Two-Channel SAR-GMTI via Fractional Fourier Transform · The Fractional Fourier transform (FrFT) is...

Two-Channel SAR-GMTI via Fractional Fourier Transform

S. Chiu

Defence R&D Canada √ Ottawa TECHNICAL MEMORANDUM

DRDC Ottawa TM 2004-172 October 2004

Two-Channel SAR-GMTI Via FractionalFourier Transform

S. ChiuDefence R&D Canada - Ottawa

Defence R&D Canada - Ottawa

Technical Memorandum

DRDC Ottawa TM 2004-172

October 2004

Her Majesty the Queen as represented by the Minister of National Defence, 2004 Sa majest́e la reine, repŕesent́ee par le ministre de la D́efense nationale, 2004

Abstract

SAR along-track interferometry (SAR-ATI) can be used to measure motion in ascene by utilizing the phase and magnitude information of an interferogram. Thephase of a target’s interferogramϕATI is, to the first order, proportional to its slantrange velocityvr at broadside. In this paper, a relatively unknown yet powerfultechnique, the so-called fractional Fourier transform (FrFT), is applied to the SAR-ATI in order to estimate moving target parameters. Due to the nature of SAR datacollection, the moving target signal is unavoidably contaminated by the scene clut-ter. The FrFT technique is first used to improve the signal-to-clutter ratio (SCR) foran enhanced ATI phase estimation, which in turn allows the estimation of target’sslant-range velocity. By mapping a target’s signal onto a fractional Fourier axis, theFrFT permits a constant-velocity target to be fully focused in a fractional Fourierdomain affording orders of magnitude improvement in SCR. Then moving targetvelocity and position parameters are derived and expressed in terms of an optimumfractional angleα and a measured fractional Fourier positionup, allowing a targetto be accurately repositioned and its velocity components computed. The proposedtechnique is applied to the data acquired by the two-aperture CV580 airborne sys-tem configured in its along-track mode. Results show that the proposed method isan effective and elegant way of accurately estimating moving target velocity andposition parameters.

Résum é

L’interf éroḿetrie longitudinale SAR (SAR-ATI) peut servir̀a mesurer le mouve-ment sur un lieu d’action en utilisant les données de phase et d’amplitude de l’inter-férogramme. La phase d’un interférogramme de cibleϕATI est, au premier ordre,proportionnellèa la vitesse distance-tempsvr à la position transversale (broadside).Dans la pŕesenteétude, une technique relativement peu connue, mais puissante,appeĺee la transforḿee de Fourier fractionnaire (FrFT), est appliquée à la SAR-ATI afin d’estimer les param̀etres d’une cible mobile.́Etant donńe la nature del’acquisition de donńees SAR, le signal de cible mobile est inévitablement conta-miné par le clutter du lieu d’action. La technique FrFT est d’abord utilisée pouraméliorer le rapport signal/clutter aux fins d’une estimation de phase ATI, laquellepermetà son tour d’estimer la vitesse distance-temps de la cible. En mappant lesignal d’une cible sur un axe de Fourier fractionnaire, la FrFT permet de foca-liser entìerement une ciblèa vitesse constante dans un domaine de Fourier frac-tionnaire, ce qui aḿeliore de plusieurs ordres de grandeur le rapport signal/clutter.Alors les param̀etres de position et de vitesse de la cible mobile sont calculés et

DRDC Ottawa TM 2004-172 i

exprimés selon un angle fractionnaire optimalα et une position de Fourier frac-tionnaire mesuŕeeup, ce qui permet de repositionner exactement une cible et decalculer les composantes de sa vitesse. La technique proposée est appliqúee auxdonńees acquises par le système áeroport́e CV580à deux ouvertures, qui est confi-guŕe selon son mode d’interféroḿetrie longitudinale. Les résultats montrent que laméthode propośee offre un moyen efficace etélégant d’estimer avec exactitude lesparam̀etres de vitesse et de position d’une cible mobile

ii DRDC Ottawa TM 2004-172

Executive summary

Background

Synthetic Aperture Radar Along-Track Interferometry (SAR-ATI) is a techniquethat can be used to measure motion in a scene. For a stationary radar, motionsensing can be measured with relative ease. The sensing radar simply takes two“pictures” of the same scene at different times and compares the two snap-shots forany changes in the scene, and this can be accomplished with only a single antenna.For airborne or spaceborne radar, the task is not as simple because the radar movesat a high speed with respect to the scene it senses. Therefore, the whole worldis moving from the radar’s perspective. So how does one detect slowly movingtargets embedded in a “moving” scene? The answer is to use two or more antennas.Additional sensors provide needed degrees of freedom for a moving sensor to detectmovers in a stationary background. Using two antennas or sub-apertures aligned inthe direction of flight allows one to capture two snap-shots of a same scene fromthe same position in space but at different times. This motion sensing techniquewas originally proposed by Dickey and Santa of General Electric [8] and is calleddisplaced phase center antenna (DPCA). Depending on the way the two images arecompared, the technique is called the subtractive DPCA if the complex images aresubtracted one from the other in order to cancel clutter or is called the ATI if thetwo images are combined interferometrically (i.e. multiplying one image with thecomplex conjugate of the other) in order to “cancel” clutter in the signal phase. TheATI approach has the advantage that it not only provides target detection capabilitybut also gives information on the target’s velocity and position from its phase.

Principal results

Moving target signals are embedded in the imaged stationary scene, which is called“clutter.” In order to accurately estimate a target’s true position and velocity param-eters, clutter contamination of the target’s signal must be minimized. For a two-aperture radar system, the additional degree of freedom provided by the secondaperture can be used to cancel clutter via the subtractive DPCA, providing movingtarget indication (MTI) information. However, the expended degree of freedom canno longer be used to estimate target parameters. The ATI, on the other hand, con-tains target parameter information but is contaminated by the scene clutter becausethe ATI method does not actually cancel clutter in the sense of removing clutterfrom the target. It only nulls clutter’s interferometric phase but does not truly can-cel the clutter as does the DPCA.

The Fractional Fourier transform (FrFT) is a relatively unknown yet powerful tech-nique, which, when used in combination with the ATI, provides significant signal-

DRDC Ottawa TM 2004-172 iii

to-clutter ratio (SCR) enhancement and at the same time allows target position andvelocity parameters to be accurately estimated. The FrFT is a generalization of reg-ular Fourier transform (FT) in that the FT transforms a signal from the time domainto the frequency domain, the FrFT transforms it into a fractional Fourier domain,which is a hybridized time-frequency domain. A constant-velocity target signal,which is a linear chirp, can be fully focused in an optimum fractional Fourier do-main. Moving target position and motion parameters are derived and expressed interms of a best FrFT angleα and positionup, which are experimentally measuredor computed from the data.

The Convair 580 radar, configured in its along-track interferometric mode, was usedto collect MTI data. Application of the FrFT to SAR-ATI to achieve both enhancedSCR and accurate parameter estimation was successfully demonstrated using thecollected airborne data. Mathematical expressions were derived using the FrFT,which relate moving target parameters to the measured FrFT parameters.

Significance of results

Moving target parameter estimation using FrFT methods, in combination with theATI, is demonstrated here for the first time. The new proposed approach is com-pared to the conventional SAR-ATI approach and is shown to be more effectiveand robust. In particular, the technique is not dependent on a target’s across-trackvelocity component (vy) or its Doppler shift, which is difficult to measure exper-imentally due to insufficient degrees of freedom. The FrFT depends only on thetarget’s Doppler rate, and this is shown to be measurable experimentally with ahigh degree of robustness. The SAR-ATI approach, on the other hand, is depen-dent not only on the target’s Doppler rate but also onvy. Moreover, the selectionof matched-filter length directly affects the measured (or the apparent) ATI phase.These additional unknowns make the SAR-ATI a less desirable method for esti-mating target parameters in comparison to the new proposed FrFT approach. TheFrFT method also allows the estimation of a target’s true azimuth position (or theso-called “broadside” position) directly from its measured position in the fractionalFourier domain without actually forming a SAR image or knowing “the position ofclosest approachtc.” The new method is elegant and simple and, most importantly,very effective.

Future Work

This work is part of the preparatory work for RADARSAT-2 MODEX project. Thesatellite is scheduled to be launched in late 2005 or early 2006. The proposedparameter estimation algorithm in the present work can be readily adapted and

iv DRDC Ottawa TM 2004-172

applied to spaceborne GMTI data such as ones to be collected by the CanadianRADARASAT-2 and the German TerraSAR-X. Both radars are equipped with twosub-apertures and can be configured in an along-track mode to sense motions. Thecontinuation of this work involves applying the technique to different terrain-typedata from the Convair 580 as well as applying it to spaceborne data when theybecome available.

S. Chiu; 2004; Two-Channel SAR-GMTI Via Fractional FourierTransform; DRDC Ottawa TM 2004-172; Defence R&D Canada -Ottawa.

