MSU LIBRARIES "I... RETURNING MATERIALS: P1ace in book drop to remove this checkout from your record. FINE§ wi11 be charged if book is returned after the date stamped below. TRANSIENT NAVEFORM SYNTHESIS FOR RADAR TARGET DISCRIMINATION By Che-I Chuang A DISSERTATION Submitted to Michigan State University in partiaT fulfiTTment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of ETectricaT Engineering and Systems Science 1983 /37—2555 ABSTRACT TRANSIENT NAVEFORM SYNTHESIS FOR RADAR TARGET DISCRIMINATION By Che-I Chuang A new scheme for radar detection and discrimination, the transient waveform synthesis method, is investigated. This scheme consists of synthesizing an aspect-independent waveform for the incident radar signal which excites the target in such a way that the return radar signal from the target contains only a single resonance mode of that target in the late-time period. When the incident waveform synthesized to excite a particular natural mode of a known preselected target is applied to a different target, the return signal will be significantly different from that of the expected natural mode. The wrong target can thus be discriminated. Three kinds of targets, a normally oriented infinite cylinder, a pair of skew-coupled wires and a system of crossed wires are investi- gated. Both integral-equation and differential-equation approaches are used to search for the natural resonance modes of the targets. Impulse responses are then computed using these natural modes and the sigularity expansion method (SEM). A complete procedure for synthesizing the required incident radar signal is developed and used to synthesize the waveform for single-mode excitation. To confirm the applicability of the waveform-synthesis scheme, the synthesized incident waveform is convolved with the impulse re- sponse of the target. Numerical results are given to demonstrate target- discrimination sensitivity based on this method. An experimental study is described later, and the results are compared with the theory. To My Grandfather Mr. Chang Chuang ii ACKNOWLEDGMENTS The author wishes to express sincere appreciation to his academic advisor, Dr. Kun-Mu Chen, for his assistance and guidance throughout this study. A special note of thanks is due Dr. Dennis P. Nyquist for his generous support and valuable comments during the course of this work. The author also gratefully acknowledges the special assistance provided by Dr. Byron C. Drachman. Finally, the author thanks his parents, Mr. and Mrs. Bee-Shang Chuang, and his wife, Hwei-Hsin, for their understanding and encourage- ment. iii Chapter l TABLE OF CONTENTS INTRODUCTION .................. l. l Singularity Expansion Method (SEN) ..... l. 2 Aspect- Independent Property of Waveform- Synthesis Method . . . . ......... DEVELOPMENT OF THE BASIC EQUATIONS ........ 2.l Linear-System Models of a Target- Discrimination System ............ 2.2 Waveform-Synthesis Scheme .......... 2.2.1 Single-Mode Excitation ........ 2.2.2 Required Signals and Output Waveforms .............. 2.3 Required Computations and Integral Equations .................. 2.3.l Required Computations ........ 2.3.2 Integral Equations .......... 2.4 Problem-Solving Procedure .......... INFINITE CYLINDER ................ 3.l Induced Current and Backscattered Field . . . 3.2 Impulse Response .............. 3.3 Incident Waveform Synthesis for Monomode Backscatter ................. 3.4 Numerical Results for Incident-Waveform Synthesis and Target Discrimination ..... SKEW-COUPLED WIRES ................ 4.l Geometry of Problem ............ 4.2 Integral Equations ............. 4.3 Induced Currents .............. 4.3.l Natural Modes ............ 4.3.2 Coupling Coefficients ........ 4.3.3 Computation of the induced currents ............... 4.4 Backscattered Field ............ 4.5 Impulse Response .............. 4.6 Numerical Results for Incident-Waveform Synthesis and Target Discrimination . . . . iv ID l2 T4 24 27 28 36 42 48 Chapter TABLE OF CONTENTS continued CROSSED WIRES .................. 5.l Geometry of Problem ............ 5.2 Integral Equations ............. 5.3 Induced Currents .............. 5.3.1 Natural Modes ............ 5.3.2 Coupling Coefficients ........ 5.3.3 Computation of Induced Currents for Antisymmetric-mode Excitation . . . . 5.3.4 Computation of Induced Currents for Symmetric-mode Excitation ...... 5.4 Backscattered Field ...... . ..... 5.4.1 Backscattered Field from the Antisymmetric-mode Excitation . . . . 5.4.2 Backscattered Field from the Symmetric-mode Excitation ...... 5.5 Impulse Responses ............. 5.5.1 Impulse Response to the Antisymmetric-mode Excitation . . . . 5.5.2 Impulse Response to the Symmetric- mode Excitation ........... 5.6 Incident-Waveform Synthesis for Single- Mode Excitation and its Application to Target Discrimination . . . ........ EXPERIMENTS . .................. 6.1 Experimental Setup .......... . . . 6.2 Operating Principle ............ 6.3 Experimental Procedure ........... 6.4 Data Processing . . . . .......... 6.5 Experimental Results ............ CONCLUSION ................... 7.1 A Target-Discrimination System Employing Waveform-Synthesis Method ..... . . . . 7.2 Some Potential Problems for Future Study ................... 7.2.1 Snythesis of required waveforms using different basis functions . . . 7.2.2 Improvements on Experiments ..... 7.2.3 Further Study on Crossed Wires . . . 7.2.4 More basic questions on SEM ..... V Page 121 121 124 132 132 141 142 144 147 148 149 150 150 152 155 163 163 165 170 172 174 186 186 189 189 191 192 193 TABLE OF CONTENTS continued Chapter Page APPENDIX A: PROGRAM FOR MODIFIED BESSEL FUNCTIONS ..... 194 APPENDIX B: PROGRAMS FOR NATURAL MODES ........... 201 APPENDIX C: PROGRAM FOR IMPULSE RESPONSES .......... 211 APPENDIX D: PROGRAM FOR.REQUIREDINCIDENT WAVEFORHS ...... 221 APPENDIX E: PROGRAM FOR CONVOLUTION .............. 223 APPENDIX F: PROGRAM FOR DATA-PROCESSING OF EXPERIMENTAL 225 RESULTS ..................... BIBLIOGRAPHY ......................... 228 vi Table 3.1 LIST OF TABLES Page Complex roots cnt = (Cr)nt + 3051-)"2 to K8(;) = 0; all roots to n = 19 for first three layers 2 = 1,2,3 ................ 35 Poles of the first layer of natural modes and corresponding residues used to compute approximated impulse response of infinite cylinder . . . . . ........ . ...... 44 vii LIST OF FIGURES Two equivalent arrangements for target discrimination . . . . . ............ The linear-system models of two equivalent synthesis schemes for target discrimination . . . A perfectly-conducting body is illuminated by a transient plane-wave, incident field . . . . A general coupled wires system . . . . ..... Configuration of an infinite, perfectly-conducting cylinder illuminated by a transient, normally- incident, transversely-polarized plane wave . . . Distribution of the roots Cnt = (gr)n2 + j(c1)n£ to K8(;) = O in the second quadrant of the complex g-plane . ........... . . . . . Integration contours in the complex-frequency plane appropriate for evaluation of hn(t) = [:1an($)}; the branch cut is appropriate for Kn(;) and /E . ........ Normalized impulse response of an infinite cylinder illuminated by a normally-incident, transversely-polarized, impulsive plane-wave field . . . . . . . . . . ....... . . . . Approximate normalized, late-time impulse response of an infinite cylinder; utilized for synthesis of incident waveform to excite monomode backscatter ........... Synthesized incident waveform required to excite monomode backscatter in the first natural mode of an infinite cylinder and the resulting monomode scattered wave along with return waveform from a target with 10% smaller radius . . . . . . . .............. Page 15 17 29 34 37 43 47 50 Figure 3.7 LIST OF FIGURES continued Synthesized incident waveform required to excite monomode backscatter in the second natural mode of an infinite cylinder and the resulting monomode scattered wave along with return waveform from a target with 10% smaller radius . . . . ..... . ....... Synthesized and scattered waveforms for the first mode excitation similar to figure 3.6 except a shorter re ........ Synthesized and scattered waveforms for the second mode excitation similar to Figure 3.7 except a shorter re ............ Synthesized incident waveform required to excite monomode backscatter in the first natural mode of an infinite cylinder with Synthesized incident waveform required to excite monomode backscatter in the second natural mode of an infinite cylinder with 1 Te = 0.594 $3 . . . . . . . . . . . . . . . Two thin wires oriented at an angle are illuminated by an incident radar signal . . . . . Partitioning of the wire for moment-method solution using pulse-function expansion . . . . Locations of the first 10 natural frequencies of the first layer of the antisymmetric modes for the two coupled wires with L/a = 200 d/L = 0.5 and for a = 0°, 30°, 60° and 90° . . . . Locations for the first 10 natural frequencies of the first layer of the symmetric modes for the two coupled wires with L/a = 200 d/L = 0.5 and for a = 0°, 30°, 60° and 90d . . . . Locations of the first natural frequencies of the symmetric and antisymmetric modes vary as functions of the orientation angle L/a = 200 and d/L = 0.5 . . . . . ....... ix Page 51 52 53 55 56 59 65 67 68 69 Figure 4.6 4.7 4.8 4.9 LIST OF FIGURES continued Locations of the second natural frequencies of the symmetric and antisymmetric modes vary as functions of the orientation angle L/a = 200 and d/L = 0.5 ........ . . . . . Locations of the first natural frequencies of the antisymmetric mode vary as functions of the spacing between wires for a = 0°, 30° 600 and 90° and with a/L= l/200 ........ Locations of the first natural frequencies of antisymmetric mode vary as functions of d/L for a = 0° and a/L=1/200. . . . . . . . . Locations of the second natural frequencies of antisymmetric mode vary as functions of d/L for a = 0° and L/a = 200 .......... Locations of the third natural frequencies of antisymmetric mode and symmetric mode vary as functions of d/L for a = 0° and L/a = 200 .................... Real and imaginary parts of0 the first natural- mode current for a = 0°, 30°, 60° and 90° with L/a = 200, d/L= 0. 5 along with those for the isolated wire . . . . . ..... . ..... Real and imaginary parts ofo the Osecond natural- and 90° with L/a= 200, d/L= 0. 5 along with0 those for the mode current for a = 0°, 30°, isolated wire . .............. Real and imaginary parts of the third natural- mode current for a = 0°, 30°, 60° and 90° with L/a = 200, d/L= 0. 5 along with those for the isolated wire . . . . . ..... . ...... Step response of current at u = 0. 5 L of a parallel wire over the ground plane with L/a = 200, d/L= 0. 5 and aspect- -angle 30° Step response of current at u = 0.5 L of an isglated wire with L/a = 200 and aspect-angle 30 ....................... Page 70 71 73 74 75 76 77 78 84 85 Figure 4.16 LIST OF FIGURES continued Impulse response of a parallel wire over the ground plane for current at u = 0.5 L with L/a = 200, d/L = 0.5 and aspect- angle 30° .................... Impulse response of current at u = 0.5 L of an isolated wire with L/a = 200 and aspect- angle 30 , both class-l and class-2 coupling coefficients are used . . ............ Geometry of equation (4.35) for radiation- zone field maintained by current in single Wire 0 o g p O o 9 000000000000000 Geometry of Equation (4.37) for radiation— zone field maintained by currents in two wires ...... . ..... . . . . ...... Backscattered-field impulse response of wire over the ground plane with L/a = 200, d/L = 0.5, a = 30° and aspect-angle 0° . , . . . Backscattered-field impulse response of a wire over the ground plane with L/a = 200, d/L = 0.5, a = 60° and aspect-angle 0° ..... Backscattered-field impulse response of a wire over the ground plane with L/a = 200, d/L = 0.5, a = 89.90 and aspect-angle 0°. The dashed line at t = 0 shows the specular- reflection response fgr the normal incidence situation when a + 90 Impulse responses of an isolated wire, a wire over the ground plane and two parallel wires with an aspect angle of 30° ........ Impulse responses of an isolated wire, a wire over the ground plane and two parallel wires with an aspects angle of 60° . . . . . . . Impulse responses of an isolated wire with L/a = 200 and aspect-angle 30° computed by using "class-l" and "class-2” coupling coefficients . . . . . ..... . ....... xi is considered . . . . . Page 86 87 88 89 95 96 97 98 99 101 Figure 4.26 4.27 4.28 4.29 4.30 4.31 4.32 4.33 LIST OF FIGURES continued Required waveforms for the incident radar signals to excite the first mode from the wire over the ground plane with Oa/L= 1/200, d/L= 0. 5 and for a 0° and 30°. The required waveform for the isolated wire is also shown for comparison .............. Required waveforms for the incident radar signals to excite the second mode from the wire over the ground plane with Oa/L= 1/200, d/L= and for a = and 30°. The required wave-5 form for the 0isolated wire is also shown for comparison ............... . . Return waveform from right target and target with 10% shorter length when the incident field is synthesized to excite the first mode of a paralled wire over the ground pland with L/a $8 and d/L= The aspect-angle m .............. Late-time backscattered fields from right and wrong targets of the case shown in Figure 4.28 .................. Late-time backscattered fields from right target and wrong target with 10% shorter length when the incident field is synthesized to excite the second mode of a paralled wire over the ground plane with L/a = 200, d/L= 0. 5. The aspect- angle m= 30o . . . . ....... Return waveform from right target and target with 20% longer length for the first mode excitation of a parallel wire over the ground plane with L/a = 200, d/L = 0.5 and aspect- angle 60° . . . . . . . . . . ...... Late-time backscattered fields from right and wrong targets for the second mode excitation of the case shown in Figure 4.31 . . . Return waveforms from right target and target with 15% shorter length when the incident field is the synthesized waveform to excite the first mode of a wire over the ground plane with L/a = 200, 03n: 0. 5 and a = 30°. The aspect- angle m= ............ xii Page 103 104 105 106 107 108 109 110 Figure 4.34 4.35 4.36 4.37 4.39 4.38 4.40 LIST OF FIGURES continued Return waveforms from right target and target with 15% shorter length when the incident field is the synthesized waveform to excite the second mode of a wire over Oghe ground plane with L/a = 200 d/L= 0. 5 and a = The aspect-angle m = 00 ..................... Required waveforms for the incident radar signals to excite the first modes from the two wires which are symmetric with respect to the incident signal, a/L = 1/200, d/L = 0.5 and for a = 00 and 30. The required wave- form for the isolated wire is also shown for comparison ................. Required waveforms for the incident radar signals to excite the second mode from the two wires which are symmetric with respect to the incident signal, a/L=1/200, d/L = and for a =0 and 30°. The required wave- form for the isolated wire is also shown for comparison ................. Required waveforms for the incident radar signals to excite the first modes from two parallel wires with a/L 3 1/200 and d/L = 0.5, when both symmetric and Anti- symmetric modes are excitable ........... Required waveforms for the incident radar signals to excite the first Antisymmetric modes from two wires wiSh a/L = 1/200, d/L= 0. 5 and for a = and 30°, when both Symmetric and Antisymmetric modes are excitable ...... . ..... . ..... Required waveforms for the incident radar signals to excite the second modes from two parallel wires with a/L = 1/200 and d/L = 0.5, when both Symmetric and Anti- symmetric modes are excitable . ......... Required waveforms for the incident radar signals to excite the second Antisymmetric modes from two wires wish a/L = 1/200, d/L = 0.5 and for a = 0 and 30°, when both Symmetric and Antisymmetric modes are excitable ..... . ............ xiii Page 112 113 114 115 116 117 118 LIST OF FIGURES continued Figure Page 4.41 Required waveform for the incident radar signals to excite the first symmetric modes from two wires with a/L = 1/200, d/L = 0.5 and for a = 00 and 30°, when both Symmetric and Antisymmetric modes are excitable . . . . . . 119 4.42 Required waveforms for the incident radar signals to excite the second Symmetric modes from two wires with a L = 1/200, d/L = 0.5 and a = 0 and 30 , when both Symmetric and Anitsymmetric modes are excitable ..... . .......... . . . . 120 5.1 A crude model of an airplane consisting of a system of crossed wires ............ 123 5.2 The side-view of the airplane along with the incident field with two types of polarizations .................. 123 5.3 Partitioning of the crossed wires for moment method, only one wing is used due to symmetry . . . . . . . ............. 132 5.4 Real parts of the first three antisymmetric modal currents on the wings . . . . . . ..... 137 5.5 Imaginary parts of the first three anti- symmetric modal currents on the wings . ..... 138 5.6 Real parts of the first three symmetric modal currents . . . . . . . . . . . . . . . ..... 139 5.7 Imaginary parts of the first three symmetric modal currents . . . . ........ . . . . . 140 5.8 Backscattered-field impulse response of a cross-wire target with a/L2 = 0.01,La = 45°, = = = = «1.: aw af a, L] + L4 L 2L2 and L4 0.6 due to the antisymmetric-mode excitation with aspect-angle ¢ 45 ........... . . . . 153 5.9 Backscattered-field impulse response of a cross-wire target with a/L2 = 0.01,La = 45°, - - = = I: w L 4 due to the symmetric-mode excitation with aspect-angle w = 450 ........ . ..... l56 Figures 5.10 U1 A [‘3 6.4 6.5 6.6 6.7 LIST OF FIGURES continued The required incident waveform to excite the first antisymmetric mode of the target described in Section 5.5 ........... Return waveforms from right target and target with 10% shorter length when these targets are illuminated by the synthesized waveform of Figure 5.10 with antisymmetric excitation . . The required incident waveform to excite the first symmetric mode of target described in Section 5.5 .............. . . . . Return waveforms from right target and target with 10% shorter length when these targets are illuminated by the synthesized waveform of Figure 5.12 with symmetric excitation . . . The required incident waveform to excite the second symmetric mode of the target described in Section 5.5 .......... . ..... Return waveforms from right target and target with 10% shorter length when these targets are illuminated by the synthesized waveform of Figure 5.14 with symmetric excitation . . . Experimental setup for measuring return signals from the target ............ Experimental arrangement for measurement of transient scattered EM waveforms ...... Illustration of the operating principle of the sampling oscilloscope . . . ......... Equivalent circuit of the receiving probe . . . Measured waveform of incident pulse transmitted by biconical antenna ....... Measured nanosecond-pulse backscatter field response of a sphere with 11" diameter to noramlly incident illumination . . . . Measured nanosecond-pulse backscatter field response of a thin, conducting cylinder to normally incident illumination . ........ XV Page 157 158 159 160 161 162 164 166 167 169 175 176 177 Figure 6.8 6.9 6.14 7.1 LIST OF FIGURES continued Measured nanosecond-pulse backscatter field response of a thin, conducting cylinder to normally incident illumination ......... Measured nanosecond-pulse backscatter field response of a wire over the ground plane with a = 90°, L/a = 200, L/c = 1.058 ns, d/L = 0.5 to normally incident illumination ........ Result of convolution between nanosecond- pulse and impulse response of wire over ground plane with a = 89.90, L/a = 200, d/L = 0.5, L/c = 1.058 ns and aspect-angle 0°. Notice that the specular reflection is not seen because a negative impulse is not shown in the impulse response ............. Measured nanosecond-pulse backscatter field response of a wire over the ground plane with a = 60°, L/a = 200, L/c = 1.058 ns, d/L = 0.5 to normally incident illumination . . . Result of convolution between nanosecond- pulse and impulse response of wire over ground plane with a = 60°, L/a = 200, d/L = 0.5, L/c = 1.058 ns and aspect-angle 0o ....... Measured nanosecond-pulse backscatter field response of a wire over the ground plane with a = 30°, L/a = 200, L/c = 1.058 ns, d/L = 0.5 to normally incident illumination . . . Result of convolution between nanosecond- pulse and impulse response of wire over ground plane with a = 30°, L/a = 200, d/L = 0.5, L/c = 1.058 ns and aspect-angle 0O ....... A proposed target discrimination system ..... xvi Page 178 180 181 182 183 184 185 187 CHAPTER 1 INTRODUCTION In recent years, research on radar target identification and discrimination utilizing transient electromagnetic waveforms has been conducted by a number of workers [1-8]. One interesting scheme is to irradiate a target with a simple waveform such as an impulse, a step or a ramp signal, and then analyze the scattered field from the target in terms of natural resonance modes of the target. It is known that the waveform of the scattered field is aspect-dependent, but the set of natural resonance frequencies extracted from the scattered field is in- dependent of the aspect angle E9-13]. Using this property, a target can be identified if the extracted set of natural frequencies is compared with the collection of known data on the natural frequencies of various targets. Two different targets can also be discriminated if the two sets of natural frequencies are compared. An inherent limitation of this scheme arises from the presence of noise in the return signal and the associated difficulty of accurately extracting natural frequencies of the target. In this thesis, an inverse scheme, to be called the "transient waveform synthesis" method, is investigated. Instead of analyzing the field scattered by the target in terms of its natural resonance modes, this new scheme sythesizes the waveform of the incident radar signal in such a way that, when it excites the target, the return radar signal contains only a single natural mode of the target. It will be shown in the following chapters that when the incident radar Signal sythesized to eXcite a particular natural mode of a preselected target is applied to a different target, the return signal will be different from that of the expected natural mode. A "wrong" target can therefore be discriminated from an "expected" target. The following sections discuss more about some theoretical background of this scheme. 1.1 Singularity Expansion Method (SEM) [9, 141 The SEM was advanced by Carl Baum as a means of treating transient and broadband electromagnetic scattering problem. This development was based on the results from many experiments in which different scatterers were illuminated by transient electromagnetic fields. It was observed during the later-time period (i.e., when the target is not under direct illumination of the exciting field) that the response of the scatterer aepeared to consist of a superposition of damped sinusoidal oscillations for which frequencies and damping constants are related to the geometry of the scatterer. The SEM was developed to explore the possibility of ex- pressing any external scattering response as a summation of damped sinusoids of which frequencies and damping constants are characterized by the scatterer in a similar way as the internal response of a cavity. By using the contour integral in the attempt to inverse-tranform the scattered field in Laplace-transform domain, it was found [9] that the time-domain scattered field can be expressed in terms of the singular- ities associated with its transform. It has been shown that for finite size objects in free space consisting of perfect conductors with constitutive parameters suitably constrained in their complex s-plane properties, the response has only poles as singularities in the s-plane [15, 16]. In this thesis all except one target are finite-size, so we consider only pole singularities which depend only upon the geometry of the target. The target in Chapter 3 is not finite-size and possesses a branch-cut singularity. It is found, however, that the response related to the branch-cut singularity can be approximated as a sum of two exponentially-decaying functions and thus belongs to the category of natural-mode response with zero frequencies. Therefore, for the targets we aim to study, the damped-sinusoids dominate the late-time response. It is extremely important to note that the natural frequency (i.e.. s = o + jw o = damping constant and w = 2n x frequenCy) depends only on the target geometry. Thus once determined, they characterize the target for any excitation and can be used for target discrimination. 1.2 Aspect-Independent Property of Waveform-Synthesis Method The simplest case of this radar waveform synthesis scheme has been studied by Chen £17] for the case of a thin wire irradiated by a radar pulse at normal incidence. For this case, it is possible to synthesize a required waveform for the incident radar signal to excite a single-mode return response at all post-incidence times. When this study is generalized to oblique incidence, difficulties are encountered in obtaining a realizable required incident waveform for exciting a single-mode, scattered field. Furthermore, the incident radar signal appears to be aspect-dependent. This difficulty arises because there exists a finite transit time for an obliquely-oriented wire, i.e., a finite time for an impulse to pass the wire. The impulse response of this wire consists of an early-time, forced response in addition to the sum of natural modes which describes a normally oriented wire. This ear1y+time, forced impulse response is difficult to approximate analytically, and consequently is responsible for problems encountered when synthesizing an incident radar signal to excite a single-mode, scattered field at all post-incidence times. To overcome this difficulty, we have concentrated on the behavior of the late-time response of targets and have found a scheme to synthe- size the required waveform for an incident radar signal of finite duration to excite a single-mode, scattered field in the late-time period (where the early-time inpulse response is not required, since that period has elapsed). More significantly, this synthesized incident radar signal is found to be aspect-independent. The details of this scheme are dis- cussed in Chapter 2. Then we apply this scheme in Chapter 3 for a target of infinite cylinder in which the exact solution exists and in Chapters 4 and 5 for coupled wires in which the integral equations are used to solve the problem. Chapter 6 discusses the time-domain scatting range for experiments related to this research. In addition to the waveform synthesis scheme, impulse response of the target is computed so that a detailed study of transient electromagnetics is complete for each example. We conclude this thesis in Chapter 7 by summarizing this scheme from the system point of view and showing some potential problems of this scheme. CHAPTER 2 DEVELOPMENT OF THE BASIC EQUATIONS This chapter is concerned with the development of fundamental equa- tions, boundary conditions, and synthesis procedure that will be used repeatedly in later chapters. Section 2.1 concerns the linear-system models for a target-discrimination system and defines the problem. Section 2.2 illustrates the scheme for target discrimation and develops the basic equations associated with it. Section 2.3 discusses the required computations involved in this problem and derives the integral equations with boundary conditions included. Finally, Section 2.4 uses the previous work to obtain a complete procedure for solving this problem numerically. 2.1 Linear-System Models of a Target-Discrimination System There are two equivalent schemes for target discrimination as depicted in Figure 2.1. The scheme on the left is the original wave- . form-synthesis method: it transmits the required incident radar signal, which is synthesized for monomode excitation, to the target. The right, expected target will yield a monomode return signal while the wrong target will not. The scheme shown on the right is the alternative implementation: the required incident signal for monomode excitation is synthesized and stored in the computer momory. An incident radar signal with some con- venient waveform (provided it possesses the desirable frequency component) 5 .coSoEetumE 588. CE 385885 22338 2: 3 8:3”. Amvoa wchHmv w a am: am as use i Aumuaaaoo a.“ vapoumv , - 953983 * vmufiavmu coausfio>coo (La? Hmcwwm cuauwu Li \ 111 \ \ \\ \\\ \ \ \ \ 58263 \ \\\ \ \\\ maaswm nu?» \“\\ Snowmen: \\\\ Hmcwum ucwwwoca R$4 vuugvou :35 X \\\ \ \k\\ 12%.? ucovwocw \\\ \\\ V“ \\ \ N \\\\ \ \\ ngfim \“\ Hmcwwm Esau.“ \ \ Esuou \“\ woumunaaoo \“ \ one... came—«u \\ \ excites the target, which yields a return signal with an irregular waveform. The return signal is convolved numerically with the stored, required in- cident signal. The convolved output signal will display a single natural mode of the target (a pure damped sinusoid) if the target is the expected one; the return signal from a different target will not produce the expected natural mode after convolution. To define the problem, consider the linear-system models in Figure 2.2. The model on the left corresponds to the scheme on the left of Figure 2.1: the input Ee(t) is the synthesized, required waveform for monomode excitation, the system is represented by the impulse response of the target, h(t), while the output, Es(t), is the backscattered electric field from the target. The input/output relation is Es(t) = Ee(t) * h(t). The model on the right is the linear-system representation of the alter- native scheme: the input, Er(t), is the radar return from the target, the system is now represented by Ee(t), which is synthesized and stored in a computer for numerical convolution, while the output, E°(t), is the result of the convolution between Er(t) and Ee(t). Therefore, the input/output relation of this model is E°(t) = Er(t) * Ee(t). ‘Ee(t)——-—+ h(t) ———>Es(t) E”(t>——> Eem ————+E°(t) (Target) (Computer) Figure 2.2. The linear-system models of two equivalent synthesis schemes for target discrimination. The problem is thus defined as follows: (1) For model on the left of Figure 2.2: Synthesize Ee(t) so that in the late-time period, the output, Ex: Es(t) = Ee(t) * h(t) will be a single natural nmde of the target. (2) For model on the right of Figure 2.2: Synthesize Ee(t) so that in the late-time period, the output, E°(t) = Er(t) * Ee(t), will be a single natural mode of the target. It is specified that Ee(t) is of finite duration Te and h(t), Er(t) are sums of natural modes in the late-time period for t 3 ZTt, where T is the one-way transit time for the t signal to pass the whole target. 2.2 Naveforem-Synthesis Scheme 2.2.1 Single-Mode Excitation For the purpose of synthesizing the required waveform for monomode excitation, we consider the model of the alternative scheme. Since Er(t) is a representation of radar return from the target, h(t) (impulse response) is a special case of Er(t) when the incident waveform is an impulse function, therefore, the second model in Section 2.1 includes the first model. From the discussion in Chapter 1, Er(t) can be expressed as Er(t) = Er(t,e) = g(t,e) + E an(e) eontCOS(wnt + ¢h(e)) (2.1) n=l where g(t,e) = forced response which exists only during the period 0 f t f 2Tt9 E an(e)e°nt Cos(wnt + wn(e)) = the sum of natural modes which "—1 exists for all t, an(e) = aspect-dependent amplitude of the nth natural mode, Wn(e) = aspect-dependent phase angle of the nth natural mode. 6 - aspect angle, on + ion = Sn = the nth natural frequency, with N + w theorectically, and finite for late-time consideration. The output, E°(t,e),can be expressed, based on the convolution theorem,as E°(t,e) = Er(t,e)* 59(t) t = J Ee(t')Er(t-t', e)dt' (2.2) o The integration limits are 0 and t respectively because both Ee(t) and Er(t,e) are causal functions. Substitution of equation (2.1) into equation (2.2) leads to n t E°(t,e) = [ Ee(t') {t(t-t',e) + I an(e) e°n 0 n=1 (t-t') - Costdn(t-t') + ¢h(e)]}dt' . For the late-time period of t 3 Te + ZTt, the upper-limit becomes Te 'since Ee(t')= o for t' 3 Te, and the forced response term does not contribute to the integral because €(t-t'.e) = o for o 5 t' < Te if t 3 Te + 2Tt. The property of g(t,e) = o for t 3 2Tt has been used. The output waveform in the late-time period then becomes 10 T N , E°(t.e) = (Ce Ee(t'){nZ1 an(e)eC"(t't )COSEwn(t-t') + ¢n(e)]}dt' (2.3) for t 3 T + 2Tt . Equation (2.3) can be rewritten as O t (e)e n {AnCos[wnt + ¢n(e)] N E°(t,e) = 2 an n=l + Bn SinEwnt + on(e)]} (2.4) where the coefficients An and Bn are given as An Te COSwnt' -0 ' = Ee(t') e ‘”t dt' (2.5) STHm t. B o n It is important to observe that An and Bn are independent of the aspect angle 9, and it is possible to choose a proper Ee(t) in such a way that all the coefficients vanish except one. By doing so E°(t,e) will consist of a single natural mode eyen though it is still aspect- dependent. 2.2.2 Required Signals and Output Waveforms Now that it is possible to choose an aspect-independnt Ee(t) to excite a single-mode E°(t,e), let's construct Ee(t) with a linear combination of basis functions as e 2N E (t) = E dmfm(t) (2.6) m—l where {fm(t)}, m = 1,2,...,2N is a set of basis functions such as pulse functions,impulse functions, Fourier cosine functions and nautral- 11 mode functions; dm are unknown coefficients to be determined based on the condition of single-mode excitation of E°(t,e). Substituting (2.6) in (2.5) leads to 2N A = 2 Mc n m=1 nm m (2.7) EN 5 B = M d . n m=l nm m where C I Mnml Te 0 t. Conwnt ’ = fm(t') e- n dt' (2-8) S Slnwnt' M 0 nm‘ C . nm incident radar pulse duration, and Te is a parameter of freedom which It is observed that M s and Mzm's are explicit functions of Te, can be varied to obtain a desirable waveform for Ee(t). The effect of changing T6 and basis functions will be examined later. Expression (2.7) can be rewritten in matrix form as An Mfim n = 1,2,...,N ...... = ------- dm (2.9) 3” him m = 1,2,...,2N . In equation (2.9), [Mum] matrix is of 2N x 2N order, and [dm] and B are two 2N column matrices. To obtain a single-mode, output waveform (e.g. the jth mode), we can set 12 Bj = l and Bn = o for n f j and An = o for all n. and solve equation (2.9) to get C Mnm An dm _ ---;-- ---- (2.10) Mnm 8" by choosing Te so that detEMnm] # 0. [dm] can then be easily determined and Ee(t) is obtained from equation (2.4) to be t E°(t,e) = aj(e)e03 Sin(wjt + ¢.(e)). (2.11a) J Similarly, we can set Aj = l and An = 0 for n f j, Bn = o for all n to get E°(t,e) = aj(e) eojt Cos(wjt + ¢j(e)) . (2.11b) It is noted that with this synthesized Ee(t), the output waveform after convolution, E°(t,e), remains single-mode for any aspect angle 6, even though the amplitude aj(6) and the phase angle ¢3(e) vary with e. In other words, when thissynthesized Ee(t) is convolved with the radar return, Er(t), the output signal contains only a single natural mode for any aspect angle as long as aj(e) is not zero. 2.3 Required Computations and Integral Equations 2.3.1 Required Computations It is obvious from Section 2.2 that search of the natural fre- quencies is an important task in synthesizing the required waveform. Natural frequencies can be obtained theoretically or experimentally. 13 In this report, the efforts are concentrated mainly on the theoretical aspects for some simple targets. For those targets which are so com- plicated that theorectical computations become almost impossible, experimetal approaches such as Prony's method 02!] are desirable. As far as the theoretical methods are concerned, there are basically two approaches; the first one is the differential-equation approach for some idealized structures while the other is the integral— equation approach for those targets that the analytical formulation is impossible. In Chapter 3 we will discuss an example of the first approach, while in Chapters 4 and 5 the second approach is used. The differential-equation approach is based on Maxwell's equations. The only difference now is that instead of using Fourier transform, we will solve the Maxwell's equations in the Lapalace- transform domain to handle the trasient nature of this problem. As for the integral-equation approach, there “h; more involved: it is necessary to match the boundary conditions to obtain the integral equation(s) and then solve it (them) numerically; and to make the numerical procedure more stable, we usually need to convert the electric field integral equation to the Hallen-type integral equation [18]. Therefore, Section 2.3.2 is devoted to the derivation of some basic integral equations and their boundary conditions which will be used repeatedly in Chapters 4 and 5. So long as we get the natural frequencies, the required excitation, Ee(t), can be determined from equations (2.6) and (2.10) with an optimal Te and proper choices of basis functions. For the waveform-synthesis, our job is done. However, to complete the transient scattering research, 14 it is desirable to compute the impulse response of the target. Once computed, any transient response can be obtained by convolving it with the incident waveform. If this impulse response is convolved with the required waveform, Ee(t), the expected response can be observed. To determine the impulse response, we apply SEM and the moment method to the integral equations. After obtaining the natural frequencies, we compute the natural mode currents and the coupling coefficients which arerelated to the residues of natural modes. Induced current is constructed based upon these coefficients and natural mode currents. Scattered field is then determined from the induced current. 2.3.2 Integral Equations In this section, we will first derive an E-field integral equation (EFIE) for transient surface current excited on a perfectly-conducting body by a transient incident-wave EM field, then use this result in a relatively general, coupled wires systems to get the coupled EFIE's. Finally we will demonstrate an easy way to convert EFIE's to coupled Hallen-type integral equations. Let's consider the geometry as shown in Figure 2.3 for a general, perfectly-conducting body illuminated by a transient, incident plane- wave, Ei(?,t), which excites, on the body surface, the induced current R(?,t) and charge, 0(r,t). The induced current and charge, in turn, maintain a scattered wave, ES(?,t). Our objective here is to derive an integral equation for the unknown current by matching the boundary condition on the surface so that the total tangential E-field on the surface is zero, “ scattered Q‘. V Figure 2.3. 15 W6 ve free space A perfectly-conducting body is illuminated by a transient plane-wave, incident field. 16 A +~i _> +5 _> + . t-(E (r,t) + E (r,t))= o ... for all r E A of perfectly-conducting body (2. Where t is the unit vector tangent to the body surface. We use Laplace transform to handle the transient behavior, and express ES(?,S) in terms of scalar and vector potentials, ES(?,s) = -v$(r,s) - s A(r,s) (2 where $(?,s) = I :(Z:; e'YR dA' = scalar potential (2. A o A(?,s) = J Mflgéfiiél-e'YR dA' = vector potential (2. A and y 2 %-= complex propagation constant. Theconservation of change in Laplace-transform domain leads to Substituting equation (2.16) into equation (2.14), we express scalar . 3 + potential in terms of source K(r,s), - 1 + + —V'-K(r;s) _ $(r,s) - J 58 4nR e YR dA' (2 0 Equations (2.17), (2.15) and (2.13) give us a relation between ~ 3 Es(?,s) and K(F,s), ~ 12+! _ ES('F,s) = 3— VEI Lfié—Q—A e YR dA' o A ~ -5 p0 [A 4nR (2. "F +' 595—51 e'YR dA'] (2. 12) .13) 14) 15) 16) .17) 18) 17 The combination of equation (2.18) and boundary condition (2.12) leads to A ~ A +' - t E‘(’F,s) = J— t VLJ 2:31:21 e’1R dA'] 560 A 4nR . g +' - su [( M e'YR dA'] for ‘Fe A, (2.19) 0 A 4nR rearranging equation (2.19) we finally get ~ ..., _ R ...} +' A 2 A + +' e Y ' AA EV °K(Y‘ ,S)(t°V) - I" t°K(Y‘ ,S)]—4*n—R- (”A = -e s t-Ei(?,s) for all T 6 A . (2.20) 0 This is EFIE for unknown R(F,s) induced on A in Laplace-transform domain. Let's consider the coupled wires system as shown in Figure 2.4, the wires may or may not be crossed. Figure 2.4. A general coupled wires system 18 Equation (2.20) becomes + + ‘ 2“ + + e-Yk ‘ +i+ ( v'-I(r',s)(t-v) - y t-I(r',s) mm: = .6 st E (r, s) (2.21) r where r is the contour of integration; note that surface integral becomes a line integral after the thin wire approximation. Since BI (“'95) A . . = 1 2 ___ = . = a v -I£(u£,s) -—7fiE:———- for z 1,2,...,K, and t v RPE- for k = 1,2,...,K, equation (2.21) can then be rewritten as - R K L 1 k2 £ 312 a _ 2 * .* . e . 221 A0 [3“; °uk Y (uk UR) 12(U2,S)] 4an£ dul _ . g1 - - '805 Uk°E (Uk,S) --- for 0 5 Uk 5 Lk’ k - 1,2,...,K (2.22) Where Rkl = R(UK’UL) for (k,£) = 1,2,...,K = l”k”k ‘ (dkt T “i“t)l = /u2 + ku '(u ou 2)- o(uu - u' 1“ 2) + d2 k k fikl ku k k2 ’ + d k2 the vector from the origin for uk to the origin for ug. To handle the thin-wire approximation at the source-point singularily u; = uk when 2 = k,dk£ = o, the wire raduis ai is included in the above formulation such that R&k(uk,u£) = ak. Then ng becomes . I 2 W2 . + . 2 2 Rk2(uk’u2) 2‘/“k "t 2Uku 2(fi k ° "2) ' 2dk2 ° (ukuk ” 2° 2>+dk2 ak (2.23) . Physically, we consider the field point uk to be on the wire surface while source point u‘ to be located along the wire axis for the thin- 2 wire approximation. The leading integral terms in equation (2.22) can 19 be modified by evaluating the integral by parts in the u; variable so 31 that IR instead of ——£- appears as unknown: auz ' 'YR -YR ' = [12' 33%(11295) 3 e k1 dul = I (u. S) 3 e k9. U2 L2. 0 aux Buk 4NRk2 R R R Buk 4flRk2 U; = 0 )LR < ) 32 e-YRkE (2 24) - I u',s q . du' . 0 R R oukauz 4flRk£ 2 If we define 'YRk£(Uk:L£) ’YRk2(”k’°) _ — e + e ”k2.(“k) z 12(L2’S) 41iR (u ,L) ' 12(0 ’5) 4nR (u ,oT' (2'25) k1 k t kt k then expressions (2.22) (2.24) and (2.25) lead to - R K L 2 Y k9. aw z . a 2 - « _ e k2 X {-J I (U'.S)L-—-——.