digital beamforming on receive: techniques and …digital beamforming on receive: techniques and...

Digital Beamforming onReceive: Techniques andOptimization Strategies forHigh-Resolution Wide-SwathSAR Imaging

NICOLAS GEBERT

GERHARD KRIEGER, Member, IEEE

ALBERTO MOREIRA, Fellow, IEEEGerman Aerospace Center (DLR)

Synthetic Aperture Radar (SAR) is a well-proven imagingtechnique for remote sensing of the Earth. However, conventionalSAR systems are not capable of fulfilling the increasing demandsfor improved spatial resolution and wider swath coverage. Toovercome these inherent limitations, several innovative techniqueshave been suggested which employ multiple receive-aperturesto gather additional information along the synthetic aperture.These digital beamforming (DBF) on receive techniques arereviewed with particular emphasis on the multi-aperture signalprocessing in azimuth and a multi-aperture reconstructionalgorithm is presented that allows for the unambiguous recoveryof the Doppler spectrum. The impact of Doppler aliasing isinvestigated and an analytic expression for the residual azimuthambiguities is derived. Further, the influence of the processing onthe signal-to-noise ratio (SNR) is analyzed, resulting in a pulserepetition frequency (PRF) dependent factor describing the SNRscaling of the multi-aperture beamforming network. The focus isthen turned to a complete high-resolution wide-swath SAR systemdesign example which demonstrates the intricate connectionbetween multi-aperture azimuth processing and the systemarchitecture. In this regard, alternative processing approachesare compared with the multi-aperture reconstruction algorithm.In a next step, optimization strategies are discussed as patterntapering, prebeamshaping-on-receive, and modified processingalgorithms. In this context, the analytic expressions for both theresidual ambiguities and the SNR scaling factor are generalizedto cascaded beamforming networks. The suggested techniquescan moreover be extended in many ways. Examples discussedare a combination with ScanSAR burst mode operation and thetransfer to multistatic sparse array configurations.

Manuscript received January 29, 2007; revised May 16, 2007;released for publication March 25, 2008.

IEEE Log No. T-AES/45/2/933006.

Refereeing of this contribution was handled by V. Chen.

This work was supported by EADS Defence Electronics, Ulm,Germany.

Authors’ address: German Aerospace Center (DLR), Microwavesand Radar Institute (HR), Muenchener Strasse 20, Wessling, 82230,Germany, E-mail: ([email protected]).

0018-9251/09/$25.00 c° 2009 IEEE

I. INTRODUCTION

Remote sensing of the Earth’s surface demandssensors that are capable of continuous global coverageand, in addition, provide detailed imagery. Importantapplications are e.g., disaster management, land andsea traffic observation, wide area surveillance, andenvironmental monitoring. Conventional syntheticaperture radar (SAR) systems cannot meet these risingdemands as the unambiguous swath width and theachievable azimuth resolution pose contradictingrequirements on system design [1]. A good azimuthresolution ±az requires a high Doppler bandwidth BDthat results from a long synthetic aperture which isilluminated by a short antenna of length La (cf. (1)).Hence, a high pulse repetition frequency (PRF) isneeded to sample the Doppler spectrum according tothe Nyquist criterion. In contrast, to unambiguouslyimage a wide swath of width Wg on ground, a largeinterval between subsequent pulses is favorable whatcorresponds to a low PRF (cf. (2))

±az ¼La2¼ vsBD

¸ vsPRF

(1)

Wg <c

2 ¢PRF ¢ sin(£i)(2)

where £i represents the incident angle, vs the sensorvelocity, and c the speed of light. To obtain a figureof merit that combines swath width and resolution, thereciprocal of (1) is multiplied with (2). This yieldsexpression (3) that does not depend any more onadjustable system parameters, which means that theSAR parameters underlie a trade-off, as the resolutioncan only be enhanced at the cost of a decreased swathwidth and vice versa

Wg±az

<c

2 ¢ vs ¢ sin(£i): (3)

Alternative SAR imaging modes push this trade-offonly further into one direction or another withoutresolving the underlying system-inherent constraint:the spotlight mode yields a high resolution, but nosufficient coverage [2] while burst modes as ScanSAR[3, 4] and TOPS-SAR [5] map a wide swath butprovide only a coarse resolution.Consequently, new system concepts are needed

to fulfill the increasing demands of future SARmissions. The most promising concepts employmulti-channel SAR systems where the receivingantenna is either split into multiple subapertureswith independent receiver channels or the receiverapertures are distributed on multiple platformsleading to a multistatic SAR. References [6]—[37]list the various approaches in chronological order.All methods are based on the simultaneous receptionof the backscattered signal with mutually displacedreceiving apertures. The additional informationpermits to overcome the aforementioned inherent

564 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 45, NO. 2 APRIL 2009

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Fig. 1. Multi-aperture system with formation of different beams(top) and corresponding block diagram with DBF on receiveprinciple (bottom). Each aperture is interpreted as individual

channel whose signals are digitized and stored before combinedcoherently a posteriori.

limitations of conventional SAR systems. The basicconcept of a multi-aperture SAR system is illustratedin Fig. 1. Each of the N receiver channel’s signalsis mixed, digitized, and stored. Then, a posteriori,digital beamforming (DBF) on receive is carried outby a joint spatiotemporal processing of the recordedsubaperture signals. In azimuth dimension this enablesa coherent combination of the N subsampled andhence aliased signals to a single output signal that issampled with N ¢PRF and free of aliasing. Comparedwith a mono-aperture system operated with PRF, theadditional samples increase the effective samplingby a factor of N and allow either for an improvedresolution or a reduction of the PRF without anincrease of azimuth ambiguities, thereby enabling themapping of a wide swath.In the following, a brief overview over different

multi-aperture systems and processing strategies isgiven. More details can be found in the correspondingreferences. Concerning classical side-lookingmulti-aperture SAR systems, the development startedwith proposing an array antenna in elevation tosuppress range ambiguous returns [9] and the ideaof splitting the antenna in azimuth direction toreduce azimuth ambiguities [10] followed by thecombination of both [11]. Then, the complexity ofthe systems increased and interest turned to moresophisticated processing strategies enabling so-called“software-defined radars,” where the multi-apertureSAR signal is processed in azimuth, elevation, or bothdimensions [13—17, 20—23].Regarding the azimuth dimension, [20], [24], and

[33] specify dedicated algorithms for multi-aperture

SAR processing. The approach presented in[24] and further elaborated in [30] introduces aphase correction that is applied to the raw data toresample the signal in azimuth, while the techniqueproposed in [20] introduces an algorithm based on ageneralization of the sampling theorem that allows forthe unambiguous recovery of the azimuth spectrumfrom multiple aliased subaperture signals. The methodis elaborated in several follow-on papers [27, 28,31, 32, 36]. Finally, [33] and [34] bring up anotherspace-time approach for application in small satelliteconstellations forming a sparse array. The basic ideais to minimize the overall noise power by a trade-offbetween a spatial filtering of the azimuth signal tosuppress Doppler ambiguities, which correspondsto the method presented in [20], and a matchedfilter.Besides, alternative concepts not directly building

on conventional SAR are brought up, as a squintedSAR configuration in combination with beamforming[8]. Further, a very general approach based ona multi-satellite constellation forming a sparselydistributed radar sensor is developed and an optimumway of processing in the space-time domain is derivedin [12] and [19]. Furthermore, [26] presents the “SARtrain” that consists of a multi-satellite constellationwhich is distributed in along-track direction and usesspread spectrum waveforms for transmission.The paper is organized as follows. Basic properties

