(“ a

LIBRARY f
Michigan State
I University

2 . j

 

 

fl

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

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU Is An Affirmative Action/Equal Opportunity Institution
amount

 

 

 

 

 

 

APPLICATION OF TRANSIENT ELECTROMAGNETIC FIELDS
TO RADAR TARGET DISCRINIINATION

By

John Edwin Ross, III

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

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1992

c. 0.x ,4

I

IJI

ABSTRACT

APPLICATION OF TRANSIENT ELECTROMAGNETIC FIELDS
TO RADAR TARGET DISCRIMINATION

By

John Edwin Ross, III

This dissertation addresses several topics associated with non-cooperative radar
target discrimination using the E-pulse and S-pulse methods. An automated discrimination
scheme based on the E-pulse and S-pulse is presented. The performance of this scheme
is evaluated for the case of wire targets in the presence of random noise. The
applicability of the automated discrimination scheme in a multiple target environment is
evaluated.

In a multiple target environment interactions between targets must be considered.
The behavior of the Singularity Expansion Method (SEM) modes of a system of targets
are investigated. Approximate methods for calculation of SEM system mode frequencies
are presented. The transient response of a system of wires is constructed using a Class
I SEM expansion and compared to results obtained using Fourier inversion of moment
method calculations. It is seen that SEM system modes are non-physical prior to the
onset of system late-time. The automated discrimination procedure is evaluated for a
system of two parallel wires and the results interpreted using the correct physical
meaning of the SEM system modes.

Accurate transient measurements are essential for implementation of an automated
discrimination scheme for complicated targets. Transient measurements can be made

directly in the time domain or synthesized from swept frequency measurements using the

inverse Fourier transform. A calibration method capable of removing systematic errors
inherent in the measurement system and background clutter is presented. Verification is
accomplished by measurement of canonical targets such as spheres and wires as well as
a wire stick aircraft model.

The calibration scheme is used for experimental study of several topics. The
response of a system of parallel wires is measured and compared to theoretical
predictions. The amount of late-time energy available for discrimination is evaluated for
realistic aircraft models. Discrimination of realistic aircraft models is demonstrated using
experimental data. It is experimentally demonstrated that coupling between aircraft

adversely affects discrimination.

Copyright by
John Edwin Ross, III

1992

In Loving Memory of My Grandmother
Dorothea R. Stine

October 14, 1901 - May 8, 1992

 

or.
u
n
.
‘5'.
v"
‘~.
~
“ 5
r
. .4
4
‘e
s
I
e.‘
n
a
u
u
b
' .
'5
u
‘ Q.
~“

ACKNOWLEDGMENTS

I would like to express my sincerest appreciation to Dr. D.P. Nyquist. Your
encouragement to continue my graduate studies at Michigan State University and your
teaching and guidance during my research will long be remembered. I am also indebted
to Dr. K.M. Chen. Thank you for affording me the opportunity to work in an
outstanding environment and for your deep concern and guidance. To Dr. Byron
Drachman, thank you for participating as a member of my guidance committee. I am
especially grateful to Dr. E]. Rothwell, the finest thesis advisor one can imagine. Your
enthusiasm, encouragement and expert technical advice will not be forgotten.

I also want to thank my fellow graduate student Ponnialr Ilavarasan whose bright
smile and persona has brightened many cloudy Michigan days. It has been a great
pleasure to work and learn with you.

To my mother and father I want to express my appreciation for their constant
encouragement and for always being there for me.

Finally, to my wife Linda, no amount of thanks for your constant love and

encouragement would be too much. It’s over, let’s get on with life.

vi

Table of Contents

Table of Contents ....................................... vii
List of Tables .......................................... x
List of Figures ......................................... xii
Chapter 1
Introduction ...................................... 1
Chapter 2
Target Discrimination Using E~Pulses and S-Pulses .............. 6
2.1 Introduction ................................... 6
2.2 Review of E—Pulse and S-Pulse Discrimination .............. 7
2.3 Quantification and Automation of Discrimination ............ 9
2.4 Performance of Automated Discrimination Scheme ........... 13
2.5 Multiple Target Discrimination ....................... 30
Chapter 3
Transient Coupling .................................. 47
3.1 Introduction ................................... 47
3.2 General SEM Formulation .......................... 50
3.3 Moment Method Solution ........................... 55
3.4 Perturbational Transcendental Solution ................... 57
3.5 Approximate Green’s Function Solution .................. 58
3.6 SEM Modes of a Single Thin Wire ..................... 61
3.7 SEM Modes of Two Identical Parallel Thin Wires ........... 75
3.8 Evaluation of Approximate Methods .................... 93
3.9 Scattered Fields of Two Identical Parallel Wires ............. 101
3.10 Scattered Fields of Two Parallel Wires of Different Length ..... 115
3.11 Summary .................................... 115

vii

Chapter 4

Discrimination of Coupled Wires ......................... 119
4.1 Introduction ................................... 119
4.2 Cross Range Separation ............................ 119
4.3 Down Range Separation ............................ 125
Chapter 5
Transient Measurements ............................... 129
5.1 Introduction ................................... 129
5.2 Time Domain Systems ............................. 131
5.3 Frequency Domain Systems ......................... 136
5.4 Calibration Procedure ............................. 141
5.5 Weighting Functions .............................. 148
5.6 Calibration Targets ............................... 149
5.7 Calibration Examples ............................. 166
5.8 Measurements with Canonical Targets using Sphere Calibrator . . . . 180
5.9 Measurement of Canonical Targets using Wire Calibrator ....... 189
5.10 High-Band Measurement Validation .................... 196
Chapter 6
Experimental Results ................................. 200
6.1 Introduction ................................... 200
6.2 Parallel Wire Systems ............................. 201
6.3 Wire Stick Airplane Model .......................... 225
6.4 Early-Time vs Late-time Energy ....................... 238
6.5 Effects of Coupling on Aircraft Model Discrimination ......... 244
6.6 Early-time Scattering from Scale Model Aircraft ............. 251
Chapter 7
Conclusions ...................................... 257
Appendix A
Frequency Domain Moment Method Formulation ............... 262
A.1 Piecewise Sinusoidal Basis Functions ................... 262
A.2 Finite Conductivity Effects .......................... 265

viii

Appendix B

SEM Formulations .................................. 268

B.l Piecewise Sinusoidal Basis Functions .................... 268

B.2 Hallen Equation for Array of Parallel Wires ............... 268

B.3 Approximate Green’s Function Solution with Sinusoidal Current . . . 271

B.4 Coupling Coefficient and Normalization Constant Calculations . . . . 274
Appendix C

Experimental Apparatus ............................... 278
Bibliography .......................................... 286

ix

Table 3.1

Table 3.2

Table 3.3

Table 3.4

Table 3.5

Table 3.6

Table 3.7

Table 6.1
Table 6.2
Table 6.3
Table 6.4

Table 6.5

Table 6.6

Table 6.7

Table 6.8

List of Tables

Comparison of the natural frequencies of a thin wire with l/a =—

200 calculated using various moment method formulations. ..... 67
SEM parameters for d/l = 1.00. ..................... 103
SEM coupling coefficients for down range separation distance
d/1=1.00. ................................... 104
Coupling coefficients and modal fields for cross range separation
d/l = 1.0 .................................... 108
SEM frequencies and normalization constants for (M = 2.00. . . . 110
Coupling coefficients and modal fields for down range separation
d/l = 2.00. .................................. 111
Coupling coefficients and modal fields for cross range separation
d/l = 2.0 .................................... 113
Rue values for model aircraft. ....................... 244

Natural frequencies of A10 model extracted using CET method. . . 245
Natural frequencies of F15 model extracted using CET method. . . 246
Natural frequencies of F16 model extracted using CET method. . . 247

SDR values computed for single targets using S-pulses for F16 and
A10. ...................................... 248

SDR values computed for coupled A10 models using S-pulses for

F16 and A10. ................................ 249
SDR values computed for single targets using S-pulses for F 15 and
A10. ...................................... 250
SDR values computed for coupled A10 models using S-pulses for
F15 and A10. ................................ 251

Table C.l
Table C.2
Table C.3

Table C.4

Equipment used for frequency domain scattering measurements. . . 278
Network analyzer parameters for low band measurements. ..... 279
Network analyzer parameters for high band measurements. ..... 279

Equipment used for time domain scattering measurements. ..... 279

xi

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

List of Figures

Noise free back-scattering response of 1.00 m (l/a = 800) oriented
at broadside. .................................

Back-scattering response of 1.00 m wire (I/a = 800) oriented at
broadside with SNR of 30 dB. ......................

Back-scattering response of 1.00 m wire (l/a = 800) oriented at
broadside with SNR of 5 dB. .......................

EDR values computed for response of 1.00 m wire (l/a = 800)
oriented at broadside. ............................

EDR values computed for response of 1.00 m wire (l/a = 800)
oriented 45 degrees from broadside. ...................

Minimum EDR values computed for 0.90 m and l. 10 m wire E-
pulses for response of 1.00 m wire oriented at broadside. ......

Minimum EDR values computed for 0.90 m and 1.10 m wire E-
pulses for response from 1.00 m wire oriented 45 degrees from
broadside. ...................................

SDR values computed for first mode S-Pulses for response of 1.00
m wire (l/a = 800) oriented at broadside. ...............

SDR values computed for second mode S-pulse for response from
1.00 m wire (l/a = 800) oriented at broadside. ............

SDR values computed for second mode S-pulse for response of
1.00 m wire (l/a = 800) oriented 45 degrees from broadside. . . .

Minimum SDR values computed for S-pulses for 0.90 m and l. 10
m wires using response of 1.00 m wire oriented at broadside. . . .

Minimum SDR values computed for second mode S-pulses for

0.90 m and 1.10 m wires using response from a 1.00 m wire
oriented 45 degrees from broadside. ...................

xii

19

20

22

23

25

27

28

29

Figure 2.13

Figure 2.14

Figure 2.15

Figure 2.16

Figure 2.17

I:igure 2.18

Figure 2.19

Figure 2.20

Figure 2.21

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Geometry for two target discrimination problem. ........... 31

Waveform produced by superposition of response from wires of
length 0.90 m and 1.20 m with l/a == 200. The response of the
1.20 m wire is delayed 25 us from the start of the 0.90 m wire. . . 34

Waveform of Figure 2.14 after convolution with E—pulse for 0.90
m wire. .................................... 36

Waveform of Figure 2.15 after convolution with E—pulse for 1.20
m wire. .................................... 37

Waveform produced by superposition of responses from wires of
length 0.90 m and 1.20 m with l/a = 200. The response of the
1.20 m wire is co-temporal with the response of the 0.90 m
wire. ...................................... 39

Waveform of Figure 2.17 after convolution with the E-pulse for
0.90 m wire. ................................ 40

Waveform of Figure 2.18 after convolution with the E-pulse for
1.20 m wire. ................................. 41

Minimum EDR vs time delay when E—pulse method is applied to
a response consisting of two 1.0 m wires. ............... 43

Minimum SDR vs time delay when S-pulse method is applied to
a waveform containing responses from two 1.0 m wires separated

by a time delay. ............................... 44
System of P perfectly conducting scatterers. .............. 51
Layering of SEM mode frequencies for thin wire (l/a = 200). . . . 63
Convergence of moment method solutions for damping

coefficient. .................................. 65
Convergence of moment method solutions for radian frequency. . . 66

Magnitude of current distribution for 3“ mode of wire with l/a =

200. ...................................... 69
Phase of current distribution for s“ mode of wire with l/a =
200. ...................................... 70

xiii

Figure 3.7

Figure 3.8

Figure 3.9

Figure 3.10

Figure 3.11

Figure 3.12

Figure 3.13

Figure 3.14

Figure 3.15

Figure 3.16

Figure 3.17

Figure 3.18

Figure 3.19

Figure 3.20

Figure 3.21

Magnitude of current distribution for 31.2 mode of wire with l/a =

200. ......................................

Phase of current distribution for s“ mode of wire with l/a =

200. ......................................

Magnitude of current distribution for SL3 mode of wire with [/0 =

200. ......................................

Phase of current distribution for 31,, mode of wire with I/a =

200. ......................................

Geometry of two parallel thin wire scatterers. .............

Trajectory of lowest order antisymmetric mode of two identical
parallel wires (l/a = 200) vs separation distance. ...........

Trajectory of first symmetric mode of two identical parallel wires
(l/a = 200) vs separation distance .....................

Trajectory of antisymmetric mode from second pole layer for two
identical parallel wires as a function of separation distance ......

Damping coefficient of lowest order principle modes of two
identical parallel wires (I/a = 200) vs separation distance. .....

Radian frequency of lowest order principle modes of two identical
parallel wires (l/a = 200) vs separation distance. ...........

Trajectory of lowest order antisymmetric mode of two identical
parallel wires (l/a=400) vs separation distance. ............

Magnitude of current distribution for lowest order antisymmetric
mode of two identical parallel wires (l/a = 200) vs separation
distance. ....................................

Phase of current distribution for lowest order antisymmetric mode
of two parallel wires (l/a = 200) vs separation distance. ......

Magnitude of current distribution for lowest order symmetric mode
of two identical (l/a = 200) parallel wires vs separation
distance. ....................................

Phase of current distribution for lowest order symmetric mode of
two identical (l/a = 200) parallel wires vs separation distance.

xiv

76

78

79

8O

82

83

85

86

89

90

Figure 3.22

Figure 3.23

Figure 3.24

Figure 3.25

Figure 3.26

Figure 3.27

Figure 3.28

Figure 3.29

Figure 3.30

Figure 3.31

Figure 3.32

Figure 3.33

Figure 3.34

Magnitude of current distribution for second layer antisymmetric
mode of two identical (I/a=200) parallel wires vs separation. . . . 91

Phase of current distribution for second layer antisymmetric mode
of two identical (l/a = 200) parallel wires vs separation
distance. .................................... 92

Damping coefficient of lowest order antisymmetric mode
frequency of two identical parallel wires (l/a = 200) vs separation
distance - approximate methods. ..................... 94

Radian frequency of lowest order anti-symmetric mode frequency
of two identical parallel wires (I/a = 200) vs separation distance -
approximate methods. ........................... 95

Damping coefficient of lowest order symmetric mode of two
identical parallel wires (l/a = 200) vs separation distance -
approximate methods ............................. 97

Radian frequency of lowest order symmetric mode frequency of
two identical parallel wires (l/a = 200) vs separation distance -
approximate methods ............................. 98

Comparison of damping coefficient calculations using the
approximate Green’s function method for a system of two identical
parallel thin wires (l/a = 400). ...................... 99

Comparison of radian frequency obtained using approximate
Green’s function method for system of two identical parallel wires
(l/a = 400). ................................. 100

Backseattered field response of two identical parallel wires (l/a =
200). ..................................... 106

Backseattered field response from two identical parallel wires (l/a
= 200) separated in cross range by d/l = 1.0. ............ 107

Backseattered field response of two identical parallel wires (l/a =
200) separated by (M = 2.0 in down range. .............. 112

Backscattered field response of two identical parallel wires (l/a =
200) separated in cross range (d/l = 2.0). ............... 114

Backscatter response of system of two different length wires (1.0m
and 0.8m) separated in down range (1.0m). .............. 116

XV

Figure 3.35

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 5.1

Figure 5 .2

Figure 5 .3

Figure 5.4

Figure 5.5

Figure 5.6

Figure 5.7
Figure 5.8
Figure 5 .9

Figure 5.10

Backseatter response of system of different length (1 .0m and 0.8m)
using SEM and moment method. .................... 117

Variation of EDR vs cross range separation distance for two
parallel 1.00 m wires (l/a=200). ..................... 121

Variation of EDR vs cross range separation distance two parallel
1.0m wires (l/a=200). ........................... 122

Variation of SDR vs cross range separation for two parallel 1.0m
wires (l/a=200). ............................... 124

Variation of EDR vs down range separation distance for two
parallel 1.0m wires (l/a=200). ...................... 126

Variation of SDR vs down range separation distance for two
parallel 1.0m wires (l/a=200). ...................... 128

Dual antenna free-field time domain transient measurement
system. .................................... 132

Single antenna free-field time domain transient measurement
system. .................................... 133

Dual antenna free-field frequency domain transient measurement
system. .................................... 137

Single antenna free-field frequency domain transient measurement
system. .................................... 139

Measurement system block diagram .................... 143

Temporal response of clutter-free calibration target

measurement. ................................. 146
Double-Gaussian pulse spectrum. ..................... 150
Double-Gaussian pulse. ........................... 151
Geometry for sphere scattering problem. ................ 153

Frequency response of a perfectly conducting sphere in back-
scatter configuration. ............................ 157

xvi

Figure 5.11

Figure 5.12

Figure 5.13

Figure 5.14

Figure 5.15

Figure 5.16

Figure 5.17

Figure 5.18

Figure 5.19

Figure 5.20

Figure 5.21

Figure 5.22

Figure 5.23

Gaussian pulse response of a perfectly conducting sphere in back-
scatter configuration. ............................ 159

Gaussian pulse response of a perfectly conducting sphere showing
second creeping wave. ........................... 160

Scattered field amplitude vs aspect angle for wire (2.0 x 0.043
inches). Moment method results computed for MSU anechoic
chamber geometry. ............................. 162

Scattered field amplitude vs aspect angle for wire (4.0 x 0.043
inches). Moment method results computed for MSU anechoic
chamber geometry. ............................. 163

Scattered field amplitude vs aspect angle for wire (6.0 x 0.043
inches). Moment method results computed for MSU anechoic
chamber geometry. ............................. 164

Scattered field amplitude vs aspect angle for wire (8.0 x 0.043
inches). Moment method results computed for MSU anechoic
chamber geometry. ............................. 165

Spectral background measurement of MSU anechoic chamber using
frequency domain system. ......................... 167

Spectral background measurement of MSU anechoic chamber using
time domain system. ............................ 168

Temporal background measurement of MSU anechoic chamber
using time domain system. ......................... 170

Temporal background measurement of MSU anechoic chamber
using frequency domain system. ..................... 171

Spectral response of 14 inch metal calibration sphere measured
using frequency domain system in MSU anechoic chamber. . . . . 173

Temporal response of 14.0 inch diameter metal calibration sphere
measured using frequency domain system in MSU anechoic
chamber. ................................... 174

Temporal response of 14.0 inch metal calibration sphere measured
using time domain system in MSU anechoic chamber. ........ 175

xvii

Figure 5.24

Figure 5.25

Figure 5.26

Figure 5.27

Figure 5.28

Figure 5.29

Figure 5.30

Figure 5.31

Figure 5.32

Figure 5.33

Figure 5.34

Spectral response of 14.0 inch calibration sphere after time domain
filtering of chamber wall reflections. Measured using frequency
domain system in MSU anechoic chamber. ............... 176

Spectral response of calibration sphere after time domain filtering
of chamber wall reflections. Measured using time domain system
in MSU anechoic chamber. ........................ 177

Theoretical response of 14.0 inch diameter metal calibration sphere
calculated using Mie series for MSU anechoic chamber configura-
tion. ...................................... 178

Transfer function of frequency domain system obtained using 14.0
inch diameter sphere as calibration standard. Results normalized
to peak value of unity. ........................... 179

Transfer function of time domain system obtained using 14.0 inch
diameter sphere as calibration standard. Results normalized to peak
value of unity. ................................ 181

Impulse response of frequency domain system obtained using 14.0
inch diameter sphere as calibration standard. Results normalized
to peak value of unity. ........................... 182

Impulse response of time domain system obtained using 14.0 inch
diameter sphere as calibration standard. Result normalized to peak
value of unity. ................................ 183

Comparison of theoretical and experimental frequency responses
of a 3.0 inch diameter brass sphere. Measurements performed
using frequency domain system in MSU anechoic chamber. . . . . 184

Comparison of theoretical and experimental transient responses of
3.0 inch diameter brass sphere. Measurements performed with
frequency domain system in MSU anechoic chamber. ........ 186

Comparison of theoretical and experimental results for the
frequency response of a stainless steel wire 6.0 inches long and
0.043 inches in diameter. Measurements performed in MSU
anechoic chamber. .............................. 187

Transient scattering response of stainless steel wire 6.0 inches long
and 0.043 inches in diameter oriented at broadside. Measurements
performed using frequency domain system. Numerical results
performed using moment method. .................... 188

xviii

Figure 5.35

Figure 5.36

Figure 5.37

Figure 5.38

I:igure 5.39

Figure 5.40

Figure 5.41

Figure 5.42

Figure 6.1

Figure 6.2

Figure 6.3

Comparison of theory and experiment for the transient response of
a 6.0 x 0.043 inch stainless steel wire oriented at 45 degrees from

broadside in the MSU anechoic chamber .................

Scattering of stainless steel wire 6.0 inches long and 0.043 inches
in diameter oriented at 45 degrees from broadside. Measurements
performed using frequency domain system. Numerical results

computed using moment method. ....................

Comparison of theory and experiment for a metal sphere 3.0
inches in diameter. A wire of length 6.0 inches and 0.043 inches
in diameter oriented at broadside was used as the calibrator for the

measured results. .............................

Comparison of theory and experiment for transient response of a
3.0 inch diameter metal sphere. A 6.0x0.043 inch wire oriented at

broadside was used as the calibrator in the measurements. .....

Comparison of theory and experiment for frequency response of
a 6.0x0.043 inch wire oriented 45 degrees off broadside.
Measurements performed using response of same wire oriented at

broadside as a calibrator ..........................

Comparison of theory and experiment for the transient response of
a 6.0x0.043 inch wire oriented 45 degrees off broadside.
Measurements performed using same wire oriented at broadside for

calibrator. ..................................

Comparison of theory and experiment transient response from a
3.0 inch diameter sphere. Measurements performed using 14 inch

diameter Sphere as a calibrator. .....................

Pulse synthesized from 1/10 cosine taper over bandwidth 1.0-7.0

GHz. .....................................

Damping coefficient of lowest order symmetric mode for two
parallel wires (l/a = 200) extracted from surface current

measurements. ...............................

Radian frequency of lowest order symmetric mode of two parallel
wires (I/a = 200) extracted from surface current measurements.

Spectral response of two parallel 6.0 x 0.043 inch wires separated

by 8.0 inches in down range. ......................

xix

190

. 191

. 192

193

. 194

. 195

. 197

. 199

. 203

.204

. 206

Figure 6.4

Figure 6.5

Figure 6.6

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

Double-Gaussian pulse response of two 6.0 x 0.043 inch wires
separated by 8.0 inches in down range. ................. 207

Spectral response of two parallel 6.0x0.043 inch wires separated
by 6.0 inches in down range. ....................... 209

Double-Gaussian pulse response of two 6.0x0.043 inch wires
separated by 6.0 inches in down range. ................. 210

Spectral response of two 6.0x0.043 inch wires separated by 4.0
inches in down range. ........................... 211

Double-Gaussian pulse response of two 6.0x0.043 inch wires
separated by 4.0 inches in down range. ................. 212

Spectral response of two 6.0x0.043 inch wires separated by 2.0
inches in down range. ........................... 214

Double-Gaussian pulse response of two 6.0x0.043 inch wires
separated by 2.0 inches in down range. ................. 215

Effects of resistive loss on wire scattering - moment method
calculations. ................................. 216

Comparison of measurements and theory including resistive
losses. ..................................... 217

Spectral response of two 6.0x0.043 inch wires separated by 2.0
inches in cross range. ............................ 219

Double-Gaussian pulse response of two 6.0x0.043 inch wires
separated by 2.0 inches in cross range. ................. 220

Spectral response of two 6.0x0.043 inch wires separated by 4.0
inches in cross range. ............................ 221

Double-Gaussian pulse response of two 6.0x0.043 inch wires
separated by 4.0 inches in cross range. ................. 222

Spectral response of two 6.0x0.043 inch wires separated by 8.0
inches in cross range. ............................ 223

Double-Gaussian pulse response of two 6.0x0.043 inch wires
separated by 8.0 inches in cross range. ................. 224

XX

Figure 6.19

Figure 6.20

Figure 6.21

Figure 6.22

Figure 6.23

Figure 6.24

Figure 6.25

Figure 6.26

Figure 6.27

Figure 6.28

Figure 6.29

Figure 6.30
Figure 6.31
Figure 6.32
Figure 6.33
Figure 6.34
Figure 6.35

Figure 6.36

Wire stick airplane model. ......................... 226

Spectral response of wire stick airplane model for nose-on (0

degrees) incidence. ............................. 228
Temporal response of wire stick airplane model for nose-on
incidence. ................................... 229
Spectral response of wire stick airplane model for 45 degree
orientation. .................................. 230
Temporal response of wire stick airplane model for 45 degree
orientation. .................................. 231
Spectral response of wire stick airplane model for broadside (90
degree) orientation. ............................. 232
Temporal response of ‘wire stick airplane model for broadside (90
degree) orientation. ............................. 233
Spectral response of wire stick airplane model for 135 degree
orientation. .................................. 234
Temporal response of wire stick airplane model for 135 degree
orientation. .................................. 235
Spectral response of wire stick airplane model for tail-on
incidence. ................................... 236
Temporal response of wire stick airplane model for tail-on
incidence. ................................... 237
Spectral response of A10 model. ..................... 240
Double-gaussian pulse response of A10 model. ............ 241
Spectral response of F15 model. ..................... 242
Double-gaussian pulse response of F15 model. ............ 243
Early time scattering by Lockheed TRlA for wing-on incidence. . 254
Early-time response of B58 Hustler for wing-on incidence. ..... 255
Early-time response of B58 Hustler for nose-on incidence. ..... 256

xxi

Figure A.l

Figure C.l

Figure c.2

Figure C.3

Figure C.4

Figure C.5

Geometry of thin-wire scatterer. ..................... 263

Michigan State University anechoic chamber configuration prior to

February 1992 ................................. 281
Michigan State University anechoic chamber after February

1992. ...................................... 282
Measured cable attenuation. ........................ 283

Gain of HP8349B Microwave Amplifier measured using HP-8720B
network analyzer. Amplifier input power = -30 dBm. ........ 284

Gain of PPL 5812 wide band amplifier measured using HP—8720B
network analyzer. Amplifier input power = -10 dBm. ........ 285

xxii

Chapter 1

Introduction

This dissertation is concerned with the application of transient electromagnetic
waves to the detection and identification of remote objects with an Ultra-Wide—Band
(UWB) or impulse radar system. Before proceeding further it is useful to put some of
the common terminology in context.

The bandwidth of a radar system can be defined in an several ways. In an

absolute sense, the spectral bandwidth W of the transmitted waveform is specified as

W=fifi an

wherej}ll and]; are the highest and lowest frequency components passed completely from

the transmitter to the receiver. Bandwidth can also be expressed in a relative sense as

B=E um

f.

where the center frequency f, is defined as

 

ft = fif' (1.3)

The term ”narrow band” refers to systems that have a fractional bandwidth B of
less than 3% and usually less than 1%. Systems with a bandwidth of 10% are considered
wideband. An UWB system on the other hand has a fractional bandwidth greater than

25% [1,2].

I)!

Conventional radars are narrow-band systems used to detect the presence or
absence of a target, measure a target’s speed, or measure a target’s position. An impulse
radar is a UWB radar system that uses very short duration pulses to achieve a high range
resolution. UWB radars can also be designed using a Frequency Modulated Continuous
Wave (FM-CW) or with a spread spectrum or noise type modulation [3].

The term ”high resolution radar” is also commonly used to describe a radar
system that provides superior resolution of target features. The exact type of technology
is ambiguous unless the purpose of the radar is specified. For mapping radars, the term
“high resolution” usually refers to a system that can resolve small areas on the ground
using Synthetic Aperture Radar (SAR) techniques. For airplane detection and tracking
radars, the term ”high resolution” refers to the range resolution and the ability to resolve

specific scattering centers along the fuselage and wings of an aircraft. It should be
pointed out that the range resolution AR of any radar is dependent only on the absolute

bandwidth W of the system [4] as

AR = —. (1.4)

Hence, a narrow-band system operating at a center frequency of 100 GHz with a
bandwidth of 2 GHz has the same range resolution as an ultra-wide band impulse radar
using the frequency range from 100 Mhz to 2.1 GHz. Thus, the term ”high resolution"
does not necessarily imply a UWB radar system.

During the past 10 years, interest in UWB radar has risen dramatically and nearly
every major meeting of electromagnetics, microwave and radar workers has sessions

devoted to topics related to UWB technology [5,6]. The UWB radar has the potential

for extracting more information from a target echo than just its speed and location.
Specifically, it is believed that UWB radars have the potential for identifying the size and
shape of a target. It has also been suggested that the UWB radar might have better
detection performance than conventional radars when the class of target to be detected
is known [7].

Much of the work presented in this dissertation is concerned with the application
of UWB radar for non-cooperative target recognition. Though there are a number of
schemes under investigation for identifying a target from its radar return, this work
focuses on resonance based methods. The natural resonances of a target are inherently
useful for target discrimination since the natural frequencies are dependent only on the
geometry of the target and not on the shape of the incident waveform or the orientation
of the target with respect to the incident wave. The aspect independence of the natural
frequencies allows for identification of complicated targets whose orientation is usually
unknown. A successful method of discrimination based on the target natural frequencies
is embodied in the Extinction pulse (E-pulse) or Single-mode extraction pulse (S-pulse)
methods [8,9,10].

Chapter 2 reviews some of the previous work on the E-pulse and S-pulse scheme.
Several advances in automating the discrimination scheme are presented. The
performance of the automated scheme for single thin-wire targets in the presence of noise
is evaluated using theoretical scattering data. This analysis serves as a baseline for
comparison for other performance evaluations.

The problem of discrimination when more than one target is present is more

difficult. When two or more targets are sufficiently separated in down range, the target

responses can be treated separately and the discrimination process applied as if there is
only a single target. These cases are discussed and analyzed in Chapter 2.

In some cases the targets may be too close together to apply range separation.
This case is more complicated since mutual interaction between targets can significantly
change the scattering response. Much effort was focused on evaluating the effect of the
mutual interactions on the E—pulse and S-pulse discrimination scheme. One effect of the
mutual interactions is to shift the natural frequencies of the group of targets away from
the natural frequencies of the individual targets. Chapter 3 is devoted to evaluating the
shifting phenomena and obtaining a better understanding of transient coupling
phenomena. It is seen that more than just the shift of natural frequencies must be
considered for a proper understanding of the effect of mutual interactions on the transient
response and on the E-pulse and S-pulse discrimination performance.

Chapter 4 presents an analysis of coupling on the E-pulse and S-pulse
discrimination schemes using theoretically generated thin wire data. The discrimination
results are interpreted according to the physical insight gained in Chapter 3.

To implement the E-pulse and S-pulse discrimination scheme, the natural
frequencies of the targets must be accurately extracted from a measured transient
response. Unfortunately, all current mode extraction procedures are sensitive to noise and
systematic errors in the measured data. Until recently, extraction procedures have been
prone to yielding natural frequencies with a non-physical positive damping coefficient
when the data is corrupted. This may be about to change with the advent of a new
method to constrain the natural frequencies to physically meaningful ranges recently

introduced by Ilavarasan [11]. In any case, the measurements must be of extremely

high quality. Further, very little is currently known of the early-time response of
complicated targets to impulse excitation. Thus accurate transient measurements are
essential in understanding and using information contained in the early-time portion of
the response. For these reasons, a great deal of effort was focused on developing and
implementing a practical transient scattering measurement system.

Chapter 5 describes in detail procedures developed and used to obtain accurate
transient scattering measurements at Michigan State University during the course of this
research. A calibration procedure is presented that can be used to eliminate most
systematic errors from both time domain measurements and measurements performed in
the frequency domain with a frequency sweeping vector network analyzer. The
ealibration is verified by measurement of canonical targets.

Chapter 6 presents numerical and experimental results associated with this
research. The experimental verification of the transient coupling effects discussed in
Chapter 3 is presented using a series of measurements of coupled thin wires. The effect
of resistive losses on the response of a system of parallel wires excited in the
transmission line mode is discussed. A wire stick airplane model is measured in several
orientations to illustrate the capability of the measurement system to accurately measure
a complicated target whose response can be theoretically calculated. The amount of
energy available for discrimination that is present in the late-time of a target response is
discussed using data from realistic scale model aircraft. The effect of coupling on
discrimination of realistic aircraft models is explored using experimentally derived data.

Finally, measurements of the early-time of several targets are presented.

Chapter 2

Target Discrimination Using E—Pulses and S-Pulses

2.1 Introduction
Though there has been considerable work published on radar target discrimination
using the E-pulse and S-pulse technique [12,13,14], there has been a continual
evolutionary improvement in the application of E-pulses and S-pulses in an automated
target discrimination scheme. This chapter briefly presents the E-pulse and S-pulse
discrimination scheme as implemented at Michigan State University. Some of the
improvements are probably useful only in a laboratory demonstration of the
discrimination scheme, however, many of the problems present in the laboratory
environment are likely to be even more difficult to solve in a practical implementation
of the discrimination scheme.
The first portion of the chapter briefly reviews the theoretical foundations of the
E-pulse and S-pulse discrimination method. Quantification of the discrimination is
discussed and problems involved in the automated decision making process are addressed.
Then, the performance of the automated scheme is evaluated for theoretical thin wire
scattering data in the presence of noise. Finally, the difficulties encountered in a multiple

target environment are discussed.

2.2 Review of E-Pulse and S-Pulse Discrimination

The E and S pulse method of target discrimination is based on the singularity
expansion method (SEM) representation of scattered electromagnetic fields first advanced
by Baum [15] in 1971. According to this theory, the transient currents and fields in

late-time can be expressed as a natural mode series

N
r(t) = 2 a. exp(ant) cos(mnt + 4)") t > T1 (2.1)

n-l
where s, = a, + jw, is the aspect independent natural frequency of the n"' target mode,
a, and ii. are aspect dependent modal amplitudes and phases, and T, is the aspect
dependent beginning of late-time. Marin later proved the validity of this expression for
the case of finite sized perfectly conducting bodies of arbitrary shape [16].
The E—pulse waveform e(t) for a particular target is defined [9] such that

e(t) . r(t) s 0 t > T, + T. (2.2)

where T, is the duration of e(t), and r(t) can be the response of the target from any

aspect angle. This definition implies that
15(3) I 9{e(t)} '5 O for s=sn and s=s;; n=l,2,3,...N (2.3)
where the s, are natural-mode frequencies of the target.

To effect target discrimination using E-pulses, the response from an unknown

target r(t) is convolved with each of the E-pulses in a database. The convolved

waveforms are denoted c‘(t) = e(t) It v(t) . If E(s) has zeros at all natural frequencies

present in the late-time portion of v(t), then c‘(t) = O in the late-time. If 15(3) does not

contain zeros at all natural frequencies in the late-time portion of v(t), then c‘(t) at 0 in

the late-time [9]. Thus, the target producing v(t) is associated with the E-pulse that yields

zero in late-time after convolution with v(t).

The sine S-pulse waveform s,‘(t) for the 1“ mode of a particular target is defined

s,'(t) t r(t) s b, exp(a, t) sine», t+6,.) t > T, + T: (2.4)

where b, and 0, are aspect dependent amplitudes and phases. The cosine S-pulse

waveform sc'(t) for the 1"" mode of a particular target is defined by

3:0) * r(t) c, exp(oit) cos(0‘t+ 4),) t > T, + r, (2.5)

where c, and 4’.- are aspect dependent amplitudes and phases, T, is the duration of the sine
and cosine S-pulses and r(t) can be the response of the target from any aspect angle. The

spectral interpretation for these definitions is

3"(3) I 20.90)} i O for s=s, and s=s,,°; n =1,2,3,...,N; "a: (2.6)

In addition, these definitions require 83.9,) = -S,'.(s,) and Sc‘(s,) = 8:13,) . The unknown

aspect dependent phases 0, and 4:,- cause difficulty in quantifying discrimination when only

a sine or only a cosine S-pulse is used. To circumvent this problem, a quadrature sum

of a cosine and sine S-pulse §‘(t) = s:(t) -js,‘(t) is defined. When the quadrature sum

S-pulse is convolved with a response waveform r(t) the following result is obtained

51“) at r(t) = b, exp(-j6l.) exp(a‘t-jwit) t >1} + T: (2.7)

Application of the Fourier transform results in

We . ' (.I‘.-0)W‘_l
y{§‘(z)*r(t)} = b,e"°' f e"‘e'1°‘d: = b,e"°' ‘ s 17(0)) (2.8)

 

0 (Si. " ‘0)

where W, is the duration of an observation window. It is clear that the frequency

variation of |.9'{§‘(t) *r(t)}| is aspect independent.

2.3 Quantification and Automation of Discrimination

The automation of the E—pulse and S-pulse discrimination method was originally
pioneered by Rothwell and Ilavarasan [17, 18]. Automated discrimination requires
a measure of the amount of signal present in the late-time of c,(t) for each E-pulse. Thus,
the E-pulse Discrimination Number (EDN) can be defined as

-1

 

 

now, 1',
EDN = f cf(t)dt [8(1) d: (2.9)
r, o

 

 

The EDN is a measure of the deviation from the expected value of zero late-time energy.
The choice of the window duration is usually based on the duration of the target response
and the Signal-to-Noise Ratio (SNR) present in the measurement of an unknown target.
Note that the late-time energy of c,(t) is normalized by the E-pulse energy. To use the
EDN for automated discrimination, the EDN is calculated for each E—pulse in the data
base. If the target producing v(t) has been cataloged with an E-pulse in the database then

the EDN for that E-pulse is zero while all other E-pulses yield a non-zero EDN. In

actual practice, noise and inaccuracies in the estimates of the target natural frequencies
used to construct the E-pulses prevent the EDN from precisely vanishing when the E-
pulse is matched to the target producing the response. Thus, the target is identified by
the E-pulse yielding the minimum EDN. As a quantitative measure of the differences in
the EDN values computed for all E—pulses in the database the E-pulse Discrimination

Ratio (EDR) can be defined as

EDR (dB) = 10.0103“, —,£"—"’—— (2.10)
mrn(EDN)

Therefore, the E-pulse yielding the smallest EDN has an EDR of 0 dB, while the EDR

produced by the other E-pulses is greater.

The S-pulse discrimination scheme is similar to the E—pulse discrimination

scheme. In this case the convolved waveforms are denoted c,(t) = s" ‘(t) :- v(t). The target

is identified by the S-pulse that causes the late-time portion of c,(t) to be a damped
sinusoid with a known complex frequency 3,. The S-pulse corresponding to the target
response produces a convolution whose late-time spectrum matches the spectrum of F(w)
to within a constant. To assign a numerical value to the quality of this match Schwartz’s

inequality

(2.11)

 

Jinx) soc) czar}2 {

ff’(x)dx fg’(x)dx

IIA

1
1; ifi’flx) =kg(x); k=con51

where [(x) and g(x) are real functions is used to define the S—pulse Discrimination

Number (SDN)

10

”c (a). -1
f|C,(w)|2do f |F(m)|’d¢.) (2.12)

SDN = l - f |C3(m)||F(w)|de

 

 

where C,(w) is the Fourier spectrum of c,(t) = “"(t) :- v(t) , F(w) is the Fourier spectrum
of the expected single mode signal taken over the finite duration W, as defined in (2.8),
and w is the radian frequency. The integration limits w, and w, are determined by the
bandwidth of the measurement system. The SDN is a measure of the error from the
expected single mode spectrum. From the Schwartz inequality it can be seen that the
SDN is zero if the spectrum of the late-time portion of c,(t) matches the expected single
mode spectrum to within a constant. As in the case of E—pulse discrimination, each of
the quadrature S-pulses in the database are convolved with the response waveform v(t).
For each convolution, the SDN is calculated. The target is identified by the quadrature
S-pulse that yields the smallest SDN. To quantify the differences in the SDN for different
quadrature S-pulses used in the discrimination we define the S-pulse Discrimination Ratio

(SDR) as

 

SDR = 10.010gm{ ,ngm} (2.13)
mm

Therefore, the SDR of the correct target is 0 dB, while the SDR of all other targets is
greater than 0 dB. If the method is working properly, the EDR and SDR values approach
infinity for E-pulses and S-pulses not associated with the target in the response. In most
applications, noise in the measurement system and inaccuracies in estimates of target
natural frequencies prevent the EDN and SDN from being zero when the E—pulses and

S-pulses match the target producing v(t).

ll

To apply the discrimination scheme, the beginning of the late-time T, for the
unknown target response v(t) must be estimated. For most targets of interest, the start
of late-time can be computed based on the maximal one-way transit time of the target T,,,
the effective pulse duration used in the measurement system T,, and an estimate of when
the wave strikes the leading edge of the target T,. For the usual case of back-scattering
measurements

7‘, = r, + T, + 2r, (2.14)

The target data base contains a value for T, as well as values of T" for each target. Thus,

only T, must be estimated from the response data.

