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Investigation of biases in Doppler centroid estimation algorithms Zhang, Tonghua 1999

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Investigation of Biases in Doppler Centroid Estimation Algorithms by Tonghua Zhang M . S c , Peking University, C h i n a , 1996 B . S . , University of Science and Technology of C h i n a , 1993  A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F THE REQUIREMENTS  FOR T H E DEGREE OF  Master of Applied Science in THE F A C U L T Y OF G R A D U A T E STUDIES (Department of Electrical & Computer Engineering) We accept this thesis as conforming to the required standard  The University of British Columbia August 1999 © Tonghua Zhang, 1999  ••.  In  presenting  degree freely  at  this  the  thesis  in  partial  fulfilment  of  University  of  British  Columbia,  I agree  available for  copying  of  department publication  this or  reference  thesis by  of this  for  his thesis  and  DE-6 (2/88)  I further  scholarly purposes  or for  her  Columbia  requirements that the  agree that  may  be  It  is  representatives.  financial  permission.  The University of British Vancouver, Canada  study.  the  gain shall  not  an  advanced  Library shall  permission for  granted  by  understood be  for  the  head  that  allowed without  make  it  extensive of  my  copying  or  my written  Abstract Synthetic Aperture Radar ( S A R ) is a microwave imaging system capable of producing high-resolution imagery from data collected by a relatively small antenna. The Doppler centroid is an important parameter in the S A R signal processing.  In principle, it is  possible to calculate the Doppler centroid from orbit and attitude data. B u t the measurement uncertainties on these parameters will l i m i t the accuracy of the estimation. Alternatively, the Doppler centroid can be estimated from the received data. In the past a few years, a number of Doppler centroid estimation algorithms have been developed.  These algorithms can be categorized as one of two kinds. The first  k i n d of algorithm utilizes the signal amplitude. T h e second k i n d of algorithm utilizes the phase of the received signal, such as the D L R algorithm, the M L C C and the M L B F algorithms. It is assumed that the estimation algorithms based on the signal phase can obtain more accurate estimates. T h e objective of this research is to examine and test the performance of the phased-based Doppler estimation algorithms w i t h different scene contrasts, S N R levels and different squint angles, and examine the sensitivity of some phase-based Doppler estimation algorithms to radiometric discontinuities to find out how the radiometric discontinuities affect these estimation algorithms. First, the signal model is carefully examined.  11  T h e effect of range sampling is  discussed. T h e three candidate algorithms, the D L R , M L C C and M L B F , are introduced. M a t h e m a t i c a l analysis of the A C C C angle and the contrast model are performed to obtain a insight of the operation of these algorithms. Experiments on simulated data w i t h different scene contrast and S N R level are performed to compare the performance of these candidate algorithms. T h e M L C C algorithm works well w i t h the E R S and J - E R S data, which normally have a low squint. However, it does not work reliably w i t h the R A D A R S A T data. Since the R A D A R S A T data has a higher squint, simulations are performed to examine the effect of squint on the D L R , the M L C C and the M L B F algorithms. Radiometric discontinuity has significant effect on the estimate of the phasedbased algorithms. T h i s thesis proposed a theory on the mechanism of how the radiometric discontinuity affects these algorithms i n different ways. T h i s thesis also proposed that, the M L B F is not affected by the radiometric discontinuity, the D L R algorithm is more sensitive to the azimuth discontinuity than the M L C C algorithm, whereas the M L C C algorithm is more sensitive to the range discontinuity than the D L R algorithm. These theories are proven by simulations and real data experiments.  in  Contents  Abstract  ii  Contents  iv  List of Tables  ix  List of Figures  xii  Acknowledgements  1  xvii  Introduction  1  1.1  Background  1  1.2  Research Objectives  4  1.3  Outline  4  iv  2  3  Model of the Received Signal 2.1  F o r m of the Received Signal  2.2  T h e Doppler Centroid  11  2.3  T h e Effect of Range Sampling  13  2.4  T h e Spectrum of the Range Compressed Signal  16  Phase-based D O P C E N 3.1  3.2  3.3  3.4  3.5  4  7 8  Estimation Algorithms  T h e Average Cross Correlation Coeff. Angle  21  3.1.1  23  Illustrating A C C C using a Point Target  The D L R Algorithm  26  3.2.1  28  Illustration of D L R w i t h a Single Point Target  The M L C C Algorithm  32  3.3.1  36  Illustration of M L C C w i t h a Single Point Target . .  The M L B F Algorithm  38  3.4.1  39  Illustration of M L B F w i t h a Single Point Target  Discussion  41  Experiments in the Low-Squint Case 4.1  19  45  Generation of Multiple-Target Simulated D a t a  46  4.1.1  46  Methodology  v  4.1.3  E x p e r i m e n t a l Database  :  48 49  Simulations w i t h Low Contrast D a t a  51  4.3  Simulations w i t h Higher Contrast D a t a  59  4.3.1  64  4.5  4.6  6  Simulated D a t a w i t h Noise .  4.2  4.4  5  4.1.2  Discussion on the Effect of Scene Contrast  Simulations w i t h Noise  74  4.4.1  76  Discussion of the Effect of Noise  Experiments on S A R D a t a  77  4.5.1  Experiments on a Low Contrast Scene  77  4.5.2  Experiments on a H i g h Contrast Scene  82  Summary  88  Higher Squint Considerations  89  5.1  Effect of Squint on the A C C C Angle  91  5.2  Effect of Squint on the Beat Spectrum  95  5.3  Single Point Target Simulations  102  5.4  M u l t i p l e Point Targets Simulation  105  5.5  Summary  108  Radiometric Sensitivities  109 vi  6.1  6.2  Experiments w i t h Simulated D a t a  Ill  6.1.1  Simulation Methodology  Ill  6.1.2  Simulation Results w i t h an Azimuth Discontinuity  118  6.1.3  Simulation Results with a Range Discontinuity  121  Discussion of Simulation Results  124  6.2.1  Effects of the A z i m u t h - D i r e c t i o n Radiometric Discontinuity  6.2.2  Comparison of the D L R and the M L C C algorithms w i t h an A Z I M U T H discontinuity  129  6.2.3  Effects of the Range-Direction Radiometric Discontinuity  131  6.2.4  Comparison of the D L R and the M L C C algorithms w i t h the R A N G E discontinuity  6.3  6.4  . . . 124  134  Experiments w i t h E R S - 1 D a t a  138  6.3.1  A z i m u t h Discontinuity i n E R S - 1 D a t a  138  6.3.2  Range Discontinuity i n E R S - 1 R a w D a t a  139  Summary  142  Conclusions  144  7.1  Summary  144  7.2  Contributions  148  7.3  Future Work  149 vii  Bibliography  150  viii  List of Tables  3.1  Parameters of the Single Point Target Simulation  4.1  A m b i g u i t y estimation results of D L R algorithm w i t h low scene contrast (correct answer = -400 Hz)  4.2  57  Fractional P R F estimation results w i t h low squint and varying scene contrast (the answer should be -400)  4.5  54  A m b i g u i t y estimation results of M L B F algorithm w i t h low scene contrast (the answers should be zero)  4.4  52  A m b i g u i t y estimation results of M L C C algorithm w i t h low scene contrast (correct answer = -400 Hz)  4.3  29  60  A m b i g u i t y estimation results w i t h low squint and varying scene contrast (the D O P C E N should be -400 and the ambiguity numbers should be zero)  61  4.6  Simulation Results of D L R and M L C C w i t h Different S N R s  74  4.7  Fractional P R F estimation results on the low-squint low-contrast Bathurst Island E R S scene  (the D O P C E N should be -159 Hz)  78  4.8  A m b i g u i t y estimation results on the low-squint low-contrast Bathurst Island E R S scene with F  os  = 1400 Hz  (the D O P C E N should be -159  Hz and the ambiguity numbers should be zero) 4.9  80  Fractional P R F estimation results on the high-contrast North Cascade Glacier E R S scene (the D O P C E N should be 485 Hz according to dtS A R , but the true fractional D O P C E N is likely 466 Hz according to these estimates)  83  4.10 A m b i g u i t y estimation results on the low-squint high-contrast North Cascade Glacier E R S scene  (the D O P C E N should be 485 Hz and the  ambiguity numbers should be zero)  85  5.1  T h e squint angles corresponding to different D O P C E N s  91  5.2  T h e specified D O P C E N i n simulations  5.3  Fractional P R F estimation results w i t h higher squint on single point target 104  5.4  A m b i g u i t y estimation results w i t h higher squint on single point target . . 104  5.5  Fractional P R F estimation results w i t h higher squint on multiple point  103  targets 5.6  106  A m b i g u i t y estimation results w i t h higher squint on m u l t i p l e point targets ( " M A E " means " M e a n Absolute E r r o r " )  6.1  Fractional P R F estimation results w i t h an azimuth discontinuity  6.2  Ambiguity estimation results w i t h an azimuth discontinuity  107  . . . 119 120  6.3  Fractional P R F estimation results w i t h a range discontinuity  122  6.4  Ambiguity estimation results w i t h a range discontinuity  123  6.5  Difference of A C C C Angles and D O P C E N s ( M L C C effect only)  136  xi  List of Figures  2.1  Geometry M o d e l of S A R Imaging  2.2  Illustrating the Sampling of Point Target Energy i n S A R Signal Space . .  14  3.1  Illustration of an A C C C Calculation  25  3.2  A C C C angles as a linear function of range frequency  31  3.3  E s t i m a t i o n of the Absolute D O P C E N i n the D L R A l g o r i t h m  31  3.4  The Spectrum of a Range Line  37  3.5  Compressed Pulses of the T w o Range Looks  37  3.6  Spectrum of the M L B F beat signal w i t h a single point target  40  3.7  Illustration of an A C C C Calculation w i t h azimuth partial exposure when PRF  8  = 1600 Hz  42  3.8  Error i n A C C C Calculation w i t h azimuth partial exposure  4.1  Convolution to Generate the Simulated D a t a of M u l t i p l e Point Targets  xii  43  .  46  4.2  P a r t i a l Coverage of Targets i n the Simulated D a t a Block  49  4.3  Generate and A d d Noise to the Simulated D a t a  50  4.4  A C C C Angles as a Function of Range Frequency ( D L R A l g o r i t h m )  4.5  A C C C angles of the two range looks i n experiment # 2  4.6  S u m and difference of A C C C angles of the two range looks i n experiment  . . .  56  #2 4.7  56  E s t i m a t i o n of D O P C E N A m b i g u i t y by M L B F algorithm w i t h low-contrast scene (note the exaggerated vertical scale)  4.8  53  58  E s t i m a t i o n of D O P C E N by M L B F A l g o r i t h m w i t h H i g h Scene Contrast of 4.8  62  E s t i m a t i o n of D O P C E N by M L B F A l g o r i t h m w i t h Scene Contrast of 1.3  63  4.10 E s t i m a t i o n of D O P C E N by M L B F A l g o r i t h m w i t h Scene Contrast of 1.6  63  4.11 A m b i g u i t y estimation error of D L R and M L C C on low contrast data  66  4.9  .  4.12 A m b i g u i t y estimation error of the M L B F algorithm on low contrast data (note the vastly different vertical scale compared w i t h Figure 4.11 — the estimate must be w i t h i n the dotted lines of ± P R F to get the correct ambiguity number)  66  4.13 A m b i g u i t y estimation error of D L R and M L C C algorithms on scenes of  increasing contrast  67  xin  4.14 A m b i g u i t y estimation error of M L B F algorithm on scenes of increasing contrast w i t h quadratic curve fitting (note the different  vertical scale  compared w i t h Figure 4.13 — the estimate must be w i t h i n the dotted lines of ± P R F to get the correct ambiguity number i n each case)  67  4.15 Overlap of T w o Targets  68  4.16 A m p l i t u d e of CPQ*  71  4.17 Phase of CPQ*  as a Function of £  as a Function of e  .  71 .  4.18 Comparison of ambiguity estimation errors of the D L R and M L C C algorithms w i t h increasing scene S N R  (low contrast scenes)  75  4.19 Illustrating the distribution of noise and signal as seen by the D L R and M L C C algorithms  76  4.20 T h e detected image of the E R S - 1 B a t hurst Island scene  79  4.21 E s t i m a t i o n Error on the Bathurst Island Scene  81  4.22 T h e detected image of the E R S - 1 N o r t h Cascade Glacier scene  84  4.23 T h e beat spectrum of data block 1 of the N o r t h Cascade Glacier E R S scene 86 4.24 E s t i m a t i o n Error on the N o r t h Cascade Glacier Scene  87  5.1  A C C C Angles w i t h D O P C E N = - 2 K H z  92  5.2  A C C C Angles with D O P C E N = - 5 K H z  92  5.3  A C C C Angles w i t h D O P C E N = - l O K H z  93  XIV  5.4  A C C C Angles w i t h D O P C E N = - 1 5 K H z  93  5.5  A C C C Angles w i t h D O P C E N = - 2 0 K H z  94  5.6  A C C C Angles w i t h D O P C E N = - 5 0 K H z  94  5.7  Beat spectrum w i t h D O P C E N = - 2 K H z  96  5.8  Beat spectrum w i t h D O P C E N = - 5 K H z  97  5.9  Beat spectrum w i t h D O P C E N = - l O K H z  98  5.10 Beat spectrum w i t h D O P C E N = - 1 5 K H z  99  5.11 Beat spectrum w i t h D O P C E N = - 2 0 K H z  100  5.12 Beat spectrum w i t h D O P C E N = - 5 0 K H z  101  6.1  Illustrating the Generation of an A z i m u t h Radiometric Discontinuity  . .  6.2  Magnitude of the Reflectivity M a t r i x w i t h an azimuth discontinuity when  113  M = 5  113  6.3  A z i m u t h Discontinuity i n the Simulated D a t a when M — 5  114  6.4  Illustrating the Generation of a Range Radiometric Discontinuity  6.5  Magnitude of the Reflectivity M a t r i x w i t h a range discontinuity when M = 5  . . . .  115  115  6.6  Range Discontinuity i n the Simulated D a t a when M = 5  6.7  Range Discontinuity i n the Simulated D a t a After Range Compression . . 116  xv  . .  116  6.8  6.9  A m b i g u i t y estimation error of the D L R and the M L C C algorithms caused by the azimuth discontinuity  128  Illustration of the effect of a range discontinuity  132  6.10 A m b i g u i t y estimation error of the D L R and the M L C C algorithms caused by the range discontinuity  133  6.11 Difference of A C C C Angles of Look 1 and Look 2  135  6.12 Illustration of the effect of a strong target on a weak target  137  6.13 T h e Detected Image of the Kamloops E R S Scene  138  6.14 T h e Detected Image of the Port A l i c e E R S Scene  140  xvi  Acknowledgements First of a l l , I would like to thank m y mother for her consistent support and encouragement throughout m y studies at U B C . W i t h o u t her love and care this work would not have been completed. I would like to thank m y supervisors, D r . I. G . C u m m i n g and D r . F . H . Wong for providing me the opportunities to do research i n the field of S A R , and for their invaluable guidance, advice and discussions during the research and the preparation of the thesis. A l l of these are greatly appreciated. I would like to thank Catherine Vigneron, M i k e Seymour and Sandor Albrecht for their great help and insightful discussions. I would like to thank Ziwei D i n g and Julong D u for their invaluable encouragement and insightful discussions throughout m y studies at U B C . I would like to thank all the members of the Signal Processing and M u l t i m e d i a Group at U B C for providing insightful discussions and friendly working environment. I a m grateful for the financial support provided for this research by the N a t u r a l Sciences and Engineering Council of Canada, by M a c D o n a l d Dettwiler and Associates, and by the Canadian Centre for Remote Sensing. TONGHUA  The University  of British  Columbia  August 1999 XVll  ZHANG  Chapter 1  Introduction  1.1  Background  Synthetic Aperture Radar ( S A R ) is a microwave imaging system capable of producing high-resolution imagery from data collected by a relatively small antenna [1, 2, 3, 4]. It is a technique for creating high resolution images of the earth's surface.  It can be  used to make remote observations of the earth during day or night, through cloud cover and even through light rain [5]. S A R images are widely applied i n many areas, such as oceanography [6], geology, forestry [7] and national defense. A S A R system consists of a microwave transmitter/receiver and an antenna mounted on a moving platform such as an orbiting satellite. T h e antenna transmits a beam of chirp signals to the ground and receives the reflected signal from the ground. T h e received signal consists of a superposition of a large amount of reflections from scatterers  1  on the ground. As in conventional radars, the high resolution in range of a SAR is attained using long transmitted pulses that are compressed to a short duration by range compression. The main advantage of S A R compared to conventional radar systems is that SAR achieves higher resolution in azimuth by coherently processing the phase history of targets in the azimuth direction [8]. SAR began with an observation by Carl Wiley in 1951 that a radar beam oriented obliquely to the radar platform velocity will receive signals having frequencies offset from the radar carrier frequency due to the Doppler effect [9].  Activities continued  on analyses and signal processing techniques and many signal processing techniques are successfully developed, such as the Range/Doppler (R/D) algorithm [10, 11], the SPEC A N algorithm [12] and the Chirp Scaling algorithm [13]. Basically the SAR signal processing consists of range processing and azimuth processing.  A n important parameter in relation to the S A R signal processing is the  Doppler centroid. The Doppler centroid is the Doppler frequency received from a given point target on the ground when the target is centered in the azimuth antenna beam pattern. Because the transmitted signal of a SAR is pulsed with the Pulse Repetition Frequency (PRF), and the P R F is designed according to the azimuth bandwidth instead of the absolute azimuth frequency, the spectrum of the azimuth signal can be aliased by several P R F ambiguities. Thus the absolute Doppler centroid can be considered to be made up of an "integer P R F part" and a "fractional P R F part". The fractional P R F part is the centroid wrapped around to the fundamental frequency range of the P R F , and the integer P R F part is referred to as the Doppler ambiguity [14].  2  Since the Doppler centroid is an important parameter i n the S A R signal processing, a lot of effort has been devoted to obtaining an accurate estimate of it.  In  principle, it is possible to calculate the Doppler centroid from orbit and attitude data. But the measurement uncertainties on these parameters w i l l l i m i t the accuracy of the estimation [14]. Alternatively, the Doppler centroid can be estimated from the received data [15]. In the past a few years, a number of Doppler centroid estimation algorithms have been developed.  These algorithms can be categorized as one of two kinds. The first  k i n d of algorithm utilizes the signal amplitude, such as the look range cross-correlation technique [16, 17], and the multiple P R F technique [18,19]. T h e second k i n d of algorithm utilizes the phase of the received signal, such as the " D L R " algorithm [20], and the M L C C and the M L B F algorithms [14]. It is assumed that the estimation algorithms based on the signal phase can obtain more accurate estimates [14]. Thus, a i n depth understanding of these algorithms is necessary. T h e squint angle is another important parameter i n the S A R geometry model. T h e M L C C algorithm works well w i t h E R S and J - E R S data, which have a low squint. However, it does not work reliably w i t h the R A D A R S A T data, which has a higher squint. T h e effect of the squint on the phase-based algorithms should be well examined. Radiometric discontinuity i n a S A R image is due to the difference of the properties of scatters on the ground surface. Some echoes of the chirp from a certain part of the ground surface are strong, whereas some echoes from another part of the ground surface are weak.  