of the spatial sampling of multi-aperture SAR systemsare summarized in Section II. The DBF algorithmin azimuth introduced in [20] is briefly recalled inSections IIIA and IIIB. The new contributions ofthe paper start in Section IIIC with an illustrativeinterpretation of the multi-aperture processing,followed by a detailed analysis on how signal,ambiguities, and noise are affected by the DBFnetwork in Sections IIID, IIIE, and IIIF, respectively.Then, in Section IV, a system design example ispresented that allows for verifying the theory derivedin Section III and demonstrates the potential ofthe multi-aperture reconstruction algorithm withrespect to different performance parameters and incomparison with alternative techniques of processingthe azimuth signal. In a next step, error sources areidentified and innovative strategies are derived inSection VA. Optimization concepts as pattern taperingon transmit (Section VB), prebeamshaping on receive(Section VC), and adapted beamforming networks(Section VD) are presented and their performanceis shown. In the case of prebeamshaping on receiveand adapted beamforming networks, the theoreticalexamination of residual errors, ambiguities, and thescaling of the signal-to-noise ratio (SNR) is extendedto the class of cascaded beamforming systems. Thepaper closes with a discussion containing an outlookon further issues like ScanSAR in multi-aperturesystems and sparse array systems.

GEBERT ET AL.: DIGITAL BEAMFORMING ON RECEIVE: TECHNIQUES AND OPTIMIZATION STRATEGIES 565


II. MULTI-APERTURE SAR SIGNAL IN AZIMUTH

In the following, this paper focuses on thecharacteristics and processing of the azimuth signalof a single platform SAR system. In order to providea better understanding, this section recalls basicrelationships of multi-aperture systems regardingspatial sampling in along-track dimension.

A. Effective Phase Center, Virtual Sample Position, andSpatial Sampling in Along-Track

The phase center of an antenna represents the“effective” position of the transmitted or receivedsignal and it is situated in the center of the respectiveantenna or aperture. As in multi-aperture systems asingle transmit antenna is combined with a numberof receiving channels, the positions of transmitter(Tx) and receiver (Rx) do in general not coincide.Such systems of spatially separated transmit andreceive apertures can be approximated by a virtualsystem where both coincide midway between thetransmitter and respective receiver and a constantphase term is added. This position is called “effectivephase center” and represents the spatial position ofthe received signal. The positions of the effectivephase centers in azimuth of a single platform systemare shown in Fig. 2 on the top. In this case thespacing of the samples is equal to half of the receiveaperture spacing ¢x. If we consider now the spatialsample positions for a train of pulses transmitted withPRF, we obtain the sampling scenario as shown inFig. 2 on the bottom. For every transmitted pulse,N signals are received whose positions in azimuthare determined by the positions of the receiverswith respect to the transmitting aperture. Hence, thegaps between samples received for a single pulseare defined by the distance between the receivingchannels, while the distance between two subsequentpulses is determined by vs=PRF. Below, the relationbetween spatial sampling and the system parametersas aperture position, sensor velocity vs, and PRF isderived and the most interesting cases are presented.

B. Uniform Sampling

In the optimum case all gathered samples aredistributed uniformly along the synthetic aperture.To obtain these equally spaced samples in a systemof N sensors moving with vs, the following relationbetween PRF and the along-track displacements xjof the receive apertures j = f2, : : : ,Ng relative to theposition x1 of receiver 1 have to be fulfilled:

xj ¡ x1 =2 ¢ vsPRF

μj¡ 1N

+ kj

¶, kj 2 Z: (4)

In a single platform system with an array antenna,the along-track displacement between adjacent

Fig. 2. Aperture position and corresponding effective phasecenter giving spatial position of sample in azimuth direction x

(top) and resulting spatial sampling (bottom) for three subsequentpulses transmitted at t¡1 (dotted), t0 (solid), and t1 (dashed).

Sample spacing for single transmit pulse is determined by aperturedistance ¢x while offset between samples of subsequent pulses

depends on sensor velocity vs and PRF.

subapertures is constant and denoted by ¢x and all kjof (4) become zero, resulting in expression (5) givingthe optimum PRF

PRF =2 ¢ vsN ¢¢x: (5)

The optimum value is referred to as “uniform”PRF as it fulfills the timing requirement for uniformlydistributed samples and yields a data array equivalentto that of a single-aperture (“monostatic”) systemoperated with N ¢PRF. Any deviation from therequirements imposed by (4) and (5) will resultin nonequally spaced samples along the syntheticaperture (“nonuniform sampling”) and the gathereddata array does no longer correspond to a monostaticsignal and cannot be processed by conventionalmonostatic algorithms without performancedegradation. Hence, classic multi-aperture systemswith split antenna in azimuth dimension underliestringent operational constraints or require appropriatepreprocessing to correct for perturbations caused bynonuniformly spaced samples (cf. Section III).

C. Special Sampling Scenarios

To complete this section, two special samplingscenarios are presented. Firstly, certain PRFs resultin an interleave of samples originating from differenttransmit pulses in a way such that again all samples



Fig. 3. Spatial sampling in azimuth direction x originating fromthree subsequent transmit pulses separated by ¢t= PRF¡1 andemitted at t¡1 (dotted), t0 (solid), and t1 (dashed) yieldinginterleaved uniform sampling (top) and spatially coinciding

samples of channels 1 and 3 (bottom).

are spaced equally, yielding a uniform sampling ofhigher order (cf. Fig. 3, top). This means that onlyafter k pulses (5) is fulfilled and no samples coincidespatially, yielding further possible PRFs of uniformsampling, PRFuni,k, given by (6). The inequality for kexcludes those values, where the timing requirement isfulfilled, but spatial samples coincide

PRFuni,k = k ¢2 ¢ vsN ¢¢x = k ¢PRFuni

(6)

k 6= p ¢Nm

, m 2 f1, : : : ,N ¡ 1g, p,k 2 N:

A second interesting case arises when samples ofdifferent transmit pulses received by apertures j and icoincide spatially as sketched in Fig. 3, bottom. Thiscondition is defined generally in (7) and simplified fora single platform system in (8), yielding the respectivePRFc specified by m and n. Note that the samplingbecomes uniform if one of the respective coincidingchannels is switched off

¢xi2+ k ¢ vs

PRFc=¢xj2+ l ¢ vs

PRFc, k, l 2 N (7)

PRFc,n,m =2 ¢ vs¢x

¢ nm=PRFuniN

¢ nm,

(8)

m 2 f1, : : : ,N ¡1g, n 2N^n < m ¢N:

III. DIGITAL BEAMFORMING ON RECEIVE:MULTI-APERTURE RECONSTRUCTIONALGORITHM

Most of the suggested systems propose to simplyoperate a multi-aperture antenna in azimuth byinterleaving the samples of the different receivingchannels without further specific processing steps[10, 11, 14, 18]. Consequently, the timing requirementof (5) has to be fulfilled to obtain a signal thatis equivalent to a monostatic signal, while in anyother case the sample positions deviate from theideal positions, but are treated as if the signal was

sampled uniformly which leads to a degradedsystem performance as discussed later. In this case,a further processing of the signal is required beforeconventional monostatic algorithms can be appliedto focus the signal. In this section, a techniquecalled Multi-Aperture Reconstruction Algorithmthat is suited to process multi-aperture signals ispresented, investigated, and compared with alternativeapproaches. This algorithm was first proposed in [20],and it is based on solving a system of linear equationsto unambiguously recover the formerly aliasedazimuth spectrum even in case of a nonoptimum PRF,i.e., nonuniformly spaced data samples. Althoughin the following focus is turned to single platformsystems, the algorithm has great potential for anymulti-aperture system, be it a distributed SAR withmultiple satellites or single antenna systems like thehigh-resolution wide-swath (HRWS) SAR [14] or thedual receive antenna mode of TerraSAR-X [25].

