W LIBRARY Michigan State University fl PLACE IN RETURN BOX to remove this checkout from your record. _ TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE J5 "mi 1 r! '90: - I" ‘, H MSU I. An Affirmative AdiorVEqual Opportunity Institution ammo-9.1 AUTOMATED RADAR TARGET DISCRINIINATION USING E-PULSE AND S-PULSE TECHNIQUES By Ponniah Ilavarasan A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 1992 ABSTRACT AUTOMATED RADAR TARGET DISCRIMINATION USING E-PULSE AND S-PULSE TECHNIQUES By Ponniah Ilavarasan The use of automated radar target discrimination using the Extinction pulse (E- pulse) and Single mode extraction pulse (S-pulse) in real applications is presented in this dissertation. A new natural frequency extraction technique is also introduced in this dissertation. Finally, the automated scheme is validated for thin wires and realistic targets, both analytically and experimentally. The performance of the automated discrimination scheme is verified in the presence of guassian noise using back-scattering theoretical responses of thin wire targets of various length-to—radii ratios. The importance of accurately extracting the natural frequencies (especially the radian frequencies) in successful discrimination is also demonstrated. Discrimination of realistic complex targets such as airplanes is only possible if their natural frequencies are extracted from their measurements with precision. The natural electromagnetic resonances of passive targets must be associated with complex natural frequencies having negative real parts. Extraction of natural resonances from measured late-time target response data, using algorithms such as Prony’s method or the unconstrained E-pulse technique, frequently lead to natural frequencies having non- physical real parts. A new constrained E-pulse technique is introduced which places constraints on the E—pulse amplitudes such that the natural frequencies have negative real parts. Accurate extraction of natural frequencies can only be expected when the signal- to-noise ratio of the measurements is high. Different types of antenna systems are investigated to improve the time-domain measurements taken inside the anechoic chamber. A general travelling—wave wire antenna is analytically studied, and antennas such as a straight wire and a V-wire are experimentally studied both as transmitter and receiver. Copyright by Ponniah Ilavarasan 1992 To my brother Elango and sister Ponmagal ACKNOWLEDGMENTS I would like to thank Drs. Byron Drachman, Dennis P. Nyquist, Kun-Mu Chen, and Edward J. Rothwell for participating in my guidance committee, all of whom made it possible to work in a friendly supportive environment. I am thankful to Dr. Nyquist who encouraged me to continue with graduate studies, and who also inspired me with his teaching. I am especially grateful to my academic advisors, Drs. Rothwell and Chen, for their generous support and guidance throughout my graduate studies. I feel fortunate to have been mentored by such a caring and knowledgeable "vaathiyar" as Dr. Rothwell. A special thanks goes to fellow graduate student and friend, Dr. John E. Ross, who helped to create an enjoyable environment in the electromagnetic laboratory at Michigan State University. I am indebted to my eldest brother, Elango, who offered constant emotional support and made my education financially possible. I deeply appreciate my family’s encouragement and patience throughout the many years of graduate studies. Finally, I cannot find the words to adequately express my thanks for all the love and support from my dear friend Melanie Atkinson. vi Table of Contents List of Tables ......................................... x List of Figures ......................................... xii List of Abbreviations .................................... xxiv Chapter 1 ............................................ 1 Introduction ...................................... 1 Chapter 2 ............................................ 6 Automated Radar Target Discrimination using E-Pulses ........... 6 2.1 Introduction ................................... 6 2.2 Preliminary ................................... 7 2.3 Definition of SNR in transient analysis ................... 8 2.4 Quantification and Automation ........................ 9 2.5 Effects of perturbed natural frequencies .................. 21 2.5.1 Analytical study ........................... 21 2.5.2 Numerical study ........................... 24 2.6 Numerical study using thin wire targets .................. 37 2.7 Conclusion .................................... 47 Chapter 3 ............................................ 49 Automated Radar Target Discrimination using S-Pulses ............ 49 3.1 Introduction ................................... 49 3.2 Preliminary ................................... 50 3.3 Quantification and Automation ........................ 51 3.4 Effects of perturbed natural frequencies .................. 5 3 3.4.1 Analytical study ........................... 53 3.4.2 Numerical study ........................... 56 3.5 Numerical study using thin wire targets .................. 65 3.6 Conclusion .................................... 71 vii Chapter 4 ............................................ 73 Natural Resonance Extraction from Transient Response ........... 73 4.1 Introduction ................................... 73 4.2 Overview of existing techniques ....................... 74 4.3 Unconstrained E—pulse Technique ...................... 76 4.3.1 Theory ................................ 76 4.3.2 Choice of E-pulse duration .................... 78 4.4 Constrained E-pulse Technique ....................... 79 4.4.1 Jury-Marden theorem ........................ 79 4.4.2 Formulation of the CET ...................... 85 4.4.3 Numerical algorithm ........................ 88 4.5 Discussion .................................... 89 4.5.1 Analysis of case I .......................... 89 4.5.2 Analysis of case 11 ......................... 97 4.5.3 Measured responses ......................... 135 4.6 Conclusion .................................... 150 Chapter 5 ............................................ 151 Free Field Measurements .............................. 151 5.1 Introduction ................................... 151 5.2 Free-field Anechoic Chamber Scattering Range .............. 153 5.3 Travelling-wave wire antenna ........................ 156 5.3.1 General Field of a Straight Wire ................. 156 5.3.2 Far-zone Specialization ....................... 161 5.3.3 Arbitrarily Shaped Wire ...................... 163 5.3.4 Numerical Results .......................... 163 5.3.5 Experimental Results ........................ 169 5.4 Travelling—wave antenna setup inside chamber .............. 175 5.5 Measurement of travelling-wave current using current loop ...... 177 5.6 V-wire Transmitter / Straight wire Receiver ............... 181 5.7 V-wire Transmitter / Horn Receiver .................... 184 5.8 Horn Transmitter / Horn Receiver ..................... 188 5.9 Deconvolution procedure ........................... 197 5.10 Conclusion ................................... 208 Chapter 6 ............................................ 210 Experimental Results ................................. 210 6.1 Introduction ................................... 210 6.2 Explanation of notations ............................ 210 6.3 Thin wire targets ................................ 211 6.4 Scale model targets ............................... 225 6.5 The effect of window duration in discrimination ............. 254 viii 6.6 Discrimination using only first two dominant modes ........... 257 6.7 Conclusion .................................... 259 Chapter 7 ............................................ 260 Conclusion ....................................... 260 7.1 Summary . . . .................................. 260 7.2 Future study ................................... 263 Bibliography .......................................... 264 Table 2.1 Table 2.2 Table 2.3 Table 4.1 Table 4.2 Table 4.3 Table 4.4 Table 4.5 Table 4.6 Table 4.7 Table 4.8 Table 4.9 Table 4.10 Table 4.11 Table 6.1 List of Tables Parameters used in creating SR #1 .................... 25 Parameters used in creating SR #2 .................... 25 Natural frequencies of SR #3 ........................ 25 Parameters used in creating response #1 ................ 90 Parameters used in creating response #2 ................ 90 Natural frequencies of response #1 .................... 90 Natural frequencies of response #2 .................... 105 Natural frequencies extracted from SR #2 with noise added. . . . . 105 Natural frequencies extracted from SR #2 with noise added. . . . . 105 Natural frequencies of 6 inch thin wire .................. 138 Natural frequencies of A—10 Thunderbird obtained via UCET using measurements separately. ......................... 143 Natural frequencies of A-10 Thunderbird obtained via CET using measurements separately. ......................... 143 Natural frequencies of A-10 Thunderbird obtained via UCET using all measurements together. ......................... 147 Natural frequencies of A-10 Thunderbird obtained via CET using measurements together. ........................... 147 Extracted natural frequencies of F-15, A-10 and B-747 obtained via UCET and CET. ............................ 227 Table 6.2 Table 6.3 Table 6.4 Squared error per point (SEPP) of F15, A10, and B747 obtained via the UCET and CET. .......................... 231 The ratio of late-time energy to total energy of F15, A10, and B747. ..................................... 232 The DL values corresponding to CET E—pulses of F15, A10, and B747. ..................................... 253 xi Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5 Figure 2.6 Figure 2.7 Figure 2.8 Figure 2.9 Figure 2.10 Figure 2.11 Figure 2.12 List of Figures Target A is F-15 Eagle and the longest dimension of F-15 is 20". ......................................... l 1 Target B is A-10 Thunderbird and the longest dimension is 6.75” . ..................................... 12 Convolutions of target B response with the E-pulses of targets A and B. ..................................... 15 The EDN values as a function of time for the convolutions of the target A response with the E-pulses of targets A and B. ....... 16 Convolutions of target A response with the E-pulses of targets A and B. ..................................... 17 The EDN values as a function of time for the convolutions of the target A response with the E-pulses of targets A and B. ...... 18 EDR values corresponding to the E—pulse of SR #1 and the newly created E—pulses, as explained in the case I. .............. 28 EDR values corresponding to the E—pulse of SR #1 and the newly created E-pulses, as explained in the case H ............... 29 EDR values corresponding to the E-pulse of SR #1 and the newly created E—pulses, as explained in the case III. ............. 3O EDR values corresponding to the E-pulse of SR #1 and the newly created E-pulses, as explained in the case IV. ............. 31 Convolutions of SR #2 with SNR of 20 dB with the first mode and all modes E—pulses. ............................. 34 Convolutions of SR #2 with the SNR of 5 dB with the first mode and all modes E—pulses. .......................... 35 xii Figure 2.13 Figure 2.14 Figure 2.15 Figure 2.16 Figure 2.17 Figure 2.18 Figure 2.19 Figure 2.20 Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5 Discrimination level of SR #2 for a varying SNR using first mode and all modes E—pulses. .......................... 36 Noise free back-scattering response of a broadside 1.00 m wire with l/a = 800. ............................... 39 Back-scattering response of a broadside 1.00 m wire with l/a = 800 and SNR of 30 dB. .......................... 40 Back-scattering response of a broadside 1.00 m wire with l/a = 800 and SNR of 5 dB. ........................... 41 EDR values computed for broadside response of 1 m wire with l/a = 800 ...................................... 43 EDR values computed for response of 1 m wire with l/a = 800 oriented 45 degrees from broadside. ................... 44 Minimum EDR values computed for wires of 0.90 m and 1.10 m for response from broadside 1.00 m wire. ............... 45 Minimum EDR values computed for wire of 0.90 m and 1.10 m for a response from a 1.00 m wire oriented at 45 degrees from broadside. ................................... 46 DL values of SR #1 with SNR of 25 dB corresponding to the 1" and 2" mode S-pulses of SR #1 and newly created S-pulses, as explained in case I. ............................. 58 DL values of SR #1 with SNR of 25 dB corresponding to the 1It and 2"I mode S-pulses of SR #1 and newly created S-pulses, as explained in case II .............................. 59 DL values of SR #1 with SNR of 25 dB corresponding to the 1It and 2'd mode S-pulses of SR #1 and newly created S-pulses, as explained in case III. ............................ 60 DL values of SR #1 with SNR of 25 dB corresponding to the 1" and 2"I mode S-pulses of SR #1 and newly created S-pulses, as explained in case IV. ............................ 61 DL values of SR #1 with SNR of 15 dB corresponding to the 1" mode S-pulses of SR #1 and newly created S-pulses, as explained in case I only for damping coefficients. ................. 63 xiii Figure 3.6 Figure 3.7 Figure 3.8 Figure 3.9 Figure 3.10 Figure 4.1 Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5 Figure 4.6 Figure 4.7 Figure 4.8 SDR values computed for first mode S-pulse for broadside response of a 1.00 m wire with l/a = 800. SDR values computed for second mode S-pulse for broadside response ofa 1.00 m wire with l/a = 800. SDR values computed for second mode S-pulse for response of a 1.00 m wire with U0 = 800 oriented at 45 degrees from broadside. Minimum SDR values computed for first mode S-pulse for 0.90 m and 1.10 m wire for broadside response of a 1.00 m wire. Minimum SDR values computed for second mode S-pulse for 0.90 m and 1.10 m wires for a response from a 1.00 m wire oriented 45 degrees from broadside. SR #1 and the waveform reconstructed using the natural frequencies obtained via the UCET. SR #1 and the waveform reconstructed using the natural frequencies obtained via the CET with q,=1.0 and pm=-1.0d- SR #1 and the waveform reconstructed using the natural frequencies obtained via the CET with qp=10. 0 and pm=-1. 0d- SR #1 and the waveform reconstructed using the natural frequencies obtained via the CET with q,=5.0 and pm=-1.0. . . . 95 SR #1 and the waveform reconstructed using the natural frequencies obtained via the CET with q,=5. 0 and pm=-1. 0d- Normalized radian frequencies of all modes obtained using the CET for different values of q, while pm=-l.0d-2. .......... 98 Normalized damping coefficients of all modes obtained using the CET for different values of q, while p"=-1.0d-2. .......... 99 Normalized radian frequencies of all modes obtained using the CET for different values of p" while q,=5.0. ............ 100 xiv Figure 4.9 Figure 4.10 Figure 4.11 Figure 4.12 Figure 4.13 Figure 4.14 Figure 4.15 Figure 4.16 Figure 4.17 Figure 4.18 Figure 4.19 Figure 4.20 Figure 4.21 Normalized damping coefficients of all modes obtained using the CET for different values of pm while q,=5.0. ............ 101 SR #2 and the waveform reconstructed using the natural frequencies obtained via the UCET. ................... 102 SR #2 and the waveform reconstructed using the natural frequencies obtained via the CET with q,=2. 0 and pm=-1. 0d- 2. ........................................ 103 SR #2 with SNR of 15 dB and the waveform reconstructed using the natural frequencies obtained via the UCET. ............ 106 SR #2 with SNR of 15 dB and the waveform reconstructed using the natural frequencies obtained via the CET with q,=2.0 and pw=-l.0d-2. ................................. 107 SR #2 with SNR of 10 dB and the waveform reconstructed using the natural frequencies obtained via the UCET. ............ 108 SR #2 with SNR of 10 dB and the waveform reconstructed using the natural frequencies obtained via the CET with q,=2.0 and pm=-1.0d-2. ................................. 109 SR #2 with SNR of 5 dB and the waveform reconstructed using the natural frequencies obtained via the UCET. .............. 110 SR #2 with SNR of 5 dB and the waveform reconstructed using the natural frequencies obtained via the CET with q,=2.0 and pw=- 1.0d-2 ...................................... 111 SR #2 with SNR of 0 dB and the waveform reconstructed using the natural frequencies obtained via the UCET. .............. 112 SR #2 with SNR of 0 dB and the waveform reconstructed using the natural frequencies obtained via the CET with q,=2.0 and p"=- 1.0d-2 ...................................... 1 13 Original SR #2 and the waveform reconstructed using the natural frequencies obtained from SR #2 with SNR of 15 dB via the UCET. ..................................... 115 Original SR #2 and the waveform reconstructed using the natural frequencies obtained from SR #2 with SNR of 15 dB via the CET with q,=2.0 and p..=l.0d—2. ...................... 116 XV Figure 4.22 Figure 4.23 Figure 4.24 Figure 4.25 Figure 4.26 Figure 4.27 Figure 4.28 Figure 4.29 Figure 4.30 Figure 4.31 Figure 4.32 Original SR #2 and the waveform reconstructed using the natural frequencies obtained from SR #2 with SNR of 0 dB via the UCET. ..................................... 117 Original SR #2 and the waveform reconstructed using the natural frequencies obtained from SR #2 with SNR of 0 dB via the CET with q,=2.0 and p..=1.0d-2. ...................... 118 Normalized radian frequencies of all modes for different time window W obtained using the UCET of SR #2 which has SNR of 15 dB (considering the whole window). ................. 120 Normalized damping coefficients of all modes for different time window W obtained using the UCET of SR #2 which has SNR of 15 dB (considering the whole window). ................. 121 Normalized radian frequencies of all modes obtained for different W using the CET with q,=2.0, pm=l.0d-2 of SR #2 with SNR of 15 dB (considering whole window). ................. 122 Normalized damping coefficients of all modes obtained for different ina the CET with q,=2.0, pw=1.0d-2 of SR #2 with SNR of 15 dB considering whole window. ............... 123 Normalized radian frequencies of all modes obtained for different W via the UCET of SR #2 with SNR of 0 dB (considering whole window). .................................. 124 Normalized damping coefficients of all modes obtained for different W using the UCET of SR #2 with SNR of 0 dB (considering whole window). ....................... 125 Normalized radian frequencies of all modes obtained for different W via the CET with q,=2.0, pn=1.0d-2 of SR #2 with SNR of 0 dB (considering whole window). .................... 126 Normalized damping coefficients of all modes obtained for different W using the CET with q,=2.0, pn=1.0d-2 of SR #2 with SNR of 0 dB (considering whole window). ........... 127 Calculated SNR of SR #2 with SNR of 0 dB (censidering the whole window) for different W. ..................... 128 xvi Figure 4.33 Figure 4.34 Figure 4.35 Figure 4.36 Figure 4.37 Relative minimum squared error between SR #2 with SNR of 10 dB and the reconstructed waveform for different number of match points. .................................... Normalized radian frequencies of all modes obtained using the UCET of SR #2 with SNR of 10 dB for different number of match points. Normalized damping coefficients of all modes obtained using the UCET of SR #2 with SNR of 10 dB for different number of match points. .................................... Normalized radian frequencies of all modes obtained using the CET of SR #2 with SNR of 10 dB for different number of match points. Normalized damping coefficients of all modes obtained using the CET of SR #2 with SNR of 10 dB for different match points. . . . Relative squared error between the original waveform and the reconstructed waveform for different T, (increment of T,=0.05). . 130 . 131 . 132 . 133 134 ................................................... 136 Relative squared error between the original waveform and the reconstructed waveform for different T, (increment of T,=0.25). ................................................... 137 Figure 4.41 Figure 4.42 Figure 4.43 Figure 4.44 Measured 6 inch wire response at broadside and the waveform reconstructed using the natural frequencies obtained via the UCET ..................................... Measured 6 inch wire response at 45° off broadside and the waveform reconstructed using the natural frequencies obtained via the UCET. Measured 6 inch wire response at broadside and the waveform reconstructed using the natural frequencies obtained via the CET with q,=5.0 and p"=-1.0d-9. ..................... Measured 6 inch wire response at 45° off broadside and the waveform reconstructed using the natural frequencies obtained via the CET with q,=5.0 and p“=-1.0d-9. ............... Expanded latter part of Figure 4.41 Figure 4.41 (via UCET). xvii OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO . 139 . 140 . 141 . 142 .144 Figure 4.45 Figure 4.46 Figure 4.47 Figure 5.1 Figure 5.2 Figure 5.3 Figure 5.4 Figure 5.5 Figure 5 .6 Figure 5.7 Figure 5.8 Figure 5.9 Figure 5.10 Figure 5.11 Expanded latter part of Figure 4.43 Figure 4.43 (via CET). . . . . 145 Measured response of A-10 at 45° off head-on and the waveform reconstructed using the natural frequencies obtained from multiple data sets via the UCET. .......................... 148 Measured response of A-10 at 45° off head-on and the waveform reconstructed using the natural frequencies obtained from multiple data sets via the CET. ........................... 149 MSU free field experimental facility and associated equipment. . . 154 Transient traveling current wave on a straight wire and local coordinate system. .............................. 157 Arbitrarily oriented wire and local coordinate system. ........ 162 General-shaped wire with traveling-wave current distribution. . . . 164 Near-zone axial component of electric field in the z = 0 plane due toastraightwireon thezaxisfora = 0, b =1m,p = 1m, 7= 1 ns. ...................................... 166 Near-zone radial component of electric field in the z = 0 plane duetoastraightwireon thezaxis fora = 0, b = 1m,p =1m, 7= 1 ns. ................................... 167 Far-zone axial component of electric field in the z = 0 plane due toastraightwireon thezaxis fora = 0, b = 1m,p = 100m, 7= 1 ns. ................................... 170 Far-zone radial component of electric field in the z = 0 plane due toastraightwireon thezaxis fora = 0, b =1m,p =100m, 7= 1 ns. ................................... 171 Experimental current pulse waveform. ................. 173 Near-zone axial field (measured and theoretical) for 63.5 cm uninsulated wire measured at p = 2.49 m, z = 0. .......... 174 Far-zone axial field (measured and theoretical) for 63.5 cm insulated wire measured at p = 2.49 m, z = 0. ............ 176 xviii Figure 5.12 Figure 5.13 Figure 5.14 Figure 5.15 Figure 5.16 Figure 5.17 Figure 5.18 Figure 5.19 Figure 5.20 Figure 5.21 Figure 5.22 Figure 5.23 Figure 5.24 Figure 5.25 Figure 5.26 Anechoic chamber with travelling-wave V-wire transmitting antenna and straight-wire receiving antenna (CT . .. Coaxial Tube). ......................................... 178 Travelling-wave current on a straight-wire antenna terminated by open load measured 12 inches from feed point. ............ 179 Travelling-wave current on a straight-wire antenna terminated by 50 ohms load measured 12 inches from feed point. ......... 180 Travelling-wave current on a left side of a V-wire antenna terminated by open load measured 12 inches from feed point. . . . 182 Travelling-wave current on a right side of a V-wire antenna terminated by open load measured 12 inches from feed point. . . . 183 Measured pulse response of the V-wire / straight wire antenna system. .................................... 185 Measured pulse response of a medium size boeing 707 aircraft model, with V-wire / straight wire antenna system response included. ................................... 186 Measured pulse response of a medium size boeing 707 aircraft model, with V-wire / straight-wire antenna system response subtracted. .................................. 187 Anechoic chamber with travelling-wave V-wire transmitting antenna and wideband horn receiving antenna. ............ 189 Measured pulse response of the V-wire / horn antenna system. . 190 Measured pulse response of a medium size boeing 707 aircraft model, with V-wire / horn antenna system response included. . . . 191 Measured pulse response of a medium size boeing 707 aircraft model, with V—wire / horn antenna system response subtracted. . . 192 Measured pulse response of the horn / horn antenna system. . . . . 194 Measured pulse response of a medium size boeing 707 aircraft model, with horn / horn antenna system response included ...... 195 Measured pulse response of a medium size boeing 707 aircraft model, with horn / horn antenna system response subtracted. . 196 xix Figure 5.27 Figure 5.28 Figure 5.29 Figure 5.30 Figure 5.31 Figure 5.32 Figure 5 .33 Figure 5.34 Figure 5.35 Figure 5.36 Figure 6.1 Figure 6.2 Figure 6.3 Figure 6.4 Theoretical bistatic scattering response of 6" thin wire. ....... 198 The measured response of 6" thin wire inside the chamber using the horn/horn antenna system. ...................... 199 FFT of the measured response of 6" thin wire. ............ 200 The system response in frequency domain obtained using 6" thin wire as a calibrator. ............................. 202 Measured response of A-lO thunderbird at 45° off head-on ...... 203 Frequency domain measured response of A-10 Thunderbird at 45° off head-on obtained via FFT. ...................... 204 The actual A-10 Thunderbird response in frequency domain after deconvolution. ................................ 205 Actual response of A-10 Thunderbird at 45° off head-on obtained via inverse FFT with rectangular and guassian cosine modulated weighting function. ............................. 206 Gaussian cosine modulated weighting function. ........... 207 The effective transmitted pulse hitting targets which is obtained by taking inverse FFT of guassian cosine modulated weighting function. ................................... 209 EDR values of convolutions corresponding to E-pulses obtained via UCET with measured responses of 8" wire at broadside, and 22.5° & 45° off broadside. ............................ 213 EDR values of convolutions corresponding to E-pulses obtained via CET with measured responses of 8" wire at broadside and 22.5° & 45° off broadside. ............................ 214 SDR values of convolutions corresponding to Q‘-S-pulses obtained via UCET with measured responses of 8” wire at broadside and 22.5° & 45° off broadside. ......................... 215 SDR values of convolutions corresponding to Q‘-S-pulses obtained via CET with measured responses of 8” wire at broadside and 22.5° & 45° off broadside. ......................... 216 XX Figure 6.5 Figure 6.6 Figure 6.7 Figure 6.8 Figure 6.9 Figure 6.10 Figure 6.11 Figure 6.12 A Figure 6.13 Figure 6.14 Figure 6.15 Figure 6.16 Figure 6.17 SDR values of convolutions corresponding to Qz-S-pulses obtained via UCET with measured responses of 8" wire at broadside and 22.5° & 45° off broadside. ......................... 217 SDR values of convolutions corresponding to Qz-S-pulses obtained via CET with measured responses of 8" wire at broadside and 22.5° & 45° off broadside. ......................... 218 EDR values of convolutions corresponding to E—pulses obtained via UCET with measured responses of 7" wire at broadside and 22.5° & 45° off broadside. ............................ 219 EDR values of convolutions corresponding to E-pulses obtained via CET with measured responses of 7" wire at broadside and 22.5° & 45° off broadside. ............................ 220 SDR values of convolutions corresponding to Q‘-S-pulses obtained via UCET with measured responses of 7" wire at broadside and 22.50 & 45° off broadside. ......................... 221 SDR values of convolutions corresponding to Q‘-S-pulses obtained via CET with measured responses of 7" wire at broadside and 22.5° & 45° off broadside. ......................... 222 SDR values of convolutions corresponding to Q’—S-pulses obtained via UCET with measured responses of 7" wire at broadside and 22.50 & 45° off broadside. ......................... 223 SDR values of convolutions corresponding to Qz-S-pulses obtained via CET with measured responses of 7" wire at broadside and 22.5° & 45° off broadside. ......................... 224 UCET and CET E-pulse of F-15 Eagle. ................ 228 UCET and CET E-pulse of A-10 Thunderbird. ............ 229 UCET and CET E-pulse of Boeing 747. ................ 230 EDR values of convolutions corresponding to E-pulses obtained via UCET with F-15 Eagle responses measured at various aspect angles. ..................................... 233 EDR values of convolutions corresponding to E-pulses obtained via CET with F-15 Eagle responses measured at various aspect angles. ..................................... 234 Figure 6.18 Figure 6.19 Figure 6.20 Figure 6.21 Figure 6.22 Figure 6.23 Figure 6.24 Figure 6.25 Figure 6.26 Figure 6.27 Figure 6.28 Figure 6.29 SDR values of convolutions corresponding to Q'-S—pulses obtained via UCET with F-15 responses measured at various aspect angles. .................................... SDR values of convolutions corresponding to Q‘-S-pulses obtained via CET with F-15 responses measured at various aspect angles. SDR values of convolutions corresponding to Qz-S-pulses obtained via UCET with F-15 responses measured at various aspect angles. .................................... SDR values of convolutions corresponding to Qz-S-pulses obtained via CET with F-15 responses measured at various aspect angles. EDR values of convolutions corresponding to E—pulses obtained via UCET with A-lO Thunderbird responses measured at various aspect angles ................................. EDR values of convolutions corresponding to E-pulses obtained via CET with A-10 Thunderbird responses measured at various aspect angles. .................................... SDR values of convolutions corresponding to Q‘—S-pulses obtained via UCET with A-lO responses measured at various aspect angles. .................................... SDR values of convolutions corresponding to Q‘-S-pulses obtained via CET with A-10 responses measured at various aspect angles. SDR values of convolutions corresponding to Qz-S-pulses obtained via UCET with A-10 responses measured at various aspect angles. .................................... SDR values of convolutions corresponding to QZ-S-pulses obtained via CET with A-lO responses measured at various aspect angles. EDR values of convolutions corresponding to E-pulses obtained via UCET with Boeing-747 responses measured at various aspect angles. .................................... EDR values of convolutions corresponding to E-pulses obtained via CET with Boeing-747 responses measured at various aspect angles. .................................... xxii . 