T h e boundary of these two parts forms a discontinuity. T h i s structure is  very common i n a S A R image, such as the boundary of ocean and land. Radiometric  3  discontinuity is an important factor that affects the performance of D O P C E N estimators, especially of the phase-based  1.2  estimators.  Research Objectives  In this research, three phased-based Doppler centroid estimation algorithms (the D L R , the M L C C and the M L B F algorithms) w i l l be examined. T h e objectives of the research project are to:  • examine and test the performance of the three candidate algorithms w i t h different scene contrasts and S N R levels, • examine the performance of these algorithms w i t h higher squint data, • examine the sensitivity of the candidate algorithms to radiometric discontinuities,  1.3  Outline  To obtain an insight of the performance of different D O P p l e r C E N t r o i d ( D O P C E N ) estimators, a mathematical model of the received data should be established and well studied. In Chapter 2, the form of the received signal after demodulation is derived. T h e n the concept of D O P C E N is introduced. T h e effect of range sampling is discussed, illustrating that the Range C e l l M i g r a t i o n ( R C M ) i n the raw data causes a shift i n the azimuth spectra, forming the dependence of D O P C E N on range time. Finally, the P r i n ciple of Stationary Phase is applied to approximate the spectrum of range-compressed 4  signal, showing that the D O P C E N  is a linear function of range frequency under this  approximation, which is the basis of the D O P C E N estimators discussed i n the following chapter. In Chapter 3, the concept of Average Cross Correlation Coefficient ( A C C C ) angle is introduced, which is used to estimate the fractional P R F part of the  DOPCEN.  T h e n the " D L R algorithm", " M u l t i - L o o k Cross Correlation ( M L C C ) algorithm" and the " M u l t i - L o o k Beat Frequency ( M L B F ) algorithm" are introduced. Single target simulations are performed to illustrate the operation of these algorithms. T h e D L R a n d the M L C C algorithm use the A C C C angle to obtain the fractional P R F part of the D O P C E N and ambiguity number, whereas the M L B F algorithm uses the beat frequency to obtain the ambiguity number. T h e sensitivity of the A C C C angle and the beat frequency to the azimuth partial exposure is discussed. In Chapter 4, experiments w i t h simulated data and real S A R data are performed to compare the performance of the D L R , M L C C and M L B F algorithms. T h e method of generating the simulated data w i t h multiple point targets i n the low squint case is introduced. Simulation results are then presented. T h e effect of scene contrast and the effect of noise are then discussed. T h e raw data of ERS-1 i n the yaw-steering mode are used to compare the performance of these algorithms on real data i n the low squint case. T h e effect of a squint mode imaging geometry on S A R signal properties is quite complicated. W h e n the squint increases, the properties of the signal structure become more complicated as the cross-coupling between the range and a z i m u t h signals increases. In Chapter 5, the effect of the squint on the A C C C angle is studied by simulations. T h e n simulations of the D L R , the M L C C and the M L B F algorithms on single point target and  5  multiple point target are performed to examine the performance these algorithms w i t h higher squint. It is found that these three algorithms still work accurately to obtain the correct estimates. T h i s proves that the approximation to the spectrum of a compressed target using the Principle of Stationary Phase is still accurate enough w i t h high squint. Radiometric discontinuity is an important factor that affects the performance of D O P C E N estimators, including the phase-based estimators. In Chapter 6, we w i l l discuss the effect of the radiometric discontinuity on the D L R , M L C C and M L B F algorithms. Simulations are first performed. T h e n the performance of the D L R algorithm and the M L C C algorithm are compared on the discontinuity i n the range and azimuth directions respectively. We explain that how the radiometric discontinuity affect the D L R and the M L C C algorithm, why the D L R algorithm is more sensitive to the azimuth discontinuity than the M L C C algorithm and why the M L C C algorithm is more sensitive to the range discontinuity than the D L R algorithm. We also explain why the M L B F algorithm is not affected by the radiometric discontinuity. These three algorithms are also performed on E R S - 1 raw data to illustrate our conclusions.  6  Chapter 2  Model of the Received Signal To obtain an insight into the performance of different D O P p l e r C E N t r o i d ( D O P C E N ) estimators, the mathematical model of the received data should be established and well understood. In this chapter, the form of the received signal after demodulation is derived. T h e n the concept of D O P C E N is introduced. The effect of range sampling is discussed, illustrating that the Range C e l l M i g r a tion ( R C M ) i n the raw data causes a shift i n the azimuth spectra, forming the dependence of D O P C E N on range time for each target. Finally, the Principle of Stationary Phase is applied to approximate the spectrum of the range-compressed signal, showing that the D O P C E N is a linear function of range frequency under this approximation. This linear relationship is the basis of the D O P C E N estimators discussed i n the following chapters.  7  2.1  Form of the Received Signal  The geometry model of S A R imaging of a single point target is shown in Figure 2 . 1 .  Figure 2 . 1 : Geometry Model of S A R Imaging The satellite carrying the S A R antenna moves with a effective velocity V relative r  to the point target. The slant range R is the distance from the antenna to the point target. Pulses are transmitted from the antenna to the ground with a constant frequency, the Pulse Repetition Frequency (PRF). In the range direction, the received signal is a analog signal as a function of range time T. The "timing" of the received signal is reset at the beginning of each transmitted pulse, to constitute the beginning of a new range line. In this way the received signals  8  form a discrete signal along the azimuth direction w i t h each new range line forming a new "sample" i n azimuth time n. T h e nominal range to the point target RQ is usually defined as the range when the point target is closest to the antenna, as the radar system passes by the point target. T h e time at which the slant range equals Ro is defined as 77 . T h e exposure time is 0  defined as the duration during which the point target is fully exposed i n the radar beam, i.e. the time required to generate the Doppler bandwidth. T h e squint is defined as the angle between the antenna pointing direction and the direction perpendicular to the velocity, as 6 shown i n Figure 2.1. After transmitting each pulse, the S A R antenna receives the reflected signals. T h e properties of the received signals are determined by parameters of the S A R system and the characteristics of the reflectors. T h e transmitted signal is a linear F M signal or a chirp, which is given by:  5,(77,  where  r ) = P(r) W(r, - n ) c o s [ 2 7 r / r + rr K (r -  /o T  c  is the  0  r  r,/2) ], 2  r =  [0,  center frequency of the transmitted chirp range time the exposure time azimuth time F M rate of the transmitted chirp azimuth time offset of beam center from zero Doppler  P(r)  envelope of chirp, and azimuth beam pattern magnitude 9  77]  (2.1)  T h e received signal from the ideal point target is assumed to be the same as the transmitted signal except for a time delay r , and is given by: d  S (v7,r) = r  P(T-T )W( -n ) d  V  COS{2TT fo(r-T )  c  +  D  7T  K (T - r - n/2) ), 2  r  d  r =  [^,^+77]  (2-2) T h e time delay due to the varying slant range from the S A R antenna to the point target at different azimuth positions is given by:  r (n) d  =  ,  c  17 = fo - t/,/2, r) + /2] c  c  (2.3)  m  where rji is the a z i m u t h exposure time, and R(r]) is the instantaneous slant range from the antenna to the point target given by:  R(ri)  = yjRo  2  + V  2  (r, - rjo)  (2.4)  2  B y expanding (2.4) i n a Taylor series around a z i m u t h t i m e 770 and selecting r}  0  as the a z i m u t h time origin, R(rj) can be approximated by:  R{V)  «  ^0 + ^  V  " '  2  (2-5)  where Ro is the closest slant range from the antenna to the point target, and V is the r  effective velocity of the antenna w i t h respect to the point target. Consider the case of demodulation to baseband,  which converts the signal to  complex form. T h i s is done by m u l t i p l y i n g the received signal by the coherent oscillator signal given by:  h{r)  = exp{-j27r/ T} 0  10  (2.6)  T h e demodulated signal is given by:  Sd(v, ) T  =  h(r) S (rj,T)  =  \ (  r  P  T  ~ d) (V - nc) ( e x p { j 27r/ (r - r ) + j 7rK {r - r - r , / 2 ) } + T  W  2  0  =  ^ ( r - r ) W( d  V  )  Vc  r  d  - r - T , / 2 ) } ) exp{-j 27r/ r}  e x p { - j 27r/o(r - T ) - ? nK (T d  d  2  r  d  0  ( e x p { - j 2nf r + j T T / ^ T -r 0  d  T,/2) } + 2  d  e x p { - j 27r/ (2r - r ) - j 7rK (r - r , - T , / 2 ) } )  (2.7)  2  0  d  r  After a low pass filter is used to remove the 2 / component, the demodulated 0  signal is given by:  r ) = ^P(T-T )W( -n )exp{-j2nf T  5,(77,  2.2  d  n  c  0  + j n K (r - r - n/2) } 2  d  r  d  (2.8)  The Doppler Centroid  F r o m E q u a t i o n (2.8), the received signal can be expressed as the product of an azimuth signal S (r)) and a range signal S (r) given by: a  r  S (n) a  S (r) r  =  fy(77 - 7 7 ) e x p { - j 2 7 r / r } c  0  (2.9)  d  = P(r-T )exp{J7rA' (r-T -T,/2) } 2  r f  r  D  (2.10)  Note that because of the range-azimuth coupling, the range signal actually is still a function of a z i m u t h time as well as of range time. However, after Range C e l l M i g r a t i o n  11  Correction ( R C M C ) , the locus of energy i n the 2-D memory is aligned w i t h the azimuth time axis. The fine range resolution is achieved by compressing the range signal, as conventional radar systems do. The high azimuth resolution of the S A R system is achieved by compressing the azimuth signal. The azimuth signal can be approximated by a chirp because of the ry relation i n (2.5). F r o m Equation (2.5) and E q u a t i o n (2.9), the azimuth 2  signal can be given by:  Sa(v)  = W(r, - r, ) e x p { - j ^ / } c e  0  e x p { - j 2n *Lf c Ko  }  2 0  V  (2.11)  Ignoring the constant part, the phase of the azimuth signal is  4>{ri) = - 2 i r ^ - f Clio  (2-12)  2 o V  Thus, the azimuth frequency or Doppler is given by:  =  M  ~7Ro  =  foT]  ( 2  "  1 3 )  Equation (2.13) shows that the azimuth signal is also a chirp signal w i t h the F M rate given by: 2 V = -^/o cKo 2  Ka  (2-14)  T h e D O P C E N is defined as the Doppler or azimuth frequency of a given target, at the pulse or azimuth time when the target lies i n the center of the beam. Thus the D O P C E N is given by: F,  dc 0  = faiVc)  = - ^rfoVc c KQ 12  (2-15)  Since the transmitted signal is repeated at the P R F , and the P R F is chosen to properly sample the azimuth bandwidth instead of the absolute azimuth frequency, the spectrum of the azimuth signal is aliased by the P R F . Expressing the D O P C E N i n units of one P R F , and considering the integer and fractional parts separately i n P R F units, the D O P C E N can be considered to be made up of an "integer P R F part" and a "fractional P R F part". T h e fractional P R F part is the centroid wrapped around to the fundamental frequency range of the P R F and is the part estimated by finding the peak of the Doppler spectrum.  T h e integer P R F part is referred to as Doppler A m b i g u i t y , and must be  estimated by other means.  2.3  The Effect of Range Sampling  After demodulation, the baseband range signal is sampled w i t h the frequency F  s  and  stored i n a 2-D memory. Each echo of a transmitted chirp forms a range line and each range sample at the same sampling time forms an a z i m u t h line. Figure 2.2 illustrates the locus of energy of an uncompressed point target i n S A R signal memory. T h e instantaneous range sampling time is shown by the vertical dotted lines. The interval between each horizontal dashed line is the time interval between each pulse, i.e. l/PRF.  T h e echo of each transmitted chirp from the point target is shown by the  solid line along the range time direction. T h e samples containing non-zero point target energy are shown by circles. Assume that the first echo arrives exactly at a sampling time, then the interval of the range time is given by:  13  Locus of Point Target Energy in Signal Memory  25  I  20  :  :  :  i  :  .  \J  <~»  W  V  \J  KJ  tj  KZ>  KJKJ  t)  l_*  VJ  • • • •  ©  A I I  KJ  •  1?  1  1  KJ  :  :  V-»  v-<  >—'  \J  \J  KJ  W  \J  «_/  v_/  KJ  f  KJ  KJ  KJ  XJ  KJ  KJ  \p  KJ  KJ  15  •  o E  • • •  ~ 10 E  'N <  U  • • •  :  •  :  C?  fc7  f  W  C3  ^  ^  W  C  d  w  t7  U  U  fcS  V  V  V  7  W  t 7 V _ /  fcT  f  w  W  V  <o* fcp  W  V  V  • • • € •• — ^  • • - \_/  V.^  —  v  V  v.  I  1  5  10 Range Time —> (cells)  •  "  "  15  20  Figure 2.2: Illustrating the Sampling of Point Target Energy i n S A R Signal Space  T =  [T (r] -r)i/2), d  c  T (r) -r]i/2)+TI]  (2.16)  2  (2.17)  d  c  where r (rj — rji/2) is given by: d  c  2Ro Td = c Let:  V + —5cKo 2  [Vc-Vi)  T = LI + T (r] - 771/2) d  c  £ = fi -  ri/2  V r  (2.18)  (2.19)  2  a  r  CRQ  14  (2.20)  and  then from E q u a t i o n  (3 = (rj - T7//2)  (2-21)  c  the 2-D phase of the demodulated signal is given by:  (2.8),  <p{n,r) = - 2 ^ / o r , + nK  [{ + a(/? - TJ )f 2  r  (2.22)  2  Thus the Doppler or the azimuth frequency is given by:  f  a  where £ =  = ,  = " 2 a [/o + K t + apKr] r  [-77/2,77/2].  Thus the term  /  + /<r£  0  r, + 2 a K rj 2  3  r  (2.23)  i n E q u a t i o n (2.23) is actually the  instantaneous range frequency. For the low squint case, such as E R S - 1 i n yaw-steering mode,  ft«2.0x  I O , (3 » - 0 . 1 5 s, and K = 0.5 GHz/s. T h e term aB K - 7  2  r  r  i n Equation  (2.23) is about 2 KHz. Compared to the range bandwidth, which is about 15 MHz, it is negligible. T h e cubic item 2a K r] 2  i n E q u a t i o n (2.23) is less than 0.02 Hz, which is  3  r  also negligible. Thus the azimuth frequency can be approximated i n low squint case by:  f  a  = -  2  a  f  V  =  -2-^-A  (2.24)  where / is the instantaneous transmitted range frequency. T h e D O P C E N is then given by:  F  dc  = - 2 - ^ - 77,/ cRo  (2.25)  F r o m E q u a t i o n (2.25), it can be seen that the D O P C E N is a linear function of the transmitted range frequency. Compared to E q u a t i o n (2.15), where there is no Range C e l l M i g r a t i o n ( R C M ) , it can be concluded that, i f there is no R C M , the D O P C E N is  15  a constant for each azimuth line. If R C M does exist, as happens i n the raw data, the D O P C E N is a linear function of range frequency i n low squint case.  2.4  The Spectrum of the Range Compressed Signal  Range compression is the first m a i n step i n many S A R processing algorithms.  Most  D O P C E N estimators work on range compressed data. Range compression is performed by convolving a matched filter w i t h each range line. Fast convolution is usually used to obtain computing efficiency. Each range line and the matched filter are transfered from the range time domain to the range frequency domain using range F F T s and are multiplied together. T h e demodulated signal i n the azimuth time, range frequency domain is given by:  +oo  /  d  + CO  /  S (n, T) e x p { - j 2TT f r } dr r  -oo  S (n) S (rj, T) e x p { - j 2TT f r } dr a  r  r  -oo +oo  P ( r - T ) e x p j j 7T K {r - r D  /  r  /  where S (r],f ) r  r  2  e x p { - j 2TT f r } dr r  (•  -i  P(T - r ) e x p j j 7T K {r d  SM  n/2) }  -oo  + °°  =  d  r  -r -  rt/2)  2  d  -oo  S (r,Jr)  - 2f r r  } dr  (2.26)  r  is the Fourier transform of S (r),T),  f  r  r  is the fundamental range fre-  quency and is given by:  fr = f-fo,  ~ BW /2 r  16  < f  r  < BW /2 r  (2.27)  where BW is the range bandwidth given by: r  BW  = K(  r  - r,/2 <  r  It has been proven that S (rj,f)  (2.28)  has no analytical expression since it involves  r  a Fresnel integral.  £ < n/2  T h e P r i n c i p l e of Stationary Phase is applied to approximate the  expression of the range spectrum. T h e P r i n c i p l e of Stationary Phase states that, when the phase Tr[K (r — r — T ; / 2 ) — 2 / r ] changes rapidly, the positive and negative values 2  r  d  of the function w i l l be approximately canceled by each other i n the integration and have negligible contribution to the integral. O n l y those parts where the phase changes slowly have the most significant contribution to the integral. B y using the Principle of Stationary Phase, the range spectrum can be approximated by [21]:  Sr(vJr)  = \K \  1 / 2  r  ^sgn(7A: )) exp ^-j TT  exp  r  ^  + 2f T r  d  +  fn r  (2.29)  T h e range matched filter of S (rj, r ) is generated according to the complex envelope r  of the transmitted chirp and is given by:  M {n) r  = =  S;(n,-u.) exp(-J7rA^.(-//-T7/2) )  (2.30)  2  A g a i n applying the Principle of Stationary Phase, the spectrum of the range matched filter is given by:  +oo  / =  M {u) exp{ -j 2TT f T} dr r  r  -oo  \K \~  1/2  r  exp | j ^ sgn(ATr) J exp ^-j 17  TT  ^ fr Tl  (2.31)  Thus the range compressed signal is given by:  S ( Jr) rc  V  =  S (n)  =  W(r] - rj ) e x p { - j 27r/ r }  a  Srinjr)  M (f ) r  r  c  W(rj -  0  n) c  exp | j ^ s g n ( ^ ) | e x p { - j 2TT f  rf  r  e x p j ^ s g n f X ) } exp{-j27r/r } d  r  r} d  (2.32)  We can see that the azimuth frequency can be given by: d  2 V = ~-^-fn cRo  2  f = -jrUri) a  or]  (2-33)  Thus the D O P C E N is given by: 2 V  2  F  r  — — dc —  r  Vcf  (2.34)  CRo D  Thus we see that from Equation (2.34), after range compression, the D O P C E N of the range compressed signal is a linear function of the range frequency.  18  Chapter 3  Phase-based D O P C E N Estimation Algorithms  In Chapter 2, the model of a point target and the concept of the Doppler centroid ( D O P C E N ) were introduced. There we conclude that, after range compression, the D O P C E N is approximately a linear function of range frequency. T h i s is the basic principle of two of the "phase-based" D O P C E N algorithms we w i l l review i n this chapter. In this chapter, the concept of Average Cross Correlation Coefficient ( A C C C ) angle is introduced, which is used i n two of the phase-based D O P C E N estimators. T h e n the m a i n phase-based algorithms are introduced: 1. the " D L R " algorithm, 2. the " M u l t i - L o o k Cross Correlation" ( M L C C ) algorithm, and 3. the " M u l t i - L o o k Beat Frequency" ( M L B F ) algorithm. 19  T h e first two algorithms, the D L R and the M L C C algorithms, use the A C C C angle to obtain the fractional P R F part of the D O P C E N and the ambiguity. T h e M L B F algorithm uses beat frequency to obtain the ambiguity. Single point target simulations are performed to illustrate the operation and properties of these algorithms. Finally, we discuss the distinction of the A C C C - a n g l e - b a s e d algorithms and the beat-frequency-based algorithm.  