works quite well.

The work which follows will be divided into several sections. First, a short review

of E-pulse and cepstral analysis theory will be presented. Second, a description of a

 

target discrimination algorithm will be discussed. Finally, numerical examples will be
presented showing the performance of cepstral analysis in regards to target discrimination.
The performance of the discrimination scheme is evaluated using both numerically

derived scattering data and experimentally measured results from practical target models.

6.2 Theoretical Background
The scattered transient return due to a short-pulse signal incident on a radar target
can be characterized by an early-time and a late-time response. The late-time response

can be modeled as a sum of N damped sinusoids in the following manner

N
f(t) = Zane°~’cos(wnr+¢n), r>:r, (6.1)
n=1

240

 

 

 

where s" =
represent the
beginning of

AnE

 

fttl. results i

Since the not
be measured
The conditic

domain. i.e.

ii‘here Ets)

frequencies

numbers sn 2.
C(i) ill a St
d€tcnnining
known. an I
genetic algo
frequencies,
E‘llulse date

Epulse in

 

 

 

 

where s" = on + j (unis the nth aspect independent natural resonant frequency, an and d)"
represent the aspect dependent magnitude and phase, respectively, and T] represents the
beginning of the late-time period.

An E-pulse e(t) is a real waveform of finite extent TE, that, upon convolution with
f(t), results in a null late-time response

C(t) = e(t)*f(t) s 0, t>TL=TI+TE (6.2)

Since the natural resonant frequencies are aspect independent, the target response f(t) can
be measured from any aspect angle and the null late-time convolution should be obtained.
The condition in (6.2) implies a particular set of zeros for the E-pulse in the frequency

domain, i.e.

E(S) = $60)} = 0 for s = Sn and s = s; (6.3)
n = 1,2,...N

 

where E(s) is the Laplace transform of e(t), and sn represent all possible natural-mode
frequencies of the particular target excited by an incident wavefomi. If the complex
numbers 3,, are known for a particular target, the E-pulse can be constructed by expanding
e(t) in a set of basis functions (usually rectangular pulses) by applying (6.2) and
determining the amplitudes of the basis functions. If the resonant frequencies are not
known, an E-pulse extraction technique based on the method of least squares [39] or
genetic algorithm [40] can be used to find the basis function amplitudes and the resonant
frequencies. An E-pulse can be constructed for each anticipated target and added to an
E-pulse data base. Next, the response of an unknown target u(t) is convolved with each

E—pulse in the data base. If the zeros of a particular E-pulse match the resonant

241

 

 

 

 

frequemies 1
indicates a in
the convoli'e

In 0“
target reSPOn
to the time (
cepstral reCC
to the trans
condition. a
discrete Spe

cepstrum is

Using the v

the minimu

The evalua'
must be su

If tl

poles and 2

All the po

 

 

 

frequencies found in the late time of the unknown target, a null convolution response
indicates a target match. If E(s) does not contain zeros at all the natural frequencies, then
the convolved waveform will be non-zero, indicating a poor match.

In order to achieve target discrimination using the E-pulse method the unknown
target response u(t) must be measured. A measured spectral response can be transformed
to the time domain using a Fourier transform. For a magnitude-only spectral response
cepstral reconstruction (see Chapter 5, section 2) can be used to form an approximation
to the transient scattered signal. Certain restrictions, known as the minimum-phase
condition, are required for an exact replication of the true time-domain signal. For a
discrete spectrum having N = 2m points defined by X [k] = lX [k] | ej ”3““, the real
cepstrum is given by

c [n] = FFT“( log [X[k] l) (6.4)

Using the window function given by

2, 1 s n < N/2
w[n] = 1, n = 0,N/2 (6-5)
0, N/2 < n sN—l

the minimum phase sequence may be computed as

X[n] z FFT—1(eFFT(W[nIc[n]) ) (6.6)

The evaluation of (6.4) and (6.6) can be done very quickly using the FFT. However, one
must be sure to avoid complications caused by aliasing.

If the original causal stable sequence is given by x[n] is minimum phase, then the
poles and zeros of its z-transform must all lie within the unit circle in the complex plane.

All the poles of any finite length sequence have poles at the origin. For most finite

242

 

 

 

 

 

 

length. samp
the sampled
decaying S€t
Chapter5.5t
is not met b
sequence. I
sequence th
minimum-pl
scheme \i’il

minimum-pl

6-3 Tar;
An I
UPOII which

angles. A]
Scattering g

COiistructio

baSIS functi

W ,

 

length, sampled sequences the zeros do not lie with the unit circle. This even applies to
the sampled values from the late-time model given by (6.1). However, for a strictly
decaying sequence it is easy to show that the minimum-phase criteria is met (refer to
Chapter 5, section 2). For sinusoidally decaying sequences, the minimum-phase condition
is not met but the reconstructed sequence may be a good approximation to the original
sequence. In this case most of the zeros lie within the unit circle. On the other hand, a
sequence that is exponentially growing has all zeros outside the unit circle and a
minimum-phase reconstruction is clearly wrong. The following target discrimination
scheme will use a simple program that calculates the cepstral coefficients and the

minimum-phase reconstructed signal.

6.3 Target Discrimination Algorithm

An E-pulse must be generated for each expected target. The expected target retum
upon which the E-pulse is based must be carefully measured over a set of different aspect
angles. An alternative would be to calculate the scattered fields for fairly simple
scattering geometries. In either case, several aspect angles are required to make sure that
all the natural resonant modes are excited and included for the E-pulse construction.
Construction of the E-pulse e(t) is performed by expanding the E—pulse e(t) in a set of

basis functions as [20]

K
e(t) = ZakgkU) (6.7)
k=1

where {gk(t)} is an appropriate set of basis functions. The convolution integral in (6.2)

 

 

 

 

 

 

 

can be expai‘

Using the be

\I'lih a set 0i

where T“ d

Usin
solution) ca:
an inhomog
the matrix 1
impulse fun

Diff
Choice of T
resonant frI
S(Flared err
“Sing the e:

t0 ininimiz

 

 

can be expanded as
TE
e(t) =fe(r)f(t—1:)dr = 0 t> TI+TE (6-8)
0

 

Using the basis functions in (6.7), substituting this into (6.8), and taking inner products

with a set of weighting function {Wm} gives

K
E akf f gk(T)f(t-t)wm(t) (It (It: = 0 (6.9)

k=1 r=0 TI+TE

 

where TW describes the end of the measurement window.
Using (6.9), a solution for oak for almost any choice of TE ("forced" E-pulse
solution) can be found by choosing M = 2N and K = 2N + 1. In this case (6.9) becomes

an inhomogeneous matrix equation with solutions for any choice of TE that does not cause

 

the matrix to be singular. The use of rectangular pulse basis functions and weighting
impulse functions causes the integration in (6.9) to become much simpler.

Different values for TE result in significantly different E-pulse waveforms. The
choice of T“ can have a significant effect on the constructed E-pulse and the extracted
resonant frequencies [4]. One suitable choice of TE is that which yields the minimum
squared error per point between the original data f(t) and a waveform f(t) constructed
using the extracted natural frequencies, amplitudes, and phases. In this case TE is chosen

to minimize the following value over the sampled interval from TI to Tl + TW

6 = llftt) -f(t) II = E ire.) -f<t.>lz (6.10)

l

244

 

 

 

 

where

The natural
amplitudes it

functions thi

ii'here Z =I
detennined
amplitude a
Of course It
\i'ai'efonn h
frequencies
In tl
for sever-a]
II‘aiisfonn \I
hit some it
Putse metht
For this st
I‘tzilization

1“ the corn

 

 

 

 

where

N A
f(t) = Z a, 6°"tcos(d>nt + (fin) (6.11)
n=1

The natural frequencies s" = 0n + jrbn can be found by solving for the E-pulse basis
amplitudes in (6.9) and then using the relation given by (6.3). For rectangular pulse basis

functions this leads to a polynomial equation of the form

2N+1
0:ka = 0 (6.12)
k=l

where Z =e’5A and A the basis function width. Once the natural frequencies are
determined a least-squares fitting routine is applied to (6.10) and (6.11) to yield the
amplitude and phase terms for the reconstructed waveform. Different values of TE will
of course lead to values of s that differ. The value of TE can be varied to yield the
waveform having the smallest value of e. In this process, the E-pulse amplitudes, natural
frequencies, and best fit reconstructed waveform are obtained.

In this chapter, E-pulses are constructed from the transient response waveforms
for several different targets. The waveforms were generated using an inverse Fourier
transform with magnitude and phase spectral information that was theoretically calculated
for some targets and measured for others. Target discrimination, performed by the E-
pulse method, requires the convolution of each E-pulse with an unknown target waveform.
For this study the unknown target waveform was generated from a minimum phase

realization of magnitude only frequency data. A measure of the amount of signal present

in the convolved late-time response e(t) is given by the E—pulse discrimination number

245

 

 

 

 

 

 

 

(EDN) [9]

where W re
is simply a
To use the
beginninga
coiIi'olutior
values of CI
for the mat
correct E-p
lowest vali
comparing

ratio (EDR

Hence, the

E‘PUIScS CI
6.4 Nu
Thi

IW0 sets 0

 

 

(EDN) [9]
EDN =[ f c2(t)dt][f5e2(t)dt]’1 (6.13)

where W represents the time window over which the signal is convolved. Equation (6.13)
is simply a measure of the deviation from the expected value of zero late-time energy.
To use the above quantity, the E-pulse is convolved with the unknown target waveform
beginning at TL. If a target exists in the data base matching the unknown waveform, the
convolution energy should be zero, whereas the other E-pulses will produce non-zero
values of convolution energy in the late time. In actual situations, the convolution energy
for the matched target will not be precisely zero due to noise, problems in constructing
correct E-pulse waveforms, or incorrect waveform construction. The EDN having the
lowest value should then be chosen as the matched target. A quantitative measure
comparing all the EDN values in the data—base is given by the E—pulse discrimination
ratio (EDR)
EDR(dB) = 101ch10 { 75% } (6.14)

Hence, the E-pulse yielding the minimum EDN value has an EDR of 0 dB while other

E-pulses contribute values greater than 0.

6.4 Numerical Results
This section will describe the application of the E-pulse discrimination scheme to

two sets of data. The first set of scattering data was generated. from a simple thin-wire

 

 

 

 

 

 

 

scattering p
series of sc
consists of
again. E-pi
tested for t;

Bac
9.5. l0.0. 1
domain me
she appro
fixed at a r
at 256 equ:
spectrum \
GHz, T = ,
obtain a ti
results in t

The pulse.

Where I :

maximum

each Wire

 

scattering program. E-pulse waveforms were generated for this data set, after which a
series of scattering targets were tested for identification matching. The second data set
consists of scaled aircraft and missiles measured in the anechoic chamber at MSU. Once
again, E-pulses were constructed for these targets and several "unknown" targets were
tested for target matching.

Backseattering data was generated for simple thin wires of length 1 = 8.0, 8.5, 9.0,
9.5, 10.0, 10.5, 11.0, 11.5, and 12.0 cm. Scattering data was generated using a frequency-
domain method—of-moments solution. Piecewise-sinusoidal basis functions and the thin
wire approximation were employed with Galerkin’s method. The length-to-radius was
fixed at a ratio of 1000 for each wire. The complex backscattered fields were calculated
at 256 equally spaced frequencies between .05 GHz and 12.8 GHz (Af = .05 GHz). The
spectrum was windowed with a Gaussian modulated cosine (GMC) function (fc = 0.0
GHz, T = .1 nsec, see Appendix A) and then inverse Fourier transformed using a FFT to
obtain a time-domain waveform. The effect of windowing in the frequency-domain
results in the backscatter transient response from a Gaussian pulse for the incident wave.

