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This is to certify that the

dissertation entitled

Scattering of Transient Electromagnetic Waves
and Radar Target Discrimination

presented by

Weimin Sun

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Electrical

 

 

Engineering

’ MM

Major professor

Date 7/2’l/J/l7

 

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MSU to An Affirmative Action/Equal Opportunity Institution 7
eMMS-m

 

SCATTERING OF TRANSIENT ELECTROMAGNETIC WAVES

AND RADAR TARGET DISCRIMINATION

By

Weimin Sun

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1989

(0043b'i5X

ABSTRACT

TRANSIENT SCATTERING OF ELECTROMAGNETIC WAVES
AND RADAR TARGET DISCRIMINATION

By

Weimin Sun

A new scheme for radar target discrimination and identification, known as the
Extinction-Pulse (E-pulse) technique, has been developed recently at Michigan State
University. Some important characteristics and practical applications of this E-pulse
technique are investigated in this thesis. The important characteristics investigated are
the aspect-independency and the noise-insensitivity of the technique. The appilcation
of the technique to discriminate radar targets coated with lossy materials and the

implementation of the technique using various antenna systems are also studied.

The aspect-independency of the technique is attributed to the fact that the syn-
thesis of the E-pulse waveform is based entirely on the natural frequencies of the tar-
get. The characteristic of noise-insensitivity is benefited from the convolutions of the
E-pulse waveform with the scattering responses of the targets; an averaging and
smoothing process. These two important characteristics have been experimentally

investigated and are reported in this thesis.

Recently it was found that the E-pulse technique can be applied to discriminate

radar targets coated with a layer of lossy material. A theoretical study has been

conducted to investigate the resonance modes of an infinitely long conducting cylinder
coated with a layer of lossy material. The trajectory of the poles of resonance modes
varying as a function of the parameters of the lossy coating and other geometrical fac-
tors is investigated. An experimental study was also conducted to discriminate various

rectangular conducting plates coated with lossy materials.

Since there is no existing method which can be used to predict accurately the
natural frequencies of a thin rectangular plate, a new method based on a new set of
coupled integral equations for the induced surface current, was developed. This set of
integral equations is more rigorous and numerically better behaved when compared
with some existing equations. These new integral equations were solved numerically
and theoretically predicted natural frequencies of the rectangular plates of various
dimensions were then compared with those experimental results extracted from the

scattered fields of the plates. Agreement between theory and experiment was found to

be very good.

ACKNOWLEDGMENTS

The author wishes to express his heartfelt appreciation to his academic advisor,
Dr. Kun-Mu Chen, for his sincere guidance and encouragement throughout the course
of this work. He also wishes to appreciate Dr. D. P. Nyquist and Dr. E. J. Rothwell

for their generous support and constructive suggestions during the period of this study.

A special note of thanks is due Dr. Byron C. Drachman for his special assistance

and concern to the author.

In addition, the author thanks his wife, Xiaoyi Min, his parents Mr. and Mrs.
Jiantang Sun, and his parents-in-law, Mr. and Mrs. Longqiu Min for their continued

love, sacrifice and encouragement in his research years.

The research reported in this thesis was supported by Defense Advanced Research
Projects Agency and monitored by Office of Naval Research under Contract No.

N00014-87-K-0336.

iv

Table of Contents

LIST OF TABLES ...................................................................................................

LIST OF FIGURES .................................................................................................

CHAPTER 1 INTRODUCTION ...........................................................................

CHAPTER 2 THEORY FOR TRANSIENT SCATTERING OF EM

WAVES ............................................................................................

2.1 Introduction .............................................................................................
2.2 Complex Domain Surface E-field Integral Equation ............................
2.3 The Singularity Expansion Method .......................................................
2.3.1 Expansion of Current in Complex Frequency Domain ............

2.3.2 Derivation of Coupling Coefficients .........................................

2.3.3 Current Expansion in Time Domain .........................................

2.3.4 Later Developments ...................................................................

2.4 Determination of Natural Modes of A Scatterer ...................................
2.4.1 General Projection Method ........................................................

2.4.2 Eigenvalue Method ....................................................................
2.4.3 Variational Principle Method ....................................................

2.5 Extraction of Natural Frequencies from A Measured
Response .................................................................................................

2.5.1 Prony’s Method ..........................................................................
2.5.2 Least Square Method .................................................................
2.5.3 Continuation Method .................................................................
2.5.4 E—pulse Method ..........................................................................

CHAPTER 3 ASPECT-INDEPENDENT AND NOISE-INSENSITIVE

TARGET DISCRIMINATION USING E-PULSE
TECHNIQUE ..................................................................................

3.1 Introduction .............................................................................................
3.2 The E-pulse Technique ...........................................................................
3.2.1 The Time Domain Synthesis .....................................................

viii

iv

6

6

8
13
14
15
18
18
2O
20
21
22

24
25
26
28
30

34
34
36
37

3.2.2 The Frequency Domain Synthesis ............................................. 41

3.2.3 Calculation of E-pulse Amplitudes ........................................... 44
3.3 Aspect-independence of the E-pulse Technique ................................... 46
3.3.1 Aspect-independence of E-pulse Synthesis ............................... 46
3.3.2 Experimental Demonstration ..................................................... 48
3.3.3 Examples of Two Different Models .......................................... 50
3.3.4 Examples of Multiple Models ................................................... 53
3.4 Noise Characteristics of the E-pulse Technique ................................... 92
3.4.1 Error Estimation on the Natural Mode Extraction ................... 92
3.4.2 The E-pulse Convolution with Natural Modes Perturbed
................................................................................................................ 96
3.4.3 Noise Performance of the E-pulse Convolution ....................... 109
3.4.4 Noise Testing on Experimental Data ........................................ 112
3.4.5 Conclusion .................................................................................. 119
CHAPTER 4 THE NATURAL OSCILLATIONS OF AN INFINITELY
LONG CYLINDER COATED WITH LOSSY
MATERIAL ..................................................................................... 122
4.1 Introduction ............................................................................................. 122
4.2 Derivation of Characteristic Equation ................................................... 124
4.2.1 Perfectly Conducting Cylinder in Free Space .......................... 130
4.2.2 Perfectly Conducting Cylinder in Lossy Medium ..................... 130
4.2.3 Lossy Cylinder in Free Space ................................................... 131
4.2.4 Perfectly Conducting Cylinder Coated with Lossy
Material ................................................................................................. 133
4.3 Numerical Algorithm .............................................................................. 134
4.4 Numerical Results of the Pole Distributions ......................................... 138
4.5 Effects of Lossy Parameters on An Exterior Mode .............................. 152
4.6 Discrimination of Conducting Plates and Conducting Plates
Coated with A Lossy Layer Using E-pulse Technique ........................ 155
4.7 Conclusion .............................................................................................. 170
CHAPTER 5 DETERMINATION OF THE NATURAL MODES OF A
RECTANGULAR PLATE ............................................................. 171
5.1 Introduction ............................................................................................. 171

vi

5.2 New Derivation of An Existing Formulation ........................................ 172

5.3 New Coupled Electric Field Integral Equations .................................... 180

5.4 Method of Moments Solution to the New Formulation ....................... 192

5.5 Numerical Convergence of the New Formulation ................................ 199

5.5.1 Pole Convergence in the Thin Strip Limit ................................ 199

5.5.2 Pole Convergence with More Basis Functions ......................... 202

5.6 Numerical Results ................................................................................... 206

5.6.1 Pole Location and Trajectory .................................................... 207

5.6.2 Modal Current Distributions ...................................................... 213

5.7 Experimental Verification ....................................................................... 230

5.8 Conclusion .............................................................................................. 236
CHAPTER 6 CONCLUSIONS ............................................................................. 237
BIBLIOGRAPHY ..................................................................................................... 241

vii

3.1

3.2

3.3

4.1

5.1
5.2

5.3

5.4

5.5

List of Tables

The first five natural frequencies of a 30 cm thin wire extracted
from its impulse response via a continuation method. The impulse
response is constructed with the first five modes and is

contaminated with a Gaussian white noise ...............................................

The first four odd natural frequencies of a 12" thin wire extracted

from its time domain measured response via a continuation method.

Only the odd modes are excited and the measured response is

contaminated with extra uniform white noise ...........................................

The comparison of the signal to noise ratios before and after the
impulse responses of a 30 cm thin wire are convolved with its E-
pulses. The first five natural frequencies are used to construct
the impulse responses and to synthesize the E-pulses of the thin—
wire. The c is the standard deviation of the added noise, and

the 0 is the incident angle of the excitation w.r.t. the

thinwire .......................................................................................................
Natural Frequencies of Rectangular Plates ................................................

Current Symmetry Features .......................................................................

Natural frequencies of a rectangular plate with various

aspect ratios ................................................................................................

Natural frequencies of a rectangular plate with various

aspect ratios ................................................................................................

Natural frequencies of a rectangular plate with various

aspect ratios ................................................................................................

Natural frequencies of a rectangular plate with various

aspect ratios ................................................................................................

viii

99

100

114

157

189

214

215

216

217

2.1

3.1
3.2

3.3

3.4

3.4

3.4

3.4

3.5

3.5

3.5

3.5

3.6

3.6

List of Figures

Description of transient scattering of EM waves in free space ......................

Decomposition of an E-pulse waveform ..........................................................

Geometry of two airplane models. a) Boeing 707 Model termed B-
707; b) McDonnell Douglas F-18 model termed F-18 ..................................

The E-pulse waveforms synthesized for B-707 model and F—18
model .................................................................................................................

The scattered waveform of B-707 model at aspect angle of 45° and
its convolved response with the E-pulse waveform of B-707 model
a) The scattered waveform; b) The convolved response ................................

The scattered waveform of B-707 model at aspect angle of 90° and
its convolved response with the E-pulse waveform of 8-707 model
c) The scattered waveform; d) The convolved response ................................

The scattered waveform of B-707 model at aspect angle of 135° and
its convolved response with the E-pulse waveform of B-707 model
e) The scattered waveform; f) The convolved response .................................

The scattered waveform of B-707 model at aspect angle of 180° and
its convolved response with the E-pulse waveform of B-707 model
g) The scattered waveform; h) The convolved response ................................

The scattered waveform of F-18 model at aspect angle of 45° and
its convolved response with the E-pulse waveform of B-707 model
a) The scattered waveform; b) The convolved response ................................

The scattered waveform of F-18 model at aspect angle of 90° and
its convolved response with the E-pulse waveform of B-707 model
c) The scattered waveform; d) The convolved response ................................

The scattered waveform of F-18 model at aspect angle of 135° and
its convolved response with the E-pulse waveform of B-707 model
e) The scattered waveform; f) The convolved response .................................

The scattered waveform of F-18 model at aspect angle of 1800 and
its convolved response with the E-pulse waveform of B-707 model
g) The scattered waveform; h) The convolved response ................................

The scattered waveform of F-18 model at aspect angle of 45° and
its convolved response with the E-pulse waveform of F-18 model
a) The scattered waveform; b) The convolved response ................................

The scattered waveform of F—18 model at aspect angle of 90° and

ix

4O

51

52

54

55

56

57

58

59

6O

61

62

its convolved response with the E-pulse waveform of F-18 model
c) The scattered waveform; d) The convolved response ................................

3.6 The scattered waveform of F-18 model at aspect angle of 135° and
its convolved response with the E-pulse waveform of F-18 model
e) The scattered waveform; f) The convolved response .................................

3.6 The scattered waveform of F-18 model at aspect angle of 180° and
its convolved response with the E-pulse waveform of F-18 Model
g) The scattered waveform; h) The convolved response ................................

3.7 The scattered waveform of B-707 model at aspect angle of 45° and
its convolved response with the E-pulse waveform of F-18 model
a) The scattered waveform; b) The convolved response ................................

3.7 The scattered waveform of B-707 model at aspect angle of 90° and
its convolved response with the E-pulse waveform of F-18 model
c) The scattered waveform; d) The convolved response ................................

3.7 The scattered waveform of B-707 model at aspect angle of 135° and
its convolved response with the E-pulse waveform of F-18 model
e) The scattered waveform; f) The convolved response .................................

3.7 The scattered waveform of B-707 model at aspect angle of 180° and
its convolved response with the E—pulse waveform of F-18 model
g) The scattered waveform; h) The convolved response ................................

3.8 Geometry of three airplane models. a) Simplified model T-15; b) Big
B-707 model termed BB-707; c) Big F-18 model termed BF-18 .................

3.9 The scattered waveform of B-707 model at aspect angle of 0° and
its convolved response with the E-pulse waveform of B-707 model
a) The scattered waveform; b) The convolved response ................................

3.9 The scattered waveform of B-707 model at aspect angle of 45° and
its convolved response with the E-pulse waveform of B-707 model
c) The scattered waveform; d) The convolved response ................................

3.9 The scattered waveform of B-707 model at aspect angle of 90° and
its convolved response with the E-pulse waveform of B-707 model
e) The scattered waveform; f) The convolved response .................................

3.9 The scattered waveform of B-707 model at aspect angle of 180° and
its convolved response with the E-pulse waveform of B-707 model
g) The scattered waveform; h) The convolved response ................................

3.10 The scattered waveform of F-18 model at aspect angle of 0° and
its convolved response with the E-pulse waveform of B-707 model
a) The scattered waveform; b) The convolved response ................................

3.10 The scattered waveform of F-18 model at aspect angle of 45° and

63

64

65

66

67

68

69

71

72

73

74

75

76

3.10

3.10

3.11

3.11

3.11

3.11

3.12

3.12

3.12

3.12

3.13

3.13

its convolved response with the E-pulse waveform of B-707 model
c) The scattered waveform; d) The convolved response ................................

The scattered waveform of F-18 model at aspect angle of 90° and
its convolved response with the E-pulse waveform of B-707 model
e) The scattered waveform; t) The convolved response .................................

The scattered waveform of F-18 model at aspect angle of 1800 and
its convolved response with the E-pulse waveform of B-707 model
g) The scattered waveform; h) The convolved response ................................

The scattered waveform of T-15 model at aspect angle of 0° and
its convolved response with the E-pulse waveform of B-707 model
a) The scattered waveform; b) The convolved response ................................

The scattered waveform of T-15 model at aspect angle of 45° and
its convolved response with the E-pulse waveform of B-707 model
c) The scattered waveform; d) The convolved response ................................

The scattered waveform of T-15 model at aspect angle of 90° and
its convolved response with the E-pulse waveform of B-707 model
e) The scattered waveform; f) The convolved response .................................

The scattered waveform of T-15 model at aspect angle of 1800 and
its convolved response with the E-pulse waveform of B-707 model
g) The scattered waveform; h) The convolved response ................................

The scattered waveform of BB-707 model at aspect angle of 0° and
its convolved response with the E-pulse waveform of B-707 model
a) The scattered waveform; b) The convolved response ................................

The scattered waveform of BB-707 model at aspect angle of 45° and
its convolved response with the E-pulse waveform of B-707 model
c) The scattered waveform; d) The convolved response ................................

The scattered waveform of BB-707 model at aspect angle of 90° and
its convolved response with the E-pulse waveform of B-707 model
e) The scattered waveform; f) The convolved response .................................

The scattered waveform of BB-707 model at aspect angle of 1800 and
its convolved response with the E-pulse waveform of B-707 model
g) The scattered waveform; h) The convolved response ................................

The scattered waveform of BF-18 model at aspect angle of 0° and
its convolved response with the E-pulse waveform of B-707 model
a) The scattered waveform; b) The convolved response ................................

The scattered waveform of BF-18 model at aspect angle of 45° and
its convolved response with the E-pulse waveform of B-707 model
c) The scattered waveform; d) The convolved response ................................

xi

77

78

79

80

81

82

83

84

85

86

87

88

89

3.13

3.13

3.14

3.15

3.16

3.17

3.18

3.19

3.20

3.21

3.22

3.23

3.24

3.25

The scattered waveform of BF-18 model at aspect angle of 90° and
its convolved response with the E-pulse waveform of B-707 model
e) The scattered waveform; f) The convolved response .................................

The scattered waveform of BF—18 model at aspect angle of 180° and
its convolved response with the E-pulse waveform of B-707 model
g) The scattered waveform; h) The convolved response ................................

The norm of the matrix S (loglo) when three modes are extracted from
an impulse response of a 30 cm thin wire ( 0 = 60° ) ....................................

The norm of the matrix S (loglo) when five modes are extracted from
an impulse response of a 30 cm thin wire ( 0 = 60° ) ....................................

The impulse response of a 30 cm thin wire with a 60° angle w.r.t. to
the incident direction of the exciting wave .....................................................

The impulse response of a 30 cm thin wire ( 0 = 60° ) is convolved
with the E-pulses which are synthesized based on the noise-perturbed
natural frequencies of the thin wire .................................................................

The impulse response of a 30 cm thin wire with a 30° angle w.r.t. to
the incident direction of the exciting wave .....................................................

The impulse response of a 30 cm thin wire ( 0 = 30° ) is convolved
with the E-pulses which are synthesized based on the noise-perturbed
natural frequencies of the thin wire .................................................................

The E-pulse waveforms synthesized for a 30 cm thin wire based on its
first five natural frequencies. The five natural frequencies are
perturbed with Gaussian white noise ...............................................................

Comparison of the signal to noise ratios before and after the impulse
responses of a 30 cm thin wire are convolved with its E-pulse. The first
five natural frequencies are used to construct the impulse responses

and to synthesize the E-pulses of the thin wire ..............................................

(a) The measured scattered waveform of B-707 model at aspect angle
of 90°; (b) its convolved response with the E-pulse
waveform of B-707 model ...............................................................................

(a) The noise contaminated (20% of maximum amplitude) scattered
waveform of B-707 model at aspect angle of 90°;

(b) its convolved response with the E-pulse waveform of

B-707 model ......................................................................................................

(a) The measured scattered waveform of F-18 model at aspect angle of
90°; (b) its convolved response with the E-pulse
waveform of B-707 model ...............................................................................

(a) The noise contaminated (20% of maximum amplitude) scattered

xii

90

91

97

98

104

105

106

107

108

113

116

117

118

4.1
4.2

4.3

4.4

4.5
4.6

4.7

4.8

4.9

4.10

4.11

4.12

4.13

4.14

4.15

waveform of F-18 model at aspect angle of 90°;
(b) its convolved response with the E-pulse waveform of
B-707 model ......................................................................................................

Geometry of a lossy cylinder coated with a lossy layer .................................

The appropriate branch cuts for the complex wavenumber and Bessel
functions ............................................................................................................

The pole distribution of a perfectly conducting cylinder coated with an
air layer .............................................................................................................

Trajectory of the first two modes of a coated conducting cylinder versus
the conductivity. (a) the first mode; (b) the second mode ...........................

The pole distribution of a perfectly conducting cylinder ................................

The first layer of poles of a perfectly conducting cylinder immersed in
lossy medium ....................................................................................................

The pole distribution of a lossy dielectric cylinder ( o conductivity,
8, relative permittivity, 1] wave impedance and a the radius of
the cylinder ) .....................................................................................................

The pole distribution of a lossy dielectric cylinder ( o conductivity,
8, relative permittivity, n wave impedance and a the radius of
the cylinder ) .....................................................................................................

The radial amplitude dependence of t-component of E-field. (a-d)
the interior modes; (e-f) the exterior modes ....................................................

The pole distribution of a coated perfectly conducting cylinder (0
conductivity, 6, relative permittivity, n wave impedance, a the radius
of the cylinder and b the radius of the coating layer) ....................................

The pole distribution of a coated perfectly conducting cylinder (6
conductivity, 8, relative permittivity, 1] wave impedance, at the radius

of the cylinder and b the radius of the coating layer) ....................................
The pole distribution of a coated perfectly conducting cylinder (6
conductivity, 8, relative permittivity, 1] wave impedance, a the radius

of the cylinder and b the radius of the coating layer) ....................................
The radial amplitude dependence of o-component of E-field inside

the cladding region. (a-d) the interior modes; (e-t) the

exterior modes ...................................................................................................
Trajectory of the second exterior mode of a coated perfectly conducting
cylinder as the coating thickness is varying ....................................................

Trajectory of the first exterior mode of a coated perfectly conducting
cylinder as the coating thickness is varying ....................................................

xiii

120

125

136

139

140
142

143

144

145

147

148

149

150

151

153

154

4.16
4.17

4.18

4.19

4.20

4.21

4.22
4.23

4.24

4.25

4.26

5.1
5.2
5.3

5.4

5.5

Radar response of a conducting plate (4"x10") at end-on incidence .............

Waveforms of E-pulses for rectangular plates (solid line is for the plate
of 15"x6" and dash line is for the plate of 10"x4") ........................................

Output of the convolution between the E-pulse synthesized for conduct-

ing plate (4"x10") and radar response from the same conducting plate
(4"x10") at normal incidence ...........................................................................

Output of the convolution between the E-pulse synthesized for conduct-
ing plate (6"x15") and radar response from the conducting plate
(4"x10") at normal incidence ...........................................................................

Output of the convolution between the E-pulse synthesized for conduct-
ing plate (6"x15") and radar response from the same conducting plate
(6"x15") at normal incidence ...........................................................................

Output of the convolution between the E-pulse synthesized for conduct-
ing plate (4"x10") and radar reSponse from the conducting plate

(6"x15") at normal incidence ...........................................................................
Radar response of a conducting plate (4"x12") at normal incidence .............

Radar response of a conducting plate (4"x12") covered with a lossy
layer at normal incidence .................................................................................

Output of the convolution between the E-pulse synthesized for conduct—
ing plate (6"x15") and radar response from the conducting plate

(4"x12") covered with a lossy layer ................................................................
Output of the convolution between the E-pulse synthesized for conduct-
ing plate (4"x12") and radar response from the same conducting plate
(4"x12") covered with a lossy layer ................................................................
Output of the convolution between the E-pulse synthesized for conduct-

ing plate (4"x12") and radar response from the conducting plate
(6"x15") .............................................................................................................

Geometry of a rectangular plate .......................................................................
Partition scheme of a rectangular plate ............................................................

Convergence of four natural modes to their thinwire counterparts as

aspect ratio is varying. The squares show the locations of the

first four thin wire modes selected from the first layer ..................................
Convergence of the third mode ( I, = (e ,e) & I, = (0,0) ) to its thin

wire counterpart as aspect ratio is varying. The result is based on the
existing formulation with one quadrant divided to 4x2, 4x3,

and 4x4 zones. .................................................................................................

Different partition schemes are applied to the first mode (I, = (e ,e)

xiv

156

159

160

161

162

163
165

166

167

168

169

174
193

200

5.7

5.8

5.9

5.10

5.11

5.12

5.13

5.14

5.15

5.16

5.17

5.18

5.19

& Iy = (0 ,0 )) with various aspect ratios ..........................................................

Different partition schemes are applied to the second mode (I, = (e ,e)
& Iy = (0,0 )) with various aspect ratios ..........................................................

Different partition schemes are applied to the third mode (I, = (e ,e)
& I), = (0 ,0 )) with various aspect ratios. ........................................................

Pole locations of symmetry (1, = (e ,e) & I, = (0,0)) with various
aspect ratios .......................................................................................................

Pole locations of symmetry (1, = (0 ,0) & I, = (e ,e)) with various
aspect ratios .......................................................................................................

Pole locations of symmetry ( I, = (0 ,e) & I, = (e ,0 )) with various
aspect ratios .......................................................................................................

Pole locations of symmetry (1, = (e ,0) & I, = (0 ,e )) with various
aspect ratios .......................................................................................................

Amplitude distributions of x- and y-components of surface currents
associated with the first mode ( I, = (e ,e) & Iy = (0 ,0 ), b/a= 0.75 and
sa/c=-O.8087+j2.51 l) .......................................................................................

Amplitude distributions of x- and y-components of surface currents
associated with the second mode ( IJr = (e,e) & I, = (0 ,0), b/a= 1.0 and
sa/c=-1.254+j6.336) ..........................................................................................

Amplitude distributions of x- and y-components of surface currents
associated with the second mode ( I, = (e ,e) & I, = (0 ,0), b/a= 0.50
and sa/c=-2.990+j12.41) ...................................................................................

Amplitude distributions of x- and y-components of surface currents
associated with the third mode ( I, = (e ,e) & I), = (0 ,0 ), b/a= 0.75
and sa/c=-l.644+j9.777) ...................................................................................

Amplitude distributions of x- and y-components of surface currents
associated with the fourth mode ( I, = (e ,e) & I, = (0 ,0 ), b/a= 1.0
and sa/c=-l.770+j10.84) ...................................................................................

Amplitude distributions of x- and y-components of surface currents
associated with the first mode ( I, = (0 ,0) & Iy = (e,e), bla= 0.60

and sa/c=-2.311+j3.623) ...................................................................................
Amplitude distributions of x- and y-components of surface currents
associated with the second mode ( I, = (0 ,0) & I, = (e,e), bla= 0.60

and sa/c=-1.068+j7.111) ...................................................................................
Amplitude distributions of x- and y-components of surface currents
associated with the first mode ( I, = (e ,0) & I, = (0 ,e ), b/a= 0.60

and sa/c=-l.073+j4.739) ...................................................................................

XV

203

204

205

208

209

210

211

219

220

221

222

223

224

225

226

5.20 Amplitude distributions of x- and y-components of surface currents

associated with the second mode ( I, = (e ,0) & I, = (0 ,e ), b/a= 1.0

and sa/c=-l.209+j10.08) ................................................................................... 227
5.21 Amplitude distributions of x- and y-components of surface currents

associated with the first mode ( I, = (0,e) & Iy = (e ,0), b/a= 1.0

and sa/c=-1.391+j6.383) ................................................................................... 228

5.22 Amplitude distributions of x- and y-components of surface currents
associated with the second mode ( II = (0 ,e) & Iy = (e ,0), b/a= 1.0

and sa/c=-1.409+j7.985) ................................................................................... 229
5.23 Experimental measurement and the transient scattering response
waveform from a 4"x10" rectangular plate ..................................................... 231

5.24 Natural mode extraction from the measured response of a 4"x10"
rectangular plate. (a) The solid line is the original waveform in the
late-time and the dotted line is the waveform reconstructed based
on the extracted modes. (b) Comparison of natural frequencies
between experiment and theory ........................................................................ 232

5.25 Experimental measurement and the transient scattering response
waveform from a 4"x16" rectangular plate ..................................................... 233

5.26 Natural mode extraction from the measured response of a 4"x10"
rectangular plate. (a) The solid line is the original waveform in the
late-time and the dotted line is the waveform reconstructed based
on the extracted modes. (b) Comparison of natural frequencies
between experiment and theory ........................................................................ 234

xvi

Chapter 1

INTRODUCTION

In recent years, the subject of transient electromagnetic wave scattering and, in
particular, radar target identification and discrimination has gained considerable atten-
tion of researchers in this field. An urgent demand for a practical target identification
system becomes evident after a tragic event on July 2, 1988 when an Iranian passenger

airplane was mistakenly shot down by the US. Navy.

Electromagnetic target identification is a synthesis procedure, rather than an
analysis problem [1]. In analysis, one is concerned with deterrrrining the scattered field
pattern when a target is known. This case is normally termed the "forward problem".
Conversely, the "inverse problem" is related to synthesis, in which one is concerned
with obtaining information about a scatterer when the scattered field response is pro-
vided. The inverse scattering problem involves the complete determination of the
scatterer including its geometry and composition, based on a set of measurements of
the electromagnetic scattered fields. Since the true inverse problem is forrrridable in its
complexity and difficulty, the most promising implementation is parametric inverse
scattering. Here parameters are determined from the experimental data to give incom-
plete information of the scatterer. The E-pulse radar identification scheme [2], which
is the subject of this research, uses as its basis estimates of a target’s resonant features

from the scattered responses.

Attempts to identify objects using electromagnetics from a remote location are
certainly not new. Various radar applications and remote sensing are typical examples.
But there is a fundamental distinction between a conventional radar and a target
identification scheme. The former is used to detect the existence of a target, while the

latter is used to identify the target. In practical radar applications, identification is

 

almost impossible without the development of a new scheme. Recently the research

in radar target discrimination has achieved considerable advance.

In the early 1970’s, electromagnetic researchers began to consider the electromag-
netic identification problem at a very basic level. They conceded the overwhelming
difficulty in obtaining a quick solution to the practical problem and returned to an
examination of the basics. In this regard, researchers were able to build on mathemati-
cal scattering theory and apply it to problems in modern physics. Lax and Phillips [3]
have shown that for the scalar wave equation, scattering from objects with Dirichlet
boundary conditions is described by a meromorphic function of complex frequency.
This work implies that the scattering can be expressed as the sum of a temporal series
of complex exponentials, which describe the resonances of the object, and a temporal

response limited to early time.

In 1971, baum [4] proposed the scattered wave expansion called the Singularity
Expansion Method. Two years later, Marin [5] showed by a clever use of Fredholm
determinants, that the meromorphic characteristics of scattering were applicable to the
vector wave equation for perfectly conducting scatterers. The stage thus was set for
workers to examine how complex resonances could be extracted from the scattered
responses. It was then hoped that the resonances would form a basis for target
identification and discrirrrination .

After the introduction of SEM, many efforts were made to adapt Prony’s method
to determine the natural resonances of a scatterer from its scattered response and to use
a direct comparison of the natural frequencies of the targets for target discrimination
[5-8]. Experimental results have been disappointing. A late-time signal to noise ratio
of 15-20 dB is required for the accurate extraction of natural frequencies from a meas-
ured target response [9]. The sensitivity of Prony’s method to noise restricts its use

for target discrimination purposes.

 

An improved frequency comparison method called the complex resonance model-
ing method was proposed by a group at the University of Arizona [1]. The basic idea
in the complex resonance modeling of electromagnetic scattering is to use interactive
algorithms [10] and filtering techniques to identify the poles and the residues involved
in the complex exponential expansion of a target response to an impulse excitation.

This method, as expected, also requires a high signal to noise ratio.

The problems associated with Prony’s method reduce the usefulness of a natural
frequency comparison scheme for target discrimination. In 1981, Kennaugh [11]
introduced the "K-pulse" concept with the idea of suppressing the resonances of the
target by providing zeros in the Laplace transform of the excitation waveform to can-
cel them and thereby measure them. By using a polynomial approximation to the
Laplace transform of the K-pulse, he was able to estimate the natural resonances of
structures such as thin wires and spheres by locating the zeros of the polynomial. The
intension of the K-pulse is to approximate the resonances of a target by geometrical
diffraction theory and to generate a time-limited unique waveform corresponding to a

particular target.

A related concept called the E-pulse scheme, proposed by Chen in 1981 [2], is a
more appropriate discrimination technique which uses the target resonance frequencies
(determined by electromagnetic modeling or scale model measurements) in the con-
struction of time-limited excitation waveforms. The E—pulse is a time-limited
waveform which, upon convolution with any scattered response from the target, pro-
duces an expected spectrum in the late-time of the convolution response. E-pulse
waveforms are based only on the natural frequencies of a target, and thus lead to an

aspect-independent and noise-insensitive scheme, as will be demonstrated in chapter 3.

A related technique involves the use of the late-time target response, with an

optimization method [12]. The optimization method uses an approach based on an

energy maximization of the received signal. In this optimization , the ratio of the
energy in some time interval to the total energy contained in the scattered response is
maximized. The motivation for this method is provided by synthesizing the waveform
without the knowledge of natural frequencies. However, it only allows the discrimina-
tion of targets with known late-time responses and provides no information about the

target’s physical features.

This thesis covers a variety of topics pertinent to the experimental investigation of

the E—pulse properties and some basic transient scattering problems.

Chapter 2 provides the basic theoretical background in transient electromagnetics.
The Laplace transform domain electric field integral equation is established and the
Singularity Expansion Method, which is a very popular analytical method of treating
the transient problem by exploiting the natural resonances of a conducting scatterer, is
reviewed. Subsequently, a procedure to extract the natural frequencies of a target

from the complex frequency domain integral equation, or from a measured response is

described.

Chapter 3 investigates various properties and more realistic applications of the E-
pulse technique, including aspect-independence, noise-insensitivity and application to
scale aircraft models. The aspect-independency comes from the fact that the synthesis
of the E-pulse waveform is based only on the natural frequencies of targets, and the
noise-insensitivity results from the convolution of the E-pulse waveform with the
scattering responses of the targets. These two crucial properties have been experimen-

tally demonstrated

Chapter 4 describes the applicability of the E-pulse technique to targets coated
with lossy layers. A study of the natural resonant modes of an infinitely long cylinder

coated with a lossy layer is conducted both theoretically and numerically. The pole

distributions and the trajectories of a few fundamental modes versus the variations of
lossy parameters and geometry are displayed. It is shown that various rectangular
plates coated with lossy foam are well discriminated experimentally by the E-pulse

technique.

Chapter 5 presents a treatment of transient scattering from a rectangular plate.
Little work has been done to extract the natural frequencies from a thin rectangular
conducting plate. Here a new pair of coupled integral equations is developed. This
pair of equations is more rigorous, and more rapidly convergent than those developed
by previous researchers. The resulting natural frequiencies are checked with values
extracted from a measured scattered response of the plate. The agreement between the

theory and experiment is shown to be very good.

Chapter 2

THEORY FOR TRANSIENT SCATTERING OF EM WAVES

2.1 Introduction

Recently, significant attention has been paid to the development of transient elec-
tromagnetic theory. In a general sense, transient may refer to any nonmonochromatic
electromagnetic problem [13]. However, sometimes nonmonochromatic problems are
analyzed or solved experimentally using single frequency concepts applied to some
narrow band of frequencies. The term "transient" here is used interchangeably with
"broad band" since the time and the frequency domain are closely related by an

integral transformation.

In the nineteenth century, early investigations of basic electromagnetic phenomena
provided no discussion of transient versus continuous wave electromagnetics. Electric
and magnetic phenomena were understood in a static sense and the time derivatives
were next introduced. All the experiments were static or transient in nature. It was
when light was shown to be an electromagnetic waves that continuous wave concepts
were introduced. Even then, experiments below the microwave region were of a tran-
sient nature. Even telegraphy was transient, and wire telephony used broad-band

transmission.

With the development of wireless telegraphy and radio, the emphasis shifted to
CW electromagnetic concepts. From that time there was some carrier or center fre-
quency which was modulated in the form of some narrow bandwidth around the carrier
frequency. The introduction of radar systems reinforced the trend toward CW elec-

tromagnetic development.

In recent decades, some new problems, which are increasingly receiving interest,
have shifted some of the emphasis back to transient electromagnetics. One class of
problems involves the protection of electronic equipment from the effects of strong
transient fields such as lightning and electromagnetic pulse (EMP). Another class of
problems concerns electromagnetic inverse scattering, in particular remote sensing and
discrimination of radar scatterers. These interests have extended to the basic
phenomenology of transient electromagnetic field generation, to antennas for measuring
the transient fields, to antennas for producing temporal and spacial electromagnetic
field distributions, and to the interaction of the field distributions with complicated

scatterers.

