C .1 STEP TRANSFORM PULSE COMPRESSION AND ITS APPLICATION TO SYNTHETIC APERTURE RADAR SYSTEMS by MARK WILLIAM SACK B . S c , E l e c t r i c a l E n g i n e e r i n g , Queen's U n i v e r s i t y , 1978 A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF A P P L I E D SCIENCE i n THE FACULTY OF GRADUATE STUDIES ( D e p a r t m e n t Of E l e c t r i c a l E n g i n e e r i n g ) We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA A p r i l 1983 Â© Mark W i l l i a m S a c k , 1983 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of ÂŁ L ÂŁ C / / C ( / C / ? C_ ÂŁ /O G IN ÂŁ C(fll fJ Q-The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6 (3/81) i i Abstract Step transform pulse compression is a matched f i l t e r i n g technique applicable to linear FM signals. A mathematical analysis of the method i s developed and i t s application to the azimuth processing of synthetic aperture radar (SAR) target returns i s explored. In p a r t i c u l a r , several sa t e l l i t e - b o r n e SAR systems are used as examples to compare the performance of t h i s technique to other approaches. Problems peculiar to SAR azimuth processing including FM rate error, range c e l l migration, and multilook processing are considered. The computation rates, memory requirements, and complexity of the processing algorithms are also evaluated. The step transform i s studied a n a l y t i c a l l y to determine how appropriate choices of processing parameters can be made to achieve acceptable image qualit y , while minimizing the hardware requirements. A method of range c e l l migration correction, which can be integrated into the step transform process, i s proposed and the corresponding parameter r e s t r i c t i o n s are derived. The hardware requirements of the step transform for SAR applications, r e l a t i v e to some other pulse compression techniques, are shown to be less dependent on SAR system parameters and more dependent on the step transform processing parameters chosen. A computer simulation program was written to confirm the a n a l y t i c a l r e s u l t s . i i i Table of Contents Abstract i i L i s t of Tables . ^vi L i s t of Figures v i i i Acknowledgements x Chapter I INTRODUCTION 1 1 . 1 Objectives 2 1.2 Structure Of The Thesis 3 Chapter II . SURVEY OF PULSE COMPRESSION TECHNIQUES 6 2.1 Time-Domain Convolution 7 2.2 Fast Convolution 8 2.3 Spectral Analysis 9 2.3.1 Reference Function Bandwidth Limitations 11 2.3.2 Application To SAR Azimuth Processing 14 2.4 Bandpass F i l t e r Spectral Analysis 16 Chapter III THE STEP TRANSFORM 18 3.1 Mathematical Analysis 19 3.2 Alternative Case 29 3.3 The A l i a s i n g Problem 31 iv 3.4 Data Selection 36 Chapter IV CHOOSING STEP TRANSFORM PROCESSING PARAMETERS 39 4.1 Windows 40 4.2 Computation Rates 49 4.3 Memory Requirements 53 4.4 Multilooking 54 4.5 Comparison With Bandpass F i l t e r Spectral Analysis .57 Chapter V DETAILS OF SAR AZIMUTH PROCESSING 59 5.1 The SAR Azimuth Signal 59 5.2 The Antenna P r o f i l e v 63 5.3' The Simulation Program 64 Chapter VI STEP TRANSFORM PERFORMANCE ON SAR AZIMUTH SIGNALS 69 6.1 FM Rate Error ; 69 6.1.1 Data For Typical S a t e l l i t e SAR Systems 71 6.1.2 Frequency Step Mismatch 80 6.2 Range C e l l Migration 84 Chapter VII IMPLEMENTATION ON SOME SATELLITE SAR SYSTEMS 91 7.1 Summary Of Processing Parameter Restrictions 91 7.2 Parameters Chosen For Typical SAR Systems .93 V 7.3 Computation Rates And Memory Requirements 95 7.4 Comparison With Other Techniques 99 7.4.1 General Comparison 100 7.4.2 Comparisons For Specific SAR Systems 104 7.4.3 Remarks On Computation Rates 107 7.5 Step Transform Processor Architecture ...109 Chapter VIII CONCLUSIONS 112 8 .1 Discussion 112 8.2 Directions For Further Research 114 8 .3 Summary- 116 BIBLIOGRAPHY 119 APPENDIX A - DEFINITION OF DATA WINDOWS 123 APPENDIX B - SYMBOLS AND ACRONYMS USED 126 APPENDIX C - RADAR PARAMETERS 128 APPENDIX D - COMPUTER SIMULATION OF THE STEP TRANSFORM ..130 j v i L i s t of Tables I. Mainlobe Width at Highest Sidelobe Level for Various Windows 43 II. Overlap Required and Sidelobe Levels for Various Windows 45 II I . Attenuation of Main Lobe at 0.5 bins from Centre ...47 IV. FM Rate Errors for Typical SAR Systems 72 V. FM Rate Error Simulation Results for SEASAT Parameters 73 VI. FM Rate Error Simulation Results for JPL Nominal Parameters 74 VII. FM Rate Error Simulation Results for COMSS/LASS Parameters 75 VIII. RCM Limits on Coarse Resolution Aperture Size 89 IX. Memory Requirements and Computation Rates 98 X. Memory Requirements and Computation Rates for Alternative Approaches v i i i L i s t of Figures 1. Reference and Target Return Signal in Spectral Analysis 12 2. Time-Frequency Plot After Mixing 15 3. Bandpass F i l t e r Spectral Analysis 17 4. Positions of Target Return and Reference Signals 21 5. Reordering the Data for Input to the Fine Resolution FFT 25 6. Input Data to the Fine Resolution FFT for 2 Point Targets 32 7. Spacing of Data Input to Fine Resolution FFT 34 8. Data Input and Output for the Fine Resolution FFT ....41 9. SAR Geometry 60 10. Flowchart of Simulation Program 65 11. % Broadening vs. FM Rate Error From Simulation Results ix 77 12. E f f e c t of FM Rate Error on Addition of Looks for SEASAT Case 79 13. Generalized Computation Requirements as a Function of Signal Aperture Extent 102 14. Step Transform Processor Architecture 111 X Acknowledgement I would l i k e to thank Dr. M. R. Ito for providing much guidance and encouragement throughout the course of th i s research. I would also l i k e to thank the people at MacDonald, Dettwiler, and Associates for providing the detailed information on SAR systems and research in the area, much of which i s not available in published l i t e r a t u r e . In p a r t i c u l a r , the assistance of Dr. Ian Cumming, in reviewing drafts of t h i s paper and providing useful suggestions to improve i t , is very much appreciated. I g r a t e f u l l y acknowledge the f i n a n c i a l support of the Natural Sciences and Engineering Research Council in the form of a Postgraduate Scholarship. I also acknowledge the f i n a n c i a l assistance of a Teaching Assistantship from the University of B r i t i s h Columbia. 1 I. INTRODUCTION The step transform [28] i s a technique for pulse compression 1 of linear FM waveforms. It has important applications in radar signal processing where pulse compression techniques are used to achieve the f u l l resolution inherent in the sig n a l . The linear FM or "chirp" signal i s in common use in radar systems because i t i s easy to generate, provides both good resolution and high energy, and i s easy to process [31]. In pa r t i c u l a r , synthetic aperture radar (SAR) signals have a Doppler history which i s a lin e a r FM waveform in the azimuth d i r e c t i o n . This fact i s determined by the physical properties of SAR systems, and so a l l SAR systems require some form of line a r FM pulse compression. The step transform technique of pulse compression i s based on the concept of spectral analysis. In essence, the linear FM signal to be compressed (target return) i s multi p l i e d by a reference signal with the same FM rate but opposite slope. The product i s a constant frequency signal whose frequency i s dependent on the r e l a t i v e positions of the reference and target return signals. A Fourier transform of the product resolves the target return data into the f i n a l output image. (Spectral analysis i s described in greater d e t a i l in Chapter II.) The step transform i s a refinement of thi s technique which uses a series of short reference ramps. The product i s a series 1 Pu lse compression i s e q u i v a l e n t to matched f i l t e r i n g in radar t e r m i n o l o g y . 2 of constant frequency signals of increasing frequency, which are each resolved using an i n i t i a l Fourier transform whose duration is the same as the short ramp. The output data of these coarse resolution transforms i s then reordered and applied to a second fine resolution transform which achieves the f u l l resolution inherent in the s i g n a l . 1.1 Objectives The objective of t h i s thesis i s to analyze the step transform method of pulse compression and understand more f u l l y i t s properties, l i m i t a t i o n s , and a p p l i c a b i l i t y to signal processing problems. Its application to the azimuth processing of SAR signals i s of interest and w i l l be examined in d e t a i l . S a t e l l i t e SAR systems present some p a r t i c u l a r l y challenging problems in signal processing and w i l l be the prime application area focussed on. Memory requirements, computation requirements, and control complexity w i l l be examined. Methods of dealing with p a r t i c u l a r SAR problems such as range c e l l migration and multi-look processing are described. The effect of errors in the estimation of the FM rate i s also examined. An analysis of the performance of the step transform r e l a t i v e to other techniques i s presented. 3 1.2 Structure Of The Thesis The approach used in the study of the step transform was to develop a basic mathematical understanding of the technique. The concepts demonstrated in t h i s t h e o r e t i c a l framework were used to write a computer simulation program which could perform step transform processing on a linear FM si g n a l . At the same time, a basic understanding of SAR systems was developed. The f i n a l stage of the research was to examine how the step transform would be used in the SAR azimuth processing problem and to examine i t s performance r e l a t i v e to some other approaches. Thus, the various issues related to the thesis topic w i l l be addressed in the following manner. Chapter II describes the commonly used techniques for lin e a r FM pulse compression as applied to azimuth SAR processing, including time-domain convolution, fast convolution, and spectral analysis. Their strengths and weaknesses and common areas of application are discussed. Spectral analysis i s presented in some d e t a i l . Chapter III describes the step transform in d e t a i l . A mathematical analysis is performed to determine the form of the output for a single point target. Some considerations for choosing the various processing parameters are discussed. In pa r t i c u l a r , i t i s shown how a l i a s i n g of adjacent targets at the same frequency i s prevented by a proper choice of parameters. Chapter IV examines the considerations used to choose processing parameters in greater d e t a i l . The data windows chosen to reduce sidelobe l e v e l s can have a s i g n i f i c a n t e f f e c t 4 on the processing requirements. The properties of the windows which p a r t i c u l a r l y a f f e c t the step transform technique are described and examples are given for some common windows. Formulae are derived to determine the actual computation rates required. Since most d i g i t a l SAR systems require multilook processing, the eff e c t of thi s requirement on the computation rates and other aspects of the step transform i s also examined. F i n a l l y , a comparison i s made between the step transform and bandpass f i l t e r spectral analysis. Chapter V presents the major d e t a i l s of SAR azimuth processing which w i l l allow us to v i s u a l i z e how the step transform f i t s into t h i s a p p l i c a t i o n . The format of the SAR azimuth signal i s presented. The eff e c t of the antenna p r o f i l e on processing of the signal i s discussed. The basic formulae which determine the FM rate of the signal and the extent of range c e l l migration are presented. A simulation program, written to perform the pulse compression operation on a single l i n e of target return data using the step transform, i s described. Chapter VI s p e c i f i c a l l y considers two problems related to the azimuth processing of SAR signals using the step transform, FM rate error and range c e l l migration. Basic t h e o r e t i c a l analysis i s supported by results obtained using system parameters for several d i f f e r e n t SAR systems. The eff e c t of FM rate errors on the image qu a l i t y i s analyzed and simulation results are presented. The ease with which range c e l l migration corrections are performed i s determined largely by the azimuth 5 pulse compression technique chosen. The performance of the step transform in thi s regard i s examined. After evaluating a l l of the above considerations, i t w i l l become apparent that there are certain requirements placed on the processing parameters which could p o t e n t i a l l y c o n f l i c t with each other. In Chapter VII, a l l of these considerations are drawn together and lead us to choose processing parameters for some t y p i c a l SAR systems. Computation rates and memory requirements are evaluated and compared to the equivalent requirements for some other pulse compression techniques. F i n a l l y , some comments are made about possible hardware implementations of the step transform. Chapter VIII presents some conclusions about the strengths and l i m i t a t i o n s of step transform pulse compression and i t s a p p l i c a b i l i t y to SAR processing. In addition, there are four appendices which present additional information. Appendix A sp e c i f i e s some data windows which might be used in the step transform process. Appendix B l i s t s the various acronyms, abbreviations, and symbols used in the paper. Appendix C presents the system and processing parameters for the three SAR systems which are used as t y p i c a l examples of poten t i a l step transform applications. Appendix D gives a basic functional s p e c i f i c a t i o n of the computer simulation program. 6 I I . SURVEY OF PULSE COMPRESSION TECHNIQUES In t h i s chapter, some commonly used approaches to performing pulse compression on radar signals w i l l be described. Each w i l l be evaluated on the basis of i t s a p p l i c a b i l i t y to the processing of SAR azimuth signals, although some have a much broader range of applications. One of the major c r i t e r i a for choosing a pa r t i c u l a r algorithm i s the computation rate required and the control complexity of the algorithm. These are determined to a large extent by the parameters of the pa r t i c u l a r radar system. Ty p i c a l l y , airborne systems have f a i r l y modest computation requirements and do not require complicated corrections for range c e l l migration (RCM) e f f e c t s . Therefore the processing algorithm chosen would l i k e l y not be the most e f f i c i e n t in terms of the number of f l o a t i n g point operations performed, but would possess a simple control structure. S a t e l l i t e SAR systems, on the other hand, require a large computational capacity and must have complicated correction factors applied to obtain acceptable image q u a l i t y . The high computation rates mean that i t may be worthwhile to implement an algorithm with a complex control structure. The gain in computational e f f i c i e n c y may be small on a percentage basis, but is magnified because of the scale of the system. Also, because of the various corrections which must be applied, i t i s l i k e l y that a sophisticated control structure w i l l be required in any event. In p a r t i c u l a r , most s a t e l l i t e SAR systems require special range c e l l migration correction (RCMC) techniques to 7 correct the quadratic component of RCM (QRCM). In order to perform RCMC â€˘ e f f i c i e n t l y , i t i s necessary either to i s o l a t e a number of targets in the same area or to separate the azimuth signal into d i s t i n c t sections. The algorithms which allow t h i s to happen are pointed out. Kirk [20] has presented an assessment of d i g i t a l algorithms for airborne SAR systems. Bennett et a l [6] and Wu [36] have considered the a p p l i c a b i l i t y of various pulse compression algorithms to s a t e l l i t e SAR processing. In the following sections, we w i l l consider the following pulse compression techniques and their a p p l i c a b i l i t y to SAR azimuth processing: 1) time-domain convolution, 2) fast convolution, 3) basic spectral analysis, and 4) bandpass f i l t e r spectral analysis. Basic spectral analysis i s described in greater d e t a i l since i t provides the basis for the step transform. 2.1 Time-Domain Convolution The time-domain convolution technique i s e s s e n t i a l l y a straightforward implementation of the convolution concept without any shortcuts. Thus i t i s the most accurate technique and exhibits a r e l a t i v e l y simple control structure. However, the computation rate increases as (R/A)-M where M = the number of azimuth samples for a single point target and R/A = length of the input data vector. 8 It becomes prohi b i t i v e for large M. The algorithm also requires that RCMC be performed for each target i n d i v i d u a l l y . If quadratic RCMC is required, the technique becomes extremely i n e f f i c i e n t . Computationally, time domain convolution is the most e f f i c i e n t algorithm for M < 32. However, the simple control structure makes i t applicable for situations where M < 50. Because of the large number of re p e t i t i v e operations and inherent parallelism, time-domain convolution also becomes more a t t r a c t i v e i f a custom VLSI implementation i s considered f e a s i b l e . This approach was chosen by JPL for the development of s a t e l l i t e SAR systems with on-board processing, as described by Arens [1] and Wu [37]. Kuhler [21] and Tyree [34] assessed VLSI technologies to determine the v i a b i l i t y of such an approach, with a major focus on CCD's. 2. 2 Fast Convolution The fast convolution technique performs the convolution operation in the frequency domain. It involves the following steps: 1) translate the input signal to the frequency domain using a Fourier transform, 2) multiply the result by a reference signal (in the frequency domain), 3) translate the result to the time domain using an inverse Fourier transform. 9 This technique i s computationally more e f f i c i e n t than the time-domain method since the number of computations required in step 2 increases as (R/A)â€˘log 2{M}. The forward and reverse Fourier transforms can be performed using an FFT algorithm which requires MÂ«log 2{M}/2 complex mu l t i p l i c a t i o n s i f M i s a power of 2. The computational e f f i c i e n c y i s increased dramatically for large M. Quadratic RCMC can be performed more e f f i c i e n t l y than in the time-domain method. When the input signal i s translated to the frequency domain, a l l targets at the same slant range are aligned on top of each other in signal memory. Thus the correction for a l l targets at the same slant range can be performed simultaneously. It i s also possible to separate out portions of the azimuth sig n a l . Implementations of t h i s method are described by Bennett and Cumming [4], van de Lindt [35], and E l l i s [14], among others. If the SAR processor being designed i s required to perform both range and azimuth pulse compression, this technique makes i t possible to perform the complete operation at once using two dimensional FFT's. 2.3 Spectral Analysis The fundamental basis for the step transform technique i s the concept of spectral analysis [24]. The spectral analysis technique of pulse compression w i l l be considered from both a the o r e t i c a l and p r a c t i c a l viewpoint. Some of the problems and 1 0 proposed solutions w i l l be considered. Consider an input signal s(t) = exp{-J7rKt 2} , -T/2 < t < T/2. Sampling at a rate F = 1/A, the discrete form of the signal i s s(k) = exp{-J7rKA 2k 2} , k = -T/2A . . . -1,0,1 . . . T/2A - 1 and for a signal with a time delay of k, samples s(k) = e x p { - J 7 r K A 2 ( k - k 1 ) 2 } , k = -T/2A + k, . . . -1,0,1 . . . T/2A + k, - 1 A matched f i l t e r operation i s performed on this signal by performing the following computation: T/2A + k, - 1 m(r) = Z s(k-k 1) s*(r-k) k = -T/2A + k, = Z s(k-k,) exp{ J T T R A 2 ( r 2 - 2 r k + k 2) k = exp{ jirKA 2r 2} Z s(k-k,) exp{ j TTKA 2 k 2 } exp{ - j 2TTKA 2 r k } k Since the objective i s to detect the magnitude of the output, the phase term outside the summation can be discarded. 2 With the r e s t r i c t i o n that T = 1/KA, the summation then represents the discrete Fourier transform of the product of the input signal and a r e p l i c a with the opposite slope. Computing th i s product, P(k) = exp{-J7rKA 2 (k-k,) 2} exp{ JTTKA 2 k 2 } = exp{-J7rKA 2k, (k, - 2 k ) } Thus P(k) i s a constant frequency signal and the actual 2 This i s not the case for the step transform, where t h i s phase information i s used to perform a second stage of processing. 11 frequency i s dependent only on the time delay, k,, of the input s i g n a l . The above analysis suggests an equivalent method of performing the matched f i l t e r i n g operation would be to multiply the target return by a reference function of opposite slope (FM modulation) and then perform a DFT.3 The major advantage of t h i s concept, i s that only one FFT i s required, as opposed to the fast convolution technique which requires both forward and inverse FFT's. However, spectral analysis i s applicable only to linear FM signals, whereas both time-domain convolution and fast convolution can be used for other types of signal coding. 