MSU LIBRARIES \— RETURNING MATERIALS: PIace in book drop to remove this checkout from your record. FINES wiII be charged if book is returned after the date stamped below. I I‘- to k fm-r ,2 s ‘ I RADAR TARGET DISCRIMINATION USING K-PULSES FROM A "FAST" PRONY'S METHOD By Lance Lynwood Webb A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science 1984 203J‘right l Lance Lyn-u: 1fl:'! ‘34? Copyright by Lance Lynwood Webb 1984 IEC'T"ZJE l5 ‘ I“ ’ ‘ g _1 a >9 53%.. ’ ' o I rEern In“: ‘ AF“! “4’3”: '5! 3‘ A .‘ A‘ r. l V - :5!" 15"]- 1S ”Else sensiti Origi WEE” ot:eC Mar target I u . for this WP: ABSTRACT RADAR TARGET DISCRIMINATION USING K-PULSES FROM A "FAST" PRONY'S METHOD By Lance Lynwood Webb An aspect-angle independent and range independent processing technique is disclosed which is based upon the SEM (Singularity Expansion Method) model solution of the EFIE (E-field integral equa- tion) for transient electromagnetic scattering. The derived kill- pulse convolution forms conveniently decompose the radar target return into aspect-angle independent parameters and aspect-angle dependent parameters. A near real-time "fast" Prony's method algorithm is performed on empirical data to exploit the inherent noise sensitivity of the ordinary Prony's method to obtain a second K-Pulse form. Original contributions of this dissertation in addition to the primary objective of developing a range and aspect-angle independent radar target discrimination technique potentially compatible with ”quiet" radar, are four original analytical tools developed solely for this purpose: a“. Lance Lynwood Webb 1. “Fast Prony's method algorithm" for real-time invariant parameter calculation of 4-dimensional radar data. 2. "Prony K-Pulse" for calculating SEM coupling coefficients from retarded scattered E-field sampled data. 3. ”Polar mode A-scope" display file processing replacing part of the conventional radar target- independent matched filter. 4. "Mode ratio discrimination detectors" for auto- matic radar target trigger and identification channels. The radar target discrimination technique is evaluated on experimental radar data obtained from NSWC. To my parents Maurine and Lynwood Webb ii l'ne aut' academic adviso' ior research at; Aspecial note ' in-de;th criti : The aut his comittee, I“ tosroents. Lastly, INS“; Dahl grer ;. . WSAEd peas ACKNOWLEDGMENTS The author wishes to express strong appreciation to his academic advisor, Dr. Kun-Mu Chen, for providing interesting topics for research and a very exciting and worthwhile dissertation goal. A special note of thanks is due to Dr. Dennis P. Nyquist for his in-depth critiques and analyses before this final form. The author wishes to express his thanks to other members of his comittee, Dr. Byron Drachman and Dr. S. Sayegh for their help- ful comments. Lastly, the author wishes to thank Dr. Bruce Hollmann of NSWC, Dahlgren, VA for collaboration on several papers for which he furnished measurement data also used herein. LIST OF TABLES [1370? FISL’REE hater 1. INTRCDL‘CI 2. TRAJISIEF. 2.1 Elel , in Id'l 2. Si' 2 F 2 .3 As; Sc: TABLE or CONTENTS LIST OF TABLES LIST OF FIGURES . Chapter 1. 2. INTRODUCTION TRANSIENT ELECTROMAGNETICS FOR RADAR TARGETS . 2.1 Electric Field Integral Equation for Radar Targets . . 2.2 Singularity Expansion Method (SEM) Notation 2.3 Aspect-Angle Dependency of Radar Target Scattered E- Field . . 2.4 Equipment Set- -Up for Transient Electromagnetic Reception . . . . . . 2.5 Processing by Digital Clutter Map. PROCESSING TECHNIQUES FOR EXTRACTING NATURAL MODE WAVEFORMS ' (.0 ._| Reception of a Radar Target E-Field Sampling the Radar Antenna Response Processing a Scattered E-Field Reception Excitation of a Single Natural Mode Waveform "Polar Mode A- Scope" Displays . Measurement Performance of Single Mode Waveform Extraction Technique . Scaling of the Invariant Radar Target Para- meters . . Observation of the Absence of a Natural Mode Waveform . . Extension of the Model .to Unknown Natural Frequencies . . . . . . PRONY'S METHOD AND THE K-PULSE w OJ on 0000000000 \0 co \1 0101th 4.1 The Original Prony' 5 Method. 4. 2 The "Prony K- Pulse" ‘. 4. 3 Prony' 5 Method and the K- Pulse Derivation iv Page vi vii 30 3O 32 33 43 50 53 74 80 82 82 89 ‘ ab :31 o t” o ‘c - l .5 Chapter 7. Skip Sampling in Prony' 5 Method . . Zeros in the Data Matrix and “Class 2 Prony Series" . . . . . . The Extended Prony' 5 Method. 4. 7 Complex Root Degrees of Freedom :5 13-h Eh 01-h THE NFAST" PRONY'S METHOD .1 Performance Enhancements for Radar Target Discrimination . . . . . . . . . Part 1--The K-Pulse . . Part 2--Roots for Excitation . Part 3--Amplitudes and Coupling Coefficients Computational Comparisons for Part 1 . Special Cases for SEM Computations 0101010101 01 O C O O . 0301-th RADAR TARGET DISCRIMINATION TECHNIQUE .1 Requirements for Automatic Radar Target Discrimination . . . Discrimination Algorithm for Radar Targets . Part A--Target Library Prony K- Pulse Con- volution . . Part B--"Fast Prony Convolution Algorithm" . Part C--Dual "Polar Mode A- -Scope" Displays . Part D--Target Trigger and Identification 6. 7 Empirical Illustration . 050505 0105 01 0101-5 (JON CONCLUSIONS . APPENDICES A. LaPlace Transforms and Z- Transforms for Prony Series . . . . . . . . . K. . . LaPlace Transform Convolution Theorem for the Sampler . . . . Couplets and the K-Pulse Singularity Theorem . Computer Code for "Fast Prony' 5 Method Algorithm" Parts and other Programs . REFERENCES Page 96 107 112 117 117 119 130 135 138 143 156 156 157 160 163 164 165 168 175 178 179 199 208 224 241 Taale 4-1 Prony's 5' 4-2 Late-Time 5-1 K-Pulse '. 5-2 imputati 5-3 Calculat‘ 6'] Radar Ta." I 6-2 Mode Rat Sh'Tary Table 11-2 5-a1 5-i3 5-.1 6-fi2 LIST OF TABLES Prony's Method Tests . . . Late-Time Model Test of Prony's Methods . K-Pulse Termination Flags Computation Comparison for Part 1 . Calculations for the "Fast" Prony K-Pulse Radar Target Discriminant Mode Ratio Discrimination Detector Performance Summary . . . . . . . . . vi Page 115 116 129 141 142 159 167 Figure 1-1 1-2 2-1 2-3 2-4 2-5 2-6 2-7 2-8 3-2 3-3 3-4 3-5 3-5 3-7 LIST OF FIGURES Typical Radar Environment and Target RCS Top Views of Aspect-Angle Dependencies. of Object Waveforms of Speech and Radar . . A Perfectly Conducting Target Illuminated by a Transient Incident E- field. . . . . Illustration of Assured Late-time Response of Retarded Scattered E-field . . . . . Antenna Plots of the Magnitude of Four of the Far- field Modes of a Thin Cylinder . . . . Equipment Set-Up for Transient EM Reception Transmitter Pulse Observed in Transmission Line . Radar Receiving TEM Horn Antenna . Measured Pulse Response of a Cylindrical Target of 12" Length 12" Above a Ground Plane Illuminated at an Aspect Angle of 45 Degrees . . . Clutter-reduced 12" Skew Wire Radar Target Clutter-reduced 18" Spherical (imaged) Radar Target Return . TEM Horn Receiving Antenna Used for Spherical Radar Targets . . . . . . . 2nd Cosine Mode Convolution 2nd Sine Mode Convolution 2nd Mode Envelope Radar A-Scope 2nd Mode Rotation Radar A-Scope Spherical Radar Target Response by Deconvolving Figure 3—2 from Figure 3-1 . . . . vii Page 16 20 22 23 25 27 28 44 45 47 47 49 49 52 Figure 13 {:3 u) 3-10 s~24 Ist M: Cylinc Ist Pi: Cylin: lst M Cylin Figure Page 3-8 lst Mode Waveform Excitation of a 19” Diameter Sphere 54 3—9 2nd Mode Waveform Excitation of a 19" Diameter Sphere 55 3-10 3rd Mode Waveform Excitation of a 19" Diameter Sphere 56 3-11 4th Mode Waveform Excitation of a 19" Diameter Sphere 57 3-12 Clutter-Reduced 18. 6" Radar Target Antenna Terminal Response . . 59 3-13 lst Mode Waveform Excitation ofCorrectly- Sized Thin Cylinder Baseline Radar Target . . . 61 3-14 lst Mode Waveform Excitation of 10% Undersized Thin Cylinder Radar Target . 63 3-15 lst Mode Waveform Excitation of 10% Oversized Thin Cylinder Radar Target . . . . . . . 64 3-16 3rd Mode Waveform Excitation of Correctly-Sized Thin Cylinder Baseline Radar Target . 66 3-17 3rd Mode Waveform Excitation of 10% Undersized Thin Cylinder Radar Target . 68 3-18 3rd Mode Waveform Excitation of 10% Oversized Thin Cylinder Radar Target . 69 3-19 5th Mode Waveform Excitation of Correctly- Sized Thin Cylinder Baseline Radar Target. . - 70 3-20 5th Mode Waveform Excitation of 10% Undersized Thin Cylinder Radar Target. . . 71 3-21 5th Mode Waveform Excitation of 10% .Oversized Thin Cylinder Radar Target. . 72 3-22 7th Mode Waveform Excitation of Correctly- Sized Thin Cylinder Baseline Radar Target. . . . 73 3-23 7th Mode Waveform Excitation of 10% Undersized Thin Cylinder Radar Target. . . . . . 75 3-24 7th Mode Waveform Excitation of 10% Oversized Thin Cylinder Radar Target. . 76 viii fight 3-25 126 127 42 4-3 4-4 46 4-7 48 4.9 2nd Mod Cylinde 4th Hod Cylinde 16th He Cylinde Undeter Major S Display Clutter (18"). Convolu First 4 5414 Se Convel. Figure the Re: Convol~ Figure Target Third ‘ Convai‘ flgure radar ‘ [05701: Figure large: Synthe Incide “flier F401 : AIQSrf Figure 3-25 3-26 3-27 4-1 4-2 4-3 4-4 4-6 4-7 4-8 4-9 4-10 5-1 2nd Mode Waveform Excitation of Baseline Thin Cylinder Radar Target at Normal Incidence' . 4th Mode Waveform Excitation of Baseline Thin Cylinder Radar Target at Normal Incidence 10th Mode Waveform Excitation of Baseline Thin Cylinder Radar Target at Normal Incidence Undetermined Coefficients of Prony's "Essai. . A. " Major Steps in the Generation of Radar A-Scope Displays . . . . . . . . . . . Clutter-Reduced Radar Response of a Thin Cylinder (18"), the Prony K-Pulse from Skip Samples, the Convolution, and Its Envelope in Decibels First Mode Waveform Excitation Synthesized from Skip Samples of a Thin Cylinder Radar Target Convolution of the First Mode Excitation Vector of Figure 4- 4 with a New Set of Skip Data Points in the Radar Target Response File . . . Convolution of the First Mode Excitation Vectors of Figure 4- 4 with all of the Data Points in the Radar Target Response FTTe . . Third Mode Waveform Excitation Synthesized from Skip Samples of a Thin Cylinder Radar Target. . Convolution of the Third Mode Excitation Vectors of Figure 4- 7 with a New Set of Skip Data Points in the Radar Target Response File . Convolution of the Third Mode Excitation Vectors of Figure 4- 7 with all of the Data Points in the Radar Target Response FTTe . . . . Synthetic "Class 2 Prony Series" at 1 Milliradian Incidence . . . . . . . . . . . . Major Parts of Prony's Method Summarized Flow Chart for K- Pulse Part of "Fast Prony' 5 Method Algorithm" . . . . . . ix Page 78 79 81 87 93 94 98 100 102 103 105 106 114 118 131 Figure 64 M Figure 5-3 5-4 5-10 5-11 5-12 6-1 6-2 6-4 6-5 A-l A-2 A-3 B-1 "Fast Prony's Method Algorithm" Block Diagram "Fast Prony' 5 Method Algorithm" Substituted in Figure 4- 6 . - "Fast Prony' 5 Method Algorithm" Substituted in Figure 4- 9 . - Prony K-Pulse Worksheet #1 . Prony K-Pulse Worksheet #2 . Non-Late-Time Prony K-Pulse Worksheet #1 Non-Late-Time Prony K-Pulse Worksheet #2 Two Mode Prony K-Pulse Worksheet #1 . Two Mode Prony K-Pulse Worksheet #2 . Two Mode Prony K-Pulse Worksheet #3 . Radar Target Discrimination Algorithm Summary "Unknown Target" Sampled Data Files . Dual "Polar Mode A- Scope" Displays for lst Target Mode . . Dual "Polar Mode A- -Scope“ Displays for 2nd Target Mode . . Dual "Polar Mode N Scope" Displays for 3rd Target Mode . . Forecasted Radar Target Discrimination Capability Approximating Function from Sampled Data Change of Variable from z-plane to s-plane . Approximating Function for Fixed m Contour Integration for V(z,m) Region of Absolute Convergence for a Product in Time Domain . . . . . . . . . . Page 134 I39 140 145 146 148 150 151 152 153 158 169 171 172 174 176 182 188 191 198 205 Figure 5.2 Counter 8-3 Clockwi t-l Single [-2 Single [-3 Two Ter Figure Page B-2 Counterclockwise Contour Closure . . . . . . . 206 8-3 Clockwise Contour Closure . . . . . . . . . 206 C-1 Single Term "Prony Series,f v(t) . . . . . . . 213 C-2 Single Term "Prony series.9 w(t) . . . . . . . 214 C-3 Two Term ”Prony Series," s(t) . . . . . . . . 217 xi i'TC'l IS PL .1. a: . 3 q: nn' j. 5. C538" (d JIF r. t: tne re: A “ .4 .se tar ;-AA 7 .52“ 8:41~ n \a «flu 3. A : “4'51 Ti"- r» CHAPTER I INTRODUCTION A radar target discrimination technique will be developed which is machine implementable. Figure l-l illustrates a typical radar set with a radar target cross-section plot. In the top sketch the transmitted waveform is propagated from the antenna to the radar target in the presence of ground clutter. In the middle sketch the scattered electric field from the target and clutter is propagated to the radar receiving antenna. The received signal at the antenna terminals is very highly dependent upon the location and orientation of the target relative to the radar site. This is illustrated by the radar target cross-section plot. A positive target identification based solely on the target radar skin return is not an easy problem. The radar target discrimination solution disclosed here will be radar target aspect-angle independent and target range independent. Figure l-2 illustrated an analogy with human brain processing of a rotating speaker. The crucial characteristic of the target required is to radiate a reasonable portion of its energy in one or more natural mode waveforms. All conducting vehicles with sharp edges are believed to adequately satisfy this requirement. In Chapter 2 we will develop the transient electromagnetic foundation of the technique. We start with the electric field integral :k’e F‘s-mu ”runaways... *_ fin“ “.135“ MM“ Figure 1-1. Typical Radar Environment and Target RCS. . S o' I I" C. .l .I -l a. l I ‘ a I \ ‘. I \ \ H “ \ 1 I . L , . / I . 1. I ‘~ I l/’ I s " ’0 \ c’? .— 0.. - o c- -— --. ' v: -I‘ ‘3» o v _’ 'a \‘ a I ' I \ I \. \\ \ s‘ I \ \ I ' ‘ ’ I I ‘0 I a \ 0 Top Views of Aspect-Angle Dependencies of Figure 1-2. Object Waveforms of Speech and Radar. equation 4 proaelled Method (SE mflel sol. aszect-an; then prese thegrcuri Whit in t is not pla In extracting ccHection equation (EFIE) which is extremely difficult to solve for any self- propelled radar target. We introduce the Singularity Expansion Method (SEM) as a model solution to the EFIE. We show that this model solution has, even for a simple scatterer, all the extreme aspect-angle dependencies observed in the empirical radar data. We then present the equipment set-up for obtaining radar data and then the ground clutter map processing. Note that we cannot have Doppler shift in this stationary set-up. For clarity Doppler exploitation is not placed in the formulation. In Chapter 3 we introduce the processing techniques for extracting a single natural mode waveform from a large but finite collection of natural mode waveforms. We then introduce one of the primary contributions of this dissertation, rectangular and polar representations of natural mode waveforms. The polar representation is important enough that we shall call it a radar "polar mode A-scope" display. The data from this display will be used in the automatic radar target discrimination algorithm. The data of these displays replace part of the conventional radar target-independent matched filter and detector techniques traditionally used for detection and tracking. In this chapter we use empirical radar data which always presents difficulties to the discrimination process much in excess of that presented by synthetic data artificially corrupted by "white noise." The observed defects are primarily due to equipment distor- tions, uncancelled clutter, thermal noise, and quantitization errors. In Chapter 4 we introduce our primary technique to be used when the radar target possesses a complexity which is obviously bey’ond t'ne ca Prony's metnr of the SE”. n; duce E. Kenna contecslated fomlation 0 first part of our "Prony's NSCr‘Ininati: original Pry rotation der‘ rich for la‘ Ctsi‘itianea . ‘ imitate a ' 3“ STE-11“. ta‘ WI 5 net “navalar t'r the 7Win “'- beyond the capabilities of analytical calculation. Our technique, Prony's method, can be used to experimentally obtain the parameters of the SEM model of the radar target's EFIE. We will also intro- duce E. Kennaugh's K-Pulse concept. His transmit formulation is not contemplated to ever be used. However it is shown that a receive formulation of the K-Pulse can be generated as an extension of the first part of Prony's method. This K-Pulse, properly specified as our "Prony's K-Pulse,“ is a crucial building block in our radar target discrimination technique. In this chapter we will introduce the original Prony's method, and then perform the now standard matrix notation derivation. This lends to standard matrix computer routines which for large numbers of sampled data values are increasingly "ill conditioned." To minimize the size of the matrix to be used, we introduce a skip sampling technique and illustrate its performance on empirical radar target data. We then introduce the extended Prony's method or least-squares formulation which is probably more popular than the regular Prony's method. At this point we introduce the important concept of "complex root degrees of freedom." This concept appears to be quite useful for our 4-dimensional electro- magnetic problem. A dramatic example is presented which illustrates the superiority of the regular Prony's method over the extended Prony's method in certain cases which will be used in our radar target discrimination technique. Further, the regular Prony’s method does not constitute an irreversible process as do all least—squares processing techniques. This is an important consideration for the design of modern "quiet" radars. In C". an ariginal C is desired it naré‘uare reSC seed by as;r near real-ti~ and its Su'ESE :r1:essin;. — 4 — — _ — _ use of the . In Chapter 5 we present the "fast“ Prony's method, which is an original contribution of this dissertation. The "fast" algorithm is desired for two reasons. The first is to fully exploit the "com— plex root degrees of freedom“ concept to the maximum with minimal hardware resources. The second is to increase the computational speed by approximately an order of magnitude in order to permit near real-time calculation of a dual "polar mode A-scope" display and its subsequent processing for automatic target discrimination processing. The structure of the unsymmetric Toeplitz matrix is explored for the first time and there is an additional by-product of the "fast Prony's method algorithm": it produces two solutions instead of one solution. Both are usable, but a comparison of the two appears to be a useful analytical tool. In Chapter 6 we present the automatic radar target discrim- ination algorithm. Several of the crucial original concepts are the use of the "2nd K-Pulsed convolution" and the dual "polar mode A- scope" displays, where for the first time we exploit the well known sensitivity of Prony's method to noise by performing a Prony's method a second time. Further, the format of the “polar mode A-scope" dis- plays make possible the original "mode discrimination ratio detectors" suitable for automatic processing in a monopulse-like fashion. In Chapter 7 we shall make some forecasts on the technique's implementation and uses and the impact of Doppler shift. Appen transforms. Z ; . . .ornctation Apaen the analytica 42:94 .11 ' 4 Use STFCJ r‘ ' ..ea51ly de Assen useful Appendix A is a complete section for review of Laplace transforms, z-transforms, modified z-transforms, and consultation for notation used throughout this dissertation. Appendix B is the Convolution Theorem for the Sampler. It is the analytical tool necessary for the handling of the linear, time: varying operation of the sampler. Appendix C is an introduction to time domain couplets and the K-Pulse Singularity Theorem. These are simple but powerful tools not easily derivable in the continuous time domain. Appendix D is the collection of computer programs found useful. 2.1 Electric Ecuation for 3. In th" bmtdary val“ be the founda: 141:1 we will here already c Let u< 4 1” FlSure 2-1 EWits. 14,; trament, in: tarset sum,i dE'fiSitY, :(T \ 1 meme a SC; ““04. that Conductor HES 94mm (2., FIRr M; ~r CHAPTER II TRANSIENT ELECTROMAGNETICS FOR RADAR TARGETS 2.1 Electric Field_Integral Equation for Radar Targets In this section we shall develop the transient electric field boundary value problem for a perfectly conducting object. This will be the foundation for the electromagnetic response of our radar target which we will assume is highly conducting at radar frequencies. We have already described the physical formulation in Figure 1-1. Let us examine a simple 3-dimensional radar target shape as in Figure 2-1 in order to define our electromagnetic quantities and concepts. The perfectly conducting radar target is illuminated by a transient, incident plane wave, Einc(F,t), which excites, on the target surface, the induced current density NLF,t), and charge density, o(F,t). These induced current and charge densities, in turn, generate a scattered wave, ES(F,t). It is the summation of the inci- dent and scattered wave which must satisfy the familiar boundary con- dition, that the tangential electric field at the surface of a perfect conductor must be zero for all time. This relation is expressed by equation (2-1). E-(E‘"°(‘F,t) + 15505;» = o for all “F e A (2-1) 3‘53) Kit) space (“0 is. ,6 ‘ O) c- (He Eel. Figure 2-1. A Perfectly Conducting Target Illuminated by a Transient Incident E-field. In equation radar target .p r. Ne w the retarded (M). ii" ‘ g t lrw/ These retarc'r 10 In equation (2-1), A denotes the set of all external points of the radar target and t is any unit tangent vector to the targer surface at + I”. We wish to express the scattered field, E5(F,t), in terms of the retarded scalar and vector potential (ref 2-1) given by equation (2-2). E5 (r, t) = -V¢( r ,t) - —-JA (r ,t) (2-2) These retarded potentials are given by equations (2-3) and (2-4). a(?,t) =13”? ”Marv , R = IF'SFI (2-3) A Amt) =ffififrggt‘R/“dn', R = ("P—Fl (2-4) A lie will ultimately use conservation of charge given by equation (2-5) when it is convenient to eliminate o(F,t). vim») = - goat) (2-5) Next we shall take the Laplace transforms of equations (2-1) through (2-5) to obtain equations (2-6) through (2-10) 2 (El"c(?,s) + ES(?,s)) = o for F e A (2-6) Es(?,s) = - V¢(r,s) - si(?,sr (2-7) In ecuati radar tar + l‘. the retar (2-2). These ret. a» ‘IvI ) r4- rvil m 10 In equation (2-1), A denotes the set of all external points of the radar target and t is any unit tangent vector to the targer surface at n) r. We wish to express the scattered field, E5(F,t), in terms of the retarded scalar and vector potential (ref 2-1) given by equation (2-2). Es(r, t) = -V¢( r ,t) - MA (2-2) These retarded potentials are given by equations (2-3) and (2-4). 6,4 11R a(?,t) ajfo r' t R/ClaA' , _ Iii-Fr (2-3) A Amt) =[H‘fi1gg t R/c)dA', R - [F'IFI (2-4) A We will ultimately use conservation of charge given by equation (2-5) when it is convenient to eliminate o(F,t). - 5%0(F, t) (2-5) 4 7<+ ‘Ql 1+ V II Next we shall take the Laplace transforms of equations (2-1) through (2-5) to obtain equations (2-6) through (2-10) E (E‘"c(?,s) + ES(?,s)) = o for F e A (2-6) ES(?,s) = - va( r, s) - sA( (r ,5) (2-7) 7" Sabstituti (2-6), we 11 + - (11 - R/ . (12(Y‘,S) _ fig—02."; EXP( S c)dA (2-8) A Kat’s) =fm§£ESSIEKD(-SR/CldA1 (2_g) - A V-R(F,s) = - so(?,s) (2-10) Substituting equation (2-10) into (2-8) and equation (2-9) into (2-6), we obtain equation (2-11). + -> “. inc 4 vtk(r;s)epr-sR/Ql . t (E (N5) + Vf 58041111 —dA A A Equation (2-11) rewritten as in equation (2-12) is often called the EFIE (Electric Field Integral Equation). -> A A + —> - , A THC '* I(vfkfia)(t-maztmugsnezgé SR/ClaA = -eost-E (r,s) A for all ‘r’eA (2-12) where y = s/c 2.2 Singularity Expansion Method (SEM) Notation The EFIE, equation (2-12) is not particularly easy to solve for the physical radar targets. The method of attack used in this dissertat? nated for mtivatior the apcear cosinusoic Sit) = .45 is wel' given by 4 4(6):; .ne rigrt 1411' .. n C'l 12 dissertation is called the Singularity Expansion Method (SEM), origi- nated for electromagnetics by C. E. Baum (ref.2-2). The physical motivation for SEM is the observation that radar returns signals have the appearance of being a sum of damped cosinusoids. A single damped cosinusoid will be represented as in equation (2-13). S(t) = 1A(exp(slt) + exp(sft)) with Re(sl)<0 (2-13) As is well known, the Laplace transforms of a complex exponential is given by equation (2-14) L(exD(51t) = (5'51)-19 RE(S) > 51 (2_14) The right-hand side of equation (2-14) displays a simple pole singular- ity in the s-plane from which the SEM obtains part of its nomenclature. There are two issues to be kept in mind about using equation (2-12) in the physical radar problem that SEM must address. One, it is a problem which must be simultaneously solved in 3 space and one- time variables. Two, the simplest physical radar excitations which could be observed are not impulses in space and time, but plane wave excitations. Hence the aspect-angle dependencies need to be present, yet separable, if the solution actually models the radar target. Without loss of generality, we will assume only simple poles (rather than multiplicities of finite degree as developed in ref.2-2). For simple poles the SEM solution to the EFIE is given by equation (2-15). 13 R (F,s) = Z na(fl,s) 3:(?) f;(s) (s-sm)’1 + fiK(fl’F 5) (2-15) p 04 where a is the index for all of the target natural frequencies 5 is a natural frequency of the radar target (s-scm).1 is the s-plane pole at $0 $K(?) is the current density natural mode of a the radar target f (s) is the pulse shape of the incident radar p plane wave n (4,5) an entire function is the coupling coef- 9 ficient from the incident plane wave to the natural mode of current density p denotes one of the possible incident polarizations T is the aspect-angle dependent vector Wr(a ? S) is the entire function which may be required for convergence It should be observed that equation (2-15) appears similar to a partial fraction expansion of a (meromorphic) function with a finite number of pole singularities. The less usual term, the entire function of the far right, WK, is due to the infinite summation index, a. The Mittag-Leffler theorem (ref. 2-9) warns us that this may (or may not) be required. An entire function is analytic everywhere in the finite s-plane, but may have poles at infinity. For our radar target natural modes with Re(s) < 0, we could expect the existence of a representation of the entire function with its poles at infinity on the left-hand side of the s-plane. In general, the entire function will not be ob definEdl- For an entire “mm this entire fl“ interested l“ rewrite Wear In 9%! de;endencies ' coefficient W target's uni"; It is twain since containing 5 tine dona i n. have is purel one convoluti coupling coef tion called a yields the "l 14 will not be observable in what we shall callthe'qate-time" (to be defined). For any finite summation on a, we shall formally drop the entire function. For the infinite summation we shall suppress writing this entire function, WK, recalling its presence only when we are interested in "early-time" results. With this in mind, we shall now rewrite equation (2-15) and use equation (2-16) from now on. K (KS) = Z na(il.5) 33(7) fp45) (s-sml'1 (2—16) In equation (2-16) it should be noted that all aspect angle dependencies for an individual natural mode occur only in the coupling coefficient which is a factor in the complex amplitude of the radar target's unique natural mode of current density and natural frequency. It is not quite so simple to write equation (2-16) in the time domain since in each term of the series there are 3 distinct factors containing 5 dependencies. These factors become convolutions in the time domain. For clarity, we will assume that the incident plane wave is purely impulsive, making f(s) = I and we are left with only one convolution which when performed is called an SEM I'class 2" coupling coefficient for the current density. There is a simplifica- tion called an SEM "class 1" coupling coefficient which accurately yields the "late time" value of the natural mode current density as expresed by equation (2-17). ?