Application of the Momentary Fourier Transform to Radar Processing

Transcription

1 Budpest University of Technology nd Economics Deprtment of Telecommunictions Appliction of the Momentry Fourier Trnsform to Rdr Processing Sndor Albrecht Ph D Disserttion Advisors Dr In Cumming Deprtment of Electricl nd Computer Engineering University of British Columbi, Vncouver, BC Dr Lszlo Pp Deprtment of Telecommunictions Budpest University of Technology nd Economics Budpest,

2 Abstrct The discrete Fourier trnsform (DFT) is widely used tool in signl or imge processing nd its efficiency is importnt There re pplictions where it is desirble to use reltively smll, successive, overlpped DFTs to obtin the spectrum coefficients The momentry Fourier trnsform (MFT) computes the DFT of discrete-time sequence for every new smple in n efficient recursive form In this thesis we give n lternte derivtion of the MFT using the momentry mtrix trnsform (MMT) Recursive nd non-recursive forms of the inverse MFT re lso given, which cn provide efficient frequency domin mnipultion (eg filtering) Discussion on the properties nd exmples of the usge of the MFT is lso given, followed by survey on its efficiency In this work we investigte the pplicbility of the MFT to synthetic perture rdr (SAR) signl processing, nd in prticulr show wht dvntges the MFT lgorithm offers to the SPECtrl Alysis (SPECA) method nd burst-mode dt processing In the SPECA lgorithm, the received signls re multiplied in the time domin by reference function, nd overlpped short length DFTs re used to compress the dt The zimuth FM rte of the signl vries in ech rnge cell, which leds to the issue of keeping the zimuth resolution nd output smpling rte constnt After the introduction to SPECA, we show wht dvntges nd disdvntges the MFT hs compred to the FFT lgorithms When SAR system is operted in burst-mode, its zimuth received signl hs segmented frequency-time energy in its Doppler history It requires tht IDFTs be locted t specific points in the spectrl domin to perform the zimuth signl compression First we give detiled description of the burst-mode dt properties, thn we show how the short IFFT (SIFFT) lgorithm hs the requirement of rbitrrylength, highly-overlpped IDFTs to process burst-mode dt, in which cse the IMFT is shown to hve computtionl dvntges i

3 Contents Abstrct i Contentsii List of Tbles v List of Figures vi Acknowledgements viii Introduction Bckground Thesis objectives nd outline 3 Theory nd Properties of the Momentry Fourier Trnsformtion 4 Introduction 4 The Momentry Fourier Trnsformtions derived form the Recursive Momentry Mtrix Trnsformtion 4 The Recursive Momentry Mtrix Trnsformtion 5 The digonl form of the MMT 7 3 Inverse of the digonlized MMT 8 4 Momentry Fourier Trnsform 9 5 The non-recursive Inverse MFT 3 Properties of the MFT 3 Cosine windows using MFT 3 Implementtion of MFT 4 3 Softwre Implementtion of the MFT 4 3 Hrdwre Implementtion of the MFT 6 33 Exmple of MFT Usge 8 4 Computing Efficiency of MFT 4 Arithmetic of MFT 4 Comprison of MFT to FFT lgorithms 43 Advntges nd Uses of MFT 7 3 Fundmentls of SAR Processing 8 3 Introduction 8 3 Overview of Rdr Remote Sensing 8 3 Rel Aperture Rdr 9 33 Synthetic Aperture Rdr 3 33 SAR Geometry 3 34 Idel point-trget model 3 35 SAR signl compression 34 4 Appliction of MFT to SPECA SAR Processing Algorithm 37 4 Introduction 37 4 The SPECA Algorithm 37 4 Wht is Dermping? 38 4 DFT Opertions 39 ii

4 43 Anlyticl Bckground of SPECA Grphicl explntion of SPECA 4 45 Multi look processing in SPECA The SPECA Algorithm Using the MFT 48 5 Appliction of MFT to Burst-mode SAR Dt Processing 59 5 Introduction 59 5 Overview of burst-mode SAR processing 6 53 Chrcteristics of burst-mode dt 6 53 Effect of the Doppler centroid on the Doppler spectrum Effect of circulr convolution on the Doppler history Odd number of bursts vs even number of bursts in the synthetic perture The SIFFT lgorithm nd its properties 7 54 Input nd output trget spce of the SIFFT lgorithm Idel trget simultion of burst-mode processing using the SIFFT lgorithm Rel dt simultion of burst-mode processing using the SIFFT lgorithm 8 55 Efficiency of SIFFT using the IMFT nd the IFFT lgorithms Arithmetic of the SIFFT lgorithm Efficiency of the SIFFT lgorithm vs zimuth DFT length Efficiency of the SIFFT lgorithm when it is used to process Envist dt Effect of vrying SAR prmeters nd SR/efficiency trdeoffs Efficiency clcultion 9 6 Conclusions 99 6 Summry 99 6 Future work Appendix A Computing Efficiency of MFT A Shift between DFTs when the MFT is more efficient A Arithmetic of MFT nd Rdix- FFT when q MFT = A3 Arithmetic of MFT nd Rdix- FFT when q MFT = 4 B The SPECA Algorithm Using the MFT 3 B Arithmetic of SPECA zimuth compression with different DFT lgorithms spceborne cse3 B Arithmetic of SPECA zimuth compression with different DFT lgorithms irborne cse 4 B3 The output smpling rte of the SPECA lgorithm spceborne cse 5 B4 The output smpling rte of the SPECA lgorithm irborne cse6 C Efficiency of SIFFT using the IMFT nd IFFT lgorithms 8 C Arithmetic of the IFFT nd IMFT lgorithm vs zimuth DFT length minimum IFFT length is used8 C Arithmetic of the IFFT nd IMFT lgorithm vs zimuth DFT length 5 IFFT lengths to choose from9 iii

5 C3 Arithmetic of the SIFFT lgorithm when pplied to Envist IS swth, FFT = 3 - minimum IFFT length is used C4 Arithmetic of the SIFFT lgorithm when pplied to Envist IS swth, FFT = 3-5 IFFT lengths to choose from C5 Arithmetic of the SIFFT lgorithm when pplied to Envist AP burst mode opertion Bibliogrphy 3 iv

6 List of Tbles Tble Memory requirement of MFT (words) 6 Tble Rel opertions in MFT for c coefficients Tble 3 Resolution versus DFT length in SPECA for C-bnd stellite SAR 46 Tble 4 Spceborne nd irborne SAR prmeters for SPECA rithmetic clcultion 5 Tble 5 Reduced nd full MFT versus mixed-rdix FFT 53 Tble 6 Prmeters of idel trget simultion 78 Tble 7 Input nd output spce of eqully spced idel trgets 79 Tble 8 ERS- prmeters for rel dt simultion 83 Tble 9 Input nd output trget spce of ERS- dt processing 83 Tble Envist swth prmeters 9 Tble Minimum nd mximum burst bndwidth of the seven swthes 9 Tble The length of the IFFTs nd the corresponding dsr 93 v

7 List of Figures Figure Windowing of the discrete-time function 5 Figure Block digrm of the full MFT lgorithm 7 Figure 3 Block digrm of the full MFT lgorithm without the modulo- FIFO 7 Figure 4 Hrdwre structure of one MFT block 8 Figure 5 FSK signl nlysis using MFT 9 Figure 6 Signl detection using MFT Figure 7 Shift between DFTs when the MFT is more efficient 3 Figure 8 Arithmetic of MFT nd Rdix- FFT when q MFT = 4 Figure 9 Arithmetic of MFT nd Rdix- FFT when q MFT = 4 5 Figure Floting Point Opertions of DFT lgorithms 5 Figure Fst Convolution with MFT nd Rdix- FFT 6 Figure Synthetic perture rdr geometry 3 Figure 3 Doppler history of the idel zimuth received signl 4 Figure 4 Dermping of single liner FM signl 4 Figure 5 Dermping of multiple trgets 43 Figure 6 Processing regions nd the plcement of successive DFT blocks in single look cse 44 Figure 7 Division of the good output smples into looks nd the loction of the DFT opertions in multi-look processing 48 Figure 8 Azimuth FM rte nd the DFT length with vrying rnge 49 Figure 9 Arithmetic of SPECA zimuth compression with different DFT lgorithms 5 Figure The rithmetic of SPECA when the MFT is more efficient 54 Figure The rithmetic of the Rnge-Doppler lgorithm nd the SPECA lgorithm using the MFT nd FFT lgorithms 56 Figure The output smpling rte of the SPECA lgorithm 57 Figure 3 Burst-mode opertion in -bem ScnSAR cse 6 Figure 4 Burst-mode processing of 6 consecutive nd fully exposed trget in one rnge cell 6 Figure 5 Burst-mode processing of 5 evenly spced fully nd prtilly exposed trget in one rnge cell 63 Figure 6 Effect of the Doppler centroid on the spectrum of fully exposed trget 64 Figure 7 Effect of the Doppler centroid on the trgets Doppler history 65 Figure 8 Effect of the circulr convolution on the Doppler history 67 Figure 9 Doppler history of consecutive trgets when there re odd or even number of bursts in the synthetic perture 7 Figure 3 How minimum IFFTs re plced to compress fully exposed trgets 7 Figure 3Shift between two consecutive IFFTs 73 Figure 3 Input trget spce minimum zimuth DFT is used 74 Figure 33 Input trget spce in zimuth time domin 76 vi

8 Figure 34 Input trget spce vs zimuth DFT length 77 Figure 35 Input nd output spce of eqully spced idel trgets 8 Figure 36 The Doppler history of rel burst-mode dt 8 Figure 38 SLC products of ERS dt 85 Figure 39 IFFT nd IMFT window length vs zimuth DFT 87 Figure 4 Arithmetic of the IFFT nd IMFT lgorithm vs zimuth DFT 88 Figure 4 Burst bndwidth nd dsr of IS swth 9 Figure 4 Arithmetic of the SIFFT lgorithm when pplied to the IS swth, FFT = 3 95 Figure 43 Arithmetic of the SIFFT when it is pplied to Envist AP burst mode opertion 97 vii

9 Acknowledgements First of ll, I would like to thnk my mother nd my wife for providing consistent support nd encourgement throughout my reserch nd studies Without their love nd cre this work would not hve been completed I would like to thnk Dr Jozsef Duds for the momentry Fourier trnsform nd his guidnce in my scientific creer I would like to thnk Dr In Cumming, for his supervision nd cdemic guidnce, s well s providing the opportunity to continue my reserch of momentry Fourier trnsform in the field of synthetic perture rdr processing I m grteful to Dr Miklos Bod nd Dr Lszlo Pp for the cdemic nd personl support throughout my work viii

10 Chpter Introduction Bckground Liner trnsformtions, such s the discrete Fourier trnsform (DFT) re frequently used in digitl signl processing, nd their efficiency is very importnt In pplictions where the DFT is pplied to signl, it is often desirble to use successive, possibly overlpping DFTs of smller extent thn the full length of the signl to obtin the spectrum coefficients These trnsformtions re normlly off-line opertions on blocks of dt, requiring smples of the signl before the trnsformtion cn be computed The momentry Fourier trnsform (MFT) which is derived here is method of computing the DFT of sequence in incrementl steps It cn be computed using n efficient recursive formul, nd it is useful in cses where the detiled evolution of the spectr of discrete series is wnted, nd in cses where only few Fourier coefficients re needed The spectrum components of the MFT cn be clculted independently nd only one complex multipliction nd two complex dditions re needed to updte ech spectrum component The inverse momentry Fourier trnsform (IMFT) is the dul of the MFT nd shres the sme property, while the non-recursive form of the IMFT requires only dditions to obtin smple of the time sequence from its spectrum with smples dely The computtionl order of the MFT to updte n -point DFT is, fctor of log improvement over the rdix- FFT lgorithm if ll incrementl results re needed If only sub-set of the trnsform domin components re needed, the computing lod of the MFT cn be further reduced, clculting only the coefficients of interest The MFT does not rely upon on being power of two to obtin its efficiency, in contrst to stndrd FFT lgorithms In this thesis we further develop the theory of MFT, exmine its pplictions nd in prticulr, see wht dvntges it offers to synthetic perture rdr dt processing

11 A synthetic perture rdr (SAR) is powerful sensor in remote sensing which is cpble of observing geophysicl prmeters of the Erth s (or nother plnet s) surfce, regrdless of time of dy nd wether conditions [3, ] SAR systems re extensively used for monitoring ocen surfce ptterns, se-ice cover, griculturl fetures nd for militry pplictions such s in the detection nd trcking of moving trgets A SAR trnsmits rdr signls from n irborne or spceborne ntenn which is perpendiculr to the flight direction of the pltform which is trvels t constnt velocity The bck-scttered signl is collected by the ntenn nd stored in rw formt Extensive signl processing is required to produce the output SAR imge The SPECtrl Alysis (SPECA) SAR processing lgorithm ws developed in 979 by McDonld Dettwiler nd Assocites, s multi-look version of the dermp-fft method of pulse compression In SPECA, the received signls re multiplied in the time domin by reference function, nd overlpped short length DFTs re used to compress the dt In contrst, precision processing lgorithm such s the Rnge Doppler (RD) method requires both forwrd nd inverse DFT opertions, thus it is less computtionlly efficient SPECA is n efficient lgorithm for moderte to low resolution processing nd generlly implemented in quick look processors for viewing of mgnitude detected imgery dt Burst-mode opertion is used in SAR systems, to imge wide swths, to sve power or sve dt link bndwidth Severl spceborne remote sensing missions employ the ScnSAR mode in ddition to other opertionl modes for rdr imging Cnd s Rdrst stellite, which ws successfully lunched in 995, is sophisticted Erth observtion system developed to monitor environmentl chnges The imging pltform supports vrious SAR operting modes, including ScnSAR mode for the low-resolution (~m) imging of ground regions of width 5 km An Advnced SAR (ASAR) system will be flown on the Envist stellite polr pltform to be lunched in October by the Europen Spce Agency This system will be ble to operte in three burst-modes: lternting polriztion mode (AP), wide swth mode (WS) nd globl monitoring mode (GM) Alternting polriztion mode provides high resolution in ny swth with polriztion chnging from sub-perture to sub-perture within the synthetic perture This results in two imges of the sme scene in different polriztion combintion with pproximtely 3 m resolution In the wide swth mode the ScnSAR technique is used providing imges of wider swth (45 km) with medium resolution (5 m)

12 Thesis objectives nd outline The objective of this reserch is to further develop the theory of the MFT, exmine its properties nd pplictions, nd in prticulr, see wht dvntges it offers to SPECA processing nd to the short IFFT (SIFFT) burst-mode processing lgorithm Chpter presents the theory nd properties of the momentry Fourier trnsform Here, we introduce the recursive form of the momentry mtrix trnsform (MMT), nd show when the MMT tkes the form of the DFT or the IDFT, the resulting MFT nd IMFT hve n efficient computtionl structure The properties nd computing efficiency of the MFT is lso discussed in this chpter In Chpter 3, n overview of SAR processing is given, where the conventionl compression method of the SAR signls is studied This chpter gives bckground knowledge for the reserch of the SPECA lgorithm nd burst-mode dt processing The zimuth FM rte of the received signl vries in ech rnge cell, which leds to the issue of keeping the zimuth resolution nd output smpling rte constnt After the introduction of the SPECA lgorithm in Chpter 4, we show wht dvntges the MFT method offers vs the FFT lgorithms when they re pplied to the SPECA SAR processing lgorithm In Chpter 5, the ScnSAR opertion mode will be introduced, nd the received burstmode dt properties will be nlyzed We will show the effect of the Doppler centroid nd the circulr convolution on the Doppler history After the properties of the input nd output trget spce re given results of idel nd rel dt simultions will be shown Surveys on the rithmetic of the SIFFT lgorithm using IMFT nd IFFT will be given, where we show how the IMFT lgorithm cn improve the computtionl efficiency of the SIFFT lgorithm Finlly in Chpter 6, conclusions of the efficiency nd pplicbility of the momentry Fourier trnsform to SAR processing will be drwn, nd suggestions for possible future work will be given 3

