# The generalized spatial correlation algorithm for self-calibration of microwave antenna arrays

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Order Number 9321359 The generalized spatial correlation algorithm for self-calibration o f microwave antenna arrays Borsari, Geordi K enneth, Ph.D. University of Pennsylvania, 1993 Copyright ©1993 by Borsari, Geordi Kenneth. All rights reserved. UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. THE GENERALIZED SPATIAL CORRELATION ALGORITHM FOR SELF-CALIBRATION OF MICROWAVE ANTENNA ARRAYS Geordi Kenneth Borsari 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 1993 r y ' Um / \ Bq^nardD. Steinberg. Supervisor of Dissertation Sohrab Rabii Graduate Group Chairperson Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. COPYRIGHT © Geordi Kenneth Borsari 1993 - 11- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TO MICHELE, MADELINE, AND MY PARENTS, KEN AND DIANA -iii- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A ck n o w le d g m e n ts My utmost thanks, appreciation, and regards go to my advisor Dr. Bernard D. Steinberg. His confidence and belief in me over the past four years has truly helped to make it possible to see this work to completion. I also wish to thank my fellow members of the Valley Forge Research Center, namely Dr. Rich Pauls, Randy Perlow, and Dr. Qing Zhou, all of whom freely gave their time to discuss various aspects of this research. Special thanks go to Randy Perlow for the multitude of discussions we had regarding all aspects of this work. I also want to thank Shirley Levy for all of her administrative help and Donald Carlson and Walter Borders for all of their help with the experimental data sets. More thanks than I could ever say go to my wonderful wife, Michele, who lived through this with me and always stood by and supported me through both good and bad times. I know it's been a long road. Lastly, I want to thank my parents, Ken and Diana, who always supported me through all of my studies. They sacrificed a lot to allow me to attain this goal. To all of you, Thank You. -IV - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT THE SPATIAL CORRELATION CLASS OF SELF-CALIBRATION, PHASE CORRECTING ALGORITHMS Geordi K. Borsari Bernard D. Steinberg High resolution microwave imaging systems generally require the use of large aperture array antennas. Such large aperture systems inevitably experience phase errors in the measured data. The Spatial Correlation class of algorithms is one class of algorithms that attempts to remove these phase errors from the recorded data set. This work develops a general algorithm (GSCA) that characterizes this class and allows its characteristics and properties to be studied. The generalized theory is applied to successfully self-calibrate experimental data sets that could not be calibrated successfully with existing theory. The GSCA reveals that the Spatial Correlation class is divided into two sub-classes of algorithms. Extensive simulations are used to compare performances of algorithms from each sub-class in the presence of receiver noise and element position errors. The performance curves presented as a result of the performance study represent the first such set of performance curves available and can be used as design curves for system designers. In the last chapter the GSCA is extended for self-calibration with near-field data. -v- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents C hapter 1: Introduction — 1 1.0 Objective...................................................................................................1 1.1 Contributions............................................................................................2 1.2 History of the Spatial Correlation Class of Algorithms....................... 5 1.3 The M icrowave Imaging System................................................... 8 1.4 The Experimental D ata............................................................................ 12 1.5 Dissertation Outline..................................................................................12 1.6 Dissertation Summary............................................................................. 14 C hapter 2: Existing Spatial Correlation-Based Algorithm s — 15 2.0 In tro d u c tio n ...........................................................................................15 2.1 The Spatial Correlation Class Defined.......................................... 17 2.2 The Spatial (Auto)Correlation Function of an Incoherent Source Distribution............................................................................................. 17 2.3 The Muller-Buffington Algorithm..........................................................19 2.4 2.5 2.6 2.3.1 The Image Sharpness Function..............................................19 2.3.2 Maximizing the Image Sharpness Function in the A perture D om ain.............................................................. 22 The Unit-Lag Spatial Correlation Algorithm..................................24 2.4.1 Practical Implementation of the Unit-Lag S C A ....................26 2.4.2 An Iterative Unit-Lag SCA..................................................... 27 The Multiple-Lag Spatial Correlation Algorithms................................. 28 2.5.1 The Least-Squares Multiple-Lag S C A .................................. 29 2.5.2 The Iterative Multiple-Lag SCA..................................... 33 2.5.3 The Full and Partial Multiple-Lag SCA's..............................35 Other Spatial Correlation-Based Algorithms..................................36 2.6.1 The Shear Averaging Algorithm............................................ 36 - vi - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.7 Chapter 3: 2.6.2 The Phase Gradient Algorithm................................................37 2.6.3 The Flax and O'Donnell Algorithm................................. 38 2.6.4 Phase Closure From Radio Astronomy..................................39 S u m m ary .................................................................................................40 The G eneralized Spatial Correlation A lgorithm — 42 3.0 In tro d u c tio n ........................................................................................... 42 3.1 The Energy Conservation Algorithm...................................................... 43 3.2 3.3 3.1.1 Development of the E C A ......................................................... 43 3.1.2 Performance of the ECA with Simulated D ata.......................46 3.1.3 Performance of the ECA with Experimental Data.............. 47 The Generalized Spatial Correlation Algorithm..................................... 57 3.2.1 Re-Interpreting the ECA Objective Function.......................... 57 3.2.2 Concepts of the Generalized Spatial Correlation Algorithm..................................................................................58 3.2.3 Development of the Generalized Spatial Correlation Algorithm................................................................................. 61 3.2.4 The Sub-Classes of the Spatial Correlation C lass................. 65 The Relationship of the GSCA to Existing Spatial Correlation Algorithms................................................................................................68 3.3.1 Revisiting Tsao's Energy Conservation Algorithm.............68 3.3.2 Use of A Narrow Interval of Integration with Experimental D ata........................................................... 71 3.3.3 Deriving the Modified Muller-Buffington Algorithm From The GSCA ............................................85 3.3.4 The Unit-Lag SCA and the G SC A .........................................86 3.4 Commonality of the Maximization Procedures........................................87 3.5 S u m m ary ...................................................................................................88 Reproduced with permission of the copyright owner. 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C h ap ter 4: Noise and Element Position Error Performance C om parision — 90 4.0 In tro d u c tio n ..........................................................................................90 4.1 Element Position Errors..........................................................................91 4.1.1 Introduction.............................................................................. 91 4.2.1 The Element Postion Error Simulation................................... 93 4.1.3 Main Beam Loss Due to Element Position Errors................. 98 4.1.4 An Approximation to the SCA Performance Curves............. 107 4.2 Receiver Noise Analysis........................................................................110 4.3 Analysis of the Solutionsof the Parametric Sub-Class and theM M B.................................................................................................122 4.4 C h ap ter 5: 4.3.1 Linearization of the MMB and the Parametric Sub-Class..............................................................122 4.3.2 The GSCA and the MLSCA.............................................131 S u m m ary ................................................................................................. 133 Self-C alibrating with the GSCA Using N ear-F ield Data — 135 5.0 In tro d u c tio n ........................................................................................... 135 5.1 The Near Field of a Linear A rray............................................................136 5.2 Elimination of the Effects of Near-Field Phase Curvature on the GSCA............................................................................................138 5.2.1 Mathematical Characterization of the Near-Field Effects on the GSCA Objective Function........................................... 138 5.2.2 Robustness of the Unit-Lag Spatial Correlation Algorithm................................................................................. 143 5.2.3 The Near-Field Weighting Function.................................. 144 5.2.4 The Effect of a Dominant Scatterer on the Near-Field Weighting Function.............................................. 146 5.2.5 Use of the Near-Field Weighting Function with Practical High-Resolution Imaging Systems.........................153 5.3 Performance of the GSCA with Near-Field D ata...................................157 5.4 S u m m ary ..................................................................................................163 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h ap ter 6: C o n clu sio n s — 165 6.0 S u m m ary ................................................................................................. 165 6.1 Conclusions.............................................................................................. 165 6.2 Suggestions for Future W o rk .................................................................. 170 A p p en d ices — 171 A: Derivation of the Variance of the Correlation Phase Noise Resulting From Additive Receiver Noise.............................................. 173 B: Main Beam Relative Gain Curves For Array Element Position E rro rs .......................................................................................................180 R e fe re n c e s .................................................................................................................. 202 In d e x 206 - ix - Reproduced with permission of the copyright owner. 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LIST OF TABLES Table 3.1 Summary of Results with Simulated D ata............................. 76 Table 4.1 Parameters of Main Beam Loss and the Values Used in the Simulations.......................................................... 96 Table 4.2 Largest Correlation Lag Necessary to Achieve Peak Performance.............................................................................97 Table 4.3 Summary of the Utility of the Spatial Correlation Algorithms.............................................................................104 Table 4.4 Performance of the ECA and M M B .....................................112 Table 4.5 Algorithm Preferences for Various Situations..................... 113 Table 5.1 Order of Magnitude Approximation of System Parameters.............................................................................154 Table 5.2 System Parameters for VFRC ISAR Imaging System.... 154 Table 5.3 System Parameters for the VFRC 83-Meter Imaging System.................................................................... 154 Table 5.4 Results of Calibration with the GSCA and Unit-Lag SCA with no Dominant Scatterer and a Data Set Originating in the Near F ield.............................................. 157 Table 5.5 Results of Calibration with the GSCA and Unit-Lag SCA with a Small Dominant Scatterer and a Data Set Originating in the Near F ield...............................................160 -x- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF ILLUSTRATIONS Figure 1.1 Diagram of Imaging System ......................................................................... 9 Figure 1.2 Diagram of a typical experimental data set in matrix form. The vertical axis represents elements in the phased array. The horizontal axis represents range to the target. Consequently, any row represents a sampled range profile received at a particular element and any column represents the data received across the array from a particular range.............................................................................................. 10 Figure 1.3 The self-calibration scheme: p's represent channel phase errors, jV s represent estimates of the phase errors, w's represent standard beamsteering weights....................................................................................11 Figure 2.1 Illustration of the geometry used in deriving equation (2.3). (Adapted from Reference 2 )............................................................... 18 Figure 2.2 Illustration of the Muller-Buffington optical phase correcting scheme. (Reprinted from Reference 24).................................................... 20 Figure 2.3 Illustration of the performance of the Muller-Buffington phase correcting scheme using simulated data. (Reprinted from Reference 2 4 )....................................................................21 Figure 2.4 Illustration of the performance of the Muller-Buffington phase correcting scheme using experimental data. (Reprinted from Reference 24)................................................................................................ 21 Figure 2.5 Illustration of the correlation measurement between elements n and (n+1).............................................................................................................. 27 - xi - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2.6 Illustration of correlation measurements of lags Z j, Z2>^3 Figure 2.7 The form of the matrix A and vector b from equation Figure 3.1 Illustration of image formation with no phase errors present in the 29 (2.9).................33 imaging system...............................................................................................45 Figure 3.2 Illustration o f image formation with phase errors present in the imaging system...............................................................................................45 Figure 3.3 Images of rangebin #1 of simulated far-field data set. (a) Image using undistorted data, (b) Image using distorted data, (c) Image using Figure 3.4 calibrated d ata.............................................................................48 Images of rangebin #4 of simulated far-field data set. (a) Image using undistorted data, (b) Image using distorted data, (c) Image using Figure 3.5 calibrated data............................................................................. 49 Images of rangebin #6 of simulated far-field data set. (a) Image using undistorted data, (b) Image using distorted data, (c) Image using Figure 3.6 calibrated...d ata............................................................................. 50 2-D images from an experimental data set. (a) Image obtained using the Unit-Lag SCA. Au = 30.5 mrads. (b) Image obtained using the ECA with (c) Image obtained using the ECA with Au = 0.66 m rads..........................................................................................52 Figure 3.7 Cross-range images of the dominantscatterer rangebin of the ISAR data set using the (a) Unit-Lag SCA. (b) ECA with Au = 30.5 mrads. (c) ECA with Au = 0.66 mrads.......................................................53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.8 Cross-range image of the dominant scatterer rangebin of a SAR data set using the (a) Unit-Lag SCA. (b) ECA with Au = 30.5 mrads. (c) ECA with Au = 0.66 mrads.................................................................... 55 Figure 3.9 Average image intensity of the undistorted simulated data......................... 58 Figure 3.10 Illustration of the result of the convolution of a rectangle function with a sinc-squared function.........................................................................67 Figure 3.11 Image of rangebin #1 of the simulated data after calibration with Au = 17.24 mrads.........................................................................................70 Figure 3.12 Average image intensity of the undistorted simulated data when a dominant scatterer exists in the data set....................................................... 73 Figure 3.13 Image of rangebin #1 with a dominant scatterer included in the averaging process.......................................................................................... 75 Figure 3.14 Image of rangebin #1 of the simulated data set with Au=16.7 mrads and a dominant scatterer present...................................................................75 Figure 3.15 Average image intensity after calibration.......................................................76 Figure 3.16 Images of three rangebins of simulated data with integration interval equal to [13,180] mrads................................................................................ 77 Figure 3.17 Images of three rangebins of simulated data with integration interval equal to [-167,0] m rads....................................................................... 78 Figure 3.18 Average image intensity of the ISAR data when the dominant scatterer rangebin is included in the averaging process.............................. 80 - xiii - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.19 Average image intensity of the ISAR data when the dominant scatterer rangebin is not included inthe averaging process.................... 80 Figure 3.20 Cross-range image of the dominant scatterer rangebin of ISAR data with Au=0.66 mrads and the dominant scatterer rangebin absent from the averaging process................................................................. 82 Figure 4.1 Theoretical relative gain curve for the DSA with the results o f four experiments showing agreement with the theory. (Reprinted from Reference [45]).............................................................................................. 94 Figure 4.2 Simulated relative gain curve produced by the simulation of Section 4.1.2, showing very good agreement with the theoretical DSA curve and the experimental results of Figure 4.1..................................................94 Figure 4.3 Relative gain curves for various numbers of correlation lags and a dominant scatterer present in the data set. (a) Relative gain curves for non-parametric sub-class, (b) Same as (a) with L=1 curve removed, (c) Relative gain curves for the parametric sub-class for the same case as (a).......................................................................................97 Figure 4.4 Relative gain curves for various numbers of correlation lags and no dominant scatterer present in the data set. (a) Relative gain curves for non-parametric sub-class, (b) Relative gain curves for the parametric sub-class for the same case as (a)............................................98 Figure 4.5 Main beam gain curves for ufov = 0.017 rads., o Ax = X. (a )L = l and (b) L=29................................................................................................. 100 Figure 4.6 Main beam gain curves for ufov = 0.167 rads., a Ax = X. (a) L=1 and (b) L=29................................................................................................. 104 - xiv - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.7 Difference between the DSA and SCA relative gain curves for fieldsX X of-view simulated and for (a) g rms position error and (b) - rms X position error and for (c) ^ nns position error and (d) X rms p o sition e rro r............................................................................................... 106 Figure 4.8 The phase error profile across the array used in the noise analysis simulations...................................................................................................... 108 Figure 4.9 RMS residual phase error after calibration for (a) 23 dB SNR (b) 13 dB SNR (c) 3 dB SNR with no dominant scatterer present in the data set.............................................................................................................113 Figure 4.10 Main beam relative gain after calibration for (a) 23 dB SNR (b) 13 dB SNR (c) 3 dB SNR with no dominant scatterer present in the data set.............................................................................................................114 Figure 4.11 Average sidelobe level after calibration for (a) 23 dB SNR (b) 13 dB SNR (c) 3 dB SNR with no dominant scatterer present in the data set.....................................................................................................................115 Figure 4.12 Peak sidelobe level after calibration for (a) 23 dB SNR (b) 13 dB SNR (c) 3 dB SNR with no dominant scatterer present in the data set.....................................................................................................................116 Figure 4.13 RMS residual phase error after calibration for (a) 23 dB SNR (b) 13 dB SNR with a dominant scatterer present in the data set...........................117 Figure 4.14 Main beam relative gain after calibration for (a) 23 dB SNR (b) 13 dB SNR with a dominant scatterer present in the data set...........................118 Figure 4.15 Average sidelobe level after calibration for (a) 23 dB SNR (b) 13 dB SNR with a dominant scatterer present in the data set.................................119 - xv - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.16 Relative difference between the GSCA and the MLSCA solutions 131 Figure 5.1 Illustration of the effect of sever near-field phase curvature...................140 Figure 5.2 Error-corrected phases of the correlation lags 1 through 40. These lags were computed from the dominant scatterer rangebin of the experimental data after calibration had been applied........................... 147 Figure 5.3 Error-corrected phases of the correlation lags 41 through 70 and 100 through 110. These lags were computed from the dominant scatterer rangebin of the experimental data after calibration had been applied..............................................................................................................148 Figure 5.4 Phase of averaged correlation coefficient of lag 1. Averaged over rangebins 30-60. Indeed, dominant scatterer bin dominates the averaging process........................................................................................... 149 Figure 6.5 Unwrapped undistorted phases of correlation lags 1 - 10 of the simulated data with a dominant scatterer present in the data set............. 150 Figure 6.6 Unwrapped error-corrected phases of the correlation lags 1 - 10 of the simulated data with a dominant scatterer present in the data set........... 150 Figure 6.7 Unwrapped error-corrected phases of lags 1 - 10 of the simulated data without a dominant scatterer present in the data set.............................151 Figure 5.8 Correlation function and weighting function taken from experimental data with a dominant scatterer and FOV=0.66 mrads. Solid line represents correlation function from experimental data set. Dashed line represents weighting function with FOV=0.66 mrads. Dasheddotted line represents weighting function with full FOV=30.5 mrads............................................................................................................... 154 - xvi - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 5.9 GSCA image with rectangular weighting function o f extent Au=0.167. There is no dominant scatterer present and the data set originates in the near field..............................................................................157 Figure 5.10 Unit-Lag SCA image with no dominant scatterer and a near-field data set............................................................................................................. 157 Figure 5.11 Estimated correlation function and the sine weighting function when no dominant scatterer is present in the data set................................... 158 Figure 5.12 (a) GSCA image with ECA approximation. Dominant Scatterer am p(l)=1.9 Field-of-view = 0.01724 (b) Unit-Lag SCA Image with Dominant Scatterer amp(l)=1.9 FOV = 0.01724......................... 160 Figure 5.13 Estimated correlation function and the sine weighting function when a dominant scatterer is present in the data set...............................................160 - xvii - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Preface A Brief Description of the Chapter Contents. The first section of Chapter 1 contains an outline of the significant contributions of this work. The remainder of the chapter provides a background of the development of the Spatial Correlation Class along with details concerning the experimental data sets used in this research. Chapter 2 describes the algorithms known to belong or be related to the Spatial Correlation class prior to this research. This chapter explains each algorithm for reference in future chapters and establishes notation to be used throughout the remaining chapters. Chapter 3 is the main body of this work. In this chapter the Generalized Spatial Correlation Algorithm (GSCA) is introduced, derived, and shown to characterize the entire Spatial Correlation class. The GSCA is used to show that Tsao's Energy Conservation Algorithm (ECA) is a member of the Spatial Correlation class. The GSCA is shown to successfully self-calibrate experimental data when the ECA failed. The reasons for the ECA failure is found in the GSCA theory. Chapter 4 compares the performances of algorithms from each sub-class in the presence of receiver noise and random element position errors through extensive simulations. The plots of the expected relative gain curves in Appendix B represent the first compilation of such curves that can be used for system design purposes. Chapter 5 addresses the problem of performing self-calibration using near-field data sets with multiple-lag correlation algorithms. It is shown in this chapter that near-field data - xviii - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sets are not as restrictive as originally thought and in most practical systems, modification of the algorithms for near-field data sets is not necessary. It is shown that the presence of a dominant scatterer is sufficient to eliminate any adverse effects on the calibration process caused by near-field data. Chapter 7 is a summary of the conclusion drawn from this work along with suggestions for future work. - xix - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Introduction 1.0 Objective The first working procedure for se lf calibration of a distorted radar phased antenna array was the Dominant Scatterer Algorithm (DSA), which required the existence of a scatterer with small physical size and large radar cross section (RCS) somewhere in the field-of-view [45]. This limitation is eliminated in algorithms that are based on the autocorrelation function of the measured wavefront of the backscatterered radiation field. This work develops a new self-calibration algorithm for a distorted radar phased antenna array that: 1) subsumes all other published correlation-based self-calibration procedures; 2) includes an algorithm, based upon Parseval's Theorem, hitherto believed to be the basis of another type of self-calibration procedure; - 1 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3) shows that the performance of these algorithms parallels that of the DSA and is as good as the DSA when a dominant scatterer is present in the data set. 1.1 Contributions The primary contributions of this research are as follows: 1) Development o f the Generalized Spatial Correlation Algorithm (GSCA) (from Ch. 3) M aximization of this algorithm's objective function provides a means for determining the estimates of the phase and element position errors present in the imaging system. The GSCA is shown to include all published algorithms that operate upon the spatial correlation measurements of the wavefront. These procedures are called spatial correlation algorithms. Examples are the Modified Muller-Buffington Algorithm [49], the Unit-Lag Spatial Correlation Algorithm [1,3], the Shear Averaging Algorithm [20], and the Least-Squares Multiple Lag Algorithm [48]. Each of these algorithms is described in Chapter 2. It also includes Tsao's Energy Conservation Algorithm (ECA) [55,56] which is based upon Parseval's Theorem and therefore had hitherto been thought to operate upon different properties of the wavefront statistics. The analysis herein discloses that spatial correlation algorithms can be divided into model-based and non-model-based groups called parametric and non-parametric sub classes. The ECA belongs to the former and the Modified Muller-Buffington belongs to the latter. Analysis in Chapter 4 shows that the parametric sub-class provides slightly superior performance, provided that the model is correctly chosen. Lastly, because the - 2 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. GSCA objective function embodies all of the known spatial correlation algorithms, new inter-relationships between these algorithms are revealed. 2) Study o f Element Position Error and Receiver Noise Tolerances (from Ch.. 4, and Appendix B) The GSCA objective function is a highly nonlinear function of the phase error estimates and the phase noise of the measured correlation values. Analysis shows that spatial correlation algorithms subsumed within the GSCA yield least-squares solutions for the phase error estimates when the phase noise variance is small. This makes algorithms from the parametric sub-class particularly attractive since they yield the same solution as the MLSCA under these conditions without the phase unwrapping process required by the MLSCA. Computer simulations lead to design data showing effects of noise and element position errors on mainbeam gain and sidelobe level as functions of field-of-view size, scan angle, rms element position error, and the number of correlation lags used in the calibration process. Performance curves are provided in Appendix B for design engineers. It is found that the main beam reconstruction performance of the GSCA is equivalent to the DSA when a dominant scatterer is in the field-of-view, and is ^ to 1 dB poorer without it. As a consequence the mainbeam gain performance can be estimated at any scan angle based on DSA theory plus a single measurement made at boresight. Additional contributions are: 3) Modification o f the GSCA fo r Self-Calibration with Near-Field Data (from Ch. 4) Since the GSCA is a multiple-lag algorithm, the stationarity of the random complex radiation field measured in the array aperture must be addressed. The algorithm must be modified to account for the non-stationarity of the random complex field measured in the array aperture that occurs when the source distribution is located in the near-field of the -3- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. array. The modification is adapted from Subbaram's work [49] with the Modified MullerBuffington multiple-lag algorithm. It is shown that under certain conditions it may be necessary to include a weighting function (termed the "near-field weighting function") in the GSCA objective function to remove the effects of the near-field phase curvature present in the recorded data set. Additionally, this chapter establishes that near-field data sets are not as restrictive as originally thought and in most practical systems modification of the algorithms (i.e., the use of the near-field weighting function) for near-field data sets is not necessary. 4) The Commonality o f the Maximization Procedures (from Ch. 3) Each of the spatial correlation algorithms referred to above was independently invented or derived for differing circumstances and different conditions. Understandably, the methods of weight vector adaptation are also different. Since all of the published spatial correlation based algorithms are subsumed within the GSCA, only one efficient method is needed to maximize the objective function. A generic non-linear optimization package can be used. However, the methods developed specifically for individual algorithms are now known to be generally applicable to all algorithms belonging to the GSCA class. These methods exploit Hermitian symmetry properties that exist in the objective functions, thereby producing efficient optimization methods. Whereas previously a different efficient optimization method had to be implemented for each algorithm, now with the GSCA theory only one method need be implemented and various spatial correlation algorithms can be used by changing the weighting function in software. 5) Analysis o f the Spatial Correlation Sub-Classes (from Ch. 4) The GSCA objective function is a highly nonlinear function of the phase error estimates and the phase noise of the measured correlation values. Because of this no significant mathematical theory regarding level of additive receiver noise or the level of -4- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. element position error that can be tolerated by spatial correlation algorithms can be derived. Analysis of algorithms from the two sub-classes shows that both algorithms yield approximate least-squares solutions. It is shown that the least-squares problem solved by the parametric algorithm is the same as the one solved by the MLSCA. However, using the parametric algorithm instead of the MLSCA eliminates the phase unwrapping requirement of the MLSCA. The analysis also shows that Attia's ULSCA is the unit-lag algorithm of the parametric class and is not a degenerate version of the MMB as previously claimed in [29]. Lastly, the analysis predicts the performance of the parametric algorithm to be superior to the performance of the non-parametric algorithm. All of these results are observed in the computer simulations presented in the same chapter. 