The pulse, p(t), as a function of time can be written as

pm = ewe/e (645)

where r = .l nsec was chosen giving a pulse width of about .2 nsec between the 4% of
maximum points. Figure 6.1 shows the magnitude of the window function applied to

each wire—scattering spectrum. Figure 6.2 shows a time—shifted incident pulse

 

 

 

 

 

 

 

correspondi
For
angles: (1)
(2) 45 deg
discussed 3
angle calcu
of the spe
generatedt
then becai
comparism
(using an I
wire for bi
good mate
the match
An
SCheme.
Subsequen'
when the I

Pulse acro

as

 

 

corresponding to the inverse Fourier transform of the spectrum shown in Figure 6.1.
For each wire, the backscattering responses were computed for two incident
angles: (1) broadside incidence with the E-field parallel to the long—axis of the Wire, and
(2) 45 degrees off of broadside with the E-field in the same plane of incidence. As
discussed above, a set of E—pulses was generated for each wire (based on the two incident
angle calculations) using a hybrid E-pulse/least-squares method [39] with the magnitude
of the spectrum used to generate the E-pulses. A set of transient waveforms was
generated using the minimum phase reconstruction algorithm. These transient waveforms
then became the unknown target responses. Figure 6.3 and Figure 6.4 shows a
comparison between the transient response generated using both magnitude and phase
(using an IFFT) and generated from the minimum phase reconstruction for the 10.0 cm
wire for broadside and 45 degrees from broadside incidence. Figure 6.3 shows a very

good match between the two reconstructions in both late and early time. In Figure 6.4

 

the match is not quite as good.

An E-pulse data base library was generated for the radar target discrimination
scheme. Table V shows the E—pulse identification and description to be used in. the
subsequent figures. Ilavarasan and Ross [9] determined the start of late time based Upon
when the wave strikes the leading edge of the target (Tb), the maximal transit time of the
pulse across the target (T,,), and the pulse width (Tp). The late time was then calculated
as

T, = r, + T], + 2T, (6.16)

Their target data base consisted of values for TD and T1r for each target. To calculate Tb

248

_;

 

 

 

 

in the prese
not considf
analyze thi
responses (
Figi
broadside
generated I
As discuss
compare th
also were g
square icor
curve repre
trai-’efonn
target usin;
Fig
the EDR it
Figure 6.7
incidence.
minimum
magnitude,
Tht

10.0 cm w

 

 

in the presence of noise, a threshold detector scheme was used. The present analysis did
not consider noise, and late time was visually estimated. The purpose here was not to
analyze the automation process but to determine if the minimum-phase reconstruction
responses could be used in the discrimination scheme.

Figure 6.5 shows the EDR values for an unknown target generated from the
broadside transient response of the 10.0 cm wire. The values in this figure were
generated by convolving the unknown target response with each E-pulse in the data base.
As discussed earlier, a matched target will be indicated by an EDR value of 0 dB. To
compare the effect due to a target represented only by spectral magnitude, the EDR values
also were generated for the Fourier transform (mag/phase) reconstruction. The curve with
square icons show the EDR response for the magnitude/phase reconstruction whereas the
curve represented with the triangular icons is the magnitude only reconstruction. For both
waveform reconstructions the correct target is identified. As expected, the unknown
target using the minimum phase reconstruction has a lower response curve.

Figure 6.6 and Figure 6.7 show similar curves for different cases. In Figure 6.6
the EDR response for the 10.0 cm wire for an incident angle of 45 degrees is shown. In
Figure 6.7 the EDR response for the 9.0 cm wire is shown for broadside angle of
incidence. In both cases, the correct target in the data base has been identified. Also, the
minimum phase reconstructed target does not match the data base as well as the
magnitude/phase reconstruction.

The resonant frequencies created from. the E-pulse construction algorithm for the

10.0 cm wire are shown in. Figure 6.8. Also shown are the natural resonant frequencies

249

 

 

 

 

 

 

 

extracted fr
frequencies
numeric va
natural resr
noise wasc
minimum r
Very well.
A s
target scat
measured i
measureme
HPSTZOB '
response 0'
hand used '
field is sli;
aircraft wi
measured 1
incidence (
Were next :
reSpouse It
the models

f0HOWIIIg l

 

extracted from the minimum phase reconstructed waveforms. In both cases, the resonant
frequencies were derived using aspect angle calculations at 0 and 45 degrees. The
numeric values along the plotted symbols indicate the late-time energy order of each
natural resonant frequency. A value of 1 indicates a higher energy than 4. Since no
noise was considered in this analysis, the E-pulse construction was quite accurate and the
minimum phase reconstruction was also good. Therefore, the natural frequencies agree
very well.

A similar target discrimination simulation was performed on data from measured.
target scattered—field responses. Five scaled models of similar physical size were
measured in the MSU anechoic chamber. Figure 6.9 shows the scale models used in the
measurements. All measurements were performed in the frequency domain using an
HP87ZOB network analyzer. The system response was removed using the theoretical
response of a l4-inch diameter calibration Sphere (refer to Appendix A). The frequency
band used was from .5 to 5.5 GHz at a frequency step size of .0125 GHz. The scattering
field is slightly bistatic and the incident electric field is polarized. in the plane of the
aircraft wings or along the roll axis of the missile. Five angles of incidence were
measured for each model: 0°, 30°, 45°, 60°, and 90°. An angle of 0° indicates a nose-on
incidence direction whereas 90° indicates broadside. The frequency spectrum responses
were next scaled to reflect actual aircraft size ratios. For this analysis the F-14 frequency
response was not scaled and the other model responses were scaled to reflect changes in
the models physical size. The model’s physical size would need to be scaled in the

following manner to reflect the correct ratios: B-58 - 1.34x, f18 - .9x, missile #1 - .27x,

 

 

 

 

 

 

 

 

missile #2 -
scaling was
The
measureme
phase was i
added for
significantl
requires in'
model was
Fourier trai
centered at
.268. PM _
for the puls
for each I;
Windowing
pulse widtl
408.and
Fig
Waveform
the nose c
COITESponc

loss of hi i

 

 

missile #2 - .3 8x. It should be noted that the missiles were generic and the intent of their
scaling was to create a missile smaller than the aircraft.

The low frequency band containing no data ( 0 GHz to the low end of the
measurement ) was quadratically interpolated to zero amplitude at zero frequency. The
phase was interpolated using a minimization linear fit scheme. The extrapolated data was
added for two reasons. First, the effect of subsequent windowing can be reduced
significantly ( less ripple in the time domain). Second, the minimum phase algorithm
requires information down to zero frequency. Next, the transient response from each
model was weighted by a GMC window and then inverse transformed using the fast
Fourier transform or minimum phase reconstruction algorithm. The GMC window was
centered at fc = 0.0 GHz and the pulse width parameter T for each target was: B-58 -
.268, F14 - .2, F18 - .18, missile #1 - .054, and missile #2 - .076 nsec. Different values
for the pulse width parameter were chosen to use as much of the scattered field. spectrum

for each target as possible. The shape of the incident pulse formed as a result of

 

windowing is similar to that in Figure 6.2. The selected values of T give a time-domain
pulse width for each target as (sec (6.15)): B-58 - .536, F14 — .4, F18 - .36, missile #1 -
.108, and missile #2 - .152 nsec.

Figure 6.10 and Figure 6.11 show the windowed frequency response and transient
waveform for the BS8 aircraft respectively. The incident angle was measured 45° from
the nose of the aircraft. The frequency response shows at least two resonant peaks
corresponding to the natural resonant frequencies of the target. Figure 6.10 shows the

loss of high frequency information lost due to windowing. To use higher frequency

 

 

 

 

 

 

 

informatior.
or measure
Both the IF
reconstruct
between tlI
Figi
from the 01
between th
the comple
to work. 1
response g
reconstruct
the missile
Usi
data base I
E~pulse w:
modes W0]
the missile
resl30nses
measurem.
transforme

Phase rec

 

 

information one can either modify the window (with subsequent time-domain problems)
or measure over a wider bandwidth. The time-domain response is shown in Figure 6.11.
Both the IFFT response (using magnitude and phase information) and the minimum phase
reconstruction are shown for comparison. In this case there is a fairly good match
between the two responses - especially in the late time.

Figure 6.12 through Figure 6.19 show the frequency and time-domain responses
from the other targets. The transient responses for the F14 and F 18 show a poor match
between the magnitude/phase reconstruction and the cepstrum reconstruction. However,
the complex frequencies may have enough similarity for the target identification scheme
to work. Figure 6.17 and Figure 6.19 show excellent agreement between the transient
response generated from the magnitude/phase reconstruction and the magnitude only
reconstruction. This is probably due to the simple nature of the scattering geometry of
the missile.

Using the time-domain transient responses generated from the IFFT, an E-pulse

data base was constructed. The data base consisted of the five measured targets. Each

 

E-pulse was constructed. with five different incident angles to insure that all resonant
modes would be represented. However, only the broadside measurements were used for
the missile models. To determine if the discrimination scheme would work, the measured
responses for the B-58, F-l4, and F—l8 at 45° incident angle and broadside missile
measurements were used for the unknown target. The frequency-domain response was
transformed to the time domain using both the full spectrum IFFT and the minimum

phase reconstruction. Table VI shows the E-pulse discrimination ratio for each

 

 

 

 

 

"unknown”
response w
is doubtful
some noise
F-l4 respo:
of 3.5 dB
reconstruct
the models
missiles.
aircraft. T
modes wit
resonant in
Fig
the magnit
if the tim
match. A:
the late-tin
in each fig
(real parts
0n
E‘FUISC d2

Pulse Wav

 

 

 

"unknown" target as a function of the E-pulse in the target data base. In each case the
response was matched with the correct target. However, due to the low EDR values it
is doubtful whether the procedure would work in the presence of more noise, although
some noise is already in the measurement process. One can see this for the case of the
F-14 response, where the F-18 was nearly identified as the correct target with a margin
of 3.5 dB for the IFFT response and a margin of only 1.7 dB for the cepstral
reconstruction. This probably could be expected due to the similar size and geometry for
the models. The table also clearly shows that the aircraft will not be identified as the
missiles. However, there is a higher chance that the missiles could be identified as
aircraft. This is most likely caused by some overlap of the missile’s natural resonant
modes with the higher resonant modes of the aircraft. On the other hand, the low
resonant modes of the aircraft will not overlap any of the mssile’s natural resonant modes.

Figure 6.20 through Figure 6.24 show the natural resonant frequencies found using
the magnitude/phase reconstructed response and the magnitude only transient response.
If the time-domain responses matched exactly, then the resonant modes should also
match. As shown in the earlier figures, the transient responses do not match exactly in
the late-time period, and therefore one shouldn’t expect an exact natural frequency match.
In each figure, the complex frequencies are almost equal; however, the damping factors
(real parts) do not match as well.

One of the difficulties in using the E-pulse discrimination scheme is creating the
E-pulse data base. A great deal of time can be spent trying to generate a good set of E-

pulse waveforms, but difficulties arise when using any type of measured data having some

 

 

 

 

 

noise. One
zero - a P0
uungthele
reconstruct
positive da‘
order lTlOdt
would be a
frequency-«
notably get
the comput
the resona
become tra
However. t

in every t)

65 Co
ThI
Several nu

in regards
evaluated
data from

for the the

 

 

noise. One of the most troublesome areas deals with keeping the damping factor less than
zero - a positive value being non-physical. However, in creating an E-pulse waveform
using the least-squares method, higher order modes may be used to obtain a better fitting
reconstruction. Usually this gives a better fit, but at the cost of adding modes with
positive damping factors. Clearly, positive damping factors are not correct, but if higher
order modes could be added with negative damping coefficients the resulting E-pulse
would be a better candidate for the discrimination data base. Several constrained natural
frequency—extraction routines have been studied by researchers at the EM Lab, most
notably genetic algorithms [40]-[42]. The major problem with the genetic algorithm is
the computationally expensive algorithm resulting in a large amount of time to calculate
the resonant frequencies. On the other hand the genetic algorithm usually does not
become trapped in any type of global minimum or does not really require an initial guess.
However, one must gain some experience in selecting some of the correct parameters as

in every type of minimization problem.

6.5 Conclusions

The use of cepstral analysis has been applied to the target discrimination problem.
Several numerical examples were presented to show the performance of cepstral analysis
in regards to target identification. The performance of the discrimination scheme was
evaluated using both numerically derived scattering data and experimentally measured
data from scaled aircraft and missile models. Cepstral analysis worked extremely well

for the theoretically generated data. In the case of the measured data there was some

 

 

 

 

 

degradatior
with E-puif
the E-pulst

obtaining a

 

 

degradation due to the difficulty in constructing a good E-pulse. Difficulties associated
with E-pulse construction will also affect any type of target discrimination scheme using
the E-pulse method. This case shows the importance of the measurement process in

obtaining a good data free from unwanted noise or other signal components.

