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Secondary range compression for improved range/Doppler processing of SAR data with high squint Schmidt, Alfred Rudolf 1986

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FOR SECONDARY RANGE COMPRESSION IMPROVED RANGE/DOPPLER PROCESSING OF SAR DATA WITH HIGH SQUINT by ALFRED RUDOLF SCHMIDT B.Sc. Engineering Physics, Queen's University, 1981 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of E l e c t r i c a l Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1986 © Al f r e d Rudolf Schmidt, 1986 In presenting t h i s thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of E l e c t r i c a l Engineering The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: September 1986 Abstract This thesis examines a new algorithm, to be c a l l e d secondary range compression (SRC), for s i g n i f i c a n t l y improving the range resolution of the range/Doppler synthetic aperture radar (SAR) processing algorithm when the radar antenna i s s i g n i f i c a n t l y squinted away from the zero Doppler d i r e c t i o n . The algorithm was recently introduced by J i n and Wu [13] for application to the SEASAT SAR sensor. S i g n i f i c a n t extensions of the i r algorithm and models are presented. F i r s t the model of range broadening in the basic range/Doppler algorithm i s extended by using a more general form for the range compressed p r o f i l e . A mathematical theory i s developed to examine more c l o s e l y the approximations involved in both basic range/Doppler processing and SRC and to explore alternate SRC implementations. The theory i s used to derive the SRC algorithm as a matched f i l t e r d i r e c t l y from the point target response model rather from the si m p l i f i e d range compressed response used by J i n and Wu. Two new discrete implementations (azimuth SRC and range SRC) are developed for both single-look and multilook processing. In addition two new alternate methods of multilook SRC are presented : fixed SRC and look-dependent SRC. The s e n s i t i v i t y of the SRC algorithms to parameter errors i s investigated. Extensive simulations are developed to quantify the image q u a l i t y produced by each algorithm for a variety of i i processing parameters. The simulation results with nominal RADARSAT parameters show that the SRC algorithms can s i g n i f i c a n t l y extend the range of squint angles which can be processed with the range/Doppler type of algorithm. Table of Contents 1. Introduction 1 1 .1 Background 1 1 . 2 Objectives 3 1.3 Structure of the Thesis 6 2. The Synthetic Aperture Radar (SAR) Concept 9 2.1 Model of SAR Geometry 14 2.2 Image Quality Measurements 22 3. Basic Range/Doppler Compression 25 3.1 Point Target Response Model 26 3.2 Range Compression 28 3.3 Simulation of the Range Compressed P r o f i l e 31 3.4 Single-Look Azimuth Compression 34 3.5 Simulation of Single-Look Azimuth Compression ...42 3.6 Simulation Results of Basic Range/Doppler Compression 52 4. Analysis of Broadening in Range/Doppler Compression .75 4.1 Broadening Model for Range/Doppler Compression without SRC 75 4.2 Broadening Simulations and Measurements 83 5. Secondary Range Compression (SRC) 102 5.1 Theory of Azimuth Matched F i l t e r i n g and SRC ....102 5.2 Azimuth SRC 106 5.3 Simulations of Azimuth SRC 112 5.4 Range SRC 129 5.5 Simulations of Range SRC 132 5.6 Summary of Single-Look SRC 142 6. Multilook Range/Doppler Processing with SRC 144 6.1 Multilook Processing with SRC 145 iv 6.2 Simulations of 4-Look Processing without SRC ...154 6.3 Simulations of 4-Look, Fixed and Look-Dependent, Azimuth SRC Processing 169 6.4 Simulations of 4-Look, Fixed, Range SRC Processing 183 6.5 Summary of Multilook SRC 188 7. Eff e c t s of SRC FM Rate Errors 189 7.1 S e n s i t i v i t y Analysis of the SRC FM Rate 189 7.2 Simulations of SRC FM Rate Error Broadening ....198 8. Summary and Conclusions 206 8.1 Recommendations for Further Research 209 Bibliography 211 v L i s t of Symbols and Abbreviations FM frequency modulation FFT fast Fourier transform IRW impulse response width ISLR integrated sidelobe r a t i o PRF pulse re p e t i t i o n frequency RCM range c e l l migration RCMC range c e l l migration correction SAR synthetic aperture radar SRC secondary range compression TBP time-bandwidth product A ^ f ) azimuth spectrum broadening function A 2 ( f ) azimuth reference function broadening function 0 A azimuth Kaiser-Bessel parameter 0j SRC/RCMC Kaiser-Bessel parameter p\ incidence angle /3R range Kaiser-Bessel parameter B r range -3dB chirp bandwidth c propagation v e l o c i t y of radar pulse c, slope of RCM curve in azimuth time domain c 2 slope of RCM curve in azimuth frequency domain D azimuth antenna length f azimuth frequency f^. azimuth beam center (Doppler centroid) frequency f azimuth look center frequency vi f p g W azimuth processed bandwidth f range frequency F s r complex range sampling rate g(t) SRC range f i l t e r g c ( t , f ) combined SRC/RCMC range f i l t e r h g s a t e l l i t e a l t i t u d e h ( t , 7 j ) point target response before processing h A ( t , 7 j ) azimuth point target response hp(t,T?) azimuth reference phase function h„T.T)(tlr)) range bandlimited h_(t rrj) h R ( t , 7 ? ) range point target response h R C ( t , 7 ? ) range compressed point target response h R C p ( t ) 1-D range compressed p r o f i l e I # of SRC/RCMC f i l t e r versions I0{ } zeroth order modified Bessel function K, azimuth FM rate K R range FM rate KRM modified range FM rate for range SRC K r SRC/RCMC FFT array length K S R C SRC FM rate L length of SRC/RCMC f i l t e r versions X wavelength v i i M l e n g t h o f r a n g e F F T N l e n g t h o f a z i m u t h F F T 77 a z i m u t h t i m e 7 7 p B W a z i m u t h p r o c e s s i n g i n t e r v a l A q ^ g -3dB o n e - w a y a z i m u t h a n t e n n a t i m e w i d t h p s l a n t r a n g e c e l l s i z e ( = c / [ 2 F ] ) r^. b e a m c e n t e r r a n g e r g e a r t h r a d i u s a t e q u a t o r r 0 s l a n t r a n g e o f c l o s e s t a p p r o a c h r ( r j ) R C M c u r v e i n a z i m u t h t i m e d o m a i n r ^ ( f ) R C M c u r v e i n a z i m u t h f r e q u e n c y d o m a i n s T ( t ) b a s e b a n d r a d a r p u l s e t r a n g e t i m e T r a n g e s a m p l i n g i n t e r v a l T A a z i m u t h s a m p l i n g i n t e r v a l T r a n g e p u l s e w i d t h 6^ o r b i t i n c l i n a t i o n a n g l e v ^ b e a m v e l o c i t y v e a r t h r o t a t i o n a l v e l o c i t y e v e q u i v a l e n t s a t e l l i t e v e l o c i t y V g g r o u n d v e l o c i t y v s a t e l l i t e o r b i t a l v e l o c i t y s v s u b - s a t e l l i t e o r b i t a l v e l o c i t y v i i i w a (7 j ) azimuth antenna weighting function W. (f) azimuth window function A W R(f r) range window function * 2-D convolution * azimuth convolution * range convolution ix L i s t of Tables Table 1. Nominal RADARSAT parameters x L i s t of Figures Figure 2.1. Spherical earth model for v e l o c i t y c a l c u l a t i o n . 16 Figure 2.2. Flat earth model for l o c a l geometry 18 Figure 2.3. Slant range plane 19 Figure 3.1. Simulated range compressed p r o f i l e 35 Figure 3.2. Range c e l l migration curve in azimuth time domain 37 Figure 3.3. Range c e l l migration curve in azimuth frequency domain 40 Figure 3.4. Range c e l l migration curve in discrete azimuth frequency domain 50 Figure 3.5. Simulated azimuth antenna weighting, and antenna plus RCM weightings in range c e l l nearest to beam center range for 5° squint 56 Figure 3.6. Azimuth spectrum before RCMC in range c e l l nearest to beam center range for 5° squint. . 57 Figure 3.7. Windowed range interpolator including 16 f r a c t i o n a l l y s h i f t e d versions 58 Figure 3.8. Azimuth spectrum aft e r RCMC for 5° squint. . 59 Figure 3.9. Azimuth reference phase f i l t e r spectrum for 5° squint 60 Figure 3.10. Azimuth Kaiser-Bessel window for 5° squint. 61 Figure 3.11. Azimuth spectrum a f t e r RCMC, matched f i l t e r , and windowing 62 Figure 3.12. F u l l y compressed range p r o f i l e for 0° and 5° squint 63 Figure 3.13. F u l l y compressed azimuth p r o f i l e for 0° and 5° squint 64 Figure 3.14. Range broadening of point target response without SRC 65 Figure 3.15. Range broadening of point target response (expanded) without SRC 66 Figure 3.16. Azimuth broadening of point target response without SRC 67 xi Figure 3.17. 1-D range integrated sidelobe r a t i o s without SRC 68 Figure 3.18. 1-D azimuth integrated sidelobe ra t i o s without SRC 69 Figure 3.19. 2-D integrated sidelobe r a t i o s without SRC. 70 Figure 3.20. 1-D range peak sidelobe r a t i o s without SRC. 71 Figure 3.21. 1-D azimuth peak sidelobe r a t i o s without SRC. 72 Figure 3.22. 2-D peak sidelobe ratios without SRC. . . . 73 Figure 3.23. Degradation of peak magnitude without SRC. . 74 Figure 4.1. Range broadening of the simulated, t h e o r e t i c a l , range broadening function 85 Figure 4.2. Predicted and actual azimuth spectra for 0° squint 86 Figure 4.3. Predicted and actual azimuth spectra for 1° squint 87 Figure 4.4. Predicted and actual azimuth spectra for 5° squint 88 Figure 4.5. Predicted and actual azimuth spectra for 10° squint 89 Figure 4.6. Predicted and actual azimuth spectra for 15° squint 90 Figure 4.7. Measured azimuth spectrum broadening in the azimuth frequency d i r e c t i o n in the near, beam center (mid), and far range c e l l s 92 Figure 4.8. Measured azimuth spectrum broadening in the azimuth frequency d i r e c t i o n in the near, beam center (mid), and far range c e l l s (expanded). 93 Figure 4.9. Measured azimuth spectrum broadening in the range time d i r e c t i o n at the lower, the beam center, and the upper processed bandwidth frequencies 94 Figure 4.10. Measured azimuth spectrum broadening in the range time d i r e c t i o n at the lower, the beam center, and the upper processed bandwidth frequencies (expanded) 95 x i i Figure 4 . 1 1 . Points on the azimuth frequency domain RCM curve used for spectrum broadening measurements. 96 Figure 4.12. Measured azimuth time-bandwidth product (TBP) versus squint angle 98 Figure 4.13. Azimuth broadening predicted by decrease in azimuth processed bandwidth 100 Figure 4.14. Range c e l l migration over processed aperture. 101 Figure 5 . 1 . Magnitudes of the SRC/RCMC f i l t e r s of length 16 for squint angles of 0°, 5°, 10°, 15°, and 20°. 111 Figure 5.2. 1-D range p r o f i l e s after SRC compression for 5° squint and various f i l t e r lengths 115 Figure 5.3. 1-D range p r o f i l e s after SRC compression for 10° squint and various f i l t e r lengths 116 Figure 5.4. 1-D azimuth p r o f i l e s a f t e r SRC compression for 5° squint and various f i l t e r lengths 117 Figure 5.5. 1-D azimuth p r o f i l e s a f t e r SRC compression for 10° squint and various f i l t e r lengths. . . .118 Figure 5.6. Percentage range broadening with SRC as a function of squint angle 119 Figure 5.7. Percentage range broadening with SRC as a function of squint angle (expanded scal e ) . .120 Figure 5.8. Percentage azimuth broadening with SRC as a function of squint angle 121 Figure 5.9. 1-D range ISLR as a function of squint angle for various f i l t e r lengths 122 Figure 5.10. 1-D azimuth ISLR as a function of squint angle for various f i l t e r lengths 123 Figure 5.11. 2-D ISLR as a function of squint angle for various f i l t e r lengths 124 Figure 5.12. 1-D range PSLR as a function of squint angle for various f i l t e r lengths 125 Figure 5.13. 1-D azimuth PSLR as a function of squint angle for various f i l t e r lengths 126 Figure 5.14. 2-D PSLR as a function of squint angle for xi i i various f i l t e r lengths 127 Figure 5.15. Peak compressed magnitude with SRC as a function of squint angle for various f i l t e r lengths 1 28 Figure 5.16. 1-D range compressed p r o f i l e s a f t e r range compression with range SRC for 5° and 10° squint 134 Figure 5.17. 1-D range p r o f i l e s a f t e r azimuth compression with range SRC for 0°, 5°, and 10° squint and length 16 RCMC f i l t e r 135 Figure 5.18. 1-D azimuth p r o f i l e s after azimuth compression with range SRC for 0°, 5°, and 10° squint and length 16 RCMC f i l t e r 136 Figure 5.19. Percentage range broadening with range SRC and a length 16 RCMC interpolator as a function of squint angle 137 Figure 5.20. Percentage azimuth broadening with range SRC and a length 16 RCMC interpolator as a function of squint angle 138 Figure 5.21. Range, azimuth, and 2-D integrated sidelobe ra t i o s with range SRC and a length 16 RCMC interpolator as a function of squint angle. .139 Figure 5.22. Range, azimuth, and 2-D peak sidelobe ratios with range SRC and a length 16 RCMC interpolator as a function of squint angle 140 Figure 5.23. Peak magnitude degradation with range SRC and a length 16 RCMC interpolator as a function of squint angle 141 Figure 6.1. Division of the azimuth frequency-domain aperture into 4 looks 146 Figure 6.2. Corresponding time-domain looks 147 Figure 6.3. Interpolated 1-D range p r o f i l e s a f t e r 4-look compression without SRC for squint angles of 0' 5°, and 10° 157 Figure 6.4. Interpolated 1-D azimuth p r o f i l e s a f t e r 4-look compression without SRC for squint angles of 0°, 5°, and 10° 158 Figure 6.5. Range broadening for single-look and 4-look compression without SRC 159 xiv Figure 6.6. Range broadening for single-look and 4-look compression without SRC (expanded sca l e ) . . .160 Figure 6.7. Azimuth broadening for single-look and 4-look compression without SRC 161 Figure 6.8. 1-D range integrated sidelobe r a t i o s for single-look and 4-look compression without SRC. 162 Figure 6.9. 1-D azimuth integrated sidelobe r a t i o s for single-look and 4-look compression without SRC. 163 Figure 6.10. 2-D integrated sidelobe r a t i o s for single-look and 4-look compression without SRC 164 Figure 6.11. 1-D range peak sidelobe r a t i o s for single-look and 4-look compression without SRC 165 Figure 6.12. 1-D azimuth peak sidelobe r a t i o s for single-look and 4-look compression without SRC. 166 Figure 6.13. 2-D peak sidelobe r a t i o s for single-look and 4-look compression without SRC 167 Figure 6.14. Peak magnitude rat i o s for single-look and 4-look compression without SRC 168 Figure 6.15. Interpolated 1-D range p r o f i l e s after 4-look compression with both fixed and look-dependent SRC for squint angles of 0°, 5°, and 10°. . .171 Figure 6.16. Interpolated 1-D azimuth p r o f i l e s after 4-look compression with both fixed and look-dependent SRC for squint angles of 0°, 5°, and 10°. . .172 Figure 6.17. Range broadening for 4-look compression with both fixed and look-dependent SRC and various SRC/RCMC f i l t e r lengths 173 Figure 6.18. Azimuth broadening for 4-look compression with both fixed and look-dependent SRC and various SRC/RCMC f i l t e r lengths 174 Figure 6.19. Comparison of range -3dB widths at 0° squint for 4-look compression with both fixed and look-dependent SRC and various SRC/RCMC f i l t e r lengths 175 Figure 6.20. 1-D range integrated sidelobe r a t i o s for 4-look compression with both fixed and look-dependent SRC and various SRC/RCMC f i l t e r lengths. . .176 xv Figure 6.21. 1-D azimuth integrated sidelobe r a t i o s for 4-look compression with both fixed and look-dependent SRC and various SRC/RCMC f i l t e r lengths 177 Figure 6.22. 2-D integrated sidelobe r a t i o s for 4-look compression with both fixed and look-dependent SRC and various SRC/RCMC f i l t e r lengths. . .178 Figure 6.23. 1-D range peak sidelobe ra t i o s for 4-look compression with both fixed and look-dependent SRC and various SRC/RCMC f i l t e r lengths. . .179 Figure 6.24. 1-D azimuth peak sidelobe ratios for 4-look compression with both fixed and look-dependent SRC and various SRC/RCMC f i l t e r lengths. . .180 Figure 6.25. 2-D peak sidelobe r a t i o s for 4-look compression with both fixed and look-dependent SRC and various SRC/RCMC f i l t e r lengths 181 Figure 6.26. Peak magnitude rat i o s for 4-look compression with both fixed and look-dependent SRC and various SRC/RCMC f i l t e r lengths 182 Figure 6.27. Range and azimuth broadening for 4-look compression with range SRC and a length 16 RCMC interpolator .184 Figure 6.28. Range, azimuth, and 2-D integrated sidelobe r a t i o s for 4-look compression with range SRC and a length 16 RCMC interpolator 185 Figure 6.29. Range, azimuth, and 2-D peak sidelobe ra t i o s for 4-look compression with range SRC and a length 16 RCMC interpolator 186 Figure 6.30. Peak magnitude degradation for 4-look compression with range SRC and a length 16 RCMC interpolator 187 Figure 7.1. SRC band-edge phase error in the range frequency domain for squint angles of 1°, 5°, 10°, 15°, and 20° as a function of beam center frequency error 1 92 Figure 7.2. SRC band-edge phase error in the range frequency domain for squint angles of 1°, 5°, 10°, 15°, and 20° as a function of r 0 error 193 Figure 7.3. Equivalent SRC band-edge phase error in the range frequency domain as a function of squint angle 196 •xvi Figure 7.4. Actual and predicted range broadening without SRC as a function of equivalent range band-edge phase error. . . 197 Figure 7.5. Range broadening with single-look azimuth SRC at 5° of squint with various SRC/RCMC f i l t e r lengths as a function of range band-edge phase error 201 Figure 7.6. Range broadening with single-look azimuth SRC at 10° of squint with various SRC/RCMC f i l t e r lengths as a function of range band-edge phase error 202 Figure 7.7. Range broadening with multilook azimuth SRC at 10° of squint with various SRC/RCMC f i l t e r lengths as a function of range band-edge phase error 203 Figure 7.8. Range broadening with multilook azimuth SRC at 10° of squint with various SRC/RCMC f i l t e r lengths as a function of range band-edge phase error 204 Figure 7.9. Range broadening with single-look range SRC at 5°, 10°, 15°, and 20° of squint with a length 16 RCMC interpolator as a function of range band-edge phase error 205 xvi i Acknowledgements I (fish to thank my supervisor, Dr. M.R. Ito, for his help and encouragement throughout the course of thi s research. I also wish to thank Dr. I.G. Cumming of MacDonald, Dettwiler and Associates (MDA) for suggesting the thesis topic and for providing numerous i n s i g h t f u l suggestions. Dr. M. Vant of Communications Research Centre (CRC) also provided helpful discussions. I am grateful for the f i n a n c i a l assistance provided by MDA, the University of B r i t i s h Columbia, CRC, and Computing Devices Company (ComDev) of Ottawa. xvi i i 1. INTRODUCTION This thesis discusses the application of a new secondary range compression (SRC) algorithm to the range/Doppler synthetic aperture radar (SAR) processing algorithm. Part i c u l a r consideration i s given to the application of SRC to Canada's proposed RADARSAT SAR sensor. 1.1 BACKGROUND The range/Doppler algorithm [6,18,19,24,26] i s a well known method for the e f f i c i e n t compression of SAR data. The compressed point target response of the algorithm becomes severely broadened in the range d i r e c t i o n when the -3dB azimuth time-bandwidth product (TBP) becomes small (less than u n i t y ) . SRC i s a new e f f i c i e n t algorithm which s i g n i f i c a n t l y reduces the range broadening and associated image qu a l i t y degradations. The TBP has a multitude of d e f i n i t i o n s depending on the desired use. For t h i s thesis the TBP i s defined as the product of the actual -3dB widths in the time and frequency domains. The TBP of any signal has a fixed lower bound which is somewhat less than 0.5 using the current d e f i n i t i o n . A small azimuth TBP may occur in SAR when there i s large range c e l l migration (RCM). RCM i s the migration of a point target's energy through more than one range resolution c e l l over the antenna illumination period. Large RCM may occur in spaceborne SARs, such as Canada's proposed RADARSAT SAR [22] or NASA's SEASAT SAR [11], when the equivalent squint angle 1 2 of the antenna r e l a t i v e to the zero Doppler d i r e c t i o n i s large. Large RCM causes the azimuth time exposure of a point target in a given range c e l l to decrease since the target energy migrates rapidly across range c e l l s . Under large azimuth TBP conditions, the time and frequency domain azimuth signals exhibit an approximate one-to-one correspondence. Thus the decrease in azimuth timewidth results in a decreased bandwidth and consequently a decreased azimuth TBP. The time-frequency correspondence i s predicted by the p r i n c i p l e of stationary phase [17] for large TBP signals. When the azimuth TBP f a l l s below unity, the correspondence is no longer v a l i d and the azimuth spectrum becomes broadened r e l a t i v e to i t s predicted bandwidth. This spectrum broadening appears after the azimuth FFT, which i s used for fast azimuth convolution in the range/Doppler algorithm. Since the point target energy l i e s along a sloped curve (the RCM curve) in both range-time/azimuth-time and range-time/azimuth frequency space, the azimuth spectrum broadening causes the point target energy to be s i m i l a r l y broadened in the range time d i r e c t i o n . If the range broadening i s l e f t uncorrected as in the basic range/Doppler algorithm, the f i n a l compressed point target response becomes broadened in range. SRC was f i r s t proposed by J i n and Wu [13] in 1984 as a method for extending the maximum squint angle which can be 3 p r o c e s s e d b y t h e r a n g e / D o p p l e r a l g o r i t h m . T h e i r p a p e r p r e s e n t e d a m o d e l f o r t h e r a n g e b r o a d e n i n g o f t h e r a n g e c o m p r e s s e d p o i n t t a r g e t r e s p o n s e a f t e r t h e a z i m u t h F o u r i e r t r a n s f o r m . T h e i r i m p r o v e d m o d e l a c c o u n t e d f o r t h e b r o a d e n i n g i n m a g n i t u d e a n d p h a s e o f t h e a z i m u t h s p e c t r u m . H o w e v e r t h e m o d e l i d e a l i z e d t h e r a n g e c o m p r e s s e d p r o f i l e a s a s i m p l e i n f i n i t e d u r a t i o n s i n e f u n c t i o n . T h i s a p p r o x i m a t i o n d o e s n o t i n c l u d e t h e e f f e c t s o f t h e r a n g e a n d a z i m u t h w i n d o w s w h i c h a r e u s u a l l y a p p l i e d i n t h e r e s p e c t i v e f r e q u e n c y d o m a i n s t o c o n t r o l s i d e l o b e l e v e l s . A l s o s e v e r a l a p p r o x i m a t i o n s r e q u i r e d t o d e v e l o p t h e m o d e l w e r e n o t f u l l y s t a t e d o r v a l i d a t e d . A c o n t i n u o u s t i m e , i n f i n i t e d u r a t i o n , a z i m u t h c o m p r e s s i o n f i l t e r m a t c h e d t o t h e a p p r o x i m a t e r a n g e c o m p r e s s e d s i g n a l m o d e l wa s d e r i v e d . T h e f i l t e r i n c l u d e d a s e p a r a t e S R C f i l t e r w h i c h w a s a p p l i e d a s a c o n t i n u o u s r a n g e t i m e c o n v o l u t i o n t o t h e a z i m u t h s p e c t r u m . T h e d e t a i l s o f t h e d i s c r e t e i m p l e m e n t a t i o n o f t h i s f i l t e r w e r e n o t p r e s e n t e d . Q u a l i t a t i v e s i m u l a t i o n r e s u l t s w e r e s h o w n f o r S E A S A T SAR p a r a m e t e r s . 