For this analysis a computer model of a threshold detector is used to determine
T,. The point at which v(t) is greater than a threshold voltage V, is taken to be T, The
threshold voltage V, must be set small enough to detect weak signals, but large enough
limit the false alarm rate to an acceptable level. For Gaussian noise, the mean time

between false alarms I}, for a given V, is given by [4] as

= L LIT. (2.15)
T’“ 8,6”sz

where 8,, is the IF bandwidth of the measurement system, and ‘0, is the mean-square

value of the noise voltage. For the analysis to be presented in the next section, the

threshold voltage was chosen so that Vr = 8 m . This resulted in 1}, = 21.36 hours for

the 1.022 GHz bandwidth used. Note that the determination of T, is essential for proper

operation of the automated discrimination scheme. As such, it may require a vastly more

12

sophisticated method to accurately estimate T, in actual practice than the simple threshold
detector used in this analysis.

It should be pointed out that the problem of E-pulse and S-pulse synthesis has not
been pursued in this work. Though the synthesis problem has already been extensively
investigated by Rothwell [9] and further refined by Ilavarasan [19], there is still a great
deal of work necessary in this area. For the purposes of this research however, it is
sufficient to say, that synthesis of the E—pulse and S-pulse requires knowledge of the
target natural frequencies. Unfortunately, the natural frequencies can be accurately
calculated for only a few simple geometries. Thus, the natural frequencies of practical
targets such as missiles and airplanes must be extracted from measured transient
scattering data. It should also be mentioned that there are ways to construct E-pulses
directly from the transient response and the natural frequencies obtained as a by product
[20]. The problem of extraction of the natural frequencies is also very difficult and
requires more research in both the measurement technology and the extraction

procedures. For the latest in the extraction procedure consult Ilavarasan [19].

2.4 Performance of Automated Discrimination Scheme

This section describes the application of the automated E-pulse and S-pulse
discrimination scheme to theoretical scattering data for thin wires. Scattering data was
generated using a frequency domain method-of-moments solution as described in
Appendix A. A Galerkin technique employing piecewise sinusoidal basis functions and
thin-wire approximations was used. The back-scattering responses of wires of length

0.80, 0.85, 0.90, 0.95, 1.00, 1.05, 1.10, 1.15, and 1.20 meters were calculated for

13

length-to-radius ratios (l/a values) of 100, 200, 400, and 800. The complex field values
were calculated at 512 equally spaced frequencies between 0.002 GHz and 1.024 GHz.
These results were subsequently inverse Fourier transformed using a fast Fourier
transform (FFI') to obtain the transient response. A Gaussian pulse was used for the

incident wave p(t). The temporal variation is given by
_ -4t2
pa) - CXp(-—2-) (2.16)
‘l’

where r = 1 ns was chosen to give a pulse width of approximately 2 ns between the 2%
of maximum points.

Calculations were performed for all wires at both broadside-incidence orientation
and 45 degrees off broadside. The wire natural frequencies were extracted from the
ealculated data using a hybrid E-pulse/least squares method [21] as would be done in
the case of measured responses. In this case, the extracted natural frequencies were found
to be very close to those computed from a singularity expansion method (SEM)
formulation as described in Chapter 3 and Appendix B. The E-pulses and quadrature S-
pulses were synthesized for each of the wire lengths and (la values. To simulate noise
encountered in a practical implementation, the calculated transient responses were
corrupted by adding varying amounts of Gaussian noise. The automated discrimination
process was applied to the noisy data.

In transient analysis, the usual Continuous-Wave (CW) definition of SNR is
inappropriate. It is more useful to interpret the SNR within a specified time window, the

choice of which affects the SNR value. For this analysis the window duration W, was

14

 

chosen as the minimum duration window that contains 99% of the total energy in the

noise free data. The ratio of the signal energy to the noise energy within this window is

rfv2(t)dt
w, (2J7)

_ ‘l’owd

 

SNR (dB) = 10.010gm‘

 

where v(t) is the noise free signal. A zero-mean Gaussian white noise model is used
throughout the analysis. To illustrate the level of noise corruption used in the analysis,
several examples are shown. Figure 2.1 shows the noise free back-scattering response
of a broadside 1.00 m wire with l/a = 800. Figure 2.2 shows the same response with
noise added to provide a SNR of 30 dB. Figure 2.3 shows the same response with noise
added to provide a SNR of 5 dB.

To test the sensitivity of the E-pulse method to SNR and differences in target
dimensions, the E—pulses for all wires with I/a = 800 were used to discriminate the
response of a 1.00 m wire with l/a = 800. The response waveform was corrupted with
varying amounts of noise with the SNR varying from 5 dB to 30 dB. EDR values for
each of these noisy signals were computed. Figure 2.4 shows the results of this test when
the 1.00 m wire is in the broadside orientation. If the 1.00 m wire is identified correctly
the EDR of the 1.00 m wire E-pulse is zero. This is the case for all values of SNR
computed. As expected, the EDR is largest for all wrong target E-pulses when the SNR
is high. When the SNR is low, the EDR for all wrong target E-pulses is less. Also, E-
pulses for targets similar to the 1.00 m wire (ie. 0.95 m and 1.05 m) provide smaller
EDR values than E—pulses for targets significantly different from the 1.00 m wire (ie.

0.80 m and 1.20 m). Figure 2.5 shows a similar result when the response is from the

15

1.00

.0
or
o

.0
o
o

 

 

 

 

l
.0
UI
o

 

 

 

Relative amplitude

L1111r1111411111411l11111i1|41441111144l

 

I
b
O

I

0.00 20.00 40.00 50.00 80.00 100.00
Time (ms)

Figure 2.1 Noise free back-scattering response of 1.00 m (l/a = 800) oriented at
broadside.

16

1.00

.0
u:
o

 

 

 

 

Relative amplitude
I ~ 0

 

1111rrrrrlrrrrrrrrrlrrl11111111111111111

 

 

 

_1.00 lllllleIII IIIIIIIIITTWIIl—FITIIIIIIIIIIIITIIIfi]

0.00 2 .0 40.00 60.00 80.00 100.00
Time (as)

Figure 2.2 Back-scattering response of 1.00 m wire (I/a = 800) oriented at broadside
with SNR of 30 dB.

17

 

 

 

1.00

 

 

 

 

 

 

 

 

 

 

 

 

 

i
0) 0.504;"
'0 1
3 w =
+1 j l
a 1 ll
I . “ 1‘ ‘ l
g 0.00 l . 1w 1 11 I '
1 Wll
8 l 1
.2
4.)
3 |
(jg-0.50 l
O:
-1.00 WWW-mm
0.00 20.00 40.00 60.00 80.00 100.00

Time (as)

Figure 2.3 Back-scattering response of 1.00 m wire (l/a = 800) oriented at broadside
with SNR of 5 dB.

18

Response: 1.00 m wire, broadside

       
 

 

- 99999080 m wire E—pulse
30-00 ‘ 88886085 m wire E-pulse
1 W090 m wire E-pulse
u 00690 0.9 m wire E-pulse
~ W100 m wire E—pulse
2500 j -H-H-+1.05 m wire E—pulse
- x—N—H—K 1.10 m wire E-pulse
- ”rt-H 1.15 m wire E-pulse
A2000 _‘ H...“ 1.20 m wire E-pulse
CD I
'0 _
v -
1500 — 7
0C 2
D _
LIJ -
10.00 F
5.00 -
1
0.00 ~
—5.00 -wwmfi1TnTrwmm1Trm1TrrrrnTnfiTrrrrn1fi1fiTrrrrn1Tmm
0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00

SNR (dB)

Figure 2.4 EDR values computed for response of 1.00 m wire (l/a = 800) oriented
at broadside.

19

Response : 1.00 m wire, 45° orientation
35.00

   
 

   

 

 

j

3 99999080 m wire E-pulse

30.00 1 139999085 m wire E-pulse

I met-8090 m wire E—pulse

- 00600095 m wire E-pulse

- M100 m wire E-pulse

25-00 ‘_ -H—H-+ 1 05 m wire E-pulse

- W 1 10 m wire E-pulse

1 MW 1 15 m wire E—pulse

A2000 _ W1 20 m w1re E-pulse
CD I
'0 _
v _
15.00 -
0: I
D _
L1J _
10.00 -:
5.00 -
0.00 —
1
—5.00 -

0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00

SNR (dB)

Figure 2.5 EDR values computed for response of 1.00 m wire (l/a = 800) oriented
45 degrees from broadside.

20

1.00 m wire in the 45 degree orientation. This illustrates the aspect independence of the
E-pulse method.

It is interesting to determine if the effectiveness of target discrimination using E-
pulses and S-pulses is a function of target Q. Note that thinner wires (those with higher
l/a ratios) have higher Q values. Thus, several tests were run using wires of different
thickness. Wires with l/a values of 100, 200, 400, and 800 were used. The evaluation
proceeds as follows: The E-pulses for all wires (1 = 0.80, 0.85, 0.90, 0.95, 1.00, 1.05,
1.10, 1.15, 1.20) with l/a = 100 were used to discriminate a 1.00 m wire with l/a =
100. Similarly, the E-pulses for all wires with l/a = 200 were used to discriminate a
1.00 m wire with l/a = 200. This process was repeated for the cases of (la = 400 and
(la = 800. In each case, 1.00 m wire was oriented at broadside and the EDR of the E-
pulses corresponding to wires that are 110% (ie. wires of length 0.90 m and 1.10 m)
were calculated. The minimum of the two EDR values was plotted vs SNR for each of
the (la eases. Figure 2.6 shows that wires with I/a = 800 have higher minimum EDR
values for a given SNR than thicker wires. This effect is seen for other aspect angles as
shown in Figure 2.7 where the wire is oriented 45 degrees from broadside.

The same tests were performed for the case of S—pulse discrimination. Figure 2.8
shows the results when using quadrature S-pulses based on the first fundamental
resonance frequency of the wires with I/a = 800. Again, v(t) is chosen as the broadside
response of a 1.00 m wire with (In = 800. The SDR results are similar to those obtained
using the E-pulse method, however, the SDR values are significantly larger than the EDR
for all values of SNR. Figure 2.9 indicates that S-pulses based on the second resonance

frequency do not yield large SDR values regardless of SNR. This is due to the fact that

21

 

35.00

 

 

j Response: 1.00 m wire. broadside
30.00 -'
: 9999eI/o = 100
- 99999:;0 = 388
- M a =
25-00 : 06660 l/o = 800
A2000 1
CD 1
U 1
V 3
15.00 -
01 I
O .
LIJ ..
10.00 t
5.00 -
0.00 3
-5.00 rrrrrrrrrlrrlrllrrIIIIrl Irrrlrrrr
0.00 5.00 10.00 15 00 20. 00 25. I00 30. 00 35. I00

 

SNR (dB)

Figure 2.6 Minimum EDR values computed for 0.90 m and 1.10 m wire E-pulses for
response of 1.00 m wire oriented at broadside.

22

 

 

 

35.00 '3 Response: 1.00 m wire, 45° orientation
30.00 3
- eeeeeI/o = 100
: aaeaa:/o = E88
25.00 - were 0 =
j 000001/8 = 800
A20.00 3
CD 1
“o -
v .1
15.00 -
05 I
D a
L1J .
10.00 '2
5.00 3
0.00 3
“5.00 IIIIIIIIIWFITjTIII IWWFIIIII[ITII]
0.00 5.00 10 00 15 00 20. I00 25. 00 30.00 35.00
SNR (dB)

Figure 2.7 Minimum EDR values computed for 0.90 m and l. 10 m wire E—pulses for
response from 1.00 m wire oriented 45 degrees from broadside.

23

Res onse: 1.00 m wire, broadside
Firs fundamental mode S—Pulse

  

     

      

 

999990.80 m wire S-pulse
589990.85 m wire S-pulse ,
35-00 '_ amaOSO m wire S-pulse
3 000000.95 m wire S-pulse ’
.. M100 m wire S-pulse e/
30 00 ; H—H—i- 1.05 m wire S—pulse a
‘ .1 x-H-N-x 1.10 m wire S-pulse
« W 1.15 m wire S-pulse ,,
1 W120 m wire S-pulse
25.00 1
1
cl
A2000 "1
m 2
'0 -1
v .1
15.00 -
0: :
Q q
(f) -
10.00 -
5.00 -
.1
0.00 3
..
-5.00 -
0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00

 

SNR (dB)

Figure 2.8 SDR values computed for first mode S-Pulses for response of 1.00 m wire
(l/a = 800) oriented at broadside.

24

35.00

 

 

- Response: 1.00 m wire, broadende
: Second fundamental mode S—pulse
30'00 - 99999080 m wire S—pulse
‘ 89880085 m wire S—pulse
I crate-A090 m wire S-pulse
25.00 ~ 00000 0.95 m wire S-pulse
1 M100 m wire S-pulse
j ++-+-H-105 m wire S—pulse
_ x—H-M-x 1 10 m wire S-pulse
A2000 - W 1 15 m wire S-pulse
m : W120 m wire S—pulse
'0 _
V -
15.00 -
03 I
O _
(f) _
10.00 -
5.00 -
0.00 -
—5.00 -
000 5.00 10.00 15.00 20.00 25.00 30.00 35.00

SNR (dB)

Figure 2.9 SDR values computed for second mode S—pulse for response from 1.00 m
wire (l/a = 800) oriented at broadside.

25

the second resonance of the 1.00 m wire is excited for broadside incidence. Thus, the
expected mode must be present in the response waveform for the S-pulse method to work
effectively. Figure 2.10 shows that the second mode S-pulse can be used effectively for
the ease of 45 degree orientation since the second mode is strongly excited.

The effect of target Q is evaluated for S-pulse discrimination as was done
previously in the ease of E-pulse discrimination. Figure 2.11 shows the results when
using first mode S-pulses for the broadside response of the 1.00 m wire. The results are
consistent with the E-pulse results except that the SDR values are larger than the EDR
values. Further evidence of this phenomenon is shown in Figure 2.12 for the case of
second mode S-pulses and a response from a 1.00 m wire at 45 degree orientation.

This section has presented an automated discrimination scheme based on the E-
pulse and S-pulse methods. The scheme can be readily applied in applications where
many targets are considered and a computer is used to make the discrimination decisions.
The performance of the discrimination scheme is evaluated for thin wire targets in the
presence of Gaussian random noise.

Several conclusions can be made from the numerical results. For the cases
considered, the S-pulse discrimination scheme is successful in situations where the targets
are 110% different in length even when the SNR is only 10 dB, while the E—pulse
scheme is successful where the targets are :1; 10% different in length with an SNR of 15
dB. Also, the discrimination is better for higher Q targets with both the E-pulse and S-
pulse schemes. Discrimination of very similar targets whose natural frequencies are also
similar requires higher values of SNR to obtain a given EDR or SDR value. On the other

hand, for targets that are vastly different in size and shape (with natural frequencies that

26

 

Response: 1.00 m wire, 45° orientation

 
   
  
 
 
 
 
    

 

35.00 3 Second fundamental mode S—pulse
.1
I 999990.80 m wire S-pulse
30.00 '3 896990.85 rn wire S-pulse
. mat-9990,90 m wire S-pulse
3 00000 0.95 m wire S-pulse
4 W100 m wire S-pulse
25.00 1 -1-H—1-1-1.05 m wire S-pulse
.. W 1.10 m wire S-pulse
- 111-Hr“ 1.15 m wire S-pulse
A2030 _: W120 m wrre S—pulse/
CD I
'0 3
v .1
15.00 .5
01 I
D -
m .
10.00 :
5.00 3
1
0.00 ‘ fi———* t +—-*—-—*
I
35.00 3

 

Wmmwwmmm
0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00
SNR (dB)

Figure 2.10 SDR values computed for second mode S-pulse for response of 1.00 m
wire (l/a = 800) oriented 45 degrees from broadside.

27

Response: 1.00 m wire, broadside
First fundamental mode S-pulse

     

oeeeoI/o =
GBBBEII/o 3.-
A-aeeel/o =
25.00 000001/8 =
A2000
(D
‘0
v
15.00
0:
C)
U)

Llllll1111141lllLlJLlJJlLlLllJlllJLlllll

 

 

[IlllliiillijllIIIITIIIII

.00 15.00 20.00 25.00 30.00 35.00
SNR (dB)

0
o
o
913
o
o
0‘

Figure 2.11 Minimum SDR values computed for S-pulses for 0.90 m and l. 10 m wires
using response of 1.00 m wire oriented at broadside.

28

Response: 1.00 m wire, 45° orientation

     

 

 

555-00 j Second fundamental mode S—pulse
30.00 3
: 99999I/0 =
_ 99999 l/a =
3 M l/a =
25-00 : 00000 I/a =
A2000 3
an I
‘0 _
v ..
15.00 -
CE I
Q _
(f) ..
10.00 -_
5.00 —_
0.00 3— — — -
—5.00-111111III[fiIITIIIIIIIIIII—rITI[1111]
0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00
SNR (dB)
Figure 2.12 Minimum SDR values computed for second mode S-pulses for 0.90 m and

1.10 m wires using response from a 1.00 m wire oriented 45 degrees from
broadside.

29

are also vastly different) it should be possible to obtain a sufficient EDR or SDR value
for discrimination even in a low SNR environment.

For systems design, curves such as those presented in this section could be
generated for a group of targets. The transmitter, receiver, and antenna requirements

could then be estimated based on the desired discrimination level and SNR environment.

2.5 Multiple Target Discrimination

The applicability of the E-pulse and S-pulse method to a waveform where there
is only a single target response has been well documented [9, 22]. The case where the
waveform may contain more than one target response is more difficult and has not yet
been adequately addressed. This section qualitatively addresses some of the problems
associated with the multiple target situation. The analysis in this section assumes that the
targets are electrically uncoupled. The case when the targets are electrically coupled is
significantly more complicated and is the subject of Chapter 3 and 4. Since the number
of possible configurations for a multiple target system can become very large and
complieated, the analysis here of this section is limited to the simpler case of two targets.

Assuming a mono-static radar, the geometry of a two target situation is diagramed

in Figure 2.13. As shown in the diagram, target 1 is located at i"l and target 2 is located

at 1’, relative to the antenna. There are several distinct situations with regard to the

loeations of the targets.

First, let li’zl = I711 + 6 so that the response from target 1 appears first while the

response from target 2 is delayed by t,=2e/c from the start of the target 1 response.

30

target 1

fi - 2 target 2

1

 

 

0%

Figure 2.13 Geometry for two target discrimination problem.

31

Then consider the case when Ii"2 —i‘,| is very large so that multiple interactions between

the targets are sufficiently delayed and diminished that the targets can be considered
isolated. The exact distance at which this approximation is valid depends on the size and
type of targets, the incident field waveform, and receiver bandwidth. The received

waveform r(t) can then be written as the superposition of two responses

v(t) = v,(t) + v2(t) (2.18)

where v,(t) and v,(t) are the responses from targets 1 and 2 assuming that the targets are
independent and hence uncoupled. Now, the discrimination procedure can be applied
easily if the response from target 1 is significantly diminished prior to the response of

target 2 and

1, > W, + T, +1,“ + 21,," (2.19)

where W, is the observation window, 1', is the pulse width, T,_ is the maximum E—pulse

duration in the database and T, is the maximum transit time in the database. In this

situation, the response of target 1 is distinctly separated in time from the response of

target 2 and the response of target 1 that is present during the late-time interval of target

2 is assumed to be negligible. A slightly more difficult case occurs when

Tp+Tcm+ZTvm < t, < W+Tp+Th+2Tnn

(2.20)

in which case the target responses overlap. In this case, discrimination proceeds as
follows: First, identify the response from target 1 by applying the method with a reduced
observation window W, sufficient to prevent inclusion of the target 2 response. With

target 1 identified, compute the convolution 19(1) = v(t) ’ e,(t). Note that the convolution

32

operation eliminates any overlap of the resonance response of target 1 onto the response

of target 2. The discrimination procedure can then be applied to the waveform w(t) as

usual except that the start of late-time for target 2 T,z is

T=T

,2 ,2 + T,l + TD + 2T, + T, (2.21)

where T”: is the beginning of the response from target 2, T,I is the E-pulse duration for
target 1, and T, and T, are the transit time and assumed E-pulse duration of target 2.
Unfortunately, if the modes excited in target 2 are degenerate with the modes of target
1 the method fails since the late-time of target 2 is zero in the w(t) waveform. This is not
as bad as it seems since at least one of the targets is identified and the identity of the
second target is limited to a smaller number of possible choices.

As a graphical example of the preceding process, consider the response shown in
Figure 2.14. Targets l and 2 are simulated with wires of length 0.90 m and 1.20 m with
I/a = 200. The start of the response from the 1.20 m wire is delayed 25 ns from the
start of the response from the 0.90 m wire. After convolving this waveform with the E-
pulse for a 0.90 m wire the waveform shown in Figure 2.15 is obtained. The modes of
target 1 are annihilated indicating that discrimination can be effected for target 1. Next,
this waveform is convolved with the E-pulse for the 1.20 m wire. The resulting
waveform is shown in Figure 2.16. Note that the late-time of target 2 is nearly zero in

this waveform indicating that discrimination can be effected.

33

 

 

 

 

 

 

 

 

1-25 ': Superposition of Wire Responses
1.00 3
0.75 3
0.50 3
a 3
.2 0.25 3:
I." e-
2 _
e a... -_\J
v "
81 I
z: -
g .-
-O.$O '_',
-0.753
31.00 3
- Wire Ien the are 0.90 m and 1.20 m
: l/o I 2 for each wire
-1925 Till['71IIIIIIIIIIIIlIIIIIIIIrITIIIIIITIUITTIIIllI
0.00 20.00 40.00 60.00 80.00 100.00

 

Time (ns)

Figure 2.14 Waveform produced by superposition of response from wires of length
0.90 m and 1.20 m with I/a = 200. The response of the 1.20 m wire is
delayed 25 ns from the start of the 0.90 m wire.

34

 

Next, consider the case of small time delays when

1,, < T, + Te... + 2T"... (2.22)

where there is an insufficient window available for evaluation of target 1 separately from
target 2. By linearity, the late-time of the received waveform v(t) is composed of the
natural frequencies contained in v,(t) and v,(t). This suggests that discrimination can be
accomplished by successively convolving the waveform with combinations of E-pulses
since

c,u(t) = v(t) *e,(t) *€2(t) = 0 t 2 T, + T,I + T,2 (2.23)

when the E-pulses e ,(t) and e,(t) match the targets producing the waveform. Note that T,
must be chosen as the point after which both target 1 and target 2 are in late-time. The
evaluation of the EDN must be limited to the late-time portion of the convolved
waveform. This could lead to inaccurate determination of target 1 depending on the exact
spacing of the targets, the Q of target 1 and the transit time of target 2. To implement
this scheme, the number of targets and the location of target 2 must be established prior
implementing the discrimination process. This information might be obtained by using
another radar system capable of resolving the targets. The identity of the targets could
then be determined with an impulse radar system using the multiple E-pulse scheme.
Alternatively, the impulse radar could be designed with sufficient angular resolution to
resolve the targets without need for a secondary tracking radar or even the multiple E-
pulse technique. Unfortunately, achieving sufficient angular resolution with an impulse
radar will probably be significantly more difficult than with a conventional radar unless

great advances are made in the area of highly directive UWB antennas.

35

Superposition of Wire Responses After
Convolution with E-pulse for 0.90 m Wire

 

 

 

 

 

 

 

 

 

0.50 T.
3 I Late Time
0.40 — |
2 l
0.30 3 l
:1 l
7 l
0.20 3
- |
a 3 1
> 0.10 1
a: a |
o 9
'5 31
a 0.00 3—4‘, lK-L—N,
8 3 1
z - 1
g -
30.20 3 l U
I l
30.30 3 l
: l
30.40 3 '
- | Wire len the are 0.90 m and 1.20 m
- l/o - 2 0 for both wires
" l
-o.50 TIIIIIIITIIIIIIIIIIIIfiIIIIIIITIIIUTIFIIIIITIIIIII
0.00 20.00 40.00 60.00 80.00 100.00

Time (ns)

Figure 2.15 Waveform of Figure 2.14 after convolution with E-pulse for 0.90 m wire.

36

Superposition of Wire Responses After
Convolution with E-pulse for 0.90 m Wire
and E-pulse for 1.20 m Wire

0.50

°-‘° Late Time

Wire len the are 0.90 m and 1.20 m
1/0 - 2 0 for both wires

0.30
0.20

0.10

 

 

 

0.00

-0.10

 

 

Voltage (relative)

 

-0.20

 

-0.30

 

 

-0.40

31 E

lllllllllllllllllllIIIJJIIJIJJLULIIIIIIIIIIIlLllJ

 

-0150 IIIIIIIIIIITUTTIIIIIIIIIIIIIIIIIIIIIIIIIITIIIIII

0.00 20.00 40.00 60.00 80.00 100.00
Time (ns)

 

Figure 2.16 Waveform of Figure 2.15 after convolution with E-pulse for 1.20 m wire.

37

To graphically illustrate the use of multiple E-pulses for discrimination, consider
the response shown in Figure 2.17. The targets are the same wires used previously
except that the response of the second wire has not been time shifted. Clearly one could
not presume to know that there are actually two target responses in this waveform. After
convolution of this waveform with the E-pulse for the 0.90 m wire the result as shown
in Figure 2.18 is obtained. Note that the response is not zero in the late-time since the
modes of the 1.20 m wire were not annihilated by the E-pulse for the 0.90 m wire.
Interestingly, it may be possible to determine if there are two targets present by
comparing the signal after convolution with the first E-pulse to the expected result of
zero. Of course if the targets were identical this method would fail. Next, this waveform
is convolved with the E—pulse of the 1.20 m wire. This result is shown in Figure 2.19
and it is clear that all natural modes are eliminated in the late-time.

In applying this method, accurate knowledge of e is essential since the
discrimination scheme is predicated on knowing the start of late-time in the measured
response. As discussed previously, the start of late-time is estimated based on the start
of the response, the maximal transit time of the target, and the pulse width. If the
beginning of the response of target 2 is unknown then the estimate of the start of late-
time for this target is in error. Unfortunately, even if target 2 is known to be present but
6 remains unknown, the estimates are such that the discrimination scheme is erroneously
be applied to early-time data. This can cause the method to fail since the early-time
response is not annihilated by the E-pulse. The best way to avoid this problem is to
design the impulse radar with angular resolution to sufficient to minimize the likelihood

of encountering this problem. Unfortunately, this does not solve all multiple target

38

1.50

 

 

 

 

 

 

 

 

T. Superposition of Wire Responses
1.25 a
1.00 3
I
0.75 E
0.50 3,
a I l
.2 0.25 3
H c-
2 I
V _,
8.30.25 {3: U
D .1
: CI
g -0.50 «3
1
-0.75 1
d
31.00 3
I
31.25 3 .
: Wire len the are 0.90 m and 1.20 m
- l/o - 2 for each wire
-1.50 1
0.00 20.00 40.00 60.00 80.00 100.00
Time (ns)

figure 2.17 Waveform produced by superposition of responses from wires of length
0.90 m and 1.20 m with I/a = 200. The response of the 1.20 m wire is
co-temporal with the response of the 0.90 m wire.

39

Superposition of Wire Responses After
Convolution with E-pulse for 0.90 m Wire
0.50

Late Time
0.40

0.30

0.20

 

I
I
I
I
l
I
I
0.10 I
l

irulrrrrlrrrrlrrlrlrrrrl

 

 

0.00 m A—A

-0.10

 

Voltage (relative)

-0.20

 

 

 

 

-0.40

Wire engths are 0.90 m and 1.20 m
Va 8 200 for both wires

-0050 IIIIITTIIIIIIIIIIIIIIIIIIIIIIlll'll'TUl'l'lUrjle]

0.00 20.00 40.00 60.00 80.00 100.00
Time (ns)

rrrrlrrrrlrrrrlrirrlrrrr

l
I
I
I
I
l
l
I

 

 

Figure 2.18 Waveform of Figure 2.17 after convolution with the E-pulse for 0.90 m
wire.

40

Superposition of Wire Responses After
Convolution with E—pulse for 0.90 m Wire
and E-pulse for 1.20 m Wire

0.50

0'40 Late Time
0.30

0.20

0.10

 

 

0.00

 

-0.10

 

Voltage (relative)

-0.20

c',
8

lljllllljlllllllllllIlllllllllILlllJllllJJllJI

 

 

Wire len the are 0.90 m and 1.20 m

-0-40 l/a - 2 0 for both wiree

 

LlLl

-0.50
0.00 20.00 40.00 60.00 80.00 100.00

Time (ns)

Figure 2.19 Waveform of Figure 2.18 after convolution with the E-pulse for 1.20 m
wire.

41

problems.

The preceding method of using multiple E-pulses relied on the fact that two
targets were known to be present and the exact start time of the target 2 response was
known. If the number of targets and their positions are not known prior to using the E-
pulse and S-pulse method, the discrimination is greatly degraded. To illustrate this point
consider the ease of two identical targets separated by a small amount in range. To
simulate this situation the discrimination scheme was applied to a series of waveforms
composed of the responses from two 1.0 m wires with the response from the second wire
time shifted by varying amounts. Since the targets are identical, one would hope that the
discrimination scheme would choose the 1.0 m wire as the target producing the response
even though the discrimination scheme only assumes one target is present in the
response. Figure 2.20 shows that for small values of time delay, the response is
identified as belonging to the 1.0 m wire. Here the minimum EDR is very large
indicating that the method works well. Unfortunately, the EDR decreases sharply when
the time delay increases beyond a certain value and the method ultimately fails. The
sudden decrease in EDR and failure of the method is due to the presence of the early-
time response of the second wire. Since only one target is assumed, the early-time
response of the second wire is misconstrued as being the late-time response from the first
wire. The S-pulse method yields similar results as shown in Figure 2.21. In both the E-
pulse and S-pulse results, the deterioration of the discrimination is not as severe for the
45 degrees orientation as compared to the broadside orientation. This is most likely due
to the weaker early-time response of the 45 degree orientation responses as compared to

the broadside responses. These curves suggest that discrimination might be possible for

42

30.00

   

 

 

: Minimum EDR for 1.0 m Wire Responses
25.00 -
I .
20.00 H
: W Egooodside ff 8
‘ e944. '
815-00 .3 agree: a roadside
'0 an:
v .-
-l
g .
LrJ 10.00 -1
5.00 -
0.00 -
q
-500 IlllIIIIIIIII[TllllllIFIIIIIIIIIIIIIIII[IIIIIIIIII
0.00 5.00 10.00 15.00 20.00 25.00

Time delay (ns)

Figure 2.20 Minimum EDR vs time delay when E-pulse method is applied to a
response consisting of two 1.0 m wires.

43

30.00

     

 

 

: Minimum SDR for 1.0 m Wire Responses
- Usnng First Mode S-Pulses
25.00 -
20.00 —
: M 35°06‘53“ 11 B d 'd
.C“‘
815-00 _, egreee 0 rec 81 e
'0 -1
v q
a: '1
O '1
m 10.00 3
d
5.00 3
.1
0.00 3
—5'oo UIIIIIIIllijTIU1TIII7IITIfIIIIIIIUIIIIIIIIIIUIIIT]
0.00 5.00 10.00 15.00 20.00 25.00

Time delay (ns)

Figure 2.21 Minimum SDR vs time delay when S-pulse method is applied to a
waveform containing responses from two 1.0 m wires separated by a time
delay.

44

identical targets if the time delay is not so large as to place significant early-time signal
components into the assumed late-time portion of the convolution (assuming no coupling
between targets). The curves also show that discrimination is not possible when the
targets are identical and have an intermediate range spacing (ie. too large for the early-
time portions to be excluded and too small for the targets to be clearly separated in range
and identified as two separate targets). The method can also be shown to completely fail
when the responses are from different targets and the fact that two targets exist is not
known prior to applying the discrimination scheme.

Next, consider the case of two targets closely spaced in both range and cross-
range. For the military, this is. a very practical situation to consider since aircraft
squadrons are routinely flown in closely spaced formations. It is also important in the
case of a target flying close to the ground or sea since the target images into the highly

conducting surface and electrically appear as two targets. In this situation both a and

If, - ?,I are small and the scattering phenomena becomes very complicated. The first

portion of the late-time before the field scattered from target 1 has reached target 2 is
simply the superposition of the responses as previously considered. After the field
seattered from target 1 has interacted with target 2 and returned to the observer, the
received waveform is no longer simply the superposition of responses. The transient
interaction among targets and the ground or sea causes some interesting effects. The
received waveform can be described by the natural frequencies of the individual
(uncoupled) scatterers prior to the interaction, and a different set (coupled) characteristic
of the system of scatterers after the interaction has occurred. The system natural

frequencies are dependent on the size and shape of the individual scatterers and their

45

 

locations and orientations. This problem is extremely important in the case of target
discrimination since any change of the natural frequencies has the potential to reduce the
effectiveness of the E—pulse and S-pulse discrimination method. The initial course of this
research effort was to establish how the natural frequencies of a target are affected by
the transient mutual coupling. Chapter 3 presents the analysis used to evaluate this
problem and the physical interpretation necessary to understand the transient coupling
problem. Chapter 4 evaluates E-pulse and S-pulse discrimination effectiveness for thin

wire targets when coupling has been included.

46

Chapter 3

Transient Coupling

3.1 Introduction

To preface the problems discussed in this chapter, it is useful to highlight some
of the relevant literature. Tesche [23] was the first analyze the current induced on a
thin wire by a transient field using SEM. In a later paper [24] Tesche examined the
far-zone transient response of a linear antenna using SEM. Umashankar, Shumpert and
Wilton [25] were the first to perform the SEM analysis of a thin wire parallel to an
infinite perfectly conducting ground plane. Later papers by Shumpert and others have
considered the special case when the wire is very close to the ground plane [26] and
when the ground is lossy [27]. Chuang [28] considers the case of a thin wire at an
arbitrary angle located above a perfectly conducting ground. Baum, Shumpert and Riggs
consider the perturbation of the SEM poles for an arbitrary object above a perfectly
conducting ground plane and present results for the case when the object is a thin wire
at an arbitrary angle [29]. Nitch, Baum and Sturm [30] used SEM to evaluate the
effect coupling on multi—conductor transmission lines. Also, in the paper by Riley, Davis,
and Besieris [31] SEM is applied in the analysis of scattering from a random
distribution of scatterers.

Most of the above mentioned papers emphasize the calculation of the SEM mode
frequencies for an object above a ground plane. Several papers present plots of the 8-
plane trajectory of the SEM mode frequencies as the distance above the ground plane is

47

varied. The trajectory is a complicated spiral-like curve that revolves about the point
associated with the natural frequency of the object when it is isolated in free space.
Unfortunately, the significance of these natural frequencies within the context of the
temporally observed currents and fields is not discussed in these papers. Such an
understanding is essential if target discrimination and identification methods based on the
SEM mode frequencies are to be properly implemented in practical systems where more
than one scatterer may be present. Only this author [32] has investigated the effect of
coupling on the performance of the discrimination methods prior to this writing.

The primary shortfall of the above papers is that none discuss the time interval
over which the SEM modes of the coupled wire system is valid and how these system
modes are related to the modes of the isolated wire. Part of this problem is that several
of these papers were published prior to a proper understanding of the use of SEM in the
representation of the scattered field response. The use of SEM in describing the scattered
field response is discussed by Morgan [33, 34], Pearson [35], Felson [36],
Dudley [37], and most recently and most completely by Baum [38]. The essence of
their publications is that the Class I SEM pole series expansion with constant coefficients
is only valid after the global modes have been established. Prior to this, the SEM pole
series must be augmented by some means with terms that result from the inverse
transform of an entire function.

There are many means of attempting to determine the entire function contribution.
The Class II coupling coefficients first put forth by Baum [39] were an attempt to
account for the entire function by expanding the early-time in a series of time modulated

damped sinusoids with frequencies identical to those used in the Class 1 expansion. The

48

Class II coefficients, however, are non-physical since they attempt to represent the early-
time response, which is essentially local information, using a series expansion that is
based on global resonances. There are also attempts to augment the class I coefficient
expansion using physical optics and other complex beam/ray methods. All of these
endeavors are important for future early-time detection and discrimination schemes.
However, for present discrimination schemes based on natural resonance frequencies, the
analysis must be confined to the portion of the response that can be represented as a
Class I pole series expansion.

This chapter attempts to clarify the meaning and significance of the SEM system
modes of a group of scatterers in relation to the temporal response of a system at various
observation positions and under varying excitations. For simplicity, the case of two thin
wires is considered. The theoretical part of the study is presented in this chapter and
begins with a general formulation for the SEM mode frequencies, currents and Class I
coupling coefficients. The behavior of the SEM mode frequencies and currents for a
system of coupled wires are investigated using several methods of varying accuracy and
computational efficiency. Then, the SEM Class I coupling coefficients are calculated and
the temporal scattered field is evaluated using the SEM approach. The SEM results are
compared to the solution obtained via the inverse discrete Fourier transform of frequency
domain method-of-moments calculations. Comparisons are also made to the response of
a single wire to illustrate how causality is important in the SEM interpretation. The
experimental portion of the study is presented in later chapters. The measurement
methodology is discussed in Chapter 5 and measured results for systems of coupled wires

are presented in Chapter 6.

49

3.2 General SEM Formulation
The analysis of the SEM mode frequencies begins in a general fashion and is
subsequently specialized for various cases of interest. Consider a system of P arbitrarily

shaped perfectly conducting scatterers as shown in Figure 3.1. The time varying electric

field E ‘ impressed upon the system generates currents 30") on the surface of the

conductors. These currents in turn generate a scattered electric field E" such that the

total electric field tangential and upon the surface of the conductors is everywhere zero.

Thus,

ip-[E’(r,1)+E‘(r,1)] = 0 (3.1)

where 5,, is a unit vector tangent to the p“ scatterer, F is the location vector, and t is the

time. The double sided Laplace Transform of a time function f(t) is defined as
9100)} . ff(t)e"dt (3.2)

where the transform variable s is the complex frequency. Applying the Laplace transform

to (3.1) results in

5,-ri‘rr91—E‘1rm = o (M)

where the transform domain field quantities are

E"(F,s) s 9,13"(?J)} (3.4)

In the transform domain, the electric field scattered by a conducting object can be written

using the dyadic Green’s function formalism as [40]

50

E1033) 13330.3)

 

2‘

Figure 3.1 System of P perfectly conducting scatterers.