- + v (u -u )J—-—-dU' + (11)} 2:] o z 2 eukauz k 2 4thi 2 auk k = -eosak.E‘(uk,s) for o _<_ uk 5 Lk, k = l,2,...,K . (2.26) This is the basic set of coupled EFIE's we will be using in Chapters 4 and 5. Examing closely equation (2.26), we can see that the kernel function of this EFIE involves a second partial derivative. This term, when applying the moment method solution, will introduce discontinuity in the basis function of charge, and thus cause some undesirable features such as the sensitivity to changes in the number of partitions and the initial guess in root searching. To avoid the~unstablecharacteristics of EFIE, we derive, in the following,the Hallen-type integral equations in which the kernel functions possess no derivatives. Equations in (2.26) are integro-differential equations. They can be reduced to pure integtal equations of the Hallen type by first 20 converting them to an inhomogeneous ODE which can be solved to provide the desired result. In the 2 = k integral term of the coupled system of EFIE's, -vR -YR 32 e kk - - 82 e kk eukauk 4ank Hui 411Rkk so that this term is singled out for the special attention, 'YR (“weenie-31%“— win 5 {—“L'Q— o auk " kk 2=l k ( ) L“ ( ) 32 2(" ‘) 9”sz ' 1’6 [ I U',S E“: + Y U 'U ]— du'} 2k 0 2 2 aukau£ k 2 4thg 2 - 31 = -eoS Uk- E (Uk,S) (2-27) where 52k = o for 2 # k;<5gk = l for 2 = k. The trick is to modify the differential operator of the second integral term (by adding and subtracting an appropriate factor) to indentify an operator which is common with that of the first integral term, [ ___—82 + Y2“) 0f] )] ___—e-YRkR' = - [___—32 - Y2] -e——-—-—-YRk£ (fi .fi ) eukaui k 2 4th2 aui 4an£ k 2 + 3 E 3 + (a .g )]._;:EE£ k 2 411R and 21 (fi .fi )3 9::i55.= d e-Ysz [ a + _§_.(fi .Q )]R k 2 4anl de2 4an£ aufl auk k 2 k2 “2+“ Ax+xxx “2'”k(”k'”2) + dk2°u2 T [”k‘“2(“k'”2) ' dk2'”k3(“k'”2) d 4nR 'YRk2 R k2 k2 I " " 2 " A A " uREl-(uk-ul) J + dk£-[u£-uk(uk-u£)] de2 4nR R k2 k2 By defining 9k2(”k’”2 equation + -YR 2 + . . . . e ) J + dkg-[ug-uk(uk-u£)] Rk2 d k2 uiEl-(uk-u k2 4"Rk2 $1 .s) s dR (2.28) (2.27) can be converted to 'Yng £_____ 4erk2 L 2' I " A l I£(u£,s) (uk-u£)du£ L 39 (u ,U'.S) 2 . k2 k 2 . £5] ' (1'52k) Jo 12(u2’s) auk du2} 'EOS Gk- E1(Uk,5) . (2'29) Recall that an inhomogeneous ODE in the following form 2 (it; - iznw) = nu) 3U can be solved [l9 3 as u C cosh yu + C sink ru + 1- f(g) sinh y(u-§)dg l 2 y o 2(U) = Therefore (2.29) can be solved as inhomogeneous ODE to be lg [Ll ( €4ng . I u',s) —————-— (a -u )du' 1:] 0 2 2 4thR k 2 3 k _ I I o 1 . - C1k cosh yuk + C 2k Sinh Yuk + ;-Jo ngInh[v(u-g)] N L as (a u' s) I k2, , 9,, { X [(1-6 ) [ 2 I (u ,s) ————— du' £=1 k2 o 2 2 ea 2 3w (5) ~- k A + - 8&2 J - eos uk-E1(g,s)} (2.30) The two terms involving §%- can be integrated by parts to give: 1 Uk _ > 1 r L2 ' agk£(gaui9s) I 3Wk2(g)_ Y Jo dg SlnhLY(U'€)J{£Z]L(]-§k2) I I£(u£,s) —3E—- du2 - —§E—- J} - o N L 9 (E,U'.S) sinh v(u -£) a = u = z I i dug 12(ui,s)(l-6k£) [ kg 1 k k 2=l 0 Y a = 0 uk K wk2(g)sinh v(uk-g) g = uk + I gk£(g,ué,s) cosh v(uk-§)d€] - X o 2=l Y 5 = o "k + Jo wk£(£) cosh v(uk-€)d€J (2.3l) where 9k£(€.ui.s) sinh Y(Uk-€) E = uk gk£(o,u£,s) sinh Yuk Y g = 0 Y . _ g = u . wk£(g) Slnh Y(uk g) k = - wk£(o) Slnh vuk (2 32) Y g = o Y 0 . The terms associated with expressions in (2.32) and last term in (2.3l) are simply proportional to cosh vuk and sink Yuk, and therefore we can redefine constants C1k and C2k to be 01k and C2k‘ Equation (2.30) is thus reduced to 23 5 [Li ( ) e-YR“ (“ “ ) ”k I u',s [— u-u)-(l-6 I g (mus) cosh Y(U 4;) £=1 0 2 2 4thg k 2 k2 0 k2 2 k dgldui . as “k, 3i = Clk cosh yuk + C2k Slnh yuk - —$—-J uk-E (5,5) sinh v(uk-g)dg (2.33) -YRk ° . . . _ e g * 2 uk . Defining Kk£(uk|u£,s) : Z;§;;—-(uk-u£) - (1'6k2) J0 gk£(€.u£,s) cosh y(u-g)dg (2.34) leads to K L2 "2] Jo I£(u£,s) Kk£(uk|u£,s)du£ U ~. 0 for o _ A C x IA r x. and k II _4 0 N u v 7Q (2.35) This set of integral equations has kernel functions which possess no derivatives, however, there are integration terms involved. This is a more stable set of IE's but with the increased cost of computer execution time and storage since the integration terms not only take much more computation, but also destroy the symmetry of the matrix which is ob- tained from employing the moment nethod. With this trade-off we use Hallen-type IE's to find the natural modes while EFIE's are used to determined the coupling coefficients. There are unknown constants introduced in equation (2.35). The way to determine them is to exploit the boundary conditions. For those wires with no cross, the constants are considered as 2K unknowns with 2K currents on the wire ends vanishing and thus dropped out of the 24 unknowns. The case of crossed wires is more complicated, we will discuss more in Chapter 5. Basically the boundary conditions used to determine unknown constants are: [ZOJ l) continuity of scalar potential across the junctions, 2) continuity of vector potential across the junctions, 3) zero current at the wire ends, 4) Kirchhoff's current law at junctions. Conditions l), 3) and 4) are sufficient to solve the problem, while condition 2) will further simplify the problem for the case with wire segments aligned in the same line. 2.4 Problem-solving Procedure. The procedure used in later chapters to solve the problem and to synthesize the required waveform, Ee(t), is outlined as follows. l. Based upon the Maxwell's equaitons and boundary conditions to form the appropriate differential equations or integral equaitons in the Laplace-transform domain, set E1(?,s) o for natural response. 2. Solve the differential equations analytically or use the moment method to form a matrix equation, AI = o . (2.36) 3. Set det(A) as a function of s, then use Muller's or Newton's method to determine its zeros, which are the natural frequencies. Solve (2.36) after finding natural frequencies to determine the natural mode current, vn(u). for nth mode. 4. Using SEM [ l4] to compute the coupling coefficients as 25 S(u,s)v (u)du P n l an(s) = (2.37) ( Jvn(u)vn(u'){g§[K(u|u',s)]} du'du r r s = s n where ~. K 3w (u ) = . .+‘ _ k2 k S(u,s) -eos uk E (uk,s) 2;] auk for o 5 uk 5 Lk = forcing function or source function? K(ulu',s) is the kernel function in (2.26), T represents the whole contour of integration in equation (2.26), the induced current is then expressed as I(u,s) = g an(s)vn(u)(s-sn)-1 "=1 (2.38) 5. The impulse response of scattered field is computed from the vector potential which is maintained by the impulse response of the in- duced current. We will discuss this more in Chapters 4 and 5. 6. The required waveform is synthesized by the process described in Section 2.2. 7. To check the results, we use discrete convolution or FFT to convolve Ee(t) with h(t). 8. Perform the transient EM experiments which will be discribed in Chapter 6, to check the impulse response. 1In Baum's formulation, the denominator has un(u) instead of v (U). n where pn(U) is the solution of JA = o; un(u) = vn(u) for the case of symmetric matrix A. This is true for those cases we want to discuss if we consider EFIE's. 2It can be easily seen from equation (2.26). 26 These are the major steps used to solve this problem. We will solve a differential equation for an infinite cylinder in which an exact analytical solution exists, in Chapter 3. In Chapter 4, a skew coupled wires system with no cross is considered. We then consider a system of crossed wires in Chapter 5 as a crude model of airplane. The experimental study of this problem is discussed in Chapter 6. CHAPTER 3 INFINITE CYLINDER The waveform-synthesis method is applied here to a target consisting of a thick, perfectly-conducting, infinite cylinder illuminated by a transient, normally-incident, transversely-polorized plane wave. Using a spectral approach in the Laplace-transform domain, the current induced on the cylinder and its backscatter-field transfer function are first calculated in Section 3.l. By inverse transforming its transfer function, the impulse response of the target is obtained in Section 3.2. It is found that this response consists of a discrete spectrum comprised of a residue series in natural resonance modes augmented by a series of continuous-spectrum terms arising from a branch-cut integration; the impulse response of the infinite cylinder can not be constructed as a pure SEM series. The late-time impulse response is subsequently ap- proximated in closed form in Section 3.3 and used to obtain the late- time backscattered field excited by an incident field with arbitrary waveshape. Based upon the latter representation of the backscattered field, the incident waveform required to excite a monomode return radar signal is synthesized. It is demonstrated in Section 3.4 that an optimal incident radar signal can be synthesized which excites (by convolution with the impulse response) a monomode return signal from the cylinder in its late-time period. When an optimal signal,synthesized to excite a particular natural 27 28 mode of a given cylinder, illuminates a cylinder of slightly different radius, the resulting return signal is found to differ from the expected monomode response. The "wrong“ cylinder is therefore sensitively discriminated from the "expected" one. Applicability of the radar wave- form synthesis method to implement target identification is therefore demonstrated. 3.l Induced Current and Backscattered Field An infinite, perfectly-conducting cylinder of radius "a" is illuminated by a normally-incident, transient, plane-wave radar signal with its electric field polorized perpendicular to the cylinder axis as indicated in Figure 3.l. The incident field is expressed as +i E (F,t) = 9 uCt-(x+a)/c]F[t-(x+a)/c] (3.1) where F(t) is an unknown waveform function to be synthesized subject to the criterion that it excite single, natural-mode backscatter from the cylinder. Laplace transforming yields E'(‘F,s) = L{E'(F,t)} = ‘ “’F(s)e'Y(x"a) (3.2) where f(s) = L{F(t)} and y = s/c is the complex propagation constant. The total EM field excited about the cylinder by 21 consists of a wave, transverse-magnetic (TM) to its direction of propagation, with E(?,s) r Er(r,w,s) + W E¢(r,¢,s) (3.3) Was) = 222nm) where 'E(r,s) and 'fi(r,s) satisfy Maxwell's equations in Laplace- transform domain, 29 2 ”It (Lil) .— N f a T(r,¢,z) ‘/, \~.+,/ l l l I 1 | l ' YEy(X1t) l transient incident I waveform 0'2” / : ,r’ z / l l /l<-- J > / \ l y ’<\.¢ \\ I \ r I \\ _ I’ ——\\| // x l I ‘7' k- . ,J -.l l Figure:21 Configuration of an infinite. perfectly-conducting cylinder illuminated by a transient. normally-incident. transversely-polarized plane wave. 30 ~ Vx%(r,s) = -pos'fi(r,s)} (3-4) N -> 1") va(?,s) = 605 E(r,s) Equation (3.4) leads to 7 l anz Lr - Eosr §§_ (3 5) E = _:J_ ifié. W 605 r and 1 3 ~ 3 ~ _ N F L5“; (rECP) - 32P- EY‘] " -L1 SHZ (3 6) Subsitute Equation (3.5) into equation (3.6), we get vzfi - yzfiz = o, (3.7) Total field E can be expressed as E = E + E , where E iS‘uwascattered field maintained by induced surface current excited on the cylinder, and satisfies the boundary condition mil 2. (r=a,o,s) = o. (3.8) Incident fields E;, H; can be expressed in cylindrical coordinates by a plane-wave expansion [21 J in the cylindrical-wave function solutions to equations (3.5) and (3.7) which are bounded in the origin as E; = F(s) e'Ya e'Yrc°S¢Cos¢ (3.9) Ia(Yr) [An(s)Cos(n¢)J ll IIM 8 n o with unknown Fourier coefficients An(s). Exploit orthogonality to determine An(s), 31 16(vr) Ei-An(s) = E(s) e-Ya [in e'YrC°S¢CosoCos(no)do (3.l0) where En is Neumann's number (60 = l; 6" = 2, n > a). From L22 1, In(z) = %-J: ezcoseCos(ne)de (3.ll) by differentiation, 15(2) = %—J: ezcoseCoseCos(ne)de. (3.12) Therefore, equation (3.l0) becomes, (Yr) El-An(s) = E(s) e'YaZnIA(-vr) II n n An(s) can be determined as An(s) =-(-l)nénE(s) e“C (3.13) h d (1)” WM) b '1 f t' W ere C 5 ya an - - = as can e ea51 y seen rom equa ion I$(yr) (3.l2). E; and E; are thus expressed as E; =-E(s) e'C E (-l)"€n I$(yr)Cos(n¢) "=0 . -C m } (3'14) 2; = E 309 X (.1)"en In(yr)Cos(n¢) n=o - 8 where Zo - (“o/Ea) . A similar expansion of the scattered field, in cylindrical- wave-function solutions to equiations (3.5) and (3.7) which satisfy the radiation condition, provides E: ":0 an(s)K8(yr)Cos(no) (3.l5) 32 with unknown Fourier coefficients an(s). Satisfaction of boundary condition (3.8) requires E:(a,w,s) =4E;(a,¢,s), which yields upon substitution of expressions of (3.14) and (3.15) IIM8 ( K' c = ? “C 0 a? 5) n(c) os(n2) (S)e n 11MB (-l)n&nIfi(q)Cos(no) n 0 leading to coefficients (-1)"e I'm _ ~ "C n n an(s) — F(s)e Ka(c) (3.16) Therefore the scattered electric and magnetic fields can be determine form equations (3.15), (3.16) and (3.5) as n 1 (-1) énIn(c) E: = E(s)e'c n20 K6(C) K(yr)Cos(no) ( ) 3 17 ~ _ e -c w {-1)6 I'(c) ' H: ___i%%§__. n20 K5(c) K(yr)Cos(n¢) Induced current excited on the cylinder by E1 is obtained as A A 1' E(¢,s) = f x Z ”2(3’9’5) = -¢P[Hz (a,W,s) + fi:(a,w,s)], which provides _ m I ( )K'( )-I'( )K ( ) 2,123) =fl—L—fioe C ")0 MM E" C n 231;) C n C ”WW 2 1 en E(s)e F X ( )n (3.18) '—___—(_) Cos( (no) ZoC n= 0 where in the latter expression the Wronskian for modified Bessel functions In(e)k5(e) - Ia(;)Kn(;) = -;" has been exploited. The radiation- zone scattered field is finally obtained as tg'lncms) = fih-wofihs) “) en 1' n(c) ZHR‘ n= 20 vQ-K' n(1;) F(s)e C(R*‘) Cos(no) (3.19) 33 where R = r/a is a normalized radial coordinate. Natural-mode solutions are those E4 # o and ECP f 0 which can exist as solutions to the homogeneous problem when E(s) = o. It is clear from the expressions (3.18) and (3.19) that normalized natural frequencies c satisfy the characteristic equaiton n2 Kn(cn2) = o (3.20) for the n2'th natural mode, where Cn2 is the 2'th complex root of K6 = a. Natural frequencies Sn2 are subsequently recovered as Sn2 = C §n2/a' Complex roots to “5(C) = a can be counted by Watson's [23 J method, and were found by Luke [24 J to number n+[l-(-1)n]/2. Note that k$(g) possesses no roots. Coefficients in the power series re- presentation for “6(F) are real for c not on its branch cut, con— sequently [25 J the roots occur in complex-conjugate pairs. It follows from the fundamental form of the modified Bessel's equation that these roots are simple zeros. Details on computation of the Cn2 (Using Newton's method and computing K6 from integral representations of Kn and In) were reported in [26 J. Other method (Using Muller's method and computing K6 from a software of Bessel function [27 3) also yields exactly the same results. All such roots are found to have negative real parts, and the distribution of approximately 200 of the Cn2 in the second quadrant of the complex c-plane is displayed in Figure 3.2; a symmetric distribution exists in the third quadrant. The roots are observed to be distributed along layers of constant 2 which are ordered by index n as indicated. Table 3.1 displays all roots to n = 19 34 X = J m nu O n a B D = on an n IIIIIIIIII J 60050033939 2 1 DEBBDDUDODIB . ..aeaaaauonauua In 0» . II I J . In I 4 I I B. a II I DO . I la, . I U 0. IIIII II I 4 IIIII I I I 2 D'a‘aou ..0 a . o = ulna II I I I I. II II I d IIII II I - . IIII III II I J pit III I I 0 UP. IIIII II I ... 1mm IIIIII III II I I. I . 0 3 III-B .- 5 II 0. q . 8 I C at x 1 ..a 1 - mm IIII mm as». u o 06 III h cs '6. II u. .3. . _. IIII 0. 0 O I I-IIlIO-Id11-Idluiull‘ld11ii1-1I11-!1IddqddlquI 9 ov a «w coon 0.0» 9..“ Gena noun 9.. 9.0 28835135....” fie complex c-plane. =(C) rn ft Crc Rem - Re(sa/c) K6“) -0 in the second quadrant 0 Figure 3.2 Distribution of the roots £111 35 Table 3.]. Complex roots cnf (cr)M+J(c1)M to Kr"(£)=0; all roots t0 n=19 for first three layers“ fl=1,2,3. i=1 ' 1=2 7:3 roots of 1'st branch roots of Z’nd branch roots of 3’rd branch .. + _ + _ ... n Cr -61 Cr -§1 Cr -61 1 .64355 .50118 2 .83455 1.4344 3 .96756 2.3739 1.9816 .44080 5 1.1612 4.2769 2.8037 2.2119 3.3098 .43637 5 1.2383 5.2366 3.1082 3.1094 3.8394 1.3104 7 1.3071 6.2002 3.3730 4.0142 4.2871 2.1891 3 1.3694 7.1667 3.6087 4.9252 4.6784 3.0733 10 1.4797 9.1069 4.0176 6.7625 5.3453 4.8574 11 1.5293 10.080 4.1935 7.6876 5.6367 5.7565 12 1.5759 " 11.059 4.3572 . 3.5152 5.9069 6.6597 13 1.6200 12.030 4.5255 9.5480 6.1592 7.5667 14 1.6618 13.007 4.6749 10.483 6.3962 8-4772 15 1.7017 13.985 4.8165 11.420 6.6200 9.3908 17' 1.7763 15.943 5.0798 13.301 7.0345 11.226 18. 1.8114 16.924 5.2029 14.245 7.2275 12.140 19 1.8452 17.905 5.3210 15.190 7.4124 13.072 36 for the first three layers 2 = 1,2,3. 3.2 Impulse Response The backscattered field along a = n can be expressed from equation (3.19) as E5b(r,s) = 9 E;r(r,n,s) = y E(s) ifi-e'C(R'1)H(s) (3.21) where transfer function H(s) is defined as H(s) = X éan(s) (3.22a) n=o with 8-2C15(c) Hn(s) = —————————- (3.22b) /E Ka(c) In expression (3.22), the ratio of Bessel functions behaves asymptotically for large C as 16(c)/Kfi(c) « exp(2§); the time-shifting factor exp(-2;) has been included in equation (3.22) to annul that behavior at g + m and thus facilitate the inverser transformation of Hn(s). Physically this introduces in equation (3.20) the right time-shifting factor exp[-c(R-l)] which corresponds to the time-delay between the "turn-on" times of incident field and backscattered field observed at normalized radial coordinate R. Apart from pure amplitude and time-shift factors, the normalized impulse response of the cylinder is obtained from expression (3.20) with F(s) = l as :- A H V II rm 1 —l fl :1: A m v H4 II II M 8 m 3' A re- V (3.23) with 37 W branch cut for complex s-plane Knm and 1f;— 5”,)“, - 41w §-= sa/c CL=Br+ CLw+ 1.1+ C€+ L2 , CR =Br+CR°° ‘ /"""" , // \\\ // x \ C C Loo y snflx Br 4‘ Ron I x C6 \ 2 x... \ Ra, .x I, x ‘/ I \ x / \ / \ // \\-—..—J" Figure3.3 Integration contours in the complex-frequency plane appropriate for evaluation of hn(t)=.2'1{Hn(s)}.- the branch out is appropriate for Kn“) andi/f . 38 _ -1 hn(t) — L _ 1 st {Hn(s)} - 2.3 (Br Hn(s) e ds . (3.24) The appropriate Bromwich contour and associated integration contours in the complex s-plane are indicated in Figure 3.3. For t < 0, Br is closed in the right half plane along CRm; since CR encloses no singular points then h (t) =~—l— 1im J H (s) eSt ds = o --- t < o (3 25) n 2nj n ’ ' vanishing of equation (3.25) can be easily shown using large-argument asymptotic forms of modified Bessed functions. For t > 0, Br is closed along CL..+ L1 + C6 + L2 to form the closed contour CL. It is easily demonstrated that the contribution from C and C6 vanishes, since L- 1im I H (s) e ds = lim J H (s) eSt ds 0 c ” c ” Roo+0° Lw 6+0 ‘ E by large and small argument approximations respectively. CL encloses all the Simple poles Sn2’ n > a, at which Kn(cn2) = a such that h (t) = ) Rn - 1. (J Hn(s) est ds + Hn(s) est ds] --- t > o 2 i L1 L2 (3.26) where Rn2 is the residue of the simple pole at Sng 39 St ‘2; I C(T-Z) . R =9 e In(C) _ .9. e In(;) n2 d a d -—— /"k' -—- ' d5): C n(C)] 2;:an dCE/C—Kn(c>] Q =Cn£ C( ‘2) 1 = E_ e T In(C) VEK3(C)+ -l-K6(c) 2&— mm Cn£(T'2) c e 15(cn2) = (a) -—— (3.27) an chng) with rormalized frequency §n2 = s a/c and normalized time T = t/(a/c), n2 -l—-Ka(;) + 0 since K6(;n£) = a. From modified Bessel equation 2/5 c=cn£ [22 J. 22K3(z) + z K6(z) - (z2 + n2)K (z) = a, we get the following, (62 + nz)K (c ) - cn2K5(cn) n = ”Q n "2 Kn(5n2) 2 Cn2 = [1 + (14212 (c ) (3.28) c n n2 n2 therefore, ( 2) C T- c e n2 In(§n2) Rnl = (a) n 2 /—— . (3.29) "*‘E;;) 1 cn£Knitngl Exploiting appropriate analytic continuation [22 J of I; and K5, the contributions from line integrals along the branch cut of /E and Ka(;) can be evaluated as 40 . ~2; , 0+JA e In(c) st H (s)eSt ds = I e ds ---- as A + o+ -w+jA /E K6(c) - t-2 . (E) [w e C( )In('C) dc a o VCE'Kfi(-c) +1 m 'C(T'2)Ii (g9 L:llfl___[ e "(C) dc J o /zt<-1)""K5(e)+3115(e)3 H (s)eStds . -2; -w-JA e I' (c) J ——————D———- eStds --- as A + o+ 0-3A /3 K5(c) m ”C(T'2)II _ -(§9 e n( a) dc 0 (TE K5(-c) 9 e'C(T'2)Ia(c) dc = .