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Performance Improvement for

Constellation SAR using Signal

Processing Techniques

ZHENFANG LI

ZHENG BAO, Senior Member, IEEE

HONGYANG WANG

GUISHENG LIAO, Member, IEEE

Xidian University

China

A new concept of spaceborne synthetic aperture radar (SAR)

implementation has recently been proposedthe constellationof small spaceborne SAR systems. In this implementation,

several formation-flying small satellites cooperate to perform

multiple space missions. We investigate the possibility to

produce high-resolution wide-area SAR images and fine ground

moving-target indicator (GMTI) performance with constellation

of small spaceborne SAR systems. In particular, we focus on

the problems introduced by this particular SAR system, such

as Doppler ambiguities, high sparseness of the satellite array,

and array element errors. A space-time adaptive processing

(STAP) approach combined with conventional SAR imaging

algorithms is proposed which can solve these problems to some

extent. The main idea of the approach is to use a STAP-based

method to properly overcome the aliasing effect caused by the

lower pulse-repetition frequency (PRF) and thereby retrieve

the unambiguous azimuth wide (full) spectrum signals from

the received echoes. Following this operation, conventional SAR

data processing tools can be applied to focus the SAR images

fully. The proposed approach can simultaneously achieve both

high-resolution SAR mapping of wide ground scenes and GMTI

with high efficiency. To obtain array element errors, an array

auto-calibration technique is proposed to estimate them based

on the angular and Doppler ambiguity analysis of the clutter

echo. The optimizing of satellite formations is also analyzed,

and a platform velocity/PRF criterion for array configurations

is presented. An approach is given to make it possible that almost

any given sparse array configuration can satisfy the criterion by

slightly adjusting the PRF. Simulated results are presented to

verify the effectiveness of the proposed approaches.

Manuscript received January 2, 2004; revised July 13, 2005;released for publication September 14, 2005.

IEEE Log No. T-AES/42/2/876423.

Refereeing of this contribution was handled by P. Lombardini.

Authors current addresses: Z. Li, Z. Bao, G. Liao, National Lab of

Radar Signal Processing, Xidian University, Xian 710071, China,

E-mail: ([email protected]); H. Wang, Chinese University of

Hong Kong, China.

0018-9251/06/$17.00 c 2006 IEEE

I. INTRODUCTION

A new conceptual implementation of spacebornesynthetic aperture radar (SAR) systems wasproposed by Air Force Research Laboratory(AFRL) in 1997 [1, 2], that is, the constellationimplementation of small spaceborne SAR systems.In this implementation, several small satellites (ormicro-satellites) fly in a formation and operate as a

single virtual satellite.Constellation SAR systems have many advantagesover conventional spaceborne SAR systems. Theycan provide multiple, long baselines in single-passobservation mode, thus greatly improving theperformance of interferometric SAR (InSAR) andground moving-target indicator (GMTI). The coherentcombination of several SAR images obtained fromdifferent observing angles can improve the imageresolution and provide accurate geometric information[3, 4]. Furthermore, combining a broad illuminationsource with multiple small receiving antennas placedon separate formation-flying micro-satellites, we can

obtain high-resolution SAR images of wide areas[57]. Another reason for using the constellationSAR systems is to mitigate the cost, fabrication,and deployment issues associated with placing alarge satellite SAR into orbit [5]. Furthermore,the likelihood of system failure can be reducedsince failure generally occurs only to individualmicro-satellites, instead of a large satellite carryingan entire system.

However, several challenging problems are alsointroduced by the constellation SAR regime at thesame time. These problems should be carefullyconsidered when building the constellation SAR

system. The following briefly reviews these problemsand introduces the corresponding solutions.

A. Highly Sparse Array

Due to the high sparseness of the distributedsatellite array (to avoid the risk of collision, theinterval between any two satellites must be at least100 m) and the limited number of satellites, the beampattern formed by this array will have high sidelobelevels and even grating lobes which can degrade theperformance of GMTI (i.e., they can cause excessiveblind speed regions in which ground moving targetscannot be detected). It is challenging work to use thishighly sparse array to suppress the clutter and detectground moving targets with fine speed response.

A method for improving the beam pattern is thateach satellite respectively transmits an orthogonalsignal (multi-frequency/coded signal), and the echoesfrom every satellites transmission are coherentlyprocessed [1]. A technique of pattern synthesis inangle-frequency space for GMTI has been proposedin [8] which requires that all satellites be spaced

436 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 42, NO. 2 APRIL 2006

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regularly to achieve regular grating lobes distribution.A so-called scanned pattern interferometric radar(SPIR) approach has been proposed in [9] whichuses the high angular variability of a sparse arraypoint spread function (PSF) to collect sufficientdata from the ground return so that the clutter andground moving targets can be separated withouta priori assumption of the clutter statistics. Thedisplaced phase center antenna (DPCA) technique

can also be used to cancel the clutter [10, 11].Unfortunately, if the DPCA condition is met, thesatellite formation cannot be used to resolve the clutterDoppler ambiguities for unambiguous SAR imagingof ground scenes (the detailed explanation is given inSection V).

In this paper we first use temporal processing(temporal degrees of freedom) to divide the echospectra into many short chips (all of the chipsoccupying the same Doppler frequency are withinthe same Doppler bin). Thus, within each Dopplerbin, each of the spectrum chips (including those ofthe clutter and ground moving targets) is confined

into a narrow angle space; these spectrum chipsare completely separated from each other in spatialdomain (angle domain). Then we can easily usespatial beamforming techniques (even for the highlysparse array) to process the spectrum chips.