1.2 O B J E C T I V E S T h e o b j e c t i v e s o f t h i s t h e s i s a r e s u m m a r i z e d a s f o l l o w s : 1. T h e r a n g e b r o a d e n i n g m o d e l o f J i n a n d Wu i s t o b e e x t e n d e d t o a c c u r a t e l y m o d e l t h e r a n g e c o m p r e s s e d p r o f i l e i n c l u d i n g t h e e f f e c t s o f r a n g e w i n d o w i n g . A l l a p p r o x i m a t i o n s a n d a s s u m p t i o n s a r e t o b e e x p l i c i t l y 4 stated and examined for their range of v a l i d i t y . 2. J i n and Wu state that the range compressed point target response may be used as the azimuth matched f i l t e r reference function (see equation (20) in [13]). However t h i s i s only true when the range compressed p r o f i l e i s approximated by a sine function and no range bandlimiting or windowing i s applied during range compression. The azimuth f i l t e r i s to be reformulated for the more general case by matching the f i l t e r d i r e c t l y to the point target response before range compression. 3. The basic SRC algorithm i s to be rederived using the extended azimuth matched f i l t e r model. Alternate methods for implementing SRC are to be examined and evaluated in terms of e f f i c i e n c y and accuracy. 4. The SRC algorithm derived by J i n and Wu consisted of a range time convolution with a continuous-time SRC f i l t e r of i n f i n i t e time duration. Methods of implementing th i s f i l t e r as a discrete f i n i t e length SRC f i l t e r are to be developed. In p a r t i c u l a r , combinations of the SRC f i l t e r with the frequency domain RCMC interpolator are to be explored. This algorithm w i l l be c a l l e d azimuth SRC. 5. J i n and Wu state that i t i s possible to perform SRC 5 d u r i n g range compression but do not d e r i v e the theory or an a l g o r i t h m . The p o s s i b i l i t i e s of implementing SRC dur i n g range compression are to be explored and ev a l u a t e d . T h i s a l g o r i t h m w i l l be c a l l e d range SRC. 6. SAR images are prone to high l e v e l s of speckle noise due to the coherent nature of the radar i l l u m i n a t i o n . To reduce the speckle n o i s e the processed a p e r t u r e i s o f t e n s u b d i v i d e d i n t o separate " l o o k s " which are then i n c o h e r e n t l y summed. New methods of implementing SRC with m u l t i l o o k range/Doppler p r o c e s s i n g are to be developed. 7. The s i g n a l parameters used i n p r a c t i c a l SAR systems may c o n t a i n e s t i m a t i o n e r r o r s or e r r o r s caused by the use of constant parameters i n block p r o c e s s i n g . The s e n s i t i v i t y of the SRC f i l t e r to parameter e r r o r s i s t o be examined. 8 . In order to choose the necessary p r o c e s s i n g a l g o r i t h m s and parameters f o r a given image q u a l i t y requirement, q u a n t i t a t i v e design curves of expected image q u a l i t y are needed. Computer s i m u l a t i o n s a re to be performed i n order t o q u a n t i f y the f o l l o w i n g items with p a r t i c u l a r c o n s i d e r a t i o n being given t o the RADARSAT SAR sensor. The s i m u l a t e d image degradations and improvements are to be measured as a f u n c t i o n of s q u i n t angle: 6 a. measure the range and azimuth broadening of the compressed p o i n t t a r g e t response of the b a s i c range/Doppler a l g o r i t h m to determine the sq u i n t l i m i t a t i o n s of both s i n g l e - l o o k and m u l t i l o o k a l g o r i t h m s b. determine the accuracy of the range broadening model i n comparison with a c t u a l broadening measurements c. q u a n t i f y the image q u a l i t y improvements f o r the new SRC a l g o r i t h m s f o r s i n g l e - l o o k and m u l t i l o o k p r o c e s s i n g with v a r i o u s p r o c e s s i n g parameters d. examine the image degradations caused by SRC parameter e r r o r s f o r a l l a l g o r i t h m s . 1.3 STRUCTURE OF THE THESIS The t h e s i s i s d i v i d e d i n t o s e v e r a l s e c t i o n s . Chapter 2 i n t r o d u c e s the b a s i c concept of SAR image formation as a 2-D matched f i l t e r i n g o p e r a t i o n of a r e c e i v e d radar s i g n a l which i s approximately l i n e a r FM i n both dimensions. A model of spaceborne SAR geometry i s developed to d e f i n e the v a r i a t i o n s of t a r g e t range with azimuth time and other key s i g n a l parameters. The image q u a l i t y measurements of i n t e r e s t are i n t r o d u c e d and the measurement procedures are o u t l i n e d . 7 Chapter 3 examines the theory and l i m i t a t i o n s of basic range/Doppler compression without SRC. A model of the signal returned from a point target i s developed to use as a reference function for the matched f i l t e r . The model i s also used in the simulations for creating a simulated range compressed point target response. The results of extensive simulations with nominal RADARSAT parameters are summarized. Chapter 4 develops a mathematical model of the azimuth spectrum broadening and range broadening which occurs with the basic range/Doppler algorithm at large squint angles. A simple accurate model of azimuth broadening i s also presented. Chapter 5 extends the theory used in chapter 4 to develop an improved matched f i l t e r which includes SRC. Two new alternate techniques for the discrete implemention of SRC (azimuth SRC and range SRC) are presented. Extensive simulation results are discussed to evaluate the new algorithms. Chapter 6 examines the application of SRC to multilook range/Doppler compression. The new concepts of fixed (look-independent) and look-dependent SRC f i l t e r s are presented. Simulations of multilook compression (with 4-looks) with and without SRC are performed to quantify the improvements. Chapter 7 examines the e f f e c t s of SRC parameter estimation errors and block processing invariance regions on image q u a l i t y . A model i s developed to relate these errors 8 to equivalent phase errors which occur in processing without SRC. F i n a l l y chapter 8 presents f i n a l conclusions and suggests areas for further research on SRC. 2. THE SYNTHETIC APERTURE RADAR (SAR) CONCEPT Synthetic aperture radar processing i s a method of obtaining image resolutions much finer than the along-track beamwidth of the radar antenna from a moving platform. It has been successfully applied to both airborne and spaceborne radars to provide all-weather high resolution imaging c a p a b i l i t i e s . Strip-map mode sensors, such as RADARSAT and SEASAT, orient the boresight of the antenna perpendicular to the d i r e c t i o n of travel of the platform, i . e . , in the cross-track d i r e c t i o n , and off to one side of the ground track. An azimuth (along-track) antenna aperture much larger than the size of the physical antenna i s synthesized by properly combining the received radar pulses over a coherent integration period with appropriate weighting. The azimuth resolution i s inversely proportional to the synthesized aperture length. The returns from point targets at d i f f e r e n t ground positions are resolved in range (cross-track) by differences in the time delay of the transmitted radar pulses and in azimuth by t h e i r Doppler s h i f t . The radar pulse i s t y p i c a l l y a l i n e a r l y frequency modulated (FM) pulse with large TBP. In range/Doppler processing the received pulses are compressed in range using standard pulse compression techniques to get a range compressed signal with a small TBP. The azimuth signal has a similar modulation (approximately linear FM) due to the changing distance between the sensor and target. By applying 9 10 pulse compression techniques in the azimuth d i r e c t i o n a well resolved image can be obtained. An additional complication of the processing occurs when the change in range to a point target over the azimuth integration period i s larger than the range c e l l s i z e . This e f f e c t , c a l l e d range c e l l migration (RCM), causes the signal energy from a point target af t e r range compression to migrate across several range c e l l s . Consequently the azimuth compression becomes a 2-D operation. The basic range/Doppler algorithm separates t h i s 2-D azimuth operation into two 1-D operations : 1. Range C e l l Migration Correction (RCMC) in which the range compressed point target energy i s interpolated and shi f t e d in range so that the energy l i e s along a single azimuth l i n e . 2. Azimuth c o r r e l a t i o n with a 1-D reference phase function. For computational e f f i c i e n c y these operations are performed in the azimuth frequency domain via fast convolution. For large azimuth TBP, linear FM type signals, the magnitude and phase c h a r a c t e r i s t i c s of the azimuth frequency domain signals can be simply related to the azimuth time domain signals by a linear scale factor using the p r i n c i p l e of stationary phare [17]. In such cases, which occur when there i s l i t t l e RCM and a small squint angle, a 11 zero-phase sinc-type interpolator can be used for RCMC. Block processing e f f i c i e n c y can be achieved since the tr a j e c t o r i e s of point targets which are adjacent to each other in azimuth time follow a common RCM curve in the azimuth frequency domain. This allows RCMC and azimuth compression to be applied to many targets simultaneously. However when the azimuth time exposure in a range c e l l becomes small due to RCM and the azimuth TBP f a l l s below unity, the magnitude and phase c h a r a c t e r i s t i c s of the frequency domain signal become broadened r e l a t i v e to their corresponding time domain signals. The broadened frequency domain signal can be properly compressed by applying a secondary compression in either the range time or azimuth frequency d i r e c t i o n . Since the width in samples of the broadened function i s much smaller in range than in azimuth, the f i l t e r i s more e f f i c i e n t l y implemented in range, hence the name secondary range compression (SRC). The SRC f i l t e r can be viewed as convolution with a quadratic phase range f i l t e r which recompresses the broadening which occurs in the azimuth Fourier transform. Two e f f i c i e n t implementations of thi s secondary range f i l t e r have been investigated : 1. Azimuth SRC in which the secondary range f i l t e r i s combined with the RCMC interpolator during azimuth compression. 12 2. Range SRC where the secondary range f i l t e r i s combined with the range frequency domain, reference function during range compression. The next section discusses a model of the spaceborne SAR sensor geometry which i s used to derive the range c e l l migration equation ( i . e . , the v a r i a t i o n of range with azimuth time) and other signal parameters. For the simulations in l a t e r sections, a set of nominal RADARSAT parameters has been chosen. These are l i s t e d in table 1. 13 Parameter Symbol Value Units range -3dB chirp bandwidth Br 17.28 MHz single-look azimuth Kaiser-Bessel * A 1.5 parameter multilook azimuth Kaiser-Bessel 2.7 parameter SRC/RCMC Kaiser-Bessel parameter 01 2.5 incidence angle Pi 20 deg range Kaiser-Bessel parameter 0R 2.7 azimuth antenna length D 14.0 m azimuth processed bandwidth (0 S=O°) fPBW 942 Hz complex range sampling rate Fsr 19.872 MHz s a t e l l i t e a l t i t u d e h s 1007.4 km # of SRC/RCMC f i l t e r versions I 16 SRC/RCMC FFT array length K r 128 length of SRC/RCMC f i l t e r versions L 4,8,16,32 length of range FFT M 2048 length of single-look azimuth FFT N 1024 wavelength X 0.05656 m azimuth processing i n t e r v a l P^BW 0.513 s pulse r e p e t i t i o n frequency PRF 1 177.9 Hz slant range c e l l size ( c / [ 2 F s r ] ) Psr 7.543 m earth radius at equator re 63716 km slant range of closest approach r 0 1 072. 1 km range pulsewidth T 36.4 *xs orbit i n c l i n a t i o n angle «i 99.5 deg beam v e l o c i t y V b 7.4575 km/s earth r o t a t i o n a l v e l o c i t y v e 0.4638 km/s s a t e l l i t e o r b i t a l v e l o c i t y v s 7.35 km/s equivalent s a t e l l i t e v e l o c i t y v e q 7.4575 km/s azimuth oversampling factor 1 .25 range oversampling factor 1.15 Table 1. Nominal Radarsat Parameters 14 2.1 MODEL OF SAR GEOMETRY The production of high resolution SAR images requires an accurate model of the physical geometry of the space platform with respect to the earth's surface. This study uses a s i m p l i f i e d f l a t earth geometric model which exhibits the e s s e n t i a l properties of SAR signals. A more sophisticated spherical earth, c i r c u l a r orbit geometric model i s used to derive accurate estimates of the actual signal parameters which are then applied to the s i m p l i f i e d model. The spherical earth, c i r c u l a r o r b i t geometric model i s used to derive an equivalent r e l a t i v e v e l o c i t y between the s u b - s a t e l l i t e point and the surface of the earth. After mapping into the slant range plane (the plane containing the platform v e l o c i t y vector and the vector joining the platform and a point target on the ground), the equivalent v e l o c i t y is applied to a l o c a l l y f l a t model of the region of the earth under the s a t e l l i t e . Second order e f f e c t s caused by l o c a l curvature of the earth, or variations in the earth's radius or s a t e l l i t e height above the surface are excluded from the simulations since the added complexity adds l i t t l e insight into the range broadening process. These secondary e f f e c t s can usually be accounted for by using modified estimates of the signal parameters. Vant [23] provides a good discussion of a similar spherical earth model. For s i m p l i c i t y , the s a t e l l i t e p o s i t i o n i s a r b i t r a r i l y chosen to be above the equator since 15 the rotation of the earth t y p i c a l l y has i t s greatest effect there. Figure 2.1 shows the geometry of the s u b - s a t e l l i t e point in an i n e r t i a l frame of reference. The s u b - s a t e l l i t e point moves with v e l o c i t y v and the earth's surface moves at tangential v e l o c i t y v g beneath i t , where v = v / C (1 ) ss s a and v g i s the tangential v e l o c i t y of the s a t e l l i t e in i t s o r b i t . The factor C , the r a t i o between the s a t e l l i t e v e l o c i t y and the s u b - s a t e l l i t e point v e l o c i t y , i s given by C a = ( r e - h s ) / r e (2) where r i s the radius of the earth and h i s the height of e s 3 the s a t e l l i t e above the earth's surface. The equivalent r e l a t i v e ground v e l o c i t y , V g , between the s u b - s a t e l l i t e point and the earth's surface i s given by : v = v - v g ss e v I = [ v 2 + v 2 - 2v v cos(0.) ] 1 / 2 g i ss e ss e 1 (3) (4) where 8^ i s the i n c l i n a t i o n angle of the s a t e l l i t e o r b i t . This ground v e l o c i t y i s translated back into an equivalent s a t e l l i t e v e l o c i t y , v , in the slant range plane as eq 16 Figure 2.1. Spherical earth model for velocity calculation. 17 v = v C (5) eq g a In order to simulate the azimuth antenna weighting function, i t i s necessary to know the v e l o c i t y at which a point target travels through the antenna beam in the antenna azimuth d i r e c t i o n . For convenience, the beam v e l o c i t y , v^, w i l l be assumed to be constant and equal to v eq The derived equivalent v e l o c i t i e s are applied to the f l a t earth model shown in figure 2.2. The s a t e l l i t e travels with v e l o c i t y v ^ at a height h g above the ground. The slant range of closest approach, r 0 , i s determined by the incidence angle, p\ , as r 0 = h_ / cos( B. ) (6) The antenna boresight may be squinted away from the r 0 d i r e c t i o n by the squint angle 6 in the slant range plane. The squint angle i s defined to be posit i v e when the antenna is pointing behind the zero Doppler d i r e c t i o n r e s u l t i n g in a negative Doppler beam center frequency. Azimuth time, v, i s measured r e l a t i v e to the ground position of closest approach as shown in figure 2.3. From t h i s figure, the following quantities can be deduced r(rj) = [ r 0 2 + ( v E G T } ) 2 ] L / 2 (7) T J c = r 0 tan(0 s) / v e g (8) F i g u r e 2.2. F l a t e a r t h model f o r l o c a l g e o m e t r y . Figure 2.3. Slant range plane. 20 r c = r(rj c ) (9) where r(r/) i s the range migration equation which defines the instantaneous range to a point target with range of closest approach, r 0 , and rj c and r c are the beam center azimuth time and range respectively. It i s useful at t h i s point to examine an approximation to the RCM equation in greater d e t a i l . The RCM curve can be expanded into a Taylor series form about the azimuth beam center crossing time, r)r, as follows : The f i r s t term represents the range to the beam center point which i s constant in the spherical earth/orbit model. The second term i s the linear component of RCM and i s commonly referred to as range walk. The higher terms w i l l be c o l l e c t i v e l y referred to as range curvature. In most sate l l i t e - b o r n e systems such as RADARSAT range walk i s the dominant component of RCM. For nominal RADARSAT parameters, range walk increases almost l i n e a r l y with squint angle from zero at 0°, to 8.9 range c e l l s at 1°, to 88 range c e l l s at 10°. Range curvature i s comparatively small being an approximately constant 0.23 range c e l l s . When range walk and/or curvature exceed the range resolution, some form of RCM correction (RCMC) i s required to maintain good azimuth r(r?) = r{nr) + r ' (j?r) ( r ? - 7 j r ) + r"(rj c) (T7-TJ c ) 2 /2 + . . . (10) 21 and range resolutions. Range walk increases approximately l i n e a r l y with wavelength whereas range curvature increases approximately as the square of the wavelength. Consequently, for longer wavelength sa t e l l i t e - b o r n e SAR's such as SEASAT, range walk may be several times larger and range curvature may be an order of magnitude larger. Terms up to the quadratic are usually s u f f i c i e n t for characterizing RCMC whereas higher order terms may be necessary to accurately represent azimuth phase. By dropping terms higher than the quadratic and substituting for r'(j? c) and r " ( 7 j ^ ) in terms of the instantaneous frequency f ^ and frequency rate K A at the beam center crossing time, the Taylor series can be written as : r ( 7 j ) * r ( T ? C ) - ( X / 2 ) [ f C ( T ? - T ? C ) + K a ( T J - T J c ) 2 / 2 ] (11) - ( X / 2 ) [ f , T } + K A T ? 2 / 2 ] (12) where = r(r? c) + ( X / 2 ) [ f c T j c - K A 7 j c 2 / 2 ] (13) f (14) This approximate form o! the RCM equation w i l l be used in subsequent sections to define the azimuth phase response and 22 i t s Fourier transform. 2.2 IMAGE QUALITY MEASUREMENTS This section discusses the relevent image quality measures of the point target response and outlines the methods used in the simulations for th e i r measurement. Several measures computed from the point target response are commonly used to determine the quality of a SAR image. These are as follows : 1. Impulse Response Width (IRW). The -3dB impulse response widths in both range and azimuth are standard measures of resolution. They are measured by interpolating in the range and azimuth directions by a factor of 128. The peak magnitude in each d i r e c t i o n i s determined. Then the distance between the -3dB points i s computed using linear interpolation between the already interpolated samples. 2. Integrated Sidelobe Ratio (ISLR). The I SLR i s the r a t i o of the integrated energy in the sidelobe region to the integrated energy in the mainlobe region. The sidelobe region i s defined as a l l samples inside of a rectangle whose sides are located at the measured -3dB positions in range and azimuth. The sidelobe region i s defined as a l l samples outside of a rectangle which i s 3 times the size of the mainlobe rectangle and which i s centered at the same pos i t i o n . 23 The 2-D ISLR i s measured on a 2-D array of size 256x256 which has been interpolated by a factor of 8 in both directions from a 32x32 array. The integrations are performed by summing squared magnitudes. Although this array does not extend out to the ends of the sidelobe regions, i t contains most of the sidelobe energy and was chosen because of memory constraints. Summing the sidelobe region over a limited area i s a good approximation when l i t t l e broadening occurs, i . e . , for low squint angles. However, the approximation breaks down for large broadening as w i l l be shown in the simulations. Fortunately, the approximation i s v a l i d over the broadening lev e l s of interest in the simulated system. The ISLR i s also measured on 1-D arrays in range and azimuth. These provide indications of the broadening in each d i r e c t i o n . The 1-D ISLR i s measured on an array of length 4096 which has been interpolated by a factor of 128 from a length 32 array. 3. Peak Sidelobe Ratio (PSLR). The PSLR is the r a t i o of the magnitude of the largest sidelobe in the sidelobe region to the magnitude of the peak of the point target response. In 1-D, range or azimuth, the PSLR i s measured on an array which has been interpolated by a factor of 128. In 2-D, the peak sidelobe is measured in the 2-D sidelobe region of an array which has been interpolated by a factor of 8 in 24 both d i r e c t i o n s . 