(99 i:llfl:1. ( a '3 o /zt(-1)"*'Kg(c)-jn15(c)1 is the real variable t = oa/c (jw = a on real axis) in the resulting integrals. The impulse response finally becomes h(T) = E h (T) n20 n n n/2(even n) c e (n+l)/2(odd n) enge = u(e)(5)[ X Z 2Re{aa£e 1 + 15(4)] (3 3a) n=o 2=l Where the sum over 2 includes only those Cn2 with positive imaginary parts and -2; e e “we ) a' n n "g (3.31) n2 = n 2 [1+(EEE) J/ZggKn(cn£) 41 e I'(c)K’(c) I'(r) = -fl- " " 'C("2)d (3.32) n n J: /Z[Kr'12(c) + 0216(6)] e C It is observed that this impulse response consists of a discrete-spectrum series of pure natural modes EXP[(Cr)n£TJ Cos[(ti)n£r + ¢n21 augmented by the series of continuous-spectrum integral terms Ifi(z) which comprise non-oscillatory functions having an essentially decaying-exponential nature. ‘The impulse response is computed for T > 2, where the various series can be appropriately truncated. The series of continuous-spectrum integral terms is found to converge rapidly, and retention of only the leading 10 terms provides adequate accuracy. In these integral terms, contributions from the neighborhood of the singularity at c = o are calculated analytically using small-argument approximations of required Bessel functions, while the upper integral limit is truncated for the remaining numerical integration because all significant contributions from the integrand are found to occur for t < 5. All significant con- tributions to the discrete-spectrum residue series of natural modes are provided by the 2 = 1 layer of complex natural frequencies c , while n2 layers with 2 > 1 provide negligible contribuiton for r > 2. The I latter series is computed by summing the first 19 terms numerically while obtaining an approximate representation for the remaining terms to n = m. Layer index 2 = l is subsequently dropped for brevity. A study of natural frequencies and the associated residue coefficients fin a5 for large n indicates that the frequency difference A; = tn - Cn-l = (-o,034 + j o, 98) and residue ratio A = aA/aa_1 = 1,0607 exp (j67.55°) approach constant values for n 3 19, i.e., the roots of layer 2 = l 42 are distributed approximately along a straight line with equal spacing. Those terms having n > 19 can therefore be approximated as a geometric progression and summed in closed form. The predominant 2 = l residue series therefore leads to' t r 19 c T aig e 19 A eACT Z a' e n + A (3.33) n=l " l-Ae CT 00 C T )1 ZReiar']1 e n1 } z 2Re{ n: with aig exp(t]91) = 3.320 exp (394.94°) exp[(-l.845 + 317.9)41. This approximation was described in [28 ]whereit was used to quantify the impulse response of a conducting sphere. Figure 3.4 indicates the normalized impulse response and its constituent components. Table 3.2 shows the poles of the first layer and their corresponding residues. The series of continuous-spectrum integral terms provides an important contribution during the early-time period, where it largely annuls the contribution by the discrete natural-mode residue series which has opposite sign during that period. The specular reflection behavior near I = o is thus obvious (Figure 3.4 has the phase inverted). The residue series provides the anticipated, well- known creeping-wave contribution. It is noted that the approximate sum . of terms for n > 19 is an important contribution in the latter series, since the leading 19 terms alone result in an impulse response having an incorrect superposed oscillation. The present impulse response agrees well with the earlier approximation by Moffatt [29 1 over the time interval considered. 3.3 Incident Waveform Synthesis for Monomode Backscatter The late-time or free-response period in the backscatter signal from a target illuminated by a transient waveforem of finite duration 43 g- complete impulse response I _____ contribution due to discrete .l natural-mode residue seriesw CF 5‘ .....___2_ contribution due to series of 53 ;\ continuous-spectrum integral terms x \ .—. ‘31 58% ‘F j \\ (Do 80¢ \ CD 5: . \ q 23 .. \, 2' \\ - 8 . \ creeping wave 23 1 £1 £5 13 a) £3 23 L. (D C 9. 2-- - .. e- e. .-2 ..fi., I j—T‘ frv j}! 111v I v ‘40 3.0 7.0 8-0 9.0 10.0 2.0 V V 310' ' 130' normalized time r=t/(a/c) Figure:14 Normalized impulse response of an infinite cylinder lluminated by a normally-incident. transversely- polarized. impulsive plane-wave field. 44 Table 31.2 Poles of the first layer .. of natural modes and corresponding residues used to compute approximated impulse response of infinite cylinder. poles of 1'st residues at c branch (i=1) '. 1 n1 Cn1=°n1"3“'n1 an1=an1+3an1 n anl wnl 8:1 ail 1 -.6435 .5012 .2407 -.4415 2 -.8345 1.434 .6099 .0932 3 -.9676 2.374 .1442 .7440 4 -1.073 “3.322 -.7707 .4583 5 -1.161 4.277 -.8034 -.6535 6 -1.238 5.237 .3760 -1.114 7 -1.307 6.200 1.318 -.0495 8 l-l.369 7.167 .5769 1.346 9 -1.427 8.136 -1.151 1.131 10 -1.480 , .107 -1.612 -.7188 11 -1.529 10.08 .0790 -1.920 12 -1.576 11.05 1.962 -.6911 13 -1.6201 12.03 1.484 .1.685 14 -1.662 13.01 -1.080 2.158 15 -1.702 13.98 -2.578 -.2002 16 -1.740 14.96 -.8475 2.630 17 -1.776 15.94 2.250 -l.898 18 -1.811 16.92 2.779 1.440 19 -1.845 17.90 -.2857 3.308 45 Te is well defined, e.g., Jones [30 J. If the initial response (arising from the incident wavefront first striking the target) occurs at t = 0, then the late time period begins at t = Te + 2Tt where Tt is the one- way transit time for the wavefront to sweep across the target. Thus the late-time period of the impulse response begins at T = 4 in the present problem. It is found that during, and just prior to, the late-time period, for T > 3, the series of real, decaying, continuous-spectrum integral terms in the impulse response (3.30) can be approximated by two real- exponential terms while the discrete natural-mode series for n > l9 can be approximated by a pair of damped-sinusoidal terms; consequently that response can be expressed as N a T 11(1) = u(i) in): an e " COS(wni+n) +1(—,) + Rn) + 0(6)] . (3.34) The residue series include Nm(Nm = 19 for all numerical results sub- sequently presented) terms arising from the first layer (2 = l) of natural-frequency roots from Figure 3.2. Natural modes contributed by the higher-order layers (2 > 1) are insignificant during that late- time period due to their rapid exponential decay. Empirical approximations I to the integral sum and the residue series for n > Nm, valid for T > 3, are -0.95(T-3) -0.083 e -0 025 e'0'35(1'3) (3.35) H A H v ll -0.026 e"°°9(“3) Cos[18.85(t-3)] -0.0l6 e"'22(T‘3-DCosti7.95(e-3.i)1 (3.36) :0 A ...q V II while C(T) is a correction term required for T < 3 to compensate 46 for the approximations inherent in 1(1) and R(T). The approximated impulse response with C(T) z o assume for the late-time period is compared in Figure 3.5 to the relatively accurate representation obtained from equation (3.30) and displayed previously in Figure 3.4. The two real-exponential terms in 1(2) can be regarded as limiting natural modes having vanishing angular frequency and subsequently be included in the residue series along with the damped sinusoids contributed by R(T). If N = Nm + 4, then the impulse response becomes N a T h(t) = u(.) 2 an e " Cos(w T + 45) + 0(4) (3.37) n The backscattered-field waveform ES(T) excited by an incident field waveform Ee(T) having finite dimation Te is obtained through the convolution theorem as T ES(T) = [ eEe(T') h(I-T') dt' for T > Te 0 '[e e N On(T-T') = J E (T')U(T-T'){ E a e Cos[w (r-i') + W J + C(r—t')}di o ":1 n n n Since C(r-r') z o for T > Te + 3, then during the late-time period the previous expression becomes T N a (t-t') S _ e e 1 n 1 1 E (T) - (a E (T ) n2] ane costen(T-T ) + 4516. N OnT . = "g1 ane [AnCOS(wnt + W") + BnSln(wnt + Wn)] (3.38) valid for T > Te + 3, where FA 1 Cos... e' ( = Ee(r') e ” dt' (3.39) B o Sinw 1' 47 O c, 3- complete. accurate impulse FESDClSE I 4 ER 3 ...... approximated impulse response (39 terms of >< 93 natural-mode reSIdue series augnented by 1: a: 4-term approx1mation) 5. I 91 Si 9;- : 1 o d 4 8 3 creeping waie (D 3‘ a) a) 4 3 i a o‘ \— E 6. U 1 a: ‘, N 94 :: .5. E : ~< / A 1 1. 9‘ 8 9:: a) '4 H 1 O E o" 32 2: O 1 ‘- d D. ,4 5% 9: 8 .....,. .-. r..- ..- .-..,.- .....-... ...... .. .. ... . '0.0 1.0 2.0 5.0 4.0 5.0 51.0 71.0 a'.o 97.0 normalized time r=t/(a/c) Figur83.5 Approximate normalized, late-time impulse response of an infinite cylinder; utilized for synthe515 of incident waveform to excite monomode backscatter. 48 It is desired to synthesize an incident waveform Ee(T) which excites only a single natural-mode scattered-field response ES(T) from equation (3.38) during the late-time period. The desired Ee can be expanded in terms of some basis functions as indicated in Section 2.2. The basis functions used are pulse functions, the results of numerical computations will be discribed in the next section. 3.4 Numerical Results for Incident-Waveform Synthesis and Target Discrimination Incident waveforms required to excite monomode backscatter consisting of purely the first or second natural modes of the infinite- cylindrical target are synthesized according to the procedure described in Chapter 2. The finite duration of the incident wavefore is chosen initially, based upon experience with thin-cylinder targets [13 l, as one normalized period of the cylinder's first natural mode; this choice leads to Te = l/f1 = Zn/(ti)H = 23/0.5012 = 12.54. The late-time response, upon which the synthesis procedure was based, occurs during T 3 Te + 3 = 15.54; numerical results for the late-time, backscattered- field response are therefore presented for T 3 15.5. The incident signal required during 0 f T 5 12.54 to excite a purely first-mode (r,1 = -.6435 + j.5012) response is indicated in Figure 3.6 along with the resulting monomode response for r 3 15.5. It is noted that the return signal, which was obtained by convolving the synthesized incident waveforem with approximate impulse response (2.34), indeed consists of a first natural mode in the late-time period. The early-time return signal (not constrained by the synthesis procedure) exhibits an irregular waveform, and is omitted for the sake of clarity. 49 Also shown in the same figure in dashed line is the return signal from a wrong-cylinder target with 10% smaller radius, it is found that the response of the preselected cylinder consists of the desired first natural mode while that from the wrong target cannot be identified as a single natural mode. Figure 3.7 indicates similar results for the incident waveform required (with Te = %-) to excite purely second- 1 mode (22 = -.8345 + jl.434) backscatter and the resulting late-time return signals from right and wrong targets. For Te = %- as chosen 1 in Figures 3.6 and 3.7, it is found that the late-time response occures after T = Te + 3 = 15.54 while the negative real part of the natural frequencies are so large that the signal is very small after such a long time. This may cause serious problem when, in the practical situation, the signal is contaminated by noise. Therefore, it is desirable to synthesize the required incident waveform with shorter duration. We tried different I and found that the waveform may be optimal when e ie is in the order of (0.55 - 0.70)l/f1: If Te required waveform oscillates quite rapidly and the matrix in equation (2.9) is too small, the has a large condition number [31] indicating an ill- , conditioned situation in synthesis procedure while a large re causes the small signal-to-noise ratio as indicated above. The require incident waveform with‘ Te = §1%1 and the resulting late-time return signals from right and wrong targets are indicated in Figure 3.8 for the first mode excita- tion and in Figure 3.9 for the second mode excitation. It is clear that with this Te’ the return signal is much stronger than that from Te = %3 and thus easier for us to indentify and discriminate the target. The required waveforms with Te = 0.594-% to excite the first mode and the l ‘l N a. 50 9" a, 9. Incident Wave .0 G 3 4.) "a 'r 5 i" O) > .... N +1 a- (5 1 15 1x 9 a- I a. 1 ID a T T 6‘ t/(a/C) l"““ / “"- 9. ’ o I I I I Q I 9‘ Scattered Waves 2 Right Target m 1 -----; Wrong Target U 3 O +’ 6 'F' ‘- ,_ 1 Q E “ 9 9 9- .5 (U E "-'.' o: 8- I 9 8.. l 9 a '0.0 100 12 ' ' I r ' fi' . .0 14.0 10.0 10.0 20.0 22.0 24.0 t/(a/C) Figure 3.6. Synthesized incident waveform required to excite monomode backscatter in the first natural mode of an infinite cylinder and the resulting monomode scattered wave along with return waveform from a target with 10% smaller radius. '3. a 51 ‘1 94 O m . 'O :5 a . :3 ;~ ___. Inc1dent Wave 3 E < .. 9 ‘5‘ 33 CU '- 9 .2 93 9 ad I o: 1 T 1 1 1 1 7 1 60.0 2.0 4.0 MI 0.0 10.0 12.0 441.0 10.0 6‘ t/(a/c) 9 .d 9- a Scattered Waves 6 -—- Right Target m 9 - - - Wrong Target ‘0 3 4.1 I: 9- D- - E < CD > 9,. I) a - - - a - - I‘ ‘ - : 1‘5 '6 tr 9 71-1 9 “a I 9 = I ‘o.0 10.0 12.0 11.0 13.0 15.0 20.0 22.0 74.0 Figure 3.7. t/(a/C) Synthesized incident waveform required to excite monomode backscatter in the second natural mode of an infinite cylinder and the resulting monomode scattered wave along with return waveform from a target with 10% smaller radius. 3. V/”‘~————— Incident Wave Relative Amplitude 0 1 T 1 1 000 2-0 400 .00 0.0 10.0 12-0 14.0 1U-0 t/(a/C) 20.8 - l 51 Scattered Waves Right Target 2‘ --—' Wrong Target Relative Amplitude -t0.0 O c1 _1 - r 1 r 1 .0 10-0 12.0 [6.0 10.0 13.0 20.0 22.0 24.3 t/(a/C) Figure 3.8. Synthesized and scattered waveforms for the first mode excitation similar to Figure 3.6 except a shorter Te' a- M ,- 53 9- 0 a- 9- Incident Wave a) O 'U 3 4.! e i~ <: 0.) > 9 IS 9‘ (O '6 m a o'- l T 1 1 1 1 1 ‘1 0 0 2.0 4 0 a 0.0 10.0 12-0 14.0 16.0 a t/(a/c) 5 9 2 9 ._ Scattered Waves 2. -——q-.Right Target g --- Wrong Target :9 a .5 4‘ D. E <: (D o .i '3‘ .1.) (6 £2 9- O 9 N I 9 7"“ ' ' l I I l 1 a . 10.0 12.0 14.0 10.0 10 0 20-0 22.0 24.0 t/(a/Ci Figure 3.9. Synthesized and scattered waveforms for the second mode excitation similar to Figure 3.7 except a shorter re. 54 second mode are shown in Figures 3.10 and 3.11 for comparision. It is obvious that they are very similar to those with Te = % %-, and therefore 1 the resulting radar returns are not computed. The possibility of using different basis functions will be discussed later. 55 Relative Amplitude " .0 ”600 l l 1 0.0 11.0 21.0 3.0 4.0 5.0 6.0 7.0 t/(a/C) Figure 3.10. Synthesized incident waveform required to excite monomode backscatter in the first natural mode of an infinite cylinder with Te = 0.594 $3. 56 i 1 l Relative Amplitude 0.0 1 1 *“‘1 0.0 1.0 2.0 3.0 4-0 510 810 710 8.0 t/(a/C) Figure 3.ll. Synthesized incident waveform required to excite monomode backscatter in the second natural mode of an infinite cylinder with Te = 0.594 ?.. l CHAPTER 4 SKEW-COUPLED WIRES We apply the waveform-synthesis method and SEM to a target, con- sisting of a pair of skew-coupled, perfectly-conducting thin wires, which is illuminated by a transient, obliquely-incident, plane wave. The geometry of the problem is defined in Section 4.1 that also specifies the incident field. The integral equations descussed in Section 2.3.2 are applied to this geometry and described in detail in Section 4.2. The induced currents on the two wires are decomposed into symmetric and antisymmetric components to reduce the coupled integral equations into single integral equation for each mode. The numerical computation is thus simplified. The integral equations are solved in Section 4.3 to obtain the induced currents. Section 4.3.1 concerns with the natural modes by solving the homogeneous integral equations while Section 4.3.2 uses these natural modes to compute the coupling coefficient associated with each mode. The induced currents are obtained by carrying out the numerical computations of coupling coefficients and inverse Laplace— transform in Section 4.3.3. These induced currents are used in Section 4.4 to compute the vector potentials which, in turn, generate the back- scattered field. Some general formulas are derived first in Section 4.4 and then the specializations of parameters are made in Section 4.5 to determine the impulse response of a few special cases. Some of the cases are related to the experiments which will be described in Chapter 6. Finally, Section 4.6 demonstrates the numerical results of waveform- 57 58 synthesis and its application to the radar target discrimination. Com- puter simulations by numerically convolving the synthesized incident waveforms with the impulse responses of the right and wrong targets are shown to indicate applicability of this waveform-symthesis scheme in practical situations. 4.1 Geometry of Problem A pair of skew-coupled, perfectly-conducting thin wires with radii "a",lengths L, orientation angles a and distance 2d are illuminated by an obliquely-incident, transient, plane-wave radar signal at an angel 0 as depicted in Figure 4.1. The incident field is ex- pressed as ‘51 E" (it) = a F(t- 5;— ) (4.1) the unit polarization vector of the incident field, where E = gxx + gyy + sz F 2 = the position vector, 11 X X .1. ‘< '~< .1. N k = k x + ky y + k2 2 - the unit pr0pagation vector, with k - i = o; and F(t) is an unknown waveform function to be synthesized based on the requirement that E1 (?,t) excites a single-mode scattered field in the late-time period. The tangential components of E1 (?,t) on the wires, in their Laplace-transform, are ~1 _ j , ~ E tanj(uj’s) - (Cy cos a-(-1) CZ s1n a) F(s) j . - r . - - + . exp{ y_kyuJ cos a ( l) kz(d uJ s1n a)21 2) (a. e o 5 uj g L, j = 1,2 for wire #1 and #2 respectively, 59 Figure 4.1. Two thin wires oriented at an angle are illuminated by an incident radar signal. 60 where E(s) = L{F(t)} and y =-% is the complex propagation constant. These electric fields excite transient induced currents on the wires, and induced currents, in turn, generate a transient backscattered electric field. Our goal is to synthesize an aspect-independent waveform F(t) for monomode excitation. 4.2 Integral Equations To simplify this problem, we decompose the induced currents into symmetric and antisymmetric components, i.e., I1 = I5 + Ia (4.3) where IS is the symmetric current which is the same in both wires while I8 is the antisymmetric current which flows in opposite directions with equal amplitude in both wires. After this simplification, each mode needs only one integral equation instead of two coupled integral equations in two unknown currents, I1 and 12. By matching the boundary condition on the perfectly-conducting wire surfaces so that the total tangential electric field, E = E1 + Eian’ equals zero there, we obtain the electric tan tan field integral equations (EFIF) from equation (2.26) as g L ) 32 2 " " e-YR.£ {- I I (U'. S' [————-n + y (u -u )] du' £=1 0 2 2 aujauz j 2 4nRj£ 2 8W.£ ~i . + SE.l_.(uj)} = -50 s Etan (uj,s) ... for o 5 0J. f L J = 1.2 (4-4) 3 J - R. ., - . ., Y J2(uJ L) + e 7123201J 0) where W. (u.) s I (L',s) e -I (o .S) = 0, (4.5) 31 J 1 4nRj2(uj,L) 2 4nRj£(uj,0) 61 since I£(L',s) = I£(o+,s) = a at the wire ends. .. __ ~i .= - By defining Sj(uj,s) = cos Etanj (uj,s), J 1,2 (4.6) as forc1ng functions, and dropping j,2 subscripts, we can rewrite equation (4.4) as L L J I](u,s) K(ulu',s)du' + I 12(u,s) K2(u|u',s)du' = S](u,s) o o L L } J I1(u,s) K2(ulu',s)du' + I I2(u,s) K(ulu',s)du' = 52(u,s) o o | - a + 2 e-YR(U9U') - where K(UIU ,S) - - [W y 1W - self-kernel, 2 2 -iR2(U.U') (4'8) K (u|u',s) = -[—§——T-+ y C05 20] e . = Za-coupling 2 auau 4nR2(u,u ) kernel, 1 with R(u,u') = [(u-u')2 + a2]2 , (4.9) R2(u,u') = {(u-u')2 Cos2 a + [2d + (u+u')sin 612 + 62}%. From addition and subtraction of two equations in (4.7) and equation (4.3), we get the following equations, L I Is(u,s)K (ulu',s) du' = SS(u,s) o S for symmetric modes L (4.10) J I (u,s) K (ulu',s)du' = S (u,s) o a a «1 . for antisymmetric modes U E LosLjs where Ks(u|u',s) (4.11) K(ulu',s) + K2(u|u',s) } Ka(u|u',s) K(u|u',s) - K2(u|u',s) , 62 and SS(U.S) = 8 [S](u,s) + 52(U.S)] (4.12) Sa(uss) = 1/2 [51(1195) ' 52(U,S)] Coupled EFIE's in (4.7) are thus decoupled. For the convenience of numerical computation, we obtain Hallen- type integral equations from (2.35) as L L [a I](u',s) Kh(u|u',s)du' + A0 12(u',s) Kh2(u|u',s)du' :05 u ~i = C Cosh yu + C sinh yu - ——— E (g,s)sinh y(u-g)dg 0 tan L L (4.13) (o I](u ,s) Kh2(u|u ,s)du + (0 12(0 ,5) Kh(u|u ,s)du . E:05 u ~i . = 012 cosh yu + C22 Slnh yu --j;— (a Etan2(g,s) Sinh y(u-g)dg for o f u f L e"YR where Kh(u|u',s) = 4?R_' = self Hallen-type kernel, e'YRz u (4.14) Kh2(u[u',s) = EE§E—-Cos Za - (a 92(£,u',s) Cosh y(u-£)dg = 20 - coupling Hallen-type kernel . . = e u'sin a + sina ,_ as a "'t“ 92 ‘5’” ’5) dR2(£.U') 44221:.u'1 R214.u') (4-15) Similar to the process of decoupling EFIE's, by adding and subtracting two equations in (4.13) and using definitions in (4.3) and (4.6),we get the following decoupled Hallen-type integral equations, 63 L J I (u',s) Khs(u|u',s) = C cosh yu + CZS sinhi u U + if $565.5) sinh i(u-4)d£ o (4.16) for symmetric modes L Jo Ia(u ,s) Kha(u|u ,s) = Cla cosh yu + CZa 51nh y u i “ . + §-Jo Sa(g,s) 51nh y(u-§)d§ for antisymmetric modes u e [o,L], C + C C - C = 11 l2 = 11 12 l where Cls ___—IT.—__ Cla -———7r——-—— , k = arbitrary constants c = C21 + C22 c = C21 ‘ C22 1 25 2 2a 2 ’ Khs(u|u',s) = Kh(u|u',s) + Kh2(u|u',s) (4.17) Kha(u|u',s) = Kh(u|u',s) - Kh2(u|u',s). Equation5(4.l6) are the integral equations to be used for finding the natural modes while equations (4.10) are used to compute the coupling coefficients [9 l which are necessary in obtaining the impulse response. 4.3 Induced Currents 4.3.1. Natural Modes Natural-mode solutions are those modes which exist as the solutions to the homogeneous problem with E(s) = o. By applying moment method [321,the integral equations in (4.16) are converted to a pair of matrix 64 equations as As(s) IS = 0 (4.l8) Aa(s) Ia = 0 = ’ = ' 1 where IS C1) Ia C1 I2 I2 I3 I3 INP INP LC . C 2 s L 2. a for symmetric and antisymmetric modes respectively. Note that in this solution the wire is partitioned into NP partitions with I1 = INP+l = 0 for the current near the wire ends and subsequently dropped from the unknown colunm matrix while C1 and C2 are included as unknowns. The detail of partitioning the wire is shown in Figure 4.2. For the purpose of getting nontrivial solutions for I and S Ia, matrices As(s) and Aa(s) must be singular, and therefore the natural frequency is that s which satisfies det [As(s)] 0 for symmetric mode } (4.19) e, det [Aa(s)] 0 for antisymmetric mod Both Newton's and Muller's methods are used to search for the roots of equations (4.19), NP = lOn is used for the nth mode. Once we've found Figure 4.2. 