B. Range/Doppler Ambiguities

A basic limitation for the design of spaceborneSAR systems is the minimum antenna area constraint[1215] which can be expressed as

Amin

= 4vsr tan'=c (1)

where vs is the velocity of spaceborne platforms, is the wavelength, r is the slant range, c is the lightspeed, and ' is the incidence angle. The requirementarises because the illuminated area of the groundmust be restricted so that the radar does not receiveambiguous returns in range or/and Doppler. As aresult, all of the current spaceborne SAR systemshave huge antennas, such as ERS-1/2 (with antennaaperture of 10 m 1 m), Radarsat-1 (with antennaaperture of 15 m 1:5 m), and SIR-C/X-SAR (withantenna aperture of 12 m 4:4 m). Unfortunately, alarge antenna leads to the failure of the conventionalspaceborne SAR systems to map a wide area withhigh azimuth resolution either in the stripmap orspotlight mode. On the other hand, it is impracticalto put such a large radar antenna on a micro-satelliteplatform. Therefore the minimum antenna areaconstraint cannot be satisfied by individual smallantennas. However, we should emphasize that the totalreceiving antenna area (i.e., the sum of all physicalreceiving aperture areas) in the constellation mustsatisfy the minimum antenna area constraint to obtain

unambiguous fine-quality SAR images, which isassumed here. Under this assumption we investigateapproaches to wide-swath and high-resolutionSAR image generation from the constellation ofsmall spaceborne SAR systems. Since an individualantenna does not meet the minimum antenna areaconstraint, the range and/or Doppler ambiguitieswill be introduced into the echoes received by eachsatellite. To avoid range ambiguity, a lower PRF is

assumed throughout this paper, so we only need toovercome the Doppler ambiguity effects.

The maximum likelihood (ML) filter and minimummean-squared error (MMSE) filter (using the entiretime, frequency, and spatial measurements) have beenused to resolve the problem of the range-Dopplerambiguities and to achieve the wide-area SAR image[5]. However, calculating the ML and MMSE filterswould require the inversion of an extremely largematrix, which is unrealistic. The authors of [5] havealso suggested that the SAR processing be firstperformed for each receiving channel, and then spatialprocessing be used to reject ambiguous targets foreach image pixel. However, several problems stillremain unsolved, such as heavy computational burden(due to the multiplication of SAR processing bythe number of satellites) and more spatial degreesof freedom required to reject the more ambiguoustargets (the number of ambiguous targets within eachpixel is twice the number of Doppler ambiguities,see [6]) within each pixel of the full-resolution SARimages. A reconstruction algorithm based on thesampling theorem has been investigated to recoverthe unambiguous Doppler spectrum [6]. Aguttes hasintroduced a totally different approach to solve the

problems of the Doppler ambiguity and nonuniformspaced satellites by using spread spectrumwaveforms [7].

In this paper we present a novel STAP-basedapproach to resolve the Doppler ambiguities byretrieving the unambiguous full spectrum of the clutterand wide spectra of ground moving targets. Thus theground moving target signals can be separated fromthe clutter. Then conventional SAR data processingtools can be directly applied to focus fully the SARimages of ground scenes and ground moving targets.Using this approach, we can simultaneously achievethe unambiguous high-resolution SAR images of wide

areas and GMTI with high efficiency.

C. Array Element Errors

To combine the echoes coherently from everysatellite, we need to know the array manifold exactly.However, it is almost impossible for the currentmeasurement techniques to collect the informationof each array element accurately, such as gain-phaseerror, position uncertainties, timing uncertainty, etc.,

LI ET AL.: PERFORMANCE IMPROVEMENT FOR CONSTELLATION SAR USING SIGNAL PROCESSING TECHNIQUES 437

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especially for the highly sparse satellite array withoperating frequency at X band.

A data-dependent array auto-calibration techniqueis proposed here to estimate the array element errorsbased on the space-time analysis of the clutter echo,which possesses both the advantages (such as fastconvergence) of source-calibrations [16, 17] and those(such as no need for artificial pilot sources) of selfcalibrations [18, 19]. As mentioned above, the clutter

echo has multiple Doppler ambiguities; the clutterspectrum chips (after processed by using azimuthfast Fourier transform (FFT)) within a Doppler binare used as the array calibration sources in our arraycalibration technique. To distinguish the differenterror contributions to array steering vectors, wemodel the three-dimensional position errors as twophase errors. One is fixed for all echoes from theantenna main beam; thus it can be incorporated intothe inherent element gain-phase error. The other isvariable for different spectrum chips; therefore itmust be estimated, respectively, for different spectrumchips. If the timing uncertainty can also be modeled

as a fixed phase error, the auto-calibration techniquehere can also provide an estimation of the timinguncertainty. The proposed array auto-calibrationtechnique can converge quickly, and the correspondingcomputational burden can be extremely decreased.

The succeeding sections of this paper are arrangedas follows. Section II presents the mathematical modeland characteristics of echoes. In Section III, theprocessing approach to SAR and GMTI is presentedin detail. The array auto-calibration technique ispresented in Section IV. The satellite formationoptimization is discussed in Section V. Section VIshows the simulated results with our conclusions in

Section VII.

II. ECHO MODEL

A. Coordinate System

The coordinate system of a constellation is definedas follows: X-axis is in the direction of the satellite(reference satellite) velocity vector; Z-axis is awayfrom the center of the Earth; Y-axis is along theorbital angular momentum vector. The (X, Y, Z)directions can also be referred to as (along-track,

cross-track, radial) or (front, left, up). The spaceborneSAR systems usually look at the right side. Thecoordinate system is shown in Fig. 1. The angles , ',and shown in Fig. 1 are referred to as the azimuthangle, incidence angle, and cone angle, respectively,and their relation can be given by

sin = sin sin': (2)

The Earth is assumed to be stationary in thiscoordinate system, i.e., the coordinate system is an

Fig. 1. Coordinate system of constellation SAR systems.

Earth reference frame. To simplify the analysis, weassume that all satellites in the constellation only havean identical along-track velocity vs. In fact, the othervelocity components are very small relative to thealong-track velocity if appropriately designing the

orbits [7], and the effects caused by them can alsobe easily compensated. The objective of the workpresented here is to present our processing approachesin an easy manner, so these practical considerationsare neglected temporarily.

To simplify the mathematical model, it is importantto give the definition of equivalent phase center firstly.