1.2 History of the Spatial Correlation Class of Algorithms In high-resolution microwave imaging, each element of a phased array antenna samples the received complex field generated by the source or scatterer distribution. Phase errors are inherently present in the signals in each receiver channel because of element position errors, propagation medium distortion, multipath, electrical mistunings, etc. These errors degrade the mainlobe gain of the array, increase its sidelobes and sometimes cause multiple lobing. The result is severe distortion of the image. A crucial step in the imaging process is self-calibration of the system for the purpose of removing the phase errors. Removal of the phase errors can be accomplished by the signal processor of the imaging system. If information regarding the system phase errors is known, the signal processor can account for the phase errors when processing the received data. If no such information about the sytem phase errors is known (as is generally the case), then -5- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. measurements made of the wavefronts received from one or more beacons or sources positioned at a known locations can disclose the phase errors present in the system. Once the phase errors are known the signal processor can remove them when processing the data. This is a survey type of calibration; it requires the positions of the sources to be known . In the general radar situation a priori information of this nature is not available. Instead, self-calibration must be employed. Self-calibration, also known as adaptive beamforming, is an adaptive procedure in which calibration is accomplished by extracting information concerning the channel phase errors directly from the measured data set, i.e., the received complex field measured at every element in the array. Any algorithm designed to self-calibrate an array system must operate on the error-corrupted data set received by the distorted array. The idea of self-calibrating optical telescopes to compensate for atmospheric turbulence was introduced in 1953 by Babcock [4], In 1954 Green [23] proposed an analogous method in the field of X-ray crystallography. Rogstad [38], in 1968, presented another technique to correct for atmospheric distortion in an optical interferometer. The spatial correlation type of algorithm was introduced in 1974 by Muller et al. [31] who proposed a real-time self-calibration method to correct atmospherically degraded telescope images; they reported the first observatory results using their method in 1977 [12], The Muller-Buffington technique involved operations upon the image intensity distribution; it was shown to be equivalent to spatial correlation operations in the aperture plane in 1977 [25]. Since the early 1980's self-calibration techniques have been used in radio astronomy to compensate for wavefront distortion induced by spatial variations in the refractive index of the propagation medium [10], [11], [17],[37]. The Valley Forge Research Center (VFRC) independently developed methods to perform the task of self-calibration for high-resolution microwave imaging systems - 6 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [ 1 , 3 , 2 8 ,2 9 , 4 0 ,4 1 , 4 2 ,4 5 , 4 8 ,4 9 , 5 4 ,5 6 ] . The m easurem ent instrum ent is known as the "radio camera" and was first described by Steinberg in 1973 [41]. The 1973 Steinberg paper described a technique to self-calibrate a large phased array used to obtain high resolution images of microwave scatterers/sources. The first radio camera experiment demonstrating the feasibility of microwave self-calibration was performed one year later [41]. The algorithm upon which most subsequent radio camera experiments are based was published in 1981 [42] with further details and examples in [28, 29,44, 45, 48, 52]. It is known as the Dominant Scatterer Algorithm, (DSA). Based on the analysis by Hamaker et al. [25] of Muller's theorem [31], Attia, in 1984, developed the (Unit-Lag) Spatial Correlation Algorithm (ULSCA) [1 ,3 ,4 8 ]. Unlike the DSA, which operated upon the wavefront from a rangebin having a dominant scatterer, it exploits the properties of the spatial correlation function of the received complex field generated by the source distribution being imaged. Other algorithms that do so are the Muller-Buffington method [29], the Modified Muller-Buffington Algorithm (MMB) [29],[49], the Multiple-Lag Spatial Correlation Algorithm [48],[49], and the Energy Conservation Algorithm (ECA) [55],[56]. Until this work the ECA was not known to belong to the Spatial Correlation Class. Muller et al. showed that an incoherent source distribution can be imaged without errors by a distorted telescope objective compensated by information derived from the intensity distribution in the image plane. However, the source distribution, or scene, to be imaged in radar is usually coherent, meaning that the phase relationships between scatterers is unchanged from pulse to pulse, provided that the transmitter and receiver locations remain fixed. Therefore, a method of emulating an incoherent source distribution had to be devised to apply Muller's concept to radar imaging situations. Two methods to "decohere" the source distribution have been developed at VFRC. The first procedure is range bin diversity, developed by Tsao [54] and Attia [3]. It is based -7- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. on the assumption that some portion of the radar rangebin data is statistically homogeneous. The radiation field received from each such rangebin can be considered as snapshots of a radiation field originating from an equivalent incoherent source distribution. In this manner multiple rangebin radar data can emulate data received from an incoherent source distribution. The second procedure is transmitter location diversity, which was developed by Subbaram [51] in 1986. In this technique successive radar pulses are transmitted from different antenna elements in the array. Consequently, a scatterer is illuminated by a different phase from pulse to pulse. As a result, the re-radiation phase relationships vary and the received waveforms will appear as if they originated from an incoherent source distribution. 1.3 The Microwave Imaging System The goal of the radio camera is to obtain high angular resolution images at microwave wavelengths. Since angular resolution is proportional to the wavelength divided by the length of the array, large aperture arrays are required for microwave systems to possess high angular resolution. For example, to obtain an image with the same resolution of optical cameras, a microwave "camera" aperture must be 3 to 6 orders of magnitude larger than that of an optical camera. Practicalities prevent building a monolithic antenna such as a dish antenna with the necessary size; consequently distributed, sparse phased arrays or synthetic apertures are required. Figure 1.1 illustrates the relevant phased array geometry. The array is assumed to be linear (one-dimensional) and lie within an x-y plane with origin located at some reference element within the array. The x-axis contains the array and the y-axis is in the transverse direction. The angle of arrival 0 of energy from a source is measured from the - 8- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. u fo v Illuminated Target Field Transmitter Signal Processor Display Figure 1.1 Diagram of Imaging System array normal. The echoes from a transmitted pulse are collected over several rangebins in all the receive channels. The transmitter is assumed to be separate from the phased array (i.e., bistatic radar) and it illuminates an angular sector called the field-of-view (FOV) significantly larger than the beamwidth of the array. One can envision a set of received data in the form of a matrix as shown in Figure 1.2. The signal processor is responsible for -9- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. self-calibration and forming the phase-corrected image. The system is assumed to be narrowband and consequently complex exponential notation is appropriate. Echo Range N Elements Figure 1.2 Diagram of a typical data set in matrix form. The vertical axis represents elements in the phased array. The horizontal axis represents range to the target. Consequently, the nth row represents the sampled range profile received by the nth element and the klh column represents the data received across the array from the k* rangebin. The combined effects of element position errors, electrical mistunings, etc., can be grouped into an undesired complex gain present in each channel, provided that the field-ofview is not excessively large. The complex gains are spatially random but temporally constant (i.e., the gain in any given channel is constant while the received complex radiation field is being measured). Steinberg [47] showed that amplitude variations have a significantly smaller effect on image quality than phase variations. Consequently, the complex gain errors reduce to channel phase errors (3n, n = 1,2,...,N. - 10 The signal - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. processor estimates the (3n's and subtracts the estimates (3n's from the recorded data. Figure 1.3 illustrates this model. i,y V V VV V V V w Figure 1.3 The self-calibration scheme: P's represent channel phase A errors, P's represent estimates of the phase errors, w's represent standard beamsteering weights. The corrected data set approximates samples o f the radiation field measured by an undistorted phased array. Therefore standard electronic-scanning beamsteering weights wn may be applied to scan the beam over the region of interest and produce a high-resolution image without distortion. If the phase errors are not estimated exactly, residual phase errors exist in each channel and some image distortion will remain. However, the level of - 11 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. distortion will be significantly decreased by self-calibration . One objective of this work is to study the accuracy of the phase error estimation for spatial correlation algorithms. 1.4 The Experimental Data All of the experimental radar data sets were obtained at the field site of the Valley Forge Research Center. The inverse synthetic aperture radar (ISAR) [15] data sets were obtained at 9.6 GHz (A. =3.123 cm) with a 4-ft parabolic transmit/receive antenna. The data sets were echoes from airplanes flying into the Philadelphia International Airport. The experiment is reported in [14],[53] and more details can be found in [13],[15]. The radar was a low power (250 watts peak), short pulse (7 ns or 1 m) fully coherent radar. Each echo trace was sampled at a 200 MHz rate following coherent quadrature demodulation, and quantized to 8 bits in each receiving channel. The short pulse length produced a range cell of 1.05 meters and the system recorded 120 range bins, spaced by 0.75 meters, from 512 radar echo waveforms. Other details concerning specific data sets are presented where necessary. 1.5 Dissertation Outline This dissertation is divided into 6 chapters, three of which present new work. This first chapter is an introduction to the self-calibration problem. A brief history of self calibration techniques is presented. Additionally, array geometry, notations, and the structure of a general imaging system is also presented in this chapter. The second chapter provides a detailed background of self-calibration based upon the spatial autocorrelation function and describes all the known spatial correlation based algorithms found in the literature. Several algorithms described were developed at the - 12 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Valley Forge Research Center (VFRC). They include Atria's Unit-Lag Spatial Correlation Algorithm, Subbaram's extensions to Attia's unit-lag algorithm, the Multiple-Lag Spatial Correlation Algorithm, and the Modified Muller-Buffington Algorithm. Other spatial correlation algorithms described are the basic Muller-Buffington Algorithm (optics and radio astronomy), the Shear Averaging Algorithm (SAR imaging), the Phase Gradient Algorithm (SAR imaging), the Flax-O'Donnell algorithm (medical ultrasound imaging) and phase closure (radio astronomy). This establishes a set o f spatial correlation algorithms and this background is used in the development of the generalized algorithm presented in Chapter 3. In Chapter 3 Tsao's Energy Conservation Algorithm (ECA) is discussed. An important limitation of the ECA discovered experimentally during this research provides the insight that leads to the development of the Generalized Spatial Correlation Algorithm (GSCA) derived in this chapter. The GSCA includes the ECA along with the algorithms described in Chapter 2. Hitherto, Tsao's algorithm was not believed to be a spatial correlation algorithm. Chapter 4 provides performance results from computer simulations of the effects of random element position errors and additive element receiver noise. Analysis of the objective functions of algorithms of the two sub-classes explains the superior performance of the parametric algorithm over the non-parametric algorithm and explains the performance difference between the unit-lag algorithms of the two sub-classes. The analysis also indicates that the parametric algorithm is favorable over the MLSCA in the presence of small variance phase noise in the measurement of the spatial correlation values. Appendix B supplements this chapter with 40 plots of expected main beam relative gain versus scan angle for the two algorithms. The GSCA assumes a set of incoherent sources in the far field of the phased array. However, in most practical high-resolution microwave imaging situations, the sources are -13- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in the near field of the array. All self-calibration algorithms focus the array as well as correct errors. The backscatterered radiation field is a spatially non-stationary random process in the near field, however, and therefore, care must be taken in how the GSCA is applied. In Chapter 5 the effects o f having the source distribution in the near field are presented. It is found that the low-order correlation lags may still be used while the higherorder lags are unsuited for use in the self-calibration process. The suitability of a particular correlation lag depends on a variety of parameters, the most important of which is the size of the field-of-view being illuminated by the transmitting antenna. Chapter 6 contains a summary and conclusions drawn from this research along with suggestions for future work in this area. 1.6 Dissertation Summary The objective of this work is to understand the use of measurements of the spatial autocorrelation function of a spatially random complex radiation field received in the aperture of a phased array antenna to self-calibrate the array antenna of a high resolution microwave imaging system. This work investigates properties relating to all known spatial correlation algorithms. The algorithms in this group are all relatively scene independent and are designed to work in situations where there is no dominant scatterer present in the data set. Because of the strong similarities between all these algorithms a unifying theory exists which firmly classifies them into one class and characterizes the class. This theory is developed and presented throughout this document. -14- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 Existing Spatial Correlation-Based Algorithms In 1974 Muller et al. [31] described in the optical literature a class of image sharpening functions that, upon maximization, yield a set of phase corrections that cancel the channel phase errors. In optics the phase errors are caused mainly by atmospheric turbulence. One particular objective function which Muller had significant success with ([16],[17]) is the integral over the image plane of the squared image intensity, i.e., Jj* I2(x,y)dxdy (2.1) Muller proved that this function reaches its absolute maximum when all phase errors have been eliminated [31]. A much simpler and more insightful proof is presented by Hamaker et al. in [25]. Hamaker showed that the maximum of (2.1) will be obtained when the random radiation field measured in the array aperture is a spatially stationary random process. The beauty of Muller and Buffington's objective function (2.1) is that it does not depend on the characteristics of the source distribution being imaged. There are no 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. constraints or assumptions made on the characteristics of the source distribution in either proof ([25] or [31]). However, the Muller-Buffington algorithm is restricted to use with incoherent radiation fields and in radar imaging the received radiation field is rarely incoherent. The rangebin diversity and transmitter location diversity techniques, developed by Attia [3] and by Subbaram [49] respectively, can be used to successfully emulate an incoherent radiation field. These techniques, along with Hamaker's proof, open the way toward a scene independent self-calibration algorithm for use in high-resolution microwave imaging. This chapter describes all of the spatial correlation algorithms known in the literature. These algorithms are subsumed within the GSCA in Chapter 3. The first section defines the Spatial Correlation class of algorithms for use throughout the remainder of the document. The second section derives the spatial correlation function of the received random radiation field and its Fourier transform relationship with the source intensity distribution. The third section presents Hamaker's proof that maximizing (2.1) is equivalent to requiring the received random radiation field to be a stationary random process. The ideas of these two sections are then combined in the fourth, fifth, and sixth sections where the ULSCA, the MLSCA, the Shear Averaging, the Phase Gradient, the Flax-O'Donnell, and the Phase Closure algorithms are discussed. The latter four algorithms stem from the fields of SAR imaging, ultrasound imaging, and radio astronomy. 2.1 The Spatial Correlation Class Defined In this document a "class of algorithms" is defined as: 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. any set of algorithms that make use of a common property or principal to perform a specific task. The Spatial Correlation class of algorithms is defined as: the set of algorithms that exploits information embedded in the measured spatial correlation values to estimate phase errors present in a phased array system. 2.2 The Spatial (Auto)Correlation Function of an Incoherent Source Distribution Figure 2.1 shows a source distribution, J(0), at range pD from the coordinate system origin. The source distribution is assumed to cover an angular extent A0 shown by the dashed lines in the figure. The radiation field suitably measured along the x-axis is given by the scalar diffraction integral as g-jkp(x)^ /• e(x) = d0 J(0) PM (2.2) J 0 provided all of the sources produce the same polarization. The distance p(x) is given by the law of cosines as p2(x) = p2 + x2 - 2xp0c o s(| -0) and p(x) = ‘s j p2 + x2 -2x p o sin(0 ) since cos(^ -0) = sin(0). If p0 > x, then the source distribution is said to be located in the far-field of the antenna and the distance p(x) can be well approximated as p(x) = p0 - xu L 2 where u = sin0. Any distance satisfying p0 > meters ( L ^ denotes the length of the array in meters) is generally considered to be in the far field [46]. Additionally, if A0 <3 1 then u = 0 and (2.2) can be written as 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. e(x) = J(u) eikxu du (2.3) Au after dropping a complex constant term. If J(u) is a deterministic quantity, the radiation field is also deterministic and (2.3) states that a Fourier transform relationship exists between the measured radiation field and the far-field source distribution. However, if J(u) is a random quantity, then the measured radiation field is also a random quantity and the field measured along the x-axis is a spatially random process. The autocorrelation function of this random process is given by R(x1,x2) = E{e(x1)e*(x2)} f E{J(u)J*(v)} e>k[xl u ' X2V1 du dv (2.4) Incoherent Source Distribution P(x) Figure 2.1 Illustration of the geometry used in deriving equation 2.3. (Adapted from [3]) 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. If the source distribution is composed of incoherent sources E{J(u)J (v)} I(u)5(u-v) where I(u) represents the intensity distribution of the sources. Then I(u)S(u-v) R( x j , x 2) = " X2V^ du dv = f I(u) eJk[xl ' x2]u du = R(X l- x 2) = R(X) (2.5) where X = xj - x2 . Since (2.5) shows only a dependence on the difference between positions xj and x2, the measured random radiation field is a wide sense stationary random process when the source distribution is located in the far field. Equation (2.5) also shows that the spatial autocorrelation function is proportional to the Fourier transform of the source intensity distribution, I(u). This Fourier relationship between the angular intensity distribution and the spatial correlation function is commonly known as the Van-Cittert Zemike Theorem [6] and is analogous to the Weiner-Khinchine theorem. 2.3 The Muller-Buffington Algorithm 2.3.1 The Image Sharpness Function The concepts involved in the Muller-Buffington Algorithm are illustrated in Figure2.2. Shown in the figure is an undistorted radiation field that suffers unknown random phase perturbations from atmospheric turbulence, spatial variations in the index of refraction, etc. The imaging lens corrects for geometrical phase differences but does not correct for the random phase perturbations that consequently propagate through the system to the adjustable phase shifters. Without the bank of adjustable phase shifters the image 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. would be completely distorted. If the phase shifters can adjust the phase of each ray appropriately, then the unperturbed phasing will be restored and an undistorted image will be formed. Phase restored Image Plane Phase disturbed Adjustable Phase Shifter Controller J j = JJ I2(x,y) dxdy Figure 2.2 Illustration of the Muller-Buffington optical phase correcting scheme. (Reprinted from [31]) Muller et al. measured the image sharpness defined by (2.1) and adjusted the bank of adjustable phase shifters until a maximum image sharpness was obtained. They proved that when the phase shifters were adjusted so that a maximum image sharpness was obtained, all the random phase perturbations had been removed from the radiation field and the resulting image was undistorted. Figures 2.3 and 2.4 show examples of the results that Muller et al. obtained with this algorithm. Figure 2.3 shows results from simulated data. The distorted image is shown in the top part of the graph and the phase corrected image is underneath it. In the simulation three point source were simulated and phase perturbations were added. The distorted image does not show the presence of the three point sources. However, after 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. maximizing (2.1), the resulting image clearly shows the three point sources. Figure 2.4 shows a similar graph of experimental data obtained from Leuschner Observatory in California. Both the upper and lower images are of the star Sirius. As in Figure 2.3, the upper image contains phase perturbations and is formed with no phase corrections applied. The lower image is what was obtained after (2.1) had been maximized and the phase corrections had been applied. The image clearly shows the presence of a point source indicating the presence of the star which can not be seen in the upper image. l o ) F t i d b e c k OFF 1 2 3 3 0 PST) (o) Spackla potlirn lb) Fiidbock ON 12300 PST) tb) Corr*et»d Imoga -OB -0.6 -0.4 -0 2 0 0.2 0.4 0.6 0B P osition In im ag o p lan* ( a r c i o c ) Taken from JOSA Septem ber 1974. pp. 1200-1210 P e t i t i o n in t h t im o g * p l o n * , s e c o n d * of ore Figure 2.4 Figure 2.3 2.3.2 M axim izing the Image Sharpness Function in the Aperture D om ain 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In [31] Muller and Buffington proved that (2.1) is maximum when all of the random phase errors are removed from the incident radiation field. In 1977 Hamaker et al. [25] made the connection between maximizing (2.1) in the image domain and enforcing a stationarity constraint in the aperture domain. Consider the Muller-Buffington objective function of (2.1) in one dimension. It is well known from Fourier optics [6] that the intensity, I(u), is a Fourier transform pair with the product of the visibility function, V(X) in the aperture, and the optical transfer function, T(X). The optical transfer function is related to the pupil function, P(X), through T(X) = P(x)P*(x+X) dx ( 2 .6 ) Since this Fourier relationship exists, Parseval's Theorem states I2(u) du = f IV(x)l2 IT(x)l2 dx (2.7) V(X) is determined solely by the source intensity distribution and consequently maximizing (2.1) is equivalent to maximizing T(X) for all X. Let the undistorted pupil function, P0, be real and assume that the atmospheric distortions affect only the phases of the measured radiation field as assumed in [31]. Therefore, P(x) can be expressed as P(x) = P0(x)eiP(x) ( 2 .8 ) and the optical transfer function, T (X ), can be expressed as T(X) = f P0(x)P0(x+X )ej [ PW ' P(X+X)] dx (2.9) 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Equation (2.9) shows that T(X) is a summation of phasors each with a phase given by the argument of the complex exponential. The magnitude (and therefore the square magnitude) of T(X) will be maximum when all the phasors are cophased. This requires the complex exponential to be independent of the variable x. The consequences of requiring the argument of the complex exponential to be independent of x can be seen by expanding P(x) in a Taylor series about some arbitrary point, "a", as p(x) = a + p'(x)|a (x-a) + p”( x )|a + .. . (2.10) The quantity P(x) - (3(x+X) will be independent of x if P(x) is composed only of a constant and linear term. A constant phase error does not affect the imaging system and is therefore of no consequence. The presence of a linear phase error across the imaging system introduces a pointing error (a shift in image location) but does not distort the image. Therefore, maximization of (2.1) yields a diffraction limited image with a possible shift in image location. Since (2.9) is a summation of complex exponentials and requiring that the quantity P(x) - P(x+X ) be independent of x demonstrates that measurements of the same visibility value between different pairs of elements should be equal. An unperturbed radiation field originating from an incoherent source distribution should yield the same correlation values independent of the location in the array at which they are measured. This means the unperturbed radiation field measured in the aperture should be a stationary random process. In the optics field, Muller's algorithm provided a method of enforcing such a constraint on the radiation field without directly measuring the correlation values. In the microwave imaging field it provided the basis for the Spatial Correlation class of self-calibration algorithms. 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.4 The Unit-Lag Spatial Correlation Algorithm The simplest algorithm in the Spatial Correlation class is the Unit-Lag SCA [3]. The basis o f the algorithm lies in the realization that the phases of the spatial correlation values measured in the aperture contain information regarding the channel phase errors. As (2.4) and (2.5) show, the spatial correlation values measured in the aperture depend only on the difference between the positions used in making the measurement when the source distribution is incoherent, in the far field of the array, and the imaging system is free of errors. Assume that s(u) is a source distribution satisfying both the incoherence and farfield requirements. The intensity distribution of s(u) is I(u) and the corresponding spatial correlation function in the antenna aperture is R (X ) where X is a continuous variable. However, the phased array is a sampled aperture and the corresponding correlation function is also sampled. Denote the sampled spatial correlation function as R(jt) = R(jtd) where t (an integer) represents the lag index and d represents the interelement spacing (i.e., the sampling interval). Such notation implies that the array is periodic. This is a sufficient, but not necessary, condition for the unit-lag SCA to work. In general, the following unit-lag algorithm can also calibrate an aperiodic array. If the source distribution, s(u), is symmetrical about the optical axis, then the correlation function will be real except for a linear phase component associated with the pointing direction of the beam. The correlation function develops an imaginary component when channel phase errors exist in the imaging system. Therefore, the phases of the correlation measurements contain all the necessary information about the channel phase errors to calibrate the system. Consider an N-element array antenna. The N signals present in each channel are in error because of any or all of the problems mentioned in Chapter 1. These corrupted signals can be expressed as e(n) = e (n )e'^ n where e(n) represents the signal that would 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. be obtained in the nlh channel if no phase errors were present. Therefore, any weight vector of the form q(n) = e*Pn will correct the signals and any procedure that yields the values of Pn solves the calibration problem. The unit-lag spatial correlation coefficient of the error-free sample between the nth and the (n+ l)111elements is R (l) =E{e(n)e*(n+l)} (2.11) which is assumed to be real in [3]. With phase errors present in the imaging system the measured correlation value is R’(n ,l) = E{e(n)e!i!(n+l)ej[Pn+1' Pn] } = E{e(n)e*(n+l)}ei[Pn+rPn] and = R ( l) e P n+rPn] (2.12) arg{ R (n,l) j = (3n+1 ~(3n (2.13) In an N-element array there are N -l lag-1 correlation values available but there are N unknown phase errors. This is enough information to calibrate the array because one can freely choose the phase of any one element in the array. Denote the reference element as element #1 and let the phase error (ij be zero. From (2.13) (32 = arg ( r '( 1 ,1 ) ) , P3 = arg ( r '( 2 , 1)) + arg ( r ’(1,1)), etc. (2.14) The phase error in the n1*1 channel is n a r g ( R '( p - U ) ) Pn = £ (2.15) p=2 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.4 .1 P r a c tic a l Im plem en tation o f the U nit-Lag SCA Equations (2.12) through (2.15) demonstrate that the system can self-calibrate provided an ensemble exists over which an expected value can be taken and second-order statistics can be defined. This requires the measurement of a radiation field that is a random process and a sequence of snapshots will represent an ensemble of sample functions of that process. However, in radar applications the backscattered field is rarely a random process because the echoes from successive transmissions are substantially identical. A method to obtain such an ensemble is to consider the radiation fields from successive rangebins as different realizations of a random process. Attia showed that this can be successfully done provided the source distributions in the rangebins being considered are statistically similar [3]. If the scene being imaged satisfies this criterion, then the imaging radar should collect as many rangebins as it can and the spatial average in range K (2.16) can be formed to approximate the expected value in (2.12). In (2.16) the subscript "Ic” is the rangebin index and "K" is the total number of rangebins averaged. This process is shown in Figure 2.5. As long as each correlation measurement is independent, in range, of the other correlation measurements, then (2.16) is an unbiased and consistent estimator t » of R (n ,l) and will converge to R (n ,l) as K—> °°. In essence, the more rangebins collected from a statistically homogeneous scene, the better the Unit-Lag SCA should perform. 2 .4 .2 A n I te r a tiv e U nit-Lag SCA 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. When the array is periodic an iterative algorithm can be used to calibrate the system. In a periodic array the distance between neighboring elements is constant and the received radiation field is spatially sampled at constant intervals. In this case, when the unit-lags are computed, the correlation distances corresponding to each computed value are equal. Since the radiation field is wide sense stationary (as shown by (2.5)) the correlation values must PnO O P „ +l t Figure 2.5 Illustration of the correlation measurement between elements n and (n+1). depend only on the correlation distance and not on the position in the array at which the value is computed. Therefore, in an undistorted (P; = 0 , for all i) periodic array , all unitlag correlation values must be equal, i.e., r '(1,1) = R (2 ,l) = . . . = r '(N-1,1) (2.17) As seen from (2.12) the phase errors alter the individual measurements. Any method that forces convergence to (2.17) will calibrate the imaging system. One such method is to maximize the objective function [1] 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. N-l JA = X RW)e ‘K P n + l ‘ Pn) (2.18) n=l It is easily verified from (2.12) and (2.18) (and proved in [1]) that J A will be maximum A A when Pn = Pn modulo 2rc. In practice the maximization is done by substituting R (n,l) a of (2.16) for R (n ,l) in (2.18) and adjusting the phase error estimates, Pn, until the global maximum is attained. This algorithm is termed "iterative" because maximization of (2.18) is usually not a one-step process, but instead requires changing the values of the parameters (the pn's) several times until no change in the value of the objective function (2.18) is observed. 2.5 The M ultiple-Lag Algorithms Spatial Correlation In addition to the unit-lag correlation values there are N-2 other correlation lags measurable in an N element array and all provide information regarding the phase errors. Figure 2.6 illustrates the measurement of the higher-order lags. In an ideal environment (zero noise and K = °o) these lags provide no more information than is already contained in the unit-lag measurements and are therefore of no use in the self-calibration process. However a limited sample size (K < °°) produces a statistical variability to the correlation estimates and additive noise further increases the output variance. 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2.6 Illustration of correlation measurements of lags Lj, L^, Element receiver noise can corrupt the correlation measurements in two ways. First, if the signal-to-noise ratio (SNR) is low the received signals in each channel will be phase rotated by the channel noise and the resulting correlation measurements will be in error. Additionally if the true correlation value given by (2.11) is close, or equal, to zero then channel noise will certainly corrupt the measurement of the phase of this value. Redundancy offered by the higher-order correlation lags reduces the effects of element receiver noise and finite rangebin averaging. The current multiple-lag algorithm of Subbaram [48],[49] exists in two forms. One form establishes a least-squares problem and the other form maximizes an objective function. These two forms are discussed in the two following sections. 