20  3.1  The Average Cross Correlation Coeff. Angle  Most of the early D O P C E N estimators used the distribution of spectral energy to form the estimate [10, 15]. Recently, several estimators have been developed which make use of the phase of the received signal to get more accurate D O P C E N estimates. In 1989, Madsen [22] used an algorithm based on using a z i m u t h phase increments i n raw signal data to estimate the fractional P R F part of D O P C E N . T h e phase increments are estimated by taking the average cross correlation coefficient ( A C C C ) between adjacent azimuth samples . 1  F r o m Equation (2.32), after demodulation and range compression, the azimuth signal i n the azimuth-time, range-frequency domain can be written as:  S (v) a  =  W{n - rj ) e x p { - j 2 7 r / r }  =  W^-rn)  c  d  expj-j^/2fo)j  ^  (3.1)  Ignoring the large constant RQ term i n the range equation and using the quadratic approximation (2.5), the range compressed signal can be expressed as:  SM where K  a  = W{n-n )  expj-JTr^n } 2  c  (3.2)  is the azimuth F M rate: 2 V f cRr J 2  rs  (3.3)  Q  The average cross correlation coefficient A C C C is defined as the average correlaMadsen actually used an approximation to the ACCC by taking the sign of the data samples. This was done for computing efficiency. 1  21  tion between one a z i m u t h sample a n d the next, a n d c a n b e c o m p u t e d b y the s u m over azimuth time:  (3.4)  where  Ar] =  1/PRF =  F r o m Equation  1/F  a  (3.2),  is t h e t i m e between consecutive a z i m u t h samples.  the A  of the range compressed a z i m u t h signal is given  C C C  by:  C  =  Sl  rf} exp{J7r K {r]  - Vc) W(ri + &r]- Vc) exp{-jirK  Wi  a  a  +  A / 7 )  2  }  v ~  £  W (r] 2  - r] ) e x p { j 2 7 r K c  a  rj Ar]}  v =  where  £  ( ^ ) c o s ( 2 7 r A ' 77 A 7 7 ) + j £  M  a  M(rj) sm(2n K 77 A 7 7 )  W (r] — r] ) a n d i s s y m m e t r i c w i t h r e s p e c t t o 2  M(T7) =  (3.5)  a  c  ?7 . C  Thus the  angle  A C C C  is g i v e n b y :  arg{C7} t a n -1  En M{r]) sin(27r K 77 A 7 7 ) a  En M ( T 7 ) COS(2TT K  77 A T ? )  a  =  t a n -1  t a n -1  M(T7 -  8 ) sm(2nK (r]  M(?7  Si) cos{2nK (r]  -  X  a  c  a  sin(27r/\" 77 A 7 7 ) ' 0  c  - SJAr])  + M(T7  - SJArj)  + M(?7  C  C  +  S ) sm(27rK (r]  +  COS{2TTK {r}  t  a  c  a  c  + S )Ar ) + . 1  1  + S^An) +  c  cos(27r K r] A 7 7 ) a  27T K  a  by  symmetry  c  7]  C  (3.6)  F  a  where  Si, S ... 2  a r e t h e a z i m u t h t i m e i n t e r v a l s nAr], s y m m e t r i c w i t h r e s p e c t t o ?7  C  22  T h e relationship between the D O P C E N and the A C C C angle would be given by:  F  dc  =  K  a V c  =  ^cf>  (3.7)  Z7T if <f> could be computed without the wraparound of the arc tan function.  B u t this  estimate is affected by the wraparound, and so E q u a t i o n (3.7) can only estimate the component of the D O P C E N lying i n the frequency range (0, F ) or ( — F /2, F /2), a  a  fractional P R F part. We denote the fractional P R F part as Fd  ca  Fdca =  3.1.1  F  a  -Z-[4>] mod 2n Z7T  a  i.e. the  which is given by:  (3-8)  Illustrating A C C C using a Point Target  Consider a single point target that exhibits a quadratic a z i m u t h phase history, as shown in the solid line of Figure 3.1(a).  T h e target duration is chosen so that the target  sweeps through Doppler frequency [—0.5F /O , 0.5F /O ] a  s  a  s  plus an offset given by the  selected Doppler centroid. In Figure 3.1, F = 1600 Hz, the D O P C E N is -200 Hz and a  the oversampling ratio O = 1.1. s  T h e phase increments per a z i m u t h sample are examined. These phase increments are drawn i n the dashed line i n Figure 3.1(a).  W h e n the current a z i m u t h sample is  m u l t i p l i e d by the complex conjugate of the previous sample, a complex number is obtained whose amplitude represents the square of the signal amplitude, and whose phase represents the phase increment <p discussed above. These complex numbers can be drawn as vectors i n the complex plane, as shown i n Figure 3.1(b). T h e line marked Z represents the product at the zero Doppler point,  23  where the phase increment is zero. Note how the length of the vectors is small at the end of the target exposure, at points A and B , and is larger near the middle of the exposure. This effect is due to the azimuth antenna pattern W(r] — r) ). c  W h e n the vectors of Figure 3.1(b) are added together coherently, i.e. they are averaged, the long vector shown i n Figure 3.1(b) is obtained, indicating the preferred direction of the phase increments.  Thus the angle of the long vector represents the  fractional part of the D O P C E N . In this case, it is estimated as -200 Hz.  24  (b) Individual and average phase increment  2.5  i  •  Estim<ated DoDcen = -200 H;7 1.5  i cc o_ CD CO  1 0.5  E  A JBBllll? B IS^^P^  -0.5 •1 -2.5  —J  z  -2  -1.5  -1  i  -0.5 0 Real part  i  0.5 >  1  Figure 3.1: Illustration of an A C C C C a l c u l a t i o n 25  1.5  2.5  3.2  The D L R Algorithm  In 1991, the G e r m a n Aerospace Establishment ( D L R ) developed a D O P C E N estimator [20] based upon the property that the absolute or unaliased D O P C E N is a function of the frequency of the transmitted signal, as we discussed i n Chapter 2. F r o m Equation (2.34), the unaliased D O P C E N is given by: 2 V  2  F  = --jrVcf  dc  where k = — 2 V n /c R c  (3.9)  is the slope of this linear function of range frequency.  2  r  = kf  0  F r o m the Chapter 2, the D O P C E N can be considered to be made up of two parts, the fractional P R F part and the Doppler ambiguity, as given by: F  dc  = F  dca  + mF  (3.10)  a  where m is the Doppler ambiguity number. After demodulation by / n , the complex signal has the range frequency:  Thus the absolute Doppler centroid is: F  dc  = kf  = k(f  0  + f) r  = kf  0  + kf  (3.12)  r  One of the m a i n purposes of D O P C E N estimators is to estimate the average value of Fdc, i-e- to estimate the D O P C E N at the centre range frequency f 2  given b y : 2  0  T h e D O P C E N is also a f u n c t i o n o f range t i m e , a n d is u s u a l l y e s t i m a t e d at several ranges. However,  we w i l l not address that aspect o f the e s t i m a t i o n i n this r e p o r t .  26  Fdcfl = kf  0  = -^-r,cfo cK  (3-13)  0  F r o m the S A R system specifications, the centre range frequency fo can be obtained accurately. Thus we only need to estimate the slope k. F r o m E q u a t i o n (3.10): F  dca  = F -mF dc  = kf  a  + kf  0  -  r  mF  a  (3.14)  = k  (3.15)  Thus the slope k can be estimated using:  ^ J T  = J7-( fo k  + kf  -  r  mF ) a  T h e D L R algorithm works i n the range frequency domain, i n which the various range frequencies i n the demodulated signal can be accessed by F F T i n g the compressed range s i g n a l . A s we discussed i n the last section, the algorithm can use the A C C C 3  angle to estimate the fractional P R F part of D O P C E N . T h e A C C C between adjacent azimuth samples at each range frequency cell is computed, then the slope k of the A C C C angles as a function of range frequency is estimated. T h e n the Doppler ambiguity can be obtained from:  m = round ( k f  0  -  F ) dca  (3.16)  and the unaliased D O P C E N is obtained from the ambiguity number m and the estimation of the fractional P R F part F  dca  using (3.10).  In the real case, due to the variation of the D O P C E N i n the along and across track directions, the data must be segmented i n blocks at the beginning of this algorithm. 3  Or  n o t p e r f o r m i n g the range I F F T j u s t yet i n range c o m p r e s s i o n .  27  T h e required segmentation, on the other hand, raises the problem of partially covered chirps, both i n range and azimuth.  Therefore, the data must be range compressed  before segmentation, whenever the range segments are smaller or i n the order of a range chirp [20]. T h e D L R algorithm is a Doppler A m b i g u i t y Resolver ( D A R ) applied to different range frequencies.  Hence, its accuracy can be predicted using the well-established  theories for D O P C E N estimators [22, 23].  3.2.1  Illustration of D L R with a Single Point Target  To illustrate the principle of the D L R algorithm, a single point target simulation is performed. In the point target simulation, parameters close to those of the R A D A R S A T and E R S - 1 S A R systems are used. However, to keep the simulation to reasonable array sizes, the parameters are modified slightly, and are given i n Table 3.1.  Compared to the real S A R parameters, we use somewhat lower range and azimuth bandwidths.  T h e T i m e B a n d w i d t h Products ( T B P s ) we use i n the simulations are  340 and 311 i n the range and azimuth directions respectively. A l t h o u g h the T B P s are reduced i n both the range and azimuth directions, they are still large enough to ensure the accuracy of the simulation by using the P r i n c i p l e of Stationary Phase i n the low squint case. Range compression is performed on each range line.  Next the A C C C of each  azimuth line is computed according to Equation (3.4) i n the range frequency, azimuth time domain.  A C C C angles are then computed as a function of range frequency, as  28  . Symbol  Parameter  Value  Units  nominal slant range  Ro  850  Km  effective radar velocity  V  7050  m/s  radar centre frequency  fo  5.26  GHz  BW  17  MHz  F  20  MHz  r  range bandwidth  r  range sampling frequency  s  chirp duration  20  Doppler bandwidth  BW  800  Hz  F  960  Hz  Fdcfl  -400  Hz  a  PRF  a  DOPCEN  Table 3.1:' Parameters of the Single Point Target Simulation shown i n Figure 3.2. A straight line is then fitted to the high-energy regions of Figure 3.2, and is shown after conversion to Hz units i n Figure 3.3. F r o m the straight line fit, the mean value is 4  found to be -400.1 Hz  and the slope is estimated as -82 mrad/MHz.  Thus the  fractional  P R F part of the D O P C E N is estimated as -400.1 Hz , and the integer P R F part of the D O P C E N is estimated as -431 Hz , which are both very accurate for their respective purposes. In Figure 3.3, the absolute centroid is estimated by projecting the sloped line back to the radar centre frequency, at which point it should intercept the vertical axis at Fd -  5  c  Note only those values within the dashed line in Figure 3.2 are used to estimate the slope since this part is the region within which the main energy of the signal is located. 4  Note that in the present simulation, we have not included any variation of beam pointing angle with range frequency, so the bias F often found in the DLR and MLCC estimators is zero. 5  0i  29  T h e ambiguity number is then obtained from (3.16), and the correct value of m = 0 is found.  30  Fit the ACCC Angle in a 1-Order Polynomial, DOPCEN = -400 Hz  -8  -6  -4  -2 0 2 4 Range Frequency —> (MHz)  6  8  Figure 3.2: A C C C angles as a linear function of range frequency  -3991  1  Estimation of DOPCEN, DOPCEN = -400 Hz 1 1 1 1 1 '—  -399.2 r  l 5250  ) 8  1 5252  1 5254  1 1 1 1 1 5256 5258 5260 5262 5264 Range Frequency —> (MHz)  1 5266 5268  ;ure 3.3: E s t i m a t i o n of the Absolute D O P C E N i n the D L R A l g o r i t h m  31  3.3  The M L C C Algorithm  In 1996, a combined S A R D O P C E N estimation scheme based on signal phase was published [14]. This scheme uses two complementary Doppler estimation algorithms, both utilizing the phase information embedded i n the radar signal. In each algorithm, upper and lower parts of the available range bandwidth of the received signal are extracted to form two range looks. T h e first algorithm, called M u l t i l o o k Cross Correlation ( M L C C ) , computes the average cross correlation coefficient between adjacent azimuth samples for each of the two looks and then takes the difference between the A C C C angles of the two range looks. T h e Doppler ambiguity is determined from the angle difference, again projecting the difference back to zero range frequency. T h e fractional P R F part is also determined from the cross correlation coefficients.  In this section, the M L C C algorithm will be  introduced. T h e second algorithm, called M u l t i - L o o k Beat Frequency ( M L B F ) will be reviewed i n the next section. T h e principle of these algorithms is to generate two independent range looks to emulate two S A R systems imaging the same region of the earth's surface. E a c h system works at a slightly different frequency given by: h  = fo ~ ^  (3.17)  h  = fo + ^  (3.18)  where fo is the centre frequency of the real S A R system and A / is the separation of the centre frequencies of the two emulated S A R systems.  32  T h e operation of the M L C C algorithm is illustrated by considering two range compressed looks of a single point target. F r o m E q u a t i o n (2.32), ignoring the target's complex amplitude and range envelope, the range compressed signals of the two looks are given by:  Si{r])  = W(T] - r] ) e x p { - j 2 7 r / i T } c  =  S (rj) 2  D  W( -ri ) n  exp{-j  e  4 n f l R { r , )  }  (3.19)  }  (3.20)  = W(r] - rj ) exp{-j27r f r } c  =  2  W(r,-rj )  exp{-j  c  4  7  d  V  f  2  m  T h e phase arguments i n Equations (3.19) and (3.20) give the azimuth phase history of the target, which are different between looks 1 and 2 because of the frequency separation A / . B y expanding the instantaneous slant range R(r}) i n Taylor series at rj = 770, as shown i n Equation (2.4), and ignoring the constant phase terms, the two range look signals can be expressed as:  Si(v)  = W(ri-r} )  5-2(77) = W(rj  where K  ai  and K  a2  exp{-J7r/W}  c  - 77 ) e x p { - J 7 r K C  rj } 2  a2  (3.21)  (3.22)  are the azimuth F M rates of the two looks given respectively by:  =  2V|/, c RQ  2V? ,.-A//2) cRo  =  U  33  y  a  ggA  =  »V?(/.  =  C/lo  A//2)  +  C/to  T h e difference between the a z i m u t h F M rates is given by: 2V Af  Af  2  ,  K«I - Kai = —V^- = ^ - r  (- )  A  where A "  a 0  3  25  is the a z i m u t h F M rate for the a z i m u t h signal centered at / : 0  Kao =  =  cRo  \ (Kai + K ) 2 a2  (3.26)  T h e A C C C s of look 1 and look 2 are given by:  &  = J2 SM  S;(r) + An)  (3.27)  C  = ^  S* (n + An)  (3.28)  2  S() 2  V  2  v  F r o m E q u a t i o n (3.6), the A C C C angles of look 1 and look 2 are given by:  4>LI =  to  argld] =  HZL^J?£  = arg[C ] =  (3.29)  (3.30)  2  F r o m E q u a t i o n (3.13) and (3.25), the difference of the A C C C angles of the two range looks is given by:  A(f> =  <t>L2 - (j>Ll 2n(K a2  34  K i)n a  e  —  Z7T  _  =  Af  K r]c  fo  F  2  7  a0  r  a  ^ % ^ J0 K  (3.31)  Since Acj) is much less than a radian, angle wrap around is not a problem and thus the absolute D O P C E N at centre frequency / _ F d c  is given by:  0  f F A<j) 0  a  '° ~ ~ ~ 2 T A T  (  }  In practice, the value of Fd ,o determined by E q u a t i o n (3.32) may not be accurate c  enough. T o improve its accuracy, the fractional P R F part F  dca  is determined directly  from the A C C C angle [14], and the estimation of E q u a t i o n (3.32) is used only to obtain the Doppler ambiguity m. F r o m E q u a t i o n (3.29), (3.30) and (3.26), the fractional P R F part is obtained from the average (aliased) phase increments by:  F d c a  _  F  ~  2ir  =  a  (fa  + <b ) L2  2  -K <3 Vc  (3.33)  a  This method is similar to that proposed by Madsen [22], except that the sign approximation is not made here. T h e error tolerance i n the two A C C C angles is relatively robust, as an error as high as 5° i n the A C C C angles causes an error of only 0.0014 F  a  in F  dca  [14]. The Doppler ambiguity is then estimated by:  o u n d ( *°>°- *»\ \ "a F  m  =  r  35  (3.34)  F  '  Finally, the absolute D O P C E N is then obtained by:  Fdc = F  + mF  dca  (3.35)  a  T h e accuracy of the M L C C method depends on the range look bandwidth and the look separation. T h e optimal separation of the looks is found to be Af and the o p t i m a l look bandwidth is W /3, s  where W  s  =  2W /3 s  be the range bandwidth of the  signal [14].  3.3.1  Illustration of M L C C with a Single Point Target  To illustrate the operation of M L C C algorithm, the same point target which is used in the D L R algorithm simulation is used. T h e range b a n d w i d t h is 17 MHz , thus the separation of the two range looks is set to 11 MHz  and the bandwidth of each look is  set to 5.6 MHz . Figure 3.4 shows the range spectrum of a compressed range line. After range look extraction, I F F T s are performed on each look to transfer the signal back to the range time domain. Figure 3.5 shows the compressed pulses of the two range looks. A C C C angles of each look are computed and the D O P C E N is obtained. In this simulation, the estimation of the fractional  P R F part of the D O P C E N is -401 Hz and the integer part is -425 Hz .  Since the principle of the M L C C and D L R algorithms is same, it is easy to predict that their estimation properties w i l l be almost the same.  36  The Spectrum of a Range Line, DOPCEN = -400 Hz  -10  -5  0 Range Frequency —> (MHz)  5  Figure 3.4: T h e Spectrum of a Range Line  300  (a) Compressed Pulse of Look 1, DOPCEN = -400 Hz  100  150  Range Time (sample) 300  (b) Compressed Pulse of Look 2, DOPCEN = -400 Hz  100  150  Range Time (sample) Figure 3.5: Compressed Pulses of the T w o Range Looks  37  3.4  The M L B F Algorithm  In the second algorithm, called M u l t i l o o k Beat Frequency ( M L B F ) , the two range looks are m u l t i p l i e d together to generate a beat signal. T h e beat frequency is then estimated and the Doppler ambiguity is determined from the beat frequency. T h e operation of M L B F algorithm can also be understood by examining what happens to a single point target. F r o m E q u a t i o n (3.21) and (3.22), the resultant beat signal Sb(rj) for a point target is given by:  S (rj) = b  =  Because K \ a  and K  a2  StWSfa) \W(ri - r? ) | exp{JTv(K 2  c  a2  - K) al  rf}  (3.36)  are quite close to one another and the Doppler bandwidth  is l i m i t e d , the frequencies of the beat signal are confined to a narrow bandwidth. Thus, a distinct beat frequency is discernible and the average beat frequency /& is given by:  fb  =  -  =  {Ka.2 ~  K  a0  Kal)Vc  — rj Jo  c  ~ ^ F  d  c  f  i  JO  (3.37)  T h e beat frequency is estimated by taking the F F T of the beat signal, often finding the peak only to the nearest cell. T h e absolute D O P C E N is then estimated by:  Fdc,o =  — - ^ y fb  38  (3.38)  3.4.1  Illustration of M L B F with a Single Point Target  T h e operation of M L B F algorithm also can be illustrated by the single point target simulation.  T h e two range looks generated i n the M L C C algorithm can be used to  generate the beat signal according to Equation (3.36). T h e n F F T s are performed on the beat signal to estimate the average beat frequency. Figure 3.6 shows the spectrum of the beat signal. Since the envelope of the beat signal is approximately symmetric w i t h respect to rj — r? , the peak of the spectrum c  can be identified. In the plot, the peak is located at the 3rd range frequency sample, corresponding to a beat frequency of 0.9 Hz. F r o m E q u a t i o n (3.38), the integer part of the D O P C E N is estimated to be -457 Hz .  39  Spectrum of Azimuth Beat Signal with true D O P C E N  n  -400  1  1  1  = -400 Hz  1  -300 -200 -100 0 100 Beat Frequency (Hz)  r  200 >  300  400  E x p a n d s c a l e to find location of peak  A I I  I  1  Peak location = 3 cells f = 0.9 Hz  0.8  T3  F  1,0.6 CO  b dC,0=-  4  57 Hz  Ambiguity =  0  ).09 degrees Squint = C  10.4 ta E  |  0.2 0  -15  -10  -5 0 Beat Frequency (Hz)  5  >  10  15  Figure 3.6: Spectrum of the M L B F beat signal w i t h a single point target  40  3.5  Discussion  T h e operation of the D L R , M L C C and M L B F algorithms have been introduced i n this chapter.  