236 . 237 . 238 . 239 . 240 . 241 . 243 .244 . 245 . 246 . 247 . 248 Figure 6.30 SDR values of convolutions corresponding to Q‘-S-pulses obtained via UCET with B-747 responses measured at various aspect angles. ..................................... 249 Figure 6.31 SDR values of convolutions corresponding to Q'-S-pulses obtained via CET with B-747 responses measured at various aspect angles. ..................................... 250 Figure 6.32 SDR values of convolutions corresponding to Qz-S-pulses obtained via UCET with B-747 responses measured at various aspect angles. ..................................... 251 Figure 6.33 SDR values of convolutions corresponding to QZ-S-pulses obtained via CET with B-747 responses measured at various aspect angles. ..................................... 252 Figure 6.34 DL values of A-lO Thunderbird measured at various aspect angles for different values of W, with all E-pulses. .............. 255 Figure 6.35 DL values of Boeing-747 measured at various aspect angles for different values of W, with dominant mode S-pulses .......... 256 Figure 6.36 DL values corresponding to convolutions of measured responses of F 15, A10, and B74 at all aspects with E-pulses created with first two dominant CET modes. ..................... 258 xxiii SR SNR DL Q‘-S-pulse SDN SDR UCET CET HO SEPP List of Abbreviations Synthetic response Signal-to-noise ratio E-pulse discrimination number E—pulse discrimination ratio Discrimination level i‘“ mode quadrature sum S-pulse S-pulse discrimination number S-pulse discrimination ratio Unconstrained E-pulse technique Constrained E—pulse technique Broadside Head-on Squared error per point xxiv Chapter 1 Introduction Radar target identification and discrimination has become more important since recent developments in the world have emphasized the use of conventional weapons instead of nuclear weapons. It is necessary to have a scheme to discriminate friendly targets from enemy targets in a war zone. The shooting down of the misidentified passenger aircraft which took place over the Persian Gulf could have been avoided if there was a system that could discriminate targets. Existing techniques can spot targets, but cannot identify them. Many researchers have been working on schemes such as polarization techniques [1,2,3] multiple frequency measurements [4,5,6] and ramp response imaging [7,8] which would identify radar targets. However, the main drawback of these techniques is aspect dependence. In order to overcome aspect dependency, many techniques such as natural resonance based ramp response [9,10,11] natural frequency comparison [12,13,14, 15,16,17,18] Kill pulse (K-pulse) [19,20,21,22,23] and Extinction pulse (E-pulse) [24,25 ,26,27,28] based on the natural frequencies were shown to be aspect independent. Although, the natural resonance based ramp response technique is aspect-independent, this technique requires the use of a predictor-correlator method. The predictor-correaltor method is fairly complicated, and is sensitive to the choice of the sampling interval. Many sampling interval values might nwd to be tried to provide adequate discrimination. Also, a multi-frequency radar system is needed to measure the response. The natural frequency comparison technique requires accurate estimation of the natural frequencies of targets. The target discrimination is done by comparing the extracted natural frequencies of an unknown target with the library of natural frequen- cies. The target is identified corresponding to the target natural frequencies closest to the extracted natural frequencies. Since extraction of natural frequencies from the late-time target responses is an inherently ill—conditioned numerical procedure, very large SNR is required in the transient return. It has therefore been concluded that this method for the direct discrimination of differing targets is impractical. The singularity expansion method (SEM) introduced by C. E. Baum in the early 1970’s and proved by L. Marin [29] that the scattered field of a conducting object in the late-time can be represented as a sum of damped sinusoids [30]. The frequencies of the late-time oscillations depend only on the geometry of the conducting object, not on the excitation waveform. The E-pulse is defined such that when it is convolved with the corresponding response, it produces zero signal in the late-time. The E-pulse technique, based only on the natural frequencies and developed at Michigan State University by K. M. Chen, had successfully shown that discrimination was possible for complex targets. If the E-pulse is chosen such that 1. all complex natural resonances (CNR) are annihilated 2. there are no zeros in the E—pulse spectrum other than those which coincide with the CNR’s of the target. 3. the duration of the E-pulse is minimal then the E-pulse is the same as the K-pulse [19]. Previous work done at Michigan State University [24,31] identified the correct target visually by inspecting all convolutions displayed at a computer screen. In real applications, it would be very difficult for the human eye to identify the correct target. Therefore, it requires an automated scheme to identify the correct target. The automated E-pulse discrimination scheme is presented in Chapter 2. Chapter 2 investigates the effect of perturbed natural frequencies in discrimination. The analysis is verified with synthetic data sets. Chapter 2 also explains the reason why discrimination is strongly effected by small changes in the radian frequency compared to small changes in the damping coefficient. Finally, the effects of varying amounts of noise on E—pulse discrimination is studied. This analysis is done by adding white guassian noise with zero mean to the numerical responses of thin wires of various lengths, and length to radii ratios. The single mode extraction pulse (S-pulse) is defined such that when the S-pulse is convolved with the corresponding response, it produces a single damped sinusoid in the late-time. Discrimination using the S-pulse [24] was initially demonstrated by K.M.Chen [32,33,34]. Cosine and Sine S-pulses are synthesized for each target [24]. The target is identified by visually inspecting all the cosine or sine S-pulse convolutions. The visual inspection is very difficult since it requires the identification of a single damped sinusoid among many sinusoids. To overcome this difficulty, both cosine and sine S-pulses are convolved with unknown responses to obtain signals Ca“) and c,,(t). A complex signal c,(t) = 6,0) + j 0,,(0 is subsequently constructed. The logarithm of c,(t) is then taken. The slopes of the real and imaginary parts of that logarithm are calculated. 3 The target is identified by visually inspecting both of the slopes which have minimum error from the expected slopes. In search of a better method, the error between the expected damped sinusoid and the corresponding convolutions was calculated. Then, the S-pulse corresponding to the minimum error was identified as the correct target. However, these methods were abandoned due to difficulty in calculating the error. An automated S-pulse discrimination scheme has been recently developed and it is presented in chapter 3. Chapter 3 also investigates the effect on discrimination due to perturbed natural frequencies. Finally, the effect on S-pulse discrimination of varying amounts of noise is studied. This analysis is done by adding white guassian noise with zero mean to the numerical responses of thin wires of various lengths, and length to radii ratios. An accurate estimation of the natural frequencies is required when radar target discrimination is based on them. The natural frequencies of simple targets such as thin wires and wire stick models can be estimated theoretically. The theoretical estimation of the natural frequencies of complex targets is very difficult. The natural frequencies of complex targets are obtained by extraction from measurements. Existing techniques such as Prony’s method, the E—pulse technique, etc. [ll,l3,35,36,37] do not put any physical constraints on the damping coefficients. The damping coefficients should be negative. If a measurement SNR is low, the natural frequencies extracted using the unconstrained E-pulse technique (U CET) [35] may have some modes with positive damping coefficients. A new extraction technique has been implemented by placing constraints on the amplitudes of the E-pulse such that the damping coefficient is restricted to negative values. This technique is called the constrained E—pulse technique (CET). Both the UCET and the CET are discussed in chapter 4. It is important to make accurate measurements with high SNR in order to achieve accurate estimations of the natural frequencies of complex targets. The E-pulse technique was successfully demonstrated using ground plane measurements, since these measure- ments had a high SNR [31]. The high SNR environment allowed accurate measurements. However, targets could not be illuminated at arbitrary aspects and polarizations. In order to simulate a realistic environment, free field range was built within an anechoic chamber at Michigan State University. The chamber allows a greater range of aspects and polarizations. A travelling-wave wire antenna is studied analytically in chapter 5 [38]. Travelling-wave wire antennas are used for transmission and reception inside the chamber. Three combinations of transmitting and receiving antennas are studied experimentally. Chapter 5 also briefly discusses how a frequency domain deconvolution procedure can be applied to time domain measurements. Chapter 6 presents evidence to validate the automated E-pulse technique using the E-pulses obtained via the UCET and CET on thin wire measurements, and the measurements of F-15 Eagle, A-10 Thunderbird, and Boeing 747 aircraft models. Chapter 6 also presents evidence to validate the automated S-pulse technique using S- pulses obtained via the UCET and CET on thin-wire measurements, and measurements of F-15 Eagle, A-10 Thunderbird, and Boeing 747 aircraft models. Finally, chapter 6 discusses the effect on discrimination of the window width used for calculating the error between the correct target and wrong targets using the E-pulse and S-pulse. Chapter 2 Automated Radar Target Discrimination using E-Pulses 2.1 Introduction Previous work by Chen et. a1. has developed the theory of the extinction pulse (E-pulse) [25,32,33,39]. The E—pulse is synthesized to annihilate, when convolved with a band-limited late-time target pulse response, all natural modes present in that response. The E—pulse concept was shown to be independent of the illumination aspect. At the beginning the efficacy of the E-pulse method was verified with noise-added artificial data sets. These results showed that the discrimination of simpler targets such as thin wires, and complex targets such as airplanes would be possible if the natural frequencies of the targets were accurately extracted from measured data. The E—pulse technique was successfully demonstrated using the measurements of a few half complex model airplanes done on a ground plane [31], and the measurements of full complex model airplanes done inside an anechoic chamber [27]. Previously, identification of the correct target was done by visually inspecting all the convolutions of a response with all of the E—pulses available in the database. This was possible when the number of E-pulses in the database was small. However, identification by visual inspection was very difficult when the number of targets in the database was large. In real applications, an automated scheme is necessary to avoid an error in the decision making process. From past work it was observed that a small perturbation of the extracted damping coefficients, such as 5 percent from the actual values, did not have a large effect on the radar target discrimination. However, a small perturbation of the radian frequencies, such as 5 percent from the actual values, had a considerable effect on the radar target discrimination. This observation is analytically studied and verified with artificial data sets. This chapter will quantify discrimination using the E—pulse in a way that is suitable for use in an automated scheme. Examples of the performance of this automated scheme are presented using theoretical scattering data for thin wire targets with varying amounts of additive noise. 2.2 Preliminary It is well known that the response of a conducting target to a band-limited transient excitation in the late-time can be written as N ’0) = 2 a. 61p(ont) cos(w't+¢n) r > TL (2.1) I'I where s, = a, + for, is the aspect independent natural frequency of the n“ target mode, a, and 41,, are aspect dependent modal amplitudes and phases, and TL is the aspect dependent beginning of late-time. The E-pulse waveform e(t) for a particular target is defined such that e(t) *r(t) I! 0 t > Tu: = TL + T, (2.2) where T, is the duration of e(t), and r(t) can be the response of the target from any aspect angle. This definition implies that E(s) - 21140} a 0 for s=sn and s=s;; n=1,2,3,...N (2-3) To affect target discrimination using E—pulses, the response from an unknown target v(t) is convolved with each of the E-pulses in a database. The convolved waveforms are denoted c‘(t) = e(t) *v(t). If E(s) has zeros at all natural frequencies present in the late-time portion of v(t), then c¢(t) = 0 in the late time. If E(s) does not contain zeros at all natural frequencies in the late-time portion of v(t), then c¢(t) .e O in the late time. Thus, the target producing v(t) is associated with the E-pulse that yields zero in the late-time after convolution with v(t). 2.3 Definition of SNR in transient analysis In order to do the noise analysis, the signal-to—noise (SNR) should be defined. In transient analysis, the usual Continuous-Wave (CW) definition of SNR is inappropriate. It is more useful to interpret the SNR within a specified time window, the choice of which affects the SNR value. For this analysis, the window duration W was chosen as the minimum duration window that contains 99% of the total energy in the noise free data. The ratio of the signal energy to the noise energy within this window is fv2(t)dt SNR (dB) = 10.0 log ’V 10 ‘l’oW (2.4) where v(t) is the noise free signal and 1:0 is the mean-square value of the noise voltage and is defined as W ‘l’o = -,1;,fv3(0dt <15) 0 where v,(t) is the noise voltage. The definition of the SNR holds throughout the whole dissertation unless otherwise specified. 2.4 Quantification and Automation A radar target return is convolved with all of the E-pulses in the database. Then, the correct target is found to be the one which has zero signal in the late-time. Automated discrimination requires a measure of the amount of signal present in the late- time of c,(t) for each E—pulse. Two automated schemes were tried. However, the first scheme failed. For the first automated scheme, a quantity called the E-pulse Discrimination Number (EDN) was defined as T '1 fcfdr o I fcfdt 0 EDN(t) = (2") where T, is the usable time window in which the target response is normally above the noise level. The EDN of each convolved waveform is calculated as a function of time r. If the target producing v(t) has been catalogued with an E-pulse in the data base, the EDN would have reached unity at an earlier time t ( = T“), while all of the other EDN would have reached unity at a later time t (> T“). In actual practice, noise, inaccuracies id in the estimates of the target natural frequencies, and the limited number of modes used to create E-pulses will prevent the EDN from precisely reaching unity at the start of the late-time of the convolution corresponding to the E-pulse of the target producing the response. Thus, the target is identified by the E-pulse yielding the minimum time t=T,,g, for the EDN to reach a prechosen number between 0.95 to 1.00. However, the scheme failed when two targets in the data base had a large difference in dimensions. The failure of the scheme is illustrated with an example. To apply the automated discrimination scheme, the beginning of the late-time TL for the unknown target response v(t) must be estimated. The late-time can be computed based on the maximal one way transit time of the target T” (maximal one way transit time of the target is equal to the longest dimension of the target divided by the speed of light), the effective pulse duration used in the measurement system 1;, and an estimation of when the wave strikes the leading edge of the target T,. For the usual case of back- seattering measurements, TL is defined as TL = To +Tp+2Ta (2.7) In the case of forward scattering measurements, TL is defined as TL = T, + T, + T, (2.8) Two targets A and B are shown in Figure 2.1 and Figure 2.2. Target A is F—15 Eagle and target B is A-10 Thunderbird. The longest dimensions of target A and B are 20 and 6.75 inches. Targets A and B have one way transit-times of T: and T: , and T: is almost three times that of T: . The minimum natural E-pulse duration is pit/0‘ [27], 10 Figure 2.1 Target A is F-15 Eagle and the longest dimension of F—15 is 20". ll Figure 2.2 Target B is A-10 Thunderbird and the longest dimension is 6.75”. 12 where p is twice the number of modes extracted, and 00,, is the largest radian frequency among the extracted modes. In general the duration of the optimum forced E—pulse is slightly larger compared to the minimum natural E-pulse duration. The E—pulse duration of target A (T: ) is large compared to the E-pulse duration of target B (T,'). In general, the absolute value of Q of the higher order modes are large compared to the lower order modes. So, only first few modes can be extracted for each target. The radian frequency of the highest order mode excited for longer targets is small compared to smaller targets. Therefore, the largest radian frequency extracted from the responses of target A is smaller to the largest radian frequency extracted from the responses of target B. The late-times of the convolved waveforms are TI} for target A, and T11. for target B, and are given by 1;} = T,+rp+2r:+r,‘ a * a a (2'9) Tu, = Tb+Tp+2T,, +T, At that time, T, was visually estimated from the measured responses. When the response of target B was convolved with the E-pulse of target A, the convolution was non-zero in the late-time Tl}, as expected. This is shown in Figure 2.3. When the response of the target B was convolved with the E-pulse of target B, the convolution was zero in the late- time 1,1,, as expected. This is shown in Figure 2.3. Figure 2.4 shows the EDN values corresponding to the E-pulses of target A and B as a function of time t. From Figure 2.4,the EDN corresponding to the E—pulse of target A reached closer to unity at t=18.0 nsec, while the EDN corresponding to the E- 13 pulse of target B reached closer to unity at t=12.5 nsec. As expected, this scheme identified target B as the correct target. However, this scheme failed while trying to identify target A, and it is illustrated below. When the response of the target A was convolved with the E-pulse of target A, the convolution was zero in the late-time Tl}, as expected. This is shown in Figure 2.5. The convolution of the response of target A with the E-pulse of target B was non-zero in the late-time T12, as expected. This is shown in Figure 2.5. From Figure 2.6, the EDN corresponding to the E-pulse of target B reached closer to unity at t= 14.0 nsec, while the EDN corresponding to the E-pulse of target A reached closer to unity at t=15.0 nsec. If this scheme was used in the identification, it would have identified target B as the correct target instead of target A. This scheme failed because the beginning of the late-time of the convolution corresponding to the E- pulse of target A was large compared to the convolution corresponding to the E-pulse of target B. The scheme faltered when the larger dimension targets nwded to be identified among small and medium dimensions targets. The scheme was abandoned since the success rate of identifying the correct target was not high enough. In search of a better method, the E—pulse Discrimination Ratio (EDN) is defined as TLOW‘ T. '1 EDN = f c}(r)dr fame (24°) 1', 0 14 time of A Lote— Late—time of B E-pulse of A E—pulse of B a a: get 8 '------ Target 8 Tor ll1TIIIIIITIIITIHIIIIrrVTIIIIIITITTIIII] qd-q—quqq-qu-u—u-qua-«dd—qqdquqqqd—-qdud-qdd 0 0 0 0 o. s o. s 1 0 O n_0 83:an m>:o_om -l.OO 10.0 15.0 20.0 Time (ns) 5.0 0.0 Convolutions of target B response with the E-pulses of target A and B. Figure 2.3 15 1.0 0.8-5 0.6-3 2 E Q . DJ 2 0.4-: 0.25 3 I". :' Target 8 * E—pulse of A I ' ------ Torget B * E-pulse of B 0.0 djjIIIIITrrlllIITTII[IIIITIIIIITHIIITIIII 0.0 5.0 . 10.0 15.0 20.0 Time (ns) Figure 2.4 The EDN values as a function of time for the convolutions of the target A response with the E-pulses of targets A and B. 16 A f o 0. 10 m n... w. a . f l m . - L m m U HO. _ .5 e '1 B l m. a A '6’ L I ff .I 00 r ee n 1616 r uu IO. WP r0 _ :1 EE r u u n AA n tt ee .. mm. b. 00 .5 TT r . n . . I . _l c a I. r0 uddu—qq-q-qdu—udJuuqqu—qq-qq-ddq—q-q-ud-d- o. O O O 0 O 0. 5 0. 5. O. 1 0 0 n_u ._1 £03:an 350.3”. Time (ns) Figure 2.5 Convolutions of target A response with the E-pulses of targets A and B. 17 1.0 0.8 0.6 Z O LrJ 0.4 0.2 Target A * E-pulse of A ' ------ Target A at E-pulse of B 0.0 [ITIIII TT1HTHII,IIITIrIjIIITIIIITITIIIII] 0.0 5.0 10.0 15.0 20.0 Time (ns) Figure 2.6 The EDN values as a function of time for the convolutions of the target A response with the E—pulses of targets A and B. 18 The EDN is a measure of the deviation from the expected value of zero late-time energy. The choice of window duration (W,) is usually based on the duration of target response and SNR present in those measurement of an unknown target. W, is uniquely defined once the global maximum transit-time of all of the targets (1'3”) and the maximum E- pulse duration of all of the E-pulses (Tfim) in the database are known. W, is given by w, = r,-r,-r,-21::“—r:m (2.11) Note that the late-time energy of c,(t) is normalized by the E-pulse energy. To use the EDN for automated discrimination, the EDN is calculated for each E-pulse in the data base. If the target producing v(t) has been cataloged with an E—pulse in the data base then the EDN for that E-pulse will be zero, while all other E-pulses will yield a non-zero EDN. In actual practice, noise, inaccuracies in the estimates of the target natural frequencies, and the limited number of modes used to create E—pulses will prevent the EDN from precisely vanishing when the E—pulse is matched to the target producing the response. Thus, the target is identified by the E-pulse yielding the minimum EDN. As a quantitative measure of the differences in the EDN values computed for all of the E- pulses in the data base, the E—pulse Discrimination Ratio (EDR) is defined as EDR = 10.010g,,{—%—)} (2.12) m Therefore, the E-pulse yielding the smallest EDN has an EDR of 0 dB, while the EDR produced by other E—pulses is greater. To implement the automated discrimination scheme, the beginning of the late—time TL for the unknown target response v(t) must be evaluated. The target data base will 19 contain a value for 1;, as well as values of T,, for each target. Thus, only T, must be estimated from the response data. For this analysis a computer model of a threshold detector is used to determine T,. The point at which v(t) is greater than the response voltage V, is taken to be T,. The threshold voltage V, must be set small enough to detect weak signals, yet large enough to limit the false alarm rate to an acceptable level. For guassian noise, the mean time between false alarms I}, for a given V, is given by [40] V2 Tfa = icxp T BIF 2",0 (2.13) B,, is the IF bandwidth of the measurement system. For the analysis done in section 2.6, the threshold voltage was chosen so that V, = 8m. This resulted in 1},=21.36 hours for the 1.022 GHz bandwidth used in the numerical analysis. The discrimination level (DL) is defined as the EDR value corresponding to the target closest to the correct target. To have confidence in the discrimination, the discrimination level should preferably be 10 dB or above. This may not be possible when two nearly identical targets need to be discriminated. This difficulty may be overcome by using the S-pulse method in the discrimination, in addition to the E-pulse method. The S‘Plllse method is discussed in chapter 3. 2O 51 2.5 Effects of perturbed natural frequencies To study the effects of the damping coefficients and the radian frequencies on the radar target discrimination, the extracted natural frequencies are assumed to differ from the actual values by a chosen percentage. 2.5.1 Analytical study Assume e(t) is an E—pulse constructed using extracted natural frequenciess: Which differ from the actual natural frequencies of the target according to s: = s" r as. . A convolution of a response r(t) given by (2.1) with an E-pulse in the late-time is given by N Ct“) = 2 “ulE(su)|€°"cos(w.t+lll.) t) TLE (2.14) n-l E(s) is Laplace spectrum of the E-pulse, and ill, is the aspect dependent phases. The cOnvolution is non zero since E (s) t 0 . Expanding E(s) by the Taylor series around the natural frequencies 3 = s: gives E(su) = E(s,‘)ess,§£ (2.15) Where higher order terms are ignored since the difference between the extracted and the aCtual natural frequencies are assumed to be small. The first order partial derivative of E(8) with respect to s is non-zero at s = s: = o;+ 00:. 21 Substituting (2.15) into (2.14) and using E(s:) = 0 yields N ("(0 = 2“» u-l M.%(SJ) e°"cos(w,t+¢,) r> Tu, (2.16) where c,(t) is the first order error term due to a perturbation of the extracted natural frequencies. To study the effects of the damping coefficients and the radian frequencies separately on the radar target discrimination, two different studies are conducted. In the first case, when the extracted frequencies have only a small perturbation from the actual values of the damping coefficients, i.e. Am, =0, the first order error term becomes N c,'(t) = Z a. %(s:) u-i e"'cos( u‘tulru) t > Tu: (2.17) .- an an It is apparent that the first order error term of the convolution in the late-time primarily depends on the magnitude of the damping coefficient perturbations. In the second case, when the extracted frequencies have only a small perturbation from the actual values of the radian frequencies, i.e. A a,=0, the first order error term becomes N c,"(t) = Ea. n-l e "'cos(co.t + ‘1'.) t > Tu. (2°18) ‘- 10,, to” c'f—afts.‘) It is obvious that the first order error term of the convolution in the late-time primarily depends on the magnitude of the radian frequency perturbations. It can be concluded that the effect on the discrimination due to a small perturbation of the damping coefficients is smaller compared to the perturbation of the radian frequencies, since the absolute value of the radian frequencies is larger than the 22 absolute value of the damping coefficients. For example high Q targets, the absolute value of the radian frequency is normally ten times or above the absolute value of the damping coefficient. It may not be true for the low Q targets such as sphere. The class of complex targets considered in our laboratory reveals that the absolute value of the radian frequency is typically ten times or above the absolute value of the damping coefficient. In general, the amount of error in the convolution can be reduced since the radian frequencies can be extracted with good precision, while the damping coefficients cannot be. This observation will be confirmed in chapter 4. There is a possibility that the first order error from different extracted modes may add up to decrease the total error of the convolution in the late-time. However, the error will not vanish unless the extracted and the actual natural frequencies are the same. The EDN due to the first order error of the convolution in the late-time is EDN“ = " as T" w‘ 7‘. (2 19) £0, E(sgas. I e 'cos(w.t+Yl)I fez“) ° 1121 1" o where T,“ is the E—pulse duration corresponding to the extracted natural frequencies. The EDN“, value corresponding to the E—pulse of the extracted natural frequencies, will be equal to EDN”, value corresponding to the E-pulse of the actual natural frequencies, if the first order error term vanishes. The DL due to the error in the mode extraction is decreased by EDN 121., = 10.0rog,o{EDN“} (2.20) ea 23 The discrimination level is decreased by a significant amount when the extracted radian frequencies vary from the actual values. However, it is not true for the case of the extracted damping coefficients. In the following section an attempt to verify this claim is done with artificial data sets. 2.5.2 Numerical study Artificial data sets #1 (SR #1) and #2 (SR #2) are created with the parameters given in Table 2.1 and Table 2.2, respectively. The natural frequencies of artificial data #3 (SR #3) are given in Table 2.3. The natural DC E-pulses of SR #1, SR #2, and SR #3 are created. SR #1 is convolved with the E—pulses of SR #1 and SR #3, and the corresponding EDR values are calculated. The BL is then evaluated. To see the effects of the damping coefficients and the radian frequencies in the DL, four cases are examined. case I: All of the extracted damping coefficients of SR #1 are perturbed by :5, 1:10, 3:15, 1:20, and 125 percent from the actual damping coefficients, while all the radian frequencies of SR‘ #1 remain the same as the actual radian frequencies. All of the E- pulses are created for the perturbed natural frequencies. To see the effect of the damping coefficients on the DL, one of the newly created E-pulses and the E-pulse of SR #3 are convolved with SR #1 and the corresponding EDR are calculated. Then, the DL is evaluated for SR #1. All of the new E-pulses undergo this procedure. The roles of the damping coefficients and the radian frequencies were then exchanged in order to see the effect of the radian frequencies. 24 Table 2.1 Parameters used in creating SR #1 Natural frequency Amplitude Phase (degrees) -0.05 +j 1.00 1.00 -180 I 0.10 + j 2.00 0.50 90 I I -0.20 + j 3.00 Table 2.2 Parameters used in creating SR #2 0.25 135 I Natural frequency Amplitude Phase (degrees) il -0.06 +j 1.10 1.00 -90 I -0.13 +j 1.70 0.60 0 #025 + j 2'79...