51

3’03) = :3, f GG‘f’Jrflf’MdV’ + Elf-Si) (3.5)

VI'VO

where G(i’,?’) is the dyadic Green’s function, .7(?’,s) is the current density and I: is a

source dyadic that is dependent only on the geometry of the principle volume V,. Since
the scatterers are assumed immersed in free space, the free space electric dyadic Green’s

function is used. It is given by

5(i’f’s) = 111—1 i,vv - i) 60:? .0 (3.6)
41! y
where the complex wavenumber y 5% and the scalar Green’s function is

In the limit of perfectly conducting scatterers, the volume current density .7(i",s) is
replaced by a surface current density I?(F,s) and the integration is carried out over the

surface of the scatterer S. Note that Z is not specified since it is dependent only on the
on the shape of the principle volume V, used in the calculation. To simplify notation, a

principal value surface integration is implied throughout the rest of this general analysis.
The field from a system of P perfectly conducting scatterers can be written as
P

it?» = )3 Me’s-Itxr’sm’ (3.8)

=1
P S,

The boundary condition (3.3) can be applied everywhere on the surface of each scatterer

to obtain a system of electric field integral equations (EFIE),

52

P
i, )3 friars) -,,i('(r’,s)ds’ = -£ -‘i5'(r,s~) res, q=1,2,...,P (3.9)

P1 s,

From SEM, the late-time portion of the current at any point on the surface of the p‘"
scatterer can be expressed as a sum of exponentially damped sinusoidal waveforms which

can be represented as

1?,(8t)= 2a,,1?,,(r.s~,)e’-‘ (3.10)

u--N

where a” is the coupling coefficient for the 11" natural mode current Kym-9.) and s, is

the n‘I complex natural mode frequency of the system of P scatterers. The current thus

has the form of a complex pole series in the transform domain. So

N

I? (m) 3 gag—Lu“) (3.11)
n- -N 3'."
no

Since the time domain current waveform must be real, one requires that s_, = s: and

a”K”(i’,s,,) = [arr K’._,,(?,s_n)]‘ . Then, substituting (3.11) into (3.9) yields

  

P N
5,?“ 2;“ 13an”; “-.s) I? m(i",s,,)dS’ = 3? ~‘E(f,s) res, q=1,2,...,P (3,12)
3:0”

The natural solutions are found by solving the equations with E‘G’Ls) = 0 . To guarantee

the finiteness of the quantity on the left hand side one must require that

53

rel

P
, E f Cifir’s) 1?,,,("’ s,)ds’ = 0 res, q=1,2,...,1D (3.13)

P'1 s
This system of homogenous EFIE’s defines the natural mode frequencies 3,, and the

natural mode currents 1?,(r,s,) of the system of scatterers.

To compute the transient currents and fields using the SEM, the current coupling

coefficients must be determined. Rewriting inhomogeneous EFIE (3.12) gives

g R .3. A '
1- [601?» 2 War 3 -t'E'(?,s) 795 (3.14)
s 3"" S-Sn
MO

where, to simplify notation the integration is carried out over the entire collection of P

seatterers. Next, replace the operation 5- by 3.6;.) -. Note that 13.0) is tangent

everywhere to the surface. Now, integrate over the entire surface of all scatterers to

obtain

 

s"f1‘1‘,, (u, ) [60}? ,s) 13(r',s,)ds’ds= 3f12,(r,s_)-‘E(r,s)ds (315)
8"" .3 I s
neo

Then, interchange i‘ and F’ and use the reciprocity property of the Green’s dyad to

rewrite the previous equation as

CI.

 

          

,3/mn)ds’ds=f1?(r,s_)i(rs)d8 (316)

    

        

 

null! 3-5.
MO

Taking the limit as s approaches s, gives

54

limo. .. “,3- um -,...
S‘SIE‘IE'O. ,3.) [ant/,9) Kn(r.S,,)dS/ds = S'°Sn -£1?.(r,s,,) E (133)“ (3.17)

where all terms on the left :1 #m are zero by definition of the m"I mode frequency (3.13).

Using L’hospitals rule on the left and solving for a, gives
_ l 1 ..
aa - :- [12,0598 (“was (3.13)
n s

where the normalization constant c,I is

c, = g [12,099 -f G(f',f",s) oK,(r’,.s~,)ds’ds |,_,_ (3.19)
S S

3.3 Moment Method Solution

To solve the homogenous EFIE for the natural mode frequencies and currents,
the method of moments is usually employed. Since the moment method is widely used,
only the outline of the solution method is mentioned here while the details are presented
in Appendix B. The case of the EFIE in (3.13) is slightly more difficult than most
moment method problems since both the frequency and the current are unknown. To
solve the problem, the matrix form of the system of integral equations is obtained by

expanding the unknown currents in an appropriate series of basis functions I,

ll
1'50”) = Z c.f.(f') (3.20)

and

Entire domain basis functions such as rims and cosines or sub-sectional basis functions

such as piece-wise sinusoids and pulse functions can be used to represent the current. The

55

equations are then tested using an appropriate weighting function. The testing operation

has the form

f 0,0") ds- (3.21)
S

where typical weighting functions w, include Dirac delta functions, pulse functions and
piecewise sinusoids. If the expansion functions are of the same form as the weighting
functions then the method is called a Galerkin method. If piecewise sinusoidal basis
functions are chosen, the EFIE can be solved easily since there is an exact analytical
solution for the field scattered from a thin wire with sinusoidal current as shown in
Appendix A. After a sufficient number of weighting functions have been applied, a
matrix equation is obtained

A(s=sn) c = 0 (3-22)
where the A matrix contains the overlap integrals given by

A... = [0,00 - f car's) 7.1?) ds’ds (3.23)
S S

and the vector c contains the coefficients of the basis functions representing the unknown
current. Note that equality holds only if s=s,. The natural frequencies of the system of
scatterers can be determined by rooting the determinental equation
det A(s) = 0 (3-24)
The natural mode currents of the system can be determined to within a multiplicative
constant by calculating the null space of the matrix A(s,,).
Rooting of the determinental equation is accomplished numerically. The Mueller

method was used for this research [41]. Since even simple structures have a large

56

number of possible modes, deflation can be used to prevent the search procedure from
wasting valuable computer time searching for modes that are already catalogued.
There were several formulations of the moment method solution used in this
work. All are based on the assumption that the thin wire approximations are valid. This
assumption is appropriate for the class of problems addressed and simplifies the analysis
by eliminating the source point singularity. The first formulation uses a Hallen
formulation for the problem of a group of vertical monopoles connected to an infinite
perfectly conducting ground screen. A pulse function expansion in the current was used
along with point matching. A second more general Galerkin formulation based on the
EFIE and a current expansion in piecewise sinusoidal basis functions was also used.

These approaches are detailed in Appendix B.

3.4 Perturbational Transcendental Solution

A perturbational solution can be obtained for the natural mode frequencies of a
group of scatterers by assuming that the modal current distribution on a scatterer in the
presence of the ground plane or other similar scatterers is nearly the same as the modal
current present on the scatterer when it is isolated in free space. This approximation can

be written as
3,459,.) s a” 13,1535.) (3.25)

where a, is a constant, 13,1653 and s; are the modal currents and frequencies of the

n“ mode of the p“I scatterer when it is isolated in free space. Using this approximation,

the homogenous EFIE becomes

57

P
afG(?,i",s) my ,spbds’ = 0 res, q=1,2,...,P (3.26)

par

Now, replacing the 1" operation by the weighting function f K "(r r,s,,,,) 1.. .ldS results 1n
3.

a matrix equation as before. Note that this procedure is essentially a moment method

approach employing current expansion with a single term entire domain basis function.

The advantage of the perturbational method is that the current is constrained and the

numerical root search is simplified and the modes are easier to characterize.

3.5 Approximate Green’s Function Solution

A closed form approximate solution for the natural mode frequencies of a simple
system of two nearly identical scatterers can be also be obtained. The assumption of
nearly degenerate coupling (ie. the resultant system modes are nearly identical to the
modes of the isolated scatterers) is used and the Green’s function is expanded in a Taylor
series about the natural mode frequency of the isolated scatterers. The Green’s function

is then approximated by retaining only the first two terms in the series expansion. Thus,

60;? .s) .. ewe ,9,,)+ (s 1;) 300,? ass—4L"; (3.27)
The EFIE then becomes
2a,..{é(rf.r .r’.s,..)+ (s -s,:.)——— ”(J—F ”I”, } K ”'0' ,gds' = res, (3.2s)
'p81

58

This relation can be simplified by retaining only the leading nonvanishing terms. Note
that for p=q, the definition of the natural mode currents and frequencies for the isolated

bodies requires that

[afar/3:0) .ip:(F/’sp:)dsl A O FESP (3.29)
s,

For ptq, assuming that

66058.0

a. 1...; < 6080.0 030)

gives

a A Baffle) o .. o P e .. e e e
a,.(s-s,,,)t,'!T (“5200 ,9,,)ds' + g; a,,:,-!é(ri's,) -I?F,(s,,)dS’ = 0
r ”a r

(3.31)

Note that the electric field associated with the n‘II mode of the p‘" isolated scatterer is

Egm a ffifi’fi) .R;(r’s,1)ds’ res, (3.32)
8'
To convert the EFIE to matrix form, replace 3, ~ with the weighting function

figure; ot..}ds (3.33)
S.

to get the matrix equation

A(s=s,,)c = 0 (3.34)

where the elements of the A(s) matrix are

59

(s #47.) 5,, q=p

f 12,219,") ~E’,‘:.(0ds cm: (3’35)
S.

and

1?°(r’,s,‘;)ds’ds (3.36)

3";

“r! K:(F,S';,) r! 566.7] __:S_ag.)|

8.
As with the other methods, the natural frequencies of the system of scatterers are
determined by rooting a determinental equation. The special case of only two conducting

seatterers results in the determinental equation:

(3 '3 15511 “12
det __
“21 (S "320)‘122

= o (3.37)

where s," and 32" are the modal frequencies of scatterers 1 and 2 when each is isolated
in free space. This determinant has the form of a quadratic equation in s
(s-s,°)(s-sz°) (Tub-2, - aua21 = 0 (3.33)

The resulting solutions for s are

s = §:JA+62 (3'39)

where E is the mean of the isolated mode frequencies and A is half the difference

between the natural mode frequencies

5'10 +320 310’s; (3,40)
2 2

 

 

52

60

 

'
')'£ ’0
1 -5 A b.

u .
:. ‘;l h
i—ob ..

1.3

i:

E. {‘1 \-
“W-~. ,‘ l
‘ o
‘
e . ‘

 

while 6 is a ratio

5 , lfn‘fgt (3.41)
“1102

These results imply that the natural frequencies of two nearly identical scatterers are

shifted from the natural frequencies of the individual isolated scatterers. Next, consider

the case of two identical scatterers. Here 5 = .r0 = s," and A = 0 so the natural

frequency of the system of identical scatterers is

a
s = .9018 = :3 (3.42)

Thus, in the case of a degeneracy in the modes of the isolated scatterers, the natural
frequency of the system of identical scatterers splits into two modes that are shifted from

the modes of the isolated scatterer. This is an extremely important result.

3.6 SEM Modes of a Single Thin Wire

This section presents results obtained via the moment method for the SEM mode
frequencies and currents of a single thin wire. The analysis of a thin wire is important
both for testing and validating the computer programs as well as providing a framework
for understanding the more complicated case of two wires considered in later sections.
A comparison of the accuracy and convergence of the solutions for the SEM mode
frequencies using two different formulations is presented. The first formulation is based
on a Galerkin method using piecewise sinusoidal basis functions to solve the EFIE. This

method is denoted EFIE-PS. The second formulation is based on the Hallen equation with

61

a rectangular pulse function expansion and point matching. This method is denoted
HALLEN.

Before proceeding further, it is also useful to review some general characteristics
of the SEM mode frequencies that have been previously established by Tesche [23] and
others. Most importantly, all poles occur in complex conjugate pairs (ie. s,=w, + jw,,
s,=w,,-jw_). Thus, it is sufficient to search for modes only in the second quadrant of the
complex 5 plane. Further, for thin wires, spheres, infinite cylinders, and other canonical
structures, the SEM pole frequencies seem to occur in layers in the complex s plane. An
example of this is seen in Figure 3.2. Here, some of the modes of a thin wire of Va =
200 are plotted. The values shown were computed using the EFIE-PS formulation. The
first layer poles lie close to the jar-axis and occur in roughly integer multiples of 0.1 =
tell. The second layer poles also appear with approximately integer multiplicity but are
placed further to the left of jw—axis than the first layer poles. The third and higher layer
poles are even further from the jar-axis. Because of this layering phenomena, Tesche
established a special notation to identify the modes. The modes are denoted sh, where
(denotes the layer number and n denotes the mode number as counted from the a-axis.
This notation is used throughout this document. It has been observed that the late-time
transient response of most structures can be adequately described using only the first
layer poles since the contribution of poles from the other layers is usually very small due
to their much larger negative damping coefficients. Tesche has also shown that the poles
of the thin wire migrate to the left as the length-to-radius ratio of the wire is decreased.

When performing numerical analyses with the moment method it is prudent to

examine the convergence of the solutions with respect to the number of basis functions

62

1000 l l l l l L l l

 

1

Complex s—plane

1°
(3
c:
1

layer 1

00
c:

c:
L,

T4
O
c:
l 21

a
layer 2 \

l

S”
c:

c:
I

Radian Frequency, w/(nc/I)
§
1

 

 

 

I I I ' I '

35.0 -4.0 -3.0 —2.0 3i .0
Damping Coefficient, a/(rrc/l)

Figure 3.2 Layering of SEM mode frequencies for thin wire (l/a = 200).

63

used in the current expansion. To test the convergence, the case of a unit length wire
with l/a = 200 was considered. The natural frequency of the lowest order SEM mode
was computed for differing numbers of basis functions using both formulations.
Figure 3.3 illustrates the convergence of the damping coefficient 0.1 as a function of the
number of ”segments" per wavelength used in the calculation while Figure 3.4 shows the
convergence of the radian frequency 011. From these plots it is evident that the EFIE-PS
formulation converges much faster than the HALLEN formulation.

To test the accuracy of the formulations the ten lowest order natural frequencies
of a thin wire were computed and compared to the results obtained by Singaraju, Giri,
and Baum [41]. The results of this comparison are shown in Table 3.1. The values
obtained by Singaraju and those obtained using the EFIE-PS formulation are in agreement
to within approximately 1%. The EFIE-PS calculations were performed using segments
of length M20. The small differences are most likely due to the different approach taken
in the solution and the fact that Singaraju implemented the angular integration over the
surface of the wire whereas the author’s program neglected integration in the interest of
computation spwd. The results using the HALLEN formulation are also in agreement
with those of Singaraju to within approximately 2%. The HALLEN calculations were
performed using segments of length M80. As previously demonstrated, these results are
much more sensitive to the size of the segments than the EFIE-PS formulation. Note that
natural frequencies of the even modes could not be calculated with the HALLEN
formulation since it was specialized to the case of monopoles above an infinite perfectly
conducting ground plane. This limitation was accepted in the interest of decreased

memory requirements and decreased execution time of the computer program. This

64

Convergence of SEM Mode Frequency

 

 

43-080; First Mode of Thin Wire (l/a=200)
1
A “ . _ ----
< :.'D-a-o--0'“°"° '0' 3°
0 J
1: .
v
8 3 - 3 2 °
-0.085 - (r r "
I ,o’ ’
E " I z A
.9 j e
.9 - /
a... /
”8 i x
9 I
U1 .1 /
.E -1 P
0. ..
I
E ~ ,
8 j , cacao HALLEN
- DDDDD EFIE-PS
-0.095 -1 ,6
3 I
" I
j 1
3 I
"‘ I
j 1
-o°1oo IIéIIIITIlITITIITTTITTIIIIIIIITIIIIIllI'
O 50 100 150 200

Segments per wavelength

Figure 3.3 Convergence of moment method solutions for damping coefficient.

65

u. wer\
.u“

L100

 

 

31 Convergence of SEM .Mode Frequency
3 First mode of thin wrre (l/a=200)
c 3
\ 2
E ‘ ?
V1.050 -: I
\ '1 l
3 q ,
I 1
.- 3: ‘ cacao HALLEN
5‘ 3 8 ocean EFIE—PS
c: - \
,g 1000 1 \
g 2 b\
L - \
LL 3 q
‘ \
C : \
.3 0.950 1 ‘0‘ ‘
o - ‘9‘ ~ g
m : e. ‘ ~ ’ _ ‘
3 T m — o
- a.
1 0.03. -0- 3° “'0" '9 - — «a
.1
0.900 1
:
00850-III'TTjT'TjT'TTijfi'IIFIITiIIIIITjFIIYUI
0 50 100 150 200

Segments per wavelength

Figure 3.4 Convergence of moment method solutions for radian frequency.

66

Table 3.1

 

Comparison of the natural frequencies of a thin wire with l/a

calculated using various moment method formulations.

200

 

Natural frequencies of a thin wire s,/(1rc/l) for l/a=200.

 

Singaraju, Girl,
Baum

EFIE-PS

HALLEN

 
     

 

   

—0.0828+j0.9251

 

  

-0.0820+j0.9166

-0.0868 +j0.9409

  

 

 

-0.1212+j1.912

-0. 1207+j 1.888

 

 

 

30.149l+j2.884

-0.l485+j2.865

—0.1519+j2.886

 

 

-0.1713+j3.874

-0.17ll+j3.845

 

  

-0.1909+j4.854

0.1905 +j4.827

-0.1926+j4.844

 

 

-0.2080+j5.845

-0.2079+j5.809

 

 

 

 

 

 

 

 

 

 

 

30.2240+j6.829 -0.2238+j6.792 -0.2288+j6.848
-0.2383+j7.821 30.2385 +j7.776 -
-0.2522+j8.807 —0.2523+j8.779 -0.2562+j8.815
-0.2648+j9.800 -0.2653+j9.766 - I

 

 

67

specialization was also useful in the study of the multiple wire case since the number of
possible modes was reduced hence making the mode search simpler.

The SEM modal current distributions for a single thin wire are also of interest in
this research. As shown previously, the perturbational methods use the current
distribution on a single scatterer to estimate the natural frequencies of a group of
scatterers. The magnitude of the current as a function of position along the wire 7/! for
the first principal mode of a thin wire of length I and l/a = 200 is shown in Figure 3.5.
Note that the HALLEN results are only plotted for one half of the wire length since this
program was specialized for monopoles. The phase of the current as a function of
position is shown in Figure 3.6. The EFIE-PS results were obtained using segments of
length M20 while the HALLEN results were obtained using segments of length N80.
The magnitude of the current computed using the moment method is nearly the same as
the cosine distribution that is expected for this mode. Figure 3.7 and Figure 3.8 show
the current distribution for the second principal mode of the wire. Note that the results
for the HALLEN formulation are not available for this mode due to the programming
implementation specified previously. The current distribution for third principal mode is
available from both programs. The magnitude of the current as a function of position for
this mode is shown in Figure 3.9 while the phase is shown in Figure 3.10. In each case,
the agreement between the expected cosine variation and the moment method

formulations are good.

68

1.10
First Principal Mode — Magnitude

8

   
 
     

.0
o
o

.0
on
o

.0
x:
o

P
a:
o

.0
an
o

.0
a.
O

P
u
o

 
 

00.909 HALLEN
09pm: EFIE-PC
Cosine

Current (relative magnitude)

0.20

0.00
-o.so -o.4o -o.30 -o.2o -o.1o 0.00 0.10 0.20 0.30 0.40 0.50
Position, z/l

Figure 3.5 Magnitude of current distribution for 3“ mode of wire with (In = 200.

69

First Principal Mode - Phase
-0.00 I’.—-00~cg°
H

- 1.25 ,’ \X

-1.50 ’

Phase (degrees)

-1.75 0

-2.00 ‘

0090.9 HALLEN
-2.25 099.99 EFIE-PS

 

-2-50 |IIIIIIIII[IIIIII1IrllllrilIIIIIIIrIIIIIIIIIIIIIII]

-0.50 -0.40 -0.30 -0.20 —0.10 0.00 0.10 0.20 0.30 0.40 0.50
Position, z/l

Figure 3.6 Phase of current distribution for 3“ mode of wire with (In = 200.

70

 

Second Principal Mode - Magnitude
1.00 p ' ., -.’ - 11‘
f‘ b. y '1‘
0.90 ’ ‘ ' ‘
I¢ h I O‘
’ \‘ I, \\
$0.80 9 - ~ ’9‘
.0 I ‘ I. \
3 o “ I l
at ' \ l \
c o 70 9 - b
m ' ‘ ’l \‘
o 0' “ l l
E 0 60 .’ ‘ " "
P " ' b
0 a “ I \
.2 ' ‘ at ‘t
*5 0.50 'l .‘ .' .|
— , ‘ a t
‘1’ 0 '.
V0.40 ; ‘.
44 l ‘
8 : x
t 0.30 .' .‘
:3 'i' ‘3
0 ,' opppp Eng-PS .
0.20 ,' Cosme "
: ‘.
. l
0.10 .' i
0 I
0.00

 

IIIIIIIll]IIWTTTIWITTITTT1IIIIIIIITIIIIIIIIITITI
-O.50 --0.40 -0.30

-O.20 -0.10 0.00 0.10 0.20 0.30

0.40 0.50
Position, z/l

Figure 3.7 Magnitude of current distribution for 8'“ mode of wire with l/a = 200

71

 

200-00 Second Principal Mode — Phase

91-01-00-qu

1 60.00
1 20.00
80.00
40.00
0.00

OO-I-O'I-OO—BO-O-O'B-OO-IO-I-Q

-40.00

Phase (degrees)

09°09 EFIE-PS
-80.00

-120.00

-160.00

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
I
I
I

o—o-o-a-oo-ob

lllllllIlllllllLlllIlllIlllIllJIlllIlllI

 

 

-2m-oo IIIIIIIIIIIIIIIIIIIIIIIIIII—IIIIIIIIIII IIII [IIIT]

I I
-0.50 -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50
Position. z/l

Figure 3.8 Phase of current distribution for sh, mode of wire with l/a = 200.

72

8

Third Principal Mode - Magnitude

 

P ' q ’
'I' ‘9 fl -. \
0.90 P i . ' J :
I q I .. \
I [I \
9' “ I I \I
1? 0.80 ,l :3 "
'U I “ ”I It
3 ' , \I
+; ¢ I .' n
.EIIWJ r ‘ n
o» .' 0“ I l‘.
E 0 60 I “ I: I
‘ " r
q; .' Q‘ I l“
2?: i ‘. i ‘1
8 ? ~
V 0.40 ,' i ' ‘l
- : z ' h.
C . I |
a) ‘ I
t 0 30 '6 - g
a : ‘- . ' .
‘ I I I
0.20 f t‘ ; . ..
I
i - 00.000 HALLEN t g
' . . opppp EFIE-PC , .
0.10 ‘, : Cosine ,l '
' r
0.00

 

IIIIIIIIIIIIIIIIIII[ITTTIITII[IIIIIIIIIrTIIIIIIII
-0.50 -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50
Position, z/l

Figure 3.9 Magnitude of current distribution for 31.: mode of wire with M: = 200

73

 

 

 

200.00 —
.. Third Princupal Mode — Phase
175.00 3 p0°°°'"
2 cl
: 5 s
150.00 ~ g l
2 : l
i l l
A125.00 - l l
tn 2 l I
a.) .. ' l
2 ~ i t
01100.00 - :5 p
0 " I I
'0 r : l:
3 75.00 " : |:
0 ~ 3 i
‘3 i : :
0' 50.00 - : E
- l r
d ' I
.. I ‘
"' I
j ." 8
0.00 3 ....-al°°°°°° 00.009 HALLEN "as...“
- 0" 00090 EFIE-PS
’25-00 rIII[IIII]lllrlrllijlllIllIIIIIIIr11r111rll11111fl

-0.50 -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50
Position, z/l

Figure 3.10 Phase of current distribution for 31.: mode of wire with (In = 200.

74

 

3.7 SEM Modes of Two Identical Parallel Thin Wires

This section presents an SEM analysis of the problem of two identical thin wire
scatterers. The natural mode frequencies of two identical parallel wires are computed
using the moment method formulations. Comparisons are made between the moment
methods and results available in the literature. The current distributions for a system of
wires obtained via the moment method are shown to be in agreement with the current
distribution assumed in the approximate analyses for the principal first layer modes.

To begin, the geometry of the problem of two parallel wires of length I radius a
and separation distance d is shown in Figure 3.11. The problem of two parallel wires
was chosen for several reasons. First, the thin wire is a high Q structure that is easily
built, measured, and theoretically modeled. There is also a large body of literature
available for the case of wire scatterers and radiators. The case of parallel wires was
chosen based on the belief that maximum shifting of modal frequencies would occur for
two objects oriented for maximum coupling in the traditional CW sense. For the target
discrimination study, it was initially thought that an estimate of the maximum shift likely
to be encountered would be sufficient to determine if transient coupling effects would be
a factor in the performance of the E-pulse and S-pulse methods.

From the literature and from the perturbational analysis presented in Section 3.4,
it is lmown that when the mode frequencies of two wires are degenerate, the modes of
the system of wires splits from the mode frequencies of the isolated objects. Effectively,
each mode of the isolated wire is replaced by a pair of modes. One mode is symmetric
and the other is antisymmetric based on the direction of current on the wires. If the

current is in the same direction for each wire the mode is symmetric. If the current is in

75

208

 

 

 

 

 

 

 

k 0%

Figure 3.11 Geometry of two parallel thin wire scatterers.

76

 

 

 

 

 

 

g

opposite directions the mode is antisymmetric. In the spirit of Tesche’s notation, the
modes of a two wire system can be denoted as s”, m, where the subscripts a or s denote
either antisymmetric or symmetric modes. In the case of two parallel wires with small
separation distance, the s,, 1.. mode is also referred to as the transmission line mode.

In the literature concerned with the analysis of parallel wires the natural
frequencies are usually plotted as a trajectory in the complex s plane as a function of the
separation distance d. The trajectory of the first antisymmetric mode as a function of
separation distance is shown in Figure 3.12. The results were computed using the
moment method with the HALLEN formulation. For close spacing, the antisymmetric
mode has a small damping coefficient a resulting in a high Q structure. As the separation
distance is increased, the pole spirals clockwise about a point near the mode of the
isolated wire frequency. However, instead of continuing to spiral toward this point, the
mode suddenly begins to move away from the isolated mode frequency and eventually
begins moving toward the origin. This behavior has been observed previously by
Umashankar and others. This is very puzzling behavior since one would intuitively think
that as the separation is increased the mode frequency should tend toward that of the
isolated wire. This puzzling behavior is the focus of this chapter. The trajectory of the
first symmetric mode as a function of separation distance is shown in Figure 3.13. The
Q of this mode is low for small values of separation distance. As the separation distance
is increased, the mode frequency spirals clockwise about the isolated mode frequency and
eventually moves toward the origin in a manner similar to that of the antisymmetric
mode. Indeed, this is peculiar behavior. An even more startling result is shown in

Figure 3.14. Here the trajectory of an antisymmetric mode that is originally in the second

77

1.00 Antisymmetric Mode

 

: '2;Q—* ..... ~ 0.1 It
‘ ,1’ Isolated
- \ 3.0 I ”I
90.90 - 4o ,¢"-‘-----:="c:::1®‘ ":f '''' ’
\ : d/l value °’,*’ ‘ '*
o _ x
t: _
v I
\ 0.80 -" ’1
3 : ,1
5: 070 -
C ' . 5.0 {K
0 I
3 1 a
0' j :
33 0 60 - '.
L“ 'i 60 3,
C j t‘
.9 - .
103 0.50 : 7.0 ‘8
a: . \
: 8.0 \
0.40 -i \
- 9.0 x
0.30 -
1
0.20 -WnTrrm1fiTmeWTmefiTrrrrrrrmfi-nwn-rn
-0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00

Damping Coefficient, o/(nc/l)

Figure 3.12 Trajectory of lowest order antisymmetric mode of two identical parallel
wires (l/a = 200) vs separation distance.

78

 

 

"00 : Symmetric Mode _3,---1.0,.-
. I” ,. cr-----._:‘ \
q ,’ ,” Isolated “. ‘,
d |\ “ 0 249 I,
Q 0.90 -‘ \\‘ ‘ ~~~~~ I: ‘1
\ - ?;Qo&l*'--
o - ,x
I: " ,I
v -' I,
\ 0.80 -‘ ,’
3 " ’I
"i I
5.; 7 : d/l value 6.0.;
5 0. 0 j :0
3 I l
8' - 7.0;
— I
L: O 60 j I‘
c _ 8.0‘.
.9 - l,
.0 0.50 "‘ I
O 9.0
a: I I,
0.40 -‘
I
0.30 -*
0.20 ‘Wmmfiwmm
-0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00

Damping Coefficient, o/(nc/l)

Figure 3.13 Trajectory of first symmetric mode of two identical parallel wires (l/a =
200) vs separation distance.

79

2.20 Antisymmetric second layer made

w/(nc/l)

-.
\I
O

a
fi
00

‘5

Radian Frequency,
8

JrLrilirrilriirlrirrlrlrrJirtilrirtlriI1J1
I
U
I
5"
O

 

0.95 Isolated 29‘ ‘. 5.0
' o r ' 6.
7O f”7.0
0. ’1
, 8.0
r 9.0
I
0.45
0.20 WWW-WWW
-0.50 -0.40 -0.30 -0.20 -0.10 0.00

Damping Coefficient, o/(nc/l)

Figure 3.14 Trajectory of antisymmetric mode from second pole layer for two identical
parallel wires as a function of separation distance.

80

layer of poles for small values of separation distance is plotted. It is clear that for small
values of separation distance this mode damps very quickly. For larger values of
separation distance, the pole moves from the second layer into the first layer near the
frequency of the isolated wire. This mode seems to replace the first symmetric mode as
it moves away from the isolated pole frequency. A discussion on the significance of this
result is deferred until later in this section.

Since the behavior of the SEM mode frequencies is so unusual it is necessary to
check the results against those available in the literature before procwding further.
Chuang [27] computed the natural frequencies of two identical wire scatterers using a
Hallen formulation similar to the one used here. Figure 3.15 shows the damping
coefficient as a function of separation distance for both the symmetric and antisymmetric
modes. The HALLEN results are seen to be in agreement with those computed by
Chuang. Figure 3.16 shows the radian frequency as a function of separation distance for
both the symmetric and antisymmetric modes. Again there is excellent agreement
between the HALLEN results and those obtained by Chuang. Note: the point d=4.01
obtained by Chuang for the case of the antisymmetric mode is probably in error. The
error most likely resulted from confusion with one of the poles moving in from another
layer. Unfortunately, this error causes Chuang to arrive at the wrong conclusion as to
the nature of the system modes' ”as the distance becomes larger and larger, the coupling
between the wires decreases, and subsequently the root becomes closer and closer to that

of an isolated wire”. Chuang’s inability to detect the eventual outward spiral is most

 

Chuang Che-I, ”Transient Waveform Synethesis For Radar Target Discrimination,”
Doctoral Dissertation, Michigan State University, p. 72, 1983.

81

 

 

 

 

0.00 -: ‘0. Damping Coefficient
I
-o.02 3
A :1
b -0.05 ‘3
t: .l
‘< ‘
b -0.07 '2
E: I
9 -O.10 :-
0 I
E -
0.) -0.13 ':
o .-
0 2
O -0.15 f:
.E I
a u
E -0.18 f:
O I
O '3 ---- HALLEN Antisymmetric ‘,
”0°20 '1 00000 Chuang Antisymmetric 0‘
- HALLE Symmetric \
j AAAAA Chuang Symmetric \
-023 . 00000 Umoshonkor Antisymmetric \
-0.25 ':
I
-0428 IlrIIIIIIIIIIIIIIIIlelIIIIIIITIIIIIIIIIIIIIIIIIII
0.00 1.00 2.00 3.00 4.00 5.00

Separation Distance, d/l

Figure 3.15 Damping coefficient of lowest order principle modes of two identical
parallel wires (I/a = 200) vs separation distance.

82

1 .00

 

 

 

: Radian Frequency
2 A A
A 0098 j “ A O A O
< d ‘I o o a
g - r
. I OI \\ O A
V 0.95 - | A a X
: ‘I I. \‘O A
S -I ‘I A i \\ A
0 0.93 - ‘9 .’ x
C. .. \ \ A
Q) _ ‘ O" A \ O
3 - \ ‘
0' j \o {l A \\
e" O 90 " \\ I, A b “ O O on
LI- - A \\ 0 ol’ A “v. I, I
\ --a
g 1 \ ‘0 O ,” ‘0
gow- :
a: : ---- HALLEN Antisymmetric '.
‘ ooooo Chuon Antisymmetric ',
_ HALLE Symmetric I
085 _ AAAAA Chuang Symmetric . “
' . Umoshonkor Antlsymmetnc 1‘
II I
J I
.. ‘.
0.83 - ‘.
0'80 IIIIIjerIIIIIIIIIIIIIIIIIIIIIIITTIIITTTIIIITIIIll
0.00 1 .00 2.00 3.00 4.00 5.00

Separation Distance, d/l

Figure 3.16 Radian frequency of lowest order principle modes of two identical parallel
wires (l/a = 200) vs separation distance.

83

likely due to a large step increment in a tracking algorithm. The fact that the modes
should continue to move away from the isolated mode frequency is supported by the
results of Ummashankar [24] and data transcribed from this paper is in agreement with
the outward spiral results computed by this author using the several different
formulations. Another interesting phenomena associated with the spiralling modes is
observed when the mode trajectories are computed for a system of wires which have
different length-to—radii ratios. Ummashankar and others have empiracally observed that
the mode trajectory of thin wires complete more spirals about the isolated mode pole
frequency than thicker wires. The trajectory of the antisymmetric mode for l/a=400 is
shown in Figure 3.17. It is seen that the separation distance at which the mode trajectory
breaks from the spiral path about the isolated mode frequency has been increased from
approximately d/l =4 to d/l= 6. Even though this phenomena has been observed
previously by Umashankar and Baum they provided no physical interpretation of the
phenomena. As of this writing there is still no adequate physical interpretation of this
phenomena.

Since the behavior of the mode frequencies for the coupled wire system is so
complicated, it is useful to examine the SEM mode current distribution associated with
the various system modes. One expects the current distribution on the wires to be nearly
sinusoidal in order to satisfy the boundary condition of zero current at the ends of the
wires. A plot of the magnitude and phase of the current distribution for the first
antisymmetric mode for different values of separation distance is shown in Figure 3.18
and Figure 3.19. Only the current on one of the wires is considered since the other wire

is identical except for a negative sign as expected for the antisymmetric mode. It is clear

84

 

0'98 Antisymmetric Mode
d/I value 0.1,
A 0-97 l/a=4OD j
< 3.9.---, ‘ g
2 0.96 ’x ,-.-- “ '
v $00,.v’ ‘s‘ \ '1
E I?" \ \ I
0.95 4” \.\ .\ I.
. 8 \ ‘ ’
>\ I? ‘ I 1'
U ’1, ‘* 1 I
5 0.94 :: , '. f
3 I: t I I
o- I' ‘ I
a) '1‘, f .' ,’
E 0.93 1"; 5.0,; f
\ I
C “ \\ f" ’1’
.9 0.92 ‘~ ‘~ 3-94’ .’
.8 \.‘ \\ ’3’! I”
m \\ ‘s\ 0" I,
0.91 \. ‘~.-:,/’ /
\ -l ’
.,)\—“" ’1'
’I” “\ 1"
0.90 6.0 v’ ‘ ’
‘ ‘~‘~.-Lo“‘,'
0.89
0-88lrrrlirllfirfrfirllrirlrIIIITIIII
-0.15 -0.10 -0.05 0.00

Damping Coefficient, o/(nc/l)

Figure 3.17 Trajectory of lowest order antisymmetric mode of two identical parallel
wires (l/a=400) vs separation distance.

85

 

 

1.20
Antisymmetric Mode — Magnitude
1 am ’A
a), “\
o t
a t
A ”a }\
8 7’ “g
,3 o 80 {:5 3,.
C 'I’ \I
0" a” {I
o ,7 I,
E " ‘39
1!" ‘
.g 0 6° 0'," {.0
4-4 'I ‘I
2 07 “.1
0 ‘I
g 0'." “.0
o'.’ ‘39
E 0.40 ‘7 ‘,:.
8 ”0 ”‘0
L 01 ‘I
:3 ,7 99999 d/I - 0.1 I,
0 a: comp d/l - 1.0 \i
,1' 09999 d/l - 2.0 1‘.
0.20 .. 99999 d/l = 3.0 ‘1,
.7 99999 d/l a 4.0 1
."
l
O
I
I
0'00 IIIITIIII'IITIITIIIITIIIWIIIIIIIIIIIIIIIIIIIIITI
-0.50 -0.30 -0.10 0.10 0.30 0.50

Position, z/l

Figure 3.18 Magnitude of current distribution for lowest order antisymmetric mode of
two identical parallel wires (I/a = 200) vs separation distance.

86

 

 

1.00
Antisymmetric Mode — Phase
-0.00 o—-—--o-----O*:‘-‘-:z?3r 933;;lZ-0-----o----.°
".5:1' I! K‘ \“:\&‘
’cc’u’ ’1 \\ \A‘s‘::‘
,A’I’ ’, [I \\ s“ “‘
a In ” I’ \‘ \\ .\‘\‘
I, ’1’ I, f *\ ‘ ‘\‘ ‘
-1 00 0 z A I, \ ‘\ “ 0
A ’ I, I ‘
U) I I I X ‘\ D
0 ,I x \ \
Q) l I \ \
I— ” I ‘\ \\
0' 5 I’ \ k
m g ‘0
3 -2 00 x x
’ \
Q) ,’ ‘
ID I ‘\
O I’ \
g I ‘\
I
-3.00 ,'
I, \‘
I, \I
t 99999 d/l :- 0.1 t
99999 d/I - 1.0
099» d/l a 2.0
-4.00 0.9.9.99 d/l II 3.0
09999 d/l . 4.0
'5-00 [IIIIlIIIIIIIIIIIIII[IIIlIIIIIIIIIIIIIII'IIIIIIIII]
-0.50 -0.30 -O.10 0.10 0.30 0.50

Position, z/l

Figure 3.19 Phase of current distribution for lowest order antisymmetric mode of two
parallel wires (I/a = 200) vs separation distance.

87

from these figures that the current is nearly sinusoidal regardless of the separation
distance. Similar results are obtained for the magnitude and phase of the symmetric mode
current distribution as shown in Figure 3.20 and Figure 3.21.

The next issue concerning the coupled wire systems is the distribution of current
present on the modes that migrate from the second layer of poles to the first layer.
Figure 3.22 and Figure 3.23 show that the current distribution of the first antisymmetric
mode of the second layer changes as a function of separation distance. The amplitude
distribution becomes more sinusoidal as the separation distance is increased and the mode
frequency moves toward the isolated mode frequency and the jar-axis. The phase
difference between the center and ends of the wire is large for small separation distances.
As the separation distance is increased, the phase becomes more uniform across the wire
which is the opposite of Figure 3.19. This is characteristic of the mode of an isolated
wire scatterer. It appears then that as the second layer pole moves closer to the isolated
pole, the current distribution becomes more sinusoidal so as to smoothly replace the first
layer pole that begins to move away from the pole of the isolated scatterer. It is
anticipated that this same effect is present for other poles as they sweep past the poles
of the isolated scatterer.

It is important to note that what is referred to as the first principal mode may not
in actuality be the 8",, of the configuration for a given spacing. Instead, the mode is

tracked from the first principal mode for close spacings.

88

A...:.:—c~::.. .w).......s~v ..s.--..v .1
.900.
P...

 

 

1-20 Symmetric Mode — Magnitude
1 .00 ””*‘\\\
I’.’ \.\\
I, ‘\
A " ‘I
a) I \
8 .3 ‘~
.4: 0.80 ”It, ‘06
8‘ I“, “‘9‘
O I" “\
E ’0' ‘0‘\
02-) 0.60 [:20 9..“
+1 ’0 M
2 [:0 0.x
0 ’II' ‘9‘
L n I‘\
v I. h
.9 0 40 " '19
C 18' '\
a) ll 0‘
L. I: N
L. '0 It
:1 ,7 09999 d/I - 0.1 I;
O I. 99999 d/I - 1.0 1‘,
3' 99999 d/l - 2.0 ‘1‘
0.20 : 09999 d/l a 3.0
.'
D
I
I
l
0-00 IIIIIIIIIIIIllIIIIIIIIIIIIITIIIIIIIIIIIIIIIIIT
-0.50 -0..30 -0.10 0.10 0.30 0.50
Position, z/l
Figure 3.20 Magnitude of current distribution for lowest order symmetric mode of two

identical (l/a = 200) parallel wires vs separation distance.

