LIBRARY Michigan State University PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE AUG M 2621962005 6/01 C'JCIRC/DateDue.p65-p.15 NONDESTRUCTIVE EVALUATION OF LAYERED MATERIALS USING THE E-PULSE TECHNIQUE By Garrett J. Stenholm A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Electrical and Computer Engineering 2002 ABSTRACT NONDESTRUCTIVE EVALUATION OF LAYERED MATERIALS USING THE E-PULSE TECHNIQUE By Garrett J. Stenholm Certain materials can be layered in configurations such that they absorb radiated energy at selected frequencies. The selection of frequencies is dependent on the electrical and conductive properties and thickness of the material layers. During routine use, these properties can be unintentionally changed and the ability to detect these changes is necessary to ensure the material absorbs the energy at the intended frequency. Using the extinction pulse (E-pulse) to evaluate layered materials is an extension of its original use as a method to discriminate radar targets. Application of the E-pulse method to the diagnosis of layered materials is examined for four layered material configurations. These configurations include a single conductor backed layer of lossless material, a single conductor backed layer of lossy material, a single conductor backed layer of magnetic radar absorbing material (MagRAM), and a single conductor backed layer of lossless material covered by an impedance sheet. Copyright © 2002 by Garrett J Stenholm To my wife Michelle and to my children Joseph and 206 for their patience and continued support iv ACKNOWLEDGEMENTS I would like to acknowledge and extend my gratitude to Dr. Edward J. Rothwell for his guidance. I would also like to thank Dr. Dennis P. Nyquist and Dr. Leo C. Kempel for their outstanding instruction. TABLE OF CONTENTS LIST OF TABLES ........................................................................................................... viii LIST OF FIGURES ............................................................................................................. x ABBREVIATIONS .......................................................................................................... xvi INTRODUCTION ............................................................................................................... 1 Chapter 1 ............................................................................................................................. 3 Overview of the E-Pulse Technique .................................................................................... 3 1.1 E-Pulse Technique for Radar Discrimination .............................................................. 3 1.1.1 Time Domain Analysis ......................................................................................... 5 1.1.2 Transform Domain Analysis ................................................................................ 6 1.1.3 Solution to the Integral Equation .......................................................................... 8 1.1.4 Forced and Natural E-pulses ................................................................................ 9 1.2 E-Pulse Technique for Layered Material Evaluation ................................................. 10 1.2.1 Late-time Response as a Sum of Damped Sinusoids ......................................... 11 1.2.2 E-pulse Diagnostics for Layered Materials ........................................................ 19 1.2.3 Demonstration of Principle — E-pulse Construction and Convolution ............... 19 Chapter 2 ........................................................................................................................... 27 Single Lossless Layer ........................................................................................................ 27 2.1 Determination of Natural Frequencies for Non-Conductive Materials ...................... 27 2.2 Computation of the EDNa and EDRa ........................................................................ 29 2.2.1 Change in Layer Thickness ................................................................................ 31 2.2.2 Change in Layer Relative Permittivity ............................................................... 53 2.2.3 Change in Aspect Angle ..................................................................................... 64 2.2.4 Change in the Number of Natural Modes .......................................................... 84 2.3 Effects of Noise on EDNa .......................................................................................... 95 2.3.1 Noise Model ....................................................................................................... 95 2.3.2 EDNa/EDRa vs. Signal—to-Noise Ratio .............................................................. 96 Chapter 3 ......................................................................................................................... 1 17 Single Lossy Layer .......................................................................................................... 117 3.1 Theoretical Determination of Natural Frequencies of Conductive Material ............ 117 3.2 Computation of EDNa and EDRa ............................................................................ 120 3.2.1 Change in Layer Thickness .............................................................................. 120 3.2.2 Change in Layer Relative Permittivity ............................................................. 136 3.2.3 Change in Aspect Angle ................................................................................... 151 3.2.4 Change in Conductivity .................................................................................... 172 3.3 Effects of Noise on EDNa ........................................................................................ 188 3.3.1 EDRa vs. Signal-to-Noise Ratio ....................................................................... 188 vi Chapter 4 ......................................................................................................................... 194 Magram - Single Layer ................................................................................................... 194 4.1 Extraction of the Natural Frequencies and E-Pulses from Response Waveforms 194 4.2 Magram Materials — Experimental Results .............................................................. 198 Chapter 5 ......................................................................................................................... 214 Impedance Sheet Over Single Layer ............................................................................... 214 5.1 Theoretical Determination of Natural Frequencies .................................................. 214 5.2 Construction of the E-pulse ...................................................................................... 218 5.2.1 Change in Surface Resistance .......................................................................... 224 Chapter 6 ......................................................................................................................... 236 Conclusion ....................................................................................................................... 236 6.1 Summary and Conclusions ....................................................................................... 236 6.2 Future Work ............................................................................................................. 237 APPENDICES ................................................................................................................. 238 A. Overview — NAT_FREQ.cpp .................................................................................... 238 A.1 Source Code — N AT_FREQ.cpp ........................................................................ 238 B. Overview — KPNAT.for ............................................................................................ 241 B.1 Source Code - KPNAT.for ................................................................................. 241 C. Overview — PCONV2.for .......................................................................................... 249 C] Source Code — PCONV2.for ............................................................................... 249 D. Overview - MLPREF.for .......................................................................................... 265 DJ Source Code - MLPREF.for ............................................................................... 265 E. Overview — NFSIG.for .............................................................................................. 271 El Source Code — NFSIG.for ................................................................................... 271 BIBLIOGRAPHY ........................................................................................................... 276 vii LIST OF TABLES Table 1-1 Natural frequencies of layered material, 8r 2 960,110, 0' = 0,d =10mm. ....... 22 Table 2-1 Material properties of the baseline layered material. ....................................... 33 Table 2-2 Material properties for changing thickness ...................................................... 40 Table 2-3 EDNa calculations for change in thickness. .................................................... 51 Table 2—4 Material properties for changing permittivity .................................................. 54 Table 2-5 EDNa calculations for change in permittivity. ................................................ 62 Table 2-6 Material properties for changing angle, parallel polarization. ......................... 66 Table 2-7 Material properties for changing angle, perpendicular polarization. ............... 67 Table 2-8 EDNa calculations for change in aspect angle for parallel polarization .......... 81 Table 2-9 EDNa calculations for change in aspect angle for perpendicular polarization. ................................................................................................................................... 82 Table 2-10 E-pulses generated using different modes. .................................................... 86 Table 2-11 EDNa calculations for change in the number of modes. ............................... 93 Table 2—12 SNR of response waveforms .......................................................................... 98 Table 2-13 EDNa for numerous SNR and thicknesses. ................................................. 111 Table 2-14 EDRa for numerous SNR and thicknesses. ................................................. 111 Table 2-15 EDNa for numerous SNR and values of relative permittivity. .................... 112 Table 2-16 EDRa for numerous SNR and values of relative permittivity. .................... 112 Table 3-1 Material properties for changing thickness .................................................... 122 Table 3-2 EDNa calculations for change in thickness of lossy material. ....................... 133 Table 3-3 Material properties for change in relative permittivity. ................................. 138 viii Table 3-4 EDNa calculations for change in permittivity of lossy material .................... 148 Table 3—5 Material properties for changing angle, parallel polarization. ....................... 153 Table 3-6 Material properties for changing angle, perpendicular polarization. ............. 154 Table 3-7 EDNa calculations for change in aspect angle for parallel polarization. ....... 168 Table 3—8 EDNa calculations for change in aspect angle for perpendicular polarization. ................................................................................................................................. 169 Table 3-9 Material properties for changing conductivity ............................................... 174 Table 3-10 EDNa calculations for change in conductivity of material. ......................... 186 Table 3-11 EDRa vs. SNR for several thicknesses. ....................................................... 190 Table 3-12 EDRa vs. SNR for several values of permittivity. ....................................... 190 Table 3-13 EDRa vs. SNR for several values of conductivity. ...................................... 190 Table 4-1 EDNa values for the three Magram samples. ................................................ 213 Table 4-2 Natural frequencies for single lossless layered case. ..................................... 213 Table 5-1 Material properties of impedance sheet. ........................................................ 220 Table 5-2 EDNa for change in surface resistance of the impedance sheet. ................... 234 ix LIST OF FIGURES Figure 1-1 Interrogation of simple layered material by transient excitation. ................... 12 Figure 1-2 Typical response waveform of a layered material, ......................................... 21 Figure 1-3 E-pulse generated from natural frequencies of layered material. ................... 23 Figure 1-4 Convolution of E-pulse with correct response waveform. ............................. 24 Figure 1-5 Response waveform for layered material, ...................................................... 25 Figure 1-6 Convolution of E-pulse with incorrect response waveform. .......................... 26 Figure 2-1 Magnitude data for response generated using MLPREFfor. ......................... 34 Figure 2-2 IFFI‘ of response data without windowing function. ..................................... 35 Figure 2—3 Frequency domain data with window function applied. ................................ 36 Figure 2-4 IFFT of response data with windowing function. .......................................... 37 Figure 2-5 Response waveform for material ‘A’. ............................................................ 41 Figure 2-6 Response waveform for material ‘P’. ............................................................. 42 Figure 2-7 Comparison of response waveforms for materials ‘G’, ‘H’, and ‘1’. ............. 43 Figure 2-8 E-pulse generated from the baseline material properties. .............................. 44 Figure 2-9 Convolution of baseline response with baseline E-pulse. .............................. 45 Figure 2-10 Convolution of material ‘A’ response with baseline E-pulse ....................... 46 Figure 2-11 Convolution of material ‘P’ response with baseline E-pulse. ...................... 47 Figure 2-12 Convolution of material ‘G’ response with baseline E-pulse ....................... 48 Figure 2-13 Convolution of material ‘1’ response with baseline E-pulse. ....................... 49 Figure 2-14 EDNa vs. change in thickness for single lossless layer. ............................... 52 Figure 2-15 Response waveform for material ‘A’. .......................................................... 55 Figure 2-16 Figure 2-17 Figure 2-18 Figure 2-19 Figure 2-20 Figure 2-21 Figure 2-22 Figure 2-23 Figure 2-24 Figure 2-25 Figure 2-26 Figure 2-27 Figure 2-28 Figure 2-29 Figure 2-30 Figure 2-31 Figure 2-32 Figure 2-33 Figure 2-34 Figure 2-35 Figure 2-36 Figure 2-37 Figure 2-38 Response waveform for material ‘P’. ........................................................... 56 Comparison of response waveforms for materials ‘G’, ‘H’, and ‘I’. ........... 57 Convolution of material ‘A’ response with baseline E-pulse ....................... 58 Convolution of material ‘P’ response with baseline E-pulse. ...................... 59 Convolution of material ‘G’ response with baseline E-pulse ....................... 60 Convolution of material ‘1’ response with baseline E-pulse. ....................... 61 EDNa vs. change in relative permittivity for single lossless layer. .............. 63 Response waveform for material ‘A’. .......................................................... 68 Response waveform for material ‘AA’. ....................................................... 69 Response waveform for material ‘F’. ........................................................... 70 Response waveform for material ‘FF’ .......................................................... 71 Response waveform for material ‘S’. ........................................................... 72 Response waveform for material ‘SS’ .......................................................... 73 Convolution of material ‘A’ response with baseline E-pulse ....................... 74 Convolution of material ‘AA’ response with baseline E-pulse. ................... 75 Convolution of material ‘F’ response with baseline E-pulse. ...................... 76 Convolution of material ‘FF’ response with baseline E-pulse. .................... 77 Convolution of material ‘S’ response with baseline E-pulse. ...................... 78 Convolution of material ‘SS’ response with baseline E-pulse. .................... 79 EDNa vs. change in aspect angle for single lossless layer. .......................... 83 E-pulse generated using 1 mode. .................................................................. 87 E-pulse generated using 5 modes. ................................................................ 88 E-pulse generated using 10 modes. .............................................................. 89 xi Figure 2-39 Convolution of baseline response with E-pulse ‘A’. .................................... 90 Figure 240 Convolution of baseline response with E-pulse ‘E’. .................................... 91 Figure 2-41 Convolution of baseline response with E-pulse ‘J’. ..................................... 92 Figure 2-42 EDNa vs. change in the number of natural modes in the E-pulse. ............... 94 Figure 243 Response waveform ‘A’. .............................................................................. 99 Figure 2-44 Response waveform ‘B’. ............................................................................ 100 Figure 2-45 Response waveform ‘E’. ............................................................................ 101 Figure 2-46 Response waveform ‘H’. ............................................................................ 102 Figure 2-47 Convolution of response waveform ‘B’ with baseline E-pulse. ................. 103 Figure 2-48 Convolution of response waveform ‘E’ with baseline E-pulse. ................. 104 Figure 2-49 Convolution of response waveform ‘H’ with baseline E—pulse .................. 105 Figure 2-50 Noise-free convolution of material ‘D’ with baseline E-pulse. .................. 106 Figure 2-51 Convolution of material ‘D’ with baseline E-pulse with SNR=15dB. ....... 107 Figure 2-52 Convolution of material ‘D’ with baseline E-pulse with SNR=2dB. ......... 108 Figure 2-53 Comparison of EDNa vs. SNR for several thickness. ................................ 113 Figure 2-54 Comparison of EDRa vs. SNR for several thicknesses. ............................. 114 Figure 2-55 Comparison of EDNa vs. SNR for several values of permittivity. ............ 115 Figure 2-56 Comparison of EDRa vs. SNR for several values of permittivity. ............. 116 Figure 3-1 Response waveform for material ‘A’ ........................................................... 123 Figure 3-2 Response waveform for material ‘P’ ............................................................ 124 Figure 3-3 Comparison of response waveform for materials ‘G’, H’, and ‘I’. ............. 125 Figure 3-4 E-pulse generated from the baseline material properties. ............................ 126 Figure 3-5 Convolution of baseline response with baseline E-pulse. ............................ 127 xii Figure 3-6 Convolution of material ‘A’ response with baseline E-pulse. ...................... 128 Figure 3-7 Convolution of material ‘P’ response with baseline E-pulse. ...................... 129 Figtrre 3—8 Convolution of material ‘G’ response with baseline E-pulse ....................... 130 Figure 3-9 Convolution of material ‘1’ response with baseline E-pulse. ....................... 131 Figure 3-10 EDNa vs. change in thickness for a single lossy layer. .............................. 134 Figure 3-11 EDNa vs. change in thickness for lossless and lossy layers. ...................... 135 Figure 3-12 Response waveform for material ‘A’. ........................................................ 139 Figure 3-13 Response waveform for material ‘P’. ......................................................... 140 Figure 3-14 Comparison of response waveform for materials ‘G’, H, and ‘I’. ........... 141 Figure 3-15 Convolution of baseline response with baseline E-pulse. .......................... 142 Figure 3-16 Convolution of material ‘A’ with baseline E-pulse .................................... 143 Figure 3-17 Convolution of material ‘P’ with baseline E-pulse. ................................... 144 Figure 3-18 Convolution of material ‘G’ with baseline E-pulse .................................... 145 Figure 3-19 Convolution of material ‘I’ with baseline E-pulse. .................................... 146 Figure 3-20 EDNa vs. change in relative permittivity for lossy layer. .......................... 149 Figure 3-21 EDNa vs. change in relative permittivity for lossless and lossy layers. ..... 150 Figure 3-22 Response waveform for material ‘A’. ........................................................ 155 Figure 3-23 Response waveform for material ‘AA’. ..................................................... 156 Figure 3-24 Response waveform for material ‘F’. ......................................................... 157 Figure 3-25 Response waveform for material ‘FF’ ........................................................ 158 Figure 3-26 Response waveform for material ‘8’. ......................................................... 159 Figure 3-27 Response waveform for material ‘SS’ ........................................................ 160 Figure 3-28 Convolution of material ‘A’ with baseline E-pulse .................................... 161 xiii Figure 3-29 Convolution of material ‘AA’ with baseline E-pulse. ................................ 162 Figure 3-30 Convolution of material ‘F’ with baseline E-pulse. ................................... 163 Figure 3-31 Convolution of material ‘FF’ with baseline E-pulse. ................................. 164 Figure 3-32 Convolution of material ‘8’ with baseline E-pulse. ................................... 165 Figure 3-33 Convolution of material ‘SS’ with baseline E-pulse. ................................. 166 Figure 3-34 EDNa vs. change in aspect angle for lossy layer. ...................................... 170 Figure 3-35 Comparison of EDNa vs. aspect angle for lossless and lossy layers .......... 171 Figure 3-36 Response waveform for material ‘I’. .......................................................... 175 Figure 3-37 Response waveform for material ‘A’. ........................................................ 176 Figure 3-38 Response waveform for material ‘8’. ......................................................... 177 Figure 3-39 Response waveform for material ‘H’. ........................................................ 178 Figure 3-40 Response waveform for material ‘J’ ........................................................... 179 Figure 3-41 Convolution of the baseline response with the baseline E-pulse. .............. 