A l l of these algorithms are based on the principle that the D O P C E N is ap-  proximately a linear function of the range frequency. Single point target simulations are performed to illustrate the operations of these algorithms. F r o m the simulations, the estimates are almost the same since they are based on the same principle. However, the D L R and the M L C C algorithm use the A C C C angle to obtain the fractional part of the D O P C E N and the ambiguity, whereas the M L B F algorithm uses the beat frequency. T h i s makes the D L R and the M L C C algorithm more sensitive than the M L B F algorithm to azimuth partial exposures. W h e n a target is not fully exposed i n the azimuth beam or it is partially covered by the azimuth beam, we say it is partially  exposed. Figure 3.7 illustrates the effect of  azimuth partial exposure on the A C C C angles. T h e D O P C E N we used here is 0 Hz . In Figure 3.7, the azimuth partial exposure is emulated by taking away some A C C C vectors at the end of the exposure.  T h e A C C C angle is biased significantly.  Figure 3.8 shows the error i n the A C C C angle introduced by the a z i m u t h partial coverage. W h e n 3/4 of the A C C C vectors are taken away, the error is about —496 Hz , which is about 30% of the P R F . T h e M L B F algorithm uses the beat frequency. T h e change of the beat frequency along the azimuth exposure is very small. For example, using the same parameters in the above experiment, the average beat frequency at the first 1/4 of the a z i m u t h exposure is about —0.7 Hz , leading to an estimate error of about 324 Hz , which is about 20% of  41  A I I I I  n  co  Percent used = 100 2.5 Est: Dopcen ;= 0 Hz 2  Percent used = 88 : 2.5 Est Dopcen = -25 Hz 2  1.5  1.5  1  1  CL 0.5 DJ C O  E  A I I I I  •c C O a. DJ  co  E  ii ii  A  •E co  E  0.5  Wll//// r%  0  0 -0.5  -0.5 Percent used = 76 : 2.5 Est: Dopcen. = -81 z H 2  Percent used = 64 : 2.5 Est.; Dopcen '= -163 Hz  1.5  1.5  1  1  0.5  0.5  0  0  -0.5  -0.5  2  Percent used = 52 : 2.5 Est: Dopcen ;= -263 Hz 2  Percent used 40 : 2.5 Est: Dopcen = -383 Hz  1.5  1.5  0  :  2  /  1  CL 0.5 DJ C O  :  1 0.5  j||§§|&  0 -0.5  -0.5  -1  A I I I I  co Q. DJ C O  E  Percent used = 28 : 2.5 Est: Dopcen - -496 rHz 2  Percent used = 16 : 2.5 Est: Dopcen ;= -606 h 2  1.5  1.5  1 0.5  ;  1  \ ill  0.5  0  0  -0.5  -0.5  ;  \ i  - 1 0 Real part  0 1 Real part -  1  >  2  Figure 3.7: Illustration of an A C C C C a l c u l a t i o n w i t h a z i m u t h p a r t i a l exposure when PRF  = 1600 Hz .  42  Effect of partial azimuth exposures on A C C C angles  10  20  30  40 50 60 70 80 Percent of target exposure captured  90  100  Figure 3.8: Error i n A C C C Calculation w i t h azimuth partial exposure  43  the P R F . T h e average beat frequency at the last 1/4 of the a z i m u t h exposure is about 0.7 Hz , leading to an estimate error of about —324 Hz , which is also about 20% of the P R F . T h e change of the beat frequency is 1.4 Hz . B o t h estimates can give the correct ambiguity number. Note that the beat frequency tolerance for ± 0 . 5 P R F is about ± 0 . 9 Hz . Thus, the M L B F algorithm is less sensitive to partial exposure than the D L R and the M L C C algorithm, which are based on the A C C C angle. T h i s conclusion will be used to explain the effect of the radiometric discontinuity on these algorithms i n Chapter 6. In the next chapter, we w i l l see that the A C C C angle gives the most accurate estimate of the fractional P R F part of the D O P C E N . T h e D L R and the M L C C algorithms use the A C C C angle, whereas the M L B F algorithm does not.For this reason, the M L B F algorithm should not be used to estimate the fractional P R F part.  44  Chapter 4 Experiments in the Low-Squint Case In this chapter, experiments w i t h simulated data and real S A R data are performed to compare the performance of the D L R , M L C C and M L B F algorithms when the antenna squint angle is relatively low. First, the method of generating the simulated data w i t h m u l t i p l e point targets is outlined. Simulation results are presented, and the effects of scene contrast and noise on the algorithm performance are then discussed. Finally, experiments are done w i t h low-squint S A R data to illustrate and compare the performance of the algorithms. E R S data is used, as it has low squints when operated in yaw-steering mode. In most yaw-steering scenes, the E R S D O P C E N is close to -400 Hz .  45  4.1  Generation of Multiple-Target Simulated Data  4.1.1  Methodology  To compare the performance  of the three algorithms, simulated data are generated.  Considering the computer memory and computing efficiency, we generate a IK  x  IK  data block, w i t h the same length of 1024 samples i n the range and azimuth directions. To obtain the most realistic simulation, each image sample w i l l have an independent target, whose amplitude and phase we can specify. T h e n a 2D convolution w i l l be used to generate the received signal from these targets, w i t h particular attention paid to obtaining a symmetrical set of partially exposed targets i n the final simulation array. Figure 4.1 illustrates the simulation methodology.  1024+2nr-2  1024+2nr-2  c  •  CONV  0  nr-1  CM I  CO  c +  0  Tf  CM  CC  c +  CM Tf  o  CM O  W  ^  1024 Tf  CM O  •  CM  1024+2na  i  1024+nr-1  na-1  024+2na-2  1024+2nr-2 nr  0 Single Point Target  Reflectivity Matrix  Simulated Data  Figure 4.1: Convolution to Generate the Simulated D a t a of M u l t i p l e Point Targets  46  F i r s t , the same single point target is generated as used i n the last chapter. Therefore, the specified D O P C E N is -400 Hz, corresponding to a squint angle of about 0.1°, which is a low squint case. T h e length of a single point target i n the range direction is denoted as nr samples and i n the azimuth direction is denoted as na. T h e received data used to form a S A R image is the superposition of signals from a distribution of a single point target [24]. A m a t r i x of reflectivity  coefficients is generated to describe the  distribution of scatterers on the ground surface and to emulate the ground reflection of the microwave energy. T h e amplitude of the reflectivity coefficients can be controlled to emulate the ground w i t h selectable levels of contrast or radiometric  discontinuities.  The  phase of the reflectivity coefficients is generated randomly w i t h a normal distribution. T h e size of the reflectivity m a t r i x is 1024 + nr — 1 samples i n the range direction and 1024 -f na — 1 samples in the azimuth direction, as shown i n Figure 4.1. T h e simulated data is generated by convolving the expanded single point target w i t h the reflectivity m a t r i x . For computational efficiency, a 2D fast convolution is used. T h e single point target and the reflectivity m a t r i x are zero-padded to the size of 1024 + 2nr — 2 by 1024 + 2na — 2 in the range direction and the a z i m u t h direction respectively. 2D F F T s are performed on the zero-padded point target and the reflectivity m a t r i x . T h e product of the single point target and the m a t r i x is then transformed back to the range and azimuth time domain w i t h a 2D I F F T . This method has an obvious physical meaning.  It assumes that i n each range  and azimuth cell, there is a single effective point target. T h e properties of the ground surface can be described by the elements of the reflectivity m a t r i x . T h e amplitude of each coefficient describes the amplitude of the ground reflectivity and the phase of each coefficient is (47T R/X) od2n, m  where R is the range to the effective scattering center of  47  each pixel and A is the radar wavelength. B y controlling the amplitude of the reflectivity coefficients, we can generate a simulated data block w i t h different scene contrast and radiometric discontinuities. To simulate the partial coverage of targets at the edge of the radar beam, only a square at the center of the product m a t r i x w i t h a size of 1024 x 1024 is selected as the simulated data.  Since the zero-padding is enough, no wrap-around errors occur i n the  fast convolution. T h e partial coverage of the antenna beam at each edge is simulated by throwing away both sides of the convolution result i n the a z i m u t h direction w i t h a size of na — 1 azimuth samples. T h i s ensures that all possible partial exposures are included, from 1 azimuth sample to na — 1 azimuth samples, as shown i n Figure 4.2. In Figure 4.2, Target 1 only has its last sample covered, Target 4 only has its first sample covered, Target 2 only has its first sample excluded and Target 3 only has its last sample excluded. In addition, partial target coverage is also included i n the range direction, although D O P C E N estimators usually work on data after range compression, i n which case partial range targets are only caused by uncorrected range cell migration.  4.1.2  Simulated Data with Noise  Simulated data w i t h different Signal to Noise Ratios ( S N R ) are required to compare the performance of the algorithms under different noise levels. To generate simulated data w i t h a specified S N R , a noise m a t r i x is generated. We assume the noise is complex, white, Gaussian noise. T h e power of the noise can be  48  CD c CO  A  CC  • Azimuth  Figure 4.2: P a r t i a l Coverage of Targets i n the Simulated D a t a Block controlled according to the specified S N R . T h e n the noise is added to the signal to form a simulated data block w i t h specified S N R . T h i s process is shown i n Figure 4.3.  4.1.3  Experimental Database  A database for multiple target simulations is generated.  F i r s t , 10 low contrast scenes  are generated using a constant amplitude but different random phase for each element (target) i n 10 reflectivity matrices. T h e n we create 10 higher-contrast scenes by setting the amplitude of every 50th data point to a higher value (constant w i t h i n each scene) from 10 to 100 w i t h a step of 10. T h e same phase noise is used i n each of the 10 scenes, so the effect of contrast can be seen. W h i l e these big targets have constant amplitude and spacing, the key to a  49  Data block  1024  Noise  Simulated data with noise  1024  +  1024 1024  1024  1024  Figure 4.3: Generate and A d d Noise to the Simulated D a t a correct simulation is to use random phases for each target. T h e contrast of each scene is measured by the commonly-used formula [22]: < I > , „ < I > 2  Contrast =  2  (4.1)  where / is the magnitude of each pixel and < . > is the expected value. We also specified 10 different noise levels and generated 10 noise matrices. B y adding them to the same low contrast scene, we have 10 low contrast scenes w i t h different SNRs.  50  4.2  Simulations w i t h L o w C o n t r a s t  Data  D L R Low-Squint, Low-Contrast Simulations F i r s t , the D L R algorithm was run on the 10 simulated data blocks with low contrast. Table 4.1 gives the simulation results. A s an example, the estimation on data block 2 is shown i n Figure 4.4 where the A C C C angles as a function of the range frequency is given, as well as a straight line giving the best linear fit to the A C C C angle. Note that only the A C C C angles w i t h i n the dotted box are used i n the estimation of the straight line, since the A C C C angles are noisy outside this high-energy portion of the range spectrum. The fractional part of the D O P C E N is estimated by the average height of the fitted line (the value at zero range frequency). T h e Doppler ambiguity (in Hz) is estimated by projecting the slope of the fitted line to the radar center frequency. T h e ambiguity number is then found by dividing by the P R F and rounding to the nearest integer.  We see that the D L R algorithm works well on a l l 10 of these multi-target, lowcontrast scenes. It gives an estimate of the fractional part which is almost perfect. It gives ambiguity D O P C E N estimates which are w i t h i n a few 10's of Hz of the correct value of -400 Hz, well w i t h i n the limits needed to estimate the correct ambiguity number on a l l of the 10 data blocks. T h e average estimate of the ambiguity D O P C E N is -413.9 Hz, and the standard deviation is 24.3 Hz.  51  Data  Frac. part  Ambiguity  Error  Ambiguity  Block  Hz  Hz  Hz  Number  1  -400  -401  -1  0  2  -400  -438  -38  0  3  -400  -390  +10  0  4  -400  -444  -44  0  5  -399  -458  -58  0  6  -400  -399  +1  0  7  -400  -424  -24  0  8  -400  -385  +15  0  9  -400  -390  +10  0  10  -400  -411  -11  0  Mean  -400  -414  -14  0  0  24  24  0  St.Dev.  Table 4.1: A m b i g u i t y estimation results of D L R algorithm w i t h low scene contrast ( rect answer = -400 Hz)  52  Fit the ACCC Angle in A Line, DOPCEN =-400 Hz  Figure 4.4: A C C C Angles as a Function of Range Frequency ( D L R Algorithm)  53  M L C C Low-Squint, Low-Contrast Simulations The M L C C algorithm was then performed on the same data set. Table 4.2 gives the simulation results.  A s an example, the A C C C angles found on data block 2 are  shown i n the following two figures. Figure 4.5 shows the measured A C C C angles of range look 1 and range look 2 for data block 2. Figure 4.6(a) shows the sum of the A C C C angles of the two range looks, from which the fractional part of the D O P C E N is estimated by finding the mean A C C C angle. Figure 4.6(b) shows the difference of the A C C C angles of the two range looks, from which the Doppler ambiguity is found by estimating the slope of the data.  Ambiguity  Error  Ambiguity  Hz  Hz  Hz  Number  1  -401  -396  +4  0  2  -400  -421  -21  0  3  -400  -384  +16  0  4  -400  -443  -43  0  5  -400  -418  -18  0  6  -400  -445  -45  0  7  -400  -430  -30  0  8  -399  -391  +9  0  9  -400  -402  -2  0  10  -400  -389  +11  0  Mean  -400  -412  -12  0  Data  Frac.  Block  St.Dev.  part  0  21  21  .  0  Table 4.2: A m b i g u i t y estimation results of M L C C algorithm w i t h low scene contrast (correct answer = -400 Hz)  We see from Table 4.2 that the M L C C algorithm also works well on all of the 54  10 scenes.  It gives near-perfect fractional D O P C E N estimates, and gives the correct  Doppler ambiguity on all of the 10 data blocks. T h e average estimate of the ambiguity D O P C E N is -412.0 Hz (a bias of-12.0 Hz), and the standard deviation is 21.3 Hz. The bias and standard deviation are a little lower than i n the D L R algorithm.  55  (a) ACCC Angles of Look 1, DOPCEN=-400 Hz T 1 1 1  Q O  I  I  I  20  <  40  I  I  I  I  1  1 1 1 180 200  1 160  I  60 80 100 120 140 Range Look Time (sample)  r  (b) ACCC Angles of Look 2, DOPCEN=-400 Hz  Range Look Time (sample) Figure 4.5: A C C C angles of the two range looks i n experiment #2 =5"  (a) Sum of Lookl and Look2, DOPCEN=-400 Hz  Range Look Time (sample) 1. < > j 0.6 F  C D  | §  I  1  (b) Difference of Lookl and Look2, DOPCEN=-400 Hz '  i 20  1  1  i  40  i  1  i  1  i  1  i  1  i  60 80 100 120 140 Range Look Time (sample)  r  i  160  1 1—I 180 200  Figure 4.6: S u m and difference of A C C C angles of the two range looks i n experiment #2  56  M L B F Low-Squint, Low-Contrast Simulations The simulation results of the M L B F algorithm on the same data set are given i n Table 4.3. A s an example, the estimation on a data block 1 is shown i n Figure 4.7. F r o m this figure, it is seen that there is no clear single beat frequency.  In this case, we fit  a quadratic curve to the data i n Figure 4.7 using the M A T L A B polyfit function, and find the peak of the parabola. T h e quadratic curve fitting is used whenever the peak beat spectrum energy is less than 1.4 of the average energy. B u t i n general, the M L B F algorithm w i l l be less reliable when a clear beat frequency is not observable.  DOPCEN  Ambiguity  Data  Beat Freq.  Block  Hz  KHz  No.  1  -38  +18.7  19  2  -32  + 15.5  16  3  -67  +32.4  34  4  +43  +21.0  22  5  +62  -30.1  -31  6  -23  +10.9  11  7  +7  -3.2  -3  8  -65  +31.5  33  9  +43  -21.0  -22  10  -38  +18.7  19  Mean  -11  +9.4  10  St.Dev.  47  21  22  Table 4.3: A m b i g u i t y estimation results of M L B F algorithm w i t h low scene contrast (the answers should be zero)  57  1.8  Spectrum of the Azimuth Beat Signal, DOPCEN = -400 Hz  1.7  Peak location = -41 Beat freq. = -38.4 Hz Fdc = 18718.8 Hz Ambiguity = 19  1.6 a>1.5 •o c 1.4 D5 CO  5 1.3 CD N  |  1.2  o z  1.1 1 0.9 0.8 -500  Azimuth Frequency —> (Sample)  500  Figure 4.7: E s t i m a t i o n of D O P C E N A m b i g u i t y by M L B F algorithm w i t h low-cont: scene (note the exaggerated vertical scale)  58  4.3  Simulations with Higher Contrast Data  A set of data w i t h different scene contrasts is used to compare the performance of the three candidate algorithms, w i t h only the magnitude of the large targets varied to change the contrast. T h e D L R , M L C C and M L B F algorithms are run on the same data sets. The results are summarized i n Tables 4.4 and 4.5. F r o m the tables, we can see that the D L R and M L C C algorithms work well for low contrast scenes, but begin to take on substantial biases when the scene contrast increases. O n the other hand, the M L B F algorithm works well w i t h high contrast scenes, but breaks down when the scene contrast is low. These properties have been observed previously [14], but have not been quantified i n this manner before . 1  F r o m Tables 4.4 and 4.5 we note that the M L C C algorithm gives better fractional D O P C E N estimates than the D L R algorithm, but the reverse is true for the ambiguity estimates.  1  N o t e t h a t the results o f these s i m u l a t i o n s are s o m e w h a t p e s s i m i s t i c , because  are u s u a l l y used in the p r o d u c t i o n processors.  59  4Kx4K  d a t a arrays  DLR Scene  Frac.  Part  MLCC Error  Frac. part  Error  Contr  Hz  Hz  Hz  Hz  1.1  -398.9  1.1  -399.3  0.7  1.2  -396.2  3.8  -398.2  1.8  1.3  -392.5  7.5  -396.7  3.3  1.6  -388.5  11.5  -395.6  4.4  2.0  -384.9  15.1  -394.6  5.4  2.4  -381.8  18.2  -393.8  6.2  2.9  -379.1  20.9  -393.0  7.0  3.5  -377.0  23.0  -392.4  7.6  4.1  -375.2  24.8  -391.7  8.3  4.8  -373.8  26.2  -391.1  8.9  Mean  -384.8  15.2  -394.6  5.4  8.9  8.9  2.8  2.8  St.Dev.  Table 4.4: Fractional P R F estimation results w i t h low squint and varying scene contrast (the answer should be -400)  60  DLR  MLBF  MLCC  DCen  Amb  Hz  Hz  No.  0  -12.2  5941  6  -39  0  -24.5  11931  12  -59  341  0  4.7  -2283  -2  0  173  573  1  0.0  0  0  375  0  160  560  1  0.9  -456  0  83  483  1  533  933  1  0.9  -456  0  2.9  218  618  1  583  983  1  0.9  -456  0  3.5  336  736  1  626  1026  1  0.9  -456  0  4.1  439  839  1  666  1066  1  0.9  -456  0  4.8  527  927  1  703  1103  1  0.9  -456  0  51  451  0.5  249  649  0.7  -2.7  1285  1.6  330  330  0.5  446  446  0.5  8.9  4315  4.2  Scene  DCen  Error  Amb  DCen  Error  Amb  Contr  Hz  Hz  No.  Hz  Hz  No.  1.1  -433  -33  0  -454  -54  1.2  -340  60  0  -439  1.3  -251  149  0  1.6  -41  359  2.0  -25  2.4  Mean St.Dev.  BF  Table 4.5: A m b i g u i t y estimation results w i t h low squint and varying scene contrast (the D O P C E N should be -400 and the ambiguity numbers should be zero)  Figure 4.8 shows the spectrum of the beat signal when the scene contrast is 4.8. The  beat frequency is easily identified i n this case and the M L B F algorithm gives a  correct estimate of the Doppler ambiguity, which is 0 i n this case. More interesting, however, is how the M L B F algorithm behaves w i t h the two scenes of contrast 1.3 and 1.6, which are the contrast values at which the M L B F algorithm changes from not working to working. In Figure 4.9, the contrast 1.3 case is shown, where it can be seen that the expected beat frequency is hidden by the m u l t i p l e beating of many  61  Spectrum of the Azimuth Beat Signal, D O P C E N = -400 Hz  Peak location = 0 Beat freq. = 0.0 Hz Fdc = -0.0 Hz Ambiguity = 0  ^ -500  0  l\  i 500  Azimuth Frequency —> (Sample)  Figure 4.8: E s t i m a t i o n of D O P C E N by M L B F A l g o r i t h m w i t h H i g h Scene Contrast of 4.8 targets of almost equal size. B u t when the contrast is raised to 1.6, the extra contrast between strong and weaker targets allows the expected beat frequency to be observed, and the correct ambiguity number is obtained.  