# MICROWAVE IMAGE UNDERSTANDING AND ITS APPLICATION TO RADAR CROSS-SECTION MANAGEMENT

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Dissertation contains pages with print at a slant, filmed a s received_________ 16. Other_______________________________________________________________________ . Text follows. University Microfilms International Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. MICROWAVE IMAGE UNDERSTANDING AND ITS APPLICATION TO RADAR CROSS SECTION MANAGEMENT Hsueh-Jyh Li A DISSERTATION in Electrical Engineering Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy. 1987 Supervisor of Dissertation Graduate Group Chairperson Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In remembrance o f My parents and my brother Hsueh-Jei Li - ii - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgements There are so many people to whom I would like to express my gratitude for making this dissertation possible. My deepest gratitude goes to my advisor, Professor Nabil Farhat, who initiated my study in the realm of microwave imaging and other related research areas and is the source for many of the ideas discussed in this disser tation. He has also provided the financial support in the course of the study. I thank .my fellow graduate students in the Electro-Optics and Microwave Optics Laboratory, in particular, Dr. Yuhsyen Shen, Dr. Charlie Werner, Zon-yin Shae, Baocheng Bai, Ken Schultz, and Kang-Suk Lee, who have provided stimulating discussions and incisive criticisms, and friendships as well, all of which made my study at Penn a pleasant experience. I also wish to thank Prof. Jaggard for stimulating discussions. Sitting in his class is really an enjoyment I especially thank to my beloved wife, Su-Jean, without her patience, under standing, and encouragement, this work would not have become a reality. Finally, I must express my apology for the many hours of neglect my son, Harry. He deserves definitely more from me. ui - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT MICROWAVE IMAGE UNDERSTANDING AND ITS APPLICATION TO RADAR CROSS SECTION MANAGEMENT Hsueh-Jyh Li Nabil H. Farhat In this dissertation the basic scattering properties of a perfectly conducting object are briefly reviewed. In the high frequency region, the scattered field of a complex shaped object can be attributed to a combination of several mechanisms. The advent of high resolution radar enables us to consider each scattering mechanism separately. The microwave image of a conducting object is interpreted from a new approach, based on the understanding of the scattering mechanism and the image reconstruction algorithm. The connection between various scattering mechanisms and their reconstructed images is then established. From this we can interpret what the image represents and predict what the image will look like for given spectral and angular windows. Several numerical and experimental examples have been included to support this new interpretation approach. A new algorithm to extrapolate the available data into the missing bands is dev ised. Both simulation and experimental results have shown the effectiveness of this method in microwave diversity radar imaging. A significant application of this new algorithm is in identifying the hot spots of a target when the bandwidth of the imag ing system is too narrow' to give acceptable resolution. The satisfactory interpretation of microwave images and the effectiveness of the extrapolation algorithm devised are fundamental to the study of RCS management and target recognition. A new term, "diaphanization", defined as the techniques of reducing RCS and the techniques of obscuring an image is introduced. RCS management is considered not only from a detection perspective, namely the reduction of the target’s RCS to elude radar detection, but also from the image point of view where disguising of a - iv - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. target’s appearance to impede recognition by an imaging radar is sought. The pro cedure which employs microwave diversity imaging to diaphanize a target over prescribed spectral and angular windows is given and the robustness of the diversity imaging system to the Gaussian noise is demonstrated. Some rules for distorting an image are proposed. These rules are: create artificial discontinuities, create multiple reflections and make the reflectivity at any given point a function of time. Traditional techniques for reducing the RCS are appliedJ o RCS management studies. These techniques are by covering absorber material, target shaping, and impedance loading. The theory pertaining to absorber-covered bodies is reviewed and the theoretical background for obscuring an image by absorber covering based on the physical optics approximation is established. Certain types of absorbers are used to cover metallic objects. Their effect on RCS reduction, range profile modification, and image distortion are experimentally studied and discussed. A concept concerning the absorber covering patterns is proposed, where the reflections from the boundaries between the covering absorbers and conductors are also considered. By using suit able covering patterns the RCS can be reduced more effectively. Some possible effects of loading an object with lumped impedances are exam ined. A thin straight wire is used as a test object. Plots of range profiles show that the surface traveling wave is an important scattering mechanism of a straight wire. Microwave images of a loaded wire are both numerically and experimentally obtained. The specific scattering mechanism of the thin wire makes its image unique in appearance. It is found that nonlinear loading and time varying loading can cause the spectra of the back-scattered field to spread as compared to the spectrum of the incident wave, which may produce several interesting phenomena, such as providing a false doppler frequency, making the receiver unable to phase lock to the frequency of the incident wave, generating unexpected peaks in the range profiles, and distort ing the reconstructed image. -v- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents Acknowledgements................................................................................................. iii A b stra c t................................................................................................................... iv Table of C ontents................................................................................................... vi List of T a b le s.......................................................................................................... ix Ltet of Figures ........................................................................................................ x Chapter 1 In troduction........................................................................................ 1 1.1 Overview of the Dissertation ....................................................................... 6 1.2 Contribution................................................................................................... 8 Chapter 2 Basic Scattering Theory of a Perfectly Conducting Object 10 2.1 Integral Equation Solutions........................................................................... 11 2.2 Approximation Solution ................................................................................ 15 2.2.1 Geometrical Optics .................................................................................. 16 2.2.2 Physical Optics ........................................................................................ 19 2.2.3 Geometrical Theory of Diffraction ......................................................... 21 2.3 Scattered Field of Some Simple Shaped Objects........................................ 24 2.3.1 Sphere....................................................................................................... 26 2.3.2 Cylinder.................................................................................................... 28 2.3.3 Wedge with Finite Length...................................................................... 30 2.3.4 Finite Plate ............................................................................................... 32 2.3.5 Comer Reflector...................................................................................... 37 2.4 Scattering Mechanisms of a Complex Shaped O bject................................ 38 2.4.1 Specular points........................................................................................ 39 2.4.2 Surface Discontinuities........................................................................... 41 2.4.3 Surface Derivative Discontinuites.......................................................... 41 2.4.4 Creeping Waves ...................................................................................... 42 - vi - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.4.5 Surface Travelling Waves ....................................................................... 43 2.4.6 Multiple Scattering ................................................................................. 44 2.5 Polarization Effect ........................................................................................ 44 2.6 RCS Measurement........................................................................................ 48 Chapter 3 Image Understanding in Microwave Diversity Imaging ............. 54 3.1 Image Reconstruction of Point Scatterers ................................................... 55 3.2 Formulation of Microwave Imaging for Conducting Objects Based on the Physical Optics Approximation ....................................................................... 61 3.3 Interpretation of the Reconstructed Images.................................................. 64 3.4 Examples....................................................................................................... 71 3.5 Discussions ................................................................................................... 81 Chapter 4 Extrapolation of Available Data into Missing Bands and Its Application to Radar Imaging ............................................................................ 83 4.1 The New Iterative algorithm for Extrapolation of Data Available in Multiple Restricted Regions ................................................................................... 86 4.2 Results........................................................................................................... 90 4.2.1 Range Profiles......................................................................................... 90 4.2.2 Reconstructed Im ages............................................................................. 99 4.3 Discussion..................................................................................................... 102 Chapter 5. Radar Cross Section Management Employing Microwave Diversity Im aging.................................................................................................. 105 5.1 Diaphanization .................. 106 5.2 Radar Cross Section Reduction ................................................................... 109 5.3 Effect of Gaussian Noise on Microwave Diversity Imaging...................... 114 5.4 Fundamental Concepts for Distorting an Image ......................................... 116 5.5 Discussions ........................................ 123 Chapter 6. Diaphanization by Absorber Covering and Target Shaping ................................................................................................................................... 124 6.1 Absorber-covered Bodies -- Theoretical Considerations ............................ 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.1.1 Semi-Infinite Plate ................................................................................... 127 6.1.2 Physical Optics Approach for an Absorber-Covered Metallic O bject....................................................................................................................... 129 6.2 Radar Absorbing Materials (RAM) ............................................................. 136 6.3 Experimental Results for Absorber-Covered Bodies .................................. 138 6.3.1 Broadband Absorber-Covered Bodies.................................................... 140 6.3.2 Surface Current Absorber-Covered Bodies ........................................... 146 6.4 Target Shaping............................................................................................... 156 6.5 Combination of Absorber Covering and Shaping....................................... 165 Chapter 7 Diaphanization by Impedance L o ad in g......................................... 173 7.1 Linear Impedance Loading .......................................................................... 175 7.1.1 Analysis of a Loaded N-port Scatterer.................................................. 176 7.1.2 Formulation of a Straight Wire with Impedance Loading.................... 181 ........ 187 7.2 Nonlinear Impedance Loading............................................................... 200 7.1.3 Scattered Fields and Images of a Thin Rod Scatterer 7.2.1 Analysis of an N-port Nonlinear Loaded Scatterer............................... 205 7.2.2 Scattering Properties of a Nonlinearly Loaded Thin Rod Scatterer 208 7.3 Time Varying Loading................................................................................. 214 7.4 Discussion...................................................................................................... 224 Chapter 8 Conclusion ......................................................................................... 228 8.1 Future Research ............................................................................................ 230 Appendix ................................................................................................................. 232 Bibliography ........................................................................................................... 235 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables 5.1 Effect of a Dominant Scatterer ............................................................... 113 5.2 Effect of Working Harder on the Dominant Scatterer ........................... 113 5.3 Effect Selective Elimination of Scatterer................................................ 113 5.4 Effect of Reduction When All Have the Same Amplitude ................... 113 6.1 Characteristics of the GDS absorbers...................................................... 149 6.2 Hierarchy of Scattering Shapes ............................................................... 158 -ix- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures 2.1 An astigmatic ray tube..................................................................................... 17 2.2 Reflection at a curved surface....................... 18 2.3 Geometry of a back-scattering problem.......................................................... 20 2.4 Geometry for a three-dimensional diffraction problem.................................. 23 2.5 Diffraction coefficient versus azimuth angle for various included angles......................................................................................................................... 25 2.6 Diffraction coefficient versus diffraction angles............................................. 26 2.7 Normalized RCS of a conducting sphere versus k a ...................................... 28 2.8 Geometry of a finite wedge diffraction problem............................................ 31 2.9 Plot of back diffraction fields versus elevation angle.................................... 33 2.10 Plot of back diffraction field versus azimuth angle....................................... 34 2.11 An arbitrary polygonal plate with N edges.................................................... 35 2.12 Procedures of calculating the diffraction field of an arbitrary edge 37 2.13 Geometry of a dihedral comer reflector......................................................... 38 2.14 Creeping wave concept of diffraction by a curved surface........................... 43 2.15 Diffraction coefficients of an infinite edge vesus azimuth angle.................. 46 2.16 Block diagram of the RCS measurement system........................................... 51 3.1. Simplified imaging geometry projected to the X - Y plane............................ 56 3.2 An example of target range reflectivity together with its successive differentials................................................................................................................ 68 3.3 Positions of the scattering centers and images of a cylinder......................... 73 3.4 Geometry and reconstructed images of a hexagonal plate........................... 75 3.5 Differential range of multiple reflection path versus rotation angle and simulated images ofa dual-cylinder object ............................................................. 3.6 76 Range profiles and reconstructed images of a conducting sphere obtained by various spectral windows................................................................... 78 3.7 80 Sketch of the B-52 airplane and the reconstructed images.......................... - x- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.1 Available data in multiple regions.................................................................. 4.2 Schematic diagram of the new iterative extrapolation method pro posed.......................................................................................................................... 87 89 4.3 Comparison of extrapolation errors and range profiles for different various extrapolation methods with spectral window f mjn = 6 GHz to /max = 16 GHz......................................................................................................... 4.4 92 Comparison of extrapolation errors and range profiles for different various extrapolation methods with spectral window / mm = 6 GHz to f max = ^0 GHz........................................................................................................ 94 4.5 Comparison of extrapolation errors and range profiles for different various extrapolation methods with spectral window f ^ = 6 GHz to f max - 12 GHz......................................................................................................... 95 4.6 Measured fields, extrapolation errors, and range profiles of a scale model B-52 at the broadside.................................................................................. 97 4.7 Measured fields, extrapolation errors, and range profiles of a scale model B-52 at the view line #64 ........................................................................... 98 4.8 Measured fields, extrapolation errors, and range profiles of a scale model B-52 at the view line #85 ........................................................................... 100 4.9 Images of the metallic scale model B-52 aircraft reconstructed from various spectral bands............................................................................................. 101 4.10 Reconstructed images of the metallic scale model of a B-52 aircraft using cross-polarized waves..................................................................................... 102 5.1 Image of the scale model B-52 before and after diaphanization................. Ill 5.2 Probability of detection for a sine wave in noise as a function of the S/N and the probability of false alarm.................................................................... 115 5.3 Effect of Gaussian noise on the signals, range profiles and images. 121 6.1 Geometry of an absorber-covered body illuminated by a plane wave. 126 6.2 Geometry of a semi-infinite conducting plate covered with a homo geneous plane layer.................................................................................................. - xi - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 128 6.3 Reflection coefficients Rp and Rv of a semi-infinite plate covered with an absorber layer versus frequency......................................................................... 130 6.4 Magnitude and phase of Rv and Rp versus incident angle............................ 131 6.5 The two equivalent structures of an absorber-covered body illuminated by a plane wave......................................................................................................... 133 6.6 Geometries of the measurement arrangement of a conducting plate. ................................................................................................................................... 139 6.7 Typical performance of the Emerson product AN absorbers........................ 140 6.8 Attenuation in dB down from metal plate when covered with absorber AN72 in normal way and inside out........................................................................ 141 6.9 Measured RCS pattern, sinograms and reconstructed images of a con ducting plate before and after diaphanization with geometry 1............................. 145 6.10 Mean RCS patterns, sinograms, and reconstructed images of the B-52 airplane before and after diaphanization.................................................................. 148 6.11 Covering configurations of a plate and their mean RCS patterns, range profiles and reconstructed images............................................................................ 152 6.12 Geometry, mean RCS, sinogram, and reconstructed image of a tube. ................................................................................................................................... 155 6.13 Geometry and mean RCS patterns of a serrated plate using geometry 1 162 6.14 Mean RCS and images of a flat plate and a serrated plate using geometry 2................................................................................................................. 164 6.15 Geometry, mean RCS patterns, and range profiles of triangular plates with sharp tips and round comers arranged in geometry 2.................................... 166 6.16 Picture, mean RCS patterns, and images of a scale model B-l air plane........................................................................................................................... 167 6.17 Covered patterns, mean RCS patterns and reconstructed image of a plate arranged in geometry 2 with tilted angle 0 = 90°.......................................... 169 6.18 Sketch of the absorber covered portions of the scale model B-l and its RCS and reconstructed images before and after diaphanization....................... 171 7.1 176 Equivalent network of an N-port loaded scatterer.......................................... - xii - Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 7.2 Geometry of astraight wire............................................................................. 7.3 N segment loaded wire and its equivalent circuit for the m th port. ................................................................................................................................... 7.4 185 Piecewise sinusoidal expansion function and set of overlapping piece- wise sinusoidal expansionfunctions.......................................................................... 7.5 182 188 Extreme values and no-load values of the back-scattered field of a straight wire scattered versus length in wavelength at various angles of incidence.................................................................................................................... 191 7.6 Impedance required to nullify the back-scattered fields for angle of incidence equal to 9=45°.................................................................. 192 7.7 Required passive impedance to maximize and to minimize the back- scattered fields for angle of incidence equal to 0=45°............................................ 192 7.8 Extreme values and no-load values of the broadside back-scattered fields of a straight wire versus length in terms of wavelength............................. 193 7.9 A straight wire illuminated by an impulsive plane wave............................ 195 7.10 The numerical range profiles of a straight wire at various angles of incidence .................................................................................................................. 197 7.11 The experimental range profiles of a straight wire at various angles of incidence .................................................................................................................. 198 7.12 Numerical and experimental fringe patterns and images of a thin rod. ................................................................................................................................... 199 7.13 Numerical range profiles of a straight wire with three loading points at various angles of incidence................................................................................. 201 7.14 Experimental range profiles of a straight wire with three loading points at various angles of incidence...................................................................... 202 7.15 Numerical and experimental fringe patterns and images of a thin rod with three loading points.......................................................................................... 203 7.16 Comparison of the measured mean RCS patterns of the thin rod with and without gaps........................................................................................................ 204 7.17 Equivalent network of an N-port nonlinear loaded scatterer......................... 207 7.18 Time domain fields and their spectra of a nonlinearly and linearly Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. loaded wire scatterer.................................................................................................. 212 7.19 Time domain scattered fields and spectra of a nonlinearly loaded wire scatterer with three loading points............................................................................ 213 7.20 Spectra of a time varying loaded scatterer..................................................... 217 7.21 The magnitude and phase of the back-scattered field and the range profile of the loaded straight wire............................................................................. 220 7.22 The magnitude and phase of the back-scattered field, and the range profile of the loaded straight wire, each impedance is randomly resistively varied between (0, 100)0.......................................................................................... 221 7.23 The magnitude and phase of the back-scattered field, and the range profile of the loaded straight wire, each impedance is reactively randomly varied between (-7 50, y'50)Q.................................................................................... 222 7.24 The range profiles of a randomly loaded straight wire................................. 223 7.25 Real part and imaginary part of the polar hologram of the randomly reactively loaded straight wire with nL = 7............................................................ 224 7.26 Optical image and microwave image of a wire model airplane ................. 226 7.27 A loaded conducting wedge............................................................................ 226 A l. 234 Multiple reflections of a two-cylinder object................................................. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1 INTRODUCTION T h e inverse or indirect scattering problem is to deduce features of source or scatter ing objects from the emitted or scattered radiation that has propagated to a detector [1]. The characteristic descriptors of a scattering object are estimated from experi mental data, utilizing the laws that relate these characteristic parameters to the experi mental data in a given situation. In the electromagnetic inverse scattering problem, the size, shape, and constitutive characteristics of an unknown scattering target are usually the parameters to be recovered based on the knowledge of the incident field and the resulting scattered field data [2]. Imaging is a technique to solve inverse scattering problem. The shapes, or the constitutive characteristics of objects are presented by pictorial representation or images. One of the main design objectives of an imaging system is to have good image resolution. The available recording aperture conventionally restricts the resolution obtainable. This constraint is known as the Rayleigh resolution criterion. The basic approach to improving resolution in microwave imaging system is to extend the effective area of the physical recording aperture through clever data acquisition schemes. These schemes are called synthetic aperture systems. The synthetic aper ture systems are usually categorized according to the relative motion between the recording systems and the object and can be grouped into synthetic aperture radar (SAR) [3,4], inverse synthetic aperture radar (inverse SAR), and spotlight-mode syn thetic aperture radar (spotlight-mode SAR) [5,6,7]. In SAR, the data are collected by means of an airborne or space-borne radar which illuminates the target area from different aspect angles and the position of the antenna remains fixed relative to the flying vehicle. In inverse SAR, the recording antenna is stationary while the target is moving to synthesize the effective aperture. The spotlight-mode SAR steers the phy sical antenna so that the same target area remains illuminated during the data collec tion interval. In the above synthetic aperture systems, the angular resolution or the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 2 - cross-range resolution of the image is determined by the effective aperture syn thesized in terms of the wavelength, while the range resolution is determined by the pulse width of the signal or the equivalent bandwidth of the signal. The concept of synthesizing an imaging aperture by frequency sweeping was first analyzed in [8]. It was found that a one-dimensional aperture can be synthesized with a single coherent receiver by sweeping the frequency of the incident illumina tion under certain circumstances. Additionally, a two-dimensional imaging aperture can be synthesized by a linear array of coherent receivers with one dimension of the aperture frequency synthesized. The use of frequency sweeping or the frequency diversity technique to achieve super-resolution in the imaging of three-dimensional perfectly conducting objects was studied and demonstrated by computer simulation [9]. By utilizing the intrinsic freedom of an imaging system, extremely highresolution projective and tomographic imaging of 3-D perfectly conducting objects has been achieved by combining angular, spectral, and polarization diversities [10,11]. In the frequency diversity imaging systems above, the object is seated on a rotating pedestal controlled by a computer. The resolution limit over a given specified angular aperture and spectral band has been analyzed in [9,11,12]. The advantage of broad-band illumination over monochromatic illumination was discussed in [8,13]. These advantages include the avoidance of object resonances [14], increased information about the geometrical and material properties of a scattering object, and the reduction of coherent image artifacts. Multi-frequency imaging tech niques have also been applied in many other areas [15,16,17,18]. The appearance of an image depends on the scattering properties of the object and the imaging technique utilized. The images obtained by visible light, infrared light, microwaves, and acoustical waves all look different [19]. Image interpretation is defined as the act of examining images for the purpose of identifying objects and judging their significance [20]. Interpretation is an essential process through which information can be retrieved from the images. However, interpretation of the images should be based on the understanding of the laws that govern the input (the incident field) and the output (the reconstructed image). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 3 - The targets of interest in this dissertation are metallic objects. The scattering properties of metallic objects at all frequency regimes have been extensively analyzed [21]. The frequencies are usually divided into three regions according to the charac teristic dimensions of the object. These are: a low frequency region, a resonant fre quency region, and a high frequency region. In microwave imaging applications, the high frequency region is most interesting. At high frequencies, the scattering prob lems are usually analyzed by approximation approaches, which include geometrical optics (GO) physical optics (PO), and the geometrical theory of diffraction (GTD) [21]. To relate the geometrical shape of a conducting body to its scattered field, the PO approximation is usually adopted. It was shown by Bojarski [22] and Lewis [23 ], that a three-dimensional Fourier transform relationship exist.: between the shape of a perfectly conducting object and its back-scattered far field under the PO approxima tion. The shape of a convex body can be retrieved if the back-scattered field can be measured for all frequencies and all directions. However, the spectral and angular windows for the data are usually restricted by practical constraints. Radar people model the object with a target reflectivity function [24], which is defined in terms of a range coordinate and a cross-range coordinate measured from the center of the syn thetic aperture. Basically, the target reflectivity function is also derived from the PO approximation. The principle of frequency diversity imaging systems [9] is also analyzed based on the PO approximation. An object scattering function was first defined and then it was shown that the object scattering function and the range corrected field are a 3-D Fourier transform pair [10,11]. However, the PO approximation is inadequate for scattering problems where edge diffractions, multiple reflections, creeping waves, and traveling waves are the dominant scattering mechanisms. The above scattering mechanisms are important contributors to the scattered field of a complex shaped conducting object [21]. The microwave image of an object also depends on the imaging technique util ized. Different data acquisition schemes may produce different images. The imaging technique used in this dissertation is the microwave diversity imaging scheme achieved by combining angular, spectral and polarization diversities, developed at the Electro-Optics and Microwave-Optics Laboratory of the University of Pennsylvania. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 4 - There is a common property between the data acquisition arrangements of microwave diversity imaging and the well-known X ray imaging technique, computer-aided tomography (CAT), which has been used to obtain the image of a slice of the object and has wide applications in medical imaging [25,26]. Both systems involve rotation of objects. This similarity suggests that the reconstruction algorithms used in CAT may be applied to microwave diversity imaging. The two reconstruction methods used in CAT are the Fourier inversion method [27,28] and filtered back-projection method [26]. These two methods can yield equivalent results [28]. Microwave tomographic and projective images reconstructed by these two methods have been obtained [10,11,29]. The imaging system involving rotation of the object has been analyzed independently using the range-Doppler principle by radar people [6,30,31,32]. But the same scheme can be interpreted as a tomographic reconstruction problem and can be analyzed using the projection-slice theorem from CAT [7]. Imaging workers usually model the object consisting of discrete points and assume that each point radiates isotropically regardless of the aspect angle. Accord ingly, they analyze the imaging system by the point-spread function [31]. However, a conducting object cannot be modeled in this manner. Even though the conducting object can be viewed as a combination of discrete scattering centers at a specific aspect angle, the locations of the scattering centers may migrate and their scattering strength can change as the aspect angle is changed. Therefore, no fixed point-spread function can be precisely defined in the conducting object imaging systems because the point-spread function is space-variant and target dependent. In this dissertation, we will investigate microwave imaging of metallic objects employing frequency, angular, and polarization diversities, with particular emphasis on image interpretation and prediction based on analysis of the scattering mechanisms and the filtered back-projection algorithm utilized in image retrieval. We will ela borate on the connection between the various scattering mechanisms and the recon structed images; interpret what the images represent; and then predict what the image will look like over given spectral and angular windows. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 5 - After the microwave images are satisfactorily interpreted, one may retrieve valu able information from the images. It will be shown later that the brightness of each image cell reflects the average scattering strength of the object over the specified spectral and angular windows. This suggests that high resolution microwave diver sity imaging can be applied to the study of radar cross section (RCS) reduction. RCS is a measure of the equivalent size of a target as seen by the radar. RCS studies create interest because of the unique applications of RCS. A special issue on RCS was published in the Proceedings of IEEE in 1965. Since then, the technology has steadily advanced. However, due to the interests of national security, some aspects of radar cross section measurement, prediction and reduction are not pub lished [33]. RCS is a function of aspect angle, frequency, and polarization. In some applica tions, it is desirable to enhance the RCS over some specified angles. On the other hand, in other applications it is often desirable to reduce or minimize the RCS over some spectral and angular windows to avoid target detection [21], RCS management is a general term for obtaining the RCS of a scattering object by manipulating the distribution and strengths of hot spots or flare spots of a target over prescribed spectral and angular windows and states of polarization. Hot spots represent all those portions that have major contribution to the received scattered fields. To reduce the RCS, the hot spots must be located and identified. Several approaches can be used to achieve this purpose. Examples include simply engineering guesses, numerical calculations, and experimental measurement [33]. The advent of high resolution imaging radar can aid in the identification of scattering centers. In this dissertation we will employ the high resolution microwave diversity imaging sys tems to locate and identify the hot spots of a complex object. After the hot spots have been located, the next problem is to remove or suppress them. Techniques of reducing RCS have been extensively analyzed in [21] and are grouped into three classes: covering the object with absorbing material or substituting the metallic material with resistive or composite material, shaping the target, and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 6 - impedance loading. These techniques will be applied in this dissertation to reduce the RCS while microwave diversity imaging will be employed as a new tool to examine the effectiveness of the RCS reduction technique. Radar designers try to make radars capable of detecting very weak signals and recognizing or identifying the target. In contrast, target designers try to reduce the target’s RCS to elude radar detection, and to disguise the target’s appearance to impede radar recognition. From the electronic counter measures point of view, the optimistic techniques should possess the capability of reducing a target’s RCS as well as obscuring the target’s images. In this dissertation we will consider RCS manage ment from both detection and imaging view points. We will examine how the images are changed after the traditional RCS reduction techniques are applied. We will also propose some fundamental rules for obscuring an image based on the under standing of the scattering mechanisms and image formation. Microwave diversity imaging has been extensively studied in the Electro-Optics and Microwave-Optics Laboratory of the University of Pennsylvania employing a unique experimental microwave imaging facility. This facility has proved to be a very valuable tool in the understanding of electromagnetic (EM) scattering and diffraction and the understanding of pertinent mechanisms in the formation of a microwave image. 1.1 Overview of the Dissertation The whole dissertation is structured into seven chapters. Each chapter is sum marized as follows: The complex field of a coherently illuminated object is the basic measured quantity for all scattering problems, because all the parameters of interest can be derived from the scattered field. In chapter 2 the basic scattering theory of a perfect conducting object will be reviewed, starting from the exact integral equation solution of Maxwell’s equation followed by its approximate solution. The scattered field of some simple objects and the scattering mechanism of a complex object will also be summarized. The polarization properties of the scattered field contain the feature Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 7 - information of the object. At high frequencies the scattered field can be expressed as from a combination of some discrete components and the polarization status of each component is independent or slowly varies with frequency. With a high resolution radar, these discrete components can be resolved and can be considered separately. The technique of RCS measurement and the calibration procedures will be discussed at the end of this chapter. In chapter 3 the image formation for microwave diversity imaging will be explained by investigating the scattering mechanism of the object and the procedures of the reconstruction algorithm. The images reconstructed from a specified angular window and spectral window will be predicted and interpreted. Several typical numerical and experimental examples will be demonstrated to show the validity of microwave images that can be generated under various conditions. These examples serve to illustrate the image’s dependence on data acquisition parameters and the scattering mechanism and prove that care must be taken to achieve a proper interpre tation of the image. To locate the hot spots of an object, fine resolution of the object detail in the reconstructed image is essential. The range resolution of a frequency diversity imag ing system is inversely proportional to the bandwidth coverage of the measurement system. In practical situations, due to the limitation of the measurement system or restriction of bandwidth allocation, the observed data can lie in a narrow band or in multiple restricted spectral regions and the bandwidth of each region can be too nar row to give acceptable resolution. In chapter 4 an algorithm will be used to extrapo late data in the missing bands from the data available and apply it to the radar imag ing. In chapter 5 a new term "diaphanization" will be introduced, which is defined as the techniques of reducing RCS and techniques of obscuring the image. The pro cedure of employing the microwave diversity imaging system to diaphanize a target will be described. RCS management is treated not only from the detection point of view but also from the image perspective. Some fundamental rules of obscuring an image will be proposed. The effect of Gaussian noise on the reconstructed image will also be examined. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 8 - Diaphanization by an absorbing material covering and target shaping will be stu died in chapter 6. Theoretical analysis of absorber-covered bodies will be reviewed and formulated under the physical optics approximation. Properties of radar absorb ing material (RAM) will be briefly described. Certain types of RAM will be used to cover the object. Their effects on the RCS and images will be examined. A hierar chy of scattering shapes provides a fundamental rule for shaping a target. The RCS of some object shapes and their frequency dependence will be experimentally stu died and discussed. In chapter 7 diaphanization by impedance loading will be numerically examined and studied. Analysis of a linearly and nonlinearly loaded N-port scatterer will be reviewed. The moment method will be used to numerically calculate the scattered field of a loaded straight wire scatterer and some interesting phenomena of a loaded wire will be discussed. These phenomena include the scattering properties, the spread spectrum of a nonlinear loading and a time varying loading, the range profiles and the reconstructed images. 1.2 Contribution The significant contributions of this dissertation are summarized as follows: 1. The image from a microwave diversity imaging system is understood by investi gating the scattering mechanisms of the object and the image reconstruction algorithm. The effect of the different scattering mechanism on the reconstructed image has been established. To our knowledge such an approach has not been reported before. Several numerical and experimental results can be satisfactorily explained. The image of a metallic object obtained by prescribed spectral and angular windows can be predicted accordingly. This successful interpretation is fundamental to research in RCS management studies and target recognition. 