89

 

0-50 Current Distribution : Symmetric Made
-000 ,yfim‘at.
”’:’” :.:O
a??? ‘x‘ftx
I‘IU,’ \ h ‘\‘\
1”, , I \.‘\ ‘0
— . ”’ II p \\ ‘1‘
050 ,4’ x I, \“ \ {X
[’1’ I, I ‘ \ \\‘
I” i I, ‘ \ \\
8 III,’ IF / \\\ h \\:\‘
0 I I I l \ ‘\ \ \
a _1.00 ill II I, \b \\ \fi
a, A II ’ \ \\ A
D II II \\ ‘
I I \ ‘
a; I, I, \‘ X
g I \ b
In ..
o 1.50 I; \
.C I \\
CL ,’ \
’I ‘I
l \
—2.00 a I
09999 d/l a 0.1
09999 d/l - 1.0
99999d/1 - 2.0
99999 d/l - 3.0
-2.50 099990/1 ,- 4.0

 

-3-OojTTTIIIT7IIIIIIITITTWIIIIIITIIIIIIWIIIIIIIITTTIIIII

-0.50 -0.30 -0.10 0.10 0.30 0.50
Position, z/l

Figure 3.21 Phase of current distribution for lowest order symmetric mode of two
identical (l/a = 200) parallel wires vs separation distance.

90

1.20

1 .00

P
on
o

0.60

Current (relative magnitude)

I, I
, 0
0.40 ,4 .
O l’
' I
I, '
I ,’ S5
I I I
I, ’
(‘1. I’
0.20 ., 6
II, ’
I I’
d
l: 1’
V"
’1
0.00
-0.50

Second

Layer Mode - Magnitude

\
” \:\
ll \\
.0; x 9..
I, \
I, I b \
A, o \A
’ I I \ ‘\\
’1 I \ \
[P I \ b‘
’ I \ \\
’I I \ \\
’1 ¢ ‘\
I” Q \‘
I I I \ \\
I, i ‘ ‘\
I \ \
5’ I \ \‘

I \ \
I \ \ \
I \ \ \
I \ \ ‘
I \ \|\
9 91 a ‘,
I \ \ \
\ \
\ \ 0
\ ‘ \
\ \ \
‘ \
ooooo d/l-2.0 8‘ \ .
cacao (VI-3,0 \ \“
99999 d/ls4,0 “ I31
\I
b 0
\ \\
\ \l
s u
I
‘0‘ ..
\
0‘ 9
\\
\\

 

IIIIIIIITIIIIIIIIIIIIIIIIIIIIIITjIIIIIIrIIIIIIIII

-0.30

-0.10 0.10
Position, z/l

0.30 0.50

Figure 3.22 Magnitude of current distribution for second layer antisymmetric mode of
two identical (I/a=200) parallel wires vs separation.

91

 

20.00
0.00
-2o.oo
A
(D
Q)
‘3 -4o.oo
as
Q)
.0
v
3 -60.00
a
.2
CL
-ao.oo
-1oo.oo
—120.oo
-o.50

Second Layer Mode — Phase

_ ..... am,- -:QpHW-r :-9»----
o--" ‘ ‘,-a-' , ' ~ ‘ ‘~o--- ‘ 9

.v' gr ‘0‘ ON.

6 1i
I |

l' “

I 00000 d/I-2.0 ‘.
,' cacao d/l=3.0 i
.1 99999 d/l84.0 ‘|
I “

' |
:' ‘g
o' a,

-0.30 -O.1O 0.10 0.30 0.50

Position, z/l

Figure 3.23 Phase of current distribution for second layer antisymmetric mode of two
identical (I/a = 200) parallel wires vs separation distance.

92

3.8 Evaluation of Approximate Methods

This section presents a comparison of the SEM mode frequencies obtained using
the moment method to those of the approximate methods outlined in sections 3.4 and 3.5.
From the last section, it is apparent that the SEM mode currents for a system of coupled
wires is nearly identical to the SEM mode currents of the single wire. This is necessary
if either of the approximate methods are to provide accurate results.

To illustrate applicability of the approximate methods to this type of problem, the
approximate methods were used to solve for the mode frequencies of the system of two
identical parallel wire considered previously. Figure 3.24 and Figure 3.25 show the
damping coefficient and radian frequency for the first antisymmetric mode of the system
of system of two identical parallel wires as a function of separation distance. Moment
method calculations obtained using the HALLEN formulation are shown for comparison.
The perturbational results were obtained by using the current distribution calculated from
the HALLEN moment method solution for a single wire. The results for the approximate
Green’s function method were obtained by assuming a sinusoidal current and the Hallen
formulation of the integral equation. It is apparent in both plots that the agreement
between the moment method and the perturbational method is very good as the separation
distance is increased. The approximate Greens function method also provides good results
for small separation distances. However, the approximation of the Green’s function by
only two terms in the Taylor series is apparently invalid for large values of separation
distance. The breakdown of this approximation is evidenced in the large error between
the approximate method and both the moment method solution as well as the

perturbational method that does not approximate the Green’s function. It is especially

93

 

0.00 . .
: Identical Parallel Wires (l/a=200)
J \ First Antisymmetric mode
I\
C.‘ - ‘\ / \ \
\ d ‘\ f \ ‘ l \
U .. \ /n / \ / \
i: _ l ~. I l
V ‘ \‘ / \ I \
\ " \ \ I \
b g l , / \ / \
.5 —O.lO - ‘. \ / \ / \ I \
8 J \ '.\ / \ / \ I \
.9 d s / " / ‘
H- ‘ \ ‘ \
“a - x - 1
I
O ., ‘J i
0 - '.
oi - '.
.S - l
g- .. ‘.
O -O.20 "' z ",
O - “ x'“
' Moment Method - HALLEN
a ---- Perturbation - Transcendental
- — — Perturbation - Approximate G
-0-30 IIITTIIII]IllIllIIIITTTTITIIII—[rrlllllIIIIIIIIITW

 

 

0.00 2.00 4.00 6.00 8.00 10.00
Separation Distance, d/l

Figure 3.24 Damping coefficient of lowest order antisymmetric mode frequency of two
identical parallel wires (l/a = 200) vs separation distance - approximate
methods.

94

1.20

Identical Parallel Wires (I/a=200)
First Antisymmetric mode

0 d
on O
O O

Radian Frequency, w/(nc/l)
p
O!
O

 

 

rurrrrrrIrirrjirrrlrrriillrrerlrrerrl111111rrrl

 

 

0.40
Moment Method - HALLEN
---- Perturbation - Transcendental
- — Perturbation - Approximate G
0'20 rrlllllllllllllllllllllllllllllllllllll'llllllili1
0.00 2.00 4.00 6.00 8.00 10.00

Separation Distance. d/l

Figure 3.25 Radian frequency of lowest order anti-symmetric mode frequency of two
identical parallel wires (l/a = 200) vs separation distance - approximate
methods.

95

ll»

«L:

‘-I

51.

interesting to note that the approximation on the Green’s function breaks down at the
point where the mode breaks away from the spiral path about the isolated mode
frequency. Similar effects are observed for the case of the lowest symmetric mode as
evidenced by Figure 3.26 and Figure 3.27. Again, the approximation on the current is
valid for most all values of separation distance while the approximation on the Green’s
function is invalid for separation distances greater than approximately 4 wire lengths. It
is anticipated that the range of validity for the approximate Green’s function method is
a function of the wire length-to-radius ratio or effectively the Q of the isolated target
mode. This conjecture is based on the previously observed effect of wire radius on the
trajectory of the spiral path and the fact that the approximate Green’s function method
appears to break down for separation distances where the actual mode breaks from the
circular path near the isolated mode frequency. The effect can be illustrated by using the
case of a wire with l/a=400 whose trajectory was previously shown in Figure 3.17. The
damping coefficient and radian frequency computed using the approximate Green’s
function is shown in Figure 3.28 and Figure 3.29. In both cases the approximate Green’s
function method produces erroneous results for separation distances greater than
approximately 6 wire lengths, the same point at which the mode begins to break from the
spiral path about the isolated mode frequency. Since the range of validity for lower Q
targets is for smaller separation distances, one would expect that this method could be
even more limited when dealing with more complicated lower Q structures such as
aircraft.

It is conjectured that this same effect allowed Hanson [42] to use the

approximate Green’s function method to model the currents and radiation from integrated

96

Damping Coefficient. a/(nc/l)

 

Identical Parallel Wires (I/a=200)
First Symmetric Mode

 

 

3
I g r \ I
-l
-0.05 3 /‘ \ / « r \ / \ l
j \\\ -x I \ l \ I
3 “l ‘\ ’ \ ’ ‘ l
: \‘ ’I \ l \ I \ I
-0.10 - \ " \ I \ I \ I
I ‘ / \
+ L: ‘. ’ ‘ I
: I . ' z \ J \
: 1 l . x \/
.l | ' |
-o.1s - ‘. ’l l] ,
: X f i I
- ‘\ I! “
1 ‘.
-o.2o E ,,,,,,,
1 Moment Method - HALLEN """"
" ---- Perturbation - Transcendental
1 - - Perturbation - Approximate G
.‘l
-0‘25 IIIIrilillllIIIIIII]IIITIIIIIIIITIIIIIIIVII—UlI'll]
0.00 2.00 4.00 6.00 8.00 10.00

Separation Distance, d/l

Figure 3.26 Damping coefficient of lowest order symmetric mode of two identical
parallel wires (I/a = 200) vs separation distance - approximate methods.

97

1.20

Identical Parallel Wires (l/a=200)
First Symmetric Mode

‘
'o
o

Damping Coefficient, a/(nc/l)
O O
i» a»
O O

 

 

 

0.40
Moment Method - HALLEN
---- Perturbation — Transcendental
- - Perturbation - Approximate G
0-20 IITIIIIIIIlllI1IIIIIIIIIIIIIIIIrIITITIII'IIIIIIIII]
0.00 2.00 4.00 6.00 8.00 10.00

Separation Distance, d/l

Figure 3.27 Radian frequency of lowest order symmetric mode frequency of two

identical parallel wires (l/a = 200) vs separation distance - approximate
methods.

98

 

 

°'°° : Antisymmetric Mode
- ‘\‘
\
‘ l a=4OO
c - a /
\ -0.02 4 ‘,
0 d I
K
v " ’5‘
s
I \ I,“ 'I ‘\
J -0 05 - I” “t [x ‘\ 'l ‘\ 3: “t
C \ ' .
0 - ‘. ’ I, 5' “ : |
.6 I‘ \ ‘ I ‘\ t “
c ‘| ‘t ‘l I! ‘t i ‘g
“0.) "' z ‘ “ l “ I ‘
0 -0.08 - ‘. “ ‘. i ‘ i i
Q | \\ t : .g ' “
q ‘. | \‘ ’ | : .
g -' | ‘\ ‘ ,‘ || : ‘t
.a “ “ X I. “ p. “
E .‘t ' x I, ‘\ " “\ 'I "
8 -O.1O " t‘ j I \\ i; a
- K ’ ‘
-0.13 -"
j — EFIE-PS
- ------ Approxnmate G
-l
‘0-15 IIIITIIIIIIlllflfiijllIrtlrIIIIrIIrTTTII1IIIIIIII]
0.00 2.00 4.00 6.00 8.00

10.00
Separation Distance, d/l

Figure 3.28 Comparison of damping coefficient calculations using the approximate

Green’s function method for a system of two identical parallel thin wires
(l/a = 400).

99

1'00 Antisymmetric Mode

l/a=4OO

038

9
m
u

 

Radian Frequency, w/(nc/l)
§ §

errlrrrrirrrrlrrrrlrrrrlirrrljirrlirlrl

 

0.85
0.83 —- EFIE-PS

------- Approximate (3
0.80

 

IIIIITTTFIIIIIII]I'll!IIIIIIIITV1IIIUITIIIIIIIUTII

0.00 2.00 4.00 0.00 8.00 10.00
Separation Distance, d/l

Figure 3.29 Comparison of radian frequency obtained using approximate Green’s
function method for system of two identical parallel wires (l/a = 400).

100

circuit dipoles. The dipoles considered were etched onto a thin dielectric film over a
perfectly conducting substrate. The close spacing of the dipole and its "image" effectively
caused the system of dipoles to work as a transmission line resonator with a very high
Q. This high Q allowed the approximate Green’s function method to produce good results
over the range of distances encountered in the printed circuit environment. One might
expect that the method may not yield good results for other types of lower Q elements
such as patch antennas. Verification of these conjectures however, is beyond the scope
of this work. It should be noted that the solution using the approximate Green’s function
has been performed for values of separation distance greater than those shown in the
plots. It is observed that the solution never spirals back toward the correct result and in
fact the predicted mode frequencies spiral outward from the isolated mode pole toward

infinity and eventually enters the right half of the complex s-plane.

3.9 Scattered Fields of Two Identical Parallel Wires

To more fully understand the phenomena of transient coupling and the SEM
modes of a multiple scatterer system, the transient response of a system of two parallel
wires is investigated. The transient response calculated using a SEM Class I formulation
is compared to the response obtained by the inverse FFT of frequency domain moment
method calculations for both the two wire and one wire scattering problems. The
objective of the comparison is to illustrate when the temporal response constructed using
the system modes is applicable and to determine which system modes are significant in

a Class 1 SEM expansion.

101

 

Calculation of the transient response using the SEM formulation requires
calculation of the coupling coefficient and mode fields for all significant modes. The
SEM mode frequencies and currents and modal fields were computed using the EFIE-PS
formulation. The general formulation for the coupling coefficient is presented in Section
3.2 while specifics of the formulation are contained in Appendix B.

For purposes of comparison, all results presented in this section are effectively
band-limited by convolution of the impulse response with a Gaussian pulse of duration
0.6l/c. The duration is taken as the interval between the 2% of maximum points. The
bandwidth of the Gaussian pulse includes first layer SEM modes up to s“ for the case
of a single wire. Since the two wire system has first layer modes that are near the modes
of the single wire, the SEM modes up to s“, and 31.5,, are used to construct the transient
response.

Since the vast majority of radar implementations are mono—static, only the
backscattering response is considered. Excitation is chosen as a plane wave with the
impressed electric field parallel to the axis of the wires. These assumptions do not limit
the validity of the general conclusions.

To begin the discussion, consider the case of two identical parallel wires
(l/a=200) that are separated in down range by d/l= 1.0. The principal layer poles
computed for this case along with the normalization constant a for a distance of 1001 are
listed in Table 3.2. The coupling coefficients and modal field strengths for down range
separation are shown in Table 3.3. It is clear by examination of the coupling coefficients
that the antisymmetric modes are dominant in this configuration. The antisymmetric

modes dominate in this configuration since the incident pulse does not excite the wires

102

 

 

  

 

 

  

 

  

 

  

 

  

 

  

 

  

 

  

 

  

 

 

Table 3.2 SEM parameters for d/l = 1.00.
Mode 0 x109 np/s 0: x109 rad/s Re{CIII F
1,1,a -0.0656 0.8274 0.0510 0.1862
1,1,s -0.0795 0.9158 0.0037 0.1638
1,2,a -0. 1538 1.7977 0.0570 0.2646
1,2,s -0.0830 1.7707 0.0725 0.3443
1,3,a —0. 1307 2.7143 0.0569 0.4383
1,3,3 -0. 1466 2.6985 0.1055 0.4064
1,4,a -0. 1613 3.6447 0.1279 0.4995
1,4,s -0.1657 3.6159 0.1710 0.5284
1,5,8 -0. 1592 4.5425 0.1652 0.6645
-0.1998 4.5705 0.1409

  

 

 

 

 

103

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 3.3 SEM coupling coefficients for down range separation distance d/l=1.00.
Mode RetAnl Im{A.} Re{E..} Im{E.}
1,1,a 0.038271 -0. 13028 -0.33590 1.16595
1,1,s 0.009256 -0.01961 -0.08121 0.16834
1,2,a 1.02x10“ -7 .79x10'17 -9.18x10‘° 5.97x10“
1,2,s -9.18x10“ -2.15x10‘“ 8.15x10’13 1.92x10l3
1,3,a 0.058433 -0. 13872 -0.50490 1.24539
1,3,s 0.052523 -0.03116 -0.46391 0.27568
1,4,8 2.88x10“ 4.20x10'“ -2.57x10tl3 -3.73x10‘3
1,4,s -2.82x10“ -1.l6x10“ 2.51x10l3 1.03x1013
1,5 ,a 0.075527 -0. 144720 -0.64593 1.30580
1,5,3 0.069919 -0.043287 -0.61446 0.39014

 

 

 

 

 

 

104

 

at the same time and in the same direction. The backscattered field response is shown in
Figure 3.30. Note that the moment method calculations for a single wire and the system
of wires are identical until the response from the second wire is received and is a clear
demonstration that causality is included in the moment method calculations. The response
obtained from the SEM formulation using the system modes clearly does not agree with
the moment method results until the incident wave has passed the second wire. The point
on the graph labeled system late-time indicates the point at which a Class I expansion
using the SEM system modes should be valid according to the Baum’s criteria [38]. This
criteria determines the time at which the scattered field can be represented solely by a
convergent Class I SEM expansion with no entire function contribution. Essentially, this
criteria states that the scattered field can not be guaranteed to be a convergent Class I
expansion until the response from the last excited portion of the scatterer traverses to all
points on the scatterer and then to the observer. It is also important to note that the SEM
response appears to be accurately represented for this case by using only the first layer
system poles.

Next, consider the case when the same wires are separated in cross range by
d/I= 1.0. The coupling coefficients and modal fields for this case are shown in Table 3.4.
It is clear from the table that the dominant system modes in this case are symmetric. The
strong symmetric mode excitation is the result of the incident field exciting the currents
on both wires at the same time and in the same direction. The backscattered field
response is shown in Figure 3.31. Again, it is clear that the response from two wires
separated in cross range is identical to that of a single wire until the wires have had time

to interact. In this case, the SEM results based on the system modes seem to provide a

105

 

 

 

 

 

 

 

 

1309005 " Gaussian Pulse Response
‘1 , | o _
a Down: range separation d/l—1.0
q : ',
I
_ I Moment method
"009.006 : »---- SEM (10 modes)
" : ~ - - Moment method (1 wire)
1 :'
l
5.00E+005 - :
A ‘ .' ..
s . :
'4: - .'
2 q ;
9 0.00E+000 - ’
V d
a -l
O, -
o
g a:
o - ..
> 5.00E+005 ,
-1.00E+006 A .
1 System Late Time
-1050E+m6 1IIIIIIIIIIIIIIIIIIIIIIIIIIUIIITTIIIIII]
2.00 4.00 10.00

6.00
Time, t/(l/c)

Figure 3.30 Backseattered field response of two identical parallel wires (l/a = 200).

106

 

 

 

 

2.009005 '3 Gaussian Pulse Response
3 Cross range separation d/I=1.0
-I "\
1.50E+006 - .' ~,
: i ‘\ Moment method
.. .0 \ s‘ """ SE“ (10 mOdQS)
. .4 l ‘, ‘ - - Moment method (1 wire)
1.00E+006 - : : '~
: z .
2 .' I x’ \
A 5.00r:+oos a .' l I ‘
Q) _. I \ "
> I I I /
._ . . \
+1 .- , I I I
.9 J I I I \ ’
'2 0.00E+000 - -’ . I ‘ I
v . \
. I I ’ \ I
Q) .. | I I \ I
g " I I I \ I
: -5.00£+oos d ‘ I I ~ ’
s : . . ‘ ,
—1.00£+006 ~ I ' ’ ’
-I | l
I I '
- I I .
-1.50E+006- , System Late Time
‘ l
‘Z-OOE+0°5 II1IITiIfirIIIIIIIIIIIIIIITIII|

 

 

2.00 4.00 00 8.00

Time, t/(l/c) 6.

Figure 3.31 Backseattered field response from two identical parallel wires (l/a = 200)
separated in cross range by d/I = 1.0.

107

= ilL. I. hilhlsllhIIL..llL

~ N N N \\

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 3.4 fraupling coefficients and modal fields for cross range separation (M =
Mode Re{A.} Im{A..} Re{E.} Im{E.}
1,1,a -3.62x10‘° -2.88x10‘7 -0.00759 -0.00106
1,1,s -0.01090 0.135075 0.08388 -1.21551
1,2,a 3.91x10‘7 -6.80x10°‘7 -3.09x10‘° 5.53x10“
1,2,s -6.53x10" 1.61x10“ 5.83x10" -1.66x10‘3
1,3,a 1.66x10‘u 8.13x10‘l3 -0.02407 ~0.00450
1,3,s ~0.013574 0.146805 0.07812 -1 .32391
1,4,a 2.33x10'“ 1.49x10“ -2.18x1013 -l.19x10"3
1,4,s -1.63x10‘“ 2.98x10"5 l.44x10"3 -3.68x10“
1,5,a 3.54x10“ 1.31x10‘5 -0.04249 -0.00448
1,5,s -0.021844 0.142170 0.12397 -1.28824

108

 

good approximation for times prior to the beginning of system late-time. It is believed
that this is purely a case of luck and does not represent physical reality in most cases.
The reason for the reasonably good fit of the SEM system modes is that the modes
closest to the modes of the isolated scatterers dominate the response. These system modes
are thus similar enough to represent the response of the isolated scatterers for a short
time span. It is also important to note that only first layer poles are required to accurately
represent the system late-time.

Next consider the case when the wires are separated by d/l=2.0 in down range.
The SEM modes and normalization constants are listed in Table 3.5. The coupling
coefficients and modal fields for the down range separation are shown in Table 3.6. As
for all down range separations the antisymmetric modes are dominant. The backscattered
field response is shown in Figure 3.32. Again, the SEM modes fail to accurately describe
events prior to the start of system late-time.

For completeness, consider the case of cross range separation d/I=2.0. The
coupling coefficients and modal fields for this case are listed in Table 3.7. The
backscattered field response is shown in Figure 3.33. This figure gives a better indication
of the inapplicability 0f the system modes prior to the time where the wires have

interacted.

109

 

Table 3.5 SEM frequencies and normalization constants for (M = 2.00.

  

 

  

 

 

 

 

 

 

 

 

 

 

Mode 0 x109 np/s 0: x109 rad/s Re{Cn x104} Im{Cll x108}
0.9022 -0.00994 0.15252
0.8449 0.05516 0.19512
1.7726 0.08644 0.27371
1.7887 0.05349 0.34251
2.7168 0.07154 0.39778
2.6957 0.12998 0.41144
3.6447 0.12794 0.49951
3.6160 0.17101 0.52845
4.5508 0.17241 0.58681 .

0.11768 0.61886 :

 

 

 

 

 

110

III blinl LIIIILIIIDIII FIIISIIFLIII [— t<l N 1.1IIKIIL'\\

Table 3.6

 

Coupling coefficients and modal fields for down range separation (M =

 

 

 

 

 

 

 

 

 

 

 

 

R6{An} Re{An} R8056} 1m{E.} 11

0.037546 0.049997 -0.32796 0.429997 ‘
-0.064562 0.132913 0.56451 -1. 18430
-3.29x10‘° 2.58x10“ 2.93x10‘” -2.09x10"’
8.36x10‘” 3.191(10m -8.35x10‘16 -2.68)t10"5

0.13674 -0.046731 -1 . 19751 0.41143

0.11905 -0. 152031 -1 .03267 1.13636
1.90x10" 8.82x10‘“ -1.67x10’13 -7.77x10'l3
-3.42x10" 4.39x10“ 3.017(1013 3.86x10‘3

0.22144 0.024776 -1.94584 -0.00036

-0. 17575 0.14929

 

111

 

1.52140

 

 

-l.3594l

 

  

 

 

 

 

 

 

 

1-5°E+°°5 : Gaussian‘ Pulse Response
I‘ .
a Down rqnge separation d/l=2.0
'1 l '1‘ Moment method
'1 ‘ : \ ----- SEM (10 modes)
10054005 .1 \I.| : ‘, . - - Moment method (1 wire)
7 l l
- I I
I I
'1 l i
‘ l l
0 5.0054005 - . g
.2 ' ' l
32' 7 E :
o ‘ l l
h -4 I :
. 0.00E+OOO - E :
2 4 . .
.22 . l l
.. . a s
.3 ~ :
+4 -5.00E+005 "1 ' '
o I I
0 ‘ 1 i
LTJ -l t O
7 I : :
" i .'
-1.00E+006 d g .
.1 : .’ System Late Time
7 .'
u I
I
_, l
-1-50E+005 ‘WW‘FFFFW‘IWH‘W
0.00 2.00 4.00 6.00 8.00 10.00

Time. l/c

Figure 3.32 Backseattered field response of two identical parallel wires (l/a = 200)
separated by (M = 2.0 in down range.

112

.- 1. .l I .1. .3,L|I.IFIII~||..~INII N IL 111.\\

Table 3.7

 

 

 

!

R6{An}

Im{An}

  

‘

Coupling coefficients and modal fields for cross range separation d/l =
2.0.

 

  

3.05x10"

3.19x10“

0.035973

0.005850

 

  

 

 

 

 

 

1,1,s 0.008085 0.12532 0.036107 4.12952
1,2,a 2.94111017 2.22111016 -4.93x10"° 4.791110“
1,2,s 1.18x10’“ 1011:1011s 1431:1015 4.451110“

 

 

   

1.05x10“

5.46x10“

0.11914

0.014623

 

 

  

0.003837

-0.11769

-0.074618

1 .05 670

 

   

2.33x10“

1.49x10“

-2.38x10"3

-7.15x10“

 

 

 
 

 

1.63x10-14

2.89x10‘”

1.371(1013

-6.70x10“

 

 

 

-l.36x10‘"

-5.54x10l3

   

0. 19929

 

 

   

0.020386

 

 

 

-0.011935

    

 

0.115884

113

 

   

 

-0.07532

-1.03891

 
    

 

      

   

 

 

 

 

 

 

 

 

 

1.509006 '; Gaussian Pulse Response
1 Cross range separation d/l=2.0
.1 :80“?th methjod)
----- m0 es
‘-°°E+°°5 j - - - Moment method (1 wire)
5.0084005 -«
0 .4
.2 '7
«no-J cl
2 0.0084000 -
m H
L
j d I
0 -
E 4008400521 i ,
o - l '
e: -1 i l
.0.;
Q ‘l
2 -1.00E+006 ‘1 I
to - |
I l
q l
—1.5084006- 1" System Late Time
-2°OOE+006 IllIlIllIIITIIIIIIIIITIITIIIIIIIII]lllIl
200 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00
Time. t/(l/c)
Figure 3.33 Backscattered field response of two identical parallel wires (I/a = 200)

separated in cross range (d/l = 2.0).

114

3.10 Scattered Fields of Two Parallel Wires of Different Length

Another interesting case is backscattering from wires of different length. The case
of wires of length 1.00 m and 0.80 m is considered. The wires are parallel and axially
separated by 1.00 m. The GMC pulse from the previous section is used to smooth the
transient data.

The results for down range separation (the wave strikes the 0.80 m wire first) are
shown in Figure 3.34. It is again clear the agreement between the SEM calculation and
the moment method are good for times after the pulse has left both targets. For early-
time, the SEM mode series does not represent physical reality.

The results for cross range separation are shown in Figure 3.35. Here it is again
demonstrated that the Class I expansion in system modes provides a reasonably good fit
for times prior to the onset of system late-time even though they strictly do not apply
there. Baum has already demonstrated that the Class I series does not converge during
this time, however, a limited number of poles may represent approximately over a

limited time span the response.

3.11 Summary

This Chapter presents the SEM analysis of coupled wire targets. The variation of
the SEM system modes has been studied both from the standpoint of a parameter
variation and interpreted using the transient backscatter response. It is seen that as
separation distance is increased that the poles spiral about the isolated mode frequency.

The poles eventually move to the left away from the isolated mode frequency and are

115

    

 

 

 

 

 

 

 

‘-5°E+°°5 1 Cpussian Pulse Response
, 1
I
j 5 1 wire 1: 1.0x0.005
.. l 1| Wire 2: O.8x0.004
1.009005 ._ 5 1 down range sep = 1.0
I E 3
I
7 i ; —— Moment Method
7 1 : ------ SEM (10 modes)
A 5.00E7I'005 -‘ 1 1
3 7 i E
u 77 ,‘ 1
2 - 1 :
0 a, I
L. "‘ , 1
V 0.0084000 - ’ f
8 1 l 5
i: . g
0 '1 1 I
': -5.0084005 - l :
""' 1 1
U “ 1
.32 - '.
LIJ “ 1 1
J u : '1’
-1.0084008 - l . ,
.. :5 System Late Time
-l 1‘;
-1°m+m6 TTIFIIIUI'TIIIIIITII[TUTIIIlirWTTTTIIITj
0.00 10.00 20.00 30.00 40.00
Time (ns)

Figure 3.34 Backscatter response of system of two different length wires (1.0m and
0.8m) separated in down range (1.0m).

116

 

 

 

 

 

 

 

 

 

 

2009005 '3 Gaussian Pulse Response
q
1 wire 1: 1.0x0.005
. wure 2: O.8x0.004
a corss range sep = 1.0
.l
1.0084006 -
4 Moment Method
: ------ SEM (10 modes)
A
a, d
.2 ‘
,3 _
2 -
0 .1
L. d
V 0.0084000 -
2 I
.9 -
1... 1
.l
.9 «
L- .1
H
o .1
2 .-
uJ -‘I.OOE+006 -
7 l
1 System Late Time
1
-2-mc+m6 IIIIITjTIIIIIIIIIIIlIUIIIIIUTTTTTjjIIII]
0.00 10.00 20.00 30.00 40.00
Time (ns)

Figure 3.35 Backscatter response of system of different length (1.0m and 0.8m) using
SEM and moment method.

117

replaced by other poles that migrate from the left. The current distribution of the
dominant SEM system modes is nearly sinusoidal. It is seen that the current distribution
of the second layer poles become more sinusoidal as they migrate toward the poles of the
isolated scatterer. These poles dominate the transient response when they are near to the
poles of the isolated scatterers. It is demonstrated that the transient response of the
system of wires can be represented by the SEM system modes only in the late-time of
the system. Some portions of the response of the system prior to the onset of system late-
time can be represented by a Class I expansion in the SEM modes of the isolated
scatterers. This is an important observation and demonstrates the concept of a sub-
resonance which may be useful in discrimination for some classes of targets.

This chapter also presents approximate formulations for the SEM mode
frequencies of nearly identical bodies. These approximate methods were shown to

provide good results if used within limits imposed by their approximations.

118

Chapter 4

Discrimination of Coupled Wires

4.1 Introduction

This chapter elaborates on the problem of discrimination using E—pulses and S-
pulses in a multiple targets situation as discussed in Chapter 2. The special case of two
parallel wires is considered and the discrimination results are interpreted using the
physical picture obtained in Chapter 3. The transient scattered field response from two
parallel wires is obtained via the inverse Fourier transform of frequency domain moment
method calculations as discussed in Chapter 2 and detailed in Appendix A. Since the
effect of noise on discrimination has already been investigated in Chapter 2, this chapter

deals strictly with noise free data.

4.2 Cross Range Separation

This section describes the application of the E—pulse and S-pulse discrimination
method to theoretically generated transient scattering data for a system of two identical
parallel wires separated in cross range.

To investigate the effect of coupling on discrimination, the response of a system
of two identical parallel 1.0m wires (l/a=200) is computed for different values of cross
range separation distance. These responses are then discriminated using 9 different F.-

pulses constructed for isolated wires of lengths 0.80m, 0.85m, 0.90m, 0.95m, 1.00m,

119

1.05m, 1.10m, 1.15m, 1.20m. The I/a is chosen as 200 for each of the isolated wires.
Note that in this analysis the discrimination process assumes that there is only one target
present. Thus, the start of late-time is estimated based only on the maximal two-way
transit times of the wires. The EDR values obtained for various values of separation
distance are shown in Figure 4.1. It can be seen from this graph that the discrimination
levels are a function of the separation distance. More importantly, the discrimination
method gives the wrong result for some values of separation distance. As a reminder, the
target with an EDR of zero is the target selected by the discrimination procedure. To
more clearly show this effect, the EDR values of wires of length 0.95m, 1.00m, and
1.05m are shown in Figure 4.2. In this figure it is seen that for d/l < 0.3, the 1.05m
wire is incorrectly chosen since the EDR for the 1.05m wire E-pulse is zero. The 0.95m
and 1.05 m wire E-pulses however provide small EDR values. When the separation
distance is between 0.3 and 0.6 the 1.00m wire is correctly chosen. When the separation
distance is between 0.7 and 1.3, the 0.95m wire is chosen as the correct target. To
interpret these results it is useful to refer to Figure 3.16. This figure shows the variation
of the radian frequency of the symmetric system mode as a function of separation
distance. It is clear that for d/l between 0.7 and 1.3 that the frequency of this mode is
significantly higher than the frequency used for constructing the 1.00m E-pulse. In fact,
these frequencies are very nearly the same as the ones used for the 0.95m E-pulse.
Moreover, for d/l < 0.3, the frequencies are closer to those of the 1.05m wire than the
1.00m wire. While for (M in the 0.3 to 0.7 range, the radian frequency of the system of
wires is similar to the 1.00m wire and the discrimination process selects the 1.00m wire

as the correct target. Thus, oscillations of the system made frequency about the mode of

120

1111‘

0
0
6
t a A +
five I I; a vmo a a 0a i
a e; I v a e; .4 ya u
a; If a; co yr ’ a I * O
eeeeeeeee O ’39 I .71 ‘9 . nu.
Bbbbbhbbb . 4. ,7 ’4 I: s .
uuuuuuuuu . e .I 7 «A: . 5
$3334 . ., . ,, .. ,, .
I
EEEEEEEEE G. a 4th. 4% Q; \+ .1
s o s I
eeeeeeee . . . e .4 , t .
cm a" I" a" a“ a" a" 0" a" C Q I \ If, \ II \ \ H O
wwwwwwwww .. .4 so; u s) . O
I e
mmmmmmmmm .. .. .... m 4
s s a s I \ I
050505050 . . 4 \
880900142 e.M w 9. «.4 .Ow .r
c
. 3

XXXKK
teeee
W
9.
II
1?.
. 21‘?“
It 'I'~
#387
39$
7
1's
°. 1"
3 00

Response: Two 1.00.m wires, parallel, broadside,
1.00

cross range separation
00000,
at..:'
8‘.
X9 9
I x65:
8 :
‘H
200

 

011 0111 x t O
‘4. - dd d !u .4 u -d u q - Wu d ~+1d fi- ud q 1d 1 u q q—Oe
0 O 0 0 O 0 0 00
O. O. O. 0. 0. O. O. 0.
O 5 0 5 0 5 O 5
3 2 2 _

33 «8

Separation Distance, d/l
Variation of EDR vs cross range separation distance for two parallel 1.00
121

m wires (l/a=200).

Figure 4.1

 

e.
..m
S
w 1*; 9’ fl
0 ’0 II n
r a I
b eee .7.- ..«v +
- huh. .- .- .
.4. mm... x...... .
w E... 9. M4 +
.o ............ x u
P w w w \t. u
. m m m . .
s \\ II
m fimw .+t xv t
.1 011' \ II \\
w WWW x. .X.
+\II'- *H' I
n - -
m.w +--+-..t9.-0.-
t \ 111
W 0 .----0....0 1m.“
r I‘IQ‘ \*\
1. a 0W.- K...-
D. {for- III--
e .0 -
O s 0 X..-
w 90. +141
T 8 \gl
9 1.. Av:-
.. n \1 0
8 1111.1 -111
S 0 1111111 + 111111111 *1
r 1111.?1 1111*1
n 4. ....... 8.....-
o s 0+ IIIIIIIII It. IIIIIII
015 IT lllllll at]:w lllll t
S o II.I+ I
e r M30.
R C .0. -
-0.-
°\“ *\
O 0 O O 0 O 0 O O
O 5 O 5 O. 5 O 5 O
0 7 5 2 0 7 5 2 O
2 1 4| 11 «I
83 m8

3.00 4.00 5.00 6.00
Separation Distance, d/l
122

2.00

1.00

wires (l/a=200).

0.00

-2.50

—5.00
Figure 4.2 Variation of EDR vs cross range separation distance two parallel 1.0m

the isolated 1.00m wire natural frequency result in the discrimination procedure
incorrectly identifying the target. The results shown in Figure 4.3 illustrate that the first
mode S-pulse is affected in the same manner as the E-pulse. Thus, coupling between
identical parallel wires separated in only in cross range causes the response to appear
similar to that of wires that are approximately 5% different. This causes the both the E-
pulse and S-pulse discrimination scheme to fail for some values of separation distance
and reduces the discrimination levels for most values of separation distance. It is seen
that discrimination is still possible even for small separations distances provided that the
targets are at least 10% different in length. It is also seen that discrimination improves
as the separation distance is increased. The discrimination scheme provides the correct
result for most cases for d/l>4 even for wires that are only 5% different in length. It
is expected that the discrimination continues to improve as separation distance is
increased. This is due to two different effects that were observed in Chapter 3. First, as
the separation distance is increased the response of the system is dominated by the SEM
modes closest to the first layer modes of the isolated scatterer. Secondly, as the
separation distance is increased the interactions between wires is sufficiently delayed and
the discrimination is effectively performed only on the isolated wire response. If the
delay between wire interactions is larger than the observation window W” the targets can
be considered as isolated as far as the discrimination process is concerned. The
simulation performed here used a window duration appropriate for the discrimination of
a single wire target. This choice is appropriate if it is not known beforehand that two
targets are present. Unfortunately, the observed reduction of discrimination level for the

high SNR theoretical data leads to the conclusion that discrimination of similar targets

123

Response: Two 1.00 m wires, parallel, broadside,
cross range separation

30.00

 

 

- m wire S-pulse
- m wire S-pulse
- m wire S-pulse
25 00 _ m wire S-pulse
' . m wire S-pulse
_ m wire S-pulse
m wire S-pulse
7 m wire S-pulse
20 00 77 m wire S-pulse
' -1
q
-4 ’0‘
d I ’l ”q ‘\
$1500 A ,q‘_:,,.--4:\\ ’49
3 d .- I ’A‘ ‘\‘\b:/ .5
d ---6 """ "#\~ \fi ’0
g " \‘7.’ ."
‘1 ‘’I \ ’I
11110.00 - "‘~~. ,' ‘. ’ o
-l 7‘X"“‘“" v’
.. I \ ‘\ I” ’4-
- ’I \‘ A ’1’
d I’ “ +I’
5.00 - ‘1: ‘1.’
-l ’ \ ‘1
I \ I‘
.1 \‘>”*‘ ’I “
: a"’ \‘\ 1"“ 0'...0
0.00 - ‘4’ ‘18-th
-5.00 ”WWW
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00

Separation Distance, d/l

Figure 4.3 Variation of SDR vs cross range separation for two parallel 1.0m wires
(l/a=200).

124

 

 

is difficult if there is more than one target and the targets are separated in cross range.

4.3 Down Range Separation

This section considers the case of E—pulse and S-pulse discrimination of two
identical parallel wires separated in down range. This case is similar to the time shifting
of two wire responses previously discussed in Chapter 2 except the effect of coupling is
included to make the analysis more accurate when the targets are not separated by large
distances in cross range. This assumption was was required in Chapter 2 to meet the
meet the criteria that the wires are electrically uncoupled.