13 Chpter Theory nd Properties of the Momentry Fourier Trnsformtion Introduction Uses of the incrementl DFT were introduced by Ppoulis in 977 [], nd by Bitmed nd Anderson in 98 [5] A detiled derivtion of the momentry Fourier trnsform ws given by Dudás in 986 [6] In 99, Lilly gives similr derivtion, using the term moving Fourier trnsform, nd uses the MFT for updting the model of time-vrying system [7] We give n lterntive derivtion of the recursive form of the MFT using generl momentry mtrix trnsform in Section [9] Section 3 gives survey on the properties of the MFT, nd in Section 4 discussion on its computtionl efficiency is given [5], [6] nd [7] The Momentry Fourier Trnsformtions derived form the Recursive Momentry Mtrix Trnsformtion In this section, we introduce the mtrix form of the momentry trnsform, nd show tht it hs recursive form We lso show tht when the momentry mtrix trnsform tkes the form of the DFT or the inverse DFT, the resulting MFT hs n efficient (recursive) computtionl structure In the lst prt of the section, the inverse of the MFT is introduced, s well 4

14 The Recursive Momentry Mtrix Trnsformtion Let x i be smple of n rbitrry complex-vlued sequence of one vrible The sequence will be nlyzed through n -point window, ending t the current smple i In subsequent nlyses, the window will be dvnced one smple t time At smple i x i enters the window, while x i- leves the window, s shown in Figure Amplitude Window t time i Window t time i- Smples x i- x i-(-) x i- x i Figure Windowing of the discrete-time function At smples i- nd i, the windowed function cn be represented by the following two column vectors: x =, i- x x i- i = i( Let T be n x nonsingulr mtrix, which represent liner trnsformtion nd hs the inverse T - The sequence of index vectors is trnsformed by T t ech smple: x i x x i x i ) () 5

15 6,,, i i i i- x T y Tx y = = () Let P be the x elementry cyclic permuttion mtrix: = P (3) When the vector x i- is multiplied by P, one-element circulr shift is performed, such tht the index of ech element is incresed by one, nd the first element becomes the lst one: = i i ) i-(i x x x x P (4) Then the x i vector cn be expressed by the shifted x i- vector nd with n djustment vector x i mde in the lst row for the difference between the smples entering nd leving the window: i x P x x + = + = i i i i i ) i-(i x -x x x x (5)

16 Substituting eqn (5) into the trnsformtion ssocited with the ith window in eqn () nd using the inverse trnsform x i- = T - y i-, the following reltionships re obtined: y i [ P xi + xi ] = T P T yi + T xi = T xi = T (6) Eqution (6) expresses the recursivity of the momentry mtrix trnsforms (MMT), since clcultion of the newly trnsformed index vector y i is obtined from the previously trnsformed vector y i- nd the difference between smples entering nd leving the window The digonl form of the MMT The momentry mtrix trnsform is prticulrly efficient nd the elements of y cn be clculted seprtely only if the similrity mtrix trnsform TPT - in eqn (6) is digonl The P mtrix hs distinct eigenvlues (λ,, λ - ) which re the nth k j k/ complex unit roots, λ k = w = e π, k =,,, There re linerly independent eigenvectors tht correspond to ech eigenvlue: - w - - λ = w = s= ; λ= w s= w ; w -(- ) λ = w k -k s = k w w w -k -(- )k -k -- ; ; λ - = w -(- ) s - = w w w -(- ) -(- )(- ) (7) 7

17 8 If the eigenvectors re chosen to be the columns of the inverse of the T mtrix, then TPT - is digonl mtrix, with the eigenvlues of P long its digonl: = = λ λ λ λ ] [ ] [ = s s s P s s s PS S TPT (8) where S is the eigenvector mtrix of P mde up of the indicted column vectors, s given in eqn (7) The digonlizing mtrix S is not unique An eigenvector s k cn be multiplied by constnt, nd will remin n eigenvector [] Therefore the columns of S cn be multiplied by ny nonzero constnts nd produce new digonlizing S There is lso no preferred order of the columns of S [] The order of the eigenvectors in S nd the eigenvlues in the digonl mtrix is utomticlly the sme Therefore ll T mtrixes which stisfy the bove mentioned properties will digonlize the momentry mtrix trnsform: ( ) i- i - i m l k i x x + = T y y λ λ λ, (9) where k, l, m {,,,-} nd T - is the lst column of the mtrix T 3 Inverse of the digonlized MMT If y i is vilble t ech smple nd the columns of T re the eigenvectors of P, n efficient implementtion of the inverse of the MMT cn be obtined The inverse MMT (IMMT) t time i: i T y x - i = ()

18 9 = = i i,i, i, i i- ) i-(- i y y y x x x ] [ y s s s x () The first row of T - contins only ones (eqn (7)), so the oldest element of x i cn be computed using dds only: = = k i,k ) ( i y x () from which the elements of the input sequence (x i-(-) x i ) cn be computed from the trnsform domin sequence y i with - smple dely In summry, the recursive form of the MMT is generl The following Section shows tht the DFT/IDFT is the only trnsform which hs the efficient digonl form (eqn (8)), s result of its column vectors being the eigenvectors in eqn (7) in specific order 4 Momentry Fourier Trnsform The mtrix of the Discrete Fourier Trnsform (DFT) nd the Inverse Discrete Fourier Trnsform (IDFT) hve the properties described in Section, thus their columns re the eigenvectors of the mtrix P Choosing specific order of the eigenvectors of P (columns of S): ) ( ) ( ) ( ) ( - w w w w w w w w w = = = 4 S F DFT (3)

19 = 4 ) ) ( ( ) ( ) ( ) ( ) ( - w w w w w w w w w = = = S F IDFT 4 ) (- ) )( ( ) (- w w w w w w w w w (4) Using the fct tht w is the th complex root of unity (ie w -k = w -k ), it cn be seen tht the columns of the IDFT mtrix re the sme s the DFT mtrix, but they re in reverse order from the second column onwrds eqn (4) Therefore if T performs the DFT eqn (5) or the IDFT eqn (6), digonl forms of the MMT cn be obtined: ( ) i- i ) ( i ) -( - - i i- - i x x w w w w w w = + = + y y F P F y x F (5) ( ) ( ) i i ) -(- ) ( i ) -(- ) -(i- i i i - i- - i y y w w w w w w = y y w w w + w w w = + = + x x F x F x y P F (6)

20 Eqution (5) expresses the recursive eqution of the momentry Fourier-trnsform (MFT) [6, 7, 8, 9] The -element vector y i contins the Fourier coefficients of the - point sequence x i ending t smple i ote tht ech spectrl component y i,k cn be clculted independently, y i,k ( y + x x ) k = w i,k i i (7) which increses efficiency if only few frequency components need to be computed, s in the zoom trnsform On the other hnd, eqn (6) is the dul of the MFT, the recursive inverse momentry Fourier-trnsform (IMFT), where the -element vector x i contins the -point time sequence nd y i contins Fourier coefficients ending t frequency bin i ote tht ech smple in x i cn lso be obtined independently nd the sme twiddle fctors, but in different order, cn be used to clculte both the MFT nd IMFT Thus it hs been shown tht if the DFT or the IDFT performs the momentry mtrix trnsform of sequence the elements of the trnsformed sequence cn be computed recursively nd independently using complex multiplies nd + complex dds (dditionl computtionl svings re vilble if the input sequence is rel-vlued) 5 The non-recursive Inverse MFT The non-recursive inverse momentry Fourier trnsform [6] cn be expressed using eqn () nd (4) s follows: x i ( ) = k= y i,k (8) from which ech smple of the input sequence x i cn be computed using dds only from the two-dimensionl time-dependent spectrum y i with - smple dely In this wy the MFT- non-recursive IMFT trnsform pir eqn (5) nd (8) cn provide n efficient frequency-domin mnipultion method (eg filtering), especilly if mny of the DFT coefficients re not needed If the elements of x i re rel, tking dvntge of the conjugte symmetry of the spectrum, the oldest element cn be computed using only the rel prt Re of the spectrum:

21 x i ( ) = k= Re { y } i,k (9) It hs been showed in [6] tht if x i is rel, the Hilbert trnsform H{} of x i-(-) cn be obtined by summing only the imginry prt of the spectrum components: H{x i ( ) } = k= Im { y } i,k () In this cse the MFT - non-recursive IMFT pir cn be useful for different signl processing pplictions where the in-phse nd qudrture component of the signl is needed (ie communictions nd rdr systems) 3 Properties of the MFT In this section, some properties of the MFT re given Section 3 shows how to implement cosine windows in the frequency domin using the MFT In Section 3, discussion on the softwre nd hrdwre implementtion of the MFT lgorithm is given, while section 33 gives n exmple of the use of the MFT 3 Cosine windows using MFT The -point DFT trets the dt sequence s if it hs periodicity of smples, x i = x i+k, for ll integer k In prctice mny signls do not hve the bove periodicity If the boxcr window is pplied to such signl, the DFT will tret it if there were discontinuities t its edges Ringing effects ner the edges of filtered signls my occur s result of these spurious discontinuities [4] Such effects cn be reduced by pplying more pproprite window In ddition to selecting portion of the input sequence, the window modifies this portion to mke it continuous t the edges when regrded s periodiclly repeted Severl types of window hve been described in the literture [], [4] This section introduces how the Hnning, Hmming nd Blckmn window cn be implemented using the MFT Given the discrete-time sequence x i, we wish to clculte the MFT of the windowed dt x w,i = w i x i t time i, where w i is the window function The ith element of the window my be expressed s follows:

22 #"! )(' #"! #"! #"! &% $ #"! )(' #"! &% $ π i Hnning: w i = 5 - cos Hmming: Blckmn: = w i = w i cos 5 cos π i π i 4π i + 8 cos The derivtion for the Blckmn window is given below: π i 4π i xwi= w ixi= 4 5 cos + 8 cos x π i π i = 4 xi 5 exp j + exp -j xi + 4π i 4π i 4 exp j + exp -j xi i = () () Tking the DFT of ech prt of (7) the spectrum of the windowed dt t time i: y wi, k = 4y 5 y + y i, k- + 4 y + y i, k- i, k i, k+ i, k+ (3) Therefore, the MFT of the windowed dt cn be obtined simply by mintining the MFT of the non-windowed dt nd pplying weighted verge in the spectrum Similr results cn be esily derived for the other cosine windows These re: Hnning: = 5 y 5 [ y y ] y i,k i,k + + i,k wi,k Hmming: = 54 y 3 [ y y ] y i,k + + i,k wi,k i,k (4) Although only generlized cosine windows cn be pplied esily with the MFT, rbitrry windows cn be pproximted if enough cosine terms re used ote, tht the memory requirement of the MFT lgorithm gets lrger, while its efficiency drops s the number of terms increses 3

23 The edge effect of the boxcr window cn lso be compensted if the non-weighted moving verge of the spectrum components is used The moving verge of spectrum coefficient t times i, for L (L < ) consecutive smples cn be expressed s: y = L mvg_i,k y j,k j= i-l+ i (5) It lso hs recursive form where, the clcultion of the verged spectrum coefficient needs the previously clculted verge nd the difference between the spectrum coefficient entering nd leving the verging window: y mvg_i+,k = y mvg_i,k + y i+,k L y i-l+,k The bove defined moving verge with the long-term verge (6) of the MFT coefficients cn lso be useful for sttisticl nlysis of the input discrete sequence y k,vg = M M i= y i,k L, M >> (6) (7) 3 Implementtion of MFT In this section, discussion on the softwre nd hrdwre implementtion of the MFT lgorithm is given Section 3 shows the computer coding of the MFT with its memory requirement, while the principle hrdwre structure of the MFT is given in Section 3 3 Softwre Implementtion of the MFT As it ws shown erlier, the spectrum components in the MFT lgorithm cn be clculted independently of ech other Thus, the MFT cn be implemented using sequence of identicl blocks, where block computes the spectrum t single frequency using eqn (7) The softwre implementtion of one MFT block cn be obtined using the trigonometric form of the eqution: y i,k ( y + x x ) k = w i,k i i 4

24 Re{y Im{y i,k i,k } } = = cos cos w k = e j πk where = cos Φ ( Φ ) + j sin ( Φ ) ( Φ )( Re{y } + Re{x -x }) sin ( Φ ) k i-,k k k k π k = (8) ( Im{y } + Im{x -x ( Φ )( Im{y }+ Im{x -x }) + sin ( Φ )( Re{y } + Re{x -x }) k i-,k i i i- i- k k i-,k i-,k i i i- i- (9) }) Equtions (8) nd (9) corresponds to the kth MFT block for complex x i, where y i,k is the kth spectrum component t smple i The MFT blocks cn be orgnized in for loop to clculte the needed DFT coefficients The following pseudocode segment illustrtes the computer coding of the MFT lgorithm, ssuming the sine nd cosine rrys (twiddle fctors) hve been precomputed: clculte(x i -x i- ); for k = strt to (strt + c -) do MFTblock(k); % computed s in eqn(9) endfor where C is the number of spectrum coefficients to be computed, C The index of the for loop indictes tht it is possible to compute only smll group of the DFT coefficients If c < or if nother subset of the coefficients re to be clculted, we refer to the MFT s reduced-mft, where significnt computtion svings cn be relized if not ll the spectrl coefficients re needed While, if the prmeter strt is zero nd c =, then ll the DFT coefficients re going to be clculted Within the MFTblock procedure, the previously computed spectrum coefficient should be stored in n rry for the computtion of the current one If the clcultion of the spectrum coefficients is off-line, the difference of the entering nd leving smples of the window cn be clculted for the whole dt set nd stored in file or n rry in the memory If it is on-line, modulo- rry is needed to clculte x i -x i- Tble gives the memory requirements of the MFT lgorithm The memory requirements depend upon the number of clculted spectrum coefficients If the whole spectrum is computed, n 8 word memory is needed for the computtion Assuming tht the input signl hs 6-bits of precision, nd 6-bit ccurcy is required t the output, the MFT rithmetic should be done with word length of t lest 4 5