255

 

 

 

 

Fig

Fig

 

 

 

fif‘" _-..;__-..._.._.._. '— -. . . 1'

1.20

1.00

.0 .0
as oo
o o

Magnitude (Relative)
.0
.p.
O

0.20

lllIIIlllIIllllllLLIJllllllllIIIIIIIIIIIlIIIlIlIIIlllIIllllI

 

 

0.00 llIIIIllllmll—I—IlFTIIT—IIIIIIITIIIfljlfl

0.00 4.00 8.00 12.00 16.00
Frequency (GHz)

Figure 6.1 Frequency response magnitude for gaussian input pulse.

1.00

0.80

 

(D

U

:3

.‘L‘.’

CL
0.60

E

<

'0

(D

-_-"—‘_ 0.40

O

E

L.

0

Z

0.20

 

0.00

'0LIIIIIIIIIILLLIIIIIIIIIIIIIiIIIIIIiIIIIIIIIIIIIIiII

0 0.20 0.40 0.60 0.80 1.00
Time (nsec)

0

Figure 6.2 Normalized gaussian incident wave pulse.

256

 

 

 

o. aw
_
0200.05 conflict/s

Figure 6.1

0.

 

 

1.00 -

 

 

 

 

 

 

 

 

 

 

 

:l
: I e
050-:
”a : t f
.2 ‘ I I”
+, _
2 3 ,’
Q) _.
L _
v 0.00 -; A.
(D ‘ i
U : |
3 _ I
t d V
a : J
E_ :
<E 0'50 — I; —— Mag. 8c Phase
3 ' ---- Mag. Only
_1900 IIII IIrIerIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIII
4.00 6.00 8.00 10.00 12.00

0.00 2.00
Time (nsec)

Noise-free backscattering response of a broadside 10.0 cm wire with l/a =
1000.

Figure 6.3

257

 

 

Figure 6.

O. 0. m0
AoZfiBoIQ ouafificc<

 

 

-—..__ _.__._
-— — — -m‘- _._- - ,

1.00

 

 

 

 

 

 

 

 

0.50 3
A :
(1) _
._>_ ‘
4g _
2 I
(l) _
L _.
V 0.00 _
G) _
‘0 I
3 _
:5 _
E1 :
<1: _O°50 : ‘ —— Mag. 8c Phase
: ---- Mag. Only

IIII

 

Ill

_1oOO IIIIIITIITIIIIIIIIIIIIIIIIIIIITIIIIIIIIIIIIIIII I
0.00 2.00 4.00 6.00 8.00 10.00
Time (nsec)

Ilfl
12.00

 

Noise-free backscattering response of a 10.0 cm wire with l/a = 1000.

Figure 6.4
Incident electric field 45 degrees from axis of wire.

258

 

Table 6.1

 

 

 

 

 

 

 

Table 6.1 E-pulse identifiers and descriptions.

E—pulse identification E-pulse description I

 

 

A 8.0 cm wire E-pulse

 

8.5 cm wire E-pulse

 

9.0 cm wire E-pulse

 

 

9.5 cm wire E-pulse

 

10.0 cm wire E-pulse

 

10.5 cm wire E—pulse

 

11.0 cm wire E—pulse

 

EQTIITJUOUU

11.5 cm wire E-pulse

 

 

I 12.0 cm wire E-pulse

 

 

 

 

 

 

259

 

50.

 

40.

1:igure 6.

 

 

50.00 —

_ lit-BEES Mag. 8c Phase
_ a Are-eta Mag. Only
40.00 - “ "
$30.00 - ‘ '
U -.I
V _
D: _
Q ..
M 20.00 —
10.00 -
0.00 I r l i "" l i i i

 

 

 

 

E—pulse id

Figure 6.5 EDR values for broadside response of 10 cm wire with l/a = 1000.

260

50

 

40

O O

3 2
Amov mam

 

 

 

50.00 -

 

 

 

_ BEBE-El Mag. & Phase
_ we Mag. Only
40.00 — ‘ = 5 a
830.00 — "
‘0 .—
v _
D: _. I:
Q _
Li—l 20.00 -
10.00 —
0.00 I I I I '“‘ I l i I i

 

E—oulse id

EDR values computed from response of 10 cm wire with l/a = 1000

Figure 6.6 .
oriented 45° from broadsrde.

261

 

 

FiguI‘e I

 

 

 

60°00 BEBE-El Mag. 8c Phase

Are-AIM Mag. Only

50.00

H
II

H

40.00

 

EDR (08)

20.00

30.00 —_ .
10.00 —_

 

0.00 I I '“' i r i l l I T

 

E—oulse id

Figure 6.7 EDR values for broadside response of 9 cm wire with l/a = 1000.

262

 

C CC C E» f.\ .L\ ( C\ II\ 1‘
4 3 3 2 2 i I e
m\UOW_lO C_ \AocijmCoe £030.6me qus
wrr

 

 

 

40.0

 

 

 

I 4
350—3 0 *
c0 :
\ 5
U :
03300-3
l E :5
o E 9
525.0 E ***** Mag/Phase
" E 00000 Mag only
>4 :
g 5
0,200 E 2
3 :
O' : 9
(IL) :
4.15.0—2‘
C E
.9 E
510.0 E 1
'6 E a
o :
a: :
5.0-E
0.0.-IIIIIIIII'IIIIIIIIIIITIITIIIIIIIIIIIIIIIIIIIIII
—2.80 —2.40 —2.00 -—1.60 —1.20 —0.80

Damping coefficient in G—Np/s

Figure 6.8 Complex frequency locations for 10.0 cm wire scattering.

263

 

 

 

 

F-14 _

/.
_ "Cm
Figure 6.9 Planar view of scaled models used for target measurements.

264

 

 

0.

Figure I

O. flu.
mustcmwOE 0>$96K

 

 

 

 

1.00

 

0.80

.0
on
o

 

 

 

0.40

 

Relative Magnitude

 

0.20

 

IlllllllLIlllllllllIIlllllILlIllllllIIIIIIIIIIIIII

 

 

 

II IIIIIIIII IIIIIIIII]Illllllllllllllllrll
O000.06' ' ' ' "1.60 2.60 3.00 4.00 5.00

Frquency (GHz)

Figure 610 Frequency domain scattered-field response of B-58 aircraft model
measured at 45 degree incident angle.

265

 

 

I .

Figure I

C

_
®UJEQE< ®>300m

C

 

 

1.00

 

 

 

 

 

 

 

 

 

 

0.50 f
G) .—
U I
3 _
If: _
a I
E _
< 0.00 —_- -
<1) I
.Z I
4; —t
_g _
<1) : t: i
m —0.50 —_ I '
: Mag. & Phase Reconstruction
j ----- Mag. Only Reconstruction
_1oOO — I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I
0.00 10.00 20.00 30.00

Time (nsec)

Figure 6.11 Transient response of B-58 aircraft model. Dashed line indicates the

minimum-phase reconstructed response.

266

 

 

0.

Figure (

nU. flu.
®U3tC®O§ ®>.LO_®K

 

 

1.00

0.80

 

 

 

mt

 

.0
on
O

 

7?)

 

 

0.40

Relative Magnitude

0.20

llllllllIlllllllllIlllllllllIlllllllllIlllllllll

MM-w

_'l
0'00 1.00 2.00 3.00 4.00 5.00 6.00

 

0.00
Frquency (GHZ)

Figure 6 12 Frequency domain scattered-field response of F - 14 aircraft model measured
at 45 degree incidence.

267

 

fl .

Figure (

/l\\

®D3taE< ®>$O~®K

 

 

1.00

 

 

 

 

 

 

 

 

 

 

 

 

 

0.50 —:
G) .—
U I
3 _
.4: _
a I
E _
< 0.00 _ *“t‘I
<1) I l
> - l
A: : l
g _ .
<D : l'
“5 —o.so : ::
- I
I I
3 —- Mag. 8c Phase Reconstruction
: ----- Mag. Only Reconstruction
_1.00 I I I I I I1 I I I I I I I I I I I I I I I I I I I I ITTI
0.00 10.00 20.00 30.00

Time (nsec)

Figure 6.13 Transient response of F-l4 aircraft model. Dashed line indicates the

minimum-phase reconstructed response.

268

 

0
Figure 6

0.

nu. nu.
mostcmoE ®>$9®K

 

 

W ¢\ 1' ..

1.00

 

 

 

 

 

 

 

 

 

0.80 5
<1) 3 I?
“O : N
.3 :
E 0.50 :
OW : i?
O _
2 :
f>f 0.40 —:
.4.) _
__O :
(D :
O: :
o.2o —:
0.00 I7IIIIIIIIIIIIIIIIIIIII[AIM
0.00 2.00 4.00 6.00

Frquency (GHZ)

Figure 6.14 Frequency domain scattered-field response of F - 1 8 aircraft model measured
at 45 degree incident angle.

269

 

( I _

oUDEQE< o>zouom

Figure (

 

 

,m —. . 7.—
J-Rhg. 4.. 4.;

1.50

 

 

 

 

 

 

 

 

 

 

 

 

: l g
1.00 : II: I \ \//\
_ ll ‘ /
(D : III I; \2750\ ///
‘0 : " / j
3 : is ///i [J
.1; 0.50 -; ;. ’ 0’ F48
O. _—_ II
E : i:
<1: - 'I '
a; 0.00 S-x/BJ. ‘\
'4: I III
2 : :.:
(l) : III
0: - III
" l
—o.5o —j if
" l
: I —— Mag. & Phase Reconstruction
E r ----- Mag. Only Reconstruction
—1.00 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIj
0.00 10.00 20.00 $0.00

Time (nsec)

Figure 6.15 Transient response of F-18 aircraft model. Dashed line indicates the

minimum phase reconstructed response.

270

 

0 0 0 0.
@UJtmeOE ®>.§O~®K

Figure l

 

 

1.00

0.80

 

.0
a)
O

 

 

Relative Magnitude
E

0.20

llllllllIlllllllllIlllllllllIlllllllllIlllllllll

 

 

 

OOO IIIIIIIIIIIIIIIIFTIIIIIIIIIIIIIIIIIIIIIIIIIIIIIITI—I

0.00 5.00 10.00 15.00 20.00 25.00
Frquency (GHZ)

Figure 6 16 Frequency domain scattered-field response of missile #1 for a broadside

measurement.

271

 

_
ooB:aE< o>.;o_om

Figure

 

 

 

' .u‘_'._.___;_ . -
__ H_ _._ __ . __

6.00 -

l

 

 

 

 

 

 

 

 

 

4.00 : l i
3 i l
: E i
(l) : \
U 2.00 :- anal
3 — 1
.-+_-: : I
E. E I
E :
< 0.00 : A: w...
<1) 3
.2 :
-+—’ :
g -—2.00 -;- I
(D :
D: : '
—4.oo é
: —— Mag. & Phase Reconstruction
E ----- Mag. Only Reconstruction
—6.00 IIIIIIIIIIIIIIIIIIIITIIIIIIIIIIIIIIIIIIIIIIIlllm
0.00 2.00 4.00 6.00 8.00 10.00

Time (nsec)

Figure 6.17 Transient response of missile #1. Dashed line indicates the minimum

phase reconstructed response.

272

 

Figure

0

O O
mostcoo§ o>:o_om

 

 

 

 

1.00

0.80

 

.0
a)
O

 

Relative Magnitude
'2

0.20

lllllllllIllllllIllIlllllllllIlllllllllIlllllllllI

 

 

 

0.00 TFIITIITfiIIIIFI—II—WI—IT—I—I—III—IIIIIIIIIIIII—I

0.00 4.00 8.00 12.00 16.00
Frquency (GHZ)

Figure 6.18 Frequency domain scattered-field response of missile #2 for a broadside
measurement.

273

 

 

 

 

_
003:0E< 0>Eo~0m

_

 

Figure

 

 

 

 

 

 

 

 

 

 

 

4.00 :
2.00 —:
m 3
TO ..
3 I
of; 0.00 _
Q- :
E :
<E :
i; —2.00 E '
.4.) ..
2 I '
<1) I
D: :
—+00{
3 —-- Mag. 8c Phase Reconstruction
Z ----- Mag. Only Reconstruction
—6.00-TTTIIIIjIIIIjTIIIFITr—Ijllllllj
0.00 5.00 10.00 15.00

Time (nsec)

Figure 6.19 Transient response of missile #2. Dashed line indicates the minimum
phase reconstructed response.