DRDC Ottawa TM 2004-172 v

Sommaire

Contexte

L’interf éroḿetrie longitudinale SAR [recherche et sauvetage] (SAR-ATI) est unetechnique qui peut servir̀a mesurer le mouvement sur un lieu d’action. Dans lecas d’un radar fixe, la d́etection du mouvement est relativement facileà mesurer.Le radar de d́etection n’a qu’̀a prendre deux “instantanés” d’un m̂eme lieuà desmoments diff́erents et̀a les comparer pour estimer le changement intervenu surle lieu d’action, et cela peut se faire au moyen d’une seule antenne. Dans le casdes radars áeroport́es ou spatiaux, la tâche n’est pas aussi simple, car le radar sedéplaceà grande vitesse par rapport au lieu d’action. Par conséquent, du point devue du radar, tout est en mouvement. Alors comment peut-on détecter des ciblesà d́eplacement lent qui sont elles-mêmes emportées dans le mouvement du lieu oùelles se trouvent ? La solution consisteà faire appel̀a deux antennes ou plus. Descapteurs supplémentaires fournissentà un d́etecteur mobile les degrés de libert́enécessaires pour la détection d’entit́es en mouvement sur un fond fixe. L’utilisa-tion de deux antennes ou sous-ouvertures alignées dans la direction du vol permetde produire deux instantanés d’un m̂eme lieuà partir d’une m̂eme position dansl’espace, mais̀a des moments différents. Cette technique aét́e propośee à l’ori-gine par Dickey et Santa de la Géńerale lectrique [8] et s’appelle DPCA (displacedphase center antenna = antenneà centre de phase déplaće). Selon la manière dontles deux images sont comparées, on parle soit de DPCA soustractive, si les imagescomplexes sont soustraites l’une de l’autre pour annuler le clutter ; soit d’ATI si lesdeux images sont combinées par interf́eroḿetrie (c.-̀a-.d. en multipliant une imagepar la conjugúee complexe de l’autre) afin d’ “annuler” le clutter dans la phasedu signal. L’approche ATI a l’avantage non seulement de permettre la détection decibles, mais aussi de fournir de l’information sur leur vitesse et leur positionà partirde la phase du signal de la cible.

Principaux r ésultats

Les signaux de cibles mobiles sont incorporésà l’image du lieu fixe, appelée “clut-ter ”. Afin d’estimer avec exactitude les véritables param̀etres de position et devitesse de la cible, il faut réduire le plus possible la contamination du signal de lacible par le clutter. Dans le cas d’un système radar̀a deux ouvertures, le degré deliberté suppĺementaire offert par la deuxième ouverture peutêtre exploit́e pour an-nuler le clutter suivant l’approche DPCA soustractive, en fournissant des donnéesMTI (visualisation des cibles mobiles). Toutefois, cette liberté accrue ne peut plusservir à estimer les param̀etres de la cible. L’interf́eroḿetrie longitudinale (ATI),par contre, fournit de l’information sur ces paramètres, mais elle est contaminéepar le clutter du lieu d’action, car elle n’annule pas véritablement le clutter, au

vi DRDC Ottawa TM 2004-172

sens òu il serait “enlev́e” de la cible. La ḿethode ATI ne fait qu’annuler la phaseinterféroḿetrique du clutter ; elle n’annule pas le clutter comme le fait la DPCA.La transforḿee de Fourier fractionnaire (FrFT) est une technique relativement peuconnue, mais puissante, qui, associéeà l’ATI, améliore sensiblement le rapport si-gnal/clutter et, en m̂eme temps, permet d’estimer avec exactitude les paramètresde position et de vitesse. La FrFT est une géńeralisation de la transforḿee de Fou-rier ordinaire (FT) : alors que la FT transforme un signal en le faisant passer dudomaine temporel au domaine fréquentiel, la FrFT le fait passer dans le domainede Fourier fractionnaire, qui est un domaine temps-fréquence hybrid́e. Un signal decibleà vitesse constante, qui est une impulsion linéaire compriḿee (chirp), peut̂etreentìerement focaliśee dans un domaine de Fourier fractionnaire. Les paramètres deposition et de mouvement d’une cible mobile sont calculés et expriḿes selon lemeilleur angle FrFT a et la positionup, qui sont exṕerimentalement mesurés oucalcuĺesà partir des donńees. Le radar du Convair 580, configuré en mode ATI, aservi à l’acquisition des donńees MTI. L’efficacit́e de l’application de la FrFT̀a latechnique SAR-ATI pour obtenir̀a la fois un rapport signal/clutter amélioré et uneestimation exacte des paramètres áet́e d́emontŕeeà l’aide des donńees recueilliespar radar áeroport́e.À l’aide de la FrFT, on a obtenu des expressions mathématiquesqui relient les param̀etres des cibles mobiles aux paramètres FrFT mesurés.

Port ée

L’estimation des param̀etres de cibles mobiles̀a l’aide de ḿethodes FrFT, associéesà l’ATI, est ici démontŕee pour la première fois. La nouvelle approche proposéeest compaŕeeà l’approche SAR-ATI traditionnelle et elle s’avère plus efficace etrobuste. En particulier, la technique ne dépend pas de la composante de vitesselongitudinale de la cible (vy) ni de son d́ecalage Doppler, qu’il est difficile de me-surer exṕerimentalement en raison de l’insuffisance des degrés de libert́e. La FrFTdépend seulement du taux de variation Doppler de la cible, et il est montré que cettevaleur est mesurable expérimentalement avec un haut degré de robustesse. L’ap-proche SAR-ATI, par contre, dépend non seulement du taux de variation Dopplerde la cible, mais aussi devy. En outre, la longueur du filtre adapté choisi influe di-rectement sur la phase ATI mesurée (ou apparente). Ces inconnues supplémentairesdiminuent l’attrait de la ḿethode SAR-ATI pour l’estimation des paramètres decible, comparativement̀a nouvelle approche FrFT proposée. Cette dernière permeten outre d’estimer la position d’azimut vrai de la cible (appelée aussi la position“transversale”- en anglais broadside) directementà partir de sa position mesuréedans le domaine de Fourier fractionnaire sans former réellement une image SARou connatre “la position de rapprochement maximaltc”(position of the closest ap-proachtc). La nouvelle ḿethode est́elégante et simple, et, surtout, très efficace.

DRDC Ottawa TM 2004-172 vii

Recherches futures

Ces travaux font partie des préparatifs du projet MODEX associé à RADARSAT-2.Le lancement du satellite est prévu pour l’automne 2005 ou le printemps 2006. L’al-gorithme propośe ici pour l’estimation des param̀etres peut facilement̂etre adapt́eet appliqúe à des donńees de radars spatiaux GMTI comme ceux que doit acquérirle satellite canadien RADARSAT-2 et le satellite allemand TerraSAR-X. Les deuxradars sont dotés de deux sous-ouvertures et peuventêtre configuŕes en mode longi-tudinal pour la d́etection de mouvements sur des lieux d’action. La poursuite de cestravaux visera entre autresà appliquer la techniquèa différents types de données deterrain provenant du Convair 580 ainsi qu’à des donńees de radars spatiaux lorsquecelles-ci seront disponibles.

S. Chiu; 2004; Two-Channel SAR-GMTI Via Fractional FourierTransform; DRDC Ottawa TM 2004-172; R & D pour la défenseCanada - Ottawa.

viii DRDC Ottawa TM 2004-172

Table of contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Résuḿe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Executive summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Sommaire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Table of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Moving Target Signal Model . . . . . . . . . . . . . . . . . . . . . . 2

3 Fractional Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 3

4 Airborne Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4.1 Raw-Data Target Detection . . . . . . . . . . . . . . . . . . . 7

4.2 Velocity/Position Estimation . . . . . . . . . . . . . . . . . . 7

5 Other Effects Affecting ATI Phase . . . . . . . . . . . . . . . . . . . 11

6 A Comparison to Matched-Filter Approach . . . . . . . . . . . . . . 17

7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Annexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

A Derivation of ATI Phase for The FrFT Approach . . . . . . . . . . . . 23

B Derivation of ATI Phase for The SAR Approach . . . . . . . . . . . . 28

DRDC Ottawa TM 2004-172 ix

List of figures

1 A target moving in a flat-earth geometry with motion components inx andy direction. The radar moves in positivex directions. . . . . . . 2

2 Time-frequency plot, using short-time Fourier transform, of targetT8 showing the Doppler-shifted linear chirp of the target overlaid onstationary back ground clutter (3 dB beamwidth also shown). . . . . . 4

3 Illustrating a target energy focused via the FrFT. The insert shows anactual moving target’s (T10) signal compressed in the fractionalfrequency domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Range-compressed raw-data DPCA output. The horizontal lines aredue to motion compensation artifacts. . . . . . . . . . . . . . . . . . 8

5 (a) - (c) are clutter-corrupted ATI signals for targets T4, T9, and T10respectively; (d) - (f) are the same targets’ ATI signals after the FrFTfiltering or focussing. . . . . . . . . . . . . . . . . . . . . . . . . . . 10

6 Track history of target T1 extracted from Figure 4. . . . . . . . . . . . 11

7 Detected moving targets in the SAR image context. The circlesrepresent target azimuth-shifted positions and the squares areazimuth-corrected positions. . . . . . . . . . . . . . . . . . . . . . . 13