2 .5 .1 The L e a s t Squ ares M u ltip le-L a g SCA The discussion regarding (2.12) and (2.13) can be generalized to show R W ) = lR0t)lej[V*+Pn+* 'Pn] (2.19) 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where the quantity V/, denotes the true phase of the lag Z correlation value given by R (Z ) = lR ( Z ) le ^ = E {e(n)e*(n+Z )}. Let <|)(n+Z,n) represent the phase of the lag Z correlation value R (n,Z) such that <Kn+Z,n) = arg (R (n ,/.)) = + p n+* - p n (2.20) When noise is present and the finite spatial average in (2.16) is formed to approximate the ensemble average, the measured correlation lag is corrupted and equations (2.19) and (2.20) become R (n,Z ) = |R(Z)lej[V*+Pn+rPn+5(n+*,n)] and (2.21) $(n+Z ,n) = arg (R (n,Z )) = V/, + Pn+Ji - Pn + S (n + Z ,n ) (2.22) where 8(n+Z,n) represents the phase noise induced by the element receiver noise and the rangebin averaging. The unit-lag correlation phase, can be estimated as A V! = kdu0 « V i where uQ is the direction towards which the transmitter is pointed, but the remaining correlation phases, Yjt, L > 1, are unknown quantities and must be removed [48, pg. 243]. This can be accomplished by forming the quantity y(n,jt) = $ (n + Z + l,n + l) - $(n+Z,n) (2.23) = Pn+jt + l ■ Pn+1 - Pn+jt + Pn + 8(n+£ + l,n + l) - 8 (n + /.,n ) In an N-element array there are N-jt measurable lag-/, correlation values. Therefore, N-Z -1 equations of the form of (2.23) can be formed for any particular lag. The equations resulting from (2.23) can be combined with the N-l unit-lag equations to produce an overdetermined system of equations of the form 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. $( 2 , 1) - ^ <J>(3,2) - = 0 2 - 1$! = $3 - $2 Y(l>2) = $4 ' $2 ' $3 + $1 (2-24) y(l»N-2) = (3n - $2 - $n- 1 + Pi When phase noise is not present (i.e., S(n+j£,n) = 0) the equations generated by (2.23) are linear combinations of the unit-lag phases [48, pg. 248] and they provide no new information. However, in the presence of noise these equations are no longer consistent with the unit-lag equations. Consequently, the higher-order lags do provide additional information and a least-squares solution can be found. This solution effectively averages over the additionalobservations provided by the measurements of the high-order lags and smoothes the effects of the noise. Care must be taken when forming the overdetermined system of equations. Since the channel phase errors, Pn, can take on any value within the range -it to 7t, significant problems can occur because of modulo 271 ambiguities when the quantity y(n, t ) is calculated. Additionally the quantity $(n+jt,n) itself may wrapped by 2% when measured. Any wrapping of the measured phases creates an inconsistency in the set of equations that exists even when there is no noise present. This inconsistency can introduce more noise into the system and therefore the wrapping must be eliminated before finding the leastsquares solution. Steinberg and Subbaram present a method of unwrapping the 2n ambiguities in Ref. [48] that is summarized in the following paragraph. Phase unwrapping can be accomplished by noting that the higher-order correlation phases are linear combinations of the unit-lag correlation phases. The equality between y(r\JL) and the unit-lag phases can be shown to be 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. y(n,jt) = $ (n + jt+ l,n + jt) - <j)(n+l,n) + 2nm - 8 ( n + jt+ l,n + /) + S (n+l,n) + 8(n + jt+ l,n + l) - 8(n+jt,n) (2.25) When the noise term is smaller than jt, the integer m in (2.25) can be determined as a y(n,jt) - [$(n+jt + l,n+jt) - $(n+l,n)] m = ---------------2k round (2.26) where m is the estimate of m and "round" implies rounding to the nearest integer. Once m is determined from (2.26), y(n ,t) can be unwrapped as Y ( n / ) l u = y (n ,jt) - 2 j t m (2.27) Once a consistency between all of the equations is established by unwrapping the 2n ambiguities, the least-squares solution can be found from the equation A Ap = b Let L (2.28) represent thehighest-order lag used in (2.28). Let every lag for I < L be included in (2.28). Then A is an ^ £ nl a x N - l j matrix containing the coefficients of the a P's in equation (2.24), P is an (N-l x 1) vector containing the estimates of the phase A rJ' errors (i.e., P = [ P 2,P3,P4 , . . . , P n ] )> and b is an ^ j^NL - x vector containing the unwrapped values of the phases of the measured spatial correlation lags. A is a sparse matrix containing only l's, -l's and 0's and possessing a lot of structure. The general form of A and b are shown in Figure 2.7 for lags one and two. The system of equations in (2.28) contains only N-l unknowns (p2 through pN) because the phase error in the reference channelisassumed to be zero. The least-squares solutionto (2.28) is P = A# b = (A ^ 'V b (2.29) 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where A $ represents the pseudoinverse of A . The negative of (3 should be applied to the 0 0 ... 0 0 0 0 1 0 0 ... 0 0 0 0 o o o o -1 0 o 1 o phased array to calibrate the imaging system. • • • • • • • • ••• • • • • • • • ♦ 0 0 0 0 . .. 0 0 -1 1 -1 1 1 0 . .. 0 0 0 0 1 . .. 0 0 0 0 1 -1 -1 • • _ 0 • • 1 • • • 0 0 0 Figure 2.7 1 -1 -1 Vi A 0(3,2) - Vj A 0(4,3) - Vi • b = $(N,N-1) - V, Y(2,2)Iu • • • • • • • . .. $(2,1) - 1_ y(N-3,2))a The matrix A and vector b from equation 2.9 This algorithm is known as the Multiple-Lag SCA (MLSCA) and was developed by Subbaram in [48]. 2 . 5 . 2 The M o d if ie d M u ller-B u ffin gton A lg o rith m (M M B) The least-squares algorithm described in the previous section is non-iterative and the estimates of the phase errors are computed in one step. The pseudoinverse of A can be computed once and stored since it is independent of the measured data. However, for arrays containing a large number of elements the calculation of the pseudoinverse can be extremely difficult and time consuming. For this reason Subbaram's MMB algorithm may be more attractive. 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The iterative algorithm follows Muller's work very closely. However, instead of maximizing an objective function in the image domain, an equivalent objective function is maximized in the lag (correlation) domain. The derivation of this algorithm is similar to the work of Hamaker et al. in [25] since they originally established the connection between maximization of (2.1) and the equivalent effects in the lag domain. Consider Muller's objective function and the derivation of the spatial correlation function of a far-field incoherent source distribution presented in Section 2.1. It was shown that the spatial (auto)correlation function and the intensity distribution are Fourier transform pairs. Therefore Muller's objective function can be written as I2(u) du = 7{I(uM (u)}|x=o = R(X) *R(X)lx=0 R2(X ) dX (2.30) where J denotes the Fourier transform, * represents convolution, and the equality of the second line in (2.30) is a result of the convolution property of the Fourier transform. When the aperture is discrete and not continuous, as with a phased array, (2.30) becomes w i >t=\< « i 2 <2-3» where L represents the maximum lag included in the calibration process (L < N-l). Because of the phase errors the measured values of R(Z) are not equal and depend on both the element index, n, and the lag index, i , as shown in (2.21). An estimate of R(Z) in (2.30) can be formed by averaging the N-Z different values measured in the array for a particular lag, Z, as 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A ROO = X A (2.32) * W ) e'J'(Pn+* ’ Pn) n=l Substituting (2.32) into (2.31) yields a lag domain objective function to be maximized that is the analogous to Muller's image domain objective function L MMB .« N' z £ | 1=1 X a , 8 (n jl)e 'j(P” +* ' Pn> | n=l Using R ( n /) from (2.21) and temporarily neglecting the noise term, 8(n+Z,n), it is easily A verified that (2.33) reaches its maximum when (3n exactly equals the phase error Pn. A Therefore applying a complex weight with phase -pn to the nth channel successfully calibrates the array. W hen the noise term, 8 ( n + / , n ) , A is not neglected T (3 « (3 = [ P2>P3»P4>*• *’P n I at the maximum of (2.33) and the effects of the noise will be smoothed by the inclusion of the higher-order correlation lags in (2.33). The procedure A is iterative because the phase errors are obtained by adjusting and re-adjusting the p's until the maximum of (2.23) is reached. The iterative procedure used to maximize (2.33) requires an initial estimate o f the A values of the P's. The estimates that the unit-lag algorithm yields are generally used for this initial guess. The function JMMB is maximized by any desirable method until the change in the value of the (2.33) is less than some prescribed tolerance. 2 .5 .3 The F u ll a n d P a r tia l M u ltiple-L ag SCA's Measurement of the N-l unit-lag correlation phases is sufficient to perform self calibration. Measuring any high-order lags in addition to the unit-lags provides redundant measurements that can be used to smooth noise effects. It is not necessary that all the available correlation lags be incorporated into the multiple-lag self-calibration process. As many lags as are needed to satisfactorily self-calibrate the system may be used in the 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. multiple-lag algorithms. The number of lags to incorporate into the algorithms generally depends on several factors, such as the strength o f the undistorted correlation values and the SNR per array element. The integration of these (and other) factors into determining the number of lags to use in the multiple-lag algorithms is discussed in more detail in the following chapters. Throughout the remainder of this report the term "full multiple-lag algorithm" will denote the use of all available correlation lags (L = N -l) with any of the multiple-lag algorithms. The particular algorithm under discussion will be apparent from the context. Likewise the term "partial multiple-lag algorithm" will denote the use of only a selection of the available correlation lags (L < N -l) and the algorithm and lags being discussed will be apparent from the context. 2.6 Other Spatial Correlation-Based Algorithms 2 .6 .1 The S h ea r A v e ra g in g A lg o rith m In 1989, J. R. Fienup published another spatial correlation-based algorithm called the Shear Averaging Algorithm [20] for use with SAR imaging systems. This algorithm closely resembles Attia's unit-lag algorithm and degenerates to exactly the Unit-Lag SCA when the rangebin data are range compressed prior to using the algorithm. Assuming the data have been range-compressed the Shear Averaging Algorithm forms the quantity K R ( n , 0 = X £k(n + ^ k ( n) (2-34) k=l which yields Attia's algorithm when i = 1 which is the case generally used. If the data in the azimuth direction are highly oversampled, then using I > 1 may be beneficial. Like all the spatial correlation algorithms, this algorithm assumes that a one-dimensional phase 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. error exists which is independent of the range dimension. The estimate of the phase error at the nlh dem ent is found from ,n<N-l (2.35) p=i if the first array element is indexed as the zeroth element. 2 .6 .2 The P h a se G ra d ie n t A lg o rith m The Phase Gradient Algorithm (PGA) was developed by Eichel et al. [19] in 1989 at Sandia National Laboratories primarily for the self-calibration of SAR images corrupted by unknown phase errors. The PGA solves for the derivative of the phase error profile across the array and integrates along the array to determine the phase error at the nlh element. The algorithm iterates between the image domain and the aperture domain until convergence is reached. One iteration of the algorithm includes the following steps. The image of each corrupted rangebin is formed and the most prominent scatterer in each rangebin is shifted to the origin. A processing window symmetric about the origin is used to window the image of each rangebin and the inverse Fourier transform is taken to obtain aperture domain data (2.36) where ^ ( n ) is the target related phase of the k th rangebin. For a general complex function y(x) = |y (x )|eJ')W (2.37) the derivative of the phase is given by the identity 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. d-o(x) _ Im {y*(x)y"(x)} 5------dx |y(x)f (2.38) Equation (2.38) is used to form a least-squares estimate o f the derivative of the phase error profile £ l m { g;(x)g;(x)} P'(x) = -* i a xk(x k (2.39) In the case of an array (sampled aperture) (2.39) becomes £ l m { g ‘(n)g (n + 1)} ji(n +1) - jl(n) = 5-------- (2.40) I g 4(“ ) k The right-hand side of (2.40) can be shown [2] to reduce to the result of the Unit-Lag SCA when the phase errors are small enough that sin((3n+1 - (Jn ) » (3n+1 - (3n. 2 .6 .3 The F le a a n d O 'D o n n ell A lgorith m In 1988 Flax and O'Donnell [21] introduced a self-calibration algorithm into the ultrasound literature that makes use of the unit-lag spatial correlation measurements to form narrow transmit and receive beams on a phased array. In the ultrasound imaging field, phase errors arise because of the different velocities of sound in the different tissue layers of the human body. The Flax and O'Donnell algorithm calculates the channel phase errors using the unit-lag SCA, applies the conjugates of these phase errors to the phased array and re-transmits. On receive, the unit-lag algorithm is again used to calculate the phase errors and the process is repeated. Flax and O'Donnell claim that with each transmission and application of the Unit-Lag SCA the shapes of the transmit and receive beams improve towards a narrow beam. 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Using the notation of [21], the authors form the quantity C(x) = (A(x)A*(x - Ax)) (2.41) where A(x) represents the signal received at position x in the array, < > denotes an ensemble average and Ax is taken to be the inter-element spacing. The phase of (2.41) contains information regarding the difference in time of arrival of the signals at two adjacent sensors. To calibrate the ith channel, the time-of-arrival differences are summed from the reference element to the i1*1 element and the total time difference is converted to radians, conjugated and applied to the ith channel. Although it is never specifically stated that the unit-lag spatial correlation measurements are being used to calibrate the array, equation (2.41) is certainly nothing more than (2.11) re-expressed using different notation. Consequently each stage of the Flax and O'Donnell algorithm utilizes the Unit-Lag SCA to determine the phase errors. 2 .6 .4 P h a se C lo su re F rom R a d io A stron om y Phase closure is a concept widely recognized in the radio astronomy field and was originally coined by Jennison [26,27]. The term refers to a linear combination of observed correlation phases around a closed loop of three or more antenna elements. The simplest example of a closure phase is * i j k = V ' * V + '*'ki (2-42) where 4 ^ = 4*^ + [3j - (3k is the phase of the correlation value measured between the j1*1 and the kltl elements; 4*^ is the phase of the true correlation value betweenthe j th and the k ^ elements and(3j is the phase error in the channel of the j 111antenna element. It is easy to verify that ^k^ijk^ij+ ^jk+ ^ki (2.43) 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Many self-calibration algorithms in radio astronomy make use of the closure phases to calibrate an imaging system. One such algorithm due to Schwab [39] implicitly makes use of the closure phases by minimizing the mean square misfit between the measured (noisy) correlation values and those predicted from a model, as modified by the phase estimates R s - nI Im AXn,m _ e K P»~Pu,)R AXn,m 2 w„.m (2.42) The weights can be used to favor measurements that have good signal-to-noise ratios or exclude those correlation values that are theoretically weak. Minimizing (2.42) is equivalent to maximizing S ' =2I I n <2- « ) m which can be re-expressed as S '= 2 X R n.m e"j(^ 1-N <£<N -1 J (2-44) n where the weights, wn m, and the model components, R^, have been assumed to depend only on the difference n-m=£ and the Hermitian symmetry of the modeled and measured correlation values has been exploited. Equation (2.44) describes a multiple-lag algorithm of a form similar to Subbaram's MMB algorithm. 2.7 Summary The goal of this chapter has been to establish the existence of a set of algorithms that utilize information embedded in the spatial correlation values of a measured radiation field to self-calibrate an array system. This is a prelude to developing the generalized algorithm that includes all the algorithms as special cases. The use of correlation-based algorithms is widespread and encompasses many fields of imaging including microwave 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (radar) imaging, ultrasound imaging, radio astronomy, and optics. All of the published self-calibration algorithms known to the author that utilize information contained in the measured spatial correlation values have been discussed in this chapter. The remainder of this dissertation is devoted to generalizing this set into a general form and studying the performance of this set of algorithms in the presence of additive noise and element position errors. When there are no errors in the measurements of the spatial correlation values, there is no need to use a multiple lag algorithm. However, when the correlation measurements are in error due either to the effects of receiver noise or insufficient rangebin averaging, the redundant measurements of the higher-order correlation lags provide an additional dimension of smoothing. Consequently, the effect of the noise is lessened by incorporating the higher-order lags into the calibration process. This is the benefit of the multiple lag algorithms. This chapter also provides an understanding of the existing spatial correlation algorithms. The concept of the random radiation field received from a set of incoherent sources being a stationary random process and Hamaker's proof connecting Muller's objective function to the aperture domain are very important to the operation o f the entire class of algorithms. Together they provide a basis for self-calibrating a microwave imaging system. The notation developed in the discussions of the Unit-Lag SCA and the MultipleLag SCA will be followed throughout this report. The forms of the both the Unit-Lag and Multiple-Lag algorithms presented in this chapter are fundamental to the generalized algorithm developed in Chapter 3. Additionally, the problems discussed in this chapter, such as the modulo 2k ambiguities encountered with the multiple-lag algorithm, are also inherent in the generalized algorithm and are discussed further in the following chapters. 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 The Generalized Spatial Correlation Algorithm Each algorithm discussed in Chapter 2 is (1) spatial-correlation based and (2) has slightly different desirable or undesirable characteristics for a particular imaging scenario. This chapter develops a generalized spatial correlation algorithm (GSCA) that allows the class to be characterized and studied as a whole instead of on a individual algorithm-toalgorithm basis. The impetus for the development of the GSCA was the failure of the Energy Conservation Algorithm (ECA) to perform under certain conditions. Studies show it to fall into the spatial correlation class of algorithms. This chapter discusses the development of the GSCA and shows how the existing spatial correlation algorithms discussed in Chapter 2 are characterized by the GSCA objective function. The first section o f this chapter digresses slightly to present a seemingly different self-calibration algorithm; the ECA. It proceeds in this order because the study of this algorithm leads to the development of the GSCA objective function. The second section develops a generalization of the ECA and shows it to be a generalized spatial correlation-based algorithm which is now called the Generalized Spatial Correlation -42- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Algorithm. The third section demonstrates how the ECA and other already existing spatial correlation algorithms are embodied by the GSCA and the fourth section discusses the need for only one efficient maximization procedure for use with these algorithms. The fifth section is a chapter summary. 3.1 The Energy Conservation Algorithm 3 .1 .1 D e v e lo p m e n t o f the E n erg y C o n serva tio n A lg o rith m The ECA was developed by Tsao [55],[56] in the early 1980's. The algorithm is based on Parseval's Theorem which relates the total energy in one domain to the total energy in another domain provided the two domains are related by the Fourier transform. In time-series analysis the familiar relationship is + oo + oo J Ix(t)l2 dt = J |X(co)|2 dco - oo (3.1) - oo where X(co) is the Fourier transform of x(t). In imaging applications where the received radiation field from the k th rangebin is the Fourier transform of the active source distribution, s^ (u), the relationship is 4* -f oo 00 J le^(x)l2 dx = J ls^(u)l2 du - oo (3.2) - oo When a phased array is used to measure the radiation field the discrete nature and finite extent of the array are represented by a weighting function w(x) = ^ w ( n ) 8(x-xn) where w(n) is the weight of the n ^ element located at position xn and 8(x-xn) represents the Dirac delta function. These weights generally include beamsteering weights. Another weighting function, q(x), is added for calibration purposes. When a passive source distribution is -43- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. illum inated by a transm itter with beamwidth A9 = Au = u2 - u l5 equation (3.2) becomes + N 00 | I e^ (x)w(x)q(x)|2 dx = ^ le^ (n)w(n)q(n)|2 - 00 n=l N =X n=1 w here i(n) rep resen ts the J l ^ ( n) l 2 = l ^ ( u ) l 2 du (3.3) Au current obtained in the nlh channel and A q( x) = q( n)8 (x-xn) = £ e n8 (x-xn). The term s^ (u) is the image of source distribution s^ (u) as seen through the combination of the transmitter and phased array radiation patterns and can be expressed as h = X ®k (x n)e'-i(kuxn+^n) = ^ n=l e^ (x n) e ^ n e'j(kuXn+^n) (3.4) n=l following the notation of section 2.3. Figures 3.1 and 3.2 illustrate theconcepts behind the ECA.When there are no phase errors in the imaging system, allthe energy inthearray aperture will transform to within the Au angular sector in the image domain as shown by (3.3). However, when phase errors are present some of the energy in the image domain will leak outside the Au angular sector and the final equality in (3.3) will not exist. The ECA objective function ( 3 ,, k = l n=l k= 1 Au -44- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Array Image Processor With no phase errors energy is located withn the illuminated angular sector in the image domain Image Plane Figure 3.1 Illumination Beamwidth Au Array Image Processor Energy tails outside Au because of aperture Imperfections Image Plane Figure 3.2 -45- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. m easures the in eq u ality . W hen there are no p h ase erro rs present A Jeca = otherwise JECA > 0. The vector of phase error estimates, (3, is iteratively A tuned by the maximization procedure to the solution, P0, given by Po = 1 P I J ECA ^ m inim ized } (3.6) A The total energy in the aperture does not change with p because the amplitudes of the currents are unchanged. Therefore, the summation in (3.5) over the square magnitudes of the aperture currents is constant. Consequently, minimizing (3.5) is analogous to maximizing K J EE CC AA,, = , S 1 h (")l d u (3.7) *=1 ' Au and (3.6) is equivalent to P0 = t P j 1ECA2 *s maximized } (3.8) The procedure used to maximize (3.7) is given in [48]. The maximization procedure A requires an initial value of the vector P Q. The choice of the initial estimate determines convergence time. Experimentation shows that convergence times can vary by a factor of 2 or 3. The Unit-Lag SCA provides a good initial estimate keeping convergence time to a minimum. 3 .1 .2 P erfo rm a n ce o f the E C A w ith S im u la ted D a ta Prior to this work, simulations were made by Tsao in 1983 [55] and in 1986 [56], Subbaram in 1990 [50], and Patrick in 1990 [33], Figures 3.3 through 3.5 illustrate the -46- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. performance of the ECA in the controlled environments of simulations. In each figure part (a) shows the undistorted image of a simulated rangebin, part (b) shows the image with distorted data, and part (c) shows the image after self-calibration. The rangebin of Figure 3.3 contains a single point source located at uQ = 0. Two nearly equal and closely spaced point sources are used in Figure 3.4; their locations are u0 i = 0 .0 6 1 4 and u02 = 0.0372. The performance is excellent; both sources are fully resolved. Figure 3.5 shows the performance for two sources that differ by 20 dB in strength. The weaker source is in the sidelobes of the stronger source (u0i = 0.0456, u02 = -0.0500). In this case, too, the performance is highly satisfactory. For consistency of comparison with the earlier work, the simulation used to generate the rangebins is similar to the one used by Tsao (and later by Patrick). A 30element linear phased array with an inter-element spacing of two wavelengths was simulated. A far-field data set consisting o f two point targets uniformly distributed between Uj = -0.0835 and u2 = 0.0835 in each of ten rangebins was generated. Target amplitudes were random variables uniformly distributed between 0 and 1 except in the first rangebin where the second target was assigned an amplitude of zero. Random element position errors were introduced to distort the array. The element position errors were uniformly distributed, random variables independent in both the x- and y-directions and also from element-to-element. The rms element position errors in Figure 3.3 - 3.5 are 7 ; O X X similar results were obtained for rms element position errors of ^ and 3 .1 .3 P erfo rm a n ce o f the ECA w ith E x p erim en ta l D a ta Experiments disclosed that the ECA performs differently with high resolution experimental data obtained from the VFRC laboratory. There are two primary reasons for this performance. The first is the use of simulated far-field data in all the simulations. The second is the assumption that only a restricted angular sector of angular extent Au is -47- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. R angebin 01 (a) u = sin (0) Rangebin #1 -10 (b) •20 .25 -30 -0.3 -01 0.3 - 0 .1 u = sin (0 ) Rangebin 01 (c) -0.1 0 0.1 u = sin (0 ) Figure 3.3 Images of rangebin #1 of simulated data set. (a) Image using undistorted data, (b) Image using distorted data, (c) Image using calibrated data. -48- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. R angebin # 4 (a) •25 -0.3 -0.2 LflJL -0.1 0 0.1 02 0.1 02 u = sin (0 ) Rangebin #4 •10 (b) •20 •25 -30 - 0.1 u 0.3 sin (0 ) Rangebin #4 -10 (C) O & is -15 •20 -25 •30 -0.3 - 0 .2 0.2 - 0.1 0.3 u = sin (0 ) Figure 3.4 Images of rangebin #4 of simulated data set. (a) Image using undistorted data, (b) Image using distorted data, (c) Image using calibrated data. -49- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Rangebin #6 •10 (a) I & •20 •30 -0.3 02 u = sin(0) Rangebin #6 •10 (b) -20 •25 •30 -0.3 0.3 • 0.1 u = sin(0 ) Rangebin #6 -10 (c) -20 -25 -30 -0.3 0.2 -0 .1 0.3 u = sin(0 ) Figure 3.5 Images of rangebin #6 of simulated data set. (a) Image using undistorted data, (b) Image using distorted data, (c) Image using calibrated data. 50- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. illuminated and therefore re-radiates energy. In practice all transmitters have finite-extent apertures and therefore possess sidelobes which illuminate targets at all angles of arrival. Even when the illuminated target is in the clear and smaller in angular extent than the transmitter beamwidth, the illuminated angular sector is the angular size of the target and not the transmitter beamwidth. These issues complicate the ECA. The far-field versus near-field issue is covered in Chapter 5, where it is shown that the ECA does have the capability to calibrate the array using near-field data provided a mild set of restrictions are satisfied. The effects of the illuminated angular sector is the focus of this section. These effects did not manifest themselves until the ECA was applied to experimental data. Recognizing these effects leads to the development of the GSCA. The ECA was applied to two types of experimental data. One type was ISAR data obtained from a Boeing 727 aircraft [13] and the second is a set of high resolution radio camera data from Phoenixville, Pennsylvania. To produce the ISAR image, a 128-element array was synthesized with an inter-element spacing of 0.37 meters and 80 rangebins were collected. The transmitting antenna beamwidth was 30.5 milliradians. The second type of data set was obtained from a bistatic system with a 1.2 meter transmitting dish and a synthetic-on-receive, 83-meter phased array. A single receiving antenna was time-shared at 330 locations with an inter-element spacing of 0.2515 meters; 76 rangebins were available. The transmitting antenna beamwidth also was 30.5 milliradians and it illuminated several streets containing houses in the town. Both systems operated at X-band with a wavelength of 3.125 centimeters. Figure 3.6 shows 2-D ISAR images. Part (a) used the SCA to calibrate, (b) the ECA with Au, the integration interval, made equal to the transmitting beamwidth. The Unit-Lag SCA phases were used as the initialization phases. Subtle differences exist between these two images, differences that become more noticeable when a 1-dimensional -51 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.6 (a) Figure 3.6 (b) Figure 3.6 (c) -52- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. slice is taken along one of the rangebins. Figures 3.7a and 3.7b show such slices from rangebin #39 which contains the dominant scatterer of the data set located in an engine slightly forward of the tail section. The dominant scatterer appears in the image as the large lobe near u = 0, and the left wing of the plane appears to its left. One can see from the images that there should be an interval in bin 39, between the dominant scatterer and the wing, in which there is no airplane mass. This "null" interval can be seen in the SCA image of Figure 3.7(a) in the close-in left-hand sidelobes that are between -20 and -30 dB relative to the main lobe. This region appears about 10 dB higher in the ECA image. In general, the ECA has filled-in the entire image within the integration interval. This is exactly what the ECA was designed to do - maximize the image energy within the specified angular sector. However, this algorithm has implicitly assumed that the radiation pattern of the transmitting antenna is rectangular across the illuminated sector and that the angular scatterer distribution is uniform. This is never the case in reality. Because o f this implicit assumption, the ECA tries to push all of the image energy into the illuminated sector, and because the objective function is unweighted in this interval, to distribute it evenly throughout the sector. This is undesirable since all antennas radiate energy outside of this sector through the sidelobes and targets that have structure, such as the airplane, violate the assumption of uniform scatterer distribution. The target edges are lost or at least blurred, and thus the target shape becomes difficult to distinguish because of the assumed uniform angular distribution of transmitter energy. Figure 3.7(c) shows an experiment in which Au is matched to the image width of the dominant scatterer in a diffraction-limited image. In this case Au = 0.66 mrad, which is on the order of one synthetic aperture beamwidth. Comparison with (a) shows the lobe due to the dominant scatterer has been streamlined and the "null" region between the dominant scatterer and the wing has been broadened. This results in a sharper, crisper, image. The distant sidelobe level is about -30 dB relative to the mainlobe in both (a) and (c). -53- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Uml-Lag SCA Image L ow i n te n sity re g io n b e tw e e n e n g in e a n a wing aE (a) •0.0s -ao4 -ao3 -o.Q2 -aoi aoi ao2 ao3 aos u = sin (0 ) ECA Image AU aE (b) -0.04 -0.03 -ao2 aoi aoi 0.02 ao3 acw u= sin(0) ECA Image E 1 •0.025 -0.02 -0.015 -0.01 -0.005 Figure 3.7 Cross-range images illustrating the importance of matching the angular integration interval in the ECA to the target size. Dominant scatterer rangebin of 1SAR data set. (a) Unit-Lag SCA (b) ECA with Au = 30.5 mrads (c) ECA with Au = 0.66 mrads -54- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The same phenomena occur with the Phoenixville, PA land data. When the integration interval is the same size as the beamwidth of the transmitting antenna, there is no improvement over the SCA image. Figure 3.8(a) shows the image of rangebin #76 (the dominant scatterer bin) using the SCA. The ECA image obtained by using an integration interval of 30.5 milliradians is shown in Figure 3.8(b). It can be seen in Figure 3.8b that the energy within the integration interval has been increased over the SCA image. However, the distribution of the energy is incorrect for a dominant scatterer. The nearly uniform distribution of energy within Au, except in the mainlobe, is evident. Figure 3.8(c) shows the image with an integration interval of 0.66 milliradians. In this case the calibration has improved, as evidenced by a gain of between 2 and 3 dB in the peak of the image. The images obtained with a narrow integration interval are desirable when a strong isolated scatterer in one rangebin dominates all the other echoes. In this case the simple DSA is the algorithm of choice. However, there is no way to know a priori when such a scatterer exists; hence the DSA is not generally applicable. In the next section the ECA is shown to belong to the class of spatial correlation algorithms; a class that is suited to a broader set of source distributions. -55- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Unil-Lag SCA Image JS £ (a) S - 0.06 -a o 2 0.02 u=si n( 0) ECA Image AU (b) •0.06 0.02 u=si n( 0) ECA Image AU (c) * 0.02 ■0.06 u=si n( 9) Figure 3.8 Cross-range images illustrating the importance of matching the angular integration interval in the ECA to the target size. Dominant scatterer rangebin of Limerick data set. (a) UnitLag SCA (b) ECA with Au = 30.5 mrads (c) ECA with Au = 0.66 mrads -56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.2 The Generalized Spatial Correlation Algorithm 3 .2 .1 R e-In terp retin g the EC A O b je c tiv e F unction The anomalies between the simulation results and the experimental results can be explained by rearranging the expression of the ECA objective function (3.7) and re interpreting the meaning of the objective function. Interchanging the order of integration and summation leads to K (3.9) Au k =1 Equation (3.9) can be scaled without affecting the solution given by (3.7) Therefore, the ECA can be characterized by maximizing the objective function K J (3.10) ECA Au k =1 The integrand in (3.10) is the average image intensity over all the measured rangebins. Integration over the sector Au can be replaced with an integration over all u provided that a rectangular windowing function is included in the integrand K rect(Au) ^ ^ Is^ (u)l d (3.11) Jk = 1 where rect(Au) is unity within the range of Au and zero elsewhere. The integral (3.11) is the image energy within the sector Au. It is maximized when the total image energy falls within Au. In other words, JECA is maximized when the phase distortion in the aperture is -57- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. corrected by the compensation weight vector such that the total image energy is confined to the illuminated sector. However, this alone does not guarantee that the image shape is a good copy of the source distribution. The evidence from experiment indicates that such an objective function molds the average image intensity to a compromise between the true intensity distribution and the rectangle function , i.e., K ^ X ^*(u)l Jfc= 1 ~ rect(Au) (3.12) Examples are shown in Figures 3.7 and 3.8, the ECA is trying to "push" all of the received energy into the specified angular sector in a manner that approximates a rectangular function within the sector. Because in practice, sidelobes allow some amount of energy to leak outside the mainlobe of any antenna, any algorithm that tries to restrict all the received energy to be within a particular angular sector will not successfully calibrate the array system. Recognition of this problem leads to the development that follows. 