DEFINITION Two separate transmitting and receivingantennas can be equivalent to one transmitting andreceiving antenna, defined as the equivalent phasecenter herein, which is located at the midpointbetween the separate transmitting and receivingantennas, after compensating a constant phaseexpfj2d2=(4r)g (where d is the distance betweenthe transmitting and receiving antennas, r is theslant range from the equivalent phase center to theilluminated ground surface, and is the wavelength).In practice, the phase compensation can be performedfor range segments one by one. In each rangesegment, the phase factor can be approximated tobe a constant, i.e., r can be approximated to beconstant. This definition holds only when the distancebetween transmitter and receiver is small enoughcompared with their distance to the surface of theEarth. It is assumed here that this definition is valid

for typical constellation SAR system parameters. Forexample, even if the distance between transmittingand receiving antennas is as large as 1000 m, themaximum phase error caused by this approximationis still less than 1 for typical spaceborne SAR systemparameters (as listed in Table I).

Based on the definition of equivalent phasecenter, it is assumed throughout this paper that thecoordinates of each satellite are given according toits equivalent phase center, i.e., each equivalent phase


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center serves as both transmitter and receiver, althoughonly one satellite is used as the transmitter in theconstellation.

B. Mathematical Model

Assume that the coordinates of the mth (m =1,2, : : : ,M and M is the number of satellites inthe constellation) satellite are (xm,ym, zm) at time

t = 0 (for the reference satellite, its coordinates are(x1,y1,z1) = (0,0,0)), and (xm + vst,ym,zm) (vs is thesatellite along-track velocity) at time t. The clutterecho received by the mth satellite at time t can bewritten as

sc,m(, t) =

Z Z(x,y)h

2rc,m(x,y,z, t)

c

g

t

x xmvs

ej4rc,m(x,y,z,t)=dxdy

(3)where

rc,m(x,y, z, t) =p

(x xm vst)2 + (y ym)

2 + (z(x,y) zm)2

(4)

h() = a()ejkr2

(5)

and t and denote the azimuth slow-time andrange fast-time, respectively, c is the light velocity, is the wavelength of the carrier, (x,y) is thecomplex reflectivity per unit area (surface scattering)at a ground reflecting cell whose coordinates are(x,y,z(x,y)) (in the following, we directly use thenotation z instead of z(x,y)), rc,m(x,y, z, t) is theslant range from the equivalent phase center of themth satellite to the ground cell, h() is the complextransmitted pulse (linear FM waveform), a() isthe complex envelope of the transmitted pulse,and g(t) represents the antenna pattern and othertime-variant characters (identical to all the receivingantennas).

Assume that the constellation SAR system operatesin the side-looking mode, and the along-track lengthof the illuminated ground region is significantly lessthan the slant range, i.e., rc,m(x,y, z, t) jx vst xmj.Moreover, we assume here that ym and zm are as smallas possible, such as only 10 m or smaller (i.e., the

SAR train as called in [7]). Thus rc,m(x,y,z, t) can beapproximated as

rc,m(x,y, z, t) y ymp

(x xm vst)2 +y2 + z2

+z zmp

(x xm vst)2 +y2 + z2

+p

y2 + z2 +(x vst xm)

2

2p

y2 + z2

= ym cosc,m(x,y, z, t)sin'c,m(x,y, z, t)

zm cos'c,m(x,y,z, t)

+p

y2 + z2 +(x vst xm)

2

2p

y2 + z2(6)

where

sinc,m(x,y,z, t) =x xm vst

p(x xm vst)2 +y2

x xm vstjyj

(7)

sin'c,m(x,y,z, t) =

p(x xm vst)

2 +y2p(x xm vst)

2 +y2 + z2

jyjp

y2 + z2

=sin 'c(x,y, z): (8)

and c,m(x,y, z, t) and 'c,m(x,y,z, t) are theinstantaneous azimuth and incidence angles (as shown

in Fig. 1) of the clutter cell (x,y,z) relative to the mthsatellite, respectively. The error can be neglected fortypical spaceborne SAR system parameters.

Using (6) in (3) gives

sc,m(, t) =

Z Z(x,y)h

2rc,m(x,y,z, t)

c

g

t

x xmvs

ejc,m(x,y,z,t)dxdy (9)

where

c,m(x,y,z, t)

=4

(ym cosc,m(x,y, z, t)sin 'c(x,y, z) zm cos'c(x,y, z))

+4

py2 + z2 +

2v2s

t

x xmvs

2p

y2 + z2: (10)

We can see from (10) that the echo signal from theclutter cell at (x,y,z) is also a linear FM signal in theazimuth time domain with the azimuth chirp rate of

kdm(x,y,z) = 2v2s

p

y2 + z2: (11)

Using (11), the instantaneous Doppler frequencyfdm(x,y, z, t) of the clutter cell at (x,y,z) relative to themth antenna can be expressed as

fdm(x,y,z, t) = kdm(x,y, z)

t

x xmvs

= 2v2s

p

y2 + z2

t

x xmvs

: (12)

According to (12) we can obtain the instantaneousazimuth time tm(x,y,z,fd) corresponding to the


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Doppler frequency fd as

tm(x,y,z,fd) =x xm

vs

p

y2 + z2

2v2sfd: (13)

Using (13) in (7) gives

sinc,m(x,y,z,fd) =p

y2 + z2

2vsjyjfd

=

2vs sin'c(x,y, z)fd

=sin c(x,y, z,fd): (14)

c(x,y, z,fd) denotes the instantaneous azimuth angleof the clutter cell (x,y, z) relative to each satellite(i.e., identical to all satellites) when the instantaneousDoppler frequency is fd. If the illuminated groundis flat (or ym and zm are very small), we can directlyobtain 'c(x,y,z) (or by approximating) accordingto the range bin . That is, if the range-Doppler binfd is given, the incidence angle 'c(x,y,z) andthe azimuth angle c(x,y,z,fd) can be calculated

according to (8) and (14), respectively. The incidenceangle and the azimuth angle are independent of thealong-track position of each clutter cell, i.e., all theclutter cells within the same range-Doppler bin have(approximately) identical azimuth and incidenceangles. Therefore, for the sake of simplicity, we usethe notations 'c() (instead of 'c(x,y, z)) and c(,fd)(instead of c(x,y, z,fd)) to denote the azimuth andincidence angles corresponding to the range-Dopplerbin fd, respectively.