8 A simulated target’s interferometric phases with significant centeroffsetδt in its track history. The target has velocity componentsvx = −25 m/s andvy = 5 m/s. The “ideal” phase withδt = 0 isshown as solid line. The phase deviation from the ideal value issymmetric with respect toδt = 0. . . . . . . . . . . . . . . . . . . . 16

9 Plotting the simulated target’svy vs. vx computed using (34) and(16). The new method takes into account the effect of offsetδt onATI phase and yields correct parameter estimates. . . . . . . . . . . . 17

x DRDC Ottawa TM 2004-172

10 A simulated deterministic target withvy = 5 m/s andvy = −25 m/sin a background Gaussian clutter and using a matched filter perfectlymatched tovrel but not tovy (v′y = 0). Assumeδt = 0. The idealATI phase is110.4 ˚ . The FrFT and SAR results are denoted by blueand red lines, respectively. (a) Voltage ratioVs/Vc = 0.06 or SCR =−24.4 dB; (b)Vs/Vc = 0.08 or SCR =−21.9 dB; (b)Vs/Vc = 0.08or SCR =−21.9 dB; (c)Vs/Vc = 0.1 or SCR =−20.0 dB; (d)Vs/Vc = 0.5 or SCR =−6.0 dB. Both signal and matched-filterlengths are equal to 3 ˚ beam width. . . . . . . . . . . . . . . . . . . 20

A.1 Derivation oftb from frequency-time geometry of target parameters. . 26

B.1 Factorization of rect[(τ − x)/T ][rect(τ/Tr)] . . . . . . . . . . . . . . 29

List of tables

1 Estimated interferometric phaseϕATI and ground velocityvg ofeighteen detected targets. . . . . . . . . . . . . . . . . . . . . . . . . 12

DRDC Ottawa TM 2004-172 xi

This page intentionally left blank.

xii DRDC Ottawa TM 2004-172

1 Introduction

Canada’s RADARSAT-2 commercial SAR satellite, to be launched in Fall 2005 orSpring 2006, will have an experimental mode (called MODEX for Moving ObjectDetection EXperiment) that will allow two portions of the full antenna aperture tobe used with two parallel receivers to define two independent data channels [1].These two sub-apertures, arranged to lie along the flight path, enable one to detecttargets with non-zero radial velocity by providing essentially two identical views ofthe observed scene but at slightly different times. It can be shown that a moving tar-get with a slant range velocityvr causes a differential phase shiftϕATI = 4πvrτ/λ(τ is the time between two observations), which may be detected by interferometriccombination of the signals from a two-channel along-track SAR system [2].

Moving target signals are embedded in the imaged stationary scene, which is called“clutter.” The interferometric phaseϕATI is often used to estimate a target’s radialvelocity and azimuth shift, without considering the fact that its phase is corruptedby the overlapping stationary clutter (cf. [3]). This may lead to serious errors in ve-locity and position estimates. In order to accurately estimate a target’s true positionand velocity, clutter contamination of the target’s signal must be minimized. For atwo-aperture radar system, the additional degree of freedom provided by the secondaperture can be used to cancel clutter via the subtractive DPCA, providing movingtarget indication (MTI) information. However, the expended degree of freedom canno longer be used to estimate target parameters. The ATI, on the other hand, con-tains target parameter information but is contaminated by the scene clutter becausethe ATI method does not actually cancel clutter in the sense of removing clutterfrom the target. It only nulls the clutter’s interferometric phase but does not trulycancel the clutter as does the DPCA.

The Fractional Fourier transform (FrFT) is a relatively unknown yet powerful tech-nique, which when used in combination with the ATI, provides significant signal-to-clutter ratio (SCR) enhancement and at the same time allows target position andvelocity parameters to be accurately estimated. The FrFT is a generalization of reg-ular Fourier transform (FT) in that the FT transforms a signal from time domain tofrequency domain, the FrFT transforms it into a fractional Fourier domain, which isa hybridized time-frequency domain. A constant-velocity target signal, which is alinear chirp, can be fully focused in an optimum fractional Fourier domain. Movingtarget position and motion parameters are derived and expressed in terms of a bestFrFT angleα and positionup, which can be experimentally measured or computedfrom the data.

DRDC Ottawa TM 2004-172 1

)0,2

,2

( 20

0020

00 ta

tvyta

tvxy

yx

x ++++

Target)0,,( 00 yx

x

y

z

Rad

ar

),0,( htva

),0,0( hva

)(tR

)0(R

Figure 1: A target moving in a flat-earth geometry with motion components in xand y direction. The radar moves in positive x directions.

2 Moving Target Signal Model

The phase history of a moving target in a flat-earth geometry, as shown in Figure 1,can be modelled as

ϕ(t) = −2kR(t)

= −2k√

[x0 + (vx0 − va)t+ ax0t2/2]2 + [y0 + vy0t+ ayt2/2]2 + h2,(1)

whereR(t) is the range history of the moving target,k = 2π/λ (λ is wavelength),v anda are target velocity and acceleration, respectively. Subscript “0” denotesvelocity at time zero, andx andy are along-track and across-track components,

2 DRDC Ottawa TM 2004-172

respectively. The target velocity and position at broadside timetb are therefore

vxb = vx0 + ax0tb,

vyb = vy0 + ay0tb,

xb = vatb = x0 + vx0tb + axt2b/2,

yb = y0 + vy0tb + ayt2b/2.

(2)

The Taylor expansion of (1) about broadside timetb can be written as

ϕ(t) ≈ −2k{Rb + vybγ(t− tb)

+1

2Rb(v2rel + ybay)(t− tb)2

+1

2Rb[(vxb − va)ax + vybay(1− γ2)](t− tb)3 + . . .},

(3)

whereRb are target’s slant range at broadside and

γ = yb/Rb,

v2rel = (vxb − va)2 + v2yb(1− y2b/R2b),(4)

The signal of a moving target in “slow-time” can, therefore, be modelled as

s = rect

[t− tbT

]ejϕ(t), (5)

which is a finite-time linear chirp with durationT centered at timet = tb. In realsignals, the “rect” function is also modulated by the antenna pattern.

Figure 2 shows the time-frequency (TF) plot, obtained via the short-time Fouriertransform (STFT) technique [7], of a moving target’s range-compressed signal (T8;see Figure 4) superimposed on a stationary background clutter. The target’s range-compressed signal is a linear chirp in slow-time, which appears as a slanted linein the TF plane. Because of the target’s across-track velocity componentvyb, thesignal is shifted in frequency (in this case in the negative direction). If the targethas non-negligible along-track and/or across-track velocity components (vxb andvyb) with respect to platform velocityva (i.e. vrel 6= va), then its signal will berotated slightly in the TF-plane with respect to a stationary point target. The signalremains a linear chirp as long as the target has no acceleration components,ax anday, or other higher order terms.

3 Fractional Fourier Transform

The fractional Fourier transform (FrFT) with rotational angleα of a signalf(t) isdefined as [5] [6]

Fα(u) =

Slow Time (Bin)

Fre

quen

cy (

Hz)

0 1000 2000 3000 4000 5000 6000 7000 8000

−300

−200

−100

0

100

200

300

Target signal outside clutter band

+116.36

Clutter Bandwidth

−116.36

Figure 2: Time-frequency plot, using short-time Fourier transform, of target T8showing the Doppler-shifted linear chirp of the target overlaid on stationary background clutter (3 dB beamwidth also shown).

where, forα not equal to zero or a multiple ofπ, the kernelKα(t, u) is given by

Kα(t, u) = cej2πa[t2+u2−2but], (7)

and

a =cotα

2, b = secα, c =

√1− j cotα. (8)

The FrFT with parameterα can be considered as a generalization of the conven-tional Fourier transform (FT). Thus, the FrFT forα = π/2 andα = −π/2 reducesto the conventional and inverse FT, respectively. Multiplying both sides of (6) bye−j2πau

2, one obtains

e−j2πau2

Fα(u) = F̃α(u) = c

∫ ∞−∞

f(t)ej2πa(t2−2but)dt. (9)


Substitutingν/2ab for u in (9), it becomes

F̃α

(ν

2ab

)= c

∫ ∞−∞

f(t)ej2π(at2−νt)dt

=

∫ ∞−∞

{cf(t)ej2πat2}e−j2πνtdt

=

∫ ∞−∞

gα(t)e−j2πνtdt

= Gα(ν).

(10)

The second last line in (10) shows that the FrFT is a variation of the standard FT.As such, many of its properties, such as its inverse formula and sampling theoremsfor band-limited and time-limited signals, can be easily derived from those of theFT by a simple change of variable. With respect to the parameterα, the FrFT iscontinuous, periodic (

Figure 3: Illustrating a target energy focused via the FrFT. The insert shows anactual moving target’s (T10) signal compressed in the fractional frequency domain.

hand, mainly introduces a Doppler-rate change in the chirp and, therefore, does notlead to significant nonlinearity [9].