“— 0.25 J 45 _ Table 2.3 Natural frequencies of SR #3. a0.05 +j 1.40 Natural frequency || 015 +j 2.90 -0.30 +j 4.00 I“ 25 case 11: The extracted damping coefficient of the dominant mode (first mode) of SR #1 is perturbed by :5, 1:10, 3:15, 1:20, and :25 percent from the actual dominant damping coefficient, while the radian frequency of the dominant mode of SR #1 and the other natural frequencies of SR #1 remain the same as the actual values. All of the E- pulses are created for the perturbed natural frequencies. To see the effect of the dominant damping coefficient on the DL, one of the newly created E-pulses and the E-pulse of SR #3 are convolved with SR #1, and the corresponding EDR are calculated. Then the DL is evaluated for SR #1. All of the new E-pulses undergo this procedure. The roles of the damping coefficients and the radian frequencies were then exchanged in order to see the effect of the radian frequencies. case 111: The extracted damping coefficient of the weakest mode (third mode) of SR #1 is perturbed by :5, 1:10, 1:15, :20, and :25 percent from the actual weakest damping coefficient, while the radian frequency of the weakest mode of SR #1 and the other natural frequencies of SR #1 remain the same as the actual values. All of the E—pulses are created for the perturbed natural frequencies. To see the effect of the weakest damping coefficient on the DL, one of the newly created E-pulses and the E—pulse of SR #3 are convolved with SR #1, and the corresponding EDR are calculated. Then the BL is evaluated for SR #1. All of the new E—pulses undergo this procedure. The roles of the damping coefficients and the radian frequencies were then exchanged to see the effect of the radian frequencies. 26 case IV: All of the extracted natural frequencies are perturbed by :5, :10, 1:15, i20, and 1:25 percent from the actual natural frequencies. All of the E-pulses are created for the perturbed frequencies. To see the effect of the natural frequencies in the DL, one of the newly created E-pulses and the E-pulse of SR #3 are convolved with SR #1, and the corresponding EDR are calculated. Then the DL is evaluated for SR #1. All of the new E—pulses undergo this procedure. The SNR of the original SR #1 is very high. To study the actual effect in the real applieation, white guassian noise with zero mean was added to the SR #1 resulting in SNR of 25 dB. Figures 2.7 through 2.10 show the DL values of SR #1 with SNR of 25 dB corresponding to the E-pulse of SR #1 and the newly created E—pulses for case 1, case 11, case 111, and case IV. The normalized quantity plotted in the x-axis is a measure of the deviation from the actual values of the damping coefficients and radian frequencies. From Figure 2.7 and Figure 2.10 it is clear that the effect on the DL due to perturbation of the radian frequencies of all of the modes is almost the same as the effect on the DL due to perturbation of all of the natural frequencies. So, the effect in the DL due to perturbation of the damping coefficients is negligible. Also, it can be concluded that if the natural frequencies of targets are separated only by the damping coefficients, discrimination will not be possible because the DL is not high enough for adequate identification. However, if the natural frequencies of targets are separated by the radian frequencies, discrimination is possible. The difference between the radian frequencies Should be at least 3 percent for adequate identification in a SNR of 25 dB enviroment. The minimum difference between the dimensions of targets which need to be 27 35.0 Radian frequency (all modes) ------- Damping coefficnent (all modes) 30.0 25.0 10.0 5.0 11111111111111lrrrilrrrrlrrrrlr||J_l 0.0 ITIIIIIII'IIIIIIIIlllIIIIIITIIIITITHHIIIIVIIIIIITI 0.75 0.85 0.95 1.05 1.15 1.25 Normallzed a or w Figure 2.7 EDR values corresponding to the E-pulse of SR #1 and the newly created E—pulses, as explained in the case I. 28 35.0 Radian frequency (1't ode) ------- Damping coeffic1ent (1’ mode) 30.0 25.0 10.0 lIILIIILJLLIJIIJIJIIIIJIII1111111[I 5.0 0.0 IlllIllerIIlIlll[I]llITlellflllllIIIllilTHIlllll 0.75 0.85 0.95 1.05 1.15 1.25 Normalized a or 0) Figure 2.8 EDR values corresponding to the E-pulse of SR #1 and the newly created E—pulses, as explained in the case 11. 29 35.0 30.0 3 25.0 3 820.0 3 '0 I V 5’ 15.0 - 10.0 3 I Radian frequency (3rd r ode) 5.0 - ------- Dampmg coeff1c1ent (3 mode) 0.0 - IITIlIlIIIflfT'II'I'TI'HHrIIIIIIIIIIIIIIIIIII'IIII 0.1'5 0.85 0.95 1.05 1.15 1.25 Normalized a or 0) Figure 2.9 EDR values corresponding to the E-pulse of SR #1 and the newly created E-pulses, as explained in the case 111. 30 35.0 30.0 25.0 DL (dB) 10.0 5.0 Jrrlrrrrlrrrrlrrrilrrr11111111114] 00 IIIITIIIIIIIIIIIIlrllllllITIIIIIIIIIIIITIITHIIIIII ° 0.75 0.85 0.95 1.05 1.15 1.25 Normallzed natural frequenc1es Figure 2.10 EDR values corresponding to the E-pulse of SR #1 and the newly created E-pulses, as explained in the case IV. 31 discriminated can be roughly estimated from knowing the minimum difference between the radian frequencies at a given SNR. In the example shown here, the minimum difference between the dimensions of targets should be equal to 3 percent or above (depending on the Q of targets) to have a BL of 10 dB. If the SNR is low, the minimum difference between the radian frequencies will be higher for adequate discrimination and the minimum difference in the target dimensions will also be higher. This is discussed in section 2.5 using numerical responses of thin wires as targets. From Figure 2.7, when the normalized radian frequency of the all modes is 0.95, the DL is 14.5 dB. From Figure 2.8, when the normalized radian frequency of the first mode is 0.95, the DL is 15.5 dB. From Figure 2.10, when the normalized natural frequencies is 0.95, the DL is 14.5 dB. For the example shown here, it can be concluded that the DL primarily depends on the dominant mode of SR #1, since the change in the DL due to the perturbation of dominant mode is nearly equal to the change in the DL due to the perturbation of all of the modes of SR #1. However, it may not be true for complex targets since they may have more than one mode to be dominant depending on the illumination aspect. From Figure 2.9, the perturbation of the weakest mode damping coefficient has a negligible effect. Also, the negative perturbation of the weakest mode radian frequency has a minimum effect on the DL. However, the positive perturbation of the weakest mode radian frequency has a considerable effect on the DL, since the positive perturbation of the weakest mode radian frequency of SR #1 is approaching one of the radian frequencies of SR #3. 32 From Figures 2.7 through 2.10, one may conclude that the discrimination can be done using only the dominant mode, since the discrimination level mainly depends on the dominant mode. These results also indicate the necessity to extract the dominant modes with good precision, especially the radian frequencies. The following study is performed to see if discrimination can be done using the dominant mode only , rather than all of the modes. The E-pulses of SR #1, SR #2, and SR #3 are created by using only the first mode. The first mode E—pulses of SR #1, SR #2, and SR #3 are convolved with the corrupted SR #2, and the corresponding EDR values are calculated. Then, the BL is evaluated for SR #2. A similar procedure is performed for all of the modes E-pulses of SR #1, SR #2, and SR #3. Figure 2.11 and Figure 2.12 show the convolution of SR #2 with SNR of 20 and 5 dB with the first mode E-pulse, and the E-pulse constructed using all of the modes. As expected, the convolution corresponding to the first mode E-pulse is non-zero in the late- time, since the first mode E-pulse does not have zeros at all of the poles of SR #2 and the noise presents in the response. Figure 2.13 shows the DL of SR #2 when the first mode and all of the modes E-pulses are used in the discrimination for varying SNR. The next closest target to SR #2 is SR #1 as expected, since the natural frequencies of SR #1 are closer to SR #2. It is interesting to note that the difference between the DL corresponding to the all modes E—pulses and the DL corresponding to the first mode E- pulses is large when the SNR is high. However, this is not true when the SNR is low. The latter observation can be clearly seen in Figure 2.12. The decrease in the DL difference between the DL corresponding to the first mode E—pulse and the DL corresponding to all of the modes E—pulses is due to the 33 0. O 0 e e um. WP 0. __ 0 EE 8 9 need 0 d om mt 0.\./ 8 08 “r 5n 0.,“ 90.. l\ .m.m 6 SS uu oim. O “ 4T - - u . 0. 0 111111 2 mum 111111111111111111111111111111 O 1...—144..fidq._4__..+-q._lq_q_-q..rro.0. 5 5 B % 2 7. 353.:an m>:o_mm Figure 2.11 Convolutions of SR #2 with SNR of 20 dB with the first mode and all modes E-pulses. 34 0.75 0.25 Relative amplitude .c'> 8 -0.75 Figure 2.12 J 1 1 1 fit us1ng all modes E—pulse : :1. - ------ usmg first mode E—pulse 11 -—1 II .. 11 11 7 1'1 - ' ' | I -. l t l 1 ‘1 l t I I - I 1 1 I -1 I I I 1 -1 I . _‘ I I 1 1 1 :1 " I ‘ a I I '. ' . 5 . - , 1. . ' ,“ : ‘ 1‘ . 1“ . : u. . 1. .~ ~ - . .1 -1 l I .‘U o 1 ’ . I V | I 1 . 11 ‘ 1 I \ 1 1 - : : ' ' U l ‘ .6 ‘. -1 I .1 , 1 1 1 l _. : I: .1 1: ‘J I 1 1 1' 1: V I I. 1 V 1 ‘3 1 _, 1 1 d U 1 - 1 .1 1 1 1 .. |: - I. ‘1 -1 | .1 IITIIITITIIIIIIIIIIITIlllIIITrIrIITIIflIIIIIIIIHII 0.0 20.0 40.0 60.0 80.0 100.0 T1 me (ns) Convolutions of SR #2 with the SNR of 5 dB with the first mode and all modes E-pulses. 35 25.0 Z W using all modes E-pulses I W usmg fll’St mode E—pulses 20.0 1 A -' {I} I '0 -l v 2 E Z > 15.0 a a) "1 _1 1 A C I .9 1 +J I O 10.0 _ .E 1 E 1 “C ‘ 0 2 ,9 5.0 f: D : I 0.0—IIIITHIFIIIIIIIIIII[IIIIITITI] -5 5 25 1 SNR (dB) Figure 2.13 Discrimination level of SR #2 for a varying SNR using first mode and all modes E—pulses. 36 presence of noise in SR #2. An example of this observation follows: SR #2 with SNR of 20 dB is convolved with the E—pulse of the first mode and the all modes E-pulses. The late-time energy is 5 .0e-4 for the all of the modes E-pulse, while it is 2.258—2 for the first mode E-pulse. Then, the E-pulses of the first mode and all of the modes are convolved with SR #2 with SNR of 5 dB. The late-time energy is 2.0e—2 for the all modes E—pulse, while it is 5.70e-2 for the first mode E-pulse. The drastic increase in the late-time energy corresponding to the all modes E-pulse is due to the presence of noise in SR #2. The DL difference between the E-pulses of the first mode and all of the modes decreases from 15 dB to 5 dB when the SNR of SR #2 changes from 25 dB to 5 dB. From the example considered, it may be concluded that adequate discrimination can not be done by using only the dominant mode. The latter observation will be verified for complex targets in chapter 6. For good discrimination results, all of the natural frequencies excited by the incident waveform should be extracted with high accuracy, especially the radian frequencies. In chapter 4, two schemes are discussed for mode extraction. 2.6 Numerical study using thin wire targets Scattering data for thin wires was generated using a frequency domain method-of- moments solution. A Galerkin technique employing piecewise sinusoidal basis functions and thin-wire approximations was used [41]. The back-scattering responses of wires of length 0.80, 0.85, 0.9, 0.95, 1.0, 1.05, 1.10, 1.15, 1.20 meters were calculated for length-to—radius ratios (l/a) values of 100, 200, 400, and 800. The complex field values were calculated at 512 equally spaced frequencies between 0.02 GHz and 1.024 GHz, 37 These results were subsequently inverse Fourier transformed using a fast Fourier transform (FFT) to obtain the transient response. A Gaussian pulse was used for the incident wave p(t). The temporal variation is given by _ 2 ptt) = exp(—4;‘—) (2.21) 1‘ where r = 1 ns was chosen to give a pulse width of approximately 2 ns between the 2% of maximum points. Calculations were performed for all wires at both broadside-incidence orientation and 45 degrees off broadside. The wire natural frequencies were extracted from the calculated data using a hybrid E-pulse/least squares method [26,35] as would be done in the case of measured responses. In this case, the extracted natural frequencies were found to be very close to those computed from a singularity expansion method (SEM) formulation [41]. The E—pulses were synthesized for each of the wire lengths and (la values. To simulate noise encountered in a practical implementation, the calculated transient responses were corrupted by adding varying amounts of Gaussian noise. The automated discrimination process was applied to the noisy data. A Gaussian white noise model is used throughout the analysis. To illustrate the level of noise corruption used in the analysis, several examples are shown. Figure 2.14 shows the noise free back-scattering response of a broadside 1.00 m wire with l/a = 800. Figure 2.15 shows the same response with noise added to provide a SNR of 30 dB. The window duration W used for SNR ,given by (2.4), calculation is 38.65 nsec (from 12.30 nsec to 50.95 nsec). Figure 2.16 shows the same response with noise added to provide a SNR of 5 dB. 38 1.00 1 I 3 a) 0.50 ‘ ‘0 1. fl 3 -1 .4: : (1 '5. 1 E 1 O 0.00 E—fl q) I .2 I U 4..) a O I E -0.50 1 V 01 1 1 —1.00 3 0.00 20.00 40.00 60.00 80.00 100.00 Tlme (ns) Figure 2.14 Noise free back-scattering response of a broadside 1.00 m wire with l/a = 800. 39 1.00 1 O 01 O Relatlve amplitude O . b . O IJJIIILLLIliLlllljlllllLLLllIlllllll l O U" 0 I b 0 L1 0.00 20.00 40.00 60.00 80.00 100.00 Tlme (ns) Figure 2.15 Back-scattering response of a broadside 1.00 in wire with l/a = 800 and SNR of 30 dB. 40 1.00 llllllllLlllll (1) 0.50 ‘0 3 l .‘t.’ O 0.00 ‘ |H ' r I ,1‘ Q) l} l .2 1 ' -.—J 2 a) —0.50 1 O: -1.00 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIITIIUII 0.00 20.00 40.00 60.00 80.00 100.00 ' Time (ns) Figure 2.16 Back-scattering response of a broadside 1.00 m wire with l/a = 800 and SNR of 5 dB. 41 To test the sensitivity of the E-pulse method to SNR and differences in target dimensions, the E-pulses for all wires with I/a = 800 were used to discriminate the response of a 1.00 m wire with l/a = 800. The response waveform was corrupted with varying amounts of noise with the SNR varying from 5 dB to 30 dB. EDR values for each of these noisy signals were computed. Figure 2.17 shows the results of this test when the 1.00 m wire is in the broadside orientation. If the 1.00 m wire is identified correctly the EDR of the 1.00 m wire E-pulse should be zero. This is the case for all values of SNR computed. As expected, the EDR is largest for all wrong target E-pulses when the SNR is high. When the SNR is low, the EDR for all wrong target E-pulses is less. Also, E-pulses for targets similar to the 1.00 in wire (i.e. 0.95 m and 1.05 m) provide smaller EDR values than E—pulses for targets significantly different from the 1.00 m wire (i.e. 0.80 m and 1.20 m). Figure 2.18 shows a similar result when the response is from the 1.00 m wire in the 45 degree orientation. This illustrates the aspect independence of the E-pulse method. To evaluate the effect of the quality factor Q of the target modes, the E-pulses for all wires with (In = 100 were used to discriminate a 1.00 in wire with l/a = 100. Similarly, the E—pulses for wires with l/a = 200, 400, 800 were used to discriminate a 1.00 m wire with (In = 200, 400, 800 respectively. In each case the broadside response of the 1.00 m wire for the given l/a was used. The EDR of the E-pulses corresponding to wires that are 110% different in length were calculated for the given l/a value. The minimum EDR values of the :l: 10% E-pulses are plotted vs SNR for each of the (la cases. Figure 2.19 shows that wires with l/a = 800 have higher EDR values for a given SNR than thicker wires. This effect is seen for other aspect angles as shown in 42 35.00 Response: 1.00 m wire, broadside 999990.80 m wire E—pulse 550-00 369560.85 m wire E-pulse Met-A090 m wire E-pulse 000000.95 m wire E-pulse M100 m wire E-pulse 25-00 -H-H-l-1.05 m wire E-pulse W1.10 m wire E—pulse aWWI-41.15 m wire E-pulse A2000 m... 1.20 m wire E CD '0 v 15.00 CE 0 Lu 10.00 5.00 0.00 I11111111111111111111111111111111111114_1_l -5.00 TrrnrrTrnTrranrTrrrrrrrrnfiTrrqurrrrmanfiTrrrnTmnn 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 SNR (dB) Figure 2.17 EDR values computed for broadside response of 1 m wire with l/a = 800. 43 Response : 1.00 m wire, 45° orientation 35.00 —_ 3 99999080 m 30.00 - 1399913085 m I marten 0 90 m 1 09500088 m 2500 '1 H—1—1—1~ 1 05 3 j W 1 10 m : “+1114 1 15 m A2000 — ***** 1 20 m CD I ‘0 _ V -1 15.00 F CK : D . LIJ _ 10.00 —_ 5.00 —_ 0.00 — .. .. .. .. .. -5.00 :memrrmmfin 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 SNR (dB) Figure 2.18 EDR values computed for response of l m wire with l/a = 800 oriented 45 degrees from broadside. 35.00 j Response: 1.00 m wire, broadside 30.00 3 : oeeeeI/o = 100 .1 88886:;0 = 388 -1 M a = 25.00 1 W 1/0 = 800 A20.00 : CD , 'o . v .1 15.00 - 06 I Q .. L1J - 10.00 '2 5.00 - 000§---- “5.00111[[IllrlrfiIIIIIIIIIlllllrfillll11] 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 SNR (dB) Figure 2.19 Minimum EDR values computed for wires of 0.90 m and 1.10 m for response from broadside 1.00 m wire. 45 EDR (dB) 35.00 30.00 25.00 20.00 15.00 10.00 5.00 0.00 -5.00 Response: 1.00 m wire, 45° orientation oeeeeI/o = 100 Bessel/a = 200 Arsenal/o = 400 OOOOOI/a = 800 14111L1414L111111411111111111111411| llll HIIITTITTIIIIIITIII lll'fIlHllTTll 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 Figure 2.20 SNR (dB) Minimum EDR values computed for wire of 0.90 m and 1.10 m for a response from a 1.00 m wire oriented at 45 degrees from broadside. 46 Figure 2.20 where the wire is oriented at 45 degrees from broadside. The SNR values of 1 m wire for l/a = 800 and 100 calculated using (2.4) is nearly identieal when the late-time portion of the responses is considered within the defined window duration W. W is 38.65 and 25.3 nanoseconds for l m wire with I/a = 800 and 100, respectively. The higher Q targets have a longer usable late-time window duration compared to the low Q targets. Therefore, the low Q targets are discriminated based on a small window duration, and information within this window duration may not be enough to give a large DL value. 2.7 Conclusion This chapter has presented an effective automated discrimination scheme based on the E—pulse method. The scheme can be readily applied in applications where many targets are considered, and where a computer is used to make the discrimination decisions. Furthermore, the performance of the discrimination scheme is evaluated for thin wire targets in the presence of Gaussian random noise. Several conclusions can be made from the numerical results. For the cases considered, the E—pulse scheme is successful with an SNR of 15 dB with the discrimina- tion of 10 dB. Also, the discrimination is better for higher Q targets. Discrimination of very similar targets will require higher values of SNR, while discrimination of very different targets is possible in a low SNR environment. For systems design, curves such as those presented in the section 2.5 could be generated for a group of targets. The transmitter, receiver, and antenna requirements could then be estimated based on the desired discrimination level and SNR environment. 47 This chapter has also demonstrated the importance of extracting the natural frequencies with good precision. In addition, it confirms that successful discrimination cannot be done using only dominant mode information. 48 Chapter 3 Automated Radar Target Discrimination using S-Pulses 3.1 Introduction The S-pulse is an E—pulse synthesized to annihilate all but one of the natural modes of a target. When the S-pulse is convolved with the target response a single natural mode emerges [24] . Initially, the S-pulse technique was not successfully used in radar target discrimination at MSU due to implementation difficulty. However, at the same time, the E-pulse technique was successfully used in radar target discrimination. The response from an unknown target is convolved with a number of S-pulses waveforms and identification is accomplished by determining which convolved waveform has a late-time response composed of a single damped sinusoid with a specific frequency and damping coefficient. Visual inspection is very difficult even when only a few targets are considered, since it is difficult to distinguish the correct single damped sinusoid among many sinusoids. An automated scheme was recently developed to overcome the above difficulty and to avoid an error in the decision making process. The effect of the automated S-pulse technique on the radar target discrimination due to perturbed natural frequencies is analytically studied and verified using synthetic data sets. 49 This chapter will quantify discrimination using the S-pulse in a way that is suitable for use in an automated scheme. Examples of the performance of this automated scheme are presented using the same theoretical scattering data used in chapter 2 for thin wire targets with varying amounts of additive noise, to examine any improvement over the E-pulse technique. 3.2 Preliminary A response r(t) of a conducting target to a band-limited transient excitation in the late-time can be expressed by (2.1). The sine S-pulse waveform s,‘(t) for the 1'" mode of a particular target is defined by s,‘(r) e r(t) a b, e°"siu(u, 1+8) 1 > T“. = T, + T, (3-1) and the cosine S-pulse waveform s,‘(t) for the i“ mode of a particular target is defined by s:(t) . r(t) a b,e°*‘eos(u,1+e,) r> Tu = T, + T, 01) where b, and 0, are aspect dependent amplitudes and phases, T, is the duration of the sine and cosine S-pulses and r(t) can be the response of the target from any aspect angle. The duration of both the cosine and sine S-pulses are the same since the duration is determined by the highest extracted radian frequency (0,.) and the number of modes (N) extracted. T, is equal topic/1.1,, with p =2N-1 and add one morein thecase ofDC S- pulses. The spectral interpretation for these definitions is 50 SIS) 3 591371)} E 0 for s=su and s=s:; n =1,2,3,,,,,N; "a: (3.3) In addition, these definitions require s,‘(s,) = -s,‘ '(s,) and s,‘(s,) = s: '(s,). The unknown aspect dependent phase 0, causes difficulty when using single sine or cosine S-pulses. The aspect angle information in the late-time makes it difficult to calculate the error between the expected damped sinusoid and the damped sinusoid obtained from the convolution of r(t) with either the cosine or the sine S-pulse. To circumvent this problem, a quadrature sum of a cosine and sine S-pulse is defined as: §‘(t) = s,‘(t) -js,'(t). When the quadrature sum S-pulse is convolved with a response waveform r(t) we obtain 51(5) * r(t) = b‘e'jo‘e(a“'j0‘o t>Tu (3.4) Applying the Fourier transform results in -jO W, ’0‘ _. '19 6‘31.'0)Wl_1 ~7{§‘(t)*r(t)} = b,e ‘fe ‘ e ’"‘dr = b,e ‘ 5 F00) (3.5) o (Si-w) where W,, as defined in (2. 10), is the duration of an observation window. It is clear that the frequency variation of [3111' l'(t) 1: mm is aspect independent since the phase angle is not involved. 3.3 Quantification and Automation The S-pulse discrimination scheme is similar to the E-pulse discrimination scheme. In this case the convolved waveforms are denoted c,(t) = 3'" ‘(t) t v(t) , where v(t) is the response from an unknown target. The target is identified by the S—pulse that 51 causes the late-time portion of c,(t) to be a damped sinusoid with a known complex frequency s,. The S-pulse corresponding to the target response should produce a convolution whose late—time spectrum matches the spectrum of F(w) to within a constant. To assign a numerical value to the quality of this match Schwartz’s inequality (3.6) Um .12 {a ff’(x)dx from. = 1: ifflxkkscx); k=const where fix) and g(x) are real functions, can be used. The S-pulse Discrimination Number (SDN) is then defined as -1 f1C,(m) Izdw f |F((..))|2 dd) (3.7) l l SDN=1- f IC,(w)l |F(w)| do]? where C,(w) is the Fourier spectrum of c,(t) = s"(1) .. v(t) , 1%)) is the Fourier spectrum of the expected single mode signal taken over the finite duration W,, and w is the radian frequency. The integration limits w, and to, are determined by the bandwidth of the measurement system. The SDN is a measure of the error from the expected single mode spectrum. From the Schwartz inequality it can be seen that the SDN is zero if the spectrum of the late-time portion of c,(t) matches the expected single mode spectrum to within a constant and thus describes discrimination in the same manner as the EDN. Each of the quadrature S-pulses in the database are convolved with the response waveform v(t). For each convolution, the SDN is calculated. The target is identified by the quadrature S-pulse that yields the smallest SDN. To quantify the differences in the SDN for different quadrature S-pulses used in the discrimination, the S-pulse Discrimination Ratio (SDR) is defines as 52 SDR = 10.0 1°810{;;Sg%6} (3.8) Therefore, the SDR of the correct target is 0 dB, while the SDR of all other targets is greater than 0 dB. If the method is working properly, the SDR values approach infinity for S-pulses not associated with the target in the response. In most applications, noise in the measurement system and inaccuracies in estimates of target natural frequencies prevent the SDN from being zero when the S-pulses match the target producing v(t). To apply the discrimination scheme, the beginning of the late-time T, for the response v(t) must be estimated. The start of the late-time is estimated by the same procedure used in chapter 2. 3.4 Effects of perturbed natural frequencies To study the effects of the damping coefficients and the radian frequencies on radar target discrimination using the S-pulse technique, the extracted natural frequencies are assumed to differ from the actual values by a chosen percentage. 3.4.1 Analytical study Assume 5‘ is an 1"" mode quadrature sum S-pulse (Q‘-S-pulse) constructed using extracted natural frequencies 5': which differ from the actual frequencies of the target according to s: = s” :1 as. . A convolution of a response r(t) given by (2.1) with the Q—S- pulse in the late-time is given by 53 N 0 0,0) = 2 an|§(s.) e"'e 'fi' t> Tu (3.9) 11-1 where S (s) is the Laplace transform of the Q‘-S-pulse, and 0. is the aspect dependent phases. The convolution given by (3.9) is not the expected single damped sinusoid with known complex frequency since the 5(5.) 1* 0 for all n. Expanding 5(5) by the Taylor series around the natural frequencies 3 = 5: gives 51:.) = s(s;)rss,§§- (3.10) & an: where higher order terms are ignored since as, is assumed to be small. The first order partial derivative of 5(3) with respect to s is non-zero at s = s: = 0:+j(1):. Substituting (3.10) into (3.9) and using 5(s:) = 0 n=1,2,...,N; net yields sgtsww W,, on) , N c,(t) = a,|§(s,‘) e" e'fl‘ + 2 an n-l The first order error from the expected single damped sinusoid, cf" (t) , due to the perturbed extracted natural frequencies is, c,(t) -c:"’ (t) , (3.12) T 8:: do. 8 e L5 t> ~ ~ 8' -10 N 315 9:0) = 01(15(51.)1‘|S(S;)1)¢ ‘9 '+ 20. 43.3843...) In! As is demonstrated in the E-pulse case in chapter 2, the first order error term of the convolution in the late-time depends on the magnitude of the damping coefficients and 54 the radian frequencies. Also, the first order error depends on the difference in the spectral magnitude of the S-pulses evaluated at s = s, and s = s,'. It can be concluded that the effect on radar target discrimination due to a small perturbation of the damping coefficients is smaller than that due to a small perturbation of the radian frequencies, as is shown for the E-pulse case in chapter 2. The SDN corresponding to the extracted 1"“ mode S-pulse is SDN“ = 1- f|c,(u)| |F"‘(m)|dwr "‘ (3.13) 111, 111‘ '1 x f|c,(o)|2dof|rw(o)|’do where C,(1.)) is the Fourier spectrum of c,(t), and 17““(10) is given by (s"-u)1v _ F‘”(w) = (1.11.5"(.1't')|e-’e ‘1‘ ‘ ‘ 1 (3.14) l (S." - w) The SDN“ will be equal to SDN“, if the first order error term vanishes, where the SDN,“ is the value corresponding to the Q’-S-pulses of the actual natural frequencies. The DL due to the error in the mode extraction is decreased by SDN DLd = 10.010g10{-§B—Ng} (3.15) M The discrimination level is decreased by a significant amount when the extracted radian frequencies vary from the actual values. However, this is not true for the case of the extracted damping coefficients. In the following section an attempt to verify this claim is done with artificial data sets. 55 3.4.2 Numerical study All the analysis will be done using the same artificial data sets and natural frequencies in chapter 2. The DC Q’-S-pulses of SR #1, SR #2, and SR #3 are created. The SNR of the original SR #1 is very high. To simulate a real application, a white guassian noise with zero mean was added to SR #1 resulting in a SNR of 25 dB. SR #1 with SNR of 25 dB is convolved with the Q‘-S-pulses of SR #1 and SR #3, and the corresponding SDR values are calculated. The DL is then evaluated. To see the effects of the damping coefficients and the radian frequencies in the DL, four cases are studied. In all the cases the extracted natural frequencies are perturbed from the actual values in a procedure similar to that used in chapter 2. case I: All of the Q'"-S-pulses are created for the perturbed natural frequencies. To see the effect of the damping coefficients on the DL, one of the newly created Q1 'z-S-pulses and the Qm-S-pulse of SR #3 are convolved with SR #1 and the corresponding SDR are calculated. Then, the DL is evaluated for SR #1 . All of the new Q"’-S-pulses undergo this procedure. Theroles of the damping coefficients and the radian frequencies are then exchanged in order to see the effect of the radian frequencies. case II: All of the Q’"—S-pulses are created for the perturbed natural frequencies. To see the effect of the dominant damping coefficient on the DL, one of the newly created Q”- S-pulses and the Qm-S-pulse of SR #3 are convolved with SR #1 , and the corresponding SDR are calculated. Then the DL is evaluated for SR #1. All of the new Q"’-S-pulses 56 undergo this procedure. The roles of the damping coefficients and the radian frequencies are then exchanged in order to see the effect of the radian frequencies. case 111: All of the Qm-S-pulses are created for the perturbed natural frequencies. To see the effect of the weakest damping coefficient on the DL, one of the newly created Q'-2-S- pulses and the Q’"-S-pulse of SR #3 are convolved with SR #1, and the corresponding SDR are calculated. Then the DL is evaluated for SR #1. All of the new Qm-S-pulses undergo this procedure. The roles of the damping coefficients and the radian frequencies are then exchanged to see the effect of the radian frequencies. case IV: All of the Q’"—S-pulses are created for the perturbed frequencies. To see the effect of the natural frequencies in the DL, one of the newly created Q’ 'z-S-pulses and the Q”- S-pulse of SR #3 are convolved with SR #1, and the corresponding SDR are calculated. Then the DL is evaluated for SR #1. All of the new Q1 'z-S-pulses undergo this procedure. Figures 3.1 through 3.4 show the DL values of SR #1 corresponding to the Q”- S-pulse of SR #1 and the newly created Qm-S-pulses for case 1, case 11, case 111, and ease IV. The normalized quantity is a measure of the deviation from the actual values of the damping coefficients and radian frequencies. Figures 3.1 through 3.4, it can be concluded that the DL obtained using the dominant mode Q—S-pulse due to perturbed frequencies is large compared to the DL obtained using 2"" mode Q—S-pulse due to perturbed frequencies. From Figure 3.1, the effect on the DL due to damping coefficient perturbations is considerable when the dominant mode Q-S-pulse is used, while this is not true for the case of the E-pulse. 