Transforming (9) into the Doppler domain (usingthe stationary phase method), we can obtain

Sc,m

(,fd

) = ej(4=)dc,m(,fd )Ac,m

(,fd

) (15)

where

Ac,m(,fd) =

Z ZS0c,m(x,y, z,fd)dxdy (16)

S0c,m(x,y,z,fd) = (x,y)h

2rc,m(x,y, z,fd)

c

G(fd)ej 0c(x,y,z,fd ) (17)

0c(x,y,z,fd) = 2fdx

vs+

4

py2 + z2

p

y2 + z2

2v2sf2d

(18)d

c,m(,f

d) = x

msin

c(,f

d)sin '

c()

+ym cosc(,fd)sin 'c() + zm cos'c()

(19)

rc,m(x,y,z,fd) = ym cosc(,fd)sin 'c() zm cos'c()

+p

y2 + z2 +2p

y2 + z2f2d8v2s

(20)

and G(fd) is the Fourier transform of g(t) andrc,m(x,y,z,fd) (substituting (13) into (6)) denotes

the slant range from the clutter cell (x,y, z) tothe mth antenna when the instantaneous Dopplerfrequency is fd. Because rc,m(x,y, z,fd) are differentfor different satellites, we must coregister theclutter echoes received by all satellites beforeperforming the following coherent processing. Wecan easily coregister the clutter echoes by shifting therange-Doppler output Sc,m(,fd) of the mth satelliteby ym cosc(,fd)sin'c() + zm cos'c() in the range

direction. After envelope is coregistered, the followingrelationships will hold:

rc,1(x,y, z,fd) = rc,2(x,y,z,fd) = = rc,M(x,y,z,fd)

= rc(x,y,z,fd) (21)

Ac,1(,fd) = Ac,2(,fd) = = Ac,M(,fd)=Ac(,fd):

(22)

As a result, there is only one phase difference

ej(4=)dc,m(,fd)

between the mth satellite and the

reference satellite from the same range-Doppler bin

output. It is important to note that the first term in theright side of (19) dose not arise from the plane waveapproximation. In other words, even if xm is as largeas possible, (19) will still be valid. Since only onesatellite is used as the illuminator and all the satellitesoperate in the side-looking mode, we must restrict thealong-track baseline lengths to ensure that the beamsof all the satellites can share enough common regions.However, the second and third terms do arise from theplane wave approximation which requires that ym andzm must be as small as possible (such as 10 m). Wecan note from (15) that there are numerous groundclutter cells within each range-Doppler bin fd,and they have the identical azimuth angle c(,fd) andincidence angle 'c(), i.e., the identical array steering

vector

1, ej(4=)dc,2(,fd ), : : : , ej(4=)dc,M(,fd )T

(where ( )T

denotes the vector transpose operator). Therefore, wecan simultaneously process all the ground clutter cellswithin each range-Doppler bin using array processingtechniques.

The echo from a ground moving target can bewritten as

st,m(, t) = th

2rt,m(x,y, z, t)

c

g

t x xmvs

ej4rt,m(x,y,z,t)= (23)

where t is the complex reflecting magnitude (RCS)of the ground moving target and rt,m(x,y, z, t) is theslant range from the mth satellite to the groundmoving target at (x,y,z). We can write rt,m(x,y,z, t) as

rt,m(x,y,z, t) =p

(x xm vst vatt)2 + (y ym)

2 + (z zm)2

vctt cosc,m(x,y, z, t) (24)


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where vat and vct are the along-track and cross-trackvelocities of the ground moving target, respectively.Similarly as what we have done for the clutter echo,we also transform (23) into the Doppler domain andcoregister the echo envelopes; then we can obtain

St,m(,fd) = ej(4=)dt,m(,fd )At(,fd) (25)

where

At

(,fd

) = t

h 2rt(x,y,z,fd)c

Gt

(fd

)ej0t(x,y,z,fd)

(26)dt,m(,fd) = xm sint(,fd)sin't()

+ym cost(,fd)sin 't() + zm cos't()

(27)

Gt(fd) = G

fd

2vct cost(,fd)

(28)

0t(x,y, z,fd) = 2

fd

2vct cost(,fd)

x

vs + vat

+4

p

y2 + z2 py2 + z22(vs + vat)

2

fd

2vct cos t(,fd)

2(29)

rt(x,y, z,fd) rc

x,y, z,fd

2vct cost(,fd)

(30)

t(,fd) c

,fd

2vct cos t(,fd)

(31)

't() = 'c() (32)

t(,fd) c,fd 2vct cosc(,fdc)

(33)and fdc (fdc = 0 in the side-looking mode) is theclutter Doppler center frequency.

The outputs from the range-Doppler bin fdmay include the clutter, the noise, and possibly aground moving target, which can be written in asimple form as follows:

Sm(,fd) = Sc,m(,fd) + St,m(,fd) + nm(,fd)

= ej(4=)dc,m (,fd )Ac(,fd)

+ ej(4=)dt,m(,fd )At(,fd) + nm(,fd)

(34)where nm(,fd) is the additive complex white noise.It is the phase terms in (34) that are used in thefollowing spatial processing.

C. Characteristics of Echoes

Before presenting the processing approaches, it ishelpful to first introduce the characteristics of echoes(including the clutter and ground moving targets)received by each small antenna.

Fig. 2. Space-time spectra of clutter and ground moving target.

(a) Unambiguous. (b) Doppler-ambiguous.