This paper will not further consider acceleration effects, and from here on the sub-scriptb will be dropped from the velocity terms to indicate that they are constant.

4 Airborne Experiment

As part of the preparatory work for the RADARSAT-2 MODEX, airborne experi-ments were conducted to acquire SAR-ATI data for typical RADARSAT-2 resolu-tions and incident angles. The data set used in this study was obtained at CanadianForces Base (CFB) Petawawa on November 5, 2000, by the Environment CanadaCV 580 C-band SAR configured in its along-track interferometer mode. For moredetails on the experimental site, see [10]. The study focuses on targets of opportu-nity (TOOs) on Highway 17, which runs through the experimental site. The high-way was monitored by two video cameras 600 m apart set up along a stretch of the


highway to measure TOO speeds. The highway has a speed limit of 90 km/h, butmost TOOs monitored were 10-20 km/h over the speed limit. The video camerasmonitored over 47 vehicles during the data acquisition period, and their monitoredground speedsvg varied from 83 km/h to 120 km/h.

4.1 Raw-Data Target Detection

Targets were detected in the range-compressed raw data domain via a combinationof the DPCA technique [8], which subtracts the aft-aperture image from the co-registered fore-aperture image to suppress the background clutter, and the raw-datatarget detection algorithm developed by Gierull and Sikaneta [11] [12]. The DPCAis first applied to the pair of fore and aft channel range-compressed signal data toobtain an “image” of target track history as shown in Figure 4. Since the targets areazimuthally uncompressed, their energies spread over the slow time. Since thesetargets are moving on Highway 17 along the across-track direction, their track his-tories are slanted in range as seen in Figure 4, indicating range walk. Uncancelledbright stationary targets also show up in the image, as seen in the upper right-handcorner of Figure 4 (where a small town is located), but their tracks are always ver-tical. In order to estimate the target parameters, Gierull and Sikaneta’s automaticdetection algorithm [11] is first applied to extract their signal tracks. Eighteen mov-ing targets are detected and analyzed.

4.2 Velocity/Position Estimation

Having detected the targets and extracted their tracks, each target is individuallyanalyzed via the following parameter estimation algorithm. The FrFT is first ap-plied to each target to maximize its SCR by mapping its energy onto a fractionalDoppler domain. This is done by scanning through the angular parameterα in (6)to maximize the SCR. Then the ATI phase is computed for this optimum fractionalangleα to get the “best” (or the least contaminated) ATI phaseϕATI . The radialvelocityvr at the broadside (also equal to the slant range velocity) is estimated fromthe interferometric phaseϕATI using the relationship [13]

ϕATI =4πτvrλ

=kdvyγ

va, (12)

whereτ is the time delay between the pair of received signals andd is the distancebetween the centers of two sub-apertures. The ground range velocityvy is calcu-lated fromvy = vr/ sin η, whereη is the incident angle andγ = sin η = yb/Rb.The along-track velocityvx, on the other hand, is estimated from the fractional (orrotational) angleα, which can be obtained when the target is best focused in the


Range Bin

Azi

mut

h B

in

100 200 300 400 500 600 700 800 900 10000

1000

2000

3000

4000

5000

6000

7000

8000

T1 T2,3

T4 T5

T6 T7 T8

T9−12 T13,14

T16

T15 T17

T18

Figure 4: Range-compressed raw-data DPCA output. The horizontal lines are dueto motion compensation artifacts.

fractional Fourier domain, that is, by setting (6) equal to a “sinc” function:

Fα(u) =

∫ ∞−∞

rect

(t− tbT

)ejϕ(t)Kα(t, u)dt

=

∫ tb+T/2−tb−T/2

ejϕ(t)Kα(t, u)dt

= κsinc[µ(u− up)]ejϕ′(u),

(13)

whereκ andµ are constants,T is signal length centered at broadside timetb, ϕ′(u)is signal’s phase in fractional Fourier domain andup is the target position on thefractional Fourier axis, which can be shown to be

up =1

4πab

(−2kγvy +

2kv2reltbRb

)(14)

Note that (13) is equal to a sinc function only whent2 term in the exponential of


the integrand is set to zero, which leads to

α = acot

{2N

Rbλf 2s

[(vx − va)2 + v2y(1− γ2)

]}(15)

or

vx = va −

√Rbλf 2s cotα

2N− v2y

(1− γ2

), (16)

whereN is sample length andfs is the sampling (or pulse repetition) frequency.The f 2s /N factor is introduced to convert frequency/time units to frequency/timesamples. The target azimuth shift correction∆x can be derived by setting thederivative of phase history (3) to zero and solving for (tc − tb), which yields theazimuth shift correction,∆x = va(tc − tb),

∆x =vyvayb

(vx − va)2 + v2y(1− γ2)≈ vyyb

va=vrRbva

, (17)

where the approximation (≈) is valid only whenva � vx. In order to calculatetb from (17), one still needs to knowtc, the time of closest approach. This can beestimated only in the image domain and only when the matched-filter is correctlymatched to target’svrel. An alternate way to estimatetb without resorting to thisadditional processing step is via the FrFT method, which is by solving (14) fortb:

tb =Rb

2kv2rel

(2πupsinα

+ 2kγvy

). (18)

This new approach permits parameter estimation (vx, vy, andxb = vatb) in the frac-tional Fourier domain directly from range-compressed raw data without actuallyforming a SAR image.

Figure 5 shows the polar plots of three (T4, T9, and T10) of the eighteen detectedtargets’ interferograms. Figures 5(a)-5(c) are interferograms obtained from the reg-ular SAR-ATI processing (i.e. from the fore and aft SAR images processed by astationary terrain matched filter). Interferograms for the same targets obtained fromthe FrFT processing are shown in Figures 5(d)-5(e). The signal points near the zerophase radial belong to the stationary clutter. As can be seen, ATI phases obtainedfrom the regular SAR processing are severely contaminated by the interfering clut-ter and their values are consistently lower than those obtained from the FrFT, whichnot only gives much cleaner and well defined phases but also correct values com-pared to their true phases. The “true” phase is here defined to be theϕATI that givesthe correct azimuth correction to bring a target back onto the road (or Highway 17).

The analysis results of all eighteen targets are summarized in Table 1, showing tar-gets’ interferometric phasesϕATI and their estimated ground velocitiesvg. The


10000000

20000000

30000000

40000000

30

210

60

240

90

270

120

300

150

330

180 0

(a)

10000000

20000000

30000000

40000000

50000000

30

210

60

240

90

270

120

300

150

330

180 0

(b)

10000000

20000000

30000000

40000000

30

210

60

240

90

270

120

300

150

330

180 0

(c)

2000

4000

6000

8000

30

210

60

240

90

270

120

300

150

330

180 0

(d)

2000

4000

6000

8000

10000

30

210

60

240

90

270

120

300

150

330

180 0

(e)

2000

4000

6000

8000

30

210

60

240

90

270

120

300

150

330

180 0

(f)

Figure 5: (a) - (c) are clutter-corrupted ATI signals for targets T4, T9, and T10respectively; (d) - (f) are the same targets’ ATI signals after the FrFT filtering orfocussing.

velocities and azimuth shifts of 10 targets are correctly estimated by the FrFT ap-proach compared to only three by the regular SAR approach.

Since all targets are either moving almost directly towards or directly away from theradar, their radial velocitiesvr are high and ambiguous due to phase wrapping. Thevelocity ambiguity can be resolved by using the track history of each target, as theone shown in Figure 6 for target T1. By converting azimuth samples to azimuth time(i.e. multiplying the number of azimuth samples by the pulse repetition frequency)and range samples to range distance (i.e. multiplying the number of range samplesby c/fA/D, wherec is the speed of light andfA/D is the A/D sampling frequency),one obtains an averagevy of +109 km/h for target T1. Therefore, the ATI phase isexpected to wrap around2π, which occurs atvr = 13.15 m/s (47.33 km/h). In fact,the measured ATI phases for all targets under consideration must be unwrapped toyield target speeds that fall within the monitored speed range (83 to 120 km/h). Forinstance, T1’s−1320 phase is in fact+5880, which yields avr = +77.3 km/h, or aground velocity of+106.2 km/h after taking into account the imaging geometry.


Range Samples

Azi

mut

h S

ampl

es

70 75 80 85 90 95 100 105 110

2000

2500

3000

3500

4000

4500

5000

5500

6000

6500

32x4m = 128 m

3750

xTpr

i = 5

.8 s

Figure 6: Track history of target T1 extracted from Figure 4.

Figure 7 plots the targets’ azimuthally corrected positions estimated from the pro-posed FrFT approach. The circles represent targets in their azimuth-shifted po-sitions and the squares are their corrected locations. As can be seen, ten out ofeighteen targets are correctly positioned on Highway 17. The other targets are offthe highway because their SCRs remain small (. −3 dB) even after the FrFT pro-cessing.