25 bits, s the multiple stges of the MFT lgorithm will crete more roundoff noise thn the log stges of the FFT Arry type Twiddle fctors (sine nd cosine rrys) Modulo- FIFO for complex x i -x i- Spectrum coefficients t time i- - y i-,k Spectrum coefficients t time i - y i-,k Size c c c Tble Memory requirement of MFT (words) 3 Hrdwre Implementtion of the MFT The trigonometric form of the MFT for one spectrum component (9) cn be esily implemented in hrdwre From the bsic blocks of MFT prllel hrdwre structure cn be built for the computtion of the DFT coefficients Figure illustrtes the block digrm of the concurrent implementtion of the MFT blocks of the full MFT lgorithm ote, the updting time of the fully concurrent implementtion is equivlent to the propgtion time of one MFT block, regrdless of the number of the clculted spectrum coefficients In Figure the rchitecture of the full MFT contins modulo- FIFO register to obtin x i- If ll the spectrum coefficients re computed, the leving smple of the window t time i cn be expressed using the IMFT lgorithm: x i = k = y i-,k (3) Substituting eqn (9) to eqn (), the recursive eqution of one MFT coefficients becomes the following: y i,k - k = w yi,k + xi k = y i,k (3) 6

26 y i-, D x i + x y i, MFT Block # y i-, D x i + x y i, FIFO - x i- + x i - x i- MFT Block # w - y i-,- D x i- + x y i,- MFT Block #- w -(-) Figure Block digrm of the full MFT lgorithm In eqn (3), the dt smple t time i nd ll the spectrum coefficients t time i- re needed to obtin y i,k The memory requirement of the MFT lgorithm is reduced by B bits, becuse there is no need to sve the input dt smples in FIFO, while the rithmetic of the MFT incresed by - rel opertions due to the clcultion of IMFT The block digrm of the MFT corresponding to eqn (3) is shown in Figure 3 MFT Block # y i, MFT Block # x i - x i- x i + - y i, x i- + x i-(-) MFT Block #- y i,- D x Figure 3 Block digrm of the full MFT lgorithm without the modulo- FIFO / 7

27 The detiled hrdwre structure of one MFT block for complex x i is given in Figure 4 This implementtion contins four multipliers (MPY) nd four dders (ALU) to obtin the complex rithmetic of the MFT The twiddle fctors nd the previously clculted spectrum coefficients re stored in registers Becuse of the prllel computtion of the rel nd the imginry prt of the MFT coefficients, the updting time of the spectrum coefficients is limited only by the propgtion time of two multipliers nd two dders Lter in this thesis, the pplicbilty of the MFT to SAR processing is investigted SAR dt compression is n off-line procedure, done on the ground, so rel-time properties of the MFT lgorithm nd its hrdwre implementtion is not discussed in detil in this thesis MFT Block # k cos(kπ/) sin(kπ/) MPY MPY MPY MPY ALU ALU ALU ALU + + Re{Y i-,k } Im{Y i-,k } Re{x i -x i- } Re{Y i,k } Im{Y i,k } Im{x i -x i- } Figure 4 Hrdwre structure of one MFT block 33 Exmple of MFT Usge To illustrte the usge of the incrementl form of the MFT, frequency shift key (FSK) modulted sinusoidl signl of length 4 smples is used Using n nlysis 8

28 window length =, nd two frequencies of 5 cycles per window nd 9 cycles per window, the mgnitude of the evolving spectrum is shown in Figure 5, when the MFT is incremented by one smple t ech nlysis stge Figure 5 FSK signl nlysis using MFT The MFT begins with the initil conditions of y = This is equivlent to hving zeros precede the dt vector In Figure 5 note how the energy in the spectrum rises from zero to mximum in the first smples Also note how spectrl lekge is observed in the first - time smples, becuse the sinusoidl signl does not hve n integer number of cycles per window over this time At time, there is n integer number of cycles per window, so ll the energy in the spectrum lies in one bin For the next - smples, lekge occurs gin s the window sliding towrds to the next frequency component of the signl The spectrl energy of the 5th frequency bin decys to zero while the spectrl energy of the 9th bin rises to its mximum This spectrum energy swpping between the two frequency bins is repeted s the window is moving through the two frequency components In Figure 6, the sme FSK signl is nlyzed in the present of noise (SR=dB) The spectrum energy swpping between the two frequency bins is lso noticeble, which shows how the MFT cn be useful for signl detection in noise environment The MFT cn be useful here if the FSK switching times re not known 9

29 Figure 6 Signl detection using MFT 4 Computing Efficiency of MFT This section exmines the computing efficiency of the MFT compred to trditionl DFT nd FFT implementtions Although computing efficiency hs mny rmifictions, we will restrict our ttention to the number of rel signl processing opertions (multiplies nd dds) required to implement the lgorithms The MFT is prticulrly efficient compre to FFT lgorithms, when successive DFTs with high overlp rtio re to be computed or when only few spectrum coefficients re needed Exmples of pplictions of the MFT to signl processing is lso given, here 4 Arithmetic of MFT The previously derived eqution for one spectrum component of n smples long MFT t time i: y i,k ( y + x x ) k = w i,k i i

30 The twiddle fctors (w -k ) cn be clculted only once nd stored in n rry before the MFT procedure This computtion is not included in the rithmetic of MFT The difference of the smple entering nd leving the window - x i -x i- - cn be preclculted t ech time moment nd used for the clcultion of ll spectrum coefficients In this cse, the spectrum of x i cn be updted from the spectrum of x i- using only complex multiplies nd + complex dds, if x i contins complexvlued dt If x i is rel, / complex multiplies nd /+ rel dds re needed to obtin the / new spectrum coefficients Tble gives summry of the number of rel opertions for these cses when only c coefficients re clculted ( c for complex dt nd c / for rel dt) Input dt Rel Multiplies Rel Adds Rel Opertions Rel 4 c 3 c + 7 c + Complex 4 c 4 c + 8 c + Tble Rel opertions in MFT for c coefficients ote, the number of opertions in ech cse cn be reduced with one, if the DC component is clculted, becuse in tht cse the twiddle fctor equls to (w k = when k=) 4 Comprison of MFT to FFT lgorithms Consider the cse where point DFTs re used to nlyze n M-point complexvlued dt record If the window is shifted by q smples between ech DFT ppliction, where q, then M + DFTs re needed to spectrum nlyze q the record, in the cse of FFT If the MFT is pplied, M MFTs re needed, becuse the spectrum coefficients hve to be clculted in ech time smples, irrespectively of the vlue of q Then, when rdix- FFTs re used: M q [ 5log ( )] OPS FFT = + (3) rel opertions, while in the cse of MFT:

31 OPS MFT M [ 8 + ] = c (33) rel opertions re needed to nlyze the whole record In this wy, the MFT becomes efficient reltive to the FFT when the shift q is smll From eqn (3) nd eqn (33) the number of shift between DFTs when the MFT is more efficient thn the rdix- FFT cn be expressed: q MFT [ 5 log ()] (M ) < M ( 8 - ) - 5 log c () (34) As we cn see from eqn (33), q MFT is function of the length of the dt record (M), the size of the window () nd the clculted MFT spectrum coefficients ( c ) In Figure 7 nd in Appendix A, the shift between DFTs when the MFT is more efficient is shown s function of the window length, with two vlues of M nd c umber of shift between DFTs when MFT is more efficient thn FFT 8 Totl smples nlyzed = 5 /4 MFT Shift between DFTs [smple] Full MFT Window size [smple] () totl smples nlyzed = 5

32 umber of shift between DFTs when MFT is more efficient thn FFT Shift between DFTs [smple] Totl smples nlyzed = 5 Full MFT /4 MFT Window size [smple] (b) totl smples nlyzed = 5 Figure 7 Shift between DFTs when the MFT is more efficient The full MFT is more efficient compred to the rdix- FFT, if the shift between DFTs is very smll (q MFT 5), while for the reduced MFT ( c = /4), the MFT is more efficient even for lrger vlues of shift ote, if the dt record is longer (Figure 7 (b)), the vlues of q MFT re lrger for ll window sizes The computtionl lod for smll mount of shifts is illustrted in Figure 8 nd 9 (Appendix A nd A3) Arithmetic of MFT nd Rdix FFT Shift Between DFTs = smple(s) Totl smples nlyzed = 5 umber of opertions (millions) 5 5 Rdix FFT Full MFT /4 MFT Window size [smple] () totl smples nlyzed = 5 3

33 Arithmetic of MFT nd Rdix FFT Shift Between DFTs = smple(s) Totl smples nlyzed = 5 umber of opertions (millions) Rdix FFT Full MFT /4 MFT Window size [smple] (b) totl smples nlyzed = 5 Figure 8 Arithmetic of MFT nd Rdix- FFT when q MFT = The rithmetic of MFT is liner with the number of the computed spectrum coefficients ( c ) nd the length of the dt record (M) For given record size (eg Figure 8 () nd Figure 9 ()) the MFT rithmetic remins the sme, with vrying shifts, while the FFT rithmetic drops down considerbly s the vlue of shift gets lrger Arithmetic of MFT nd Rdix FFT Shift Between DFTs = 4 smple(s) 5 Totl smples nlyzed = 5 umber of opertions (millions) 4 3 Rdix FFT Full MFT /4 MFT Window size [smple] () totl smples nlyzed = 5 4

34 Arithmetic of MFT nd Rdix FFT Shift Between DFTs = 4 smple(s) 3 Totl smples nlyzed = 5 umber of opertions (millions) 5 5 Rdix FFT Full MFT 5 /4 MFT Window size [smple] (b) totl smples nlyzed = 5 Figure 9 Arithmetic of MFT nd Rdix- FFT when q MFT = 4 In Figure the floting point opertion per DFT is shown for rdix- FFT, mixedrdix FFT, MFT nd the direct DFT, when the consecutive windows re overlpped by / smples (ie q MFT = /) The rithmetic of the mixed-rdix FFT in ll clcultions throughout the thesis ws estimted using the Mtlb's fft nd flops functions The rdix- FFT is very efficient if the DFT length is power of two, while the MFT is more efficient when is non-composite number (eg prime) Floting Point Opertions per DFT 8 Shift between DFTs = / umber of opertions (millions) DFT Full MFT Mixedrdix FFT Rdix FFT Window size [smples] Figure Floting Point Opertions of DFT lgorithms 5

35 When there is no overlp between the two consecutive DFTs (ie q MFT = ) during the spectrum nlysis, the MFT hs to be pplied consecutive times to obtin the next vlid DFT In this cse, the rithmetic of the MFT becomes the sme s the direct DFT in Figure, while the rithmetic of the other lgorithms remins the sme As n exmple of efficiency of the MFT, consider n -point frequency domin filter with fst convolution method pplied to the complex-vlued dt record The filter coefficients re pre-clculted nd stored nd the filter is pplied with rdix- FFT (or MFT), n rry multiply, then n inverse FFT (or IMFT) When rdix- FFT is used to obtin the convolution ( ), where FFT COPS FFT = M 5 log 6 FFT FFT > (35) rel opertions re needed In the cse of the MFT the number of rel opertions needed re: COPS MFT = M [ c +] (36) The comprison of efficiencies is shown in Figure : Arithmetic of Frequency Domin Convolution 5 Totl smples nlyzed u = 4 o of frequency points needed = /8 umber of opertions (millions) 5 Full MFT 5 Rdix FFT /8 MFT Window size [smples] Figure Fst Convolution with MFT nd Rdix- FFT If the MFT is used to compute ll the spectrum coefficients, then the rdix- FFT is more efficient for most of the DFT lengths But if only subset of the spectrum 6

36 coefficients need to be computed (ie sub-bnd filtering), the MFT/IMFT trnsformtion pir cn be more efficient for mny vlues of 43 Advntges nd Uses of MFT The computtionl order of the MFT to recursively clculte the coefficients of n - point DFT is, fctor of log improvement over the FFT If only subset of the spectrum components re needed, the computing lod of the MFT cn be further reduced, clculting only the frequency coefficients of interest The MFT does not rely upon on being power of two to obtin its efficiency, in contrst to stndrd FFT lgorithms In this wy, the MFT cn provide more efficient computtion of the DFT when ny or ll of the following conditions pply: DFTs re highly overlpped only few Fourier coefficients re needed specific, non-composite DFT length is needed Concerning the bove properties of the MFT, we cn sy tht it cn be useful in different pplictions of signl processing such s: on-line computtions in rel-time spectrl nlysis on-line signl identifiction nd detection speech processing rdr nd sonr processing 7

37 Chpter 3 Fundmentls of SAR Processing 3 Introduction In this chpter the bsic geometry of the synthetic perture rdr (SAR) system, the mthemticl form of the idel received signl of point trget nd the trditionl SAR compression lgorithm, the rnge-doppler lgorithm re introduced 3 Overview of Rdr Remote Sensing Remote sensing refers to the monitoring of plnet s surfce or tmosphere from distnt loction Remote sensing cn be performed either pssively or ctively Pssive systems do not provide trget illumintion, but operte by mesuring the mount of certin type of rdition from the plnet s surfce or tmosphere The most useful pssive remote sensing tools re opticl multispectrl which operte in the visible nd infrred regions Such systems need only to be equipped whith receiver Active systems, such s lser nd rdr imging systems, provide controlled source of trget illumintion These systems must be equipped with both trnsmitter nd receiver A potentilly very useful ctive imging tool is the synthetic perture rdr (SAR), which opertes by trnsmitting microwve bem, receiving, nd mesuring the strength nd time dely of the echo from the surfce The reflection coefficient of the ground, tht is, the power of the returned pulse, depends on the geometric nd dielectric properties of the medium Digitl signl processing techniques re used on the returned signl to extrct the groung reflectivity mp The imge obtined in this mnner is mesure of the physicl properties, including orienttion, roughness, topogrphy nd moisture content of the objects on the imged surfce 8

38 Opticl systems re widely used becuse of the very high resolution they provide Opticl imges re mesure of the chemicl composition of the imged surfce Becuse of their very smll wvelength, however, they re useless t night nd through cloud cover Microwves re not ffected by tmospheric moisture, mking them idel for imging plnet surfces through cloud cover In the Mgelln mission, synthetic perture rdr ws used to imge the surfce of Venus through its very dense tmosphere Rdr remote sensing by stellite offers wide coverge of lnd t fixed intervls regrdless of wether, clouds or time of the dy The Europen Remote-Sensing stellite (ERS-), for exmple, hs different orbitl ptterns with repet periods of 3, 35 nd 76 dys where ech individul orbit in the pttern will lst pproximtely minutes This includes coverge of res which re inccessible by other mens due to politicl, or geogrphicl difficulties 3 Rel Aperture Rdr Rdr hs been used for mny yers in militry pplictions for trcking trgets, in ir trffic control, speed mesurement nd other uses From militry pplictions the term trget hs been used to refer to the ircrft or ship being trcked In remote sensing, trget refers to the re of ground from which the bcksctter is received A rdr (s described by Tomiysu in []) is n electromgnetic wve sensor hving pulsed microwve trnsmitter, ntenn nd receiver The ntenn cn be time shred between the trnsmitter nd receiver A pulse is trnsmitted, reflected off the trget nd received The ntenn is crried by the ircrft nd produces fn bem to the side of the flight trck In rel perture rdr (RAR) the ntenn is unfocused becuse the receiver does not ccurtely conserve the phse of ech reflected pulse For this type of imging system high resolution will be obtined by trnsmitting very nrrow bem, or effectively, by hving smll wvelength, lrge ntenn, nd smllest possible distnce to trget Wvelengths of microwve bems re from 3 cm to 3 cm, nd the distnce from the stellite to ground is on the order of hundreds of kilometers Thus the only wy to increse the resolution is to increse the size of the ntenn The resolution of both RAR nd opticl systems is of the order of the bemwidth, λr/l Where R is the rnge to the trget, λ is the wvelength, nd l is the dimter of the ntenn perture or lens For exmple, for resolution of m, with λ=3 m nd R=7 km, the ntenn length would hve to be pproximtely m! Such n ntenn would be extremely costly, nd would hve power nd stbility problems in spce Opticl systems, however, cquire good resolution with resonble sized ntenns becuse of the very smll wvelength of the electromgnetic energy 9