180 Figure 3-42 Convolution of material ‘A’ response with the baseline E—pulse. .............. 181 Figure 3-43 Convolution of material ‘8’ response with the baseline E-pulse. .............. 182 Figure 3—44 Convolution of material ‘H’ response with the baseline E-pulse. .............. 183 Figure 3-45 Convolution of material ‘J’ response with the baseline E—pulse. ............... 184 Figure 3—46 EDNa vs. change in conductivity. .............................................................. 187 Figure 3-47 Comparison of EDRa vs. SNR for several thicknesses. ............................. 191 Figure 3-48 Comparison of EDRa vs. SNR for several values of permittivity. ............. 192 Figure 3-49 Comparison of EDRa vs. SNR for various values of conductivity. ........... 193 Figure 4—1 Magnitude plot of scattered response from unknown Magram materials. 202 Figure 4-2 Phase plot of scattered response from unknown Magram materials. ........... 203 xiv Figure 4-3 Frequency domain response plots for single layer lossy materials. ............. 204 Figure 4-4 Extrapolated magnitude plots of scattered responses. .................................. 205 Figure 4-5 Extrapolated phase plots of scattered responses ........................................... 206 Figure 4-6 Time domain response of unknown Magram samples. ................................ 207 Figure 4-7 Time domain response for extrapolated data ................................................ 208 Figure 4-8 E-pulse generated from Sample A response using 5 modes ......................... 209 Figure 4-9 Convolution of the sample A response with the sample A E-pulse. ............ 210 Figure 4-10 Convolution of the sample B response with the sample A E-pulse. .......... 211 Figure 4-11 Convolution of the sample C response with the sample A E-pulse. .......... 212 Figure 5-1 Layered material covered by impedance sheet ............................................. 217 Figure 5-2 Frequency domain scattered response of Salisbury screen. ......................... 221 Figure 5-3 Response waveform for baseline material .................................................... 222 Figure 5-4 E-pulse generated for impedance sheet response. ........................................ 223 Figure 5-5 Response waveform from material ‘A’. ....................................................... 226 Figure 5-6 Response waveform from material ‘V’. ....................................................... 227 Figure 5-7 Comparison of response waveforms from materials ‘J’, ‘K’, and ‘M’. ....... 228 Figure 5-8 Convolution of response from material ‘K’ with the E-pulse. ..................... 229 Figure 5-9 Convolution of response from material ‘A’ with the E-pulse. ..................... 230 Figure 5-10 Convolution of response from material ‘V’ with the E-pulse. ................... 231 Figure 5-11 Convolution of response from material ‘J’ with the E-pulse. .................... 232 Figure 5-12 Convolution of response from material ‘M’ with the E-pulse. .................. 233 Figure 5-13 EDNa vs. surface resistance. ...................................................................... 235 XV ABBREVIATIONS E—Pulse — Extinction Pulse EDN - E-Pulse Discrimination Number EDNa - E-Pulse Discrimination Number (version a) EDR - E-Pulse Discrimination Ratio EDRa - E-Pulse Discrimination Ratio (version a) xvi INTRODUCTION The basis of the E-pulse technique lies in the fact that the late-time scattered response of a conducting target to a transient, wide-band pulse is a sum of damped Sinusoids. These damped Sinusoids are composed of an infinite number of complex natural frequencies. Given knowledge of these natural frequencies, an E—pulse can be generated that when convolved with the response yields zero late-time energy. The natural frequencies of a conducting target are unique and, therefore, so are the E-pulses for each target. The E-pulse can be looked at as a filter which is target specific and can be used to discriminate that target. The response of a layered material to a transient, wide-band pulse is also a sum of damped Sinusoids. Therefore, the E—pulse technique can be extended to layered materials. The complex natural frequencies which compose the damped Sinusoids depend on the material properties, such as the thickness, permittivity, permeability, and conductivity of the layers. Application of the E—pulse method to the diagnosis of layered materials is examined for four layered material configurations. These configurations include a single conductor backed layer of lossless material, a single conductor backed layer of lossy material, a single conductor backed layer of magnetic radar absorbing material (MagRAM), and a single conductor backed layer of lossless and lossy material covered by an impedance sheet. Chapter 1 opens with an overview of the E-pulse method. Included in this overview is a brief description of the application of the E-pulse technique as a method of discriminating radar targets. This is followed by an analysis of the mathematics behind the E—pulse, as well as a discussion of different types of E—pulses. The late-time natural frequencies of a target are the basis of the E-pulse technique so a discussion on the determination of the natural frequencies is presented. Chapter 1 then discusses the application of the E-pulse to layered material. It is shown that the response of lossless layered materials to transient excitations is a sum of damped Sinusoids. The chapter closes with an example as a proof of principle. Beginning with chapter 2, each material configuration is explored beginning with a discussion of the determination of the natural frequencies for each specific configuration. For each material configuration, results are generated for noise free measurements followed by an exploration into the effects of noise on the measurements. Chapter 2 investigates the effects of changing thickness and permittivity of the layers of lossless materials, as well as the effects of aspect angle and the number of modes used to generate the E-pulses. A method to quantify the results is then discussed and is followed by the presentation of the results. Chapter 3 considers the effects of changing thickness, permittivity, permeability, and conductivity of the layers of lossy materials. Chapter 4 opens with a discussion of the method used to extract the natural frequencies from a scattered response waveform. Following is an investigation into the applicability of the E-pulse technique for analyzing changes in the material properties of MagRAM. Chapter 5 investigates layered materials covered by an impedance sheet. This includes using the E-pulse to detect changes in the surface resistance which depends on the conductivity and thickness of the impedance sheet. Chapter 6 closes the thesis with conclusions that include a summary and discussion of future works. Chapter 1 Overview of the E-Pulse Technique Initial applications of the E-pulse technique were in the area of radar target discrimination. As a general technique, its methods can be applied to layered material evaluation to determine changes in material parameters. A brief description of the original use for the E-pulse will be discussed followed by its application to layered material evaluation. 1.1 E-Pulse Technique for Radar Discrimination It has been shown [1] that the late-time response of a conducting radar target to a band-limited transient excitation can be decomposed into a sum of damped sinusoids as follows, N r(t)=Zanea’ltcos(a)nt+¢n), t>Tl, (1.1) n=1 where s" = 0'" + jwn is the aspect independent natural frequency of the nth target mode, an and (1),, are the aspect dependent amplitude and phase of the nth mode, T1 = 2T (T is the one—way transit time to the target) is the beginning of late time, and N modes are assumed to be excited by the incident electric field. A waveform, e(t) , can be generated using the natural frequencies of the target, 3,, = 0",, + ja)" , which when convolved with the target response, results in a null solution, c(t)=e(t)*r(t)=0, t>TL=Tl+Te (1.2) where Teis the duration of e(t). This waveform is called an extinction pulse or E-pulse. Two scenarios exist when determining the E-pulse for a target. For the first scenario, the natural frequencies are known, either through prior knowledge of the target characteristics or through theoretical determination. Theoretical determination of the natural frequencies is practical only when considering targets with simple geometry as the natural frequencies are based solely on the geometry of the radar target [1]-[4]. Given the natural frequencies of a target, the response can be modeled by substitution into equation (1.1). Consequently, the E-pulse is determined by substituting the modeled response, r(t) , into equation (1.2) and finding the roots of the equation. Theoretical determination of the natural frequencies for non-conductive and conductive materials will be shown in the later chapters. The second scenario supposes that the natural frequencies are not known, but that the actual response waveform of a target is known, as would be the case when attempting to identify an unknown target. The natural frequencies of the target can be extracted numerically by substituting this target response waveform directly into equation (1 .2) as r(t) , and again solving for the roots of the equation. Since the natural frequencies are dependent on the geometry of the targets, the E- pulses generated from these unique natural frequencies are also unique and thus target specific. A database of E-pulses can be built for all known targets and can be referenced against all incoming radar responses to identify radar targets. Determining the E-pulse can be done in the time domain or through the use of the Laplace transform equations, F(s)=£{f(t)}= )f(t)e“"dt, (1.3) and f(t)=£_‘{F(s)}=?fl7 F(s)e"ds. (1.4) Br Both cases are considered in the following sections. 1.1.1 Time Domain Analysis Given the first scenario, that the natural frequencies are known, an E-pulse can be generated using equation (1.2). Writing the convolution equation (1.2) in integral form, substituting (1.1) and simplifying yields the following [1], [2], e(r)r(t—r)dr e( (7)2 a,,ea"( ‘ T ),cos[a) —r)dr+¢n] (1.5) 0 N = Z a e0’1‘ [An cos(a),,t + Q, ) + 8,, sin (rant + (on )1, r> TL where srn (ant {3"}=Tje(r)e—U"T{C.Oswnr}dr. (1.6) 0 :8 = , lSnSN, (1.7) where N is the number of modes composing the E-pulse. Therefore, solving the integral equation, T6 je(r)e"’"’{:§f$?f)dr=o, ISnSN (1.8) 0 for e(z') gives the E-pulse that will force the convolution with the target response, r(t) , to zero for t > TL. 1.1.2 Transform Domain Analysis Again, this analysis assumes the complex natural frequencies, {Sn} , are known. Transforming the convolution equation (1.2) using the Laplace transform yields, £{c(t)}=£{e(t)*r(t)}, (1.9) £{c(t)} = £{e(t)}£{i a,,e0"‘ cos(a),,t + (21,, )), (1.10) n=l N . (3)2ng {e J¢ne (011+lel) +e‘j¢ne(031_lwn)’}, (1.11) _ 2 N . . * E:E(s)i%[£{em"es"t)+2(6—10’1es’1‘H, (1.12) N a el¢n+ e—j¢n (5)8624; , (1.13) 2 s * — Sn where E (s) the Laplace transform domain representation of e(r). Equation ( 1.13) is the Laplace transform domain equivalent of equation (1.2). To find C(t) = 0, equation (1.13) is transformed back to the time domain. Using the inverse Laplace transform equation (1.4) and invoking Cauchy’s residue theorem to solve the resulting integral gives [1], err) - i—“AM WW. .4: stir—m.) _ 2 " "1)6 ° n=l * Therefore, C(t) in (1.14) is equal to zero when E(s,, ) = E(s,, ) = 0, so Te E(s,,)-£{e(t)}— Je s“‘e(t)dt=0 lSn1, then 21 < Z0 and thus F < 0. For the lossless case (0:0) with parallel polarization, Z now/poemll—sinzai (141) 0 = ' (”J/(050 = ”m/r—srnz e,- (1.42) and i ’M) l - 2 (0 111050 :1— 81, '8111 0i 21 = (1.43) ("x/#081 _ 770‘} Sir —sin2 61' _ . (1.44) Elr And again, it can easily be shown that since 81, > 0 then Z1 < 20 and thus F < 0 . Therefore, for both polarizations, F < O a fact that will be referenced later. These interfacial reflection coefficients are used to find the total reflected wave as shown in [5]. The total reflection coefficient is, _ r—P2 R(w)'1—rp2(w)’ (1.45) 14 in which P(a)) = [1‘sz (1.46) is called the propagation factor, d is the thickness of the material, and the wave vector is kw = 71.12 4.3 sin2 a, (1.47) 2%(/£1, —sin26,- . (1.48) To further analyze the reflected wave, it is recognized that s = ja) and thus a) = —js . Through substitution the total reflected wave can be described in s-plane notation, —j2d[::—S)\’£1r —sin2 6i R(S)= r—e _. . (1.49) —j2d(i)(/£1,-sin26i l—Fe c _ —sr l—I‘eTST where 2d\/6‘1, -sin2 91’ 2' = , (1.51) c and c is the speed of light, 1 C: . «#08 As described in [4], the singularity expansion method (SEM) is a way to represent (1.52) the solution to electromagnetic scattering problems in terms of the singularities in the complex frequency domain. The pole terms associated with a scattered response determine the natural frequencies of the scatterer. Therefore, the pole terms of (1.50) determine the natural frequencies of the layered material. In the time domain this leads to 15 a representation of the scattered response as a sum of damped sinusoids. Expanding the denominator of ( 1.50) using the binomial series, (1—z)"=Zz"=1+2+zz+z3+m, (1.53) n20 gives, °° n R(s) =(r—e’”) Z (re—5’) (1.54) n=0 ‘F 0° rn —snT 00 n *(n+1)ST — X e - 2 F e (155) n=0 n=0 : r+ i Fn+lg—snt‘ _ i Fnt—le-nzsr (1.56) n=l m=1 _ 2__ 0° n-l -sn‘r _r‘+ r 121‘ e (1.57) n=l Transforming to the time domain using the inverse Laplace transform (1.4) and the time- shifting theorem gives, R(t) = I‘é’(r)+(l“2 —-1) i Fn—l§(t—n2') n=1 (1.58) =F§(t)+u(t-T)(F2—l) Z 1“""§(t—nr) (1.59) l—rz 00 n =F6(t)+u(t—r) Z —r 6(t—nr). (1.60) nz—oo Using 16 —r" ——1”, 1.61 I. ( ) ( ) gives (l-F’) °° Irr" R(t)=F6(t)+u(t—r) 1‘ Z 1‘ (—1)"c5(t—nr). (1.62) nz—oo Since (I‘m = (-l)" , this becomes (l-I") °° 10"" - R(t)= Fo’(tt)+u( —r) r 2 r e‘f"”6(z—nr) (1.63) 2 t (1—r) (7“) -11” .. =F§(t)+u(t—T) F2 [Fl e T 250—122). (1.64) n:— Using the Poisson-Sum formula [6], Z 6(z+nr)=-‘- Z ef’m’O’, (1.65) 11:"00 n=—-oo 27: where (00 = -— leads to, 2' R(t)= 1‘6(r.t)+u( —r) 2 |1'| e 1" Z 6 r (1.66) F 2 ,,=.oo (1_1-2) [%+1)°° (211—l)?! =F§(t)+u(t—2) 2 |r| 2 e r. (1.67) F 2 ,,__._oo Separating the summation into two components gives 1‘ )I‘1_2(2'+ 1] °° j(—2n l)f[ j(2m— l)72':_ Ze“ + 2 e (1.68) Ifizz-11:1 m- _ —oo R(t)=F6(r)+u(t—2') 17 By making the power of the exponent negative in the second summation, it’s range is then the same as the first summation and the two can be combined giving I 1-1“2 [tr-+1] °° j(2m—071': -j(211-l)fl'i R(t)=F6(t)+u(t—T) 2 |r| 2 e r+e r (1.69) F 7 n=1 Using (1+) (t+r)lln|l‘| r e r =|r| (1.70) gives 1 t t -2 r+2—lF°° '2—17r— —'2—lrr— R(r)=F§(r)+u(t—2)l f e( Mi '2 e" " ) T+e " " ) r (1.71) F 7 n=1 and recognizing that llanlr er =|1‘| (1.72) leadsto 1_I“2 llnIFlt °° j(2n—l)fl£ —j(2n—1)7ti R(t)=F§(t)+u(t—2’) F2' 7 e T+e 1' . (1.73) n=1 Finally, recognizing that the quantity within the summation notation can be expressed as a cosine leads to the final form of the scattered response, - 2 llnrz °° 1 F er H Zcos(2n—l)£t. (1.74) T n=1 R(r) =I‘§(t)+u(r—2) 1‘2" It can be seen from (1.74) that the scattered response is thus made up of a sum of damped sinusoids. 18 1.2.2 E-pulse Diagnostics for Layered Materials Having shown that the scattered response of a lossless layered material is equal to a sum of damped sinusoids the E-pulse technique can be used as a diagnostic tool. This is also true for a conducting material (0790), but this is considerably more difficult to show. The goal of applying the E-pulse technique to layered materials is to give the ability to determine changes in material parameters, (e. g., thickness, permittivity, permeability, and conductivity). As with radar target identification, the E-pulse technique begins with identifying the natural frequencies of the intended targets and using this history as a baseline to sense changes. Periodic testing of layered materials using the E-pulse can determine any changes in the material parameters. Since the convolution of an E-pulse with the response waveform used to generate it gives a zero result in the late time, any result other than zero shows that there has been some change in the properties of the material from the baseline measurement. 1.2.3 Demonstration of Principle - E-pulse Construction and Convolution As a demonstration of how the E-pulse technique works, Figure 1-2 is the response of a layered material with ,u = #0, 5 = 980, 0' = 0, and d 2 10mm to a transient excitation. The natural frequencies of this material, as shown in Table 1-1, are calculated from the material properties using the software program NAT_FREQ.cpp, which is included in appendix A, and a natural E-pulse, shown in Figure 1-3, is generated using the Fortran program KPNAT.for, which is included in appendix B. When the response is convolved with this E-pulse, using the Fortran program PCONV2.for, included in 19 appendix C, the resulting late time convolution will be zero as required by (1.2). The resulting convolution waveform is shown in where the late time is indicated by the arrow. It can be seen that the convolution is zero in the late time. To contrast this, the relative permittivity of the material is changed from 9 to 10 and the resulting response waveform, shown in Figure 1-5, is convolved with the E-pulse generated from the original waveform. The late time response, as indicated by the arrow in the convolution waveform, Figure 1-6, is by inspection, not equal to zero. This would indicate a change in the material properties. 20 0.6 d 0.4 E 0.2 { Relative Amplitude —4 4 -1.2 I I I I I I I T T l I I I I T I T T T I T T T T T I T T I I T I T I 0 0.5 1 1.5 2 2.5 3 Time (ns) Figure 1-2 Typical response waveform of a layered material, ,u=,uo,£=9€0,0'=0,andd=10mm. 21 3.5 N on x 109 (0n (Gr/sec) 1 -3.463 15.697 2 -3.463 47.092 3 -3.463 78.487 4 -3.463 109.882 5 -3.463 141.227 Table 1-1 Natural frequencies of layered material, 8,. = 960, 120,0 = 0, d = 10mm . 22 E-Pulse Magnitude 2.5 _ _ 0.5— O T 4 T r r , T r . , r , r , . , . 7 0.00 0.02 0.04 0.07 0.09 0.11 0.13 0.16 0.18 0.20 Time (ns) Figure 1-3 E-pulse generated from natural frequencies of layered material. 23 Relative Amplitude O —‘l 0 {————> 1 Late time ., -< -0.07 I I I r l r l r l 1 I 1 I I l r I T r T T I r I l I I I r l I I I I 0 0.5 1 1.5 2 2.5 3 Time (ns) Figure 1-4 Convolution of E-pulse with correct response waveform. 24 3.5 Relative Amplitude I .0 ON 1 14 l '1.2 I I I I I T 1' I I I I r I I I I I I Iffi T I I T I I I I I I I T I 0 0.5 l 1.5 2 2.5 3 3.5 Time (ns) Figure 1-5 Response waveform for layered material, p=a0,e=10£0,or=0,andd=10mm. 25 Relative Amplitude 0.02 0.01 3 -0.01 1411 -002 { —0.03 f 1111 -0.04 1 -005 : -0.06 3 O L411111 Late time 007W... 0 Figure 1-6 Convolution of E-pulse with incorrect response waveform. 0.5 I I T I 1.5 I T T T T I 2 Time (ns) 26 2.5 T T ‘r 7 F T 3.5 Chapter 2 Single Lossless Layer 2.1 Determination of Natural Frequencies for Non-Conductive Materials The success of the E-pulse technique lies with the ability to determine the natural frequencies of the target under investigation. As has been previously mentioned, the natural frequencies for simple layered materials can be determined theoretically. For a more complex structure, the natural frequencies can be extracted from its scattered response. A simple structure will be defined as a single layer, conductor backed material with real permeability and either real or complex permittivity. A complex structure will be defined as either a single layered material with both complex permittivity and permeability or a multiple layered material. This chapter will discuss the most simple case of a single layer, conductor backed material with real permittivity and permeability (0:0). Determining the natural frequencies of simple layered material with no conductivity is done by finding the poles of ( 1.50). Thus, o(s)=1-re’"=o, (2.1) or re'("+j“’)’ =1, (2.2) or (re-”Me'W) : 1. (2.3) 27 —or . Since Fe 18 real, then {Janis real and thus equals 1 or -1. Therefore F eTUT also has to equal 1 or -1 to satisfy (2.3). It was shown previously that F < 0, so both quantities must be equal to -1, and Since—1 S F < 0 , the right hand side of (2.4) is positive and greater than zero and so 01' Also, and so 01' l 0'2'=-ln—, III 1 0'=—ln|F|<0. 2' e_ij=—l, a)2'=i(2n+l)lt, n =0,1,2,~~, 72' T a); ,=i(2n+1) Therefore, the natural frequencies are s" =$ln|F|ij(2n+l)§, n=0,l,2,---. (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) (2.10) Note that these frequencies match those in (1.74) and can be calculated given that the relative permittivity, E, , thickness of the material layer, (1, angle of incidence, t9,- , and the 28 polarization of the excitation are known. These natural frequencies can then be used to generate E-pulses for the layered material. 2.2 Computation of the EDNa and EDRa Detecting changes in the properties of materials is done by evaluating the energy in the late-time portion of the convolved waveform. Evaluation of the energy in the late- time convolution can be done by inspection when large changes in material properties occur. However, to determine minor changes in the material properties it is necessary to implement a scheme to quantify the changes. Thus, a threshold level can be determined and a quantitative comparison can be performed for those changes that are not evident by inspection. The determination of changes in the measurements can then be automated and human subjectivity can be eliminated. This quantitative analysis of the convolution late—time energy was first demonstrated by P. Ilavarasan, et.al. in [7]. Ilavarasan defined the E—Pulse discrimination number (EDN) as, -—l TL+W Te EDN: [C3(t)dt Je2(t)dt , (2.11) TL 0 where [c3 (t)dt is the energy in the convolution waveform, [e2 (t)dt is the energy in the E-pulse waveform, TL is the start of late-time, W is some time later than TL determined such that most of the energy of the late-time is considered, and Te is the duration of the E—pulse. Essentially, the EDN is the late-time energy of the convolution waveform normalized by the E-pulse energy. 