2. A new iterative algorithm has been devised to extrapolate the data available into missing bands. As a consequence of extrapolation, resolution of the microwave diversity imaging for a given bandwidth can be improved, and the system’s abil ity to pin-point the distribution of hot spots is increased. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 3. 9 - A new term "diaphanization", which is defined as the techniques of reducing RCS and techniques of obscuring the image, has been introduced. RCS manage ment is treated not only from the detection point of view but also from the imaging perspective. This opens a new direction for future research in RCS management studies. 4. The robustness of the microwave diversity imaging to the Gaussian noises has been demonstrated. Some fundamental rules of obscuring an image have been proposed and experimentally studied. 5. A way of covering metallic objects with absorbing materials, which is also con sidered from the absorber shaping perspective, is proposed and experimentally studied. This technique is a combination of absorber covering and shaping. This technique creates artificial discontinuities in surface impedances and causes appearance of unexpected detail in the image, which might deceive the radar’s "eye" and make it unable to recognize the object. 6. Some phenomena of a lumped linearly and nonlinearly loaded straight wire have been numerically and experimentally examined. These phenomena include the scattering properties, the effectiveness of providing false Doppler frequency by time varying loading, and the effect of fixed loading and random loading on the range profiles and the reconstructed images. Plots of range profiles show that the surface traveling wave is also an important scattering mechanism of a straight wire even though the incident angle is not close to the end-on direction. The effect of impedance loading on the reconstructed image has not been reported before. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 10- CHAPTER2 BASIC SCATTERING THEORY OF A PERFECTLY CONDUCTING OBJECT W h en an electromagnetic (EM) wave impinges on a conducting body, it induces oscillating charges and currents on its surface. These induced charges and currents are the sources of the scattered field. The total field at an observation point in the presence of a conducting body is comprised of the scattered field and the incident field, which is the field without the object and is assumed known. Therefore, the scat tered field is defined as the difference between the total field and the incident field. The theoretical formulations for scattering problems rest on Maxwell’s equa tions. Wave equations can be derived from Maxwell’s equations. Closed form solu tions for wave equations are available only when the scattering geometry coincides with one of the few separable coordinate systems. For any shape of scattering geometries, exact integral equations can be formulated. However, these formulations were not considered useful until the advent of computer age. Numerical techniques can be applied to a body of arbitrary shape and generally are only limited by the object to wavelength ratio. When the object size is large compared to a wavelength, approximation solutions are usually used to overcome the computing difficulty. In this chapter, we will briefly review the basic scattering theory of a perfectly conducting object starting with Maxwell’s equations. The material to be described will be confined to those topics which will be used later in this dissertation. In Sec. 2.1, the integral equation solution will be formulated assuming that the time dependence of the field is in sinusoidal harmonics. In practical situations, the high frequency region is of great importance. Asymptotic approximation solutions including those for geometrical optics, physical optics, and the geometrical theory of diffraction will be summarized in Sec. 2.2. A complex shaped object can usually be divided into several objects of simple shapes. The scattered field of some simple Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -11 - objects will therefore be discussed in Sec. 2.3, where these simple shapes will be used as examples in later chapters. In the high frequency region, the scattered field of a complex shaped object can be attributed to a combination of several scattering mechanisms. The scattering properties of each mechanism is fundamental to the understanding of the scattering of a complex shaped body and will be discussed in Sec. 2.4. Polarization of the scattered field carries important information about the target. In Sec. 2.5 we will express the scattered field as a combination of some discrete components and show that the polarization status of each component is independent or slowly varying with frequency. Precise measurement of the scattered field is very important in obtaining a high quality microwave image. Magnitude and phase are both required in the measurement. In Sec. 2.6, techniques of RCS measure ment and the calibration procedures needed to achieve high accuracy will be dis cussed. 2.1 Integral Equation Solutions Assume that all of the sources are in what is otherwise free space, Maxwell’s equations for harmonic time dependence e can then be expressed as, V xfiV ) = - j ^ o f t F ) = ycoeofV) + TV) (2.1) (2.2) V-£V) = p(7rf)/e0 (2.3) V F tff) = 0 (2.4) and the continuity equation is ?Q*) = -j(op<f) (2.5) where |i.0 and £q are the permeability and permittivity of free space respectively, r* is theposition vector, T is the current density and p is thechargedensity. It is con- venientto introduce the scalar potential <(>and vector potential A as Ft(r) = — VxAV) M-o Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.6) - 12and £(r>) = -V<$><r)-j (2.7) If the two potentials are related by the Lorentz gauge condition V -^ V ) + j coHoEo^C^ = 0. (2.8) the wave equations can be written as V t f + k 2t = - p / (2.9) V2<|>+ k 2((>= -p/eo (2.10) and where k 2 = co2|ioeo. and k is called the wave number or propagation constant in free space. The solution of X can be expressed as X = \io jf(f)G (?,?*) dv' (2.11) where dv' is the volume element in the source region, G is called the Green’s function and is the solution of the following function V 2G + k 2G = -5(r» - r ) (2.12) where 8 is the Dirac delta function. G is given by G ( r r ') = T4nR T (2<13) with r = i/?i = \ r - r \ (2.i4) By substituting Eq.(2.13) and Eq.(2.11) into Eq.(2.7), and after laborious vector operation, one can show that the field scattered by a conductor with surface current density Ts (/*) is [34] { - y c o p /,( n + — S’ [Ts ( r y i R]tR [ - k 2 + ^ U k + - h ] J tutQ j(jk + j)? sr))G (? , K K r) ds' Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.15) - 13 and /fa ) = HoJ (ik + \) G S’ K (?, r ) f ( f ) x lR ds' (2.16) where, I _ R \t-r\ =£ n 17s ( r } ds' is the area element, and S' is the conductor surface. From the above equations, it is apparent that the scattered fields at any position can be determined if the current source distributions are exactly known. When a metallic object is illuminated by a plane wave, current will be induced on the surface and the induced current will become the source of the reradiated or scattered field. However, the induced current distribution usually cannot be expressed in closed form except for those objects whose surfaces coincide with the surfaces of orthogonal cur vilinear coordinates. To find the source distribution, boundary conditions have to be applied. For the scattering problem involving a perfect conductor, the boundary con ditions on the conducting surface requires that hxM101 = f i x ^ + i ^ ) = 0 (2.18) and n x ft101 = nx(Fts + tf inc) - Ts (2.19) where l? ot and Fttot are the total electric field and the total magnetic field in the presence of the conductor; j?s and Ffs are the scattered electric field and scattered magnetic field due to the induced surface current, ^ inc and Jrtinc are the incident electric field and incident magnetic field, and h is the unit outward normal to the sur face. As a result, we have the following integral equations n x ^ C r ) = -2n x j { -y c o p ^ r) + v <7ytR]lR [-k 1 + \ ( j k + -U ] I u)£n K -T-C i k + ± ) J s r ) } G ( ? , n ds' R ''" R and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. K (2.20) - 14Ts (f) = 2n x f tinc + 2 n x | ( j k + j ) G ( f , r ) Ts ( f ) x lr ds' (2.21) Eq.(2.20) is called the Electric Field Integral Equation (EFEE), and Eq.(2.21) is called the Magnetic Field Integral Equation (MFIE). Note that in Eq. (2.20) and Eq.(2.21), a factor 2 has been multiplied to account for the presence of the perfectly conduct ing surface. For an object of general shape, the solution must be based on numerical tech niques. Numerical techniques can be applied to a body of arbitrary shape and gen erally are only limited by the wavelength size of the body. The moment method is the most popular numerical method for treating the scattering problem of a perfectly conducting object. This method has been extensively studied [35,36,37] and will be used to solve the scattering property of a straight wire loaded with impedance in chap.7. Once the surface current is determined, the scattered fields can be determined by Eq.(2.20) and Eq.(2.2l). If only the far fields are of interest, these equations can further be simplified. Under the far field condition, the distance R in the phase term and denominator term can be approximated by R ~ r - lr r* J_ _ J_ R ~ r where r is the distance between the observation point and the originorthereference point, lr is the unit vector of the line connecting the origin and theobservation point and is parallel to lR. The far zone vector potential X can be expressed as A V ) = ~ r ~— f £ ( r * V Wr‘^ ds' 4nr Js' (2.22) Furthermore, all the terms in the integrand of Eq.(2.15) with order higher than or equal to MR can be neglected. The far fields are then reduced to g(i*)= ri y * 47tr .j ( T .c n - f i o n - h t , ] eill' r Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. <2.23) - 15 (2.24) It can be shown that X is related to X by X = - ja X j where XT is the transverse component of (2.25) X along the direction lr . Eq.(2.23) and Eq.(2.24) will be the fundamental equations used to calculate the far field of a con ducting scatterer. The previous analyses are based on the time harmonics assumption. Current developments in high resolution radar and EMP technology have created interest in the radiation and scattering of transient waveforms from conducting bodies of various types. Transient scattering gives more insight into the scattering mechanism of a scatterer [38]. There are two independent techniques for solving transient scattering problems. The first involves the computation of the frequency domain response of the structure, which is then Fourier transformed to yield the time-domain response. The second is based on a direct formulation of the integral equation in the time domain and its subsequent solution using a time-stepping rather than matrix inversion algo rithm [39]. In chap.7 the transient behavior and the spread spectrum behavior of a straight wire scatterer with nonlinear loading will be studied. The first of the two techniques mentioned above will be used. Computational aspects of these two methods have been summarized in [37,40]. 2.2 Approximation Solutions As the dimension of an object gets larger, the solution obtained by a series expansion becomes poorly convergent. Additionally, the moment method is restricted to objects with dimensions not large in terms of a wavelength. The high frequency approach or the asymptotic approximation method can then be used to overcome computer limitation. When an object is large compared to a wavelength, the scattering and diffraction are found to be essentially a local property of the object [41]. Examples of local scattering regions include points of specular reflection, shadow boundaries, tips, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 16 - edges, etc. The high frequency approximations to be discussed are geometrical optics, physical optics, and the geometrical theory of diffraction. 2.2.1 Geometrical Optics Geometrical optics was originally developed to analyze the propagation of light where the frequency is so high that the wave nature of light need not be considered. In this approximation, the optical laws may be formulated in the language of geometry'. The energy may then be regarded as being transported along certain curves or light rays [42]. Some of the principal properties of a geometrical field are [43]: 1. The geometrical rays may be defined as a family of curves which are every where normal to the constant geometrical wavefronts. In a homogeneous medium, all rays are straight lines. If the refractive index is not a constant, the rays are curved, and the rays always bend toward the regions of higher refrac tive index. 2. The variation of the amplitude of the geometrical optics field within a ray tube is determined by the law of energy conservation. For example, an astigmatic ray tube is shown in Fig. 2.1. The principle radii of curvature of d a 0 are p! and p2, while the principle radii of curvature of d a are (pj+S) and (p2+S). The law of energy conservation requires that da0 da P1P2 (Pi+S)(p 2+S) (2.26) The magnitude of the electric field £ at d a can then be expressed in terms of the field £ 0 at d a 0 by the relation (2.27) (Pi+S)(P 2+S) 3. The phase delay along a ray is given by the optical length of the ray. The phase delay along all rays connecting any two equi-phase surfaces is the same. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 17 - line caustic 2 Fig. 2.1 An astigmatic ray tube. [44] Combining 2 and 3, the geometrical optics field Z? at S can be written as *<S) = m e - ^ y j (pij ; ; ^ 5) . * • (2.28) where \j/(0) is the phase at d Cq. It is important to find an expression for the radii of curvature of the wavefront of the reflected wave in terms of the geometrical radii of curvature of the object sur face and the parameters of the incident waves. Reflection at a curved surface is shown in Fig. 2.2. The geometrical optics reflected field at a distance 5 away from the reflection point can be expressed as [41] t ( S ) = El 1(Qr )R A / - ■- Pl' P2 — V (pf+S)(p 2'+S) (2. 29) in which p f and p£ are the principal radii of curvature of the reflected wavefront at the reflection point QR, £f* (QR) is the incident field at QR, and R is a dyadic with R =e'p e rp - el e[ (2.30) where e rv , el are unit vectors perpendicular to the plane of incidence, and e lp , e rp are unit vectors parallel to the plane of incidence so that Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 18 - ep = ev xs (2.31) where £ is the unit vector in the direction of propagation. *k Fig. 2.2 Reflection at a curved surface[41]. The general relationship between pf, p£ and the parameters of the surface and the incident wave can be found in [41]. For the two dimensional case, i.e., an infinite line source parallel to the axis of a convex cylinder of arbitrary cross-section, pr can be expressed as [44], — = — + ----pr p‘ r CCOS0; (2.32) where p' is the principal radius of curvature of the incident wavefront at QR, and r c is the radius of curvature of the surface at QR. The geometrical optics field may require no correction when to is sufficiently large. This is the case for back-scattering from smooth curved surfaces with radii of curvature very large in terms of a wavelength. However, geometrical optics fails in the case of a cylinder and flat plate, where one of or both of the principal radii of the surface approach infinity. It also fails when the specular point approaches a shadow boundary on the surface. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 19- 2.2.2 Physical Optics Physical optics is an approximation that expresses the induced surface current density in terms of the incident field. When the object dimensions are much larger than a wavelength, physical optics assumes that the field at the surface of the scatter ing body is the geometrical surface field. This implies that, at each point on the illuminated side of the scatterer, the scattering takes place as if there were a fictitious infinite tangent plane at that point. While over the shadow region, the field at the sur face is zero. For a perfect conductor, the surface current is assumed to be * n x f t tot (ry) = 2/1 ( f') x ff1(7*) 7*on the illuminated surface Z<r> = ' 0 elsewhere (2.33) Let the incident plane wave at the differential surface ds' be & < ?) = it° e jldi r where it0 is (2.33A) the incident field at the origin and is a constant vectorperpendicular to thedirection of incidence, and lt is the unit vector in the direction of incidence. Substituting the above expressions into Eq.(2.23) and (2.24), one has A*(r>) = — \fi ( O x t f V j e jk ^ 2 n rJ & (? > = 2nr - yr ds J { /iC ^ x tfV ) ~ J Jill (2.34A) [ n ^ x F P r y i r V , ) ■eik{f '~ ^ r ds' ff* (j*) - J h L ^ L f ^ (j*) x 2 nr D sai (2.34) x lrejk{lr~ii)r ds' (2.35) where s m refers to the illuminated surface. If the surface can be expressed as z' = / {x ,y'), the normal vector to the surface, n , and the element area, d s', on the surface become a - - 0 / f ix ' )* - O f f iy ' )y+ ? V i + o / / a / ) 2+ o / / a / ) 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.36) -20and ds' = dx'dy' V1 + 0 / fix' )2+(df Idy' )2 (2.37) The general expressions of the far electric and magnetic fields can be obtained by substituting Eqs. (2.36) and (2.37) into Eqs. (2.34) and (2.35). Consider the special case of back-scattering, where /) = - l r . Without loss in generality, assume that the z axis is along the lr direction, and the geometry is as shown in Fig. 2.3. It can be proved then that Eq. (2.34) and Eq.(2.35) become Eis( f) = Jkf jkr ^ f (f -a (r^)exp( - j 2kz' )ds' 2tcr aJm = ~1k2^ (z n OOJexp( - j 2kz' )ds' antenna T z= L \z -z - shadow boundary shadowed side Fig. 2.3 Geometry of a back-scattering problem. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.38) (2.39) -21 - It is noted that £* and Z?°, Fts and Ft0 are in the same directions. Furthermore, Cn-z)ds'=nzds' is the projection of the element of the surface area ds' to the x '- y ' plane; hence, nz ds' =ds'z =(ds'z Idz' )dz' where s'z is defined as the projection of the surface area for z>z' onto the x' —y' plane. Therefore, Eq.(2.38) and Eq.(2.39) become & <T) = | c * v H 2 k z ') ^ d z ' (2.40) and Ft*r)= J exp(-y 2kz' )~ ^~ d z' (2.41) where L is the maximum length of the object in the z direction, and the xf - y' plane is taken to be behind the shadow boundary (we have defined s'z=0 for z' > L and s'z = s 'max for z' < 0). Note that the scattered field is proportional in magnitude to the Fourier transform of ds'z Idz' . It is worth noting also that the equations obtained from physical optics for the scattered field from a conducting body often reduce to the equation of geometrical optics in the high frequency limit. 2.2.3 Geometrical Theory of Diffraction Neither Physical optics nor geometrical optics can predict a nonzero field in the shadow region. However, geometrical optics may be extended to include a class of rays, called diffracted rays, which permits the calculation of fields in the shadow region of a scatterer. This theory is called geometrical theory of diffraction (GTD) and was developed by Keller [45]. In 1953, Keller introduced diffraction coefficients and related them to the asymptotic expression developed by Sommerfeld. Keller made the following postula tions [45]: 1. The diffracted field propagates along ray paths that include points on the boun dary surfaces. These ray paths obey the principle of Fermat, known also as the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 22 - principle of shortest optical path. 2. Diffraction, like reflection and transmission, is a local phenomenon at high fre quency. That is, it depends only on the nature of the boundary surface and the incident field in the immediate neighborhood of the point of diffraction. 3. A diffracted wave propagates along its ray path so that (a). Power is conserved in a tube of rays, and (b). Phase delay equals the wave number times the distance along the ray path. The diffracted field can be written by a symbolic expression [ E d ] = [ D 1 [ E l ] A Cs)e~Jks (2.42) where [ E d ] and [ E l ] are column matrices consisting of the scalar components of the diffracted and incident field respectively, [ D ] is a square matrix of the scalar diffraction coefficients, s is the distance from the observation point to the diffraction point, and A (s) is the spreading factor. Shown in Fig. 2.4 is the geometry for a three-dimensional diffraction problem, where s' is the source point, s is the observation point, (2-n )k ( 0 <, n <2 ) is the included angle of the wedge, and Q is the diffraction point which is determined by the Fermat principle. It is more convenient to describe the diffraction system using ray-fixed spherical coordinates with Q as the center. The plane containing the incident ray and the edge of the wedge will be referred to as the plane of incidence; while the plane containing the diffracted ray and the edge will be referred to as the plane of diffraction. The unit vector s ' is in the direction of incidence and the unit vector s is in the direction of diffraction. Using this ray-fixed coordinate system, E}{s) /•— s1 * . 0 o i 1 Eq.(2.42) can be expressed as E ‘'(Q) -D v EH Q ) A (s)e ^ (2.43) where p ' and v' refer to directions parallel and perpendicular to the plane of incidence respectively, and p and v refer to directions parallel and perpendicular to the plane of diffraction respectively. Keller’s diffraction coefficient may then be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -23- written as plane l o f diffra< t i o n plane o f in c id e n c e Fig. 2.4 Geometry for a three-dimensional diffraction problem. ^ P • e jW V sin(7t/rt) B)2 rts i„ 9 ' 1 cos 7t/n - cos (<j>-<t)')/n 1 cosrc/n - cos(<jH-<J>,)/rt (2.44) The above expression is valid when the observation point is apart from the shadow or reflection boundaries, and will become singular as a shadow or reflection boundary is approached. To overcome the singular property of the Keller’s diffraction coefficient, Kouyoumjian and Pathak [41] proposed a uniform theory of wedge diffraction, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -24- known as UTD. The UTD can be used to treat different types of waves, including plane waves, cylindrical waves, conical waves and spherical waves, while avoiding the singularity problem near the boundary regions. However, the UTD formulas are much more complicated. The formula in the UTD requires calculating the diffraction coefficient and the spreading factor can be found in [41]. The computer program used for calculating the diffracted field using the UTD has been given in the appendix of Ref [44]. Edge diffractions will become the major contributor to the scattered field of a complex shaped object in the absence of specular points. It will be very helpful if the diffraction coefficients can be numerically plotted. In the following, all the diffraction coefficients were calculated using the UTD. For an infinite wedge, back-scattered fields exist only when 6 = 90°. The plots of Dp and Dv versus the azimuth angle <|> (<M>0 for various exterior angles n are shown in Fig. 2.5. From these figures, one can observe the strong dependence of the diffraction coefficient on the azimuth angle. This is the main difference between the diffracted fields due to an edge and the scat tered fields of a thin wire. The back-scattered field of a thin wire is independent of the azimuth angle. It is also noted that the dependence of the diffraction coefficients on the exterior angle is very small except at those angles approach to <|>= n n. If the incident angle is fixed and the diffraction angle is varied, i.e., the bistatic angle is changed, the diffraction coefficients Dp , Dv versus the diffraction angle for a given set of parameter (n=2, 0, =0d=9O°) are shown in Fig. 2.6. 2.3 Scattered Field of Some Simple Shaped Objects A complex shaped object can usually be divided into several parts of simple shapes. It is important to know the RCS properties of some commonly encountered simple objects. In the following, the scattered field of some simple objects will be summarized. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -25- diffrac. coef. in dB 20 10 • -10 ■ -20 45 90 angle in degree (a) 45 90 135 180 135 180 135 180 diffrac. coef. in dB 0 -20 0 diffrac. coef. in dB (b) -10 -20 45 90 angle in degree (c) 0 Fig. 2.5 Diffraction coefficient versus azimuth angle, (a), n =2.0, (b). n=1.6, (c). n=1.4. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 45 90 135 180 angle in degree Fig. 2.6 Diffraction coefficient versus diffraction angles, where 0i = Qd = 90°, n=2. 2.3.1 Sphere Consider a conducting sphere with radius a being illuminated by a plane wave. The wave is incident in the - z direction and the incident field If* is polarized in the x direction. In this geometry, the homogeneous Helmholtz equation of Eq. (2.9) can be written in spherical coordinate form, and its eigenfunctions are called the spherical wave functions. A plane wave can be expressed in terms of the spherical wave func tions [46]. The scattered field can then be expressed as a superposition of the infinite spherical wave functions. Coefficients of the infinite series expressions are deter mined by the boundary conditions which are E Q(a ,9,<j)) = £^(0 ,0,0) = 0. The scattered field can be expressed in terms of 0 and <]> components. For the special case of back-scattering, (i.e., 0=0°,0=0°), the far back-scattered field can be expressed as £* = E q s 0 = Exx (2.45) 2n+ l with Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -27a n B n where = f / y» 2n+l j *(ka) K J) » 0| + 1) h ? \k a ) (2.46) ( / r *1 2» +1 W n W Y n (n + l) [lcah£l\ka )]' differentiation, denoted with a (2.47) prime, is with respect ka , to h^l\ x ) = j n(x) + jyn(x), where j n(x) and yn(x) are the spherical Bessel functions of the first and second kinds with order n, respectively. The radar cross section is, (2.48) To numerically calculate the RCS of a sphere, it is not possible and not practical to calculate the infinite series. The required number of terms to calculate the RCS of a sphere has been discussed in [21]. However, if is very large, the convergence rate will be very slow, and the numerical calculation of the Hankel function and its derivative becomes difficult because more significant digits are required. In that case, an asymptotic expression will be more convenient [21]. The RCS of a sphere normalized with respect to its geometrical cross section versus ka is shown in Fig. 2.7. The frequency regions are usually divided into three areas according to the value of ka. According to [21], the low frequency region (or Rayleigh region) involves ka < 0.4; the high frequency region (or geometric optics region ) involves ka > 20; and the resonant region involves 0.4 < ka < 20. In low and high frequency regions, an approximate solution involving a very simple expres sion [21] can replace the series expansion solution. The normalized RCS is propor tional to (ka)4 for ka<0A and approaches a constant value for ka>20. In the resonant region, the oscillatory nature is due to the interference of the direct reflection wave and the "creeping wave" that circles the rear of the sphere and is launched back in the direction of the radar. The contribution of the creeping wave decreases as the fre quency increases, which can be seen by the damped interference pattern. The effect of creeping waves on the reconstructed image will be discussed in a later chapter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -28 - 4 (N * 3 CO u a* 2 •8 N 1 o s 5 0 10 k tfi Fig. 2.7 Normalized RCS of a conducting sphere versus ka . 2.3.2 Cylinder For the geometry of an infinite cylinder, Eq.(2.9) can be written in cylindrical coordinates and the eigenfunctions are called cylindrical wave functions. A plane wave can also be expressed in terms of cylindrical wave functions [46]. Consider an infinitely long circular cylinder with an axis in the z direction being illuminated by a plane wave. The arbitrarily polarized incident wave can be resolved into two com ponents: a TE component with the E-field parallel to the cylinder axis and a TM component with the H-field parallel to the same axis. In this two-dimensional case, the axial incident electric field will produce only an axial scattered E field, while the axial incident H field will produce an axial H field. Therefore, the scattered field of an arbitrary polarized incident wave can be found by solving the scattered field from the TE and TM components separately. The boundary condition is Ez (p=a) = E^(p=a) = 0. The scattered far field for the two polarization cases are 'gjikp-n/4) oo ^ Jn(ka) E STE — £ l TE v 7C V£p Bt l H ? \k a ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.49) -29i(ikp-Ji/4) cos/t<)> (2.50) where primes indicate the derivative with respect to k a , and 1 e«= 2 n=0 n *0 (2.51) The Hankel function is a linear combination of the Bessel functions of the first kind and second kinds, H « \k a ) = Jnm + iYn(ka) (2.52) In the above equations, p is the distance from the cylinder axis to the observation point, and <j>is the bistatic angle. The scattering property of an infinite cylinder is usually expressed as scattering width, which is defined as the RCS per unit length. The scattering width of the two cases are (2.53) COSH 0 I COSn I (2.54) The number of terms required to numerically calculate the RCS of an infinite cylinder has been discussed in [21]. For ka c l and ka"> 1, Eq.<2.49) and (2.50) can be substituted by an approximate expression to avoid the difficulty of numerical evaluation. In practice, the objects of interest are of finite dimension. For the finite case, exact analytic expressions are not available, and one should resort to approximate solutions. If the dimension of the object is large compared to a wavelength, the scat tered fields are usually found by physical optics approximation. The expressions can be found in [47]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -30- 2.3.3 Wedge with Finite Length For the case of an infinite conducting wedge, the diffracted field exists only in the forward direction, i.e, 0 = tc- 0 ' (see Fig. 2.8). However, if the length of the wedge is finite, the diffracted field can be observed in any arbitrary direction due to the effect of finite length. The diffracted field of a finite edge for arbitrary incident and diffracted angles has been treated [48], where the concept of equivalent electrical current and equivalent magnetic current has been applied. The equivalent current concept states that the source of the diffracted field is ascribed to a fictitious equivalent current, both electric and magnetic, flowing along the edge. Consider the finite wedge, with excluded angle n and edge length L , directed in the z direction as shown in Fig. 2.8. An arbitrary point along the edge can be chosen as the origin, but it is convenient to set the origin at the center of the edge. The x axis is directed along the normal to the edge lying on the front face of the edge, the y axis coincides with the normal to this face. The incident wave is in the s ' direction and the diffracted wave is in the £ direction. The angle between the front face and the edge-fixed plane of incidence containing the vectors s ' and £ is 0'. The corresponding angle for the edge-fixed plane of observation is <j>. The incident field I?1 can be separated into TE and TM components (or parallel and perpendicular components) with respect to the plane of incidence, i.e., £* = £&§' + £ i 0' = ^ + f i J w (2.55) The diffracted wave can also be separated into TE and TM components with respect to the plane of observation, i.e., £?* = £ g 0 -l- £ 1 0 = £ j £ + £ 7^ (2.56) Denote the equivalent electric current and equivalent magnetic current on the edge as I and M, the scattered field will then be sin0 j I (z)e]k2cos0 dz Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.57) -3 1 - J observation T point V source Fig. 2.8 Geometry of a finite wedge diffraction problem. £ = v \ £o 4jc - ■sin9 r f M (z)c-,tecos0 dz * (2.58) The expressions of I and M for arbitrary incident and observation angles have been derived in [48]. For the special case of a plane conductor (i.e., n =2 ) and backscattering case (9 = —0' <j>= I and M can be simplified to [48] Vm/ep (l+cos<|>) M b ) = H ‘(z) - - - ■ ---- — jk sin20cos<|> Hz) = (2.59) _ H i( t ) z c o s e a - K ^ ) y/fcsin20cos<() yit sin^sin(J) where //z‘ and Ez‘ are the z components of the incident fields along the edge. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2 6 0 ) -32- The back-scattered fields can be obtained by substituting Eq.(2.59) and Eq.(2.60) into (2.57) and (2.58) and are (2.61) E, = Y /COUo » -} * 47t r sin0 M (0) L sin(fcL cos0) kL cos0 (2.62) The major difference between the finite wedge and the infinite wedge can be summarized as follows: 1. The diffracted fields of a finite wedge can be observed at any direction via the s\n(kL cosQ) term, . If then the diffracted field can be observed only kLcosO when 0 = 90°. 2. For the infinite wedge, a TE incident wave can produce only a TE diffracted wave and a TM incident wave can produce only a TM diffracted wave. That is, the diffraction coefficient matrix is diagonal. However, the diffraction coefficient matrix of the finite wedge is generally not diagonal. The equivalent current den sity depends on the z component of both the incident H field and the incident E field as seen in Eq. (2.60). Since the derivations of the equivalent current sources are based on the geometric theory of diffraction, singularity occurs when the observation point is on the shadow boundary or reflection boundary. The back-scattered fields versus the elevation angle of a finite wedge with length equal to 10 wavelengths, exterior angle equal to 2 ti , and incident azimuth angle equal to 60° are plotted in Fig. 2.9. The back-scattered fields versus the azimuth angle for L=10A., n=2%, and 0=85°, are plotted in Fig. 2.10. 2.3.4 Finite Plate Exact expressions of the field scattered by a conducting finite plate do not exist, because the boundary condition cannot be written as ux = constant, where ux is an orthogonal curvilinear coordinate. The geometrical optics approximation cannot be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -33- flQ 3 0 <u a -15 Q. -30 -45 20 55 90 125 160 125 160 angle in degree e u • mm 1 - 15 I -30 -45 20 55 90 angle in degree (b) Fig. 2.9 Plot of back diffraction fields versus elevation angle for n =2k , <j>=60°, L = 10X (a), parallel polarization, (b). vertical polarization. applied in the case of a plate. When the dimension of a plate is large compared to a wavelength, both physical optics and the diffraction of a finite wedge as discussed in the previous section can be used however to evaluate the RCS. In this section, both methods will be described. With the physical optics (PO) approximation, the induced current density is Ts = I n x lt' and the scattered field is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.63) - 34- horizontal vertical f 0 *3. I -W -30 132 175 angle in degree Fig. 2.10 Plot of back diffraction field versus azimuth angle for L=10X, n =2n, 0=85°. -y com-o e-j* — [(nx/7‘ ) - (rixtf‘' /*)4 ] j 27160 plate ds' (2.64-a) (2.64-b) plate where n is the unit normal vector of the plate, and (? is the factor preceding the integral in Eq. (2.64-a). An arbitrary polygonal plate with N edges lying on the z=0 plane is shown in Fig. 2.11, where 7- is the vector from the origin to the z'th node point, and &,■ is the unit vector directed from node i to node z+1. The scalar scattered field E s is a two dimensional Fourier transform of the shape of the plate. Define the function s(x,y) as 1 (x,y) inside the plate 0 (x,y) outside the plate (2.65) and T - x'x + y'y k (lr -/)) = k (sin0r cos<t>r -sin0,- cos0t-)Jc+k (sin0r sin<pr -sin0,- sin(j)t-)y Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.66) - 35 - > x Fig. 2.11 An arbitrary polygonal plate with N edges lying on the z=0 plane. + k (cosBr-cosQi)z = id + v y + w z = lc (2.67) Substitute Eqs.(2.65), (2.66), (2.67) into (2.64), it* can be expressed as £* = ? JJ dx dy = ^ 5 (u,v) (2.68) plate The expression of S(u,v) can be proved to be [49] l S (u ,v) = — £ x &n ejtr * /„ j eiam**mdt if v*0 (2.69) 0 JV n = 1 where /„ is the length of the nth edge, t is a temporary variable, Eq.(2.69) can be further simplified as -1 S (u ,v) = — N I J V n=\ jk & n «■<*„)■ jit-K Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.70) -36- S (u ,v) can also be expressed as follows if u*0 gJP-Z+i _ eJ?< jJc &n S(u,v) = -7- X (yA»)’ inefi* . JU n=1 The significance of Eq. (2.69)is: the scattered if lc -&n * 0 if ?&„=<> (2J1) field of a conducting plate is mathematicallyequivalent to the summation of the scattered fields of eachedge. The contribution of each edge is proportional to -------- /„ v sin(lnJc&n) —------- . While the lnk-&n significance of Eq. (2.70) and (2.71) is: the scattered field of a conducting plate is mathematically equivalent to the summation of the scattered field of each vertex. The (*•& „) -> contribution of each vertex is proportional to (— - — )/jk-6in. The other approximation method is to consider the scattered field of a plate as a summation of the diffraction fields of each edge. The contribution of each edge is found by the previous section (Sec. 2.3.3). In this approximation, the extent of the plate normal to the edge direction is assumed infinite. For each edge, one can specify a set of edge-fixed coordinates for that edge and then find the incidence angle, diffraction angle, TE and TM components of the incident wave and diffracted wave in terms of the edge-fixed coordinates. Finally, one can use coordinate transformation to convert the scattered field in the edge-fixed coordinate system to the laboratory coordinate system. The procedures are shown in Fig. 2.12. The difference of the back-scattered field of a plate using the two approaches above has been numerically and experimentally compared [57]. It was concluded that the PO approximation method matches experimental results very well in the proximity of main-lobe regions but deviates noticeably from these results at wide angles from the specular direction. Outside the main-lobe regions, the second method is more effective than the PO method. In a later chapter, we will use the second method to numerically calculate the scattered field of a hexagonal plate over spectral and angular windows of interest. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 37 - f i x e d coord. r ectan gular coord. 4- lo c a l coord. s ph er ica l coord. r e ct an gu la r coord. lo c a l coord. sp he r ic al coord. f i x e d coord. re cta ngu la r coord. s p he r ic al coord. r e c t angular coord. I s p he r ic al coord. ■V + Fig. 2.12 Procedures of calculating the diffraction field of an arbitrary edge. 2.3.5 Corner Reflector Comer reflectors or retro-reflectors are designed to provide a large RCS over a wide range of aspect angles and frequencies. The most popular comer reflectors are dihedral and trihedral comer reflectors, which are formed by the intersection of two and three flat plates respectively. The angle required between the plates to achieve the optimum cross section is a function of the bistatic angle. In the case of a dihedral comer reflector, the required included angle is given by |3 = 90°+a (2.72) where a is the bistatic angle. The high frequency RCS patterns can be estimated by using geometrical optics and taking into account multiple scattering. The approximate back-scattered cross section of the dihedral comer reflector is given by [21] 167W2fc2sin2(jiV4 + b) ~ A*q Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 38 - where a , b , and <(>are defined in Fig. 2.13. Fig. 2.13 Geometry of a dihedral comer reflector. The dominant contribution to the back-scattered field is due to the double bounce mechanism. Except for polarization either in the plane of incidence or per pendicular to the plane of incidence, the polarization of the scattered field will be different from the polarization of the incident field due to multiple reflections. 2.4 Scattering Mechanism of a Complex Shaped Object When the object becomes complex in shape, no exact analytic solutions are available. The moment method, which uses the grid wire model, has been applied to find the scattered field of complex shaped objects [50]. However, the computing time is extensive especially when the object is large compared to a wavelength. Computer simulation using the PO approximation has been conducted and SAR images of air planes were obtained [24]. However, the RCS of a complex object obtained by the PO approximation method is valid only for certain range aspect angles because edge diffractions and other scattering mechanisms are also major contributors to the scat tered field.. To a first order approximation, the object is separated into a number of geometric shapes or segments, and the scattered fields from each shape are then Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -39- coherentJy superimposed to form the total fields. However, care must be taken because some shapes may be obscured by other shapes for certain aspect angles. Thus the analytic components of the cross section of particular geometric shapes may not be the same when the shape is in one part of the complex body as they are in an isolated scatterer [21]. In the high frequency region, i.e., when the body dimensions are large in terms of a radar wavelength, the scattered fields can be attributed to a combination of the following components [21]: 1. Specular scattering points. 2. Scattering from surface discontinuities - edges, comers, tips etc. 3. Scattering from surface derivative discontinuities. 4. Creeping waves or shadow boundary scattering. 5. Traveling wave scattering 6. Multiple scattering points. In the following, we will briefly discuss each of these scattering mechanisms. 2.4.1 Specular Points The specular points for a given transmitter and receiver can be determined by the ray optics, where the reflection law must be satisfied. The geometrical optics cross section of specular points on a conducting surface is given by o = 7tp!p2 (2.74) whereplt and p2 are the principle radiiof curvature at thespecular points. If the sur face atthe specular points can be described by z = f ( x , y ), and theincident wave is in the -z direction, then a is given by [21] 0 2/ /ax2>02/ (2.75) /dy2) - (d2f / d x d y ) It is noted that the cross section of a specular point is independent of frequency and depends only on the curvature at that point. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -40- Specular point reflections can also be obtained by the asymptotic solution of the physical optics expressions. In Eq. (2.34A), as k goes to infinity, by applying the sta tionary phase method [51], the result is A k j t j r) = - ^ e ~ jkr E (2.76) where 7j' are the vectors such that dwr-it)-r] da' Ir *i =0 (2.77) and //2| n _ t \.rn = -<'■ J | J da'2 (2.78) J where d Ida' is derivative with respect to the surface curvature. The points corresponding to the solutions of Eq. (2.77) are called stationary points, equi-phase points, specular points, or scattering centers. The absolute value of the term h (Tj Ox^o is called the scattering strength for that scattering center. It is seen that the locations of the scattering centers depend on the directions of lr and /, as well as the shape of the metallic surface. The above analysis illustrates that the objects in the high frequency regime can be regarded as of discrete form, consisting of point scattering centers. Consider the back-scattering case. Orient the coordinate system so that the z axis lies along the back-scattering direction and passes through a point on the surface with its normal pointing in the z direction. Define the surface of the object as z —f {x ,y) and the coordinates of the specular point as [0, 0, y (0,0)]. By applying the stationary phase method to Eq. (2.39), one can obtain the following asymptotic scat tered field solution [21], it’(7, = ^ » * [ r- 2 /(0 .0 )H v ) v ^ /?o (2.79) where \jr is either zero, re/4, or k/2 depending on whether the surface in the vicinity of the specular point is convex, saddle or concave; pj and p2 are the maximum and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 41 - minimum radii of curvature of the surface at the specular point. The back-scattering cross section is then Hs O = 47tr2 l — I2 = 7lp!p2 (2.80) which is identical to that predicted by geometrical optics. 2.4.2 Surface Discontinuities When one of the radii of the surface curvature goes to zero, an edge is formed. If both radii of the surface curvature go to zero, a tip is formed. Geometric optics cannot predict the scattered field of edges and tips because the RCS would be zero as expressed in Eq. (2.74). In the absence of specular reflections, contribution from edge diffraction becomes more important. The contribution due to edge diffractions can be estimated using GTD as described in the previous section. The contribution from a finite wedge can be obtained by the equivalent current method. The procedures of calculating the scattered field were shown in Fig. 2.13. An edge can be a straight edge or a curved edge. The dependence of the backscattered field on frequency and size depends on the type of edge (straight or curved) and angle of incidence (at or away from normal incidence). 