2.3.1 Reference Function Bandwidth Limitations It i s necessary to consider the e f f e c t of a f i n i t e time aperture and sampling in the time domain. A time-frequency plot of the sampled reference function i s shown in Figure 1a. Due to sampling of the waveform at the pulse r e p e t i t i o n frequency (PRF) the plot forms a sawtooth pattern with a period of F/K seconds. This leads to the question of what happens to a point target return which straddles the discontinuity as in Figure 1b. After the mixing operation, the signal to the l e f t of the "discontinuity" has a frequency KA2k,. To the right of the discontinuity, the reference function has been shifted in time F/K seconds. Therefore P(k) = s(k-k,) s*(k - 1/KA) 3 This concept i s also used in the design of frequency synthesizers, as described by Darby and Hannah [12]. 12 Figure 1 - Reference and Target Return Signal in Spectral Analysis â€˘ Time a. The Sampled Reference Function REFERENCE â€˘ Time TARGET RETURN Time -F/2 b. Target Return in Relation to Reference 13 = exp{-J7rKA 2 (k-k, ) 2} exp{jirKA2 (k - 1/KA)2} = exp{-J7rKA 2 (k 2 - 2'kk, + 2k/KA - 1/K2A2)} Thus the frequency on both the l e f t and right hand sides of the discontinuity are the same. The discontinuity i s at k = 1/2KA. The phase on the right side of the discontinuity i s - 7 r K A 2 (k 2 + 2k/KA - 1/K2A2) = - 7 r K A 2k 2. This i s exactly the phase on the l e f t side of the discontinuity. Thus, the reference function discontinuity does not have any ef f e c t on the DFT processing, since there i s no apparent discontinuity after the mixing operation. This result can be explained by considering the addition of frequencies in the mixing operation to be done with modular arithmetic because of the sampling of the waveforms. Al t e r n a t i v e l y , one can consider the reference function to continue upward in frequency instead of taking the sawtooth pattern shown, bounded by the sampling frequency F. In some papers on the step transform [28][29], a rudimentary description of the spectral analysis technique i s given. The implication i s made that the reference function must be of a f i n i t e duration of F/K seconds and, because the target return does not usually overlap exactly with the reference, there i s an energy loss during the mixing operation. In fact, the above results show that the reference function does not have to be of f i n i t e duration, contradicting the assertions made in the above c i t e d papers. There are problems, however, because of the f i n i t e length of the FFT aperture. 1 4 2.3.2 Application To SAR Azimuth Processing The fact that the target return signal occurs only over a f i n i t e aperture imposes a more s i g n i f i c a n t l i m i t a t i o n on the basic spectral analysis approach. Adjacent;- targets do not occupy exactly the same aperture after the mixing operation, as shown in Figure 2. (In fact, a parallelogram-shaped processing region i s formed.) As a result, using a single FFT to compress both targets results in a loss of information ( i f the FFT i s too short) or a l i a s i n g of adjacent targets F/K seconds away ( i f the FFT i s too long). There are several ways of dealing with t h i s problem. The most obvious is to simply overlap the FFT's so each target i s covered s i g n i f i c a n t l y by at least one FFT aperture. FFT output points which contain a s i g n i f i c a n t amount of energy from two d i f f e r e n t targets are rejected. Thus, a crude f i l t e r i n g operation i s performed to prevent a s i g n i f i c a n t degree of a l i a s e d energy from adjacent point targets being resolved. For the production of lower resolution images, t h i s method works very well. By performing t h i s f i l t e r i n g operation in a more sophisticated manner, the computation rate can be reduced s i g n i f i c a n t l y for high resolution systems. For many SAR systems, i t i s also necessary to is o l a t e small groups of targets in the same area to perform e f f i c i e n t range c e l l migration correction, as discussed in Chapter VI. For these reasons, alter n a t i v e s to the basic spectral analysis approach are sought. 1 5 Figure 2 - Time-Frequency Plot After Mixing TARGET RETURNS BEFORE MIXING â€˘ Time TARGET TARGET RETURNS AFTER MIXING Time 16 2.4 Bandpass F i l t e r Spectral Analysis One such alternative i s to d i r e c t l y f i l t e r the output of the mixer into several d i s t i n c t passbands, as shown in Figure 3 and described in [24]. Thus the processing region i s divided up into several subregions, in each of which the energy loss or a l i a s i n g is minimized. An FFT i s performed on each subregion to resolve the point targets. Since each passband i s down-converted to baseband and subsampled, the number of FFT points to be processed is approximately the same as i f there was no overlapping of the FFT's. The decrease in the number of FFT points largely compensates for the additional computations required to perform the bandpass f i l t e r i n g operation. Since each f i l t e r separates out targets in azimuth proximity to each other, i t i s possible to perform a single quadratic RCMC operation on a group of targets, allowing t h i s correction to be performed e f f i c i e n t l y . This method i s c a l l e d bandpass f i l t e r spectral analysis (BPF SPECAN). In succeeding chapters, some analogies w i l l be drawn between the BPF SPECAN and step transform approaches. 17 Figure 3 - Bandpass F i l t e r Spectral Analysis A Frequency F i l t e r Channels 18 I I I . THE STEP TRANSFORM In th i s chapter, a rigorous theore t i c a l analysis of the step transform i s presented in order to gain some insight into the fundamental l i m i t a t i o n s of the technique. This analysis i s useful for any linear FM pulse compression application, not just SAR azimuth processing. Step transform pulse compression i s performed in several stages of computation and data manipulation. I n i t i a l l y , the target return signal i s mixed with a series of short overlapping reference signals. The reference signals have the same FM rate but opposite slope to that of the target return. The time extent of each short reference signal i s a coarse resolution aperture. An appropriate window function i s also applied to each coarse resolution aperture to reduce the sidelobes on the FFT output. The amount of overlapping of the coarse resolution apertures i s determined by the width of the mainlobe of the window function chosen. For a single point target, the data in each aperture represents a constant frequency s i g n a l . The exact frequency i s dependent on the position of the point target (see Sec. 2.3) and the frequency from one aperture to the next i s incremented by a constant value. An FFT i s applied to each coarse resolution aperture to compress the information from a single point target into a small number of FFT output bins. Thus, the relevant data from a single point target l i e s in d i f f e r e n t output bins in each successive coarse resolution FFT. The data i s related by a line a r phase term dependent on the target return and a quadratic 19 phase term dependent on the position of the short reference ramp. The quadratic phase term i s removed and the data i s aligned sequentially in a fine resolution aperture. This aperture now contains a constant frequency signal, the frequency of which i s dependent only on the point target position. A suitable data window (to reduce sidelobes in the f i n a l output image) and FFT are applied to each fine resolution aperture. Due to overlapping of the coarse resolution apertures and possible a l i a s i n g between coarse resolution output pulses, not a l l of the fine resolution FFT output data i s valid, and, therefore, a data selection process must be included. Since the coarse resolution FFT output function is sampled at a d i f f e r e n t p osition for each target, i t may be necessary to perform a scaling operation to produce a high quality output image. 3.1 Mathematical Analysis I n i t i a l l y , the step transform w i l l be considered s t r i c t l y on a theore t i c a l l e v e l to gain a better understanding of the mechanisms involved." The target return signal i s represented by the function s(t) = exp{ j (7rB/T)t 2} and i s v a l i d over the region -T/2 < t ^ T/2. The reference signal i s represented by a series of ramps of the following form: 4 Preliminary portions of t h i s derivation e s s e n t i a l l y p a r a l l e l the analysis presented by Perry and Martinson [29]. 20 s*(t) = exp{-j (7rB/T)t 2} , -T'/2 < t < T'/2 The i n i t i a l analysis uses only one reference ramp at position t = nA multiplied by a target return signal at position t = -mA, as shown in Figure 4a. 5 The resu l t i n g expression i s as follows: s(t+mA) s*(t-nA) = exp{ j (7rB/T) (t+mA) 2 } exp{-j (7rB/T) (t-nA) 2} Let the sample interval be A and k be an integer sample number over the time aperture of the reference ramp, 0 < k < T'/A - 1. Therefore t = kA + nA - T'/2 and s(t+mA) s*(t-nA) = s(k,m) s*(k,n) exp{ j (TTB/T) (kA + nA - T'/2 + mA)2} â€˘ exp{-j (TTB/T) (kA + nA - T'/2 - nA) 2} exp{ j (TTB/T) (2kA 2 (m+n))} â€˘ exp{jUB/T)(m 2A 2 - T'Am)} â€˘ exp{ j (TTB/T) (n 2A 2 - T' An )} â€˘ exp{jUB/T)(2mnA 2)} The f i r s t factor, which includes the variable k, i s resolved by the coarse resolution FFT applied over the time aperture of the reference ramp. This phase factor determines the frequency of the signal to be resolved by the FFT. The other phase factors are constant within the time aperture of the ramp and are dealt with at la t e r stages of the step transform process. Therefore l e t <t> = (TTB/T) ((m 2A 2 - T' Am) + (n 2A 2 - T'An) + (2mnA2)) (1) 5 Figure 4 shows the target and reference signals as being p a r a l l e l . In r e a l i t y , the slopes are exactly opposite. The drawing i s meant to i l l u s t r a t e that the two signals have the same FM rate. 21 Figure 4 - Positions of Target Return and Reference Signals A Frequency Time a. Target Return Signal and Reference Ramp Frequency Time b. Multiple References Spaced at Regular Intervals 22 so that s(k,m) s*(k,n) = exp{j<5} exp{ j (JTB/T) (2kA 2 (m+n))} The next stage requires that an FFT of length T'/A be applied to the product calculated above. A data window (represented by W,(k)) may also be applied to the signal to reduce sidelobes. The resulting output i s a sequence of values determined as follows: T'/A - 1 s(r) = Z W,(k) s(k,m) s*(k,n) exp{-j ( 2TTA/T') kr} , k=0 r = 0 . . . T'/A - 1 In order to simplify the mathematics i n i t i a l l y , a boxcar window w i l l be used so that W,(k)=1 for a l l k. Thus T'/A - 1 s(r) = exp{jc6} Z exp{ j (TTB/T) (2kA 2 (m+n))} exp{-j (2TTA/T' )kr} k=0 T'/A - 1 = exp{j<6} Z exp{ j27rkA(BA(m+n)/T - r/T')} k=0 Evaluating the summation using the series X sin{(X+1)0/2} Z exp{jk0} = exp{jX0/2} , 0 * 0 sin{Â©/2} k=0 = X + 1 , Â© = 0 such that X = T'/A - 1 and 0 = 27rA(BA(m+n)/T - r/T'), the output of the coarse resolution FFT may be expressed as 23 s(r) = exp{j0} exp{j(T'/A - 1 ) (7rA( BA(m+n) /T - r/T'))} sin{7rT'(BA(m+n)/T - r/T')} . f r jt T'BA(m+n)/T sin{7rA(BA(m+n)/T - r/T')} = T'/A , r = T'BA(m+n)/T Thus s(r) i s a pulse centred at r = T'BA(m+n)/T. Note that for most point targets, t h i s w i l l not evaluate to an integer value of r. It i s now necessary to consider the processing of data from many reference ramps. The ramps are spaced NA time units apart at n = N(i+a), i+a = . . . -2,-1,0,1,2, . . . (Figure 4b). The variable i i s used to index data within each fine resolution aperture while a denotes the coarse resolution aperture from which the f i r s t data point i s picked. Substituting t h i s formula for n, the centre position of the coarse resolution FFT output pulses may be expressed as r = T'BA(m+N(i+a))/T. Thus the number of c o e f f i c i e n t s separating pulses in successive coarse resolution apertures i s H = BT'AN/T. This value i s used to increment r when data is reordered for input to the fine resolution FFT. Since r i s an integer, the increment H must also be a po s i t i v e integer value or close to i t to minimize undesired amplitude modulations. (Alternatively, H may be the inverse of an integer value. In that case, fine resolution aperture data would be picked from every 1/H coarse resolution apertures. The derivation for t h i s case i s presented in the next section.) Since the range of values for r i s 0 . . . T'/A - 1, the number of samples in each fine resolution 24 T'/A aperture i s = T/BA2N.6 A mathematical expression for the H input data to the fine resolution FFT is derived by substituting n = N(i+a) and r = iBT'AN/T into the expression for s ( r ) . Thus BA(m+n)/T - r/T' = BA(m+N(i+a))/T - iBT'AN/(TT') = BA(m+Na)/T Leaving the <t> phase factor alone for the moment, we see that the expression for s(r) becomes s(i,a) = exp{jc6} exp{j(T'/A- 1 ) (7rA2B(m+Na)/T) } sin{7rT'BA(m+Na)/T} â€˘ , i = 0 . . . (T/BA2N) - 1 sin{7rA 2B(m+Na)/T} Since the variable a i s a constant over the complete fine resolution aperture, the function s(i,a) has a constant amplitude for each value of i . The phase i s modified only by the factor exp{jc6}. The reordering process i s better understood with the help of Figure 5. In t h i s i l l u s t r a t i o n , each column represents the data output of one coarse resolution FFT. Some of the pulses for a single point target are drawn in their r e l a t i v e positions. The objective of the reordering process i s to assemble a l l of the samples from the same target and with the same magnitude, as shown by the diagonal l i n e s , in sequence. This i s possible 6 Perry and Martinson [29] c i t e an example in which the coarse resolution FFT contains 32 samples while the fine resolution FFT contains 64 samples (corresponding to 2 coarse resolution apertures). The analysis presented here suggests that such an implementation would not work very well, since there would be a d i s t i n c t phase and magnitude discontinuity in the middle of the fine resolution FFT data. Further comments w i l l be made in Sec. 4.4, where multilooking i s discussed. 25 F i g u r e 5 - R e o r d e r i n g t h e Data f o r Input t o the F i n e R e s o l u t i o n FFT 26 because the output of each coarse resolution FFT i s exactly the same, except for an appropriate s h i f t in frequency and phase. Note also the triangular regions of unused data at each end. The s i t u a t i o n where H = 1 i s quite easy to v i s u a l i z e in terms of the processing algorithm and mathematical analysis being described. However, the si t u a t i o n where H > 1 i s not quite so simple. In th i s case, the reordering process i s modified so that data for H fine resolution apertures i s picked from the same set of coarse resolution apertures. The range of values for i for each of the fine resolution apertures would be as follows: 1st - 0 . . . T/BA2N - 1 2nd - 1 . . . T/BA2N Hth - H-1 . . . T/BA2N - 2 + H The variable a is incremented by 1 after every Hth fine resolution aperture. The l i m i t a t i o n on the value of H i s explained in another way by following the reasoning of Elachi et a l [13] in their treatment of the step transform. A condition i s imposed that the number of coarse resolution apertures over the return signal be equal to the number of frequency resolution elements in the output of the coarse resolution FFT, which cover the bandwidth of the return signal. This statement may be expressed in the following equation 27 T T' BT'AN â€” = â€” â€˘ BA or = 1 NA A T Note that the left-hand side of the second equation i s simply H, the value used to increment r. Elachi uses the s i m p l i f i e d case of no overlap, i . e . T' = NA, to conclude that T' = SQRT{TÂ«B}. Clea r l y , t h i s i s in error and should read T' = SQRT{T/B}. This statement by Elachi may have been inspired by Martinson's assertion [25] that the time-bandwidth product of a single reference ramp should be approximately equal to the square root of the time-bandwidth product of the return s i g n a l , which i s l i k e l y also in error. The variable 0 , as defined in equation (1), has three main components. The term (7rB/T) ( 2mnA2 ) = (irB/T) ( 2mN( i+a) A 2 ) increments the phase of s(i,a) by a constant value for each succeeding term. This represents a CW signal whose frequency i s dependent on m, the position of the target return s i g n a l . The fine resolution FFT resolves the information presented in t h i s phase factor. The term (7rB/T)(n 2A 2 - T'An) represents a quadratic phase factor which i s dependent on n, the position of the coarse resolution aperture and i s independent of point target p o s i t i o n . It must be removed prior to processing the fine resolution FFT, to make the input to t h i s FFT a CW s i g n a l . Note that removal of t h i s factor can be done at any time prior to processing of the fine resolution FFT, including during the m u l t i p l i c a t i o n of the reference and target return signals. The term (7rB/T)(m 2A2 - T'Am) represents a quadratic phase factor which i s constant across a l l of the target return data and 28 therefore does not affe c t processing of the data. Thus the input data to the fine resolution FFT takes the following form: s(i, a ) = exp{j ( T T B / T ) (2mNiA2 )} â€˘ exp{ j (T TB/T) (A2m) (m-1 )} exp{j (7rB/T) (ANa) (2mA-A+T' )} sin{ (T TB/T) (T' A) (m+Na)} (2) sin{ (T TB/T) (A 2 ) (m+Na)} The output of the FFT i s obtained by evaluating the following expression: T/BA2N -1 s(c,a) = Z W 2(i) s(i,a) exp{-j27r(BA2N/T) ic} , i=0 c = 0 . T/BA2N - 1 W 2(i) i s a data window chosen to minimize sidelobes in the f i n a l output. For the purposes of t h i s analysis, i t i s assumed that W 2(i)=1, i . e . a boxcar window i s used. Thus, s(c,a) = exp{ j (T TB/T) (A2m) (m-1 ) } â€˘ exp{ j (T TB/T) (ANa) (2mA + T' - A)} sin{ ( T T B / T ) (T' A) (m+Na) } sin{(7rB/T) (A 2 ) (m+Na)} T/BA2N - 1 Z exp{ j ( T T B / T ) (2mNiA2 ) } exp{-j ( T T B / T ) ( 2NicA 2 ) } i = 0 The summation evaluates to the following expression: s(c,a) = exp{J7rm} exp{ j (7rB/T) (A2m) (m-N-1 ) } â€˘ exp{ j (7rB/T) (ANa) (2mA + T* - A)} sin{ (T TB/T) (T'A) (m+Na) } sin{ (irB/T) ( A 2 ) (m+Na)} exp{-J7rc }exp{ j (rrB/T) ( C N A 2 )} 29 sin{7r(m-c)} sin{ (T TB/T) (NA2) (m-c)} and sin{ ( T T B / T ) (T'A) (m+Na)} |s(c,a)| = sin{ (T TB/T) (A 2) (m+Na)} sin{7r(m-c)} . (3) sin{ (T T B / T ) (NA2 ) (m-c) } The magnitude of s(c,a) is the data from which the f i n a l output image i s obtained. In order to prevent multiple images i t may be necessary to reject some data. An explanation of the data selection procedure i s provided in Sec. 3.4, after the need to produce th i s data in the f i r s t place has been j u s t i f i e d . Suitable data windows on the coarse and fine resolution apertures are required to produce images with sidelobes lev e l s acceptable for most SAR applications. The f i r s t l i n e of equation (3) would be modified according to the window chosen for the coarse resolution aperture, W,, while the window chosen for the fine resolution aperture, W2, would result in appropriate modifications to the second l i n e . 3.2 Alternative Case For the case where H < 1, the number of fine resolution aperture samples i s T'/A. Substituting n = N(i+a)/H and r = i into the equation for s(r) and noting that H = BT'AN/T results in the following expression: BA(m+n)/T - r/T ? = BA(m+N(i+a)/H)/T - i/T* 30 = BAm/T + a/T' Therefore the fine resolution aperture data may be expressed as follows after removing the quadratic phase factor: s(i,a) = exp{ j (jrB/T) (m2A2 - T' Am) } â€˘ exp{ j (7rB/T).(2mN(i+a)A2/H)} â€˘ exp{j(T'/A " l)(irA(BAm/T + a/T'))} sin{;rT'(BAm/T + a/T')} s i n {it A (B Am/T + a/T')} Applying an FFT to thi s data as before, T'/A - 1 s(c,a) = Z W 2(i) s(i,a) exp{-j27ric (A/T')} i = 0 Assuming W 2(i) = 1 for a l l i , s(c,a) = exp{ j (7rB/T) (A2m(m - 1 + 2Na/H)) } â€˘ exp{j;ra(1 - A/T')} sin{irT'(BAm/T + a/T')} sin{7rA(BAm/T + a/T')} exp{ J7r(m - c) (1 - A/T' )} sin{7r(m-c)} sin{7r(m-c) A/T' } and | s(c,a)| sin{7rT ' (BAm/T + a/T')} s i n { 7rA (BAm/T + a/T')} sin{7r(m-c)} sin{7r(m-c)A/T' } This, of course, i s very similar to the expression derived in the preceding section and the same comments apply. In fact, for 31 H = 1, they turn out to be i d e n t i c a l . The reordering process has b a s i c a l l y been described in the previous section. The variable a i s incremented by 1 for each fine resolution aperture but successive fine resolution data points are taken from every 1/H apertures. 