In trying to understand the transient electromagnetic problem people attempt to
look at the subject in new ways. In 1971, Baum postulated the Singularity Expansion
Method [4,14] for representing and calculating efficiently the transient electromagnetic
responses of antennas and scatterers. Since then the conceptual basis of SEM, namely
mode representations [15], has been employed as the framework for electromagnetic
systems identification. Since the complex resonance frequencies of the natural current
modes on the scattering body are determined only by the structure geometry and com-
position, the poles of the corresponding natural mode scattered fields are unique innate
functions of the target. This has led many researchers to believe that a viable target
identification technique can be based on the extraction of the natural frequencies of the

target from the target’s radar echo.

The E-pulse technique [16,17], which will be depicted in the next chapter, is a
particularly useful target discrimination scheme based on the natural resonances of tar-
gets. Before proceeding to introduce various aspects of the E-pulse technique, this
chapter is dedicated to providing the theoretical basis for later chapters. Section 2.2

derives a general complex frequency domain surface electric field integral equation

formulation of transient scattering problems. Section 2.3 briefly reviews the introduc-
tion and development of the Singularity Expansion Method. The key stage in apply-
ing the SEM to a transient scattering problem is the extraction of the pole terms in the
SEM representation. Thus section 2.4 formulates the theoretical or numerical extrac—
tion of natural modes via a complex domain integral equation. The last section in this
chapter describes some of the well developed numerical algorithms which are used to

extract natural resonance frequencies from a measured response.

2.2 Complex Domain Surface E-field Integral Equation

Consider a perfectly conducting body with finite dimensions in free space, as
illustrated in Fig. 2.1. Let this body have a surface S which bounds a volume V.
Assume E‘(r,t) is the transient impressed field which excites a surface current K(r,t) on
the surface of the body. The scattered field, which is maintained by the induced sur-
face current K(r,t), is correspondingly denoted by E‘(r,t). The total field E(r,t) is the

sum of the impressed and induced fields.

In the Laplace transform domain, the incident and the induced fields are denoted

E‘(r, s) = L [E‘(r,t)] (2.2.1)

and

Es(r, s) = L [Es(r,t)] (2.2.2)

The integral equation is established based on matching boundary conditions on the sur-

face of the conducting body. The null tangential component of the E-field results in

t - E(r,t) = t - [E‘(r,t) + ram-,0] = o r e 5 (2.2.3)
where t is a unit vector tangential to the surface. Applying the Laplace transform on

(2.2.3) the complex domain E-filed boundary condition leads to

£0 , ”0 Conductor

/

  

Incident direction

\.

 

Fig. 2.1 Description of Transient Scattering of EM Waves in Free Space

10

t ‘ E‘(r,s) = — t - E‘(r,s) r e S (2.2.4)
It is a common practice to represent the scattered field using retarded potentials

E’(r, t) = — VCD‘"(r, t) — %A‘(r, t) (2.2.5)

where the scalar potential is related to the surface charge by

1 ma t—fl)
C

“505 R

 

<D"(r, t) = 4 dS’ (2.2.6)

and the vector potential is related to the surface current by

K(r’, ti)
ASO.’ t) .__. TRIS-76—

with R being the distance between the source point and the observation point.

dS’ (2.2.7)

Invoking the Laplace transform, the scattered field and potentials are written as

 

E‘(r, s) = - Vd>‘(r, s) — sA’(r, s) (2.2.8)
and
1 , e'IR ,
<I>’(r, t) = E15930“ , S) R d5 (2.29)
s _ Ho , e'” ,
A (r, s) _ 4—15-ISKO.’ s)—R- dS (2.2.10)

where y = s/c is the complex wavenumber.

To derive the integral equation, the current continuity equation must be used

V - K(r, t) + ipsu, t) (2.2.11)
3:
The transform of (2.2.11) is
V ~ K(r, s) = - sp,(r, s) (2.2.12)

Employing the results of (228-2.2.11), the scattered filed in the complex fre-

quency domain is represented as

0'7 R
R

 

_ 1 I 1 Silo I e—YR I
E’(r, s) _ — EISV p,(r, s) as — 71-:- [SKn- , s)—-R—- dS

ll

 

 

_ 1 y. I 8-YR I Sl‘l’O I e-JYR
_ 4 [SW K(r,s) R as 4“ ISK(r.s) R dS’
_ 1 , , , e‘IR ,

Using the boundary condition (2.2.3) with (2.2.13) yields the electrical field

integral equation

R .
3:1? dS’ = — seot - E‘(r, s) (2.2.14)

Equation (2.2.14) is a general formulation which will be specialized in Chapter 5 to

 

t - [st V’-K(r’, s) V - 72 K(r’, 3)]

treat a rectangular plate problem. For notational purpose it is more convenient to
inu'oduce a Green’s function into the integral equation. There are two commonly used

Green’s functions. The scalar one satisfies the wave uation
“I

( V2 - 72 ) G(r,r’;s) = — 6(r — r’) (2.2.15)
with appropriate boundary conditions. And the dyadic one obeys the equation

( V x V x + y’) G(r,r’;s) =T8(r - r') (2.2.16)
and appropriate boundary conditions. In (2.2.16) T is a unit dyadic. The explicit
representation for the scalar Green’s function, which satisfies the radiation condition, is
easily shown to be

, e‘IR
G(r,r ;s) = 2?]?- (2.2.17)

where R is ir - r’l. The usual electric dyadic Green’s function can be explicitly

expressed by

G,(r,r';s) = (T — -:2—V ) G(r,r’;s) (2.2.18)

Sometimes it is necessary to exclude the strong singularity at the source point from the

above representation. The form of the dyadic Green’s function is then modified to

8031-23) = P.V. fie(l‘,r';s) — gal-:2? (2.2.19)

12

where “E is the radius of an arbitrarily small circular path surrounding the source point
and RV. means principle value. Now, going back to the scattered field formulation of

(2.2. 13),

 

V’K’ eflR—V’K G( ")
0.0 4”? — 0.0 ms

the first term in (2.2.13) is expressed in Green’s function notation as
V [S V’ - K(r’, s) G(r,r’;s)dS’ = V [s V’ - [K(r, s) G(r,r’;s)]dS’
+ V L V G(r,r’;s) . K(r’, s)dS’ (2.2.20)
Both integrals on the right hand side are improper. They should be evaluated in a

principle value sense by excluding a small circular path of radius ‘6’ about the point of

r = r’ in the limit as ‘6 —> 0. Subsequently, (2.2.20) results in

V [s V’ - K(r’, s) G(r,r’;s)dS’ = J— K(r, s) + j W G(r,r’;s) . K(r’, s)dS’ (2.2.21)
47: 3

One can write the scattered electric field in terms of Green’s function as

E‘(r, s) = - Silo PM I [‘1’ - 5557-] G(r,r’;s) - K(r’, s)dS’ + K r" 5 (2.2.22)
5 r 4m
or in terms of the dyadic Green’s function as
E’(r, s) = — stro [s G(r,r’m) - K(r’, s)dS’ (2.2.23)

Then the electrical field integral equation of (2.2.14) is written in terms of dyadic

Green’s function as

Silo [s t - G(r,r';s) - K(r’, has = t . E‘(r, s) r e 5 (2.2.24)

It is noted that equation (2.2.24) is a special form of the more general volume
integral equation of

Iv T'(r,r’; s) - J(r’, s)dV’ = I’(r, s) r,r’ e V (2.2.25)

where T'(r,r’; s) is the dyadic kernel of the integral equation and the I’(r, s) is the fore-

ing function. For convenience the integral equation (2.2.25) can be further written

13

using a symmetric product notation

< F(r,r'; s) ; J(r’, s) > = I’(r, s) (2.2.26)

as commonly used in the development of SEM.

2.3 The Singularity Expansion Method

Since the Singularity Expansion Method is closely related to the subject of target
identification and forms the foundations of various natural frequency based target
discrimination schemes, it is proper to dedicate this section to the introduction of the

SEM.

The Singularity Expansion Method was first introduced by Baum in 1971 [4]. It
provides a compact means of representing broad-band transient electromagnetic
phenomena on resonant bodies in terms of the complex natural resonances of the body
in the complex Laplace transform domain. The development of the SEM was stimu-
lated by experimental observation of the general characteristics of typical transient
responses in experiments on various complicated scatterers. Transient response
waveforms appear to be dominated by a few damped sinusoids in the late-time period.
The frequencies of these resonances were seen to be dependent on the physical proper-
ties of the scatterers, but not on the excitation. Since the Laplace transform of a
damped sinusoid corresponds to one pole or one pair of poles in the complex plane,
and a broad-band pulse excites many natural resonance modes, one can ask what part
of the complete solution these damped resonances represent. This question leads to

the Singularity Expansion Method.

Subsequent to its introduction, the SEM has been applied to many problems. In
fact in his introductory paper Baum constructed the formal SEM solution for the per-

fectly conducting sphere [4]. Marin conducted an analytical solution for a prolate

l4

roidal scatterer [l8]. Tesche implemented the first numerical SEM solution to a
cylindrical antenna [18]. Umashankar applied SEM to the circular loop antenna
L-shaped wire [20]. Marin and Latham made an important theoretical contribution
EM by demonstrating that in the complex plane the only singularities in fields for
: perfectly conducting bodies are poles [21]. In addition, various workers applied
nethod to several coupled cylindrical problems[22], and the rectangular plate prob-

[23].

In the early years of the SEM development, there were relatively few efforts con-
ed with SEM description of the scattered fields and the early time response. Later
Heyman and Felsen considered a hybrid representation of induced currents and
ered fields for an infinite cylinder [24,25]. The early-time behavior is described
geometrical diffraction theory while a natural mode description is used for late-
. Morgan suggested a unified natural mode representation of induced currents and
scattered fields based on fundamental causality [26]. Pearson used the temporal
ace transform and an exponential entire function to include the early-time in the

rec current natural mode expansion [27].

To have a better insight to the SEM, we limit our scope to a simpler situation.
I the late-time behavior is of concern and only the first order poles of a finite per-

y conducting scatterer are considered.

1) Expansion of Current in Complex Frequency Domain

It is conjectured that the surface current induced by delta function excitation on
e-size perfectly conducting bodies in free space can be represented in the complex

uency domain as

K(r, s) = z nan) Ka(r)(s - sa)‘”'°' + W(r, s) (2.3.1)

 

 

 

15

where ma is an integer presenting the order of the or’th pole and W is an entire function
(which has no singularity in finite complex plane). The notation na(s) indicates the

coupling coefficient of the or’th natural current mode of Ka(r).

One important feature of the singularity expansion is the separation of object and
incident waveform singularities when they are not coincident. Assuming the incident
field waveform is shaped with fls), the normalized response of current to the incident
waveform is

V(r, s) = fls) K(r, s)
= Vo(r, s) + V,,,(r, s) (2.3.2)
Restricting consideration to the case of the first order poles in (2.3.1) and neglecting

the entire function W, the object and waveform parts are respectively

votr. s) = ms.) nuts) Kamts — sq)“ (2.3.3)
Vw(r, s) = 2 710(5) Kamw + possible entire function (2.3.4)

where the second term is found as a singularity expansion over the singularities of the
incident waveform and the first term is a singularity expansion over the singularities of
the object. By this separation it is exhibited that the natural frequencies and modes of
the object are independent of the incident waveform, which provides the basis for tar-

get identification in the following chapters.

2.3.2) Derivation of Coupling Coefficients

Starting with the integral equation (2.2.26)

< T’(r,r’; s) ; K(r’, s) > = I’(r, s) (2.2.26)
the natural frequencies and modes are first found by equating

< T'(r,r'; sa) ; K(r’, sa) > = 0 (2.3.5)

 

16

ulation of the natural frequencies using an integral equation is discussed in the
section. Having found the natural frequencies and modes, one can find the cou-

; vector which is defined by

< p.(r) ; F(r,r’; so.) > = 0 (2.3.6)

numerical solution to (2.3.6) can be obtained by applying a moment method.

Now the kernel, the current and the excitation are each expanded in a power

5 around the point s = so, as

T‘(r,r'; s) = E; (s - 50,), may) (2.3.7)
[=0
13(00- r’) = -1— i fir r" s) I (2 3 8)
or r I! as! r v szsa - °
I(r, s) = E; (s - sa)’ 15,? (2.3.9)
[=0
(I) = i _3.’_
r, I! 851 (r, s) 'm. (2.3.10)

, the response of the current in (2.3.1) is written by separating the pole term of Sr:

y the first-order poles considered) as

K(r. s) = noise.) Ka(r)(s - so)“1 + K’(r. s) (2.3.11)
re K’(r, s) is analytic in the neighborhood of Sa-

then, these expansions around s = Sa are substituted into the integral equation
.26) and the results are grouped according to powers of s — 50- The coefficient of

term (s - sq)"l is evaluated at 50

< rift”); no.0.) Ka(r’) > = 0 (2.3.12)

 

 

 

17

that by (2.3.8)

Dyan) = F(r,r’; s) 'ma

; (2.3.12) is consistent with (2.3.5). The coefficient of the term (s — 3:00 gives at

< fifkrr’); K’(r’, so, ) > + < I‘SKnr’); na(sa) Ka(r’) > = 190). (2.3.13)
'ating on the left by the coupling vector u(r) and noting that the first term is set to

by (2.3.6), it is found that

< Mr): “Jim-3; 11.0..) Kan) > = < 110); 15.90) >. (2.3.14)
coupling coefficient at 30‘ is then solved as

< Mr); 185%) >

2.3.15
< u(r); Tight); Ka(r') > ( )

 

“0150.) =

The essence of this derivation is based on the expansion near sa. Equation
15) gives only the coupling coefficient at s = Sm but provides the current residue
1e pole of s = 301- For other value of s the coupling coefficient can have various
is chosen for convenience and desired convergence of the sum. Since any entire
tion with a zero at s = 3a can be added to nets“) to satisfy the (2.2.26), the choice

particular form of na(sa) is related to the entire function W in (2.3.1).

As is known, an entire function arises from the Laplace transform and has no
ularities in the finite complex frequency plane. It must rise and fall in the time
ain faster than an exponential function. Such a time function is then limited in

rrtance to early times. The damped sinusoids will dominate at late times.
There are two classes of coupling coefficients which are defined in [14]. The first
simplest one of these, referred to as "class 1", is given for turn on time at t’ by

(30,- s)!’ < “(1'); 1590') >

2.3.16
< u(r); WING; K010") > ( )

 

"(1(5) = e

 

 

 

18

where t’ is chosen for convenience and convergence. The "class 2", or convolution

form coupling coefficients are given by

 

= < pm; Itr. S) > 2.3.17
110.0) < u(r);1‘9)(r.r’); Ka(r') > ( )

The s dependence of na(s) is from the excitation of I(r, s).

Class 2 coupling coefficients are more complicated to calculate than class 1 cou-
pling coefficients. However, class 2 coupling coefficients give smoother early-time
results for a finite number of poles. As an example, if the class 1 coupling coefficients
are used, the representation of (2.3.16) in terms of the surface electrical integral equa-

tion (2.2.24) is shown to be

(3,. .y [.E‘tr. s.) ' K(r. 045
8G r,r’;s .
sauoj'sdsmr, s) - jS—SéS—llma - K(r, s)dS

 

“(2(5) = e (1.318)

2.3.3) Orrrent Expansion in Time Domain

The expansion of the surface current in time domain is obtained from (2.3.1) by
an inverse transform. If the class 1 coupling coefficients are used, the time domain

correspondence of (2.3.3) is then written as

v,(r, t) = u(t — r’) Zflsa) nan“) Ka(r) e‘“’ (2.3.19)

where only the first order poles have been assumed and u is the Heaviside step func-

tion.
The class 2 coupling coefficients give the time domain response of (2.3.3) as
V0(r, t) = u(t - t’) 2 flsa) [< u(r); 128 )(r,r’); Ka(r’) >]“1
at
[< utr); Km 0 >1*[u<r)e‘“’1 Katr) (2.3.20)

where an asterisk * indicates convolution operation. The individual terms are seen to

 

19

be more complicated than for class 1, but the early time response is smoothed.

2.3.4) Later Developments

One of the early extensions of the SEM concept was to the calculation of the
field radiated or scattered fields into the space surrounding the conductor. Until recent
years, some investigators proposed field expansions which utilize physical causality to
include the early-time response. A series of papers [26-28] considers cases under
which an entire function can be avoided, based on asymptotic estimates of terms
appearing in the integral equations. Nevertheless, at early times, convergence troubles
occur. The scattered fields do require an entire function in their representation [29,30].
The entire function remains somewhat controversial for a while, but it is not critical

for many practical applications.

Another extension of the SEM concept was the introduction of the more general
eigenmode expansion method (EEM) to find more properties of the SEM. The natural
frequencies in the SEM are related to the eigenvalues in the EEM. The vanishing
eigenvalues are the natural frequencies. However there are many mathematical prob-
lems to be considered concerning completeness, root vectors, sense of convergence,

BIC.

After the SEM concept was established, it was noted that the natural frequencies
of a scatterer were independent of the exciting fields. This property offered a potential
use for target identification purposes. Notable developments in this subject are still
progressing.

Since the original impetus toward the SEM came from the observation of the gen-
eral properties of the transient electromagnetic response, it is understood that the SEM
should be applied to experimental data. Many investigators have suggested many tech-

niques to determine the natural frequencies from the experimental data (this will be

 

discussed in the next two sections).

While one path of the developments is toward the application of the SEM, the
other path is toward the investigation of the mathematical foundations [31]. The SEM
is mathematically justified in the asymptotic sense. Mathematicians are pursuing a
rigorous exploration of the SEM theory with a view to define the precise limits of
applicability.

Research on the topics of branch contributions, connection with high-frequency

representations and SEM synthesis of equivalent circuits have also made advances

[32].

2.4 Determination of Natural Modes of a Scatterer

It is seen that the pole valus are very important in the singularity expansion of
surface currents on a finite conducting body. This section illustrates three approaches

to determine the natural poles of a scatterer.

There is an open question regarding whether the complex poles of the scatterer
are simple or not. It has been proved that for a spherical or linear scatterer the poles
are sirnple [31,32]. But no conclusive arguments have been given for a generic finite
body. Thus it is only a conjecture that a finite conducting body has only simple poles.

Therefore we will presume the simplicity of the poles for any finite conducting body.

2.4.1) General Projection Method

As was stated previously, the determination of natural modes of a scatterer can be

performed via the homogeneous integral equation of (2.3.5).

< F(r,r’; sq) ; K(r’, 3a) > = 0 (2.3.5)
In MoM form (2.3.5) yields

 

21

(rnm( Sa ) ) ' (Kn )a = 0 (2.4.1)

or, as an operator equation
A x = 0 (2.4.2)
where A is viewed as an operator. The operator A projects the space vector x on to a

null space. In fact, the complex poles of a scatterer are the points at which the opera-

tor A is not invertible ((2.4.2) has a nontrivial solution).

Let the { f,- } be a basis of space L2 and

N
x = 2 cjf} (2.4.3)

j=1
If P,l is projection on the linear span of (fl, f2, fN ), the projection of (2.4.2)

P, A P, x = 0 (2.4.4)
leads to
N
2 00(3) Cj = 0 , I S l S N (2.4.5)
i=1
where
0,, = <A f,- , f? (2.4.6)

The system equation (2.4.5) has a nontrivial solution if and only if

det [ 0,-1- ] = 0 (2.4.7)
The left hand side of (2.4.7) is an entire function of S. If 352° is the m’th zero of
the determinant of the operator A projected onto a linear span of (f,, f2, fN), then it

is shown [31] that the set of

s”, = lim SE2!) (2.4.8)

N —D co
coincides with the set of points at which A is not invertible, that is the union of the
set of complex poles of the scatterer. This proposition justifies the current numerical

method widely used for calculation of the complex poles.

 

22

2.4.2) Eigenvalue Method

In 1975, after the introduction of his Singularity Expansion Method, Baum intro-
duced the eigenmode expansion to find more properties of the SEM[15]. The eigen-

values and eigenmodes are defined by

< f'(r,r'; s) ; Ka(r’, s) > = Aa(s) Ka(r, s) (2.4.9)

and

< lla(r. S) :I'(r.r’; S) > = MS) ua(r’. S) (2.4.10)
where Ka(r, s) and ua(r, s) are right and left eigenmodes and Ms) is the associated

eigenvalue. The eigenmodes can be generally biorthonormalized as

< ua(r, s) ;KB(r, s) > = {(1) 038;?“ (2.4.11)

It is apparent that the poles of a scatterer are the points where the eigenvalues of
1.0,(s) = 0. It is proved mathematically that the set of the poles of a scatterer coincide
with the set of the complex zeros of the eigenvalue functions 1.,(s). Thus the calcula-
tion of the poles can be performed by evaluating the eigenvalue functions and finding

their complex zeros. The eigenvalues 1.0,(s) can be found by means of a projection

method.

2.4.3) Variational Principle Method

This method is also related to the eigenmode expansion. If a pole is denoted by

so, from the above argument we have

Ms) —) 0, as s —) so (2.4.12)

The zero eigenvalue is associated with the eigenvector which is identical to the natural

resonant mode at the pole. That is

K(r, s) -~> K(r, so), as s —> so (2.4.13)

 

23

In practice, computation of the eigenvalues Ms) resorts to numerical schemes.
: computational costs for tracking the behavior of a single eigenvalue as a function

r are expensive.

The third method here is based on the generalized Rayleigh quotient [33]

_ < ua(r, s) ; P(r,r’; s) ; Ka(r, s) >
Ads) — < IlaCI’. S) ; K00“. S) > (2.414)

 

ich is easily derived from (2.4.10). This form can be shown to be a stationary form
Ms). Its explicit matrix form is
iu?(S)l+ia.,(S)llKj-‘(S)l

[11?(3)]*[K?(s)]
ere + means transposed conjugate. The equality (2.4.15) or (2.4.14) holds to the

 

MAS) = (2.4.15)

t order only. When the matrix [090)] is Herrnitian, then [u?(s)] = [K?(s)] and

1.14) becomes the Rayleigh quotient. In this case the eigenvalues are real.

The stationary form (2.4.15) may be used to approximate the vanishing eigen-
ue provided the estimates for both right and left eigenvectors are available. This
thod dependents on a reasonable initial estimate for the pole location, and subse-
:ntly the eigenvectors. This estimate may be determined from physical insight, from
:king the pole trajectory with respect to a geometry parameter, or from argument

ection of a region of the left complex s plane until a pole is located.

One of the estimation techniques suggested by [33] is to triangularize the matrix
(3)] in (2.4.15) and its transposed conjugate by Gaussian elimination with pivoting.
the triangular form, the zero of the determinant [a,j(s)] is manifested as a zero in

low right position of the matrix. The natural mode can be determined by back-
ving the homogeneous matrix equation. One means to obtain estimates for [u?(s)]
l [K?(s)] results from the observation about the triangularized matrix in a neighbor-

)d of a pole So- It is observed that the triangularized matrix varies slowly in s.

 

 

 

24

Therefore the lowest right entry can be replaced by a zero. This replacement allows
us to backsolve the resulting triangular homogeneous equation for an estimate of the
natural mode [K?(s)]. By the same process, [u?(s)] is estimated from the transposed

conjugate of the matrix [0,,(s)].

The eigenvalue is estimated by (2.4.15) when the estimates of eigenvectors are
obtained. The zeros of the eigenvalue function are then searched by any standard
algorithm. Note that better accuracy may be gained by iteratively perfonning the

above estimates.

It is seen this method is variational in character and its accuracy rests on the abil-
ity to estimate the natural modes of a scatterer. This estimate-dependence introduces
some risk into the search for poles. As a result, the poles obtained by the method will

have to be verified or refined by determinant calculation.

2.5 Extraction of Natural Frequencies from a Measured Response

After the introduction of the SEM for the representation of transient and broad-
band electromagnetic interaction with general complicated targets, there has been con-
siderably attention given to associated analysis of electromagnetic-response experimen-
tal data to find the natural frequencies. The very basic requirement for the implemen-
tation of a radar target identification scheme based on natural resonances of targets is
the knowledge of the natural frequencies of a wide variety of targets. For most realis-
tic complicated targets, it is impossible to determine theoretically the natural frequen-
cies. Thus it is very vital to be able to extract the natural frequencies from a measured

response scattered from a scale target.

Extraction of natural frequencies is accomplished by applying numerical tech-

niques to various experimental sample data. In the last decade, many numerical

25

methods have been proposed with considerable success. This section depicts some of
them, which are significant in the development of the numerical techniques for extrac-

tion of natural frequencies from a measured response.

2.5.1) Prony’s Method [6,8,34]

Prony’s method is an old but well respected technique for experimental data
regression by a finite sum of complex exponentials. Suppose one is given transient

response data uniformly sampled as a set of ordered pairs (yk, tk; k = 0, n) with

tk =10 + k A k = 0,1,...JI (2.5.1)
It is desired to represent the data in terms of a finite sum of complex exponentials
given by
ym=zmfi' can
i=1

where a,- and s,- are both complex. At the sample points, (2.5.2) can be written as

gw=§mé as»
where
b,- = aies‘ '° (2.5.4)
and
g=gA as»

One wishes to use given transient response data in conjunction with (2.5.3) to
find the unknowns b,- and 2,-. To implement a solution, the following algorithm by
Prony is used. Let the z,-’s be roots of a polynomial

fl+2adfl=0 05$

i=1

If we define a0 = 1, (2.5.6) becomes

26

E; a,- 2"” = 0 (2.5.7)
5:0

Multiplying the k+j’th equation of (2.5.3) by am_j and adding the results together
yield

K0301». +Y(’k+r)am—r + ' ' ‘ 4" Y(tk+m)0‘0
=Eazzklam+am—12+'“+aozml
i=1

= 0 (2.5.8)

Since (10 = 1, (2.5.8) above can be written as

Z a,- y<tk....,-) = 001....) (2.5.9)

j=1

The same scheme is employed for any k in the interval [0,n-m], we can have a
total of n-m+1 equations in m unknowns of [09-]. Equation (2.5.9) is a matrix equa-
tion which can be solved using standard techniques. Once the unknowns of [09-] are
obtained, the roots of polynomial equation (2.5.6) are solved. Using (2.5.5), the

natural frequencies in the representation of (2.5.2) are given by

In Z,‘
S,‘ = T (2.510)

Note that for y(t) to be calculated uniquely, there must be at least 2m sampled
response points. Then y(t) can be made to fit exactly through those points. If fewer
points are used the resulting complex frequencies are not good estimates of natural fre-
quencies. Prony’s method is extremely simple to use, but it has many shortcomings
which greatly reduce its usefulness under practical circumstance. The worst of them is
its sensitivity to noise. A modified algorithm is implemented by a classical least
square procedure. If (2.5.9) is solved using a least square criteria, the resulting
coefficients aj are less influenced by random noise. However, Prony’s algorithm is

inherently ill-conditioned, any modification can only give marginal improvements.

 

27

2.5.2) Least Square Method

The classical natural estimate of natural frequencies contained in a transient

response can be determined by finding the minimum norm of the residue function of

the form
E(x) = 1120012 = 35 12,300 (2.5.11)
i=1
where
RA") = F(x, ‘1') - Kit) (25-12)
with
x = [abolxohol , - - - . a~,oN,(uN,¢N,K]T (2.5.13)
and
N O t
F(x, t) = K + 2 an e " cos((u,,t + on) (2.5.14)
n31

Minimization of (2.5.11) is accomplished by setting the partial derivatives with

respect to each components of x to zero. The gradient of the error function E is

f(x) = V,E(x) = {YR(x)TR(x) = 0 (2.5.15)

where the Jacobian matrix of R(x) is given by

BRA")
8x-

1

men.) = (2.5.16)

The solution to nonlinear equation (2.5.15) is supplied using Newton’s algorithm

x = x0 — [fo(x0)]"f(x) (2.5.17)
This equation is then iterated successively to converge to the desired solution from

some initial guess for x. The gradient of f(x) is shown to be expanded as
Vxfixo) =iiktxrii£<x> +£3.00 (2.5.18)

where

'S’R(x) = i R,(x) Vigor) (2.5.19)

i=1

 

28

with

3280‘)

2 . . -_-_ —_
[ Ver(x) 11,1 axlaxl

(2.5.20)

It is obvious that extensive computation is required to solve (2.5.17) from an ini-
tial guess. There are two problems encounted when the Newton’s method is used
here. First, local convergence of Newton’s method requires very good initial guesses,
which are difficult to obtain in practice. Second, this least square analysis is an ill-
conditioned problem since the best fitting function is represented by a set of damped

sinusoid functions whose linear independency is not satisfactory.

2.5.3) Continuation Method

Since the nonlinear least square problem established above is an ill-conditional
problem, no algorithm can be depended on to produce the true solution when the data
used is from a measured response. The general technique for dealing with this kind
of problems in numerical analysis is the "Regularization", which transforms the origi-
nal problem into a related well-conditioned problem with a solution which approxi—
mates the desired solution. One of the regularization schemes is termed a "continua-

tion method" [35] (more generally a homotopy method).

Instead of minimizing the norm of the residue vector in (2.5.11), consider the

minimization of the parametric function with a parameter 1:
G,(x) = r IR(x)l2 + (1 — r) IW(x)I2 (2.5.21)
where W(x) is given by

W(x) = 355 — 1 (2.5.22)

and x° is a initial guess for x. Normally the function W(x) is called a "penalty func-

tion" which is designed to keep x from wandering too far away from the initial guess

 

 

29

and to improve the conditioning of the problem.
Minimization of (2.5.21) with respect to x for a given 1 needs
V,G,(x) = 1: ‘j’R(x)TR(x) + (1 - r) ‘j’w(x)TW(x) = 0 (2.5.23)
where the Jacobian matrix of W(x) is given by

8W,(x)
8x -

I

[‘j’w(x)TW(x) 1,, = (2.5.24)

Note that t = 1 (2.5.23) gives the original ill-conditioned least square problem. How-
ever, with 1: = 0 (2.5.23) reduces to a well-conditioned problem which has the trivial

solution x = x°.

The continuation method is an iterative procedure, beginning with 't = 0 and x° as
an initial guess. At each successive iteration t,- is increased by A 'r and (2.5.23) is
solved using a standard equation solver. The A r is assumed very small to assure the
convergence and the procedure continues until I = If. If 1f: 1, then the original prob-
lem is solved. However, it is often necessary to stop at rf< 1 to avoid (2.5.23) becom-
ing too ill-conditioned. The consequent solution is a good approximation to the origi-

nal problem.

The continuation method outlined above eases the initial requirement for the
minimization, however it may not guarantee the convergence since the step size A: is
fixed and is always increased. A modified alternative algorithm can be developed by
tracing the path of (x, 1) leading from (x, 0) to (x, If). The path extends from the initial
guess to the best approximation for the desired solution. It is informative to
parametrize the path by the arc length. Then, the chain rule is used to establish a

differential equation. If we define

H(x(s), 1:(s)) = Vth(x) (2.5.25)
then taking the derivative of (2.5.23) results in

 

 

 

30

 

A ___ _d_!5_ dH(X(S). 10)) fl =
dsH(x(s), 12(5)) V,H(x(s), 1(3)) ds + d1: ds 0 (2.5.26)

Equation (2.5.26) can be solved by a standard ordinary differential equation follower,

leading to the desired solution at r = If.

Another modification can be performed by using variable step size which

increases at points with small curvature and decrease at points with sharp turns.

One of the important steps taken in this algorithm is to decide the “If where the
iteration procedure is terminated. Similar to (2.5.17), the solution to (2.5.26) by

Newton’s method can be presented as

Sm = x.- - 1V.H(x.~. 1,.)1“ H(x.-. 1,.) (2.5.27)
where 1,, is the predicted value of 1: at the step 1'. At each iteration the condition

number of the matrix

[VXH(X“, 1%)] (2.5.28)
is checked to determine whether the algorithm is terminated. If the condition number

is too big, termination is demanded. The solution with the value of t, at which the

iteration is stopped is used as the approximation to the desired solution.

The continuation method is easy to implement and due in great part to its noise-
insensitivity its use is highly recommended. The shortcomings with this method are
attributed to the computation cost which sometimes can be reduced by an alternative

method.

2.5.4) E-Pulse Method

To reduce the computation cost of the continuation method, a new proposed
method for extraction natural frequencies is constructed by using the E-pulse concept
Which will be described in chapter 3. The E-pulse is a waveform with finite duration

and is designed to produce a zero late-time period in its convolution with the response

31

from the target, based on whose natural frequencies the E-pulse is synthesized. As the
response will certainly be contaminated by noise, application of an E-pulse will pro-
duce a small nonzero late-time response. Thus the proposed E—pulse method [36] is an
iterative procedure which searchs for Optimal parameters to construct the E-pulse and

consequently to yield a minimum late-time response.

Let the convolution response between a transient scattered response and its
corresponding E-pulse be written as
C(t) = e(t)*r(t) t > T, + TL (2.5.29)
where e(t) is the waveform of E-pulse and r(t) is the scattered response. The T, indi-
cates the period of the E-pulse and the TL indicates the beginning of the late-time in
the scattered response. It is assumed the late-time scattered response can be
represented by the sum of damped sinusoids

N
r(t) = z a,e°" 605(0),, z + 0,) t > TL (2.5.30)

n=1

and the E—pulse waveform can be expanded over a set of basis functions

K
60‘) = Zak fatt) (2-5-31)
Ic=1

The E-pulse convolution demands the late-time response of C(l’) to be zero. Using

(2.5.29) we obtain

N
c(:) = z IE(s,,)ra,,e°" 605(0),, z + 0,) t > TL + T, (2.5.31)
n=l

where E(s) is the Laplace transform of the E-pulse e(t)

TI
E(s) = j e(t) e“ ‘dt
0

K

akfaa) e" ‘dt (2.5.32)
H

chin. :i

 

 

 

 

 

Co

32

If the natural frequencies of a target are known, an E-pulse is synthesized by

demanding

E(sn) = 150;) = 0 1 s n s N. (2.5.33)
Conversely, if e(t) is deternrined, the natural frequencies annulled by the E—pulse can

be found by locating the roots to

K
E(s) = F1(s) e' A z orke'“ A = 0 (2.5.34)
k=SI

where the expansion of (2.5.31) has been inserted into (2.5.32) with the basis functions

defined by

t — k-l A _
fin) = {10“ I 1 ) 0‘ 61mg? (2.5.35)
The roots of (2.5.34) is found by solving the polynomial
K
2 (1" Z" = 0 (2.5.36)
i=1
where
Z = e“ A (2.5.37)

There are two approaches to implement the E-pulse method for extracting natural

modes. One approach is to minimize the late-time convolution of

c2(t) = [e(t)*r(t)]2 (2.5.38)
with respect to the natural frequencies, and the other approach is to minimize (2.5.38)

with respect to the amplitudes of basis functions in (2.5.31). If the minimization is
done w.r.t. the natural frequencies, the iteration procedure is taken to update the
natural frequencies, to synthesize the E-pulse via (2.5.32) and to convolve the E-pulse
and the scattered response until a minimum point is reached. The natural frequencies
are available when the minimum point is located. If the minimization is w.r.t. the

amplitudes of basis functions used to expand the E-pulse, the iteration is taken to

 

33

update the amplitudes in the E-pulse expansion and to convolve the E—pulse with the
given scattered response until a minimum late-time convolution is obtained. Then

(2.5.36) and (2.5.37) are employed to extract the corresponding natural frequencies.