39 33 Synthetic Aperture Rdr The principles of synthetic perture rdr (SAR) were first introduced by Crl Wiley in 95 with the observtion tht rdr crried on bord moving pltform with its bem oriented t n ngle to the pltform velocity would receive signls which re offset from the rdr frequency due to the motion of the rdr which cuses the Doppler effect By filtering this signl the bemwidth cn be mde effectively nrrower (or equivlently, the perture will be longer) This techniques enbles very high resolution imges to be obtined without need to increse the ntenn s physicl size In SAR the demodultion of the signl return to bsebnd is perfomed very ccurtely to conserve the phse between djcent smples This phse history is used to filter the dt in the long-tck dimension, wheres in RAR filtering is only performed in the cross-trck dimension Other thn two-dimensionl terrin imging, SAR dt is used in polrimetry nd interferometry Polrimetry is bsed on the principle tht different types of terrin nd ground cover hve different rdr reflectivity bsed on the polriztion of the trnsmitted bem Most SAR ntenns re linerly polrized, tht is, they send nd receive the sme polriztion In polrimetric SAR the complete polriztion response of the trget is mesured by interleving orthogonlly polrized rdr pulses on trnsmission nd mesuring the mplitude nd phse of the co-polr nd cross-polr returns on reception The rtion of like to cross polrized return cn be used to extrct significnt geophysicl nd biophysicl prmeters such s surfce roughness, moisture content of soil, biomss content of forest, or thickness of thin se ice Interferometry is the process of estimting terrin topogrphy from SAR imges by using two complex imges of the sme scene tken from different ngles After the imges hve been shifted ppropritely to lign identicl points, the phse difference between the two imges (clled the interferogrm) is clculted nd used long with other prmeters such s the ngle nd position shift of the sensor to clculte the height of ech pixel 33 SAR Geometry The min components of the SAR system re the ntenn, pulsed trnsmitter, nd coherent receiver, crried on pltform which is either stellite, or n irplne The stellite or irplne crrying the SAR ntenn moves cross the surfce of the erth t constnt velocity while trnsmitting microwve pulses t regulr intervls nd receiving the returned pulse 3

40 Assume, the irplne or stellite crrying the SAR ntenn trvels cross the surfce of the erth t constnt velocity (V r ) while trnsmitting microwve pulses t given pulse repetition frequency (PRF) to the ground with squint ngle Θ s it is shown in Figure The direction of trvel of the SAR ntenn is clled the zimuth direction while the direction of trvel of the trnsmitted pules is referred to s the rnge direction ( zimuth nd rnge directions re perpendiculr to ech other) The trnsmitted pulses trvel t the speed of light (c = 3x 8 m/s) which is much fster thn the velocity of the ntenn In this wy, the ntenn cn be treted s sttionry from the time when it sends out one pulse nd receives the reflections from the ground trgets Then the ntenn moves to one position to the next zimuth position, sending out nother pulse nd receiving the bck sctter gin Becuse of the lrge disprity in time durtion of the two directions, zimuth is referred to s the slow time, η xis while rnge s the fst time, t xis stellite trck ltitude η V r η ntenn Θ R R zimuth bem foot print trget swth rnge Figure Synthetic perture rdr geometry 3

41 The received dt cn be modeled s mtrix where ech row is the echo from one trnsmitted pulse These trnsmissions build up in the zimuth direction to form the mtrix Ech row, or pulse echo, is function of t nd ech column is function of η The imge is formed by mesuring the rnge distnce of the echo Becuse the ntenn nd the imged swth re on different plnes, the distnce mesured is in slnt rnge, R s seen in Figure 34 Idel point-trget model The two-dimensionl point trget eqution will be developed in the this section The point trget eqution is the received signl if only one sctterer were present on the ground The SAR is liner system to which the principle of superposition pplies, therefore the process developed for point trget cn be pplied to the rel received signl where the received signl is modeled s the sum of mny point trgets The rdr signl must be pulse with enough power tht the received signl hs good signl-to-noise rtio An impulse like wveform would hve these qulities, but would require very high pek power In SAR systems lrge bndwidth dispersed-energy pulse is trnsmitted nd the received signl is processed to nrrow pulse by pulse compression The trnsmitted pulse is generlly chirp signl, exp( jπ K r t ), nd the idel received SAR signl from point trget cn be written s two dimensionl signl [3]: s ( t, η) = P( t) A( η) exp jπ K r t R( η) c 4R( η) jπ λ In eqn (37) P(t) is the pulse envelope in rnge direction, A(η) is the zimuth ntenn pttern, K r is the rnge FM rte, nd λ is the rdr wve length The received signl s(t,η) cn be seprted to the rnge signl s r (t,η) nd the zimuth signl s (η) s follows: (37) s ( t, η) = s s ( t, η) = P( t) r ( t, η) s s ( η) = A( η) exp r ( η) R( η) exp jπ K r t c 4R( η) jπ λ (38) 3

42 After the chirp trvels through the slnt rnge R(η) nd bck, the received signl s r hs time dely t d = R(η)/c In this cse, for the sme trget, but in different zimuth position, the time dely of the received chirp is different, cusing rnge cell migrtion in the dt memory [] From the geometry of Figure, the closest slnt rnge of the trget (R o ) is t zimuth time η = η When the trget is t some rbitrry position with respect to the ntenn, the slnt rnge R cn be expressed s: R ( η) = R o + Vr ( η η ) (39) Becuse R >> V r (η-η ), eqn (39) cn be pproximted by prbol, expnding the eqution in second order Tylor series round η : Vr η η R( η) R + R ( ) (4) Combining eqn (38) nd (4) together, the received zimuth signl cn be written s: s ( η) = A( η) exp 4 R jπ λ exp Vr jπ ( η η ) R λ ( jπ K ( η η ) ) 4 R = A( η) exp jπ exp λ (4) where the zimuth FM rte K is K V = λr r (4) ote, tht the vlue of R chnges for ech rnge cell, therefore the zimuth FM rte chnges lso, nd this must be tken into ccount when processing the dt from different rnge loctions The zimuth or Doppler frequency, f Dop cn be obtined by differentiting the phse in eqn (4): 33

43 f Dop dφ( η) ( η) = = K ( η η ) dη (43) ote tht s (η) hs constnt phse -4πR /λ which is proportionl to R This constnt phse must be preserved or recovered fter the zimuth compression It is needed for further SAR processing pplictions, such s SAR interferometry (InSAR) The time vrible η in the zimuth signl is vlid within the exposure period of the trget, which is determined by the ntenn pttern A(η) When the ntenn length is L, the foot print width of the ntenn bem t the trget is λr /L, so the trget exposure time is λ R Te = L Vr (44) Let η c represent the zimuth time when the bem center crosses the trget Then, η c = R tn(θ/v r ), nd the vlid intervl of η for s (η) is: Te η c η < Te ηc + (45) 35 SAR signl compression The received SAR dt in both rnge nd zimuth cn be modeled s the convolution of liner FM chirp nd the ground reflectivity Using the form of the idel received signl in eqn (38) mtched filter cn be derived for ech dimension nd pulse compression cn be performed on the received dt The pulse compression rerrnges the energy received from ech ground trgets into single focused pulse The loction of the mximum energy of the pulse corresponds to the loction of the trget in rnge nd zimuth, while the strength of the pulse represents the reflectivity of the trget The Rnge/Doppler (RD) lgorithm is trditionl, highly ccurte nd efficient method for compressing SAR dt It consist of the following mjor stges [3]: rnge FFT, rnge mtched filter multipliction, 34

44 rnge IFFT, zimuth FFT, rnge cell migrtion correction (RCMC), zimuth mtched filter multipliction nd zimuth IFFT In the RD lgorithm, the rnge nd the zimuth signl re compressed seprtely using different mtched filters In either cse, the mtched filtering cn be implemented vi time domin convolution or frequency domin multipliction In the rest of this section, the pulse compression is introduced through the frequency domin zimuth compression The received zimuth signl in eqn (4) cn be simplified without loss of generlity by ignoring the phse term exp(-jπ4r /λ) nd the ntenn pttern A(η): ( π ( η j K ) s ( η) exp η = ) (46) The spectrum of the signl in eqn (46): f Fdc f S ( f ) = rect exp jπ fη K T K e (47) In eqn (47), F dc is the Doppler centroid which is the zimuth frequency or Doppler shift when the point trget is in the center of the bem nd F dc = -K (η c -η ) The Doppler bndwidth of the zimuth signl F BW V r = Te K = L (48) The zimuth resolution ρ in time units is the reciprocl of the bndwidth Therefore the zimuth resolution in distnce is: L ρ = Vr = F BW (49) 35

45 The bove result zimuth resolution is equl to the hlf of the ntenn length - is n pproximtion in tht we ssumed the ntenn pttern in the zimuth frequency domin to be rectngulr window spnning the width of the 3 db bndwidth The ctul resolution of SAR system cn be derived only experimentlly The mtched filter in the zimuth frequency (Doppler) domin is the complex conjugte of S (f): * f ) S ( f ) M ( = η = The frequency domin compressed signl is the product of the spectrum of the mtched filter nd the spectrum of the zimuth signl: (5) Sc ( f ) = M ( f ) S( f ) f Fdc = rect K T e exp( jπ f η ) (5) The compressed signl in the time domin is the inverse Fourier trnsform of S c (f): s c ( η) = F{S c(f)} = K Te exp( jπ Fdc ( η η )) sinc ( K Te ( η η )) (5) In eqn (5), the compressed pek is t the point of the trget s closest pproch (η ) This compressed pek loction cn be chnged to other position, such s the trget s strting time or Doppler centroid loction, by chnging the formt of the mtched filter In the following two chpters, survey on the pplicbility of the Momentry Fourier Trnsform to SAR signl processing lgorithms is given Chpter 4 shows how the MFT cn be pplied to the SPECA SAR processing lgorithm, while in Chpter 5 it is shown wht dvntges the MFT offers when it is used for burst-mode dt processing 36

46 Chpter 4 Appliction of MFT to SPECA SAR Processing Algorithm 4 Introduction As it ws mentioned in the previous chpter, SAR signl compression in rnge nd zimuth cn be ccomplished by cross-correltion in the time domin using time domin convolution, or in the frequency domin, using the fst-convolution vrint Rnge-Doppler method Alterntely, dvntge cn be mde of the liner FM structure of signls by replcing the cross-correltion opertion with frequency filtering opertion performed by DFT This method is clled SPECtrl Alysis (SPECA) [] nd [5] In this chpter fter the theory of the SPECA lgorithm, we introduce reduced- MFT lgorithm for the SPECA SAR processing lgorithm nd exmine wht dvntges the MFT hs compred to the FFT lgorithms when they re pplied to SPECA [7] nd [5] 4 The SPECA Algorithm The SPECA lgorithm consists two mjor computtionl steps: Dermping DFT extrction Dermping is the opertion of multiplying liner FM signl with complex conjugte reference signl with the sme FM rte, but opposite FM slope The dermping opertion turns the liner FM trget signls into constnt frequency sine wves, with frequency proportionl to zimuth position The next step in the 37

47 processing is to seprte the trget energy into different output cells, corresponding to their zimuth position This is done by performing short-length DFTs cross the dermped dt These opertions will be explined in greter detil in the following Sections 4 Wht is Dermping? The concept of dermping in the time domin is derived from the, so-clled, stretch technique The stretch technique llows liner mnipultion of the time nd bndwidth coordintes of liner FM signl by mens of mixing with nother signl of different frequency-time slope in order to slow down, speed up, or time reverse the signl A liner FM signl hs the following form: ( jπ K ( t τ ) ) T s( t) = exp s s - t < T (53) Where K s is the FM rte of the signl, T is the time extent of the pulse, nd τ s is the time position of the zero frequency of the signl The zero frequency of the trget is the sttic phse point, the time t which the phse rte goes from positive to negtive (or vice vers) If the phse switch-over occurs in the middle of the signl time durtion, the zero frequency will coincide with the center frequency The instntneous frequency, f s of the signl s(t) is given by differentiting the phse in eqn (53): f dφs ( t) t) = = K dt s ( s s ( t τ ) (54) If the signl in eqn (53) is multiplied with nother liner FM signl r(t) which hs different FM rte K r, nd is shifted in time with respect to s(t), the product will be liner FM signl with FM rte (K s +K r ): s( t) r( t) = exp ( jπ K ( t τ ) ) T r( t) = exp r r - t < ( jπ ( K + K ) t ) exp( jπ ( K τ + K τ ) t) exp( jπ ( K τ + K τ )) s r s s r r T s s r r (55) 38

48 The first time term on the right hnd side of eqn (55) represents liner FM signl with FM rte equl to K s +K r, the second is constnt frequency signl, while the third is constnt complex number In SPECA the sme technique is pplied However, to trnsform the trgets from liner FM chirps into constnt frequency signls, K r is chosen to be K s Then the dermped signl, d(t) is d( t) = exp ( jπk ( τ τ ) t) exp( jπk ( τ τ )) The instntneous frequency, f d of the dermped signl is, s r s s s r (56) f dφd ( t) t) = = K dt d ( s r s ( τ τ ) (57) where we cn see tht f d does not vry with time In SPECA the FM rte of the trnsmitted pulse is used to generte the conjugte of the idel received signl which then dermps the received dt by complex multipliction in the time domin The generted function is the SPECA mtched filter nd is clled the reference function 4 DFT Opertions After dermping, ech trget will hve constnt frequency vlue proportionl to the time position of its zero Doppler frequency with respect to tht of the reference function s seen from eqn (57) A DFT performed on this signl will confine most of the energy of ech trget to single frequency bin The reltive positions of these detected energies in the frequency domin will be proportionl to the reltive positions of the trgets time loctions, therefore the trgets will be detected t the correct positions reltive to ech other Short length DFT blocks will be plced long the dermped signl such tht ech trget s trjectory is included in t lest one DFT input vector The DFT block plcement will be explined in more detil lter in this Chpter 43 Anlyticl Bckground of SPECA As we sw in Chpter 3 compression of rdr signls is ccomplished by cross correlting the received signl with the conjugte of the idel received signl In this 39

49 section we will show tht the cross correltion opertion is equivlent to dermping followed by DFT opertion The smpled zimuth time received signl from single point reflector is liner FM signl of the form ( jπ K ( k ) ) s( k) = exp η - k < when the trget exposure is centered upon zero Doppler, thus η =η c = Figure shows the Doppler history of s(k) In the bove eqution K is the zimuth FM rte in Hz/s, is the number of smples in signl exposure nd η is the zimuth smple spcing in s (58) frequency K η η η zimuth time K η Figure 3 Doppler history of the idel zimuth received signl The compressed output, c(k) is derived by cross correlting the received signl, s(k) nd the conjugte of the idel received signl, s * (k) c * ( k) = s( i) s ( i + k) = s( i)exp η i= i= ( jπ K ( k + i) ) (59) 4