3 .2 .2 C o n cep t o f th e G en e ra lize d S p a tia l C o rrela tio n A lg o rith m A more general calibration algorithm can be obtained by generalizing (3.11). The ECA performed well in the simulations for several reasons. First, all of the simulated radiation fields originated from within a very well defined angular sector. Second, 90% of the simulated point sources possessed amplitudes of the same order of magnitude and therefore the effects of any one source did not dominate the combined effects of all the sources. Thus the pronounced effect of the dominant scatterer was absent from the simulation problem. Third, the point sources were distributed throughout the entire fieldof-view (FOV). These three factors combined to yield an average intensity distribution that -58- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. was approximately rectangular. Figure 3.9 shows the undistorted average intensity distribution of the simulations. It is apparent that the energy is confined within the illuminated angular sector and, except for the region near u = ±0.04, the squared amplitude (intensity) is roughly of uniform level. Consequently, the undistorted average intensity distribution roughly approximates a rectangle function and the presence of the rectangle function in (3.11) is effective. A verage Im age Intensity 120 100 *oo 3 •0.05 0 0.15 0.05 0.2 0.25 u = sin(0 ) Figure 3.9 Average image intensity of the undistorted simulated data This suggests that the rectangle function in (3.11) should be generalized to K J GSCA lA Is P(u) 1 Y k =1 , p(u) g(u) du (u )l du (3.13) (3.14) where the function g(u) represents the average intensity distribution obtained from K rangebins and p(u) is a general weighting function particular to the characteristics of the environment being imaged. The algorithm described by this objective function proves to be -59- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a desirable self-calibration algorithm when each rangebin of the complex source distribution comprising the scene can be modeled as a realization of a random process. Let the expected value of the random process, s(u), be denoted by E[s(u)]. Since K realizations of s(u) are available an average intensity distribution, K ■ 2 (3.15) g(u) = ^7 £ I Sfc(u) K A=1 2 can be calculated to estimate the expected value of | s(u) | . The expected value represents a hypothetical intensity distribution derived from a hypothetical incoherent source distribution, sh(u). In other words, the quantity Ejjs(u)|2 j represents an intensity distribution that would be obtained from an equivalent single rangebin composed of incoherent sources. If data were collected from such a source distribution, the radiation field measured in the aperture would be a stationary random process and its spatial correlation function would be independent of the position in the array at which it is calculated. Since g(u) is an unbiased, consistent estimator of E^' 74 ' j, g(u) will approach E |js(u )|2 j as K —> «». Assume K is large enough that g(u) = E[js(u)|2 j. Since E^|s(u)|2 j represents the intensity distribution of an unknown distribution of incoherent sources, its inverse Fourier transform, 1F_ 1 | e | | s ( u ) | 2 j | , measured in the aperture. is the spatial autocorrelation function of the radiation field Consequently, J - 1{g(u)} closely approximates the autocorrelation function of the radiation field originating from sh(u) and is also approximately independent of the position in the array at which it is calculated. When phase errors are present g(u) * E |js(u )|2 j regardless of how large K is, in which case J -1{ g( u) } will not be independent of the position in array. Consequently, the measured radiation field is no longer a stationary random process. -60- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. If the phase errors are somehow removed, then g(u) once again will equal E[js(u)|2 ]. Likewise, if by any process, g(u) can be forced to approximately equal E |js(u)[2 J, then the phase errors will have been removed by that process. Consequently, two separate but equivalent methods can be described to remove the phase errors. Equation (3.13) can be used to mold g(u) to E^ |s(u)|2 j or the measured radiation field can be forced to be stationary. The mathematical connection between these two concepts and the method of forcing stationarity on the measured radiation field is described in the following section. 3 .2 .3 D e velo p m en t o f the G en era lized S p a tia l C o rre la tio n A lg o rith m Let S denote the set of rangebins (realizations) being imaged. If S is illuminated by a transmitting antenna having a radiation pattern fT(u), the angular reradiation distribution from the k th rangebin is fT(u)s|t (u) and the received radiation field, (x), in the array aperture is ^7"-1 {fT(u)Syt ( u) }. Let fQ(u) denote the beam pattern of the undistorted receiving array and let f(u) represent the beam pattern of the array with phase errors present. The corresponding array weight vectors are the inverse Fourier transforms of the radiation patterns, i.e., wQ(x) = j T ( f 0(u)) and w(x) = J ’ Hfiu)}. The distortion-free current in the aperture is iQfc(x) = w 0(x)e^ (x) = w0( x ) J ' 1{fT(u)sJt (u)) (3.16) The image is related to these currents through the Fourier transform as s k (u) = j { i fc(x)} = J { wQ(x) ^ _1 (fT(u)s^ (u))} = f0( u ) * f T(u)s*(u) where * means convolution. (3.17) The unaberrated intensity distribution is the squared magnitude of (3.17) -61 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. l ok <u ) = |f0(u) * fT(u)sjk(u)|2 Since (3.18) (u) is a realization of a random process, the intensity distribution is a random quantity described by I0(u) = | f0(u) * fT(u)s(u) | 2 (3.19) Define p(u) as the expected image intensity distribution, i.e., P(u) = E { I0( u)} = E { I fQ(u) * fT(u)s(u) |2 } (3.20) Since (3.13) tends to mold g(u) to p(u), defining p(u) in this manner will cause the algorithm to adjust the phase corrections until the average image intensity distribution is as similar to the undistorted expected undistorted intensity distribution as possible. This definition of p(u) eliminates the effects demonstrated by Figures 3.6(b), 3.7(b) and 3.8(b). Now express (3.13) as an inverse Fourier transform * J = p(u) g(u) du = J ' 1{ p(u)g(u)}1 ^ (3.21) J By the convolution property of the Fourier transform J ' ^ p M g t u ) ) ! ^ = ( J _1{p(u)} * J ' M g(u) } )ljt=0 (3.22) The first right-hand term is the autocorrelation function of the undisturbed received radiation field, which is given by J ' M p ( u ) } = 7 ' 1 { E { | f 0( u ) * f T(u)s(u)| 2 } } (3.23) -62- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Evaluation of (3.23) requires knowledge of the second-order statistics of s(u). In general these statistics are never known exactly and approximations need to be made. These approximations are discussed in Sections 3.2.4 and 3.3.3. The second right-hand term in (3.22), 7 ~ l { g (u )}, is K 7 _1 { g( u)) = J _1 { ^ X | s k (u)|2 } k =1 X = £ ^ ( i s ^ u ) ! 2} (3.24) *=l 2 where |s^ (u)| is the measured intensity distribution of the image. In the absence of phase errors, the inverse Fourier transform of |s^ (u)| 2 is the autocorrelation function of the measured current distribution i^ (x) in the array aperture. As shown in Chapter 2, the radiation field measured in the aperture is a stationary random process, when the source distribution is located in the far field, and the corresponding correlation function does not depend on the location of the measurement in the array. Consequently, the correlation values of a particular lag are the same no matter where in the array they are measured. When phase errors are present the measured radiation field is no longer a spatially stationary random process and the spatial correlation values depend on both the lag number and the element position number at which the they are measured. Therefore, (3.24) can be expressed as K 7 ' l i g( n) } = ^ X M k =1 n ’Z) = ^ (n’/ ) (3-25) A w here R ( n , j t ) is the generalization o f (2.16) to multiple lags and A A^ Rk (n,jt) = e^ (n+Z.)e k (n). Since (3.25) should be independent of element position, n, -63- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. when there are no phase errors, an estimate of the unaberrated value can be obtained by averaging over n as in section 2.4.2 to obtain R (it) = — T R ( n /) e i(Pn+£ ' W (3.26) The image domain objective function of (3.21) can now be converted to an aperture domain objective ^GSCA = * R^ l L=Q (3.27) by combining (3.21), (3.22), (3.23), (3.25) and (3.26), where wp(jt) = J 'M p( u ) } is the weighting function expressed in equation (3.23). When the convolution operation is explicitly written (3.27) simplifies to = Z w p(jt) * <*) = l =-L Z W p (o £ t=-L R ( n / ) e j(P n+i ' ^ n) (3.28) n=l A where the Hermitian symmetry of R (-t') has been used to eliminate the negative argument. Therefore maximizing (3.13) is the same as maximizing (3.28) which forces the random radiation field to be stationary. Equation (3.28) shows that J GSCA is a weighted sum of the spatial correlation values measured in the array aperture. The function is a general form of the objective functions of all of the currently published spatial correlation-based algorithms. -64- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The weighting function Wp(jt) defines the specific algorithms and determines how well the algorithms perform. To implement any algorithm characterized by (3.28) Wp(Z) must be determined by either (1) assuming a form for p(u) or (2) estimating p(u) from the measured data. These two choices define two sub-classes within the Spatial Correlation class which are discussed in the following section. 3 .2 .4 The S u b -C la sses o f the S p a tia l C o rrela tio n C lass The weighting function, wp(jt), in (3.28) is the inverse Fourier transform of p(u). Recall from (3.20) that p(u) = e {| fQ(u) * fT(u)s(u) I }. The convolution can be explicitly written to yield p(u) = E { Jf0 (x1)fTs(u-x1) dX] | f*0(x2)f*s(u-x2) dx2 } (3.29) For compactness, the notation has been altered such that frs(u ) = fT(u)s(u) and * * * likewise, fTs(u) = fT(u)s (u). The product of the integrals in (3.29) can be combined into the double integral p (u ) = jJ f0 (Xi )f* (x2)fx(u-Xi)fj(u-x2)E { s(u-x1)s*(u-x2) } dX}dx2 (3.30) Let tj = u-Xj and let ^ = u-x2 such that dtj = -dXj and dt^ = -dx2 . Therefore p(u)=j J V u -tl ) fo (u"^)fT (tl)fT<^)E t s( tl ) s* ('2 )} d ti d i2 (3 .3 i ) The simplest source model of radar clutter is a random distribution, independent in cross range, in which case E{s(tj)s*(<2)} ~ I(ti)8(t]-t2) and p(u) can be expressed as -6 5 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. P (u) = JJ f0 (u -t1)f*(u-i2)fT( ti) 4 (t2)I(t1)5(t1-t2) d f 1 dt 2 = | I f0(u-*!) l2lfT( ti) l2 I(t1) d t 1 = lfT(u)l2 1(u) * lf0(u)I2 (3.32) from which the weighting function, wp(Z), is obtained by taking the inverse Fourier transform of (3.32) w p(Z) = -1 { p (u )} = f ~ l { lfT(u)l2 I(u ) * lfg(u)12 ) = [ y 1 {lfT(u)l2 } * J ' H I(u) 1 ] J ' l { lf0(u)l2 ) = [ J _1( lf T(u)l2 } * R (Z) ] 7 ~ x { lf0(u)l2 } (3.33) The weighting function depends on the spatial correlation function R(j£), which is normally unknown. At this point two paths can be taken toward further evaluation of the weighting function. One route is to assume a particular form of the intensity distribution and therefore of R(jt). Another route is to substitute an estimate of R(Z) into (3.33). Both of these methods yield viable algorithms each possessing advantages and disadvantages over the other. These advantages and disadvantages along with the particular algorithms obtained by the different methods are discussed in Section 3.3. Each method forms a sub class of algorithms as shown in Figure 3.10A. These two sub-classes have been termed the parametric and non-parametric sub-classes respectively since one requires the choosing o f a model (the form of the intensity distribution) and the other makes no such assumptions. Briefly, the performance of the model-based form is somewhat superior provided that the model is accurate. Future references in this document to these two sub classes refer to the manner in which (3.33) is evaluated. - 66 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. SCA Class Parametric Sub-Class Non-Parametric Sub-Class A model is assumed for the intensity distribution, I(u), to determine Wp(jt). No model is assumed. Instead an estimate ofR O t) is used. Examples MMB ECA ULSCA Shear Averaging Phase Closure Flax-O'Donnell Figure 3.10A The structure of the Spatial Correlation class as revealed by the GSCA and examples of existing algorithms in the particular sub-classes. -6 7 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3 The Relationship of the GSCA to Existing Algorithms 3 .3 .1 R evisitin g T sao's E n erg y C o n serva tio n A lg o rith m The fact that (3.13) represents a general form of the objective function used by Tsao and Patrick can be easily seen by setting p(u) equal to a rectangle function. The keen reader will note that the resulting objective function, (3.10), does not necessarily imply a rectangular transmit pattern. That is, the convolution of fQ(u) with the product of fT(u) and s(u) is rectangular, but the transmitter pattern is not necessarily rectangular. Consider the general form of p(u) from (3.32). If the intensity distribution is nearly uniform within the angular sector illuminated by the transmitter and the transmitting antenna possesses a rectangular radiation pattern then the product lfT(u)l2 I(u) will also be nearly rectangular with an extent equal to the extent of the transmitting antenna radiation pattern. In this case (3.32) becomes p(u) = rect(AuT) * lf0(u)l2 = lfT(u)l2 * lfQ(u)I2 (3.34) High-resolution imaging requires the receiving aperture to be large enough to produce a very narrow receive beam. Generally, this receive beam is at least an order of magnitude narrower than the angular sector illuminated by the transmitting antenna. Thus, in the convolution in (3.34), the undistorted receiver array pattern fQ(u) will act essentially as an impulse in u, in which case rect(AuT) * lfQ(u)I2 « rect(A uT) = lfT(u)l2. This effect is demonstrated in Figure 3.10 where a rectangle function (dotted line) has been convolved with a sine2 function (dashed line). The result is shown by the solid line in the figure. The mainlobe of the sine2 function is approximately one-tenth the extent of the rectangle. The convolution produces a function nearly rectangular in shape. - 68- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. O f course, even in simulation the function f0(u) is not exactly an impulse, and consequently, the function p(u) is not precisely a rectangle function. In fact the complex source distribution s(u) will modulate the rectangular transmitter pattern, and the convolution with the finitely narrow, undistorted receive beam f0(u), will cause the rectangular transmit pattern to spread in angle and deviate in shape from a rectangle. However, because the spreading of the transmitter pattern is negligibly small due to the narrowness of the undistorted receive pattern, and the magnitude of the distortion is small, the function p(u) can be closely approximated by a rectangular function with an angular extent equal to that of the transmitter pattern. This approximation yields a weighting function p(u) = lfT(u)l2. Assuming that fT(u) is normalized, replacing p(u) with p(u) in (3.13) yields the ECA objective function given by (3.10). If we view (3.13) as a GSCA objective function, it is not quite the correct objective function for the particular simulations of Tsao and Patrick. However, the sidelobes of the receive antenna power pattern produce negligible leakage of energy from the mainlobe of the transmit power pattern and (3.10) proves to be an excellent approximation to (3.13) in this case. Therefore, the results observed by Tsao and Patrick do not contradict the GSCA and, in fact, should be expected even though the objective function used is not the exact objective function predicted by the GSCA theory. This theory is supported by part of the discussion of the ECA in [48]. In the process of deriving a method of maximizing (3.10) the authors manipulate the function J eca int0 form J ECa =Z Z P(m.D ei( ? " ' m=l n=l ^ (3.35) -6 9 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Convolution of N arrow Sine Squared with W ider Rectangle 1.4 1.2 0 1 0.8 0.6 £ 0.4 0.2 0 20 40 60 80 100 120 140 S am ple Index . 2 Figure 3.10. The result of the convolution of a rectangle function with a sine' function. The solid line represents the final result. where K A /, \vA>* * , , n( \ \ j Ok (u2‘ ul) (xrTxm) • fk(u2"u l) (x n"xm)l V » p(m ,n) = (u2-u ,) e 2 sin e ----------^-------------2*i e * (xm)e * (xn) 1 J k=l (3.36) If a ^ scale factor is introduced into (3.35) then p(m,n) can be expressed as j \ k (uj-uj) (xn-xm) . |'k(u2-u1)(xn-xrn)i a sine R(xm, xn) p(m,n) = C e 2 (3.37) by letting C = (u2-Uj) and K A R( x m- x n) = k S k=\ (3.38) The function described by (3.35) can be re-mapped to include a lag index in the following manner. Let m = n+1 such that I = m-n. Therefore, equation (3.35) becomes -7 0 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. N J'ECA eca = N-n 1 I P(n+Z,n) ej(l^n ' ^ n+l) (3.39) n=l ji=l-n The terms in (3.39) can be re-grouped and the objective function, JEq a , can be re expressed as . J g c A - l i ec < ej 1 <“«><*»-»> ■ *»> 1=1 n=l (3.40) N-l N -i =X w ( / ) X R (n, / ) 1=1 n=i N-l =S L=l (3.41) N-l w O O R (/) = ]L wp(Z)R (Z) > (3.42) 1=1 where the weighting function wp( / ) is a sine function which is the inverse Fourier transform of the rectangle function rect(Au) in (3.10) as predicted by the GSCA theory. Consequently, evidence that the ECA is a member of the Spatial Correlation class has existed since the writing of [48] although it has not been recognized. 3 .3 .2 U se o f A N a rro w In terva l o f In teg ra tio n w ith E x p erim en ta l D a ta Another anomaly that has been previously observed and discussed in section 3.1.2 is the high quality images obtained from the experimental data when the interval of integration in (3.14) is reduced to the size of an undistorted receive beamwidth. The ECA requires the integration interval to be the size of the transmitter beamwidth. However, the images obtained by reducing the integration interval to the size of an undistorted receive -7 1 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. beamwidth (nearly two orders of magnitude smaller) are superior to those obtained from the ECA integration interval (Figures 3.6 (b) and (c)). One of the most perplexing aspects of this phenomenon was that it was observed with experimental high resolution microwave data, but not in the simulated data used by Tsao and Patrick. In fact, when the integration interval was reduced in the simulations, the images obtained were noticeably inferior to the ones obtained when an integration interval the size of the transmitter beamwidth was used (see Figure 3.3 part (c), and Figure 3.11). This effect can easily be explained by again considering the GSCA objective function (3.13). I m a g e o f R a n g e b i n #1 , F O V = 0 . 0 1 7 2 4 -10 -30 -35 ' 4-0.25 -0.2 -0.15 -0.1 -0.05 0.05 0.15 0.2 0.25 u = sin(0) Figure 3.11 Image of rangebin #1 of simulated data after calibration with Au = 17.24 mrads. The transmitter beamwidth is 167 mrads. Consider the final form of (3.32) and let I(u) = a + ^rect(Au') over the angular interval Au > Au' where Au represents extent of I(u) and Au' represents the extent of the rectangle function of amplitude !A. p ( u ) = |f T(u)| [ a + j?rect(Au') ] * | f 0(u)| -7 2 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ^rect(A u') If A > a then (3.33) can be approximated by p(u) = ^rect(Au') |fT(u)| * If0(u)I (3.44) Equations (3.43) and (3.44) model a situation where a Au' sector of the intensity distribution is significandy stronger in square magnitude than the rest of the distribution. If the width of the Au' sector is extremely small such that rect(Au') = 8(u-b) then the function p(u) becomes (3.45) which shows that the appropriate weighting function is a scaled form of the product of the power patterns of the undistorted receiving array and the transmitter array pattern. Ignoring the scale factors the weighting function can be taken to be the power pattern of the undistorted receiving array. This situation models a data set containing a dominant scatterer. To calibrate such a data set p(u) should take the form of the power pattern of the undistorted receiving array steered to the location of the dominant scatterer, which is not known. It turns out that the dominant scatterer location is not necessary to successfully selfcalibrate the array. Recall that (3.13) can be converted into an aperture domain objective function through the Fourier transform and that the weighting function wp(jt) is the inverse Fourier transform of p(u). When a dominant scatterer is present in the data set wp(jt) evaluates to -7 3 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. = ^ if T(b)|2 eJDfc f l { |f0(u )T ) (3.46) If the true location of the dominant scatterer is u=b0 and not u=b, then g(u) = lf0(u-b0)|2 and {g(u)} becomes = e jb°^ J l { If0(u )|2 } (3.47) and the aperture domain objective function develops a linear phase shift in t which does not affect the calibration procedure. Examples of the effects of a dominant scatterer on the function g(u) are shown in Figure 3.9 and Figure 3.12. Figure 3.9 shows the error-free average image intensity, g(u), of the simulated data of Tsao and Patrick in which all of the sources are of comparable amplitudes. The extent of g(u) is approximately the field-of-view of the transmitting antenna. If p(u) is approximated as a rectangle function the extent of p(u) should be made equal to the transmitter beamwidth, yielding the ECA. The results of Tsao and Patrick corroborate this theory. However, when one of the sources dominates the other sources, the averaging process in g(u) is captured by the image of the dominant source and takes the shape shown in Figure 3.12. Clearly, the rectangle that now best approximates the average image intensity is one with a width equal to one beamwidth of the receive array. In this case the GSCA theory yields an algorithm that is not the ECA. -7 4 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Average Image Intensity 120 100 o T3 3 D. E < •S •0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 U = si n ( 0 ) Figure 3.9 Average image intensity of the undistorted simulated data with the original source amplitudes. (Reprinted from page 58) A v e r a g e I ma g e I nt ens i t y, g(u), wi t h Or i g i na l So u r c e A m p l i t u d e s 12000r O. 6000 S3 4000 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 u = sin(9) Figure 3.12 Average image intensity of the undistorted simulated data with a dominant scatterer. Amplitude is 13 times the next largest scatterer. -7 5 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The amplitude of source #1 in the simulated data was originally 0.9 corresponding to the average image intensity in Figure 3.9. After calibration using an narrow integration interval (17.24 mrads = one array beamwidth) its image is shown in Figure 3.11. When the amplitude of 0.9 was changed to 10.9 corresponding to the average image intensity in Figure 3.12, the image improves as shown by Figure 3.13. If the source amplitude is maintained at 10.9 and the integration interval is increased to the size of the transmitter beamwidth (167 mrads) the resulting image (Figure 3.14) is again inferior to the image obtained with a narrow integration interval (Figure 3.13). The wide integration interval represents a p(u) with extent the size of the transmitter beamwidth and the algorithm molds g(u) to best approximate this wide extent. Figure 3.15 shows the final average image intensity. This image more closely approximates the wide rectangular p(u) than does the function of Figure 3.12. Table 3.1 summarizes these results. Items in boldface are results of combinations predicted from GSCA theory. Table 3.1 Summary of Results with Simulated Data Type of Intensity Distribution Narrow Integration Interval Wide Integration Interval Dominant Scatterer in FOV Good Image Poor Image Poor Image Good Image Uniform Intensity Distribution Figures 3.16 and 3.17 demonstrate that the angular coordinate of the dominant scatterer need not be known to successfully self-calibrate the array. Figure 3.19 shows images of rangebin #'s 1,4, and 6 of the simulated data with the rectangular function p(u) centered about u = 0.0965. In Figure 3.17 the interval is centered at u = -0.0835. The array self-calibrates in both cases and the images are similar. -7 6 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I m a g e o f R a n g e b i n #1 w i t h D o m i n a n t S c a t t e r e r • 10 ‘5 CO Q -25 -30 -35 -40 •0,1 -0.05 0.05 0.1 0.15 0.2 0.25 u = sin(0) Figure 3.13 Image of rangebin #1 with a dominant scatterer included in the averaging process I m a g e o f R a n g e b i n #1 wi t h D o m i n a n t S c a t t e r e r -10 -15 •20 -25 -30 •35 .4 0 1 11.11 II II.. II 11 U II -0.25 -0.2 -0.15 I u J -0.1 -0.05 .-------------tJ — L IL 1 L J_ 1 L .1 1 0 0.05 0.1 0.15 11, 11 II 11 II 0.2 0.25 u = sin(0) Figure 3.14 Image of rangebin #1 of the simulated data set with Au = 16.7 mrads and a dominant scatterer present in the data set. 77- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A v e r a g e I m a g e I n t e n s i t y A f t e r Ca l i b r a t i o n 7000 6000 *2 5000 4000 % 3000 2000 1000 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.15 0.2 0.25 u = sin(0) Figure 3.15 Average image intensity after calibration. The function g(u) has been molded to better resemble the rectangle function p(u). -7 8 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Rangebin # 1 *10 •15 (a) 8 Q -20 -25 •35 •0.05 U = Q 0.05 sin(0) 0.15 02 025 0.15 02 025 0.15 02 025 R a n g e b in #4 •10 (b) •25 •30 -35 .4oM h i ii ii u u i u y ii i i -0.25 -02 -0.15 -0.1 -0.05 y 0.05 u ss s i n ( 0 ) R a n g e b i n #6 -10 -15 (C) ^ O -20 *25 -30 -35 -40 •0.05 sin Figure 3.16 Images of three rangebins of simulated data with integration interval equal to [13,180] mrads. 79- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. R a n g e b i n #1 -10 (a) •25 •35 •0.05 Q 0.05 u = sin(0) 0.15 02 025 0.15 02 025 0.15 0.2 025 R a n g e b i n #4 •10 (b) -25 •30 •35 .4 0 U 1-H II, II! I, I -0.25 -0 2 -0.15 ---------------■--------- I I I I. I -0.1 -0.05 0 0.05 u = sin(0) I II, I 0.1 R a n g e b i n #6 •10 (C) ’§ Q •20 -25 -30 •35 -40 sin Figure 3.17 Images of three rangebins of simulated data with integration interval equal to [-167,0.0] mrads. - 80 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The same results are found with the experimental data. The data set consists of radar returns from a Boeing 727 aircraft flying into Philadelphia International Airport and was obtained using inverse synthetic aperture techniques described in Sections 1.3 and 1.4 and in [40], at a range of approximately 2.7 km. A very strong specular echo comes from an engine near the tail of the plane. Figure 3.18 shows the average error-corrected image intensity of the experimental data set. This figure demonstrates that a dominant scatterer is present in the data set. There is approximately an order of magnitude difference between the intensity of the dominant scatterer and the next strongest source. When the dominant scatterer rangebin is not included in the averaging process, the average error-corrected image intensity is as shown in Figure 3.19. The strongest source is now only approximately 2.5 times as large as the next strongest source in intensity. In this case the angular extent of g(u) has become more appropriately the width of the total field-of-view since the intensity distribution is more closely modeled by the distribution of Section 3.3.1. Figure 3.20 shows the calibrated image of the dominant scatterer rangebin of the ISAR data set. This image was obtained after calibrating the system using an integration interval of Au = 0.66. The dominant scatterer rangebin was excluded from the calibration process. The need to reduce the integration interval is evident when Figure 3.20 is compared with Figure 3.7(c). Both figures used the same integration interval but Figure 3.7(c) includes the dominant scatterer rangebin in the calibration process while Figure 3.20 does not. Note that the lack of knowledge of the location of the dominant scatterer did not hamper the self-calibration process. -81 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xio3 A v e r a g e I m a g e I n t e n s i t y o f E x p e r i m e n t a l Data -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 u = sin(0) Figure 3.18 Average image intensity of the ISAR data when the dominant scatterer rangebin is included in the averaging process xio5 A v e r a g e I m a g e I n t e n s i t y o f E x p e r i m e n t al Data 0.01 0.02 0.03 0.04 0.05 u = s in ( 0 ) Figure 3.19 Average image intensity of the ISAR data when the dominant scatterer rangebin is not included in the averaging process - 82 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I ma ge o f D o m i n ^ ^ ^ ^ ^ e r e r Ra ngebi n -0.025 -a ra -0.01s o.oi -0.005 o 0.005 0.01 a o is 00 2 0025 u = sin(0) Figure 3.7(c) Cross-range image of the dominant scatterer rangebin of ISAR data set with Au = 0.66 mrads and the dominant scatterer rangebin included in the averaging process. (Reprinted from page 53) I ma ge o f D om in an t Sc att ere r Ra ngebi n 1---------- .---------- .---------- ,---------- .---------- ,--------- 7 0 1--------- 1 ----------- -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 u = sin(0) Figure 3.20 Cross-range image of the dominant scatterer rangebin of ISAR data set with Au = 0.66 mrads and the dominant scatterer rangebin absent from the averaging process -83- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Both the experimental and the simulated data agree with the theory of this section; i.e., when a dominant scatterer is present in the field-of-view, the integration interval should be reduced to the size of the receive array beamwidth and the image quality is enhanced. The GSCA explains the high quality images obtained from the experimental data along with the low quality images obtained from the simulated data when a narrow integration interval is used. It also suggests a method of determining when to reduce the integration interval to one array beamwidth. The method is borrowed directly from the DSA. Since the absolute location of the dominant scatterer does not need to be known, the problem is detecting only the presence of a dominant scatterer. This can be done by measuring the normalized amplitude variances of each rangebin as [29] 2 / (3.48) k where N is the number of elements in the array. A rangebin with < 0 . 1 2 contains a k dominant scatterer [24] and the integration interval must be reduced to the size of one array beamwidth. The DSA often is also applicable when (3.48) is less than 0.12. It is also much simpler to implement. However, algorithms belonging to the Spatial Correlation class are applicable under broader conditions than the DSA. The GSCA theory shows that any algorithm within the Spatial Correlation class can be realized by altering the weighting function Wp(jt). This is discussed in Section 3.4. Also, Chapter 4 shows that DSA performance is attained by the spatial correlation algorithms even when a dominant scatterer is present. Therefore, since the spatial correlation algorithms are applicable in more general -84- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. circumstances than the DSA, and since there is no loss in performance level when a dominant scatterer exists, use of the spatial correlation algorithms is preferable. 