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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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SECONDARY RANGE COMPRESSION FOR IMPROVED RANGE/DOPPLER PROCESSING OF SAR DATA WITH HIGH SQUINT  by ALFRED RUDOLF SCHMIDT B.Sc.  E n g i n e e r i n g P h y s i c s , Queen's U n i v e r s i t y , 1981  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in  THE  FACULTY OF GRADUATE STUDIES  Department of E l e c t r i c a l  We accept  Engineering  t h i s t h e s i s as conforming  to the r e q u i r e d standard  THE  UNIVERSITY OF BRITISH COLUMBIA September 1986  ©  A l f r e d Rudolf  Schmidt, 1986  In p r e s e n t i n g  this  requirements f o r  thesis  an advanced  B r i t i s h Columbia, I  scholarly Department  reference  for extensive  purposes or  by  f i n a n c i a l gain  and  study.  copying of  be  granted  his  or  her  shall  be allowed  Department of E l e c t r i c a l The U n i v e r s i t y of B r i t i s h 2075 Wesbrook Place Vancouver, Canada V6T 1W5  1986  Engineering Columbia  the  s h a l l make I further  the  Head  of it  agree  this thesis  representatives.  permission.  Date: September  by  or p u b l i c a t i o n of not  of  the U n i v e r s i t y  the L i b r a r y  may  understood that copying  fulfilment  degree at  agree that  f r e e l y a v a i l a b l e for that p e r m i s s i o n  in p a r t i a l  of  for my  It  is  this thesis  for  without  my  written  Abstract T h i s t h e s i s examines a new a l g o r i t h m , t o be c a l l e d secondary range compression  (SRC), f o r s i g n i f i c a n t l y  improving the range r e s o l u t i o n of the range/Doppler s y n t h e t i c a p e r t u r e radar (SAR) p r o c e s s i n g a l g o r i t h m when the radar antenna  is significantly  s q u i n t e d away from the zero  Doppler d i r e c t i o n . The a l g o r i t h m was r e c e n t l y  i n t r o d u c e d by  J i n and Wu [13] f o r a p p l i c a t i o n t o the SEASAT SAR sensor. S i g n i f i c a n t e x t e n s i o n s of t h e i r a l g o r i t h m and models are presented. First  the model of range broadening i n the b a s i c  range/Doppler a l g o r i t h m i s extended by using a more general form f o r the range compressed p r o f i l e . A mathematical theory i s developed t o examine more c l o s e l y the approximations i n v o l v e d i n both b a s i c range/Doppler p r o c e s s i n g and SRC and to e x p l o r e a l t e r n a t e SRC implementations. The theory i s used to d e r i v e the SRC a l g o r i t h m as a matched f i l t e r  directly  from the p o i n t t a r g e t response model r a t h e r from the simplified  range compressed response used by J i n and Wu.  Two new d i s c r e t e  implementations (azimuth SRC and range  SRC) are developed f o r both 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 . In a d d i t i o n two new a l t e r n a t e methods of m u l t i l o o k SRC are presented : f i x e d SRC and look-dependent SRC. The s e n s i t i v i t y of the SRC a l g o r i t h m s t o parameter errors i s investigated. E x t e n s i v e s i m u l a t i o n s are developed to q u a n t i f y the image q u a l i t y produced by each a l g o r i t h m f o r a v a r i e t y of i i  p r o c e s s i n g parameters. The s i m u l a t i o n r e s u l t s with nominal RADARSAT parameters show that the SRC a l g o r i t h m s 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 a l g o r i t h m .  Table of Contents 1.  Introduction  1  1 .1 Background  1  1 . 2 Objectives  3  1.3 S t r u c t u r e of the T h e s i s 2.  3.  The S y n t h e t i c Aperture  6  Radar  (SAR) Concept  9  2.1 Model of SAR Geometry  14  2.2  22  Image Q u a l i t y Measurements  Basic Range/Doppler 3.1  Point Target  Compression  25  Response Model  26  3.2 Range Compression 3.3 S i m u l a t i o n 3.4  Single-Look  3.5 S i m u l a t i o n  28  of the Range Compressed  5.  of Single-Look  4.1  Broadening Model without SRC  4.2  Broadening S i m u l a t i o n s  34  Azimuth Compression Range/Doppler  A n a l y s i s of Broadening i n Range/Doppler f o r Range/Doppler  Compression  (SRC)  Theory of Azimuth Matched F i l t e r i n g  and SRC  of Azimuth SRC  of Range SRC  M u l t i l o o k Range/Doppler 6.1  83  ....102  112 129  5.6 Summary of Single-Look 6.  75  106  5.4 Range SRC 5.5 S i m u l a t i o n s  .75  102  5.2 Azimuth SRC 5.3 S i m u l a t i o n s  ...42 52  Compression  and Measurements  Secondary Range Compression 5.1  31  Azimuth Compression  3.6 S i m u l a t i o n R e s u l t s of B a s i c Compression 4.  Profile  SRC  P r o c e s s i n g with SRC  M u l t i l o o k P r o c e s s i n g with SRC iv  132 142 144 145  6.2 S i m u l a t i o n s of 4-Look P r o c e s s i n g without SRC ...154  7.  6.3 S i m u l a t i o n s of 4-Look, F i x e d and Look-Dependent, Azimuth SRC P r o c e s s i n g  169  6.4 S i m u l a t i o n s of 4-Look, F i x e d , Range SRC Processing  183  6.5 Summary of M u l t i l o o k SRC  188  E f f e c t s of SRC FM Rate E r r o r s  189  7.1 S e n s i t i v i t y A n a l y s i s of the SRC FM Rate  189  7.2 S i m u l a t i o n s of SRC FM Rate E r r o r Broadening 8.  ....198  Summary and C o n c l u s i o n s  206  8.1 Recommendations f o r F u r t h e r Research  209  Bibliography  211  v  L i s t of Symbols and A b b r e v i a t i o n s FM  frequency modulation  FFT  fast Fourier transform  IRW  impulse response width  ISLR  integrated sidelobe  PRF  p u l s e r e p e t i t i o n frequency  RCM  range c e l l m i g r a t i o n  RCMC  range c e l l m i g r a t i o n c o r r e c t i o n  SAR  s y n t h e t i c a p e r t u r e radar  SRC  secondary range compression  TBP  time-bandwidth  A^f)  azimuth spectrum broadening  A (f)  azimuth r e f e r e n c e f u n c t i o n broadening  0  azimuth K a i s e r - B e s s e l  2  A  ratio  product  0j  SRC/RCMC K a i s e r - B e s s e l  p\  i n c i d e n c e angle  /3  range K a i s e r - B e s s e l  B  range -3dB  R  r  function function  parameter parameter  parameter  c h i r p bandwidth  c  p r o p a g a t i o n v e l o c i t y of radar p u l s e  c,  s l o p e of RCM  curve i n azimuth time domain  c  s l o p e of RCM  curve i n azimuth frequency domain  2  D  azimuth antenna  length  f  azimuth frequency  f^.  azimuth beam c e n t e r (Doppler c e n t r o i d ) frequency  f  azimuth look c e n t e r frequency  vi  fpg  azimuth processed bandwidth  W  f  range frequency  F  complex range sampling r a t e  s r  g(t)  SRC range  g (t,f) c  combined  h  g  satellite  filter SRC/RCMC range  filter  altitude  h(t,7j)  p o i n t t a r g e t response before  h (t,7j)  azimuth point t a r g e t response  hp(t,T?)  azimuth reference  phase  h„ . (t r))  range b a n d l i m i t e d  h_(t rj)  h (t,7?)  range p o i n t t a r g e t response  h (t,7?)  range compressed  h  1-D range compressed  profile  # of SRC/RCMC f i l t e r  versions  A  T  T)  l  R  R C  R C p  (t)  I I{ 0  }  azimuth FM r a t e  K  range FM r a t e  R  p o i n t t a r g e t response  RM  modified  K  SRC/RCMC FFT a r r a y  K  K  r  S R C  function  r  z e r o t h order m o d i f i e d  K,  processing  Bessel  function  range FM r a t e f o r range SRC length  SRC FM r a t e  L  l e n g t h of SRC/RCMC f i l t e r  X  wavelength  vii  versions  M  length  of  range  N  length  of  azimuth  77  azimuth  77  p B W  FFT FFT  time  azimuth  processing  A q ^ g  -3dB  one-way  p  slant  r^.  beam  r  g  earth  radius  r  0  slant  range  range  azimuth  c e l l  center  interval  antenna  size  (  =  c/[2F  at  of  closest  approach  in  azimuth  time  r^(f)  RCM c u r v e  in  azimuth  frequency  s (t)  baseband  t  range  time  T  range  sampling  A  azimuth  radar  interval  sampling  range  pulsewidth  6^  orbit  i n c l i n a t i o n  v^  beam  v  earth  rotational  equivalent  Vg  ground  v  s a t e l l i t e  v  interval  angle  velocity  v  s  domain  pulse  T  e  )  equator  RCM c u r v e  T  ]  range  r(rj)  T  timewidth  velocity  s a t e l l i t e  v e l o c i t y  v e l o c i t y  orbital  s u b - s a t e l l i t e  v e l o c i t y  orbital  v i i i  velocity  domain  w (7j)  azimuth antenna weighting f u n c t i o n  W. ( f ) A  azimuth window f u n c t i o n  W (f )  range window f u n c t i o n  *  2-D c o n v o l u t i o n  *  azimuth c o n v o l u t i o n  *  range c o n v o l u t i o n  a  R  r  ix  List  Table  of T a b l e s  1 . Nominal RADARSAT parameters  x  L i s t of F i g u r e s F i g u r e 2.1. S p h e r i c a l e a r t h model f o r v e l o c i t y  calculation. 16  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 geometry  18  F i g u r e 2.3. S l a n t range plane  19  F i g u r e 3.1. Simulated range compressed  profile  F i g u r e 3.2. Range c e l l m i g r a t i o n curve i n azimuth time domain  35 37  F i g u r e 3.3. Range c e l l m i g r a t i o n curve i n azimuth frequency domain 40 F i g u r e 3.4. Range c e l l m i g r a t i o n curve i n d i s c r e t e azimuth frequency domain 50 F i g u r e 3.5. Simulated azimuth antenna weighting, and antenna p l u s RCM weightings i n range c e l l nearest t o beam center range f o r 5° squint 56 F i g u r e 3.6. Azimuth spectrum before RCMC i n range c e l l nearest t o beam c e n t e r range f o r 5° s q u i n t . . 57 F i g u r e 3.7. Windowed range i n t e r p o l a t o r  i n c l u d i n g 16  f r a c t i o n a l l y shifted versions F i g u r e 3.8. Azimuth spectrum a f t e r RCMC f o r 5° s q u i n t . F i g u r e 3.9. Azimuth r e f e r e n c e phase  58 . 59  f i l t e r spectrum f o r 5°  squint  60  F i g u r e 3.10. Azimuth K a i s e r - B e s s e l window f o r 5° s q u i n t .  61  F i g u r e 3.11. Azimuth spectrum a f t e r RCMC, matched f i l t e r , and windowing 62 F i g u r e 3.12. F u l l y compressed range p r o f i l e f o r 0° and 5° squint 63 F i g u r e 3.13. F u l l y compressed azimuth p r o f i l e f o r 0° and 5° squint 64 F i g u r e 3.14. Range broadening of p o i n t t a r g e t response without SRC  65  F i g u r e 3.15. Range broadening of p o i n t t a r g e t response (expanded) without SRC  66  F i g u r e 3.16. Azimuth broadening of p o i n t t a r g e t response without SRC  67  xi  F i g u r e 3.17. 1-D range i n t e g r a t e d s i d e l o b e r a t i o s without SRC 68 F i g u r e 3.18. 1-D azimuth i n t e g r a t e d s i d e l o b e r a t i o s without SRC 69 F i g u r e 3.19. 2-D i n t e g r a t e d s i d e l o b e r a t i o s without SRC.  70  F i g u r e 3.20. 1-D range peak s i d e l o b e r a t i o s without SRC.  71  F i g u r e 3.21. 1-D azimuth peak s i d e l o b e r a t i o s without SRC. 72 F i g u r e 3.22. 2-D peak s i d e l o b e r a t i o s without SRC.  . . . 73  F i g u r e 3.23. Degradation of peak magnitude without SRC. . 74 F i g u r e 4.1. Range broadening of the simulated, t h e o r e t i c a l , range broadening f u n c t i o n 85 F i g u r e 4.2. P r e d i c t e d and a c t u a l azimuth s p e c t r a f o r 0° squint  86  F i g u r e 4.3. P r e d i c t e d and a c t u a l azimuth s p e c t r a f o r 1° squint  87  F i g u r e 4.4. P r e d i c t e d and a c t u a l azimuth s p e c t r a f o r 5° squint  88  F i g u r e 4.5. P r e d i c t e d and a c t u a l azimuth s p e c t r a f o r 10° squint  89  F i g u r e 4.6. P r e d i c t e d and a c t u a l azimuth s p e c t r a f o r 15° squint  90  F i g u r e 4.7. Measured azimuth spectrum broadening i n the azimuth frequency d i r e c t i o n i n the near, beam center (mid), and f a r range c e l l s 92 F i g u r e 4.8. Measured azimuth spectrum broadening i n the azimuth frequency d i r e c t i o n i n the near, beam center (mid), and f a r range c e l l s (expanded). 93 F i g u r e 4.9. Measured azimuth spectrum broadening i n the range time d i r e c t i o n a t the lower, the beam c e n t e r , and the upper processed bandwidth frequencies  94  F i g u r e 4.10. Measured azimuth spectrum broadening i n the range time d i r e c t i o n a t the lower, the beam c e n t e r , and the upper processed bandwidth f r e q u e n c i e s (expanded) 95  xi i  F i g u r e 4 . 1 1 . P o i n t s on the azimuth frequency domain RCM curve used f o r spectrum broadening measurements. 96 F i g u r e 4.12. Measured azimuth time-bandwidth product versus s q u i n t angle  (TBP) 98  F i g u r e 4.13. Azimuth broadening p r e d i c t e d by decrease i n azimuth processed bandwidth 100 F i g u r e 4.14. Range c e l l m i g r a t i o n  over processed  aperture. 101  F i g u r e 5 . 1 . Magnitudes of the SRC/RCMC f i l t e r s of length 16 f o r squint angles of 0°, 5°, 10°, 15°, and 20°. 111 F i g u r e 5.2. 1-D range p r o f i l e s a f t e r SRC compression f o r 5° squint and v a r i o u s f i l t e r lengths 115 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 compression f o r 10° squint and v a r i o u s f i l t e r lengths 116 F i g u r e 5.4. 1-D azimuth p r o f i l e s a f t e r SRC compression f o r 5° squint and v a r i o u s f i l t e r lengths 117 F i g u r e 5.5. 1-D azimuth p r o f i l e s a f t e r SRC compression f o r 10° squint and v a r i o u s f i l t e r l e n g t h s . . . .118 F i g u r e 5.6. Percentage range broadening with SRC as a f u n c t i o n of s q u i n t angle  119  F i g u r e 5.7. Percentage range broadening with SRC as a f u n c t i o n of s q u i n t angle (expanded s c a l e ) .  .120  F i g u r e 5.8. Percentage azimuth broadening with SRC as a f u n c t i o n of s q u i n t angle  121  F i g u r e 5.9. 1-D range ISLR as a f u n c t i o n of squint angle f o r v a r i o u s f i l t e r lengths 122 F i g u r e 5.10. 1-D azimuth ISLR as a f u n c t i o n of squint f o r v a r i o u s f i l t e r lengths  angle 123  F i g u r e 5.11. 2-D ISLR as a f u n c t i o n of squint angle f o r v a r i o u s f i l t e r lengths  124  F i g u r e 5.12. 1-D range PSLR as a f u n c t i o n of squint f o r v a r i o u s f i l t e r lengths  125  angle  F i g u r e 5.13. 1-D azimuth PSLR as a f u n c t i o n of s q u i n t f o r v a r i o u s f i l t e r lengths  angle 126  F i g u r e 5.14. 2-D PSLR as a f u n c t i o n of squint angle f o r xi i i  various  filter  lengths  127  F i g u r e 5.15. Peak compressed magnitude with SRC as a f u n c t i o n of squint angle f o r v a r i o u s f i l t e r lengths  1 28  F i g u r e 5.16. 1-D range compressed p r o f i l e s a f t e r range compression with range SRC f o r 5° and 10° squint  134  F i g u r e 5.17. 1-D range p r o f i l e s a f t e r azimuth compression with range SRC f o r 0°, 5°, and 10° s q u i n t and length 16 RCMC f i l t e r 135 F i g u r e 5.18. 1-D azimuth p r o f i l e s a f t e r azimuth compression with range SRC f o r 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 i n t e r p o l a t o r as a f u n c t i o n of squint angle 137  F i g u r e 5.20. Percentage azimuth broadening w i t h range SRC and a l e n g t h 16 RCMC i n t e r p o l a t o r as a f u n c t i o n of squint angle 138 F i g u r e 5.21. Range, azimuth, and 2-D i n t e g r a t e d s i d e l o b e r a t i o s with range SRC and a l e n g t h 16 RCMC i n t e r p o l a t o r as a f u n c t i o n of s q u i n t a n g l e . .139 F i g u r e 5.22. Range, azimuth, and 2-D peak s i d e l o b e r a t i o s with range SRC and a l e n g t h 16 RCMC i n t e r p o l a t o r as a f u n c t i o n of s q u i n t angle 140 Figure  5.23. Peak magnitude d e g r a d a t i o n with range SRC and a l e n g t h 16 RCMC i n t e r p o l a t o r as a f u n c t i o n of squint angle 141  F i g u r e 6.1. D i v i s i o n of the azimuth aperture i n t o 4 looks  frequency-domain  F i g u r e 6.2. Corresponding time-domain  146 looks  147  F i g u r e 6.3. I n t e r p o l a t e d 1-D range p r o f i l e s a f t e r 4-look compression without SRC f o r squint angles of 0' 5°, and 10° 157 F i g u r e 6.4. I n t e r p o l a t e d 1-D azimuth p r o f i l e s a f t e r 4-look compression without SRC f o r squint angles of 0°, 5°, and 10° 158 F i g u r e 6.5. Range broadening f o r s i n g l e - l o o k and 4-look compression without SRC xiv  159  Figure  6.6. Range broadening f o r s i n g l e - l o o k and 4-look compression without SRC (expanded s c a l e ) . . .160  Figure  6.7. Azimuth broadening f o r s i n g l e - l o o k and 4-look compression without SRC 161  Figure  6.8. 1-D range i n t e g r a t e d s i d e l o b e r a t i o s f o r s i n g l e - l o o k and 4-look compression without SRC. 162  Figure  6.9. 1-D azimuth i n t e g r a t e d s i d e l o b e r a t i o s f o r s i n g l e - l o o k and 4-look compression without SRC. 163  Figure  6.10. 2-D i n t e g r a t e d s i d e l o b e r a t i o s f o r s i n g l e - l o o k and 4-look compression without SRC 164  Figure  6.11. 1-D range peak s i d e l o b e r a t i o s f o r s i n g l e - l o o k and 4-look compression without SRC 165  Figure  6.12. 1-D azimuth peak s i d e l o b e r a t i o s f o r s i n g l e - l o o k and 4-look compression without SRC. 166  Figure  6.13. 2-D peak s i d e l o b e r a t i o s f o r s i n g l e - l o o k and 4-look compression without SRC 167  Figure  6.14. Peak magnitude r a t i o s f o r s i n g l e - l o o k and 4-look compression without SRC  168  Figure  6.15. I n t e r p o l a t e d 1-D range p r o f i l e s a f t e r 4-look compression with both f i x e d and look-dependent SRC f o r squint angles of 0°, 5°, and 10°. . .171  Figure  6.16. I n t e r p o l a t e d 1-D azimuth p r o f i l e s a f t e r 4-look compression with both f i x e d and look-dependent SRC f o r s q u i n t angles of 0°, 5°, and 10°. . .172  Figure  6.17. Range broadening f o r 4-look compression w i t h both f i x e d and look-dependent SRC and v a r i o u s SRC/RCMC f i l t e r lengths 173  Figure  6.18. Azimuth broadening f o r 4-look compression with both f i x e d and look-dependent SRC and v a r i o u s SRC/RCMC f i l t e r lengths 174  Figure  6.19. Comparison of range -3dB widths a t 0° squint for 4-look compression with both f i x e d and look-dependent SRC and v a r i o u s SRC/RCMC f i l t e r lengths 175  Figure  6.20. 1-D range i n t e g r a t e d s i d e l o b e r a t i o s f o r 4-look compression with both f i x e d and look-dependent SRC and v a r i o u s SRC/RCMC f i l t e r l e n g t h s . . .176 xv  F i g u r e 6.21. 1-D azimuth i n t e g r a t e d s i d e l o b e r a t i o s f o r 4-look compression with both f i x e d and look-dependent SRC and v a r i o u s SRC/RCMC f i l t e r lengths 177 F i g u r e 6.22. 2-D i n t e g r a t e d s i d e l o b e r a t i o s f o r 4-look compression with both f i x e d and look-dependent SRC and v a r i o u s SRC/RCMC f i l t e r l e n g t h s . . .178 F i g u r e 6.23. 1-D range peak s i d e l o b e r a t i o s f o r 4-look compression with both f i x e d and look-dependent SRC and v a r i o u s SRC/RCMC f i l t e r l e n g t h s . . .179 F i g u r e 6.24. 1-D azimuth peak s i d e l o b e r a t i o s f o r 4-look compression with both f i x e d and look-dependent SRC and v a r i o u s SRC/RCMC f i l t e r l e n g t h s . . .180 F i g u r e 6.25. 2-D peak s i d e l o b e r a t i o s f o r 4-look compression with both f i x e d and look-dependent SRC and v a r i o u s SRC/RCMC f i l t e r lengths 181 F i g u r e 6.26. Peak magnitude r a t i o s f o r 4-look compression with both f i x e d and look-dependent SRC and v a r i o u s SRC/RCMC f i l t e r lengths 182 F i g u r e 6.27. Range and azimuth broadening f o r 4-look compression with range SRC and a l e n g t h 16 RCMC interpolator .184 F i g u r e 6.28. Range, azimuth, and 2-D i n t e g r a t e d s i d e l o b e r a t i o s f o r 4-look compression with range SRC and a l e n g t h 16 RCMC i n t e r p o l a t o r 185 F i g u r e 6.29. Range, azimuth, and 2-D peak s i d e l o b e r a t i o s for 4-look compression with range SRC and a l e n g t h 16 RCMC i n t e r p o l a t o r 186 F i g u r e 6.30. Peak magnitude d e g r a d a t i o n f o r 4-look compression with range SRC and a l e n g t h interpolator  16 RCMC 187  F i g u r e 7.1. SRC band-edge phase e r r o r i n the range frequency domain f o r squint angles of 1°, 5°, 10°, 15°, and 20° as a f u n c t i o n of beam c e n t e r frequency error 1 92 F i g u r e 7.2. SRC band-edge phase e r r o r i n the range frequency domain f o r squint angles of 1°, 5°, 10°, 15°, and 20° as a f u n c t i o n of r e r r o r 193 0  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 frequency domain as a f u n c t i o n of s q u i n t angle 196 •xvi  Figure  7.4. A c t u a l and p r e d i c t e d range broadening without SRC as a f u n c t i o n of e q u i v a l e n t range band-edge phase e r r o r . . . 197  Figure  7.5. Range broadening with s i n g l e - l o o k azimuth SRC a t 5° of squint with v a r i o u s SRC/RCMC f i l t e r lengths as a f u n c t i o n of range band-edge phase error 201  Figure  7.6. Range broadening with s i n g l e - l o o k azimuth SRC at 10° of squint with v a r i o u s SRC/RCMC f i l t e r lengths as a f u n c t i o n of range band-edge phase error 202  Figure  7.7. Range broadening with m u l t i l o o k azimuth SRC at 10° of squint with v a r i o u s SRC/RCMC f i l t e r lengths as a f u n c t i o n of range band-edge phase error 203  Figure  7.8. Range broadening with m u l t i l o o k azimuth SRC at 10° of squint with v a r i o u s SRC/RCMC f i l t e r lengths as a f u n c t i o n of range band-edge phase error 204  Figure  7.9. Range broadening with s i n g l e - l o o k range SRC at 5°, 10°, 15°, and 20° of squint with a length 16 RCMC i n t e r p o l a t o r as a f u n c t i o n of range band-edge phase e r r o r 205  xvi i  Acknowledgements I (fish t o thank my s u p e r v i s o r , Dr. M.R.  Ito, for h i s  help and encouragement throughout the course of t h i s research.  I a l s o wish to thank Dr. I.G. Cumming of  MacDonald, D e t t w i l e r and A s s o c i a t e s  (MDA) f o r suggesting the  t h e s i s t o p i c and f o r p r o v i d i n g numerous i n s i g h t f u l suggestions. (CRC)  Dr. M. Vant of Communications Research Centre  a l s o provided  helpful discussions.  I am g r a t e f u l f o r the f i n a n c i a l a s s i s t a n c e provided by MDA,  the U n i v e r s i t y of B r i t i s h Columbia, CRC, and Computing  Devices Company (ComDev) of Ottawa.  xvi i i  1. INTRODUCTION T h i s t h e s i s d i s c u s s e s the a p p l i c a t i o n of a secondary range compression  new  (SRC) a l g o r i t h m t o the  range/Doppler s y n t h e t i c a p e r t u r e radar (SAR)  processing  a l g o r i t h m . P a r t i c u l a r c o n s i d e r a t i o n i s given to the a p p l i c a t i o n of SRC  to Canada's proposed RADARSAT SAR  sensor.  1.1 BACKGROUND The range/Doppler a l g o r i t h m method f o r the e f f i c i e n t compressed  [6,18,19,24,26] i s a w e l l known  compression of SAR d a t a . The  p o i n t t a r g e t response of the a l g o r i t h m becomes  s e v e r e l y broadened  i n the range d i r e c t i o n when the  azimuth time-bandwidth than u n i t y ) . SRC  product (TBP) becomes small  i s a new  e f f i c i e n t algorithm  -3dB (less  which  s i g n i f i c a n t l y reduces the range broadening and a s s o c i a t e d image q u a l i t y d e g r a d a t i o n s . The TBP has a m u l t i t u d e of d e f i n i t i o n s depending on the d e s i r e d use. For t h i s t h e s i s the TBP product of the a c t u a l -3dB  i s d e f i n e d as the  widths i n the time and frequency  domains. The TBP of any s i g n a l has a f i x e d lower bound which i s somewhat l e s s than 0.5 small azimuth TBP may cell  using the c u r r e n t d e f i n i t i o n . A  occur i n SAR  m i g r a t i o n (RCM). RCM  when t h e r e i s l a r g e  i s the m i g r a t i o n of a point  t a r g e t ' s energy through more than one range r e s o l u t i o n over the antenna  i l l u m i n a t i o n p e r i o d . Large RCM  may  NASA's SEASAT SAR  cell  occur i n  spaceborne SARs, such as Canada's proposed RADARSAT SAR or  range  [22]  [11], when the e q u i v a l e n t squint angle  1  2  of  the antenna  r e l a t i v e t o the zero Doppler d i r e c t i o n i s  large. Large RCM causes the azimuth time exposure of a p o i n t target  i n a given range c e l l  to decrease s i n c e the t a r g e t  energy migrates r a p i d l y a c r o s s range c e l l s .  Under l a r g e  azimuth TBP c o n d i t i o n s , the time and frequency domain azimuth s i g n a l s e x h i b i t an approximate  one-to-one  correspondence. Thus the decrease i n azimuth timewidth r e s u l t s i n a decreased bandwidth and consequently a decreased azimuth TBP. The time-frequency correspondence i s p r e d i c t e d by the p r i n c i p l e of s t a t i o n a r y phase [17] f o r l a r g e TBP s i g n a l s . When the azimuth TBP f a l l s below u n i t y , the  correspondence i s 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 p r e d i c t e d  bandwidth. T h i s spectrum broadening appears a f t e r the azimuth FFT, which i s used f o r f a s t azimuth c o n v o l u t i o n i n the  range/Doppler  algorithm.  