(?.t) = u-1(t-t') z na(“,sa) ogri) exp(sat) (2’17) 01 The n unique, it Sl ling coeffici is the time l coefficient f as well as tr causality to thin cylinder be sonennere assured that In general, l “‘49 Pmpagai ObsEl'vation I titration (2- 15 The new constant t' is a new parameter which need not be unique, it simply states when (in time) we may use a simplified coup- ling coefficient. We will be interested in a still later time t" which is the time lag for which we may use the simplified "class 1" coupling coefficient for the calculation of the retarded scattered E-field as well as the current density. In the general case we will invoke causality to derive a parameter we will use later. Let us consider a thin cylinder radar target as in Figure 2-2. Let our observation point be somewhere on the negative z-axis (e.g., z = 0-). In order to be assured that the simplified coupling coefficient may be used, we must, in general, have a delay long enough for the incident plane wave to have propagated clear of our target as viewed in retarded time at our observation point. This assured value of time, t", is given by equation (2-18) for an impulsive plane wave on a very thin cylinder. t" = max {L/c + L/c cose} = max {2-way transit time} (2-18) 89¢ 9.4 For some shapes (e.g., a perfect sphere), this is known to be too conservative (ref. 2-2). But his a priori knowledge of its complete shape is not a permissible part of a radar discrimination problem. From causality we would not know for sure if there existed a defect on the far side of the sphere. Propagation at the speed of light will require, in general, a two-way transit time of the object by the incident impulsive plane wave in order to use the simplified coupling coefficient for the damped cosinusoid responses in the retarded scattered E-field. T‘flxvc...“ \ / 16 we mmcoqmwm we?» vasexxz_vc¢.:£. can. :5 3.938“ > asea .upmaa1m cmgmuumum umugmpmm mum; cmcsmm< mo coaumcamaapH .~-N mgzmaa veo.:acJo 33w Jung/b (will A. N .\ «A. E\4 28.33% 103 2.2.3 3.3 in .Pu055. ‘31 $8 I . . N lllllllllllllllll o / I AI 0!" I .a.afwA~W/,/ Keepii coefficients ' or t > t". we the "late-ti” Sill time-done radar discrir tinnal technt 'class 2" ty; a'class 2" a difference 1 node. This and identity ”‘9 ASPECt-e 0” the basis titre of the “‘J ‘ -A \ He Slibstj tUtir “late the the z‘ilisis frm EN. fiua‘.‘ .17 Keeping in mind that both the “class 1“ and "class 2" coupling coefficients will give identical time domain results in the “late-time" or t > t", we need several more observations. First it is only in the "late—time" region that the aspect-angle dependent factors in the SEM time-domain expansion are constant. Our aspect-angle independent radar discrimination technique will exploit this effect. Our computa- tional technique, however, will be convolutional and will resemble a "class 2" type calculation. We will now look at the difference between a "class 2" and "class 1" representation. In the time domain the difference is obviously a time limited function for each individual mode. This is because of causality prior to the incident plane wave and identity of the two representations in the "late-time." Therefore, the aspect—angle dependence of the individual modes, potentially lasts on the basis of causality for a duration of t", the two-way transit time of the radar target by an incident plane wave. 2;3 Agpect-Angle Dependency of Radar Target Scattered E-Field We have used the EFIE to conceptually calculate the radar target current density. After solving for the current density, we may calculate the scattered E-field, ES(F,t), anywhere in space by substitution into equation (2-7). In fact, only the far-field E-field for one or two polarizations are all that are desired. We will cal- culate the E9 far-field pattern for a wire-like target aligned with the z-axis. The numerical value of the scattered field is calculated from equation (2-7) to give equation (2-19). iron aquatic associated s observance far-field p [2-9) reooi in eqaatior ’FE‘f‘e 18 iii-WES) - 5630.5) -‘6-V¢~(F,s) 5% ems) (2-19) It file—- + s Sine AZ(F,s) - From equation (2-19) we note the additional inverse distance associated with the scalar potential will, in general, eliminate its observance in the far-field pattern. Before we define the normalized far-field pattern, note that the presence of exp(-sR/c) in equation (2-9) requires our introduction of the retarded scattered field as in equation (2-20). A Eiet(?,s) = exp(leI/c) ES(F,S) (2-20) We will not define the far field Ee total scattered radiation field as given by equation (2-21). A Ee(e.o.5) = lirlI-rfl FENCES )75 (2-21) Now using only the first term in equation (2-19), we obtain equation (2-22). Ee(6;¢,s) = limit IFIS Sine AZ(F,s ) exp(s [Fl/c) lrl+00 =- a. s,51nefi—§r-f-K(F:s lexp(-SAR/c)dA' 42-22) A where AR = limit {IF'-?|-IFI} lrl+00 low the SE44 sc‘ that each mode scattered far . role as in Eq. £2435) = =s Si O. For a far fiel its naximn va the coupling c l” ion the Sur Emlln’i coeff localized pat I'7 a {1 L.) \ /-~ p _., ‘ C \_,/ l0r f 1 hi . M We see It sh the II d FA 19 Now the SEM solution for K(F,s) was given by equation (2-15). Noting that each mode has its own pole singularity, we can obtain the scattered far field of a single mode by the Cauchy residue about that pole as in Equation (2-23). Eg(6,¢) = Residue Ee(e,o,s) 3'23) (1 = sassne Tia]{00(i.sal$:(r')f(sa)exp(-saAR/C)dA' (2'24) A For a far field pattern, we normalize equation (2-24), typically by its maximum value. Note carefully that the pulse shape, f(s), and the coupling coefficient are not functions of the integration variable r' (on the surface, A). Assuming only that the pulse shape and coupling coefficient are nonzero, they will not be part of the normalized pattern Egn(e’¢) given by equation (2-25). Eula») _, e“ (e 4) = 9 = Sine g. thr1exp(-s AR/c)dA' On ’ a -—r—— a a max Ee(9.¢) sine A e m for f(sa)#0, na(n,sa)#0 (2'25) As an example, (ref 2-3,pp. 1609), the patterns of four modes of a thin wire scatterer are illustrated in Figure 2-3. It should be observed that the individual patterns are dis- tinctly different as a function of aspect-angle after the nonharmonic time dependence has been surpressed. For actual reception of a radar 20 180 target return i dazendence nusl based on its e 2.4 Eui er". Electroragneti The tr, fro“. an electr ties due to ti plane wave or rent is optic" radar pulse sr it is not in g or target. He FRI-allied tile 2' iii-u: for trg data lll this cl lifillmm at Ill The {J Figure 2.5, h" .3 n5 fall “I fix tEfidlng appwl alumni sel; All; I ““439. 4 FOP '1 ”OCES S 1 r. :lar . We Have 0r. 21 target return the nonharmonic time dependence and the aspect angle dependence must be decoded simultaneously to identify a radar target based on its electromagnetic invariant parameters. 2.4 Equipment Set-Up for Transient Electromaggetic Reception The theory of SEM has shown (Ref. 2-3) that the radar return from an electromagnetic scatterer can be decomposed into singulari- ties due to the scatterer and those originating with the radar pulsed plane wave or transmitted signal. A radar receiver in a benign environ- ment is optimized to enhance the signal-to-noise ratio of the returned radar pulse shape. It may also have provisions to reject clutter, but it is not in general optimized for the return of a specific scatterer or target. Hence we require a different reception technique to enhanced the discrimination process. Figure 2’4 shows the equipment set-up for transient electromagnetic reception used for the initial data in this dissertation. The data were collected by Dr. Bruce Hollmann at the Naval Weapons Center at Dahlgren (ref. 2-5). The transmitted pulse originates from an impulse generator, Figure 2-5, with approximately .2 ns rise time, .7 ns duration, and .3 ns fall time. The transmit antenna is an imaged conical antenna extending approximately 10 feet over an aluminum ground plane. These equipment selections are made to avoid adding singularities to the pulse shape, f(s), of the incident plane wave on the radar target. For processing simplicity it is desirable to approximate an impulsive plane wave on the radar target. . . (22% \ «2.5.... \ l (Eh! I 241-M .\ 22 as... Q _ .coapnmumm zu ucmamcmg» Lea unluom unmanascm .enm mgamau «0000 001319: A «55:3 LEE _ 3.5:: .8... .Paafiao. g: as ma 8» A — . _ 93... 9.33:3 2:65» — $32k 1"l” \\ I, Dcc G C M035 aqmtm RMPLITUDE 0-6 0-8 d 0-4 23 PULSE 0-2 0-0 T T I I I 1-0 2-0 3-0 4-0 5-0 6-0 TIME IN NRNOSECONDS Figure 2-5. Transmitter Pulse Observed in Transmission Line plane. gain, ar an the r satplin; response the anal: at 256 pr tires, 1C are quant only Sign uniform q "UFO-duce Eras 24 The receiving antenna is a TEM horn also imaged on the ground plane. This choice is made as a trade-off among usable bandwidth, high gain, and minimum blockage and distortion of the incident plane wave on the radar target. The radar receiver in this equipment set-up is a microwave sampling oscillosc0pe. The sampling heads of the oscilloscope have a response into the 12.5 GHz range. A microcomputer actually samples the analog output waveforms of the sampling oscilloscope. The microprocessor stores sampled data values of radar return at 256 precise discrete time values. At each of these 256 discrete times, 100 radar return sampled data values are averaged. All values are quantized to 8 bits. The quantized 100 sample averaging is the only signal-to-noise enhancement performed by the equipment. For uniform quantitization, the rms value of the quantitization noise introduced is (ref. 2-6) given by equation (2-26). 2 2-2 bits =-—— = (-6 bits - 10.8). 1 10 dB (2'25) 5!” 2 = E{nq} So the quantitization rms noise is at least -58.8 dB for an 8 bit A/D conversion. The spatial orientation of the transmit antenna, radar target, and receive antenna is important. For the bistatic radar data used in this dissertation, they are configured approximately as an equi- lateral triangle. Figure 2—5 shows the output of the impulse gen- erator directly into the sampling oscilloscope. Figure 2-6 is the time response waveform for the TEM horn directly viewing the transmit « lrlr... . r l I a lull c._ C.— UCDN ~ 1.07:... m. D O 1. Li... a. Cl Ally RMPLITUDE 0-0 25 TEMHORN 0-0 1 1 1 1 2.0 4.0 8.0 8.0 10.0 12.0 TIME IN NRNDSECONDS Figure 2-6. Radar receiving TEM horn antenna. antenna. fort. in 1 vaiefortr. 2.5 Proc amounts c noise. 1 repeatab‘, ground rs cuently l SIOTEge ( called a 26 antenna. It may be seen that even with the care provided, the wave- form in the receiving antenna transmission line is distorted from the waveform in the pulse generator transmission line. 2.5 Processing by Digital Clutter Map Any radar equipment set-up can be expected to have large amounts of radar clutter (Ref. 2-7, pp. 9) in addition to thermal noise. In the short-time period, the clutter is deterministic and repeatable. An early method of clutter rejection is to store the back- ground returns for a radar without a target present and then subse- quently perform a comparison with radar target might be present. The storage of the background radar return without a target is often called a "ground clutter map" (ref. 2-8, p. 403). Figure 2-7 contains three traces each of which is 256 sam- pled data points. One of the top two traces is labeled target plus clutter and noise. The other trade is labeled clutter. The target is physically absent during the measurement of this file of clutter plus noise. This clutter trace constitutes our ground clutter map in the direction of the target to be measured. The 8 bit quantitization is not obvious from these two traces. The processing performed is to directly substract the clutter map from the target plus clutter and thermal noise file. The bottom trace is the processed difference. Note that any drift in DC level or gain drift of the analog amplifiers can cause distortion. Figure 2-8 is the same processed file, but with an expanded scale. Now the quantitization and drift is visible in the file. It can also be calculated that the maximum number of bits used ll 27 .mmmgmmo me we upmc< uumnm< cm um umumcaszppn mcmpa ucsoao m m>oa< =NH spasm; =~H ac Hannah pmuagucapxu a mo mmcoammm om—aa umgammoz mazouumozcz zu uzuh o.mn o.mn o.w~ o.m~ 8.». o.m. cum o.o mmcoammg mmpaa r 9’0 BODIITJUU amuuapu C I 0'! Louuapu ucm mmcoammc mmpaa S‘I .a-~ masmaa 28 .ammcaa Lava“ «La: zaxm =NH uauauaa-aaaa=_u mozouumozcz z— E: .— .m-~ acsmaa 9mm 9....» 9m» 9...: 9m. 3.: own 9.... -m W .n. owe u o «35.. 33.3 a J. 3 Hoomnc\4~mz_z zmxw :N~ by the process end is only 3 46.8 dB to - The qr sarples adjace to conoletely or order of 0. 29 by the processed radar target return is a peak of 6 bits, but near the end is only 3 bits. Hence the quantitization noise varies from -46.8 dB to -28.8 dB. The quantitization is further exaggerated by differencing of samples adjacent in time. The rationale for the time differencing is to completely eliminate DC bias in the data and to reduce the degree or order of other distortions which may be present. 3,1 Resectic‘l In th far-field re: will receive antenna temi the antenna 1 scattered fie ingantenna'g lolarization Pair of polar miscllaPter CC"fitment. Hes “lid retarc taraft) The TA? (.2 ’3 Equall'On (2‘ CHAPTER III PROCESSING TECHNIQUES FOR EXTRACTING NATURAL MODE WAVEFORMS 3.1 Reception of a Radar Target E—Field In the previous chapter we developed expressions for the far-field retarded scattered E—field. The radar receiving antenna will receive a portion of this scattered E-field and deliver it at the antenna terminals to a transmission line. The response observed at the antenna terminals depends upon the polarization match of the scattered field and the radar receiving antenna as well as the receiv- ing antenna's response for a particular direction. We can account for polarization effects if we always calculate or measure an orthogonal pair of polarizations for the scattered E-field and the antenna. In this chapter we shall perform these calculations only for a e-polarized component. The orthogonal component procedes identically. We shall denote the e-polarized far-field radar target scat- tered retarded E-field by equation (3-1). tare(t) = L'1(TAR6(B,¢,S)) (3-1) The TARe (e,¢,s) is the frequency domain expression we obtained in equation (2-21) for this particular radar target scatterer. 30 In a 5 plane wave in: aTEllhorn. b: 0 tenrzit) = This tine 13v. antenna temi from directio antenna bores isoulsive res the frequency 43-1). It st illicit is lllCi tili‘oljgh the E by performS llll sinultar transmitted l and nojSE FEE “litter“??? Giver. by enue 31 In a similar notation, we shall represent the 9-polarized plane wave impulse response of our particular radar receiving antenna, a TEMehorn, by teme(t), in equation (3—2) teme(t) = L'1(TEM6(6',¢',S)) (3-2) This time TEM6(9',¢',s) is the frequency domain expression for the antenna terminal response to a unit e-polarized impulsive plane wave from direction (e',¢'). We shall hereafter assume that the radar antenna boresight is oriented toward the radar target. That the impulsive response of a planar aperture antenna can be obtained from the frequency domain antenna response is well developed in reference (3-1). It should be observed that the transmitted plane wave pulse which is incident on the radar target should also be received at least through the antenna sidelobes by the radar receiving antenna. However, by performing the ground clutter map processing of Section 2—5, we will simultaneously eliminate the sidelobe leakage of the originating transmitted plane wave pulse, Einc(?,s), from the target plus clutter and noise response of the radar receiving antenna. We will denote the clutter-suppressed radar target antenna terminal response by v(t) as given by equation (3-3). +oo v(t) ajr teme(t)tare(t-T) dT = L'lfTEMe(e',¢',s) TAR6(6,¢,S)} (3-3) line will receptior upon the the inci: 3.2 Sac radar t2 tine-in option CH‘JCTU lm’arfa 47:30 0‘ 32 As we can observe from equation (3-3), the antenna transmission line will contain natural mode waveforms from the radar target, the reception antenna, and less obviously from the plane wave incident upon the radar target. In our notation the possible natural modes of the incident plane wave pulse were not separated from TAR6(9,¢,s). 3.2 Sampling_the Radar Antenna Response So far we have characterized both the scattered E-field of the radar target scatterer and the receiving antenna response as linear, time-invariant processes with Laplace transforms. This allows us the option of computing the composite response either by a time domain convolution or a transform domain product. There is one process we shall use that is not "linear, time- invariant." It is our sampling process or A/D conversion. The opera- tion of sampling is characterized as a multiplication or modulation (ref. 3-5, pp. 30). As a mathematical computation there is nothing new. Only the roles of the time domain and the frequency domain have been switched. This time domain product becomes a convolution in the frequency domain. For uniform sampling, we might characterize the system as linear, frequency-invariant or linear periodically time- varying. Although taken alone the sampler may be mathematically simple, we must use it in combination with linear time-invariant processing and electromagnetic scatterers. This makes both the time domain and frequency domain representations more complex conceptually but Tw" - - generall ing conc reviewir (3-4) 1'. mltipl; Equatio terina ETC 33 generally more convenient computationally. For the purpose of gain- ing conceptual fluency, Appendix B is recommended reading after reviewing the notation of Appendix A. There are two equations we will use extensively. Equation (3-4) is simply the definition of the sampler of period T which will multiply the antenna terminal response. 400 6T(t) = _z S(t-nT) (3-4) n_-w Equation (3-5) is the output of this sampler placed at the antenna terminals for our radar target scatterer. antT(t) = 6T(t)(teme(t) * tare(t)) (3-5) 3.3 Processing a Scattered E-field Reception Our next objective is by means of simple processing of the scattered field observed at the terminals of the reception antenna to obtain a single, j-th mode waveform. This could be either a purely real natural mode waveform representation of one of the complex natural mode phasor waveforms of an equivalent representation. Our processing at the antenna terminals will be a weighted summation of N sampled- data values of the retarded far-field reception antenna response. The form of this process for a conventional radar is illustrated in Figure 3-1. In our experimental set-up of Figure 2-4, the sampling occurs in the sampling head of the sampling oscilloscope. Any con- tinuous representations of the retarded far-field antenna response In ecu. of the functi: antenna in; fur Present Proce 5 U1 34 are stretched and/or filtered versions of these sampled data values. We may represent this process by equation (3-6). . N-l . vJ(t) = p(t) * z ai 6(t-kT) * antT(t) (3—6) In equation (3-6), the function p(t) represents the final smoothing of the output waveform after or during the processing. The impulse function S(t-kT) is, of course, synchronized with the sampler on the antenna terminals. For convenience, we will always choose the smooth- ing function, p(t), of finite duration in time. When the retarded scattered E-field from the radar target is present at the antenna terminals, we may represent the j-th mode processed antenna response, v3(t), as in equation (3-7). - NT+ o VJ(t) =f eJ(T) antT(t-t) (1': (3'7) The excitation function, eJ(t), is easily identified from equation (3—6) and is given by equation (3-8). - N-l . ejct) = ego) * P(t) = p(t) * z a k=0 6(t-kT) (3-8) In a similar manner we will define the pulse shape independent sam- pled mode response vg(t) by equation (3-9) and its transform by equation (3-10). V: 35 N 1 vg(t) = 2 a3 S(t-kT) * ant(t) (3-9) k=0 A- N‘l - A V%(s) = 2 ai exp(-sTk) ANTT(s) (3-10) k=0 Before we make any simplifications in the sampled antenna terminal response, we need to make a generalization which is necessary in circumstances when more than one file or data set are used together. Typical cases which require synchronization of data are deconvolution and comparison of more than one data file. For the real radar problem we do not know the synchronization of the sampler relative to the analog retarded antenna terminal response. To specify this synchroni- zation, we shall use the modified z-transform notation of Appendix A. The use of m as an argument in parenthesis shall denote the use of this parameter. Equations (3-9) and (3-10) become equations (3-11) and (3-12). . N-l . v3(t,m) = z afl C(t-kT) * antT(t,m) (3-11) k=0 . N-1 . _k VJ(z,m) = Z aJ z ANT(z,m) (3'12) k=0 k 3.4 Excitation of a Single Natural Mode Waveform The objective of this section is to determine the conditions under which the output convolution of equation (3—13) can be a single natural mode waveform using N coefficients {ailt;3. ww 1 JI; v i" I d“ 5,! At this pc response 1 using n = r art,. “ills szr ) - 4 -r' it) Flew-1 actually ch. . enae 0 s44... 5'“ "Ip- . - ,.,_ ‘5 afi- I *PEnde 36 N-l X J (t. l = vd m (k=0 aia(TekT))'antT(t-T,m) a. (3-13) At this point we will simplify the sampled scattered antenna terminal response by assuming that the receiving antenna is nondistorting and using m = 1 as given by equation (3-14) and (3-15). antT(t,m=1) = 6T(t)tare(t) ' (3'14) . N-1 . vg(t,m=1) = (kgoaio(t-kT))*(6T(t)tare(t)) (3-15) Now tare(t) is more succintly known by equation (2-22) for a wire- like target. A U -> —> —+ TARe(e,¢,s) = s Sine EN] K(r',s) exp(-sAR/c)dA' (2-22) A Using SEM the model form of the solution is given by equation (3-16). TAR646.¢.S)= Z (s-sal'l E(e,¢,s) (3-16) (I The Laplace transform of the sampled antenna terminal response is actually the z-transform of the antenna terminal response after the change of variable from s to 2. Due to the simple pole form of the SEM model solution, the evaluation is performed term-wise as in Appendix A to yield equation (3-17). lie Wll ‘ drain in the (1' hate ti -' I .J Crate; 37 TARe(G,¢,z) = i (1.20/2)’1 E(e,¢,z) (3-17) We will now rewrite the simplified equation (3-13) in the frequency domain to illustrate what excitation of the single j-th mode implies in the frequency domain as given by equation (3-18). A. N-1 x var.) = 2 as exp(-sTk) 2 (l-exp((s -s)T))’1 c(e,o.s) (3-18) k=0 k a a Note that we have suppressed the entire function which may be asso- ciated with the infinite index set, a. From equations (2-27) and (2-18) we know that in the time domain, tare(t), can be calculated from the simplified current density coupling coefficient in the "last- time" of the retarded far-field scattered E-field. We note that tare(t) is composed of terms of the form given by equation (3-19). -1 -1“ _ u _ L {(s-sa) Ce(e,¢,s)} — u_1(t-t )Ca(e,e)exp(sat) (3 19) We shall use this new representation in tare(t) in the "late-time". Also, we have used this opportunity to change from the SEM vector containing the aspect angle dependency of the incident plane wave on the radar target with the more conventional spherical angles (9,¢). We then obtain equation (3-20) for the "late-time" representation of equation (3-18) now in the time domain. 38 N-I vg(t) =kEO ai 6(t-kT)* g ut1(t-t")exp(sat) Ca(e,¢)r6T(t) for t-nT > t“, n an integer (3-20) We will not be able to achieve our desired objective with equation (3-20) as it stands. The change we shall make is to trun- cate the infinite summation on the index set a. This is a reasonable approximation in the late-time for all targets which yield “class 1" coupling coefficients in the retarded scattered far-field which are absolutely convergent. In this case we obtain equation (3-21) which is the fundamental equation for signal processing of the radar target return. We will denote this finite sunmation by denoting the index set by m instead of a. . N-1 . N J _ 1 - . vd(t) - a ak 6(t-kT)* a u_1(t-t )exp(smt) lee.o)5T(t) k-O m-I t - nT > t" (3-21) Recall that our star notation denotes the convolution operator in the time domain. For t in the "late-time“ the step function u-1(t-t") disappears and this allows a simple evaluation of the integral opera- tor. In order to formalize in a matrix form we shall calculate the convolution of the processing with the coefficients of the sampler output to obtain equation (3922). He no ofa. I0 de 39 (t-kT)*exp(smt) .[6(T-kT)exp(Sm(t-T))dT exp(smt)Mmk (3-22) The scalar constants Mmk are given by equation (3-23). Mmk = exp(-smTk), m = 1, . . ., N; k = 0, . . ., N-l (3-23) We note, in passing, that these scalar constants are functions only of a single natural frequency and the sampling time kT. Next we wish to define a natural mode waveform Cm(e,¢,t) by equation (3-24). Cm(e,¢,t) = Cm(e,¢)exp(smt) m = 1, . . .,N (3-24) We will now rewrite equation (3-21) in terms of the quantities defined by equations (3-22) through (3-24) to give equation (3-25). . N-l . N vg(t) = u_1(t-t") Z a: Z Cm(9.¢.tl Mmk6T(t)’ t=nT > F (3-25) k=0 m=1 Let us now suppose that we desire the processed (pulse-independent) radar target return to be samples of a single "late-time" natural mode waveform of the form given by equation (3-26). 3 - _ u . vd(t) - Cj(e.¢,t)5T(t) t_nT > t + NT, h an 1nteger (3-25) We note that in equation (3-27) we have exactly N unspecified con— stants to be used. If use exactly N sampled data values of the received waveform, we can potentially solve for the unknowns {afl}:=o. In or: rota: 50 in ”rich 40 In order to facilitate a matrix formulation, we need to compress our notation a little further as given by equations 0, 1, . . ., N-l (3-27) v: = vJ(nT), n C = C ( , ,nT), n O, 1, . . ., N-l; m = 1, . . ., N (3-28) IlITI Ill We may now write N sampled data values of v3 as a column vector [3%] nd express equation (3-25) as the matrix equation (3-29). ('3' .' ' J' - LVn: [Chm] Mmk [3k] (3 29) Upon observing the structure of equation (3-29), we observe that we will obtain equation (3-30) if equation (3-31) is satisfied. j - = - . ‘ _-_ _ vn - an, n 0,1,. . .,N 1, J 1,...,N/2 (3 30) no“ 0 M‘ ' k j . . I = m [ak] (one 1n j-th row) (3-31) 1 Loul So in order to excite the desired natural mode wave form, we need concern ourselves with the matrix of frequency/sampling constants Mmk which are independent of aspect angle. exci coat 4‘ r 1 5E. 4: Sim” 41 Hopefully, the reader will recognize the form of equation (3-31) and note that the {a3}:;3 may be obtained as the j-th column of the inverse matrix of Mmk' Writing out all N natural mode wave form excitation vectors in matrix form, the relationship is more evident in equation (3-32). .