2.4.3 Surface Derivative Discontinuities As shown in Sec. 2.2 the back-scattered field using the physical optics approxi mation can also be expressed as (2.41) _ -jke-J* j 2 nr (2.81) where / is the integral in Eq. (2.41). The RCS is then (2.82) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -42- / can also be expressed as [21] N j =i av, dns' (-D fl expO‘2/:/. ) (------------------) n~ ( 2 j k ) n ' a/;+ a/;- (2.83) where lj* and ly are points at which discontinuities in the derivative of the projected area exist. It is noted that the frequency dependence of the n th surface derivative discontinuities to / is proportional to k~n as can be seen in Eq. (2.83). Therefore, the higher the frequency is, the less the contribution of the surface derivative discon tinuity will be. The above equations will be used later to account for the interpreta tion of microwave diversity imaging. 2.4.4 Creeping Waves Creeping waves are characterized by the wave striking at the shadow boundary of a smooth, curved, perfectly conducting surface to propagate along the body surface in the shadow region, and emerge at the opposite shadow boundary. The concept of a creeping wave can be illustrated by Fig. 2.14. In this figure, the incident wave illuminates the scatterer and is diffracted at the shadow boundary at point A, at which the incident ray is tangent to the surface. At this point, a por tion of the energy propagates on the surface of the scatterer, shedding energy by radi ation as it progresses in a direction tangent to the surface at point B. The creeping wave can be described by a launching coefficient at the point of capture, by a diffraction coefficient at the point of radiation, by an attenuation factor which accounts for the rate of radiation, and by a description of the path on the scatterer traversed by the creeping wave [44], The creeping wave loses energy in proportion to the distance traveled in terms of wavelength behind the smooth surface. Consequently, as frequency increases, the contribution of creeping waves lessens. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -43- E1' Fig. 2.14 Creeping wave concept of diffraction by a curved surface. 2.4 J Surface Traveling Waves When long thin bodies, such as wires, ogives, and prolate spheroids are illuminated near nose-on incidence, the back-scattered field exhibits a large peak if there is a component of the incident electric field tangential to the surface and in the plane of incidence. The back-scattered field are dominately contributed by the rear of the body [21]. The body acts as a traveling wave antenna in the end-fire mode excited by the incident field. This scattering contribution is called the surface travel ing wave. The analogy between the operation of end-fire antennas and the surface wave contribution to the scattered field on long bodies has been extensively analyzed in [52]. The back-scattered cross section of a long, thin body due to a traveling wave is related to a reflection coefficient determined by the rear shape of the scatterer [21]. However, there is no analytical means available for estimating the reflection coefficient. The traveling wave phenomenon of a thin wire can be clearly observed by Fourier transforming the broad-band frequency response and will be discussed in Chap.7. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -44- 2.4.6 Multiple Scattering For concave bodies or a body with comers, the contribution of multiple scatter ing to the RCS becomes more important. To determine the position of the multiple reflection points, ray optics can be used. The ray should satisfy Snell’s law, i.e., the angle of incidence should be equal to the angle of reflection. Assume now that the multiple reflection points have been found, then the scattered field due to a set of two multiple reflection points may be approximated by Ak) = 2 KRr lcy S 2 F { ) x f i 2(k)ejk{(f'-f *)r{] (2.84) where f f tf ) = h Z E g l (2.85) tjjc ) = (2.86) 27C/C21 kySi where /? 2i is the distance between the two scattering points, f 2i is the unit vector connecting scattering point 1 to point 2, and "f{ are the position vectors of the two multiple scattering points, S 1 and S2 are the second derivatives at points 1 and 2 as defined in Eq.(2.78), and h ( f{ ) and h t f { ) are the normal vectors at positions T { and r 2/, respectively. 2.5 Polarization Effect Polarization dependence of a target is usually expressed in terms of the polariza tion scattering matrix, which relates the polarization of the incident electric field to the polarization of the measured scattered field. Mathematically, the relationship between the scattered fields and the incident field can be written by “ • El E\ a ll a \2 II E s2 d 2\ Oft i or Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -45 - [E* ] = [S ] [E* ] (2.88) where the subscript 1 and 2 represent any two general orthogonal polarization com ponents, and dij elements are, in general, complex quantities. The matrix [ S ] is called the scattering matrix. The most commonly used orthogonal polarization sets are parallel and perpen dicular, right-hand circular and left-hand circular. If the transmitting antenna and receiving antenna have the same polarization state, the measurement is called a co polarized system. On the other hand, if the transmitting antenna and receiving antenna have orthogonal polarization, the measurement is called a cross-polarized system. For a conducting reflector, a wave which is right circularly polarized becomes left circularly polarized ( and vice versa) upon reflection. Therefore, the co-polarized cases include: a parallel (or perpendicular) polarized transmitted field and a parallel (or perpendicular) polarized received field, and a right-hand (left-hand) circularly polarized transmitted field and a left-hand (right-hand) circularly polarized received field. The cross-polarized cases include a parallel (or perpendicular) polar ized transmitted field and a perpendicular (or parallel) polarized received field, and a right-hand (or left-hand) circularly polarized transmitted field and a left-hand (or right-hand) circularly polarized received field [21]. It is noted that the definition of cross-polarized terms for the circular polarization case is totally reversed in some papers [53]. In this dissertation, we will adopt the definition defined in [21]. In general, for targets possessing an asymmetrical shape, the scattered field is polarized differently from the incident field. This phenomenon is known as "depolari zation” or "cross-polarization". However, depolarization is difficult to define in the bistatic case, because the orthogonality of two polarization vectors at two different spatial positions has no unanimous definition. Under the physical optics approximation, it has been shown in Sec. 2.2.2 that the back-scattered field of a conducting object is polarized in the same direction as that of the incident field. Therefore, physical optics predicts no depolarization. Multi ple reflections usually cause the state of polarization to change. The monostatic diffraction coefficients, Dv and Dp , of an infinite edge have different values as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -46- shown in Fig. 2.5. If the incident wave has right-hand circular polarization, the co polarized (left-hand circularly polarized) and the cross-polarized (right-hand circularly polarized) components of the diffraction coefficients versus the azimuth angle are shown in Fig. 2.15. It is interesting to note that the cross-polarized component of the diffraction coefficient is nearly constant over the whole aspect angles. co T3 3 TEL -15 -30 135 180 angle in degree Fig. 2.15 Diffraction coefficients of an infinite edge vesus azimuth angle. The illuminating waves are circularly polarized, 0j =0^=90°, ty=$d ,n=2. Diffraction at a finite edge will usually cause depolarization except when the inclination angle 0 is equal to 90°. A nonzero <|> component can exist in the backscattered field when the incident field is in the 0 direction. This can be examined in Eqs. (2.59) to (2.62), where both the equivalent electric and magnetic currents and both the two orthogonal components, E q and E^, of the diffracted fields remain nonzero in the equations when 0 is not equal to 90°. Furthermore, the ratio of the two orthogonal components is a function of incident angles (0 and <|>) only and is independent of the wedge length and frequency if the assumptions of the equivalent current method hold. Assume both the transmitting and receiving antennas are circu larly polarized. The ratio of the co-polarized component to the cross-polarized com ponent of the back-scattered waves can be measured for various angles of incidence. Conversely, if the ratio of the two components is measured, one can roughly estimate Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -47 - the incident angles with respect to the edge-fixed coordinates. In other words, the polarization status of the back-scattered field provides feature information about the wedge. At high frequencies, as described in the previous section, the scattered field of a complex object can be viewed as contributed to by discrete sources with different scattering mechanisms. Usually, the major contributors are specular reflections, edge diffractions, and multiple reflections. Accordingly, the scattered fields due to these discrete sources can be expressed as Z ( k ) = 2 3 (k)ej/f'r‘ + 2 tjihye***' + 2 ^m(k)ej f ^ i j m (2-89) where P = k (lr - lL), a) is the scattering vector of the ith scattering center, 5* is the diffraction vector of the j th edge, and l?m is the scattering vector of the /n th multiple reflection. Under the high frequency assumption, all 3}’s are independent of fre quency and have the same polarization as the incident wave; all by’s are independent of frequency and their polarizations can be different from each other and different from that of the incident wave; 7?m is independent of frequency but its polarization is dependent on the ray paths traveled and is usually different from that of the incident wave. The advent of high resolution radar makes the expression of Eq. (2.89) very valuable, because high resolution radar has the capability to separate each discrete scattering center on the target. From the polarization status of each source, one can estimate its scattering mechanism and its significant feature. Although the polariza tion of each source is independent of frequency, the polarization of Ps (k) can changes with frequency. The polarization of J?s (k) is determined by the complex ratio of the two orthogonal components of the total field. The ratio is a function of frequency because of the terms e P 'r . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -48- 2.6 RCS Measurement The RCS of a target is the equivalent area of the target assuming the target rera diated the incident power isotropically. It is a function of the target’s geometry, fre quency, aspect angle, and polarization. Mathematically, the RCS a is defined by c(* ,04,0',40 = lim4itf2 (2.90) The multiplication by 4nr2 is to make a independent of r. Definition of the coordinate systems used are required to describe the RCS measurement. When static scattering measurements are made, the illuminating antenna is usually fixed and the object is rotated. A stationary coordinate system is defined relative to a fixed reference point Body motions relative to the reference sys tem can then be described by three angles, the yaw angle, the pitch angle and the roll angle. Scattering data for all directions of illumination can be obtained by fixing either the pitch angle or the roll angle and by varying the remaining two angles [21]. Once the coordinate system is defined, the polarization of the antennas and the scattering matrix can then be determined. The measured received signal Vs can be expressed by Vs = c [hr ] [ U r 1 [ S ] [ U ] [ ht ] (2.91) where [hr] and [ht \ are the polarization vectors of the receiving antenna and transmitting antenna respectively, [S] is the scattering matrix as defined in Eq.(2.87), [£/] = cosy siny -siny cosy is the transform matrix with y the polarization angle (see Ref. [21], p.896). If the same antenna is used for both transmitting and receiving, then a 12 = a i\- if the target possesses a plane of symmetry, and the matrix elements are referenced to this plane, then a 12 =^2i =0- The calculation of the amplitude and phase of the elements of the scattering matrix has been discussed in [21]. Intrinsic to the definition of RCS is the assumption that the incident field at the scatterer is a plane wave and the scattered field varies as 1/r. If the observation dis tance is long enough, the conditions above can usually be satisfied. The distance Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -49- criteria are usually developed according to the phase and amplitude of field variation along the transverse and longitudinal direction. The constraint that the transverse phase variation between the incident field at the edge and center of the target extent is within tc/8 results in the requirement that the distance Rm must be greater than 2L2/Xq, where L is the maximum dimension of the target as projected in a plane per pendicular to the direction of observation. In reality, the distance in the RCS meas urement conducted either indoor or outdoor sometimes cannot satisfy the required conditions. To reduce the separation requirement and to obtain a nearly uniform plane wave illumination, some "compact range" techniques have been proposed and implemented [54], The fundamental concept of the compact range technique is usage of a device which converts a spherical or cylindrical wave from the feed into a quasi-plane wave. This conversion process is usually accomplished with the use of a lens or a reflector [55]. If a full-scale target is not available for measurement or if the effect on RCS of a design variation before the final design is chosen needs to be determined, building a scale model of the target is useful. The theory of modeling EM systems has been established [56]. Relations between the quantities in the full and scale systems are found in [21], It is theoretically possible to model an EM system using scale factors to make absolute measurements of parameters in the model system, and to relate the measured quantities to the corresponding parameters in the full scale system. For example, let the full-scale system be represented by unprimed quantities, and the model system by primed quantities. Assume the permittivity and permeability of the model medium is the same as those of the full-scale system. If the length / is scaled by p times, i.e., I' = lip, the conductivity in the model system must be p times the conductivity in the full-scale system, and the frequency be / ' = p f in order that the two EM systems be equivalent. It is noted that the required far field distance can be lessened p times for the scale model system, because „ . *- 2/ '; - x 2 - 21* Vp ' P\ - p <292) Therefore, scale models are frequently used so that the far field condition can be met by the measurement system. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -50- The block diagram of the system used in out work for RCS measurement is shown in Fig. 2.16 [10]. A test object is mounted on a rotating pedestal controlled by a MINC-23 microcomputer. The transmitting antenna is a 36" parabolic reflector fed by a Transco 9C 27400-2 righthand circularized spiral antenna. The return signal is received by a dual polarization EM Systems A6100 2-18 GHz dual polarization horn antenna. A Norsal 4115 90° hybrid combines the horn antenna outputs to meas ure the circularly polarized signal. The received signal is amplified by an Avantek AWT-1803911 FET amplifier and then fed to an HP 8350 network analyzer. The signal is generated by a HP 8340A synthesized sweeper, which covers frequency from 10 MHz to 26.5 GHz, and is then amplified by a Varian VZM-6991K3 TWT amplifier. The computer controls the synthesizer and collects amplitude and phase data from the network analyzer through the IEEE-488 interface bus. Therefore the whole measurement is fully automatically controlled by a micro-computer, where a,b, and $ are defined in Fig. 2.13. To obtain the absolute RCS of an object, an absolute calibration procedure has to be used. In the environment of an anechoic chamber, the calibration procedures are as follows [10]: 1. In the absence of the object, measure the clutter fields, which include reflection from the ground, absorber, supporting holder etc., and denote the clutter field as N( f ) . 2. Choose an object as the reference object, whose RCS must be known or can be theoretically calculated. Denote the RCS as aR ( f ). The reference object can be a sphere, a cylinder, or a comer reflector. Measure the field in the presence of a reference object. Assume that the presence of the reference object won’t disturb the clutter field, the measured field will then be SR( f ) = N ( f ) + R( f ) , where R ( f ) denotes the field scattered by the reference object only. 3. Remove the reference object and replace it by the test object, the measured field will then be Sa( f ) = N ( f ) + 0 ( f ) , where 0 ( f ) is the scattered field due to the object only. The RCS of the object a0 ( f ) is then o0( f ) = ~ ~ ^ ~ <5r ( f ) \R(f)\2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. COHERENT RECEIVER -T I RECEIVER REFIRENCE FREQUENCY U) SOURCE TRANSMITTER COMPUTER PEDESTAL CONTROL Fig. 2.16 Block diagram of the RCS measurement system. -52- ' S0( f ) - N ( f ) ,2 a93) If a complex field measurement is desired, the phase relative to the reference point (usually the rotation center of the target) is needed. In our measurement sys tem, a long metallic cylinder with its front rim placed to coincide with the rotation center is found to be a good way to obtain a phase reference. However, the exact RCS of a long cylinder is difficult to evaluate over a wide frequency range because the beamwidth of the transmitting antenna is not constant and cannot cover the whole cylinder. Therefore, the measured magnitude can not reflect the true RCS of the cylinder. A conducting sphere with proper radius will be more suitable as a standard reference object because the sphere can be within the main-lobe of the transmitting antenna for each frequency. The magnitude of the measured field can then be com pared with the theoretical value, from which the frequency response of the measure ment system can be obtained. However, it is more difficult to calibrate the phase reference by using a sphere because the center of the sphere is difficult to align with the rotation center. In this situation, we recommend the following calibration pro cedure: 1. Measure the clutter noise N( f ) . 2. Mount a long cylinder with its front rim coincident at the rotation center. Meas ure the scattered field Sc ( f ). 3. Replace the cylinder with a sphere with known radius. Denote the scattered field as Ss ( f ) and the theoretical RCS of the sphere as os ( f ). 4. Replace the sphere by the test object. The scattered field is St ( f ). The complex scattered field of the object with respect to the rotation center is then " \Sc( f ) - N ( f ) Et ( f ) = Ss ( f ) ~ N( f ) I ’ ' 7) (2-94) and the absolute RCS of the target is IE ,(H I2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.95) -53- Clutter is the reflection from all objects other than the test object itself. The above analysis assumes that the background scattering is not affected by the presence of the object. In some situations, the presence of the object did affect the background scattering. For example, a tilted plate might reflect the incident wave to the holder and then reflect back to the receiving antenna. In that case, coherent subtraction won’t improve the results. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3 IMAGE UNDERSTANDING IN MICROWAVE DIVERSITY IMAGING T he appearance of an image depends on the scattering properties of the object and the imaging technique utilized. For example, the microwave scattering properties of an object are different than those noted using optical waves. Optical wavelengths are usually smaller than the surface irregularity of most objects. Optical illumination is scattered diffusely from such rough surfaces yielding image information about the whole surface. However, a surface with the same roughness can be smooth compared to microwave wavelengths and the scattered waves may not be seen in most aspect angles. Therefore, microwave images can be entirely different from their optical counterparts. The imaging scheme used directly determines and affects image qual ity. For example, the image resolution obtained by conventional incoherent sidelooking radar is much poorer than that obtained by synthetic aperture radar [3]. Therefore, understanding and interpretation of an image should be based on both knowledge of the scattering properties and the image formation technique imple mented. Basic scattering theory of a perfectly conducting object has been reviewed in the previous chapter. In this chapter, we will relate these scattering properties to the reconstructed images obtained by a specific imaging scheme, microwave diversity imaging, achieved by combining angular, spectral and polarization diversity. In this imaging system, the object is seated on a rotating pedestal and the back-scattered field is measured. A synthetic imaging aperture is achieved in this fashion costeffectively without resort to an array aperture. A very famous imaging technique which also involves rotating the object is computer aided tomography (CAT), which is concerned with the reconstruction of a section of the human body by means of Xray taken at different orientations. Basically CAT is a problem of image reconstruc tion from its projections. This problem has wide applications including medical Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -55- imaging, geophysical signal processing, astronomy, and microwave imaging. In this chapter, we will start by reviewing the image reconstruction of an object consisting of point scatterers. We assume each scatterer scatters isotropically for each aspect angle and no mutual coupling exists among them. This analysis is helpful in illustrating the connection between reflection type microwave imaging and transmission type CAT. Determining the shape of a perfectly conducting object from a finite number of measurements of its scattered fields is one of the purposes of microwave diversity imaging. The PO approximation is usually applied to relate the geometrical shape of a conducting convex body to its scattered field. In Sec. 3.2, we will summarize the formulation of microwave imaging for conducting objects based on the PO approxi mation. As we indicated before, the PO approximation is inadequate for the scattering problem of a complex shaped metallic object The point-scatterer model is also not suitable for describing the scattered field of a perfect conductor. However, we still apply the reconstruction algorithm used in CAT to reconstruct microwave images. Therefore in Sec. 3.3 we will give an alternate interpretation to the reconstructed images based on the understanding of the scattering mechanisms and the reconstruc tion algorithm. Several typical numerical and experimental examples will be demonstrated in Sec. 3.4. These examples will illustrate the influence of the various scattering mechanisms, angular windows, spectral windows, and polarizations on the recon structed image. 3.1 Image Reconstruction of Point Scatterers Assume that the image of an object consisting of point scatterers is to be obtained. Each point scatterer scatters isotropically for each aspect angle and there is no mutual coupling among the scatterers. A plane wave with wave number k is used to illuminate the object and the scattered wave is received. The object is then rotated and the measurement is repeated. Let O be the reference point and the rotation center Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -56 - of the object, z axis be the rotation axis, and a be the bistatic angle. The lines con necting the rotation center to the transmitter and receiver are both perpendicular to the z-axis, that is, the antennas are in the x - y plane. The imaging geometry pro jected to the x - y plane is shown in Fig. 3.1. Let gt be the reflectivity of the ith point scatter at (* ;,? /, z,-)» then the three-dimensional (3-D) reflectivity function of the object becomes u x y V r receiver - i transmitter Fig 3.1. Simplified imaging geometry projected to the X - Y plane. g(x, y, z) = £ gt 8(x- x i,y - y t ,z-z{) (3.1) i When the object is rotated through an angle <|>,the range and crossrange of the point (x ,y) becomes u and v respectively. The relationbetween (u ,v) and (x ,y) is u = xcos<t> + ysin<(> (3.2a) v = - x sin<(> + y cos<|) (3.2b) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -57- Therefore, x = wcos(J) - vsin<t> (3.3a) y = Ksin<j> + vcos<|) (3.3b) The scalar scattered field with rotation angle 0 can be expressed as £*(**) = J L f f f **(U’V’Z) e-jW<n**'<n) 4jc£JJJ R*Sf)Rr( f ) _ jk y g(|>(“ |»Vi»Zi ) ^-j^Rl+RD 4TO 7 (3.4) /?//?/■ where R ‘(7*) and R r (F*) are the distances between the point T^and the transmitter and receiver respectively, similarly, R f a n d R f are the distances between the ith scatterer and the transmitter and receiver respectively. Under the far field approxima tion, the denominators R l (r*), R l (?) become r ‘ (j *)= r ‘ - (3.5) R ‘ (ry) ~ R r and the phase terms simplify to R ‘ <f) + R r( D ~ R t + R r - 2kucos(oU2) (3.6) where R ‘ and R r are the distances between the rotation center and the transmitter and receiver respectively. By substituting Eqs. (3.5) and (3.6) into (3.4), E s(k,$) can be expressed by £ a(*,0) = Jk6~m t +Rr ) 'L g ^ u i ,vi ,zi)ej2kUiCOs(aj2) 4jteR‘R r i (3.7a) = C( k) ^ g ^ u i ,vi ,zi )ej2kUiCOs(aJ2) I (3.7b) Where C( k) denotes the term left to the summation in Eq. (3.7a). Let 8 1«)>(«, v ) = J g ^ u ,v ,z) dz £ 2$(m) = / / g$(u,v,z) dzdv = j g ^(u,v) dv (3.8) (3.9) whereg i0(w,v) is the projection of the object reflectivityfunctiononto the u -v plane at (u,v) and g 2q(u ) is the projection of thereflectivityfunction along the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -58 - v-axis at range u. Eq. (3.7) can be rewritten as £ 5(*,<|>) = C ( k ) j g 2<)(n )e '2tocos(a/2) du (3.10) If the term in front of the integration is removed, the range-corrected field E^ (k ,<)>) becomes Es' ( k S = jg^(u)ej2kucosandu (3.11) It is noted that E* (k,<)>) and g ^ i u ) are a Fourier transform pair. E* (k,<J>) is referred to as the Fourier space data. If the Fourier space data are available over sufficient bandwidth, i.e., over a sufficient range of wavenumber k , the projected data g ^ u ) can be estimated by a FT of the range-corrected fields or Fourier space data E* (fc,<j>). There are some similarities and differences between CAT and microwave diver sity imaging systems. Both systems involve rotation of the object However, the measured quantities in CAT are the projection data along the rays; while the meas ured quantities in the microwave diversity imaging systems are the Fourier space data which are the FT of the projection data along constant ranges. If we inverse FT the Fourier space data for each aspect angle in the microwave diversity imaging system, we have the estimated projection data along constant ranges at each aspect angle. At this point the two imaging systems are equivalent because both call for image recon struction from an object’s projections. The reconstruction algorithms used in CAT can then be applied to this imaging system. Two methods have been used to recon struct the computer tomographic image: the direct Fourier method and the backprojection method. These two methods yield equivalent results, as proven by the central-slice theorem or the projection slice theorem [27], and will be described below. Consider the two-dimensional case, let g'(x,y) be the density function to be reconstructed from its projections. The FT of g' is defined as G' ( X, Y) = J { g, (x,y)e-^xX+y^dxdy so that Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.12) J g' ( x, y) = / G 'Q C 'Y y e jW -W d X d r —«o (3.13) oo The projection of %’ along v -direction at angle <|>is given by oo P$(u)= J g'(ncos(|>-vsin(l>, nsin<(M-vcos<j))dv (3.14) The function p$(u) represents a series of such line integrals for each value of <|>. The 1-D FT of p$(u) is given by oo P*(U)= j p $(u)e~juUdu oo = J oo = oo j g'(ncos<|>-vsin<J>, Msin<(H-vcos<t))e-^ d n d v ■ oo / J —o o g' (X ,y )e-j(Uxcos^ uy**#>dxdy oo = G' (Ucos<(), U sin<{>> (3.15) The statement of the projection-slice theorem is P ^ U ) = G '(U cos<J), U sin<(>), (3.16) that is, the FT of the projection at angle <|> is a "slice" of the 2-D transform G'(X ,Y) taken at an angle <|>with respect to the X axis. If the FTs of the projections (or the Fourier space data) at various angles <(> are available, then the density function can be reconstructed by inverse FTs of the Fourier space data. When a reconstruction is obtained by using this central-slice theorem, the method is referred to as a direct Fourier method. In microwave fre quency and angular diversity system, the Fourier space data are usually in polar for mat. To make use of the fast algorithm of FT, one should convert the polar format into rectangular format [58]. The FT relation in Eqs. (3.12) and (3.13) can also be written in polar coordinate form. Let (r, \j/) and (p, q) be the polar coordinates of the space (x, y) and Fourier Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -60- space ( X , Y ) respectively, the relationships between these pairs are x = rcosy, X = pcosrj, y = r siny Y = r sinrj (3.17) The polar expression of Eq. (3.13) is then 2k oo g '(r,y ) = J J pG '(p,ri)e-/2ro’pcos(11-v)rfpdTj o o 2k oo = J o jpG ' (p,T\)ej2nrpcos<^ v)dp dr\ o = j F [pG' (p,Ti)](r cos(T|-\|0)d q o (3.18) where F [ ] is the FT of the bracketed quantity with respect to p. The above equa tion is the basis of the filtered back-projecdon algorithm. The procedures of the algorithm are as follows: 1. Fourier transform the product of the Fourier space data at each angle <j> and the function p. The function p is also called the rho -filter [26]. This output is called the filtered projection. 2. The output is then uniformly back-projected in the v-direction, this backprojection line is called filtered back-projection line. Since the operation of computing involves starting with a 1-D function to obtain a 2-D one, this opera tion is referred to as back-projection. 3. The contributions to a pixel at (r ,\|/) from different filtered back-projection lines are added coherently, i.e., polarity is taken into account. It is noted that the multiplication of the Fourier space data by p will make the point-spread function a delta point if the Fourier space data are available for all fre quencies and all directions. Compare the above 2-D analysis to the microwave frequency diversity imaging, we have the following dualities: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 61 - g'(x,y)-+-+. gi(x,y) U ■*—+- 2kcosal2 Therefore, the reconstructed image in the microwave diversity imaging is the estimated reflectivity function g\(x,y), which is an image projected onto the z =0 plane. In practice, the spectral band and the angular window of a microwave im agin g system are usually restricted by practical constraints. In that case, the point-spread function will not be a point but is space-variant and angular window dependent. The effect of the various angular windows on the point-spread function has been dis cussed in [12]. 3.2 Formulation of Microwave Imaging for Conducting Objects Based on the PO Approximation Practically, a conducting object does not consist of discrete point scatterers. In particular, a conducting object has obstruction effects, where various of its parts are seen for certain viewing angle only. In addition, polarization is also an important scattering factor. Therefore, the point-scatterer model is not suitable for a metallic object because the scattering properties of the latter are dependent on frequency, aspect angle, and polarization. In this section, we will investigate the formulation of microwave imaging for a metallic object based on the PO approximation. Under the PO approximation and far field condition, the back-scattered field of a convex metallic object has been expressed in Eq. (2.38) and is repeated (3.19) •>ia If we divide the object into two regions, lit region 1 and shadow region 2, and express the boundary cross section as A , then by the divergence theorem, that is, the closed surface integral of a vector B is equal to the volume integral of the divergence Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -62- of the vector, = jV&dv, (3.20) we can write Eq.(3.19) as j eJ™' % d ? f - j e j2kf' H r d?' Sill + A jke~jkr go 2nr region 1 jke~jkr go 2nr region 1 JJJ V ie j7ki' r lr )dv - fe i7jd'r lr d?' jjj A j2Jcej7jd' r dv - je i2kir'r ir <&’ A e 'j * (3.21) where jjj j2 keiW ’*‘ dv - region 1 (3.22) A is called the normalized scattered field. If the transmitter/receiver is placed opposite to /, direction or the object is reversely placed, then the lit region and the shadow region will be interchanged, and the conjugate of E/ ( k , - l r) becomes Es (k, - lr ) = 2k j j j - j2 k e jVd, r dv - j e j2Jc(-l' y r {-lr)-{-(&') (3.23) region 2 A By adding E/ ( k, l r) and E /* ( k ,-lr), one has E /(k ,lr ) + E / \ k - l r) = e ^ d V (3.24) Use the relation 2klr = P , the above equation can then be written in the following scalar form = j j j e ^ d 3/ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.25) -63- Define the characteristic function y o f the scatterer as 1 In for all T*' e v (3.26) A lc P U /llA rP and Ti p) = -k[Es (P )+E ^-P)] / k 2 (3.27) then Eq.(3.25) can be rewritten as oo T(P) = JJJ (3.28) dv' It is seen that T(P) and yfj*) are a 3-d Fourier transform pair. The above relation is called the "Bojarski Identity". If V(P) can be measured for all P , i.e., all frequencies and all directions /r , it is possible to reconstruct the geometric shape of the object from the inverse FT of T(P), that is, oo 7(0 = JJJr {?)e-jp r d p (3.29) It is noted that the FT pair is between y(7*) and T(P) but not between yff') and E ',(P ) or E's (-P ). A A For the bistatic case, i.e., lt * -/r , define P = k(lr - l t ) (3.30) and define an object scattering function ^ (J* jn f ) ^ as - [(/z (?)xfi?)jr]ir for all r on Siu A = jo elsewhere <3-31> After removing the term in front of the integral, Eq.(3.19) can be rewritten as PS\ P ) = J P)e~j P r ds' (3.31) Sill It was stated that PS(P) and yCr*, P ) comprise a 3-D FT pair [10,11]. However, Y'JP) is not only a function of is also a function of P. T4', but also a function of lr and Pi or I,, that is, it Therefore, the 3-D FT relation is not exactly satisfied Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -64- between Ps (P) and the geometric shape. All the above relations are derived under the physical optics approximation. However, the PO approximation suffers from some difficulties. First, the PO approxi mation is valid only for high frequency, while the inverse FT relation in Eq.(3.29) requires integration over all frequencies, including the low frequency and resonant regimes, for which the physical optics approximation is not valid. Second, Eq.(3.29) requires integration over all frequencies and all aspect angles. In most practical situa tions, however, only a limited set of frequencies and angular windows are available. In [59], a scattering information aperture function was defined as unity or zero for all values of P for which T(P) is known or unknown respectively. An integral equation for incomplete scattering information was developed and a regularized analytic closed form for that integral equation was suggested. In the next section, we will use a different approach to consider the effect of finite spectral and angular windows on the images. Furthermore, the PO approximation is inadequate for scattering problems of a complex shaped object. The scattering mechanism of a complex shaped object has been discussed in Sec. 2.4 The main contributors to the scattered field are specular reflections, edge diffractions, multiple reflections, and creeping and traveling waves. Although the scattering properties of a metallic object are different from those of an object consisting of point scatterers, we will still use the same reconstruction algo rithms, i.e., the direct FT method and the back-projection method. However, we will give an alternative interpretation of the images reconstructed by these methods. 3.3 Interpretation of the Reconstructed Images In the following, we assume the lines connecting the reference point to the transmitter and receiver are both perpendicular to the rotational axis. As mentioned in Sec. 3.1, the direct FT method and the back-projection method yield equivalent results. The back-projection method, however, provides more physical insight into the image formation process of microwave diversity imaging. As stated in Sec. 3.1., the two basic procedures in the back-projection algorithm are first to obtain filtered pro jected data and then to back-project the outcome. In the following, we will rename Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 65 - the projected data along constant ranges as the range profile. Mathematically, the range profile at a specific angle is defined as the FT of the frequency response of the range-corrected scattered fields at that aspect angle. Because the scattered fields are measured only over a finite bandwidth and there is a lack of low frequency data, the range profile obtained by a FT of the passband data will be different from that obtained by a FT of the infinite data. Besides, the range profile and the filtered range profile, which is the FT of the product of the range-corrected scattered field and a ramp weighting function, have similar shapes if the ratio of the highest frequency to the lowest frequency is not much greater than 1 as will be elaborated upon below. In the following we will first discuss the physical properties of the range profiles. As mentioned in Sec. 2.4, to a first order approximation, the scattered field of a complex object can be expressed as the linear superposition of the fields scattered from discrete specular points (or scattering centers), fields diffracted from the "visi ble" edges, and fields scattered from the multiple reflection points, creeping waves and traveling waves, etc. The range-corrected scattered field contributed by the scattering centers or spec ular reflections can be written as = (3 -3 2 ) i where is the scattering strength of the i th specular reflection point and is propor tional to the surface curvature at that point. A FT of the above equation with respect to frequency will give peaks at the differential ranges of the scattering centers with amplitudes equal to their scattering strength. Where an edge is the dominant contributor, the range-corrected scattered fields due to the edge have been expressed in Eq. (2.61) and Eq. (2.62) and are repeated here, r Eq = • o f / m r Sin(fct,COS0) sine 1(0) L — - -------— kL cosG . jk L cos0 „ - jk L cos0 = s i n e I (0 ) L -~ —n 2jkLcosO Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3 .3 3 ) -66- E , = ^ s i n 8 M(0H Sin< f C° ^ v kLcosQ _jkL cos0_ _ ,___________ -yAL cos0 = ^ s in 8 M (0 )te (3.34) where both / ( 0) and M(0) are functions of 6 and <(>, where 6 and $ are defined in the edge-fixed coordinate system. The FT of Eq.(3.33) and Eq.(3.34) over finite bandwidth will give two peaks, located at range about ±(L/2)cos0, which are at the differential ranges of the end points of the edge and have an amplitude roughly pro portional to l/(£Lcos6). Where multiple reflection points are the major contributors, the scattered field due to multiple reflections expressed in Eq.(2.84) can be used. The FT of Eq. (2.84) will give peaks at a range equal to the differential ray path length covering the multi ple reflection points. The expressions of the scattered fields due to creeping and traveling waves are more complicated and are dependent on the geometry of the object and measurement arrangement. We do not attempt to discuss these conditions mathematically, but will show their effect through use of examples. From the above analysis, it can be concluded that locations of peaks or dom inant features of the range profiles correspond to the differential ranges of specular points, the end points of visible edges, and the differential ray path length covering the multiple reflection points. The alternative expressions of the corrected back-scattered field using the PO approximation have been shown in Eq. (2.81) and (2.83) and are repeated here, £s _ y * » ■ where j exp(j2lcl )( y S a e w i r ^ 0 , x _ j^A iL L ) a r$ 35) if- and lj- are points at which discontinuities in the derivative of the projected area A (/) exist. dA(l)/dl is also called range reflectivity by radar worker [24]. If the bandwidth available is finite, and the ratio of the highest frequency to the lowest frequency of the measurement system is not much larger than 1, then the denomina tor can be approximated by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where k ^ , k max, and k0 are the minimum, maximum, and central wavenumber of the measurement system respectively. Substituting Eq.(3.36) into Eq.(3.35), the FT of Eq.(3.35) will give peaks at ranges of those points with nonvanishing discontinu ous derivatives. For example, assume the range reflectivity is shown in Fig. 3.2. The scattered field, which is the Fourier transform of the range reflectivity, can first be obtained by successive differentiation of the range reflectivity with respect to range, and then by applying some fundamental theorems to find the FT. From the fundamental Fourier transform theorems, we have the following FT pairs: F ( co) /( /? ) /'(/? ) h - * . j v F ( © ) f " ( R) -« -* --(02 F ( g ) ) We also know the FT of S(R - R a ) is By using the above relationship, we can easily obtain the scattered field, which is -jRi<a P-}R& p-jR*o p-jR*<o E(a)=a' ^j tor +ar—r z- o r - a^~r- o r - a‘Hyo) <3 - 3 7 > where a 2 and a 3 are the slope of segment 2 and 3 respectively. If the ratio of the highest frequency to the lowest frequency of the measurement system is not much greater than 1, then the denominator can be approximated by 1 to 1 CO2 1 _ 1 0.5(comax + (amin) co0 1 Q)2 (3.38) (3.39) By first substituting Eqs. (3.38) and (3.39) into (3.37) and then taking the FT of Eq. (3.37), the estimated target reflectivity (or range profile) is then / ( / ? ) = [ a LS(/?-/?!) + a 2S(/?-/?2) - a 38(R-R3) - a & R - R J ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 68 - 0 differentiation a1 \ f” ! i » i i * « i • i i i i i i i i i 1* -a $ differentiation a2 >i f -a 3 Fig. 3.2 An example of target range reflectivity together with i differentials. successive Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where * represents convolution, AR is the range resolution of the measurement sys tem and is inversely proportional to the bandwidth available. The simple example and analysis above show that the range profile estimated from passband data accentuates discontinuities of the reflectivity function or object scattering function. From this perspective, we can also conclude that the position of the peaks of the range profiles correspond to the differential ranges of those points with discontinuous derivatives in the range reflectivity. All range profiles for all aspect angles can be represented as a sinogram. Sino gram representation has been used in CAT [26] and will be applied to represent the range profiles of various aspect angles. A sinogram is a 2-D intensity varied display with an abscissa of differential range along with the range profile, an ordinate of aspect angle, and intensity or brightness proportional to the magnitude of the range profile. For an isotropic point scatterer, the sinogram is a sinusoidal curve because the peak of the range profile at aspect angle <|> is at a distance equal to rcos(0-<}>), where r is the distance from the point scatterer to the rotation center, 6 is the angle between the x-axis and the line connecting the point scatterer and the rotation center, and <|> is the rotation angle which is the angle between the x-axis and u axis. The peak of the range profile for each aspect angle has the same magnitude in the point scatterer case. However, the sinogram of a conducting object has different proper ties. When the object is rotated or the receiver is moved, the scattering centers will migrate to those points which satisfy Eq. (2.77). For example, the sinogram of a conducting sphere with its center located at the rotation center is a straight line but not a sinusoidal curve, because the symmetry of the sphere makes the range profiles independent of aspect angle. The sinogram of an edge is also different from that of two point scatterers each located at the ends of the edge. Both cases have sinograms that consist of two sinusoidal curves, but the intensity of the curves in the former sinogram span a very large dynamic range because the equivalent scattering strength of the end points of an ecige is highly dependent on aspect angle as explained earlier in Sec. 2.3.3; and scattering from an edge can contain a very strong single peak at Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -70- aspect angle corresponding to broadside. After the sinogram is obtained, a 2-D image can be produced by the technique of back-projection as discussed below. The implementation of the back-projection algorithm should satisfy the follow ing requirements [26,29]: (1). The back-projection line should contribute to those pixels which it intersects and to no others. (2). The contribution of the back-projection line to a pixel must be proportional to the scattering strength. (3). The contributions tc a pixel from different filtered range profiles must add "coherently" such that the bipolar nature of the filtered range profiles is preserved. If only the scattering centers are considered, after back-projection, the image reconstructed can be interpreted as "collections of the projected scattering centers that are visible over the angular window used in data acquisition." To predict what the image will look like over the angular window, one can formulate the equations that describe the target surface, use Eq.