When a spaceborne SAR system operates in theside-looking mode, the relation (i.e., (14)) between theDoppler frequency fd of a ground clutter cell and thecorresponding cone angle (in order to simplify the

equations, we substitute the notation for c(,fd))can be given by

fd = 2vs sin=: (35)

Assume that the azimuth mainlobe width of anindividual small antenna equals 0.015 rad, the satellitevelocity vs = 7000 m/s, and the wavelength =3 cm; then the clutter Doppler frequency comingfrom the antenna mainlobe is confined within[3500 Hz,+3500 Hz]. The clutter Doppler frequency

fd is directly proportional to sin , so the clutterspectrum can be shown by the thick line in the

fd-sin plane (also referred to as the space-time plane)

of Fig. 2(a).The PRF fr is determined by the total receivingantenna length (i.e., the sum of the physical receivingantenna lengths) Ltotal in azimuth, and it must satisfy

fr 2vs=Ltotal: (36)

To avoid range ambiguity, a lower PRF such asfr = 1400 Hz is assumed here. The lower PRFwill introduce azimuth (Doppler) ambiguities. Inthis example, there will exist five azimuth anglesfor the same Doppler frequency, as shown by thefive thick lines in Fig. 2(b). We cannot obtain asatisfactory SAR image without resolving the Doppler

ambiguities.For a ground moving target, its Doppler frequency

fd and cone angle are related by the followingequation:

fd =2(vs + vat)

sin +

2vct cos

: (37)

The second term in the right side of (37) is anadditional Doppler frequency shift caused by thecross-track motion of the ground moving target. Thisterm indicates that for the same cone angle , the


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Fig. 3. Processing procedures for unambiguous SAR mapping of ground scenes and GMTI.

Doppler frequency of the ground moving target hasan additional offset relative to that of the clutter; in

other words, for the same Doppler frequency fd, theground moving target is at a different cone angle fromthe clutter. The spaceborne SAR system has somefeatures which can be used to simplify the processingof GMTI. First, the satellite velocity is very high.Assuming that the satellite velocity vs = 7000 m/s, theslant range r = 1000 km, the wavelength = 0:03 m,and the antenna length (in azimuth) D = 2 m, thenthe full aperture time illuminating a ground movingtarget is only r=(vsD) 2 s. Thus, we can make theassumption that the velocity of a ground target doesnot change during the whole dwell time. Second, theazimuth mainlobe width is so narrow that the secondterm in the right side of (37) can be approximatedas a constant term in the side-looking mode. Third,the Doppler chirp rate of a ground moving target isapproximately equal to that of the clutter becausethe along-track velocity of a ground moving targetis extremely small compared with that of a satelliteplatform. Therefore we can use the Doppler chirprate of the clutter to focus the ground moving targetechoes. The thin lines shown in Fig. 2 represent thespace-time spectrum of a ground moving target. Thespace-time spectrum of a ground moving target isseparated from that of the clutter due to the Doppler

frequency shift arising from the cross-track velocity,as shown in Fig. 2. Note that the position of theclutter spectrum in the space-time plane is fixed, whilethe spectrum of a ground moving target may be at anyposition depending on its cross-track velocity.

III. PROCESSING APPROACH

We first give the definition of spectrum componentwhich is used extensively in the following.

DEFINITION The space-time spectrum lines shown inFig. 2(b) can be divided into many short segments

(chips) by using the azimuth FFT, each of whichis defined as a spectrum component (i.e., spectrumchips) in this paper, such as those in the circlesmarked in Fig. 2(b).

The proposed approach first extracts all spectrum

components (including those of the clutter and groundmoving targets) from the space-time plane as shown

in Fig. 2(b) by means of space-time beamforming(in practice, the spatial beamforming and temporalbeamforming are performed separately). Thus, thespectrum components of ground moving targets

can be separated from those of the clutter. Then theresolved clutter spectrum components are rearrangedto obtain the unambiguous full spectrum of the

clutter as shown by the thick line in Fig. 2(a),and the resolved spectrum components of ground

moving targets are also rearranged to obtain the fullspectra of ground moving targets as shown by thethin line in Fig. 2(a). Finally, the wide-swath andhigh-resolution SAR image can be obtained from

the unambiguous clutter full spectrum by usingconventional SAR imaging operations. To improvethe output signal-to-noise ratio (SNR), the full spectra

of ground moving targets can also be processed

by using SAR imaging operations followed byconstant-false-alarm-rate (CFAR) operation. The basicprocedures are illustrated in Fig. 3.

The proposed approach is comprised of the

following four steps.Step 1 Temporal ProcessingThis step divides the echoes received by each

satellite into many space-time spectrum components

(i.e., spectrum chips), which is necessary to thefollowing spatial processing.


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An azimuth FFT is used to divide the echoesreceived by each satellite into many spectrumcomponents. Within each Doppler bin, each of thespectrum components (including those of the clutterand ground moving targets) is confined into a narrowangle space. Moreover, these spectrum componentsare completely separated from each other in the spatialdomain (angle domain). In the example shown inFig. 2(b), there are five spectrum components of the

clutter and five ones of a ground moving target withineach Doppler bin.

As given in (16), the spectra of all clutter cells(within the same range bin) are superimposed in thespace-time plane after performing the azimuth FFT,with linear phase differences corresponding to theiralong-track positions. That is, these clutter cells withineach spectrum component have an identical arraysteering vector. As a result, the spatial processing ofeach spectrum component is equivalent to processingall the clutter cells within this spectrum component,which is one of the reasons why the approach is moreefficient.

No other processing needs to be performed beforethe azimuth FFT. The SAR imaging operations (suchas range migration compensation, range compression,etc.) will be performed after the unambiguous fullspectra of the clutter and ground moving targets areretrieved.

Step 2 Spatial ProcessingThis step extracts all spectrum components from

the space-time plane.Spatial beamforming techniques are used to

extract all spectrum components from the space-timeplane. To extract a clutter spectrum component,

we steer the array mainlobe (i.e., the array steeringvector) to the direction of the clutter spectrumcomponent and place nulls in the directions of allthe other clutter spectrum components within thecorresponding Doppler bin; thus the desired clutterspectrum component can be extracted. The otherclutter spectrum components within this Dopplerbin can also be extracted by using the same method.Similarly, to extract a spectrum component located inthe ground moving target region, the array mainlobeis steered to it and all the clutter spectrum componentswithin the corresponding Doppler bin are nulled out;thus the desired moving target spectrum component

can be extracted. To extract a spectrum component,we only need to constrain the array beam pattern inthe five or six spectrum component directions withinthe corresponding range-Doppler bin, which canbe easily realized even for a highly sparse satellitearray. Because the coherent processing is requiredin the following steps, the phase and amplitudeinformation of all the spectrum components mustbe kept unchanged in this step. The way to preservethe phase and amplitude information of all the

resolved spectrum components will be detailed in thefollowing.