62  Spectrum of the Beat Signal, Scene Contrast = 1.3  -500  Azimuth Frequency —> (Sample)  500  Figure 4.9: E s t i m a t i o n of D O P C E N by M L B F A l g o r i t h m w i t h Scene Contrast of 1.3 Spectrum of the Beat Signal, Scene Contrast = 1.6  -500  Azimuth Frequency —> (Sample)  500  Figure 4.10: E s t i m a t i o n of D O P C E N by M L B F A l g o r i t h m w i t h Scene Contrast of 1.6  63  4.3.1  Discussion on the Effect of Scene Contrast  F r o m the experiments, we can see the scene contrast has an important effect on the candidate algorithms. M L C C algorithms on  Figure 4.11 shows the estimation errors of the D L R and the  low contrast data blocks. T h e x-axis is the number of the data  block, each w i t h the same low contrast. T h e y-axis is the estimation error. Figure 4.12 gives the error of the M L B F algorithm on the same data set. Tables 4.1, 4.2 and 4.3 give the average error and standard deviation of the D L R , M L C C and the M L B F algorithms on the 10 low contrast data blocks processed i n Section 4.2.  We see that the D L R and M L C C performance is similar, but the  MLBF  algorithm performs very badly. W h e n the scene contrast becomes h i g h , the performance of the three algorithms changes. Figure 4.13 shows the estimation errors of the D L R and M L C C algorithms as the scene contrast changes from low to high. T h e x-axis is the scene contrast. The y-axis is the estimation error. Figure 4.14 gives the error of the M L B F algorithm on the same data set. Note that only when the estimation error is w i t h i n the two dashed lines i n Figures 4.11 - 4.14 is the estimation of the Doppler ambiguity correct. T h e limits of the allowed error are [—F /2, F /2]. a  a  In the present simulation, the limits are [—480,480] Hz.  The scene contrast of the low contrast data set is equal to 1. T h e D L R algorithm and  the M L C C algorithm work well on this set of data.  T h e M L B F algorithm does  not work on the low contrast data set. W h e n the scene contrast increases, the M L B F algorithm begins to work well while the error of the D L R algorithm and the M L C C  64  algorithm becomes large, leading to an incorrect estimation of the Doppler ambiguity. T h i s can be explained by considering the A C C C when m u l t i p l e point targets are involved. In Chapter 3, we proved that the average increment of the azimuth frequency could be estimated by the A C C C angle i n the case that only single point target is involved.  W h e n multiple targets are involved, the correlation between these targets  will affect the A C C C angle, thus affect the estimation of the increment of the azimuth frequency.  65  Estimation Errors on the Data Set with Low Contrast  No. of Data Block  Figure 4.11: A m b i g u i t y estimation error of D L R and M L C C on low contrast data 401  Estimation Errors on the Data Set with Low Contrast 1  1  1  1  No. of Data Block  Figure 4.12: A m b i g u i t y estimation error of the M L B F algorithm on low contrast data (note the vastly different vertical scale compared w i t h Figure 4.11 — the estimate must be within the dotted lines of ± P R F to get the correct ambiguity number)  66  Estimation Errors on the Data Set with High Contrast  2.5 3 3.5 Scene Contrast  1.5  4.5  Figure 4.13: A m b i g u i t y estimation error of D L R and M L C C algorithms on scenes of  increasing contrast Estimation Errors on the Data Set with High Contrast 14000 12000  x  10000  "N"  A 8000 i o  ilj 6000 c g | 4000 o> HI  2000  -2000  1.5  2.5 3 3.5 Scene Contrast  4.5  Figure 4.14: A m b i g u i t y estimation error of M L B F algorithm on scenes of increasing contrast with quadratic curve fitting (note the different vertical scale compared with Figure 4.13 — the estimate must be within the dotted lines of ± P R F to get the correct ambiguity number i n each case)  67  Figure 4.15: Overlap of T w o Targets  Consider two point targets P and Q which overlap i n the S A R signal domain i n the a z i m u t h direction, as shown i n Figure 4.15 [14]. rjp is the a z i m u t h time for closest approach for target P. T]Q is the a z i m u t h time for closest approach for target Q . e is the separation between target P and target Q . We assume target P and target Q in the same range cell and they have the same power, thus their envelopes have the same magnitude. T h e a z i m u t h signal is given by:  S(V)  = SP( ) + Sgirj)  (4.2)  V  where Sp(n) and SQ(TI) are given by:  S (n)  = PW(n-  S()  = Q W( - )exp{-j7r  P  Q  V  V  r, ) exp{-J7TK (r, c  a  Vc  - T^) }  K (n a  2  )} 2  VQ  (4.3)  (4.4)  where P is the complex envelope of target P, Q is the complex envelope of target Q and W(rj — rjc) is the a z i m u t h antenna profile. F r o m E q u a t i o n (3.4), the A C C C is given by:  68  + Q(V)] [S (v  + ATy) + S (  =  £[SPW  =  Cpp* + CQQ- + CPQ* + Cgp*  S  P  Q  V  +  An)}  (4.5)  The first two items, Cpp* and CQQ*, are the A C C C s of target P and target Q respectively:  arg{Cpp.} = arg{C Q.} = Q  2nK rf a  c  (4.6)  F  a  They have the same phase angle, which is the desired one. T h e other two terms, CpQ'  and CQP>, are due to the overlap of the exposures of target P and target Q . If  there is no overlap between target P and target Q , these two items w i l l be zero and the phase angle of the A C C C of S(rj) w i l l be the desired one. However, overlap does happen in the real case. Consider the third term, CPQ>. To generalize the analysis, suppose target Q is fixed and target P moves from left to right. One important parameter is the separation e between target P and target Q . If the origin of the azimuth time axis is set to r/Q, the third term i n Equation (4.5) is given by:  C . PQ  =  PQ*Y W(n-n -e)W(n-n )ex {J7:K (2n  =  PQ*£M(n,e)exp{j4>(77,£)}  t  where  c  c  P  + Ar -e)(AT  a  I  + e)} (4.7)  Af (n, e) = W(r) — rj — e) W(rj — rj ) c  69  1  c  (4.8)  (f)(rj, e) = TT A' (2r? + Ar? - e) (Ar? + e)  and  (4.9)  a  Note that M(r?,e) and <f>(rj,e) are both symmetrical about r) = r? + e/2. s  c  Assume r?; is the length of the azimuth aperture, we have:  -m  < £ < Vi  (4-10)  and e < 0 :  r? - r? /2  e>0:  r? + e - r ? ( / 2 < r? < r? + r?//2  c  ;  < r? < r? + e + r?,/2 c  c  c  After a mathematical derivation, the phase angle of Cpq* is given by:  arg{C  P Q  . } = </>(T?„£) = 7rAT (2r? + A 7 ? - £ ) ( A r ? + £) a  c  T h e amplitude of CPQ* is shown i n Figure 4.16 as a function of e. angle of CPQ> is shown i n Figure 4.17 also as a function of s.  70  (4.11)  T h e phase  Magnitude of the ACCC of Target P and Target Q  -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 Separation between target P and target Q —> (sec.) Figure 4.16: A m p l i t u d e of CPQ> as a F u n c t i o n of e  Phase of the ACCC of Target P and Target Q  2000  -0.8  -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Separation between target P and target Q —> (sec.)  Figure 4.17: Phase of Cpg« as a F u n c t i o n of e  71  F r o m Figure 4.16 and Figure 4.17, we can see that the amplitude of the A C C C s between target P and target Q is symmetric about e = —Ar/. T h e phase of the A C C C s is an odd function of s.  T h e A C C C is complex conjugate about e.  Thus the sum  of all A C C C s between target Q and all its neighbor targets i n the same range cell is approximately a real number. In a ideal case, the reflection of all point target has the same power and random phase.  In this case, for each point target, we have a sum of  A C C C s between it and all its neighbor targets i n the same range cell and all the sums have the same amplitude and random phase. T h e mean of all A C C C s due to overlap is zero. Thus the A C C C angle should be unbiased i n the ideal case, which has the lowest scene contrast of 1. T h i s is why the D L R algorithm and M L C C algorithm work well on low contrast scenes. W h e n the scene contrast is higher than 1, i n which case the reflectivity of targets in the same range cell are likely to have different powers, the mean of all A C C C s due to overlap may not be zero, leading to a random bias i n the estimation of the phase increment at each range cell. T h i s random bias w i l l affect the estimation of the slope, which is a important parameter i n the D L R and M L C C algorithms, leading to a estimation error i n the D O P C E N . T h i s is i n fact because of insufficient averaging of the A C C C s due to overlap since the strong targets have dominant effect over weak targets.  The  higher the scene contrast becomes, the worse the insufficient average is. T h i s is why the estimation error of the D L R and the M L C C algorithms becomes large when the scene contrast becomes high. Reference [14] explains why the M L B F does not work on low contrast scenes and why it works well on high contrast scenes i n details. In summary, when two targets are present, there are three peak frequency components i n the F F T of the beat signal, one  72  corresponding to the Doppler centroid, and two due to the cross beating between the two targets. W h e n more than two significant targets are present i n the same range cell, the distortion of the peak beat frequency gets worse. A s the number of dominant targets increase, the power due to the cross beating can eventually mask out the required beat frequency, as we observed i n Figure 4.7. Thus, it can be concluded that, the D L R and the M L C C algorithm work well on low contrast scenes, while the M L B F algorithm works well on high contrast scene. Since the D L R and the M L C C algorithms are based on similar principles, the performance of these two algorithms on low contrast scenes is almost the same. F r o m Figure 4.13, the M L C C algorithm is more sensitive to the scene contrast than the D L R algorithm. This effect is explained i n Chapter 6.  73  Simulations with Noise  4.4  As the D L R and M L C C algorithms behave almost the same in terms of scene contrast, it is interesting to also compare them i n terms of their response to scene noise. To perform this comparison, we generate a set of simulated data w i t h different noise levels. The signal to noise ratio is defined as: SNR  =  10 log ^  (4.12)  where P is the power of the signal and N is the power of the noise. T h e simulation results are shown in Table 4.6. T h e first row indicates the different noise levels in units of dB. T h e second row is the estimation results of the D L R algorithm and the third row is the estimation results of the M L C C algorithm. Figure 4.18 shows the trend of the estimation error of the D L R and M L C C algorithms as the noise level decreases. 5  4  2  3  1  SNR  (dB)  30  20  10  DLR  (Hz)  -401.2  -397.2  -384.0  -383.1  -369.9  -369.5  -369.1  -363.7  MLCC  (Hz)  -396.5  -393.9  -379.6  -377.1  -375.4  -386.3  -381.3  -358.0  Table 4.6: Simulation Results of D L R and M L C C w i t h Different S N R s  74  Performance of DLR and M L C C under Noise  45  0  5  10  15  20  25  30  SNR —> (dB) Figure 4.18: Comparison of ambiguity estimation errors of the D L R and M L C C al] rithms w i t h increasing scene S N R (low contrast scenes)  75  4.4.1  Discussion of the Effect of Noise  We see from Figure 4.18 that the estimation error decreases when the S N R increases. Note that even when the S N R is low, the D L R and the M L C C algorithms still work reliably, as the ambiguity estimation errors are well below P R F / 2 . Thus we can conclude that the D L R algorithm and the M L C C algorithm are robust to white noise on low contrast scenes. T h e estimation errors of the D L R and the M L C C algorithms on same noise level are almost the same. Consider the spectrum of a range chirp, as sketched i n Figure 4.19. T h e dashed line is the power level of the white noise. From the figure, it can be seen that the ratio of the signal power to the noise power either in the whole range bandwidth or i n the bandwidths of look 1 and look 2 are the same under the assumption that the noise is white. Thus the performance of the D L R algorithm and the M L C C algorithm i n white noise is essentially the same.  A  Look 1  Look 2  I L  J  J-  i  !  :  Noise Level  A  1  :  i  Range Frequency  Figure 4.19: Illustrating the distribution of noise and signal as seen by the D L R and M L C C algorithms  76  4.5  4.5.1  Experiments on S A R Data  Experiments on a Low Contrast Scene  Real S A R data is used to test the performance of the three candidate algorithms. A n ERS-1 Bathurst Island scene in yaw-steering mode is used to test the algorithms under low-squint conditions.  T h e acquisition data is Oct.  25, 1995 and the orbit/track is  22365/1539. Figure 4.20 shows the detected image of this scene. T h e left part is snow and rock and the right part is sea ice. It is noticed that the scene contrast is fairly low. T h e absolute D O P C E N in the middle of the scene is -159 Hz as obtained by M D A ' s d t S A R processor. T h i s reference D O P C E N is used i n the processing of the raw data. It is a good estimate since the detected image has a high quality. T h e horizontal direction is the range direction w i t h near range on the left. T h e azimuth direction is vertical, w i t h azimuth time increasing i n the upward direction. We extracted 6 consecutive data blocks along the azimuth direction. Each data block is 4096 x 4096 samples i n the range and azimuth directions. T h e overlap of each block is 1696 azimuth samples. Table 4.7 gives the fractional P R F estimates of the D L R and M L C C algorithms. Table 4.8 summarizes the ambiguity estimation results of the D L R , M L C C and M L B F algorithms. Note, i n the real case, there is a frequency offset ( F O S ) i n the estimation of the D L R algorithm and the M L C C algorithm. T h e F O S does not exist in the estimation of the M L B F algorithm [14]. T h i s effect is probably caused by a frequency dependency of the beam squint angle [20]. T h e frequency offset i n the estimation on E R S - 1 yaw-steering  77  mode data is about 1400Hz [25]. In the table, the F O S value has been subtracted from the estimation of the D L R and the M L C C algorithms.  DLR  MLCC  Block  Scene  Fract.  Error  Fract.  Error  No.  Contr  Hz  Hz  Hz  Hz  5  1.144  -162  -3  -158  1  4  1.145  -151  8  -150  9  1  1.145  -147  12  -152  7  3  1.146  -147  12  -148  11  2  1.153  -158  1  -155  4  6  1.164  -149  10  -148  11  Mean  1.150  -152  7  -152  7  St.Dev.  0.008  6  6  4  4  Table 4.7: Fractional P R F estimation results on the low-squint low-contrast Bathurst Island E R S scene (the D O P C E N should be -159 Hz )  78  Figure 4.20: T h e detected image of the E R S - 1 Bathurst Island scene  79  DLR  MLCC  MLBF  Block  Scene  DCen  Err  Am  DCen  Err  Am  DCen  Err  Am  No.  Contr  Hz  Hz  No.  Hz  Hz  No.  Hz  Hz  No.  5  1.144  -232  -73  0  -315  -156  0  -403  -244  0  4  1.145  280  439  0  206  365  0  -403  -244  0  1  1.145  -210  -51  0  -119  40  0  -403  -244  0  3  1.146  2  161  0  -41  118  0  -403  -244  0  2  1.153  85  244  0  28  187  0  -201  -42  0  6  1.164  202  361  0  139  298  0  -403  -244  0  Mean  1.150  21  180  0  -17  142  0  -369  -210  0  St Dev  0.007  192  192  0  171  171  0  82  75  0  Table 4.8: A m b i g u i t y estimation results on the low-squint low-contrast Bathurst Island E R S scene w i t h F = 1400 Hz (the D O P C E N should be -159 Hz and the ambiguity numbers should be zero) os  A l l three algorithms give the correct estimate of the Doppler ambiguity on the 6 data blocks. Note that the M L C C algorithm has a lower mean error, and a lower standard deviation than the D L R algorithm. Though the scene contrast is fairly low, the M L B F algorithm still works reliably. Thus, the M L B F algorithm is reasonably robust to scene contrast i n real S A R data. Figure 4.21 shows the estimation error of the three algorithms i n different scene contrasts. Note that the scene contrast is computed on the detected image and is different form the contrast metric used i n the simulation, which is computed from the reflectivity coefficients.  T h e estimation error increases when the scene contrast increases, as we  have seen i n the simulations, but the range of contrast is too narrow to draw definite  80  conclusions.  Figure 4.21: E s t i m a t i o n Error on the Bathurst Island Scene  81  4.5.2  Experiments on a High Contrast Scene  A n E R S - 1 N o r t h Cascade Glacier scene in yaw-steering mode, which has a high scene contrast, is used to test the algorithms. The acquisition data is Sep. 23, 1995 and the orbit/track is 21916/385. Figure 4.22 shows the detected image of this scene. This scene mainly consists of mountains, including forest, snow and glaciers. The absolute D O P C E N i n the middle of the scene is 485 Hz as obtained by M D A ' s d t S A R processor.  This value is the best independent indication we have to check the  accuracy of our estimates. However, as we do not know how accurate the d t S A R estimate really is, the best indication of the accuracy of our algorithm is the standard deviation of the estimates from block to block. This is because the true D O P C E N changes very slowly w i t h azimuth. The horizontal direction is the range direction w i t h near range on the left. T h e azimuth direction is vertical, w i t h azimuth time increasing in the upward direction. We also extract 6 consecutive data blocks along the azimuth direction. Each data block is 4096 x 4096 samples i n the range and azimuth directions. The overlap of each block is 1696 azimuth samples. Table 4.9 gives the fractional P R F estimates of the D L R and M L C C algorithms. Table 4.10 summarizes the ambiguity estimation results of the D L R , M L C C and M L B F algorithms. In the table, the F O S value, which is about 1400 Hz, has been subtracted from the estimation of the D L R and the M L C C algorithms. T h e results of Table 4.10 suggests that the F  os  is wrong at least for this scene. If we set F  03  = 2477 Hz , the mean  error i n the estimate of the ambiguity is close to zero, and the D L R and the M L C C algorithm w i l l give the correct estimate for every block.  82  DLR  MLCC  Block  Scene  Fract.  Error  Fract.  Error  No.  Contr  Hz  Hz  Hz  Hz  4  1.333  467  -18  465  -20  5  1.332  462  -23  461  -24  1  1.328  470  -15  467  -18  3  1.309  469  -16  466  -19  6  1.284  463  -22  462  -23  2  1.256  466  -19  467  -18  Mean  1.307  466  -19  465  -20  St.Dev.  0.029  3  3  2  2  Table 4.9: Fractional P R F estimation results on the high-contrast North Cascade Glacier E R S scene (the D O P C E N should be 485 Hz according to d t S A R , but the true fractional D O P C E N is likely 466 Hz according to these estimates)  83  Figure 4.22: T h e detected image of the E R S - 1 N o r t h Cascade Glacier scene  84  DLR  MLBF  MLCC  Block  Scene  DCen  Err  Am  DCen  Err  Am  DCen  Err  Am  No.  Contr  Hz  Hz  No.  Hz  Hz  No.  