S i n c e the p o i n t t a r g e t energy l i e s along a sloped curve (the  RCM curve) i n both range-time/azimuth-time and  range-time/azimuth  frequency space, the azimuth  spectrum  broadening causes the p o i n t t a r g e t energy to be s i m i l a r l y broadened  i n the range time d i r e c t i o n . I f the range  broadening i s l e f t  u n c o r r e c t e d as i n the b a s i c  a l g o r i t h m , the f i n a l becomes broadened SRC  compressed p o i n t t a r g e t  range/Doppler  response  i n range.  was f i r s t proposed by J i n and Wu [13] i n 1984 as a  method f o r extending the maximum squint angle which can be  3  p r o c e s s e d by t h e r a n g e / D o p p l e r  algorithm.  Their  paper  p r e s e n t e d a model f o r the range broadening of the compressed  range  point target response a f t e r the azimuth  transform. T h e i r improved model accounted f o r the in magnitude  Fourier broadening  and phase of the azimuth spectrum. However  model i d e a l i z e d the range compressed infinite duration  p r o f i l e as a  the  simple  sine f u n c t i o n . This approximation does  i n c l u d e the e f f e c t s of the range and azimuth windows  not  which  are u s u a l l y a p p l i e d in the r e s p e c t i v e frequency domains to control sidelobe levels. Also several  approximations  r e q u i r e d to d e v e l o p the model were not f u l l y s t a t e d or validated. A continuous time, i n f i n i t e duration, compression compressed  f i l t e r matched s i g n a l m o d e l was  s e p a r a t e SRC  filter  time convolution  to the approximate d e r i v e d . The  w h i c h was  azimuth  range  filter  included  a p p l i e d as a c o n t i n u o u s  to the azimuth spectrum.  d i s c r e t e implementation of t h i s f i l t e r  range  The d e t a i l s of  were not  SAR  parameters.  OBJECTIVES  The o b j e c t i v e s of t h i s t h e s i s a r e summarized  as  1.  i s t o be  The  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  extended profile  to a c c u r a t e l y model the range  follows:  compressed  i n c l u d i n g the e f f e c t s of range windowing.  a p p r o x i m a t i o n s and assumptions  a r e t o be  the  presented.  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 were shown f o r SEASAT  1.2  a  All  explicitly  4  2.  s t a t e d and  examined f o r t h e i r  J i n and Wu  s t a t e that the range compressed p o i n t  response may reference  range of v a l i d i t y .  be used as the azimuth matched  function  (see equation  (20)  target  filter  i n [ 1 3 ] ) . 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 f u n c t i o n and bandlimiting  no  range  or windowing i s a p p l i e d during  compression. The  azimuth f i l t e r  f o r the more general  i s to be  range  reformulated  case by matching the  filter  d i r e c t l y to the p o i n t t a r g e t response before  range  compression.  3.  The  b a s i c SRC  algorithm  i s to be r e d e r i v e d using  extended azimuth matched f i l t e r for  implementing SRC  terms of e f f i c i e n c y  4.  The  SRC  algorithm  model. A l t e r n a t e methods  are to be examined and and  Wu  as a d i s c r e t e f i n i t e length SRC  developed. In p a r t i c u l a r , with the  5.  J i n and Wu  This algorithm  filter  implementing filter  this  are to  combinations of the SRC  frequency domain RCMC i n t e r p o l a t o r are  explored.  in  c o n s i s t e d of a  with a continuous-time SRC  of i n f i n i t e time d u r a t i o n . Methods of filter  evaluated  accuracy.  d e r i v e d by J i n and  range time c o n v o l u t i o n  the  w i l l be c a l l e d azimuth  s t a t e that i t i s p o s s i b l e to perform  be  filter  to  be  SRC.  SRC  5  during an  range c o m p r e s s i o n  a l g o r i t h m . The p o s s i b i l i t i e s  during  range c o m p r e s s i o n  evaluated.  6.  b u t do n o t d e r i v e t h e t h e o r y o r  to the coherent  into  incoherently  to high  nature  reduce the s p e c k l e subdivided  a r e t o be e x p l o r e d a n d  This algorithm w i l l  SAR images a r e p r o n e  o f i m p l e m e n t i n g SRC  be c a l l e d  levels  of t h e r a d a r  noise  separate  r a n g e SRC.  of speckle  n o i s e due  i l l u m i n a t i o n . To  the processed  aperture  i s often  " l o o k s " which a r e then  summed. New methods o f i m p l e m e n t i n g SRC  with multilook  r a n g e / D o p p l e r p r o c e s s i n g a r e t o be  developed.  7.  The s i g n a l contain constant of  8.  parameters  quantitative  t o parameter  f o r a given  design  n e e d e d . Computer  curves  processing  image q u a l i t y of expected  algorithms  requirement,  image q u a l i t y a r e  s i m u l a t i o n s a r e t o be p e r f o r m e d i n  t o q u a n t i f y t h e f o l l o w i n g items being  The  image d e g r a d a t i o n s  simulated  by t h e u s e o f  e r r o r s i s t o be e x a m i n e d .  consideration  be  SAR s y s t e m s may  i n b l o c k p r o c e s s i n g . The s e n s i t i v i t y  t o choose t h e necessary  parameters  order  in practical  e s t i m a t i o n e r r o r s or e r r o r s caused  t h e SRC f i l t e r  In o r d e r and  p a r a m e t e r s used  given  with  particular  t o t h e RADARSAT SAR  sensor.  a n d improvements a r e t o  measured a s a f u n c t i o n of s q u i n t  angle:  6 a.  measure t h e range compressed p o i n t range/Doppler limitations  and a z i m u t h target  broadening  response  of the b a s i c  a l g o r i t h m to determine  of both  single-look  of the  the squint  and m u l t i l o o k  algorithms  b.  determine in  c.  comparison  quantify SRC  t h e a c c u r a c y o f t h e range  and m u l t i l o o k  with v a r i o u s p r o c e s s i n g parameters  errors  1.3 STRUCTURE OF THE  matched  filtering  into  variations  of t a r g e t  outlined.  sections.  o f SAR  FM  range  The  Chapter  2  image f o r m a t i o n a s a  2-D  radar s i g n a l which  i n b o t h d i m e n s i o n s . A model o f  geometry i s d e v e l o p e d  parameters.  interest  several  concept  approximately linear SAR  SRC  for a l lalgorithms.  o p e r a t i o n of a r e c e i v e d  spaceborne  by  THESIS  i s divided  introduces the basic  signal  measurements  examine t h e image d e g r a d a t i o n s c a u s e d  The t h e s i s  model  improvements f o r t h e new  algorithms for single-look  parameter  is  broadening  t h e image q u a l i t y  processing  d.  with actual  broadening  to d e f i n e the  with azimuth  image q u a l i t y  t i m e and o t h e r key  measurements of  a r e i n t r o d u c e d and t h e measurement  procedures are  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 s i g n a l r e t u r n e d from a point t a r g e t i s developed t o use as a r e f e r e n c e f u n c t i o n f o r the matched f i l t e r .  The model i s a l s o  used i n the s i m u l a t i o n s f o r c r e a t i n g a s i m u l a t e d range compressed  p o i n t t a r g e t response. The r e s u l t s of e x t e n s i v e  s i m u l a t i o n s with nominal RADARSAT parameters are summarized. Chapter 4 develops a mathematical model of the azimuth spectrum broadening and range broadening which occurs with the  b a s i c range/Doppler a l g o r i t h m at l a r g e squint a n g l e s . A  simple a c c u r a t e model of azimuth broadening i s a l s o presented. Chapter 5 extends the theory used i n chapter 4 to develop an improved matched f i l t e r  which  i n c l u d e s SRC.  Two  new  a l t e r n a t e techniques f o r the d i s c r e t e implemention of  SRC  (azimuth SRC and range SRC)  are presented. E x t e n s i v e  s i m u l a t i o n r e s u l t s are d i s c u s s e d to e v a l u a t e the  new  algorithms. Chapter 6 examines the a p p l i c a t i o n of SRC range/Doppler compression. The new  to m u l t i l o o k  concepts of f i x e d  (look-independent) and look-dependent  SRC  filters  p r e s e n t e d . S i m u l a t i o n s of m u l t i l o o k compression  are (with  4-looks) with and without SRC are performed to q u a n t i f y the improvements. Chapter 7 examines the e f f e c t s of SRC  parameter  e s t i m a t i o n e r r o r s and block p r o c e s s i n g i n v a r i a n c e r e g i o n s on image q u a l i t y . A model i s developed to r e l a t e these e r r o r s  8  to e q u i v a l e n t phase e r r o r s which occur i n p r o c e s s i n g without SRC. F i n a l l y chapter 8 presents f i n a l c o n c l u s i o n s and suggests areas f o r f u r t h e r r e s e a r c h on  SRC.  2 . THE  SYNTHETIC APERTURE RADAR (SAR)  CONCEPT  S y n t h e t i c a p e r t u r e radar p r o c e s s i n g i s a method of obtaining  image r e s o l u t i o n s much f i n e r than  the  along-track  beamwidth of the radar antenna from a moving p l a t f o r m . I t has been s u c c e s s f u l l y a p p l i e d to both a i r b o r n e  and  spaceborne radars to provide a l l - w e a t h e r high r e s o l u t i o n imaging c a p a b i l i t i e s . Strip-map mode sensors, such as RADARSAT and  SEASAT, o r i e n t the b o r e s i g h t of the antenna  p e r p e n d i c u l a r to the d i r e c t i o n of t r a v e l of the p l a t f o r m , i.e.,  i n the c r o s s - t r a c k d i r e c t i o n , and o f f to one  s i d e of  the ground t r a c k . An azimuth  ( a l o n g - t r a c k ) antenna aperture much l a r g e r  than the s i z e of the p h y s i c a l antenna i s s y n t h e s i z e d p r o p e r l y combining the r e c e i v e d radar p u l s e s over a i n t e g r a t i o n p e r i o d with a p p r o p r i a t e weighting. resolution  i s i n v e r s e l y p r o p o r t i o n a l to the  aperture l e n g t h . The  The  by coherent  azimuth  synthesized  r e t u r n s from p o i n t t a r g e t s at d i f f e r e n t  ground p o s i t i o n s are r e s o l v e d i n range ( c r o s s - t r a c k ) by d i f f e r e n c e s i n the time delay of the t r a n s m i t t e d p u l s e s and The modulated  i n azimuth by t h e i r Doppler  shift.  radar pulse i s t y p i c a l l y a l i n e a r l y (FM)  pulse with l a r g e TBP.  radar  frequency  In range/Doppler  p r o c e s s i n g the r e c e i v e d p u l s e s are compressed i n range using standard p u l s e compression techniques compressed s i g n a l with a small TBP. s i m i l a r modulation  (approximately  The  azimuth s i g n a l has a  l i n e a r FM)  changing d i s t a n c e between the sensor  9  to get a range  and  due  to the  t a r g e t . By a p p l y i n g  10 p u l s e compression techniques i n the azimuth d i r e c t i o n a w e l l r e s o l v e d image can be o b t a i n e d . An a d d i t i o n a l c o m p l i c a t i o n of the p r o c e s s i n g occurs when the change i n range t o a p o i n t t a r g e t over the azimuth i n t e g r a t i o n p e r i o d i s l a r g e r than the range c e l l effect, called  size. This  range c e l l m i g r a t i o n (RCM), causes the s i g n a l  energy from a p o i n t t a r g e t a f t e r range compression to migrate a c r o s s s e v e r a l range c e l l s . Consequently the azimuth compression becomes a 2-D o p e r a t i o n . The b a s i c  range/Doppler  a l g o r i t h m s e p a r a t e s t h i s 2-D azimuth o p e r a t i o n  i n t o two 1-D  operations :  1.  Range C e l l M i g r a t i o n C o r r e c t i o n range compressed  2.  (RCMC) i n which the  p o i n t t a r g e t energy i s i n t e r p o l a t e d and  shifted  i n range so that the energy l i e s along a s i n g l e  azimuth  line.  Azimuth c o r r e l a t i o n with a 1-D r e f e r e n c e phase f u n c t i o n .  For computational e f f i c i e n c y these o p e r a t i o n s are performed  i n the azimuth frequency domain v i a f a s t  c o n v o l u t i o n . For l a r g e azimuth TBP, l i n e a r FM type s i g n a l s , the magnitude and phase c h a r a c t e r i s t i c s of the azimuth frequency domain s i g n a l s can be simply r e l a t e d t o the azimuth time domain s i g n a l s by a l i n e a r s c a l e f a c t o r the p r i n c i p l e of s t a t i o n a r y phare  using  [17]. In such c a s e s , which  occur when t h e r e i s l i t t l e RCM and a small s q u i n t angle, a  11  zero-phase s i n c - t y p e i n t e r p o l a t o r can be used f o r RCMC. Block p r o c e s s i n g e f f i c i e n c y can be a c h i e v e d s i n c e the t r a j e c t o r i e s of point t a r g e t s which are adjacent to each other  i n azimuth time f o l l o w a common RCM curve i n the  azimuth frequency domain. T h i s a l l o w s RCMC and azimuth compression t o be a p p l i e d t o many t a r g e t s s i m u l t a n e o u s l y . However when the azimuth time exposure  i n a range  becomes small due to RCM and the azimuth TBP f a l l s  cell  below  u n i t y , the magnitude and phase c h a r a c t e r i s t i c s of the frequency domain s i g n a l become broadened  r e l a t i v e to t h e i r  c o r r e s p o n d i n g time domain s i g n a l s . The broadened domain s i g n a l can be p r o p e r l y compressed  frequency  by a p p l y i n g a  secondary compression i n e i t h e r the range time or azimuth frequency d i r e c t i o n . S i n c e the width i n samples broadened  function  i s much smaller i n range than i n azimuth,  the  filter  i s more e f f i c i e n t l y  the  name secondary range compression The SRC f i l t e r  q u a d r a t i c phase  of the  implemented  i n range, hence  (SRC).  can be viewed as c o n v o l u t i o n with a  range f i l t e r  which recompresses the  broadening which occurs i n the azimuth F o u r i e r t r a n s f o r m . Two e f f i c i e n t  implementations of t h i s secondary range  have been i n v e s t i g a t e d :  1.  Azimuth SRC i n which the secondary range f i l t e r i s combined  with the RCMC i n t e r p o l a t o r d u r i n g  compression.  azimuth  filter  12  2.  Range SRC where the secondary range f i l t e r  i s combined  with the range frequency domain, r e f e r e n c e  function  d u r i n g range compression.  The next s e c t i o n d i s c u s s e s a model of the spaceborne SAR  sensor geometry  which i s used t o d e r i v e the range c e l l  m i g r a t i o n equation ( i . e . , the v a r i a t i o n of range with azimuth time) and other s i g n a l parameters. For the simulations in l a t e r  s e c t i o n s , a set of nominal RADARSAT  parameters has been chosen. These are l i s t e d  in table  1.  13  Parameter  range -3dB c h i r p  Symbol  Value  Units  Br  17.28  MHz  *A  1.5  bandwidth  s i n g l e - l o o k azimuth K a i s e r - B e s s e l parameter m u l t i l o o k azimuth K a i s e r - B e s s e l parameter SRC/RCMC K a i s e r - B e s s e l parameter  2.7  i n c i d e n c e angle range K a i s e r - B e s s e l  parameter  azimuth antenna l e n g t h azimuth processed bandwidth  (0 =O°) S  altitude  K  s r  e a r t h r a d i u s at equator s l a n t range of c l o s e s t approach range p u l s e w i d t h o r b i t i n c l i n a t i o n angle beam v e l o c i t y  equivalent  orbital  PBW  f  I  r e p e t i t i o n frequency range c e l l s i z e ( c / [ 2 F ] )  satellite  20  s  l e n g t h of SRC/RCMC f i l t e r v e r s i o n s l e n g t h of range FFT l e n g t h of s i n g l e - l o o k azimuth FFT wavelength azimuth p r o c e s s i n g i n t e r v a l  earth r o t a t i o n a l  Pi  Fsr h  # of SRC/RCMC f i l t e r v e r s i o n s SRC/RCMC FFT array l e n g t h  pulse slant  2.5  0R D  complex range sampling rate satellite  01  velocity velocity  satellite  velocity  azimuth oversampling f a c t o r range oversampling f a c t o r Table 1. Nominal Radarsat Parameters  r  L M N X ^PBW PRF Psr  e r  r  deg  2.7 14.0 942  m Hz  19.872  MHz  1007.4  km  16 128 4,8,16,32 2048 1024 0.05656 0.513  m s  1 177.9 7.543  Hz m  63716  km km *xs deg  «i  1 072. 1 36.4 99.5  V  b  7.4575  km/s  v  e  0.4638  km/s  v  s  7.35  km/s  v  eq  7.4575  km/s  0  T  1 .25 1.15  14  2.1  MODEL OF SAR GEOMETRY  The  production  accurate  of high r e s o l u t i o n SAR images r e q u i r e s an  model of the p h y s i c a l geometry of the space  p l a t f o r m with  respect  uses a s i m p l i f i e d  t o the e a r t h ' s  surface. This  study  f l a t e a r t h geometric model which e x h i b i t s  the e s s e n t i a l p r o p e r t i e s of SAR s i g n a l s . A more sophisticated spherical earth, c i r c u l a r orbit model i s used t o d e r i v e accurate  estimates  geometric  of the a c t u a l  s i g n a l parameters which are then a p p l i e d t o the s i m p l i f i e d model. The  s p h e r i c a l e a r t h , c i r c u l a r o r b i t geometric model i s  used t o d e r i v e an e q u i v a l e n t  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 p o i n t and the s u r f a c e of the e a r t h . A f t e r mapping i n t o the s l a n t range plane  (the plane c o n t a i n i n g the  p l a t f o r m v e l o c i t y v e c t o r and the v e c t o r and  j o i n i n g the p l a t f o r m  a p o i n t t a r g e t on the ground), the e q u i v a l e n t 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  e a r t h under the s a t e l l i t e . Second order l o c a l curvature  of the e a r t h , or v a r i a t i o n s i n the earth's  r a d i u s or s a t e l l i t e h e i g h t from the s i m u l a t i o n s insight  e f f e c t s caused by  above the s u r f a c e a r e excluded  s i n c e the added complexity  i n t o the range broadening p r o c e s s .  Vant  modified  of the s i g n a l parameters. [23] p r o v i d e s  a good d i s c u s s i o n of a s i m i l a r  s p h e r i c a l e a r t h model. For s i m p l i c i t y , position  little  These secondary  e f f e c t s can u s u a l l y be accounted f o r by using estimates  adds  the s a t e l l i t e  i s a r b i t r a r i l y chosen t o be above the equator  since  15 the r o t a t i o n  of the e a r t h t y p i c a l l y has  there. Figure point  i n an  2.1  shows the geometry of the  inertial  at t a n g e n t i a l v e l o c i t y v  and  v  = v  ss  / C  s  the earth's  s u r f a c e moves  beneath i t , where  g  (1 )  a  f a c t o r C , the r a t i o between the  v e l o c i t y and  = ( r  a  where r  e  e  - h  s  ) / r  h  = v  v  g  ss  - v  I = [ v i  where 8^ i s the  3  s u r f a c e . The  2  ss  of  equivalent  sub-satellite  i s given by :  (3)  e  + v  2  e  - 2v  v cos(0.) ] ss e 1  1  /  (4)  2  i n c l i n a t i o n angle of the s a t e l l i t e  T h i s ground v e l o c i t y satellite  surface  i s the height  s  r e l a t i v e ground v e l o c i t y , V g , between the  g  by  (2)  above the e a r t h ' s  the e a r t h ' s  i s given  e  i s the r a d i u s of the e a r t h and  p o i n t and  in i t s  satellite  the s u b - s a t e l l i t e p o i n t v e l o c i t y ,  the s a t e l l i t e  v  sub-satellite  i s the t a n g e n t i a l v e l o c i t y of the s a t e l l i t e  g  o r b i t . The  C  and  effect  sub-satellite  frame of r e f e r e n c e . The  p o i n t moves with v e l o c i t y v  v  i t s greatest  i s translated  velocity, v  back i n t o an  , i n the s l a n t eq  orbit.  equivalent  range plane as  16  Figure 2.1. Spherical earth model for v e l o c i t y  calculation.  17 v  = v  eq  In  C a  g  (5)  order to simulate the azimuth antenna weighting  function,  i t i s necessary t o know the v e l o c i t y at which a  p o i n t t a r g e t t r a v e l s through the antenna beam i n 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^, will  be assumed to be c o n s t a n t and equal to v eq The d e r i v e d e q u i v a l e n t v e l o c i t i e s are a p p l i e d to the  flat  e a r t h model shown i n f i g u r e 2.2. The s a t e l l i t e  with v e l o c i t y v ^ at a h e i g h t h  g  travels  above the ground. The s l a n t  range of c l o s e s t approach, r , i s determined by the 0  i n c i d e n c e a n g l e , p\ , as r  = h_ / cos(  0  )  B.  (6)  The antenna b o r e s i g h t may be s q u i n t e d away from the r d i r e c t i o n by the squint angle 6  i n the s l a n t  0  range p l a n e .  The s q u i n t angle i s d e f i n e d t o be p o s i t i v e when the antenna i s p o i n t i n g behind the z e r o Doppler d i r e c t i o n  resulting  in a  negative Doppler beam c e n t e r frequency. Azimuth time, v, i s measured r e l a t i v e t o the ground p o s i t i o n of c l o s e s t  approach  as shown i n f i g u r e 2.3. From t h i s f i g u r e , the f o l l o w i n g q u a n t i t i e s can be deduced  r(rj)  =  TJ = r c  [ r  0  2 0  +  ( v  E  G  T } )  tan(0 ) / v s  e  2  g  ]  L  /  2  (7)  (8)  F i g u r e 2.2.  Flat  e a r t h model f o r l o c a l  geometry.  F i g u r e 2.3.  S l a n t range p l a n e .  20 r  c  = r(rj )  (9)  c  where r(r/) i s the range m i g r a t i o n equation which d e f i n e s the instantaneous range t o a p o i n t t a r g e t with range of c l o s e s t approach, r , and r j and r 0  and range It  c  c  are the beam center azimuth time  respectively.  i s u s e f u l a t t h i s p o i n t to examine an approximation  to the RCM equation i n g r e a t e r d e t a i l . The RCM curve can be expanded  i n t o a T a y l o r s e r i e s form about the azimuth beam  center c r o s s i n g time, r) ,  as f o l l o w s :  r  r(r?)  =  r{n )  +  r  r'  (j? ) r  r"(rj ) ( T 7 - T J ) / 2 2  c  The f i r s t p o i n t which  c  (r?-7j ) r  +  +  ...  (10)  term r e p r e s e n t s the range to the beam c e n t e r  i s c o n s t a n t i n the s p h e r i c a l e a r t h / o r b i t  model.  The second term i s the l i n e a r component of RCM and i s commonly r e f e r r e d t o as range walk. The higher terms w i l l be collectively  r e f e r r e d to as range c u r v a t u r e .  In most s a t e l l i t e - b o r n e systems such as RADARSAT walk  i s the dominant  component of RCM. For nominal  parameters, range walk  range  RADARSAT  i n c r e a s e s almost l i n e a r l y with s q u i n t  angle from zero at 0°, t o 8.9 range c e l l s at 1°, t o 88 range c e l l s a t 10°. Range c u r v a t u r e  i s c o m p a r a t i v e l y small being  an approximately c o n s t a n t 0.23 range c e l l s . When range  walk  and/or c u r v a t u r e exceed the range r e s o l u t i o n , some form of RCM c o r r e c t i o n  (RCMC) i s r e q u i r e d t o m a i n t a i n good  azimuth  21 and range  resolutions.  Range walk  i n c r e a s e s approximately l i n e a r l y w i t h  wavelength whereas range c u r v a t u r e i n c r e a s e s approximately as the square of the wavelength. Consequently, f o r longer wavelength s a t e l l i t e - b o r n e SAR's such as SEASAT, range may  walk  be s e v e r a l times l a r g e r and range c u r v a t u r e may be an  order of magnitude  larger.  Terms up t o the q u a d r a t i c a r e u s u a l l y s u f f i c i e n t f o r c h a r a c t e r i z i n g RCMC whereas higher order terms may be necessary t o a c c u r a t e l y represent azimuth phase. By dropping terms h i g h e r than the q u a d r a t i c and s u b s t i t u t i n g and  f o r r'(j? ) c  i n terms of the instantaneous frequency f ^ and  r " ( 7 j ^ )  frequency r a t e K  A  a t the beam center c r o s s i n g time, the  T a y l o r s e r i e s can be w r i t t e n as :  r(7j)  *  r ( T ?  C  )  -  (X/2)[  -(X/2)[  f  ( T ? - T  C  f , T } +  ?  C  )  K ( T J - T J ) / 2 2  +  K T? /2  a  c  (11)  (12)  ]  2  A  ]  where  = r(r? ) + c  f  (X/2)[  f T j c  c  -  K 7j A  2 c  /2  ]  (13)  (14)  T h i s approximate form o! the RCM equation w i l l be used i n subsequent s e c t i o n s t o d e f i n e the azimuth phase response and  22 i t s Fourier  2.2  transform.  IMAGE QUALITY MEASUREMENTS  T h i s s e c t i o n d i s c u s s e s the r e l e v e n t image q u a l i t y measures of the p o i n t t a r g e t response and o u t l i n e s the methods used i n the s i m u l a t i o n s  f o r t h e i r measurement. S e v e r a l measures  computed from the point t a r g e t response are commonly used to determine the q u a l i t y of a SAR  1.  Impulse Response The -3dB  Width  image. These are as f o l l o w s :  (IRW).  impulse response widths i n both range and  azimuth are standard  measures of r e s o l u t i o n . They are  measured by i n t e r p o l a t i n g i n the range and azimuth d i r e c t i o n s by a f a c t o r of 128. The peak magnitude i n each d i r e c t i o n i s determined. Then the d i s t a n c e between the -3dB p o i n t s i s computed using l i n e a r i n t e r p o l a t i o n  2.  between the a l r e a d y  i n t e r p o l a t e d samples.  Integrated  Ratio  The I SLR  Sidelobe  (ISLR).  i s the r a t i o of the i n t e g r a t e d energy i n the  s i d e l o b e region to the i n t e g r a t e d energy i n the mainlobe r e g i o n . The s i d e l o b e region  i s d e f i n e d as a l l samples  i n s i d e of a r e c t a n g l e whose s i d e s are l o c a t e d at the measured -3dB  p o s i t i o n s i n range and azimuth. The  s i d e l o b e region  i s d e f i n e d as a l l samples o u t s i d e of a  r e c t a n g l e which i s 3 times the s i z e of the mainlobe r e c t a n g l e and which i s centered  at the same p o s i t i o n .  