29 This definition of the EDN is appropriate for applications in target discrimination where one target response is convolved with a database of many E-pulses for known targets. However, in the case of layered material diagnostics, Ilavarasan’s definition of the EDN is not appropriate. This leads to the definition of version ‘a’ of the E-Pulse discrimination number (EDNa), TL+W Te TL+W EDNa: ICE(I)dt Ie2(z)d: j r2(t)dt , (2.12) TL 0 TL where Irz (t)dt is the energy in the response waveform. When analyzing layered materials there is only one E-pulse generated from a baseline response measurement of the material and from this an EDNa is calculated. The E-pulse is then convolved with the scattered response of the material at a later time and a new EDN a is calculated and compared to the baseline EDNa. In order to account for any possible changes in the amplitude measurement of the response, the EDN must be normalized by the response waveform energy as well as by the E—pulse energy. Through the use of the EDN a, minor changes in the material properties can be detected. An initial baseline response measurement can be used to generate an E-Pulse for a specified layered material. Using these two waveforms, the EDNa is calculated and should be zero. However, practical measurements will contain some level of noise and inaccuracies in determining the exact natural frequencies will prevent the EDNa from completely vanishing. To allow consistent comparisons of changes in the EDNa, a variant of the E-Pulse discrimination ratio (EDR) described by Ilavarasan in [7] is defined as 30 (2.13) EDNa EDRa(dB) : 10 loglo {baseline EDNa} . If the material does not undergo any changes from its baseline, the calculated EDNa should not change from its baseline measurement and the EDRa should yield a OdB reading. On the other hand, if the material undergoes a change in one of its properties, the EDNa measurement will increase and consequently so will the EDRa. Depending on the how much change is tolerated in the layered material properties, a threshold can be established and the EDRa measurement can be used to determine whether the threshold has been exceeded. The EDRa will not be utilized for the noise-free data cases. Theoretically, since the data is noise free, the EDNa for the baseline case will be zero. Therefore, calculating the EDRa for all cases except the baseline will be indeterminate since the denominator in (2.13) will be zero. The EDRa will be used only when analyzing the noise corrupted data. In those cases, the baseline EDNa will never be zero, due to the noise, so an appropriate EDRa value can be calculated. 2.2.1 Change in Layer Thickness The first material property to be examined will be the thickness of the material layer. The following examples involve noise free responses constructed using the baseline material properties shown in Table 2-1. Several scattered responses of layered materials were simulated using the Fortran program MLPREF.for, which is included in appendix D. The properties of the layered materials used to create the responses are shown in Table 2-2. Input for the MLPREF.for program is the material properties (e.g., number of layers, thickness of layers, 31 permittivity, permeability, etc.). Output from MLPREF.for is the scattered response data presented in the frequency domain. Analysis of the response data requires it to be in the time domain so another software program, UTILZBexe, was used to process the frequency domain data. The magnitude data for the frequency response generated using MLPREF.for for a typical layered material is shown in Figure 2-1. The frequency content is a user defined input in MLPREF.for and the frequency range 0—18GHz was chosen for this case. Since the layer is backed by a conductive plate the magnitude of the frequency response is one, as expected. Since there is a discontinuity in the data at 18 GHz, the inverse Fourier transform (IFFI’) data will include many high frequencies, as shown in Figure 2—2. This makes it difficult to analyze the time domain data. In order to remove the effects of the discontinuity, the frequency domain data was processed with a Gaussian modulated cosine windowing function with a width, 2' = 0.06 ns , and a center frequency of 0 GHz, as seen in Figure 2-3, before running the [EFT program. The result, shown in Figure 2-4, is a cleaner time domain response waveform containing the same pertinent information as Figure 2-2, but without the unnecessary high frequency content. 32 Thickness of layer d=10mm Permittivity of layer 8:980 Permeability of layer pzpo Conductivity of layer 0:0 Angle of incidence of excitation 0=0° Polarization of excitation Parallel polarization Number of modes used to calculate E-pulse 5 modes Table 2-1 Material properties of the baseline layered material. 33 Amplitude. 1.2 .0 .o .o A O\ 00 1 1 l 1 1 1 1 1 1 1 1 1 l 0.2 0 A I I I I I I I I I I I . I I I O 5 10 15 Frequency (GHz) Figure 2-1 Magnitude data for response generated using MLPREF.for. 34 20 Amplitude. 0.6 0.4 : 0.2 5 .'O 9'3 9'3 o A N I 1 l I .O OO 111—Ll1 I p— 1111 0 1111111 TTrTT‘FTI' I y—a C N 1111 1.5 TTW Ffil I T 2 Time (ns) 2.5 3.5 Figure 2-2 [F FT of response data without windowing function. 35 Amplitude. 1.2 0.6 3 0.2 3 O I I I I I I I I I I I I I I I T 0 5 10 15 Frequency (GHz) Figure 2-3 Frequency domain data with window function applied. 36 20 Amplitude. 0.6 - 0.4 0.2 11111 L411_LI 11 l 1111 Time (ns) Figure 2-4 IFFT of response data with windowing function. 37 It would not be practical to present every waveform simulated, so a few are shown as examples. Referring to Table 2-2 for the material reference for each response, the extremes of the measurements, material ‘A’ and material ‘P’, are shown in Figure 2-5 and Figure 2-6. Each spike in the plots is a reflection of the incident wave. It can easily be seen that the thicker the material, the farther apart each reflection is separated. It is very difficult to see any difference in the responses between the materials that have relatively minor changes in their thickness. This is shown by plotting the response waveforms for materials ‘G’, ”,H and ‘I’ on the same plot, Figure 2-7. To evaluate these response waveforms for changes in the thickness of the layer, each response is convolved with an E-pulse, shown in Figure 2-8. This E-pulse is generated using the Fortran program KPNAT.for. The program generates the E-pulse from the natural frequencies of the baseline material, shown in Table 1-1, using (2.10). This E-pulse will be known as the baseline E-pulse. Figure 2-9 shows the convolution of the material H response with the baseline E-pulse. It can be seen that the energy in the late time, TL = 2T,, + Tp + Td, (2.14) where Tt, is the one-way transit time through the layer (which varies depending on properties of the layer and the aspect angle), TI) is the incident excitation pulse width (0.1ns), and Td is the time before the excitation pulse strikes the face of the layer (0.5ns), is zero as expected. In comparison, the convolutions of the other responses are shown in Figure 2-10, Figure 2-11, Figure 2-12, and Figure 2-13. It is evident in all of these plots, even when the thickness is only changed by 2.5%, as is the case for both material ‘G’ and material ‘I’, that the late-time energy is greater than zero. This signifies a change in the 38 thickness. 39 Material 8 Ir 6 Thickness Polarization Angle of (mm) incidence, 91 A 980 110 0 5 Parallel 0 B 980 llo 0 6 Parallel 0 C 980 110 0 7 Parallel 0 D 980 110 0 8 Parallel 0 E 980 I10 0 9 Parallel 0 F 980 I10 0 9.5 Parallel 0 G 980 I10 0 9.75 Parallel 0 H (baseline) 980 I10 0 10 Parallel 0 I 980 110 0 10.25 Parallel 0 J 980 110 0 10.5 Parallel 0 K 980 I10 0 1 1 Parallel 0 L 980 po 0 12 Parallel 0 M 980 pg 0 13 Parallel 0 N 980 I10 0 14 Parallel 0 P 980 110 0 15 Parallel 0 Table 2-2 Material properties for changing thickness. 40 Relative Amplitude 0.6 q r 0.4 ~ 0.2 f -0.4 5 -0.6 f -0.8 f 0 f i -0.2 j T 0.5 1 Time (ns) Figure 2-5 Response waveform for material ‘A’. 41 1.5 2.5 0.6 ~ 0.4 E l 0.2 5 -02 5 -0.4 3 Relative Amplitude -0.6 — I -O.8 : Time (ns) Figure 2-6 Response waveform for material ‘P’. 42 Material 'H' - — - Material '0' """" Material '1' Q '6 3 '3. E < E -0.4j E : 5'2 -063 I i -0.8 ~ -1 J '1.2«TlT'T'IIIrIIIrIrTyrl,,T,TT71T1 0 0.5 1 1.5 2 2.5 3 3.5 Time (ns) Figure 2-7 Comparison of response waveforms for materials ‘G’, ‘H’, and ‘1’. 43 E-Pulse Amplitude. 2.5 1.5 i 0.5 - O 7 T I T I l I F I I I l ' l I l I l 0.00 0.02 0.04 0.07 0.09 0.11 0.13 0.16 0.18 0.20 Time (ns) Figure 2-8 E-pulse generated from the baseline material properties. 44 Relative Amplitude 0.01 _ -0.0l j 002 f -003 f 1 -0.04 3 0.05 f -0.06 { _. .4 0 _ ... .. .. —1 —’ Late time —0.07‘.... 0 Figure 2-9 Convolution of baseline response with baseline E-pulse. 0.5 l I I T T T I T I I l I I l T T l 1.5 2 Time (ns) 45 2.5 3.5 Relative Amplitude 0.02 0.01 : 111111 '0-01 Late time 1J_11_ -0.02 -003 ‘ 1 C -004 U ‘0-05 I I I I l I T T T T T T T I f I I I I I Time (ns) Figure 2-10 Convolution of material ‘A’ response with baseline E-pulse. 46 Relative Amplitude 0.03 . 0.02 1 114141 0.01 1 LJ -001 f g ' Late time -002 f 1 -003 3 i -004 { -0.05 1 Y ' I I I I I I 1 T 1 r r 1 I I I Y T F 0 l 2 3 4 Time (ns) Figure 2-11 Convolution of material ‘P’ response with baseline E-pulse. 47 Relative Amplitude 0.01 q 0 ~——( fl * ‘ Late time -0.01 E -002 3 111 -0.03 l -0.04 -0.05 1 1 1 -0.06 e.,.Tj 0 0.5 1 1.5 2 2.5 3 Time(ns) Figure 2-12 Convolution of material ‘G’ response with baseline E-pulse. 48 Relative Amplitude 0.01 -0.01 111 -002 -003 -004 0.05 ' ~ Late time .1 —4 -0.06 T 7 F T I T 1 T T Figure 2-13 Convolution of material ‘1’ response with baseline E-pulse. 1.5 2 Time (ns) 49 To quantify the results, the EDNa is calculated. These values for each of the materials examined are shown in Table 2-3. Since this data is noise free, theoretically the EDNa should be zero for the baseline material. This is not the case due to numerical error in generating the response waveforms and in calculating the convolutions. Using only 5 modes to generate the E-pulse also contributes to the error. Figure 2-14 shows a plot of the tabulated EDNa calculations. It can be seen that as the change in thickness from the baseline increases the EDNa increases as well. Determining how much change in thickness can be detected using the EDNa plots depends on the noise level. This will be discussed later in this chapter. 50 Material Thickness EDNa A 5 7.3OE-03 B 6 6.63E-03 C 7 5.15E-03 D 8 3.10E-O3 E 9 9.73E-04 F 9.5 2.6 lE-O4 G 9.75 6.78E-05 H 10 3.70E-07 I 10.25 6.24E-05 J 10.5 2.51E-04 K 1 1 9.54E-O4 L 12 3.06E-03 M 13 4.94E-03 N 14 5.95E-03 P 15 6.33E—03 51 Table 2-3 EDNa calculations for change in thickness. EDNa 0.008 _ 0.007 0.006 0.005 0.004 f 0.003 0.002 f 0.001 f Figure 2-14 EDNa vs. change in thickness for single lossless layer. T I T I T T 1.0 Thickness (mm) 52 12 14 16 2.2.2 Change in Layer Relative Permittivity The ability to diagnose changes in the relative permittivity of the material layer is now examined. The following examples involve noise free responses and the baseline material properties are the same as for changes in thickness and are shown in Table 2-1. Several scattered responses of layered materials were again simulated using MLPREF.for. The properties of the layered materials used to create the responses are shown in Table 2-4. Again only a limited selection of examples was chosen. Citing Table 2-4 for the material reference for each response, the extremes of the measurements, material ‘A’ and material ‘P’, are shown in Figure 2-15 and Figure 2-16, respectively. It can be seen that the permittivity affects the separation of the reflections just as the thickness of the layer does. It is very difficult to see any difference in the responses between the materials that have relatively minor changes in their permittivity. This is shown by plotting the response waveforms for materials ‘G’, ‘H’, and ‘I’ on the same plot, Figure 2-17. Evaluation of these response waveforms for changes in the permittivity of the layer is done exactly as before with changes in thickness. Each response is convolved with an E-pulse, shown in Figure 2-8, which is generated from the natural frequencies of the baseline material, shown in Table 1-1. This will again be known as the baseline E- pulse. The same convolution, as seen previously in Figure 2-9, of the material ‘H’ response with the baseline E-pulse shows the energy in the late time as zero. In comparison, the convolutions of the other responses are shown in Figure 2-18, Figure 2-19, Figure 2-20, and Figure 2-21. It is evident in all of these plots, even when there is relatively minimal change in the permittivity, as is the case for both material ‘G’ and 53 material ‘I’, that the late-time energy is greater than zero. This signifies a change in the permittivity of the material layer. Material 8 Ir 0 Thickness Polarization Angle of (mm) incidence, 0, A 480 110 0 10 Parallel 0 B 580 110 0 10 Parallel 0 C 680 110 0 10 Parallel 0 D 780 pg 0 10 Parallel 0 E 880 P0 0 10 Parallel 0 F 8.580 110 0 10 Parallel 0 G 8.7580 110 0 10 Parallel 0 H (baseline) 980 I10 0 10 Parallel 0 I 9.2580 90 0 10 Parallel 0 J 9.580 110 0 10 Parallel 0 K 1080 I10 0 10 Parallel 0 L 1180 I10 0 10 Parallel 0 M 1280 110 0 10 Parallel 0 N 1380 I10 0 10 Parallel 0 P 1480 I10 0 10 Parallel 0 Table 2-4 Material properties for changing permittivity. 54 Relative Amplitude 0.4 0.2 14111 #11111 11111 L 1 I 7 IT r T F l T I T T 0.5 1 Figure 2-15 Response waveform for material ‘A’. Y TiT T 1.5 r T I I T I 2 Time (ns) 55 3.5 0.8 d 0.6 5 0.4 f 0.2 1141111 0 3 I l -0.2 . -0.4 f Relative Amplitude O6 2 -0.8 5 Time (ns) Figure 2-16 Response waveform for material ‘P’. 56 0.6 _ ; Material 'H' 0'4 j - - - Material 'G' ------ Material '1' 11144 0.2 Relative Amplitude ’1.2d fTTIIIIIIIIIIIIIIIIIITIfIr11IIIrI 0 0.5 1 1.5 2 2.5 3 3.5 Time (ns) Figure 2-17 Comparison of response waveforms for materials ‘G’, ‘H’, and ‘I’. 57 Relataive Amplitude 0.02 0.01 5 14411411 -001 ~ -002 E -003 f -0.04 : > Late time -005 ’ Figure 2-18 Convolution of material ‘A’ response with baseline E-pulse. TTTFTTfIITIIITTITTTITTTTT 0.5 1 1.5 2 Time (ns) 58 2.5 3.5 0.03 _ 0.02{ i 0.01 : I, : ‘3 0: 1 a 3 WI 5 -0.01 9 Late time o : .2 . % -0.023 a: I -0033 0045 -o.05‘.........,....22E,,11 0 1 2 3 4 Time(ns) Figure 2-19 Convolution of material ‘P’ response with baseline E-pulse. 59 0.01 -0.01 j Late time 0 1 1': B Z i .002 « E -< < 0 ‘1 {E 003 ~ .2 I Q 9‘ 1 -0.04 a I -005 ] 006’T4....,....-.--.T-,,.,,,,,,,,,,,p, 0 0.5 l 1.5 2 2.5 3 Time (ns) Figure 2-20 Convolution of material ‘G’ response with baseline E-pulse. 60 Relative Amplitude Late time -0.07 I r fiT T r T ‘1' fi I I I I I 0 0.5 1 Figure 2-21 Convolution of material ‘1’ response with baseline E-pulse. 1.5 I I I I I I T I T I 2 Time (ns) 61 2.5 3.5 To quantify the results, the EDNa values are calculated. The values of EDNa for each of the materials examined are shown in Table 2-5. Figure 2-22 shows a plot of the tabulated EDNa calculations. It can be seen that as the change in thickness from the baseline increases the EDNa calculations increase as well. Determining how much change in permittivity can be detected using the EDNa plots depends, again, on the noise level. Material 8, EDNa A 4 4.80E-03 B 5 3.89E-03 C 6 2.6 lE-03 D 7 1 .29E-03 E 8 3.40E-04 F 8.5 8.65E-05 G 8.75 2.25E-05 H 9 3.7OE-O7 I 9.25 1.9OE-05 J 9.5 7.69E-05 K 10 3.01E-O4 L 1 l 1.09E-O3 M 12 2. 14E-03 N 13 3 .23E-03 P 14 4.22E-03 62 Table 2-5 EDNa calculations for change in permittivity. 0.006 L441 0.005 3 0.004 3 1 4 EDNa 0.003 5 0.002 3 0.001 3 0 T T I F r j I I T I T I T T r I I I I I F F F‘IT r fiTiT I 3 5 7 9 11 13 15 Relative Permittivity Figure 2-22 EDNa vs. change in relative permittivity for single lossless layer. 63 2.2.3 Change in Aspect Angle The next property to be investigated is the angle of incidence of the transient excitation. One of the advantages of the E-pulse in discriminating radar targets is that it is independent of the aspect angle of the incident field. In applications to layered material it is expected that the aspect angle should have some effect on the EDNa, although it is hoped that the effect is small if the change in angle is small. Table 2-6 and Table 2-7 list the material properties used to create the response waveforms with changing aspect angle. The following examples involve noise free responses. The baseline material is again the same as for the two previous cases of changing thickness and permittivity. These properties are shown in Table 2-1. Several scattered responses of layered materials were again simulated using MLPREF.for. Since the polarization of the incident wave affects the response from the material both perpendicular and parallel polarizations are considered. The properties of the layered materials for parallel polarization are shown in Table 2-6. The properties of the layered materials for perpendicular polarization are shown in Table 2-7. For each polarization, several angles were simulated beginning from 0° to 10° in increments of 1° and then from 10° to 45° in 5° increments. For both parallel and perpendicular polarizations, the angles 0°, 5° and 45° are shown as examples. The baseline material response for parallel polarization, material ‘A’, is shown in Figure 2-23. The baseline material response for perpendicular polarization, material ‘AA’, is shown in Figure 2-24. The material responses for an aspect angle of 5°, material ‘F’ and material ‘FF’ are shown in Figure 2-25 and Figure 64 2-26, respectively. The material responses for an aspect angle of 45°, material ‘8’ and material ‘SS’ are shown in Figure 2-27 and Figure 2-28, respectively. Each response is convolved with the baseline E-pulse shown in Figure 2-8. Figure 2-29 and Figure 2-30 show the convolution of the baseline material, material ‘A’ response and material ‘AA’ response, respectively, with the baseline E-pulse. It can be seen that the energy in the late-time is again zero as expected. In comparison, the convolutions of the other responses are shown in Figure 2-31, Figure 2-32, Figure 2-33, and Figure 2-34. It can bee seen in Figure 2-31 and Figure 2-32 that for a small change in the aspect angle the late-time energy stays relatively low. However, as the angle gets very large the late-time energy increases. Since the natural frequencies of a layered material depends on 61-, the E-pulse method is not independent of the aspect angle for layered materials. This is because a layered material is not a finite—sized body and does not allow the excitation to completely pass beyond the material, as in the case of a radar target. 65 Material 8 p 0 Thickness Polarization Angle of (mm) incidence, 9r A(baseline) 980 po 0 10 Parallel O B 930 P0 0 10 Parallel 1 C 980 pg 0 10 Parallel 2 D 980 po 0 10 Parallel 3 E 950 pg 0 10 Parallel 4 F 980 po 0 10 Parallel 5 G 980 po 0 10 Parallel 6 H 980 P0 0 10 Parallel 7 I 980 [.1 0 0 10 Parallel 8 J 980 1.10 0 10 Parallel 9 K 980 no 0 10 Parallel 10 L 980 no 0 10 Parallel 15 M 980 po 0 10 Parallel 20 N 930 HO 0 10 Parallel 25 P 980 pg 0 10 Parallel 30 Q 980 pg 0 10 Parallel 35 R 980 po 0 10 Parallel 40 S 980 P0 0 10 Parallel 45 Table 2-6 Material properties for changing angle, parallel polarization. 66 Material 8 u 6 Thickness Polarization Angle of (mm) incidence, 9r AA(baseline) 980 [.10 0 10 Perpendicular 0 BB 980 pg 0 10 Perpendicular 1 CC 980 no 0 10 Perpendicular 2 DD 980 no 0 10 Perpendicular 3 BB 980 pg 0 10 Pemendicular 4 FF 980 po 0 10 Perpendicular 5 GG 920 no 0 10 Perpendicular 6 HH 980 1.10 O 10 Perpendicular 7 II 980 no 0 10 Perpendicular 8 J] 980 pg 0 10 Perpendicular 9 K 930 no 0 10 Perpendicular 10 LL 980 po 0 10 Pemendicular 15 MM 930 no 0 10 Perpendicular 20 NN 980 P0 0 10 Perpendicular 25 PP 980 po 0 10 Perpendicular 30 QQ 980 pg 0 10 Perpendicular 35 RR 980 po 0 10 Perpendicular 40 SS 980 pg 0 10 Perpendicular 45 Table 2-7 Material properties for changing angle, perpendicular polarization. 67 lllLLJ_LJllll lllJ :> i -0.2 f -0.4 ; RelativeAmplitude -0.6 lrLarlrlr I F r114441 _: .1o2dl"'ll'lllT'l'ITlTYI1TTTTITII 0 0.5 1 1.5 2 2.5 Time (ns) Figure 2-23 Response waveform for material ‘A’. 68 Relative Amplitude '0.8 q 0.6 f 0.4 : 0.2 f 0.5 1 1.5 2 2.5 Time (ns) Figure 2-24 Response waveform for material ‘AA’. 69 r l r 1 r l 1 l r r r l r r T T r r l l r 1 3.5 Relative Amplitude 0.8 . 0.6 3 0.4 5 0.2 O I .O O\ lrir T r T l I l T T T T rj T T I T T T T I T T T T T T l T TTfi T T 0.5 1 1.5 2 2.5 3 Time (ns) Figure 2-25 Response waveform for material ‘F’. 70 3.5 0.8 1 0.6 5 0.4 lLl_Lllll Relative Amplitude b N '1.2 T“'TYTTTTVYTYTITTTTTWIIIrrlr 0 0.5 l 1.5 2 2.5 Time (ns) Figure 2-26 Response waveform for material ‘FF’. 71 0.8 0.6 r l LL44 1111 0.4 0.2 f O D l -0.2 j .04 : Relative Amplitude -0.6 1 -0.8 3 u “ -1,2 TTIT[llllllllf[TrlfTTlflIllTlllTTTTTTTT 0 0.5 l 1.5 2 2.5 3 3.5 Time (ns) Figure 2-27 Response waveform for material ‘8’. 72 0.8 , 0.6 j 0.4 3 0.2 cut u -r I q —1 Relative Amplitude b N -0.6§ -O.8~i “u -12 m T .Tf... . m M. 0 05 1 15 2 25 3 35 Time (ns) Figure 2-28 Response waveform for material ‘SS’. 73 0.02 0.01 5 05—10— . ——> ' _0.01 1 Late trme -0.02 ‘ 003 j ml11414 I Relative Amplitude -0.04 llJLllli —0.05 q -0.06 f 007T o 0.5 1 1.5 2 2.5 3 Time (ns) Figure 2-29 Convolution of material ‘A’ response with baseline E-pulse. 74 0.02 . 0.01 3 Oj—jfl ._ 001 ~ —* Late time LJ L1 -0.02 3 U -o.03 3 Relative Amplitude -o.04 f -0.05 f -0.06 5 007- 0 0.5 1 1.5 2 2.5 3 Time (ns) Figure 2-30 Convolution of material ‘AA’ response with baseline E-pulse. 75 Relative Amplitude 0.02 q 0.01 5 .1 (is—j fl —. -0.01 : “———* Late time 4 -002; i H -0.03 2 1 —o.04 3 3 -0.05 : 1 .J -0.06 { -O-O7-‘TTT'TTTTTTIYTYIIITYTTYTY] rrrrrrrrr 0 0.5 1 1.5 2 2.5 3 Time (ns) Figure 2-31 Convolution of material ‘F’ response with baseline E-pulse. 76 002, 0013 4 E101“ -0.01 ~ —" Late time -002} -0035 Relative Amplitude -oo4§ .1 -0055 -0063 -007“ ... .,., .....v.s. --,. ,.., ,,,, .,,...-. o 05 1 15 2 25 3 Time (ns) Figure 2-32 Convolution of material ‘FF’ response with baseline E-pulse. 77 Relative Amplitude 0.02 q 0.01 f -0.01 : Late time -0.02 1114;1111L1 -0.03 11111 -0.04 -0.05 f '0.06 TTTTTTTTTT7T7TFTTTTTTTYiTTTTTTTYTYrTTTYY 0 0.5 1 1.5 2 2.5 3 3.5 Time (ns) Figure 2-33 Convolution of material ‘8’ response with baseline E-pulse. 78 (102 , (101 1 1 0 g -001 Late time —0.02 -0.03 llllLL+llJ_l_LlLL1_Llll Relative Amplitude 4104 5 l 4105-3 4106-3 4107 ‘. .. ... ... ., ,., ,, ,.. .. ,,, ,, ,,, ,, ,., ,2.21. 0 (15 1 15 2 :15 3 :15 Time (ns) Figure 2-34 Convolution of material ‘SS’ response with baseline E-pulse. 79 4 Table 2-8 shows the calculations for EDNa versus aspect angle for parallel polarization and Table 2-9 shows the calculations for EDNa versus aspect angle for perpendicular polarization. A plot of the EDNa values for both polarizations is shown in Figure 2-35. It can be seen in the plot that there is minimal change in the EDNa value up to 10°-15° (depending on the definition of minimal). This suggests that a moderate change in aspect angle is tolerable. Typical applications of the E-pulse technique for layered material evaluation would generally be such that minimal deviation in aspect angle from the baseline angle would be seen. A measurement device built to implement this technique could easily be designed to limit changes in the aspect angle of less than a couple of degrees. Thus, the problem of the aspect angle dependent nature of the EDNa could be eliminated. 80 Material Aspect Angle, 01 EDNa A 0 3.70E-07 B 1 3.72E-O7 C 2 3.78E-07 D 3 3.90E-07 E 4 4. 13E-O7 F 5 4.51E-07 G 6 5.1 1E-07 H 7 6.02E-07 I 8 7.32E-07 J 9 9.14E-07 K 10 1.16E-06 L 15 3.88E-06 M 20 1.08E-05 N 25 2.48E-05 P 30 4.93E-05 Q 35 8.75E-05 R 40 1.43E-04 S 45 2.17E-O4 Table 2-8 EDNa calculations for change in aspect angle for parallel polarization. 81 Material Aspect Angle, 91 EDNa AA 0 3.70E-O7 BB 1 3.72E-O7 CC 2 3.78E-O7 DD 3 3.92E-07 EE 4 4.15E-O7 FF 5 4.55E-O7 GG 6 5.17E-O7 HH 7 6.10E-O7 II 8 7 .45E-07 JJ 9 9.31E-07 KK 10 1.18E-O6 LL 15 3.99E-O6 MM 20 1.13E-05 NN 25 2.69E-05 PP 30 5.58E-05 QQ 35 1.06E-04 RR 40 1.87E-04 SS 45 3.17E-04 Table 2-9 EDNa calculations for change in aspect angle for perpendicular polarization. 