A. Multi-Aperture Impulse Response and QuadraticApproximation

In a multi-aperture configuration consisting ofa receiver j separated by ¢xj from the transmitter,the azimuth response hs,j(t) for a point-like targetat azimuth time t= 0, slant range R0 and for acarrier wavelength ¸ is described by (9), for the timeneglecting the azimuth envelope of the signal. Thephase is determined by the lengths of transmit andrespective receive path which are approximated by thesquare roots assuming a straight sensor trajectory

hs,j(t,¢xj)

= exp

·¡j 2¼

¸

μqR20 + (vs ¢ t)2 +

qR20 + (vs ¢ t¡¢xj)2

¶¸:

(9)

Quadratic approximation of (9) yields (10) thatallows for relating the multi-aperture point targetresponse to the monostatic response hs(t) by aconstant phase shift ¢'j =¡¼ ¢¢x2j =(2¸ ¢R0) anda time delay ¢tj =¢xj=(2vs), what corresponds tothe effective phase center position midway betweentransmitter and receiver as introduced in Section IIA.In Doppler frequency (‘f’) domain, this relationbetween a monostatic impulse response Hs(f) andthe multi-aperture response of channel j, Hs,j(f), isdescribed by the filter Hj(f) (cf. (11) and (12))

hs,j(t)»= exp

·¡j 4¼

¸R0

¸¢ exp

"¡j ¼ ¢¢x

2j

2¸ ¢R0

#

¢ exp"¡j 2¼ ¢ v

2s

¸

(t¡ (¢xj=2vs))2R0

#

= hs(t¡¢tj) ¢ ej¢'j (10)



Fig. 4. Left: Block diagram of multi-channel SAR system with “reconstruction” of N subsampled channels by filters Pj(f).Right (zoom): Each Pj (f) consists of N bandpass filters Pjm(f) valid on a sub-band of width PRF.

Hs,j(f) =Hj(f) ¢Hs(f) (11)

Hj(f) = exp

"¡j ¼ ¢¢x

2j

2¸ ¢R0

#¢ exp

·¡j ¼ ¢¢xj

vs¢f¸:

(12)

The multi-aperture response can hence beseparated into the influence of the multi-aperturesystem described by Hj(f) and the conventional SARimpulse response given by Hs(f). Further, for singleplatform systems the same azimuth antenna patterna(t) and A(f), respectively, can be assumed for allchannels. This yields the block diagram of Fig. 4,where us(t) and Us(f), respectively, describe the scenereflectivity. Filtering of the scene with A(f) ¢Hs(f)yields the monostatic SAR signal in Doppler domain,U(f), with its time domain representation u(t). Themulti-aperture SAR signal of channel j is then givenby Uj(f) and uj(t), that are related to U(f) and u(t)by Hj(f) according to (13) and its time domainrepresentation hj(t), respectively

Uj(f) =Hj(f) ¢U(f): (13)

B. Theoretical Background

The multi-aperture reconstruction algorithm isfounded on a generalization of the sampling theoremaccording to which N independent representations ofa signal, each subsampled at 1=Nth of the signal’sNyquist frequency, allow for the unambiguous“reconstruction” of the original signal from the aliasedDoppler spectra of the N representations. This meansthat a band-limited signal U(f) is uniquely determinedin terms of the responses Hj(f) of N linear systemswith input U(f), sampled at 1=Nth of the Nyquistfrequency. The functions Hj(f) may be chosen ina quite general way, but not arbitrarily [38, 39].Transferred to a multi-aperture SAR system, U(f)gives the monostatic SAR signal while the functionsHj(f) represent the ‘channel’ between the transmitterand each receiver j with respect to the monostaticimpulse response (cf. Fig. 4 and Section A). Notethat in principle the complete multi-channel SARsignal model of (9) including the two-way patterns

of the respective channels can be used for a completereconstruction of the scene reflectivity, but for reasonsof simplicity only the quadratically approximatedsystem model of A is considered in the following.The received signals are sampled in azimuth by

PRF and hence the maximum signal bandwidth isN ¢PRF. A compact characterization of the wholesystem is then given by the matrix H(f), that containsall channel representations Hj(f) shifted by integermultiples of the PRF according to (14)

H(f) =

266664H1(f) ¢ ¢ ¢ HN (f)

H1(f +PRF) ¢ ¢ ¢ HN (f +PRF)

.... . .

...

H1(f +(N ¡ 1) ¢PRF) ¢ ¢ ¢ HN (f +(N ¡ 1) ¢PRF)

377775 :(14)

Then, as shown in [39], the inversion of H(f)yields a matrix P(f) that contains in its rows Nfunctions Pj(f) that are decomposed by the columns inN functions Pjm(f) defined on sub-bands m of widthPRF that make up the Doppler spectrum (cf. Fig. 4and (15)). As we will see later, the scaling by N isreasonable from an energy point of view. Note thatin the considered case H(f) is invertible as long assamples of different receive apertures do not coincidespatially

P(f) =N ¢H¡1(f)

=

266664P11(f) P12(f+PRF) ¢ ¢ ¢ P1N(f+(N ¡ 1)PRF)P21(f) P22(f+PRF) ¢ ¢ ¢ P2N(f+(N ¡ 1)PRF)...

.... . .

...

PN1(f) PN2(f+PRF) ¢ ¢ ¢ PNN(f+(N ¡ 1)PRF)

377775 :(15)

The aliased frequencies in the Doppler spectraof the individual channels are then suppressed andconsequently the original signal U(f) is recoveredby filtering each of the multi-aperture channels jwith its appropriate “reconstruction” filter Pj(f) andsubsequent coherent combination of all weightedreceiver channels (cf. Fig. 4). To complete Fig. 4, theconventional monostatic SAR focusing filter Pmf(f) isincluded.



C. Illustration of Principle

In this section, an illustrative approach to explainthe principle and limitations of the multi-aperturereconstruction algorithm is given. Consider a systemaccording to Fig. 4 with exemplary N = 2 aperturesand U(f) describing a monostatic SAR signalwhile Uj(f) =Hj(f) ¢U(f) represents the signal atreceiver j. At first, assume U(f) spectrally limitedto [¡PRF,PRF]. Then, Fig. 5 on the left shows thespectrum of the band-limited signal “seen” at thereceiver j before sampling (top) and its periodiccontinuation after sampling with PRF (middle)yielding Uj(f), given in (16). For convenience weintroduce the notation Ujk(f) =Uj(f+ k ¢PRF) wherethe index k indicates a shift by k ¢PRF in Dopplerdomain

Uj(f) =1X

k=¡1Uj(f+ k ¢PRF) =

1Xk=¡1

Ujk(f)

=1X

k=¡1U(f+ k ¢PRF) ¢Hj(f+ k ¢PRF):

(16)As can be observed, for any frequency, the

subsampled and aliased signal consists of not morethan two (in general N) shifted and superimposedspectra, as not more than one spectrum (in generalN ¡ 1 spectra) of the periodic continuation overlapswith the original spectrum. Hence, the spectra U1(f)and U2(f) can be weighted and combined in sucha way, that the component of the original spectrumis recovered (cf. (17), top), while the back-foldedcomponent is cancelled (cf. (17), bottom). With Pjm(f)denoting the reconstruction filter for receiver j on theDoppler frequency interval m (cf. Fig. 4), this requiresthe following equations to hold in the interval I1 (cf.Fig. 5, middle, left):