57 Hal-M1: mode S-pulse (radian frequency) W 1 d mode S—pulse damping coefficient) W2; mode S-pulse radian frequency) W2 mode S—pulse damping coefficient) 40.0 35.0 lIJJIIIIJIlllllllllIIJLIIIIJIIIIJIJIIIII l I 0.0 I'llIIITIIIII]IIIIIIITIIIIIIIIIIIllllll'lillIIIII] 0.75 0.85 0.95 1.05 1.15 1.25 Normallzed a or a) Figure 3.1 DL values of SR #1 with SNR of 25 dB corresponding to the 1‘ and 2"I mode S-pulses of SR #1 and newly created S-pulses, as explained in case I. 58 W 1': mode S-pulse (radian frequency) met-t 1;, mode S—pulse damping coefficient) WZM mode S-pulse radian frequency) W2 mode S-pulse damping coefficient) 40.0 35.0 .01 o N O O L.1411111111111111111111111111111114111111| J: .0 o O 0.85 0.95 1.05 1.15 1.25 Normallzed a or 0) Figure 3.2 DL values of SR #1 with SNR of 25 dB corresponding to the 1" and 2" mode S-pulses of SR #1 and newly created S-pulses, as explained in case 11. 59 40.0 7. 55.0 - 30.0 3 25.0 9, A CD I ‘C .. V200 : __l -1 C3 2 15.0 - I 10.0 S W 1:: mode S-pulse radian frequency) 3 M 1"cl mode S-pulse damping coefficient) - W 2nd mode S—pulse radian frequency) 50 : W 2 mode S-pulse damping coefficient) 2 010 .- II[ITIIIIITHIUHIIIIIUIIIIIIIIIIUIIIIIT'ITIfi'[[111 0.75 0.85 0.95 1.05 1.15 1.25 Normalized a or c.) Figure 3.3 DL values of SR 111 with SNR of 25 dB corresponding to the 1* and 2.. mode S-pulses of SR #1 and newly created S-pulses, as explained in case 111. =lI-IlI-IlI-1lI-1lr1't mode S-pulse 40°C W2“ mode S-pulse 35.0 5.0 / /' 0.0 . '5 0.85 0.95 1.05 1.15 1.25 Normahzed natural frequencues N O O 1111J11111111111111114111111111111111111 O \I Figure 3.4 DL values of SR #1 with SNR of 25 dB corresponding to the 1" and 2" mode S-pulses of SR #1 and newly created S-pulses, as explained in case IV. 61 However the SNR of SR #1 for the case considered is 25 dB. In order to examine whether the latter observation holds for a smaller SNR, the dominant mode Q—S-pulse of SR #1 and the dominant mode Q-S-pulses created only for the damping coefficient perturbations are convolved with SR #1 having SNR of 15 dB. The corresponding DL values are calculated. This is shown in Figure 3.5. The effect on the DL has been reduced; however, the DL is almost 7 dB when the normalized perturbed damping coefficient is 0. 80. From the example shown here, the observations conclude that when using the dominant mode Q-S-pulse, it may be possible to discriminate targets with a DL of 5 .0 dB or above obtained if the difference in the damping coefficients is 20 percent or above at a SNR of 15 dB or above and the radian frequencies do not change. From Figure 3.1 and Figure 3.2, it is clear that the effect on the DL obtained using the dominant mode Q-S-pulse due to perturbation of all of the radian frequencies is nearly the same as the effect on the DL obtained using the dominant mode Q-S-pulse due to perturbation of the dominant mode radian frequency. Also, the latter observation holds for the case of the damping coefficient. However, the effect on the DL obtained using 2" mode Q-S-pulse due to perturbation of all of the radian frequencies is large compared to the effect on the DL obtained using 2“ mode Q—S-pulse due to perturbation of the dominant mode radian frequency. Also, the latter remark applies in the case of the damping coefficient. From Figure 3.3, the perturbation of the weakest mode damping coefficient has a negligible effect on the DL. The negative perturbation of the weakest mode radian frequency also has a negligible effect on the DL. However, the positive perturbation of the weakest mode radian frequency has a considerable effect on the DL, since the 62 I 25.0 - A .. CD -1 '0 -l v -1 _1 I Q d 20.0 4 15.0 ‘lIII—IITTIIllfi—IllllI—rTHIHIrTII[IIIIIIIIIIIIIIIIIIII 0.15 0.85 0.95 1.05 1.15 1.25 Normalized a Figure 3.5 DL values of SR #1 with SNR of 15 dB corresponding to the 1" mode S- pulses of SR #1 and newly created S-pulses, as explained in case I only for damping coefficients. 63 positive perturbation of the weakest mode radian frequency of SR #1 is approaching one of the radian frequencies of SR #3. A similar effect is noticed in the case of the E—pulse. From Figure 3.4, it is clear that the effect on the DL due to perturbation of the radian frequencies of all of the modes is almost the same as the effect on the DL due to perturbation of all of the natural frequencies. However, the effect in the DL obtained using the dominant mode Q-S-pulse due to the damping coefficient perturbations is also considerable, while it is not true for the E—pulse discrimination. These results clearly indicate the necessity to extract the dominant modes including the damping coefficient with good precision. If the natural frequencies of targets are separated by a small percent in radian frequencies at a high SNR, discrimination is possible. The difference between the radian frequencies should be at least 2 or 3 percent for adequate identification in a SNR environment of 25 dB environment when Q’ -S-pu1se or Qz-S-pulse is used. The minimum difference between the dimensions of targets to be discriminated can be roughly estimated from knowing the minimum difference between the radian frequencies at a given SNR. In the example shown here, the minimum difference between the dimensions of targets should be equal to 2 percent or above (depending on the Q of the targets) to have a DL of 10 dB. If the SNR is low, the minimum difference between the radian frequencies will be higher for adequate discrimination, and the minimum difference in the target dimensions will also be higher. This is discussed in the following section using numerical responses of thin wires as targets. To have good discrimination results using the S-pulse technique, all of the natural frequencies excited by the incident waveform should be extracted with high accuracy, 64 especially the radian frequencies and dominant mode damping coefficients. In chapter 4, two schemes are discussed for mode extraction. 3.5 Numerical study using thin wire targets The study done in this section is conducted using the scattering data from chapter 2. The Q‘-S-pulses are created for all the thin wire targets. Varying amounts of guassian noise are added, and the performance of the S-pulse discimination is studied, as was done for E-pulse discrimination in chapter 2. Figure 3.6 shows the results of using Q-S-pulses based on the first fundamental resonance frequency of the wires with l/a = 800. As in chapter 2, v(t) is chosen as the broadside response of a 1.00 in wire with I/a = 800. The SDR results are similar to those obtained using the E—pulse method; however, the SDR values are significantly larger than the EDR for all values of SNR. Figure 3.7 indicates that Q—S-pulses based on the second resonance frequency do not yield large SDR values regardless of SNR. This is because the second resonance of the 1.00 in wire is not strongly excited for broadside incidence. Thus, the expected mode must be present in the response waveform for the S-pulse method to work effectively. Figure 3.8 shows that the second mode Q-S- pulse can be used effectively for the case of 45 degree orientation since the second mode is strongly excited. The effect of target Q is evaluated for S-pulse discrimination as was done previously in the case of E—pulse discrimination. Figure 3.9 shows the results when using first mode S-pulses for the broadside response of the 1.00 in wire. The results are consistent with the E—pulse results except that the SDR values are larger than the EDR 65 Res onse: 1.00 m wire, broadside Firs fundamental mode S-Pulse 09660080 m wire S-pulse 93699085 m wire S—pulse 35'00 —_ Ate-#4090 m wire S—pulse - 000000.95 m wire S—pulse - M100 m wire S-pulse 30 00 j -H-H-+105 m wire S—pulse ' _ W 1 10 m wire S—pulse - W 1 15 m wire S—pulse : M 1 20 m wire S—pulse 25.00 - J A20.00 —‘ m I “o _ v _ 15.00 - 05 I D _ (f) _ 10.00 - 5.00 - 0.00 : e e e r.- e e -1 .1 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 SNR (dB) Figure 3.6 SDR values computed for first mode S-pulse for broadside response of a 1.00 m wire with l/a = 800. 35.00 , Response: 1.00 m Wire, broadside 3 Second fundamental mode S—pulse 30'00 - 999990.80 m wire S-pulse ‘ 866600.85 m wire S-pulse : Meant-0.90 m wire S-pulse 25,00 3 00000 0.95 m wire S-pulse - W100 in wire S-pulse ; -l-l-H-l-1.05 m wire S—pulse . W 1.10 m wire S-pulse A20°OO "‘ W 1.15 m Wire S-pU1se m : M120 m wire S-pulse 'D .. v - 15.00 ‘ CK 2 C) .. (f) . 10.00 '1 5.00 - 0.00 - -5.00 — mm 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 SNR (dB) Figure 3.7 SDR values computed for second mode S-pulse for broadside response of a 1.00 in wire with l/a = 800. 67 Response: 1.00 m wire, 45u orientation 3500 j Second fundamental mode S—pulse - 999990.80 m wire S-pulse 30-00 ‘_ 888800.85 m wire S-pulse _ M030 m wire S-pulse - 069900.95 m wire S-pulse - W100 m wire S-pulse 2500 j +++H 1.05 m wire S—pulse - W 1.10 m wire S-pulse - Hm 1.15 m wire S—pulse A2000 _‘ W120 rn wire S-pulse/ m ; '0 - V - 15.00 - 05 I D _ (f) - 10.00 - 5.00 - 0.00 - t 1 t 1 "A“ 1k i -5.00 - 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 SNR (dB) Figure 3.8 SDR values computed for second mode S-pulse for response of a 1.00 m wire with (In = 800 oriented at 45 degrees from broadside. 68 Response: 1.00 m wire, broadside 555-00 2 First fundamental mode S-pulse 1 30.00 3 : eeeeoI/a = . BBB‘B’EJI/O = .. M I/a = 25.00 : WWI/a = A20.00 3 CD I '0 - v - 15.00 - 05 I D d (f) . 10.00 -_ 5.00 3 I oooE—-—— “5.00 Illl|Il rfiTII—IIII IIII111rITIII1—l 0.00 5.00 1000 1500 20. I00 25. 00 30. 00 35. 00 SNR (dB) Figure 3.9 Minimum SDR values computed for first mode S-pulse for 0.90 m and 1.10 m wire for broadside response of a 1.00 m wire. 69 Response: 1.00 m wire, 45° orientation 3500 Second fundamental mode S—pulse 30.00: , : oeeeel/o = 100 ‘ - Bessel/o = 200 " - AMI/a = 400 25-00 -: WI/a = 800 A20.00 : m I ‘0 .. V .. 15.00 ~ 0: 1 o - (f) . 10.00 - 1 I 500-: 1 0.00 :—--— -5.00‘r 111111 1111 11Tr11111 1111 111F] 000 5.00 10.00 15 oo 20. oo 25. I00 30. I00 35.100 SNR (dB) Figure 3.10 Minimum SDR values computed for second mode S-pulse for 0.90 m and 1.10 m wires for a response from a 1.00 m wire oriented 45 degrees from broadside. 70 values. Further evidence of this phenomenon is shown in Figure 3.10 for the case of second mode S-pulses and a response from a 1.00 m wire at 45 degree orientation. This phenomenon occurs due to similar reasons given in chapter 2 for the E-pulse case. 3.6 Conclusion This chapter has demonstrated an automated discrimination scheme based on the Q—S-pulse methods. The scheme can be readily applied in applications where many targets are considered and a computer is used to make the discrimination decisions. This chapter has also demonstrated the importance of extracting the natural frequencies with good precision. From the analytical study of the perturbed frequencies, it is concluded that if the SNR is high enough, it will be possible to discriminate targets whose damping coefficients are separated by 20 percent or more while the radian frequencies remain the same. The DL is at least 5 .0 dB when using the dominant mode QS-pulse. The SDR values corresponding to the dominant mode QS-pulse are always large compared to the EDR values at a given SNR. Furthermore, the performance of the discrimination scheme is evaluated for thin wire targets in the presence of Gaussian random noise. For the cases considered, the S-pulse discrimination scheme is successful even when the SNR is only 10 dB, while the E-pulse scheme is only successful with an SNR of 15 dB. For the same reasons given in chapter 2, the discrimination is also better for higher Q targets with the S-pulse schemes, as was true for the E-pulse technique. Discrimination of very similar targets will require 71 higher values of SNR, while discrimination of very different targets is possible in a low SNR environment. If a target has more than one dominant mode, more than one Q-S-pulses can be created for the target. Therefore, complex targets such as airplanes, missiles, etc. can be effectively discriminated using more than one Q-S-pulses for each target in addition to the E—pulse. In order to use the S-pulse scheme effectively, it is necessary that the response of the target contains the mode used to create the S-pulse. For systems design, curves such as those presented in section 3.5 could be generated for a group of targets. The transmitter, receiver, and antenna requirements could then be estimated based on the desired discrimination level and SNR environment. 72 Chapter 4 Natural Resonance Extraction from Transient Response 4.1 Introduction Radar target discrimination based on natural resonances requires an accurate and efficient means of extracting natural frequencies from measured data. The natural resonances of realistic complex targets can only be extracted from scale or full model measurements since it is difficult to determine them theoretically. The natural electromagnetic resonances of passive targets must, by the physical constraint of power balance, be associated with complex natural frequencies having negative real parts. Extraction of those natural resonances from measured late-time target response data, using algorithms such as Prony’s method or the unconstrained E-pulse technique (UCET), frequently lead to natural frequencies having non-physical real parts. This occurs because the algorithms attempt to fit a natural-mode series to the measured data without imposing any physical constraints on the natural frequencies; noise or other aberrations in the measured data result in the best fit occurring for some resonant frequencies with positive real parts. This is particularly true for high-Q targets where the physical damping coefficients are small in absolute value. Intuitively, it is expected that allowing the real part of the natural frequencies to take the wrong sign might also adversely affect the associated radian frequencies; this 73 was found to indeed be typical. The UCET algorithm for natural mode extraction first computes an E—pulse which annihilates the late-time target response [26]; since the frequency content of that E—pulse must vanish at each target natural frequency, the Laplace transform of the E-pulse is equated to zero and the resulting expression is solved for the resonance frequencies [35]. A new constrained E—pulse technique (CET) has been developed in which the physical constraint requiring natural frequencies with negative real parts has been implemented. The constraints are imposed by using a constrained optimization algorithm to construct an E—pulse which optimally annihilates the late-time target response while roots of its Laplace transform possess only negative real parts. This chapter mainly focuses on mode extraction using the unconstrained E-pulse technique (UCET) and the constrained E-pulse technique (CET); however all other techniques are briefly discussed. This new CET algorithm and the UCET have been tested using artificial late-time responses which were corrupted by additive Gaussian noise. Finally, efforts to overcome the ill-conditioned nature of frequency extraction from noisy measured data [42,43] are exploited with the use of multiple data sets. 4.2 Overview of existing techniques Early researchers interested in experimentally determining natural frequencies concentrated on Prony’s method [1 1 , 13,44]; however, it had many drawbacks. The first drawback is that the natural resonances cannot be extracted under low signal-to-noise conditions since it is inherently an ill-conditioned algorithm [45] in its basic form. The second drawback is that if the number of modes are underestimated, the extracted natural resonances vary from the actual values (examples are given in [24]). Several recent 74 improvements of Prony’s method have made the scheme more robust [46] while tech- niques have also been devised for estimating pole content [36,37]. A variety of other techniques for resonance extraction have been introduced, including the pencil-of-function methods [47 ,48,49] and several nonlinear [50,51,52] and combined linear and nonlinear [5 3,54] least squares approaches. In addition, Ksienski [42] has outlined the benefits of using multiple data sets, while Baum has stressed the importance of incorporating a priori information about the scatterer [55]. Initially at Michigan State University, the natural frequencies of measured transient responses of complex targets were effectively extracted using the continuation method in a resonably low signal-to-noise (SNR) enviroment. The continuation method works fairly well in the presence of random noise, and it extracts the natural resonances within reasonable error even if the number of modes are underestimated. The main drawback is that it requires good initial estimations of the natural resonances, and these estimations are obtained by applying FFT on measured responses. Estimations of the natural resonances are quite difficult in very low SNR environments using FFT. If estimation of natural resonances are far away from the actual values, the numerical algorithm converges very slowly to the actual values. Particularly suited for radar target applications are a group of resonance extraction techniques which synthesize the discriminant waveform directly from measured data, and provide the natural frequencies as a by product of the algorithm. Several authors have developed algorithms around this approach [35,56,57] and typical is the E—pulse scheme. Two types of E—pulse schemes are described, the Unconstrained E—pulse Technique (U CET) and the Constrained E-pulse Technique (CET). 75 4.3 Unconstrained E-pulse Technique 4.3.1 Theory The scattered field response of a conducting object at an aspect angle i can be written in the late-time as a sum of damped sinusoids N r,(t) = 2 a”: "cos(cout+¢~.) t> TL “-1) vi where TI. is the beginning of the late-time response, a“ and 4’.» are the aspect dependent amplitude and phase of the n“ mode at aspect angle 1', and s. = a. + j o. is the aspect independent 1:“ natural resonance of the target, and only N modes are assumed excited by the incident waveform. An E-pulse, e(t), is defined as a waveform of finite duration 1‘, which satisfies 1' c,(t) - e(t)*r,(t) = {mow-Hwy = 0 ”TH". (4.2) 0 If this integral equation can be solved for the unknown E-pulse waveform, then the complex natural frequencies contained in r, (t) are the solutions {3”} to E (s) = 0 , where E(s) is the Laplace transform of e(t). A solution to (4.2) can be obtained by using the method of moments. Expanding K e(t) = 2 abfka) (4.3) l-l where {ft} is an appropriate set of basis functions, substituting into (4.2), and taking inner products with a set of weighting functions {$9.} gives 76 g 1'. Try a (t’)r (t-t’)w.(t)dtdt’ = 0 ‘2'; til-‘0 HIT? ‘ (4.4) m = 1,...,M i = 1,...,I Where T, is the total time window where the signal is above the noise level, and M and I, respectively, denote the number of matching points forced to be zero in the late-time (t> TL+T‘) and number of data sets used while extracting natural resonances. In the standard E-pulse technique, the selection of M =K =2N, and I =1 resulting in a "natural " E-pulse, makes (4.4) a homogenous matrix equation. Solutions for {(1,} thus exist for discrete values of T. which cause the matrix to be singular. An alternative approach is to choose M=2N, K =2N+1, and I=I resulting in a ‘fforced" E-pulse. Then (4.4) becomes an inhomogeneous matrix equation, with solutions corresponding to any choice of T, which does not cause the matrix to be singular. A new method chooses M to be greater than (2N+1) and I > 1, and it causes (4.4) to be an overdetermined matrix equation, A a = 0. The dimensions of the matrix equation is I xM by K. K is 2N+2 for a DC forced E-pulse. The last column of the overdetermined matrix equation is moved to the right, and the equation is solved using the least squares method (normal equations). It may be ill-conditioned if the choice of I XL! is too large compared to K. The natural frequencies in r,I (t) can be found most easily by using subsectional basis functions of width A in (4.3), since E(s) = 0 then reduces to the polynomial equation 77 l 2 «,2" = o (4-5) r-r where Z = e "‘. To maximize the computational efficiency, rectangular basis functions are used in (4.3) while impulses are used for weighting (point matching). The integral on t in (4.4) is then trivial, while the integration on t’ is done using the trapezoidal rule. 4.3.2 Choice of E-pulse duration In theory, the technique should work for any choice of T,. However, there are practical limitations on the range of T,. The lower bound of T, is determined by the sampling interval used to measure r,(t) , and the upper bound is determined by Tw - TL from the limits of integration of (4.4). The choice of the natural E-pulse duration may not exist in this limitation. The best choice of forced E-pulse duration can be made within this limitation by choosing to minimize the squared error 1-1 31 I e = 2 [3) MO) -I’,(t) l2 (4'6) where e, is the late-time energy of the i "' data set, and the f,(t) is the reconstructed waveform N f,(t) = Zafie°-‘cos(ont+$fi) (4.7) ad 78 A and the sum is over sampled values between TL and T". Here {5. = a" +10”) are the solutions to (4.5), and {65.3%,} minimize a with T, and {3,} fixed. 4.4 Constrained E-pulse Technique The poles of the target response r,(t) are the roots of the polynomial E(s) = 0. If the roots lie in the left-half plane, Le, a < 0, then |Z| >1. The CET imposes constraints on E-pulse technique such that either the modulus of the roots of the polynomial E(s) = 0 is greater than one or the extracted natural frequencies of the target response r,(t) have negative real parts. The Jury-Marden theorem can identify how many roots of a polynomial with real coefficients (real polynomial) are inside or outside or on the unit disk. First, the theorem [58] is stated and a few examples are shown to verify and avoid the singularity occurring in some cases with real polynomials. Then, the problem is formulated using the E-pulse technique such that all natural frequencies have negative real parts. 4.4.1 Jury-Marden theorem A real polynomial (coefficients of polynomial are real) is considered as P(x) = aox'+a,x"‘+...+a x+a. ao>0 (4-8) 3-1 The real polynomial P(x) is convergent if all its roots have modulus less than one. Then, construct an array having initial rows {c,,,cu,...,cwl} = {aw¢,,...,a.} (4.9) {d,,,du,...,d, r1} = {a.,a._,,...,ao} 79 and subsequent rows defined by C_ c_ + 6,, = ”'1 ”J l i = 2,3,...,n+1 (4.10) di-m di-le do' = claw-1+3 (4.11) The real polynomial P(x) is convergent if and only if the first column elements satisfy 11,, > 0, d“ < 0, i = 3, 4,..., n+1. Provided that the array is regular, i.e., du it 0, for all i, then P(x) has no roots with unit modulus, and there are k and n-k roots inside and outside the unit disk lxl < l, where k is the number of negative products in the sequence R, = (—1)‘d2,d3,...d,,,., i -- l,2,...,n (4.12) A few examples are considered to verify the Jury-Marden theorem. A third order real polynomial has roots 0.5,-O.5,-1.5 so I’(x) becomes P(x) = 8x3+12x2-2x-3 (4-13) and array elements 0,, and d‘, are c”. 8 12 -2 -3 d” -3 -2 12 8 c” 20 9o 55 dz, 55 90 20 (4.14) c,, -3150 -2625 d,, -2625 -3150 cu 3031875 11,, 3031875 Then the sequence R, is 80 R, -55 < 0 R2 -144375 < o (4.15) R, = 4377.4011 > 0 As expected, it indicates that there are two roots inside and one root outside the unit circle. It is also interesting note that the value of higher order constraints are very large, and it may cause numerical problems while solving for higher order polynomial. As a second example, a third order real polynomial has roots -0.5,0.5,4.0 so P(x) becomes as P(x) = 4x3 - 16x2 -x +4 (4-16) and array elements c9 and d, are c”. 4 -16 -1 4 d” 4 -1 -l6 4 c2, 60 60 0 d” 0 60 60 (4.17) cg 0 d4] 0 The computed array is not regular because the elements (1,, is zero for i = 2 and 4. If one of the elements (1,, is zero for i) 2, then the location of the roots of the real polynomial P(x) cannot be identified because the sequence R, is zero for some 1'. Some of the roots of the real polynomial P(x) may lie on the unit circle and may cause the polynomial PM not to converge. From the Jury-Marden theorem, there can be two types singular cases. 81 singular case 1: Here no complete row of zeros is obtained, implying that P(x) has no roots on the unit circle. We introduce a small perturbation. This is done by replacing x in P(x) by (l +a)x, and using the fact that [(l +e)x]" - x"(l +ke). By applying the theorem to the perturbed polynomial, it is thus possible determine the number of roots inside the disk of radius 1 + e , and hence to deduce the number of roots inside the unit disk itself. The perturbed polynomial P,’(x) becomes P,(x) = 4(1+3e)x3-16(l+28)x2-(1+e)x+4 (4-13) and modified array elements Cr: and (1,, for the perturbed polynomial P,’(x) are given as CU 4(1 +32) -l6(l+26) -(l+e) 4 (IM 4 -(l+e) -l6(1+2e) 4(1+3¢) ‘21 60 +112: -60 -3l6e 24a dz: 24s -60 -316e 60 +112: (4.19) c,, -3600-24240e 3600+134402 4,, 3600+13440e -3600-24240e cv 77760000e d” 777600008 Notice that in the construction of the array, terms in e’ have been ignored at each step. The reason for ignoring r:2 is that none of the whole rows are zero; however 22 cannot be ignored if any of the whole row is zero, and this is illustrated in the next example. For this modified array , the sequence R, is R1 = -24e R2 = 24: (3600+13440e) (4.20) R3 = - 1.86624le (3600 + 13440:):2 We compute the signs of R‘ 82 Provided that e is sufficiently small, then irrespective of its sign, the sequence contains two negative terms. As expected, it can he therefore concluded that there are two roots inside and one root outside the unit disk. A third order real polynomial has roots -2.0,(l:1j)l(/2 so P(x) becomes P(x) = x3+(2-./2)x2+(1-2,/§)x+2 (4.21) and array elements c, and d” are c1] 1 242 1-2,/2 2 d” 2 1-2,/2 2-(/2 1 c2, -3 3/2 -3 -3 3./2 -3 (4.22) c3} 0 0 d3, 0 0 C41 0 :1q 0 since the whole row of the array elements d1“ is zero for i = 3 and 4, then the array is not regular for similar reason explained in the previous example. From the Jury-Marden theorem, this is a second type singular case. singular case 2: Here at some stage a complete row of zeros is obtained. This implies that there are some roots on the unit circle, and/or roots such that xx, = I . If the real polynomial 83 has a complex root (xi), then it’s complex conjugate (x,) is also a root. If both the complex roots lie on the unit circle, then x2, is equal to one. The perturbation method just described can still be used. Suppose that with e assumed positive, the modified array is used to find that there are p, and n-p, roots inside and outside the disk of radius (1 + a). By changing the sign of e , we can similarly find that there are p2 and n-p2 roots inside and outside the disk of radius (1 - 8). It therefore follows that there are p,-p2 roots actually on the unit circle, with p2 inside and n-p, outside. We therefore construct array with perturbed polynomial as done in the previous example and modified array elements c, and do' are cu (1+3e) -(2-./2)(1+2e) (1-2,/2)(1+e) 2 d” 2 (1-2,/2)(1+e) (2-,/2)(1+2e) (1+3e) c2, -3 -4(1+,/2)e 3(/2+(8-(/2)e -3+6e dz, -3+6e 3/2+(8-/2)e -3-4(l+(/2)e (4.8) c,, -6e(4+./2) 12e(5+2,/2) d3, 128(5 +2,/2) -6e(4+,/2) c,, 7282(57+36,/2) d4] 7282(57+36,/2) Notice that because the singularity occurs in the third pair of rows of the original array, 2 we must retain terms 8 in subsequent rows of the modified array. The sequence R, is R1 3 -68 R2 12(5 +2f2)e(-3+6e) (4.24) R3 = 864(57+36,/2)e3(-3 +62) We compute the signs of R, 84 R, + + R, ' + R, " + Thus by Jury-Marden theorem, we have p, = 2, p2 = 0, so P(x) has two roots on the unit circle and one outside, as expected. 4.4.2 Formulation of the CET The aim is to minimize the norm of the convolution of the target response with the corresponding E-pulse while requiring the extracted poles of a target lie in the left half-plane. The function which must be minimized is r F(ak) = 2 [oil (4.25) M where T. TU c, = a f(t’)r(t-t')wm(t)dtdt’ g; tel-‘0 HIT. k l (4.26) m = 1,2,...,M i = l,2,...,l The poles of the target response r,(t) are the roots of the polynomial E(s) = 0. If the roots lie in the left-half plane i.e o < 0 , |Z| > 1. A new polynomial is defined, so Jury- Marden theorem can be used to find the constraints on the amplitudes of the E-pulse waveform. The new polynomial is defined as 85 x P(w) = Z akw‘b“) = 0 hi where w -- Z " . Expanding the (K - I)“ order real polynomial P(w) gives P(w) = a,w“’” + azw“'2)+a3w“'3) +... +01,,,_,)w+1zx The first row array elements c, and d, are chosen to be {c,,,c,2,...,cu} = {a,,a2,...,ax} {d,,,d,2,...,du} = {a‘,...,a2,a,} and subsequent rows are defined by cr-r,r c1-1J+l CU = r = 2,3,.. ,K dr-r,1 di-le and du = C;.,.,.; (4.27) (4.28) (4.29) (4.30) (4.31) According to the Jury-Marden theorem, the real polynomial P09) is convergent if and only if (1,, > 0, and d,, < 0 for i = 3,4,...,K. To have all the roots of the real polynomial P(w) inside the unit disk, the number of negative products in the sequence R, should be K-l. The whole sequence R, is negative if (1,, > 0, and 11,, < 0 for i = 3,4,...,K. Thus, the function F(a,) needs to be minimized with the constraints 11,, > 0, and 11,, < 0 for 1' = 3,4,...,K, for all the extracted target poles to be in the left half- plane. 86 The optimum E-pulse duration is chosen, as in the case of the UCET, when the squared minimum error between the original and reconstructed waveform using the extracted natural frequencies is minimum. The constraints are nonlinear and require sophiscasted minimization routines. To overcome the difficulty a new function is defined as . x T(a,) = F011,) +qpf3: 22: H ((1,1) (432) ‘- where q, is a constant (supplied by the user), ff: is the minimum value ofF(a,) calculated without constraints (in the range of values of T, considered in finding optimum E-pulse duration), and the Heaviside unit step function is defined as l ; d2150,du20 4.33 HM") = {0 ; d,,>0,du<0 ( ) The newly defined function adds a penalty whenever the constraints are not met. It is important to note that the penalty function should not be large compare to the function F( 01,) . Adding the huge penalty would drive the function T( “1) away from the function F( (1,) , and the sudden jump caused by the penalty function makes it difficult to find the proper global minimum of F( 01,) satisfying the constraints. Therefore the value of q, must be chosen such that the penalty function should not be large compared to F( 11,) . The minimization of T( “1) can be easily performed since the problem is now unconstrained. 