82 4.0E-04 1 * — perpendicular polarization 3513-04 3 +para11e1 polarization 3013-04 { 2.5E-O4 : 2.0E-04 j EDN a 1.5E-04 : 1.0E-04 { 5013-05 3 0.0E+OO ‘ 50 Aspect Angle (degrees) Figure 2-35 EDNa vs. change in aspect angle for single lossless layer. 83 2.2.4 Change in the Number of Natural Modes The final parameter to be investigated for the noise-free data is the number of natural modes, or natural frequencies, that are used to generate the E-pulse. As shown by (2.10), there are an infinite number of natural frequencies, or modes, that can be calculated for a layered material. It is obvious that only a finite number of these modes can be used to reconstruct a response and generate an E-pulse. Therefore, it is necessary to determine the optimum number of modes to use. Because using the natural modes to reconstruct the response waveform is an estimation, it is most likely that using a greater number of modes would yield a more accurate estimate. However, not only do the modes reconstruct the response, they are also used to generate the E-pulse. Presently, the investigation is to determine the optimum number of modes used to generate an E-pulse which will give the most accurate EDNa calculations. To determine the optimum number of modes needed to generate an E-pulse, a baseline response will be convolved with many natural E-pulses that have been generated with KPNAT.for using different numbers of modes. Table 2-1 shows the material properties used to reconstruct a baseline response waveform using (1.1) and MLPREF.for. Figure 2—4 shows a plot of this waveform. Ten different E-pulses were generated, each with a different number of modes, as shown in Table 2-10. Examples of three of these E-pulses are shown in Figure 2-36, Figure 2-37, an Figure 2-38. Figure 2-36 shows an E-pulse generated with only one mode, Figure 2-37 shows an E—pulse generated with 5 modes, and Figure 2-38 shows an E-pulse generated with 10 modes. As the number of modes increases the E-pulses are seen to become narrower and when 84 convolved with the response waveform yield a smaller value of the convolution. The convolution of the baseline response with these three E-pulses can be seen in Figure 2-39, Figure 2-40, and Figure 2-41, respectively. 85 Number of Modes 1 \OOOQOLA-DDJN B C D E F G H I J p—n O Table 2-10 E-pulses generated using different modes. 86 2.5 2 2 Q 4 “g - E 1.5 - a .. E < . Q) .2. 1 3 1 ~ 6) -< a: 0.5 2 O . T 0.00 0.20 Time(ns) Figure 2-36 E-pulse generated using 1 mode. 87 2.5 1.5 a Relative Amplitude l 0.5 O T 1 I [ 1 1 l 1 7 l T ‘1— T T I T I l v 0.00 0.02 0.04 0.07 0.09 0.11 0.13 0.16 0.18 0.2 Time(ns) Figure 2-37 E-pulse generated using 5 modes. 88 2.5 i 23. Q 3 “g 3 £1.5— Q‘ .. E . < . Q .2. ‘ 3 1- o . a: 0.54 O VIIIYIIIIIIIIIITITTTTTITTTVTIII 0.00 0.03 0.06 0.09 0.13 0.16 Time (ns) Figure 2-38 E-pulse generated using 10 modes. 89 Relative Amplitude 0.01 . i -001 f 1 —0.02 f -003 f -004 f -0.05 f .4 -0.06 3 ”T’WN ‘——' Late time -007 ‘ 0 Figure 2-40 Convolution of baseline response with E-pulse ‘E’. l I T T T T l l T T T T T T T l T T T T l T T T T 0.5 1 1.5 2 2.5 3 Time (ns) 91 3.5 0.01 q -001 1 -0.01 —0.02 5 Relative Amplitude -002 1 —0.03 3 .J 0.00 j——————W ‘—' Late time —o.03 ‘ . Figure 2-41 Convolution of baseline response with E-pulse ‘J’. ITIIIIFfTI 1 1.5 T fj T 2 Time (ns) 92 As done previously, evaluation begins by tabulating the EDNa values, as seen in Table 2-11. These EDNa values are plotted against the number of modes in Figure 2-42. It can be seen from both Table 2-11 and Figure 2-42 that all of the E-pulses except the one generated using only one mode give a value very close to zero. This evidence suggests that using 5 modes to generate the E-pulses is more than sufficient. A quick way to check that the proper number of modes is being used is to compare the imaginary parts of the natural frequencies, (1),, , to the frequency range used to generate the responses. By comparing the natural frequencies in Table 1-1 to the windowed frequency data in Figure 2-3 it can quickly be seen that only the first 5 modes are within the frequency range of the response data. E-pulse Number of Modes EDNa A 1 3.32E-02 B 2 2. l2E-O4 C 3 3.26E-O6 D 4 6.63E-07 E(baseline) 5 3.70E-07 F 6 2.76E-07 G 7 2.27E-O7 H 8 1 9515-07 I 9 l .72E-O7 J 10 1.55E-07 Table 2-11 EDNa calculations for change in the number of modes. 93 0.04 _ 0.035 0.03 0.025 0.02 5 EDNa 0.015 0.01 11L441111L 0.005 f Number of Modes Figure 2-42 EDNa vs. change in the number of natural modes in the E-pulse. 94 2.5 a Q 'g . 5 1.5 — a .1 E . < . Q) .2 r 3 1— — q.) . a: - , 0.51 O I I I I I I I I I I I I I I I T I I I 0.00 0.02 0.04 0.07 0.09 0.11 0.13 0.16 0.18 0.20 Time(ns) 88 Figure 2-37 E-pulse generated using 5 modes. 2.5 1 24.. J O i "S 1 51.5— a‘ a E < _ Q) .2. - 31— Q m 4 0.5] 1 O lllJTTTTVTTTTTTTTTTTTTJTTT‘T'TTT' 0.00 0.03 0.06 0.09 0.13 0.16 Time (ns) Figure 2-38 E-pulse generated using 10 modes. 89 Relative Amplitude 0.02 . -002 g —0.04 5 -0.06 f 1 -0.08 g -0.1 f -012 5 0.14 3 .1 -O.16 5; 05—1 2 Late time -0.18 d 0 T T T T T T T T T j j I T T T T I j T T T l 1.5 2 2.5 3 Time (ns) Figure 2-39 Convolution of baseline response with E-pulse ‘A’. 90 3.5 Relative Amplitude 0.01 ‘ -001 i -0.02 f 0.03 -0... g -0.06 5 +4 —> Late time -0.07 ‘ Figure 2-40 Convolution of baseline response with E-pulse ‘E’. TfT TTTTTTTTT T T T TTTTTTijfTTTrffrT T T T 0.5 l 1.5 2 2.5 3 Time (ns) 91 3.5 Relative Amplitude 0.01 q 0.00 3 —0.01 L11 -0.01 11111 -0.02 111 -0.02 111 —0.03 { —‘2 Late time -0.03 Figure 2-41 Convolution of baseline response with E-pulse ‘J’. 1 1.5 2 Time (ns) 92 3.5 As done previously, evaluation begins by tabulating the EDNa values, as seen in Table 2-11. These EDNa values are plotted against the number of modes in Figure 2-42. It can be seen from both Table 2—11 and Figure 2-42 that all of the E-pulses except the one generated using only one mode give a value very close to zero. This evidence suggests that using 5 modes to generate the E-pulses is more than sufficient. A quick way to check that the proper number of modes is being used is to compare the imaginary parts of the natural frequencies, a)” , to the frequency range used to generate the responses. By comparing the natural frequencies in Table 1-1 to the windowed frequency data in Figure 2-3 it can quickly be seen that only the first 5 modes are within the frequency range of the response data. E-pulse Number of Modes EDNa A 1 3.32E-02 B 2 2. 12E-O4 C 3 3.26E-06 D 4 6.63E-07 E(baseline) 5 3.70E-07 F 6 2.76E-07 G 7 2.27E-O7 H 8 l .95E-O7 I 9 l .72E-07 J 10 1.55E-07 Table 2-11 EDNa calculations for change in the number of modes. 93 0.04 . 0.035 0.03 3 0.025 0.02 3 EDNa 0.015 rlmurrlrr 0.01 0.005 j O 14 L 1 ’OooosqTTTTTTTTTTDTTTTITTITIIIIIYITI 0 2 4 6 8 10 12 Number of Modes Figure 2-42 EDNa vs. change in the number of natural modes in the E-pulse. 94 2.3 Effects of Noise on EDNa So far it has been shown that the E-pulse technique can diagnose changes in material properties for noise-free models. Practical measurements, however, are not noise—free. Thermal noise from the test equipment as well as background noise from the atmosphere are typical sources of interference that must be considered. The noise model applied to the simulation data is discussed. Then this noise model is used to evaluate how well the E-pulse technique works to diagnose changes in layer thickness and permittivity using noisy response data. 2.3.1 Noise Model A zero—mean gaussian noise model with probability density function, 1 ex2/20'2 , 2 —oo0, (2.15) fx (x): 2720' where 0'2 is the variance, was used to corrupt the response data. In the case of zero- 2 mean gaussian noise, a is also called the time average power [8]. For transient signals we define the time-average power as, I s2 (t) dt Signal power 2 ET , (2.16) where 5(1) is the noise-free response signal and W is chosen as the minimum duration time window that contains 99% of the total energy of the noise-free data. Thus, the signal-to-noise ratio (SNR) is defined as, (2.17) 2 SNR(dB) =1010g10{ 0' Signal power} 95 Using (2.17) to determine the time-average power of the noise , 02 , yields, [52 (t)dt 2 __ W 0' -WIOSNR/10° (2.18) By substituting values for the SNR, the time-average power of the noise can be determined. It should be noted that in order to improve the statistics and ensure an accurate Gaussian model, 500 random samples of noise data were generated using (2.18). Each of these samples was then added to the response data and, following convolution, the EDNa resulting from each noisy response was calculated. The mean of these 500 EDNa values was then calculated and used as the EDNa. 2.3.2 EDNa/EDRa vs. Signal-to-Noise Ratio Since the effects of changing the material properties on the EDNa using noise-free data was shown in previous sections, the major discussion here will be to show the effects of noise on the EDNa. The effects of noise on the data for EDNa versus change in thickness and EDN a versus change in permittivity will be shown. However, the number of data points presented will be limited (e.g., only 5 layer thicknesses will be evaluated compared to 15 in the noise free case). This is in consideration of the amount of data that would be generated to evaluate every data point as in the case of the noise free data. For both the change in thickness and the change in permittivity the same baseline response will be used as an example. Seven different levels of noise were added to this baseline response waveform, seen in Figure 2-43. The levels of noise that were added to this response are shown in Table 2-12. The baseline response waveform is the same one 96 shown in Figure 24. One fact to note is that the duration of the response waveform in Figure 2-43 is less than the response waveform in Figure 2-4. The duration is determined such that the response waveform contains 99% of the energy contained in the original waveform. Plots of the noise corrupted response waveforms for three of the SNR values shown in Table 2-12 are shown as examples. Figure 2-44, Figure 2-45, and Figure 2-46 show the baseline response corrupted by noise such that the SNR is 30dB, 15dB, and 2dB, respectively. An E-pulse generated from the natural frequencies of the baseline response is shown in Figure 2-8. The resulting waveforms generated by convolving these three responses with this E-pulse are shown in Figure 2-47, Figure 2-48, and Figure 2-49. As expected, the late-time convolutions of the noise corrupted waveforms contain more energy. As the SNR decreases the late-time energy increases. It is intuitively obvious that as more noise is added to the data, it will become harder to detect changes in the thickness or even to identify a response with no changes. It is possible to detect changes in thickness or permittivity up to a certain level of noise corruption. As an illustration of how noise can affect a convolution waveform for a material, other than the baseline, consider Figure 2—50, Figure 2—51, and Figure 2—52. Figure 2-50 shows a plot of the noise-free convolution of material ‘D’ from Table 2-2 in section 2.2.1 with the baseline E-pulse. Figure 2-51 and Figure 2-52 show the same convolution with SNR of 15dB and 2dB, respectively. The effects of noise are evident in the figures. As the noise increases it becomes increasingly more difficult to determine whether the late- time energy is caused by a change in material properties or by noise. 97 Response Waveform Signal-to-Noise (SNR) Noise-free 30 25 20 15 10 5 :EQTlmUOw> 2 Table 2-12 SNR of response waveforms. 98 .0 .o .0 N b O\ 1 1 144 1 l 1 1 1 1 O 1111 Relative Amplitude -1.2 T I I I T I I T I l I I T T T T I I I l I I I T T T T I f Tfi I 0 0.2 0.4 0.6 0.8 1 1.2 Time(ns) Figure 2-43 Response waveform ‘A’. 99 Relative Amplitude .o .o .o N 13 O\ 1 1 O 1111111 -02 3 -04 : -O.6 { —0.8 ~ -1.2 ‘ TTTTTTTTTTTTTTTjTTTT 0.2 0.4 0.6 0.8 Time(ns) Figure 2-44 Response waveform ‘B’. 100 1.2 Relative Amplitude 0 0.2 0.4 0.6 0.8 1 1.2 Time(ns) Figure 2-45 Response waveform ‘E’. 101 0.7 , 0.5 1 1 Relative Amplitude 0 0.2 0.4 0.6 0.8 1 1.2 Time(ns) Figure 2-46 Response waveform ‘H’. 102 0.02 4 0.01 { 0 E -001 ‘ Late time 1 -0.02 111 -0.03 Relative Amplitude 1 -0.04 I -0.05 5 U i ‘0.06 I I I I I I I I I I I I I I T I I I I I I I I I I I I T r T I I f I 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Time(ns) Figure 2-47 Convolution of response waveform ‘B’ with baseline E-pulse. 103 0.02 0.01 f -0.01 Late time -0.02 f -003 3 Relative Amplitude -0.04 111111 -0.05 111 C -0.06 mmtmrm1114..........,....,.... 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Time(ns) Figure 2-48 Convolution of response waveform ‘E’ with baseline E-pulse. 104 0.02 0.01 -0.02 Relative Amplitude -0.04 -0.05 ’ -0.06 -001 Z -0.03 —1 j —‘ .1 q _ .1 ._I .1 _. .1 _, .. q .1 Late time 0 I T I I I I T I I . I l I I I 0.2 0.4 0.6 0.8 1 1.2 Time(ns) IITIIIIIIfI‘II r# 1.4 Figure 2-49 Convolution of response waveform ‘H’ with baseline E-pulse. 105 Relative Amplitude 0.02 0.01 3 111111 -0.01 1 -0.02 11111 -0.03 111 -0.04 l -0.05 U Late time -0.06 ‘ Figure 2-50 Noise-free convolution of material ‘D’ with baseline E-pulse. 0.2 0.4 T T f T 0.6 Time (ns) 106 1.2 0.02 q 0.01 § -0.01 111111 4 Late time 111 —0.02 -0.03 Relative Amplitude 1114141L1111 0 0.2 0.4 0.6 0.8 1 1.2 Time (ns) Figure 2-51 Convolution of material ‘D’ with baseline E-pulse with SNR=15dB. 107 0.02 . 0.01 o i -1101 i -002 1 Relative Amplitude -003 j -004 f -005 { Late time -0.06 ° 0 7 f7 T T T T T T T T T T j T T T T T T 0.2 0.4 0.6 0.8 1 Time (ns) 1.2 Figure 2-52 Convolution of material ‘D’ with baseline E-pulse with SNR=2dB. 108 Table 2-13 shows the EDNa values for several SNR and thicknesses. Table 2-14 shows the EDRa values calculated from those EDNa values in Table 2-13. Table 2-15 shows the EDNa values for several SNR and values of permittivity. Table 2-16 shows the EDRa values calculated from those EDNa values in Table 2-15. Following the tabulated EDNa and EDRa values are their plots. Figure 2-53 shows a plot of the EDNa versus SNR for several thicknesses and Figure 2-54 shows the EDRa versus SNR for several thicknesses. Figure 2—55 shows a plot of the EDNa versus SNR for several values of the permittivity and Figure 2-56 shows the EDRa versus SNR for several values of permittivity. It can be seen in the EDNa plots, Figure 2-53 and Figure 2—55, that as the SNR increases, the EDNa value for the baseline case approaches zero while the EDNa values for the other cases get larger. This shows that as the SNR improves it becomes easier to detect changes in the thickness and permittivity. In the EDRa plots, Figure 2-54 and Figure 2—56, a similar observation can be made. As the SNR increases, the EDRa value for all of the cases except the baseline increases. The baseline case, of course, will always have an EDRa of zero. Again, this increase in EDRa suggests that as the SNR improves it becomes easier to detect changes in the thickness or permittivity. A particular amount of change in the thickness or permittivity would determine an EDRa threshold value. For example, if it was desired to detect a change of 1mm or greater in thickness one could use Figure 2-54 to determine what threshold value of EDRa to set for a particular SNR. While it is possible to detect changes in material properties for good SNR, it is anticipated that at very low SNR it will be impossible to 109 detect small changes. 110 Thickness 2dB SdB 10dB 15dB 20dB 25dB 30dB (mm) SNR SNR SNR SNR SNR SNR SNR 8 2.49E-03 2.92E-03 3.34E-03 3.5 lE-03 3.58E-03 3.60E-03 3.60E-03 9 1.09E-03 1.13E-03 1.18E-03 1.21E-03 1.22E-03 1.22E-03 1.22E-03 10 3.50E-04 2.07E-04 8.47E-05 2.89E-05 1.01E-05 4.28E-06 2.31E-06 11 1.13E-03 1.17E-03 1.19E-03 1.21E-03 1.22E-03 1.22E-03 1.22E-03 12 2.6 lE-03 3 .03E-03 3.4 113-03 3.57E—03 3.63E-03 3.64E-03 3655-03 Table 2-13 EDNa for numerous SNR and thicknesses. Thickness 2dB SdB 10dB 15dB 20dB 25dB 30dB (mm) SNR SNR SNR SNR SNR SNR SNR 8 8.52 1.155+01 1.60E+01 2.08E+01 2.5513+01 2.921~:+01 3.191~:+01 9 4.93 7.37E+00 1.14E+01 1.62E+01 2.08E+01 2.4SE+01 2.72E+01 10 0.00 0.00 0.00 0.00 0.00 0.00 0.00 l l 5.09 7.52E+00 1.15E+01 1.62E+01 2.08E+01 2.45E+01 2.72E+01 12 8.73 1.17E+01 1.60E+01 2.0913+01 2.56E+01 2.93E+01 3.20E+01 Table 2-14 EDRa for numerous SNR and thicknesses. 111 a, 2dB SdB 10dB 15dB 20dB 25dB 30dB SNR SNR SNR SNR SNR SNR SNR 1.1 1E-O3 1.25E-03 1.38E-03 1.43E-03 1.45E-03 1.46E-03 1.46E-03 8 4.96E-04 4.52E-04 3.98E-04 3.99E-04 3.92E-04 3.90E-04 3.89E-04 2.28E-04 2.03E-04 8.1 113-05 29213-05 1.02E—05 3.57E-06 2.22E-06 10 6.12E-04 5.24E-04 4.43E-04 4.08E-04 3.96E-04 3.93E~04 3.92E-04 1 1 1.23E-03 1.36E-03 1.3lE-03 1.36E-03 1.36E-03 1.38E-03 1.37E-03 Table 2-15 EDNa for numerous SNR and values of relative permittivity. a, 2dB SdB 10dB 15dB 20dB 25dB 30dB SNR SNR SNR SNR SNR SNR SNR 6.87 7.89E+00 1.23E+01 1.69E+01 2.15E+01 2.61E+01 2.82E+01 3.38 3.48E+00 6.91E+00 l.l4E+01 1.58E+01 2.04E+01 2.24E+01 0.00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 10 4.29 4.12E+00 7.37E+00 1.15E+01 1.59E+01 2.04E+01 2.25E+01 11 7.32 8.26E+00 1.2 lE+01 1.67E+01 2.12E+01 2.59E+01 2.79E-1-01 Table 2-16 EDRa for numerous SNR and values of relative permittivity. 112 0.004 0.0035 : 0.003 5 0.0025 0.002 f EDNa 0.0015 { +8mm +9mm + 10mm -0- 11mm +12mm 0.001 5 0.0005 { 0 5 10 15 SNR (dB) _ .I .I O TTTTTTTjTT IT I TTj’TTTTTT 35 Figure 2-53 Comparison of EDNa vs. SNR for several thickness. 113 40 EDRa(dB) 35 : +t=8(mm) : +t=9(mm) 3O? +t=10(mm) j +t=11(mm) 25: +t=12(mm) 4 20f 15{ 10} 5? 0:**T—F¥TTTT*TTTT*TTTT*—r—TTT¥TTTT¥I TIT 0 5 10 15 20 25 30 SNR(dB) Figure 2-54 Comparison of EDRa vs. SNR for several thicknesses. 114 35 0.0016 4 3 -B- 8r=7 0.0014 - -)(—er:8 : + 8r=9 0.0012 9 + 81:10 + 81:11 0.001 -+ a 2 0.0008 ~ In 1 0.0006 1 0 5 10 15 20 25 30 35 40 SNR (dB) Figure 2-55 Comparison of EDNa vs. SNR for several values of permittivity. 115 EDRa(dB) ()4 TfiwIXIrer—r—r—w—I—fi-v—v—r—r—K—r—r—i—X—v—r—H—X—f—v—r—r—r 0 5 10 15 20 25 3O 35 SNR(dB) Figure 2-56 Comparison of EDRa vs. SNR for several values of permittivity. 116 Chapter 3 Single Lossy Layer The goal of this chapter is to explore how the conductivity of the material layer affects our ability to determine changes in layered material properties. The chapter begins with an explanation of how to determine the natural frequencies of conductive material. A similar approach to that used in chapter 2 is used, the difference being that for conductive material the permittivity is complex. The remainder of the chapter shows the results of using the E-pulse to determine changes in the thickness, permittivity, angle of incidence, and conductivity for noise-free data. The last section will show the effects of noise on the use of the EDNa. 3.1 Theoretical Determination of Natural Frequencies of Conductive Material To calculate the natural frequencies for materials with conductivity not equal to zero requires a slightly different evaluation. Begin by letting .0' 81(0))=8081,-j-(; ...0'¢0'(a)) (3.1) or, 0’ 81(s)=£081,+—S-. (3.2) Therefore, 117 [ ’70 , ...perpendicular polarization 0 . 2 \[61r+———sm 61 J 350 z = 1 £1, +1-sin2 6,- 350 . . 770 a , ...parallel polarization €Ir+_ i 550 Once again, the poles of (1.50) determine the natural frequencies. Therefore, 1— “fie—”(3) = O where d . 2 __ +__ 9. 1(5) CJEI, S80 8111 , —2dF(s) c where 01' 21 (s)+Zo ‘(Zl (s)—Zo)e_sr(s) =0 The solution to (3.9) for the natural frequencies depends on the polarization of the incident wave. Given the incident wave has perpendicular polarization produces the following expressions; (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) (3.10) 20 = ——I70 , cos 6,- 2l (s)+ZO -(zl (s)—Zo)e’”(5) = 0. Thus, ’70 + 770 _ 770 _ 770 e—sz‘(s):0 F(s) cosfli F(s) c0319,- 01' c0565- + F(s) -(cos(9i — F(S))e-sr(s) O . Given the incident wave has perpendicular polarization produces the following expressions; F(S) Z _ , 20:770cos61i, 0' G = +—, (S) Elr 580 Zl(S)+ZO—(ZI(S)—Zo)e—ST(S) =0. Thus, F s F s _ 770Gésg+nocosfii—(UOG—E:Y—;—nocosfli]e 510) :0 or F(s)+G(s)cos€i —(F(s)—G(s)cos61i)e-ST(3) =0 If the incident angle of the excitation is zero, then 119 (3.11) (3.12) (3.13) (3.14) (3.15) (3.16) (3.17) (3.18) (3.19) (3.20) H(s)=F(s)l6i:O= 0(5) (3.21) and (3.14) and (3.20) reduce to the identical expression 1+H (s)-[1—H(s)]e“"(‘) (3.22) where 7(5) =EH(S). (3.23) C Equation (3.14), (3.20), and (3.22) cannot be solved in closed form but can be solved numerically. The numerical solution to these equations is performed using the software program NFSIG.for, included in appendix E, written by Dr. Ed Rothwell. 3.2 Computation of EDNa and EDRa Quantifying the results in chapter 3 is done exactly the same way as was done in chapter 2. The EDNa is used to quantify the results for the noise-free data and the EDRa is used to quantify the results for the noise-corrupted data. 3.2.1 Change in Layer Thickness The material analyzed in this section is essentially the same material analyzed in section 2.2.1. The difference is that the material analyzed in this chapter has a non-zero conductivity (0' = 0.5). The properties of the materials evaluated are shown in Table 3-1. The thickness ranges from 5mm to 15mm with the baseline material having a thickness of 10mm. The baseline material is used to generate the E-pulse and is the material used for comparison in determining any changes in the thickness. Keeping in mind the number of plots that would be required to illustrate all of the material configurations, only a representative sample was chosen. The two extreme cases 120 of change in thickness as well as the two cases with minimal change in thickness were chosen to illustrate the response curves and the convolution curves. Figure 3-1 and Figure 3-2 show the response waveforms of the extreme cases while Figure 3-3 shows a comparison of the responses of the two cases with thickness nearest to the baseline with the baseline response. It can be seen that when the thickness changes by more than 5mm in either direction the response waveform changes significantly. A good example of the effects of the conductivity can be seen by comparing Figure 3-2 with Figure 2-6. The response waveform in Figure 2-6 is that of a material with zero conductivity, but with all of the same properties otherwise as the material whose response waveform is shown in Figure 3-2. This comparison clearly shows that the material with conductivity exhibits signal loss as the pulse travels through the material. As mentioned, the baseline material properties were used to generate the E—pulse, shown in Figure 3-4. This E—pulse was convolved with all of the response waveforms and the late-time energy of the convolution waveforms were used to determine the EDN a. Convolution of the E-pulse with the baseline response waveform is shown in Figure 3-5. It can be seen that, as expected, the late-time energy is near zero. The fact that it is not exactly zero can be attributed to the use of the number of modes to generate the E-pulse as well as round-off error. The convolutions of the E—pulse with the remaining illustrated cases are shown in Figure 3-6, Figure 3-7, Figure 3-8, and Figure 3-9. The materials with large deviation in thickness show a significant amount of late-time energy in their convolutions. In addition, late-time energy is seen in the Figure 3-8 and Figure 3-9 for the cases where the thickness was only slightly changed. 