P11(f) ¢H1(f) ¢U(f) +P21(f) ¢H2(f) ¢U(f)!=N ¢U(f)

P11(f) ¢H1(f+PRF) ¢U(f+PRF)+P21(f)¢H2(f+PRF) ¢U(f+PRF)!=0:

(17)

For a uniform sampling, no processing needs tobe applied, i.e., the samples of all signals have tobe simply interleaved and hence it seems reasonableto set the arbitrary normalization factor equal to N,as this ensures a magnitude equal to 1 of the filtersPjm in this special case. Setting up the correspondingequations on the interval I2 and shifting them to I1yields (18) which allows for setting up the linear

Fig. 5. Signal spectrum at receiver j before (top) and after(middle) sampling with PRF and after reconstruction (bottom)

taking into account all receivers. Left shows signal band-limited to[¡PRF,PRF] that is correctly recovered while the bandwidth onthe right exceeds [¡PRF,PRF] entailing erroneous contributions ek .

system of (19) that corresponds to (15)

P12(f+PRF) ¢H1(f+PRF) ¢U(f+PRF)+P22(f+PRF) ¢H2(f+PRF) ¢U(f+PRF)!=N ¢U(f+PRF)

P12(f+PRF) ¢H1(f) ¢U(f) +P22(f+PRF) ¢H2(f) ¢U(f)!=0 (18)

H(f) ¢P(f) =N ¢ 1, P(f) =N ¢H¡1(f): (19)

In a next step, we analyze a scenario wherethe bandwidth of the signal exceeds N ¢PRF. ForN = 2, an example for the spectrum of such asignal is given in Fig. 5 on the right, before (top)and after (middle) sampling. In contrast to theband-limited case, the sampled signal Uj(f) consistsof up to three contributions, as the spectra of theperiodic continuation may overlap. For the generalcase, this means that more than N spectra maycoincide at a certain Doppler frequency. From themathematical point of view this results in a linearsystem of equations that is underdetermined andconsequently the original spectrum can in generalnot be reconstructed exactly. As shown in [28], acomplete suppression of the contributions from theshifted spectra is not achieved, because the filtersPj(f) are determined as if the signal bandwidth waslimited to §N ¢PRF=2. Hence only the ambiguous



energy within the band [¡N ¢PRF=2,N ¢PRF=2] ofthe original signal is cancelled by the reconstruction.All energy outside this band is not well suppressedand finally gives rise to ambiguous contributions ekin the reconstructed signal (cf. Fig. 5 bottom right)that is specified in the following section. How theambiguous energy can be reduced, e.g. by selectingappropriate sub-bands or weighting of the azimuthspectrum during the reconstruction is discussed inSection V.

D. Signal Power

In the following, the signal power is definedas the mean energy of the unambiguous signal,which is limited by the system bandwidth in azimuthIs =N ¢PRF and eventually further confined bythe processed Doppler bandwidth BD · Is. Todetermine the signal power, at first the output signalpower of the reconstruction network is determined.Remember that the filters Pj(f) are chosen such thatthe “original” signal received by a single apertureU(f) is reconstructed. Recall furthermore that thespectral weighting of U(f) depends only on itsenvelope defined by the joint pattern A(f) of thetransmitter and a single receiver element. Thismeans that we basically obtain a signal that is notdependent on the reconstruction filter network up toa constant scaling factor of N as given in (17). Thisfactor accounts for a digital summation yielding anamplitude gain increased by a factor of N with respectto a single channel and consequently the power isamplified by N2 compared with the input signal powerps,el. Then, the signal power ps is determined by themean squared value of N ¢U(f) limited to the systembandwidth in azimuth IS = [¡N ¢PRF=2,N ¢PRF=2](cf. (20)). The limitation to IS is expressed by therectangular window function rect(f=Is) and thecalculation of the mean value within this interval isindicated by the operator E[:]1

ps =N2 ¢E[jU(f) ¢ rect(f=IS)j2] =N2 ¢ps,el: (20)

If the signal is focused with a defined bandwidthBD only this part of IS is used for the compression,what can be understood as an additional lowpass filterof width BD that is applied to the signal. This filteringis equivalent to a fixed integration time resulting in anazimuth spectrum of width BD. For the case wherethe system is operated with a constant duty-cycle,the average transmit power per time is constant.This means that the resulting energy after focusingis independent from the PRF, as e.g. the decreasing

1Note that for deterministic signals as U(f) the operator E[:] isidentical to an integration over frequency normalized by the intervalwidth, while for stochastic processes E[:] describes the expectationvalue. As both give a measure of the power, in the following thesame operator symbol is used to provide a consistent notation.

energy per sample for a higher PRF is compensatedby the fact that more samples are gathered within theintegration time and then combined during azimuthcompression. We obtain (21), giving ps,BD thatdescribes the signal power after focusing with BD,where the mean value is calculated on the originalinterval Is and a “white” scene is assumed whichmeans that jU(f)j can be described by the signalenvelope A(f)

ps,BD =N2 ¢E[jU(f) ¢ rect(f=BD)j2]

=N2 ¢E[(A(f) ¢ rect(f=BD))2]: (21)

E. Residual Reconstruction Error, Error Power, andAzimuth Ambiguities

Section C illustrated that energy outside theband [¡N ¢PRF=2,N ¢PRF=2] of the original signalis not cancelled by the algorithm and disturbs theunambiguous reconstruction of the multi-apertureSAR signal. With knowledge of the overallconfiguration and the antenna patterns, this sectionderives the spectral “error” that remains from thesealiased parts after the system of reconstruction filters.The further development considers the sampledsignal at receiver j in (16) and focuses on the kthcontinuation of the original spectrum Uj(f+ k ¢PRF) =Ujk(f). It is now of interest, how the ambiguousparts of U(f), i.e., the contributions outside theoriginal frequency band [¡N ¢PRF=2,N ¢PRF=2],are folded back by the sampling. As the furtherprocessing of Ujk(f) by the filters Pjm(f) is definedon sub-bands m of width PRF, the original band issplit into respective intervals Im = [(¡N=2+m¡ 1) ¢PRF,(¡N=2+m) ¢PRF], where m= 1, : : : ,N. Theambiguous contributions of Ujk(f) are located atfrequencies that deviate more than §N ¢PRF=2from the center frequency ¡k ¢PRF, i.e., jf+ k ¢PRFj>N=2 ¢PRF. Further, only contributions within[¡N ¢PRF=2,N ¢PRF=2] are of relevance, yieldingthe following expressions for the bands where theambiguous parts are located:

N

2PRF¡ k ¢PRF· f · N

2PRF, k > 0 (22)

¡ N2PRF· f ·¡N

2PRF¡ k ¢PRF,

k < 0: (23)

Consequently, depending on k, certainsub-bands of the spectrum are not cancelledby the reconstruction algorithm and have to beconsidered when determining the error. Assuminga symmetrical pattern, i.e., A(f) = A(¡f), theproblem is symmetrical and it is sufficient toconcentrate on k > 0 and (22). In this case,sub-bands Im up from index m0 = maxfN ¡ k+1,1gcontribute to the error. Note that choosing m0 = 1



does not lead to wrong results as the respectivesummands are zero due to the inverse filtering byPjm(f). In a next step the processing of the signal istaken into account by considering the weighting of thesignal Ujk(f) with the filters Pjm(f) over all sub-bandsIm where the signal is not properly reconstructed.Then we sum over contributions from all receiverchannels j. This yields ek(f) that describes thespectrum of the remaining output error due to aliasingof the input signal caused by the kth backfoldedspectrum (cf. (24) and Fig. 5 bottom right). Uj(f)is expressed by the monostatic SAR signal U(f)multiplied by the respective Hj(f). Again, index kindicates a shift from f to f+ k ¢PRF due to thesampling and the signal is confined to the sub-bandIm by a rectangular window of width PRF arounda center frequency f0,m = (¡N=2+m¡ 1=2) ¢PRFrepresented by rect((f¡f0,m)=PRF)

ek(f) =NX

m=m0

NXj=1

Uk(f) ¢Hjk(f)| {z }Ujk(f)