The effect of coupling on the EDR values for down range separation is shown in
Figure 4.4. It is clear in almost all cases that the effectiveness of discrimination is
severely degraded for all values of separation distance shown. It is especially unfortunate
that the EDR values are severely reduced for close spacing since it was shown previously
that discrimination could be accomplished for close spacing in down range provided that
the targets were effectively uncoupled by a large cross range separation. Overall, an
effect similar to the cross range results are obtained with respect to the variation of the
discrimination levels with separation distance. In this case however, the response of the
system is dominated by the antisymmetric mode for most values of separation distance.
Also, the strong early-time contribution from the second wire tends to obscure the
physical picture of the simpler case of cross range separation. The early-time contribution
also prevents the discrimination from improving for larger separation distances as was
observed for the case of cross range separation. Unfortunately, the S-pulse method yields

similar results as evidenced by the plot in Figure 4.5. Thus, for down range separation

125

Response: Two 1.00 m wires, parallel, broadside,
down range separation

    

 

30.00 -

- 000000.80 m wire E-pulse

2500 _, 00m 0.85 m wire E-pUISO

# 00000 0.90 m w1re E-pulse

00000 0.95 m wire E-pulse

5 0000-11 1.00 m wire E-pulse

"‘ ++++~+1.05 m wire E-pulse

4 "4.4.4- 1.10 m wire E-pulse

20.00 - “a“ 1.15 m wire E—pulse

. “e00 1.20 m wire E-pulse
815.00 4
'0 -l
V .l

0:

o 1
Lu 10.00 -1
-l
5.00 -
0.00 q
q
-5.00 -7

 

“WWW—“WWW”
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00
Separation Distance, d/l

Figure 4.4 Variation of EDR vs down range separation distance for two parallel 1.0m
wires (l/a 77—7200).

126

 

it is observed that the discrimination is severely degraded for all values of separation
distance. For close spacing, the degradation is due only to the effect of coupling, while
for greater separation distances, the degradation is primarily due to the presence of the
early-time response from the second target. It now seems that an effective
countermeasure for the E—pulse and S-pulse methods is to fly aircraft in a close V
formation to cause both coupling among aircraft and to intersperse early-time effects into
the late-time response of the leading aircraft. This method will probably also be effective
even for lower Q targets and targets that are dissimilar since the principle of corrupting

the late-time response with early-time data will still be present.

127

Response: Two 1.00 m wires, parallel, broadside,
down range separation

     

30.00 -
- 00000 0.80 m wire S-pulse
2500 _ 09999 0.85 m w1re S-pulse
_ 40000 0.90 m wire S-pulse
00000 0.95 m wire S-pulse
‘ new-01.00 m wire S-pulse
‘ 444441.05 m wire S—pulse
- 1.414.414. 1.10 m wire S-pulse
20,00 — «40¢ 1.15 m wire S-pulse
- «04“ 1.20 m wire S-pulse
.1
a 1500 l,
3 ‘t
" e
a: ‘ "
O III
(I) 10.00 -7 -
7 t
- 1
.. 7"‘~’1’""‘“‘-a
7 7.11 3
5.00 - ‘.\ ‘\” ,(“\;\‘\
-' \vnvl \\ “‘
-=. ..... =.:\..:>:s.=.:~:§
" I If; ‘51
7 l’ ‘37";
0.00 - +0":- 4. -------- ‘ l
q
-5.00 -

 

WWW
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00
Separation Distance, d/l

Figure 4.5 Variation of SDR vs down range separation distance for two parallel 1.0m
wires (l/a=200).

128

Chapter 5

Transient Measurements

5.1 Introduction

As discussed previously, accurate measurement of the late-time portion of the
transient response of a complicated scatterer is essential for implementation of the E/S-
pulse discrimination method. Extraction of complex modal frequencies from measured
data with good accuracy is difficult [9]. There have been many papers written concerning
mode extraction and spectral estimation. The performance of all extraction methods is
degraded with the addition of random noise to the desired waveform. Typically, signal-
to-noise ratios on the order of 10 to 15 dB are required for accurate mode extraction.
Transient measurements of this quality are difficult to achieve for the ten-to-one
bandwidth desired. It has been shown that mode extraction is sensitive to additive random
noise [9], but qualitative assessment of the effect of other types of systematic errors in
the data have not been performed. However, it is safe to surmise that the effects of non-
random noise and distortion of the measured waveform may be detrimental to the
accuracy of the mode extraction. Usually, the performance degradation manifests itself
in the shifting of the extracted modal frequencies from their true values and the addition
of extraneous non-physical modes induced by the measurement system or extraction

procedure. The extraneous modes prevent accurate extraction of target modes.

129

 

Extraneous modes are especially troublesome when the extraction procedure requires an
initial guess as to the number of actual modes contained in the data.

The early-time portion of the transient response is receiving more attention for
use in detection and target identification. Analysis of the early-time response of
complicated targets using current theoretical and numerical models is difficult and time
consuming despite the advanced computer technology available today. Thus, the accurate
measurement of the early-time response of a target is important for the advancement of
any early-time detection or discrimination schemes using complicated targets. The
capability to make accurate transient measurements is also necessary in the verification
of current and future numerical models.

This chapter describes two different schemes for performing transient
measurements. The first scheme uses traditional time domain reflectometry methods
while the second scheme uses a Fourier synthesis approach. The advantages and
disadvantages of the schemes are discussed. A method of calibration and deconvolution
of the measurement system response is presented along with examples of measured
spectral and temporal responses. Characteristics of various calibration targets are
discussed. Theoretical expressions for bi-static scattering from a conducting sphere are
presented. Examples of scattered field measurements obtained using different methods
and ealibration standards illustrate the level of fidelity obtainable using different schemes

after calibration. The equipment used in the experiments is described in Appendix C.

130

 

5.2 Time Domain Systems

Traditionally, transient measurements have been performed directly in the time
domain [43,44,45]. The principal component of a time domain system is a wide-
band oscilloscope, usually of the digital sampling type. The most important part of the
sampling oscilloscope is the sampling head. The rise time specification for the sampling
head sets the effective bandwidth of the oscilloscope. The second important component
is a very stable pulse generator. The pulse output should be of short duration and have
a high peak voltage. The pulse repetition rate should be relatively high to provide good
performance. Also required is a wide-band transmitting antenna and a wide-band probe
or receiving antenna. The antennas and probes are usually mounted inside an anechoic
chamber or over a large conducting ground screen. An example of a time domain
measurement system using separate receive and transmit antennas inside an anechoic
chamber is shown in Figure 5.1. Recently, the performance of anechoic chamber systems
have been enhanced by using several transmit antennas along with power dividers and
amplifiers [46]. Methods such as these increase the usable bandwidth and signal-to-
noise-ratio of the system. It is also possible to construct a time domain system that uses
a single antenna for both transmit and receive as shown in Figure 5 .2. However, the
unavailability of suitable wide-band circulators limits the usefulness of this approach. A
version of this one antenna system was tested during this research. A wide-band (05-15
6112) 20 dB directional coupler was used in place of the circulator. The results were
disappointing due to the attenuation and distortion induced by the coupler. The use of a
wide-band transmit-receive switch in place of the circulator would also be possible, but,

the author has not tested this configuration.

131

 

 

 

 

 

l l
: Transmit Target :
l |
l l
l |
I Foam I
| Receive Pedestal :
l l
L _________________ _l

Anechoic Chamber

 

 

 

 

 

 

Delay
Sampling Line
Scope

 

Pulser

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

V

_ Trlgger

 

 

/\ /\

 

Figure 5.1 Dual antenna free-field time domain transient measurement system.

132

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Wide—Band [— ———————————————— "I
Circulator I l
| Target |
O ‘7’ '
| I
l |
I Wide—Band |
l Antenna |
| Foam |
l Pedestal |
L ________________ _|
Anechoic Chamber
IVA,
. Delay
Samplmg Line
Scope Pulser
\
/ \ Trtgger
/
\

 

 

 

Figure 5.2 Single antenna free-field time domain transient measurement system.

133

Direct time domain transient measurements have several inherent advantages over
continuous wave measurements. The most important advantages are associated with time
gating and the speed of the measurements. Time domain systems have the ability to
easily reject extraneous reflections. Common sources of these extraneous reflections are
the walls of the anechoic chamber, the antenna support structures, and the edges of the
ground screen. The extraneous reflections can only be gated out if the measurement
system has been properly designed and the target has been properly placed.

Another advantage of the time domain system is that the measurements are direct.
In some modern ultra wide-band oscilloscopes the waveform is recorded completely and
instantaneously after transmitting only one pulse. This type of measurement has no
limitations concerning the linearity and/or time invariance of the target. Digital sampling
oscilloscopes require transmission of several hundred pulses to acquire a waveform. This
requirement still allows for non-linearities in the target, but the target must not change
position or size within the waveform acquisition time. Digital averaging of many
waveforms is usually employed to reduce random noise in the final measurement.
Averaging requires more time but the process is still relatively fast. Typically, a 512
point waveform using 100 waveform averages is acquired in under 30 seconds with the
Michigan State University system.

Unfortunately, time domain systems also suffer from several inherent
disadvantages. The most important limitations being limited dynamic range and stability.
Most waveform processing oscilloscopes measure the voltage in discrete steps with an
analog-to-digital (A-D) converter. The finite word size used in A-D converters leads to

truncation error. An A-D converter with N bits can represent 2N different voltage values.

134

 

. 15

3.5»

Pb.

.51.-

. ll.-

 

Assuming a uniform digitization, the measurement cannot be more accurate than one part
in 2". This leads to a dynamic range of 2". A 10 bit converter provides about 30 dB
dynamic range. This is not very good if a deconvolution procedure, as described later,
is to be employed. It should be noted that waveform averaging can be employed to
extend the dynamic range of measurements that are limited by the word length used in
the A-D converter. The extent to which averaging can improve scattering measurements
depends on the stability of the pulse generator and triggering circuitry.

A problem inherent in all systems, but particularly troublesome in time domain
measurement systems, is equipment stability. Stability of the oscilloscope and pulse
generator is limited. Slight variations in pulse generator output and noise in the time base
circuitry of the oscilloscope lead to timing jitter. The overall effect is to produce a slight
time shift from one measurement to another. In situations where the background clutter
must be subtracted from the target measurement this effect induces a systematic error that
is approximately proportional to the derivative of the background clutter. For systems
where the clutter is large this causes great difficulty with mode extraction.

For cases where the time domain system is not located in a sealed anechoic
chamber there is a problem of reception of extraneous signals. The wide bandwidth
required for good fidelity in the transient measurement unfortunately provides equally
good fidelity for reception of television, radio, computer emissions and other types of
undesired signals. This characteristic of time domain systems limits their effectiveness
if there is insufficient transmitter power. In the case of the transient scattering range at
Michigan State University, the extraneous signals do not seem to be significant since it

is located inside a metal building.

135

 

Finally, the shape of the pulse waveform is not adjustable in most time domain
systems. The effectiveness of short pulses from the generator is usually limited by the
relatively long duration of the system impulse response. The calibration and
deconvolution scheme described later can be employed to eliminate this distortion.
However, the stability, dynamic range, and pulse bandwidth of most time domain

systems are usually insufficient to effectively use this scheme.

5.3 Frequency Domain Systems

Recently, the application of computer control to high quality solid state
microwave equipment has resulted in a new generation of automated vector network
analyzers. Vector network analyzers are capable of measuring both the magnitude and
phase of the S-parameters of two port networks over a very wide frequency range. This
capability gives the vector network analyzer the ability to synthesize the transient
response of a target via the inverse discrete Fourier transform. The extremely wide
dynamic range of vector network analyzers allows frequency domain deconvolution of
the transmit-receive system response from the target measurement with good results. The
wide frequency sweep allows for synthesis of very short duration pulses provided the
deconvolution procedure is done accurately. A typical free-field dual antenna frequency
domain measurement system using a network analyzer is shown in Figure 5.3. With a
dual antenna configuration, the transmit port is chosen as port 1 and the received signal
is directed into port 2 of the analyzer. A measurement of S21 then provides a measure of
scattering from port 1 to port 2 from both the scatterer and the environment. The

measurement parameter 8,; can be selected to reverse the situation. This is useful for bi-

136

 

 

 

 

 

 

 

[— ——————————————— _l
l / l
: Receive Target :
I \ E] |
l I
| _/ l
l I
i . Transmit |
I Foam I
I _\ Pedestal |
L_ _______________ _l

Anechoic Chamber

 

 

Vector Network

AM 821 Analyzer

Port 1 Port 2

 

 

 

 

 

 

 

 

.6

 

 

 

Figure 5.3 Dual antenna free-field frequency domain transient measurement system.

137

 

static measurements since the target can be positioned once and two different illumination
angles obtained. In a dual antenna configuration amplifiers can be added to both the
transmit and receive channels to help boost the dynamic range of the measurements. Of
course, this results in the loss of the bi-static advantage. Some researchers have also
added a frequency up—converter on the transmit channel and a reciprocal frequency down-
oonverter on the receive channel to increase the bandwidth of the network analyzer
systems into the millimeter-wave portion of the spectrum [47]. Such additions provide
range resolution for narrow band remote sensing applications but have yet to be used in
ultra-wide band measurements.

A frequency domain transient scattering system employing the network analyzer
and a single antenna for both transmit and receive is shown in Figure 5.4. Here, port 1
of the analyzer is connected to the antenna without using any amplifier stages. By
measuring S“, the reflection from the target can be made with a single antenna. A single
antenna system is very attractive since one is usually interested in a backscatter
measurement which is difficult to approximate in small anechoic chambers with two
antennas. Further, the interactions of a two antenna system are much greater than a
single antenna system. Unfortunately, the use of a single antenna system is limited by the
large reflections that occur within the antenna itself. The scattered field due to a target
only a few meters away is usually many dB lower than the antenna feed point reflection.
The target component is actually beyond the measurement accuracy of the network
analyzer for the TEM horns used in the MSU system in most instances.

There are several limitations to the use of a network analyzer for transient

measurements. The first limitation is characteristic of all frequency domain systems. The

138

 

 

 

 

l— ——————————————— fl
I |
: / Target :
| Transmit a, I
I Receive I
V l \ l
| |
| Foam I
: Pedestal :
|_ _______________ __l

Anechoic Chamber

 

Vector Network

N 811 Analyzer

Port 1 Port 2

 

 

 

 

 

 

 

 

 

é—

Figure 5.4 Single antenna free-field frequency domain transient measurement system.

139

target must be linear and time invariant for the inverse transform of the frequency
domain measurements to represent the actual transient response. The second limitation
is due to limitations of the equipment typically available. The measurement of the S-
parameter at several hundred different frequencies takes slightly longer than the time
domain systems. Typically, the time required to measure 401 frequency points between
400 Mhz and 4.4 Ghz with an averaging factor of 10 and an IF bandwidth set to 30 Hz
is about 2.5 minutes. Finally, the use of time gating, while still allowed is somewhat
limited.

To synthesize the transient response from data measured in the frequency domain
care must be used in selecting the number of points N, and the bandwidth B, used in the
measurement. Of fundamental importance is the sampling rate of the frequency domain

measurement f;

f, = fl , (5.1)

To use the frequency domain approach all transient events must be diminished to the
noise level within a duration equal to the sampling rate f,. Otherwise, transient events
that occur after t = j; overlaps onto the responses at t = t - m 1;, where m is an integer.
This effect is referred to as alaising and is well understood in the usual sense of sampling
a band limited signal in the time domain where the sampling rate must larger than twice
the highest frequency contained in the signal.

This sampling interval constraint limits the use of the network analyzer system
when the antennas are not inside an anechoic chamber. This is one of the reasons that

attempts to use a single antenna system employing the conical antenna on the MSU

140

 

ground screen were not successful. The reflections from the edges of the ground screen
and the laboratory ceiling caused alaising.

It should be noted that just as the time domain system allows direct gating of the
extraneous reflections, the frequency domain system allows gating of extraneous
frequency components from the measurement. This may seem of little consequence until
one realizes that some of the strongest signals received in the time domain system are due
to frequency components are contaminated from multi-path effects or are merely direct
antenna-to—antenna coupling. In time domain systems, the oscilloscope vertical amplifier
gain must be set to prevent clipping of these large undesired signal components. This
results in fewer bits of resolution for the smaller desired signal components and reduces

the dynamic range.

5.4 Calibration Procedure

To take full advantage of the pulse bandwidth of a time domain system or the
wide frequency sweep available with a network analyzer, the clutter and system transfer
function must be removed from the measurement. There are many methods for
deconvolution in the literature, but the process is in general very sensitive to noise and
somewhat ill-conditioned [48]. Thus, to make effective use of deconvolution, the
original data must be as accurate and noise free as possible. If the signals are measured
in the time domain, deconvolution can be performed directly in the time domain if the
system impulse response is known [49]. This procedure has been applied in time

domain systems that use a ground screen and a vertical electric field probe.

141

 

A method to obtain the system impulse response and perform deconvolution in
the frequency domain is detailed in this section. The method is similar to that used by
Morgan [45]. The method can be applied to data obtained in the frequency domain or
time domain with the obvious limitations incurred in the dynamic range, stability and
bandwidth for the case of the time domain system. It is assumed throughout the following
development that all data is sampled and discrete transforms are implied.

The objective of any calibration procedure is to determine and remove any
repeatable systematic errors from the raw measurement. In the case of transient
measurement systems these errors are associated with anechoic chamber clutter, antennas,
transmission lines, and probes as shown in Figure 5 .5 . The first step in the calibration
procedure is to measure the empty chamber or ground plane. This is called the

background measurement and can be modeled by

Rbm = swam +H,(t)} + me (5.2)

Here 11.0) models the transfer function of the direct coupling from the transmit antenna
to the receive antenna, H,(t) models the transfer function of the coupling from the
transmit antenna to receive antenna via the anechoic chamber, antenna supports, and

target mount. M0) is random noise and $0) is the system transfer function
50) = 11,0) 114;) ram (5.3)

where 11,0) and H,(/) are the transfer functions of the receive and transmit antennas from
the transmission line into the free-field environment while EU) represents the spectral
content of the pulse or CW source. The next step is to measure a known calibration

target. Calibration targets are discussed more extensively in the next section. The

142

 

V

 

Transmitted Field
®—>Ht (f) >

 

 

 

 

 

 

 

Antenna Mutual Chamber
Source: I Coupling I Scatterer / Interactions / Clutter
PUB? 0" Transmitting

CV Antenna

 

 

 

H, U) H, 0)

SC

 

 

 

 

 

 

 

H (f)

 

 

 

 

 

R(f)
R H,(f)

Recelvlng
Receiver Antenna

NU)

 

 

 

 

Received Fiel

 

 

Norse

Figure 5.5 Measurement system block diagram.

143

d

 

 

H, (f)

 

 

l

 

I

 

calibration target measurement gives
Wm = sth.w+H.(/) min + .209} + N""(f) (5-4)
where N’“(f) is random noise, H,‘(/) is the transfer function of the calibrator target

which is assumed to be known, and H50) is the transfer function of the interaction of
the calibration target with the anechoic chamber. Note that the interaction term is causal
in the time domain; it cannot not occur prior to the time required for the wave scattered
from the target to reach the chamber walls and return to the receive antenna. Note also
that the interaction term is represented here as an additive effect.

The next step is to measure the desired target, giving

R""(/) = sm {Ham +Hc(f) +H,'(t) +1130} + N""(f) (55)
where N+"(l) is random noise, H,‘(f) is the transfer function of the unknown target and H320)
is the transfer function due to the interaction of the unknown target with the anechoic
chamber. Again, this term is causal in the time domain.

Next, the background measurement is subtracted from the measurement of the
calibration target and the unknown target. Note that in actual practice, the background
is measured more than once and the subtraction is performed immediately in the
oscilloscope or in the computer just prior to saving the raw data to disk. Measuring the
background and performing the subtraction for each measurement in this fashion reduces
the time between measurements and minimizes the effect of equipment instabilities.

Subtracting gives

Rem = Red’m‘Rbm (5.6)
Rim = Rt+bm_Rb(f)

144

 

 

 

The clutter free calibration and desired target measurements are thus obtained:

R‘(/) = smly,‘m+H;m} + my) (5.7)
W) = sth.‘<n+H.‘.m} + 1W)

The noise terms are as

New = Nubm _ me (5.8)
N,(f) = N“"(t) -N"(f)

The next step in the calibration depends on the quality of the anechoic chamber. If the

anechoic chamber is of very high quality, then the interaction terms are small and can

be neglected along with the noise terms giving

mm = S (017:0) (5.9)
1W) = so It?»

Now, solving for S(/) in terms of the known calibration target transfer function gives

 

so) = R cm (5.10)
Hfm

Next, computing the unknown target transfer function leads to

' =.R_'(Q (5.11)
”:03 305

For anechoic chambers of lesser quality, the interactions terms can be too large to be

neglected. To solve this problem, R‘(f) can be transformed to the time domain and

windowing can be used to eliminate any interaction terms that are sufficiently delayed
beyond the end of the calibration target response. To help visualize the response of the
calibration target and the interaction terms in time refer to Figure 5 .6. Again the noise

terms are neglected.

145

 

r(t)

UV th/

-1 -1

9 gsm Hung 9 § Sm Hang

 

Figure 5.6 Temporal response of clutter-free calibration target measurement.

146

Define the time response of the calibration measurement as
r‘(r) = T‘tR‘m} (5.12)
Multiply by a window function w(t) to exclude interaction terms
r‘”(t) = r‘tt) w(t) (5.13)

If w(t) is defined properly (all of the interaction terms are excluded), and if

.9‘“{S(f)H,‘(f)} is approximately time limited and not truncated by the window function

the following approximation is valid:

R cWar) = $0) 173;) (5.14)

where R“"(f) = fir c"(t)}. Then, the system transfer function 50) can be obtained as

 

50) = Rm“) (5.15)
H,‘(f)
Thus, from (5 .6)
r : = R‘tf) (5.16)
H,(f) +me TV)

Finally to isolate H,'(f) the inverse Fourier transform is applied to obtain
Rt
Him +H.‘.(t) = VII—8%)} (5-17)
If the target impulse response is approximately time limited, and the interaction term is

causal and delayed beyond the end of the target impulse response, then the target impulse

response can be isolated using time gating as before.

147

5.5 Weighting Functions

Before moving on to other topics, the subject of weighting functions and FFF’s
must be addressed. Weighting functions are used to eliminate the effect an edge
discontinuity has on a spectrum or time response after transformation. It is not the
purpose of this work to expound upon all possible weighting functions. It is important
to be aware of what effect the choice of weighting function has on the outcome of the
calibration. The choice of weighting function is subjective and varies according to the
emphasis of the data.

For this research three weighting functions were used extensively. The most often
used weighting function was the rectangle function. This function is well known and was
used primarily to eliminate interaction terms from the calibration measurement. The
second function used was the Gaussian Modulated Cosine (GMC). This weighting
function is versatile since the center frequency and spectral width can be adjusted

independently. The GMC waveform is given by

we) = cosmf) exp(-rr(t/t)2) (5.18)

and its corresponding spectrum is
WKO‘) = r{exp(-rr(f-fc)t2) + exp(-1t(f+f;)tz)} (5-19)

where f, is the frequency of the cosine and 1 is the shape factor.

The Double-Gaussian (DG) weighting function was also used in the research. A
DG waveform is obtained by adding two Gaussian waveforms with different shape factors
and amplitudes. The amplitude factors are usually adjusted to give a waveform with zero

mean. The spectral content of the zero DC DC waveform is

148

 

 

= 1 _<_nf_)i _ 1 -993 5.20
W480) a alexp( a gazexfi a ) ( )

where

(5.21)

and

a, ._. __E_ ,2 . _¢_“_;_ (5,22,

J07; - F: E - Ila—2
The spectrum of the DG waveform used most often in the dissertation research is shown
in Figure 5.7. This spectrum works well for the bandwidths typically used in the low
band configuration of the frequency domain measurement system. The shape factors used

for this research are

1." = 0.25x10'9 s 1'2 = 0.5x10'9 3 (5-23)

The equivalent DG temporal response is shown in Figure 5 .8.

5.6 Calibration Targets

There are a number of objects suitable as calibration targets for various
applications [50]. A calibration target for a transient measurement system must have
a frequency response that is known in both magnitude and phase over a very wide range
of frequencies. Ideally, the response is not be a function of aspect angle or highly
resonant. The response should be large compared to the clutter to give high signal-to-

nose ratios in the calibration measurement. For radars that use the same linear

149

1 .20

 

 

Double—Gaussian Pulse Spectrum

 

 

1.00 ‘-
—I
I; -l
.> "'l
- —l
E O 80 ._.
a)
L I-l
v q
C) -l
‘3 0.60 —
.‘t'i -
c d
0’
o .4
2 a
q, 0.40 —
C5 —l
o
t d
o .-
> -
0.20 #
.4
0.00
0.00

TUITTTYIUTTTTTTUTUUTVIIIUII“[IIUTU'III

UIVIVIFII

' I
1.00 2.00 3.00 4.00 5.00
Frequency (GHz)

Figure 5.7 Double-Gaussian pulse spectrum.

150

 

 

 

 

 

 

 

 

1.20 : Double—Gaussian Pulse
1.oo :-
I:
0.80 —
0.60 -:
q
A0 2
.2 0.40 :2
H -I
2 t:
E.’ 0.20 —
v "‘
a. = .

o 0.00 - i
g -
o —.
> -o.2o E
-o.4o E
I
-O.60 :
:

-0-80 UIrI‘ltlU'lU[TlFll'lerII'TIUIIIUU'IUII'IYUIIYIm

0.00 2.00 4.00 5.00 8.00 10.00

Time (ns)

Figure 5.8 Double-Gaussian pulse.

151

polarization for both transmit and receive, a metal calibration sphere is the most common
calibration target. For polarimetric radars, a metal sphere is used for the like polarized
components while the cross-polarized components require a target that depolarizes the
incident field. Ideally, the target return should provide both horizontal and vertical field
components that are approximately the same strength. A common depolarizing target is
a thin wire or an array of thin wires oriented at a 45 degree angle with respect to the
incident electric field vector. There are other calibration targets in use; however, many
of them are suitable only for an amplitude calibration. This is not a problem for (CW)
radar cross section (RCS) measurements where phase is not important. However, for
time domain systems the phase information is necessary. The metal sphere and thin-wire
were used as calibrators in this research and are discussed in this section.

The sphere is a canonical structure in electromagnetics. Scattering from a sphere
was first solved by Mie in 1908 [51]. Since then, the analysis of sphere scattering has
appeared in many graduate electromagnetics texts such as Stratton [52] and Harrington
[53]. The notation used in course notes by Chen [54] is presented here.

Consider a plane electromagnetic wave incident on a perfectly conducting sphere

as shown in Figure 5 .9. The incident field is assumed to have only an 3 component and
is propagating in the 2 -direction. Thus

5 “(a = 2 E, exp(-jkz) (5.24)

The exact scattered field without far-zone approximations can be written as

E’m= -5 Zt- )" 27%” {a ,ijiteM) + 1b,. fifiiteM} (5-25)
ml

152

 

 

 

 

inc 1?
H a: .>
k
a sphere
3;

Figure 5.9 Geometry for sphere scattering problem.

153

I

 

where, the spherical vector wave functions are given as

fif’ (6M) = o-lzfikkk) P.”(cose> {Sim} 6
0M srnB

 

cosm¢ ( 5.2 6)
zi‘lkk) [gamma] {33:} «l»
(a) n_(__n+l) zto cosrn¢ ~
N;”(6.¢.R)= m “(km "(oooe){sinm 4,}R
+ (o _ gsmtb} (5.27)
klR an [R (MIL; P Wow)“ 6
m (o sinmdt}
is.” m6 -a——R[R 2,. (WP, ”(case){ 6

where P:(cose) represents the associated Legendre polynomials and the z,f°(kR) rep-

resents the various kinds of spherical bessel functions.

z,‘"(kR) = j..( . 290*) mm (5.23)
zf’acm = hfi"<kR). zi"(kR) = 1:90:10

The spherical wave expansion coefficients are given as

a .
a = J.(ka) b = 3R R (5.29)

hf’aca) 3316‘” 1:90:10) I“

 

 

The spherical vector wave functions can be approximated in the far-zone (kR> >1) as

jn+l

(_ 'kR) {12(0086) .
kR exp 1

”Show - cost 6 - (5% Pltoose))sin¢ 6} (5'30)

 

 

jn+l

snn¢<l>

 

 

P.3(cos6). - (5.31)
sinB

N336“) ~<-1) CXP('J’<R){(%P,(¢OSG))COS¢ é -

154

 

The far-zone scattered electric field can then be written as

l
_ _ e_______xp( -ij) 2n+1 P,,(cose) + (i r } (5.32)
55m 1'5 m .23 nT—n+1){"~ .——sin6 b. 66 P.(Cose)) coso

1
exp__(__ 7“) 2n+l _ P.(¢°sel} . (5.33)
5,0) ’5“ Id? §n(n+1){a'(aep"(°°s)) b“ sine 8”

For the back-scatter case, the following relations are useful in the evaluation of the Mic

series:
.1 u a _1
P:(COSO) I0,“ = -(—2?—n(n+l) Epkcosfl) l0=x = Lz—rnoul) (5.34)

The evaluation of the previous expressions to obtain the response of the sphere
is numerically difficult and there has been considerable work on the evaluation of the
series [55]. The introduction of personal computers based on 32 bit microprocessors
and efficient hardware floating point units has greatly reduced the cost of performing
computations such as the Mie series evaluation. Availability of such computer power
allows the programmer to spend less time on code development and still have a program
which provides useful and accurate results in less time than was possible only a few years
ago. The expressions shown in (5.32) and (5.33) were programmed with the use of
widely available generalized Bessel function subroutines. Most routines used in the
program were written or modified by Anton Tijus of the University of Delft,
Netherlands. These routines are very accurate and efficient.

To test the validity of the program results, the special case of back-scattering was

considered. The result showing the normalized back-scattered electric field versus

155

normalized frequency for a perfectly conducting sphere a distance of 10 sphere radii

away is shown in Figure 5.10. Here the first sphere resonance
1;, = —£— (5.35)

is used as a normalization. Note that most authors plot the back-scattering RCS instead
of electric field. The field is presented in this thesis since the calibration procedures
require the field values. The relationship between normalized electric field and back-
scattering RC8 is given by
0b = 41: R1{i§.:(_e_=fl1}2 (5.36)
IE ”"I
The back-scattering RCS of a sphere in the high frequency limit is the area the

sphere projects onto a plane that is parallel to the incident wavefront:

ab = rt a2 (5.37)
Thus, for the case of a unit radius sphere the normalized scattered field at a distance of
10 sphere radii is

|E‘(B=rt)|

IE“‘|

1
fi (5.33)
0.05 for R= 100

The curve in Figure 5.10 approaches this limiting value for high frequencies. According
to Skolnik [4] the RC8 of a conducting sphere at the first resonance is approximately 5.6
dB larger than the RC8 of the sphere in the optical limit. To use this number as a check
of the results in Figure 5.10 the normalized scattered field at resonance due to a sphere

10 radii away is calculated as

156

 

 

 

 

 

r-'-.'-"."'-'--fl'--..'--"H-"---..'--"-'-'.. ............... r--..--“-'..'--l1
. . . . . . . . - . .
—--q-'fi--d-"«--' "-'d ...... J ....... J ................. ‘-J--d--J---jr
. . c . . . . . o . c a
. . . . . . . o . . . .
---‘-"-'J"-*--' .."4 """" J ........ J ............... 1-J'-‘-'J--'A'
. . . . . . . . . . . .
. . . . . . . . . . n .
. . . . . . . . . . q .
.‘-P|'r'-‘---r"' ----b """" £ ....... £ ............... .r-‘--.--L---A'
. . - . . . . . . . . .
. . . . . - . . . . . u c
. . . . . - . . . . . . -
— . . . . lr . . . . . . .
. . - . . . . . . . . u
. . . . . . . . . . c c
. . . . . 1 . . . _ u . .
---‘--"'d--"-' --'4 ..... J ......... J ................ .1-‘--d-'J---Al
-"’--"-"--‘-I- U -" ...... J ......... J ................ .1“--‘-'J---l'
-"r--r-'L---r-' -" ...... L ......... L ............... L'|‘-'."L'-'j'
. o - . - . . .
—--q--fi--d--'q' ---'d ...... J ......... J ............... ”'-d---h--J----jl
"-'-P-'-r--b----w ' ----h ..... L ........ k ............... L'-b--b--L---l'

. . . . - . .
---r-'r--‘---r ---'h ..... L """""" L ................ L.-L--b-'L-'-L'
. . . . . . _ n . . . .
. - . . . . -
---q-'fi"d--'q--' ..... J ......... J ................ u'-d---h--J."-AI
. . . . A . . . n o . . . .
. . . . . . . . . . . .
---‘--"-4--"---+---' "-J """"" J-"-'---'-'.0---*-J--d--4--'1'
. . . g c . . . . . .
. . . . . . . . . . . . .
. . — . . . . . O . . c .
.I’r"r-'b-" I -h ...... L ......... L ........... :"--.'-L"b-'L"'l'
. . . . . . . . . - .
. . . . - . . . . . . .
. . . . . . . = . . - .
. . . . . . . . . . .
. . . . . . . . . . . c
. . . - u - . . . . o .
. . . . . . . . . R . . . .
.‘fi--‘-'d'--4-"*----d ...... J ........ J .............. .1'J--d--J--'l
.. -‘1'-"--*--'+-'-" ..... J ......... J ................ .1"--‘-I‘-'-l'
--'P- --‘-'-r--'L.--"* ..... B ......... ‘ ................ .r-L--h--L---l'
. . . . . - . — . . c .
.-'q"fi-'d - "-.. .... d ...... J ........ J ................ .--d--d--J-'-‘I
--'F-'r-'b-'-p-- '---h ..... L ......... L ................. r-h--b--L---ll
- . . . . . . . . . - . .
--'r--r--b-'-p--'L'-'-' I"L ........ L ................ r'b-'b--L--"'
. . . . . . . . . — . .
. . . . . o . . u . . a .
-'-Q--‘--J-'-*--'J'-"-d ..... J ..... J ............... 1-‘--‘--J'--l'
. . - . . . o . . u o u
. - u . . . . - . . . c .
.-"--‘--4'l""-+-'--‘ """" J ......... J ............ +-‘--d--J--'l'
- . . . . . . . - - p .
. . . . . . - . . . c .
. . . . . . . . . - .
P b b b P p b b b h n b P
— W q q q A q u u — q d u
1 1
O. O

0
.
3Q 23... 2:80 82.0532

(f/fo)

10

Normalized frequency.

1

157

Frequency response of a perfectly conducting sphere in back-scatter

configuration.

Figure 5.10

 

£~*(0=n) = 0.9527
E inc R (5.39)

0.09527 for R=10a

The results in Figure 5.10 are in agreement with these calculations.

The transient field response versus normalized time of a sphere in the back-scatter
configuration is shown in Figure 5.11. The transient response was synthesized by
applying a GMC weighting function to the spectrum previously shown and then applying
an inverse FFl‘. The GMC had a center frequency of 0 Hz and a damping factor of r =
0.025 to. The strong specular reflection is followed by the well known creeping waves.
Since the second creeping wave is very small the scale was magnified in Figure 5 . 12 to
show the second creeping wave.

The thin wire was also used as a calibration target in this research. The thin-wire
has been extensively studied in the literature. Although the scattering properties of a thin-
wire cannot be expressed in closed form as in the case of the sphere, it is well
understood and can be numerically modeled. For purposes of this research the thin-wire
was modeled using the method of moments. A Galerkin method was employed using
piece-wise sinusoidal sub-sectional basis functions. The details of this formulation are
described in Appendix A.

There are several difficulties encountered when using a wire for calibration over
an ultra-wide bandwidth. One of these difficulties is that the thin-wire response is a
function of orientation angle. This aspect dependency is not a problem in the theoretical
formulation, but it can become difficult to place the wire on the styrofoam support inside

the anechoic chamber in exactly the same position in a repeatable fashion. Any

158

 

0.0012
0.0010 .
Goussuon Pulse (‘1' = 0.025)
0.0008
0.0006

0.0004

0.0002

 

 

Electric Field (relotive)

0.0000

 

43.0002
I? II ICLC) a

1111111111rrrrrtrirlrrrrirrrrlrrrrtirrrlrr111rIrilrrrrrrrrllrrrlrrrrrlrrrrrrrrrl

 

 

-0-0004 TIIITTIIUIIIIrTITI 'IIIIITIIjW

1.00 2.00 3.00 4.00
Time (relotive) (t to)

Figure 5.11 Gaussian pulse response of a perfectly conducting sphere in back-scatter
configuration.

159

N
o
'l‘
8
on
I

1.05-005

irrrruitrrrrrrrr

is

6.8E-021

Electric Field (relative)

-1.0£—005
Gaussian Pulse (‘1' = 0.025)

R 310.00

 

lllLllllllllllJllLl

 

 

-2°o£-005 fT1IIIITIITIIIIIII lT‘TITjIII]

1.00 2.00 [3.00 4.00
Time (relative) (t/to)

Figure 5.12 Gaussian pulse response of a perfectly conducting sphere showing second
creeping wave.

160

discrepancies between the actual aspect angle and the angle used in the calculations cause
errors in the deconvolution procedure. The problem of aspect dependency is not as severe
for wires that are small with respect to a wavelength, but for long wires the errors can
be considerable. Unfortunately, short wires do not provide a strong signal. The use of
short wires for calibration then leads to a loss of signal-to-noise ratio in all measurements
subsequent to deconvolution.

To illustrate the aspect dependency problem, a series of numerical simulations
was performed. The spectral response of several different length wires was calculated for
the anechoic chamber configuration used at Michigan State University. A frequency
range of 0.400 to 4.400 Ghz was chosen for the calculations to coincide with the values
usually used in the low band configuration for target measurements.

The calculations were done with the method-of-moments for various orientation
angles. The configuration of the chamber modeling for these calculations is shown in
Appendix C. Figure 5.13 shows the response of a stainless steel wire 2.0 inches long and
0.043 inches in diameter for various aspect angles. The response is only slightly different
at the highest frequencies. As the wire length is increased to 4.0 inches, the variation
with angular position is more pronounced as shown in Figure 5.14. Similar effects are
observed for wires of length 6.0 and 8.0 inches as seen in Figure 5.15 and Figure 5.16.

It should be noted that calibration using the theoretical field values at a point is
only valid if far-zone conditions are enforced in the measurement system. This
requirement means that the variation of the transmitted field over the target should be
small. Similarly, the variation of the scattered field over the aperture of the receive

antenna should be small as well. The results in this dissertation were obtained by using

161

0.50
—— Broadside
------- 5 degrees off
-- -- 10 degrees off

0.40 .-
l
I \
I \
a ! \
'48 0.30 I \
l
T) I \
L. r
E I. \
.9 . \
U- 020 / \
o .
'5 l \
8 II \ ‘<~
In ,/ \“
0.10 ./
’/
/
/
0.00
0.00 1.00 2.00 3.00 4.00 5.00

Frequency (GHZ)

Figure 5.13 Scattered field amplitude vs aspect angle for wire (2.0 x 0.043 inches).
Moment method results computed for MSU anechoic chamber geometry.

162

1.00

 

 

 

 

I
0.80:-
1
j l
A I I
Q) q
..>. - 1
460.60- .
— .. l —— Broadsude
Q) ._.
b - I ------ 5 degrees off
: - — 10 degrees off
‘9 .. l
.9 I I
LI-0.40-
o _, l
': q
H d
8 1
[I] i
0.20:
1
l
1
0-00 IITIITTITIIIITIIIIIIITITIIIITITIIIIFTTileIITTIUII
0.00 1.00 2.00 3.00 4.00 5.00

Frequency (GHz)

Figure 5.14 Scattered field amplitude vs aspect angle for wire (4.0 x 0.043 inches).
Moment method results computed for MSU anechoic chamber geometry.

163

1.60

 

 

 

 

 

 

1
q
1.20 - A .
j —— Broodsade
A - Il ------ 5 degrees off
a) a — —
.2 d II 10 degrees off
,_, -
2 ._.
a) c:
B -l
v0.80 -
E I
.93 -
u. -
.9 3
b 4
U -l
2 _,
Lu 0.40 -
q
1
4
1
0'00 IIIIITWITFIIITIIITIII—TTIllT—TIIIIITIIITHTTIIIIIIIII
0.00 1.00 2.00 3.00 4.00 5.00

Frequency (GHZ)

Figure 5.15 Scattered field amplitude vs aspect angle for wire (6.0 x 0.043 inches).
Moment method results computed for MSU anechoic chamber geometry.