121 Material 2: u 0 Thickness Polarization Angle of (mm) incidence, 0. A 980 o 0.5 5 Parallel 0 B 980 pg 0.5 6 Parallel 0 C 980 110 0.5 7 Parallel 0 D 980 p0 0.5 8 Parallel 0 E 980 p0 0.5 9 Parallel 0 F 980 [.10 0.5 9.5 Parallel 0 G 980 p0 0.5 9.75 Parallel 0 H (baseline) 980 p0 0.5 10 Parallel 0 I 980 p0 0.5 10.25 Parallel 0 J 980 [.10 0.5 10.5 Parallel 0 K 930 p0 0.5 11 Parallel 0 L 980 [10 0.5 12 Parallel 0 M 930 p0 0.5 13 Parallel 0 N 980 110 0.5 14 Parallel 0 P 980 p0 0.5 15 Parallel 0 Table 3-1 Material properties for changing thickness. 122 .O .O 45 C\ r r 1 r J 1 r .9 N l 1 111_rlrrrr Relative Amplitude .é N #14411 -1.2 1 l T T T T l T T I T T I F I Time (ns) Figure 3-1 Response waveform for material ‘A’ 123 2.5 0.2 . 0 j ""\ -0.2 fl -o.4 -0.6 3 Relative Amplitude -1.21111.4...,...,, 0 0.5 l 1.5 Time (ns) Figure 3-2 Response waveform for material ‘P’ 124 0.4 _ - Material 'G' 0.2 q - - - Material 'H' ------ Material '1' O A" A a 1 l 3 -0 2 ~ . E: ' : < -0.4: '| I g - l {-3 -063 .. a: j n -0 8 _ -1 - 1 -1.2 ‘ , . . . T T 0 0.5 1 1.5 2 2.5 Time (ns) Figure 3-3 Comparison of response waveform for materials ‘G’, ‘H’, and ‘I’. 125 l9 LII 411111 E-pulse Amplitude. N 0.5 f 0.00 0.04 0.09 0.13 0.18 Time (ns) Figure 3-4 E-pulse generated from the baseline material properties. 126 Relative Amplitude 0.01 q o f -001 -002 -003 -0.04 - -0.05 j -0.06 j -007 -0.08 j fl —* Late time -0.09 ‘ 0 Time (ns) Figure 3-5 Convolution of baseline response with baseline E-pulse. 127 Relative Amplitude 0.02 . 0.01 - 1 0 fl ~ ’0-01 ‘ ' Late time -002 5 -0.03 n -0.04 f -0.05 : -0.06 E —0.07 11 h -0.08dwvraTIrwrrtwfif,..r., 0 0.5 1 1.5 2 Time (ns) Figure 3-6 Convolution of material ‘A’ response with baseline E-pulse. 128 Relative Amplitude 0.02 q 0.01 5 0 f -001 i N -002 i -003 -004 5 -005 i -0.06 -007 L1 :: 11111 -0.08 Late time 0.0931,... Figure 3-7 Convolution of material ‘P’ response with baseline E-pulse. Time (ns) 129 2.5 Relative Amplitude 0.01 q 0: \ -0.01 f —0.02 -003 -004 -005 -0.06 5 -0.07 1 u -0.08 : Late time -009, 1 1 . . 1 0 0.5 Figure 3-8 Convolution of material ‘G’ response with baseline E-pulse. Time (ns) 130 2.5 0.01 , 111111 -0.01 1111 -0.02 -0.03 1_L_111_LL1 -0.04 f -0.05 i 1 -0.06 — Relative Amplitude -0.07 j 008 f ( WC: 0 ‘—’ Late time -0.09 d 0 Figure 3-9 Convolution of material ‘1’ response with baseline E-pulse. Time (ns) 131 2.5 To quantify the results of the convolution late-time energy, the EDN a method of section 2.2 was used. Using (2.12), the EDNa values were calculated for all of the materials in Table 3-1. The resulting EDN a values are shown in Table 3-2. As expected, the baseline material had the smallest EDNa and as the thickness deviated from further from the baseline the EDNa increased. The EDNa values are plotted against the thickness in Figure 3-10. One distinction to note is the asymmetric property of the plot. This is due to the loss of the signal caused by the material. As the material becomes thicker the loss increases and thus the late-time response energy decreases. This results in a decrease in the late-time energy and thus a lower EDN a value. Figure 3—11 shows a comparison of the EDNa values calculated in this section to those calculated in section 2.2.1 for change in thickness of lossless material (0' = 0). Again, the effects of the material loss can be seen in the EDNa. When the material causes signal loss the EDNa values decrease overall. This fact will become very important in chapter 4 when magram materials are discussed. 132 Material Thickness EDNa A 5 l .92E-03 B 6 l .66E-O3 C 7 1.30E-03 D 8 6.82E-04 E 9 1.84E-04 F 9.5 4.59E-05 G 9.75 l. 16E-05 H 10 2.16E-07 I 10.25 9.98E-06 J 10.5 3.83E-05 K 1 1 1.35E-04 L 12 3 .76E-04 M 13 5.33E-04 N 14 5.69E-04 P 15 5.41E-04 133 Table 3-2 EDNa calculations for change in thickness of lossy material. EDNa 2.00E—03 , 1.80E-03 j 1.60E—03 1.401303 3 1.20E-03 j 10012-03 8.00E-04 6.00E-04 40013-04 2.00E-04 : 0.00E+00_wrrv.Irrtwrtv..jrr......... 4 6 8 10 12 14 Thickness (mm) Figure 3-10 EDNa vs. change in thickness for a single lossy layer. 134 EDNa 90013-03 . 8.00E-03 7.001303 : 6.00E-03 l 5.00E-03 40013-03 30013-03 20013-03 10013-03 f 0.00E+00 ‘ 4 Figure 3-11 EDNa vs. change in thickness for lossless and lossy layers. 8 10 12 Thickness(mm) 135 —- 0:0.0 -‘3- 6:0.5 T T r T T T T T 14 16 3.2.2 Change in Layer Relative Permittivity Once again, the materials analyzed in this section are similar to those in section 2.2.2 except this section examines lossy materials (0’ = 0.5). The material properties for the samples in this section are shown in Table 3-3. The relative permittivity of the material samples range from 4 to 14. The baseline material used to generate the E—pulse and used for comparison has a relative permittivity of 9. The materials used to illustrate the response and convolution waveforms are the extreme values 4 and 14 as well as the values immediate to the baseline value, 8.75 and 9.25. Figure 3-12 and Figure 3-13 show the response waveforms for the extreme values, materials ‘A’ and ‘P’. For the smaller value, the permittivity is approaching that of air. Thus, the response is nearer to what would be expected from a metal plate. The initial reflection is off the material layer and air interface. This is followed by a reflection from the metal backing. Since the permittivity is low, most of the energy of the pulse is reflected by these two reflections. Any remaining energy within the layer is quickly able to escape the material layer. The layer with the higher permittivity initially reflects more energy at the material layer and air interface. Energy that enters the material layer takes longer to escape, hence the additional reflections that are seen. Figure 3-14 shows a comparison of the baseline response to the responses from materials with the smallest deviation in permittivity, materials ‘G’ and ‘1’. Virtually no difference can be seen in these response waveforms. An E-pulse is generated using the baseline material properties. This is the same E-pulse generated in the previous section and can be seen in Figure 3-4. This E-pulse was convolved with each material response. Figure 3—15 shows the resulting convolution 136 using the baseline response. As expected, the late-time energy is nearly zero. Figure 3-16 and Figure 3-17 show the waveform resulting from the E-pulse being convolved with the responses of the extreme values of permittivity, materials ‘A’ and ‘P’. The late- time energy in both of these instances is large because of the large change in permittivity. The last two convolutions are shown in Figure 3-18 and Figure 3-19. These are the E- pulse convolved with materials ‘G’ and ‘I’, the materials with the least change in permittivity. While the late-time energy is smaller than the extreme value convolutions, it is still evident by inspection. 137 Material 8 u 0 Thickness Polarizatio Angle of (mm) 11 incidence, 01 A 480 110 0.5 10 Parallel O B 580 110 0.5 10 Parallel 0 C 680 110 0.5 10 Parallel 0 D 780 110 0.5 10 Parallel 0 E 880 110 0.5 10 Parallel 0 F 8.580 110 0.5 10 Parallel 0 G 8.7580 110 0.5 10 Parallel 0 H (baseline) 980 110 0.5 10 Parallel 0 I 9.2580 110 0.5 10 Parallel 0 J 9.580 110 0.5 10 Parallel 0 K 1080 110 0.5 10 Parallel 0 L 1180 110 0.5 10 Parallel 0 M 1280 110 0.5 10 Parallel O N 1380 110 0.5 10 Parallel 0 P 1480 110 0.5 10 Parallel 0 Table 3-3 Material properties for change in relative permittivity. 138 0.2 q Relative Amplitude Time (ns) Figure 3-12 Response waveform for material ‘A’. 139 Relative Amplitude Time (ns) Figure 3-13 Response waveform for material ‘P’. 140 2.5 0.4 g 3 Material 'G' 0.2 — ---Material 'H' j ----- Material '1' O v. .1» 1 1 3 -02 4 ‘2'. . < -0.4f 4» . .2 . :3 -065 °‘ ‘ 1 -0.8 1 j _1 I l 1 '1.2 I T 1 l fi 0 0 5 1 1.5 2 2 5 Time (ns) Figure 3-14 Comparison of response waveform for materials ‘G’, ‘H’, and ‘I’. 141 Relative Amplitude 0.01 1 o i -0.01 i -0.02 -0.03 -004 —0.05 -0.06 3 -007 -0.08 E fl ‘—'* Late time -0.09 d 0 Figure 3-15 Convolution of baseline response with baseline E-pulse. I T T T T T T TIT Y I T T 0.5 1 1.5 Time (ns) 142 r r T 2.5 Relative Amplitude 0.01 _ 0 f -0.0l : -0.02 j 003 -004 i -0.05 j ~0.06 j -007 i -0.08 5 Late time -0.09 l 0 Figure 3-16 Convolution of material ‘A’ with baseline E-pulse. Time (ns) 143 2.5 O W 9 Late time Relative Amplitude Time (ns) Figure 3-17 Convolution of material ‘P’ with baseline E-pulse. 144 2.5 Relative Amplitude 0.01 . —1 0 d -1 -4 -001 5 11111 -0.02 11L11 -0.03 1111 —0.04 -0.05 1111;111 -0.06 f -0.07 1 -0.08 - ——* Late time -0.09 d 0 Time (ns) Figure 3-18 Convolution of material ‘G’ with baseline E-pulse. 145 Relative Amplitude 111114111111 1111 fl Late time Figure 3-19 Convolution of material ‘I’ with baseline E-pulse. 0.5 1 1.5 2 Time (ns) 146 2.5 The late-time energy contained in the convolution waveforms is quantified using the EDNa. Table 3-4 shows the values of EDNa calculated for each material configuration. It can be seen that the material with the lowest EDNa value is the baseline material ‘H’. Figure 3-20 shows the EDNa values plotted against the permittivity. An asymmetry exists in the plot similar to that in the EDNa plot for changing thickness. This is due to the ability of the energy to more easily escape the material layer with low permittivity causing a lower late-time energy relative to the convolution late-time energy of the materials with high permittivity. Figure 3-21 shows a comparison of this EDNa plot with that of the EDNa plot versus permittivity for the lossless case, shown in Figure 2-22. This plot shows the material loss effects the late-time energy for the change in permittivity just as it did for the change in thickness. Again, this fact will be further considered in chapter 4. 147 Material Permittivity EDNa A 4 6.01E-04 B 5 4.73E-04 C 6 3.22E-04 D 7 1.69E-04 E 8 4.77E-05 F 8.5 1.24E-05 G 8.75 3.39E-06 H 9 2. 16E-07 I 9.25 2.95E-O6 J 9.5 l. l8E-05 K 10 4.71E-05 L 1 1 1.78E-04 M 12 3.64E-04 N 13 5.69E-04 P 14 7.66E-04 148 Table 3-4 EDNa calculations for change in permittivity of lossy material. 9.00E—04 , 1 8.00E-04 : 7.00E-04 11411111 6.00E-04 _ 5.001504 3 EDNa 4.00E-O4 3 1 3.001504 5 20013.04 5 1.001304 5 0.00E+OO J l I I I 1 T J T I 7 T T T I l r T I T T T T T f fl f T T 3 5 7 9 ll 13 Relative Permittivity Figure 3-20 EDNa vs. change in relative permittivity for lossy layer. 149 5.00E-O3 , - — =0.0 4.50E-03 : +6=0.5 4.00E-03 - 3.50E-O3 -; 3.00E-03 111L444111 2.50E-03 EDNa 2.001303 3 15013-03 5 1001303 5 H 5.00E-O4 — 0.00E+00—1111..... 3 5 11 13 15 \1—111 \0 Relative permittivity Figure 3-21 EDNa vs. change in relative permittivity for lossless and lossy layers. 150 3.2.3 Change in Aspect Angle This section will discuss the effects of angle of incidence on the EDNa for lossy material layers. Table 3-5 shows the properties of the material layers investigated when the incident field had parallel polarization and Table 3-6 shows the properties when the field had perpendicular polarization. The angle of incidence was tested to a maximum of 45° for both polarizations. These material properties are the same as those in section 2.2.3 with the exception being the conductivity is not zero in this section. Three different angles of incidence are used to illustrate the response and convolution waveforms. Figure 3-22 and Figure 3-23 show the response waveforms for 0° angle of incidence for both parallel and perpendicular polarization, respectively. Figure 3-24 and Figure 3-25 show the response waveforms for 5° angle of incidence. Finally, Figure 3-26 and Figure 3-27 show the response waveforms for 45° angle of incidence. All of the response waveforms appear very similar, and thus indicate there is not a large effect to the response from the angle of incidence. However, the real test will be the difference in the EDNa values. Each response waveform was convolved with the E-pulse shown in Figure 3-4 and the late-time energy was determined. The convolution waveforms for the example responses are shown in Figure 3-28 through Figure 3-33. Figure 3-28 and Figure 3-29 show the convolution of the E-pulse with the response waveforms for material ‘A’ and ‘AA’. Figure 3-30 and Figure 3-31 show the convolution of the E-pulse with the response waveforms for material ‘F’ and ‘FF. And finally, Figure 3-32 and Figure 3-33 show the convolution of the E-pulse with the response waveforms for materials ‘5’ and ‘SS’. While the extreme cases of 45° angle of 151 incidence shows a significant amount of late-time energy the baseline case and the case of 5° angle of incidence show near zero late-time energy. By inspection it would not be possible to discriminate the 0° and 5° convolutions. The EDNa values will show the difference in the late-time energies. 152 Material 8 u 0 Thickness Polarization Angle of (mm) incidence, 91 A(baseline) 980 110 0.5 10 Parallel O B 980 110 0.5 10 Parallel l C 980 110 0.5 10 Parallel 2 D 980 110 0.5 10 Parallel 3 E 980 o 0.5 10 Parallel 4 F 980 110 0.5 10 Parallel 5 G 980 p0 0.5 10 Parallel 6 H 980 110 0.5 10 Parallel 7 I 980 110 0.5 10 Parallel 8 J 980 110 0.5 10 Parallel 9 K 980 110 0.5 10 Parallel 10 L 980 110 0.5 10 Parallel 15 M 980 110 0.5 10 Parallel 20 N 980 110 0.5 10 Parallel 25 P 980 110 0.5 10 Parallel 30 Q 980 go 0.5 10 Parallel 35 R 980 110 0.5 10 Parallel 40 S 930 110 0.5 10 Parallel 45 Table 3-5 Material properties for changing angle, parallel polarization. 153 Material 8 p 0 Thickness Polarization Angle of (mm) incidence, 0, AA(baseline) 930 110 0.5 10 Perpendicular 0 BB 980 p0 0.5 10 Perpendicular 1 CC 980 p0 0.5 10 Perpendicular 2 DD 980 110 0.5 10 Perpendicular 3 BB 980 110 0.5 10 Perpendicular 4 FF 980 110 0.5 10 Perpendicular 5 GG 980 110 0.5 10 Perpendicular 6 HH 980 110 0.5 10 Perpendicular 7 II 980 110 0.5 10 Perpendicular 8 JJ 980 110 0.5 10 Perpendicular 9 KK 950 p0 0.5 10 Perpendicular 10 LL 980 110 0.5 10 Pgendicular 15 MM 980 110 0.5 10 Perpendicular 20 N N 980 110 0.5 10 Perpendicular 25 PP 980 110 0.5 10 Perpendicular 30 QQ 980 110 0.5 10 Perpendicular 35 RR 980 110 0.5 10 Perpendicular 40 SS 980 110 0.5 10 Perpendicular 45 Table 3-6 Material properties for changing angle, perpendicular polarization. 154 °~ "1 .02 5 O -04 5 Relative Amplitude -0.6 { 0 0.5 1 1.5 2 Time (ns) Figure 3-22 Response waveform for material ‘A’. 155 2.5 Relative Amplitude 0.4 . 0.2 3 111111 -0.2 11111 -0.6 Time (ns) Figure 3-23 Response waveform for material ‘AA’. 156 2.5 0.4 d 0.2 § Relative Amplitude b A 111111 L411111l -0.6 -0.8 3 -1 l U -1.2 i . . - r . O 0.5 1.5 Time (ns) 157 Figure 3-24 Response waveform for material ‘F’. 2.5 0.4 11411 0.2 j 02 3 —0.4 ~ Relative Amplitude -0.6 f -0.8 3 Time (ns) 158 Figure 3-25 Response waveform for material ‘FF’. 2.5 Relative Amplitude c': h -0.6: 08% l 1 1 -124 T T T T T T T 0 05 1 15 2 Time (ns) Figure 3-26 Response waveform for material ‘8’. 159 2.5 Relative Amplitude 9': 9': O\ A 1 1111111 #3 00 1 Time (ns) Figure 3-27 Response waveform for material ‘SS’. 160 2.5 Relative Amplitude 0.02 -002 3 -004 3 \ fl {H Latetime Figure 3-28 Convolution of material ‘A’ with baseline E-pulse. 0.5 1 1.5 2 Time (ns) 161 2.5 Relative Amplitude 0.02 -002 1 -0.04 -0.06 -0.08 Figure 3-29 Convolution of material ‘AA’ with baseline E-pulse. 1 \ fl {_’ Late time 0.5 1 1.5 2 Time (ns) 162 2.5 Relative Amplitude 0.02 -002 3 -004 3 fl —* Late time Figure 3-30 Convolution of material ‘F’ with baseline E-pulse. T 7 T T T T T T T r r T 0.5 1 1.5 2 Time (ns) 163 2.5 Relative Amplitude 0.02 -0.02 -0.04 -0.06 -0.08 Figure 3-31 Convolution of material ‘FF’ with baseline E-pulse. 1 1 \ fl ‘——* Late time 0.5 l 1.5 2 Time (ns) 164 2.5 0.02 0 3 1 3 0 Late time .3 -0.02 ~ 3 73" . <2 -0.04 ~ 0 . .2 E . Q - a: -0.06 - U 008 3 ‘0 1 T T f w T 1 T I 0 0 5 1 1 5 2 Time (ns) Figure 3-32 Convolution of material ‘8’ with baseline E-pulse. 165 2.5 Relative Amplitude O 1 j l O Late time Time (ns) Figure 3-33 Convolution of material ‘SS’ with baseline E-pulse. 166 2.5 Quantification of the late-time energy was accomplished using the EDN a values. Table 3-7 and Table 3-8 show the EDNa values for each aspect angle for both the parallel and perpendicular polarization. These tables show that there is very little difference in the EDNa values up to 10°. A graphical representation of this data is shown in Figure 3-34. Here again the data shows relatively little change in the EDNa up to 10°. Figure 3-35 shows a comparison of the EDNa values calculated in this section for lossy materials to those calculated in section 2.2.3 for lossless materials. Consistent with previous discussions on the lossy material EDNa values, this comparison shows that the lossy material results in lower EDNa values overall compared to the lossless material values. 167 Material Aspect Angle, 0. EDNa A 0 2.16E-07 B 1 2.16E—07 C 2 2.18E-07 D 3 2.20E-07 E 4 2.25E-07 F 5 2.34E-07 G 6 2.47E-07 H 7 2.68E-07 I 8 2.98E-07 J 9 3.39E-07 K 10 3.96E—07 L 15 1.04E-06 M 20 2.80E-06 N 25 6.67E-06 P 30 1.43E-05 Q 35 2.83E-05 R 40 5.30E-05 S 45 9.6lE-05 Table 3-7 EDNa calculations for change in aspect angle for parallel polarization. 168 Material Aspect Angle, 91 EDNa AA 0 2. 16E—07 BB 1 2. l6E-07 CC 2 2.17E—07 DD 3 2. 18E-07 EE 4 2.21E-O7 FF 5 2.26E-07 GG 6 2.35E-07 HH 7 2.48E-07 II 8 2.69E-07 J J 9 2.98E-07 K 10 3.39E-O7 LL 15 7.91E-07 MM 20 l .94E-06 N N 25 4.18E-06 PP 30 7.89E-06 QQ 35 1.33E-05 RR 40 2.04E-05 SS 45 2.88E-05 polarization. 169 Table 3-8 EDNa calculations for change in aspect angle for perpendicular EDNa 1 .00E-04 9.00E-05 8.00E-05 7.00E-05 6.00E-05 5.00E-05 4.00E-05 3.00E-05 2.00E-05 1 .00E-05 O O O [Tl + O C Figure 3-34 EDNa vs. change in aspect angle for lossy layer. .1 .. .. _. _ — perpendicular polarization 2 —0— parallel polarization ... ... .1 .4 ..4 _( —. .1 .1 q _. ... _. —1 _. _. _ _, _. 1+111LL11L141L141114 G M1 10 l 20 30 Aspect Angle (degrees) 170 50 3.50E-04 q : + o=0.0, perpendicular polarization 3.00304 3 +o=0.0, parallel polarization 1 -— 020.5, perpendicular polarization 2.50304 3 +0205, parallel polarization 20013.04 3 a 3 .. Z . a : 1.50E-O4 : 1.0013-04 '1 50015—05 /': 0.00E+003--3' -'-'-“'37i ..TTT4T., T O 10 20 30 40 50 Aspect Angle (degrees) Figure 3-35 Comparison of EDNa vs. aspect angle for lossless and lossy layers. 171 3.2.4 Change in Conductivity This section will explore whether a change in the conductivity of the material layer can be detected using the EDN a values. The material properties of the various sample materials evaluated are shown in Table 3-9. The response and convolution waveforms for five values of conductivity are shown as examples. The response waveform for the baseline material, material ‘I’, is shown in Figure 3-37. The response waveforms for the extreme cases of change in conductivity, materials ‘A’ and ‘S’ are shown in Figure 3-39 and Figure 3-36. There is an obvious difference between the baseline material response and these responses so this should reflect a difference in EDNa. It can be seen in Figure 3-38 that when the conductivity gets as high as 5, virtually all of the incident pulse gets reflected by the material layer and air interface. The material layer acts nearly as though it were a conductive plate. Lastly, Figure 3-40 and Figure 3-38 show the response waveforms for the materials with the least change in conductivity from the baseline, materials ‘H’ and ‘J’. The response waveforms look very similar to the baseline response so the EDNa values should be closer. The EDNa values are calculated using the late-time energy in the convolution waveforms. Thus, each material response is convolved with a baseline E-pulse. The baseline E-pulse used is shown in Figure 3-4. The convolution of the baseline E-pulse with the baseline response is shown in Figure 3-41 and as expected the late-time energy is near zero. Figure 3-42 and Figure 3-43 show the convolution of the baseline E-pulse with the response waveforms for materials ‘A’ and ‘S’. In the first figure it can be seen that the late-time energy is relatively large while in the second it is not so obvious. Figure 3-44 and Figure 172 3-45 show the convolution of the baseline E-pulse with materials ‘H’ and ‘J’. Again, by inspection it is not clear how large the late-time energy might be in relation to the baseline. Thus it is necessary to calculate and compare EDNa values. 173 Material 8 p 0 Thickness Polarization Angle of (mm) incidence, 91 A 980 po 0 10 Parallel 0 B 980 p0 2e-5 10 Parallel 0 C 980 p0 2e-2 10 Parallel 0 D 980 110 0.1 10 Parallel 0 E 980 110 0.2 10 Parallel 0 F 930 110 0.3 10 Parallel 0 G 980 110 0.4 10 Parallel 0 H 980 110 0.45 10 Parallel 0 I (baseline) 980 p0 0.5 10 Parallel 0 J 980 110 0.55 10 Parallel 0 K 980 p0 0.6 10 Parallel 0 L 980 110 0.7 10 Parallel 0 M 980 110 0.8 10 Parallel 0 N 99,0 110 0.9 10 Parallel 0 P 980 110 1.5 10 Parallel 0 Q 980 p0 2.0 10 Parallel 0 R 980 110 2.5 10 Parallel 0 S 980 po 5 10 Parallel 0 Table 3-9 Material properties for changing conductivity. 174 Relative Amplitude b A Time (ns) Figure 3-36 Response waveform for material ‘I’. 175 2.5 0.6 , 0.4 3 1 0.2 j L.» D 1 Relative Amplitude Time (ns) Figure 3-37 Response waveform for material ‘A’. 176 0.2 O 11L4_1111 .6 .<'= 9': O\ J§ N 1 1 1 1 1 1 1 Relative Amplitude is 00 1 '1 . 2 D I I I l I T 7 Y T r V T T T T r T Time (ns) Figure 3-38 Response waveform for material ‘8’. 177 Relative Amplitude 111111 1 Time (ns) Figure 3-39 Response waveform for material ‘H’. 178 Relative Amplitude Time (ns) Figure 3-40 Response waveform for material ‘J’. 179 0.01 — -0.01 1 1 1 —0.03 3 -005 3 Relative Amplitude -0.07 a 1 1 fl '—* Late time -009 3 Time (ns) Figure 3-41 Convolution of the baseline response with the baseline E-pulse. 180 0.04 _ 0.02 4102 3 Late trme 11111 -0.04 Relative Amplitude -0.06 111111 -0.08 3 Time (ns) Figure 3-42 Convolution of material ‘A’ response with the baseline E-pulse. 181 0.01 ‘ -0.01 1 -0.03 1 Late time 1 L 1 1 1 Relative Amplitude -0.05 3 -0.07 1 #11 1 1 1 -0.O9TTTTTTTTTTT,..,.-,,Tfi, Time(ns) Figure 3-43 Convolution of material ‘8’ response with the baseline E-pulse. 182 0.02 v Late time .3 -0.023 B E‘ < 004 ~ 0 _ .2. E a - a: 006 ~ -0083 1 -0.1 T T T T 0 0.5 1.5 Time (ns) 2.5 Figure 3-44 Convolution of material ‘H’ response with the baseline E-pulse. 183 0.01 -0.01 —0.03 —0.05 Relative Amplitude -0.07 -0.09 1 1 1 1 1 1 0 {—-> Late time Time (ns) Figure 3-45 Convolution of material ‘J’ response with the baseline E-pulse. 184 Table 3-10 shows the EDNa values calculated for the various values of conductivity. It is not so evident as in previous EDNa tables how effective the EDNa is in this instance. These EDNa values are seen plotted in Figure 3-46. While at first glance it might appear that the EDNa effectively discriminates changes in conductivity, by comparing these EDN a values to those in previous EDNa plots, there is in fact a much smaller change in EDNa for changes in conductivity. This would imply that for small changes in conductivity (e.g., from a baseline of 0.5 to 0.75), it might be very difficult to detect a change. A very low EDNa threshold would have to be set relative to those that might be set to detect changes in thickness or permittivity. This will become more evident in the next chapter when noise is added to the data. 185 Material Conductivity, EDNa 0( S/ m) A 0 9.48E-04 B 2e-5 9.48E-04 C 2e-2 8.39E-04 D 0. 