274

 

 

 

 

 

 

 

Table 6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 6.2 E-pulse discrimination ratio (dB) for each library E-pulse as a function of
input waveform response.
Target data base response
Input Response B-58 F-l4 F-18 misl misZ
B-58 0.0 7.8 13.2 29.2 26.5
F-14 19.0 0.0 3.5 40.4 40.5
F-18 32.7 21.9 0.0 44.7 41.9
misl 14.6 8.2 6.8 0.0 15.3
mis2 20.7 18.9 11.4 33.2 0.0
B-58" 0.0 7.8 13.0 24.1 25.7
F-l4i 16.7 0.0 1.7 31.8 32.1
F-l8* 21.6 12.0 0.0 36.2 35.1
misl" 14.0 7.2 5.9 0.0 14.1
mis2i 17.7 17.3 9.5 31.3 0._(_)__

 

 

 

 

 

 

 

* = indicates a minimum phase reconstruction

275

 

 

 

Figure

w\UOm|O C_ \AOHODUOLC CO_MW_DOK

 

 

 

 

 

 

 

———-“ -_._..-__-_.; —"" ' ‘

 

 

8.00 -_
: a: O
a) _
\ :
U _
D? 6.00 -:
I ..
O .-
C _
._ : * O
>\ _
g -.
(D 4 00 j
3 _ *
C- - 0
CL) _.
“F _ * o
C —
.9 I
E 2.00 '2
'0 _
C _
Q: _
— ***** Mag/Phase
3 00000 Mag only
0.00—II—InTrrlfiTF1ITIfiTIITI—rWIIIII]
-0.3 —0.2 -0.1 0.0

Damping coefficient in G—Np/s

Figure 6.20 Complex frequency locations for late~time response of B—58

276

 

 

 

 

AI

CO.LO._UOK

Figure

mVoomloz E kopwnoot

 

 

 

 

 

6.00

*0

2.00

***** Mag/Phase
ooooo Mag only

Radiation frequency in

b
o

(:4
O
O
lllllllllIlllllllllilllllllilIlllllllllIIllllllllIlllllllllI
*

 

 

0.00 TIFIIIFIIIIWFFIWI‘IIFIITWITIFIIWIIIIITIIIIIWIIIIIIIIIIII]

—-O.5 -0.4 —0.3 —0.2 .—O’1 0.0 0.1
Damping coefficient In G—Np/s

Figure 6.21 Complex frequency locations of late-time response for F-14.

277

 

 

 

 

Figure

m\UOmlO r: \AOCODUOIC CO.:~O._UOK

 

 

 

 

 

10.00

 

.5. o *
m 9.00 E
\ :
“o :
0C2 :
l 8.00 E
o :
: or:
.E E
>‘ 7.00 '3
o :
C :
(D :
3 _
o- .:
a) 6.00 :
Lt :
I * 0
g 3
.9 :
To] :
: o
01 _: *
4°00 : ***** Mag/Phase
E ooooo Mag only
3.00 IIIIIIIIIIIIITIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIITI_I

 

 

—1.0 —O.8 —O.6 —0.4 —0.2 0.0 0.2
Damping coefficient in G—Np/s

Figure 6.22 Complex frequency location for late-time response using F-18.

278

 

1| 1

WVUOKIWA r: \AODWDUOLu. COEEUOK

Figure

 

 

 

 

 

 

 

 

35.00

N
.0
o
o

15.00

***** Mag/Phase
ooooo Mag only

Radiation frequency in

.0
o
o

IlJllllllLIlllllLLLlIlILllllILJIIIILLLIIILJJIILLLIIILLIIIILII

 

.0“
o
o

errrn-ijrnTrrrrrnfifimTrrrmfiTrrrrrrrrrrrrrranTrmTrmj
—3.5 —:5.0 -2.5 -2.0 —1.5 —1.0 —0.5 -0.0
Damping coefficient in G—Np/s

Figure 6.23 Complex frequency for late-time response of missile #1.

 

 

 

 

 

 

Figure

 

m\UOvi._lO r: \AOCODUOIC COEEUOK

 

 

 

 

 

 

 

25.00

i *o
(0 I
%20.00 :
o _
01 : ***** Mag/Phase at
('9 j ooooo Mag only
E 15.00 E * 0
>\ _
0 :
C ..
G) ..
a- : a
“d 10.00 E o
“— .. x:
C :
O —r
143 I
.9 g
g 5.00 :
01 L‘
3
0.00 —

 

mmmmmmmm
—3.5 —3.0 --2.5 -2.0 -1.5 —1.0 —0.5 -—0.0
Damping coefficient in G—Np/s

Figure 6.24 Complex frequency locations for late-time response of missile #2.

 

 

 

 

 

 

 

 

 

 

7.1

\Videb:
detecti
detecti
skimn‘.
introdl
impler
E-puls
transie
Lab at
to the
larger
detecti
like su

an infi

requiri

2) a lie

 

 

 

Chapter 7

Conclusions

7.1 Summary

This thesis has presented a number of topics related to the application of ultra-
wideband (UWB) radar to the detection and discrimination of radar targets. Both the
detection and discrimination algorithms are based on the E-pulse method. A new
detection technique has been introduced which can be used to detect the presence of sea-
skimming missiles in a sea-clutter environment. The new detection technique has been
introduced in order to overcome some of the difficulties associated with previous
implementations of the E—pulse detection method. A discrimination scheme, based on the
E-pulse method, has been developed which uses the late-time portion of a target’s
transient signal. This scheme differs from previous work done by researchers in the EM
Lab at MSU by using the methods of cepstral analysis to construct a close approximation
to the late-time portion of a target’s transient response using only the modulus of a
target’s spectral return. One problem often encountered in the theoretical study of
detection techniques is the generation. of theoretically calculated radar returns for a sea-
like surface. Therefore, a new numerical method to compute the scattered return from
an infinite, perfectly-conducting, periodic surface has been developed.

Scattering from most types of sea-like surfaces is very computationally expensive,
requiring large amounts of computer memory and fast processing capabilities. In Chapter

2, a new numerical-scattering method has been developed and tested. This new method

 

 

 

 

 

 

caknfl:
0f inf
dunen
crests
been l
numer
llsuig

in 0rd:

 

by rec
apprO)
genera

Thein

use uvi

inade:
can be
COlnpu

lherne

 

 

 

i3 Sign
Period
minai

times ]

 

 

 

 

calculates the scattered field in the far zone from a perfectly-conducting, periodic surface
of infinite extent. The periodic surface, characterized by surface roughness in one
dimension, is illuminated by a plane wave whose electric field is parallel to the surface
crests (TE excitation). Due to the periodicity of the surface, a periodic current model has
been used to develop an electric field integral equation. The integral equation was
numerically solved using the method of moments with rectangular-pulse basis functions.
Using several test cases, this new algorithm was compared to several established methods
in order to validate the new technique. The performance of the new method was tested
by recording the combined array fill and solve time as a function of the number of
approximation terms used in the algorithm. Finally, the new technique was used to
generate the scattered field from a sinusoidal surface having a wavelength of 1 meter.
The importance of this new algorithm is that it can be used to generate scattering data for
use with different target detection schemes.

Several important conclusions in regard to computational considerations can be
made for the new scattering algorithm. First, the scattering from a finite-length surface
can be approximated by the periodic current method. The new method decreases the
computation tim by a factor of 10 to 20 as compared to the non-periodic solution using
the method of moments for the EF IE. Second, the amount of computer memory required
is significantly reduced due to the periodic nature of the solution and the use of a single
period of the scattering surface. Due to the increase in computational speed and reduction
in memory this new method can be applied to a surface having a dimension nearly ten

times larger than used in the past. For research in the lab at MSU most surface have

 

 

 

 

 

 

 

 

 

 

been r
l to 2
freque
surfacr
possib
about

foras

be the

is an t
have t
target
using :
3 was
finite-l
since 2
length
l‘ectang
alSO at
at the
Pl‘Oble:

COmpu

 

been restricted to approximately 1000 surface segments and an overall surface length of
1 to 2 meters. For a 2 meter surface length consisting of 10000 surface segments the
frequency range must also be restricted to a maximum value of about 15 GHz (if 10
surface segments are used per wavelength). Using the new methodology it becomes
possible to compute the scattered field from a surface with an individual wavelength of
about 1 to 2 meters in the same frequency range and using about 1000 surface segments
for a single period. In this case the computational time and memory requirements will
be the same as before, however a much larger surface period can be used.

The detection of a sea-skimming missile in a heavily cluttered sea enviromnent
is an extremely important problem for the navy. Previous researchers in the EM Lab
have tackled the problem using the E—pulse method. Chapter 3 reviewed the E-pulse
target detection scheme. The theory for this method was developed and several test cases
using non-baseband scattering data were presented. Another topic reviewed in Chapter

3 was the numerical solution to the electric field integral equation for scattering from a

 

finite~length, perfectly-conducting, 2—dimensional surface. This discussion was included
since a large number of scattering calculations were performed in this thesis for finite
length surfaces. The computation of the electric field uses the method of moments with
rectangular pulse basis functions. The use of the spatial decomposition technique was
also applied. to the basic scattering algorithm. Although this method was not developed
at the EM Lab, the implementation allowed the author to solve larger scale scattering

problems. Implementation of the spatial decomposition algorithm did not change the

 

computational speed, but the technique was shown to be very useful for problems

283

 

 

 

 

 

 

 

 

 

 

requiri
the me

requin

genera
eradic:
target.
One p:
target/
to clut
metho
transn‘

to ma)

numer
These
CRTVl
discus:

the gel

The fit

model

 

requiring large amounts of computer memory. Several examples were used to illustrate
the method and data was presented showing the amount of memory and computer time
required to successfully run different size scattering problems.

In the E-pulse detection technique, a finite duration waveform (E—pulse) is
generated which, when convolved with a target/clutter response signal, effectively
eradicates the clutter component of the signal to allow for detection of the embedded
target. A great amount of care and experience is needed to create an effective E—pulse.
One problem often encountered is that the convolution of the B-pulse with the combined
target/clutter response will attenuate both the clutter and target, resulting in a poor target
to clutter ratio. Chapter 4 discussed a new detection algorithm, based on the E-pulse
method, that overcomes this problem. The new algorithm generates a clutter reducing
transmit waveform (CRTW) which is designed not to eradicate the clutter altogether but
to maximize the target to clutter energy ratio.

The theory for this new CRTW technique was presented in Chapter 4. The
numerical calculation of an effective CRTW becomes a global maximization problem.
These problems are inherently difficult to solve so a genetic algorithm was applied to the
CRTW construction problem. A detailed implementation of the genetic algorithm was
discussed including computational considerations. The chief problem facing the use of
the genetic algorithm is its inherently slow convergence time.

To test the effectiveness of the CRTW algorithm several examples were presented.
The first example used the measured clutter return from a perfectly conducting sea-surface

model in conjunction with the measured scattered return from a scaled missile model.

284

 

 

 

 

 

 

 

 

 

A.CI{
This f
for a s
realist
the sc
CRTV
intvhi
techni

the CI

and ar
schem
the C]
variati
Changl
CRTV
missil:
Cl’eater
CRTV
and di

target

 

A CRTW was constructed and applied to a combined sea-surface/target—return signal.
This first example was designed to show the effectiveness of this new detection technique
for a static situation. A second example considered an evolving sea surface using a more
realistic sea-surface model. A time simulation of an evolving sea surface was created and
the scattered fields were numerically calculated. Using a measured missile model, a
CRTW was calculated for the initial sea-surface state. A simulation was then performed
in which a missile traveling over the evolving sea surface was detected using the CRTW
technique. Results of that simulation showed the effects of an evolving sea surface on
the CRTW technique and the need to update the CRTW.

A CRTW is created from the measured return of a clutter producer (sea surface)
and an anticipated target. The effect of return from different target types on the detection
scheme was also studied in Chapter 4. Early detection of an antiship missile depends on

the CRTW method being tolerant to variations in a missile’s scattered return. These

 

variations can be due to roll-stabilized flight-control systems, missile configuration
changes, or variation in the incident and scattering angles. To test the tolerance of the
CRTW method to different target geometries, the scattered returns from several different
missile models were measured in the anechoic chamber at the EM Lab. A CRTW was
created from the scattered return of a sea surface and one of the measured targets. The
CRTW detection algorithm was then applied to the combined return of the sea surface
and different targets. In every test the CRTW detection algorithm was able to detect each
target with only slight variations in the calculated convolution energy ratio.