164

 

 

 

 

 

 

 

2.00-
q
1.50-
“ l
—l I
A - :'
Q) - '1
.2 - i'
H _I '11
C _ I; .
T) . :1 -— Broodsnde
93100 :1 II ------ 5 degrees off
. . — — 10 degrees off
.2 j :1
.2 . . -
L1... .1 5 1
.9 i :
L l
H T I
o - :
2 “ r
m0.50- I
-l i
1
"1
0°00 T1IUIIIITIIIIIIIIIIIUUIllflTTlIIIUIIIIIIIIIIIUIIU]
0.00 1.00 2.00 3.00 4.00 5.00

Frequency (GHZ)

Figure 5.16 Scattered field amplitude vs aspect angle for wire (8.0 x 0.043 inches).
Moment method results computed for MSU anechoic chamber geometry.

165

a field point near the throat of the TEM horn antennas.

5.7 Calibration Examples

This section presents experimental and theoretical results to demonstrate the
calibration procedure described earlier. Examples of the various waveforms and spectra
encountered in the measurement procedure are presented to provide a baseline of
expectations for future workers. Significant characteristics of these waveforms and
spectra are discussed.

The first step in the calibration procedure is to measure the background clutter.
The background measurement can reveal problems with a measurement system. The
spectral representation of the background measurement made with the HPS720B Network
Analyzer inside the MSU Anechoic Chamber is shown in Figure 5.17. This measurement
shows that coupling between transmit and receive antennas is strongest at the lower
frequencies. A similar curve is obtained by taking the FFT of a background measurement
using the time domain system. This result is shown in Figure 5.18. This curve should
not be identical to the one obtained using the frequency domain system since the power
spectral density of the pulse used in the time domain system is different from that
generated by the frequency domain systems.

Strong coupling is indicative of both direct free space antenna-to antenna coupling
and coupling via reflections from the chamber walls and antenna support structures.
Coupling via the chamber walls is usually stronger at low frequencies since the efficiency
of most absorber decreases as the frequency is decreased. Note, however, that absorber

efficiency also decreases if the frequency is increased into the millimeter-wave regime.

166

 

Background Measurement
Frequency Domain System

0

O

O

a)
111lll+LllllLlllllllllllllJlJlIll]

 

 

000° _‘J.AA‘
- IIIIIITITrIlIIIIIIIITIIIFIIII[ll

0.00 1.00 2.00 3.00 4.00 5.00
Frequency (GHz)

Figure 5.17 Spectral background measurement of MSU anechoic chamber using
frequency domain system.

167

1.20
Background Measurement

Time Domain System

1.00

.0
on
o

 

.0
an
o

Magnitude (relative)
.0
g
0

0.20

 

0.00 ‘ “m-‘e‘n-e -
0.00 0.50 1.00 1.50 2.00 2.50 3.00
Frequency (GHZ)

Figure 5.18 Spectral background measurement of MSU anechoic chamber using time
domain system.

168

Higher quality absorber on the chamber walls, on the antenna supports and between the
antennas can reduce the direct coupling. Reduction of direct coupling is probably the
most efficient way to improve the quality of the scattering measurements. Ideally, the
background response is very small as compared to the target response. Clutter subtraction
is then only a minor correction. In practice, this is not the case and the clutter is of the
same order or larger than the target response. A larger clutter signal limits the dynamic
range of the measurement system when subtraction is necessary. Even though the
network analyzer has a tremendous dynamic range, the ultimate accuracy of any
measurement is no better than .01 dB in magnitude and .1 degree in phase. The same
effect is present in time domain systems as well.

Another perspective on the background measurement is seen in the time domain.
Figure 5.19 shows the temporal background measurement made using the time domain
measurement system. The direct coupling is easily differentiated from the scatter of the
anechoic chamber. Since the transmit and receive antennas are located close together the
direct coupling occurs before the transmitted wave has time to interact with the chamber
and return to the receive antenna. Thus, the first part of the signal is due to direct
coupling while the scatter from the chamber walls starts at a later time. A similar result
is obtained by Fourier transforming the result obtained from the frequency domain

system as shown in Figure 5.20.

169

 

60.00
Background Measurement

Time Domain System

40.00

20.00 H n n

8

 

 

 

 

 

 

 

 

 

-20.00 U U

 

 

Voltage (relative)

-40.00

-60.00

-80000 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 T I I

0.00 20.00 40.00 60.'00
Time (ns)

Figure 5.19 Temporal background measurement of MSU anechoic chamber using time
domain system.

170

 

 

 

 

 

 

 

 

1.00 -
3 Background Measurement
- Frequency Domain System
0.50 E
r; I
.2 0.00 T.
H _
2 d
0 I
L
V -
81 2
o -0.50 :
~_-' -
O G
> 2
1
fi
-1.00:
1
-1.50 1r!IIITTIITF1IIIIIIjIIIfiTTIII]
0.00 20.00 40.00 60.00

Time (ns)

Figure 5.20 Temporal background measurement of MSU anechoic chamber using
frequency domain system.

171

 

The next step is to measure the calibration standard. The spectral response of a
14 inch diameter metal calibration sphere measured using the frequency domain system
is shown in Figure 5 .21. The Inverse Fourier Transform of this measurement is shown
in Figure 5.22, A direct time domain measurement of the same sphere is shown in
Figure 5.23. The portion of the response due to the interaction of the target with the back
wall is clearly identifiable at approximately 40 us. The interaction term is windowed out
of the time domain response and the Fourier Transform operation performed. The
spectrum of the calibration sphere without the interaction term obtained from the
frequency domain system is shown in Figure 5.24. Note that the frequencies outside the
original measurement bandwidth must be discarded here to prevent introduction of
extraneous noise. The spectrum of the windowed calibration sphere measurement
obtained using the time domain system is shown in Figure 5.25.

The theoretical response of the 14 inch diameter sphere for the MSU anechoic
chamber configuration is shown in Figure 5.26. These results were computed by
numerical evaluation of the Mie series as outlined in the previous section. Note that for
pure backscatter the sphere does not depolarize, however, for the bistatic configuration
of the anechoic chamber there is some depolarization. In this case both components of
scattered field must be used.

The system transfer function was computed using the theoretical sphere response
and the windowed sphere measurement as shown in (5.15). The result is shown in
Figure 5.27. The result has been normalized to a peak value of unity. Normalization is
used since the waveshape, and not the wave amplitude, contains the desired information.

A plot of the transfer function of the time domain system using the same procedure is

172

0'0035 Calibration Sphere

Frequency Domain System

0.0030

0.0025

0.0020

0.0015

0.0010

Magnitude 821 (relative)

0.0005

lllllILlJlLlllllLJlllllllllllllllll

 

 

0.0000 IIIIIIIII]llIlllllllIIIITIIIIITIIIIIIIIIIIIIIIIII]
0 1.00 ‘0

Frequency (0H2)

Figure 5.21 Spectral response of 14 inch metal calibration sphere measured using
frequency domain system in MSU anechoic chamber.

173

Calibration Sphere
5-05’004 Frequency Domain System

4.0E—004
3.0E .. 004
2.0E-004
1.0E-004

5.4E-019

 

 

-i .0E-004

-2.0E-004

 

-3.0E-004

Voltage (relative)

—4.0£-004

-5.0E-004

 

-6.0E-004
-7.0E-004

0.00 10.00 20.00 30.00 40.00 50.00
Time (ns)

Figure 5.22 Temporal response of 14.0 inch diameter metal calibration sphere
measured using frequency domain system in MSU anechoic chamber.

174

 

Calibration Sphere

 

 

 

 

 

 

 

4°-°° : Time Domain System

1
-l
q
d
20.00 E
1

9 0.00 EW—T
f: 1
a) -1
C" I
U ..
z _
o - ..
> 20.00 I
1
-40.00 -

-60000 IIIIITTIT'IWIIIIIIIIITIIFIIIIIIIIIIIIIIIIIIIITTII]
0.00 10.00 20.00 30.00 40.00 $0.00
Time (ns)

Figure 5.23 Temporal response of 14.0 inch metal calibration sphere measured using
time domain system in MSU anechoic chamber.

175

0.0035

 

 

 

 

T Windowed Calibration Sphere
- Frequency Domain System
0.0030 d
..
00025 ~
1: l
3 0.0020 -
2 -l
0 cut
L d
v
Q) ..
c”0.0015 -
o
g -4
o .1
> -
0.0010 -
0.0005 -
000000 IIIIIIIIIIIIIIIIIIIIIIIIIIIfiIIIIIIIIIIIIIIIIIIII]
0.00 1.00 2.00 3.00 4.00 5.00

Frequency (GI-l2)

Figure 5.24 Spectral response of 14.0 inch calibration sphere after time domain
filtering of chamber wall reflections. Measured using frequency domain
system in MSU anechoic chamber.

176

 

”000 Windowed Calibration Sphere
Time Domain System

1.0000

0.9000

.0 .o o o
U 0’ \J O
O O O O
O O O 8
O O O

 

 

0.4000

0.3000

Voltage Magnitude (relative)

0.2000

0.1000

 

 

0.0000 ‘ ‘ ‘
0.00 1.00 2.00 3.00 4.00 5.00

Frequency (GHZ)

Figure 5.25 Spectral response of calibration sphere after time domain filtering of
chamber wall reflections. Measured using time domain system in MSU
anechoic chamber.

177

 

 

°°°4°° Calibration Sphere

Mie Series

0.0350

0.0300

0.0250

Ive)

 

 

0.0200

Itude (relat

0.0150

Magn

0.0100

0.0050

0.0000

 

JilllllllllJJLJlllJJLlllll[J]1111141llllJ

awn—WWW
1.00 2.00 3.00 4.00 5.00

Frequency (CH2)

'8

Figure 5.26 Theoretical response of 14.0 inch diameter metal calibration sphere
calculated using Mie series for MSU anechoic chamber configuration.

178

 

System Transfer Function
“000 Frequency Domain System

1 .0000
0.9000
0.8000

0.7000 14.0 inch Sphere

 

0.6000

0.5000

0.4000

Voltage (relative)

 

0.3000

0.2000

0.1000

 

0.0000 IIIIIIIIITIIIIIIIIIIIIIIIIIIII[IIIIIIIIWIIIIIIIIII]

0.00 1.00 2.00 3.00 4.00 5.00
Frequency (CH2)

Figure 5.27 Transfer function of frequency domain system obtained using 14.0 inch
diameter sphere as calibration standard. Results normalized to peak value
of unity.

179

shown in Figure 5.28. Note that there are peaks in the spectrum. These peaks are most
likely due to resonances of the individual horn antennas. The Inverse Fourier Transform
of the system transfer function is known as the system impulse response. The band-
limited system impulse response of the frequency domain measurement system is shown
in Figure 5.29. It is clear that the extra oscillations that would be induced into an
uncalibrated measurement are not negligible for mode extraction or early-time studies.

The impulse response of time domain system is shown in Figure 5 .30.

5.8 Measurements with Canonical Targets using Sphere Calibrator

With the system transfer function determined, other target measurements can be
accurately performed. For validation purposes, the response of the targets must be known
by either theoretically or through some different experimental procedure. This section
presents measurements of canonical targets using a sphere as a calibrator.

A good verification target is another calibration sphere with a diameter different
from the one used to perform the initial calibration. Unfortunately, other calibration
spheres of known high quality were not available for this research. However, a small
hollow brass sphere 3.0 inches in diameter was available. This sphere was originally
intended for other purposes and had a 3/ 8 inch hole on its surface. A comparison of the
corrected spectral response of this sphere obtained using the I-IPS720—B Network
Analyzer with the theoretical Mie series solution is shown in Figure 5.31. The calibrated
experimental results and the theory agree well. The uncalibrated sphere measurement is
also shown to illustrate how calibration improves the accuracy of the measurement. The

largest errors are present at the highest frequencies. This is probably due to the hole in

180

1'20 System Transfer Function

Time Domain System

1.00

.0
00
o

.0
or
o

 

Magnitude (relative)
.0
fi
0

 

0.20

 

 

0.00
0.00 1.00 2.00 3.00 4.00 5.00
Frequency (GHZ)

Figure 5.28 Transfer function of time domain system obtained using 14.0 inch

diameter sphere as calibration standard. Results normalized to peak value
of unity.

181

Approximate System Impulse Response
1-20 ‘ Frequency Domain System

1.00 ‘ n
0.80 ""

0.60 -

14.0 inch Sphere

0.40 '-

0.20 -

 

,c': .o
8 8
l

 

-0.40 -

Voltage (relative)

-0.60 -

 

 

-0.80 - H

-1.00 -‘

d

 

 

-1.20 rillIIIIIIITIIIIIIIIIITTIllrrfi

5.00 10.00 15.00 20.00
Time (ns)

Figure 5.29 Impulse response of frequency domain system obtained using 14.0 inch
diameter sphere as calibration standard. Results normalized to peak value
of unity.

182

System Impulse Response

1.00 Time Domain System

0.50

.0
o
o

 

-0.50

Voltage (relative)

 

 

-1.00

’I.50 [IIITFIIIIIIIIIIIIII'IIIIIIIIIIITIIIIIII]

0.00 5.00 10.00 15.00 20.00
Time (ns)

Figure 5.30 Impulse response of time domain system obtained using 14.0 inch

diameter sphere as calibration standard. Result normalized to peak value
of unity.

183

 

 

 

 

1.20 '-
- Frequency Response
100 _ , Mie Series .
_ ’ \ ----- HP87208 (l4 in..Sphere Cal.)
“:1 t - - - HP87208 no calibration)
-l l, “‘
A "' a ‘
a) _ a I , Metal Sphere .
.2 f ‘ \ diameter a 3.0 in.
H 0.80 .. y
2 l
0 .1 r, 1
L - I I
v . I
T
g _ Ii '1 1‘
I "\
3 0.60 -4 ,0 ‘ x‘ \
r: - ,I x
c I I “
OI '- \‘ lI \‘ I ‘I
o - \ ’I \\ I, ‘I
2 - I“ t ” “"- r
I I
Q) 0.40 I‘ I, r.
m --I '\ \ I, I
O I ‘ I I
L" " \ n
O - I
> g \
0.20 4 ‘
.1 I I F «I I
‘14 ‘ J "‘1
.. “(JV 1 0‘“
-i ‘0',”
_ ‘um'tn‘
0°00 IIIIIIIIIIIIIIIIIIIrIIIIIIIIIIIIIIIIIIIIIIIITIIII]
0.00 1.00 2.00 3.00 4.00 5.00

Frequency (GHz)

Figure 5.31 Comparison of theoretical and experimental frequency responses of a 3.0

inch diameter brass sphere. Measurements performed using frequency
domain system in MSU anechoic chamber.

184

the sphere and the effects of windowing of extraneous system noise from the measured
response. The transient response of the sphere to the DG waveform described previously
in Figure 5.8 is shown in Figure 5.32. The agreement between theory and experiment
is quite good.

Another good test target is the thin wire. As previously discussed, the theoretical
response of these targets can be obtained via the method-of-moments and used to evaluate
the usefulness of the calibration. A limited number of comparisons are shown here for
single wires, while a large number of comparisons are made for the case of coupled
wires in the next chapter.

The spectral response of a single stainless steel wire 6.0 inches long and 0.043
inches in diameter oriented at broadside is shown in Figure 5.33. There is excellent
agreement between the theoretical model and the calibrated network analyzer
measurement. Note that the response is dominated by the lowest order mode. The higher
order modes, especially the even ones, are not excited as strongly in the broadside
orientation. The even modes would not be excited at all except that there is a slight bi-
static angle in the measurement apparatus. The bi-static angle was included in all
theoretical calculations. The response of the wire measured using the time domain system
is also shown on this plot. This measurement agrees well within the very limited
bandwidth of the time domain measurement system. The transient response to the DG
waveform of Figure 5 .8 is shown in Figure 5 .34. Again the agreement between theory
and experiment is very good. The temporal response measured using the time domain
measurement system is not shown since the DG waveform is unsuitable for the limited

bandwidth of that system. The spectral response of the same wire oriented at 45 degrees

185

 

 

 

 

 

 

 

 

1.20 —
100 ” Double-Gaussian Pulse Response
0'80 - —— Mie Series
- ---- HP87208 (14 in. Sphere Col.)
0.60 —
0.40 —
7c? I
.2 0.20 - A
4n!
2 ‘ /| . .
8 0.00 - ‘~ ‘
v
3-020 -
O -
z:
o _ ..
> 0.40
a- I
-O.60 - U
—O.80+
‘ Metal Sphere
-190 _. diameter - 3.0 in.
-1-20 IIIIII—IrIIIIIIIIIIIITIIIIIIIIlIIIIIIIIIlTIfiIIIII]
0.00 2.00 4.00 6.00 8.00 10.00

Time (ns)

Figure 5.32 Comparison of theoretical and experimental transient responses of 3.0 inch
diameter brass sphere. Measurements performed with frequency domain
system in MSU anechoic chamber.

186

1.20

 

 

 

 

 

_ Frequency Response
F
1.00 .1 Moment-Method
_ ---- HP87208 (14 in. Sphere Col.)
- - Time Domain (14 in. Sphere Col.)
.4
A -1
0) wire a 6.0 x 0.043 in.
,2 “ orientation = broadside
H 0.80 —
2 .4
0)
g u
V
0)
T
gaw—
: q
C
05 ..
g _
I
Q) 0.40 '-
05 .-
C
t -
° 7
> "'1
0.20 -«
.l
0'00 IIIIIIIIIlIIIIIIIII—IIIIjIIIIIlITIIIIIIITIIrITIITI—l
0.00 1.00 2.00 3.00 4.00 5.00

Frequency (GHz)

Figure 5.33 Comparison of theoretical and experimental results for the frequency
response of a stainless steel wire 6.0 inches long and 0.043 inches in
diameter. Measurements performed in MSU anechoic chamber.

187

1.20

 

 

 

 

 

 

 

 

 

 

J .
‘00 Double—Gaussran Pulse Response
0'80 - Moment—Method
- I. ---- HP87208 (14 in. Sphere Cal.)
0.60 - .
0.40
A
Q)
.2 0.20
no.)
2
E.’ 0.00
v
83-020
0
t
o -
> 0.40
-O.60 J
-O.80 —
' wire I 6.0 x 0.043 in.
-190 _l orientation - broadside
“1.20 IjTIIIIIIjIIIITIIIIIIIIIIIIIIIIIIIIIIIfl
0.00 5.00 10.00 15.00 20.00

Time (ns)

Figure 5.34 Transient scattering response of stainless steel wire 6.0 inches long and

0.043 inches in diameter oriented at broadside. Measurements performed
using frequency domain system. Numerical results performed using
moment method.

188

from broadside is shown in Figure 5.36. In this response, the second mode is more
strongly excited than the first, while the third mode also has significant energy content.
The transient response of the wire in the 45 degree orientation is shown in Figure 5.35.
Again, the agreement between theory and experiment is good. Discrepancies between
theory and experiment are most likely due to differences in the angle used in the

experiment and the angle used in the theoretical model.

5.9 Measurement of Canonical Targets using Wire Calibrator

A thin wire can also be used as a calibration target. To demonstrate the use of a
thin wire as a calibration target, the 6.0 x 0.043 inch wire was measured for broadside
orientation. This response was taken as the calibration standard. Then, the response of
the 3.0 inch sphere was computed using this calibration measurement. The frequency
domain result is shown in Figure 5 .37 while the time domain result is shown in
Figure 5.38. Again, the agreement between theory and experiment is very good
indicating that a thin wire is also a valid calibration target. Note that great eare was used
in placing the 6.0 x 0.043 inch wire in the broadside position so that angular effects
previously discussed were minimized.

A thin wire can also be used as a calibrator for another wire target. In this case
the broadside response from the 6.0 x 0.043 inch wire is used as a calibrator for the
same wire oriented 45 degrees off broadside. The frequency domain result is shown in
Figure 5.39 while the time domain result is shown in Figure 5.40. The theory and
experiment agree very well except for the highest frequencies used in the measurement.

These discrepancies are probably the result of a slight error from the assumed 45 degree

189

1.20 -

 

 

 

 

 

 

 

100 ‘ Double-Gaussian Pulse Response
.1
0'80 A Moment-Method
- ---- HP87208 (14 in. Sphere Col.)
0.60 -
- .:
ll
0.40 - ,g
A -I H
a) , .l
.2 0.20 — . g l ,
+4 l ' I
2 J 'l .. z .
- I I
82 0.00 E 1: ‘:‘. ;‘. n
V d ' :, . r' \' r
' v
8.4.20 - ' ll '
o _ I:
: :l
o _ _ l
a I
I
-0.60 -
-0.80 "1
" wire I 6.0 x 0.043 in.
-100 .0 orientation - 45 degrees off broadside
-1.20 irrrrlITFlrrrrllllllrlrrrrrllllrfirTrrll]
0.00 5.00 10.00 15.00 20.00
Time (ns)

Figure 5.35 Comparison of theory and experiment for the transient response of a 6.0
x 0.043 inch stainless steel wire oriented at 45 degrees from broadside in
the MSU anechoic chamber.

190

1.20

1.00

.0 P
or 00
o 0

Voltage Magnitude (relative)
p
p
O

0.20

-l
0.00

Frequency Response

 

Moment-Method
..... “533720 (14 in. Sphere Cal.)

 

 

I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I

 

"‘“J‘n ’\
, a wire - 6.0 x 0.043 in. .0. \‘pl '
’ orientation - 45 degrees off broadside

 

Figure 5.36

IIIIIIIIIIIIIIIIIIIIIIITIIIIlleIIIIIIIIITIIIIIIIl

1.00 2.00 3.00

4.00
Frequency (GHz)

5.00

Scattering of stainless steel wire 6.0 inches long and 0.043 inches in
diameter oriented at 45 degrees from broadside. Measurements performed

using frequency domain system. Numerical results computed using
moment method.

191

1.20 —

- Frequency Response

1.00 —

Metal Sphere
,’ diameter - 3.0 in.

.0
m
O
111

l

.0

or

o
1

Voltage Magnitude (relative)
p
4.»
O

 

 

 

 

0.20 -
" r
d i Mie Series
q ' ---- HP87208 (6.0 x 0.043 in wire calibrator)
0°00 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIfl
0.00 1.00 2.00 3.00 4.00 5.00

Frequency (GHz)

Figure 5.37 Comparison of theory and experiment for a metal sphere 3.0 inches in
diameter. A wire of length 6.0 inches and 0.043 inches in diameter
oriented at broadside was used as the calibrator for the measured results.

192

 

 

 

 

 

 

 

 

1.05 ; Double—Gaussian Pulse Response
1
0.80 -: :fi
.. l I
‘ l 1 Metal Sphere .
0.55 .1 : 1 diameter = 3.0 in.
2 : '.
. l ‘
. : ',
A 0.30 -‘ l 1
9 : : . .
e - H. ‘
_ ‘ I
8 005: '1 5 . r \ Arugvvuflfian {. _ V-
V -‘ I 1 ‘
Cl 1 ' 1 l
o - ‘ 1
(gs-0.20 - l 1 l '
“ l
B " : I 1
B I . l . «
> -0.45 - i i i ‘
2 l: 1U
_ l‘ l
- :r
-0.70 - g:
3 l
- ‘ Mie Series
”51 ---- HP872OB (6.0 x 0.043 in. wire calibrator)
-I.20 IIIIII—ITIIIIIIIIIIITIIIIIIITIIIIIIIIIIIIIIIIITIW—l
0.00 2.00 4.00 6.00 8.00 10.00

 

Time (nS)

Figure 5.38 Comparison of theory and experiment for transient response of a 3.0 inch
diameter metal sphere. A 6.0x0.043 inch wire oriented at broadside was
used as the calibrator in the measurements.

193

12° " Frequency Response

wire - 6.0 x 0.043 in.
orientation - 45 degrees off broadside

.0

m

00
l

l

.0
00
o
l

.0
\I
N
L r

 

O

or

-.
IJLJ

.0
p
00
l

l

 

Voltage Magnitude (relative)
O O O
is '3 8
r 1

.°
N
g
1411
I

l
a

 

Moment-Method "\\’\ ' x n
---- HP872OB (broadside 6.0 x 0.043 in ’ ‘
_ wire col.)

9
O
m
l
\

 

0.00

 

IIIIIIIII'IIIIIIIIIIIIjIIIIIIlIIIIIIIIjjIIIIIIIIIj

0.00 1.00 2.00 3.00 4.00 5.00
Frequency (GHZ)

Figure 5.39 Comparison of theory and experiment for frequency response of a
6.0x0.043 inch wire oriented 45 degrees off broadside. Measurements

performed using response of same wire oriented at broadside as a
calibrator.

194

 

 

 

 

 

 

 

 

 

 

: Double—Gaussian Pulse Response
1.05 E
2 wire - 6.0 x 0.043 in.
0.80 _. orientation I 45 degrees olf broadside
0.55 E
I
’3 0.30 :1
.2 e
4.; I
2 .1
a, 0.05 E
h I
v q
8. —0.20 E
o a .
2': 2
o .
> -0.45 a
-0.70 E
_o 95 I Moment-Method
' . ' ---- HP87208 (Broadside 6.0 x 0.043 in wire col.)
1
“1.20 jIIIIIIIIIIIIIIIIIIIITIIIIIIIIIIerIIII]
0.00 5.00 10.00 15.00 20.00

Time (nS)

Figure 5.40 Comparison of theory and experiment for the transient response of a
6.0x0.043 inch wire oriented 45 degrees off broadside. Measurements
performed using same wire oriented at broadside for calibrator.

195

angular orientation of the wire within the chamber.

5.10 High-Band Measurement Validation

All of the measurements shown previously were performed using the MSU ”low-
band” configuration. The "low-band" configuration uses a bandwidth of 0.400 to 4.400
GHz. The lower frequency limit is set by the antennas and anechoic chamber walls,
while the upper limit is set by the bandwidth of a Picosecond Pulse Labs 10 dB Amplifier
used on the output of the analyzer. A second ”high-band” configuration is also available
at MSU. This configuration uses a HP8349B microwave amplifier having a gain of
approximately 20 dB over the 1.0 to 20.0 GHz band. This allows the measurement band
of 1.0 to 7.0 GHz to be used. with good success. In this case, the lower limit is set by
the amplifier while the upper limit is set by limitations in the antennas and the feed
cables. Several modifications were made to the MSU anechoic chamber in an attempt to
increase the upper frequency limit beyond 7.0 GHz. The equipment and anechoic
chamber configurations are described more fully in Appendix C. The measurements
presented here were performed prior to the anechoic chamber upgrade of February 1992.
Beeause of the higher power available in the high-band configuration, the results in the
new configuration promise to be even better than the low band measurements previously
presented.

As an example of the potential quality of the high-band configuration the 3.0 inch
sphere was measured using the 14.0 inch sphere as a calibration standard. Figure 5.41
shows the transient response obtained by applying a 1/10 cosine taper weighing function

to the measurements prior to the IFFI‘. The equivalent pulse of the UN cosine window

196

0.0015
Metal Sphere (3 inch diameter)

0.0010

1/10 Cosine Taper
1.0 - 7.0 CHZ

0.0005

0.0000

 

Voltage (relative)

-0.0010

-0.0015

 

 

 

Mie Series
----- HP-87208 (14.0 in. sphere col.)

l
O
(’0
IJIILLLJLJIIIIIIJJJIIIIllllll111111

 

 

-0'0020 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIleTIIIIII1

2.00 3.00 4.00 5.00 6.00
Time (ns)

Figure 5.41 Comparison of theory and experiment transient response from a 3.0 inch
diameter sphere. Measurements performed using 14 inch diameter sphere
as a calibrator.

197

 

 

is shown in Figure 5.42. It should be noted that the data has not been normalized to the
peak value in this case indicating that the measurements can be made very accurately
relative to an absolute standard such as the 14 inch calibration sphere. The same

conclusion holds true for the low-band configuration as well.

198

 

0.1500
Synthesized Pulse

 

 

 

0.1000

0.0500
A
o
.Z
.0.)
2
9 0.0000
v
0
or
o
4:
o -
> 0.0500

—0.1000 1/10 Cosme Taper
1.0 - 7.0 GHz
-0'1500 [IIIIIIjIIIIIIIIfirIIIlIIIIIIIIIlfiIITIIII11
4.00 4.50 5.00 5.50 5.00
Time (ns)

Figure 5.42 Pulse synthesized from 1/10 cosine taper over bandwidth 1.0-7.0 GHz.

199

 

Chapter 6

Experimental Results

6.1 Introduction

This chapter discusses experimental results obtained during the course of this
research. The calibration method discussed in Chapter 5 is applied to make accurate
measurements of the both the early-time and late-time responses of canonical targets as
well as complicated aircraft models.

The results can be grouped into several categories. The first group of results is
concerned with the topics discussed in Chapter 3. Specifically, the scattered fields and
temporal currents of a system of two parallel wires are measured. The scattered field
measurements compare favorably with those computed using the frequency domain
moment method and thus validate conclusions of Chapter 3. The SEM mode frequencies
were extracted from the late-time portion of the transient current response measured
using a surface current probe. The extracted mode frequencies are seen to be in
agreement with results predicted using the SEM formulation.

Some of the capabilities of the frequency domain measurement system are
illustrated for the case of a wire stick airplane model. This is currently the most
sophisticated target that can be easily and accurately modeled using available computer
programs. In fact, the model measurements indicated areas of improvement in the

computer program used for the theoretical calculations.

200

A third group of results is concerned with experimental determination of the
effects of coupling on discrimination using the E-pulse method. To do this, the transient
scattered field of isolated and coupled model aircraft are measured using the frequency
domain measurement system.

Another group of results is concerned with the amount of energy contained in the
late-time vs early-time portion of an aircraft response. Accurate frequency domain
measurements are used to determine the fraction of the total energy in practical responses
that can be used for target discrimination. Lastly, the measured early-time response of

several scale model aircraft are shown and discussed.

6.2 Parallel Wire Systems

The SEM mode frequencies and currents of a system of two parallel wires have
been extensively discussed and analyzed in Chapter 3. This section presents
measurements supporting the theoretical calculations performed in Chapter 3 regarding
the SEM mode frequencies, transient currents, and scattered fields of a coupled wire
system. The measurements presented in this section represent, to the best of the author’s
knowledge, the first ever accurate measurements of transient coupling effects. The
importance of measurements such as these should be apparent after the discussions of the
theoretical difficulties presented in Chapter 3.

At the start of the investigation of transient coupling, the only way to obtain
reliable transient measurements was to use the ground plane transient scattering range.
To measure the effect, one monopole was constructed with a slot so that a loop type

magnetic flux probe [56] could be attached. A second identical monopole was placed

201

the same distance from the transmitting antenna and the distance between the monopoles
was varied in increments of 5 cm. For each value of separation distance the transient
current was recorded. Then, the SEM mode frequencies were extracted from the
measured transient current response using the Hybrid E-Pulse/Least Squares technique
[21]. In the extraction process, great care was taken to avoid inclusion of any part of the
response that occurred prior to the time when the wires could have interacted. The
damping coefficient and radian frequencies obtained from this process for the first mode
are plotted in Figures 6.1 and 6.2. Results obtained from the HALLEN formulation of
the moment method discussed in Chapter 3 are shown for comparison. The accuracy of
the results is quite good, especially for the damping coefficient which is typically very
difficult to accurately extract from measured data.

To investigate the effects of coupling on the transient response of multiple wire
systems several configurations were measured with the frequency domain measurement
system which was discussed in Chapter 5. Two identical stainless steel wires were
considered. The wire dimensions are 6.0 x 0.043 (diameter) inches. A 14 inch diameter
metal sphere was used as a calibration standard. The measurements were performed in
the old anechoic chamber configuration prior to February 1992 as shown in Appendix
C. The ”low band” (0.400-4.4 GHz) configuration was used throughout. The double
gaussian weighting function described in Chapter 5 was used to smooth the impulse
responses. For purposes of comparison between theory and experiment, all curves are
normalized to a peak value of unity. This normalization is appropriate since the
discrimination and detection schemes used in this thesis are dependent only on the

spectral content and the shape of the waveform and not on the overall amplitude of the

202

0.00

Identical Parallel Wires (l/o=200)
First Symmetric Mode

-0.05

o/(rrc/l)

-0.10

-0.15

 

Damping coefficient,

-0.20

 

Moment Method
00000 Measurement

LirrrrrrrlrrrrirrrrLerrrrrrrirLirrrrrrlrrrrruhriJ

 

 

-o-25 IIIIIIIIIIIIIIIIIII'IIIIIITII

I II
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Separation Distance, d/l

Figure 6.1 Damping coefficient of lowest order symmetric mode for two parallel
wires (I/a = 200) extracted from surface current measurements.

203

 

1.10

Identical Parallel Wires (l/a=200)
First Symmetric Mode

'0
o

.0
to
o

 

Radian frequency, w/(irc/l)

0.80

 

Moment Method
0000:: Measurement

errrrrrrlrrrirarrLl4Lerrrrrlrrrrrrrrr]

 

 

0°70 IIIIIIIIITIIIIIIIIrTfIII—[IIITIIIII III

I I
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.
Separation distance, d/I

l
00

Figure 6.2 Radian frequency of lowest order symmetric mode of two parallel wires
(l/a = 200) extracted from surface current measurements.

204

response.

Figures 6.3 and 6.4 show the near backscatter spectral and temporal responses of
the two wire system when the wires are separated 8.0 inches in down range. The wires
are in the broadside orientation. Several features are seen in the temporal response. The
first response is the initial early-time specular reflection from the first wire. This impulse
like response occurs at approximately 1.5 ns. This is followed by the late-time damped
sinusoidal response of the first wire. The damped sinusoidal behavior is interrupted by
the early-time specular reflection of the second wire. This is the impulse like response
that occurs at approximately 2.8 ns. This response is followed by the late—time damwd
sinusoidal response of the system of wires.

The temporal components are represented in the spectral measurement in different
ways. It is immediately obvious that the spectral response has a number of peaks and
nulls that appear at approximately integer multiples of one another. This effect is caused
by the presence of the two specular responses that are embedded in the temporal

response. An approximate temporal representation of the early-time specular response is

v¢(t) = {0(1) - 00-1)} *th) (6.1)

where t is the separation between the responses, and the convolution of f,(t) represents

approximately the difference between a Dirac delta function and the actual specular

response. The Fourier transform of this relation is

V,(i) = W(f){l+e'1°‘} = We”? {eh-””73 (6.2)

The magnitude of this spectrum is

205

1.20

1.00

p p
or 00
o 0

Voltage Magnitude (relative)
p
A
0

Frequency Response

 

Moment—Method
---- HP87208 (14 in. Sphere Col.)

2:5

wire 1 I 6.0 x 0.043 in.
wire 2 I 6.0 x 0.043 in.
range separation I 8.0 in.
cross-range sep. I 0.0 in.
orientation I broadside

 

 

 

 

" . '1 flfi‘v I
,’ ' \‘I' '0' "n :
I. I .‘I . .l
0.20 . - ' ', ' ,r
I. " \ I I I
'J ' 3
I , I
I: ‘r' ‘
0.00 TIIIIIIIIIIIIIIIHIIIIIIIIIIIIIIIIIIrIIIIIIIIIIIIr]
0.00 1.00 2.00 3.00 4.00 5.00
Frequency (GHz)
Figure 6.3 Spectral response of two parallel 6.0 x 0.043 inch wires separated by 8.0

inches in down range.

206

120‘ Double—Gaussian Pulse Response

1.00 -‘

3,—

0-80 "‘ Moment-Method
.. ---- HP87208 (14 in. Sphere Cot)

0.60 -

 

—v"——

0.40 -

0.20 - I

“J
0.00 4 ’

 

' 'v'

 

 

 

 

 

Voltage (relative)

l I
.o .o
# N
O O
l

1 r
a

 

 

-0.60 -

_I wire 1 I 6.0 x 0.043 in.
-0.80 wire 2 I 6.0 x 0.043 in.
range separation - 8.0 in.
cross-range sep. I 0.0 in.

 

 

-100 - orientation I broadside
4
-‘°20 IIjIIIIIIlTIIIIIIIIIIIIIIIIIIIIIIIIIIII]
0.00 5.00 10.00 15.00 20.00
Time (ns)

Figure 6.4 Double—Gaussian pulse response of two 6.0 x 0.043 inch wires separated
by 8.0 inches in down range.

207

IV.(f)l = Ithzcos-‘i’g- (63)
which has nulls and peaks separated by

Af = l (6.4)
t

For 8.0 inch down range separation this calculation gives 750 MHz between nulls or
between peaks. This number is consistent with the measurements.

Also apparent in the spectral response are peaks associated with the late-time
portions of the response. The peak at approximately 800 MHz is associated with the first
principal mode of the system of wires as well as that of the first wire prior to interaction
with the second wire. For close wire spacings however, the spectral response is
dominated by the late-time response of the system. For down range separations, the wires
currents are excited at different times. This leads to the antisymmetric system mode
dominating the response. It is interesting to observe how the response changes as a
function of down range separation. As the separation distance between the wires is
decreased, the delay between the specular reflections decreases and the distance between
the nulls in the spectral response is increased. This effect is clearly seen by comparing
the previous results with those of Figures 6.5 and 6.6 where a separation distance of only
6 inches is used and those of Figures 6.7 and 6.8. Note also that the height of the peak
associated with the first principal mode has increased relative to the rest of the spectrum,
while the duration of the temporal response is correspondingly increased.

Finally, consider the case when the wires are separated in down range by only 2.0
inches. These results are shown in Figures 6.9 and 6.10. There are no nulls within the
measurement bandwidth due to the specular reflections in the spectral response. It is seen

208

1-20 Frequency Response

1'00 Moment-Method

---- HP872OB (14 in. Sphere Cal.)

 

._-----m

.0
00
0

wire it I 6.0 x 0.043 in.
wire 2 I 6.0 x 0.043 in.

range separation as 6.0 in.
cross-range sep. I 0.0 in.
orientation .. broadside

 

.0
or
o

P
as
O

 

Voltage Magnitude (relative)

 

0.20
0.00 1
0.00 1.00 2.00 3.00 4.00 5.00
Frequency (GHz)
Figure 6.5 Spectral response of two parallel 6.0x0.043 inch wires separated by 6.0

inches in down range.

209

 

 

 

 

 

 

 

 

 

 

 

'20 - Double-Gaussian Pulse Response
1.00 -
0.80 _.
_I
I
0.60 - '
0.40 - I q
A ...
a) l
.2 0.20 — , I
4.4 , l
2 J : .
8 0.00 — g I
V . I I l
I I
8.4.20 - , I I
o I | I I
4.: .. . :1 U
o _. ._. .
> 0'40 ' — Moment-Method
~ ' ---- HP87208 (14 in. Sphere Col.)
-O.60 E
A
wire 1 I 6.0 x 0.043 in.
-0.80 - wire 2 I 6.0 x 0.043 in.
_ range separation I 4.0 in.
cross-range sep. I 0.0 in.
-190 ._. orientation I broadside
‘1
-1.20 rrrIIIIII[W11IIIIIIIIIIIIIIIIIIITIIIIII1
0.00 5.00 10.00 15.00 20.00
Time (nS)
Figure 6.6 Double-Gaussian pulse response of two 6.0x0.043 inch wires separated by

6.0 inches in down range.

210

1-29 '1 Frequency Response

 

1.00 -
'1
4 I

Moment—Method
---- HP872OB (14 in. Sphere Cal.)