4. Peak Magnitude Degradation. As more energy i s spread into the sidelobes, the magnitude of the point target response peak decreases causing a decrease in signal-to-noise r a t i o (SNR). The decrease in peak magnitude has been measured as a function of squint angle. However the SNR i s not d i r e c t l y related to the measured peak magnitude since a normalization based on the noise power d i s t r i b u t i o n must be used. In the simulations the peak magnitudes are normalized to the sum of squares of the RCMC or combined SRC/RCMC f i l t e r c o e f f i c i e n t s . This normalization i s appropriate for a white d i s t r i b u t i o n of noise over a l l range c e l l s . The peak magnitudes are also normalized by the azimuth processed bandwidth which decreases with increasing squint angle. The processed bandwidth i s proportional to the noise power i f the noise in the azimuth signal i s white. 3. BASIC RANGE/DOPPLER COMPRESSION This section presents a theory to describe the basic range/Doppler compression algorithm. Discrete implementations of the range and azimuth compression operations are presented. Extensive simulations are used to quantify the image degradations which occur for large squint angles. The range/Doppler algorithm, also known as a frequency domain interpolation algorithm or a frequency domain corr e l a t i o n algorithm, has been described in several good papers [2,6,20,21,24,26]. The theory developed here provides further insight into the approximations involved in deriving the basic range/Doppler algorithm as a f i l t e r matched to the point target response. The approximate one-to-one correspondence between the time and frequency domain azimuth signals which i s v a l i d for large azimuth TBP signals i s used. Later sections provide a refinement of t h i s approximation which accounts for the spectrum broadening process and provides the basis for the SRC algorithm. A model of the return from a point target which was presented by J i n and Wu [13] i s extended to include the range window. The model i s used in the simulator to generate a 1-D point target return signal which i s subsequently compressed in range. Range compression i s performed by range matched f i l t e r i n g and windowing in the range frequency domain to produce a 1-D range compressed p r o f i l e . From t h i s , a 2-D range compressed signal i s simulated by s h i f t i n g the 25 26 range p r o f i l e peak in range along the RCM curve defined by r(rj) and multiplying by the azimuth phase coding which i s approximately linear FM. Azimuth compression, which i s also performed as a fast convolution in the frequency domain, consists of applying an azimuth fast Fourier transform (FFT), performing RCMC, multiplying by a 1-D azimuth reference phase function and window function, and transforming back to the azimuth time domain using an inverse FFT. The image quality of the simulated compressed point target responses are measured for several squint angles. These measurements provide a baseline for comparison with l a t e r simulations using SRC. 3.1 POINT TARGET RESPONSE MODEL The complex received signal after quadrature demodulation from a point target with range of closest approach r 0 can be modelled in continuous range and azimuth time as [26] h(t,7?) = h A ( t , T j ) * h R ( t , r ? ) (15) h A(t,T?) = w a ( t j ) exp [ - j 4 7 T r(T7)/X] 5[t - 2r(r?)/c] (16) h R ( t , i ? ) = 8(T?) s T ( t ) (17) where t i s c o n t i n u o u s range time measured from the time of t r a n s m i s s i o n of the p u l s e of i n t e r e s t , W ^ T J ) i s the a z i m u t h 27 antenna function, c i s the speed of l i g h t , and * denotes 2-D convolution. The function h R(t,rj) represents the response in the range d i r e c t i o n after quadrature demodulation and can be clo s e l y approximated by the transmitted complex modulation function, s T ( t ) . This approximation i s v a l i d when the Doppler s h i f t i s much smaller than the transmitted bandwidth as i s usually the case. The function h A ( t , 7 ? ) represents the hypothetical continuous azimuth response to an impulse-type radar pulse assuming that the radar does not move appreciably during the tr a n s i t time of the pulse. A good discussion of the v a l i d i t y of t h i s stop-start approximation in which the sensor i s assumed to be stationary during pulse transmission and reception i s given by Barber [2]. Although azimuth time i s actually sampled at the pulse re p e t i t i o n frequency (PRF) of the radar, the continuous azimuth time model i s v a l i d i f the PRF i s chosen s u f f i c i e n t l y high to prevent s i g n i f i c a n t a l i a s i n g of the azimuth s i g n a l . Optimum SAR processing in a least mean squared error sense consists of f i l t e r i n g the return signal with a 2-D matched f i l t e r which is matched to the point target si g n a l . Therefore the ideal matched f i l t e r impulse response can be written as : h*(-t,-7j) = h*(-t,-r?) * h*(-t,-77) (18) 28 The f i l t e r consists of two parts : * 1. a range matched f i l t e r , h R ( - t , - T j ) , which compresses the coded range pulse * 2. and an azimuth matched f i l t e r , h A ( - t , - r j ) , which compresses the azimuth phase coding and compensates for RCM. Additional range and azimuth f i l t e r i n g in the form of frequency domain windows i s often applied in order to control the tradeoff between sidelobe levels and impulse response widths. In addition the azimuth weighting due to the antenna aperture i s dropped from the azimuth matched f i l t e r since i t s ef f e c t i s similar to the azimuth window. The following sections describe the approximations required to derive the range and azimuth matched f i l t e r s of the basic range/Doppler algorithm. 3.2 RANGE COMPRESSION Range compression may be viewed as a convolution of the * received signal with a f i l t e r , h R ( - t , - 7 j ) , matched to the transmitted pulse and a window function, w R ( t ) , which i s designed to reduce the energy in the sidelobes. The continuous 2-D range compressed signal may be written as h R C ( t , 7 ? ) = h ( t , 7 j ) * h * ( - t , - T ? ) * [ 6 ( r j ) w R ( t ) ] (19) 29 = h A ( t , 7 j ) * [6(7}) h R C p ( t ) ] (20) where h R C p ( t ) = 8 T ( t ) * t S*(~t) \ W R ( t ) (21) i s the 1-D range compressed p r o f i l e which i s usually similar in shape to a sine function, and * f c denotes convolution in range time. In t h i s form the range compressed signal i s expressed as a range time convolution of the 1-D range compressed p r o f i l e with a 2-D phase function, h A ( t , r j ) , which i s non-zero only along the RCM curve. It should be noted that the time o r i g i n of the range compressed signal has been selected so that the range compressed p r o f i l e i s symmetric about t=0. For RADARSAT and most other s a t e l l i t e SAR's, the transmitted signal is a linear FM pulse which can be represented at baseband as where a(t) i s the amplitude function, <p(t) i s the phase modulation function, T i s the pulsewidth, and K R i s the range l i n e a r FM rate. s T ( t ) = a(t) e x p [ - j * ( t ) ] (22) a(t) = rect(tA) (23) *(t) = - * K R t 2 (24) 30 After some manipulation, the range compressed p r o f i l e can be shown to have the form of a weighted sine function [23] : h R C p ( t ) = { ( T - | t | ) r e c t ( t / 2 T ) s i n c [ i r K R t ( T - | t | ) ] } * t w R(t) (25) The range compression convolution i s performed more e f f i c i e n t l y in the range frequency domain using fast convolution. Fast convolution i s computationally e f f i c i e n t when the length of the convolution kernel in samples is a power of 2 larger than about 32. The fast convolution method consists of : 1. Fourier transforming the range matched f i l t e r (which i s the complex conjugate of the Fourier transform of the transmitted pulse) and the received pulse (for the simulations the received pulse i s assumed to be the same as the transmitted pulse with appropriate delay); 2. multiplying together the received signal, the matched f i l t e r , and the sidelobe control window; 3. and inverse Fourier transforming the r e s u l t . In continuous time and frequency theory, the fast convolution range compression operation can be expressed as 31 h R c p ( t ) = f l W R(f R) |S T<f R>|* } (26) where w R ( f R ) a n o" S T ^ R ^ a r e fc^e F o u r i e r transforms of w R ( t ) and s T ( t ) respectively, 3.3 SIMULATION OF THE RANGE COMPRESSED PROFILE This section describes the method used in the simulations to generate a discrete, range compressed p r o f i l e , h R^ p(m). Where necessary, the symbol " w i l l be used to denote discrete signals. A discrete, l i n e a r FM modulation function, s T(m), i s formed in the range time domain as s T(m) = a(mT) exp[-j^CmT)] , -(M/2) < m < (M/2) (27) where T i s the range sampling period and a(mT) i s the rectangular pulse envelope of width T . This function i s transformed with a range F F T of length M where M > T / T . The frequency samples are squared and a m u l t i p l i c a t i v e sidelobe control window i s applied to get H R C p ( k ) = WR(k) | S T ( k ) | 2 , -(M/2) < k < (M/2) (28) where S T(k) i s the F F T of s T(m), w R ( k ) i s the frequency domain window function, and k i s the frequency index. Many window functions are available for c o n t r o l l i n g the sidelobes. The p r i n c i p a l window vsed in the simulations i s a Kaiser-Bessel window defined as [12] 32 WR(k) = / 0{/3 R[l-(2k/M) 2] l / 2} / /0{/3R) , -(M/2) < k < (M/2) (29) where 0 D is a window parameter which controls the amount of weighting. As 0 R i s increased, the mainlobe width of the range compressed p r o f i l e increases whereas the energy in the sidelobes and the magnitude of the peak sidelobe decrease. The zeroth-order modified Bessel function of the f i r s t kind, I 0 , i s approximated by the power series : P 7 0(x) = Z [ (x/2) P / pl V (30) p=0 The number of terms used in the simulations (P=15) provides an accuracy of about 14 s i g n i f i c a n t figures for the Bessel function. Since the simulated range compressed p r o f i l e defines the weighting along each azimuth l i n e of the simulated 2-D range compressed si g n a l , a close approximation of the continuous p r o f i l e i s desired. Thus the discrete p r o f i l e is interpolated by a factor I (1=16 in the simulations) by zero padding the frequency array to a length of MI before transforming back into the time domain . This e f f e c t i v e l y performs interpolation [2] with a time-aliased sinc(x) function. Except for the small differences caused by a l i a s i n g errors, the interpolated samples provide a good 33 simulation of the samples which would be obtained by compressing a set of time delayed return pulses. In order to interpolate properly, the zero padding must be performed at the ends of the pulse spectrum, i . e . , in the middle of the FFT array, as follows H R C p ( k ) , -(M/2) < k < (M/2) (31 ) 0 , -(MI/2) < k < -(M/2) and (M/2) < k < (MI/2) where ' i s used to denote the interpolated s i g n a l . After an inverse FFT of length MI i s applied, the interpolated, range compressed p r o f i l e , h R C p(m'), -MI/2 < m' < MI/2, i s obtained where m' i s the interpolated array index and the sampling period i s T/I. The peak of the p r o f i l e occurs at m'= 0. Since the FFT i s being used to perform a linear convolution, the FFT length, M, must be large enough to exclude i n v a l i d samples which occur because of the FFT's c i r c u l a r convolution. If the pulsewidth i s T and the desired number of v a l i d compressed samples before interpolation is Q, M must s a t i s f y M > (r/T) + Q - 1. The length M i s usually chosen to be the next larger power of 2 to allow the use of e f f i c i e n t FFT algorithms. After interpolation, v a l i d samples occur for -(QI/2) < m' £ (QI/2). Figure 3.1 shows the simulated range compressed p r o f i l e a f t e r interpolation for the nominal RADARSAT parameters H R C p ( k ) -34 given in Table 1. The window parameter, 0 R = 2.7, was chosen to produce a 1-D peak sidelobe r a t i o (PSLR) of -21.7 dB and a 1-D integrated sidelobe r a t i o (ISLR) of -21.0 dB. 3.4 SINGLE-LOOK AZIMUTH COMPRESSION This section describes the theory and approximations used in developing the basic range/Doppler azimuth compression algorithm for single-look processing. As described e a r l i e r , the basic range/Doppler algorithm makes use of the approximate s i m i l a r i t y between the time and frequency domain signals of large TBP linear FM type signals. Azimuth compression consists of convolving the range compressed signal with an approximation to the azimuth matched f i l t e r , h A(-t,-r?), and an azimuth window to control sidelobes. The antenna function w (rj) i s usually dropped from the azimuth f i l t e r since i t s ef f e c t i s sim i l a r to that of the azimuth window. Excluding the azimuth window for the time being, an azimuth reference function (the time-reversed complex conjugate of the azimuth matched f i l t e r ) can be written as : h p(t,r/) = exp[-j47rr (Tj)/X] 6 [ t-2r ( T J ) / C ] (32) In order to understand the discrete implementation of th i s approximate matched f i l t e r , the f i l t e r must be bandlimited in range to the range sampling frequency, F s r -This excludes frequencies which would be a l i a s e d by range R a n g e C o m p r e s s e d P r o f i l e betar = 2.7 16 -12 -8 - 4 0 4 8 12 16 Range Sample Number 36 sampling and also provides the basis for interpolation in the range d i r e c t i o n . Continuous time signals w i l l be used to develop the algorithm. These can be d i s c r e t i z e d in range and azimuth and time-aliased according to the length of the FFT 1s in order to provide a discrete model of the algorithm. The ideal rectangular range bandlimiting f i l t e r has the form of a sine function. Thus the range bandlimited reference function can be formulated as : h F R B ( t , r j ) = h p ( t , T j ) * t sinc( 7 r F s r t ) (33) = exp[-j4?Tr(T?)/X] sinc( ?rF s r[ t - 2 r ( i j ) / c ] ) (34) The shape of t h i s 2-D reference function i s shown in figure 3.2. Cross-sections of the function in the azimuth time d i r e c t i o n exhibit a lin e a r FM type of phase c h a r a c t e r i s t i c which i s the same as the phase along the RCM curve in the previous i n f i n i t e bandwidth reference function, h p ( t , 7 j ) . The envelope of t h i s signal in the azimuth d i r e c t i o n is a sine function centered at the RCM curve with a time-warping e f f e c t created by the range curvature terms of the RCM equation, r{n). Since the RCM over the azimuth time interval defined by the azimuth antenna beamwidth i s predominantly l i n e a r , e s p e c i a l l y for RADARSAT parameters, the time-warping of the sine envelope i s small. Figure 3.2. Range c e l l migration curve in azimuth time domain. 38 The approximate azimuth timewidth of the reference function can be determined by using a linear aproximation to rin) and determining the -3dB azimuth times of the sine envelope. Using the Taylor series expansion of chapter 2 with only f i r s t order terms and evaluating at the beam center range time, t = 2r(r}^)/c, the approximate azimuth envelope i s sine (7rF s rXf c [ T J - 7 J c ] / c ) . The -3dB timewidth of this envelope is given by : Ar, 3 d B - 0.884 c / U f ^ ) (35) When the squint angle (and therefore the beam center frequency, f^) i s small, the TBP in the azimuth d i r e c t i o n near the beam center range i s large. For a large azimuth TBP signal, the Fourier transform of the azimuth signal i s similar in phase and magnitude to the azimuth time domain signal except for a scaling constant. Using the p r i n c i p l e of stationary phase [17], the scaling between the time and frequency axes can be determined by expressing the instantaneous frequency as a function of azimuth time : f ± (77) = -(2/X) r'(rj) (36) Substituting for r'(r?) with the derivative of equation (7) and rearranging gives the inverse mapping r ^ U ) = r 0 / { v e g [ ( 2 v e g / ( X f ) ) 2 - 1 ] i / z } (37) 39 where 7?^(f) i s the azimuth time corresponding to the instantaneous azimuth frequency f. Substituting back into equation (7) gives the approximate frequency domain RCM curve : r . ( f ) = r d j . l f ) ) = r 0 d + 1 /[ ( 2 v & q / ( Xf)) 2 - (38) Thus the azimuth Fourier transform of the range bandlimited reference function excluding amplitude constants can be approximated as : H F R B ( t f f ) * h F R B ( t , r ? i ( f )) (39) = exp[-j4irr. (f )/X] sine d r F f t-2r . (f )/c ]) (40) The shape of thi s azimuth frequency domain reference function i s shown in figure 3.3. This approximate equation forms the basis for the azimuth frequency domain, fast convolution implementation of azimuth compression in the basic range/Doppler algorithm. The approximate azimuth matched f i l t e r i s the complex conjugate of thi s frequency domain reference function. An azimuth sidelobe control window i s also applied in the azimuth frequency domain. The basic range/Doppler azimuth compression algorithm i s expressed as : 40 Figure 3.3. Range c e l l migration curve in azimuth frequency doma i n. 41 a ( t , T ? ) = F' 1{[ H R C ( t , f ) * t H * R B ( - t , f ) ] W A(f-f c)} (41) = F- 1{exp[ j47rr i(f )/X] W A(f-f c) • H R C ( t ' , f ) s i n c ( 7 r F s r [ t ' - t - 2 r i ( f )/c])dt'} (42) where a(t,rj) i s the f i n a l compressed point target image. The procedures may be summarized as follows : 1. Apply an azimuth Fourier transform (approximated by an FFT) to the range compressed point target return to get H R C ( t , f ) . 2. Interpolate in range time with a sine function matched to the range sampling frequency in order to extract the energy at the range defined by the RCM curve. In practice the RCMC interpolation i s performed as a range time convolution with a short windowed sine function to minimize the number of computations. A Kaiser-Bessel window (^=2.5) i s used in the simulations. 3. Multiply by the complex conjugate of the azimuth frequency domain reference phase function and the azimuth sidelobe control window. Rather than using the approximate reference function above, a closer approximation i s formed by computing the FFT of a discrete time domain reference phase function of unity magnitude. This i s the approach used in the simulations. 42 4. Apply an inverse azimuth Fourier transform (approximated by an inverse FFT) to obtain the f i n a l compressed image. 3.5 SIMULATION OF SINGLE-LOOK AZIMUTH COMPRESSION This section develops a simulation model of single-look azimuth compression for the basic range/Doppler algorithm. The steps involved in generating and compressing a simulated 2-D range compressed azimuth signal are described. The main steps of the simulation are : 1. generation of the azimuth time domain phase function 2. generation of a frequency domain, single-look, reference phase f i l t e r 3. simulation of azimuth weighting due to the azimuth antenna function and RCM 4. azimuth FFT 5. frequency domain RCMC 6. m u l t i p l i c a t i o n by the frequency domain azimuth reference phase f i l t e r and window 7. azimuth interpolation performed by zero-padding in the frequency domain 43 8. inverse azimuth FFT to produce a time domain point target image The azimuth time domain phase function can be expressed in discrete azimuth time as p(n) = exp[jtf(n)] , 1 < n < N (43) <Mn) = -(4ir/X) r( TJ C + ( 2 n-N - 1)T A / 2 ) (44) where T A i s the azimuth sampling period, n i s the azimuth time index, and N i s the length of the azimuth FFT (a power of 2 ) . For convenience, the beam center time, TJ^,, i s placed at the center of the FFT at n = (N+l ) /2 . The hyperbolic RCM equation given in equation (7) i s used throughout the simulations instead of the Taylor series approximation. A single-look, azimuth reference phase f i l t e r in the azimuth frequency domain i s formed by computing the FFT of it p(n) and taking i t s complex conjugate to get P (k), 1 < k < N. Each azimuth l i n e of the discrete 2-D range compressed signal, hR(,(m,n), i s created by applying two forms of weighting to p(n). The f i r s t form of weighting i s the azimuth antenna function, w (n). In actual SAR's, the a azimuth time width of the antenna function varies slowly with squint angle and becomes s l i g h t l y asymmetrical. Since t h i s complicates the geometric model and introduces small 44 variations which are not of interest here, the antenna function w i l l be assumed to be constant in time width and shape for the squint angles considered. The two-way azimuth antenna function i s approximated by a sincMx) type^ function, as produced by a uniform, continuous aperture antenna, and i s defined in discrete azimuth time as w a(n) = sinc 2[irDv b(2n-N-1 )T A/(2Xr 0) ] , 1 < n < N (45) where D i s the azimuth antenna length and the beam center time occurs at n = (N+l)/2. The two-way -6 dB width of the antenna function i s The second form of weighting i s due to RCM. The weighting i s applied by determining the range distance between each azimuth sample and the range migration curve and selecting the nearest amplitude from the interpolated, range compressed p r o f i l e . The distance between the range migration curve and the desired azimuth l i n e in uninterpolated range samples i s computed as A r j 6 d B = 0.884 Xr 0/(Dv f a) (46) d(m,n) = m - 1 - (2/cT) [ r ( T J C + [ 2n-N-1 ]T A/2) - r m i n ] , 1 < n < N 1 < m < M max (47) where m i s the range c e l l index, and M max i s the number of 45 azimuth l i n e s being generated. The positioning of the azimuth l i n e s in range time i s a r b i t r a r y since i t depends only on the phase of the range sampling clock. Therefore the azimuth l i n e s are a r b i t r a r i l y positioned so that the nearest l i n e (m=l) corresponds to range time 2 r m ^ n / c , and the farthest l i n e (m=M_= ) corresponds to range time 2 r m = / c . These ranges, r . and r , are the ranges of the nearest 3 mm max 3 and farthest azimuth l i n e s required for processing. They are determined by the length of the interpolator, the amount of RCM over the processed bandwidth, and the number of desired azimuth l i n e s in the output image. In order to retrieve the nearest sample from the interpolated range compressed