- 1 N . r q [Mmk] a0 ... 60 l: 10 ... 0 E E 0 1 ... 0 (3-32) 1 N : LaN-l aN-L L0 0 co. 1.: Hence if we can obtain all of the natural mode waveform excita— tion vectors, we have, in fact, the inverse matrix of Mmk which is evident from equation (3-32). We may now conclude that if the matrix of frequency/sampling constants is invertible, then we can obtain excitation coefficients for each of the finite number, N, of natural mode waveforms. In fact, we are able to separate them in the "late- time". It should be observed that whether or not the frequency/ sampling constants matrix is invertible is determined by the collection of natural frequencies and sampling time and is independent of the aspect-angle of the target or the amplitude of the received waveform. We will later observe that we may be able to obtain the excitation coefficients for natural mode waveforms even when the matrix of fre- quency/sampling constants is not invertible. The procedure is similar to the K-Pulse Singularity Theorem of Appendix C and is performed automatically in the "fast" Prony's method of Chapter 5. ecstatic 42 For the case when 1:"ka is invertible, the j-th mode excitation coefficient vector is given by equation (3-33). We shall call the vector on the far right of equation (3-33) with the 1 in the j-th row as the selection vector. .It simply selects on the columns of the inverse matrix of frequency/sampling constants. , -1 0- j k 0 . (3'33) [ak] = m 3 (one in j-th row) 1 L6. Now suppose we desire to extract a real natural mode waveform. We may, for example, select the lst cosine natural mode excitation by remembering the order in which we placed conjugate natural frequen- cies in the matrix of frequency/sampling constants. In our case they are index staggered by N/2, so we obtain equation (3-34). feel . [er] = [4.11 ‘31 am. shim/2F .. [as] a». L 33 A similar relation holds for the lst sine natural mode excitation. Similarly, we may obtain the lst sine natural mode excitation by equation (3-35). (A) (J1 43 n! X’ H M L___J II I—'—7 3 a x. L__-J l ...: ----...... r 0 II or n! X- H L___l I to“ Km 1 ...—a + z \ c__'3. A w l 0.2 01 V Superscripts s and c shall denote these real excitation coefficients for the ej(t) of equation (3-8). 3.5 "Polar Mode A-Scope" Displays One of the fundamental radar displays is called a radar A-scope. This is a retarded transmitter triggered display of the processed received target response as a function of time, generally at baseband (carrier removed). The significance of the display is that for a target much stronger than clutter, the envelope of the transmitter pulse is often viewable although it is altered by the radar system response and the radar target. The time delay of the triggered radar return behind a replica of the transmitter pulse envelope gives the radar range estimate. lf_it were not for the extreme aspect-angle dependence of the radar return, one would be tempted to use the amplitude of the radar return as an estimate of target size. Figure 3-1 is a radar A-scope display of the response of our experimental radar system, Figure 2-4, to a hemispherical radar target imaged on a conducting ground plane. For completeness, the TEM horn antenna associated with this particular radar target data is given by Figure 3-2. HERSUREMENT RHPLITUDE IN VOLTS 44 19“ SPHERE - 1.0 2:0 3:0 4:0 5:0 Bl-O 7.0 TIME IN NRNOSECONDS Figure 3-1. Clutter—reduced 18" Spherical (imaged) Radar Target Return. MERSUREHENT RMPLITUDE 45 8-0 TEM HORN FILE: NINETEEN7O 4-0 2.0 l-O | I I I 3-0 I 1 r’ T 0-0 1.0 2.0 4.0 5.0 8.0 7.0 0-0 TIME IN NRNOSECONDS Figure 3-2. TEM Horn Receiving Antenna used for Spherical Radar Targets. plays 4 cririne for i first ratil 46 We are now ready to develop the special radar A-scope dis- plays which are destined to play a crucial role in the final dis- crimination technique of this dissertation. In order to describe the display, we shall in this section ggly use a synthetic data file for our target response at the antenna terminals. The data is created by equation (3-36) which we shall call a "prony series". S(t) = mgl cm exp(smt), t = nT > O, n an integer (3-36) For the natural frequencies in equation (3-36), we shall choose the first 10 complex conjugate pairs of a 6" wire with a length-to-radius ratio (L/a) of 400. Hence N is 20 and our excitation technique of section 3.4 yields an excitation coefficient vector [Efl:l of 20 ele- ments. Convolving by equation (3-37) the 2nd cosine mode excitation (j=2) with the synthetic "prony series" we obtain Figure 3-3. There is another real natural mode waveform which looks quite similar to this. Convolving by equation (3-38), the 2nd sine mode excitation with the synthetic "prony series" we obtain Figure 3-4. NT ' . . (3-37) I each) S(t-T) (IT Aj(t) T+ is . ([N e (T) S(t-T) dT (3-38) Bj(t) EMPLITUDE DHPLITUDE iill W) 6" HIRElL/Rz4003 2ND MODE SIGNHL 3 I F'*' late-time period convolved output A(t); .N\ synthesized signal for.~ exc1t1ng c051ne mode .700 1:3 4:0 0:0 01.0 1T0.—0 15.0 14.0 318-0 ' HDRHRLIZED TIHEITIIL/CI) Figure 3-3. 2nd Cosine Mode Convolution. 5" HIREIL/Rz4DD) 2ND MODE SIDNRL I 2'1 I... late-time , period 24 i ii 24 convolved output s(t) a "II l3 .4 l as E v synthesized signal for -. cl exciting sine mode émo zfo Jo 0:0 010 10.0 15.0 11.0 15.0 NDRHRLIZEO TIMEITIIL/CII Figure 3-4. 2nd Sine Mode Convolution. 48 It is important to note that each of these convolutions appear as the expected damped cosinusoids starting at t = 2.04 normalized time. It takes that length of normalized time to accumulate 20 samples of the "Prony series" in order to satisfy the conditions of independent data samples under which equation (3-32) was derived. Also visible as pulses on the left-hand side of Figures 3-3 and 3-4 are the j-th mode excitation coefficient vectors for the 2nd mode. The two convolved waveforms Aj(t) and Bj(t) are what we shall call a rectangular j-th mode radar A-sc0pe plot. We shall now perform a seemingly simple transformation of these two rectangular mode radar A-scope plots into a polar form as given by equations (3-39), (3-40), and (3-41). C(t) = Alt) - jB(t) (3-39) env(t) = Re(clog(c(t)) (3-40) rot(t) = Im(clog(C(t))- conts. phase (3-41) where clog() is the complex logarithm with phase made continuous. Figure 3-7 shows the envelope, env(t) as a function of normalized time with a straight line of slope equal to the damping coefficient of the 2nd natural frequency of the synthetic "Prony series". Note that the envelope does not meet with the straight line until the 20 nonzero samples have been accumulated. The last equation of the transformation yields what we shall call "rotations" which is an analytically continued phase of the com- plex logarithm. Only the continuation operation is not easily NRFIERS ROTHTIONS -5000 4.0 000 l 49 6" HIREIL/R:400) 2ND MODE SIONRL \ Rennlcmli tui=Mo-54ui (fizt ‘ I r 1 I r 1 0.0 5.0 10-0 15.0 20.0 26-0 30-0 35.0 WURHRLIZED TIHElT/(L/Cll Figure 3-5. 2nd Mode Envelope Radar A-SCOpe. 5" NIREIL/R:4DOI 2N0 MODE SIGNHL 1 l E? n 6mm I I T r I l’ 0 .0 5.0 10.0 15.0 20.0 35.0 30.0 35.0 NORHRLIZEO TIHEtT/tL/Cil Figure 3-6. 2nd Mode Rotation Radar A-Scope. Dita? ”or; d)!- y. w 0; *‘er5i 557:15 50 implementable in either analog or digital hardware. However, we do not need the exotic phase unwrapping algorithms necessary for cepstral analysis (ref. 3-6). This is because we only have a single mode remaining rather than a large sum of complex exponentials whose phase is to be continuous. To continue the phase we make only a trivial logical comparison of the previous sample throughout this dissertation. It should be noted that only the starting point is not unique for "rotations." The radar A-scope plots of rotations yields some interesting information not observable in the previous three plots. There appears to be ng_early time transient in this plot. Starting with sample one, the "rotations" plot align with the straight line whose slope equals the minus imaginary part of the second natural frequency of the syn- thetic “Prony series." The display of equations (3-40) and (3-41) shall hereafter be called the "polar mode A-sc0pe" display. We shall expand upon this concept in Chapter 6. 3.6 Measurement Performance of Single Mode Waveform Extraction Technigue We shall now attempt to extract a single natural mode waveform from the radar target scatterer of Figure 3-1. Our objective is to obtain the j-th mode real natural mode waveforms Aj(t) and Bj(t) which from our synthetic data of Figures 3-3 and 3-4 we now can resemble damped cosinusoids for normalized time for which only nonzero data samples are used in equations (3-37) and (3-38). These results are for tin r) as r F\U u. .... 51 for a "Prony series" which we cannot expect unless the following three conditions are met on the incident plane wave plus shape, the receiving antenna response, and the "late-time" target far-field response: (1) f(t) (2) tem (t) 6(t) from equation (2—15) s(t) from equation (3-14) (3) tret > t" (2-way transit time) from equation (2-18) All three of these conditions are troublesome. The "late-time" condi- tion would be easy if the first two were satisfied. Figure 2-5 shows the originating pulse shape and Figure 3-2 shows the receiving TEM horn antenna response. We will perform a slightly defective correction to meet these conditions. A digital deconvolution of the receiving antenna response, Figure 3-2, will be performed on the clutter-reduced radar target response, Figure 3-1. Deconvolution is difficult on measurement data because one cannot obtain the sampling synchronization (which is auto- matic with synthetic files with subsequent additive noise) which is required by equation (3-13). The deconvolution technique we will use is the discrete time Least-squares Wiener filtering of reference 3-7. The results of our deconvolution are shown in Figure 3-7 for a pre- whitening parameter of 5%. A good error analysis of deconvolution is given in reference 3-8 of the "Deconvolution" collection of the Geo- physics reprint series. This technique for extracting the natural mode waveforms depends upon a priori knowledge of the natural frequencies. Used in 52 0.3 19 " SPHERE: IQLHMBDRS 0.2 0-1 HHPLITUDE '? 02 -9 l -003 l 9 O . l T I T l T ‘l 0.0 2.0 4.0 0.0 0.0 10.0 12.0 14.0 NORHRLIZED TIHE Figure 3-7. Spherical Radar Target Response by Deconvolving Figure 3-2 from Figure 3-1. the pair decc ham nat' na: re: lid i‘ /-:3’ ID 53 the synthesis of these excitation waveforms are 19 natural frequency pairs from reference 3—9. Figure 3-8 shows the results for the deconvolved 19" spherical radar target of Figure 3-7. 0n the left- hand side in sub-figures (a) and (b) we have the desired waveforms A1(t) and Bl(t) and also just below on the same plots, the corre- sponding excitation vector, [eié:land [EKIS , for each of the desired waveforms. 0n the right hand side we have the plots we have defined as the "polar mode radar A-scope." In these two plots we can clearly see that in the time period (of nonzero samples in the convolution), the output data displayed is clearly parallel to the correct damping line olt in the envelope display and the correct rotation line wlt in the rotation display. This is positive confirmation of a pure natural mode waveform in this time period. The results for the 2nd natural mode waveforms are in Figure 3-9. Again the "polar mode radar A-scope" plots give a confirmation of the desired natural mode waveform in the output convolutions. Figures 3-10 and 3-11 give positive confirmations for the 3rd and 4th mode natural mode wave- forms. As this was one of our earliest targets, more desirable longer data sequence was not taken. 3.7 Sgaling of the Invariant Radar Target Parameters In the previous two sections we have presented displays of the invariant parameters of a radar target for the cases when the invariant parameters were known a priori and we excited these known parameters. We must also be interested in what happens when a '0‘ J (a) '0‘ L late-time period k_¢. 0.0 A In. 0 1. l I I 54 ... '4 ‘4 I do. 000. l (C) I» I . k~‘\\\\::f; -1 late-time . “We“? I period Sé‘ convolved 3;‘ '“u 2_ output 2 L \ a. A(t) ;. Re{ln[C(t)]) .. , C(t) = A(t) - .i B(t) V“, excitation vector ?‘ ' phasor OUtPUt for lst cosine 3.! .5 "ll'mode waveform g. I .. H H' 0. 1.0 010 0.0 05 030 03.0 00.0 00.0 00.0 '0.0 010 0'; 030 010 00.0 00.0 01.0 . NIIRRLHEO '"fli MREU "M 2- (b) 0.. (d) 2‘ convolved ' 1 2‘ 'r_... ate-time ' output . period 2" s(t) | 2“. I ' . .' _,4 late-time go go. . periOd 5°" ' g t‘ a; a. 0' id ' H lluuw \ .- “Vs \ ' q l excitation vector , “\\\‘ 0‘ for lst sine 0‘ ~w]t mode waveform ' 3 '3. 10.0 0.0 T0 0:: 030 03.0 05.0 0540 03.0 70.0 030 03 01.0 01.0 7.0 30.0 05.0 0'0 ”RMLIIEO TIRE ”MILHEO "M t/(a/c) t/(a/c) Figure 3-8. lst Mode Waveform Excitation of a 19" Diameter Sphere. 55 3‘ (a) 3- (C) 2- 2! ab " . convolved \fib‘. I 1 ' - 0. ate-time .“ output 0" 0‘50 “"2 period A(t) 4.? . 9 .04 :1 g !-—o- late-time g. 33 :‘l I PeriOd 2'" 'a l ' g i Re{ln[C(t)]} E 2. I ' i :4 \\ o I z “‘5. -* I vi 00:) = A(t) - 3 S(t) .2. -. excitation vector , ' P0350r OUtP0t 3. for 20d cosine g. mode waveform g 5... .2. .'.. 2. .5 r... .3. a. .3. -... .1. .1. .1. .1. 6.. .1. 6... MM"!!! "ll! "BIL"!!! "It! 9- (b) -. (d) a g i. - 6‘ convolved .4 -—v’, ‘3“ time output ;: / S(t) ; g . .—-- late-time g. 3 2‘ period 5.5. ... g. E 2- in z :- 3 \ excitation vector 0 for 20d sine 2 mode waveform 3 7010 050 050 7.0 05 7.0 00 .0 01.0 00.0 ‘0; 0:0 0:0 0:0 0'.0 We 00.0 01.0 MHILIIID "I‘ll MRILIIEO "ll! t/(a/C) t/(a/C) Figure 3-9. 20d Mode Waveform Excitation of a 19" Diameter Sphere. 56 2' (a) . ‘ (C) .. . ' _ 4% _ convolved . ‘\\\\ «9% .. / a \\ 0 I .0. A... " . output 7 qhh, ”’2 late-time A(t) g 1 period E !._..-. late-time 00 : :3. | period §;. . g . s ‘y. z- ; Re(ln[C(t)]} \\ 0" 0. l C(t) = A(t) - J S(t) = phasor output -\\; q z. - a °3t for 3rd cosine . mode waveform . z T I i T W U I . I l ‘I r v.. .;'. 1... ‘0.0 0.0 0.0 0.0 {2.20 "an 00.0 00.0 00.0 0.0 0.0 0.0 0.0 ”01:“ it»? 0 2- (b) e. (C) 9 convolved a“.. °' output ' t late-time s(t) 9 0 period .0. '11 . !—-0— late-time 3:4 I periOd ;' E E °.' 1;. .- . “a, .\<'“bt v <0? ‘K\\ excitation vector 3. for 3rd sine mode waveform 3 T . . . . 4. '03 030 3.0 0.0 {2:0 "3.0 30.0 «.0 0L0 0.0 0.0 ... '83....x'2‘... "auto-o 00.0 «.0 t/(a/C) ' t/(a/C) Figure 3-10. 3rd Mode Waveform Excitation of a 19" Diameter Sphere. 57 2' (a) 3‘ (c) 2‘ convol ved 3i}. ' / - output ’4. . . .u A(t) l q -"1 late-time ' '-——' late-time ... ' period ! period . 0 a: . - l *- g. | 3.. : 3." . ”V' I E ' 0&3 \ l ., e . 7‘ r Re{1n[C(t)]} ' " '. g. . \ 7‘ ~ c t = A1: - B \ excitation vector ( ) ( ) J (t) g. for 4th cosine ; " phasor OUtPUt o t mode waveform 4 \ 7.0.» {.0 if. 01.0 010 00.0 00.0 01.0 00.0 70.0 030 010 01.0 01.0 05.0 00.0 03.0 WIIED "It! ' man "I'll 3' (b) 3. (d) convolved . - 2- / output :4 "'"l 3303.... 3(t) I : 1.". -—v- late-time -.1 s" I period ' : § ' § ' ..e. :‘73. «Q» : g .2- ‘0, "\ -u t '2 9 4 I V €0,‘\\/ Wt .'\\ ° ,2 \. \ excitation vector . ' 3. for 4th sine :1. mode waveform s 2 ?0.0 050 0.0 010 010 00.0 0L0 01.0 00.0 '0.0 07 30 0:0 010 05.0 00.0 03.0 0 WLIHO "HI mum "It! t/(a/c) t/(a/c) Figure 3-11. 4th Mode Waveform Excitation of a 19" Diameter Sphere. 58 natural mode waveform is present, but it is not the one of the target modes that we are looking for. Our derivation so far gives us no clues in this case. The only thing we know for certain is that when we are exciting the j-th mode waveform, the (j+1)-th waveform will not be passed. The "late-time" suppression of the (N-l) modes not excited depends on many items. For our implementation this is known to be approximately 210 dB. Only on the sensitive envelope plot of the “polar mode A-scope" is the extent of this suppression viewable. Again, it must be emphasized that if the undesired mode waveform is not exactly one of the natural frequencies used in the matrix of frequency/sampling constants, equation (3-34) we have not yet developed a method of calculating its suppression. In this section we will examine the effect of small departures of the natural frequencies from the values used in the frequency/ sampling constants matrix. For our baseline we will use an 18" wire (inclusive of the image) for a radar target scatterer. The particular wire is actually making good electrical contact to our conducting ground plane at normal incidence. The natural frequencies of the wire are less damped than that for the sphere. This means that the natural mode waveforms of the wire target are relatively strong in the "late- time" making deconvolution of the receiving TEM horn antenna of dubious value. Figure 3-12 shows our clutter-reduced radar target antenna terminal response. The length-to-radius of this wire target (L/a) is 400. We may readily calculate by method of moments 10 pairs of natural frequencies which we shall use in the frequency/ sampling constants matrix of equation (3-34). As a note to be 59 .wmcoammm pmcwscm» mccmuc< “match Leena =m.mfi umu:uma-gmuu=pu mozoumwozmz 2” quh o.._ o.~— o.od ope ova ow. o.~ .NHIm mszmwm 9.9 f'Z' 9'1- Hoovn¢\jmmHz ..w.mH BflnlITJHU 1 9'0 9'1 l 9‘2 60 justified in the latter chapters, we choose the sampler period to be such that the total length of the excitation vector is greater than t", the maximal 2-way transit time of the target, of equation (2-18). Figure 3-13 shows the radar A-scope plots for the lst natural mode waveform excitation of the correctly-sized (and no Doppler) target. The rectangular plots on the left-hand side illustrate the expected damped cosinusoids starting at the retarded time for which nonzero data for the full length of the excitation vector occurred. The polar plots on the right-hand side give a very sensitive indicator of the purity of the lst natural mode waveforms. The double triangle pointers on the dashed damping mode linear, separated by exactly one excitation vector time length. From the envelope plot in the upper right hand corner, we can identify nearly pure natural mode waveform occurring starting at approximately 0.5 normalized time units past the end of the excitation vector. There are about 0.5 normalized time units of zeros at the beginning of this file. Next to be observed is the rotations plot on the lower right-hand corner. In this case, we can detect a constant slope which is close to the exact rotation line which is dashed. The measured rotation line is slightly more negative. From a comparison with data to follow, we could estimate the wire to be about 2% shorter than the length we are using for our frequency/sampling constants matrix. A 2% error in reading the ruler is not unreasonable. This constitutes our baseline A-scope plots for the lst mode excitation of a correctly-sized cylindrical radar target. 18.6' "L"!!! 61 WIREiL/R:400) IST MODE 18.6" HIRElL/R:400) lST HODE I I-0 18.6" "L"!!! v v v 1 ... On. Is. I-. NORMLIZEO "HEN/HIE!) r 0.0 HIRElL/R:400) lST "005 00101 I” -!.0 -0.0 ..o. a .2 .0 '-r. Y I I I 1 .0. Cu. '0. ... 1.. mum TlfllilllL/CH ‘bqa.s~ HIREIL/R:4DD) 151 none ‘0‘. .-1P-0 T .0 l-O Figure 3-13. V T fl T f I 0.0 0.0 0.0 0.0 0.0 7.0 0.0 0 muzeo TlntlT/lL/CH lst Mode Waveform Excitati " I ‘ Y 0.0 do 0.0 0.0 0.0 7 .0 mum "MU/(L10!) I .I to. on of Correctly-Sized Thin Cylinder Baseline Radar Target. 62 Next we shall scale the clutter-reduced radar target antenna terminal response by 10% physical undersize, but retain the excitation vectors for the correctly sized radar target we have just viewed. The undersized (and oversized) radar target antenna terminal response files we will be using are created from the original correctly sized sampled data files. The first step is rescaling the time coordinate of the samples. Next cubic splines (references 3-4 and 3-10) are used to obtain sample data points in synchronism with the excitation vector samples. Figure 3-14 displays the lst mode excitation for the original sized target processed on the 10% undersized clutter-reduced radar target antenna terminal response. The rectangular radar A-scope dis- plays are different by heuristic examination by eye, but not by a quantifiable amount. The polar mode radar A-scope gives a more quan- tifiable identification of a dominant natural mode waveform. It should be noted that the rotation plot of Figure 3-14 has a constant "late-time" slope. However, this constant slope differs from the expected slope for the correctly-sized radar target by a slight, but definitely greater, negative slope. The envelope plot has more ripples than that for the baseline, Figure 3-13, for the lst mode. Let us now examine Figure 3-15 which is for a 10% oversizing of the original baseline response of Figure 3-13. The rectangular plots of the radar A-scope are again similar but difficult to inter- pret. They are difficult to interpret because one cannot use either the amplitude of the mode waveform or the time delay of the mode 63 -. 0. on on. '51 3‘ :4 o '1 o 8 a 9 0 2: 9-4 .. ° é’ ' 0 ¢ ". .4 I °. 7'1 "3 --i I e e A" ‘r fir v 1’ 1 v Y fl :. 1 I I Y T Y Y I 0-0 1.: 8-0 0.0 0.0 0.0 0.0 IL0 0-0 0.0 1.0 0.0 0.0 0.0 0-0 0.0 7.0 0-6 wonneleto TIHEll/lL/Cll wonnnLIZEO llflfll/lL/Cli c: 0 0w- .“ '2 o _‘ 511‘ ° '2 ..'.'l ’4 '2 0“ --i u s i — q _ '. 01-: o- h .l g s o. 00 9-4 .1 ‘ 0 a. o 0 0 e 9 N. 7‘ ..1 x e -. ' 1’ T I T 7 Y Y fl T Y Y T V Y I T ‘ 0-0 1.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 LO [.0 0 0 0.0 0.0 0.0 7.0 LC NDRHHLIIED YIHEIT/iL/Cll Figure 3-14. Cylinder Radar Target. nonnatizto rlntttle/CI) lst Mode Waveform Excitation of 10% Undersized Thin 4 64 0.0 |.0 0:0 0:0 0:0 0T0 “O'HQLIIID 'lfl‘lY/iL/C)) “HPLI'UOF IOTITIOIS fl 0.0 41:; 0:0 0:0 0:0 070 40:0 v.0 IURHHLllCP TIHCCY/(L/CI) 0.0 :10 0:0 0:0 0:0 0:0 NORHR-IZED llflfii/(L/CI) Figure 3-15. Cylinder Radar Target. .0 40:0 0:07 0:0 Filo 0:0 0:0 1:0 nonnatlzto Tlfliil/lL/Cll lst Mode Waveform Excitation of 10% Oversized Thin 65 waveform for target discrimination. One cannot use the amplitude because both range dependence and aspect-angle dependence of the target destroy its use as an aspect angle independent radar target discrimination techinque. One cannot use time delay for discrimina- tion for two reasons: one is its obvious range dependence and the second is that even if a radar range estimate is available from other processing algorithms, its resolution will not, in general, be pre- cise enough to calibrate individual modes. The rotations plot is not quite as clear as before. The slope does not appear to be as constant as the two preceding cases, but there is a slight trend to be less negative than either of the two preceding rotation plots. The envelope plot for the 10% oversized radar target is more lumpy than the plot for the correctly-sized radar target antenna terminal response. We shall temporarily skip the second mode processing. The reason for this is that for the normal incidence on the thin cylinder, no even natural mode waveforms would be excited. This follows from symmetry arguments. Figure 3—16 is the 3rd mode waveform excitation processing on the correctly-sized 18.6" (L/a = 400) radar target antenna terminal response. The rectangular radar A-scope plots clearly indicate the presence of a waveform of at least approximately correct frequency. The rotation plot clearly indicates a constant slope, but more nega- tive than the exact rotation line. This would positively correlate with our lst mode estimate of a 2% undersizing of the original data 18.6" HIRE( L/R:400l 3RD MODE 66 i 18.6" HIREIL/R=400) 3RD NUDE q : ' g. *- 3.. i- 3i 3 0. 70 Co 0:0 010 fino 01.0 07; 7:0 010 "0.0 {.0 0:0 070 0T0 0.0 010 0:0 IORHRLIIED ilflfll/(L/C)! ”O‘HILIIED llflfllllLICI) :- :~ 18.6" HIREIL/Flz400) 3RD HODE 18.6” HIRE(L/R:400) 3RD H005 :4 2* 1‘ Q . . g Q o" T 3. I Q. '6‘ i =7 b E_ l a, is 1 1 ' 0. 7‘ I 7 o . .. \l 'i" 5* e 2 10 {.0 0T0 0:0 0:0 0:0 0:0 110 0'.0 '0.0 C0 010 0T0 :0 010 0:0 0:0 Figure 3-16. InlflflLllED TlfltlT/lL/Cll IOIHRLIIED TlflilfllL/Cl) 3rd Mode Waveform Excitation of Correctly-Sized Thin Cylinder Baseline Radar Target. 67 file either by a ruler measurement or by method of moments natural frequency calculation. Another good confirmation with the lst mode is in the envelope plot. The 3rd mode envelope plot has a "late- time" relatively constant slope that also starts approximately 0.5 normalized time units behind the end of the excitation vector. This is the same as for the lst mode envelope. For the 10% undersized radar target antenna terminal response processed for the correctly-sized 3rd mode waveform, we obtain Figure 3-17. The rectangular 3rd mode plots are not indicating the same waveform purity as before. The polar mode radar A-scope plots show observable departures from the expected constant "late-time" behavior. For the 10% oversized radar target antenna terminal response processed for the correctly-sized3rd mode waveform on Figure 3-18, the rec- tangular plots differ by a difficult to quantify amount. In the polar mode radar A-scope plots the difference from the 3rd mode base- line is most obvious. Figure 3-19 constitutes our baseline for fith mode waveform processing with the correctly-sized radar target. Only the rotation plot can give a waveform confirmation for this extremely weak mode waveform. Figure 3-20 gives the 10% undersized plots. In this case, the fifth mode waveform cannot realistically be observed and a dis- tinction can be made from Figure 3-19. A similar situation holds for the 10% oversized plots of Figure 3-21. Figure 3-22 constitutes our baseline for the 7th mode waveform processing with the correctly-sized radar target. Only the rotation plot can give a waveform confirmation for this extremely weak mode 68 wnPLlluot 0.0 A ‘ 9 ?d '. 9i 0 O a 9'4 0. 0. Tho C0 £0 £0 Co 05: 03 am £0 ‘00 :0 £0 £0 30 C0 C0 {0 0} wonnfltlltc TIHEITIIL/Cll wonnntlzco TlflEll/lL/Cll C O ..1 .'1 9J °..J O N .J . o' o 0:. 3. O I g o g .. '_", 0" .8- T b i a - 0. .4 Q . 9 0'4 ?- o O Q! 9" and . Q _' fl '0 1 fl V f 7 fl 7 I . I v v y Y T Y 00 am am am am 00 00 10 00 oa no em 00 am so 00 to r NORHRLIZED IlnEtT/IL/CI) wonnntxzto YlHElT/(L/Cl) Figure 3-17. 3rd Mode Waveform Excitation of 10% Undersized Thin Cylinder Radar Target. ‘ 901- L l [001 0.0 69 3. a” .4 g -. e = T T T f r 1' 1 ‘ v v T T T v '1 I '0.0 t 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 '0.0 0.0 0.0 0.0 0.0 0.0 0.0 no 0 NORMLIIEO flout/tut!) wonnnuzw flour/tutu O O 6" o" :d 2: O O .‘i 61 ~ 0. ."i “I" 8 a.) $.- - . '- C J . 0 fl 3 a D. D 9‘ 1’ v. 0. 9* a, - 0. 6. 2 I I 0 e 5 Y fl Y Y T V f I u I T 1' f V V V '0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 '0.0 3.0 0.0 0.0 0.0 0.0 0.0 1.0 0 NIMLIIED “Hill/(LIE)! MMLIIED YllIUT/iL/Cl) Figure 3—18. 3rd Mode Waveform Excitation of 1 % Oversized Thin Cyliner Radar Target. ‘ IE .6" HIRElL/R:4OOJ 5TH HODE "LIT“! '9 70 18.6" HIREIL/R:400) 5TH "ODE 9 . 