3.3 The A l i a s i n g Problem With the above background information, i t i s possible to consider some of the parameter l i m i t a t i o n s in order to be able to resolve individual point targets. Again, the situations for H < 1 and H ÂŁ 1 must be dealt with separately. Considering the situation where H > 1 f i r s t , the magnitude of the input data to the fine resolution FFT as a function of a i s determined by the data window for the coarse resolution aperture, W,. From equation (2) sin{ (irB/T) (T'A) (m+Na) } | s ( i , a ) | = sin{ (T TB/T) (A 2) (m+Na) } for a boxcar window. The set of input data for each fine resolution FFT has a frequency determined by the phase factor exp{ j (7rB/T) (2mNiA 2)}. Consider the situation where both point targets of Figure 6 have the same frequency as seen by the fine resolution FFT, i . e . (7rB/T) (2m,NA2) = (7rB/T) (2m2NA2) + 27rl , 1 = any integer Since the target return information i s spread over many coarse resolution apertures and the FFT does not concentrate the energy from a target into a single output c o e f f i c i e n t , there i s a 32 F i g u r e 6 - Input Data to the Fine R e s o l u t i o n FFT f o r 2 P o i n t T a r g e t s Magnitude Target a t t = -m,A Â» a 33 potential a l i a s i n g problem between the two targets. For the two closest such point targets 1=1 and m,-m2 = T/BNA2. The number of coarse resolution apertures separating the two targets i s given by a 2 - a 1 = (m2-in1)/N = T/BN2A2 = (T'/NA) â€˘ (T/BT'AN) = (T'/NA) / H T'/NA = ( a 2 - a,) â€˘ H Thus the overlap r a t i o , T'/NA, exactly defines the number of samples separating the coarse resolution FFT output pulses for the two targets and the number of fine resolution apertures between them. In order to minimize t h i s a l i a s i n g e f f e c t in processing the fine .resolution aperture, processing parameters must be chosen to ensure that such interference occurs only at the sidelobe l e v e l s of the windows chosen for the coarse resolution aperture. The spacing between main lobes of p o t e n t i a l l y i n t e r f e r i n g targets can be expressed as the product H(a,-a 2) and must be kept above a certain minimum. Since T'/AN i s d i r e c t l y proportional to the amount of computation required in processing the coarse resolution apertures, i t i s desirable to minimize t h i s value. A c r i t i c a l parameter in determining the minimum-permissible value of T'/AN i s the width, W, of the main lobe of the window chosen for the coarse resolution aperture. Consider the three situations depicted in Figure 7. The image quality which can be gained by using a main lobe separation as shown in 34 F i g u r e 7 - Spacing of Data Input to F i n e R e s o l u t i o n FFT Magnitude ^ a b. B o r d e r l i n e I n t e r f e r e n c e i n Main Lobe Magnitude Next t a r g e t at same frequency +~ a >- a c. D e f i n i t e I n t e r f e r e n c e i n Main Lobe 35 Figure 7a i s minimal. This i s especially true for windows with constant sidelobe l e v e l s , eg. Dolph-Chebyshev. It is actually permissible to allow some interference of the main lobes as long as the dimension a'^1 (Figure 7c). This ensures that there i s at least one fine resolution aperture for each point target which does not experience interference from other targets at the same frequency (other than at sidelobe l e v e l s ) . For this condition, T'/NA > (W+l)/2. For the situation where H < 1, the frequency of the fine resolution aperture i s determined by the factor exp{ j (7rB/T) (2mNiA2/H)}. For two point targets with the same f requency ( T T B / T ) (2m,NA2/H) = (irB/T) (2m2nA2/H) + 2*1. For the two closest targets m, - m2 = TH/BNA2 The number of coarse resolution apertures separating the two targets i s given by a, - a 2 = T'/NA In choosing processing parameters for t h i s s i t u a t i o n , i t must be considered that the coarse resolution FFT output function is a c t u a l l y sampled at a rate 1/H times that of the situation where H > 1. Therefore, the mainlobe target data i s contained in 1/H times as many fine resolution apertures and the processing parameters are r e s t r i c t e d by the r e l a t i o n s h i p (T'/NA)-H > (W+H)/2 Substituting H = BT'AN/T, the relationship becomes BT' 2/T > (W+H)/2.. By substituting in some actual values, i t can be. demonstrated that, i f T' i s large enough to s a t i s f y the above relationship, 36 t h e n H > 1. V a l u e s f o r some t y p i c a l SAR s y s t e m s a r e s p e c i f i e d i n A p p e n d i x C a n d t h e m e t h o d o l o g y u s e d t o c h o o s e t h e p r o c e s s i n g p a r a m e t e r s i s d i s c u s s e d i n C h a p t e r V I I . The m a i n l o b e w i d t h , W, a l o n g w i t h o t h e r f i g u r e s of m e r i t f o r some t y p i c a l w i n d o w s , i s d i s c u s s e d i n t h e n e x t c h a p t e r . 3.4 D a t a S e l e c t i o n I t i s now a p p r o p r i a t e t o d i s c u s s t h e p r o c e s s by w h i c h r e d u n d a n t o r a l i a s e d d a t a i s r e m o v e d t o p r o d u c e t h e f i n a l o u t p u t image. From t h e d e r i v a t i o n s o f S e c . 3.1 and 3.2, i t i s o b s e r v e d t h a t t h e r e i s a d i r e c t c o r r e s p o n d e n c e b e t w e e n t h e t a r g e t r e t u r n p o s i t i o n i n t h e i n p u t a n d o u t p u t d a t a of t h e s t e p t r a n s f o r m . T h u s , t h e o u t p u t p u l s e p o s i t i o n , i s d i s p l a c e d one s a m p l e f o r e a c h u n i t t i m e s a m p l e d i s p l a c e m e n t o f t h e i n p u t t a r g e t r e t u r n s a m p l e . ( I n t h e f o l l o w i n g d i s c u s s i o n , i t w i l l be a s s u m e d t h a t t a r g e t s may o n l y be p o s i t i o n e d a t u n i t s a m p l e s p a c i n g s t o s i m p l i f y t h e e x p l a n a t i o n . ) T h i s means t h a t e a c h FFT o u t p u t b i n c o r r e s p o n d s t o a s i n g l e t a r g e t . The o v e r l a p r a t i o T'/NA a l s o s p e c i f i e s t h e r a t i o o f t h e t o t a l d a t a p r o d u c e d t o t h e number o f p o i n t t a r g e t s . T h e r e f o r e , t h e number o f d a t a p o i n t s r e t a i n e d f r o m e a c h FFT must T'/AH be â€” = N/H, w h i c h i s t h e r a t i o o f t h e l e n g t h o f t h e f i n e T'/NA r e s o l u t i o n FFT t o t h e o v e r l a p r a t i o . T h i s e n s u r e s t h a t o n l y one d a t a p o i n t i s r e t a i n e d f o r e a c h t a r g e t . The a c t u a l d a t a s e l e c t i o n p r o c e s s i s s e t up by c h o o s i n g a s i n g l e t a r g e t a s a r e f e r e n c e . The most a p p r o p r i a t e c h o i c e f o r a 37 reference target i s one which results in sampling of the coarse resolution FFT output exactly at the peak of the pulse. By ca l c u l a t i n g the frequency of the fine resolution aperture data for that target, the position of the output pulse in the fine resolution FFT output data stream can be predicted. The FFT from which the data i s picked i s the one which used the data sampled from the peak of the coarse resolution FFT output pulse as input. The best available data ( i . e . minimum alia s i n g ) for the adjacent N/H - 1 targets i s found in the adjacent N/H - 1 samples in the same FFT. (If the overlap r a t i o i s not large enough for the coarse resolution aperture window, there may be no unaliased data for some targets. The above procedure w i l l ensure that the data with the least amount of a l i a s i n g i s retained.) For the next N/H targets, N/H output samples are picked from the next fine resolution FFT, s t a r t i n g at the bin after the one l e f t off at in the previous FFT. The data i s scaled by ca l c u l a t i n g the Fourier transform of the coarse resolution window in the bin around the peak of the function and using that data to scale the N/H samples picked from each fine resolution FFT. The above procedure can also be i l l u s t r a t e d by the use of an example, as follows. The length of the coarse resolution FFT is 128 and H = 1. The spacing of the coarse resolution apertures i s 42 samples. Therefore, the length of the fine resolution FFT i s 128 and 42 data points must be picked from each. If the reference target chosen produces "good" data in fine resolution FFT #3 at bin 78, data must be picked as 38 follows: 1) from FFT #1, bins 102-127 and 0-15, 2) from FFT #2, bins 16-57, 3) from FFT #3, bins 58-99, 4) from FFT #4, bins 100-127 and 0-13, 5) from FFT #5, bins 14-55, etc. It should be noted that an average of N/H points must be picked from each FFT. If N/H = 40.5, for example, the procedure must be set up so that 40 points are picked from the f i r s t FFT and 41 from the next. In situations where the fine resolution aperture i s padded with extra zeros, there are two e f f e c t s . F i r s t l y , the frequency resolution of the fine resolution FFT i s increased so that each output bin does not correspond to a single point target. In e f f e c t , the f i l t e r output i s oversampled. Secondly, the formula N/H can no longer be used to determine the number of points retained from each FFT. It i s necessary to use the actual length of the fine resolution FFT in c a l c u l a t i n g t h i s value. 39 I V . CHOOSING STEP TRANSFORM PROCESSING PARAMETERS The l a s t c h a p t e r o u t l i n e d t h e b a s i c o p e r a t i o n s o f t h e s t e p t r a n s f o r m p r o c e s s . I n t h i s c h a p t e r , some i n f o r m a t i o n w i l l be p r e s e n t e d w h i c h i s u s e f u l i n d e t e r m i n i n g t h e p r o c e s s i n g p a r a m e t e r s u s e d i n an a c t u a l d e s i g n . The q u a l i t y o f t h e o u t p u t image i s d e t e r m i n e d i n p a r t by t h e d a t a windows u s e d . The d a t a window u s e d f o r t h e c o a r s e r e s o l u t i o n a p e r t u r e a l s o p l a y s a s i g n i f i c a n t r o l e i n d e t e r m i n i n g t h e c o m p u t a t i o n r a t e r e q u i r e d . The p r o p e r t i e s o f windows w h i c h a r e c r i t i c a l t o t h e s t e p t r a n s f o r m p r o c e s s w i l l be e x a m i n e d a n d f i g u r e s o f m e r i t w i l l be p r e s e n t e d f o r s e v e r a l common w i n d o w s . The s t e p t r a n s f o r m w i l l t h e n be e x a m i n e d t o d e t e r m i n e t h e amount o f c o m p u t a t i o n r e q u i r e d a t e a c h s t a g e . F o r m u l a e a r e d e r i v e d w h i c h s p e c i f y t h e t o t a l c o m p u t a t i o n r a t e p e r o u t p u t p i x e l . A s i g n i f i c a n t f a c t o r i n d e t e r m i n i n g c o m p u t a t i o n r a t e s i n SAR p r o c e s s i n g i s t h e r e q u i r e m e n t f o r m u l t i l o o k i n g . A method f o r p e r f o r m i n g t h i s o p e r a t i o n i s p r e s e n t e d a n d i t s e f f e c t on t h e c o m p u t a t i o n r a t e i s d e t e r m i n e d . B a s i c memory r e q u i r e m e n t s f o r t h e s t e p t r a n s f o r m p r o c e s s a r e a l s o d e t e r m i n e d . F i n a l l y , h a v i n g s t e p p e d b a c k t o t a k e a g l o b a l v i e w o f t h e s t e p t r a n s f o r m t e c h n i q u e , i t w i l l be a p p r o p r i a t e t o compare i t w i t h a v e r y s i m i l a r t e c h n i q u e , b a n d p a s s f i l t e r s p e c t r a l a n a l y s i s . B e c a u s e o f t h e s i m i l a r i t i e s b e t w e e n t h e t w o , i t i s p o s s i b l e t o co m p a r e i n d i v i d u a l s t e p s i n e a c h t h a t h a v e s i m i l a r f u n c t i o n s . 40 4.1 Windows Data windows serve the very important function, in most pulse compression algorithms, of reducing the amount of energy contained in the sidelobes to acceptable l e v e l s . In the step transform process, both the coarse and fine resolution apertures must be windowed to obtain images of acceptable quality, i . e . sidelobes lower than the nominal 13 dB for the boxcar window. The window chosen for the fine resolution aperture only has an effect on the output image in close proximity to the f i n a l compressed pulse, i . e . within the same fine resolution aperture. Therefore, i t s effect on the f i n a l output can be d i r e c t l y observed from the properties of the window. For example, the 3 dB width of the output pulse w i l l be approximately the same as the 3 dB width of the window chosen. The usual figures of merit, such as integrated side lobe r a t i o and 3 dB width, can be used in choosing an appropriate window. The window chosen for the coarse resolution aperture, however, a f f e c t s the image over many fine resolution apertures. As shown in Figure 8, the magnitude of the input data to each fine resolution aperture corresponds to some point on the Fourier transform of the window function chosen for the coarse resolution aperture. Since t h i s data has i t s e l f been windowed and Fourier transformed, one spurious peak w i l l appear at the same position in each fine resolution aperture. Since the output function of the coarse resolution aperture i s actually sampled to obtain input data for the fine resolution aperture, the peak sidelobe levels w i l l not actually be observed for a l l 42 point target positions. In addition to sidelobe l e v e l , the width of the mainlobe must also be considered in choosing a window function for the coarse resolution aperture. As was noted in Sec. 3.3, where the a l i a s i n g problem was discussed, the width of the mainlobe is d i r e c t l y related to how much the coarse resolution apertures must be overlapped to prevent a l i a s i n g . For the best image qu a l i t y , i t i s desired to reduce the sidelobe l e v e l s as much as possible. However, in general, windows with lower sidelobes have wider mainlobes. Thus, for a p a r t i c u l a r application where the step transform i s to be used, the highest acceptable sidelobe levels should be s p e c i f i e d . A coarse resolution window which meets that s p e c i f i c a t i o n and minimizes the mainlobe width is then chosen. Following t h i s procedure ensures that acceptable image qu a l i t y w i l l be maintained, while at the same time minimizing the amount of processing required. Relevant figures of merit were measured for several types of window, as s p e c i f i e d in Appendix A. The data was obtained by evaluating the Fourier transform of each window at 0.01 bin inter v a l s in the region of the main lobe. The only windows which gave questionable results were the Dolph-Chebyshev and Barcilon-Temes windows. In p a r t i c u l a r , the sidelobe lev e l s obtained for these windows do not match those sp e c i f i e d by Harris [16]. (For the Dolph-Chebyshev window, the sidelobe l e v e l should be -20a.) Also, the mainlobe width follows some s l i g h t l y eccentric variations as a function of the aperture length. This might be due to the fact that the procedure to 43 Table I - Mainlobe Width at Highest Sidelobe Level for Various Windows Window Length of Window a 8 16 32 64 128 256 512 Hamming 3.70 3.80 3.84 3.84 3.84 3.84 3.84 Minimum 3-term Blackman-Harris 5.72 5.80 5.86 5.90 5.90 5.90 5.90 3-term Blackman-Har r i s 5.26 5.28 5.34 5.38 5.38 5.38 5.38 Minimum 4-term Blackman-Harris 7.86 7.88 7.88 7.88 7.88 7.88 4-term Blackman-Har â€˘ris 6.44 6.32 6.30 6.30 6.30 6.30 6.30 Rectangle 1 .62 1 .62 1 .62 1 .62 1 .62 1 .62 1 .62 4-sample Kaiser-Bessel 3.0 6.10 6.16 6.16 6.16 6.16 6.16 6.16 Exact Blackman 6.20 5.98 5.90 5.88 5.88 5.88 5.88 Blackman 5.92 5.66 5.64 5.66 5.64 5.64 5.64 Gaussian 2.0 3.20 3.12 3.10 3.10 3.10 3.10 3.10 Gaussian 2.5 5.86 5.88 5.90 5.90 5.90 5.90 Gaussian 3.0 6.80 6.72 6.70 6.70 6.70 6.70 Gaussian 3.5 9.90 9.90 9.90 9.90 9.90 9.90 Gaussian 4.0 11.2 1 1 .2 11.1 11.1 11.1 11.1 Kaiser-Bessel 2.0 4.22 4.28 4.30 4.30 4.32 4.32 4.32 Kaiser-Bessel 2.5 5.38 5.28 5.26 5.26 5.26 5.26 5.26 Kaiser-Bessel 3.0 6.22 6.24 6.22 6.20 6.22 6.22 6.22 Kaiser-Bessel 3.5 7.16 7.18 7.18 7.18 7.18 7.18 7.20 Kaiser-Bessel 4.0 8.12 8.14 8.16 8.16 8.16 8.16 a i s a parameter used in some windows to govern the tradeoff between sidelobe l e v e l and mainlobe width. See Appendix A for complete d e f i n i t i o n of windows. 44 Table I - continued Window a Length of Window 8 16 32 64 128 256 512 Dolph-Chebyshev 2.0 3.24 3.22 3.22 3.22 3.22 3.20 Dolph-Chebyshev 2.5 3.82 3.94 3.94 3.94 3.94 3.94 Dolph-Chebyshev 3.0 4.48 4.78 4.74 4.74 4.74 4.80 Dolph-Chebyshev 3.5 5.20 5.42 5.54 5.56 5.54 5.62 Dolph-Chebyshev 4.0 5.58 6.04 6.26 6.30 6.24 6.04 Barcilon-Temes 2.0 3.26 3.46 3.54 3.56 3.54 3.56 3.52 Bare ilon-Temes 2.5 3.56 3.82 3.94 3.98 3.98 3.96 3.98 Barcilon-Temes 3.0 3.82 4.44 4.60 4.64 4.64 4.64 4.72 Barcilon-Temes 3.5 4.10 5.10 5.32 5.38 5.42 5.40 5.62 Barcilon-Temes 4.0 4.40 5.68 6.06 6.16 6.16 6.18 6.06 calculate the window function e s s e n t i a l l y involves specifying the Fourier transform (for the Dolph-Chebyshev window). The Fourier transform function i s sampled and an inverse FFT i s performed to obtain the time domain data c o e f f i c i e n t s . Possibly, the Fourier transform function sampling i n t e r v a l must be decreased in order to obtain agreement with Harris' r e s u l t s . Also, there are cert a i n errors in the window s p e c i f i c a t i o n s presented by Harris, as noted in Appendix A. In Table I, the mainlobe width (W as depicted in Figure 6), i s presented. In Table I I , the mainlobe width i s converted to the amount of overlapping of the coarse resolution apertures required for H i 1 by u t i l i z i n g the r e s t r i c t i o n on the value of the overlap r a t i o derived in Sec. 3.3. This figure i s presented 45 Table II - Overlap Required and Sidelobe Levels for Various Windows Window a Sidelobe Length of Window r a l l o i I (dB/oct) 8 16 32 64 128 256 512 Rectangle -6 1.31 -13 1 .31 -13 1 .31 -13 1.31 -13 1 .31 -13 1 .31 -13 1.31 -13 Gaussian 2.0 -6 2.10 -30 2.06 -31 2.05 -32 2.05 -32 2.05 -32 2.05 -32 2.05 -32 Dolph-Chebyshev 2.0 0 2.12 -34 2.11 -34 2.11 -33 2.1 1 -34 2.1 1 -33 2.10 -34 Barcilon-Temes 2.0 -6 2.13 -28 2.23 -34 2.27 -38 2.28 -38 2.27 -38 2.28 -39 2.26 -38 Barcilon-Temes 2.5 -6 2.28 -32 2.41 -38 2.47 -40 2.49 -39 2.49 -41 2.48 -40 2.49 -41 Hamming -6 2.35 -34 2.40 -40 2.42 -42 2.42 -42 2.42 -43 2.42 -43 2.42 -43 Gaussian 2.5 -6 3.43 -42 3.44 -43 3.45 -43 3.45 -43 3.45 -43 3.45 -43 Dolph-Chebyshev 2.5 0 2.41 -39 2.47 -45 2.47 -44 2.47 -44 2.47 -44 2.47 -44 Kaiser-Bessel 2.0 -6 2.61 -43 2.64 -44 2.65 -45 2.65 -45 2.66 -46 2.66 -46 2.66 -46 Barcilon-Temes 3.0 -6 2.41 -35 2.72 -45 2.80 -48 2.82 -49 2.82 -48 2.82 -48 2.86 -51 Dolph-Chebyshev 3.0 0 2.74 -45 2.89 -55 2.87 -54 2.87 -54 2.87 -54 2.90 -52 Gaussian 3.0 -6 3.90 -54 3.86 -56 3.85 -56 3.85 -56 3.85 -57 3.85 -57 Barcilon-Temes 3.5 -6 2.55 -37 3.05 -53 3.16 -56 3.19 -58 3.21 -57 3.20 -56 3.31 -63 4-term Blackman-Harri s 1 -6 3.72 -60 3.66 -57 3.65 -57 3.65 -57 3.65 -57 3.65 -57 46 Table II - continued Window Sidelobe P a 1 1 n f f Length of Window CL r a i i o n (dB/oct) 8 16 32 64 128 256 512 Kaiser-Bessel 2.5 -6 3.19 -60 3.14 -59 3.13 -58 3.13 -58 3.13 -58 3.13 -58 3.13 -58 Blackman -18 3.46 -74 3.33 -58 3.32 -58 3.33 -58 3.32 -58 3.32 -58 3.32 -58 Dolph-Chebyshev 3.5 0 3.10 -50 3.21 -59 3.27 -64 3.28 -63 3.27 -63 3.31 -60 3-term Blackman-Harri; -6 3.13 -49 3.14 -54 3.17 -58 3.19 -61 3.19 -62 3.19 -62 3.19 -62 Exact Blackman -6 3.33 -54 3.38 -60 3.42 -65 3.43 -67 3.44 -68 3.44 -68 3.44 -68 4-sample Kaiser-Bessel 3.0 -6 3.55 -62 3.58 -66 3.58 -66 3.58 -66 3.58 -66 3.58 -66 3.58 -66 Barcilon-Temes 4.0 -6 2.70 -39 3.34 -59 3.53 -66 3.58 -68 3.58 -67 3.59 -69 3.53 -62 Minimum 3-term Blackman-Harrii -6 3.36 -57 3.40 -62 3.43 -67 3.45 -70 3.45 -71 3.45 -71 3.45 -71 Kaiser-Bessel 3.0 -6 3.61 -67 3.62 -70 3.61 -69 3.60 -68 3.61 -70 3.61 -70 3.61 -70 Dolph-Chebyshev 4.0 0 3.29 -55 3.52 -66 3.63 -74 3.65 -74 3.62 -73 3.52 -71 Gaussian 3.5 -6 5.45 -68 5.45 -70 5.45 -71 5.45 -71 5.45 -71 5.45 -71 Kaiser-Bessel 3.5 -6 4.08 -77 4.09 -82 4.09 -81 4.09 -81 4.09 -81 4.09 -81 4.10 -83 Gaussian 4.0 -6 6.1 1 -83 6.08 -87 6.05 -87 6.04 -87 6.04 -88 6.04 -88 Minimum 4-term Blackman-Harri s -6 4.43 -90 4.44 -92 4.44 -92 4.44 -92 4.44 -92 4.44 -92 Kaiser-Bessel 4.0 -6 4.56 -92 4.57 -93 4.58 -94 4.58 -94 4.58 -94 4.58 -94 47 Table III - Attenuation of Main Lobe at 0.5 bins from Centre Window a Length of Window 8 16 32 64 128 256 512 Hamming 1.8 1 .8 1.8 1 .8 1 .8 1 .8 1 .8 Minimum 3-term Blackman-Harris 1 .1 1 .1 1 .1 1 .1 1 .1 1 .1 1 .1 3-term Blackman-Hai â€˘ris 1 .3 1.3 1.3 1 .3 1.3 1.3 1.3 Minimum 4-term Blackman-Harris 0.8 0.8 0.8 0.8 0.8 0.8 4-term Blackman-Hai -r i s 0.9 1.0 1.0 1.0 1.0 1.0 Rectangle 3.9 3.9 3.9 3.9 3.9 3.9 3.9 4-sample Kaiser-Bessel 3.0 0.8 0.9 1.0 1.0 1.0 1 .0 1.0 Exact Blackman 0.9 1.0 1 .1 1 .1 1 .1 1 .1 1.2 Blackman 0.8 1.0. 1.0 1 .1 1 .1 1 .1 1 .1 Gaussian 2.0 2.1 2.1 2.1 2.1 2.1 2.1 2.1 Gaussian 2.5 1.6 1.6 1.6 1.6 1 .6 1 .6 Gaussian 3.0 1 .2 1 .2 1.2 1.2 1.2 1.2 Gaussian 3.5 0.9 0.9 0.9 0.9 0.9 0.9 Gaussian 4.0 0.7 0.7 0.7 0.7 0.7 0.7 Kaiser-Bessel 2.0 1 .4 1.5 1.5 1.5 1.5 1 .5 1 .5 Kaiser-Bessel 2.5 1 .2 1.2 1.2 1.2 1 .2 1 .2 1.2 Kaiser-Bessel 3.0 1.0 1.0 1.0 1.0 1.0 1 .0 1.0 Kaiser-Bessel 3.5 0.9 0.9 0.9 0.9 0.9 0.9 0.9 Kaiser-Bessel 4.0 0.8 0.8 0.8 0.8 0.8 0.8 48 Table III - continued Window a Length of Window 8 16 32 64 128 256 512 Dolph-Chebyshev 2.0 2.1 2.0 2.0 2.0 2.0 2.0 Dolph-Chebyshev 2.5 1 .8 1 .7 1.7 1.7 1 .7 1.7 Dolph-Chebyshev 3.0 1 .5 1 .5 1 .4 1 .4 1 .4 1 .4 Dolph-Chebyshev 3.5 1 .4 1.3 1.2 1 .2 1.2 1.3 Dolph-Chebyshev 4.0 1 .2 1 .1 1 . 1 1 .1 1 .1 1 . 1 Barcilon-Temes 2.0 2.2 2.0 1 .9 1.9 1.9 1.9 1.9 Bare ilon-Temes 2.5 1.9 1.7 1 .6 1.6 1.6 1 .6 1 .6 Barcilon-Temes 3.0 1.7 1 .4 1 .4 1.3 1.3 1.3 1 .4 Barcilon-Temes 3.5 1.6 1.3 1 .2 1.2 1.2 1.2 1.2 Bare ilon-Temes 4.0 1.5 1.2 1 .1 1.0 1.0 1.0 1 . 