3 .3 .3 D e riv in g th e M o d ifie d M u ller-B u ffin gton A lg o rith m fr o m the G SC A The objective function of (3.13) is a generalized form of the objective function of the Modified Muller-Buffington Algorithm [49] J MMB = t | R « ) f = i R « ) R ‘ (<) f=l (3.49) f=l which is the same as (3.28) if wp(Z) is set equal to R(Z). An important concern is what assumptions are made in arriving at (3.49). The function p(u) is a reference intensity distribution to which the average measured intensity distribution, g(u), is to be matched. The reference intensity distribution is obtained from the hypothetical incoherent source distribution, sh(u), as discussed in Section 3.2.2. This intensity distribution contains no phase errors and is affected only by the windowing of the array aperture. Consequently, p(u) is a diffraction-limited intensity distribution of incoherent sources having an inverse Fourier transform that is the spatial correlation function, Rh(Z), of the random radiation field. Therefore, Rh(Z) is the proper weighting function, wp(jt), for the objective function of (3.28), i.e., R h( / ) = wpU ) = 7 ’ 1{ E { | f 0( u ) * f T(u) s(u)| 2 } } (3.50) and (3.26) can be expressed as -85- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. L L £ w pOOROt) = X t= L R hOO R ( l ) (3.51) i=-L In practice Rh(j£) is not likely to be known a p rio ri. In the MMB Rh(jt) is estimated from the data as R(Z) and its substitution for wp(jt) in (3.28) yields the MMB objective function. Therefore, the MMB can also be derived from the same generalized objective function and is an example of an algorithm belonging to the non-parametric sub-class. No assumptions are made on the form of p(u) or wp(jt). Instead wp(Z) is estimated from the data set thereby requiring less information regarding s(u) than any algorithm from the parametric sub-class. The trade-off is a performance that is slightly inferior to that of a parametric algorithm, as shown in Chapter 4. However, the performance difference spans only 1 to 2 dB. Consequently the MMB is one of the more attractive algorithms in the spatial correlation class since it is widely applicable and performs well. 3 .3 .4 The U n it-L ag S C A a n d the G SC A The spatial correlation class as described by the GSCA objective function is inherently a set of multiple-lag algorithms. Within the GSCA structure the ULSCA materializes in one of two ways; (1) as a member of the parametric sub-class with (3.52) or (2) as an algorithm resulting from compensation for near-field conditions. The latter situation is discussed in detail in Chapter 5 while the former is the subject of this section. The function wp(jt) can be arbitrarily truncated to conform to (3.52). Since there is no redundancy in this algorithm the noise performance is inferior to the multiple-lag algorithms but is computationally simpler than any multiple-lag algorithm. - 86- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It is important to note that the ULSCA belongs to the parametric sub-class. The non-parametric sub-class also contains unit-lag algorithms. These are obtained by setting L = 1 in (3.28). It is shown in Chapter 4 that these algorithms do not yield the ULSCA solution. 3.4 Commonality of the Maximization Procedures GSCA theory shows that a single objective function can be used for all the spatial correlation algorithms. Yet each algorithm was developed with a customized method to determine the estimates of the phase errors. The ULSCA forms a "running summation" [3], the MLSCA makes use of a least squares solution [48], and the MMB maximizes an objective function [29], as does the ECA [48]. There is no need, however, to customize a different maximization procedure to each algorithm. Any of the existing procedures can be used to maximize the objective function of any of the spatial correlation algorithms. A procedure that exploits the nearly sinusoidal variation of the ECA objective function as a function of the phase error estimates Pn and efficiently solves for the set of estimates is given in [48]. Since the ECA is now known to belong to the spatial correlation class, all of the objective functions characterized by the GSCA objective function can be maximized by this efficient procedure. The maximization procedure designed for the MMB can also be used to efficiently maximize the GSCA objective function. In conclusion, a single efficient maximization procedure can be hardwired without restricting the set of spatial correlation algorithms available to the signal processor. The change from one spatial correlation algorithm to another can be implemented simply by changing the general weighting function wp(jt). -87- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.5 Summary This chapter shows that, with the exception of those algorithms that use strong point scatterers to cophase antennas, all published self-calibration algorithms belong to a common (spatial correlation) class and that a single objective function leads to the estimates of the phase errors. Anomalies between simulation and experimental results with the Energy Conservation Algorithm have indicated the theory of Tsao to be inadequate to self-calibrate an array imaging system under general conditions. While simulation results verify that the ECA is a viable self-calibration algorithm when the transmit radiation pattern is rectangular and the field-of-view is uniformly filled with scatterers, the experimental results demonstrate its weakness under more general and realistic conditions. The ECA objective function (3.8) can be manipulated into the more general form (3.10). The two components, p(u) and g(u), are interpreted as an expected image intensity distribution and an average image intensity distribution respectively. The generalized, image domain objective function can be Fourier transformed to an aperture domain objective function composed of a general weighting function wp(jt) and an estimate of the A spatial correlation function of the measured radiation field, R (i). Therefore, the ECA and its generalization are spatial correlation algorithms with a weighting function that depends on the statistical properties of the source distribution. The objective function is a general form of the objective functions of all of the existing spatial correlation algorithms. Any particular algorithm within the Spatial Correlation class is determined by the form of the general weighting function wp(jt). The ECA, the MMB algorithm, and the ULSCA have all been shown to be specific cases of the GSCA. - 88- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The GSCA theory has been used to explain the anomalies between the simulated and experimental data. It has been shown that the ECA weighting function is incorrect when a dominant scatterer is present as in the experimental data. The theory tells us that the correct weighting function in u-space is one proportional to the receive array power pattern. A simple method of determining when to change the ECA weighting function has been presented. This method borrows the technique of detecting the presence of a dominant scatterer from the DSA. Since it is only necessary to detect the presence of a dominant scatterer and its location is not needed this is also a viable technique for determining when to alter the ECA weighting function. Tests with both simulated and experimental data have shown this theory to be correct. -89- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4 Noise and Element Position-Error Performance Analysis In this chapter computer simulations and first-order analysis of the GSCA objective function are used to assess the merits of algorithms within the two spatial correlation sub classes. The algorithms are evaluated based on their performances in the presence of: (1) element position errors and (2) element receiver noise. The evaluation is used to determine the usefulness of the algorithms from one sub-class relative to algorithms in the other sub class. It is found that although the performance of the MMB is inferior to the performances of the parametric algorithms, the lack of need for a priori information generally outweighs the performance difference and makes the MMB widely applicable. Section 4.1 presents the results of the element position error evaluation. Such errors induce phase errors that are functions of angle of arrival and therefore cause a loss in the main beam gain that depends on the angle to which the beam is scanned. The loss in the main beam gain is also a function of several other variables, namely the size of the fieldof-view illuminated by the transmitting antenna, ufov, the rms position error in the locations of the array elements, a Ax, and the number of correlation lags L incorporated into the -90- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. calibration process. Consequently, the algorithms are evaluated by the maximum angle to which the reconstructed beam can be steered without incurring a designated loss. The loss is taken as 1 dB following Steinberg's evaluation of the DSA in [45]. Section 4.2 presents the results of the element receiver noise evaluation. The algorithms were tested with two different source distributions; one corresponding to the uniformly filled field-of-view used in the simulations of Chapters 3 and 4, and the other corresponds to the uniformly filled field-of-view with a dominant scatterer present. The latter source distribution corresponds to the dominant scatterer distribution also used in Chapters 3. Section 4.3 presents an analysis of the GSCA objective. The analysis is used to explain the results obtained from the element receiver noise evaluation. At the same time, the analysis connects the algorithms of the parametric sub-class with the MLSCA of [48] when the receiver noise power is small. Under this condition the algorithm from the parametric sub-class with weighting function w0( l ) yields the same solution as the MLSCA with a weighting matrix G (discussed in Section 2.5.1) composed of the weights of w0(/) . 4.1 Element Position Errors 4 .1 .1 In tro d u ctio n If the exact relative locations of the elements of an array are known then appropriate phase shifts can be calculated to form and scan a beam. If the exact element locations are not known then only an estimate of the appropriate phase shifts can be made based on the assumed relative locations of the array elements. Using the assumed locations instead of the exact locations introduces phase errors into each channel that will degrade the quality of the main beam and raise the sidelobes. -91 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The relative phase error introduced into the n1*1channel from a point source located at an angle 0 measured from broad-side is calculated from the array geometry as [45] cpn(0) = kAxn sin(0) + kAyn cos(0) (4.1) where Axn and Ayn are the differences in the x and y directions, respectively, between the assumed relative location of the nth element and its actual location. The wavenumber, k, converts the magnitude of the distance error to a phase error. When more than one point source is present in a rangebin the total phase error (i.e., the difference between the phase of the signal that would have been received if the element was located at the assumed location and the phase of the signal that is actually received) in the n111channel is a nonlinear function of the phase errors produced by the individual point sources (4.1) and the dependence on 0 shown in (4.1) causes the total phase error to vary with 0 . If the illuminated sector (i.e., the field-of-view) is sufficiently small, the cosine term in (4.1) will be approximately unity for all point sources regardless of their angular location. The sine term in (4.1) will vary approximately linearly with 0 with a slope of unity. Therefore, if 0 varies over a small range, sin(0) also varies over a small range and the phase error in the nlh channel will appear as a constant phase error corrupted by noise. Therefore any of the spatial correlation algorithms are able to calibrate the imaging system to yield an improved image since an effective constant phase error exists in each channel. Also, because the effective phase errors will appear embedded in noise, any of the multiple-lag algorithms provide better performance than any algorithm employing only unit-lag correlation measurements. The effect of element position errors on the spatial correlation class of algorithms has not been previously studied. Because of the nonlinear relationship between the element position errors, Axn and Ayn, and the total phase error in the channel, mathematical analysis is limited and does not yield useful design information. This section provides a -92- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. quantitative description of the performances of the two algorithms of the Spatial Correlation class through the use of extensive computer simulations. In studying the effects of random element position errors on the performance of the DSA, Steinberg [45] considered the expected relative loss in the main beam gain as a figure of merit and derived a theoretical curve for this loss as a function of scan angle and the rms element position error in the x-direction. Following [45], the expected loss in the main beam is also taken as the figure of merit in this section for the spatial correlation algorithms. With a dominant scatterer the performance of the parametric algorithm is identical to the theoretical performance curve o f the DSA even when the unit-lag algorithm is used. The non-parametric algorithm performance is inferior and requires the use of multiple lags to achieve the DSA performance level. Without a dominant scatterer neither algorithm achieves the DSA performance level but the performance curves of both algorithms can be approximated by the DSA performance curve shifted by a constant loss term. 4 .1 .2 The E lem en t P o sitio n E rro r Sim ulation The simulation used throughout Section 4.1 was used earlier in Chapter 3. The elements formed a linear array with an interelement spacing of two wavelengths. Each element was displaced from its assumed position by a random amount, Axn and Ayn . The quantities Axn and Ayn are assumed to be independent random variables uniformly distributed within X X 2 ’ 2 Y Y and ’ 2 2 respectively, where X and Y are determined by ’ specifying the desired rms position errors, o Ax and o Ay, and using the relation <JAxV l2 = X for zero mean, uniformly distributed random variables. A pair of iid random variables (representing the x and y element position errors) were generated for each element in the array and for each realization. Twenty-five realizations were generated in each -93- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. simulation and an estimate of the expected relative gain curve was calculated by averaging over the 25 realizations. In each realization an algorithm from the parametric and non-parametric sub-classes was used to estimate the phase errors present in the imaging system. The MMB algorithm was used as the representative of the non-parametric sub-class. This algorithm was chosen because it forms an unbiased estimate of wp(jt) and is a multiple-lag algorithm. Its performance should be one of the best of the algorithms within this sub-class. From the parametric sub-class a set of algorithms must be chosen since wp(jt) must change with the intensity distribution. Since the field-of-view was filled with uniform scatterers and sidelobes were not simulated on the transmitting antenna, the function p(u) was set equal to a rectangle function with extent equal to the field-of-view. This corresponds to a sine function for wp(Z). The estimates of the phase errors were then subtracted-out of the respective data set and images were formed of a point source located at 10 angular positions equally spaced between 0.0 radians and 0.1 radians. The peak of the main lobe in each image was averaged over the 25 realizations and the estimate of the expected loss was taken as the normalized averaged value. To check the sufficiency of averaging over 25 realizations to approximate the expected value, the simulation was first run using the DSA with parameter values corresponding to those of an actual experiment performed at the VFRC field site in Valley Forge, PA. The experiment, described in [45, pg. 193], was conducted at L-band (30 cm wavelength). A 27 meter array composed of 15 randomly located elements was used as the receiving aperture. The variance of the x-direction element position error (the y-direction position errors affect the calibration process but the x-direction position errors predominate the scanning process) was calculated to be 11.1 ft2. At 30 cm wavelength this corresponds to an rms position error of approximately 3.33L The experimental results have been presented in [45, pg.195] and are reprinted in Figure 4.1. As indicated on the graph, the -9 4 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. solid line represents the theoretical normalized expected main beam relative gain predicted from the equation [45] AG= - 4 .3 k 2c ^ x62 dB (4.2) where 0 is the scan angle in radians and k represents the wavenumber. Figure 4.2 shows the results of the simulation obtained by averaging 25 realizations of element position errors. This figure is in very good agreement with both the theoretical curve predicted by (4.2) and the experimental data of Figure 4.1. The average percent error is 5.06%. Because of the agreement between Figure 4.1 and 4.2, 25 realizations were used to approximate the expected main beam relative gain curves. To obtain Figure 4.2 a dominant scatterer had to be created in the simulated data. Therefore, to check the simulation using the DSA the amplitude of one of the simulated point scatterers was increased to 10.9 amplitude units, which is 12.5 times larger than the amplitude of the next strongest point source. For the evaluation of the spatial correlation algorithm performances, the amplitude was diminished to 0.9 amplitude units, only 1.03 times larger than the next strongest source. Equation (4.2) shows that the expected main beam relative gain for the DSA is a function of two variables, (JAx and 0. However, the expected main beam relative gain for the Spatial Correlation class of algorithms is a function of four independent variables; the rms element position error, o Ax, the scan angle, 0, the number of correlation lags, L, used in the calibration process, and the size of the field-of-view, ufov. Simulations were run for various combinations of these parameters. Table 4.1 lists the four parameters and the values for which simulations were run. The variable 0 was varied over the interval [0.0,0.1] radians and was sampled at ten -9 5 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ELEMENT POSITION ERRORS -2 s3 -3 e 3 P redicted loss a. f r o m th a o r y -6 -7 S u n sngle (degrees) Figure 4.1 Theoretical relative gain curve for the DSA with the results of four experiments showing agreement with the theory. (Reprinted from Steinberg [45]) Theoretical and Simulated DSA Main Beam Loss Curves Simulation R esults Theoretical R esults m ■o _c a <3 o> a o oc -5 - 0.0 2.0 1.0 Scan Figure 4.2 A n g le 3 .0 4 .0 (d e g re e s) Simulated relative gain curve produced by the simulation of Section 4.1.2 showing very good agreement with the theoretical DSA curve and the experimental results of Figure 4.1. -9 6 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. equally spaced intervals yielding a total of 11 sample points. The rms element position error, c Ax, was varied over one wavelength and sampled at four points corresponding to X X X —, —, —, and X. The field-of-view was varied from 17 mrads to 167 mrads; the latter 8 4 2 represents a field-of-view of almost 9.6 degrees which is very large for practical high resolution radar imaging systems. For the variable L, only L = 1 and L = 29 were simulated representing the use of a small number of lags and a large number of lags. These two cases represent the extremes for the simulated thirty element array. Performance between these two cases is expected to follow the Table 4.1 Parameters of Main Beam Loss and the Values Used in the Simulations Ax 0.017 0.04 0.08 0.12 0.167 0 rads performance of "constant-phase-error-in-noise" model studied in Section 4.2. This model assumes that the phase error in the nlh channel is composed of a time invariant term plus a time varying term. The simulations of Section 4.1.3 verify this by exhibiting the same characteristics as the noise performance curves of Section 4.2. Although L = 29 is the largest set of lags simulated, this does not limit the use of the performance data since most practical systems will not use more lags for two reasons. First, recall that the measurement of high-order lags introduces a redundancy into the -9 7 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. GSCA objective function. As shown in Section 4.3 the weight this redundant measurement carries is proportional to the squared magnitude of the measurement (for the non-parametric sub-class) or the theoretical value (for the parametric sub-class). In either case the magnitude will always decrease with increasing lag number and the contribution of the measurement to the solution will become negligible. Consequently, adding additional array elements to increase L will, at some point, become useless. Figures 4.3 and 4.4 indicate that L < 5 is needed when a dominant scatterer is present and that 5 < L < 10 is useful when no dominant scatterer is present but beyond 10 lags the performance improvement vanishes. Table 4.2 summarizes these results. Table 4.2 Largest CoiTelation Lag Necessary to Achieve Peak Performance Dominant Scatterer Parametric Algorithm Non-Parametric Algorithm Present 1 1 -5 Absent 5 -1 0 5 -1 0 Secondly, practical considerations must also be taken into account when considering implementation of an algorithm from the Spatial Correlation class. The computation time required to calibrate using 29 correlation lags is significantly greater than the time required using only a low-order, multiple-lag algorithm. Consequently, any system employing an array containing more than thirty elements will most likely be limited to the use of a lowerorder multiple-lag calibration algorithm due to processing time constraints. Therefore the need for performance curves with L > 29 is seen as a rare situation. 4 .1 .3 M a in B ea m L o ss D u e to E lem en t P o sitio n E rro rs Figures 4.3 and 4.4 show relative gain curves for both the parametric and nonparametric sub-classes with o Ax =-g and ufov = 167 mrads and a dominant scatterer present. Figure 4.3(a) contains several curves from the non-parametric sub-class -9 8 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. R e la tiv e 0.0 (a) G a in C u rv e s N on-P aram etric DS Class MMB Lag 29 MMB Lag 10 MMB Lag 5 MMB Lag 1 O •0.2 « -0.3 0 .0 0 0 .0 2 0 .0 4 Scan 0.00* to r th e A lgorithm 0 .0 6 Angle 0 .0 8 0 .1 0 0 .1 2 (radians) R elative Gain C u r v e s for the N on-P aram etric A lgorithm •0.01 . c (b) 3 -0.02 o> <3 Ea> -0.03 -0.04 0 .0 0 DS Class MMB Lag 29 MMB Lag 10 MMB Lag 5 0 .0 2 0 .0 4 Scan 0.00 0 .0 6 A ng le 0 .0 8 0 .1 0 0 .1 2 (radians) R elative Gain C u r v e s for the P a r a m e t r i c A lg o r i t h m CO (c) ■u -0.01 • c <o u a> « -0.02 • c® c -0.03 0 .0 0 DS C lass ECA Lag 29 ECA Lag 10 ECA Lag 5 ECA Lag 1 0 .0 2 0 .0 4 Scan Figure 4.3 0 .0 6 Angle 0 .0 8 0 .1 0 0 .1 2 (radians) (a) Relative gain curves for the non-parametric sub-class for different values of L with a dominant scatterer in the simulated data set. (b) Same as (a) with ordinate scale expanded. (c) Relative gain curves for the parametric sub-class for the same case as (a). -9 9 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Relative Gain Curves for the Non-Parametric Algorithm 0.2 0.0 CD U -0.2- Domscat MMB Lag 29 MMB Lag 10 MMB Lag 5 MMB Lag 1 _c (3 o o > n o -0.4- -0.6 - CC -0.8 - 0.00 0.02 0.04 Scan 0.06 A n g le 0.08 0.10 0.12 (radians) Relative Gain Curves for the Parametric Algorithm 0.0 -O D— Q D1 ] F =e— a — r - f i — ^ =3 ; CD ■8 c Domscat ECA Lag 29 ECA Lag 10 ECA Lag 5 ECA Lag 1 -0.2 (S (b) (5 © > 0 cc -0.3 - -0.4 -0.5 0.00 -a— b— h 0.02 0.04 Scan 0.06 A ng le 0.08 — i---- r~ 0.10 0.12 (radians) Figure 4.4 (a) Relative gain curves for the non-parametric sub-class for different values of L with no dominant scatterer in the simulated data set (b) Relative gain curve for the parametric sub-class for the same case as (a) -1 0 0 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. corresponding to the different number of correlation lags used into the calibration process. Part (b) of Figure 4.3 shows the same curves as part (a) with the L = 1 curve removed to show the curvature of the remaining curves. Part (c) shows the corresponding curves from the parametric sub-class. Note that this graph contains the L = 1 curve. The curves of (b) and (c) are virtually identical and the performance variation as a function of L is the same as the performance with additive noise shown later in Section 4.2. Here, as in Section 4.2, a significant improvement occurs in the non-parametric algorithm when the number of correlation lags is increased from L = l . Also, the parametric algorithm yields performance nearly identical to the DSA performance, even for L = 1. This is also the case in Section 4.2. Figures 4.4(a) and (b) show the performance curves for the same parameter values as Figure 4.3 when a dominant scatterer is not present. Again, the performance follows the results of Section 4.2 where a significant improvement in the performance is obtained around L = 10 in both sub-classes. The performance in the presence of element position errors as a function of L follows the same characteristics as the additive receiver noise case between L = 1 and L = 29. Like the DSA, all of the algorithms belonging to the Spatial Correlation class are phase correcting algorithms. Therefore it is expected that the SCA main beam gain curves will possess a dependency on the scan angle, 0, similar to the DSA gain curve. Figures 4.5(a,b) and 4.6(a,b) make the comparison for two different FOV's. SCA and DSA gain curves are plotted. The DSA gain curve is used as a benchmark for the spatial correlation algorithms because it represents an ideal calibration process. The parametric curve is always plotted using squares for plot symbols and is listed as "P Sub-Class" in the legends. Similarly, the nonparametric sub-class is listed as "NP Sub-Class " in the legends and its curve is plotted using a diamond shaped plot symbol. -101 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves o.o P Sub-Class NP Sub-Class DS Class m -0.5 ■o c eg C3 ID > « ID DC -1.5 - -2.0 0.00 0.02 0.04 0.06 S can A ng le 0.08 0.10 0.12 (ra d ia n s ) Main Beam Gain Curves o.o P Sub-Class NP Sub-Class DS Class ca -0.5 u © > R> ■*- S> CC -2.0 0.00 Figure 4.5 0.02 0.04 0.06 Scan A ng le 0.08 0.10 0.12 (ra d ia n s ) Main beam gain curves from simulated data for ufov = 17 mrads., o Ax = A. (a) L = 1 (b) L = 29 - 102 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Following Steinberg [45], 1 dB is taken as an acceptable main beam loss and the angle, 9, at which the relative main beam gain is -1.0 dB is the maximum allowable scan angle, 0max, of the calibrated system. Since the beam can be steered in either the positive or negative theta direction, the scan range of the calibrated system is 20max. Figure 4.5 shows the relative gain curves for a small field-of-view , u fov = ^ torads, G ^x = X meters, and L = 1 and 29 lags. Such a field-of-view represents a single point source being illuminated and seen through a diffraction limited system or a data set containing a dominant scatterer. The performance difference of the parametric algorithm is negligible between (a) and (b) and equal to the DSA performance. This is true of all the parameter combinations from Table 6.1 using ufov = 17 mrads (see Appendix B). Because of the complexity of L = 29 relative to L = 1, the parametric unit-lag algorithm is the preferred algorithm in this situation. To achieve DSA performance with a non-parametric algorithm, a multiple-lag algorithm must be used. Part (a) shows 17.2% of the total allowed main beam loss is lost simply by using the non-parametric, unit-lag algorithm. The loss in 0 max due to implementing the non-parametric unit-lag algorithm instead of the parametric unit-lag algorithm is 15 mrads or a loss of 30 mrads in scan range. The performance loss can be overcome by incorporating more correlation lags into the calibration process. Figure 4.5(b) shows that for L = 29, the performances of the two algorithms are essentially identical to each other and to the DSA. Therefore DSA performance can be achieved when a dominant scatterer is present by implementing a multiple-lag spatial correlation algorithm or the parametric unit-lag algorithm. When a multiple-lag algorithm is used the non-parametric algorithm is preferred since it does not require the additional step of testing the data set for the presence of the dominant scatterer before calibration. Overall, the parametric unit-lag is the preferred algorithm with a dominant scatterer. - 103 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The curves of Figure 4.6 indicate that the performance characteristics remain unchanged by variations of the field-of-view. However, with increasing field-of-view size the performance difference between the two unit-lag algorithms also becomes increasingly important. For example, two imaging systems operating with the performance curves of Figure 4.5(a) have scan ranges that differ by 16.05 mrads while the same two systems operating with the performance curves of Figure 4.6(a) have scan ranges that differ by 100 mrads. Although the scan range difference of 16.05 mrads may be acceptable, the 100 mrads difference in scan range is most likely unacceptable. In this situation, again, the parametric unit-lag algorithm is the preferred algorithm. Another consequence of increasing the field-of-view is the number o f correlation lags needed to achieve DSA performance in a non-DSA environment. With a small fieldof-view or in the presence of a dominant scatterer only 1 lag is needed with the parametric algorithm. However, as the field-of-view is increased, the unit-lag algorithm is no longer sufficient to achieve DSA performance (Figure 4.6(a)). With a large field-of-view the 29lag parametric algorithm still does not achieve DSA performance (Figure 4.6(b)). However, the scan range difference between the parametric algorithm and the DSA has been reduced from 57.6 mrads to 17.5 mrads by including the higher-order lags. The 29lag non-parametric algorithm is competitive with the parametric algorithm since the scan ranges differ by 12.5 mrads. Provided this difference is acceptable, the non-parametric multiple-lag algorithm is now the preferred algorithm since it does not require assumptions to be made regarding the intensity distribution. Table 4.3 summarizes the utility of the spatial correlation algorithms. All algorithms listed have a non-zero scan range relative to 1 dB expected relative gain loss for the particular case of interest. Algorithms in bold italics represent the preferred algorithms as discussed in this section. - 104 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.3 Situation Dominant Scatterer Applicable Algorithm Parametric } xr • r Non - ParametncJ Parametric 1 N on-Param etncJ No Dominant Scatterer Large FOV = 167 mrads U n it-L ag Multiple-Lag Parametric Unit-Lag Parametric 1 ^ M u ltip le-L a g Non - ParametricJ -1 0 5 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves o.o P Sub-Class NP Sub-Class DS Class -0.5 CD 5. -1.0 c « O © > -2.0 -2.5 -3.0 0.00 0.02 0.04 0.06 Scan Angle 0.08 0.10 0.12 (radians) Main Beam Gain Curves P Sub-Class NP Sub-Class DS Class (dB) -0.5 - c (b) ■ -1.0 - CO O 0) > -1.5 *-* n 0) DC -2.0 - —i 1---- 1---- 1--------1-1-------- 1-1-------- 1-1---- 1-2.5 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Scan Figure 4.6 Angle (radians) Main beam gain curves from simulated data for ufov = 83.5 mrads., g Ax = X (a) L = 1 (b) L = 29 -106- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 .1 .4 A n A p p ro x im a tio n to th e S C A P erfo rm a n ce C u rves The data of Figures 4.5, 4.6, and Appendix B indicate that wheno Ax < X or ufov < 120 mrads, the relative gain curves can be approximated by the DSA curve shifted by a small constant downward. Furthermore, the differences are vanishingly small (< 0.1 dB). Figure 4.7 plots this shift for all of the parameter combinations used in Figures 4.5, 4.6, and Appendix B. Therefore, the relative gain curve changes from (4.2) to AG = - 4 .3 k 2 o 2 02 - C Ax dB (4.3) where the constant C is obtained experimentally or through simulation by a single measurement on boresight. Beyond one wavelength rms the difference is better approximated by a linear function and (4.3) must be modified to AG = - 4 . 3 k 2 o 2 62 - C ,9 - C„ dB Ax 1 2 (4.4) where two measurements must be made to determine Cj and C2. -107 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Difference Between DSA and SCA Relative Gain Curves 0.1 0 CD 0.08 i o o c ok . 0) 0 .0 6 - Q 0 .0 4 - Field-of-view (radians) 0.167 0.12 0.08 0.04 0.017 _c a O ra QC 0.00 "3 -0.02 0 .0 0 0.0 2 0 .0 4 Scan 0.06 A ngle 0.08 0 .1 0 0.12 (ra d ia n s ) Difference Between DSA and SCA Relative Gain Curves 0.10' ~ o o 0.08 0) 0.0 6 - c Field-of-view (radians) 0.0 4 - (b) 0.167 -*—*—-*—ft—ft—ft—a_ -a—ft—* i 0.12 0.08 0.04 0.017 c (9 o 0) > I 0) cc 0.02 J 0.00 -0.02 H— i— i— >— i— ■— i— ■— i— ■— i— r0 .0 0 0 .0 2 0 .04 Scan Figure 4.7 (a) and (b) 0 .0 6 A ngle 0.08 0 .1 0 0 .1 2 (radians) Difference between the DSA and SCA relative gain curves for fields-of-view X X o 4 simulated and for (a) ^ rms position error and (b) 7 rms position error -108 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Difference Between DSA and SCA Relative Gain Curves 0.20 m •o o o c 0 0 Q c 0 o 0 > 2. 0 cc 0.15Field-of-view (radians) 0.10- 0.167 0.12 0.05- 0.08 0.04 0.017 0.00 -0.05 - -0.10 0 .0 0 0 .0 2 0 .0 4 Scan 0 .0 6 A ngle 0 .0 8 0.10 0.12 (radians) Difference Between DSA and SCA Relative Gain Curves 0.6 *D 0 o c 0 i. 0 0 .4 Field-of-view (radians) 0.2- 0.167 0.12 D c a 0.08 0.04 0.017 0.0 (3 0 > 0 0 -0.2- DC -0.4 0 .0 0 0 .0 2 0 .04 Scan Figure 4.7 (c) and (d) 0.06 A ngle 0 .0 8 0.10 0.12 (radians) Difference between the DSA and SCA relative gain curves for fields-of-view X simulated and for (c) - rms position error and (d) \ rms position error -1 0 9 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.2 Receiver Noise Analysis The simulation of Section 4.1 was modified to study the effect of element receiver noise on the spatial correlation algorithms. A spatially random element phase-error profile x with an rms value of 0.78 radians ~ g wavelengths was added (Figure 4.8). Zero mean complex Gaussian noise was added to the received signal at each element from each of the 10 rangebins. The real and imaginary components of the noise were independent from element to element and from rangebin to rangebin with equal variances inversely proportional to the desired signal-to-noise ratio. Twenty-five noise realizations were generated and the array was calibrated using one algorithm from each sub-class. The residual phase error on each element was calculated as the difference between the known phase error at that element (Figure 4.8) and the estimate of the phase error as determined by the two algorithms. Each set of residual phase errors was recorded for each Phase Error Profile Across the Array 1 .5 0 .5 S3 os - 0 .5 - 1 .5 10 15 20 25 30 Array Element Number Figure 4.8 The phase error profile across the array used in the simulations of Section 4.2 - 110- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. noise realization. The rms residual phase error, the expected mainbeam relative gain, and the expected average and peak sidelobe levels were estimated from the 25 realizations. The plots of Figures 4.9 through 4.15 illustrate the performances of algorithms of the two sub-classes in terms of the residual phase error, mainbeam gain, average and peak sidelobe levels as a function of correlation lag. The number of elements is N = 30 and the SNR per element is 3, 13, and 23 dB. Figures 4.9 - 4.12 exclude a dominant scatterer; in Figures 4.13 - 4.15 one is included. The validity of the simulations can be checked by comparing the measured mainbeam relative gain and average sidelobe level to the values predicted from theory. The mainbeam relative gain is related to the phase error variance by [45] AG = - 4 .3 ap dB (4.5) where o^ represents the variance of the residual phase errors, (3n, n = 2 , . . . , N . The values shown in Figures 4.10 and 4.14 are within 2.6% of the values predicted by (4.5). The average power pattern of a uniformly weighted array distorted by phase errors only is given by [46] 2 e| f(u)f*(u) j = | E je-^j 1 r i l E{l e jP} Jpl l2 ) f Q(u)f*(u) + — af l((ll--|E N (4.6) where a is the magnitude of the element weights and f0(u) is the undistorted radiation pattern. Assuming a = 1, the average sidelobe level is then given by ASL=l k ) I E(f<u)f*(u)}du i2 E{ejfJ}| l- |E { e jP] L - L J L j f o(u) f o(u)du - — l - l - i XM i 0 0 N (4.7) - Ill - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where U denotes the sidelobe region (all u-space exclusive of the mainlobe) and xCV) represents the size o f the sidelobe region. The first term of (4.7) is the ASL of the normalized undistorted power pattern (ASL0) scaled by the quantity |E |e ^ J j . The second term represents an increase in the ASL due to phase errors. For small phase errors such that sin (5 = (5 and cos (3 = 1 - ^ p 2 , |E |e ^ || can be approximated by lE{cil f " ’ - i 6 ! 