5 Other Effects Affecting ATI Phase

When examining target track histories, one observes that they are not always com-plete, having breaks and missing portions (see Figure 4), at which times targetsappear invisible to the radar. As a result, the centers of the tracks may not alwayscorrespond to the broadside times of these targets. As can be seen in Figure 4,most tracks are not centered on the highway. Thus, one cannot determine the targetbroadside timetb from the track history. But more importantly (as will be shown


Table 1: Estimated interferometric phase ϕATI and ground velocity vg of eighteendetected targets.

Target SARϕATI [◦] FrFTϕATI [◦] “True” ϕATI [◦]ID / vg[km/h] / vg[km/h] / vg[km/h]T1 −142/104.5 −132.7/106.2 −131/106.5T2 −42/118.9 −42.8/118.7 −43/118.7T3 0/0 −87/110.7 −157.7/98.4T4 +32/118.7 +80/110.4 +79/110.6T6 −60/110.4 −71/108.5 −71.3/108.5T7 +106/100.8 +103/101.3 +100/101.8T8 +88/103.2 +85.4/103.6 +85/103.7T9 −26/110.7 −65/104.5 −63.5/104.7T10 −30/109.9 −85/101.1 −84/101.3T11 −5/113.8 −70/103.4 −104.5/98.0T12 0/0 −135/93.0 −137/92.7T13 −60/104.2 −10/112.1 −83/100.6T14 −47/106.0 −48/105.9 −86.5/99.8T15 +30/124.7 +45/122.0 +44/122.2T16 −32/124.4 −130/106.7 −150/103.1T17 0/0 −82/115.0 −153.5/102.2T18 0/0 +44/120.8 +98.5/111.0


Range Bin

Azi

mut

h B

in

100 200 300 400 500 600 700 800 900 1000

1000

2000

3000

4000

5000

6000

7000

8000

T1

T2

T3

T4

T6

T7 T8

T9

T10

T11

T12

T13,14

T15

T16

T17

T18

Figure 7: Detected moving targets in the SAR image context. The circles representtarget azimuth-shifted positions and the squares are azimuth-corrected positions.

below), this time offsetδt of the target track center fromtb, together with a non-negligiblevx (with respect tova), causes the estimate of the target interferometricphaseϕATI to deviate from its true value.

If the phase of the signal received at channel 1 is

ϕ1(t) = −2kR(t) = −2k[Rb + vyγ(t− tb) +v2rel2Rb

(t− tb)2], (19)

then the phase of the signal at channel 2 in far-field approximation is

ϕ2(t) = ϕ1(t)− kµ(t)d, (20)

whereµ(t) ≈ vx − va

Rb(t− tb) (21)

is a Taylor series expansion of direction cosine “cosψ” about broadside timetb.To co-register the two channels, the signal received at channel 2 is time-shifted by


t = d/2va to yield a phase:

ϕ2(t+ d/2va) =− 2kRb − 2kγvy(t− tb)− kγvyd

va− kv

2rel

Rb(t− tb)2

− kdRbve(t− tb)

(22)

where

ve =(vx − va)2 + (1− γ2)v2y

va+ (vx − va)

=v2relva

+ (vx − va).(23)

If the signal track is of lengthT and is not centered attb (i.e. t = tb + δt), thenco-registered signals at apertures 1 and 2 can be written, respectively, as

s1 = rect

[t− tb − δt

T

]ejϕ1(t) (24)

and

s2 = rect

[t− tb − δt

T

]ejϕ2(t+d/2va). (25)

Substituting (24) into (6) and integrate to maximize the target signal in the fractionalFourier domain, one obtains

Fα1(u) = ej�1

∫ +∞−∞

rect

[t− tb − δt

T

]ejξ1tdt

= ej�1∫ tb+δt+T/2

tb+δt−T/2ejoξ1tdt

= ej[�1+ξ1(tb+δt)]Tsinc

(ξ1T

2π

),

(26)

where

�1 = −2kRb + 2kγvytb −kv2relt

2b

Rb+ 2πau2 (27)

and

ξ1 = −2kγvy +2kv2reltbRb

− 4πabu. (28)

Similarly, the optimum fractional Fourier transform of the co-registered signal re-ceived at channel 2 can be shown to be

Fα2(u) = ej[�2+ξ2(tb+δt)]Tsinc

(ξ2T

2π

), (29)


where

�2 = �1 −kγvyd

va+ktbdveRb

(30)

and

ξ2 = ξ1 −kdveRb

. (31)

Therefore, the phase of interferogram (ϕATI = arg[Fα1(u)Fα2(u)∗]) evaluated atoptimum fractional Fourier angleα can be shown to be equal to

ϕATI =kdvyγ

va+kdveRb

δt, (32)

which indicates thatϕATI is also sensitive tovx if the target track is not centered attb and its sensitivity is dependent on the degree of offsetδt and platform velocityva.

Figure 8 shows the ATI phase of a simulated moving target with its track historycentered att = tb + δt. The “true” phase (i.e. whenδt = 0) is shown as a solid line.The target moves at a constant velocity (vx = −25 m/s andvy = 5 m/s) in a flatearth geometry as depicted in Figure 1. The dominant term inve in (32) is along-track velocityvx. Therefore, largeδt andvx values lead to significant deviation ofATI phase from the true value. It is also noted that the phase deviation is symmetricwith respect toδt = 0.

The estimation procedure proposed in the previous section yields good positionalestimates for the 18 targets examined, even though a majority of the target tracksare not centered at broadside. This is because most targets examined are movingmainly toward or away from the radar with relatively smallvx component, usuallyin the range of 2 to 5 m/s. This is especially true for targets T1 through T12.Therefore, track center offsets do not significantly affect thetb estimates.

In more general cases, however,δt cannot be ignored and, therefore, the estima-tion procedure proposed in the previous section can no longer be applied directlyto obtain velocity and position estimates, particularly when bothδt andvx are non-negligible. In order to resolve this, an additional independent equation is needed.This can be obtained by making use of “angle-of-arrival” information. When un-registered signals from two identical channels are combined to form an interfero-gram, its phase (or angle) provides angle-of-arrival information of a target, insteadof its velocity information as in the co-registered-signal case. When the target is atbroadside to the antenna beam, the distances between the target and the two receiv-ing sub-apertures should be identical and, therefore, the phases of the two signalsreceived at two channels should also be identical, which leads to a zero ATI phase.However, whenδt 6= 0 the ATI phase computed from the two unregistered signals


2

4

6

8

30

210

60

240

90

270

120

300

150

330

180 0

129.19o offset = 250/va

91.61o offset = −250/va

110.40o offset = 0

Figure 8: A simulated target’s interferometric phases with significant center offsetδt in its track history. The target has velocity components vx = −25 m/s andvy = 5 m/s. The “ideal” phase with δt = 0 is shown as solid line. The phasedeviation from the ideal value is symmetric with respect to δt = 0.

is no longer zero but can be shown to be

ϕATIu =k(vx − va)d

Rbδt, (33)

which is a function of bothvx and δt. Substituting (33) into (32) to eliminateunknown parameterδt and solving forvy results in

vy =vakγd

{ϕATI − ϕATIu

[Rbλf

2s cotα

2Nva(vx − va)+ 1

]}, (34)

which has only two unknowns,vx andvy. Target velocity components can, there-fore, be obtained from (34) and (16), and its azimuth position from (18). Therevised parameter estimation procedure is applied to the above simulated target andits velocity components are estimated and plotted in Figure 9. Using this revisedapproach, the target parameters are now correctly estimated.


−30 −20 −10 0 10 20 30−30

−20

−10

0

10

20

30

vx (m/s)

v y (

m/s

)

(vx, v

y) = (−25, 5 ) m/s offset = − 250/va

offset = 250/va

Equation (16)

Equation (34)

Figure 9: Plotting the simulated target’s vy vs. vx computed using (34) and (16).The new method takes into account the effect of offset δt on ATI phase and yieldscorrect parameter estimates.

6 A Comparison to Matched-FilterApproach

One would like to compare the FrFT method (32) with the matched-filter (MF)approach (i.e. via SAR azimuth compression). If one assumes a matched-filterthat matches exactly to the Doppler rate orvrel (i.e. the second coefficient in theTaylor expansion) of a constantly moving target but does not match correctly toits Doppler shift orvy (i.e. the first coefficient of the Taylor expansion), then thereference function for azimuth compression can be written as

h(t) = rect

[t

Tr

]ejϕr(t). (35)

where

ϕr(t) = −2k(vryt+1

2Rbv2relt

2) (36)


andvry andTr in (36) are not necessarily equal tovy andT , respectively, in (24) and(25). The resulting ATI phase obtained by solving

ϕATImf (t) = arg [s1(t) ∗ h∗(−t)][s2(t) ∗ h∗(−t)]∗ (37)

can be shown to equal

ϕATImf =kvyγd

va− kγdve

2v2rel(vy − vry) +

kdveRb

[δt

2∓ Tr − T

4

], (38)

where “∓” corresponds to cases where(tc − tb − δt) > 0 and(tc − tb − δt) < 0,respectively, andtc is the azimuth-shifted-time position of target, which is also thetime of closest approach if the MF is matched to the target’s Doppler rate. Whenvry = vy (i.e., when the reference function matches exactly to bothvx andvy), (38)reduces to

ϕATImf =kdvyγ

va+kdveRb

(δt2∓ Tr − T

4

). (39)

The MF approach requires matching tovrel andvy simultaneously, which is moredifficult to accomplish than matching only tovrel as in the FrFT approach.