.0
00
o
1

wire 1 I 6.0 x 0.043 in.
wire 2 = 6.0 x 0.043 in.
range separation I 4.0 in.
cross-range sep. I 0.0 in.
orientation I broadside

.0

01

o
I

 

Voltage Magnitude (relative)
g
r

 

 

 

 

0301
-I
I
I'..\
0-00 I11Iiifiaflrrrirliijiirriirli[iiiiiilr11IIIIIIIII]
0.00 1.00 2.00 3.00 4.00 5.00

Frequency (GHZ)

Figure 6.7 Spectral response of two 6.0x0.043 inch wires separated by 4.0 inches in
down range.

211

1.20

Double-Gaussian Pulse Response

 

 

 

 

 

 

 

 

1.00 A
0.80 -
0.60 —
s
0.40 —
A -l
0 l
.2 0.20 - ‘ ', ,' ‘. .
‘6 ' ' 'i :' ‘.
-l l ‘ \
E J I: : i : E I, ‘. p, ‘\ I “ ‘\ t" s ‘ ’ ‘
50.00-l '01:.‘l‘.:\‘:""l\’ .’\J’ I“ ‘-
l l . ‘ l ‘ I I l
u: || .‘ I“ I
3 -o.2o « ' ‘
O ..
t
O .. _
> 0'40 Moment-Method
-' ---- HP87208 (14 in. Sphere Cal.)
-O.60 -
‘ wire 1 - 6.0 x 0.043 in.
-O.80 - wire 2 - 6.0 x 0.043 in.
d range separation - 4.0 in.
cross-range sep. - 0.0 in.
-1.00 .4 orientation - broadside
”1.20 IIIITIIIl[IIIIIIIIIIIIIITITIIIIITII1I111
0.00 5.00 10.00 15.00 20.00
Time (ns)

Figure 6.8 Double-Gaussian pulse response of two 6.0x0.043 inch wires separated by
4.0 inches in down range.

212

that the peak associated with the first resonance is very narrow, or equivalently, the
temporal response rings for a long time at the first resonant frequency. The increase in
Q of the antisymmetric mode is consistent with the calculations of Chapter 3 that show
the antisymmetric mode pole moving very close to the jw-axis. Note that for this case the
moment method predicts a narrower resonant peak than is measured. This is in contrast
to the other measurements where the agreement is relatively good. It is believed that the
difference between theory and experiment is primarily due to resistive losses in the
stainless steel wires. These losses are not severe except when the Q of the resonance
becomes very high. To test this conjecture, the moment method was modified to account
for a distributed resistive loss and the scattering was recalculated for this case. The
theory used to account for the loss is well known and is detailed in Appendix A. Since
the actual conductivity of the stainless steel wires was unknown, the calculations were
performed for several different values that correspond to readily available materials. The
resulting curves are shown in Figure 6.11. It is apparent from the figure that resistive
losses affect primarily the response at resonance while the off resonance response is not
strongly affected. The radian frequency of the peak is relatively unaffected, but as
expected, the resistive losses alter the damping coefficient of the mode. Figure 6.12
shows a comparison of the measurements to the curve obtained using a = 1.5‘7x107 S/m.
The agreement is much better than for the case of a perfect conductor.

Now, consider the case of the same two wires separated in cross range. In this
situation, the specular responses from the wires arrive at the receiver at the same time.
Since the currents on the wires are excited in unison by the incident electric field, the

dominant system resonances are of the symmetric type. From Chapter 3 it was observed

213

 

 

 

 

 

‘-20 '1 Frequency Response
fi
1.00 -" Moment-Method
---- HP87208 (14 in. Sphere Cal.)
A .1 I
v :
.2 .
4.: 0.80 — g
2 . wire 1 - 6.0 x 0.043 in.
8 : wire 2 - 6.0 x 0.043 in.
v - 0 range separation - 2.0 in.
: cross—range sep. - .0.0 in.
cu I | orientation - broadsnde
l
gaw- H
a“: H |
r: ,. l
8‘ - ' '= °
2 :i-
0 0.40 -*
O‘
O
I: q
0
>
0.20 +
0-00 "WWW
0.00 1.00 2 00 3.00 4.00 5.00

Frequency (GHz)

Figure 6.9 Spectral response of two 6.0x0.043 inch wires separated by 2.0 inches in
down range.

214

 

 

 

 

 

 

1-20 - Double-Gaussian Pulse Response
1.00 d
1
0.80 -
0.60 -
0.40 .. l
e 1
.2 0.20 -l , . . i .
H I ,i " | ‘ , . ‘
% - ‘ " I'.‘ :i 0.: Int :‘t “ ‘I .‘\ " ’.‘ " 'i M
I l I ‘ I | \ \
b 0.00 -' ’ \ |. 'l “ i t‘ I ‘. " “l \ '1' ‘ y |\: '0 \l ‘1 \. \1
-l t .H t" " I I "
0 ' I
as —O.20 4 ,
O -l N
1’
O .. _
> 0‘40 , — Moment-Method
- ---- HP87208 (14 in. Sphere Cal.)
-O.60 -
q wire 1 I 6.0 x 0.043 in.
-O.80 - wire 2 I 6.0 x 0.043 in.
_ range separation I 2.0 in.
cross-range sep. I 0.0 in.
-100 .. orientation I broadside
-1.20 Iriirtillllllliilllltrtlrrrltl1rllrllll]
0.00 5.00 10.00 15.00 20.00
Time (ns)

Figure 6.10 Double-Gaussian pulse response of two 6.0x0.043 inch wires separated by
2.0 inches in down range.

215

0.02
Resistive Loss Effects

 

asinf (PE
..... a=6.17x10c7) A )
. .. .— 0:3.72x107 Al
_ _ 0-0.70x10 solder)

0.02

 

wire 1 I 6.0 x 0.043 in.

't wire 2 I 6.9 x 0.043 in.
H range separation I 2.0 in.
cross-ran e esp. I . in.
g 00
l “ orientation I broadside

0.01

 

0.01

E-field, Relative Magnitude

1:141:11:lrrtirrttrlrritlrrltltrrrrritrl

 

 

 

0'00 rjIIIIIIIIITIrIIIlIrllWIllllI

1
0.50 1.00 1.50 2.00
Frequency, GHZ

Figure 6.11 Effects of resistive loss on wire scattering - moment method calculations.

216

1.20

1.00

.0
an
o

0.60

Electric Field (relative)
.0
.p
O

0.20

0.00

Resistive Loss Effects

sO—pour-----'--
'—

- - - - HP87208

 

14 in. Sphere Cal.)
a=1.57x10 (brass) Moment method

wire 1 I 6.0 x 0.043 in.
wire 2 I 6.0 x 0.043 in.
range separation I 2.0 in.
cross-range sep. I 0.0 in.
orientation I broadside

 

|IIITTITIIIIIIIIIIII[IIIIIIIIIIIIITIIIIIIIIIIIIIII]

1.00

2.00 3.00 4.00 5.00

Frequency (GHz)

Figure 6.12 Comparison of measurements and theory including resistive losses.

217

that for close spacing, the dominant symmetric mode has a large negative damping
coefficient. This is easily seen in Figures 6.13 and 6.14 which show the spectral and
temporal response when the wires are separated by 2 inches in cross range. The
difference between this case and the case of 2 inch down range separation is impressive
and illustrates the need for detection and discrimination schemes that are insensitive to
the shape of the response waveform. As the cross range separation distance is increased,
the Q of the symmetric modes is increased and the response is dominated more and more
by the early-time response of the system which is characteristic of the response of the
isolated wires. These properties are illustrated in the responses for cross range separation
of 4.0 inches which are shown in Figures 6.15 and 6.16 and those with a cross range

separation of 8.0 inches which are shown in Figures 6.17 and 6.18.

218

 

 

 

 

 

 

:U
0
AMv>:~U_.m..LV

E

.0. .0..
WUDHWCOOE WOOu~O>

Fe.

figure

 

1.20 'n

Frequency Response

1.00 -‘

 

Moment-Method
.. . ---- HP87208 (14 in. Sphere Cal.)

0.80 -

 

.0

a:

o
I

 

 

Voltage Magnitude (relative)

wire 1 I 6.0 x 0.043 in.
0.20 - wire 2 I 6.0 x 0.043 in.
range separation I 0.0 in.
cross-range sep. I 2.0 in.
_ orientation I broadside

 

 

0.00 IIIIII1TUIIUIII1TTrITIlllllIIlirITllIIIIIIIIIIIIII

0.00 1.00 2.00 3.00 4.00 5.00
Frequency (GHZ)

Figure 6.13 Spectral response of two 6.0x0.043 inch wires separated by 2.0 inches in
cross range.

219

 

Voltage (relative)

Figure 6.14 Double-Gaussian pulse response of two 6.0x0.043 inch wires separated by

‘20 Double-Gaussian Pulse Response
1.00
0.80
0.60
I
0.40
0.20
l
.0. J , to. w. -
*:
-O.20
-O.4O Moment—Method
---- HP87208 (14 in. Sphere Col.)
-O.60
wire 1 I 6.0 x 0.043 in.
-O.80 wire 2 I 6.0 x 0.043 in.
range separation I 0.0 in.
cross-range sep. I 2.0 in.
-1.00 orientation I broadside
-1020 [lllllllIllIlIIIIITTUUIITITFT'IFIIUIIIW
0.00 5.00 10.00 15.00 20.00
Time (ns)

 

 

 

 

 

 

 

2.0 inches in cross range.

220

 

  
 
  
  
    

 

 

 

 

1-20 7 Frequency Response
fi
1'00 q Moment-Method
---- HP87208 (14 in. Sphere Cal.)

A "'
Q)
.2 wire 11 I 6.0 x 0.043 in.
+2 0.80 I wire 2 = 6.0 x 0.043 in.
2 range separation I 0.0 in.
e cross-range sep. I 4.0 in.
v s orientation I broadside
G)
g 0.60 -l
:‘1
C
C5 ..
O
2
q) 0.40 -‘
0‘
O
t .4
O
>

0.20 -

0°00 TIIIIII[I]!IIIII111I1TIIIIIYUIITIIIUIIUITTUIIIIIT—l

0.00 1.00 2.00 3.00 4.00 5.00

Frequency (GHz)

Figure 6.15 Spectral response of two 6.0x0.043 inch wires separated by 4.0 inches in
cross range.

221

120" Double-Gaussian Pulse Response

1.00 -
0.80 ..
0.60 -
0.40
0.20

0.00

 

 

Voltage (relative)

I I
.o .o
& N
O O
I

 

 

 

-— Moment-Method
H ---- HP87208 (14 in. Sphere Cal.)

-O.60 -

wire 1 I 6.0 x 0.043 in.
-0.80 I wire 2 I 6.0 x 0.043 in.
range separation I 0.0 in.
cross-range sep. I 4.0 in.

 

 

-100 .. orientation I broadside
-1.20 IIIIIIIIIIIIITIIIII[IIIIIIIIIrrIIIIIIIh
0.00 5.00 10.00 15.00 20.00
Time (ns)

Figure 6.16 Double-Gaussian pulse response of two 6.0x0.043 inch wires separated by
4.0 inches in cross range.

222

 

 

 

   

 

 

1-20 ‘ Frequency Response
1.00 -
4 i
A I.
Q) I . Moment-Method
,2 080 :: ---- HP87208 (14 in. Sphere Cal.)
4.; _
_g ' f'.
m : ‘.
L | '
v -i l I
l I
Q) : i wire 1 = 6.0 x 0.043 in.
'0 060 _J I . wire 2 I 6.9 x 0.043 in.
3 : : range separation I 0.0 in.
-- I . crass-range sep. I .8.0 in.
g, : ; orientation . broadside
:3 ‘ l l
5 i' l
q) 0.40 '4 i
0‘ .‘
o
t -
o
>
0.20 -
0.00 llllllll[III—riIleilllIIIIIIIIITWIIIVVTIIIIIIIIII
0.00 1.00 2.00 3.00 4.00 5.00
Frequency (GHz)
Figure 6.17 Spectral response of two 6.0x0.043 inch wires separated by 8.0 inches in

 

cross range.

223

120‘ Double-Gaussian Pulse Response

 

 

 

 

 

 

 

 

 

 

 

 

1.00 I
0.80 I
0.60
0.40 ,
A I
Q) l
.2 0.20 ;
'9 l
2 :
2 0.00 g
V I
I
I
85-020 :
o l
I :
O ..
> 0.40 E
l
O 60 _ ' Moment-Method
’ - ---- HP87208 (14 in. Sphere Cal.)
-O.80 I
_ wire I I 6.0 x 0.043 in.
wire 2 I 6.0 x 0.043 in.
-100 _ range separation I 0.0 in.
cross-range sep. I 8.0 in.
‘I orientation I broadside
-1020 TIIIIIllTrTIIIIITITjIIITIIIIIW—TTIIIIITfi
0.00 5.00 10.00 15.00 20.00
Time (ns)

Figure 6.18 Double-Gaussian pulse response of two 6.0x0.043 inch wires separated by
8.0 inches in cross range.

224

6.3 Wire Stick Airplane Model

This section presents a comparison of theory and measurement for a wire stick
airplane model. The wire stick model is diagramed in Figure 6.19. The model was made
from solid copper wire that was pre-stressed to increase stiffness by stretching it beyond
the plastic limit. The wires were attached using tin-lead solder. The size of the joints was
kept small by filing away the excess solder. The joints were made as small as possible
while maintaining the physical and electrical integrity of the contact.

The wire model was measured in the anechoic chamber using the frequency
domain scattering measurement system. The low-band measurement (C.4-4.4 GHz)
configuration was used for all measurements. It was anticipated that the measurements
would be repeated for the high-band configuration but this was not possible due to the
timing of equipment malfunctions and the anechoic chamber upgrade. The measurements
were verified by comparison to results obtained from a frequency domain moment
method calculation. Unlike other results in this thesis, the theoretical calculations in this
section were obtained using a computer program developed by Dr. E]. Rothwell. This
program uses piecewise sinusoidal basis functions and testing functions as previously
discussed. The program is general and can handle arbitrarily oriented wires and junctions
between wires. The junction is accounted for by placing a half segment on each wire at
the junction. Kirchoff‘s current law is enforced at the junction.

The incident field is assumed to be a plane wave and the scattered field is
computed according to the geometry of the anechoic chamber prior to February 1992.

The aircraft model was measured for five aspect angles between nose-on and tail-on.

225

0.906

_lr

3.203

 

0.984

wire radius = 0.018

All dimensions in inches

Figure 6.19 Wire stick airplane model.

226

This series of measurements is presented in this section. The spectral and temporal
responses of the nose—on response are shown in Figures 6.20 and Figure 6.21. As before,
the theoretical and experimental results are normalized to a peak value of unity. The
double gaussian pulse is also used as before. Again, the theoretical and experimental
results are in excellent agreement for both the spectral and temporal responses. The
spectral response indicates that nose-on response is dominated by the SEM modes of the
wing and tail.

When the airplane model is illuminated 45 degrees from nose-on, the response
becomes more complicated as shown in Figures 6.22 and 6.23. When the fuselage is
parallel to the incident field, the response is dominated by the lower fiequency fuselage
and wing-fuselage modes. The broadside (90 degree) case is shown in Figures 6.24 and
6.25. For an orientation angle of 135 degrees from nose-on, the response is again more
complicated. The responses for this orientation are shown in Figures 6.26 and 6.27. For
tail-on incidence, the response is similar to the nose-on incidence. The response is

dominated by the wing mode as shown in Figures 6.28 and 6.29.

227

1.20 Wire Stick Airplane Model

1.00 0 Degrees (nose—on)
I
'i
$0.80 ‘.
> '. —— Theory
2; ' ______ E t
o ; xperimen
T) '.
t, i
0.50 i,
'0 ‘i
76 i
E ‘i‘
.9 i. i
b 0.40 ‘ "I.
8 z a
[I] “.’r\’\ ' '
\‘I‘ I,
0.20 ‘ .\"‘l‘f\"~’\-\ ‘ I"
0.00 i
0.00

1.00 2.00 3.00

4.00
Frequency (GHz)

5.00

Figure 6.20 Spectral response of wire stick airplane model for nose-on (0 degrees)
incidence.

228

1.00

 

 

 

 

 

 

 

 

 

 

 

q
1 Wire Stick Airplane Model
.4
0.75 ~
j 0 Degrees (nose—on)
3 :
0.50 -‘ I
- 3 -———- Theo
j : ------ Experiment
’3? - i i
3 0.25 - :
H d
2 ..
8 d
v 000 ._..M/i
'0 -‘ I
75 j I'
U.- - i: i
1
.2 -0.25 I ‘:
g. _ l
4.1
U ‘l
2 ..
._.; -0.50 - ii
q
.1
-0.75q g
-1.00 Iiil[iiiilii1iliiilfiiiTliiil]
0.00 2.50 5.00 7.50 10.00 12.50 15.00

Time (ns)

Figure 6.21 Temporal response of wire stick airplane model for nose-on incidence.

229

 

 

 

 

 

1'20 2 Wire Stick Airplane Model
100 J , 45 Degrees
Z ,‘|
: v: -— Theory
: ' , ------ Experiment
: i. i
A030 - :. '
(D I i‘
> _ I: g
:0: .. [I I i'
C : i 1 I
B - i i 1
L 2 l l‘ ' I
V0.60 - : . . t‘.
‘ l l I I I
U I 1 I ‘ K I
a, : ' l :
e... - “ I .
LL c-i ' I
: ' ‘ . ' '
.2 - l 3.. E
b 040 j I ' ‘I‘H a
0 d | I. ‘ \\
2 : ‘i : ‘. i
q I
LJJ j v .| '0‘. I“ 5
I1 .I I, ‘\ :
0.20 "J ' I I '
.. i,\ , I I
-i I I “I
s :.
o-OOdIIIIIWITTTIIIIIIIITTIIIII[IIIIIIIIIIITIIIIIIIIIII]
0.00 1.00 2.00 3.00 4.00 5.00

Frequency (GHz)

Figure 6.22 Spectral response of wire stick airplane model for 45 degree orientation.

230

1.00

 

 

 

 

 

 

 

 

: Wire Stick Airplane Model
0.75 -i
2 45 Degrees
.I
0.50 —
- -———— Theo
j ------ Experiment
’0‘ -
,2 0.25 -
._.. -
2 ._.
a) -i
b - 1
0.00 - 1
29 ‘ l
.2 I 1 1
LI. _ 1
0 -0.25 - l I
.C : I
.5;
o -1
-“—’ ' i
._.; -0.50 I
" l
-0.75 -
.i
1
-1.00 lllIIIIIIIIIIllTlIIIIT11111TIIIIITIIIIIj

 

 

0.00 2.50 5.00 7.50 10.00 12.50 15.00 17.50 20.00
Time (ns)

Figure 6.23 Temporal response of wire stick airplane model for 45 degree orientation.

231

1.20 Wire Stick Airplane Model

 

 

 

 

1.00 ,
90 Degrees (broadside)
1
$0.80
._>_ . -— Theo
E ------ Experiment
0 l
‘- 1
v0.60 .
E l
.9.) .'
LL 1
U
'i: 0.40
4.]
U
.9
La
0.20
0'00 IIIIIIIIIIrITTIIIIIIIITITIIIIIIIIIjIIIIIIIITIIIIII]
0.00 1.00 2.00 3.00 4.00 5.00

Frequency (CH2)

Figure 6.24 Spectral response of wire stick airplane model for broadside (90 degree)
orientation.

232

 

 

 

 

 

 

 

 

 

 

 

 

 

1 Wire Stick Airplane Model
0.80 —
3 - 90 Degrees (broadside)
0.55 I
I —— Theory
0.30 i . , ------ Experiment
A _J U I‘
Q) _l II
‘°"’ Ii 1 '
2 0.05 - I .
Q) - 1
L —1 I
v j ‘ :
'o ‘ 1
.72 -0.20 -_ 3 :i 1
LL. " t :
_i l
E -0.45 - i;
y q
0 -
l) a
L” —0.70 -
-0.95 ..
-1.20 riil1lili[111iiiiiilr1iiiriiilriliiliiil
0.00 5.00 10.00 15.00 20.00
Time (ns)

Figure 6.25 Temporal response of wire stick airplane model for broadside (90 degree)
orientation.

233

1.20 Wire Stick Airplane Model

 

 

 

1.00 135 Degrees
—— Theory
1 .. -
r -- -- Experiment

AO.80 ‘
i0
.2 l
.0.)
2 r
a) l
h i
V0.60
E
.2
LL.
.2
L 0.40
.0.-I
U
2
Lu

0.20

l
0.00
0.00 1.00 2.00 3.00 4.00 5.00

Frequency (GHZ)

Figure 6.26 Spectral response of wire stick airplane model for 135 degree orientation.

234

 

 

 

 

I Wire Stick Airplane Model
I.OO -‘ I
075-? 135 Degrees
- -——— Theo
0,50 1 ------ Experiment
2: .
.2 3
fl .
2 0.25 -
a) -
L d
V d
D d
.7.) 0.00 -
i... -« ‘
U -l
'C -0.25 I
.H -I
U .1
g -
L” -0.50 — ’
I
-0.75 I
-I.00 TiiriiiiIIIIiiriiiiliiiliiiTiliirTiiiii]

 

 

0.00 5.00 10.00 15.00 20.00
Time (ns)

Figure 6.27 Temporal response of wire stick airplane model for 135 degree
orientation.

235

1.20 Wire Stick Airplane Model

 

 

1.00
180 Degrees (tail—on)

$0.80 .

.2 '. -— Theory

§ i. ------ Experiment
0 i
b .

0.60 . g
E 1
.92 ‘
LL
0
'1: 0.40
.0.;
U
.93
Lu

0.20

0.00 ’

0.00 1.00

2.00 3.00 4.00
Frequency (GHZ)

5.00

Figure 6.28 Spectral response of wire stick airplane model for tail-on incidence.

236

Wire Stick Airplane Model
1.00

 

0.75 i 180 Degrees (tail—on)
Theory
050 i ------ Experiment
)1

0.25

0.00

 

 

-O.25

 

 

 

Electric Field (relative)

-O.50

 

 

 

-0.75

LlllLllllllLilIlllJllllllllLllJlJIllIlllllL]

 

 

-1°oo I I I l I I I I T I I I I I I l I I I l I I I I T U l U r l j T 1 T I I I 1 T1
0.00 5.00 10.00 15.00 20.00
Time (ns)

Figure 6.29 Temporal response of wire stick airplane model for tail-on incidence.

237

6.4 Early-Time vs Late-time Energy

The E-pulse and S-pulse discrimination schemes are based on the late-time
response of the target. Thus, it is of great importance to determine how much of the
received signal energy is contained in the late-time for practical targets. This section
presents results of an experimental study which examines this question.

To quantify the amount of energy in the late-time response of the target, the ratio
of the energy in the late-time to energy in the received waveform is defined as

f v,(t)2dt
R," = h“ “'" (6.5)

fv,(t)2dt

 

where v,(t) is the received signal.

This ratio is expected to be a function of the bandwidth of the pulse that is used
to excite the target. For purposes of this study, only pulses whose spectrum includes the
lower target resonances were considered.

For the experimental study, three aircraft models were used. They are:

(l) 0.8. Air Force A10 Thunderbolt (6.75 inch wing span)

(2) U.S. Air Force F15 E (7.5 inch nose—to-tail length)

(3) U.S. Air Force F16 (6.375 inch nose-to-tail length)

The response of these models was obtained using the frequency domain measurement
system described in Chapter 5. The low-band (C.4-4.4 GHz) configuration was used and
a 14 inch diameter sphere used as the calibration standard.

An example of the spectral response of the A10 model for three different
incidence angles is shown in Figure 6.30. The resonant nature of the response is clearly

238

 

demonstrated in the lower half of the measured spectrum. An example of the double
gaussian pulse response of this model is shown in Figure 6.31. It is clear that for both
nose-on and broadside responses the response in late-time is very small while the 45
degree response has more significant late-time oscillations.

The energy available for target discrimination is a function of the aspect angle.
This means that the ability to discriminate based on natural resonances is aspect
dependent whereas the natural resonance frequencies are aspect independent. Similar
observations can be made for the case of the F15 model. The frequency response is
shown in Figure 6.32 while the double-gaussian pulse response is shown in Figure 6.33.

To provide a measure of the amount of energy available for discrimination, the
R,“ values are calculated for the three models for 5 different aspect angles. The results
are shown in Table 6.1. It is clear from this table that the fraction of total signal energy
present in the late-time varies greatly with aspect angle. There are however clearly cases
when the late-time response contains a significant percentage of the total received energy.
Unfortunately, there are also cases where the late-time energy is very small in
comparison to the total received energy. This leaves some worries about the ability of

the E—pulse and S-pulse schemes to provide good discrimination for these aspect angles.

239

0.030

0.025

.0
O
N
O

0.015

0.010

Electric Field (relative)

0.005

JlllllllllllIllllllllllllllllllllllllllllllllllllllllllllljl

U.S. Airforce A10 Thunderbolt Model
(6.75 inch wing span)

I
/

 

J

 

nose-on

------ 45 degrees

- - - broadsude

I“
~\\ ’\ J
/ ”\ l
I“ r
I, \ \
\
\ I l
[\K \ \I" l
I:
\ At
'II‘ fix" ’ ft" vi“ I
0' . I; "N V). |
I II I '| |
I ‘ t
I \ I
ll, ‘
N
I
\\‘
\
‘I‘o

 

 

0.000

O

.00

1.00

2.00 3.00
Frequency (GHz)

Figure 6.30 Spectral response of A10 model.

240

I I I I I I I I I l I I I I I I I I I I I I I I I III I I l I I I I I I I I I l I I I’I’I I I I I I

4.00 5.00

U.S. Airforce A10 Thunderbolt Model

 

 

 

 

 

 

 

 

‘-°°E+°°7 j (6.75 inch wing span)
I nose—on
5005+006-1 l ...... :5 degrees 0‘.
. q ~ - — - roadSide
l 2

A ' I:

a) s 'I

> - :i

:1 ‘4 II

a . :n

E q H:

L. .. A l i

V o.ooc+ooo - :

2 : t i

o . \I i

i: . i: ;

.2 I l '

h _

.0.:

o -I

2 -

m -5.00E+006 -

-1-mE+oo7 IIIIIIIIIIIITIIIIITIIIIIrIIrIlIIIIIIIII]
0.00 5.00 10.00 15.00 20.00

Time (ns)

Figure 6.31 Double-gaussian pulse response of A10 model.

241

0.040

 

 

 

 

- U.S. Airforce F15 Model
3 (7.5 inch nose-tail)
' nose-on
0.030 4 ~
- n ------ 45 degrees
3 I \ - - - broadsude
2? ~ , \
.2 ‘ ‘ l
E _ l \
cu . l \
b ‘ l \
0.020 - g \
a r . . ~
.- .. l \ '
L‘— -I ' I‘ \ ”
-I '.
o _ I.’ ‘~ ‘
'C - l‘ ‘. (a I‘
-o-' ' “ JI \
U " I: “ I‘ I
2 " ‘l i n “
Lu 0.010 - ' ‘ .' \ i
u I: I ‘0. I \ ‘
‘ l.’ ‘. W I \“
I l ‘. i‘i" \ "\ ’l /\
- I. ‘i r: ‘\I l'l V7.3; I‘“ VJ“: ‘ V
.. ' v\“.
.. '. ' )1
q ‘ ‘1' ‘ \ f /" I'I“I\
V I g V q
" x J \ V"
00000 IIIIIIIIIIIIIrIIIIIIIIIIIIT’IIIIIITIIIIIIIrIIIIIm
0.00 1.00 2.00 3.00 4.00 5.00

Frequency (GHz)

Figure 6.32 Spectral response of F15 model.

242

1.OOE+007 j U.S. Airforce F15 Model

 
  
  

 

nose-on
------ 45 degrees

5.00E+006 — - - bl’OOdSlde

0.00E+000

 

 

 

Electric Field (relative)

-5.00E+006

lljllLllIJLllLllLlIJllllllllIllllJlJJ

 

 

—1-OOE+OO7 I I I I I I I I I l 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 l
0.00 5.00 10.00 15.00 20.00
Time (ns)

Figure 6.33 Double-gaussian pulse response of F15 model.

243

Table 6.1 Rhe values for model aircraft.

 

 

 

   
 

 

 

 

 

 

ll Angle (degrees) F15 Model A10 Model =
0 (nose-on) 0.32 0.05
22.5 0.20 0.13 0.10 I
45.0 0.11 0.35 0.17 I
67.5 0.12 0.15 0.02 I
90.0 0.08 0.04 J03 I

 

 

 

 

 

6.5 Effects of Coupling on Aircraft Model Discrimination

This section extends the discussion of Chapter 4 to investigate the effect of
coupling on E-pulse and S-pulse discrimination schemes for the case of realistic aircraft
models separated in cross range. Since realistic aircraft models are difficult to model
theoretically, the evaluation is made using experimentally derived data.

For this study, two identical A10 models, an F16 model, and an F15 model are
used. These models are the same as those used in the previous section. The individual
models were measured using the same technique as previously described. The constrained
E-pulse technique (CET) developed by Ilavarasan ['2] was used to extract the natural
frequencies of the A10, F15 and F16 models from the measured transient responses. The
extraction was performed using responses from three aspect angles for each target.

Due to limitations on the early version of the CET computer program a maximum

of 9 modes could be reliable extracted from measured data. The aircraft models are very

244

complicated structures and the transient data apparently contained more than 9 modes
over the measured bandwidth from 0.4 to 4.4 GHz. To allow use of the CET program,
the data was filtered using a GMC weighting function with parameters 0 = 1.3 ns, 60 =
1.3 GHz. This window function reduces the effects of spectral truncation on the transient
response and eliminates enough modes so that the CET program can be used. The natural
frequencies extracted for the F15 model are shown in Table 6.2, while those of the A10
are shown in Table 6.3, and the F16 results are listed in Table 6.4. Note that the lowest
frequency mode common to all targets is most likely the result of low frequency

truncation of the frequency spectrum.

Table 6.2 Natural frequencies of A10 model extracted using CET method.

 

 

 

 

 

 

 

 

 

 

Mode 0 x 109 np/s w x 10’ rad/s I
1 -0.4184 9.8037 I
2 -4.586 x 107 9.4300 I
3 0.2728 8.2461 I
4 0.0207 7.0993 I
5 0.2727 6.0708
6 0.1275 5.1669 n
7 -8.2958 x 107 4.7827 1'
8 0.3552 4.0202
1 9 0.0477 3.2012

 

 

 

 

 

245

Table 6.3 Natural frequencies of F15 model extracted using CET method.

 

 

 

 

 

 

 

 

 

 

Made a x 109 np/s w x 109 rad/s N
1 -0.4413 9.6503
2 -0. 1673 9.8805
3 .02944 8.3826
4 -0.1734 7.7460
5 -0.0398 6.8470
6 -0.03OO 5.3380
7 -0.0520 5.1525
8 -O.2831 4.2018
9 -0.0003 3.1904

 

 

 

 

 

 

 

Originally, it was anticipated that the E—pulse method would be used to
demonstrate the effects of coupling for measured data from realistic aircraft. However,
after much effort it was found that the S-pulse method provides better discrimination than
the E-pulse method. Apparently, measured data from complicated targets contains more
modes than can currently be used effectively. Also, there are also unavoidable errors in
the extracted natural frequencies due to underestimating the number of modes, noise, and
systematic errors in the measurements due to slight inaccuracies in the deconvolution
procedure. Since the E-pulse attempts to extinguish all modes, any errors in the natural

dominant mode natural frequencies results in slight variations from the idealized zero

246

result. Unfortunately, these slight variations drastically affect the discrimination ability
of the E—pulse.

In the case of S-pulse discrimination, the dominant mode of the response is
usually chosen for discrimination. Thus, the convolution operation results in a waveform
that is dominated by a single mode signal. For the S-pulse, however, errors in the natural
frequencies cause incomplete cancellation of only the weaker modes and hence a smaller
overall deviation of the convolution from the expected single mode spectrum. It is
conjectured that this may be a real and significant advantage of the S-pulse in actual

applications where measured data must be used.

Table 6.4 Natural frequencies of F16 model extracted using CET method.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Mode 0 x 109 np/s w x 109 rad/s I
1 0.2919 10.158 I
2 0.3460 9.2589
3 0.4020 8.3495
4 -2.47 x 10*5 7.4621 I
5 0.0744 6.4881
6 0.3020 5.6535
7 -O.1046 4.9759 I
8 0.3473 3.9691 I
9 0.0784 3.2080 _l

 

247

To evaluate the effect of coupling on discrimination, the S-pulse discrimination
scheme was applied to the targets individually and to the coupled A10 targets. For the
F15 model an S-pulse was constructed for mode 3, while for the A10 model mode 5 was
used, and mode 3 was used for the F16. Note that mode 3 of the F15 and F16 are very
similar. This similarity prevented the S-pulse method from providing good single target
discrimination results when all three models were included in the discrimination process.
Thus, a binary discrimination problem was evaluated. The first binary discrimination
problem performed used the F16 and the A10 model responses and S-pulses. The

discrimination results for single aircraft are shown in Table 6.5. Here it is seen that

Table 6.5 SDR values computed for single targets using S-pulses for F16 and A10.

 

 

 

 

 

 

 

 

 

 

 

Ii Target Orientation F16 (dB) A 10 (dB)
1 F16 nose-on 0.0 17.42
; F16 45 degrees 0.0 16.37
i F16 broadside 0.0 4.38
I A10 #2 nose-on 0.0 4.73
2 A10 #2 45 degrees 15.07 0.0
: A10 #2 broadside 13.49 0.0
l

i A10 #2 " 30 degrees 4.84 0.0
i A10 #2 "‘ 60 degrees 20.23 0.0
l A10 #1 * 30 degrees 7.89 0.0
I A10 #1 * 45 degrees 13.34 0.0

 

 

 

248

discrimination is very good for most aspect angles. The waveforms denoted with * are
independent of the mode extraction procedure. Discrimination with these waveforms
confirm the validity of the extracted modes and the overall results. This is important to
prevent a circular argument that is possible when using the same waveforms for both
made extraction and demonstration of discrimination.

The discrimination process using the same two S-pulses was applied to measured
responses of two A10 models separated vertically in cross range. (In this configuration
one model appears to be flying directly above the other with an aspect of approximately
45 degrees from nose-on.) The results of this study are shown in Table 6.6. It is seen
that the SDR values are significantly lower than those obtained when only a single A10

aircraft is present at a 45 degree orientation.

SDR values computed for coupled A10 models using S-pulses for F16 and
A10.

Table 6.6

 

 

 

 

 

 

Target Separation F16 (dB) A10 (dB) I
2 - AlO’s 1.75" 4.19 0.0 I
2 - AlO’s 3.00” 3.18 0.0
2 - AlO’s 4.25” 0.0 0.17
2 - AlO’s 5.50” 0.16 0.0
2 - AlO’s 6.25” 1.41 0.0

 

 

 

 

 

249

The second binary discrimination problem considered the case of the F15 and A10
S-pulses. The results of this study for single targets are shown in Table 6.7. It is clear
that excellent discrimination is possible for single targets in this binary case. Finally, the
discrimination process was performed using measured data for two coupled A10 aircraft
as before. The results of this study are shown in Table 6.8. The results in this case
indicate that discrimination while still possible is significantly degraded as compared to
the single target scenario.

The results of this section demonstrates using realistic aircraft models that

coupling can degrade the discrimination performance of the S-pulse method. It is

Table 6.7 SDR values computed for single targets using S-pulses for F15 and A10.

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

Target Orientation F 15 (dB) A10 (dB) I
F15 nose-on 0.0 3. 75
F15 45 degrees 0.0 12. 50
F15 broadside 0.0 4. 68
A10 #2 nose-on 0.0 1.40
A10 #2 45 degrees 18.32
A10 #2 broadside 17.57
A10 #2 * 30 degrees 10.72
A10 #2 * 60 degrees 22.23 €31
A10 #1 r 30 degrees 10.91 0. 0 I
A10 #1 "‘ 60 degrees 16.47 0.0 I

 

 

 

250

Table 6.8 SDR values computed for coupled A10 models using S-pulses for F15 and

 

 

 

 

 

 

 

A10.
Target Separation F15 (dB) A10 (dB)
2 - AlO’s 1.75" 7.24 0.0
2 - AlO’s 3.00” 7.49 0.0
2 - AlO’s 4.25” 3.47 0.0
2 - A10’s 5.50" 4.91 0.0
l 2 - AlO’s 6.25" 4.36 0.0 .4

 

 

 

 

 

expected that coupling will also degrade the performance of the E—pulse method. Of
course, the amount of degradation is a function of the number and types of targets to be
discriminated. Because of these complexities, the results of this section can only be
interpreted as a demonstration that coupling affects discrimination results and not as a

general quantitative measure of the amount of degradation to be expected.

6.6 Early-time Scattering from Scale Model Aircraft

This section discusses the use of early-time scattering phenomena in target
detection and discrimination. The measured responses of several 1/48 scale aircraft
models are used to illustrate early-time scattering mechanisms. The measurements were
performed in the MSU anechoic chamber using the vector network analyzer in the
frequency band from 1.0 to 7.0 Ghz. The calibration procedure of Chapter 5 was used

to correct the raw measurements for systematic errors introduced by the measurement

251

system. A metal calibration sphere 14 inches in diameter was used as a calibrator in all
measurements. To achieve a narrow pulse without introducing excessive windowing
effects into the transient data a 1/ 10 cosine taper weighting function was used. The
effective pulse for this weighing function was shown previously in Chapter 5 and is not
be repeated here.

The scale model targets were constructed from plastic model kits that are widely
available in hobby shops and department stores. Since models are constructed of plastic,
they were carefully coated by hand with aluminum foil to more closely simulate the
actual conducting aircraft. The foil was attached using a cyanoacrylate adhesive
commonly available as " super glue" in the department stores. The models were simplified
by omitting certain features such as some externally attached weapon systems and
missiles and by leaving all landing gear in the up positions. All flaps and movable wing
parts were permanently fixed into position to eliminate any chance that the electrical
properties of the models would change during or between measurements. The models
used in this chapter include: N .A.S.A. Lockheed TRl (atmospheric research plane -
updated version of U2 spy plane) and a U.S.A.F. B58 Hustler (obsolete supersonic
strategic bomber).

One of the most obvious and interesting phenomena associated with early-time
scattering is the fact that response can be seen to be caused by specific localized
"scattering centers" at distinct locations on the aircraft. This is in marked contrast to the
late-time scattering response which is of a global nature. This can clearly be seen in
Figure 6.34 which shows the synthesized transient response of the Lockheed TRl model

for nearly "wing-an" incidence. Note that the size of the aircraft drawing has been scaled

252

to the time axis so that the association between the response peaks and the physical
locations on the aircraft model are more evident. The incident wave is applied from the
left for this interpretation. From this figure it is clear the that response is dominated by
two strong reflections. One reflection is from the vicinity of the left electronics pod and
left edge of the tail wing. The other strong reflection is from the main fuselage of the
aircraft. A smaller reflection occurs when the pulse strikes the edge of the wing from the
left. It should be pointed out that there is little late-time energy associated with this target
response. This due primarily to the fact that the lowest frequency and hence highest Q
natural resonances of this aircraft model are well below the 1.0 GHz lower limit used in
the measurement and are not excited. Unfortunately, even use of the low-band
measurement configuration does not resolve this problem since the low frequency limit
is set by the anechoic chamber performance.

As a further illustration of the scattering center phenomena, consider the wing-on
response of the B58 Hustler model shown in Figure 6.35. Again, the scattering centers
are clearly associated with the engines and main fuselage of the aircraft. Note that the
scattering centers on the right hand side of the aircraft are not as distinct as those on the
left hand side. This is most likely due to a shadowing of these discontinuities by the
engines on the left and the main fuselage.

The aspect dependency of the early-time scattering can be seen by examining the
nose-on response of the same B58 Hustler model as shown in Figure 6.36. Here, the
scattering centers appear to be associated primarily with the current discontinuities the

engine mounting junctions.

253

 

 

 

 

 

 

 

 

 

 

1 h
.1
0.002 -

A ‘

Q) ..