87 4.4.3 Numerical algorithm The powell method [59] is used to the find the minimum of T( “‘19- All minimization routines require an initial guess for the unknown E-pulse amplitudes. First, the E-pulse amplitudes are solved using UCET and the roots of the polynomial E(s) =0 are found. If any of these roots lie in the right half-plane, the real part is multiplied by a negative number p", while imaginary part is kept the same. Intuitively, in general, if the real parts of the roots are greater than zero, the constraints will then force them towards zero. 80, the p“, is chosen to be negative while remaining closer to zero (in the order of -l.0d-2 to -l.0d-9). Using these modified roots, new amplitudes of the E-pulse waveform are calculated and they are used as the initial guesses for the function T(a,) to be minimized. The positive real part of a root is multiplied by p", because the initial guesses of the E-pulse amplitudes should satisfy the constraint requirement in order to find the proper minimum [60]. The range in which the optimum E-pulse is found is decided by roughly estimating the highest order natural frequency excited by the incident waveform. Once the estimation of highest excited radian frequency (”1.) is known, the duration of the natural E-pulse is T = L" (4.34) where p is twice the number of requested in the mode extraction. The range for searching T, is normally taken to be between 0.5 T,, and 2T,,; however if the optimum forced E- 88 pulse duration is smaller than the natural E-pulse duration, very poor discrimination results. 4.5 Discussion The performance of both UCET and CET are tested on two types of artificial responses, and measured target responses inside the anechoic chamber. Responses #1 and #2 are created with the parameters shown in Table 4.1 and Table 4.2 respectively. case 1: One of the modes in synthetic response #1 (SR #1) has a positive damping coefficient to test whether mode extraction using the CET forces the positive damping coefficient toward zero. It is also interesting to find the effect of the positive damping coefficient on the other natural frequencies with negative damping coefficients. case 11: All modes used in creating synthetic response #2 (SR #2) have negative damping coefficients. A varying amount of guassian noise is added to SR #2, and the natural resonances are extracted using both the UCET and the CET schemes to test their noise sensitivity. 4.5.1 Analysis of case I The natural resonances of SR #1 are extracted via the UCET, and the resulting natural frequencies are presented in the first column of Table 4.3. The second column of Table 4.3 displays the natural frequencies of the SR #1 via the CET with q,=2.0 and p"=-l.0d-2. As expected, the positive damping coefficient is forced to zero when the 89 Table 4.1 Parameters used in creating response #1 Table 4.2 Parameters used in creating response #2 : -0.050 + j 1.00 1.00 [ Natural frequency Amplitude Phase (degrees) —0.05 +j 1.00 1.00 -180 0.10 + j 2.00 0.10 0.0 -0.10 +j 3.00 1.50 180 0.075 + j 2.00 0.80 0.100 + j 3.00 0.60 -0.125 +j 4.00 0.40 Table 4.3 Natural frequencies of response #1 145 0.10 + j 2.00 0.0000 + j 2.00 90 0.10 +j 3.00 0.1005 +j 3.00 I CET is used. This is due to the fact that the squared error between the original waveform and waveform reconstructed using the natural frequencies obtained via the CET should be minimum. This can only happen by forcing positive damping coefficients closer to zero since the CET gives a heavy penalty to positive damping coefficient. The damping coefficients of the other modes have shifted slightly; however the radian frequencies have not changed much. Figure 4.1 shows SR #1 and the corresponding waveforms reconstructed using the natural frequencies obtained via the UCET. As expected, the reconstructed waveform is almost same as SR#1 since the extracted natural frequencies is same as the natural frequencies used in creating SR #1. The constants q, and p", are supplied by the user, and there is no formula or definition what the values should be given. In order to see the effect of these two parameters, two types of studies are done. First q, was changed from 1.0 to 10.0 while p“ remains the same, and then p", was changed from -1.0 to -l.0d-9 while q, remains the same. Figure 4.2 and Figure 4.3 show SR #1 and the corresponding waveforms reconstructed using the natural resonances obtained via the CET with q,=1.0 and 10.0 while pn=-l.0d-2. Figure 4.4 and Figure 4.5 show SR #1 and the corresponding waveforms reconstructed using the natural resonances obtained via the CET with p,,,=- 1.0 and -l .0d-9 while q,=5.0. For the example considered, the reconstructed waveforms are the same for SR #1 regardless of which values of q, and p“, are used while extracting the natural resonances. It is very difficult to establish a range of values for q, and p... The normalized radian frequencies and damping coefficients are defined as 91 1.0 1 H 0.5 3 il a) - '0 I 3 - .4: 4 i a 1 E - O 0.0 3: a) u > d '43 I U 2 “l 0) 4 U 0: -0.5 -_ 3 U original waveform : ------ reconstructed waveform “-1.0 filerTIIYIIIIIlllr111111111r111111rT1—l 0.0 10.0 20.0 30.0 40.0 Time (ns) Figure 4.1 SR #1 and the waveform reconstructed using the natural frequencies obtained via the UCET. 92 1.0 - h 0.5 3 ll 0) _. '0 I 3 _ h , .4: 4 ' a j, 3 I" o O 0 '2 3' a) - ~’ .2 1 U 4" _l .9 ‘1‘ \, (l) 3 U [r —0.5 -+ , - v 3 H' original waveform : , ------ reconstructed waveform -1.0 1111r11—Tll11111111111111IIIIFITII11111r] 0.0 10.0 20.0 30.0 40.0 Time (ns) Figure 4.2 SR #1 and the waveform reconstructed using the natural frequencies obtained via the CET with q,=l.0 and pm=-l.0d-2. 93 1.0 ‘ ~ 0.5 .4 ll 0) c- '0 I i D _ n n :1: s .5. : It 3‘ O 0.0 -_ " 1 a) -l \’ .2 I "6 1‘ a :l U 05 -o.5 - , I 11 original waveform j l: ------ reconstructed waveform —1.0 1111111j1111111711111111IIIUIIIIIIITIIII 0.0 10.0 20.0 30.0 40.0 Time (ns) Figure 4.3 SR #1 and the waveform reconstructed using the natural frequencies obtained via the CET with q,=10.0 and p,,,=-l.0d-2. 94 1.0 '1 h fi 0.5 ': a g : :3 1 " a; + l . o. 2 ‘ ' O 0.0 :3 " 0) ~ , " ._>. I ~ - l 2 -l ‘1’ J 0: -0.5 3 U .l .( I U’ original waveform j , ------ reconstructed waveform -1.o TIII—IIIITIIHIITIIIIUIUIllITlllIIlller] 0.0 10.0 20.0 30.0 40.0 Time (ns) Figure 4.4 SR #1 and the waveform reconstructed using the natural frequencies obtained via the CET with q,=5.0 and pm=-l.0. 95 0.5 3 H a) .- ‘0 j , l 3 , h . .‘t’ 4 ' a - a a. E 3 3 ‘3 ~“ ‘ - I‘ l‘ O 0.0 _ - a) .1 \I .2 3 U 4.; ° l a) 3 U v 01 -o.5 3 , j 0 3 H original waveform : i, ------ reconstructed waveform -1.0 j111711j1111111111111111111i11111111111| 0.0 10.0 20.0 30.0 40.0 Time (ns) Figure 4.5 SR #1 and the waveform reconstructed using the natural frequencies obtained via the CET with q,=5.0 and p,,,=-l.0d-9. 96 (0‘ 0. (4.35) 3:. norm 0: where (s, = a, + it») is an extracted natural frequency obtained either via the UCET or the CET, and (s, = a, + ij is the expected natural frequency. Figure 4.6 and Figure 4.7 show, respectively, the normalized radian frequency and damping coefficient as a function of q, with p,,,=-l.0d-2. Figure 4.8 and Figure 4.9 show, respectively, the normalized radian frequency and damping coefficient as a function of p", with q,=5.0. From these results, it is seen that the change in radian frequencies from the actual values is negligible, while the two negative damping coefficients are within five percent the expected value. The error in two negative damping coefficients is due to the fact that the algorithm forces the positive damping coefficient to be zero or negative. The error in natural frequencies is surprisingly low. From this example, it is concluded that the values of q, in the range 1.0 to 10.0 and p", in the range of -l.0d-2 to -l.0d-9 will give good estimations of natural frequencies. 4.5.2 Analysis of case 11 The natural resonances of SR #2 are extracted via the UCET, and the resulting natural frequencies are presented in the first column of Table 4.4. The second column of Table 4.4 displays the natural frequencies of SR #2 via the CET with q,=2.0 and p"=-1.0d-2. Figure 4.10 and Figure 4.11 show SR #2 and the corresponding waveforms reconstructed using the natural resonances obtained via the UCET and the CET with q,=2.0 and p,,,=-1 .0d-2, respectively. The natural frequencies obtained using 97 1 .0020 I W 1: mode : WZM mode 3 W3 mode >\ I U -l 0C) . 31.0010 '2 O- .l a) .. L - (,_ _ C I O - 161.0000 * 1* 1* 1* "i o I L - '0 2 Q) .- .'2‘ I '6 .. 0.9990 4 E i L— '1 0 I Z . 1 0.9980 H11j111111|1111111111111111111'1111111111r111111111 0.00 2.00 4.00 6.00 8.00 10.00 qP Figure 4.6 Normalized radian frequencies of all modes obtained using the CET for different values of q, while p,,,=-l.0d-2. 98 E 5: 5* 5: 5‘ .. - 0.95 : 4—J -l C : .93 0.85 -_ .2 : “‘4: 0.75 -‘ Q) I 0 : 0 0.65 1 0’ E Hall-OH: 1'1 mode .5 0-55 t W 2"" mode 0- E W 3" mode E 0.45 3, 0 4 '0 1 0.35 1 '0 . 0) : ,[1’ 0.25 1 o a E 0.15 «3 L .. O : Z 0.05 f: j t a t t a a t—*—*——* —0.05 1 0.00 2.00 4.00 6.00 8.00 10.00 qp Figure 4.7 Normalized damping coefficients of all modes obtained using the CET for different values of q, while pm=-1.0d-2. 1.0100 3 W 1:; mode - W2“! mode ‘ W3 mode >\ .. U .. C a) -l 3 C! U" "l 0) 1.0050 4 L d H.- C q .9 ‘ .0 - 0 CI L. . 8 .3. ”1.0000, 2; a a a W B .. E '1 L q 0 - Z _, 0.9950 ‘w-ITm-n-n-n-rrn-rnfi-rnfifi-rrn-n-mwm -10.00 -8.00 -6.00 -4.00 -2.00 -0.00 10g10( _pneg) Figure 4.8 Normalized radian frequencies of all modes obtained using the CET for different values of pm while q,=5 .0. 100 \ 1 .05 8* [F1- 18* near nut-11 int-li- {-1 0.95 0.85 0.75 0.65 W 1:; mode W 2“l mode W3 mode 0.55 0.45 0.35 0.25 Normalized damping coefficient * t t i i i i t t 1 1411lunJrrrrlrrrrJrJJrlrr“1er111114111111111111111] WWW-”m -10.00 -8.00 -6.00 -4.00 -2.00 -0.00 log10( -pneg) Figure 4.9 Normalized damping coefficients of all modes obtained using the CET for different values of pm while q,=5.0. 101 1.0 .0 u: rrrrLlrrrrerrrl : Relative amplitude O O —0.5 3 3 orginal waveform j ------ reconstructed waveform T. -1.0 11111111111111111111111111111]ll1111111|11r111111| 0.0 20.0 40.0 60.0 80.0 100.0 Time (ns) Figure 4.10 SR #2 and the waveform reconstructed using the natural frequencies obtained via the UCET. 102 1.0 .0 01 11111111111411111 : Relative amplitude O O I .0 UI orginal waveform ------ reconstructed waveform 111111111111111111 -100 W111llllr111rllllllliillllllllIUIUIIIIUIIIIIIIIII] 0.0 20.0 40.0 50.0 80.0 100.0 Time (ns) Figure 4.11 SR #2 and the waveform reconstructed using the natural frequencies obtained via the CET with q,=2.0 and p,,,=-l.0d-2. 103 both the UCET and CET is same as the natural frequencies used in creating SR #2 since the signal-to-noise of SR #2 is almost infinity (no noise added). Therefore, the reconstructed waveforms using the natural frequencies obtained via both the UCET and CET is same as the original SR #2, as expected. To simulate noise encountered in a practical setup, SR #2 was then corrupted by adding varying amount of white gaussian noise with zero mean. SNR values of the corrupted responses are calculated using the definition given by (2.4). The natural resonances of SR #2 of SNR with 0, 5, 10, 15 dB are extracted via the UCET, and the resulting natural frequencies are presented in the first and third column of Table 4.5, and the first and third column of Table 4.6. The second and fourth column of Table 4.5, and second and fourth column of Table 4.6 display the natural frequencies of SR #2 with SNR of 0, 5, 10, 15 dB via the CET with q,=2.0 and pm=1.0d-2. The natural frequencies obtained using the UCET and the CET match the expected values when the SNR is high. The natural frequencies obtained using the CET match fairly well the expected values especially the radian frequencies while the natural frequencies obtained using UCET do not match the expected values when the SNR is low. Four modes were requested in the mode extraction. The UCET only gave four modes when the SNR is 15 dB, and three modes when the SNR is 10 dB or below, while the CET gave four modes for all the SNR. Figure 4.12, Figure 4.14, Figure 4.16, and Figure 4.18 show of SR #2 with SNR of 15, 10, 5, 0 dB and the corresponding waveforms reconstructed using the natural frequencies obtained from corrupted SR # 2 via the UCET. Figure 4.13, Figure 4.15, Figure 4.17, and Figure 4.19 show SR #2 with SNR of 15, 10, 5, 0 dB and the 104 Table 4.4 Natural frequencies of response #2 ” via UCET via CET I 0.050 + j 1.00 0.050 + j 1.00 0.075 +j 2.00 0.075 +j 2.00 0.100 +j 3.00 0.100 +j 3.00 0.125 + j 4.00 0.125 + j 4.00 Table 4.5 Natural frequencies extracted from SR #2 with noise added. I SNR 008, UCET SNR 0dB, CET SNR 508, UCET 4.5e-4 + jl.044 0.049 + j1.012 0.051 + jl.008 0.046 + j0.985 0.075 + jl.903 0.047 + j2.019 0.005 + j1.959 0.071 + j2.008 0.426 + j2.720 0.068 + j2.816 0.048 + j3.006 0.255 + j2.913 0.352 + j3.976 1 0.396 + j3.813 Table 4.6 Natural frequencies extracted from SR #2 with noise added. I SNR lOdB, UCET SNR lOdB, CET 0.047 + j1.007 0.064 + j0.998 0.051 + j0.998 SNR lSdB, UCET SNR 1508, CET -0.048 + j1.001 . 0.066 + j1.999 0.082 + j1.999 0.080 + j1.987 0.079 + jl.975 ‘ 0.118 + j2.970 0.162 + j2.816 0.061 + j2.982 0.062 + j2.975 ; 0.352 + j3.976 105 0.222 + j4.153 0.126 + j4.026 ‘ 1.0 .0 U! '1- I .0 u: corrupted or inal waveform ------ reconstructe waveform Relative amplitude O O rrrrrmrrrlrLerrrlllllrriirrrlrjrrrrrrrl -1.0 IIIIrIIIITIII1TIIITIITITITIIITTIIIjIIIIITWTTIIIII 0.0 20.0 40.0 60.0 80.0 100.0 Time (ns) Figure 4.12 SR #2 with SNR of 15 dB and the waveform reconstructed using the natural frequencies obtained via the UCET. 106 .0 u: 11114111111L114J ‘. Relative amplitude O O l .0 UI corrupted orginal waveform ------ reconstructed waveform 11111111114111111 -1.0 111II11IIIVTIIIIIIIIIIUIII[11'11111IIUIITIIrIIIIII 0.0 20.0 40.0 60.0 80.0 100.0 TI me ( ns) Figure 4.13 SR #2 with SNR of 15 dB and the waveform reconstructed using the natural frequencies obtained via the CBT' with q,=2.0 and p,,,=-l.Od-2. 107 1.0 2:10- .0 u: 11111111111111141 Relative amplitude O O I .0 u: corrupted or inal waveform ------ reconstructe ’ waveform I 'o I P rrrrrrrrrlrrrrrrr O 20.0 40.0 60.0 80.0 100.0 Time (ns) Figure 4.14 SR #2 with SNR of 10 dB and the waveform reconstructed using the natural frequencies obtained via the UCET. 108 1.0 0.5 3 a) u '9 j l :3 - l .t’ - l .5. ' i l l l , | CE) 00 1 ll ' t 3 a) > .4: 2 0) 01 -0.5 corrupted or inal waveform ------ reconstructe waveform —1.0 |111T11111|Tl11111fl11111111111r111111111111111111] 0.0 20.0 40.0 60.0 80.0 100.0 Time (ns) Figure 4.15 SR #2 with SNR of 10 dB and the waveform reconstructed using the natural frequencies obtained via the CET with q,=2.0 and p"=-l.0d-2. 109 .0 U: :27.— 1111111111111111111] —-—-—‘_ -----—_ - -___ .__. - ’— o - o a- - Relative amplitude O O _l -0.5 '- 3 corrupted orginal waveform j ------ reconstructed waveform -100 IIIIfIlII]ll11111111111111111|1111111111111111111] 0.0 20.0 40.0 50.0 80.0 100.0 Time (ns) Figure 4.16 SR #2 with SNR of 5 dB and the waveform reconstructed using the natural frequencies obtained via the UCET. 110 'o O U" Relative amplitude .0 . 0 g 1L11L11111111111||||11111111111111111111 I .0 u: corrupted orginal waveform ------ reconstructed waveform l 'o I 20.0 40.0 60.0 80.0 100.0 TI me (ns) Figure 4.17 SR #2 with SNR of 5 dB and the waveform reconstructed using the natural frequencies obtained via the CET with q,=2.0 and p"=-l.0d-2. 111 1.0 .0 or 111111411111111111 .0 o Relative amplitude I .0 or corrupted or inal waveform ------ reconstructe waveform —1.0[111111111]IrIIITIIUIIIIIIrilIIIIIUITTTIIIITIIIIIT] 0.0 20.0 40.0 60.0 80.0 100.0 Time (ns) Figure 4.18 SR #2 with SNR of 0 dB and the waveform reconstructed using the natural frequencies obtained via the UCET. 112 1.0 O U' 114111111111111111 (D .0 3 I .t.’ '. . . E l l I 1‘ . l l‘ I ‘ 1 IL, 1 l o 00 ‘1' l t’,[|l',i ,3 ll "l 0) A . 3' i .2 +J ,1 2 l G) Cl1-05 corrupted orginal waveform ------ reconstructed waveform -100 r111lllllTlrrrlTTIrTIIIlllelIllIlIIIII1IIIIIIIT'3F' 0.0 20.0 40.0 60.0 80.0 100.0 Time (ns) Figure 4.19 SR #2 with SNR of 0 dB and the waveform reconstructed using the natural frequencies obtained via the CET with q,=2.0 and p"=-l.Od-2. 113 corresponding waveforms reconstructed using the natural frequencies obtained from corrupted SR # 2 via the CET with q,=2.0 and p,,,=-1.0d-2. From Figure 4.16 and Figure 4.18 it can be observed that the UCET attempts to fit the non decaying tail-end of SR #2, giving natural resonances with positive damping coefficients. Figure 4.17 and Figure 4.19 it can be concluded that the CET does not attempt to fit the non decaying tail-end of SR #2. The UCET, by attempting to fit the non decaying tail-end of the waveform, perturbs natural resonances from the expected ones; however, the CET does not. The waveforms reconstructed using the natural frequencies obtained from the corrupted waveform SR #2 via the UCET and the CET is compared with the original SR #2. If both the waveforms match fairly well, the natural frequencies obtained from the corrupted SR #2 will be a good estimates. Figure 4.20 and Figure 4.22 show the original SR #2 and the waveform reconstructed using the natural frequencies obtained from the corrupted SR #2 with SNR of 15 and 0 dB via the UCET. Figure 4.21 and Figure 4.23 show the original SR #2 and the waveform reconstructed using the natural frequencies obtained from the corrupted SR #2 with SNR of 15 and 0 dB via the CET with q,=2.0 and p,,,=-1.0d-2. Figure 4.20 and Figure 4.21 show that results from both techniques are quite similar when the SNR is high. Figure 4.22 and Figure 4.23 show that the CET gives much better results compare to the UCET when the SNR is low. Since 99 % of the total energy of SR #2 is within the window between 0 and 25 nanoseconds, it may be deleterious to use the whole time window while extracting the natural resonances because only the noise information is added after 25 nanoseconds. The 114 1.0 orginal waveform with no noise added ------- reconstructed waveform using natural frequencues of corrupted waveform .0 tn 3- lllllllIllllllllLJ Relative amplitude O O I .0 tn ‘— 11111111111111111 I 'o .0 I o 20.0 40.0 60.0 80.0 100.0 Time (ns) Figure 4.20 Original SR #2 and the waveform reconstructed using the natural frequencies obtained from SR #2 with SNR of 15 dB via the UCET. 115 1.0 orginal waveform with no noise added ------- reconstructed waveform usmg natural frequencres of corrupted waveform .0 tn llllllllllllllllll Relative amplitude O O I .0 tn l b I p lllLllllJIllllLJl O 20.0 40.0 60.0 80.0 100.0 Time (ns) Figure 4.21 Original SR #2 and the waveform reconstructed using the natural frequencies obtained from SR #2 with SNR of 15 dB via the CET with q,=2.0 and p,,,=1.0d-2. 116 1.0 orginal waveform with no noise added ------- reconstructed waveform usrng natural frequencres of corrupted waveform .0 tn II ' Relative amplitude O O l .0 u: IIJiiIIIIlIIIItII "-1.0 WTITIIIIIjllllllllllUIlI‘IIUllIITIIUIITIIIIIIIIIII 0.0 20.0 40.0 60.0 80.0 100.0 Time (ns) Figure 4.22 Original SR #2 and the waveform reconstructed using the natural frequencies obtained from SR #2 with SNR of 0 dB via the UCET. 117 1.0 orginal waveform with no noise added ------- reconstructed waveform usmg natural frequenCies of corrupted waveform .0 (II "“-:=- lllLlllIllllllllLI .0 o ' Relative amplitude I .0 tn -100 [111111111]IIIIIIIIITIIIIIIIIIIIIIIIIIIIIIIIIUIIIUI 0.0 20.0 40.0 60.0 80.0 100.0 Time (ns) Figure 4.23 Original SR #2 and the waveform reconstructed using the natural frequencies obtained from SR #2 with SNR of 0 dB via the CET with q,=2.0 and p,,,=l.0d-2. 118 natural resonances of SR #2 ,which has a SNR of 15 and 0 dB if the whole time window is considered, are extracted using both the UCET and the CET with q, =2.0 and pw=- 1.0d-2 for a shorter time window. Figure 4.24, Figure 4.25, Figure 4.28, and Figure 4.29 show the normalized radian frequency and damping coefficient extracted from SR #2 using the UCET, which has 3 SNR of 15 and 0 dB (considering the whole time window), as a function of time window duration W. Figure 4.26, Figure 4.27, Figure 4.30, and Figure 4.31 show the normalized radian frequency and damping coefficient extracted from SR #2 using CET with q,=2.0 and p,,,=-1.0d-2, which has a SNR of 15 and 0 dB (considering the whole time window), as a function of time window duration W. As expected, the natural resonances are better for small W since the SNR is higher for a smaller window compared to a bigger time window and this effect can be clearly seen in Figure 4.32. For example, a SNR is 0 dB for W= 100 nsec and a SNR is approximately 7.6 dB for W=20 nsec. The UCET gives a good estimate of the radian frequencies and damping coefficients for high SNR; however it produces the radian frequencies within 10% error and non-physical damping coefficients (> 0) for low SNR. The CET gives a good estimate of the radian frequencies and damping coefficients for high SNR, and also gives a fairly good estimate of the radian frequencies (within 5 % error) and physically possible damping coefficients (< 0), as required by the constraints, even for low SNR. The relative error in the damping coefficients obtained via the CET are very high (in the order of 300%) under low SNR; however this high error does not seem to effect the estimate of the radian frequencies as much as it does in the UCET since the damping coefficients are forced to be negative. 119 1.20 W 1’; mode W 2; mode W 30. mode W4 mode 0 lJilllllLiLJlJlLlllIlllllllllJlllllLllII 1 .00 Normalized radian frequency 0 to O 0.80 11'1r1111l111111IITITITIIIIIIITIIIIITIlllll'lf‘fq 10.0 50.0 50.0 70.0 90.0 110.0 W (nsec) Figure 4.24 Normalized radian frequencies of all modes for different time window W obtained using the UCET of SR #2 which has SNR of 15 dB (considering the whole window). 120 2.00 1.50 Normalized damping coefficient —I O O llJLlJlllIlllllllllIJllllllllllllllLlJJJ 0.00 1j111111]1111IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 10.0 30.0 50.0 70.0 90.0 110.0 W (nsec) Figure 4.25 Normalized damping coefficients of all modes for different time window Wobtained using the UCET of SR #2 which has SNR of 15 dB (consider- ing the whole window). 121 1-20 W 1" mode : m2: mode - W3,” mode 2 W4 mode x -l 0 1 0C) .. 31.10: 0' -i 8 I u— -l C I O - 91-00. 5 W O . L -I U : a) -l .t’ I 6 1 0.90- E - L .. O I Z - q .1 -l —( 0.80 T1T1111111111111llllllfill111|11111111F11W111111—| 10.0 30.0 50.0 70.0 90.0 110.0 W (nsec) Figure 4.26 Normalized radian frequencies of all modes obtained for different W using the CET with q,=2.0, p,,,=1.0d-2 of SR #2 with SNR of 15 dB (considering whole window). 122 3.00 '2 HIM-#1:; mode 2 WZM mode : W3“ mode y 2 W4 mode C I .2 I ‘4- 1 ~— .. 8 : 2.00 o '3 I O : .E 2 04.503 E : O I '0 : “01.00 I — — ..-.- a) - .L‘.‘ 3 _ 6 E r. - - :1" €0.50 -; O 1 Z a 0.00 ‘jTllllljljiTIllIlFIIIIrlIlllllIIIIIIIITITIIIIIIII] 10.0 30.0 50.0 70.0 90.0 110.0 W (nsec) Figure 4.27 Normalized damping coefficients of all modes obtained for different ina the CET with q,=2.0, pm=l.Od-2 of SR #2 with SNR of 15 dB considering whole window. 123 1-20 W 1" mode W 2:: mode W3“ mode W4 mode 0 1111111111111nmilliliiiiiiiiiliriiiili1| 1 .00 - ’7‘ r' \ . I" Y Normalized rodion frequency 0 in O 0.80 U1]lrfilTllleTIIlI[WITIIIIIIU'IIIIIIIIIITTTITI] 10.0 30.0 50.0 70.0 90.0 110.0 W (nsec) Figure 4.28 Normalized radian frequencies of all modes obtained for different W via the UCET of SR #2 with SNR of 0 dB (considering whole window). 124 2.00 +4 1.003 C : .0.) E 0 E iii-0.00: u.- .. 0) E O E U-1.oo—; 0‘ E .E a 7 O--2.oo-; ‘ E : 8 E -3.oo€ , .0 E O‘ a) 1 N E ‘-‘=-4.00: O : ‘ E 5 WELmodje L : W moe o-s.ooa ms: mode 2 3 W4 mode —6.00-llllITIII][ITIIIITIIIIIIlIIIIIIIIIIIIIIIIIITllllI—l 10.0 30.0 50.0 70.0 90.0 _ 110.0 W (nsec) Figure 4.29 Normalized damping coefficients of all modes obtained for different W using the UCET of SR #2 with SNR of 0 dB (considering whole window). 125 1.20 W 1" mode W 2': mode M 3m (mode W4 mode d d O llJllJLLLJlllllllJlJ :\ A; 1 .00 », .. ‘5“ Normalized rodion frequency -. H Y IIV A 0.905 I 1 '1 0.80 IllrIT'jl'llliilIflllllll‘lll'IIIIIIIII—[Iljl'rjm 10.0 30.0 50.0 70.0 90.0 110.0 W (nsec) Figure 4.30 Normalized radian frequencies of all modes obtained for different W via the CET with q,=2.0, pm=1.0d-2 of SR #2 with SNR of 0 dB (considering whole window). 126 5-00 HHHHI: 1': mode W 2'; mode my mode W 4‘" mode 4.00 3.00 E" o o JiliiJiilLiiliilJiliiilitllnliiliiiLill “K 15‘” Normalized damping coefficient '0 O A 5/ V 0.00 1lllilefi]llllrjllilllIIIII1I—[1Ill'l'IrIT—rlifjllil 10.0 30.0 50.0 70.0 90.0 1 10.0 W (nsec) Figure 4.31 Normalized damping coefficients of all modes obtained for different W using the CET with q,=2.0, pm=l.Od-2 of SR #2 with SNR of 0 dB (considering whole window). 127 8.00 6.00 4.00 SNR (dB) 2.00 0.00 IIIIIIIIIIIIIIIIIIIIIIlllITlI—ll11r1I111] .o 40.0 60.0 80.0 100.0 W (nsec) O Lilliiiiiliilliiililliiitiliilijtiiiiiil‘ N Figure 4.32 Calculated SNR of SR #2 with SNR of 0 dB (considering the whole window) for different W. 128 It has been noticed that the number of match points (M) being smaller or larger compare to the number of modes (N) requested in the mode extraction results in a bad estimate of the natural resonances. To investigate the sensitivity of the E—pulse technique to the number of match points used (M), the natural resonances of SR #2 with SNR of 10 dB are extracted using both techniques for different number of match points. The minimum value of M is twice the number of modes requested plus one for a forced E- pulse, i.e. M 2 2N +1, and one more for DC forced E—pulse, i.e M 2 2N +2. The natural resonances of SR #2 with SNR of 10 dB are extracted using both the UCET and the CET with q,=2.0 and pm=-l .Od-2 for various numbers of match points. Figure 4.33 shows the relative minimum squared error between SR #2 with SNR of 10 dB, and the reconstructed waveform using the natural frequencies obtained via the UCET and the CET for various numbers of match points. The error is comparably small for both techniques when M is between 40 and 60, and four modes are extracted. Figure 4.34 and Figure 4.35 show the normalized radian frequencies and damping coefficients obtained from SR #2 with SNR of 10 dB using the UCET for various M. Figure 4.36 and Figure 4.37 show the normalized radian frequencies and damping coefficients from SR #2 with SNR of 10 dB using the CET with q,=2.0 and pm=-l.0d-2 for various M. From 4.34 through 4.37, it can be observed that the extracted natural frequencies using the UCET and CET are pretty good when M is between 40 and 80. Thus, it is hypothesized that for the general case, when M is between 4(2N +2) and 6(2N +2) for the forced DC E-pulse, estimation of the natural resonances works better, and it may be good idea to extract the natural resonances for different values of M within this defined range. 129 3.00 L d O I L -l 0‘) . 2.00- 3 I E : .. a) ‘ \\ A .2100 ‘ 4—1 4 2 -l q; J o: - : a|'--l-*|~l--'|=using UCET 1 Wusmq CET 0.00 IIIIIIllllllllIrI1ill1lll11llllllllleIIIIIIIIIIj—l o 20 40 so 80 100 Number of match paints Figure 4.33 Relative minimum squared error between SR #2 with SNR of 10 dB and the reconstructed waveform for different number of match points. 130 1.10 1 W 1:: mode a W 2a mode - W3; mode 3 W4 mode -l 1 " R fig 1.05 : q ‘ A 1.00 4: v V Normalized radian frequency 0 (.0 U! LlJlllllilllllllllL 0.90 llIllllTlllllllIIIITTIIIIIIIIIT1IIIIIIIIIITIIIIII| o 20 40 so so 100 Number of match pomts Figure 4.34 Normalized radian frequencies of all modes obtained using the UCET of SR #2 with SNR of 10 dB for different number of match points. 131 2.00 1.50 llLlllllllllllllllll 1.00 A A H ' «A W 1'; mode W 2':“ mode W 3 mode Normalized damping coefficient llllllllLllllllllll W4“ mode 0.00 1rlIIIITITITlIIIIIIIIIIIllIITI1WIWIIIIIIFTIIIIII] O 20 4O 50 80 100 Number of match points Figure 4.35 Normalized damping coefficients of all modes obtained using the UCET of SR #2 with SNR of 10 dB for different number of match points. 132 1'10 W1“ mode firm 2': mode W3.“ mode W4 mode '0 UI lllllllllllLllllJlll 1.00 / 7‘ t Normalized radian frequency 0 (.0 0| JlllllllllLllllllll 0.90 jflilijlllIIIIIIITTITTIITrITIIIIIIIIIIIIUIITIIIIW o 20 40 so so 100 Number of match paints Figure 4.36 Normalized radian frequencies of all modes obtained using the CET of SR #2 with SNR of 10 dB for different number of match points. 133 2.50 +, I C I Q) .1 -0 2.00 3 i; I H— _ Q) _. O I 0 : 05150 j .E : A 0- : v‘ 1 7 ' g 1 ’ ‘ ‘\-§ 1‘ “V D 1.00 1 A "£le -l _ s = " .. - V g :1 is o I E 0.50 1 g : W1; mode 0 2 m2“ mode Z . W3". mode 2 W4 mode 0.00dIIIIIllI‘erTTIIj—[IITIrTIIIIUIIIITIIIlIrUTI—IIIIII] O 20 40 60 80 100 Number of match points Figure 4.37 Normalized damping coefficients of all modes obtained using the CET of SR #2 with SNR of 10 dB for different match points. 134 While searching for the optimum E—pulse, it is necessary to step up in a small increment of T, because the E-pulse technique may find the local minimum squared error instead of the global minimum. Figure 4.38 shows the relative squared error for T, ranging from 0.85 to 2.85 in the increment of 0.05. The error corresponding to the lower and upper end of T, is considerably large compared to the error corresponding to T, around the natural E-pulse duration. This implies that the extracted natural frequencies corresponding to the lower and upper end of T, are incorrect. Figure 4.39 shows the relative squared error for T, ranging from 0.85 to 2.85 in the increment of 0.25. From Figure 4.39, It can be clearly seen that the increment of T, is large, the E-pulse technique skips the T, value corresponding to point A in Figure 4.38 and finds T, value corresponding to point B in either Figure 4.38 or Figure 4.39 as the optimum E-pulse duration. Therefore, the increment of T, should be small, and the experience has shown that the increment of T, should be in the order of 0.01 to 0.05 and not to exceed 0.1. 4.5.3 Measured responses Responses of a 6 inch thin wire at broadside and 45° off broadside were measured in frequency domain inside the anechoic chamber, and the deconvolution is applied to measured responses to remove the system response, as explained in chapter 5 (for more details see [41]). The experimental SNR is roughly estimated to be around 15 dB. The natural frequencies of the 6 inch thin wire were extracted, using both techniques, from broadside and 45° off broadside measurements. The theoretical natural resonances of 6 inch thin wire obtained using SEM [41] are presented in the first column of Table 4.7, while the second column of Table 4.7 displays the natural resonances of the 6 inch thin 135 0.0004 L - O —l t .. (90.0003: '0 q a) u: L - O - 3 - O' m .. a) .. ”20.0002- 4.: .. 