In the example shown in Fig. 2(b), within eachrange-Doppler bin there are five clutter spectrumcomponents and five spectrum components of aground moving target, and each spectrum componentcomes from a different cone angle. The array steeringvectors of the clutter and ground moving targetspectrum components can be given by

pic(,fd) =h

1, ej(4=)dc,2(,fd +ifr ), : : : , ej(4=)dc,M(,fd+ifr )iT

(38)and

pit(,fd) =h

1,ej(4=)dt,2(,fd +ifr), : : : ,ej(4=)dt,M(,fd +ifr)iT

(39)

respectively, where i = I,I+ 1, : : : ,I 1,I (2I is thenumber of the Doppler ambiguities), fr (fr=2 fd 2 > > 2I+1 2I+2 = = M =

2n , and um

is the eigenvector associated with the eigenvalue m.

2) Each column ofB=P is orthogonal to each

column of matrix U = [u2I+2, u2I+3, : : : , uM].

The estimate algorithm comprises two steps. Inthe first step, fixing dim (initial value is zero), weestimate the gain-phase error gme

j0m of each arrayelement. In the second step, using the result of thefirst step, i.e., holding the gain-phase error fixed, weestimate xm coso ym sino. The algorithm iteratesalternatively between the two steps to find the finalsolution, as summarized below.

Step 1 Estimating the gain-phase error [18]Definition of cost function:

Jc =IX

i=I

kUHpik (68)

where U represents the estimate of the matrix U. If

U were a perfect estimate of U then the minimumvalue of Jc would be achieved for the true and p

i.Thus, the minimization of Jc can provide estimates of (i.e., gme

j0m ) and pi (i.e., xm coso ym sino).

The estimate of can be obtained by minimizingthe cost function Jc:

min

(Jc) = min

(IX

i=I

piH

HUUHpi

)

= min

(IX

i=I

HQi

H

UUHQi

)

= min

(

H

"IX

i=I

QiH

UUHQi

#

)(69)

where

= [11, 22, : : : ,MM]T (70)

Qi = diagfpig: (71)

and diagfpig denotes a diagonal matrix whosediagonal is the vector pi. To place the unknownparameters at the most outer sides of the aboveexpression, the equation pi = Qi is used in (69).Therefore, the optimum solution of the abovequadratic minimization problem under linear


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constraint Hw = 1 (where w = [1,0,0, : : : ,0]T) can bedirectly obtained by

= Z1w=(wTZ1w) (72)

where

Z=

IXi=I

QiH

UUHQi: (73)

Then the gain-phase errors gm

ej0m are given by

= diag(): (74)

Thus, we can compensate the gain-phase error(including the inherent gain-phase error and thosearising from the timing and relative positionuncertainties) at each array element according to thediagonal element of .

We usually choose the clutter spectrumcomponents in the Doppler center bin (i.e., fd = 0in the side-looking mode) as the calibration sources.From the computer simulations we find that evenif we ignore the effect of xm coso ym sino(i.e., assume xm coso ym sino = 0 in the firststep), the estimate accuracy of the phase error ej

0m

is still satisfactory (less than 1). Furthermore, tofurther reduce the effect of noise, we can average theestimates obtained from other Doppler bins.

Step 2 Estimating xm cos o ym sinoHolding fixed, we then estimate xm coso

ym sino.The array steering vectors pi of the calibration

sources can be decomposed into the following twoparts:

pi = pi0 + Pi1d (75)

where

pi0 =h

1, ej4di2o

=, : : : , ej4di

Mo=iT

(76)

Pi1 = diag

1,j

4 sini sin'

ej4d

i2o

=, : : : ,

j4 sini sin'

ej4d

iMo

=

(77)

d =

0, x2 coso y2 sin o, : : : ,

xMcoso yMsinoT

: (78)

We rewrite the cost function of (68) as follows:

Jc = kE Fdk (79)

where

E = [eTI eTI+1 e

TI ]

T (80)

F = [fTI fTI+1 f

TI ]

T (81)

ei = UHpi0 (82)

fi = UHPi1: (83)

The least square solution of (79) can be obtained by

d = realf(FHF)1FHEg: (84)

If the accuracy of Taylor-series expansion of(55) cannot meet practical requirements, however,much higher estimation accuracy will be achieved byincreasing iteration number, and this can also furthermitigate the rigorous requirement on the measurementaccuracy.

The optimum estimates of gmej0m and xm coso

ym sino can be directly obtained by (72) and(84). Therefore the computational burden is reducedsignificantly.

V. SATELLITE FORMATION OPTIMIZATION

To retrieve the unambiguous full clutter spectrumfrom the Doppler-ambiguous ground echoes byusing the processing approach in Section III, thesatellite arrays must meet some constraints. Thissection is to investigate the array configuration

requirements.Throughout this paper, a satellite array with very

short cross-track baselines (on the order of 10 m) isassumed. In fact, if the illuminated ground surfaceis flat or it can be known accurately, the proposedapproaches can still apply to an array with longcross-track baselines. However, the illuminatedground terrain is usually not flat and usually notknown accurately, which means that the arraysteering vectors of spectrum components are alsounknown for an array with long cross-track baselines.Another problem introduced by an array with long

cross-track baselines is the envelope coregistration inthe presence of Doppler ambiguities. Therefore, theprocessing approaches suffer seriously from a largethree-dimensional array. The shorter the cross-trackbaselines, the less effect of a rugged terrain. Forsuch a reason, a sparse linear array (with veryshort cross-track baselines) in along-track directionrelative to the ground is the ideal formation for SARmapping and GMTI applications. However, evenfor an along-track sparse linear array, there are stillsome configuration constraints that must be satisfiedfor retrieving the unambiguous full clutter spectrum.The mathematical analysis of the array configurationconstraints is given in the following.