50 Multiplying the right-hnd side of eqn (59) by exp(jπk η k)*exp(-jπk η k) the following result cn be obtined: ( jπ K η ( k k) ) s( i)exp ( jπ K η i ) c( k) = exp exp jπ K k η ( i + i= The first term in the right-hnd side of eqn (6) is phse rottion term nd cn be ignored if the next opertion is the detection opertion which removes the phse informtion The second term of the summtion is the conjugte of the idel received signl, s * (k) Therefore eqn (6) cn be rewritten s c( n) = i= s( i) s * ( i)exp jπ K n η ( i + ) Let p(k) denote the product of the signl nd its complex conjugte, p(k) = s(k)s * (k), nd chnge in limits of the summtion, then c(k) is ) (6) (6) ( jπ K k i) c( k) = p( i )exp η i= The number of smples in the reference function is = K η in eqn (6) we cn further simplify the compressed output: c( k) = i= p( i )exp = exp( jπ k) i= jπ ki p( i)exp jπ ki (6) Substituting for K (63) The exp(jπk) term before the summtion is phse rottion term which gin cn be ignored if the next opertion is detection Eqution (63) shows tht the result of crosscorrelting the input signl with the conjugte of the idel received signl is equivlent to performing DFT on the product of the received signl nd the reference function The bove derivtion cn only be crried out if the signls hve liner FM form, therefore the SPECA lgorithm is only vlid for liner FM signls 4

51 44 Grphicl explntion of SPECA Greter understnding of the SPECA method is possible by grphiclly exmining the dermping nd the DFT opertions This section will use the geometry of idel point reflectors s they undergo the opertions of SPECA compression to show how the lgorithm determines trget loctions frequency () Frequency - time history of point trget shifted in time zimuth time frequency (b) Frequency - time history of the reference function zimuth time frequency (c) Frequency - time history of the dermped trget before lising zimuth time frequency (d) Frequency - time history of the dermped trget zimuth time Figure 4 Dermping of single liner FM signl 4

52 The reference function is generted using the negtive of the FM rte of the idel received signl (eqn (58)) with bndwidth equl to the zimuth smpling frequency, F If the trget nd the reference function hd equl time loctions for their zero Doppler frequencies the frequency of the product would be zero Hz This would be the cse in eqn (57) for τ s =τ r However, in the generl cse the zero Doppler frequency of the trget is displced in time from tht of the reference function (seen in Figure 4 (d)) proportionl to the displcement s seen in eqn (56) Becuse the trget is shifted in time with respect to the reference function, the reference function is repeted in time to encompss the entire length of the trget This cuses frequency jump in the dermped signl (Figure 4 (c)), so lising is needed to restore the signl (Figure 4 (d)) zimuth frequency zimuth frequency () Frequency - time history of set eqully spced point reflectors zimuth time F Hz F Hz (b) Frequency - time history of the reference function zimuth time zimuth frequency F Hz (c) Frequency - time history of the product function when the signl in () is multiplied by the reference function (b) zimuth time Figure 5 Dermping of multiple trgets 43

53 The opertions in Figure 4 must be extended to n entire line of received dt The received signl, which is modeled s the convolution of the trnsmitted pulse nd the ground reflectivity [3], cn be represented s sum of point reflectors s it is shown in Figure 5 () The reference function is repeted in time to encompss the whole length of received dt The swtooth frequency-time history of the reference function is shown in Figure 5 (b) The result of the dermping of eqully spced point reflectors is shown in Figure 5 (c) After dermping, the trgets whose zero Doppler frequency is locted within one reference function cycle will hve frequencies rnging from -F / to F / These trgets will constitute one processing region The formtion of prllelogrm shped regions from the dermped trgets is shown in Figure 5 (c) ote tht ech incrementl step in the time direction in Figure 5 () results n incrementl step in the frequency direction in (c), nd tht the frequency continuity is reset by F Hz (ie f = nd f = F re connected) = input DFT smples gurd bnd F 8 7 zimuth frequency g 3 4 slope = K [Hz/s] 9 /βm M p = (-β)m DFT DFT /βm zimuth time M : G good DFT output smples : discrded DFT output smples Figure 6 Processing regions nd the plcement of successive DFT blocks in single look cse 44

54 Consecutive processing regions re seprted by lines of slope of the zimuth FM rte (K ) nd constitute prllelogrm shped re Figure 6 shows one processing region in more detil The prmeters defined on the figure re clculted s follows: Vr Azimuth FM rte: K = [Hz/s] λ R One cycle of the reference function: F M = [smples] K The processed region of the totl Doppler spectrum: = ( β )M, where prmeter β denotes the gurd bnd Velocity of sub-stellite point: V r [m/s] Wvelength: λ [m] Closest slnt rnge: R [m] Smpling frequency of the zimuth signl: F [Hz] Ech dermped trget in the prllelogrm hs unique frequency rnging from -F / to +F / It is this unique frequency which defines the position of ech trget with respect to other trgets in the sme prllelogrm The prts of the energy of ny trget which originte ner the ends of ech trjectory (ie ner the sloped lines) hs poor SR becuse they rise from the low gin prt of the zimuth bem profile nd becuse of the reltively high presence of lised energy there The gurd bnd is represented by dshed lines in Figure 6 The totl width of the gurd bnd in the time direction is β times the distnce between the sloped lines This mens tht only (-β) frction of the totl vilble Doppler spectrum of ech trget trjectory is processed The next step in the processing is to seprte the trgets into different energy cells with regrd to their position in the time domin This is done by performing short length DFTs cross the dermped dt The plcement of the DFTs is lso shown in Figure 6, where the DFT length is smples The loction of the first DFT block (DFT ) is rbitrry, but for the ske of illustrtion it is plced so tht the lst smple corresponds to the bottom right corner of the processing region The first DFT rectngle is divided into two prts by horizontl line where the left-hnd side of the rectngle touches the right side of the gurd bnd The upper section of the rectngle contins invlid output smples which must be discrded becuse the trgets in tht region re not fully exposed, while output points corresponding to the lower section of the rectngle re kept s the good DFT output points If = αm, then the unused portion of the vilble time xis is (-α-β) M smples long nd the height of the vlid prt of the rectngle is M p 45

55 G = (-α-β) (64) DFT output smples G is the number of good points retined from the DFT opertion The next DFT (DFT ) must be pplied so tht the frequency of the lowest frequency cell corresponding to the good output region is exctly one cell higher thn the highest frequency cell of the current DFT This plcement is shown in Figure 6, s well The gp between successive DFTs in the input time domin: g = (-α-β)m (65) The DFT length, is governed by the desired zimuth resolution ρ, by the reltion []: 89Vr F 89 F λ R = = σ ρ K σ ρvr (66) where σ is the weighting fctor used in the ppliction of DFT such tht σ is the effective number of smples used in the DFT input From eqn (66) it is seen tht the DFT length () is directly proportionl to the rnge (R ) nd inversely proportionl to the resolution (ρ ), while the vribles V r, F nd K re defined by the SAR system The zimuth FM rte strongly depends upon the rnge while V r depends weekly upon R for stellite systems nd constnt for irborne systems The zimuth resolution cn be expressed from eqn (66): 89Vr F 89 F λ R ρ = = σ K σ V r (67) Tble 3 gives vilble zimuth resolutions for vrious DFT lengths, with the following C-bnd SAR stellite prmeters: K = Hz/s, F = 65 Hz, V r = 68 m/s, σ = 68 DFT length - Resolution - ρ 8 45 m 56 3 m 5 m Tble 3 Resolution versus DFT length in SPECA for C-bnd stellite SAR 46

56 45 Multi look processing in SPECA The processing scheme of Figure 6 is single look processing becuse the zimuth compression opertion produces only one output point for ech zimuth loction A multi look processing scheme is introduced s follows The lrge gps between DFTs in Figure 6 shows tht much of the input dt is not used, which is indictive of the excess zimuth bndwidth vilble when this length of DFT is used This excess bndwidth cn be used to generte multiple looks Extr looks cn be generted by dividing the G good output points from single DFT into l equl groups, nd ssigning ech group to different look number l is the number of looks The next DFT is then plced so tht the beginning of its good output points re contiguous with the end of the first look of the first DFT Such division into looks nd plcement of the second DFT is illustrted in Figure 7 for the fourlook cse In this cse the trget is fully exposed in four consecutive DFTs, nd the fifth DFT would hve the sme loction s the second DFT in Figure 6 Using the good points, obtined from ech DFT output eqn (64), the number of good output points per look is L=IT G ( α β ) = IT l l ote tht G/ l is not in generl n integer so tht usully L l < G nd tht smll frction of the good output points re not used L must be n integer so tht the looks fit together evenly t look summtion time for ech successive DFT (68) The shift between successive DFTs is: L F q= K ( - β ) R λ F 89 = ρ V σ ρ r (69) In generl q is not n integer, nd the nerest integer must be chosen in order to decide where to plce the second DFT 47

57 F q zimuth frequency G Look 4 Look 3 Look Look 4 Look 3 Look Look L Look DFT zimuth time DFT Figure 7 Division of the good output smples into looks nd the loction of the DFT opertions in multi-look processing For complete efficient dt utiliztion q σ, so the number of looks tht re normlly tken cn be expressed s follows: = l F σ K σ F ρ 89V ( - β ) = ( - β ) r σ F ρ V r (7) ote tht l is linerly proportionl to ρ nd does not depend on R 43 The SPECA Algorithm Using the MFT The zimuth FM rte of the received SAR signl is inversely proportionl to rnge, so it chnges s the rnge vries in ech cell In order to keep the resolution nd output smpling rte constnt cross the swth, there is need to choose different DFT lengths, with the DFT length incresing one smple t time s rnge increses The ffect of the vrying rnge on the FM rte nd the required DFT length for spceborne nd irborne cse re shown in Figure 8 48

58 Azimuth FM Rte nd DFT Length in SPECA 7 3 Azimuth FM Rte [Hz/s] 5 3 Azimuth FM Rte DFT length Window size [smples] Rnge [Km] () Spceborne SAR processing Azimuth FM Rte nd DFT Length in SPECA Azimuth FM Rte [Hz/s] 6 DFT length 3 Window size [smples] 4 Azimuth FM Rte Rnge [Km] (b) Airborne SAR processing Figure 8 Azimuth FM rte nd the DFT length with vrying rnge The rdr prmeters for the two cses re given, in Tble 4 ote tht in the spceborne cse there is need for only 6 different DFT sizes to keep the resolution constnt through the whole swth, while in the irborne cse wide rnge of window length is needed 49

59 Rdr prmeter Spceborne SAR Airborne SAR Units Velocity (V r ) 7 [m/s] Wvelength (λ) [m] Weighting prmeter (σ) Gurd bnd (β) Slnt rnge (R) [km] Smpling frequency (F ) 68 3 [Hz] umber of looks ( l ) 6 - Azimuth resolution (ρ ) 5 4 [m] Tble 4 Spceborne nd irborne SAR prmeters for SPECA rithmetic clcultion The rdix- FFT lgorithm cn be only used when the DFT length is power of two In other cses of window length mixed-rdix FFT lgorithm is used to chieve efficiency only for highly composite It mens, for ech different DFT different FFT lgorithm should be implemented within the SPECA processor, which mkes the rchitecture rther complex when mny different FFT lengths re needed In contrst to FFT lgorithms, the structure nd the efficiency of the MFT does not rely upon on the size of the DFT The sme simple lgorithm cn be used to clculte ll of the necessry DFTs during the zimuth compression The number of rel opertions of MFT to process n M smples long region, when ll of the spectrum coefficients re computed ( c = ) is given in eqn (33) During the DFT extrction only portion of the spectrum coefficients - good output smples (G) - re used t the sme time to obtin the compressed dt Although the mount of these spectrum components remins the sme through the processing, but their position chnges with the DFTs So, the conventionl form of the reduced-mft lgorithm cnnot be used in this cse The sub-bnd of the clculted spectrum coefficients hs to chnge its positions in the frequency domin in phse with position of the good output smples The rithmetic of the required reduced-mft lgorithm is introduced below For the first MFT (DFT in Figure 7) ll of the good output smples re computed for the first time, so using eqn (33) the rithmetic of the first MFT is: [ 8 ] MFTOPS DFT = G + (7) 5

60 For the next MFT (DFT in Figure 7) the position of the sub-bnd of the good output points shifted towrds to the higher frequencies with L smples (the good output points in look) Thus, there re L new frequency components to clculte in ddition to the (G-L) reclculted ones The number of rel opertions needed for the L new coefficients of the second DFT is: MFTOPS EW [ 8 ] DFT = L + (7) The rithmetic of the previously computed (G-L) spectrum coefficients of the second DFT is: MFTOPS OLD [ 8( G ) ] DFT = q L + (73) The shifting of the intervl of the good output points in the frequency domin continues during the whole zimuth processing, which mens tht the 'new' nd the M 'old' spectrum coefficients hve to be computed + times through the whole q region Using the previously introduced equtions MFTOPS reduced = MFTOPS DFT + M + q ( MFTOPS + MFTOPS ) DFT EW DFT OLD = M q [ 8G + ] + + ( [ 8L + ] + q [ 8( G L) + ] ) (74) rel opertions re needed to process the whole processing region with the reduced MFT Although eqn (74) looks rther complex, the implementtion of the bove described reduced MFT lgorithm is the sme s the full MFT lgorithm, except the timing nd synchroniztion of the sub-bnd of the spectrum coefficients Figure 9 (Appendix B nd B) shows the number of opertions of the SPECA zimuth compression for spceborne nd irborne cses 5

61 DFT Opertions in SPECA 6 DFT umber of Opertions (millions) Full MFT Reduced MFT Mixedrdix FFT Window size [smples] () Spceborne SPECA processing DFT Opertions in SPECA umber of Opertions (millions) DFT Mixedrdix FFT Full MFT Reduced MFT Rdix FFT Window size [smples] (b) Airborne SPECA processing Figure 9 Arithmetic of SPECA zimuth compression with different DFT lgorithms The DFT lgorithms used to obtin the DFT output smples re the direct DFT lgorithm, the full nd reduced MFT, the mixed-rdix nd rdix- FFT Although the rdix- FFT lgorithm is the most efficient, it cn be used only once during the process, when the DFT length is 56 smples (Figure 9 (b)) As the window length gets lrger (K smller) the size of the processing region lso gets lrger, thus more 5