Hz  Hz  No.  4  1.333  1837  1352  1  1295  810  0  201  -284  0  5  1.332  1601  1116  1  1845  1360  1  -201  -686  0  1  1.328  1897  1412  1  1363  878  1  -201  -686  0  3  1.309  1505  1020  1  1333  848  1  201  -284  0  6  1.284  1149  664  0  1890  1405  1  -201  -686  0  2  1.256  1269  381  0  1871  1386  1  -201  -686  0  Mean  1.307  1543  1058  1  1599  1145  1  -67  -552  0  St Dev  0.029  273  273  270  270  0.4  189  190  0  0.5  Table 4.10: A m b i g u i t y estimation results on the low-squint high-contrast North Cascade Glacier E R S scene (the D O P C E N should be 485 Hz and the ambiguity numbers should be zero)  O n l y the M L B F algorithm gives the correct estimate of the Doppler ambiguity on the 6 data blocks since for each data block, a clear dominant frequency was always observed i n the beat spectrum. A s an example, Figure 4.23 shows the beat spectrum of data block 1. F r o m Figure 4.23, we found the peak or dominant beat frequency, which in this case is 0.4 Hz . T h i s beat frequency estimate must be accurate to ± 1 . 6 Hz i n order for the correct ambiguity to be found. The D L R and M L C C algorithm do not work reliably on this high-contrast scene. Thus, the M L B F algorithm is reasonably robust to scene contrast i n real S A R dataFigure 4.24 shows the estimation error of the three algorithms i n different scene contrasts.  85  Spectrum of Azimuth Beat Signal 1.1 -  u  -2000  Peak location = 1 Beat freq. = 0.4 Hz Fdc = -201 Hz  i  -1500  i  -1000  i  i  i  -500 0 500 Beat Frequency (sample)  i  1000  i  1500  u  2000  Figure 4.23: T h e beat spectrum of data block 1 of the N o r t h Cascade Glacier E R S scene  86  Estimation Error on the North Cascade Glacier Scene 2000 1500h 1000  500 h  -500 h -1000 1.25  1.3  Scene Contrast  Figure 4.24: Estimation Error on the N o r t h Cascade Glacier Scene  87  1.35  4.6  Summary  Since the M L C C algorithm works best with scenes of low, uniform contrast, while the M L B F algorithm works best w i t h scenes of high contrast, and since they all need the processing of range look extraction, these two algorithms can be efficiently combined together to form a reliable D O P C E N estimator over large ranges of scene contrast. The criteria which have been found useful for selecting the best of the M L B F and the M L C C results are "Normalized Correlation", "Consistency of estimate" and the "Scene Contrast" [14].  88  Chapter 5  Higher Squint Considerations T h e effect of a squint mode imaging geometry on S A R signal properties is quite complicated [26]. However, as far as the D O P C E N estimation is concerned, when the squint is low, the approximation using the Principle of Stationary Phase is accurate enough. W h e n the squint increases, the properties of the signal structure become more complicated as the cross-coupling between the range and a z i m u t h signals increases [27]. T h i s may effect the performance of the D L R and the M L C C algorithms, which use the A C C C angle, and the performance of the M L B F algorithm, which uses the beat frequency estimation. Since the computation of the spectrum of a chirp involves the calculation of Fresnel integrals, for which no analytical expressions can be obtained, we cannot obtain an analytical expression for the spectrum of a compressed chirp. Therefore, the properties of the A C C C angles and the beat frequency should be studied i n detail in the higher squint case using non-analytical means.  89  In this chapter, the effect of the squint on the A C C C angle and the effect of the squint on the beat spectrum is studied by simulations. T h e n simulations of the D L R , the M L C C and the M L B F algorithms on single point target and multiple point target are performed to examine the performance of these algorithms w i t h higher squint.  It  is found that these three algorithms still work accurately to obtain correct estimates. T h i s proves that the approximation to the spectrum of a compressed target using the Principle of Stationary Phase is still accurate enough w i t h high squint (see Chapter 3). Note that the simulations are ideal in the sense that we assume that the beam pointing angle does not vary with range frequency, i.e. , F  os  90  = 0.  5.1  Effect of Squint on the A C C C Angle  T h e D L R , the M L C C and the M L B F algorithms were found to work reliably and accurately w i t h E R S and J - E R S data, which normally have a low squint. W h e n the squint increases, the properties of the signal structure become more complicated as the crosscoupling between the range and azimuth signals increases [27]. M D A found that M L C C algorithm did not work reliably w i t h R A D A R S A T data, which generally has a higher squint. Since the standard beam of R A D A R S A T has a higher squint than that of E R S , we w i l l examine the effect of squint on the A C C C angle, which is the most important parameter that the M L C C and the D L R algorithms use. To examine the effect of squint on the A C C C angles, computer simulations are performed. T h e point target we use in the simulations is the same as used i n the last chapter. We can change the squint angle by specifying the D O P C E N when the target is generated. Table 5.1 gives the corresponding squint angle of different D O P C E N values used in the experiments of this chapter. T h e radar parameters used are those adapted from the E R S and R A D A R S A T systems, and are given i n Table 3.1. Figures 5.1 - 5.6 show the A C C C angles as a function of the range frequency w i t h the 6 different squint values given i n Table 5.1. DOPCEN Squint  [KHz) (degree)  -2  -5  -10  -15  -20  -50  0.46  1.56  2.31  3.50  4.62  11.42  Table 5.1: T h e squint angles corresponding to different D O P C E N s  91  ACCC Angles vs. Range Frequency with DOPCEN =-2 KHz  CO-0.2  T3  jg -0.4 O)  oc. O < O <  •0.6  p*-  *4  —  Q.  2  c  -1 -1.2 -10  ^5  0 Range Frequency -> (MHz)  5  Figure 5.1: A C C C Angles w i t h D O P C E N = - 2 K H z  ACCC Angles vs. Range Frequency with DOPCEN = -5 KHz 1  1  1  Range Frequency -> (MHz) Figure 5.2: A C C C Angles w i t h D O P C E N = - 5 K H z  92  IO  ACCC Angles vs. Range Frequency with DOPCEN = -10 KHz  -5  0 5 Range Frequency -> (MHz)  Figure 5.3: A C C C Angles w i t h D O P C E N = - l O K H z ACCC Angles vs. Range Frequency with DOPCEN = -15 KHz  -5  0 Range Frequency -> (MHz)  5  Figure 5.4: A C C C Angles w i t h D O P C E N = - 1 5 K H z  93  10  ACCC Angles vs. Range Frequency with DOPCEN = -20 KHz  | 0.8 c 0.6-  Range Frequency -> (MHz) Figure 5.5: A C C C Angles w i t h D O P C E N =  -20KHz  ACCC Angles vs. Range Frequency with DOPCEN = -50 KHz 0.21  1.2' -10  :  1  1  1  '  '  '  -5  0  5  1  10  Range Frequency -> (MHz) Figure 5.6: A C C C Angles w i t h D O P C E N = 94  -50KHz  F r o m the simulations, we can see that, with the increase of the squint, the linearity of the A C C C angles as a function of the range frequency remains unchanged.  Thus,  we can conclude that the approximation of the chirp spectrum using the Principle of Stationary Phase holds in the sense that the A C C C angle is a linear function of the range frequencies when the squint increases.  5.2  Effect of Squint on the B e a t S p e c t r u m  To examine the effect of squint on the beat spectrum, computer simulations are performed.  T h e point target we use in the simulations is the same as used in the last  chapter. T h e same D O P C E N s described in Table 5.1 are used i n the simulations. T h e radar parameters used are those adapted from the E R S and R A D A R S A T systems, and are given i n Table 3.1. Figures 5.7 - 5.12 show the beat spectra w i t h the 6 different squint values given i n Table 5.1. F r o m the simulations, we can see that there is always a dominant frequency in the beat spectrum even with a very high squint (DOPCEN  = —50 KHz ). We also  found that the beat frequency bandwidth increases w i t h the squint. T h i s is because of the short duration of the azimuth signal (or the beat signal) w i t h i n any one range cell. T h e beat frequency bandwidth is inversely proportional to the duration of the signal i n one range cell.  95  Spectrum of Azimuth Beat Signal with true DOPCEN = -2 KHz 1  i 0.8 CD TJ  1  1  1  1  1  r  1  \  CO  I 0.41 E |0.2  r  -400  j  -300  i  -200 -100 0 100 Beat Frequency (Hz)  i_  200 >  300  400  Expand scale to find location of peak i  I  i  -40  1  i  -30  1  i  -20  1  1  1  i  i  i  -10 0 10 Beat Frequency (Hz)  1  i  20 >  1  i  30  Figure 5.7: Beat spectrum w i t h D O P C E N = - 2 K H z  96  r  i  40  u  50  Spectrum of Azimuth Beat Signal with true DOPCEN = -5 KHz T  A I I  1  1  1  1  1  0  100  1  1  r  300  400  40  50  Ih  ' 0.8r  CD  -a fo-6r co 5  |0.4f To E |o.2  r  -400  -300  -200  -100  Beat Frequency (Hz)  200  >  Expand scale to find location of peak  -30  -20  -10  0  10  20  Beat Frequency (Hz)  >  30  Figure 5.8: Beat spectrum w i t h D O P C E N = - 5 K H z  97  Spectrum of Azimuth Beat Signal with true DOPCEN = -10 KHz  -400  -300  -200 -100 0 100 Beat Frequency (Hz)  200 >  300  400  Expand scale to find location of peak T  i  -20  1  i  -10  1  i  0  1  1  1  i  i  i  10 20 30 Beat Frequency (Hz)  1  i  >  40  1  i  50  Figure 5.9: Beat spectrum w i t h D O P C E N = - l O K H z  98  r  L_  60  S p e c t r u m of A z i m u t h B e a t S i g n a l with true D O P C E N T  A I I I I  1  1  1  1  1  1  = -15 K H z —1  1  0.8  CD T3  CO  |o.4r E -5 0.2[  -400  -300  -200 -100 0 100 B e a t F r e q u e n c y (Hz)  200 >  300  Figure 5.10: Beat spectrum w i t h D O P C E N = - 1 5 K H z  99  400  Spectrum of Azimuth Beat Signal with true DOPCEN = -20 KHz  -400  -300  -200 -100 0 100 Beat Frequency (Hz)  200 >  300  400  1  r  Expand scale to find location of peak  i  1  0  10  j  i  1  i  20  '.  1  1  1  i  i  i  30 40 50 Beat Frequency (Hz)  1  i  60 >  i  70  Figure 5.11: Beat spectrum w i t h D O P C E N = - 2 0 K H z  100  L_  80  Spectrum of Azimuth Beat Signal with true DOPCEN = -50 KHz  -400  -300  -200 -100 0 100 Beat Frequency (Hz)  200 >  300  400  Expand scale to find location of peak  60  70  80  90 100 110 Beat Frequency (Hz)  120 >  130  Figure 5.12: Beat spectrum w i t h D O P C E N = - 5 0 K H z  101  140  In the next two sections, we w i l l examine the accuracy of the slope of the A C C C angle vs. range frequency by performing the D L R and the M L C C algorithms on single point target and multiple point targets. T h e M L B F algorithm is also performed on single point targets and high contrast, multiple point targets data to examine the accuracy of the peak of the beat frequency under high squint..  5.3  Single Point Target Simulations  In this section, the D L R and the M L C C algorithms are performed on single point targets with the D O P C E N increasing from -2 KHz to -50 KHz to examine the accuracy of the slope that the D L R and the M L C C algorithm use.  T h e M L B F algorithm is also  performed to examine the accuracy of the peak of the beat frequency. T h e method to generate the single point is the same as described i n Chapter 4 except that the D O P C E N is specified corresponding to different squint shown i n Table 5.1. T h e P R F we use is still 960 Hz . T h e specified D O P C E N , the fractional P R F part of the D O P C E N and the ambiguity number are given i n Table 5.2. Table 5.3 gives the estimates of the fractional P R F part of the D O P C E N from the D L R and the M L C C algorithms. Table 5.4 gives the estimates of the ambiguity by the D L R , the M L C C and the M L B F algorithms.  102  DOPCEN KHz  Frac. part  Ambiguity  Hz  -2  -80  -2  -5  -200  -5  -10  -400  -10  -15  360  -16  -20  160  -21  -50  -80  -52  Table 5.2: T h e specified D O P C E N i n simulations Using the correct values given in Table 5.2, the estimates of the fractional P R F part of the D O P C E N given by the D L R and the M L C C algorithms are very accurate (See Table 5.3).  Thus, we can conclude that the average of the A C C C angles is still  accurate when the squint is high. F r o m Table 5.4, the estimates of the ambiguity given by the D L R and the M L C C algorithms are still accurate even when the squint is high. F r o m the single point target simulations, we can see that, when the squint is high, the slope of the A C C C angle is accurate enough for ambiguity estimation. Also the estimates of the ambiguity given by the M L B F algorithm is accurate even when the squint is high. Thus, we can conclude that the location of the peak frequency i n the beat spectrum is still accurate when the squint is high. In the next section, we use multiple point targets to demonstrate that the D L R , the M L C C and the M L B F algorithms work reliably when the squint is high.  103  DLR DOPCEN  Frac.  Error  Frac. part  Error  Hz  Hz  Hz  Hz  -2  -80  0  -80  0  -5  -200  0  -200  0  -10  -400  0  -400  0  -15  362  2  361  1  -20  160  0  161  1  -50  -79  1  -80  0  KHz  Part  MLCC  Table 5.3: Fractional P R F estimation results with higher squint on single point target  DLR  MLBF  MLCC  DOPCEN  DCen  Error  Amb  DCen  Error  Amb  DCen  Error  Amb  KHz  Hz  Hz  No.  Hz  Hz  No.  Hz  Hz  No.  -2  -1999  1  -2  -1998  2  -2  -1940  60  -2  -5  -5067  -67  -5  -5025  -25  -5  -5022  -22  -5  -10  -10051  -51  -10  -10026  -26  -10  -9930  70  -10  -15  -15007  -7  -16  -15023  -23  -16  -14952  48  -16  -20  -19982  18  -21  -20084  -84  -21  -19974  26  -21  -50  -49954  46  -52  -50070  -70  -52  -49879  121  -52  Table 5.4: A m b i g u i t y estimation results with higher squint on single point target  104  5.4  Multiple Point Targets Simulation  To demonstrate that there is no bias i n the estimation of the D L R , the M L C C and the M L B F algorithm when the squint is high, multi-target simulations are performed. T h e D L R and the M L C C algorithm are performed on 6 low-contrast data blocks w i t h the scene contrast of 1. T h e M L B F is performed on 6 high-contrast data blocks with the scene contrast of 4.8. T h e D O P C E N s we use are shown i n Table 5.1. These data blocks are generated using the same method described i n Chapter 4. Table 5.5 gives the estimates of the fractional P R F part of the D O P C E N from the D L R and the M L C C algorithms performed on the 6 low-contrast data blocks. Table 5.6 gives the estimates of the ambiguity by the D L R , the M L C C and the M L B F algorithms. The D L R and the M L C C algorithms are performed on the 6 low-contrast data blocks, and the M L B F algorithm is performed on the 6 high-contrast data blocks. In the simulations, the estimates of the fractional P R F part given by the D L R and the M L C C algorithms are very accurate. A l l the D L R , the M L C C and the M L B F algorithms gave the correct estimates of the ambiguity. Looking at the mean absolute error for each estimator, we note that the M L C C algorithm has a slightly better accuracy than the D L R algorithm on the low contrast scenes. E v e n w i t h the high contrast scenes, the M L B F algorithm has a higher mean absolute error than the D L R and the M L C C algorithms have on low contrast scenes. T i l l now we can conclude that, the linearity of the A C C C angle as a function of the range frequency holds when the squint is high. T h e slope of this line is accurate enough for the D L R and the M L C C algorithms to obtain the correct ambiguity number.  105  We can also conclude that, the peak of the beat frequency will not be masked when the squint is high. T h e location of the peak frequency is accurate for the M L B F algorithm to get a correct estimate of the ambiguity number.  DLR DOPCEN  Error  Frac. part  Error  Hz  Hz  Hz  Hz  -2  -80  0  -80  0  -5  -200  0  -200  0  -10  -400  0  -400  0  -15  360  0  361  1  -20  159  1  160  0  -50  -81  -1  -79  1  KHz  Frac.  Part  MLCC  Table 5.5: Fractional P R F estimation results w i t h higher squint on multiple point targets  106  DLR  MLBF  MLCC  DOPCEN  DCen  Error  Amb  DCen  Error  Amb  DCen  Error  Amb  KHz  Hz  Hz  No.  Hz  Hz  No.  Hz  Hz  No.  -2  -1988  . 12  -2  -1993  7  -2  -1826  174  -2  -5  -5030  -30  -5  -5032  -32  -5  -5022  -22  -5  -10  -9987  13  -10  -10017  -17  -10  -9587  413  -10  -15  -15028  -28  -16  -14989  11  -16  -15532  -532  -16  -20  -20019  -19  -21  -20013  -13  -21  -19632  368  -21  -50  -49961  39  -52  -49972  28  -52  -49765  235  -52  MAE  24  18  290  Table 5.6: A m b i g u i t y estimation results w i t h higher squint on multiple point targets ( " M A E " means " M e a n Absolute Error")  107  5.5  Summary  In this chapter, we examined the linearity of the A C C C angle as a function of the range frequency when the squint increases. W h e n the squint becomes high, this linearity still holds. Single point target simulations and multiple point target simulations are performed. F r o m the performance of the D L R and the M L C C algorithms under high squint, we conclude that the linearity and the slope is accurate enough for these algorithms to obtain correct estimates of the ambiguity number. We also examined the spectrum of the azimuth beat signal when the squint i n creases. W h e n the squint becomes high, the bandwidth of the beat frequency increases. However, the peak frequency i n the beat spectrum is not masked and the location of the peak frequency is still accurate enough for the M L B F algorithm to obtain correct estimates of the ambiguity number. The M L C C algorithm works well on E R S data, but may not work on R A D A R S A T data, which has a higher squint. F r o m the simulations i n this chapter, we can conclude that the origin of any bias i n the M L C C algorithm on R A D A R S A T data is not due to the high squint.  108  Chapter 6  Radiometric Sensitivities Radiometric discontinuities i n a S A R image are due to the difference of the reflectivity of scatterers on the ground surface. Echoes of the chirp from a certain part of the ground surface are strong, whereas echoes from another part are weak. T h e boundary of these two parts forms a discontinuity i n the S A R data after compression. These discontinuities are very common i n S A R images, such as at the boundaries of water and land. Radiometric discontinuities have a significant effect on the performance of D O P C E N estimators, including the phase-based estimators. In this chapter, we w i l l discuss the effect of radiometric discontinuities on the D L R , M L C C and M L B F  algorithms.  Simulations are first performed to investigate the effect of the discontinuities. We use single-direction discontinuities, first i n the azimuth direction, then i n the range direction, and compare the performance of the three algorithms w i t h each discontinuity. For the D L R and M L C C algorithms, we use low-squint, low-contrast data sets, except for the contrast change introduced by the radiometric boundary. For the M L B F algorithm, we  109  use low-squint, high contrast data sets. We explain how each radiometric discontinuity affects each algorithm, showing why the D L R algorithm is more sensitive to the azimuth discontinuity and why the M L C C algorithm is more sensitive to the range discontinuity. We also explain why the M L B F algorithm is not affected by the radiometric discontinuity. T h e three D O P C E N estimation algorithms are also run on E R S - 1 data to illustrate our conclusions.  110  6.1  Experiments with Simulated Data  "6.1.1  Simulation Methodology  In this part, the methodology of simulations is introduced.  