152} = (4-8) and (4.7) can be written as ASL » f 1 1 1 ^ 2 °P _2 ° ASL + - 1 o N (4.9) For a 30 element uniformly weighted array with a = 1, ASLQ =0.0035 or -24.56 dB and Op of Figures 4.9 and 4.13, the ASL's of Figures 4.10 and 4.14 are within 6.0% of the values predicted by (4.9). For medium and high SNR's ( > 13 dB, Figures 4.9(a,b)) there is little difference between the performances of the two algorithms of the two sub-classes until L > 25 lags. Both algorithms require L > 10 to substantially increase their performances over the respective unit-lag performance levels. As L is increased beyond 10 the MMB exhibits a constant performance level while the ECA performance continues to improve. At L = 29 the MMB rms residual phase error remains unchanged from its value o f 0.13 radians at L = 10 while the ECA rms residual phase error is reduced to 0.035 radians. Consequently, the parametric algorithm is the preferred algorithm when a large number of lags are available for use. - 112- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The difference in peak sidelobe performance is also important (Figure 4.12). At L = 29 the peak sidelobe level of the ECA is 2 dB below that of the MMB. This is due to the leveling-off of the performance of the MMB for L > 10 lags. A trade-off exists when using the MMB algorithm; the price paid for not knowing characteristics of the source distribution is a lower limit on the performance of the algorithm. This is evident by the constant performance level exhibited by the MMB algorithm at lags > 10 in Figures 4.9(a,b). The performance limit is reached within 10 lags and the use of any more lags, regardless of their strengths, does not improve performance. Figure 4.9(c) shows that at low SNR's (= 3 dB) the gap between the ULSCA and the ECA or MMB narrows and the preferred algorithm is less definitive. Table 4.4 illustrates the weak performances of both algorithms. The multiple-lag algorithms do provide some improvement in performance which the designer may or may not find adequate for the increased complexity. Quantitatively the preferred algorithm is still the parametric algorithm since this algorithm provides the greatest improvement over the ULSCA performance level. However, the need to model the source distribution can arguably out-weigh the marginally superior performance provided by the parametric algorithm. Table 4.4 Performance of the ECA and MMB Residual Phase Error Relative to the ULSCA Level for 3 dB SNR Change From 23 dB SNR Parametric Algorithm (ECA) 61.3 % 46.3 % Non-Parametric Algorithm (MMB) 80.4 % 32.3 % -1 1 3 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. With a dominant scatterer present (Figures 4.13 - 4.15) all of the multiple-lag algorithms perform equally well and yield performances equal to that of the ULSCA. The unit-lag algorithm corresponding to L = 1 in the MMB objective function yields considerably worse performance. Therefore the ULSCA (the simplest) is the preferred unit-lag algorithm and the MMB (non-model based) algorithm is the preferred multiple-lag algorithm. Table 4.5 summarizes the algorithm preferences for the various cases discussed in this section. Table 4.5 Situation Preferred Algorithm Dominant Scatterer Parametric Unit-Lag No Dominant Scatterer Non-Parametric Multiple-Lag -114 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. RM S R esid u al Phase Error (SN R = 23 dB ) 0.4 0 .3 - •3 SCA ECA MMB 0.2- (a) 0.0 0 10 20 30 Maximum Correlation Lag RMS Residual Phase Error (SNR = 13 dB) 0.4 0.3- -* — -* — -O 0.2(b) 0.0 0 10 20 SCA ECA MMB 30 Maximum Correlation Lag RMS Residual Phase Error (SNR = 3 dB) 0.4 0 .3 - SCA 0.2- ECA MMB (c) o.o 0 10 20 30 Maximum Correlation Lag Figure 4.9 (a) rms residual phase error with a 23 dB SNR (b) rms residual phase error with a 13 dB SNR (c) rms residual phase error with a 3 dB SNR - 115 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. M ain Beam Gain (S N R = 23 dB) 0.0 c •a - 0.1 - 0.2 o SCA ECA MMB -0.3 (a) CQ c -0.4 -0.5 - 0.6 0 10 20 30 Maximum Correlation Lag Main Beam Gain (SNR = 13 dB) o.o -0.1 03 3 c ■a o -0.2 SCA ECA MMB -0.3 (b) PQ c •a -0.4 -0.5 - 0.6 10 0 20 30 Maximum Correlation Lag Main Beam Gain (SNR = 3 dB) -0.1 -0.2 a •a a (c) cq c I -0.3 -K— -tr -0.4 SCA ECA MMB -0.5 -0.6 0 10 20 30 Maximum Correlation Lag Figure 4.10 (a) Main beam gain with a 23 dB SNR (b) Main beam gain with a 13 dB SNR (c) Main beam gain with a 3 dB SNR 116 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A verage Sidelobe Level (SN R = 2 3 dB) •21 O 5 £o (a) 73 .* 2 c/3 o o2> < -22 SCA ECA -23 MMB •24 0 10 20 30 Maximum Correlation Lag Average Sidelobe Level (SNR = 13 dB) -21 CQ T3. C >J (b) £O 73 2 c/3 0bo a -22- SCA ECA MMB -2 3 - -2 4 - 1 -25 0 10 20 30 Maximum Correlation Lag Average Sidelobe Level (SNR = 3 dB) -21 CQ ■o o> (C) £o "53 •o c?3 « S2 > < -2 2- SCA ECA -2 3 - MMB -2 4 - -25 0 10 20 30 Maximum Correlation Lag Figure 4.11 (a) Average sidelobe level with a 23 dB SNR (b) Average sidelobe level with a 13 dB SNR (c) Average sidelobe level with a 3 dB SNR - 117 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Peak S id elo b e L evel (SN R = 23 dB) -80 % (a) SCA ECA MMB .”2 c/3 1 Dm -12•13 0 20 10 30 Maximum Correlation Lag Peak Sidelobe Level (SNR = 13 dB) OQ S' 15 > (b) $ ££ ■o C/3 I Cl. SCA ECA -10 MMB •11- -13 0 10 20 30 Maximum Correlation Lag Peak Sidelobe Level (SNR = 3 dB) CQ (c) £0 SC A ECA MMB -10 c/3 1 Oh -12 -13 0 20 10 30 Maximum Correlation Lag Figure 4.12 (a) Peak sidelobe level with a 23 dB SNR (b) Peak sidelobe level with a 13 dB SNR (c) Peak sidelobe level with a 3 dB SNR - 118 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. RM S R esid u al Phase Error (S N R = 23 dB) 0.08 0 .0 6 - 0 .0 4 - 0 .0 2 - 0.00 0 10 20 30 Maximum Correlation Lag RMS Residual Phase Error (SNR = 13 dB) 0.08 0 .0 6 - vi s (b) ’•CQ § SCA ECA MMB 0.04 - OS 0 .0 2 - 0.00 10 20 30 Maximum Correlation Lag Figure 4.13 (a) rms residual phase error with a 23 dB SNR and a dominant scatterer (b) rms residual phase error with a 13 dB SNR and a dominant scatterer - 119 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. M ain Beam Gain (S N R = 23 dB ) 0.005 i 0.000 -0.005 •I o - 0 .010 - 0.020 -0.025 0 20 10 30 Maximum Correlation Lag Main Beam Gain (SNR = 13 dB) 0.005 0.000 e •a -0.005 a (b) SCA ECA MMB - 0.010 - 0.020 -0.025 0 20 10 30 Maximum Correlation Lag Figure 4.14 (a) Main beam gain with a 23 dB SNR and a dominant scatterer (b)Main beam gain with a 13 dB SNR and a dominant scatterer - 120 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A verage Sid elob e Level (SN R = 23 dB) -24.00 O -24.50 ■ -24.75 - -25.00 0 10 20 30 Maximum Correlation Lag Average Sidelobe Level (SNR = 23 dB) -24.00 — -24.25 - -M -A — -o — -24.50 - (b) SCA ECA MMB t/5 | -2 4 .7 5 - -25.00 0 20 10 30 Maximum Correlation Lag Figure 4.15 (a) Average sidelobe level with a 23 dB SNR and a dominant scatterer (b) Average sidelobe level with a 13 dB SNR and a dominant scatterer -121 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3 Analysis of the Solutions of the GSCA and the MMB The GSCA objective function is a highly nonlinear function of the variables and consequently general closed-form analysis is intractable. However considerable insight regarding several of the results of the previous two sections can be obtained through firstorder analysis of the GSCA objective function. Such analysis is made in this section and is valid provided good initial estimates of the phase errors are used and the variance of the phase noise, n+^, is small. It is shown that linearization leads to a weighted linear least- squares problem and that the two sub-classes are asymptotically equivalent as the measurement noise goes to zero, but otherwise the parametric sub-class is superior. In addition, it is shown that the MMB does not degenerate to the ULSCA when only the unit lag measurements are used as claimed in [29]. 4 .3 .1 L in ea riza tio n o f the M M B an d th e P a ra m e tric S u b -C la ss To demonstrate the linearization of the parametric sub-class of Spatial Correlation algorithms, consider a 5 element array example. The GSCA objective function can be expressed as 4 •>GSCA= L= - 4 = |wp( l) |2cos(p2+512-ft2) + |wp( l) |2cos(p3-P2+52 3-^3+^2) + |w p( l ) |2COs(p4-p3+53)4-$4+$3)+ |w p( l ) |2COS(p5-p4+54j5-$ 5+p4) + Iwp(2) |2cos(P3+8] 3-^3) + |wp(2)|2cos(p4-p2+52)4-$4+$2) + |wp(2)|2cos(p5-p3+53)5-ft5+(33)+ |wp(3)|2cos(p4+51)4-$4) -122 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. + |w p(3 )|2cos(p5-P2+82)5-$5',$ 2 )+ |w p(4)|2cos(p5+51)5-$ 5) (4.10) Following the notation of Chapter 2, 8n n+/, is the phase noise in the measurement of the correlation between the n111 and the (n+j£)th elements. The weights wp(Z) are complex in general. The phase of wp(jt) ideally represents the true phase, , of the true spatial correlation function, R^. Consequently, the true correlation phase can be assumed canceled and any mismatch resulting from incorrect modeling of the source distribution can be absorbed into the phase noise term, 5njn+ /,. This is the reason that only the magnitude o f Wp(Z) is included in (4.10). The magnitudes are squared because the magnitude of the product [wpOO R "(>t ) ] is |w (Z)|l Rz | = |w pOO|2 provided the weighting function is selected correctly. Equation (4.10) can be written more compactly in the form = |w p( l ) |2cos(r12) + |w p( l ) |2cos(r2 3 ) + | wp( l ) |2cos(r34) + |w p( l ) |2cos(r4 5 ) + |w p(2 )|2C0S(r13) + |wp(2)|2cos(r24) + | wp(2)|2cos(r3)5) + |w p(3 )|2cos(r14) + |w p(3)|2cos(r2 5 ) + |w p(4 )|2cos(r15) (4.11) where rn n+jt represents the phase residual of the product formed between the nth and (n+Z)th elements. If the algorithm is initialized with the unit-lag phases then the residuals in (4.11) will be relatively small, the cosine terms in (4.11) can be approximated by the first two terms of its Taylor series expansion «*<w > " 1 - 2 w <4 J 2 ) and (4.11) can be approximated by -123 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 ^ w pO t)R * (/) 1= - 4 X ( 5- ^ ) | w p W | 2 - \ ( I Wp (1) | Z l 2 + | Wp(l) | Z £3+ I Wp (1) | Z l 4 + | Wp(l) 22 r4.5 ^=1 Wp(2) + Wp(3) fU + r M+ Wp (2 ) Wp(3) ' 2 , r 2,4 r 2,5+ wp(2) wp(4) 3,5 1,5 (4.13) and generalized to an N-element array by J GSCA= X f=-N+l 2 N -l » £ ( N - 0 wp(/) f=l N -l N - f -X X wpW f=l m=l m,m+f (4.14) Since all of the quantities in the double summation are positive, (4.14) will be maximized when the weighted sum of the squares of the residuals is minimized. This implies that the GSCA objective function yields a weighted least-squares solution and the various algorithms characterized by the GSCA by their different weighting functions of wp(Z) can be analyzed using least-squares theory. Least-squares theory indicates that the measurements should be weighted inversely proportional to their variances [18], [30], [34]. Appendix A derives the variance of 5n n+jt under an assumption of strong wavefront correlation. The variance is inversely 2 proportional to | R(^) | . Consequently, multiplying the residuals of lag L by | R(^) | properly weights them. - 124 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 It is easy to show that a unique the least-squares solution exists. Define matrices Ap and Gp for the 5 element array example w <1)1 <1)1 1 0 0 O' w U)| -1 1 0 0 -1 0 1 0 0 0 -1 1 0 -1 1 0 1 0 0 1 w w P(i)| wp(2) G P = Diag wp(2) 0 0 0 -1 0 wp(3) -1 0 0 0 0 0 wp(2) wp(3) 0 1 (4.15) 0 1 1 wp(4) and two vectors, one containing the phase error estimates and another containing the measured correlation phases P2 + 8 ,i2 P3 ~ P2 + ^2,3 P4 - P3 + 834 P5 ~ P4 + ^ 4,5 P3 + $1,3 P2 A Pp = P3 P4 - P2 + 82j4 bp - P4 P5 - P 3 + 53,5 (4J6) P4+814 LP5 P5 ~P2 + ^2,5 P5 + ^1,5 and find the solution to the least-squares problem A A GpApPp ~ bp (4.17) -125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It is easy to verify that the first four rows of A are linearly independent; consequently Ap is full rank and a unique least-squares solution exists. In fact, the first N -l rows of Ap for an N-element array represent unit-lag correlation measurements of exactly the same form as the ULSCA. Since the ULSCA induces a full rank mapping, the addition o f more equations to the ULSCA system cannot decrease the rank of A p . Additionally, the product, G PAP, is also full rank since G P is square and full rank [32]. Consequently, a unique least-squares solution will always exist even for an N-element array of arbitrary size. The linearization of the MMB algorithm is accomplished in much the same manner as with the parametric sub-class. The MMB objective function of a 5-element array is 4 I A X |o A * A A * A A* A A* A I ROOT = 2R ( l) R ( l) + 2R (2)R(2) + 2R (3)R(3) + 2R (4)R(4) 1= - (4.18) 1 A * A A* A A where the factor of 2 arises because R (-1 )R(-1) = R (1 )R(1) and R(jt) is an average over the N-jt lag L measurements available. For the 5-element example the terms of (4.18) evaluate to 2 R * ( l ) R ( l ) = 4 | R 1| 2 + 2 | R 1| 2c o s (p 1- 2 p 2+ p 3 + 5 2)3 - 5 1)2- ^ 1+ 2 ^ 2- ^ 3) + 2 | r J cos(P2-2P3+P4+83)4-82)3-$2+2^3-$4) + 2 |R j| COS(P3-2P4+p5+S4)g-83)4-P3+2P4-P5) + 2 | r J COS(Pj -P2-P4+P5+84 g-8j 2-Pj+ p2+P4-p5) + 2 | r J COS(P]-P2-P3+P4+83j4-S ^ 2-$ |+ $ 2+$3-$4) + 2 | r J cos(P2-P3-p4+P5+84)5-S2)3-P2+P3+$4-P5) -126 (4.19) - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2R*(2)R(2) = 4 |R 2| 2 + 2 |R 2| 2cos(pr p2-P3+P4+52)4-5 1)3 ^ 14 24 3 ^ 4 ) + 2 | r 2| cos(Pj-2P3+P3+83 3-5jj3-pj+2P3-p3) 2 R (3)R (3) = 4 | r 3| + 2 | r 2| cos(P2-P3-P4+P5+53 5-§24-^2+^3+^4-^5) (4.20) + 2 | r 3| c o s t p ^ ^ + p s + S ^ - S ^ p ^ + V f e ) <4-21) Following (4.12) and (4.13), the cosines are expanded into the first two terms of their Taylor series and the residual terms are collected. Maximization of the objective function (4.18) results in weighted least-squares solution. In this case the matrix problem A A (4.22) G NpA Np P Np ~ b NP is characterized by matrices GNP and ANP, and vectors PNp and b Np of the form lR i -2 1 0 O' 1 -2 1 0 0 1 -2 1 -1 0 -1 1 -1 -1 1 0 1 -1 -1 1 R2 -1 -1 1 0 |r 3 0 -2 0 1 1 -1 -1 1 -1 0 -1 1 lR i lR i lRi lR: G Np - D i a g R2 Am NP lR3 (4.23) |r 4 and - 127 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 P 2+ P3+52,3'5 1,2 P2-2P3+P4+53)4-82>3 p3-2p4+p5+64)5-83j4 P2 A (3 _ HNP “ % A P4 -P2-P4+P5+54,5'81,2 -P2-P3+P4+834-812 A b NP = (A n A\ P2 - P3 - P4 + P5 + 84 5 - 82)3 - P2 - P3 + P4 + 82 4 - S1>3 ^5 ~2p3 + p5 + S3j5 - S1>3 P2 "P3 ‘ P4 + P5 + ^3,5 ' &2,4 - p2 - P4 + p5 + 82>5 - S1>4 The generalization of (4.24) to an N-element array is straightforward but cumbersome. Note that the matrix ANP is rank deficient by one. For the five-element array ANP in (4.23) has rank 3 although it has 4 columns. In general, ANp will have N -l columns and a rank of N-2. Because ANP is rank deficient an infinite number of solutions to the least-squares problem exist. The particular solution obtained by maximizing the MMB objective function cannot be determined without more information. The two vectors, 6 P and 6 NP, can each be decomposed into the sum of two vectors. One vector in each decomposition will lie in the range space of the appropriate mapping matrix (Ap for ftp and ANp for 6 NP), and the other vector will lie outside of it. A A The vectors bNP and b p can be expressed as b NP “ b NP + ^ b NP (4.25) bp — b p + 8bP (4.26) where the vectors bNP and b p lie within the range space of ANP and Ap, respectively, and the vectors 8bNP and 8 b p do not lie within the range space of their respective A -1 2 8 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. matrix. If there is no system noise, both systems o f equations are consistent and the resulting vectors lie within the range space of the respective A matrix. Therefore, the vectors bNP and bp can be readily identified as -2p2+p3 p2-2p3+p4 -p2-p4+p5 P2 P3’ P2 P 4'P3 -P 2 ‘ P4+ P5 P 5'P4 ' P 3 ' P4 + P 5 P2 - P3 ' P4 + P5 P2 bp P3 — - P2 - P3 + P4 -2P3 + p5 P2 (4.27) P4'P2 P5‘ P3 P4 ‘ P 3 ' P4 + P 5 P 5'P 2 - P2 - P4 + P5 - and the vectors lying outside of the range spaces of ANp and Ap are, respectively, 52 ,3'5 1,2 8 1,2 83,4‘ 82,3 S2,3 84,5’ 83,4 5 3,4 84,5‘ 8 1,2 S4,5 5 3,4"5 1,2 5 1,3 5bp = 84,5 ' 82,3 52,4 52,4 " 5 1,3 83,5 8 3,5 ' 8 1,3 8 1,4 8 3,5 ' 82,4 82,5 (4.28) 82,5 ' 5 1,4 Likewise the vectors f ^ P and $ pcan also be decomposed as -129 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. &NP “ P + % P (4.29) ftp = P + e P (4.30) where (3 is a vector of the actual element phase errors and £NP and £p are error vectors. From equations (4.25) through (4.27) it can be seen that G NpA NpP = G Np b N p (4.31) G pA pp = G pbp (4.32) and consequently the least-squares problems of (4.179) and (4.22) become G NpANp6Np = G NpSb^ p (4.33) G p A p £ p ~ G NP8 b P (4.34) At this point the hinderance of the rank deficiency of G NPANP8NP to the analysis of the estimation errors in each sub-class becomes apparent. Because ANp is rank deficient, G NpANp is also rank deficient and, therefore, the pseudoinverse does not exist and a generalized inverse must be used to solve (4.33). However, without more information, the generalized inverse that yields the same solution as the maximization of the MMB objective function, cannot be determined. The superior performance of the parametric algorithm over the MMB can be explained from this analysis. The measurement vector, 8bNp, has twice the variance of the measurement vector Sbp. Thus the advantage of having a priori knowledge is that the phase error estimates of the parametric algorithm will generally be better than the estimates provided by the MMB algorithm. Further consider (4.10) and (4.18) with the lag index, I , running between -1 and +1. Both objective functions now incorporate only the unit-lag measurements into the -1 3 0 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. calibration procedure. In [29] the claim is incorrectly made that the MMB degenerates to the ULSCA. However the ULSCA equations are obtained from (4.17) by retaining only the unit-lag terms. Equation (4.17) derives from the parametric sub-class. Since the MMB belongs to the non-parametric sub-class its unit-lag algorithm is not the ULSCA and will yield performance inferior to that of the ULSCA. 4 .3 .2 The G SC A a n d the M LSC A The analysis o f this section reveals a connection between the algorithm characterized by the GSCA objective function and the MLSCA. The connection is revealed through the linearization of the parametric sub-class. The linearization shows that the phase error estimates provided by the parametric algorithms are the same, within a first-order approximation, as the estimates provided by the MLSCA. The importance of this is three fold. First, the performance of the MLSCA can be obtained through the parametric sub class of algorithms and because of the commonality of the maximization procedures, discussed in Section 3.4, this performance can be achieved at will by simply altering the weighting function wp(Z) in software. Secondly, the MLSCA performance can be obtained without the burden of phase-unwrapping. Thirdly, the MLSCA is a parametric algorithm and is therefore not scene independent. For best results, the measurements in the least-squares problem should be weighted inversely to their variances. Therefore, knowledge of the intensity distribution must known a priori . The first and second conclusions are the most significant. The least squares problem that results from linearizing the parametric sub-class is the same problem solved by the MLSCA. However because of the cosinusoidal nature of the parametric objective function, modulo 2k ambiguities do not affect the parametric algorithms. Since the parametric and non-parametric algorithms can be implemented with one efficient -131 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. maximization procedure, the MLSCA can also be implemented with the same maximization procedure. The conditions for which the parametric algorithms yield the MLSCA solution must be recognized. It is important that the initial guesses of the phase errors used to initialize the maximization procedure be good guesses and that the variances of the random variables, 5n n+^ ,be small. Figure 4.16 shows the average ratio of the 2-norm of the error vector to the 2-norm of the least-squares solution averaged over 50 realizations. Therefore % is P g sc a " P m lsca £ = --------- (4.35) MLSCA where PGSCA is the vector of phase error estimates determined from the GSCA algorithm and PMLSCA is the vector determined from the MLSCA. The results of Figure 4.16 were obtained from a 5 element and a 7 element array and the cosine terms in the GSCA objective function were given equal weights. Even if 5n ( has variance of ^ radians, the parametric solution will differ from the MLSCA solution by 18%. The ULSCA provides good initial estimates for the pn's and can be used to satisfy the first condition. The second condition is less controllable. Excluding lags with small theoretical amplitudes from the calibration process will help to meet the second condition. -132 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Comparison of GSCA and MLSCA Solutions 0.20 0.15- 7 Elements 0.10- 5 Elements 0.05- O 0.00 0.0 0. 2 0 .4 0. 6 0 .3 1.0 Noise Variance Figure 4.16 Relative difference between the GSCA and the MLSCA solutions. 4 .4 S u m m a ry The set of expected relative gain curves presented in this chapter and Appendix B is the first published study of element position error tolerances of spatial correlation algorithms and the performance plots for the additive noise case are the first to study the performance over various SNR's. Element position tolerances of the spatial correlation algorithms have never been studied. The relative gain curves compiled in Appendix B are the basis for the mathematical approximation of the relative gain curves presented in Section 4.1.4 and provide quantitative tolerances for system designers. When a dominant scatterer is present the relative gain curve should be approximated by the DSA relative gain curve given by (4.2). When there is no dominant scatterer present -133 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and a Ax < X meters or ufov < 0.12 radians, (4.3) approximates the relative gain curve to within 10% error. Since the 0 dependence of the relative gain is known through (4.2) and (4.3) there is no need to make measurements of the relative gain curves at multiple scan angles. For L < 10 the parametric algorithm is more desirable to implement since it exhibits twice the relative gain (in dB) of the non-parametric algorithm. As L increases both algorithms experience a large improvement in performance between L = 5 and L = 10. For L > 10 the performance difference is negligible and in this case the nonparametric algorithm should be implemented. Processing time constraints and computation complexity must also be assessed versus the required performance of the main beam gain or scan range. If processing time and computation complexity are not severe constraints then a high-order multiple-lag nonparametric algorithm should be implemented to achieve the desired performance with no a priori knowledge regarding the intensity distribution, However, if processing time and computation complexity are severe constraints then it is worth obtaining a priori knowledge regarding the intensity distribution to allow the use of a low-order multiple-lag parametric algorithm. Quantitatively, the parametric algorithm out-performs the non-parametric algorithm in the presence of element receiver noise. The performance difference is small and if this difference is tolerable the non-parametric algorithm is preferred. However, for L > 10 the performance of the parametric algorithm continues to improve with increasing lag while the performance of the non-parametric algorithm remains constant. If this performance is required, then the parametric algorithm with L > 10 must be used. -1 3 4 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 Self-Calibrating with the GSCA Using Near-Field Data The GSCA has been derived for a far-field source distribution for which the L o rr distance is nominally > - ~ L where Larr represents the length of the array. In general the A high-resolution array aperture is so large that the source distribution is in the near field instead, meaning that the wavefront from a point source appears spherical rather than planar. Since the GSCA is derived under far-field assumptions, the near-field phase curvature has not been taken into account. It is shown in the first section of this chapter that the GSCA objective function can be modified to self-calibrate the array using near-field data. It also has the ability to eliminate the quadratic terms of the polynomial expansion of the phase curvature by treating such terms as system phase errors. The second section mathematically characterizes the near-field effects on the spatial correlation algorithms and proposes a method of eliminating them by adding another -1 3 5 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. weighting function termed the "near-field weighting function". The inclusion of this weighting function into the GSCA objective function makes the GSCA useful for near-field self-calibration. Section 5.2.3 discusses the necessity of this weighting function in general and under certain specific conditions. The third section o f this chapter discusses the performance of the various spatial correlation algorithms using near-field data and the last section provides a summary of the chapter. 5.1 The Near Field of a Linear Array Beginning with Section 2.1, the received radiation field as given by (2.2) is e(x) = fe 'jkP (x )l J (0 ) { Expanding p(x) = '\J de ( 2 .2 ) j PM + x2 -2xposin(0) in a binomial expansion and preserving only the linear and quadratic terms, (2.2) can be written as -lk(ux e(x) = J(u) e *2/1_n2\ — L) 2Po du (5.1) Au where, as before, u=sin8 and constant multipliers are ignored. If the source distribution is concentrated at u0 (i.e., J(u) = 8(u-u0)), the radiation field has the form x2(l-u 2) 2(1-Uq) ) xr x 'jk ( u x ' ~ 2 ^ ~ ) a ' jk(uox ' 1 T o(u-u0) e zPo du = e zPo e(x) = (5.2) Au and is no longer the Fourier transform of the source distribution because of the quadratic term in the phase. The quadratic term is the focusing term. It is necessary to eliminate it to -1 3 6 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. focus the array. It is also necessary to eliminate it to apply the GSCA, which invokes a Fourier relationship between source distribution and the radiation field. If the quadratic phase term were be eliminated then the curvature of the phase-front is removed and the Fourier relationship between the radiation field and the source distribution is restored. Furthermore, when J(u) is a random function, then elimination of the quadratic term makes the radiation field a stationary random process and its spatial correlation function is independent of the measurement location in the array. Consequently, the Fourier relationship between the spatial correlation function and the intensity distribution is restored. If Au is small then the quadratic kernel becomes approximately independent of u and (5.1) reduces to (5.3) J Au which reduces to (5.4) for a point source. The second term is a plane wave propagating from u = uQ. The first term can be viewed as a deterministic phase distortion term A(|)(x) = k X . It can be 2p0 treated as an error that appears in the aperture. This is exactly the type of error the self calibration procedure is designed to eliminate. Consequently, the GSCA removes the phase curvature of near-field data. Therefore only standard Fourier beamforming weights are necessary to scan the beam and image the source distribution after self-calibration has taken place. The requirement of Au being small is not restrictive since most high-resolution systems operate with a small field-of-view. -137 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For example, both the synthetic-on-receive and ISAR systems used to collect the experimental data presented in this document employed a 4 ft parabolic dish for the transmitting antenna. The beamwidth was 1.75 degrees and the total variation of the (1u2) = cos20 term in (5.1) is less than 0.05% of its maximum value of unity. 5.2 Elimination of the Effects of Near-Field Phase Curvature on the GSCA 5 .2 .1 M a th em a tica l C h a ra cteriza tio n o f the N e a r-F ie ld E ffects o n the G SC A O b je c tiv e F unction The quadratic phase distortion term makes the radiation field a nonstationary random process and the corresponding autocorrelation function dependent on the measurement position in the array. Let Ku) = X aiSC u-U i) (5.5) Then x2 ( l - u ? ) -jk e(x) = £ a; e X U . -------------------- — 1 2p ro (5.6) x2 ( l - u 2 ) jk XU 0 - o Let the array be focused by multiplying (5.6) with a focusing term e Then -jk x (u .-u E[e(x)e*(x + X)} = ]T a 2e i )— - — (u 2 - u ? ) o' 2o *0 0 1 jk ( x + X ) ( u . - u ) - ( x + X ) i o' 2p o (u 2 - u 2 ) 0 1 -138 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. which reduces to jk £d(u.-u ' 1 o')- f2d2(u2-u?) v 0 1' 2P„ -jk nW2(u2-u?) v 0 1' (5.7) =X*i where the discrete nature of the array has been accounted for by substituting x = nd and X = (n^ - npd = PA with d representing the inter-element spacing, n the element index, and I the correlation lag index. The nonstationarity appears in the second exponential of (5.7). The phase of this term is a random quantity due to the presence of the random variable u;. However, some lags are approximately stationary if the expected value of the nonstationary _ U. phase term is less than J q radians [29]. If Uj is uniformly distributed between uq — u and u + —^ 0 2 then k n & i V - u 2)1 o 1 kn£d2u?fov 12pr (5.8) For a particular lag, t , the largest that (5.8) can be is kn^d2(u2 - u 2) 1 o I k ( N - l ) l d 2u2Qv _____ (5.9) max 12p„ Forcing (5.9) to be less than or equal to ^ radians establishes a quadratic inequality in the lag variable, L, given by 2 (0.6)?ipo Z2 -(N )(* )+ p° > 0 2 j2 (5.10) ufo v d where ufov = an§ular extent half of the field of view d = spacing between array elements p0 = range to which the array is focused -139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. N = total number of elements in the array X = operating wavelength of the imaging system for the largest approximately stationary lag to be used in the calibration process. The solution of the strict equality of (5.10) yields the highest-order lag that has a nonstationary term no greater than ^ radians. This result suggests that an additional weighting function should be included in (3.26). The weighting function should be zero for all lags beyond the largest allowed lag determined from the strict equality of (5.10) and may be rectangular or tapered for smaller lags. Therefore the objective function of (3.26) should be modified to L J4 = Z wnf (*)w p(/)R (Z ) i =-L (5.11) where wNF0O represents the necessary weighting of the correlation coefficients to account for the nonstationarity of the received radiation field. It is interesting to examine Tsao's ECA in the near field. The ECA was founded on Parseval's Theorem, which describes a Fourier Transform relationship, implying that the received data set has originated in the far field of the array. The objective function (3.7) can be Fourier transformed to the form of equation (3.26) in which the dependence on the spatial correlation lags is explicitly shown. Equation (3.26) requires that all the measured correlation lags be calculated and summed. To test the effects of a nonstationary correlation function on the objective function of (3.26), the simulation of Tsao and Patrick was modified to include a quadratic phase curvature in the data to simulate near-field conditions. The results showed that the image improved when the highly nonstationary lags were removed from the calculation of the objective function; see Figure 5.1. Each plot is an image of a point source located at 0 radians 8 meters away from a 1.74 meter array putting the source in the near field of the -140- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. array. This is a very severe test. The array was calibrated using two algorithms; one that made use of all of the available lags and another that used only the first three lags. After calibration with each algorithm a point source located at zero radians was imaged. The top plot is the image obtained when all the lags available in the array are used; in the bottom plot only lags 1, 2, and 3 are used. Although neither image is a high quality, the lower image is superior in its definition of the mainlobe, the level of its first sidelobe, and in the asymptotic sidelobe level. -141 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ECA Performance In New Field with 29 Correction Lags (a) S •20 *40. • 0.3 • 0.2 • 0.1 0.2 0.3 ECA Performance In Netr Field with 3 Correlation Lagt (b ) 8 • Figure 5.1 0.2 • 0.3 0.1 Illustration of the effect of sever ncar-field phase curvature, (a) calibration using all 29 lags. Mainlobe hardly evident. First sidelobe - 4 dB. (b) only the first 3 lags. Mainlobe is evident. First sidelobe i s -10 dB. - 142- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The experimental data set originates in the near field of the array. When the phase curvature of the near-field data is approximated even by a simple quadratic approximation, the spatial correlation function is no longer stationary in the array aperture. The value of a particular correlation lag now depends on the location at which it was measured within the aperture. Therefore, averaging a particular lag over locations is meaningless. Because the GSCA is a multiple lag correlation algorithm, the stationarity of the radiation field must be a concern. This concern is not encountered in the Unit-Lag Spatial Correlation Algorithm [2] because generally the unit-lag correlation exhibits a negligibly small nonstationary term even in near-field conditions. This property is discussed further in the following section. 5 . 2 . 