The benefits of the E-pulse method resides with its lower noise-sensitivity and
fewer parameters utilized in the minimization procedure. The drawbacks with this
method may be oriented to the requirement for the knowledge of the number of modes

before they are extracted.

,-

. \...]

a
'\
o
O

.c'..- '0
.10 \.'u t! 4"
v- ' . C
”$33.1...1'51. 0 °

tr ire (
..m‘- get...
.5‘“.

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- 3311’

 

Chapter 3

ASPECT-INDEPENDENT AND NOISE-INSENSITIVE TARGET
DISCRIMINATION USING E-PULSE TECHNIQUE

3.1 Introduction

Radar target identification methods using the time domain response of a target to
a transient incident waveform have generated considerable interests recently. Among
various available radar detection schemes , the E-pulse technique has demonstrated a
great potentiality. The E-pulse technique developed by our group has been applied
successfully to various simulated targets, including wire structures and some compli-
cated aircraft models. To demonstrate further the applicability of the E-pulse tech-
nique to complex targets, and to explore the inherent important properties of the E-
pulse scheme, extensive experimental studies with various complex targets in a labora—

tory environment have been conducted.

The E—pulse technique consists of synthesizing discriminant signals, including the
Extinction—pulse (E-pulse) and single-mode extraction pulses (S-pulses), based on the
natural frequencies of a target, and convolving the discriminant signals with radar
returns from the targets [37-40]. When the discriminant signals of a target are numeri-
cally convolved with the late-time response of the expected target, zero or single-mode
late-time responses are produced in the convolved outputs. However, when the
discriminant signals of a target are convolved with radar return from a different target,
the convolved outputs are significantly different from zero or single-mode responses in

the late-time period. Thus the differing targets are discriminated.

The most important and inherent properties of the E-pulse technique are charac-

34

35

terized by the aspect-independence and the noise-insensitivity. These two properties

are also most important requirements for a successful candidate of radar identification
system.

The aspect-independence is attributed to the fact that the synthesis of the E-pulse
is based merely on the natural resonances of a target via the singularity expansion
method. The time domain electric field scattered by the target is divided into an
early-time, forced response period when the excitation waveform is traversing the tar-
get, and a late-time, free oscillation period that exists after the excitation waveform has
passed the target. The late-time response can be decomposed into a sum of damped
sinusoids oscillating at frequencies determined by the geometry and the material of the
target. An E—pulse is a transient, finite duration waveform which annihilates the con-
tribution of a select number of these natural resonances in the late-time response.
This leads to the aspect-independence of the E-pulse. Since the target resonance fre-
quencies are totally independent of the excitation waveform, the E-pulse will eliminate
the desired natural modal contents of the late-time scattered field regardless of the

orientation of the target with respect to the transmitting and receiving antennas.

The noise-insensitivity is attributed to the convolution process involved in the
implementation of the E-pulse scheme, because the integration procedure tends to
smooth the random noises. It is easier to investigate this property via statistical esti-

mation, and it will be discussed in the section 3.4.

This chapter is devoted to investigate the aspect-independence and the noise-
insensitivity of the E-pulse scheme. Section 3.2 provides an introduction to the E-
Pulse technique in both time domain and frequency domain. Section 3.3 concentrates
on the study on the aspect-independence of the E-pulse technique. Section 3.4 deals
With the noise-insensitivity of the E-pulse technique. These two properties of the E-

Pulse are positively proven by extensive experimental results.

36

3.2 The E-pulse Technique

The extinction pulse technique is one of the target discrimination schemes based
on the target’s natural frequencies, that involves illuminating a radar target with an
appropriate time varying waveform and then analyzing the scattered electromagnetic
field. The time-domain scattered field response of a conducting target has been
observed to be composed of an early-time forced period, when the excitation field is
interacting with the scatterer, and a late-time period during which the target experi-
ences a free electromagnetic oscillation. The late-time portion can be decomposed into
a finite sum of damped sinusoids (assuming the incident field waveform has a finite
usable bandwidth), which oscillate at frequencies determined purely by target geometry
and material. The natural resonance behavior of the late-time portion of the scattered
field response can be utilized to provide an aspect-independent scheme for radar target
discrimination. Various details about the concept and the synthesis of E-pulses have
been reported in [16,17,41]. Here, only the basic principles are introduced in this sec-
tion.

An extinction pulse (E-pulse) is defined as a finite duration waveform which,
upon an interaction with a particular target, eliminates a preselected portion of the
target’s natural mode spectrum. By basing the E-pulse synthesis on the natural fre-

quencies, the E-pulse waveform is made aspect-independent.

The target discrimination of the E-pulse technique is implemented by convolving
an E-pulse waveform with the measured late-time scattered field response of a target.
If the scattered field is from the expected target, the convolved response will be com-
posed of an easily interpreted portion of the expected natural mode spectrum. While if
the target is different from the expected, a portion of unexpected spectrum will be

manifested, resulting in an unexpected convolved response.

37

Since the target’s scattered field response is the convolution of the incident field
waveform and the target’s scattered field impulse response, the E-pulse technique can
be implemented using two approaches. The first transmits the E-pulse directly and
uses a frequency domain approach, and the second is a more tractable scheme which
convolves the E-pulse waveform with the measured scattered field waveforrrr and uses

a time domain approach.

3.2.1) The Time Domain Synthesis

Assume that the measured scattered field of a conducting radar target can be
represented during the late-time period as a finite sum of damped sinusoids:
N

r(t) = z a,e°"cos (0),: + ¢n)9 t > T, (3.2.1)

n=l

where a, and 4),, are the aspect dependent amplitude and phase of the n’th mode, T,
describes the beginning of late time, and only N modes are assumed to be excited by
an incident field waveform. Then the convolution of an E-pulse e(t) with the measured

response becomes

00) = €(‘)*"(t)

T.

g e(t’) r(t—t’) dt’

= £31 a,e°"[A,,cos (cunt + 4),.) + anin (am + 0,)1 (3.22)
Where
t > T, = T, + T,
T.
(3:): i ‘3‘") 5°", (2?: 3’5} 4’ (3.2.3)

and T, is the finite duration of e(t).

As defined above, the E-pulse will eliminate a preselected portion of natural mode

 

 

 

 

38

spectrum. Two normally constructed waveforms are introduced here. One is con-

structing e(t) to make C(t) = 0, t > T,. It is required that

A, = B,I = 0, 1 S n .<. N. (3.2.4)
In addition, e(r) can also be constructed so that C(t) is composed of just a single mode.
In this case, the e(t) is termed a "single mode extraction pulse," and the e(t) is syn-

thesized by demanding

A,=B,,=o, 1$nSN,n¢m (3.2.5)

to excite only the m’th natural mode. Thus, requiring

A,=B,=0, lSnSN,n¢m
and
A,” = 0 , B", at 0 (3.2.6)
results in
c(:) = c,(:) = ameWBmsin (wmt + 0,) (3.2.7)

Similarly, requiring

A,==B,,=O, SnSN,n¢m
and
8,,I = 0 , Am rt 0 (3.2.8)
yields
C(t) = ,(t) = ameo'JAmcos (tomt + 0,.) (3.2.9)

The E-pulse synthesized on the basis of (3.2.5) is called a "sin/cos" single mode
extraction pulse since the convolved response C(t) contains both sine and cosine com-
ponents. Similarly, (3.2.6) results in a "sine" and (3.2.8) a "cosine" single mode

extraction pulse.

A physical interpretation of the E-pulse can be facilitated by decomposing the

39

excitation waveform as shown in Fig. 3.1 as
e(t) = e’(:) + e‘(t) (3.2.10)
where J0) is an excitatory component, nonvanishing during 0 s t < Tf, the response to

which is subsequently extinguished by the extinguishing component e‘(t) which exists

during Tfs t S T,. Substituting (3.2.10) into (3.2.3) yields

1, T,
.00., cos (0,,t’ -6..t’ cos (1),!
16060 {sin wn{}q/ = —‘[ef(r’)e {sin (”JFK (3.2.11)
I
The extinguishing component of the E-pulse necessary to eradicate the response due to

a preselected excitatory component can be constructed as an expansion over an

appropriately chosen set of linearly independent basis functions as

 

 

 

 

 

2N
e‘(t) = Z c,,f,,,(t). (3.2.12)
m = 1
It is easy to show that (3.2.11) leads to
21V
2 Mffncm = 47", 1 s I s N (3.2.13)
m = l
where
r, {
cos (0
Mi}; = Ifm(r’)e-°’ {{sin ruff}! (3.2.14.a)
I
T, t,
, cos a),
1:?9 = lame-O" ’{sin wn{}dt’. (3.2.14.b)
The matrix notation is written as
FC, - ”F, q
MC . .
[ 1”] . = — . . (3.2.15)
Mlm . .
1C7”. .F’”.
The solution of the C1. C2~ determines the extinction components in (3.2.12) and

thus the E-pulse.

e(t)

f
e (t) e°(t)

 

 

 

 

Fig. 3.1 Decomposition of an E-pulse Waveform

41

It is instructive to identify two fundamental types of E—pulses. If the Tf> 0 the
forcing vector on the right side of (3.2.15) is nonzero and a solution for e‘(t) exists for
almost any choice of T,. This type of E—pulse has a nonzero excitatory component and
is termed a "forced" E-pulse. In contrast, when Tf= 0, the forcing vector vanishes and

a homogeneous equation results. The solutions for e‘(t) exist only when the deter-
minant of the coefficient matrix vanishes, i.e., when det [M] = 0. These solutions
correspond to discrete values for the E-pulse duration T,, which are determined by
rooting the deterrninantal characteristic equation. Since there is no excitatory com-
ponent, this type of E-pulse is viewed as extinguishing its own excited field and is

called a "natural" E-pulse.

3.2-2) The Frequency Domain Synthesis
As defined, an E—pulse is a time-limited pulse and its Laplace transform can be

Presented by
on T.
E(s) = L [ e(t) 1: ,[ e(t) e‘“ (1!: ! e(t) 6‘“ dt (3.2.16)
To view the E—pulse synthesis in the frequency domain, (3.2.4) can be further

manipulated as:

1', ,
A, — 13,, = j; e(t’) e‘°"[ costo,t’ - jsintunf 1 dt’ = E(sn) = 0 (3.2.17)
TO
A, + )3, = L e(t’) e‘°~’1 cow + jsinmnt’ 1 av = 50;) = 0. (3.2.18)
Two equivalent requirements for E-pulse synthesis are written as:
E(s,)=E(s;)=0 lsnSN (3.2.19)
In addition, a sin/cos m’th mode extraction waveform can be synthesized via
E(s,)=E(s;)=0 ISnSN, natm (3.2.20)

While a sine m’th mode extraction waveform requires

..._.
1...

' -r ‘! _"“’.1"‘-'5..
—. ‘ efi;“mt..

 

.‘i“ l .3". -' ~00» _
J 39-] ~"o'.%“00:J-r" I a. ,

o- o
O...

i.
0.3!

42

E(s,)=E(s;)=0 lSnSN, natm
E( s", ) = —E( s; ) (3.2.21)

and a cosine m’th mode extraction waveform necessitates

E(s,)=E(s;)=0 1_<_nSN, n¢m
E( s,,, ) = E( s; ) (3.2.22)
With the help of (3.2.17), a physical insight into the convolution of (3.2.2) is pro-
vided by

N
e(t) = 2 a,e°"'[A,,cos ((1),: + 11),) + anin (0),,t + 0,)]
n = 1
N O
= 2: a..lE(s,.)I e "‘cos (014+ w.) t > r, (3.2.23)

n=l

where

v. = <1). — tan“( 3,74,, )

It is seen that the frequency domain and time domain requirements for synthesiz-
ing an E—pulse are identical. A significant benefit of using a frequency domain
approach comes from the increased intuition allowed by (3.2.23). When an E-pulse
Waveform is convolved with the measured response of an unexpected target, the ampli-
tudes of the resulting natural mode components are determined by evaluating the mag-
nitllcle of the spectrum of e(t) at the natural frequencies of the target ( a result of the
Cauchy residue theorem ). Thus, the E-pulse spectrum becomes the key tool in

Predicting the success of E-pulse discrimination.

To implement the E-pulse requirements in frequency domain, the E-pulse

waveform is represented mathematically by two components as in (3.2.10),

e(t) = 0’0) + e"(t) (3.2.10)
The forcing component is chosen freely, while the extinction component is determined

by expanding it in a set of basis functions

2N

e‘(t) = Z c,,,f,,,(t). (3.2. 12)

m=l

43

and then employing the appropriate E-pulse conditions from (3.2.20-22). For an E-
pulse designed to extinguish all of the modes of the measured response, using (3.2.20)

results in the matrix equation

 

 

 

 

 

 

’F1<s1)F2<s1> Farm” -E’<S1>W
.61.
. . - 1 - . c2 -
F1010 F2010 Fran.) —Ef(s~) 3 224
F101) F201) Farsi) ' —E’<si) (' ° )
bCZNJ
first) F201) Faust) —Ef(s}i>

where

Fmts) -- [ram
5’0) = [eta]

and the matrix is chosen to be a square. Similar equations can be constructed to

acCommodate the requirements given by (3.2.21) or (3.2.22).

As in time domain synthesis, two types of E-pulse can be identified. When
edit) at 0, the forcing vector on the right hand side of (3.2.24) is nonzero, and solutions
fol‘ E-pulse amplitudes exist for any choice of E-pulse duration, T,, which does not
cEll-lse the matrix to be singular. In contrast, when ef(t) = 0 the matrix equation
b(ECtrmes homogeneous, and solutions for e‘(t) exist only for a specific duration T,,

which are calculated by solving for the zeros of the detenninantal equation. The

fOl‘rrrer type of E-pulse is termed "forced" and the latter "natural".

The frequency domain approach make it possible to visualize an improved E~

Plllse waveform whose spectrum has been shaped to accentuate the response of a

known target. For example, by using damped sinusoids or Fourier cosines as basis
functions in the E-pulse expansion, it is possible to concentrate the energy of the E-
pulse near prechosen frequencies, and to enhance the single mode response of a partic-

ular target.

3-2-3) Calculation of E-pulse Amplitudes

It is possible to choose any number of basis sets to present the E-pulse waveform.
However, a judicious choice must take the following into consideration: simplicity of
representation and calculation, noise averaging quality, possibility of continuous
waveform representation, completeness, and frequency domain shaping. A basis set
consisting of very complicated functions would need to display some important alter-

nate quality to outweight the first two considerations.

The normally used basis sets for representation of E-pulse are polynomial set,
darnping sinusoid set, Fourier cosine basis set, pulse function set, and impulse function
Set- To limit the scope of this section, only the pulse function set is applied as an

example to show the basic steps in the synthesis of an E-pulse.

The most useful basis set, due in most part to its great simplicity, is pulse func-

tion set, which is composed of sub-sectional pulse functions as

g(t—(m—1)A) (m—1)A .<. t 5 mA
fm(t) =

(3.2.25)
elsewhere
Where g(t) is some arbitrary function and A is the pulse width. Then
7'.
Fm(s) = J g(t—[m—1]A)e"‘" dt
-.-. Fl(s)e’Ae-*"‘A (3.2.26)

arid the matrix equation (3.2.24) can be written for the case of the natural E-pulse as

45

 

 

 

 

1 z1 (202 (ZOZN-l
.61.
. c2
1 ZN (ZN)2 ° ' ' (Zn/)ZN-l .
1 z: (2'?)2 (20%"l - =0 (32°27)
hem-t
1 27v (2702 (2302”-1
where
z, = 6"“ (3.2.28)

Equation (3.2.27) is homogeneous, and thus has solutions only when the determinant

Of the matrix is zero. As the determinant is of the Vanderrnond type, the condition for

a singular matrix can be calculated as

A = If?" , p = 1,2,.... 1 s k _<_N. (3.2.29)
1:

Thus, the duration of the E-pulse depends only on the imaginary part of one of the
natural frequencies. With A determined, the basis function amplitudes can be solved

uSing Cramer’s rule and the theory of determinants as

Cm = (-1)mP(2Jv-1)-(m-r) (32.30)

WherePH- is the sum of the products It — r' at a time, without repetitions, of the quanti-
ties 21, 2;, 22, ..., 23,. For example
i t t I! I! t
P5_2 = 212122 '1' 212122 '1’ 212123 '1' 212222 + 212223 '1' 212223
+ 2.1222; + 21252:; 4' 212323 ‘1' 222323
It is worth noting that g(t) does n0t appear in the analysis, and the resulting pulse
amplitudes are independent of the individual pulse shapes. However, when discrim-

inating between different targets, g(t) manifests itself through the term F,(s).

46

As is seen that the crucial requirement for synthesis of an E—pulse is the
knowledge of the natural modes of a target. In the past decades, many researchers
have devoted to extract theoretically the natural modes of various conducting targets.
But the number of theoretically solvable geometries is very limited, thus, the numeri-

cal techniques have to be applied to extract the natural frequencies from a measured
late-time response. Unfortunately, the numerical extraction of natural frequencies from
a late-time target response is very sensitive to noise. Some well developed algorithms
need to be selected to extract the natural frequencies from the measurements in a
laboratory where the late-time pulse responses of the scaled target models are meas-
ured in a signal S/N ratio controlled environment. Discrinrinant signals are then syn-
thesized and stored in a computer as data files, and subsequently convolved numeri-
cally with actual transient target radar returns. Since the convolution operation is
numerically well behaved (a smoothing integral operation), the S/N ratio required for

Practical radar return is less restricted.

3-3 Aspect-Independence of the E-pulse Technique

One of the most important merits of the E-pulse technique is attributed to its
asFeet-independence. This section will study the aspect-independence of the E-pulse
technique for target discrimination. It is shown that the synthesis of an E-pulse
Waveform is independent of the external excitation (aspect-dependent). Various air-
Plane scale models are used in experiments. The aspect-independence is convincingly

demonstrated by extensive experimental results.

3.3.1) Aspect-independence of E-pulse Synthesis

The synthesis of an E-pulse requires merely the knowledge of the natural frequen-

cies of the given target. In fact, the natural frequencies of a target depend only on its

47

geometry and electrical properties, but not on the orientation of the external forcing
fields. Therefore the determination of an E-pulse waveform is definitely independent

of the aspect angles in which the excitation field is applied.

It is presumed in the last section that the response of the measured time domain
scattered field of any conducting radar target can be represented in terms of a finite
sum of damped sinusoids as given in (3.2.1), where the amplitudes and phases are all
excitation-dependent (aspect-dependent) parameters, but the damping coefficients and
radian frequencies are aspect-independent inherent parameters of targets. The convolu-
tion of an E-pulse waveform with the measured radar return is represented by (3.2.2).
The result of convolution is also a sum of damping sinusoids. To understand the

aspect-independence of the E-pulse waveform, (3.2.2) is duplicated here

6‘0) = e(t)*r(t)

T.

j e(t’) r(t—t') dt’

N
z a,e°"[A,cos ((11,: + 1),) + anin (0,: + (1,)1. (3.2.2)

nal

Where the new amplitudes of damped sinusoids are contributed by a,’s and A,’s or
3:38. It was specified that the contribution of a,’s is from the incident field, while

An’s and B,’s relate to the E-pulse by (3.2.3)

7
A, ' , . cos 0),,t’
{8n}: 1! e(t) e—O’“ {sin 03’1“} dt’ (3.2.3)
The unknown E-pulse waveform e(t) is evaluated via the preassigned A,’s and B,’s.

But A,’s and B,’s are preassigned to be 0 or 1 only. Thus, the E-pulse waveform is

Completely determined once the natural frequency spectrum is specified.

It is seen that the amplitudes and phases of the scattered field, which are aspect

dependent, are proportional to the amplitudes of the convolved outputs. This may be

48

Misunderstood that the convolved result is aspect dependent since its amplitudes are
related to the aspect dependent parameters. The key point to observe the aspect
independency of the E-pulse scheme is the pre-selection of the natural modes to be
excited. The E-pulse is designed to annul certain natural resonances of one target.
The fate of any natural resonance is either excited or not excited. The targets are

identified based only on the existance of the pre-selected natural resonances, not on the
strengths of those resonances. For instance, an E-pulse is to extinguish all natural
modes of a target, the zero amplitude will be expected in the late-time period of the
convolution with any radar return of the anticipated target, but the significant nonzero
amplitude will be seen in the late-time period if it is convolved with radar response

from a different target.

3 - 3 - 2) Experimental Demonstration

It is a common knowledge that any experimentally measured response of a target

is always contaminated by the environmental and system noise. Because of these
nDiscs we can not expect a perfect zero output, or a pure natural oscillation in the
late-time convolved output of an E-pulse with the response from a right target. Conse-
quently the late-time convolved output is somewhat influenced by the amplitude and
the phase of the incident field. Effects of the noise on the E-pulse technique will be
investigated more carefully in the next section. Fortunately, the contribution of the

nOise is not to any particular natural mode, rather to a frequency band. As long as the

Signal to noise ratio is high enough, the discrimination of the target is not affected.

Thus, it is essential to be able to verify experimentally the aspect independence of
the E-pulse technique. A extensive experimental study has been conducted to measure
the near field pulse responses of various airplane models, on a ground plane, time-

domain scattering range at Michigan State University. The aspect-independence of the

49

E~pulse technique, which utilizes both a waveform designed to eliminate all the natural
modes of a target and a set of waveforms designed to extract various individual target

resonances, has been positively demonstrated by the experimental results.

When using E-pulse waveforms it is usually difficult to tell which E-pulse
waveform discriminates better than others. Before we proceed to show some typical
experimental results with different targets, it is appropriate here to specify a few terms

which are useful in quantifying of the discrimination by an E-pulse waveform.

First, the late-time portion in a target’s scattered response is defined as the free
oscillation period when the excitation waveform has passed the target. If the early
time period of a measured response is determined by

T,,- = Tw + 27‘,
Where Tc is the maximal one way transit time of the target and TW is the duration of
the incident waveform, the late-time will start from T5. For the early-time period of a
convolved output, T,, the finite duration of the E-pulse waveform must be added to
this TE to satisfy the requirements of the convolution. The obvious point here is that

Tc must always be taken to be finite, or else there would be no late-time period in the

cOnvolution waveform.

Second, the discrimination ratio is introduced and defined as the ratio of the
energy contained in the late-time portion of the convolved waveform, to the total
el'lergy contained in the whole convolved waveform. As is well known that a transient
Scattered response of a radar target contains most of its energy in the early time with
olily about few percents of energy in the late-time period. Thus, the discrimination
ratios of targets are usually within few percents, but there is a significant difference

between the discrimination ratios of a right target and a wrong target.

50

3.3.3) Examples of Two Different Models

Fig. 3.2 depicts two aircraft scale models with similar dimensions. One model
(shown in Fig. 3.2.a) is a scaled aluminum aircraft model of Boeing 707 which is
noted as B-707, and the other (shown in Fig. 3.2.b) is a scaled aluminum aircraft

model of McDonnell Douglas F—18 noted as F-l8. Note that the two models are con-
structed to be of similar dimensions eventhough they are vastly different in actual

sizes-

The scattered responses of these two models are measured at different aspect
angles. A continuation method or a hybrid E-pulse method is applied to extract the
natural modes from the measured responses. It is observed that one particular mode of
a model is excited at some aspect angles, but may not be excited at the other aspect
anglees. Also worth noting is that the extracted natural frequencies of one target from
measurements in different aspect angles are not completely consistent because of the
experimental error. In our study, the extracted natural frequencies of one model from
the measurements at various measured aspect angles are averaged over using some
weighting rules. The simplest rule is weighting equally in averaging natural models.

The final extracted natural frequencies are also listed in Fig. 3.2.

Based on the extracted natural modes, the natural E-pulse waveforms are syn-
tl-letsiZed using rectangular pulse basis functions with the minimum pulse duration. The
E‘la‘llse for B-707 is constructed to eliminate the first five natural modes of B-707
I110(1e1 and the E-pulse for F-18 is to eliminate first three natural modes of F-18 model.
me E-pulse waveforms are normalized with respect to one of the rectangular basis

Du
lses. Fig. 3.3 shows the natural E-pulse waveforms for both B-707 and F-18 aircraft
In Clels.

The synthesized E-pulse waveforms are then stored in a computer, and convolved

51

THE FREQUENCIES USED FOR E-PULSE SYNTHESIS

B707 Model
—o.273 j2.58 (a) 1 8 cm
-o.441 j6.3l [7 j
‘0' 150 j9°95 / ///////////';'///////// U
—o.430 jll.86 -——33 cm———
—o-337 j12.31
F18 Model (b)
P ll? cm

-0-418 j3.36 / L
-0- 440 j9-50 / //”////77////////////V/4/
-0-612 j11.75 33.5 °"' '1

F-

10:3- 3.2 Geometry of Two Airplane Models. a) Boeing 707 Model termed B-707;

 

 

 

 

 

 

 

 

CDonnell Douglas F—18 Model termed F-18

52

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 __
E-pulse for F18
1 _ . ....... .
Relative .
1'1de 0—_- ...... ;.--__.__._.-_.-. ...................... 1L---..
"""" E-pulse for B707
_1 §
—2 a, ' ’
r\
x l n
0 1 2 3
Time in ns

Fig. 3 -
3 .l‘lre E-pulse Waveforms Synthesized for B707 model and F-18 Model

 

53

with the real-time scattered responses later. Figs. 3.4.a-h show the scattered responses,

of the B-707 model at aspect angle of 45°, 90°, 1350 and 180°, and their convolved
results with the E-pulse waveform for B-707 model. Though the late-time portions of
these convolved responses are not identically zero, they are very small. The E-pulse is
seen to successfully eliminate the natural modes. In contrast, Figs. 3.5.a-h show the
scattered responses of the F-18 model measured at different aspect angles and their
convolved results with the E-pulse waveform of B-707 model. Now the late-time por-
tions of the convolved outputs are seen to be quite large in magnitude. Discrimination
between these two targets can thus be accomplished convincingly by comparing the

late-time portions of each of the convolved responses.

It can be observed that the discrimination ratios of the right target are quite
different from that of the wrong target. For these two targets, the discrimination ratio
for the right target is less than 3% for all aspect angles, while it is about 10% for the

Wrong target at all aspect angles.

Figs. 3.6.a-h and 3.7.a—h show the convolved results of the scattered waveforms
from these two airplane models with the E-pulse waveform synthesized for the F—18
airci‘aft model. Comparing the late-time portions of the convolved results, one can see
a Significant difference in discrimination ratios between the two models. Again the
discrimination ratio of the right target is lower than 3% (except for the case of 180°
aspect angle when that is 3.6%), and that for the wrong target is near 10% (except for

th
e Case of 135° aspect angle when that is 4.6%).

3 -
3 ‘4) Examples of Multiple Models

The results of the preceding section proved convincingly the feasibility of the B-
Pulse

any

scheme for discriminating two aircraft models. But our motivation is to identify

particular target among many other targets. This section will give the

54

 

Aspect angle 45°

20—

Relative _
Amp litude 7

 

 

 

 

—20 a :
;<——— late-time
‘ L L 1 1 (a)
-40
O 2 4 6 8 10
Time in us
20 — discrimin. ratio = 1.7%

    

 

 

 

5 e
Relative
Plitude --------
F— late-time
‘10 -( I
(b)
.25 L 1 I 1
0 2 4 6 8 10

Time in us

Its Fig. 3.4 The Scattered Waveform of B-707 Model at Aspect Angle of 45° and
ter Q‘Drrvolved Response with the E-pulse Waveform of B-707 model a) The Scat-
Waveform; b) The Convolved Response

55

 

Aspect angle 90°

 

20—

 

Relative

 

 

 

 

Amplitude O -
.20 _
1 (o)
1 1 1
—40 0 2 4 6 8 10
Time in us
20 _ discrimin. ratio = 2.4%

Relative
Amplitude

—10 ~

 

(d)

 

 

—
h

.2 1 1
5 O 2 4 6 8 10

Time in ms

 

I t Fig. 3.4 The Scattered Waveform of B-707 Model at Aspect Angle of 90° and
t s Convolved Response with the E-pulse Waveform of B-707 model c) The Scat-
ere<1 Waveform; d) The Convolved Response

56

 

Aspect angle 135°

20—

 

 

 

 

 

     

 

 

 

Relative 0 _
Amplitude
.20 _
1 (e)
1 1 1
40 O 2 4 6 8 10
Time in ns
20 - discrimin. ratio = 2.1%
5 ..
It'i‘rlative ___________
plitudc """""""""
§<— late-time
-10 - :
(f)
.2 1 1 1 1
5 0 2 4 6 8 10
Time in us
I t Fig. 3.4 The Scattered Waveform of B-707 Model at Aspect Angle of 135° and
t 8' Convolved Response with the E-pulse Waveform of B-707 Model e) The Scat-

ed Waveform; t) The Convolved Response

 

57

 

Aspect angle 180°

20—

Relative 0 _

 

 

 

 

 

    

 

 

 

Amplitude
—20 _ <—— late-time
1 (g)
I l l
‘40 O 2 4 6 8 10
Time in us
20 - discrimin. ratio = 1.3%
5 —J
Etel ative __________
Plitude
:<— late-time
-lO — :
(h)
.25 1 1 1 I
O 2 4 6 8 10

Time in ns

I t Fig. 3.4 The Scattered Waveform of B-707 Model at Aspect Angle of 180° and
tes Convolved Response with the E-pulse Waveform of B-707 Model g) The Scat-
recl Waveform; h) The Convolved Response

 

58

 

Aspect angle 45°

20—

 

Relative 0 A _
Amplitude

 

-20 ..

 

 

 

 

 

 

20 .4 discrimin. ratio = 12.7%

Plimdc

-10 ...

(b)

 

 

 

Time in ns

QQ Fig. 3.5 The Scattered Waveform of F~18 Model at Aspect Angle of 45° and Its
t1Volved Response with the E-pulse Waveform of B-707 model a) The Scattered
Veform; b) The Convolved Response

59

 

20—

Relative
Amplitude

 

Aspect angle 90°

 

 

 

 

10

 

 

 

 

.20 _
1 (C)
‘ 1 1 1
_40 O 2 4 6 8
Time in us
20 — discrimin. ratio = 12.8%
5 -+
RelatiVe
Plitude " " '“ "' """" "' "' "" " """
—lO - E .
:+—— late-time
(d)
.25 1 1 1 1
O 2 4 6 8 10
Time in ns

C

E1"'itform; d) The Convolved Response

Q Fig. 3.5 The Scattered Waveform of F-l8 Model at Aspect Angle of 90° and Its
V'V “Volved Response with the E—pulse Waveform of 8—707 model c) The Scattered

 

4O " Aspect angle 135°

20—

Relative
Amplitude 0 —

 

 

 

 

 

 

 

 

-20 - :
§<—— late-time
1 1 1 L (e)
—40
O 2 4 6 8 10
Time in ns
20 - discrimin. ratio = 10.8%
5 _
Relati‘Ve
plitude --------------- T -------------- - q
-10 fl : .
3— late-time
(f)
-25 l l 1 1
2 4 6 8 10
Time in ns

Its Fig. 3.5 The Scattered Waveform of F-18 Model at Aspect Angle of 135° and
te QOnvolved Response with the E-pulse Waveform of B-707 Model e) The Scat-
rfid Waveform; f) The Convolved Response

-61

 

40 Aspect angle 180°

20—

 

Relative
Mplitude 0 —

 

 

 

 

 

 

 

 

-20 — g
;<— late-time

1 1 1 1 (g)

40 0 2 4 6 8 10
Time in us
20 _ discrimin. ratio = 11.5%
5 -1
Plitudc """""""""""""""""""
—10 — ; .
34— late-time
(h)

-25 1 1 1 1

2 4 6 8 10

Time in us

Its Fig. 3.5 The Scattered Waveform of F-18 Model at Aspect Angle of 180° and
ter QOrlvolved Response with the E-pulse Waveform of B-707 Model g) The Scat-
Waveform; h) The Convolved Response

62

 

Aspect angle 45°

20—

 

 

Relative 0 A _

 

 

 

 

 

 

 

 

 

 

Amplitude
-20 - ;
§¢—— late-time
1 L 1 (a)
1
_40 0 2 4 6 8 10
Time in us
20
discrimin. ratio = 2.01%
10 —
RelativC
Plimdco' ' ""' " ' ' -- """""" ‘
—10 —1 <—— late-time
(b)
_20 l l l l
O 2 4 6 8 10
Time in us

Q0 Fig. 3.6 The Scattered Waveform of F-18 Model at Aspect Angle of 45° and Its
W nVOlved Response with the E-pulse Waveform of F-18 model a) The Scattered
Véform; b) The Convolved Response

63

 

Aspect angle 90°

 

20 A
Relative
Arnplitude

db

.20 _

 

 

 

 

2O

 

discrimin. ratio = 2.02%

10-

Relativc
Amplitude 0 q

 

 

-10 d

f<~—— late-time

 

(d)

 

 

_20 1 1 1 1
O 2 4 6 8 10

Time in us

 

Q9 Fig. 3.6 The Scattered Waveform of F-l8 Model at Aspect Angle of 90° and Its
W t“Iolved Response with the E-pulse Waveform of F—18 model c) The Scattered
a~‘Veform; d) The Convolved Response

 

40 Aspect angle 135°

20-

 

Relative
Arnplitude O -

—20 a

 

 

 

 

 

discrimin. ratio = 1.6%

10-1

 

 

 

 

 

 

Relative O
Plitude ‘
-10 _
.20 1 1 1 1
2 4 6 8 10
Time in ns

Fig. 3.6 The Scattered Waveform of F-18 Model at Aspect Angle of 135° and
nvolved Response with the E-pulse Waveform of F-l8 Model e) The Scattered
VCform; t) The Convolved Response

65

 

40 Aspect angle 180°

20—

 

Relative
Amplitude O —

 

 

 

 

 

 

 

 

 

 

 

 

—20 -— .
51—— late-time
1 4 1 1 (g)
‘40 0 2 4 6 8 10
Time in ns
20
discrimin. ratio = 3.6%
10 -
Relative 0
Plitude ‘ ' '
'10 - :4— late-time
(h)
_ 4 1 1 1
20 2 4 6 8 10

Time in as

Its Fig. 3.6 The Scattered Waveform of F-18 Model at Aspect Angle of 180° and
ter COnvolved Response with the E-pulse Waveform of F-18 Model g) The Scat-
Qd Waveform; h) The Convolved Response

 

66

 

 

 

 

 

 

 

 

4o
Aspect angle 45°
20 -
Relative O _ _ _____________________ _ - - .
Amplitude '
—20 - :
‘— late-time
1 1 (a)
1 1
‘40 0 2 4 6 8 10
Time in ns
20
discrimin. ratio = 8.6%
10 q
Relative O- _ _______ _ _ ____ ___" ___--- ____- ..
Amplitude ' ..
-10 _ :<—-— late-time
(b)
.20 1 1 1 1
O 2 4 6 8 10

Time in us

Fig. 3.7 The Scattered Waveform of B-707 Model at Aspect Angle of 45° and
Its Convolved Response with the E-pulse Waveform of F- 18 model a) The Scat-
tered Waveform; b) The Convolved Response

67

 

Aspect angle 90°

20-

 

 

 

 

 

 

 

 

 

 

Relative O __
Amplitude
.20 __ :
«_— late-time
1 1 (o)
1 L
-40 0 2 4 6 8 10
Time in ns
20
discrimin. ratio = 13.4%
10 a
Relative O- __ __ __ ___- __-___- --
Amplitude

 

      

—10 — f<——— late-time

(d)

 

 

 

.20 1 1 L 1
0 2 4 6

Time in ns

Fig. 3.7 The Scattered Waveform of B-707 Model at Aspect Angle of 90° and
Its Convolved Response with the E—pulse Waveform of F—18 model c) The Scattered
Waveform; d) The Convolved Response

68

 

Aspect angle 135°

20 --

Relative 0 _
Amplitude

.20 ..