1 along with the highest sidelobe l e v e l measured for each window so that the appropriate tradeoffs can be made. A secondary consideration in the choice of a window i s the rate of sidelobe f a l l o f f , also presented in Table I I . This also a f f e c t s the amount of energy in the sidelobes and therefore the image q u a l i t y . As was mentioned previously, the Fourier transform of the coarse resolution window i s ac t u a l l y sampled to obtain input data for the fine resolution apertures. The fine resolution aperture containing the f i n a l output pulse for a p a r t i c u l a r point target may obtain i t s input data up to 0.5 resolution bins from the peak of the coarse resolution FFT output pulses. This w i l l r e s u l t in attenuation of the output image for certain point 49 targets. The severity of this attenuation can be measured by determining the l e v e l of the coarse resolution window at 0.5 resolution bins from the centre of the pulse. This data i s presented in Table I I I . If the attenuation i s severe enough, i t may be necessary to perform a scaling operation on the output data from the fine resolution FFT. 4.2 Computation Rates The processing parameters chosen to perform the step transform process play a major role in determining the computation rate required. The effect of cer t a i n parameters, such as the coarse resolution aperture overlap r a t i o T'/NA, i s obvious. However, there are other contributing factors whose role i s not quite so obvious. The approach taken here w i l l be to break the step transform down into the major computational steps and i n i t i a l l y examine the computational requirements of each stage. The o v e r a l l process w i l l then be examined so that an indication of the optimal values for the processing parameters can be obtained. In succeeding chapters, when the step transform technique i s applied to SAR azimuth processing, additional constraints w i l l be imposed on the processing parameters. No attempt w i l l be made here or in l a t e r chapters to account for every single computation which must be performed. Instead, a measure of the computation rate w i l l be obtained by c a l c u l a t i n g the number of complex mu l t i p l i c a t i o n s required. 50 Some stages do not involve complex computations, but require a s i g n i f i c a n t number of real m u l t i p l i c a t i o n s to be performed. These are included by assuming that 1 complex m u l t i p l i c a t i o n i s equivalent to 4 real m u l t i p l i c a t i o n s and making the appropriate conversion. The step transform process can be broken down into the following major steps: 1) m u l t i p l i c a t i o n of reference and target return signals, 2) coarse resolution FFT (windowing of the aperture can be done as part of the m u l t i p l i c a t i o n - step 1), 3) quadratic phase correction, 4) re-ordering of the data for the fine resolution FFT, 5) windowing of the fine resolution aperture, 6) fine resolution FFT, 7) data selection, magnitude evaluation and sca l i n g . The factors which d i r e c t l y a f f e c t the computation rate and the stages they impact are as follows: 1) length of the coarse resolution FFT, T', step 2, 2) amount of overlapping of the coarse resolution apertures, expressed as the r a t i o T'/NA, steps 1-3, 3) the length of the fine resolution FFT, T/BA2N, steps 5 and 6.7 Note that the above factors are cl o s e l y i n t e r r e l a t e d and, in general, a f f e c t the image qual i t y as well. 7 It i s assumed that the window c o e f f i c i e n t s are predefined and, therefore, the type of windows used w i l l not af f e c t the computation rate. 51 As mentioned in the previous chapter, i t is only necessary to consider the situation where H > 1. For e f f i c i e n t FFT processing, the length of the coarse resolution FFT, T'/A, i s r e s t r i c t e d to be a power of 2. For similar reasons, the fine resolution FFT length, T'/AH, should also be a power of 2. Thus there i s a further r e s t r i c t i o n on H, that i t be a power of 2 as well. For situations where H i s not a power of 2, the fine resolution FFT would be somewhat longer. Consider a processing region of length R time units and R/A samples. The number of complex mul t i p l i c a t i o n s which must be performed at each stage of the step transform process i s as follows: 1) M u l t i p l i c a t i o n - (R/A)(T'/AN) complex m u l t i p l i c a t i o n s 2) Coarse Resolution FFT - there are R/AN FFT's each requiring T'â€˘log 2(T'/A}/2A complex m u l t i p l i c a t i o n s Total = RT'â€˘log 2(T'/A}/2A 2N 3) Quadratic Phase Correction - (R/A)(T'/AN) complex mul t i p l i c a t i o n s 4) Windowing the Fine Resolution Aperture - 2T'(R-T)/(A 2N) real m u l t i p l i c a t i o n s or T'(R-T)/(2A 2N) complex mul t i p l i c a t i o n s 5) Fine Resolution FFT - there are (R/A - T/A)(T'/AN) points to be processed using FFT's of length T/BA2N which requires (R-T)T'B/T FFT's. Each FFT requires T-log 2{T/BA 2N}/2BA 2N complex m u l t i p l i c a t i o n s . Total = (R-T)T'â€˘log 2{T/BA 2N}/2A 2N 6) Output scaling - (R-T)/A real m u l t i p l i c a t i o n s or (R-T)/4A complex m u l t i p l i c a t i o n s . 52 Total computations = (T'/A 2N)(2R) + (T'/2A 2N)(R.log 2{(T'/AN)(T/BA 2)}) - (T'/2A 2N)(T.log 2{T/BA 2N}) + (T'/2AN + 1/4)(R-T)/A To fi n d the amount of computation per input data point, CP, divide by R/A. CP = 2T'/AN + (T'/2AN).log2{T'/A} + (T'/2AN)(1 - T/R)â€˘log 2{T/BA 2N} + (T'/2AN + 1/4)(1 - T/R) (4) Assuming T << R CP = (T'/AN)(2.5 + log 2{(T'/A)(T/BA 2N}/2) + 1/4 = (T'/AN)(2.5 + log 2{(T'/AN)(T/BA 2)}/2) + 1/4 Thus, for H > 1, the only means of varying the computation rate is by changing the overlap r a t i o T'/NA. For the si t u a t i o n where T is close to the same length as R, the amount of data rejected during the reordering process (see Figure 5) plays a s i g n i f i c a n t role in determining the computation rate. The most s i g n i f i c a n t computation done on thi s data i s the coarse resolution FFT. The actual amount of data rejected i s dependent only on the overlap r a t i o , T'/NA. Thus, in t h i s s i t u a t i o n , i t becomes desirable to minimize the length of the coarse resolution aperture. Overall, i t would appear that the situation where H = 1 seems to minimize the computation requirements. 53 4.3 Memory Requirements Before attempting to assess the memory requirements of the step transform process, i t i s necessary to do some basic analysis of the architecture which might be used to implement the technique. Martinson, in his description of a step transform processor implementation [25], considers t h i s issue in some d e t a i l . The analysis presented here w i l l b a s i c a l l y follow the same l i n e s of reasoning, but w i l l incorporate the additional results presented in previous sections. Note that some of the basic' issues related to SAR azimuth processing, such as RCMC, which requires that data across several range c e l l s be accessible at the same time, w i l l not be considered at this point. The step transform processor described by Martinson uses a pipeline architecture, and i s designed to perform the pulse compression operation in real-time as the data i s received. Two memory banks are used so that data can be written into one module from the coarse resolution FFT output while data i s read to the fine resolution FFT input from the other memory module. When unit slope diagonalization (H = 1) i s required, t h i s method allows both memory modules to be in use at a l l times. For the sit u a t i o n where H > 1, the only solution which was found to the problem of reading and writing at the same time, i s to provide a separate bank for each FFT output bin. The actual amount of memory storage required i s determined by the number of coarse resolution apertures which must be used to obtain data for one complete fine resolution aperture. Since 54 each coarse resolution aperture contains T'/A samples, the dimensions of the matrix are T'/A by T'/AH. The t o t a l number of data elements i s therefore T' 2/A 2H. Since H = BT'AN/T, Total memory words TM = T'T/BA2N = (T'/AN) â€˘ (T/BA 2) Thus, the amount of memory i s independent of a l l processing parameters except the overlap r a t i o T'/NA. For the si t u a t i o n where H = 1, i t can be demonstrated that only (T'/2A)(T'/A -1) + 1 memory locations in the reorder memory contain v a l i d data at any one time. It may be possible to develop an address sequencing algorithm such that the actual size of the reorder memory can be reduced to this amount, as described by Martinson. No attempt w i l l be made to develop such an algorithm here. Therefore memory requirements w i l l be s p e c i f i e d approximately as being in the range TM/2 to TM. 4.4 Multilooking An additional consideration which af f e c t s the computation rate i s the requirement to reduce speckle noise in the output image. The phenomenon of speckle occurs in a l l imaging systems which use phase information to reconstruct the image and was f i r s t observed in holographic experiments. The s t a t i s t i c a l properties of speckle have been described in d e t a i l by Dainty [11]. There are two basic methods of speckle noise reduction. In the f i r s t category, speckle noise i s reduced by smoothing the 55 f i n a l image data [22]. In the second category, the azimuth signal i s divided up into several sections or looks. Each section i s processed separately and the results are added incoherently by throwing away the phase information [30]. The second method must be integrated into the pulse compression operation and only the impact on step transform processing w i l l be considered here. A somewhat subjective judgment must be made to determine the number of looks required, usually on the basis of the type of t e r r a i n and the application for which the f i n a l image i s to be used. Ford has done a survey of user preferences [15] which suggests that the optimum number of looks may vary from 2 to 32 for geological applications. In the step transform technique, multilook processing can be performed as follows. The output data of the coarse resolution FFT's i s partitioned according to the number of looks required. A fine resolution FFT i s performed on each look and the data from these FFT's i s added incoherently. The data selection process described in Sec. 3.4 i s modified so that N/HL points are selected from each FFT, where L i s the number of looks. Since the length of the fine resolution FFT's i s changed, the computation rates w i l l change as well. Ideally, each look should contain completely independent data. However, a small amount of overlapping w i l l not seriously a f f e c t the speckle noise reduction properties of this process [8]. Therefore, i t i s possible to vary the length of the fine resolution FFT's to a certain extent and the r e s t r i c t i o n that H be a power of 2 may be removed. Note that the length of the 56 fine resolution FFT's cannot be allowed to become too short or else there w i l l be a l i a s i n g e f f e c t s due to the wraparound properties of the FFT. In general, the FFT length should not be shorter than 16 samples. Also, look overlap should not be greater than 50%. Normally, i t i s expected that the output resolution i s reduced by the number of looks. The e f f e c t on the computation rates i s dependent, to a large extent, on the number of looks required. If the fine resolution aperture can be greater than approximately 16L samples, there is not l i k e l y to be a radic a l change in the computation requirements. The single fine resolution FFT considered in Sec. 4.2 i s replaced by L FFT's of length T'/AHL samples. The t o t a l number of complex m u l t i p l i c a t i o n s is reduced to LÂ«TÂ»log 2{T/BA 2NL}/2BA 2N for the fine resolution FFT stage. However, the looks must then be combined incoherently in a look summation stage. The length of the fine resolution aperture can be increased to accommodate a greater number of looks by increasing the size of the coarse resolution aperture while maintaining H constant. 8 This results in an increase in the overlap r a t i o T'/NA and a corresponding increase in the computation rate. In general Perry and Martinson [29] lengthen the fine resolution FFT by concatenating two apertures together. As discussed in Sec. 3.1, this would not seem to work very well. One might speculate that t h i s i s done to provide greater f l e x i b i l i t y in the number of looks. However, since the data in the apertures comes from exactly overlapping portions of the target return, the speckle noise would not be reduced by t h i s additional processing. 57 terms, i t means that there is a certain number of looks which can be handled without a s i g n i f i c a n t change in computation rates. Above that value of L, some s i g n i f i c a n t increase in the overlap r a t i o i s required with a corresponding increase in the computation rate. 4.5 Comparison With Bandpass F i l t e r Spectral Analysis BPF SPECAN pulse compression i s similar in many ways to the step transform. Both employ e s s e n t i a l l y the same stages of processing conceptually, although the implementation of the concepts i s quite d i f f e r e n t . After performing the i n i t i a l mixing operation, the BPF SPECAN method uses f i l t e r s to separate groups of adjacent targets. In the step transform, t h i s stage i s implemented using the coarse resolution FFT's which separate groups of adjacent targets into separate FFT output bins. Both techniques then employ an FFT to perform the f i n a l output resolution. The major difference between the two techniques i s in the degree of f l e x i b i l i t y afforded. The step transform employs a very s p e c i f i c coarse resolution f i l t e r , the FFT. The number of targets included in a s p e c i f i c FFT output bin is determined largely by the requirements of the step transform process, as outlined in Chapter I I I . There i s . not a great deal of f l e x i b i l i t y to adapt to the requirements of the individual SAR system. BPF SPECAN, on the other hand, allows a great deal of f l e x i b i l i t y in the choice of processing parameters. The number 58 of f i l t e r s required and the type of f i l t e r may be adapted to the pa r t i c u l a r application. As w i l l be seen in l a t e r chapters, th i s proves to be a d i s t i n c t advantage in the SAR azimuth processing applica t i o n . 59 V. DETAILS OF SAR AZIMUTH PROCESSING In this chapter, the process of f i t t i n g the step transform into the SAR azimuth signal processing problem i s begun. In order to do t h i s , the coding of azimuth signal w i l l be b r i e f l y described and some of the problems which occur when one attempts to process t h i s signal using a d i g i t a l processor w i l l be pointed out. It w i l l be shown that the linear FM rate of the azimuth signal is not constant, but varies as a function of target range. The azimuth signal for a single point target may also migrate between range c e l l s and does not maintain a constant amplitude. The signal is actually attenuated as a function of the antenna p r o f i l e . The performance of. the step transform with regard to many of these e f f e c t s cannot be predicted mathematically, since i t i s not possible to obtain a closed form mathematical solution as was done in previous chapters. A simulation program, which was used to observe the performance of the step transform in a quantifiable manner, i s described. 5.1 The SAR Azimuth Signal The basic p r i n c i p l e s of synthetic aperture radar have been thoroughly investigated in the l i t e r a t u r e [9][10][33]. Bennett et a l [5] have provided a description of the encoding of the azimuth signal which i s p a r t i c u l a r l y useful for the topics under discussion in t h i s chapter. Consider a radar platform moving above the earth's surface as depicted in Figure 9. With reference to the geometry of Figure 9, the instantaneous slant 60 Figure 9 - SAR Geometry Earth's Surface 6 1 9 range to a point target can be approximated by the function ( V r j ) 2 r ( i j ) = r 0 + (5) 2 r 0 where r 0 = slant range of closest approach 7] = elapsed time since r(r?) = r 0 V = v e l o c i t y of platform. 9 The demodulated phase of the radar return signal i s -2 -2 V 2 7 ] 2 tf>(i?) = â€” r ( r j ) = â€” ( r 0 + ) X X 2 r 0 where X i s the radar wavelength. The Doppler frequency i s deft 2V 2T? f ( T?) = â€” = - Hz. d7? Xr 0 The received azimuth signal i s 47T j s ( r 0 , 7 ? ) = A ( 7 ? - 7 7 0 ) exp{ r(rj)} X where A(7j) = the antenna p r o f i l e function 7}0 = the azimuth time o f f s e t of the beam centre from from zero Doppler. This i s a linear FM signal with a frequency sweep r a t e 1 0 -2V 2 K = ( 6 ) Xr 0 For airborne SAR systems, V i s the actual v e l o c i t y of the plane. For satellite-borne SAR's, the earth's curvature and rotation must be taken into account and V i s a derived parameter c a l l e d the radar v e l o c i t y . 1 0 In the the o r e t i c a l analysis of Chapter I I I , the FM rate i s defined as the r a t i o B/T. If B i s defined to be the PRF for the SAR azimuth processing s i t u a t i o n , then T i s the time the target return signal takes to sweep between the frequencies -PRF/2 and PRF/2. 62 Thus, the FM rate varies as a function of range. To achieve acceptable image quality, and unless processing i s done only over a very narrow range, the reference function of the pulse compression algorithm must be changed p e r i o d i c a l l y to take into account th i s varying FM rate. From the function rirj), i t i s clear that i t i s possible for the azimuth signal to migrate between range c e l l s , depending on the parameters of the pa r t i c u l a r radar involved. In the case of most s a t e l l i t e radars, the azimuth signal l i e s on a parabolic curve in signal memory which may cross several range c e l l s . In such situations, some form of range c e l l migration correction (RCMC) i s c l e a r l y required, since the azimuth signal must l i e in a single l i n e in signal memory in order to be processed. The variable attitude of s a t e l l i t e s also - presents some problems in processing SAR azimuth signals. E s s e n t i a l l y , i t means that the antenna may not be pointing broadside to the dir e c t i o n of platform motion. This phenomenon not only i n t e n s i f i e s the range c e l l migration problem, as w i l l be seen in Chapter VI, but also causes the antenna beam centre to be shifted s i g n i f i c a n t l y away from the zero Doppler l i n e . This i s s i g n i f i c a n t since the processing region i s positioned around the beam centre and w i l l change the frequency region occupied by the processed portion of the azimuth s i g n a l . Alluding to the theo r e t i c a l analysis of Chapter I I I , t h i s means that the output of the coarse resolution FFT's w i l l be rotated in di r e c t i o n and amount corresponding to the s h i f t of the beam centre. This error may cause problems during the reordering process. It i s 63 e a s i l y corrected by rotating the FFT outputs the appropriate amount. 5.2 The Antenna P r o f i l e For most situations, the antenna p r o f i l e may be represented by a sine function so that s i n 2 {krj} A ( T J ) = (k7?) 2 where k i s some constant determined by c a l i b r a t i n g the antenna. Thus, the actual signal to be processed i s somewhat d i f f e r e n t than that presented in the t h e o r e t i c a l development of Sec. 3.1. Not only does the signal not have a constant magnitude but i t also does not have a f i n i t e time duration. This implies that the Doppler frequency of the azimuth signal also i s not bounded. Since the azimuth signal i s sampled at a rate determined by the PRF, there w i l l be some aliase d energy in the s i g n a l . The SAR parameters must be chosen to minimize th i s a l i a s i n g effect and ensure good image quality [19], One problem which must be handled by the pulse compression algorithm i s that the energy towards the edges of the signal returns i s decreased s i g n i f i c a n t l y from that at the centre. This attenuation, caused by the antenna p r o f i l e , may be so severe that the signal information i s obliterated by noise. In such cases, i t i s necessary to f i l t e r out that portion of the signal return. In the step transform, t h i s may be done by discarding appropriate coarse resolution output bins, since there i s a d i r e c t 64 correspondence between the time position of the coarse resolution aperture and the frequency seen by the coarse resolution FFT. The data region rejected i s c a l l e d the guardband and may be expressed as a f r a c t i o n , /3, of the PRF. Since some good signal information i s discarded as well, t h i s does have the effect of broadening the f i n a l output pulse. This technique is also used in non-SAR applications, where the input signal is normally sampled at a rate somewhat greater than Nyquist. In that s i t u a t i o n , the coarse resolution FFT output bins at frequencies above Nyquist do not contain v a l i d data and are rejected. 