2 R o b u stn ess o f the U nit-L ag S p a tia l C o rrela tio n A lgorith m The unit-lag algorithm emerges from the generalized objective function by maximizing (3.26) with L = l. As discussed in Chapter 3, this situation can occur in one of three different ways. First, the algorithm can be arbitrarily truncated to include only the unit-lag measurements. This is strictly done to achieve minimal computational complexity. Second, the weighting function, wp(Z), may fall-off very rapidly such that it is essentially zero for all lags of order higher than one. This situation is solely a result of the scene being imaged, the characteristics of which may dictate that such a weighting function is appropriate. The third manner in which the unit-lag algorithm manifests itself is when the imaging geometry and the parameters of the imaging system dictate that a near-field weighting function must be applied to the general objective function. If the imaging is deep enough into the near field that all correlation lags above the unit-lag must be eliminated from the objective function, then the unit-lag algorithm becomes the appropriate algorithm for calibration. Very little about the properties of the Spatial Correlation class can be learned from the first two situations just described. In the first case the lags are arbitrarily truncated -143 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without any consideration given to the GSCA theory. The manifestation of the unit-lag in the second case is due only to the characteristics of the scene being imaged. However, the third situation does reveal an important property of the unit-lag algorithm. As the source distribution comes further into the near field more of the higher-order correlation lags have to be eliminated from the objective function. Only the lower-order lags will be sufficiently "stationary" to remain in the objective function. Obviously, at a certain point only the unitlag will remain included in the objective function and beyond this none of the correlation lags are suitable to calibrate the array. The unit lag is the last correlation lag to breakdown and, consequently, the unit-lag algorithms are the most robust near-field spatial correlation algorithms within their respective sub-classes with respect to near-field calibration capability. 5 .2 .3 The N e a r-F ie ld W eigh tin g Function The near-field weighting function wNF(Z) arising in (5.11) as a result of the nonstationarity of the correlation function turns out to be unnecessary in most highresolution microwave imaging scenarios. The reason for this lies in the scaling and duality properties of the Fourier Transform and the structure of the data. Consider an arbitrary GSCA objective function of the form (5.12) When (3.4) is substituted into (5.12) only the functions p(u) and e"^kuxn^ are functions of u. Therefore the integral over u degenerates to (5.13) Recognizing that xm - xn = Zd, it is easily seen that (5.13) is a Fourier transformation of the function p(u) from the variable u to the variable Z. The scaling property of the Fourier -144 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Transform tells us that as the spatial extent of the p(u) in u-space increases, the width of the "mainlobe" of the corresponding transformed function in lag-space decreases. Since this function weights the correlation values of various lags, only the lags within the mainlobe contribute significantly to the objective function. Alternately when the extent of the source distribution is small in u-space, and therefore the integration interval is narrow, the weighting function in lag-space expands and more correlation lags can contribute significantly to the objective function. In summary, when the field-of-view is large the mainlobe of the weighting function extends for only a few lags and the objective function is minimally affected by near-field phasefront curvature. The higher-order lags, which are the most nonstationary, get multiplied by very small weights. Only the low-order lags are given significant weights. Provided that these lags have small nonstationary components no significant deterioration in the image will be noticeable. If more lags than just the unit lags have negligibly nonstationary components then the resulting final image will be superior to the image produced by the ULSCA. However, at extremely short range such that even the unit lags have significant nonstationary components, the GSCA will not produce a superior image over the ULSCA since any contribution o f the higher-order lags introduces more nonstationarity into the objective function. For example, p(u) = rect(ui,u2) yields the ECA for a rectangular unaberrated intensity distribution located between Uj and u2 in u-space. Following (5.13) the lagdomain weighting function is w (£) = sine (5.14) of which the mainlobe extends i (U2-Uj )d (5.15) -145 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. If the solution to (5.10) is greater than (5.15) there is no need for the near-field weighting function in (5.14). 5 . 2 . 4 The E ffect o f a D o m in a n t S c a tte re r on the N e a r-F ie ld W eigh tin g Function The situation changes when either a dominant scatterer is in the data set or only a narrow field of scatterers exists. An example is an aircraft target free of clutter. When a dominant scatterer dominates the averaging process, the error-free averaged image closely resembles the error-free image of the dominant scatterer rangebin. In this case the most appropriate integration interval is a very narrow interval on the order of one receive array beamwidth (discussed in Section 3.3.2). A second interesting case is a narrow field of scatterers, typified, for example, also by an airplane target free of clutter. In this case a narrow integration interval is also the most appropriate. Due to the Fourier relation of (5.12), the correlation-domain weighting function will have a broad mainlobe. This will cause many of the higher-order correlation lags to carry significant weight in the calculation of the objective function. This is a potentially dangerous situation when self-calibrating with near-field data since the higher-order lags are the most nonstationary lags. This problem is remedied when a dominant scatterer is in the field of view. When phase errors are present the correlation value between the signals at the nlh array element and the (n+Z)1*1 array element after focusing the array to a distance p0 and angle u0 is R „.„« = e‘ i<li»w " li“ > E = '? e i<T® t V i (5.16) which is (5.7) modified by the phase error term e\s\up5(-j(P^s\do4(n+jt) - p\s\do4(n))). The phase terms T§ and T ^ § represent the stationary and nonstationary terms, respectively, -1 4 6 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -k{d/.(Uj - u ) - d2( l 2 - 2/.)(u? - u q )} T s = -------------!— 5----------------------!------- 5— (5.18) 2 Po and Uj is the direction of the ith point source in the source distribution. From (5.17) we can see two alternative ways to make TNS negligible. One is to keep the product nI small, ( 2 2\ which was the objective of section 4.2.1. Another way is to make the difference vuj ' u o ' very small. This is the situation when a dominant scatterer is present. Let a dominant scatterer located at u = u; ‘d s be in the data set and let the ULSCA phases be used to initialize the GSCA. Then the array will always be approximately focused and steered toward the dominant scatterer, i.e., u„ = u; 0 . In this case the term ‘d s i = iD§ in (5.16) dominates the summation. Both T§ and Tj^§ are approximately zero 2 2 because each term has as a factor either (u; - uQ) or (u; - uQ) and the range o f u is small, by definition, for this case. Therefore, the undistorted correlation values of all lags of the dominant scatterer rangebin should be zero phase. Additionally, the undistorted rangebin averaged correlation values used in (3.27) will also be approximately zero phase since the dominant scatterer bin will dominate the averaging process. Consequently, no nonstationary effects will appear in the objective function and no near-field weighting function is needed. In this case the higher-order lags now included in the objective function through the wide mainlobe of the sine weighting function can be used to provide a better estimate of the system phase errors than would be obtained by the unit lag SCA, and a superior image will result. This predicted effect is verified by the results of the application of the GSCA, using the rectangular approximation, to the experimental data. Figures 5.2 and 5.3 show the - 147 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. error-corrected phases of most of the lags from rangebin #39 (the dominant scatterer bin of the ISAR data set). It is evident that all of the phases of the correlation lags have approximately zero slope as a function of array element number, n. The deviations from zero of the individual phases are caused by the residual phase errors after phase correction and also by the fact that uQ only approximately equals ujDS. Even the higher-order correlation lags have nearly zero phase slope. Figure 5.4 shows the phase of the lag 1 correlation value as a function of array element number averaged over 30 rangebins. Due to the influence of the dominant scatterer the average correlation value exhibits zero phase slope. No significant nonstationary effects are observed. Consequently, no near-field weighting function is required when applying the GSCA with the rectangular approximation to this data set. The image produced by incorporating all of the correlation lags into the objective function is verified in Figure 3.6(c) to be superior to the one produced by incorporating only the unit lag (Figure 3.6(a)). -148 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Phases of Lags 1 -10 Phases of Lags 11 -20 Radians M § 3d a Array Element Number Array Ebmcnt Number Phases of Lags 31 -40 Radians Phases of Lags 21 -30 § 39 Array Element Number Figure 5.2 Array Elemenl Number Error-corrected phases of the correlation lags 1 through 40. These lags were computed from die dominant scatterer rangebin of the experimental data after focusing had been applied. - 149 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Phases of Lags 51 -60 4 3 2 I } 0 0 ■2 •3 -4 0 Array Element Number 20 40 60 80 100 120 140 A m y Element Number Phases of Lags 100 -110 Phases of Lags 61 -70 (0 § 60 100 Array Elcmmt Number Amy Element Number Figure 5.3 80 Error-corrected phases of the correlation lags 41 through 70 and 100 through 110. These lags were computed from the dominant scatterer rangebin of the experimental data after focusing had been applied. - 150- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 Phase o f Lag 1 of Rangebin Averaged Correlation Coefficient 3 2 1 0 1 2 3 ■4 0 20 40 60 80 100 120 140 Array Element Number Figure 5.4 Phase of Averaged Correlation Coefficient of Lag 1. Averaged over rangebins 30 - 60. The zero slope indicates the dominant scatterer bin dominates the averaging and nonstationary effects are eliminated. The same results are observed in the simulated near-field data. Rangebin #1 contains only one point source at u = 0. Figure 5.5 shows the phases of the first ten correlation values computed from the data before the simulated phase errors were added to the data. The non-zero slopes of the curves in this figure illustrate the nonstationarity of the correlation phases. Figure 5.6 shows the same correlation phases computed from the error-corrected data when the point source in rangebin #1 is a dominant scatterer. Figure 5.7 shows the same correlation phases also computed from error-corrected data when there is no dominant scatterer present in the data set. The zero slope can be observed in the case of a dominant scatterer (Figure 5.6) but not in the case when a dominant scatterer is absent (Figure 5.7). The correlation function is clearly stationary when a dominant scatterer is present and nonstationary when a dominant scatterer is not present. -151 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Radians Unwrapped Phases of Correlation Lags 1 - 1 0 -10 Array Element Number Figure 5.5 Unwrapped undistorted phases of correlation lags 1 -10 of the simulated data without a dominant scatterer present. The non-zero slopes indicate the nonstationarity of the correlation function. Slopes increase with lag number. Unwrapped Phases of Correlation Lags 1 -1 0 3- 2 - 1 - -1 - -2 - -3 - 4 . ------- .------ .-------------- ,---------------,-------------- ,--------------0 5 10 15 20 25 30 Array Element Number Figure 5.6 Unwrapped error-corrected phases of the correlation lags 1 - 10 of the simulated data with a dominant scatterer present. The zero slopes indicate the slationarity of the correlation function. -1 5 2 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. U nw rapped P hases o f Correlation L ags 1 - 1 0 C /3 £ os - 10. Array Element Number Figure 5.7 5 .2 .5 Unwrapped error-corrected phases of lags 1 -10 of the simulated data with a dominant scatterer not present. The non-zero slopes indicate that the correlation function is still nonstationary. U se o f the N e a r-F ie ld W eighting F unction w ith P r a c tic a l H igh -R esolu tion Im agin g System s The solution of the strict equality of (5.10) yields the largest correlation lag that should be included in the GSCA objective function when a near-field data set is being calibrated. It turns out that for most practical high-resolution imaging systems the solution of (5.10) is greater than the largest lag measurable in the array aperture. Consequently there is generally no need to apply a near-field weighting function. This can be verified by considering the parabola defined by "fov d2 -153 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. = (x -C !)2 + c2 where Ci = -z , c2 = (5.19) / ctXp0 N2 \ 6 0 -=— r - -r, and a = — when the upper bound on V uf o v d ) 3 (5 .9 ) is — (C3 = 10 yields equation (5.10)). The minimum of this parabola occurs at c3 N L = c j = ^ and the minimum value is ymin = c2. If c2 > 0 then the equality o f (5.10) is satisfied for all lags and there is no need to include the near-field weighting function in the GSCA objective function. When c2 < 0 equation (5.19) will have two roots and the smallest root represents the maximum lag that should be included in the objective function. However, for most practical high-resolution imaging systems c2 is positive. Since c2 is dependent on several parameters, it can become negative in many different ways. However, these cases generally represent impractical situations. Consider the dependence of c2 on p0. For c2 to be positive requires “ 1 Po ~ 2 ur fov N2 „ ~ 7 L' T > 0 d <5 -20> and, therefore, p0 to satisfy P o >v4f^ (f)“L ^ NA aJ V4 « y where q = ^ represents the number of inter-element wavelengths. Consider a 1000-element linear array operating at 1 cm wavelength, an interelement spacing of 10 cm, and a field-of-view o f 20 mrad, as Po > (^ )(0 .2 5 )(1 0 6)(10)(4x10’4)(10-1) = ^ listed in Table 5.1. T herefore, meters = ( ^ ) c 3 » 17c3. Choosing a nominal value of 10 for c3 yields p0 > 170 meters. For targets beyond this distance no near-field weighting is required. -154 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 5.1 Order of Magnitude A jproximation of System Parameters Parameter X Order of Magnitude 3 10 elements _2 10 meters d 10'1 meters ufov 2x1O'2 Po 170 meters N These values are order-of-magnitude parameters for most VFRC experiments. Tables 4.2 and 4.3 list the parameter values and p0 for the X-band ISAR system and the 83-meter X-band phased array at VFRC. Table 5.2 System Parameters for VFRC ISAR Imaging Systei 1 Parameter | Parameter Value N 128 elements P 11.84 wavelengths d 0.37 meters ufov 0.0305 radians Po 27 meters Table 5.3 System Parameters for the VFRC _________ 83-Meter Imaging System_______ Parameter Parameter Value N 330 elements P 8.04 wavelengths d 0.2515 meters Ufov 0.0305 radians Po 85 meters - 155 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. All VFRC targets observed with these systems are at ranges greater than p 0 and consequently there is no concern for adverse effects from the near-field phase curvature. This prediction is verified by the image shown in Figure 3.6(c), which is a high quality image obtained by calibrating a near-field data set using the GSCA with no near field weighting function, and p(u) set to a very narrow rectangle function as described in Chapter 3, Section 3.3.2. This causes wp(jt) to be a sine function with a very wide main lobe as shown by the dashed line in Figure 5.8. In this case all of the measurable correlation lags are within the main lobe and all contribute significantly to the GSCA objective function. C o r r e l a t i o n F u n c tio n E s t im a te o f E x p erim en tal D a ta o Z 0.2 0 20 40 60 80 100 120 140 Lag Figure 5.8 Correlation function and weighting function taken from experimental data (Run 3) with a dominant scatterer and integration interval of 0.66 mrads. Solid line shows correlation function from experimental data set. Dashed line shows weighting function with integration interval of 0.66 mrads. Dashed-dotted line shows weighting function with integration interval of 30.5 mrads. - 156 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.3 Performance of the GSCA with Near-Field Data The problem of the nonstationarity of near-field data has been shown to disappear when the backscattered radiation arises from a compact source such as a point radiator (a dominant scatterer) or when a second weighting function is applied to the objective function as shown in (5.11). What has not been discussed is the strength of the correlation values. The radiation field of a source distribution must be highly correlated in space for the sampled correlation function to have large values at the higher order lags. Because of the inverse relationship between lag-space and u-space, a complex source distribution which radiates a highly correlated electric field must be very narrow in extent in u-space. These are the characteristics of a dominant scatterer. They are also the characteristics of a data set obtained from a system with a narrow transmit antenna pattern. However as the field-ofview increases, the extent of the source distribution in u-space increases and the extent of the spatial correlation function in lag-space decreases. This means fewer correlation lags have strong values and are reliable enough to use to estimate the system phase errors. One can easily think of the limiting case where the field-of-view becomes large enough that only the unit lag is significant. In such a case the GSCA degenerates to the ULSCA and exhibits no superiority over the ULSCA. Because of this the GSCA will yield its best performance when a dominant scatterer is present in the data set or when a narrow transmitting pattern is employed, thus allowing multiple correlation lags to be used. For this reason the GSCA proves to be a superior algorithm to the Unit-Lag SCA in cases where a dominant scatterer is present, a narrow field-of-view exists, or the system parameters are such that near-field effects are not significant. These characteristics are demonstrated in Figure 5.9 and Figures 5.10 through 5.13. Figures 5.9 and 5.10 are prime examples of the ability of the GSCA to produce images superior to the ULSCA when there are higher order correlation lags of significant -157 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. strength in the spatial correlation function. Figure 5.9 shows the error-corrected image of the first rangebin of the simulated data set, as obtained from the GSCA, with p(u) set to a rectangle function. There are no dominant scatterers present in the data set. Figure 5.10 shows the ULSCA image of the same rangebin. The ULSCA image is fair, while the GSCA image is close to a diffraction-limited image. The quantitative superiority of the GSCA image is measured by the peak and average sidelobe levels, shown in Table 5.4, which are 4 and 2 dB better respectively. Two factors contribute to the GSCA's ability to remove the artifacts of the ULSCA image. First, the number of correlation lags is large. Another important factor is that the GSCA possesses more information. This information is imbedded in the weighting function wp(jt). As shown in Table 5.4 Results of Calibration with the GSCA and Unit-Lag SCA with no Dominant Scatterer and a Data Set Originating in the Near Field Main Beam Gain Average Sidelobe Level (dB) (dB) Peak Sidelobe Level (dB) GSCA Image (Figure 5.8) -0.011 -24.5 -12.38 Unit-Lag SCA Image (Figure 5.9) -0.32 -22.5 -8.15 Section 3.3.2, to obtain a weighting function p(u) that is narrower than the field-of-view, an assumption must be made on the shape of the expected intensity distribution. Consequently, the GSCA is being implicitly told what to expect the mean intensity distribution to look like whereas the ULSCA, and also the multiple-lag MMB, are given no such information. This effect is analogous to the experience in the spectral estimation field where parametric spectral estimation techniques perform well when the parametric model and its -158 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. G S C A Im age o f R an geb in #1 -10 -15 -20 -25 -30 - 0.05 0.05 0.15 0.2 0.25 u = sin ( 0 ) Figure 5.9 GSCA Image with rectangular weighting function of extent Au = 0.167. There is no dominant scatterer present and the data set originates in the near field Unit-Lag SCA Image of Rangebin #1 -10 .8 -15 -20 -25 ■301.11 fl. I l l n II I III I I II LLI U I - 0.25 - 0.2 - 0.15 - 0.1 - I 0.05 0 0.05 0.1 0.15 0.2 0.25 u = sin ( 0 ) Figure 5.10 Unit-Lag SCA image with no dominant scatterer and a near-field data set. - 159 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. parameters are chosen correctly. These methods have been called "super-resolution" techniques based on their ability to resolve very closely spaced frequencies. However, if the model and/or parameters are not chosen correctly, the techniques fail badly. When assumptions are made on the expected intensity distribution p(u) the procedure effectively becomes a model-based self-calibration technique. When the assumptions are correct, the calibration is better than ULSCA or the MMB. However, when the assumptions are incorrect, the resulting calibration is virtually useless. Figure 5.11 shows an estimate of the correlation function obtained from the GSCA error-corrected data set of the simulation. Also shown (dashed line) is the sine weighting function used in (3.26) corresponding to a rectangular p(u). The weighting function applies large weights to the correlation function where the function is large and applies small weights to the function where the correlation is weak. It is apparent that at least two and perhaps as many as 5 or 6 lags contribute to the GSCA objective. Thus the GSCA has Amplitude of Correlation Function and Sine W eighting Function ’o0 3 1 0.6 1 0.4 0.2 \/ Lag Figure 5.11 Estimated Correlation Function and the Sine Weighting Function when no Dominant Scatterer is present. -1 6 0 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. access to better estimates of the system phase errors than the SCA, provided that strong correlation exists between array elements at higher-order lags. Figures 5.12(a) and (b) show that the same effect occurs when the point source in rangebin #1 is just barely dominant (the amplitude has been increased to 1.9). The GSCA image has lower sidelobes than the ULSCA in the neighborhood of the main lobe. This is demonstrated by the measurements shown in Table 5.5 that show the disparity between the main beam gains and average sidelobe levels have largely vanished. However, the peak sidelobe levels are still significantly different and because of this the GSCA image is still the preferred image. Table 5.5 Results of Calibration with the GSCA and Unit-Lag SCA with a Small Dominant Scatterer and a Data Set Originating in the Near Field Main Beam Gain Average Sidelobe Level (dB) (dB) Peak Sidelobe Level (dB) GSCA Image (Figure 5.10(a)) -0.016 -24.5 - 12.88 Unit-Lag SCA Image (Figure 5.10(b)) -0.13 -23.6 -9.73 As the strength of the point source grows, the superior performance of the GSCA over the Unit-Lag SCA becomes less noticeable. Figure 5.13 shows that the weighting function Wp(Z) (dashed line), "over-weights" the correlation lags (solid line) since the higher-order lags have not substantially increased in strength. However, as the point source increases in strength, the normalized correlation function flattens and it levels off at unity as the point source amplitude grows without bound. As the strength of the source grows, the weighting function better matches the correlation function. Similar results are obtained with the experimental data. The solid line of Figure 5.8 illustrates the estimated correlation function obtained from the phase corrected data after application of the GSCA. The dashed line represents the sine weighting function - 161 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. G S C A Im a g e ( E C A ) ■10 (a) -20 '8Q -O.OS 0 0 05 0.1S 0.2 0.25 0.2 025 u = sin (0 ) T he U n k -L ag SCA Im age •10 •15 (b) -20 •30 -35 •0.05 0 0.05 0.15 u = sin (8 ) Figure 5.12 (a) GSCA image with ECA approximation. Dominant Scatterer amp(l)=1.9 integration interval 17.24 mrads (b) Unit-Lag SCA Image with Dominant Scatterer amp(l)=1.9 C o r r e la ti o n E s t im a te S in e W e i g h t in g F u n c tio n 0.8 ■auo f 0.6 ■§ Z 0.2 0 5 10 20 15 25 30 Lag Figure 5.13 Estimated Correlation Function and the Sine Weighting Function when a Dominant Scatterer is present. - 162 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. obtained when an integration interval the width of one array beamwidth is applied. The dashed-dotted line represents the sine weights obtained from an integration interval with a width of the entire field-of-view. Knowing that this data set contains a dominant scatterer, we expect the sine weights corresponding to an interval the width of one array beamwidth to be a more appropriate weighting function than the weighting function corresponding to the entire field-of-view. This hypothesis is clearly verified by Figure 5.13. Figures 3.6(a) and 3.6(c) illustrate the improvement in image quality (even when the ECA approximation is used) of the GSCA image over the Unit-lag SCA image. 5.4 Summary This chapter extends the GSCA to the near field. It is shown that the algorithm can successfully self-calibrate an array imaging system using data acquired in the near field of the array provided that the nonstationarity of the received radiation field is compensated by a near-field weighting function in the GSCA objective function. This weighting function either removes or tapers higher-order correlation lags, thereby removing those lags that exhibit nonstationarity. The Unit-Lag Spatial Correlation Algorithm emerges from the GSCA class when the near-field weighting function has such a form that only the unit-lag weight is non-zero, at which point the stationarity constraint of Section 4.2 is enforced only on the unit-lag measurements. Since the non-zero extent of the near-field weighting function will diminish as the source distribution is moved further into the near field of the array, the unit-lag weight is the last weight to become zero. Consequently, the unit-lag algorithms are the most robust to near-field effects in the Spatial Correlation class since it is the last lag to suffer the effects of the nonstationarity of the received radiation field. -163 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The effects of the nonstationarity of the received radiation field are not seen in most practical applications. Generally, most high-resolution imaging systems illuminate small enough sectors at large enough ranges that all of the measurable correlation lags in the aperture are not functions of measurement position in the array aperture. Additionally, a dominant scatterer in the field of view has been shown to eliminate the near-field effects on the algorithms. Inclusion of the higher-order correlation lags is desirable provided that the redundancy they offer smoothes the phase-error estimations more than the noise they introduce disrupts the estimation process. The GSCA with an appropriate weighting function produces superior images than the ULSCA even when the calibration is done using near-field data. -164 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 Conclusions The failure of the ECA to successfully self-calibrate an array imaging system using experimental data provides the motivation for this work. The search for a modification of the ECA results in the development of the GSCA. The GSCA explains why the ECA failed to calibrate the imaging system with the experimental data and provides a solution to achieve successful calibration with that data. Performance characteristics of the two spatial correlation sub-classes revealed by the GSCA are studied in the presence of element position errors and element receiver noise through simulations and first-order analysis of the GSCA objective function. Additionally, the GSCA, which is developed under far-field assumptions, is extended to calibrate in near-field conditions. The following is a list of the conclusions of this research. C onclusions (From Chapter 3) The ECA has been shown to be inadequate to successfully self-calibrate the experimental data obtained at VFRC. The search for a modification has led to the -165 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. development of the GSCA. The GSCA has been used to explain the unsuccessful results of the ECA with the VFRC experimental data and has also been used to determine the appropriate change in the ECA to obtain successful calibration. In general, the weighting function should match the theoretical spatial correlation function. This requires a priori knowledge regarding the intensity distribution and spatial correlation algorithms requiring such knowledge have been shown to define a sub-class of the Spatial Correlation class termed the parametric sub-class. It has been shown that the ECA is a spatial correlation algorithm and, in particular, belongs to the parametric sub class. The presence of a dominant scatterer has adverse effects on the ECA. The assumptions implicit in the ECA theory are not appropriate when a dominant scatterer is present. In such a case the weighting function determined from the ECA does not match the theoretical correlation function and calibration is unsuccessful. It has been shown from GSCA theory that when such a scatterer is present the mainlobe of the weighting function must be narrowed to the size of one receive-array beamwidth. If a priori knowledge regarding the intensity distribution is not available then algorithms from the non-parametric sub-class must be used to self-calibrate. These algorithms estimate the theoretical correlation function from the data and use these values for the weighting function; consequently, they are scene independent. The MMB algorithm has been shown to belong to this sub-class. Algorithms belonging to both the parametric and non-parametric sub-classes are all characterized by the GSCA objective function. The manner in which the weighting function is determined separates the two sub-classes and the particular weighting functions separate the individual algorithms. Because the GSCA objective function is the general form of all of the algorithms, only a single, efficient maximization procedure needs to be implemented to maximize the objective function. This means that an imaging system can have one efficient maximization procedure hardwired and still have access to all of the - 166 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. spatial correlation algorithms, both parametric and non-parametric, by simply changing the weighting function in software. (From Chapter 4) The discovery of two sub-classes prompted a performance comparison between algorithm s from the sub-classes. Simulations were established to compare the performances of a parametric algorithm and a non-parametric algorithm with a particular source distribution in the presence of element position errors and element receiver noise. In both cases the parametric algorithms performed better than the non-parametric algorithms. However the performance difference is generally minimal (1 to 2 dB) and in many cases the need for a priori knowledge of the intensity distribution outweighs the superior performance of the parametric algorithm. In these cases the non-parametric MMB algorithm is the preferred algorithm. Element position error performance curves measured as a function of scan angle for the ECA, the MMB, and the ULSCA exhibit the same shape as the performance curves measured and derived by Steinberg [45] for the DSA. The curves for the spatial correlation algorithms are shifted vertically by a constant (over scan angle) relative mainbeam loss term. From this a quick and easy approximation to the performance curves of the spatial correlation algorithms has been presented. This approximation requires the measurement of the relative mainbeam gain after calibration at only a single scan angle. For simplicity, this can be taken on boresight. This measurement is used to vertically shift the DSA performance curve, yielding an approximation to the performance curve of the particular spatial correlation algorithms used for the calibration. It has been shown that this approximation is accurate to within a tenth of a decibel provided o Ax < X wavelengths or ufov < 120 mrads. - 167- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. When a dominant scatterer is present the performance curves of the parametric and non-parametric multiple-lag algorithms are essentially identical to the DSA performance curve. Consequently, there is no loss from DSA performance by implementing a multiplelag spatial correlation algorithm instead of the DSA when a dominant scatterer is present. It has also been shown that DSA performance is attained by the ULSCA when a dominant scatterer is present. In this case the ULSCA is the preferred algorithm since it is computationally simple and does not depend on the shape of the intensity distribution. First-order analysis o f the GSCA objective function for both sub-classes has provided evidence that the ULSCA belongs to the parametric sub-class and is not equivalent to the MMB algorithm with L = 1. This explains the performance difference between the ULSCA and the MMB with L = 1 and the superior performance of the ULSCA over the unit-lag MMB illustrated by the performance curves for these algorithms. The first-order analysis has shown that phase error estimates from non-parametric algorithms have more noise than estimates from parametric algorithms. The poor performance of the unit-lag MMB is explained by this analysis since the unit-lag MMB belongs to the non-parametric sub-class. Lastly, the first-order analysis has shown that algorithms from the parametric sub class solve nearly the identical least-squares problem solved by the MLSCA when the phase noise variance is small. Therefore, essentially the same solution obtained by the MLSCA can be obtained from a parametric algorithm. The significance is that because of the cosinusoidal nature of the parametric algorithm objective functions modulo 2k ambiguities in phase measurements are not important. However, the same phase measurements must be unwrapped before the MLSCA can be used. - 168 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (From Chapter 5) When the source distribution lies in the near field of the array imaging system, the received radiation field is a nonstationary random process. Consequently, the theoretical spatial correlation function will depend on the position in the array at which it is measured. This is a problem for the spatial correlation algorithms since they all assume any nonstationarity is due solely to the presence of phase errors. It has been shown that the GSCA objective function must be modified to account for the nonstationarity that occurs with near-field source distributions. The solution is the inclusion of a near-field weighting into the GSCA objective function that truncates, or re weights, the correlation lags used for calibration. Analysis o f the origin of the nonstationarity has shown that the higher-order lags are the most severely nonstationary while the lower-order lags can generally still be used for calibration. Consequently, the unit-lag algorithms are the most robust to near-field effects of all of the spatial correlation algorithms, since the unit-lag is the last lag to become nonstationary. The analysis also indicates that the nonstationarity problem is not a serious as it was previously thought to be. The reason for this is two-fold. First, the analysis has shown that the presence of a dominant scatterer eliminates the nonstationarity and the need for the near-field weighting function. Second, because of the parameters of most practical highresolution microwave imaging systems the nonstationarity o f the radiation field is not perceivable and the near-field weighting function is, again, not needed. However, because the GSCA is inherently a multiple-lag algorithm, the effects of near-field source distributions must be considered and the need for the near-field weighting function must be checked. -1 6 9 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.2 Suggestions for Future Work The intention of this research was to "close the book" on the Spatial Correlation class of algorithms. As mentioned in Chapter 1, to a large extent, this goal has been accomplished. However, some loose ends still exist and these are cited in this section as suggestions for future work. There are four such topics and these are discussed briefly below. 