(e)

 

 

 

Time in us

20

 

discrimin. ratio = 4.6%

10—

Relative
Amplitude

 

 

    

 

-10 - -<—— late-time

 

(f)

 

 

1 1
2 4 6 8 10

Time in ns

_

 

Fig. 3.7 The Scattered Waveform of B-707 Model at Aspect Angle of 135° and
Its Convolved Response with the E-pulse Waveform of F-18 Model e) The Scattered
Waveform; t) The Convolved Response

69

 

20—

Relative

Amplitude 0 ‘

-20 —

 

Aspect angle 180°

3.1—_— late-time

(g)

 

 

20

1 l L
4 6

Time in ns

10

 

10—

Relative 0 _
Amplitude

—1O —

 

discrimin. ratio = 8.8%

;4—- late-time

(h)

 

 

Time in ns

10

Fig. 3.7 The Scattered Waveform of B-707 Model at Aspect Angle of 180° and
Its Convolved Response with the E-pulse Waveform of F-18 Model g) The Scat-
tered Waveform; h) The Convolved Response

70

experimental with five different airplane models.

Two of the five aircraft models are B-707 and F—18 aircraft models which have
been shown in the proceding section. The dimensions and shapes of the other three
models are depicted in Fig. 3.8. One is a bird-like model called T-15 which is arbi-
trarily constructed and named. Other two models depicted in Fig. 3.8 are enlarged 1.5
times aircraft models of the existing B-707 and F-18. We call them BB-707 and BF-
18 to differentiate them from the other models of B-707 and F-l8.

The scattered responses from the five aircraft models are measured at different
aspect angles. Then the measured waveforms are convolved with the E-pulse
waveform of any one model. Figs. 3.9.a - 3.13.h show the convolved results of the
scattered waveforms of different models with the E-pulse synthesized for B-707 model.
Figs. 3.9.a-h show the scattered field waveforms from the B-707 model measured at
different aspect angles and their convolved results with the B-707 E-pulse waveform.
Very small amplitudes in the late-time portions of the convolved results indicate the
right target. Discriminantion ratios of less than one percent for all aspect angles

definitely support the judgement.

Figs. 3.10.a—h show the scattered field waveforms of the F-18 model measured at
various aspect angles and their convolved results between the scattered waveforms and
the same B-707 E-pulse waveform. Figs. 3.11.a-h show the similar results with the
T-15 model, and Figs. 3.12.a-h and Figs. 3.13.a-h show the results for the case of the
big B-707 and big F-18 models. Relatively large amplitudes in the late-time portions
of the convolved results and the much larger discrimination ratios compared with the

case of the right target are obvious indications of the wrong targets.

71

(a)
19 cm

 

// /////////3 /.~'//‘//// '7’
-——31 cut—fl

 

 

 

(b) 32. 8 cm

 

/".”,"./,: .// , I //////"' / /// /
“ 64.5 cm

 

 

(C) T
28.2 cm

I

 

/////'/’///////////x/“71//}"//// //, // W

l————— 72.4 cm fi

 

Fig. 3.8 Geometry of Three Airplane Models. a) Simplified Model T-15; b) Big B-
707 Model termed BB-707; c) Big F-18 Model termed BF-18

72

 

20 -

 

Relative

Amplitude O -

 

(a)

 

 

 

é
AL—

Time in us

30

 

discrimin. ratio = 0.47%

15—

Relative 0
Amplitude _

 

§¢—— late-time

-15—

 

 

 

(b)
__ L 1 1
3O 0 2 4 6 8 10

Time in ns

Fig. 3.9 The Scattered Waveform of B-707 Model at Aspect Angle of 0° and Its
Convolved Response with the E-pulse Waveform of B-707 model a) The Scattered
Waveform; b) The Convolved Response

73

 

Aspect angle 45°

20-

 

 

Relative

Amplitude O '—

 

(C)

 

 

 

25

 

discrimin. ratio = 0.86%

15—

5 .-

Relative
Amplitude
.5 _

-15—

 

 

 

 

Time in us

Fig. 3.9 The Scattered Waveform of B-707 Model at Aspect Angle of 45° and
Its Convolved Response with the E-pulse Waveform of B-707 model c) The Scat-
tered Waveform; d) The Convolved Response

74

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

30
Aspect angle 90°
15 -
Relative 0 ‘Jh _ _ _ .
Amplitude
-15 ..
_. 1 1 1 1
30 0 2 4 6 8 10
Time in us
20
discrimin. ratio = 0.6%
10 .1
Relative
Amplitude
—10 _,
. (f)
.20 1 1 1 1
2 4 6 8 10

Time in us

Fig. 3.9 The Scattered Waveform of B-707 Model at Aspect Angle of 90° and
Its Convolved Response with the E—pulse Waveform of B-707 Model e) The Scat-
tered Waveform; t) The Convolved Response

75

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4o
Aspect angle 180°
20 4
Relative 0 __
Amplitude
—20 -« :
34-— late-time
‘ (g)
.40 l L l I
O 2 4 6 8 10
Time in us
30
discrimin. ratio = 0.8%
15 -
Relative 0
Amplitude _,
§«—— late-time
-15 _ :
(h)
-30 1 1 1 1
O 2 4 6 8 10

Time in us

Fig. 3.9 The Scattered Waveform of B-707 Model at Aspect Angle of 180° and
Its Convolved Response with the E-pulse Waveform of B-707 Model g) The Scat-
tered Waveform; h) The Convolved Response

76

 

Aspect angle 0°

20-

 

 

Relative

Amplitude 0 - -

--20 f

 

 

 

 

 

 

 

 

 

1 1 L l
40 0 2 4 6 8 10
Timein ns
30
discrimin.ratio=3.5%
15—
Relative O __ __ ___ _ ___ ___ ___ _- _- _-- .
Amplitude " " " ' " =
—15q i..— late-time
(b)
-30 1 1 1 1
O 2 4 6 8 10
Time in us

Fig. 3.10 The Scattered Waveform of F-18 Model at Aspect Angle of 0° and Its
Convolved Response with the E-pulse Waveform of B-707 model a) The Scattered
Waveform; b) The Convolved Response

77

 

Aspect angle 45°

20-

 

 

Relative 0 _
Amplitude

 

 

-20 —

 

 

 

 

30

 

discrimin. ratio = 3.8%

15—

 

 

 

 

 

 

Relative O
Amplitude ‘
_15 ._
(d)
-30 1 1 1 1
0 2 4 6 8 10

Time in us

Fig. 3.10 The Scattered Waveform of F—l8 Model at Aspect Angle of 45° and
Its Convolved Response with the E-pulse Waveform of B-707 model c) The Scat-
tered Waveform; d) The Convolved Response

78

 

 

 

 

 

 

 

 

     
   

 

 

 

 

 

40 ‘ Aspect angle 90°
20 —
Relative
Amplitude 0 _
.20 _
1 1 I 1
’40 o 2 4 6 8 10
Time in ns
30
discrimin. ratio = 4%
15 —
Relative 0 _ ___ ___ ___ __
Amplitude _-
;1-— late-time
-15 -— :
(f)
-30 1 1 1 L
0 2 ~ 4 6 8 10
Time in us

Fig. 3.10 The Scattered Waveform of F-18 Model at Aspect Angle of 90° and
Its Convolved Response with the E-pulse Waveform of B-707 Model e) The Scat-
tered Waveform; f) The Convolved Response

79

 

Aspect angle 180°

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4°“ l
20-
Relative
Amplitude O _
-20— H
late-time (8)
1 g l L
‘40 0 2 4 6 8 10
Time in ns
discrimin. ratio = 5%
25 -
10 -
Relative _ __ __ _ ___ __ __ __ _ -_ -_ _- ---
Amplitudes " ‘ " ' '
§—— late-time
.20 ._
(h)
_35 L i l l
O 2 4 6 8 10

Time in ns

Fig. 3.10 The Scattered Waveform of F-18 Model at Aspect Angle of 180° and
Its Convolved Response with the E-pulse Waveform of B-707 Model g) The Scat—
tered Waveform; h) The Convolved Response

80

 

 

 

 

 

 

 

 

 

 

 

 

 

 

40
Aspect angle 0°
20 - 1
Relative 0 J - _ _
Amplitude '1
.20 ..
.40 l 1 l I
0 2 4 6 8 10
Time in ns
30 4 discrimin. ratio = 6%
15 -—
Relative
Amplitude 0.. - - ...................... - --_- ---
_15 _ :<——— late-time
(b)
.30 1 1 1 1
0 2 4 6 8 10
Time in us

Fig. 3.11 The Scattered Waveform of T-15 Model at Aspect Angle of 0° and Its
Convolved Response with the E-pulse Waveform of B-707 model a) The Scattered
Waveform; b) The Convolved Response

81

30

 

Aspect angle 45°
20 —

10—

Relative 0 _
Amplitude

 

_10 _.

 

-20 ..

late-time

 

 

 

-30 1 1 1 1
0 2 4 6 8 10

Time in ns

 

3O

 

discrimin. ratio = 6.8%
20 -

10 -1

Relative
Amplitude
0 - ----------------------------------- 1

-10- a .
g——— late-time

 

 

 

(d)
.20 1 n 1 1
0 4 6 8 10

Time in ns

Fig. 3.11 The Scattered Waveform of T-15 Model at Aspect Angle of 45° and
Its Convolved Response with the E-pulse Waveform of B-707 model c) The Scat-
tered Waveform; d) The Convolved Response

82

 

 

 

 

 

 

 

 

 

 

10

 

 

 

 

 

 

 

Aspect angle 90°
30 .1
Relative _
Amplitude
_10 _.
§¢—— late-time
' (c)
- 1 1 1
3O 4 6 8
Time in as
20
discrimin. ratio = 9%
10 _
Relative 0
Amplitude ‘
-10-« :<—— late-time
(f)
-20 1 1 1
4 6 8
Time in ms

10

Fig. 3.11 The Scattered Waveform of T-15 Model at Aspect Angle of 90° and
Its Convolved Response with the E-pulse Waveform of B-707 Model e) The Scat-
tered Waveform; f) The Convolved Response

83

 

 

     

 

 

 

 

 

 

 

25 a Aspect angle 180°
Relative
Amplitude
_5 _
. late-time (g)
- 1 1 1 1
20 0 2 4 6 8 10
Time in us
15 a discrimin. ratio = 9%
Relative ‘
Amplitude
_5 _ .
¢— late-time
(h)
-15 1 1 1 1
0 2 4 6 8 10

Time in us

Fig. 3.11 The Scattered Waveform of T-15 Model at Aspect Angle of 180° and
Its Convolved Response with the E-pulse Waveform of B-707 Model g) The Scat-
tered Waveform; h) The Convolved Response

84

 

 

 

 

 

 

 

 

 

 

    

 

 

 

 

 

40 _ Aspect angle 0°
20 -
Relative
Amplitude 0 __ _ _ _
-20 1 {V §<——— late-time
(a)
.40 1 1 1 1
0 2 4 6 8 10
Time in ns
discrimin. ratio = 6.1%
25 .4
10 -
Relative ........................
Amplitude_5 A
u—— late-time
-20 —l
(b)
_3 5 1 1 1 1
0 2 4 6 8 10

Time in us

Fig. 3.12 The Scattered Waveform of BB-707 Model at Aspect Angle of 0° and
Its Convolved Response with the E-pulse Waveform of B-707 model a) The Scat-
tered Waveform; b) The Convolved Response

85

 

 

 

 

 

 

 

 

 

 

 

 

o
30 _ Aspect angle 45
10 -—
Relative ' '
§«—— late-time
-30 -
U (C)
_ L 1 1 1
50 2 4 6 8 10
Time in us
25 1 discrimin. ratio = 6%
10 -
Relative ------------------ - ................. .
Amplitude—5 ..
<—-— late-time
—20 - '
(d)
-35 1 l 1 1
0 2 4 6 8 10

Time in us

Fig. 3.12 The Scattered Waveform of BB-707 Model at Aspect Angle of 45°
and Its Convolved Response with the E-pulse Waveform of B-707 model c) The
Scattered Waveform; d) The Convolved Response

86

 

 

 

 

 

 

 

 

 

30
Aspect angle 90°
15 -
Relative 0— __‘___ ___________________________
Amplitude
-15 -
§¢— late-time
: (e)
_ o 1 1 1 1
3 0 2 4 6 8 10
Time in ns
discrimin. ratio = 7.8%
20 -
5 .1
Relative _____ ___ __ _ ___ -__ _ -___ ......
Amplitude
-10 —
§<—— late-time
1 1 1 1 (f)
-25
0 2 4 6 8 10

Time in us

Fig. 3.12 The Scattered Waveform of 83-707 Model at Aspect Angle of 90°
and Its Convolved Response with the E-pulse Waveform of B-707 Model e) The
Scattered Waveform; f) The Convolved Response

87

 

 

 

 

 

 

 

 

30
Aspect angle 180°
15 -
Relative 0 _ ________________________________
Amplitude ‘
_15 _.
late-time
(g)
__30 L L L I
0 2 4 6 8 10
Time in ns
discrimin. ratio = 8%
20 -4
5 _
Relative , _____________________________
Amplitude - 1
-10 _. 3 .
,‘——— 1ate-t1me
(h)
_25 l J 1 l
2 4 6 8 10

Time in us

Fig. 3.12 The Scattered Waveform of BB-707 Model at Aspect Angle of 180°
and Its Convolved Response with the E-pulse Waveform of B-707 Model g) The
Scattered Waveform; h) The Convolved Response

30

88

 

154

Relative
Amplitude

—15 1

o-..

 

 

 

 

Aspect angle 0°

 

(a)

 

 

20

10

 

10~

Relative

Amplitude O .-

—10 ~

 

 

discrimin. ratio = 7.5%

 

 

 

Time in ns

10

Fig. 3.13 The Scattered Waveform of BF—18 Model at Aspect Angle of 0° and

Its Convolved Response with the E-pulse Waveform of B-707 model

tered Waveform; b) The Convolved Response

a) The Scat-

89

 

Aspect angle 45°

25—

10—

Relative
Amplitude 5

 

—1

 

-20 J

 

(C)

 

 

 

Timein ns

30

 

discrimin. ratio = 5.1%

15—

Relative

Amplitudeo'" ----_- -- ...............

"'15 “ #— late-time

(d)

 

 

 

Time in us

Fig. 3.13 The Scattered Waveform of BF-18 Model at Aspect Angle of 45° and
Its Convolved Response with the E-pulse Waveform of B-707 model c) The Scat-
tered Waveform; d) The Convolved Response

9O

 

 

 

 

 

 

 

 

 

 

 

30
Aspect angle 90°
15 -
Relative "pm --., -- _ --- --.. "_ --., - -
Amplitude
—15 _ U .
#— late-time
' (e)
_ 1 1 I L
30 ° 2 4 6 8 10
Time in ns
20 m
discrimin. ratio = 7.7%
10 -
Relative
Ammitude O J ---------------------------- s
_10 7 <— late-time
(f)
—20 l 1 1 1
2 4 6 8 10

Time in us

Fig. 3.13 The Scattered Waveform of BF-18 Model at Aspect Angle of 90° and
Its Convolved Response with the E—pulse Waveform of B-707 Model e) The Scat-
tered Waveform; f) The Convolved Response

91

 

 

 

 

 

 

 

 

 

 

20
Aspect angle 180°
10 4
Relative 0 _ _ _
Amplitude
_10 ..
(g)
-20 ' ' ‘ '
2 4 6 8 10
Time in ns
20
discrimin. ratio = 7%
10 -
Relative 0 _________________________________
Amplitude _ -----
-10 _1 :<——— late-time
(h)
_20 l I I I
0 2 4 6 8 10
Time in us

Fig. 3.13 The Scattered Waveform of BF-l8 Model at Aspect Angle of 180° and
Its Convolved Response with the E-pulse Waveform of B-707 Model g) The Scat-
tered Waveform; h) The Convolved Response

92

3.4 Noise Characteristics of the E-pulse Technique

Another important characteristic of the E-pulse technique is its noise-insensitivity.
This section will explore all aspects concerning noise-insensitivity of the E-pulse tech-
nique for target discrimination. We will first look for a fairly good error estimate on
the extraction of the natural modes from a measured target response. Then the noise
analysis is performed on the convolution procedure. The last subsection shows the

experimental proof of the noise insensitivity of the E-pulse technique.

3.4.1) Error Estimation on the Natural Mode Extraction

The implementation of the E~pulse target discrimination scheme, which is based
on natural resonances of the target, needs accurate information about the natural fre-
quencies of a target. Unfortunately, for most realistic targets, theoretical and numeri-
cal determination of the natural resonances is impossible. Thus, the determination of
natural modes requires extracting the natural frequencies from measured responses of a
scale model. Since no experimental measurement can avoid envirenmental noise, it is
prudent to obtain an error estimate of the extracted natural frequencies from the noise

contaminated measured responses.

Several numerical algorithms have been introduced in Chapter 2. The most popu-
lar technique is the least square parameter estimation, including minimization of a
regularized ill-condition least square equation and minimization of the norm of the
late-time convolution between the measured response and the E-pulse waveform. The
least square problem involved here is a nonlinear problem, a strict estimate is rather
difficult. To make problem easier, only a simplified and linearized estimation will be

pursued.

It was stated previously that the late-time response of a target can be represented

93

by a sum of damping sinusoids. The natural candidate for the choice of the fitting
functions takes the form

N
g( x, t) = Z anew cos(0),,t + (In) (3.4.1)

n=l

where x is the vector of fitting parameters

X = [01,...,aN;01 ..... ONKDI ..... OJN;¢1 ..... ¢NIT (3.4.2)

The measured sample data vector is written in the form

 

 

)1“
f2
F = I (3.4.3)
.an
and the fitting function vector is defined as
SOHO-I
G(x) = I . (3.4.4)
Lg(x;tM).

 

 

The function to be minimized is given by the summation of the squared residues

M
h<x> = % 21f.- - g(XJt-NZ = gar - G)T(F — G) (3.4.5)
i=1

where T denotes the transposition of a matrix, and the derivative of (3.4.5) is denoted

as

. a ,
ax] h(X)

Dh(x) = i . (3.4.6)

8
. 31m hml

The j’th component of (3.4.6) is written in an explicit form as

 

 

 

94

i=1

— —a— - M . - . _—a- .
(Dh(X))j— ax] h(X) - E (f. 8(x.tt))( ax, 80‘”)
It is easy to show that

Dh(x) = DGT(x) (G(x) — F) (3.4.7)

The necessary requirement for minimization of the function h(x) is reduced to
Dh(x) = DGT(x) (G(x) - F) = 0 (3.4.8)
Now if the sample data is somehow perturbed, the regressed parameters of x will
be slightly different. Assuming the x, is the solution with sample data F1 and the x2 is
the solution with sample data F2, (3.4.8) is satisfied for both X, and x2.
Dh(x1) ..—. DGT(x1) (G(xl) — F1) = 0 (3.4.9)
Dh(x2) = DGT(x9 (G(xg — F?) = 0. (3.4.10)
Subtracting (3.4.10) from (3.4.9) yields

DGT(x1) (G(xl) — F.) — DGT(xQ (G(xz) - F2) = 0 (3.4.11)
An equivalent form is provided by

[DGT(x1)—DGT(X7_)][G(X1)-F1] + DGT(x2)10<x1)—G(x2) + Fz-Fn = 0. (3.4.12)
A close observation reveals that (3.4.12) is a nonlinear equation. The exact estimation
of difference between X, and x2 is difficult, but if the perturbation of the sample data
is small, an approximate estimation based on (3.4.12) can be performed. Now the

second order derivative of the fitting function vector is used to present the differences

of both the fitting function and its first order derivative.

DGT(x1)—DGT(x2) z DZGT(x2) (x1 — x2)

where 026 is a three-dimensional matrix and

can) — 00"?) = 070m) (x1 - x2) + g- DDTG<x9<x1 - xzxxt — x2)

Then, (3.4.12) results in

020709) (xl - x2)(G(x1) — F,) + DGT(x2)1 DTG(x2)(x1 — x2)

95

+ -;- DDTG(X2)(x1 — x2)(x1 - x7) + F2 — F1] = 0 (3.4.13)
If the perturbation of sample data F is small, it is meant

11F1 - F2"
“F,"

The approximate expressions (which are higher order perturbation)

(it — x1)(x1 - x;)=0
and

(X1 " X2)(G(X1) - 170:0

hold. (3.4.13) is thus linearized into a form

DGT(x2)[ 07009904 - x2) + F2 — F,] = 0 (3.4.14)

An estimation for Xr—Xz is then made as

x, - x2 = 100%,) DTG(x7)]’1 DGT(x,)(Fl — F7) (3.4.15)
with the upper limit being
11x, - x2" = 11100732) 070mm“ DGT(x2)II "(Fl — F2)" (3.4.16)
It is true that (3.4.16) is created based on the square function of (3.4.5) and it is
not the function used in the numerical algorithm described in the chapter 2. With the

help of (3.4.16), the estimation of XI — x2 based on the function of a regularized form

1 M 2 1 M 02
1200 = — r 21:- - g(x.t.->I + — (I- t) 21x. — .1
2 i=1 2 i=1
= i- 1: (F - G)T(F — G) +% (1 — x) (x — xo)T(x - x0) (3.4.17)
where x0 is a known vector, is easily obtained as

1 DGT(x2)[ DTG(x2)(xl - x2) + F2 — F1] + (1 — t)(x1 — x,) = o
11x, — lel 5. 1111007012) 0700(2) + (1-1)! 1“1 DGT(x2)II “(Fl — F7)"
=nsu 11F, - F2” (3.4.18)

with I a unit matrix and S defined as:

96

S = t[DGT(x2) DTG(x2) + (1—1:)l 1‘1 DGT(x2) (3.4.19)

Fig. 3.14 shows the norms of the matrix S when the G(X) is a three-mode fitting
function for an impulse response of a 30 cm thin wire. The impulse response is con-
structed based on the first five natural modes of the thin wire which has a 60° angel
w.r.t. the direction of the incident wave. The M associated with the x axis is the
length of the vector F. Fig. 3.15 is the same plot as Fig. 14 except five natural modes
are assumed in the fitting function of G(X). It is seen that the norm of the matrix S is
increased as the ’t approaches 1. The norm of S is much bigger when five modes are
expected. This implies that we may lose all significant digits when more than a reson-
able number of modes are expected. The oscillations of the curves with 1: close to 1
reflect the ill-condition of the original least square problem since when t is 1, the reg-
ularized equation reduces to the original least square problem. The norm of S is only
the maximum error estimation for the extracted natural frequencies. The practical

application of the least square method to (3.4.5) can result with much smaller error.

To see how much the extracted natural frequencies can be perturbed when the
sample data are perturbed, Table 3.1 displays the first five natural frequencies of a 30
cm thin wire extracted by means of a continuation method from an impulse response
of the thin wire. The impulse response is constructed based on the first five natural
nodes. It is seen that when the standard deviation of the added white noise is less than
10% of the maximum amplitude of the data, the extracted modes are very close to the
true values. Table 3.2 shows the results with the same method applied to a real
response scattered from a thin wire. The theoretical values of the natural modes are
compared to the extracted ones. They are not affected very much even when extra

noise is added.

3.4.2) The E-Pulse Convolution With Natural Modes Perturbed

97

 

 

 

 

5
x\
K [I s‘
\ I \
4 [J\ at I, \\
— \\\\ /\ f \\
\\ I, \\ ,I ‘\
E ’ ‘ ,I \
\\ ’1' \ I \*
\‘\ 1
‘ Q I ‘ I,
\ ‘I \ I
O ‘ [‘9 ‘ I,
LogllSll x ,I \g 11,.
\ \I ‘s
o 1' ~..
3— ‘\0 g\€~g
“0- re-
. Q‘Q-Q 9‘9.
‘Q- _ 'Cl
\\ ~° ‘0"‘4-0
\
‘c‘ __.__.__.__.
*--o--o--o--0-'.".
.. t=1.0
2 D ‘L'=O.7
.1
0 1:0.4
0 ’C=0.1
I I I I
32 64 96 128
Sample points ( M )

Fig. 3.14 The norm of the matrix S when three modes are extracted from an impulse
response of a 30cm thinwire ( 6 = 60° )

98

 

 

 

 

10
’I‘
ll
’1
l1
’ 1
’ 1
’ 1
8.5- / 1,
I
1 1
* 1
1
1
1
1
7 f3
.4 , ,1
Z] l“ [1“ I’X\
I 1 \ ’1 6 \
~ I, L)\’ \s‘
I
LogllSlI , \ x
El D ‘
‘0 \‘ ‘\
5.5— 9’ \ ‘9“ “
\ ‘9- ~.*
’4 \\ . 9.9... ‘J
0’ ‘ ‘0- Q-‘Q-Q-
\\ ‘9-.g__°__o_-° ‘CI
\ --o--o--o-.o
V--O--o--o--o--o-+--O--O--O
4-1
* t=1.0
1:1 1:=O.7
0 1:0.4
0 1:0.1
I I I L
32 64 96 128
Sample points ( M )

Fig. 3.15 The norm of the matrix 5 when five modes are extracted from an impulse
response of a 30cm thinwire ( 6=60°)

99

 

 

 

 

 

mode # theory no noise 5% noise 10% noise
S 1 -0.2601+j2.906 -0.2601+j2.906 -O.2595+j2.901 -0.2667+j2.894
$2 -0.3808+j6.007 -0.3808+j6.007 -O.3862+j6.051 -0.3912+j5.988
S 3 -0.4684+j9.060 -0.4684+j9.060 -0.4136+j8.960 -0.2322+j9. 193
S4 -0.5381+j12. 17 -0.5415+j 12.17 -1.1 l78+j 12.02 -0.4702+j13.40
$5 -0.5997+j15.24 -0.5997+j15.25 -0.4716+j14.98 -0.6789+j15.09
Table 3.1 The first five natural frequencies of a 30 cm thin wire extracted

from its impulse response via a continuation method. The impulse response is
constructed with the first five modes and is contaminated with a Gaussian white
noise.

100

Natural Frequencies of 3 Thin Wire

 

 

 

 

 

 

mode # theory no noise 5% noise 10% noise
S 1 -0.1254+j 1.401 -0.1493+j 1.409 -0. 1440+j 1.409 -0.1476+j 1.415
52 -0.2258+j4.366 -0.2685+j4.389 -0.2754+j4.399 -0.2953+j4.327
S3 -0.2891+j7.349 -0.2877+j7.653 -0.5851+j7.374
S 4 -0.3392+j 10.34 -0.3336+j 10.69 -0.3358+j 10.70 -0.3300+j10. 15
Table 3.2 The first four odd natural frequencies of a 12" thin wire extracted

from its time domain measured response via a continuation method. Only the odd
modes are excited and the measured response is contaminated with extra uniform
white noise.

101

It was implied in the last section that we can not extract the exact natural modes
from a measured response even though the results may be close to the true values. As
most practical E-pulses are synthesized based on the extracted natural frequencies, it
is appropriate to ask the question of whether the E-pulse can eliminate the natural
resonances of late-time response scattered from an expected target, or how the E-pulse

convolution will be affected by the shifting of the expected natural frequencies.

From (3.2.23) the convolution of an E-pulse with the scattered field response
from a target is represented by

N
C(t) = z a,IE(s,,)1e°~‘cos(co,,: + 11),) t > TL

n=l

where the E(s) is the spectrum of the E-pulse given by

no TC
E(s) = l e(t) e‘"“ dz: (I e(t) e"~" dt (3.2.16)
and E(sn) is E(s) evaluated at 3,, where 3,, is the n’th natural frequency of the target.
Now assume that a target is characterized by the natural frequencies of 31, ~ ~ - , sN,
while the extracted natural frequencies from its scattered response are 3?, - - - , s2,

The E-pulse synthesized for this target satisfies

T.
E(sg) = g e(t) (3.32! dt = 0 (n=l.2,...,N) (3.4.20)
But the spectrum of the E-pulse is not zero at frequencies of SI, - - - , sN. Conse-

quently the late-time convolution of C(t) of (3.2.23) is not zero either. The error esti-

mate can be performed as follows. Let’s define
s, = s2 + As, (n21,2,...,N) (3.4.21)
Evaluating (3.2.16) yields

T.

E(sa) = I e(t) 6"" d:

102

To
= I e(t) e—s‘thAs"I dt (3-4-22)

Suppose that the extracted 32 may be very close to the true value of s". The value of

As, will then be very small and the relationship of

As, T, 4: 1 (n=1,2,...,N) (3.4.23)
holds. Equation (3.4.22) results in

T

E(sn) = ! e(t) [‘3‘ (1 — A3,, t) d:

To
°1

=-As,, I e(t)e"~:dz

= A s, 33- E(s2) (3.4.24)

Thus the convolution of the E-pulse with a late-time scattered response from an

expected target has a nonzero amplitude of

N
C(t) = Z gum-5’; E(sbl e°~‘cos (19,14. 111,.) :> TL (3.4.25)
11 -—- 1

It is seen that if the differences of Ash... , 13s,, are sufficiently small, the late-time
convolved output is negligible. Extensive experimental results in the section 3.3
confirmed this finding. In that section E-pulse was synthesized based on the natural
frequencies which were extracted from a measured response and the amplitude of the
late-time convolved output between the E-pulse and a scattered field response from a

right target was found to be small.

In the preceeding section, it is shown that the extracted natural frequencies will
be perturbed if the experimental data is contaminated by noise. Since the synthesis of
the E-pulse is based on the natural frequencies extracted from measured responses, the
extracted frequencies are always perturbed from the exact frequencies. To view the

tolerable range of the perturbation on the natural frequencies, a numerical experiment

103

is performed on the impulse response of a thin wire.

Fig. 3.16 shows the backscattered impulse response of a thin wire oriented 45°
w.r.t. the incident wave. The impulse response is constructed with the first five modes.
It is observed that the early-time part of the impulse response is oscillatory. This
early-time response is faulty because an early-time response of a target can not be con-
structed with a sum of natural modes with second-kind coupling coefficients. We have
ignored this faulty early-time response, because only the late-time response, which is

correct as depicted in Fig. 3.16, is needed for our analysis.

The first five natural frequencies are then perturbed by white Gaussian noise with
various deviations.This perturbation is directly applied to the narural frequencies. Sub-
sequently, the E-pulses are synthesized based on the noise perturbed natural frequen-
cies. Thereafter, the synthesized E-pulses are convolved respectively with the impulse
response of the thin wire shown in Fig. 3.16. The convolution result is shown in Fig.
3.17. Fig. 3.18 and Fig. 3.19 are the results from the test with the impulse response
of the same thin wire but orientated 30° w.r.t. the incident wave. It is obvious that
when the noise perturbation is more than 5%, the convolved responses in the late time
are significantly different from the expected null response. Fig. 3.20 shows the vari-
ous E-pulses which are synthesized from the noise perturbed natural frequencies of the
thin wire.

It is observed that the E-pulse synthesis and the E-pulse convolution are quite
sensitive to the perturbation of the natual frequencies. These results consequently
suggest that the natual frequencies must be extracted from scale-model targets in a
noise-controlled laboratory envirenment. However, it should be remembered that this
sensitivity of the E-pulse to the perturbation of the natural modes provides the E-pulse

technique with the potential to discriminate two similar sized targets.

104

 

 

 

 

 

 

 

 

 

 

s1 = -O.2601+j2.906
10 - $2 = -0.3808+j6.007
$3 = -0.4684+j9.060 x109 radian
s4 = —0.5381+j12.17
55 = —o.5997+115.24
5 _.
Relative 0 ...............
Amplitude _. ....................................................
_5 _1
1 1 1 1
o 3 6 9 12 15

Time in ct/L

Fig. 3.16 The impulse response of a 30cm thinwire with a 60° angle w.r.t. to the incident
direction of the exciting wave.

105

 

 

 

 

 

 

 

 

 

 

1- no noise perturbed
. ----- 1% noise perturbed
.' . ---------- 5% noise perturbed
:1
c E
: 1
0.5.. g g
1
r '1
r 1
5 1
Relatrve 0.. r g
Amplitude w 1
’1 :
1 .5
1. I
-0.5— E 1 - late time
11'?
L I:
(.15
is
-12 23
1 1 1 1
0 3 6 9 12 15

Time in ct/L

Fig. 3.17 The impulse response of a 30cm thinwire ( 0 = 60° ) is convolved with the E-

pulses which are synthesized based on the noise-perturbed natural frequencies
of the thinwire.