5.3 The Simulation Program The program was written in FORTRAN on the MTS computer system at UBC. The basic modular structure of the program i s presented in the flowchart in Figure 10. Appendix D contains a basic functional s p e c i f i c a t i o n of the program. It follows the stages of the step transform process as outlined in previous chapters. However, there are some additional steps. At the beginning, the processing parameters are chosen either from preset values or information entered by the user. At the end, some analysis of the output i s c a r r i e d out and the result may be plotted, i f requested by the user. The following SAR system and processing parameters may be altered by the user: 1) data windows for coarse and fine resolution apertures, 6 5 Figure 1 0 - Flowchart of Simulation Program Set I n i t i a l Parameter Values I Allow User to Change Parameter Values I Determine Additional Parameters eg. H, Fine Resolution FFT Length, Adjust N to Closest Integer Value I Form Target Return Signal and Reference Ramp I Mix Target Return and Reference Ramp I Coarse Resolution FFT's and Quadratic Phase Correction Reorder Data into Fine Resolution Apertures I Fine Resolution FFT's, Look Summation, Scaling and Data Selection Analyze Data and Plot 66 2) coarse resolution time extent and overlapping, 3) -number of looks, 4) guardband, 5) antenna attenuation p r o f i l e , 6) FM rate error and time extent of target return signal, 7) point target position, 8) resolution of the f i n a l output image. A l l time apertures are s p e c i f i e d in terms of the number of samples. The user i s able to specify a factor by which the fine resolution FFT's can be lengthened (padded with zeros) to obtain greater resolution in the f i n a l output image. The f i n a l output data i s analyzed using an interpolation routine which uses exponential functions and variable "tension" parameters. The routine i s c a l l e d SMOOTH and i s resident on the MTS computer system [27], The antenna constant, k, was chosen i n d i v i d u a l l y for each system. For the SEASAT SAR, the attenuation p r o f i l e of the antenna used on the s a t e l l i t e was chosen, such that k = 142.37. For the JPL Nominal and COMSS/LASS systems, no such data was avai l a b l e . Therefore, an antenna p r o f i l e was chosen such that the signal was attenuated by -6 dB at the ends of the aperture. Although the user has a large degree of freedom in choosing the processing parameters, there are some r e s t r i c t i o n s . The time extent of the coarse resolution apertures i s r e s t r i c t e d to values such that the number of samples i s a power of 2. This i s done for e f f i c i e n t processing of the coarse resolution FFT. Secondly, the amount of overlapping i s adjusted to ensure the 67 frequency steps between coarse resolution apertures are as close as possible to integer values when the FFT outputs are computed. Because of these two r e s t r i c t i o n s , i t i s not possible to ensure that d i f f e r e n t SAR systems have i d e n t i c a l processing parameters, such as the amount of overlapping. Since overlapping of the coarse resolution apertures i s a large determinant in the computational e f f i c i e n c y of the step transform, i t i s possible that the computation rates w i l l vary from one system to another. The data window for the coarse resolution apertures was chosen to obtain the minimum sidelobe levels without introducing a l i a s i n g problems for that p a r t i c u l a r SAR system. The processing parameter values used to simulate each p a r t i c u l a r SAR sytem are spe c i f i e d in Appendix C. As w i l l be described in the next chapter, in actual SAR systems, there i s a necessity to provide for non-integer sample spacing of the reference ramps. No attempt was made to provide this c a p a b i l i t y in the simulation program, since t h i s would add the complication of requiring the computation of multiple reference ramps. However, t h i s meant that the system parameters had to be adjusted to obtain ideal pulse compression. From the experience with the simulation program, some observations can be made about the step transform technique. For the software implementation, large amounts of memory space are required because of the overlapping of the coarse resolution apertures. The mixing operation, coarse resolution FFT, and quadratic phase correction cannot be done in place, i . e . in the data space where the target return signal i s stored. For t h i s 68 reason, two large complex number arrays are required. Array SIGNAL contains the i n i t i a l point target return si g n a l . The output of the mixing operation is stored in array UNORD, where in-place FFT routines and quadratic phase correction are performed. In order tp perform the reordering operation to assemble the fine resolution FFT input data, a t h i r d large memory space i s required. However, since the target return data i s not required anymore, array SIGNAL i s used for t h i s purpose. The fine resolution FFT and subsequent processing i s performed in array SIGNAL. 69 VI. STEP TRANSFORM PERFORMANCE ON SAR AZIMUTH SIGNALS SAR azimuth compression i s a special case of radar signal processing which, in many cases, requires special compensations to produce high qu a l i t y images. In p a r t i c u l a r , s a t e l l i t e radars present special problems due to the variable platform v e l o c i t y , large distance from the earth's surface, and variable attitude. Two of the major factors which must be dealt with in s a t e l l i t e SAR signal processing are s i g n i f i c a n t variations in the FM rate and migration of the azimuth signal between range c e l l s . In th i s chapter, the discussion on azimuth coding of the SAR signal i s continued by examining how the step transform i s able to deal with the above problems. The simulation program described in the previous chapter was used to obtain quantitative results on the performance of the step transform. 6.1 FM Rate Error The e f f e c t s of FM rate errors w i l l i n i t i a l l y be considered at a theoreti c a l l e v e l by returning to the mathematical analysis of Sec. 3.1. Consider a target return signal whose FM rate i s mismatched to the reference function by a fraction b. Thus s(t) = exp{ j(irB/T) ( 1+b)t2} After the mixing operation, 70 s(k,m) s*(k,n) = exp{ j (JTB / T ) (1+b) (2kA2 (m+n))} â€˘ exp{ j (TTB/T) (1+b) (m2A2 - T' Am)} â€˘ exp{ j (TTB/T) (1+b) (n 2A 2 - T' An) } â€˘ exp{ j (TTB/T) (1+b) (2mnA2)} â€˘ exp{j(trB/T) (b) (k 2A 2 + T' 2/4 - kAT')} (7) When an FFT i s applied to the above data, the series takes the form T*/A - 1 s(r) = Z W,(k) exp{j(c,k + c 2 k 2 ) } k=0 where c, and c 2 are constants. Since t h i s i s no longer a geometric series, a closed form evaluation cannot be obtained. Therefore i t i s not possible to predict, using an exact formula, the e f f e c t s of the frequency mismatch on image q u a l i t y . However, i t i s possible to make some observations based on the previous insights gained into how the step transform works. In the f i r s t place, one observes that the coarse resolution aperture after the mixing operation no longer contains a constant frequency s i g n a l . It now contains a residual FM rate as s p e c i f i e d in the l a s t l i n e of equation (7). This w i l l result in broadening of the coarse resolution FFT output pulses. This could result in increased a l i a s i n g problems and may require increased overlapping of the coarse resolution apertures. Secondly, the frequency increments between successive coarse resolution apertures has been altered. (The frequency of the coarse resolution aperture i s s p e c i f i e d in the f i r s t l i n e of equation (7).) Thus, when data i s reordered for input to the 71 fine resolution FFT's, there w i l l be some amplitude modulation of the s i g n a l . This has the e f f e c t of broadening the output pulse. The frequency of the input signal to the fine resolution FFT w i l l also be modified, shown in the fourth l i n e of equation (7). L astly, from the t h i r d l i n e of equation (7), i t w i l l be noted that the quadratic phase factor is also modified and during the process w i l l not be removed completely. This w i l l also modify the input signal to the fine resolution FFT so that i t sweeps across some range of frequencies. This w i l l result in some misregistration of looks, since the FFT's w i l l resolve d i f f e r e n t portions of the target signal to d i f f e r e n t frequenc ies . Thus, i t can be seen that a l l stages of the step transform process are affected to some extent by mismatching of the FM rates. In order to quantify these e f f e c t s , i t i s necessary to perform some simulations. 6.1.1 Data For Typical S a t e l l i t e SAR Systems In order to present simulation results which are meaningful, i t i s f i r s t necessary to determine how quickly the FM rate changes for t y p i c a l SAR systems. Using equation (6) of Sec. 5.1, we see that 72 dK 2V2 - Hz2/m d r 0 Xr2, Converting t h i s formula to more convenient units, dK p = â€” â€˘ 100 % FM rate error / range c e l l d r 0 r 0 where p = slant range c e l l width Typical values were computed using the data in Appendix C and are presented in Table IV. Note that there i s a residual error in the autofocus estimation of the FM rate of approximately Â±0.06 % for the JPL Nominal case and Â±0.1% for the SEASAT case. These figures should be added to the data of Table IV when considering the maximum possible error. Table IV - FM Rate Errors for Typical SAR Systems System Slant Range km FM Rate Error Hz2/m %/range c e l l %/!00 range c e l l s SEASAT 860 810 0.00059 0.00067 0.00077 0.0081 0.08 0.08 JPL Nominal 550 500 0.0016 0.0020 0.0019 0.0021 0.19 0.21 COMSS/LASS 636 586 0.0040 0.0047 0.00090 0.00098 0.090 0.10 Simulation runs to determine FM rate error e f f e c t s were carri e d out for the SEASAT, JPL Nominal and COMSS/LASS s a t e l l i t e 73 SAR systems using the processing parameters s p e c i f i e d in Appendix C. The step transform f i l t e r output was interpolated by expanding the fine resolution FFT by a factor of eight. The resul t i n g data i s summarized in Tables V, VI and VII and shows a wide d i s p a r i t y in the results obtained for the three systems. Table V - FM Rate Error Simulation Results for SEASAT Parameters % Rate Error 1 Look 4 Looks 3 dB Width % Broadening Peak Magnitude 3 dB Width % Broadening Peak Magnitude 0.0 1 .75 0.0 0.0 dB 1 .36 0.0 0.0 dB 0.02 1 .76 0.6 -0.03 dB 1 .37 0.7 -0.01 dB 0.04 1 .80 2.9 -0.2 dB 1 .38 1.5 -0.06 dB 0.06 1 .87 6.9 -0.3 dB 1 .39 2.2 -0.1 dB 0.08 1 .97 12.6 -0.6 dB 1 .40 2.9 -0.3 dB 0.10 2.10 20.0 -0.9 dB 1 .42 4.4 -0.4 dB 0.12 2.27 29.7 -1.2 dB 1 .45 6.6 -0.6 dB 0.14 2.47 41 . 1 -1.6 dB 1 .48 8.8 -0.8 dB 0.16 2.69 53.7 -2.0 dB 1 .51 11.0 -1.0 dB 0.18 2.92 66.9 -2.3 dB 1 .56 14.7 -1.3 dB 0.20 3.15 80.0 -2.6 dB 1 .61 18.4 -1.6 dB Note that % Rate Error = 100-b where b i s the variable defined in Sec. 5.2. There are several d i f f e r e n t factors which influence the r e s u l t s , each of which w i l l be dealt with i n d i v i d u a l l y . The Tables 74 Table VI - FM Rate Error Simulation Results for JPL Nominal Parameters % Rate Error 1 Look 4 Looks 3 dB Width % Broadening Peak Magnitude 3 dB Width % Broadening Peak Magnitude 0.0 1 .64 0.0 0.0 dB 1 .36 0.0 0.0 dB 0.02 1 .66 1.2 -0.09 dB 1 .37 0.7 -0.03 dB 0.04 1 .75 6.7 -0.4 dB 1 .39 2.2 -0.1 dB 0.06 1 .92 17.1 -0.7 dB 1 .41 3.7 -0.3 dB 0.08 2.16 31.7 -1.3 dB 1 .44 5.9 -0.6 dB 0.10 2.50 52.4 -1.9 dB 1 .49 9.6 -0.9 dB 0.12 2.88 75.6 -2.5 dB 1 .56 14.7 -1.3 dB 0.14 3.24 97.6 -3.1 dB 1 .64 20.6 -1.7 dB 0.16 3.62 120.7 -3.5 dB 1 .75 28.7 -2.2 dB 0.18 4.03 145.7 -3.9 dB 1 .90 39.7 -2.8 dB 0.20 4.44 170.7 -4.3 dB 2.08 52.9 -3.3 dB specify the results obtained for certain measurements made on the main lobe of the output data of the processor. The 3 dB width of the main lobe i s s p e c i f i e d in terms of the number of output samples. The % broadening i s calculated using the 3 dB width of the main lobe at 0% rate error as a reference. For the 4-look s i t u a t i o n , because of overlapping of the looks, i t was possible to use a data window of the same length as the FFT. Since a Hamming window was used on the fine resolution aperture, the minimum 3 dB width i s ac t u a l l y 1.33 bins i f the eff e c t of the antenna p r o f i l e i s not considered. For the single-look 75 Table VII - FM Rate Error Simulation Results for COMSS/LASS Parameters % Rate Error 1 Look 4 Looks 3 dB Width % Broadening Peak Magnitude 3 dB Width % Broadening Peak Magnitude 0.0 1 .64 0.0 0.0 dB 1 .38' 0.0 0.0 dB 0.1 1 .66 1 .2 -0.05 dB 1 .39 0.7 -0.01 dB 0.2 1 .70 3.7 -0.2 dB 1 .39 0.7 -0.07 dB 0.3 1 .76 7.3 -0.4 dB 1 .40 1 .4 -0.2 dB 0.4 1 .87 14.0 -0.7 dB 1 .42 2.9 -0.3 dB 0.5 2.01 22.6 -1.0 dB 1 .44 4.3 -0.4 dB 0.6 2.21 34.8 -1.4 dB 1 .47 6.5 -0.6 dB 0.7 2.44 48.8 -1.9 dB 1 .50 8.7 -0.8 dB 0.8 2.71 65.2 -2.3 dB 1 .54 11.6 -1.1 dB 0.9 2.97 81.1 -2.7 dB 1 .59 15.2 -1.3 dB 1.0 3.23 97.0 -3.1 dB 1 .65 19.6 -1.6 dB case, however, the data window i s reduced by a factor of (1-/3) with respect to the length of the FFT and, therefore, the window function i s sampled at a rate of 1/(1-/3). This accounts for the d i s p a r i t y between the single-look and 4-look cases at 0% rate error. The reference used for the peak magnitude of the main lobe i s the value at 0% rate error. I n i t i a l l y , some simulations were done to determine the e f f e c t of point target placement on the f i l t e r response. Placement was done for 2 cases such that the coarse resolution FFT output was sampled exactly at the peak of the main lobe and 76 as far off to the side of the main lobe as possible. The antenna p r o f i l e was turned o f f . It was observed that as the FM rate error increased, there was far less magnitude va r i a t i o n in the fine resolution FFT input data for the case where the coarse resolution aperture was sampled at the peak of the main lobe. However, there was no observable difference in the 3 dB width or attenuation of the main lobe between the two cases. This suggests that the most important information for the pulse compression algorithm is contained in the phase of the input si g n a l , not the magnitude. The analysis of Chapter III and the fact that the antenna p r o f i l e i t s e l f introduces considerable magnitude v a r i a t i o n tends to support t h i s conclusion. For the data presented here, the point targets were positioned so that the coarse resolution FFT output was sampled at the peak of the main lobe. The single most s i g n i f i c a n t observed ef f e c t of the rate error was to change the fine resolution aperture data from a CW signal to a linear FM si g n a l . There was no measureable s h i f t in the centre frequency and, therefore, there was no s i g n i f i c a n t displacement of the output p u l s e . 1 1 The only s i g n i f i c a n t e f f e c t for the single look case was some attenuation of the peak of the output pulse and broadening of the main lobe, as can be seen from the data in Tables V, VI and VII and shown in Figure 11. For multi-look processing, the residual FM component in the 1 1 This result i s only v a l i d for situations where the antenna beam centre is aligned with the zero Doppler l i n e . 77 Figure 11 - % Broadening vs. FM Rate Error From Simulation Results % Broadening Rate 0.0 0.05 0.10 0.15 0.20 Error 78 fine resolution aperture data has some additional e f f e c t s . Since each look covers only a portion of the aperture, the data in each look does not cover as broad a portion of the frequency spectrum as for the single-look case. Therefore, the % broadening within each look i s less than for the single-look case. However, the centre frequency of the individual looks i s displaced causing misregistration of the output pulses between individual looks. This contributes to the broadening of the f i n a l output pulse when the looks are summed together. The e f f e c t i s i l l u s t r a t e d in Figure 12. Because the resolution of the f i n a l output image i s decreased for the multi-look s i t u a t i o n , the actual % broadening of the output pulse i s s i g n i f i c a n t l y less for that case. The next most s t r i k i n g fact about the results obtained i s that there i s a great v a r i a t i o n between the three SAR systems used. This can be attributed to the variation in the extent of the processed signal aperture in terms of the number of samples, T/A. Due to the exact time-frequency correspondence of the linear FM signal, the actual frequency displacement of the azimuth signal at the ends of the aperture i s much greater for the JPL Nominal system than for the COMMS/LASS system. Therefore, a given % FM rate error has a much more severe impact on the image quality obtained from the JPL Nominal system. FM rate error e f f e c t s can be more e a s i l y studied a n a l y t i c a l l y using the BPF SPECAN pulse compression approach as a vehicle and t h i s has been done by MacDonald, Dettwiler and Associates [24]. Because the BPF SPECAN and step transform techniques e s s e n t i a l l y 79 Figure 1 2 - Eff e c t of FM Rate Error on Addition of Looks for SEASAT Case Magnitude Look 3 Summed Output Look 4 - 2 . 0 Time (samples) 80 perform the same transformation on an input signal, i t i s expected that FM rate errors would have very similar e f f e c t s on the output images of both. Although the r e s u l t s presented here and in [24] cannot be compared exactly, because d i f f e r e n t data windows were used and d i f f e r e n t approaches were taken to the measurement and analysis of the data, similar trends are demonstrated. 6.1.2 Frequency Step Mismatch Another factor which must be considered when examining the ef f e c t s of FM rate error, is that the step transform processing parameters may need to be changed as well. In p a r t i c u l a r , as the FM rate changes, the value of H or the frequency stepping between coarse resolution apertures w i l l change as well. As was noted in Chapter I I I , i t i s necessary to maintain these steps close to an integer number of output bins. The p r i n c i p l e means of making adjustments to compensate for t h i s error i s by adjusting the time, NA, between successive coarse resolution apertures. However, th i s adjustment can only be made in increments of the sampling period (determined by the PRF) or integer values of N. For e f f i c i e n t FFT operation, the number of samples in the coarse resolution aperture must be a power of 2 and, therefore, cannot be used to do t h i s finetuning. The p r i n c i p l e question which must be asked i s whether adjustment of NA by integer values of N provides a s u f f i c i e n t degree of accuracy. In Sec. 3.1, i t was shown that the stepping 81 between coarse resolution apertures i s BT' AN H = output bins. T Substituting the FM rate K = B/T, H = KT'AN output bins. For two d i f f e r e n t coarse resolution aperture separations H H N, - N 2 = K,T'A K 2T'A H K 2 ~~ K 1 ( ) (8) T'A K,K2 As explained before, the minimum adjustment for N i s 1, and, therefore the minimum value of N, - N 2 i s 1. Thus K ^ T ' A K 5 - K1 -H Assuming K 2 = K, and substituting K, = K 2 = K on the righthand side, K2-K, KT'A K H KT'A 100 % change in FM rate. H Noting that H = KT'AN and assuming N, = N 2 = N, the % change in FM rate can also be expressed by the following equation K2-K, 1 = - â€˘ 100 % K N Using the data in Appendix C, we see that the change in FM rate i s 1) for SEASAT, 2.4 % 2) for JPL Nominal, 1.7 % 82 3) for COMSS/LASS, 6.3 %. It can be seen that incrementing N by 1 does not provide a fine enough adjustment for FM rate change compensation. In order for t h i s method to be used, the value of N would have to be on the order of 1000. (For N=1000, (K 2-K,)/K = 0.1%). This i s c l e a r l y not feasible, since i t i s required that (T'/NA) > 2 to prevent a l i a s i n g e f f e c t s . For example, in the COMSS/LASS case, T/A = 1081 samples. There are also upper l i m i t s on the value of T' imposed by the RCMC requirements, as is discussed in the following section. Thus i t i s apparent that, at least for the three s a t e l l i t e SAR systems described here, i t i s not possible to rely on t h i s simple method of parameter adjustment for FM rate errors. An a l t e r n a t i v e method might be provided as follows. It i s possible to give the appearance of non-integer values of N by recalculating the reference function for each new position so that the ramp occupies a s l i g h t l y displaced time segment, but s t i l l has i t s zero frequency at the proper time p o s i t i o n . In t h i s way, the reference ramps are spaced at non-integer values of N and, thereby, achieve the correct value of H. This would not aff e c t the theore t i c a l analysis of Chapter I I I , since one end of the ramp i s being extended and the other end deleted. The important phase factors, i . e . the linear FM factor for input to the fine resolution FFT and the quadratic phase correction term, are s t i l l present. Perry and Martinson, in the i r t h e o r e t i c a l analysis of the step transform [29], suggest that such an adjustment may be necessary in some cases. 83 However, a s i g n i f i c a n t price i s paid for t h i s solution in that the reference must be recalculated for each instance or a number of d i f f e r e n t reference ramps must be stored. For example, i f the required value of N i s 38.2, there would be 4 possible reference ramp positions between samples and 5 d i f f e r e n t reference functions would be required. Therefore, the accuracy required of N must be determined in order to get some idea of the number of reference functions which must be used. Thus, returning to equation (8), H K 2 â€”K 1 N i " N 2 = ( ) T 1 A K 2 K 2 -K, = N( ) K The inverse of t h i s value i s the number of.reference functions required. In the previous section, i t was demonstrated that, for a given % broadening of the main lobe, the three SAR systems exhibit very d i f f e r e n t FM rate accuracy requirements. If i t is assumed that 10% broadening i s the most that can be tolerated, the following results are obtained for the 4-look case: 1) for SEASAT, 0.06 % FM rate accuracy or approximately 17 d i f f e r e n t reference functions, 2) for JPL Nominal, 0.04 % FM rate accuracy or approximately 17 d i f f e r e n t reference functions, 3) for COMSS/LASS, 0.8 % FM rate accuracy or approximately 8 d i f f e r e n t reference functions. 