The detection variable known as the convolution energy ratio paramaterizes the

 

 

 

 

 

 

 

 

effiect
a targ
backg
proper
Chapt
threat
the pr
the (I
value

SC Ellie]

the se
mhun
anhch
asintr
the tar
see flu
surfac
anuso
surfac
measu

diam

 

effectiveness of the CRTW detection algorithm. A large value indicates the presence of
a target. For no target this variable has a value of 0 dB (for a non-evolving clutter
background). The calculation of the convolution energy depends on the selection of a
proper energy window size. The topic of energy window size was also discussed in
Chapter 4. The use of a narrow window allows for finer range resolution of the target
threat. However, a narrow window can lead to a higher probability of false detection in
the presence of an evolving sea surface. Nonetheless, a narrow window can be used if
the CRTW is updated at a high rate. A wider energy window will broaden and lower the
value for the convolution energy ratio, including the maximal baseline value from the
scattering surface.

An important topic for sea-surface scattering is the multipath interaction between
the sea surface and the target. The creation of the CRTW involves using the sea-clutter
return having no target contribution and a calculated or measured return from an
anticipated target. In the detection process, the target/clutter return is not derived from
a simple combination of the individual returns. There can be significant coupling between
the target and surface, including multipath effects, which effect the measured return. To
see the effect on the detection algorithm, the scattered return from several different sea-
surface models was measured. These models included the Stoke’s surface and the double
sinusoid. The response from a small missile model was also measured. With the sea-
surface and. missile return a CRTW was created. Next, the missile and target were
measured together. Several measurements were performed by placing the missile at

different locations with respect to the wave crests and at different heights above the sea-

 

 

 

 

 

 

 

surfac
and ti
algori‘
Howe
wavef

able tr

knowr
been
algorit
were
proces
retum

techni.

at this
amoun
period
this “I
detecti
methot

CRTu

 

surface models. Using these measurements, the CRTW detection algorithm was applied
and the convolution energy ratio was calculated. In every test, the CRTW detection
algorithm was very effective in finding a target that was located over a wave trough.
However, the effect of the interaction was clearly evident in the convolution energy
waveforms. For a target located close to a wave crest the detection algorithm was not
able to find the target.

A final topic covered in Chapter 4 was the use of a clutter suppression technique
known as coherent processing clutter reduction. This technique, discussed by Iverson, has
been used for detecting moving targets on a static clutter background. The basic
algorithm was modified for a non-stationary clutter background. Several simulated tests
were performed for a missile flying over an evolving sea surface. The coherent
processing clutter reduction technique was shown to be very effective if the sea-clutter
return could be measured at a high rate. The most appealing characteristic of this
technique is the simple nature of the algorithm, making it ideal for real-time applications.

Several important conclusions regarding the new detection algorithm can be made
at this point. First, the new CRTW algorithm is extremely robust but suffers from the
amount of computer time required to construct the CRTW. Second, the CRTW must be
periodically updated for a changing sea surface. Using the Kinsman sea-surface model
this update rate must occur approximately once a second. Third, the new CRTW
detection scheme is quite tolerant to different target types. This makes the CRTW
method very good for target detection but a poor candidate for target discrimination. The

CRTW method is also very effective in detecting targets even in the presence of the

 

 

 

 

 

 

 

 

multi]
well i
in tim
this n

algori

frequc
or me
to the
metho
the m

116C685

of mu
were

respor
time it
late-ti]
early

cepstrz
respon

Giam

 

multipath phenomena. Finally, the modified coherent processing technique works quite
well for target detection in the presence of a changing background clutter. At this point
in time the author has some reservations about the robustness of this method. However,
this method can easily be used in a real-time application due to the simplicity of the
algorithm.

Construction of the radar transient response can be done using the method of
frequency domain synthesis. In this method, the frequency domain response is calculated
or measured over some bandwidth. Next, an inverse discrete Fourier transform is applied
to the frequency-domain data to generate the transient response. In Chapter 5, the
methods of cepstral analysis were used to reconstruct the transient response using only
the modulus of the spectral response. The purpose of Chapter 5 was to provide the
necessary background for the target discrimination analysis in Chapter 6.

A discussion of cepstral analysis theory was presented in Chapter 5. The topics

 

of minimum-phase conditions and energy relations were discussed, and several examples
were presented illustrating the use of minimum-phase reconstruction. The transient
response from an UWB radar can be characterized by an early-time response and a late-
time response. A discussion of the minimum-phase reconstruction for the early-time and
late-time responses was presented. Also, some time was devoted to the separation of the
early and late-time transient response using cepstral analysis. For most signals the
cepstral reconstruction of the early-time response is unpredictable. The late-time
response, although not minimum phase, has an energy-concentration nature similar to that

of a minimum-phase signal. Several examples were presented showing the reconstruction

 

 

 

ofthe
the n
uniqu
scane

hue-u

deveh

perfor

 

the di

and e

 

worke
be ca
dificr
schni
the co

10W.

12

 

Thh s

Presen

 

 

 

 

of the late-time waveform using only the magnitude of its frequency spectrum. One of
the most important properties for target detection using the E-pulse method is the
uniqueness of a target’s complex frequencies in the late-time portion of the transient
scattered signal. Several examples were used to show that a cepstral reconstruction of the
late-time response contains a good match to the actual complex frequencies.

A target identification scheme using the E-pulse method and cepstral analysis was
developed in Chapter 6. Several numerical examples were presented showing the
performance of cepstral analysis in regard to target discrimination. The performance of
the discrimination scheme was evaluated using both numerically derived scattering data
and experimentally measured results from practical target models. Cepstral analysis
worked very well for the numerically generated data since a very accurate E-pulse could
be calculated. Construction of a good E-pulse from measured data is a much more
difficult task. For the measured data presented in Chapter 6 the cepstral analysis
technique worked quite well. In every case the detection algorithm was able to identify
the correct target, although in several cases the E~pulse discrimination ratios were quite

low.

7.2 Topics For Further Study

There are several t0pics in this thesis which could be the basis for future work.
This section will address several of them.

The new CRTW technique was presented in Chapter 4. Several examples were

presented showing the effectiveness of this new method. The use of data from an actual

 

 

 

 

 

 

 

 

UWB
agon
cenai

duac

cmnp
kw :
cmnp
man.

pmbh

Onep
orera
an u

Valuai

Signal:
area,

fined
and ex
resona

l‘econs

 

UWB radar would be extremely useful for additional testing of the new detection
algorithm. Not only would this data present the effects of noise and multipath, but it
certainly would provide information about how often to update the CRTW. Also, this
data could be used to thoroughly test the coherent processing detection algorithm.

One of the real problems with the CRTW detection algorithm is the amount of
computer time required to generate a CRTW. Since the CRTW must be updated eveiy
few seconds, the computation of the CRTW must be quite fast. Therefore, a
computationally fast algorithm must be designed if this detection technique is to be
realized. This problem may be very difficult due to the global minimization nature of the
problem.

The effect of multipath and sea-surface/target interaction is a very interesting topic.
One possible area of work is designing a scattering algorithm for a 3-dimensional target
over a 2—dimensional sea-surface waveform. Once again, this could be a difficult problem
and would be computationally slow. However, the generated results will be quite
valuable for use with different target-detection algorithms.

Two chapters were devoted to the application of cepstral analysis to UWB radar
signals and target identification. There are several possibilities for future work in this
area. First, a target identification algorithm could possibly be designed. by working
directly in the cepstral domain. This project would require a great deal of knowledge
and experience with cepstral analysis. Second, more research is needed. in regards to the
resonant frequencies extracted from the late-time portion of the minimum—phase

reconstructed signal. Several examples in this thesis showed that the minimum-phase

 

 

 

 

 

 

 

 

 

 

extrac
Howe
greatt
filters

again.

 

extracted frequencies match the magnitude/phase reconstructed frequencies quite well.
However, there is a significant amount of material here if the problem is analyzed in
greater detail. A final area of research would be in the design and testing of various
filters to separate the early and late-time signal components using cepstral analysis. Once

again, this could possibly be done in the cepstral domain.

291

 

 

 

 

 

 

 

 

 

APPENDIX

 

 

 

A.1

ngna
use 0
may 1

Henc

 

analy

thne

hequ
Incas
exhac
ngnal
recon

magn

 

based
becon
t0 acc

Geosu

 

 

 

—_———~ - - ,___ _ a.“ __——.~— __-_..... - - —— - -

Appendix A

Scattering Measurements

A.1 Introduction

A target identification scheme is based upon a set of target characteristics or
signatures. If these signatures have been derived from scattering measurements, then the
use of an accurate measurement system is essential. The use of noisy or inaccurate data
may be detrimental to the identification process, even making the scheme nearly useless.
Hence, it becomes vital to a detection scheme using the E-pulse method or cepstral
analysis to collect the best possible data.

An E—pulse discrimination scheme requires an accurate measurement of the late-
time portion of the transient response from a scattering target. Extracted resonant
frequencies are derived from the late-time portion of the signal. Hence, poor
measurements will lead to an inaccurate characterization of the scattering target. The
extracted modes become very difficult to calculate from a noisy or inaccurate late-time
signal and may lead to wrong or non-physical values for the resonant modes. The
reconstruction of the late-time portion of a target response using only the spectral
magnitude is also highly susceptible to inaccurate measurements. A discrimination system
based upon the extraction of resonant frequencies from the reconstructed. waveform will
become useless if poor measurements are made. Therefore, it becomes very important
to accurately measure complicated scattering structures in order to verify the utility of

cepstral analysis in conjunction with an E-pulse detection scheme.

 

 

 

 

 

 

 

perfio
discu
at hit
order
of w

@8861]

AHZ

durat
udde
of a

unte-
then

select
pulse
scahe
Ofelt
the g:
measr

Calcui

 

 

This appendix will describe the frequency-domain synthesis approach [43],[44] for
performing transient measurements. The frequency-domain synthesis method will be
discussed as well as the principal components of the measurement system in the EM lab
at MSU. A discussion of system deconvolution and calibration will be presented. In
order to illustrate some of the concepts an example will be presented. Finally, the topic
of windowing or weighting functions will be examined. Discussion of this topic is

essential since the concept of frequency windowing is used throughout this thesis.

A.2 F requency-Domain Synthesis and Scattering Measurements in the EM lab
In a frequency-domain synthesis system, the scattered response due to a short-
duration time pulse is synthesized using a frequency—domain measurement made over a
wide bandwidth. The basic idea behind this method is to measure the scattering response
of a stationary target at a large number of frequencies. Once this has been done, the
time-domain response is formed by applying a weighting function to the measure data and
then transforming to the time domain by using an IFFT. The measured bandwidth and
selected weighting function determine the duration and shape of the equivalent input
pulse. Unfortunately, the measured response in a real system does not consist of a
scattered return from just the target under investigation. The measured data also consists
of clutter, noise, and effects due to the system response. To eliminate any effect due to
the system response, a measurement is made of a known scatterer (calibrator). The
measured scattering field from the calibrator can then be compared to a theoretically

calculated response of the calibrator to determine the system response. From this

 

 

 

 

 

 

 

 

 

 

 

 

infion

pyrar
[flace
chant
hequ
Incas
Anah
'lown
ngna
a 10(
Prech
[flach
(Hi2)
Packs
anahr
the r
Config
P002

0Deni

 

information the system response can be eliminated from the desired target measurement.