(2.77) and Eq.(2.78) to calculate the locations and the scattering strengths of the scattering centers for each aspect angle, and then use back-projection to reconstruct the images. If the surface of the object is smooth, and the rotational angle increment is small, i.e., the angular spacing between adjacent range profiles is small, then the resultant image will be "the projected outline of the scattering centers of the object over the angular window", which usually reflects the projective outline of the shape of the object. The image of an edge of finite length L can also be interpreted through backprojection considerations. To simplify the discussion, only the back-scattering case is considered. For a specific aspect angle, the positions of the peaks in the range profile of an edge are located at the projections of the end points to the line of sight. Amplitudes of the peaks are functions of 0 and <|> and are proportional to sine ( kL cos 0 ), where 0, 0 are defined in the edge-fixed coordinate system. After back-projection, the contributions of this specific range profile to the reconstructed Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -71 image will be two parallel lines oriented in the image plane at the aspect angle. All similar back-projected line pairs for various aspect angles will pass through one of the end points, intensifying the brightness of the end points. However, when the aspect 0 = 90° is within the angular window (i.e., the aspect at which the incident wave is normal to the edge is contained within the angular window) , the backprojection due to this range profile will be a single bright line because Lcos0 =0, and sine ( kL cos 6 ) = 1. The bright line coincides with the location of the edge, and hence an edge will be well portrayed in the final image. The above discussions infer that the microwave image of an edge will be two bright points whose locations correspond to the end points of the edge if the normal aspect angle is not included in the angular window; otherwise a line joining the two points and representing the edge will appear in the image. 3.4 Examples Some examples of reconstructed images and their interpretations are given next. In the following numerical examples, the frequency coverage is from 6 GHz to 16 GHz. (a). Effect of monostatic and bistatic imaging system In the first example, we show the dependence of the image on the data acquisi tion geometry (monostatic and bistatic schemes). Consider an infinite metallic cylinder with radius 20 cm. The cylinder’s axis is parallel to the rotational axis. The transmitting and receiving antennas have identical linear polarization which are paral lel to the axis of the cylinder. The fields scattered by the cylinder are calculated using the formula shown in Sec. 2.3.2. The reconstructed images are obtained by rotating the target over 360° and by changing the bistatic angle between the transmitter and receiver over 360°. The scattering center of a cylinder in the monos tatic case is at a point (actually a vertical line) on the surface where the unit normal vector to the surface is parallel to the directive of the incident wave, while the scattering center in the bistatic case is at a point (a vertical line) of the surface where the unit normal vector parallel to the bisector of the angle between the transmitter Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -72 - and receiver. The bisector is lying in the azimuthal plane. The scattering strength of the scattering center for the bistatic case is proportional to cos (a/2), where a is the bistatic angle [21], Over the specified angular window, projective image of the scattering centers is a circle for the monostatic case and is a semicircle for the bis tatic case. The positions of the scattering centers on the cylinder resulting from object rotation (or equivalently by rotating the transmitter/receiver) and from independent receiver rotation while the transmitter is fixed and the sketches of the back-projection for these two cases are shown in Figs. 3.3(a) and 3.3(b) respectively. The numerically simulated images are shown in Figs. 3.3(c) and 3.3(d). (b). Effect of Angular Window In the second example, we show the effect of the angular window on the recon structed image. The geometry of a hexagonal plate shown in Fig. 3.4(a) is chosen for this purpose. The inclination angle between the plate and the rotational axis is 6 = 85°. The polarizations of the transmitting and receiving antennas chosen are right-hand circularly polarized and left-hand circularly polarized respectively, which means that the image is formed from the co-polarized scattered field. It is assumed that the scattered fields of the hexagonal plate are contributed by the six edges. The field scattered from each edge is calculated using the equivalent current method as described in Sec. 2.3.3. Additionally, the procedures of the calculation were described in Fig. 2.12. The sinogram over angular window from <J>= 0° to <j) = 180° is shown in Fig. 3.4(b). In the sinogram display the bottom line represents the range profile of the first aspect angle while the top line represents the range profile of the last aspect angle. The sinograms are displayed in linear scale. The dynamic display range of the sinogram has been properly chosen so that weak signals will not be overridden by strong signal. It is noted that the sinogram contains several intensityvaried sinusoidal curves. The images reconstructed from angular window 1 ( <|> = 0° to 180° ), window 2 ( 0 = 0° to 110° ), window 3 ( <p = 70° to 180° ), and window 4 ( <|) = 0° to 87 °) are shown in Figs. 3.4(c) to (f) respectively. In window 1, each edge of the plate becomes normal to the line of sight during rotation of the object and therefore all the edges are visible in the image of Fig. 3.4(c). In window 2, edges Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -73ROTATKIG OBJECT ( o r r o t a t i n g T/R) back p r o j e c t i o n T/R T/R T/R T/R (a) bistatic diversity back p r o j e c t i o n T Fig. 3.3 Positions of the scattering centers of a cylinder in its projective image and sketch of back-projection for the case of (a), rotating both the transmitter/receiver; and (b). rotating the receiver only; over 360°. (c). the numerically reconstructed image of case (a), and (d) is the numerically reconstructed image of case (b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -74- 3 and 6 are never normal to the line of sight and therefore are not visible in Fig. 3.4(d), but the end points of the edges remain visible. The image of Figs. 3.4(e) and (0 can be interpreted in a similar manner. (c). Effect of Multiple Reflections and Bistatic Angles In the third example we will show the effect of multiple scattering and the effect of different bistatic angles in the data acquisition system on the images obtained. An object consisting of two infinite separated parallel cylinders is shown in Fig. 3.5(a). The radius of each cylinder is 4 cm, and the distance between the two cylinders is 60 cm. Both the transmitting and receiving antennas are right-hand cir cularly polarized which means that the image is formed from the cross-polarized sys tem. The multiple scattering phenomenon is more pronounced for this polarization case, because the specular reflection alters the state polarization from RHCP to LHCP and vice versa and will be rejected by this antenna arrangement Experimental and numerical results concerning multiple scattering of this structure have been discussed in [13]. Series expansion solutions were used to calculate the scattered fields, and the computing time necessary to generate the field data is very lengthy. However, under the high frequency approximation, multiple reflection points for a given set of aspect angles and bistatic angles can be found by ray optics. The details of the derivation are shown in the Appendix A. When the object is rotated, the differential ray path (relative to the round trip from the rotation center) will also be changed. The plots of differential path length versus rotation angle <(> and the numerically reconstructed images with bistatic angle a = 0° , a = 16°, and a = 40° are shown in Fig. 3.5(c), (d) and (e) respectively. In the bistatic case, there are two sets of multiple reflection points which satisfy the Snell’s reflection law as shown in the Appendix A. The differential path lengths of these two sets are different and are also functions of the bistatic angle, which explains the difference between these plots of differential ray path and the reconstructed images. It is noted that multiple reflections have produced artifact in the image that does not correspond to the physical shape of the object. (d). Effect of Creeping Waves Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. :o id) is) it) Fig. 3.4 (a). Geometry of a hexagonal plate and the measurement arrangement, (b). Sinogram of the hexagonal plate over angular window (0° to 180°). Image recon structed from different angular windows: (c). window 1 (0° , 180°), (d). window 2 (0°, 110°), (e). window 3 (70°, 180°), (f). window 5 (0°, 87°). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Transmitter a 0 d ruiif.i-' i " 6o ) J0 *’ 0 r o t a t i o n angle -k| ) 3 .5 r o t a t i o n angle ,0 *•5 ■* 5 rocacion angle Fig. 3.5 (a). A dual-cylinder object, and plots of the its numerically calculated differential range of multiple reflection path versus rotation angle and simulated images obtained by using different bistatic angles equal to (b). 0°, (c). 16°, (d). 40°. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. >I -77- The theoretical scattered field of a conducting sphere has been expressed in Eq. (2.45). The oscillatory nature in the RCS plot as shown in Fig. 2.6 shows the existence of the creeping waves. In this example, we will examine the spectral dependence of the creeping waves and its effect on the reconstructed images. As stated earlier range profiles usually give more insight into the scattering mechanisms. The location of the peaks in the range profile gives the information about the path traveled by the incident wave during scattering and the magnitude of the peaks in the range profile represents the average scattering strengths of that scattering mechanism encountered by the wave over the specified spectral window. The damped interfer ence pattern of the RCS plot in Fig. 2.6 is an evidence that the contribution of creep ing wave decreases as frequency increases. This fact can also be examined by con sidering the range profiles obtained with different frequency windows. The range profiles of a conducting sphere obtained from different frequency coverages are shown in Fig. 3.6. The first peak of the range profile is at differential range -a from the center, which corresponds to the specular reflection from the front point of the sphere; while the second peak is at a distance of about na/2 from the center, which corresponds to the effect of creeping waves which travel around the shadow part of the sphere before emerging to travel in the direction of the receiver. Examining the height and width of the lobes in the response obtained with different spectral windows, one can have the following observations: 1. The widths of the lobes corresponding to the lower frequency window are broader than those corresponding to the higher frequency window. 2. The height difference between the two peaks increases as the central frequency of the spectral window increases. It is noted that the above figures are plotted on a dB scale. In the microwave regime, the surface curvature of targets of interest is usually large compared to the radar wavelength. One can conclude that the contribution of the creeping wave to the range profile is much smaller than that of the specular reflection and can usually be neglected. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. S ca 45 c 30 15 0 (c) distance in terms o f radius (a) 60 ip h t r e ; k a ^ l 3 3 t o 46 n x > 66 ; d l t p ( 6 6 .3 6 ) 23 T3 C 5) 3 o. 45 30 E a 15 15 ■ 0 j # ik. - 3 - 2 - 1 0 1 A 2 3 distance in terms o f radius (b) (d) Fig. 3.6 Range profiles and reconstructed images of a conducting sphere obtained by various spectral windows, (a), window 1 (ka =0.2 to 13.3) (b). window 2 (ka =26.7 to 40) (c). window 3 (ka =0.2 to 26.7) (d). window 4 (ka=13.3 to 40) -78 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 -79- The reconstructed image of the conducting sphere formed using different spec tral windows are also shown in Fig. 3.6 for an angular window spanning from 0° to 360°. As shown in Example 1, the contribution of a constant range line in the sino gram to the image is a ring. Therefore, the expected image will be two-concentric circles. The inner circle has radius a and the outer circle has radius na/2. However, the outer circle can hardly be seen in the image because of the finite dynamic range of display and film used. From this example, we reach the conclusion that the effect of the creeping waves on the reconstructed image can usually be neglected. (e). Images of a Complex Shaped Object The above numerical results can be related to and used in the interpretation of more complex shaped objects. An experimental example of the reconstructed image of a complex object is given next The test object a metalized 1:100 scale model of a B-52 aircraft with 79 cm wing span and 68 cm long fuselage was mounted on a computer-controlled positioner situated in an anechoic chamber environment. 201 equal frequency steps covering the 6.1 to 17.5 GHz range were used to obtain the frequency response of the object The transmitting and receiving antennas are both right-hand circularly polarized. The sketch of the B-52 airplane and angular, sino gram, and reconstructed image of the test object using data collected in an angular window of 90° extending from head-on to broadside in 128 views is shown in Fig. 3.7(a), (b) and (c) respectively. The intensity or brightness of each pixel is propor tional to the average scattering strength of that portion over the angular window. This fact suggests that microwave diversity imaging can be applied to the study of radar cross section (RCS) management, which will be discussed in the subsequent chapters. Over this angular window, there exist certain aspects when the edges of the right wing and right tail are perpendicular to the bisector of the transmitter and receiver. However, there is no such aspect angle in the window used when the bisec tor is perpendicular to the edges of the left wing and tail. Therefore, the edges of the right wing and right tail can be seen clearly in the reconstructed image, while only the end points of the left wing and tail are visible in the image, as shown in Fig. 3.7(c). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) it f I (b) (c) w Fig. 3.7 (a). Sketch of the B-52 airplane and the angular windows used. (b). Sino gram of the B-52 airplane over an angular window (0° to 90°). (c). Image recon structed over angular window (0° to 90°). (d). Image reconstructed over angular window (18° to 90°). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 81 - If the angular window is reduced from 18° to 90° as shown in Fig. 3.7(a), the reconstructed image becomes as shown in Fig. 3.7(d). Over this reduced angular win dow, there is no specular reflection from the fuselage and engines. This explains the difference between Figs. 3.7(c) and 3.7(d). From the above experimental examples, we can conclude that the microwave images represent a projection image of the "visi ble scattering centers, visible edges when the angular window contains views where the edges are perpendicular to the line of sight, or end points of the visible edges when these edges are never perpendicular to the line of sight". Note for bistatic imaging we can define the line of sight for data acquisition as being along the bisec tor of the angle a between the transmitter and receiver. The above examples show the validity of image understanding and interpretation as applied to microwave diversity images generated under different conditions and serve to illustrate that the image is dependent on the data acquisition parameters (monostatic, bistatic, polarization state, frequency range, and angular window) and hence care must be taken for proper interpretation of the image. 3.5 Discussion The analysis and experimentation described in this chapter is based on far field assumption. If the far field condition is violated, the constant phase line can no longer be approximated by a line perpendicular to the bisector of the angle between the transmitter and receiver, and the Fourier transform relation between the scattered field and object function as shown in Eq. (3.28) or Eq. (3.31) is no longer valid. To overcome the difficulties caused by near field imaging, the matched filtering method has been proposed in the acoustical multi-frequency imaging [16]. This concept has been applied to imaging of rotating objects and near field images can be recon structed by this method [17]. However, the computing time is very time consuming because no fast numerical algorithm can be applied to this problem. Near field acoustic image has also been experimentally studied [60]. All the above discussion is based on the point scatterer model. For a complex shaped conducting body, the expressions of the near fields are more complicated. A thorough theoretical analysis on the near field frequency diversity imaging of metallic objects has not been Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 82 - reported and should prove to be a challenging research topic. This aspect of research is particularly fruitful for microwave nondestructive evaluation of dielectric and com posite structures. Interpretation of the microwave images conducted in this chapter also assumes that the transmitter and receiver both lie in the azimuthal plane perpendicular to the rotational axis. In this situation, the image is a tomographic and projective image. However, if the bisector is not perpendicular to the rotational axis, the image of a point not on the x - y plane will not be a point but an unfocused ring [12,61] if the same reconstruction algorithms are applied. Mathematical analysis can be found in [12]. However, this phenomenon can also be easily predicted from the sinogram representation [12]. Successful interpretation and prediction of the microwave image are fundamen tal to research in several areas, including target identification, classification, and radar cross section management. Its application to radar cross section management studies will be further discussed in the subsequent chapters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 83 - CHAPTER 4 EXTRAPOLATION OF AVAILABLE DATA INTO MISSING BANDS AND ITS APPLICATION TO RADAR IMAGING Fine resolution of object detail in the reconstructed image is necessary to locate hot spots of an object over the prescribed spectral and angular windows employing microwave diversity imaging. It is known that the range resolution of a frequency diversity imaging system is inversely proportional to the bandwidth covered by the measurement system. However, due to limitations of the measurement system or a restriction of the bandwidth allocation, the observed data can lie in a narrow band or multiple restricted spectral regions known as pass bands. Several methods of extra polating the measured data beyond the observed regions have been proposed and tested [62,63,64] in an attempt to achieve complete information in the unrestricted spectral range. In these methods a prior knowledge of the maximum dimension of the object is assumed, and an iterative procedure is applied. The use of linear predic tion for the interpolation and extrapolation of missing data and data gaps has also been reported recently [65]. To increase the resolution obtained from spectral data of such a limited extent, techniques of nonlinear power spectrum estimation have been used with notable suc cess [66]. These include the maximum likelihood method (MLM) [67], the Pisarenko method [68], the Prony’s method [69], multiple signal classification (MUSIC) [70], autoregression (AR) [71], linear prediction (LP) [72], and the maximum entropy method (MEM) [73]. The autoregressive model is a predictive, all-pole model. It yields information about the time series outside the known interval and has applica tions in many diverse problems. The basic idea of the MEM is to choose the spec trum which corresponds to the most random or the most unpredictable time series whose autocorrelation function agrees with a set of known values. This condition is equivalent to an extrapolation of the autocorrelation coefficients of the available time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • 84- series by maximizing the entropy of the process [73]. Linear prediction is used to predict the next sample value in terms of the present and past values, and the linear prediction parameters are found by minimizing the error power [74]. The basic philo sophies of AR, MEM, and LP are different. However, these three methods are equivalent for the case of a stationary Gaussian process [66]. Most nonlinear spectrum estimation techniques are developed to process the data in the time or frequency domain. However, there is an analogy between the timefrequency domain and the space-spatial-frequency domain. In microwave diversity imaging, for a given aspect angle, the frequency response of the scattered fields corresponds to a set of time domain data, while the square of the absolute value of the range profile corresponds to the power spectrum. The linear prediction method is especially suited for those cases when the spec tra are discrete. Under the high frequency condition, it has been shown in chap. 2 that the scattered fields of a complex target can be ascribed to a few discrete scatter ing centers and edge diffractions. Additionally, the locations of the scattering centers and their scattering strengths are independent of the operating frequency for a given transmitter/receiver pair. This is equivalent to saying that the spectra (or range profiles) of the scatterer are also discrete. These phenomena provide the motivation for application of the linear prediction method to microwave diversity imaging. Although the spectra estimated by the MEM or AR can be very sharp and well resolved, this may not be an advantage in a microwave imaging system. If the data are not sampled densely enough in the spectral domain, the sharp well resolved com ponent may be missed, and the results may not faithfully reflect the actual spectral amplitudes. As stated in chapter 3, image reconstruction from microwave diversity imaging systems involves coherent superposition of the data in the spectra, or range profiles, of the scatterer (obtained at different aspect angles), where these are estimated from partial data available in segmented bands. If the estimated amplitudes of the range profiles obtained by the MEM or AR depart from the desired values because of undersampling, image degradation will result. Furthermore, we are interested not only in the magnitude of the range profile, but also in the phase of the range profiles as required in coherent superposition. To overcome the dense Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 85 - sampling requirement and retain the phase information of the range profiles, it may be preferable to extrapolate the data available from the various pass bands into the vacant bands before the spectra or range profiles are formed. An intuitive method of extrapolating the data beyond the observed region is to predict the exterior data by using the same parameters obtained by the linear predic tion model. One of the most popular approaches to linear prediction parameters esti mation with N data samples is the Burg algorithm [75]. For a given number of data samples in a given observation interval, in order to separate the discrete spectra (in this chapter, spectrum is defined as the Fourier transform of the observed data), the required model order in the linear prediction method increases as the separation of spectra decreases. In addition, for a given model order and given number of sampling points, it is easier to distinguish the two close spectra components (scattering centers) by a data set with longer observation interval than that by a data set with shorter observation interval. It was also suggested that the model order should be constrained to no more than half of the number of data points for short data segments so that spurious peaks in the spectral estimate will be avoided [66]. From the above obser vations, one can conclude that it would be more difficult to resolve two closer point targets with short data band than a longer data band. If all the observed data within multiple restricted regions can be fully utilized, better resolution can be expected. In the next section, a new iterative method which uses the Burg algorithm to find the linear prediction parameters and then an iterative procedure to modify these parameters is proposed and tested with both simulated and realistic measured data. With this algorithm, one can obtain acceptable extrapolation beyond the observed region if the spectra are in discrete forms and the separation of the spectra is not too small. Both simulation and experimental results will be presented in Sec. 4.2 to demonstrate the effectiveness of the method in microwave diversity radar imaging. More discussions on this algorithm and others and their applicability to the extrapola tion of data from a finite angular aperture to an exterior angular aperture will be given in Sec. 4.3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -864.1 The New Iterative Algorithm for Extrapolation of Data Available in Multiple Restricted Regions Assume the data sequence available is { X\..... xN }.If one wishes to predict xn on the basis of the previous p samples, theforward linear predictor will have the usual form ~ ~ ^ i^ p k ^ n -k (4 .1 ) *=1 where apk is the prediction coefficient. Denote the forward prediction error for a p th-order linear predictor as epn, then ep n ~ * n ~ = apkx n -k k =0 (4 .2 ) where ap0 = 1. One can also back predict xn_p on the basis of samples xn_p+l,...jcn, i.e., %n-p = ~ ^ i apk*n-p+ k (4 .3 ) *=1 where * denotes the complex conjugate. Denote the backward prediction error for a p-th-order linear predictor as bpn, then bp n — * n - p ~ % n-P ~ ( 4 .4 ) ^Ld ^ p k ^ n - p + k k=0 An approach to linear prediction parameter estimation was introduced by Burg [75]. The linear prediction parameters are obtained by minimizing the sum of the forward and backward prediction error energies ep , defined by A/-1 = 2 n =p „ N-1 1V. 1 + S n - p 1 bpn 1 ’ <4-5) subject to the constraint that the prediction parameters satisfy the Levison recursion relationship [66], Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 87 Qpp —Q p-ijc (4.6) **ppt* p - l,p - k for all orders from 1 to p. If one is going to extrapolate from the available data beyond the observed region, a straight forward way is to use the estimated prediction parameters apk and the measured data in the following equations, %N +j ~ *-j= &pk * N +j - k k=1 J ** 0 » (4.7) £ a %k x - j + k k=1 J >0 , (4.8) where Adenotes the estimated value. If the data available are confined to multiple separate spectral regions or passbands of equal width as illustrated in Fig. 4.1, and one tries to extrapolate from the observed data to the vacant bands, an intuitive method is to divide the inner vacant band into two parts of equal width and to extrapolate into the left part by using the prediction parameters obtained from the data set of region I and extrapolate into the right part by using the model parameters obtained from the data set of region region I n. ^ region II o Fig. 4.1 Available data in multiple regions. Passband (shaded region) surrounded by vacant bands. If the data sequence can be correctly expressed by the prediction parameters, then the extrapolation error, which is defined as the absolute value of the complex difference between the actual values (either computer generated or measured values) and extrapolated values, would be very small. However, if the prediction parameters cannot model the sequence correctly, the error of extrapolation may accumulate. We Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 88 have found that the linear prediction model which characterizes the data sequence is more accurate for longer data strings and larger model orders, especially in the pres ence of noise. However, the model order should not exceed half of the number of the samples because the estimated spectrum will produce spurious peaks [66]. In order to utilize the information available in different regions, a new iterative algorithm using the Burg algorithm to estimate the prediction parameters is proposed. The procedure illustrated in Fig. 4.2 is as follows: 1. Divide the inner vacant band into two parts of equal width. Extrapolate into the left part by using the prediction parameters obtained from the data set of region I and extrapolate into the right part by using the prediction parameters obtained from the data set of region H If the bands are not equal in width, unequal divi sion of the vacant intervening bands may be appropriate. 2. With the "vacant band'V data together with the observed data, use the Burg algorithm to find a new set of prediction parameters. 3. Using this set of prediction parameters and the data of region I, extrapolate into the left part of vacant bands, and using the same set of prediction parameters and the data in region II, extrapolate into the right part of the vacant bands. 4. Using this set of parameters together with the extrapolated data, estimate the data in the observation region I and II. Calculate the error energy between the measured data and the estimated data in the observation regions. The error energy is denoted by E j and is given by: Ei = £ i " * i |2 + '*/ - * 'i I2 = £ \ei \ 2 + \bi I2 , i (4.9) where xt are the measured data, £f are the forward estimation of jq , x \ are the backward estimation of x, , e-t is the forward prediction error, and £>,• is the backward prediction error. 5. With the measured data together with the estimated "vacant band'V data, use the Burg algorithm to find a new set of prediction parameters. From the meas ured data and this new set of prediction parameters, extrapolate the "vacant band'V data as described in step 3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ITERATION ALGORITHM 1. * AR PARAMETERS ESTIMATION Uj}i --------------J--------------- Available data In regions I & II are used eo extrapolate Into regions III & IV. Use data in regions I + II + III + IV to estimate the parameters (aj}i, where 1 represents the iteration number. Use data in I and faj >i to extrapolate Into region III. Use data in II and {aj}i to extrapolate into region IV. 1. Convergence Test a. Use data In III and {aj}l to estimate new data values in region I. Use IV and (3 4 ) 1 to estimate new data values in region II. * « •» # -1 II — Calculate error fii ’ EI + EII For the resulcant data Is in atep 3: If e^ < e j ^ , i+1 ■» i, go from step 3 to 2 otherwise iteration stopped. Fig. 4.2 Schematic diagram of the new iterative extrapolation method proposed. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -90- 6. Use the same procedure of step 4 to calculate the new error energy of the passbands, call it E 2. 7. Compare E \ with E 2, if E 2 is smaller than E t, replace the error energy Ej by E 2 , repeat step 5. 8. If E 2 is greater than E x, stop the iteration, and take the extrapolated data of the previous loop as the final result. In step 1, if the width of a single band (band I and/or band II) is not large enough, the extrapolation errors produced by the prediction parameters obtained from single passband data may be very large, in that case, we can set the data in the vacant bands to zero. The iterative method above can be easily applied to the case where only one single data band is available. The procedures are almost the same except that only one data band sequence is used to extrapolate to the exterior bands and to calculate the extrapolation errors. 4.2. RESULTS In this section, the performance of the new algorithm proposed will be evaluated using both simulated and realistic data. 4.2.1 Range Profiles First, assume for simplicity an object consisting of n point scatterers located at ( rQ + yj ) is illuminated by a plane wave, where r0 is the distance between the transmitter/receiver and a reference point of the object and yj is the differential range of the j th scatterer (range relative to ra). Under the far field condition and ignoring multiple scattering, and considering for simplicity a scalarized field, the range-corrected scalar field can be expressed as E' s (k) = ' Z a j e jllcy> j Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.10) - 91 - In the following simulation, the theoretical values of E's (k) are calculated in 200 equally spaced frequency steps covering the frequency range f { = 6 GHz to /200 = 16 GHz, with signal to noise ratio set to 40 dB. These values anticipate the realistic experimental data utilized in testing the algorithm. Assume the available (computed) data are in the following pass-band (/3 0 >/ so ) ( / 120 i/ 170 )• We want t0 extrapolate the data to the vacant bands ( / l » / 2 9 )• ( / s i >f 119 )> and ( / 171 , / 2oo )• The range resolution obtained by the DFT method using the whole bandwidth ( / 1 ./2 0 0 ) *s about 1.5 cm. The resolution using a single frequency band is about 5.5 cm. The resolution using both frequency bands is about 2.0 cm, however, very high side-lobe level will be produced. We con sider a scattering object with seven point scatterers, the location and scattering strength for each point scatterer are ( r j = - 30 cm, ay = 0.5), ( r 2 = - 20 cm, a 2 = 0.5), ( r 3 = - 10 cm, a 3 = 0.5), ( r 4 = - 2 cm, a 4 = 1), ( r s = 10 cm, a 5 = 0.25), ( r 6 = 20 cm, a 6 = 0.25), ( r 7 = - 30 cm, a 7 = 0.25). The values of the field at each sampled frequency f j are calculated using eq. (4.10). Define the extrapolation error at frequency f j as £ ( / , ) = I E's ( f j ) - E ' s ( f j ) I , (4.11) where E' s( f j ) is the extrapolated value at each f j . The extrapolation errors for different algorithms are compared and shown in Fig. 4.3(a). The bold solid curve is the amplitude of the theoretically computed fields E's ( f j ), the thin solid curves are the extrapolation error after 100 iterations using the algorithm proposed in [63], the dashed curves are obtained by using the Burg algorithm to find the prediction param eters from the respective pass-band, and using this set of parameters together with data in each pass-band to extrapolate to the outside regions (bands HI and IV). The dotted line curves are obtained using this new algorithm with one iteration and with model order 25. The algorithm proposed in [63] involves basically application of the Gerchberg algorithm to data in the multiple restricted regions. However, no numerical or experimental results are given in that paper. It is clear from the results obtained here that the algorithm in [63] seems to be ineffective in the case considered as the errors can exceed the amplitude of the theoretical fields. The extrapolation obtained Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -92- 1 30 80 120 170 200 SAMPLE POINT (o) 1.00 Ul a 3 t Z Si s 0.25 * 75 -75 ?5 O (c) 75 0 75 Id a 3 H Z o < 2 -75 (d) DISTANCE (cm) (e) Fig. 4.3 (a) Magnitude of theoretical fields and comparison of extrapolation errors of different methods, / 1 = 6 GHz, / 2oo = 16 GHz. ===== magnitude of theoretical fields. extrapolation error from a single passband, no iteration. extrapolation errors from new iterative algorithm. extrapolation errors from algorithm proposed in [3]. (b) FFT of the whole band data. (c) FFT of the passband data. (d) FFT of the passband and extrapolated datawith 1 iteration. (e) FFT of the passband and extrapolated datausingalgorithm proposed in [63]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -93 - fro m a sin g le p a ss-b a n d are n o t g o o d in th is e x am p le, b ecau se the m o d el o rd e r is not su fficien t to m o d el th e d a ta se ries in th e p re sen c e o f n oise. T h e n e w m eth o d pro p o se d is se e n to p ro d u c e sm all e rro r a fte r o n e iteration . T h e F o u rie r tra n sfo rm s (F T ) o f th e a ll-b an d d a ta (i.e, d a ta in reg io n I to IV ), p a ss-b a n d d a ta o n ly , p a ss-b a n d p lu s e x tra p o la ted d a ta u sin g th e a lg o rith m in [63] and p a ss-b a n d p lu s e x tra p o la te d d a ta w ith th e n e w m eth o d p ro p o se d a re sh o w n in Figs. 4 .3 (b ) to 4 .3 (e ). T h e se tran sfo rm s y ield th e ra n g e p ro file o f th e scatterin g o b je c t It is c le a r th a t a F T u sin g p a ss-b a n d d a ta o n ly (F ig. 4 .3 (c )) h as v e ry h ig h sid e-lo b e stru ctu re, th e F T o f th e e x tra p o la te d d a ta u sin g th e a lg o rith m in [3] (F ig. 4 .3 (e)) is to ta lly d iffe re n t fro m th e o rig in a l o f F ig . 4 .3 (b ). T h e re su lt o b ta in e d b y F o u rier tra n sfo rm in g th e d a ta g e n e ra te d b y th e p ro p o se d a lg o rith m is sh o w n in F ig . 4 .3 (d ). It e x h ib its e x c e lle n t a g re e m e n t w ith th e re su lt o f F ig. 4 .3 (b ). T h e m ag n itu d es o f the p e a k s in F ig s. 4 .3 (b ) a n d 4 .3 (d ) d e p a rt fro m the o rig in a l a ssig n e d v a lu e s b ecau se o f th e z ero p a d d in g u se d in th e fa st F o u rie r tran sfo rm (F F T ) a lg o rith m . T h is lac k o f fid elity in sc a tterin g stre n g th re co n stru c tio n d o e s n o t h a v e a d isc e rn ib le d eg rad in g e ffe c t on th e q u a lity o f th e im a g e re c o n stru c te d as w ill b e illu stra te d b elo w , b u t is im p o rta n t a n d m u st b e d e a lt w ith w h en a q u a n titativ e a n aly sis o f sc a tterin g stren g th s is n eed ed . th e fre q u e n c y c o v e ra g e is in cre ased to ( / i = 6 GHz, / 200 = 20 GHz) w ith the n u m b er o f sa m p lin g p o in ts b e in g fix ed a t 2 0 0 an d the p a ss-b a n d s are k e p t at ( / 3 0 » / so ) a n d ( / 1 2 0 - f i 7o)> c o m p u ted fields a n d the e x tra p o la tio n e rro rs w ill be as sh o w n in F ig. 4.4 . It is seen th at the ex tra p o la tio n e rro r in d ic ate d by the d ash ed lin e b eco m es sm a lle r d u e to th e in cre ased freq u en cy c o v erag e. I f th e freq u en cy c o v e rag e is d e c re a se d to ( f \ = 6 G H z, /2 0 0 = 12 G H z), th e re su lts w ill be a s sh o w n in F ig. 4.5. It is seen th a t th e e x tra p o la tio n e rro rs in d ic ate d by th e d a sh e d an d do tted cu rv e s are n o w b o th h ig h . T h e F F T s o f the w h o le b an d d ata, th e p a ss-b a n d d a ta only, an d th e e x tra p o la te d p lu s p a ss-b a n d d a ta u sin g this m eth o d are sh o w n in F ig s. 4.5(b) to 4 .5 (d ) re sp ec tiv e ly . T h e re su lts in F igs. 4 .4 an d 4 .5 in d icate th e d esirab ility o f u sin g se g m en ted sp ectral d a ta sp a n n in g w id e r sp ectral ran g es. If Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -94- SAMPLE POINT Fig. 4.4 Magnitude of theoretical fields and comparison of extrapolation errors with and without iteration, / 1 = 6 GHz, / 200 = 20 GHz. — — magnitude of theoretical fields. extrapolation error from respective passband, no iteration. ...............extrapolation errors from new iterative algorithm. Although the above algorithm is an iterative one, it was found that extrapolation errors usually decrease significantly after the first iteration, and further iterations do not seem to improve the results. Therefore, it is practical and usually sufficient to use only one iteration. The performance of the algorithm using realistic data is also evaluated. The test object, a metalized 1:100 scale model of a B-52 aircraft with a 79 cm wing span and 68 cm long fuselage was mounted on a computer-controlled elevation-over-azimuth positioner situated in an anechoic chamber environment. 201 equal frequency steps covering the / t = 6.1 to / 2oi = 17.5 GHz range were used to obtain the frequency response of the object. The target is positioned for a fixed elevation angle of 30° while the azimuth angle was altered between 0° and 90° in steps of 0.7° for a total of 128 angular aspect views. The pass-bands are first defined as ( / 30 , / 80) and ( / 120 >/ no)- The measured values and the extrapolated errors of the broad-side perspective which is 90° from the head-on orientation are shown in Fig. 4.6(a). The solid line curve is the amplitude Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -95- MAGNITUDE 4 3 2 1 0 80 120 SAMPLE POINT 200 170 (a) MAGNITUDE 1.00 0.75 1 - - 0.50 0.25 - j Il 0.00 -75 0 (b) A 75 -75 0 75 -75 DISTANCE (cm) (c) 0 75 (<3) Fig. 4.5 (a) Magnitude of theoretical fields and comparison of extrapolation errors with and without iteration, f \ = 6 GHz, / 2oo = 12 GHz. — -magnitude of theoretical fields. extrapolation error from respective passband, no iteration. extrapolation errors from new iterative algorithm. (b) FFT of the whole band data. (c) FFT of the passband data. (d) FFT of the passband and extrapolated data with 1 iteration. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. -96- of the range-phase corrected field. The dashed curve represents the extrapolation error resulting from extrapolating from each single band (bands I, II) with model order 25 as described in step 1 of the proposed algorithm. The dotted line curves are obtained using the new algorithm with 1 iteration and model order 25. The extrapolation error for measurement is defined in a manner similar to the definition of error in the numerical simulation as the magnitude of the difference between the corrected meas ured fields and the extrapolated fields. The Fourier transforms from the whole band data, the pass-band data only, and the pass-band together with extrapolated data are shown in Figs. 4.6(b), 4.6(c) and 4.6(d) respectively. A Fourier transform of the corrected scattered fields will give the range profile of the target in that view. In this figure, it is seen that the extrapolation errors do not improve after one iteration. The reason stems from the plot of the range profile shown in Fig. 4.6(b). In this view direction, the major contributions to the scattered fields are due to the fuselage and primarily those engines and fuel tanks which are on the illuminated side. Specular scattering from these points are well separated in time or distance and their number is small. Hence the linear prediction parameters obtained from a single pass-band are sufficient to model the data sequence. The extrapolation errors are not as small as those obtained by simulations. The reason for this is that the applicability of the linear prediction model to the extrapolation of scattered fields of a metallic object is based on the high frequency approximation. In the measurement data, however, polarization effects, edge diffraction, multiple scattering and the failure to satisfy the high frequency approximation in the lower region of the frequency band utilized in the measurement will degrade the performance of the algorithm. For the view line #64, which is 45° away from broadside, the measured fields, the extrapolation errors, and the range profiles reconstructed from the whole band data, passband data only, and (passband + extrapolation data ) after one iteration are shown in Fig. 4.7. It is seen that there are many peaks in the range profile and the separation between the the peaks is small. At this aspect, no strong specular points exist. The major contributions to the scattered fields are from the weak edge diffractions. Furthermore, no edge is normal to the bisector at this aspect. The scattering strength due to the other mechanisms is usually a function of frequency. It Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. MAGNITUDE - 97 - SAMPLE POINT (o) MAGNITUDE too 0.75 0.50 0.25 0.00 -45 45 -45 0 45 -45 45 DISTANCE ( c m ) ( b) (c) ( d) Fig. 4.6 Measured Fields, Extrapolation Errors, and Range profiles of view line #1. (a) Magnitude of the measured fields and comparison of extrapolation errors without and with 1 iteration — magnitude of theoretical fields. extrapolation error from respective passband, no iteration. extrapolation errors from new iterative algorithm. (b) FFT of the whole band data. (c) FFT of the passband data. (d) FFT of the passband and extrapolated data with 1 iteration. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ■ ,t. 6.0 J , 8.9 i_________________ 11.8 14.6 17.5 frequency in G H z (a) -65 0 distance in c m 65 -65 (b) 0 65 (c ) d istan ce in c m a. -15 -65 -65 distance in cm (d) (e ) distance in cm Fig. 4.7 Measured Fields, extrapolation errors and range profiles of view line #64. magnitude of theoretical fields. extrapolation error from respective passband, no iteration. extrapolation errors from new iterative algorithm. (a) magnitude of measured fields and comparison of extrapolation errors without and with 1 iteration. (b) FFT of the whole band data (c) FFT of the passband data (d) FFT of the passband and extrapolation data with 1 iteration. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. -9 9 - is known that the linear prediction model or equivalently, the AR model, is an all pole model, which assumes that the observed fields are contributed by discrete sources, and their scattering properties (positions and scattering strength) are indepen dent of frequency. However, the physical realities violate the fundamental assump tions of the model at this aspect. Therefore, small extrapolation errors are not expected from this aspect. Shown in Fig. 4.8 are the measured fields, the extrapolation errors and the range profiles of the view line #85. The front wing is normal to the bisection line in this aspect The discrepancy between the range profiles estimated from the whole-band data (Fig. 4.8(b)) and from the (passband + extrapolated data) (Fig. 4.8(d)) is small, illustrating the effectiveness of the algorithm in this set of data. 4.2.2 Reconstructed Images The reconstructed images of the test object using data collected in an angular window of 90° extending from head-on to broadside in 128 aspects and different fre quency bands are shown in Fig. 4.9. A co-polarized transmitting/receiving system was used. Figure 4.9(a) is obtained by using the whole band data; Fig. 4.9(b) is obtained by using the pass-band data alone. Figures 4.9(c) and 4.9(d) are obtained by extrapolating without iteration and after one iteration respectively. The model order used is M = 25 in both cases. If the pass-band is defined as ( / 65 , / 130), the reconstructed images obtained by using the pass-band data alone and by extrapolation without iteration and after 1 iteration are shown in Figs. 4.9(e), (f), and (g) respectively. The model order used is also M = 25. It is seen that the image resolution of Figs. 4.9(c), (d) and Figs. 4.9(f), (g) has been improved a great deal compared with that of Fig. 4.9(b) and Fig. 4.9(e). The ringing phenomenon appearing in Fig. 4.9(b) is due to the use of multiple restricted bands. For a filled antenna array, the angular resolution of the array is approximately proportional to the total length of the array. If the array is thinned or is empty in some portions, the resolution is still proportional to the total length of the physical Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 100- 15 "3 3 10 5 0 6.0 8.9 14.6 11.8 17.5 frequency in G H z (a) 25 CO •3 C u •a 3 0 distance in cm a■a 65 0>) -65 0 (c ) distance in cm 25 15 ■ua _______ 1 Q. E 3 -5 -15 distance in cm •65 i ih ii (e ) 0 65 distance in cm Fig. 4.8 Measured Fields, extrapolation errors and range profiles of view line #85. ■■1 magnitude of theoretical fields. extrapolation error from respective passband, no iteration. extrapolation errors from new iterative algorithm. (a) magnitude of measured fields and comparison of extrapolation errors and with 1 iteration. (b) FFT of the whole band data (c) FFT of the passband data (d) FFT of the passband and extrapolation data with 1 iteration. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. <e> (f) (g ) Fig. 4.9 R e co n stru c te d im ag es o f th e m etalize d scale m o d el B -52 a ircraft u sin g an a n g u la r w fn d o w o f 9 0 ° e x te n d in g fro m h ead -o n to b ro ad -sid e in 128 lo o k s a n d d iffere n t sp ectral co v erag e. R eco n stru ctio n s from : (a) e n tire b a n d w id th i f x , f z o x ) (b) p a ssb a n d ( / 30 > / so)» ( / 120 • / no)(c) p a ssb a n d ( / 3 0 . / so)> ( / 120 • / no) a n d ex tra p o la tio n d ata (ex trap o lated d a ta in to em p ty ban d s) w ith o u t iteratio n . (d) p a ssb a n d ( / 30 , / 80), ( / 120 , / n o ) a n d e x tra p o la tio n d a ta w ith 1 iteration . (e) p a ssb a n d ( / 65 , f l30). (f) p a ssb a n d ( f 65 , / 130) and ex trap o latio n d a ta w ith o u t iteration . (g) p a ssb a n d ( / 65 , / 130) an d ex trap o latio n d a ta w ith 1 iteratio n . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 102- array, but high sidelobe levels or grating lobes will appear. The image reconstructed from the passband (/65»/i3o) shown in Fig. 4.9(e) looks fuzzy due to the insufficient bandwidth. The reconstructed images without and with 1 iteration as shown in Fig. 4.9(f) and Fig. 4.9(g) look almost the same. This fact indicates that extrapolation from a single band is sufficient to give good images for microwave imaging applications. The above results are obtained using the co-polarized data. When the same extrapolation algorithm is applied to the cross-polarized data, reconstructed images as shown in Fig. 4.10 are produced. These images illustrate the effectiveness of apply ing the extrapolation method to radar imaging regardless of the polarization being used. Fig. 4.10 Reconstructed images of the metallic scale model of a B-52 aircraft using cross-polarized waves. (a) Reconstructed from wholeband. (b) Reconstructed from passbands (f 30, / so) ( f 120 / 170) extrapolation data without iteration. 4.3 Discussion Three parameters can be varied in the iterative extrapolation algorithm. They are: the model order number used in the extrapolation from a single band, the number of iteration, and the model order number used in the iteration procedures. The selection of a suitable combination of these three parameters poses an interesting Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 103- problem as it is difficult to have a rule of thumb. Relevant factors include the pro perty of the scattering source, the width of the passband, and the signal to noise ratio. If the number of discrete sources is large, separation between sources is small, or the SNR ratio is low, then the model order number should be large. If the number of discrete source is small and the separations are far apart, then small model order is sufficient to model the data sequence. A criterion for the error energy has been tested in the iterative algorithm. It was found that the iterations stopped upon an increase in error energy. It was also found that extrapolation errors may not decrease even though the error energy does, and the speed of convergence may be slow. Therefore, it is better to set a maximum number of iterations. Simulations show that extrapolation errors can be much improved after only 1 iteration. Further iterations do not improve the extrapolation error a great deal. Therefore, from the practical point of view, 1 iteration is sufficient to obtain better result. From the reconstructed image shown in Figs. 4.9(c) and (d), u may lead to a conclusion that iterative algorithm is not very helpful in the application of radar imaging or extrapolation of scattered fields into unavailable bands. It has to be pointed out that the iterative method is necessary only when the data sequence can not be modeled properly from the single band. The application of the linear model or all-pole model to the extrapolation of scattered field into unavailable bands is based on the assumption that the scattering properties of the scattering mechanisms are not a function of frequency. This assump tion is valid only when the wavelength is much smaller than the dimension of the tar get. The test target used is a 1:100 scale model B-52 airplane, and the frequency range used is from 6.0 to 17.5 GHz. In a real imaging system, however, the fre quency coverage will not be so broad. For example, the center frequency and bandwidth of a practical radar imaging system can be about 2 GHz and 200 MHz respectively. The scattering mechanisms and their scattering properties will not change a great deal over that frequency coverage. Therefore, the extrapolation algo rithm with or without iteration will probably work better in practice than in this test ing system. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 104- The previous examples show the extrapolation of the scattered field into the exterior bands for a given transmitter and receiver. One may ask if the same algo rithm can be used to extrapolate the complex field over a given angular aperture into the exterior aperture. If the object consists of discrete point sources and each point source radiates isotropically, then the source model can be considered as an all-pole model. One may deduce the field outside the aperture from the known field in the given aperture. However, the all-pole model is not valid for a metallic object over all space, because the locations of the scattering centers and their scattering strength are highly dependent on the aspect angle. The shielding effect of the conductor prevents the information of the object in the shadow region from being detected. Therefore, the application of the linear prediction method for extrapolation of the scattered field in a given aperture to a wide exterior aperture is not recommended. The image quality of the reconstructed images using extrapolation data looks as good as that obtained from the whole band data. However, there is a difficulty in the definition of "good". The quality of an image is subjective. To measure the effect of the extrapolation on the reconstructed image, one may subtract the image obtained from extrapolation data from that of the whole band data, pixel by pixel, using com puter and then compare the difference. However, this kind of measure is not effective because the significant features of an image can be extracted only from the whole picture but not from a single pixel or a single line of the image. Even though the difference obtained by subtracting pixel by pixel may not be small, the images may still look similar. On the contrary, the extrapolation errors in the frequency domain can be defined clearly because they can be compared with the experimental data exactly. The development of a technique which quantitatively measures the extrapola tion algorithm in object space would significantly enhance the analysis of this research. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 105- CHAPTER5 RADAR CROSS SECTION MANAGEMENT STUDIES EMPLOYING MICROWAVE DIVERSITY IMAGING A s mentioned in Chap. 2, radar cross section (RCS) is a measure of the equivalent size of a target as seen by the radar. RCS is a function of aspect angle, frequency, and polarization. For certain applications, it is desirable to enhance the RCS over some specific range of aspect angles. For example, strong return signals are required to track a missile or an aircraft in flight. In contrast, for other applications it is often desirable to reduce or minimize the RCS over specified spectral and angular windows so the target is less likely to be detected. RCS management is a general term for obtaining the RCS of a scattering object by manipulating the distribution and strengths of the hot spots or flare spots of a tar get over prescribed spectral and angular windows and states of polarization. Hot spots represent all those areas that have major contributions to the received scattered fields. There are several ways to determine the site of hot spots. An experienced engineer can roughly point out the possible locations of the hot spots; a computer code can be used to estimate the locations of the scattering centers and their scatter ing strength if the equations that describe the surface of the object can be formulated. However, all such estimates need experimental verification. Microwave diversity imaging systems using angular, wavelength, and polariza tion diversity have been discussed in Chap. 3 and have been used to obtain images of metallic objects with nearly optical resolution. It was stated in Chap. 3 that the pixel intensity or brightness of the reconstructed image is proportional to the mean scatter ing strength at the corresponding position averaged over the specified angular win dow and spectral window. This fact suggests microwave diversity imaging can be employed to determine the hot spots and to study RCS management. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 106 - Two steps are usually involved in the study of RCS management. First the hot spots must be identified. Certain techniques are then utilized to manipulate the hot spots. Fine resolution of object detail in the reconstructed image is essential for locating the hot spots of an object over the prescribed spectral and angular windows and the given states of polarization of the transmitter and receiver. The range resolu tion of a frequency diversity imaging system is proportional to the bandwidth of the measurement system. If the bandwidth is too narrow to give acceptable resolution, the algorithm for the extrapolation of data available in multiple restricted bands or in a single narrow band developed in the previous chapter can be applied to locate the hot spots. Previously, radar workers usually treated the RCS from the detection perspec tive. Therefore, techniques of reducing RCS were developed to reduce the signal to noise ratio (S/N) so that the target can more easily elude detection from radar. How ever, as will be shown later a low S/N is not a sufficient criterion for eluding detec tion when microwave diversity imaging systems are employed where signals from many broad-band sensors are combined to form an image. In this dissertation we will consider therefore the RCS management from the imaging point of view, that is, we consider principles and techniques for making the image unrecognizable by radar. In this chapter we will introduce a new term, "diaphanization", and describe how to use microwave diversity imaging to diaphanize an object. Effects of Gaussian noise on the reconstructed image will be discussed. Some fundamental concepts for distort ing the image will be proposed. 5.1 Diaphanization According to the Webster dictionary, “diaphanous” means: 1. so fine or gauzy in texture as to be transparent or translucent 2. vague or indistinctive, airy. Accordingly, we introduce a new term “diaphanization”, which can be used to describe: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 107- 1. All techniques of reducing RCS in order to render a target invisible to radar detection. 2. All techniques of obscuring the image so that the image is vague, indistinctive or not recognizable by an imaging radar Therefore, diaphanization is a technique which will influence either detection jor image formation and recognition. From the detection perspective, diaphanization will generally lower the target’s signal to noise ratio (S/N), thus decreasing the likelihood of detection [76]. There fore, it is desirable to minimize the RCS. From the imaging point of view, however, low S/N is not a fatal problem for microwave diversity imaging systems. It will be shown later that microwave diversity imaging is very robust to Gaussian noise, even though the noise is stronger than the signal. Therefore, techniques that distort the reconstructed image to make it unrecognizable should be considered. In the follow ing, we will demonstrate how to apply microwave diversity imaging to diaphanize an object and to explain the motivation for studying image distortion. The procedure of utilizing microwave diversity imaging to diaphanize a target for prescribed spectral and angular windows is as follows: 1. Obtain the image of the target before diaphanization. 2. Identify the hot spots which are the bright portions in the image. 3. Use certain techniques to diaphanize the object, i.e. to minimize reflection from the hot spots. 4. Obtain the image of the target following diaphanization. 5. Compare the images before and after diaphanization. An example will be given below to demonstrate the above procedure. In the following experimental example, 201 equal frequency steps covering the 6.0 to 17.5 GHz range were used to obtain the frequency response of the object for different aspect angles. The magnitude of the scattered fields are measured relative to a refer ence cylinder. The RCS defined in this chapter is the square of the measured relative field and therefore the unit of RCS is a relative level but not an absolute value. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 108- measurement is referred to as co-polarized when both the transmitting and receiving antennas have the same sense of circular polarization; and is referred to as crosspolarized when the transmitting antenna and receiving antenna have an opposite sense of circular polarization. The test object, a metalized 1:100 scale model of a B-52 airplane with 79 cm wing span and 68 cm long fuselage was mounted on a computer-controlled positioner situated in an anechoic chamber environment. A sketch of the B-52 and the recon structed image using data collected in an angular window of 90° extending from head-on to broadside in 128 looks with cross-polarized waves are shown in Figs. 5.1(a) and (b) respectively. The intensity or brightness of each pixel is proportional to the average scattering strength of that portion over the angular and spectral windows. The bright spots are the hot spots of the target. To reduce the RCS, we cover the hot spots with broad-band absorbers (Emerson and Cumming AN-72). Portions of the B-52 model covered with broad-band absorber are indicated by the dark lines in the sketch of Fig. 5.1(c). These parts consists of the right hand sides of the fuselage, all edges of the right wing, and the right sides of the two right engines and fuel tank. The RCS of an object is usually reduced after diaphanization, but it is not reduced for every frequency because the scattered field is a result of interference and is a function of frequency. Therefore, it is not proper to measure the effectiveness of a diaphanization on RCS at a specific frequency if a prescribed spectral range is of interest. We have adopted a more reasonable measure — that of comparing the mean scattered fields before and after diaphanization. The mean RCS is obtained by averaging the RCS at each frequency over the prescribed spectral window for a given aspect angle. The mean RCS versus aspect angle before and after diaphanization using cross-polarized waves are shown in Figs. 5.1(d). From these plots, we find that the RCS has been reduced for most aspect angles. The reconstructed image after diaphanization are shown in Figs. 5.1(e). The images before and after diaphanization have the same dynamic display range. One can see that diaphanization reduces the brightness of the hot spots. However, if we lower the dynamic display range of the post diaphanization image, the image becomes as shown in Figs. 5.1(f). Clearly, the nature of the target is still recognizable. From a detection point of view, covering Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 109 - the hot spot portions did reduce the RCS. However, from the imaging perspective, the image is still recognizable. 5.2 RCS Reduction Reduction of a target’s RCS will lower the S/N and therefore decrease the detection probability and increase the false alarm rate [76]. For example, Shown in Fig. 5.2 is the probability of detection for a sine wave in noise as a function of the S/N and the probability of false alarm. Furthermore, the detectable distance of a tar get will be halved if the RCS of the target is reduced by 12 dB, as derived from the radar equation [76]. Therefore, it is desirable to make the RCS as small as possible. The practical question persists, to what extent can the RCS be reduced? The follow ing RCS reduction numerical examples are instructive in the answering of this ques tion. At high frequency, the scattered field at a given aspect can be attributed to several discrete sources or scatterers as stated in Chap. 2. The RCS at a given fre quency is then a ( k ) = I ' Z ' f c e j2kri I'2 = 1 i 1 2 ( 6. 1) i where a,-, r ; , and <{>,■ are the RCS, relative range, and relative phase of the i -th scatterer respectively. It is noted that Eq. (6.1) is a coherent summation. In general, a, and rv are aspect dependent but are independent of frequency or only vary slightly with frequency as discussed in Chap. 2. However, the total RCS fluctuates rapidly with respect to aspect angle and frequency because of the coherent sum of the phase term ejt>i, which is highly frequency and aspect dependent. If the RCS of interest is analyzed over a wide spectral window, it is not proper to compare the RCS before and after diaphanization at a single frequency. It is far better to compare the mean RCS averaged over the window of interest. The phase angles are then equally prob able because of the frequency stepping or sweeping and the mean RCS can be statist ically characterized as the noncoherent sum <*m = Z Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. <6-2) - 110- 45 before amplitude in dB 35 after 25 15 5 128 i i ne number d Fig. 5.1 (caption see next page) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - Ill - ® f Fig. 5.1 (a). Sketch of the B-52 and the angular window. (b). Reconstructed image of B-52 (c). Sketch of the portion (dark area) covered with broad-band absorber. (d). The mean scattered field versus aspect angle before and after diaphanization. (e). The reconstructed image after diaphanization displayed with the same dynamic range as that of (b). (0- Reconstructed image display with a lower dynamic range. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 112- The mean RCS pattern then represents the averaged scattering characteristics of the target as a function of aspect angle averaged over the specified spectral window. The noncoherent sum is helpful in illustrating the extent to which the mean RCS can be reduced. Numerical examples have been given in [33] to show the RCS reduction number game. Assume a simple target consisting of three contributors, a strong one with 200 m 2 in area and the remaining smaller two 20 m 2 in size. Several strategies were considered and the resultant RCS reductions were compared. The effect of reducing the returns of one or all of the contributors, the effect of working harder on the dominant scatterer, and the effect of selective elimination of scatterers are shown in Table 5.1 to 5.3 respectively. If all three contributors have the same amplitude, the effect of reduction is as shown in Table 5.4. The examples above illustrate the increased difficulty associated with reducing the RCS of an object when multiple contributors are present. The total RCS reduc tion can have the expected results only if all the contributors have been reduced effectively. For this reason each decrement in RCS is obtained at a successively higher cost. Techniques of reducing the RCS can be grouped into three classes [21]: cover ing the object with absorbing material or substituting metallic parts with resistive or composite material; shaping the target; and impedance loading. The first method is based on absorption of the incident radiation. The second method is based on redirecting the incident waves, i.e., scattering the incident illumination into as wide a solid angle as possible or into a direction away from the probing system. The third method, also known as passive cancellation, is based upon the concept to introduce an echo source whose amplitude and phase can be adjusted to cancel another echo source [33]. In this paper, we will give examples showing how microwave diversity imaging is used in RCS management and diaphanization studies of a complex object. It is important to emphasize that RCS reduction is a study of compromise in which advantages are balanced against disadvantages [33]. For example, a reduction in RCS at one aspect is usually accompanied by an enchancement at another aspect if Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 113 Tablef-! Effect of a Dominant Scatterer [33] R educe R educe 0t by 10 dB 0 |. 02 by lOdB 20 20 20 20 2 20 20 2 2 U ntreated 200 0i 02 Reduce <72 , 0j by 10 dB 0 \, O} 20 TOTAL 240 60 42 24 dB reduction 0.0 6.0 7.6 10.0 Tablet-2 rooi Effect of Working Harder on the Dominant Scatterer L J R educe a t . 02 R educe ot by 15 dB. 02 by 10 dB. by 13 dB . ot by 10 dB 02 by 10 dB U ntreated R educe at by 15 dB 02 200 20 20 6.3 20 .0 20.0 6.3 2.0 20.0 6.3 2.0 _10 T O TA L 240 46.3 28.3 10.3 0 7.1 9.3 13.7 0\ 02 dB reduction Table f '3 Effects of Selective Elimination ofScatterers ^ J Untreated Ot Elim inate o\ and ot 0\ 200 20 20 0 20 _20_ 200 0 _0 T O TA L 240 40 200 dB reduction 0.0 7.8 0.8 E lim inate o\ O'. Table 5-4 Effect of Reduction When AH Have the Same A mplitude £33] R educe ot by R educe Reduce Ot, 02 01, 02. Ot by 10 dB Untreated 10 dB by 10 dB 20 20 20 2 2 20 TOTAL 1“ 60 2 20 20 42 24 6 dB reduction 0.0 1.6 4.0 10.0 Ot O'. Ot • 2 2 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 114 - the surface of the object is reshaped. The use of absorbing material will increase the weight, volume, and cost of manufacture and maintenance. Therefore, trade offs have to be made according to the requirements for the system design and its mis sions. 5.3 Effect of Gaussian Noise on Microwave Diversity Imaging As shown in Fig. 5.2, a low S/N will decrease the detection probability and increase the false alarm rate. For a practical radar detection system, S/N above 15 db is usually required. It is also known that coherent summation will produce a coherent gain or superposition gain. That is, if a signal contaminated by Gaussian noise are coherently summed n times, the resultant S/N will increase n times [76]. The recon struction algorithm of microwave diversity imaging involves obtaining the filtered range profiles and back-projecting them in the image plane. Both procedures involve coherent summations and hence result in coherent gain. Therefore, the robustness of microwave diversity imaging to Gaussian noise is expected. Microwave diversity imaging usually uses stepped frequency response measure ment For each frequency step, the bandwidth of the receiver is kept the same. The standard deviation of the Gaussian noise at each frequency step is therefore the same. Under the high frequency approximation, at a specific aspect the field scattered from a metallic object can be expressed as the superposition of the scattered fields of N discrete scattering centers such that the scalar fields at frequency f m is given by Es i f m) = E ai exP0 2jt/m ri) i=1 2 (6-3) where a,- is the scattering strength of the / -th scattering center, r-t is the differential range of the i -th scattering center with respect to the reference point, / j and / 2 are the lowest and highest frequencies of the measurement system respectively, Np is the number of frequency points. Assume the noise power level of the receiver at each frequency step is N, then the S/N due to the i -th point scatter is Ia,12 / N . Assume the point scatterers are well separated apart so that at range rt the contributions due to the side-lobe of other scatterers can be neglected. The signal (sampled value at r,- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 115- Q 2 9 0 0 0 0 0 0.9999 0.9995 0.999 0.998 0.995 0.99 0.98 | 0.95 U I 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 Probability ot to n * alarm — 10“ 1or1 lO^Kr'iaW' 0.05 4 6 8 10 12 14 (V * ),, jignal-to-noist ratio. 48 16 18 20 Fig. 5.2 Probability of detection for a sine wave in noise as a function of the S/S and the probability of false alarm. in the FT domain or range profile) to noise ratio will become Np Ia,-12 /N since the Fourier transform is a process of coherent summation. Therefore Np is the coherent gain or superposition gain for this example. Back-projection is a process of coherent summation of the contributions of the back-projection lines obtained from the filtered range profile of each aspect angle. This additional coherent process will make the reconstructed image more robust to the Gaussian noise. If the object consists of isotropic point scatterers, i.e., the scat tered fields of these point scatterers are isotropically distributed, then the overall coherent gain will be A!aNp if data from Na aspect angles are used to reconstruct the image. In practice for a metallic object for example, the locations and scattering strengths of the scattering centers and end points of edges will change or migrate Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 116 - when the object changes aspect (rotation) during data acquisition. Therefore, the sig nal (image intensity at a pixel) to noise ratio in the image space cannot be expressed in the above simple form. The following example illustrates the effect of Gaussian noise on the recon structed image. Figure. 5.3(a) shows the mean RCS after diaphanization of the B-52 scale model versus aspect angle. We used the computer to generate Gaussian noise with two lev els N i and N 2 (see Fig. 5.3(a)) and add those to the measured fields to get two sets of noisy data. The corresponding S/N levels of N i and N 2 316 about 3 dB and -2 dB respectively relative to the mean RCS. The frequency response of the first aspect angle (broadside view) without and with noise levels N j and N 2 are shown in Figs. 5.3(b), (c), and (d) respectively. The range profiles corresponding to the data in Figs. 5.3(b), (c), and (d) are shown in Figs. 5.3(e), (f)» and (g) respectively. If we stack the range profiles of the first 64 view angles together in a perspective representation, the range profiles without and with the two levels of Gaussian noise will appear as shown in Figs. 5.3(h), (i), and (j). The images reconstructed from these three sets of range profiles are shown in Figs. 5.3(k), (m) and (n) respectively. The results presented in Fig. 5.3 show clearly that Gaussian noise affects the frequency response most, the range profile less, and the reconstructed image least because of coherent or superposition gain. 5.4 Fundamental Concepts for Distorting an Image From the above examples, it can be seen that low S/N or strong Gaussian noise is not a sufficient criterion for totally obscuring the images. In order to develop tech niques for making the image unintelligible or obscured, one must resort to under standing of the scattering mechanism and the reconstruction algorithm involved. Radiation can originate from several locations on a scatterer. These include excitation regions, impedance loads, sharp bends, and open ends [38]. The finite passband data and finite range resolution make the range profile accentuate the discontinuities in reflectivity function, and the image can be interpreted as a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 117 - 20 amplitude in dB no i s e l ev e l N 15 10 n o i s e l ev e l N, 5 0 1 32 64 96 128 1ine number a Fig. 5.3 (caption see page 121) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.0 8.9 11.2 14.6 17.5 FREQUENCY IN GHz 20 LU O Z> h- 6.0 8.9 11.2 14.6 17.5 FREQUENCY IN GHz c I 8.9 11.2 14.6 17.5 FREQUENCY IN GHz Fig. 5.3 (caption see page 121) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 119 - MAGNITUDE 4.0 3.0 1.0 -130 -6 5 0 65 130 DISTANCE IN cm MAGNITUDE 4.0 — 1----- ‘i 0 65 3.0 2.0 1.0 -130 4.0 MAGNITUDE 1— -65 130 DISTANCE IN cm " ■■■ r - - —r — .... . r 3.0 2.0 1.0 0 -130 -65 0 65 130 DISTANCE IN cm s Fig. 5.3 (caption see page 121) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 120 - 2 . 2e*0l •€.38e+0i -3 .3 2 e» C i‘ 8.0:j' 3 .3 * * 0 t . 35e*vl' " 3 . 32e*0l* C. 0*+00 j Fig. 5.3 (caption see page 121) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 5.3 (a). Averaged scattered fields of the scale model B-52 after diaphani zation versus aspect angle and the generated Gaussian noises with level iVj and N 2. Frequency response of the first view angle (broadside) (b). without adding noise, (c). with noise level N h and (d) with noise level N 2. (e). The range profile of (b); (f). The range profile of (c); (g). the range profile of (d); The stacked range profiles of the first 64 view angles (h) without adding noise; (i) with noise level N \; and (j) with noise level N 2. The reconstructed image (k) without adding noise; (m) with noise level and (n) with noise level N 2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 122 - collection o f the discontinuities of reflectivity function [5]. The reconstruction algo rithm followed by include steps: obtaining filtered range profiles followed by backprojection. Both steps as pointed out earlier are coherent summation processes. Based on these observations we will propose some fundamental concepts or rules for obscuring the image. 1. Creation of artificial discontinuities: The image is a collection of the discon tinuities of the reflectivity function. For a set of given objects, their images can be obtained and stored to act as the data base for subsequent recognition pur pose. If unexpected discontinuities were created, the reconstructed images of the target will be different from their original ones. In that case, visual recognition will be more difficult or may fail. 2. Creation of multiple reflections: Locations of peaks in the range profile due to multiple reflections correspond to the differential ray path between the multiple scattering points on the object. Such peaks correspond to as physical detail on the object and help therefore to distort and obscure information about object. If the object is shaped so that multiple reflections are deliberately produced to be the dominant contributions to the scattered fields, then the image might be obscured or distorted. 3. Manipulation of local reflectivity: Here we can make the reflectivity at a given point a function of aspect angle and frequency or randomly modulate it in time. The contributions to a pixel from different filtered range profiles obtained at different aspect angles are added coherently. If the reflectivities of a point obtained at different aspect angles have the same phase, that image point will be intensified after back-projection. If the phase of reflectivity at that point is ran domly varied for each frequency and aspect angle, the range profile might pro duce random peaks and the peaks in the range profile corresponding to those reflection points might become smaller because their reflectivities at each fre quency is not in phase. After back-projection, the intensity at that image point might become smaller and the image might be contaminated by random noise. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 123 - In the following chapters we will apply the above ideas to study some diaphani zation techniques which might be useful in reducing the RCS or distorting the images and making them intelligible. 5.5 Discussions There are three hierarchies in high resolution imaging radar. By frequency diversity, one can obtain the range profile, which gives the range information of the hot spots at that aspect. However, there are ambiguities in determining the hot spot positions in the directions of the cross-range and the rotational axis. By applying angular diversity in the azimuthal direction, the positions of the hot spots in the direction of cross-range can be located. However, the images obtained are projective images. The locations of the hot spots in the direction of the rotational axis are still ambiguous. If another angular diversity in the elevation direction is employed, by applying the weighted projection theorems [9], a 3-D tomographic image can be obtained and the hot spots can be pinpointed more accurately. Frequency swept tomographic imaging of 3-D perfectly conducting objects has been demonstrated by computer simulation [9] and experiment [60]. It was stated that by multiplying the 3-D space data by a weighting function, a tomographic image at any selective plane can be obtained. The analysis is based on the the PO approximation and is valid for point-scatterer objects. However, a more convincing interpretations for 3-D imaging of a complex shaped metallic object needs to be developed. The RCS of a target at a certain frequency can be very small due to the coherent interference of the scattering centers. In that situation, the detectability of the target will be very small. But its RCS may not be small at many other frequen cies. The detection probability and false alarm shown in Fig. 5.2 was analyzed based on a sinusoidal wave at a single frequency. If frequency diversity is employed, The modification methods for detection criterion utilizing the frequency diversity proper ties so that the radar is more powerful in detection are worth investigating. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 124 - CHAPTER 6 DIAPHANIZATION BY ABSORBER COVERING AND TARGET SHAPING A s defined in the previous chapter, diaphanization refers to techniques of reducing RCS and techniques of obscuring an image. In this chapter, we will consider and dis cuss RCS modification by covering the target with absorbing iayers and by target shaping. Metallic bodies are good reflectors and scatterers of electromagnetic waves. However, high reflectivity is not a desirable property in some applications. For exam ple, the large metallic structure on ships or airplanes makes them easily detected by radar, masts, beams, towers, hangers, and other shipboard structures are not only obs tacles to the functioning of communication systems, but can also be sources of interference, which may degrade the performance of shipboard communication gear and distort antenna radiation patterns. Application of radar absorbing material (RAM) to metallic bodies can reduce the RCS and eliminate the undesired interfer ence by partially absorbing incident EM energy, suppressing thereby reflections and scattering. To reduce the RCS of a target with a coating of RAM is, in effect, to reduce the reflection coefficient of the air-RAM interface. Single-layered and multilayered coated surfaces have been analyzed using transmission line modeling [21], and will be summarized in Sec. 6.1. But this approach fails to predict the correct field over a wide range of aspect angles, since the transmission line method does not consider diffraction phenomena. Scattering from resistive sheets and resistive strips has also been reported in the literature [76], a high frequency solution for diffraction by a strip with two arbitrary surface impedances when illuminated at edge-on incidence is obtained by a spectral extension of the geometrical theory of diffraction [78]. A solu tion for the problem of EM scattering by a half pilane with two different surface Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 125 - impedances for both normal and oblique incidence was developed [79]; an asymptotic high frequency estimation of monostatic RCS of a finite planar metallic structure coated with lossy dielectric was made theoretically and compared with experimental results [80]. In this chapter, we do not attempt to numerically calculate scattering from a plate covered with RAM using the above developed formulations. The electri cal properties ( e, ( i ) of the RAM used differ from the theoretical assumptions of the cited literatures. Therefore, we will concentrate more on experimental studies. RAMs can be divided into four groups: magnetic material absorbers, resonant absorbers, broad-band absorbers, and surface current absorbers [81]. In Sec. 6.2, we will briefly discuss the properties of these four RAMs. Experimental results of the RCS and images of metallic objects before and after covering by RAM will be presented and discussed in Sec. 6.3. Target shaping is another mean for reducing RCS. The basic notion of target shaping is different from covering by absorbing material in that it is nonabsorptive. A commonly observed phenomenon is any attempt to reduce RCS by a nonabsorptive method over a range of aspect angles is promptly and assuredly met by an enhance ment of the RCS over some other range of aspect angles. The above phenomenon is sometimes considered as the "radar cross section conservation" principle, although it is not a principle in the strict sense of the word [82]. In some applications, only a certain range of aspect angles is of interest. Therefore, the object is designed to minimize the RCS within the angular window of interest. To investigate the effect of target shaping on the RCS, it is helpful to compare the RCS of simple objects over prescribed angular and spectral windows. The RCS of some simple objects have been analyzed in [21]. Radar cross section reduction using cylindrical segment has been reported [82]. In Sec. 6.4, we will compare the RCS for objects of various shapes and state the basic rules for target shaping con siderations. We have concluded in Chap. 3 that the reconstructed image represents "discon tinuities in the reflectivity function and its derivatives." If we coat the target surface inhomogeneously, i.e, we coat the surface only in certain areas, then we creat Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 126- artificial discontinuities in the reflectivity function. These discontinuities can be reflected in the reconstructed image. If the boundaries of the covered regions and conductors are suitably shaped, the total scattered field may be more effectively reduced. In Sec. 6.5, we will examine how the combination of absorbing material covering and shaping affects RCS and imaging. 6.1 Absorber-Covered Bodies - Theoretical Considerations Consider a metallic object covered with a homogeneous absorbing material with complex permittivity E2, permeability (J.2, and illuminated by a source 7q as shown in Fig. 6.1. Vm and Sm denote the volume and the boundary surface of the conductor, Va and Sa denote the volume and the outer surface of the absorber respectively, and V0 denotes the volume of the flree space. The general formulation of Maxwell equa tions for this structure lead to V0 To f Fig. 6.1 Geometry of an absorber covered body illuminated by a plane wave. ( V2 + Icq ) (r*) = T0(f) ( V2 + k l ) & z ( f) = 0 where k§ = (D^Eo, k \ = w2M-2e2>^1 space, for 7* in free space (6.1) for r e Va (6.2) is the total electric field at point r* in free is the total field at point r* inside the absorbing material. The solution Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 127 - must satisfy the following boundary conditions nm( t ) x ^ 2(f) ~ 0 for r*on Sm na(r)i<h? 2 ~ (n) = 0 t on SA na(r)x(!Pl ( t ) - i t t2(t)) = 0 to n SA na ( t ) i \ h A ~ H i# i (/*)) = 0 t o n SA na<?)x(A(?) - (r*)) = 0 t o n SA where nm (t) is the unit vector normal to the surface Sm at t , and ha (t) is the unit vector normal to the surface Sa at t . In the absence of the metallic object and absorbing material, that is, the whole space being free space, the solution of Maxwell’s equations is the incident wave and is denoted by E?(t). The scattered field from the absorber covered body is then defined as 2*0*) = ^ < ? )-& (? ) (6.3) In general, the exact solutions of (t) are not available except for those boundary surfaces which coincide with the surfaces of curvilinear coordinates. Exact solutions of the scattered field from a coated sphere and coated infinite cylinder have been analyzed in [21]. In the following, we will review the properties of a semi infinite plate covered with RAM and then use the PO approximation to study the absorber covered body. 6.1.1 Semi-infinite Plate The scattering problem of a plane wave incident at an arbitrary angle on a medium which consists of an arbitrary number of plane layers, each of which is homogeneous and has specific dielectric properties, has been analyzed in [21]. A sin gle homogeneous layer which covers an infinite conducting plate in otherwise free space is a special case and its geometry is shown in Fig. 6.2. In this figure, Gj is the angle of incidence and reflection, and (J.2 are the complex permittivity and com plex permeability of the layer respectively, and d is the thickness of the layer. By using the plane-wave and transmission line analogy or by applying suitable boundary Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 128 - conditions, the reflection coefficients Rp and Rv can be shown to be [21] a = Fig. 6.2 Geometry of a semi-infinite conducting plate covered with a homogeneous plane layer. R _ K _ ~Hy -T1o-yz!tanK2d H‘ T|0-yz§ tamc2d p /\y _ E; _ -r\Q-jzv2vmK2d ™^ EJ where tj0 = ViVeo ^ (6. 5) 710-yz$tanK2d characteristic impedance of free space, k2 = *s the complex wave number of the layer, the subscripts p and v represent parallel and per pendicular polarizations respectively, z2 and z£ are the equivalent complex charac teristic impedance of the layer and are defined by z v2 = — a / ^ " = ^ l/e2 cos02 \ e2 V 1 - (MoEo/^e^sin^! (6.6) for an incident electric field vector perpendicular to the plane of incidence, and z \ = cose2V f = V lV ^V l - ((ioeo/M-2e2)sin20i (6.7) for an incident electric field vector lying in the plane of incidence. 02 in Eqs. (6.6) and (6.7) is the refraction angle in the layer (medium 2 in Fig. 6.2). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 129- For a lossy material, the permeability and permittivity are complex and are given by M-2= 1*2 + iV-2 (6.8) e2 = e2' + ye2" (6.9) If the operating frequency / , incident angle 8^ permittivity e* permeability and thickness d of the absorbing layer are given, the reflection coefficient can be deter mined. It is noted that the reflection coefficients for vertical polarization and horizon tal polarization are generally different at oblique incidence. Therefore, the reflected wave of a circularly polarized incident wave is, in general, ellipdcally polarized. Shown in Fig. 6.3 are examples of the magnitude and phase of R v and R p versus frequency for a given set of parameters with = 1 + y'0.4, p.2 = 1 + y'O, 9t = 0° (normal incidence), and thickness d=0.8 cm. R v is equal to R p at each fre quency for this special case of 9i=0°. For the other incidence angles, IR p I is in general not equal to l/?wI if the layer is lossy. Shown in Figs. 6.4(a) and (b) are the magnitude and phase of R v and R p versus the incidence angle with parameters e2 = 1 +j 1 .0 ,112 = 1 + yO, /= 1 1 GHz, and d - 0.6 cm. The parameters chosen are to simulate the broadband absorber to be used later. It is seen that the magnitude and phase of l/?v I are not equal to those of \RP I for every incident angle. The magni tude and phase of the reflection coefficients for the co-polarized waves and crosspolarized waves with the same parameters are shown in Figs. 6.4(c) and (d) respec tively. 