1 4.92E-04 E 0.2 2.24E-O4 F 0.3 8.03E-05 G 0.4 1.64E-05 H 0.45 2.84E-O6 I 0.5 2.16E-07 J 0.55 3.78E-06 K 0.6 1.00E-05 L 0.7 3.23E-05 M 0.8 5.82E-05 N 0.9 8.35E-05 P 1.5 1.67E-O4 O 2.0 l .71E—04 R 2.5 1.55E-04 S 5.0 8.05E-05 186 Table 3-10 EDNa calculations for change in conductivity of material. EDNa (dB) 10013-03 , 90013.04 { 8.00E-04 1 7.00E-04 j 6.00E-04 : 111 5.00E-04 L1 4.00E-04 1 1 3.00E-04 : 2.001304 3 10013.04 5 0.00E+00 3 T T T T 0 Conductivity (S/m) Figure 3-46 EDNa vs. change in conductivity. 187 3.3 Effects of Noise on EDNa The same noise model as that used in chapter 2 for lossless materials was applied to the lossy materials discussed in this chapter. The noise was added to the response waveforms before convolution and for each case 500 different samples were generated and then averaged to arrive at the EDNa. Since noise corrupts all of the signals, including the baseline, none of the convolutions are expected to have zero late-time energy. To allow consistent comparisons of changes in the EDNa, the EDRa was calculated and used as the quantitative tool for noisy data. 3.3.1 EDRa vs. Signal-to-Noise Ratio The goal of using the EDRa values is to be able to determine changes in material properties even in data with low SNR. Ideally, we would like to be able to identify changes in properties regardless of noise, but that is obviously not practical. The discussion in chapter 2 on noise corruption showed examples of the noise corrupted response waveforms and resulting convolutions. The response waveforms and convolutions are essentially the same so the these examples will not be shown in this chapter. Instead, the focus of discussion will be the EDRa values and their plots. The three properties examined are the thickness, permittivity, and conductivity. It has been shown thus far in this chapter that when the material layer is lossy the EDNa values calculated are lower than when the material is lossless. With the addition of noise it is expected that discrimination of material property changes will become increasingly more difficult. Table 3-11 shows the EDRa values for changes in thickness. The baseline material has a thickness of 10mm and thus its EDRa is zero for all SNR, as it should be 188 according to (2.13). Consistent with discussion on noise corruption in chapter 2, SNR’s of 2, 5, 10, 15, 20, 25, and 30dB were considered. Table 3-12 shows the EDRa values for changes in permittivity and Table 3-13 shows the EDRa values for changes in conductivity. One interesting thing to note about Table 3-13 is that some of the EDRa values are negative. This means that the EDNa value for the baseline was not the smallest value for that particular configuration of SNR and conductivity. This implies that the changes in the material properties cannot be detected at this particular level of noise. The EDRa values are shown plotted in Figure 3-47, Figure 3-48, and Figure 3-49. Figure 3-47 shows the EDRa versus SNR for changing thicknesses. It can be seen that for all SNR’s investigated some threshold EDRa value could be set to determine changes in the thickness. Figure 3-48 shows the EDRa versus SNR for changing values of permittivity. Again, it can be seen that a threshold EDRa value could be set to determine changes in permittivity. Where the difficulty begins it with Figure 3-49 which shows the EDRa versus SNR for changing conductivity. What this plot shows is that as the SNR decreases it becomes more difficult, even impossible, to determine small changes in the conductivity. It should be noted that for larger changes in conductivity (e. g., a change from 0:05 to 15:04) it is still possible to set a threshold to detect changes. 189 Thickness 2dB SdB 10dB 15dB 20dB 25dB 30dB (mm) SNR SNR SNR SNR SNR SNR SNR 8 2.90 4.42 8.45 13.3 18.1 22.7 27.4 9 1.72 1.76 4.31 8.17 12.6 17.1 21.7 10 0 0 0 0 0 0 0 11 1.15 1.09 3.49 7.12 11.4 15.8 20.4 12 2.09 3.03 6.49 10.9 15.6 20.1 24.8 Table 3-11 EDRa vs. SNR for several thicknesses. Permittivity 2dB SdB 10dB 15dB 20dB 25dB 30dB SNR SN R SNR SNR SNR SNR SNR 7 0.757 1.69 3.93 7.47 11.9 16.4 21.2 8 0.176 0.657 1.65 3.52 6.94 11.1 15.9 9 0 0 0 0 0 0 0 10 0.848 1.29 2.03 3.96 7.23 11.2 15.9 11 1.09 2.44 4.75 7.94 12.3 16.7 21.6 Table 3-12 EDRa vs. SNR for several values of permittivity. Conductivity 2dB SdB 10dB 15dB 20dB 25dB 30dB (S/m) SNR SNR SNR SNR SNR SNR SNR 0.3 1.46 1.57 3.05 5.25 9.17 13.9 18.2 0.4 0.865 0.522 1.05 1.91 3.97 7.73 11.6 0.45 0.478 0.451 0.765 0.798 1.51 3.46 6.0 0.5 0 O 0 0 0 0 O 0.55 0.285 0.0158 0.336 0.228 0.899 2.68 5.22 0.6 0.561 -0.129 0.537 0.788 2.61 6.06 9.71 0.7 -0.572 -0.0159 0.875 2.58 5.76 10.2 14.4 Table 3-13 EDRa vs. SNR for several values of conductivity. 190 EDRa(dB) 30 +t=8(mm) . +t=9(mm) 25 3. +t=10(mm) +t=ll(mm) 20 : +t=12(mm) 153 103 Figure 3-47 Comparison of EDRa vs. SNR for several thicknesses. 10 15 20 25 SNR(dB) 191 O-HTTXLfTT—TXmTTTrXTTTrFr—IW—r-fi-i—F-I—r-X—H—r—F- 30 35 1 +sm7 3 +8I:8 vs 20 3 +ar=9 ' +am10 +8r=ll " a ‘5? / 3 a: 1 ,‘ I 1 , g: _ [£1 10 2 / : '-‘ _/ 5 ~ / O i I ¥ T r ¥ rfi 1 0 15 20 25 30 SNR(dB) 35 Figure 3-48 Comparison of EDRa vs. SNR for several values of permittivity. 192 25 +o=0.4 4 +o=O.45 20 ~ +o=0,5 . +o=0.55 j +o=0.6 a 157. '3 a . m 1 g . {-7-} 10 2 5 _ 0 5 10 15 20 SNR (dB) TTTT AT T T. 25 30 35 Figure 3-49 Comparison of EDRa vs. SNR for various values of conductivity. 193 Chapter 4 Magram - Single Layer The previous chapters investigated the use of the E-pulse in diagnosing simple, single layered materials. Extending the E—pulse technique to more complicated layered materials will be discussed in this chapter and in chapter 5. The focus of this chapter will be on single layered Magram materials. Analysis of material properties using the E-pulse is dependent on the natural frequencies of the scattered response waveform. For simple layered materials where the properties were known it was possible to theoretically determine the natural frequencies. Given the scattered response of a layered material without knowledge of the material properties, the natural frequencies must be extracted from the response waveform numerically. 4.1 Extraction of the Natural Frequencies and E-Pulses from Response Waveforms A solution of (1.22) for e(t) is only possible given that the natural frequencies are already known. When the natural frequencies are unknown they must be extracted from the experimental response waveform. We begin by expanding the E-pulse into a sum of basis functions, em=§amm. an m=l where {fm (1)} is an appropriate set of basis functions and Am is the amplitude of the m’th basis function. Substituting (4.1) into (1.2) gives, 194 M C(t):zAmfm(t)*re(I)’ t>TL+Te (4'2) m=l Te M = JZAmfm(z')re(t—T)d2'=0, I>TL+Te, (4.3) 0 m=l where re (t) is the experimental response waveform. A solution to (4.3) for {Am} can be found using the moment method technique. Taking the inner product of (4.3) with a set of weighting function { w," } gives, M Te TW 2 Am] I f," (2')re(t-z')wk (t)dtdz'=0, k =1,2,3,~-,K. (4.4) ”1:1 0 TL+Te Here fm (t) is chosen to be a set of sub-sectional basis functions in the E-pulse expansion, 1, (m—l)AS.t.<.mA f’" (I) = {0, otherwise (4'5) where A is the pulse width. The solution to (4.3) for{Am} is not unique. In fact, depending on the choice of M, there are infinitely many solutions. By choosing M = K = 2N , (4.4) becomes a set of homogeneous linear equations, and the resulting E- pulse is the “natural” E-pulse discussed previously. Sets of E-pulse amplitudes,{Am}, that result in c(t) = O , occur only at discrete values of Te. By choosing M = 2N +1, K = 2N , (4.4) results in a set of linear equations that depend on a forcing function and the E-pulse is the “forced” E-pulse discussed previously. Since c(t) = 0 does not depend 195 on the determinant matrix being zero, solutions for {Am} occur for any Te except those that give solutions for the “natural” E-pulse. Numerous E-pulses with different Te are generated by forcing (4.2) to be zero. Within this set of E-pulses there exists one which will allow the most accurate discrimination. In other words, many E-pulses are generated that force (1.2) to be very nearly equal to zero, but not exactly zero. As will be shown, the natural frequencies extracted are merely estimates with some degree of error. To find the optimum E-pulse, first the natural frequencies are extracted from each E-pulse and a set of estimated A response waveforms, {r(t)} are created using (1.1). Next, a comparison of each ;(I) to the original response, re (t) using a least squared error technique gives the r(t) that most nearly represents re (t). Given this r(t) , the corresponding E-pulse used to generate it is then the most optimum for that specific response waveform, re (t). Assuming a “natural” E-pulse has been generated, extraction of the natural frequencies from the generated E-pulses begins by taking the Laplace transform of (4.1), 2{e m=l M E(s,,)= Z A,,,F,,,(s,,), n =1,2,3,...,N. (4.7) m=l It was shown in section 1.1.2 that E(Sn ) = E(s; ) = O to satisfy c(t) = 0. Therefore, by finding the roots of (4.7), the natural frequencies sn = 0'" + jwn can be determined. Using the Laplace transform (1.3), 196 OO Fm(s)= Ifm(z)e‘“dt, 0 where fm(t) was defined by (4.5). Thus mA Fm(s,.)= I 1e“”’dt (m-1)A - A 3" —le-snmA Sn Substituting (4.10) into (4.7) gives, (ca—5"A -l) M ) ____ 2 Ame.snmA = 0 , ’1 m=l E(sn = s the roots of which are found by solving the polynomial equation, 2 Amzl’zn :0 m: l where __-sA Zn—e ". Knowing 2,, allows one to solve for the natural frequencies, 1 Sn =-Zln(Zn), where the pulse width, A , has been shown [2] to equal, A=p—”, p=1,2,3,---, ISkSN. wk (4.8) (4.9) (4.10) (4.11) (4.12) (4.13) (4.14) (4.15) These natural frequencies can then be substituted into (1.1) to determine the estimated response, ;(I) , for each E-pulse. 197 Once a set of estimated responses is found for each E-pulse, they can all be compared to the original response waveform to determine the optimum E-pulse. This is done by minimizing the squared error, A 2 . 2 £=|r(t)—r(t)| =Z|r(r,-)—r(ti)l , (4.16) r' where the sum is over sampled values in the late-time of the responses. This squared error is the error between the original response waveform and the set of estimated response waveforms reconstructed from the generated E-pulses. The reconstructed response which gives the minimum error determines the optimum E-pulse. 4.2 Magram Materials — Experimental Results This section deals with the scattered response waveforms from Magram materials. The goal is to extract the natural frequencies from the scattered response waveforms, generate an E-pulse from these natural frequencies, and determine if the generated E- pulses are sufficient to detect changes in the magram material properties. Due to the proprietary nature of magram materials it was not possible to obtain any for testing. However, the scattered response frequency domain data from three unknown Magram material samples was provided for analysis. These samples will be referred to as sample A, sample B, and sample C. For each sample the magnitude and phase data was provided in the frequency range of 2-18 GHz. Figure 4-1 shows the magnitude plots of the scattered response from each sample and Figure 4-2 shows the phase plots from each sample. The magnitude plots can be compared to the magnitude plot for the single layer lossless materials shown in Figure 2-1 and the magnitude plots for the single layer lossy materials shown in Figure 4-3. It can be seen that the Magram responses show decreased 198 response for certain selective frequencies while the other materials do not exhibit this frequency selection. Magram materials behave similar to frequency selective surfaces. A difference in the frequency responses that was addressed is the frequency range of the experimental data, 2-18GHz. This differed from the range of the theoretical data which had a frequency range of 0-18 GHz. This additional discontinuity at 2 GHz contributes additional error when the IFFT function is performed. To eliminate this error and to keep the results consistent with previous frequency domain responses, the experimental data was extrapolated to 0 GHz. This extrapolation was performed using the Microsoft Excel spreadsheet program. The results of these extrapolations are shown in Figure 4-4 and Figure 4-5. To extract the natural frequencies from the response data, the frequency domain data was transformed into the time domain. This procedure was performed using the IFFT program in UTIL2B.exe. Before running the IFFT program a Gaussian modulated cosine windowing function was applied with width, 2' = 0.06 , and center frequency of 2 GHz for the original data and 0 GHz for the extrapolated data. Both the original data and the extrapolated data were processed with the IFFT function and the resulting time domain waveforms are shown in Figure 4-6 and Figure 4-7. Figure 4-6 shows the time domain response waveforms for the original data. The error due to the discontinuity at 2 GHz is seen by the rise in amplitude before the initial reflection from the layer. This waveform should be zero until the first reflection. Figure 4-7 shows the time domain response waveform from the extrapolated data and it shows that this error has been improved, but there is still some error present. 199 Generating E-pulses from these time domain scattered response waveforms turns out not to work using the program EP.for. While the program does generate an E-pulse, the result does not give the proper results when trying to discriminate between scattered responses. Figure 4-8 shows an E-pulse generated from the sample A scattered response waveform using 5 modes. Each sample’s response waveform was convolved with this E- pulse to see if the resulting convolution waveform would show an ability to discriminate between the three sample responses. Figure 4-9 shows the convolution of the sample A response waveform with the E-pulse. It is obvious that the late-time energy in the convolution is not zero as we would expect. Figure 4-10 shows the convolution of the sample B response waveform with the E-pulse. Figure 4-11 shows the convolution of the sample C response waveform with the E-pulse. All three of the convolution waveforms contain a large amount of energy and are all very similar in shape. The EDNa values were calculated for each convolution and can be seen in Table 4-1. It is evident that the EDNa is not an appropriate way to determine changes to Magram materials. The reason that the E-pulse does not work for Magram materials is most likely due to the nature of the material. It is designed to absorb energy at certain frequencies. Since the Magram materials absorb the incident wave, seemingly very efficiently, there is not enough scattered response to allow extraction of the natural frequencies. In addition, the very frequency range from which we are trying extract the natural frequencies from is the same range these particular sample responses are designed to absorb energy in. For example, in the single lossless layer case, the natural frequencies that were calculated and used to generate the B-pulse for one particular case can be seen in Table 4-2. The frequency range of the scattered response investigated was 0-18 GHz and 200 according to Table 4-2 each natural mode in the E-pulse is within that range. Therefore, when the E-pulse is convolved with the scattered response these natural frequencies cancel. However, in the case of the Magram materials, there is not enough scattered response signal to extract the correct natural frequencies and even though EP.for generates an E-pulse, it is not made up of the correct frequencies with which to cancel the scattered response frequencies and thus the convolution is not zero and the technique does not work. 201 0 9 + Sample A ' -°- Sample B 0.8 a + Sample C 0.7 - Relative Magnitude .0 .0 .O O N L») A LII l r l l l 1 1 l O p—d r r Ll_r_r O .1 a O p—n O p—n U] 20 Frequency (GHz) Figure 4-1 Magnitude plot of scattered response from unknown Magram materials. 202 250 . + sample A 200 : -°- sample B 150 j + sample C 100 5 505 _1 -50 j Phase Angle (deg) o -100 j -150; -200 5 -250 q I T t I t 1 pm r r 1 r 1 1 I ‘ T t T— _r O 5 10 15 20 Frequency (GHz) Figure 4-2 Phase plot of scattered response from unknown Magram materials. 203 1.2 ~ - - - sigma=0.4 : ------ sigma=0.5 1 : sigma=0.6 .g 0.8 3 3 . 'E an as E 0.6 ~ , E 3 i “- i. g : I ' i : |‘_ : |> ' a: 0.4 . ‘l r i l , I u l - ’, . . 0 '- . . . f O 5 10 15 20 Frequency (GHz) Figure 4-3 Frequency domain response plots for single layer lossy materials. 204 0.6 9 Relative Magnitude .9 A r 1 r 0.2 : + sample A + sample B + sample C 10 15 Frequency (GHz) Figure 4-4 Extrapolated magnitude plots of scattered responses. 205 20 Phase angle (deg) 250 , 200 i 150 § 100 3 50% -50 E -100 5 -150 5 -200 3 + sample A -°- sample B + sample C -250 * f T T T T T T T T T T T T T T T T T T 5 10 l 5 Frequency (GHz) Figure 4-5 Extrapolated phase plots of scattered responses. 206 2O 0.2 —O.2 a Q) '5 3 i -0.4 ~ E < O .5 -0.6 .‘3 Q) a: -0 8 - : # sample A _1 I - - - sample B i ----- sample C “1.2 A I 1 r T u T I r r O 1 2 3 4 Time (ns) Figure 4-6 Time domain response of unknown Magram samples. 207 0.2 . -O.2e o 1 '6 5 1 2,-0.4 E < 0 g; —O.6~ .‘S . Q . ‘3‘ i -O.8e - “ sampleA _1Z ---sampleB ----- sampleC -l.2 Hm.Hm.Tfirrrrmrrm........ O 0.5 1 1.5 2 2.5 3 3.5 Time (ns) Figure 4-7 Time domain response for extrapolated data. 208 Ampitude. 100 _ 80 g 60% 4o; 20- Dag. _- _ l I I IDI I'I'Tllml'I' 0 0.25 0.5 0.75 I I oo ox i: r12 0 O O O O rrlrrrrlrr'rlrerllrrr rr -100 " Time (ns) Figure 4-8 E-pulse generated from Sample A response using 5 modes. 209 Relative Amplitude > Late time 0 0.5 1 1.5 2 2.5 3 3.5 Time (ns) Figure 4-9 Convolution of the sample A response with the sample A E-pulse. 210 N rlrrr Relative Amplitude o > Late time I N 111111 '6 T T I T T T T r T T T T T T T I T I I T I T T T T T T T T I T T Trj 0 0.5 1 1.5 2 2.5 3 3.5 Time (ns) Figure 4-10 Convolution of the sample B response with the sample A E-pulse. 211 8 .1 6 a 4 q o . .5 . = -4 f: q E. 2: E . < . a 4 l a: 0 v S _ é _ ~ > Late time -2 d -4 a 1 -6 r r T T T r r r r I r t T r T r r 1 r T T r r T 1 r r r T T r r fiT O 0.5 1 1.5 2 2.5 3 3.5 Time (ns) Figure 4-11 Convolution of the sample C response with the sample A E-pulse. 212 le EDNa A 3.80e-2 B 3.70e-2 C 3.54e-2 Table 4-1 EDNa values for the three Magram samples. Mode 0 (S/m) to (Grads/sec) f (GHz) 1 -3.466 15.71 2.5 2 -3.466 47.12 7.5 3 -3.466 78.54 12.5 4 -3.466 110.0 17.5 5 —3.466 141.3 22.5 Table 4-2 Natural frequencies for single lossless layered case. 213 Chapter 5 Impedance Sheet Over Single Layer Previous chapters have discussed single layer conductor backed materials. The discussion will now explore a single layer conductor backed material with a thin resistive sheet placed on the surface of the layer. This thin sheet, or impedance sheet, will help reduce the backscatter from the layered materials at certain frequencies and is called a Salisbury Screen [12]. The surface resistance of the impedance sheet is defined in terms of the thickness and conductivity of the sheet, R =—, (5.1) where 0' is the conductivity in Siemens per meter and t is the thickness of the impedance sheet in meters. The purpose of this chapter will be to show how changes in the surface resistance of the impedance sheet will effect the EDNa. Several scattered response waveforms for layered materials with impedance sheets with different surface resistances will be simulated using the program MLPREF.for. The changes in surface resistance will be due to changes in the conductance; thus, the thickness of the impedance sheet will remain constant. 5.1 Theoretical Determination of Natural Frequencies The first step in using the E-pulse as a means to identify changes in material properties is to determine the natural frequencies. Previous cases did not consider the configuration of a layered material with an impedance sheet, so the mathematics for this configuration must first be verified to be consistent with previous findings. Figure 5-1 214 shows the path of the wave in a layered material with an impedance sheet. To find the natural frequencies of the scattered response begin with the time domain scattered response given by, Rm = my) +7] (—1)T2f(t—2T) +T1(—1)I‘2(—1)T2f(t—4T)+~-, (5.2) where T is the transit time across region 1. The transmission coefficients T1 and T2 are the transmission coefficients into and out of the impedance sheet, respectively, while F1 and F2 are the respective reflection coefficients. The function f (I) is the incident pulse that is used to excite the material. Equation (5.2) can be rewritten as 00 R(t)=I‘1f(t)+TlT22(—l)" 3f(t—2nr) (5.3) n=1 or R(t)=f(t)*h(t), (5.4) where h(t) = r,5(z)+nrzi(—r)" rgao—znr) (5.5) n=l is the unit impulse function of the layered material [13]. By letting (—1)” r369 —2nT) = (—1“2 )" 6(t —2nT) = eat§(t-2nT) (5.6) then e26” = -r2 (5.7) and solving for alpha gives 215 1 =—l —I‘ 6’ 2T “( 2). Equation (5.8) can then be substituted into (5.5) giving h(t) = 1‘16(t)+T1T2em i 5(t—2nT). n=l Using i (so—m) = ..(z —2r) i (so —2nT) n=l nz—oo and the Poisson-Sum formula [6] where (00 = 375, (5.9) then becomes 2' 00 1,15: Iz(t)=F16(t)+T1T2emu (t—2T )% Z e "T nz—oo =F16(t)+%TTlu(t 2T) 2 WNW") . (5.8) (5.9) (5.10) (5.11) (5.12) (5.13) where a)" = I—ITJE and an = a. The impulse response, (5.13), can finally be expressed as ht=() F16(t)+—7:2§2-u(t- 2T) em +2Zemcos(a)nt)), n=1 and the natural frequencies are given by . 1 rm Sn =a+jwn =37—_III(— F2)+j7 . 216 (5.14) (5.15) —> X Region 0 (50.)410) Region 1 (£1,710) (PEC) \\\\\\\ i g .5] A \\\\\\\\\ \\\\\\\\\\ k\\\\\\\\\\ — — - _ _ .L l | l I l \\\\\\\\\ L. Impedance sheet Figure 5-1 Layered material covered by impedance sheet. 