¢Pj(f) ¢ rectμf¡f0,mPRF

¶| {z }

Pjm(f)

=Uk(f) ¢NX

m=m0

NXj=1

Hjk(f) ¢Pjm(f): (24)

The complete ambiguous spectrum due to aliasingis obtained by summation of all individual ambiguouscontributions after reconstruction, ek, yielding e§(f) in(25), where the factor 2 accounts for both signs of k

e§(f) = 2 ¢1Xk=1

ek(f)

= 2 ¢1Xk=1

0@Uk(f) ¢ NXm=m0

NXj=1

Hjk(f) ¢Pjm(f)1A :(25)

Expression (25) can be divided into thecharacteristics of the azimuth signal and the overallconfiguration. The signal spectrum Uk(f) is definedby size and tapering of the transmit antenna and thesingle receive aperture, while the second part of (25)is defined by the products Hjk(f) ¢Pjm(f) that dependon the relative positions of all apertures. Relatingthe ambiguous power given by the mean squaredamplitude of e§(f) with the signal power of (20)results in an equivalent azimuth ambiguity-to-signalratio for multi-aperture SAR systems (AASRN)that takes into account the weighting and coherentcombination of the individual channels by thebeamforming network

AASRN =E[je§(f)j2]

ps: (26)

So far the AASRN was calculated for the completeDoppler bandwidth given by N ¢PRF. In a next step,

the respective processed bandwidth and the associatedlowpass filtering are included, yielding the AASRN,BDfor the image. Therefore e§(f) is band-limited toBD by a rectangular window rect(f=BD) before theambiguous power is calculated

AASRN,BD =E[je§(f) ¢ rect(f=BD)j2]

ps,BD: (27)

In order to obtain the response of the residualambiguity of order k in the image, the respectiveambiguous contribution ek defined by (24) has to befocused by the used SAR processor.

F. SNR Scaling in Digital Beamforming Networks–©bf

In this section, the influence of the DBF networkon signal and noise power is investigated and aparameter that characterizes the impact of a networkof digital filters on the SNR is derived. As undercertain conditions the SNR is improved by theprocessing, we abandon the term “noise figure” buttalk about “SNR scaling” in beamforming networksand denote it by ©bf to avoid confusion with existingterminology.Consider the multi-aperture system as a linear

system of N channels j with an input signal uj(t) andadditive white Gaussian receiver noise componentsnj,B(t) that are mutually uncorrelated between thechannels. The corresponding spectra are given byUj(f) and nj,B(f) and the signal-to-noise ratio isdenominated by SNRel,j . The system model of a singlechannel j is shown in Fig. 6. After reception, thesignal is amplified in the low-noise amplifier (LNA)that is characterized by its power gain Gj , amplitudegain gj , and its noise figure with respect to power (Fj)and amplitude (fj). Under the realistic assumption ofa sufficiently high Gj the nj,B(f) are the dominatingsources of thermal noise and hence all other thermalnoise sources afterwards are neglected. Afteramplification the signal is sampled in azimuth with aratio of PRF yielding a subsampled and consequentlyaliased time-discrete signal. The quantization errorinduced by the sampling is modelled as an additivenoise source nqj and assumed to be uniformlydistributed and spectrally white as proposed in [40].Further, we assume the granular error dominant andconsequently no SNR loss caused by clipping isencountered. The signal of each channel j is thendigitally filtered by a Doppler frequency dependentfilter Pj(f) that is defined on a bandwidth of N ¢PRF.Finally, the outputs uout,j[k] of all N branches arecombined coherently yielding the output signal uout[k](cf. Fig. 6). As the system is linear and only additivenoise sources are considered, signal and noise powercan be analyzed separately, when investigating theinfluence of the reconstruction network on the SNR.Note that in the following, “power” is used to describe



Fig. 6. System model for single channel of multi-aperture system. After reception of Uj(f), receiver noise is added. Then RF signal isamplified and digitized before filtering and coherent combination of all channels.

the average energy per time or frequency, respectively.The normalisation to the respective interval widthis included in the mathematical operator E[:] thatrepresents the mean value for deterministic signalsand the expectation value for stochastic processes. Forreasons of simplicity, considerations are carried outin frequency domain and after sampling in Dopplerfrequency domain, both designated by f. Aftersampling, only Doppler frequency is considered, asreconstruction by the filters Pj(f) is carried out inDoppler domain.First, consider how the beamforming network

affects the noise power. In channel j, the thermalnoise contribution after sampling is gj ¢fj ¢ nj(f)while nqj(f) accounts for the quantization noise.After digitization, filtering with Pj(f), and coherentsummation of all channels j, (28) describes the overallnoise n(f) in the reconstructed data. The output noisepower pn is then defined by the mean square value ofn(f) given by expression (29)

n(f) =NXj=1

Pj(f) ¢ (nj(f) ¢ gj ¢fj + nqj(f)) (28)

pn = E[jn(f)j2]

= E

264¯¯ NXj=1

Pj(f) ¢ (nj(f) ¢ gj ¢fj +nqj(f))¯¯2375 :(29)

As nqj and nj are uncorrelated [40], the overallnoise power pn simplifies to the sum of the noisepowers induced by thermal receiver noise componentspn,rx,j and the quantization (pn,q). Assuming mutuallyuncorrelated nj , the squared sum representing thereceiver noise power in (29) simplifies to the sum ofsquared values (cf. (30))

pn =NXj=1

E[jnj(f) ¢Pj(f)j2] ¢Gj ¢Fj| {z }Thermal noise pn,rx,j

+E

24¯¯NXj=1

Pj(f) ¢ nqj(f)¯¯235

| {z }Quantization noise pn,q

=NXj=1

pn,rx,j +pn,q: (30)

The power of nj(f) can be expressed by the powerat the point of reception, pn,el,j , i.e., the power ofnj,B(f), as the sampling changes only the powerspectral density of the noise without affecting itspower or spectral appearance

pn,el,j = E[jnj,B(f)j2] = E[jnj(f)j2]: (31)

Finally, we assume all subaperture elements jto be identical, which means that we expect thesame characteristics for Gj , Fj , gj , fj and thermalnoise of same power pn,el,j for all elements. Further,the assumption of spectrally white receiver noiseallows for separating the noise power and its spectralweighting given by the Pj(f), which yields thefollowing expression for the system’s output noisepower pn that consists of pn,rx and pn,q

pn = pn,el ¢G ¢F ¢NXj=1

E[jPj(f)j2]| {z }Thermal noise pn,rx

+E

24¯¯NXj=1

Pj(f) ¢ nqj(f)¯¯235

| {z }Quantization noise pn,q

= pn,rx+pn,q: (32)

Due to the mutual correlation of the quantizationerrors in the channels j, the scaling of the error by thenetwork can be stronger than the amplification of thethermal noise. Generally, we expect that the numberof bits will be chosen such that the quantization noisein the output signal will be negligible compared withthe thermal noise, which means that the overall noisecan be approximated by pn,rx defined by the inputnoise power and an amplification factor determinedby the reconstruction filter network (cf. (32)).In a next step, the influence of the reconstruction

network on the signal power is investigated. Takinginto account the power gain G of the LNA andconsidering the scaling of the signal power by N2 bythe beamforming network (cf. Section D), the signalpower after reconstruction ps is given by scaled inputnoise power ps,el according to (33)

ps = ps,el ¢G ¢N2: (33)