23 The 2-D ISLR i s measured on a 2-D array of s i z e  256x256  which has been i n t e r p o l a t e d by a f a c t o r of 8 i n both d i r e c t i o n s from a 32x32 a r r a y . The i n t e g r a t i o n s are performed by summing squared magnitudes. Although  this  a r r a y does not extend out to the ends of the s i d e l o b e regions,  i t c o n t a i n s most of the s i d e l o b e energy and was  chosen because of memory c o n s t r a i n t s . Summing the s i d e l o b e region over a l i m i t e d area approximation when l i t t l e low  i s a good  broadening occurs,  squint a n g l e s . However,  i . e . , for  the approximation breaks  down f o r l a r g e broadening as w i l l be shown i n the simulations. Fortunately,  the approximation i s v a l i d  over the broadening l e v e l s of i n t e r e s t i n the  simulated  system. The ISLR i s a l s o measured on 1-D a r r a y s and  azimuth. These provide  i n range  i n d i c a t i o n s of the broadening  in each d i r e c t i o n . The 1-D ISLR i s measured on an a r r a y of l e n g t h 4096 which has been i n t e r p o l a t e d by a f a c t o r of  3.  128 from a l e n g t h 32 a r r a y .  Peak Sidelobe  R a t i o (PSLR).  The PSLR i s the r a t i o of the magnitude of the l a r g e s t sidelobe  in the s i d e l o b e region  to the magnitude of the  peak of the p o i n t t a r g e t response. In 1-D,  range or  azimuth, the PSLR i s measured on an array which has been i n t e r p o l a t e d by a f a c t o r of 128. In 2-D, sidelobe  i s measured i n the 2-D  the peak  s i d e l o b e r e g i o n of an  a r r a y which has been i n t e r p o l a t e d by a f a c t o r of 8 i n  24 both  4.  directions.  Peak Magnitude Degradation. As more energy  i s spread i n t o the s i d e l o b e s , the  magnitude of the p o i n t t a r g e t response peak  decreases  c a u s i n g a decrease i n s i g n a l - t o - n o i s e r a t i o  (SNR). The  decrease i n peak magnitude has been measured as a f u n c t i o n of squint angle. However the SNR  i s not  d i r e c t l y r e l a t e d to the measured peak magnitude since a n o r m a l i z a t i o n based on the n o i s e power d i s t r i b u t i o n must be used. In the s i m u l a t i o n s 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  a p p r o p r i a t e f o r a white d i s t r i b u t i o n of n o i s e over a l l range c e l l s . The peak magnitudes are a l s o normalized by the azimuth processed bandwidth which decreases with i n c r e a s i n g squint angle. The processed bandwidth i s p r o p o r t i o n a l to the n o i s e power i f the n o i s e i n the azimuth  s i g n a l i s white.  3. BASIC RANGE/DOPPLER COMPRESSION T h i s s e c t i o n p r e s e n t s a theory t o d e s c r i b e the basic range/Doppler compression a l g o r i t h m . D i s c r e t e implementations of the range and azimuth  compression  o p e r a t i o n s a r e p r e s e n t e d . E x t e n s i v e s i m u l a t i o n s are used to q u a n t i f y the image d e g r a d a t i o n s which occur f o r l a r g e squint angles. The range/Doppler a l g o r i t h m , a l s o known as a frequency domain i n t e r p o l a t i o n a l g o r i t h m or a frequency domain c o r r e l a t i o n a l g o r i t h m , has been d e s c r i b e d i n s e v e r a l good papers [2,6,20,21,24,26]. further  insight  The theory developed here p r o v i d e s  i n t o the approximations i n v o l v e d i n d e r i v i n g  the b a s i c range/Doppler a l g o r i t h m as a f i l t e r matched to the p o i n t t a r g e t response. The approximate  one-to-one  correspondence between the time and frequency domain azimuth s i g n a l s which i s v a l i d  f o r l a r g e azimuth TBP s i g n a l s i s  used. Later s e c t i o n s p r o v i d e a refinement of t h i s approximation which accounts f o r the spectrum broadening p r o c e s s and p r o v i d e s the b a s i s f o r the SRC a l g o r i t h m . A model of the r e t u r n from a p o i n t t a r g e t which was presented by J i n and Wu [13] i s extended t o i n c l u d e the range window. The model i s used i n the simulator t o generate a 1-D point t a r g e t r e t u r n s i g n a l which  i s subsequently  compressed  i s performed by range  i n range. Range compression  matched f i l t e r i n g  and windowing i n the range frequency  domain t o produce a 1-D range compressed a 2-D range compressed  p r o f i l e . From t h i s ,  s i g n a l i s s i m u l a t e d by s h i f t i n g the  25  26 range p r o f i l e peak i n range a l o n g the RCM curve d e f i n e d by r(rj)  and m u l t i p l y i n g by the azimuth phase coding which i s  approximately l i n e a r FM. Azimuth compression, which i s a l s o performed as a f a s t c o n v o l u t i o n i n the frequency domain, c o n s i s t s of a p p l y i n g an azimuth f a s t F o u r i e r transform (FFT), performing RCMC, m u l t i p l y i n g by a 1-D azimuth r e f e r e n c e phase  f u n c t i o n and window f u n c t i o n , and  t r a n s f o r m i n g back t o the azimuth time domain using an i n v e r s e FFT. The  image q u a l i t y of the s i m u l a t e d compressed  point  t a r g e t responses are measured f o r s e v e r a l squint a n g l e s . These measurements provide a b a s e l i n e f o r comparison with l a t e r s i m u l a t i o n s using SRC.  3.1 POINT TARGET RESPONSE MODEL The complex r e c e i v e d s i g n a l a f t e r quadrature demodulation from a p o i n t t a r g e t with range of c l o s e s t approach r  0  can be  modelled i n continuous range and azimuth time as [26]  h(t,7?) = h ( t , T j ) * h ( t , r ? )  (15)  h (t,T?) = w ( t j ) e x p [ - j 4 7 T r ( T 7 ) / X ] 5[t - 2r(r?)/c]  (16)  h (t,i?)  (17)  A  A  R  R  a  =  8(T?)  s (t) T  where t i s c o n t i n u o u s r a n g e t i m e m e a s u r e d f r o m t h e t i m e o f transmission of thepulse of i n t e r e s t , W ^ T J )  i s the azimuth  27 antenna f u n c t i o n , c i s the speed of l i g h t , and * denotes 2-D convolution. The  f u n c t i o n h ( t , r j ) r e p r e s e n t s the response i n the R  range d i r e c t i o n a f t e r quadrature demodulation and can be c l o s e l y approximated by the t r a n s m i t t e d complex modulation f u n c t i o n , s ( t ) . T h i s approximation i s v a l i d when the T  Doppler s h i f t  i s much s m a l l e r than the t r a n s m i t t e d  bandwidth  as i s u s u a l l y the case. The  function h ( t , 7 ? ) A  r e p r e s e n t s the h y p o t h e t i c a l  continuous azimuth response t o an impulse-type radar p u l s e assuming transit of  that the radar does not move a p p r e c i a b l y d u r i n g the time of the p u l s e . A good d i s c u s s i o n of the v a l i d i t y  t h i s s t o p - s t a r t approximation i n which the sensor i s  assumed t o be s t a t i o n a r y d u r i n g p u l s e t r a n s m i s s i o n and reception actually  i s given by Barber sampled  [ 2 ] . Although azimuth time i s  a t the p u l s e r e 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  PRF  i s chosen s u f f i c i e n t l y h i g h t o prevent  a l i a s i n g of the azimuth  sense c o n s i s t s of f i l t e r i n g which  significant  signal.  Optimum SAR p r o c e s s i n g  matched f i l t e r  i f the  i n a l e a s t mean squared e r r o r the r e t u r n s i g n a l with a 2-D  i s matched to the p o i n t t a r g e t  T h e r e f o r e the i d e a l matched f i l t e r  signal.  impulse response can be  w r i t t e n as :  h*(-t,-7j) = h*(-t,-r?) * h*(-t,-77)  (18)  28 The f i l t e r  c o n s i s t s of two p a r t s : *  1.  a range matched f i l t e r ,  h (-t,-Tj), R  which compresses  the  coded range p u l s e * 2.  and an azimuth matched f i l t e r , compresses  h (-t,-rj), A  which  the azimuth phase coding and compensates  for  RCM. A d d i t i o n a l range and azimuth f i l t e r i n g  i n the form of  frequency domain windows i s o f t e n a p p l i e d  i n order to  c o n t r o l the t r a d e o f f between s i d e l o b e l e v e l s and impulse response widths. In a d d i t i o n the azimuth w e i g h t i n g due to the  antenna a p e r t u r e i s dropped from the azimuth matched  filter  since i t s effect  The  i s s i m i l a r to the azimuth window.  f o l l o w i n g s e c t i o n s d e s c r i b e the approximations  r e q u i r e d to d e r i v e the range and azimuth matched f i l t e r s the  of  b a s i c range/Doppler a l g o r i t h m .  3.2 RANGE COMPRESSION Range compression may  be viewed as a c o n v o l u t i o n of the *  r e c e i v e d s i g n a l with a f i l t e r ,  h (-t,-7j), R  matched to the  t r a n s m i t t e d pulse and a window f u n c t i o n , w ( t ) , which i s R  designed to reduce the energy i n the s i d e l o b e s . continuous 2-D h  R C  (t,7?)  =  range compressed h(t,7j)  *  s i g n a l may  h*(-t,-T?)  *  [ 6 (rj)  The  be w r i t t e n as w  R  (t)]  (19)  29 = h (t,7j) A  *  [6(7})  h  R C p  (t)]  (20)  ( t )  (21)  where  h  R  C  p  ( t )  = 8  T  ( t ) *  t  S*(~t)  i s the 1-D range compressed  \  W  R  p r o f i l e which i s u s u a l l y  in shape t o a sine f u n c t i o n , and *  fc  similar  denotes c o n v o l u t i o n i n  range time. In t h i s form the range compressed  signal i s  expressed as a range time c o n v o l u t i o n of the 1-D range compressed  p r o f i l e with a 2-D phase f u n c t i o n , h ( t , r j ) , A  which  i s non-zero only along the RCM c u r v e . It should be noted that the time o r i g i n of the range compressed  s i g n a l has been s e l e c t e d so t h a t the range  compressed  profile  i s symmetric  about t=0. For RADARSAT and  most other s a t e l l i t e SAR's, the t r a n s m i t t e d s i g n a l  isa  l i n e a r FM p u l s e which can be r e p r e s e n t e d a t baseband as  s ( t ) = a(t) exp[-j*(t)]  (22)  a(t) = r e c t ( t A )  (23)  *(t) = - * K t  (24)  T  2  R  where a ( t ) i s the amplitude f u n c t i o n , modulation f u n c t i o n , range l i n e a r FM r a t e .  <p(t) i s the phase  T i s the p u l s e w i d t h , and K  R  i s the  30 After  some manipulation,  the range compressed p r o f i l e  can be shown t o have the form of a weighted sine f u n c t i o n [23] :  h  R C p  (t)  = {(T-|t|)rect(t/2T)sinc[irK t(T-|t|)]} R  *  t  w (t) R  (25)  The  range compression c o n v o l u t i o n  efficiently  i n the range frequency  c o n v o l u t i o n . Fast c o n v o l u t i o n  i s performed more  domain using  fast  i s computationally  when the l e n g t h of the c o n v o l u t i o n  efficient  k e r n e l i n samples i s a  power of 2 l a r g e r than about 32. The f a s t c o n v o l u t i o n method c o n s i s t s of :  1.  Fourier transforming the complex conjugate  the range matched f i l t e r of the F o u r i e r transform  t r a n s m i t t e d pulse) and the r e c e i v e d p u l s e s i m u l a t i o n s the r e c e i v e d pulse  2.  of the  ( f o r the  i s assumed t o be the same  as the t r a n s m i t t e d pulse with a p p r o p r i a t e  delay);  m u l t i p l y i n g together  the matched  filter,  3.  (which i s  the r e c e i v e d s i g n a l ,  and the s i d e l o b e c o n t r o l window;  and i n v e r s e F o u r i e r t r a n s f o r m i n g  In continuous convolution  time and frequency  the r e s u l t .  theory,  the f a s t  range compression o p e r a t i o n can be expressed  as  31 h  where and  (t)  R c p  w R  (f ) R  = f  a n o  l W (f ) R  " T^R^ S  |S <f >|* }  R  a  r  e  T  fc  ^  e  F  (26)  R  o  u  r  i  e  r  transforms of  w R  (t)  s ( t ) respectively, T  3.3 SIMULATION OF THE RANGE COMPRESSED PROFILE T h i s s e c t i o n d e s c r i b e s the method used i n the s i m u l a t i o n s to generate a d i s c r e t e , range compressed p r o f i l e ,  h ^ (m). R  p  Where necessary, the symbol " w i l l be used t o denote d i s c r e t e s i g n a l s . A d i s c r e t e , l i n e a r FM modulation f u n c t i o n , s ( m ) , i s formed i n the range time domain as T  s (m) = a(mT) exp[-j^CmT)] , -(M/2) < m < (M/2) T  (27)  where T i s the range sampling p e r i o d and a(mT) i s the rectangular  pulse envelope of width T . T h i s f u n c t i o n i s  transformed with a range F F T of l e n g t h 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 s i d e l o b e c o n t r o l window i s a p p l i e d t o get  H  R C p  ( k ) = W (k) R  |S (k)|  2  T  , -(M/2) < k < (M/2)  where S ( k ) i s the F F T of s ( m ) , T  T  w R  (28)  ( k ) i s the frequency  domain window f u n c t i o n , and k i s the frequency  index.  Many window f u n c t i o n s a r e a v a i l a b l e f o r c o n t r o l l i n g the s i d e l o b e s . The p r i n c i p a l window vsed i n the s i m u l a t i o n s K a i s e r - B e s s e l window d e f i n e d as [12]  isa  32 W (k)  = / {/3 [l-(2k/M) ] 2  R  0  R  l / 2  } / / {/3 ) , 0  R  -(M/2) < k < (M/2)  where 0  D  i s a window parameter which c o n t r o l s the amount of As 0  weighting.  i s i n c r e a s e d , the mainlobe width of  R  range compressed p r o f i l e s i d e l o b e s and The  (29)  the  i n c r e a s e s whereas the energy i n the  the magnitude of the peak s i d e l o b e  decrease.  z e r o t h - o r d e r m o d i f i e d B e s s e l f u n c t i o n of the  kind, I , 0  first  i s approximated by the power s e r i e s :  P 7 (x) 0  Z  =  [  (x/2)  P  / pl V  (30)  p=0  The  number of terms used i n the s i m u l a t i o n s (P=15) p r o v i d e s  an accuracy  of about  14 s i g n i f i c a n t  f i g u r e s f o r the  Bessel  function. Since the simulated the weighting  range compressed p r o f i l e  along each azimuth l i n e of the simulated  range compressed s i g n a l , a c l o s e approximation continuous  profile  transforming  of  2-D  the  i s d e s i r e d . Thus the d i s c r e t e p r o f i l e i s  i n t e r p o l a t e d by a f a c t o r I (1=16 padding the  defines  frequency  i n the s i m u l a t i o n s ) by  a r r a y to a l e n g t h of MI  zero  before  back i n t o the time domain . T h i s e f f e c t i v e l y  performs i n t e r p o l a t i o n f u n c t i o n . Except  [2] with a t i m e - a l i a s e d s i n c ( x )  f o r the small d i f f e r e n c e s caused by  a l i a s i n g e r r o r s , the i n t e r p o l a t e d samples p r o v i d e a good  33 s i m u l a t i o n of the samples which would be obtained compressing  a set of time delayed  by  return pulses.  In order to i n t e r p o l a t e p r o p e r l y , the zero padding  must  be performed at the ends of the p u l s e spectrum, i . e . , i n the middle  of the FFT a r r a y , as f o l l o w s  H H  R C p  R C p  (k)  , -(M/2) < k <  (M/2)  (k) -  (31 )  0  , -(MI/2) < k < -(M/2) and  (M/2)  < k < (MI/2)  where ' i s used to denote the i n t e r p o l a t e d s i g n a l . A f t e r i n v e r s e FFT of l e n g t h MI compressed p r o f i l e , h where m'  R C p  i s a p p l i e d , the i n t e r p o l a t e d , range  ( m ' ) , -MI/2  < m'  < MI/2,  i s the i n t e r p o l a t e d a r r a y index and  p e r i o d i s T/I. The Since the FFT  an  the  i s obtained sampling  peak of the p r o f i l e occurs at m'= i s being used to perform  0.  a linear  c o n v o l u t i o n , the FFT l e n g t h , M, must be l a r g e enough to exclude  invalid  samples which occur because of the FFT's  c i r c u l a r c o n v o l u t i o n . If the pulsewidth  i s T and the d e s i r e d  number of v a l i d compressed samples before Q, M must s a t i s f y M > (r/T) + Q - 1. The chosen to be the next  after  interpolation, valid  samples  shows the simulated range compressed  profile  f o r -(QI/2) < m' F i g u r e 3.1  length M i s usually  l a r g e r power of 2 to allow the use of  e f f i c i e n t FFT a l g o r i t h m s . A f t e r occur  interpolation is  £  (QI/2).  i n t e r p o l a t i o n f o r the nominal RADARSAT parameters  34  given i n Table 1. The window parameter, 0 to  produce a 1-D peak s i d e l o b e r a t i o  a 1-D i n t e g r a t e d s i d e l o b e r a t i o  3.4 SINGLE-LOOK AZIMUTH  = 2.7, was chosen  R  (PSLR) of -21.7 dB and  (ISLR) of -21.0 dB.  COMPRESSION  T h i s s e c t i o n d e s c r i b e s the theory and approximations used i n d e v e l o p i n g the b a s i c range/Doppler azimuth compression algorithm for single-look  processing.  As d e s c r i b e d e a r l i e r , the b a s i c range/Doppler a l g o r i t h m makes use of the approximate s i m i l a r i t y between the time and frequency domain s i g n a l s of l a r g e TBP l i n e a r FM type s i g n a l s . Azimuth compression c o n s i s t s of c o n v o l v i n g the range compressed matched f i l t e r ,  s i g n a l with an approximation t o the azimuth h ( - t , - r ? ) , and an azimuth window t o c o n t r o l A  s i d e l o b e s . The antenna f u n c t i o n w (rj) i s u s u a l l y from the azimuth f i l t e r of  since i t s e f f e c t  dropped  i s s i m i l a r t o that  the azimuth window. E x c l u d i n g the azimuth window f o r the  time being, an azimuth r e f e r e n c e f u n c t i o n  (the t i m e - r e v e r s e d  complex conjugate of the azimuth matched f i l t e r ) can be w r i t t e n as :  h (t,r/) = exp[-j47rr(Tj)/X] 6 [ t-2r ( T J ) / C ]  (32)  p  In  order to understand the d i s c r e t e implementation of  t h i s approximate matched f i l t e r ,  the f i l t e r  must be  b a n d l i m i t e d i n range to the range sampling frequency,  F s r  -  T h i s excludes f r e q u e n c i e s which would be a l i a s e d by range  Range Compressed  Profile  betar = 2.7  16  -12  -8  -4  0  4  Range Sample Number  8  12  16  36 sampling and a l s o p r o v i d e s the b a s i s f o r i n t e r p o l a t i o n i n the  range d i r e c t i o n . Continuous time s i g n a l s w i l l be used t o  develop the a l g o r i t h m . These can be d i s c r e t i z e d  i n range and  azimuth and t i m e - a l i a s e d a c c o r d i n g t o the l e n g t h of the FFT s 1  i n order t o p r o v i d e a d i s c r e t e model of the a l g o r i t h m . The  i d e a l r e c t a n g u l a r range b a n d l i m i t i n g f i l t e r  form of a s i n e f u n c t i o n . Thus the range  has the  bandlimited  r e f e r e n c e f u n c t i o n can be formulated as :  h  F R B  (t,rj) = h (t,Tj) * p  t  = exp[-j4?Tr(T?)/X] s i n c (  The  sinc( 7 r F t s r  )  (33)  ?rF [ t - 2 r ( i j ) / c sr  shape of t h i s 2-D r e f e r e n c e f u n c t i o n  ] )  (34)  i s shown i n  f i g u r e 3.2. C r o s s - s e c t i o n s of the f u n c t i o n  i n the azimuth time  d i r e c t i o n e x h i b i t a l i n e a r FM type of phase which  characteristic  i s the same as the phase along the RCM curve i n the  previous i n f i n i t e  bandwidth  envelope of t h i s s i g n a l  reference function, h p ( t , 7 j ) .  i n the azimuth d i r e c t i o n  The  i s a sine  f u n c t i o n c e n t e r e d a t the RCM curve with a time-warping e f f e c t c r e a t e d by the range c u r v a t u r e terms of the RCM e q u a t i o n , r{n).  Since the RCM over the azimuth time  interval  d e f i n e d by the azimuth antenna beamwidth i s predominantly linear, especially of  f o r RADARSAT parameters, the time-warping  the s i n e envelope i s s m a l l .  F i g u r e 3.2. Range c e l l m i g r a t i o n curve i n azimuth time domain.  38 The  approximate  azimuth  f u n c t i o n can be determined rin)  timewidth of the r e f e r e n c e  by u s i n g a l i n e a r aproximation t o  and determining the -3dB azimuth  times of the sine  envelope. Using the T a y l o r s e r i e s expansion with o n l y f i r s t c e n t e r range envelope  order terms and e v a l u a t i n g at the beam  time, t = 2r(r}^)/c, the approximate  i s sine ( 7 r F X f sr  t h i s envelope  Ar,  3 d B  of chapter 2  c  [  TJ-7J ]/c). c  azimuth  The -3dB timewidth of  i s given by :  - 0.884 c / U f ^ )  When the squint angle  (35)  (and t h e r e f o r e the beam c e n t e r  frequency, f ^ ) i s s m a l l , the TBP i n the azimuth near the beam center range  i s l a r g e . For a l a r g e azimuth TBP  signal,  the F o u r i e r t r a n s f o r m of the azimuth  similar  i n phase and magnitude t o the azimuth  s i g n a l except  direction  signal i s time domain  f o r a s c a l i n g c o n s t a n t . Using the p r i n c i p l e of  s t a t i o n a r y phase [17], the s c a l i n g between the time and frequency axes can be determined  by e x p r e s s i n g the  instantaneous frequency as a f u n c t i o n of azimuth  f (77) = -(2/X) r'(rj)  (36)  ±  Substituting  time :  f o r r'(r?) with the d e r i v a t i v e of equation (7)  and r e a r r a n g i n g g i v e s the i n v e r s e mapping  r^U)  = r  0  / { v  e  g  [(2v /(Xf)) e g  2  - 1]  i  /  z  }  (37)  39 where 7?^(f) i s the azimuth time c o r r e s p o n d i n g t o the instantaneous azimuth frequency f . S u b s t i t u t i n g back equation  (7) g i v e s the approximate  into  frequency domain RCM  curve :  r.(f) = rdj.lf)) = r  0  d + 1 /[ ( 2 v / ( Xf))  2  & q  -  (38)  Thus the azimuth F o u r i e r t r a n s f o r m of the range b a n d l i m i t e d r e f e r e n c e f u n c t i o n e x c l u d i n g amplitude c o n s t a n t s can be approximated as :  H  F R B  (t f) * h f  F R B  (t,r  ? i  ( f ))  (39)  = e x p [ - j 4 i r r . (f )/X] s i n e d r F f t - 2 r . (f )/c ])  The  shape  function  of t h i s azimuth frequency domain  (40)  reference  i s shown i n f i g u r e 3.3.  T h i s approximate e q u a t i o n forms the b a s i s f o r the azimuth  frequency domain, f a s t  azimuth compression  convolution  i n the b a s i c  range/Doppler  The approximate azimuth matched f i l t e r conjugate of t h i s  implementation of algorithm.  i s the complex  frequency domain r e f e r e n c e f u n c t i o n . An  azimuth s i d e l o b e c o n t r o l window i s a l s o a p p l i e d  i n the  azimuth frequency domain. The b a s i c range/Doppler compression a l g o r i t h m  i s expressed as :  azimuth  40  F i g u r e 3.3.  Range c e l l m i g r a t i o n curve i n azimuth doma i n.  frequency  41  a(t,T?)  = F' {[  H  1  R C  (t,f)  = F- {exp[ j47rr (f )/X] 1  i  •  H (t',f) R C  *  t  H* (-t,f) R B  ] W (f-f )} A  c  (41)  W (f-f ) A  c  s i n c ( 7 r F [ t ' - t - 2 r ( f )/c])dt'} s r  i  where a(t,rj) i s the f i n a l compressed  point target  (42)  image. The  procedures may be summarized as f o l l o w s :  1.  Apply an azimuth F o u r i e r transform  (approximated by an  FFT) t o the range compressed  t a r g e t r e t u r n t o get  H  2.  R C  point  (t,f).  Interpolate  i n range time with a s i n e f u n c t i o n matched  to the range sampling frequency i n order t o e x t r a c t the energy a t the range d e f i n e d  by the RCM c u r v e . In  p r a c t i c e the RCMC i n t e r p o l a t i o n time c o n v o l u t i o n  i s performed as a range  with a short windowed s i n e f u n c t i o n t o  minimize the number of computations. A K a i s e r - B e s s e l window (^=2.5) i s used i n the s i m u l a t i o n s .  3.  M u l t i p l y by the complex conjugate of the azimuth frequency domain r e f e r e n c e  phase  f u n c t i o n and the  azimuth s i d e l o b e c o n t r o l window. Rather than using the approximate  reference  f u n c t i o n above, a c l o s e r  approximation i s formed by computing  the FFT of a  d i s c r e t e time domain r e f e r e n c e  f u n c t i o n of u n i t y  magnitude.  phase  T h i s i s the approach used i n the s i m u l a t i o n s .  42 4.  Apply an i n v e r s e azimuth F o u r i e r transform (approximated by an i n v e r s e FFT) t o o b t a i n the f i n a l compressed  3.5 SIMULATION OF SINGLE-LOOK AZIMUTH  image.  COMPRESSION  T h i s s e c t i o n develops a s i m u l a t i o n model of s i n g l e - l o o k azimuth compression f o r the b a s i c range/Doppler  algorithm.  The steps i n v o l v e d i n g e n e r a t i n g and compressing a simulated 2-D range compressed  azimuth s i g n a l are d e s c r i b e d .  The main steps of the s i m u l a t i o n are :  1.  g e n e r a t i o n of the azimuth time domain phase f u n c t i o n  2.  g e n e r a t i o n of a frequency domain, s i n g l e - l o o k , phase  3.  filter  s i m u l a t i o n of azimuth weighting due to the azimuth antenna  f u n c t i o n 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 phase f i l t e r  7.  reference  reference  and window  azimuth i n t e r p o l a t i o n performed by zero-padding i n the frequency domain  43  8.  