6.1.2 Physical Optics Approach for an Absorber-Covered Metallic Object The reflection coefficient derived in the previous sub-section is valid only when the plate is infinite and when the receiver is in the forward direction, i.e., the normal of the plate is in the direction bisecting the angle between the transmitter and receiver. The problem of an arbitrarily shaped metallic object covered with an absorber is very difficult to analyze. However, if the radii of curvature of the object is large compared to a wavelength, the physical optics approximation may be used to analyze the problem. m ---------------Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 130- logl/?v l =logl/?* 0.75 log scale U TJ 0.50 c. I -30: ■ 0.25 linear scale J O .O -40 6.0 8.9 11.8 14.7 17.5 frequency in GHz (a) 180 eo <u TS e 93 t/j 2 "a. -90 -180 6.0 8.9 14.7 11.8 frequency in GHz 17.5 (b) Fig. 6.3 Reflection coefficients Rp and Rv of a semi-infinite plate covered with an absorber layer versus frequency The parameters used are e2 = 1 + y 1, ji2 = 1 + y'0, 0 1 = 0° (normal incidence), and 4=0.6cm. (a). Magnitude of Rv and Rp in linear scale and in log scale. (b). Phase of Rv and Rp . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 •88 -44 0 44 incident angle in degree (a) 88 180 SO lR -180 0 -88 44- 88 incident angle in degree (b) 0.75 ■38 .B 0.50 K , IRa l -88 0 88 44 incident angle in degree 180 i -90 ■ -180 L . -88 , -44 0 . 4 4 incident angle in degree — 88 Id) Fig. 6.4 M agnitude and phase o f Rv and Rp versus incident angle. (a). M agn itu d e o f R v and R n . (b). Phase o f Rv and Rp . (c). M agn itu d e o f reflection c o e ffic ien t for c o -p o la r ize d and c r o ss-p o la rized w a v es. (d ). P h ase o f reflectio n c o e ffic ien t for c o -p o la r iz e d and c r o ss-p o la rized w a v es. T h e param eters u sed are: e : = 1 -i- ; 1 . |4; = 1 + ; 0 , / = U G H z. and d=0.6cm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 132 When an absorber-covered body is illuminated by a plane wave, a field will be induced inside the absorber. If only the field external to the absorber is of interest, the original structure can be replaced by an equivalent structure through the equivalence principle [45]. The two equivalent structures are shown in Fig. 6.5, where the internal field has been set to zero, and an equivalent electric surface current density Js and an equivalent magnetic surface current density Ms have been applied to the surface SA. To support the original field outside the absorber, Le., to make these two structures equivalent, die equivalent current density must satisfy the relations (6.10) Ts = n x l F and (6.11) where n is the outward normal, and I F and FF are the total original fields over S q. The equivalence principle does not help us in solving the problem, because eF and FF remain unknown. It only helps us to view the problem from another per spective. However, if IF and FF can be obtained or approximated through other reasonable assumptions, then the equivalence principle will be a very valuable con cept. If we assume that the reflection on the surface takes place as if the point of reflection lies on a fictitious infinite tangent plane, then the total field at a point r* on the boundary surface SA could be expressed as £^(7=*) = £*> *)+ £*(?*) = t T* F ) + ? Ty<F) + '£rv:<F) + £ rTM<F) (6.12) i F ( T ) = t f 'c n + f f r <F) = /T * ^ ) + + #'*(/'•') + FT™ {t*) (6.13) where the superscripts i and r represent the incident and reflected waves respec tively, and the superscripts TE and TM represent the TE and TM components respectively. The incident field and the reflected field on the boundary surface can Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 133- F- Eq '• ? = ri x f f ‘ S* l? ,= ? x n Fig. 6.5 The two equivalent structures of an absorber covered body illuminated by a plane wave. be related to the reflection coefficients Rp and Rv and are defined as F 7* = R vg iTE (6.14) s 'x F ™ =Rp s ix ? TU (6.15) where s l and s T are the unit propagation vectors of the incident wave and reflected wave respectively. Rv and Rp are obtained as if the surface at the reflection point was an infinite tangential plane. Their expressions are shown in Eqs. (6.4) and (6.5). It is noted that Rp and Rv are functions of frequency, angle of incidence, absorber’s composition and the object’s geometry. If the surface of the absorbercovered body and its equivalent surface impedance can be mathematically described, the reflection coefficients Rp and Rv and accordingly the total fields ET, H T and the equivalent current densities Ts , can then be determined. The back-scattered fields can finally be calculated by Eqs. (2.23) and (2.24). Without loss in generality, consider an incident field propagating in the z direc tion and describe the surface of the absorber-covered body by the equation z' = f ( x ' , y ' ) . The geometry is the same as that shown in Fig. 2.3 with the notable difference of the absorber covered surface substituted for conducting surface. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 134- Expressions for the surface’s normal vector n and the surface element area ds' are shown in Eqs. (2.36) and (2.37) respectively. The expressions of the back-scattered field for various incident field polarizations can be found in [21]. For the special case of an x -polarized incident wave, that is, the back- scattered field is given by [21] . (d fld x 'j1R. - 0 f / d y ' ) % , 0 f / t e ) 2 + 0 // 3 y ') 2 (Rp+Rv)(dfldx'Xdf/dyf ) , ■ej2kof dx'dy' @ f/dx')2 + (df/d y')2 (6.16) -Wo =■ £= = [0^+ 0^] V4jt/?o (6.17) Expressions of Rp and Rv are functions of the incidenceangle6;, which expressed in terms of the integration variables x' and y' asgivenbelow cos0: = n z = sin9 m m [21] . * ---------- .■ Vl H d f/dxf)2 + 0 //d y ') 2 m M can be (6.18) m V l+@f fix ')2 + (d fldy')2 It is noted that the back-scattered field is depolarized because in general a 12 does not vanish. Eq. (6.16) can also be expressed as i Sui + f j g 2< / S S ) e ™ dx'dy' Sm ( 6 .20 ) Let’s elaborate the first term of the above equation and rewrite it as B = j g l ( x ' y j ) e j2kaf dx'dy' Sm = j g x(x' y / )ei7k* (n i)ds' Sm Define Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (6.21) - 135- A '(z') = \g \(x ' ,y\z') ctfdty' Sm (6.22) where A '(z') is the projection of the weighted surface area for z > z' onto the x ' -y ' plane. Eq. (6.21) can then be rewritten as B = f ex p t-y ^o z') — ~ — dz' o °z (6.23) By comparing Eq. (6.23) and Eq. (2.40), one notes that the single difference is the definition of s' (z' ) and A' (z'). The former is the projection of the surface area while the latter is the projection of the weighted surfacearea. It has been pointed out that Eq. (2.40) canberewritten as Eq. (2.83). Similarly, Eq. (6.23) can also be written as N ;=i J J d Z L exp0.2M ;.) ( ^ n=i m r ' ) . W a/;- i ) a/;- (6.24) where Z;+ and Z- are points at which discontinuities in the A '(z') or its derivatives exist. The reflection coefficient of a metallic object is always -1 for all frequencies. The range profile of a metallic object renders range information for those points at which s' (z') and its derivatives are discontinuous. These discontinuities usually give information about the object’s shape. However, the range profile of an absorbercovered body is more complicated. No simple statement is sufficient to describe the properties of that range profile because the absorber-covered surface reflection coefficient is a function of frequency as well as a function of the constitutive parame ters of the absorber. However, if we assume that multiple reflections between the absorber cover and the conductor’s surface can be neglected and the thickness of the absorber is much smaller than the range resolution of the imaging system, we can obtain reflection coefficients varying slowly with respect to frequency. In that case the absorber-covered body’s range profile will have peaks at those ranges where A '(z') and its derivatives are discontinuous. It is noted that A'(z') can be decreased by absorber claddings which reduce the surface reflection coefficients. Furthermore, discontinuities in A'(z') can be artificially created by use of absorbing materials at Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 136 - unexpected regions of the metallic surface. These created discontinuities may distort the object’s shape information and deceive visual recognition systems. The above analysis gives theoretical background for the RCS reduction and image distortion by absorber covering. 6.2 Radar Absorbing Materials (RAM) Radar absorbing materials (RAMs) are those materials which can absorb the incident radar wave energy. RAM was first developed by Germans during the latter part of World War II to overcome radar detection of their submarines. After 40 years of development, many different types of RAM have been fabricated. RAMs are used in a wide variety of applications, examples are: improvement of antenna patterns, reduction of undesirable reflections from objects and devices, achievement of reflectionless environments (anechoic chamber), and reduction of radar cross section. Basically, RAM can be divided into four groups [81]: magnetic absorbers, resonant absorbers, broadband absorbers, and surface current absorbers. In the following we will briefly describe each of these. A. Magnetic Absorber In magnetic absorbers, the value of the complex magnetic permeability is in the same range as the complex permittivity. Therefore, the characteristic impedance of these media is close to that of free space, and reflection from the absorber’s surface will be small. If the thickness is great enough to attenuate the incident wave to a significant degree.before it reaches a metallic surface, the overall reflection will be small. Examples of this group are ferrites. The reflection coefficient is reduced with these material from 10 to 25 dB in the frequency range of 30 MHz to 1000 MHz [81]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 137 - B. Resonant Absorbers At microwave frequencies, most materials have higher permittivity than permea bility in magnitude. The characteristic impedance of the materials is therefore smaller than that of the free space. The difference in characteristic impedance causes a high reflection from the air / absorbers interface. However, it is possible to cancel the reflection by suitably introducing another reflection from the interface of the backside of the absorbers and the object. RAMs have been classically regarded as resonant when they are a quarter-wavelength thick. However, the actual absorber thickness used to minimize the reflectance of a material is rarely a quarter wavelength. If the material is dominantly magnetic in character, the required thick ness for minimum reflection may even exceed a half wavelength [83]. C. Broadband Absorbers The basic concept of broadband absorbers is to devise a material which has |ir = er at the front surface and has thickness great enough to attenuate the reflection from the back interface. Ferrite can accomplish this in the VHF and lower UHF range. The other concept is accomplished by varying the impedance of an absorber gradually from its incidence surface to a lower impedance at its rear surface. The dielectric constant is very low at the front surface and increased to very high at the rear. Therefore, the front face reflection is greatly reduced and the required thickness is also reduced. Broadband absorbers are divided into two types: the flat-faced type and the geometrically shaped type. The first type is usually made up of stacked layers, while the second type achieves the impedance gradient by a geometrical tran sition from free space into a lossy media. The geometrically shaped type usually takes the form of carbon-loaded, flexible foam pyramids or wedges. This is the kind of absorber widely used in the construction of anechoic chambers. D. Surface Current Absorbers In contrast to the three groups above, which are usually designed for reducing specular reflection or main-lobe reflection, surface current absorbers are designed to attenuate non-specular or side-lobe energy. Non-specular returns occur when currents Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 138 - encounter discontinuities, such as gaps, edges, or comers. Surface current absorbers attenuate the currents before they encounter such discontinuities. Analytical method for predicting surface current absorber performance are based on the dielectric slab guided wave analysis [33]. The surface impedance is the key variable in relating the performance to the material configuration. Surface current absorbers can be in the form of a thin sheet, or a paint-on version. They usually have a high permeability. 6.3 Experimental Results for Absorber-Covered Bodies As an aid for presenting our results, some common measurement notations will be defined first. The transmitting antenna used in the following experiments is a righthand circularly polarized antenna. The measurement of the scattered field will be referred to as co-polarized if the receiving antenna is lefthand circularly polarized; and is referred to as cross-polarized if the receiving antenna is righthand circularly polarized. The RCS of an object treated in this chapter is either the RCS at a certain frequency or the mean RCS averaged over a specified spectral band. Three spectral bands are studied in this chapters. They are : band 1 ( 6 GHz to 11.25 GHz), band 2 (11.25 GHz to 16.5 GHz), and band 3 (6 GHz to 16.5 GHz). The calibration pro cedure stated at the end of Sec. 2.6 has been used to obtain the object’s absolute RCS at each frequency. The RCS of a given conducting sphere is theoretically cal culated and experimentally measured. The absolute RCS of the object is then found through the calibration procedure. The unit of the absolute RCS used in this chapter is in dB square centimeter ( dB scm), because the conventional unit dB square meter (dB sm) is too big for the RCS of objects used in this dissertation. Characteristics of a plate covered with an absorber are of primary interest in this chapter. The experimental arrangement of the plate referred in this chapter is represented in two geometries as shown in Figs. 6.6(a) and 6.6(b). The transmitting and receiving antennas are both lying in the azimuthal plane. In Fig. 6.6(a), the plate is perpendicular to the azimuthal plane and two edges of the square plate are always normal to the line of sight. In a bistatic system we define the line of sight as the bisector of the angle between the transmitter and receiver (T/R) during the course of rotation. A zero degree aspect angle is defined as the angle at which the normal of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 139 RECEIVER TRANSMITTER (a) RECEIVER^ TRANSMITTER b Fig. 6.6 Geometry of the measurement arrangement of a conducting plate. (a). Geometry 1— the plate is perpendicular to the azimuthal plane. Zero degrees is defined as the specular direction. (b). Geometry 2— the plate is tilted with an angle defined as the angle between the plate and the rotational axis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 140 - the plate coincides with the line of sight In Fig. 6.6(b) the plate has a tilt angle 0, defined as the angle between the rotational axis and the plate. The arrangements in Figs. 6.6(a) and 6.6(b) will be referred to as geometry 1 and geometry 2 throughout this chapter. In this section, we will discuss some experimental results pertaining to RCS and images of an object covered with absorbers. The two types of absorbers used are broadband absorbers and surface current absorbers. 6.3.1 Broadband Absorber-Covered Bodies The broadband absorber used is the Emerson Eccosorb AN72. AN72 is a light foam sheet absorber. Typical electrical performance of the Eccosorb AN series is shown in Fig. 6.7. TYPICAL ELECTRICAL PERFORMANCE OF ECCOSORB AN 0 8 w N F R 0M M E T A L P L. A T E 100 FREQUENCY G Ht • « lo w e s t FREQUENCY f o r i* m a x im u m RE FLEC TIV ITY Fig. 6.7 Typical performance of the Emerson product AN absorbers. The dielectric constant of a broadband absorber is usually tapered to match the characteristic impedances of air and conductors. The AN72 absorber has different Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 141 - surface resistance on the two sides and is an inhomogeneous material. The two sides of the AN72 are different in color. The front surface is a white-gray color and is designed to match the characteristic impedance of air, while the back surface is a bronze color and is designed to match that of a conductor. In our work AN72 was used to cover a conducting plate (30 cm x 30 cm). The measured co-polarized attenuation in reflectivity of the metal plate when covered with AN72 in the normal way (bronze side being attached to the conductor) and in the reversed way (whitegray side being attached to the conductor) in the broadside direction are shown in Fig. 6.8. The attenuation factor is about 8 to 15 dB in this frequency range when the plate is covered in the normal way and is much smaller when covered with the absorber reversed. i i I V 0.00 E-01 a covered inside out r i t u e d i n covered in normal way d b b. 'J0E+00 • FPEQUEMCY IN GHZ Fig. 6.8 Attenuation in dB of a metal plate covered with absorber AN72 in the normal way and in the reversed way, 0 dB being the response of the plate without absorber covering. Next, the square plate is covered with AN72 on both sides. The measurement arrangement is that of geometry 1. The co-polarized and cross-polarized RCS patterns of the plate before and after diaphanization at frequency 11.25 GHz are shown in Figs. 6.9(a) and (b) respectively. The co-polarized and cross-polarized mean RCS patterns of the plate before and after diaphanization versus rotation angle averaged Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 142- over band 1 and band 2 are shown in Figs. 6.9(c) and (d) respectively. It is noted that the co-polarized mean RCS has been reduced a great deal for small rotation angle and the amount of reduction decreases as the rotation angle increases. The mean RCS after diaphanization are even greater than those before diaphanization when the rotation angle is greater than 65°. A possible reason for the above phenomenon is that AN72 (12 mm in total thickness for two sides) is much thicker than the thickness of the metallic plate (1.6 mm). The effective area facing to the T/R after application of the absorber covering is much greater than that of the con ducting plate when the plate is rotated close to the edge-on direction. Besides, AN72 is not designed for attenuating side-lobe energy. However, the cross-polarized mean RCS after diaphanization is much smaller than that before diaphanization for every aspect angle. This phenomenon is contradictory to the co-polarized case and its rea son needs more investigation. The co-polarized sinograms and cross-polarized sinograms before and after diaphanization are shown in Figs. 6.9(e) and (f) respectively. For each sinogram display the bottom line represents the range profile of the first aspect angle while the top line represents the range profile of the last aspect angle. The range profile is obtained by Fourier transforming the range-corrected field over band 3. The magni tude of the range profile is proportional to the brightness of the display. The sino grams are displayed in linear scale. The dynamic display range has been suitably chosen so that weak signals will not be overriden by strong signals. The maximum value of the reconstructed sinogram and its dynamic display range are shown in the attached caption. The rotation angle coverage is also noted in the Figures. From the sinograms illustrated, one can conclude that the main contributors to the scattered field of a conducting plate arranged in geometry 1 are the two vertical edges. After diaphanization the equivalent scattering strength of the rear edge (the one farther away from the T/R) has been reduced a great deal for every rotation angle as can be seen from the reduced brightness in the sinogram. However, the co-polarized scatter ing strength of the front edge (the one closer to the T/R) has been increased when the plate is rotated close to the end-on direction. The co-polarized images reconstructed over an angular window from <f>= 0° to 90° before and after diaphanization are shown Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 143 J without absorber 9 .i t E -il v jv y vV1 * with absorber AN72 t.ttE -tl 2 .2 !E -il — rotation ancle in decree (a) S .tt E«tl 3.78 £♦•1 2. St E -tt U without absorber k ................... .. ■ with absorber AN72 1.25 E « tl v f t t ¥ f \ i ROTATION ANCLE IN DECREE (b) s .t t E - il conducting plate, co-polarized with AN72, co-polarized . . . . . . . conducting plate, cx-polarize with AN72, cx-polarized 3 .St £-tl 2. t t E -tt • ‘A.1.........?.*! S .t t E’ t t 9 .0 9 E -tl 2 .2SE-91 A .S tE -tl — ro ta tio n ancle :n decree 6 ..'1 E -il (c) t.t* E-tl 3. St E -tt conducting plate, co-polarized with AN72. co-polarized * conducting plate, cx-polarized — with AN72, cx-polarized i ‘V 2 .tt E -t l I A ... w fi S .tt E -tt t», i.itE -9 1 2.2SE-tt ».!♦£.#1 ROTATION ancle in oecree C.75E-9I (d) F ig . 6 .9 (cap tion se e page 145) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. AN72 cowered p l a t e ; co-p o lo r max:18; d ls p ( 1 .0 ) ; (0.90)deg e co nductin g pi a t e ; c x - p o l a r ; (0,90) —max—13;d 1s p (1 ,0 ) p l a t e w ith A N 72;cx-polar;(0,90) max=2.2; d l s p ( 1 ,0 ) f v e r t i c a l c o n d u ctin g p l a t e • a x : 89; d l» p (8 5 » G 5 );(0 » 9 0 )deg . v e r t i c a l PK72 co v ered p la te max: 83; d lip < 0 6 .6 5 ); (0 .9 0 )d e g g Fig. 6.9 (caption see next page) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 145 v e r t ic a l p la te ; ( 3 0 .9 0 ) d e g v e t le a l PN72 c o v e re d ; max: 7 9 ; d ls p ( 7 5 .5 5 ) max: 7 8 ; d ls p ( 7 5 . 5 5 ) n c o n d u c tin g p l a t e ; c x - p o l a r •o x :‘7 2 ; d l s p ( 7 0 .5 0 ) ; (3 0 ,9 0 ) d e g PN72 c o v e re d p l a t e '; c x - p o l a r max 5 9 , d ls p ( 7 0 . 5 0 ) ; ( 3 0 , 9 0 ) d e g Fig. 6.9 Measured RCS pattern, sinograms and reconstructed images of a conducting plate before and after diaphanization with geometry 1. (a). Co-polarized RCS pattern at frequency 11.25 GHz. (b). Cross-polarized RCS pattern at frequency 11.25 GHz. (c). Co-polarized and cross-polarized mean RCS patterns averaged over band 1. (d). Co-polarized and cross-polarized mean RCS patterns averaged over band 2. (e). Co-polarized sinograms of the plate before (above) and after (below) diaphanization. (f). Cross-polarized sinograms of the plate before (above) and after (below) diaphanization. (g). Co-polarized images of the plate reconstructed over an angular window from <|>= 0° to 90° before (left) and after (right) diaphanization. (h). Co-polarized images of the plate reconstructed over an angular window from <[)= 30° to 90° before (left) and after (right) diaphanization. (i). Cross-polarized images of the plate reconstructed over an angular window from <J)= 30° to 90° before (left) and after (right) diaphanization. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 146 - in Fig. 6.9(g). The brightness of the images is displayed in log scale. The maximum value and the dynamic display range of the images are also noted in the accompany ing captions of these figures. The co-polarized and cross-polarized images recon structed over an angular window from 0=30° to 90° before and after diaphanization are shown in Figs. 6.9(h) and (i) respectively. The difference in appearance between the images reconstructed over different angular windows can be explained by backprojecting the sinogram into the image space as described in Chap. 3. It is also noted that the brightness in images has been reduced after diaphanization. Next, we used AN72 to cover a complex shaped object. A scale model B-S2 airplane is partly covered with absorber. The covered portions have been shown in Fig. 5.3. The co-polarized and cross-polarized mean RCS patterns averaged over band 3, sinograms, and images of the scale model B-52 before and after diaphaniza tion are shown from Figs. 6.10(a) to Fig. 6.10(f). The above results show that the broadband absorber is effective at attenuating main-lobe energy but not effective at reducing side-lobe energy. In the next subsec tion we will use an alternative absorbing material to cover a metallic object. 6.3.2 Surface C urrent Absorber-Covered Bodies The surface current absorber used is the Emerson product Eccosorb GDS, a thin (0.8 mm in thickness), high ioss, silicone rubber sheet that can be bonded to curved surfaces. The characteristics of GDS are shown in Table. 6.1. GDS will lower the reflection by attenuating surface waves and multiple bounces. GDS is used to cover a conducting plate (30 cm x 30 cm). Two covering configurations are tested. The first configuration covers the whole plate by GDS on both sides as shown in Fig. 6.11(a). The second configuration covers only the verti cal edges as shown in Fig. 6.11(b). It has been found that the induced surface current density of a conducting strip when illuminated by a plane wave is highest in the neighborhood of an edge [77]. Covering only the edges with GDS is therefore expected to reduce most of the return energy. When the measurement is arranged in geometry 1 with the covered edges parallel to the rotational axis, the co-polarized Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 147 - 4. 3. 2 1. '> line number comparison o f b 5 2 r l . p and b 5 2 r l . c ! 0 ; a v e .3 I I V a n fi t e+91 u d e i n d b l.OOO0e+00 > lina number com pariso n o f b S 2 r r . p and b 5 2 r r . c ! 0 ; a v e . 3 b 5 2 r 1 . p ; m a x : 4 1 ; d l s p ( 3 . 0 ) ; ( 0 ,9 0 ) d e g n b 5 2 r r . p ; m a x > 7 ; d l» p ( 3 .0 ) j( 0 » 9 0 ) d e g r. ' 1 b 5 2 r 1 . c ;m a x : 5 ; d l s p ( 3 . 0 ) ; ( 0 . 9 0 ) d e g f- A\:- c r b S 2 rr.c .rn ax :2 .7 ;d i» p (3 .0 );(0 .9 0 )d d Fig. 6.10 (caption see next page) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 4 8 - b S a^-J.p ; w ith o u t a b t o r b e r max 8 9 ; d 1« p ( 0 4 ,6 0 ) ; ( 0 , 9 0 )d e g bSZrrt.pi c x - p o la r ; . • a x <63; d l » p ( 9 4 . 6 0 ) ; (9 « 9 0 )d e g b52M c ; w ith o b to r b e r . max BS, d l* p (B 4 .G 0 ); C 0 ,9 0 )d eg b S 2 r r .c ; w ith a b s o r b e r ) c x -p o la r max ^ ; d l t p ( 9 4 , G 0 ) ; ( 0 ,9 O ) d e g F ig . 6 .1 0 M e a n R C S p a tte rn s, sin o g ra m s, a n d re c o n stru c te d im a g es o f the B -5 2 a irp lan e b e fo re a n d a fte r d iap h a n iz a tio n . (a). C o -p o la riz e d m e a n R C S p a tte rn s a v era g ed o v e r b an d 3. (b). C ro ss-p o la riz e d m ea n R C S p a tte rn s a v era g ed o v e r b a n d 3. (c). C o -p o la riz e d sin o g ra m s b efo re (ab o v e) a n d a fte r (b elo w ) d iap h a n iz a tio n . (d). C ro ss-p o la rize d sin o g ra m s b e fo re (ab o v e) a n d a fte r (b e lo w ) d iap h a n iz a tio n . (e). C o -p o la riz e d im a g e b e fo re (left) an d a fte r (rig h t) d ia p h a n iz a tio n re c o n stru c te d o v e r an a n g u la r w in d o w fro m 0 = 0 ° to 9 0 °. (f). C ro ss-p o la riz e d im a g e b e fo re (left) and a fte r (rig h t) d ia p h a n iz a tio n re c o n stru c te d o v e r an a n g u la r w in d o w fro m (|)=0o to 9 0 °. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 149- Table 6.1 Characteristics of the GDS absorbers Frequency (GHz) 3 8.6 Dielectric const k' 20 20 Dielectric loss tangent tanS 0.67 0 Magnetic permeability Km' 3.5 1.2 Magnetic loss tangent tanSm 0.4 0.4 Attenuation dB/cm Relative Impedance IZ l/Z0 l 1.1 0.4 45 0.4 and cross-polarized mean RCS patterns of the plate without and with GDS covering averaged over band 1 and band 2 are measured and shown from Fig. 6.11(c) to Fig. 6.11(f). It is noted that the mean RCS has been reduced after applying a GDS cover ing, and the amount of reduction increases as the rotation angle increases. This confirms the effectiveness of GDS at reducing side-lobe energy or surface waves. The amount of reduction over band 1 is smaller than that over band 2, implying GDS is more effective at higher operating frequencies. One can also find that covering edges is not as effective at reducing RCS as covering the whole plate. An explana tion for this observation can be obtained via an examination of the range profiles which render information on scattering mechanisms. The co-polarized and cross-polarized sinograms of the GDS-covered plates are shown in Figs. 6.11(g) and (h) respectively. By examining the co-polarized sino grams of Fig. 6.11(g), one notes two more traces in the sinogram of the second configuration. These extra traces are reflected from the boundaries of the absorber and the plate. Surface impedance is discontinuous at these boundaries. This phenomenon verifies the statement that reflections occur at those regions with discon tinuous surface impedance or reflectivity. However, the cross-polarized sinogram of the second configuration as shown in the lower part of Fig. 6.11(h) has a different appearance. No extra trace is found in the front edge. This phenomena needs more investigation. The co-polarized and cross-polarized projection images of these two GDS-covered plates are shown in Figs. 6.11(i) and (j) respectively. It can be seen Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 150- GDS GD S aluminum plate *■ aluminum plate (a) ; j V a * ? i t Q d a i n <b) S . 90 E+01 conducting plate edge covered w ith GD S whole covered with GDS 3 .9 0 E+01 2.00 E+01 5 .00 E+00 d b ♦ . 50E+01 . 00E-01 2.25E+01 — ROTATION ANGLE IN DEGREE 6 . 75E+01 (c) 5 .0 9 E+01 conducting plate edge covered with GDS whole covered with GDS ROTATION ANGLE IN DEGREE (d) Fig. 6.11 (caption see page 152) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 151 - conducting plate ***** edge covered with G D S whole covered with GDS 3 It d E+00 2.25E+01 ROTATION ANGLE IN DEGREE (e) r conducting plate ******* edge covered with GDS whole covered with GD S V -V V ______________ ________ E+00 -J, 00E-01 2.25E+01 4.S0E+01 - - > ROTATION ANGLE IN DEGREE 6.75E+01 (0 GDS uihole cowered; c x -p o la r- ; m a x : 4 .8 ; d i s p ( l < 0 ) ; (0 ,9 0 ) d e g ‘ GDS e d g e co w ered ; c o - p o l a r max : 6 . 9 ; d l t p ( l . O ) r ( 0 , 9 0 ) d e g Fig. 6.11 (caption see next page) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6.11 Covering configurations of a plate and their mean RCS patterns, range profiles and reconstructed images, (a). Both sides are covered with GDS. (b). Only a pair of edges are covered with GDS. (c). The mean co-polarized RCS averaged over band 1. (d). The mean co-polarized RCS averaged over band 2. (e). The mean cross polarized RCS averaged over band 1. (f). The mean cross-polarized RCS averaged over band 2. (g). Co-polarized sinograms of the GDS wholly covered plate (above) and edge covered plate (below), (h). Cross-polarized sinograms of the GDS wholly covered plate (above) and edge covered plate (below), (i) Co-polarized images of the GDS wholly covered plate (left) and edge covered plate (right) reconstructed from an angular window from <J)=30° and 90°. (j) Cross-polarized images of the GDS wholly covered plate (left) and edge covered plate (right) reconstructed from an angular win dow from <t>=30° and 90°. with permission of the copyright owner. Further reproduction prohibited without permission. - 153- that brightness of the rear edge has been reduced as compared to the left images of Figs. 6.9(h) and 6.9(i), which are the images of the same plate without absorber cov ering. This fact implies that diffraction from the rear edge has been suppressed because the brightness of an image pixel is proportional to the mean scattering strength of that portion averaged over the specified angular window and spectral win dow as stated earlier. If we compare the RCS plots and the reconstructed images, we can conclude that covering the whole plate is more desirable from the detection avoidance point of view. However, covering the edge portions is more effective at distorting the image because the discontinuities in the edge covered piate give rise to twin projection images at each edge, distorting thereby the image as compared to that of the wholy covered plate. The second object tested was an aluminum tube with one end covered by a cap and the other end open. Its geometry and dimensions are shown in Fig. 6.12(a). The motivation for studying this object stems from its likeness to an engine intake. The tube is placed with its axis perpendicular to the rotational axis. The direction normal to the cap is defined as zero degrees. The co-polarized mean RCS pattern of the tube averaged over band 3 is shown in the solid curve of Fig. 6.12(b). Its sinogram is shown in the upper part of Fig. 6.12(c). Examining the sinogram, one can find that multiple reflections are important contributors to the back-scattered field when the rotation angle exceeds 110°, because extra peaks can clearly be seen beyond the maximum dimension of the tube. Besides, the range distance of these extra peaks decreases as the rotation angle increases, which can be explained by the plot of the ray paths as shown in Fig. 6.12(d). The incident ray can be reflected to the receiver in the following ways: (1). It can impinge on the rim of the empty mouth and then be diffracted to the receiver. (2). At a later time it may hit the rim of the back cap and be diffracted. (3). It may enter into the mouth, bouncing between the walls, be reflected back by the cap, and finally emerge from the mouth and propagate directly to the receiver. (4). The ray may reflect back from the cap, impinge on the rim of the mouth, and be diffracted to the receiver. The ray path lengths of cases (3) and (4) decrease as the rotation angle is increased, which explains the appearance of the sino gram. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. alum inum tube r 7.5 c m capped —* !4 c m |* - ~ H 4cm h - 30 cm (a) receiver <t>= 0° <t> = 180' transm itter <)> = 90' w ith ou t absorber w ith absorber ♦ . 3OE»01 - ■ ROTATION ANCLE IN DEGREE (b) F ig. 6 .1 2 (cap tio n see n ex t p ag e) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 155t u b e ; ~'(Q, 180) max:14; d l s p (1,0) tub e ; with abs or be r ; ( 0 ,i© 0 ) max: S . 7 ; d l s p ( 1 , 0 ) r •• i d tu b e ;d = 3 " , L = 1 2 "; ( 1 0 0 . 1 8 0 )d eg max =0$<.^dj*p ( 7 5 .5 0 ) . a b t o r b e r c o v e re d . t u b * ; ( 1 0 0 .1 0 0 ) max =6 9 ; d l s p ( 6 5 .5 0 ) e Fig. 6.12 Geometry, mean RCS, sinogram, and reconstructed image of a tube. (a). Geometry of a tube and the sketch of the absorber covered regions, (b). Co-polarized mean RCS patterns before and after absorber covering, (c). Sinograms before (above) and after (below) diaphanization. (d). Ray tracing of an incident wave travelling inside a hollow tube with a smaller rotation angle (left) and larger rotation angle(right). (e). Images before (left) and after (right) diaphanization reconstructed over an angular win dow from <t>=100° to 180°. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 156- Assume the angular window of interest is from <(>=100o to 180°. Multiple reflections and rim diffractions are the dominant scattering mechanism over this win dow. To reduce multiple reflections and edge diffractions, we cover both sides of the cap with an AN72 broadband absorber, and cover the inner and outer rings of the mouth and the outer ring of the other end with a GDS absorber. The covered pat terns are also sketched in Fig. 6.12(a). The mean RCS pattern after application of the absorber covering is shown in the dashed curve of Fig. 6.12(b). It is seen that the RCS after diaphanization has been reduced noticeably over the specified window. The sinogram after diaphanization is shown in the lower part of Fig. 6.12(c). It is seen that multiple reflections have been suppressed and the edge diffraction strength has also been reduced. The co-polarized images of the tube before and after diaphanization are shown in Fig. 6.12(e). 6.4 Target Shaping The purpose of shaping is to manipulate or deform the target surface so as to reflect the incident energy in directions away from radar. However, a reduction in RCS at one viewing angle is usually accompanied by an enchancement at another angle when the surface is reshaped. The success of shaping depends on the existence of angular sectors over which low RCS is less important. Shaping is a trade-off between structural and electromagnetics requirement. The general procedure of shaping a target is as follows [33]: 1. Define the threat sectors, i.e., the solid angle relative to the target over which RCS is to be reduced. 2. Take advantage of shaping without resorting to absorbing materials — Shape must be chosen to optimize the mission of the vehicle, and the mission itself determine the blend of RCS reduction versus structural characteristic. This requires iterations of RCS and aerodynamic performance predictions for a set of shapes and then verification by testing. 3. If the final configuration cannot meet the requirement, use absorber cladding provided this is aerodynamically permissible. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 157 - To study the effect of target shaping on the RCS, it is helpful to examine the dependence of the various analytic components of the RCS on frequency. At high frequencies, the RCS is dependent on the surface curvature of the equi-phase points if specular reflections are involved (see Sec. 2.4.1). The RCS is proportional to f 2 for a flat surface; proportional to / for a single-curved surface; and independent of / for a double-curved surface. For example, the specular RCS of a plate with dimen sion w x l , of a cylinder with radius a and length /, and of a sphere with radius a are as follows: a = 47t/2W2 X2 plate 2ttal2 X cylinder 7ia2 (6.25) sphere When one of the radii of the curvaturegoes to zero, an edge is then created. The diffraction fields of an edge with length / can be written as follows (see Sec. 2.3.3): E d = j k D i k f i j i))/sny p ° se) kl cosQ (6.26) where D is the diffraction coefficient, whose expression has been shown in Eq. (2.44), and D is proportional to 1I k. If the inclination angle is 0=90°, i.e., normal incidence to the edge, then sinc(£/cos0) = 1. Therefore, E d is independent of k and is proportional to /. In the side-lobe region, for a given 0, E d is proportional to sinf/WcosQt e jk lc o sQ -jk lc o s Q — ----«— = • - ------, the equivalent diffraction of the end points (or tips) xcos0 2jk cos0 e ± jk lco sB } is proportional to I------------1 = ----- -— —. Therefore, the RCS of a tip is propor2jkcosQ 2y£cos0 tional to 1/A:2 and is independent of /. In the case of surface discontinuities, the fre quency dependence of the n th surface derivative discontinuities is k~2(-n~l\ as seen in Eq. (2.83). In short, the RCS of a body is dependent on the operation of several kinds of mechanisms. These mechanisms can be categorized according to their strength and can be listed in hierarchy according to their dependence on the fre quency and dimension. Shown in Table. 6.2 is the hierarchy of scattering shapes Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 158 - Table 6.2 Hierarchy of Scattering Shapes [52] G eom etry Type Freq. Dep. Size Dep. Formula M aximum square trihedral com er retro-reflector right dihedral com er reflector a = 12jca4/X2 M aximum o = 8ita2b2/X2 m axim um Flat plate a = 4ica2b2/\2 norm al incidence m axim um C ylin d er norm al incidence Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 159 - Table 6.2 Hierarchy of Scattering Shapes (continued) [52] Freq. Size Type Geom etry Formula Dep. Dep. sphere F° L 2 m axim um a= jta 2 norm al incidence Straight edge normal incidence k /(9 . F° V L 2 L 2 0 - aspect K Curved edge normal incidence / ( 6 . e te) a l / 2 F -i L & 1 0,/u * interior dihedral angle between faces meeting at edge X2 g (a Apex F -2 L° a > \ ,(3,0,0) . P - interior angles of tip 0, 0 - aspect angles a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 160 - [52]. This table is very helpful, because it provides a fundamental rule for shaping a target. It is obvious that in order for the RCS to become smaller with increasing fre quency, no specular contribution to the scattered field should exist. The body should appear as a tip or an edge at the aspect at which the RCS is to be minimized. It is interesting to note that the frequency dependence of the RCS of a rectangu lar plate is affected by the aspect angle. In the specular direction, the RCS is propor tional to &2 and the square of the area; at the aspect angle with incident wave normal to a pair of edges, the RCS is independent of frequency but is proportional to the square of the length of that edge; at side-lobe regions, the RCS is proportional to 1/k2 and is independent of the edge length. As discussed earlier, a reduction in RCS over a range of aspect angles is usually accompanied by an enchancement in another range of angles when the surface is reshaped. Some examples will be given next to verify the above statement A square conducting plate (40 cm x 40 cm) is reshaped into a serrated square plate by serrating its edges as shown in Fig. 6.13(a). The scattering properties of the serrated plate and the unserrated plate which we will simply refer to as the plate will be studied and compared. First, we use geometry 1 to measure the scattered field. In this arrangement the line of sight is normal to the two vertical edges of the plate but not perpendicular to each serrated edge. Therefore these two vertical edges are the dominant contributors to the back-scattered field of the flat plate, while the tips are the major contributors to the scattered field of the serrated plate. The mean co polarized RCS of the plate and serrated plate averaged over band 1 and band 2 versus rotation angle are shown in Fig. 6.13(b) and Fig. 6.13(c) respectively. Com paring Figs. 6.13(b) and (c), one can find that the mean RCS patterns of the two bands differ very little for the flat plate case except that the mainbeam width is more narrow in band 2. However, the mean RCS patterns of the serrated plate are greater in band 1 than in band 2. The above results are consistent with what was predicted in Table. 6.2., which shows the scattered field from an edge with normal incidence is proportional to the edge length but independent of frequency while the contribution from a tip is independent of edge length but inversely proportional to frequency. The sinograms and images of the plate and the serrated plate are shown in Figs. 6.13(d) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 161 - T —•1 6cm )•— (a) flat plate serrated plate a n ? i t •d u e i n d D a .eeE -at DOTATION ANGLE in degree (b) flat plate serrated plate v a « P 1 i t u d 9 I n d b ROTATION ANGLE IN DEGREE (c) Fig. 6.13 (caption see next page) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. v e r tic a l serg ated p la te ; max:55; d l s p ( 1 , 0 ) ; (0 ,9 0 ) d e g d ■ E e Fig. 6.13 Geometry, and mean RCS patterns of a serrated plate using geometry 1. (a). Geometry of a serrated plate, (b). Mean co-polarized RCS patterns of a flat plate and a serrated plate averaged over band 1. (c). Mean co-polarized RCS patterns of a flat plate and a serrated plate averaged over band 2. (d). Sinogram of the flat plate (above) and serrated plate (below), (e). images of the flat plate (left) and the serrated plate (right) reconstructed from data collected over an angular window from <j)=30° to 90°. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. - 163 - and (e) respectively. Next, the measurement is arranged in geometry 2 with a tilted angle 0=90°. Zero degrees is defined as the angle at which a pair of edges the plate are normal to the bisector. The co-polarized mean RCS patterns of these two plates averaged over band 3 are shown in Fig. 6.14(a). Away from the broadside, the major contributors to the back-scattered field are the tip points. There are more tips in the serrated plate than in the flat plat Additionally, part of the serrated edges are normal to the bisec tor at <(>=450 and 0=135°. These facts explain why the mean RCSs of the serrated plate are higher than those of the fiat plate in most aspects and why two peaks appear in the dashed curves at 0=45° and 135°. The mean RCS patterns shown in Fig. 6.13 and Fig. 6.14 show evidence of the fundamental concept of target shaping, that is, a reduction in RCS at one viewing angle is usually accompanied by an enhancement at another viewing angle. The co-polarized sinogram and image reconstructed over an angular window from 0= 0° to 360° for a serrated plate mounted as in geometry 2 with a tilted angle 0= 60° are shown in Figs. 6.14(b) and (c) respectively. In this experimental arrange ment the field scattered from the styrofoam support cannot be subtracted effectively. The two straight vertical lines in the sinogram and the blurred part at the center of the image are both from the styrofoam supporter. Therefore we dot not compare the mean RCS patterns with this measurement arrangement. In the following example we will study the effect of a sharp tip and a round comer on the RCS. The objects are a triangular plate with sharp tips and a triangular plate with two round comers. These two structures are shown in Figs. 6.15(a) and (b) respectively. The measurement is arranged in geometry 2 with a tilted angle 0=90°, i.e., the plates are lying in the azimuthal plane. 90 degrees is defined as the nose-on angle. These structures are used to crudely simulate a delta wing of an air plane. The co-polarized mean RCS and sinograms of these two plates averaged over band 3 are shown in Figs. 6.15(c) and (d) respectively. By comparing the traces in the sinograms of Fig. 6.15(d), one can find that the energy reflected from a round comer is much less than that from a sharp tip when the edge is not normal to the bisector. The sinograms also illustrate some other scattering mechanisms which Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 164- 21 v E+Ol flat p late 3 M serrated p late fi t a d a E-01 i n d b 1.35E+02 ROTATION ANGLE (a ) dag t i l t e d e e r g a te d p l a t a i 3 9 m a x i GSi d la p ( £ 5 . 