array, hR^.p(m'), the distance d(m,n) must be converted to an interpolated index as m'(m,n) = round[ I d(m,n) ] (48) where the function round[x] rounds x to the nearest integer. Combining the antenna and RCM weightings, the discrete 2-D range compressed signal can be expressed as h(m,n) = w a(n) p(n) h R C p ( i r i j (m,n) ) (49) Once a l l the required azimuth lines are generated, each l i n e i s transformed to the frequency domain with an azimuth FFT of length N to get the range compressed frequency domain signal, H(m,k) , 1 < k < N. 46 As with the range FFT in the previous section, the azimuth FFT must be long enough to produce the desired number of v a l i d compressed samples. However, since the azimuth time domain signal does not f a l l abruptly to zero due to the s i n e 2 form of the antenna function, an a r b i t r a r y processing time interval containing most of the signal energy must be chosen. Denoting the processing i n t e r v a l as 7 j p B W , and the desired number of compressed azimuth samples after convolution as R, the FFT length must s a t i s f y N ^ * ? p B W / T A + R " 1 { 5 0 ) to prevent wraparound errors due to the c i r c u l a r convolution. For the current simulations, the processing in t e r v a l i s set equal to the two-way -6 dB antenna width. Before the azimuth reference phase f i l t e r can be applied to H(m,k), i t i s necessary to correct for RCM in the azimuth-frequency, range-time domain using a range interpolator. This correction, RCMC, e f f e c t i v e l y straightens the range migration curve so that the matched f i l t e r need only be applied to a single azimuth l i n e to produce a single azimuth l i n e of the f i n a l image. The ideal range interpolator for a discrete range signal i s a sine function which i s range time alias e d according to the length of the range compression FFT. In order to reduce the length of the interpolator and thereby reduce the number of computations, a f i n i t e length 47 approximation i s usually used. Also, rather than generating a d i f f e r e n t set of interpolator c o e f f i c i e n t s for each azimuth time, several shifted versions of the interpolator are precomputed for a set of equally spaced f r a c t i o n a l range sample s h i f t s and the nearest closest version i s used. In the current simulations, the approximate interpolator i s formed by applying a discrete Kaiser-Bessel window to a sine function to get h j d r q ) = sinc [7r(q+lL)T] w :(q+lL) , -(L/2) < 1 < (L/2) , 0 < q < Q-1 (51) W j ( i ) = / 0 { ^ I [ l - ( 2 i / Q L ) 2 ] l / 2 } / /ot/Jj} , -(QL/2) < i < (QL/2) (52) where Q i s the number of f r a c t i o n a l l y shifted versions of the interpolator, q denotes the f r a c t i o n a l s h i f t , L i s the length of each s h i f t e d version, 1 i s the index within each shi f t e d version, and i s the window weighting factor. These parameters were a r b i t r a r i l y chosen to be Q = 16, 0j = 2.5, and L = 4, 8, 16, or 32. To extract the peak energy at each azimuth sample time for a given point target with a range of closest approch, r 0 , the interpolator peak i s shifted in range so that i t s peak coincides with the frequency domain RCM curve defined in equation (38). This s h i f t i s implemented in two steps. F i r s t the interpolator i s moved an integer number of samples 48 so that i t s peak i s less than one sample away from the RCM curve. Secondly, one of the Q interpolator versions i s chosen such that the chosen.interpolator version has i t s peak closest to the actual position of the range migration curve. This e f f e c t i v e l y performs a f r a c t i o n a l s h i f t of the interpolator. The range l i n e and the interpolator are then mult i p l i e d to complete the approximate interpolator convolut ion. RCMC and azimuth reference phase m u l t i p l i c a t i o n are performed only over the processed bandwidth, f p B W r which corresponds to the processing i n t e r v a l , *?pBW^ This bandwidth i s centered at the beam center frequency, f r , given by The processed bandwidth i s computed from the azimuth processing i n t e r v a l , r ? p B W r with the assumption that phase terms higher than the quadratic are small over the processing i n t e r v a l , as f c = f.<„c> (53) f PBW PBW (54) K a ( T ? ) = - ( 2 / X ) r"(r?) = - ( 2 v * / [Xr(rj)]) [ 1 - (v » / r (rj) ) 2 ] (55) (56) where K. i s the azimuth li n e a r FM rate which i s A approximately constant over the processing i n t e r v a l . Since a 49 discrete azimuth signal i s used, the processed band i s aliased by the PRF. Figure 3.4 shows the form of the range compressed signal after the azimuth FFT including the spectrum a l i a s i n g which is caused by azimuth sampling. RCMC straightens the point target range migration curve into a single azimuth frequency l i n e . This l i n e i s compressed by m u l t i p l i c a t i o n with the frequency domain, azimuth reference phase f i l t e r and a sidelobe reduction * window. As stated previously, the f i l t e r i s P (k). The azimuth window i s computed over the processed bandwidth and set to zero outside. A Kaiser-Bessel azimuth window containing N p B W = f p B WN/(PRF) nonzero samples i s computed over frequency indices k = 1 to N as : WA(k) - / 0{/3 A[l-(2[k-1 ]/N) 2]} / / O(0 A} , 1 < k < N p B W/2 +1 70{/3A[ 1-(2[k-1-N]/N) 2]} / 7 0{j3 A} , N-NpBW/2+2 < k < N 0 , otherwise (57) Before m u l t i p l i c a t i o n , the window i s c i r c u l a r l y s h i f t e d modulo N so that the peak of the window function i s at frequency sample k^ ., which i s the nearest sample equal to or less than the ali a s e d beam center frequency. 50 F i g u r e 3 . 4 . R a n g e c e l l m i g r a t i o n c u r v e i n d i s c r e t e a z i m u t h f r e q u e n c y d o m a i n . 51 A s i n g l e l i n e o f t h e f i n a l c o m p r e s s e d i m a g e i s p r o d u c e d b y a p p l y i n g a n i n v e r s e F F T o f l e n g t h N t o t h e s p e c t r u m . T h i s l i n e c o n t a i n s t h e c o m p r e s s e d r e t u r n s f r o m t a r g e t s w i t h t h e s a m e r a n g e o f c l o s e s t a p p r o a c h , r 0 , b u t d i f f e r e n t a z i m u t h p o s i t i o n s . T h e p r o c e s s o f RCMC, a z i m u t h r e f e r e n c e p h a s e m u l t i p l i c a t i o n a n d w i n d o w i n g i s r e p e a t e d f o r e a c h o f t h e d e s i r e d o u t p u t r a n g e c e l l s . I n t h e o r y , a d i f f e r e n t RCM c u r v e w i t h d i f f e r e n t v a l u e o f r 0 s h o u l d b e u s e d f o r e a c h r a n g e . H o w e v e r , i f t h e r a n g e e x t e n t i s s m a l l , a s i n t h i s c a s e w h e r e we a r e o n l y i n t e r e s t e d i n t h e i m m e d i a t e r e g i o n o f a p o i n t t a r g e t r e s p o n s e , t h e same r a n g e m i g r a t i o n c u r v e c a n b e u s e d f o r a l l l i n e s b y s h i f t i n g i n r a n g e b y a n a p p r o p r i a t e n u m b e r o f r a n g e c e l l s . When l a r g e r r e g i o n s a r e p r o c e s s e d , a c a r e f u l a n a l y s i s i s r e q u i r e d t o d e t e r m i n e t h e r a n g e i n v a r i a n c e r e g i o n w h i c h i s t h e d i s t a n c e i n r a n g e o v e r w h i c h t h e c o m p r e s s i o n f i l t e r s d o n o t v a r y a p p r e c i a b l y . I n p r a c t i c e , t h e e n t i r e i n v a r i a n c e r e g i o n i s p r o c e s s e d a s a b l o c k t o i n c r e a s e e f f i c i e n c y . T h e i s s u e o f r a n g e i n v a r i a n c e o f t h e a z i m u t h r e f e r e n c e p h a s e f u n c t i o n h a s b e e n e x a m i n e d i n o t h e r r e p o r t s . C h a p t e r 7 e x a m i n e s r a n g e i n v a r i a n c e f o r t h e new SRC f i l t e r f u n c t i o n . F i n a l l y , i t i s d e s i r a b l e t o i n t e r p o l a t e t h e c o m p r e s s e d a z i m u t h s i g n a l f o r t h e p u r p o s e s o f i m a g e q u a l i t y m e a s u r e m e n t a n d t o d e c r e a s e t h e l o s s o f i n f o r m a t i o n i n t h e s u b s e q u e n t d e t e c t i o n o f t h e c o m p l e x s i g n a l . T h e m e t h o d u s e d i s z e r o p a d d i n g i n t h e a z i m u t h f r e q u e n c y d o m a i n b e f o r e t h e i n v e r s e 52 FFT. F i r s t the spectrum i s c i r c u l a r l y s h i f t e d so that the beam center frequency l i e s nearest to the f i r s t FFT array sample. The array i s then padded with zeros in the middle to form a length NJ array where J i s the interpolation factor. By applying an inverse FFT of length NJ, the desired interpolated signal i s obtained. 3.6 SIMULATION RESULTS OF BASIC RANGE/DOPPLER COMPRESSION This section presents and discusses the resu l t s of quality measurements of simulated point target responses which were produced for a range of squint angles. The simulation programs were implemented on an Amdahl 470 V/8 computer in RATFOR (ra t i o n a l FORTRAN) under the MTS (Michigan Terminal System) operating system. RATFOR i s a structured precompiler which produces FORTRAN code. The f i r s t step in the simulation was the production of a range compressed p r o f i l e as in figure 3.1. The second step in the simulation was the production of a simulated range compressed azimuth sig n a l . Figure 3.5 shows the azimuth time domain magnitudes of the simulated antenna weighting, and the antenna plus RCM weightings for a squint angle of 5.0°. The magnitudes are shown for the range c e l l closest to the beam center range. The azimuth spectrum produced by the azimuth FFT i s shown in figure 3.6. Before RCMC the azimuth bandwidth i s quite small. The f i n i t e length interpolator used for RCMC i s shown in figure 3.7. The figure shows the length 16 53 i n t e r p o l a t o r and i n c l u d e s a l l 16 f r a c t i o n a l l y s h i f t e d v e r s i o n s . F i g u r e 3.8 shows the c o r r e c t e d spectrum a f t e r RCMC i n which the processed bandwidth i s c l e a r l y seen. The azimuth r e f e r e n c e phase f i l t e r spectrum f o r a 5.0° squ i n t angle i s shown i n f i g u r e 3.9. The f i l t e r i s generated i n the azimuth time domain without weighting and then transformed with an azimuth FFT. The azimuth K a i s e r - B e s s e l window with 0 A = 1.5 i s shown i n f i g u r e 3.10. The point t a r g e t azimuth spectrum a f t e r RCMC, azimuth r e f e r e n c e phase f i l t e r i n g , and windowing i s shown i n f i g u r e 3.11. The spectrum r i p p l e s are c h a r a c t e r i s t i c of the azimuth r e f e r e n c e phase f i l t e r spectrum. Upon a p p l i c a t i o n of an i n v e r s e FFT, the f i n a l compressed p o i n t t a r g e t response i s produced. Sample 1-D c r o s s - s e c t i o n s of the compressed response are shown i n f i g u r e s 3.12 and 3.13 f o r squ i n t angles of 0° and 5°. The azimuth window f a c t o r was chosen to produce comparable s i d e l o b e l e v e l s i n the range and azimuth d i r e c t i o n s f o r small s q u i n t angles as would be done i n a p r a c t i c a l system. F i g u r e 3.12 shows that severe range broadening occurs f o r a sq u i n t angle of 5.0°. The range and azimuth broadening are summarized i n f i g u r e s 3.14 to 3.16 f o r a range of squ i n t a n g l e s . Range broadening i n c r e a s e s r a p i d l y f o r squint angles above 4° with 5% and 10% broadening o c c u r i n g at about 3.65° and 4.23° r e s p e c t i v e l y ( f o r L = 16). However, azimuth broadening i s r e l a t i v e l y i n s i g n i f i c a n t remaining below 2% f o r s q u i n t angles up to 6°. 54 The 1-D and 2-D ISLR measurements are summarized in figures 3.17 to 3.19. It shows that both ratios increase s i g n i f i c a n t l y as the squint angle increases indicating that the point target response not only becomes broader in range, but also becomes f l a t t e r spreading more energy into the sidelobes. The ISLR measurement i s seen to be l i m i t e d to less than 5° since a large amount of the sidelobe energy l i e s outside of the f i n i t e integration region for larger squint angles. This causes the ISLR to drop sharply above 5°. Another r a t i o of i n t e r e s t , the PSLR, i s summarized in figures 3.20 to 3.22. This shows that the peak range sidelobe i s usually the peak 2-D sidelobe as well. The discrepancy at small angles between the three figures is due to the higher interpolation factor used in measuring the 1-D PSLR. At some of the higher angles, the range PSLR dips well below the 2-D curve. This occurs since the closer range sidelobes merge with the mainlobe causing sidelobes further out to be measured as the peak sidelobe. This behaviour can be seen in figure 3.12. The f i n a l measurement of interest i s the degradation of the peak magnitude which i s plotted in figure 3.23. This measurement was normalized to the sum of squares of the interpolator c o e f f i c i e n t s and the azimuth processed bandwidth as would be appropriate for a white noise model with nqise equally d i s t r i b u t e d over the range and azimuth c e l l s . Since the actual noise d i s t r i b u t i o n may be somewhat 55 d i f f e r e n t , care should be taken in r e l a t i n g the peak magnitude degradation to changes in signal-to-noise r a t i o . The figure shows a reduction in peak magnitude with increasing squint angle caused by poor compression. At the 5% and 10% range broadening squint angles, the degradations are approximately 0.47dB and 0.83dB respectively. The degradation r i s e s rapidly above t h i s . 56 (8P) ftpnjiufiDN Figure 3.5. Simulated azimuth antenna weighting, and antenna plus RCM weightings in range c e l l nearest to beam center range for 5° squint. 57 (SP) •pnuufiDN Figure 3 .6. Azimuth spectrum before RCMC in range c e l l nearest to beam center range for 5° squint. Figure 3.7. Windowed range interpolator of length 16 including 16 f r a c t i o n a l l y s h i f t e d versions. iQ C * Azimuth Spectrum after RCMC i** squint=5.0 cleg , betqa=1.5 , betar=2.7 3 0 0.2 0.4 0.6 0.8 1 Frequency normalized to PRF Azimuth Matched Filter Spectrum squint=5.0 deg - . . . . , . , , . Illin iii i irtH 1 1 r-- C V 1'" •" T - T "' "1 " i 0 0.2 0.4 0.6 0.8 1 Frequency normalized to PRF Azimuth Kaiser-Bessel Window squfnt=5.0 deg . betaq=1.5 c 0.3 0.2 0.1 '• :  0 H H 1 1 1 1 1 1 1 h 0 0.2 0.4 0.6 0.8 1 Frequency normalized to PRF 62 ( 8 P ) •prujuCDN F i g u r e 3 . 1 1 . A z i m u t h s p e c t r u m a f t e r RCMC, matched f i l t e r , and w i n d o w i n g . 63 ( g p ) • p r u j u f i D H igure 3.12. F u l l y compressed range p r o f i l e without SRC for 0° and 5° squint using a length 16 RCMC interpolator. 64 (8P) •pnimfiDW Figure 3.13. F u l l y compressed azimuth p r o f i l e without SRC for 0° and 5° squint using a length 16 RCMC interpolator. 65 1 1 1 1 1 1 1 1 1 T" ° O O O O O O Q O O O O 6uiu»pDOjq »6uDg % Figure 3.14. Range broadening of point target response withdut SRC for various RCMC interpolator lengths. 66 M II c 3 or </>«, II fiujudpciojq efiuDH % Figure 3.15. Range broadening of point target response (expanded) without SRC for various RCMC interpolator lengths. 67 fiuiuopDojq mnuijzv % Figure 3.16. Azimuth broadening of point target response without SRC for various' RCMC interpolator lengths. 68 Figure 3.17. 1-D range integrated sidelobe r a t i o s without SRC for various RCMC interpolator lengths. 69 CO o M II - * CO ro g o 2 3 tr - CM N I II CO 7 •n to T in in o" N I I in N I M I (BP) tnsi Figure 3.18. 1-D azimuth integrated sidelobe r a t i o s without SRC for various RCMC interpolator lengths. 70 Figure 3.19. 2-D integrated sidelobe r a t i o s without SRC for various RCMC interpolator lengths. 71 « - T - N < N C N < N N C M N N N N > 0 I I I I I I I I I I I I I (ap) aisd Figure 3.20. 1-D range peak sidelobe r a t i o s without SRC for various RCMC interpolator lengths. 72 tf CO -p o tf CO -p N < ft I <0 M II <0 g o c 3 II - cs _ l o M I in o M I M I in cs I CS I in cs cs I K) cs I in *n cs I cs I (SP) tnsd Figure 3.21. 1-D azimuth peak sidelobe r a t i o s without SRC for various RCMC interpolator lengths. 73 Figure 3.22. 2-D peak sidelobe r a t i o s without SRC for various RCMC interpolator lengths. 74 Figure 3.23. Degradation of peak magnitude without SRC for various RCMC interpolator lengths. 4. ANALYSIS OF BROADENING IN RANGE/DOPPLER COMPRESSION This section presents a theory which characterizes the broadening, primarily in range, which occurs at high squint angles in basic range/Doppler processing without SRC. The theory extends the range broadening model of J i n and Wu [13] to include the ef f e c t s of the range sidelobe control window. The p r i n c i p l e of stationary phase, which was used in the previous chapter to relate the azimuth spectrum to the azimuth time domain signal, i s replaced by an approximate Fourier transform which accounts for the spectrum broadening which occurs in low TBP linear FM signals. Measurements of the simulated azimuth TBP and spectral broadening in both the azimuth frequency and range time directions are presented. Azimuth broadening i s shown to be accurately predicted by the decrease in processed bandwidth with increasing squint angle, a qual i t y which i s inherent to the simulation models. L i t t l e , i f any, azimuth broadening i s caused by the azimuth spectral broadening. 4.1 BROADENING MODEL FOR RANGE/DOPPLER COMPRESSION WITHOUT  SRC The previous chapter presented a theory of range and azimuth compression for the basic range/Doppler algorithm. It was shown that the 2-D range compressed signal can be expressed as : 75 76 h R C ( t f „ ) = h A ( t f , ) * t h R C p ( t ) (58) where h A(t,rj) = w (7?) exp[-j47rr ( T ? ) / X ] 6[t-2r (T?)/C ] (59) and h R C p ( t ) is the 1-D range compressed p r o f i l e which i s usually similar in shape to a sine function. The f i r s t step after range compression in basic range/Doppler azimuth compression i s the computation of the azimuth fast Fourier transform (FFT) of the range compressed s i g n a l . In continuous-time theory t h i s i s replaced by a continuous azimuth Fourier transform. In general, the azimuth Fourier transform of the range compressed signal cannot be represented exactly in closed form, or even in a separable form. A major approximation i s now made which allows the azimuth spectrum to be expressed in a form which r e s t r i c t s the range-azimuth coupling to a delta l i n e function as in the azimuth time domain signal in equation (58). A similar approximation was o r i g i n a l l y presented in [13] but i t s v a l i d i t y was not f u l l y discussed. The convolution in range is approximated by a convolution in azimuth as : 77 hRC ( t' T j ) * h A ( t ' , » ) % h R C P ( C l T ? ) (60) = { WJTJ) exp[-j (47r/X)r(7j) ] 6 [ r?-r - 1 (ct/2) ] } a * h 7? RCP (C 1 T?) (61) where r _ 1 ( r ) i s the inverse function of the RCM equation, r(ri), and c, i s the slope of the RCM curve at the beam center time given by : The approximation can be viewed as a li n e a r projection of the range p r o f i l e about the RCM curve into azimuth time with the position of the peak of the p r o f i l e corrected to l i e along the RCM curve. Two major assumptions have been made : 1. The shape of the amplitude p r o f i l e in the azimuth d i r e c t i o n i s assumed to be the same as the range compressed p r o f i l e with the exception of a scaling constant. In r e a l i t y the azimuth p r o f i l e i s s l i g h t l y asymmetric due to range curvature. - - X f c / c (62) 2. The scaling constant, which i s the slope of the RCM curve, i s assumed to be constant over the azimuth processing i n t e r v a l . Actually the slope varies slowly 78 over the processing i n t e r v a l again due to range curvature. The f i r s t assumption seems reasonable when the azimuth timewidth i s small, i . e . , when the amount of range curvature over the -3dB azimuth timewidth i s much less than the range resolution. Fortunately, t h i s condition occurs at larger squint angles where spectrum broadening i s of most interest. The assumption may break down at small squint angles where l i t t l e broadening occurs. The second assumption may be more sensitive to range curvature since l i n e a r i t y i s assumed over the entire azimuth processing i n t e r v a l . For RADARSAT parameters range curvature is very small and the slope of the RCM curve does not vary appreciably over the processed azimuth aperture. However for longer wavelength SARs, the slope may vary somewhat. The chapter on multilook processing addresses t h i s assumption in more d e t a i l . Applying the azimuth Fourier transform and using the convolution theorem leads to the following equation for the range compressed spectrum : H R C ( t , f ) = W g(f) * f F{ exp[-j(47r/X)r(rj) ] } * f [ F{6[n-r-'(ct/2)]} • F { h R C p ( c , 7 J ) } ] (63) where W (f) i s the Fourier transform of the azimuth antenna 79 weighting function, w (r?). In thi s form the azimuth spectrum consists of the convolution of three functions in azimuth frequency. The second two functions w i l l be evaluated e x p l i c i t l y . The middle function i s the Fourier transform of the phase along the RCM curve. Using the quadratic approximation of equation (12), i t can be evaluated in closed form [17] as : F{ exp[-j(47r/X)r ( T 7 ) ] } * (l//|K A |) exp[ jtf(f) ] (64) where iMf) = - w t f - f , ) 2 / ^ + U/4)SIGN(K A) + * 0 (65) \jj0 = -4rrr 1/X (66) and SIGN(x) denotes the sign of x. Although the quadratic approximation to the azimuth phase i s used here, the exact transform w i l l be substituted back after extracting a broadening function. The delta function in the t h i r d term contains the range-azimuth coupling. Its transform i s a linear phase complex exponential in which the rate of change of phase i s a function of range : F{ Sin-r- 1 (ct / 2 ) ] } = exp[ -j2irfr- 1 (ct /2 ) ] (67) 80 F i n a l l y the transform of the scaled range compressed p r o f i l e including the range window i s given by : F{ h R C p ( C l T } ) } = (1/|c,|) H R C p ( f / C l ) (68) The following substitutions and approximations were not shown in the paper by J i n and Wu but are essential to understanding the broadening model. Using the above transforms the second convolution in equation (63) can be rewritten as : 1 H R C p ( f 7 c J e ^ ( f " r ) e - J 2 * f ' r - ' ( c t / 2 > d r |c,| v/|KA| ( 6 g ) = e W f > f ! H R C p ( f V C l ) e ^ V ^ | c , | • l ^ l ej2irf ' [ ( f - f ,)/K A - r - 1 ( c t / 2 ) ] flf, ( ? 