i n v e r s e azimuth FFT to produce a time domain p o i n t target  image  The azimuth time domain phase f u n c t i o n can be expressed in d i s c r e t e azimuth time as  p(n) = exp[jtf(n)] , 1 < n < N  = - ( 4 i r / X ) r( T J +  <Mn)  where T  C  A  (43)  (2n-N-1)T /2 A  (44)  )  i s the azimuth sampling p e r i o d , n i s the azimuth  time index, and N i s the l e n g t h of the azimuth FFT  (a power  of  2 ) . For convenience, the beam c e n t e r time, T J ^ , ,  at  the c e n t e r of the FFT at n = ( N + l ) / 2 . The h y p e r b o l i c  equation g i v e n i n equation (7)  i s placed RCM  i s used throughout the  s i m u l a t i o n s i n s t e a d of the T a y l o r s e r i e s approximation. A s i n g l e - l o o k , azimuth r e f e r e n c e phase f i l t e r azimuth frequency domain i s formed by computing  i n the  the FFT of  it  p(n) and t a k i n g  i t s complex conjugate to get P ( k ) , 1 < k <  N. Each azimuth l i n e of the d i s c r e t e 2-D  range compressed  s i g n a l , h ,(m,n), i s c r e a t e d by a p p l y i n g two forms of R(  weighting t o p ( n ) . The azimuth antenna  first  form of weighting i s the  f u n c t i o n , w ( n ) . In a c t u a l SAR's, the a  azimuth time width of the antenna  function varies  slowly  with s q u i n t angle and becomes s l i g h t l y asymmetrical. Since t h i s c o m p l i c a t e s the geometric model and i n t r o d u c e s small  44 v a r i a t i o n s which are not of i n t e r e s t here, the antenna f u n c t i o n w i l l be assumed t o be c o n s t a n t i n time width and shape f o r the squint angles c o n s i d e r e d . The two-way azimuth antenna f u n c t i o n  i s approximated by a s i n c M x ) type^  f u n c t i o n , as produced by a uniform, continuous a p e r t u r e antenna, and i s d e f i n e d i n d i s c r e t e azimuth time as  w (n) = sinc [irDv (2n-N-1 ) T / ( 2 X r ) ] , 1 < n < N  (45)  2  a  b  A  0  where D i s the azimuth antenna l e n g t h and the beam center time occurs a t n = (N+l)/2. The two-way -6 dB width of the antenna f u n c t i o n i s  Arj  6 d B  = 0.884  Xr /(Dv ) 0  (46)  fa  The second form of weighting i s due to RCM. The weighting i s a p p l i e d by d e t e r m i n i n g the range d i s t a n c e between each azimuth sample and the range m i g r a t i o n curve and s e l e c t i n g the nearest amplitude from the i n t e r p o l a t e d , range compressed  p r o f i l e . The d i s t a n c e between the range  m i g r a t i o n curve and the d e s i r e d azimuth l i n e i n u n i n t e r p o l a t e d range samples  i s computed as  d(m,n) = m - 1 - (2/cT) [ r ( T J  C  +  [  2n-N-1 ] T / 2 ) - r A  1 < n < N  where m i s the range c e l l index, and M  max  m i n  1 < m < M max  ] , (47)  i s the number of  45 azimuth l i n e s being generated. The p o s i t i o n i n g of the azimuth l i n e s  i n range time i s a r b i t r a r y  s i n c e i t depends  only on the phase of the range sampling c l o c k . T h e r e f o r e the azimuth l i n e s a r e a r b i t r a r i l y p o s i t i o n e d so that the nearest line  (m=l) corresponds t o range time 2 r ^ / c , and the m  farthest l i n e  (m=M_  =  These ranges, r . mm 3  n  ) corresponds t o range time  2r  m =  /c.  and r , a r e the ranges of the nearest max 3  and f a r t h e s t azimuth l i n e s r e q u i r e d f o r p r o c e s s i n g . They are determined by the l e n g t h of the i n t e r p o l a t o r , the amount of RCM over the processed bandwidth,  and the number of d e s i r e d  azimuth l i n e s i n the output image. In order t o r e t r i e v e the nearest sample from the i n t e r p o l a t e d range compressed  a r r a y , h ^. (m'), the d i s t a n c e R  p  d(m,n) must be converted t o an i n t e r p o l a t e d index as  m'(m,n) = round[ I d(m,n) ]  (48)  where the f u n c t i o n round[x] rounds x t o the nearest i n t e g e r . Combining  the antenna and RCM weightings, the d i s c r e t e 2-D  range compressed  s i g n a l can be expressed as  h(m,n) = w (n) p(n) h a  R C p  ( i r i j (m,n) )  (49)  Once a l l the r e q u i r e d azimuth l i n e s a r e generated, each line  i s transformed t o the frequency domain with an azimuth  FFT of l e n g t h N t o get the range compressed s i g n a l , H(m,k) , 1 < k < N.  frequency domain  46 As with the range FFT i n the p r e v i o u s  s e c t i o n , the  azimuth FFT must be long enough to produce the d e s i r e d number of v a l i d compressed  samples. However, s i n c e the  azimuth time domain s i g n a l does not f a l l a b r u p t l y to zero due to the s i n e processing  form of the antenna  2  f u n c t i o n , an a r b i t r a r y  time i n t e r v a l c o n t a i n i n g most of the s i g n a l  energy must be chosen. Denoting the p r o c e s s i n g 7 j  p  B  W  ,  and the d e s i r e d number of compressed  a f t e r c o n v o l u t i o n as R, the FFT l e n g t h must  N ^ *?p  / T B W  A  +  R  "  i n t e r v a l as  azimuth  samples  satisfy  1  {  5  0  )  to prevent wraparound e r r o r s due t o the c i r c u l a r c o n v o l u t i o n . For the c u r r e n t s i m u l a t i o n s , the p r o c e s s i n g interval  i s set equal  Before  to the two-way -6 dB antenna width.  the azimuth r e f e r e n c e phase  filter  can be  a p p l i e d to H(m,k), i t i s necessary to c o r r e c t f o r RCM  i n the  azimuth-frequency, range-time domain using a range i n t e r p o l a t o r . T h i s c o r r e c t i o n , RCMC, e f f e c t i v e l y  straightens  the range m i g r a t i o n curve so that the matched f i l t e r  need  only be a p p l i e d to a s i n g l e azimuth l i n e to produce a s i n g l e azimuth l i n e of the f i n a l interpolator  image. The  range  f o r a d i s c r e t e range s i g n a l i s a sine f u n c t i o n  which i s range time a l i a s e d a c c o r d i n g range compression In order  ideal  to the l e n g t h of the  FFT.  to reduce the l e n g t h of the i n t e r p o l a t o r and  thereby reduce the number of computations, a f i n i t e  length  47 approximation i s u s u a l l y used. A l s o , r a t h e r than a different  generating  s e t of i n t e r p o l a t o r c o e f f i c i e n t s f o r each  azimuth time, s e v e r a l s h i f t e d v e r s i o n s of the i n t e r p o l a t o r are precomputed f o r a s e t of e q u a l l y spaced f r a c t i o n a l sample s h i f t s and the nearest the c u r r e n t  range  c l o s e s t v e r s i o n i s used. In  s i m u l a t i o n s , the approximate i n t e r p o l a t o r i s  formed by a p p l y i n g a d i s c r e t e K a i s e r - B e s s e l window t o a sine f u n c t i o n t o get  hjdrq)  = sinc[7r(q+lL)T]  w (q+lL) , :  -(L/2) < 1 < (L/2) , 0 < q < Q-1  Wj(i) = / { ^ [ l - ( 2 i / Q L ) ] 2  0  I  l / 2  } / /ot/Jj} ,  -(QL/2) < i < (QL/2)  where Q i s the number of f r a c t i o n a l l y the  (51)  (52)  s h i f t e d v e r s i o n s of  i n t e r p o l a t o r , q denotes the f r a c t i o n a l  s h i f t , L i s the  l e n g t h of each s h i f t e d v e r s i o n , 1 i s the index w i t h i n each s h i f t e d v e r s i o n , 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 e x t r a c t the peak energy a t each azimuth sample time f o r a given p o i n t t a r g e t with a range of c l o s e s t approch, r , 0  the i n t e r p o l a t o r peak i s s h i f t e d  i n range so that i t s  peak c o i n c i d e s with the frequency domain RCM curve d e f i n e d in equation First  (38). T h i s s h i f t  i s implemented i n two s t e p s .  the i n t e r p o l a t o r i s moved an i n t e g e r number of samples  48 so that i t s peak i s l e s s than curve.  one sample away from the RCM  Secondly, one of the Q i n t e r p o l a t o r v e r s i o n s i s  chosen such that the c h o s e n . i n t e r p o l a t o r v e r s i o n has i t s peak c l o s e s t  to the a c t u a l p o s i t i o n of the range m i g r a t i o n  curve. T h i s 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  i n t e r p o l a t o r . The range l i n e and the i n t e r p o l a t o r a r e then m u l t i p l i e d t o complete the approximate i n t e r p o l a t o r convolut i o n . RCMC and azimuth r e f e r e n c e phase m u l t i p l i c a t i o n are performed only over  the processed  corresponds t o the p r o c e s s i n g i s centered  f The  c  a t the beam center  p  B  W  r  which  *?p ^ T h i s bandwidth  frequency,  BW  f , g i v e n by r  (53)  c  processing  bandwidth i s computed from the azimuth  interval,  terms higher processing  than  r  ( T ? )  ?  p  B  W  r  with the assumption that phase  the q u a d r a t i c a r e small over the  i n t e r v a l , as  f PBW  a  interval,  f  = f.<„ >  processed  K  bandwidth,  (54)  PBW  =  -(2/X)  (55)  r"(r?)  = - ( 2 v * /[Xr(rj)])  [ 1 -  (v  » / r (rj) )  2  ]  (56)  where K. i s the azimuth l i n e a r FM r a t e which i s A approximately  constant  over  the p r o c e s s i n g  i n t e r v a l . Since a  49 d i s c r e t e azimuth  signal  i s used, the processed band i s  a l i a s e d by the PRF. F i g u r e 3.4 shows the form of the range compressed s i g n a l a f t e r the azimuth FFT i n c l u d i n g the spectrum a l i a s i n g which i s caused by azimuth RCMC s t r a i g h t e n s the p o i n t i n t o a s i n g l e azimuth  sampling.  t a r g e t range m i g r a t i o n curve  frequency l i n e . T h i s 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  r e f e r e n c e phase f i l t e r  and a s i d e l o b e r e d u c t i o n *  window. As s t a t e d p r e v i o u s l y , the f i l t e r  i sP (k).  The azimuth window i s computed over the processed bandwidth and s e t t o zero o u t s i d e . A K a i s e r - B e s s e l azimuth window c o n t a i n i n g N  p B W  = f  p B W  N / ( P R F ) nonzero  samples i s  computed over frequency i n d i c e s k = 1 t o N as : W (k) A  / { / 3 [ l - ( 2 [ k - 1 ]/N) ]} / / ( 0 } , 2  0  A  O  A  1 < k < N  p B W  /2 1 +  7 {/3 [ 1-(2[k-1-N]/N) ]} / 7 {j3 } , 2  0  A  0  A  N-N  pBW  /2+2 < k < N  0 , otherwise  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  modulo N so that the peak of the window f u n c t i o n  (57)  shifted i s at  frequency sample k^., which i s the nearest sample equal t o or l e s s than the a l i a s e d beam c e n t e r frequency.  50  Figure  3.4.  Range  c e l l  frequency  migration domain.  curve  in  discrete  azimuth  51  A single l i n e of the f i n a l compressed  image  i s produced  by 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 the compressed  r e t u r n s from t a r g e t s with the  same r a n g e o f c l o s e s t a p p r o a c h , r  0  , but different  azimuth  positions. 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 and windowing d e s i r e d output range c e l l s . with different value of r  0  i s repeated f o r each of the I n t h e o r y , a d i f f e r e n t RCM c u r v e s h o u l d be used f o r each  range.  However, i f t h e range e x t e n t i s s m a l l , as i n t h i s case where 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  region of a point  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 be u s e d f o r a l l l i n e s by 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 number 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 determine t h e range i n v a r i a n c e region which  i s the d i s t a n c e i n range over which the  compression f i l t e r s do n o t v a r y a p p r e c i a b l y . In p r a c t i c e , the entire i n v a r i a n c e region i s processed as a block t o increase e f f i c i e n c y . The issue o f range i n v a r i a n c e of the azimuth r e f e r e n c e phase  f u n c t i o n h a s been examined  r e p o r t s . Chapter 7 examines filter  i n other  r a n g e i n v a r i a n c e f o r t h e new S R C  function.  Finally,  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 the compressed  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 image q u a l i t y  measurement  and t o decrease t h e loss of i n f o r m a t i o n i n t h e subsequent d e t e c t i o n of t h e complex  s i g n a l . The method used i s zero  padding i n t h e azimuth f r e q u e n c y domain  before the inverse  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 c e n t e r frequency l i e s nearest t o the f i r s t FFT a r r a y sample. The a r r a y i s then padded with zeros i n the middle t o form a l e n g t h NJ a r r a y where J i s the i n t e r p o l a t i o n  factor.  By a p p l y i n g an i n v e r s e FFT of l e n g t h NJ, the d e s i r e d interpolated signal  i s obtained.  3.6 SIMULATION RESULTS OF BASIC RANGE/DOPPLER COMPRESSION T h i s s e c t i o n p r e s e n t s and d i s c u s s e s the r e s u l t s of q u a l i t y measurements of simulated p o i n t t a r g e t responses which were produced f o r a range of s q u i n t a n g l e s . The s i m u l a t i o n programs  were implemented  on an Amdahl 470 V/8 computer i n  RATFOR ( r a t i o n a l FORTRAN) under the MTS (Michigan Terminal System) o p e r a t i n g system. RATFOR i s a s t r u c t u r e d precompiler which produces FORTRAN code. The f i r s t  step i n the s i m u l a t i o n was the p r o d u c t i o n of  a range compressed  p r o f i l e as i n f i g u r e 3.1. The second s t e p  i n the s i m u l a t i o n was the p r o d u c t i o n of a simulated compressed  range  azimuth s i g n a l . F i g u r e 3.5 shows the azimuth time  domain magnitudes  of the simulated antenna weighting, and  the antenna p l u s RCM weightings f o r a s q u i n t angle of 5.0°. The magnitudes beam center  a r e shown f o r the range c e l l c l o s e s t to the  range.  The azimuth spectrum produced by the azimuth FFT i s shown i n f i g u r e 3.6. Before RCMC the azimuth bandwidth i s q u i t e s m a l l . The f i n i t e  l e n g t h i n t e r p o l a t o r used f o r RCMC i s  shown i n f i g u r e 3.7. The f i g u r e shows the l e n g t h 16  53 interpolator  a n d i n c l u d e s a l l 16 f r a c t i o n a l l y  versions. Figure in  which The  squint in  3.8 shows t h e c o r r e c t e d  spectrum a f t e r  t h e p r o c e s s e d bandwidth  i sclearly  azimuth  filter  r e f e r e n c e phase  a n g l e i s shown i n f i g u r e  shifted  seen.  spectrum  3.9. The f i l t e r  f o r a 5.0° i s generated  t h e a z i m u t h t i m e domain w i t h o u t w e i g h t i n g a n d t h e n  t r a n s f o r m e d w i t h an a z i m u t h F F T . The a z i m u t h window w i t h 0 target  spectrum phase  A  azimuth  filtering,  = 1.5 i s shown i n f i g u r e spectrum a f t e r  and windowing  ripples  filter  compressed  point  small  response  of t h e compressed  3.12 a n d 3.13 f o r s q u i n t  levels  squint Figure  in figure  3.11.  was c h o s e n  reference  3.12 shows t h a t  a squint  are  summarized  the f i n a l  i s p r o d u c e d . Sample  1-D  r e s p o n s e a r e shown i n a n g l e s o f 0° a n d 5 ° . The t o produce  comparable  i n t h e r a n g e and a z i m u t h d i r e c t i o n s f o r  a n g l e s a s would  for  be done i n a p r a c t i c a l  a n g l e o f 5 . 0 ° . The r a n g e and a z i m u t h i n figures  broadening  3.14 t o 3.16 f o r a r a n g e o f s q u i n t rapidly  f o r squint  a b o v e 4° w i t h 5% and 10% b r o a d e n i n g o c c u r i n g 4.23° r e s p e c t i v e l y  system.  s e v e r e range b r o a d e n i n g o c c u r s  a n g l e s . Range b r o a d e n i n g i n c r e a s e s  ( f o r L = 1 6 ) . However,  broadening  i s relatively  for  a n g l e s up t o 6 ° .  squint  phase  The  of the azimuth  o f an i n v e r s e FFT,  target  a z i m u t h window f a c t o r sidelobe  reference  spectrum.  cross-sections figures  i s shown  Kaiser-Bessel  3.10. The p o i n t  RCMC, a z i m u t h  are characteristic  Upon a p p l i c a t i o n  and  RCMC  insignificant  a t about  angles 3.65°  azimuth  r e m a i n i n g below  2%  54 The  1-D  and  f i g u r e s 3.17  2-D  to 3.19.  ISLR measurements are summarized i n I t shows that both r a t i o s  s i g n i f i c a n t l y as the s q u i n t angle i n c r e a s e s  increase  i n d i c a t i n g that  the p o i n t t a r g e t response not only becomes broader i n range, but a l s o becomes f l a t t e r s i d e l o b e s . The  spreading  more energy i n t o the  ISLR measurement i s seen to be l i m i t e d  l e s s than 5° s i n c e a l a r g e amount of the l i e s o u t s i d e of the f i n i t e squint a n g l e s .  to  s i d e l o b e energy  i n t e g r a t i o n region  for larger  T h i s causes the ISLR to drop s h a r p l y above  5°. Another r a t i o of i n t e r e s t , the PSLR, i s summarized in f i g u r e s 3.20  to 3.22.  T h i s shows that the peak range  s i d e l o b e i s u s u a l l y the peak 2-D discrepancy  at small angles  to the higher  between the three  The  f i g u r e s i s due  i n t e r p o l a t i o n f a c t o r used i n measuring the  PSLR. At some of the higher below the 2-D  s i d e l o b e as w e l l .  angles,  curve. T h i s occurs  the  range PSLR d i p s w e l l  s i n c e the c l o s e r range  s i d e l o b e s merge with the mainlobe c a u s i n g  sidelobes further  out to be measured as the peak s i d e l o b e . T h i s behaviour be  seen i n f i g u r e The  can  3.12.  f i n a l measurement of i n t e r e s t i s the degradation  the peak magnitude which i s p l o t t e d in f i g u r e 3.23.  This  measurement was  the  normalized  to the sum  i n t e r p o l a t o r c o e f f i c i e n t s and  of squares of  the azimuth  bandwidth as would be a p p r o p r i a t e  processed  f o r a white n o i s e model  with nqise e q u a l l y d i s t r i b u t e d over the range and cells.  1-D  Since  the a c t u a l n o i s e d i s t r i b u t i o n may  azimuth  be somewhat  of  55 d i f f e r e n t , care should be taken  i n r e l a t i n g the peak  magnitude degradation t o changes i n s i g n a l - t o - n o i s e r a t i o . The  f i g u r e shows a r e d u c t i o n i n peak magnitude with  i n c r e a s i n g s q u i n t angle caused by poor compression. 5% and 10% range broadening are approximately  At the  squint a n g l e s , the degradations  0.47dB and 0.83dB r e s p e c t i v e l y . The  d e g r a d a t i o n r i s e s r a p i d l y above t h i s .  56  (8P) ftpnjiufiDN  F i g u r e 3.5. Simulated azimuth antenna w e i g h t i n g , and antenna p l u s RCM weightings i n range c e l l nearest to beam c e n t e r range f o r 5° s q u i n t .  57  (SP) •pnuufiDN  F i g u r e 3.6.  Azimuth spectrum b e f o r e RCMC i n range c e l l nearest to beam c e n t e r range f o r 5° s q u i n t .  Figure  3.7.  Windowed range i n t e r p o l a t o r of l e n g t h 16 i n c l u d i n g 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 .  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  Frequency normalized to PRF  0.8  1  Azimuth Matched Filter Spectrum squint=5.0 deg -  .  Illin iii  .  .  ,  .  .,,  .  irtH  i  1  1  0  r--  0.2  C  V  0.4  1'"  •" T  -  T  0.6  Frequency normalized to PRF  "'  "1  0.8  "  i  1  Azimuth Kaiser-Bessel Window squfnt=5.0 deg . betaq=1.5  c 0.3 0.2 0.1 0  ' •  H 0  H  1  0.2  1  1  0.4  :  1  1  1  0.6  Frequency normalized to PRF  1  0.8  h 1  62  ( 8 P ) •prujuCDN  F i g u r e 3 . 1 1 . Azimuth spectrum a f t e r and windowing.  RCMC, m a t c h e d  filter,  63  (gp)  igure 3.12.  •prujufiDH  F u l l y compressed range p r o f i l e without SRC f o r 0° and 5° s q u i n t using a l e n g t h 16 RCMC interpolator.  64  ( 8 P ) •pnimfiDW  F i g u r e 3.13.  F u l l y compressed azimuth p r o f i l e without SRC f o r 0° and 5° s q u i n t using a l e n g t h 16 RCMC interpolator.  65  O  1 O  1 O  1 O  1 O  1 O  1 Q  1 O  1 O  O  1 O  T" °  6 u i u » p D O j q »6uDg %  F i g u r e 3.14. Range broadening of p o i n t t a r g e t response withdut SRC f o r v a r i o u s RCMC i n t e r p o l a t o r lengths.  66  M  II  c 3  or  </>«, II  fiujudpciojq efiuDH %  F i g u r e 3.15.  Range broadening of p o i n t t a r g e t response (expanded) without SRC f o r v a r i o u s RCMC interpolator lengths.  67  fiuiuopDojq  F i g u r e 3.16.  mnuijzv %  Azimuth broadening of p o i n t t a r g e t response without SRC f o r various' RCMC i n t e r p o l a t o r lengths.  68  F i g u r e 3.17.  1-D range i n t e g r a t e d s i d e l o b e r a t i o s without SRC f o r v a r i o u s RCMC i n t e r p o l a t o r l e n g t h s .  69  CO  M II  o  -*  ro go  CO  3  tr  2  - CM  N  II  I  CO  7  •n to  T  in o"  in  (BP) tnsi  N I  I  in  M  N I  I  F i g u r e 3.18. 1-D azimuth i n t e g r a t e d s i d e l o b e r a t i o s without SRC f o r v a r i o u s RCMC i n t e r p o l a t o r l e n g t h s .  70  F i g u r e 3.19.  2-D i n t e g r a t e d s i d e l o b e r a t i o s without SRC v a r i o u s RCMC i n t e r p o l a t o r l e n g t h s .  for  71  «  I  -  T  I  -  N  I  <  I  N  C  I  N  I  <  N  I  N  C  I  M  I  N  N  I  N  I  N  >  I  0  I  (ap) aisd  F i g u r e 3.20.  1-D range peak s i d e l o b e r a t i o s without SRC f o r v a r i o u s RCMC i n t e r p o l a t o r l e n g t h s .  72  <0  tf  M  CO  II  -p o  <0  tf g o  CO  c 3  -p  - cs  II  _ l  N  < ft  I  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  F i g u r e 3.21. 1-D azimuth peak s i d e l o b e r a t i o s without f o r v a r i o u s RCMC i n t e r p o l a t o r l e n g t h s .  SRC  73  F i g u r e 3.22.  2-D peak s i d e l o b e r a t i o s without SRC v a r i o u s RCMC i n t e r p o l a t o r l e n g t h s .  for  74  F i g u r e 3.23. Degradation of peak magnitude without SRC f o r v a r i o u s RCMC i n t e r p o l a t o r l e n g t h s .  4. ANALYSIS OF BROADENING IN RANGE/DOPPLER COMPRESSION T h i s s e c t i o n p r e s e n t s a theory which c h a r a c t e r i z e s the broadening, p r i m a r i l y  i n range, which occurs a t h i g h squint  angles i n b a s i c range/Doppler p r o c e s s i n g without SRC. The theory extends the range broadening model of J i n and Wu [13] to i n c l u d e the e f f e c t s of the range s i d e l o b e c o n t r o l window. The p r i n c i p l e of s t a t i o n a r y phase, which was used i n the p r e v i o u s chapter to r e l a t e the azimuth spectrum t o the azimuth time domain s i g n a l ,  i s r e p l a c e d by an approximate  F o u r i e r t r a n s f o r m which accounts f o r the spectrum broadening which occurs i n low TBP l i n e a r FM s i g n a l s . Measurements of the simulated azimuth TBP and s p e c t r a l broadening i n both the azimuth frequency and range time d i r e c t i o n s a r e presented. Azimuth broadening i s shown t o be a c c u r a t e l y  predicted  by the decrease i n processed bandwidth with i n c r e a s i n g squint angle, a q u a l i t y which i s inherent t o the s i m u l a t i o n models. L i t t l e ,  i f any, azimuth broadening i s caused by the  azimuth s p e c t r a l broadening.  4.1 BROADENING MODEL FOR RANGE/DOPPLER COMPRESSION WITHOUT SRC The p r e v i o u s chapter presented a theory of range and azimuth compression f o r the b a s i c range/Doppler a l g o r i t h m . I t was shown that the 2-D range compressed expressed as :  75  s i g n a l can be  76  h  R C  (t „) = h (t ,) * f  A  f  t  h  R C p  (58)  (t)  where  h (t,rj) = w A  and h  R C p  (7?)  exp[-j47rr  ( t ) i s the 1-D  usually similar The  first  (T?)/X]  6[t-2r (T?)/C ]  range compressed  i n shape to a sine  p r o f i l e which i s  function.  step a f t e r range compression  range/Doppler azimuth compression  (59)  in basic  i s the computation of the  azimuth f a s t F o u r i e r transform (FFT) of the range  compressed  s i g n a l . In continuous-time theory t h i s i s r e p l a c e d by a continuous azimuth F o u r i e r t r a n s f o r m . In g e n e r a l , the azimuth F o u r i e r t r a n s f o r m of the range compressed cannot be r e p r e s e n t e d e x a c t l y separable  signal  i n c l o s e d form, or even  in a  form.  A major approximation i s now  made which a l l o w s the  azimuth spectrum to be expressed i n a form which  restricts  the range-azimuth c o u p l i n g to a d e l t a l i n e f u n c t i o n as i n the azimuth time domain s i g n a l i n equation (58). A s i m i l a r approximation was v a l i d i t y was  o r i g i n a l l y p r e s e n t e d i n [13] but i t s  not f u l l y d i s c u s s e d . The c o n v o l u t i o n  i s approximated by a c o n v o l u t i o n  i n azimuth as :  i n range  77  h  RC  ( t  '  T j )  *  h  ( t A  ' » ,  %  )  h  RCP  = { WJTJ) e x p [ - j  (60)  ( C l T ? )  (47r/X)r(7j) ] 6 [ r?-r - (ct/2) ] } 1  a  *  where r  _ 1  (r)  7?  h RCP (C  1  T?)  (61)  i s the i n v e r s e f u n c t i o n of the RCM  equation,  and c, i s the slope of the RCM curve at the beam  r(ri),  c e n t e r time given by :  - - X f  The of  c  (62)  / c  approximation  the range p r o f i l e  can be viewed as a l i n e a r p r o j e c t i o n  about the RCM curve  i n t o azimuth time  with the p o s i t i o n of the peak of the p r o f i l e l i e along  c o r r e c t e d to  the RCM curve. Two major assumptions have been  made :  1.  The shape of the amplitude direction  profile  i n the azimuth  i s assumed t o be the same as the range  compressed p r o f i l e with the e x c e p t i o n of a s c a l i n g constant.  In r e a l i t y the azimuth p r o f i l e  is slightly  asymmetric due t o range c u r v a t u r e .  2.  The s c a l i n g c o n s t a n t , which i s the slope of the RCM curve,  i s assumed t o be constant  processing i n t e r v a l .  over  the azimuth  A c t u a l l y the slope v a r i e s slowly  78 over the p r o c e s s i n g i n t e r v a l again due t o range curvature.  