106

 

 

 

 

 

 

 

 

 

 

 

S 1 = -O.2601+j2.906
4o _ 52 = -O.3808+j6.007
S 3 = -0.4684+j9.060 x109 radian

54 = -0.5381+j12.17

I 55 = -0.5997+j15.24
20 —.

Relative o 5 _
Amplrtude " ' ' ' ' '
-20 _
—40 —
1 1 1 1
0 3 6 9 12 15

Time in ct/L

Fig. 3.18 The impulse response of a 30cm thinwire with a 30° angle w.r.t. to the incident

direction of the exciting wave.

107

 

 

1 _. , no noise perturbed
' ----- 5% noise perturbed
.......... 10% noise perturbed
l
0.5 — F

Relative 0
Amplitude ‘

h

 

 

..--

 

 

 

 

 

 

-0.5 .4 late time
_1 _,
1 1 1 1
0 3 6 9 12 15

Time in ct/L

Fig. 3.19 The impulse response of a 30cm thinwire ( 0 = 300 ) is convolved with the E-

pulses which are synthesized based on the noise-perturbed natural frequencies
of the thinwire.

108

 

 

 

 

 

 

 

 

 

 

 

 

2
1J :......: I“
s 3 ' '
. 1 I
1 I
. I '
2 ........ I
("17'") i r"; I
I
O— 1
Relative
Amplitude
—l— I .......
E
i
F
E
E
E
E
E
_2_ 1 nonoise
i
----- 1%noise
............. 5% noise
_3 l l I I l
O 0.5 l 1.5 2 2.5 3

Time in ct/L

Fig. 3.20 The E-pulse waveforms synthesized for a 30cm thinwire based on its first

five noise-perturbed natural frequencies

109

It is interesting to note that if the natural frequencies of a target are assigned to
be second order zeros of E(s), so the derivative of the E-pulse complex spectrum is
also zero at any natural frequency of the target, then the convolved output of (3.4.25)
will remain zero. However when discrimination between two targets whose natural
frequencies are located close to each other in the complex plane is desired, the

difference between two late-time convolved outputs may not be large enough.

3.4.3) Noise Performance of The E—Pulse Convolution

This subsection is devoted to investigate the statistical noise estimation when the
E-pulse technique is applied to the scattered response which is contaminated with

noise. As a preliminary analysis, only white additive stationary noise is considered.

The typical parameters used to describe random noise are mean, autocorrelation
function, variance, standard deviation and power spectrum. When zero-mean station-

ary white noise of n(t) is considered, it is mathematically implied that [42]

E,,[n(t)] = 0 (3.4.26)
"0
11,,(1) = T15(1) (3.4.27)
and
5,0) = % (3.4.28)

where E denotes the mean, R denotes the correlation function, S denotes the power
spectrum of the noise and No is the amplitude of the power spectrum. Now a target

scattered response waveform is assumed to be

r(t) = r00) + n(t) (3.4.29)

with ro(t) being uncontaminated response. The convolution of the response (3.4.29)

with an E-pulse waveform is given by

110

e(t') r(t - t’) dt’

T

e(t’) ro(t—{)dt' + tea) n(r-t’)dt’

e(t)

d—L:~lh:~i

T.
= 000) + tea) n(t—t’)dt’ (3.4.30)

The mean of the E-pulse convolved response is presented by
T.
E[6(t)l = 5141(2)] + 51 tea) n(:—-z’)d/1

T.
= co(t) + tea) E[n(t—t’)] dt’

= C00) (3.431)
and the autocorrelation function is given by

RM. ‘2) = 51 600002) I

T

= 51 00000007) 1 + Ercoao z[e(r) nor/)1 4!
T, T. T.
+ 1511:1112) t e({) n(:,—1’)1 dt’ + E[£e(t’) n(1,—z’) dt’ ted) n(12—i')1 d?
Tc
= Co(‘1)¢o(‘2) + 000018“) Elma-")1 d/

r, T. T.
+ 0002) ten) E[n(tl—{)] dt’ + gem (11465111014) nor-b] d? (3.432)

Using (3.4.26) and (3.4.27) leads to

T, T,
N .. - ..
11,01, :2) = co(t1)co(tz)+70 t[e(t) dt’ t[e(t) 8(11—1’ — 12+» dt
NOT
= co(t1)co(t2) + 3- oteu’) e(12—:,+:') dt’ (3.4.33)

The variance of the E-pulse convolved response waveform takes the value of

111

62(1) = R.(:. t) - 5214111

T.

= 660) + J? ([620) dt’ - 660)

= [:2 p, (3.4.34)

where P, is the energy contained in the E-pulse waveform.

The noise model used here is an ideal one. More practical would be white noise
band limitted by the band-width of the system being considered. As an example, an
ideal low-pass system is considered. Its transfer function is

1 1 fl < w
wm ={ (3.4.35)

0 elsewhere

Then, the power spectrum of the noise output from system is

3,0!) = 3;)— IW(f)l2 (3.4.36)

the associated autocorrelation function is

R.<1:)= % I lelzexm—flnft) df

No W
= —2— L cosZrtft df

= WN sin211Wt . .
°_211Wt (3 4 37)

The variance of the noise is easily obtained as

6,2, = R,,(0) — 5111(1)] = R,(0) = wzv0 (3.4.38)
with (3.4.32), the autocorrelation function of the E-pulse convolved response for the

case of band-limited white noise is shown to be

T T ..
‘ ‘ .. Sin21tW(t1-t2+t'-t) -
R,t,r = t r +WN t’ dt’ t ,, dt 3.4.39
(1 2) Co(1)¢o( 2) 0£e() £30 211W(t1-t2+{—t) ( )

 

112

The variance takes the form

 

0%=R.(t.t)
7. r. , f ,
=w~ ( dt’ " “"27““ :34“ 3.4.40
ota ) tee) 2mm) 1 ( >

One of the most often used parameters to specify noise is termed the standard
deviation which is the square root of a variance. If a noise is specified with a standard
deviation, the variance can be easily found. With the help of (3.4.38), the parameter
No is obtained as long as the system bandwidth is given. The signal noise ratio before

the E-pulse convolution is thus written as

(S/Mo = 1010g10%g—%| (3.4.41)

while the signal to noise ratio after the E-pulse convolution is as

E8

(SW), = 1010g1u T. T. (3.4.42)

WNO t e(t’) dt’ t e(t) “gig/pg}? d?

 

 

where the bar denotes the average in a time period.

The behavior of the noise performance of the E-pulse convolution is described by
Table 3.3 in which the signal-to-noise ratios are compared before and after the convo-
lution of the noise-contaminated impulse responses of a thin wire with its E-pulses.
The signal-to-noise ratios for different aspect angles are evaluated with (3.4.41) and
(3.4.42). The noise deviation associated with the x axis is the percentage of the max-
imum amplitude in an impulse response. Fig. 3.21 depicts the result of Table 3.3.
The 0 is the angle between the wave incident direction and the thin wire axis. An

ennhancement of about 20 dB after the E-pulse convolution has been achieved.

3.4.4 Noise—Testing on Experimental Data

113

 

 

 

 

 

 

50 — after convolution
----- before convolution
40 —«
30 -—
S/N

( dB ) 2° "
10 —
o _

'0

a

1 1 1 1 1

0.05 0.1 0.15 0.2 0.25

Noise deviation

Fig. 3.21 Comparison of the signal to noise ratios before and after the impulse responses
of a 30cm thinwire are convolved with its E-pulse. The first five natural frequencies are
used to construct the impulse responses and to synthesize the E-pulses of the thinwire.

114

 

 

 

 

 

 

9:600 9:450 9:300

0’ before * after * before * after * before * after *
0.05 32.34 49.94 23.48 43.05 21.01 41.11
0.10 18.48 38.08 9.617 29.19 7.146 27.25
0.15 10.37 27.97 1.508 21.08 -0.96 19.14
0.20 . 4.613 22.21 4.25 15.32 -6.72 13.39
0.25 0.151 17.75 -8.71 10.86 -11.2 8.926

Table 3.3 The comparison of the signal to noise ratios before and after the

impulse responses of a 30 cm thin wire are convolved with its E-pulses. The first
five natural frequencies are used to construct the impulse responses and to syn-
thesize the E-pulses of the thinwire. The a is the standard deviation of the added
noise, and the 0 is the incident angle of the excitation w.r.t. the thinwire.

115

In the preceeding analysis, only the constructed thin wire responses are used as
examples. But the analysis is applicable to any response. To show the applicability,
extensive experiments with measured responses from airplane models have been con-

ducted. However only a few examples can be shown.

In the experiments conducted, very noisy radar responses are created by adding
extra white Gaussian noise to the measured responses of the targets. These noisy
responses are then convolved with the E-pulses of the targets. It is found that the E-
pulses are still effective in smothering the noise and are capable of discriminating

between the expected and unexpected targets from these noisy responses.

Fig. 3.22.a shows the pulse response (the response excited by an incident Gaus-
sian pulse) of a B—707 aircraft model measured at 900 aspect angle, without extra noise
added. Fig. 3.22.b is the convolved output of the pulse response of Fig. 3.22.a with
the E-pulse synthesized for the B—707 model. As expected, a very small output is pro—
duced in the late-time portion of the convolved response. This identifies the pulse

response of Fig. 3.22.a to be from the expected B-707 model target.

Subsequently, a noisy response is generated by purposely adding Gaussian white
noise to the measured pulse response of the B-707 model shown in Fig. 3.22.a. The
standard deviation of the noise is as large as 20% of the maximum amplitude of the
measured response. The noise contaminated response is shown in Fig. 3.23.a. When
this noisy pulse response of Fig. 3.23.a is convolved with the E-pulse waveform of a
B-707 model, a satisfactory convolved output, as shown in Fig. 8.b, is obtained. This
convolved output resembles that of Fig. 3.23.a. The late-time response still remains
small. The signal-to-noise ratio is enhanced from -1.12 dB to 23.7 dB after the E-
pulse convolution. This confirms that the B-707 model may be identified from the
noisy pulse response of Fig. 3.23.a.

116

 

20—

Relative

Amplitude 0 _

-20 ..

 

Aspect angle 90°

 

 

 

 

 

 

-40

 

 

20—

5 _
Relative
Amplitude

-10 ..

(b)

 

 

 

 

Time in ns

10

Fig. 3.22 The noise-free scattered waveform of B-707 model at aspect angle of
90° and its convolved response with the E-pulse waveform of B-707 model a) the
uncontaminated waveform; b) the convolved response

117

 

Aspect angle 90°
S/N = -1.12 c 3

20—

 

 

   

 

 

 

 

 

 

 

Relative 0 _
Amplitude
-20 .3
(a)
.40 1 1 1 1
0 2 4 6 8 10
Time in us
20 - S/N = 23.7 dB
5 4
Relative ____ __ ____ ___ _ __ _ ___ -_ - .
Amplitude '
§.__ late-time
-10 _ :
(b)
_25 l I L I
0 2 4 6 8 10
Time in us

Fig. 3.23 The noise contaminated scattered waveform of B-707 model at aspect
angle of 90° and its convolved response with the E-pulse waveform of B-707 model
a) the noise contaminated waveform (the standard deviation of noise is 20% of the
maximum ammitude); b) the convolved response

118

 

 

 

 

 

 

 

 

 

 

Aspect angle 90°
40 _
20 —
Relative
Amplitude
-20 _
(a)
0 2 4 6 8 10
Time in ns
20 —
5 ..
Relative _ _ _ _ - _ _ .....
Amplitude -- -- -..- -- - ----- -- "- --
—10 —< 5 .
1<-—— late-time
(b)
0 2 4 6 8 10

Time in us

Fig. 3.24 The noise-free scattered waveform of F-l8 model at aspect angle of
90° and its convolved response with the E-pulse waveform of B-707 model a) the
uncontaminated waveform; b) the convolved response

119

Next, an attempt is made to discriminate an unexpected F-18 target model by
applying the E—pulse for the B-707 model to the noisy pulse responses of the wrong
target. Fig. 3.24.a is the pulse response of the F-18 model measured at the same
aspect angle of 90° without extra noise added. When the response of Fig. 3.24.a is
convolved with the E-pulse of the B-707 model, the convolved output is shown in Fig.
3.24.b. It is seen that a relatively large late-time response is obtained. This is the

indication of the wrong target.

As done previously for the response of the B—707 model, the pulse response of
an F—18 model is contaminated with white Gaussian noise. The standard deviation is
kept to be 20% of the maximum amplitude of the measured response of the F-18
model. Fig. 3.25.a shows the resulting waveform. The noisy pulse response of Fig.
3.25.a is subsequently convolved with the E-pulse of the B-707 model. The convolved
output is shown in Fig. 3.25.b. As compared to the result of Fig. 3.24.b, the con-
volved output of Fig. 3.25.b presents a relatively unchanged early-time response fol-
lowed by a still large and somewhat noisy late-time response. This late-time response
is sufficiently large to indicate that the noisy pulse response of Fig. 3.25.a is scattered
from an unexpected target other than the B-707 model. In this case the signal-to-noise

ratio is enhanced from -2.43 dB to 30.1 dB after the E-pulse convolution.

Based on this experimental investigation, we can see the E—pulse convolution
enhances the signal to noise ratio by more than 20 dB. This confirms the analysis
conducted in the previous section and results the conclusion that the E-pulse target

discrimination scheme is noise-insensitive.
3.4.5 Conclusion

Several aspects concerning the noise characteristics of the E-pulse technique for

120

 

40 -1

204

Relative

 

Amplitude 0

-20 -

 

 

Aspect angle 90°
S/N = —2.43 dB

 

 

 

 

 

20—

5 —
Relative
Amplitude

—10 -4

 

SIN = 30.1 dB

ge— late-time

(b)

 

 

Time in ns

10

Fig. 3.25 The noise contaminated scattered waveform of F-18 model at aspect
angle of 90° and its convolved response with the E-pulse waveform of B-707 model
a) The noise contaminated waveform (the standard deviation of noise is 20% of the
maximum amplitude); b) the convolved response

121

radar target discrimination have been investigated. An error estimate on the extraction
of the natural frequencies of a target from measured responses has been given. It has
been shown that if a set of natural modes of a target is perturbed more than 5%, the
corresponding E-pulse waveform and the consequent convolution are significantly
different. This property provides the E-pulse with the potential to discriminate two
similar sized targets, and strongly suggests that the extraction of the natural frequen-
cies from measured responses must be done on a scale model in the laboratory
environment. Also, the signal-to-noise ratio of a response has been demonstrated to
be enhanced 20 dB by the E-pulse convolution. Thus the noise insensitivity of the E-

pulse convolution has been demonstrated.

Experimental results with the measured responses from scale airplane models
have verified the analysis in the chapter. The scattered responses from two similar

sized airplane models with 20% extra white noise added can be used to discriminate

the targets.

Chapter 4

THE NATURAL OSCILLATIONS OF AN INFINITELY LONG
CYLINDER COATED WITH LOSSY MATERIAL

4.1 Introduction

The radar cross section of a metallic target can be greatly reduced by coating its
surface with a layer of lossy material. When it is necessary to discriminate such a tar-
get with a target discrimination scheme, such as the E-pulse technique, which is
entirely based on the target’s natural frequencies, it is of the first importance to know

the effects of lossy coatings on the natural frequencies of the target.

In the last few years, the Singularity Expansion Method has been applied to iden-
tify the natural frequencies of a perfectly conducting cylinder [43], a radially inhomo-
geneous lossy cylinder [44] and a conducting sphere coated with lossy layer [45]. But
considerably less attention has been devoted to the SEM analysis of the coated per-
fectly conducting cylinder. The coated cylinder is selected as an example to investi-
gate the effects of the lossy coating on the natural oscillations of the cylinder. The
significance of this analysis is directly related to the available potential of the E—pulse

technique in discrimination of targets coated with lossy material.

An infinitely long conducting cylinder coated with a layer of lossy material and
an infinitely long lossy homogeneous cylinder are considered here. The occurence of
the lossy regions in current problems has resulted in the introduction of the concept on
the "exterior" mode and the "interior" mode [50]. The main difference between an
exterior mode and an interior mode resides at the radial behavior of the scattered field
in the lossy region. The field distribution associated with an exterior mode behaves

as a standing wave in the lossy region, while the field associated with an interior mode

122

123

is attenuated into the center of the cylinder.

It is shown that the natural frequencies of exterior modes are substantially shifted
on the complex plane only when the coating thickness is comparable with the radius of
the cylinder. When the coating material has a parameter of (ma > 100 (where o is the
conductivity, 1] is the wave impedance of free space and a is the radius of the
cylinder), it has little effect on the natural frequencies of the exterior modes. In con-
trast, the natural frequencies of the interior modes, whose existance is attibuted to the
imperfect conducting properties of the cylinder or the lossy coating, are greatly depen-
dent on coating thickness and parameters. They are shifted upward on the complex
plane when the coating thickness is reduced and they are shifted left-ward when the
conductivity is increased. As a rough estimation, the interior modes are no longer
dominant when the conductivity satisfies the condition 011a > 100, or when the coating

thickness is less than 10 percent of the radius of the cylinder.

In this chapter a generic characteristic equation for the extraction of the natural
frequencies of a coated cylinder is derived. The pole distributions and the pole trajec-
tories of a number of dominant resonant modes are presented. In addition, the experi-
mental investigation into the discrimination of rectangular plates coated with lossy
material is described to demonstrate that the E-pulse technique is capable of identify-

ing targets coated with lossy material.

Section 4.2 presents the theoretical study on the coated cylindrical structure. A
generic characteristic equation for formulating natural frequencies is derived. The

equations characterizing a few special geometries are deduced from the generic one.

Section 4.3 discusses the algorithms needed to select a proper branch cut, to
evaluate the modified Bessel functions, to normalize the characteristic equations, to

search zeros on the complex plane, and to check numerical consistency of the pro-

124

gramming.
Extensive numerical results are presented in section 4.4 to reveal the effects of

the coating thickness and parameters on the natural modes. The pole distributions and

the typical field distributions are presented in this section.

The further study on a few dominant exterior modes received additional discus-
sion in section 4.5. The pole trajectories of the dominant modes are traced versus a

variety of lossy parameters to show the common effects of the lossy materials on the

natural modes.

As a preliminary experimental investigation into the application of the E-pulse
technique to targets coated with lossy layers, some experimental results on the
discrimination of rectangular plates coated with a lossy foam are presented in section
4.6. It appears that the E—pulse technique is not impaired by the presence of the lossy

coating.

The conclusion of this chapter is given in section 4.7.

4.2 Derivation of Characteristic Equation

Consider a lossy cylinder coated with a lossy layer, the geometry is shown in Fig.
4.1. The cylindrical coordinate system is chosen with the z-axis in the axial direction

and the space is naturally divided into three regions.

Assume the incident wave reaches the coating surface at the origin of the time

and the waveform of incident electric field is shaped as:

E‘(r,t) = y U(t—(x+b)/c) F(t—(x+b)/c) (4.2.1)

In the Laplace transform domain,

E‘(r,s) = y F (s) e- W”) (42-2)

125

 

 

 

El
[=1
~<

l

-----I.--------

 

 

\

Fig. 4.1 Geometry of a lossy cylinder coated with a lossy layer.

126

i
c

with 70 = and F(s) is the single-sided Laplace transform of F(t), where F(t)
is the shape function of the incident waveform and U(t) is the step function.

In cylindrical coordinates, the incident E-field is decomposed into r and 6 com-
ponents. For TE—polarized excitation, the H-field has only the 2 component.

E‘(r,s) = r E,(r,¢,s) + 1,» E,(r,¢,s) (4.2.3)

and

H‘(r,s) = z H,(r,¢,s) (4.2.4)

Once the z-component of H-field is known, the E-field is determined by:

1 a
Er : -—.—-—Hz 4.2.5
6 sr all) ( )
_ _ _Li
E, — e‘s 8r H, (4.2.6)
where
VZH, — 72 Hz = 0 (4.2.7)
72 = 52115 (4.2.8)
8* = E + -:—,- (4.2.9)

The complex wavenumber y and complex permittivity 8* have been introduced.
In order to simplify matching boundary conditions, the incident fields are
represented in terms of modified Bessel functions

sb

E, = —F(s)e c z(—1)"e, 1',(yor) cos(n¢) (4.2.10)
"=0
:2 ..

H; = %e C 56(4)" 8,, 1,,(‘yor) cosn¢ (4.2.11)

where the identity ( see 9.6.34 in reference [8] )

e‘ ...e = 10(2) + 2331,12) cosk0 (4.2.12)
b=l

127

has been used along with 1,,(s), which denotes the first kind modified Bessel function,

and the constant,

8 _ 1 n=0
" " 2 n>0
The scattered fields in different regions are expressed using the following

modified Bessel function series:

Within region III:

H: = —Ea,,(s) K,(yor) cosnd) (4.2.13)
"=0
191. = xl—udeozaxs) K’.(Yor) cosn¢ (4.2.14)
"=0

where K,(s) is the second kind modified Bessel function.

Within region 11 :

H: = -irb.(s)l.<rzr)cosm1 + c.<s)K.(12r>cosn¢1 (4.2.15)
F0

5‘8 = 12- ilb.(S)l’n(1'2r)cosn¢ + C..(S)K’.(1'2r)cosn¢l (4.2.16)
823 "=0

Within region I :

H; = - Ears) 1.010 008110 (4.2.17)
n=0

1:; = i?— Edgs) I’,,(ylr) cosn¢ (4.2.18)
815 "=0

Here the complex wavenumber and complex permittivity are defined:

 

 

e; = 62 + ozls (4.2.19)
12=Vs2u£2+u02s (42-20)
a} = 81 + 01/3 (4.2.21)
71 = \lszuel + 1.1013 (4.2.22)

Four independent boundary conditions on two interfaces are formulated by

Hj(r = a') = H“;(r = a+) (4.2.23)

128

E30 = a‘) = E’ O(r— - a‘“) (4.2.24)
H§(r = b )= H‘Xr- -— b”) + H‘ (r- — b+) (4.2.25)
Ear = b') = 5‘30 = b+) + 510 = H“) (4.2.26)

From equations (4.2.10—26), four simultaneous equations for the unknowns
a,,(s),b,,(s),c,,(s) and d,,(s) are derived by matching boundary conditions and using the
orthogonality of the functions { cosmt), n=0,1,2 ...... ].

4.611.614 = 12.610024 + c 8.18.624) (4.2.27)
3.1- .(s)!’.(11a>- - —.—1 12.611964) + c.(s)K'.62a) 1 (4.2.28)
813 823

_b:
-—F§le ‘(— -1) e (Velma. <s>K<y >=b,. (s)1(1' b>+c. (016.6 I» (4.2.29)
me "In It 0b n 2 2

sb

—F(s)eT(-1)n8nl’n(yob) + “HO/eoan(s)K’n(‘YOb) = ésl[bn(S)I’n(72b) + Cn(S)K’n(y2b)]

        

 

 

 

 

................................ (4.2.30)
The matrix form of these equations is:
16.6012) 4.6212) 46.6212) 0 ‘_ q
‘1 udeoK'.(rob) -.— 13.021) ..(rzb) 0 Ms)
825 82s bn(s)
0 111(720) KnCYZa) -ln(71 0) 011(5)
4.
0 71362072 3.3—Kama) «Ii-1mm) . (s).
L 823 825' 818
. F -32 .
41 e .(_,)., ’.< )
We; "’ 7"”
_sb
F (1)6 ° (—1)" 8.1.001?)
= 0 . (4.2.31)
L O .1

 

 

The natural modes are contributed by nonzero a,(s),b,,(s),c,,(s) and d,(s) when

F(s)=0, and are identified from the determinant of the coefficient matrix of the above

129

equation:
lamb) -l.(12b) 4.6212) 0
W300?) -— :35 13.01213) :2: lamb) o
d“ 01.682) lama) —I.(1.a) = 0 (4232)
0 £13024 {iv-maze —é§-I’.<rla)

 

 

By using the Laplace expansion of 2 by 2 factorization, the above equation can

be simplified to low order determinants:

K .(Yob) -1.(Yzb) K .0260 -1.(71 0)
det det 12—
\IWEOK’ .(rob) - . 13021)) -:- K'.(rza) -—.-f.(rra)
€48 843 as

Is

 

K.(Yob) - .(Yzb) I.(Yza) - .(71 a)
— det 72 - det 72 = 0 (4.2.33)
duo/comm)» 58.6.12) 3:130ch ) :-;-s I’.(rra)
l

The further expansion of the determinant yields this general characteristic equa-

tion:

11.6.6110 soK' .(rob)— :.K.(1Iob)l’ (1212)] 11.01160—L2 2s .(Yza)—I’.(rr 041-19020]

81S

1.6.a)I’.(vza)—-Y.‘—I.(12a)1'.(vla)1 = 0
82S 61S

~14 uo/EoK'.—.—(rob)K.(rzb)— :K.(vob)K'

         

............. (4.2.34)

This is an equation whiCh characterizes a lossy cylinder coated with a lossy layer
in free space. Some simplified equations for special cases can be deduced by the

selection of a parameter set.

130

4.2.1) Perfectly Conducting Cylinder in Free Space

If we let a=b, 01:02 and ohoz—m , the generic characteristic equation (4.2.34)

gives rise to a very simple form:

From equations (4217-18) and (4221-22), we have 72 = y] & e} = a; and:

 

12- : .11- : (521L81+11015)U2 ___ ‘jj < ‘IIL—o/E—o (4 2 35)
82‘ 8“ (81+fl)s (81+fl)"2
S S
—Y;2—ln(‘yla) 1,4720) — 1:14 "(720) 1,4710) = 0 (4.2.36)
82S 813

Now if Wronskians’s formula is used [8],

l’,,(z)K,,(z) - n(z)K',,(z) = l/z (4.2.37)

we can obtain:

I , 'Y
1.1(710) 3?- K .(vza) - 1.010) —.‘- 16.6.4)

 

 

823 81s
= 1.1.1 1,6,.) Karma) - l’.(rra) lama» = - .1 (42-38)
813 Elsa
Put (4.2.35-38) into (4234):
1 1.0/2b) \Illo/Eo K3601» — 3.3- n(Yob) lamb) 1-(— .1 )
82S elsa
= .0112) \lllo/Eo K'.(rob)'(- -:1-) = 0 (4.2.39)

813“
Since a, = 02—900 and 71-)... , then I,(y,b)—>oo exponentially, and the only choice

is :
K’,(yoa) = 0 (4.2.40)

4.2.2) Perfectly Conducting Cylinder in Lossy Medium

If we let yo -) y, everything is the same except that the outside free space of a

perfectly conducting cylinder is replaced by lossy medium. Equation (4.2.40) is

131

modified to

K’ ,(ya) = 0 (4.2.41)
The poles of (4.2.41) are directly mapped from the free space ones by a quadratic

transformation.

52118 + 1,108 = sole (4.2.42)
.170 = {-011 - [(01])2 + 4(s0/c)28,]m 1/22, (4.2.43)

4.2.3) Lossy Cylinder in Free Space

If we let a = b, and 01: 02 at 0, an equation characterizing a lossy cylinder is

derived from (4.2.34)

Since 01 = 62 , we still have 71 = 72 . and equations of (4.2.36) and (4.2.38) are

 

valid.
72 , 71 .
“111—Inch“), "(720)—T1n(72a)l nCYla) = 0 (4.236)
823 813
72 , 71 , 1
—."n(Yra)K "(720) — "T‘l n(Yra)Kn(720)) = . (4.2.38)
823 813 Elm

Starting from (4.2.34), after the zero terms are eliminated, we should have:

11.62wa EoK'n(Yob) — ‘Z.2"Kn(Yob)1'n(Yzb)]'(“ —.‘—-) = 0 (4.2.44)
828 elsa
The equivalent equation will be :
/.(Yzb)\fllo’86K'.(Yob) - g:- .<vob)1'.(12b) = 0 (4.2.45)

Now that a=b in this special case, from (4.2.42) we have:
l.(72a)\/uo’_ eoK'.<voa) - g 81011111624 = 0 (4.2.46)
For easy computer programming, the parameters are normalized as :

70a = 5f- = 2 (4.2.47)

132

yla = (5211151 + 1.101s)“2a = \fgzsa/c = 21 (4.2.48)

with e}, = e], + 01/580 = 81, + $3 and n is the wave impedance in free space.
2

If we denote

72 71 = 0211814101st

15:3- 818 (81+G1/S)S
\Illo/Eo
#81,

Equation (4.2.43) with normalized parameters now becomes:

 

= 6.746.. + 0.erer =

 

(4.2.49)

VET: K’.(z) 1.421) — 1',(z,) K.(z) = 0 (4.2.50)

It is constructive here to introduce some relations for Bessel functions: ( see

9.6.26 of reference [8] )

K',(z) = —-21-1 K,_,(z) + K,,,(z)1 (4.2.51)
I’.(z) = -;-1 1,..(z)+ I.+1(2)l (4.2.52)

The equation (4.2.50) now is transformed to:

‘Efi [Ct-1(2) + K.+1(Z)l 1.421) + I ln—1(21) + In+1(21)l KAI) = 0 (4253)

When 01 is very large the 21 will be large too and to avoid the numerical

difficulty of blowing up caused by 1,,(21) —> e:tzl as zl—)oo , we can let:

F(z) = 427 K'.<z) I.(21) - I'.<z.) K.(z) (4.2.54)
F,(z) = F(z) at" (4.2.55)

where :1: sign is determined by argument of 21 .
Because F1(z) & F(z) have the same roots, solving 171(2) = 0 is equivalent to solv-

ing F(z) = 0 .

F1(2) = 72.71 K 4(2) + K...<z)1I.(z.)e*"
+ 11,101)?" + I,,,(z,)e*"1 K,(z) = 0 (4.2.56)

133

1

Now the value of 1,,(21) eiz is limited to a finite range even though 2, -—> very

large.

In some situations, the root searching algorithm needs the derivative of the

searching function, the derivative of F,(z) can be expressed as

z 0
me) = F’(z) e:1 :t efilgnz) (4.2.57)

4.2.4) Perfectly Conducting Cylinder Coated With Lossy Material
We will let a at 0, 62¢ 0, 01 —-) oo; this case is the most interesting case.

From (4.2.34):

 

 

32 + "2
3+ = ( “£1 :00) (4.2.35...)
81S (€1+'—1‘)S
S
Therefore
2 + o 112
£35. ___ “ “£2 :22” (4.2.35.b)
2 (82+—)s
s
We have the relation:
.3}- < 3,3— (4.2.58)
818 805'

Use (4.2.58) in (4.2.34), and notice that I’,(y,a), 1,,(yla) are very large, but it is
about the same order as y, -> oo (see (48)). and K’,(yza), K,('yza), I’,(72a), 1,,(72a) are all

finite since 72 is finite. Using above notations, we obtain from (4.2.34):

3.2—1.6413620- 41.687168) —> 41.682368) (4.2.59)
61-1 618 82s
72 Yr , 72
TI n(Yla)K' 11(72a) - TI n<Yla)Kn(Y2a)) '9 Tln(71a)K' 11(720)
323 £13 625'

.................. (4.2.60)

134

Put (4.2.59-60) into (4.2.34):

 

%In(7ra)IK'n(720) < I.(7.b)W’.(rob) - :3: K000136212) )
— I’.(r.a) («Jud/soK'.(rob)K.(12b)— gamma.» )1 = 0 (4.2.61)

Rearrange the terms:

¥I.(rra)1¥K.(rob) («302013620 + I’.(12a)K'.<12b) )
823 823

+ \Illo/EoK'nWob) ( K'.(72a)l.(72b) - 1'.(720)K.(72b) )l = 0 (42.62)

LX710) iS exponentially increasing as long as Rem) is nonzero and 71—)“, ,

neglecting the nonzero factor:

72

“83:15.00” I -K'.(72a)/'n(72b) +1'.(720)K'n(72b) ]
+ “JO/80 K’nCYOb) I K’n(72a)ln(72b) - l’n(72a)Kn(72b) I = 0 (45263)

Similar to (4248-50) the normalized parameters can be written as:

7oa=fl=2
C

72a = (5211.82 + 1102s)ma = V32,— sa/c = 22 (4.2.64)
.12. = We

‘
€25

 

(4.2.64.a)

a

with e;=ez,+ozlseo=ez,+ 233
Equation (63) with normalized parameters becomes

£819.06!» 1 -K’.(72a)l’.(12b)+ 1’»(720)K'n(72b) 1

+ Vito/£0 K'..(70b) I K'.(72a)1.(72b) - 1' .(720)Kn(72b) l = 0 (42-65)

4.3 Numerical Algorithm

The zero search of equation (4.2.34) has to be numerical due to its nonlinear

135

nature and complexity. Special consideration in this section is devoted to the selection
of branch cut, evaluation of Bessel functions with complex arguments, pole location

algorithm and the numerical consistence checking.

The occurrence of the complex permittivity in parameters within the arguments of
modified Bessel functions leads to branch points in the complex 8 plane. An examina-
tion of the complex wave number 72 suggests the existence of branch points at s = 0
and s = 42/84. The associated branch cut extends from the origin to the point
s = -02/84 along the negative real axis in the 3 plane [46]. However, an appropriate
branch cut for the modified Bessel function with a complex argument lies along the
entire negative real axis in the 7; plane [47]. Figure 4.2 shows the mapping of the s
plane to the 72 plane. As long as the point P does not move across the negative real
axis beyond the point of s = 42/54 in the 3 plane, its image P’ will not cross the nega-
tive real axis in the 72 plane. Thus the negative real axis in the s plane is selected as
the branch cut, which is appropriate for evaluation of Bessel functions. By referring to
[47], integer order Bessel functions are regular in the s plane out along the negative
real axis; this branch cut is also appropriate for the characteristic equation (4.2.65).
The integration contour for the inverse Laplace transform will be closed on the princi-
pal Reimann sheet; consequently only poles residing on the first sheet are implicated.
Since the conjugate symmetry of (4.2.65) gives rise to conjugate symmetry of the
poles, the poles have to be located only within the second quadrant without crossing

the cut.

An easy way to computate the modified Bessel functions with complex arguments
is to represent the Bessel functions using their integral representations [47,48]. Now
that the numerical evaluation of an integral can be very efficient and accurate, the cal-
culation of the Besel functions can reach high accuracy too. An associated advantage

with this approach is its capability of using a normalization procedure involved in the

136

25:05: _ommom can

.38::o>a3 5388 2: ..8 33 £285 oaatmoama 2F Né .wE

328:3 _ommom com So

gtv 0“ O \

2w

 

E .5 :
25.9%».

 

3 3.

“38:52;

5388 Sm So \

E 8H .