9 ,1 -: e r r i. .0 To 0'.0 01.0 030 0'4 01.0 11.0 C0 7.0 .0 :70 £0 03 01.0 01.0 070 0:0 MMLXIED YIHEIYIIL/CH “ml!“ tlflllT/(L/CH 18.6" HIREIL/fl:400) 5TH HODE 18.6" HIREIL/R:400) 5TH HODE :- 2 g 5.. i: v.- 1‘ \ .— -: 8 q 9‘ $1 8”, \\ ., 9 “004‘ \_ ,‘ ?" %r4’b \..‘~' ., 9 ?" ,4 z 3 70 .0 1:0 0:0 010 0'0 030 010 7.0 T04 C0 010 030 010 0'4 07.0 130 0 . . 0:0 ”MIIED TIMI Y/lL/C H Figure 3-19. MIIED "REIT/(LIEU 5th Mode Waveform Excitation of Correctly-Sized Thin Cylinder Baseline Radar Target. 71 "3 .1 =11 '. °< ‘3 °C ‘5 O ',.1 9. . 9'4 0'5. " . '0:0 1:0 0:0 0:0 0:0 0‘0 0'0 1'0 0 0 5. 0 Y ' Y ' ' ' ' ‘ 0 . 0 n ‘0. '0. .0. .0. ‘0. .00 1. .( woanaLlltD Tlntt T/lL/C H IDIMLIHD 1111:. l/lL/Cll 0 . a . 6‘] 2 't 0. D .‘ ' - a s. O .' 3 _, 8. 3‘." g? ‘. 30 y. g I e C. 9-1 201 2 0. 9‘ h e 0. ? T r T l 0.0 1.0 0.0 0 0 ' ' ' ' v . . . r v 1 . ' ‘ . 0.0 0.0 0.0 1.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 7.0 0.: IMHRLIIED TlflElT/lL/Cll MMLIICO "RENAL/CH Figure 3-20. 5th Mode Waveform Excitation of 10% Undersized Thin Cylinder Radar Target. ' 72 9- 9. O o . :4 .‘iL .. .- ‘x o4 1" \‘\ u 9 .0 1" 3 00 a. 3: - ." - .i f I . 2 ,. e -. 9i I11 - 0. .d ,I -. 0. 70.0 11.0 07.0 01.0 a: 0:0 0:0 0:0 0'.0 '0.0 110 01.0 0:0 01.11 0:0 0T0 1:0 0 MMLIIIO Hanna/CH IMMLIICD YanY/IL/CH v 0. 0 :q 2* E ‘ 6" :1 :‘ : common: 0 O l_ 00'0"” -0 0 l 0‘ O -. I 4 '00-. 0 I .‘t. 6‘)” ........ .. 0. “00 """" ‘5‘ 2d € 00 ‘ ~‘s 0 ’4’ 1 /% '1 0. 9i Q4 - 0. 7 fl V I I Y 1' 1 2 V V V V Y 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 010 '0.0 11.0 :10 0.0 0.0 0.0 0.0 1.0 0 wonnnuzto Hunt/tut!) IMMLIIED nnumucvi Figure 3-21. 5th Mode Waveform Excitation of 10% Oversized Thin Cylinder Radar Target. ... 1 18.6" HIREKL/R:400) 7TH MODE 73 ‘ 10.5" HIREiL/R=4OD) 710 0005 flL, ‘5‘ =4 ““““- ’7‘” 44%30 94%? § : 5* :2. g . ‘ O 1 ‘“x :< ,. 9 e 9‘ f' a z. 5 2 10.0 110 0:0 0:0 0:0 . 0:0 1:0 44010 ‘0.0 110 40:07 0:0 010 0.0 010 1:0 IDRHHLIIED Tlflttl/(L/Cll ”ml!" ”HEN/(LIEU 18.5" HIRElL/R:400) 7TH NUDE 18.6" HIREiL/fl:400) 7TH NUDE a s a z z s. g g 9 ‘1“. D- O. .— . ~ 0 " o = '1" "‘x.‘ .- 5, 2q ‘2; " ¥d #00 ‘ 640 1 . )' 7‘ §‘ 4&0 ‘0‘ lg. 1 -: '. .. 0. e e ? ' ' r 1 I . 1 1 1 I I I T 04 10 0a 04 am 00 ‘00 1a 04 0a 04 00 00 1m £0 0a 0a wonnntxzeo TlflE(l/(Ll£ll Figure 3-22. “ML”!!! TIRE! TliL/C I l 7th Mode Waveform Excitation of Correctly-Sized Thin Cylinder Baseline Radar Target. 74 waveform. Figure 3-23 gives the 10% undersized plots. In this case, the seventh mode rotation plot completely breaks down. A negative confirmation is easy. Figure 3-24 gives the 10% oversized plots for the seventh mode excitation. In this case the rotation plot (note different scale) breaks down in another manner for this mode. A negative confirmation of this mode is easy. 3.8 Observation of the Absence of a Natural Mode Waveform From the preceding two sections we developed techniques to recognize or detect specific natural mode waveform. For the standard radar problem always contending with thermal noise, propagation scin- tillation, target glint, false targets, receive nonlinearities, etc., our initial technique is at most half a useful procedure for a radar. If these were not critical issues, we could use synthetic radar target files exclusively and ignore the difficulties of empirical radar target data which we will deal with exclusively. Complementary to the detec- tion of specific natural mode waveforms is the detection of "false alarms". In fact, the signal-to-noise ratio of a radar system is sometimes characterized in terms of its probability of detection versus its "false alarm" (reference 3-10). In this section we shall observe the absence of a radar target natural mode waveform in three cases. The data we shall use are our baseline correctly-sized radar target antenna terminal response of the previous section. Note that it is not deconvolved f0r either the antenna response (H* for the incident plane wave pluse shape. We will be observing target mode excitations for which no radar target natural 75 N r :lL.‘ ‘20. 1 q \. 0" 7 - 9 6" 0.“ a 9 .1‘ «:4 E “ “ 3 e B '- ‘3 - ° _ .1 .J 0 '2 t a 9 a 9-4 -.'- 1 ' '- 01 7" l I 34 '7 ' °. 9w 1 1 fl v 1' 1 I - I 1 fl 1 T V 'O.'.‘ l-0 2.0 3.0 6-0 6.0 0-0 1:0 0.0 '0.0 L0 0.0 31.0 0.0 0.0 0-0 7-0 wORchIZED ”NEH/(LIE!) NORHALIZED Hunt/tutu “ 9 :2“ g- i 6‘, ' 3+ ‘ '1 0 ° 6 . I 9. ‘3‘ 0'" .0. U a “X .8... g: ‘0‘ :;o z 7‘ .- 0- t c 8 . 5.9 -. .. . 9 ‘0 1‘ 3n 00 ‘3‘ 6“ ‘~ .. . 04 . - ) a" §~ ’04. .9 . 8 - 9 9i 5- ~ 9 4% g , I v r v 0 1 1 T W . I 1 0 1 I ' 0.0 1.0 0.0 0.0 4.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0 0 0.0 0.0 7.0 Figure 3-23. NORHRLIZED llflill/[L/Cli NDRHRLIZED llfiElT/lL/Cll 7th Mode Waveform Excitation of 10% Undersized Thin Cylinder Radar Target. II 76 RHPLIIUOE -I .0 0.0 1._ J 1111! I"! -:.. "0. ': : ° ? ..‘4 g: . o4 .‘ I I | 00 ‘ ° 0:- *7 V v v T 1 1 'lTi I Y V ' ' ‘ ' .°~C ha [-0 3.0 4.0 0.0 01.0 7.0 ... '0.0 1.0 2.0 3.0 0.0 3.9 0.0 7-9 NORHRLXIEO Tlflfll/(L/Cil NORHHLIIEO 'IHEil/IL/Cil ‘5- 9 t . 0‘} K - o I a 0* u-I I U a 0 9 g 0 = - .. 9. j ‘\ -a *I‘ 1 \ .1 c t . x o . a. c ‘. e 9 \ ° 0-4 6“ ' 0 183" — . a a. :3 ~ ' e 11‘ _ ' a. a 2. x‘ I I 0 Q I 9. . ~ 1 'I Y Y Y I I I l 1 I I 0 o 0.0 0.0 0'.0 1.0 0.0 '0.0 1.0 0. 0.0 0 0 0.0 1.0 . 0 31.0 . NORHRLIIED TlfiEil/lL/Cll IORHRLIIED TlHEiT/(L/C)l Figure 3—24. 7th Mode Waveform Excitation of 10% Oversized Thin Cylinder Radar Target. 77 mode waveforms are present. We may do this because there are physi- cal configurations for which the coupling coefficients of certain current density natural mode must be zero. For the thin cylinder illuminated by a normally incident plane wave, there can be no even natural modes excited. This is due to the perfect symmetry of this particular radar target with respect to the incident E-field. In Figure 3-25 we observe the second mode waveform processing of our baseline correctly-sized radar target antenna terminal response. From the rectangular plots we observe a strong transient located during the first nonzero samples of the excitation vector, but followed by no corresponding natural mode waveform response in the expected "late- time" portion observed in the odd mode waveforms. The envelope plot shows a pulse-like envelope of time length equal to the excitation vector time length. (Remember that our envelope plot are on a com- pressed logarithmic scale.) We will observe this pulse-like shape in the envelope plot again in the next chapter and we will call its aspect-angle independent excitation vector a K-Pulse for a specific radar target. In Figure 3-26 we observe the fourth mode waveform processing of our baseline correctly-sized radar target antenna terminal response. There should be no waveform originating from the radar target at this frequency. The rotation plot does, however, indicate a waveform with a rotation slope slightly lower to that of the correct fourth natural mode waveform. The envelope plot appears as a pair of staggered pulse shapes and is not explainable as a target mode waveform. The rectangu- lar plots indicate perhaps the existence of two distinct ringing II 78 18.6" HIREIL/Fl:400l 2ND MODE 18.6“ HIREIL/R:400) 2ND HOUE ”1.111100 .0 0 ’d ?0.0 1:0 010 0.0 :0 0:0 0:0 110 T0 70.0 110 010 01.0 070 0:0 010 11.0 ”Mil“! TIMIY/(L/CH mmuto TIHElI/lL/CH 18.6" REL/9:400) 2ND HODE 18.6" HIREIL/R=400) 2ND "ODE 01111111100 J l Y I j V V I V T V V 0.0 1.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 1.0 mun TlnEIT/IL/cn v 0 n. I 0. . 0. 0:0 0:0 0:0 110011101100 I"!!! TllL/CH Figure 3—25. 2nd Mode Waveform Excitation of Baseline Thin Cylinder Radar Target at Normal Incidence. 79 3. 9. 18.6" HIRElL/R:400) 4TH ”005 18.6" HIREIL/Fl=400) 4TH NUDE ;- :IL\ '3‘ h L \\%\”o o ' a“ an” :- ¥~ \ “b \\ g- :9 k“ E.- 20.»: H: ‘ I 2 F a a!) -.- e r 2 3 T0.0 110 a 010 07.0 010 03 1:0 0'.0 "0.0 13 010 0:0 T0 07 0'.0 1:0 IMMkleD HHEII/(L/CH maul!!!) "REIT/IL/CH 18.6" HIRE(L/R:400) 4TH HOUE 18.5" HIREIL/Fl:400) 4TH HODE 2+ 5 a I :9 g... g. E? ‘ 2 .4 s. :1. z 0 “ 2 2 10.0 1.0 010 01.0 T0 0'0 030 1:0 010 ‘0.0 110 0:0 0:0 010 7.0 010 0T0 Figure 3-26. NORMLIIED ”HUI/(L10) l WWII“) ”It! TllL/C I I 4th Mode Waveform Excitation of Baseline Thin Cylinder Radar Target at Normal Incidence. 80 waveforms, one in the early-time and one of small amplitude starting at 3; normalized time units. It is believed this is a mode originating within the radar system set-up, rather than from thermal noise. This is suggested by the coherence displayed by the rotation plot. In Figure 3-27 we observe the tenth mode waveform processing of our baseline correctly-sized radar target antenna terminal response. There does not appear to be any coherent signal energy at this natural frequency since the rotation plot fails to match the expected slope. These plots are what we should expect to be generated from thermal noise within the radar system. 3.9 Extension of the Model to unknown Natural Frequencies So far everything we have done is based upon a priori exact knowledge of the invariant radar target parameters--the natural fre- quencies. There are no flyable airborne radar targets for which the natural frequencies are known in an analytically closed form. We have also seen that the radar system itself may contribute some natural frequencies to the radar return. Further we should not expect to observe all of the system natural frequencies on a specific radar target return. However, if we can obtain these radar target invariant parameters by another route, we will use them in exactly the same manner we would use equation (2-22). In the next chapter we shall initiate a reliable procedure for obtaining these natural frequencies from measurement data. So we must add another goal in our quest for an aspect-angle independent radar target discrimination technique. 18.6" HIREKL/R:400) 10TH HODE InPLlTUOE 0.0 81 18.6" HIREIL/Rz400) 10TH HODE -. -. '0.0 110 01.0 010 0:0 050 0'.0 1:0 0 .0 T0 .0 1 .0 01.0 070 0:0 0'.0 01.0 910 IORHRLXZCO TIHElI/(L/Cll IOIHRLIIEO IIHEII/lL/Cll 18.6" HIRElL/R:4OC) 10TH MODE 18.6" HIREIL/R:400) 10TH MODE 3. Q . q. 9 00 9" “ x“ e. 3 § 3 in \‘ S 2 - 0': '24 a \ ‘. . r i 4., .. g at? <5 .\ . T‘ 4b» “x ‘1’) ‘-,‘ O. 'o 04. .... g. 6‘ g1 #1 ‘0‘ x 2 z '0.0 1:0 0:0 0:0 0T0 0'.0 0:0 20 '0.0 110 0:0 0'.0 010 0'0 01.0 710 IOIHRLIICD IIflEll/lL/Cl) Figure 3-27. NOIHRLIIED TIHEIY/(L/Cll 10th Mode Waveform Excitation of Baseline Thin Cylinder Radar Target at Normal Incidence. CHAPTER 4 PRONY'S METHOD AND THE K-PULSE 4.1 The Original Prony's Method Almost 200 years ago (1795) a measurement-based technique using a finite series of complex exponentials was developed by R. Prony (ref. 4-1). Although the original technique was developed for evaluating the temperature dependence of vapor pressures over liquids, we shall find it of use to use in the radar target discrim- ination problem. We have already used in equation (3-36) what we have called a "Prony series". With a few exceptions, Prony's method will perfectly match a Prony series to equally spaced continguous sampled data points of a continuous function. For example, for 2N sampled data points, Prony's method will solve for the unknown constants of an N term Prony series. In addition there is a least-squares formulation of the com- plex exponential matching technique called the "extended Prony's method" (ref. 4-2). Similar to the "extended Prony's method" are many recent techniques such as linear prediction, maximum likelihood, .and maximum entropy method which also have a least-squares formula- tion (ref. 4-3, 4-4). These latter techniques are often expressed in terms of an analysis spectrum or of a synthesis filter. However in 1795, Gauss had not yet disclosed the famous least-squares technique. Hence Prony's method is the only one of the above techniques not 82 83 influenced by the least-squares technique. Least-squares techniques are not reversible to the original data. A "reversible operation" like Prony's method is a desirable building block feature for a modern "quiet“ radar design. We shall start the derivation of Prony's method with a defini- tion of a Prony series of N complex exponentials in continuous time by equation (4-1). 01-1) W O N v(t) = E Ck exp(skt) , t k 1 The invertible sampled data version of equation (4-1) is given by equation (4-2). V(t,m) (4'2) N kilck exp(Sk(nT+mT-T)) where t nT + mT-T n an integer O < m': 1 lde have used the notation of the modified z-transform described in Appendix A. We will not revert to a more common sampled data notation (If the ordinary z-transform notation, equation (4-3), by picking a specific synchronization, m = 1. v(t,m=1) = (4'3) fltfl:z N C exn(s Tn) = z c {e T " k 1 k k k=1 k xp(sk )} , t=nT, n=O,1,.... 84 In order to prepare for matrix notation, we will compress the nota- tion of equation (4-3) into equation (4-4). v = " k II II M2 ('0 z: , n=0,1,...,2N-1 (4-4) "(‘12 Ck{exp(skT)}n k 1 1 Let us consider an arbitrary sequence of 2N equally spaced sample data values {vn}§261. We should observe that in equation (4-4), there are exactly 2N unknowns on the right hand side to match the 2N sampled data values on the left hand side. Prony's method is a procedure to determine the unknown constants of the series of complex exponentials. Prony's method consists of a procedure we shall divide into three parts. The first part always appears to be the least motivated on physical grounds. Remember that we are going to determine the 2N constants of the 0 term "Prony series" of equation (4-4) by 2N sampled data values of the empirical data. The simultaneous determination of the 2N constants of the N term "Prony series" is not a linear problem such as the 2N term Fourier series based upon the same 2N sampled data values. Conceptually, suppose the natural frequencies have been deter- mined. Then the complex amplitudes for each term of the "Prony series" could be determined from any N sampled data values of equation (4-4). We will, for example, take the first N sample values and put them into matrix notation of equation (4-5). Hence the crucial step is to obtain the natural frequencies. This will occur in two steps. If we possessed a differential equation, we would take the Laplace trans- form and obtain a polynomial in frequency. The roots of the 85 r1 1 1 - F01. on ‘- exp(slT) exp(szT) exp(sNT) C2 v1 . 3 : = : (4-5) exp(slT(N-1))exp(szT(N-1))...exp(sNT(N-1))J 6N v" 1 _ — a c. “J homogeneous polynomial would be our natural frequencies. Equation (4-6) gives the Laplace transform of the "Prony series" and equation (4-7) is the modified z-transform for m = l. A N V(s) = § ck (s-sk)'1, Re(s) > m:x(Re(sk)) (4-6) 1 'C 1 V(z,m=1) = k k (1-zk/z)-1,|2| > m:X(le‘) (4-7) IIMZ Instrinsic to Prony's method is the exclusive use of sampled data values. So we do not observe continuous waveforms characterized by differential equations, but we do have a characterization of dif- ference equations. From 2N sampled data values we could form a poly- nomial of degree 2N-1. But this is too large, since we wish to match it to the N term "Prony series". We shall be looking for some func- tion a(nT) such that when convolved with v(nT) will yield an output with no natural mode waveform, o(t) in equation (4-8), remaining. Alternatively, 0(2) in equation (4-9), is an entire function. o(t) = a(t)*v(t) . t=nT, n is an integer (4'8) 0(2) = A(2)V(2) (4-9) 86 But equation (4-8) can be expressed as equation (4-10), and equation (4-11) must hold. 0 T = n-l dz _ (n ) '4g A(z)V(z)z §;3-, n> 0 (4 10) unknown, n=0,1,...,N-1 o(nT) = 0 , n=N,N+1,...2N-1 (4-11) 0 , n 2 2N if V(z) has only N poles The significant feature which A(z) possesses is that it possesses zeros exactly where V(z) possesses poles. For A(z) to possess N zeros, it must be of degree N, with in general N + 1 terms. This means that a(nT) will be of finite length, (N + 1)T. Figure 4-1 shows the page in Prony's “Essai . . ." where N equations in N + 1 unknowns were originally set up. We show this for a number of reasons: (1) it is different from modern matrix presentations, (2) he could solve it by hand, (3) our “fast" Prony's method in the next chapter will resemble it more than the modern matrix presentations. In Prony's notation [zi]::;1 represent the sampled data values. Equation (4-12) is a matrix representation of the Prony system of N equations in N + 1 unknowns [Ai]?=0 -Ir - -. F20 21 ... zN_1 ZN A0 I F0 i1 ?2 °°' iN ?N+1 A1 9 I I 2 : = : (4-12) ZN-2 zN-l °°' Zen-3 Zen-2 AN-l 0 z z ... z z A N-l N - _ _ 2N2 NIH-NJ _0J 87 . gas}. 1, ° 30 a s s A x“ ' ’ : . " ’3 valeurs correspomhntcs dex. . . .o ; x,; xx, . . .. .ns-,; (- + 129: (n+s)x,...'..(zu— 1):, % ies quantités z“, z, . z, . &c. doivem former une suite recurielise dons ii fist trouver I'échclie dc relation ; soient A“. A,. A. . . . . Am des 00%. . indélerminés, leis qu'on ait ies equations dc condition ' A92.» + A, z, +11, z,+.....+A,., 3,, =0-- A02, +A, z, + A, z,,—+-.....-|-A,., gm”; 6.- A0 z, +A,z,,,+A, z,,+.....+A,,, a.+,,=:.o “Adm-#42» +46 +~~~~+A~ z..+.;=°‘ ........ 00.00.00...0.0.0.0....COOOOOCOCO0.... . o ............OOOOQCOOCOQOOOCOOC0.000.000.000000 Ac 2,.-.) + A,z,,, + A, z,_,,,,,+. . . "+‘4r-2 a.,,_,, = o. On pouns, poo: plus de commodité, suppose: AN = s «has les sppflasi. nume’riques. . , . ‘ . A, A ‘ ' . Ces équauons étant en nombre n donneront ies n nppomT; A' z m N A. . A(u- ' ) 44(0) ' . . . . A(.) on aura qui composem l'écheile de relation demandée .‘et Poumzt A,:A,=—z,:z, . Vsleut de'duise de deux observations. (,1... - 1.1., A0: Au=——— lot. "' LL A, : A" =____.,z. - 1’2" '- Z, I. Vsleurs déduixes do quite observations. Pourn=s A :A _ +(ann—LZ'V)Z~+ (Z,I"+i.z~)z"+ (ta (...-‘1 z-)t' ° "' -(z.z.-z.zu)z. -(z,zn-z.z.)z, -(z.z.-z.z-)z. - +(z.z~— 2.2.)2- + (LL— mo-)z-~+(z.z-—zz)zv Pour”: A :A = ' ' . 3 I III —(z’z"—Lz.)z,— (Ll. -.l.zn)l.‘-(i.i-—I.Z.)lo A 'A __ +(z.z. — z.z-)z. + (Lz-—z.z.)z~ + (LL’Z-Lh' ,. ,,,_ —- - (Ll. - z.z..)z. - (m. - can. - (z.t.- I. 1.)!» Vsicurs déduites de six obsexvsuons. 52C. kc. 8m. Figure 4-1. Undetermined coefficients of Prony's "Essai. . . ." 88 Hopefully there are enough matrix entries that the data matrix can be identified as a Hankel matrix. In a Hankel matrix all entries along a diagonal from lower left to upper right possess an identical value. This is the case here even though equation (4-12) is not a square matrix. By solving equation (4-12) we will obtain a polynomial, in terms of A(z), which we can root. From these N roots, we can obtain the N natural frequencies of the "Prony series". Then we can solve equation (4-5) for the complex amplitudes. Then we will possess the values of all 2N constants of the "Prony series" of equation (4-1). We shall close this introductory section and discontinue the Prony notation. In the next section, we shall define the "Prony K-Pulse". After defining the "Prony K-Pulse", we shall solve the three parts of Prony's method using the now more common matrix notation and techniques. 4.2 The "Prony K-Pulse" The original "kill-pulse" or K-Pulse concept (ref. 1-2) is a time-limited excitation waveform, k(t), (like our a(t) in equation (4-8))which when convolved with the radar target scatterer yields no natural mode waveform in the scattered E-field. Alternatively, we can specify that k(s) be an entire function. It appears that the original K-Pulse concept is solely a transmitting formulation. Hence the K-Pulse, k(s), could be the plane wave "pulse shape", f(s) in equation (2-15), of our basic SEM model solution of the EFIE. 89 We shall decline to advocate the transmit formulation of the K-Pulse if we are confined to use either analog radar transmitters or analog transmitting antenna. We will find extensive use for the K- Pulse in our receiving formulation of our radar problem solution. We shall use it as fluently as one might use a multi-dimensional impulse function for a specific radar target. There are two useful facts we will close this section on. First. the entire function, WK(T,F,S), must be in the time domain of duration less than the maximal one-way transit time of the radar target itself. This is based upon the hypothesis that there exists a retarded time for which the "class 1" current density coupling coefficients are valid. Its contribution in the far-field scattered E-field may last as long as a maximal 2-way transit time of the inci- dent plane wave over the radar target. Second the K-Pulse can be as short as a maximal 2-way transit time of the plane wave over the radar target. This is not to say that shorter K-Pulses are not possible for some aspect-angles and some targets, but we are stating the general, aspect-angle independent case. The "Prony K-Pulse" is defined as one of the kill vectors, a(nT), of equation (4-8) or equation (4-9) whose time length is not less than t" of equation (2-18). 4.3 Prony's Method and the K-Pulse Derivation We have already defined a "Prony series" by equation (4-1). We shall now recast part 1 of Prony's method by recasting equation (4-12) in terms of our standard sampled data values [vi]§261 by (4-13)- h— v2N—1 v2N-2 °'° vN-1 — 90 (4-13) {he modern method of solving this equation is to note that the vector [a%], can be divided by as without alteration of the solution and then moving the left hand column to the right hand side to obtain equation (4-14) which possesses the same a O ' normalized solution. Note that in euqation (4-13) we performed a trivial reflection of the sampled data matrix which we will exploit in the next chapter. r- vN-1 VN-z vN vN-1 V2N-2 v2N-3 "° L_. VN‘l—J V L.2N-1_ (4-14) This equation may be solved for the unknown vector [as] by either standard matrix arithmetic or by the fast "Covariance method" (ref. 4-5). N «91:0 {1,a1,a2,... ,aN} Our "Prony K-Pulse" is now given by equation (4-15). (4-15) 91 Part 2 of Prony's method is to find the N roots {21.}?=1 of the Prony polynomial P(z). We shall define the Prony polynomial from the solution set {ai}§=0 (a0 = 1) by means of equation (4-15). N i .z = H (z-z ) , a = 1 (4-16) 0 1 k=1 k 0 "NZ CD P(z) = . 1 As one would expect from equation (4-4), we may obtain the natural frequencies by taking the complex logarithm of the calculated roots. These roots are obtainable by standard computer library routines. For polynomials of very large degree, this is a difficult problem of active interest. Equation (4-17) gives us the nonunique natural fre- quencies. He shall normally take the branch with the smallest numeri- cal value. -1 = T clog(zk), k = 1,2,. . .,N (4-17) 5k Part 3 of Prony's method is to solve for the complex amplitudes of the "Prony series" now that the natural frequencies are known. This is readily solved by use of equation (4-18). __ fl _. .1 _ .7 . V 1 1 1 C1 0 . .' . _ . (4'13) .N-I .N-l N-l 21 22 ZN CN VN_1 L. _ .. .. L... _J. 92 Equation (4-18) is solved for the unknown amplitudes, Cm, either by standard matrix arithmetic or by methods more suitable to Vandermonde matrices (ref. 4-6, 4-7, 4-8). At this point all of the 2N constants of the Prony series of equation (4-1) are now known. Our technique of radar target identifi- cation will use these parameters and further processing of equation (4-15), the Prony K-Pulse. Figure 4-2 summarizes the major steps we will perform to obtain our plots of the radar A-scope displays. Note that the first step is the generation of a "Prony K-Pulse". This by itself will be found to be inadequate for radar target discrimination. Ne form individual mode excitations by deleting one root from the Prony poly- nomial. Alternatively this may be done in the time domain by couplet convolution and deconvolution as in Appendix C. Using couplets as in equation (4-19), then the j-mode excitation vector would be numerically evaluated by deleting the j-th couplet from this convolution, leaving only N-1 convolutions as in equation (4-20). [kn 2:0 = (1,-Zl)*(1.-Zz)*...*(1,-zj)*...*(1,-2N) (4-19) [6,3, 2;}, = (la-Zl)*(1.-22)*...*(1,-zN) (4-20) Figure 4-3 is a diplay of the original clutter-reduced radar target antenna terminal response file on the upper left with the computed K-Pulse on the upper right. The output K-Pulse convolution With the sampled data file is given on the lower left corner. Note 93 {3%, Part 1--The K-Pulse f2? From 2N sampled data values ééé Obtain solution set [ii $31 /. 2: Yielding Prony polynomial, P(z) gég Yielding the ”Prony K-Pulse" 257 /”’ ”’422/¢%2¥%%ég2i Part 2--Roots for Excitation y/ From solution set // Obtain zeros of polynomial /;: Obtain excitation vectors by / deleting a specific root from the ”Prony K-Pulse" Part 3--Radar A-Scope ??5 Convolve all sampled data values /, by Specific excitation vectors Display rectangular and polar mode radar A-scope plots // // /7' ’ / ¢7%/ //// ////////2///M Figure 4-2. Major Steps in the Generation of Radar A-Scope Displays. 94 .mpmawuma cw maoFm>cm muH ecu .cowuzpo>:ou on» .moPaEmm awxm scam empamux acoca on» .A=m~v cmvcvpzu cpzp a mo mmcoammm sauna umuaumaicmpaapo .m-e mcsmpm mozouuwczcz z“ u2~b mozcuumozcz z~ uzuh ...: ...: 9.2 .... a.» a... a.» o... 9.2 92 9.. .... 3 o.~ 0.0 . . . p p p _ .r r . p _ . _ m. o O mu 0 rm.” “av x im m. / _ _ _ m. ....m N am 0 a o ..m. 3:33.13. ..m L .. x n ..l 0 U I \ In.” [x1 1W 3 . 2:32. n... .m rm ..u D mozouumozcz z~ uz_h mozouumozcz zu unuh a... 06 a.” a; . o.» a; o; a... ...: 9.2 a... a.» .... 3 o... .r F p F L h u v.0 r :mohi: Saz. r . 1.. O 9 +w row 0 .d t .d «I 11 u u n n .0 u mu 0 0 3 Au v > fiO 3 Z .O .0 1! waDmix *20 n. mt: ..m: 3838 $33 3.5.38. 9;: ..m~ _ ’ z o In. 95 that the "late-time" response (remember that the "Prony K-Pulse" is defined so that "Class 1“ assumptions should be satisfied for the retarded scattered E-field of this target) is approximately zero. However, using the compressed logarithmic scale for the envelope radar A-scope plot on the lower right corner, we see that the envelope is suppressed about 250 decibels for a two-way transit time and then is suppressed by only about 15 decibels. If we possessed all the radar target natural frequencies within our K-Pulse, noise permitting we would continue the 250 dB suppression with increasing retarded time. We will increase our radar target discrimination tools by next examining single mode excitation vectors and their convolutions with our radar target file. There is an important concept that does not arise much in the literature that we need to understand. In the absence of data noise and numerical "ill conditioning," if the N-th order matrix equation (4-14) of 2N sampled data points is nonsingular, but the (N+1)-th order matrix equation (4-14) of 2(N+1) nonzero sampled data points is singular, we say that we have "identified a Prony series“ which has only a finite number of terms, namely N. We should expect all subsequent time values, t > 2NT, of data from a continuous physical process to be perfectly predicted by this "Prony series." Finally, we shall provide the motivation for the skip sampling of the next section. If we use standard matrix arithmetic as was done on all empirical data in this chapter, the standard matrix equa- tion solvers may generate errors in the solution set. Certain, even Snail, errors can be catastrophic to Prony's method. A particular 96 error observed which destroys our target discrimination technique is the splitting of a root. If the root corresponds to a target natural frequency of large amplitude. we are defeated. The solution used in this chapter is to reduce the order of the matrix and hence the degree of the Prony polynomial. Since the "Prony K-Pulse" has a minimum time length. to use the sampled data values, we must methodically skip some data. We call this skip sampling. 