The f i r s t  assumption seems reasonable when the azimuth  timewidth i s s m a l l , i . e . , when the amount of range curvature over the -3dB azimuth timewidth i s much l e s s than the range r e s o l u t i o n . F o r t u n a t e l y , t h i s c o n d i t i o n occurs a t l a r g e r squint angles where spectrum broadening i s of most  interest.  The assumption may break down at small squint angles where little  broadening o c c u r s .  The second assumption may be more s e n s i t i v e to range curvature since l i n e a r i t y  i s assumed over the e n t i r e azimuth  p r o c e s s i n g i n t e r v a l . For RADARSAT parameters range curvature i s very small and the slope of the RCM curve does not vary a p p r e c i a b l y over the processed azimuth a p e r t u r e . However f o r longer wavelength SARs, the slope may vary somewhat. The chapter on m u l t i l o o k p r o c e s s i n g addresses t h i s assumption i n more d e t a i l . A p p l y i n g the azimuth F o u r i e r t r a n s f o r m and using the c o n v o l u t i o n theorem range compressed  H  R C  leads t o the f o l l o w i n g equation f o r the  spectrum :  ( t , f ) = W (f) * g  f  F{ exp[-j(47r/X)r(rj) ] } *  [ F{6[ -r-'(ct/2)]} n  • F{h  R C p  f  (c,7 )} J  ]  (63)  where W ( f ) i s the F o u r i e r t r a n s f o r m of the azimuth antenna  79 weighting  f u n c t i o n , w (r?). In t h i s form the azimuth spectrum  c o n s i s t s of the c o n v o l u t i o n of three f u n c t i o n s i n azimuth frequency. The  second two f u n c t i o n s w i l l be evaluated  explicitly.  The middle f u n c t i o n i s the F o u r i e r transform of the phase along the RCM curve. Using the q u a d r a t i c approximation of equation  F{  (12), i t can be evaluated  exp[-j(47r/X)r(T7)]  i n c l o s e d form [17] as :  } * ( l / / | K | ) exp[ jtf(f) ] A  (64)  where  iMf) = - w t f - f , ) / ^ 2  \jj  0  +  U/4)SIGN(K ) + * A  (65)  0  = -4rrr /X  (66)  1  and SIGN(x) denotes the s i g n of x. Although approximation  the q u a d r a t i c  t o the azimuth phase i s used here, the exact  transform w i l l be s u b s t i t u t e d back a f t e r e x t r a c t i n g a broadening The  function.  d e l t a f u n c t i o n i n the t h i r d term c o n t a i n s the  range-azimuth c o u p l i n g . I t s transform  i s a l i n e a r phase  complex e x p o n e n t i a l i n which the r a t e of change of phase i s a f u n c t i o n of range :  F{  Sin-r-  1  ( c t / 2 ) ] } = exp[ -j2irfr- ( c t / 2 ) 1  ]  (67)  80 Finally profile  F{  the t r a n s f o r m of the s c a l e d  range  compressed  i n c l u d i n g the range window i s given by :  h  R  The  C  p  (  C  l  } = (1/|c,|) H  T } )  following  R C p  (f/  )  C l  (68)  s u b s t i t u t i o n s and approximations were not  shown i n the paper by J i n and Wu  but are e s s e n t i a l to  understanding the broadening model. Using the above transforms the second c o n v o l u t i o n rewritten  i n equation (63) can be  as :  1  H  R  C  p  (f7cJ  e ^  (  f  "  r  ) e  -J2*f'r-'(ct/2>  |c,| v/|K | A  =  (  eW >  f ! H  f  |c,|  C  p  (fV  C  l  )  e  ^  V  leading  j 2 i r f ' [ ( f - f ,)/K  substituted  - r- (ct/2)] 1  A  flf  ,  (  ?  0  )  curve i n equation (64). be  back.  remaining i n t e g r a l d e f i n e s the amplitude broadening  and phase d e v i a t i o n change of v a r i a b l e ,  =  )  ^  Thus the exact time domain phase f u n c t i o n w i l l  U,  g  e x p o n e n t i a l can be r e c o g n i z e d as the t r a n s f o r m  of the azimuth phase along the RCM  The  6  • l ^ l e  The  R  d r  f  /  K  A  along each azimuth l i n e . Using the  81  the i n t e g r a l can be expressed as :  A,(f)  *  6[f-fi~K r- (ct/2)]  (72)  1  f  A  where  A,(f) - (1/|c |)  H  a  („/c ) e-^ A^ K  R C p  2  2  3 * » t dr, i 2  e  fl  (73)  =  (l/|c |) H 2  R C p  ( i / c ) e ^V*'* ? 1  2  }  ( ? 4 )  and  c  2  As  =  c,/K  A  =  (75)  -Xf /(cK ) c  A  i t stands, the broadening  convolution  i n azimuth  i s expressed as a  frequency which  range. To express the broadening  i s dependent on  i n terms of range  time,  A , ( f ) must be p r o j e c t e d back i n t o range using the approximately constant s l o p e , c , of the RCM 2  azimuth  frequency domain. T h i s second p r o j e c t i o n of the  s i g n a l model about and Wu.  curve i n the  The  the RCM  curve was  not d i s c u s s e d by J i n  same c o n d i t i o n s apply f o r t h i s p r o j e c t i o n  b e f o r e except that the s i g n a l frequency domain. The  result  i s now is :  i n the  azimuth  as  82  A,(f)  *  5[f-fi"K r- (ct/2)] 1  f  A  -A^t/cj)  *  t  6 [ t - ( 2 / c ) r ( [ f - f , ]/K ) ]  (76)  A  Since the azimuth antenna  function  i s a slowly  varying  f u n c t i o n of azimuth time and the remainder of equation (63) i s an approximately l i n e a r FM s i g n a l , the  i t s c o n v o l u t i o n with  above terms can be approximated by a m u l t i p l i c a t i o n [17]  with a s c a l e d v e r s i o n of the antenna  f u n c t i o n . The s c a l i n g  i s d e f i n e d by the azimuth frequency t o azimuth time mapping, r j ^ ( f ) . T h i s g i v e s the f i n a l range compressed  H  R C  form of the azimuth transformed  s i g n a l as :  ( t , f ) = w ( r ( f ) ) • F{ e x p [ - j ( 4 7 r / X ) r ( 7 j ) ] } a  •  ? i  { 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 alter  the range d i s p e r s i o n d e f i n e d by equation (77) s i n c e  azimuth compression occurs p r i m a r i l y p a r a l l e l to the RCM curve. T h e r e f o r e the approximate range p r o f i l e a f t e r compression  i s determined by A , ( t / c ) which 2  azimuth  i s the i n v e r s e  F o u r i e r t r a n s f o r m of a weighted l i n e a r FM s i g n a l . When the width of the l i n e a r FM s i g n a l  i s small  when the phase at the -3dB p o i n t s i s much l e s s than r a d i a n s ) , the q u a d r a t i c phase e x p o n e n t i a l produces l i t t l e  (i.e., it/2  i n equation (74)  broadening of the i n v e r s e t r a n s f o r m . T h i s  83  o c c u r s when c are  2  and t h e r e f o r e a l s o f ^ and the s q u i n t angle  s m a l l . Thus f o r small squint a i g l e s , the range  profile  b e f o r e and a f t e r the azimuth F o u r i e r transform i s approximately the same and l i t t l e a f t e r compression without  4.2  range broadening appears  SRC.  BROADENING SIMULATIONS AND  MEASUREMENTS  T h i s s e c t i o n p r e s e n t s the r e s u l t s of s i m u l a t i n g the range broadening f u n c t i o n developed i n the p r e v i o u s s e c t i o n . are  These  compared to measurements of the broadening of the range  compressed  point target  response used i n the p r e v i o u s  s i m u l a t i o n s . These s i m u l a t i o n s are s i m i l a r to those p r e s e n t e d by J i n and Wu.  However the range window has  added and d e t a i l e d q u a n t i t a t i v e are  been  image q u a l i t y measurements  performed. Measurements of the azimuth TBP are presented to r e l a t e  measurements of azimuth s p e c t r a l broadening to the decrease i n azimuth  TBP.  A p r e d i c t e d azimuth broadening curve i s shown which i s based on the decrease i n processed bandwidth w i t h i n c r e a s i n g squint angle. The range broadening f u n c t i o n , A , ( t / c ) , p r o v i d e s a 2  model f o r the range broadening of the azimuth spectrum which o c c u r s i n the azimuth F o u r i e r t r a n s f o r m . T h i s f u n c t i o n  has  been s i m u l a t e d with nominal RADARSAT parameters with the same range window parameter  (0 =2.7)  p r e v i o u s s i m u l a t i o n s . The -3dB  R  that was  used i n  range widths have  been  84 measured f o r v a r i o u s squint angles and are summarized i n figure  4.1.  The simulated broadening f u n c t i o n was  generated  a c c o r d i n g to equation (74) with f r e p l a c e d by t / c i n v e r s e F o u r i e r transform i n t e g r a l approximated i n v e r s e FFT. The range compressed H £ ( TJ ,/c 2 ) , was R  p  range  2  and the  by an  spectrum,  simulated as having zero phase with a  magnitude d e f i n e d by the range window f u n c t i o n . T h i s m u l t i p l i e d by the q u a d r a t i c phase broadening term.  was  The  f u n c t i o n s were generated i n a l e n g t h N (N=2048) d i s c r e t e frequency domain a r r a y with maximum frequency equal to the range sampling r a t e . An  i n v e r s e FFT was a p p l i e d to form a  d i s c r e t e , range time domain, broadening f u n c t i o n .  The  r e s u l t i n g response width was measured with the image q u a l i t y measurement programs d i s c u s s e d e a r l i e r . The range broadening p r e d i c t e d by the broadening f u n c t i o n agrees w e l l with the a c t u a l range broadening r e s u l t s shown i n f i g u r e 3.14  of the p r e v i o u s c h a p t e r .  A d d i t i o n a l broadening occurs with s m a l l e r RCMC i n t e r p o l a t o r s , e s p e c i a l l y at l a r g e squint a n g l e s , due to the i n t e r p o l a t o r windowing. Since the range broadening model i s based upon p r e d i c t i o n s of azimuth spectrum broadening, the shapes of the azimuth time and frequency domain s i g n a l s were examined. F i g u r e s 4.2  to 4.6  show the r e l a t i o n s h i p between the azimuth  timewidth of the s i m u l a t e d range compressed bandwidth  s i g n a l and i t s  a f t e r the azimuth FFT. The upper graphs  show the  85  O  m  O  1  O  1 tn  O  1 M  O  1  Ff O O  *-  F i g u r e 4.1. Range broadening of the s i m u l a t e d , t h e o r e t i c a l , range broadening f u n c t i o n without SRC.  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  1  —i  100  -80  1  1  1  -60  1  1  1  1  1  1  1  1  1  1  -40 -20 0 20 40 Predicted relative frequency ( H z )  Azimuth Amplitude  1  60  1  1  r-  80  100  Spectrum  squint = 0 deg, br = 2.7  0.1  i  1 11 1 1 1  0  J  -0.1 -0.2 -0.3  Mr  i  -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1 •100  -80  -60  -40 Frequency  -20  O  20  40  60  80  relative to beam center ( H z )  ure 4.2. Predicted and actual azimuth spectra for 0' squint.  1 00  Azimuth Time-domain Amplitude squint = 1 deg, br = 2.7  m •D TJ 3 C t> O  2  0  20  40  60  Predicted relative frequency ( H z )  Azimuth Amplitude Spectrum squint = 1 deg, br = 2.7  1 0 -1 -2  m  -3  TJ  >—' TJ 3 C  o>  -4 -5  O  -6 -7 -8 -9 — 10 -| -100  1 1 1 1 1 1 1 1 -80  -60  -40  -20  r 0  20  1 oo  Frequency relative to beam center ( H z )  F i g u r e 4.3. P r e d i c t e d and a c t u a l azimuth s p e c t r a squint.  f o r 1°  Azimuth Time —domain Amplitude squint — 5 deg, br — 2.7  s  -100  -80  -60  -40  -20  0  20  40  60  80  100  Predicted relative frequency (Hz)  Azimuth Amplitude Spectrum squint = 5 deg, br = 2.7  —40 -| T 1 -100 -80  Figure 4 . 4 .  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -60  -40 -20 O 20 40 60 Frequency relative to beam center (Hz)  P r e d i c t e d and a c t u a l azimuth s p e c t r a squint.  80  for 5°  100  Azimuth Time-domain Amplitude squint = 10 deg, br = 2.7  m TJ  •  TJ 3 C t» O  Predicted relative frequency (Hz)  Azimuth Amplitude Spectrum squint — 10 deg, br — 2.7  0 -5 -10  m TJ  -15  TJ 3  "E  -20  D  -25 -30 -35 —40 -i -100  1 1 1 1 1 1 1 1 -80  -60  -40  r  -20  Frequency relative to beam center (Hz)  F i g u r e 4.5. P r e d i c t e d and a c t u a l azimuth s p e c t r a f o r 10° SQUin t •  Azimuth Time-domain Amplitude squint = 15 deg. br — 2.7  5 0  /\  -5 -10  ffi TJ  -15  T> 3  *E  J  -20 -25  if  -30  /ni  -35  AA  A  -40 -100  -80  -60  -40  -20  0  20  40  60  80  100  Predicted relative frequency (Hz)  Azimuth Amplitude Spectrum squint = 15 deg, br = 2.7  5 0 -5 -10 CO TJ  -15 TJ 3  C t»  -20  a  2  -25 -30 -35 -40  -i 1 1 1 1 1 20 40 60 -40 -20 Frequency relative to beam center (Hz)  —1 1 1 1 1 1 1 1—  -100  -80  -60  F i g u r e 4.6. P r e d i c t e d and a c t u a l azimuth s p e c t r a squint.  1 1 r~ 80  100  f o r 15°  91 timewidths with the time s c a l e converted to p r e d i c t e d frequency using the azimuth FM r a t e at the beam center as follows :  *p r e dji-c t ejd " f ^C - Kjrrr?-,) A C  (78)  t  The lower graphs show the a c t u a l bandwidths. The graphs are p l o t t e d f o r squint angles of 0, 1, 5, 10, and small squint angles (0° and  15 degrees. At  1°) the time and frequency  domain s i g n a l s 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 e v i d e n t . At 10° and 15° the broadening i s so severe that the a c t u a l s p e c t r a 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 i n f i g u r e s 4.7 to 4.10.  These measurements were performed a f t e r the azimuth  FFT but before RCMC. Broadening i n both the azimuth frequency and range time d i r e c t i o n s was measured at three p o i n t s on the RCM  curve as shown i n f i g u r e 4.11  lower azimuth processed bandwidth cell); the  : at the  frequency (the f a r range  at the beam center frequency (the range c e l l  nearest  beam center range); and at the upper azimuth processed  bandwidth  frequency (the near range  cell).  The spectrum broadening measurements i n the azimuth frequency d i r e c t i o n at low squint angles are i n a c c u r a t e due  Figure  4.7.  Measured azimuth center  azimuth frequency  (mid),  and  spectrum  broadening  direction far  range  in  the  c e l l s .  in  near,  the beam  93  F i g u r e 4.8. Measured azimuth spectrum broadening i n the azimuth frequency d i r e c t i o n i n the near, beam c e n t e r (mid), and f a r range c e l l s (expanded).  Figure  4.9.  Measured range  azimuth  time  center,  spectrum  direction  and  frequencies.  the  upper  at  broadening  the  lower,  processed  in  the  the beam  bandwidth  95  F i g u r e 4.10.  Measured azimuth spectrum broadening i n the range time d i r e c t i o n a; the lower, the beam c e n t e r , and the upper processed bandwidth f r e q u e n c i e s (expanded).  96  Figure 4.11.  P o i n t s on the azimuth frequency domain RCM curve used f o r spectrum broadening measurements.  97 to the l a r g e spectrum r i p p l e s . At l a r g e r squint angles the azimuth spectrum i s smoother a l l o w i n g more a c c u r a t e measurements. The broadening measurements agree w e l l with the a c t u a l broadening r e s u l t s with l a r g e RCMC i n t e r p o l a t o r s and the range broadening p r e d i c t e d by the broadening model. The broadening i s somewhat g r e a t e r a t the low frequency ( f a r range) end of the RCM curve and s l i g h t l y frequency  smaller at the high  (near range) end. T h i s i s due to the small change  in RCM slope over the processed bandwidth caused by range curvature. As the squint angle i n c r e a s e s , the slope of the RCM curve i n the beam center range c e l l  increases causing a  decrease i n the azimuth time width and TBP. For small TBP's, the shape of the azimuth spectrum a f t e r the azimuth FFT broadens and no longer c l o s e l y  resembles the azimuth time  domain s i g n a l . Azimuth -3dB time width measurements of the range compressed  s i g n a l were performed i n order t o c a l c u l a t e  the azimuth time-bandwidth p r o d u c t s (TBP) i n the three range c e l l s noted above as a f u n c t i o n of s q u i n t a n g l e . The r e s u l t i n g c u r v e s , f i g u r e 4.12, can be used to r e l a t e broadening t o the azimuth TBP. From a l i n e a r  range  interpolation  of the t e s t p o i n t s , the beam c e n t e r azimuth TBP's c o r r e s p o n d i n g t o 5% and 10% range broadening a r e 0.76 and 0.57  respectively. Azimuth broadening i s much smaller than range  broadening. The azimuth broadening which does occur can be a t t r i b u t e d t o the decrease i n azimuth processed bandwidth  Measured Azimuth TBP's before SRC  99 with i n c r e a s i n g squint angle which i s inherent to the s i m u l a t i o n model of the azimuth antenna azimuth  -6dB  (two-way) antenna  f u n c t i o n . Since the  time width was  kept constant  over changes i n squint angle whereas the azimuth  frequency  r a t e v a r i e d , the processed bandwidth c a l c u l a t e d by equation (54) decreased with i n c r e a s i n g s q u i n t a n g l e . S i n c e the azimuth  r e s o l u t i o n i s i n v e r s e l y p r o p o r t i o n a l t o the  processed bandwidth,  the percentage azimuth broadening  can  be p r e d i c t e d by c a l c u l a t i n g the percentage decrease i n processed bandwidth as shown i n f i g u r e 4.13.  The  predicted  azimuth broadening agrees very c l o s e l y with the measured results. Very l i t t l e spectrum  azimuth broadening  i s caused by the  broadening which causes range broadening  d i s t o r t i o n of the azimuth phase spectrum TBP  under low  c o n d i t i o n s i s very small over the -3dB  bandwidth.  Since azimuth compression  the phase spectrum, Finally,  little  f i g u r e 4.14  s i n c e the azimuth  azimuth  i s mainly a f u n c t i o n of  azimuth broadening o c c u r s . shows how  the t o t a l amount of  RCM  over the processed a p e r t u r e v a r i e s with squint angle f o r the given set of RADARSAT parameters. RCM  i n c r e a s e s almost  l i n e a r l y with squint angle as i s expected f o r a predominantly q u a d r a t i c curve.  100  CuiuftpDOjq mnujizv %  F i g u r e 4.13.  Azimuth broadening p r e d i c t e d by decrease i n azimuth processed bandwidth.  Range Cell Migration (RCM) versus Squint Angle  T  2  1  1  4  1  1  6  1  1  8  1  1  10  Squint angle  1  1  12 (deg)  1  1 14  1  1  16  1  1  18  1—  20  5.  SECONDARY RANGE COMPRESSION (SRC)  T h i s chapter i n t r o d u c e s a new secondary range compression  (SRC) a l g o r i t h m which compensates f o r the range  broadening which occurs i n the azimuth FFT and which i s not accounted f o r i n the b a s i c range/Doppler a l g o r i t h m . A mathematical theory f o r the SRC a l g o r i t h m i s developed directly  from the p o i n t t a r g e t response model i n chapter 3 .  Two new d i s c r e t e  implementations a r e developed : azimuth SRC  and range SRC. I t i s shown with s i m u l a t i o n s that the SRC algorithms provide excellent which  recompression of the energy  i s spread by the azimuth FFT at l a r g e squint angles  (small azimuth TBP's) f o r nominal RADARSAT parameters.  5.1  THEORY OF AZIMUTH MATCHED FILTERING AND SRC  T h i s s e c t i o n extends the theory of azimuth compression developed i n chapter 3 to show how SRC can be used to recompress the energy which i s d i s p e r s e d by the azimuth Fourier transform. The  i d e a l matched f i l t e r  f o r a point target signal  h*(-t,-rj) = h * ( - t , - 7 ? ) * h*(-t,-rj)  is:  (79)  Chapter 3 d e s c r i b e d the theory of range compression i n which the  range compressed  s i g n a l h ( t , 7 ? ) i s formed by c o n v o l v i n g R C  * with the range matched f i l t e r used t o c o n t r o l  h ( - t , - T ? ) and a range window  sidelobes.  102  R  103 Azimuth compression c o n s i s t s of c o n v o l v i n g the range compressed s i g n a l with an azimuth r e f e r e n c e phase *  filter,  h p ( - t , - 7 ? ) , and an azimuth window t o c o n t r o l s i d e l o b e s . The i d e a l i z e d azimuth r e f e r e n c e phase f u n c t i o n was given previously  i n equation  (32) as :  h ( t , 7 j ) = exp[-j47rr (T?)/X] 6[ t - 2 r ( T?)/C ]  (80)  F  Its  azimuth F o u r i e r transform can be e v a l u a t e d using the  Fourier  transforms  from chapter  4 :  H ( t , f ) = F{exp[-j47rr (T?)/X]} * p  J<M> f  V/|K |  *  F  F{ 6 [ t-2r (r?) /c ]}  (81 )  - j 2 7 r f r - (ct/2) 1  f  (82)  A  00  — oo  =  e ^ j  ( f )  r "  J<Mf-f)  e  e  -3  ( f f  / A K  e  ) f  -j27rf'r-  '  1  (ct/2)  (83)  df'  2  J2jrf'[(f-f')/K -r- (ct/2)] 1  e  A  df' (84)  Using  the same change of v a r i a b l e s as b e f o r e , 7j = f ' / K , and 1  s u b s t i t u t i n g back the exact phase, the f i l t e r  becomes :  A  F o u r i e r t r a n s f o r m of the azimuth  104  H (t,f) p  = F{  exp[-j47rr(T?)/X] }  • { A (f) *  6[f-fi-K r" (ct/2)] }  (85)  1  2  f  A  where  2  A (f)  = |K |  2  e  A  = F" { 1  = •IK.I  |K | A  e  e  e~i* h ' K  ^/V  f  J27rfT},  drj.  (86)  }  V 2  2 e  (87)  -J(»/4)SIGN(K )  (88)  A  The azimuth spectrum broadening f u n c t i o n A ( f ) i s s i m i l a r i n 2  form to A ^ f ) of chapter 4 except that no range windowing i s applied. The  r e f e r e n c e phase  filter  d e f i n e d by e q u a t i o n  c o u l d be a p p l i e d to the range compressed convolution  (85)  response as a  i n the azimuth frequency d i r e c t i o n . However,  s i n c e the width of the azimuth spectrum i s u s u a l l y l a r g e r i n the azimuth d i r e c t i o n  than i n the range time d i r e c t i o n ( i n  terms of the number of samples), i t i s more e f f i c i e n t apply the f i l t e r  i n the range time d i r e c t i o n . T h i s i s  accomplished by p r o j e c t i n g the f i l t e r the approximately l i n e a r domain RCM  to  curve :  i n t o range time using  slope, c , of the azimuth frequency 2  1 05  H (t,f)  F{ e x p [ - j 4 7 r r ( T j ) / X ]  p  • { A (t/c ) 2  2  *  }  6[t-(2/c)r([f-f,]/K )] }  t  A  T h i s p r o j e c t i o n uses the same assumption  (89)  of a l i n e a r RCM  curve which was used i n chapter 4. This  filter  can now be a p p l i e d  to the range  compressed  spectrum along with an azimuth window f u n c t i o n W ^ f - f ^ ) to perform azimuth compression with SRC. The peak of the window function  i s c e n t e r e d a t the beam c e n t e r  frequency, f , c  proper windowing. T h i s produces the f o l l o w i n g azimuth compressed  for  form f o r the  frequency domain s i g n a l :  a(t,rj) = F-'{ [ H  R C  = F- { [ H ( t , f ) *  t  1  R C  (t,f)  *  t  H (-t,f) F  ] W (f-f ) A  c  }  (90)  g ( t , f ) ]F*{exp[-j47rr(Tj)/X]}W (f-f )} c  A  c  (91 ) where  g ( t , f ) = g(t) * c  t  6[t+(2/c)r([f-f,]/K )] A  = g( t + ( 2 / c ) r ( [ f - f , ] / K ) A  g(t) =  (93)  )  (94)  k* (-t/c ) 2  2  = V/|K | J ( ^ / 4 ) S I G N ( K ) A  e  (92)  A  e  -J7rK  S R C  t  2  (95)  106  K  SRC  =  l  /  (  Equation  K  A * C  }  =  K  A  [  c  /  (  X  f  )  P  (  9  6  )  C  (91) d e s c r i b e s one form of SRC a l g o r i t h m i n  which the SRC f i l t e r ,  g ( t ) , i s applied during  azimuth  compression as a range time c o n v o l u t i o n . Instead of a p p l y i n g the  SRC f i l t e r ,  [13],  g ( t ) , s e p a r a t e l y as suggested by J i n and Wu  i t can be combined  a combined  with the RCMC i n t e r p o l a t o r t o form  SRC/RCMC f i l t e r ,  g ( t , f ) . T h i s type of c  implementation w i l l be c a l l e d azimuth SRC. A l t e r n a t i v e l y the SRC f i l t e r range compression of  can be implemented  during  i n the range frequency domain. T h i s type  a l g o r i t h m w i l l be c a l l e d range SRC. I f the t r a n s m i t t e d  range p u l s e i s l i n e a r FM, the SRC f i l t e r with the range compression matched f i l t e r  can be combined by simply  modifying the l i n e a r FM rate of the f i l t e r .  Both of these  implementations w i l l be examined i n f o l l o w i n g  sections.  5.2 AZIMUTH SRC T h i s s e c t i o n d e s c r i b e s the azimuth SRC implementation. Equation  (91) d e s c r i b e s the b a s i c  form of the azimuth SRC  azimuth compression a l g o r i t h m . A f t e r t r a n s f o r m a t i o n i n t o the azimuth frequency domain, the 2-D range compressed H  R C  (t,f),  filter, result the  i s convolved i n range with a combined  g (t,f), c  which  signal,  SRC/RCMC  i s azimuth frequency dependent. The  i s a 1-D azimuth s i g n a l . T h i s i s then m u l t i p l i e d by  azimuth r e f e r e n c e phase  filter  and the azimuth window.  Upon i n v e r s e t r a n s f o r m a t i o n with an i n v e r s e azimuth FFT, a  107 s i n g l e compressed  azimuth l i n e  The SRC/RCMC f i l t e r ,  i s obtained.  g (t,f), c  i s formed by the  c o n v o l u t i o n of two components :  1.  the SRC f i l t e r ,  g ( t ) , which compresses  the d i s p e r s i o n  caused by the azimuth F o u r i e r t r a n s f o r m i n both range and azimuth.  2.  a range-azimuth coupled d e l t a f u n c t i o n which  extracts  energy along the RCM curve, i . e . , performs RCMC.  Since the range compressed  s i g n a l e x i s t s only f o r  d i s c r e t e range and azimuth time, d i s c r e t e SRC and RCMC f i l t e r s a r e r e q u i r e d . These can be formed by b a n d l i m i t i n g the  i d e a l continuous f i l t e r s t o the range and azimuth  sampling  rates.  There a r e s e v e r a l ways of implementing the f i l t e r s . For SRC p r o c e s s i n g i n the azimuth frequency domain, i t i s most efficient  t o combine the SRC f i l t e r  T h i s combined f i l t e r , filter,  g^,(t,f),  g ( t ) , but i s s h i f t e d  and RCMC i n t e r p o l a t o r .  