‘

 

a £9-

 

N\&N.O+m

25...-..“

137

pole location. As we know, the number range in any computer system is very limited
if an exponential function with a rather big argument needs to be expressed. The
Bessel functions with complex arguments behave as exponential functions when the
argument is quite big. An invesigation of the effects on the poles of lossy parameters,
varying within a considerably large range, is impossible if a pre~normalization is not

attempted.

As mentioned previously the pre-normalization is performed by multiplying an
exponential function to the pole-searched equation. The adding of the extra factor to
the characteristic equaton does not gives rise to new extra poles, since the integral
evaluation of the Bessel functions multiplied by an exponential function needs no extra

computational efforts.

The natural frequencies are found by searching the zeros of the characteristic
equation. The approximate locations of the zeros are determined by using the algo-
rithm which relates the argument change of an analytic function integrated along a
closed contour to the number of the zeros and poles of the function inside the contour
[49]. When there are only zeros in the searched region, the algorithm can be

mathematically described by

313,15” F’(s)/F(s) ds = 2(st (4.3.1)
where F(s) is the function searched for zeros, C is the integrated contour and s, is the

nth zero of the function F(s) inside the C.

After the approximate zeros are determined, they are improved in accuracy by
calling a NAG subroutine of COSPBF which is based on the Powell method. For the
trajectory of a pole, the Newton method can even be applied as long as the varying

rate of the parameters is slow enough.

138

Because of the numerical complexity of the characteristic equations, the numerical
consistency has to be validated before any attemption is initiated. One easy numerical
test is made by coating an air-layer on a perfectly conducting cylinder. Intuitively this
coated cylinder is exactly the same as a perfectly conducting cylinder. As we
expected, the poles located numerically for the air-coated cylinder are distributed pre-
cisely on the locations where the poles of a perfectly conducting cylinder are supposed
to lie. Fig. 4.3 shows the pole disribution located numerically on the complex plane
for an air-coated perfectly conducting cylinder. The domed line is drawn to indicate
the positions expected and Star lines serve as the result of the numerical implementa-

tion on the characteristic equation for a coated cylinder.

Another test is followed by varying the conductivity of the coated layer on a per-
fectly conducting cylinder from a finite number to an infinite one. Physically, as the
conductivity is increased to very large, the coated cylinder approaches a good conduct-
ing cylinder with a radius b. Fig. 4.4.a shows the trajectory of the first mode of a
coated conducting cylinder with conductivity varying from a small value to a quite
large value. The conductivity is increased by a factor of 1.22 in each step. The
upper square gives the location of the first mode of the perfectly conducting cylinder
with radius a, and the lower square shows the position of the first mode of a perfectly
conducting cylinder with radius b. It is observed that as the conductivity is increased,
the pole is moved toward the location of a thicker perfectly conducting cylinder. The

test on the second mode is shown in Fig. 4.4.b.

4.4. Numerical Results of the Pole Distributions

Based on the equation (4.2.34) and the algorithm discussed above, the zeros of
the characteristic equations (4.2.40. 4.2.52, 4.2.61) are located at a variety of parameter

SCtS.

(oa/c

50

139

The Poles of A Coated Conducting Cylinder

 

 

 

o
b/ a =-- 1.10 I
i
f a
‘ a
i
t 3 i
* t f
40 _‘ t . 't .
a * i I
. ¢ .
‘ ‘ t ‘ §
.‘ . t ‘ ‘
O. . . . ‘.
3' ’. a. 1' 5
.I. .. ‘ e .
-, e e ‘
. a ‘ b ‘
e 3 . t ‘
30 -4 o " 1 i i
o f . t "
fi ‘ t C ‘ f
* . ‘ Q .
c . a
t " ‘5 g ‘ “
g 3 ‘ . t ‘
a it ‘ 1. 't it
e 3' " ix in
‘ .. . f i o.
t '0 “ t a. 3.
t '* * ‘3 a a.
20 —" II 'fi 3' . 3. it
t .t ‘ t 't g.
t ‘ ._ ‘ t .t
t t f ‘. 3, -
i ‘ t ‘ i i
4- * 1" ‘. a. .
e 't O f it- "
t 't 't "'_ . "
e 't "t ‘ ". t
. .0 .. T. f. ‘
e 't 't t _ i.
10 “ e 'e 'e 't ‘f a
t ‘e 't 't f g
e '13-. “a 'e f 1‘
‘ ‘. "E "' “. a.
" *. ”i 3‘ ’. i
e I. t. 't t. *
e t y t t .
e "e '1. ’6. 't 'f
0 t_ t t. h. 1
= -_-_- O 't O ’t V
ona 0 , e, l . . . -.,,_ ,, .
A A: A. A A
l l l
-15 —10 —5 0
Ga / c

 

Fig. 43 The pole distribution of a perfectly conducting cylinder coated with an
air layer.

140

.0008 0:008 05 SV ”0.008 E: 05 A8 53:03:00 05
0032, 000530 9:800:00 000000 0 .«0 00005 030 .05 05 .«o b08070; v.0 .wE

 

 

 

 

0:06
. 3.0. 00. 2.0. 0.0- 2.0. 00.
_ _ _ — _ N.—
.D v Nu; x 030. 00....
...,...fi .0._ ... .0
r 3..
Q Q
Q
d
0 c c m
d
a c manna—B
i ¢
a a 0 in.
Q Q
Q
Q d
...
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d
3.03:ka
.. 9...
2

20¢ 7.030... 0:000m 2C.

 

 

200
2.0. 0.? 8.0. 00.
_ _ p
. x 00;. 00:
An. N- .
:33 .o._ u .0
c .l v.0
c
<
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< < 030.25
c c <
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5.0.26 100

 

 

0.0; 30050005 2:.

141

The zeros of equation (4.2.40) are exactly the poles of natural modes. It is well
known that the poles of a perfectly conducting cylinder are positioned in layers. Fig.
4.5 shows the first 250 dominant poles. Later, these poles are served as the reference

for the studying of more complicated geometries.

The zeros of equation (4.2.41) are mapped from the zeros of equation (4.2.40) by
a quadrature transformation suggested in equation (4.2.42). The result is shown in Fig.
4.6. Only the first layer of the poles of a perfectly conducting cylinder in free space,
is transformed with five lossy parameter sets. It is seen that as the conductivity is
increasing, the poles are shifted toward a negative direction. Note that the damping
coefficients of the lower order modes are shifted more if compared with the higher

modes.

The pole locations for a lossy cylinder are quite different. Fig. 4.7 presents a typ-
ical pole distribution with a normal lossy parameter. The dotted lines are drawn to
show the positions where the poles of a perfectly conducting cylinder are located and
to serve as the reference locations. It is seen that some poles are distributed in layers
close to dotted lines, but some poles lying in arcs are close to the imaginary axis. We
call those poles, close to the imaginary axis, interior modes, while we call those in
layers exterior modes. Further study reveals that the fields of an exterior mode inside
the cylinder attenuate radially from the cylinder surface to the center and the fields of
an interior mode inside the cylinder behave radially as a standing wave. The physical
insight to this difference is that the energy losses of an exterior mode are mainly attri-
buted to the radiation of the surface current. In contrast, the energy losses of an inte-
rior mode are mainly attributed to the power loss inside the cylinder. This main

difference exists in the case of a coated perfectly conducting cylinder.

Fig. 4.8 shows the poles of a lossy cylinder with a rather different lossy parame-

ter, compared to Fig. 4.7 the normalized conductivity is changed from 0.2 to 20. The

142

The Poles of A Conducting Cylinder

 

 

 

50

tone
0
0
.-
t
to
to o
00 e
t t t
e t
to. t t
to
0.. 0
tot e 0
tot. .- t
to e
not.
not... to t c
#00000 0 0
no t
to t e
0 e t
one o
e o t
tote #0 0
tot: t t
at. t 4
to e e
to e
to 0 t
to. t
to. it e
t e
e
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t
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to. 0
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to e
t e
o o
c
to t
to t
at.
to
at
o
0
e
e
at
at
o _ _
4 w 0 0
2 1

(Dale

 

-l0

—l5

oa/c

Fig. 4.5 The pole distribution of a perfectly conducting cylinder.

143

 

J . ’ ..
(ma, 5 . 0.0
2900“ I anal/5::- 1.0
A j A (ma/e --- 5.0 . ‘ ' ’
A I r I ‘ . C
. 4 I 0173‘s r- 10.0 ’ I I
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V ‘ ° ‘ I I
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~ ‘ I I
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--22.oo —1é.5o -1"7.oo -14.50 -12.00 -9'.50

 

rT r T r r—T'fr 7"
--7.00 -4.50 —2.00

Damping coefficient ( Rc(sa/v) )

Fig. 4.6 The first layer of poles of a perfectly conducting cylinder immersed in
lossy medium.

144

The Poles of A Lossy Dielectric Cylinder

 

 

 

50

I I II II n I I II
IIII‘ IIIIC IIII0I IIII III ‘I III IIIIIIIIIIXIII: I2‘ I
...
I I I I I IIIIIIIII.¢.I.I.
IIIIIIIIIIIII II II II II I ..........
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IA
IIIIIIIIIIIII I.I
..I
3
I.I.I ..I
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I.I .
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I I I I .I ...... .I .
I I I I ........ I.I .I
I I I I I ...... I. . .
O. . . . ........... ..‘. ... .‘
IIIIIII .... I.I
.I.I I.I
I.I. .I
I.I. .I .
. .... ... ..
I . .
.... I. It
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.... ... .
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I.I.I I n.
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. I
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.I.I. I I a
I.I.I m
..
A _ _ _
0 0 0 0 0
4 3 2 1

(Dale

 

-10

—15

(Ia/c

Fig. 4.7 The pole distribution of a lossy dielectric cylinder ( a conductivity, 5,

relative permittivity. n wave impedance and a the radius of the cylinder ).

145

The Poles of A Lossy Dielectric Cylinder
so

 

I. O
.9
I.I O "

I.I.I..

I I G '
II
'Ii

     
    
    

40—1

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I
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.5

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I
...

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upturn”, coupes»...

10 . . '. 'o. 3.
—q

 

 

 

Fig. 4.8 The pole distribution of a lossy dielectric cylinder ( o conductivity, 5,
relative permittivity, 1] wave impedance and a the radius of the cylinder ).

146

exterior modes have not been affected very much, but the interior modes are shifted
quite a lot in a more negative direction. Physically the interior modes have more

power loss inside the cylinder with a higher conductivity.

Fig. 4.9 illustrates the amplitude of the o component of E-field. The angular vari-
ation order is chosen as n=6. The amplitudes of 4 interior modes and 2 exterior
modes of E—field are plotted. The standing wave behavior of interior modes and the

attenuation behavior of exterior modes are very clearly observed.

Fig. 4.10 is the pole distribution of a coated perfectly conducting cylinder with a
coating thickness of 50% of the radius. The dotted lines are still drawn here to serve
as the reference. It is observed that the exterior modes are in layers, and interior
modes are in arcs and close to the imaginary axis. Now examine the effects if the
coating thickness is kept same, but the normalized conductivity is changed. The result
is shown in Fig. 4.11. The interior modes are all shifted to the left. The exterior
modes are dominant now since they are lower in radian frequencies and have a smaller
damping coefficient. If the coating thickness is changed instead of the conductivity,
the effect of the coating thickness on the interior modes is shown in Fig. 4.12. The
first are of interior modes is moved far upward when the coating thickness is dropped

to 10% of the radius. Once again the exterior modes are seen as dominant.

Similar to lossy cylinder in free space, a coated perfectly conducting cylinder has
interior modes too. Their fields behave as standing waves in the cladding region. Fig.
4.13 shows the amplitude of the o component of E-field inside the cladding region.
The angular order is selected with n=8. Four interior modes and two exterior modes
are illustrated. The difference of the field distribution inside the coating layer between
the exterior modes and the interior modes are very apparent. In addtion it is seen that
the amplitude of E-field of an exterior mode inside the coating layer is relatively very

small. That is why the exterior mode is substantially affected only when the coating

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a: a? a:
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r— O”- Qg owl U run

 

 

148

The Poles of A Coated Conducting Cylinder

 

 

 

 

15
b/a = 1.50
t
to— i
I
'. t
.
'. '. t
(Ga/C 1 1: O\.
. , , .
I!"
t 1 t
2‘ .
s—r ‘ 2‘
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h“
‘ fi 2 t
" I
# 'i T I. t
I '- I.
‘3 T ‘4 I
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. 2’. ‘l ’. . ft .
cma=2. e,=s .. .. . = r
I 2‘ I. n- "
0 l l
—10 -5 o

(Ia/c

Fig. 4.10 The pole distribution of a coated perfectly conducting cylinder ( a
conductivity, 6, relative permittivity, n wave impedance, a the radius of the
cylinder and b the radius of the coating layer ).

149

The Poles of A Coated Conducting Cylinder

 

 

 

 

lS
b/a = 1.50 :‘
ro_ ‘ g
.3
. 2
O .2
O . I.
‘ '.
I : ’
CDC/C ‘ O ',I
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t‘ I
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- I
O 0 ¢ . . g
. ‘ o .I . 0
ma = 20 , E, = 5 fi’ . '2. . C
0 l I
—10 -S 0
O’a/c

Fig. 4.11 The pole distribution of a coated perfectly conducting cylinder ( a
conductivity, 8, relative permittivity, 1] wave impedance, a the radius of the
cylinder and b the radius of the coating layer ).

150

The Poles of A Coated Conducting Cylinder

 

 

 

 

15
.
.
b/a = 1.10 .
.
t
.
‘
O
i
‘ I
. g .
I ' . I
10-1 .I ‘l I
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l' I
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5... I
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I - I 'l' ‘
I .I * ‘
I «.I I *
0110 = 2 , 8, =5
O I '.I I t
l I
—10 -5 0
00/6

Fig. 4.12 The pole distribution of a coated perfectly conducting cylinder ( a
conductivity, 5, relative permittivity, 11 wave impedance, a the radius of the
cylinder and b the radius of the coating layer ).

151

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152

thickness is comparable to the radius.

4.5 Effects of Lossy Parameters on An Exterior Mode

Since a perfectly conducting scatterer has only exterior modes and the normally
used coating layer is not very thick, it is only necessary to investigate the effects of
lossy parameters on the exterior modes. The extensive pole trajectories by varying the

lossy parameters have been pursued. But only a few examples are specified.

Fig. 4.14 shows the trajectory of the second mode. The pole is traced as the
coating thickness is varied with a step size of 0.04. The trajectories with four sets of
parameters are shown here. The circle line serves as the reference line which is the
location of the second pole of a perfectly conducting cylinder with radius b. As the
coating thickness is increased, the radius of the out layer b is increased and the circle
line positions the pole of the cylinder which is increasing in radius. The star line
corresponds to a normalized conductivity of 80. After a few steps, the star line comes
close to the perfectly conducting cylinder line, and after more steps the triangle line,
which corresponds to a conductivity of 15. The last one coming close is the square
line with conductivity of 5. In Fig. 4.14, we can see that when the conductivity is
higher, the pole of a coated cylinder behaves more like that of a perfectly conducting
cylinder, and when the coating thickness is thicker, the pole of a coated cylinder
behaves more like that of a perfectly conducting cylinder.

Fig. 4.15 presents the first pole traced by varying the permittivity. The pole is
traced as the coating thickness is increased with a step size of 0.04. Three trajectories
for three different parameter sets are given. It is seen that when the coating thickness
is small the poles with different permittivities are located close to each other, however

when the coating is thick, they are more separated.

Radian Frequency ( Irn(sa/v) )

153

 

 

 

 

1.50 +
: Iii/0-1.0
.1 l3!
... O on I a
1.35:J o A D
D
j o I A o
: . . °
120-: o u A n
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I D
1 o I A
4 0 fi 0
1.054 ° ' A
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.4
-4 0° '. A D
0.90‘ O I A C
.1 o I i
.r o I A a
_4 Do I. A D
-< O I A D
0 75 4 °o 'uA 0:
' . x er: 10.0 ana= 80.0 °o{:.. 3i
j A er: 10.0 ma= 15.0 ‘4 g!
.1 Cl 8 = 10.0 a'na= 5.0 A g
i o er= 10.0 a'qa== on a
0.60 T—T—T 2T 1’ T Y FT Y T I I I r T I T T I I r 1 T I I I 1
-O.90 -O.80 -0.70 —O.60 —O.50 —O.4O —O.3O -O.

Damping coefficient ( Re(sa/v) )

20

Fig. 4.14 Trajectory of the second exterior mode of a coated perfectly conduct-
ing cylinder as the coating thickness is varying.

( Im(sail/V) )

Radian Frequency

154

 

0.55

 

 

 

.J
-1 b/a=1.0
'1
a I
I 0
-l b! b o
0.48: AOA'o b (in o o c:
_j I
A I
Z A " o
a A I
a . a
0.41 — A I.
q A o
'i A ‘
I A ,, o
a A
.. A ‘ o
0.34— A I o
4 A I
-l AA ' 0
: A I D
-4 AA ‘ U
1 AA 1" °
0.27“4 A I o
I 'f. 0
0
j A 8r 50.0 Una: 15.0
., D 8r 20.0 ana= 15.0
0.20 f Tilt: 15.0r war: 115.9 I 1 T' T T r r T I r
—O.7O -O.57 -O.45 - .32

Damping coefficient ( Re(sa/v) )

—O.20

Fig. 4.15 Trajectory of the first exterior mode of a coated perfectly conducting
cylinder as the coating thickness is varying.

155

4.6 Discrimination of Conducting Plates and Conducting Plates Coated with a

Lossy Layer Using E-Pulse Technique

One of the most important properties of the E-pulse technique is its ability to
discriminate targets at multiple aspects. Of particular interest is its ability to discrim-
inate aircrafts when the excitation field illuminates the wings of the aircrafts at normal
or end-on incidence. This section will presents experimental results demonstrating a
successful application of E—pulse technique to discriminate rectangular plates simulat-
ing aircraft wings [51].

There is recent interest in the discriminating aircrafts coated by imperfectly con-
ducting materials to reduce their radar cross-sections. This section also presents the
results demonstrating that the E-pulse technique is suitable for discriminating rectangu-
lar plates coated with microwave absorber. As revealed in the sections above, the
natural modes are not substantially affected when the lossy coating is not thick. Since
the E-pulse technique is only based on the natural frequcies of the targets, it implied
that the success of the E-pulse scheme is not impaired by the presence of lossy coat-
ing.

Fig. 4.16 shows the transient scattered field response of a 4"x10" aluminum plate
measured above a conducting ground screen, at end—on incidence (electric field parallel
to plate edge, magnetic field perpendicular to plate face). Similar results have been
obtained for normal incidence (electric and magnetic fields parallel to plate face) and
for oblique incidence, and for a 6"x15" plate at these same three aspect angles. The
natural frequencies contained within the late-time portions of these waveforms have
been extracted using both a continuation method [35] and an E-pulse method [36], and
the typical results are shown in Table 4.1. It is apparent from the results that different
subsets of the natural mode spectra of the plates are excited at different aspect angles.

However, it is also seen that the natural frequencies of the modes excited at different

Relative Amplitude

35 00

27 $0

30 00

‘12 50

S 000

‘3 300

-17 50

156

 

1

 

 

 

 

 

 

000

1.000

r

2.000

3.000

V 000 3 000 6 000

trrne'

IFW F15

P 000

8.000

9 000

Fig. 4.16 Radar response of a conducting plate (4"x10") at end-on incidence.

10 00

157

Table 4.1 Natural Frequencies of Rectangular Plates

 

 

 

 

 

“19““?“0” 10 x 4 " Plate 15 x 6 " Plate
direction

Sl 0.2703 + j 1.500 Sl 0.0574 + j 0.996

Face . 82 0.2742 + j 4.921 82 0.2757 + j 3.245
Illuminated . .

S3 0.0126 + J 7.474 S3 0.3303 + J 7.131

S4 0.0693 + j 9.799 S4 -1.0359 + j 10.59

81 0.2739 + j 1.526 S1 0.2322 + j 1.017

S2 0.2235 + j 4.450 32 0.2282 + i 3.237

Side _ S3 0.2737 + j 6.107 83 0.1708 + j 5.401
Illumrnated .

84 0.0140 + j 9.859 34 -0.0463 + J 6983

86 -0.3318 +j 11.86

81 -0.2731 + j 1.527 SI -0.2616 + j 1.039
Tiltly . .

. 0. 21 .1 0. .
S4 0.1428 + j 9.702 S4 0.0110 + j 9.810
SS 0.0120 + j 6.629 85 -24077 + j 12.02

 

 

 

158

aspects are consistent between the two measurements.

The natural frequencies of the entire set of modes excited at the three aspect
angles for which measurement were made are used to construct E-pulse waveforms for
the 4"x10" and 6"x15" plates. These waveforms are shown in Fig. 4.17. Discrimina-
tion between the two plates is undertaken by convolving the E-pulses with the meas-

ured responses of the two plates at various aspect angles.

Fig. 4.18 shows the convolution of the E-pulse synthesized for the 4"x10" plate
with measured response of the 4"x10" plate, at normal incidence. the small amplitude
of the late-time segment of the convolved response confirms that the measurement is
indeed from the 4"x10" plate. In contrast, Fig. 4.19 shows the convolution of the E-
pulse of the 6"x15" plate, with the measured response of the 4"x10" plate. Here the
relatively large late-time component of the convolved response indicates that the meas-

ured response is not that of the 6"x15" plate.

Similarly, Fig. 4.20 shows the convolution of the 6"x15" E-pulse with the meas-
ured response of the 6"x15" plate, at normal incidence. As expected, the small late-
time component of the convolved response indicates the target is a 6"x15" plate. Fig.
4.21 shows the convolution of the 4"x10" E-pulse with the 6"x15" plate response, and
the large amplitude of the late-time portion indicates that the target is not a 4"x10"

plate.

The results of Figs. 4.18-4.21 provide convincing evidence that two conducting
plates, simulating the wings of two different aircrafts, can be discriminated using the
E-pulse technique at normal incidence. Equally convincing results have been obtained

at end-on and oblique incidence.

The main purpose of this chapter is to show the applicability of the E-pulse tech-

nique to targets coated with lossy layers. It is desirable to obtain equally convincing

Relative ampl1tude

S 000

3.930 *

3.073 ‘

1.013 ‘

7500 7

‘ 3123 ‘

-1 37s .0— _-

-2 “38 1

-3 500
0

159

 

 

 

 

 

 

 

 

 

 

 

 

”___-Am-_-—————.-—-—

 

 

 

 

 

l
I
__E_

 

 

I. _ — ..

I
l
l
l
l
1
|
I
l

 

 

 

T

000 5000

Fig. 4.17 Waveforms of E-pulses for rectangular plates (solid line is for the
plate of 15"x6" and dash line is for the plate of 10"x4").

r

1.000

1

r

500

2 000

1 1n”?

2 500

1r1 F15

3 000

3.500

9 000

__f

H 500

5 000

Relative Amplitude

160

 

35.00
2‘19“
10.731
10.63‘

2.5001

‘3 623 ‘

f late-time
-13 75 ~ '

-21 98 "

 

 

 

'30.00 t r r r r T r I f
0 000 1.000 2.000 3 000 “l 000 3 000 6 000 7 000 8 000 9.000 10 00

Ilm? to OS

Fig. 4.18 Output of the convolution between the E-pulse synthesized for con-
ducting plate (4"x10") and radar response from the same conducting plate

(4"x10") at normal incidence.

0...“.qu J u _ «fig

 

AU)!

. 1.5.:

35 00

20.00

10.73

Relative Amplitude

‘13 75 ‘

-21 88 ‘

-30 00

10 63“

2.500

'5 623 ‘

161

 

1

1

l

 

late-time

    

 

 

Y T
000 1.000 2.000

Fig. 4.19 Output of the convolution between the E-pulse synthesized for con-
ducting plate (6"x15") and radar response from the conducting plate (4"x10") at

normal incidence.

r

3 000 H 000 3 000 6 000 7 000 0 000

11m? 1n ns

T

9 000

10 00

Relative Amplttude

05.00

30.30 4

00.00 <

7.300 4

-S.000 ‘

‘17.30 4

'30 00 4

“42.501

-55.00

162

 

..- -1-0- --

late-time

 

 

f r r

 

0.000 1.000 22000 3.000 H 000 5 000 6 000 7 000 0.000 9 000 10 00

film? 1n US

Fig. 4.20 Output of the convolution between the E-pulse synthesized for con-
ducting plate (6"x15") and radar response from the same conducting plate
(6"x15") at normal incidence.

WEIJ J - _nurt C .05.. a a .1 ~ raw.

Relat rue nmpl ttude

163

 

35 00

20.00 a

10.73 ‘

10 ”J

2.500 a

‘3.623 ‘

‘13 75 4

-21 88 7

 

 

 

 

 

-—_4D__

 

 

late-time

 

 

 

-30.00
0 000

r
1.000

T T
2.000 3 000

T f T

H 000 3 000 6.000 7 000 0 000 9 000

trme in ms

10 00

Fig. 4.21 Output of the convolution between the E-pulse synthesized for con-
ducting plate (4"x10") and radar response from the conducting plate (6"x15") at

normal incidence.

164

results from the rectangular plates coated with lossy materials. To demonstrate that
the presence of a lossy layer will not adversely affect the ability to discriminate using
the E-pulse method, measurements have been made of the response of a 4"x12"
aluminum plate at normal incidence, and an E-pulse for this target has been con-
structed. Subsequently, the same 4"x12" plate has been coated with a one half-inch
thick layer of microwave absorber, and its response again measured at normal
incidence. The resulting measured waveforms are shown in Figs. 4.22-4.23. It is

observed that these two waveforms are quite different.

With this data, an attempt is made to discriminate the 6"x15" plate from the lossy
foam coated 4"x12" plate, assuming the information is only available for the 4"x12"
plate without lossy coating. Fig. 4.24 shows the convolved response of the 6"x15"
plate E-pulse with the response of the coated 4"x12" plate. As expected, the late-time
portion of the convolved response exhibits a large amplitude. In contrast, Fig. 4.25
shows the convolved response of the E-pulse synthesized for the uncoated 4"x12" plate
with the measured response of the coated 4"x12" plate. The small amplitude of the
late-time response demonstrates that the coating has little effect on the E-pulse
discrimination. Lastly, Fig. 4.26 shows the convolution of the E-pulse for the
uncoated 4"x12" plate with the response of the 6"x15" plate. Here the late-time por-
tion has a large amplitude. Similar results have been obtained using waveforms meas—

ured at end-on or oblique incidence.

The implication of these results is that discrimination between radar targets using
the E-pulse technique is possible even if the targets are coated with lossy material,
and the information is available only for the uncoated targets. The reason for this is
that the E-pulse waveforms are based entirely on the natural frequencies of the targets,
and the natural frequencies of the resonance region are not perturbed greatly by the

addition of the lossy layer. As investigated in the last few sections, when the coating

35 oo

27 so

an 00
0.1
'3

.5 12 so
CL
E

C s 000
c.
>

2 -a :00
a
c:

-‘.o oo

-'.7 so

-23 00

o

165

 

l

l

l

l

 

 

 

 

 

— — ___ ———J—--—— —— — —-i
000 1 000 2.000 3.000 H.000 3 000 6 000 7 000 0 000 9 000

t 1m?

10 HS

Fig. 4.22 Radar response of a conducting plate (4"x12") at normal incidence.

10 00

166

 

35 00

27 30

30 00

12 30

‘2 300

Relalrve Qmpltlude

'10 00
-17 50

-25 00
0

1

l

1

soul

 

.4

T r 1T

 

 

 

000 1.000 2.000 3 000 H 000 5 000 6 000 7 000 0 000 9 000

11m? 1n OS

Fig. 4.23 Radar response of a conducting plate (4"x12") covered with a lossy
layer at normal incidence.

10 00

Rolatrve Qmplttude

25 00

10.13'1

11 254

‘0 373-1

'2.500

'9 375 1

167

 

d

-16 as 1 late-time

-23 13

~30 00
0

q

 

 

 

r T—

V T T T T T T
.000 1.000 2.000 3 000 H 000 5 000 6.000 7 000 0 000 9.000

ttme 1n US

Fig. 4.24 Output of the convolution between the E-pulse synthesized for con-

ducting plate (6"x15") and radar response from the conducting plate (4"x12")
covered with a lossy layer.

10 00

Relative Amplitude

168

 

35.00

A

26.25

17.30 ‘

‘0 730-4

1

'0.730 ‘

1

'17 50
late-time

-26 25 1

 

 

 

‘33 00 I r 1 r f r W I r
0.000 1.000 2.000 3 000 H.000 S 000 6.000 7 000 8 000 9 000 10 00

time in ns

Fig. 4.25 Output of the convolution between the E-pulse synthesized for con-
ducting plate (4"x12") and radar response from the same conducting plate

(4"x12") covered with a lossy layer.

Relative amplitude

169

 

25 00

10.13‘

11.33‘

* 373‘

-2 500 ‘

‘3 375 ‘

‘16 35 ‘

~23 13 ‘

 

l late-time

 

 

‘30 00 f f r Y
0 000 1.000 2.000 3 000 H 000

77 f

3.000 6.000 7 000 8.000 9 000

film? in ms

Fig. 4.26 Output of the convolution between the E-pulse synthesized for con-

ducting plate (4"x12") and radar response from the conducting plate (6"x15").

10 00

170

thickness is not thick the exterior modes are dominant, and the natural frequencies of
the exterior modes are determined primarily by the geometry of the underlying con-

ducting plate.

It is interesting to note that although the measured waveforms of the 4"x12" plate
and the 4"x12" coated plate (Fig. 4.22-4.23) are quite different, the natural frequencies
contained in each are nearly identical. Thus, the presence of the lossy layer serves
mostly to perturb the amplitudes and the phases of the natural modes. An obvious
scenario is as follows. The natural frequencies of a group of aircrafts are determined
by scale model measurements, and a set of E-pulses constructed. However, in actual-
ity, each of the aircraft is operated carrying an additional coating of lossy material, for
the purpose of thwarting high frequency radar. Since the E-pulse technique is based
on resonance region frequencies, the presence of the coating does not affect the ability

to accurately discriminate the targets in practical situations.

4.7 Conclusion

The pole distribution of a coated cylinder structure has been investigated in this
chapter. The special attention has been paid to the effects of the lossy coatings on the
dominant natural modes. It has been shown that the natural frequencies of exterior
modes are substantially shifted on the complex plane only when the coating thickness
is comparable with the radius of the cylinder. When the coating material has a param-
eter of ona > 100, it has little effect on the natural frequencies of the exterior modes.
The natural frequencies of the interior modes are greatly dependent on coating thick-
ness and parameters. As a rough estimation, the interior modes are no longer dom-
inant when the conductivity satisfies the condition of cm > 100, or when the coating

thickness is less than 10 percent of the radius of the cylinder.

Chapter 5

DETERMINATION OF THE NATURAL MODES
OF A RECTANGULAR PLATE

5.1 Introduction

Subsequent to the introduction of the Singularity Expansion Method by Baum in
1971 [4], considerable emphasis has been devoted to the analysis of various perfectly
conducting scatterers such as sphere, prolate sphere, infinitely long cylinder and wire
structures [18-20]. However the SEM analysis of a rectangular plate has not received
much attention of researchers, even though a rectangular plate is a very fundamental
geometry for many realistic scatterers. The knowledge of its natural modes is then of

importance in the application of the SEM to many transient scattering problems.

The natural modes of a perfectly conducting body can be obtained numerically by
means of the method of moments solution to an electric field or magnetic field integral
equation formulation. In early 1974, Samii and Mittra [52,53] proposed an integral
equation for formulating the scattering problem of a rectangular plate illuminated by a
plane wave. Later on this integral equation was modified by Pearson [54] to extract
the SEM parameters of a plate in the complex domain. The formulation used by Pear-
son was essentially the same except the real frequency was directly replaced by a com-
plex frequency. The controversy on the completion of the modified formulation has

kept the consequent numerical results from being published.

As is well known the singularity of the surface current occurs at all edges. The
special numerical treatment, which uses basis functions containing the correct edge
singularity in subdomains near edges, is more adequate to represent the currents with

singularity at edges [55]. But to simplify the programming, the uniform piecewise

171

172

constant functions are employed as basis functions. The edge effects remain under

such a simpler treatment.

This chapter will develop a new set of coupled electric field integral equations to
determine the natural modes of a thin rectangular plate, and present the results of the

method of moments solution to this equation set.

Section 5.2 introduces an independent derivation of an previously used formula-
tion which was adapted by Pearson in 1976. Even though it has been a controversial
formulation in regard to its completion, it remains as a good approximate solution to

this specified problem for the first few lowest dominant modes.

Section 5.3 puts forward a new set of coupled electric field integral equations for
a thin rectangular plate, and reveals the symmetry relationships in the distribution of a

natural modal current.

Section 5.4 presents the numerical procedure based on the method of moments
solution to the new set of EFIEs. The numerical convergence tests, which include the
thin-strip limit and the solution with more basis functions, are performed in section 5.5

to validate the solution procedure and computer programming.

Section 5.6 shows numerical results of the natural frequencies and the modal
current distributions. Section 5.7, the last section, presents the comparison between the
natural frequencies predicted theoretically and that extracted from the measured

response to verify our theory.

5.2 New Derivation of An Existing Formulation

As a brief historic overview, the formulation proposed by Samii and Mittra [52]
is derived in this section. What is introduced is an independent derivation but the final

result is exactly the same. A new formulation for the problem will be derived in next

173

section.

Figure 5.1 is the geometry of a thin rectangular plate. Following the basic steps

in reference [52,53], one can easily obtain the integro-differential equation of

(i+iZ-+kZ)IKd>dx’d)/=-zxfl (5.2.1)

8x2 3y2 32

where K is the current density on the surface of the plate, 2 is an unit vector in z

direction, k is the wave-number and the Green function is

<r>= :11; (I'm-"U lr-r’l

It should be indicated that the above equation is valid only in the range of

J < x < 9- and -—2 < y < %, and the components of K, & K, are uncoupled with

2 2 2

respect to the incident polarization. But since an appropriate solution should have
K, & Ky coupled together, the boundary conditions have to be imposed again when a
solution is pursued. Thus the redundancy is consequently created which was demon-

strated in the numerical procedure.

Since our purpose here is to find a nontrivial excitation-free solution in the com-
plex domain, the preceedin g equation is modified by replacing the real wavenumber

with complex frequency:

a_2_+ a__’-_2

- K <1) dx’ = 0 5.2.2
<— 83+ <- :2) ) I dy ( )
In order to simplify the notations in the coming derivation, the vector potentials
are introduced:
Ar: 4 —-°-J'1<,"— (5.2.3.a)
Ho
Ay ___ 71192—3... (5.2.3.b)

174

row x

 

I
MIO‘

MIG

 

 

 

 

 

Fig. 5.1 Geometry of a rectangular plate.