62 opertion is needed to process one rnge cell The envelope of the plots in Figure 9 () nd (b) represents the tendency of the growing rithmetic of SPECA ote tht the rithmetic of the full nd reduced MFT is more uniform in contrst with the rithmetic of the mixed-rdix FFT lgorithms Also note, tht the rithmetic of the mixed-rdix lgorithm is equl to the direct DFT if the window length is noncomposite number (ie prime) In Tble 5 the rtio between the mximum nd minimum of rel opertion of the reduced-mft nd mixed-rdix FFT re given, followed by the rtio between the verge opertion of reduced nd full-mft nd mixed-rdix FFT of the full swth Rtio of flops Spceborne SAR Airborne SAR Mx of reduced MFT Min of reduced MFT Mx of mixed rdix FFT Min of mixed rdix FFT reduced MFT per swth mixed rdix FFT per swth full MFT per swth mixed rdix FFT per swth Tble 5 Reduced nd full MFT versus mixed-rdix FFT From Tble 5 we cn see tht in the spceborne cse there is less thn % difference between the minimum nd mximum of the reduced-mft flops, while the mximum of the rel opertions of the mixed-rdix FFT more thn 8 times bigger thn the minimum In the irborne cse, the zimuth FM rte chnges more drmticlly, thus the difference between the minimum nd mximum flops re much lrger When the MFT is pplied to SPECA there is fctor of difference between the mximum nd minimum rithmetic, while in the cse of the FFT the mximum of flops is 4 times bigger thn the minimum This lrge difference in the processing rithmetic mkes the timing of the dt flow in SPECA rther difficult when FFT lgorithms re implemented ote, the results in Tble 5 re strongly depend on the zimuth resolution nd on the width of the swth In the SPECA lgorithm, the resolution is inversely relted to the DFT length, thus lrger DFTs re needed to obtin finer resolution As the trnsformtion length is getting longer, the intervl of the good output points will shorten, therefore more DFT 53

63 blocks with higher overlp rtio will be needed to cover the processing region (Figure 6) In this cse, the SPECA lgorithm requires more computtion to obtin the zimuth compression Figure nd illustrte the rel opertions per input smples of the trditionl Rnge-Doppler (RD) lgorithm nd the SPECA lgorithm with the MFT nd rdix- FFT lgorithms for spceborne SAR system The used system prmeters for the nlysis re the sme s in Tble 4, except the number of looks, l = in this cse Shift [smples] Shift Between DFTs Window size [smples] Azimuth Resolution 68 resolution [m] Window size [smples] () 5 Opertions/Input smples vs DFT length in SPECA 45 Opertions/Input smples (millions) Rdix FFT Full MFT Reduced MFT Window size [smples] (b) Figure The rithmetic of SPECA when the MFT is more efficient 54

64 As it ws shown in Section 4, the MFT lgorithm gets more efficient s the rdix- FFT when the shift between successive DFTs re not bigger thn five smples Smll mount of shift between the DFTs nd the corresponding resolution re shown, in Figure () The shift between DFTs is smller thn six smples only for lrge window lengths (8, 8, 83, 84, 85 smples), close to the mximum ( mx = (-β)m = 86) Thus the MFT lgorithms re more efficient thn the Rdix- FFT lgorithm only for very fine resolutions In Figure (b) the rel opertions per input smples of the SPECA lgorithm with the full-mft, reduced-mft nd rdix- FFT re shown Figure () illustrtes the rithmetic of the RD lgorithm nd the SPECA with different DFT implementtions As the DTF length gets lrger, thus the zimuth resolution gets finer (ie > 7 smples, ρ 85 m) the RD lgorithm gets more efficient, even if the rdix- FFT is pplied in SPECA (Figure (b)) Thus, when fine resolution is needed, the RD lgorithm is used for SAR signl compression From Figure (b) nd (b) it cn be seen tht the FFT lgorithms re more efficient thn the full- or reduced-mft for ll trnsformtion lengths usully used in SPECA, thus the MFT does not improve significntly the computtionl efficiency of the SPECA lgorithm, except t the finest resolutions Opertions/Input smples (millions) Opertions/Input smples vs DFT length in SPECA nd RD Full MFT Reduced MFT Rdix FFT RngeDoppler Window size [smples] () 55

65 4 Opertions/Input smples vs DFT length in SPECA nd RD 35 Opertions/Input smples (millions) SPECA with Rdix FFT 5 RngeDoppler Window size [smples] (b) Figure The rithmetic of the Rnge-Doppler lgorithm nd the SPECA lgorithm using the MFT nd FFT lgorithms Beside the complexity nd computtionl efficiency, nother importnt issue in the SPECA lgorithm to keep the output smpling rte constnt In other words, trgets which re T seconds prt in zimuth input time must pper T seconds prt in the output dt It ws shown in [] tht the zimuth output smple rte is K F out = F The output smpling rte strongly depends on the zimuth FM rte K, so when it chnges trough the swth, there is need for slowly vrying DFT length to keep the output smpling rte constnt Figure (Appendix B3 nd B4) shows F out s the function of rnge, when MFT nd rdix- FFT is pplied in the SPECA lgorithm to spceborne nd irborne systems ote when the MFT lgorithm is pplied, the output smpling rte is more uniform for both cses During the ppliction of the rdix- FFT only two trnsformtion lengths - 56 nd 5 - cn be used for the irborne cse, while only one, = 5 cn be used for the spceborne cse This is the reson of the lrge migrtion of the output smpling rte when the rdix- FFT is pplied to the SPECA Hz (75) 56

66 Output Smpling Rte 65 Rdix FFT: = 5 6 Output Smpling Rte [Hz] MFT Rnge [Km] () Spceborne cse Output Smpling Rte 75 7 Rdix FFT: = 56 nd 5 Output Smpling Rte [Hz] MFT Rnge [Km] (b) Airborne cse Figure The output smpling rte of the SPECA lgorithm In this chpter, the pplicbility of the MFT to SPECA SAR processing lgorithm hs been investigted Although, the MFT does not improve the computtionl efficiency of the SPECA lgorithm, except t the finest resolutions, it hs the following dvntges over the rdix- nd mixed-rdix FFT lgorithms when they re pplied to SPECA: The MFT hs consistent computing lod s the DFT length chnges 57

67 It is esier to implement the MFT lgorithm for vrible window length The rchitecture of SPECA processor using MFT is less complex, becuse the sme MFT lgorithm cn be used for the different window lengths The full rdr resolution cn be chieved, becuse the full Doppler spectrum of the trgets cn be used for the compression by using high-overlpped DFTs The output smpling rte of SPECA is more uniform 58

68 Chpter 5 Appliction of MFT to Burst-mode SAR Dt Processing 5 Introduction Burst-mode opertion is used in SAR systems, such s RADARSAT or EVISAT, to imge wide swths, to sve power or to sve dt link bndwidth In this opertionl mode the received dt hs segmented frequency-time energy in its Doppler frequency domin There re severl lgorithms for burst-mode dt processing: one of them is the Short IFFT (SIFFT) lgorithm which ws proposed by Dr Frnk Wong t MDA [8] SIFFT's stiching error cross the stiching points, nd the effect of the IFFT length on the compressed imge SR re discussed in [] The phse-preserving property of burst-mode processing lgorithms nd the SIFFT is discribed in detil in [9], [] nd [] In this chpter, fter the introduction of burst-mode opertion, properties of burstmode dt processing nd the SIFFT lgorithm re discussed The effect of the frequency domin convolution on the Doppler history of burst-mode dt is described We show how the Doppler history is ffected when the perture contins odd or even number of bursts The shift between the spectr of two consecutive compressed trgets nd two consecutive IFFTs in the Doppler history re derived [6], [7] nd [8] Explicit formuls for both the input nd the output trget spce of the SIFFT lgorithm re given [8] At the end of this chpter, explicit formuls for the rithmetic of the SIFFT lgorithm using the IMFT nd mixed-rdix IFFT lgorithms re given [6], [7] nd [8] It is shown how the rithmetic depends on the zimuth DFT length [6], [7] nd [8], followed by survey on the efficiency of the SIFFT lgorithm when it is pplied to Envist burst-mode dt [5] 59

69 5 Overview of burst-mode SAR processing Burst-mode is commonly used in SAR systems in ScnSAR mode, where the bem is switched between two or more swths to mximize the imged swth width A -bem ScnSAR mode is illustrted in Figure 3 Stellite trck Antenn Altitude R Azimuth sub-swth burst gp rnge cell Rnge Figure 3 Burst-mode opertion in -bem ScnSAR cse In this opertion mode, the rdr bem scns through one sub-swth for certin time intervl, nd switches to the next one After scnning through the second sub-swth, the rdr switches bck to the first one to strt the next burst cycle The burst cycle hs to be short enough to mke sure ech trget is fully exposed in t lest one burst The dt from one sub-swth hve to be processed seprtely from other sub-swths, becuse the rdr bem covers different ground re in different sub-swths In the zimuth direction, the dt is segmented into discrete bursts (shded re in Figure 3), while in rnge the signl of ny sub-swth is continuous, thus the dt is not cquired in discrete rnge bursts In this Chpter, -bem burst mode processing is investigted nd described with the following ssumptions: the zimuth DFT strts with burst, there re integer number of bursts nd gps in the synthetic perture, the totl number of bursts nd gps in the synthetic perture is odd, burst nd gp length re the sme nd the length of the zimuth DFT cn be chosen rbitrrily 6

70 53 Chrcteristics of burst-mode dt A typicl -bem burst-mode dt collection pttern is shown in Figure 4 Dt from sixteen consecutive, fully exposed trgets in one rnge cell re shown, where the burst nd gp length re the sme, nd % of the perture length Dshed lines show the zimuth exposure time of ech trget if the SAR is operting in continuous mode, nd the solid prts of ech line show tht prt of ech trget ctully exposed in burst mode These re the vlid trgets of continuous-mode processing: they hve complete frequency-time history nd get compressed using the full Doppler perture (F ) mtched-filter (eg in the RD lgorithm) The prt of the trget's exposure cptured in burst mode vries with ech trget, which is illustrted in the frequency-time digrm of Figure 4 Ech successive trget is received t lower Doppler frequency within given burst, but is lter cptured t higher frequency in the next burst, s long s it stys within the bem The slope of the lines in this prt of the figure is given by the zimuth FM rte K, F = T K [Hz] (76) where F is the rdr pulse repetition frequency (PRF) nd T is the time tken for trget to generte Doppler frequency spn of F (which refer to s the synthetic perture time or the totl bem exposure time) In the given exmple, T in the timedomin consists of five burst lengths (3 burst nd gps), so F in the frequencydomin lso consists of five burst bndwidths (Figure 4) Using eqn (76), the synthetic perture length in zimuth smples is = T F F = K (77) Using the reltionship between the time nd the frequency domin given in eqn (76), the shift between two consecutive trgets spectrl energy (q tr, Figure 4) cn be obtined s follows: K q = tr Hz = Tsmpl K [Hz] F q Hz FFT K FFT q = tr tr bin round = round qtr Hz = round [frequency bins] f F F (78) 6

71 In the eqution bove, T smpl is the smpling time or the shift between two consecutive trgets, f is the distnce in Hz between two consecutive frequency bins, nd FFT is the length of the zimuth DFT in zimuth time smples ote, q tr is proportionl to K nd FFT, so the shift vries with rnge nd the length of the DFT The bndwidth of the burst in Hz nd in frequency bins cn lso be obtined using eqn (76): F = ceil F F burst Hz = T burst = ceil b K = F burst Hz b FFT burst b in f F K K [Hz] [frequency bins] (79) where b is the burst length in zimuth time smples zimuth time Azimuth exposure of 6 fully exposed trgets in one rnge cell (input trget spce) Azimuth DFT Length Synthetic Aperture (T ) T smpl 6 T burst burst burst burst 3 Frequency-time digrm of 6 trgets q tr F burst Continuous signl Burst signl slope = K [Hz/s] Trget burst 4 Trget 6 Doppler history of 6 trgets F Doppler frequency trgets Trget Trget Trget 3 Trget 4 Trget 5 Trget 6 Trget 7 Trget 8 Trget 9 Trget Trget Trget Trget 3 Trget 4 Trget 5 Trget 6 burst burst burst 3 burst 4 -F /+F dc F /+F dc F dc Figure 4 Burst-mode processing of 6 consecutive nd fully exposed trget in one rnge cell 6

72 The Doppler history of the sixteen trgets is lso shown in Figure 4, where it cn be seen tht it tkes up to two bursts (eg burst nd 4) to cover ll of them The Doppler history is the decomposition of the spectrum of trgets fter the mtched filter multipliction It shows the distribution of ech trget s spectrl energy tht would pper if DFT ws tken over the full four bursts nd four gps nd thn multiplied with the mtched filter ote tht the spectrl energy distribution of trgets is the sme before nd fter the MF multiply, becuse the MF is the conjugte complex of the idel trget In the Doppler history, some trgets pper in two full bursts (eg Trget 6), some pper in three full bursts (eg Trget ), while others pper in two full nd one prtil burst In this cse, the verge number of trget exposures or bursts per perture is 5 The number of bursts/perture in ScnSAR systems is typiclly between 5 nd 3 zimuth time Azimuth DFT Length Azimuth exposure of 5 trgets in one rnge cell (input trget spce) Synthetic Aperture (T ) f f f3 f4 f5 f6 f b9 b8 b7 b6 b5 b4 b3 b b burst burst burst burst Frequency-time digrm of 5 trgets slope = K [Hz/s] b9 b8 b7 b6 b5 b4 b3 b b f f f3 f4 f5 f6 f7 Doppler history of 5 trgets F Doppler frequency continuous signl burst signl from trgets before synthetic perture fully exposed trgets from trgets fter synthetic perture trgets f f f3 f4 f5 f6 f burst burst burst 3 burst 4 b9 b8 b7 b6 b5 b4 b3 b burst burst burst 3 b -F /+F dc F dc F /+F dc Figure 5 Burst-mode processing of 5 evenly spced fully nd prtilly exposed trget in one rnge cell 63

73 Besides the fully cptured trgets, there re lso prtilly exposed trgets both t the leding edge nd triling edge of the zimuth DFT These trgets re incomplete, so they re discrded during continuous-mode processing The frequency-time digrm nd Doppler history of prtilly exposed trgets re shown in Figure 5 The light gry region corresponds to signls from trgets tht begin previous to the strt of the DFT (trgets b-b9), while the drk gry region corresponds to signls from trgets which end fter the end of the DFT (trgets -9) ote tht most of the prtilly exposed trgets re lso completely cptured by one or two bursts, so they cn be fully compressed using one burst width of their spectrum In this wy more trgets with lower resolution cn be fully compressed in the sme synthetic perture s in the continuous-mode for given DFT length If single-look complex processing is to be done, then there is choice of which bursts to use for ech trget ormlly the trget exposures closest to the Doppler centroid (F dc ) will be selected, s shown by the hevier lines in the Doppler history of Figure 4 nd 5 However, other bursts my be chosen, when the dt is processed for InSAR purposes Doppler spectrum of fully exposed trget Doppler spectrum of fully exposed trget, centered round F dc 45 burst 3 burst burst 45 burst burst burst mplitude 5 mplitude F/ F/+Fdc Fdc F/ Doppler frequency () F/+Fdc Fdc F/+Fdc Doppler frequency (b) Figure 6 Effect of the Doppler centroid on the spectrum of fully exposed trget 53 Effect of the Doppler centroid on the Doppler spectrum After zimuth DFT, the spectrum of trgets is centered round zero frequency nd periodic on the smpling frequency which is the pulse repetition frequency (F ) in this cse [4, 9, ] So, the spectrum nd the Doppler history of trgets centered round the Doppler centroid only if it is equl to zero (F dc =) Figure 6 () shows the 64