Combining (32) and (33), the following expressionfor the scaling of the SNR by the network is obtained



[31]ps,el=pn,elps=pn

=SNRelSNRout

=F ¢PN

j=1E[jPj(f)j2]N2

: (34)

To concentrate on the effect of the reconstructionnetwork, we define the SNR scaling factor ©bf ofthe DBF network by (34) normalized to the optimumvalue of F=N that is achieved for uniform sampling

©bf :=SNRel=SNRout

(SNRel=SNRout)jPRFuni=

PNj=1E[jPj(f)j2]

N:

(35)Note that ©bf considers only the relation between

SNRel and SNRout at the respective PRF. Thisincludes the gain by oversampling but does notaccount for a possible variation of the SNRel with thePRF as it is e.g. encountered for a constant duty cycle.From a matrix theoretical point of view, the

scaling of the receiver noise power in the data asquantified by (35), can be expressed by the sum ofthe eigenvalues ¸j(f) of the matrix P(f) ¢PH(f),where PH represents the conjugate transpose of P.For a given PRF the sum over all eigenvalues isconstant and consequently the SNR scaling factor©bf is not frequency dependent. The equality of therepresentations of the SNR scaling factor in (35)and (36) is derived using the relation between thetrace of a matrix and the sum over its eigenvalues incombination with the Hilbert-Schmidt norm

©bf =

PNj=1¸j(f)

N: (36)

In 2006 Wang and Bao [41] have shown thatfor the considered system and the approximationsproposed in [20] the scaling of the noise by thereconstruction network as given in (34) can beexpressed explicitly by harmonic functions as followsin (37), where the sample positions of receivers p andr are described by their respective sample times tpand tr normalised by 1=PRF. Basically the numeratorgives the deviation of the samples from the optimumregular spaced positions given by 2¼ ¢ n=N, while thedenominator takes the mutual distances between thereceivers into accountNXj=1

E[jPj(f)j2]

=1N

NXp=1

PN¡1n=0

QN

r=1,r 6=p(1¡ cos(2¼ ¢ n=N ¡ 2¼ ¢ tr ¢PRF))QN

r=1,r 6=p(1¡ cos(2¼ ¢PRF ¢ (tp¡ tr))):

(37)In a further step, we take the focusing of the

data into account to describe the noise in the image,pn,BD. Based on the assumption of spectrally whitereceiver noise, the power spectral density of thenoise decreases with increasing PRF while thenoise power remains constant [42]. Hence, to derive

TABLE IMission Performance Requirements

Carrier wavelength ¸ 0.031 mOrbit height hs 500 km—700 kmCoverage (incident angle) μi 20±—55±

Swath width Ls 100 kmGeometric resolution in azimuth &range

±az , ±rg · 1 m

Range ambiguity suppression(distributed targets)

RASR ·¡21 dB

Azimuth ambiguity suppression(distributed targets)

AASRN ·¡21 dB

Noise equivalent sigma zero NESZ · 19 dB

an expression for the noise power that remains in theimage, the noise power spectral density (defined bynoise power and PRF) in combination with a lowpassfilter of bandwidth BD has to be considered after thesignals are filtered by the Pj(f). This yields (38) forthe noise power after focusing, which is derived from(32) extended by the rectangular window rect(f=BD)limiting the Doppler spectrum

pn,BD = pn,el ¢G ¢F ¢NXj=1

E[jPj(f)j2 ¢ rect(f=BD)]:

(38)The dependency on the PRF cannot be seen

explicitly, but it follows implicitly from the restrictionof the filters Pj(f) to BD and the definition of thePj(f) on a bandwidth of N ¢PRF over which themean value is calculated. Hence an increasing PRF incombination with a constant BD means a decreasingspectral part–with respect to N ¢PRF–thatcontributes to the noise power. For BD =N ¢PRF,(38) is equivalent to (34). The SNR scaling factor withrespect to the image ©bf,BD cannot be given explicitly,as the relation between input and output signal powerdepends on the shape of U(f) in combination withBD as defined in (21), but of course ©bf,BD can becalculated using (21) and (38).Note that receiver noise and ambiguous power

(cf. Section E) are mutually independent, and thusthe resulting interfering power in the data and theimage can be simply determined by addition of therespective expressions.

IV. SYSTEM DESIGN EXAMPLE

A. Requirements and Timing

In the following, an X-band high-resolutionwide-swath SAR system is designed with the basicmission performance requirements of Table I and fortwo different swath definitions. First, a conventionalapproach is chosen that is suitable to cover therequired incident angle range from 20± to 55± withswaths of 100 km and a variable overlap of 10—20%.This case takes into account the interference ofthe transmit event and the nadir echo. The detailed



Fig. 7. Top: Timing analysis for orbit height of 580 km suitableto cover incident angle range from 20—55± with 6 distinct swathsof width 100 km taking nadir echo (horizontally streaked) andinterference of transmit events (vertically streaked) into account.

Bottom: Timing analysis with nadir echo assumed to besufficiently suppressed by DBF in elevation. Dashed line indicates

center of swath.

relation between swaths and incident angles is givenin Table III. Second, a system is regarded, wherethe beamforming capability in elevation is sufficientfor suppressing the nadir echo and consequently thetiming depends only on the transmit events. In thiscase the coverage requirement is modified to enableimaging of a 100 km swath centered on an arbitraryangle of incidence. A detailed timing analysis witha duty cycle of 15% yields a suitable orbit heightof 580 km §10 km, leading to a sensor velocity of7560 m/s. Fig. 7 on the bottom shows the result forthe suppressed nadir echo, while Fig. 7 on the topaccounts for the conventional scenario. In this casethe small mutual shift of swaths of the same referencenumber is due to the variable overlap of adjacentswaths.

B. System Parameters

The timing shows that for the nominal orbitheight a PRF range from 1250 Hz up to 1350 Hzis sufficient to ensure the coverage of all six swaths(cf. Fig. 7, top). If the height deviates by 10 km,the minimum PRF decreases to » 1240 Hz. Inthe case where the nadir is suppressed, the sameminimum PRF is chosen. To allow for the imagingof an arbitrarily centered swath, the suitable PRFdecreases with increasing incident angle as the dashed

TABLE IISystem Parameters

Orbit height hs 580 km§ 10 kmSlant Range R0 604 km—1112 kmSensor velocity vs 7560 m/sPRF range PRF 1240—1470 HzReceive aperture size in azimuth daz,rx 1.6 m

Number of Rx channels in azimuth N 7Overall antenna length in azimuth La 11.2 mTransmit antenna size in azimuth daz,tx 3.0 m

Reconstruction Network SNRScaling Factor

©bf · 0:7 dB

Receive aperture size in elevation del,rx 0.08 m

Number of Rx apertures in elevation Nel 25Overall antenna height in elevation Ha 2 mTransmit antenna size in elevation del,tx 0.19 m—0.41 m

Maximum transmit antenna gain Gtx 38.7 dB—42.1 dBMaximum gain of single receiverchannel

Grx,j 46.2 dB

Transmit peak power Ptx 5 kWDuty cycle dc 15%Boltzmann constant k 1:38 ¢ 10¡23 J/KSystem temperature T 300 KLosses (atmospheric, system,receiver noise, 2-way)