The expression (38) also indicates that if one does not correctly choose avry for thereference function (i.e.vry 6= vy), the azimuth processing may actually lead to abiased or erroneous ATI phase because of its “vy − vry” dependence. A simulationrun of a moving target withvy = 5 m/s andvy = −25 m/s in background Gaussianclutter and using a matched filter perfectly matched tovrel but not tovy (vry = 0),clearly shows that the target’s ATI phase, computed via the MF approach, devi-ates strongly from the ideal phase value computed using (12), as shown in blue inFigure 10. The ideal ATI phase is calculated to be110.4 ˚ , but the SAR approachconsistently yields biased results for various SCRs. The SCR is here defined to bethe target’s signal-to-clutter power ratio before pulse compression. Even when theSCR is sufficiently large, the ATI phase is still biased at101.2 ˚ , which can be pre-dicted and confirmed by (38). Moreover, the MF method is dependent on “Tr −T ,”in addition to its dependence onδt and “vry − vy.”

Also shown in the figure are the results computed using the FrFT method (in red).Without atb offset (i.e. δt = 0), the technique yields the ideal ATI phase, exceptwhen the SCR is very small and the ATI phase becomes severely contaminated bythe interfering clutter. It is clear, therefore, that the FrFT approach to parameterestimation is superior to the MF method because the FrFT approach only requiresmatching to target’s Doppler rate, which is a relatively straight-forward and robustprocess. The MF, on the other hand, requires not only matching to the target’sDoppler rate but also to its Doppler shift, a process proven to be difficult and unre-liable due to insufficient information.


7 Conclusions

Moving target parameter estimation using FrFT methods, in combination with theATI, is demonstrated here for the first time. The new approach is compared to theconventional SAR-ATI approach and is shown to be more effective and robust. Inparticular, the technique is not dependent on a target’s across-track velocity com-ponent (vy) or its Doppler shift, which is difficult to determine due to insufficientdegrees of freedom. The FrFT depends only on target’s Doppler rate, and this isshown to be measurable experimentally with a high degree of robustness. The SAR-ATI approach, on the other hand, is dependent not only on a target’s Doppler ratebut also onvy. Moreover, the selection of matched-filter length directly affects themeasured (or the apparent) ATI phase. These additional unknowns make the SAR-ATI a less desirable method for estimating target parameters than the new proposedFrFT approach. The FrFT method also allows the estimation of target’s true az-imuth position (or the so-called “broadside” position) directly from its measuredposition in the fractional Fourier domain without actually forming a SAR imageor knowing “the position of closest approachtc.” The new method is elegant andsimple and, most importantly, very effective.


−20 0 20 40 60 80 100 120 140 160 1800

50

100

150

200

250

300

ATI Phase (degree)

PD

F

Vs/V

c = 0.06

SCR = −24.4 dB

(a)

−20 0 20 40 60 80 100 120 140 160 1800

50

100

150

200

250

ATI Phase (degree)

PD

F

Vs/V

c = 0.08

SCR = −21.9 dB

(b)

−20 0 20 40 60 80 100 120 140 160 1800

50

100

150

200

250

ATI Phase (degree)

PD

F

Vs/V

c = 0.1

SCR = −20 dB

(c)

−20 0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

ATI Phase (degree)

PD

F

Vs/V

c = 0.5

SCR = −6.0 dB

(d)

Figure 10: A simulated deterministic target with vy = 5 m/s and vy = −25 m/s in abackground Gaussian clutter and using a matched filter perfectly matched to vrelbut not to vy (v′y = 0). Assume δt = 0. The ideal ATI phase is 110.4 ˚ . The FrFTand SAR results are denoted by blue and red lines, respectively. (a) Voltage ratioVs/Vc = 0.06 or SCR = −24.4 dB; (b) Vs/Vc = 0.08 or SCR = −21.9 dB; (b)Vs/Vc = 0.08 or SCR = −21.9 dB; (c) Vs/Vc = 0.1 or SCR = −20.0 dB; (d)Vs/Vc = 0.5 or SCR = −6.0 dB. Both signal and matched-filter lengths are equal to3 ˚ beam width.


References

1. Livingstone, C.E. 1998. The Addition of MTI Modes to Commercial SARSatellites. In Proc. of 10th CASI Conference on Astronautics, Ottawa, Canada,26-28 October 1998.

2. Raney, R.K. 1971. Synthetic aperture imaging radar and moving targets. IEEETransactions on Aerospace and Electronic Systems, AES-7(3), pp. 499-505.

3. Gierull, C.H. 2002. Moving target detection with along-track SARinterferometry. Technical Report TR 2002-084, Defence R&D Canada -Ottawa, August.

4. Chiu, S. 2003. Clutter effects on ground moving target velocity estimationwith SAR along-track interferometry. In IGARSS’2003, Proceedings of theInternational Geoscience and Remote Sensing Symposium, Toulouse, France,21-25 July.

5. Almeida, L.B. 1994. The fractional Fourier transform and time-frequencyrepresentations. IEEE Trans. Signal Processing, Vol. 42, No.11, pp.3084-3091.

6. Zayed, A.I. 1996. On the relationship between the Fourier and fractionalFourier transforms. IEEE Trans. Signal Processing Letters, Vol. 3, No. 12, pp.310-311.

7. Boashash, B. 2003. Heuristic formulation of time-frequency distributions. InTimeFrequencySignalAnalysisandProcessing, Elsevier, pp. 38-41.

8. Dickey, F.R., Jr., Santa, M.M. 1953. Final report on anticlutter techniques,General Electric Company Rept. R65EMH37, 1 Mar. 1953.

9. Sharma, J.J. and Collins, M.J. 2004. Focusing accelerating ground movingtargets in SAR imagery. In this conference (EUSAR 2004).

10. Livingstone, C.E., Sikaneta, I., Gierull, C.H., Chiu, S., Beaudoin, A.,Campbell, J., Beaudoin, J., Gong, S., Knight, T.A. 2002. An airborne SARexperiment to support RADARSAT-2 GMTI. Can. J. Remote Sensing, Vol. 28,No. 6, pp. 794-813.

11. Gierull, C.H. and Sikaneta, I.C. 2003. Raw Data based Two-Aperture SARGround Moving Target Indication. In IGARSS’2003, Proceedings of theInternational Geoscience and Remote Sensing Symposium, Toulouse, France,21-25 July 2003.


12. Gierull, C.H. and Sikaneta, I.C. 2004. Ground moving target parameterestimation for two-channel SAR. In this conference EUSAR 2004.

13. Raney, R.K. 1992. Special SAR techniques and applications. In AGARDLecture Series 182 on Fundamentals and Special Problems of SAR,AGARD-LS-182, pp. 10-1 to 10-15.


Annex ADerivation of ATI Phase for The FrFTApproach

The full derivations of a moving target’s ATI phase (32) and motion/position pa-rameters are given here for the fractional Fourier transform approach.

If phases of received signals at channels 1 and 2 from a moving target are

ϕ1(t) = −2kR(t)

= −2k[Rb + γvy(t− tb) +

(vx − va)2 + (1− γ2)v2y2Rb

(t− tb)2]

= −2kRb − 2kγvyt+ 2kγvytb −kv2relRb

(t2 − 2tbt+ t2b)

(A.1)

and

ϕ2(t) = −2kR(t)−kd(vx − va)(t− tb)

Rb, (A.2)

respectively, then to register channel 2 signal with that of channel 1,ϕ2(t) is for-ward time-shifted byd/2va to yield

ϕ2(t+ d/2va) =− 2kRb − 2kγvy(t− tb +

d

2va

)− 2k v

2rel

2Rb

(t− tb +

d

2va

)2− k(vx − va)d

Rb

(t− tb +

d

2va

)=− 2kRb − 2kγvy(t− tb)−

kγvyd

va− kv

2rel

Rb(t− tb)2

− kv2reld

Rbva(t− tb)−

��

��>

0kv2reld

2

4Rbv2a− k(vx − va)d

Rb(t− tb)

−��

��

��*0

k(vx − va)d2

2Rbva

= ϕ1(t)− kγvyd

va− kv

2reld

Rbva(t− tb)−

k(vx − va)dRb

(t− tb)

(A.3)

where the strike-through terms are considered negligible. If the signal track is cen-tered att = tb + δt, then co-registered signals at apertures 1 and 2 can be written,respectively, as

s1 = rect

[t− tb − δt

T

]ejϕ1(t) (A.4)


and

s2 = rect

[t− tb − δt

T

]ejϕ2(t+d/2va). (A.5)

Substituting (A.4) into (6) and integrate to maximize the target signal in fractionalFourier domain, one obtains