.2

H d

‘D ..

L

v _I

m q

0’ a

O

t -l

o - —1

> 0.002 -
.l
I l LOCKHEED 131 Model (1:48)
. Wing-on lncrdence
'l HP87208 1.0-7.0 60-12 1/10 cosine taper
4

-0.006 111-WWW

1.00 2.00 3.00 4.00 5.00 6.00 7.00

Time (ns)

Figure 6.34 Early time scattering by Lockheed TRlA for wing-on incidence.

254

 

 

 

 

 

\.
Jlli

l [11:10

 

 

 

 

 

 

 

 

(L004

0.002

 

 

 

Voltage (relative)

 

 

U 858 Hustler 1:48

Wing-on Incidence
-0.ooe

HP87206 1.0—7.0 6H2 1/10 cosine taper

111111111111111111111111111111rr111111111111111111111111141]

 

 

-0'008 I I I TNT 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 I I fl I I I I I I I I I I
1 .00 2.00 3.00 . 6.00
Time (ns)

Figure 6.35 Early-time response of B5 8 Hustler for wing-on incidence.

255

   

0.002

 

 

 

 

I
'l
0.001 1
I
1
J
’5? ‘
.2 -0.000 1
H ..
2 d
0 I
L
v ‘
g I
Z .1
§ 1
I
2 858 Hustler 1:48
'°-°°2 2‘ Nose-on lncndence
I
: HP87208 1.0-7.0 CH: 1/10 cosine taper
-o.003 1
1.00 2.00 3.00 4.00 5.00 5.00 7.00
Time (ns)

Figure 6.36 Early-time response of B58 Hustler for nose-on incidence.

256

Chapter 7

Conclusions

This dissertation has presented a number of tapics and ideas related to the
application of UWB radar to the detection and discrimination of targets. In Chapter 2,
the performance of an automated discrimination scheme using the E-pulse and S-pulse
method is evaluated using transient scattering data for thin-wire targets in the presence
of varying amounts of additive random gaussian noise. The results of this analysis serve
as a baseline for comparison of other discrimination methods as well as the effects of
multiple targets on the effectiveness of the discrimination. Some initial considerations for
applying the discrimination scheme in a multiple target environment are discussed in the
last section of Chapter 2. In examining the multiple target environment, questions arise
concerning the effects of interactions on the SEM mode frequencies of the target. To
attempt to answer these questions, the detailed examination of the SEM analysis of a
system of coupled thin wire targets was undertaken. Several methods of varying accuracy
and computational requirements were used to calculate the SEM mode frequencies of the
coupled wire system. The behavior of the SEM modes of the system of wires is very
complicated. The physical significance of the system modes had not been previously
published. Chapter 3 attempts to provide an increased understanding of the transient
coupling phenomena and provide the correct physical picture for interpretation of the

SEM formulation. In Chapter 4 this new physical picture is used as a guide to interpret

257

the results from an empirical study of the performance of the E-pulse and S-pulse
discrimination scheme as applied to analytically generated scattering data for a system
of two identical thin wire targets. Specifically, it is seen that the discrimination scheme
is adversely affected when two thin wire targets are in close proximity. It is found that
for cross range separation that the principle mechanism for degrading the discrimination
is the shift of the natural frequencies due to the target interactions. In the case of down
range separations, the principle degrading mechanism is the large early-time response of
the second target.

A great deal of effort was expended during the course of this research in attempts
to improve the transient measurement capability at the Michigan State University
Electromagnetics Laboratory. This improvement was motivated by the need to have
accurate transient scattering data from complex targets such as aircraft that cannot be
modeled theoretically. In particular, the application of the E-pulse and S-pulse methods
require that the natural frequencies of the targets be measured in a laboratory setting
prior to implementing the discrimination scheme. Since the extraction algorithms in
current use are sensitive to noise as well as systematic errors induced by the
measurement systems, the transient measurements must be made as accurately and
carefully as possible. Further, efforts are currently underway to use the early-time
portion of the transient response of the target to enhance detection as well as for target
discrimination and imaging applications. Thus, acquiring high quality measurements of
the early-time response of complicated targets is essential for proof of concept
applications and for use in verifying the accuracy of theoretical models. Chapter 5

presents a method for making accurate transient scattering measurements using a

258

frequency stepping vector network analyzer. The accuracy of the measurements is due
both to the inherent advantages of the network analyzer such as high stability and wide
dynamic range and to the use of a robust procedure for removing clutter and
deconvolving the effects of the measurement system from the raw measurements. It is
seen that the deconvolution process also improves the results obtained using direct time
domain measurements. The accuracy of the measurements is demonstrated using several
canonical targets such as the sphere and the thin wire.

Chapter 6 presents further demonstration of the quality of the measurement system
as applied to measurement of the scattering from a system of coupled thin wires. These
measurements are of sufficient quality to verify the theoretical calculations and physical
pictures associated with the transient coupling phenomena. The SEM modes extracted
from measurements of the transient surface current on a system of coupled wires further
demonstrates the coupling phenomena. It is believed that the measurements of coupled
wires presented here are accurate and are possibly the first such measurements published.
Chapter 6 also presents comparisons of the measurements of a wire stick aircraft model
to theoretical calculations to further demonstrate the measurement capability.

It has been widely accepted in the literature and EM community that the early-
time portion of the response contains most of the energy in the scattered waveform.
Measurements performed at Michigan State University have for a number of years
suggested that this may not be the case in actual measurements of realistic targets excited
in the resonance region. Chapter 6 settles this issue by making use of the accuracy of the
transient measurements to calculate the amount of energy that is present for scale models

of realistic aircraft. For the targets considered, it is demonstrated that the early-time does

259

indeed contain the larger fraction of total backscattered signal energy, but that there are
instances when the late-time energy content is significant and useful for discrimination
via the E-pulse and S-pulse.

The measurement capabilities are also used to illustrate early-time scattering
measurements of practical 1/48 scale model aircraft. It is seen that the early-time
response is composed of responses from distinct scattering centers along the fuselage of
the aircraft. These scattering centers are generally associated with discontinuities and
sharp bends. It is seen from measurements that the early-time response is also very
aspect dependent. This aspect dependency places constraints on potential discrimination
and detection schemes using the early-time response.

In summary, this dissertation has addressed a number of topics pertinent to the
application of UWB radar to target discrimination and should serve to enhance the
understanding of the transient scattering phenomena. In particular, the improved
measurement schemes should continue to pay large dividends in future research at
Michigan State University.

Though a number of problems associated with resonance based target
discrimination have been addressed in this dissertation, many interesting and useful
problems associated with UWB radar remain. Specifically, an SEM formulation that
makes use of substructure resonances to describe the early-time response should be
investigated further. Effective implementation of the E—pulse and S-pulse schemes will
also require more exhaustive studies of aircraft responses and characterization of these
responses using SEM. It would also be useful to translate the EDR and SDR numbers

into a measure of confidence in the discrimination results. This could be extended to the

260

problem of confidence that the indicated target is contained in the library of E-pulses and
S-pulses.

Detection is also a very important aspect of UWB radar. It is anticipated that a
study of transient polarimetrics would be useful for reduction of sea and ground clutter
in relation to the target response. Of course, a study of this sort would require accurate
measurements of the polarimetric response of both targets and typical clutter mechanisms

such as sea surfaces and vegetation.

261

Appendix A

Appendix A

Frequency Domain Moment Method Formulation

A.1 Piecewise Sinusoidal Basis Functions

Many results in this dissertation were generated using a moment method
formulation based on the EFIE and piecewise sinusoidal basis functions. As such, it is
appropriate to provide a summary of this formulation.

The geometry of a thin wire segment with half length A and radius a is shown in

Figure A. 1. Assume the current distribution on the wire is

1(2) = Insinkm - Izl) (A.l)
where I. sinkA is the amplitude of the current at the center of the wire and k=wlc is the

wavenumber. Note that it is common practice to normalize the current by the constant
factor sinkA . To simplify notation, this factor is suppressed throughout this Appendix.

It is well known that the fields of the thin wire with a sinusoidal current distribution have

an analytic expression [57]. The (i) component of magnetic field is

B¢(P.Z) = “fit-33k 'ju' + e'fl‘k’ - (2coskA)e""’} (A.2)
where
p = x2+y2 r = p24,;2 (A-3)
and

262

\
\
l\

 

 

 

Figure A.l Geometry of thin-wire scatterer.

263

R1 = l/Pz+(Z’A)2 R2 = Jp2+<z+m2 (M)

The a component of electric field is obtained from

 

 

 

 

1 BB.
E = _ _ (A.5)
,(p.z) jams 62
which yields
I - “I“: '1“: db
E(p.z) =jn-12—A‘ + “4‘ ~2¢osm£‘_ (M)
" 41: R1 p R2 A r

e
where n = —° is the impedance of free space. The 2 component of electric field is
J “a

 

 

obtained from
6(pB
E,(p.z) = . 1 t) (M)
Jouep 69
which yields
-' I ‘1“: '1“: 4»
559.2) = 111 "It +e -2coskA£-—- (A-8)
41: 1 R1 R2 1'

The boundary condition on the tangential electric field at the surface of a perfect

conductor is
{XE-'0?) = .‘.E"(;.') (A3)

If a structure is modeled by a number of thin wire segments then the scattered field can

be written as

264

N
E’m = 2 £3?) (A.10)

u-l

where 33?) is the field scattered by the nm wire segment. This field can be calculated

from the previous expressions and can be extended to segments that are not parallel to
the z-axis by appropriate vector transformations. If all segments are parallel to the z-axis,

the EFIE can be converted to matrix form by applying the Galerkin testing function

f sink(A- |z-z,,l)£-t..}dz (A.ll)

agar

where z, is the center of the m‘“ segment. The resulting matrix equation has the form
A] = E (A.12)

where I is the column vector containing the amplitudes of the current basis functions, the

entries in the A matrix have the form

A” = f sin(A-|z—z_|)£-E:(F)dz (A.l3)

”(I

and the column vector E has entries of the form

E... = - f sinktA-Iz-Z.I)z‘°3‘(r‘)dz (A.14)

saga:

A.2 Finite Conductivity Effects
The finite conductivity of the wire is accounted for by using the boundary

condition

i-E‘m + £~i3"m +200) = o (A.15)

265

where the impedance of the wire 2‘ can be approximated by the skin effect resistance

from [5 8] as

-_1.
= ”(T/a) ; T=m 2W (ll-16)
2nan0(Ta)

zi

 

where J, is the Bessel function of order 0 and 10’ is the derivative of J, with respect to
the argument, a is the radius of the wire, a is the conductivity, and It is the permeability.
Using the previous piecewise sinusoidal weighting function, the matrix elements for the

case when the weighting function and basis functions overlap (m =n) are augmented with

A
r, = z‘ fsin2k(A-|z|)dz (A.l7)
-A
The integral can be rewritten as
A A 0
fsin’ktA—IzIMz = fsin’km -z)dz + fsin2k(A+z)dz (A.18)
-A 0 -A

Making the change of variables z=-z in the second integral results in

A A
[sinzktA -lz|)dz = 2 f sin2k(A -z)dz (A.19)
-A 0

After a further change of variables u=-—:- -z, this integral is in tabulated form [59]. The

resulting additive term is

t, = Z‘{A - lama} (A.20)
2k

266

For the cases when the weighting function only overlaps half of the basis function, the

matrix elements are augmented by

D

2, = sink(A-|zl)sink(A-lz-A|)dz

O

which can be rewritten as

C

A
t = fsink(A -z)sinkzdz
0

then using sinusinB =écos(a -p) -%cos(a +9) this becomes

A a
1 1
t =— A- -— askA
c ~2£cask( 21)dz 2oc dz

Evaluating the integrals gives the following result

:, = z" -AcoskA + ism/m
2 2k

267

(A.21)

(A.22)

(A.23)

(A.24)

Appendix B

Appendix B

SEM Formulations

B.l Piecewise Sinusoidal Basis Functions

For the computations of the SEM mode frequencies, currents and scattered fields,
the piecewise sinusoidal basis function as presented in Appendix A can be used. The
formulation must be generalized to complex frequency. This is easily accomplished by

substituting a: with -js in the expressions for the current and scattered fields.

B.2 Hallen Equation for Array of Parallel Wires
An electric field integral equation for the special case of an array of 1’ wires

parallel to the z-axis and centered at z=0 can be written as [60],

l
.l
p 2
2 f {—,IP(Z’)— '12! p(Z’)}gm(z’z ’,S)dZ’ = 7%‘5": (193) (3.1)
r! z
-1
2

where I, is the length of the p"I wire, I, is the axial current on the p“ wire, 7 is the

complex wavenumber, n is the impedance of free space, E;(z,s) is the 2 component of

the impressed field on the qm wire, and g" is the scalar Green’s function under the thin
wire approximations which is given by

e "R"

RN

(B.2)

 

8,,(z.z ’5) =

268

 

where under the thin wire approximations

R = fiz-z’ful; (B.3)

P9

 

where d" is the radius of the p‘” wire for p=q, and d" is the axial spacing between wires
for prq . For the purpose of computing the SEM mode frequencies and currents only the

homogenous form is required. The equation can be written as

s ”k...

I,(z’)g,,(z.z’,9)dz’ = 0 (3'4)

62 - P
(622 72);;

”hr

To avoid the numerical differentiation, the previous expression is rewritten in the Hallen

form as

I,(z’)g,,(f’.i”.s)dz’ = A,sinhyz + choshyz q=l,2,...,P 03-5)

"5 ”ho--

P
2
p-l

tel-$-

By specializing to the case of P monopoles above and perpendicular to a perfectly
conducting ground plane (z=0) with their bases at z=0, the homogenous equation

becomes

" (3.6)
)3

p-1

I,(2’)§,,,(z.z’.s)dz’ = choshvz q=l.2.....P

:3“. ”Ir."-

where

ENW'J) = 8,4(21’5) +g"(z,-z’,s) (3.7)

269

accounts for the image of the monopoles. Since I, and C(i’i ,s) are even functions of z,

the unknown coefficient A, must be zero. A moment method solution can be implemented

by expressing the current on each monopole in a pulse function expansion

"P”

u?
I,(z) = g; c II (2) 03-3)

where M, is the number of basis functions on the p“I monopole and the pulse functions

are defined as

HMO.) = { 1 if (ZW—A’) S 2 S (2"+A') (3.9)

0 otherwise

where 2,, is the center of the rn'll basis function on the p‘II wire, and A, is the half width

of the pulses on the p‘ll wire. The centers of the pulses are usually chosen as

I
z,“ = (2m-1)A,; A = -25 (B.10)

_1-
1»

2M,
Then substituting the current expansion in to the integral equation and applying the

weighting function

I
J
2
.11
fw,,(z){...}dz q=l,2,...,P n=l,2,...,M, (B )
0
results in the equations
11 3e .‘1
2 p M, 2 - 2 $.12)
quna) 2 2 c,_ [11,,(2 ’)g,,(i°,i",s) dz ’dz = B, f w,,(z) eoehyzdz
0 P" "1 0 0

Passing the testing function through the sums and integrations results in

270

I:

w (z)§,,(z.z ’J)dzdz ’ = a, f wq,(z)coshyzdz
0

,5

)3 c (3.13)
par

p-l

11,.(1’)

4"

6", ”Hr
cg ml:~

!
a...

If point matching is used then, w,,,(z) =6(z-z,,,) where the match points are chosen to be
at the center of the pulse basis functions. Then using the sifting property of the Dirac

delta function,

3

r _ =1 ,... (E14)
2 Cr»- II”(z’)g,q(zq,,z’,s)dz’-choshyzfl = 0 tqr=l§,...fl,

p-l

!
._-

OK, Nb"

The above choice of testing functions results in the same number of testing points
(equations) as basis functions per monopole. There must then be P additional equations
to solve the problem due to the presence of the 3,. These constraints are obtained by

using the boundary condition on the current at the ends of the monopoles
I,(z=l,) = 0 p=l,2,...,P (3.15)

Thus, for the case of pulse basis functions ch”, = 0. Hence, the number of unknowns
is reduced by P and the resulting square matrix has zero determinant for s=s,,. A

numerical procedure is used to locate these zeros as discussed in Chapter 3.

B.3 Approximate Green’s Function Solution with Sinusoidal Current

This section contains details of the formulation of the approximate Green’s
function method of Chapter 3 for the case of two identical parallel wires. The
formulation follows that used by Chuang and Nyquist [61]. Recall from Chapter 3 that
for the special case of two identical scatterers, one need only calculate two constants to

evaluate the shift in the natural frequency.

271

a
s = if: a = ._.13 (B.16)

where s: is nIll natural mode frequency of the isolated scatterer. From Chapter 3,
a12 = f 1?:0) - f caress ~I?,°(r')ds'ds (3,17,
31 32

where 1?: is the n“I natural mode current of the isolated scatterer. This can be specialized

for the case of two identical wires parallel to the z-axis as

I I

“12 = f1..°(z)qu(z.z’.s,,°)I,°(z’)dz’dz (B.18)
0 0

where I is the length of the wire, 6,, is given by

 

612(2’2/5130) = { 62
az

all

+ (732}3 12(11 59.?) (3°19)

and
’73::

R12

 

It12 = «(z-2’)2 +de (330)

 

8129-1990 = e

where (1,, is the axial distance between the wires. Then,

a!
azaz 6”

 

I I I I
012 = {{I:(Z)I:(zl) (2,1 ’fiufidz’dz + {£I:(Z)I:(Z I)(Y:)2gu(z’z [93361:le (B.21)

The first term is integrated by parts twice, once on z’ and once on 1, to remove the

spatial derivatives from 3,2(z,z’,s,,°). Note that I,°(z=0,l) = 0 so that

272

 

’ ' r r
”’31 51:09 62 (u “341 'dz = f [1: [(161.7 '(z)gu(z.z 'J.°)dz’dz 03-22)
0 0 0 0

6262’ng

where the prime on the currents denotes a derivative with respect to the argument.

Hence,

II

“12 = ff{1no’(ll)luo’(l) + 032132 01.0(Z)}8u(z,z',v:)dz’ dz

00

The second constant has the general form
5., = [13:0) ~f £6027 .9) I,.,;°I?:<r’)d8’ds
SI S]
which can be written as

I. I
3n = {{1.°(z)1.°(25%61,(zz CS) Is-S:dz [dz

 

 

 

where
a2
Gum“) = {az’az + Y’}8u(z.z’.s)
with
I can!" 2
8,,(z.z .3) = R R“ = \/(Z'Z')2 +0
II

where a is the wire radius. Then,

 

I I a:
5,, = f f1,°(z)1,°(z’) 62
00

& £811(z’z Ins) Lug: dz Idz +

I

I I
ffl.°(z)1:(z’)3(73281,(z.z’,s)|,.,~42’dz
o o as '

273

(13.23)

(3.24)

(3.25)

(3.26)

(3.27)

(3.28)

The first integral can be integrated by parts as before to obtain
1 r I ’ a
f [1: (z )1." (agency. as) I,.._-dz’dz
o o
The second integral can be expanded to obtain
r r

0 0 0 2 a
I {1. (2)1, (z w, wag-guru as) I...;dz’dz

.YAI o o o
and combining terms gives

 

Noting that £81131 ',S) I”: = -‘

 

- l t 0’ 0I o 2 o o e-Y'R I
“11 = {10‘ {In (1),). (7") " Yn(E‘Y:)I. (2)]. (2’)} dz

If the current on the wires is assumed to be sinusoidal,

I,°(z) = sinylt—z; OszsI; n61

then the derivatives on the current can be performed analytically to yield

00 I

 

I o 2 I
an- ff{(_;_t_)2c ash—1:2“ “"32 + Y.('RT _YDsinnrltz sinnrtz }

  

B.4 Coupling Coefficient and Normalization Constant Calculations

II
=f {smmsinm (v 2)’ + (3’7") cosmwsflz}gu(u .s.°)dz’dz

e ’73"

 

(B.29)

(13.30)

(3.31)

(13.32)

(3.33)

(3.34)

This section describes some details associated with calculation of the coupling

coefficient a, and normalization constant c,. The Pocklington form of the EFIE for wires

parallel to the z-axis can be written as

274

f1,(z)6(r.r’.s)dz’ = 753537;) (3.35)
P

where G-(Fi’s) is the Pocklington form of the Green’s function under the thin wire

approximations
Ear/,3) = {_a: _ 72}; (13.36)

and R is the distance between the source point and field point. Using the thin wire

approximations

 

R(z.z’) = «(z-2r + 42 “*3”

where and d is the wire radius when the source point and field point are on the same
wire and d is the axial spacing between the wires when the source point and field point

are on different wires. Then, the expression for c, of Chapter 3 is written as

c. = {1.12) i 9g? r_(z') dz’dz (B.38)

where the % operator has been passed through the surface integrations. Here, I, is the
current distribution of the n“ mode. Numerical differentiation is avoided by analytieally

evaluating £612; ’,s).

2 I = 1 ii?“ - a 2"" (1339)
636%: ’3) 4n{azzas R as(7 R

a e’" e'“ . .
Note tha Err-T = -—c-. Then, the first term rn braces rs

275

 

r = 522.51 "" ... 123.53.“
1 322 «19 R c azaz R
Note that in - .215 and ie-vn = fizz-w ms,
dz R 62 R
T = 1.2 ___-y(z-z’)e..,R
1 (:62 R
. . . a l _ z-z’ .
Expanding usrng the product rule and the relation EOE) - -—R;- grves

I
T1 = _l _a_(_Z_"Z’)e—1R+£_'_§__ie-yk
€67. R R dz

 

-1R _ 2
T1 = “1‘ {—+(z- -z’)d(- 112) I—E—EQ-v}
c R

 

T = _ e""l_ (z-z’)’ Y(___2.-z")2
1 Y C R R3 R2

The second term in braces from (B.39) is

-18 2
T2=iyz.e__ = fl—Le"!
as R CR 6

Then after further manipulation,

 

_ _ (z-z’)2 7(z-z’)’_
;G(z,z ’5) Y— :{R+ R3 R2 ‘1}

(13.40)

(13.41)

(3.42)

(13.43)

(13.44)

(3.45)

(15.46)

This result is useful for computation of the normalization constants. Due to the

formulation used here, the coupling coefficient of Chapter 3 must be modified by the

factor of 7/1) is given by

276

a, = -—1— 11f!,(z’)E,‘(r.s,)dz (3.47)
P

0,,11

277

 

Appendix C

Appendix C

Experimental Apparatus

This appendix summarizes the equipment used for the experimental portion of the

research. The equipment used in the frequency domain scattering measurements is listed

in Table C.0. There were two different configurations used in the frequency domain

scattering measurements. In both cases, the dual antenna configuration of ? was used.

The ”low-band” configuration used the PPL 5812 amplifier to amplify the signal from

port 1 of the network analyzer. The pertinent network analyzer parameters used in this

configuration are shown in Table CO. The ”high-band” configuration used the HPS349B

amplifier to amplify the signal from part 1 of the network analyzer. The parameters used

in this configuration are given in Table C.O. The equipment used in the time domain

scattering measurements are summarized in Table C.O.

Table C.l

(1)
(1)
(1)
(2)

(1)
(1)
(1)
(1)
(1)

Equipment used for frequency domain scattering measurements.

Hewlett-Packard HP8720B Vector Network Analyzer

Hewlett-Packard HP8349B Microwave Amplifier

Picosecond Pulse Labs PPL 5812 lOdB Broadband Amplifier

American Electronics laboratory AEL H-1734 Wideband TEM Horn
Antennas

23.5 foot RG-9B cable with N-Type Connector

22.5 foot RG-9B cable with N—Type Connector

B&K Precision D.C. Power Supply 1610 (for PPL Amplifier)

IBM PC/AT compatible microcomputer

HP-82335 HPIB interface and software library

278

Table C.2

Network analyzer parameters for low band measurements.

Measurement Parameter: S21
Frequency Sweep: 0.4 - 4.4 GHz
Number of points: 401
IF Bandwidth: 100 Hz
Number of Averages: 10
Sweep Time: 30 seconds
Table C.3 Network analyzer parameters for high band measurements.

Measurement Parameter: S21
Frequency Sweep: 1.0 - 9.0 GHz
Number of points: 801
IF Bandwidth: 100 Hz
Number of Averages: 10
Sweep Time: 60 seconds
Table C.4 Equipment used for time domain scattering measurements.

( 1) Tektronix 7854 Digital Sampling Oscilloscope
( 1) Picosecond Pulse Labs PPL lOOOB Pulse Generator

(1) Tektronix Type 113 Delay Cable

(1)
(1)
(2)

(1)
(1)

23.5 foot RG-9B cable with N—type connectors

22.5 foot RG-9B cable with N-type connectors

American Electronics laboratory AEL H—1734 Wideband TEM Horn
Antennas

IBM PC/AT compatible microcomputer

HP-82335 HPIB interface and software library

279

An anechoic chamber was used extensively throughout the research and is
essential for performing frequency domain scattering measurements. The Michigan State
University anechoic chamber is constructed of wood with 6 inch pyramidal foam
absorbers covering the inside walls. The dimensions of the chamber and approximate
locations of targets and antennas for the configuration used throughout most of this
research is shown in Figure G]. A new configuration was implemented near the
completion of this research in an attempt to reduce the direct coupling of antennas,
scattering from the antenna supports, and cable losses. This new configuration is shown
in Figure C.2.

To accurately characterize the measurement system, the network analyzer was
used to measure the gain of the amplifiers and attenuation of the cables over a wide range
of frequencies. The results of these measurements are provided here to indicate the
limitations of the measurement system, areas for possible improvement, and baseline
measurements to assess future equipment changes. The small signal gain of the HP8349B
amplifier is shown in Figure C.4. The gain of the PPL 5812 amplifier gain is shown in
Figure C.5. Cable losses for the system prior to February 1992 are shown in Figure Q3.
The new chamber configuration makes use of the old receive cable after being cut into
two sections. Thus, the total cable losses in the new system are effectively those of the

old receive cable.

280

Anechoic Chamber Configuration

_e p—M T

K g Transmitter ’
target center 1 2 Top View
K... G)

 

 

 

l

5.5

 

 

 

 

 

target center

gnarnmmer O®T 1 2 Side View
5.5

d
u

Shaker

——>l

I.»
U.

,é_.
“(_—

 

 

 

24

 

 

All Dimensions in Feel

Figure C.1 Michigan State University anechoic chamber configuration prior to
February 1992.

281

Anechoic Chamber Configuration

 

 

 

 

 

 

ill _ "cumin" iorgoi censor T

ZTOERocoivor ® 12 TOP View
5.0

J

\ & doors 2

w — Transmitter target cantor T

ifgnmfl ® 12 Side View
5.0

i

 

 

 

 

 

i< 24 9|

All Dimensions in Foot

Figure C.2 Michigan State University anechoic chamber after February 1992.

282

Mognfiude 821 (dB)

Coble Attenuation

    

-10.00

-20.00

-30.00

-40.00

ILLIIl[JIIIJLJIIIIIIJII111111111]lllllu

 

 

-50.00 ‘
“‘
'*.
-60.00 . ‘
—— Tronemut Coble
------- Receive Cable
-70.00
‘80-00 TIIIIIIIIIIIII1TrIIjrlllllIIIITIIIIIFrI
000 2.50 5.00 7.50 10.00 12.50 15.00 17.50 20.00

Frequency (CH2)

Figure C.3 Measured cable attenuation.

283

40.00

30.00

20.00

10.00

 

0.00

 

Com (dB)
ilflllninllnitiliilanlnantlliini11111114¢111111J

 

 

-10000 Iilrlllll IllIUIIIIIIIIIITTIUITTjTIIIITj

I
0 5.00 10.00 15.00 20.00
Frequency (GHz)

9
0

Figure C.4 Gain of HP8349B Microwave Amplifier measured using PIP-8720B
network analyzer. Amplifier input power = -30 dBm.

284

15.00

 

10.00 1

1

5.00 3

A .-
m .-
‘O u
v “
.E q
o :1
o 0.00 -
I

I

--5.00 J

A

q

A

—10.00 :memm
0.00 1.00 2.00 3.00 4.00 5.00 6.00

Frequency (GHz)

Figure C.5 Gain of PPL 5812 wide band amplifier measured using HP-87ZOB network
analyzer. Amplifier input power = -10 dBm.

285

Bibliography

10.

11.

12.

Bibliography

J.D. Taylor, ”Introduction to Ultrawideband Radar Technology,“ Draft of notes
presented at the First Ultrawideband Radar Symposium, Los Alamos National
Laboratory, March 1990.

C.W. Davis, 111, N.M. Tomljanovich, J.D.R. Kramer, Jr. and LL. Poirer,
”Characteristics of Ultra Wide Band Radars Pertinent to Air Defense,” Los
Alamos Symposium on Ultra-Wideband Radar, 1990.

I. LaHaie, R. Harger, S. Robinson, and J. Miller, "An Evaluation of
Nonsinusoidal Radar Techniques," Report to U.S. Government from
Environmental Research Institute of Michigan, June, 1985.

M.L. Skolnik, WW5, New York: McGraw-Hill, 1980.

1992 IEEE International Microwave Symposium Program Guide, Albuquerque,
NM, June 1992.

1992 Joint Symposia, IEEE APS International Symposia, URSI Radio Science
Meeting, Nuclear EMP Meeting - Call for Papers.

Alfonso Farina and Flavio A. Studer, "Detection with High Resolution Radar:
Great Promise, Big Challenge," Microwave Journal, pp. 263-273, May 1991.

L. Webb, "Radar Target Discrimination Using K—pulses from a ’Fast’ Prony’s
Method, " Doctoral dissertation, Michigan State University, May 1984.

13.1. Rothwell, ”Radar Target Discrimination Using the Extinction-Pulse
Technique,” Doctoral dissertation, Michigan State University, 1985.

W. Sun, "Scattering of Electromagnetic Waves and Radar Target
Discrimination,” Doctoral dissertation, Michigan State University, 1989.

P. Ilavarasan, E. Rothwell, K.M. Chen, and DP. Nyquist, ”Natural Resonance
Extraction with Physical Constraints,” to be presented at the 1992 National Radio
Science Meeting in Chicago, IL, June 1992.

K.M. Chen, D.P. Nyquist, E.J. Rothwell, L.L. Webb and B. Drachman, ”Radar
Target Discrimination by Convolution of Radar Returns with Extinction-Pulses
and Single-Mode Extraction Signals,” IEEE Transactions on Amennas and
Propagation, vol. AP-34, no. 7, pp. 896-904, July 1986.

286

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

E]. Rothwell, D.P. Nyquist, K.M. Chen and B. Drachman, “Radar Target
Discrimination Using the Extinction-Pulse Technique," IEEE Transactions on
Antennas and Propagation, vol. AP-33, no. 9, pp. 929-937, September 1985.

13.]. Rothwell, K.M. Chen, D.P. Nyquist and W. Sun, ”Frequency Domain E-
Pulse Synthesis and Target Discrimination," IEEE Transactions on Antennas and
Propagation, vol. AP-35, no. 4, pp. 426-434, April 1987.

GE. Baum, ”On The Singularity Expansion Method For The Solution of
Electromagnetic Interaction Problems," Interaction Notes, Note 88, December
1971.

L. Marin, "Representation of Transient Scattered Fields in Terms of Free
Oscillations of Bodies," Proceedings of the IEEE, pp. 640-641, May 1972.

P. Ilavarasan, E. Rothwell, J. Ross, D. Nyquist and K.M. Chen, “Time Domain
Radar Target Discrimination Using S-Pulse Waveforms,” presented at the 1991
North American Radio Science Meeting, London, Ontario, Canada, June 22-25,
1991.

P. Ilavarasan, LE. Ross, EJ. Rothwell, D.P. Nyquist and K.M. Chen,
”Performance of an Automated Radar Discrimination Scheme using E-Pulses and
S-Pulses,‘ submitted for review, IEEE Transactions on Antennas and
Propagation.

P. Ilavarasan, ”Automated Radar Target Discrimination Using E-Pulse and S-
Pulse", Doctoral Dissertation, Michigan State University, 1992.

E]. Rothwell, K.M. Chen, and D.P. Nyquist, "Extraction of the Natural
Frequencies of A Radar Target from A Measured Response Using E-Pulse

Techniques, " IEEE Transactions on Antennas and Propagation, vol. AP-35, no.
6, pp. 715-720, June 1987.

5.1. Rothwell and K.M. Chen, ”A Hybrid E—Pulse/Least Squares Technique for
Natural Resonance Extraction," Proceedings of the IEEE, pp. 296—298, vol. 76,
no. 3, March 1988.

W. Sun, ”Scattering of Transient Electromagnetic Waves and Radar Target
Discrimination,” Ph.D. Dissertation, Michigan State University, 1989.

RM. Tesche, ”On the Analysis of Scattering and Antenna Problems Using the
Singularity Expansion Technique,” IEEE Transactions on Antennas and
Propagation, v01. AP-21, no. 1, pp. 53-62, January 1973.

RM. Tesche, ”The Far-Field Response of a Step-Excited Linear Antenna Using

SEM, " IEEE Transactions on Antennas and Propagation, v01. AP-23, no. 6, pp.
834-838, November 1975.

287

25 .

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

K.R. Umashankar, T.H. Shumpert and DR. Wilton, ”Scattering by a Thin Wire
Parallel to a Ground Plane Using the Singularity Expansion Method,“ IEEE
Transactions on Antennas and Propagation, vol. AP-23, no. 2, pp. 178-184,
March 1975.

T.H. Shumpert and DJ. Galloway, ”Finite Length Cylindrical Scatterer Near
Perfectly Conducting Ground. A Transmission Line Mode Approximation,” IEEE
Transaction on Antennas and Propagation, vol. AP-26, no. 1, pp. 145-151,
January 1978.

L.S. Riggs and T.H. Shumpert, ”Trajectories of the Singularities of a Thin Wire
Scatterer Parallel to Lossy Ground,” IEEE Transactions on Antennas and
Propagation, vol. AP-27, no. 6, pp. 864-868, November 1979.

Che-I Chuang, ”Transient Waveform Synthesis For Radar Target
Discrimination," Doctoral Dissertation, Michigan State University, 1983.

C.E. Baum, T.H. Shumpert and LS. Riggs, ”Perturbation of the SEM-Pole
Parameters of an object by a mirror object," Electromagnetics, 8: 169-186, 1989.

J. Nitch, C.E. Baum and R. Sturm, "Splitting of Degenerate Natural Frequencies
in Coupled Two-Conductor Lines by Distance Variation," Interaction Note 477,
July 1989.

DJ. Riley, W.A. Davis and I.M. Besieris, "The singularity expansion method
and multiple scattering,“ Radio Science, vol. 20, no. 1, pp. 20—24, January-
February 1985.

J. Ross, E. Rothwell, D.P. Nyquist and K.M. Chen, ”Multiple Target
Discrimination using E-pulse Techniques,“ 1990 IEEE/APS Symposium Digest,
Dallas, TX, June 1990.

M.A. Morgan, ”Singularity Expansion Representations of Fields and Currents in
Transient Scattering, " IEEE Transactions on Antennas and Propagation, vol. AP-
32, n0. 5, pp. 466-473, May 1984.

M.A. Morgan, ”Response to Comments Regarding SEM Representations," IEEE
Transactions on Antennas and Propagation, vol. AP—33, no. 1, p. 120, January
1985.

L.W. Pearson, ”A Note on the Representation of Scattered Fields as a Singularity

Expansion, " IEEE Transactions on Antennas and Propagation, vol. AP-32, no.
5, pp. 520-524, May 1984.

L.B. Felson, ”Comments on Early Time SEM,” IEEE Transactions on Antennas
and Propagation, vol. AP-33, no. 1, pp. 118-119, January 1985.

288

37.

38.

39.

40.

41.

42.

43.

45.

46.

47.

48.

D.G. Dudley, "Comments on SEM and the Parametric Inverse Problem,“ IEEE
Transactions on Antennas and Propagation, vol. AP-33, no. 1, pp. 119-120,
January 1985.

C.E. Baum, ”Representation of Surface Current Density and Far Scattering in
EEM and SEM With Entire Functions,” Interaction Notes, Note 486, February
1992.

C.E. Baum, ”On the Singularity Expansion Method for the Solution of
Electromagnetic Interaction Problems," Interaction Notes, Note 88, 1971.

AD. Yaghjian, "Electric Dyadic Green’s Functions in the Source Region,”
Proceedings of the IEEE, vol. 68, no. 2, pp. 248-262, February 1980.

B.K. Singaraju, D.V. Giri and C.E. Baum, ”Further Developments in the
Application of Contour Integration to the Evaluation of the Zeroes of Analytic
Functions and Relevant Computer Programs,’I Mathematics Notes, Note 42,
March 1976.

G.W. Hanson, ”The Singularity Expansion Method for Integrated Electronic
Circuits," Ph.D. Dissertation, Michigan State University, 1991.

C.L. Bennett, "The Numerical Solution of Transient Electromagnetic Scattering

Problems,” in W, ed. by P.L.E. Uslenghi, Academic
Press, 1978, pp. 393-428.

Morgan M.A. , ”Time-Domain Scattering Measurements,” IEEE Antennas and
Propagation Newsletter, vol. 6, pp. 5-9, August 1984.

M.A. Morgan and B.W. McDaniel, ”Transient Electromagnetic Scattering: Data
Acquisition & Signal Processing,” IEEE Transactions on Instrumentation and
Measurement, vol. 37, no. 2, pp. 263-267, June 1988.

M.A. Morgan and N .J . Walsh, 'Ultra-Wide-Band Transient Electromagnetic

Scattering Laboratory, " IEEE Transactions on Antennas and Propagation, vol.
AP-39, no. 8, pp. 1230-1234, August 1991.

FT. Ulaby, T.F. Haddock, J. East, and M. Whitt, ”A millimeter-wave newtork
analyzer based scatterometer, " IEEE Transactions on Geoscience and Remote
Sensing, vol. GE-26, no. 1, pp. 75-81 January 1988.

B. Drachman, "Two Methods to Deconvolve: L,-Method Using Simplex
Algorithm and Lz-Method Using Least Squares and a Parameter,” IEEE
Transactions on Antennas and Propagation, vol. AP-32, no. 3, pp. 219-225,
March 1984.

289

49.

50.

51.
52.

53.

54.

55.

56.

57.
58.

59.

61.

E.J. Rothwell and W.E. Sun ”Time Domain Deconvolution of Transient Radar
Data,” IEEE Transactions on Antennas and Propagation, vol. AP-38, no. 4, pp.
470-475, April 1989.

Rfl 'i rm -T hni li ' ,ed.NicolasC.
Currie, Artech House, 1989.

Mie, Ann. Physik, 25, 377, 1908.

J .A. Stratton, Elgtremagnetie Theeg, New York and London: McGraw-Hill,
1941, pp. 563-573.

R.F. Harrington, Time Harmenie Electromagnetic Fields, New York: McGraw
Hill, 1961.

K.M. Chen, EE83QElmtremagnetie Waves 1; Qeurse Netes, Michigan State
University, 1989.

D. Deirmedjian. WW USAF
Project Rand Report R-456-PR, April 1969.

N. Gharsallah, ”Measurements of Transient Electromagnetic Fields with Surface
Probes,” Master’s Thesis, Michigan State University, 1987.

RS. Elliot, Antenna jtjhmg and Design, New York: Prentice Hall, 1981.

5- Ram- J-R- Whinnery and T- Van Duw. Eiddmdflaxcsiacmmnnicafim
Monies, 2nd ed.; New York: Wiley, 1984.

CRC SIANDARD MATHEMATICAL IABLES, ed. by W.H. Beyer, 25m ed.;
Boca Ratan, Florida: CRC Press, 1980.

D.P. Nyquist, EE226§4227 - Antenna Thmty Qegrse Netes, Michigan State
University, 1989.

CL Chuang and D.P. Nyquist, ”Perturbational formulation for nearly degenerate
coupling," 1984 National Radio Science Meeting, Boulder, CO.

290