4 5 ) 4- V / S N\ VV / ’ c Fig. 6.14 Mean co-polarized RCS and images of a flat plate and a serrated plate using geometry 2. (a). Co-polarized mean RCS a flat plate and serrated plate averaged over band 3. (b) Sinogram of the serrated plate with a tilted angle 0=60° and an angular window from 0° to 360°. (c). Image of the serrated plate reconstructed from <))=0o to 360°. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 165 - require more investigation. A good example of target shaping is the B-l bomber, which appears to be designed to have little specular reflection over a range of viewing angle relevant to practical encounters with probing monostatic radar system. A metalized 1:100 scale model of the B-l was mounted on a rotating pedestal with inclination angle 20° as shown in Fig. 6.16(a). The origin of the azimuthal rotation angle $ is defined as the broadside direction and a 90° rotation angle is in the head-on direction. The mean RCSs averaged over band 3 using co-polarized and cross-polarized waves versus the rotation angle are shown in Fig. 6.16(b). The sinograms and projection images recon structed over an angular window from <j)=0° to 90° of these two polarizations are shown in Figs. 6.16(c) and (d) respectively. For this object structure and measurement arrangement, multiple reflections between the flat bottom and the fuel tanks are important contributors to the RCS for those aspects free of strong specular reflections. The difference between the copolarized and cross-polarized mean RCS is small for many aspects. This phenomenon is different from that of seen with the B-52, where the co-polarized mean RCS is much stronger than the cross-polarized mean RCS in most aspects. The effect of multiple reflections on the images generated is the blurring in the image of the bottom structure as shown in Fig. 6.16(d). 6.5 Combination of Absorber Covering and Shaping If the final shape of an object is determined and its RCS cannot meet the requirement, then one should resort to absorber covering. When an absorber is coated on a metallic object, the boundaries of the covered regions will introduce discontinui ties in surface impedance and cause radiations. Scattering hierarchy also exists in the shape of the coating boundaries. The total scattered field may be more effectively reduced if the coating boundaries are suitably shaped or tapered. In the optical region one can paint an object’s surface different colors and pat terns so that its appearance differs from the original one. This process is called camouflage. In the microwave region, if the surface of the object is coated with Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 16630cm -< VO (b) r ,(a) w ith sharp tip w ith round c o m e r s 1.35E+02 ROTATION ANGLE IN DEGREE (c) - — — l J. : t r i a n g u l a r p l a t e ; ro u n d c o r n e r max: 3 . 0 ; d l a p C . 1 . 0 ) ; ( 0 . 100)deg (d) Fig. 6.15 Geometry, mean RCS patterns, and sinograms of triangular plates with sharp tips and rounded tips arranged in geometry 2. (a). Geometry of a triangular plate with sharp tips. (b). Geometry of a triangular plate with round comers, (c). Co-polarized mean RCS patterns of the two plates averaged over band 3. (d). Sinograms of the plates with sharp tips (above) and rounded tips (below). Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. I V co-polarized a M e+01 T i ******* cx-polarized t u d e i n E+01 d b 0 .0 0 E -0 1 2.26E+01 > ROTATION ANGLE IN DEGREE 6.77E+01 (b) b i u r l ; amx: 1 3 ; d i s p ( 1 , 0 ) ; ( 0 , 9 0 ) d e g ' . JB S fr b l h r l ; c © - p o la r ; (O ,9 0 )d e g a a x -8 0 i d l* p C 0 5 /G 0 ) b lu rr; b l u r r ; c x - p o l a r s ( 0 .9 0 ) d e g mox^ 6 6 ; d l« p ( 9 5 .G O m ax :8 ;d l» p ( l > 0 ) ; (0 ,9 0 )d e g (d) Fig. 6.16 Picture, mean RCS patterns, and images of a scale model of B-l airplane, (a). Picture of a scale model B-l airplane mounted on a rotating pedestal, (b). Mean co-polarized and cross-polarized RCS patterns averaged over band 3. (c). Co-polarized (above) and cross-polarized (below) sinograms over an angular window from <j>=0° to 90°. (d). Co-polarized (left) and cross-polarized (right) images reconstructed from data collected over angular window <j)=0° to 90°. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 168 - absorbing materials in different thicknesses or patterns, its microwave images may also be altered and may become unrecognizable. If the absorber covering is con sidered with shaping of the covering boundaries, the technique is referred to as a combination of absorber covering and shaping. We will give two examples based on the above considerations. A square con ducting plate (30 cm x 30 cm) is covered with GDS and the covering pattern is shown in Fig. 6.17(a). The measurement is arranged in geometry 2 with a tilted angle 6=90°. Zero degrees is defined as the angle at which a pair of edges are nor mal to the line of sight We shape the covering absorbers in a curved manner in order to avoid any specular reflections which might be generated by an edge normal to the line of sight. The co-polarized mean RCS pattern of the plate with and without an absorber covering are shown in Fig. 6.17(b). It can be seen that the RCS has been reduced for most aspect angles. Their sinograms over an angular window from <]>=0o to 360° are shown in Figs. 6.17(c) and (d). One can find that the equivalent scattering strength of the edges and tips has been effectively suppressed. The reconstructed images of these two plates are shown in Fig. 6.17(e), where both images have the same dynamic display range. The brightness of the edges and tips after diaphanization has been reduced. If the display dynamic range of this coated plate’s image is lowered, the resultant image is shown in Fig. 6.17(f). The next example is the scale model B-l covered with a blend of GDS and AN72. The picture of the absorbor-covered scale model B-l is shown in Fig. 6.18(a). We cover those regions which might cause specular reflections with broad band absorbers, and cover the edges and comers with GDS to reduce edge diffractions and multiple reflections. The measured mean co-polarized and crosspolarized RCS averaged over band 3 before and after absorber covering are shown in Figs. 6.18(b) and (c). It is seen that the amount of RCS reduction is not great in most aspect angles for the co-polarized waves, while the amount of reduction is pro nounced for the cross-polarized waves. The co-polarized and cross-polarized sino grams before and after absorber covering are shown in Figs. 6.18(d) and (e) respec tively. Comparing the sinograms one can find that the absorbers can suppress cross polarized waves more effectively, which is in agreement with the plots of Figs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 169 - firDA i V w ith o u t absorber ;- w ith ab sorb er G D S a M ?i t u d e i n d b 0 . 00 E-01 •— > ROTATION ANGLE IN DEGREE (b) f l a t p la t e ; th eta= 9 0 m a x : 0 . 9 4 ; d l s p ( . 0 S . 0 ) ; ( 0 . 360)deg GDS c o a t e d p l a t e ; th e t a = 9 0 m a x : 0 . 3; d i « p ( , 0 5 , 0 ) ; (0 ,3 S 0 )d e g • t i l t e d p l a t e ; th e ta = 9 0 m ax:S 3; d l t p (S3* 3 0 ) ' t i l t e d c o a te d p l a t e ; t h e ta » 9 0 m ax:4 9 ; d l t p (53 * 3 6 ) -rrr' I c o a te d t i l t e d p l a t e ; t h e t a - 9 0 max: 4 9 ; d l t p ( 4 9 ,3 0 ) F ig . 6 .1 7 C o v e re d p a ttern s, m ean R C S p a ttern s an d re c o n stru c te d im a g e o f a p late arra n g e d in g e o m e try 2 w ith tilted an g le 9 = 90°. (a) C o v ered p a tte rn s o f a plate, (b ) C o -p o larize d m ean R C S p a tte rn s w ith o u t an d w ith a b so rb e r c o v e rin g av erag ed o v e r b an d 3. (c) S in o g ra m o f th e c o n d u ctin g p late o v e r an a n g u la r w in d o w fro m <j>=0° to 3 6 0 °. (d) S in o g ra m o f th e a b so rb er-c o v ere d p la te o v e r a n a n g u la r w in d o w fro m <|)=0o to 360°. (e). Im ag e s o f th e p la te b e fo re (left) an d a fte r (rig h t) a b so rb er c o v erin g . B o th im a g es h a v e the sa m e d y n a m ic d isp lay ran g e, (f). Im ag e o f the a b so rb e r-c o v e re d p late w ith a lo w e re d d y n a m ic d isp lay ran g e. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 170- (a) w ith o u t ab so rb er w ith ab sorb er 0 . 00 E-01 2.26E+01 — • ROTATION ANGLE IN DEGREE 6. 77E+01 (b ) — a ■+*++ w ith ab sorb er n T i t u d w ith o u t ab sorb er E+01 E+01 a i n E+01 d b 0OL-01 -> ROTATION ANGLE IN DEGREE (C) Fig. 6.18 (caption see next page) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. b l u r m ; w ith o u t a b t o r b e r • a x - 6 2 ; d l t p ( 0 6 ,9 6 ) b l u r r c ; a b t o r b e r c o v e re d max: d i t p (6 0 ,5 6 ) '4 # ~ "rLW v i \ Fig. 6.18 Pictures of the absorber-covered scale model B-l and its RCS and recon structed images before and after diaphanization. (a) Pictures of the absorber-covered scale model B -l, top view (left) and bottom view (right), (b) Mean co-polarized RCS patterns before and after diaphanization averaged over band 3. (c) Mean cross polarized RCS patterns before and after diaphanization averaged over band 3. (d) Copolarized sinograms of B-l before (above) and after (below) diaphanization. (e). Cross-polarized sinograms of B-l before (above) and after (below) diaphanization. (f). Co-polarized images of B-l before (left) and after (right) diaphanization. (g). Cross polarized images of B-l before (left) and after (right) diaphanization. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 172 - 6.9(c), 6.9(d), 6.11(e), and 6.11(f)- The co-polarized and cross-polarized images before and after absorber covering are shown in Figs. 6.18(f) and (g) respectively. By examining Fig. 6.18(g), one can see that multiple reflections have been suppressed for the cross-polarized waves. However, the images are still recognizable. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 173 - CHAPTER 7 DIAPHANIZATION BY IMPEDANCE LOADING Results in the previous chapter confirm that covering an object with absorbers reduces the RCS and altering the object shape can direct the field away from a searching radar. Images can somewhat be distorted by the introduction of artificial discontinuities in the surface impedance of an object. However, the shape of a com plex object can still be recognized even when the scattering strength of the hot spots has been reduced. In this chapter we will investigate another approach, impedance loading, and examine its effect on RCS and images. The scattered field from a conducting body with impedance loading can be expressed as the sum of a component independent of the load value and a loaddependent component [21]. Accordingly, it is possible to specify the impedance so as to minimize the back-scattered field for a particular frequency and aspect angle. In other words, impedance loading is a technique which introduces an echo source to cancel the back-scattered field. However, the required impedance value is highly dependent on the structure of the target, location and distribution of the loads, operat ing frequency, and observation aspect. Therefore, fixed linear impedance loading is not effective at reducing the RCS over broad spectral windows and wide aspects, and this technique has been discarded in RCS reduction application. Nevertheless,’ impedance loading can cause other interesting effects, for exam ple, it might change the natural frequency of a target [84] or alter the range profile and image’s appearance. Recent advances in technology make it possible to control the load electronically. This time varying loading can introduce other phenomena. Examples are: making the receiver in a coherent radar unable to phase lock to the frequency of the incident wave, shifting the apparent frequency of the scattered field to provide a false Doppler shift [85], and spreading the spectrum of the scattered field to decrease energy within the bandwidth of the receiver. This latter Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 174- phenomenon can be viewed as thermalization of the incident wavefield. The degree that the scattered field is changed by introducing modifiable loads will clearly deter mine how pronounced the impedance loading affects the above phenomena. A load can either be linear or nonlinear and is characterized by the voltagecurrent curve (v - / curve) across its terminals. Linearly and nonlinearly loaded Nport scatterers have been analyzed [86,87,88,89,90,91]. Numerical techniques are usually required to theoretically calculate the field scattered from a loaded scatterer of arbitrary shape; the moment method is the most popular of these. It is well known that the computing time of the moment method is exponentially proportional to the dimension of the object in terms of the operating wavelength. For microwave diver sity imaging, the object is usually large compared to a wavelength and multiple fre quencies have to be used. This is equivalent to adding another dimension to the problem. It is therefore impractical to numerically analyze the effect of loading on the scattered field and image of a 2-d or 3-d object using frequency diversity imag ing. If the object is confined to a thin rod, however, the computing time will be tolerable if the numerical analysis program is sufficiently sophisticated. Furthermore, edges of a complex object may be approximated by thin wires in certain situations and will be further discussed later in this chapter. Besides, when a thin rod is illuminated near.^he end-on direction, the scattered field is contributed by a special scattering mechanism, a surface traveling wave [21]. It would be interesting to examine the image created by this special scattering mechanism. For the above rea sons, a thin rod will be the primary object to be numerically and experimentally stu died in this chapter and will serve as a vehicle for gaining insight in the difficult problem of nonlinear loading of a complex body for the purpose of modifying its RCS. Monochromatic imaging of a monopole antenna has been studied holographi cally by Iizuka [14], who was interested in visualizing resonance effects. However, we are interested in a wire scatterer rather than an antenna and frequency diversity imaging instead of a monochromatic holographic image. In this chapter we will review the scattering theory of scatterers with linear loading, nonlinear loading, and time-varying loading.. A thin rod will be used as the test object. Numerical and experimental results pertaining to the scattered field and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 175 - images of a thin rod will be demonstrated and discussed. 7.1. Linear Impedance Loading The effect of linear impedance loading on the back-scattering of a conducting object has been extensively studied. Chen and Liepa [92] presented a theoretical and experimental study on the minimization of back-scattering of a thin cylinder with central loading. Yu and Shen [93] used multiple impedance loading to modify the scattered fields of a thin wire. Exact expressions of current distribution on the wire were formulated. For targets of other shapes, the field scattered from a conducting object with impedance loading can be expressed in terms of a load-independent quan tity and a load-dependent quantity by utilizing N-port network theory. Harrington et al. analyzed the scattering from loaded N-port scatterers [86] and determined the characteristic modes of a loaded scatterer [90]. They applied the concept of charac teristic modes of a loaded body to control of radar scattering by reactive loading [87,88]. They also gave a procedure for obtaining the reactive loads of an N-port loaded scatterer which maximize the radar cross section [89]. Although the back-scattered fields could be maximized or minimized by lumped impedance loading, the load values are highly dependent on the frequency, loading positions, number of loading points, and aspect or viewing angle. Loading can be lumped or distributed. Wu and King [94] reported that an antenna can be made reflectionless if it is resistively loaded such that its internal impedance continuously changes from the center-fed point to the ends and satisfies a certain form. Their theoretical results .were verified experimentally by Shen [95], who approximated the continuous resistive loading by a linear stepped-function approximation. Rao et al. [96] demonstrated experimental results of the broadband characteristics of a cylindri cal antenna with exponentially tapered capacitive loading. However, the problem we are dealing with involves a scatterer rather than an antenna, with a lumped loading instead of distributed loading. In the following, an N-port network analogy of a loaded N-port scatterer will be summarized. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 176 - 7.1.1 Analysis of a Loaded N-port Scatterer A loaded scatterer is one which has one or more ports terminated by admittance elements. An N-port scattering system can be represented by a general network diagram as shown in Fig. 7.1. Denote the mutual admittance between port k and port j at angular frequency CD as Ykj (to), which is defined as the short circuit current at port k divided by the voltage at port j with all other ports being short circuited, that is, > i N-port Linear Netwoik System b Fig.7.1 Equivalent network of an N-port loaded scatterer. = /*(«» i k (« ) = ° for all i * j (7.1) If the voltage at each port is Vj((o), then by superposition, the current at port k will be h «Q) = " £ Ykj (co)V, (co) - lgkE(co)Einc (co) (7.2) where Igkg is the short circuit current at port k due to an applied unit incident field Einc (co) with all ports being short circuited. The scattered fields of the loaded Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. scatterer can be written as N Es (co) = X Egj (CO)Vj (co) + EgE(co) >=l (7.3) where Egj (co) is the scattered field due to a unit voltage applied at port j with all other ports being short circuited, Vj (co) is the voltage across port j , and EgE (co) is the scattered field of the scatterer without loading. For linear impedance loading, the port current and port voltage are related by 4 (co) = y*(co)V*(co) (7.4) where y*(co) is the load admittance at port k. Substituting Eq.(7.4) into Eq.(7.2) we have the following matrix relation (7.5) where [ Y ] is a matrix with elements l*; (co), [ y ] is a matrix with elements ym(co) m=n (7.6) m*n and [ lE ] is a vector matrix with element (7.7) ImFM = -/*«£(©)•£,« (®) From Eq.(7.5), the voltage vector [V] can be obtained from the relation, [V] = {[K] + Lv]}-1 [IE] (7.8) Substituting Eq.(7.8) into Eq.(7.3), the scattered field becomes Es(co) = [Eg]T-{[Y] + [y])"1^ ] + EgEm (7.9) where [ Eg ]T is the transpose of the vector [ Eg ] with element Egl (co). It is noted that the quantities [F], [Eg], [IE], and ^ ( c o ) are independent of the loaded impedance, the only load-dependent quantity is the matrix [y]. If only one loading point is present, the scattered fields can be simplified as (7.10) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 178where F (co) is the input admittance of the scatterer at the loading port, and y/,(co) is the loaded admittance. Eq.(7.10) can be abbreviated in the following form E.(co) =X(a>)A (to) + B (co) (7.11) = vy (f») 7 T+ — TT yL (to) <7-12) A (co) = IgE (co)Einc (co)Eg (co) (7.13) with fi(co) = £g£(to) £ ,( co) can be nuUified if the real part of the solution of yE(co) in Eq.(7.10) is nonnegadve. If the real part of the solution is negative, then an active load is required to nullify the scattered field. Assume the load is constrained to be passive, £,(co) can have both maximum and minimum values if yL (co) values are properly chosen. Green expressed the scattered fields of a scatterer loaded with ZL at one-point in a different form [97] Es (Zl ) = Es (Z*) + r;/(Z * )£ r (7.14) =ES(Z*) + r v V(Z*)Er' (7.15) where ES(Z*) is the field scattered by the object with a conjugate matched load, E r is the field scattered by the object if excited by a unit current source at the load point, Z* is the conjugate of the equivalent impedance seen when looking into the load terminals, I(Z*) is the current flowing through a conjugate matched impedance at the load terminal when the object is in the presence of the incident field, V(Z*) is the voltage across the port terminated with conjugate matched impedance, E r is the field scattered by the object when excited by a unit voltage source at the load point, and Tj and IV are the reflection coefficients at the loading point and are given by r* r, = ZL ~ Za ~~— ~ ~ -a + Zt Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (7 .1 6 ) - 179 and rv - C7.it > Z*(Za +ZL) It is seen that Ej(Za),/(Z a), Er , V(Za), and E / are independent of the load impedance. The only load-dependent terms are T/ and Fv . For a passive load the absolute value of the reflection coefficient is less than or equal to 1. If I ES(Z*) I 5 V{Z*)E/ , then the scattered field can be nullified by a load with reflection coefficient ES(Z*) - r v = (7-18) V(Z*)E/ If IE*(Za*)l > W i Z ^ E ' I tlicn l£A.I can not be nullified.However, minimized. In die following, we the Mattered field can bemaximized or willshow that theextremevalues are on the ITI = 1 circle, which implies that the load is purely reactive. Let ES(Z*) = a x + j b x V(Z*)Er' = a 2 + j b 2 r v = * +Jy where a x, b i, a 2, b 2, x , and y are all real. We then have the following problem: Find x and y , such that \Es (Zl )\ = f { T v ) = f ( x , y ) = I (a x+jb {)+(x+jy )(a2+jb2) I (7.19) has extreme values subject to x 2 + y 2<\. The solutions are as follows: 1. If a y + b y < a 2 + b 2 , the minimum value of f ( x , y ) is zero, which occurs when ~(aya2+byb2) X° = --------2TT2— ’ aj+ bj Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (7,20) - 180 and aib2-a2bi y° = 2. a iT+TbTj T (7-21) If a 2+b2>a2 +b2 , the extreme values occur when the following boundary con ditions hold i n = ljt2+y2 l 1/2= 1. (7.22) In this case, the load is purely reactive. Let y 2 = I - * 2, then f 2 = \ E s \2 = (a 2+b 2 )+(a 2 +b2 ) + 2(a ta 2+b \b 2>x+2 (ai~a i&^ l - x 2 (7.23) Take the partial derivative of/ 2 with respect to x» and set it to zero, we then have d f2 „ mi i_ \ — = 2(0^2+^ 1^ 2) ----------- 1 ax \ n ” ® (7.24) (g 1a 2+ b 1b 2)2 !)(a|+b2Z) i ^ ( 2 L ) ' m a x = m a x ( / ( * o ) » / ( “ ■*())) ( 7 -2 6 ) = min( f ( x 0) , f ( - x 0)} (7.27) From the solutions of x 0 and y 0, we can find the required ZL from Eq.(7.17). It is noted that the solution of ZL which minimizes or maximizes the scattered field depends on the locations of the loading points. Es is a function of N complex variables if the number of loading points is N. There are N degrees of freedom to modify the value of Es and the solutions of the null fields or of the extremes may not be unique. If all the impedance values of each loading point are restricted to be the same, Eq.(7.9) is then a function of one complex variable. For that case we can find the extreme values of the scattered fields of the loaded scatterer. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 181 - To obtain the scattered fields of the loaded scatterer, the quantities [Eg], [T], [IE], and EgE in Eq.(7.9) must be determined. All those quantities are dependent on the geometry of the body. Exact expressions of these quantities can be obtained only for certain simple objects. Numerical techniques, however, can be applied to a body of arbitrary shape and are generally only limited as stated earlier by the wavelength size of the body. The moment method is one of the most popular numerical techniques to solve this EM problem. In this section we will numerically investigate the scattering properties of a linearly loaded scatterer. A thin straight wire is the simplest geometry and will be used as an example. A review of the formula tion for a straight wire with impedance loading follows. 7.1.2 Formulation of a Straight Wire with Impedance Loading Consider a cylindrical wire with radius a and extension from z= -/ to z=l along the z-axis as shown in Fig.7.2. Assume the radius is small compared to a wavelength so that the only significant component of current on the wire is the axial component. Denoting the current on the wire as I ( / ) , the z-component scattered field due to this current distribution can be written as [44]: El = — f[-^ (Z-’Z ) + * V (z.z')]/0 O dz' ycoeo -/ dz2 (7.28) where V|/(z ,z ) is the free space Green’s function, with k the wavenumber in free space; R is the distance between the observation point (x ,y ,z) and the source point (x’ ,yf ,z'), and is given by R = ^ ( x - x ' ) 2+ ( y - y ) 2+(z-z' )2 (7.30) If the incident field is Zf , the boundary conditions require that the tangential com ponent of the total field, (i.e., the sum of the scattered field and the incident field) along the wire surface be zero (£/(z) = - E ‘(z)), or f/(z') ytoeo-/ dz2 + kMz,z')]dz' = - £ '( z ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (7.31) - 182z i> A 1 z . zn zb ^ 1 " T ' ' I = 1 o II J Z = -l -H I* a Fig.7.2 Geometry of a straight wire. The above equation is called Pocklington’s Equation [44]. Ez‘(z) is usually known or can be evaluated. The only unknown parameter to be obtained is the current distribu tion 7(z'). For convenience, we rewrite Eq.(7.31) in the following form z \l(z)K (z ,z') dz ' = - E zl (z) -z (7.32) Assume that the current I (z ) is approximated by a series of expansion functions Fn such that I (z ') = h n W ) (7.33) n -1 where /„ is a complex expansion coefficient, and Fn(z') is an expansionfunction which is nonzero only for z' in Az„. SubstitutingEq.(7.33) intoEq.(7.32), and letting z =zm, one obtains £ / „ j Fn (z' )K (zm,z' )dz = -Ei(zm) n=1 Az, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (7.34) - 183 - where the subscript m on zm indicates that the integral equation is being enforced at segment m. Expressing the integral part of Eq.(7.34) as Z ^ , and the right part as Vm, one has the following compact form = Vm m = 1,2 (7.35) M n=l It is noted that the EM integral equation has been reduced to a set of linear simul taneous equations. Expressing Eq.(7.35) in matrix form we have [zm ] [ ',] = [^] (7.36) The matrices f z ^ 1, j/„ j, and jym j are refened to as the generalized impedance, current, and voltage matrix respectively. However, the units of Z ^ ,, /„, and Vn need not necessarily be ohms, amperes, and volts respectively. The boundary conditions can be enforced at a point on each segment or in a weighting average sense over each segment. Denote Wm(z) the m ,h weighting func tion. The boundary condition becomes jWm (z )[E/(z) + E ‘(z )]dz = 0 m =1 ,...A (7.37) By substituting Eq. (7.34) into Eq. (7.37), one has N 1 1 Z A ,J w«(z) i W ) K ( 2 ,z')dz' dz = - j W m(z)E‘(z) dz 71=1 -/ Azm (7.38) -I which can be abbreviated by = Vm (7-39) 71=1 If the weighting function and the expansion function a.e in the same forms, the pro cedure is called the Galerkin method [35]. It is noted that integration over z results in the elements of the generalized impedance matrix and generalized voltage matrix having volts as their units. It is noted that Z ^ is independent of the incident waves; it is only a function of the geometric structure of the scatterer or antenna, expansion function, and weighting function being used. The Choice of a desired expansion Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 184- function and testing function has been widely discussed[36]. The most popular ones are pulse expansion with impulse weighting, 3-term trigonometric expansion with impulse weighting, overlapping triangle expansion and weighting, and overlapping piecewise sinusoidals [37]. The physical meaning of the element Zmn can be interpreted as "the weighted scattered fields integrated over the segment Azm due to a source at segment Az„ with weighting function Wm(z) and source expansion function F„(z')." Once [Z ^ ] and [Vm] are known, the current distribution along the wire can be calculated by [/] = [zm j '1 [ v . ] = [rm ] [v „ ] where (7.40) j = jZfm J 1 and is called the generalized admittance matrix. The physi cal meaning of the elements of Jy^, j is the mutual admittance between port i and port j defined as the short circuit current at port i due to a unit voltage across the port j when all other ports are short circuits. For the specific geometry of a straight wire, the generalized impedance matrix has a special property. All the values of the N 2 matrix elements are contained in any one row of jz^ , j, say the first row, if all segments Azm are of equal length. All other rows are merely a rearranged version of the first row. The elements satisfy the following relationship Zm n= Z \,\m-n i+l n2 1 (7.41) Such a matrix is called a toeplitz matrix. Efficient algorithms are available to calcu late the inverse of a toeplitz matrix. The computing time and storage required for solving a toeplitz matrix is much less than that for solving a non-toeplitz matrix [44]. The inverse of a toeplitz matrix, however, is not in toeplitz form [85]. A scatterer or an antenna with N segments can be viewed as an N-l port net work. At each port, a generator and/or impedance (active or passive) can be con nected in series or in parallel to the port. The N-segment wire and its equivalent circuit are shown in Fig.7.3. If a load Zm and a generator V* is inserted into the m junction having a current lm, the total voltage at that port is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. th - 185- L EC EC EC port port 2 port 3 m th port port 4 Fig.7.3 N segment loadedwire (a) N-l port tenninals pairs, (b). Equivalent circuit for the m th port. Vv m = Vs y m - i1m Z Cjm (7.42) The m th equation in a system of N linear equations will become iz -./.-v a n- Imzm (7.43) 1 or 2 (7.44) = vi n- 1 where Zm-,, ^nvn (7.45) = Zmm nun “I" Zm m Z__' nvi —Z__ tnn for m&n (7.46) It is seen that the effect of lumped loading is simply that of adding a load impedance Zm to the corresponding diagonal elements in the impedance matrix. The impedance matrix, however, is no longer toeplitz, and the efficient program for solving a toeplitz matrix cannot be applied directly. Usually, the number of loading points is much less Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 186 - than the number of segments. The new solution for the current vector can be expressed in terms of the current solution without loading and the loaded impedance [85]. Suppose there are K loading points with impedance serially connected to the m ‘ih port. Denote Z the generalized impedance matrix without loading, Z ' the matrix with element Z zm' form e {m,} ' = ' 0 otherwise Zmn ~ 0 (7-47) f°r m * n (7.48) We have (Z + Z ')7 = V (7.49) Let 7° be the current solution vector without loading, that is, Z7° = V * (7.50) 7° = Z~lV =YV (7.51) where7° is the vectorof the current solution without loading. Multiplying Eq.(7.49) by T, we have Y(Z+Z')I = YV (7.52) (U+YZ')I (7.53) =YV where U is the N.xN unit matrix. After expansion of Eq.(7.53), we have the follow ing linear equation ( l + n , ^ ' wwV«, + £ r « , ^ . / 1^ = / » i*i In = 7„ ° - £ YnmZ m.m.lmi /= n not in {mxrn%..., mK} (7.54) (7.55) i= 1 It isnoted that the number of unknowns in the linearly simultaneousequations of Eq.(7.54) is K (the number of loading points) instead of N (the total number of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 187- segments). The computing time required for solving K unknowns is much less than that for solving N unknowns. After are solved, the current solution /„ ( n not in {m, }) can be expressed in terms of the {/„.} and elements of [ 7° ], the current solution vector without loading. 7.1.3 Scattered Field and Images of a Thin Rod Scatterer In the following, we will use the moment method to find the effect of impedance loading on the scattered fields of a straight wire scatterer and its recon structed images. The piecewise sinusoidal Galerkin method is used. The piecewise sinusoidal function is defined as sin[A: (z — x)] s in tk ^ -z ,,.!)] sin[k(zw+1-z)] *«<*> = sin[fc (zn+1-z „ )] zn - 1 (7.56) zn<z<zn+1 (7.57) and is shown in Fig. 7.4. The elements of the generalized impedance matrix using the above expression and weighting function can be shown to be [44] ^mn ~ *: SW i z - z ^ Q ] + Y sin[k(zm+1-z)] '-1 sin(fcAzm) * sin(&Azm) Zm (7.58) -jkR. -jk R . Rn - 1 j 30 sin(fcAz_) - 2cos(kAzn)Rr R n+1 dz where R n -1 = ^Ja2 + (z-zn_i ) 2, Rn = R n -1 = ^ a 2 + ( z - z n )2 ^la 2 + (7.59) (z-zn+1)2 = zn ~ zn -1 = zn+l ~ z n » ^ zm ~ zm ~ zm -l = zm+l — zm (7.60) When the angle of incidence is 0, the elements of the generalized voltage vector become Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 188- Fig.7.4 (a) Piecewise sinusoidal expansion function, (b) Set of overlapping piecewise sinusoidal expansion functions. Vm = (e'/tem‘lCOS0 —eji*mGOS0[cos(/: Azm) - j cos0sin(fc Azm)] s\n(k Azm)k (l-cos20) + ejh m+1cose _ e jkxmcosejcos^ Azm) + y cos0sin(itAzm)]} (7.61) The impedance dependence of the field scattered from a loaded straight wire will be demonstrated. The maximum and minimum scattered fields can be obtained by suitably choosing an impedance value for a given number of loading point and loading position as previously shown. Noted in Fig. 7.5 are the maximum and minimum values of the back-scattered field, which are obtained by loading a suitable passive impedance at the center point, versus the rod length in terms of incident wavelength at several angles of incidence. The scattered fields without loading (i.e., short circuits) are also shown in the figures. In the broadside (see Fig. 7.5(a)), the difference between Esmax (co) and Esmin (co) is not very great when the wire length is greater than 1.4 wavelengths. Apart from the broadside, scattered fields can be nullified for some frequencies at some aspect angles. This can be explained from Eq. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 189- (7.10). If the second term EgE (co) of Eq. (7.10) is much greater than the maximum value of the first term, the effect of loading on the scattered field will be very small. Apart from the broadside, i.e., in the side-lobe region, EgE(co) can be very small. Therefore, it is more possible to nullify the back-scattered field in the side-lobe region than in the broadside. The required impedances (real part and imaginary part) to nullify the back-scattered fields for angle of incidence equal to 9 = 45° are shown in Fig. 7.6. If the real part of the required impedance is negative, an active load is required to nullify the field. Shown in Fig. 7.7 are the passive impedances required to maximize and minimize the back-scattered field at 0 = 45°. As analyzed in Sec. 7.1.1, the impedance required to optimize the field is purely reactive if the backscattered field cannot be nullified. Observing the plots of the above figures, one notes that the required impedance is highly dependent on frequency. Plots of the required impedance at other 6 (not shown here) also shows the same frequency dependence. From the above figures, one can conclude that fixed impedance loading is effective at reducing or enchanting the RCS only at certain frequencies and aspect angles. As shown in Fig. 7.5, the back-scattered fields can be reduced a great deal through a central passive loading for most frequencies when the aspect angle differs from broadside. At broadside, the difference between the maximum value and the minimum value of the scattered field is small (see Fig. 7.5(a)). Now, we will increase the number of loading points and study their effect on the broadside back-scattered fields. We assume that each loading impedance is of the same value and is reactively varied between -j200Q and y'200Q in 10Q steps. The maximum, minimum, and noload values of the broadside back-scattered field are shown in Fig. 7.8 for the number of loading points nL equal to 3, 5, and 7. The positions of the loading points are also shown in the accompanying figures. From the figures shown, one can note that additional equi-spaced and equi-amplitude loads are fairly ineffective at changing the broadside RCS. Next, we will examine the effect of impedance loading on the range profiles and images. Range profiles can give useful insight into the scattering mechanism. Radia tion can originate from several places on an arbitrarily shaped wire object. These Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 190 2.0 o -3 3 C. 0 .5 r" \ . 2 .3 1.1 3 .5 4 .8 6.0 length in terms of wavelength (a) .3 2 .2 4 •8 .2L i. E * .0 8 3 .5 2 .3 1.1 4 .8 6.0 length in terms of wavelength (b) 05 amplitude ^m ax .0 3 7 5 E (Z / .0 2 5 .0 1 2 5 0 1.1 ^ m in 1 ' v •' .S ’ / ------------- .S C ./- ....... \ A VA -' _____ 3.5 / 7 V y 2 .3 = 0) l 0 / / 4 .8 AV Aw. 6 .0 ' length in terms of wavelength CO F ig. 7 .5 (cap tion se e n ex t p a g e) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 191 - E(Zl = 0) max mm a 4.8 2.3 3.5 length in terms of wavelength 6.0 (d) 2.4 1.2 0.6 1.1 4.8 2 .3 3.5 length in terms of wavelength 6.0 (f) 1.2 =0) ^mix A E{Zl 4> ■aa 0.9 | "^ m infi\ 0.6 0.3 l.l 4.8 3.5 2.3 length in terms of wavelength (e) 6.0 F ig.7.5 Extreme values and no-load values o f the back-scattered field o f a straight wire scattered versus length in w avelength at angle o f incidence equal to (a) 9 0 °, (b) 75°, (c) 60°. (d) 45°. (e) 30°. (f) 15°. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 192 - real part imaginary part V) £ JZ o oo a ea T3 & 2.3 3.5 4.8 length in terms of wavelength Fig.7.6 Impedance required to nullify the back-scattered fields for angle of incidence equal to 0=45°. 400 •3 .S 200 i -400 1.1 2.3 3.5 4.8 6.0 length in terms of wavelength Fig.7.7 Required passive impedance to maximize and to minimize the back-scattered fields for angle of incidence equal to 9=45°. Real part of ZL(Emin) Imaginary part of ZL (E ^ Imaginary part of ZL(Emax) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 193 - Cl 0.5 1.1 4.8 2.3 3.5 length in terms of wavelength (a) 6.0 2.0 o 1.0 2 1* s CQ 1-0 0.5 1.1 2.3 3.5 4.8 6.0 length in terms of wavelength (b) 0 E{Z, = 0) 2.3 3.5 length in terms of wavelength Cc) F ig .7 .8 E x trem e v a lu e s an d n o-load v a lu e s o f the b road sid e back -scattered fie ld s o f a straight w ire v ersu s len gth in term s w a v elen g th w h en the num ber o f lo a d in g p o in ts is (a). nL= 3, (Z[ = 0 .5 / = - z 3 z 2= 0 ) (b). nL= 5 . (Z] = 0 .6 7 / = -Z j, z 2 = 0 .3 3 / = - z 4_ z 3 = 0 ). (c). nL=7, ( z [ = 0 .9 5 / = - z 7. = 0 .6 7 / = - z 6 z 3 = 0 .3 3 / = - z 5 z 4 = 0 ). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 194 - include the excitation region, an impedance load, a change in radius, a sharp bend, a smooth curve and an open end [38]. Consider a straight wire illuminated by an impulsive plane wave with angle of incidence 0 as shown in Fig. 7.9. In this scatter ing arrangement the only places which cause radiation are the end points of the wire. The pulse traveling in free space will impinge on the upper end point first. Part of the incident energy will be re-radiated and the remaining energy will continue to travel along the wire. This traveling pulse will be partly re-radiated when it reaches the lower end and partly reflected upward along the wire. This process of radiation and reflection will continue until the pulse dies out. The original pulse propagating in free space will hit the lower end point some time after it impinges on the upper end point. The process of radiation, reflection, and guiding propagation along the wire will then occur just as in the case of the upper end point. The differential path lengths of radiation occurring at each instant relative to the path length when the impulsive illumination hits the center point of the wire are as follows: at 1 I j = - h cos0 - h cos0 = - 2 h cos0 at 1' l { = hcosB + h cos0 = 2/icos0 at 2 l 2 = -hcosQ + 2 h + /icos9 = 2h at 2' 12 = /icos9 + 2h - /icos9 = 2h at 3 / 3 = -/tcos0 + 4h - /tcos9 = 4h - 2/tcos9 at 3' / 3' = heos0 + Ah +hcosd = 4h + 2Acos0 at 4 I 4 = -/icos9 + 6 h + /icos9 = 6 h at 4' 1/ = ftcos0 + 6 h - hcosQ = 6 h Let the length of the wire be 30 cm, and the frequency coverage be from 6GHz to 16 GHz. In other words, the length in terms of wavelength is from 6 to 16. The polarization of the incident field is assumed to be 0-polarized. Shown in Fig. 7.10 are the magnitude of the range profiles for angle of incidence equal to 30°, 45°, 60°, and 75°. If we carefully examine the range profiles shown in Fig. 7.10, we can find that the peaks marked with 1 and 1' depart more from the center as the angle 0 decreases, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 195 - incident illumination illumination z —-h z = -h Fig.7.9 A straight wire illuminated by an impulse. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 196 - while the peaks marked with 2 and 2', and 4 and 4' remain at the same position and thus are independent of the angle of incidence. Those peaks marked with 3 move toward 2, while those marked with 3' move toward 4' as 0 increases. These obser vations verify the above analysis. It is interesting to note that the ratio of the ampli tude of peak 1 to that of peak 1' decreases as 6 decreases. A real thin rod with length 12" and diameter 1/8" is used as a test object to experimentally verify the above numerical results. The frequency coverage is from 6 GHz to 16.5 GHz and the waves used are co-polarized. It’s range profiles at several aspects are shown in Fig. 7.11. It is noted that a fixed bistatic angle about 16° exists between the transmitting antenna and receiving antenna in the experiment. In this bistatic case the differential path length of path 2 and path 2’ will not be equal. This fact explains the discrepancy between the experimental and numerical range profiles. From the previous analysis and the range profiles shown in Fig. 7.10, one can see that the phenomenon of traveling waves is quite evident in the straight wire case. If the rotation center is chosen at the center point the effect of the constant ranges (2 h , 6 h , etc.) on the reconstructed image will be a ring with constant radius as explained in Chap. 3. Shown in Figs. 7.12(a) and (b) are the numerical and experi mental fringe patterns of the real part and imaginary part of the scattered fields respectively, while Figs. 7.12(c) and (d) are the numerical and experimental images reconstructed over an angular window with 0 from 20° to 80° respectively. It is seen that a ring appears in the images and the end points are intensified. This example shows that the presence of traveling waves usually degrades the image. If three lumped resistors, each with resistance 50£2, are added at z {=0.5l, z2=0, z 3= - 0 .5/, these loading points will cause extra reflections. Both the incident wave impinging on the loading points and the waves traveling along the wire arriving at the loading point will cause additional reflections. This fact results in additional peaks in the range profiles. Shown in Fig. 7.13 are the range profiles of the 3-loading-point wire at several angles of incidence. Examining these plots, one finds that more lobes appear and the lobe produced by the loading point is not as nar row as those produced by the end points. Furthermore, the number of lobes between 1 and I ' is not necessarily equal to the number of loading points (for example, see Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 197 - 0.075 S 0.10 0 37 -37 distance in cm distance in cm (b) (a) 0.1 •u . 0.1 0.075 . . . - 1" u 0.075 • ■a / a U 005 St 0 05 • $ 03 a 0.025 § 0.025 . -37 -37 distance in cm (c) — -r i 0 II J l J 3 ' A* ' 37 74 distance in cm (d) F ig .7 .!0 T h e nu m erical ran g e profiles o f a straight w ire for angle o f incid en ce equal to (a) 30°, (b) 45 °, (c) 6 0 °. an d (e) 75°. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. F ig .7 .11 T h e ex p erim en ta l ra n g e p ro files o f a straight w ire fo r angle o f in cid en ce equal to (a) 3 0 ° , (b ) 4 5 ° , (c) 6 0 ° , and (d) 7 5 ° . -199 - d i p . 0 ; < 0 .8 0 )d e g . max : 2 . 0 ; d l * p ( . 5 , - . 0 6 ) a r^ n b p o d . 0 ; (0 / 9 0 ) d e g max; 7 . 2 ; d l o p ( 2 . - 2 ) " r**l podi (it.T fttfe g w < E E i dl*p (66.30) d i p . 0 ; <0,00)deg m a x ;6 .l;d i« p (,5 .-.0 6 ) r o d , 0 ; <0.90)deg max: 1 4 ; d i s p ( 2 , - . 2 ) tlaiUtad P6di (le.TVldM «*x>45j ditp (4s.a»> F ig .7 .12 N u m e ric a l a n d e x p e rim e n ta l frin g e p a ttern s an d im a g e s o f a th in ro d . (a) R e a l p a rt (left) a n d im a g in a ry p a rt (rig h t) o f th e n u m e ric a lly b a c k -sc a tte re d field p a t te rn . (b ) R e al p a rt (le ft) an d im a g in a ry p a rt (rig h t) o f th e m e a su re d b a c k -sc a tte re d fie ld p a ttern , (c). N u m e ric a l im a g e re c o n stru c te d o v e an a n g u la r w in d o w fro m 0 = 2 0 ° to 8 0 °. (d). E x p e rim e n ta l im a g e re c o n stru c te d o v e an a n g u la r w in d o w fro m 0 = 2 0 ° to 8 0 °. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 200- Fig. 7.13(c), with 0=45°). These additional lobes are due to reflections of the travel ing waves. To experimentally examine the loading effect, we divide the thin rod mentioned into three sections with a 1 mm gap between sections. These gaps are expected to produce a loading effect. However, it is difficult to assign a loading value in each gap. Additionally, the equivalent loading impedance is also a function of frequency because the gap distance in terms of wavelength is changed with frequency. The experimental range profiles for several angles of incidence are shown in Fig. 7.14. More peaks appear in the range profiles due to the additional discontinuities in the gaps. However, the magnitude of these peaks differs from the counterparts of Fig. 7.13. The numerical and experimental fringe patterns of the real part and imaginary part of the scattered fields from the above loaded scatterers are shown in Fig. 7.15(a) and (b) respectively. Their numerical and experimental images reconstructed from an angular window from 0=20° to 80° are shown in Fig. (c) and (d) respectively. It is seen that the loading impedance and the surface traveling waves have distorted the images. By comparing the images of Fig. 7.15 and Fig. 7.13, one can conclude that the images have been successfully distorted by impedance loading. However, the price paid is an increase in the RCS. The measured mean RCS patterns with and without gaps averaged over the whole bandwidth are shown in Fig. 7.16. 