217 5.2 Construction of the E-pulse The current version the program EP.for does not implement the math in section 5.1 correctly and is unable to properly extract the natural frequencies and generate a correct E-pulse. In the interest of time, the E-pulse used to generate results in this chapter was generated heuristically. Twenty-one layered materials with varying values of surface resistance were compared. IN each case region 1 is considered to be a foam layer with parameters of free space 8 = 5 , ,u = ,0' = O . Table 5-1 shows the material properties of the impedance 0 #0 sheet for each layered material configuration. Material ‘K’, with a surface resistance of 377 (82/ sq.) , is considered the baseline material. The response waveform from material ‘K’ was used to generate the E—pulse and for comparison to determine changes in surface resistance. This baseline case is a Salisbury screen that produces a low reflected field at when the impedance sheet is located a quarter wavelength from the conductive backing, d = %’ or f = 5.9 GHz . This is clearly seen in Figure 5-2 computed using MLPREF.for. The purpose of the E-pulse is to give a zero result when it is convolved with a response waveform from which the E-pulse is generated. To get the correct E-pulse, first the ratio of consecutive reflections was determined. This ratio was used to determine the amplitudes of the E-pulse pulses. Each pair of consecutive reflections should have the same ratio, from (5.5), this ratio is seen to be ~F2. As an example, Figure 5-3 shows the time response for material ‘K’ with f (1‘) resulting from windowing the frequency data generated by MLPREF.for using a Gaussian modulated cosine windowing function with 218 z" = 0.06 ns and a center frequency of 0 GHz. In this figure, the ratio of reflection #2 to reflection #1 should be equal to the ratio of reflection #3 to reflection #2, and so on for each remaining pair. This ratio was determined to be, —F2 = 0.33 for the baseline response. Therefore, an E-pulse with two pulses with amplitudes in this ratio was generated. The separation of the two pulses is equal to the separation of the reflections which again should be equal for each pair of reflections. This was determined to be 0.085 (ns) which is equal to twice the one-way transit time, T , through the layer. In order for the E-pulse to generate a zero late-time convolution the overlap area of each pulse with a reflection must cancel one another. Therefore, the second pulse of the E- pulse was made negative. The resulting E-pulse is shown in Figure 5-4. This E-pulse was convolved with the response waveform from each layered material configuration and the resulting EDNa were recorded. 219 Material s p a Rs Impedance sheet (S/m) (Q/sq) thickness (mm) A 80 P0 400 50 0.05 C so po 133.3 150 0.05 D so po 100 200 0.05 E so po 80 250 0.05 F 80 p0 66.7 300 0.05 G 80 110 57.1 350 0.05 H so po 55.6 360 0.05 J 80 P0 54.1 370 0.05 K(baseline) so po 53.1 377 0.05 M so uo 51.9 385 0.05 P so uo 50 400 0.05 Q so po 44.4 450 0.05 R 80 [.10 40 500 0.05 S 80 P0 36.4 550 0.05 T so 110 33.3 600 0.05 U 80 90 30.8 650 0.05 V 80 p0 28.6 700 0.05 Table 5-1 Material properties of impedance sheet. 220 Magnitude (dB) H O I I I N V-‘ t—‘ l O Ur O U] C Ur 11111111rrlrrrrlrrrrlrrrrlrrr llil I N LI] 1 —— 2000 I O rrlrrrr Z '35 I I I I I Frequency (GHz) Figure 5-2 Frequency domain scattered response of Salisbury screen. 221 12 Relative Amplitude 0.2 -1.2 “ ; I \ ‘ 3 2 3 I * '\ 1 0 0:5 I I I I I r ' T T 1:5 r Time (ns) Figure 5-3 Response waveform for baseline material. 222 0.6 3 LL41 l 0.4 Amplitude. 0.2 { 0 0.085 0.17 Time (ns) Figure 5-4 E-pulse generated for impedance sheet response. 223 5.2.1 Change in Surface Resistance Of the twenty-one material configurations shown in Table 5-1, the response and convolution waveforms for five configurations will be shown as examples. The baseline material is material ‘K’ with a surface resistance, Rs = 37752. The scattered response for this material is shown in Figure 5-3. In comparison, Figure 5-5 and Figure 5-6 show the extreme cases of change in surface resistance. Figure 5-5 shows the scattered response waveform for material ‘A’ with RS = 5052 and Figure 5-6 shows the waveform for material ‘V’ with R5 = 70052. From (5.1), as the conductivity becomes smaller, the surface resistance becomes larger. It can be seen in Figure 5-5 that when the surface resistance is small (0’ is large) the impedance sheet acts as a good conductor and reflects a large portion of the incident energy at the first reflection. This first large reflection is followed by many smaller reflections as the wave dies out. As seen in Figure 5-6, when the surface resistance is large (0' is small) the impedance sheet acts as a poor conductor and very little energy is reflected initially. Also, very little energy is confined within the space between the conductive backing and impedance sheet. Most of the energy is reflected off of the conductive backing which is seen in the second reflection of Figure 5-6. One can assume that the large difference between these responses and the baseline response will yield large differences in the EDNa values. In contrast to the extreme changes in surface resistance, Figure 5-7 shows a comparison of the baseline response to the responses of material ‘J’ and ‘M’ which have minor changes in surface resistance, R5 = 37052 and RS = 38552 , respectively, from the baseline. As can be seen, these waveforms are very similar and nearly indistinguishable 224 from one another. It is expected that there will be little difference in the value of EDNa between these responses. Using PCONV2.for, each of the responses for the materials listed in Table 5-1 were convolved with the constructed E-pulse and the EDN a was calculated from the late- time energy of the resulting convolution waveform. Figure 5-8 shows the convolution of material K, the baseline material, with the E-pulse. This can be compared to Figure 5-9, the convolution of material ‘A’ with the E-pulse, and Figure 5-10, the convolution of material ‘V’ with the E-pulse. It is evident that there is more late-time energy in the latter two convolutions, as expected. This difference in late-time energy is not evident in Figure 5-11 and Figure 5-12. These are the convolution waveforms for materials ‘J’ and ‘M’, respectively. N 0 large difference in EDN a is discemable from these figures. For that matter, it would be a challenge even to determine which convolution was from the baseline material. To determine this, the EDNa values that were calculated are shown in Table 5-2. While there is little difference in the EDNa values between ‘J’, ‘K’, and ‘M’ one could easily determine which was the baseline material from these numbers. The EDNa values are plotted in Figure 5-13. This plot shows that the E-pulse does have the ability to discriminate changes in the surface resistance to a certain degree. 225 Relative Amplitude 0.2 . -1.2 ‘ 111 l I l T T I if I I T I I 0.5 l 1.5 Time (ns) Figure 5-5 Response waveform from material ‘A’. 226 Relative Amplitude 0.2 -0.2 E —0.4 -O.6 1 r r r r 1 r f r T 1 0.5 1 1.5 Time (ns) Figure 5-6 Response waveform from material ‘V’. 227 0.2 - 0 W -0.2 3 Q . g . E Z ‘3. -0.4 - n E I < . q.) . .5 -O.6— .5. I O .. 1 U -0.8 : 1 RS237052 U -1 3 ----- Rs=37752 — — - Rs=38552 "l.2 I r Y I I t t I t I T T ff r r T 0 0.5 1 1.5 2 Time (ns) Figure 5-7 Comparison of response waveforms from materials ‘J’, ‘K’, and ‘M’. 228 Relative Amplitude 0.01 0 : \ -0.0l 5 Late-time —0.02 i -0.03 —0.04 3 U 1 -o.05‘ . . . . . . . . . . . . . . , 0 0.5 1 1.5 Time (ns) Figure 5-8 Convolution of response from material ‘K’ with the E-pulse. 229 Relative Amplitude 0.02 q 0.01 3 0 i : N .0.01 ; Late-time -0.02 f —0.03 f -0.04 f -o.05 f U -0.06 4 . r . . . . . . . T . . . . , . . T O 0.5 1 1.5 Time (ns) Figure 5-9 Convolution of response from material ‘A’ with the E-pulse. 230 Relative Amplitude 0.01 . 'O'Ol I Late-time 11111 -0.02 l r —0.03 —0.04 -0.05 3 P4 1 -0.06......244,,,,,rfi Tmie(ns) Figure 5-10 Convolution of response from material ‘V’ with the E-pulse. 231 Relative Amplitude 0.01 o J \\ '0'01 Late-time 11L —0.02 3 l -0.03 '0.05 -‘ I I T T I I I I I I IT T I I I O 0.5 l 1.5 Time (ns) Figure 5-11 Convolution of response from material ‘J’ with the E-pulse. 232 Relative Amplitude 0.01 r l r r r -0.01 -0.02 l l -0.03 L. -0.04 3 Late-time -0.05 I I T T I T T I T I T Time (ns) Figure 5-12 Convolution of response from material ‘M’ with the E-pulse. 233 Material Surface EDN a Resistance, Rs (9 I Sq) A 50 8.95E-05 B 100 2.28E-04 C 150 2.44E-O4 D 200 l .73E-04 E 250 8.92E-05 F 300 3.00E-05 G 350 3. l4E-06 H 360 1 . l7E-O6 J 370 1.70E-O7 K 377 l .34E-08 M 385 1.84E-08 N 390 7. 18E-08 P 400 3 .37E-07 Q 450 7.99E-07 R 500 2.28E-06 S 550 1 .88E-05 T 600 4.55E-05 U 650 7.76E-05 V 700 l. 12E-O4 Table 5-2 EDNa for change in surface resistance of the impedance sheet. 234 0.0003 0.00025 3 0.0002 : 0.00015 EDNa 14111111 1 0.0001 0.00005 1 O“IIIIITTTIIIIVTIIIIIIITIYIIIITIIIIIrrttt 0 100 200 300 400 500 600 700 800 Surface Resistance (52/sq) Figure 5-13 EDNa vs. surface resistance. 235 Chapter 6 Conclusion 6.1 Summary and Conclusions This thesis has demonstrated the effectiveness of the E-pulse in detecting changes in the material properties of simple layered materials. Not only was the E-pulse effective for noise-free responses it was shown to be effective when the scattered responses were corrupted by noise. Many configurations of simple layered materials were investigated in this thesis. They included single lossless layered materials, single lossy layered materials, and a single lossless layered material with an impedance sheet. For the simple cases it has been shown that the E-pulse can be used for determining changes in certain properties. These properties included the layer thickness, relative permittivity, conductivity, and surface resistance of the impedance sheet. Just as importantly, this thesis has demonstrated the ineffectiveness of the E-pulse for detecting changes in Magram material properties. The cause of failure of the E-pulse for the Magram material was due to the effective design of the material. Magram material is designed as a radar absorber, its purpose being to absorb the radiated energy of incident waves. Although the effect is somewhat narrow-band, the temporal response is primarily specular, and lacks the multiple reflections needed for useful E-pulse discrimination. It has also been shown that the angle of incidence and the polarization of the incident field do not adversely effect the E-pulse technique. One important practical 236 issue addressed in this thesis was the effect of noise on the E-pulse technique. It was shown that for moderate levels of noise the EDRa is very effective tool in quantifying the late-time energy. . 6.2 Future Work Some ideas to consider for future research include determining the effectiveness of the E-pulse in determining changes in material properties for multiple layered materials. The impedance sheet layered material can be investigated for changes in the thickness of the impedance sheet and for changes in the thickness of the layer itself. A combination of the multiple layer and the impedance sheet can be considered. Some work has been done with a Salisbury screen covered by a loaded core. One last area to be investigated would be the use of statistical decision theory in combination with the E- pulse to help determine changes in material properties in the presence of significant levels of noise. 237 APPENDICES A. Overview — N AT_FREQ.cpp This program determines the natural frequencies of a layered material given the material properties. The material properties given as inputs are the thickness, the relative permittivity and the angle of incidence. The user can define the number of frequencies to calculate and write them to a file. A.1 Source Code — NAT_FREQ.cpp ”********************************************************************** // Garrett J Stenholm // Date: 6/7/01 // Name: nat_freq // // Purpose: This program will determine the natural frequencies of a single layer, conductor backed slab // // Input: Slab thickness, t // Material dielectric constant, Er // Angle of incidence, Qi // User defined filename for output, file // Number of frequencies to calculate, num_freq // // Output: A file with the natural frequencies written in a two column, num_freq rows format // ”********************************************************************** #include #include #include #include #include using namespace std; 238 char file[30], answer; ifstream in, exist; ofstream out; int num_freq, checkzl; double thickness, epsilon_r, inc_angle, tau, gamma, sigma; const double c=2.998e8, pi=3. 14159265359; void main() { // user input cout << "\nEnter the slab thickness (in mm): "; cin >> thickness; thickness = thickness * 1e-3; cout << "\nEnter the dielectric constant: "; cin >> epsilon_r; cout << "\nEnter the angle of incidence: "; cin >> inc_angle; cout << "\nEnter the number of frequencies to calculate: "; cin >> num_freq; valarray omega(num_freq); // calculate natural frequencies // calculate tau tau = (2 * thickness * sqrt(epsilon_r - pow( sin(inc_angle),2) )) / c; // calculate gamma gamma = (sqrt(l - pow(sin(inc_angle),2) ) - sqrt(epsilon_r - pow(sin(inc_angle),2) )) / ( sqrt(l - pow(sin(inc_angle),2) ) + sqrt(epsilon_r - pow(sin(inc_angle),2) ) ); // calculate sigma if(gamma<0) { gamma = gamma * -l; } sigma = (l/tau)*log(gamma); // calculate omega for(int n=0; n<=num_freq-1; n++) { omega[n] = ((2*n+1)*pi)/tau; } // write frequencies to file // check for existing file and Open file while(check==1) 239 { cout << "\nEnter the name of the output file: "; cin >> file; exist.open(file); if(exist.good()) { cout << "\nThis file already exist, do you want to overwrite(y/n)? "; cin >> answer; if(answerz-z'y') { check=0; exist.close(); exist.clear(); out.open(file); } else { checkzl; exist.close(); exist.clear(); } } else { check=0; exist.close(); exist.clear(); out.open(file); } } // write to file for(int i=0; i<=omega.size()-1; i++) { out << fixed << showpoint << setprecision(3); out << sigma* 1e-9 << " " << omega[i]*le-9 << endl; } // close file out.close(); 240 B. Overview - KPNAT.for This program generates a natural E-pulse from the natural frequencies. The program reads the natural frequencies from a data file and outputs an E-pulse to a file. B.l Source Code - KPNAT.for program kpnat parameter (mmx=15, nmx=32) character*30 fnm, ftype real b(nmx), diag(nmx) integer ipvt(nmx) common /sfill/ a(nmx,nmx), nmodes, nb, dt, iexcit, nexcit, idc common /freqs/ qm(mmx), sg(mmx) pi = 4.*atan(l.) write(*,*) '*****************************************************' write(*,*) '** Program: KPNAT.FOR **' write(*,*) '** **1 write(*,*) '** Purpose : Construct natural E/S Pulse from **' write(*,*) '** natural frequencys. **' write(*,*) '** **1 write(*,*) '** Update : March 19, 1991 **' write(*,*) '** **I write(*,*) '** Programmers : Dr. E]. Rothwell **' Wfit€(*.*) '** Ponniah Ilavarasan **' write(*,*) '** John E. Ross III **' write(*,*) '*****************************************************' write(*,*) write(*,*) 'Natural frequency file name ?' read (*,5) firm 5 forrnat(a) call rdfrq(fnm, n) nmodes = n write(*,*) 'Enter option # :' write(*,*) ' 0. E-pulse' write(*,*) ' l. Sine S-pulse' write(*,*) ' 2. Cosine S-pulse' 241 write(*,*) ' 3. Combined S-pulse' read(*,*) iexcit c nexcit = O if(iexcit .eq. 0) goto 200 c c Display modes c Write(* *) '********************************************' write(*,*) ' # sigma omega' do 195 i = 1, nmodes write(*,*) i, sg(i), qm(i) 195 confinue write(* 31‘) '********************************************' 9 c write(*,*) 'Enter number of mode to excite' read(*,*) nexcit sigma = sg(nexcit) omega = qm(nexcit) c 200 write(*,*) 'Kill dc (l=y O=n) ?’ Read(*,*) idc c c Find highest frequency given c imode = l qmmax = qm(imode) do 205 i = 1, nmodes if(qm(i).gt.qmmax) then qmmax = qm(i) imode = i endif 205 confinue c c Set degeneracy of modes (ie number of multiple modes) c No degeneracy if mmode = 1. c mmode = 1 c c Calculate time interval and number of basis functions c dt = mmode*pi/qm(imode) nb = 2*nmodes-1 if(iexcit .eq. 0) nb = nb+1 if(iexcit .eq. 3) nb = nb-l if(idc .eq. 1) nb = nb+1 242 90 call fill nsize = nb-l istart = imode if(iexcit .eq. 2 .or. iexcit .eq. 3) istart = istart-l do 100 i = istart, nsize do 90 j = 1, nb a(i,j) = a(i+1,j) confinue 100 confinue C do 110 n = l, nb b(n) = -a(n,nb) 110 confinue C call decomp(nsize,a,ipvt,diag) call solve(nsize,a,ipvt,b) b(nb) = 1. call normal(b,nb,dt,iexcit,nexcit) pdur = dt*float(nb) write(*,*) write(*,*) 'Pulse duration = ',pdur write(*,*) 'Number of basis functions = ',nb dtpul = pdur / nb write(*,*) ' # Amplitude' do 210 i = l, nb write(*,*) 1, b(i) 210 confinue C if(nexcit.eq.0) ftype = 'Time' if(nexcit.ne.0) ftype = 'Spulse' write(*,*) 'Filename for results' read(*,S) fnm call wrdata(fnm,ftype,dtpul,nb,sigma,omega,b) end 243 c RDFRQ - Reads in the natural frequencies from a disk file. c subroutine rdfrq(fnm, nmodes) parameter (lun=10, mmx=15) c character*30 fnm c common /freqs/ qm(mmx), sg(mmx) c open (unit:lun, file=fnm, statusz'old') c i = l 10 read(lun,*,end=99,err=999) sg(i), qm(i) i = i + 1 goto 10 c 99 nmodes = H close(lun) return c 999 write(*,*) '*** Error in RDFRQ ***' close(lun) stop c end c c c c WRDATA - writes data to disk in ndf format c subroutine wrdata(fnm,ftype,dt,np,sigma,omega,b) parameter (lun=10, nmx=32) c character*30 fnm, ftype real b(nmx) c open(lun, file=fnm, statusz'unknown') c write(lun,5) ftype 5 format (a) c if (ftype.eq.'Spulse') write(lun,*) sigma, omega write(lun,5) '0.0' c write(lun,*) dt 244 C do 20 i = 1, np 2O write(lun,*) b(i) c close(lun) return c end c c c c FILL - Fills the coefficient matrix c subroutine fill parameter (mmx=15, nmx=32) common /sfill/ a(nmx,nmx),nmodes,nb,dt,iexcit,nexcit,idc common /freqs/ qm(mmx),sg(mmx) nm2 = 2*nmodes do 110 n = l,nmodes do 100 m =1, nb tl = float(m-1)*dt tu = tl + dt a(n,m) = rcos(n,tu) - rcos(n,tl) a(n+nmodes,m) = rsin(n,tu) - rsin(n,tl) 100 confinue llO confinue c if(iexcit .eq. 0) goto 500 if(iexcit .eq. 2) goto 190 nr = nmodes + nexcit nt 2 nm2 - 1 do 150 n = nr, nt do 140 m = 1, nb a(n,m) = a(n+1,m) 140 confinue 150 confinue c 190 if(iexcit .eq. 1) goto 500 c nr = nexcit nt = nm2 - 1 do 250 n = nr, nt do 240 m = 1, nb 245 a(n,m) = a(n+1,m) 240 confinue 250 confinue c 500 if(idc .eq. 0) goto 600 c do 520 m = 1, nb a(nb,m) = l. 520 continue c 600 return end c c c c RSIN - Evaluate integral of exp(x)*sin(x) c 00000 00000 real function rsin(n,x) parameter (mmx=15, nmx=32) common /freqs/ qm(mmx) ,sg(mmx) rsin = exp(-sg(n)*x)*(-sg(n)*sin(qm(n)*x) - qm(n)*cos(qm(n)*x)) return end RCOS - Evaluate integral of exp(x)*sin(x) real function rcos(n,x) parameter (mmx=15, nmx=32) common /freqs/ qm(mmx) ,sg(mmx) rcos = exp(—sg(n)*x)*(-sg(n)*cos(qm(n)*x) + qm(n)*sin(qm(n)*x)) return end NORMAL - normalizes E-Pulses subroutine normal(b,nb,dt,iexcit,nexcit) 246 parameter (nmx=32) c real b(nmx) c if((iexcit .eq. O).or.(iexcit.eq.3)) goto 500 c monn = 0. do 50 l = l, nb 1] = float(l-l)*dt tu = tl + dt if(iexcit .eq. 2) morm = rnorm+b(l) * & (rcos(nexcit,tu) - rcos(nexcit,tl)) if(iexcit .eq. 1) morm = morm+b(l)* & (rsin(nexcit,tu) - rsin(nexcit,tl)) 50 continue c do 1001 = 1, nb b(l) = b(l)/rnorm 100 continue c 500 return end c C - c c SOLVE - solves matrix eqn ax=b with decomposed by decomp subroutine solve(n,a,ipvt,b) parameter (nmx=32) integer n,ipvt(nmx) integer nm1,k,kb,kpl,km1,m,i reala(nmx,nmx),b(nmx),s c c Forward elimination c if (n.eq.1) goto 30 nml = n - 1 do 10 k = 1, nml kpl = k + l m = ipvt(k) s = b(m) b(m) = b(k) b(k) = 8 do 10 i = kpl, n 10 b(i) = b(i) + a(i,k)*s 247 c c Back substitution c do 20 kb =1, nml kml = n - kb k = kml + l b(k) = b(k)/a(k,k) s = -b(k) do 20 i: l,km1 20 b(i) = b(i) + a(i,k)*s c 30 b(1)= b(1)/a(1,1) c return end DECOMP - decomposes matrix a to triangular form From : Linear Algebra by Gilbert Strang 000000 subroutine decomp(n,a,ipvt,diag) parameter (nmx=32) integer n, ipvt(nmx) integer nm1,i,j, k, kp1,m real a(nmx,nmx), diag(nmx) real p, t ipvt(n) = 1 if (n .eq. 1) Go to 70 nml = n-l do 60 k =1, nml kpl =k+1 0 Find pivot p m = k do 10 I = kpl, n 10 if (abs(a(i,k)) .gt. abs(a(m,k))) m = i ipvt(k) = m if (m .ne. k) ipvt(n) = -ipvt(n) p = a(m,k) a(m,k) = a(k,k) a(k,k) = p 248 diag(k) = p if (p .eq. 0.0) goto 60 c c Compute mutipliers c 20 do 30 i = kpl, n 30 a(i,k) = -a(i,k)/p c c Interchange rows and columns c do 50 j = kpl, n t= a(mJ) a(m.j) = a(k,j) a(k,j) =t if (t .eq. 0.0) goto 50 do 40 i = kpl, n a(iJ) = a(i.j) + a(i,k)*t 40 continue 50 continue 60 continue c 70 diag(n) = a(n,n)*float(ipvt(n)) c return end c C---- C C. Overview - PCONV2.for This program is the program which does most of the work when using the E-pulse technique. It convolves the scattered response waveforms with the E-pulses. It gives the user the ability to either use noise-free response signals or add noise to the signal. It outputs the value of the EDNa and writes the convolution waveform to a file. C.1 Source Code - PCONV2.for PROGRAM PCONV2 C C PERFORMS TRAPEZOIDAL RULE CONVOLUTION OF DATA SET C AND PULSE FUNCTION BASED E-PULSE C C VERSION OF 29 May 2001 C Revised by Garrett J Stenholm C REAL hah(2050,2050) REAL H(2050,2050) ,T(2050), AN(2050), EDNA(512), SNR(512), Er REAL ED(32), EPS(32), th(32), ang(32), TEMP1(32), TEMP2(32), d, ai CHARACTER*32 FILENAM, dummy, NOISEl, NOISE2, temp, EPULSE, RESPON character*32 filnam(20), filnamc(20), COMMA, array CHARACTER*32 FILE, blah integer in, FILES, CH, pos, size, file_flag, update, element C COMMON /BLOCK1/ RM(2050),B(2050),DT,DLT,AL(50),NP COMMON IBLOCK2/ CON(2050),TCON(2050),DTC C IJ=1 in = 500 FILES = 1 c EPULSE = 'epulleh.ndf‘ c RESPON = 'respth.ndf‘ c Ttr = 0.1 c Tp = 0.1 c Tb = 0.5 NC = 250 c INOISE = l Cccccccccccccccccccccccccccccccccccccccccccccccccccccc C Create files for (response + noise) signal Cccccccccccccccccccccccccccccccccccccccccccccccccccccc filnam(l) = 'r' filnam(2) = 'respb.dat' filnam(3) = 'respc.dat' filnam(4) = ’respd.dat' filnam(5) = 'respe.dat' filnam(6) = 'respf.dat' filnam(7) = 'respg.dat' filnam(8) = 'respb.dat' filnam(9) = 'respi.dat' filnam( 10) = 'respj .dat’ filnamc(l) = 'c' filnamc(2) = 'convb.dat' filnamc(3) = 'convc.dat' filnamc(4) = 'convd.dat' filnamc(5) = 'conve.dat' 250 10 filnamc(6) = ‘convf.dat' filnamc(7) = 'convg.dat' filnamc(8) = 'convh.dat' filnamc(9) = 'convi.dat' filnamc( 10) = 'convj.dat’ WRITE (*,*) 'ENTER NAME OF E-PULSE FILE (.ndf)’ READ (*,1) EPULSE format(a) OPEN (lO,FILE=EPULSE,STATUS='UNKN OWN’) read (10,1) dummy read (10,*) start read (10,*) dd np=l READ (10,*,end=10) AL(np) np=np+l go to 5 np=np-1 write (*,*) 'no. pulses=',np tk=np*dd write (*,*) 'tk=',tk CLOSE (10) CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C Determine the energy in the E-pulse CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC ll 15 Ee=0 do 11 J=l,np Ee=Ee+dd*(AL(J)**2) confinue write (*,*) 'E-pulse energy =',Ee print 1 SIZE = 1. WRITE (*,*) 'ENTER NAME OF RESPONSE FILE (.ndf)’ READ (*,1) RESPON OPEN (l0,FILE=RESPON,STATUS='UNKNOWN') read (10,1) dummy read (10,*) start READ(10,*) DT I = l READ(10,*,END=20) H(1,I) T(I) = FLOAT(I-1)*DT l: 1+ 1 251 GO TO 15 20 NPTS = I-l WRITE(*,*) 'NPTS = ',NPTS CLOSE (10) Ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C Determine the energy in the response file Ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc EFO do 21 1:1 ,NPT S Er=Er+DT*(H(1,I)**2) 21 confinue Cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C Enter times to determine the late-time,TL, for convolution energy calculation Cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc write (*,*) 'Enter the one-way transit time: ' read (*,*) Ttr write (*,*) 'Enter timee delay: ' read (*,*) Tb write (*,*) 'Enter pulse width: ' read (*,*) Tp Ccccccccccccccccccccccccccccccccccc C ADD NOISE IF DESIRED Ccccccccccccccccccccccccccccccccccc WRITE(*,830) 830 FORMAT(1X,'ADD NOISE?