2 1 (D -l B cg _ q r A 0.0001 TIU'IIllllll'll'llljlr'lIIll'IlIIIUIIIUI 0.75 1.25 1.75 2.25 2.75 Te Figure 4.38 Relative squared error between the original waveform and the reconstruct- ed waveform for different T, (increment of T,=0.05). 136 0.0004 .7 1 L - O u- t - (”0.0003- .0 .. a) .1 L. .. 0 a I) U— "l (i) W a) _. .200002- +4 - _c_J - a) a B m q 1 .1 0.0001 IIITTIIIIfiF'IITIT—FlIIUIIIfIIIIU1IIIIII] 0.75 1.25 1.75 2.25 2.75 Te Figure 4.39 Relative squared error between the original waveform and the reconstruct- ed waveform for different T, (increment of T,=0.25). 137 Table 4.7 Natural frequencies of 6 inch thin wire. Theory (SEM) Measured (U CET) Measured (CET) 0477 + j 5.728 -O.536 + j 5.601 -0.568 + j 5.602 I -0.694 +j 11.77 -0.584 +j 11.49 -O.654 +j 11.43 I -0.848 + j 17.84 0.327 +j 17.91 —0.767 +j 17.28 Lows + j 23.93 0.974 +j 8.874 -0.855 +j 24.01 wire obtained via the UCET. The third column of Table 4.7 displays the natural frequencies of the 6 inch thin wire obtained via the CET with q,=5.0 and p,,,=-1.0d-9. It can be concluded that the first three radian frequencies obtained using both schemes are close to the theoretical values; however the damping coefficients obtained via the CET are better than the UCET. The fourth mode obtained using the CET is close to theoretical values while the fourth mode obtained using the UCET does not have any physical meaning. The CET gives a better estimation of natural resonances for the 6 inch thin wire than the UCET. Figure 4.40 and Figure 4.41 show the measured responses of a 6 inch thin wire at broadside and 45° off broadside and the waveforms reconstructed using the natural frequencies obtained from both measurements via the UCET. The natural frequencies of the 6 inch thin wire is extracted using both the measurements simultaneously, i.e. I =2, rather than obtaining the natural frequencies of the 6 inch thin wire separately, i.e. I =1 for each case, and averaging the closest. Figure 4.42 and Figure 4.43 show the measured response of the 6 inch thin wire at broadside and 45° off broadside and waveform reconstructed using natural frequencies obtained from both measurements using the CET 138 0.3 .0 Relative amplitude I .0 u original waveform ------ reconstructed waveform O ‘ lIJIUJJIJJILIJJIIlllllllllllljlllllLlllllllllllLJ -0.5 l‘IIIlIll'fTIITijlllllllllIWTfllllllllIlITTIIIIII[[1111 2.0 4.0 so 8.0 10.0 12.0 Time (ns) Figure 4.40 Measured 6 inch wire response at broadside and the waveform recon- structed using the natural frequencies obtained via the UCET. 139 0.5 -_ 0.3 E A " c0 3 : ‘0 - 3 I E 0.1 — ' Q_ 3 O 1 a) 1 ' > -O.1 -: J 1,: . .9 1 (D 2 01 : -0.3-3 1 original waveform Z ------ reconstructed waveform -0.5 .1 llIIITITIIIIIIIIjIIIIIIIIIlI—IlTlljllTififiTTTllifillIll 2.0 4.0 6.0 8.0 10.0 12.0 Time (ns) Figure 4.41 Measured 6 inch wire response at 45° off broadside and the waveform reconstructed using the natural frequencies obtained via the UCET. 140 1 5. 0.35 (D I ‘O .. 3 I E 0.1 -_'_ Q : E 4 ~ - - O I ~ ‘- ‘ ' q (1) .1 > -O.1 '1 ‘ , 44 -‘ \' 2 2 CD 2 01 : —0.3-§ 2 original waveform I ------ reconstructed waveform -0.5 -ITrIIIIIIIIlTTIIITII1IIIIIIITITITITIITIIITIIIITTIIIIITW 2.0 4.0 6.0 8.0 10.0 12.0 Time (ns) Figure 4.42 Measured 6 inch wire response at broadside and the waveform recon- structed using the natural frequencies obtained via the CET with q, =5.0 and p,,,=-l.0d-9. 141 0.5 : 0.5-3 <0 3 1 ‘0 - 3 I E 0.1 :- 4 ‘1 : E : O -l l 0) 1 > -0.1 1 '43 4 o g l <19 I 01 : —0.3-3 1 original waveform I ------ reconstructed waveform ‘0.5 3T111lllll]ITWIIFIIIIIfiTfiIII[IIIIIllllllllIIIIIIIIIITI 2.0 4.0 6.0 8.0 10.0 12.0. Time (ns) Figure 4.43 Measured 6 inch wire response at 45° off broadside and the waveform reconstructed using the natural frequencies obtained via the CET with q,=5.0 and p,,,=-1.0d-9. 142 Table 4.8 Natural frequencies of A-10 Thunderbird obtained via UCET using measurements separately. I E Head-on (HO) 45° off HO 90° off HO 0.065 + j 2.395 0.061 +j 2.309 -0.039 + j 2.290 0.205 + j 3.322 0.166 + j 3.227 0.133 + j 3.284 0.225 + j 4.375 0.155 + j 4.481 0.253 +j 5.251 0.215 + j 5.243 -0.278 + j 4.503 0.057 + j 4.980 0.204 + j 6.520 0.414 + j 5.877 === 0.339 + j 5.885 I] Natural frequencies of A-10 Thunderbird obtained via CET using measurements separately. Table 4.9 90° off H0 0064 + j 2.395 0.157 + j 3.448 0.447 + j 4.581 II 0.057 + j 5.435 0.351 + j 6.062 Head-on (HO) -1.2e-8 + j 2.221 -0.208 + j 3.243 -6.9e-8 + j 4.273 -0.521 + j 5.025 -3.4e-7 + j 5.482 45° off HO -3.e-4 + j 2.274 -0.193 +j 3.167 -0.020 + j 4.408 -0.296 + j 5.265 -0.464 + j 5.809 with q,=5.0 and pm=-1.0d-9. From Figure 4.41, it can be observed that the positive damping coefficient in one of the natural frequencies is caused by trying to fit the latter part of the 6 inch thin wire response; however it does not occur when the CET is used. This observation can be clearly seen from the expanded latter part of the Figure 4.41 and Figure 4.43 as given by Figure 4.44 (via UCET) and Figure 4.45 (via CET). Responses of an A-10 Thunderbird at head-on (HO), and 45° and 90° off HO were measured in the time domain inside the anechoic chamber, and deconvolution applied to 143 m . ' o l f x e .. 4! v l d m M Tu \\\ r .H O f- .md 1 II V e T I ad I ’\I\ w U II I r 1 I\ It . t- a s I .m n . 3. .mw . Ill r -0 o are 3 fit!\ I IIIIIII . . .l t . . .I III . I . I 1 f I 1 r T T. i 1 1 3 5 w 100 m 0. 0. 0 0 0. 0 n_u n_u mu 09::an m>zo_mm 13.0 11.0 9.0 7.0 Time (ns) Figure 4.44 Expanded latter part of Figure 4.41 Figure 4.41 (via UCET). 144 0.05 : A 0.05 E m 1 ‘0 4 3 i E 0.01 :1 ’1‘ Q. . I’ ‘1‘ E : ' x , ’~ - . O 2 l, ‘\ \,I’ \\ I ‘ ’ 7‘ -3 T 0) 3 Z -0.01 2‘ ‘ .H —I 2 i (D 1 01 1 -0.05 S 2 original waveform I ------ reconstructed waveform -0.053ITITj—ITTWIjTIjTIIIIIITIllillj] 7.0 9.0 11.0 13.0 Time (ns) Figure 4.45 Expanded latter part of Figure 4.43 Figure 4.43 (via CET). 145 the measured responses. The natural frequencies of the A—10 were extracted using both techniques from each aspect angle measurement separately and all measurements simulta- neously (multiple data sets, I =3). The first, second, and third columns of Table 4.8 and Table 4.9 are the natural frequencies obtained from the measurements of HO, 45° and 90° off HO separately using the UCET and the CET. The first column of Table 4.10 and Table 4.11 is the average natural frequencies of the closet natural frequencies of three aspect angle (all the columns of Table 4.8 for UCET and Table 4.9 for CET), while the second column of Table 4.10 and Table 4.11 is the natural frequencies obtained using all the three aspect angle measurements using the UCET and the CET. Figure 4.46 and Figure 4.47 show the measured response of the A-10 Thunderbird at 45° off HO and the waveform reconstructed using the natural frequencies obtained from multiple data sets using the UCET and the CET with q,=5.0 and p,,,=-1.0d-9. In past work the natural frequencies were obtained separately from each aspect angle measurement, and the closest natural frequencies were grouped and averaged to obtain a global set of target natural frequencies. It is evident from the examples shown here that the previous method has a potentially high error, and there is a possibility that nearly identical natural frequencies might be assumed to be from the same mode. The natural frequencies were grouped considering the radian frequencies, even if the damping coefficients differed significantly, resulting in very bad estimation for damping coefficients. The new method using multiple data sets for extracting natural frequencies overcomes all the difficulty mentioned above, and gives much better discrimination results. From Figure 4.46 and Figure 4.47, it can be concluded that estimation of natural frequencies obtained using the CET are better than the UCET. 146 Table 4.10 Natural frequencies of A-10 Thunderbird obtained via UCET using all measurements together. Table 4.11 Averaging Multiple 0.029 + j 2.331 0.042 + j 2.246 0.168 + j 3.278 0.189 +j 3.192 0.219 + j 4.453 0.114 + j 4.291 0.175 + j 5.158 0.631 + j 4.882 0.377 + j 5.881 0.080 + j 5.600 0.204 + j 6.520 3.860 + j 6.490 measurements together. Natural frequencies of A-10 Thunderbird obtained via CET using Averaging 0.021 + j 2.297 Multiple -0.061 + j 2.254 0.186 + j 3.286 0.221 + j 3.130 0.156 + j 4.421 4.6-11 + j 4.10 0.409 + j 5.145 0.900 + j 4.746 0.029 +j 5.459 0.121 + j 5.458 II 0.408 + j 5.936 147 0.316 + j 6.312 I Relative amplitude 0.010 , 1 0.005 3 I , I A ” ’ ... l r ‘ ,t I‘ I‘ 4 "1 0“ :‘, '1‘, ,’\ j l ', a 1 ‘ " \ '1 \\ 0.000: it: .‘i‘ll‘;‘,\;\‘ 3 '1," ,5 '1": ‘4‘," " -i l I \' .. 'l ‘ '1 I I U -0.005 1 -0.010 {f _ l 1 original waveform I ------ reconstructed waveform —0.015 TITITTIIIIIIIITIIIIITTITIIIWUfi 10.0 15.0 20.0 25.0 Time (ns) Figure 4.46 Measured response of A-10 at 45° off head-on and the waveform recon- structed using the natural frequencies obtained from multiple data sets via the UCET. 148 0.010 0.005 1 '1 J A g 1 l‘ l\ I 3 3 ’ 1, ; ‘t ,9, I; 0.000 -_ ‘ , 1 ' ‘, . O- : I, ‘11 “J " ‘\ E : o 3 \J U ‘3 —0.005 5 1,: - 2 2 <1) '1 [K .4 —0.010 f 1 original waveform I ------ reconstructed waveform I —O.015-1ITTIITTTTIWITIIITTTIIIITIITII] 10.0 15.0 20.0 25.0 Time (ns) Figure 4.47 Measured response of A-10 at 45° off head-on and the waveform recon- structed using the natural frequencies obtained from multiple data sets via the CET. 149 4.6 Conclusion It is concluded that the CET gives better estimates for the natural resonances than the UCET for low SNR responses, and the CET always gives natural resonances with negative damping coefficients as required; however it is important to note that UCET is much better scheme than many other well known techniques [24]. The proper choice of q, and p", is important to obtain good estimates for the natural frequencies. There are sophiscated minimization routines does not require to input these two parameters. The powell method is used in our discussion since the advanced routines are not available in our college computing facility. It has also been demonstrated that using multiple data sets to obtain a global set of natural frequencies gives much better result rather obtaining the natural frequencies for each aspect measurement separately and averaging the closest. It is important to have smaller increment for T, while searching for the optimum E-pulse duration. 150 Chapter 5 Free Field Measurements 5.1 Introduction Chapters 2 and 3 reveal the importance of accurately extracting the natural frequencies of complex targets from their measurements in order to have good discrimination results. Chapter 4 demonstrates that the natural frequencies cannot be extracted with high accuracy if the signal-to-noise ratio (SNR) of the measurement is very low. The natural frequencies of complex targets are usually obtained from measured responses of scale model or full model targets in a laboratory environment. A time domain chamber has been established to demonstrate the E/S pulses technique in a free-field environment. The chamber provides a simulation of the free- space radar environment where realistic scale-model targets can be illuminated at arbitrary aspect angles and polarizations. Further details about the chamber and the measurement scheme are given in the following sections. The duration W,,, as defined in (2.11), should be at least 25 percent or more of T, (defined in chapter 2) to have successful discrimination. The duration of T, is normally determined by the length of time the target response rings above the noise level. So, the duration of T, is primarily dictated by the SNR and the Q of the target. For the 151 cases considered (target size ranging from 6.0 to 7.25 inches), if T, is 20-ns, W, should be at least 4-ns or more. It was empirically observed that if W, was 3-ns or less, the information contained within the duration W,, of the late-time convolutions would not be enough for successful discrimination. The latter observation will be confirmed in Chapter 6. The duration W,, can be maximized by either having T, as large as possible, or by transmitting a short duration pulse (1;). The other parameters T, and T, (defined in chapter 2) cannot be controlled since they are target dependent. However, the duration of the effective transmitting pulse can be controlled by the choice of a transmitting and a receiving antenna. In addition, the receiving antenna should also have a ”quiet period" so that a smaller target response can be amplified without clipping the signal. In this chapter, travelling-wave wire antennas are analyzed in time-domain [38]. Also, travelling-wave antennas are experimentally studied to see whether they satisfy the requirements for both the transmitting and receiving antennas. The travelling-wave current is also measured on the wire antennas. Three combinations of transmitting and receiving antennas are experimentally studied inside the time-domain chamber. 1. V travelling-wave antenna as a transmitter / straight-wire as a receiver 2. V travelling-wave antenna as a transmitter / broadband horn as a receiver 3. Broadband horn as a transmitter and a receiver. The performance of the above antenna systems is given in sections 5.6-5.8. The frequency-domain deconvolution method is currently used to remove the system response from time-domain and frequency-domain measured responses. A brief 152 description of the deconvolution procedure is discussed in section 5.7 (for more details please refer to [41]). 5.2 Free-field Anechoic Chamber Scattering Range Before the free-field anechoic chamber scattering range was built, radar target discrimination had been done with measurements of a few half-model targets on a ground plane [31]. The ground plane measurements had a high SNR, and these measurements allowed successful discrimination. However, targets could not be illuminated at arbitrary aspects and polarizations. The chamber, on the other hand allows a greater range of aspects, and polarizations. Figure 5.1 shows a schematic diagram of the present time-domain free-field experimental facility. The chamber is 24’ long by 12’ wide by 12’ high and is lined with 6" pyramid absorber. A Picosecond Pulse Labs model 1000B-01 pulse generator provides quasirectangular pulses of l-ns duration and amplitude 40 V to an American Electronic Laboratories model H-1734 wideband horn (0.5-6.0 GHz) which can be resistively loaded to reduce the inherent oscillations, and the field is received by an identical horn antenna. The receiving antenna is connected to a Tektronix 7854 waveform processing oscilloscope via S2 sampling heads (75-ps rise-time). At present, neither the receiving or transmitting horn antenna is resistively loaded since the deconvolution procedure is used to remove the system response. The system response includes the inherent oscillations from both the receiving and transmitting antennas. A microcomputer controls a waveform processing oscilloscope which acquires the received signal and passes it to the computer for processing and analysis. 153 lulu- m 3f Tait In 754 one: lacuna t"12°" A s Q 00 18A PC-XT user-0° uncut-r- #95127: fr... Figure 5.1 MSU free field experimental facility and associated equipment. 154 The longest dimension of a target measured inside the chamber without any interference from the absorber is dictated by the performance of the absorber at low frequencies. The absorber used inside the chamber does not perform well under 400 or 500 MHz, so the longest dimension of the target should be less than 8 inches. If the dimension of the target exceeds the above limit, one or more target resonances might be less than 400 MHz. Therefore, the target scattered field contributions of low frequencies will not be absorbed when it hits the absorber. The actual target response will be contaminated by the multiple reflections off the chamber walls due to direct incident and target scattered field. The majority of the low frequency contributions by the direct incident field come from the back wall of the chamber since the antennas are directive. This contribution is eliminated by proper time gating. A target response is usually obtained by the following procedure. Proper timing of the waveform processing oscilloscope is accomplished by using the variable trigger delay internal to the pulse generator. One hundred waveforms are used during each measurement and averaged in real-time. The target is placed on a styrofoam stand inside the chamber, which is approximately 8’ away from both of the antennas (located side by side with a vertical separation of 2’), and the measurement is taken. Then, a clutter measurement is taken after removing the target from the chamber. The clutter measurement is subtracted from the target measurement. The resulting measurement is the target response contaminated with the system response. Then, the actual target response is obtained by deconvolving the system response from the measurement. 155 5 .3 Travelling-wave wire antenna Transient fields produced by traveling-wave currents on linear antennas have become an increasingly important research topic [61,62,63,64,65,66,67]. Most work (see for example [68]) is done in the frequency domain with transform techniques. Although this type of approach gives accurate solutions, it seems to obscure the physical pictures available in a purely time—domain analysis. The time-domain impulse response of a straight wire supporting a traveling-wave current is obtained via potential theory, and the superposition concept yields an expression for general excitation. Moreover, the dielectric coated wire with current-wave velocity v less than c is treated. A closed form solution is obtained for the rectangular pulse response. 5.3.1 General Field of a Straight Wire Consider a transient traveling current wave on a straight wire (Figure 5.2). Assume the current arises at point z = a and is completely absorbed at z = b after traversing the wire. If the wire is thin, the current wave can be modeled using 30:0 = 2601601010) [urz-a) -u(z-b)1 (5-1) where v is the wave velocity along the wire and ((0 is the current waveform and u(t) is the unit step function. The retarded vector potential at an observation point in free space is then .._,, _ :1 = (pa/41:)I‘Kr JR RIC) dV’ (5.2) v where R = IF- F’ | , whereas I is related to scalar potential (I) through the Lorentz gauge 156 \ / (x,y,z) V ‘< Figure 5.2 Transient traveling current wave on a straight wire and local coordinate system. 157 v.1 = -poeoa¢lat (5.3) The resulting electric field is 123‘ = -“p-a,i'/at (5.4) The field can be calculated using time-domain superposition. Replacing 1(1) with 6(t) , the dirac delta function, gives the electric field impulse response I? ”(F, t) . The field due to a general waveform [(0 is then given by the convolution 13‘0”») = f 1(t’)i"(r,t-:')dt' (5.5) Using (5 .2) the vector potential impulse response is calculated as D I‘m) = (no/4102‘ f :5 [G(z’)]g(z’)dz’ (5-6) a where G(z’) =t-z’lv-R/c and 8(Z/)=1IR. Making a change of variablesu = G(z’) gives 0(5) A.“ = -(uol4n) f 5(u)F[G"(u)]du (5.7) 6(a) where F(z’) =g(z’)lG’(z’) and G’(z’) =dG(z’)ldz’. This leads to h _ -F[G"(0)] for G(b)s0sG(a) 5.8 A‘ - ("0, 41:) {0 otherwise ( ) where 6(a) = t-a/v-R‘lc, G(b) = t-b/v-Rblc, Ra = [p2+(z-a)2]m, and 158 R, = [p2 +(z -b)’]"2 . (Note that p is the cylindrical coordinate radial variable). Here G"(O) is the value :5, which satisfies 6(20') = O, or z; = (1/m’)[(vt-n2z):nm (5.9) where D1 =(vt-z)2+p2m2, m2 = l-nz, and n = v/c. Substituting this value gives H25) = v/[-Ro+n(z-zo’)], where R, = Rug). Using 6(20’) = t-zglv-Ro/c = 0 yields H25) = :v/D (5.10) Since D is always positive, the minus sign in (5.10) is chosen so that the vector potential (5.8) is in the same direction as the current. Putting (5 .10) into (5.8) then gives the vector potential impulse response A: = (prov/4n D)[u(t- T, -R,/c) -u(:-T,-R,/c)1 (5-11) where T, = a/v, and T, = b/v. Note that the vector potential exists only within a time window dictated by the current wave transit time and the propagation delay from the wire ends. The scalar potential is determined by substituting (5.11) into (5.3). Taking care to properly evaluate step function derivatives results in the impulse response 4>’* = flow" -In '1 —(z-a>/R.1 x[(a+nRa-z)2+p2m2]'m)u(t-Ta-Ralc) -33 —1- -1- _ 4“ (020) [n (z b)/R.,] x[b+nR,-z)2+p’m21-m)u(:-T,-R,/c) (5.12) where n is the free-space intrinsic impedance. 159 Using (5 .11) and (5 . 12) in (5 .4) gives the electric field impulse response R2 i5" 3 --2—cu(t-Ta-Ralc) m2[Ra[v(t-Ta)cosBa-Ra]-6¢v(t-T‘)sin6.]+[fia] 1t { [v(t - To) -Racosfla]2 + mznfsinze, }”2 “ “ - ~ .1 + gun-1,3,1...) m2[R,,[V(t-Tb)cosfib-Rb]-6bv(t-Tb)smeb]+[Rb] (5 3) {[v(t - T9 - Rbcose ,12 +mzR§sin¢e,}3’2 Rf +—n— 6 6(t-T -R lc) ”Sine“ -6 5(t-T -R /c) nsinfl, 4n ‘ ‘ ‘ Ra(l meow“) b b " Rb(l -ncos0b) Here conversion from global to local coordinates is given through 2 Rucosegb - 6.,» sine...» (5.14) fi = Qbsmefib + ea'bCOSead, where sine,I = le‘, 0080‘ = (z-a)/Ra, sineb = p/Rb, and cow, = (z-b)/Rb. Finally, from (5.5), the general electric field is given by ‘T.-R C A “ ' E = _ “cm: ‘ I“ I(t’) R,[v(t-Ta-z’)cosea-R,]-e,v(z-T,-t’)smea dt, 4" { [v(t - To - z’) -Racosea]2 +m2RZsinzea }”2 -T -R c A . + ncm’ ‘ bf *’ 1(t’) R,[v(:-T,-z’)cose,-R,] -é,v(:-r,-:’)sme, , 41'! {[‘(t-Tb-t’) -Rbcosflb]z+m2R:sin26b}m (5.15) +1 era-r -Rlc) "Sine“ -é,I(t-T,-R,/c) "he" 41: ‘ ‘ ‘ Rad-noose“) Rb(1-neos0,) -27: (RaqU-T‘ — R,/c)IR3 4540- T» 4510/“: ) where the charge 4 is given by I = dq/dt. It is interesting to note that when v = c (i.e., the uncoated wire) this reduces to 160 E = _'L{éal(t-Ta-Ralc)( "Sine“ J 41: Ra(l -ncosOa) - é I(t-T -R,/c) "Sine” (s 16) b b Rb(l #100865) ' - 1‘2 (R¢q(t-Ta-R¢lc)/R: 4n - “,q(t-T,-R,/c)/Rf) These results are found to match previously published expressions. The 6 components in (5.16) are those furnished by Manneback [69], while the radial components match those in Schelkunoff [70]. 5.3.2 Far-zone Specialization The use of global coordinates in (5.15) allows the far-zone specialization for an arbitrarily oriented wire (Figure 5.3). Neglecting terms which decrease faster than NR, and using R>L, saw,- 0,1?“ -§,-R, and 6,-6, - 0, gives E(fit) = L's—[2017(0) [I(t - To -Ra/c) - I(t- To -L/v -—Rb/c)] (5.17) where To is the time at which the current wave enters the wire segment, and 17(0) = min!) [(1 -ncosO). Expanding the current functions in a Taylor series about t-To-Rlc and using :2 =1?st -dsino, it is found that the leading terms cancel whereas the second terms combine to give E = (ho/4n)(L/R)Rx(itxa)1’(t-To-R/c) (5.18) 161 (x,y,z) A S A u b O R R0 Q L/P. O 3' we Figure 5.3 Arbitrarily oriented wire and local coordinate system. 162 where 12 is the unit tangent to the straight wire, and higher-order terms can be neglected for pulse-width much larger than L/v. 5.3.3 Arbitrarily Shaped Wire Treatment of a traveling-wave current distribution on an arbitrarily shaped wire (Figure 5 .4) uses (5.18) and superposition to give 13" = (n/4nc)fnxnx’ (‘ "1’?” R/c)du’ (5.19) I‘ for a line current along curve I‘. 5.3.4 Numerical Results The straight wire behavior can be best illustrated through numerical examples. The special case of a rectangular pulse current waveform yields a closed-form answer for E, and is used in subsequent examples. Letting I (t) = Io[u(t) -u(t — 7)] in (5.15), and using 0 ift-Ta’b-RM/c<0 p“ = 7 If y < t-TM-Ra’blc ‘ (5-20) t-Tfib-Ra’blc ify >t-Tu-Ra‘blc leads to 163 X / (x,y,z) C) IJL 3L C Figure 5.4 General-shaped wire with traveling-wave current distribution. 164 if — 4 6 I t-T -Rl) "Sine“ -é I(t-T -R/ ) "Sine‘ ' W ”)[ “( ° ‘ c Ra(l-ncosfla) " " " c R,(1-neose,) -(nc/4n>I,rR,p,/R3-R,/R§p,,1 [anzvcose -R v2(t-T )] _ 2 ° 2 0 a a 0 (11cm l4n)Io{Rasm Oa[ A‘(t_T‘) -[n2R3vcosfla -Rav2(t- T,l - [30)] Aa(t-Ta-fla) , [[RavcoseaU-Ta) -RZ(1 -n2siuze,)] -9‘vsrn6‘ Aa(t-Ta) (5.21) _ [R‘vcoseau-T‘ - pa) -R3(1 -n2sinzea)] Aa(t-Ta-Ba) [nzafveose,-R,v2(t-T,)] s,(t-T,) +(ncm214n)I,{R,sinze,[ _ [nzvaeese,-R,v2(t-T,- p,)] A“: - Tb - [3,) . [R veose (t-T)-R2(1-nzsin26 )1 -ébvmeb[ b b A b(t-;) b b b _ [R,veose,(t-T,- Bb)-R:(1-n28in29g)] AbU- Tb - 6,) where A‘.b(t) = m1V2R3,b8in26,.b{[vt "Rama“? + 0121?}..an 6...}"2 (5.22) Figure 5.5 and Figure 5 .6 show the near-zone axial (E) and radial (5,) components of E(z =0,t) caused by a straight wire on the z-axis, computed using (5 .21) 165 1O ° *1 A E > --10 V G) .0 g-zo L ‘a 3% -30 E .9 LL. -4o v/c = 1.0 ------ v/c = 0.7 -50IIIIIIIIIITIIUWUIIITIlIrIIITT'ITIIIIIIIIIIIUIIIIIWYIITUII‘] o 2 4 6 8 10 12 Time (ns) Figure 5.5 Near-zone axial component of electric field in the z = 0 plane due to a straightwireon thezaxis fora = O, b = 1m,p = 1 m, 7= lns. 166 N T 0.. ”e 3 \ -2: y" i > , , : v , : ‘D -4— E 5 '0 .r : i .3 : : a _6: L._ : ..... E - 5 <1: - : E —8-: - E .2 - ------ . |_._ - _10._ - v/c = 1.0 - ------- v/c = 0.7 -12 I l l I T I I I I 1 l I I I fl l I 1 l I 1 I 0 2 4 6 8 10 12 Time (ns) Figure 5.6 Near-zone radial component of electric field in the z = 0 plane due to a straightwireon thezaxisfora = O, b = 1m,p = 1 m, ‘y= lns. 167 with pulse duration 7 = 1 ns (which is less than the wave transit time L/v along the wire). In this case, there is no 1?, component contribution to the axial field. Thus when v = c, the charge term in (5.15) is not implicated; the first axial field event exactly duplicates the current pulse, as revealed by (5.15). This field originates when current appears at z = a. The second event seen when v = c is due to current being absorbed at z = I); both transit time UV and the path length difference from the wire ends account for the later starting time. Since both 6,, and Rb contributions to the field exist, the pulse is not exactly duplicated. The slope visible here is due to the charge accumulation described in (5.15). The total accumulation remains as residual static charge; thus? remains constant and nonzero after the second event. When v t c , the first term of equation (5.15) augments contributions from the current waveform; thus the first event is not an exact duplicate of [(0. This term accumulates, so the second event merges into the first, but begins later due to the greater transit time L/v. The same residual E results as with the v = c case, because v does not affect total charge accumulation. The radial field component (Figure 5 .6) exhibits similar behavior. As E, lacks a 6. component, when v = c only the charge term contributes to (5.15); this is the leading edge of the field waveform. The constant slope is due to charge accumulating at z = a. When the current pulse has completely passed 2 = a, a constant field is maintained by accumulated charge until the current reaches 2 = b. Then both 6, andR, 168 components contribute to E, and form the second event in Figure 5.6. A constant residual field remains after the end of the second event, caused by the total accumulated charge at both 7. = a and z = b. When v at c, the constant E produced by charge accumulating at z = a is augmented by the first term in (5 .15); therefore the first and second events merge. Far-zone outcomes are shown in Figure 5.7 and Figure 5 .8 for the same [(0. Terms in E9 vary inversely with distance-squared from the wire, whereas E: has a term which varies inversely with distance; this accounts for the vertical scale difference between these two graphs. Each event in the axial field plot corresponds to the current wave passing a wire end, and replicates [(0. Due to decay of inverse-square terms, no residual field appears. The amplitude difference between the cases v = c and v at c is given by the ratio n = We in (5.15). The radial field is quite similar to its near-zone counterpart. However, the effect of the accumulating term in (5.15) is reduced, and the two events are separate. It is interesting to note the absence of a residual field after the second event; radial component cancellation in the far-zone static fields is responsible. 5.3.5 Experimental Results To verify the theory presented above, an experiment was performed using a straight-wire transmitting antenna above a conducting ground plane. The antenna was fabricated as an extension of the center conductor of a 50 O coaxial cable with the outer conductor contacting the ground plane. The antenna was fed by a Picosecond Pulse Labs 169 0.4 0.2 5"": A n i E E : \ : = > g '. v . r O) 0.0 —-'1 ----- -—'—-— i ..... 'o :3 . :0: I B. E-o.2 ..... <2 '52 .92 “-0.4 v/c = 1.0 ------ v/c = 0.7 -0.6 330 332 334 . 336 338 340 342 Time (ns) Figure 5.7 Far-zone axial component of electric field in the z = 0 plane due to a straight wire on thez axis for a = O, b = l m, p = 100 m, 7= lns. 170 0.003 0.002 - /'\ .. ’I : E - :’ : \ - : l a - s s '0 t l D - I I r: - : t “a - E : E 0.000 - —§— _ ‘ EU. 4.) ‘ II 2—10- :: a) ‘ l: 0: -l _15_. I: —— Measured waveform .. ------- Calculated waveform -20tfi1]IfillllltlflltIIIrr] 0 2 4 . 10 12 Time (ns) Figure 5.10 Near-zone axial field (measured and theoretical) for 63.5 cm uninsulated wire measured at p = 2.49 m, z = 0. 174 re-reflected, but with a positive coefficient. In this case the wire and image fields cancel; only a small event is seen. The third event is created by the second reflection from the wire end. The significant difference between field amplitudes generated by the first and second reflections is due to radiation loss. These results are quite similar to those observed by Schmitt et al. [71] using a similar experimental setup. The dotted line shown in Figure 5.10 represents the time derivative of theoretical E, computed using (5.15). Here, I(t) is the pulse shown in Figure 5 .9, while propagation velocity is chosen as v = 0. 9986. The theoretical curve is developed by superimposing straight-wire solutions with current and wave directions chosen to match the multiple reflections of the antenna/image system. The reflection coefficient at the antenna end is taken to be -1 for the first reflection and 0 for the second reflection; at the feed +1 is used. The theory agrees quite well with experiment. Similar results are shown in Figure 5.11 for an insulated wire transmitting antenna. Here, the reflections occur at later times than with the uninsulated wire; thus v < c. The dotted line in Figure 5.11 shows the theoretical curve corresponding to v = 0.993c. Again good agreement is attained. 5.4 Travelling-wave antenna setup inside chamber Figure 5 .12 shows the schematic diagram of the measurement setup inside the chamber when a V-wire antenna was transmitter, and a straight-wire antenna was a receiver. The left-arm (AC) was connected to the outer conductor of coaxial tube #1 (CI‘ #1), and the other end of the left-arm (AC) was connected to the inner conductor of CT 175 O 'l n 1 -l 5.. 