84 6.2 Range C e l l Migration The range c e l l migration problem arises due to the fact that the azimuth signal does not l i e in a single range c e l l , as shown in equation (5) of Sec. 5.1. Depending on the extent of the migration, i t may not be possible to complete the step transform processing without some corrections. Ideally, each target would be corrected i n d i v i d u a l l y in situations where RCMC is required. However, t h i s results in excessive amounts of processing. Therefore, ways of i s o l a t i n g groups of targets which can be corrected at the same time are sought. In the step transform process, groups of targets are isolated by the coarse resolution FFT. Each FFT output bin contains phase and magnitude information r e l a t i n g to a certain number of targets which are clos e l y grouped together. Furthermore, th i s information i s isolated to a known segment of the azimuth signal because of the unique time-frequency re l a t i o n s h i p of the line a r FM signa l . It may be possible to move the data in each FFT output bin by an appropriate number of range c e l l s to achieve the desired correction. Admittedly, t h i s w i l l not correct each target exactly. However, i f the errors are s u f f i c i e n t l y small so that image quality remains acceptable, the computation savings w i l l c e r t a i n l y make i t worthwhile. In order to determine the f e a s i b i l i t y of performing RCMC in such a manner, i t i s f i r s t necessary to analyze the range c e l l migration phenomenon in greater depth. Often for s a t e l l i t e systems, there i s no mechanism to adjust the alignment of the antenna beam centre with respect to the zero Doppler l i n e . 85 However, i t i s possible to determine the offset of the beam. Equation (5) of Sec. 5.1 may be modified to express target range as a function of azimuth time with respect to the beam centre crossing. Substituting r\ = r\ + T?0 into equation (5) we obtain BX V 2 V 2 V 2 r(7? ) = r 0 + ril + â€”r)0n + r\2 BX 2r 0 r 0 BX 2 r 0 BX where r) = azimuth time referenced to beam centre crossing BX 7j 0 = azimuth time offset of beam centre from zero Doppler The f i r s t two terms represent the target range to the beam centre. Since the time offset of the beam centre, TJ 0, remains constant over r e l a t i v e l y long periods of time, both these terms can be considered to be constant. The t h i r d term i s known as range walk and represents a linear o f f s e t which can be corrected by a simple skewing operation p r i o r to step transform processing. This process has been described in d e t a i l for a BPF SPECAN implementation of a SAR processor [24], The same considerations apply to the step transform and the skewing operation i s equally applicable, since i t can be done pr i o r to any other processing. Perry and Kaiser [28] b r i e f l y describe a similar RCMC method for their step transform processor. Since the application i s an airborne SAR system, no further RCMC i s required in that case. The fourth term represents the quadratic component of RCM and must be considered more c l o s e l y . In deciding whether i t would be possible to correct for quadratic RCM after processing the coarse resolution FFT, there are two major issues which must 86 be considered. F i r s t l y , within any one coarse resolution aperture, there must be enough signal from the point target so that the FFT can adequately f i l t e r the signa l . Therefore, the QRCM should never be so severe that the azimuth signal migrates across more than one range c e l l in a single coarse resolution aperture. This implies that there is a maximum length for the coarse resolution aperture. Secondly, each coarse resolution FFT output bin contains data on targets at various range and azimuth positions in the same general area. In r e a l i t y , each target follows i t s own path through signal memory and should be corrected i n d i v i d u a l l y . The errors introduced by applying the same correction to a l l targets in an output bin must be determined. This places an upper l i m i t on the frequency range covered by. each coarse resolution output bin and e f f e c t i v e l y places a lower l i m i t on the length of the coarse resolution aperture in terms of the number of samples. Each of these l i m i t s w i l l now be derived in a quantitative fashion. The most severe QRCM occurs at the ends of the processed aperture, where r? = T(l-/3)/2. Thus, QRCM at that point may be BX expressed as V 2 QRCM, = (T ( l - | 3 ) / 2 ) 2 2 r 0 V 2 T z = (1-/3)2 8R0 At the other end of that coarse resolution aperture 7? = T(1-j5)/2 - T' and BX 87 V 2 QRCM2 = (T(l - 0 ) / 2 - T ' ) 2 2r 0 V 2 (T 2(l-/3) 2/4 - T'T(1-/J) + T' 2) 2r 0 Therefore the extent of QRCM over that single aperture i s V 2T' QRCM, - QRCM2 = (TO-/3) - T') m 2r 0 V 2T' = (TO-/3) - T') range c e l l s 2r 0p Assuming T' Â« T( 1-/3), 1 2 2r 0p T' = (QRCMT-QRCMJ) V 2T(l-/3) To prevent undue pulse broadening in the range d i r e c t i o n , the QRCM should be kept to less than 1 range c e l l over a coarse resolution aperture. Therefore, QRCM, - QRCM2 < 1 and 2r 0p T' < V 2T( 1-/3) or T' 2r 0p â€” < samples A V2T(1-/3)A Now, an expression for the lower l i m i t on T' w i l l be derived. There are T'/A coarse resolution FFT output bins, each covering a bandwidth of KTA/T' Hz or target information over a time period TA/T'. Again, the situa t i o n where the QRCM i s the 1 2 The e f f e c t s of thi s and another similar approximation are demonstrated at a lat e r point. 88 most severe must be considered, i . e . the time r? = T(l-/3)/2 to BX T? = T(l-/3)/2 - TA/T'. Therefore, BX V 2T 2 QRCM, = (1-/3)2 8 r 0 and V 2T 2 QRCM3 = (0-/3) 2/4 - (1-/3)A/T' + A 2/T' 2) 2r 0 so that, V 2T 2A QRCM, - QRCM3 = (1 - 0 - A/T') m 2r 0T' V 2T 2 A = (1 - /3 - A/T') range c e l l s 2r 0pT' Assuming A/T' Â« 1 - /? V 2T 2A(l-/3) QRCM, - QRCM3 = m 2r 0T' V 2T 2A( 1-/3) = range c e l l s 2r 0T'p Again, for QRCM, - QRCM3 < 1 range c e l l s , V 2T 2A( 1-/3) T' > s 2r 0p or T' V 2T 2(l-/3) â€” > samples A 2r 0p The permissible range of values for T' for some t y p i c a l SAR systems i s presented in Table VIII. Since some appoximations were made to obtain the range of values for T', the actual QRCM 89 Table VIII - RCM Limits on Coarse Resolution Aperture Size Units SEASAT JPL Nominal COMSS/LASS Slant range km 860 550 636 - T' Minimum - T'/A s samples 0.0246 40 0.0139 36 0.00122 2 QRCM in sample for minimum T' range c e l l s 0.97 0.96 0.47 - T* Maximum - T'/A s samples 0.0801 132 0.0806 210 0.324 535 QRCM over aperture for maximum T' range c e l l s 0.97 0.96 0.41 corresponding to each l i m i t has been calculated. As can be seen from the figures, the approximations a c t u a l l y produce somewhat conservative estimates. There are some situations where i t may not be possible to accept the above r e s t r i c t i o n s placed on the value of T' including: 1) the two ranges of values s p e c i f i e d for T' by the above cal c u l a t i o n s do not inte r s e c t , 2) the amount of uncorrected RCM produces unacceptable degradation of image q u a l i t y , 3) the range of values for T'/A does not include a power of two (for e f f i c i e n t FFT computation). In the above sit u a t i o n s , a greater frequency resolution i s required for a given amount of target return data. This 90 resolution can be obtained by padding the aperture with zeros. Note that the coarse resolution FFT output pulses w i l l be broadened by using this technique. Care must be taken to provide s u f f i c i e n t overlapping of the apertures so that there are no a l i a s i n g problems with the fine resolution FFT data. As an extension of t h i s concept, an attempt could be made to define an optimum value of T' using the c r i t e r i o n that the product (QRCM, - QRCM2)(QRCM1 - QRCM3) be a minimum. It turns out that there i s no absolute minimum for that product as a function of T'. However, i t does have a negative slope and, therefore, lesser values are found for larger values of T'. 91 -VII. IMPLEMENTATION ON SOME SATELLITE SAR SYSTEMS In t h i s chapter, an attempt w i l l be made to draw together the conclusions which have been made about how individual factors affect the step transform technique. The process of parameter selection for the step transform w i l l be described and the solutions arrived at for some t y p i c a l SAR systems w i l l be examined. Using that data, some observations are made about the performance of the step transform in terms of computation and memory requirements. This information allows us to make quantitative comparisons with other pulse compression techniques. F i n a l l y , some ideas are presented on what an implementation of a step transform SAR processor might look l i k e . 7.1 Summary Of Processing Parameter Restrictions The f i r s t parameter chosen i s the length of the coarse resolution aperture. Table VIII s p e c i f i e s upper and lower l i m i t s on i t s value, in terms of time units and number of samples for the three SAR systems considered, based on RCMC considerations. For e f f i c i e n t FFT processing, the r e s t r i c t i o n that the number of samples must be a power of 2 i s added. This s t i l l leaves several d i f f e r e n t values which can be chosen. Counterbalancing t h i s i s the consideration that there must be no s i g n i f i c a n t a l i a s i n g between FFT output samples because the FFT is too short. This applies to the fine resolution FFT as well and, in that case, the number Of looks, L, must be taken into 92 account. For t h i s reason, T'/AL should be 16 or greater. A secondary consideration i s that the value of T* i n d i r e c t l y a f f e c t s the number of reference functions which must be stored by determining the value of N.' The relationship between N and the number of reference functions i s described in Sec. 6.1.2. Once a value of T' has been chosen, i t i s possible to determine the spacing between the reference ramps using the formula H = BT'AN/T, noting that H must be an integer. The number of possible values for N i s reduced by noting that the overlap r a t i o T'/NA should be s u f f i c i e n t l y large to prevent any s i g n i f i c a n t a l i a s i n g of the coarse resolution FFT output data. However, i t i s desirable to minimize T'/NA since i t plays a very s i g n i f i c a n t role in determining the computation rates and memory requirements. After choosing the value of N, and thereby specifying the overlap r a t i o T'/NA, i t i s possible to choose an optimal window for the coarse resolution aperture using data such as that presented in Table I I . Note that Table II gives only a p a r t i a l selection of possible windows since some, eg. Kaiser-Bessel, actually o f f e r a continuum of choices using the parameter a. The data window for the fine resolution aperture i s determined by the requirements of the f i n a l output pulse in terms of the mainlobe width and sidelobe l e v e l s . 93 7.2 Parameters Chosen For Typical SAR Systems Since each of the example SAR systems chosen for consideration i s somewhat unique in i t s properties and the challenges i t presents, they w i l l be discussed on an individual basis. In each case, the slant range was reduced to obtain parameters which would result in an integer value of N for use in the simulation program. The slant range was not increased since Appendix C sp e c i f i e s the maximum slant range for each system. SEASAT The most challenging problem posed by the SEASAT SAR system for any azimuth cor r e l a t i o n algorithm i s the s i g n i f i c a n t amount of QRCM. As can be seen from Table VIII, the coarse resolution aperture size i s b a s i c a l l y limited to two choices, 64 and 128 samples. Both choices permit four-look processing, which i s what SEASAT i s designed for. To determine the spacing of the reference ramps at the maximum slant range, the formula H = BT'AN/T i s used and shows that for T'/A = 128 samples, N = 41.7 and H = 1. The system parameters were modified as spec i f i e d in Appendix C to allow an integer value of N to be used in the simulation program. A 64 sample aperture cannot be used because i t results in a sit u a t i o n where H < 1 and there i s a l i a s i n g of the data in the fine resolution FFT's. 94 JPL Nominal This system i s quite similar to the SEASAT SAR, except that the RCM problem turns out to somewhat less severe, as demonstrated in Table VIII. On the other hand, the FM rate changes much more quickly than in the other cases. Also the data rate i s greater, which has a s i g n i f i c a n t effect on the processing rate. There are s t i l l only two di f f e r e n t choices for the size of the coarse resolution aperture. Using the system parameters for the maximum slant range, for T'/A = 128 samples, N = 58.9 and H = 1. The modified system and processing parameters for this case are sp e c i f i e d in Appendix C. Once again, the 64 sample aperture cannot be used because of a l i a s i n g e f f e c t s . COMSS/LASS The COMSS/LASS system i s the least demanding of the three systems, in that the RCM problem i s not nearly as great as in the other two systems. This means that much less overlapping of the processing swaths i s required, with a corresponding decrease in the excess computation capacity and memory requirements. Using the same method as for the previous two systems, the following results apply at the maximum slant range: 1) for T'/A = 128 samples, N = 33.5 and H = 4, 2) for T'/A = 64 samples, N = 16.8 and H = 1. Again the case where H = 1 i s chosen, since the fine resolution aperture contains 64 samples versus 32 for the other case. This allows greater f l e x i b i l i t y in choosing the number of looks. 95 Note from Appendix C that the chosen processing parameters for the COMSS/LASS system result in a high overlap r a t i o . Because of the limited choice of values for H, the next smallest overlap r a t i o which could be used was approximately 1.9. From Table II, i t w i l l be seen that there are no windows l i s t e d except the boxcar which can be used with such a small overlap r a t i o . From the data presented, i t i s surmised that any such window would have sidelobe lev e l s of 25 - 30 dB. The step transform would produce processing a r t i f a c t s at those levels and the image qual i t y would not be acceptable. This decision has a s i g n i f i c a n t e f f e c t on the computation rates and memory requirements discussed in the next section. 7.3 Computation Rates And Memory Requirements The information presented in Chapters V and VI reveals that there are some additional considerations which must be taken into account in addition to those presented in Sec. 4.2 and 4.3 when determining the computation and memory requirements. One of the s i g n i f i c a n t aspects of SAR signal processing i s that the output i s a two dimensional image. Thus the calculations made previously in Chapter IV must be expanded to take account of those aspects. Because of the RCM phenomenon, i t i s necessary to have several azimuth l i n e s of data stored in memory in order to compress even a single point target. In practice, azimuth processing i s done in range subswaths which are of a manageable size in terms of memory storage 96 requirements, yet allow azimuth processing to be completed for a s i g n i f i c a n t number of azimuth l i n e s . Thus, the amount of overlapping of the range subswaths does not become a s i g n i f i c a n t burden. In order to produce a standardized comparison, the computation rates, calculated for the three SAR systems sp e c i f i e d in Appendix C, w i l l be expressed quanti t a t i v e l y in terms of the number of complex mul t i p l i c a t i o n s per second (cmps) required to produce an image of 128 azimuth l i n e s in real time. The s p e c i f i c computational stages considered w i l l be those r e l a t i n g to the pulse compression operation i t s e l f . Linear RCMC and look summation requirements w i l l not be considered since they are the same for a l l approaches. The requirement to correct QRCM w i l l be considered quantitatively only in the amount of overlapping required in the range subswaths. 1 3 Thus, processing i s performed over (128 + QRCM) azimuth l i n e s before RCMC and 128 azimuth l i n e s after RCMC. Another factor which must be considered i s the azimuth time extent of the image to be produced. It w i l l be necessary to validate or reject the assumption made in Chapter IV that the processed portion of a point target return signal i s small in comparison to the f u l l time extent of the processing region, i . e . the assumption that the regions of rejected data in Figure 5 are small. This w i l l be done by computing the number of data points not used and the amount of processing done on them. 1 3 In general, i t i s necessary to perform some sort of interpolation to reduce pulse broadening s u f f i c i e n t l y in the range d i r e c t i o n . This must be done for each input point to the fine resolution FFT. 97 Thus, the processing requirements at each stage are as follows: 1) Reference Function Multiply - FÂ«(128 + QRCM)â€˘(T'/NA) cmps, 2) Coarse Resolution FFT - F-028 + QRCM)â€˘(Tf/NA)â€˘log2{T'/A}/2 cmps, 3) Quadratic Phase Correction - FÂ«(128 + QRCM)â€˘(T'/NA) cmps, 4) Fine Resolution FFT Window - 64F â€˘ (T' /NA) â€˘ (1 - /3) cmps for the single look case and 64F-(T'/NA) cmps for the multi-look case, 5) Fine Resolution FFT - 64FÂ»(T 1/NA)â€˘log 2{T'/AL} cmps, 6) Scaling - 64F/L cmps. The amount of rejected data i s e s s e n t i a l l y equal to the time extent of the processed signal aperture, T/A. The coarse resolution aperture FFT and reference function multiply are the only computations which must be performed on t h i s data. Therefore, the number of complex m u l t i p l i c a t i o n s i s (128 + QRCM)(T/A)(T'/NA) for the multiply operation and (128 + QRCM)TT'â€˘log2{T'/A}/2N for the FFT. Memory requirements may be computed as follows, assuming that each complex number occupies 1 word of memory. A single coarse resolution aperture must be stored ahead of the reference function multiply requiring (128 + QRCM)T'/A words of memory. For a l l of the systems considered here, processing parameters have been chosen such that H = 1. The reorder memory, therefore, requires approximately T' 2/2A 2 words per range l i n e or (128 + QRCM)T'2/2A2 words per swath. Look summation memory requirements are not included since they are the same for a l l 98 Table IX - Memory Requirements and Computation Rates SEASAT JPL Nominal COMSS/LASS Looks 1 4 1 4 1 4 Computation Rates (x 10s cmps) Reference Function Multiply 0.7 0.7 0.8 0.8 0.9 0.9 Quadratic Phase Correction 0.7 0.7 0.8 0.8 0.9 0.9 Coarse Resolution FFT 2.5 2.5 2.8 2.8 2.6 2.6 Fine Resolution Aperture Window 0.3 0.3 0.3 0.4 0.4 0.4 Fine Resolution FFT 2.3 1.7 2.6 1 .8 2.5 1.7 Scaling 0.1 - 0.2 - 0.1 -Total 6.6 5.9 7.5 6.6 7.4 6.5 Complex M u l t i p l i e s on Rejected Data (x 10 s) Reference Function Multiply 2.3 2.3 2.3 2.3 0.6 0.6 Coarse Resolution FFT 8.1 8.1 8.1 8.1 1.7 1.7 Total 10.4 10.4 10.4 10.4 2.3 2.3 Memory (x 10 s words) Coarse Resolution Aperture 0.02 0.02 0.02 0.02 0.01 0.01 Reorder Memory 1 .1 1 .1 1 .1 1 .1 0.3 0.3 Total 1.12 1.12 1.12 1.12 0.31 0.31 99 approaches. Examining Table IX, i t can be seen that the computation requirements are amazingly similar for a l l three systems, in spite of the wide dis p a r i t y in sampling rate and number of samples in the processed signal aperture. This i s due, in large part, to the moderating effect of the overlap r a t i o . Note that there i s a s i g n i f i c a n t d i s p a r i t y in memory requirements, which are determined by the number of samples in the coarse resolution aperture. In ca l c u l a t i n g the amount of computation on the rejected data, i t is seen that the JPL Nominal and SEASAT systems exhibit similar requirements, due to the ef f e c t of the overlap r a t i o . However, for the COMSS/LASS si t u a t i o n , the processed aperture i s much shorter and i s the overwhelming influence. 7 .4 Comparison With Other Techniques The results of the previous section demonstrate that the hardware requirements to perform step transform pulse compression do not always turn out as one would expect for a given set of system parameters, but are also heavily influenced by the processing parameter requirements of the step transform. In order to arr i v e at some general conclusions, i t i s necessary to derive some results which demonstrate a general trend and to support those results with some s p e c i f i c examples. Comparisons w i l l be made with three other pulse compression techniques described in Chapter II; time-domain convolution, fast 100 convolution and bandpass f i l t e r spectral analysis. I n i t i a l l y , a generalized comparison w i l l be done to show the major trends. Then, the computation rates w i l l be calculated for the three systems sp e c i f i e d in Appendix C, using the same assumptions as in the previous section for the step transform. 