To examine the effect of  radiometric discontinuity on the D L R , t h e . M L C C and the M L B F algorithms, we examine the discontinuity i n each of the azimuth and range directions separately. The method of generating low-contrast simulated data for the D L R and the M L C C algorithm is the same as that described i n Chapter 4. T h e discontinuity is generated in the reflectivity coefficient matrix. Figure 6.1 illustrates the pattern of the azimuthdirection discontinuity. After an uniform reflectivity m a t r i x is generated, i n which the magnitude of each coefficient is equal to 1 and the phase of each coefficient is random, we create a discontinuity at the central azimuth cell. We keep the magnitude of all coefficients in Part A the same and amplify the magnitude of all coefficients i n Part B by a factor M. define the Magnitude R a t i o M  We  as:  M  = y-  (6.1)  where IA is the magnitude of the coefficients i n Part A and IB is the magnitude of the coefficients i n Part B . Figure 6.2 shows an azimuth-direction  discontinuity in the  central azimuth cell when M = 5. Note that the plot is subsampled for plotting efficiency. After the reflectivity m a t r i x is generated, it is convolved w i t h the expanded point target array by fast convolution. Since the magnitude of the coefficients in the matrix represents the strength of the scatterers, a radiometric discontinuity is generated in the  111  simulated data.  Figure 6.3 shows the discontinuity i n the simulated data, illustrating  that the azimuth data encoding has smoothed the edge, as a z i m u t h compression has not taken place yet. To generate the radiometric discontinuity i n the range direction, the same method is used except that the boundary between Part A and Part B is at the central range cell, as shown i n Figure 6.4. Figure 6.5 shows the magnitude of the reflectivity matrix which contains a radiometric discontinuity i n the range direction. Note that the plot is also subsampled. Figure 6.6 shows the range discontinuity i n the simulated data after the convolution w i t h the point target. Note that this data w i l l be range compressed before the D O P C E N algorithms are run, so that the discontinuity becomes sharp again, as shown in Figure 6.7.  112  PartB  Part A E  N <  Reflectivity Matrix Range  Figure 6.1: Illustrating the Generation of an A z i m u t h Radiometric Discontinuity  Reflectivity Matrix, M = 5  Figure 6.2: Magnitude of the Reflectivity M a t r i x w i t h an azimuth discontinuity when M = 5  113  Azimuth Time (sample)  0  0  Range Time (sample)  Figure 6.3: A z i m u t h Discontinuity i n the Simulated D a t a when M = 5  114  Figure 6.4: Illustrating the Generation of a Range Radiometric Discontinuity  Reflectivity Matrix, M = 5  Azimuth  0  0  Range  Figure 6.5: Magnitude of the Reflectivity M a t r i x w i t h a range discontinuity when M — 5  115  Azimuth Time (sample)  °  0  Range Time (sample)  Figure 6.7: Range Discontinuity in the Simulated D a t a After Range Compression  116  To generate high-contrast  simulated data for the M L B F algorithm, the same  method is used except that, before generating the radiometric discontinuity, high contrast is generated i n the reflectivity m a t r i x , as described i n 4. In the simulations, the scene contrast before generating radiometric discontinuities is 4.8. T h i s guarantees that the M L B F algorithm works well without radiometric discontinuities so that our research can focus on the effect of the radiometric discontinuity.  117  6.1.2  Simulation Results with an Azimuth Discontinuity  Nineteen low-contrast simulated data blocks containing a radiometric discontinuity in the a z i m u t h direction are generated w i t h the Magnitude R a t i o changing from small to large. T h e D L R and the M L C C algorithms are run on these data blocks, The estimation results of the fractional P R F part of D O P C E N are shown i n Table 6.1. The estimation results of the Doppler ambiguity are shown in Table 6.2.  In the ambiguity case, the  results are given i n Hz units, which is the intercept when the A C C C slope is projected to the radar frequency. In a similar fashion, another set of simulated data consists of nineteen highcontrast simulated data blocks is generated w i t h the Magnitude R a t i o changing from small to large to test the M L B F algorithm. T h e results are also shown i n Table 6.2. Note that the parameters we used in the simulations are given in Table 3.1, with a true D O P C E N of -400 Hz . They represent low-squint data, except for the contrast change at the radiometric discontinuity.  118  DLR M  Frac.  Part  MLCC Error  Frac.  part  Error  Hz  Hz  Hz  Hz  1.0  -400  0  -400  0  1.1  -394  6  -395  5  1.2  -389  11  -390  10  1.3  -385  15  -386  14  1.4  -381  19  -382  18  1.5  -377  23  -379  21  2.0  -365  35  -367  33  3:0  -355  45  -356  44  4.0  -350  50  -351  49  5.0  -348  52  -346  54  10.0  -345  55  -345  55  20.0  -346  54  -346  54  30.0  -344  56  -344  56  40.0  -345  55  -344  56  50.0  -345  55  -344  56  60.0  -346  54  -345  55  80.0  -345  55  -345  55  100.0  -346  54  -345  55  120.0  -345  55  -343  57  Table 6.1: Fractional P R F estimation results with an azimuth discontinuity  119  DLR M  MLCC  MLBF  DCen  Error  Amb  DCen  Error  Amb  DCen  Amb  Hz  Hz  No.  Hz  Hz  No.  Hz  No.  1.0  -402  -2  0  -404  -4  0  0  0  1.1  -354  46  0  -380  20  0  0  0  1.2  -489  -89  0  -482  -82  0  0  0  1.3  -450  -50  0  -417  -17  0  0  0  1.4  -549  -149  0  -520  -120  0  0  0  1.5  -501  -101  0  -464  -64  0  0  0  2.0  -106  294  0  -239  161  0  0  0  3.0  16  416  0  -2  398  0  0  0  4.0  -100  300  0  -205  195  0  0  0  5.0  -760  -360  0  -680  -280  0  0  0  10.0  7  407  0  -208  192  0  0  0  20.0  -1090  -690  -1  -1064  -664  -1  0  0  30.0  -655  -255  0  -639  -239  0  0  0  40.0  949  1349  1  419  819  1  0  0  50.0  -601  -201  0  -551  -151  0  0  0  60.0  -1331  -931  -1  -760  -360  0  0  0  80.0  742  -1142  1  7  407  0  0  0  100.0  1427  1827  2  1421  1821  2  0  0  120.0  1592  1992  2  641  1041  1  0  0  Table 6.2: Ambiguity estimation results w i t h an azimuth discontinuity  120  6.1.3  Simulation Results with a Range Discontinuity  Nineteen low-contrast simulated data blocks containing a radiometric discontinuity in the r a n g e direction are generated w i t h the Magnitude R a t i o changing from small to large. T h e D L R and the M L C C algorithms are run on these data blocks, The estimation results of the fractional P R F part of D O P C E N are shown i n Table 6.3. The estimation results of the Doppler ambiguity are shown i n Table 6.4.  In the ambiguity case, the  results are given i n Hz units, which is the intercept when'the A C C C slope is projected to the radar frequency. In a similar fashion, another set of simulated data consists of nineteen highcontrast simulated data blocks is generated w i t h the Magnitude R a t i o changing from small to large to test the M L B F algorithm. The results are also shown i n Table 6.4.  121  DLR M  Frac.  Part  MLCC Error  Frac.  part  Error  Hz  Hz  Hz  Hz  1.0  -400  0  -400  0  1.1  -400  0  -400  0  1.2  -400  0  -400  0  1.3  -400  0  -400  0  1.4  -400  0  -400  0  1.5  -400  0  -400  0  2.0  -400  0  -401  1  3.0  -400  0  -401  1  4.0  -399  1  -401  1  5.0  -399  1  -401  1  10.0  -399  1  -402  2  20.0  -399  1  -402  2  30.0  -399  1  -402  2  40.0  -399  1  -402  2  50.0  -399  1  -402  2  60.0  -399  1  -402  2  80.0  -399  1  -402  2  100.0  -399  1  -402  2  120.0  -399  1  -402  2  Table 6.3: Fractional P R F estimation results w i t h a range discontinuity  122  DLR M  MLBF  MLCC  DCen  Error  Amb  DCen  Error  Amb  DCen  Amb  Hz  Hz  No.  Hz  Hz  No.  Hz  No.  1.0  -402  -2  0  -404  -4  0  0  0  1.1  -403  -3  0  -394  6  0  0  0  1.2  -399  1  0  -387  13  0  0  0  1.3  -404  -4  0  -389  11  0  0  0  1.4  -396  -4  0  -380  20  0  0  0  1.5  -395  5  0  -382  18  0  0  0  2.0  -389  11  0  -371  29  0  0  0  3.0  -370  30  0  -366  34  0  0  0  4.0  -358  42  0  -350  50  0  0  0  5.0  -330  70  0  -317  -83  0  0  0  10.0  -301  99  0  -200  200  0  0  0  20.0  -266  134  0  -253  147  0  0  0  30.0  -248 .  152  0  -42  358  0  0  0  . 40.0  -241  159  0  -113  287  0  0  0  50.0  -198  202  0  -147  253  0  0  0  60.0  -145  255  0  -11  389  0  0  0  80.0  -100  300  0  38  438  0  0  0  100.0  7  407  0  21  421  0  0  0  120.0  161  461  0  264  664  1  0  0  Table 6.4: A m b i g u i t y estimation results w i t h a r a n g e discontinuity  123  6.2  Discussion of Simulation Results  F r o m the simulation results, we can see that the radiometric discontinuity has different effects on the D L R and the M L C C algorithms in the range and i n the azimuth direction. However,%&th the azimuth and the range discontinuities have little effect on the M L B F algorithm. In this section, we discuss the following four aspects of the results: • How the azimuth discontinuity affects the D L R , the M L C C and the M L B F algorithms, • W h i c h A C C C - b a s e d algorithm performs better w i t h the azimuth discontinuity, • How the range discontinuity affects the D L R , the M L C C and the M L B F algorithms, and • W h i c h A C C C - b a s e d algorithm performs better w i t h the range discontinuity.  6.2.1  Effects of the Azimuth-Direction Radiometric Discontinuity  F r o m Table 6.1 and Table 6.2, we can see that the discontinuity in the azimuth direction has a large effect on both the D L R and the M L C C algorithms.  In the case of the  fractional part, both estimators suffer a bias of about 55 Hz fairly independent of the Magnitude R a t i o . In other words, the average A C C C angle is biased. In the case of the Doppler ambiguity, the results are much more random, indicating that the slope of the A C C C vs. range frequency has a random error. In some cases, the  124  error is large enough to create an ambiguity error of one P R F .  1  The reason for this behavior of the estimators is easy to understand. In Chapter 4, we concluded that, when only one target is considered, the A C C C angle is given by Equation (3.6). This is the desired value which can be used to get the correct estimation of the Doppler ambiguity and the fractional P R F part of the  DOPCEN.  W h e n m u l t i p l e point targets are involved, the A C C C angle will have an error component due to the cross correlations between two overlapped targets, as shown in Equation (4.5).  In Chapter 4, we concluded that, for a fixed target, the sum of the  cross correlation coefficients between the fixed target and its neighboring targets due to overlap is a random complex number. O n l y when the power of a l l targets is the same, does the mean of a l l the sums for each target due to overlap become zero, and the error in the A C C C angle tends to average out. W h e n the strength of targets are not the same, the cross correlation coefficients due to overlap cannot be totally averaged out, leading to a error in the A C C C angle estimates. In this simulation, targets in Part A are weak while targets in Part B are strong. T h e n near the boundary of Part A and Part B , the sum of a l l A C C C s between a target and all its neighboring targets in the same range cell is a random complex number. A l l the sums for each target cannot be averaged out due to the difference of the target strength near the boundary, and w i l l cause a random error i n the estimation of the phase increment. T h i s random variation i n the estimation of the A C C C angles at each range cell introduces a random error into the estimation of the slope, thus leading to a random ^ote that the production estimators are usually applied to 4Kx4K data blocks, which will reduce the random component of the error considerably. Here we use a l K x l K data block. The larger data block may not improve the bias, unless the larger data block contains many discontinuities of different sizes and directions, so that their effect cancels out.  125  error i n the estimation of the D O P C E N ambiguity. In addition, an azimuth radiometric discontinuity can be modeled as an azimuth partial exposure of the weak target since a strong target w i l l hide or upset the angle of a small target. Since the A C C C angle is very sensitive to the a z i m u t h partial exposure, as explained i n Chapter 3, the partial exposure of the weak target introduces a bias in the estimate. F r o m the estimation of the fractional P R F part of the D O P C E N shown i n Table 6.1, we can see that the mean of the random variation i n the A C C C angles at each range cell is not zero, thus the estimation of the fractional P R F part of the D O P C E N is significantly biased. Figure 6.8(a) shows the estimation error i n the estimates for Doppler ambiguity of the D L R and M L C C algorithms as a function of magnitude ratio. Figure 6.8(b) shows a zoomed view on the Magnitude R a t i o from 1 to 5. Note only when the error falls in between —PRF/2  and PRF/2,  which is represented by the two dotted lines i n the  figure, may the estimator gives a correct estimate of the D O P C E N ambiguity. We can see that the error is random, although it seems to increase w i t h very large Magnitude Ratios. T h e M L B F algorithm is not affected significantly by the a z i m u t h discontinuity. T h i s is easy to understand.  In Chapter 4, we conclude that, when more than one  significant target is present i n the same range cell, the peak beat frequency becomes masked by cross-beating. A s the number of dominant targets increase, the power due to the cross beating can eventually mask out the required beat frequency, as we observed in Figure 4.7.  Since the azimuth discontinuity does not increase the number of the  126  dominant targets, it does not affect the performance of the M L B F algorithm, as we have seen in the simulations. In addition, the azimuth discontinuity behaves like a partial azimuth exposure. We have explained i n Chapter 3 that the beat frequency is relatively independent of partial azimuth exposure. This can also explain why the M L B F algorithm is not significantly affected by the azimuth discontinuity.  127  (a) Estimation Error  4000 £  3000  t  2000  LU c i  1000  co  LU  •1000  40  60 80 Magnitude Ratio  100  (b) Zoomed View  Magnitude Ratio Figure 6.8: A m b i g u i t y estimation error of the D L R and the M L C C algorithms caused  by the azimuth discontinuity  128  6.2.2  Comparison of the D L R and the M L C C algorithms with an A Z I M U T H discontinuity  In addition to the effect of partial exposure on the A C C C - b a s e d vs. the M L B F algorithms, there is also a distinction between the D L R and the M L C C algorithms due to the different operation of these two algorithms. F r o m Figure 6.8, we can see that, when estimate the Doppler ambiguity, the D L R algorithm is more sensitive to the azimuth discontinuity than the M L C C algorithm. Assuming that the bias in the A C C C angle due to the discontinuity is a random variable 7, the measured A C C C angle can be written as: K  2TT 4> =  n  a  c  —pT—  +  7  (6.2)  Since 7 is a random variable, it introduces a random error at each range cell or each range frequency. This leads to an error i n the estimation of the slope, which can lead to an error i n the estimated ambiguity number w i t h either algorithm. For the M L C C algorithm, the biased A C C C angles of range look 1 and range look 2 is given by: 2?T  Kgi  T]c  ( R  _v  fa  2nK rj 4>L2 = = a2  c  a fa  + 72  (6.4)  where 71 and 72 are the errors i n the A C C C angles of look 1 and look 2 due to the azimuth discontinuity. The difference of <pn and ^  &<p =  <f>L2 - <PL\ 129  2  is given by:  27r{K -K )r) a2  al  c  F  a  2ir(K -K ) a2  al Vc  + (72 -71) + A  F  7  (6.5)  a  where A 7 is equal to 72 — 71. Since the two range looks image the same scatterers on the ground, we can assume that the random variables, 71 and 71, are correlated w i t h each other. T h e n the random variable, A 7 , has a smaller variation than 7 i n Equation (6.2). If the correlation is significant, the error i n the ambiguity estimate of the M L C C algorithm should be smaller than that i n the D L R algorithm. In the simulations, the average estimation of the D L R algorithm is -140 Hz and the standard deviation is 771 Hz . T h e average estimation of the M L C C algorithm is -238 Hz and the standard deviation is 550 Hz , which supports the above theory. Thus we see that the process of finding the difference of the A C C C angles of look 1 and look 2 i n the M L C C algorithm reduces the variation of the error due to the azimuth discontinuity. T h e lack of this step i n the D L R algorithm makes the D L R algorithm more sensitive to the azimuth discontinuity than the M L C C algorithm.  130  6.2.3  Effects of the Range-Direction Radiometric Discontinuity  F r o m Table 6.3 and Table 6.4, we can see that, when we estimate the Doppler ambiguity, the discontinuity i n the range direction has effects on the D L R algorithm and the M L C C algorithm, especially on the M L C C algorithm. However, when we estimate the fractional P R F part of the D O P C E N , the range discontinuity has negligible effect on either the D L R or the M L C C algorithm. In Chapter 3, when we discuss the azimuth signal after range compression, as shown i n Equation (3.2), we ignored the effect of range cell migration ( R C M ) . In fact, the azimuth signal after range compression is a curve instead of a line confined to one range cell. T h e higher the squint is, the more obvious the curvature is. In a low contrast scene, simulations show that this effect can be ignored. However, when there is a large discontinuity i n the range direction, a part of the azimuth signal of a strong target extends into the weak region, and affects the phase of the weak targets. Figure 6.9 illustrates this situation. In Figure 6.9, the thick line represents a strong target near the boundary and the t h i n line represents a weak target near the boundary., since both the D L R and the M L C C algorithms work in the azimuth direction, the strong target w i l l affects the estimate of the phase of the weak target. This seems to affect the distribution of the errors i n the A C C C angle, in such a way that it biases the slope estimate without affecting the average A C C C angle over the whole range time or frequency domain. T h e nature of the slope bias is shown i n Figure 6.10(a), which shows the error of the ambiguity estimate of the D L R and M L C C algorithms as a function of the magnitude ratio. Figure 6.8(b) shows a zoomed view on the Magnitude R a t i o from 1 to 5.  131  T h e M L B F algorithm is not affected by the range discontinuity. This is because that, the estimation of the M L B F algorithm uses the average of the spectrum of the beat signal at each range cell. The effect of the range discontinuity on the the beat signal near the range discontinuity is averaged out, as we have seen i n the simulation results. In addition, the partial exposure, or weak targets masked by strong targets, have a large effect on the A C C C angle but little effect on the beat frequency. This can also explain why the D L R and the M L C C algorithm are affected by the range discontinuity, whereas the M L B F algorithm is not.  Strong Part • Weak Part 13  E N <  Range Figure 6.9: Illustration of the effect of a r a n g e discontinuity  132  (a) Estimation Error  N  DLR MLCC  1000  I  111  cg  500  "•4—»  CC  E  CO  HI  -500 20  40  60 80 Magnitude Ratio  100  120  (b) Zoomed View  200  2  3 Magnitude Ratio  4  Figure 6.10: A m b i g u i t y estimation error of the D L R and the M L C C algorithms caused by the r a n g e discontinuity  133  6.2.4  Comparison of the D L R and the M L C C algorithms with the R A N G E discontinuity  Even though the m a i n difference in reaction to range discontinuities is between the A C C C - b a s e d algorithms and the M L B F algorithm, there are differences between the two A C C C - b a s e d algorithms. F r o m Figure 6.10, we can see that the M L C C algorithm is more sensitive than the D L R algorithm to the range discontinuity. B o t h algorithms are biased by the range discontinuity, as discussed above. T h e range discontinuity also causes another bias only in the M L C C algorithm, which makes the M L C C algorithm more sensitive to the range discontinuity than the D L R algorithm. To illustrate this effect, we first consider a single point target.  