L ¢F 5.7 dB

Azimuth loss Laz 2.7 dB

line representing the swath’s center indicates (cf.Fig. 7, bottom). When reaching the minimum PRFof 1250 Hz, a “jump” to a PRF of 1470 Hz becomesnecessary to guarantee continuous coverage (cf.Fig. 7, bottom, dotted line).To achieve the required resolution in azimuth,

simulations have shown that a processed bandwidthBD = 7:6 kHz is needed. In combination withthe minimum PRF this determines a minimumnumber of receive apertures to ensure the necessaryeffective sampling ratio on receive. In the presentcase we obtain N = 7. Further, the overall antennalength is to be chosen to set the optimum PRFwithin the required PRF range. It turns out that anoverall length La of 11:2 m = 7 ¢ 1:6 m fulfils therequirements concerning the SNR scaling factor of thereconstruction network (©bf) caused by the recoveryof the nonuniform sampling. The respective value of©bf resulting from the PRF for each swath is listed inTable III. An analysis of the ambiguous energy shows,that on transmit an aperture antenna of length daz,tx =3 m and without tapering results in a sufficiently highsuppression (cf. Fig. 8, top). For a detailed analysis ofthe impact of different antenna lengths and tapering,refer to Section V. Table II summarizes the systemparameters and dimensions.To ensure a slant range resolution of 1 m, a chirp

bandwidth of 150 MHz is necessary and hence theappropriate bandwidth BRg for a resolution of 1 mon ground is 150 MHz=sin(μi), where μi denotesthe incident angle. Consequently, the required



bandwidth in range depends on the minimum μi ofthe respective swath. Further, to illuminate the wholeswath for all incident angles, the effective heightdel,tx of the transmit antenna is varied to guaranteea coverage of 100 km on ground within the 3 dBbeamwidth angle μ3 dB. The value of μ3 dB in degreesis approximated by 51 ¢¸=del,tx according to [43].The selection of the respective swath can be eitherachieved by an electrical steering of the transmitbeam or a mechanical steering of the antenna by aroll-maneuver of the satellite. In the following, weassume a mechanical steering that ensures an optimumpointing of the antenna to the region to be imaged.Table III lists the respective values for the necessarychirp bandwidth, effective transmit antenna height,and corresponding maximum antenna gain Gtx,max.On receive, the antenna in elevation consists of alarge number of independent elements that allow forbeamforming in elevation which ensures a high gainand a sufficient suppression of range ambiguities. Forthis, one may apply the scan-on-receive (SCORE)technique suggested in [14] and [15] that uses areal-time beamforming to scan the reflected pulse as ittravels over the ground. Problems may occur with thistechnique by topography and it might be necessaryto apply a more sophisticated approach to cope withthe problems arising from height variations withinthe scene [44]. In order to guarantee the steeringcapability of the elevation beam to cover a swathof 100 km, a scan angle up to §4:5± with respect toantenna boresight is necessary. By scaling the solutionfor a swath width of 80 km in [35], the necessaryelement spacing in elevation to avoid grating lobesis chosen to 8 cm. To obtain a sufficiently high gainon receive and achieve the required low NESZ, areceiving antenna of height of 2 m is used, whichresults in a number of 25 subantenna elements.Table II and Table III summarize the parameters inelevation dimension.

C. System Performance: Azimuth AmbiguitySuppression, Resolution, NESZ

In a first step, the suppression of azimuthambiguities is simulated within the relevant PRFrange and compared with its analytic predictionin (27) derived in Section IIIE. The results showgood agreement with a deviation of less than 0.1 dB

TABLE IIISwaths and Respective Parameters

Swath μi BRg(μi,min) del,tx Gtx,max PRF ©bf

1 20.0—29:0± 439 MHz 0.19 m 38.7 dB 1340 Hz ¡0.92 dB2 27.5—35:6± 325 MHz 0.22 m 39.4 dB 1250 Hz 0.06 dB3 34.9—41:9± 262 MHz 0.26 m 40.1 dB 1350 Hz ¡0.96 dB4 40.9—47:0± 229 MHz 0.30 m 40.7 dB 1260 Hz ¡0.12 dB5 46.4—51:7± 207 MHz 0.36 m 41.5 dB 1330 Hz ¡0.86 dB6 50.3—55:0± 195 MHz 0.41 m 42.1 dB 1260 Hz ¡0.12 dB

Fig. 8. Top: Simulated (solid) azimuth ambiguous energysuppression (AASRN ) versus PRF and its prediction (diamonds)for BD = 7:6 kHz. Bottom: AASRN achieved by reconstructionalgorithm (solid) compared with phase correction (dashed) [24],

simple interleaving of samples (dotted) [10], and patternnull-steering (diamonds) [33]. PRF region of interest limited by

shaded areas.

(cf. Fig. 8, top). A closer analysis of (27) revealsthat the amplification of the ambiguous energyoutside the frequency interval [¡N=2 ¢PRF,N=2 ¢PRF] rises with increasing deviation from theoptimum PRF, which is caused by the increasingmean squared value of the filter functions Pj(f).Next, the multi-aperture reconstruction algorithm iscompared with alternative algorithms for processingthe multi-aperture azimuth signal. As a first referencecase, the samples of the multiple azimuth channelsare simply interleaved yielding the output signalwithout further processing as proposed in [10]. In amore complex approach, the method of [24] takesthe properties of the SAR signal into account asit compares the multi-aperture signal’s phase withthe phase of a monostatic and uniformly sampled



Fig. 9. Characteristic of SNR scaling factor ©bf versus PRF dueto signal reconstruction with consideration of processed Dopplerbandwidth of 7.6 kHz (solid) and for whole bandwidth (dashed).

signal. This yields a Doppler frequency-dependentphase difference between the desired and theactual signal that is corrected by the processing.Finally, a space-time approach is evaluated that isbased on a frequency-dependent adjustment of theweighting coefficients of the azimuth channels tosteer nulls in the joint antenna pattern to the anglescorresponding to the ambiguous Doppler frequencies.This null-steering corresponds to a spatial filteringof the data to suppress ambiguous frequencies inthe azimuth signal [27, 33]. Fig. 8 on the bottomshows that the best suppression is achieved bythe reconstruction algorithm and the null-steeringapproach. As shown in Appendix C, under certainapproximations this approach is already included inthe reconstruction algorithm. For a more detailedcomparison of different processing methods, refer to[36].Regarding the azimuth resolution, all algorithms

remain below 1 m for the optimum PRF of 1350 Hz,but only the multi-channel processing approaches(phase correction, null-steering, reconstructionalgorithm) provide a constant resolution over thewhole PRF range, while the resolution degradesfor increasing offset from the optimum PRF if nodedicated processing is applied and the samples arejust interleaved.Next, the noise equivalent sigma nought (NESZ) is

determined. In a preliminary step, the SNR scalingfactor ©bf , which describes the variation of theSNR caused by the DBF network in dependencyof the PRF, is evaluated (cf. Fig. 9). ©bf worsenswith increasing deviation from the optimum PRFdue to the rising mean square value of the Pj(f)that tends to infinity when the PRF entails spatiallycoinciding samples and the whole system bandwidthin azimuth is considered. This effect is mitigated bythe increasing oversampling caused by rising PRFvalues in combination with a constant BD entailingfor the image a reduced part of the input noise power(cf. Fig. 9, solid line). The improvement for uniformsampling is directly given by 10 ¢ log[BD=(N ¢PRF)] =¡0:95 dB.