0 ) The leading exponential can be recognized as the transform of the azimuth phase along the RCM curve in equation (64). Thus the exact time domain phase function w i l l be substituted back. The remaining integral defines the amplitude broadening and phase deviation along each azimuth l i n e . Using the change of variable, U, = f / K A 81 the integral can be expressed as : A , ( f ) * f 6 [ f - f i ~ K A r - 1 ( c t / 2 ) ] (72) where A,(f) - (1/|c a|) H R C p ( „ / c 2 ) e - ^ K A ^ 2 e3 2* f l»t dr, i (73) = ( l / | c 2 | ) H R C p ( i ? 1 / c 2 ) e ^V*'* } ( ? 4 ) and c 2 = c,/K A = -Xf c/(cK A) (75) As i t stands, the broadening i s expressed as a convolution in azimuth frequency which i s dependent on range. To express the broadening in terms of range time, A,(f) must be projected back into range using the approximately constant slope, c 2 , of the RCM curve in the azimuth frequency domain. This second projection of the signal model about the RCM curve was not discussed by J i n and Wu. The same conditions apply for t h i s projection as before except that the signal i s now in the azimuth frequency domain. The result i s : 82 A,(f) * f 5 [ f - f i " K A r - 1 ( c t / 2 ) ] - A ^ t / c j ) * t 6 [ t - ( 2 / c ) r ( [ f - f , ]/KA) ] (76) Since the azimuth antenna function i s a slowly varying function of azimuth time and the remainder of equation (63) is an approximately linear FM signal, i t s convolution with the above terms can be approximated by a m u l t i p l i c a t i o n [17] with a scaled version of the antenna function. The scaling i s defined by the azimuth frequency to azimuth time mapping, rj ^ ( f ) . This gives the f i n a l form of the azimuth transformed range compressed signal as : H R C ( t , f ) = w a ( r ? i ( f ) ) • F{ exp[-j(47r/X)r(7j) ] } • { A,(t/c 2) *fc 6 [ t - ( 2 / c ) r ( [ f - f , ] / K A ) ] } (77) Azimuth compression without SRC does not s i g n i f i c a n t l y a l t e r the range dispersion defined by equation (77) since azimuth compression occurs primarily p a r a l l e l to the RCM curve. Therefore the approximate range p r o f i l e after azimuth compression i s determined by A,(t/c 2) which i s the inverse Fourier transform of a weighted linear FM si g n a l . When the width of the linear FM signal i s small ( i . e . , when the phase at the -3dB points i s much less than it/2 radians), the quadratic phase exponential in equation (74) produces l i t t l e broadening of the inverse transform. This 83 occurs when c 2 and therefore also f^ and the squint angle are small. Thus for small squint a igles, the range p r o f i l e before and after the azimuth Fourier transform i s approximately the same and l i t t l e range broadening appears af t e r compression without SRC. 4.2 BROADENING SIMULATIONS AND MEASUREMENTS This section presents the results of simulating the range broadening function developed in the previous section. These are compared to measurements of the broadening of the range compressed point target response used in the previous simulations. These simulations are similar to those presented by J i n and Wu. However the range window has been added and detailed quantitative image qual i t y measurements are performed. Measurements of the azimuth TBP are presented to relate measurements of azimuth spectral broadening to the decrease in azimuth TBP. A predicted azimuth broadening curve i s shown which i s based on the decrease in processed bandwidth with increasing squint angle. The range broadening function, A , ( t / c 2 ) , provides a model for the range broadening of the azimuth spectrum which occurs in the azimuth Fourier transform. This function has been simulated with nominal RADARSAT parameters with the same range window parameter (0 R=2.7) that was used in previous simulations. The -3dB range widths have been 84 measured for various squint angles and are summarized in figure 4.1. The simulated broadening function was generated according to equation (74) with f replaced by t / c 2 and the inverse Fourier transform integral approximated by an inverse FFT. The range compressed range spectrum, H R £ p ( TJ ,/c 2), was simulated as having zero phase with a magnitude defined by the range window function. This was multiplied by the quadratic phase broadening term. The functions were generated in a length N (N=2048) discrete frequency domain array with maximum frequency equal to the range sampling rate. An inverse FFT was applied to form a discrete, range time domain, broadening function. The re s u l t i n g response width was measured with the image quality measurement programs discussed e a r l i e r . The range broadening predicted by the broadening function agrees well with the actual range broadening results shown in figure 3.14 of the previous chapter. Additional broadening occurs with smaller RCMC interpolators, e s p e c i a l l y at large squint angles, due to the interpolator windowing. Since the range broadening model is based upon predictions of azimuth spectrum broadening, the shapes of the azimuth time and frequency domain signals were examined. Figures 4.2 to 4.6 show the relationship between the azimuth timewidth of the simulated range compressed signal and i t s bandwidth aft e r the azimuth FFT. The upper graphs show the 85 1 1 1 1 Ff O O O O O O O m tn M *-Figure 4.1. Range broadening range broadening of the simulated, t h e o r e t i c a l , function without SRC. 0.1 0 -0.1 -0 .2 -0 .3 -0.4. -0 .5 -0 .6 -0 .7 -0 .8 -0 .9 -1 Azimuth Time—domain Amplitude squint — 0 deg, br — 2.7 0.1 0 -0.1 -0 .2 -0 .3 -0 .4 -0 .5 -0 .6 -0 .7 -0 .8 -0 .9 -1 ure — i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r -100 - 8 0 - 6 0 - 4 0 - 2 0 0 20 40 60 80 100 Predicted relative frequency (Hz) Azimuth Amplitude Spectrum squint = 0 deg, br = 2.7 1 11 i 1 1 1 J M r i •100 - 8 0 - 6 0 - 4 0 - 2 0 O 20 40 60 Frequency relative to beam center (Hz) 80 1 00 4.2. Predicted and actual azimuth spectra for 0' squint. Azimuth Time-domain Amplitude m •D TJ 3 C t> O 2 squint = 1 deg, br = 2.7 0 20 40 60 Predicted relative frequency (Hz) Azimuth Amplitude Spectrum m TJ >—' TJ 3 C o> O 1 0 -1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 squint = 1 deg, br = 2.7 — 10 -| 1 1 1 1 1 1 1 1 r -100 - 8 0 - 6 0 - 4 0 - 2 0 0 20 Frequency relative to beam center (Hz) Figure 4.3. Predicted and actual azimuth spectra for 1° squint. 1 oo Azimuth Time —domain Amplitude squint — 5 deg, br — 2.7 s  -100 - 8 0 - 6 0 - 40 - 2 0 0 20 40 60 80 100 Predicted relative frequency (Hz) Azimuth Amplitude Spectrum squint = 5 deg, br = 2.7 —40 -| T 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -100 - 8 0 - 6 0 - 4 0 - 2 0 O 20 40 60 80 100 Frequency relative to beam center (Hz) Figure 4 . 4 . Predicted and actual azimuth spectra for 5 ° squint. Azimuth Time-domain Amplitude m TJ • TJ 3 C t » O squint = 10 deg, br = 2.7 Predicted relative frequency (Hz) Azimuth Amplitude Spectrum m TJ TJ 3 "E D 0 - 5 -10 -15 -20 -25 -30 -35 squint — 10 deg, br — 2.7 —40 -i 1 1 1 1 1 1 1 1 r -100 - 8 0 - 6 0 - 4 0 - 2 0 Frequency relative to beam center (Hz) Figure 4.5. Predicted and actual azimuth spectra for 10° SQUin t• ffi TJ T> 3 *E CO TJ TJ 3 C t» a 2 5 0 - 5 -10 -15 -20 -25 -30 -35 Azimuth Time-domain Amplitude squint = 15 deg. br — 2.7 -40 / \ J if /i n A A A5 0 - 5 -10 -15 -20 -25 -30 -35 -40 -100 - 8 0 - 6 0 - 4 0 - 2 0 0 20 40 60 80 100 Predicted relative frequency (Hz) Azimuth Amplitude Spectrum squint = 15 deg, br = 2.7 —1 1 1 1 1 1 1 1— -100 - 8 0 - 6 0 - 40 - 2 0 -i 1 1 1 1 1 1 1 r~ 20 40 60 80 100 Frequency relative to beam center (Hz) Figure 4.6. Predicted and actual azimuth spectra for 15° squint. 91 timewidths with the time scale converted to predicted frequency using the azimuth FM rate at the beam center as follows : * j - t j " f^ - Kjrrr?-,) (78) predicted C A C The lower graphs show the actual bandwidths. The graphs are plotted for squint angles of 0, 1, 5, 10, and 15 degrees. At small squint angles (0° and 1°) the time and frequency domain signals are very s i m i l a r . At 5° s i g n i f i c a n t spectrum broadening i s evident. At 10° and 15° the broadening is so severe that the actual spectra no longer resemble the predicted spectra. The amount of range and azimuth broadening of the simulated range compressed azimuth spectrum was measured to determine the accuracy of the range broadening model. The broadening measurements are summarized in figures 4.7 to 4.10. These measurements were performed after the azimuth FFT but before RCMC. Broadening in both the azimuth frequency and range time directions was measured at three points on the RCM curve as shown in figure 4.11 : at the lower azimuth processed bandwidth frequency (the far range c e l l ) ; at the beam center frequency (the range c e l l nearest the beam center range); and at the upper azimuth processed bandwidth frequency (the near range c e l l ) . The spectrum broadening measurements in the azimuth frequency di r e c t i o n at low squint angles are inaccurate due F i g u r e 4 . 7 . M e a s u r e d a z i m u t h s p e c t r u m b r o a d e n i n g i n t h e a z i m u t h f r e q u e n c y d i r e c t i o n i n t h e n e a r , b e a m c e n t e r ( m i d ) , a n d f a r r a n g e c e l l s . 93 Figure 4.8. Measured azimuth spectrum broadening in the azimuth frequency d i r e c t i o n in the near, beam center (mid), and far range c e l l s (expanded). F i g u r e 4 . 9 . M e a s u r e d a z i m u t h s p e c t r u m b r o a d e n i n g i n t h e r a n g e t i m e d i r e c t i o n a t t h e l o w e r , t h e b e a m c e n t e r , a n d t h e u p p e r p r o c e s s e d b a n d w i d t h f r e q u e n c i e s . 95 Figure 4.10. Measured azimuth spectrum broadening in the range time d i r e c t i o n a; the lower, the beam center, and the upper processed bandwidth frequencies (expanded). 96 Figure 4 . 11 . Points on the azimuth frequency domain RCM curve used for spectrum broadening measurements. 97 to the large spectrum r i p p l e s . At larger squint angles the azimuth spectrum i s smoother allowing more accurate measurements. The broadening measurements agree well with the actual broadening re s u l t s with large RCMC interpolators and the range broadening predicted by the broadening model. The broadening i s somewhat greater at the low frequency (far range) end of the RCM curve and s l i g h t l y smaller at the high frequency (near range) end. This i s due to the small change in RCM slope over the processed bandwidth caused by range curvature. As the squint angle increases, the slope of the RCM curve in the beam center range c e l l increases causing a decrease in the azimuth time width and TBP. For small TBP's, the shape of the azimuth spectrum after the azimuth FFT broadens and no longer c l o s e l y resembles the azimuth time domain si g n a l . Azimuth -3dB time width measurements of the range compressed signal were performed in order to calculate the azimuth time-bandwidth products (TBP) in the three range c e l l s noted above as a function of squint angle. The resul t i n g curves, figure 4.12, can be used to relate range broadening to the azimuth TBP. From a lin e a r interpolation of the test points, the beam center azimuth TBP's corresponding to 5% and 10% range broadening are 0.76 and 0.57 respectively. Azimuth broadening i s much smaller than range broadening. The azimuth broadening which does occur can be att r i b u t e d to the decrease in azimuth processed bandwidth Measured Azimuth TBP's before SRC 99 with increasing squint angle which i s inherent to the simulation model of the azimuth antenna function. Since the azimuth -6dB (two-way) antenna time width was kept constant over changes in squint angle whereas the azimuth frequency rate varied, the processed bandwidth calculated by equation (54) decreased with increasing squint angle. Since the azimuth resolution i s inversely proportional to the processed bandwidth, the percentage azimuth broadening can be predicted by c a l c u l a t i n g the percentage decrease in processed bandwidth as shown in figure 4.13. The predicted azimuth broadening agrees very c l o s e l y with the measured r e s u l t s . Very l i t t l e azimuth broadening i s caused by the spectrum broadening which causes range broadening since the d i s t o r t i o n of the azimuth phase spectrum under low azimuth TBP conditions i s very small over the -3dB azimuth bandwidth. Since azimuth compression i s mainly a function of the phase spectrum, l i t t l e azimuth broadening occurs. F i n a l l y , figure 4.14 shows how the t o t a l amount of RCM over the processed aperture varies with squint angle for the given set of RADARSAT parameters. RCM increases almost l i n e a r l y with squint angle as i s expected for a predominantly quadratic curve. 100 CuiuftpDOjq mnujizv % Figure 4.13. Azimuth broadening predicted by decrease in azimuth processed bandwidth. Range Cell Migration (RCM) versus Squint Angle T 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1— 2 4 6 8 10 12 14 16 18 20 Squint angle (deg) 5. SECONDARY RANGE COMPRESSION (SRC) This chapter introduces a new secondary range compression (SRC) algorithm which compensates for the range broadening which occurs in the azimuth FFT and which i s not accounted for in the basic range/Doppler algorithm. A mathematical theory for the SRC algorithm i s developed d i r e c t l y from the point target response model in chapter 3. Two new discrete implementations are developed : azimuth SRC and range SRC. It i s shown with simulations that the SRC algorithms provide excellent recompression of the energy which i s spread by the azimuth FFT at large squint angles (small azimuth TBP's) for nominal RADARSAT parameters. 5.1 THEORY OF AZIMUTH MATCHED FILTERING AND SRC This section extends the theory of azimuth compression developed in chapter 3 to show how SRC can be used to recompress the energy which is dispersed by the azimuth Fourier transform. The ideal matched f i l t e r for a point target signal i s : h*(-t,-rj) = h*(-t , - 7 ? ) * h*(-t,-rj) (79) Chapter 3 described the theory of range compression in which the range compressed signal h R C ( t , 7 ? ) i s formed by convolving * with the range matched f i l t e r h R ( - t , - T ? ) and a range window used to control sidelobes. 102 103 Azimuth compression consists of convolving the range compressed signal with an azimuth reference phase f i l t e r , * h p ( - t , - 7 ? ) , and an azimuth window to control sidelobes. The id e a l i z e d azimuth reference phase function was given previously in equation (32) as : Its azimuth Fourier transform can be evaluated using the Fourier transforms from chapter 4 : h F(t,7j) = exp[-j47rr (T?)/X] 6[ t-2r ( T?)/C ] (80) H p ( t , f ) = F{exp[-j47rr (T?)/X]} * F F{ 6 [ t-2r (r?) /c ]} (81 ) J<Mf> * -j27rfr- 1 (ct/2) (82) f V/|KA| 00 eJ<Mf-f) e-j27rf'r- 1 (ct/2) df' (83) — oo = e j ^ ( f ) r " e-3 ( f f/ KA ) f' 2 e J 2 j r f ' [ ( f - f ' ) / K A - r - 1 ( c t / 2 ) ] df' (84) Using the same change of variables as before, 7j1 = f'/K A, and substituting back the exact Fourier transform of the azimuth phase, the f i l t e r becomes : 104 H p ( t , f ) = F{ exp[-j47rr(T?)/X] } • { A 2 ( f ) * f 6 [ f - f i - K A r " 1 ( c t / 2 ) ] } (85) where A 2 ( f ) = |KA| e 2 e J 2 7 r f T } , d r j . (86) = F" 1{ |KA| e~i*KhV'2 } (87) = •IK.I e ^ / V f 2 e - J ( » / 4 ) S I G N ( K A ) (88) The azimuth spectrum broadening function A 2 ( f ) i s similar in form to A ^ f ) of chapter 4 except that no range windowing i s applied. The reference phase f i l t e r defined by equation (85) could be applied to the range compressed response as a convolution in the azimuth frequency d i r e c t i o n . However, since the width of the azimuth spectrum i s usually larger in the azimuth dir e c t i o n than in the range time d i r e c t i o n (in terms of the number of samples), i t i s more e f f i c i e n t to apply the f i l t e r in the range time d i r e c t i o n . This i s accomplished by projecting the f i l t e r into range time using the approximately linear slope, c 2 , of the azimuth frequency domain RCM curve : 1 0 5 H p ( t , f ) F{ exp[-j47rr(Tj)/X] } • { A 2 ( t / c 2 ) * t 6 [ t - ( 2 / c ) r ( [ f - f , ] / K A ) ] } (89) This projection uses the same assumption of a linear RCM curve which was used in chapter 4. This f i l t e r can now be applied to the range compressed spectrum along with an azimuth window function W^f-f^) to perform azimuth compression with SRC. The peak of the window function i s centered at the beam center frequency, f c , for proper windowing. This produces the following form for the azimuth compressed frequency domain signal : a(t,rj) = F-'{ [ H R C ( t , f ) * t H F ( - t , f ) ] W A ( f - f c ) } (90) = F- 1 { [ H R C ( t , f ) * t g c ( t , f ) ]F*{exp[-j47rr(Tj)/X]}W A(f-f c)} (91 ) where g c ( t , f ) = g(t) * t 6 [ t + ( 2 / c ) r ( [ f - f , ] / K A ) ] (92) = g( t + ( 2 / c ) r ( [ f - f , ] / K A ) ) (93) g(t) = k*2(-t/c2) (94) = V/|KA| eJ(^/4)SIGN(K A) e - J 7 r K S R C t 2 (95) 106 KSRC = l / ( K A C * } = KA [ c / ( X f C ) P ( 9 6 ) Equation (91) describes one form of SRC algorithm in which the SRC f i l t e r , g ( t ) , i s applied during azimuth compression as a range time convolution. Instead of applying the SRC f i l t e r , g ( t ) , separately as suggested by J i n and Wu [13], i t can be combined with the RCMC interpolator to form a combined SRC/RCMC f i l t e r , g c ( t , f ) . This type of implementation w i l l be c a l l e d azimuth SRC. Alte r n a t i v e l y the SRC f i l t e r can be implemented during range compression in the range frequency domain. This type of algorithm w i l l be c a l l e d range SRC. If the transmitted range pulse i s linear FM, the SRC f i l t e r can be combined with the range compression matched f i l t e r by simply modifying the linear FM rate of the f i l t e r . Both of these implementations w i l l be examined in following sections. 5.2 AZIMUTH SRC This section describes the azimuth SRC implementation. Equation (91) describes the basic form of the azimuth SRC azimuth compression algorithm. After transformation into the azimuth frequency domain, the 2-D range compressed signal, H R C ( t , f ) , i s convolved in range with a combined SRC/RCMC f i l t e r , g c ( t , f ) , which is azimuth frequency dependent. The result i s a 1-D azimuth sig n a l . This i s then mult i p l i e d by the azimuth reference phase f i l t e r and the azimuth window. Upon inverse transformation with an inverse azimuth FFT, a 107 single compressed azimuth l i n e i s obtained. The SRC/RCMC f i l t e r , g c ( t , f ) , i s formed by the convolution of two components : 1. the SRC f i l t e r , g ( t ) , which compresses the dispersion caused by the azimuth Fourier transform in both range and azimuth. 2. a range-azimuth coupled delta function which extracts energy along the RCM curve, i . e . , performs RCMC. Since the range compressed signal exists only for discrete range and azimuth time, discrete SRC and RCMC f i l t e r s are required. These can be formed by bandlimiting the ideal continuous f i l t e r s to the range and azimuth sampling rates. There are several ways of implementing the f i l t e r s . For SRC processing in the azimuth frequency domain, i t i s most e f f i c i e n t to combine the SRC f i l t e r and RCMC interpolator. This combined f i l t e r , g^,(t,f), i s the same as the 1-D SRC f i l t e r , g ( t ) , but i s shi f t e d in range by an amount which varies with azimuth frequency. Although azimuth frequency i s discrete, the range time s h i f t required for the SRC/RCMC f i l t e r can take on any continous value due to the coupling between range and azimuth. To avoid creating a new shifte.d f i l t e r version for each discrete azimuth frequency, the continuous s h i f t may be 108 approximated by an integer s h i f t and a f r a c t i o n a l s h i f t as was done with the RCMC interpolator in chapter 3. Integer range sample s h i f t s are handled by s h i f t i n g the entire f i l t e r the required number of samples. Fractional s h i f t s are approximated by choosing the best of several precomputed versions of the f i l t e r each of which i s sh i f t e d by a frac t i o n of a range sample. The sh i f t e d versions are produced by interpolating the f i l t e r by an integer factor I and then extracting the di f f e r e n t phases. This i s the same approximation that was previously used to form the basic RCMC interpolator except that the combined f i l t e r now has a small quadratic phase and i s broadened in amplitude. Several steps are required to produce a suitable discrete SRC/RCMC f i l t e r : 1. Form an a n a l y t i c a l SRC f i l t e r in the continuous range frequency domain. 2. Bandlimit the function to the range sampling frequency, F , to prevent a l i a s i n g . G ( f r ) = F{ g(t) } (97) = (Xf c/c) e ^ f r / K SRC (98) rect( f / F c r ) G(f ) (99) 109 3. Sample the continuous range frequency function with sample spacing 1/T = F /K to get K samples. T i s one period of the corresponding range time domain function. It must be chosen to be much larger than the -3dB range timewidth of H R^.(t,f) to prevent serious time domain a l i a s i n g . G(k r) = r e c t ( k r / K r ) GU r/T) , -Kf/2 < k r < Kr/2 (100) 4. Zero pad the array on both ends to a length of IK r where I i s the interpolation factor which defines the number of f r a c t i o n a l l y s h i f t e d versions of the f i l t e r . G j(k r) = G(k r) , -Kr/2 < k r < Kr/2 0 , -IK r/2 < k r < -Kr/2 , Kr/2 < k r < IK r/2 (101) 5. Apply an inverse range FFT of length IK . gjdnj) = FFT-H G(k r) } , -IK r/2 < m1 < (102) 6. Multiply by a length IL window, w , ^ ) , to get a f i l t e r of minimum length where L i s the length of each f i l t e r version. g J(m I) w^nij) -IL/2 < mx < IL/2 (103) 110 7. Extract the f r a c t i o n a l l y s h i f t e d versions of the f i l t e r . g^m) = gjdnl + i) w,(ml + i) , 0 < i < 1-1 , -L/2 < m < L/2 (104) In the current simulations, 16 versions (1=16) and four d i f f e r e n t f i l t e r lengths (L = 4, 8, 16, or 32) are used. Figure 5.1 shows the magnitudes of several SRC/RCMC f i l t e r s of length 16 after interpolation by a factor of 16 for several squint angles. It i s seen that the f i l t e r s resemble the sine type of interpolator for small angles but broaden for larger angles. Consequently, larger squint angles also require longer f i l t e r s to gather a l l the dispersed energy. In basic range/Doppler processing without SRC, g^(m), is approximated by a zero phase f i n i t e length interpolator which corresponds to the zero squint SRC/RCMC f i l t e r . The approximation holds for small squint angles since the nonlinear phase v a r i a t i o n of G(k r) approaches zero as | f ^ | and the squint angle approach zero. Consequently, G(k r) approaches a rectangular signal with constant phase and gj(mj) approaches a time a l i a s e d sine function (or sampling function). The RCMC interpolator i s shortened by mu l t i p l i c a t i o n with a f i n i t e length window, such as a Kaiser-Bessel window, to minimize the amount of computations while also minimizing the spreading and a l i a s i n g of the interpolator spectrum. F i g u r e 5 . 