74 spectrum of fully exposed trget (f on Figure 5), while Figure 7 () shows the periodic Doppler history fter the zimuth DFT ote tht the spectrum of the mtched filter (MF) used in trget compression hs the sme properties s the received trgets, regrding the Doppler centroid Sequence of trgets spectrl energy in burst is discontinued t F /+F dc (Figure 7 ()), so the Doppler history centered round zero frequency does not reflect the order of the input trgets Therefore, the spectrum hs to be centered round F dc before zimuth compression In theory, it mens tht from the periodic spectrum the [- F /+F dc ;F /+F dc ] intervl hs to be chosen for further processing In prctice, the dt record contining the discrete spectrum hs to be re-rrnged, thus the first F dc * fft /F spectrum coefficients hve to be moved to the end of the rry contining the spectrum The re-rrnged spectrum nd Doppler history centered round F dc re shown in Figure 6 (b) nd 7 (b), respectively F burst 3 burst burst burst 4 burst 4 burst burst burst 3 burst burst -F / -F /+F dc F dc F / F /+F dc Doppler frequency () Doppler history of trgets fter zimuth DFT F burst burst burst 3 burst 4 burst burst burst 3 -F / -F /+F dc F dc F / F /+F dc Doppler frequency (b) Doppler history of trgets centered round the Doppler frequency ( F dc ) Figure 7 Effect of the Doppler centroid on the trgets Doppler history 65

75 53 Effect of circulr convolution on the Doppler history It ws mentioned erlier tht the Doppler history is the decomposition of the spectrum of trgets fter the mtched filter multipliction (ie frequency-domin convolution) As it is shown in Figure 5, the region of fully exposed trgets is between the two tringulr shpe prtilly exposed regions in the zimuth time nd in the frequencytime digrm This reltionship between the regions is not vlid in the Doppler history: lthough, both prtilly exposed regions lso hve tringulr shpe, they complete ech other to rectngulr below the fully exposed region Originlly, the position of the light gry region (trgets previous to perture) is before the region of the fully exposed trgets, corresponding to the position of trgets to the synthetic perture in zimuth time domin The chnge in the position of the light gry region is cused by the wrp-round properties of the frequency domin convolution trgets burst trgets burst burst burst 3 burst burst burst 3 burst burst 3 -F /+F dc F dc F /+F dc Doppler frequency Doppler -F /+F dc F dc F /+F dc frequency () zimuth DFT length is equl to the synthetic perture nd 3 bursts nd gps long trgets burst burst burst 3 trgets burst burst burst burst 3 burst -F /+F dc F dc F /+F dc Doppler frequency burst 3 -F /+F dc F dc F /+F dc (b) zimuth DFT length is 3 bursts nd 3 gps long Doppler frequency 66

76 trgets burst burst burst 3 trgets burst burst 4 burst burst burst 3 burst 4 burst burst 3 -F /+F dc F dc F /+F dc Doppler frequency -F /+F dc F dc F /+F dc (c) zimuth DFT length is 4 bursts nd 3 gps long Doppler frequency trgets burst burst burst 3 trgets burst burst 4 burst burst 3 burst burst 4 burst burst 3 -F /+F dc F dc F /+F dc Doppler frequency -F /+F dc F dc F /+F dc (d) zimuth DFT length is 4 bursts nd 4 gps long Figure 8 Effect of the circulr convolution on the Doppler history Doppler frequency Figure 8 shows the originl prllelogrm region of the Doppler history nd how the light gry region gets wrpped-round, when frequency-domin circulr convolution 67

77 [4] is used insted of time-domin liner convolution Trgets in the fully exposed region get compressed into different output cells, while when the top tringle gets wrpped-round different trgets re compressed into the sme cell depending on which Doppler sub-bnd (eg burst or burst 3) is used This property of the wrpped-round trget Doppler history hs to be tken under considertion, when extrcting trgets during burst-mode signl compression ote tht distnce between the lst burst nd the first burst t the border of the tringulr regions in the Doppler history depends on the zimuth DFT length nd hs to be considered during burst-mode processing Also note, the number of trgets previous to perture re independent from the length of the zimuth DFT, while the number of exposed trgets nd the number of trgets fter perture depends on how mny bursts nd gps in the zimuth DFT This property of burst-mode dt is described in more detil in Section 54, when trgets re compressed by the SIFFT lgorithm 533 Odd number of bursts vs even number of bursts in the synthetic perture As it ws mentioned previously, usully trget exposures (burst bndwidths) closest to the Doppler centroid re used for trget extrction in burst-mode dt processing In this section, discussion on the properties of the Doppler history when the perture contins odd or even number of bursts is given Figure 9 () nd (b) shows the Doppler history of the sme perture when it is divided into five or seven burst-bndwidths (F burst bin ) nd the length of the zimuth DFT is equl to the synthetic perture ( fft = ) The processing region in the Doppler frequency domin is 3 burst-bndwidth wide (3 F burst Hz ) round the Doppler centroid independently of the number of burst in the perture The strt nd the end point of this region re P P strt end = = k 5 F k + 5 F bursthz bursthz = k = k strt end F F bursthz bursthz [Hz] [Hz] where k equls to the number of burst lengths (bursts + gps) in the perture (ie F = k F burst Hz ) ote, it is ssumed tht there re integer number of bursts nd gps in the perture nd the first intervl is burst It cn be shown tht if there re odd number of bursts in the perture thn k strt is odd, while if the perture contins even (8) 68

78 number of burst thn it is even This lso mens tht right round the F dc there is burst in the even cse nd there is gp in the odd cse This property effects the pttern of trgets exposure in the processing region in the following wy: If there re odd number of complete bursts in the perture, thn the first group of bursts strts in the middle nd ends t the beginning of the processing region (Figure 9 ()) The second group strts t the end nd scns through the whole intervl The lst burst group (from trgets previous to perture) lso strts t the end but it lsts till only the middle of the region ote, the first nd the lst group contin hlf number of trgets thn the other groups If there re even number of complete bursts in the perture, thn ll trget groups hs the sme length nd strt t the end nd lst till the beginning of the processing region (Figure 9 (b)) ote, the difference between the Doppler history of the odd nd the even cse is cused by the wrp-round effect of the circulr convolution (Figure 8) If trgets from before the perture would not get wrpped round thn the two cses would hve the sme burst pttern, thus ll trget groups would hve the sme length nd would strt t the end nd would lst till the beginning of the processing region Also note tht the burst pttern is independent of the length of the zimuth DFT nd hs to be considered during trget extrction trgets processing region burst burst burst 3 burst burst burst 3 F burst F burst (F dc ) 3F burst 4F burst 5F burst (-F /+F dc ) (F /+F dc ) Doppler frequency () odd number of bursts (3 bursts) in the synthetic perture 69

79 trgets processing region burst burst burst 3 burst 4 burst burst burst 3 burst 4 F burst F burst 3F (F dc ) burst 4F burst 5F burst 6F burst 7F burst (-F /+F dc ) (F /+F dc ) Doppler frequency (b) even number of bursts (4 bursts) in the synthetic perture Figure 9 Doppler history of consecutive trgets when there re odd or even number of bursts in the synthetic perture 54 The SIFFT lgorithm nd its properties Most SAR processing lgorithms re bsed on the fst convolution principle where frequency-domin mtched filter is used in the zimuth or Doppler frequency domin When this method is pplied to burst-mode dt, the inter-burst gps re filled with zeros nd ll the bursts re compressed t once using full length mtched filter followed by n IFFT However, the compressed trgets re then left with burstinduced modultion The SIFFT lgorithm differs from the conventionl fst convolution lgorithm in tht short, overlpped IFFTs re tken fter the mtched filter multiply in the Doppler domin [9,, ] When one burst of trget is fully cptured by the IFFT, little or no energy from djcent bursts of the sme trget is present in the sme IFFT In this wy, ech IFFT compresses group of trgets without interference (modultion) from 7

80 other bursts, nd n ccurte impulse response is obtined The IFFT cts like bndpss filter to extrct trget energy from the segmented form of the trgets spectr The filter is time vrying in the sense tht ech successive IFFT is pplied to different frequency bnd To cpture trget fully, the length of the IFFT must be t lest s long s the length of the bndwidth of one burst of tht trget The minimum IFFT length cn be clculted using eqn (79): b K FFT IFFT min = Fburst bin = ceil [frequency bin] F The IFFT cnnot be longer thn the bndwidth of one burst plus one gp, so tht fully-exposed trget is not contminted by prtil exposure of the sme or nother trget t different frequency The length of the gp is equl to the burst length, so the mximum length IFFT is b K FFT IFFT mx = Fburst bin = ceil [frequency bin] F In prctice, these length limits must be modified little becuse of the spreding of trget energy in the frequency domin - ie gurd bnd is used when locting the IFFTs ote, IFFT mx nd IFFT min re proportionl to K nd FFT The effect of this property is discussed in Section 55, where the rithmetic of the SIFFT is given IFFT mx is usully smller thn FFT, so less thn the whole Doppler spectrum is used for zimuth compression This mens tht the output resolution of the SIFFT lgorithm is smller thn the mximum vilble by fctor of IFFT / FFT, thus (8) (8) ρ = SIFFT ρ mx IFFT FFT (83) Loctions of IFFT min to compress trgets in the processing region re shown in Figure 3 Only fully exposed trgets re shown in the cse when there re four bursts nd four gps in the zimuth DFT In Figure 3, both the input nd output trget spce re shown, from where it cn be seen tht only every 5th trget from the input trget spce (trgets f-f6) gets compressed (trgets O-O3) by the SIFFT lgorithm becuse the IFFT min is /5th of the full Doppler spectrum (ρ SIFFT = /5) The shift between two consecutive compressed trgets spectr (q outr ) cn be obtined s follows, 7

81 K q = FFT FFT outr Hz = qtr Hz [Hz] IFFT F IFFT qoutr Hz FFT K qoutr bin = round = round qoutr Hz = round f F F FFT IFFT [frequency bin] (84) q ifft q outr q tr zimuth time - output trget spce O O O3 O4 O5 O6 O7 O8 burst burst burst 3 f f6 f f6 f f6 f3 f36 zimuth time - input trget spce O9 f4 O f46 O f5 O f56 O3 f6 burst 4 -F /+F dc IFFTmin F dc F /+F dc Doppler frequency IFFTmin processing region Figure 3 How minimum IFFTs re plced to compress fully exposed trgets It cn be seen tht IFFT Min cptures the complete energy of single burst spectr of trgets O5 (f) nd O3 (f6) For these trgets, IFFT Min does not extrct ny energy from other bursts spectr, so their impulse response is not corrupted by modultion Similrly, IFFT Min cptures the complete energy of single burst spectr of trgets O4 (f6) nd O (f56) In order to form continuous output imge, the results of successive IFFTs re stitched together [, ] If only bursts with the 7

82 highest energy re used to compress trgets, then ech output trget gets to different output cell ote, the first IFFT extrcts trgets with the highest indexes (O5 nd O3) from ech trget group As the IFFT is shifted towrds higher frequencies in the Doppler spectrum, trgets with lower indexes re compressed The lst IFFT, t the end of the processing region extrct trgets with the lowest indexes from the trget group The shift between two consecutive IFFTs (q ifft ) is lso indicted in Figure 3, nd cn be obtined s follows All IFFTs strt t the beginning of burst s spectr of trget, so q ifft cn be divided into integer number of q outr (Figure 3) The number of q outr in the shift between two consecutive IFFTs is Q outr IFFT F = floor q outr bin burst bin + (85) Using the eqution bove the shift between two consecutive IFFTs is IFFT Fburst bin q ifft = qoutr bin floor + [frequency bin] q outr bin (86) ote, when IFFT Min is used, the shift between IFFTs is equl to the shift between output trgets (q ifft = q outr ) When IFFT gets longer, q ifft lso gets longer, thus fewer IFFTs re needed to compress ll trgets (ie one IFFT extrcts more trgets) q ifft q outr IFFT IFFT -F /+F dc F /+F dc F dc Doppler frequency Figure 3Shift between two consecutive IFFTs 73

83 54 Input nd output trget spce of the SIFFT lgorithm In this section the input nd output trget spce of the SIFFT lgorithm is described, thus it is shown how mny trgets from n zimuth DFT cn be compressed using the SIFFT lgorithm During the investigtion it is ssumed tht trgets hving complete burst exposure in the processing region re being compressed The zimuth exposure, frequency-time digrm nd the un-wrpped Doppler history of the cse when the minimum zimuth DFT length is used nd the perture consists of three bursts nd two gps re shown in Figure 3 Trgets with complete burst exposure in the processing region re considered s the input for trget compression, s it is indicted by hevier lines nd bold numbering in Figure 3 Azimuth exposure of trgets Frequency-time digrm of trgets zimuth time Azimuth DFT length ( FFT ) Synthetic Aperture ( ) burst gp burst gp burst 3 f trget group 3 trget group trget group trgets compressed by SIFFT b4 b8 b b6 b b4 f b b4 b6 b8 b b b4 b6 b8 b b b4 trgets compressed by SIFFT trget group trget group trget group b4 b b b8 b6 b4 b b b8 b6 b4 b f burst 4 6 gp 8 burst 4 6 gp 8 burst 3 4 Un-wrpped Doppler history of trgets (F +F burst )/ processing region (F +F burst )/ Doppler frequency F dc -F / F F dc dc +F / Figure 3 Input trget spce minimum zimuth DFT is used Trgets compressed by SIFFT cn be divided into trget groups A trget group is formed from different trgets burst exposure from the sme burst, thus trgets in 74

84 trget group re extrcted from the sme burst, nd different trget groups correspond to different bursts (eg trget group - burst, trget group burst, etc) ote, ech trget hs only one complete burst exposure in the processing region, except trgets t the edge of trget groups (trget b5 nd 5) nd the number of trgets in trget group is equl to b It ws shown in Section 53 tht the input trget spce corresponding to n zimuth DFT length consists of three regions (trgets from before nd fter perture, nd fully exposed trgets) First it is shown how mny trgets re compressed in ech region, then the chrcteristic of the whole input trget spce is given The beginning of the burst exposure of the lst compressed trget from the previous to perture region is (F +F burst )/ Hz wy from the edge of the Doppler history (Figure 3) The number of trgets in the (F +F burst )/ Hz intervl equls to the number of compressed trgets tht begin before the perture (ITS before ): ITS before = ( F + F ) q burst tr / + b = (87) ote, ITS before is independent from the zimuth DFT length ( FFT ) The number of extrcted fully exposed trgets (ITS fully ) is the sme s the number of trgets compressed in continuous mode processing, nd depends linerly on the zimuth DFT length: ITS fully = FFT + (88) When the minimum FFT is used, the number of extrcted trgets from fter perture (ITS fter ) is equl to ITS before (Figure 3) As the zimuth DFT gets lrger, ITS fter gets smller until nother full gp nd full burst is not covered by the DFT During this b intervl, ITS fter depends linerly on FFT, nd cn be obtined s follows: ITS fte r = + b b FFT frction Figure 33 () shows the number of compressed trgets in zimuth time domin when the zimuth DFT length is lrger with nother gp It is seen from the figure tht the number of extrcted fully exposed trgets gets lrger, the number of compressed trgets fter perture gets smller while the totl number of processed trgets b (89) 75