Fα1(u) = ej�1

∫ +∞−∞

rect

[t− tb − δt

T

]ej(ξ1t+ν1t

2)dt

= ej�1∫ tb+δt+T/2

tb+δt−T/2ejξ1tdt = ej�1

ejξ1t

jξ1

∣∣∣∣tb+δt+T/2tb+δt−T/2

= ej[�1+ξ1(tb+δt)]2

ξ1

ejξ1T/2 − e−jξ1T/2

2j

= ej[�1+ξ1(tb+δt)]Tsin(πξ1T/2π)

πξ1T/2π

= ej[�1+ξ1(tb+δt)]Tsinc

(ξ1T

2π

),

(A.6)

where

�1 = −2kRb + 2kγvytb −kv2relt

2b

Rb+ 2πau2, (A.7)

ξ1 = −2kγvy +2kv2reltbRb

− 4πabu, (A.8)

and

ν1 = −kv2relRb

+ 2πa = 0. (A.9)

Note the target signal is maximized in the fractional Fourier domain whenν1 isequal to zero, which yields a ”sinc” function in (A.6) and the optimum fractionalangleα (a = cotα/2):

α = acot

(kv2relπRb

). (A.10)

Because the discrete FrFT deals with frequency and time samples rather than Hertzand seconds, one must convert them to frequency and time samples, which yields

α = acot

[kv2relπRb

(N

f 2s

)]. (A.11)

Similarly, the optimum fractional Fourier transform of the signal received at chan-nel 2 can be shown to be

Fα2(u) = ej[�2+ξ2(tb+δt)]Tsinc

(ξ2T

2π

), (A.12)


where

�2 = �1 −kγvyd

va+ktbdveRb

(A.13)

and

ξ2 = ξ1 −kdveRb

. (A.14)

Therefore, the phase of interferogram (arg[Fα1(u)Fα2(u)∗]) or the ATI phase (ϕATI)evaluated at the optimum fractional Fourier angleα yields

ϕATI = �1 + ξ1(tb + δt)− [�2 + ξ2(tb + δt)]= �1 − �2 + (ξ1 − ξ2)(tb + δt)

=kdvyγ

va− kdve

Rbtb +

kdveRb

(tb + δt)

=kdvyγ

va+kdveRb

δt,

(A.15)

which indicates thatϕATI is also sensitive tovx if the target track is not centered attb and its sensitivity is dependent on the degree of offsetδt and along-track velocitycomponentvx.

The positions of a moving target in the fractional Fourier axis for channels 1 and 2can be derived by settingξ1 = 0 andξ2 = 0, that is, evaluating at the maxima ofthe “sinc” functions in (A.6) and (A.12), which yields

u1 =1

4πab

(−2kγvy +

2kv2reltbRb

)(A.16)

and

u2 =1

4πab

(−2kγvy +

2kv2reltbRb

− kv2reld

Rbva− k(vx − va)d

Rb

). (A.17)

The third and fourth terms in (A.17) are negligible compared to the first two terms,and‖u1 − u2‖ is smaller than the spatial resolution of “sinc” functions, implyingspatial overlap on the fractional Fourier axis. Therefore one can setu1 = u2 = upand the target azimuth positionxb (= tbva) can be obtained directly from (A.16) bysolving fortb:

tb =Rbv2rel

(πupk sinα

+ γvy

). (A.18)

Converting Hertz-second to frequency-time samples, (18) becomes

tb =

(f 2sN

)Rbv2rel

(πupk sinα

+ γvy

). (A.19)


f

t

ffr

fD

tb tc

α

SignalChirp

α

up

Figure A.1: Derivation of tb from frequency-time geometry of target parameters.

(A.19) can also be derived from the geometry of the signal chirp in the frequency-time plane as shown in Figure A.1. From the figure, the time of closest approachtcis equal to

tc =up

cosα, (A.20)

andtanα equal to

tanα =tc − tbfD

. (A.21)

Combining (A.20) and (A.21) gives

tb = tanα(up/ sinα− fD), (A.22)

which can be shown to be exactly equal to (A.19) when substituting for

fD = −kvrπ

= −kvyγπ

(A.23)

and

tanα =f 2sRbλ

2Nv2rel=f 2sRbπ

Nv2relk. (A.24)


For unregistered channels, the ATI phase can be similarly derived. The phase of asignal at channel 2 is

ϕ2(t) = ϕ1(t)−k(vx − va)d

Rb(t− tb), (A.25)

and the fractional Fourier transform of the signal at channel 2 is

Fα2(u) = ej�2u

∫ +∞−∞

rect

[t− tb − δt

T

]ej(ξ2ut+ν2ut

2)dt

= ej[�2u+ξ2u(tb+δt)]Tsinc

(ξ2uT

2π

),

(A.26)

where

�2u = �1 +k(vx − va)tbd

Rb, (A.27)

ξ2u = ξ1 −k(vx − va)d

Rb, (A.28)

andν2u = 0. Therefore the unregistered ATI phase is

ϕATIu = �1 + ξ1(tb + δt)− [�2u + ξ2u(tb + δt)]= �1 − �2u + (ξ1 − ξ2u)(tb + δt)

= −k(vx − va)tbdRb

+k(vx − va)d

Rb(tb − δt)

=k(vx − va)d

Rbδt.

(A.29)

(A.29) is used to eliminate the additional unknown parameterδt due to the offset intb, the broadside time. Substituting

δt =RbϕATIu

k(vx − va)d(A.30)

into (A.15), one obtains

ϕATI =kγvyd

va+

vevx − va

ϕATIu ; (A.31)

and solving forvy yields

vy =vakγd

{ϕATI − ϕATIu

[Rbλf

2s cotα

2Nva(vx − va)+ 1

]}. (A.32)

Parametersα, ϕATI , andϕATIu are measured experimentally. Therefore,vx andvycan be solved using (A.11) and (A.32).


Annex BDerivation of ATI Phase for The SARApproach

The full derivation of a moving target’s ATI phase (38) is shown here for the SARprocessing approach.

The reference function for azimuth compression can be written as

h(t) = rect

(t

Tr

)ejϕr(t), (B.1)

wheretb in (B.1) is set to zero:

ϕr(t) = −2k(Rb + vyγt+1

2Rbv2relt

2) (B.2)

The azimuth compressions of the registered signals are accomplished via correla-tion with h(t):

z1(t) = =

∫ ∞−∞

s1(t+ τ)h∗(τ)dτ

=

∫ ∞−∞

[rect

(t+ τ − tb − δt

T

)ejϕ1(t+τ)

][rect

(τ

Tr

)ejϕr(τ)

]∗dτ

=

∫ ∞−∞

rect

(τ + t− tb − δt

T

)rect

(τ

Tr

)ej[ϕ1(t+τ)−ϕr(τ)]dτ

(B.3)

and

z2(t) =

∫ ∞−∞

s2(t+ τ)h∗(τ)dτ

=

∫ ∞−∞

[rect

(t+ τ − tb − δt

T

)ejϕ2(t+τ+d/2va)

][rect

(τ

Tr

)ejϕr(τ)

]∗dτ

=

∫ ∞−∞

rect

(τ + t− tb − δt

T

)rect

(τ

Tr

)ej[ϕ2(t+τ+d/2va)−ϕr(τ)]dτ

(B.4)

Using Figure B.1 as a visual aid, one can see that “rect[(τ − x)/T ][rect(τ/Tr)]” in(B.3) and (B.4), wherex = −t+ tb + δt, can be factored into

rect

(τ − xT

)rect

(τ

Tr

)= rect

(τ −M±

L

)rect

(x

Tr + T

), (B.5)


LL

M− M+ τx x

Figure B.1: Factorization of rect[(τ − x)/T ][rect(τ/Tr)]

whereM± = x2 ±Tr−T

4andL = Tr+T

2− |x|. Then the matched-filter processing

leads to

z1(t) = rect

(x

Tr + T

)ejε1

∫ M±+L/2M±−L/2

ejζ1τdτ

= rect

(x

Tr + T

)ejε1

ejζ1τ

jζ1

∣∣∣∣M±+L/2M±−L/2

= rect

(x

Tr + T

)2

ζ1

ej(ε1+ζ1M±)(ejζ1L/2 − e−jζ1L/2

)2j

= rect

(x

Tr + T

)ej(ε1+ζ1M±)L

sin(π ζ1L

2π

)π ζ1L

2π

= rect

(x

Tr + T

)Lej(ε1+ζ1M±)sinc

(ζ1L

2π

)

(B.6)

and

z2(t) = rect

(x

Tr + T

)Lej(ε2+ζ2M±)sinc

(ζ2L

2π

), (B.7)


whereε1, ε2, ζ1, andζ2 are derived below. Therefore, the phase of interferogram,arg[z1(t)z∗2(t)], is equal to

ϕATI = ε1 − ε2 + (ζ1 − ζ2)M±. (B.8)

Expressingϕ1(t), ϕ2(t), andϕr(t) more generally as

ϕ1(t) = β0 + β1(t− tb) + β2(t− tb)2, (B.9)

ϕ2(t) = β0 + (β1 + β3)(t− tb) + β2(t− tb)2, (B.10)and

ϕr(t) = β0 + βr1t+ β2t

2, (B.11)

where

β0 = −2kRb,β1 = −2kvyγ,βr1 = −2kvryγ,β2 = −kv2rel/Rb,β3 = −k(vx − va)d/Rb.