1 . M a g n i t u d e s o f t h e S R C / R C M C f i l t e r s o f l e n g t h 16 f o r s q u i n t a n g l e s o f 0 ° , 5 ° , 1 0 ° , 1 5 ° , a n d 2 0 ° . 1 1 2 5.3 SIMULATIONS OF AZIMUTH SRC This section describes the results of computer simulations of the azimuth SRC algorithm. The algorithm i s similar to the basic range/Doppler azimuth compression algorithm except that the RCMC interpolator is replaced by a combined SRC/RCMC f i l t e r . Range compression i s performed as in chapter 3 to produce a 1-D range compressed p r o f i l e . This p r o f i l e is used by the azimuth compression simulation routine to form a simulated 2-D range compressed s i g n a l . This signal i s Fourier transformed in azimuth using an FFT of length 1024 for RADARSAT parameters. The combined SRC/RCMC f i l t e r i s then applied as a discrete range time convolution to compensate for the dispersion caused by the azimuth FFT. The res u l t i n g 1-D azimuth signal i s multiplied by the FFT of the exact azimuth reference phase f i l t e r and a Kaiser-Bessel window to control sidelobes. F i n a l l y the 1-D azimuth signal is passed through an inverse FFT to produce one azimuth l i n e of the compressed image. The processing i s repeated for each desired azimuth l i n e . The processing parameters are as in Table 1. The res u l t i n g compressed range and azimuth p r o f i l e s are shown in figures 5.2 to 5.5. It i s seen that very l i t t l e broadening occurs in range when L is large. However severe broadening can s t i l l occur for smaller f i l t e r s since an appreciable amount of energy i s dispersed beyond the width of the shorter f i l t e r s . The length of the SRC/RCMC f i l t e r 113 has l i t t l e e f fect on the azimuth p r o f i l e . The -3dB percentage broadening measurements are summarized by figures 5.6 and 5.7 in range and figure 5.8 in azimuth. The results are shown for the four d i f f e r e n t lengths of SRC/RCMC f i l t e r . The azimuth broadening results are the same as the results without SRC. The predicted azimuth broadening curve of chapter 4, figure 4.13, agrees closely with the actual r e s u l t s . The range broadening measurements show that for squint angles below about 7° with a length 16 f i l t e r the percentage range broadening and azimuth broadening are comparable and small (below 3%). Above 7° the range broadening quickly r i s e s since s i g n i f i c a n t energy i s dispersed outside of the 16 sample width of the f i l t e r . It is seen that longer f i l t e r s produce less range broadening. Figures 5.9 and 5.10 summarize the 1-D ISLR measurements. From these i t i s seen that the azimuth ISLR remains almost constant whereas the range ISLR decreases (improves) as squint increases. The decrease i s more rapid for the shorter f i l t e r lengths. This decrease in range ISLR for shorter f i l t e r s corresponds to the larger range broadening. It appears that the windowing applied to shorter f i l t e r s causes the range spectrum to be tapered. The result is more range broadening with lower range sidelobes. Figure 5.11 summarizes the 2-D ISLR measurements. Whereas the ISLR without SRC deteriorated with increasing squint angle, the ISLR with SRC improves slowly. 1 1 4 The peak sidelobe ratios (PSLR) in range, azimuth, and 2-D are summarized in figures 5.12 to 5.14. As with the ISLR's, the azimuth PSLR's are v i r t u a l l y constant with squint angle while the range PSLR's decrease with increasing squint angle and decreasing f i l t e r length. By comparing graphs, i t can be seen that the decreases in ISLR and PSLR are less than about 3dB for squint angles smaller than the angle at which the range broadening i s 5%. In order to compare the peak magnitudes after compression, the peaks were normalized to the sum of the squares of the SRC/RCMC c o e f f i c i e n t s and the azimuth processed bandwidth. The results are summarized in figure 5.15. Whereas the peak s t r i c t l y decreases without SRC, the normalized peak actually increases s l i g h t l y for small squint angles before decreasing at larger squints. The reason for t h i s increase is not c l e a r l y understood. However the smaller variations in peak magnitude with SRC indicate that an improved signal-to-noise r a t i o i s achieved. ( 9 P ) epnuufiDfl i g u r e 5 . 2 . 1-D r a n g e p r o f i l e s a f t e r SRC c o m p r e s s i o n f o r s q u i n t a n d v a r i o u s f i l t e r l e n g t h s . 116 (3P) •pnuuCDN F i g u r e 5.3. 1-D range p r o f i l e s a f t e r SRC c o m p r e s s i o n f o r 10° s q u i n t and v a r i o u s f i l t e r l e n g t h s . i£J C fD Ol Ol — o I a cn £) 0) 3 3 rt C rr 0) 3* 3 n < o O <D C in cn 0) t-h t-h •-• rr ^ CO n- i (D n OT 50 I—J o fD 3 O vQ o rf 3 3 " 0 in i • fD in in w O 3 m TJ TJ 3 C o 2 SRC Compressed 1-D Azimuth Profiles -5 -10 -15 -20 -25 -30 -35 -40 squint = 5 deg., br = 2.7, ba = 1 .5 L= 4 , 8, /6 , 32. 16 L=32 Azimuth sample number - L=16 L=8 L=4 SRC Compressed 1—D Azimuth Profiles squint = 10 deg., br = 2.7, ba = 1.5 -16 -12 -8 -4 0 4 8 12 16 L=32 Azimuth sample number L=16 L=8 L=4 119 o CM Figure 5.6. Percentage range broadening with single-look azimuth SRC as a function of squint angle and f i l t e r length. Figure 5 . 7 . Percentage range broadening with single-look azimuth SRC as a function of squint angle and f i l t e r length (expanded sca l e ) . 121 —I—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I—\—#- ° ^ O f t B M O l f i ^ W N ^ O f t B M S l f l ^ l O N r - O CuiudpDOjg % Figure 5 . 8 . Percentage azimuth broadening with single-look azimuth SRC as a function of squint angle and f i l t e r length. 122 (ap) aisi Fiqure 5.9. 1-D range ISLR with single-look azimuth SRC as a Figure f u n c t i o ^ o f s q u i n t angle for various f i l t e r lengths. 123 Figure 5.10. 1-D azimuth ISLR with single-look azimuth SRC as a function of squint angle for various f i l t e r lengths. 124 (ap) *nsi Fiaure 5 11. 2-D ISLR with single-look azimuth SRC as a Figure 5 - , 1 - f ^ t i o n o f s q u i n t a n g l e for various f i l t e r lengths. 125 N N N N N N N N N N K ) K ) I O K ) « I O I O I O I O I I I I I I I I I I I I I I I I I I I (ap) cnsd Figure 5.12. 1-D range PSLR with single-look azimuth SRC as a function of squint angle for various f i l t e r lengths. 126 o M CO ID M II o o* « 3-- to - cs II o M I in d CS I cs I in cs I M CM I CS I »0 CS I m CM I CM I (ap) a i s d Figure 5.13. 1-D azimuth PSLR with single-look azimuth SRC as a function of squint angle for various f i l t e r lengths. 127 Figure 5.14. 2-D PSLR with single-look azimuth SRC as a function of squint angle for various f i l t e r lengths. 128 *- d d 1 ^ ' ts 1 I I I (9P) *pn|iu6DW Figure 5.15. Peak compressed magnitude with single-look azimuth SRC as a function of squint angle for various f i l t e r lengths. 129 5.4 RANGE SRC This section describes an alternate implementation of the SRC f i l t e r c a l l e d range SRC. This method performs SRC in the range frequency domain during range compression. The SRC f i l t e r i s combined with the range compression f i l t e r by simply a l t e r i n g the li n e a r FM rate of the range compression f i l t e r . Since the r e s u l t i n g modified range compression f i l t e r has the same number of c o e f f i c i e n t s , no additional computations are required. In addition a shorter RCMC interpolator can be used for the subsequent azimuth compression even at large squint angles since the azimuth spectrum remains well compressed aft e r the azimuth FFT. One complication of th i s method is the range invariance of the SRC f i l t e r . Depending on the radar parameters and size of the range swath, the range swath may need to be subdivided into smaller range invariance regions for the purpose of range compression. The issue of invariance regions, which i s also of concern in azimuth SRC, i s examined in chapter 7. The modified range compression f i l t e r can be expressed in the range time domain as : h R M ( t ) = s*(-t) * t g(t) ( 1 05 ) = [ r e c t ( t A ) e^V ] * t g(t) ( 1 0 6 ) The range compression f i l t e r i s implemented using fast 1 3 0 convolution in the range frequency domain. Applying the range Fourier transform to the f i l t e r gives the following form : H R M ( f r ) = F{ r e c t ( t A ) e^V } G ( f f ) (107) Since the f i r s t term i s the Fourier transform of a large TBP linear FM signal, i t may be approximated using the p r i n c i p l e of stationary phase as : F{ r e c t ( t A ) e J 7 r KR f c 2 } « rect(f /[K_T]) e 3 W F R 2 / K R (108) The Fourier transform of the SRC f i l t e r , g ( t ) , was given previously as : G( f r ) = (Xf c/c) e J 7 r f r 2 / K S R C (109) Substituting back these transforms and dropping the constant magnitude terms gives the following form for the frequency domain combined f i l t e r : H R M ( f r ) - r e c t ( f r / [ K R r ] ) e " ^ r V K R M (110) where K r m - K R/[ 1 - ( K R / K S R C ) ] (111) 131 Since K g R C >> KR, the modified reference function i s also a large TBP si g n a l . Therefore the p r i n c i p l e of stationary phase can again be used to evaluate the inverse Fourier transform of the modified range compression f i l t e r : h R M ( t ) =* r e c t ( t A ) e j , r KRM t 2 (112) This equation shows that the combined SRC/range compression f i l t e r i s a lin e a r FM pulse with a modified FM rate, The modified FM rate i s a function of both the range of closest approach of the point target, r 0 , and the squint angle. The range of closest approach primarily a f f e c t s the azimuth FM rate, K., and to a lesser extent the beam center frequency, f c . The squint angle mainly a f f e c t s the beam center frequency and to a smaller extent the azimuth FM rate. As the squint angle approaches zero, f^ also approaches zero and K A approaches a constant. Therefore K S R C = K A [ c / ( X f c ) ] 2 approches i n f i n i t y and the modified FM rate, KRM' a P P r o a c n e s t n e unmodified FM rate, K R. The range SRC algorithm i s es s e n t i a l l y the same as the basic range/Doppler algorithm presented in chapter 3 except that the linear FM rate of the range compression f i l t e r i s modified according to the squint angle. Range SRC i s superior to azimuth SRC since the equivalent range time domain SRC f i l t e r i s much longer than p r a c t i c a l SRC/RCMC f i l t e r s used in azimuth SRC. Since the SRC f i l t e r in range SRC i s applied over the entire range bandwidth, the 132 equivalent range time domain f i l t e r i s very long, much longer than the p r a c t i c a l SRC/RCMC f i l t e r lengths of between 8 and 16 samples used in azimuth SRC. The longer equivalent f i l t e r allows more of the dispersed energy to be recompressed (eventhough the recompression i s implemented as a p r e f i l t e r ) . 5.5 SIMULATIONS OF RANGE SRC This section presents the results of simulating the range SRC algorithm with nominal RADARSAT parameters. The same simulation programs that were used for basic range/Doppler compression are used except that the modified l i n e a r FM rate, developed in the previous section i s used in the range compression f i l t e r . A single RCMC interpolator length, L = 16, was used for a l l the simulations. Figure 5.16 shows the broadening of the range compressed p r o f i l e after range SRC for squint angles of 5° and 10°. The broadening occurs because of the mismatch of the linear FM rates of the transmitted range pulse and the modified SRC/range compression f i l t e r . This predistorton becomes recompressed by the azimuth FFT. Figures 5.17 and 5.18 show the range and azimuth p r o f i l e s after azimuth compression. It i s seen that very l i t t l e broadening occurs in range whereas some broadening does occur in azimuth as predicted by the decrease in processed bandwidth. The percentage range and azimuth broadening with range SRC i s summarized in figures 5.19 and 133 5.20. For squint angles below 15°, the range broadening i s very small, less than 0.55%. The ISLR and PSLR measurements are summarized in figures 5.21 and 5.22. Since l i t t l e broadening occurs, both the ISLR and PSLR are approximately constant with squint angle. In fact the range ISLR and PSLR improve slowly with increasing squint angle. The peak magnitudes are compared in figure 5.23. The absence of large variations indicates that the signal-to-noise r a t i o remains r e l a t i v e l y constant with changes in squint angle. C "I fD 10 o o -C 9 I •"••TJ O r t fD • tn Oi Ul 3 O fD 3 o * o rt-TJ tr n fD •-« tn OJ in 3 fD n> TJ t/i i » o o m O fD f-« tn ui n» o i-t> r t 0> fD 3 n n — OJ O 3 ouQ fD Range Compressed Profiles w/Range SRC m TJ TJ C 0 3 L = 16, squinr = 5, 10 deg. •4 0 Sample number 1 6 to it* 135 (SP) frprniufiDfl Figure 5.17. 1-D range p r o f i l e s after azimuth compression with single-look range SRC for 0°, 5°, and 10° squint and a length 16 RCMC f i l t e r . 136 (BP) •pruiufiDfl Figure 5.18. 1-D azimuth p r o f i l e s a f t e r azimuth compression with single-look range SRC for 0 ° , 5°, and 10° squint and a length 16 RCMC f i l t e r . 1 3 7 N O j w f s i o t f J ^ i q CN »- a> to r» <o in KI cs o o o o d d d o o o d I Figure 5.19. Percentage range broadening with single-look ramje SRC and a length 16 RCMC interpolator as a function of squint angle. 138 • ^ o f h e o r N t e i n ^ K i c M ^ o o o o r o t D i n ^ t O M ^ o CuiudpDOjg % Figure 5.20. .Percentage azimuth broadening with single-look range SRC and a length 16 RCMC interpolator as a function of squint angle. 139 I I I I (SP) aisi •Figure 5.21. Range, azimuth, and 2-D integrated sidelobe r a t i o s with single-look range SRC and a length 16 RCMC, interpolator as a function of squint angle. i I I (ap) aisd Figure 5.22. Range, azimuth, and 2-D peak sidelobe ratios with single-look range SRC and a length 16 RCMC interpolator as a function of squint angle. 141 (SP) »pn4!u6DN > J D*J Figure 5.23. Peak magnitude degradation with single-look range SRC and a length 16 RCMC interpolator as a function of squint angle. 142 5.6 SUMMARY OF SINGLE-LOOK SRC This chapter has presented the results of investigations into the use of an SRC algorithm for improving the range/Doppler compression algorithm at large squint angles. An SRC f i l t e r was developed by projecting the ideal matched f i l t e r into the range dimension after applying the azimuth FFT. This projection assumes l i n e a r i t y of the RCM curve over the processing i n t e r v a l . The approximate quadratic phase form of the azimuth phase is used to derive the SRC f i l t e r . Two implementations were developed and simulated. The simulations show that both implementations work well at recompressing the energy dispersed by the azimuth FFT. Since the dispersion increases with squint angle, longer SRC/RCMC f i l t e r s are required to recompress the dispersion at higher squint angles. The range SRC implementation provides an SRC f i l t e r which i s e f f e c t i v e l y much longer than the SRC/RCMC f i l t e r s used in azimuth SRC. Consequently at large squint angles range SRC performs much better than azimuth SRC. For a length 16 f i l t e r the azimuth SRC algorithm extends the squint angle which causes 5% range broadening from 3.65° to 8.03°. For 10% range broadening, the squint angle i s increased from 4.23° to 9.29°. In addition the range sidelobes are actually improved by the SRC algorithm indicating excellent compression. For the range SRC algorithm with a length 16 RCMC interpolator, the range broadening remains very small over a l l the squint angles which were simulated being less than 143 1.3% for squint angles up to 20°. Thus the range SRC implementation provides better compression than the azimuth-SRC algorithm. In practice t h i s improved performance must be weighed against the possible complications caused by range invariance of the SRC f i l t e r . This issue i s discussed further in chapter 7. 6. MULTILOOK RANGE/DOPPLER PROCESSING WITH SRC In t h i s chapter, methods of implementing SRC for multilook ( s p e c i f i c a l l y 4-look) processing are examined. In multilook processing the aperture i s divided into several looks which are compressed separately and then summed incoherently in order to reduce speckle noise. Two new methods of implementing SRC are proposed and investigated : 1. Fixed Multilook SRC. This method uses the same SRC f i l t e r for each look. The f i l t e r i s matched to the center frequency of the f u l l aperture, i . e . , the Doppler centroid. This i s the same f i l t e r that was used previously for single-look processing. 2. Look-Dependent Multilook SRC. In t h i s method, a dif f e r e n t SRC f i l t e r i s used for each look. This compensates for any changes in the slope of the range c e l l migration (RCM) curve between looks since each f i l t e r i s matched to each look center frequency. Simulations of multilook processing of a point target response are performed with and without the above SRC algorithms to quantify the improvements in image quality possible with multilook SRC. 144 145 6.1 MULTILOOK PROCESSING WITH SRC This section further extends the theory of azimuth compression presented in chapter 5 to describe multilook range/Doppler processing both with and without the use of two new multilook SRC algorithms. Multilook azimuth compression with the range/Doppler algorithm i s e s s e n t i a l l y the same as f u l l aperture range/Doppler processing except that the processed bandwidth i s divided into separate, often overlapping, processing bands or looks, which are compressed i n d i v i d u a l l y and then incoherently summed. Due to the approximate correspondence between the time and frequency domains of the azimuth signal each frequency domain look corresponds to d i f f e r e n t intervals of the azimuth time domain aperture. For a point target signal, each look contains data c o l l e c t e d from d i f f e r e n t time inte r v a l s which cover d i f f e r e n t ranges of incidence angles, or look angles. This change in look angle causes the speckle noise to have l i t t l e c o r r elation between looks. Thus incoherent summation of the compressed looks reduces the speckle noise l e v e l . Figure 6.1 shows how the processed bandwidth i s divided into separate looks (in t h i s case, 4 looks). Figure 6.2 shows the corresponding time domain look angles. In multilook processing, range compression i s performed as in single-look processing to produce a 2-D range compressed s i g n a l . In chapter 5, single-look azimuth compression was developed as an approximation to an exact 146 received Doppler spectrum with two-way azimuth antenna weighting i U l l l \ processed bandwidth 820 samples at f c - PRF/2| 0 dB (PBW) 0° squint -6 dB f c + PRF/2 1024 point FFT 256 samples , Look 2 ( 256 samples 256 samples Figure 6.1. Di v i s i o n of the azimuth frequency domain aperture into 4 looks. 147 beam center crossing time, T ? C i i i Look 3 | Figure 6.2. Corresponding time domain looks. 1 48 matched f i l t e r implemented in the azimuth frequency domain. For multilook processing, a similar compression algorithm is applied to each frequency band or look. The formulation of the azimuth compression f i l t e r for each look proceeds as in chapter 5 up to equation (85) which expresses the 2-D compression f i l t e r in the azimuth frequency domain. For single-look ( f u l l aperture) processing, the f i l t e r is usually truncated near the -3dB azimuth frequencies with some form of window function. The exact selection of processing bandwidth is a trade-off between factors such as ambiguity errors, a l i a s i n g noise, the desired azimuth resolution, and the type of window used. For the current research, the f u l l aperture processed bandwidth has been a r b i t r a r i l y chosen to be the -3dB one-way (-6dB two-way) azimuth antenna bandwidth. As stated e a r l i e r , multilook processing divides the processed bandwidth into several looks which are often overlapping. For ease of implementation and computational e f f i c i e n c y of the inverse azimuth FFT's, each look has been chosen to be 256 samples long. For 4-look processing, which has been simulated here, these looks are overlapped to f i t into the processed bandwidth. Since the processed bandwidth varies slowly with squint angle, the number of frequency domain samples in the processed bandwidth varies from 820 samples at zero squint down to 680 samples at 20 degrees of squint. The corresponding percentage overlaps range from 149 26.6% to 44.8% respectively. A Kaiser-Bessel azimuth window function i s applied separately to each look to control the azimuth sidelobe l e v e l s . Therefore the e f f e c t i v e combined window for the f u l l aperture, which is the summation of the individual look windows, extracts more energy from the outer looks than an equivalent single-look Kaiser-Bessel window applied to the f u l l aperture. Since the bandwidth of each look is fixed, the compressed resolution of each look i s approximately constant regardless of squint angle. The antenna weighting has l i t t l e e f fect over the smaller look bandwidths since for the inner looks the amount of weighting i s less and for the outer looks the weighting does not have a central maximum and i s approximately l i n e a r . In order to convert the above azimuth look compression f i l t e r from an azimuth frequency convolution to a more computationally e f f i c i e n t range convolution,' the f i l t e r is projected into range as in chapter 5 using the approximately li n e a r slope of the RCM curve. However there i s now a choice of slopes to use for the individual look f i l t e r s . For a fixed azimuth SRC implementation, the same SRC/RCMC f i l t e r is used for a l l looks. Thus the slope at the center of the f u l l aperture is used for the projection as in single-look SRC. This may introduce very small errors in the outer looks since the slope of the RCM curve in the outer looks d i f f e r s s l i g h t l y due to range curvature. For systems such as RADARSAT in which the range curvature i s small over 150 the f u l l aperture (about 0.23 range c e l l s ) , the error i s extremely small as w i l l be demonstrated in the simulations. For systems with larger range curvature such as longer wavelength SAR's including SEASAT, the error may be more s i g n i f i c a n t . In such cases a look-dependent SRC implementation i s possible in which the differences in RCM slope between looks are compensated by designing a separate SRC/RCMC f i l t e r for each look. For t h i s method, the slope used for projecting the f i l t e r into range i s taken as the slope at the look center frequency rather than the f u l l aperture center frequency. This complicates the control and memory requirements for the azimuth processor since several SRC/RCMC f i l t e r s (in t h i s case 4 of them) need to be precomputed and stored. For single-look SRC, the slope of the azimuth frequency domain RCM curve at the center frequency of the f u l l aperture, or the beam center frequency, f c , was expressed as : c 2 = C i / R A = " X f C / ( c K A ) ( 1 1 3 ) where K A i s the azimuth frequency rate at the beam center frequency. This equation can be used to calculate the slope at any a r b i t r a r y look center frequency, f L« To do t h i s accurately, i t i s necessary to know the azimuth frequency rate at that frequency. This is accomplished by using equations (37), 151 (38), and (56) : r j ^ f ) = r 0 / { v e g [ ( 2 v e g / ( X f ) ) 2 - 1 ] i / z } (114) r. ( f ) = r 0 / [ 1 - ( X f / ( 2 v e g ) ) 2 (115) K.