175

where y = f and R = [(x-{)2 + (y—y’)2]1’2. The alternative forms for (5.2.2) are written

as:
82 82 s 2
—_ .I— — — A = 0 5.2.40
82 82 s 2 _

There are several methods to seek a proper homogeneous solution to equation
(5.2.4). We used the method by which the vector potentials are expanded in series of
cylindrical harmonic functions. It is assumed that the cylindrical harmonic function set
i 1001)). 11(yp)sin¢, 11(YP)COS¢, 1201330124): 12(YP)C‘032¢: 13(YP)Si"3¢» 13(YP)COS3¢. ------- I is
complete for the representation of any function that satisfies (5.2.4). Thus, the general

homogeneous solutions for A, and A, are:

A, = Z (ancos no + b,sin no) 1,,(yp) (5.2.5.a)
Ay = 2 (c,cos no + dnsin no) 1,,(yp) (5.2.5.b)

where p and 0 are cylindrical coordinates of the observation point on the plate with
x = pcos d), and y = psin ([1. Since the A, and Ay are coupled, the coefficients of
a,’s, b,’s, c,’s and d,’s are not all independent. An equation is generated when the
boundary condition of H - n = 0 is applied on the plate surface where the n is the unit

vector in the normal direction of the plate:

2 - V x A = 0
That is
a - _a_
By A, — axA’ (5.2.6)

In the cylindrical coordinate system, the derivatives with respect to x & y can be

expressed as:

i = gimp i + COS —-a— (5.2.7.3)

8y 39 P «M

176

— = _ — fl—
8 cos (pa 8 (5.2.7.b)
Using (5.2.5) and (5.2.7), we obtain:

aiyA’ = Z (a,sin¢cosn¢ + bnsimbsinmb) 7 [KW P)

+3 2 n (-—a,,cos¢>sinn¢ + bncosocosno) 7 M? P) (533)

It rs proper to introduce some identities here to simplify the above representation
Referring to [47], we have:

l,’(z)=l,_1(z)- 31,,(2)

(5.2.9.a)
I,’(:) = I,,,(z) + f ,(z) (5.2.9.b)
Letting z = yp, the modified identities are expressed as
7 1mm = 7 map) — firm (5.2.10.a)
7 1mm = r lump) + {Di/Mp) (5.2.10.b)
With (5.2.10), we are able to rewrite (5.2.8) as:
say—A, = —;-£{a,[sin(n+l)¢ - sin(n—-1)¢] + b,[cos(n-l)¢ — cos(n+l)¢]} YA’CYP)
+-1— Dr {a,[-sin(n+l)¢ — sin(n—1)¢] + b,[cos(n—l)¢ + cos(n+1)¢]} 7!,(yp)
Pi"; ansin(n+1)¢[71..'(YP) - g-MYPH- ansin(n-1)¢[Yln'(YP) + €- ”(WM
+ anOS(n-1)¢[Yln'()p) + f "(1])” - busin(n-1)¢[Yln'(lp) - 'EIMPH
= "it 2 a,[ sin(n+l)¢l,+1('yp) - sin(n—1)¢I,_1(yp) 1
+ bn[ COS(n-1)¢1...1(W) - 605(n+1)¢1n+1(YP) ] (5.2.11)

Followrng the same steps, we can take the derivative of A with respect to x

£14, = g(cneosocoan) + dnCOS¢Sinn¢) Y HOP)

177

-11. D;(-¢,sirr¢sinn¢ + d,sin¢cosn¢) Y 1,.(YP)

= —;-2{c,[cos(n+l)¢ + c0s(n-l)¢] + d,[sin(n—1)¢ + sin(n+1)¢]} yI,’(yp)

mg- Dr{-c,[cos(n—1)¢ - cos(n+1)¢] + d,[sin(n+l)¢ — sin(n—1)¢]} 'YMYP)

=§2 c.cos(n—1)¢w1.’(rp) + % mm] + c.cos(n+1>¢IrI.’(w> - film”

+ d.sz'n(n+1)¢[ W(Yp) - imp) 1 + d.sin(n—1)¢[ W(rp) + % .(rp) l

= 321 )3 c.[ costn-1)¢I..1(vp> + COS("+1)¢’n+1(YP) 1

+ d,[ sin(n—1)¢l,_l(yp) + sin(n+1)¢l,+,(yp) ] (5.2.12)
If the relation of SEEM: %A‘ is enforced now, coefficients in (5.2.11) and

(5.2.12) are not easily related since the summations are not uniformly ordered. By
reordering the (5.2.11) and (5.2.12) according to the indices of { I, }, (5.2.12) is

altered to the form:

aa—x-A’=12(C"’1 + C’H’l) COSIIOIRCYP) + (dn-l + dM-l) Sinn¢ln(W) (5.2.13)

with c,-1 & d,_1 = 0 if n=0
and (5.2.11) to the form:

a—ayA" = %E (an—l - an+l) sinn¢l,(yp) + (bn+l " bn-r) COS’W’AYP) (52-14)

with a,_, a b,,_, = 0 if n=0.

aA =aA we

Now using (5.213) and (5.214), to satisfy the relation of— 8— —a , ,
1A y

obtain:

Cn-t + Cn+1 = bn+1 " bn—r (5-2-15-3)
dn—l + dn+l = an_1 — a,“ (5.2.15.b)

178

If we put (5.2.15) back into (5.2.5), it is still difficult to see the the coupling
between A, and A, since the coefficients in (5.2.5) are not coupled in a one-to~one

form. But if we try to change the coefficients in (5.2.5) to

a, = aH’ + a,,+1’ (5.2.16.3)
b, = -b,,.1’ — bfil’ (5.2.16.b)
c, = c,_1’ - ch’ (5.2.16.0)
d, = b,_1’ — dfil’ (5.2.16.d)

with a_1’ = b_,’ = 0-1 = d_1’ = 0. We then have

Cn—l + CH = Cn—z' - 0;: + 0n, - Cn+2' = fin-2' ‘ Cn+2'
bn+l — bII-I = _bn+2’ " bu, + n, + bit—2’ = bn-Z’ _' bn+ I
an—l - an+l = all-2’ + an, " an, " 0M2, = all-2’ " all-+2,
dn—l + dn-t-l = din-2’ — dn’ + dn’ _ dirt-2’ = din—2’ - n+2,

Equations (5.2.15) are now equivalent to

Cn_2’ - Cn+2’ = bn_2’ — burg, (5.2.17.3)
arz’ - “n+2, = drz’ - dn+2' (5.2..17b)
If the above equations are formed as a matrix equation, it can be shown that the

unique solution will be

a,’ = d,’ (5.2.18.a)

I

,, = c,’ (5.2.18.b)
Finally, equations (5.2.18) are substituted into (5.2.5):
Ax = 2 (61.608 m? + busin n¢)l.(rp)
= 2‘, (a...1' + a.+1’)cos n¢l.(vp) — (b._1’ + b..+1’)sin mam) (5.2.19.a)
A, = £1(c,,cos no + dnsin n¢)l,(yp)
= E; (cu—1’ - calficos n¢1.(vp) + (d..1’ — n+1')sin n¢1,(yp) (5.2.19.b)

It is easy to compare now if the above summations are re-sequenced according to

the indices of a,’. b,’. c,’ & dn'

179

A, = Za,’[cos(n+1)¢ Ifllflp) + u,_1cos(n—l)¢ 1,.1(yp)]
— b,’[sin(n+1)¢ I,+1(yp) + u,_lsin(n-1)¢ I,_1('yp)] (5.2.20.3)
A, = 2c.'[cos<n+1>¢ Imam) - u..1cos<n—1>¢ I.I(vpn
- d,’[sin(n+1)¢ I,+1(yp) — u,_lsin(n—l)¢ 1,,_1(7p)] (5.2.20.b)
with
“=0 n<0
m=l n20
Using (5.2.18), we have

A. = Ea.’IwS(n+1)¢ 1mm) + u._1COS(n-1)¢ [HUN]
:b,’[sin(n+l)¢ 1,,+,(yp) + u,_lsin(n—1)¢ I,_1(‘yp)] (5.2.21.a)

Ay = an’lcos(n+1)¢ 1mm) - u.._1COS(n-1)¢ 1.40m]
:an'ISin(n+1)¢ 1n+1(YP) - un-tsin(n-1)¢ 1.409)] (52-2113)
It is worth noting that the above equations are equivalent to the representations

for A, & A, which were used by Pearson in [54]. Referring to [54],

A. = 'tuZJM‘dSI cos(n+1)¢J.+1(-jSp/C) - un—1603(n—1)¢J._1(-jsp/C) ]

-f'd;[ sin(n+1)¢/n+1(—jsp/c) — u,_1sin(n—l)¢.l,_1(—jsp/c) ] (5.2.22.a)
A, = Lsmzf‘m sintn+1>w...(—jsp/c) + uHsintn—I)¢J..1<-jsp/c) 1

—j"d;[ cos(n+1)¢.l,,+1(—jsp/c) + u,_1cos(n—1)¢J,,_1(—jsp/c) ] (5.2.22.b)

It is well known [47] that

Ml) = (-D" NI) (5123)
In(-Z) = (*1)"’n(2)

and it is easy to deduce the following relation:

j“J,,(—jsp/c) = (—1)"I,,(-sp/c) = 1,,(sp/c) (5.2.24)

180

Replacing the Bessel functions in (5.2.22) with modified Bessel functions results

in
A. = fimmt cos(n+1)¢1.+1(sp/C)+ u.-1cos<n-1)¢I.-.<sp/c) 1
+jd;[ sin(n+l)¢l,+1(sp/c) + u,,_,sin(n—1)¢I,,_1(sp/c) 1 (5.2.25.a)
A, = Jf—zdzt sm<n+1>¢l,..1<sp/c) - u...1sin<n—1>¢I.-1(sp/c> 1
-J'J.I[ 005(n+1)¢1.+1(sP/C) - un-rcOS(n-1)¢l —1(SP/C) ] (5.2-25b)

Now we can see that (5.2.25) and (5.2.21) are exactly the same as long as we set

s/c = y, 1%d: = a,’ & flgjd; = -b,,’.

5.3 New Coupled Electric Field Integral Equations

In this section, a new set of coupled electric field integral equations is developed.
We will start with the general electric field integral equation in the complex domain
and then aim to obtain a Hallen-type formulation because the Hallen-type equation has

a better numerical convergence and stability, according to linear antenna theory.

Consider a thin rectangular plate located on the x-y plane, as shown in Fig. 5.1.
As introduced in Chapter 2, a homogeneous electric field integral equation in the com-
plex domain is satisfied by a modal surface current distribution on the surface of a thin

plate. Mathematically it is described as:

 

R
[ [ V'-K(r’.s)t-V - 721mm) 1:; ds’ = 0 (5.3.1)

where

R=lr’—rl

181

r’ and r are the coordinates of the source point and the observation point, s is the
natural frequency of a modal current, K is the modal surface current density and t is
the unit vector in the tangential direction on the plate surface. For the geometry
shown in Fig. 5.1, the surface current K is two-dimensional and it is natural to apply
a rectangular coordinate system here. From (5.3.1) two decomposed and coupled

scalar equations result:

     

     

I{[-£—,— —K,(x’,y’ ,s) + 83—7 K’,(){,y J)]—72-—1(,,()(,y’3)}‘1;q= O (5.3.2.a)
[{l-ag; KAX’Hy’S) + 33—, ,(A’y ’.s)]— - 72K (10’s m} (5.3.2.b)

In order to simplify the above equations, some further steps are followed:

 

dydsR
[33; mm, was

=ij/2df I: —Kx(1.y’.S)aa—x (iTR-Mf

Integrating by parts leads to

I

     

e—qR
I37 Km ”31:41:12

           

 

{Ta Kg’ 32 (e-qR
=_I:d’/K 41m A. n. I ’( ”57—3.: me)“

It is useful to note that the current has no normal component at the edges of the
plate, and R is only the function of I x — X I & I y — y’ I. Thus the following equations
are introduced:

K,(-a/2, y, s) = 0 (5.3.3.a)
K,( all y, s) = 0 (5.3.3.b)

R 2 R
32 t“ )= «Ii-(i) (5.3.4)

 

 

afax 41tR 3x2 41tR

182

If (5.3.3) and (5.3.4) are applied, it follows that:

b/2a/2

a I a e-IYR I
I2 .528)! Xxx.) .s) ax( 41m) dx’dy

b/2 a/2

didI/I KIO’J’M—a-ze 2R ’

8x2 4KR

 

   

= 1 [gr ,y (5.3.5)

             

3:241tR

In a similar way, we can get:

a a e-IYR I
Iay’ K,(Jc’,y’,s)a x4ans

I

 

 

   

R
=I flail”: —K,()(,y’ ,s)-—- 3y 4:]?

y’=b/2

           

   

43R y’=—b/2
32 ‘7”

e
axaxI 4m W]

012
-— I Ice/52s)
—a/2

 

As denoted in (5.3.4), the edge conditions for the y-Component current are

K,( x, -b/2, s) = o (5.3.6.a)
K,( x, b/2, s) = o (5.3.6.b)

Applying (5.3.6) to the preceeding equation results in:
£337 K,(J(,y’.s)ax -e—-—4_1:gds

a/2 H2

_ a2 a“ ,

 

 

       

=II:(1.yS) (5.3.7)

ax 33y( 41tR
Parallel to (5.3.5) and (5.3.7), we can obtain:

     

I31 KAJ/J'J):aye MRR

 

         

= [ K,(.(,y’,s) 8x; 4“,, (5.3.8)

183

 

IWKAW’) ”A

 

     

     

ay 2:an
“I, KI‘J'J)::2( 4,). (5.3.9)
We can use (5.3.5)-(5.3.9), to transform (5.3.2) into:
, a;
IIKMMHB x242) +K ,(x ,-——y’,s) ,xa aleay 41m (5.3.10.a)
KN“ — Ki 5.3.10.1)
II (.y’as)xay+K, ,(y’sxa‘: ( )

       

It is noted that the differential operators inside (5.3.10) are performed on the field
coordinates rather than the source coordinates, and that they can be taken out of the

integrals.

a
a—a—xy

£45K IK V(r,) .S)—-1c;ds’= o (5.3.ll.b)

2 R
(i-fl I w.r.sfi—I—

3331”“

As we did 1n the last section, it is proper to introduce vector potentials again.

K,(x’

            

(5.3.11.a)

            

        

             

A,(x. y, s) = MIX,“ ’ (5.3.12.a)
A,(x. y, (5.3.12.b)
With representation of (5.3.12), (5.3.11) are simplified to the form of:
) A — A, = 0 5.3.13.a
122283;); ( )
-y’ A, +—— 82 =0 (5 3 13 b)
(:a:_2 ) 8 MW . . .

It is interesting to know that with one more operation, the above equations are

easily transformed to the same integro-differential equations which were developed by

2
Samii and Mitrra [52]. If the operator (3?;42) is operated on (5.3.13.b) and Gal—72)

a)?

on (5.3.13.a), the results are:

184

(2%)(g-x-2- -v’)A A.+ (8:245My — A, = o (5.3.14.a)
~12) ( a:2-1'2)+ A (3:242) 35; A, = o (5.3.14.b)
From (5.3.14.a)
32 82 32 a4
-- A -— —— A = 0
Byzaxz “5%,-fl +-;—,>+v‘1 . way, .
Since 7 is nonzero, it leads to:
%;+ £72, — v2 ) A, = 0 (53.15.21)
From (5.3.14.b), we can have:
2 2
(;7+ 337 - )2 ) A, = o (5.3.15.b)

It is pointed out that the two equations represented by (5.3.13.a) and (5.3.13.b)
are apparently coupled. Our aim is to seek a homogeneous solution to (5.3.13) associ-
ated with the edge conditions for normal components of unknown currents. In fact
(5.3.13) are partial differential ones, the complete homogeneous solutions to them are
not easy to find. However, a u’eatment that is often used in the linear antenna theory
is applied here. The solution is determined in one dimensional case first. Now if
(5.3.13.a) is taken as an example, we can write a forced ordinary differential equation

with variable y being a parameter.

 

82
(I3 — 12M A.= My A, (5.3.16)
Let
32 31
T(x.y) - - 6:3; A (5. . 7)

be the exciting force for A, . Instead of a partial differential equation, ( 5.3.16) is con-

sidered as an ordinary differential equation with variable y as a parameter.

--——— A =T , 5.3.18
(2:; 1?) (xy) ( )

185

The solution to (5.3.18) is searched in the Fourier tramsformed domain. Let the

Fourier transform of A, be A,(§, y) :

A.<x.y> = 31,; I Axeywdt (5.3.19)

The Fourier transformation is applied to (5.3.16):

 

A~x(§. y) = G(t) 71:. y) (5.3.20)
with
6(2) = — 1
:2 + 12
As a matter of fact, G(é) is the transfer function. Its representation in the space
domain is:

1 .. cg;
— d
21: .1. (E. -J‘Y)(§ +17)

The above inverse transform can be evaluated by using the Cauchy Residue

g(x) = -

 

g (5.3.21)

Thoerem. It was stated previously, 7 is related to the modal natural frequency of the
current on the plate and it lies on the left half complex plane since a natural frequency
has a negative real part. Therefore jy lies on the lower half plane and —jy on the upper

half plane.

If the x > 0, the Bromwich contour goes around the upper plane and the —j'y is the

only singular point to be considered in the calculation of residues.

 

g(x) = «Lam—411 = 6" (5.3.22.a)
21c -2j7 27

If the x < 0 , the Bromwich contour goes around the lower half plane, and the

usable singular point of concern is n. This time the contour path has a clockwise

direction now, and the sign has to be changed when the Cauchy Theorem is applied.

g(x) = —-‘—-(—2u1)[fl:1 = 84" (5 3 22 b)
2n 2jy 27 ' ' '

 

 

186

The solution for A, is the inverse transform of (5.3.20):

A.(x. y) = I Tony) g(x-x’) cw

_ _L x (x—z) .. (H)
- ”(In T(J(.y)e7 d! + I T(%.y)e“' dx’ 1

The integration paths in the above equation can be decomposed into two parts.

1 0

I(-)dx/= I (~>d¥+I(-)dfi

—oo —oo

and
I(-)ax=I(W—I(-)dz
The decomposed integral representation for A, is now

I

0
A,(x, y) = 3171 I T(J(,y)e7 (“1)ch + I T(:(,y)ey (”W

0° x

+ I T(J(,y)e‘7 (“0dr — I T(J(,y)e‘7 (’5‘ch 1
= .1. 7x -7x
27 [awe + b0)e ]

X

+ 3‘; I mam e"! W — eY 0”") W

= A(y)coshyx + B(y)sinhyx — I- IT(£,y)sinhy(2(—x) dz (5.3.23)

where A0) and 80») are unknown functions of y . From (5.3.17), the excitation force is

 

axay A,

The solution to (5.3.13) is then obtained as

KL )0 =

X

_ - .1 __32 - _
A,(x, y) - A(y)coshyx + B(y)smh‘yx + Y I May A,(J(,y)smhy(x’ x) (1%

Integrating by parts on the coupling term yields

187

A,(x, y) = A(y)coshyx + B(y)sinh'yx + :1; [%A,(J(,y)sinhy({-x) H24)

1

- I—a— A,(x’,y)coshy(1(-x) (11

By
= A(y)c0shyx + B(y)sinhyx - :1; -%A,(0,y)sinhy(—x)

X

— I73; A,(J(,y)cosh‘y(x/—x) dx’ (5.3.24)

The third term in the right hand side of (5.3.24) can be written as

% %A,(0,y)sinhy(—x) = C10)sinhyx (5.3.25)

with C10) being the unknown function. Therefore, (5.3.25) can be combined with the
other terms in (5.3.24) and since the coefficients in this equation are all unknown func-

tions of variable y, we have

A,(x, y) = P(y)coshyx + B(y)sinhyx - I%A,(x’,y)cosh‘I(JK-x)d.{ (5.3.26.a)

and a similar formulation for A,(x, y)

y

A,(x, y) = C(x)coshyy + D(x)sinhyy — I§;A,(x,y)cosW—y)dy (5.3.26.b)

where the P(y), B(y), C(x) and D(x) are unknown coefficients, but they are functions
of only one variable. Equations (5.3.26) represent the basic fo1rnulation for extracting
the natural frequencies of a rectangular plate for any modes. The x component and y
component of modal currents are closely coupled through the coupling integrals. This
formulation is verified to possess good numerical stability and convergence. The only
drawback is more computation for the evaluation of the coupling integrals. More

details about the numerical properties will be discussed in the next section.

A natural frequency is found when the complex frequency 3 is determined at a
certain discrete value where there are nonuivial currents of K, and K,, and the associ-

ated functions P(y), B(y), C(x), D(x), which satisfy the excitation-free homogeneous

188

equation (5.3.15). The concomitant values of s are poles of a rectangular plate. The
vanishing of excitation dependence and the symmetry of geometry give rise to the
symmetry of the current distribution in (5.3.15). By using the symmetry relations to
the solution procedure, we gain a significant computational saving in the numerical

solutions for all the natural modes.

As shown in Fig. 5.1, it is expected from the geometrical symmetry that a modal
current is symmetrically or antisymmetrically distributed with respect to the x-axis and
the y-axis. But the respective symmetries for K, and K, are not arbitrary. They must

be compatible with each other since the current continuity equation has to be satisfied:

v . K(x, y, t) + %p(x, y, 1) = 0 (5.3.27)

In complex domain, it has a form:

a 8 _
3190‘, y, s) + Ely—Ky“, y, s) — —’y c p(x. y. 3) (5328)

From the right hand side of (5.3.28), it is apparent that there are only four possi-

ble symmetries for charge p with respect to the x and y axes. Consequently, the sym-

metries for %K, and -%-K, are also limited to four cases. For example if p(x,y) is

symmetric with respect to the x-axis and antisymmetric with respect to the y-axis, then

the derivative of 384,; must be symmetric with respect to the x-axis and antisymmetric
x

with respect to the y-axis. Therefore the current of K, should be antisymmetric with
respect to the x-axis and antisymmetric with respect to the y-axis. The same analysis
results in the current of K, which is symmetric with respect to the x-axis and sym—
metric with respect to the y-axis. Table 5.1 summarizes the four symmetry cases

which will be used in the following sections.

By means of this table, the information about possible symmetries can be used to

reduce the complexity in the numerical evaluation of the integral equation (5.3.26).

 

189

 

 

 

 

 

 

 

Table 5.1 Current Symmetry Features
Kx (x. y) Ky (x. y)
POSSible Sym w rt Sym wrt Sym rt Sym wrt
. ... . .. .W.., . ...
Symmetry x-axis x-axis l x-axis x-axis
Kx = (e. e) ;
Ky = (o, 0) even even odd odd
Kx = (0’ 0) 3 odd odd even even
Ky = (e. 6)
Kx = (0’ e) ; odd even even odd
Ky = (e. 0)
Kx = (e, o) ;
Ky = (o, e) even odd odd even

 

 

 

 

 

 

 

 

 

190

Before we develop the simplified formulations for each symmetry case, some conclu-

sions deduced from symmetries of modal currents need to be specified.

First the vector potential of A, has the same symmetry properties as that of the
current of K,, and the symmetry properites of the vector potential of A, are the same as
that of K,. This is easy to prove, for example, by considering four symmetrically dis-
tributed point current sources which contribute symmetrically or antisymmetrically to
the vector potential at any four symmetric observation points. Summation of the con-
tributions from all the point current sources reveals the same symmetric or antisym-
metric properties.

Second, the symmetry properties of the vector potentials can be stated mathemati-

cally ( refer to Fig. 5.1 ).

If K,(x, y) is symmetric w.r.t. the x-axis and symmetric w.r.t. the y-axis, K,(x, y)
must be antisymmetric w.r.t. the x-axis and antisymmetric w.r.t. the y-axis. They are

denoted as K,(x, y) = (e, e) & K,(x, y) = (o, o) . Some equations can be established:

1A,,(0, y) = -§-A,(x, 0) = A,(o, y) = A,(x, 0) = o . (5.3.29.a)
8x 8y

If K,(x, y) = (0, o) & K,(x, y) = (e, e) ,

J2-A,(O, y) = 1A,“, 0) = A,(0, y) = A,(x, 0) = 0 . (5.3.29.b)

81: By
IfK,(x,y)=(e,o)&K,(x,y)=(0,e),
iAm y)=-11m 0)=A(0 y)=A(x 0)=o (5329c)
ax I ’ 8y y t y 9 X 9 . . . .
And if K,(x,y)=(o, e) & K,(x, y)= (e, 0),

32.
By
With (5.3.29), the number of unknown functions in equations (5.3.26) are reduced

A,(x, 0) = %A,(0, y) = A,(x, 0) = A,(O, y) = 0 . (5.3.29.d)

by a factor of two. As an example, we analyze one symmetry case here. If

191

K,(x, y) = (e, e) & K,(x, y) = (o, o) , we have from (5.3.29.a) and (5.3.26),

A,(x, 0) = C(x) = 0

and
$1.10.» = 7 Bo) — 1A (o. y) = 0
8x 8y ’
Since A,(O, y) = 0 for this symmetry case, %A,(0, y) = 0 is also satisfied. It fol-
lows that

730’)=0-

Therefore if K,(x, y) = (e, e) &K,(x, y) = (o, 0) (5.3.26) are reduced to the form

A,(x’ , y)coshy(1( —x)dt’

d—ak

elm“ elm

Ax(x. y) = A0) coshvx -

A,(x. y) = DOC) sinth - A,(x. ”WSW-web"

If K ,(x, y) = (0, 0) &K,(x, y) = (e, e) ,

I

Ax(x. y) = 30') sinhYx - A,(f. y)cosh't(¥-x)dx’

elm

y

A,(x. y) = C(x) 008th - 537/120. ”WSW-”(6’ .
If K,(x, y) = (e, 0) &K,(x, y) = (o, e)
A,(x, y) = A(y) coshyx — -%—A,(J(, y)cosh'y(J(—x)d.{
y
Aye. y) = C(x) coshvy - gay—Axe. y'>cosmo'—y>dy' .
And if K,(x, y) = (o, e) &K,(x, y) = (e, 0)
A,(x, y) = 80) sinhyx - %A,(x’, y)coshy(x’-x)dx/
)1
A,(x, y) = D(x) sinhyy — %A,(x, y’)coshy(y’-y)dy’ .

(5.3.30.a)

(5.3.30.b)

(5.3.3l.a)

(5.3.31.b)

(5.3.32.a)

(5.3.32.b)

(5.3.33.a)

(5.3.33.b)

192

5.4 Method of Moments Solution to the New EFIE

The integral equations of the forms (5.3.30)-(5.3.33) for each of the four sym-
metry cases can be respectively discretized by a method of moments [56]. Here the
two-dimensional subsectionally constant expansion functions are used with collocation
testing. The zones are equally divided respectively along the x-axis and the y-axis. A

typical zoning scheme as shown in Fig. 5.2 has 10 by 10 zones on the whole plate.

As mentioned above, the unknown currents K, and K, are expanded in piecewise
constant functions. The unknown coefficients of P(y), B(y), C(x) and D(x) are also
expanded with the same basis functions. Notice that the number of bases for these
unknown coefficients is equal to the number of edge zones, which are preassigned to

zeros for the edge conditions to be enforced.

The major contribution to the calculation efficiency by the pulse basis functions is
that the evaluation of vector potentials can be performed one-time. The individual ker-
nel integral terms for all argument combinations, e.g., different distance combinations,
are computed first. The subsequent steps are then taken to pick by subscript entries
from a stored matrix and to construct the matrix equation according to the symmetry

conditions being assigned.

For the implementation of a method of moments solution to (53.30-53.33), some
further steps have to be carried out in detail. As an example, the symmetry of
K,= (e, e) & K,=(o, o) is assumed. By (5.3.30) the integration regions of vector
potentials on the left hand side are shrunk to the first quadrant.

al2 b/2

,, = (11 ,x, “R
A(xy) _IZ _IQK( y’)

41cR

 

dyl
a/2 b/2
= I aw I xxx/52s) [G(x—m—y’r) + G(m/y—y’s) (54-1)

+ G(x—x’,y+y’,s) + G(x+1(,y+y’,s)] dy’

193

a/2

 

 

 

 

 

 

 

 

 

-b/2

b/2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 5.2

-a/2

Partition scheme of a rectangular plate

194

where G is the Green function

6., [(Jr-Jt’)’+(>*-)")21"2

 

G _ I, _ I, =
(x x y y S) 4n[(x-2/)2+0-y’)2]“2

The integration region for A, is shrunk to

a/2 b/2

Aye. y) = Idx’ I K.(x’.y'.v>16(x-r.y—y'.s)— G(xdy—yfis)

- G(x—X,y+y’,s) + G(x+)(,y+y’,s)] dy’

The coupling term in (5.3.30.a) is transformed to

              

 

I£A( A,( ,y)coshy(£—x)d.x’
" , ,, _a_e ,,
= M" X ” "" ay4 dy
X 8‘7 R
=Id£ IK WW“ 3,6) 4M, coshxr-x) dr'dy”

where

R = l «@202 + (Y”-y’)2 1‘”
Integrating by parts again leads to:

Iggy-A,(x, y)cosh)({—-x)d)(

‘ “’2 R y”=b/2
= I arcoshw—x) IR d1”[—K,(.{’,
-a/2

     

yII=_ b a

I KM

     

”1:128 727 I

Once again the edge conditions of (5.3.6) are used:

%A,(X, y)cosh'y(x’ -x)d1(
x a/2 b/2

=Id£coshy(1’—x)-I2 ch’ I —§—

_a__y ,, KWLRR y”

(5.4.2)

(5.4.3)

(5.4.4)

 

195

When the currents are expanded in terms of pulse basis functions with the pulse

widths of Ax and Ay, we let

b K,(x. y) = 223a,... P..<x> Pmo) (5.4.5.a)
a K,(x, y) = 22b”, P,(x) Pm(y) (5.4.5.b)
where
_ 1 (n—1)A S 5 Mn
mm) ’ {0 ll casewhere

The derivative of current K, with respect to y is expanded as:

.3.
3y

Equation (5.4.4) now has the discretized form of

,(x. y) = £2:sz am 1 so - (m-1)Ay) - so - mm 1 (5.4.5.c)

N

Bay/1,0! , y)coslry(1/—x)dx’

x W

= 211.22 b”, IcoshyoK—mr [ F(x/X’,y,m—1) — F(x/X’,y,m) 1 dr' (5.4.6.a)

(II-LA:

with

{rut-11")” + My?!“
41t[(A’-Jt”)2 + (v-mAy)2]"2
The same expansion is applied to (5.3.30.b) and it gives:

 

F (1 4’ Cm) =

IaiAAx. y’)coshr(y’-y)dy’
X

y mm
= %22 a,”I Icoshfiy'-y’)dy’ I I F (y’,y”,x,n—1) - F (y’ ,y”,x,n) ] dy” (5.4.6.b)
n m (m-1)Ay

The evaluation for vector potentials expanded with pulse basis functions is
straight forward:
mAy nAx

M ) 122a I AKA-AAA-Am
x, y = —
x b n m M! (m—1)Ay (rt-LAX 41t[(x’-x)2 + w-Y)2]1,2

 

dx/dy’ (5.4.7.a)

 

196

"'A’ "I“ (HOV-1r)2 + (r—yflm

 

A,(x, y) = Izzy)”, dx’dy’ (5.4.7.15)

n m (m—l)Ay (Ir— ”14““! -X)2 + 0’_y)2] 1,2

It is convenient to define some notations here:

 

fl ) W m “Ky-”2+ WW ardy’ (5 4 8 )
x,y,n.m = . . .a
(m—l)Ay (rt-LA: 41t[(1’-X)2 + 0"“)92] m
and
1 W
F ,(x,y,n,m) = Icosh‘Kf-xflf I F (x’X '.y.m) df' (5.4.8.b)
(.._ )Az

Substitution of (5.4.1-5.4.8) to (5.3.30) yields neat complementary numerical
forms of (5.3.30):

22% amflx.y.n.m)=£Bumo>coshvx—zz% b...1F,(x,y.n.m—1)—p,(x,y,n.m)1 (5.4.9.a)
ZZZ-,1,- bmflx.y.nm)=2CnPn(x)sin’rYy-ZE-I-aMIFx(x.y.m.n-1)—F,(x,y,m.n)] (5.4.9.b)

Matching points at the centers of partition zones and shrinking the number of
unknown amplitudes to one quarter, and with symmetry properties of the currents, one
moment matrix equation is obtained as

[ GUM lquqp [ Qijm lqpxqp [ an!» 1
.. = 0 5.4.10
[ Rijmm lquqp [ Tijm lquqp [ bum ] ( )
where 2q is the partition number along the x direction and 2p is the partition number

along the y direction. And n,i S q, mJ s p .

1
Gym = 3 mx,.yj.nm)+flxg.yj.2q—nm)+f(x,-.y,-.n.2p-mHflx1.y,-.2q—n.2p-m)l
T,,-M = — [flx,-,y,-,n,m)—f(x,-,y,~,2q—n,m)—f(x,-,y,~,n,2p—m)+f(x,-,y,-,2q—n,2p—m)]

Q1)”. = [F ,(x;.y,-.n.m-1)—F,(x1.yj.n.rn)-F ,(x;.y,-.2q-n.m-1HF,(x,-.y,-.2q—n.m)l

alt—IQ"

[—F,(x,-,y,-,n,2p—m-l )+F,(x,~,y,,n,2p—m )+F,(x,-,y,-,2q—n,2p—m—l )

-F,(x1.y,-.2q-n.2p-m)]

 

 

 

197

Rim. = 211-[qu mun-1)-Fx(x1.y,°.m,n)+F,(X.-.y,-Jn.Zq-n-1)-Fx(x.-. pm.Zq—n)l
[F,(x,-,y,,2p—m,n—l)-—F,(x,-,y,-,2p—m,n)+F,(x,-,y,-,2p—m,2q—n—1)

-Fx(xby,-.ZP-m.Zq-n)]

Zim=a,,,,,
Emar-bum

x,- = 0.5a - (i—0.5)*Ax
y,- = 0.515 - (j-0.5)*Ay

When n=1:
G i,- M = -P,,,(y,-)coshyx,-
Rij,nm = 0
aunt = Bm
When m=1:

T,,, = —P,,(x,)sinhyy,-

11
Qijm = 0

b,,,,, = C,

The matrix in (5.4.10) is a function of complex frequency, a complex resonance
occurs when this matrix has a zero determinant. The condition to guarantee the
existence of natural resonances therefore is

det lg?” 1]”qu [[gwm 11"”qu =0 (5.4.11)
um qpxqp um 4pm”:

It can be seen that this determinant is an analytic function in the complex plane
since the original integral equation is differentiable on the surface of the plate. Some
well developed root search algorithm can be directly used. The SEARCH subroutine
[49] is called to locate all preliminary dominant poles in the complex plane, and then
the NAG library is used to improve the accuracy of this locating. All the other sym-

metrical cases can be numerically analyzed in the same way.