1) Completion o f the Linearization o f the GSCA Objective Function Chapter 5 shows that a linearization of the GSCA objective function leads to a weighted least-squares problem. However, the rank deficiency of the non-parametric mapping currently prevents the derivation of any mathematical analysis useful in comparing the performances of the two sub-classes. The least-squares theory is exhaustive, as is the theory of generalized inverses for matrices. At this time, it appears as though one piece of information is missing from the least-squares problem of the non-parametric sub-class. The crucial piece of information may be found by re-visiting the non-parametric objective function and extracting some information describing the manner in which the phase estimates are obtained through maximization of the function. This information needs to be incorporated into the non-parametric system of equations to boost the mapping matrix to full-rank status. At that point the comprehensive theory of least-squares could be used to compare the variances of the phase estimates from the two sub-classes. The theory of generalized inverses of matrices may provide the solution to this problem. Another route to a mathematical comparison of the performance of the two sub classes may be the development of upper and lower bounds on the performance measures. Many upper and lower bounds exist for various matrix norms (for example, see [50]). Additionally, the mapping matrices of both sub-classes possess an immense amount of -170- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. structure. Possibly, this structure, combined with the appropriate bounds on an appropriate matrix norm, could yield a useful mathematical performance measure. 2) The Performance Limits o f the Two Spatial Correlation Sub-Classes It is very possible that this topic goes hand-in-hand with the previous topic. In Chapter 6 the observation was made that the performance of the non-parametric sub-class appeared to level-off as the number of incorporated into the calibration process was increased. At the same time the parametric sub-class appeared to continue to approach the diffraction limit. It would certainly by useful to know what are the performance limits of the two sub-classes and if the performance limit of the non-parametric sub-class is below that of the parametric sub-class. It seems very possible that if the linearization problem discussed in 1) is solved, this question may also be simultaneously answered. Consequently, the best attack on this problem may be to attack the problem discussed in 1). 3) Experimental Verification o f the Tolerance Curves As mentioned in Chapter 6, until this research no extensive tolerance study had been published and no research at all had been published regarding the tolerances of the element position errors. The tolerance curves presented in Chapter 6 and Appendix B should be good approximations to the actual curves. However, experimental verification would certainly be gratifying. Such verification would allow the system designer to unhesitatingly employ the approximations (7.1) and (7.2) to the relative gain curves as a function of scan angle, q, suggested from the simulation results of Chapter 6 (and Appendix B). 4) Expanding the Spatial Correlation Class to Include the DSA This last suggestion is probably the most challenging, and exciting, topics of the four listed in this section. Now that the Spatial Correlation Class has been expanded and -171 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. its scope appears to be established, the question of further expansion arises: namely, "Can the Dominant Scatterer Algorithm be incorporated into this class"? There appears to be several things in favor of this pursuit. First, it can be argued that since the DSA is also a phase-correcting algorithm it might also be subsumed by another expansion of the Spatial Correlation Class. Second, the tolerance curves presented in Chapter 6 and Appendix B indicate that the algorithms of the parametric sub-class and the multiple-lag algorithms of the non-parametric sub-class yield relative gain curves nearly identical to the DSA element position error tolerance curve. A breakthrough in this area would certainly be exciting and probably carry with it many ramifications. -1 7 2 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix A Derivation of the Variance of the Correlation Phase Noise 2 The quantity Cj can be evaluated by considering a signal-plus-noise model for the received complex field at each array element. T ^n,n+jt = T S ^et,ne^^n + Tlt)n)(et,n+£e^ n+* + Tlt,n + ^ t=l 3 = ^n,n+jt + S ^Pn n+jt^ Pn,n+jt (A .2 ) ^.n+ Z The sum of the terms containing noise in (A.l) is represented in (A.2) as the sum of three complex numbers with each complex number defined as T •^1 ^ r i n,n+£ ~ T ^ n,n+j£ rlt,ne t,n+jde ( A . 3) T %2 .e^r2n.n+^ = T ^ ^ t ,n + jd et,ne^ n n,n+i ’ t=l (A -4) T •^ 3 n,n+£ ^ 3n,n+Z = j ^ ^t.n^n+Z (A-5) t=l » and Rn n+jt the noiseless estimate of the phase distorted true correlation value =k£ et,nHn+1£<=K|in ' Iin^ ) (A.6 ) t=l In this model complex Gaussian noise is assumed to be present at each array element. The noise observations are assumed to be independent both spatially and temporally. -173 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Additionally, the I and Q components of the noise are also assumed to be independent. Each component (I and Q) is zero mean and has a variance of o^. The phase of n+ji can be found from (A.2) as + Pn - P n + * ] + ^ p V . = T an' 1 j?Pn,n+ZS in ( r Pn,n+jt) = 1 n.n+t n ,n + £ | r J c 0s [ v < + f n - | W ] + £ (A.7) J’P n ^ i C0S(rf » , +i ) Pn,n+jt1 p Tan' 1 = v * + Pn - Pn+Jt + = 1 n.n+fc R- t i + (A.8) 5 > p „ , t i cos(!w Pn fl+t where £p = r p - + Pn - Pn+jtl- T°r elements possessing a strong correlation with a moderate signal-to-noise ratio the inverse tangent function in (A.8) is approximately linear with its argument. Therefore Pn - P n + i + i | (A.9) £ ^ P = p n,n+jt = / 1 and the error, 5n n+^, due to noise is approximately 3 X 5n’n+* ~ | R I 1 p ^Pn,n+Jt Sln(^Pn,n+i) = (A. 10) 1 n,n+jt The quantity ^ n,n+jt sin(4, a, n,n+£ n,n+i A ) can be considered as the imaginary part of the quantity ,»♦/ = t=l -174 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. t=l Likewise, the quantities % , sin (t2 n ,n + t ,) and A-i n,n+j i . sin(E2 n,n+X. ,) represent the n ,n+ Z imaginary parts of * A , . - n,n+i t=i T t=l and * 3 respectively. n,n+i e j 6 3 „ ,„ + i = i X — ’ 1 . , n 1l . t n t f ' i [ V ' + |J " ‘ |5 n ^ ’ ’ 1 <A ' 1 3 > Express the noise in terms of inphase and quadrature components as Tl»IjlIn = nhIjlln + j n<fil^lln where two components are independent of each other and also between rangebins and elements. Similarly express the data in terms of inphase and quadrature components as e( n = eit n + je q t n . Equation (A.l 1) can now be expressed as T = T - S ^nit,n + j ntH ,n^eH,n+j6 + j e<^t,n+Z^ [cos(x*//.+ Pn) ' j sin('*fJt +Pn)] (A. 14) 1=1 = T X [Ci+jC^]leosCVjt+Pn) - jsin(Vz+pn)] (A.15) t=l T = T X C 1cos(Vjt+pn)+C2sin(Vjt+pn) + j(C 2cos(Vjt+pn)-C 1sin(Vjt+pn)) (A.16) t=l - 175 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Therefore the imaginary part is T Im a g U j e ^ 1n,n+Z} = ( C2cos(Vji+pn)-C 1sin(Vjt+pn) ) n,n+jt (A.17) j where C 1 = n*t,neit,n+/ - nV .,eV + £ (A. 18) C 2 = " V 'Y n + t + ni.,ne V t Z (A. 19) By defining C, = nit n+jte it „ - n q ^ e q , „ and C2 = nql n+jteil n + n y n+l e\ n the imaginary part of (A. 12) evaluates to T I m a g l^ n,n+jt eJ^2n,n+jt} = ^ T , ( C2Cos(Vjt-pn+jt)-C’1sin(Vjt-pn+^)) t=l (A.20) The imaginary part of (A. 13) is more easily determined to be T Im a g U 3 e \ n+t | 4 l ( n,n+jt t~l ncU,nnW + nit,nnCW ^ (A.21) Following equation (A. 10) 8 n ,n +1 “ Ima§ t e^2n,n+1 } + eJ^n,n+Z) + Imag {^ n,n+jl n,n+jt Im ag{^3 eJ^3n,n+jt} (A.22) n,n+jt The variance of 8nin+J£ can now be computed as E[8„Vi] E[lmag2{^, L e^Y.n+z} + e [~Imag2] ^ n,n+jt L [ n,n+jt 2 Imag [M-, jEo e Vn+jt} (A.23) n,n+Z -176- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Since the noise components are independent from element to element, all of the cross-terms are zero under the expectation operator. Since each term in (A.22) represents a sample mean averaged over T rangebins, each term in (A.23) represents the variance of the correspondng estimator in (A.22). Consider the variance of the first term E [im ag2 {.3, L e^n.n+jl}! n,n+jt T = ^2 X J T I K ti=l t2=l cos(Vjt+pn)-C1 sin(Vjt+pn) ) ( c 2t c o sO ^ + P n )-^ s i n ^ + p j ) 1 1 2 2 (A.24) T = ^2 S E [ C 2cos2(Vjt+pn)]+ E [C isin 2(Vj6+pn) ] - E [ C 1C2cos(Vjt+pn)sin(Vjt+pn) ] t=l (A.25) It is tedious, but fairly simple to verify that E [ c 2cos2(Vj£+pn) ] = eit,n+/ E [ nClu n ]COs2^ +Pn) + eClt,n2^ E [ nit!n]COs2^ +Pn) + 2eit,n+Ze(lt,„+/ E[ nit,n]E[nclt,n]cos2^ +Pn) (A-26) 2 If the I and Q components of the noise are zero mean and have equal variances, o^, then (A.26) becomes E [ c 2cos2(Vjt+pn) ] = 0 2 Icl n+jL l2cos2(Vjt+pn) (A.27) Following the same approach it can be shown that E [ c 2sin2('l/2+Pn)] = letn+jt P s i n ^ + p ,,) (A.28) -177 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and e (A.29) [ C 1C2cos(Vjt+pn)sin(Vjt+pn) ] = 0 Therefore (A.25) evaluates to Imag2{ ^ y \ , n +*}] let,n+/ n,n+Z t=l = T 1( t Z l e t,n+J 2 ) t=l (A.30) 4 < k J 2> where < • > denotes the spatial (rangebin) average. The same approach can be followed to evaluate E Im a g 2 } ^ e J^2n , n + / ) l . n,n+jt J All that is required is to interchange the positions of the n and n+ i subscripts and the manipulations remain the same. This yields e|" Imag2 {^ L e ^ 2n,n+jt} n,n+jt - ^ < 1 l2> (A.31) The last term in (A.23) evaluates to r 2 , , Imag [A, e Jn,n+z} n.n+jt _ - AT (A.32) Substituting (A.30), (A.31), and (A.32) into (A.23) yields n+jt|2> + <1 et,n 2 2 an > + “t (A.33) - 178 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ^ [ < K ,J 2>+<I e,,nP>+2o^] (A.34) The terms | e t n+l | 2 and | e t n |2 are estimates of the zero-lag correlation value, R(0), and should nearly equal provided a sufficiently large number of rangebins are averaged. Thereofore (A.34) can be approximated by :[ 5n ,n .i] ” °T| ( R(0) T |R /P V [1+ 0n] (A.35) clearly showing that E ^ 8 2)n+Jj J is inversely proportional to | R ^ | - 179- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix B Main Beam Relative Gain Curves For Array Element Position Errors The plots presented in this appendix represent approximations to the expected main beam relative gain for all combinations of the variable values listed in Table 4.1. Figures 4.5 and 4.6 have been taken from this set as representative plots to illustrate the general characteristics exhibited by all of the 40 plots generated. In addition to displaying the general characteristics discussed in Chapter 4, this set can, and should, be used quantitatively as very good approximations to the expected main beam relative gain curves as a function of scan angle, 0, for various conbinations of system specifications. The plots are presented here mostly for this purpose in addition to supporting the claims made in Chapter 4 ffom based on the plots of Figures 4.5 and 4.6. The appendix is organized in the following manner. It is assumed that the design engineer will either be given an existing radar sytem possessing a specific transmitting antenna or the size o f the field-of-view will be set by some higher-priority system specification. In either case the size of the field-of-view is assumed to be pre-determined and canno be altered. The remaining free variables then are L, the number of correlation lags incorporated into the calibration process and o ^ , the rms element position error in the x-direction. Therefore, the plots are organized so that plots deriving from the same size field-of-view are clustered. Each page contains two plots corresponding to the same fieldof-view and rms element position error, but different number of correlation lags. The succession of plots begins with the smallest field-of-view simulated, U fo v = 0.017 radians, and ends with the largest, Ufov = 0.167 radians. In all of the following plots, the -180- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. field-of-view values represent the full field-of-view. In some plots the dashed line representing the DSA curve is not visible. This is because it is overlaid with a solid curve from one of the sub-classes o f the Spatial Correlation Class. The DSA curve should always be one of the upper curves in the plots since it always passes through the point (AG=0.0,0=0.0). -181- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves m ■a P Sub-Class NP Sub-Class DS Class o > ra o cc •*— - 0.2 - -0.3 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Scan Angle (radians) Main Beam Gain Curves 0.00 - 0.01 - - 0.02 - a oc P Sub-Class NP Sub-Class DS Class -0.03 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Scan Angle (radians) Figure B.l Main beam gain curves for ufov = 0.017 and a Ax = g (a) L = 1 (b) L = 29 - 182 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves 0 . 0 -O ' m •o c cs (5 - 0.2 P Sub-Class NP Sub-Class DS Class - 0) > ■*-> o v -0.3 CC -0.4 0.00 0.02 0.04 0.06 Scan 0.08 0.10 0.12 Angle (radians) Main Beam Gain Curves 0.00 0.02 - ffi 2. -0.04 C 0 0) > «= -0.06 -0.08 fl> P Sub-Class NP Sub-Class DS Class CC -0.10 - 0.12 0.00 Figure B.2 0.02 0.04 0.06 Scan Angle 0.08 0.10 0.12 (radians) Main beam gain curves for ufov = 0.017 and ctAx = (a) L = 1 (b) L = 29 - 183- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves OQ 2. -0.5 c ‘5 (5 P Sub-Class NP Sub-Class DS Class 0 .0 0 0 .0 2 0.04 0.0 6 0.08 0 .1 0 0.12 Scan Angle (radians) Main Beam Gain Curves o.o «> -0.3 J5 0) P Sub-Class NP Sub-Class DS Class -0 .4 - -0.5 0.00 0 .0 2 0.04 0.0 6 Scan Angle Figure B.3 0.08 0 .1 0 0.12 (radians) Main beam gain curves for ufov = 0.017 and a Ax = ^ (a) L = 1 (b) L = 29 - 184- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves CD -0.5 - ■o c 15 0 ID > P Sub-Class NP Sub-Class DS Class - 2.0 0.0 0 0 .0 2 0.04 0 .0 6 0.0 8 0.10 0.12 Scan Angle (radians) Main Beam Gain Curves o.o CD -0.5 - TJ c BJ 0 0) > (0 a> oc P Sub-Class NP Sub-Class DS Class - 2.0 0.0 0 0 .0 2 0.04 Scan Figure B.4 0 .0 6 0.0 8 0.10 0.12 Angle (radians) Main beam gain curves for ufov = 0.017 and a Ax = X (a) L = 1 (b) L = 29 -185- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves -0.05 ■ CD ~o c P Sub-Class NP Sub-Class DS Class -0 .1 0 - B O g -0.15- _ra o> DC - 0.20 - -0.25 0 .0 0 0.02 0.04 0.0 6 Scan Angle 0.08 0.10 0.12 (radians) Main Beam Gain Curves 0.00 O •o - 0.01 c o -0.02 - 4) > B | -0.03 - -0.04 0.00 P Sub-Class NP Sub-Class DS Class 0.02 0.04 0.0 6 0 .0 8 0.10 0.12 Scan Angle (radians) Figure B.5 Main beam gain curves for ufov = 0.04 and c Ax = (a) L = 1 (b) L = 29 - 186 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves o.o ■& m ■o - 0.2 - P Sub-Class NP Sub-Class DS Class cs 0 v > -0.8 0 .0 0 0 .0 2 0.04 0 .0 6 Scan Angle 0 .0 8 0.1 0 0.1 2 (radians) Main Beam Gain Curves 0.00 - 0.02 - -0.04 -0.06 -0.08 - 0£ - 0.10 P Sub-Class NP Sub-Class DS Class - -0.12 0.0 0 0 .0 2 0.04 Scan Figure B .6 0.0 6 Angle 0 .0 8 0.10 0.12 (radians) A. Main beam gain curves for ufov = 0.04 and o Ax = — (a) L = 1 (b) L = 29 -187- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves -0.5 P Sub-Class NP Sub-Class DS Class O 0) > 0.00 0 .0 2 0.04 0.0 6 Scan Angle 0 .0 8 0 .1 0 0.12 (radians) Main Beam Gain Curves o.o m TO c -0.2‘«o (3 ® -0.3 - © P Sub-Class NP Sub-Class DS Class -0.4- -0.5 0.00 0 .0 2 0.04 0.06 Scan Angle Figure B.7 0 .08 0 .1 0 0.12 (radians) Main beam gain curves for ufov = 0.04 and a Ax = ^ (a) L = 1 (b) L = 29 -188- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves o.o-t? GQ -0.5 -o c O <D > « 8) cc P Sub-Class NP Sub-Class DS Class -2.0 0 .0 0 0 .0 2 0.04 0 .0 6 Scan Angle 0.08 0 .10 0 .12 (radians) Main Beam Gain Curves o.o GQ -0.5 T3 (3 O v > jO o cc P Sub-Class NP Sub-Class DS Class -2.0 0 .0 0 0 .0 2 0.04 0 .0 6 0.08 0 .10 0.12 Scan Angle (radians) Figure B .8 Main beam gain curves for ufov = 0.04 and o Ax = k (a) L = 1 (b) L = 29 -189- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves o.o m T3 £ a (3 o > o -0.2© P Sub-Class NP Sub-Class DS Class cc -0.3 0 .0 0 0.02 0.04 0.0 6 0.08 0.10 0.12 Scan Angle (radians) Main Beam Gain Curves 0.00 - 0.01 - to T3 c -0.02 1 (0 (5 § -0.03 - P Sub-Class NP Sub-Class DS Class -0.04 - -0.05 0.00 0.02 0.04 0.0 6 Scan Angle Figure B.9 0.08 0.10 0.12 (radians) Main beam gain curves for ufov = 0.08 and <rAx = g (a) L = 1 (b) L = 29 -190- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves o.o -0.1 ST £ ‘°-2 co O P Sub-Class NP Sub-Class DS Class «> -0.3 JS2 o tt -0.4 -0.5 0 .0 0 0.0 2 0.04 0.06 Scan Angle 0.08 0 .10 0 .1 2 (radians) Main Beam Gain Curves 0.00 0.02 - ffi 2- ra (3 -0.04 -0.06 > | -0.08 v CC - - P Sub-Class NP Sub-Class DS Class 0.10 0.12 0 .0 0 0 .0 2 0.0 4 Scan Figure B.10 0 .0 6 Angle 0.08 0 .1 0 0 .1 2 (radians) Jl Main beam gain curves for ufov = 0.08 and c Ax = j (a) L = 1 (b) L = 29 -191- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves 0 .0 1» CQ 2. -0.5 - C P Sub-Class NP Sub-Class DS Class *3 C o > n v cc 0 .0 0 0 .0 2 0.04 Scan 0.06 0.08 0 .1 0 0 .1 2 Angle (radians) Main Beam Gain Curves o.o co ■o .5 'to (5 -°'2 ' P Sub-Class NP Sub-Class DS Class -0.5 0 .0 0 0 .0 2 0.04 Scan Figure B.l 1 0.06 0.08 0.10 0 .12 Angle (radians) Main beam gain curves for ufov = 0.08 and o Ax = X ^ (a) L = 1 (b) L = 29 -192- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves o.o ffi -0.5 *o c C O O 0) > M 4) DC P Sub-Class NP Sub-Class DS Class -2.0 0.00 0.0 2 0.04 Scan 0 .0 6 0.0 8 0.10 0.12 Angle (radians) Main Beam Gain Curves o.o ffi -0.5 •o c C O O 4> > « 0> DC P Sub-Class NP Sub-Class DS Class -2.0 0.00 0.0 2 0.04 Scan Figure B.12 0 .0 6 0 .0 8 0.10 0.12 Angle (radians) Main beam gain curves for ufov = 0.08 and a Ax = X (a) L = 1 (b) L = 29 -193- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves o.o CD T3 C *(0 O - 0.2 - P Sub-Class NP Sub-Class DS Class o > -r* £ -0 .3- -0.4 0 .0 0 0.0 2 0.04 0 .0 6 Scan Angle 0.08 0.10 0 .1 2 (radians) Main Beam Gain Curves 0.00 0.02 - - m •o C -0.04 - (0 O o> > -0.06 - P Sub-Class NP Sub-Class DS Class -0.08 - - 0.10 0.00 0 .02 0.04 0 .0 6 Scan Angle Figure B.13 0.08 0.10 0 .1 2 (radians) Main beam gain curves for ufov = 0.12 and c Ax = g (a) L = 1 (b) L = 29 -194- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves o.o m T3 £ -0.2ID o o > P Sub-Class NP Sub-Class DS Class «) a -0 .4 - -0.5 0 .0 0 0 .0 2 0.04 0 .0 6 Scan Angle 0 .0 8 0 .1 0 0 .1 2 (radians) Main Beam Gain Curves 0.00 -0.05 c 'n O -0.10- ffi > _n ffi CC -0.15 - - 0.20 0 .0 0 P Sub-Class NP Sub-Class DS Class 0 .0 2 0.04 Scan Figure B.14 0.06 0.0 8 0 .1 0 0 .12 Angle (radians) Main beam gain curves for ufov = 0.12 and a Ax = (a) L = 1 STILL NEED (b) L = 29 -195- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves 0 .2 P Sub-Class NP Sub-Class DS Class 0.0 m O = -0-2 ' cs <5 " -0.4 jO a> a - - 0.6 • 0.8 0.00 0.02 0.0 4 Scan 0 .0 6 Angle 0.08 0.10 0.12 (radians) Main Beam Gain Curves o.o m 2. - 0.2 - C 'co C3 o> > n> -0.4 - "55 P Sub-Class NP Sub-Class DS Class CC -0.5 - - 0.6 0.00 Figure B.15 0.02 0.0 4 0 .0 6 Scan Angle 0.08 0.10 0.12 (radians) X Main beam gain curves for ufov = 0.12 and a Ax = ^ (a) L = 1 (b) L = 29 -196- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves o.o -0.5 - m u c (0 O §> -1 .5- ro o * -2.0-2.5 0 .0 0 P Sub-Class NP Sub-Class DS Class 0 .0 2 0.04 0.06 Scan Angle 0 .0 8 0.10 0.12 (radians) Main Beam Gain Curves o.o m -0.5 -o 0) > to 0) DC P Sub-Class NP Sub-Class DS Class - 2.0 0 .0 0 0 .0 2 0.04 0.06 Scan Angle Figure B.16 0 .0 8 0 .10 0.12 (radians) Main beam gain curves for ufov = 0.12 and o Ax = X (a) L = 1 (b) L = 29 -197- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves o.o P Sub-Class NP Sub-Class DS Class m ■o 0) - 0.6 ■ £ *■» n u CC - 0.8 - 0 .0 0 0.02 0.04 0.06 Scan Angle 0.0 8 0.10 0.12 (radians) Main Beam Gain Curves 0.00 0.02 - ffi 2- P Sub-Class NP Sub-Class DS Class -0.04 c (3 -0.06 - 0) B -0.08 0) *-• CC - 0.10 - 0.12 00 Figure B.17 0.02 0.04 0 .0 6 Scan Angle 0.08 (radians) Main beam gain curves for ufov = 0.167 and o Ax = g (a) L = 1 (b) L = 29 -198- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves o.o 0.2 - c '5 - -0.4 ■* (3 ® - 0.6 - P Sub-Class NP Sub-Class DS Class CO O - 0. 8 - a 0.0 0 0 .0 2 0.04 0 .0 6 Scan Angle 0 .0 8 0 .1 0 0 .1 2 (radians) Main Beam Gain Curves 0.00 .N •0.05 - [ c *3 C -0 .1 0 - <u > V cc -0.15 - - P Sub-Class NP Sub-Class DS Class 0.20 0.0 0 0.0 2 0.04 Scan Figure B.18 0 .0 6 Angle 0 .0 8 0 .1 0 0 .1 2 (radians) Main beam gain curves for ufov = 0.167 and a Ax = j (a) L = 1 (b) L = 29 -199- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves 0.0 CD ■o -0.5 - c *5 O o > Q *3 P Sub-Class NP Sub-Class DS Class 0 .0 0 0 .0 2 0.04 Scan 0 .06 Angle 0 .0 8 0 .1 0 0.12 (radians) Main Beam Gain Curves o.o m 2. C (0 O - 0.1 - 0. 2 - -0.3 - o > B - 0 .4 V P Sub-Class NP Sub-Class DS Class DC -0.5 - - 0.6 0.00 0.0 2 0.04 0.0 6 Scan Angle Figure B.19 0.0 8 0 .1 0 0.12 (radians) Main beam gain curves for ufov = 0.167 and a Ax = ^ (a) L = 1 (b) L = 29 - 200 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main Beam Gain Curves o.o -0.5 CD *o c <0 CD > ra - 2.0 - ® CC P Sub-Class NP Sub-Class DS Class -2.5 -3.0 0.00 0.02 0.04 0.0 6 0 .0 8 0 .10 0.12 Scan Angle (radians) Main Beam Gain Curves o.o -0.5 CD ~o c C O CD <D > 2 o E P Sub-Class NP Sub-Class DS Class -2.0- -2.5 0.00 0 .0 2 0.04 0.06 0.08 0 .1 0 0.12 Scan Angle (radians) Figure B.20 Main beam gain curves for ufov = 0.167 and a Ax = X (a) L = 1 (b) L = 29 - 201 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. REFERENCES [1] Attia, E. H., "Phase Synchronizing Large Antenna Arrays Using the Spatial C orrelation Properties of Radar Clutter", Ph.D. D issertation, Univ. of Pennsylvania, 1984. [2] Attia, E. H., and K. Abend, "Algorithms for Array Self-Phasing/Self-Cohering", Lecture Notes, Naval Weapons Center, China Lake, CA, March 1991. [3] Attia, E. H., and B.D. Steinberg, "Self-Cohering Large Antenna Arrays Using the Spatial Correlation Properties of Radar Clutter", IEEE Trans, on Antennas and Propagation, Vol. 37, N o.l, January 1989. [4] Babcock, H.W., "The Possibility of Compensating Astronomical Seeing", Publ. Astron. Soc. Pacific, Vol. 65, 1953. [5] Bates, Kenneth, "Tolerance Analysis for Phased Arrays", Acoustical Imaging. Vol. 9, Keith Y. Wang, ed., Plenum Press, 1980, pp.239 - 262. [6] Born, Max, and Wolf, Emil, Principles of Optics. Pergamon Press, New York, 1980. [7] Borsari, G. K.and B. Steinberg, "The Source Statistics Algorithm", Valley Forge Research Center Report # 16-91,1991. [8] Borsari, G. K., "Comparison of the Performance of the Energy Conservation Algorithm Using Simulated vs. Experimental Microwave Data", Valley Forge Research Center Progress Report #60, 1990, pp. 11 - 22. [9] Borsari, G. K., "The Source Statistics Algorithm", Vailey Forge Research Center Progress Report #61, 1991. [10] Bracewell, R.N., and Werneck, S.J., "Image Reconstruction over a Finite Field of View", Journal of the Optical Society of America, Vol. 65, No. 11, pp. 13421346, November 1975. [11] Bruck, Y.M., and Sodin, L.G., "On the Ambiguity of the Image Reconstruction Problem", Optics Communications, Vol. 30, No. 3, pp. 304-308, September 1979. [12] Buffington, A., F. Crawford, R. Muller, and C. Orth, "First Observatory Results with an Image-Sharpening Telescope", Journ. Opt. Soc. Am., March 1977, pp. 304 - 305. [13] Carlson, D., et.al., "ISAR Program for Aircraft Imaging", Valley Forge Research Center Quarterly Progress Report, No. 49, Dept, of Electrical Engineering, University of Pennsylvania, pp. 81-91, October-December, 1985. - 202 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [14] Carlson, D., et.al., "VFRC Array Systems", Valley Forge Research Center Quarterly Progress Report, No. 48, Dept. Electrical Engineering, University of Pennsylvania, pp.1-15, July-Sept, 1985. [15] Carlson, D., "Research on Inverse Synthetic Aperture Radar with Adaptive Beamforming", Valley Forge Research Center Final Report on Contract No. PO#478903-71 for Unisys Corp., Great Neck, New York, Dept, of Electrical Engineering, University of Pennsylvania, July 1987. [16] Chen, C., and Andrews, H., "Target-motion-induced-radar imaging", IEEE Trans, on Aerospace and Electronic Systems, Vol. AES-16, pp.2-14, January 1980. [17] Cornwell, T.J., "The Application of Closure Phase to Astronomical Imaging", Science, Vol. 245, pp.263-269, July 1989. [18] Deutsch, R., Estimation Theory. Prentice-Hall, Englewood Cliffs, NJ 1965. [19] Eichel, P. H., D. C. Ghiglia, and C. V. Jakowatz, Jr., "Speckle Processing Method for Synthetic-Aperture-Radar Phase Correction", Journal of the Optical Society of America, Vol. 14, January 1989, pp. 1-3. [20] Fienup, J., "Phase Error Correction by Shear Averaging", Technical Digest Series, Volume 15 - Signal Recovery and Synthesis II, pp. 134-137, June 1989. [21] Flax, S.W., and M. O'Donnell, "Phase-Aberration Correction Using Signals From Point Reflectors and Diffuse Scatterers: Basic Principles", IEEE Trans, on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 35, November 1988. [22] Goodman, J.W., Introduction to Fourier Optics. McGraw-Hill, New York, 1968. [23] Green, D. W., V. M. Ingram, and M. F. Perutz, Proc.Roy. Soc. Vol. A225, 287, 1954. [24] Halat, J., "A Survey of Image-Embedded Passive Beamforming Sources for a Radio Camera", Masters Thesis, Dept, of Elec. Eng., University o f Pennsylvania, December 1985 [25] Hamaker, J., J. O'Sullivan, and J. Noordam, "Image Sharpness, Fourier Optics, and Redundant-Spacing Interferometry", Journ. Opt. Soc. Am., August 1977, pp. 1122 - 1123. [26] Jennison, R. C., Ph. D. Dissertation, Victoria University of Manchester, 1953. [27] Jennison, R. C., Monthly Notices of the Royal Astronomical Society, Vol. 118, pp. 276 - ?? [28] Kang, B., H. Subbaram, and B. Steinberg, "Improved Adaptive-Beamforming Target for Self-Calibrating a Distorted Phased Array", IEEE Trans, on Antennas and Propagation. Feb. 1990, pp. 186 - 194. [29] Kang, Bongsoon, "Research on Self-Calibrating Algorithms for High-Resolution Microwave Imaging Systems", Ph.D. Dissertation, Drexel University, 1989. -2 0 3 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [30] M osteller, F., and John W. Tukey, D ata A nalysis and R e g r e s s io n . Addison-Wesley, Reading, MA, 1977. [31] M uller, R., and A. Buffington, "Real-Time Correction of Atmospherically Degraded Telescope Images Through Image Sharpening", Journal of the Optical Society of America, Sept. 1974 pp. 1200 - 1210. [32] Nering, E. D., Linear Algebra and Matrix Theory, John Wiley & Sons, New York, 1970. [33] Patrick, Sherry, "Validation of Tsao's Energy Conservation Algorithm for SelfCalibration of Distorted Phased Arrays", Master's Thesis, Univ. of Pennsylvania, 1990. [34] Plackett, R. L., Principles of Regression Analysis. Clarendon Press, Oxford, 1960. [35] Pollaine, S., A. Buffington, and F. Crawford, "Measurement of the Size of the Isoplanatic Patch Using a Phase-Correcting Telescope", Journ. Opt. Soc. Am., January 1979, pp.84 - 89. [36] Powers, E.N., "Adaptive Arrays for Microwave Imaging", Ph.D. Dissertation, Dept, of Electrical Engineering, University of Pennsylvania, 1974. [37] Readhead, A.C.S., "Radio Astronomy by Very Long Baseline Interferometry", Scientific America, pp. 53-61, June 1982. [38] Rogstad, D.H., "A Technique for Measuring Visibility Phase with an Optical Interferometer in the Presence of Atmospheric Seeing", Applied Optics, Vol. 7, No. 4, April 1968, pp. 585-588. [39] Schwab, F. R., "Adaptive Calibration of Radio Interferometer Data", Proceedings of the 1980 International Optical Computing Conference, Washington D. C., 1980. [40] Steinberg, B., and B. Kang, "1-D High Resolution Airplane Imagery", Valley Forge Resarch Center Progress Report #56,1988, pp. 23-32. [41] Steinberg, B.D., "Design Approach for a High-Resolution Microwave Imaging Radio Camera", Journal of the Franklin Institute, pp.415-432, December 1973. [42] Steinberg, B.D., "Radar Imaging from a Distorted Array: the Radio Camera Algorithm and Experiment", IEEE Trans, on Antennas and Propagation, Vol. AP-29, No.5, pp.740-748, September 1981. [43] Steinberg, B. D., "A Modification to the Energy Conservation Algorithm", Valley Forge Research Center Progress Report #60, 1990, pp. 23 - 29. [44] Steinberg, B. D., "Microwave Imaging of Aircraft", Proc. of IEEE. Dec. 1988 pp. 1578 - 1592. [45] Steinberg, B. D., Microwave Imaging with Large Antenna Arrays. John Wiley & Sons, New York, 1983. -2 0 4 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [46] Steinberg, Bernard D., Principles of Aperture and Array System Design. John Wiley & Sons, New York, 1976. [47] Steinberg, B.D., "A Theory of the Effect of Hard Limiting and Other Distortions Upon the Quality of Microwave Images", IEEE Transactions on Acoutics, Speech, and Signal Processing, Vol. 35, October 1987. [48] Steinberg, B.D., and Subbaram, H.M., Microwave Imaging Techniques. John Wiley & Sons, New York, 1991. [49] Subbaram, H., and B. Steinberg, "Scene Independent Self-Calibration o f Phased Array Antennas", Accepted for publication by IEEE Trans, on Ant. and Prop., 1988. [50] Subbaram, H., et. al., "Integrated Design of Smart Phased Array Systems", Interspec Phase 1 SBIR Final Report, March 1990. [51] Subbaram, H., "Artifact Suppression and Superresolution Techniques for Coherent Microwave Imaging", Ph.D. Dissertation, Univ. of Pennsylvania, 1986. [52] Taheri, S., and B. Steinberg, "Tolerances in Self-Cohering Antenna Arrays of Arbitrary Geometry", IEEE Trans, on Ant. and Prop., Sept. 1975, pp. 733 - 739. [53] Taheri, S.H., and B. D. Steinberg, "ISAR Program for Aircraft Imaging", Valley Forge Research Center Quarterly Progress Report, No. 48, Dept, of Electrical Engineering, University of Pennsylvania, pp. 65-69, July-September, 1985. [54] Tsao, J., "A Position Error Correction Algorithm and a Revised Clean Technique for Random Thinned Array Imaging Systems", Ph.D. Dissertation, Moore School of Electrical Engineering, University of Pennsylvania, Philadelphia, PA, 1983. [55] Tsao, J., "An Ad Hoc Approach to the Solution of the Distorted Random Array Problem", Valley Forge Research Center Seminar Series, Philadelphia, Feb. 11, 1983. [56] Tsao, J., "Phase Array Beamforming by the Parseval's Theorem", IEEE AP-S International Symposium, Philadelphia,PA, June 1986. [57] Van Loan, Charles F. and Gene H. Golub, Matrix Computations. Johns Hopkins University Press, Baltimore, MD, 1985. -2 0 5 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. IN D E X angle of arrival 89 necessity of 143 use w/ practical imaging systems 152-156 relationship to existing algorithms 67, et seq. weighting function of 64-65 average image intensity 56, 87 beam width, see resolution bistatic radar 9 calibration 6 contributions 2-5 image sharpness 20 incoherent source distribution 19,23 Dominant Scatterer Algorithm 1,7,92, ISAR 12 100, 102 main beam relative gain curve 94 least squares see GSCA , linearization of objective function ECA87 linearization of objective function of non-parametric sub-class 125127 of parametric sub-class 121-125 element position error analysis 9 0 ,, et seq. phase errors due to 91 simulation of 92-100 Energy Conservation Algorithm 2, 7,13, MMB algorithm 87 139 Modified Muller-Buffington 86 experimental data, description of 12 imaging system, description of 12 Modified Muller-Buffington Algorithm 7, 13, 84-85, 121, 129, 157, 159 multiple-lag 29 far-field 18 of linear array 17 multiple-lag algorithm full 36 iterative 33 least-squares 29-33 partial 36 field-of-view 9,10, 14 Fourier relationship of radiation field and intensity distribution 18 near field of linear array 135-136 future work 167-170 Generalized Spatial Correlation Algorithm noise analysis 109,, et seq. 2, 42, 57, et seq. development of 60,64 linearization of objective function 121,, et seq. of non-parametric sub-class 125127 of parametric sub-class 121-125 near-field weighting function 135, 142, 162 effect of dominant scatterer on 145-152 non-parametric sub-class see Spatial Correlation Class of algorithms,sub classes of parametric sub-class see Spatial Correlation Class of algorithms, sub-classes of Parseval's Theorem 43 radar cross section 1 -2 0 6 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. radio camera 7, 8 relative gain curves of main beam non-parametric sub-class 9 7 ,, et seq. parametric sub-class 9 7 ,, et seq. resolution 8 scan angle 92, 94, 100, 102 self-calibration algorithms 1 history of 6 spatial correlation algorithms, class of 7 function 7, 59, 65, 84, 142 derivation of 17-19 Spatial Correlation Class of algorithms sub-classes of non-parametric sub-class 65 parametric sub-class 65 transmitter location diversity 8 Van-Cittert Zemike Theorem 19 -207- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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