(f) = -(2v 2 /[Xr. (f )]) [1 - (v T J. (f )/r. (f ) ) 2 ] (116) The f i r s t two equations determine the azimuth look center time, T j . ( f r ) , and look center range, r - ( f r ) . These are 1 Li 1 Li substituted into the t h i r d equation to determine the look center frequency rate, K (f ). F i n a l l y the slope, c , of the A Li LI azimuth frequency domain RCM curve at the look center frequency i s calculated by substituting into the equation for c 2 : c L = - X f L / [ c K A ( f L ) 3 (117) The f i l t e r i s projected into range as in equation (89) of chapter 5 with c 2 replaced by c L : H p ( t , f ) ~ F{ exp[-j47rr(r?)/X] } • ( A 2 ( t / c L ) * f c M t - ( 2 / c ) r ( [ f - f , ] / R A ) ] } (118) where denotes the azimuth frequency rate at the beam center frequency, i . e . K ( f r ) . For fixed SRC, c. i s computed 152 using the beam center frequency, f^,, while for look-dependent SRC, the look center frequency of each look is used. The resulting algorithm for compressing each look i s given by equations (90) to (96) with c 2 again replaced by the appropriate c L . As in single-look processing, there are several alternative methods for implementing the SRC f i l t e r . Fixed range SRC can provide better compression due to the longer e f f e c t i v e length of the SRC f i l t e r . However the range SRC method becomes less e f f i c i e n t i f look-dependent SRC i s required because a separate range compression would be required for each look. Look-dependent azimuth SRC i s more e f f i c i e n t since only the additional look-dependent f i l t e r s need to be generated. For the azimuth SRC implementation, the combined SRC/RCMC f i l t e r , g^,(t,f), is implemented as a range convolution with the azimuth spectrum. As in single-look azimuth SRC, the f i l t e r i s approximated by precomputing 16 interpolated versions of the SRC f i l t e r g ( t ) . The convolution i s implemented by computing the integer and f r a c t i o n a l range c e l l s h i f t s for RCM correction (RCMC) and using the appropriately shifted version of the precomputed f i l t e r . The window, w A ( f " f L ) ' * s s n i f t e c * t 0 t n e appropriate look center frequency for each look. The time domain image for each look is computed by applying an inverse azimuth FFT of length 256. 153 Before look detection and look summation, each look image, which i s s t i l l in-complex form, must f i r s t be interpolated in order to reduce a l i a s i n g of the increased bandwidth of the detected s i g n a l . Although an interpolation factor of 2 i s s u f f i c i e n t , the simulations interpolate each look by a factor of 8 in both range and azimuth before detection and look summation as part of the image qua l i t y measurement process. The interpolated look images are detected and summed together to form the f i n a l multilook image. For simulations of multilook processing without SRC, the combined SRC/RCMC f i l t e r g ( t , f ) was replaced by the zero phase RCMC interpolator of chapter 3 which i s the same for each look. For multilook range SRC, only fixed SRC was simulated since look-dependent multilook range SRC was i n e f f i c i e n t . The same modified linear FM rate was used for the combined SRC/range compression f i l t e r as in single-look range SRC. The following sections describe the results of simulations of the both fixed and look-dependent multilook SRC algorithms as well as simulations of multilook processing without SRC which are used as a baseline for compari son. The simulations were performed using the nominal set of RADARSAT parameters l i s t e d in Table 1. The azimuth Kaiser-Bessel window parameter was changed from 1.5 to 2.7. More weighting i s required in multilook processing since the 154 antenna weighting has much less e f f e c t over the reduced look bandwidths. The value of 2.7 was chosen to produce azimuth sidelobe l e v e l s which are comparable in size to the range sidelobes as in e a r l i e r simulations. 6.2 SIMULATIONS OF 4-LOOK PROCESSING WITHOUT SRC This section discusses simulations of 4-look, range/Doppler processing without SRC. Simulations were performed with a length 16 RCMC f i l t e r . The simulation results are summarized by figures 6.3 to 6.14. Figures 6.3 and 6.4 show the 1-D range and azimuth p r o f i l e s a f t e r 4-look azimuth compression for squint angles of 0°, 5°, and 10°. The simulations were performed as outlined in the previous section including look detection, and look summation. The p r o f i l e s were interpolated by a factor of 8 before detection by zero padding in the frequency domain to preserve the signal bandwidth after detection and to increase the accuracy of the image quality measurements. The range p r o f i l e s are very similar to the single-look p r o f i l e s shown e a r l i e r . This i s expected since the range broadening i s primarily due to the azimuth FFT which i s applied in both single-look and multilook compression. Small differences are expected because of small variations in range width over the azimuth spectrum before azimuth compression which are caused by range curvature. 155 The azimuth p r o f i l e s are also similar to the single-look r e s u l t s . However the mainlobe widths are approximately 3.3 times wider than the single-look r e s u l t s . This i s primarily due to the smaller bandwidth of each look compared to the f u l l aperture processed bandwidth. The azimuth sidelobe le v e l s d i f f e r s l i g h t l y because of the larger azimuth window parameter. The range and azimuth broadening results for both single-look processing and 4-look processing are shown in figures 6.5 to 6.7 for squint angles of 0 to 6 degrees. The 4-look range broadening results are the same as the single-look r e s u l t s . As explained above, th i s i s expected since range broadening occurs primarily in the azimuth FFT. Azimuth broadening for single-look processing increases with squint angle due to the reduced processed bandwidths. In contrast, there i s very l i t t l e azimuth broadening for 4-look processing since each look contains e s s e n t i a l l y the same bandwidth. The azimuth antenna weighting has less broadening effect on the reduced bandwidths of the indi v i d u a l looks. Figures 6.8 to 6.10 show that the range, azimuth, and 2-D integrated sidelobe ra t i o s (ISLR) for 4-look and single-look processing behave s i m i l a r l y with increasing squint angle. The range ISLR curves increase with increasing squint angle due to the spreading of energy from the mainlobe to the sidelobes. The apparent drop in ISLR past 5 degrees i s due to the f i n i t e integration area of the image 156 qu a l i t y measurements. For squint angles over 5 degrees, the range broadening i s over 25%. This causes a s i g n i f i c a n t amount of energy to l i e in the sidelobes outside of the integration area. Consequently the ISLR measurements are inaccurate for large amounts of broadening. The azimuth ISLR curves show very l i t t l e v a r i a t i o n with squint angle whereas the 2-D ISLR curve i s a composite of the range and azimuth curves. The peak sidelobe r a t i o (PSLR) measurements are shown in figures 6.11 to 6.13. Again the single-look and 4-look results are similar with the range results varying with squint angle and the azimuth results e s s e n t i a l l y constant. The 2-D measurements are somewhat lower than the 1-D measurements since they are measured with a much coarser sample spacing (interpolated by 8) than the 1-D measurements (interpolated by 128). F i n a l l y the peak magnitude r a t i o s , which compare the peak compressed magnitude to that obtained for zero squint, are shown in figure 6.14. The peak magnitudes are normalized by the sum of squares of the RCMC f i l t e r c o e f f i c i e n t s as would be appropriate for white noise i d e n t i c a l l y d i s t r i b u t e d over the range c e l l s . In r e a l i t y the noise i s only approximately evenly d i s t r i b u t e d over range c e l l s so that care must be used in r e l a t i n g the peak magnitude results to signal-to-noise r a t i o s . 157 (SP) •pnjiuftDW Figure 6.3. Interpolated 1-D range p r o f i l e s a f t e r 4-look compression without SRC for squint angles of 0 ° , 5°, and 10° and a length 16 RCMC interpolator. 158 (8P) •pnuufion Figure 6.4. Interpolated 1-D azimuth p r o f i l e s a f t e r 4-look compression without SRC for squint angles of 0°, 5°, and 10° and a length 16 RCMC interpolator. 159 6uiu©pD0jg % Figure 6.5. Range broadening for single-look and 4-look compression without SRC using a length 16 RCMC interpolator. 160 gure li.6. Range broadening for single-look and 4-look compression without SRC using a length 16 RCMC interpolator (expanded scale). 161 Figure 6.7. Azimuth broadening for single-look and 4-look compression without SRC using a length 16 RCMC interpolator. 162 o o o r 9 •D C o 3 CT </>» o o • Figure 6.8. 1-D range integrated sidelobe r a t i o s for single-look and 4-look compression without SRC using a length 16 RCMC interpolator. 163 on O - P c6 tf o CO » , 0 + - > - a Cd g 0) o N T - M o o 9 •o c o ~c 3 CT o o in CO 0) T in o M I in d CM I CM I m CM I CM <M I (8P) dlSI Figure 6.9. 1-D azimuth integrated sidelobe r a t i o s for single-look and 4-look compression without SRC using a length 16 RCMC interpolator. 164 Figure 6.10. 2-D integrated sidelobe r a t i o s for single-look and 4-look compression without SRC using a length 16 RCMC interpolator. 165 0 w- CM tn •<* «n <o CM CM CM CM CM CM CM 1 I I I I I I (ap) *nsd Figure 6.11 . i - D rarge peak sidelobe ratios for single-look and 4-look compression without SRC usinq a length 16 RCMC interpolator. 166 o o ov 9 TJ 9 CD C o - CM 3 CT V) •» o o CO T in co T in o M I in o CM I CM I in CM I CM CM I (BP) dlSd Figure 6.12. 1-D azimuth peak sidelobe r a t i o s for single-look and 4-look compression without SRC using a length 16 RCMC interpolator. 1 (ap) aisd Figure 6.13. 2-D peak sidelobe r a t i o s for single-look and 4-look compression without SRC using a length RCMC interpolator. 168 Figure 6.14. Peak magnitude r a t i o s for 4-look compression without RCMC inter p o l a t o r . single-look and SRC using a length 16 169 6.3 SIMULATIONS OF 4-LOOK, FIXED AND LOOK-DEPENDENT, AZIMUTH  SRC PROCESSING The next section discusses simulations of 4-look processing performed with both fixed and look-dependent azimuth SRC. The results for the two algorithms were i d e n t i c a l within the l i m i t s of the measurements. Consequently only one set of results are presented. This s i m i l a r i t y shows that look-dependent processing i s not necessary for RADARSAT. The results are almost i d e n t i c a l since the slope of the RCM curve varies very slowly over the aperture for RADARSAT parameters. Consequently the slopes at the individual look center frequencies are v i r t u a l l y the same. As stated e a r l i e r , systems such as SEASAT which exhibit larger range curvature may have larger variations in RCM curve slope over the aperture requiring look-dependent SRC processing. The results of the simulations are contained in figures 6.15 to 6.26. A smaller selection of squint angles was used than in the single-look azimuth SRC simulations since the results are very s i m i l a r . Simulations were performed for squint angles of 0, 1, 5, and 10 degrees and for SRC/RCMC f i l t e r lengths of 4, 8, 16, and 32. Figures 6.15 and 6.16 show the range and azimuth p r o f i l e s a f t e r azimuth 4-look compression for squint angles of 5 and 10 degrees. As in the previous section the processing included interpolation, look detection, and look summation. The range p r o f i l e s show the broadening which occurs.with smaller length SRC/RCMC f i l t e r s at larger squint 170 angles. The azimuth p r o f i l e s show ne g l i g i b l e changes with f i l t e r length. Figures 6.17 and 6.18 summarize the range and azimuth broadening results respectively. The broadening figures are computed from the -3dB response widths r e l a t i v e to the zero squint width for each f i l t e r length. Differences between the impulse response widths at zero squint for d i f f e r e n t f i l t e r lengths are very small (less than 0.2%) and are shown in figure 6.19. The multilook broadening results can be compared to the single-look azimuth SRC results in figures 5.6 and 5.8. The 4-look range broadening results agree very clo s e l y with the single-look r e s u l t s . As in the simulations without SRC, almost no azimuth broadening occurs for 4-look processing. The ISLR and PSLR measurements are shown in figures 6.20 to 6.22 and figures 6.23 to 6.25 respectively. The results agree clo s e l y with the single-look azimuth SRC curves. The azimuth sidelobes are lower by about 0.7dB due to the larger azimuth window parameter. Figure 6.26 shows the peak magnitude variations with squint angle with the peaks normalized by the sum of the squared SRC/RCMC f i l t e r c o e f f i c i e n t s . 171 (8P) •pniiufiBW Figure 6.15. Interpolated 1-D range p r o f i l e s after 4-look compression with both fixed and look-dependent SRC for squint angles of 0°, 5°, and 10° and a length 16 f i l t e r . 1 7 2 (ap) ftprmufiDw Figure 6.16. Interpolated 1-D azimuth p r o f i l e s after 4-look compression with both fixed and look-dependent SRC for squint angles of 0°, 5°, and 10° and a length 16 f i l t e r . 173 fiuiuftpDojg % F i g u r e 6 . 1 7 . R a n g e b r o a d e n i n g f o r 4 - l o o k c o m p r e s s i o n w i t h b o t h f i x e d a n d l o o k - d e p e n d e n t S R C a n d v a r i o u s S R C / R C M C f i l t e r l e n g t h s a s a f u n c t i o n o f s q u i n t a n g l e . 174 o o - CO II o rt £ rt E a> % cd •*-O u ,rt -p - to 'to - * CD 3 cr " t o II - ts c s < r > c o r > » < c i n ^ > o c s * - . - » c q h . < o i n ' * K i t s ^ : o ^ : o o o o o o o o o o I F i g u r e 6 . 1 8 . A z i m u t h b r o a d e n i n g f o r 4 - l o o k c o m p r e s s i o n w i t h b o t h f i x e d a n d l o o k - d e p e n d e n t SRC a n d v a r i o u s SRC/RCMC f i l t e r l e n g t h s a s a f u n c t i o n o f s q u i n t a n g l e . ro o 3 o i Q ?>r r t I rr a tn ro • TJ a> a a» r t in W o o> 3 CL < 0) »-•• o c cn t/J W O \ w o X o ro c rt) cn 0 o n o 3 1 0) O M. O (A X - O 3 O o o •a ft) o> cn 3 cn io O 3 I U> < a r t tr K O* a o r t r t rr rr tn t-f« Ol t->- r t X fD O O J o 0) cn ' 3 iQ Q i C D r t II t> 'c o Comparison of Zero Squint Broadening for various SRC/RCMC filter lengths 0.9 0.8 0.7 0.6 0.5 0.4 • 1 6 SRC/RCMC filter length. L azimuth + range 32 176 . - M C M C M N N N N C M I I I I I I I I I (ap) aisi Figure 6.20. 1-D range integrated sidelobe r a t i o s for 4-look compression with both fixed and look-dependent SRC and various SRC/RCMC f i l t e r lengths as a function of squint angle. O O I d CO n-t <-> * | 2 00 T - « - <o CM II go c rr ll - CM II in T in o CM I in o CM I CM I in CM I CM CM I (ap) aisi Figure 6.21. 1-D azimuth integrated sidelobe r a t i o s for 4-look compression with both fixed and look-dependent SRC and various SRC/RCMC f i l t e r lengths as a function of squint angle. 1 78 I I I I (SP) insi Figure 6.22. 2-D integrated sidelobe r a t i o s for 4-look compression with both fixed and look-dependent SRC and various SRC/RCMC f i l t e r lengths as a function of squint angle. 1 79 M to II go 3 rr II O « - N t O ' + « r > « 0 r H O D O > O ' -( M C S C S C S C S C S C S C N C M C S t O K ) I I I I I I I I I I I I (ap) ansa Figure 6.23. 1-D range peak sidelobe ra t i o s for 4-look compression with both fixed and look-dependent SRC and various SRC/RCMC f i l t e r lengths as a function of squint angle. 180 W O •i—» O " o N I o CN I - 00 - <0 M to II 9~ g< 3 CT V)«0 - CM tf) O CM I M I in CM I CM CM I in CM CM I to CM I in to CM I CM I (ap) aisd Figure 6.24. 1-D azimuth peak sidelobe r a t i o s for 4-look compression with both fixed and look-dependent SRC and various SRC/RCMC f i l t e r lengths as a function of sguint angle. 181 o » n — i n t s i n i o i n ' * /VI • #VJ • ISJ /SJ CS 1 CS 1 CS 1 CS I I I I (ap) aisd Figure 6.25. 2-D peak sidelobe r a t i o s for 4-look compression with both fixed and look-dependent SRC and various SRC/RCMC f i l t e r lengths as a function of squint angle. 182 ^ o d i ^ i I l (ap) •prniufiDH Figure 6.26. Peak magnitude r a t i o s for 4-look compression with both fixed and look-dependent SRC and various SRC/RCMC f i l t e r lengths as a function of squint angle. 183 6.4 SIMULATIONS OF 4-LOOK, FIXED, RANGE SRC PROCESSING This section presents the results of simulations of 4-look processing with a fixed range SRC algorithm. Since the range SRC algorithm uses an SRC f i l t e r which is e f f e c t i v e l y much longer than the f i l t e r used in azimuth SRC, the range broadening caused by windowing of the SRC f i l t e r i s expected to be much smaller. This was shown with the single-look range SRC simulations and i s also true of the multilook implementation. Since very l i t t l e broadening occurs, the range and azimuth compressed p r o f i l e s are veri-similar to the zero squint, multilook p r o f i l e s shown e a r l i e r and are therefore not shown here. Simulations are performed with a length 16 RCMC f i l t e r . The measurements of range and azimuth broadening are summarized in figure 6.27. For squint angles up to 20° the range broadening i s less than 1.8%. The integrated sidelobe rat i o s shown in figure 6.28 show that there i s l i t t l e v a r i a t i o n in ISLR with squint angle (less than -1 dB for squint angles up to 20°). S i m i l a r l y the peak sidelobe ratios summarized in figure 6.29 show an even smaller v a r i a t i o n . F i n a l l y the peak magnitude measurements shown in figure 6.30 display only a small variation with squint angle. This indicates that there i s l i t t l e change in SNR for changes in squint angle. 184 Figure 6.27. Range and azimuth broadening for 4-look compression with range SRC and a length 16 RCMC interpolator as a function of squint angle. 185 (ap) cnsi Figure 6.28. Range, azimuth, and 2-D integrated sidelobe r a t i o s for 4-look compression with range SRC and a length 16 RCMC interpolator as a function of squint angle. (ap) disd Figure 6.29. Range, azimuth, and 2-D peak sidelobe ratios for 4-look compression with range SRC and a length 16 RCMC interpolator as a function of squint angle. 187 Figure 6.30. Peak magnitude degradation for 4-look compression with range SRC and a length 16 RCMC interpolator as a function of squint angle. 188 6.5 SUMMARY OF MULTILOOK SRC This chapter has presented two forms of SRC algorithms to be used with multilook range/Doppler compression: fixed azimuth SRC, and look-dependent SRC. Both methods have been shown to be e f f e c t i v e in reducing the range broadening which occurs at large squint angles. Comparisons of the simulation re s u l t s of both multilook SRC algorithms for 4-look processing have shown that there are no measurable differences in image quality for RADARSAT parameters. As a re s u l t , fixed azimuth SRC should be used since i t requires less memory and computation for i t s single SRC/RCMC f i l t e r . In fact the 4-look results are very similar in range to the single-look results indicating that the number of looks does not a l t e r the effectiveness of the SRC algorithm. The multilook SRC simulations show that use of the azimuth SRC algorithm can s i g n i f i c a n t l y reduce the point target response broadening which occurs at larger squint angles. The improvements increase with larger SRC/RCMC f i l t e r lengths since the larger f i l t e r s can c o l l e c t more of the energy which has been spread out by the azimuth FFT. For a nominal length 16 f i l t e r the 5% and 10% range broadening squint angles can be extended from 3.65° to 8.0° and from 4.23° to 9.3° respectively. With the multilook range SRC algorithm the e f f e c t i v e f i l t e r length i s much longer. Thus the broadening i s much less than with azimuth SRC. For squint ancjles up to 20° .the range broadening i s less than 1.8%. 