65 L,i=Mn+l uni = An(i-l) 0, 1 = 1 z 7m“: {I}. L I ""‘U:= O 1 1---“. Np = # of partitions = Mn for the nth mode A .—__L_=L n Mn lOn Partitioning of the wire for pulse-function expansion. moment-method solution using 66 5n = on + jwn to be a root, its complex-congugate, 5:, is also a root, since equations in (4.19) have real coefficients. Therefore, for those numerical results we'll show below, only those roots which are in the second quadrant are demonstrated explicitly, and the natural modes are in the form of An eOnt Cos(wnt + m"), where An and en depend on the aspect-angle (which, in turn, depends on E and k). In the following, some natural-frequency distributions with L/a = 200 are demonstrated with different parameter varied. Figures 4.3 and 4.4 are the distributions of the first 10 natural frequencies in the first layer with d/L = 0.5 and a = 0°, 30°, 60° and 90° together with the first 10 natural frequencies for the isolated wire. Figure 4.3 is for antisymmetric modes and Figure 4.4 for symmetric modes. The dis- tributions of roots with different angles are so close to the roots of the isolated wire that it is not easy to make any conclusion about the coupling effect due to different angles. In Figure 4.5, we plot the trajectories of the first antisymmetric and symmetric roots with L/a = 200, d/L = 0.5, with changing a. It is obvious that they are converging to the first root of the isolated wire as a increases; this is rea- sonable since increasing 8 reduces the coupling between the wires. The other interesting observation is that the root of the isolated wire is roughly the average of the roots of antisymmetric and symmetric modes. We see about the same property appears in Figure 4.6 for the second modes. The effect on the first antisymmetric mode for L/a = 200, a = 0°, 30°, 600 and 900 by changing d is reflected in Figure 4.7; there are spiral- like trajectories with a largest "radius" for the case of a = 00 and the a = 900 case smallest. This is again a demonstration of a smaller coupling for a larger angle. Another important observation is that as 67 ‘9 ‘31 o I 4.1 ___: Isolated w1re X 1 a = 00 C): a = 300 .31 x: 01 = 60° (:1: a = 90° ‘9 3' A o .4 u 3 p 0" O) o» .3 ID ‘3- fl' ‘8- N 1: ' I T 1 i l c.4000 $600 -5000 .'2600 -2000 -1500 -1000 -500 040 o gE'X 10' Figure 4.3. Locations of the first 10 natural frequencies of the first layer of the antisymmetric modes for theotwo Souplsd wires0 with L/a = 200, d/L = 0.5 and for a = 0 , 30 , 60 and 90 . O 68 -—-—: Isolated wire : a = 00 30 60° 0 E3 Dr C) X p Q n 0 ID :3 °-36 .0 Figure 4.4. l I l 1 -;0-0 -25-0 -20-0 -15-0 ~10.0 -5.0 0.0 EL.X 102 nC Locations for the first 10 natural frequencies of the first layer of the symmetric modes for thS two ocoupled wires with L/a = 200, d/L= 0. 5 and for a = 0, and 90. 69 °! 531 ll,‘ Symmetric modes 2_ 0 ° Antisymmetric modes " Isolated wire ‘52 = 00 “3.. “‘\‘ O! ‘\\ o .\ a - 10 \ \ \\ ID \ .. \ O \ A 3 c3 ___ 1 .—4 ’-"'0- ”‘0“.‘0 ’ x 1... ,r” ‘m _ 0 I A on ’0 01 90 / 3 I: /’ x // / ‘1’ / A. 2 ’ I ~— __- ".,- ’ °.'.1 o a = 909 ‘ 4‘“ m 1 \ \\\ 9- “0. a: ‘c‘ 7"-o- ........ .9 01 = 10° 01 = 0° '9- O «3 1 l 1 I I ' 0 '50 -'4 0 ”-12.0 -11.0 -1o.0 -9-0 -8-0 ‘7") ‘6' ' it. x 10' 11C Figure 4.5. Locations of the first natural frequencies of the symmetric and antisymmetric modes vary as functions of the orientation = 200 and d/L = 0.5. angle ; L/a 70 A': Symmetric modes 0 : Antisymmetric modes x : Isolated wire 10 wL X 11C -b-" 10.6 T r I 1 F T T 1 -17.0 -16.0 -1s.o -i4.0 -13.0 -12.0 -11.0 ~10.o -9.. °_. 2 “C x 10 Figure 4.6. Locations of the second natural frequencies of the symmetric and antisymmetric modes vary as functions of the orientation angle ; L/a = 200 and d/L = 0.5. L 5%5 X 10 71 2 o 33 :31 d/L = 07 E 1 I 9. (UL = 2.0 r : I ’ ’ A - .- ~ ‘ )I 94 III, ~'\\ d/L = 1" : ',:_. . , ,- ‘4 , c: ‘”' ' ;.___.d L - 2 0 I i \ I, I'_ If, . / — :4 1 1 T I X 5 ’1 “ ' i n . .7 __11 U 1" 7 1, \ . f r 31: . u - '3‘ 0 I O" .‘\. I I I \ \ I / ‘ \ .--’ 2.1 \~..~.‘_ .fir’.’ ”II . ”I~ ._—_. d/L = .l 1‘..“’., z-uJ -'1...1 -1|.0 310.0 -;.11 $.11 -;.0 -1-0 01.0 .4130 14.0 ~IrI-U 10.0 0 a .1 ~17 1 .1 0 0 CL CL ——-x 100 ——- c nc x 100 X: Isolated wire 1 o T - a = 50 a = 900 .1 CD .~ I—' .4 ’0' ‘0: '. 0.-...‘0. x . ” Q '- ‘.’ ”‘9 d = 2 . O, , . {5.9—4L 0 9 :~ ' :1 ,‘l .. . d/L = .1 3 :1 d/L = 2.0 .3 d/L = .1 :1 :4... .1...) I. o J... .‘a .37.. ... .I.‘ a“ -IO-0 “-6 48.0 0|... 00.0 -|.O 0-0 441 0-0 Figure 4.7. Locations of the first natural frequencies of the anti- symmetric mode vary as . functions of the spacing between wires for a = 0 , 30°, 60° and 90° and with a/L = 1/200. 71 3 o _ "l a = 0 a " 300 §« d/L ;1 ,4 d/L 29 t o ’ A - .- N .H‘ / ’ I / ’ .\\ d/L : 4 — .‘ I Q \ O , :.-O ~~ l. ,’ X ‘1 - ,g' «___. d/L 2.0 3*! ‘ i X 3 .f ‘. ‘ I - I [l l , ' . O —’ U l‘ I .. \ ‘ , ’ 3 1: ~14 3“ ii \ ‘0‘ o I . \ \ .--'.’.' 3‘ x ‘ w: v“ ’ x - - d/L C d -. ' .4 5* . :4... {no 43.0 no a $.11 jam 0.: ‘4“. ....1 L... J.” .5... o .1 .1: 51.; o o _q_L X 100 9.1-. X "'00 X: Isolated wire 0 3‘ a = 60 a = 900 éi 3: fl 6‘ O .‘ Ho- '— z-i " V‘ 7 ~.‘ >< . -’ (at-0“ d/L = 2.0 .’ ,9 ° «3 "3’ 3 ~ :i : ‘ ' d/L .1 i :1 d/L = 2.0 «5 d/L * , . . r j T :um at“ -ha J». .a. It .1 is an ‘-u.o -u.o «.0 10 a 4.0 -o-o «.0 4.0 Figure 4.7. wires for a 03 Locations of the first natural frequencies of the anti- symmetric mode v functions oof the spacing between ”60 and 900 and with a/L= l/ZOO. 72 the distance becomes larger and larger, the coupling between the wires decreases, and subsequently the root becomes closer and closer to that of an isolated wire. This figure is compared with the result in [33] in which only a wire over ground plane and thus only antisymmetric modes exist. Figures 4.8, 4.9 and 4.10 are demonstrations of how symmetric and antisymmetric natural frequencies change as d changes for the first, the second and the third mode, respectively, with a = 0°. It is found that for higher order modes the "radii" of spirals become smaller and smaller and converge faster to the root of the isolated wire. To complete the natural-mode solution, we must compute the natural- mode currents associated with the natural frequencies. The way to com- pute them is to substitute the roots we've found into (4.18) and solve the homogeneous equations by eliminating one equation (one row of matrix) and setting a particular segment of the current (the best choice is the segment which has the maximum current; if this choice happens to be zero- current segment, then the solution will blow up) to be one and moving the corresponding column to the rightluuuiside with the negative sign. Some results for different modes, different angles with L/a = 200, d/L = 0.5 are shown in Figures 4.ll - 4.13, they are also compared with pure sinusoidal current distribution which is approximately the case for the isolated wire.It is seen that the imaginary part of natural-mode currents are affected more by the coupling. Compare Figures 4.5 and 4.6 for the coupling effect. 73 Antisymmetric Mode ' a 2‘ 9d/L = 0 . I 5‘ d/L = 2.0- f /. O... ‘ ! , ’ ‘ ~o ,.' / o-J / / \ . ° 0 \ d/L = .l / ' O / l/ I “ I t. i , , O l / >< QJ \ _J t.)0 \ o\ O 3 F \ \ l/ \\ \ O] Q :d O\ ‘ \\ V‘ \ -0 0’ l/ o . \ o , \ ‘o ————— o 9.4 o C 6 1 1 1 r 1 . fi 1, .‘.00 -‘400 -1200 -‘000 - ..00 15.0 "00 -200 000 X: Isolated Wire 0L ‘2 :5 X 100 9.1 a Symmetric Mode é-l .... "' f _ ~ \ ~. O, \ 0-1 If \ \ / \ / \ / .. ..-' - s ‘ \ ‘\ / / ‘o\ \ I- / I \\ \ o 4// I ‘\ \ .— 7 , \ 0‘ x t. ’ L l ' _s u. | O ‘ l 3 t: I 1 x b o ' I l \ , I \ \ / .o\ \ / \ \ .1 a \ ‘0, x/ 0d \ \ z.' \o C = ‘. d/L .1 o a l 1 r 1 r I T . 40.0 “‘00 -12-0 40-0 ' “.00 08.0 «.0 ~2oo O- oL/nC x lOO Figure 4.8. Locations of the first natural frequencies of antisymmetric mode vary as functions of d/L for a = 0° and a/L = l/ZOO. 5%? 74 ‘1 e Antisymmetric Mode , S3 3* e x . f .' d/L = .1 ...-J U 9 I . , ', 3 I: 2- 1 ‘ , O O I \ \ / '2 \ k ‘ ' " 2“ \ . -. ' ,. 2" ‘ - .__ - , ’ ' f-lI-O HID-0 -IH-0 -'|2.0 40.0 -L0 4.0 4.0 -2.0 X: ISO-lated Wi re oL ——-x 100 11C El §ymmetric Mode 0‘ '3 ‘l , ’ ’- W .. ‘ I \o o . , ’ \ o-J . , \ \ Ill \ i1 ,’ \ I, , " ' - ‘0\ ° ’ x o ..J ,' ° \ , r— " I \ l I o I x l t . . I __I'U 2 1 ' l, / 3 F: .‘ ‘ 1 .1 / \ I . 1 \ ' ' / .- \ O K I / / 2“ ‘0 ‘ x -.x _ d/L = .1 £1 d/L = 1.2 2.1.... 4... -5... .1... .1... -1... -1... .1... .1... 0L ——-x lOO 11C Figure 4.9. Locations of the second natural frequencies of antisymmetric mode vary as functions of d/L for a = 0° and L/a = 200. x 10 11C" ‘Y 2 mm x l0 nC 75 Antisymmetric Mode ’ —." ' .. _ I -‘OIO x 100 1 41-0 0L 11C T Y 41 -|.-0 -|i-0 'IO-O X: Isolated wire §ymmetric Mode '2..° Figure 4.l0. T I I I I I I I '2'.” '2.00 -2100 020-0 .‘.00 40-0 40-0 -|2- %xl00 Locations of the third natural frequencies of antisymmetric mode 8nd symmetric mode vary as functions of d/L for - and L/a = 200. ‘ 76 Real Part 0) 1D :5 :fi 23 E <1: 0) > .5 f6 '6 C! 90 o o z 0'. ate o'.a 1’0 1‘2 1 . 1 a O u/L El 0 A 30 at- 60° _ . e 90° ol --u.sinusoidal Dis- Imaginary Part tribution 64 . ———-Isolated wire 0 'D 3 :2 1i 5 0) .> .5 rd '8 a: éom 0T2 01.. 0'4 013 1:0 1:2 1:. gig u/L Figure 4.ll. Real and 3mag1‘8ary 0Barts ofO the first natural- mode current for a = 0, and 900 with L/a = 200, d/L= along with those for the isolated wire. o 77 é“ . ' Real Part 0) "D :3 4.) 23 E < (D > .5 f0 15 o: :90 a 0‘2 0‘. oi or. 170 1'" 1‘0 1“ 0 ........ . «1110 u L / A300 0 3K 60 g. #909 . r , . Imaginary Pa t um-Sinuso1dal 2‘ Distribution /. _——Isolated w1re m '0 3 4.) Ti E < CD > .5 (U '6 CI u/L Figure 4.12. Real and gmagiBary Barts ofothe second natural-mode current for a = O , 30 60 and 90 with L/a = 200, d/L = 0.5 along with those for the isolated wire. 78 Real Part Q) '0 3 4.1 Ti E < d) > .5 (O '6 a: . 1: 0° 2 A 30° 1.. .1. .1. .1. .1. .1. .1. .1. *7. .14 60° u/L 7:900 --~Sinusoidal _ Distribution on ———Isolated wire Imaginary Part ,3 Z- . 3 .~ 4..) . Z . ’ o. 6‘ g / 0) c > .5 .5 f0 23 9. a a 0.0 1 u/L Figure 4.13. Real and imaginary Barts ofothe third natural-mode current for a = 0 , 3O , 60 and 90 with L/a = 200, d/L = 0.5 along with those for the isolated wire. 79 4.3.2 Coupling Coefficients From SEM, the induced current can be expressed as )-1 N I(n,s) = n21 an(s) vn(u) (s-sn Where vn(u) is the distribution of the nth mode current, an(s) is the coupling coefficient corresponding to the spatial dis- bribution vn(u) and the temperal variation is controlled by (s—sn)'1 in the Laplace-transform domain. The coupling coefficient is computed by using equation (2.37). For this problem currents are decomposed into symmetric and antisymmetric components and expressed as Ns - -l IS(U.S) - ”=1 asn(5) vsn(U) (s-ssn) (4.20) N -1 Ia(u,s) = nZN +1 aan(s) .an(u) (s-san) s where N = NS + Na and Ns = number of symmetric modes used, and Na = number of antisymmetric modes used. In the following computations NS = Na = 10 are used. The coupling coefficients are computed by L Ss(u,s) vsn(u)du a (s) = 0 sn ILL ( ) ( a | v u v u'){-—{K (u u',s)l} _ - du'du o 0 sn sn as s S-Ssn L (4 21) I S(u,s)v (u)du - _ o a an aan(s) - L L . 8 . . . I0 van(u) van(u ){E{Ka(u u ,s)JS}=S du du an 1 O 80 Coupling coefficients in (4.21) are called "class-2" coupling coef- ficients which take into account of the causal property by different "turn-on" times for the contributions from different current segments. The "class-l" coupling coefficients are sn asn(s)ls=ssn } (4.22) an an a a II 9.! A U) v U" H m We'll see that this formula is indeed true only in the late-time since it does not take care of causality. 4.3.3 Computation of the Induced Currents Referring again to Figure 4.2 with NP = Mn = lOn, the nth mode current can be expressed as M v (u) = Z I . p .(u) (4.23) Where Pni(u) l for (i - %)A 5 u 5 (i - %)A II O m —l m m 2 :- m 1 m —.';_ = -= and An - Mn , Ini 0 for 1 1 and Mn + 1. Defining uni = (i - l)An = center of the ith segment for the nth mode, we get, from equation (4.19), 3 Zn (l-%)An ( ) I . S u,s du i=2 "‘ (1%).” " an(s) = T M - (4.24) n n IIM {é%{K(u|u',s)J} du'du S=S (l-%)An[(j-%)A n n 2 '=2 I”) Inj ((i- ~3-)A (’- E): J 2 J 2 n 81 With Sn(u,s) for symmetric modes and antisymmetric modes expressed in terms of equaitons (4.12), (4.6) and (4.2), K(ulu',s) in terms of equations (4.11) and (4.8), we can carry out the integrations in (4.24) as Mn Mn G (s) _ s X In1gn1(s) ~ 2 Inign1(s) a (S) = n = :O—‘E(S) i=2 _ _ CF(S) i=2 n D 2 M 4M 609 M 7M’ n n d n n I I . Z 2 n1 nj nij Z 2 n1 nj n13 i=2 j=2 1=2 3:2 (4.25)1 where 9n1'(5) = v 9n1(5) 2(EXC05a + czsina) A = (kyCOSa + kzsina) sinh [y(kyCOSa + kzsina)'-— 2(C COSa - C Zsina) ' (Ry COSa - k zsine?) . exp{-YEkZd + (kyCOSa + k Zsina)Un A o .... O n o . SlnhLY(kyCOSa - k251na) ??Jexp{v[kzd-(kyCOSa-kZSlna)u ni]}’ (4.26) dnij(s) = 4nC dnij = I - 2y J + Y2K (4 27) nij n nij n nij’ ‘ with I _ 2e (i- j)? A2r n+a2 _e'YnI/Qi-j+l) )2A2+62 _e-Yn/(i-j-UZAZH2 nij = 120 Q is used. ...: not 0 82 -yn/(i-j)2A:cosza + [2d+(i+j-1)Ansina]2+a2 1+ {e -y /(i—j)2A2C052o + [2d+(i+j-3)A sinq12+a2 + e n n n -y /(i-j+l)2A2c052a + [2d+(i+j-2)A sina12+a2 _ e n n n 2 2 2 -yn/(i-j-l)2A§cos a + [2d+(i+j-2)Ansina] +a . - e 2 e'YR (uni’unj) e'YnR2(uni’unj) .. t n13 n R(uni’unj) R2(uni,unj) Cos 2a for C_a II D ni’unj) 2 COS 20 - ynA -YR2(U + J for i = j, n R2(uni,unj)7 n n An+/h:+a2 J = A an ( ) 2 2 n n -37———- -2 Mb + a . -A +72 +a2 “ n n -1 R(u -.u -) -1 R (u -.u -) Knij = AfiEE n n1 "3 t e n 2 n1 "3 Cos 2a J S - _fl and yn c The "f" sign from equation (4.26) to Equation (4.31): "+" should be used for symmetric modes for n e [1,NSJ "-" should be used for antisymmetric modes for n 6 [N5 + 1.N]. It is obvious from the above definitions that both d' and gr'n.(s nij are dimensionless. (4.28) i f j (4.29) (4.30) (4.31) ) Upon substitution of (4.23) and (4.25) into (4.20), we obtain symmetric and antisymmetric components of induced current as 83 N M M ~ 5 n n = -CF S [‘1 ’1 ' 15(“’5) Efié‘l‘ E .2 E Dn (S'Sn) Inilnk gni(s)pnk(u) n-l 1-2 k-2 (4.32) M M n n Nh D' = 4 D = . . '.. ; 4. ere n to n jZZ jZZ In1Iann13 ( 33) and similarly, cF(s) N M" M" -1 -1 Ia(”’5) = ’ 600 E .E E Dn (5'5 ) Inilnkgni(5)Pnk(“1 n-NS+'I 1'2 k-z (4.34) The induced currents on both wires are thus computed according to equations (4.3). The numerical result for the special case of L/a = 200, d/L = 0.5, o =00 and the wire over ground plane (therefore only anti- symmetric modes are considered) with a step-function input at u = 0.5L as a function of time is shown in Figure 4.14. This result is compared with the result shown in an existing paper [33]. Using equation (4.34) with L/a = 200, d/L = 0, a = 90° without computing the contribution from coupled wire, we can determine the current at u = 0.5L due to the step-function input applied to an isolated wire. This result is shown in Figure 4.15 and compared well with the result shown in [10]. The impulse responses of induced current at u = 0.5L are shown in Figure 4.16 and Figure 4.17 for a parallel wire over ground plane and the isolated wire. It is easy to see that the early time of the current for an aspect angle o is just one-way transit time, Tt = L cos W/c, the time for all the current segments to be "turned on". The aspect angle used in Figures 4.14 - 4.17 is 30°, which gives us Tin= /§ L/c as the early-time in which "class-l" and "class-2" impulse responses are different as shown in Figure 4.17. Relative Amplitude 0-2 I 84 (D C '1‘ . 1 0.0 100 200 Figure 4.14. Step response of current at u = 0.5 L of a parallel wire 1 3.0 over the ground plane with L/a = 200, d/L = 0.5 and aspect- angle 30°. 85 0-4 ‘ 0.2 0-1 "4 Relative Amplitude V O O '0 0 1'0 era 1 4'0 5'0 6'0 7'0 1 o o I o a o o 8.0 t/(L/C) Figure 4.15. Step response of current at u = 0.50L of an isolated wire with L/a = 200 and aspect-angle 30 . Relative Amplitude 86 \m+ -0-4 \fi* l‘——L s““390 r ‘ d Iflrler'Tr,,,,r_’_,7 I I I I I0 710 8:0 0 0 1'0 2.0 3-0 4.0 5.0 6. t/(L/c) Figure 4.16. Impulse response of a parallel wire over the ground plane for current at u = 0.5 L with L/a = 200, d/L = 0.5 and aspect-angle 30°. Relative Amplitude 87 q - e1 f I. L 309’ 9.1 . i I t 6. °: °. N ‘3') —>— Class-2 Class-l -0.4 l -008 r 0.0 1-0 2.0 3.0 4-0 5.0 6.0 7.0 8.0 t/(L/C) Figure 4.17. Impulse response of current at u = 0.5 h of an isolated wire with L/a = 200 and aspect-angle 30 , both class-l and class-2 coupling coefficients are used. 88 4.4 Backscattered Field The scattered electric field in the radiation-zone can be deter- mined from thevector potentials maintained by induced currents on the two-wires. Consider first the scattered field maintained by the cur- rent on only one wire, as shown in Figure 4.18. It is easy to show that N ES(?,S) = 0 s AS(?,s)sin e (4.35) N lJ 'Y R°° L 1 where AS = 4%- e Rm I I(z',s)eY2 COS edz', (4.36) o Figure 4.18. Geometry of equation (4.35) for radiation-zone field maintained by current in single wire. Rw’ in (4.36) is the distance between the starting end of the wire and the observation point. Using linear superposition, consider the current problem as shown in Figure 4.19, the scattered field can be expressed as x 89 kP—CL —+++—cL -—>1 u A ’27 1 02 Rm1 P (Observation point) a.) I R60 A C .. 61 4;) ‘fi.“ .112 Figure 4.19. Geometry of Equation (4.37) for radiation-zone field where and maintained by currents in two wires. 5 + _ . ~s . . ~s . E (r,s) - 61 5A1 Sln 01 + 02 5A2 s1n 02 (4.37) -YRm . ~s _ “o e 1 L . Y” cosel . A] - .4—11- T [0 11(11 ,5) e du (4.38) ~s “o e'YRmZ L ( ) yu' cos 62 A = ——- -———————-J I u',s e du' , R001 3 R0° - d C05 0 (4.39) R002 z R0° + d COS 0. A For radiation-zone backscattered electric field, r] z r2;:r = -R , 90 then Cos 0 = [2 . (-&)1 = -k2 = . . _“ = _ _ - 1 Cos 01 [u] ( k)] Ry cos a kZ s1n a (4.40) Cos 02 = [02 — (-k)] = -ky cos a + kZ sin J. Since in sphericalcoordinates 5 sin e = (2 x r) x r, by the analogy of Figure 4.19 and the spherical coordinate, the following relations can easily be derived: A 61 sin 61 = [u1 x (-k)]x(-k) = x(kxky cos a + kax s1n a) 2 . . 2 . . + y(ky cos a + kykz s1n o - cos a) + 2(kykz cos a + k2 s1n o- s1n o) A 62 1&2 x(-£)1x(-E) _ . . . 2 . - x(kxky cos a - kzkx s1n a) + y(ky cos a - kykz s1n a - cos a) 2 . . s1n 01 + s1n 01). + 2(k k cos a - k Y Z Z (4.41) Substitution of Equations (4.39), (4.32) and (4.34) into Equation (4.38) leads to M u -va+ydc056 N n ~s o e cF(s) A = ——- {- Z 2 in D' (S- -s N) 9' (S) 1 4. Rm 600 n=1F2 k_21ni1nkn (k'2oAn u'cos e P (u') eY 1 du'} (k-§)A "k 2 n ~ M Mn Cu F(s) -YR°°+YdCOS 6 N n _ = " ingn Clse e Rm { 2 2 7" D. 1(“11) 1Inilnkgni(s) Y 1 n= -1 i= -2 k= 2 yu cos 6 yo cose e nk 1 sinh (———2—L ) ~ - Rw+ dcose N Mn MnIg'1.(s) yu cos yA cos a F(s) e Y Y Ini Inkn nk q n 1 - ' ycose1 Rm n21 122 kZZ Dn(s- sn)' e s1n(————§————), (4.42) 91 similarly, ~ -YRm-ydcoso Ns N Mn Mn I I g' (5) Yu cos 6 fis_ _ Efis) e _ ) 7 2 hi nk ni e nk 2 2 YC0562 Rm n=1 néNs+1 i52 k=2 Dn(s'sn) yancosez sinh (———*:r———l (4.43) Substituting (4.42) and (4.43)into (4.37), we obtain the expression for Ebs(r, s) as M M 9' (S) g + N -wa . N n nr Ini Ink ni Ebs(r,s) = -CF(s) §“§;— {altanelechose Z X Z _ D"15 S") n=1 i=2 k=2 N N N yu cose YA C059 . - S N n n e nk lsinh(__fl_§__l)]+ eztaneze chose( ‘ ) Z - 2 n=1 n=NS+l i=2 k=2 In11nkg'1.(s) yunkcoso2 . yA C0562 l Dn(s- 537 e s1nh(—-—§————~J]}. (4.44) Ns N Notice that ( 2 - X ) in the latter part of Equation (4.44) takes n=1 n=N +l care of symmetric and antisymmetric contributions in wire #2. It is easy to see that all the parameters in Equation (4.44) have been defined. 4.5 Impulse Response To determine the impulse response, let's specify E and i in terms of aspect-angel 4. It is easy to show from Figure 4.1 that cx = o 1 cy = s1n 4 1 (4.45) CZ = COS W 1 and 92 k = o x ky = cos W (4.46) k2 = - sin W. Substitute the above two set of equations into Equation (4.26), we get A -v[u .cos(q*a)-dsin W] gn1(s) = 2 tan(4*a)sinh [vcos(¢*a) 7g-Je "1 An -v[un1 cos(4~a)+d sin 4] i'2 tan(¢-a)sinh [ycos(¢-a) 7?1 e (4.47) Substitutions of (4.45) and (4.46) into (4.40) and (4.41) provide 61sin 61 = -E sin(¢+a) ézsin 62 = -i sin(¢-a) Cos 61 = -cos(¢*a) (4.48) Cos 62 = -cos(¢-a) Cos e = sin 4 . Equations (4.47) and (4.48) together with Equation (4.44) lead to M M g N -yRm , . n n I I g' (5) -Yu cos(¢+a) Ennis) = cF(s) e R ;{tan(q»+a)e1dsmq) _, z 31.125521“ n=1 i=2 k=2 n n N M M yA cos(¢*a) s N n n g'1(s) sinh( n 2 ) + tan(¢-a)e 1d SinCP( E- f ) Z £13112ksm) n= 1 n= —Ns+1 i= -2 k= 2 n n -yU cos(¢-a) yA cos(¢-a) e ”k sinh( n 2 )} N -YR* =’:1<'1=(s)e H'(S) (4°49) Roo 93 where K' = 2c, (4.50) and 1 1 21 1 21d sinw 1 g” g" IniInk 'l(”n1+”nk)°°s(1+“) H' s = tan 4*a e . e n=1 1&2 k=2 Dn1s'sn) vAncos(w+a) sinh [ 2 1 ( 1 ( )( 95 N Mn M niInk -v[un1cos(w+a)+unkcos(4-a) + tan ¢+a tan W—a - -1—(———- {e n=1 néNS+l i=2 k=211n S'Sn) -v[un1cos(¢-a)+unkcos(4+a)] yAnCOS(4*a) yancos(¢-a) + e }sinh[ 2 JsinhE 2 J N Mn Mn I I - (u +u )cos(¢-a) 2 —2yd sin? ni nk 1 ni nk + tan (@-a)e Z Z )1 m )6 n=1 i=2 k=2 n n vA cos(W-a) 51nh I: 2 J . (4.51) For experiments, as will be discussed in Chapter 6, 4 = O0 , 2 N Mn Mn I111Ink -y(un1+unk)c05a 2 yAncosl \‘P _‘__2 * !