7.4.1 General Comparison Computation rates w i l l be expressed as the number of complex m u l t i p l i c a t i o n s required to produce a single point of output data. RCMC requirements are not taken into account, since a s p e c i f i c system i s not considered. The results are expressed as a function of the number of samples in the processed signal aperture, M, and the number of looks, L, and are displayed' graphically in Figure 13. FFT processing e f f i c i e n c y i s assumed to be equivalent to that achieved for power of 2 FFT lengths. Time-Domain Convolution The time-domain convolution approach i s generally characterized as having large computation rates and small memory requirements. For multi-look processing, frequency translation and lowpass f i l t e r i n g must be done to extract data for the individual looks. For s a t e l l i t e systems, the antenna may not always be pointing broadside. Therefore, the beam centre may not always be aligned with the zero Doppler frequency and we w i l l assume that these two stages must be performed for 101 single-look processing as well. Frequency translation requires 1 complex m u l t i p l i c a t i o n per point (cmpp) for each look or L cmpp in t o t a l . Lowpass f i l t e r i n g i s done using a 16-point non-recursive d i g i t a l f i l t e r which requires the equivalent of 2 complex mu l t i p l i e s per point or 2L cmpp. The f i l t e r output can be subsampled by a factor L. Therefore, the correlation function i s M/L2 complex words for each look and the azimuth correlator requires M/L cmpp. Therefore, the t o t a l number of computations per point i s CP = 3L + M/L cmpp. Fast Convolution If i t i s assumed that the forward FFT's are overlapped by 50% 1 4 , each FFT i s 2M/L samples long. M'log 2(2M/LJ/L complex mu l t i p l i c a t i o n s are required for each FFT aperture. The matched f i l t e r i n g stage requires 2M/L complex m u l t i p l i c a t i o n s . The inverse FFT's are of length 2M/L2 and require MÂ«log 2{2M/L 2}/L 2 complex mu l t i p l i c a t i o n s each or M'log 2{2M/L 2}/L complex m u l t i p l i c a t i o n s per aperture. Processing of each aperture produces M/L v a l i d output points. Therefore, CP = log2{2M/L} + 2 + log 2{2M/L 2} = 4 + 2-log2{M} - 3Â«log 2{L} cmpp. The FFT overlap determines the tradeoff between computation and memory requirements. If the overlap i s reduced to 25%, the computation rate i s decreased by approximately 20 - 25 % and the memory requirements are doubled. The 50% value was chosen here, since i t results in comparable memory requirements for the step transform and fast convolution methods. This i s demonstrated for some s p e c i f i c SAR systems in the next section. 102 Figure 13 - Generalized Computation Requirements as a Function of Signal Aperture Extent Fast H 1 1 log 2{M} 4 8 12 16 â€˘f 4 + 8 -+-12 -r-^-log2{M} 16 103 Bandpass F i l t e r Spectral Analysis The bandpass f i l t e r spectral analysis technique requires that the number of bandpass f i l t e r channels be chosen. The major considerations which determine the number of channels required are based on the extent of QRCM. For the purposes of this discussion, i t i s assumed that 16 channels are required. The f i r s t stage i s the reference function multiply which requires 1 cmpp. Bandpass f i l t e r i n g and subsampling are performed as a single operation and requires 6 cmpp [24]. The FFT's are approximately M/16L samples long and each require M'log2{M/16L}/32L complex m u l t i p l i c a t i o n s . There are 16L FFT's for every M points requiring log 2(M/16L}/2 cmpp. The FFT input data must also be windowed, requiring 0.5 cmpp. Therefore, CP = 7.5 + log2{M/16L}/2 = 5.5 + (log 2{M} - log 2{L})/2 cmpp. Step Transform For the step transform, i t i s necessary to make some assumptions about the processing parameters to be used in order to show a general r e s u l t , as for the previous approaches. Therefore, the following values are s p e c i f i e d : T'/NA = 2 and H = BT'AN/T = 1. For convenience, assume B = 1, T = M and A = 1. Therefore, M = T'AN = T' 2/2. The number of samples in each coarse resolution aperture i s 1 04 SQRT{2M}. Therefore, using the expression for CP as derived in Sec. 4.2, with appropriate modifications for multi-look processing, CP = 6 + log2{M} - log 2{L} + 1/4L cmpp. 7.4.2 Comparisons For Specific SAR Systems In t h i s section, computation rates and memory requirements are computed for three SAR systems, using the same c r i t e r i a as for the step transform in Sec. 7.3. The number of samples in the signal aperture i s T(1 -/3)/ A. Results are presented in Table X. Time-Domain Convolution For t h i s approach, quadratic RCMC i s done during the convolution operation. Range swaths are overlapped for frequency translation and lowpass f i l t e r i n g . Therefore, computation rates may be calculated as follows: 1) Frequency Translation - (128 + QRCM)FL cmps, 2) Lowpass F i l t e r i n g - 2FL(128 + QRCM) cmps, 3) Convolution - 1 28FÂ»T( 1-/3)/AL cmps. Memory requirements are one look extent for each range c e l l or 128T( 1-j3)/AL words. Memory requirements for the f i l t e r and correlator are n e g l i g i b l e . Fast Convolution The fast convolution approach requires an appropriate 105 Table X - Memory Requirements and Computation Rates for Alternative Approaches SEASAT JPL Nominal COMSS/LASS Looks 1 4 1 4 1 4 Computation Rate (x 106 cmps) Time-Domain Convolution 956 62 2134 137 195 14.7 Fast Convolution 6.9 6.4 8.2 7.7 4.6 4.2 Bandpass F i l t e r Spectral Analysis 2.4 2.2 4.4 4.0 2.2 2.0 Step Transform 6.6 5.9 7.5 6.6 7.4 6.5 Memory (x 10 s words) Time Domain Convolution 0.7 0.17 1.0 0.24 0.14 0.03 Fast Convolution 1 .1 0.3 2.3 0.6 0.26 0.07 Bandpass F i l t e r Spectral Analysis 0.4 0.4 0.6 0.6 0.08 0.08 Step Transform 1 .1 1 .1 1 .1 1 . 1 0.31 0.31 choice of FFT length to achieve an adequate compromise between computation and memory requirements. Here, the FFT length P i s chosen to be approximately double the length of the matched f i l t e r T(1-/3)/AL. For e f f i c i e n t processing, the closest power of 2 i s chosen. The single-look FFT lengths used are 8192 samples for SEASAT, 16384 samples for JPL Nominal and 2048 samples for COMSS/LASS. The computational requirements may be computed by considering that T(1 - /3)/AL data points must be 106 rejected. Thus, the computation rate i s calculated as follows: 1) Forward FFT - PÂ«log 2{P}/2 complex mul t i p l i c a t i o n s per FFT or F-(128 + QRCM) .P-log 2{P}/2(P - T(1-/3)/AL) cmps, 2) "Matched F i l t e r - P complex multiplications per aperture or 128FP/(P - T(1-J3)/AL) cmps, 3) Inverse FFT's - L FFT's of length P/L per aperture requiring PÂ«log 2{P/L}/2L complex mu l t i p l i c a t i o n s each or 64P-log 2{P/L}/(P - T(1-j3)/AL) cmps. Memory requirements are 1 f u l l FFT aperture per range bin or (128 + QRCM)P words per range swath. Bandpass F i l t e r Spectral Analysis Computation rates are as follows: 1) ' Reference Function Multiply - F-(.128 + QRCM) cmps, 2) Bandpass F i l t e r i n g - 6FÂ«(128 + QRCM) cmps, 3) FFT's - the length, P, of the FFT's i s 64, 256 and 512 samples respectively for the single-look COMSS/LASS, SEASAT and JPL Nominal cases respectively. There are 16L FFT's every T seconds, requiring 128â€˘16PÂ«log 2{P/L}/2T cmps per range swath. 4) Window - each FFT aperture must be windowed, requiring 128-16P/2T cmps. The main memory requirements are determined by the input buffer requirements for the FFT's. A similar scheme to that employed for the step transform may be used so that only a triangular region of data must be stored [24], Since the outputs of the bandpass f i l t e r s are oversampled by a factor of 1.33, the memory 1 07 requirements are 1.33(128 + QRCM)Â»T(1 - 0)/2A. 7.4.3 Remarks On Computation Rates The most s i g n i f i c a n t conclusions can be drawn from the general results presented in Sec. 7.4.1. Figure 13 demonstrates that, for the vast majority of systems, the BPF SPECAN approach is the most e f f i c i e n t . It i s evident that the crossover point with the fast convolution technique moves to the right as the number of looks i s increased. Thus, there are certain situations, when a large number of looks is required, where the fast convolution technique i s more e f f i c i e n t than both the BPF SPECAN and the step transform. But, the step transform v i r t u a l l y never has an advantage over the BPF SPECAN approach in terms of these c r i t e r i a . These conclusions are generally supported by the data in Table X. It is intere s t i n g to note that the r a t i o of the step transform to BPF SPECAN computation rates for those three systems i s very close to the overlap r a t i o required in the step transform process for each of the systems. A major additional consideration i s the method of performing RCMC. This i s p a r t i c u l a r l y c r i t i c a l for systems which have a s i g n i f i c a n t QRCM component, i . e . SEASAT and JPE Nominal. In th i s regard, time-domain convolution becomes extremely d i f f i c u l t to implement, since there i s no way of separating groups of point targets to achieve some sort of block processing e f f i c i e n c y as part of the pulse compression algorithm. The fast convolution technique presents a viable 108 alternative in this regard, in that when the forward FFT i s performed, a l l of the targets are superimposed on each other in signal memory and they can a l l be corrected at the same time. In the BPF SPECAN approach, the f i l t e r s are used to separate groups of targets in close proximity and block processing e f f i c i e n c y i s achieved by correcting the groups together. As was explained in Chapter IV, there are p a r a l l e l s between the step transform and BPF SPECAN approaches, in that they both separate out groups of targets. However, for cases of severe RCM, the BPF SPECAN technique holds certain advantages in f l e x i b i l i t y . As shown in Table VIII, for the SEASAT case, there is b a s i c a l l y one choice for the length of the coarse resolution aperture without making s i g n i f i c a n t s a c r i f i c e s in computation and memory requirements. In the BPF SPECAN technique, the equivalent choice which must be made is the number of bandpass f i l t e r channels. There tend to be less r e s t r i c t i o n s on the number of channels than on the coarse resolution aperture length. For each of the systems, there are additional considerations which a f f e c t the complexity of the control l o g i c . The variable antenna pointing angle of s a t e l l i t e SAR systems poses some s i g n i f i c a n t complications for the fast convolution approach, in that the matched f i l t e r output function must be recalculated as the angle changes. For the step transform, the control logic must deal with the requirement for multiple reference functions. The BPF SPECAN approach i s affected by the fact that the output sample rate varies with the FM rate and, 109 thus, slant range [24]. Considerable e f f o r t must be expended computationally to compensate for th i s e f f e c t . The above discussion gives some indications of the hardware requirements of the step transform for azimuth SAR processing r e l a t i v e to some popular alternative approaches. As can be seen from the r e s u l t s , i t i s more e f f i c i e n t than general convolution algorithms in many situations, but i s not competitive with bandpass f i l t e r spectral analysis, which i s also s p e c i f i c a l l y designed to f i l t e r linear FM signals. 7.5 Step Transform Processor Architecture The basic design concept of the processor which w i l l be proposed i s similar to that of Martinson [25], in that both use a pipeline architecture. However, some adjustments must be made for -the SAR s i t u a t i o n . It i s assumed that processing i s done in range subswaths which correspond approximately to the FM rate error which can be tolerated. As was demonstrated in Sec. 6.1.1, thi s varies greatly for the three systems we have considered. Each range c e l l would e s s e n t i a l l y be processed by a separate pipeline and the pipelines would converge at certain points such as the reorder memory where RCMC must be performed. The pipeline processors would be grouped in range subswaths, each group being controlled by a separate control processor. Such an architecture bears s i m i l a r i t i e s to a SIMD (single instruction stream, multiple data stream) machine, in terms of the usual c l a s s i f i c a t i o n system for multiprocessor 110 architectures. A l t e r n a t i v e l y , there could be only one group of processors which would process each subswath in turn. The control processor would be responsible for c a l c u l a t i n g the reference functions and determining the processing parameters and other variables such as rotation of the coarse resolution FFT outputs according to the beam centre o f f s e t . Figure 14 i l l u s t r a t e s t h i s configuration. This approach has a reasonable degree of modularity. Therefore, i t should be amenable to the design of basic system components which could be used in s l i g h t l y d i f f e r e n t configurations for the di f f e r e n t SAR systems and s t i l l retain b a s i c a l l y the same computational rates as calculated previously for a l l of the systems. 111 Figure 14 - Step Transform Processor Architecture 1 1 4-> 4 J U 0) U CU 0) r-i (1) l-r f-i (0 iH (0 <D U <u o to to to to CO TJ 4-> C 4 J C m to (0 (0 Q a EH EH &4 |X4 c o Â»IH in 4 J -3 H Izi O Gu in cu Â« E-> EH EH CX4 &4 CL4 Cu |X4 Ix, 1 t-l u cu cu cu X â€˘e-t 2 â€˘ 2 ' i t >. > i u 1-1 u o o o s 6 E cu cu cu 2 2 2 c o 4 J O (0 o s J e 3 tn f n Z D 4 D EH 1 12 VIII. CONCLUSIONS The step transform process has been described in great d e t a i l in the preceding chapters. In t h i s chapter, the results which have been obtained w i l l be summarized and some comments w i l l be made about further work which must be done before the step transform could be implemented in actual applications, such as the SAR systems described in Appendix C. 8.1 Discussion The step transform was o r i g i n a l l y developed for signal processing applications which present much less stringent demands than some of the SAR systems considered here. Perry and Kaiser consider use of a step transform processor in both conventional and synthetic aperture airborne radar applications [28]. Martinson describes some s p e c i f i c a t i o n s of a step transform processor implementation which has only three d i f f e r e n t SAR focussing modes [25], indicating that the FM rate must only be adjusted to three d i f f e r e n t values over the f u l l range swath. This i s far d i f f e r e n t from the JPL Nominal SAR system, which i s extremely sensitive to FM rate errors because of the large time extent of the processing aperture. These requirements s i g n i f i c a n t l y complicate the step transform as has been described in previous chapters. Table X and Figure 13 show that the step transform i s competitive with the fast convolution method in terms of computational requirements. In addition, i t s major use of the 1 1 3 very basic concepts of d i g i t a l signal processing such as FFT's, data windows, and mixers should make the design of a step transform processor r e l a t i v e l y straightforward, since a l l of the components have been studied in depth before. The major potential design problem i s probably the design of the re-order memory, which i s also used to correct QRCM when required. The challenge here i s to ensure that both operations can be carr i e d out e f f i c i e n t l y without creating a major bottleneck in the process. The actual memory requirements should also be minimized in order for the step transform to be at a l l competitive, since Table X shows that the step transform has close to the highest memory requirements of any of the approaches considered. Most other pulse compression applications present less stringent requirements. For example, in normal radar systems, the pulse generator is part of the same system and could be designed p a r t l y for optimum step transform processing. E s s e n t i a l l y , the designer only needs to have control over one of the system parameters, such as the sampling frequency or the FM rate. Therefore, one would not have to worry about problems such as frequency step mismatching as described in Sec. 6.1.2. However, i t i s possible that other approaches to pulse compression would also be s i m p l i f i e d in such applications and i t would be necessary to perform some analysis of the individual application to make some judgment on the r e l a t i v e merits of each approach. No attempts are made to do that here. It should be noted, however, that in Chapters III and IV, the fundamental 1 1 4 concepts of the step transform are presented without regard to any p a r t i c u l a r application and would be useful in such an analysis. 8.2 Directions For Further Research Many of the results presented in this thesis were derived using a n a l y t i c a l methods. A computer simulation program was written to perform the step transform on a linear FM signa l . This was useful in gaining a better understanding of how the step transform works and in confirming the a n a l y t i c a l r e s u l t s . It was also used to examine FM rate error e f f e c t s . However, a SAR processor produces a two dimensional output image and_ the system must produce acceptable pulse compression in both azimuth and range d i r e c t i o n s . The major cause of image degradation in the range d i r e c t i o n i s inadequate RCMC, so that not a l l of the energy from a single point target i s contained within a single range c e l l when the azimuth pulse compression operation i s performed. In Sec. 6.2, some analysis was done on the basic l i m i t a t i o n s which must be placed on step transform processing parameters to allow RCMC to be performed e f f i c i e n t l y . However, the actual RCMC technique proposed was not implemented in a simulation program. Therefore, the next major step in the study of the step transform would be to extend the simulation program to compress a point target with s i g n i f i c a n t QRCM in two dimensions applying the RCMC techniques described previously. Examining the side lobe lev e l s and main lobe width in the range 1 15 di r e c t i o n , i t could be determined whether or not the li m i t a t i o n s imposed on the step transform processing parameters are adequate. If not, the alternative of padding the coarse resolution aperture with zeros would have to be implemented. This could seriously a f f e c t the computation rate and memory requirements and would have some effect on the competitiveness of the step transform r e l a t i v e to other approaches. Before implementation of the step transform as a SAR processor, i t would also be useful to obtain some further a n a l y t i c a l results on the error e f f e c t s . In p a r t i c u l a r , i t i s useful to use error budgeting techniques to indicate the degree of precision required of the processing algorithm. There are cer t a i n fixed errors which remain constant such as the FM rate estimation error. Deducting errors such as thi s from the t o t a l permissible FM rate error gives an indication of the maximum allowable error contribution from the pulse compression algorithm. Because of the high computation rates required by s a t e l l i t e SAR systems, i t i s generally considered that hardware requirements for a SAR processor are dominated by the actual data processing function. Control functions are assumed to play a less s i g n i f i c a n t role in determining the v i a b i l i t y of a pa r t i c u l a r algorithm. E f f o r t s to implement algorithms which de-emphasize control requirements, eg. the JPL work on the time-domain convolution technique [1] [37], have not been very successful. In thi s case, i t was largely due to the slower-than-predicted advancement of VLSI technology. Control 1 16 structures are often implemented in software and the increasing dominance of software costs in most development projects makes i t dangerous to ignore this facet of processor design. In previous chapters, some issues related to the control structure have been mentioned, but further work on thi s aspect i s required. 8.3 Summary The step transform process and i t s application to SAR azimuth processing has been examined in some d e t a i l . The requirements placed on the processing parameters and the selection of parameters for e f f i c i e n t computation have been described. Some of the r e s t r i c t i o n s placed oh the parameters are f a i r l y stringent and lead to similar hardware requirements for SAR systems which exhibit large d i s p a r i t i e s in their requirements for most other approaches. It was demonstrated that the step transform approach i s more e f f i c i e n t than general matched f i l t e r i n g algorithms i f i t i s possible to use c r u c i a l processing parameter values, such as the overlap r a t i o T'/NA, which are close to the optimal for the step transform. This point i s i l l u s t r a t e d most c l e a r l y by the three SAR systems used to perform a quantitative analysis of the step transform r e l a t i v e to other approaches. The JPL Nominal system, with T'/NA = 2.21, and the COMSS/LASS system, with T'/NA = 4.0, exhibited approximately the same computational requirements. Yet, other approaches cut their computational requirements in 1 1 7 half for the COMSS/LASS system. The step transform can be seen, therefore, as a s i g n i f i c a n t step forward from general algorithms for the processing of li n e a r FM signals. This breakthrough i s made even more s i g n i f i c a n t in l i g h t of the advent of s a t e l l i t e SAR systems, whose high computation rates make i t extremely a t t r a c t i v e to seek speci a l i z e d processing techniques. The step transform i s representative of a class of algorithms which perform the matched f i l t e r i n g operation on a continuous stream of data, rather than processing d i s t i n c t blocks of data as in the fast convolution method. However, since the introduction of the step transform, a large amount of work has been done on the spectral analysis concept, including the development of the BPF SPECAN technique. Some p a r a l l e l s have been drawn between the step transform and the BPF SPECAN approaches in t h i s paper. In terms of the amount of computation, the step transform l i e s somewhere in between the fast convolution and BPF SPECAN approaches. For the design of multi-look SAR processors, both the BPF SPECAN and step transform have the advantage that multi-look processing i s a r e l a t i v e l y minor consideration in most cases, and only impacts one stage of the algorithm. This provides an added degree of f l e x i b i l i t y over the fast convolution technique. In comparing the BPF SPECAN and step transform, the major issue besides computation rates i s the processing of the output data. The BPF SPECAN requirement to process the output data to obtain a uniform data rate over the f u l l range swath brings i t 118 somewhat more in l i n e w i t h the s tep t r a n s f o r m , which has g reater computat iona l requirements and c o n t r o l complex i t y w i t h i n the a l g o r i t h m i t s e l f . 