(1=Y 0=N):' $) READ(*,*) INOISE Cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C Calculate 99% of energy in noise-free signal and re-determine the number of points Cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc if(INOISE .EQ. 1) then alimit=Er S=0 K=O 34 K=K+1 S=S+DT*(H(1,K)**2) if(S .LT. alirrrit) goto 34 NPT S =K endif write(*,*) 'alimit= ',alimit write(*,*) 'S= ',S write(*,*) NPTS: ',NPTS 252 C WRITE (*,*) 'ENTER NO. PTS IN CONVOLUTION:' C READ(*,*) NC C DLT = TK/FLOAT(NP) T1 = T(1) Cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C SHIFT ORIGIN OF RESPONSE Cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc DO 30 I: 1,NPTS T(I) = T(I)-Tl 30 CONTINUE IF (INOISE .EQ. 1) THEN write(*,*) 'Enter the desired SNR: ' read(*,*) SR db=SR/10 Cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C Calculate the variance and standard deviation to C determine noise energy based on the response signal energy Cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc W=NPTS*DT var=Er/(W*(10**db)) sigma=sqrt(var) Cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C Add noise to the response signal and create 'in' number of samples C 'seed' is the starting point for determining the random numbers used C to simulate gaussian noise Cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc WRlTE(*,840) 840 FORMAT(1X,'ENTER RANDOM DECIMAL SEED:' $) READ(*,*) SEED OPEN (30,FILE="temp.dat",STATUS='UNKNOWN') DO 53 IJ=1,in DO 50 I: 1,NPT S AN(I) = sigma*GAUSRV(SEED) hah(I,IJ)=AN(I) WRITE (30,*) hah(I,IJ) l=IJ+1 H(l,I) = H(l,I) + AN(I) 50 CONTINUE 253 Cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C Calculate the noise energy and SNR calculated C from the generated random noise Cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc En=0 do 710 J =1 ,NPTS En=En+DT*(AN(J)**2) 710 continue SNR(IJ)=10*log10((S/En)) C write (*,*) IJ,’ Actual SNR: ', snr Ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C Write the first 10 new noise-corrupted response signals to files cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc IF(IJ .LT. FILES+1) THEN OPEN (10,FILE=filnam(IJ),STATUS='UNKNOWN') DO 52 I: 1,NPTS WRITE (10,*) T(I), H(U+l,I) 52 CONTINUE CLOSE( l 0) ENDIF C 53 CONTINUE CLOSE(30) C ENDIF Cccccccccccccccccccccccccccccccccccccccccccccccccccccccc C CALCULATE LINEAR INTERPOLATION COEFICIENTS Cccccccccccccccccccccccccccccccccccccccccccccccccccccccc if(INOISE .EQ. 0) in=1 c do 65 U=1,l 35 NPTSMl = NPTS-1 DO 60 I: 1,NPTSM1 RM(I) = (H( 1 ,(I+l))-H( l ,I))/DT B(I) = H(1,I)-RM(I)*FLOAT(I-l)*DT 60 CONTINUE C TS = 0. TE = T(NPTS-l) DTC = (TE-TS)/FLOAT(NC-l) TT =TS DO 70 I: 1,NC TCON(I) = TT CON(I) = R(TT) 254 TT ='IT+DTC 70 CONTINUE c IF(IJ .GT. 1 .AND. U .LT. FILES+2) THEN c OPEN (lO,FIL&filnamc(IJ-l),STATUS='UNKNOWN') c DO 275 J=1, NC 0 WRITE (10,*) TCON(J)+start,CON(J) c275 CONTINUE c CLOSE(lO) c ENDIF Ccccccccccccccccccccccccccccccccccccccccccc C Determine the late-time Cccccccccccccccccccccccccccccccccccccccccccc TL = 2*Ttr+Tb+Tp TLi = nint(TL/DTC)+1 if(INOISE .EQ. 0) then write(*,*) 'Enter the name of the convolution output file: ' read (*,*) FILENAM c FILEN AM 2 'conv.dat' OPEN (10,FILE=FILENAM,STATUS='UNKNOWN') DO 285 J=l, NC WRITE (10,*) TCON(J)+start,CON(J) 285 CONTINUE CLOSE(lO) endif Ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C Calculate the energy of the response and convolution waveform Ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc Er=0 do 621 J=1,NPTS Er=Er+DT*(H(IJ,J)**2) 621 confinue Ec=0 do 71 K=TLi,(NC- 1) Ec=Ec+(TCON(K+1)-TCON(K))*(CON(K)* *2) 71 confinue Cccccccccccccccccccccccccccccccccccc C EDN calculation Cccccccccccccccccccccccccccccccccccc EDN = Ec / Ee EDNA(IJ) = Ec / (Ee * Er) 255 c write (*,*) IJ,‘ Old EDN: ',EDN, ' New EDN: ',EDNA(IJ) 65 confinue Cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C Write the ENDA 2x2 array to a file for comparison Cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc write(*,*) 'Enter the name of the file to write the EDNA array to: ' read(*,*) array open (10,FILE=array,STATUS='UNKNOWN') do 67 i=l,in write (10,*) EDNA(i+1) 67 continue close( 10) Cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C Calculate the mean and standard deviation of the EDN Cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc SUMsnr=O DO 760 I=2,in SUMsnr = SUMsnr+SNR(I) 760 CONTINUE AMEANsnr = SUMsnr/FLOAT(in-l) SUM=0 DO 770 I=2,in SUM = SUM+EDNA(I) 770 CONTINUE AMEAN = SUM/FLOAT(in-1) SUMM=0 DO 75 1:2, in SUMM = SUMM+(EDNA(I)-AMEAN)**2 75 CONTINUE VARI = SUM]VI/FLOAT(in-2) SD = SQRT(VARI) if(INOISE .EQ. 1) then WRITE (*,*) 'Response file = ’, RESPON WRITE (*,*) 'Ttr = ', Ttr WRITE (*,*) 'Number of samples = ', in WRITE (*,*) 'Files written = ', FILES WRITE (*,*) 'Mean SNR = ', AMEANsnr 256 WRITE (*,*) 'Mean EDNA = ', AlVfEAN WRITE (*,*) 'Standard deviation = ', SD elseif(INOSIE .EQ.0) then WRITE (*,*) 'Response file = ', RESPON WRITE (*,*) 'Ttr = ', Ttr WRITE (*,*) 'EDNA = ', EDNA(l) endif 800 WRITE (*,*) 'DO YOU WANT TO WRITE THE EDNA TO A FILE? ' WRITE (*,*) '1. EPSH.ON FILE' WRITE (*,*) '2. THICKNESS FILE' WRITE (*,*) '3. ANGLE FH.E' WRITE (*,*) '4. DO NOT WRITE TO A FILE' c READ (*,*) CH ch=4 803 format(f10.2, a3, F15.10) IF(CH .EQ. 1) THEN WRITE (*,*) 'ENTER Er: ' READ (*,*) Er 1:1 ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c open file and read into memory ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc OPEN (10, FILE='DELTAE.DAT', STATUS='UNKNOWN') 805 READ (10,803,END=810) EPS(I), COMMA, ED(I) I=I+l GOTO 805 810 I=I-1 CLOSE(IO) size = I cccccccccccccccccccccccccccccccccccccccccccccccc c determine if the file existed or is new cccccccccccccccccccccccccccccccccccccccccccccccc ccccccccccccccccccccccccccccccccc c file is new ccccccccccccccccccccccccccccccccc IF(size .EQ. 0) THEN file_flag = 1 OPEN (10, FILE='DELTAE.DAT', STATUS='UNKNOWN') WRITE(10,803) Er, ',', EDNA(l) CLOSE(IO) 257 ENDIF ccccccccccccccccccccccccccccccccc c file has already been created ccccccccccccccccccccccccccccccccc IF(I .NE. 0) THEN FILE='DELTAE.DAT' update =0 do 815 i=1, size if(Er .eq. EPS(i) .and. update .eq. 0) THEN call replace(EPS, ED, Er, EDNA(I), 1, size, FILE) update =1 else if(Er .lt. EPS(i) .and. update .eq. 0) then call insert(EPS, ED, Er, EDNA(I), i, size, FILE) update =1 else if(Er.gt.EPS(i).and.update.eq.0.and.Er.lt.EPS(i+1)) then call insert(EPS, ED, Er, EDNA(I), i+1, size, FILE) update =1 else if(Er.gt.EPS(i).and.i.eq.size.and.update.eq.0) then call append(EPS, ED, Er, EDNA(I), i+1, size, FILE) update =1 endif 815 confinue ENDIF ELSE IF(CH .EQ. 2) THEN WRITE (*,*) '2 IS A BETTER CHOICE' WRITE (*,*) 'ENTER thickness: ' READ (*,*) (1 1:1 cccccccccccccccccccccccccccccccccccccccccccccccccccccc c open file and read into memory cccccccccccccccccccccccccccccccccccccccccccccccccccccc OPEN (10, FILE='DELTAT.DAT', STATUS=‘UNKNOWN') 820 READ (10,803,END=835) th(I), COMMA, ED(I) I=I+l GOTO 820 835 I=I-1 CLOSE(lO) size = I CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC c determine if the file existed or is new ccccccccccccccccccccccccccccccccccccccccccccccccccccccc 258 1:} L ,1 ccccccccccccccccccccccccccccccccccccccc c file is new ccccccccccccccccccccccccccccccccccccccc IF(size .EQ. 0) THEN file_flag = 1 OPEN (10, FILE='DELTAT.DAT', STATUS='UNKNOWN') WRITE(10,803) d, ',', EDNA(l) CLOSE(lO) ENDIF ccccccccccccccccccccccccccccccccccccccc c file has already been created ccccccccccccccccccccccccccccccccccccccc IF(I .NE. 0) THEN FILE='DELTAT.DAT' update =0 do 850 i=1, size if(d .eq. th(i) .and. update .eq. 0) THEN call replace(th, ED, (1, EDNA(I), i, size, FILE) update =1 else if(d .lt. th(i) .and. update .eq. 0) then call insert(th, ED, (1, EDNA(I), i, size, FILE) update =1 else if(d.gt.th(i).and.update.eq.0.and.d.lt.th(i+1)) then call insert(th, ED, (1, EDNA(I), i+l, size, FILE) update =1 else if(d.gt.th(i).and.i.eq.size.and.update.eq.0) then call append(th, ED, d, EDNA(I), i+1, size, FILE) update =1 endif 850 confinue ENDIF ELSE IF(CH .EQ. 3) THEN WRITE (*,*) '3 IS THE BEST CHOICE' WRITE (*,*) 'ENTER angle: ' READ (*,*) ai I=1 cccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c open file and read into memory cccccccccccccccccccccccccccccccccccccccccccccccccccccccccc OPEN (10, FILE='DELTAA.DAT', STATUS='UNKNOWN') 860 READ (10,803,END=870) ang(I), COMMA, ED(I) I=I+1 259 GOTO 860 870 I=I-1 CLOSE(lO) size = I cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c determine if the file existed or is new cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c ccccccccccccccccccccccccccccccccccc c file is new ccccccccccccccccccccccccccccccccccc IF(size .EQ. 0) THEN file_flag = 1 OPEN (10, FILE='DELTAA.DAT', STATUS='UNKNOWN') WRIT E(lO,803) ai, ',', EDNA(l) CLOSE(lO) ENDIF ccccccccccccccccccccccccccccccccccc c file has already been created ccccccccccccccccccccccccccccccccccc IF(I .NE. 0) THEN FILE=DELTAA.DAT' update =0 do 890 i=1, size if(ai .eq. ang(i) .and. update .eq. 0) THEN call replace(ang, ED, ai, EDNA(I), i, size, FILE) update =1 else if(ai .lt. ang(i) .and. update .eq. 0) then call insert(ang, ED, ai, EDNA( 1), i, size, FILE) update =1 else if(ai.gt.ang(i).and.update.eq.0.and.ai.lt.ang(i+1)) then call insert(ang, ED, ai, EDNA(l), i+1, size, FILE) update =1 else if(ai.gt.ang(i).and.i.eq.size.and.update.eq.0) then call append(ang, ED, ai, EDNA(l), i+l, size, FILE) update =1 endif 890 confinue ENDIF ELSE IF(CH .EQ. 4) THEN WRITE (*,*) 'OK, GOODBYE!’ ELSE IF(CH.NE.1 .AND. CH.NE.2 .AND. CH.NE.3 .AND. CH.NE.4 ) THEN 260 WRITE (*,*) 'N OT A VALID CHOICE' GOTO 800 ENDIF 999 STOP C END C******************************************************************** C FUNCTION R(TT) CALCULATES CON VOLUTION AT TIME TT COMMON /BLOCK1/ RM(2050),B(2050),DT,DLT,AL(50),NP R = 0. DO 90 N: 1,NP 0 0 0 000 SUM = 0. TL = TT-FLOAT(N)*DLT IF (TL .LT. 0.) TL=O. TU = TT-FLOAT(N-l)*DLT IF (TU .LT. 0.) TU=0. K1 = INT(TL/DT)+2 K2 = INT(TU/DT)+1 SUM = SUM + RI(Kl-l,TL,FLOAT(Kl-1)*DT) SUM = SUM + RI(K2,FLOAT(K2-l)*DT,TU) KZM = K2-l IF(KZM .LT. Kl)CtO TO 60 DO 50 I: K1,K2M SUM = SUM + RI(I,DT*FLOAT(I-l),DT*FLOAT(I)) 50 CONTIBIUE C 60 R = R + AL(N)*SUM C 90 CONTINUE C RETURN END C******************************************************************** C FUNCTION RI(I,T1,T2) C C CALCULATES INTEGRAL OF RESPONSE 261 COMMON /BLOCK1/ RM(2050),B(2050),DT,DLT,AL(50),NP R1 = (T2-Tl )*(B(I)+ RM(I)*(T2+T1)/2.) RETURN END C******************************************************************** C FUNCTION RINT (TLTZ) C C CALCULATES INTEGRAL FROM T1 T0 T2 C COMMON [BLOCKZ/ CON(2050),TCON(2050),DTC C ISTART = INT(Tl/DTC) + 2 [END = INT(T2/DTC) + l SUM = 0. IF (IEND .GT. ISTART) THEN DO 10 I=ISTART,IEND-l SUM = SUM + 0.5*DTC*(CON(I)+CON(I+1)) 10 CONTINUE ENDIF HA = CON(ISTART- 1) HB = CON(ISTART) TA = TCON(ISTART-l) TB = TCON(ISTART) HI = HA + (Tl-TA)*(HB-HA)/(TB-TA) SUM = SUM + 0.5*(TB-T1)*(H1+HB) C HA = CON(IEND) HB = CON (IEND-H) TA = TCON(IEND) TB = TCON(IEND+1) HI = HA + (T2-TA)*(HB-HA)/(TB-TA) SUM = SUM + 0.5*(T2-TA)*(H1+HA) C RINT = SUM C RETURN END C********************************************************************** C FUNCTION GAUSRV (SEED) 262 C C CALCULATES A GAUSSIAN RANDOM VARIABLE OF MEAN 0 AND STANDARD DEVIATION l C PI =4. *ATAN( l .) SEED = 997.*SEED-AINT(997.*SEED) R1 = SEED SEED = 997.*SEED-AINT(997.*SEED) R2 = SEED GAUSRV = SQRT(-2.*LOG(R1))*COS(2.*PI*R2) C RETURN END C******************************************************************** C SUBROUTINE insert (arrayl, array2, e11, e12, pos, size, file) C C adds an element to the interior of an array C character*32 file real array1(32), array2(32), e11, e12 real templ(32), TEMP2(32) integer pos, size, quit, tempsize write(*,*) 'insert' quit = size-pos+l do 100 1:1, quit temp 1 (I)=array1 (pos+I- 1) temp2(I)=array2(pos+I- 1) 100 confinue tempsize = I-l array 1 (pos)=ell array2(pos)=e12 do 150 1:1, tempsize array 1 (pos+I) = temp 1(1) array2(pos+I) = temp2(I) 150 confinue 203 format(f10.2, a3, F1510) OPEN (10, FH.E=file, STATUS='UNKNOWN') 263 do 2501=1, size+1 WRIT E(lO,203) arrayl(l), ',', array2(I) 250 confinue CLOSE(IO) C C RETURN END C********************************************************************** C******************************************************************** C SUBROUTINE append (arrayl, array2, e11, 612, pos, size, file) C C adds an element to the end of an array C character*32 file real array1(32), array2(32), e11, e12 integer pos, size write(*,*) 'append' array1(pos)=e11 array2(pos)=e12 203 format(f10.2, a3, F15.10) OPEN (10, FILE=file, STATUS='UNKNOWN') do 250 1:1, size+1 WRIT E(10,203) array1(I), ',', array2(I) 250 confinue CLOSE( 10) C RETURN END C********************************************************************** g******************************************************************** C SUBROUTINE replace (arrayl, array2, ell, e12, pos, size, file) C C replaces an element in an array C character*32 file 264 real arrayl(32), array2(32), ell, e12 integer pos, size write(*,*) 'replace' array 1 (pos)=ell array2(pos)=e12 203 forrnat(f10.2, a3, F15.10) OPEN (10, FILE=file, STATUS='UNKNOWN') do 250 1:1, size WRIT E(10,203) arrayl(I), ‘,', array2(I) 250 confinue CLOSE(IO) C C RETURN END (:********************************************************************* D. Overview - MLPREF.for This program generates the scattered response from a layered material. The user defines the properties of the layered material including the number of layers, layer backing, and the properties of the layers. The user also defines the type of incident waveform and its properties including polarization and angle of incidence. In addition, the user defines the frequency range of the incident waveform. The output is a frequency domain representation of the scattered response to the defined layered material. D.l Source Code - MLPREF.for program mlpref C C ********** c This program reads appropriate input data for studying the reflection c of a plane wave pulse obliquely incident upon a grounded multi-layered 265 material composite. It subsequently quantifies that reflection over a specified band of frequencies by calling appropriate sub-programs for the spectrum of an incident pulse and the spectral reflection coeffic- ient of the grounded composite. ********** O O C)(3(5 0 implicit integer*4 (i-n) implicit real*8 (a-h,o-z) complex* 16 j,epsr(0:lO),mur(O:lO),k(O:10),kz(O:10),z(0:10),p(0:10) complex* 16 r(0:10),ei,ips,rc,rcm,rw dimension h(O: 10) common/b1/pi,c,j common/b2/tau,iw common/b3/tha,nl,ib,ip common/b4/h,epsr,mur,k,kz,z,p,r open(8,fi1e='m1pref.out',status='unknown') open(9,file='ips.dat',status='unknown') open(10,f1le='m1pref.dat',status='unknown’) C ********** c Data Entry Segment C ********** write(*,*) 'Enter waveform type iw (impulsezl, gaussian=2) >' read(*,*) iw if(iw.eq.2) then write(*,*) 'Enter gaussian pulse duration gpd in ps > read(*,*) gpd endif write(*,*) 'Enter polarization index ip (perpzl, pamZ) >' read(*,*) ip write(*,*) 'Enter incidence angle tha in degrees >' read(*,*) tha write(*,*) 'Enter number nl of material layers > read(*,*) n] write(*,*) 'Enter backing index ib (1=space, 2=conductor) >' read(*,*) ib do n=1,nl,1 write(*,*) 'Enter h (in mm), epsr and mur for layer number n =', + n,‘ >' read(*,*) h(n),epsr(n),mur(n) enddo write(*,*) 'Enter start,end frequencies fs,fe in GHz >' read(*,*) fs,fe write(*,*) 'Enter number nf of frequency points > read(*,*) nf C ********** c Data Output Segment 266 C ********** write(*,lO) write(8,10) 10 forrnat(/,2x,'ESSENTIAL INPUT DATA',//) if(iw.eq.l) then write(*,14) write(8,14) elseif(iw.eq.2) then write(*,lS) gpd write(8,15) gpd endif 14 forrnat(/,5x,'impulse incident waveform') 15 forrnat(/,5x,'gaussian incident waveform,',5x,'2T = ',f5.0,' ps') if(ip.eq.l) then write(*,20) tha write(8,20) tha elseif(ip.eq.2) then write(*,21) tha write(8,21) tha endif 20 fonnat(5x,'perpendicular polarization,',5x,'tha = ',f4.1, +' degrees') 21 fonnat(5x,'parallel polarization',5x,'tha = ',f4.1,' degrees') if(ib.eq.1) then write(*,25) write(8,25) elseif (ib.eq.2) then write(*,26) write(8,26) endif 25 format(5x,'free-space backing'Jl) 26 forrnat(Sx,'perfect-conductor backing',//) write(*,30) write(8,30) 3O format(2x,'n',4x,'h(n) in mm',10x,'epsr(n)',23x,'mur(n)',/) do n=1 ,nl,1 write(*,3 l) n,h(n),epsr(n),mur(n) write(8,3 1) n,h(n),epsr(n),mur(n) enddo 31 forrnat(2x,i2,4x,f5.2,5x,'(',e11.4,',',e11.4,')',5x,'(',e11.4,',' +,e11.4,')') write(*,35) fs,fe,nf write(8,35) fs,fe,nf 35 format(//,5x,'fs = ',e12.5,' GHz',/,5x,'fe = ',e12.5,' GHz‘,/,5x, +'nf = ',i4,‘ pts.',//) write(8,40) 267 4O format(/,8x,'f‘,l3x,'RWr',l1x,’RWi',llx,'RWa',l1x,'RWp',/) C ********** c Initial Parameter Computation Segment C ********** epsr(0)=dcmplx(1.d0,0.d0) mur(0)=dcmplx(1.d0,0.d0) tau=l .d-l2*gpd/2. do n=1,nl,1 h(n)=h(n)*1.d-3 enddo fs=fs* 1 .d9 fe=fe* 1 .d9 df=(fe—fs)/nf pi=4.*atan(1.d0) dpm l 80./pi c=3.d8 j=dcmplx(0.d0, 1 .d0) tha=tha*pi/180.d0 C ********** c Data Computation Segment C ********** do i=1,nf+1,1 f=fs+(i-1)*df ei=ips(f) rc=rcm(f) rw=ei*rc eim=abs(ei) eip=dpr*atan2(imag(ei),dble(ei)) rwm=abs(rw) rwp=dpr*atan2(imag(rw),dble(rw)) write(8,50) f*1.d-9,rw,rwm,rwp write(9,51) f*1.d-9,eim,eip write(lO,51) f*1.d-9,rwm,rwp 50 format(5(2x,e12.5)) 51 format(3(2x,el2.5)) enddo endfile 8 endfile 9 endfile 10 end C ********** C ********** complex*16 function ips(f) c C ********** c This subroutine sub-program computes the spectrum of the incident 268 c C c 0 (3(7 0 (3(3 pulse waveform. ********** implicit integer*4 (i-n) implicit real*8 (a-h,o-z) complex* 16 j common/bl/pi,c,j common/b2/tau,iw omeg=2.*pi*f if(iw.eq.l) then ips=dcmplx(1.d0,0.d0) elseif(iw.eq.2) then ps=sqrt(pi)*tau*exp(-(omeg*tau)**2/4.) ips=dcmplx(ps,0.d0)*exp(-j*5.*omeg*tau) endif end ********** ********** subroutine param(f) This subroutine calculates the required wave parameters for each of the material layers based upon their thickness and constitutive par- ameters. It finally calculates the interfacial reflection coeffic- ient Rn at the interface between the n-l'st and n'th layers. implicit integer*4 (i-n) implicit real*8 (a-h,o-z) real*8 lO,kO,kx complex* 16 j,epsr(0:10),mur(0: 10),k(0:10),kz(0: lO),z(O:10),p(O: 10) complex* 16 r(0: 10) dimension h(O: 10) common/b l/pi,c,j common/b3/tha,nl,ib,ip common/b4/h,epsr,mur,k,kz,z,p,r if(f.lt.1.d0) then fl = 1 .d0 else fl =f endif lO=c/f1 zi0=120.*pi k0=2.*pi/10 kx=k0*sin(tha) do n=0,nl,l k(n)=sqrt(epsr(n)*mur(n))*k0 kz(n)=sqrt(k(n)*k(n)-kx*kx) 269 O O (>(3 O C) O O O O if(ip.eq. 1) then z(n)=k0*zi0*mur(n)/kz(n) elseif(ip.eq.2) then z(n)=zi0*kz(n)/(k0*epsr(n)) endif if(n.eq.0) then r(n)=dcmplx(0.d0,0.d0) p(n)=dcmplx(0.d0,0.d0) else r(n)=(z(n)—z(n-l))/(z(n)+z(n- 1)) p(n)=exp<—j *kz gam(n)=(r(n)+gam(n+l)*p2)/(1.+r(n)*gam(n+1)*p2) end E. Overview - NFSIG.for This program determines the natural frequencies for materials which have none zero conductivity. The inputs are the material properties including the thickness, relative permittivity, permeability, conductivity, angle of incidence, and polarization. The program allows the user to define the number of natural frequencies to calculate. E.1 Source Code - NFSIG.for Program NFSIG : This program computes the natural frequencies of a grounded dielectric slab. c The slab has real, constant constitutive parameters eps, muO, sigma : This version uses values for lossless slab as guesses for lossy slab : version of 8 June 2001 c complex guess 1 , guess2,funperp,funpar,ans, g1 , g2 character*20 filnam(lO) external funperp,funpar c common /b1/ epsr,sigma,delta,epsO,clite,stheta,ctheta c pi = 4.*atan(l.) epsO = 8.854e-12 271 rmuO = 4.*pi*1.e-7 clite = l./sqrt(rmu0*epsO) etaO = sqrt(rmuO/epsO) filnam( 1 )='freq0.dat' filnam(2)='freq 1 .dat' f1 lnam(3)='freq2.dat' filnam(4)='freq3 .dat' filnam(5)='freq4.dat' write (*,*) 'Enter relative permittivity' read (*,*) epsr write (*,*) 'Enter conductivity (S/m)’ read (*,*) sig write (*,*) 'Enter layer thickness (m)' read (*,*) delta write (*,*) 'Enter incidence angle (deg)' read (*,*) thetai thetai = pi*thetai/ 180. stheta = sin(thetai) 5ch = stheta*stheta ctheta = cos(thetai) write (*,*) 'Enter: 1=perpendicu1ar polarization' write (*,*) ' 2=paralle1 polarization' read (*,*) ipol write (*,*) 'Enter number of frequencies to find' read (*,*) nf if (ipol .eq. 1) then 21 = etaO/sqrt(epsr—sth2) 20 = etaO/sqrt(1.—sth2) end if if (ipol .eq. 2) then 21 = etaO*sqrt(epsr-sth2)/epsr 20 = etaO*sqrt(l.-sth2) end if gamma = (zl-zO)/(zl+zO) tau = 2.*delta*sqrt(epsr-sth2)/clite sr = log(abs(gamma))ltau iters = 100 if (sig .gt. .5) then iters = int(sig/.005) end if dsi g = sig/iters 272 open(20,file='natural.in’,status='unknown') do n=0,nf-l open (lO,file=filnam(n+1),status='unknown') do i=l,iters sigma = i*dsi g if (i .eq. 1) then guessl = cmplx(sr,(2*n+1)*pi/tau) guessZ = 1.1*guessl gl = guessl g2 = guessZ else g2 = g1 g1 = ans guessl = g1 guess2 = g2 endif if (aimag(guessl) .lt. 0.d0) guessl=conjg(guessl) if (aimag(guessZ) .lt. 0.d0) guesschonjg(guessZ) if (ipol .eq. 1) then call zsecnt (funperp,guessl,guess2,6,100,ans,its) end if if (ipol .eq. 2) then call zsecnt (funpar,guessl,guessZ,6,100,ans,its) end if ar = rea1(ans)*1.e-9 ai = aimag(ans)*l.e-9/(2*n+1) write (10,*) sigma,ar,ai enddo close (10) write (*,*) 'n=',n write (*,*) 'sigma=',sigma write (*,*) 'freq=',ans are = rea1(ans)*1e-9 aim = imag(ans)*1e-9 write (20,*) are,',',aim end do close(20) end complex function funperp (8) complex s,f common /b1/ epsr,sigma,delta,epsO,clite,stheta,ctheta f = sqrt(epsr+sigma/(s*epsO)—stheta*stheta) 273 funperp = ctheta+f—(ctheta-f)*exp(-2.*delta*s*f/c1ite) return end complex function funpar (3) complex s,f,g common /b l/ epsr,sigma,delta,epsO,clite,stheta,ctheta g = epsr+sigma/(s*epsO) f = sqrt(g-stheta*stheta) funpar = f+g*ctheta-(f-g*ctheta)*exp(-2.*delta*s*f/clite) return end 0 SUBROUTINE ZSECNT (FUNC,Zl,22,NSIG,ITMAX,ANS,ITS) SOLVES FOR THE COMPLEX ZERO OF THE COMPLEX FUNCTION FUNC BY USING THE SECANT METHOD 21,22 = INITIAL GUESS NSIG = NO. SIGNIFICANT DIGITS DESIRED IN ROOT ITMAX = MAXIMUM ALLOWED NO. ITERATIONS ANS = ROOT ITS = NO. ITERATIONS REQUIRED COMPLEX FUN C,ANS,Zl ,22,F1 ,F2,ZNEW,DER ABSIG = 1./10.**NSIG 0 0 0000000000 ITS = 0 F2 = FUNC(ZZ) c WRITE (*,*) 'Z = ',Z2 C WRITE (*,*) 'F = ',F2 C 10 F1 = FUNC(Zl) DER = (F2-F1)/(Z2-Zl) ITS = ITS + 1 WRITE (*,*) '******************* ITS = ',ITS WRITE (*,*) 'Z = ',Zl WRITE (*,*) 'F = ',Fl WRITE (*,*) 'DERIVATIVE = ',DER WRITE (*,*) '****************************************' 000000 274 ZNEW = 21 - F1/DER TESTl = CABS(ZNEW-Zl) TEST2 = CABS(ZNEW-Z2) IF (TESTl .LT. TEST2) THEN TEST = TESTl/CABS(ZNEW) [F (TEST .LT. ABSIG) THEN ANS = ZNEW RETURN ENDIF F2 = F l 22 = Z] Z1 = ZNEW ELSE TEST = TEST2/CABS(ZNEW) IF (TEST .LT. ABSIG) THEN ANS = ZNEW RETURN ENDIF 21 = ZNEW ENDIF IF (ITS .GE. ITMAX) THEN WRITE (*,*) 'MAXIMUM ITERATIONS EXCEEDED' ANS = ZNEW RETURN ENDIF GO TO 10 END 275 BIBLIOGRAPHY [1] E]. 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Ilavarasan, “Automated Radar Target Discrimination Using E-Pulse and S-Pulse Techniques,” Ph.D. dissertation, Michigan State University, 1992. [8] A. Leon-Garcia, Probability and Random Processes for Electrical Engineering, 2nd Edition, Addison-Wesley, Reading, MA, 1994. [9] KM. Chen, D.P. Nyquist, E.J. Rothwell, L.L. Webb, and B. Drachman, “Radar Target Discrimination by Convolution of Radar Return with Extinction-Pulses and Single-Mode Extraction Signals,” IEEE Transactions on Antennas and Propagation, vol. 34, pp. 896-904, July 1986. [10] E]. Rothwell, K.M. Chen, D.P. Nyquist, “Extraction of the Natural Frequencies of a Radar Target from a Measured Response Using E-Pulse Techniques,” IEEE Transactions on Antennas and Propagation, vol. 35, pp. 715-720, June 1987. [11] 1E. Ross, “Application of Transient Electromagnetic Fields to Radar Target Discrimination,” Ph.D. dissertation, Michigan State University, 1992. [12] ER Knott, J .F. Schaeffer, M.T. 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