i s the same as the 1-D SRC  i n range by an amount which  v a r i e s with azimuth frequency. Although azimuth frequency i s d i s c r e t e , the range time s h i f t filter  r e q u i r e d f o r the SRC/RCMC  can take on any c o n t i n o u s value due t o the c o u p l i n g  between range and azimuth. To a v o i d c r e a t i n g a new shifte.d f i l t e r  v e r s i o n f o r each  d i s c r e t e azimuth frequency, the continuous s h i f t may be  108  approximated was  by an i n t e g e r s h i f t and a f r a c t i o n a l s h i f t as  done with the RCMC i n t e r p o l a t o r  i n chapter 3. Integer  range sample s h i f t s a r e handled by s h i f t i n g the e n t i r e filter  the r e q u i r e d number of samples.  approximated  F r a c t i o n a l s h i f t s are  by choosing the best of s e v e r a l  v e r s i o n s of the f i l t e r  precomputed  each of which i s s h i f t e d by a  f r a c t i o n of a range sample. The s h i f t e d v e r s i o n s a r e produced  by i n t e r p o l a t i n g the f i l t e r  and then e x t r a c t i n g the d i f f e r e n t  by an i n t e g e r f a c t o r I  phases. T h i s i s the same  approximation that was p r e v i o u s l y used to form the b a s i c RCMC i n t e r p o l a t o r except that the combined f i l t e r small q u a d r a t i c phase and i s broadened  i n amplitude.  S e v e r a l steps a r e r e q u i r e d t o produce d i s c r e t e SRC/RCMC f i l t e r  1.  now has a  a suitable  :  Form an a n a l y t i c a l SRC f i l t e r  i n the continuous  range  frequency domain.  G(f ) r  = F{  (97)  g(t) }  = (Xf /c) e ^  f  c  2.  r  /  K  SRC  B a n d l i m i t the f u n c t i o n t o the range sampling F  , t o prevent  rect(  f /F  c r  (98)  frequency,  aliasing.  ) G(f )  (99)  109 3.  Sample the continuous sample spacing  range frequency  1/T = F  /K  p e r i o d of the corresponding It  t o get K  f u n c t i o n with samples. T i s one  range time domain f u n c t i o n .  must be chosen t o be much l a r g e r than the -3dB range  timewidth of H ^.(t,f) to prevent  s e r i o u s time domain  R  aliasing.  G(k )  = rect(k /K )  r  4.  r  G U / T ) , -K /2  r  r  f  < k  r  < K /2  (100)  r  Zero pad the a r r a y on both ends to a l e n g t h of I K  r  where  I i s the i n t e r p o l a t i o n f a c t o r which d e f i n e s the number of 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 of the f i l t e r .  Gj(k ) = r  G ( k ) , -K /2 r  < k  r  0 , -IK /2 < k r  , K /2  < k  r  5.  r  <  -K /2 r  < IK /2  r  (101)  r  Apply an inverse range FFT of l e n g t h IK .  g j d n j ) = FFT-H 6.  r  < K /2  r  G ( k ) } , -IK /2 < m r  r  1  <  M u l t i p l y by a l e n g t h IL window, w , ^ ) ,  (102) t o get a  of minimum l e n g t h where L i s the length of each  filter filter  version.  g ( m ) w^nij) J  I  -IL/2  < m  x  < IL/2  (103)  110 7.  E x t r a c t the f r a c t i o n a l l y  g^m)  s h i f t e d v e r s i o n s of the f i l t e r .  = g j d n l + i ) w,(ml + i )  , 0 < i < 1-1 , -L/2 < m < L/2 (104)  In the c u r r e n t s i m u l a t i o n s , different  f i l t e r lengths  16 v e r s i o n s  (1=16) and four  (L = 4, 8, 16, or 32) are used.  F i g u r e 5.1 shows the magnitudes of s e v e r a l SRC/RCMC f i l t e r s of l e n g t h  16 a f t e r  i n t e r p o l a t i o n by a f a c t o r of 16 f o r  several squint angles.  I t i s seen that the f i l t e r s  the s i n e type of i n t e r p o l a t o r f o r small angles for  resemble  but broaden  l a r g e r a n g l e s . Consequently, l a r g e r squint angles  r e q u i r e longer  filters  t o gather  also  a l l the d i s p e r s e d energy.  In b a s i c range/Doppler p r o c e s s i n g without i s approximated by a zero phase f i n i t e  SRC, g^(m),  length i n t e r p o l a t o r  which corresponds t o the zero squint SRC/RCMC f i l t e r . The approximation holds  f o r small s q u i n t angles  s i n c e the  n o n l i n e a r phase v a r i a t i o n of G ( k ) approaches zero as | f ^ | r  and  the squint angle approach z e r o . Consequently, G ( k ) r  approaches a r e c t a n g u l a r s i g n a l with constant  phase and  gj(mj) approaches a time a l i a s e d sine f u n c t i o n (or sampling f u n c t i o n ) . The RCMC i n t e r p o l a t o r i s shortened by m u l t i p l i c a t i o n with a f i n i t e  l e n g t h window, such as a  K a i s e r - B e s s e l window, t o minimize the amount of computations while a l s o minimizing interpolator  spectrum.  the spreading  and a l i a s i n g of the  Figure  5.1.  Magnitudes for squint  of the angles  SRC/RCMC f i l t e r s of 0 ° , 5 ° , 10°,  of length 16 1 5 ° , and 20°.  11 2 5.3 SIMULATIONS  OF AZIMUTH SRC  T h i s s e c t i o n d e s c r i b e s the r e s u l t s of computer of the azimuth SRC a l g o r i t h m . The a l g o r i t h m  simulations  i s s i m i l a r to  the b a s i c range/Doppler azimuth compression a l g o r i t h m except that the RCMC i n t e r p o l a t o r i s r e p l a c e d by a combined SRC/RCMC  filter.  Range compression  i s performed as i n chapter  produce a 1-D range compressed  3 to  p r o f i l e . This p r o f i l e  i s used  by the azimuth compression s i m u l a t i o n r o u t i n e t o form a simulated  2-D range compressed  s i g n a l . This s i g n a l i s  F o u r i e r transformed i n azimuth using an FFT of l e n g t h for  1024  RADARSAT parameters. The combined SRC/RCMC f i l t e r i s  then a p p l i e d as a d i s c r e t e range time c o n v o l u t i o n t o compensate f o r the d i s p e r s i o n caused by the azimuth FFT. The resulting  1-D azimuth s i g n a l i s m u l t i p l i e d by the FFT of the  exact azimuth r e f e r e n c e phase f i l t e r  and a K a i s e r - B e s s e l  window to c o n t r o l s i d e l o b e s . F i n a l l y the 1-D azimuth s i g n a l i s passed through an i n v e r s e FFT to produce one azimuth of the compressed  image. The p r o c e s s i n g  i s repeated  line  f o r each  d e s i r e d azimuth l i n e . The p r o c e s s i n g parameters are as i n Table  1. The r e s u l t i n g compressed  range and azimuth p r o f i l e s are  shown i n f i g u r e s 5.2 t o 5.5. I t i s seen that very broadening occurs  little  i n range when L i s l a r g e . However severe  broadening can s t i l l  occur  f o r smaller  filters  s i n c e an  a p p r e c i a b l e amount of energy i s d i s p e r s e d beyond the width of the s h o r t e r f i l t e r s .  The l e n g t h of the SRC/RCMC  filter  113 has l i t t l e e f f e c t on the azimuth The -3dB  percentage broadening measurements are  summarized by f i g u r e s 5.6 azimuth. The  profile.  and 5.7  i n range and f i g u r e 5.8 i n  r e s u l t s are shown f o r the four  l e n g t h s of SRC/RCMC  different  filter.  The azimuth broadening r e s u l t s are the same as the r e s u l t s without SRC. of  The p r e d i c t e d azimuth broadening curve  chapter 4, f i g u r e 4.13,  r e s u l t s . The  agrees c l o s e l y with the a c t u a l  range broadening measurements show that f o r  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 q u i c k l y r i s e s since s i g n i f i c a n t energy i s d i s p e r s e d o u t s i d e of the 16 sample width of the f i l t e r . I t i s seen that longer f i l t e r s F i g u r e s 5.9 and 5.10  produce  l e s s range  summarize the 1-D  broadening.  ISLR  measurements. From these i t i s seen that the azimuth ISLR remains almost constant whereas the range ISLR decreases (improves) as s q u i n t  i n c r e a s e s . The decrease i s more r a p i d  for  the s h o r t e r f i l t e r  for  shorter f i l t e r s  l e n g t h s . T h i s decrease i n range ISLR  corresponds to the l a r g e r  range  broadening. I t appears that the windowing a p p l i e d to s h o r t e r filters  causes the range spectrum to be t a p e r e d . The  i s more range broadening with lower range F i g u r e 5.11 the  summarizes the 2-D  ISLR without SRC  sidelobes.  ISLR measurements. Whereas  d e t e r i o r a t e d with i n c r e a s i n g  angle, the ISLR with SRC  result  improves s l o w l y .  squint  1 14 The peak s i d e l o b e r a t i o s 2-D  (PSLR) i n range, azimuth,  are summarized i n f i g u r e s 5.12  ISLR's,  the azimuth  to 5.14.  As with the  PSLR's are v i r t u a l l y constant with  s q u i n t angle while the range PSLR's decrease with s q u i n t angle and d e c r e a s i n g f i l t e r graphs,  3dB  increasing  l e n g t h . By comparing  i t can be seen that the decreases  are l e s s than about  and  i n ISLR and PSLR  f o r squint angles s m a l l e r than the  angle at which the range broadening  i s 5%.  In order to compare the peak magnitudes a f t e r compression,  the peaks were normalized to the sum  squares of the SRC/RCMC c o e f f i c i e n t s and the processed bandwidth. The f i g u r e 5.15. SRC,  of the  azimuth  r e s u l t s are summarized i n  Whereas the peak s t r i c t l y decreases  the normalized peak a c t u a l l y  without  increases s l i g h t l y for  small s q u i n t angles before d e c r e a s i n g at l a r g e r s q u i n t s . The reason f o r t h i s i n c r e a s e i s not c l e a r l y understood. However the s m a l l e r v a r i a t i o n s i n peak magnitude with SRC that an improved  indicate  s i g n a l - t o - n o i s e r a t i o i s achieved.  ( 9 P ) epnuufiDfl  igure  5.2.  1-D r a n g e p r o f i l e s s q u i n t and v a r i o u s  a f t e r SRC c o m p r e s s i o n filter lengths.  for  116  (3P)  F i g u r e 5.3.  •pnuuCDN  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 and v a r i o u s f i l t e r l e n g t h s .  10°  i£J C  SRC Compressed 1-D Azimuth Profiles  fD  squint = 5 deg., br = 2.7, ba = 1 .5  Ol  -5  Ol — o I a  L = 4 , 8, /6 , 32.  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— o J  fD 3 O vQ o rf 3 3"0 in i • fD in in w O 3  -10 m TJ  -15  TJ 3  -20  C o  2  -25 -30 -35 -40 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 L=32  -8  -4  0  4  Azimuth sample number L=16 L=8  8  12  L=4  16  119  o  CM  Figure  5.6.  Percentage range broadening with s i n g l e - l o o k azimuth SRC as a f u n c t i o n of squint angle and f i l t e r length.  Figure  5 . 7 . Percentage range broadening with s i n g l e - l o o k azimuth SRC as a f u n c t i o n of s q u i n t angle and f i l t e r l e n g t h (expanded s c a 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 s i n g l e - l o o k azimuth SRC as a f u n c t i o n of s q u i n t angle and f i l t e r length.  122  (ap) aisi  F i q u r e 5.9. Figure  1-D f  u  n  c  t  range ISLR with s i n g l e - l o o k azimuth SRC as a ^ t angle f o r v a r i o u s f i l t e r  i  o  lengths.  o f  s  q  u  i  n  123  Figure  5.10.  1-D azimuth ISLR with s i n g l e - l o o k azimuth SRC as a f u n c t i o n of s q u i n t angle f o r v a r i o u s f i l t e r lengths.  124  (ap)  *nsi  F i a u r e 5 11. 2-D ISLR with s i n g l e - l o o k azimuth SRC as a Figure - ^ l e for various f i l t e r 5  ,  1  f  t  i  o  n  lengths.  o  f  s  q  u  i  n  t  a  n  g  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 s i n g l e - l o o k azimuth SRC as a f u n c t i o n of s q u i n t angle f o r v a r i o u s f i l t e r lengths.  126  o M CO  M II  ID  o  o*  «  3-  - to  II  - cs  o M  I  in  d  CS  I  cs I  M CM  in  I  cs I (ap)  Figure  CS  I  »0 CS  I  m CM  CM  I  I  aisd  5.13. 1-D azimuth PSLR with s i n g l e - l o o k azimuth SRC as a f u n c t i o n of s q u i n t angle f o r v a r i o u s f i l t e r lengths.  127  F i g u r e 5.14.  2-D PSLR with s i n g l e - l o o k azimuth SRC as a f u n c t i o n of squint angle f o r v a r i o u s f i l t e r lengths.  128  *-  d  d  1  I  ^  '  I  ts  1  I  (9P) *pn|iu6DW  Figure  5.15. Peak compressed magnitude with s i n g l e - l o o k azimuth SRC as a f u n c t i o n of squint angle f o r various f i l t e r lengths.  129  5.4 RANGE SRC This section describes SRC  filter  an a l t e r n a t e implementation of the  c a l l e d range SRC. T h i s method performs SRC  range frequency domain d u r i n g  range compression. The  in the SRC  filter  i s combined with the range compression f i l t e r  simply  a l t e r i n g the l i n e a r FM r a t e of the range compression  filter. filter  Since  the r e s u l t i n g m o d i f i e d  by  range compression  has the same number of c o e f f i c i e n t s , no a d d i t i o n a l  computations are r e q u i r e d . In a d d i t i o n a s h o r t e r RCMC i n t e r p o l a t o r can be used f o r the subsequent  azimuth  compression even at l a r g e squint angles s i n c e the azimuth spectrum remains w e l l compressed One c o m p l i c a t i o n of the SRC  a f t e r the azimuth FFT.  of t h i s method i s the range  f i l t e r . Depending  on the radar  parameters and  s i z e of the range swath, the range swath may subdivided  i n t o smaller  regions,  need to be  range i n v a r i a n c e regions  purpose of range compression. The  invariance  i s s u e of  f o r the  invariance  which i s a l s o of concern i n azimuth SRC, i s  examined i n chapter 7. The modified in  range compression f i l t e r  can be expressed  the range time domain as :  h  R M  (t)  = s*(-t) *  t  (105)  g(t)  = [ rect(tA)  The range compression f i l t e r  e^V  ] *  t  (106)  g(t)  i s implemented  using  fast  130  convolution  i n the range frequency domain. Applying the  range F o u r i e r transform  to the f i l t e r  gives the f o l l o w i n g  form :  H  R M  Since  ( f ) = F{ r e c t ( t A ) e ^ V  } G(f )  r  the f i r s t  f  term i s the F o u r i e r transform  l i n e a r FM s i g n a l ,  i t may be approximated using  (107)  of a l a r g e TBP the p r i n c i p l e  of s t a t i o n a r y phase as :  F{  rect(tA) e  «  The  rect(f  J 7 r K  R  }  f c 2  /[K_T]) e 3  F o u r i e r transform  W F  2 / K R  R  (108)  of the SRC f i l t e r ,  g ( t ) , was given  p r e v i o u s l y as :  G(f )  = (Xf /c)  r  c  e  J 7 r f  r  2 / K  SRC  (109)  S u b s t i t u t i n g back these transforms and dropping the constant magnitude terms g i v e s the f o l l o w i n g form f o r the frequency domain combined f i l t e r  H  R M  :  (f ) - rect(f /[K r]) e " ^ r  where  r  K  r  r  m  - K /[ 1 - ( K / K R  V  R  R  S R C  )  ]  K  RM  (110)  (111)  131 Since K  >> K , the m o d i f i e d r e f e r e n c e f u n c t i o n i s a l s o a  g R C  R  l a r g e TBP s i g n a l . Therefore the p r i n c i p l e of s t a t i o n a r y phase can again be used t o evaluate the i n v e r s e F o u r i e r t r a n s f o r m of the m o d i f i e d range compression  h  R M  ( t ) =* r e c t ( t A ) e  T h i s equation filter  j , r K  RM  filter  :  (112)  t 2  shows that the combined SRC/range  compression  i s a l i n e a r FM pulse with a m o d i f i e d FM r a t e ,  The  m o d i f i e d FM rate i s a f u n c t i o n of both the range of  c l o s e s t approach of the point t a r g e t , r , and the squint 0  angle. The range of c l o s e s t approach p r i m a r i l y a f f e c t s the azimuth FM r a t e , K., and t o a l e s s e r extent the beam center frequency,  f . The squint angle mainly c  c e n t e r frequency  a f f e c t s the beam  and t o a smaller extent the azimuth FM  r a t e . As the squint angle approaches zero, f ^ a l s o approaches zero and K = K [c/(Xf )] A  K  a  The  r  o  a  c  n  e  approaches a c o n s t a n t . Therefore  K  S R C  approches i n f i n i t y and the m o d i f i e d FM r a t e ,  2  c  RM' P P  A  s  t  n  e  unmodified  FM r a t e , K . R  range SRC a l g o r i t h m i s e s s e n t i a l l y the same as the  b a s i c range/Doppler a l g o r i t h m presented  i n chapter  that the l i n e a r FM r a t e of the range compression  3 except  filter is  m o d i f i e d a c c o r d i n g t o the squint angle. Range SRC i s s u p e r i o r t o azimuth SRC since the e q u i v a l e n t range domain SRC f i l t e r  time  i s much longer than p r a c t i c a l SRC/RCMC  f i l t e r s used i n azimuth SRC. Since the SRC f i l t e r  i n range  SRC i s a p p l i e d over the e n t i r e range bandwidth, the  132 equivalent  range time domain f i l t e r  i s very l o n g , much  longer than the p r a c t i c a l SRC/RCMC f i l t e r  l e n g t h s of between  8 and 16 samples used i n azimuth SRC. The longer filter  equivalent  a l l o w s more of the d i s p e r s e d energy t o be  recompressed  (eventhough the recompression i s implemented as  a prefilter).  5.5 SIMULATIONS OF RANGE SRC T h i s s e c t i o n p r e s e n t s the r e s u l t s of s i m u l a t i n g the range SRC a l g o r i t h m with nominal RADARSAT parameters. The same s i m u l a t i o n programs that were used f o r b a s i c  range/Doppler  compression are used except that the m o d i f i e d l i n e a r FM rate,  developed i n the p r e v i o u s s e c t i o n  range compression f i l t e r .  i s used i n the  A s i n g l e RCMC i n t e r p o l a t o r  length,  L = 16, was used f o r a l l the s i m u l a t i o n s . F i g u r e 5.16 shows the broadening of the range compressed  p r o f i l e after  range SRC f o r squint angles of 5°  and  10°. The broadening occurs because of the mismatch of  the  l i n e a r FM r a t e s of the t r a n s m i t t e d range p u l s e and the  m o d i f i e d SRC/range compression f i l t e r .  This predistorton  becomes recompressed by the azimuth FFT. F i g u r e s 5.17 and 5.18 show the range and azimuth p r o f i l e s a f t e r azimuth compression. I t i s seen that very little  broadening occurs i n range whereas some broadening  does occur i n azimuth as p r e d i c t e d by the decrease i n processed bandwidth. The percentage range and azimuth broadening with range SRC i s summarized i n f i g u r e s 5.19 and  133 5.20.  For s q u i n t angles below 15°, the range broadening  is  very s m a l l , l e s s than 0.55%. The  ISLR and PSLR measurements are summarized i n  f i g u r e s 5.21  and  5.22.  Since l i t t l e  the ISLR and PSLR are approximately  broadening  o c c u r s , both  constant with squint  a n g l e . In f a c t the range ISLR and PSLR improve slowly with i n c r e a s i n g s q u i n t angle. The peak magnitudes are compared i n f i g u r e 5.23. absence of l a r g e v a r i a t i o n s i n d i c a t e s that the s i g n a l - t o - n o i s e r a t i o remains r e l a t i v e l y constant changes i n squint angle.  with  The  Range Compressed Profiles w/Range SRC  C "I fD  L = 16, squinr = 5, 10 deg.  o o C 9 I  10  •"••TJ O r t fD • tn Oi  3  Ul  O fD  3 o * o  rt-TJ  tr n  fD •-« tn OJ in 3 fD  m  TJ  TJ  C 0  3  n>  TJ t/i i » o o m O fD f-« tn ui n» o i-t> rt  0> fD 3 n n — OJ  O3  •4  0  1 6  Sample number  ouQ  fD to it*  135  (SP) frprniufiDfl  F i g u r e 5.17.  1-D range p r o f i l e s a f t e r azimuth compression with s i n g l e - l o o k range SRC f o r 0°, 5°, and 10° s q u i n t and a l e n g t h 16 RCMC f i l t e r .  136  (BP) •pruiufiDfl  F i g u r e 5.18. 1-D azimuth p r o f i l e s a f t e r azimuth compression with s i n g l e - l o o k range SRC f o r 0 ° , 5°, and 10° s q u i n t and a l e n g t h 16 RCMC f i l t e r .  137  NOjwfsiotfJ^iq  CN »-  a> to r» <o in  KI cs  o o o d d d  o o o  o  d I  Figure  5.19. Percentage range broadening with s i n g l e - l o o k ramje SRC and a length 16 RCMC i n t e r p o l a t o r as a f u n c t i o n of squint a n g l e .  138  •^ofheorNtein^KicM^oooorotDin^tOM^o  CuiudpDOjg %  Figure  5.20. .Percentage azimuth broadening with s i n g l e - l o o k range SRC and a l e n g t h 16 RCMC i n t e r p o l a t o r as a f u n c t i o n of squint a n g l e .  139  I  I  I  I  (SP) aisi  •Figure 5.21. Range, azimuth, and 2-D i n t e g r a t e d s i d e l o b e r a t i o s with s i n g l e - l o o k range SRC and a length 16 RCMC, i n t e r p o l a t o r as a f u n c t i o n of squint angle.  i  I  I  (ap) aisd  Figure  5.22. Range, azimuth, and 2-D peak s i d e l o b e r a t i o s with s i n g l e - l o o k range SRC and a l e n g t h 16 RCMC i n t e r p o l a t o r as a f u n c t i o n of s q u i n t angle.  141  (SP) »pn4!u6DN  Figure  >JD*J  5.23. Peak magnitude d e g r a d a t i o n with s i n g l e - l o o k range SRC and a l e n g t h 16 RCMC i n t e r p o l a t o r as a f u n c t i o n of squint a n g l e .  142 5.6 SUMMARY OF SINGLE-LOOK SRC T h i s chapter has presented the r e s u l t s of i n v e s t i g a t i o n s i n t o the use of an SRC a l g o r i t h m f o r improving the range/Doppler compression a l g o r i t h m at l a r g e squint a n g l e s . An SRC f i l t e r filter  was developed by p r o j e c t i n g the i d e a l matched  i n t o the range dimension a f t e r a p p l y i n g the azimuth  FFT. T h i s p r o j e c t i o n 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 q u a d r a t i c phase  form of the azimuth phase i s used t o d e r i v e the SRC f i l t e r . Two implementations were developed and s i m u l a t e d . The s i m u l a t i o n s show that both implementations work w e l l a t recompressing the energy d i s p e r s e d by the azimuth FFT. Since the  dispersion  i n c r e a s e s with squint angle, longer SRC/RCMC  f i l t e r s are r e q u i r e d t o recompress  the d i s p e r s i o n a t higher  squint a n g l e s . The range SRC implementation p r o v i d e s an SRC filter  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 i n azimuth SRC. Consequently at l a r g e squint angles range SRC performs much b e t t e r than azimuth SRC. For  a l e n g t h 16 f i l t e r  the azimuth SRC a l g o r i t h m  extends the squint angle which causes 5% range broadening from 3.65° to 8.03°. For 10% range broadening, the squint angle i s i n c r e a s e d from 4.23° t o 9.29°. In a d d i t i o n the range s i d e l o b e s are a c t u a l l y indicating excellent For  improved by the SRC a l g o r i t h m  compression.  the range SRC a l g o r i t h m with a l e n g t h 16 RCMC  i n t e r p o l a t o r , the range broadening remains very small over all  the s q u i n t angles which were simulated being l e s s than  143 1.3% f o r s q u i n t angles up t o 20°. Thus the range  SRC  implementation p r o v i d e s b e t t e r compression than the azimuthSRC a l g o r i t h m . In p r a c t i c e t h i s weighed  improved performance must be  a g a i n s t the p o s s i b l e c o m p l i c a t i o n s caused by range  i n v a r i a n c e of the SRC f i l t e r . T h i s issue i s d i s c u s s e d further  i n chapter 7.  6. MULTILOOK RANGE/DOPPLER  PROCESSING WITH SRC  In t h i s chapter, methods of implementing SRC f o r multilook  ( s p e c i f i c a l l y 4-look) p r o c e s s i n g are examined. In  m u l t i l o o k p r o c e s s i n g the a p e r t u r e  i s divided into several  looks which are compressed s e p a r a t e l y and then  summed  i n c o h e r e n t l y i n order t o reduce speckle n o i s e . Two new methods of implementing SRC a r e proposed and i n v e s t i g a t e d :  1.  F i x e d M u l t i l o o k SRC. T h i s method uses the same SRC filter  f o r each look. The f i l t e r  center frequency  i s matched to the  of the f u l l a p e r t u r e ,  c e n t r o i d . T h i s i s the same f i l t e r  i . e . , the Doppler  that was used  previously for single-look processing.  2.  Look-Dependent M u l t i l o o k SRC. In t h i s method, a different  SRC f i l t e r  i s used f o r each look. T h i s  compensates f o r any changes i n the slope of the range c e l l migration filter  (RCM) curve between looks s i n c e each  i s matched t o each look center  frequency.  S i m u l a t i o n s of m u l t i l o o k p r o c e s s i n g of a p o i n t t a r g e t response are performed with and without algorithms  the above SRC  to q u a n t i f y the improvements i n image q u a l i t y  p o s s i b l e with m u l t i l o o k SRC.  144  145 6.1  MULTILOOK PROCESSING WITH  SRC  T h i s s e c t i o n f u r t h e r extends the theory of azimuth compression  presented  i n chapter  5 to d e s c r i b e m u l t i l o o k  range/Doppler p r o c e s s i n g both with and without two  new  m u l t i l o o k SRC  the use  of  algorithms.  M u l t i l o o k azimuth compression  with the range/Doppler  a l g o r i t h m i s e s s e n t i a l l y the same as f u l l range/Doppler p r o c e s s i n g except  aperture  that the processed  bandwidth  i s d i v i d e d i n t o separate, o f t e n o v e r l a p p i n g , p r o c e s s i n g bands or l o o k s , which are compressed i n d i v i d u a l l y and i n c o h e r e n t l y summed. Due between the time and each frequency  to the approximate  frequency  correspondence  domains of the azimuth  domain look corresponds  then  signal  to d i f f e r e n t  i n t e r v a l s of the azimuth time domain a p e r t u r e . For a p o i n t t a r g e t s i g n a l , each look c o n t a i n s data c o l l e c t e d different  time  i n t e r v a l s which cover d i f f e r e n t  from  ranges of  i n c i d e n c e a n g l e s , or look a n g l e s . T h i s change i n look causes the speckle noise to have l i t t l e  c o r r e l a t i o n between  l o o k s . Thus incoherent summation of the compressed reduces  the speckle noise l e v e l . F i g u r e 6.1  processed  angle  looks  shows how  the  bandwidth i s d i v i d e d i n t o separate looks ( i n t h i s  case, 4 l o o k s ) . F i g u r e 6.2 domain look  shows the corresponding  angles.  