4.4 Skip Sampling in Prony's Method A frame of sampled data used in the standard Prony's method consists of 2N sampled data points. For the special case of 2N+1 sampled data points, Prony differenced the original data to obtain a new seqeunce of 2N sampled data values. In either case, there are now N simultaneous equations such as Figure 4-1. Now in our matrix formulation, either equation (4-14) or (4-18) might become "singular." If equation (4-14) becomes singular, this is a highly desirable event for Prony's method. In the next chapter we shall use this criterion for terminating the length of the Prony K-Pulse. In the absence of thermal noise or quantization errors, this implies that an N-l term (or less "Prony series" will do as well as the N term "Prony series." lvhen the transposed Vandermonde matrix, equation (4-18) goes singular, 1this is a totally different situation. This often observed event is ailmost solely due to ”ill-conditioning" of the matrix system. "Ill- (:onditioning" of matrices typically increase with matrix size, all ealse remaining equal. For this reason alone, it is often in our best ‘interest to keep the size of our matrices as small as possible. Hence 97 in order to satisfy our "Prony K-Pulse" definition, and minimize the effects of "ill-conditioning," we may choose to thin or skip some of our sampled data when we construct equations (4-14) and (4-18). There are many reasons why the target "Prony series" may underspecify the empirical sampled data. Three good reasons are: 1. Thermal noise 2. "Class 1" coupling coefficient observation may not be valid for all of the sampled data. Note that our definition of the "Prony K-Pulse" assures us that at least N sampled data points are in the "class 1" observation for an impulsive plane wave incident upon the radar target 3. Incident plane wave pulse shape possesses a duration comparable to the equivalent length of important, but highly damped, natural mode waveforms In the next chapter we shall find out that the "fast" Prony's method can at least obtain the correct natural frequencies for an N-term "Prony series" if at least N sample data points satisfy the “class 1" coupling coefficient E-field observation criterion t > t" of equation (2-18). Figure 4-4 shows the radar A-sc0pe displays for the lst mode excitation; this time obtained by removing a first mode zero from the "Prony K-Pulse." The top left plot illustrates the cosine excitation vector for a target natural mode and the resulting output convolution of this excitation vector with the clutter-reduced radar target antenna terminal response. The lower left plot shows the same for the lst sine Inode excitation vector. The polar mode radar A-scope displays are shown 98 mozouumozxz z— w=~h o.“— o.m. 9.... a... a... - 9r. 9.... ...... u 0 a . Tm uoaaaa ae_u-aaap.siiio " 5.. wozouwwozcz z— wtuh ...". 9m. 9.». a... owe a“. 9+“ ......p... .i m _ Auvgm H - Auvfi< u .uvu 1% HS 8 237?:2. 0 To notmn 2.3-3.: All]. _ in AmH.m~. woo: >Zomm Fwd mmHz :mH SNOIlUlOU 91391330 .ammcmp Lanna cmucwpau swab a mo mo—a2mm amxm Soc» um~wmmcpcam comump_uxm Egohw>m3 one: umcwm mozouwmozzz z— u:—» .e-¢ stance osm— ofi: o.»- B. ow. a..." 96. HT W _ o _ ADV . . :1 o ir _ 3 . ....u _ _ v0.7.3 2.5-33le _ rr wozouumozcz z~ mc~p 9.2 92 9. as o... ...... 9... P L P F n p b . _ w _ 33o rm Aanq . I2 r — p- : __ L ..u - 1 _ ‘ o 2.7.3 2.5.3: 1w moo: >zoma Hm". mam: .m_ rw BOHLIWJHU 3001114HU 99 on the right hand side. The upper right plot is the envelope display in decibels. The rotation plot of the first mode is displayed in the lower right of this same figure. We have employed a skip sampling in obtaining this figure. Ne have selected from the response file the lst, 5th, 9th, 13th, etc., sample. We will be using every fourth sample and can select the synchronization. In Figure 4-4 the excitation vectors were selected by starting the data at the 5th sample. The same synchronization was used in the output convolutions. Now to determine if these radar A-scopes are part of a useful radar target discrimination tool, we need to determine if acceptable results can be obtained for different thermal noise and different amplitudes and phases of the natural move waveforms of the radar target. An ideal check of this technique is to use a different syn- chronization of the same skip sampling. Since our clutter-reduced radar target antenna terminal response file has not been smoothed subsequent to the receiver thermal noise, the noise samples are uncorrelated. Further, if the output conVOIUtion is properly time- tagged, the results are directly comparable with the cycle 15 syn- chronization. Figure 4.5 shows the results of changing the synchronization. This skip sampled file (cycle 14) contains none of the clutter- reduced radar target antenna response sampled data points which were used in the computation of the excitation vector we are using in the radar A-scope displays. For the first mode the performance is similar, taut definitely not perfect in the time period from 4.96 to 9.36 ns. In the polar mode A-scope plots, it can be seen that the rotations 100 .mp3; mucoammm amuse» sauna may cw mpcmoa mama nwxm we now zmz a mozoummozcz z. w:_h 0.: 0.2 9.3 o.- o.o 9.. =4. 0.0 P p . F b m P p ..l- _ W _ lahw- a mmfiuvchpwem .m .m . _ 1.0 “3.23 3.3-33 111.. o _ _ .... mozouuwozcz 2. us.» 9? 9m. 9.». WT. o..- e... am 9... 4, m _ r _ ...... Act. A. ¢:~<- <50 ...... ano a mmfiuvuvcpwmm w _ _ o _ aw _ _ _ _ vm v0.73... 3.5-32 1.11.. m. 57¢: moo: >20”... .5“ mm; :2 9040118108 31301330 mozoummozmz z E: h ewe; v-4 weaned co aouua> =o_p.u.uxm «so: 3.... any co coauaposcoo--.m-e a.=m_a 92 3.: a... a... . a“. a...“ 9... u w . E; .. Tm 3.23 3.3-32 Lil-l. . Io mozoumwozcz 2. us: 9m. 9L2 own a... . a“. 9% 9o... _. H u A V < avufiw .r : :__ m _ . _ 0 3.23 «23-32. rm moo: »zomn_ Fwd m5: :3 _ 3001113118 3001113110 101 are virtually identical, but there is a very small ripple in the envelope A-scope plot. The only visible defect in the rectangular A-scope plots on the left hand side of the figure are due to the skip sample granularity. The next test is of great interest. ,All of the data in the radar target antenna response file is convolved with the first mode excitation vector. This is done by actually exploiting the form of the modified z-transform (Appendix A) notation, v(t,m). Convolutions are performed for the four fixed values of m. Then, the outputs are pr0perly demultiplexed. Of the total number of points used, 256, only 44 or 17% were used in the computation of the excitation vector. Only 9% of the displays output convolution points are expected to be a perfect synthesis of the natural mode waveform. Another 48% of the points potentially satisfy "class 1" conditions for the waveform. The plot result of Figure 4-6 shows impressive performance. The rec- tangular A-scope plots show thermal noise and quantitization noise similar to the original data file. The polar mode A—scope identifies a rotation rate which is virtually constant from 4 to 9.4 ns. Although the envelope A-scope plot shows about a decibel of ripple in the same time interval, the mean value appears to have a constant slope. Similar calculations were performed for the 3rd Prony mode of the 18.6" wire (L/a - 400). The 3rd mode waveform is the second largest amplitude for this wire because of the normal incidence of the trans- mitted plane wave to the thin cylinder target. Figure 4-7 are the A-sc0pe displays for the 3rd mode excitation vector. The starting point for the convolution is the same as for Figure 4.4. The 102 .mpwm mmcoammm “macaw sauna on» c_ mucwoa name on» mo .flflm 53?; 4.. maamaa to maopua> accompauxu ace: “wave as» co =o_p=_o>=ou wczoummozcz z~ w: _ h 93F 9m. 9m. 9.- 91 . e“. am 9... 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W — — n- o ..u I. . - - — G 0 U I. _ m z ...... . wotua ass-331' v0.23 3.5-3: 1.11 1.. m. mozouuwozcz 5 mt: mozaummozcz 5 wt: 9: 9.: 93 9.. 9o 9. 9... 9o 9: 92 9. 9. 9. 9a 9a. _ . _ . _ P _ ..4 . _ p _ p . ... m m o Rim.“ ..3.«- :5. L. 1.. ..oa :..:.=:.5 m m Lu 0 L. m 3 Is ‘1 II 1.0 3 Im- uotmn «5.5-32 1 o 3 v0.23 3.5-32 11. s -..u ..w a MmH.mH_ moo: >zomm 0mm mmaz =w~ moo: >zomm 0mm m H: =w~ -m L: 300111-4116 300111-3148 104 excitation vector for the 3rd mode cosine excitation vector is visible in the upper left plot of this figure, starting a 0 ns and extending to 4.2 ns. The output convolution of this excitation vector with the same skip sampled clutter-reduced radar target antenna terminal response file is shown overlaid. The rectangular A-scope plot for the 3rd mode sine excitation is shown in the lower lift of the figure. The polar mode A-scope plots on the right half of the figure illus— trate that the sharp breaks in the rectangular plots between 5 and 9.4 ns were due solely to skip sample point granularity. Figure 4-8 shows the results of a different synchronization. There exist envelope A-scope variations of several decibels but the rotation rate is virtually constant in the appropriate time interval. None of the data points of this file are in common with the previous Figure 4-7. Hence the thermal and quantitization noise are uncorre- lated in the large signal region (starting at 0.5 ns.). The final illustration in this section of this technique is the convolution of all of the data in the clutter-reduced radar target antenna terminal response file with the 3rd mode excitation vectors. Figure 4-9 shows the result. Note that if Figures 4-7 and 4-8 are overlaid when scales permit, their data are included in Figure 4-9. In the polar mode A-scopes on the right side of Figure 4-9, a ripple with a 0.2 ns period is visible. The origins of this 0.2 ns ripple are well understood for the rotations plot of this particular file. The origin of the m = O synchronization is off exactly one unit (the branch cut) from the other synchronizations of the composite response file. There are easy fixes for the composite rotation plot. One is 105 .mpwm mmcoammm ummcmp sauna mg» cw mpcwoa mum: apxm mo pom :mz a cum; m-e mesmwu mo msouuo> cowuaymuxu «no: ucwsb on» we cowu=~o>cou wozouumozcz z— wt~h mozouumozcz z— u:—» 9". 9m. 9m. P.- 9... r9r. 9.. 9... 9m. 9.... 9.. oh. - a“. 9.. 9a. m _ . rw m -+ .37...:.r:::e. w A... o fr W ..i— ..u .0 I. _ O U a I. m _ uWHN . i? 3...... 2.5.3: 1 as 8:3 2.2-32 11... o -1 r. o o nonmeozcz 2_ at.» mozoummozez z~ m=_h 9.r_ 9m. 9m. 9.. 9.. 9». 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U I; mm “N ,o S . ...... weep-..., ... o mozouumozcz 2. ns.. o.~_ c.9— o.n 9.. 5.. o.~ 0.0 p P _ p o _ p h. m E: n. vb “u w n“ q q 1. a. .w o ...... ae.u-.uap ... ---- a Amfiw moo: >zomm 0mm m H: =mfi F9 ..-. ..3... 300111dHU 3001113HU 107 to compute the composite polar display from the composite rectangular A-sc0pe displays. Only one rotation origin would exist and the resulting rotation plot is extremely linear as can be observed from the limits of the rotation ripple. We do not have as simple a solu- tion for the ripples in the envelope display. The method we shall ultimately adopt is to use very large composite files. It has been shown in reference 4-9 that these output plots consisting of an elementary natural mode waveform of the radar target all by itself yield readily identifiable aspect-angle invariant para- meters of the radar target even in the presence of noise and typical radar hardware distortions. 4.5 Zeros in the Data Matrix and —:— "Class 2 Prony Series" The topic we shall discuss in this section is related to four other topics we shall discuss in other sections besides this section: 1. The "double K-Pulse" technique which is the corner- stone of our radar target discrimination technique may yield similar data types 2. The "fast" Prony's method algorithm of the next chapter is similar to the iterative solution procedure of this section 3. This is an introduction to a "class 2 Prony series" 4. The concept of "root degress of freedom" will draw from this section We shall now consider what we shall call a "class 2 Prony Series" of N complex exponentials, which is an N term "Prony series" 108 plus an entire function, H(s), whose time domain, w(t), possesses a limited time duration. This function is nonzero only for a time duration_ defined by equation (4-21). _ w(t). 0 s t=nT-< (N-1)T ' - "N(t) ' { 0 , otherwise, n an integer (4 21) He shall then define the "class 2 Prony series" of N terms by equa- tion (4-22). NI (t) + 2 CR exp(skt) , t=nT 2 0 (4-22) v(t) = w N k=1 n an integer Now for the balance of this section, we shall, for clarity and simplicity, choose a specific entire function, N(s), satisfying equa- tion (4-23) in the time domain. N. k=1 n an integer This means that the first N-I samples of v(t) will be identically zero. Substituting this particular v(t) into equation (4-14) gives the form of equation (4-24). vN vN-1 - 0 a2 vN+1 : ° = - : (4-24) V V V a V _2N-2 2N-3 N-L L N __ 2N-1_J 109 Note that the prior Prony matrix equation (4-13) is not triangular as may be seen in equation (4-25) after substitution. r- I 1* '7 vN VN-l . O 1 0 vN+1 vN 0 a1 0 . . . = . (4-25) v2N-2 v2N-3 '°° vN-l aN-1 O V2N-1 v2N_2 ... vN aN 0 L- c-Jb- bed From equation (4-25), it can easily be seen that the "Prony K-Pulse“ can easily be obtained iteratively starting with the top row or equa- tion of equation (4-25). For example a1 is given by equation (4-26). a1 = -VN/VN_1 (4-26) Equation (4-26) is then substituted into equation (4-25) and then the second row of (4-25) is used to solve for the unknown a2. For illus- tration take N' = 1 in equation (4-22), then equation (4-26) will have solved for the single necessary root 21 = exp(slT). The balance of the ai will be zero for this special case. Part 2 of Prony's method is similar. Roots are solved as before. We must be prepared to obtain less than N roots if aN = 0, etc. We shall assume here that we have obtained N zeros for Part 3. Part 3 of Prony's method is significantly different. We have 2N possible discrete time equations to solve for N complex amplitudes as before. For a "Prony series" it makes no difference which of these 110 N equations we use. So, the first N equations are always used to minimize the computational effort. Equation (4-18) displays the use of the first N equations, but we could have used the last N equations. However, for our particular "class 2 Prony series," it does make a difference which equations we use. Note the right hand side of equation (4-18) is all zeros if we choose the first N equations, but the right hand side is all nonzeros if we choose the last N equations. Hence there are potentially N sets of amplitude coefficients which we can use to fit sampled data points. One might be tempted to use a least-squares fit to obtain a "best" single fit. We choose not to do this for three reasons: 1. We are primarily interested in the t > t" region anyway 2. We do not wish to perform an irreversible operation 3. We do not mind the early-time varying complex ampli- tudes which resemble the SEM "class 2" coupling coeffi— cients. Because of (1), we shall use the last N equations for the "class 1" condition and use the time retarded equation (4-27) to minimize calculations. 2 1 C1 V” 2 1 2 ZN C2 VN+1 = ' (4-27) h-l N-l N-1 Z Z. 2 c 1 2 N v -- J 1.. N. -2N'i. 111 4.6 The Extended Prony’s Method The extended Prony's method (ref. 4-3) is a least-squares formulation of Prony's method. It is actually a quite different method although the equation formulations look almost identical. Equation (4-28) defines the "Prony K-Pulse" for the extended Prony's method. ”vN_1 ... v0 (a1 vN ‘ 12N 2 vN—l ' = ' ' ' ; . 1 (4-28) . . 1 VL+2N-2 °°° vL+N-1_ Lat-J VL+2N~14 The matrix equation (4-28) contains L simultaneous equations more than the regular Prony's method equation (4-14). Because of these addi- tional equations, we may not be able to obtain an exact solution set or "Prony K-Pulse." A standard least-squares minimization of (4-28) is used which is the reason that the extended Prony's method is an irreversible operation. Part 2 of the extended Prony's method is identical to the regular Prony's method. Part 3, the complex amplitudes, of the extended Prony's method can no longer be exactly satisfied for all of the possible values on the right hand side of equation (4-27) or (4-18). Hence, again a least-squares formulation is used again. There is an error associated with both Part 1 and Part 3. There is no known analytical method to simultaneously minimize both of these errors as formulated 112 here. There are many iterative type techniques to perform this opera- tion if desired. 4.7 Complex Root Degrees of Freedom In this section we shall introduce the important concept of complex root degrees of freedom which turns out to be an important predictive parameter ““1 forecasting radar target discrimination power. He shall use a synthetic "class 2 Prony series" for illustrating the effect of this parameter. Note that we are going to use early time values of this synthetic waveform which are zero as in the section on Prony's method on triangular data matrix. This particular data file are calculated for a 1 milliradian incidence. So the odd natural fre- quencies yield natural mode waveforms with large complex amplitudes. There are also 9 even natural mode waveforms with extremely small complex amplitudes. (The reason for 9 and not 10 is that we picked the sample spacing to alias the highest frequency.) The number of independent sample data points constitutes the information content used by Prony's methods. We shall hold this parameter constant through- out this section. We will analyze the "complex root degrees of freedom" by varying the synchronization or starting time as we use the regular Prony's method and the extended Prony's method. The number of sample data points used is 50. For the regular Prony's method, this means that the equation (4-14) matrix is 25 by 25. For the extended Prony's method we will use the 50 sampled data points in 31 rows and 19 columns, yielding a "Prony K-Pulse“ of length 19+1 or 19 roots. 113 Table 4-1 is summary log of the results for various starting times. Table 4-2 is summary of the results in terms of the complex root degrees of freedom. Here it is visible that increasing the root degrees of freedom enhances our ability to obtain accurate invariant parameters of the radar target, namely its natural frequencies. For the same information content, the extended Prony's method is inferior. NM .mucmewuc. 5......_..z H 3. .....am 9c... N ...... uwpaepcsm mozoumwozcz 2H mzmh .9... ..3m.. 0.9... emu o.m~ o.v— D.Nu 0.0.— o.o 9.0. . L _ _ _ r _ I 8393.. 29m .255 a M... 83985 80 mom.) 2 .... .9 o 0! f f 06.06 MQM fiw .y 6 ma? wuzmeHuZH Damzfi ”JMDOZ mmHz 300117dNU 115 rpm Fpm FF. Fpm H~.oH 9 mac: #5. suoH use .Fpm P—m oH.oH H mcoc PF. cuoH can .5.. PP. oo.oH N mcoc PF. Loccm wH aoscm NH om.m m mac: PP. PP. PP. om.m e mcoc FF. PP. PP. o~.m m mcoc PP. cpofi pan .P—m P—m mm.m o mcoc Fpm mcoc PF. me.m 5 use: Pym meo: PFm mm.m w «no: rpm ucoc ppm m~.m a mac: mcoc meo: PP. m~.m oH mco: mco: mco: PF. mo.w mH mean one: one: mco: mm.m mfl mpoom muoom mpoom muoom pecan cm>m uuo cm>m cue m: meowmn uumccou .smwwccou uumccou uumccou mmpa5mm wiNmMMMMmeV chumz meHm Acmpsmmcu chpmz mwxmm mzah .mumm» vogue: m.»coca .H-c u4meao5H me ope “meme op Poccmpxev geese worse-x .Leeee we: owe .xpco goo. “map 5. aFFocoPmmoeoo emoteeeo a... s“..._=mc.m o.s;»..om_< ”3.xeALO. moomm-v a-.. A. .0. mo on-v ..- u H h. V» Lem me o+ A no...pemo. 5...... oz m. oom- oo me oNN- .peeuso Leccm 5cpwgemF< meeoecoureewm OF Fpo k Aop-m_v-me x as.» xpmeeocmu535wm op FF. ..Aop-mmv-mp.x 05.. Fmeez we move: eeo OF we :ewumewmwuceew peoccoe pmewpcou xpmeeocou535wm a PP. mp x as.“ xrmaec03535wm mew: __o P o- mp A mew. .—eee5 we meeoz Loews m we :ewuoewmweceew peegcee goewrgem Anew: meepm> opo5om omv Eeeemcu-ee-meocmoo poem m— chpe5 mF x _m eonpmz m.xcecm emecegxm Anew: mmepo> epe5om omv EoceoLu-me-mmegmoo poem mm xwgpeE mm x mm eospmz m.>coge Lopemom ... m5...5.m .mp as.“ e. 55....3. PF< meeeprQ5m Loews saw: moeez ce>m m moeauwpo5e Lewm5 sew; moeez eeo op meeoz m— we Fo5ce: 5ogm pceewecH c5_ wuozpmz m.>:oLm Lo uwmk flmuoz mEWH-mHOJ .N-emxb.» CHAPTER 5 THE "FAST" PRONY'S METHOD 5.1 Performance Enhancements for Radar Target Discrimination Each of the three major parts of Prony's method shown in the block diagram of Figure 5-1 will be altered significantly computation- ally in order to permit the real time use of discrimination waveforms of up to several thousand sample data points. The suppression of noise, clutter, and radar system distortions are correspondingly enhanced. In Part 1 of Prony's method the N x N data matrix is never formed. Only 2N data storage locations are used. The algorithm will be described by means of the original N x N data matrix, but this is 3 multi- N2 only for illustrative purposes, since we do not perform the N plications and divisions of a direct matrix solution, but only multiplications and divisions. In Part 2 of Prony's method, we remove the firm requirement to solve for all of the roots. Together with the use of skip data samp- ling, close estimates of the major target roots can be obtained along with rough estimates of the spacing of nearby roots. A derivative- free accelerated root solver such as the Hooke-Jeeves method can be used to obtain fast solutions even for noise-like data. In Part 3 of Prony's method we eliminate the second matrix, the transposed Vandermonde matrix, used to solve for the complex 117 118 Part I--lhe K-Pulse From 2N sampled data values . . N Obtain solution set [adisl Yielding Prony polynomial Yielding the "Prony K-Pulse" Part 2--Roots for Excitation From solution set Obtain zeros of polynomial Yielding natural frequencies Obtain excitation vectors Part 3--Complex Amplitudes From zeros 8 sampled data values Obtain complex amplitudes Yielding coupling coefficient Yielding exact synthesis for N sampled data values Figure 5-1. Major Parts of Prony's Method Summarized. 119 amplitudes of the natural mode waveforms. Fortuitously, the fast amplitude solution derived is a ratio of discrimination waveforms which are required for the radar target identification technique derived in the next chapter. It should be observed in the derivation of the “fast" Prony algorithm, that twg solutions sets are necessary in its derivation. Only one is disclosed in the traditional Prony's method. These two different solution sets clarify the operation of the radar target discrimination technique exhibited in the next chapter. 5.2 Part 1--The K—Pulse In Chapter 4 we noted the Hankel structure of the original Prony formulation of the undetermined coefficients in Figure 4-1 and equation (4-12). In anticipation of this chapter, we performed a trivial reflection of the Hankel form into the more familiar Toeplitz form we observed in equation (4-13). The Toeplitz matrix often results from the discretization of a continuous convolution (ref. 5-9, pp. 50). The most general Toeplitz matrix has identical elements along the diagonals from upper left to lower right. Note that this matrix possesses an odd number of possibly distinct values. It will be more esthetic for us to use 2N+1 sample data values and to obtain two solutions of the undetermined coefficients which when we apply our physical constraint (equation 2-18) becomes our "Prony K-Pulse." The matrix or data form we must use is never a symmetrical Toeplitz matrix. For the symmetric Toeplitz matrix, the very popular Levison (ref. 5-1) recursion algorithm as refined by Robinson 120 (ref. 5-2) exists in published form. He will utilize the illuminating matrix description of Robinson in deriving the unsymmetricalToeplitz recursion. The algorithm which we shall derive with its pair of usable solutions leads to the radar target discrimination technqiue of the next chapter. Let us start with Prony's N equations of Figure 4-1 in our notation in equation (5-1). vN+1 + alvN + ... + an1 = O vN+2 + aIVN+1 + ... + an2 = O . o (5_1) + a v = O v2N + aIVZN-l + °°' N N In Chapter 4 we performed a standard matrix solution by moving the data values in the first column to the right side. Here our solution will procede by introducing another equation, equation (5-2). + a + . . . + an = a (5-2) VN 1 N-l o In equation (5-2) we have now used the 2N+l-th sample data value. Also, a new dependent variable, a, which we shall call the error variable, has been introduced. It may turn out that there is a solution {a1}?=1 to equation (5-1) which permits equation (5-2) to be satisfied with a = 0. With noisy data this will be highly unusual, but if it happens, we shall claim that we have identified a "Prony series." Note that if we use both equations (5-1) and (5-2) we are using 2N+1 sampled data points, but the last point is superfluous if 121 we actually have a "Prony series." From now on we shall use equa- tion (5-3) which is a composite of the N+1 equations of equations (5-1) and (5-2). vN+1 + alvN + ... + an1 = O VN+2 + alvN+1 + ... + an2 =,O (5-3) + a = O IVZN-l 4' ... + anN We shall for the convenience of our "fast" algorithm write the N+1 simultaneous linear equations of equation (5-3) as the matrix equa- tion (5-4). __ ._1 7.1.. r. __ vN VN-l V0 . ; a ’ 0 vN+1 vN v1 al I = . ° . -, ‘ 0 (5-4) V ... V a ___‘_’2N 2N-1 IL __Nj I_ _ Although from equation (5-4) it is partially obscured, we will remem- N i sampled data values. This fact is clearer by the Chapter 4 notation bar that the [a1] =1 are determined by only the last 2N of our 2N+1 for the now standard matrix notation of Prony's method given by equation (5-5). Equation (5—5) is obtained by deleting the top row of equation (5-4) and moving the left most column of the data matrix to the right hand side. 122 . __1 ._ .. ._ 1 VN VN-1 . o 0 V1 31 VN+1 v v v a V N+1 N 2 2 N+2 - - - - = - : (5-5) VZN V2N’1 ... VN aN VZN L— ... L. ..T L # Now let us start from the Prony's method of Figure 4-1 once again. This time we shall use the first 2N of the 2N+1 sampled data values. These N simultaneous equations can be written as equation (5-6) instead of equation (5-1). + =0 vaN + bN_1vN-1 + ... VO +b +...+V1=0 b N-IVN NVN+1 . (5-5) + .. bNVZN-1 + bN-IVZN-z + co. VN-1 As we did in equation (5-2) let us introduce another equation with a new dependent error variable. 8. given by equation (5-7) b vN = B (5-7) bNV2N * N-1V2N-1 * ° ' ° Again we denote the combined n+1 linear equations by another equation, equation (5-8). For the convenience of the "fast" algorithm, we shall write the N+1 simultaneous linear equations of equation (5-8) as matrix equation (5-9). For completeness and future reference, we will perform the analogous operation by which we obtained equation (5-5). This time we delete the last row of equation (5-9) and move the right most column to the right hand side to obtain equation (5-10). 123 vaN + bN-lvN-l + ... + v0 = O vaN+1 + bN-lvN + ... + v1 = O . . 1 (5-8) bNVZN-l + bN-1V2N-2 + °'° * vN-1 ' ° bN"2N + bN-1V2N—1 + °°° + vN = B -- — “'1 - 0-1 vN VN-l Vb1 bN o vN+1 vN vO bN-l ' E ‘ (5-9) v v v 1 B _2N 2N-1 ELL _ __ _ r’ “I” "l ' 7 ng vN_1 .. v1! bN V0 E VN+1 vN ' v2 bN-l = _ v1 i b v (5-10) V2N-1 VZN‘Z .. 