The frequency-domain synthesis system used at the EM lab is shown in
Figure A.1. This system consists of an anechoic chamber having floor dimensions of
12’ x 24’ and a ceiling height of 12’. The surfaces of the chamber are covered with
pyramidal absorbers having a pyramid depth of 6 inches. Various scattering targets were

placed on a pedestal which was located approximately in the middle of the anechoic

 

chamber. This pedestal is made of styrofoam and is nearly transparent to the microwave
frequencies normally used in the measurements (.5 - 18.0 GHz). The frequency-domain
measurements were performed using a Hewlett-Packard HP8720-B Vector Network
Analyzer. Two different configurations were used in the scattering measurements. The
"low-band" configuration used a PPL 5812 10dB broadband amplifier to amplify the

signal from port 1 of the network analyzer to the transmit antenna. This amplifier has

 

a 10dB gain from .1 to 2 GHz and is powered by an external power source (B&K
Precision DC. Power Supply 1610). A nearly monostatic system was designed by
placing two America Electronics Laboratory (AEL) H-17 34 TEM Horn antennas (.5 - 6.0
GHz) approximately 24 inches apart. The "high-band" configuration used a Hewlett-
Packard HP8349B Microwave Amplifier to amplify the signal from port 1 of the network
analyzer to the transmit antenna. A nearly monostatic antenna arrangement consisted of
the AEL H-1498 Wideband TEM Horn Antennas (2.0 - 18.0 GHz). In both
configurations no external amplifier was used to amplify the receiver signal connected to
port 2 of the network analyzer. The transmitting and receiving horns were placed through

openings in a. wall of the anechoic chamber approximately 24 inches apart and at a height

 

 

 

 

 

 

of60
hour
spher
exten

netwt

An3

the t
ineas
The

ineas

pl‘OCC

aneci
scane
hansl
cahbi

Iher

When

Han

 

 

of 60 inches from the floor. The unknown targets and calibrator were located 12 feet
from the horns and at a height of 60 inches from the floor. A 14 inch diameter metallic
sphere was used as the system calibrator. The network analyzer is connected to an
external IBM PC/AT compatible microcomputer which controls the fimctions of the

network analyzer, the rotating podium, and stores all measurements for later processing.

A.3 Calibration Procedures

The goal of measuring the scattered return from an unknown target is to obtain
the target’s transfer function. As discussed in the previous section, a scattering
measurement also includes transfer functions associated with other system components.
The purpose of this section is to review the calibration procedures used for the

measurements in this thesis. An example will also be included to illustrate these

 

procedures. A detailed discussion of the calibration procedure can be found in [6].
Figure A.2 shows the system block diagram for the measurement system in the
anechoic chamber at MSU. The aim of the calibration procedure is to calculate the target
scattering transfer function by using the theoretical response of a known calibration
transfer function. Here, f represents the frequency parameter. The first step in the
calibration process is to make a measurement without any target in the anechoic chamber.

The received signal R ”( f ) for the background. measurement can be modeled as

Rbtf) = S(f){H,,(f) + H.(f)} (AA)

where S (f ) represents the system transfer function, N b (f) represents random noise,

Ha (f) models the transfer function due to direct coupling between the transmit and

295

 

 

 

 

 

recei‘
chain
funct

anten

when

calibr
thesis
usefu
in the
the S

calibr

where
intera

noise.

target

 

 

receive antennas, Hc( f ) represents the interaction between the antennas via the anechoic
chamber environment, and f represents the frequency parameter. The system transfer
function can be written in terms of the transfer functions of the receive and transmit

antennas Hr(f) and Ht(f) as

S(f) = H,(f) H,(f) E(f) (A-Z)

where E( f ) represents the spectrum of the input signal.

The next step in the calibration procedure is to measure the response from a
calibration target whose theoretical response is known. For the measurements done in this
thesis a 14 inch diameter metallic sphere was used as the calibrator. The sphere is a very
useful calibrator since a closed form solution for the scattered electric field can be found
in the Mie series. Also, the scattered electric field from the sphere is not a function of
the sphere’s orientation with respect to the incident field. The response from the

calibration target measurement Rcib (f ) can be written as

R"'”(f) = S(f){Ha(f) + Hc(f) + Hjtf) + Hsi} + N“”(f) (A3)

where H;( f ) represents the known calibration transfer function, HS: (f ) represents the

interaction between the calibrator and the anechoic chamber, and N “b ( f ) is random

noise.

The last step in the measurement process is to measure the response of the desired

target R‘ ”’ (f ). This measurement can be modeled with the following equation

Wm = Stf){H,(f) + lam + H50”) + 11.2} + N””(f) (A4)

296

 

 

 

 

 

 

wher
intert

randt

both

when
that ‘
elimiz
(A6)

repre:

to tra
delay-
USCI‘ E
If the

Calibr

Next,

TCSpQ]

 

 

 

where HS'( f ) represents the target scattering transfer function, HS; (f ) represents the
interaction between the target and the anechoic chamber, and N ”b ( f ) is once again
random noise.

Following the measurement process, the background term can be subtracted from

both the calibration and target measurements. This subtraction yields

Rem = S(f){H.c(f) + Hem} + N“”’(f) - N”(f) (AS)

R‘(f) = S(f){H.’(f) + 11.20)} + N“”(f) — m f) (A6)

where R C( f ) and RI (f ) represent the clutter free calibration and target terms. Notice
that the direct and indirect antenna coupling terms Ha( f ) and Hc( f ) have been

eliminated. For a high quality anechoic chamber the random noise terms in (A.5) and

 

(A.6) can be neglected. The interaction terms Hs:( f ) and Hs:( f ) in (A.5) and (A6)
represent unwanted signal in the spectral content of RC( f ) and R’ (f ).

One method of eliminating most of the interaction in the calibration spectrum is
to transform RC( f ) into the time domain and window out any interaction term that is
delayed beyond the end of the calibrator’s response. The step requires a great deal of
user experience, including knowledge of the target and chamber scattering characteristics.
1f the inverse Fourier transform of (A.5) is taken, then the time response rc(r) of the
calibration measurement is

rem = 9'"{Rc(f)} (M)
Next, an appropriate window function w(t) is applied to (A.7), yielding a windowed time

response rcw( r) as

297

 

 

 

 

If the

and t

For a

is tin

April:

when

be wr

Next,

A tin

transt

 

 

rcwm = w(t) rem (A-S)

If the time response r C( t) is written as the sum of a calibrator only time response rsc( t)

and the interaction term rs:( r) , then (A.8) can be written as

rcwm = w(t) {rim + rs:(t)} (A-9)

For a properly chosen window, the interaction term rs:( t) can be eliminated. If r:( t)
is time limited and not truncated by the time window, then (A.9) can be written as

rCW(t) : rsc(t) (A.10)

Applying a Fourier transform to (A.10) yields

RCWo‘) = Rjtf) = so) Hftf) (All)

where R” (f ) = .9 { rcw( t) ] . With this result, the system transfer function S(f) can

be written as

so) = w (A.12)
H. (f)
Next, (A.6) can be written as
Hsttf) + Hsitf) = ”1:8? (“3)

A time-domain representation of (A.13) can be obtained by applying an inverse Fourier

transform to the above equation to obtain

11:0) + 12.20) = alt’ggg} (AM)

298

 

 

 

 

 

To isolate the target scattering response, Hst( f ), the time-domain response in
(A. 14) can be time gated as before. hst( 2‘) must be time limited and not truncated by the
window function w(t). Also, the interaction term h;.( I) must be delayed in time beyond
the target response h;( I).

An example will be used to illustrate the features of the calibration process.
Figure A.3 shows the measured response of a 14 inch diameter calibration sphere
(background subtracted) over the frequency band from 2.0 GHz to 17.0 GHz with a step
size of .01 GHz. After time windowing out the interaction terms between the sphere and
the chamber, the windowed spectral response of the 14 inch sphere is shown in
Figure A.4. Dividing by a theoretically calculated response (see (A.12)) leads to the
calibration curve shown in Figure A.5.

As a check on the calibration process, a 3 inch metallic sphere was measured in
the anechoic chamber. The background-subtracted spectral return is shown in Figure A6.
A windowed response from the 3 inch sphere is shown in Figure A.7. This spectrum was
obtained by time gating out the interaction term in the time domain and then transforming
the time response back into the frequency domain. Finally, the response in Figure A.7
was divided by the calibration curve. The resulting response of the 3 inch sphere is

shown in Figure A.8 along with the theoretically calculated response. This figure shows

quite a good match between the two curves.

 

 

 

 

 

A.4 Windowing Functions

The use of different windowing functions has been employed in every phase of
this thesis. Therefore, some time will be devoted to this topic. Due to the broad scope
of this topic, only features pertinent to this thesis will be discussed. This section will
present a brief overview of windowing functions and their use. The most common
window functions used in this thesis will be discussed and several examples will be
presented in order to illustrate some of the most import window characteristics.

A window function can be used in a number of ways to modify a time or
frequency signal. For signal processing applications involving the transformation of data
from one domain to the other, the use of proper windowing becomes a necessity. For any
broadband signal which is frequency truncated, a transformation from one domain to the
other usually introduces unwanted oscillations in the transformed domain. If the signal
to be transformed is multiplied by an appropriate window function, then the number of
oscillations in the transformed domain may be reduced. To see this, consider a
frequency-domain signal F ( to) which is multiplied by a windowing function W( 00) to ‘

yield the frequency-domain response R(o>). This is given by

R(o>) = W0») F(w) (A-15)

A transformation to the time domain yields the following convolution

r(t) = w(t) *ftt) W6)

where r(t), w(t), and f(t) are the time-domain representations of R(o)), W((0), and. F(00),

 

 

 

 

 

 

 

respectively. Consider the case where the windowing function is a simple rectangular
window of unit height. The time domain representation of a rectangular window will be
a sinc function. With this function, the convolution in (A.16) may yield a highly
oscillatory response. To overcome this problem, a proper choice for the window function
must be selected. The selection of a "good" window function is highly dependent on the
characteristics of the signal. The type of signals derived from measurements or
theoretical calculations in this thesis are usually band-limited with discontinuities at the
endpoints. These discontinuities always cause problems after applying a forward or
reverse Fourier transform. The effects of the discontinuities can be reduced by smoothing
the data at the endpoints by using a carefiilly chosen window.

The ideas in the preceding paragraph can also be used in a slightly different
manner. Take, for example, the use of an ultra-Wideband (UWB) radar. A short pulse

signal is transmitted in order to detect or identify a potential threat. The transient

 

response of the target can be represented as the convolution of the time-domain input
signal and the target’s impulse response. In the frequency domain, the target frequency
response is formed by multiplying the target’s transfer function with the Fourier transform
of the input signal. In this case, the input signal (in the frequency domain) acts as a
window on the target’s transfer function. Therefore, a window function applied to a
calculated or measured transfer function represents the Fourier transform of a short pulse
that an UWB radar would be transmitting.

Two different window functions have been used in this thesis. They are the cosine

taper and the Gaussian modulated cosine (GMC). These functions were chosen to reduce

 

 

 

 

 

 

the effects of the oscillation phenomena and to represent the Fourier transforms of the
short pulses for studies in UWB radar phenomena.
The cosine taper function ( also known as the Tukey function ) can be represented

in the frequency domain as

 

 

 

. 2 1; f-FL t-l FL
srn (ETC AF ) f> 1: FH or f< T
Wtf) = (A.17)
_ F
1 f < I 1F” and f > ——L
t

where FH and FL represent the highest and lowest frequencies in the band-limited
spectrum, A is FH - FL, and t is a shape factor which must be an even integer. The
standard notation for the cosine taper is I/‘C cosine taper. Notice that large values of 1:
result in a window that is nearly rectangular.

The GMC window can be represented in the frequency domain as

_,, _ 2 _,, + 2
WGMC(f) = T{ e [(f f.)T] + e [(f fc)T] } (A.18)

where fC is the center of the window and T controls the width of the window. The time-

domain representation of this function can be written as

wGMCU) = cos(itfct)e”‘(‘/T)2 (A-19)

With the variables fC and T, the shape and position of this window can easily be changed.
This window is ideal for baseband spectral data (data down to DC). For this case, fc can
be set to 0 and the value of T varied. to slowly taper off the higher frequency data.

The best way to illustrate the characteristics of the cosine taper and GMC windows

is through several examples. These examples will be reflective of the type of windows

302

 

 

 

 

 

used in this thesis. Figure A.9 shows the cosine taper fimction applied to the frequency
band 1.0 GHz to 5.0 GHz for several different values of I. As can be seen, large values
of ‘C form a window that is more rectangular in shape. Small values of t flatten out the
window leading to a bell shaped curve. The time-domain representation of the curves in
Figure A9 are shown in Figure A. 10. Notice that the curve associated with a large value
of ‘C is highly oscillatory. On the other hand, using a small value of I will lead to a less
oscillatory function, but there will be a loss of spectral information and energy. For a
larger bandwidth signal, the transformed cosine taper window will have an associated
pulse that is narrower. However, the parameter 1: will control the number of oscillations
in the pulse. Figure A.11 shows several cosine taper functions for the frequency band
from 2.0 GHz to 17.0 GHz. Figure A.12 shows the time-domain representations
associated with the transforms of the cosine taper windows. In this figure, the peaks are
much narrower than in Figure A.10, but the oscillations are very similar.