In m u l t i l o o k p r o c e s s i n g , range compression as i n s i n g l e - l o o k p r o c e s s i n g to produce a 2-D compressed s i g n a l . In chapter compression  time  was  developed  i s performed range  5, s i n g l e - l o o k azimuth  as an approximation  to an  exact  146  r e c e i v e d Doppler spectrum i Ulll  \  with two-way azimuth antenna weighting  0 dB  -6 dB processed  bandwidth  820 samples at f  c  (PBW) 0° squint  - PRF/2|  f  1024 p o i n t FFT  256  samples ,  Look 2 256  (  samples  256  samples  F i g u r e 6.1. D i v i s i o n of the azimuth frequency domain aperture into 4 looks.  c  + PRF/2  147  beam c e n t e r c r o s s i n g time, T ? C i i i  Look 3 |  F i g u r e 6.2. Corresponding time domain l o o k s .  1 48 matched f i l t e r For  implemented  i n the azimuth frequency domain.  m u l t i l o o k p r o c e s s i n g , a s i m i l a r compression a l g o r i t h m i s  a p p l i e d t o each frequency band or look. The f o r m u l a t i o n of the azimuth compression f i l t e r f o r each look proceeds as i n chapter 5 up t o e q u a t i o n (85) which expresses the 2-D compression f i l t e r  i n the azimuth  frequency domain. For  single-look  (full  a p e r t u r e ) p r o c e s s i n g , the f i l t e r  i s u s u a l l y t r u n c a t e d near the -3dB azimuth f r e q u e n c i e s with some form of window f u n c t i o n . The exact s e l e c t i o n of p r o c e s s i n g bandwidth  i s a t r a d e - o f f between f a c t o r s such as  ambiguity e r r o r s , a l i a s i n g n o i s e , the d e s i r e d azimuth r e s o l u t i o n , and the type of window used. For the c u r r e n t research, the f u l l a p e r t u r e p r o c e s s e d bandwidth has been a r b i t r a r i l y chosen t o be the -3dB one-way (-6dB two-way) azimuth antenna  bandwidth.  As s t a t e d e a r l i e r , m u l t i l o o k p r o c e s s i n g d i v i d e s the processed bandwidth  i n t o s e v e r a l l o o k s which are o f t e n  o v e r l a p p i n g . For ease of implementation and computational e f f i c i e n c y of the i n v e r s e azimuth FFT's, each look has been chosen t o be 256 samples  l o n g . For 4-look p r o c e s s i n g , which  has been simulated here, these l o o k s are o v e r l a p p e d to f i t i n t o the processed bandwidth. S i n c e the processed bandwidth v a r i e s slowly with squint angle, the number of frequency domain samples  i n the processed bandwidth v a r i e s from 820  samples a t zero s q u i n t down to 680 samples a t 20 degrees of s q u i n t . The c o r r e s p o n d i n g percentage o v e r l a p s range from  149 26.6%  to 44.8%  respectively.  A K a i s e r - B e s s e l azimuth window f u n c t i o n i s a p p l i e d s e p a r a t e l y to each look to c o n t r o l the azimuth s i d e l o b e l e v e l s . Therefore the e f f e c t i v e combined window for the a p e r t u r e , which i s the summation of the i n d i v i d u a l  look  windows, e x t r a c t s more energy from the outer looks than equivalent  full  an  s i n g l e - l o o k K a i s e r - B e s s e l window a p p l i e d to the  f u l l a p e r t u r e . Since the bandwidth of each look i s f i x e d , the compressed r e s o l u t i o n of each look i s approximately constant has  r e g a r d l e s s of squint a n g l e . The  outer looks the weighting i s approximately  i s l e s s and  filter  linear.  from an azimuth frequency  computationally  efficient  l i n e a r slope of the RCM  range convolution,' the f i l t e r  f o r the i n d i v i d u a l look  is  approximately a choice  filters.  implementation,  the same  i s used for a l l l o o k s . Thus the slope at the  center of the f u l l s i n g l e - l o o k SRC.  5 using the  curve. However there i s now  For a f i x e d azimuth SRC SRC/RCMC f i l t e r  compression  c o n v o l u t i o n to a more  p r o j e c t e d i n t o range as i n chapter  s l o p e s to use  f o r the  does not have a c e n t r a l maximum  In order to convert the above azimuth look  of  weighting  l i t t l e e f f e c t over the smaller look bandwidths s i n c e for  the inner looks the amount of weighting  and  antenna  aperture  T h i s may  i s used for the p r o j e c t i o n as i n  i n t r o d u c e very small e r r o r s i n the  outer looks s i n c e the slope of the RCM looks d i f f e r s s l i g h t l y due  curve  i n the  outer  to range c u r v a t u r e . For systems  such as RADARSAT i n which the range c u r v a t u r e i s small over  150 the f u l l a p e r t u r e (about 0.23  range c e l l s ) ,  extremely small as w i l l be demonstrated For  the e r r o r i s  i n the s i m u l a t i o n s .  systems with l a r g e r range c u r v a t u r e such as longer  wavelength  SAR's i n c l u d i n g SEASAT, the e r r o r may  s i g n i f i c a n t . In such cases a look-dependent implementation  i s possible  be more  SRC  i n which the d i f f e r e n c e s i n RCM  slope between looks are compensated by d e s i g n i n g a separate SRC/RCMC f i l t e r  f o r each l o o k . For t h i s method, the slope  used f o r p r o j e c t i n g the f i l t e r  i n t o range  i s taken as the  slope at the look center frequency r a t h e r than the  full  a p e r t u r e c e n t e r frequency. T h i s complicates the c o n t r o l memory requirements f o r the azimuth processor s i n c e SRC/RCMC f i l t e r s precomputed and For  several  ( i n t h i s case 4 of them) need to be stored.  s i n g l e - l o o k SRC,  domain RCM  and  the slope of the azimuth  curve at the c e n t e r frequency of the  a p e r t u r e , or the beam center frequency, f , was c  frequency  full expressed  as :  c  =  2  where K  A  C  i /  R A  =  "  X f  i s the azimuth  / ( c K C  A  )  ( 1 1 3 )  frequency rate at the beam center  frequency. T h i s equation can be used to c a l c u l a t e the slope at any arbitrary it  look center frequency, f « L  i s necessary to know the azimuth  To do t h i s  accurately,  frequency r a t e at that  frequency. T h i s i s accomplished by using equations (37),  151 (38),  The  and  (56) :  rj^f)  = r  0  / { v  r.(f)  = r  0  / [ 1 - (Xf/(2v ))  K.(f)  = -(2v  first  two  / [ X r . (f )])  2  L  r  These are  Li  the look  Li  LI  curve at the look center  i s c a l c u l a t e d by s u b s t i t u t i n g  =  i n t o the equation  -Xf /[cK (f )3 L  filter  H ( t , f ) ~ F{ p  •  center  r-(f ).  look center  i n t o the t h i r d equation to determine  A  denotes  (117)  L  i s p r o j e c t e d i n t o range as i n equation  chapter 5 with c  where  (116)  2  :  The of  (f )/r. (f ) ) ]  the azimuth 1  frequency domain RCM  frequency  c  (114)  frequency r a t e , K (f ). F i n a l l y the s l o p e , c , of the  azimuth  2  TJ.  look c e n t e r range,  A  c  }  (115)  [1 - (v  equations determine  r  substituted  i / z  2  Li  1  for  - 1]  2  e g  e g  time, T j . ( f ) , and  center  [(2v /(Xf))  e g  2  r e p l a c e d by c  L  (89)  :  exp[-j47rr(r?)/X] }  (A (t/c ) * 2  L  fc  the azimuth  Mt-(2/c)r([f-f,]/R )]} A  frequency rate at the beam  frequency, i . e . K ( f ) . r  (118)  For f i x e d SRC,  c. i s computed  152 using the beam center frequency, f^,, while f o r look-dependent SRC, is  the look c e n t e r frequency of each look  used. The r e s u l t i n g a l g o r i t h m f o r compressing each look i s  given by equations (90) to (96) with c the  a g a i n r e p l a c e d by  2  appropriate c . L  As i n s i n g l e - l o o k p r o c e s s i n g , there are s e v e r a l a l t e r n a t i v e methods f o r implementing the SRC range SRC  filter.  Fixed  can p r o v i d e b e t t e r compression due to the longer  e f f e c t i v e l e n g t h of the SRC  filter.  method becomes l e s s e f f i c i e n t  However the range  SRC  i f look-dependent SRC i s  r e q u i r e d because a separate range compression would be r e q u i r e d f o r each look. Look-dependent efficient  azimuth SRC  i s more  s i n c e only the a d d i t i o n a l look-dependent  filters  need to be generated. For  the azimuth SRC  SRC/RCMC f i l t e r ,  implementation, the  g^,(t,f),  i s implemented  combined  as a range  c o n v o l u t i o n with the azimuth spectrum. As i n s i n g l e - l o o k azimuth SRC,  the f i l t e r  i s approximated by precomputing  i n t e r p o l a t e d v e r s i o n s of the SRC convolution  i s implemented  f r a c t i o n a l range c e l l  filter  by computing  shifts  f o r RCM  g ( t ) . The the i n t e g e r and  correction  using the a p p r o p r i a t e l y s h i f t e d v e r s i o n of the f i l t e r . The window, look c e n t e r for of  w A  (f"f )' L  *  s  s  n  if  t  e  c  *  16  t  0  t  n  e  (RCMC) and precomputed  appropriate  frequency f o r each look. The time domain image  each look i s computed by a p p l y i n g an i n v e r s e azimuth FFT length  256.  153 Before  look d e t e c t i o n 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  to reduce a l i a s i n g of the i n c r e a s e d  i n order  bandwidth of the detected  s i g n a l . Although an i n t e r p o l a t i o n  f a c t o r of 2 i s s u f f i c i e n t , the s i m u l a t i o n s  i n t e r p o l a t e each  look by a f a c t o r of 8 i n both range and azimuth  before  d e t e c t i o n and look summation as part of the image q u a l i t y measurement process. The i n t e r p o l a t e d look detected and summed together  images are  to form the f i n a l  multilook  image. For s i m u l a t i o n s of m u l t i l o o k p r o c e s s i n g without SRC, the combined  SRC/RCMC f i l t e r  g ( t , f ) was r e p l a c e d by the  zero phase RCMC i n t e r p o l a t o r of chapter for  3 which i s the same  each look. For m u l t i l o o k range SRC, only f i x e d SRC was  s i n c e look-dependent m u l t i l o o k range SRC was The same m o d i f i e d  simulated  inefficient.  l i n e a r FM r a t e was used f o r the combined  SRC/range compression f i l t e r as i n s i n g l e - l o o k range SRC. The f o l l o w i n g s e c t i o n s d e s c r i b e the r e s u l t s of s i m u l a t i o n s of the both f i x e d and look-dependent SRC a l g o r i t h m s  multilook  as w e l l as s i m u l a t i o n s of m u l t i l o o k  p r o c e s s i n g without SRC which are used as a b a s e l i n e f o r compari son. The s i m u l a t i o n s were performed using the nominal set of RADARSAT parameters l i s t e d  i n Table  1. The azimuth  K a i s e r - B e s s e l window parameter was changed More weighting  from 1.5 t o 2.7.  i s required in multilook processing  s i n c e the  154 antenna weighting has much l e s s e f f e c t over the reduced look bandwidths. The value of 2.7 was chosen to produce azimuth s i d e l o b e l e v e l s which are comparable s i d e l o b e s as i n e a r l i e r  6.2  i n s i z e to the range  simulations.  SIMULATIONS OF 4-LOOK PROCESSING WITHOUT SRC  T h i s s e c t i o n d i s c u s s e s s i m u l a t i o n s of 4-look, range/Doppler p r o c e s s i n g without SRC. length  S i m u l a t i o n s were performed with a  16 RCMC f i l t e r . The s i m u l a t i o n r e s u l t s are summarized  by f i g u r e s 6.3  to  6.14.  F i g u r e s 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 f o r squint angles of 0°, 5°, and 10°. The s i m u l a t i o n s were performed as o u t l i n e d i n the p r e v i o u s s e c t i o n  i n c l u d i n g look d e t e c t i o n ,  and look summation. The p r o f i l e s were i n t e r p o l a t e d by a f a c t o r of 8 before d e t e c t i o n by zero padding i n the frequency domain to p r e s e r v e the s i g n a l bandwidth  after  d e t e c t i o n and to i n c r e a s e the accuracy of the image q u a l i t y measurements. The range p r o f i l e s are very s i m i l a r t o the s i n g l e - l o o k p r o f i l e s shown e a r l i e r . T h i s i s expected s i n c e the range broadening i s p r i m a r i l y due to the azimuth FFT which i s applied  i n both s i n g l e - l o o k and m u l t i l o o k compression. Small  d i f f e r e n c e s are expected because of small v a r i a t i o n s i n range width over the azimuth spectrum before azimuth compression which are caused by range c u r v a t u r e .  155 The azimuth p r o f i l e s are a l s o s i m i l a r t o the s i n g l e - l o o k r e s u l t s . However the mainlobe widths a r e approximately 3.3 times wider than the s i n g l e - l o o k  results.  T h i s i s p r i m a r i l y due to the smaller bandwidth of each look compared t o the f u l l a p e r t u r e processed bandwidth. The azimuth s i d e l o b e l e v e l s d i f f e r  s l i g h t l y because of the  l a r g e r azimuth window parameter. The range and azimuth broadening r e s u l t s f o r both s i n g l e - l o o k p r o c e s s i n g and 4-look p r o c e s s i n g are shown i n f i g u r e s 6.5 to 6.7 f o r squint angles of 0 t o 6 degrees. The 4-look range broadening r e s u l t s are the same as the s i n g l e - l o o k r e s u l t s . As e x p l a i n e d above, t h i s i s expected s i n c e range broadening occurs p r i m a r i l y  i n the azimuth FFT.  Azimuth broadening f o r s i n g l e - l o o k p r o c e s s i n g  increases  with squint angle due to the reduced processed bandwidths. In  c o n t r a s t , there i s very l i t t l e azimuth broadening f o r  4-look p r o c e s s i n g s i n c e each look c o n t a i n s e s s e n t i a l l y the same bandwidth. The azimuth antenna weighting has l e s s broadening e f f e c t on the reduced bandwidths individual  of the  looks.  F i g u r e s 6.8 to 6.10 show that the range, azimuth, and 2-D i n t e g r a t e d s i d e l o b e r a t i o s  (ISLR) f o r 4-look and  s i n g l e - l o o k p r o c e s s i n g behave s i m i l a r l y with i n c r e a s i n g s q u i n t a n g l e . The range ISLR curves increase with i n c r e a s i n g squint angle due to the spreading of energy from the mainlobe t o the s i d e l o b e s . The apparent drop i n ISLR past 5 degrees i s due to the f i n i t e  i n t e g r a t i o n area of the image  156 q u a l i t y measurements. For squint angles over 5 degrees, the range broadening  i s over 25%. T h i s causes a s i g n i f i c a n t  amount of energy  to l i e i n the s i d e l o b e s o u t s i d e of the  i n t e g r a t i o n a r e a . Consequently  the ISLR measurements are  i n a c c u r a t e f o r l a r g e 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 s i d e l o b e r a t i o in f i g u r e s 6.11  to 6.13.  (PSLR) measurements are shown  Again the s i n g l e - l o o k and  r e s u l t s are s i m i l a r with the range squint angle and the azimuth The  2-D  4-look  r e s u l t s v a r y i n g with  r e s u l t s e s s e n t i a l l y constant.  measurements are somewhat lower than the  1-D  measurements s i n c e they are measured with a much c o a r s e r sample spacing ( i n t e r p o l a t e d by 8) than the 1-D ( i n t e r p o l a t e d by  measurements  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 f o r zero s q u i n t , are shown i n f i g u r e 6.14. by the sum  The peak magnitudes are normalized  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 a p p r o p r i a t e f o r white n o i s e i d e n t i c a l l y over the range c e l l s .  distributed  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 i n r e l a t i n g the peak magnitude r e s u l t s to signal-to-noise  ratios.  157  (SP) •pnjiuftDW  Figure  6.3.  I n t e r p o l a t e d 1-D range p r o f i l e s a f t e r 4-look compression without SRC f o r squint a n g l e s of 0 ° , 5°, and 10° and a l e n g t h 16 RCMC i n t e r p o l a t o r .  158  (8P) •pnuufion  F i g u r e 6.4.  I n t e r p o l a t e d 1-D azimuth p r o f i l e s a f t e r 4-look compression without SRC f o r s q u i n t a n g l e s of 0°, 5°, and 10° and a l e n g t h 16 RCMC i n t e r p o l a t o r .  159  6uiu©pD0jg %  F i g u r e 6.5.  Range broadening f o r s i n g l e - l o o k and 4-look compression without SRC using a l e n g t h 16 RCMC interpolator.  160  gure l i . 6 . Range broadening f o r s i n g l e - l o o k and 4-look compression without SRC using a l e n g t h 16 RCMC i n t e r p o l a t o r (expanded s c a l e ) .  161  F i g u r e 6.7.  Azimuth broadening f o r s i n g l e - l o o k and 4-look compression without SRC using a l e n g t h 16 RCMC interpolator.  162  o o or 9  •D  C  o  3 CT  </>»  o o  •  F i g u r e 6.8.  1-D range i n t e g r a t e d s i d e l o b e r a t i o s f o r s i n g l e - l o o k and 4-look compression without u s i n g a l e n g t h 16 RCMC i n t e r p o l a t o r .  SRC  163  on O -P  c6 tf  o CO  o o  »  , 0 +->  9 •o  -a  Cd  g  0)  o  c o ~c  3  CT  -  M  o o  N  T  in CO  0)  T  in  o  in  I  CM  M  (8P) dlSI  Figure  d  I  CM  I  m CM  CM <M  I  I  6.9. 1-D azimuth i n t e g r a t e d s i d e l o b e r a t i o s f o r s i n g l e - l o o k and 4-look compression without u s i n g a l e n g t h 16 RCMC i n t e r p o l a t o r .  SRC  164  Figure  6.10.  2-D i n t e g r a t e d s i d e l o b e r a t i o s f o r s i n g l e - l o o k and 4-look compression without SRC using a l e n g t h 16 RCMC i n t e r p o l a t o r .  165  0 CM  CM  CM CM  CM  CM  CM  «n  <o  I  I  I  I  I  I  w-  1  tn  •<*  CM  (ap) * n s d  Figure  6.11 . i rarge peak s i d e l o b e r a t i o s f o r s i n g l e - l o o k and 4-look compression without SRC u s i n q a l e n g t h 16 RCMC i n t e r p o l a t o r . D  166  o  o  ov 9  TJ 9  CD C  o  3 CT V) •» - CM  CO  T  in co  T  in  o I  (BP)  F i g u r e 6.12.  in o  M  CM  I  CM  I  in CM  I  o o  CM CM  I  dlSd  1-D azimuth peak s i d e l o b e r a t i o s f o r s i n g l e - l o o k and 4-look compression without using a l e n g t h 16 RCMC i n t e r p o l a t o r .  SRC  1  (ap) aisd Figure  6.13.  2-D peak s i d e l o b e r a t i o s f o r s i n g l e - l o o k and 4-look compression without SRC u s i n g a l e n g t h RCMC i n t e r p o l a t o r .  168  F i g u r e 6.14. Peak magnitude r a t i o s f o r s i n g l e - l o o k and 4-look compression without SRC u s i n g a l e n g t h 16 RCMC i n t e r p o l a t o r .  169  6.3 SIMULATIONS OF 4-LOOK, FIXED AND LOOK-DEPENDENT,  AZIMUTH  SRC PROCESSING The  next s e c t i o n d i s c u s s e s s i m u l a t i o n s of 4-look  processing  performed with both f i x e d and look-dependent azimuth SRC. The  r e s u l t s f o r the two a l g o r i t h m s  were i d e n t i c a l w i t h i n the  l i m i t s of the measurements. Consequently only one set of r e s u l t s are presented.  T h i s s i m i l a r i t y shows that  look-dependent p r o c e s s i n g The  i s not necessary  r e s u l t s are almost i d e n t i c a l  RCM curve  v a r i e s very  f o r RADARSAT.  s i n c e the slope of the  slowly over the aperture  f o r RADARSAT  parameters. Consequently the slopes at the i n d i v i d u a l center  f r e q u e n c i e s are v i r t u a l l y  earlier,  the same. As s t a t e d  systems such as SEASAT which e x h i b i t l a r g e r range  curvature may have l a r g e r v a r i a t i o n s i n RCM curve the aperture The  look  slope over  r e q u i r i n g look-dependent SRC p r o c e s s i n g .  r e s u l t s of the s i m u l a t i o n s are c o n t a i n e d  in figures  6.15 to 6.26. A smaller s e l e c t i o n of squint angles was used than i n the s i n g l e - l o o k azimuth SRC s i m u l a t i o n s s i n c e the r e s u l t s are very squint angles filter  s i m i l a r . Simulations  were performed f o r  of 0, 1, 5, and 10 degrees and f o r SRC/RCMC  lengths of 4, 8, 16, and 32.  F i g u r e s 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 f o r squint of 5 and 10 degrees. As i n the p r e v i o u s processing  angles  s e c t i o n the  i n c l u d e d i n t e r p o l a t i o n , look d e t e c t i o n , and look  summation. The range p r o f i l e s show the broadening which occurs.with  smaller l e n g t h SRC/RCMC f i l t e r s at l a r g e r squint  170 a n g l e s . The azimuth p r o f i l e s show n e g l i g i b l e changes with filter  length.  F i g u r e s 6.17  and 6.18  summarize the range and azimuth  broadening r e s u l t s r e s p e c t i v e l y . The broadening f i g u r e s are computed from the -3dB  response widths r e l a t i v e to the zero  s q u i n t width f o r each f i l t e r  l e n g t h . D i f f e r e n c e s between the  impulse response widths at zero squint f o r d i f f e r e n t lengths are very small figure  ( l e s s than 0.2%)  filter  and are shown i n  6.19.  The m u l t i l o o k broadening r e s u l t s can be compared to the s i n g l e - l o o k azimuth SRC  results  i n f i g u r e s 5.6 and 5.8.  The  4-look range broadening r e s u l t s agree very c l o s e l y with the s i n g l e - l o o k r e s u l t s . As i n the s i m u l a t i o n s without  SRC,  almost no azimuth broadening occurs f o r 4-look p r o c e s s i n g . The ISLR and PSLR measurements are shown i n f i g u r e s 6.20  to 6.22  and f i g u r e s 6.23  to 6.25  respectively.  r e s u l t s agree c l o s e l y with the s i n g l e - l o o k azimuth c u r v e s . The azimuth s i d e l o b e s are lower by about to  The SRC  0.7dB due  the l a r g e r azimuth window parameter. F i g u r e 6.26  shows the peak magnitude v a r i a t i o n s with  squint angle with the peaks normalized by the sum of the squared SRC/RCMC f i l t e r  coefficients.  171  (8P)  Figure  6.15.  •pniiufiBW  I n t e r p o l a t e d 1-D range p r o f i l e s a f t e r 4-look compression with both f i x e d and look-dependent SRC f o r s q u i n t angles of 0°, 5°, and 10° and a l e n g t h 16 f i l t e r .  172  (ap) ftprmufiDw  Figure  6.16.  I n t e r p o l a t e d 1-D azimuth p r o f i l e s a f t e r 4-look compression with both f i x e d and look-dependent SRC f o r s q u i n t angles of 0°, 5°, and 10° and a l e n g t h 16 f i l t e r .  173  fiuiuftpDojg  Figure  6.17.  %  Range b r o a d e n i n g f o r 4-look compression with both f i x e d and look-dependent SRC a n d various SRC/RCMC f i l t e r lengths as a f u n c t i o n of squint angle.  174  o  o  o  - CO  II  - to  'to  rt £ CD  rt E  a> % -*  cd •*-  O  3  cr  "to  u  II  ,rt -p  - ts  cs<r>cor>»<cin^>ocs*-.-  Figure  6.18.  »cqh.<oin'*Kits^ o^ o o o o o o o o o o I :  :  Azimuth broadening for 4-look compression with 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 squint angle.  c rt)  Comparison of Zero Squint Broadening  cn  for various SRC/RCMC filter lengths  ro o 0 3 o n  o o  i Q ?>r rt I  3  rr a  1 0)  tn ro • TJ a>  a  a»  O M. O (A X- O  3  0.9 0.8 II  O  0.7  rt o o  in•a  t>  W ft) o> cn 3  o  cn io  0.5  o> O  3 3  CL  < 0)  »-••  o c cn  t/J W  O  I U>  < a  0.6  'c  0.4  rt  tr K  O* a  o  rt  o  rt  rr  rr tn  t-f« Ol  \  t->- r t  X  OJ  w X o fD  o ' 0)3  O  o  cn iQ  16  Qi C  ro  D rt  •  S R C / R C M C filter length. L azimuth + range  32  176  .  I  -  M  I  C  I  M  C  I  M  I  N  N  I  N  I  N  C  I  M  I  (ap) aisi  F i g u r e 6.20.  1-D range i n t e g r a t e d s i d e l o b e r a t i o s f o r 4-look compression with both f i x e d and look-dependent SRC and 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 s q u i n t a n g l e .  O  CM  II  - «  O Id  CO n-t <->  - <o  *|  go c rr  ll  2  - CM  II  00  T  in  T  in  o CM  I  (ap) aisi Figure  6.21.  in o CM  I  CM  I  in CM  CM CM  I  I  1-D azimuth i n t e g r a t e d s i d e l o b e r a t i o s f o r 4-look compression with both f i x e d and look-dependent SRC and 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 s q u i n t a n g l e .  1 78  I  I  I  I  (SP) insi  F i g u r e 6.22.  2-D i n t e g r a t e d s i d e l o b e r a t i o s f o r 4-look compression with both f i x e d and look-dependent SRC and v a r i o u s SRC/RCMC f i l t e r lengths as a f u n c t i o n of squint a n g l e .  1 79  M  to II  go 3  rr  II  O (  I  « - N t O ' + « r > « 0 r H O M C S C S C S C S C S C S C N C  I  I  I  I  I  I  I  D O > O ' M C S t O K )  I  I  I  I  (ap) ansa  F i g u r e 6.23. 1-D range peak s i d e l o b e r a t i o s f o r 4-look compression with both f i x e d and look-dependent SRC and 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 squint a n g l e .  180  W  O •— i » M to -  II  00  O  - <0 9~  g< "  3  o  CT  V)«0  N  - CM  I o CN  I  tf) O CM  I  M  I  in  CM  CM CM  I  I  (ap) aisd  F i g u r e 6.24.  in  CM CM  I  to  CM  I  in to  CM  CM  I  I  1-D azimuth peak s i d e l o b e r a t i o s f o r 4-look compression with both f i x e d and look-dependent SRC and 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 s g u i n t a n g l e .  181  o » n /VI  • CS  I  — #VJ 1  CS  •  I  i n t s i n i o i n ' * ISJ  1  CS  /SJ  1  I  CS  I  (ap) aisd  F i g u r e 6.25. 2-D peak s i d e l o b e r a t i o s f o r 4-look compression with both f i x e d and look-dependent SRC and v a r i o u s SRC/RCMC f i l t e r lengths as a f u n c t i o n of squint a n g l e .  182  ^  o  d I  (ap)  i  ^  i  l  •prniufiDH  F i g u r e 6.26. Peak magnitude r a t i o s f o r 4-look compression with both f i x e d and look-dependent SRC and v a r i o u s SRC/RCMC f i l t e r lengths as a f u n c t i o n of squint angle.  