In fact, xm can be decomposed into the followingtwo parts:

xm = kmvsTr + xm (85)

where km is an integer, Tr is the pulse repetitionperiod, and vsTr=2 xm < vsTr=2. If ym and zm arevery small (or even zero), we can rewrite the arraysteering phase ej4d

im= of the ith clutter spectrum

component of the range-Doppler bin fd as (using


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(85))

ej4dim = = ej4=(xm sin

i sin'+ym cosi sin'+zm cos')

= ej2(fd +ifr)(xm=vs)+j(4=)(ym cosi sin'+zm cos ')

= ej2fd kmTr ej(4=)xm sini sin'+j(4=)(ym cos

i sin'+zm cos')

ej2fd kmTr+j(4=)(ym coso sin '+zm cos')ej(4=)xm sin i sin':

(86)

Note that only the second exponential term in (86)relates to the ith spectrum component. That is, thearray steering vectors of clutter spectrum componentsonly depend on xm but are independent of kmvsTr.Equation (86) means that no matter how long analong-track linear array is, only its equivalent shortarray with the maximum length of vsTr is virtuallyvalid for resolving the Doppler ambiguities. Werefer here to the array with the element positions(along-track) of [0,x2, : : : , xM] as the compressedarray. Therefore, to evaluate the Doppler ambiguityresolving performance of a large sparse array is

equivalent to evaluating that of its correspondingcompressed array. Since the compressed array isalso a conventional array as indicated by (86), itsconfiguration requirements for resolving sources(i.e., spectrum components in this paper) in differentdirections are well known; that is, the array elementsshould be well distributed in the interval [0, vsTr].

In practical processing we can firstly delaythe pulses received by each satellite in slow time(azimuth time) domain to obtain the compressed arraybefore performing the SAR and GMTI processingin Section III; i.e., the pulses received by the mthsatellite are delayed by kmTr. This is equivalent to

changing the along-track position of the satellite fromxm to xm = xm kmvsTr.

In fact, the DPCA technique can also be usedfor GMTI. The DPCA technique requires that thealong-track intervals of satellites must be kmvsTr (km isan integer). Unfortunately, when the DPCA conditionis satisfied, the along-track length of the compressedarray is zero (i.e., xm = 0). As discussed above, thiskind of satellite formation does not hold the abilityto retrieve the unambiguous full clutter spectrum forSAR imaging of ground scenes. Therefore, the DPCAformation is not a good choice.

If a given satellite formation cannot meet theabove array configuration requirements, what shouldwe do? A straightforward method is to change thesatellite formation by thrusters. Another methodis to slightly adjust the PRF. From (85) we cansee that xm depends not only on xm, but also onTr, so we can change xm by slightly changingthe PRF. For example, assume that the arrayelement positions (along-track) of a given formationare 0 m, 50.0627 m, 100.3473 m, 150.3043 m,200.4931 m, 250.1518 m, 300.4489 m, 350.2959 m,

and 400.2448 m, respectively. For vs = 7481:5 m/s andfr = 1496 s (i.e., vsTr = 5:001 m), the correspondingelement positions (along-track) of the compressedarray are 0 m, 0.0526 m, 0.3272 m, 0.2742 m,0.4529 m, 0.1016 m, 0.3887 m, 0.2257 m, and0.1646 m, respectively. Because the compressed arrayis too short (only 0.5 m), it is difficult to resolve theDoppler ambiguities. If the PRF is changed as fr =1478 Hz, then the corresponding element positions

of the compressed array will become 0 m, 4.5055 m,4.1710 m, 3.5090 m, 3.0787 m, 2.1183 m, 1.7963 m,1.0242 m, and 0.3541 m, respectively. Now, thecompressed array can satisfy the configurationrequirements.

For SAR and GMTI applications, we require thesatellite formation be an along-track linear array (orwith very short cross-track baselines). Unfortunately,this kind of satellite formation cannot be used tocollect the topographic height information. Thisproblem can be solved by two along-track lineararrays with a long cross-track interval betweenthem. The ground echoes received by each linear

array are processed separately to produce twowide-swath, high-resolution, complex SAR imagesof the same ground scene, and then the obtained twocomplex SAR images can be processed by using theinterferometric processing to retrieve the topographicheight information [27].

VI. SIMULATED RESULTS

The simulated constellation comprises 9 smallsatellites, all of which have identical small antennas.In this implementation, one small satellite is usedas both illuminator and receiver, while all the other

small satellites only receive the echoed signals. Wealso assume they operate in the side-looking mode.The key simulation parameters of the constellationSAR system are listed in Table I. The measuredcoordinates of each satellite and the correspondingerrors are given in Table II (equivalent phase centers).To verify the ability of the proposed approach toresolve Doppler ambiguities and achieve high qualitySAR images, we use a real SAR image (obtained byour airborne SAR system) as the simulated groundscene (flat surface) and generate the clutter returnusing the parameters listed in Table I. The clutterreturn from the simulated ground scene is receivedby all the satellites. In the following we use the clutterto verify the techniques of array auto-calibration andunambiguous SAR mapping of ground scene.

We first verify the performance of the arrayauto-calibration technique. To estimate the covariancematrix of (65), 200 samples were used. All thesamples were obtained from 20 adjacent range bins,with 10 samples obtained from each range bin. Themethod for obtaining 10 samples from each rangebin is described as follows. The total number of pulses


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TABLE I

Main Simulation Parameters of Constellation SAR System

Orbit Altitude Incidence Angle Band Antenna Size Bandwidth PRF Satellite Velocity

750 Km 45 X-band 2 m 1 m 100 MHz 1496 Hz 7481.5 m/s(along-trackcross-track)

TABLE II

Measured Coordinates and Corresponding Errors

Measurement Measurement Measurement Error Error Error

No. of Satellite Value xo (m) Value yo (m) Value zo (m) x (m) y (m) z (m)

1 0 0 0 0 0 0

2 41.0610 2.4275 1.5671 0.2088 0:2192 0.03953 96.5630 0.2871 2:6572 0:1014 0.1624 0:1293