85 remined the sme In Figure 33 (b), the zimuth DFT contins nother full burst In this cse, ITS fter is lrger nd equl to ITS before, the number of extrcted fully exposed trgets is lrger thn in the previous cse nd the totl number of compressed trgets contins nother trget group The totl number of compressed trgets, ie the whole input trget spce (ITS) for given zimuth DFT ( FFT ) is equl to the sum of the number of processed trgets in the three regions (ITS = ITS before + ITS fully + ITS fter ), nd cn be expressed s follows: ITS = + b + + b FFT floor b (9) zimuth time Azimuth DFT Length ( FFT ) Synthetic Aperture ( ) burst gp burst gp burst 3 gp3 f f3 f b4 b b b8 b6 b4 b b b8 b6 b4 b trget group 3 trget group trget group trgets compressed by SIFFT () zimuth DFT length = 3 burst + 3 gp zimuth time Azimuth DFT Length ( FFT ) Synthetic Aperture ( ) burst gp burst gp burst 3 gp3 burst 4 f f3 f5 f7 f9 f b4 b b b8 b6 b4 b b b8 b6 b4 b trget group 4 trget group 3 trget group trget group trgets compressed by SIFFT (b) zimuth DFT length = 4 burst + 3 gp Figure 33 Input trget spce in zimuth time domin 76

86 ITS, ITS before, ITS fully nd ITS fter versus the zimuth DFT length re shown in Figure 34 In the given exmple, = 4 smple, b = 8 smple nd FFT + 7 b As it cn be seen from the figure, ITS is step function: it strts t + b + nd is constnt for b intervl Then it jumps with b when FFT contins n odd number of complete bursts nd gps ( FFT = (k+) b ) Figure 34 Input trget spce vs zimuth DFT length The output trget spce (OTS), or the number of trgets which re compressed by the SIFFT lgorithm, is proportionl to ITS As mentioned erlier, the SIFFT resolution is smller thn the mximum vilble by fctor of IFFT / FFT The number of compressed trgets in the output trget spce is OTS = floor ( ITS ρ ) SIFFT = floor + b + + b FFT The number of compressed trgets in the three regions cn be obtined similrly to OTS: b IFFT FFT (9) 77

87 OTS before = + floor b IFFT FFT OTS fully = floor IFFT ( ) + FFT FFT OTS fte r = floor + b b frction FFT b IFFT FFT (9) In the following two sub-sections more detil re shown of the properties of the SIFFT lgorithm, through idel trget nd rel dt simultion Processing prmeter Vlue Unit Azimuth smpling frequency (F ) 6733 Hz Azimuth FM rte (K ) - Hz/s Doppler Centroid (F dc ) 447 Hz Synthetic perture length ( ) 4 smples Burst length ( b ) 8 smples Points between eqully spced trgets 35 smples Tble 6 Prmeters of idel trget simultion 54 Idel trget simultion of burst-mode processing using the SIFFT lgorithm A tool to simulte the SIFFT lgorithm on eqully spced idel trgets when IFFT min ( IFFT = F burst bin ) is used ws developed in Mtlb Mjor steps of the simultion re the following: idel bursty trget genertion, zimuth FFT, zimuth mtched filter genertion nd multiply, center the multiplied trget spectrum round the Doppler centroid, determine the border of the processing region in Doppler domin, pply short IMFTs nd 78

88 extrct the compressed trgets from the output Doppler history into the output cell The rdr prmeters used during the simultions re given in Tble 6, while the input nd output trget spce re shown in Figure 35 nd Tble 7 Trgets were compressed using four different zimuth DFT lengths nd the regions of trgets before perture, fully exposed trgets nd trgets fter perture re lso indicted in the figure nd the tble FFT = 4 FFT = 68 FFT = 959 FFT = 96 ITS ITS before ITS fully ITS fter OTS OTS before OTS fully 57 3 OTS fter Tble 7 Input nd output spce of eqully spced idel trgets 6 Output trget spce zimuth DFT = 4 smples, burst = 8 smples from trgets fter synthetic perture burst 4 before perture fter perture burst mplitude from trgets before synthetic perture burst 3 3 trgets compressed by SIFFT compressed trgets, 7 smples prt () zimuth DFT length = 3 burst + gp, IFFT = 8 smples 79

89 6 Output trget spce zimuth DFT = 68 smples, burst = 8 smples from trgets fter synthetic perture fully exposed trgets burst 4 before perture fully exposed fter perture burst mplitude 8 6 from trgets before synthetic perture burst trgets compressed by SIFFT compressed trgets, 7 smples prt (b) zimuth DFT length = 3 burst + 3 gp, IFFT = 336 smples 6 Output trget spce zimuth DFT = 959 smples, burst = 8 smples fully exposed trgets burst 4 before perture fully exposed fter perture from trgets fter synthetic perture from trgets before synthetic perture burst burst 3 mplitude burst 4 3 trgets compressed by SIFFT compressed trgets, 7 smples prt (c) zimuth DFT length = 4 burst + 3 gp -, IFFT = 39 smples 6 Output trget spce zimuth DFT = 96 smples, burst = 8 smples fully exposed trgets burst 4 before perture fully exposed fter perture from trgets fter synthetic perture from trgets before synthetic perture burst burst 3 mplitude burst 4 7 trgets compressed by SIFFT compressed trgets, 7 smples prt (d) zimuth DFT length = 4 bursts + 3 gp, IFFT = 39 smples Figure 35 Input nd output spce of eqully spced idel trgets 8

90 Figure 35 () shows the FFT Min cse, thus when the zimuth length is equl to the synthetic perture length In this cse, there is only one fully exposed compressed trget nd the number of processed trgets from before nd fter perture is the sme ote, more trgets re compressed thn the length of the IFFT In the second cse (Figure 35 (b)), there re three full bursts nd gps in the FFT, thus the zimuth DFT contins one more gp t its triling edge The number of fully exposed compressed trgets re lrger, while the number of compressed trgets from fter perture is fewer The totl number of processed trgets (OTS) is the sme s in the previous cse becuse there is no other full burst in FFT The IFFT is lrger, becuse it is linerly dependent on the zimuth FFT (eqn (8) nd (8)) In the third cse (Figure 35 (c)), FFT contins one more burst except one smple Thus, the zimuth DFT still hs only three full bursts despite of the increse in its length It mens tht the input nd the output trget spce remin the sme s it ws in the previous cses ote tht the fully exposed region is lrger nd the region of trgets from fter perture is smller gin thn it ws before Also note tht the region of trgets from previous to perture remins the sme in ll cses s it is independent from FFT (eqn 87) In the lst cse, there is one more complete burst (burst 4) in the zimuth DFT compre to the previous cses So, both the input nd the output trget spce is lrger, thus more trgets cn be compressed with the SIFFT lgorithm ote tht the length of the IFFT is the sme s it ws in the previous cse (Figure 35 (c)), while the number of compressed trgets incresed with floor( b IFFT / FFT ) Also note tht the number of compressed trgets from fter perture is gin equl to the number compressed trgets from before perture s it ws in the first cse, but the number of compressed fully exposed trgets is incresed The Doppler history of the four bursts three gps cse (Figure 35 (d)) ws generted using the IMFT lgorithm is shown in Figure 36 IMFTs, with size of IFFT min were tken, t ech frequency bin, nd the result of the trnsforms were stored in mtrix Figure 36 shows the vlues of the mtrix, where the brightness of the imge corresponds to the mgnitude of the output result The brightest points represent the good output nswers, while the drker prts show the gps in the frequency domin The three trget regions nd the 3 burst-wide processing region re lso indicted in the figure ote, the output trget spce given in Figure 35 (d) ws extrcted from the processing region of this mtrix Also note tht the number of rows (output trget spce xis) is equl to the length of the IFFT ( IFFT min ) As it cn be seen from the figure there is energy lekge of the good output points in the figure When the IMFT is pplied to process dt record, it needs smples to fully contin dt So, in the Doppler frequency there is n smple long rmp before the first vlid result Similrly, fter the lst vlid result there is n smples long 8

91 rmp s the IMFT is sliding off from the dt These properties of the frequency nlysis cuse the lekge in the output trget spce Idelly, Figure 36 would look like the Doppler history in Figure 8 (c), thus it would hve the sme intervls of gps nd trgets burst burst burst 3 fully exposed burst 4 fter perture burst burst before perture processing region burst 3 Figure 36 The Doppler history of rel burst-mode dt 543 Rel dt simultion of burst-mode processing using the SIFFT lgorithm A rel dt experiment ws done to verify the bove described properties of the SIFFT lgorithm A burst-mode processor ws creted by combining the MDA/UBC dtsar processor nd the Mtlb SIFFT zimuth processing progrm introduced in the previous section In the experiment, the rw dt ingestion, rnge compression nd RCMC re done by the dtsar processor The RCMC-ed dt, long with the required processing prmeters, such s F, K nd the Doppler centroid frequency re red into the 8

92 Mtlb progrm for zimuth processing Before we cn pply the SIFFT lgorithm to the dtsar output dt, we hve to generte the burst mode signl nd correct the ntenn pttern to void sclloping in the output The ntenn pttern correction is done by summing the zimuth FFT of ech rnge cell, nd polyfitting the summed dt The burst mode dt is emulted from SAR continuous-mode dt by windowing the signls in the zimuth direction in the Mtlb progrm The prmeters of the ERS- dt nd the length of the burst re shown in Tble 8 Figure 38 shows both, continuous nd burst-mode single look complex (SLC) SAR imges, which were generted by the RD nd by the SIFFT lgorithm, respectively Processing prmeter Vlue Unit Smpling frequency (F ) 6799 Hz Doppler Centroid (F dc ) 447 Hz Burst length ( b ) 73 nd 95 smples Rnge cells 4 smples Azimuth DFT length ( FFT ) 48 smples Tble 8 ERS- prmeters for rel dt simultion Although, the synthetic perture length vries with rnge, the burst length is pproximtely /5th (Figure 38 (b)) nd /7th (Figure 38 (c)) of the perture through the whole processing region During the zimuth compression, IFFT min ws used on bursts spectr close to the Doppler centroid, so the burst-mode dt is extrcted with the best SR Continuous mode Burst-mode b = /5 = 37 Burst-mode b = /7 = 95 ITS OTS ITS OTS ITS OTS All trgets Trgets from before pert Fully exposed trgets Trgets from fter pert Tble 9 Input nd output trget spce of ERS- dt processing 83

93 The resolution of the output imge is /5th of the originl continuous SLC when b = 73 nd /7th when b = 95, becuse mximum only /5th or /7th of the trgets Doppler spectrum is used for signl compression (ie ρ SIFFT = /5 nd /7) In Tble 9, the vlue of the input nd output trget spce for ll cses re given which were clculted using the formuls derived in Section 53 As it ws mentioned erlier the number of vlid trgets of continuos processing is equl to the number of fully exposed trgets in the zimuth DFT Figure 38 () shows, the compressed continuous dt with full resolution ote, the fully exposed section in Figure 38 (b) nd (c) is exctly the sme region with different resolution s it is in the continuous cse It is seen from Tble 9, tht more trgets cn be compressed for given zimuth DFT length when the burst is lrger (Figure 38 (b)) In this cse, there re more trgets in both prtilly exposed regions input trget spce, thus the SAR system cn see further in both directions (leding nd triling edge of the perture) () continuous mode processing full resolution imge (b) burst-mode processing - burst length is /5th of the perture ( b = /5) 84

94 (c) burst-mode processing - burst length is /7th of the perture, ( b = /7) Figure 38 SLC products of ERS dt 55 Efficiency of SIFFT using the IMFT nd the IFFT lgorithms In this section, the rithmetic nd efficiency of the SIFFT lgorithm using the IMFT nd mixed-rdix IFFT lgorithms re discussed First, the rithmetic of SIFFT is given nd it is shown how it depends on the zimuth DFT length Then, n investigtion on the efficiency of SIFFT when it is pplied to Envist dt is given 55 Arithmetic of the SIFFT lgorithm During the efficiency evlution of the SIFFT lgorithm lter in this section, the IMFT nd the mixed-rdix IFFT lgorithms re used So, formul for the rithmetic of these lgorithms re derived below As it ws shown in Section 533, the processing region of the SIFFT lgorithm is three burst-bndwidth (3F burst bin ) long in the Doppler history, thus both the IFFT nd IMFT lgorithms hve to be pplied only in this region There is q ifft shift between consecutive IFFTs, so the number of IFFTs pplied in the processing region is UM IFFT 3 F bin IFFT = ceil burst + q 85 ifft (93)

95 The number of opertions needed to compress ll trgets using the IFFT lgorithm is OP UM OP ceil 3 F burst bin IFFT IFFT = IFFT = + qifft OP IFFT IFFT where OP IFFT is the number of rel opertions needed for one smple long mixed-rdix IFFT ote tht if is power of (rdix- IFFT) then OP IFFT = 5 log () As we sw in Section 4, the IMFT lgorithm with IMFT long window needs M(8 IMFT + ) rel opertions to process n M-point complex dt record In the cse of -bem burst processing M = 3F burst bin, so the rithmetic of the IMFT lgorithm is (94) OP IMFT = burst bin IMFT 3 F (8 + ) (95) ote, both formuls in eqn (94) nd eqn (95) depend on the zimuth DFT length ( FFT ) in the following wy: OP IFFT through IFFT, F burst bin nd q ifft, while OP IMFT through IMFT nd F burst bin 55 Efficiency of the SIFFT lgorithm vs zimuth DFT length During the rithmetic clcultion, prmeters of the idel trget simultion given in Tble 6 re used with the following zimuth DFT intervl: 96 FFT 59 First, we mke the IFFT nd IMFT window length to the minimum ( IFFT = IMFT = F burst bin), thus there is only one window length to choose from in the IFFT nd IMFT lgorithms Secondly, we consider the cse when the IFFT nd IMFT window length is llowed to be up to four smples longer thn the minimum (ie F burst bin IFFT nd IMFT F burst bin + 4) This llows some flexibility in choosing fvorble (more efficient) IFFT window from five different window sizes, t the expense of smll decrese in SR More detil on SR vs IFFT is given in section 55 Substituting the SAR prmeters given in Tble 6 into eqn (8) nd eqn (8), the window length of the IMFT nd IFFT cn be obtined: 39 IMFT nd IFFT 54 In the first cse, both IMFT nd IFFT window lengths re equl to the burst bndwidth, nd ll the 3 clculted window sizes re used (Figure 39 ()) In the second cse, fvorble window size (lrger thn the minimum length) cn be chosen from five consecutive window lengths The IMFT is more efficient when the window length is smller, while the IFFT is more efficient when the window length is higher composite number It cn be seen in Figure 39 (b) tht the IFFT window is lrger in 86

96 most cses thn the minimum window length, while the IMFT window length is still equl to the burst bndwidth () minimum IFFT length is used (b) 5 IFFT lengths to choose from Figure 39 IFFT nd IMFT window length vs zimuth DFT In both cses, the signl processing system of the SIFFT lgorithm is rther complex if the IFFT lgorithms re used, becuse ll the mixed-rdix lgorithms corresponding 87

97 to the different window sizes hve to be implemented In contrst, it is esier to implement the IMFT lgorithm for vrible burst nd zimuth DFT length, becuse the sme lgorithm cn be used for the different window lengths () minimum IFFT length is used (b) 5 IFFT lengths to choose from Figure 4 Arithmetic of the IFFT nd IMFT lgorithm vs zimuth DFT 88