7.2 Nonlinear Impedance Loading The property of a load can be characterized by the voltage-current curve (v-t curve) across a load. For a linear load, the v - i curve is a straight line. If the v - i is not a straight line, then the load is called a nonlinear load. Typical examples of non linear ioads are diodes for switching, thresholding, modulation, and harmonic genera tion. Nonlinear effects are important in the area of electromagnetics. For example: nonlinear effects have to be considered when an antenna system containing semicon ductor devices is illuminated by a strong lightening stroke or a nuclear Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 201 - .1 0 <i> 3 T3 i jiA a .025 /I L — 0 -37 37 (a) (b) 74 .10 - • U .075 •a 2 | 37 distance in cm .10 *, -075 ■a 0 distance in cm 3 .050 U -050 .025 .025 -37 0 37 distance in cm (c) 74 -37 0 37 distance in cm (d) F ig .7 .13 N u m erical range profiles o f a straig h t w ire w ith three load in g p oints at angle o f incidence e q u a l to ( a ) 30° (b) 45°, (c) 60°, and (d) 7 5 ”. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 -202 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - F ig .7 .14 E x p erim en ta l range p r o file s o f a straigh t w ir e w ith th ree lo a d in g p o in ts at a n g le o f in c id e n c e eq u a l to (a) 3 0 ° , (b ) 4 5 ° , (c ) 6 0 ° , and (d ) 7 5 ° . - 203 - d i p . 3 ; (0 ,7S )deg d i p . 3 ; ( 0 ,7 S ) d e g max ■1 , 8 ; d l s p ( . 5 , - . 0 6 ) max :6.2;dlsp(^S,__-p05) a r o d . 3 ; ( 0 .9 0 ) d e g m ax ;4 .3 ;d lsp (2 .-,2 ) r o d . 3 ; C0,90)deg max: 1 4 ; d l s p ( 2 . - , 2 ) b r**l rodj 4 tegM niftj tl0*70)d»g mx >GB; d t* p (58*33) d ilMul*t*d rod) fiL*3i ft»60 ohm mx >48; d l* p (48*33) c Fig.7.15 Numerical and experimental fringe patterns and images of a thin rod with three loading points, (a). Real part (left) and imaginary part (right) of the numerically back-scattered field pattern, (b). Real part (left) and imaginary part (right) of the measured back-scattered field pattern, (c). Numerical image reconstructed ove an angular window from 8=20° to 80°. (d). Experimental image reconstructed ove an angular window from 9=20° to 80°. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 204 - 3 M P 1 % U * e i n d b hV E .3 : RGD0 AND R0D3 Fig.7.16 Comparison of the measured mean RCS patterns of the thin rod with and without gaps. electromagnetic pulse [90]; a scatterer with nonlinear loading may spread the spectra of the scattered field into several harmonics, "thermalizing" thereby the scattered field.. Nonlinear effects for a wire scatterer and a dipole antenna have been extensively studied. Sarkar and Weiner [91] applied the Volterra series technique to analyze a nonlinearly loaded antenna. The advantage of that method is that individual fre quency components in the output can be determined directly without performing a fast Fourier transform on the total time domain. The Volterra series technique is use ful if the nonlinearity is not too strong. Schuman [98] used a space-time-domain integral equation to treat a nonlinearly loaded antenna. Liu and Tesche [90] used two methods for analyzing antennas and scatterers having nonlinear resistive loads. The first approach involves the use of a direct time-domain integral equation. The second uses frequency domain data to compute the time-dependent currents and voltages across the nonlinear loads. The nonlinear problems are then solved by time-stepping and convolution utilizing the solution of the linear portion of the network. Djordjevic and Sarkar [99] used a method similar to the method of [90] but with a different approach, which requires less time-domain data or a shorter duration of Green’s func tion. Laudt and Miller [85] used the time-domain integral equation to study the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 205- behavior of a wire antenna or wire scatterer loaded with nonlinear elements, the non linear load types considered include those with piecewise-linear voltage-current curves and a load with time varying resistance. In this section we wili first review the analysis of an N-port nonlinearly loaded scatterer and then numerically discuss the scattering properties of a nonlinearly loaded thin rod scatterer. 7.2.1 Analysis of an N-port Nonlinear Loaded Scatterer Consider an N-port network. Assume an ideal voltage source v; (r) is applied across the port j , and all the other ports are short circuited. The short circuit current at port k due to vy-(r) can be expressed as Ik (co) = Ykj (co)Vj (co) where V j ( ( 0) (7.62) k =1A..JV is the FT of v; (f), /*(©) is the FT of the short circuit current ik(t) at port k, Ykj (co) is the mutual admittance between port k and j , and co is the angular frequency. If Vj(co) = 1 for all co, then the short circuit current ik(t) in the time domain will be the inverse FT of Y kj (co), that is, (7.63) where the subscript gkj denotes the Green’s function corresponding to the excitation at port j and response at port k. If an arbitrary function Vj (co) is excited at port j , the response at port k will be i k { t ) = F - l { Y kj { a y V j ( ( s » } t = ig k j ( t ) * Vj(t) . (7.64) o w h ere * d en o tes co n v o lu tio n . W hen all ports are ex cited , the cu rre n t at p o rt w ritten as k can be (7.65) j =l o If th e lo ad at p o rt j is lin ear, then there is a sim p le relatio n betw een Vy(co) and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 206Ij (co), and the current at each port can be obtained from the frequency domain by solving the simultaneous equations. However, if the load is nonlinear, the relation between the load current and the load voltage can no longer be expressed in fre quency domain, but has to be treated in time domain. Let the v - i relation of the nonlinear load at port j be defined by ij(t) = Fj(Vj(t)) (7.66) where Fj is a known function. By substituting Eq. (7.66) into Eq. (7.65), one has N \ ^ jt(v jk (0 )= E jigkj(t^t)Vj(x)dx j=i o (7.67) The above equation can be expressed in discrete form, that is, the integration can be replaced by the summation Fk (vk (<?Af)) = £ £ igkj [(<7-p )&t]Vj (p At )At ;=1 p=0 *=1,..JV (7.68) where the subscript q denotes the q th instant time step. The sum in the above equa tion can be separated into two parts Fk (v* (q At)) = '£ igkj(0 )Vj(qAt)& j =i + £ qZ ig kj[(q-p)to]vj (pAt)At j =1 p=o k=\,...,N (7.69) The first term in the right part of the above equation contains the load voltage at t=qAt, while the second term contains only the the load voltage at the previous times. The terms igkj (0) and igkj ((q -p )Ar)for agivenelectromagnetic becalculated in advance. If the load voltageof the previous system can values are known, the only unknown parameter in Eq. (7.69) is the instant ioad voltage Vj (q A t) at t=qAt. This unknown can then be determined by solving the simultaneous nonlinear equa tions of Eq.(7.69) and the load currents can then be solved with Eq. (7.66). Consider a scatterer with an n-port nonlinear load being illuminated by a plane wave, its equivalent network can be expressed as shown in Fig. 7.17. Computation of the radiated field requires an additional port to include the incident field as an Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. - 2 07- r N-port linear network system Yij{ G>) » 1 8 i► C*1 1 Fig.7.17 Equivalent network of an N-port nonlinear loaded scatterer. excitation. The load current and the radiated fields can be calculated from the following equations N q N X Fk (v* (qAt)) = £ £ igkj [(q - p )At ]vj (p At )At i-\p = o (p At )At (7.70) p=0 and N q Erad (QAt) = X S y=ip=o TP ]vj(P & )Af + i ^ £ [(<7-p)Ar]EI„c(pA/)Ar p=0 (7.71) where igkE represents the short circuit current at port k due to an impulse incident field excitation; EgJ and EgE represent the Green’s functions for the scattered field when there is an impulse excitation applied at port j and an impulse incident electric Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 208 - field respectively. To reduce the required time duration of the impulse response at each port, Djordjevic and Sarkar [99] proposed a method which involves inserting a suitable resistance terminated at each port so that a shorter duration of Green’s function can be used to carry out the calculation. To deduce the original characteristic, a negative resistance of the same value is then connected in series with the inserted resistance. 7.2.2 Scattering Properties of a Nonlinearly Loaded Thin Rod Scatterer Intuitively, it is expected that nonlinear loading will generate harmonics of the incident wave, so that the energy of the fundamental mode of the back-scattered wave will be reduced. In this subsection we will examine the above statement and use a thin rod as a test object. For a straight wire scatterer with given length to radius ratio (I/a), the time functions ig^ (r), igkE(t), Egj(t), and EgE(t) of the previous subsection can be found from the inverse FT of the frequency domain data which can be obtained by the moment method as described in the previous section. These functions are independent of the loaded impedance. The characteristics of the nonlinear loads are specified by the v - 1 curve of the load. The frequently used nonlinear loads are the piecewise linear resistors, or the loads with their v - i relation which can be expressed as a summation of a power series. T h e n o n lin e a r lo ad to be c o n sid ered is p iecew isely lin ear an d is c h ara cteriz ed by (7.72) w h ere v c u t is th e c u to ff v o ltag e o f the n o n lin ear load (fo r e x am p le, a diode), y i a n d y 2 are th e fo rw ard an d rev ersed c o n d u ctan ce o f the load resp ectiv ely , and v ( r ) an d t ( r ) are th e v o ltag e acro ss the load and th e cu rre n t flo w in g th ro u g h the load re sp ec tiv e ly . In th e fo llo w in g calcu latio n s, w e w ill assu m e th at v c u t c a n alw ay s be c o m p e n sa te d by a b ias v o ltag e so that the effectiv e v c u t is eq u al to zero. T h ere is an Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 209- abrupt change in the slops when the load voltage passes through the zero value. This abrupt change might generate a dc term and higher harmonics. As seen from Eq. (7.71) the scattered field can be separated into two com ponents: a load-dependent part (t!<s first term) and a load-independent part (the second term). If the illuminated wave is a monochromatic plane wave, the loadindependent component is the field scattered from the load-free scatterer and is sinusoidally varied in the steady state. The nonlinear load only affects the first term of Eq. (7.71). In the following examples we will discuss these two components indi vidually. Consider a thin rod, with length equal to three wavelengths and a length to radius ratio equal to 100, that is illuminated by a sinusoidal plane wave from broad side. A lead is inserted at the center of the rod. If the characteristic of the load is piecewisely linear with y i = 1 and y 2 - 0.01, the time domain back-scattered field of the load-dependent part is shown in Fig. 7.18(a). Its spectrum is shown in Fig. 7.18(b), which is obtained by a FT of the time domain field. It is seen that addi tional harmonics appear in the spectral plots. If the load is linear with y i = y 2 = 1, its corresponding time-domain field and spectra are shown in Figs. 7.18(c) and 7.18(d) respectively. While shown in Figs. 7.18(e) and 7.18(f) are the counterparts when the load is linear with y j = y 2 = 0.01. Note that the scales in the above figures are not all the same. In reference to the above spectral plots, one can see that the magnitude of the fundamental mode with nonlinear loading is greater than that with _yj = jy2 = 1 but is smaller than that with yj = y 2 = 0.01. From the viewpoint of RCS reduction, fixed nonlinear loading is ineffective at reducing the RCS at a certain frequency because one usually can have a smaller RCS by suitably choosing a linear load with conductance equal to one of the two slopes of the v - i curve. The loadindependent component of the time domain field is shown in Fig. 7.18(g). It is seen that the strength of the load-independent part is much stronger than that of the loaddependent part. The spectra of the back-scattered field before and after nonlinear loading are shown in Figs. 7.18(h) and (i). The above figures tell us that a single nonlinear load has very little effect on the RCS and spectra when the wave is illuminated from the broadside direction. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -210-. l.MOOs • — ' c a n p la p o i n t fig .7 .1 8 (a) •-06 f i g . 7 . 16(b ) V > M 1 I t ud a ( er • I \ t 1.3225a*02 aampla p o i n t fig.?.l8<«? V a " f 1.8088 «-»7 1 t )) 1.2m ; .-97 ’\f*jwywv\< at 7.18(d) 7 jg Option see p3ge 212) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -211 - 9 m T l t u d 9 t r • a 1 t. lanpl* ♦'tg7.ia(9> —- po in t a n ? i t u d • Fig. 7.18 (caption see next page) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -212- 1.49994*99 • — l a x s l a p o in t flj.7 .t9 !3 ) 2.9449 4-94 1.3999 - ■ 4 -9 * 1.9494 4-94 9 . 9999 « -« 9 k. ' -7.34944*46 fij. -3 .7 S 4 « 4 * 9 9 -2 .92974-92 3.79914*99 (h) 7 .1 9 (h ) 2 .9 9 9 9 4-64 " 1.3969 -■ f 4-94 1 , t i I 1 .9 9 9 9 4 -9 4 3 .9 9 9 9 4-99 s-7.59944*49 ♦'i j. 7 , t 0 ( i ) 1 Jv- - 2 .9 2 9 7 « - * 2 3 .7 9 9 1 4 * 9 9 (i) F ig .7 .1 8 T im e d om ain field s and their spectra o f a n on lin early and lin early loaded w ire scatterer. (a) T h e load -d ep en d en t p a n o f the back -scattered field o f a non lin early lo a d ed w ire w ith y j = 1 a n d y 2 = 0 .0 1 . (b) Sp ectral p lot o f (a), (c) T h e lo a d -d ep en d en t p a n o f the back -scattered field o f a n on lin early lo a d ed w ire w ith y j = y 2 = 1* (d ) Spectral p lo t o f (c). (e ) T h e lo a d -d ep en d en t p a n o f the b ack-scattered field o f a n onlinearly lo a d ed w ir e w ith y l = y 2 = 0 .0 1 . (f) S p ec tral p lo t o f (e). (g ) T h e lo a d -in d ep en d en t p a n o f the back -scattered field o f a thin w ire, (h ) Sp ectral p lot o f (g ). (i) T h e spectra o f the back -scattered field o f a n o n lin early lo a d ed thin w ire w ith y i = I and y 2 = 0 .0 1 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 213 - i 1 . mi»*ii — : V a " If i t ( s a x p l a p o in t 4.0000 e-0 4 3.0000 e-0 4 'i 2. m i ; a-0 4 1.0000 a-0 4 - • -----------------ji -----------' i Fig.7.19 Time domain scattered fields and spectra of a nonlinearly loaded wire scatterer with three loading points, (a) The load-dependent component of the back-scattered fields, (b) The spectra of the total back-scattered field. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 214- If we increase the number of point from 1 to three and locate them at z =0, ±1 / 2 , the load-dependent component of the time domain field and the resultant spectra of the total back-scattered field are shown in Figs. 7.19(a) and (b) respec tively. Compare Fig. 7.19(a) and Fig. 7.19(b), one can see that increasing the number of loading point increases the contribution of the load-dependent component. However, the effect is still very small with broadside illumination. 7.3 Time Varying Loading The loads considered in the previous sections are fixed loads. In this section we will consider another type of load, time varying load, whose impedance value is time dependent Advances in technology have made it possible to control the load elec tronically. It has been pointed out that a time varying load might shift the apparent frequency of the scattered field to provide a false doppler shift and to make the receiver at the probing radar unable to phase lock to the frequency of the incident wave [85]. An example of a half-wave dipole scatterer with time varying resistive loading was given in [85]. In this section we will study some other interesting phenomena caused by a time varying load and will numerically examine how effective the time varying load may be in producing the expected effects. The object to be tested is a thin rod and the factors to be considered are the type of loads, length of rod, and direction of incident waves. An incident scalar wave with angular frequency co0 and magnitude £,° can be expressed as £ ,(r ) = R e t E . V 0* } (7.73) where Re{*} denotes the real part of *. The doppler frequency corf of an object moving with velocity u toward the radar is then u v>d = “ o— c (7.74) where c is the light velocity. Assume that the load value is varied with time and is denoted by ZL(t), the time domain back-scattered field is then Es (0 = Re { E's (t)e )(o)o+(OdX Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (7.75) - 215 - where E's (t) = Es [ZL (t)] (7.76) If ZL (t) is a constant, the doppler frequency which appears in radar is (ad . If ZL(t) is sinusoidally varied with frequency cot, the spectra of the scattered field will have the following components: co0 + 0)rf ± «<»!, where n is an integer. However, the magni tude of each spectral component depends on how E's [ZL(t)\ changes with respect to ZL(t). If the variation in the magnitude and phase of E's [ZL (t)\ is great during excursion of one cycle, the portions of the spectral components other than ©0+0)d will be more pronounced, which may provide the radar with a false doppler fre quency. On the other hand, if the variation is very small, the other spectral com ponents will be much smaller than the dominant component (Oq+co^. In that case, time varying loading is ineffective at providing false doppler frequencies. The effectiveness of a time varying load can be measured by a modulation index defined as _ \E's {ZL(t))\ max 11 “ IE's (ZL (t)) Imax IE's (ZL(t))\ min (7.77) \E’s (ZL(t))\ min The value of T| is between 0 and 1. If the modulation is close to 1, the scheme will be more successful in providing a false doppler shift. If T| is very small, the scheme cannot provide the desired effect. In Fig. 7.5, we have plotted the possible maximum and minimum values of the back-scattered field of a central passively loaded straight wire versus frequency at certain aspect angles. From those plots, we can predict how effective the time vary ing loading is. In the following we will examine the spectra of Es (t) of a sinusoidally varied load, either varied resistively or reactively. Assume the resistively varied load is given by ZL(t) = RL{t) = 250(1-coscojt) Q while the reactive loading is given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (7.78) - 216 - ZL (t)= j XL (t) = j 250 sinov Q (7.79) the loading point is at the center of the load. Shown in Fig. 7.20 are the spectra of E's(t) of a straight wire operated at several wavelengths and incident angles. The modulating index of each case is also shown in the accompanying figures. From Fig. 7.5 and Fig. 20, we make the following conclusions: 1. There are more sidebands in the spectral plots when the modulation index is greater. 2. Reactive loading usually has a greater modulation index than resistive loading does, which implies that reactively time varying load are more effective at pro ducing a false doppler shift. 3. The modulation index is small in the broadside case when the length of the wire is greater than 1.4 wavelengths. 4. In the side-lobe regions, the modulation index can be close to 1 for many fre quencies but not for every frequency. The above analysis and conclusions are examined from the consideration of pro viding a "false" Doppler frequency. However, the same conclusions can also be applied in the RCS consideration. A radar receiver usually has a bandwidth limita tion. If the load varying frequency is greater than the bandwidth of the receiver, the sideband energy will be rejected by the receiver. In this sense we can say that time varying loading does spread the spectrum of the back-scattered field. It is noted that the philosophy of diaphanization by nonlinear loading is also based on the spectrum spreading effect. Nevertheless, time varying loading differs from nonlinear loading in the following: a. The separation between the successive harmonics in time varying loading is the load varying frequency, while the separation between the successive harmonics in nonlinear loading is the frequency of the incident wave. b. When the modulating frequency of the time-varying load is much smaller than the carrier frequency of the incident wave, time-varying loading can be viewed as a form of linear loading. In that case time-varying loading can be analyzed in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - LIZ- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. F ig .7 .2 0 Spectra o f a tim e vary in g load ed scatterer w ith param eters (a) /=1.2X, 0 = 90°, T] = 0.21, nL = (b) / = \.2X, 0 = 90°, n = 0.82, nL = (c) / = 3 . OX, 0 = 90°, T| = 0.24, nL (d ) / =3.0X., 0 = 4 5 ° , t) = 1.0, nL = 1, 1, z= 0, resistive loading. 1, z= 0, reactive loading. 1, z = 0, reactive loading. z = 0 , rea ctiv e load in g. - 218 - the frequency domain. While nonlinear loading can only be analyzed in the time domain. Practically, it is much easier to produce a time varying loading effect than to produce a nonlinear loading effect in the application of radar scattering. An input bias energy (cutoff energy) is usually required to make a practical diode ( for exam ple, a step recovery diode) producing the desired nonlinear property. However, the incident wave energy is usually not great enough to satisfy this requirement. On the contrary, the input impedance at a loading port can be easily varied. For example, the input impedance at the loading port can be electronically controlled by inserting a digital phase shifter or a tuning stub between the loading port and the terminating loaa. We will now examine the effect of time varying loading on the range profiles and the reconstructed images. The loaded values of the examples shown in Sec. 7.1.3 are fixed for each frequency and each aspect angle. Examining the range profiles of those examples and the range profiles shown in chap.6, one can find that there are more peaks in the range profiles due to the created discontinuities. How ever, those peaks which are a result of reflection from the end points of the object can still be easily observed in the range profile even though some extra peaks have been created between them. The range distances at the right most and the left most peaks in the range profile usually provide information about the object’s dimension. One may use an active broad-band slave jammer to obscure or to distort the range information, but this is not what we wish to discuss. We try to use a passive impedance load to. achieve this goal. Impedance loading can change the magnitude and phase of the scattered fields. If the loaded values are randomly varied for each loading point, each frequency, and each aspect angle, that is. if they are randomly varied for each time instant, this randomness might cause random peaks in the range profile. However, if the number of loading points is too small, or if the range for which the impedance value is varied is not suitably chosen, the random loading may not give the desired effects. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 219 - We have concluded that reactive loading can make a more drastic change in the scattered fields (either in phase or amplitude) than resistively loading can, and increasing the number of loading points can produce greater field variation. It is also known that the reflection coefficient at a given point is a function of the input impedance and the loading impedance at that point. If the loaded impedance at a point is randomly switched between capacitive loading and inductive loading, the phase of the reflection coefficient at that point will be changed at each time instant. Consequently, the range profile may produce more random peaks and the magnitude of the peaks corresponding to the loading point may be reduced. In the following we will compare the effect of a fixed loading, a randomly resis tive loading, and a randomly reactive loading on the back-scattered field and the range profiles of a loaded straight wire. The parameters used are nL=5, and 6=45°. Each loading impedance is either fixed to SO ohms, or randomly resistively varied from 0 to 100 ohms, or randomly reactively from -j50 ohms to +j50 ohms. The magnitude and phase of the back-scattered field and the range profiles of the above three loading cases are shown in Figs. 7.21, 7.22, and 7.23 respectively. From the above figures one can find that the difference in the RCS with fixed resistive loading versus that with randomly resistive loading is small and randomly resistive loading does not create random peaks in the range profile. If the number of loading point is changed to 3 and 7 and each load is randomly varied between (-y'50, y'50)Q, the corresponding range profiles are shown in Figs. 7.24(a) and (b) respectively. It is seen that a small number of loading point is inadequate for generating random peaks and additional loading points obscure the range profile more effectively. The real part and imaginary part of the back-scattered fields and the recon structed images of the randomly reactively loaded wire with nL=7 are shown in Fig. 7.25. Comparing Fig. 7.25 with Fig. 7.17, one sees more noise in the image of the ranH nm lv rpnpHvpIv InaHpH sraftprpr Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 220 - .75 .50 o. rm 3 .25* 1.1 4.1 10.0 7.0 length in terms of wavelength (a) 13.0 180 -180 QL 1.1 u 4.1 7.0 -37 0 distance in cm (c) 13.0 10.0 length in terms of wavelength (b) .075 -3 3 a .050 .025 - -74 74 37 F ig -7 21 T h e (a) m a g n itu d e and (b ) p h a se o f the b a c k -sca ttered field , a n d (c ) the ra n g e nL= 5, z p r o file of the lo a d ed straigh t w ire [ = 0 .6 7 / = - z 5 z 2= 0 .3 3 b - z 4, z 3= 0 , an d 0 = 4 5 ° w ith zL =50Q, w h ere Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 221 - a -75 -3 ! , .5 0 1.1 4.1 10.0 13.0 length in terms of wavelength 7.0 (a) 180 00 a "O c -180 1.1 4.1 10.0 7.0 13.0 length in terms of wavelength (b) a -3 3 Q. s3 .075 - .0 5 0 .025 -74 -37 0 distance in cm 37 74 (c) F ig .7 .2 2 T h e (a ) m a g n itu d e and (b ) p h a se o f the b a ck -sc a tter e d fie ld , an d (c ) the ran ge p ro file o f th e lo a d ed straigh t w ire, e a c h im p e d a n c e is ran d om ly r e sistiv e ly v a ried b e tw e e n (0, 100)Q, w h ere nL= 5, z l=Q.67l=-2S z2=0.33/»-Z4, z3=0 , and 0=45°. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 222 « *3 3 - .75 9* .50 .25 l.i 4.1 7.0 10.0 length in terms of wavelength (a) 13.0 180 ao u •a e -180 1.1 4.1 7.0 10.0 length in terms of wavelength 13.0 (b) .075 i . .050 .025 -74 -37 0 distance in cm 37 74 (c) Fig.7.23 T he (a) m agnitude and (b) phase o f the back-scattered field, and (c) the range profile o f the loaded straight wire, each im pedance is reactively random ly varied betw een (-y 50, y 50)Q , where r t L - 5 , z l= 0 .6 7 /= -?5 i z 2=0.33/>Z4, z 3=0, and 0=45°. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 223 - v a n P 1 7.5 000 e-0 2 i t u a e — > d i s t a n c e i n cm fig . 7 . 24 (a) (a) 1.0000 e-01 7 .5 0 9 9 e-0 2 5 .0 0 9 0 a-0 2 2 .5 0 0 9 a-0 2 -7.5 0 0 9 a* 0 1 - 3 . 7646e+01 — d is t a n c e in cm -2 .3 2 9 7 e -0 1 ... fig. ' 7 . 2 4( b ) 3 . 706ie*01 ^ Fig.7.24 The range profiles of a randomly loaded straight wire with (a.) nL = 3, z x = 0.5/ = - z 3, and z2=0. (b) nL = 7 , z { = 0.951 = - z 7, z2 = 0.67/ = - z 6, z3 = 0.33/ = - z 5, and z4 = 0. Each impedance is randomly varied between (—y 50, y‘50)Q, where 0 = 45°. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 224 - d 1p r n n .7 r ; NL=7 (.96,-1.75);(.5,-.05) d l p r a n . 7 1 ; N_=7 (4.81,-3.32);(.5,-05) I (a) (b) iipoinfLB^jjfTseasp) i (ge.oo d y n a (lS 7 .0 )i(1 5 0 ,0 )' (C) ' (d) Fig.7.25 (a) Real part and (b) imaginary part of the range-corrected scattered field of the randomly reactively loaded straight wire with nL = 7, z. - 0=95/ = - z 7. z2 = 0.67/ = -Zf,. z3 = 0.33/ = - z 5, and z4 = 0. The reconstructed image using angular window over (c) (20°, 90°) and (d) (20°, 84°). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 225 - 7.4 Discussion A thin rod has been used as a test object throughout this chapter to numerically examine how linear loading, nonlinear loading, and time varying loading affect the scattering properties and images of an object A scale model Boeing 747 aircraft has been approximated by a pipe model and a thin wire model and their transient responses have been measured and compared [38]. In this section we will first present microwave images of a wire model airplane and then discuss the acceptability of a thin wire in the approximation of a finite edge. An optical image and a microwave image of a wire model airplane are shown in Fig. 7.26(a). The object was mounted on a rotating pedestal with inclination angle equal to zero degrees. The angular window used is from broadside to nose-on and the frequency coverage is from 6 GHz to 16.5 GHz. Examining the reconstructed images one can find that the image intensity of the front "edge" and rear "edge" of the right "wing" is almost of equal brightness. The above phenomena are the major differences from those of a scale model as shown in the previous chapters. The rea son for this difference may be explained as follows: most parts of this wire model object can be illuminated by the incident wave. In other words, no shadow region exists in this object. Next we will discuss the acceptability of a thin wire in the approximation of a finite edge. Consider a z directed wedge with finite length L as shown in Fig. 2.8. It has been pointed cut that the back-scattered field of an edge is a function of the azimuth angle and the elevation angle. The back-scattered field pattern is a sine function along the 0 direction and is a function of <|> in the azimuthal direction. In contrast, the back-scattered field pattern of a long wire is close to a sine function (except near the end-on direction) along the 0 direction and is isotropic in the <)> direction. If we divide the wedge into two regions by introducing an intervening insulator as shown in Fig. 6.27, the back-scattered field can be expressed as the superposition of the fields scattered from the narrow strip and the remaining plates. The above statements assumes that the narrow strip and the remaining plates are iso lated scatterers (i.e., no mutual coupling existing between them) and the scattering from the insulator is much smaller than that from the conducting parts because of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) (b) Fig. 7.26 Optical image and microwave image of a wire model airplane (a) optical image (b) Co-polarized image reconstructed over angular window (<J> = 0° to <j) = 90°) perfect conductor insulator Fig. 7.27 A loaded conducting wedge. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 227 - low reflectivity of the insulator. The narrow strip might then be approximated by a thin wire and the properties of a loaded thin wire examined in the previous sections may be applied to the narrow strip case. As indicated in Sec. 6.4, the major contributors to the scattered field of a plate are their tips when the incident wave is not normal to the edges and the equivalent scattering strength from a tip at a given aspect is not a function of the length of the edges. Therefore, the scattered field from the narrow strip is comparable to that from the remaining plates when the aspect is in the side-lobe region. For the above reason it is expected that the effect of introducing impedance loading to the narrow strip may not be overridden by the presence of the remaining plates. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 228 - CHAPTER 8 CONCLUSION In this dissertation we briefly review the basic scattering properties of a perfectly conducting object. In the high frequency region, the scattered field of a complex shaped object can be attributed to a combination of several mechanisms. The advent of high resolution radar enables us to consider each scattering mechanism separately. We have interpreted the microwave image of a conducting object from a new view point, based on the understanding of the scattering mechanism and the image recon struction algorithm. The connection between various scattering mechanisms and their reconstructed images is then established, from which we can interpret what the image represents and predict what the image will look like over given spectral and angular windows. Several numerical and experimental examples have been included to sup port this new approach to image interpretation. Based on the understanding of the scattering mechanism and the knowledge of modem spectral estimation techniques, we devise a new algorithm to extrapolate the available data into the missing band and apply this algorithm to radar imaging. Both simulation and experimental results have shown the effectiveness of this method in microwave diversity radar imaging. A significant application of this new algo rithm is in identifying the hot spots of a target when the bandwidth of the imaging system is too narrow to give acceptable resolution. The satisfactory interpretation of microwave images and the effectiveness of the extrapolation algorithm devised are fundamental to the study of RCS management and target recognition. We introduce a new term, "diaphanization", defined as the techniques of reducing RCS and the techniques of obscuring an image. We consider the RCS management not only from a detection perspective, the reduction of the target’s RCS to elude radar detection, but also from the image point of view, the dis guising of a target’s appearance to impede radar recognition. This consideration Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 229- opens a new direction for future research in RCS management studies. We give the procedure which employs microwave diversity imaging to diaphanize a target over prescribed spectral and angular windows and demonstrate the robust ness of the diversity imaging system to the Gaussian noise. We also propose some rules for distorting an image. These rules are: create artificial discontinuities, create multiple reflections and make the reflectivity at a given point a function of time. Traditional techniques for reducing the RCS are applied to RCS management studies. These techniques are absorber covering, target shaping, and impedance load ing. We review the theory pertaining to absorber-covered bodies and establish the theoretical background for obscuring an image by absorber covering based on the physical optics approximation. Certain types of absorbers are used to cover metallic objects. Their influence on RCS reduction, range profile modification and image dis tortion are experimentally studied and discussed. We propose a concept concerning the absorber covering patterns where we also consider the reflections from the boun daries between the covering absorbers and conductors. By using suitable covering patterns the RCS can be reduced more effectively. Finally we examine some possible effects of loading an object with lumped impedances. A thin straight wire is used as a test object. Plots of range profiles show that the surface traveling wave is an important scattering mechanism of a straight wire. Microwave images of a loaded wire are both numerically and experi mentally obtained. The specific scattering mechanism of the thin wire makes its image unique in appearance. We also demonstrate that nonlinear loading and time varying loading can cause the spectra of the back-scattered field to spread from the incident wave, which may produce several interesting phenomena, such as providing a false doppler frequency, making the receiver unable to phase lock to the frequency of the incident wave, generating unexpected peaks in the range profiles, and distort ing the reconstructed image. The main imaging geometry of concern in this dissertation involves rotation of objects. However, the approach to image interpretation described in Chap.3 can also be applied to bistatic angular diversity imaging, which is to synthesize the angular Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 230- aperture by varying the bistatic angle between the transmitter and receiver. One may first study the bistatic scattering properties and range profiles for various scattering mechanisms and then back-project the range profiles according to the relative motion between the transmitter and receiver. Because the information collected by bistatic diversity imaging is different from that by rotating an object, the reconstructed images are expected to be different. However, it is possible to predict the image from the sinogram of the bistatic image. 8.1 Future Research Several areas which may be further investigated are suggested below: A. Analysis of microwave diversity imaging constructed in this dissertation assumes that the observation distance is in the far field region. The scale model airplane used is 1:100. Assume the frequency operated in the real system is 2 GHz, then the frequency in the scale system should be 200 GHz. Therefore, millimeter wave imaging is more desirable in order to simulate a real system. However, the observation distance conducted in the anechoic chamber will be in the near field region for such high frequencies. A thorough analysis of near field diversity imaging for a conducting object has not been reported and should prove to be a challenging research topic. B. Interpretation of the microwave images conducted in this dissertation also assumes that the transmitter and receiver both lie in the azimuthal plane perpen dicular to the rotational axis. The image reconstructed is a projective image. If the line of sight is not perpendicular to the rotational axis, the image will be unfocused. Seeking an efficient algorithm to correct the unfocused image would be an interesting task. C. The range resolution of microwave diversity imaging depends on the bandwidth available while the cross-range resolution depends on the angular aperture in terms of wavelength. The extrapolation algorithm developed in Chap.4 is to extrapolate the scattered fields into the exterior band for each fixed aspect. The applicability of this algorithm is based on the fact that the scattering properties Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 231 - of each scattering mechanism are independent or only slightly varied with respect to frequency. In practice the angular window available may be too nar row to give acceptable cross-range resolution. Methods of increasing the crossrange resolution are worth investigating. D. The images obtained are projective images. The locations of the hot spots in the direction of the rotational axis are still ambiguous. A 3-d tomographic image can pinpoint the hot spots more accurately. Analysis of 3-d tomographic images for conducting objects was based on the physical optics approximation [9]. However, that interpretation encounters the same difficulties as mentioned in Chap. 3. Therefore, a more convincing interpretation for 3-d imaging of a complex shaped metallic object needs to be developed. E. The comparison of the images before and after diaphanization are made by the human brain. That judgement is subjective. How to combine machine vision with knowledge of scattering and image formation should prove to be very valu able future research work. Automatic machine recognition and evaluation of microwave diversity data based on models of neural networks is an active research area at the Electro-Optics and Microwave Optics Laboratory. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 232- APPENDIX Consider two identical parallel cylinders with radius a, separated with a distance 2r 0 seated on a rotating pedestal and illuminated by a plane wave as shown in Fig. 5(a). The cross section of that arrangement is shown in Fig. Al. The line connecting the centers of the cylinders passes through the rotating center and makes an angle 6 with the respect to the rotation axis. The polar coordinatesandrectangular coordi nates of the points on the surfaces of these two cylinders with respect to cylinder center are (a , each y ^ , and (a, <j>2'), (x2f, y { ) respectively. Let the bis tatic angle between the transmitter and receiver be a, the unit vectors in the direction of transmitter and receiver be I, and lr respectively. To find the stationary points of the multiple reflection, geometrical optics will be applied. The reflection law requires that the angle between the incident ray and normal fine must be equal to the angle between the reflected ray and the normal line. The stationary points of multiple reflection on the respective cylindrical surfaces for a given set of {r0, a , 6, a} are to be determined. Assume the incident ray first hit cylinder 2, reflect to cylinder 1, and then bounce back to the receiver as shown in Fig. Al(a). The notation of the angles I;, i|f2, a, 0, 0i', §2 *s defined in Fig. A 1(a). It can be shown the following relationship must be satisfied in order to satisfy the reflection law. 2 rosin0 - a [costo?' - aJ2 ) + sin(W] — H-----------.— ILf----------------- — = -cot(20, - a/2 ) 2 rocos0 + a [sin(<p2' - a/2) -cos<)>2 ] (A.l) <t)j' = q>2' - 90° - a/2 (A.2) where <|>2' is restricted to be 270° < 02 ^ 360° when -90° < 0 < 90° 180° < 02' < 270° when 90° < 6 < 270° For a given set of parameters {r0, a, 0, a }, the azimuth angle <t>2' and therefore 0i' can be determined by solving Eq.(A.l) and (A.2). The other case is that the incident ray first hits cylinder 1, reflects to cylinder 2, and then bounces back to the receiver. The geometry of this situation is shown in Fig. A 1(b). The stationary points must satisfy the following relationship Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 3 3 y 012 a ll y y a ll Fig.Al Multiple reflections of a two-cylinder object, (a). The incident wave reaches the left cylinder first, and is then reflected back by the right cylinder, (b). The incident wave reaches the right cylinder first and is then reflected back by the left cylinder. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 234 - <J>i" = <>2" - 90° + a/2 2 rosin0 - a [cos( <{>2" + a/2) + sin<|>2"] 2rocos0 + a [sin( <|>2" + a/2 ) - cos<t>2"] (A.3) = -cot(2<|>2" + a/2) (A.4) From the above derivation, it can be seen that there are two pairs of multiple reflection points for a bistatic system and only one multiple reflection point for the monostatic case. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 235 - BIBLIOGRAPHY [1] H. P. Baltes, ed., Inverse Scattering Problems in Optics, Springer-Verlag, New York, 1980. [2] W-M. Boemer, "Polarization Utilization in Electromagnetic Inverse Scattering," in Chapter 7 of Inverse Scattering Problems in Optics, H. P. Baltes ed., Spring-Verlag, New York, 1980. [3] L. J. Cutrona, E. N. Leith, L. J. Porcello, and W. E. Vivian, "On the Applica tion of Coherent Optical Processing Techniques to Synthetic Aperture Radar," Proc. IEEE, Vol. 54, No. 8, pp. 1026-1033, 1966. [4] Kiyo Tomiyasu, "Tutorial Review of Synthetic Aperture Radar (SAR) with Applications to Imaging of Ocean Surface," Proc. IEEE, Vol. 63, No. 5, pp.563-583, 1975. [5] N. H. Farhat and C. K. Chan, "Three Dimensional Imaging by Wave-Vector Diversity," in Acoustical Imaging, Vol. 8, Edited by A. F. Metherell, Plenum Press, New York, pp. 499-516,1980. [6] J. L. Walker, "Range-Doppler Imaging of Rotating Objects," IEEE Trans. Aerospace and Electronic Systems Vol. AES-16, Vol. 1, pp. 23-52, 1980. [7] D. C. Munson Jr., J. D. O’brien, and W. K. Jenkins, "A Tormographic Formu lation of Spotlight-mode Synthetic Aperture Radar," Proc. IEEE, Vol. 71, No. 8, pp.917-925, 1983. [8] N. H. Farhat, "Principles of Broad-Band Coherent Imaging," J. Opt. Soc. Am., Vol. 67, No.8, pp. 1015-1021, 1977. [9] C. K. Chan, and N. H. Farhat, "Frequency Swept Tomographic Imaging of Three-Dimensional Perfectly Conducting Object," IEEE Trans. Antenna and Propaga., Vol. AP-29, pp. 312-319, 1981. [10] N. H. Farhat, C. L. Wemer and T. H. Chu, "Prospect for Three-Dimensional Projective and Tomographic Imaging Radar Networks," Radio Science, Vol. 19, No. 5, pp. 1347-1355, 1984. [11] T. H. Chu, "Optimal Methologies in Inverse Scattering Utilizing Wavelength, Polarization and Angular Diversity," Ph.D. Dissertation, University of Pennsyl vania, Dec. 1983. [12] Yuhsyen Shen, "Dynamic Microwave Imaging," Ph.D. dissertation, University of Pennsylvania, 1987. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 236- [13] G. Tricoles and N. H. Farhat, "Microwave Holography: Applications and Tech niques," Proc. o f the IEEE, Vol. 65, No. 1, pp. 108-121, 1977. [14] K. Iizuka and L. G. Gregoris, "Application of Microwave Holography in the Study of the Field from a Radiating Source," Applied Physics Letters, vol. 17, No. 112, 15 Dec. 1970. [15] T. Sato, O. Ikeda, and K. Endo, "Combined Spectral and Aperture Synthetic Ultrasonic Imaging System," IEEE. Trans. Sonics and Ultrasonics, Vol. SU28, pp. 64-69, 1981. [16] H. Emert, and R. Karg, "Multifrequency Acoustical Holography," IEEE. Trrans. Sonics and Ultrasonics, Vol. SU-26, pp. 279-286, 1979. [17] H. J. Li, H. A. Yeuh, and G. Wade, "Using Multiple Frequency Method for Artifact Reduction," Proc. o f the National Science Council, Taipei, Republic of China, Part A: Applied Science, Vol. 7, No. 4., pp.301-312, O ct, 1983. [18] H. J. 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