4 145.5115 3:4532 4.7626 0.1463 0:1417 0.21495 199.0970 1.6433 5.0979 0.2110 0.1179 0.0567

6 252.0825 4.4777 2:1979 0.0601 0.1295 0.15097 307.8350 3:8661 1.0681 0:1595 0:0628 0:1541

8 354.5835 3.9742 4.4865 0:0865 0.1168 0:20139 413.3735 4:5444 4:8235 0.0346 0.1412 0.1642

used for estimating the covariance matrix is 1519 (i.e.,the total azimuth time is about 1.0 s). The length ofeach azimuth FFT is 1024 (pulses) and the intervalbetween two adjacent FFTs is only 55 pulses (i.e.,969 overlapping pulses). We assume that there is nothe inherent gain-phase error (due to the fact thatthe estimate of ejm is the same as the estimate ofej4dm= and the estimate of gm is unnecessary tothe proposed adaptive processing approach) at eacharray element and the clutter-to-noise ratio (withineach Doppler bin) is 10 dB. The estimate results (onetrial) of the array auto-calibration technique are shownin Table III. The estimate accuracy is on the order of

1

which can meet our practical requirements.After estimating and compensating the position

errors for each satellite, we then verify the abilityof the proposed processing approach to resolvethe Doppler ambiguities. Shown in Fig. 4(a) is theresult of SAR imaging without compensating theposition errors, while shown in Fig. 4(b) is the resultwith compensating the position errors. Fig. 4(b)demonstrates that the proposed processing approachcan recover the unambiguous SAR image of thesimulated ground scene. The performance of GMTIis also investigated with the same clutter from thesimulated ground scene. A ground moving scatterer

is placed in the ground scene with its RCS equal tothe average RCS of the clutter SAR pixels (i.e., theinput signal-to-clutter ratio is 0 dB) and its cross-trackvelocity (no along-track velocity) is in the range of[025] m/s. The input clutter-to-noise ratio (withinthe SAR image) is 45 dB (somewhat higher thanthat in practical systems, for example, 30 dB). Theechoes from the simulated ground scene (including thestationary pixels and the moving target) are receivedby all the satellites. As mentioned in Section III,

Fig. 4. SAR processing of simulated scene return using proposed

approach. (a) Without compensating the position errors.

(b) With compensating the position errors.

since the velocities of ground moving targets areusually unknown (i.e., the array steering vectors areunknown), we need to search the whole velocity range(generally [060] m/s) for ground moving targets.In this simulation, we search the velocity range of[025] m/s at the interval of 0.5 m/s, although wedo know the true velocity of the ground movingtarget. The signal-to-clutter+noise-ratio (SCNR)


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TABLE III

Estimates of Position Errors

No. of720

dm dm

720dim d

im

(i = 2)

Satellite (degree) (degree)

1 0 0

2 0.05 1.48

3 0.20 1:854 0.60 0:97

5 0:01 1.196 0.88 0.80

7 0:17 1:318 0:31 0.839 0:44 1:58

Fig. 5. SCNR improvement factor velocity response.

improvement factor (IF) versus the cross-trackvelocity is shown as the curve (also referred to asthe velocity response) in Fig. 5. The SCNR IF isdefined as the output SCNR of a system divided bythe SCNR at the input of the system.

From Fig. 5 we can see that a fine IF curve canbe achieved. Note there is a null at the cross-trackvelocity of 22.5 m/s in Fig. 5. This is because the

space-time spectrum of the ground moving targetwill be at the same position in the space-time planeas the clutter spectrum if the cross-track velocityequals to 22.5 m/s; i.e., they cannot be separated inthe space-time plane. This area is referred to as theblind velocity area in literatures. Except the blindvelocity area, fine IF can be achieved by using theapproach.

VII. CONCLUSIONS

Future spaceborne SAR system will be requiredto achieve wide area surveillance, not only mappingof wide ground scenes with high resolution but alsodetecting ground moving targets. The constellationSAR system is a promising choice. However, someproblems impede the implementation of constellationSAR systems at present, including range/Dopplerambiguities, highly sparse array signal processing,timing and position uncertainties, etc. In this paper,we use signal processing techniques to solve theproblems of high-resolution SAR imaging ofwide areas, detection of ground moving targets,

array auto-calibration, and satellite formationoptimization. The interesting characteristics of theproposed techniques are simplicity, efficiency, androbustness.

ACKNOWLEDGMENT

The authors wish to thank Dr. Hongwei Liu andMs. Shelley Crawford for revising the English and

syntax.

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Zhenfang Li received the B.S. degree in automatization and the M.S. degree inelectronic information engineering from Xidian University, China, in 1999 and

2001, respectively.He is currently pursuing the Ph.D. degree in radar signal processing at the

National Lab of Radar Signal Processing at Xidian University. His currentresearch is in SAR and GMTI signal processing of distributed small satellite SARsystems.

Zheng Bao (M80SM90) was born in Jiangsu, China in 1927. He graduatedfrom the Communication Engineering Institute of China in 1953.

Currently, he is a professor at Xidian University. His research fields arespace-time adaptive processing (STAP), radar imaging (SAR/ISAR), automatictarget recognition (ATR) and over-the-horizon radar (OTHR) signal processing.

Dr. Bao is a member of the Chinese Academy of Sciences. He has authored orcoauthored 6 books and published over 300 papers.

Hongyang Wang was born in TianJin, China, in 1976. He received the M.S. andPh.D. degrees in signal processing in 2001 and 2005, respectively, both from theNational Lab of Radar Signal Processing at Xidian University, China.

He is now working at the Chinese University of Hong Kong. His researchinterests include weak signal detection and space-time signal processing withapplications in radar and wireless communication systems.

Guisheng Liao (M96) received the B.S. degree from Guangxi University,Guangxi, China, in 1985 and the M.S. and Ph.D. degrees from Xidian University,Xian, China, in 1990 and 1992, respectively.

He joined the National Lab of Radar Signal Processing at Xidian Universityin 1992, where he is currently a vice director of the lab. His research interests aremainly in statistical and array signal processing, signal processing for radar andcommunication, and smart antenna for wireless communication.


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