The GMC window is a function of the parameters fc and T. In this thesis, the
selected value of fC depended on whether the frequency band was baseband or non—
baseband. For a baseband signal, the value of fC was set to 0. For the non-baseband
signal, fC was set centered between the high and low frequency endpoints. The
characteristics of this window can be compared to the cosine taper window using the
frequency band 1.0 to 5.0 GHz. Figure A.13 shows several different GMC windows as
a function of the window shape parameter T. More information in the frequency
spectrum is used when using smaller values of T. The corresponding time-domain

representations of the curves in Figure A.13 are shown in Figure A.13. Here, small

 

 

 

 

values of T are associated with narrow pulses. As compared to the cosine taper window,
the generated pulses can still be oscillatory, and spectral information will be lost for a
window that is too narrow.

For a baseband signal, the GMC window parameter fc is set to zero. This window
was used extensively in the cepstral analysis section (Chapter 5 and 6). A good example
of this window is shown in Figure A.15 for different values of the shape parameter T.
Here, the baseband. ranges from 0 GHz to 5 GHz. All of these windows keep the low-
frequency Spectral information but attenuate the higher components. The associated time-
dornain pulses for the windows shown in Figure A.15 are shown in Figure A.16. A nice
feature of these pulses is that all are Gaussian in shape and have no oscillations. The fact
that there are no oscillations makes this window very attractive for baseband signal
processing.

The above examples are very typical of the window functions used in this thesis,
and the frequency bands used in the examples are representative of those used in the
preceding chapters. Parameters associated with each window are explained in the main

body of the thesis.

304

 

 

 

 

 

 

 

 

 

Trans Horn

Rec Horn

 

 

 

 

 

 

I

Ampllfler :l' Ml “59203 .tl— ‘rree17::j,_i_rur
[in .._ was “i\\

 

 

 

 

 

 

 

 

 

 

 

 

 

i

 

Network Analyzer

 

Figure A.1 Anechoic chamber using a frequency-domain measurement system at
Michigan State University.

 

 

 

E (f) Transmitted Field

: : t Antenna Scatterer Mutual Chamber

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

coupling Interactions Clutter
Transmitting
Antenna
Hart) Hs(f) |H,,(f> Hc(f)
R (r) r” + +
R a ' Hr( f) Received Field
Receiver @ Ziggy?
Noise N (f)

Figure A.2 Measurement system block diagram for scattering measurement analysis.
Taken from Ross, [6].

306

 

 

 

3.0E—OO3

2.5E-003

2.0E—OO3

 

1.5E—003

1.0E—OO3

Relative Magnitude

5.0E—OO4

 

lllllllliIllllllllIllillllllIlllllllllIlllllllllIlllllllll

 

 

illlllliliij
O'OEIILOOOooHIIII”bio“”unioioi'HHH150 20.0

Frequency (GHZ)

Figure A.3 Measured frequency response of 14 inch diameter metal calibration sphere.

307

2.0E—OO3

 

 

 

 

 

 

 

 

1.55-003 5
<0 _
‘0 _
3 .—
7‘1’ I
C _
03 _
O _
2 105—003 -—
G) _.
.2 I
+J .-
g _
(D Z
0:5.OE—OO4 -—
O°OE+OOO IIIIIIIIrIrIIIIIIIIIIIIIIIIIIIIIIIIIIIj—I
0.0 5.0 10.0 15.0 20.0

Frequency (GHz)

Figure A.4 Spectral response of 14 inch diameter calibration sphere after time domain
gating of chamber wall reflections.

308

8.0E—002

6.0E—002

 

 

gnfiude

4.0E—002

 

 

 

 

Relative Mo

2.0E—002

lllllllllllllllllllllllllllllllllllllll

 

 

lllllllllll‘l
O.OE+OOOO.OFII[IITIESOIOTIIIIIII1IO[.OIIIlllll15.0 200

Frequency (GHZ)

Figure A.5 Transfer function for the frequency-domain system using the 14 inch
diameter sphere as a system cahbrator.

309

, 777 -w». x-.r._-.:-(a—;1:w»_ca ”<4

6.0E—OO4

lllllll

5.0E-OO4

4.0E—OO4

3.0E—OO4

2.0E—OO4

Relative Magnitude

1.0E—OO4

lllllilllllllllillIllllllillllllllliillllllllilll

 

 

O-OE+OOOOOIlIIlIlIIS'IIIIIIIIIIIIIIlllll1l5|dllllllj36|.o

1 . ~
Frequency (GHZ)

Figure A.6 Measured frequency response of a 3 inch diameter metallic sphere.

310

5.0E—OO4

4.0E—OO4

3.0E—OO4

 

 

 

2.0E—OO4

 

Relative Magnitude

1.0E—OO4

llllllllillllllllllllllIllllilllllllllilllllllll

 

 

[lllllllll]
O°OE+OOOool I I I I I I I [SFWT I I I [laid [TNT I '1'5.o 20.0

Frequency (GHZ)

Figure A.7 Spectral response of a 3 inch diameter sphere after time-domain gating of
the chamber wall reflections.

311

 

8.0E—003

 

 

 

 

 

7.05—003 E {i
5 l
: ' l
_ l I\
<1) 6.0E—OO3 E 1 {1 I
U : | I i ’i
" l
3 I i l I' I! l‘ “ ,\
'_ : I i \ I \ I r /\
C _ : I I l \ I I 1 \~/ \I I
35.05 003 : .‘ l‘ l
2 E i “I, i
(1) 4.05—003 E g l J
2 Z I i
+4 : I
2 E l
(D 3.05—003 5 i),
D: 5 .
: ----- Theoretical Response
3 —- Measured Response
2.0E—003 E
1°OE_003-IllIlliilrllllllllllllllllllellll[Ill—Ii
0.0 5.0 10.0 15.0 20.0

Frequency (GHZ)

Figure A.8 Comparison between theory and experiment for a 3 inch diameter metallic
sphere. The measured data has had the system transfer function removed.

312

 

1.0

.0
oo

lllllllilllllllllilllllllllilllllllllillllllllll

ive)

Amplitude (Relat
O .0
~P~ a)

.0
m

Figure A.9

 

 

 

 

 

 

 

\‘ “
\“ ‘
\‘ l l
‘ l
‘i‘ l l
‘ \
\\ “
\‘ ‘l
\
’r = 8 \ “
‘ l
——7' = 5 ‘\ l
____7- = 4 \\ ‘\‘
"""" ’7' = 2 \\ \\
\\ \
\\ \\
‘ \
\\ \\
\\\\
\\\
\\¥
llillrjl[IIITTIITTIIIIIIIIlIllilllTllll—j
1.0 2.0 3.0 4.0 5.0

Frequency (GHZ)

Cosine taper weighting curves as a function of the window shape

parameter I.

313

 

 

llllllllli

.0
01

 

 

 

 

l
.0
01

 

Amplitude (Relative)
C)
O

LlllllllllJllllllllilllljlllll

 

 

-1.0 lllllllllllllll'lllllll'lllll‘l—l—I

0.0 2.0 4.0 6.0 8.0
Time (nsec)

.0
U1

Liiiiiliiliiiiiiiiil

 

 

.0
o

 

 

Amplitude (Relative)

O
01
iiiiiiiLJiiiiLiiii

 

 

—1-O TrTTITIITTlTIlIIITTTIIIITIIIIIIBIO
0

0.0 2.0 4.0 50
Time (nsec)

Amplitude (Relative)

1.0

.0
01

lllllllllllllll!lllllllllllllllllll

Amplitude (Relative)

.0
o

l
.0
01

1111]

“all

 

 

 

 

 

1.0 1
0.0

TfrTIIIIUIIIIIITIITTIIIIIIIIII]

2.0 4.0
Time (nsec)

 

 

.0
o

01
I11 lllllJllllllllllllll

1

|
P
01

 

iillllllilii

—1.0
0.0

 

TllllllllilllIITTIIIIIITTITTII]

. 8.0
Time (nsec)

Figure A.10 Cosine taper weighting functions transformed to the time domain. Original
window bandwidth from 1.0 to 5.0 GHz.

314

 

 

 

 

 

 

 

1.0

 

 

 

 

 

 

 

1’ ‘\ \
’1’ \ \
,’ \\ \
,’ ‘\\ \
I’ ‘\ \
I, “ \
II, “ \
O . 8 I’ ‘\
/"'\ ,” \
(D ’ \\
.2 I
+2 ‘\
2 O . 6 \
<1)
D: ‘I
V \
<1)
U ‘I
:504 \
3': ‘I
Q- 'r = 8
E - - - 7' = 6
<1: - - - - 7' = 4
02 ________ 7.: 2
O . O l I l I I f I I I
2 . O . 8 . O 1 1 . O 1 4. 0

Frequency (GHZ)

Figure A.“ Cosine taper weighting curves as a function of the window shape
parameter r.

315

1.0 1.0

.0
01
.0
01

 

 

 

 

 

 

Amplitude (Relative)
O
O
Amplitude (Relative)
O
O

lllllllllllllllllIlllllllllllllllllllli

LiiiiiiiiliiiiiIiiiliiiiiiiiiliiiliiiii|

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

—0.5 —o.5
— l 1TT —1.0IIII[ll[IlIlelllll'IIlrIITll[llIIIIIIII
. [[llleTT IIITrIlIl leTIIIUI II r! j—I
1O0.0 0'5 1b 1.'5 2.0 0.0 0.5 . 1.0 1.5 2.0
Time (nsec) Time (nsec)
1.0:
: 1.0:
0.55 E
3 ‘ 5:
.2 : AO- .
.l -I
E : g 3
at 2 :6 5
v00 & .4
a) : vo0_
U _‘ -1
0)
3 : U :
Z _4 3 _
Q. _ :.:: _
E- d a :
.1 <_0u5_
_ 4 :
_ 7' : _
3 - ’r = 2
“1.0 IIIIrIIII IIIIIIIIIll""l"'l""""'l : IIIIIIII
0.0 [5 1.0 1.5 ' —1,00011IIIIII6|IIIIIIIII[IIIIIIIIIII 2lo
Time (nsec) . Time (nsec)

Figure A.12 Cosine taper weighting functions transformed to the time domain. Original
window bandwidth from 2.0 to 17.0 GHz.

316

 

 

 

 

IO
I5
n
\I
1-
‘l
\\.I-
_
\ \Elv
\\ \~~.IO
‘ .
\\ \\\\\\ l4
\\ \\\ \\\\ T )
\\\ \\\ \\\\\ I. Z
\\\\\|‘\I‘|\I|Hln \\\\\\\\\ H H
\\‘||\II\II..HI. ||||||||||| In G
\\\H|\ ||||||| .l. /l\
\‘M‘ ||||||
“III I
I0 VJ
ill! I o
/”I’I'I' I3 C
III/II IIIII n
III“. lllllll l e
ll/IIII lllllllll I. U
z III/L ......... - q
H II III, IIIIII l e
I .I III
G / / - r
02 I/ l/ 1/ I F
68 . / x
O o 011! / cl/III
. / C10.
3 __ __ __ __ //,H-2
’l
= TTTT /7
_ u m T
C . T
f _ _ n .I
_ _ u I
r
_______
_ _______________________________________________.___ AU.
6 4i
. s 4. 3 2 I 0
O O . .
O O

O O. 0.
$523: wastage

Figure A.13 Gaussian modulated cosine weighting curves as a function of the window

shape parameter T. Center frequency fc = 3.0 GHz.