0 VJ!- L_l -‘ L N_-]:_ Now we wish to reflect on the siniliarities and differences of equations (5-5) and (5—10). Note that equation (5-5) did not use one, v0, of the 2N+1 sampled data points and equation (5-10) did not use a different, VZN, one of the 2N+1 sampled data points. However, the sampled data matrix on the left hand side in each of equations (5-5) and (5-10) is identical. Only the right hand side causes the solu- tions [ai]?=1 and [bi]§=1 to be different. 124 In the popular symmetric Toeplitz recursion, the data matrix is symmetric as well as Toeplitz. Hence there are only N+1 distinct values in the 2N+1 sampled data sequence. It can be observed that for this special case from equations (5-5) and (5-10) that the two solutions [ai]?=1 and [bi]?-1 are, in fact, identical. With these initial observations out of the way, we shall derive the K-Pulse part of the "fast Prony's method algorithm" by induction or recursively. The induction will start from the middle of the 2N+1 sampled data sequence which is VN' We shall use increas- ing larger and larger numbers of the sampled data points such as in equation (5-11) for m 5_N. Ne shall by induction be assuming that solutions exist for some m as in equation (5-11). TIT fl vN vN-l vN-m ( )‘ i am m l VN+1 VN vN-m+1 a1 5 0 . = g 3 (5-11) (m) 3 ' v v ... v a T O - N m L__N+m N+m1 .... L _J L .— Simultaneously we use the same data matrix to devel0p a recursion for our second solution in equation (5-12) which is assumed to exist. —— #- ... ———-n .1 (m) V— vN vN_1 ... vN-m bm O b(m) o VN+! vN °°° vN-m+1 m-l = : (5-12) B "N+m VNim-l vN ...L: __ L_m_ 125 Note that we must superscript our solution recursions since they may vary substantially from the previous recursion, in general. The next step in the recursion is to increase the size of the data matrix by one row and one column. This given rise to two new equations (5-13) and (5-14) where cm and (Im are new dependent vari- ables resulting. " “’l r“ - "~ . vN VN-l vN-m vN-m-l. l 1 i I “m t E v v v § 2 a(m)é i O i g vN+1 N N-m+1 N-m I * 1 i i ‘ g I . . I E l = 3 I (5_13) Q v v v a(m)§ i 0 § vN+m N+m-1 N N+1 m : l I . E V V V v 0 g l C L. N+m+1 N+m N+1 N _. L_ _J L md F"' , F" “ F’ “l E vN vN-1 vN-m vN-m-l 0( ) dm i m vN+1 vN vN-m+1 vN-m bm 0 ° 3 = - 5-14 v v v b(m) O ( ) vN+m N+m-1°" N N+1 1 L_VN+m+1 vN+m "N+1 VN _. L_l - Lem; The matrix equation (5-13) uses a new sampled data point, VN+m+1' This results in the new dependent variable, cm, on the right hand side. Similarly, equation (5-14) uses a different new sampled data point, VN-m-l' This, in turn, results in the new dependent vari- able, dm, on the right hand side of equation (5-14). For measured 126 data containing “white noise," am. 8m, cm, dm will typically be non- zero for all m. Next we shall form a linear combination of equations (5-13) and (5-14) for the purpose of eliminating cm. He shall multiply equa- tion (5-14) by Km+1 and add. In order to keep the notation compact, we form equation (5-15) with the same yet to be determined constant, Km+1° "1 {1 i [-0 a(ml) E agm)§ l bém) . =l . g +Km+1 . (5‘15) - l (m+1) D g e .émil L0 .J i_1 J using the updated "Prony K-Pulse" of equation (5-15), we may more compactly write the desired linear combination of equations (5-13) and (5-14) as equation (5-16). r- fi 9' vN vN—l "' vN-m vN-m-l 1 ’am Km+1dm-1 v v v v a§m+1) 0 "° N-m+1 N-m - s N+1 .N . . . ‘ . (5-16) j . o . ; .(m+1) i 0 ivN+m VN+m-1 ... vN VN-l f a? 1) 2 , “1+ = + bN+m+1 vN+m ° vN+1 vN __é Lam+1 . Lcm Km+18m4 127 By examination of equation (5-16) we can see that if we Km+1 by equation (5-17) and the new am+1 by equation (5-18), we will have advanced the induction or recursion (Ni equation (5-11) by one step. K m+1 — 'cm/Bm (5-17) (5-13) OLm+1 = 0‘m + Km+1dm Note that equation (5-17) is well defined if and only if 8m is not zero. If “m or 8m is zero, we shall set flag 1 and terminate the K-Pulse part of this algorithm and reset the output length of the K-Pulse at m+1 instead of N+1. Again for noisy measurement data this is unlikely to occur. Similarly, we shall form a different linear combination of equations (5-13) and (5-14) this time for the purpose of eliminating dm in equation (5-14). We shall multiply equation (5-13) by Lm+1. and add. Again in order to keep the equation compact, we form equation (5-19) with the same yet to be determined constant Lm+1. WW 3 l ‘3 1 bém+1) béml agm) (5-19) 3 = E +Lm+1 3 b("(+1) bgml aém) Ll. _‘ 1 __ L9 . 128 Using the updated "Prony K-Pulse" of equation (5-19), we may more compactly write the desired linear combination of equations (5-13) and (5-14) as equation (5-20). "p '_ b(m+1)H "' ‘7 [CN vN-l '°° vN-m vN—m-l bm+1 dm+Lm+1ami 3 b(m+1); 3VN+1 vN '°° vN-m+1 vN-m bm ; 0 1 E o ‘ 0 (5-20) I: = ; ! (m+1) , 3VN+m vN+m-1 °°° vN VN-l 1 0 l vN+m+1 vN=m "' vN+1 vN ._ ..E _J 8m +CLm+1 mj By examination of equation (5-20) we can see that if we may choose L by equation (5-21) and the new 8 by equation (5-22), we will m+1 m+1 have advanced the induction or recursion on equation (5-21) by one step on the index m. Lm+l - -dm/am (5-21) em+1 = em + Lm+1cm (5-22) We now possess the (m+1)-th inductive solutions predicated only upon the existence of the m-th solution of equations (5-11) and (5-12) and also amf0#8m. However, we shall set flag C if cm=0 and flag D if dm=0. We shall also terminate the K-Pulse part of the algorithm at m for flag C just as we did for flag A. Table 5-1 summarized our conclusions on the significance of these two flag terminations. We may use flag D in the next chapter. 129 Table 5-1. K-Pulse Termination Flags. Termination Flag A Flag C Flag D Case (Singular) am=8m O #0 f0 (cm.dm) Not Applicable (051m) (cm,0) For nonzero unsymmetric Yes No No data, Are samples of a "Prony series identified? For nonzero symmetric data Yes No No are samples of a "double- sided" complex exponential series identified? Is a "class 2 Prony series" Not Applicable Yes Yes a possibility? Is a "2nd K-Pulse Convolution" Not Applicable Yes Yes a possibility? 130 The induction still requires a starting point for which there exists a solution pair. For m=O, this is easy since equations (5-11) and (5-12) become scalar equations. It is easily seen that if vN is nonzero, the solution pair exist. Figure 5-2 is a flow chart diagram for the K-Pulse part of the "fast Prony's method algorithm." For illustrative purposes we shall perform the first two recursions for m=O and m=l in detail in Section 5-6. 5.3 Part 2--Roots for Excitation In the derivation of the regular Prony's method, a Prony poly- nomial was formed as in equation (5-23). P(z) = 2N + alzN'1 + . . . + aN (5-23) N where the solution set [a1]1 =1 is the final coefficient set obtained in the first part of Prony's method. Since equation (5-23) is of degree N, it possesses N (possibly nondistinct) roots and can be factored as equation (5-24). N P(z) = H (z—z.) (5-24) As previously pointed out the same solution set can be used to obtain what we have previously defined as the Prony K-Pulse, given by equation (5-25). [Kn]fi=0 = [anfl=0 (5-25) where a 1 O 131 Start m=0 N (degree) a0=1, b =1 0 L 1" Calculate Reduce a m m Y ? Is(om-0) r N Calculate degree N + m \W/ Calculate Km+1, Lm+1 Calculate {a(m)}m 1 = (m) m {bi }i=1 k Finished. (N) N {ai }i=0 Figure 5-2. (N) N Flow Chart for K-Pulse part of $m+m+1 "Fast Prony's Method Algorithm". 132 For the third part of the "Fast" Prony's method, we shall use z-transforms of the Prony K-Pulse sequence which is easily calculated to be equation (5-26). K z-n K(z) = . n 1 (5-26) IIMZ 0 Note that the z-transform of the Prony K-Pulse contains negative powers of 2 instead of positive powers of 2 as in the Prony poly- nomial. Equation (5—27) gives the relationship between the two. 2 K(z) = z-NP(z) = H (1-zi/z) (5-27) i=1 The regular Prony's method requires the determination of all of the roots of the Prony polynomial. This is because the Vandermonde matrix of equation (4-18) requires all N roots. There are many pub- lished methods of finding all of these roots. Standard computer library routines such as INSL appear to perform quite satisfactorily up to N=IOO for the Prony algorithms. Numerical conditioning may become a significant problem for very large N. lRooting algorithms for largest degrees give this warning. For target discrimination we shall require accurate knowledge only of the target natural frequen- cies which have large amplitudes. This is because they are the invariant parameters of our radar problem. In Part 3 we alter the method of determining the complex ampli- tudes so that we relieve the requirement to solve for all of the roots 133 in Part 2 of this algorithm. This is an order of magnitude reduction in computational requirements. Figure 5-3 illustrates the iterative process by which we avoid solving for all roots in the Prony poly- nomial of very large degree. We first perform the skip sampled data "fast Prony's method algorith" solving for all roots. 'Ne determine which of these natural frequencies has large target return energy. These are the “Targeted Roots" that we shall track and find in the higher degree Prony polynomial for the dense sampled data. By using the properly scaled "target root" and initial step size, we can solve a high degree polynomial by an accelerated method such as the Hooke-Jeeves (pattern search) algorithm. Appendix D contains computer code validatable by reference 5-8. Since only the natural frequencies are invariant, we must use equation (5-28) in the rooting process, where the subscript m denotes the skip sampled data parameter and n denotes the denser sampled parameter. 1 _ = -1 - clog(zmi) - Si Tn clog(zni) (5 28) Tm Current data sequences tested are less than 256 signal sampled data Pcrints. The limit of this method is well in excess of this number. Nith the root Zni determined, the i-th mode excitation vector, 51(2) is obtained by synthetic division on the K-Pulse, K(z). Since 21- has already been determined to be a root, the division is exact. 134 PART l--The K-Pulse Skip Sampled Dense Sampled J. PART 2A-Roots for Excitation lst loop, find all roots 3 O wk PART 28--Targeted Roots & Targeted Excitations PART 3--Complex Amplitudes Skip Sampled Dense Sampled 0® ‘ ® ® Figure 5-3. "Fast Prony's Method Algorithm" Block Diagram. 135 5.4 Part 3--Amplitudes and Couplin97Coefficients The third part of Prony's method calculates the complex amplitude or residue of each complex exponential in the Prony series which is written in equation (5-29). [Vn]:=0 A1.[exp(s1.Tn)]:=0 (5'29) do ll M2 ...I Even if equation (5-29) is not true for all time, it can be satisfied exactly for a time sequence of 2N sample data point (contiguous) by allowing exactly N complex modes in equation (5-29). Ne can use Part 1 of Prony's method to obtain the Prony K-Pulse given by equation (28). We can also write it in its equivalent form of equation (5-30) even if we have not yet solved for all of the roots or natural frequencies. [kn12zo = [1,-eXP(s]T)J*.-.*[l.-exp(sNT)] (5-30) Convolving the K-Pulse with the signal sampled data sequence, we obtain equation (5-31) [kango *[vn]:;o = [1,-exp(slT)]* . . . N *[1,=exp(sNT)]*.)21A1.[exp(s1.ln)J‘I":O (5-31) 1: Neact recalling our definition of the j-th mode excitation waveform, it cart also be written in factored convolutional form as in equation (5-32). 136 j N-l N ' [an]n=0 = H [11*[l.-exp(s Tl] (5'32) . . l lfJ There is one simple identity which we shall use repeatedly. If the convolution of a single complex exponential with the inverse couplet, given by equation (5-33) [exp(siTn)]:;0*[1,-exp(siT)] = [1,0,0 . . .] (5-33) Using equations (5-31) and (5-30), equation (5-29) becomes obviously of finite duration and is given by equation (5-34). N-l [kn]*[vn]:=0 = A1IEflln=O (5-34) 2 1: l Hence the output convolution of a sampled data Prony series with its Prony K-Pulse is of length NT in time and is actually the N different j-th mode excitation waveforms weighted by the complex amplitude of the j-th mode in the original Prony series. Taking the z-transform of each side of equation (5-34), we obtain equation (5-35). N . z'”]”'1 = A1E1(z) (5-35) A.[E. _ "‘0 i-1 (ch) = K(z)5zomm pmp map: :99 moo: >zomm pm_ mm“: :99 D 9 140 .o-e o.=o.. a. oo.......=m =sgppeoope cospoz ...eoea pm... mozouumozcz z_ u:_» 9.: 9.2 9.9. 9.9 9.9 9.v 91o. 9.9. F p . . L p » ... w a... m. Tr w 0 i. 9 .w 99299999292 29 w:_p 9.: 9.2 9.9. 9.9 9.9 9.. 9.~ 9.9 r p p p p b p . .0. 0 a. 0 .w .m 1.. w .m m moo: >zomm 0mm map: =m_ SNOIlUlOH 81381330 wozouuwozcz z~ uz_h .mum mgampu o.m. o.m. o“. o“. oww ow. o.o. mozcuwwczcz 2— wtuh o.m. o.m. owml‘ o“. ow. ow. o.op ‘ a w woo: >zomm 0mm wmaz zap .OIIBOHlIWdHH 101-3001116HU 141 m.m~ 0&0 HG N + monm vwcm+ mm ANuZuANVAH+zv N\2mu z 2w.mpup ANNN ....-. .... N..paoop _ opepoeszm aocu a Popcoz m.mmm no me + was“ omv+ 9mm A.-..AN+2.~ e-M\zo+ N z~+m\mz 2N". ANNN oo.mum .... .eo>ou axmoposu xoco a Poxeoz N.omH o + N.o~H com oooapoe. m\z+pfip+zv+~\~zm+o\mz 2N". Aoum ..omv xxmopocu mono. -co>ou oxo—pmoz Hmop N-zmp+mzn+.fip+zv Amcoppopom NV p+zwup A... ....-. .... oucopgo>ou poo: o.m .o+ o.~ New 0‘ “--.“.r- --. Q ~+zm+ N z AN-. up... Amcoppopom N p+z~u. Agog. ...... cop...s . .0 mppco op cospuz Lop mcoppocmoo mo Eom omcpoomc mp mcoppocmoo ooo. xpepoz gpmcmp EospcpEup .czpoo wo mmemmouz oospmz mm—ooux mcopmp>po. mopo mcoppouppopppoz mcoppocooo Egppcom~< can: cop mcoppoeooo mo Eom .H pcoo com compgooEou coppopaosou .mcm upm>2N) are required to compute reasonable estimates (Ref. 5-4) of the processed symmetrical sample data points than are required for the "fast" Prony method 3. If the raw data are not symmetricized, the "fast" Prony method obtains twjg§_as many solutions, each of which is different and useful in the radar target discrimination problem 5.6 Special Cases for SEM Computations Numerical zeros play a crucial role in the Operation of the "fast Prony's method algorithm." First, if equation (5-40) holds, Part 1 of the algorithm must be terminated, returning the "Prony K-Pulse" of equation (5-41). v(NT) = O, of the 2N+1 sampled data (5-40) {k }” i i=0 = {1,0,0, . . . 0} (5-41) 144 This special K-Pulse is just the identity operator and is a necessary form for use in the next chapter. We shall identify equation (5-40) by noting that equation (5-42) occurred in Part 1 of the algorithm. = 0 (5-43) A much more desirable event to occur is equation (5-53) aN = 0 (5-43) This event tells us that lower order errors do, al, a2, . . ., aN-l were all nonzero and that the next iteration is now singular. This means that we have an exact Prony series solution to the 2N+1 sampled data. Note that equation (5-42) is consistent with this interpreta- tion for the 2-1 + 1 = 1 sampled data point. To illustrate what happens, we take a simple "Prony series" given by equation (5-44). vn = Aexp(51Tn) = A2: , n > 0 (5-44) 0 , otherwise This is simple enough; we can perform the computations by hand. We do so on the worksheet of Figure 5-6. Note that o1=0. This is our singularity flag A which tells us we have successfully detected a 1 term Prony Series. From the summary of Figure 5-7, we can tell that we have a (forward) Prony K-Pulse given by equation (5-45) with trailing zeros appended if desired. 145 Function Type is Natural Mode vn - o, n m=0: 2 2 [1n 'n-1][l:] , 1°o:l “0‘ "n ' A121*‘222 . . 3 3 vn+1vn 0 _c0 c0 vn+1 A111M222 1" c c ‘1 [vn vn_1][0] . do] do vn_1 A12111222 I . 2 2 Vn+1vn 1 80 80 V" A121+A222 K --c /B :-v [y 9 (A Z3+A 3 2 2 1 0 D n+1 n ' 1 1 2221/(Alz +A2251 11.-do/aoe-vn_l/vn.-(A122+A222)/(A121+A222) m=1z 3 3 F1 . [1 111 1 K111’(A1z1*:2’21(A1:1*A2’21 . 3 3 31 a] +K1 . . (A121) +(A222) +A121A222+A121A222 L 1.1 b F q H — -2 2 2 3 3 bi F0] F1 [11‘ 1"(#1430 12A1’1A2’2‘A1z1Azzz’A1z1A2221 +L I 1 .1 1 3 5L 1'; ‘35241’1A2221z1'22121'11 11n- Vn-l'n-z 1 11°1 “1'“011'K1L11‘aoeozA1’1A222121’22121'11 - vn+1vn 'Vn-l a} ‘ 0 tvn+zvn+1vn ° _‘1 ‘1'“121*A2‘3'351111zi*12231(“121*1223) ‘3631121A223121'2212"51z122 Vn vn-1'n-2 0 d1 d1"Bo (“121*‘2221 +“1"‘2 vn+1vn vn_1 b: . o =851(A1A22§+A1A223+2A121A222)=-slzilz;1 vn+2vn+lvn 1 81 81w] K29-c1/8132122 -1 -1 Figure 5-10. Two Mode Prony K-Pulse Worksheet #1. 152 Two Natural Modes Function Type is Upper left conrner of Matrix n - (:::> 0, n<0 A121+A222, otherwise m=2: r1 1 Fb'l ti'li+bi1-'(“121*A222)/Bo'z1’2‘A121*A2’2)/Bo a; - a: +K2 b: .-50 1(A121(zl+zz)+Az 22(21 +22)) 2 - 2_ . 8-(2 +2 ) ...2 0 L1 I2 K2 2122 1 2 P 2 P 2 . 2-12-1 b5] 0 1 1 b2- L2 1 2 . bi . b} +L2 a: 2 1 1_ 31(A z +A z )- -21 1 2.1(A3 z +A 2 3)/B l 1 0 bl-b1+Lza- ‘ 0 1 1 2 2 1 1 2 2 0 5- L d =_(z-+12'1) v" Vn-lvn-ZVn-Zi [1 1 {“2} “Zulu-K?» Cl1(1'22122 11221:) O ’ 2 vn+1vn vn-lvn-Z al 0 2 'n+2'n+1'n 'n-l '2 0 H4) 5 l'vmyvmzvm1v" J L0 . :2. c2- A121+A222-(213+22)(A121+A22 2 F _ T _ +212 2(A z 3+A212)— =0 1 ”q 'n-i'n-2'n-3l ° d21"2 (“121*A222)21 z2 "A1*A2)(’1*’2 ) 2-1-2 vn+1vn vn-lvn-Z b2 0 "Alzl +A222 2 vn+2vn+lvn vn-l b1 0 Vn+3'n+2'n+1'n L1 I Bel 32‘31(1'K2L2" “1‘0 L . . L K3"°2/82' Exit at m=2. L3"‘2/°2‘ Figure 5-11. Two Mode Prony K-Pulse Worksheet #2. 153 Function Type is no Natural Modes vn - 0’ n n "(0 A121+A222, ’ otherwise Upper left conrner of Matrix n - Q (m) (m) , m 6i bi “m 8m cm dm Kn+1 Ln+1 2 2 3 3 A z * A z3+A z3 A z +A z 0 1 1 A121+A222 A121+A221 lAlz 1 1 2 2 . 1 1 2 2 1 1 2 2 ‘80 ’30 -1 -T 120 A z A zi-B z 2 p8 z -1 -1 1 K (A z +A z ) 1 1 2 l 1 2 l.l 2122 21 22 1 l 1 2 2 ( )2 -1 -2“ ' 21-22 22 -1 2 l 1 0 0 ~A121 -1 1 ’(21*22) ’(21 +22 ) -Azz;1 z z 2.12.1 l 2 l 2 Figure 5-12. Two Mode Prony K-Pulse Worksheet #3. 154 K-Pulses has the correct roots with no dependency upon complex amplitudes of the original "Prony Series." Note for the shorter sequencies at m = 1, complex amplitude dependency still exists. Note carefully, if one amplitude is much larger than the other amplitude, then at m = l the shorter sequences approximately contains only one root without amplitude dependency. Now we shall observe the effect of violating the late-time ”Prony series" as we did in section 4.7. Let us alter equation (5-49) by only one sampled data point. The different point is given by equation (5-50). v = 0 (5-50) He will simply reuse Figures (5-10) through (5-12) and note the dif- ferences. First, 3(2) sequence is a perfect Prony K-Pulse, but a2 f 0. So the algorithm will not be terminated by singularity flag A. Since C2 and 03 will be zero, we will terminate at m = 2 by means of flag C. Also note that d2, L3, and b(2) are not the same as before. Next suppose that equation (5-49) is modified by both equa- tions (5-50) and (5-51) Recalculating in Figures (5-10) through (5-12) only the steps which differ, we find the short sequence a(l) is as far as we can go without significant differences. At this step if one of the two amplitudes 155 were much larger than the other, we would have approximately iden- tified one root without amplitude dependency. The pattern we are observing is that for a sequence composed of significant energy in only a small number of high amplitude modes, low "complex root degrees of freedom" will yield an approximate short K-Pulse which is amplitude independent of these high amplitude modes. This is compatible with the results of section 3.7 when the iden- tified roots were always identified in batches. All large amplitude modes together and when sufficient "complex root degrees-of—freedom" occurred, the finite remaining small amplitude modes. CHAPTER 6 RADAR TARGET DISCRIMINATION TECHNIQUE 6.1 Requirements for Automatic Radar Target Discrimination We have now developed a number of analytical tools for radar target discrimination. The last remaining tool is the radar target discriminant itself. we choose not to embue our radar target proces- sor with any learning ability or artificial intelligence. It must, however, be completely compatible with normal radar defects of clutter, thermal noise, propagation scintillations, receiver dis- tortions, coded transmitter waveforms, etc. We shall adopt the following three processor requirements for the radar target discrim- ination: 1. Criteria must be radar target aspect-angle independent 2. Criteria must be radar range independent 3. Only reversible operations may be used Requirement (1) is obviously desirable since it deletes the neces- sity of storing extremely complex radar target data files within the discrimination processor. Its implementation may not be obvious. Requirement (2) is also desirable and its implementation should be more obvious to a radar designer. Angle tracking radars are required to solve this problem. We choose a monopulse-like technique 156 157 for both requirements (2) and (1). By means of a normalizing channel, we shall develop a radar target discriminant from a "mode ratio dis- crimination detector." Requirement(3) will be achieved by avoiding least-squares techniques in our processor and analyses. 6.2 Discrimination Algorithm for Radar Targets Figure 6-1 illustrates the discrimination algorithm which we shall test on empirical radar target data. The significant features to be noted are the use of the following original analytical tools: 1. Dual "polar mode A-scope" displays 2. Double "Prony K-Pulse" convolutions 3. "Fast Prony's method algorithm" Data which must be stored in the memory of the radar target discrimina- tion processor are for gagh radar target in its library: 1. "Prony K-Pulse" for the radar target antenna terminal response 2. Natural frequencies (or invariant parameters) of the radar target which may have significant energy in the radar return signal Note that our "Prony K-Pulse" for each specific target is specifically measured (or calibrated) for our specific radar. This K-Pulse file will contain natural mode waveforms of our radar system which may be excited by the radar target. Table 5-1 gives the functions which the radar target discrim- ination processor must perform. 158 Radar Target Return AID Converter é A (TITS! Prony I.Pulse Convolution J, “Fast" Pr n Bl °’ Algorithn second Prony K.Pulse Convolution [$90. O .5 € €@ @ D C2 0' Mode Excitation Convolution Target Trigger Channel a Target Identification NA D Sm-andooifference for r) 2) Figure 6-1. Radar Target Discrimination Algorithm Summary. 159 TABLE 6-1. Radar Target Discriminant Part Procedure A 8-1 8—2 Target or lst Prony K-Pulse convolution is performed using a previously measured K-Pulse on this radar and this target. The output of the convolution should contain not signifi- cant natural mode waveforms of the "right target" or the radar system. For the "right target" only return, after the N+1 samples of "early time" response, most remaining energy will simply be thermal noise and nonsuppressed clutter. "Fast" Prony algorithm is performed to detect any residual natural mode waveforms of significant energy. The primary output of this step is the second "Prony K-Pulse." The second Prony K-Pulse Convolution is performed on the original clutter-reduced radar target antenna terminal response file. This should kill the unexpected natural mode waveforms of the unknown target and radar system, but not necessarily the natural mode waveforms which are expected and would have been killed by the first "Prony K-Pulse." We shall call this output convolution the "2nd K-Pulsed" file. We now perform mode excitation convolutions on both the original clutter-reduced radar target antenna terminal response file and the "2nd K-Pulsed" file. The mode excitations are derived from the "Prony K-Pulse" and radar target natural frequency in the manner of the "fast Prony's method algorithm," part 3. The dual "polar mode A-scope" displays are for a human radar operator discrimination. The machine radar target discriminant uses the sum-and-differences of the dual traces. The difference files must be level even in the presence of thermal noise for a "right" target. The sum file is necessary to detect target energy above a noise threshold. 160 6.3 Part A--Target Library Prony K-Pulse COnvolution Let us start the analysis without the inclusion of noise. In this case in the frequency domain at the antenna terminal we have equation (6-1) which we have simplified slightly from equation (3-5). 1.5) + § “k(s.6.d>)(S-sk)'1 (6-1) v(s) =1] ( 9 k=1 Now W( ,s) is the entire function originating in equation (2-15) and the Ak(s,6,¢) are the entire functions required by the "class 2" coupling coefficients observed in the retarded scattered E-field. After the sampler, the invertible, but more precise modified 2- transform in equation (6-2) is more appropriate. -1‘ e m -1 , L1 Ak(z,m,e,¢)zk(1-zk/z) (5 2) V(Z.m) = W (1.2,m)+z e k Stored within the radar target library of our radar target discrim- ination processor is a specific "Prony K-Pulse" of length N+1 which must contain the radar target invariant parameters for which much of the target energy will be located. Equation (6-3) is the target library K-Pulse. K(z) = (l-Zi/Z) (6'3) I ":12 1 Ida first perform a time domain convolution of the clutter-reduced radar target antenna terminal response, v(t,m) from equation (6-2) 161 with the target library K-Pulse, k(t=nT), of equation (6-3) to obtain the "first K-Pulses convolution," y(t,m), representable as equation (6-4). N Y(z,m) = K(z)lvle(1l,z,m) + z'lkZIAk(z,m,e,¢)z': n (1-2142) = wk N . -1 +z{n(1-. 13A, '"- -1 i=1 Zf/Z) k=N+1 k(z m.6,o)zk(1 zk/z) = ETe(z,m) + NMN(z,m) + LT(z,m) (5-4) We wish to discuss each of the three terms given by equations (6-5), (6-6), and (6-7) separately. ETe(z,m) = K(z)we(fl,z,m) (6-5) -1 N m NMN(z,m) = 2 Z Ak(z,m,e, (6-22) Aj(t,m) For our special case of the "right target" equation (6-22) is a com- plex zone for all time. Even small amounts of thermal noise will not greatly disturb this general flatness. We still need equation (6-21) as a trigger channel for our discrimination processor to tell us when radar target energy is present at the antenna terminals. Next we need to examine special cases with noise and "wrong target" to fully explore the power of this technique described. The baseline low noise case is summarized in Table 6-2. Now it should be obvious that thermal noise in the radar system must degrade both our target trigger channel and our target identification channel, Aj(t,m). However, since we have performed nonlinear operations, it is not clear what the effect of low level thermal noise might be. We need to return to Part B to evaluate the effect of thermal noise in the radar receiver. Equation (6-9) becomes equation (6-23). —r.-_ F— 1 [_yN-1+€N-1 yN-2+€N-2 °°' y0+€o E1 EN EN yN-1+€N-1 °'° y1"51 g2 EN+1 = - . (6-23) 8mm 6mm yN-1+€N-1 gN E2N-1 L. _J..- L... .J 167 .} .I . el’a..-‘ LTII. Anwummmc mxmzpa maopm mumcwmmu mmwa m>wpwmon FmUwaxu «up; Aacm 4F“ mamaEmu once aumcgoucw saw: mucmmmga ummcmu mo copmuwccw mamasmn mace pumccou saw: mucmmmca amuse» do copmumucm Am\nvaaam :moz unopm>=u =uomcm» pumgcoucH= =pmmgm» uumcgou: =mucmmmga Humane: mcwgau au_amwo .aLwEE=m wucmscomcma Louumuwo cowumcmsmcumwo cowumcwewgumwo xmpasou mcp mo mpcmcoasou ovumm mvoz .Nio wpnmh 168 Equation (6-23) cannot be solved by inspection. One would be tempted to speculate that relation (6-24) would hole for small noise levels. lgi| << 1, i = 1, 2, . . ., N (5-24) This does not always seem to hold, but all that is necessary is that kp(t-nT) act similar to an identity operator on v(t,m). This must be the case unless there are one or more strong complex exponential waveforms within y(t,m). This would be the case for the "wrong target," and complex root degrees of freedom are consumed to kill the complex exponential. Remember that the error variable, am, told us if we success- fully found a "class 1" Prony series. Flag C of the Part 1 algorithm told us if we successfully found a "class 2 Prony series." What happens ”success" is not complete is not theoretically clear, but empirically it appears that the complex root degrees-of-freedom will suppress some thermal noise in the NT §.t §_(2N-1)T retarded time region if not used to kill a complex exponential waveform. 