Figure 6.11(a)  shows the range compressed pulse of a range look. In this case, the location of the peak is at the 82nd range cell. Figure 6.11(b) shows the difference of A C C C angles of range look 1 and range look 2. Note that the M L C C algorithm works in the range time domain.  Table 6.5 gives the value of the difference of A C C C angles around the peak and the corresponding estimation of the D O P C E N according to E q u a t i o n (3.32). F r o m Table 6.5, we can see that only the A<p value at the peak of the target gives the correct estimate of the D O P C E N , i n this case, -400 Hz . Others around the peak are too noisy to be useful. T h i s is because, after range compression, almost all the signal energy is compressed to the peak index. Other parts of the compressed pulse have a very low S N R . Thus we can conclude that, i n t h e * M L C C algorithm, only the range  134  (a) A Range Compressed Look 150  20  40  60  80  100  120  140  160  (b) Difference of ACCC Angles of Lookl and Look2  60 80 100 Range Time (sample)  160  Figure 6.11: Difference of A C C C Angles of Look 1 and Look 2 sample at the peak location is "useful" for the estimation process. Other parts of the range compressed signal are noisy and the difference of the A C C C angles of these parts lead to a high phase noise. Now consider two targets near one another. After range compression, two peaks appear and these two peaks are next to each other i n the range time domain, as shown in Figure 6.12. F r o m the conclusion above, only the values of A<6 of the two peaks have high S N R and their average can give the correct estimation of the D O P C E N . T h e phase noise of the strong pulse w i l l affect the useful phase value at the peak of the weak pulse, depending upon the relative strengths of each target. W h e n the M L C C algorithm works on a low contrast scene, the phase noise is relatively low compared to the peak it affects. However, when there is a discontinuity in the range direction, the "noise" from the strong target w i l l effectively distort the weak  135  Range  A0  Cell No.  (mrad)  DOPCEN (Hz)  79  -8.2  614  80  0.0  0  81  3.6  -268  82  6.2  -425  83  -0.1  10  84  -2.5  187  85  -10.0  745  Table 6.5: Difference of A C C C Angles and D O P C E N s ( M L C C effect only) peak, leading to a random bias i n the average of a l l the A</> values at each peak near the range discontinuity. This results i n an additional bias i n the estimates of the M L C C algorithm. Since the D L R algorithm works in the range frequency domain, after range F F T s , the spectra of strong targets and weak targets are a l l aligned. Thus the strong targets and the weak targets near the discontinuity do not interfere w i t h each other i n the way that they do i n the M L C C algorithm. T h i s may explain why the M L C C algorithm is more sensitive to range discontinuities than the D L R algorithm.  136  Strong Target  Weak Target CD  "O •t—'  "c  CO  Range Time Figure 6.12: Illustration of the effect of a strong target on a weak target  137  6.3  Experiments with ERS-1 Data  To demonstrate the effect of the radiometric discontinuity i n real data and its effects on the D L R algorithm, the M L C C algorithm and the M L B F algorithm, E R S - 1 raw data are used. Note in this case, the estimation of the D O P C E N given by the D L R algorithm and the M L C C algorithm contains a frequency offset, which is attributed to the variation of the antenna pointing angle w i t h transmitted frequency.  6.3.1  Azimuth Discontinuity in ERS-1 Data  A n E R S - 1 scene i n yaw-steering mode is used.  T h i s image, K a m l o o p s , is a medium  contrast image, thus, a l l the three algorithms are expected work well on this scene. The acquisition data of this scene is Sep. 23, 1995 and the o r b i t / t r a c k is 21916/385. This scene contains a lake, which forms a discontinuity i n the a z i m u t h direction. We took a 1024 x 1024 data block containing this discontinuity selected from the right hand side of Figure 6.13.  Figure 6.13: T h e Detected Image of the K a m l o o p s E R S Scene T h e absolute D O P C E N in the middle of the scene is 527 Hz , obtained by M D A ' s 138  d t S A R processor.  This value is used i n the processing of the raw data to obtain the  detected image shown i n Figure 6.13. This estimate is likely reliable since the detected image has a high quality. The horizontal direction is the range direction w i t h near range on the left. T h e azimuth direction is vertical, w i t h azimuth time increasing i n the upward direction. T h e estimate of fractional P R F part of the D L R algorithm on this data block is 435 Hz . T h e estimate of the M L C C algorithm is 467 Hz . T h e estimate error of these two algorithms are -92 Hz and -60 Hz respectively. B o t h of these estimates exhibit a small bias, likely due to the azimuth discontinuity, as discussed above. T h e ambiguity estimate of the D L R algorithm on this data block is 2447 Hz . The estimate of the M L C C algorithm is 944 Hz . Since the P R F is about 1680 Hz , the M L C C algorithm can gives the correct estimate of the Doppler ambiguity and the D L R algorithm does not. This observation agrees w i t h the simulation results of Section 6.1.2. The ambiguity estimate of the M L B F algorithm is 201 Hz , which is accurate enough to get the correct ambiguity, zero. This experiment illustrates that the D L R algorithm is more sensitive to the azi m u t h discontinuity than the M L C C algorithm. It also demonstrates that the azimuth radiometric discontinuity has little effect on the M L B F algorithm.  6.3.2  Range Discontinuity in ERS-1 Raw Data  Another E R S - 1 scene i n yaw-steering mode is used which contains a radiometric discontinuity i n the range direction. T h i s image, Port A l i c e , is obtained on Sep. 23, 1995 and  139  the orbit/track is 361/999.  The left part of this scene is ocean and the right part is  land. T h e boundary between the ocean and the land forms a discontinuity in the range direction, as the ocean is smoother and has low reflectivity near the shoreline. We select a data block containing this discontinuity and the size of the data block is 1024 x 1024 samples.  Figure 6.14 shows the detected image from which this block was taken near  the ocean/land boundary.  Figure 6.14: The Detected Image of the Port A l i c e E R S Scene  The absolute D O P C E N in the middle of the scene is -385 Hz , obtained by M D A ' s d t S A R processor. T h e horizontal direction is the range direction w i t h near range on the left.  T h e a z i m u t h direction is vertical, w i t h azimuth time increasing i n the upward  direction. The fractional P R F part estimate of the D L R algorithm is -358 Hz . T h e estimate of the M L C C algorithm is -367 Hz . The estimate errors are 27 Hz and 18 Hz respectively. T h i s bias is relatively small and may be caused by the high scene contrast. Thus, this experiment demonstrates that the range discontinuity has a small effect on the estimation of the fractional P R F part of the D O P C E N . The ambiguity estimate of the D L R algorithm on this data block is -926 Hz .  140  The ambiguity estimate of the M L C C algorithm on this data block is -1116 Hz . Since the P R F is about 1680 Hz , both algorithms give the correct estimate of the Doppler ambiguity number, but the better estimate of the D L R algorithm leaves more room for error. This observation agrees w i t h the simulation results. The ambiguity estimate of the M L B F algorithm is -805 Hz . It is also accurate enough to get the correct estimate of the ambiguity, zero. Thus this experiment illustrates that the M L C C algorithm is more sensitive to the range discontinuity than the D L R algorithm. It also demonstrates that the range radiometric discontinuity has little effect on the M L B F algorithm.  141  6.4  Summary  In this chapter, we discussed the effects of radiometric discontinuities on the D L R , M L C C a n d . M L B F D O P C E N estimation algorithms. T h e azimuth discontinuity has no effect on the M L B F algorithm since it does not increase the number of the dominant targets and the M L B F algorithm is not sensitive to the resulting partial exposure.  However, the azimuth discontinuity has a significant  effect on the D L R and the M L C C algorithms since both of these two algorithms average the A C C C angle over the azimuth direction (See Chapter 3). T h e discontinuity in the azimuth direction introduces a bias i n the A C C C angles at each range cell, leading to an error i n the estimation of the slope. Thus the estimation o f the projected D O P C E N , which is used to obtain the Doppler ambiguity, can have a substantial error since small errors i n the slope causes a large error i n the estimate of the D O P C E N . The fractional P R F estimates are also significantly biased. T h e process of finding the difference of the A C C C angles of look 1 and look 2 in the M L C C algorithm reduces the variation of the bias due to the azimuth discontinuity. The lack of this step i n the D L R algorithm makes the D L R algorithm more sensitive t o . the azimuth discontinuity than the M L C C algorithm. T h e range discontinuity has no effect on the M L B F algorithm since the beat frequency is relatively independent of the position i n the azimuth exposure and the M L B F algorithm works on the average of the beat spectrum over range. Also, the range discontinuity has a small effect on the D L R and the M L C C algorithm compared to the azimuth discontinuity. This is because only the A C C C angles near the discontinuity is affected. A range-compressed signal is actually a curve instead of a straight line confined 142  in one range cell. In a low contrast scene, simulations show that this effect can be ignored. However, when there is a large discontinuity i n the range direction, a part of a azimuth signal of a weak target extends into the strong region, thus is largely reduced due to the strong targets. T h i s may introduce a bias into the A C C C angles i n the vicinity of the discontinuity, thus a bias exists in the estimation of the D O P C E N . Since the M L C C algorithm works in the range time domain, the phase noise of a strong target w i l l distort the desired phase value at the peak of a weak target. T h i s may lead to a bias i n the average of the difference of the A C C C angles at each peak near the range discontinuity. However, the D L R algorithm works i n the range frequency domain. T h e spectra of strong targets and weak targets are all aligned. Thus the strong targets and the weak targets near the discontinuity do not interfere w i t h each other in the way that they do i n the M L C C algorithm. T h i s makes the D L R algorithm more robust to the range discontinuity than the M L C C algorithm. T h e results also point out that the two algorithms use the same method of estimating the fractional P R F part of the D O P C E N , as they both simply use the average A C C C angle over the range swath. Thus the fractional P R F estimates are substantially the same in each experiment. However, it is clear that the performance in estimating the Doppler ambiguity is quite different, which we attribute to the different way i n which the data is aligned when estimating the change in A C C C w i t h range frequency.  143  Chapter 7  Conclusions  7.1  Summary  T h e objective of this research is to examine and test the performance of the phased-based Doppler estimation algorithms w i t h different scene contrasts and S N R levels, examine the performance of these algorithms w i t h higher squint data, and examine the sensitivity of some phase-based Doppler estimation algorithms to radiometric discontinuities to find out how the radiometric discontinuities affect these estimation algorithms. T h e form of the received signal after demodulation is derived. T h e effect of range sampling is discussed, illustrating that the R C M i n the raw data causes a shift in the azimuth spectra, forming the dependence of D O P C E N on range time. T h e Principle of Stationary Phase is applied to approximate the spectrum of range-compressed signal, showing that the D O P C E N is a linear function of range frequency under this approxi-  144  mation, which is the basic principle of the phase-based D O P C E N estimators. T h e operation of the D L R , M L C C and M L B F algorithms have been introduced in this chapter. A l l of these algorithms are based on the principle that the D O P C E N is approximately a linear function of the range frequency. Single point target simulations are performed to illustrate the operations of these algorithms. F r o m the simulations, the estimates are almost the same since they are based on the same principle. It also shown that the linear approximation in the low squint case is accurate enough to obtain an accurate estimate of the D O P C E N . Performance of the D L R , M L C C and M L B F algorithms are examined. T h e D L R algorithm and the M L C C algorithm work well on this set of data. T h e M L B F algorithm does not work on the low contrast data set. W h e n the scene contrast increases, the M L B F algorithm begins to work well while the error of the D L R algorithm and the M L C C algorithm becomes large, leading to an incorrect estimation of the Doppler ambiguity. As the D L R and M L C C algorithms behave almost the same i n terms of scene contrast, it is interesting to also compare them i n terms of their response to scene noise. We conclude that the D L R algorithm and the M L C C algorithm are robust to white noise on low contrast scenes. T h e estimation errors of the D L R and the M L C C algorithms on same noise level are almost the same. T h e linearity of the A C C C angle as a function of the range frequency is examined when the squint increases.  W h e n the squint becomes high, this linearity still holds.  F r o m the performance of the D L R , the M L C C and the M L B F algorithms under high squint, we conclude that the linearity and the slope is good enough for these algorithms to obtain correct estimates of the ambiguity number. T h e M L C C algorithm works well  145  on E R S data, whereas may not work on R A D A R S A T data, which has a higher squint. F r o m the simulations, we can conclude that the origin of the bias i n the M L C C algorithm on R A D A R S A T data is not due to the higher squint. Radiometric discontinuities have a significant effect on the performance of D O P C E N estimators, including the phase-based estimators.  F r o m simulations, we can see  that the discontinuity i n the azimuth direction has a large effect on both the D L R and the M L C C algorithms. In the case of the Doppler ambiguity, the results are much more random, indicating that the slope of the A C C C vs. range frequency has a random error. In some cases, the error is large enough to create an ambiguity error of one P R F . W h e n multiple point targets are involved, the A C C C angle w i l l have an error component due to the cross correlations between two overlapped targets. For a fixed target, the sum of the cross correlation coefficients between the fixed target and its neighboring targets due to overlap is a random complex number. O n l y when the power of all targets is the same, does the mean of a l l the sums for each target due to overlap become zero, and the error i n the A C C C angle tends to average out. W h e n the strength of targets are not the same, the cross correlation coefficients due to overlap cannot be totally averaged out, leading to a error i n the A C C C angle estimates. F r o m simulations, we can see that, when estimate the Doppler ambiguity, the D L R algorithm is more sensitive to the azimuth discontinuity than the M L C C algorithm. T h e process of finding the difference of the A C C C angles of look 1 and look 2 i n the M L C C algorithm reduces the variation of the error due to the azimuth discontinuity. The lack of this step in the D L R algorithm makes the D L R algorithm more sensitive to the azimuth discontinuity than the M L C C algorithm.  146  T h e M L B F algorithm is not affected by the azimuth discontinuity. W h e n more than two significant targets are present i n the same range cell, the distortion of the peak beat frequency gets worse. A s the number of dominant targets increase, the power due to the cross beating can eventually mask out the required beat frequency.  Since the  azimuth discontinuity does not increase the number of the dominant targets, it does not affect the performance of the M L B F algorithm. W h e n we estimate the Doppler ambiguity, the discontinuity i n the range direction has effects on" the D L R algorithm and the M L C C algorithm, especially on the M L C C algorithm. However, when we estimate the fractional P R F part of the D O P C E N , the range discontinuity has negligible effect on either the D L R or the M L C C algorithm. Since the D L R algorithm works i n the range frequency domain, after range F F T s , the spectra of strong targets and weak targets are all aligned. Thus the strong targets and the weak targets near the discontinuity do not interfere w i t h each other in the way that they do i n the M L C C algorithm. T h i s may explain why the M L C C algorithm is more sensitive to range discontinuities than the D L R algorithm. T h e M L B F algorithm is not affected by the range discontinuity. This is because that, the estimation of the M L B F algorithm uses the average of the spectrum of the beat signal at each range cell.  T h e distortion of the the beat signal near the range  discontinuity is averaged out. Since the M L C C algorithm works best with scenes of low, uniform contrast, while the M L B F algorithm works best w i t h scenes of high contrast, and since they all need the processing of range look extraction, theses two algorithms can be efficiently combined together to form a reliable D O P C E N estimator over large ranges of scene contrast.  147  7.2  Contributions  T h e contributions of this research work are:  • E x a m i n e d the effect of range sampling, and concluded that, i f there is not R C M , the D O P C E N is a constant for all azimuth lines. If R C M does exist, the D O P C E N is a linear function of the range frequency. • E x a m i n e d the cross correlation item i n the A C C C angle due to overlap when multiple targets are involved, and concluded that, only when the strength of all targets are same can these items be averaged out. • Understood the difference between the A C C C - b a s e d algorithms and the M L B F approach. • Implemented quadratic curve fitting to the M L B F estimator when the scene contrast is low. • Proposed the theory on how the radiometric discontinuity affects the performance of the D L R , M L C C and M L B F algorithms, and proved it by simulations and real data experiments. • Proposed the theory on why the D L R algorithm is more sensitive to the azimuth discontinuity than the M L C C algorithm, whereas the M L C C algorithm is more sensitive to the range discontinuity than the D L R ' algorithm, and proved it by simulations and real data experiments.  148  7.3  Future Work  Our results suggest the following topics for future research:  • More experiments are needed to quantify the effect of radiometric discontinuities and to improve the performance of the M L C C and the D L R algorithms. • A l l these algorithms should be examined w i t h real data, which have a higher squint, for example, R A D A R S A T  data.  149  Bibliography [1] C . 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