_ 2a1 200 1.0 30° L/a 2d/L (p: Impulse responses of an isolated wire, a wire over the ground plane and two parallel wires with an aspect angle of 30°. Figure 4.24. * 99 1 an isolated wire _, 3. E / :1 -” .—' .1 ‘_/1\/’,\ ‘ L g .. - - 51f “7’1 Z ;. 1(9' 12a .2 1 ‘ 3? ’1 L/a = 200 8 i- w = 60° .1 a wire over 1? 1 1 ground plane Ti 3 red 14— L—u.‘ a) 2‘ 1 t ‘ ' —-—T .2 . ‘2‘? 261 d *1; z. ‘\ " t 1; /9€K/9¢K?9§C/7¢/ :— '-. ' L/a = 200 r d/L = 0.5 3« W = 60° 1. ‘5‘ a two parallel @(//’{ 1 wires 8 f .11 :1 l relative ampl E%;;; O. r- \ \ l— DJ 11 ll -' N o 8 +1 ....a 61 f Ifi—u +-.. an? o- cf.- ~ 'vv“ lo. M. I-O 0-0 1.0 O O t/(L/c) Impulse responses of an isolated wire, a wire over the ground plane and two parallel wires with an aspects angle of 60°. T r L. ... 100 shown in Chapter 6. Figure 4.25 indicates the difference between "class- 1" and "class-2" responses for o = 00 and 4 = 30°, it is clear that they differ from eaCh other only in the early-time period. 4.6 Numerical Results for Incident-Waveform Synthesis and Target Discrinnnation. Incident waveforms required to excit monomode backscatters con- sisting of purely the first and the second natural modes of the skew- coupled wires target are synthesized according to the procedure de- scribed in Chapter 2. The finite duration of the incident waveform is chosen based on the experience with thin-cylinder targets L13], as one(normalized)period of the first natural mode; this choice leads to,e.g., Te = l/f1 =0T§§§§£E7I-=2.2502(L/c) for the coupled wire over the ground plane with a = 0°, d/L = 0.5, a/L = 0.005. The late-time response, upon which the synthesis procedure was based, occurs during = L cos W + 2d sin 9 c c = one-way transit time t 3 Te ‘1' 2Tt where Tt for the incident waveform to sweep acrossthe whole target in this particular case. Therefore, the late-time response begins at t = 2(cos 30° + 2x0.5x sin 30°) L/c +2.2502 L/c =4.9822 L/c for 4 = 30° and begins also at t = 2(cos 600 + 2x01551n 60°) L/c +2.2502 L/c = . _ o, = ____2!L____ 4.9822 L/c for W - 5° . 0" the °ther “a"d Te 0.9201nc/L = 2.1737(L/c) for case with a = 30°, d/L = 0.5, a/L = 0.005 and 1t = L-Egéflt- for W = 0°. Therefore, the late-time response begins at t = Te + 2Tt = (2.1737 + 2x cos 30°) %-= 3.9057 (%) for this particular case upon which we have performed one of the experiments. The incident signal required during 0 f t 5 22502 (%) to excite a pure first-mode [s1 = (-0.0734 + j0.8888)%?J response is indicated 101 q E '_ a .~‘.H— L 6‘ ‘~ 1 2a L/a = 200 =3- o = 30° N 9. {Class-l =2- JV 4" O 9 Class-2 7‘ '9 N- I 9‘ n- I 9‘ T 1 1 1 1 T 1 1 1 0.0 1.0 2.0 3.0 4.0 5.0 8.0 7.0 8.0 Figure 4.25; Impulse responses of an isolated wire with L/a = 200 and aspect-angle 30o computed by using ”class-1 and "class-2" coupling coefficients. 102 in Figure 4.26. The return signal for aspect-angle W = 30°, which is obtained by convolving the synthesized waveform with the impulse re- ponse in Figure 4.23, is shown in Figure 4.28 along with the return signal from a target consisting of a parallel wire over the ground plane with 10% shorter length. It is seen that before the late-time response begins, i.e., t 5 4.9822 L/c, the return signal exhibits an irregular waveform while for t 3 4.9822 L/c the return signal indeed demonstrates the monomode behavior. The return signal from the shorter target can not be identified as a single natural mode of this target and it can therefore be discriminated from the preselected "right" target. Figure 4.29 shows only the late-time response part for better discrimination. The required signal to excite the second-mode [ $2 = (-0.1691 + j l.9248)nc/L] backscattered field is shown in Figure 4.27. The return signals of aspect-angle W = 300 for the right target and the wrong target with 10% shorter length are indicated in Figure 4.30. It is noted that the higher order mode displays a better target-dis- crimination ability. Figures 4.31 - 4.32 demonstrate the return signals of aspect-angle W = 602,which is obtained by convolving required synthesized waveforms with the impulse response in Figure 4.24, for the right target and the wrong target with 20% longer length; Figure 4.31 is for the first-mode excitation while Figure 4.32 is for the second mode excitation. It is found that the target-discrimination ability is excellent in this case. For a skew-coupled wire over the ground plane with a = 30°, d/L = 0.5, a/L = 0.005, the incident signal during 0 5 t 5 2.1737 (L/c) to excite a pure first-mode [s1 = (-0.1156 + j 0.9201)nC/L] response is indicated in Figure 4.26 while that for the Relative Amplitude (3-9 I (1.7 (1.5 (1-3 0-1 103 ----- : Isolated wire ‘-€F-: a = 00 + : 01 = 300 '00? 0.!) Figure 4.26. l I ] 0.5 1.0 1. t/(L/C) Required waveforms for the incident radar signals to excite the first mode from the wire over the gr8und plane with a/L = 1/200, d/L = 0.5 and for a = 0 and 30°. The required waveform for the isolated wire is also shown for comparison. I 1 fl 5 2.0 2.5 3.0 Relative Amplitude oil 2-[] 1 ‘1-5 1. 0 [1.5 1 104 --.—— : Isolated wire _9_:01=00 +z o=30° -1.5 0-() Figure 4.27. 1 1 1 1 1 0.5 1.0 1.5 2.0 2.5 t(L/C) Required waveforms for the incident radar signals to excite the second mode from the wire over the ggound plane with a/L = 1/200, d/L = 0.5 and for a = 0 and 30°. The required waveform for the isolated wire is also shown for comparison. 3.() 10.0 8.0 Relative Amplitude 6.0 105 Wrong Target Right Target 0.0 Figure 4.28. m 1 1 1 1 1 1.0 2.0 3.0 4.0 5.0 6.0 710 t/(L/C) Return waveform from right target and target with 10% shorter length when the incident field is synthesized to excite the first mode of a paralled wire over the ground planeowith L/a = 200 and d/L = 0.5. The aspect- angle W = 30 . 106 _____ Wrong Target Right Target 5.0 1 Relative Amplitude ‘2000 1 1 1 r 4.5 5.0 5.5 8.0 6.5 7.0 7.5 8.0 8.5 t/(L/c) Figure 4.29. Late-time backscattered fields from right and wrong targets of the case shown in Figure 4.28. 107 e E1 D— N =2 '1 “61—11%: [‘3‘ ‘30 vi \ o ----— Wrong Target 2‘ ___. Right Target 0.) B 4-> 9.1 :1: ID D. E < 0) 0> D .5 5'1 (U B . c: 9 1.0-1 l ‘? 3.. l '9 EB- I 9‘ D ‘f 1 i f 1 1 F 1 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 Figure 4.30. t/(L/C) Late-time backscattered fields from right target and wrong target with 10% shorter length when the incident field is synthesized to excite the second mode of a paralled wire over the ground plane with L/a = 200, d/L = 0.5. The aspect-angle W = 30°. 0 (n 108 +. 1 E 9 ..- < N ‘9‘1n—. L_+1 =7——r ‘3 \60° d 1”“ W ----- Wrong Target 5 Right Target 5- / \ r‘ /\ I \ \ x l 1 .. I 1 \ / \ m \ 1 I \ :1 \ I «H \ 1 I \ A '1- ‘ I 1 r— , \ 1 I \ D. 1 1 , \ 1g \ ' ' 1 \ l ‘ 1 ' l‘ \ m 1 1 . .2 1 \ : I \ +9 l 1 \ 1 I 2 \ l \ I I / \ gs) 1 1 \ l \ D l 1 \ 1 I \ {lg-1 \ 1 ' I \ 1 l 1 1 1 I 1" I 1 1 \1 I \ I K / D. I l\/ 2.. \ I i 1 \ I =——> \/ 1 - , Late-t1me D I 22‘ l ‘? c: T ‘1 1 fi’ 1 1 1 1 1 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 t/(L/C) Figure 4.31. Return waveform from right target and target with 20% longer length for the first mode excitation of a parallel wire over the groung plane with L/a = 200, d/L = 0.5 and aspect-angle 60 . 109 +. E1 9 °l 12\~1¢-— L —>' * 7.600 E D \ 0" ”77, I I/I'r77 m ---- Wrong Target Right Target O cu «5- ‘U N 3 :2 23 E5 o < 0 Cd w v. > .5 (U '3 ‘1 ‘2- D I ‘? O—l T 9 \ I, O- \ ‘7' x,’ ‘9 O mud I 9 “7 9' 1 1 1 5‘0 6'5 7'0 715 8‘0 ' . 505 ' . . 4.0 4.5 5 ° t/(L/C) Figure 4.32. Late-time backscattered fields from right and wrong targets for the second mode excitation of the case shown in Figure 4.31. llO second-mode [$2 = (-0.l2l5 + j l.9400)wc/L] excitation is shown in Figure 4.27. The return signals for the right target and the wrong target (with l5% error in length) are shown in Figure 4.33 and Figure 4.34 for the first and the second modes, respectively. The above numerical results are based upon synthesis using l0 natural-mode basis functions. It is found that the required waveforms for the isolated wire, the wire over ground plane (therefore only anti- symmetric modes are excitable) for a = 00 and a = 300 are roughly the same as can be easily seen form Figures 4.26 and 4.27. From the experience with the isolated wire [13], due to the fact that the natural modes are nearly orthogonal, the natural-mode basis functions can well span the lO-dimensional space; different choices of basis functions like a-function basis and pulse-function basis lead to a unique required waveform. Therefore there is really no difference in using natural- mode basis or pulse-function basis. This is also true for the case in which the incident signal is symmetric with respect to two wires so that only the symmetric modes are excitable, since l0 symmetric modes are also nearly orthogonal and complete in lO-dimensional space. Figures 4.35 and 4.36 are some typical required waveforms for this case. For the general case in which both symmetric and antisymmetric modes are excitable, the matrix in (2.9) is somewhat ill-conditioned due to the fact that each symmetric natural frequency is quite close to its corresponding antisymmetric counterpart numerically. This leads to different synthe- sized waveforms for different basis functions. We will discuss more about the possibility of using different basis functions in Chapter 7. For the time being, only some results for waveform-synthesis using natural-mode basis set are shown in Figures 4.37 — 4.42. Relative Amplitude —?0.0 “1 E L L/\ k d """ Wrong Target ———-— Right Target Figure -0 1.0 2.0 T l I I I I 1 410 5.0 t/(L/c) 4.33 Return waveforms from right target and target with 15 % shorter length when the incident field is the synthesized waveform to excite the first mode of a wire over the grou- nd plane with L/a2200, d/L=O.Sand ds3dz The aspect-angle 9 'o.0 . H! E 112 L / o k d if _ ’ D- ...... Wrong Target 9 ""“ Right Target d. . 'r 1 1‘ ’\ T- I '1 ' 1 ' 1 i 1 ' 1 I i l l I ’ ‘ 1 ‘ i 3‘ ' i 1 ‘ ‘ 1 I ‘ I ‘ l I 1 l 1 \ 1 l ' 1 f o \ i ’ \ 1 I ' l l 1 \ -‘ I . I 9_ 1 ‘ 1 ‘ f ; \ 1 I \ I ’ o ‘ | ‘ ‘ l 1’ I \ \ / ' | I ‘ I l 7 \ I \ : ' ' 1 t ‘ I J 5- \I ‘ I ‘ : \l: l 1 l ‘ \ I 1 ’ 1 I \ l ‘ x ' p 1 ' 1 °.‘ ‘ l I ‘1’“ 1 1 ‘1 . ‘1 «2 d o— ' I Y ‘.3 1 1 1 1 1 .f T 1 0.0 1.0 2.0 3.0 4.0 5.0 '6.0 7.0 8.0 t/(L/c) Figure 4.34. Return waveforms from right target and target with 15 % shorter length when the incident field is the synthesized waveform to excite the second mode of a wire over the ground plane with L/a=200, d/L=0.5 anacxasdi The aspect- angle V =01 113 Relative Amplitude [1.0 -2.0 Dot) 1 (1.5 1 1 1 *1 l 1.0 1.5 2.0 2.5 3.0 t/(L/C) Required waveforms for the incident radar signals to excite the first modes from the two wires which are Symmetric with respect 30 the igcident signal, a/L = l/ZOO, d/L = 0.5 and for a = 0 and 30 . The required waveform for the isolated wire is also shown for comparison. Figure 4.35. Relative Amplitude 114 2-[) I --"-- Isolated wires “3 + (1:00 vi -1&- a = 300 1 1 1 1 1 1 0-0 0.5 1-0 1.5 2.0 2.5 3.0 t/(L/C) Figure 4.36. Required waveforms for the incident radar signals to excite the second modes from the two wires which are Symmetric with respect 80 the i8cident signal, a/L = l/200, d/L = 0.5 and for a = 0 and 30 . The required waveform for the isolated wire is also shoWn for comparison. -2-0 Relative Amplitude 20.0 ‘1000 ”2000 ”3000 40.0 115 D 51;- -----: Antisymmetric mode Symmetric mode (1.0 Figure 4.37. 1 1 1 1 0.5 1.0 1.5 2.0 216 —;10 t/(L/C) Required waveforms for the incident radar signals to excite the first modes from two parallel wires with a/L = l/ZOO and d/L = 0.5, when both Symmetric and Antisymmetric modes are excitable. Relative Amplitude 2-(1 l O-[l ‘2-0 -4-0 8.(l —4>L~ 116 30° '1’1’1’1’1’1’1’1111 -600 000 005 100 Figure 4.39. I 1 1 1 1 ._1 1‘6 200 206 t/(L/C) Required waveforms for the incident radar signals to excite the first Antisymmetric modes from two wires with a/L = 1/200, d/L = 0.5 and for a = 0° and 30°, when both Symmetric and Antisymmetric modes are excitable. (a) O Relative Amplitude 40-0 117 c3 c5.. ----- : Antisymmetric mode on Symmetric mode (J-O Figure 4.38. T I F I 1 0.5 1.0 1.6 2-0 2-5 t/(L/C) Required waveforms for the incident radar signals to excite the second modes from two parallel wires with a/L = l/200 and d/L = 0.5, when both Symmetric and Anti- symmetric modes are excitable. 3-(3 Relative Amplitude 25.0 l1 (1.0 ___-___; i ‘5-0 118 30 £2 E’. I 9 ID 1‘ 1 1 1 1 1 1 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Figure 4.40. t/(L/C) Required waveforms for the incident radar signals to excite the second Antisymmetric modes from two wiges with a/L = l/200, d/L = 0.5 and for a = 0 and 30 , when both Symmetric and Antisymmetric modes are excitable. Relative Amplitude 119 ‘? ed 0 --£>- a = 0 + 01 " 300 c? D— D ID 2" V\’¥’\’\-'\’\'9\’\’Vlw" -1000 -500 l -1500 20.0 r I 1 T l 1 0.0 0.5 1-0 1.5 2.0 2.5 3.0 t/(L/c) Figure 4.4l. Required waveform for the incident radar signals to excite the first symmetric modss from Swo wires with a/L = l/200, d/L = 0.5 and for a = 0 and 30 , when both Symmetric and Antisymmetric modes are excitable. Relative Amplitude 10-0 15.0 S-CJ (1.0 / 120 __9_ —fi&—- 30 Q 3 "M‘i'i'i'i'i'i'it‘ 3-() D 1.5 ‘2‘ 1 1 1 1 1 1 0.0 0.5 1.0 1.5 2.0 2.5 t/(L/C) Figure 4.42. Required waveforms for the incident radar signals to excite the second Symmetrig modes grom two wires with a/L = l/200, d/L = 0.5 and a = 0 and 30 , when both Symmetric and Anti- symmetric modes are excitable. CHAPTER 5 CROSSED WIRES In this chapter, a crossed-wire system is used as a crude model of an airplane to investigate the applicability of the waveform- synthesis method. The geometry of this problem is described in Section 5.l. The results of Section 2.3.2 are applied to this target and symmetry is considered to obtain both EFIE and Hallen-type IE's in Section 5.2. As it turns out, only one integral equation is needed for the antisymmetric modes while a set of two couped integral equations are required to solve the symmetric modes. Section 5.3 is devoted to the solutions of the induced currents to both symmetric- and antisymmetric- excitations: Section 5.3.1 concerns mainly the natural modes; Section 5.3.2 discusses the coupling coefficients which are used in Sections 5.3.3 and 5.3.4 to compute the induced currents for the antisymmetric- and the symmetric-excitations,respectively. Section 5.4 applies the field-current relation derived in Section 4.4 to compute the backscat- tered fields for two types of excitations. Some numerical results are shown in Section 5.5 to demonstrate the impulse responses for different polarizations of the incident waveform. The waveform-synthesis method is used in Section 5.6 to discriminate the right and wrong targets. 5.l Geometry of Problem A crude model of an airplane consisting of a "fuselage” of length L1 + L4, two "wings" each with length L2 oriented at an 121 122 angle a with respect to the fuselage is indicated in Figure 5.l. The junction of this crossed wires system is L.l from the "nose“ and L4 from the "tail". To simplify the problem, we assume that the fuse- lage and wings are constructed using thin wires with radii af and aw, respectively. The incident field is expressed as X” 1‘11») = z: F(t- 'F) (5.1) 0| where E, R and F are defined in the same way as in Section 4.l. and F(t) is an unknown waveform fucntion to be synthesized to excite a single-mode scattered field in the late-time period. There are two types of polarizations to be considered in the latter sections: (1) symmetric-mode excitation with 2 sin ¢ + x cos ¢ (5.2) mm H so that 12 = 13. (2) Antisymmetric-mode excitation with i=9 (5.3) so that 12 = -I3. and E = -2 cos ¢ + i sin o. (5.4) Notice that Rot = 0 for this plane-wave incident transient field. The tangential components of E1(F,t) on the wires, in their Laplace-transform, are A (uk.s) = (a-ak1f1s1e‘m'?) (5.5) +1 E tank Uk 6 [O’Lk]’ k = 19233; U4 6 [“L4,0]o 123 z + U1-UZ-L] u1=u4=uz_z 62 =-£ COS a + y sin 0 +12 - Il U3 = -2 cos a-y sin a x 0 @+ y Figure 5.l. A crude model of an airplane consisting of a system of crossed wires. X X + + ‘ <— 2 9 7 Z A 9’1” .y l’o CD"" A ’I E C 12.x, ; = 2 sin c + i cos 9, c = y Symmetric-mode excitation Antisymmetric-mode excitation Figure 5.2. The side-view of the airplane along with the incident field with two types of polarizations. 124 These electric fields excite transient induced currents on the wires, and induced currents, in turn, generate transient backscattered electric field. Our objective is to synthesize an aspect-independnent waveform F(t) for the monomode excitation. 5.2 Integral Equations To simplify this problem, we decompose the induced currents on the wings into symmetric and antisymmetric components; i.e., where IS is the symmetric-current component which is the same in both wings while Ia is the antisymmetric-current component which flows in opposite directions with equal amplitude in both wings. After this simplification, each mode needs only one integral equation to describe the wings instead of two coupled integral equations. However, the coupling between wings and fuselage still exists. The electric field integral equation can be obtained from Equation (2.l6) as g 1 L2 ( ) 32 2(‘ ”) e-YR” WW) .1 must—......) 1 £=1 0 R 1 aukauz k R 4an1 R auk - ”l = . _ . - - cos Etank(uk’s) --- for uk €[0,Lk], k l,2,3, U4 EE L4,0J (5.7) 125 . - e'YRk1(“k’L2) e'YRk2(”k’0) where wk£1uk) s I£(L£,s) 4“Rk2(uk’Lt) - 12(0+ ,s)e 4 RkQIuk’O) for 1 = 1,2,3 (5 8) e-YRk4(U k:0) + e'YRk4(uk"L4 (0, s)e - I [(- L ) ,s] 4 4nR Rk41uk10) 4 4 4angiuk,-L4)} for 2 = 4. . - _ _ + _ = Since I£(L£,s) — 0 for 2 f l,2,3 and I4[(-L4) ,s] - 0 for t 4 at wire ends, wk£(uk) can be rewritten as e-YRk£(uk,O) Wk£(uk) =‘I£(O+,S Se) 4flszruk,T for 9.. = 1,2,3 (5.9) ( - ) e’YRk4(uk:0) f = I O ,s or 2 = 4 4 4an4(Uk,0T If we define Iz(u,s) = I](u,s) for u e [0,L]] (5.l0) 14(u,s) for u E [-L 0] 4’ as shown in Figure 5.l, then the fuselage can be described by the following integral equation by setting k = z in Equation (5.7) and dropping the subscripts for uk and u' 2” L1 2 2 e.YRf L2 - I L I Z(u' ’S)[auau+ Y J 4an du' - J0 [12(U'.S) + 13(U'.S)J ' 4 -yR e'YRf(u 0) 2 Ylf a 2 - + [SUSGT'- Y COS a] §————-du' - 52(U15) + [12(0 ’5)'Iz (0 ’S)]% 3u4aniu, 0; 'Ya (U90) + I (o+ ) + I (0+ ) 41-9 f C 2 ’5 3 ’5 3 au 4‘R};TE‘07'(5.11) 126 1 /2 = | 2 2 where Rf [(u-u ) + af] (5.12) 1 RH = [u2 + u'2 + 2uu‘ cos a + a?]6 and Sz(u,s) = -eos Elan (u,s) = the forcing function on the fuselage. z (5.13) Because Rf(u,0) = (u2 + ai)% = R]f(u,0) from Equation (5.l2), the last two terms on the right hand side of Equation (5.ll) become ’YRf(u90) -—i e [1 (0+ 5) + 1 (0+ 5) + I (O+ s)-I (O- s)]= 0 due to EU 4anu,0 2 ’ 3 ’ z ’ z ’ ’ KCL' 1 (0‘ ) - 1 (0+ ) + 1 (0+ ) + 1 (0+ ) . z ,s - z ,s 2 ,s 3 ,s . ‘2 2 e'YRf By defining Kf(u[u ,s) = -[SE%UT-+ y J 4an = self kernel for fuselage 'YRif . _ a _ 2 e = _ . K]f(ulu ,s) - -[ y cos a] Egfilf_' a coupling kernel auau' for fuselage (5.l4) and using definitions in (5.6),the integral equation in (5.ll) can be rewritten as L l L I I I 2 I I I .— J-L4 IZ(U aS)Kf(U|U aS)dU + 2[0 IS(U ,S)K]f(u|u ,s)du — Sz(u,s) (5.15) u 6 [-L L 19 4]. Similarly, the integral equations associated with the wings can be obtained from Equation (5.7) as - R L 2 Y m L 2 l . a 2 e . 2 . a - 4 Y 4nR N 127 L2 2 e'YRZw + -1 I3(u ,s)[m+ 7 cos 2004142 du = 32(u,s) + film ,5)- 0 2w IZ(0 ,S)J -YR]w(U,0) ‘YRW(U,0) ‘YR2w(u90) 3—9 +1 (0“,s)—"3—e +1 (0" s) 19 (5.16) au 4"le(u’0) 2 au 4an(u,O) 3 ’ au 4nR2w(u,O) and L1 32 e‘YRiw L2 32 - I L Iz(u ’S)[auau' y cos “J4nR1 du - J 12(u ’S)[auau' + - 4 w 0 2 -YR2w e n y cos 2a] 4 R du " 2w L2 32 2 e.YRW + - - [0 13(u ,s)[W+ y l 4an du = 53(u,s) + [Iz(0 ,S)-Iz(0 ,S)J - - ,0 - j— e YR]W(U,0) + I (0+ )3— e YR2w(U )+ I (0+ )1 e YRw(u,O) (5 17) au 4nR (u,0) 2 ’5 au 4nR (u,0) 3 ’5 au 4nR in,0) ' lw 2w w _ . 2 2 g where Rw - [(U-U ) + aw] 1 2 .2 . 2 P le = [u + u +2uu cos a + aw]2 ) (5.l8) _ 2 .2 . 2 % R2w - [u + u - 2uu cos 2a + aw] .1, . _ _ ”i and 52(u,s) - cos Etan2(u’s) . (5.l9) 53(U,S) = -805 E1 tan3(u,s) - = = 2 The fact that Rw(u,0) - R]w(u,0) R2w(u,0) [u L + a2]2 and KCL w lead to the vanishing of the right hand side of Equations (5.16) and (5.17) except 52(u,s) and S3(u,s). 128 ‘YR If we define K (ulu' s) = -F—§E——-+ 2] §———!-= self kernel for wing \ w ’ “auau' Y 4an 32 2 e-YR1W K]w(u|u ,s) = -[53337-- y cos a] 4flle = a-coupling kernel for wing, . 32 2 e-YRZW - L K2w(u|u ,s) — ‘LSUSUT + y cos 2a] ZE§E;—-- 2a-coupling kernel for wing , 5.2 ( 0) J Equations (5.16) and (5.17) can be rewritten as L1 L2 J L Iz(u ,s)K]w(u u ,s)du + [0 12(u ,s)Kw(u|u ,s)du + - 4 L2 + [0 I3(u ,s)K2w(u|u ,s)du = 52(u,s), (5.21) and L1 L2 I L Iz(u ,s)K]w(u|u ,s)du + [0 12(u ,s)K2w(u|u ,s)du - 4 L2 + J0 I3(u ,s)Kw(u|u ,s)du = S3(u,s). (5.22) Both Equations (5.2l) and (5.22) have the domain u €[0,L2], therefore they can be added and subtracted with each other as [by using the definitions in Equation (5.6)], L L I 1 I (u',s)K1 (ulu',s)du' + I 2 I (u',s)K (u|u',s)du' = S (u s) u€[0,L _ w 0 s s J z s ’ 2 L4 for the symmetric modes, (5.23) and L2 I Ia(u',s)Ka(u|u',s)du' = Sa(u,s) u e[0,L2] for the antisymmetric modes. 