(B.12)

Then “ϕ1(t+ τ)− ϕr(τ)” in (B.3) can be shown to be

ϕ1(t+ τ)− ϕr(τ) = β0 + β1(τ + t− tb) + β2(τ + t− tb)2

− (β0 + βr1τ + β2τ 2)= ε1 + ζ1τ,

(B.13)

whereε1 = β1(t− tb) + β2(t− tb)2 (B.14)

andζ1 = β1 − βr1 + 2β2(t− tb). (B.15)

Similarly,

ϕ2(t+ τ + d/2va)− ϕr(τ) = β0 + (β1 + β3)(t+ τ + d/2va − tb)+ β2(t+ τ + d/2va − tb)2

− (β0 + βr1τ + β2τ 2)= ε2 + ζ2τ,

(B.16)

whereε2 = (β1 + β3)(t− tb + d/2va) + β2(t− tb + d/2va)2 (B.17)

andζ2 = (β1 − βr1) + β3 + β2[2(t− tb + d/2va)] (B.18)


Therefore, one can derive the general expression for “ε1 − ε2”:

ε1 − ε2 = β1(t− tb) + β2(t− tb)2 − (β1 + β3)(t− tb + d/2va)− β2(t− tb + d/2va)2

= −β1d

2va− β3(t− tb)− β3

d

2va− β2

[(t− tb)

d

va+

��

��70

d2

4va

].

(B.19)

Evaluating (B.19) at the position (t = tc) where signal or “sinc(·)” is maximum,that is, whenζ1 = 0:

ζ1 = β1 − βr1 + 2β2(t− tb) = 0 (B.20)

or

tc − tb = −β1 − βr1

2β2, (B.21)

then substituting “t− tb” for “ tc − tb” and using (B.12), (B.19) becomes

ε1 − ε2 = −β1d

2va+β3(β1 − βr1)

2β2− β3d

2va+��β2

[(β1 − βr1)d

2��β2va

]=

�2kγvyd

�2va− k(vx − va)d

��Rb

(��−2kγ)(vy − vry)

��−2kv2rel/��Rb

+�

��

��*0

k(vx − va)d2

2Rbva− �

2kγ(vy − vry)d�2va

=kγdvyva

− kγd(vy − vry)(vx − vav2rel

+1

va

).

(B.22)

Similarly, ζ1 − ζ2 can be shown to be

ζ1 − ζ2 = ��β1 − βr1 +��

2β(t− tb)− [��β1 − βr1 + β3 + 2β2(��t− tb + d/2va)]= β3 + β2d/va

=k(vx − va)d

Rb+kv2reld

Rbva

(B.23)

EvaluatingM± at tc gives

M± =x

2± Tr − T

4=−(tc − tb − δt)

2± Tr − T

4

=(vy − vry)yb

2v2rel+δt

2± Tr − T

4.

(B.24)


Finally, the ATI phase via SAR processing can be derived from (B.8) as follows:

ϕATI = ε1 − ε2 + (ζ1 − ζ2)M±

=kγdvyva

−kγd(vy − vry)ve

v2rel

+kdveRb

[(vy − vry)yb

2v2rel+δt

2± Tr − T

4

]=kγdvyva

−kγd(vy − vry)ve

2v2rel+kdveRb

(δt

2± Tr − T

4

),

(B.25)

where “±” correspond tox = −(tc − tb − δt) > 0 andx = −(tc − tb − δt) < 0,respectively.


DOCUMENT CONTROL DATA(Security classification of title, body of abstract and indexing annotation must be entered when document is classified)

1. ORIGINATOR (the name and address of the organization preparing the document.Organizations for whom the document was prepared, e.g. Centre sponsoring acontractor’s report, or tasking agency, are entered in section 8.)

Defence R&D Canada - Ottawa3701 Carling Avenue, Ottawa, Ontario K1A 0Z4, Canada

2. SECURITY CLASSIFICATION(overall security classification of the documentincluding special warning terms if applicable).

UNCLASSIFIED

3. TITLE (the complete document title as indicated on the title page. Its classification should be indicated by the appropriateabbreviation (S,C,R or U) in parentheses after the title).

Two-Channel SAR-GMTI Via Fractional Fourier Transform

4. AUTHORS(Last name, first name, middle initial. If military, show rank, e.g. Doe, Maj. John E.)

Chiu, S.

5. DATE OF PUBLICATION (month and year of publication of document)

October 2004

6a. NO. OF PAGES (totalcontaining information. IncludeAnnexes, Appendices, etc).

47

6b. NO. OF REFS (total cited indocument)

13

7. DESCRIPTIVE NOTES (the category of the document, e.g. technical report, technical note or memorandum. If appropriate, enter the type of report,e.g. interim, progress, summary, annual or final. Give the inclusive dates when a specific reporting period is covered).

Technical Memorandum

8. SPONSORING ACTIVITY (the name of the department project office or laboratory sponsoring the research and development. Include address).

Defence R&D Canada - Ottawa3701 Carling Avenue, Ottawa, Ontario K1A 0Z4, Canada

9a. PROJECT OR GRANT NO. (if appropriate, the applicable research anddevelopment project or grant number under which the document waswritten. Specify whether project or grant).

15eg

9b. CONTRACT NO. (if appropriate, the applicable number under whichthe document was written).

N/A

10a. ORIGINATOR’S DOCUMENT NUMBER (the official document numberby which the document is identified by the originating activity. Thisnumber must be unique.)

DRDC Ottawa TM 2004-172

10b. OTHER DOCUMENT NOs. (Any other numbers which may beassigned this document either by the originator or by the sponsor.)

11. DOCUMENT AVAILABILITY (any limitations on further dissemination of the document, other than those imposed by security classification)

( X ) Unlimited distribution( ) Defence departments and defence contractors; further distribution only as approved( ) Defence departments and Canadian defence contractors; further distribution only as approved( ) Government departments and agencies; further distribution only as approved( ) Defence departments; further distribution only as approved( ) Other (please specify):

12. DOCUMENT ANNOUNCEMENT (any limitation to the bibliographic announcement of this document. This will normally correspond to the DocumentAvailability (11). However, where further distribution beyond the audience specified in (11) is possible, a wider announcement audience may beselected).

Unlimited

13. ABSTRACT (a brief and factual summary of the document. It may also appear elsewhere in the body of the document itself. It is highly desirable that theabstract of classified documents be unclassified. Each paragraph of the abstract shall begin with an indication of the security classification of theinformation in the paragraph (unless the document itself is unclassified) represented as (S), (C), (R), or (U). It is not necessary to include here abstracts inboth official languages unless the text is bilingual).

SAR along-track interferometry (SAR-ATI) can be used to measure motion in a scene by utilizing the phaseand magnitude information of an interferogram. The phase of a target’s interferogram ϕATI is, to the firstorder, proportional to its slant range velocity vr at broadside. In this paper, a relatively unknown yet powerfultechnique, the so-called fractional Fourier transform (FrFT), is applied to the SAR-ATI in order to estimatemoving target parameters. Due to the nature of SAR data collection, the moving target signal is unavoidablycontaminated by the scene clutter. The FrFT technique is first used to improve the signal-to-clutter ratio (SCR)for an enhanced ATI phase estimation, which in turn allows the estimation of target’s slant-range velocity. Bymapping a target’s signal onto a fractional Fourier axis, the FrFT permits a constant-velocity target to befully focused in a fractional Fourier domain affording orders of magnitude improvement in SCR. Then movingtarget velocity and position parameters are derived and expressed in terms of an optimum fractional angle αand a measured fractional Fourier position up, allowing a target to be accurately repositioned and its velocitycomponents computed. The proposed technique is applied to the data acquired by the two-aperture CV580airborne system configured in its along-track mode. Results show that the proposed method is an effectiveand elegant way of accurately estimating moving target velocity and position parameters.

14. KEYWORDS, DESCRIPTORS or IDENTIFIERS (technically meaningful terms or short phrases that characterize a document and could be helpful incataloguing the document. They should be selected so that no security classification is required. Identifiers, such as equipment model designation, tradename, military project code name, geographic location may also be included. If possible keywords should be selected from a published thesaurus. e.g.Thesaurus of Engineering and Scientific Terms (TEST) and that thesaurus-identified. If it not possible to select indexing terms which are Unclassified, theclassification of each should be indicated as with the title).

Synthetic Aperture Radar (SAR), Fractional Fourier Transform, Ground Moving Target Indication (GMTI),Parameter Estimation

Defence R&D Canada

Canada’s leader in defenceand national security R&D

Chef de file au Canada en R & Dpour la défense et la sécurité nationale

R & D pour la défense Canada

www.drdc-rddc.gc.ca

Two-Channel SAR-GMTI via Fractional Fourier Transform · The Fractional Fourier transform (FrFT) is...

Documents

Transcript of Two-Channel SAR-GMTI via Fractional Fourier Transform · The Fractional Fourier transform (FrFT) is...