1 0) cl '0 O- - - 3 .. l .4: -l l E). ~ '. < '5: (D _ .Z ‘ 4.0 -l 2—10- (D - m .- .J _15- : —— Measured waveform . ------- Calculated waveform -20FIIIIIII1II1IIIIIIIIIIII 0 2 4 6 - 10 12 Time (ns) Figure 5.11 Far-zone axial field (measured and theoretical) for 63.5 cm insulated wire measured at p = 2.49 m, z = 0. 176 #2. The right-arm (BD) was connected to the inner conductor of the CT #1, and the other end of the right-arm (BD) was connected to the inner conductor of CT #3. The straight- wire antenna (BF) was connected to the inner conductor of CT #4, and the other end was connected to the inner conductor of CT #5. The lengths of AC and BB were 21’ while the length ofEF was 11’. 5.5 Measurement of travelling-wave current using current loop The travelling-wave current was measured on the straight-wire (BF), and both arms (AC and BD) of the V-wire using a current sensing loop. Figure 5.13 and Figure 5 .14 show the travelling-wave current of the straight wire (EF) measured at 12" from point B with open and 50 ohms loads placed at the end of Cl‘ #5. The time difference between the initial points of the first travelling-wave and the reflected travelling-wave currents is 20 us, as expected. The distance between the measurement point and the termination point is 10’, so it takes 20 ns for the wave to travel from the measurement point to the termination point and back to the measurement point assuming the wave is travelling at the speed of light. The experimental current pulse waveform is shown in Figure 5 .9. The travelling-wave current is the derivative of Figure 5 .9 since the current loop act as a differentiator. The termination with the 50 ohms load reduces the inherent oscillations coming from the reflected travelling-wave since it is a better match than an open circuit. Figure 5.15 and Figure 5.16 show the travelling-wave current on the left arm (AC) and the right arm (BD) measured at 12" from points A and B for the V-wire antenna terminated with an open load placed at the end of CT #2 and #3. The time 177 Figure 5.12 Anechoic chamber with travelling-wave V-wire transmitting antenna and straight-wire receiving antenna (CT Coaxial Tube). 178 400.0 300.0 - a) 1 7.3 200.0 - 23 1 2;: 3 ll 8. 100.0 '2 E , D I a) 0.0 T—W .2 I «J . 2 -100.0 ~ n: 2 t1: 1 -200.0 1 -300001!IUIIIVIITUIUIIIUY‘YYII’] 0 10 20 40 50 30 Time (ns) Figure 5.13 Travelling-wave current on a straight-wire antenna terminated by open load measured 12 inches from fwd point. 179 é 1 1 300.0 ': n) I T.) 200.0 - 3 I ,4: -l 0. 100.0 - E . O I 1D 0.0: > .. '53 I 2 -100.0 - 0 l 0': 1 -200.0: 1 -30000.VUTF[IITIIITII I"'V|IIII1 0 10 20 40 50 30 Time (ns) Figure 5.14 Travelling-wave current on a straight-wire antenna terminated by 50 ohms load measured 12 inches from fwd point. 180 difference between the initial travelling-wave and the reflected travelling wave currents is 40 ns, as expected. The distance between the measurement point and the termination point is 20 feet. So, the wave takes 40 us to travel back and forth from the measurement point. Figure 5 .15 and Figure 5.16 illustrate that the travelling-wave currents have opposite signs, as expected. The travelling-wave current amplitude on the left-arm (AC) is slightly larger compared to the travelling-wave current amplitude on the right arm (BD). 5.6 V-wire Transmitter I Straight wire Receiver For the case v = c, the scattered field produced by the travelling—wave antenna, from (5.15), at far-zone is the duplicate of the time limited excitation pulse. Therefore, the benefit of using the V-wire as a transmitting antenna is that it produces a duplicate of the time limited excitation pulse and directed radiation. However, the drawback is that fields radiated from the wire ends (points C and D) of the V-wire antenna due to travelling-wave current reflections re-excite the target, as is also very clear from (5.15). The re—excited fields radiated from the wire ends is also observed in Figure 5 .10 and Figure 5.11. The latter problem can be avoided by extending the wire ends as long as possible with out slacking or bending the wire. The benefit of using a straight-wire as a receiving antenna is that it produces a large amplitude received signal since the wire antenna acts as a integrator. At the time of the measurements, the performance of the absorber was not considered. For reasons similar to those indicated in section 5.2, the target signal may have been contaminated with multiple reflection signals since the received signal is dominated by low frequency 181 200.0 150.0 100.0 50.0 -50.0 Relative amplitude -100.0 O O Irirrliriilirriliriiliiilliiii11111] -150.0 .0 o 10.0 20.0 30.0 40.0 50.0 Tl me (ns) Figure 5.15 Travelling-wave current on a left side of a V-wire antenna terminated by Open load measured 12 inches from feed point. 182 100.0 50.0 -50.0 Relative amplitude -100.0 0 O lIlllllllIllIllll[JLIJJLJIJIILIJII[111111111IJIIIJ -15000 IIIIII1TIIII'1'11'I1'llllIIIIIIIIIIIIj'll'rjiTIIUI 0.0 10.0 20.0 30.0 40.0 50.0 T: me (ns) Figure 5.16 Travelling-wave current on a right side of a V-wire antenna terminated by open load measured 12 inches from feed point. 183 oscillations. However, the main drawback is that the straight-wire antenna does not have a ”quiet period". The ”quiet period" is followed by the system response where the target response can be amplified without clipping the signal. If the ”quiet period" is large, then the target response will be a small perturbation of a clutter measurement (This can be seen from Figure 5.17 and Figure 5.18.). Therefore, the target response may not be an accurate measurement. Figure 5.17 shows the background measurement of the chamber with the V-wire and straight-wire antenna system. It is very clear from Figure 5.17 that there is no “quiet period”. Figure 5 .18 shows the measurement of a medium boeing 707 including the background measurement. Figure 5.19 shows the response of the medium boeing 707 found by subtracting Figure 5.17 from Figure 5.18. The target response is contaminated with the chamber back wall reflection after 17 ns since the receiving antenna was placed approximately 9’ away from the chamber back wall. The antenna system was abandoned due to two main drawbacks. The first drawback was that the antenna system required a forward scattering arrangement. The second drawback was that there was no ”quiet period”. Therefore, target responses were only small perturbations of the antenna system response. Also, the reflection from the chamber wall was high since the received signal was dominated by the low frequency oscillation. 5.7 V-wire Transmitter / Horn Receiver The benefit of using a wideband horn as a receiver is that it is strongly directive and produces a large received signal. Also, it has a ”quiet period” followed by the horn 184 75.0 1 .1 50.0- a) -l 'D 25.0“ :3 q ,4: .. .5. - E a: O OO:J 0) - > d '4: - 2-250- 0) .. m .. -50.0'* ”75.0 thjilrllllllblilillillIIIIIIIIIIYIII'llrrIIlIITT' 0.0 10.0 20.0 30.0 40.0 50.0 Time (ns) Figure 5.17 Measured pulse response of the V-wire/ straight wire antenna system. 185 75.0 50.0 25.0 9 o llJlllLlllllllllgllllLJllllij -25.0 Relative amplitude -50.0 -75.0 --rWfi-rfiTrr1Tfi1fi-rrrr1-rrrr1-n711-nTrrnfl-rrrn-m 0.0 10.0 20.0 30.0 40.0 50.0 Time (ns) Figure 5.18 Measured pulse response of a medium size boeing 707 aircraft model, with V-wire / straight wire antenna system response included. 186 5.0 3.0 a) U l :3 :‘_:.’ 1.0 o. E o °>’-1.0 .5 2 <0 0: -3.0 -5.0'11—11111IIIIUIIIIUIII1III1IIIllIIjIIIIIIIITYIITII—UW 0.0 10.0 20.0 30.0 40.0 50.0 Time (ns) Figure 5.19 Measured pulse response of a medium size boeing 707 aircraft model, with V-wire / straight-wire antenna system response subtracted. 187 ringing period. However, the main drawback is that the strong natural oscillations of the horn may mask a target resonance of a similar frequency. The natural oscillations of the horn can be reduced by adding a series of lumped resistors. Figure 5 .20 shows the schematic diagram of the measurement setup used inside the anechoic chamber. The V-wire antenna was connected in the same way as was done in the previous section. The horn antenna was situated to receive the E-field parallel to the floor. Figure 5.21 shows the background measurement of the chamber with the V-wire and the horn antenna system. It can be clearly seen that the receiving horn antenna has a long ”quiet period" following the hom’s natural oscillations. Figure 5.22 shows the measurement of a medium boeing 707 including the background measurement. Figure 5.23 shows the response of the medium boeing 707 including the system response found by subtracting Figure 5.21 from Figure 5.22. The reflection from the chamber back wall is not present in the target response since it was eliminated by proper time gating. This measurement using the antenna system was better compared to the previous antenna system since the receiving antenna had a reasonably long ”quite period" . However, the antenna system was abandoned since it required a forward scattering arrangement. 5.8 Horn Transmitter / Horn Receiver The benefit of using a horn as a transmitting antenna is that it produces a time limited excitation pulse. The time limitation of the excitation pulse depends on the horn’s 188 L, .3. -. ‘ 4 ‘ ‘ ;, . ’ c h ‘V/' ' . . v’ \ V. ‘f‘. \ l L" ‘ ; ¢~.\ 4‘ \ ‘. Q _/ . a 0” . -, , ' 3 fill . n3“ ._ ,..' 3”; \‘ [’21 .718.» " "I’.‘\ . ‘4‘ \x ) {3'17 ,’-. ' e . 7;“. g I“ \x‘ I, . ‘. . . . . - ‘.'_ ‘4. ' “-, .1 .‘ «3 am...) Lo...» M an BELLA W “(v/9.0 e038 (no.3. MA stanza ex.» :41 .a.‘ swan ‘ {HM- Figure 5.20 Anechoic chamber with travelling-wave V-wire transmitting antenna and wideband horn receiving antenna. 189 175.0 125.01 a) -1 '0 75.0 — :3 . ,4: - E .- 0 25.0: a) - .2 '1 +, . g —25.0-4 m .1 0: '1 —75.0- 4 -125o0 lllIlIljrllllrIlUfifiUIIIIIIIlTTIIIIIIW 0.0 5.0 10.0 15.0 20.0 Time (ns) Figure 5.21 Measured pulse response of the V-wire I horn antenna system. 190 175.0 125.0‘ Q) .1 'O 75.0- 3 .. ,4: q E d O 25.0: 0) .. > a 1.: _. 2 -25.0- (D .. CE 1 .1 -75.0- “ -12500 I'lljlTjIIIIITTIIIIIIIIIIIIIIIIIIIIIIII—l 0.0 5.0 10.0 15.0 20.0 Time (ns) Figure 5.22 Measured pulse response of a medium size boeing 707 aircraft model, with V-wire I horn antenna system response included. 191 5.0 .0 o lllLIllllllJLlJJllll Relative amplitude J» O l _ZOoO TI—IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIFTl 0.0 5.0 10.0 15.0 20.0 T: me (ns) Figure 5.23 Measured pulse response of a medium size boeing 707 aircraft model, with V-wire / horn antenna system response subtracted. 192 natural oscillations, and the oscillations are reduced by adding a series of lumped resistors, as is done for the receiving horn antenna. The amplitude of the horn natural oscillations are reduced by a considerable amount when the series of lumped resistors are added to the horn antennas. Thus, the amplitude of the incident field is reduced, and late— time scattered energy of targets is reduced by a considerable amount. However, the series of lumped resistors are removed from both the transmitting and receiving horn antennas since the deconvolution procedure is used. Figure 5 .1 shows the schematic diagram of the measurement setup inside the chamber. Figure 5 .24 shows the background measurement of the chamber with the horn/horn antenna system. The first part (2-12 ns) of the measurement is dominated by the direct coupling between the horns. This direct coupling can be reduced by adding a series of lumped resistors, placing absorber at the mouth of the horn antennas, and fixing a microwave absorber in between the horn antennas. The direct coupling also can be reduced by placing the horns off diagonal, as explained in section 5.2. The measurement beyond 35 ns is invalid since it is contaminated by the low frequency contributions of absorbers from the chamber back wall. So, targets are placed inside the chamber such that the target responses are present in the time window between 14 ns and 34 ns. Figure 5.25 shows the measurement of the medium boeing 707 including the background measurement. Figure 5.26 shows the response of the medium boeing 707 found by subtracting Figure 5.24 from Figure 5.25. The target response is only valid up to 34 ns. The error in the target response between 0 and 14 us is due to subtraction. The subtraction error can be reduced by decreasing the direct coupling between the horns. However this would have no effect on the target response. 193 20.0 10.0 I .0 o Relative amplitude O O JllUiJlllJlllllJJJll11111111141141]IJJJ -20.0 HTIIIIWIIIFIIIITIIIIITTjIIIIIIIIIIlllllllthtfirfl 0.0 10.0 20.0 30.0 40.0 50.0 TI me (ns) Figure 5.24 Measured pulse response of the horn / horn antenna system. 194 20.0 : .1 I 10.0‘ a m “ I '0 I D - 1': - a . l E .. o 00': m d > "1 '43 I 2 - l 0’ I 0:-10.0: U -2000 llll'T‘IjjIIIIIrTll—rrl'rl1j—t'IIIIT'1'1IIIIIIIIIIII 0.0 10.0 20.0 30.0 40.0 50.0 Time (ns) Figure 5.25 Measured pulse response of a medium size boeing 707 aircraft model, with ham / horn antenna system response included. 195 6.0 . 4.0 d a) - '0 2.0 - l :3 - ,4: _ TD. - E 8W O 0 0 ': a) - > _ 1:; u g -2.0 - a, . 0: —. .1 -4.0 - —6.0 — 0.0 10.0 20.0 30.0 40.0 50.0 Time (ns) Figure 5.26 Measured pulse response of a medium size boeing 707 aircraft model, with horn / horn antenna system response subtracted. 196 The horn/horn antenna system is currently used to do the time and frequency [41] domain measurements. The main advantage is that the antenna system allows simulation of a bistatic scattering measurement. Thus, the deconvolution procedure can be performed on these measurements. 5.9 Deconvolution procedure Measured target response (r,(t)) is a convolution of the actual target response (r(t)) with the system response (s,(t)). The measured response in the frequency domain is R,(f) = R(f)S,(f) (5.23) The objective is to obtain the system response (S,(f)), and to remove it from the measured response (R... (f D. Then, the actual target response in the time domain can be obtained by applying inverse FFT on R( f ). A 6" thin wire is used to calibrate the system response. The bistatic scattering theoretical response (W,( f )) of the 6" thin wire oriented at broadside is calculated in the frequency domain using method of moments (MoM) for the geometry used in obtaining the measurements (for more details refer [41]). This is shown in Figure 5.27. The 6" thin wire is placed parallel to the mouth of the horn antennas (broadside) and approximately 9’ away from both the horns. The response of the 6" thin wire was measured. This is shown in Figure 5.28. Figure 5.29 shows the FFT of the measured response (W.(f)) of the 6" thin wire. So, the system response is 197 1.2 1.0 .0 oo Relative amplitude p o .0 N 0.0 [IITITIIIITITTIIIITTIIIIIIIIITIUIIIIUIIl—l 0.0 1.0 2.0 3.0 4.0 Frequency (GHZ) Figure 5.27 Theoretical bistatic scattering response of 6" thin wire. 198 15.0 10.0: i Q, I "a 5.0- i 3 - fl: - g d O 002W 1 Q) '1 .2 j 4.; _g -50- U a, . 0‘ I —10.0: 1 1 . l -15o0djTjITTIITIIIIITT'IIIITIUTUTUUIII'IIIIIT—l 0.0 5.0 10.0 15.0 20.0 Time (ns) Figure 5.28 The measured response of 6" thin wire inside the chamber using the horn/horn antenna system. 199 600.0 500.0 4:. O .0 0 Relative amplitude N U 0 O O O '0 0 100.0 0.0 Figure 5.29 A... - 0.0 1.0 2.0 Frequency (GHz) FFT of the measured response of 6" thin wire. 200 3.0 4.0 W (.0 S = "' (5.24) ,(f) WAD Figure 5.30 shows the system response in the frequency domain. The system has two natural frequencies, s, = -O.373 + j3.04 and s, = -0.401 + j5.20. If these two natural frequencies are not removed from the actual target response, it will be very difficult to extract the natural frequencies of targets and to do discrimination successfully since these two natural frequencies will be present in every measured target response. The following example shows how the deconvolution procedure is applied to the measurements of complex targets. Figure 5.31 shows the measured response of a scale model A—lO Thunderbird placed 450 off head-on (HO). Then, the measured response of the A-lO Thunderbird is transformed into the frequency domain via FFI‘. This is shown in Figure 5.32. The frequency domain measured response is then divided by the system response S, (f). The resulting response is the deconvolved response of the A-lO Thunderbird at 45° off H0 in the frequency domain. This is shown in Figure 5.33. Figure 5.34 shows the time domain deconvolved response of the A-lO Thunderbird at 45° off HO obtained via inverse FFT with a rectangular and a guassian cosine modulated weighting function. The deconvolved response of the target, resulting from the rectangular weighting function, is contaminated by the tail end of sine function. This effect is clear at the beginning and the ending of the actual response. The actual response of the target, resulting from the guassian cosine modulated weighting function, is smooth. Figure 5.35 shows the guassian cosine window used to obtain the deconvolved response (This guassian cosine modulated weighting function will be used in chapter 6 in conjunction with the deconvolution procedure unless otherwise specified). The effective 201 800.0 or o .0 o 400.0 Relative amplitude 200.0 A‘A“ - .. 0 1.0 2.0 3.0 4.0 Frequency (CH2) .0 o O - lrrrirrirrlrirrrrrrrlrrirririrlriiirLirJiriirJ k Figure 5.30 The system response in frequency domain obtained using 6" thin wire as a calibrator. 202 15.0 10.0 5.0 0.0 l .U‘ o Relative amplitude —10.0 irrrlrrrrlriirliiiilrr11114111 -15-O lllTlll[1|IllIrIIlllerTIIIIIIIIIIIIITTj 0.0 5.0 10.0 15.0 20.0 Time (ns) Figure 5.31 Measured response of A-lO thunderbird at 45° off head-on. 203 600.0 a) - 13400.0- 3 .. ,4: .. T5. - 0 I <0 - .2 ‘ up . 2200.01 0) m -. 0.0 AIFWTTITIIIlTTTIIITjTjIITTITIr-Tjillllll] 0.0 0.5 1.0 1.5 2.0 Frequency (CH2) Figure 5.32 Frequency domain measured response of A-lO Thunderbird at 45° off head-on obtained via FFT. 204 1.2 1.0 .0 co Llllllllllllllllllllllll1111111111llllllllllljllllllllllll] Relative amplitude p o .0 N 0.0 IIITTIIITTrTIleITUIIIIIITTTIIIIIIIIIIT] 0.0 0.5 1.0 1.5 2.0 Frequency (CH2) Figure 5.33 The actual A-lO Thunderbird response in frequency domain after deconvolution. 205 1 i 0.50 1 g t a) 1 E n '0 I 3 " l .4: - , n o. 1 ' E _, . O 0.00 T. Q) q > -l 2.: j . o - l T) : it -0.50 — 3 Rectangular window 1 ------ Caussran cosune wundow —1.00qIIUIITIIIIIIIIIIITITITTjIfiIIT] 0.0 10.0 20.0 30.0 Time (ns) Figure 5.34 Actual response of A—lO Thunderbird at 45° off head-on obtained via inverse FFT with rectangular and guassian cosine modulated weighting function. 206 0.6 0.5 .0 4:. Relative amplitude .0 —D 0.0 IIIITTTITl]IIITITIIIIIIIIITTII[IIIITIIII] 0.0 0.5 1.0 1.5 2.0 Frequency (CH2) Figure 5.35 Gaussian cosine modulated weighting function. 207 transmitted pulse hitting targets is determined by applying inverse FFT on the guassian cosine modulated weighting function. Figure 5.36 shows the effective transmitted pulse for the guassian cosine modulated weighting function. The duration of the effective transmitted pulse is approximately between 2.2 ns and 2.5 ns. 5.10 Conclusion The analysis of the travelling-wave antenna presents a time-domain theory for the transient fields produced by a general traveling-wave current on a straight wire antenna. A closed-form solution for a rectangular-pulse current waveform is revealed; numerical results from this permit the investigation of the differences between near vs. far zones and coated vs. uncoated wires. Experimental findings for both coated and uncoated wires show good agreement with theory. The results are also used to reach an expression for the far-zone field of a general curved wire. The travelling antennas are not used inside the chamber since it requires a forward scattering setup. If a travelling antennas are used as a receiver, it will not have a ”quiet period” . The deconvolution procedure cannot be applied on a forward scattering measure- ment. On the other hand, the horn antennas system, as a transmitting and a receiving antenna, allows a bistatic measurement. The deconvolution procedure can be performed on measured responses of complex targets to obtain the actual target responses by removing the system response. The successful application of the automated E and S pulses techniques to thin wires and complex targets measurements of the horn/ horn antenna system will be given in chapter 6. 208 1.20 ., 0.70- Q) —1 '0 I 3 .. n: . O— I O 0.20: fi 0) -1 ._>. I 4" -l g s 0’ I C11—0310- 1 T _0080 IIITIIIII'UrTUUIITIIIIIIIIIIIIIITIITITTIUIII’IIITn 5.0 7.0 9.0 1 1.0 13.0 15.0 Time (ns) Figure 5.36 The effective transmitted pulse hitting targets which is obtained by taking inverse FFT of guassian cosine modulated weighting function. 209 Chapter 6 Experimental Results 6.1 Introduction Chapters 2 and 3 verified the performance of the E and S pulses techniques using synthetic data sets and numerical thin wire data sets. In this chapter, the application of the E and S pulses techniques is tested using measurements obtained inside the anechoic chamber. The antenna system used for the measurements is two wide-band horns which act as receiver and transmitter. This chapter will verify some of the conclusions made in chapters 2-5. 6.2 Explanation of notations BS Broadside. HO Head-on. 8SIEP 8.5" (target name) E—pulse. 8SISP1 8.5 " (target name) first dominant mode quadrature S-pulse. 8518P2 8.5 " (target name) second dominant mode quadrature S-pulse. F15EP F-15 Eagle (target name) E-pulse. FISSPl F-15 (target name) first dominant mode quadrature S-pulse. FlSSP2 F-15 (target name) second dominant mode quadrature S-pulse. 210 This notations remain the same throughout this chapter. 6.3 Thin wire targets The responses of thin wires, whose lengths ranged from 6.5" to 8.5" in an increment of 0.5”, and whose diameter was 0.03125”, were obtained in time domain inside the anechoic chamber at broadside (BS), 22.5° and 45° off BS. The actual responses, i.e. target responses after removing system response from the measurements, of all the thin wires were obtained in the time domain using deconvolution, as explained in chapter 5. E-pulses of all of the thin wires are created using both the unconstrained E-pulse technique (UCET) and constrained E-pulse technique (CET) (for more detail refer to chapter 4). In both cases, while creating E—pulses, only the measured responses of 22.5° and 45° off BS were used. The BS measurement was used for a independent check during the discrimination procedure. The natural frequencies of each target were also obtained as a by product of these algorithms. Q"’-S-pulses of the thin wires were created using the natural frequencies obtained from both techniques. Figure 6.1 and Figure 6.2 show the EDR values of convolutions corresponding to the E-pulses obtained via the UCET and CET with the measured responses of the 8" wire at BS, 22.50 and 45° off BS. The discrimination scheme using the UCET or the CET E-pulses identifies the 8" wire as the correct target, as expected. The next closest targets to the correct target are either the 8.5 " or 7.5” thin wire. Figure 6.7 and Figure 6.8 show the EDR values of convolutions corresponding to the E-pulses obtained via the UCET and CET with the measured responses of the 7" 211 wire at BS, 22.50 and 45° off BS. The discrimination scheme using the UCET or the CET E-pulses identifies the 7 " wire as the correct target, as expected. The next closest targets to the correct target are either the 7 .5 " or 8" thin wire. However, the 6.5” thin wire should have been the next closest target instead of the 8" thin wire. The error may have been due to inaccurate estimation of the second mode of the 6.5" thin wire. The valid bandwidth of the time domain measurements is approximately 1.4 GHz (0.4-1.8 GHz). A theoretical estimation of the imaginary part of the second mode of the 6.5" thin wire is 1.7 GHz. The weighting function used, while transforming the deconvolved frequency domain response into the time domain response, de-emphasizes the contributions from the lower and upper end of the frequency components. Therefore, it is difficult to extract the second mode of the 6.5” thin wire. Figures 6.3 through 6.6 show the SDR values of convolutions corresponding to all of the first and second mode Q-S-pulses obtained via the UCET and the CET with the measured responses of the 8" wire at BS, 22.5° and 450 off BS. The discrimination scheme using the UCET and the CET Q"2-S-pulses identifies the 8" wire as the correct target, as expected. The next closest targets to the correct target are either the 8.5” or 7.5 " thin wire. Figures 6.9 through 6.12 show the SDR values of convolutions corresponding to all of the Qm-S-pulses obtained via the UCET and CET with the measured responses of the 7" wire at B8, 22.5° and 45° off BS. The discrimination scheme using the UCET and CET Q"2-S-pulses identifies the 7 " wire as the correct target, as expected. The next closest targets to the correct target are either the 7.5" or 6.5" thin wire. 212 30.0 25.0 : 35°: 81:... .1 0 d W 45° off as 65 - .0 15.0 ': v _ J a q L A O 10 0 - u q 5.0 - I may: 85IEP _ um 8|EP co .1 * * 4* 2222275183 . 7IEP . 651EP I -5.0 T—IIIIIIIIIIITWIIFIIIITIIIIjIIIIIIIIIIIII 0 1 2 3 4 Target Position Figure 6.1 EDR values of convolutions corresponding to E-pulses obtained via UCET with measured responses of 8" wire at broadside, and 22.5° & 45° off broadside. 213 30.0 25.0- 20-0 j 1 Broogside q 2 2205 offBS - 3 45 offBS EB ‘ " U130: v .. a: 2 010.0- LrJ - 5.0— 0.0: t t 1% fi 1 -500 ITIITIjIITITFIUrrfilrIIIIIIUIIIUTUTIIIII 0 1 2 3 4 Target Position Figure 6.2 EDR values of convolutions corresponding to E-pulses obtained via CET with measured responses of 8" wire at broadside and 22.5° & 45° off broadside. 214 30.0 25.0 - 20-0 j 1 Broa side _ 2 2205 off BS _ 3 45 off 85 ES 1 .0 15.0 -: V .1 g I 10.0 .1 (f) .1 2 5.0 a i may 851$P1 . malsm 0.0 - i a. .1, 22323751391 8 6361131 I First mode S-pulse -5.0 TUI‘UIIUUIIIUIIerIIIIIIIIIUUITIIUTUUTII 0 1 2 3 4 Target Position Figure 6.3 SDR values of convolutions corresponding to Q‘-S-pulses obtained via UCET with measured responses of 8" wire at broadside and 22.5° & 45° off broadside. 215 30.0 I 25.0 - I 6* 20.0 : 1 Broagside _ 2 2205 off BS _ 3 45 off 85 EB - .0150 j v . CD: 2 10.0 .. m . 5.0 - I WBSISPl .. “ii-#815131 0.0 - 'k a t I First mode S-pulse -500 fiWITIrrIIIllTjIllllrllllillllrtflIlUITI 0 1 2 3 4 Target Position Figure 6.4 SDR values of convolutions corresponding to Q‘-S-pulses obtained via CET with measured responses of 8" wire at broadside and 22.5° & 45° off broadside. 216 30.0 q .. 1 25.0 a 20.0 ': 1 Broaslside . 2 2205 off as .. 3 45 off 85 A _ % 150 j v _‘ 0: 1 010.0 n U) -1 5.0 - I wasnspz " mglsslggz . * mi * °° '3 21.5.23. I Second mode S-pulse -5o0 IIIIIIITIIIIIIllIIIIIIIIIIUIUIIIIIIITTIj 0 1 2 3 4 Target Position Figure 6.5 SDR values of convolutions corresponding to Qz-S-pulses obtained via UCET with measured responses of 8" wire at broadside and 22.5° & 45° off broadside. 217 30.0 25.0 -1 .1 20.0 :1 1 Broa side g 2 221.5 off 85 . 3 45 off BS 66 1 .0150: v .1 g i 1 . T (n 00 j 4!" 1 5.0 - I wasnspz 0.0 :1 t 11. a 3 Second mode S-pulse -500 rTUIIIlIIIIIIIIIIIIIIIIIUIIIUIrfifiII'll] 0 1 2 3 4 Target Position Figure 6.6 SDR values of convolutions corresponding to Qz-S-pulses obtained via CET with measured responses of 8" wire at broadside and 22.5° & 45° off broadside. 218 25.0 20.0 oagside 15.0 22205610“ BS 45 off BS A CD '0 v 10.0 c: O LiJ 5.0 MSEEP 0'0 R R A W7SIEP @759 651EP .500'I'"111'IIf'IIIII'IljiilI'll'll'IlTrTjjj 0 1 2 3 4 Target Position Figure 6.7 EDR values of convolutions corresponding to E-pulses obtained via UCET with measured responses of 7 " wire at broadside and 22.5° & 45° off broadside. 219 q j :1 __.b 20.0-j : it : 13£°§"‘it°es :. o 15.0 : :5 45° off as A : C0 : ‘O . v : 10.0: 0f- : D : DJ : 5.0-3 0.0;. A R R E _5.0-IIIITIIII]!lTrIIIITIITIIIIFTfIIIITUIIIUI 0 1 2 3 4 Target Position Figure 6.8 EDR values of convolutions corresponding to E—pulses obtained via CET with measured responses of 7 " wire at broadside and 22.5° & 45° off broadside. 220 1 20.0 - . 1 Broaslside 15.0 - 2 2205 off BS .. 3 45 off BS A m '1 ‘0 ‘ ‘A‘ "3 v "1 N 10.0 - [r .- 0 - (f) - 5.0 - 3 mam 211211 0.0 I R :61 ‘5 W75|SP1 mass. - . 1 - First mode S-pulse -5o0 TWIIIHTIU]IIIIIITIIIUIIWTTFTIIWIIIUTjII] 0 1 2 :5 4 Target Position Figure 6.9 SDR values of convolutions corresponding to Q‘-S-pulses obtained via UCET with measured responses of 7" wire at broadside and 22.50 & 45° off broadside. 221 25.0 20.0 Broa gside 15.0 220551 eff BS 45 off 85 A CD '0 v 10.0 0: D (f) WBSISPl 5.0 WBISPl 0.0 R R R First mode S-pulse _5o0 [WIITTTTIIITIIIIIFFIj—IlllTIIIFIVTIIITTIII 0 1 2 3 4 Target Position Figure 6.10 SDR values of convolutions corresponding to Q‘-S-pulses obtained via CET with measured responses of 7" wire at broadside and 22.5° & 45° off broadside. 222 20.0 : 1 15.0 1 Z 1 Broa side : 2 2205 off as : 3 45 off BS 810.0 ': '0 I v _ O: I D I (f) 5.0 ‘- .: M 85|SP2 0'0 - R R mslspz Z W756i? 3 @2323. 3 Second mode S—pulse _500.jUIIIIUHTIUlll'llIIIITIUerrI'lIU'IIIIII O 1 2 4 Target Position Figure 6.11 SDR values of convolutions corresponding to Qz-S-pulses obtained via UCET with measured responses of 7 " wire at broadside and 22.5° & 45° off broadside. 223 20.0 15.0 1 Broagside 2 2205 offBS :5 45 ones 810.0 ‘0 V 0C 0 (I) 5.0 0.0 R g, g WBSISPZ Second mode S-pulse “5.0 llIIIIIIIIIIIIIITTIIIIIIIIIIIIIIItlitrtll 0 1 2 3 4 Target Position Figure 6.12 SDR values of convolutions corresponding to Qz-S-pulses obtained via CET with measured responses of 7" wire at broadside and 22.5° & 45° off broadside. 224 The DL values corresponding to the Q2-S-pulses are not large enough for around or at BS measurements because the second mode is not excited strongly in these aspect angles. However, the DL values corresponding to Qz-S-pulses are large enough to have successful discrimination at other aspect angles. For the example considered, the DL values corresponding to both the UCET and the CET Q‘-S-pulses are large compare to the DL values corresponding to both the UCET and the CET E-pulses. The DL values corresponding to the CET E-pulses are always large compared to the DL values corresponding to the UCET E-pulses. Also, the DL values corresponding to the CET Q‘-S-pulses are always large compared to the DL values corresponding to the UCET Q‘-S-pulses. The reason for this observation is that the natural frequencies obtained via the CET is better than the UCET. This is very evident in chapter 4. It is important to note that the responses of the 8" and 7" wires at BS, which are the independent check, identify the 8" and 7" wire while discriminating targets using both the E and S pulses techniques. This illustrates that in order to obtain a complete set of natural frequencies of complex targets within the experimental bandwidth, it is not necessary to use many aspect angle measurements. 