317

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1'0: 1.0:
0-5: 0.51
”I? 3 A :
.2 I “>’ :
E 3 *5 :
51:) I 6 :
v 0.0 E“, 0.0 _'
<1) : a, :
U - '0 I
3 a 3 ..
.4: - .4: -
a : "a :
E_ _' E _'
< 0.5 _ (‘05-
; fc=3.0 GHz,T=.6 . fc=3OGHz,T=.8
—i.O I I f I I I T l I I I I I I r I —1.0 I I r T I I i F I 1 I I I I
0.0 1.0 2.0 3.0 4.0 0.0 1.0 . 2.0 3.0 4.0
Time (nsec) TIme (nsec)
1.0 :
: 1.0:
0.5 3 :
/‘\ : 0.5“
.02) : fa :
4.; _ > -
2 — -.: —
a) j 2 2
33 00 ‘ (it) ‘
' ~ v 0.0 —
s : «I I
‘ ‘o 3
.4? i 3 -
El ‘ = I
. a ‘
<§—o.5— E_05;
. <1: ' -
’ I. = 3.0 GHz, T = 1.0 g I. = 3.0 GHz, T = 1.2
"i.0 fir I I I I I I I I I I —1.0 I T ' ' i l l ' I I l ' I
0.0 “O 2.'0 3b 4-0 0.0 1.'0 2.0 3.0 4.0

Time (nsec) Time (nsec)

Figure A.14 Gaussian modulated cosine weighting functions transformed to the time

domain. Original window centered at fc = 3.0 GHz.

318

 

 

 

b
l

   
           

 

 

 

0.8 g
A E
g 3 I. = 0.0 GHZ
.4: Z ‘I l \
2 0.6 : ‘I‘ \\ \ _— T = .25
(D : i‘ \ \ — _ — T = .35
Q“, : I X \ ---— T = .50
: ‘I‘ \ \ -------- T = .75
'00) 3 i \‘ \
3 0.4 t i“ \\ \
it.” “ I \ \
a I i, \\ \
Z \
_ \‘ \ \
0.2 - I. \
: \‘ \
__ \ \
_ ‘\ \
_ \ \
_ \\ \\
0.0 1 i ~~~~ ~
0.0 1.0 2 O 3.0 4.0 5.0

Frequency (GHZ)

Figure A.15 Gaussian modulated cosine weighting as a function of window shape
parameter T. Center frequency fc = 0.0 GHz.

319

 

 

 

l I

 

 

 

 

i\>

 

 

 

 

 

 

1.0 E
1.? 5
> 0.8 E
'43 :
.9 :
<1) :
DC 2
V0.6 ':
<1) :
"O :
:5
:104 3
E ' :
E 1
<1: :
0.2 E
E ’I’l”///
O O T T‘“’:I ’T 1
0.0 1 O 3.0

. 2
Tune (nsec)

Figure A.16 Gaussian modulated cosine weighting functions transformed to the time
domain. Original windows centered at fc = 0.0 GHz.

320

 

 

 

 

 

LIST OF REFERENCES

 

 

 

 

 

 

 

[1]

[21

[31

[4]

[61

LIST OF REFERENCES

C.Phillips, P. Johnson, K. Garner, G. Smith, A. Shek, R.C. Chou, and S. Leong,
"Ultra-high-resolution radar development and test," in: Ultra-Wideband, Short-
Pulse Electromagnetics 2, Lawrence Carin and LeOpold B. Felsen, led., Plenum
Press, New York, 1999.

MP. Fargues, and WA. Brooks, "Comparative study of time-frequency and time-
scale transforms for ultra-Wideband radar transient signal detection," IEE Proa-
Radar,Sonar Navig, Vol. 142, No.5, pp.236-242, October 1995.

E]. Rothwell, K.M. Chen, D.P. Nyquist, A. Norman, G. Wallinga, and Y. Dai,
"Target detection and identification using a stepped-frequency ultra~wideband
radar," SPIE, vol 2845, pp. 26-37, Denver, Colorado, Aug. 5-6, 1996.

C. Baum, E.J. Rothwell, K.M. Chen, and DP. Nyquist, "The singularity expansion
method and its application to target identification," IEEE Proa, vol. 79, no. 10,
p. 1481-1492, Oct. 1991.

E]. Rothwell, "Radar Target Discrimination Using the Extinction-Pulse
Technique," Ph.D. Thesis, Department of Electrical Engineering, Michigan State
University, 1985.

1E. Ross, 111, "Application of Transient Electromagnetic Fields to Radar Target
Discrimination," Ph.D. Thesis, Department of Electrical Engineering, Michigan
State University, 1992.

P. Ilavarasn, "Automated Radar Target Discrimination Using E~pulse and S-pulse
Techniques", Ph.D. Thesis, Department of Electrical Engineering, Michigan State
University, 1992.

Q. Li, "Radar Target Discrimination Using Early-Time Responses", Ph. D. Thesis,
Department of Electrical Engineering, Michigan State University, 1995.

P. Ilavarasan, J.E. Ross, E.J. Rothwell, K.M. Chen, and DP. Nyquist,

"Performance of an automated radar target discrimination scheme using E pulse
and S pulses," IEEE Antennas Propagat., vol 41, no. 5, pp. 582-588, May 1993.

321

 

 

 

 

 

 

 

 

[10]

[11]

[12]

[13]

[141

[151

[16]

[21]

RR. Lentz, "A numerical study of electromagnetic scattering from ocean-like
surfaces," Radio Science, vol. 9, no. 12, pp. 1139-1146, Dec. 1974.

CL. Rino, T.L. Crystal, A.K. Koide, H.D. Ngo, H. Guthart, "Numerical simulation
of backscatter from linear and nonlinear ocean surface realizations," Radio
Science, Vol. 26, no. 1, pp. 51-71, Feb. 1991.

A. Norman, "Transient Scattering of Electromagnetic Waves in an Ocean
Environment," Ph.D. Thesis, Dept. of Electrical Engineering, Michigan State
University, 1996.

R. Chen and JG West, "Analysis of Scattering from Rough Surfaces at Large
Incidence Angles Using a Periodic-Surface Moment Method," IEEE Trans.
Geoscience and Remote Sensing, Vol. 33, No. 5, pp. 1206-1213, Sept. 1995.

RF. Harrington, Time~Harmonic Electromagnetic Fields, McGraw-Hill, New
York, 1961

 

S. Singh, W.F. Richards, J .R. Zinecker, D.R. Wilton, "Accelerating the
convergence of series representing the free space periodic green’s function," IEEE
Trans. Antennas Prop, vol. 38, no 12, pp. 1958-1962, Dec. 1990.

LS. Gradshteyn, and I.M. Ryzhik, Table of IntegralLSeries and Products, New
York, Academic Press, 1980.

 

M. Abramowitz, and LA. Stegun editors, Handbook of Mathematical Functions,
Dover Publications Inc., New York, 1965.

 

C.A. Balanis, Advanced EngineeringElectromzngnetics, John Wiley & Sons, Inc,
1989.

Blair Kinsman, Wind WaveS'LTheir Generation and Propagation on the Ocean
Surface Chap. 8, Prentice-Hall, Englewood Cliffs, NJ. 1965.

 

E. Rothwell, D.P. Nyquist, KM. Chen, and B. Drachman, "Radar target
discrimination using the extinction-pulse technique," IEEE Trans. Antennas
Propagat, vol. AP-33, no. 9, pp. 929-936, Sept. 1985.

K.M. Chen, D.P. Nyquist, E.J. Rothwell, L.L. Webb, and B. Drachman, "Radar
target discrimination by convolution of radar return with extinction-pulses and
single-mode extraction signals," IEEE Trans. Antennas Propagat, vol. AP—34, no.
7, pp. 896-904, July 1986.

K.M. Chen, E. Rothwell, D.P. Nyquist, J. Ross, P. Ilavarasan, R. Bebermeyer, Q.

322

 

 

 

 

 

 

[23]

[24]

[25]

[291

[301

[311

[32]

Li, C.Y. Tsai, and A. Norman, "Radar identification and detection using ultra-
wideband/short—pulse radars," Ultra- Wideband, Short-Pulse Electromagnetics 2,
Edited by L. Carin and LB. Felsen, Plenum Press, New York pp.535-542, 1995.

E]. Rothwell, K.M. Chen, D.P. Nyquist, A. Norman, G. Wallinga, and Y. Dai,
"Target detection and imaging using a stepped-frequency ultra-Wideband radar,"
Short-Pulse Electromagnetics 3. Albuquerque, NM, May 27-31, 1996.

E.Rothwell, K.M.Chen, D.P. Nyquist, P. Ilavarasan, J .E. Ross, R. Bebermeyer, and
Q. Li, "A general E-pulse scheme arising from the dual early-time/late-time

behavior of radar scatterers," IEEE Trans. Antennas Prop, vol. 42, no.9, pp. 1336-
1341, Sept. 1994.

K.R. Umashankar, S. Nimmagadda, and A. Taflove, "Numerical analysis of
electromanetic scattering by electrically large objects using spatial decomposition
technique," IEEE Trans. Antennas Prop., Vol. 40, No. 8, pp.867-877, August
1992.

DE. Iverson, "Coherent processing of ultra-Wideband radar signals," IEE Proc-
Radar, Sonar Navig, vol. 141, no. 3, pp. 171-179, June 1994.

 

D.E. Goldbert, Genetic Algorithms, New York: Addison~Wesley, 1989.

FM. Tesche, "On the use of the Hilbert transform for processing measured CW
data", IEEE Trans. Electromagnetic Compatibility, vol. 34, no. 3, pp. 259-266,
Aug. 1992.

A.V. Oppenheim, and R.W. Schafer, Digital Signal Processing. New York:
Prentice-Hall, 1975.

T. Ulrych, "Application of homomorphic deconvolution to seismology,"
Geophysics, vol. 36, no. 4, pp. 650-660, Aug. 1971.

W.D. Blair, J .E. Conte, and TR. Rice, "An Instructive Example of Homomorphic
Signal Processing," IEEE Trans. Educ. , Vol. 38, No. 3, pp. 211—218, Aug. 1995.

D.G. Childers, D.P. Skinner, and RC. Kemerait, "Cepstrum: A Guide to
Processing," Proceedings of the IEEE, Vol. 65, No. 10, pp. 1428-1443, Oct. 1977.

Al. Berkhout, "On the Minimum Phase Criterion of Sampled Signals," IEEE
Trans. Geosci. Elect, Vol. GE-l 1, No. 4, pp. 186-198, Oct. 1973.

P. Henrici, Applied and Computational Complex Analysis, Vol. 1, John Wiley &
Sons, Inc. 1974.

323

 

 

 

 

 

 

[351

[361

[37]

[381

[391

[401

[41]

[421

[43]

[441

T.F. Quatieri, and J .M. Tribolet, "Computation of the real cepstrum and minimum
phase," in Programs for Digital Signal Processing. New York: IEEE Press,
1979.

J. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University Press,
Oxford, England, 1965.

KB. Atkinson, An Introduction to Numerical Analysis, John Wiley & Sons, Inc.,
1978.

E.J. Rothwell, K.M. Chen, D.P. Nyquist, and W. Sun, "Frequency domain E-pulse
synthesis and target discrimination," IEEE Trans. Antennas Propagat, vol. AP—35,
no. 4, pp. 426-434, Apr. 1987.

E.J. Rothwell, and KM. Chen, "A hybrid E-pulse/least squares technique for
natural resonance extraction," IEEE Proc., vol. 76, no. 3, pp. 296-298, March
1988.

P. Ilavarasan, E.J. Rothwell, R. Bebermeyer, K.M. Chen, and DP. Nyquist,
"Natural Resonance Extraction from Multiple Data Sets Using a Genetic

Algorithm," 1994 International Symposium Digest Antennas and Propagation, Vol.
1, pp. 576-579.

P. Ilavarasan, E.J. Rothwell, K.M. Chen, and DP. Nyquist, "Natural resonance

extraction from multiple data sets using physical constraints," Radio Science, vol.
29, no. 1, pp. 1-7, Jan-Feb. 1994.

Q. Li, E.J. Rothwell, K.M. Chen, and DP. Nyquist, "Scattering center analysis of
radar targets using fitting scheme and genetic algorithm," IEEE Trans. Antennas
and Propagation, Vol. 44, No. 2, pp. 198-206, Feb. 1996.

M.A. Morgan and N]. Walsh, "Ultra-wide-band transient electromagnetic
scattering laboratory," IEEE AP—S, Vol. AP-39, No. 8, pp.1230-1234, August
1991.

M .A. Morgan, "Ultra-Wideband impulse scattering measurements," IEEE AP—S,
Vol. 42, No. 6, pp.840—846, June 1994.

324

 

 

' . HICHIGQN STQTE UNIV. LIBRRRIES
lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
. 312930156399