198

A premium on computational efficiency has to be placed in the evaluation of the
matrix and its determinant. Many such evaluations are required in the course of itera-
tion to locate roots. It is thus significant to approximate the numerical calculations of

coupling integrals and vector potentials.

Since the pulse basis functions are used to present the unknown currents, vector
potentials are contributed by every patch on the plate. When a matching point is
picked, the contribution of one patch to vector potential is only the function of dis-
tance between the source point and the observation point. The one-time computation
is conducted to fill an "interaction matrix" which is made up of the individual kernal
integral terms from all argument combinations, which is necessary in the computati-

[1011.

Different approximations are applied in the calculation of the interaction matrix.
For the self patche, i.e., the patch in which the match point sits, the integration is per-
formed analytically after the exponential function is approximated by 1. For the
patches adjacent to the matching patch, the integral is well-behaved and can be
evaluated numerically. For further separated patches, the integral is estimated approxi-

mately as the value of the kemal at the patch center is multiplied by the patch area.

The computation of the coupling integrals is much more time consuming. The
introduction of delta functions reduces a 3D integration to a 2D integration. It has
been numerically tested that the 2D integration can be further reduced to one dimen-
sional integration by exploitation of the central value theorem on the internal integra-
tion. The computation time is greatly reduced by a factor of 5, while the difference

of the results before and after this approximation is less than one percent.

 

 

 

199

5.5 Numerical Convergence of the New Formulation

The applicability of one formulation depends heavily on its numerical convergent
properties. Two tests on the numerical convergence of the new formulation are dis-
cussed in this section. One is the pole convergence in a thin-strip limitation and the

other is convergence with more basis functions.

5.5.1) Pole Convergence in the Thin-strip Limit

Intuitively a rectangular plate approaches a thin strip when the aspect ratio,
which is defined as the ratio of width to length, is very small. As is well know, a
thin strip relates to an equivalent dipole. One test is performed by observing the tra-
jectory of some typical poles when the aspect ratio is diminished. It is expected that

the convergence of those poles to thinwire modes should be apparent and uniform.

It is instructive to note that for the given coordinate system, only the modes with
symmetries of K,= (e,e); K, = (0,0) and K, = (0,e); K, = (e,0) are qualified to be
related to thinwire modes since it is nonphysical that the x-component of currents is
antisyrnrnetrical in the y coordinate in the thin strip limit.

Figure 5.3 provides some insight into the convergence of poles to thinwire coun—
terparts for a range of aspect ratios. Two modes denoted by (e, e) are from the sym-
metry case of K, = (e,e) ; K, = (0,0) , while the other two denoted by (a, e) are from
the symmetry case of K, = (0,e); K, = (e,0). The first four thinwire modes picked
from the first layer are displayed by small squares to serve as the limits of the trajec-
tories. Each solid line is a trajectory of one mode. The trajectory is started with
aspect ratio = 1 and stepped with a step size of 0.1. We observe that the four poles

being considered converge apparently and uniformly to thin-strip limits.

Compared to the result of the first fromulation, Fig. 5.4 shows the pole trajectory

for the third mode with symmetry the of K, = (e,e) ; K, = (0,0) . The aspect ratios are

200

 

 

 

 

 

 

15
(M)
W ............. {3 (1,4)
1.0 0.1
10—
(e.e)
: _, ¢ .......... {3 (1,3)
1.0 0.1
(ca/c
(0,e) ....... :1 (1,2)
__M
S_ f“ : 3 : 3 0.1
1.0
(6.6) ...a (1,1)
M1
1.0
0 1 1 r 1
-2s -2 -1.5 —1 —05 0
(Sale

Fig. 5.3 Convergence of four natural modes to their thinwire counterparts as
aspect ratio is varying. The squares show the locations of the first four thinwire
modes selected from the first layer.

201

 

 

 

 

 

_1
3 5
I. A 2.5
.1, J.
T 1 1 J
'075 '05 '025 0
0L
'5'?

. Fig. 5.4 Convergence of the third mode ( I, = (e,e) & I, = (0,0) ) to its thinwire

counterpart as aspect ratio is varying. The result is based on the existing formu-
lation with one quadrant divided to 4x2, 4x3, and 4x4 zones.

202

shown as decimal fractions. The dash line was obtained with 5x5 zones and the solid
line was obtained with 4x4 zones, or 4x3 and 4x2 zones, as the aspect ratio was
shrunk. The location with sign A —> O is the limit to which the trajectory is supposed
to converge. It is seen that this convergence is not uniform. This poor convergence in

thin strip limit is attributable to the numerical instability of the first formulation.

5.5.2) Pole Convergence with More Basis Functions

A MoM solution to the new formulation presented in the last section requires the
discretization of the electrical field integral equation. Mathematically this procedure is
an approximation of an infinite-dimension space with a finite-dimension space. The
convergence of any numerical scheme implies that the error induced by finite-
dimensional approximation is uniformly decreased as the dimensions, which are used
to discretize the integral equation, are increased. A strong numerical verification of
this convergence is beyond the available capability of any computer system, but the
convergent tests on a few dominant modes with varying amounts of basis functions

can provide us with good insight into the convergent rate.

As some typical examples, the first three modes with symmetry of
K,= (e,e); K, = (0,0) are investigated in Figs. 5.5-5.7. Three different partition
schemes are employed. They are identified as 6x6, 10x10 and 16x16 partition zones on

the whole plate.

Figure 5.5 shows the convergence of the first pole with various aspect ratios. The
comparison with different partition zones is shown as the dotted line ( 6x6 ), dashed
line ( 10x10 ) and solid line ( 16x16 ). We can see the excellent agreement between
the dashed line and the solid line for varying aspect ratios. The trajectory of the
10x10 partitions is very close to that of the 16x16 partitions. Even with 6x6 partitions

the result is close enough to that of finer partitions. It is seen that this special

203

I,=(e,e) & I,=(0,0)

 

 

 

 

 

4
3..
". 0.2
CI
(ca/c
2...
0 6X6
0 10x10
A 16x16
1 I I f
-1 —08 -O.6 —0.4 —O.2
oa/c
Fig. 5.5 Different partition schemes are applied to the first mode (

I, = (e,e) & l, = (0,0) ) with various aspect ratios.

204

 

 

   

 

 

 

14
12—
10—
(Dale
3..
6—4
0 6X6
1.0
D 10x10 .
1 16x16
4 1 T l 7
-4 —3 -2 —1 0

oa/c

Fig. 5.6 Different partition schemes are applied to the second mode (
I, = (e,e) & I, = (0.0) ) with various aspect ratios.

 

205

l,=(e,e) & l,=(o,o)

 

 

 

 

...
0.4
14 .1
12 _ '
(ha/c '.I
,9
10 — '3.
I.
1.0
0‘
‘-° w 0-2
3 _.
0 6X6
Cl 10x10
A 16x16
1 I I I 1
-3.5 -3 —2.5 -2 -l.5 —l

(Ia/c

Fig. 5.7 Different partition schemes are applied to the third mode (
l, = (e,e) & I, = (0.0) ) with various aspect ratios.

 

 

 

206

dominant mode converges very fast. Physically, we should expect such a convergent
rate, because we used 6 pulses, 10 pulses and 16 pulses within a half wavelength to

represent the currents on the plate.

Figure 5.6 shows the result of the second pole. The same notations are kept as in
Fig. 5.5. From Fig. 5.6, we still observe the good agreement between the trajectory
of 10x10 zones and that of 16x16 zones for a range of aspect ratios. One can also see
that the result of 6x6 partitions is less accurate as the aspect ratio is decreased. This
discrepancy is obviously attributed to the fact that 6 pulses are not enough to present

the currents within a range of one wavelength.

Figure 5.7 is the trajectories for the third mode. Similarly as the second mode,
the agreement between the results of the 10x10 zones and the 16x16 zones is good
enough. But the discrepancy of the trajectory with 6x6 zones is very apparent as

expected.

At this point we may conclude that for the first few dominant modes a 10x10 par-
tition sheme is adequate to discretize the integral equation. As a rule of thumb the
applicability of a partition scheme could be estimated by the criteria that more than 6

pulses are required to present the currents in a one wavelength range.

5.6 Numerical Results

Extensive computations have been conducted [57] to locate all the dominant poles
in the complex plane for all symmetry cases and to solve for the current distributions
for those natural modes. But only representive natual modes for selected poles are
presented herein for brevity. This section provides the pole trajectories of the first few

dominat modes for each symmetry case, and some typical modal current distributions.

 

 

207

5.6.1) Pole Location and Trajectory

It was stated previously that the pole location can be found from the zero search-
ing of the moment matrix determinant. The determinant is an analytic function in the
complex plane, the pole location is thus two-dimensional. If 10x10 partitions are used
to discretize the integral equation, a 50x50 final moment matrix is created. To search
for a zero on the complex plane, the determinant of the 50x50 matrix is evaluated

iteratively and the basic pole location algorithm is very important.

The method proposed by Singaraju, Giri and Baum [49] is based on the theorem
which relates the variation of the argument of a complex function integrated along a
contour, to the number of zeros and poles within the range bounded by the coutour.
This approach is used conveniently and successfully for our present problem of locat-
ing a pole preliminarily before an iterative method is used to improve the precision.
The applicability of the method is due to analyticity of the moment matrix determinant

and the locality of its poles.

Once the complex plane has been scanned for zeros by using the SEARCH sub-
routine which is the code of the above algorithm, the output poles are served as initial
guesses to be improved by calling the NAG library. The called NAG subroutine is

c05nbf and is generated based on an iterative algorithm.

The locations of poles for a rectangular plate are given in Figs. 5.8-5.11. Only
poles in the third quadrant are displayed since any physical pole has a negative real
part and all poles are arranged with conjugate symmetry which is deduced from the
conjugate property of the integral equation formulation. The poles displayed are nor-
malized with respect to the length of the plate and the light speed. Each solid line is a
trajectory of one pole. The trajectory is initiated with an aspect ratio of 1.0 and

stepped with a size of 0.1 in terms of aspect ratio.

208

l,=(e,e) & l,=(o,o)

 

15.4

(Dd/C

10—1

 

 

 

 

-10 -8 —6 —4 -2 0

Fig. 5.8 Pole locations of symmetry ( I, = (e,e) & I, = (0.0) ) with various aspect
ratios.

 

209

I,=(0,0) & l,=(e,e)

 

 

 

 

 

25
20—
15—
malc I}
1.0
10A
1 1.0
1.0 1.0
5..
0 I r I I
—10 -8 —6 -4 -2 o
(Ia/c

Fig. 5.9 Pole locations of symmetry ( I, = (0,0) & I, = (e,e) ) with various aspect
ratios.

 

210

I,=(o,e) & I,=(e,o)

 

20—

15-4

(ca/c

10—

5.,

 

 

 

 

oa/c

Fig. 5.10 Pole locations of symmetry ( I,=(o.e) & l,=(e,o) ) with various
aspect ratios.

211

I,=(e,o) & I,=(0,e)

 

 

 

 

25
20 —
1.0
03
15 -<
1.0
(Dd/C
1.0
10 ‘ 0.3
1.0
1.0
5 — 0.3
1.0
o I 1
—6 —4 -2 0

(Sale

Fig. 5.11 Pole locations of symmetry ( I, = (e,0) & I,= (0,e) ) with various
aspect ratios.

 

212

Figure 5.8 shows the pole trajectories with the symmetry of
K, = (e ,e); K, = (0,0). Where so on the horizonal axis is the damping coefficient
and a) on the vertical axis is the radian frequency. With this symmetry, the x-
component of currents is symmetric with respect to both x-axis and y-axis. Out of the
thirteen modes searched there are two special poles which real parts diminish as the
aspect ratio is decreased. As a matter of fact, they are closely related to the thin wire
counterparts. For the thin strip limit, they go to the first and the third thinwire modes.
The physical significance of their behaviors depends on their dominance over the other
modes since they have smaller damping coefficients. On the contrary, all the other
modes tend to have more negative real parts as the aspect ratio is decreased. In other
words they contribute to the scattering fields more and more when the plate approaches
a square plate. An interesting phenomenon is due to the fact that the third mode has a
"loop" trajectory. The same natural frequency is reached at two aspect ratios. The
currents with this symmetry conditions are easily excited with a plate placed on a
ground-plane and illuminated by a normally incident pulse. We will discuss this

further in the next section.

Figure 5.9 shows the pole trajectory with symmetry of K, = (0,0) ; K, = (e ,e) .
With this symmetry, the x-component of current is antisymmetric with respect to both
the x-axis and the y-axis. It was mentioned previously that the modes with this sym-
metry can not be related to any cylindrical wire modes since the antisymmetry of the
x—component current with respect to the y-coordinate is not physical in the thin strip
limit. It is straightforward that the modes belonging to the symmetry of
K, = (e ,e) ; K, = (o ,0) are identical with those belonging to the symmetry
K, = (o ,0) ; K, = (e ,e) on a square plate. This identity is equivalent to a 90 degree

rotation of the coordinate system.

213

Figure 5.10 and Fig. 5.11 show the results of the other symmetry cases. The
poles for these two cases are located within a smaller negative real range as the dom-
inant modes, which have smaller damping coefficients. The modes in "deeper" layers

have not been searched.

The natural frequencies for each symmetry case are summarized in Tables 5.2-

5.5.

5.6.2) Modal Current Distributions

In this subsection we present graphical plots intended to characterize the natural
modes associated with the poles shown in the last subsection. Since complete mode

presentation for every pole is impractical only some selected modes are presented.

Each natural mode is represented by a two-component complex-valued vector
function of two variables. Since the poles are distributed on the left-half complex
plane with conjugate symmetry, any excited mode is accompanied by its conjugate
mode. The real contribution to the resonance comes from the real parts of complex-
valued amplitudes of currents. Thus only the real parts of the complex currents solved
from the moment matrix equation are plotted in a 3D format. The amplitude distribu-

tions displayed belong to a pair of conjugate modes.

Figures 5.12-5.16 are 3D plots of selected natural modal current distributions with
the symmetry of K, = (e ,e); K, = (0 ,o). The current amplitudes of the two com-
ponents distributed on the whole plate are displayed by the z-axis and the origin on the
x-y plane has been displaced a quadrant to make the plots clear. It is noted that the
displayed amplitude values are normalized current densities, i.e. I, = K, - b, and

I, = K, - a.
Figure 5.12 is the current distrubutions of the first dominant mode. We can see

the dominance of the x-component of the current by comparing the amplitudes of two

214

Natural Frequencies of A Rectangular Plate

1,: (e, e) & 1,: (0, 0)

 

 

 

modes b/a=1.0 b/a=0.8 b/a=0.5 b/a=0.3

S1 -0.827+j2.155 -0.748+j2.237 -0.622+j2.401 -0.517+j2.562
Sz - l.128+j5.703 -l.700+j6.902 -2.69l+j1 1.17

S3 -l.659+j8.852 -l.551+j8.805 -1.528+j8.801 -l.182+j8.832
S4 -1.593+j9.754 -l.559+j10.37 -2.669+j 12.77 -3.956+j 18.37
S 5 -1.611+j12.17 -2.232+j 15.25 -3.253+j24.55

S6 -1.425+j13.85 -1.230+j15.77 -l.528+j17.26 -3.306+j20.82
S7 -1.581+j15.63 -l.300+j15.77 -1.528+jl7.26 -3.306+j20.82
S 8 -4.288+j 1.731 -4.808+j 1.898 -6.005+j2.219 -7.692+j2.595
$9 -3.914+j2.942 —4.953+j3.317 -6.920+j3.060 -7.502+j3.412
S 10 -4.609+j6.907 4.61 1+j7.453 -5.510+j7.503 -7.220+j7.285
S“ '5.238+j9.316 '6.658+j11.52

512 -5.332+jl 1.01 -6.520+j 13.09 -9.257+j 17.94

 

 

 

 

 

Table 5.2 Natural frequencies of a rectangular plate with various aspect ratios.

 

215

Natural Frequencies of A Rectangular Plate

1,: (0, 0) & 1,: (e, e)

 

 

 

modes b/a=1.0 b/a=0.8 b/a=0.5 bla=0.3
Sl -0.827+j2.155 -1.163+j2.588 -3.043+j3.383 —4.012+j2.771
S2 -1.128+j5.703 -1.005+j5.971 -1.006+j6.700 -1.200+j7.063
S3 -1.659+j8.852 —2.223+j1 1.04 -3.382+j 17.84
S4 -1.593+j9.754 -2.236+j11.81 -3.828+jl8.26
S5 -1.611+j12.l7 -1.282+j12.15 -1.314+j12.52 -1.223+j13.11
S6 -1.425+j13.85 -1.486+j14.93 -2.712+j19.59 -5.466+j31.24
S7 -1.581+j15.63 -2.024+jl9.52
S8 -4.288+j1.731 -4.743+j1.949 -5.756+j2.434 -7.010+j2.965
S9 -3.9l4+j2.942 -3.899+j3.218 -3.l69+j4.321 -4.520+j8.108
Slo -4.609+j6.907 -6.092+j7.990 -8.334+j8.077 -8.546+j9.233
S11 -5.238+j9.316 -5.200+j9.391 -5.311+j9.662 -5.344+j 10.60
512 -S.332+j1 1.01

 

 

 

 

 

Table 5.3 Natural frequencies of -a rectangular plate with various aspect ratios.

 

216

Natural Frequencies of A Rectangular Plate

1,: (e, 0) & I, = (0, e)

 

 

 

modes b/a=1.0 bla=0.8 b/a=0.5 b/a=0.3

Sl -O.553+j3.279 -O.664+j3.667 -1.279+j4.704 -3.045+j5.632
S, -1.9l9+j8.688 -1.437+j9.026 -1.l72+j9.465 -l.331+j10.l7
S3 -1.088+j9.073 -l.919+j11.02 -3.289+j17.69 -4.418+j25.21
S4 -1.462+j11.48 -1.786+j 13.05 -3.3 19+j 18.74

S5 -1.852+le.53 -1.538+j15.52 -1.300+j15.72 -1.109+j16.12
S6 -1.330+j 15.66 -2.072+j 19.51

S-, -1.300+j16.37 -l.277+j17.27 -l.934+j20.74 -5.179+j31.1o
S8 -l.442+j16.59 -1.894+j20.11 -2.941+j32.09

 

 

 

 

 

Table 5.4 Natural frequencies of a rectangular plate with various aspect ratios.

 

217

Natural Frequencies of A Rectangular Plate

I,=(0, e) & I,=(e,o)i

 

 

 

 

 

 

 

 

modes b/a=1.0 bla=0.8 b/a=0.5 b/a=0.3
S 1 -1.252+j5.745 -1.873+j6.969 -3.189+j10.67 -5.330+j19.21
$2 -1.865+j5.193 -1.539+5.417 -1.166+j5.443 -0.898+j5.616
S3 -1.266+j7.187 -1.548+j8.075 -3.344+j11.80 -5.258+j16.00
S4 -1 . 148+j 12.49 -1.608+j 13.15 -1.900+j 14.79 -4.080+j20.05
55 -1.878+j12.84 -2.396+j15.62 -3.913+j24.96 -5.093+j41.99
S6 -1.975+j12.07 -1.742+j12.20 -1.372+j12.18 -1.355+j12.15
S 7 -1.899+j 12.51 -2. 102+j 15.29 -3.298+j24.74

Table 5.5 Natural frequencies of a rectangular plate with various aspect ratios.

 

218

componets of currents. The x-component is almost 10 times larger than the other
component. The symmetric shape and the edge effect at two edges of y=0 and y=1

are apparently manifested.

Figures 5.13 and 5.14 show the the second mode for two different aspect ratios.
The y-component of currents is dominant for this mode. A comparison of Figs. 5.13

and 5.14 indicates that the y—component is more dominant with a smaller aspect ratio.

Figure 5.15 is the current distribution for the third mode. The common feature of

a three-half cycle variation along the x direction is observed.

Figure 5.16 is the fourth mode. There is more variation along the y-direction for
the x-component of current. It is seen that the smoothness begins to diminish when
the frequency is higher, which implies that more basis functions are needed for the

solutions of higher modes.

Figures 5.17-5.18 show the modes belonging to the symmetry of
K, = (e,e) ; K, = (0,0). Since the modes with this symmetry are identical to the
modes with the previous symmetry, under 90 degree rotation for aspect ratio of 1, no
currents have been shown with aspect ratio 1. Fig. 5.17 is the first mode. In contrast,

the y-component is much more dominant now. Fig. 5.18 shows the second mode.
Figures 5.19-5.20 give the modal distributions for the first and the third modes

with the symmetry of K, = (e ,o) ; K, = (0,e) . From Fig. 5.19 we observe that the x-

component of current has the same shape as the y-component for aspect ratio of 1.

Figure 5.20 shows that the similarity remains even when aspect ratio is not unity.

Figures 5.21-5.22 give the modal currents associated with the first and the second
modes with the symmetry of K, = (0,e) ; K, = (e ,o) . The one-cycle variation is very
obvious here. The shape similarity is also repeated. Figure 5.21 is the first mode.

We see one period variation in the x-direction for the x-component and in the x-

 

219

it]

 

 

 

 

 

 

 

Fig. 5.12 Amplitude distributions of x- and y-components of surface currents
associated with the first mode ( l, = (e,e) & I, = (0,0), b/a= 0.75 and sa/c=-
0.8087+j2.51 1)

 

220

83.191 \ \

. M\\‘\\

11]

 

 

I
-ea.1~ I

 

 

 

 

 

7.091‘

Ix

-S,106 1

 

-17,30 «

 

Fig. 5.13 Amplitude distributions of x- and y-components of surface currents
associated with the second mode ( I, = (e,e) & I, = (0.0). b/a= 1.0 and sa/c=-
1.254+j6.336)

 

 

221

I 1;
i.
a t?
, ’1/
W
04!
{A
41
M
as,
“4%

 

 

 

 

 

13.68-

7.0174

[x

.356“ -‘

 

'6.3OSJ

 

Fig. 5.14 Amplitude distributions of x- and y-components of surface currents
associated with the second mode ( l, = (e,e) & I, = (0,0), b/a= 0.50 and sa/c=-

2.990+j12.41)

 

 

222

'4

 

 
  
  

/

1x
8
.V
w
K

      
 

 

 

 

‘ ‘g‘lz.\{‘<;oll, ‘
\‘mtmil

 

 

 

.6563

Fig. 5015 Am . l g .

. Plltude distributrons f

assocrated ' . 0 X- and y-com ne

1.644+'9 With the thud "we ( Ix: (e,e) & I = (opo ms 3f 5mm Currents
J .777) y ’0): b/a— 0.75 and sa/C=-

 

\
Ԥ
——>"t
f3?»
«h
7;?
,/Q
:9’
11
'

 

88888

 

 

, 5555

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 5.16 Amplitude distributions of x- and y-components of surface currents
associated with the fourth mode ( I, = (e,e) & I, = (0.0), b/a= 1.0 and sa/c=-
1 .770+j 10.84)

224

 

0,000 "

 

 

 

 

 

 

,18‘081

.0615 "

IX

-,0615 "

 

 

 

 

 

Fig. 5.17 Amplitude distributions of x- and y-components of surface currents
associated with the first mode ( I, = (0,0) & 1,: (e,e), b/a= 0.60 and sa/c=-
2.311+j3.623)

 

225

It;

 

1,901 -

.6336 ‘

Ix

-,6336 1

 

-1,9ox .

 

 

 

 

 

 

Fig. 5.18 Amplitude distributions of x- and y-components of surface currents
associated with the second mode ( I, = (0,0) & l’ = (e.e), b/a= 0.60 and sa/c=-
1.068+j7.111)

 

226

It,

49.83

 

 

 

 

Ix

 

 

 

 

Fig. 5.19 Amplitude distributions of x- and y-components of surface currents
associated with the first mode ( I, = (e,0) & I, = (0,e), b/a== 0.60 and sa/c=-
1.073+j4.739)

227

It,

 

 

 

 

lx

 

 

 

Fig. 5.20 Amplitude distributions of x- and y-components of surface currents
associated with the second mode ( I,= (e,0) & I, = (0,e), b/a= 1.0 and sale:-
1.209+j10.08)

 

228

 

 

 

 

 

 

 

 

 

 

Fig. 5.21 Amplitude distributions of x- and y-components of surface currents
associated with the first mode ( I, = (0,e) & Iy = (e.o). b/a= 1.0 and sa/c=-
1.391+j6.383)

 

229

3.Ua7 a

1,1'421

It}

 

 

 

 

.9500

 

lx

 

 

 

 

 

 

Fig. 5.22 Amplitude distributions of x- and y-components of surface currents
associated with the second mode ( I, = (0,e) & I, = (e,0), b/a= 1.0 and sa/c=-
1.409+j7.985)

230

direction for the y-component. But referring to Fig. 5.22, for the second mode the one
period variation is manifested in both x-direction and y-direction.

In this section the natural modes are characterized by the pole trajectories and the
current distributions. The modes are observed generally to be consistent with physical
expectations. But a good theory should be consistent with experiment. The next sec-
tion gives an experimental investigation on the extraction of natural modes from a

measured response to verify our analytical results.

5.7 Experimental Verification

The SEM analysis for transient scattering problem is based entirely on the conjec-
ture that the late-time scattered field response of a conducting target can be completely
represented by a summation of damped sinusoidal functions. It has been more than
twenty years now since the first formulation was attempted, to author’s knowledge, no
experimental investigation on this problem has been reported. It is thus extremely pru-
dent to verify experimentally the natural resonance behavior of a rectangular plate,
and in the same time to affirm the natural resonance obtained from the current theory

by comparing them to those extracted from a measured response.

The current theory is confirmed by the experimental results shown in Figs. 5.23-
5.26. Fig. 5.23 the experimental setup and a measured scattering response of a 4"x16"
rectangular plate placed perpendicularly on the ground plane. The coordinate system
is chosen as indicated to align the x-axis along the longer dimension. The external
exciting pulse is perpendicularly polarized. Due to the image effect of the ground
plane, the equivalent dimension of this plate is 8"x16". It is interesting to note that
the image plane prevents the natural modes with symmetries of
K, = (e ,e) ; K, = (0 ,0) and K, = (0,e) ; K, = (e ,o) from being excited. The dotted

line indicates the beginning of the late-time portion of the measured response.

231

 

 

 

 

 

 

 

 

 

 

 

Relative
Amplitude

 

 

 

 

 

 

Time in ns

Fig. 5.23 Experimental measurement and the transient scattering response waveform
from a 4"x10" rectangular plate.

1.2

232

 

0.6 —l

 

Relative

 

 

 

 

 

 

 

 

Amplitude 0 '7
-0.6 -
........... angina] wavefom
—— reconstructed waveform
-1.2 L l 1 r
0 2 4 6 8
Time in ns
Experiment Current Theory L.W. Pearson
SI -0.314 5.13 SI -O.752 5.01 SI -l.010 4.94
52 -0.l90 10.0 S; -O.982 9.35 52 not avail.
S3 -O.524 3.65 S3 -0.956 3.51 5:, -0.671 3.52
54' -0.199 7.33 S4 -0.876 7.07 8,, -l .770 6.69
SS -1.060 12.7 S5 '0.971 1 1.7 SS nOt avail.

 

 

 

 

Fig. 5.24 Natural mode extraction from the measured response of a 4"x10" rectangular
plate. (a) The solid line is the original waveform in the late-time and the dotted line is
the waveform reconstructed based on the extracted modes. (b) Comparison of natural
frequencies between experiment and theory.

 

 

233

 

 

(oi

 

 

 

 

3o-

 

Relative

Amplitude 10 "

 

 

-10~

 

 

_

 

L
0 2 4 6 8 10

Time in ns

Fig. 5.25 Experimental measurement and the transient scattering response waveform
from a 4"x16" rectangular plate.

234
1.2

 

0.6 4

Relative

Amplitude O 7

-0.6 fl

 

........... original waveform
___. reconsu'ucted waveform

L l
4

Time in ns

 

 

l
6 8 10

-1.2 l

 

 

Experiment Current Theory L.W. Pearson

 

s, -O.264 1.46 5, -0.264 1.54 51 —O.285 1.55
:52 -0.228 4.59 S2 0546 5.19 s, -0590 5.01
5, 41.107 9.92 s, -O.609 9.61 s, not avail.
.94 4.130 6.19 S4 .1270 6.22 54 not avail.

 

 

 

 

 

Fig. 5.26 Natural mode extraction from the measured response of a 4"x10" rectangular
plate. (a) The solid line is the original waveform in the late-time and the dotted line is
the waveform reconstructed based on the extracted modes. (b) Comparison of natural
frequencies between experiment and theory.

235

Five natural frequencies have been extracted from the measured response by the
continuation method [35]. Out of the five modes the first two are with the symmetry
to K, = (o ,0) ; K, = (e ,e) and the other three are with the symmetry to
K, = (0 ,0); K, = (e ,e) . In order to confirm the reliability of the experimental
results, the late-time response is reconstructed by using the extracted five modes and
compared to the original data. Figure 5.24 provides the comparison between the
natural frequencies experimentally extracted and the theoretically predicted. The
agreemnet of radian frequencies between experiment and current theory is excellent.
As we might expect, the agreement of damping coefficients is not satisfactory since the
experimental extraction of damping coefficients is very noise-sensitive. At the same
time the possible modes predicted from the existing formulation are listed in Fig. 5.24.
We can see that the existing formulation works equally good for the very dominant

modes, but there are two absent modes here which are experimentally measured.

As one more example, Figs. 5.25 and 5.26 show another experiment result with a
10"x4" rectangular plate. Four natural modes are extracted from the measured
response, and they are compared with theory. The comparison between the recon-
structed late-time response and the received original response implies that more higher
order modes are required to present this response. For the chosen coordinate, only the
modes with the symmetries of K, = (e,e); K, =(o,o) and K, =(e,o); K, =(o,e)
are excited. The first mode is the most dominant mode over all natural modes, and
the consistency between theory and experiment is very apparant. The fourth mode is
actually the dominant mode with the symmetry of K, = (e ,o) ; K, = (0,e) ; it is also

well verified by experiment.

236

5.8 Conclusion

A new coupled surface integral equation formulation for extraction of natural fre-
quencies of a rectangular plate has been proposed. The numerical solution to this for-
mulation has been conducted by a method of moment and extensive numerical results
are presented. A few dedicated experiments have been performed to confirm the
natural frequencies predicted by the current theory. The agreement is good. Com-
pared to the existing formulation of this problem, the new one works better for the first
few very dominat modes, and much better for higher modes. It is more rigorous, con-
verges better in the determination of poles and their trajectories, numerically better

behaves in forming a moment matrix, and is more suitable for the solution of surface

currents.

Chapter 6

CONCLUSIONS

This thesis has studied a variety of topics pertinent to the experimental and
theoretical investigations of the E-pulse properties and some basic transient scattering

problems.

As the basic theoretical background in transient electromagnetics, the Laplace
transform domain electric field integral equation and the Singularity Expansion Method
have been discussed. Since the E-pulse technique for target discrimination is based
merely on the knowledge of the natural frequencies of a target, procedures to extract
the natural frequencies of a target from the complex frequency—domain integral equa-
tion, or from measured responses have been described in Chapter 2. The choice of a
proper numerical algorithm has been emphasized to be important in the extraction of

the natural frequencies of a target from its measured responses.

The E-pulse technique has been shown by extensive experimental results to be
aspect-independent when it is applied to discriminate various scale aircraft models.
This property is attributed to the aspect-independence of the natural modes of a target.
A linearized error estimate on the natural frequencies extracted from measured
responses of a target has been provided. It has been shown that if a set of natural
modes of a target is perturbed more than 5%, the corresponding E-pulse waveform
and the consequent convolution are significantly different. This property provides the
E-pulse with the potential to discriminate two similarly sized targets, however
strongly suggests that the extraction of the natural frequencies from measured
responses must be done on a scale model in the laboratory environment. Also, the

signal-to-noise ratio of a response has been demonstrated to be enhanced 20 dB by the

237

238

E-pulse convolution. Thus the noise insensitivity of the E-pulse convolution has been

demonstrated.

To study the applicability of the E-pulse technique to targets coated with lossy
layers, the behavior of the natural resonant modes of an infinitely long cylinder coated
with a lossy layer have been investigated. The natural oscillatory modes can be
grouped into the "interior" modes and the "exterior" modes. It has been shown that
the natural frequencies of exterior modes are substantially shifted on the complex plane
only when the coating thickness is comparable with the radius of the cylinder. When
the coating material has a large conductivity, the natural frequencies of exterior modes
are not affected. While the natural frequencies of interior modes are greatly depen-
dent on coating thickness and parameters. As a rough estimate, the interior modes are
no longer dominant when the conductivity satisfies the condition of ma > 100, or
when the coating thickness is less than 10 percent of the radius of the cylinder. A few
rectangular plates coated with lossy foam have been well discriminated experimentally
by the E-pulse technique. It is expected that the E-pulse technique can be applied to

discriminate a target coated with lossy material without substantial modification.

The SEM analysis of a rectangular plate has received special attention in this
thesis due in great part to the fact that a rectangular plate is a very fundamental
geometry for many realistic scatterers, and the knowledge of its natural modes is then
of importance in the application of the SEM to many transient scattering problems. A
new coupled surface integral equation formulation for extraction of natural frequencies
of a rectangular plate has been established. Subsequently, the numerical solution to
this formulation has been conducted by a method of moments and the related numeri-
cal results have been presented. A few dedicated experiments have been performed to
confirm the natural frequencies predicted by the current theory. The agreement is very

good. Compared to the existing formulation of this problem, the new one has been

239

shown to be better in numerical convergence, and provide better solution of higher

order modes of surface current.

The work of this thesis has reflected some progress on the development of the E-
pulse technique for target identification and discrimination. However, there remain
many topics which deserve attention in the future investigation. First, the extension of
the E-pulse scheme to the target discrimination in the presence of multiple targets
should be pursued further. Second, the discrimination between different targets
should be quantified in order to automate the procedure of discrimination with the E-
pulse scheme. Third, the effect of an anisotropic layer coated on a perfect conductor
on the natural modes of the conductor should be investigated. Lastly, effort should be
made on the investigation of target discrimination by using the. early-time portion of

the scattered responses.

 

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