6.4 Scale model targets Scale model targets used to demonstrate discrimination are an F15 Eagle (7.25" in length), an A10 Thunderbird (6.75"), and a Boeing 747 (6.0"). The responses of each of these targets were measured at HO, 22.5°, 45°, 67.5°, and 90° off H0. The system response was removed from the measurements using deconvolution, as explained in 225 chapter 5. The E-pulses of the targets were created using both the UCET and CET techniques. In both cases, while creating the E-pulses, only the target measurements of HO, 45° and 90° off HO were used. The other target measurements were used for an independent check during the discrimination procedure. The extracted natural frequencies obtained via the UCET and CET are given in Table 6.1. All the values given in Table 6.1 are needed to be multiplied by 10E+9 in order to obtain the actual values. The radian frequency of the dominant mode for the F15, A10, and B747 is approximately the same, however the damping coefficient differ significantly. The extracted CET and UCET radian frequencies of the F15, A10, and B747 do not vary drastically except the lower order radian frequencies. However, the extracted CET and UCET damping coefficients of the F15, A10, and B747 vary drastically. The UCET and CET Q"2-S- pulses of were also created using the natural frequencies given in Table 6.1. Figures 6.13 through 6.15 shows the UCET and the CET E-pulses of the F15, A10, and B747, respectively. Both the UCET and CET E—pulses are normalized such that the E-pulse energy is equal to one. In general the duration of the forced E-pulses is approximately around the duration of the natural E-pulse. The natural E-pulse duration is given by (4.34), and it only depends on the highest radian frequency of the extracted natural modes. The highest radian frequency of each target is approximately the same for both techniques. Therefore, the duration of both the UCET and CET E—pulses of the F15, A10, and B747 are approximately the same. The wave shapes of both the UCET and CET E-pulses are quite different. It is important to note that the UCET E-pulse is a DC E-pulse, while the CET E-pulse is not a DC E-pulse. The constraints placed on the amplitudes of the E-pulse do not permit to create a DC CET E—pulse. The UCET natural 226 Extracted natural frequencies of F-lS, A-10 and B-747 obtained via UCET and CET. Table 6.1 Sarag- osmium.- Saremc www.38- mm.m..+m:.- Bart-e.- m§+§~s agree.- mearmec $.32; Serge- 8.383 mnmrgx Sarge 2.238.- ...Nnran- 8.0192.- zero?- ee.e_+§.- 3.4.3.80- Rfiisc Edi-mm.- oSTmen- native.- seremec 8.3+ we..- 8.»? 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EB B-m 988 “E ‘ 227 1.0 - 0.5 - "I—r— a) - E w, ‘o 3 :1 3 _ " .‘L'.’ - -- E. : --- L--.’ ‘ E .- O 00 :_ ————————————————— a) -4 > —I '43 I 2 - a) j - CK -O.5 -_ ---= I : —— UCET F15 E-pulse :___: ------ CET F15 E—pulse —1.0 Ililllllllll'llllllljiiilllIIITIIIIIIUIIIITIIIIII] 0.0 1.0 2.0 3.0 4.0 5.0 Time (ns) Figure 6.13 UCET and CET E—pulse of F-lS Eagle. 228 0.9 -7 I 3 fl 0) 0.4 -: ---. .0 "‘ : . 3 ‘ --- "0 ---I i .4: 1 ' : l : .m E : E E : "'1' E E —————————————— :---%4 -------- 4———- 0-01E —I E E 5 a) - : : > : i i : 2; q "' : : '-__! 2 ‘ l : : 3‘; ~ . . i = -06 2. -- a 1 i : : a : "'m‘ — UCET A10 E-pulse : g ------ CET A10 E—pulse —1.1 IIIIIIIII[IIIIIIITWIIIIIIIIITIIIIIITITIIIIIIIIIIII 0.0 1.0 2.0 3.0 4.0 5.0 Time (ns) Figure 6.14 UCET and CET E-pulse of A-lO Thunderbird. 229 1.0 3 05 : —l—‘_ a) + min-3 E '0 i : : 3 _ I"' | “c1 : E E ‘ i 5 O 00 j ———————— E ———————————— - a) - : .2 f. : E 3 m Ill 05 —o 51 j 3 — UCET B747 E-pulse ; ------ CET B747 E-pulse ‘1.0 IIIIIIIIIIIIIIIIfiI[IIIIIIIFIIIIIIIITIIIIIIIIIIII1 0.0 1.0 2.0 3.0 4.0 5.0 Time (ns) Figure 6.15 UCET and CET E—pulse of Boeing 747. 230 frequencies are different from the CET natural frequencies, especially the damping coefficients, and this would definitely cause a difference in the E—pulses. The squared error per point (SEPP) of the targets is given in Table 6.2. The UCET SEPP is always large compared to the CET SEPP. This phenomenon is most apparent in the case of the F15. The SEPP using the CET is small since it does not try to fit the latter part of the responses because the damping coefficients are restricted to negative values. Therefore, the CET fits the responses well in the earlier part of the late- time where most of the late-time energy is present. The UCET attempts to fit the whole Table 6.2 Squared error per point (SEPP) of F15, A10, and B747 obtained via the UCET and CET. SEPP (UCET) F15 1.29E-3 [I A10 5.045—4 fl B747 5.305-4 response including the latter part of the response because the damping coefficients are not restricted. Therefore, in the process trying to fit the whole waveform, it does not fit the earlier part of the late-time well, and it produces a large SEPP. The ratio of the late—time energy to the total energy (R,,) is defined as T fr’(t)dt R1. = :~__ (6.1) U fr2(t)dt 0 231 Table 6.3 The ratio of late-time energy to total energy of F15, A10, and B747. I Head-on 22.50 off 45" off H0 675" off 90° off (HO) HO HO HO F15 0.055 0.036 0.024 0.011 0.014 A10 0.004 0.254 0.237 0.225 0.040 —B747 0.034 0.064 0.151 0.126 0.016 The Rh of the F15, A10, and B747 is given in Table 6.3. The Rh for H0 excitation of the F15 is the largest compared to other aspect angles, however the R, for all aspect angles considered for the F15 do not vary drastically. The Rh for 22.5° off HO excitation of the A10 is the largest compared to other aspect angles. The R, for excitation at 45° and 675° off HO for the A10 is in the same range as for the A10 at 22.5° off HO excitation. The Ru corresponding to H0 of A10 is very low compared to other aspect angles. The Ru values for excitation at 45° and 67.5° off HO of the B747 are approxi- mately the same. The Rh for excitation at other aspect angles for the B747 is low. However, they do not differ from the values corresponding to 45° or 67.5° off HO by a large amount. It will be studied how the Ru values will have an effect on discrimination. Figure 6.16 and Figure 6.17 show the EDR values of convolutions corresponding to the E-pulses obtained via the UCET and CET with the measured responses of F15 at HO, 22.5°, 45°, 67.5°, and 90° off H0. The discrimination scheme using the UCET or CET E—pulses identifies the F15 as the correct target, as expected. The DL values corresponding to both the UCET and CET E—pulses are approximately the same. 232 25.0 : 1 Heo -on « 3 {3‘16” 20-0 ‘4 4 6705‘? off HO ‘ 5 90 off H0 15.0 - A : m d .0 v " 10.0 - m .- Q . LIJ .- 5.0 - .J q 0.0 -— 9.: a e # II: - WHSEP a WAlOlEP 4 W8747EP —S.O rill]IIlllllllllllllliTrjfiIIl] 0 1 2 3 4 5 6 Target Position Figure 6.16 EDR values of convolutions corresponding to E-pulses obtained via UCET with F-15 Eagle responses measured at various aspect angles. 233 25.0 q : 1 Heo -on - 3 -?“-5'° 20-0 r 4 6705‘? off HO 4 5 90 off HO 15.0 a -i A .- m -I .0 v " 10.0 - Q: .- D a LL] .. 5.0 — 0.0 - a a s; * a J WHSEP - WMOIEP - W8747EP ’5.0 IIIIIIIIrI—rilfilllxl[liitltlifi] o 1 2 3 4 5 6 Target Position Figure 6.17 EDR values of convolutions corresponding to E-pulses obtained via CET with F-lS Eagle responses measured at various aspect angles. 234 Figures 6.18 through 6.21 show the SDR values of convolutions corresponding to the Q"’-S-pulses obtained via the UCET and the CET with the measured responses of the F15 at H0, 22.5°, 45°, 67.5°, and 90° off H0. The discrimination scheme using the UCET and the CET Qm-S-pulses identifies the F15 as the correct target, as expected. The DL values corresponding to the CET Q‘-S-pulses are the same or larger compared to the DL values corresponding to the UCET Q‘-S-pulses. The DL values corresponding to the UCET Qz-S-pulses are not large enough (DL value is chosen to be above 10 dB for clear discrimination, and DL is chosen to be 5 dB for nominal discrimination) to identify the F15 with confidence at HO and 22.5° off H0. The DL values corresponding to the CET QZ-S-pulses are not large enough to identify the F15 with confidence at HO and 67.5° off H0. The failure of the Qz-S-pulse discrimination at these aspect angles may be due to the lack of strong excitation of the mode used to create Qz-S-pulse for the F15. It is also important to note that the responses of the F15 at 22.5° and 67.50 off HO, which are the independent check, identify the F15 while discriminating using both the E and S pulses techniques. Figure 6.22 and Figure 6.23 show the EDR values of convolutions corresponding to the E-pulses obtained via the UCET and the CET with the measured responses of the A10 at HO, 22.5°, 45°, 67.5°, and 90° off H0. The discrimination scheme using the UCET or the CET E-pulses identifies the A10 as the correct target, as expected. The DL values corresponding to the CET E-pulses are considerably higher than the DL values corresponding to the UCET E-pulses except at H0. The E-pulse discrimination failed at HO for both the UCET and CET E—pulses. The failure is due to negligible amount of late-time energy (Ru=0.004). 235 25.0 1 Heo -on 3 3.3505 -?‘i-6*° 20-0 4 67°56) off HO 5 90 off HO 15.0 A U] '0 v 10.0 C! O (f) 5.0 lllllllLLiJJLllJllllJLllllJlll (10 $ # * *———————* alt-Hafiz F15$P1 WAlOISPl W8747SP1 -5.0 I I I I I I I I I I I I IjT I I I I I I I I I l I I I I I O 1 2 3 4 5 6 Target Position Figure 6.18 SDR values of convolutions corresponding to Q‘-S-pulses obtained via UCET with F-15 responses measured at various aspect angles. 236 25$) : 1 Heo -on ~ g -?"--':° 20-0 - 4 6705‘? off HO 4 5 90 off HO 15.0 — A 2 m .0 - v “ 10.0 - Q: .. Q .- m 4 5.0 - 1 0.0 - 4 4 4 4 4 - 444444-1591 - WAlOlSPl - W8747SP1 ’5.0 IIIIIIIII]IIII]I1|I|TIII|IIII| o 1 2 3 4 5 6 Target Position Figure 6.19 SDR values of convolutions corresponding to Q‘-S-pulses obtained via CET with F-15 responses measured at various aspect angles. 237 15.0 10.0 Heo -on 2205 off H0 45 (off H0 6705 off H0 90 off H0 U'I-hUN-e 5.0 SDR (dB) 0.0 as as at i- l. W F15$P2 W A1 01 SP2 W B747SP2 -S.O IIII]!IIIIIIIIIITITIIIIIIIIII] o 1 2 3 4 5 6 Target Position 111111111111111111111111111111111111111 Figure 6.20 SDR values of convolutions corresponding to Qz-S-pulses obtained via UCET with F-15 responses measured at various aspect angles. 238 15.0 a 1 Heacg-on 1 2 2205 off HO .. 3 45 (off HO d 4 6705 off HO - 5 90 off H0 1 10.0 —' A Z (I) - '0 - V .- 5.0 - 0: I Q .. (f) .- o.o 3 4 4 4 4 4 I : WHSSPZ g WA101SP2 4 W8747SP2 -5.0 IITIIIIIIIIIIIIIIII[IIIIIITII] O 1 2 3 4 5 6 Target Position Figure 6.21 SDR values of convolutions corresponding to Qz-S-pulses obtained via CET with F-15 responses measured at various aspect angles. 239 15.0 : 1 Heo -on _ 2 2205 off HO - 3 45 (off HO ‘ 4 6705 off HO : 5 90 off H0 10.0 3 A : CD . '0 .- v - 5.0 - DC I O . LrJ . I 0.0 : $.- t t at t 'i ‘ WHSEP : WMOIEP .. W8747EP “5.0 IIIWIWITI‘IIIIIIITIIIIfIIIIIIII 0 1 2 3 4 5 6 Target Position Figure 6.22 EDR values of convolutions corresponding to E—pulses obtained via UCET with A-lO Thunderbird responses measured at various aspect angles. 240 15.0 : 1 Heo -on - 2 2205 off HO - 3 45 (off HO - 4 6705 off HO 1 5 90 off HO 10.0 :- A : CD 4 ‘0 - v .. 5.0 - 01 j C) - LlJ _ 0.0 :- t it an: t v: 1 '1 WHSEP : WA1015P .. W8747EP '5.0 I I I I I T 7 I I I rj I I I 1 T I I l I I T I l I I I I 1 O 1 2 3 4 5 6 Target Position Figure 6.23 EDR values of convolutions corresponding to E-pulses obtained via CET with A-10 Thunderbird responses measured at various aspect angles. 241 Figures 6.24 through 6.27 show the SDR values of convolutions corresponding to the Qm-S-pulses obtained via the UCET and CET with the measured responses of A10 at HO, 22.5°, 45°, 67.5°, and 90° off H0. The discrimination scheme using the UCET and CET Q"2-S-pu1ses identifies the A10 as the correct target, except for the Qz-S-pulses at H0, since the mode used to create QZ-S-pulse is not excited strongly at H0. The DL values corresponding to the CET Q‘-S-pulses are large compared to the DL values corresponding to the UCET Q‘-S-pulses. The DL values corresponding to the UCET Qz-S-pulses are not large enough to identify the A10 with confidence at all aspects. The DL values corresponding to the CET Qz-S-pulses are not large enough to identify the A10 with confidence at HO and 45° off H0. The failure of the Qz-S-pulse discrimination at these aspect angles may be due to lack of strong excitation or inaccurate estimation of the mode used to create the Qz-S-pulse of the A10. It is also important to note that the responses of the A10 at 22.5° and 67 .5° off HO , which are the independent check, identify the A10 while discriminating using both the E and S pulses techniques. Figure 6.28 and Figure 6.29 show the EDR values of convolutions corresponding to the E-pulses obtained via the UCET and CET with the measured responses of the B747 at HO, 22.5°, 45°, 67.5°, and 90° off H0. The discrimination scheme using both the UCET and CET E-pulses identifies the B747 as the correct target, as expected. The DL values corresponding to the CET E—pulses are large compared to the DL values corresponding to the UCET E—pulses. Figures 6.30 through 6.33 show the SDR values of convolutions corresponding to the Q‘°2-S-pulses obtained via the UCET and CET with the measured responses of the B747 at HO, 22.5°, 45°, 67.5°, and 90° off H0. The discrimination scheme using both 242 254) : 1 Heo -on - § 443° 20-0 a 4 6705‘? off HO " 5 90 off H0 15.0 4 A j ca .0 - V - 10.0 -+ 0: - Q a (f) .- s 5.0 - 0.0 - t t t t t - WF1SSP1 - map-10191 - W8747SP1 -5.0 IIIIIIIIIIIIIIIIIIIIIIIIIIFTI] o 1 2 3 4 5 6 Target Position Figure 6.24 SDR values of convolutions corresponding to Q'-S-pulses obtained via UCET with A-lO responses measured at various aspect angles. 243 25.0 20.0 - I 15.0 -1 A d m '1 .0 . V '1 10.0 - g - U) : 1 Heo -on . 2 2205 off H0 50 q 3 45 eff HO 1 4 6705 off H0 1 5 90 off H0 .. 0.0 - t t t 1k t ‘ WF1SSP1 d WA101SP1 q W8747SP1 -5.0 llIllfijIIIITII[IIIFIIIIIIIIIII O 1 2 3 4 5 6 Target Position Figure 6.25 SDR values of convolutions corresponding to Q‘-S-pulses obtained via CET with A-lO responses measured at various aspect angles. 244 20.0 7. I 1 Heo -on - 2 2205 off HO = 3 23.410. - 4 of H 15.0 : 5 90° off H0 310.0 f '0 I v d 01 I O 1 U) 5.0 :1 I 0.0 E I? 4 t 4 t I j W F15SP2 .- WAlOlSPZ ; W8747SP2 "5.0 I I I I I I I I 1 I T I I I I I I I I I I I I I I I I I Ii 0 1 2 3 4 5 6 Target Position Figure 6.26 SDR values of convolutions corresponding to Qz-S-pulses obtained via UCET with A-10 responses measured at various aspect angles. 245 20.0 1 1 1 1 15.0 -: 310.0 E '0 2 v _ m 1 e 1 - .- m 5'0 4 2 zzosq’ off HO : 3 45 8" H0 - 4 6705 off HO : 5 90 off HO 0.0 f I 4 4 .- 4. 3 4444-1: F1SSP2 .. WMOISPZ ; 4566648747392 -5.0 I I I I I I I I I I I I I j I I I I I I I I I I I I I I I I o 1 2 3 4 5 6 Target Position Figure 6.27 SDR values of convolutions corresponding to Qz-S-pulses obtained via CET with A-lO responses measured at various aspect angles. 246 25.0 20.0 15.0 10.0 EDR (dB) 5.0 0.0 —5.0 11111111111111111111Lttllltiil Heo -on 2205 off H0 45 8” H0 6705 off H0 90 off HO Ul-hUN—h A) A W F15EP W A101 EP W B747EP *IjII IIrI O 1 I IIFIIIITIII 2 3 4 Target Position 1111] 5 IIIII 6 Figure 6.28 EDR values of convolutions corresponding to E—pulses obtained via UCET with Boeing-747 responses measured at various aspect angles. 247 25.0 : 20.0 — 15.0 - A j m —1 .0 v d 10.0 d 0: . Q - L1J _ - 1 Heo -on 5.0 - 2 2205 off HO - 3 45 eff HO _ 4 6705 off H0 ~ 5 90 off HO 0.0 .4 $4“ $4 5: 54" 4: n -* WHSEP - WMOIEP - W8747EP -5.0 ITIIIIIIIIIfiIIITIIIIIIIIIIITI] 0 1 2 3 4 5 6 Target Position Figure 6.29 EDR values of convolutions corresponding to E-pulses obtained via CET with Boeing-747 responses measured at various aspect angles. 248 25.0 2 .4 3 d 1 Heo -on - 3 31-30 20-0 j 4 6705‘? off HO 5 90 off H0 .1 15.0 a A 3 a3 .0 - v "' 10.0 - Q: —- Q .- (f) .4 5.0 ~ 0.0 — 34 34 3: 44 3: ~ 4444417155131 a WAlOlSPl . W8747SP1 -5.0 [71IIIIIIIIIIIITTTIIIIITIIIII] o 1 2 3 4 5 6 Target Position Figure 6.30 SDR values of convolutions corresponding to Q‘-S-pulses obtained via UCET with B-747 responses measured at various aspect angles. 249 25.0 1 1 Heo —on 4 3 31-30 20.0 4 4 6705‘? off HO ‘ 5 90 off HO .. 15.0 - A : an .0 .. v " 10.0 - m .. 0 CI! (f) -1 5.0 - 0.0 - 6% ‘4‘ ‘4‘ 6% 6: d WF1SSP1 - WA101SP1 - W8747$P1 -5.0 IIIIIIIIIIIIITIIITIjiIIIIIITIII O 1 2 3 4 5 6 Target Position Figure 6.31 SDR values of convolutions corresponding to Q‘-S-pulses obtained via CET with B-747 responses measured at various aspect angles. 250 20.0 15.0 —' I $10.0 E '0 : v g 0: 3 1 Heo -on D : 2 2205 off HO (f) 5.0 -1 3 45 6)” H0 1 4 6705 off HO 3 5 90 off HO 0.0 _: A: [Iran A: AA AA 3 444-141715st - mmmspz 3 W8747SP2 -5.0 IIIfifIIIIIIITIrIIIIIIIIIrIfIII o 1 2 3 4 5 6 Target Position Figure 6.32 SDR values of convolutions corresponding to Qz-S-pulses obtained via UCET with B-747 responses measured at various aspect angles. 251 20.0 : 1 Heocg-on : 2 2205 off HO - 3 45 pff HO 2 4 6705 off HO : 5 90 off HO 15.0 — 3 1 310.0 f '0 3 v _ O: I D 2 (f) 5.0 -_‘ 0.0 E 5: 1;: 5,: 4;: A; E W F15SP2 . mmmspz 1 W8747SP2 -5.0 l IfiIITfI llljiilljfiTTII [Illillj O 1 2 3 4 5 6 Target Position Figure 6.33 SDR values of convolutions corresponding to Qz-S-pulses obtained via CET with B-747 responses measured at various aspect angles. 252 the UCET and CET Q"2-S-pulses identifies the B747 as the correct target, except for the UCET Q‘-S-pulses at HO since the mode used to create Q‘-S-pulse is either not extracted with high accuracy or not excited strongly. The DL values corresponding to CET Q"2-S- pulses are the same or larger than the DL values corresponding to the UCET Q"2-S- pulses. Table 6.4 gives the 'DL values corresponding to the convolutions of F 15 , A10, and B747 responses with the CET E-pulses. From Table 6.3 and Table 6.4, it is clear that the discrimination does not work well when the Ru value is low. For example, the DL value for A10 at H0 is 1.50 dB (Ru=0.004) while the DL value is 7.50 dB for A10 at 22.5° off HO (R,,=0.254). It can be concluded that S-pulse discrimination works better than E-pulse discrimination. The above claim can be clearly demonstrated by considering the response of A10 at H0. The DL value corresponding to the UCET or the CET E—pulses is approximately 1.5 dB. The DL value corresponding to the UCET 0r CET Q‘-S-pulses is either 12 dB or 21 dB. The main drawback of the S-pulse scheme is that the mode Table 6.4 The DL values corresponding to CET E-pulses of F15, A10, and B747. Head-0n 22.5° off 45° off HO 67.50 off 90° off (HO) (dB) JIii-(1B) (dB) HO:(dB) HO (dB) F15 12.0 7.50 8.50 7.75 5.50 r A10 1.50 7.50 7.00 7.25 4.00 I B747 8.00 6.00 7.50 8.00 10.0 253 used to create the Q-S-pulse must be excited in an unknown response. If the aspect angle information is known, then the dominant mode Q-S-pulse at that aspect angle can be used to have successful discrimination. 6.5 The effect of window duration in discrimination If the duration W,,, given by (2.11), is too small, then the discrimination based on this duration may not give reliable results. As explained in chapter 5, the duration W, is restricted by many parameters. So, sometimes the discrimination may have to be performed on a smaller duration W,,. Therefore, a study has been done to examine how the effect of the duration on the DL values. Figure 6.34 shows the DL values of convolutions corresponding to the CET E- pulses with the measured responses of the A10 at HO, 22.5°, 45°, 67.5°, and 90° off H0 for different values of W, ranging from 2.0 to 5 .0 nsec. When W, is 2.0 nsec, the DL value is negative at HO while the DL has positive values at the other aspect angles. The E—pulse discrimination scheme did not identify the A10 as the correct target at HO. When W, is between 2.5 and 5 .0 nsec, the DL values do not vary by a large amount at 45°, 67.5°, and 90° off HO. However, there is a large variation in the DL values at 22.5° off HO. It is difficult to make a clear conclusion from this example, except that the E-pulse discrimination scheme fails at HO. Figure 6.35 shows the DL values of convolutions corresponding to the CET Q‘-S- pulses with the measured responses of the B747 at HO, 22.5°, 45°, 67.5°, and 90° off HO for different values of W,, ranging from 2.0 to 5 .0 nsec. The S-pulse discrimination scheme failed at HO for W, = 2.0, 2.5, 3.0, and almost at 3.5 nsec. The S-pulse 254 15.0 ' 1 Heo —on I 2 205% off HO Wrifeienfg _ 3 45 ff HO W,, = 2.5 ‘25 ‘ 4 67°5 °" “0 WW. = 3.0 ' ' 5 ° °"“° swag”. ‘ .. o-e-e-e-ew, = 4.0 - H Wd = 4.5 ' w, = 5.0 10.0 1 8 7.5 — '0 I V S 5.0 — 2.5 — d 0.0 - 1 T T T T —2.5 lllllllllllllllllll|llllllllll Torget Position Figure 6.34 DL values of A-lO Thunderbird measured at various aspect angles for different values of W, with all E-pulses. 255 25.0 20.0 15.0 DL (dB) 5.0 0.0 —10.0 Figure 6.35 Heo -on 45" 6’“ H0 6705 off H0 90 off H0 U'JiblN—e .la 11111111111111111111111111111111111 Wreference WWd = 2.0 = 2.5 = 3.0 = 3.5 = 4.0 = 4.5 = 5.0 l I l l l I l l l I l l I l I l l l I 1 l I l l l l l 6 Torget Position DL values of Boeing-747 measured at various aspect angles for different values of W, with dominant mode S-pulses. 256 discrimination scheme has DL values of 4.0 or 5.0 dB at HO for W, = 4.5 or 5 .0 nsec. The DL values do not change by a large amount at 22.5°, 45°, and 67.5° off HO for W, ranging from 4.0 to 5.0 nsec. It can be concluded that if W, is too small, discrimination may not be successful because the late-time energy of all of the E-pulse convolutions in the duration W, may not be reliable enough to distinguish the correct target among many targets. In the case of the S-pulse, the convolution corresponding to the correct target may not have enough periods of oscillations of the known damped sinusoid. Also, by incorporating a larger window duration W, the error due to inaccurate estimations of modes, using limited number of modes, and noise is suppressed in the case of the S-pulse. When W, is large, it will also have more information about wrong targets. 6.6 Discrimination using only first two dominant modes As mentioned in chapter 2, A simple study will be done to see whether it will be possible to discriminate the scale model targets with dominant modes E or S pulses. The E-pulses of the F15, A-10, and B747 are created only using the first two dominant CET modes. Then, these E-pulses are convolved with the responses of the F15, A-10, and B747 and the corresponding DL values are calculated. Figure 6.36 shows the DL values corresponding to convolutions of responses of the F15, A10, and B747 with the E-pulses created using first two dominant modes. Discrimination based on these results does not work for F15 and B747. However, A10 is identified as the correct target in all aspects with reasonable DL values, except at HO. From this example, it can be concluded that discrimination cannot be done only using the 257 10.00 WHS W A10 W B747 Cm reference 5.00 0.00 DL (dB) -5.00 Heo -on 2205 off H0 45 (off H0 6705 off H0 90 off H0 [11111111111111rrrrrLJrrirrrrrlrrrrirrrrJ O C) Ui-bUN-b O -10.00 O 2 3 4 5 6 Target Position Figure 6.36 DL values corresponding to convolutions of measured responses of F15, A10, and B74 at all aspects with E-pulses created with first two dominant CET modes. 258 dominant modes, as is shown in chapter 2. A similar study can be done using S-pulses, and it can also be concluded that discrimination results will be the same. 6.7 Conclusion The discrimination of F15, A10, and B747 aircraft models are demonstrated using both the UCET and CET E-pulses, and also using both the UCET and CET Q-S-pulses. The DL values corresponding to both the CET E-pulses and Q-S-pulses are primarily large compared to the DL values corresponding to the UCET E-pulses and Q-S-pulses. The natural frequencies obtained via the CBT is much more accurate than the natural frequencies obtained via the UCET (for more details refer to chapter 4). Therefore, the CET E—pulses and Q-S—pulses performs better than the UCET E-pulses and Q—S-pulses. The DL values corresponding to the S-pulses are large compared to the DL values corresponding to the E—pulses. From the example considered, if the window duration W, is too small, the discrimination based on the E-pulse and S-pulse may not be reliable since the information available within that window may not be enough to distinguish the correct target from wrong targets. 259 Chapter 7 Conclusion 7 .1 Summary This dissertation has effectively demonstrated an automated scheme for E and S pulse radar target discrimination. The E and S pulse schemes can be readily used in applications where many targets are considered, and where a computer is used to make the discrimination decisions. Also, a new mode extraction scheme Constrained B-pulse Technique (CET) has been developed. This scheme constrains the extracted damping coefficients to be negative. Poor discrimination will result if the natural frequencies (especially the radian frequencies) are not extracted with high accuracy. The error in the E-pulse convolution depends on small changes in the absolute value of the extracted and actual natural radian frequencies and/ or damping coefficients. In general, for the targets considered in our laboratory, the absolute values of the radian frequencies is at least ten times those of the damping coefficients. However, this may not be true for low Q targets such as the sphere. Numerical analysis based on synthetic data sets supports the above claim. In the case of the S-pulse, the error in the Q‘-S-pulse (1“ mode quadrature S-pulse) convolution also depends on small changes in the absolute value of the extracted and actual radian frequencies and/or damping coefficients, and to the change of the spectral magnitude of 260 the S-pulse evaluated at the 1‘" mode of the extracted and actual natural frequencies. It is also important to note that successful discrimination cannot be achieved using E and S pulses created using only dominant modes. This also confirms the necessity to extract all the natural frequencies with high accuracy. Discrimination is possible for very different targets at low signal-to-noise ratio (SNR). However, discrimination will be very difficult for nearly identical targets at low SNR. The discrimination of wire targets differing by 10 percent is possible with a discrimination level (DL) of 10 dB using E—pulses at SNR ranging from 18 to 23.5 dB for l/a ranging from 800 to 100. The discrimination of wire targets differing by 10 percent is possible with a DL of 10 dB using the Q‘-S-pulses at SNR ranging from 8 to 15 dB for l/a ranging from 800 to 100. These results indicate that the discrimination based on S-pulses is better than that based on E-pulses. The main drawback of the S- pulse scheme is that the mode used to create the S-pulse must be present in the measured response. The natural frequencies obtained using the CET are better than those obtained using the Unconstrained E—pulse Technique (U CET), especially for low SNR responses, as shown in chapter 4. The extracted radian frequencies obtained using the CET are closer to the noise-free values (error is within 5 percent) even when signal-to-noise ratio (SNR) is low as 0 dB. However, the damping coefficients are quite different from the noise-free values, but are always negative due to constraints placed on the amplitudes of the E—pulse. On the other hand, the extracted radian frequencies obtained using the UCET are farther from the noise-free values (error is within 10 percent), and the damping coefficients are often positive. 261 Travelling-wave wire antennas could not be used successfully to demonstrate radar target discrimination. The travelling wave antenna system required a forward scattering set up so that deconvolution could not be applied. When a travelling-wave wire antenna is used as a transmitter, the fields radiated by the reflected current waves at wire ends re—excites the target. The travelling-wave wire antenna receiver does not have a sufficient quiet period. Thus, using a travelling- wave antenna as either receiver or transmitter was abandoned. Two wide-band horn antennas were finally chosen for both time and frequency domain systems, allowing bi-static scattering measurements. The deconvolution procedure is applied to these measured responses to remove the system response. Experience has shown that the discrimination results based on deconvolved responses are better compared to those based on non-deconvolved responses. Discrimination of thin wires whose lengths range from 8.5” to 6.5” has been successfully demonstrated using E and S pulses obtained from both the CET and UCET. Also, discrimination of scale model F-lS Eagle, A-lO Thunderbird, and Boeing 747 aircraft models has been successfully demonstrated using E and S-pulses obtained from both the CET and the UCET. Most of the time, DL values corresponding to the CET E and S pulses are large compared to DL values corresponding to the UCET E and S pulses. This is due to the fact that the CET extracts the natural frequencies more accurately than the UCET. It is observed that the squared error between the original and reconstructed waveforms is always lower for the CET compared to the UCET. The DL values corresponding to the dominant mode Q-S-pulses are always large compared to DL values corresponding to the E-pulses. It is concluded that if the window 262 durations used for calculating the EDR and SDR are too small, discrimination is not reliable. 7.2 Future study Late-time scattered field energy can be very low, as is apparent from the scale model measurements. Thus, It would be wise to combine early-time information with late-time information in a discrimination scheme. However, the early time period is aspect dependent. An investigation has been recently conducted at MSU to combine the early and late-time information. An interesting future topic would be to apply the E and S pulse technique to both early and late-time. Empirical results show that if the forced E-pulse duration is less than the natural E-pulse duration, the resulting E-pulse is highly oscillatory with majority of its energy above the highest extracted radian frequency and poor discrimination results are obtained in the presence random noise. It has not been understood why this phenomenon occurs. A study should be done to explain this phenomenon. The natural frequencies obtained using the CET can be improved by using more s0phiscated minimzation routines. The procedure used in chapter 4 required an input of the constant q,. A feeling for the proper choice for q, can only be obtained with experience. If routines from the NAG and IMSL libraries are used, the quantity q, can be avoided, and only initial guesses for the E—pulse amplitudes would be required. It is also necessary to find a way to create a DC CET E-pulse. This may be achieved by placing a zero at = -1.0E-9+j0.0. 263 Bibliography 10. 11. Bibliography E. 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