6.7 Empirical Illustration In this section we shall present the empirical validation of our radar target discrimination algorithm on the radar measurement data received from NSWC. Figure 5-2 shows the two sampled data files we shall use in this section exclusively. The "right target" for which Target Library Prony K-Pulse is obtained is for a 12-inch thin cylin- der'(L/a = 200) at a 45 degree elevation angle, 12" above the "' 169 .mmme mama umpasmm gummcah czocxcng .mim mczmwu wozoumwozmz 2H MZHH . 9mm o.LmN 9mm cm: 9%: cum o.o. 8813 ..ms .bwmfi 0283/ m. 8813 ..g ..moms. Egg. 1.... C . J... 3.0me ozomz= w .Lmomcp Frog: r 0’9 300111dWU 170 conducting ground plane. The "wrong target" whose target library is not used, is a normal incidence 9" wire (18" imaged with L/a = 400). Performing the Radar Target Discrimination Algorithm of Figure 6-1 and Table 6-1, we obtained the dual "polar mode A-scope" displays for each target for the first three modes of the "right target." These are displayed in Figures 6-2, 6-3, and 6-4. In each case the "right target" A-scope displays are on the left and the "wrong target" A-scope displays are on the right. Ideally, a “right target" file with different noise samples should be used in this illustration so that the dual traces of the A-scope displays will not perfectly align. The independent supplier of the data was unable to comply with the request. The next best method was to use a slightly defective lst (or target) K-Pulse. Note that this part of the technique is performed in the original radar/ target calibration and is independent of the real-time target dis- crimination algorithm. In this case the target K-Pulse was synthe- sized using the extended (least-squares) Prony's method. This means that there could not be a perfect noise suppression for gay part of the data file. This is believed to be the reason that the "2nd K- Pulsed" file in Figure 6-2 shows a 2.6 dB processing gain over the normalizing file in the envelope display. In both the Ist and 2nd mode displays, the termal noise in the original data does not obscure our discrimination process because we are looking for differences between the dual traces. The differ- ences are alight for the"right target' but quite large for the "wrong target." In the "wrong target" envelope displays there are extended 171 .mcoz ummgmk umfi com mampqmwo =waouml< muoz gmpoa= pmzo .mum manna; mozouumczcz z- u:_» mozouwwozcz :— w:.— o..- o.~_ o.o. 9.. 9.. c.. 9.“ 9.: 9.9» o.w« o.ou o.o. o.o. 9.9 9.: p b p p . p b b . . O'Bl' 0'01- SNOIlUlOU SNOIlUlOU mozouumozcz z— utub mczcuwmozcz 2“ Utah o.o.. 9m. 3...: an. own a... own ewe» ......» emu 3.; 9w. ...». own a...» m a so 2“” z woo: >Zomm .—w~ ...—.mwms. $29.3: moo: >Zomm bW— ..Hmwmdp :55... 81391330 172 .muoz ummcmp new cow mzmpammo =wgoumi< mvoz Lupoq= pane mozouwwczcz z. w:_b 0.0— o.~— 0.6- 0.. 0.. 0.0 o.N 9.9 . p P A F L p F r 9.9- o.~— F » moo: »zomm DZN ...—moms. ozom? .¢-a acamaa mozouwmozcz :— u:_b 9.9?» 9mm 9.?“ 9m— 9.w. 9P. 9.9. ,w w ,w m a mm a an a mozouuwozcz z. u:_h 9.99 9.99 9.99 9.9. 9.9— 9.9 9.9. p h F P p TP PM a m. m 3 m u a .m woo: >zomm ozw ..Eme. :52: SNOIJHlOU '91391330 173 time intervals for which the "2nd K—Pulsed convolution" is 15 dB below the normalizing trace. In the 3rd mode display of Figure 6-5, the differences are still visible for the "wrong target.“ For the "right target? it is known that the 3rd mode is at best 11 dB lower in level from the lst mode. Discrimination in the presence of more noise would likely be difficult for this mode. This plots illustrate how the radar designer would establish thresholds for Part D of the algorithm for automatic discrimination. 174 .muoz Humane ugm Lo» mampgmwo =¢aoum-< muoz meoa= peso wozcuwwozcz z. u:_~ 9.9. 9.~_ 9.9— 6.0 9.0 0.. o.~ 9.9. p p . b P, P p l o nfiflwss W 1] L-l---IIIL¥ QZN 11 0 >220 zozkboxu um r In W rm ....u mozoummozcz z- man» 9.: 9.“. 9.99 9.9 9.9 9.. 9.9 9.9. . p . P p p b M W moo: >zomm 0mm flags: wzomz: SNOIlUlOU 91381330 .muo 9.53... mozouumozcz z— uz- O.Wm a.m~ D.ww o.m¢ o.m— own a o. m wozouwwozcz z— uz—h 9.mn 9.ww 9.m~ me— cwmu own 9.JP moo: >23: 0mm ..Eumg :63: SNOIlUlOU 91391330 CHAPTER 7 CONCLUSIONS In Chapters 2 through 5, we developed the analytical tools of the "fast Prony's method algorithm," the “Prony K-Pulse," the "polar mode A-scope," and the "mode discrimination ratio detectors." In the last chapter, using only these basic analytical tools, we developed a radar target discrimination algorithm which worked very well on empirical data from a strong clutter background without Doppler shift exploitation. Different radar operating modes and environments will require more elaborate radar target discrimination algorithms. To fully exploit these radar target discrimination principles, faster digital logic is advisable. When devices are available which even approach the speed and power of the 100 GHz controllable binary counter of reference 7-1, it is confidently believed that the forecast of the cartoon of Figure 7-1 will become reality. Original contributions of this dissertation in addition to the prinary objective of developing a range and aspect-angle independent radar target discrimination technique potentially compatible with "quiet" radar, are four original analytical tools developed solely for this purpose: 175 176 .au?Fwnmqmu :owumcwEPLUmwo “macaw sauna nmumcumcou .H-N mesmwa 177 "Fast Prony's method algorithm" for real-time invariant parameter calculation of 4-dimensional radar data. "Prony K-Pulse" for calculating SEM coupling coefficients from retarded scattered E-field sampled data. "Polar mode A-scope" display file processing replacing part of the conventional radar target- independent matched filter. "Mode ratio discrimination detectors" for auto- matic radar target trigger and identification channels. APPENDICES 178 APPENDIX A LAPLACE TRANSFORMS AND Z-TRANSFORMS FOR PRONY SERIES 179 APPENDIX A LAPLACE TRANSFORMS AND Z-TRANSFORMS FOR PRONY SERIES A-l Introduction and Notation We shall make frequent use of the Laplace transform pairs of electromagnetic quantities which we will consider in this disserta- tion to have Laplace transforms and inverse transforms for some appropriately defined region of convergence. In addition, for the purpose of the ordinary z-transform only, we must temporarily impose continuity on the time function of the electromagnetic quantity. This is because the integral transforms we shall use, essentially ignore removable points of discontinuity, but these same removable points of discontinuity will drastically effect the ordinary z-transform if they are, in fact, in the sampled data points. This additional requirement of time domain continuity is not required for the modified z-transform unless it is to be compared to the z-transform of a specific discrete sampled data sequence. All electromagnetic quantities shall be presumed to satisfy these requirements and will be denoted in this chapter by the letter v. We will employ the following consistent convention for physical quantities and func- tions: 180 181 v(t) the time function (in lower case) V(s) the Laplace transform (in upper case) V(z) the z-transform (in upper case) V(z,m) the modified z-transform (in upper case) A-2 Ordinary z-Transform and Laplace Transform The Laplace transform pairs for time waveform of a continu- ous causal electromagnetic quantity, v(t), which we shall hereafter assume possesses a Laplace transform given by equation (A-l). The constant a is called the abscissa of absolute convergence of the transform and is determined by the values of s for which the integral converges absolutely. Equation (A—Z) is the inversion formula for the inverse Laplace transform. L[v(t)] = 9(5) = f v(t)exp(—st)dt, Re(s) > a (A-l) O c+joo ‘1 = = " d A-Z L [V(s)] v(t) f. v(,s).2_sfi,c>a,t>o ( ) -joo Figure A-1 is a diagram of a typical continuous transient waveform (v(t) as a function of time. Also indicated in this figure are the discrete sample values v(OT), v(1T), . . ., v(5T). These discrete values are given exactly by equation (A-3) v(nT) - v(t for n = 0, 1, . . . (A-3) )lt-nT ’— b /‘t. . ///_,, .. \\\\\\\. V 183 Next it can be seen that v(t) can be approximated as a function of all time values by a step function of the discrete samples of equation (A-3) by va(t) given in equation (A-l). va(t) = : v(nT) pT(t-nT) (A-4) n=0 pT(t) is a rectangular pulse of area T given by equation (A-5) [Ll-(E) = 1, O LtpT(t-nT)] (A-6) n= 11: ‘The Laplace transform of the pulse function pT(t) is given by equation (A-7) and its time shifted version by equation (A—8). LEpT(t)] = L[U_1(t)-u_](t-T)] = [l-exp(-sl)]/s = PT(5) (A-7) L[PT(t-nT)J = L[pT(t)] exp((-sTn) = PT(S) exp(-sTn) (A-8) 184 This gives equation (A-7) as the Laplace transform of our step func- tion approximation, va(t), to v(t). L[va(t)l = v(nT)PT(s)exp(-sTn) = PT(s) ; v(nT)exp(-sTn) (A-9) n - 0 n—O HMS From equation (A-9) it can be seen that the Laplace transform of va(t) has two distinct now separated characteristics. PT(s) is related to the shape of the approximating pulse function of equation (ILS). The summation on the right of equation (A-9) is independent of this pulse and dependent on the sampling process. We shall now rewrite equation (A-9) to emphasize these inde- pendent characteristics as in equation (A-lO). L[va(t)] = L[pT(t)] L[vd(t)] (A-lO) Let us now examine this new function vd(t) which appears in equa- tion (A-IO). It is independent of the pulse shape, pT(t), used in the synthesis process of approximating v(t). He can obtain a formula for vd(t) by taking the inverse Laplace transform, equation (A-Z) of its transformed value in equation (A-9). This yields equation (A-ll). vd(t> = L'1[L[vd(t)ll L'1[ E v(nT)exp(-sTn)] n=0 ; v(nT)6(t-nT) (A-ll) n=0 Note that equation (A-ll) is a sum of generalized functions (Dirac delta functions). This is no problem here since from equation (A-ID) 185 it can be seen that there will always be associated with vd(t) some realizable (but perhaps undesirable) pulse shape, pT(t), which it is multiplied by in the Laplace domain or convolved in the time domain. Now we are ready to define the ordinary z-transform of v(t). He will be guaranteed its existence since we are deriving it solely from the Laplace transform, 9(5), which exists by hypothesis. We define the z-transform of v(t) by equation (A-IZ), where vd(t) is given by equation (A-ll). ' co V(z) = z[v(t)] = L[vd(t)]| _1 = Z v(nT) z.n (A-12) s=T ln(z) n=0 for Re(s=T-1ln(z)) > a Rewriting equation (A-12), we obtain an often-used equation (A-13). Q V(Z) = Z v(nT) z-n, Re(z) > exp(aT) (A-13) n=0 So we are now capable of computing the z-transform either from the complete sampled data values of v(t) and the region of convergence or from the Laplace transform L[vd(t)]. Next we wish to obtain an inversion formula for equation (A-IZ). That is, recover L(v(t)), and therefore v(t), solely from V(z). This inversion cannot be exact for the nonsampled data points of v(t). As a useful intermediary step, we shall first invert Tequation (A-13). That is, obtain sample data points [v(nT)J:=0 from \I(z). Note that we can easily obtain the second sampled data value 186 (n=1) by the use of Cauchy's integral formula on the Laurent series, V(z), as in equation (A-14). v(lT) =-§%§ .ig V(z)dz (A-14) lzi=r>exp(aT) The circular contour of equation (A-14) is in the region of converg— ence of V(z) and encloses only one simple pole of V(z) of the sampled data z-transform representation. In general the n-th sampled data point can be obtained by applying the Cauchy integral formula to the -1 function V(z)zn as fliequation (A-lS). v(nT) =.§%3 V(z)z"’1dz (A-15) Izj=r>exp(aT) Equation (A-15) consitutes an inversion formula for recovery of the time domain sampled data values from the z-transform, V(z). There is another useful form of equation (A-IS). By substituting equation (A-16) into (A—15), we obtain equation (A-17). z = exp(sT) (A-16) 1 Z=r n 1 v(nT) "’ mfi V(Z)Z dZ 2;;xp(sT)=r = EFIT—JrfgexmsT)__fl-rv(exp(st))(exp(sT))".1d(exp(sT)) sT=cT+jn -—? V(exp(sT))exp(sTn)d(sT) 2"3 sT=cT-jn (A-17) 187 Performing the last substitution in equation (A-17), we obtain equation (ArlB). T c+jn/T v(nT) = ij if. . [TV[exp(sT)] exp(sTn) ds (A-18) C-Jn Equation (A-18) constitutes an inversion formula for recovery of time domain sampled data values from the s-plane. Note that equation (A-lB) only requires values of for V(z) for one circle in the z-plane. But one circle in the z-plane is an s-place strip which is of width 2n/T in the imaginary direction. Furthermore, because the argument of V(exp(sT)) is periodic in s in the imaginary direction, any strip (parallel to the real axis) of width 2n/T can be used to close the circular contour in the region of convergence in the z-plane. As illustrated in Figure A-l, any of a number of strips in the s-plane can be used to obtain the values in the z-plane necessary to recover any of the original sampled data values [v(nT)]:=0. Now to recover the original v(t) (nonuniquely), we note that equation (A-IB) is, in fact.well defined for noninteger values of n. By substituting equation (A-19) into equation (A-18), we obtain a new function vc(t) given by equation (A-20). nT (A-19) fl ll c+jn/T vc(t) = §;3’~/p V[exp(sT)] exp(st) ds (8'20) ' c-jn/T .Ni< wgsmwu $9 .mcmiim 8. 933A sot 393.5, .3 9935 II.II:.||.n|u.|urll.||un+l.u $75311. lllll + 7 .i . . n 52".! 1111111 w I a?" \ b . . Sn . {can}... llllvn. 189 The function vc(t) is equal to va(t) and v(t) exactly only at the original discrete values of time. Taking the Laplace tranform of vc(t), which was obtained from V(z), we have a Laplace transform §c(s) which satisfies our requirements. Equation (A-21) gives the formula for calculating the Laplace transform of vc(t). V(s) = j{:(t)exp(-st)dt m T c+jn/T =[{m {V(exp(qT))lexp(qt)dq}exp(—st)dt c-jn/T T C+JNIT m “273. {V(exp(qTH exp(-(s-q)t)dt}dq c-jn/T ‘ o c+jn/T .;L_ _LLeXP(qT)) an s-q dq (A-21) C-jn/T The notation we shall use for this nonunique inversion is given by equation (A-22). C+jn/T _ T - v(s) - an J(:jn/T V(gfg(qT)) dq (A 22) A-3 Modified z-transform and the Laplace Transform Next we will derive a unique correspondence between the Laplace s-plane domain and the z-plane of the modified z-transform. Returning to Figure A-1, let us now assume that not only are the values of v(t) available at the discrete time values t = 0T, 1t, 2T, . . ., but for t defined by equation (A-23). 190 t = (n-1+m)T, n = 0, 1, 2, . . . 0 :_m < 1 (A-23) Hence v(t) is available for all values of t as in equation (A-24), but may be accessed in the noncontinuous manner of the standard z-transform. v(t) = v(n-l + m)T) (A-24) He will again synthesize an approximating step function va(t,m) defined by equation (A-ZS). a(t,m) = v((n-1_m)T) pT(t-nt- : (v(t-T+mT)6(t-nT)) n=0 n=0 *pT(t) (A-ZS) V Carefully note, if m is defined by equation (A-23), then we have the special case of equation (A-26). va(t,m) = v(t=nT-T+mT) for all t (A-26) Next let us try a different relation m = 1. We now have equation (A-27). 0v((n-1+1)T) pT(t-nT) = va(t) (A-27) va(t,m=1) = n IIM8 We now have a more general approximating step function, va(t,m), defined by equation (A-25). In this form, we can see the effect of choosing different values of fixed m. Figure A-3 is a picture of whathe are doing for m = 1/2. Note that the location of the sampled values has changed and hence the shape of the step function l ’/// :3 \\\_, .. .2 .. \\\\°:§‘ A V 192 approximation, va(t,m). Now the approximation is exact at a different set of waveform values, v(-T/2), v(T/Z), . . ., v((n-1/2)T), . . . . Next we will take the Laplace transform of va(t,m) using equation (A-25) with arbitrary m-dependence to obtain equation (A-ZB) L[va(t,m)] = exp(-sT) :0L[v(t+mT) 6(t-nT)] PT(S) (A-28) "3 Note that the -T in v(-) is a constant time shift which becomes an entire function in the Laplace domain, whereas the mT is still a variable and remains inside the operator. Also, the pulse shape, pT(t), has been represented as a time domain convolution with the time shifted o-function so that in the Laplace domain, the PT(s) can be factored out in equation (A-ZB). What remains within the last operator brackets of equation (A-28) is the product of two time functions. We will now use the convolution theorem of the Laplace transform which is given by equation (Ae29). (See Appendix B.) C+jm L[f3 (A-33) Note that if m 1, then equation (A-33) becomes equation (A-9). The part of equation (A-33) which is independent of the pulse shape we shall define to be the modified z-transform of v(t). The modified z-transform of theoriginal function v(t) will be defined by equation (A-34). V(2.m)= Z mLLLV(t)]J = L[vd(t.m)1| T,11 ( ) (A-34) S: nz The formula for computing the modified z-transform from the Laplace transform is given by equation (A-35). x t 1 V(q)exp(Tgm)]_exp(-T(5-qjjdq‘SzT-l]n(z) _ (A-35) 194 Using the inverse Laplace transform of L[vd(t,m)1. we could obtain vd(t,m) which when convolved with the pulse function, pT(t), satis- fies equation (A-36). va II M .. ("3'1” Jl. v(nT-T+mT)exp(-s(n-1+m)T)d(t=nT-T+mT) , Ogmsl 0 nT u "MB T~/. exp(ésT(1-m)) 2 v(nT-T+mT)exp(-sTn)dm 0 n=0 1 m = T‘jr 21.m Z v(nT-T+mT)z'ndm| 0 n=0 z=exp(sT) 1 T] zl-mV(z,m)dml (A-39) 0 z=exp(sT) A-4 Transforms of a Prony Series We shall now use the preceding formalism to obtain the modified z-transform of a Prony series of natural mode waveforms given by equation (A-4). M2 V(t) = A, exp(sit) , t a o, Re(s.) < o (A-40) i=1 1 The Laplace transform of equation (A-40) is easily computed from Equation A-l to be equation (A-41). A.(s-s.)'] (A-41) Substituting equation (A-41) into the formula for the modified .z-transform, equation (A-45), yield equation (A-42). 196 As a verification we will find the unique Laplace transform from this modified z-transform. Substituting equation (A-43) into the inversion formula (A-45), we obtain equation (A-44). T £214" V(z,m)dm 9(5) z=exp(sT) m 1 N z. N A. 1 1-m -1 1 l m -m = T 2 z 2 A. dm = T 2 .’~ 2.2 dm L i=1 1121“ #1141” o ‘ N A" f1 < 1 < 'lnd = T 2 ~-—-—-— exp m n 2.2 m . . z -1 _ T g Ai exp(ln(zi/z))-1 = T g A1 21/ ' i=1 l-zi/z ln(zi/z) i=11-21/2 ln(zi)-ln(z) N -Ai N Ai = T 2: = z —— _ i=1siT'ST i=1 5'51 (A 44) This is the correct answer. We may also obtain the ordinary z-transform of v(t) by substituting equation (A-40) into the formula (A-37) for the ordi- nary z-transform yielding equation (A-45). N -1 2 Ai(1-zi/z) (A-45) ' 1 V(a) = zV(z,m)| 0 = m= 1 ‘This, of course, is an expected result. 197 V(z,m) = 271,3 '1 f (fwd) \7(p)1_2:p _T'2_p dp|S=T_1]n(z)(A-42) c-Jw If the poles of 9(5) are distinct, the Cauchy formula is a particular easy method of evaluating equation (A-42). Figure A-4 is a diagram of the finite singularities of the integrand of equation (A-42). By assumptions in equation (A-40) all poles of V(p) lie in the left hand plane. The value of c is chosen to pass through an analytical strip of the integrand and in this case leave the poles of the sampling function on the right hand side of the Bromwitch countour set by c. First we will evaluate by using a contour closed on the left as shown in Figure A-4. In this case we will need to evaluate the integrand at the N poles of V(s) which are enclosed by this contour. Using the Cauchy residue formula, we obtain equation (a-43). By closing the contour to the left, the entire function exp(mpT), remains bounded even at infinity. This was the-logic of introducing (1-m)T rather than mT originally. 9(plexp(me) _ -1 . l = '1 - - \ V(plexmem) Z 2 11m1t (p'pi’l-exp(pT)exp(-ST) n.- W9,- -1 N 2'1? = z X A. -—-—~1 , 0 < m < 1 i=1 1 1-ziz zi=exp(piT) (A-43) 198 .AE.~V> so» cowumgmmucm Laoucou .e-< mczmwg X 1A.-..IAI/ X // l/ X X o x 4 X e X X . x a V 228m . A X “ AAanmvHuvaxmuH a o \ -- .6 «.28 AHX mm X A S e X F X w 3:5 VA .. VA 43 X X \ X \ \\ X flllfilll“ APPENDIX B LAPLACE TRANSFORM CONVOLUTION THEOREM FOR THE SAMPLER 199 APPENDIX B LAPLACE TRANSFORM CONVOLUTION THEOREM FOR THE SAMPLER Here we wish to derive equation (8-1) with the appropriate region of absolute convergence. 1 €4.ij A L[v(t)g(t)] =53] v(p)e(s-p)dp (3-1) c... For our purpose we shall assume v(t) is strictly causal or satisfies equation (8-2) but that g(t) is not necessarily causal. This will give us a derivation adaptable for either the unilateral Laplace transform or the bilateral Laplace transform. v(t) = {g(t)’ ”0 } , ts 0 (8-2) Suppose the Laplace transform pairs of v(t) are given by equations (B-3) and (B-4) where a1 is the axis of absolute convergence. For this strictly causal function, the region of absolute convergence is the open half plane to the right of the axis of absolute converg- ence. ZOO 201 00 LOW] = V(S) = f “Ham-515W, Rem) a1 (8-3) -1 A - cl+jm V(S)€Xp($t) d5 C >3 9 t>09 L [V(s)] = v(t) - c -3... 7.3. 1 1 (3.4) Note that we have used the bilateral Laplace transform limits in equation (B-3). However, since v(t) is strictly causal, we could obtain V(s) from the unilateral Laplace transform tables without modification. Next, the Laplace transform pairs of the double-sided function of time, g(t), are given by equations (8-5) and (8-6), where b2 and a2 are the limits of absolute convergence of (B-5). L(g(t)) = 8(5) = _/ g(t)exp(-st)dt, a2 exva1 for P(p) (3'7) So far we clearly have restrictions (B-8) and (B-9) due solely to v(t) and V(q). a1 < c1 for absolute convergence of L(v(t)) (B-B) Re(q) > a1 is the region of absolute convergence of V(q) (3-9) Continuing, we obtain equation (8-10). 1 c1+jm L(v(t)g(t)) = 5;; ‘ . V(q)G(s-q)dq. a2< Re(s-Q)< b2 C1-J°° (B-IO) Note that we have picked up another restriction (B-11) due to the region of absolute convergence of G(s-q). a2 + Re(q) < Re(s) < b2 + Re(q) (B-11) Combining the restrictions (A-9) and (A-1I), we obtain equation (B-12). 203 a1 + a2 < Pe(q) + a2 < ReIs) b2 + Re(q) (8-12) Figure A-l illustrates the region of absolute convergence of equation (8-12) and the remaining constraint (8-8). The final region of abso- lute convergence in the s-plane can be obtained from the cross-hatched region in the bottom of Figure 8-1. In the p-plane the line Re(p) = c must be chosen in this region of absolute convergence (which depends on s). The minimum value of c is given by equation (8-8). He shall now demonstrate the evaluation of the Laplace transform convolution theorem and its region of absolute convergence on typical electromagnetic quantitites. In these cases both functions are causal with proper choice of time origin. For the strictly causal v(t), we choose for evaluation a two complex mode Prony series as given by equation (8-13). N v(t) = Z Aiexp(sit), t > O, Re(sl) < a1, Re(sz) < a1 (8-13) i=1 We choose for the second function one term of the sampling function used in Appendix A. This single term is given by equation (8-14). g(t) = A3exp(s3t), r.: O, Re(s3) < a2 = 0 (8-14) The required Laplace transforms are given in equations (8-15) and (8-16). V(p) = Aim-s94. Re(p) > a1 (8-15) "MN 1 1 204 s(s-p) = A3(s-p-s3)'1. Re(s-p) > a2 (3-16) The desired transform of v(t)g(t) is given by equation (8-17) and its restrictions by equation (8-18). L(v(t)g(t)) = fwgfwwpms-pmp (3-17) c-jw a1 < c, a1 + a2 < Re(s) (3-13) Note that in (8-18) the region of absolute convergence for s in (8-17) is given by the sum of the axes of absolute convergence for V(s) and 6(5). Since only simple poles are involved, the evaluation is a simple Cauchy integral formula which must include a line in the region of absolute convergence as in Figure 8-2. Using the closed contour, C1, in Figure 8-2, equation (8-17) can be evaluated as in equation (3-19). 1 x x L(v(t)g O 0 n < 0 (C-5) Convolutions are the fundamental processing technique which we shall use. For a given finite length (or memory) processor a(t), we define the convolution output, c(t,m), as given by equation (C-6). T C(t,m) = v(t-t', m) a(t')dt' (C-6) When synchronization is not important, we obtain equation (C-7). N C = z y .a. (C-7) There is a sampled data operator which is of extreme use to use. It is the identity operator with respect to convolution, 60(t) or 50. 211 We will call it the unit impulse (or unit sample). It is defined by equation (C-8). (C-8) * = ' V" 50 V" We shall also introduce advances or delays by means of this unit impulse operator as given by equations (C-9) and (C-lO). vn * 61 = vn+1 (c-9) (C-10) * = vn 6-1 vn-l The use of these unit impulse functions permits us to represent all values of ‘h by a single summation as in equation (C-ll). (C-11) The next important concept we need is the convolutional inverse of V", which may or may not exist. We will denote it by v; and it is defined by equation (C-12) if it exists. (C-12) is a single term "Prony series“ for which we can represent all time values as equation (C-13), then the couplet, cn. given by equation (C-14) is the convolution inverse. (C-13) ....J‘ 212 C = 6 - 216 (C-14) n o 1 To verify the desired property, we convolve the two functions as in equation (C-IS). * -_- * _ * cn n 60 vn 2161 V‘n m l m i+1 = Z 2 6 Z 2. 6 . i=0 1 n-i 1:0 I n-i+1 = z z. 5 _ - 2 z a . = a (C-15) Before introducing new complexity, we shall illustrate with two examples. In Figure C-1 top we illustrate a natural frequency and sample spacing which satisfy equation (C-15). p = 21 = SlT = 0.75 (C-16) The couplet which is the convolutional inverse is illustrated imme- diately below. Similarly in Figure C-2 top we illustrate a natural frequency and sample spacing which satisfy equation (C-17). p = 21 = slT = -O.75 (C-17) The couplet which is the convolutional inverse is illustrated imme- diately below. Although we cannot conveniently illustrate for a complex root, the same relationships hold.. 213 9% ‘F RHoe 8 3‘ 1"0.75 :1 t: .J g:- c, f T T T T T X z j J Cl: 9 ID 6- . I O. [N] {RHOnN FOR N>=O( 70.0 21.0 410 8:0 810 10.0 12.0 TIME UNITS 9% w “‘3.1 D O :3 .— I] % 9 C: o “.3“ XEN]:[1.—.751 1! a RHO= 0.75 T r r l l r ' 0.0 2.0 4.0 8.0 8.0 10.0 12.0 TIME UNITS Figure C-l. Single Term "Prony Series," v(t). 214 -5 35F X00: T 0 FOR N< o RHOflN FOR N>=0 DJ Ila-1 D 9 D .— .J \ § 0 T T T X 2 c1: 9 i X Xfi m ,”,0:75 é- «W °. , I I I I I I 1 0.0 2.0 4.0 8.0 0.0 10.0 12.0 14.0 TIME UNITS ST Lu "3.. c1 9 D I: _J “- o g: 6" X[N]== [ 1 I +‘ne7ESJ a RHo=—o.75 : I l I r l I l 0.0 2.0 4.0 6.0 8.0 10-0 12.0 14.0 TIME UNITS Figure C-2. Single Term "Prony series," w(t). 