1 19 BIBLIOGRAPHY 1. Arens, W. E. "Real Time Synthetic Aperture Radar Processing for Space Applications", Real-time Signal Processing, Proc. Soc. Photo-Opt. Instr. Eng. 154, 14-21(1978). 2. Barcilon, V., and G. C. Temes. 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S p e c i a l i s t 1 20 Seminar on Case Studies in Advanced Signal Processing, I n s t i t u t i o n of E l e c t r i c a l Engineers, pp. 97-101, September 1979. 13. El a c h i , C , et a l . "Spaceborne Synthetic-Aperture Imaging Radars: Applications, Techniques, and Technology", Proceedings of the IEEE, v o l . 70, no. 10, pp. 1174-1209, October 1982. 14. E l l i s , A. B. E. "A Proposed Method for D i g i t a l l y Processing SAR Data from a S a t e l l i t e " , Int. S p e c i a l i s t Seminar on Case Studies in Advanced Signal Processing, I n s t i t u t i o n of E l e c t r i c a l Engineers, pp. 155-160, September 1979. 15. Ford, J . P. "Resolution Versus Speckle Reduction Relative to Geologic I n t e r p r e t a b i l i t y of Spaceborne Radar Images: A Survey of User Preference", IEEE Trans, on Geoscience and Remote Sensing, v o l . GE-20, no. 4, pp. 434-444, October 1 982. 16. Harris, F. J . "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform", Proceedings of the IEEE, v o l . 66, no. 1, pp. 51-83, January 1978. 17. Helms, H. D. " D i g i t a l F i l t e r s with Equiripple or Minimax Responses", IEEE Trans, on Audio and Electroacoustics, v o l . AU-19, no. 1, pp. 87-93, March 1971. 18. Helms, H. D. "Nonrecursive D i g i t a l F i l t e r s : Design Methods for Achieving Specifications on Frequency Response", IEEE Trans, on Audio and Electroacoustics, v o l . AU-16, no. 3, pp. 336-342, September 1968. 19. Hidayet, M., and P. A. Mclnnes. "On the Sp e c i f i c a t i o n of an Antenna Pattern for a Synthetic Aperture Radar", Radar-77 International Conference, I n s t i t u t i o n of E l e c t r i c a l Engineers, October 1977, pp. 391-395. 20. Kirk, J . C. "A Discussion of D i g i t a l Processing in Synthetic Aperture Radar", IEEE Trans, on Aerospace and Electronic Systems, v o l . AES-10, no. 3, pp. 326-337, May 1975. 21. Kuhler, D. "Application and Limitation of Very Large Scale Integration in SAR Azimuth Processing", Proceedings of the 1978 Synthetic Aperture Radar Technology Conference, New Mexico State University, Las Cruces, N.M., March 1978, paper V-7. 22. Lee, J. S. "A Simple Speckle Smoothing Algorithm for Synthetic Aperture Radar Images", IEEE Trans, on Systems, Man, and Cybernetics, v o l . SMC-13, no. 1, pp. 85-89, January/February 1983. 121 23. Lynch, D. "Signal Processor for Synthetic Aperture Radar", Real-time Signal Processing II, Proc. Soc. Photo-Opt. Instr. Eng. 180, 225-229(1979). 24. MacDonald, Dettwiler and Associates. The Design of a D i g i t a l Breadboard Processor for the ESA Remote Sensing S a t e l l i t e Synthetic Aperture Radar F i n a l Report, European Space Agency Report No. 276, Doc. No. 00-0636, July 1981. 25. Martinson, L. W. "A Programmable D i g i t a l Processor for Airborne Radar", IEEE International Radar Conference, A p r i l 1975, pp. 186-191 . 26. Martinson, L. W., and J. A. Lunsford. "A CMOS/SOS Pipeline FFT Processor - Construction, Performance, and Applications", IEEE National Aerospace and Electro n i c s Conference, May 1977, pp. 574-579. 27. Moore, C. UBC CURVE - Curve F i t t i n g Routines, Computing Centre, University of B r i t i s h Columbia, Vancouver, B.C., September 1981. 28. Perry, R. P., and H. W. Kaiser. " D i g i t a l Step Transform Approach to Airborne Radar Processing", IEEE National Aerospace and Electronics Conference, May 1973, pp. 280-287. 29. Perry, R. P., and L. W. Martinson. "Radar Matched F i l t e r i n g " in Radar Technology, ed. E. Brookner, ch. 11, pp. 163-169, Dedham, MA: Artech House, 1977. 30. Porcello, L. J . , et a l . "Speckle Reduction in Synthetic-Aperture Radars", Journal of the Optical Society of America, v o l . 66, no. 11, pp. 1305-1311, November 1976. 31. Purdy, R. J. "Signal Processing Linear Frequency Modulated Signals", in Radar Technology, ed. E. Brookner, ch. 10, pp. 155-162, Dedham, MA: Artech House, 1977. 32. S0ndergaard, F. "A Dual Mode D i g i t a l Processor for Medium Resolution Synthetic Aperture Radars", Radar-77 International Conference, I n s t i t u t i o n of E l e c t r i c a l Engineers, October 1977, pp. 384-390. 33. Tomiyasu, K. "Tutorial Review of Synthetic-Aperture Radar with Applications to Imaging of the Ocean Surface", Proceedings of the IEEE, v o l . 66, no. 5, pp. 563-583, May 1978. 34. Tyree, V. C. "Custom Large Scale Integrated C i r c u i t s for Space-Borne SAR Processors", Proceedings of the 1978 Synthetic Aperture Radar Technology Conference, New Mexico State University, Las Cruces, N.M., March 1978, paper V-4. 122 35. van de Lindt, W. J . " D i g i t a l Technique for Generating Synthetic Aperture Radar Images", IBM Journal of Research and Development, v o l . 21, no. 5, pp. 415-432, September 1977. 36. Wu, C. "Considerations on Real-Time Processing of Spaceborne Synthetic Aperture Radar Data", Real-time Signal Processing I I I , Proc. Soc. Photo-Opt. Instr. Eng. 241, 11-19(1980). 37. . "Electronic SAR Processors for Space Missions", Proceedings of the 1978 Synthetic Aperture Radar Technology Conference, New Mexico State University, Las Cruces, N.M., paper V-3. 38. Wu, C , et a l . "SEASAT Synthetic-Aperture Radar Data Reduction Using P a r a l l e l Programmable Array Processors", IEEE Trans, on Geoscience and Remote Sensing, v o l . GE-20, no. 3, pp. 352-358, July 1982. 39. Wu, C , K. Y. Lu and M. J i n . "Modeling and a Correlation Algorithm for Spaceborne SAR Signals", IEEE Trans, on Aerospace and Electronic Systems, v o l . AES-18, no. 5, pp. 563-574, September 1982. 123 APPENDIX A - D E F I N I T I O N OF DATA WINDOWS The f o l l o w i n g window d e f i n i t i o n s a r e t a k e n f r o m t h e s u r v e y p a p e r by H a r r i s [ 1 6 ] . R e c t a n g l e - w(n) = 1.0 , n = 0 , 1 , 2 . . . N - 1 Hamming - w(n) = 0.54 - 0.46 cos{27rn/N} , n = 0 , 1 , 2 . . . N - 1 B l a c k m a n - H a r r i s w(n) = a 0 - a, ' COS^Trn/N} + a 2Â«cos{47rn/N} - a 3'COsCeTrn/N} , n = 0 , 1 , 2 . . . N - 1 M i n 3-Term 3-Term M i n 4-Term 4-Term a 0 0.42323 0.44959 0.35875 0.40217 a i 0.49755 0.49364 0.48829 0.49703 a 2 0.07922 0.05677 0.14128 0.09392 a 3 0.0 0.0 0.01168 0.00183 4-sample K a i s e r - B e s s e l F o r a = 3.0, use B l a c k m a n - H a r r i s window w i t h t h e f o l l o w i n g c o e f f i c i e n t s : a 0 = 0.40243 a, = 0.49804 a 2 = 0.09831 a 3 = 0.00122 E x a c t B l a c k m a n Use B l a c k m a n - H a r r i s window, w i t h ' t h e f o l l o w i n g c o e f f i c i e n t s : 7938 a 0 = = 0.42659071 18608 124 9240 a, = = 0.49656062 18608 1 430 a, = = 0.07684867 18608 a 3 = 0.0 Blackman Use Blackman-Harris window with the following c o e f f i c i e n t s : a 0 = 0.42 a, = 0.50 a 2 = 0.08 a 3 = 0.0 Gaussian - w(n) = exp{-2(an/N) 2} Kaiser-Bessel I 0{7ra SQRT{ 1 . 0 - 4(n/N) 2}} w(n) = , 0 < n < N/2 where I 0 i s the zero-order modified Bessel function of the f i r s t kind. Dolph-Chebyshev cos{N cos-1{/3 cos{7rk/N}}} W(k) = cosh{N cosh" 1{0}} a where /3 = cosh{cosh" 1 {1 0 }/N} For the time-domain samples w(n), perform an inverse DFT on the samples W(k) and scale a p p r o p r i a t e l y . 1 5 1 5 Harris [16] sp e c i f i e s that a DFT be performed to obtain Dolph-Chebyshev window c o e f f i c i e n t s . Actual computation showed thi s to be in error. Results were confirmed by consulting the o r i g i n a l paper by Helms [18]. 125 Barcilon-Temes A cos{y{k}} + B(y{k}sin{y{k}}/C) W(k) = (C + AB)((y{k}/C) 2 +1.0) where A = sinh{C} B = cosh{C} a C = cosh" 1{10 } 0 = cosh{C/N} y{k} = N cos"1{/3 cos{7rk/N}} For the time-domain samples, w(n), perform a DFT on the samples W(k) and scale a p p r o p r i a t e l y . 1 6 1 6 Harris [16] sp e c i f i e s that an inverse DFT be performed to obtain Barcilon-Temes window c o e f f i c i e n t s . Actual computation showed th i s to be in error. Results were confirmed by consulting the o r i g i n a l paper by Barcilon and Temes [2], 126 APPENDIX B - SYMBOLS AND ACRONYMS USED a = coarse resolution aperture number B = Bandwidth of signal BPF SPECAN = BandPass F i l t e r SPECtral ANalysis cmpp = complex multiplies per point cmps = complex multi p l i e s per second CP = Number of computations per point A = Sampling period = 1/F 17 = Azimuth time variable r e l a t i v e to zero Doppler l i n e 7} = Azimuth time variable r e l a t i v e to beam-centre crossing BX 7j 0 = Azimuth time offset of beam centre from zero Doppler F = Pulse re p e t i t i o n frequency H = Frequency step between coarse resolution apertures in FFT output bins j 2 = -1 K = Azimuth FM rate L = Number of looks X = Radar wavelength M = Number of samples in signal aperture N = Time displacement between successive coarse resolution apertures in samples PRF = Pulse Repetition Frequency QRCM = Quadratic component of RCM RCM = Range C e l l Migration RCMC = Range C e l l Migration Correction r 0 = Slant range at closest approach SAR = Synthetic Aperture Radar 127 SPECAN = SPECtral ANalysis SQRT{x} = the square root of x T = Time extent of SAR signal in azimuth d i r e c t i o n T' = Time extent of coarse resolution aperture V = Velocity of radar platform W = Width of main lobe of window W, = Data window for coarse resolution aperture W2 = Data window for fine resolution aperture 128 APPENDIX C - RADAR PARAMETERS System Parameters at Maximum Slant Range System Parameters Units SEASAT JPL Nominal COMSS/LASS Maximum slant range of - r 0 closest approach km 860 550 636 Pulse r e p e t i t i o n frequency - F Hz 1647 2600 1650 Wavelength - X m 0.235 0.235 0.057 Slant range c e l l width - p m 6.59 10.43 5.73 E f f e c t i v e platform v e l o c i t y - V m/s 7170 7600 - 6800 Azimuth FM rate (at max. r 0 ) - K Hz/s 508 894 2550 Time period - T of signal - T/A s samples 3.24 5330 2.90 7540 0.65 1081 Extent of QRCM m range c e l l s 78.4 11.9 110.4 10.6 3.84 0.67 129 Adjusted System Parameters and Step Transform Processing Parameters System Parameters Units SEASAT JPL Nominal COMSS/LASS Azimuth - K FM rate Hz/s 517 91 1 2659 Time period - T of signal - T/A s samples 3.19 5248 2.86 7424 0.62 1024 Slant range of closest - r 0 approach km 846 540 610 Processing Parameters Coarse - T* resolution aperture - T'/A s samples 0.0783 128 0.0493 128 0.0338 64 Time between - NA coarse resolution apertures - N s samples 0.0249 41 0.0223 58 0.0097 16 Overlap - T'/NA ra t i o 3.12 2.21 4.00 Coarse resolution aperture data window - W, Kaiser-Bessel a=2.5 Dolph-Chebyshev a=2.0 Kaiser-Bessel a=3.5 Guardband - 0 No. of samples rejected 0.15 20 0.15 20 0.15 10 Bandwidth of processed signal Hz 1390 2194 1392 Frequency stepping of apertures - H 1 1 1 Fine resolution aperture - T'/AH samples 128 128 64 130 APPENDIX D - COMPUTER SIMULATION OF THE STEP TRANSFORM The program i s written in FORTRAN IV and uses single precision arithmetic. Source code for each subroutine i s stored in a separate f i l e named as follows: routine.fs. The program i s run by using the following MTS command: $RUN STEPTRAN 1=outputfile or $RUN STEPTRAN+*IG 1 = O U t p u t f i l e (the l a t t e r i f output of f i l t e r i s to be plotted on the terminal). I n i t i a l l y the user controlled parameters are set to some arb i t r a r y values. The user i s then prompted to enter a parameter name from a l i s t of possible choices and i s given the opportunity to change the value of that parameter. If a name i s spelled wrong, the user i s requested to re-enter the name. This process continues u n t i l the user types a carriage return only, which indicates that a l l parameters are set to the user's s a t i s f a c t i o n . Step transform processing i s then performed. When processing i s completed, a plot of the output may appear on the screen, depending on the value of the parameter YESPLT. The user can manipulate the plot by entering IG commands. When th i s portion of the processing has been completed, the user i s asked i f another run i s to be performed. If the answer i s affirmative, changes to parameter values are again requested. At t h i s stage, the parameter values are set to the values to which they had been changed for the previous run. Parameter names, types and meanings are spe c i f i e d in the accompanying table at the end of the appendix. The parameter values are passed to the appropriate subroutines in which they are used by the use of named COMMON blocks. The major procedural subdivisions of the program are the following subroutines: MAIN - the main procedure of the program. It declares the major data structures and i n i t i a l i z e s them. Subroutines INIT, PARAM, STEPT and DATAM are c a l l e d . INIT - sets i n i t i a l values of parameters which are user controlled. This subroutine is usually written so that parameter values correspond to the p a r t i c u l a r SAR system being studied. A l l named common blocks are used in t h i s routine. PARAM - controls user selection of parameter values. A l l named common blocks are used in t h i s routine. STEPT - oversees step transform data processing. It calculates the processing parameters not d i r e c t l y controlled by the user, such as the length of the fine resolution FFT, the frequency steps between coarse resolution output pulses, 131 and recalculates the spacing of the coarse resolution apertures, taking into consideration the user requested value. It c a l l s FORMST, MULT, FFT1, ORDER and FFT2. FORMST - forms the target return signal (with antenna p r o f i l e as requested) and reference function (with coarse resolution aperture window as requested). It c a l l s WINDOW. MULT - performs m u l t i p l i c a t i o n of target return by reference function. FFT1 - performs coarse resolution FFT operation, quadratic phase correction, clearing of guardband data. ORDER - reorders coarse resolution FFT output data into fine resolution apertures. FFT2 - performs fine resolution FFT operation on 10 apertures surrounding the pulse peak. Look summation, data selection and output scaling are also performed. It c a l l s WINDOW. DATAM - converts f i l t e r data to real values. It determines the position of the peak of the output pulse and interpolates the data to determine the 3 dB width. A plot i s produced i f s p e c i f i e d by the user. WINDOW - performs a window operation on complex data passed to i t in an array. The data window to be used i s also passed as a parameter. The major data structures used are as follows: SIGNAL - complex array of 32000 elements. FORMST uses t h i s data structure to store the i n i t i a l target return s i g n a l . ORDER uses i t to store the fine resolution apertures. UNORD - complex array of 32000 elements. The output of the multiply operation i s stored in t h i s array and the coarse resolution FFT's and associated processing are performed in-place in thi s data structure. FILTER - complex array of 512 elements used to store the reference ramp. PDATA, XDATA - real arrays of 8192 elements used to store the y-and x-coordinates respectively of the f i l t e r output. They are used mainly for interpolation and p l o t t i n g of the output function. 132 Data Output Parameters Parameter Name Type and Range of Values Meaning PLTWDH Integer Specifies the number of abscissae values around the peak of the output pulse which are to be plotted. POUT Logical T or F If TRUE, f i n a l output data from f i l t e r i s printed on unit 1. PRINTO Logical T or F If TRUE, output data from multiply operation i s printed on unit 1. PRINT1 Logical T or F If TRUE, output data from coarse resolution FFT's i s printed on unit 1 . PRINT2 Logical T or F If TRUE, output data from reordering operation, i . e . input data to fine resolution FFT's i s printed on unit 1. PRINT3 Logical T or F If TRUE, output data from fine resolution FFT's i s printed on unit 1 . PRNTER Logical T or F Controls values of PRINTO, PRINT1, PRINT2, PRINT3, and POUT. If FALSE, none of above may be set to TRUE. TENSHN Real > 0. Specifies tension parameters for interpolation of f i n a l output. 0. = cubic spline interpolation 0.+ = mixed exponential interpolation (almost polygonal for large values) YESPLT Logical T or F If TRUE, plot i s produced - on screen i f *IG is. concatenated to $RUN command - in unit 9 i f *IG i s not used 133 SAR System and Processing Parameters Parameter Name Type and Range of Values Meaning ACCURA Power of 2 > 1 Specifies factor by which fine resolution FFT's are expanded for data interpolation. ALPHA 1 Real > 0. Specifies value of parameter a for appropriate coarse resolution aperture windows. ALPHA2 Real > 0. Specifies value of parameter a for appropriate fine resolution aperture windows. FFTCNT Integer > 0 Specifies point in f i r s t fine resolution FFT at which data selection i s to s t a r t . Value i s determined empirically. GUARD Real 0.-1 . Specifies guardband, /3, or amount of rejected data from coarse resolution FFT's. Number of points zeroed at each end of FFT i s GUARD*REFWID/2 +. 0.5 NLOOKS Integer Specifies number of looks. RATERR Real Specifies percentage rate error to be used in formation of target return s i g n a l . REFWID Power of 2 < 512 Specifies number of samples in reference ramp. SRATE Real Specifies factor by which target return i s oversampled. START Integer | START| < 1000 Specifies position of target return pulse. STPSIZ Integer Specifies sample spacing between successive coarse resolution apertures. 1 34 SAR System and Processing Parameters (cont.) Parameter Name Type and Range of Values Meaning SYSTEM Integer Specifies antenna p r o f i l e to be used on target return s i g n a l . 1 = SEASAT 2 = JPL Nominal 3 = COMSS/LASS For a l l other values, no p r o f i l e i s applied. WIDTH Integer < 8000 Specifies number of samples in target return pulse. WIND1 Integer Specifies type of window used on coarse resolution aperture. WIND 2 Integer Specifies type of window used on fine resolution aperture.
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Step transform pulse compression and its application to synthetic aperture radar systems Sack, Mark William 1983
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Title | Step transform pulse compression and its application to synthetic aperture radar systems |
Creator |
Sack, Mark William |
Publisher | University of British Columbia |
Date Issued | 1983 |
Description | Step transform pulse compression is a matched filtering technique applicable to linear FM signals. A mathematical analysis of the method is developed and its application to the azimuth processing of synthetic aperture radar (SAR) target returns is explored. In particular, several satellite-borne SAR systems are used as examples to compare the performance of this technique to other approaches. Problems peculiar to SAR azimuth processing including FM rate error, range cell migration, and multilook processing are considered. The computation rates, memory requirements, and complexity of the processing algorithms are also evaluated. The step transform is studied analytically to determine how appropriate choices of processing parameters can be made to achieve acceptable image quality, while minimizing the hardware requirements. A method of range cell migration correction, which can be integrated into the step transform process, is proposed and the corresponding parameter restrictions are derived. The hardware requirements of the step transform for SAR applications, relative to some other pulse compression techniques, are shown to be less dependent on SAR system parameters and more dependent on the step transform processing parameters chosen. A computer simulation program was written to confirm the analytical results. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-04-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0064975 |
URI | http://hdl.handle.net/2429/24091 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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