0 (5.24) 129 where Ks(u|u',s) = Kw(u|u',s) + K2w(u|u',s) } (5.25) Ka(u|u',s) = Kw(u|u',s) - K2w(u[u',s) , and Ss(u,s) = %[Sz(u,s) + S3(u,s)]'} (5.26) Sa(u,5) = %[82(u,5) - 53(U.S)J It is important to notice that the symmetric modes can be solved by using the coupled integral Equations (5.15) and (5.23) while the antisymmetric modes can be solved by using only a decoupled integral Equation (5.24). This is because whenever we are dealing with the anitsymmetric modes. due to the cancellations from two wings, the forcing function on the fuselage, Sz(u,0), is zero and Iz(u,0) is thus vanishing. In other words, if Iz # O for the antisymmetric modes, then the coupling between wings and fuselage will result in non-zero symmetric current which should be zero under the requirement for the existence of only the antisymmetric modes. Therefore, antisymmetric modes can be solved much more easily. Equations (5.15), (5.23) and (5.24) are the EFIE's to be used for com- puting the coupling coefficients. To compute the natural modes, Hallen-type integral equations will be used due to the reasons discussed in Chapter 2. They are ob- tained from Equation (2.35) as L L I I l u 2 ' ' ' ' J-L Iz(u ,s)th(u|u ,s)du + (0 [12(u ,s) +I3(u ’S)]Kh]f(ulu ,s)du 4 U = C1 cosh vu + C2] sinh vu + %-J Sz(g,s)sinh y(u-g)dg (5.27) 0 130 for fuselage1 -L4 3 u 5 L], and L L [-1 I (u',s)Kh]w(u|u',S)dU' + L02 I2(U'.S)Khw(U|U',S)du' + 2 L4 L2 [0 I3(u ,s)Kh2w(ulu ,s)du . 1 u . = C12 cosh yu + C22 51nh YU + 7 [0 $2(£,S) 51nh Y (U-€)d§ 9 (5-28) L L l . , . 2 . ' ' L~L 12(u ’S)Kh1w(ulu ’S)d” + LO 12(U ,S)Kh2w(u|u ,s)du + 4 {L2 I l d I J0 I3(u ,s)Khw(u|u ,s) u u = (:13 cosh yu + (:23 sinh w + H0 S3(g,s)sinh y (Ll-mg (5.29) for wings 0 5 u 5 L2, -YRf where th(u|u',s) =—%;§——-= Hallen-type self kernel for fuselage, 1 f 'YRw Khw(u|u',s) = :"R = Hallen-type self kernel for wing, w 'YRf u cos a - J g]f(g,u',s) cosh y (u-g)dg 0 Hallen-type a-coupling kernel for fuselage, U COS a ' J 9]w(€gu',5) COSh Y (U-€)d§ 0 = Hallen-type a-coupling kernel for wing, (5.30) Due to continuity of vector potential in the z direction across this junction [20], C1] = C14 5 C1 and IE's for wires #1 and #4 can therefore be combined as a single integral equation. 1 131 -YR2W u u - e : Kh2w(u|u ,s) - 4nR2 cos 2a - J g2w(g,u ,s) cosh Y (u-€)d€ w 0 = Hallen-type Za-coupling kernel for wing. 'YR 1f . 2 d e u' Sln a and g (u,U'.S) =[ ( 1—— 1f dR1f 4nR1f R1f “YR 1w . 2 d e u' Sln a 9 (u,U',S) =[ ( )1 ——-———— (5.31) lw dR1w 4nR1w le ’YR2 2 d e w u‘ sin 2a 9 (U.U'.S) =[ ( ]—-————-— 2w dsz 4nR2w sz Continuity of scalar potential across the juction [20] leads to C k = 1,2,3,4. 2k 5 C2’ Addition and subtraction of Equation (5.28) and (5.29) thus provide L L 1 I . ' 2 ' I . ['4— IZ(U 95)Kh]w(u|u ,S)du + [0 IS(U ,S)KhS(U|u ,S)dU 4 U = c cosh yu + c2 sinh yu + l( s (5,5)5inh Y(U-E)d€ (5.32) S Y 0 S for the symmetric modes, u e [0,L2]; and L u 2 I - I I _ l - I I (u ,s)Kha(u,u ,s)du - Ca cosh yu + Y (0 Sa(€,s)51nh y(u-g)dg 0 a (5.33) for the antisymmetric modes, u €[0,L2] C + C _ 12 13 where Cs - ————§———— ( 5.34) C = C12 ‘ c13 a 2 132 and Khs(u|u',s) Khw(UIU',S) + Kh2w(”l”"s)) ) (5.35) Kha(u|u ,s) = Khw(u|u ,s) - Kh2w(ulu ,5). Using the definition of symmetric-mode current, Equation (5.27) is re- written as L L 1 I . I 2 ' ' I — L-L Iz(u ,s)th(ulu ,s)du + 2 L0 Is(u ,S)Kh]f(u|u ,s)du _ c1 cosh YU 4 U - l_ (5.36) + C2 51nh vu + Y [0 SZ(€,S)Sinh Y(u-€)d€ for u GL-L4,L]], Equations (5.32), (5.33) and (5.36) are used to search for the natural modes by setting all forcing functions zero. 5.3 Induced Currents 5.3.1 Natural Modes Natural modes are those solutions which exist when all the forcing functions are zero. We apply moment method to this problem by the partitioning as shown in Figure 5.3. Note that only one wing is needed due to symmetry. Figure 5.3. Partitioning of the crossed wires for moment method, only one wing is used due to symmetry. 133 For the antisymmetric modes only one wing is considered and only (5.33) is used. By applying moment method, Equation (5.33) becomes Aa(s)Ia = O (5.37) where (I. 1 I2 Ia = . (5.38) 1N2 C a C6 is included as unknown while I is zero at wire end and thus N2+1 dropped. For the symmetric modes, both (5.32) and (5.36) are sloved. This is a system of coupled integral equations, moment method can still be used to solve the problem, however, much more effort should be taken with extreme care. The matrix form of Equations (5.32) and (5.36) is A (S)I = O (5.39) where 134 c1 1 I2 I Wire #4 INp+1 1 Np+2 Wire #1 NZ (5.40) 2 NZ+2 Wire #2 INT+1 NT+3 NZ = Np+N], NT = Np+N +N 1 2‘ I1 = IZ(-L4) = O, INZ+1 = 12(L1) = O and INT+2 = 12(L2) = 0 are replaced by the unknown constants; .. , _ + o ) while INT+3 - 12(0 ). The last row of As(s) is to apply the boundary condition INp+l = 12( by using KCL at u = 0: I I z Np+1 = NT+3 + 2 INZ+2 Since - + + + + + 12(0 ) = 12(0 ) + 12(0 ) + 13(0 ) = 12(0 ) + 212(0 ) for symmetric modes. The boundary condition for the antisymmetric modes is automatically satisfied because 12(0') = 12(o*) + 12(o+) + 13(o+) = 12(o+) for antisymmetric modes and we don't have to apply KCL explicitly in Aa(s). 135 The natural frequencies are those roots of detIAS(s)1 = 0 and detEAa(s)] = 0 which yield the nontrivial solutions for Equations (5.37) and (5.39). The roots are computed as in Chapter 4 by Newton's method. We first search for the roots of antisymmetric modes, because it is almost the same as those of the skew-coupled wires except that d = 0 and 12(u = 0) f 0. For the testing purpose, 2%. = 0.01, a = 900 with N2 = 5n for the nth mode are used in the root-searching sub- routine. As expected, the roots found are exactly the same as those of an isolated wire with %-= 0.005 and number of partitions lOn for the nth mode except that only the first, third, fifth,---modes are found since these modes have the current distributions which are antisymmetric modes by our definition. We therefore use the roots just found as initial guesses and N2 = 5: for the nth mode to search for the roots of the special case with [§-= 0.01, a = 45°, ti- 1 [Z-= 0.6, aw = af. If we define L = L4 + L1 = 2L2 then the roots are: = 0.8 M II 1 (-.0606 + j.9743) {2 (.D II (-.2051 + j2.9720)“—LC- (I! ll - TIC 3 (-.2974 + 34.9039) -E- (I) ll . 17C 4 (-.3202 + 36.8690) 1:- (I) II (-.3719 + j8.8708) % We next search for the roots of symmetric mode. To compare with roots found in the existing literature L34] using EFIE and piecewise sinusoidal expansion, weczomputed the roots for cases with L4 + L = L-2L a =a =a E“-=005 =9o° and E-‘-=05 063101 - 29w f 9L 09a 9 4 090,39. 136 with N2 = 8 (in [34], N2 = 9) and found that the comparison is very good. We then use the results of a = 900 as initial guesses to search L _ _ l _ = = 1 + L4 — L - 2L2, ——-- 0.8, aw af 8, for the roots of a = 45°, L L 4 %—-= 0.01. Here are the roots: 2 S1 = (-.0469 + j 0.9315) nc/L $2 = (-.0769 + j 1.0418) nc/L S = (-.1358 + j 2.6620) nc/L 3 S4 = (-.1444 + j 3.3028) nc/L $5 = (-.1738 + j 3.9582) nc/L S6 = (-.2039 + j 4.7351) nC/L S7 = (-.2315 + j 5.3609) nc/L S8 = (-.2503 + j 6.8961) nC/L $9 = (-.2253 + j 7.1328) nc/L $10 = (-.2915 + j 8.0685) nc/L . Due to the tremendous computing cost, we use N2 8 for the first five roots and N2 = 12 for the rest (in [34], N 9 for all the 2 roots). We tested with N = 20 for S and found that only 1.35% 2 6 difference exists between N2 = 12 and N2 = 20. Therefore, the convergence of Hallen-type IE using pulse expansion is reasonably good and comparable to EFIE using piecewise sinusoidal expansion. Natural mode currents are found by solving the homogeneous equation by the procedure as described in Section 4.3.1. Natural-mode current distributions for the first three antisymmetric modes are shown in Figures 5.4 and 5.5 for the real and imaginary parts respectively while Figures 5.6 and 5.7 show the symmetric modes. It should be noted 137 ‘9 «I 9.. N lst mode —-~--- 2nd mode u3_ *— -- 3rd mode (I) .U 3 :2 Ti E <: a) .> .5 It! '8 a: ‘9 T I I I I I I I I 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 u/L Figure 5.4. Real parts of the first three antisymmetric model currents on the wings. 138 6.1-) 2.0 l ————— lst mode 1.5 ------- 2nd mode ---— 3rd mode 1-0 Relative Amplitude 0.5 l 1.5 ”T I I I r I I 0.0 012 0.4 0.8 0.8 1.0 1.2 1.4 1.6 u/L Figure 5.5. Imaginary parts of the first three antisymmetric modal currents on the wings. 139 Fuselage eezeepes< e>eee_em lst mode u/L 2nd mode —-—-—- -- '3rd mode (a n5 nu .1I . w 8111 . oeuo d 0.0 n.00 ee=e_eee< e>eeeeem u/L Real parts of the first three symmetric modal currents. Figure 5.6. 3“ 140 Q) U 3 4..) ?i E Fuselage < OJ > .5 f!) G) o: 5’... -5. -2... -5. -5... .. o'. 3!. u/L lst mode - ------ 2nd mode 3- —-——-3rd mode 3. (D 'U 3 4.: '8 < Wings 0) > .5 It! '8 D: a 'o.0 at: 0T: at: of. 015 of: ‘ 011 01.0 u/L Figure 5.7. Imaginary parts of the first three symmetric modal currents. ,4 3~ 140 Q) '0 3 4..) g— Fuselage < (D > .5 (U m a: ”M i... -3... -a.. -a.= .10 a. at. a. u/L lst mode - ------ 2nd mode 3- —-——-3rd mode 0) 'U 3 4.: E <: Wings CD > .5 '5 OJ :3: 7a.» at: 0T: 0:: 0'4 01.6 010 ' at: 01.0 u/L Figure 5.7. Imaginary parts of the first three symmetric modal currents. .1 141 that junction condition on KCL is satisfied. 5.3.2 Coupling Coefficients The induced current as expressed in terms of SEM is )'1 (5.41) N I(u,s) = Z] an(s)vn(u)(s-sn n: where vn(u) is the distribution of the nth natural-mode current, and an(s) is the coupling coefficient associated with the nth mode. The coupling coefficient can be computed by using (2.37) with the integration performed over all the wires involved. Using the short hand notation introduced by Baum [9 ], Equation (2.37) can be rewritten as a (s) = (5.42) n Where < , > is the inner product of functions seperated by the comma with integration over the common spatial coordinates; a symbol (dot product in (5.42))above this comma indicates the type of multiplication used and the integration limit is over the object of interest. With this notation, we can always use Equation (5.42) to compute the coupling coefficients for both symmetric and antisymmetric modes without any confusion by knowing that for antisymmetric modes the domain is u e [0,L2] while for symmetric modes the domain is over u E [0,L2] for the wing and u 6 E-L4,L]J for the fuselage. In the following sections, we will consider symmetric and antisymmetric modes seperately. The kernel function in Equation (5.42) is the kernel function for the electric field integral equation. 142 5.3.3 Computation of Induced Currents for Antisymmetric-mode Excitation For the antisymmetric-mode excitation as shown in Figure 5.2, the forcing functions are computed from Equations (5.5), (5.l3) and (5.l9) as Sa(u,s) = %-[52(u,s) - S3(u,s)] = - Eos sin a e Y2 COS w F(s) = - Eos sin a e'W COS a C05 w F(s) for 0 g u 5 L2, Ss(u,s) = 0 for 0 g u 5 L2, (5.43) and Sz(u,s) = 0 for -L4 5 u 5 L1 . The kernel function for Equation (5.42) is 2 'YR 2 'Ysz w 3 2 e + Y2 cos 2o] ___e ,(5.44) I = _ ______ ___... ._§___ Ka(ulu ’S) [auau' + Y J 4an + [auau' 4nR2w Since (5.24) is the only equation we need to use [(5.l5),(5.23) give us I2 = IS = 0], the integration limits in (5.42) are therefore 0 and L2. The detailed computation of induced currents for this particular case is very similar to that for the coupled wires as shown in Chapter 4 except that the current at u = 0 is not necessarily zero any more. If we adopt the same notations and defin$§ions used in Sfifition 4.3.3, the only difference is that instead of g ( ), we have ; ( ) + (half- segment contribution from Inl at u2 = 0) now. The coupling coefficient can therefore expressed as Nn(S) an(s) = D (5.45) n where M . n (1-%)A . N Nn(s) = X n' [ 3 n [-€os sin a e'W COS a C05 q F(s)]du i=2 ‘ (f7)a n A + Inl [in [-&05 sin a e'W COS a COS T F(s)]du _ 0 Mn -yu . cos 0 cos 4 yA = - gsggéiég—9-~(s) .2] Ini e n1 sinhE—Efl-c05Z + <12,K.|>2 + <13,K.|>2 ’ SZ(U,S) lJEE-L],L4] + + z 2, 2 3,K2>2 S2(u,s) l1€[0,L2] + <1 2 2 K>2 = S3(u,s) u€[0,L where < >z stands for the integration with respect to u' form -L4 to L1 while < >2 is the integration form 0 to L2. The reason for showing all three EFIE's without considering the symmetry at this point is that Equations (5.50) can be considered as a single EFIE which is symmetric: <1, 13> = § (5.51) - F I I ‘ where l_- -151 T 521 -E-J-_El-j---51. = 1---( = : I 121 § 1 52‘ 3"“ 5 -51.:--'f--.§---'32__ “‘ -’"'3 a u 13) {53" \K1: K2: K4 Because 5. is,as a whole, a symmetric kernel, un = Vn [definition of u is illustrated in the footnote of Equation (2.37)], and (5.42) n can therefore be used without changing the denominator to (pn(U); d I . I a§{K(01U ,S)], Vn(u )>. The computation of Equation (5.42) is now conceptually easy. It can be reduced to the following form: (note that I2 = I3 and S = S 2 for symmetric modes and KS = K + K2) 3 (Sz;vzn> + 2 (5 52) (vzn’ ds K’ vzn> + 4 + 2 an(S) = 146 where v = --- and v2“ = v3" for symmetric modes. It should be noted that d L1 2 d (Vzn; ag—K]; V2n> =[_L [0 vzn(u)v2n(u_)a§-K](ulu ,s)du du. 4 Equation (5.52) can be rewritten as (refer to Figure 5.3 with NP = M4”, N = M and N = M for the nth mode) 1 1n 2 2n Nn(S) an(s) - D (5.53) n where Nn(s) = Nzn(s) + N2n(s) = N4n(s) + N]n(s) + N2n(s) (5.54) N4n(s) E <52; Vzn> for u E [-L4,0] E <52; v4n> 1 ~ M4n (1'M4n'20An = -€os sin m F(s){ X I4ni I eYu COS W du i: (i-M - §)A 4n 2 n 0 yu cos o +14n for u E [0,L11 E <32; vln> M (i- —)A ~ ln 2 n = -605 sin m F(s) { _X Ilni f eYU cos w du 1=2 (i- %)A n A n + I1 [Zi'evu COS W du} (5.56) nl O and N2n(s) E 2 1 M2n (1' 29An = 2 605 sin w cos a F(s){ X IZni I e-W COS a C05 m du 1:2 (1‘ %)A n A n + I 2 e'Yu COS a COS m du} (5-57); 2nl 0 Dn is defined similar to Equation (5.47) with three terms involved. Induced currents can now be expressed as N Nn(s) _] Iz(u,s) = nZl Dn vzn(u) (s-sn) N Nn(s) _] 12(u,s) = 13(u,s) - n21 Dn v2n(u) (s-sn) . (5.58) 5.4 Backscattered Field The scattered field in the radiation-zone can be determined from the method discussed in Section 4.4 by linear superposition. It is therefore obvious that 148 -> _ A “S . A ~5 - A - (r,s) - e 5 AZ Sln e + 92 5 A2 Sln 92 + 63 s 43 Sln 63 (5.59) where (51,62,63) and (91,92,93) are defined in the same way as indicated in Figure 4.l9. In the following two sections we will discuss the backscattered fields of symmetric.enuiantisymmetric-mode excitations separately. 5.4.l Backscattered Field from the Antisymmetric-mode Excitation This case is very similar to that of the skew-coupled wires in Chapter 4 except d = 0 and I2 = -13 f O at u = 0. The vector polentials are expressed as [refer to (4.38) and (4.39)] "YR“ L ' A = __. I (u ,s)e du' 2 4n Rm 0 2 (5.60) ‘YRm L ' ~ p yU COS 6 A: = - 1%. e Rm [ 2 12(u',s)e 3 du' 0 ii: = 0 , where cos 62 = [62.(-£)3 = - cos o cos a } (5.61) cos 93 = [03.(-k)] = - cos 5 cos Q And from Equation (4.41), ézsinez [02x(-k)]x(-k) xsin o cos o cos a -ysina + 2 sinzo COSa (5.62) A A {u3x(-k)]x(-k) xsin m cos p cos a + ysina + z sinzoCOSa 63S‘ln93 149 Substitutions of Equations (5.60)-(5.62) into Equation (5.59) yield Eb5(?,s) = -25 K§(?,s) sin a 9 (5.63) -2 sin a S K3(?,S)g. 5.4.2 Backscattered Field from the Symmetric-mode Excitation All three wires contribute to the scattered field, this is a much more complicated case although conceptually easy. The vector potentials are expressed as [refer to Equations (4.38) and (4.39)] — Rm L ‘ ~s _ E9_e Y l . YU' cose . AZ - 4n Rm J-L Iz(u ,s)e du 4 ~ u -YRm L Yu' cos 6 ) A: = fie Rm [02 12(u',s)e 2 du' (5.64) ~ u -YRm L YU' cos 6 A5 = Egg—RT [()212(“"~°’)e 3 d“' J where Cos e = [G -(-k)l = cos 6 z (5.65) Cos 62 = cos 63 = - cos n cos a Equation (4.4l) yields 6 sin e = [fizx(-k)lx(-k) = -2 cos n sin 6 - 2 sin2 6 62 sin 92 = 2 sin o cos n cos a -9 sin a + 2 sin2 @ cos a (5.66) 63 sin 63 = x sin n cos o cos a + 9 sin a + 2 sin2 6 cos a 150 Substitute Equations (5.64)-(5.66) into Equation (5.59), we can obtain the following result: Ebs( + A . A .2 NS A . r,s) [-x cos m Sln o -z Sln ojs AZ(F,s) + 2[x Sln o cos 6 cos a N + 2 sin2 5 cos a]S A;(?,s) s sin o [x cos m + 2 sin ¢][2 cos a K;(?,s) - K:(?,s)] N s sin o [2 A§(?,s)cos a - ~ A§(F.s)na. (5.67) It is obvious that when a equals 90°, the only contribution comes from fuselage, and zero-angle incidence results in zero response for the symmetric-mode exictation. 5.5 Impulse Responses By setting F(s) = l in Equations (5.63) and (5.67), we can compute the impulse responses of the backscattered fields for both symmetric- and antisymmetric-mode excitations. In the following sections, L "' = 0 = = = = = i: a - m 45 , aw af a, L1 + L4 L 2L2, a/L2 0.0l, L4 0.6 and ._g = 0.8 are assumed. 4 r 5.5.l Impulse Response to the Antisymmetric-mode Excitation By working out the detail of Section 5.4.l, the backscattered field due to the antisymmetric-mode excitation can be expressed as 1: _, - 'YRw flfing=eAeRm Hu*mu 65m tanza where A = 4C *7— : 8C , (5.69) cos 6 151 and h(t) = Impulse response = L-]{R6[HC(S)]} ; Mn M n 'Y(U .+U )/2 yA - _ n1 nk . 2__Jl Wlth HC(S) - "Z ___—(S—S—YE 41.2: [(22 IniInk e 51nh ( 4 ) M . yAn -yAn/4 n 'Yuni/Z 2 -yAn/4 2 + 4In1s1nh(—z—)(l-e )(122 Ini e ) + In1(l-e ) ] (5.70) It is easy to see from the first term in the bracket of (5.70) that the early-time period of h(t) is t 5 23- which is the two-way transit time. From now on we would like to concentrate on the late-time response upon which the waveform-synthesis method is based, the "Class-l" coupling coefficients are therefore applied with all y'S replaced by yn for the nth mode. Physically, this means that all sources corresponding to different segments are "turned on" and the bracket quantity becomes a constant coefficient. The impulse response is thus expressed as N s t L2 h(t) ={Re n2] Cne " }---for t 3 17- (5.71) where Cn = fi5—[4 jg: kg: 1m.Inke-anniwnkvzsinh2 ( YnAn) + 4In15inh(Y":")(l-e-YnAnl4)(1E: Inie-Ynuni/Z) + I§1(l-e-Y"A”/4)21M = ft-[Z sinh (Lin) 1or):1m.e-Y”l"”'/2 + In](l-e-Y"An/4)]2. (5.72) With the natural modes computed in Section 5.3.l, the impulse response 152 of Equation 5.7l is shown in Figure 5.8. 5.5.2 Impulse Response to the Symmetric-mode Excitation The backscattered field due to the symmetric-mode excitation in Section 5.4.2 can be expressed as -YR gbS + _ A e m * E (r,s) - g B Rm F(t) h(t) (5.73) where B = 8 c tan2 4 = 8c (5.74) and h(t) = Impulse response = L']{Re[Hc(s)]} M with H (s) = g 1 [2 22” I e’YUZni/2 YAn c n=l Dfi1s-sn) i=2 2ni sinh(—Z-) -YAn M1n yA +12n](l-e T) - Z Ilni ew1nl/‘Q—sinh(—E) i=2 2/2 YAn M I -—-— 4n . yA - —l%l'(92/§ -1) - Z I4ni eYu4nl//z sinh (-—2 ) i=2 2/2 I4n(qu+1) -—-r’1 2 N ems) -—-—————————(l-e 22)] 5 ) ‘T—D ) (5.75) 2 n=1 n S'Sn where u2ni = (1-114n ulni = (i-l)An ) (5.76) The fact that C5(s) in (5.75) is square of the bracket quantity is not a coincidence, as a matter of fact, C6(s) a N§(s): this is true for all wire targets discussed in Chapter 4 and this chapter. This is Relative Amplitude o 153 9 °.. ‘9 a... I 5.. I ? T I l l l 1 T 1 0.6 1.5 2.8 3.5 4.5 6.5 8.5 7.5 8.. t/(L/c) Figure 5.8. Backscattered-field impulse resgonse of a cross-wire target with a/L2 = 0.01, a = 45 L = 2L2 and El'= 0.6 due to the antisymmetric-mode excitation with aspect-aflgle m = 45°. , aw = af = a, L1 + L4 = 154 because Nn(s) = and Eb5(?,s) 5'4(?,s) a [ I(u',s)eYU COS edu' Notice that S(u',s) a Elan(u',s) a e'Y(?'°k) the above interesting relation and this is ture only when we consider the backscattered field for which cos 9 = [u-(-k)] in the vector potential integration (for any direction scattered field -k should be replaced by the appropriate unit vector). From Equation (5.75) it is easy to see that the early-time L L +L period 1'5 t 5 2 Tt’ Tt = two-way transit time = max{7§, /2 1c 4}= L +L /2 1c 4 = J2'%- this is twice the time for the incident waveform to sweep across the fuselage: during this period it also sweeps across L the wing because it only takes %-7§- to pass the wings (this is the one-way transit time for the antisymmetric-excitation in which only the wings are involved). The late-time impulse response can therefore be expressed as N s t is used to show L h(t) = Re{ g C" e " }_-_ for t 3 7‘23 (5.77) n=l Where C = (5.73) The numerical result as computed by Equation (5.77) and the natural modes in Section 5.3.2 is plotted in Figure 5.9. 5.6 Incident-Waveform Synthesis for Single-Mode Excitation and its Application to Target Discrimination Incident waveform required to excite single-mode backscatter consisting of purely the first or the second natural modes of the crossed wires target are synthesized according to the procedure described in Chapter 2. The finite duration is chosen as one period of the first 2" = 2.0528%. The required antisymmetric mode, i.e., Te = l/f1 = 0.97435c/L waveform to excite the first antisymmetric mode is shown in Figure 5.l0. The return backscattered waveform can be computed as the convolution of waveforms in Figures 5.8 and 5.l0. Figure 5.ll is the result of this convolution along with the return waveform from the wrong target with l0% shorter wings. It is easy to see that the return from the right target displays single-mode response after the late-time begins at t = 2Tt + Te = (0.5 + 2.0528)%u However, the return from the wrong target can not be identified as the single-mode. For the symmetric-mode excitation,the required waveform to excite the first mode is shown in Figure 5.12, and Figure 5.l3 is the returns from right and wrong targets. This time the late-time begins 211 L . . . _ at t = 2Tt + Te = (/§'+Tf§§T3fl)En Sim1lar results of required wave form for the second symmetric mode and its radar returns from right and wrong targets are shown in Figures 5.l4 and 5.l5. Relative Amplitude 156 Figure 5.9. I T I 1 T T 1 2-4 3-4 4-4 5.4 6-4 7.4 8.4 t/(L/C) Backscattered-field impulse response of a cross-wire . = = 0 = = : target w1th E/L2 0.0l, a 45 , aw af a, L1 + L4 L = 2L2 and Ll'= 0.6 due to the symmetric-mode excitation 4 with aspect-angle m = 45°. 