215 Next if vn and “h are each exponential waveforms and xn is given by equation (C-lB), then xn _ vn * wn =0 k=O The convolutional inverse is given by the triplet yn defined by equation (C-19) = -1 -1 -' . - yn wn * vn ' (50 2151) * (60 2251) do - (21 + 22) 61 + 212262 (C-19) Note that observed these exact values in the two-mode Prony K-Pulse Worksheet of Figure 5-12. Hence, one interpretation of a Prony K- Pulse that it is a specific convolutional inverse of a waveform which is the convolution of elementary exponential waveforms. One must also observe that if we have a waveform which is represented by a couplet such as equation (C-13) its convolutional inverse is given by the infinite term wavefunction given by equa- tion (C-14). There is no finite length convolutional inverse for the couplet. Now we illustrate a more complex case, the linear combination (If two different exponential waveforms as given by equation (C-ZO). Sn = avn + bwn (C-ZO) 216 This waveform is illustrated in the top of Figure C-3. Not knowing the convolutional inverse of equation (C-19) by inspection, we shall formulate a triplet which resembles our "Prony K-Pulse“ which is illustrated in the bottom of Figure C-3. This potential Prony K-Pulse is given by equation (C-Zl). =.1 -1 kn wn * vn (C-21) We calculate the output of this convolution, cn, as given by equation (C-22). 0 II kn * (avn + bwn) = (W';1 * v51) * (avn + bwn) av'1* v’1 * v + bw"1 * v'1*w, n n n n n n -1 n 1* -1 "n = avn + bv (C-22) -1 -1 - w + * a n bvn wn Note that equation (C-21) is a linear combination of two different couplets. Therefore, in general, cn cannot be the convolutional identity and sn cannot have a finite length convolutional inverse because of the arbitrary amplitudes a and b of the exponential wave- form. Generalizing these results to a "Prony series" of N terms, the potential "Prony K-Pulse" is an N-plot. An N-plot is just the N convolutions of the corresponding N couplets. In this case, the output of the potential "Prony K-Pulse“ convolved with the "Prony 217 (—.651xxN RDDI+-75]xxN 3% §3~ T 3 1;: - 3: 2TJL__TL_JK__§L_;Z__JL__:;_4K {0.0 210 410 010 010 _ 16.o 15.0 TIHE UNITS K—PULSE 11"‘1 3:; g. [1,+.65] CONV[1,—.75] 2 i: z: z: 2: :2 :zesz-ir-az——ae—+z 95:0 2.0 410 810 610 15.0 15.0 11.0 rxne UNITS Figure C-3. Two Term "Prony Series," S(t)° 218 series" is, itself’ an (N-1)-plet (N nonzero sampled data values) containing the amplitude dependencies of the exponential waveforms. Lastly, the "class 2 Prony series" given by equation (C-23) V =w + Z ckzk (C-23) where n is an N-l length waveform whose transform is an entire func- tion. From Chapter 5, we know that if we possess enough "complex root degrees of freedom," in our "Prony K-Pulse," we can recover both the 2k 5 and ck s. C-2 K-Pulse Singularity Theorem In the derivation of the "fast Prony's method" algorithm of Chapter 5, we intentionally detected a flag A or singularity condi- tion. For sampled data waveforms, this sometimes occurs sooner than we might expect from an observation of only the continuous time domain. This optional theorem will give insight into why this happens and also illustrate the use of couplets in analyzing sampled data problems. First of all, we have observed that for an N term "Prony series" with known 2-way transit time, we can always obtain an N+1 length "Prony K-Pulse" which by the previous section is an N-Plet (or repetitive convolution of N couplets. Note that the length of the "Prony K-Pulse" denotes a quantized amount of information content. We shall denote this standard or N+1 values of information content as 219 the o-th order singularity. When we are able to use only N values of information content, we shall call this the I-st order singularity times. Similarly the 2-nd order singularity times exists, they will result in an N-I length "Prony K-Pulse." Still higher order singu- larity times are similarly defined. We are now ready for the state- ment of the theorem. K-Pulse Singularity Theorem: The first order singular times, T1, of the N term real "Prony series" are given by equation (C-24). T1 = P(2N)HPm (c-24) where P is a positive integer HPm is the half-period of any complex natural frequency, sm. Proof for P = I and any m e {1, 2, . . ., N}: The sample spacing is T - HPm. Now the m-th real natural mode is: . _ n00 * noo Amexp(oan)cos(men+¢m) - (Cmipm1o + Cm{p; 10) which is the sum of two complex natural modes of the identical period and nagnitude of amplitude. Expanding, co {0:}0 = {exp(om T-n)exp(jwm T . n)}o but mm T = w HP II :1 So exp(inn) = (-1)n exp(jwm TPn) n°°_ n _ .noo {omIO - {eXP(0m T-n)(-1) } - {(-lom1) 10 Similarly, {03"} = {exp(om T-n)exp(-jwn T - n)}' But -wm T = -mePn = -n So exp(-jmm T - n) = exp(-jwn) = (-1)n {0;n1: = {exp(om - T-n)(-1)n}o = {(-|pm1)n}o co IXOW’the same couplet (l, + lpm|1kills both {pg} and {pEn}o. 0 Hence, if T = HPm were used in the N term “Prony K-Pulse" with 2N+1 information values, there would be an extra (1, + Ime) couplet. Only one of the two identical couplets is necessary. The reduced 221 K-Pulse has only 2N information values. Proof for P = 2 and any m = .... II 2HPm {exp(om - ZHPm - n) - exp(jmm . 2HPm . n)}o r-H '0 3 H—J II But mm - ZHPm = 2n Then exp(jwm . 2HPm - n) = I 00 n — o o co: nm {om}o - {exp(om 2HPm 0)}o {loml }0 Similarly since - w - 2HPm = -2n *n m _ n m {pm 10 - {lonl } Hence, the couplet (1, - [pm|) kills two complex natural modes: co * co {0"} and {p "‘1 m o m o One of the two identical couplets can be deleted, yielding a reduced N-term "Prony K-Pulse" of only 2N information values instead of 2N+1 for arbitrary T. 222 Proof for P odd and any m (of N): T1= P-HPm - _ n exp(Jwm F’ HPm - n) - (-1) non. rToo {om}o - {(- Ioml) } o *n w _ n m _ {om 10 - {(-loml) }0 same result as P - 1. where [em] = exp(om 0 P ° HPm) Proof for P even and any m (of N): exp(jwm - p . HPm - n) = (+1)n n m _ n m {om10 - {1me 1 o n w n w {0* } = {lo I } same result as P = 2. m m o o ' where lel = exp(om . P - HPm) Converse Proof: Suppose T f P . HPm for any M€(1, . . ., N) then {031 f {0;n} for all m, since mm f O for all m. Hence the O O . n on *n (0 same couplet (1, -p) does not kill both {pm} and {pm }o' O 223 Suppose, however, that T is such thatco one of the 2N couplets, (1,a) of the N- Mode K-Pulse kills both {pm }: and {p2} }m,£e(1, . . ., N). 0 But if this is true, then (1,a*) which is also a member of the N-Mode * °° a, K-Pulse kills both {omnio and {p:"}o. But this means that this 1 defines a 2nd (or higher) order singularity which was deleted by hypothesis. Hence we have shown that if T f P . HPm for some m c (1, . . ., N), then either T = P r 2N - T is not a singular time or 1 T1 is singular of order greater than 1. APPENDIX D COMPUTER CODE FOR "FAST PRONY'S METHOD ALGORITHM" PARTS AND OTHER PROGRAMS 224 C t t t MULTI-RATE CONVOLUTIQNS t t t I t t t i t t i i t t t i t t i t 00 C t t * FOR SOUPLING CQEPFI {BNTS I t t i t t t t t t t t t t t i i t % c * * * 31L; PURPOSE * * FIL * PURPOSE * * * * * * * * * * * * * c LINE UNIT-9 LINE UNIT-10 C cos AMP SINE AMP C LINE UNIT-13 2 LINE UNIT-14 C DECI 4 ROTATIONs C cos EXCITATION 8 SINE EXCITATION c SAN LE SPACINCS RECORD LENGTHS C I INPUT WAVEFORM 18 EXCITATION IN E .11 1191111101001“ 1 I C t t t t i i t t t i t 9 i t t t t i t t t i t t t i i t t t t t t t 2 pROGRAM FALTUNG(INPUT,OUTPUT) 3 EHARACT R ANS*4 ALL PF 'ATTACN',‘DIENIRE','OIENIRE1§') CALL 9? 'ATTA H','DPULSE , DIFPULSE DO 2 I=117 11 2 RBWIND I PRINT*,'§NTE§: CONv :POR CONVOLUTION ' READ * ' A ' ANS CALL DICTION 1 1 CALL TEARCH . g IF ANS.88.'CONV';THEN CALL TART(1O 4 CALL CONVOLu 5 CALL DIFCLAS 9 ELSE IF(ANS.88.'PLOT')THEN PRINT* 'PL T LY CALL PLOTS END IF PRINT*,'* * * STOP OR NEW ANSWER ' 1 RETD *,'(A)')ANs g I? ANs.EQ. STOP )CALL EXIT 60 To 1 g C ttttgygtttittttttttittitittttttttiittit*itttttfitttt*ttttitttttttttttti E EEYEQ:£5eggsziggIA§§é§gg£t§Ié§IIREQ§¥9%9&t£&9I§&t9£§S%§§teeetetaetttaa SUBROUTINE DICTIQN IMPLICIT COMPLEX C) NTOT) C , IFSKEW , IESKEw TACN',;wIRE1e;, GHTEE§6 SKEW , N3 2 'bIRECT',RECL 1 ='OIRECT ,RECL- ) xgfiTATION PFN- ' PFN=,A14)')PFN méfi'v“ £51085" L sLFfi: ' STIIG, ELT,CPA,T,1,NP,NP,MLBNGTH) ttittttitttitttttfitt*ttttitttittttititt*****t*******tttt SAINP),CSA(NP),RFA(NP1 ) D I 8 N' .Q. mu '0 v SEARC? ¥XCIEEN1%3,D8LT,CFA,T,1,NP,NP,MLBNGTH) REVERS CFA,MLENGTH% RPULSE EA,NLENOTH FA) * NOE I = o xc L ZRPOLY REA,MLENGTH—1,CR,IER) T* 'NO?XCIT=',NOEX IT L gRoER cx MLENGTH-l 5) NT 'NOE?CIT=' N xc T RBTL CAOICE C5A,MLENGTN,Cx,CZI) c titit**g§§***itit*tiittttt*********t*t*****ttit*ttttttttttttttttttiit! OTS PRIN$* g’OU'I'PU'I' EXCITATION TAPBI'? ' IP1NOOT.E .0 CALL EXIT CALL BARP OT 1,NOUT, SA,NOEXCIT,T,MLENGTH) R T RN C *ttttttgttittttttittittitttttitit*ttttttttttttttttttitti*tttttttttttit ENTRY DIFCLAS ~h¢d\h¢d~h¢dOWN”OWNMGWWNMUKMflUUKMflUNMHDfiflmbhflflhfihMNthMdQNNHUWMHKnMUKNMQh»fl0NWflflHHF4flHFW4J O'UO'UO'UOOZOFJ ONOOOZPZN'UOOOOOOU'O >IU>50>IU>>O?Z m>v>wwwwwvvv>>>~> FHfigf‘Hl‘t‘I’Jl‘H CNxDQOUHNdNWKMDGhmMflbhmflfla¢thmUHMMMHCNMDQOMUWMVFCWGDNO\ ”I?! ['12 *3 :0 0d 'U W 225 226 PIsACOSé-l.) Do 3 I- 14 3 RENIND I 58A§I=I°I LE G READTII,*?TT,ENVL EEG? R3s* ET E"V5 READT12,*TTT,ROTL READ 14 * TT ROTS ROT= ROT -ROTL g=c§xP CMERX(ENV, 2. *PI*ROT)) CLAST= C 31 CONTI E CR 1 T L NGTH, 7) CALL ROTAT 1Cx 1 LE EfifTH HI CA%LRSW ,CR, 1, T LENGTH, 5) C tittttttttttttiittttttttttflii*ttttttttttt*ttttttttttii*ttttttttttttttt ENTRY START(LR) LARGER: N OT (LR PRIN NTE” SIGNAL LPN- ' READ OPEN LR PILTM ggLEESIOEAL LR, L L s, LARGER, T DELN, LENGTH) ISRIP= DELT+.0001) DELN PRéngt, COMPUTED I KIP= ',ISKIP C ititttt itt*ttii**i**t*****titttttttttttttt*tttttttttttttttttttttttttt ENTRY CONVO CALL SIGNAL 109 1 5 Ms T, DELN, LENGTH) CALL REVERSE CSA, MLEN GT H) NSKIP= MS/ISKIP PRINT* COMPTTED NSKIP- CALL EARPLOT I Do 2 II=7,16 2 REWIND II IANS=0 IF(ISKIP&NE.1)THEN PRINT 'ENTER 0 FOR NO DEMULTIPLEx ? ' READ*,IANS END IP Do 1 ISTART=1 ISKIP NSKIP=(MS+ISKIP-ISTART{/ISKIP MAxLENG=MLEN TH+NSK IP— CALL DCONV L cx CSA, MLENGTH, ISKIP ISTART,N§KIP) CATL NROUT ITRIP,Cx' MLENGTNh T,ISTART NSKIP,9 IP IANS.E8.0 T7EN CALL R TAT cx MLENGTH MAXLENG} ENDC?%L WROUT ISKIP, cx, MLENGTH T START,NSRIP,13) I CONTINUE IF(ISKIP.NE.1) IF(§$NS.NE.O THEN CALL DNURTKR ISKIP s, MST CALL DMux KR$1 ISRIP Rsfi RRR+1) CALL CPILETKR RCx CALL wm 0T 1, CR, xNLENGTI-i, T, L MS, 13) n31? ‘1 ALL LINE(C? cz I, 1 LENG H, DELT) CALL ROTAT? CR 1 'LENGTN L8R01,C2,1,TT,1,L NGTH,11) ******************ttit!*tti**********;;*;£6;***ttfitttttttifltfitfltiititi LEVEL 2A: EXCITES, EXCITE, REVERSE, *ttttttttttgtgyeg‘*§£§§§E‘ttgzézg‘it9991‘QEEQtttttttttttttttfitttttittt SUBROUTINE DCONV ngCX, CFA,M ,ISKIP, S, ISTART, NMAX) IMPLICITN COMPLE CX(NMAX+M 1) ISTART= ISTAR RT NMAX+ CFA(M) RFAIMAkS(ISKIP,NMAX).T(ISKIP,NNAX) MIN IT, M) MAX IT-'NMAx+1,1) SKIP. 15 cs1, NMEENGTH, T, NSKIP) i 1 E hxmmnqmmnmm»«manqmmMmexmmnqmmmmm»«mamqmuwmm»«mamqmwmam»«mmnqmméhm»«mmndowunmeccxxxmaooccr qqmmaommmmmammmmmmmmmmpbh9999999uwuwwuuuwwwuwnwwnnunwwpwwwwwHwooooooooooomqmmfiwuw WENSI INT 1 TT=,0 cx IT IMIN= JNIN- 227 D0 11=iM 11X(IT 1 CONTINUE C ******Qttttt*tiitfifiiitiitttttfiitfi*ti*****fi*************i**fi****t*****i ggTRYDEggITES(LW, ,DELT, CPA, T, ISKIP, NMAx, M, MLENGTN) READiLw, * ,END-23, ,ERR824gTFIRSTt C§A(1) NTLWI**END§23 ENE-24 TT CXIIT1+CFA(J)*S(ISTART, IT-J+1) LW READrH-H e veA;oa-»- N*QP*X w H pa 0 I?) O 422 C‘i‘i'flt‘l HOG ”Hsz woo ZZZHU AZZ CCZ 'UHH meva st> NNNNNNNNNNNNNNNNNNNNNNNNHHHPHHHHHHHHPHHHHHHHHHHHPHHH A'IJH ll. ta 2am TENT IgKIP K PRINT 10,1 IRACE, K, T(ISKIP, K) K, CPA(N) CONTINUE RETURN FORMAT Té' 12 ' F§2 ' Ns, CEM'S m; ,2E5.I) PRINT*,'TH RE ARE' RORLY' ' CTCLES' ON THIS QIL PRI N 'ER RRO ON RADING 'PILE- READ 109 REC-IPAGE MLENGT H SRiNTEm LEN ,IPAGE, ' CYCLE ON PILE- ' ,MLENGTN RETURN C tttttttttittttittitttt*titttttttti*iit*tttt*tttttttttttttttttitttttttt fiTRYm TRIP ET(CZI RE' ISKIP, NMAX, LW) hhfid Q #QKD cczI- ON C WRITE LN, T ISKIP,1 ,',',CMPLx 1.,0.) wRITE LW,* T ISKIP,2 ,',',CZI C 2I ggigg LN¢* T I§Ké513,,',',CZI*CCZI 52%gg*"TI ISKIP,',1)-',T(ISKIP,1) C ***********************************************ti********************fi EN RY SIGNAL(LS, ISKIP, IS, 5, NMAX, T, DELN, LENGTH) IF IS. E?.1)REWIN DLS D0 2 K= NMAX IuwmmnmvannmvwmnmuNmnmuummumunumnno k) «ma mO\ 51 RE RN c ******* fit*tiiiifiit*tfiti*tiittttt******ti*titfi*titttiittfitfitfitttttflfitt 81 C tiflittrgi*tttttfifitti******tii*iIi****.********t**t*t****fi*******i0*... 61 228 READ(Ls *,END-25,ERR-26)T(IS,K),S(IS,K) égNGTH H: mt TO PRIggaé ,‘ERROR ON READING FILE- ',LS LE PRINT*, 'L NGTH OF §¥GN?L FI&E- ',LENGTH PRINT* 'T ISTART, 2 T 15.2 CON TIN E DELN: T IS 2) -T(Is. 1) PR§N§NJ HELN(' ,1)- ',DBLN tatttat«atittattttttttttttttatttatttctttttttaa«tatttattttttatttttttttt ESTRYKWROUT(ISKIP,CX,M,T,KK,NMAX,LT) =1,NMAX éF( ETTTKR§)' .EQ. TF).AND. (LT.EQ.9))THEN PRINT* T CXI', TF, CXTF WRITE KK , , :NwRITETLT41)T§T (kT’R), ' ,BETTégé TTZK)) ~«NWOOUWMNWOOMWMMHMUU‘ tiitggzgggtftititiittttiiitttti*titttittfititt*ttttfittittttflfiittttttifii ENTRY OTAT (8x M,NMAX) AL=20 cc 1 .3 pst ROT ~¢QQ ~¢Qd hNflfi4HHhflfl44HHCKXN3DOCEKXJDUWMNNMUDWMmMNMDmCNNNMD ~¢dfl E num»4:¢nm«muuumvhmmam\mumbunmacmanouuu» in?ch1?=cx(I 1) TH ELEMENT IS 2ERo' .NE.0.)CALL CROTATE(CX(I),PI,AL,ROTSTEP,FRACTIO) «1- 1:1th EN 50*mfiwfldun’fiv A'IJHM o H 29m~2ooya V k]. V n 0 2 Ch!“ ”“1! 2 C m T N *t***fi***ittttiitfitttifitifiitt*tit*ttttttttitiitttttfiitttfiitifittitfititt ENTRY L1NE(cx C21 M NMAx DELT) PRINT* RST= T? is TF= TF TEIRSTSD gT-% MA DELT 'REs= cc tAssv(cC, czx, cx NMAX) titttttttittttittttttttttttttttttt*ttttttttttt*tttttttttttt EVERS§;CPA, ,M) CFA( §.. RETURN 1 F M iii! THEN NON: AUSAL ADVANCED BY 1 ' -CEA(K+1) “‘Wuflfi co E PRINT', POLYNOMIAL REVERSED ' END IE' ENTRY RPDL5E(CEA, M, RFA) 1-1 M REA 1 =REAL(CEA(I)) gngY BARPLOT(ISKIP, LU, CPA, M, T, NMAX) CONREWIND LUTI n-TT1N 25- -T(1 1) pRINT* 'SAMRLE SPACING= ,D,‘ NS' gngE(99, ,§8C81)D I! 0~H>mn*erMDvcnqu fl 'nmhazun>xwumwwzn< m .Anquimqrnmr4MHHz 8EW):3V>00flntchNflteZZ&* 'fl :MaHuncnx ~n~a~w+ae DC 2 D Ho-m Dznm-Jizntn :‘ 2“: HCWWMfl A(NNZMRP#U» '? 2>0vlz=~v> h» (52 Ml unaw- oE-o In; v'm-J o7" - N - 'Ux 0A" (1 r» ux»uwM»ummguwmahnmwhwmnumumwoma»umm»uwm»uwmnunmuhno on»hmmoahmmuuuunwmounmqomeumuucumn«numbum»«amnh«wma~»m~cwxndun» tnuwmawmnmvwmnmn 229 -1 U, K,LU) CA L R U, 'E.LU+1) 6 CONTINUE RETURN D C *i********iitittttfiitfititittiittattgtttttttttfittfitttttittiiiiittittttt CL§VEL 28 DMUX CFIL§ 9RD§R CHO ******&*****&* at. £R *‘t it; Etttttttttit!attttttttttttttttttttt SUEROUTINE fiT ZFIHFX*zxxx CbVZFfl‘CV" M "U Hzmu m 2 HH x a n I l.“u”““mu~’"MN.¢MWNmJUMWumflUMwflmJUMWWfimduMWNflJdUMwflmJwMWflMWMWKMflUMNmJwhWMM“WMWMJWUMW“M»U~WNJ“&WO PI: CZI-SSS PRINT ,‘POLAR ZEROS' DO 1 J=2 CZP-CPOLAR(ELOG(CZ(J-1))) PRINT 1 CFA(£)= CPA 11= . , REIT? f'ENTRRN 0 OR MODE TO 3 To BE DELETED IP?I.E8.0)mE , PRI T*, x- PULSE ELSE PRINT 10, I ,CPOLAR(CLOG(Cz(I))) CZI-CZ(I) 230 END IF DO 2 i=1 R-} CONTIW J. NE. I CALL POLYN(CFA, NR, C2(J)) IF(I. NE. 0)NR=NR 1 RBTU N FORMAT(13,' (',F7.4,',',F7.4,')') END ****ittttiittttttttttittttittt*tttt*tttttttit*tttttttittttttttittttttt £§¥§Et§§tR Spegx§1¥1N§RAQ§¥£3SR‘tEgkxy‘tSRgréIE‘ts&駧ttitttttttittttt RET RN tatttttgtatitgatttttttctaatititttttttttcttttttatttt*Rtttttatttttctatct ENTRY POLYN(CFA, NR, CV) m8 J- -1 A I FA V* 33.121321 H3“ C "M“ t****** ittitttttttttttttttttttttittittttttttt*ttitttttttttttttttttttt ENTRY SWITCH(CG,CH) CC'CG CG'CH CH=CC BTUR ititttitittttttttttittttitttttititittttitttttittt*tttttttttttttttttfitt ENTRY CROTATE(CCX, PI ,AL,ROTSTEP,FRACTIO) g8TIgC=FRA CT 0 X: LOG(C x ENVEL= REAL CCf)‘ AL FRACTIOSAIMAG CCX {2C /PI R0 INC=FRACTIO R0 IF ROTINC.GT. 0. 5 ROTSTEP=ROTSTEP1. IF ROTINC.LT. 0.5 ROTSTEP= ROTSTEP+1. ROTATIOtFRACTIO+R TST EP CCXICMPLX(ENVBL,ROTATIO) *ttttttiiifittfitfltitflitti*t***i*****t**t*t**fi****tittttttfittttflfittfififlfifi Sg¥RO TINE RECPLOT( AU, DTAU, A, MU) TE MU,* TAU, ,1, . w IT MU, TA ' ITE MU,* TAU4DT U, ' ,',A WRIEENMU, * TAU+DTAU, ' ' ,0. ND *tttttt*tittiit*i**t*ttt*ii**t*******tittifi*ttitit*iitiitifitfiittttfifiit SUBROUTINE OCLASSYéCC, ,CZI, CX, LENGTH) IMPLICIT COM DIMEN SI N CX(LENGTH) DO 2 I= ,LENGTH CK I =CC*CZI**I RETURN Ittfigt*ttttflittittittflt*tttt****t*****tt*9iit!*tfifltitttiitttttifitfitfifit OOOOOOOOGOO mo 33 C 0.}. ’II" *t** LEVE titi *tfifi *t** 231 * t t t t t t t i t t H. 1! t i K. t t t t t t t t t i t i * ' t t t i * §A§T.P§°IY.S.M§T§°9 2L§°§{I“"& :H9L38. .A§A&Y§E§ . . . . . * * F1 E PU OS * * EIL PURPOSE * * * * * * * t * SU ”PAC II PLOT SCRATCH * SYNTHETIC EREQ. {0 INPUT WAVEPORM * PRINT 1 K-PULSB * MODEL 11% SPECTRA NORMALIZED * 1% EY§¥§E¥I PUL E 117 MggPEE s : t t i t t t E EC W? t t g i i 2 t t t t i t i O * PROgRAM FPRONYD(INPUT, OUTPUT) DO I-o 8 coumb'é" “’9” WRITE(110 '(///) ) EALL PRAME ALL LOOPSET CALL ER Es CALL POLYNOM sggp gttttit*ttii*tttitttttitittttttt*ttitttttI*tittttttttttiittttttttt gt}§ta§§¢¥§‘*&99P§§t‘i§§é§§§‘tEgggggt‘it*tttitttittitttttttttttttt SUEROUTINE DICTI N IMPLICIT COMPLEX C) IMPLICLT DOUBLE PRECIS ON(w) IMPLIC.T HARA R*7 O PARA EvLR N=256 NCOLS=1,NP-Ioo,,NPP-NP+1 LR=109, Lw-LR+1) PARA‘L'LR OPRONés PR NY DIMENSION cfoP C (NP ,CAM(NP) CPA(NPP). cz(NP) DIMENbION W8 NP w} NP) 4 DIMENSION WAVEIN N; WFASI? NP+1) CZNUM(NP) : DIMENSION NPA NP NP 5 ONP REA NPP) RPASIO(NPP) EQUIVALENCE(CP(1PA 2 4 * *itttttittttti ‘ttfi * *ti**;**fittit*t**t*******t*******itfififiitflt‘ ENTRY PRAME 4 CALL PFE'ATTACH','LOA208', ,'wIREI ROOTSLOAZSB'; 6 CALL PP 'ATTACM','LOA40 ' 'NIREI ROOTSLOA4 ' RENIND 12 CALL TARGET(NM LR Lw) CALL TIME NR TTI MEC ISKIP, LR) PRINT*,'#ROWS+ CO MLUMNS- + ,NR ' RE- ENTER CROWS ' RBAD* NC NCaMAkUUH Known-o: K~wcxx§3v 8UWU>wO ”Z'UFJOMWF‘ ’OM ”HMwaw" ll ZZH C'flzouo o H 2c>rn- s o 2 «av mo 29 Iv 00-] 0: -m 2 nwuuuw'aHOmer-opmmn O 50 '1) thwu I 2, II II Hg :1 ll fi-JIIU, XWWOO- LOU-'1” :S(N+J)*A(0) EORNARDsEORwARD+s N+J- EACKNRDsEACKwRD+s N- K) CONTINUE RCFORWs- FORWARD/ERRORB RCBACK=- BACKWR?{ER RF -RCEORN*RCEACK) O RORF RORBRCF -ERRORB+RCBACK*FORWARD RC( BJ) A J sRCEOR *B(0) K= -1 s VEB=B( -K) E J-K)= J- AK)=A(K+ CONTIN ’émi} K)+ RCEACK*A(K) CFO Rw*SAVEE .» quadqqqwmmmmmma¢mmmmmmmmmum¢h¢bob....uwuwwwuwwwNNNNNMMNNNHHHHHHHHHHoooooooooowwmw mwmmwwmmwmmwmqmwummwmWNHomqmmowNO-‘omqm 238 B( )8RCBACK*A o; 77 IF ERRo .E8.N THE 7 PRINT* Ro ERIES IDENTIFIED D0E=' J 7 égéTE(110,*)¥PRONY SERIES IDENTIFIED Dor=‘,a 1 R§EURN g IF (RCFORW 38.0.).AND.(RCBACK.EQ.O.))THEN 4 RI NT* ASS 2 PRONY SERIES D0E=' J 5 wRIT8311 ,* CLASS 2 PRONY SERIES, DOE-',J g DOF= RETURN END IF 3 CONTINUE RETURN % C tttttttiiittittt*ittt'ktttttitittitittttttitittttiitittttiittttttttiitt 3 239 C tit.itItitttttttitt*ttitttttttttt*fitttii****ttttt*tttttttttttttttttttfi PROGRAM JEEVES INPUT OUTPUT) IMPLICIT COMPL x( EXTERNAL F R wI wRITE ' Nléfi ' WRITE , , zARAA a SHETTY ** CORR ECTI wON TO PAGE 280' , wRITE * OME§H D OF HOONE AND JEEv ESN 1TH DISCRETE STEPS CD1=CNPLx CDZ=CMPL cx=CMPLx2 31% 2 CALL mEgtE x CD1 CD2 ggégE 8 * NOTE ROUND- -OE ERROR IN BOOK AT ITERATION x-a J-2' c itttgggtttttittttttitttttttfittttttflititttfitttit!ttttttitttfiitittittttt SUEROUTINE HOOKE% ,Cx, CD1 CD2) INPLICITL COMPLEX EOEIAAL SSPLA C P7c§= QRBA: 1% “1 ? §f*4+( EAL(C)- 2. *AIMAG(C))**2 DATA A DELTA EPSILON= .1 ALPHA-I. CYSC? YlsF CY) J-I K81 x1=Y1 3133313" 3 C tittttttt ‘tt fittitttttt*tttt*ttttttt*tttitttttittfiii***t*t*********t DO 1 NNaI 200N IF(J.88.2 C08 D2 DELTA LS E 4 CD=CD1*DELTA 3 END IF CYP-?Y+C YP-F CYP Y2=YP IF(Y2.LT.Y1)THEN LELAG-.TRUE. CYZ-CYP ELSE LPLAG=.EALSE. CTN-Cv-CD YN=F CYN Y2=YN IF(Y2.LT.Y1)THEN SFLAG-.TRUE. CYZ-CY-CD ELSE CYZ-CY Y2=Y1 SELAO=.EALSE. em??? 1" *****tttt*tttittiti’ttttt*titt‘ktfitt*ttttit*ttttit'kttttttittitfi'kttttfittt 2 g7JIm .LTSM E2)THEN HEN ; fi¥§,28)N1 DELTA cx J, CY, CD1, CYP ; LswRITE .3 x 1,?1,Y2, AS' ERITE g, zogx1 DELTA cx J CY CD1, CYP, CYN ; wRITE 30 x1,Y1,YP, At 1N ,Av 7 END IF 7 J-2 Yl-YZ CY=EY2 GO TO ELSE IE(LELAG wRITE Hso J CY CD2, CY2 NRITE '60 Y1, EPA ELSE Av-AE IE§SELAchy=AS NR TE 8, o J,CY,CDZ,CYP,CYN 240 HRITE(8,60)Y1,YP,AF,YN,AV DE)? u.- c *tiifit‘tfittiiiiit*ifitit'lttiiittifiifiifiiiItlitftiitifiltiffiitiiiiiiflifiifi IF(E§§LgfiXI)THEN CX=CY2 CY=CX+ALPHA*(CX-CXB) xlsy Y1=F CY) IP(DELTA LEi§PSILQN)THEN wRITEIB, ' K: ,K ' s',Cx ' (x)-‘,x1 PRINT‘, K=',K, x. cx, E(x -',xI RETURN END IF DELTA-DELTA/Z. CY=CX t‘ m [*1 ttitittittliiiiitiitifiiiitiDitttt*tliifififiiilitilititii K 5x ?HDEL 4x HF(XK) 4x }HJ,4x,5HF(YI),4x,5x,2HDY, 1 12H? YJ—DEL*DY) x 1H ,E4.2,1H,,P4.2,1H),1x,11, i2?5.2,A3,4x)) ttittiititfitittfitttiiiltittifiiititiiii*i 60 C it!‘ M LEX(C) 2. **4+ REAL(C)-2.‘AIMAG(C))**2 Fl 2 U ’4 HH "III N H p44»wpm44w+m»44HHhmamuwhmmnamubwaawhwwawhuwmaowummno umuMuNkmumuNthuoHHhwauHPhmm4xmaocmxmoocmxnqmuuuanH WNHOWO<0“fiwnwflmflpflmmfiwNHCMfihflmflhUNHOCEKXMJDOOO OCKXMDOOCKXJDDOCKXXNDOCKMDOCKMDCKMDOCKNDO UfiAvZRUxHH REFERENCES 241 2-1 2-2 2-3 2-4 2-5 2-7 2-8 2-9 3-2 REFERENCES Chapter 2 R. S. Elliott, Antenna Theory and Design. Englewood Cliffs: Prentice-Hall, 1981, pp. 11-26. C. E. Baum, "The Singularity expansion method," in Transient Electromagentics. Ed., L. B. Felsen. New York: Springer- Verlag, 1976, pp. 129-179. C. E. Baum, "Emerging technology for transient and broad-band analysis and synthesis of antennas and Scatterers'," Proc. IEEE, Vol. 64, No. 11, November 1976, pp. 1598-1616. Kun-Mu Chen, "Radar waveform synthesis method--A new radar detection scheme,"IEEE Trans. Antennas & Propagation, Vol. AP-29, No. 4, July 1981. PP. 553-566. L. L. Webb, B. Drachman, K. M. Chen, D. P. quuist, C-I Chuang, Bruce Hollmann, “Convolution of synthesized radar signals for single or zero-mode excitation with experimental radar returns," 1983 National Radio Science Meeting, University of Colorado, 5-7 January 1983, CR2-6. J. G. Proakis, Digital Communications. New York: McGraw-Hill, 1983. PD. 72. D. C. Schleher, ed., MIT Radar. Dedham, Mass.: Artech House, 1978. ‘ ’ R. O'Dennell, C. Muehe, M. Labitt, & L. Carthledge, "Advanced signal processing for airport Surveillance radars," EASCON-74, 1974, pp. 71—71F, also reprinted in MIT Radar ab0ve. A. S. B. Holland, Introduction to the Theory of Entire Functions. New York: Academic Press, 1973, pp. 102. Chapter 3 D. R.Rhodes, Synthesisof Planar Antenna Sources. Oxford: Clarendon Press, 1974, pp. 10, 43-46. E. M. Kennaugh, "The K-Pulse concept," IEEE Transactions on Antennas & Propagation, Vol. AP-29, No. 2, March 1981, pp. 327-331. 242 3-3 3-4 3—5 3-7 3-8 3-9 4-1 4-3 4-4 243 Lance Webb, Byron Drachman, K. M. Chen, K. P. Nyquist, C-I Chang, Bruce Hollmann, "Convolution of synthesized radar signals for single- or zero-mode excitation with experimental radar returns," USNC/URS National Radio Science Meeting, 6 January 1983, 84-1. Lance Nebb, K. M. Chen, D. P. Nyquist, Bruce Hollmann, "Extrac- tion of single-mode backscatters by convolving synthesized radar signals with a radar return," 1982 International IEEE/APS Symposium, University of Houston, APS-8-3, May 26, 1983. R. E. Crochiere & L. R. Rabiner, Multirate Di ital Si nal Processing. Englewood Cliffs, N.J.: Prentice-Hall, g983, pp. 30. J. M. Tribolet, Seismic Application of Homorphic Si nal Processing. Englewood Cliffs, N.J.: Prentice-Ha , 1979. William K. Pratt, Digital Image Processing. New York: John Wiley & Sons, 1978. PP-‘110-415. Wayne T. Ford & James H. Hearne, "Least-squares inverse filter- ing," Geophysics, Vol. 31, No. 5, pp. 907-926 (also reprinted in G.hL.Webster, Decgnvolution. Tulse: Society of Exploration Geophysicists, 1978). K. M. Chen & D. Nestmoreland, "Radar waverform synthesis for exciting single-mode backscatters from a sphere and application for target discrimination," Radio Science, Vol. 17, No. 3, pp. 574-588. Chapter 4 R. Prony, "Essai experimental et analytiques sur les lois de la dilatabilité des fluides élastiques et sur celles de la Force expansive de la vapeur de l'eau et de la vapeur de l'alkool, a différentes temperatures," Journal de l'Ecole Polytechnique (Paris), Vol. 1, Cahier 2, Floreal et Prairial, An. III (1795), pp. 24-76. M. K. Kay, & L. M. Marple, Jr., "Spectrum analysis—-A modern perspective," Proceedings IEE, Vol. 69, No. 11, November, 1981, pp. 1380-1419. John Makhoul, "Linear prediction: A tutorial review," Proc. IEEE, Vol. 63, No. 4, April 1975. PP. 561-580. E. M. Kennough, "The K-Pulse concept,"IEEElTans A&P, Vol. AP-29, No. 2, PP. 327-331. 4-5 4-6 4-7 4-8 4-9 5-5 244 Martin Morf, et al., "Efficient solution of covariance equa- tions for linear prediction," IEEE Trans. on acousticsfi, speech & signal processing, Vol. ASSP-25, No. 5, October 1977, pp. 429-433. F-C, Chang, & H. Mott, "0n the Matrix Related to the Partial Fraction Expansion of a Proper Rational Function," Proc. IEEE, August 1974. PP. 1162-1163. L. Weiss, 8 R. N. McDonough, "Prony's method, z-transforms, and Pade approximation," SIAM Review, Vol. 5, No. 2, April 1963. PP. 145-149. Francis Brophy & A. C. Salazar, "Consideration of the Pade approximant technique in the synthesis of recursive digital filters," IEEE Trans. on Audio & Electroacoustics, Vol-21, No. 5, December 1973, pp. 500-505. L. Webb, K-M. Chen, D. P. Nyquist, B. Hollmann, "Extraction of single-mode backscatters by convolving synthesized radar signals with a radar return," IEEE/APS Symposium at University of Houston, APS-8-3, May 26, 1983. Norman Levinson, Journal of Mathematics and Physics, Vol. XXVI, No. 2, July 1947. pp. 110-119. E. A. Robinson & M. T. Silva, Deconvolution of Geophysical Time Series in the Exploration for Oil & Natural Gas," ETSevier, 1979, pp. 195-198. J. 0. Market & A. H. Gray, Jr., Linear Prediction of Speech. New York: Springer-Verlag, 1976. I. C. Gohberg & I. A. Fel'dman, Convolutional equations and projection methods for their solution," American Mathematical Society, Providence, R.I., 1974. Martin Morf, B. Dickinson, T. Kailath, & A. Vieira, "Effi- cient solution of covariance equations for linear prediction," IEEElTansactions on_Acoustics, Speech, and Signal Processing, Vol. ASSP-25, No. 5, October 1977. PP. 429-433. J. R. Westlake,l\Handbook of Numerical Matrix Inversion and Solution of Linear Equations. New York? Wiley, 1968. R. Hooke, & T. A. Jeeves, "Direct search solution of numerical and statistical problems," Journal of Association Computer Machinery, 8, pp. 212-229, 1961. Hokhtar S. Bazaraa & C. M. Shetty, Nonlinear Programming. New York: John Wiley & Sons, 1979. 7-1 A-l A-3 A-4 A-6 A-7 A-8 A-9 B-l 245 Chapter 7 C. A. Hamilton 8 F. L. Lloyd, "100 GHz binary counter based on DC SQUID's," IEEE Elec. Dev. Letts, Vol. EDL-3, No. 11, pp. 335-338. Appendix A A. Nehorai, G. Su, M. Morf, "Estimation of Time Differences of Arrival by Pole Decomposition,“ IEEE Trans. on Acoust., Speech, Signal Processing, Vol. ASSP-31, No. 6, pp. 1478-1491, December 1983. R. H. Barker, "The pulse transfer function and its application to sampling servo systems,“ Proceedings IEE (London), Vol. 99, Part IV, 1952. PP. 202-317. J. R. Ragazzina, G. F. Franklin, Sampled-Data Control Systems. Englewood Cliffs, N.J.: Prentice-Hall, 1983. A. V. Oppenheim, A. S. Willsky, Si nals and Systems. Engle- wood Cliffs, N.J.: Prentice-Hal , 1983. E. I. Jury, Sampled Data Control Systems. New York: John Wiley & Sons, 1958. E. I. Jury, "A note on the steady-state response of linear time-invariant systems," I.R.E. Proceedings1Correspondence), Vol. 48, No. 5. pp. 942-4. A. V. Oppenehim, R. W. Schafer, Di ital Si nal Processing. Englewood Cliffs, N.J.: Prentice-Hall, 1975. A. R. Bergen, & J. R. Ragazzini, "Sampled-Data Processing Techniques for Feedback Control Systems," Trans. AIEE, Vol. 73, 1954. E. I. Jury, “Snythesis and critical study of sampled-data control systems," Trans. AIEE, Part II, Paper 56-208, July 1956. Appendix B Bracewell, Fourier Integral and its Applications. New York: McGraw-Hill, 2nd edition, 1982, pp. 224.