183 6.4 SIMULATIONS OF 4-LOOK, FIXED, RANGE SRC PROCESSING T h i s s e c t i o n presents the r e s u l t s of s i m u l a t i o n s of 4-look p r o c e s s i n g with a f i x e d range SRC a l g o r i t h m . Since the range SRC a l g o r i t h m uses an SRC f i l t e r i s e f f e c t i v e l y much longer than the f i l t e r SRC, the range broadening filter  i s expected  which  used i n azimuth  caused by windowing of the SRC  to be much s m a l l e r . T h i s was shown with  the s i n g l e - l o o k range SRC s i m u l a t i o n s and i s a l s o true of the m u l t i l o o k 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 veris i m i l a r t o the zero s q u i n t , m u l t i l o o k p r o f i l e s shown e a r l i e r and are t h e r e f o r e not shown here. S i m u l a t i o n s are performed with a l e n g t h 16 RCMC f i l t e r . The measurements of range and azimuth broadening are summarized i n f i g u r e 6.27. For squint angles up to 20° the range broadening  i s l e s s than  1.8%. The i n t e g r a t e d s i d e l o b e  r a t i o s shown i n f i g u r e 6.28 show that there i s l i t t l e variation  i n ISLR with squint angle  ( l e s s than  -1 dB f o r  squint angles up to 20°). S i m i l a r l y the peak s i d e l o b e summarized i n f i g u r e 6.29 show an even smaller Finally  ratios  variation.  the peak magnitude measurements shown i n f i g u r e  6.30  d i s p l a y only a small v a r i a t i o n with squint a n g l e . T h i s i n d i c a t e s that there i s l i t t l e change i n SNR f o r changes i n squint  angle.  184  F i g u r e 6.27.  Range and azimuth broadening f o r 4-look compression with range SRC and a l e n g t h 16 RCMC i n t e r p o l a t o r as a f u n c t i o n of squint a n g l e .  185  (ap) cnsi  Figure  6.28. Range, azimuth, and 2-D i n t e g r a t e d s i d e l o b e r a t i o s f o r 4-look compression w i t h range SRC and a l e n g t h 16 RCMC i n t e r p o l a t o r as a f u n c t i o n of squint angle.  (ap) disd Figure  6.29. Range, azimuth, and 2-D peak s i d e l o b e r a t i o s f o r 4-look compression with range SRC and a l e n g t h 16 RCMC i n t e r p o l a t o r as a f u n c t i o n of squint a n g l e .  187  F i g u r e 6.30.  Peak magnitude d e g r a d a t i o n f o r 4-look compression with range SRC and a l e n g t h 16 RCMC i n t e r p o l a t o r as a f u n c t i o n of s q u i n t a n g l e .  188 6.5  SUMMARY OF MULTILOOK  SRC  T h i s chapter has presented two forms of SRC a l g o r i t h m s to be used with m u l t i l o o k range/Doppler compression: f i x e d azimuth SRC,  and look-dependent SRC.  be e f f e c t i v e at  Both methods have been shown to  i n reducing the range broadening which occurs  l a r g e squint a n g l e s . Comparisons  of the s i m u l a t i o n  r e s u l t s of both m u l t i l o o k SRC a l g o r i t h m s f o r 4-look p r o c e s s i n g have shown that there are no measurable d i f f e r e n c e s i n image q u a l i t y result,  f i x e d azimuth SRC  f o r RADARSAT parameters. As a  should be used s i n c e i t r e q u i r e s  l e s s memory and computation f o r i t s s i n g l e SRC/RCMC In  fact  the 4-look r e s u l t s are very s i m i l a r  filter.  i n range to the  s i n g l e - l o o k r e s u l t s i n d i c a t i n g that the number of looks does not  a l t e r the e f f e c t i v e n e s s of the SRC The m u l t i l o o k SRC  algorithm.  s i m u l a t i o n s show that use of the  azimuth SRC a l g o r i t h m can s i g n i f i c a n t l y  reduce the point  t a r g e t response broadening which occurs at l a r g e r a n g l e s . The improvements filter the  squint  i n c r e a s e with l a r g e r SRC/RCMC  lengths s i n c e the l a r g e r f i l t e r s can c o l l e c t more of  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  s q u i n t angles can be extended from 3.65° to 8.0° and from 4.23° to 9.3°  respectively.  With the m u l t i l o o k filter  range SRC a l g o r i t h m the e f f e c t i v e  l e n g t h i s much l o n g e r . Thus the broadening i s much  l e s s than with azimuth SRC. For squint ancjles up to 20° .the range broadening i s l e s s than  1.8%.  7. EFFECTS OF SRC FM RATE ERRORS T h i s 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 r a t e e r r o r s . SRC FM r a t e e r r o r s a r e caused by e r r o r s i n both the beam center  frequency, f ^ , and the range  pf c l o s e s t approach, r . L i m i t s on the p r o c e s s i n g 0  s i z e , or the i n v a r i a n c e  r e g i o n , and parameter  e r r o r s a r e developed f o r s p e c i f i c broadening Simulations  a r e performed to q u a n t i f y  by SRC FM r a t e e r r o r s with v a r i o u s  block  estimation limits.  the broadening caused  algorithms  and f i l t e r  l e n g t h s . Only e r r o r s i n the SRC FM rate are simulated the e f f e c t s of parameter e r r o r s on other operations  (e.g.,  The  processing  RCMC, and the azimuth reference  f u n c t i o n ) can be modelled and p r e d i c t e d  since  phase  independently.  broadening r e s u l t s are parameterized i n terms of  the band-edge phase e r r o r i n range frequency. The broadening without SRC i s s i m i l a r l y parameterized by e v a l u a t i n g the equivalent SRC  band-edge phase e r r o r caused by not a p p l y i n g an  filter.  7.1  SENSITIVITY ANALYSIS OF THE SRC FM RATE  SRC  FM r a t e e r r o r s a r i s e from two sources : parameter  estimation  e r r o r s and f i l t e r  invariance errors.  e r r o r s occur s i n c e both the the beam center and  Estimation  frequency, f , c  the range of c l o s e s t approach, r , are u s u a l l y estimated 0  from inexact measurements of the p o s i t i o n and a t t i t u d e of the radar often  platform.  The beam center  frequency estimate i s  r e f i n e d with a Doppler c e n t r o i d e s t i m a t i o n  189  algorithm  190  [143. SRC  f i l t e r  processing. block r ,  but  mismatch the  SRC  0  the  the  For  the  of  the  However  more  in  center  beam  Thus  a  error  to To  f i l t e r , as  range  must  added  0  value  of  resulting  near  dimension  of  the  edge  limit  block  is  c a l l e d  the  estimation  errors  and  when  geometric  determining  error  estimation effects  SRC FM  rate  can  is  models  across  predicted  the  model  frequency  frequency  be  examine the  r  one  range  size  block  processed  to  The  targets  Both  center  this  f^  the  the  of  the  SRC  region.  center  the  block.  region.  frequency  can  result  matched  block  be  a  across  limits  point  sophisticated  Although  effect  K  The  beam  beam  occur.  rate  approximate  the  0  center  errors  thesis,  r  only  since  -ire  varies  0  is  SRC FM  invariance  r  errors  rate  invariance  invariance  range  its  the  broadened.  range  of  SRC FM  at  in  value  processed block  become  r  The  usually  0  invariance  predict  the  not  by  in  this  independent a  error  tabulated  adding  the  f  of  slow  processed  invariance  is  used  0  variation  range can  in  c  r .  swath.  also  this  t h e s i s ,  invariance  error. of  parameter  errors  be  expressed  in  on  terms  the of  f  SRC  c  and  :  SRC  "  K  = - (2v  A  c  2 e q  2  /  (  X  f  C  )  /[Xr 3) 0  (119)  2  (c/[Xf 3) c  2  (  1 -  (Xf /[2v c  e g  3)  2  }  3/2  (120)  191 In order to normalize the a n a l y s i s to the range bandwidth, B^,  the phase of the SRC  band-edge frequency  filter  -3dB  at the  i s used as a parameter.  The  range  band-edge  phase i s given i n r a d i a n s by :  *(B /2) = 7 r ( B / 2 ) / K  (121)  2  r  r  S R C  Using p a r t i a l d e r i v a t i v e s , the band-edge phase e r r o r can expressed approximately  i n terms of the beam c e n t e r  frequency e r r o r , A f , and  r  c  A<//(B /2) = A r r  3  0  -  [,(B /2)VK r  •  S R C  0  tf(B /2) r  3r  e r r o r , A r , as : 0  + Af  c  9 3f  0  tf(B /2)  (122)  r  C  ]  {(Ar /r ) + 2(Af /f )[l 0  be  0  c  c  + (3/2)/[(2v  /[Xf ]) -1]]} 2  c  (123)  Range broadening  i s small, t y p i c a l l y  l e s s than  when the magnitude of the band-edge phase e r r o r  is less  7r/2 r a d i a n s . The c o r r e s p o n d i n g e r r o r l i m i t s on A f vary depending F i g u r e s 7.1 the range  10%,  c  and  on the s q u i n t angle and the parameters  and 7.2  show the SRC  than Ar  0  used.  band-edge phase e r r o r i n  frequency domain f o r v a r i o u s s q u i n t a n g l e s (1° to  20°) as a f u n c t i o n of A f  c  and A r . The curves use exact 0  c a l c u l a t i o n s of the phase e r r o r s r a t h e r than the  partial  d e r i v a t i v e 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  Figure  7.1  6SDL|d  eSpe-puDg  SRC band-edge phase e r r o r i n the range frequency domain f o r s q u i n t angles of 1°, 5°, 10°, 15°, and 20° as a f u n c t i o n of beam c e n t e r frequency error.  1 93  F i g u r e 7.2.  SRC band-edge phase e r r o r i n the r a n g e ^ requency domain f o r squint angles of 1°, 5°, 10 , 15 , and 20° as a f u n c t i o n of r e r r o r . 0  194 l i n e a r which agrees with the p a r t i a l d e r i v a t i v e  expansion.  The magnitude of the phase e r r o r i n c r e a s e s with  increasing  squint angle and i n c r e a s i n g parameter  e r r o r . For RADARSAT  with squint angles l e s s than 10°, the phase e r r o r  i s less  than 7r/2 f o r beam c e n t e r frequency e r r o r s l e s s than +/- 3000 Hz and r  0  e r r o r s l e s s than +/- 15%. The maximum  expected beam center frequency e r r o r  i s much l e s s  (on the  order of 100 Hz e x c l u d i n g range v a r i a n c e s ) . The maximum expected r  0  i s a l s o l e s s being on the order of +/- 5%.  More exact measurements of range broadening as a f u n c t i o n of band-edge phase e r r o r f o r p a r t i c u l a r a l g o r i t h m s and f i l t e r  lengths are performed  by s i m u l a t i o n  i n the next  sect ion. The  range broadening which occurs without SRC can be  parameterized by an e q u i v a l e n t range frequency band-edge phase e r r o r . Since the broadening  r e s u l t s i n chapter 3 were  shown as a f u n c t i o n of squint angle, i t i s s u f f i c i e n t to r e l a t e the squint angle t o the e q u i v a l e n t phase e r r o r . The e r r o r i s given by the band-edge phase of the i d e a l SRC filter  which can be computed from equation (121) by  rewriting get  i t i n terms of the squint angle, 6 , and r  0  to  :  /2) =  i//(B  IT  This r e l a t i o n broadening  -2TT(B  /[2c]) Xr 2  IT  i s plotted  tan 0  [ 1 +tan  2  0  S  2  0 S  ]  / 1  2  (124)  i n f i g u r e 7.3. The 5% and 10% range  squint angles given i n chapter 3 correspond to  195 e q u i v a l e n t range phase e r r o r s of -74° and  -100°  r e s p e c t i v e l y . Thus the 90° phase l i m i t used e a r l i e r range broadening l i m i t  as a 10%  i s reasonably c l o s e . For comparison  with the r e s u l t s i n the next s e c t i o n , the broadening r e s u l t s without SRC are r e p l o t t e d  in f i g u r e 7.4  e q u i v a l e n t range phase e r r o r  instead  as a f u n c t i o n of  of squint a n g l e .  196  Figure  7.3.  E q u i v a l e n t SRC band-edge p h a s e e r r o r i n t h e r a n g e f r e q u e n c y d o m a i n w i t h o u t SRC a s 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 «n  O  O  O »o  O M  O ^  6uju#pDojg *6uDy %  Figure  7.4. A c t u a l range broadening without SRC with a l e n g t h 16 RCMC i n t e r p o l a t o r and p r e d i c t e d range broadening as a f u n c t i o n of e q u i v a l e n t range band-edge phase e r r o r .  198 7.2  SIMULATIONS OF This  FM  RATE ERROR BROADENING  s e c t i o n present?? s i m u l a t i o n s  caused by SRC The  SRC  FM  of range broadening  rate e r r o r s with v a r i o u s  SRC  algorithms.  r e s u l t s i n d i c a t e that the band-edge phase e r r o r  frequency  i n range  i s a good general measure of the expected range  broadening. The  s p e c i f i c amount of broadening  somewhat with the  s i z e of SRC  filter  and  varies  the type of  SRC  algorithm. As in  the  noted e a r l i e r , parameter e r r o r s were only SRC  f i l t e r . The  azimuth reference  simulated  remainder of the p r o c e s s i n g  phase m u l t i p l i c a t i o n , etc.)  (RCMC,  was  simulated  without parameter e r r o r s . Thus the measured range broadening i s s o l e l y the  r e s u l t of SRC  FM  r a t e e r r o r s . The  range  broadening was  measured r e l a t i v e to the  range response width  without SRC  r a t e e r r o r s so that only  the  FM  broadening caused by  the FM  r a t e e r r o r was  additional measured. Only  n e g a t i v e band-edge phase e r r o r s were simulated s i n c e p o s i t i v e phase e r r o r The SRC  first  algorithm.  squint  angle was  from f i g u r e s 7.1 a n g l e s , 5° and  r e s u l t s are very s i m i l a r .  simulation The  involved  chosen to i n c l u d e  10°,  the  7.2  l a r g e s t expected e r r o r  of the p r e v i o u s s e c t i o n . Two  for 5° of  squint  The  range broadening measurements are  7.5  and  samples. At  f o r SRC  filter  5° of squint  azimuth  simulated f o r each  were s i m u l a t e d . The  phase e r r o r s were -17°  7.6  single-look  maximum phase e r r o r  and  the  the  maximum simulated and  -68°  for  summarized by  l e n g t h s of 4,  squint  8,  16,  and  10°. figures 32  range broadening i s l e s s than  199 2% f o r a l l f i l t e r l e n g t h s . At 10° of s q u i n t , only the longer filters  (of l e n g t h 16 and 32) were used s i n c e the shorter  f i l t e r s produced SRC  unacceptably l a r g e broadening even without  FM r a t e e r r o r s  broadening  (greater than 85%). The l a r g e s t  range  ( f o r a phase e r r o r of -68°) was l e s s than 8%.  The azimuth SRC s i m u l a t i o n s were repeated f o r m u l t i l o o k p r o c e s s i n g with almost  identical  r e s u l t s as shown i n f i g u r e s  7.7 and 7.8. T h i s i d e n t i c a l behaviour shows the independence of  the range broadening process from the look e x t r a c t i o n  process. The  next s i m u l a t i o n was performed  range SRC a l g o r i t h m . A 16 sample range for  RCMC. Since the broadening  with the s i n g l e - l o o k i n t e r p o l a t o r was used  r e s u l t s without  errors  i n d i c a t e d that range SRC c o u l d be used f o r l a r g e r angles, s q u i n t angles of 15° and 20° were used to  squint  in addition  the 5° and 10° s q u i n t s used b e f o r e . The maximum simulated  phase e r r o r s were again s e l e c t e d to i n c l u d e the maximum expected phase e r r o r s at each squint angle. The chosen v a l u e s were -17° f o r 5° of s q u i n t , -68° f o r 10° of s q u i n t , and -136° f o r both 15° and 20° of s q u i n t . The range broadening  i s summarized i n f i g u r e 7.9. The broadening  l e v e l s vary only s l i g h t l y with d i f f e r e n t maximum broadening at 10° of s q u i n t of  s q u i n t a n g l e s . The  (again f o r a phase e r r o r  -68°) i s s m a l l e r with range SRC than with azimuth SRC  ( l e s s than 5% compared with 8% f o r azimuth SRC). Since the m u l t i l o o k azimuth SRC measurements of broadening wih SRC FM r a t e e r r o r s were e s s e n t i a l l y the same  200  as the s i n g l e - l o o k r e s u l t s , the m u l t i l o o k should  a l s o be very  s i m i l a r to the s i n g l e - l o o k range  r e s u l t s . Consequently the m u l t i l o o k not  simulated.  range SRC r e s u l t s SRC  range SRC a l g o r i t h m  was  201  O  W &  CM  to II  CO  A _,_) rj)  rt '  a N  ^ N  m  II  c5M  c 3  9  2" -  £  Q. 9  £  rt  II  O  ed O u  PQ  II  rt CM  in  fiUIUdpDOjg  Figure  7.5.  %  Range ' b r o a d e n i n g 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° o f 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 a s a f u n c t i o n o f r a n g e band-edge p h a s e error.  202  fiuiuepDojg  Figure  7.6.  %  Range b r o a d e n i n g w i t h s i n g l e - l o o k a z i m u t h SRC a t 1 0 ° 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 a s a f u n c t i o n o f r a n g e band-edge p h a s e error.  203  CuiudpDojg %  F i g u r e 7.7.  Range broadening with m u l t i l o o k azimuth SRC at 5 of s q u i n t with v a r i o u s SRC/RCMC f i l t e r lengths as a f u n c t i o n of range band-edge phase error.  204  6uiu*pr>ojg $  F i g u r e 7.8. Range broadening with m u l t i 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-edqe phase error. ' ^  205  fiuiuapDOjg  Figure  7.9.  %  Range broadening with s i n g l e - l o o k range SRC at 5 ° , 1 J ° , 1 5 ° , and 2 0 ° of s q u i n t with a length 16 RCMC i n t e r p o l a t o r as a f u n c t i o n of range band-edge phase e r r o r .  8. SUMMARY AND CONCLUSIONS T h i s t h e s i s has shown that a new a l g o r i t h m , c a l l e d secondary range compression  (SRC), s i g n i f i c a n t l y  amount of range broadening which occurs at l a r g e angles i n the b a s i c range/Doppler SRC a l g o r i t h m was f i r s t for  reduces the squint  compression a l g o r i t h m . The  suggested by J i n and Wu [13] i n 1984  use with the SEASAT SAR. T h i s t h e s i s has extended the  theory of the SRC a l g o r i t h m to examine the approximations i n v o l v e d and to e x p l o r e a l t e r n a t e implementations. In a d d i t i o n t o 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 d u r i n g range compression, has been presented and examined. A l s o , two new m u l t i l o o k SRC a l g o r i t h m s have been developed f o r use i n m u l t i l o o k azimuth  compression.  Many s i m u l a t i o n s with nominal RADARSAT parameters been performed t o q u a n t i f y the image q u a l i t y  have  improvements  p o s s i b l e with SRC. A s e n s i t i v i t y a n a l y s i s of SRC with respect t o parameter  e r r o r s has been i n c l u d e d . The a n a l y s i s  i n d i c a t e s that the SRC a l g o r i t h m i s very t o l e r a n t to parameter  e s t i m a t i o n and i n v a r i a n c e e r r o r s . 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 i s r e q u i r e d over the nominal  updating  150 km. RADARSAT ground  range  swath f o r squint angles up to 15° using the range SRC implementation  (assuming an r  0  e r r o r of l e s s than +/- 5% and  a beam center frequency e r r o r of l e s s than +/- 200 H z ) . SRC p r o v i d e s a c l o s e r approximation t o exact matched f i l t e r i n g when the azimuth time-bandwidth 206  product (TBP) of  207  the  range compressed  the  range c e l l  point target  response, as measured i n  nearest the beam c e n t e r range, f a l l s  u n i t y . The SRC f i l t e r  below  i s formulated by using a q u a d r a t i c  phase approximation of the azimuth phase coding and a l i n e a r approximation t o the range m i g r a t i o n curve over the processed azimuth bandwidth. These approximations allow the azimuth F o u r i e r spectrum t o be d e r i v e d a n a l y t i c a l l y . The d e r i v e d spectrum accounts f o r the spectrum broadening which occurs with low azimuth TBP's. The b a s i c  range/Doppler  a l g o r i t h m without SRC does not account f o r azimuth spectrum broadening s i n c e i t i s d e r i v e d with the p r i n c i p l e of s t a t i o n a r y phase which i s v a l i d only f o r l a r g e TBP s i g n a l s . It  has been shown that the range b a n d l i m i t e d azimuth  matched f i l t e r  e x h i b i t s s i m i l a r azimuth spectrum broadening  under low azimuth TBP c o n d i t i o n s . When range c u r v a t u r e i s small enough that the RCM curve can be c o n s i d e r e d  linear  over the p r o c e s s e d azimuth a p e r t u r e , as i s the case f o r RADARSAT, the azimuth matched f i l t e r the  into  range time d i r e c t i o n . The r e s u l t i n g a p p l i c a t i o n of the  SRC f i l t e r  i n range i n s t e a d of azimuth a l l o w s s h o r t e r and  more e f f i c i e n t filter the  can be p r o j e c t e d  filters  t o be used. Combination of t h i s  range  with the frequency domain RCMC i n t e r p o l a t o r leads t o  new azimuth SRC a l g o r i t h m . The e f f e c t i v e n e s s of t h i s  algorithm filter.  i s p r o p o r t i o n a l t o the l e n g t h of the SRC/RCMC  An e f f e c t i v e l y  longer SRC f i l t e r  combining the SRC f i l t e r  can be formed by  with the range compression  filter  d u r i n g range compression. T h i s r e s u l t s i n a new range SRC  208 a l g o r i t h m . When the range p u l s e i s a l a r g e TBP  linear  FM  s i g n a l as f o r RADARSAT the combined SRC/range compression filter  d i f f e r s from the o r i g i n a l range compression  only by a small change i n l i n e a r FM  filter  rate.  Computer s i m u l a t i o n s with nominal RADARSAT parameters have v e r i f i e d the accuracy of the new v a r i e t y of f i l t e r  SRC a l g o r i t h m s f o r a  parameters and squint a n g l e s . For  s i n g l e - l o o k azimuth SRC p r o c e s s i n g with a 16 p o i n t SRC/RCMC filter, and  i t was  found that the squint angles which produce  5%  10% range broadening can be extended from 3.65° and  4.23° r e s p e c t i v e l y without SRC  to 8.03° and 9.29°  r e s p e c t i v e l y with azimuth SRC.  For s i n g l e - l o o k range  SRC  p r o c e s s i n g with a 16 p o i n t RCMC i n t e r p o l a t o r , the range broadening was to  20°, which  shown to l e s s than 1.3% i s the l a r g e s t  f o r squint angles up  squint angle s i m u l a t e d . The  s i m u l a t i o n s of m u l t i l o o k SRC p r o c e s s i n g showed very s i m i l a r r e s u l t s i n d i c a t i n g that the s e p a r a t i o n of looks does not g r e a t l y a f f e c t the range broadening p r o c e s s . Somewhat s u r p r i s i n g l y , the s i m u l a t i o n s 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. T h i s i n d i c a t e s that frequency domain RCMC,  with or without SRC,  adequately e x t r a c t s the azimuth phase  spectrum along the RCM  curve f o r compression  i n 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, s e v e r a l areas remain t o be examined  further.  The approximate geometric model used i n t h i s which assumes a l o c a l l y  flat  thesis,  e a r t h below the radar p l a t f o r m ,  c o u l d be r e p l a c e d with a more r e f i n e d model which accounts for  parameter v a r i a t i o n s over the curved e a r t h . One  a l t e r n a t i v e would be t o use a c o n s i s t e n t approximation t o the  range m i g r a t i o n 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 e a r t h h y p e r b o l i c equation used here, a T a y l o r  series  approximation t o the e l l i p s o i d a l model with s e v e r a l  terms  c o u l d be used. Such a model c o u l d a l s o be used t o i n c o r p o r a t e s a t e l l i t e motions o u t s i d e of the nominal The r e f i n e d model would be u s e f u l filter  f o r d e r i v i n g more a c c u r a t e  parameters, e s p e c i a l l y f o r spaceborne SARs, and f o r  determining more p r e c i s e bounds on s i g n a l parameter and v a r i a t i o n s over the range and azimuth In SRC,  orbit.  errors  swaths.  azimuth SRC and i n range/Doppler p r o c e s s i n g without  the SRC/RCMC f i l t e r  i s windowed i n the range time  domain i n order t o 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 . T h i s causes a d d i t i o n a l the  point  partially  range broadening of  t a r g e t response. The a c t i o n of the window i s only understood and i t s o p t i m a l i t y has not been  e s t a b l i s h e d . I t may be p o s s i b l e t o develop a method of compensating  fo:: the range broadening e f f e c t s of the window  by m o d i f y i n g the SRC/RCMC f i l t e r  spectrum before  windowing.  210  Since one of the e f f e c t s of the window i s t o taper the filter  amplitude spectrum, the f i l t e r  spectrum c o u l d be  p r e d i s t o r t e d by a m p l i f y i n g the higher f r e q u e n c i e s before windowing i n order t o o b t a i n a spectrum c l o s e r to the i d e a l flat  spectrum a f t e r windowing. Finally  the SRC a l g o r i t h m has been examined f o r nominal  RADARSAT parameters which i n v o l v e very l i t t l e  range  c u r v a t u r e over the processed azimuth bandwidth. Since the SRC f i l t e r  i s d e r i v e d using a l i n e a r RCM assumption and a  q u a d r a t i c azimuth phase assumption, range c u r v a t u r e and higher order e f f e c t s may be more s i g n i f i c a n t which e x h i b i t  i n systems  l a r g e r range c u r v a t u r e such as longer  wavelength spaceborne SAR's. S i m u l a t i o n s of systems  with  much l a r g e r wavelengths should be performed t o 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 i n radar imaging", IEEE Trans. V o l . AES-20, No. 4, J u l y 1984. [2] B.C. Barber, "Theory of d i g i t a l imaging from o r b i t a l s y n t h e t i c - a p e r t u r e radar", I n t . J . Remote Sensing, V o l . 6, No. 7, 1985. [3] G.A. Bendor, T.W. 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