157 ‘ 3.0 Relative Amplitude w FL 0‘ O ID a O , 1 I 1 l l 1 1 0.0 0.5 1-0 1.5 2.0 2.5 3.0 3.5 t/(L/c) Figure 5.l0. The required incident waveform to excite the first anti- symmetric mode of the target described in Section 5.5. 20.0 J 5-0 1 Relative Amplitude -?.0 000 ‘1000 -1600 20.0 .— ‘ —v ‘..“ “ d ’ —-——————-.—._‘--— ~ ‘— {Late-time |-—+— I 158 Right Target Wrong Target 0-0 Figure 5.ll. 1 2.0 I 4.0 1 1 1’ r 6-0 8.0 10.0 12.0 t/(L/C) 14-0 Return waveforms from right target and target with 10% shorter length when these targets are illuminated by the synthesized waveform of Figure 5.l0 with antisymmetric excitation. 20.0 4‘4 15-0 1 0.0 Relative Amplitude ’10-0 ‘1600 20-0 159 0-0 Figure 5.12. l I I 1 1 015 110 1.6 2.0 2.5 3.0 3.5 t/(L/C) The required incident waveform to excite the first symmetric mode of target described in Section 5.5. 4 .E 160 ‘9 :31 ‘3 531 Right Target 2.. Wrong Target ----- m l . I . cu I l * 1 +4 é-J I ‘ I I ' I N I I I I : g I ‘ I I l <: 1 , 1 1 . 1" "' . o- l I It; "' I ‘ I I: '1 ‘ I\ '3" I I I II I ‘ I I a: I I l I: I 1 I I ‘ I I, ' \‘ I} O l 1 I o l I I I I I I I I '5 I I I I I \J I I O. I 1 I II I I I I 2—1 I l I '1 ‘ \ I | I I ' I 'I I \v ' I I I .1 1 I 3'- ’ ' i ‘ 1' ‘. :Late-time I \I l——F ' O, I o T 1 1 1 1 1 1 1 0.0 2-0 4-0 8-0 8.0 10-0 12.0 14.0 t/(L/C) Figure 5.13. Return waveforms from right target and target with l0% shorter length when these targets are illuminated by the synthesized waveform of Figure 5.l2 with symmetric excitation. Relative Amplitude 15.0 161 0.0 Figure 5.l4. 1 1 1 1 1 1 0.5 1.0 1.5 2.0 2.5 3.0 t/(L/c) The required incident waveform to excite the second symmetric mode of the target described in Section 5.5. 3.6 162 Right Target Wrong Target 0 O o _ A Dead 0.0 duapw_ee< d>_pe_em _ o.o~i 1 o.cNI _ o.amn o.O¢ t/(L/c) Return waveforms from right target and target with l0% shorter length when these targets are illuminated by the synthesized waveform of Figure 5.l4 with symmetric excitation. Figure 4.15. CHAPTER 6 EXPERIMENTS An experimental facility, the time-domain scattering range, has been constructed for the measurement of time-domain, transient, scattered fields excited by radar targets which are illuminated by short-pulse incident fields. This chapter is dovoted to the description of the experimental setup, its operating principle, the experimental procedure and the data-processing along with the experimental results. Section 6.1 describes the setup of this time-domain scattering range. Section 6.2 discusses the operating principle upon which the range functions. The experimental procedure is described in detail in Section 6.3. Then, in Section 6.4, we develop a software package for processing the irregular raw data into the useful information. In the end, we conclude this chapter by showing the neumerical results from different targets including sphere, isolated wire and skew—coupled wires in comparison with the theoretical results. 6.l Experimental Setup A large ground plane composed of nine 4' x 8' modules has been constructed. Atficonical transmitting antenna (monocone over ground plane) with a length of 8 feet and a half-angle of 80 is fabricated. A short monopole (1.6 cm) is used as the receiving probe. The setup is shown in Figure 6.l. The basic experimental arrangement including 163 164 uowumu any Eoum mamcwwm cusuou wcwusmmoe pow asumm Hmucmsaumaxm .H.o muswfim , \. .w \ \ mamaa vcaouw xxx .5 \ \ \ II\VIIIIIII.\lllIIll\lllI|IIIlllla- \ x xx x x \ \ \ x x x x x x \ \ \ \ \ \ .— \ \ \ m? x x . x x \ \ \ * \ \ \ \ 0 cm x x mnoua x , u uwwumu \ x .mcmuu x waw>wmumu x x x x x \ x \ \ \ \ . \ IIIIkIIlIIIIIIklIIIIIIIIIIIIIIIIIII 165 relevant pieces of equipment is shown in Figure 6.2. Nanosecond pulses of 400 V amplitude excite the TEM biconical-horn transmitting antenna and are displayed on a sampling oscilloscope which is triggered by those same pulses. The incident field E1 illuminates both the re- ceiving probe and test target. The backscattered field E5 from the target subsequently excites the receiving probe, and can be separated from E1 due to its additional propogation time. The output signal from the short receiving probe is processed through the sampling oscilloscope. Both the horizontal sweep voltage and the sampled receiving probe signal outputs from the oscilloscope are analog-to-digital con- verted by an A-D converter which is controlled by a microcomputer. Both the time-base and receiving-probe data are stored in computer memory (RAM). The data can then be recorded on the cassette tape or the floppy disk and subsequently transferred, via telephone modem, to the MSU CYBER 750 computer system where they are placed in permanent disk-file storage. All data processings are then accomplished on the CYBER. 6.2 Operating Principle Since the impedance of the biconical horn is essentially frequency-independent, then the transmitted incident wave field nearly replicates the pulse generator output. The operating principle of the sampling oscilloscope can be easily demonstrated in Figure 6.3. The sample density determines the horizontal display rate since one sample is taken for each repetition of the lKHz pulse generator. tramps. ///A// ground screen 1 kHz rep. rate nano- pulser second o___,_._. peak I'GC. mon0pale 6 cm 1. //AI probe 55 Jipsfl target ///7 m a: g - .. / variable delay 1 .11... sampling I ‘1, oscilloscope 0-500 V -—W a): J H V atten. 9 " I J? l horiz. sweep out I 1 __<__ (11) 1(2) {<5 * ch. A vert. out VA-D converter t p . . a e 3 I _T cassette J. . microcomputer ‘— telephone ._____s,n data processing MSU CYBER-750 COMPUTER SYSTEM disk files Figure£L2.Experlmental arrangement for measurement of transient scattered EM waveforms. l 167 WAVEFORM MEASUREMENT WITH SAMPLING OSCILLOSCOPE \v(t) 3. trigger trigger . trig er T021 ns sample TR2].ms 'kTDJ 1 -¥ Tp-’-| 1 4.104 .~ R [fit R Repetitive waveform to be sampled Input to sampling scope must be perfectly repetitive at a 10W' repetition rate (usually 1-10 kHz). Sampling system acquires samples at fixed time intervals (incremented delays) following arrival of a trigger signal. One sample is taken from each repetition of the waveform being sampled. - - Total number of samples taken from measured waveform depends upon variable sample density (samples/div.). Horizontal data display rate depends upon sample density and, signal repetition rate. Figure 6.3. Illustration of the operating principle of the sampling oscilloscope. 168 The equivalent circuit of the receiving probe is shown in Figure 6.4. The voltage source, Vs’ is proportional to the electric field which illuminates the probe and therefore equals to Esh, where h is the effective length of the antenna, ZA is the probe impedance when it is used as a transmitting antenrun and R is the load resistance. L The voltage received by sampling scope, VR, is therefore v =_s_L = L (5.1) from the theory of linear antenna [35]. For the short probe, ZA(s) =-—l- + rA (6.2) CA where CA is the capacitance and rA is the resistance of the antenna acting as a transimitting element, with §%— >> rA over the main part A of the frequency range of this experiment (f f 3 GHz). Equation (6.1) can be rewritten as _ s VR(s) - sE hRLCA/Il+s(rA+RL)CAJ. (6.3) In this experiment RL = 509 for the coaxial cable and therefore 1 ——>>R SCA L (6.4) RL >> rA over the main part of the frequency range of this experiment. Based on equation (6.4), equation (6.3) now becomes 5 VR(s) z sE h RLCA, (6.5) 169 I41 Figure 6.4. Equivalent circuit of the receiving probe. 170 This implies that t) .. 9.531.111 (6.6) R( dt ' We used (6.6) as the approximation to perform our data-processing, and obtained satisfactory results. 6.3 Experimental Procedure To start an experimental run, the operator first enters the number of data points to be A-D converted for the entire horizontal display, and the computer then directs the A-D converter to begin sampling the horizontal sweep output. The operator then actuates the single-sweep mode of the scope; when the converted horizontal sweep signal exceeds a preset threshold, the microcomputer commands the A-D converter to begin taking probe data samples at a fixed rate until the previously specified number of data points have been acquired. Both the time-base and receiving-probe data are stored in RAM and later recorded on the cassette-tape. It is important to make sure that the sample density is ad- justed to yield an accurate reproduction of the measured waveform. Another parameter which must be preset before any experimental run is the time per division of the experimental waveform, tpd’ which controls the total time interval of valid data. It is necessary that E5 be measured during the time interval which precedes the arrival of clutter return from edges of the ground screen, etc. at the receiving probe. Since the A-D converter of this particular system can only record the positive voltage accurately, any negative voltage will be recorded as zero voltage, we should add a DC offset, through the use 171 of an oscilloscope, to all the output signals to the A-D converter. This DC voltage should be adjusted so that the minimum voltage to the A-D converter is positive and yet the maximum voltage does not exceed the saturation voltage of the oscilloscope. It is therefore necessary to evaluate this DC voltage in the data-processing. For this purpose, the time axis should be adjusted so that we have enough points (l0-20 points) which precede the incident waveform as the basis for evaluating the DC level. Usually, there will be a DC-level drifting during the experimental interval, and it may be necessary to make another 10 to 20 points after the retrace of the scope available. If we want to get these several points after the retrace for the evaluation of the DC level near the end of experimental time interval, it is desirable to actuate the sweeping of time base by first setting scope in the single- sweep mode and then switching to the normal mode when the experimental run begins. By doing so,data points after the retrace are recorded until we switch to the single-sweep mode again. In any scatter-field measurement, the receiving probe response is sampled, A-D converted, and stored both with and without the target present. Those responses are subsequently numerically integrated during CYBER processing to annul the differentiation introduced by the receiving probe. Finally, the reference signal (target absent) is subtracted from the total probe response to isolate the desired backscattered field. During the entire experiment, the key controller is a cassette- tape recorder: it not only records the experimental data, but also is responsible for loading the program which controls the sampling and the 172 A-D conversion of the data. To transfer the data from microcomputer to CYBER, we also need this recorder to control the action. 6.4 Data Processing The FORTRAN program for processing the experimental data is stored in a CYBER permanent file named EXPDPEW as listed in Appendix Before using this program, it is necessary to attach four input data files which are transferred from microcomputer to CYBER: TAPE 1 TAPE 2 TAPE 5 TAPE 6 horizontal sweep out (time base) raw data for the experimental run without target present vertical out raw data for the experimental run without target horizontal sweep out raw data for the experimental run with target vertical out raw data for the experimental run with target The objective of this program is to create five output data files: TAPE 3 = TAPE 7 = Data in TAPE 1 and TAPE 2 are combined together to display the measured response in volts as a function of real time in nanoseconds. This file shows all the data points before the retrace and subtracts the DC offset which appears in TAPE l and TAPE 2. This file has three columns; the first is the real time in ns, the second is the raw data of real voltage and the third is the data after integration. Same as TAPE 3 except that this file is for the raw data of the experimental run with target present. 173 TAPE 4 Data in TAPE 3 are splined using a IMSL subroutine "ICSSCU" so that the response as a continuous function of any particular time becomes available. This enables us to subtract the response without target from the response with target. TAPE 8 Same as TAPE 4 for the splined data with target. TAPE 9 Data in TAPE 4 is subtracted from data in TAPE 8, so that we have the target response along with the result after integration. Therefore, the integrated data in this file is supposed to be the backscattered field. There are four options for evaluating the DC offset: Option l - DC is the average of the first l0 points which precede this incident waveform. Option 2 - DC is the average of l0 points right after the retrace. Option 3 - With DCl = DC from Option 1 and DC2 = DC from Option 2, DC value is assumed to be drifting linearly from DCl in the beginning to DC2 in the end. Option 4 - Based upon the assumption that the transmitting antenna does not transmit DC component, [mf(t)dt = O O t should be satisfied; therefore, I (f(t)+VDC)dt 5 V 0 at.“ if t is properly chosen. This option is very sensitive to the choice of upper integral limit t. Usually, the DC level drifts a great deal during the experiment and it is not necessarily a linear drift. If the DC level is not evaluated 174 accuralely, there will be an accumulated error present after integration, i.e., an error estimation of DC results in a ramp—function type of error. It is therefore desirable to use Option l or Option 2 and then change the slope of the response using a platter and the plotting software package like SPOCS [36]. Two experimental runs (with and without target) sometimes display the incident waveforms with shifted time and drifted voltage. We handle this problem by adding another option with an index IY: if IY = l, we shift time axis and rescale the vertical axis so that. the maxima of the incident waveforms appear at the same time with the same amplitude. 6.5 Experimental Results Typical scattered field measurements are indicated in Figures 6.5 - 6.8. Figure 6.5 is the measured waveform of the incident pulse transmitted by the biconical antenna. It is clear from this result that E1 maintained by the transmitting bicone is an approximate replication of the pulse generator output. The measured scattered field response of a sphere with ll" diameter illiminated normally by this incident pulse is indicated in Figure 6.6, which displays the right creeping- wave and the specular reflection behavior [28]. The response of an isolated wire with L/a = 200, and L/c = 2;ll6 ns due to normal incidence is indicated in Figure 6.7. This latter smoothed impulse response (target impulse response to a short incident pulse) compares very well with the result obtained when the measured incident pulse of Figure 6.5 is convolved with the known theoretical impulse response. Figure 6.8 is the backscattered field response of an isolated wire with L/c = 200 175 0.5 0.2 1 d Relative Amplitude 0.1 1 ‘1 1 11’ 1 ‘1 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 time in ns Figure 6.5. Measured waveform of incident pulse transmitted biconical antenna. by Relative Amplitude "004 l --- 176 “005 0.0 Figure 6.6. l l l 1.0 2.0 3.0 4.0 5.0 6.0 7.0 time in ns Measured nanosecond-pulse backscatter field response of a sphere with ll" diameter to normally incident illumination. Relative Amplitude o 177 I n—l F'~—~D4 X')‘ M to O O L/c 2.ll6 ns 200 L/a Figure 6.7. 1 1 1 1 1 1 1 2.0 4.0 6.0 8.0 10.0 12.0 14.0 time in ns Measured nanosecond-pulse backscatter field response of a thin, conducting cylinder to normally incident illumination. 178 O L/c = 1.058 ns L/a a 200 Relative Amplitude T I I I I 'o.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 time in ns Fimne 61L Measured nanosecond-pulse backscatter field response of a thin, conducting cylinder to normally incident illumination. 179 and L/C = l.058 ns. In this figure, more "ringings" are seen because the length of the wire is one half of the previous one. The response of the skew-coupled wires with different angles are indicated in Figures 6.9, 6.ll and 6.l3. Figure 6.9 is the ex- perimental result for the case of L/a = 200, L/c = 1.058 ns, d/L = 0.5 and a = 90°; this is very similar to those in Figures 6.7 and 6.8. Figure 6.9 compares very well with Figure 6.lO which is the convolved result of the incident waveform of Figure 6.5 and the impulse response of Figure 4.22. Figure 6.ll and Figure 6.l2 are the similar results for the case of a = 600; the comparision is not as good as the case of a = 90°. This is because the "ringings" of e = 600 case is not as strong as that of a = 900 case and, therefore, the signal-to-noise ratio is not as good. Results for a = 300 case are shown in Figure 6.l3 and Figure 6.l4. The signal-to-noise ratio in this case is even worse and, therefore, a poor agreement between these two figures is observed. 180 Relative Amplitude 0.8 . I T I j 0.0 2-0 4.0 8.0 8.0 10.0 12.0 14.0 16-0 time in ns Figure 59, Measured nanosecond-pulse backscatter field response of a wire over__the ground plane with (1.900, L/a2200, L/c-1.058 ns, d/L-O.5 to normally incident illumination. Relative Amplitude Figure 6.lO. Result of convolution between 181 ‘3 1.0-I “ A k D 41 ,77777I7777777 9.1 m 9. N /“\ . / ;- ///fi\ / 94 l / I // o. / v-o-l I / | / 9 Nd I 9 "3 1 1 1 1 1 T 1 fl 0.0 1.0 2.0 3.3 4.0 5.0 6.0 7.0 8.0 time n ns nanosecond-pulse and O impulse responseof a wire over ground plane with a = 89.9 fi/gizoo, d/LaO.5, L/c=1.058 ns and aspect-angle 01 0 cs that the s ecular reflectio 1 because a negativg’impulse is not ghoSnnfg 553D impulse response. 182 a "It. 0. Relative Amplitude ”008 I T I I T I I fl 0.0 2-0 4.0 6.0 8-0 10.0 12.0 14.0 16.0 time in ns Figure 6.ll. Measured nanosecond-pulse backscatter field gesponse of a wire over the ground plane with a = 60 , L/a = 200, L/c l.058 ns, d/L = 0.5 to normally incident illumination. 183 0.5 a- -..--- --_L--._-_...- .....| 0-4 o 3 L. ‘\:> g f 3 Nd I ... a / $1 I .E f ..g.’ '74 I" ‘ / *3 ° / / /\ a: / / d . / \ /, ‘? 9.4 I “3 ‘.3 j r I 1 I I I 1 0.0 1.0 2.0 3-0 4-0 5.0 5-0 7.0 8-0 time in ns Figure 6.l2. Result of convolution between nanosecond-pulse and impulse response of'a wire over ground plane with a = 60°, L a = 200 d/L = 0.5, L/c = l.058 ns and aspeet-angle 0°. / 184 ‘S- 31' l I n L ,rI ' " --“‘_’ -05-- q k d 0-1' Ill/1171111,] i ’l I fl 3 o“ I I l l OJ D “U 3 .4: I E? ‘34 < D I ‘” I > . ‘3 N D '— D Q) '-l (I D c? o- l ' l (5.4 I c? I I I I m I I 0.0 1-0 2-0 3.0 4.0 5.0 6.0 7.0 time in ns Figure 6.13. Measured nanosecond-pulse backscatter field 5esponse of a wire over the ground plane with a = 30 , L/a = 200 L/c l.058 ns, d/L = 0.5 to normally incident illumination. Relative Amplitude N o D 185 'o.0 Figure 6.14. I I I I I 1.0 2.0 3.0 4.0 5.0 time in ns Result of convolution between nanosecond-pulse agd impulse response of a wire over ground plane with a 3 30 , L/a = 200, d/L = 0.5, L/c = l.058 ns and aspect-angle O . O) o o \I O 8 .0 CHAPTER 7 CONCLUSION It has been demonstrated that an aspect-independent, optimal incident radar waveform of finite duration Te can be synthesized to excite a target in such a way that in the late-time period of t > Te + 2Tt (Tt = target one-way transit time) its backscattered field consists of a single natural mode which can be used to identify and discriminate the target. By constraining only the late-time target response, a time-domain synthesis technique was developed which does not require the knowledge of the forced, early-time impulse response. Three types of targets are presented not only to confirm the applicabilitv of this scheme but also to study the transient electromagnetic behaviors of the three targets. A time-domain scattering range has also been constructed to perform the transient electromagnetic experiments. In this chapter, we will conclude this study by summarizing the waveform- synthesis method from the system point of view and discussing some potential problems associated with this method for the future investigation. 7.l A Target-Discrimination System Employing Waveform-Synthesis Method Depicted in Figure 7.1 is a potential target-discrimination system employing the waveform-synthesis method. In this system, all the required waveforms to excite the single-mode responses of the "friendly" targets are stored in a large computer system, one "channel" for each target. 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