7. EFFECTS OF SRC FM RATE ERRORS This chapter examines the s e n s i t i v i t y of the SRC f i l t e r to SRC FM rate errors. SRC FM rate errors are caused by errors in both the beam center frequency, f^, and the range pf closest approach, r 0 . Limits on the processing block siz e , or the invariance region, and parameter estimation errors are developed for s p e c i f i c broadening l i m i t s . Simulations are performed to quantify the broadening caused by SRC FM rate errors with various algorithms and f i l t e r lengths. Only errors in the SRC FM rate are simulated since the e f f e c t s of parameter errors on other processing operations (e.g., RCMC, and the azimuth reference phase function) can be modelled and predicted independently. The broadening results are parameterized in terms of the band-edge phase error in range frequency. The broadening without SRC i s s i m i l a r l y parameterized by evaluating the equivalent band-edge phase error caused by not applying an SRC f i l t e r . 7.1 SENSITIVITY ANALYSIS OF THE SRC FM RATE SRC FM rate errors arise from two sources : parameter estimation errors and f i l t e r invariance errors. Estimation errors occur since both the the beam center frequency, f c , and the range of closest approach, r 0 , are usually estimated from inexact measurements of the position and attitude of the radar platform. The beam center frequency estimate is often refined with a Doppler centroid estimation algorithm 189 190 [143. S R C f i l t e r i n v a r i a n c e e r r o r s - i r e a r e s u l t o f b l o c k p r o c e s s i n g . T h e v a l u e o f r 0 v a r i e s a c r o s s t h e p r o c e s s e d b l o c k b u t t h e S R C FM r a t e i s o n l y m a t c h e d t o o n e v a l u e o f r 0 , u s u a l l y a t t h e c e n t e r o f t h e b l o c k . T h e r e s u l t i n g m i s m a t c h i n t h e S R C F M r a t e l i m i t s t h e r a n g e d i m e n s i o n o f t h e p r o c e s s e d b l o c k s i n c e p o i n t t a r g e t s n e a r t h e b l o c k e d g e b e c o m e b r o a d e n e d . T h e r a n g e b l o c k s i z e l i m i t i s c a l l e d t h e S R C r a n g e i n v a r i a n c e r e g i o n . B o t h r 0 e s t i m a t i o n e r r o r s a n d r 0 i n v a r i a n c e e r r o r s m u s t b e a d d e d w h e n d e t e r m i n i n g t h e S R C r a n g e i n v a r i a n c e r e g i o n . F o r t h e a p p r o x i m a t e g e o m e t r i c m o d e l u s e d i n t h i s t h e s i s , t h e b e a m c e n t e r f r e q u e n c y i s i n d e p e n d e n t o f r 0 . H o w e v e r m o r e s o p h i s t i c a t e d m o d e l s p r e d i c t a s l o w v a r i a t i o n i n b e a m c e n t e r f r e q u e n c y a c r o s s t h e p r o c e s s e d r a n g e s w a t h . T h u s a b e a m c e n t e r f r e q u e n c y i n v a r i a n c e e r r o r c a n a l s o o c c u r . A l t h o u g h t h i s e r r o r i s n o t t a b u l a t e d i n t h i s t h e s i s , i t s e f f e c t c a n b e p r e d i c t e d b y a d d i n g t h e f c i n v a r i a n c e e r r o r t o t h e f ^ e s t i m a t i o n e r r o r . T o e x a m i n e t h e e f f e c t s o f p a r a m e t e r e r r o r s o n t h e S R C f i l t e r , t h e S R C F M r a t e c a n b e e x p r e s s e d i n t e r m s o f f c a n d r 0 a s : K S R C " K A c 2 / ( X f C ) 2 (119) = - ( 2 v e q 2 / [ X r 0 3 ) ( c / [ X f c 3 ) 2 ( 1 - ( X f c / [ 2 v e g 3 ) 2 } 3 / 2 (120) 191 In order to normalize the analysis to the range -3dB bandwidth, B^, the phase of the SRC f i l t e r at the range band-edge frequency i s used as a parameter. The band-edge phase i s given in radians by : *(B r/2) = 7 r ( B r / 2 ) 2 / K S R C (121) Using p a r t i a l derivatives, the band-edge phase error can be expressed approximately in terms of the beam center frequency error, A f c , and r 0 error, A r 0 , as : A<//(Br/2) = Ar 0 3 tf(Br/2) + A f c 9 tf(Br/2) (122) 3 r 0 3f C - [ , ( B r / 2 ) V K S R C ] • { ( A r 0 / r 0 ) + 2 ( A f c / f c ) [ l + (3/2)/[(2v / [ X f c ] ) 2 - 1 ] ] } (123) Range broadening i s small, t y p i c a l l y less than 10%, when the magnitude of the band-edge phase error i s less than 7r/2 radians. The corresponding error l i m i t s on A f c and Ar 0 vary depending on the squint angle and the parameters used. Figures 7.1 and 7.2 show the SRC band-edge phase error in the range frequency domain for various squint angles (1° to 20°) as a function of A f c and A r 0 . The curves use exact c a l c u l a t i o n s of the phase errors rather than the p a r t i a l d erivative expansion above. However the curves are almost 1 92 0> 9 "O O CS on 9 s* >s o c 3 rr •° c o E o o CO 0> 9 TJ TJ (Cap) jojje 6 S D L | d eSpe-puDg Figure 7.1 SRC band-edge phase error in the range frequency domain for squint angles of 1°, 5°, 10°, 15°, and 20° as a function of beam center frequency error. 1 93 Figure 7.2. SRC band-edge phase error in the range^ requency domain for squint angles of 1°, 5°, 10 , 15 , and 20° as a function of r 0 error. 194 linear which agrees with the p a r t i a l derivative expansion. The magnitude of the phase error increases with increasing squint angle and increasing parameter error. For RADARSAT with squint angles less than 10°, the phase error i s less than 7r/2 for beam center frequency errors less than +/- 3000 Hz and r 0 errors less than +/- 15%. The maximum expected beam center frequency error i s much less (on the order of 100 Hz excluding range variances). The maximum expected r 0 i s also less being on the order of +/- 5%. More exact measurements of range broadening as a function of band-edge phase error for p a r t i c u l a r algorithms and f i l t e r lengths are performed by simulation in the next sect ion. The range broadening which occurs without SRC can be parameterized by an equivalent range frequency band-edge phase error. Since the broadening results in chapter 3 were shown as a function of squint angle, i t is s u f f i c i e n t to relate the squint angle to the equivalent phase error. The error is given by the band-edge phase of the ideal SRC f i l t e r which can be computed from equation (121) by rewriting i t in terms of the squint angle, 6 , and r 0 to get : i / / ( B /2) = -2TT ( B / [ 2 c ] ) 2 X r 0 tan 20 [ 1 +tan 2 0 ] 1 / 2 (124) IT IT S S This r e l a t i o n is plotted in figure 7.3. The 5% and 10% range broadening squint angles given in chapter 3 correspond to 195 equivalent range phase errors of -74° and -100° respectively. Thus the 90° phase l i m i t used e a r l i e r as a 10% range broadening l i m i t i s reasonably close. For comparison with the results in the next section, the broadening results without SRC are replotted in figure 7.4 as a function of equivalent range phase error instead of squint angle. 196 F i g u r e 7.3. E q u i v a l e n t SRC band-edge phase e r r o r i n the range f r e q u e n c y domain w i t h o u t SRC as a f u n c t i o n of s q u i n t a n g l e . 197 •a- o 11 o 00 O O O O O O «n »o M ^ 6uju#pDojg *6uDy % Figure 7.4. Actual range broadening without SRC with a length 16 RCMC interpolator and predicted range broadening as a function of equivalent range band-edge phase error. 198 7.2 SIMULATIONS OF SRC FM RATE ERROR BROADENING This section present?? simulations of range broadening caused by SRC FM rate errors with various SRC algorithms. The results indicate that the band-edge phase error in range frequency is a good general measure of the expected range broadening. The s p e c i f i c amount of broadening varies somewhat with the size of SRC f i l t e r and the type of SRC algorithm. As noted e a r l i e r , parameter errors were only simulated in the SRC f i l t e r . The remainder of the processing (RCMC, azimuth reference phase m u l t i p l i c a t i o n , etc.) was simulated without parameter errors. Thus the measured range broadening i s s o l e l y the result of SRC FM rate errors. The range broadening was measured r e l a t i v e to the range response width without SRC FM rate errors so that only the additional broadening caused by the FM rate error was measured. Only negative band-edge phase errors were simulated since the p o s i t i v e phase error results are very s i m i l a r . The f i r s t simulation involved the single-look azimuth SRC algorithm. The maximum phase error simulated for each squint angle was chosen to include largest expected error from figures 7.1 and 7.2 of the previous section. Two squint angles, 5° and 10°, were simulated. The maximum simulated phase errors were -17° for 5° of squint and -68° for 10°. The range broadening measurements are summarized by figures 7.5 and 7.6 for SRC f i l t e r lengths of 4, 8, 16, and 32 samples. At 5° of squint the range broadening i s less than 199 2% for a l l f i l t e r lengths. At 10° of squint, only the longer f i l t e r s (of length 16 and 32) were used since the shorter f i l t e r s produced unacceptably large broadening even without SRC FM rate errors (greater than 85%). The largest range broadening (for a phase error of -68°) was less than 8%. The azimuth SRC simulations were repeated for multilook processing with almost i d e n t i c a l results as shown in figures 7.7 and 7.8. This i d e n t i c a l behaviour shows the independence of the range broadening process from the look extraction process. The next simulation was performed with the single-look range SRC algorithm. A 16 sample range interpolator was used for RCMC. Since the broadening results without errors indicated that range SRC could be used for larger squint angles, squint angles of 15° and 20° were used in addition to the 5° and 10° squints used before. The maximum simulated phase errors were again selected to include the maximum expected phase errors at each squint angle. The chosen values were -17° for 5° of squint, -68° for 10° of squint, and -136° for both 15° and 20° of squint. The range broadening i s summarized in figure 7.9. The broadening leve l s vary only s l i g h t l y with d i f f e r e n t squint angles. The maximum broadening at 10° of squint (again for a phase error of -68°) is smaller with range SRC than with azimuth SRC (less than 5% compared with 8% for azimuth SRC). Since the multilook azimuth SRC measurements of broadening wih SRC FM rate errors were e s s e n t i a l l y the same 200 as the single-look r e s u l t s , the multilook range SRC results should also be very similar to the single-look range SRC r e s u l t s . Consequently the multilook range SRC algorithm was not simulated. 201 O W & CO A _,_) rj) rt ' a m II c N 3 ^ 2" N -rt ed O u PQ rt in CM CM to II c5M 9 £ Q. 9 £ II O II fiUIUdpDOjg % F i g u r e 7.5. Range 'broadening w i t h s i n g l e - l o o k a z i m u t h SRC a t 5° of s q u i n t w i t h v a r i o u s SRC/RCMC f i l t e r l e n g t h s as a f u n c t i o n of range band-edge phase e r r o r . 202 fiuiuepDojg % F i g u r e 7.6. Range broadening with s i n g l e - l o o k azimuth SRC at 10° of s q u i n t with v a r i o u s SRC/RCMC f i l t e r l e n g t h s as a f u n c t i o n of range band-edge phase e r r o r . 203 CuiudpDojg % Figure 7 .7. Range broadening with multilook azimuth SRC at 5 of squint with various SRC/RCMC f i l t e r lengths as a function of range band-edge phase error. 204 6uiu*pr>ojg $ Figure 7.8. Range broadening with multilook azimuth SRC at 10 of squint with various SRC/RCMC f i l t e r lengths as a function of range band-edqe phase error. ' ^ 205 fiuiuapDOjg % Figure 7.9. Range broadening with single-look range SRC at 5 ° , 1 J ° , 1 5 ° , and 2 0 ° of squint with a length 16 RCMC interpolator as a function of range band-edge phase error. 8. SUMMARY AND CONCLUSIONS This thesis has shown that a new algorithm, c a l l e d secondary range compression (SRC), s i g n i f i c a n t l y reduces the amount of range broadening which occurs at large squint angles in the basic range/Doppler compression algorithm. The SRC algorithm was f i r s t suggested by J i n and Wu [13] in 1984 for use with the SEASAT SAR. This thesis has extended the theory of the SRC algorithm to examine the approximations involved and to explore alternate implementations. In addition to the azimuth SRC implementation presented by J i n and Wu, a new implementation of SRC, c a l l e d range SRC, which i s performed during range compression, has been presented and examined. Also, two new multilook SRC algorithms have been developed for use in multilook azimuth compression. Many simulations with nominal RADARSAT parameters have been performed to quantify the image quality improvements possible with SRC. A s e n s i t i v i t y analysis of SRC with respect to parameter errors has been included. The analysis indicates that the SRC algorithm i s very tolerant to parameter estimation and invariance errors. In p a r t i c u l a r , with a range broadening l i m i t of 5%, no SRC f i l t e r updating i s required over the nominal 150 km. RADARSAT ground range swath for squint angles up to 15° using the range SRC implementation (assuming an r 0 error of less than +/- 5% and a beam center frequency error of less than +/- 200 Hz). SRC provides a closer approximation to exact matched f i l t e r i n g when the azimuth time-bandwidth product (TBP) of 206 207 the range compressed point target response, as measured in the range c e l l nearest the beam center range, f a l l s below unity. The SRC f i l t e r i s formulated by using a quadratic phase approximation of the azimuth phase coding and a linear approximation to the range migration curve over the processed azimuth bandwidth. These approximations allow the azimuth Fourier spectrum to be derived a n a l y t i c a l l y . The derived spectrum accounts for the spectrum broadening which occurs with low azimuth TBP's. The basic range/Doppler algorithm without SRC does not account for azimuth spectrum broadening since i t i s derived with the p r i n c i p l e of stationary phase which is v a l i d only for large TBP signals. It has been shown that the range bandlimited azimuth matched f i l t e r exhibits s i m i l a r azimuth spectrum broadening under low azimuth TBP conditions. When range curvature i s small enough that the RCM curve can be considered linear over the processed azimuth aperture, as i s the case for RADARSAT, the azimuth matched f i l t e r can be projected into the range time d i r e c t i o n . The re s u l t i n g application of the SRC f i l t e r in range instead of azimuth allows shorter and more e f f i c i e n t f i l t e r s to be used. Combination of t h i s range f i l t e r with the frequency domain RCMC interpolator leads to the new azimuth SRC algorithm. The effectiveness of t h i s algorithm is proportional to the length of the SRC/RCMC f i l t e r . An e f f e c t i v e l y longer SRC f i l t e r can be formed by combining the SRC f i l t e r with the range compression f i l t e r during range compression. This results in a new range SRC 208 algorithm. When the range pulse i s a large TBP linear FM signal as for RADARSAT the combined SRC/range compression f i l t e r d i f f e r s from the o r i g i n a l range compression f i l t e r only by a small change in linear FM rate. Computer simulations with nominal RADARSAT parameters have v e r i f i e d the accuracy of the new SRC algorithms for a variety of f i l t e r parameters and squint angles. For single-look azimuth SRC processing with a 16 point SRC/RCMC f i l t e r , i t was found that the squint angles which produce 5% and 10% range broadening can be extended from 3.65° and 4.23° respectively without SRC to 8.03° and 9.29° respectively with azimuth SRC. For single-look range SRC processing with a 16 point RCMC interpolator, the range broadening was shown to less than 1.3% for squint angles up to 20°, which i s the largest squint angle simulated. The simulations of multilook SRC processing showed very similar results indicating that the separation of looks does not greatly a f f e c t the range broadening process. Somewhat sur p r i s i n g l y , the simulations showed that n e g l i g i b l e azimuth broadening i s caused by the azimuth spectrum broadening of the azimuth FFT. This indicates that frequency domain RCMC, with or without SRC, adequately extracts the azimuth phase spectrum along the RCM curve for compression in azimuth. 209 8.1 RECOMMENDATIONS FOR FURTHER RESEARCH Since the concept, of SRC i s r e l a t i v e l y new, several areas remain to be examined further. The approximate geometric model used in t h i s thesis, which assumes a l o c a l l y f l a t earth below the radar platform, could be replaced with a more refined model which accounts for parameter variations over the curved earth. One alter n a t i v e would be to use a consistent approximation to the range migration equation based on an e l l i p s o i d a l earth model such as presented by Barber [2]. Rather than the f l a t earth hyperbolic equation used here, a Taylor series approximation to the e l l i p s o i d a l model with several terms could be used. Such a model could also be used to incorporate s a t e l l i t e motions outside of the nominal o r b i t . The refined model would be useful for deriving more accurate f i l t e r parameters, esp e c i a l l y for spaceborne SARs, and for determining more precise bounds on signal parameter errors and variations over the range and azimuth swaths. In azimuth SRC and in range/Doppler processing without SRC, the SRC/RCMC f i l t e r i s windowed in the range time domain in order to reduce the number of c o e f f i c i e n t m u l t i p l i c a t i o n s . This causes additional range broadening of the point target response. The action of the window is only p a r t i a l l y understood and i t s optimality has not been established. It may be possible to develop a method of compensating fo:: the range broadening e f f e c t s of the window by modifying the SRC/RCMC f i l t e r spectrum before windowing. 2 1 0 Since one of the eff e c t s of the window i s to taper the f i l t e r amplitude spectrum, the f i l t e r spectrum could be predistorted by amplifying the higher frequencies before windowing in order to obtain a spectrum closer to the ideal f l a t spectrum after windowing. F i n a l l y the SRC algorithm has been examined for nominal RADARSAT parameters which involve very l i t t l e range curvature over the processed azimuth bandwidth. Since the SRC f i l t e r i s derived using a linear RCM assumption and a quadratic azimuth phase assumption, range curvature and higher order effects may be more s i g n i f i c a n t in systems which exhibit larger range curvature such as longer wavelength spaceborne SAR's. Simulations of systems with much larger wavelengths should be performed to determine the l i m i t s to the above assumptions. Bibliography [1] D.A. Ausherman, A. Kozma, J.L. Walker, H.M. Jones, E.C. Poggio, "Developments in radar imaging", IEEE Trans. Vol. AES-20, No. 4, July 1984. [2] B.C. Barber, "Theory of d i g i t a l imaging from o r b i t a l synthetic-aperture radar", Int. J . Remote Sensing, Vol. 6, No. 7, 1985. [3] G.A. Bendor, T.W. Gedra, "Single-pass fine-resolution SAR autofocus", IEEE National Aerospace and El e c t r o n i c s Conference, 1983, pp. 482-8. [4] J.R. Bennett, I.G. Cumming, "A d i g i t a l processor for the production of SEASAT synthetic aperture radar imagery", Symposium on Machine Processing of Remotely Sensed Data. [5] J.R. Bennett, I.G. Cumming, " D i g i t a l techniques for the multi-look processing of SAR data with application to SEASAT-A", 5th Canadian Symposium on Remote Sensing, V i c t o r i a , B.C., Aug. 1978, pp. 506-516. [6] J.R. Bennett, I.G. Cumming, R.A. Deane, "The d i g i t a l processing of SEASAT synthetic aperture radar data", IEEE International Radar Conference, 1980, pp. 168-175. [7] D.J. Bonfield, J.R.E. Thomas, "Synthetic-aperture-radar real-time processing", IEE P r o c , Vol. 127, Pt. F, No. 2, A p r i l 1980. [8] K.R. Carver, C. E l a c h i , F.T. Ulaby, "Microwave remote sensing from space", Proc. IEEE, Vol. 73, No. 6, June 1 985. [9] C.E. Cook, M. Bernfeld, Radar signals: an introduction to theory and application , Academic Press, New York, 1967. [10] I.G. Cumming, J.R. Bennett, " D i g i t a l processing of SEASAT SAR data", Proc. of IEEE ICASSP, Washington, D.C, 1979, pp. 710-718. [11] C. E l a c h i , T. B i c k n e l l , R.L. Jordan, C. Wu, "Spaceborne 21 1 212 synthetic-aperture imaging radars: applications, techniques, and technology", Proc. IEEE, Vol. 70, No. 10, Oct. 1982. [12] F.J. Harris, "On the use of windows for harmonic analysis with the discrete Fourier transform", IEEE Proc. Vol. 66, No. 1, Jan. 1978. [13] M. J i n , C. Wu, "A SAR c o r r e l a t i o n algorithm which accomodates large-range migration", IEEE Trans. Vol. GE-22, No. 6, Nov. 1984. [14] P.F. Kavanagh, "Doppler centroid ambiguity estimation for synthetic aperture radar", Master's thesis, University of B r i t i s h Columbia, Canada, Aug. 1985. [15] J.C. Kirk, "A discussion of d i g i t a l processing in synthetic aperture radar", IEEE Trans. Vol. AES-11, No. 3, May 1975. [16] C.J. Oliver, "Fundamental properties of high-resolution sideways-looking radar", I EE P r o c , Vol. 129, Pt. F, No. 6, Dec. 1982. [17] A. Papoulis, Signal Analysis, McGraw-Hill, 1977. [18] R.K. Raney, "Processing synthetic aperture radar data", URSI Open Symposium on Remote Sensing, Washington D.C, Aug. 1981. [19] M. Sack, M.R. Ito, I.G. Cumming, "Application of e f f i c i e n t linear FM matched f i l t e r i n g algorithms to synthetic-aperture radar processing", I EE P r o c , Vol. 132, Pt. F, No. 1, Feb. 1985. [20] K. Tomiyasu, "Tu t o r i a l review of synthetic-aperture radar (SAR) with applications to imaging of the ocean surface", Proc. IEEE, Vol. 66, No. 5, May 1978. [21] W.J. van de Lindt, " D i g i t a l techniques for generating synthetic aperture radar images", IBM J. Res. Develop., Sept. 1977. 213 [22]M.R. Vant, P. George, "The RADARSAT prototype synthetic-aperture radar signal processor", IGARSS '84. [23]M.R. Vant, G.E. Haslam, "A theory of 'squinted' synthetic-aperture radar", Communications Research Centre, Dept. of Communications, Canada, Report No. 1339, Nov. 1980. [24]C. Wu, "A d i g i t a l approach to produce imagery from SAR data", AIAA System Design Driven by Sensor Conference, Paper 76-968, Pasadena, CA, Oct. 1976. [25]C. Wu, B.Barkan, W.J. Karplus, D. Caswell, "SEASAT synthetic-aperture radar data reduction using p a r a l l e l programmable array processors", IEEE Trans. Vol. GE-20, No. 3, July 1982. [26]C. Wu, K.Y. Liu, M. J i n , "Modeling and a c o r r e l a t i o n algorithm for spaceborne SAR signals", IEEE Trans. Vol. AES-18, No. 5, Sep. 1982. 

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