In addition, one has to account for the azimuthloss factor Laz that considers the decay of the jointTx/Rx azimuth pattern which attenuates the recordedsignal while the added noise power remains spectrallywhite [45]. Hence, the SNR becomes dependenton the Doppler frequency. Assuming a normalisedrectangular filter of bandwidth BD for focusingin azimuth, Laz is expressed by (39), where A0(f)represents the normalised weighting of the Dopplerspectrum by the joint Tx/Rx azimuth antenna pattern.A transmit antenna of 3 m and a receiving antenna of1.6 m yield a loss of Laz = 2:7 dB

Laz =BDR BD=2

¡BD=2 jA0(f)j2df: (39)

Finally, the well-known expression for the NESZ[1] is extended by the SNR scaling factor of thereconstruction filter network ©bf,BD (cf. Section IIIF)which comprises a possible gain by oversampling,but does not account for the changed input SNRelwith the PRF. As ©bf,BD is normalized to PRFuni, anadditional compensation factor ©bf,NESZ = PRF=PRFuniis required. Effectively, this considers the noisepower at PRF with respect to PRFuni according tothe changed number of samples. Note that all otherlosses (system, atmospheric, etc.) and the receivernoise figure are summarized in the loss factor L ¢F.The description and values of all parameters can befound in Table II

NESZ =

256 ¢¼3 ¢R30(£i) ¢ vs ¢ sin(£i)¢k ¢T ¢Brg(£i) ¢©bf,BD©bf,NESZ ¢L ¢F ¢LazPtx ¢Gtx(£i) ¢N ¢Grx,j ¢¸3 ¢ c0 ¢ dc

:

(40)Under consideration of the above parameters

and the respective transmit pattern we obtain thefollowing characteristic of the NESZ versus groundrange (cf. Fig. 10, top). In the case where an arbitraryswath of 100 km can be imaged, the variation of theNESZ for a swath centred at a certain range is givenby the best and worst value within the swath (cf.Fig. 7, bottom). Note that this takes into account theadaptation of the chirp bandwidth and the effectivetransmit antenna height with varying ground range.The steps in the curve result from the steps in thePRF and the corresponding SNR scaling factor of thereconstruction network. In both cases the NESZ isbelow the required ¡19 dB.

V. OPTIMIZATION POTENTIALS

In a further step, strategies for optimizing theSAR system performance are developed. The firstsection identifies the main error sources, analyzestheir impact on the respective performance parametersand derives general strategies to minimize the errors.Then, in the following sections, specific techniquesaccording to those strategies are developed and the



Fig. 10. Top: NESZ versus ground range for coverage by 6distinct swaths. Bottom: Best (solid) and worst (dotted) NESZ forarbitrarily located swaths. Ground range gives pointing location of

transmit antenna beam.

results of the optimized system are presented. Asmain performance parameters to be optimized, wefocus on the signal-to-noise ratio, suppression ofambiguous energy, and geometric resolution. Theoptimization aims at improving the SNR and theambiguity suppression while keeping the resolutionat a constant level.

A. Error Sources and Optimization Approaches

1) Error Sources: In Section IIIE it was presentedthat the ambiguous energy in the signal afterreconstruction is determined by two aspects. First (asin any conventional SAR system) the joint antennacharacteristic of transmit and single receive aperturecontains information from azimuth angles thatcorrespond to Doppler frequencies outside the bandIs = [¡PRF ¢N=2,PRF ¢N=2]. These contributionsare subsampled and give rise to azimuth ambiguitiesin the image. In addition, the ambiguous energy isweighted and possibly amplified by the power spectraldensities of the reconstruction filter functions Pjm(f).In general, this amplification is more severe, thestronger the nonuniform sampling of the signal is.Concerning the noise power, Section IIIF showed thatthe processing by the reconstruction filter functionsmay cause a degradation of the SNR, as the filtersPjm(f) possibly amplify the noise power whilepreserving the signal power. Analogically to theambiguities, this amplification rises with increasing

nonuniform sampling of the signal due to the inversecharacter of the filter system. In this context, aninvestigation of the spectral properties of ©bf showsthat the degradation of the SNR for strong nonuniformsampling with PRF values above the optimum PRFis dominated by the spectral sub-bands m of order 1and N, i.e., the lowest and highest sub-band within thesystem bandwidth Is. Consequently, the singularityof the noise and ambiguity scaling for spatiallycoinciding samples (cf. Section IVC) is caused bythese sub-bands, while the contribution of the othersub-bands is not critical. This seems logical from aninformation theoretical point of view as the remainingsamples when skipping the coinciding channelsstill yield an effective sampling ratio that fulfils theNyquist criterion with respect to the bandwidth of theinner sub-bands.2) Optimization Potential: The above analysis

of error sources yields two main areas of possibleoptimization. First, the minimization of ambiguousenergy in the received signal allows for reducing theambiguities in the image. This means that antennapatterns on transmit and receive are adapted tooptimally confine the desired Doppler band, whichis the subject of a detailed analysis following inSection B that introduces the idea of tapering ontransmit, while Section C includes the adaptationof the receive characteristic. Besides, the azimuthprocessing shows potential for improving theperformance by minimizing the error amplificationcaused by the reconstruction filters. In a first simpleapproach, the ratio between PRF and BD can be set ina way to benefit from a large oversampling and allowfor eliminating frequency bands that primarily causethe degradation of the performance. Basically thiscorresponds to an appropriate lowpass filtering of thesignal to achieve acceptably moderate levels of SNRdegradation even near to singular PRFs. Accordingto the first paragraph it is necessary to filter the noutermost bands at the lower and upper border of theDoppler spectrum to suppress completely the strongrise in noise scaling at the nth singular PRF. Hence,the sampling ratio N ¢PRF has to exceed the processedbandwidth BD by a factor of N=(N ¡ 2 ¢ n) to keep theSNR degradation low:

BD · (N ¡ 2 ¢n) ¢PRF: (41)

Although a very effective way of reducing theSNR degradation, this requires a relatively high PRFto guarantee the necessary oversampling with respectto the processed bandwidth BD. This might causeproblems with the timing. Simply decreasing BDfor a given PRF represents the easier way to ensure(41), but is at the cost of a deteriorated geometricresolution. Nevertheless, a strongly improvedSNR might be obtained by a moderately degradedresolution. So, finally this results in a trade-offbetween azimuth resolution and SNR.



Fig. 11. Basic principle of pattern tapering on transmit.Suppression of aliased energy in combination with optimization of

signal energy.

Furthermore one can aim at improving theconditions for the processing by matching thephase centres and the PRF to obtain a data arraythat is sampled as uniform as possible. This can beeither achieved by an adaptive management of thePRF to obtain an improved spatial sampling, butthis approach shows only great potential in sparsearray systems and is hence only treated shortly inAppendix D. In single platform systems, it is morepromising to adapt the phase centers to the PRF byadding a reconfigurable preprocessing network tothe conventional beamforming obtaining a cascadednetwork as introduced and elaborated in Section C.In a next stage, the processing itself might be

modified, mitigating the negative effects of theinverse filter by introducing a trade-off betweenambiguity suppression and SNR optimization inthe processing strategy. This comprises for examplean adapted beamforming approach that merges theclassical ambiguity suppression strategy with a beamsteering approach that optimizes the signal power (cf.Section D). As one will see, the transition from thetechniques in Section C to Section D is smooth, asboth approaches are based on cascaded processingnetworks.

B. Pattern Tapering on Transmit

As derived in Section IIID, all spectral energyoutside the band [¡N ¢PRF=2,N ¢PRF=2] causesaliasing in the reconstructed signal and finally givesrise to ambiguities. This can be avoided by confiningthe Doppler bandwidth of the signal to N ¢PRF byan appropriate antenna pattern. In a very simpleapproach, one could just enlarge the dimension of thetransmit antenna resulting in a narrower pattern. Butas this is achieved only at the expense of resolution,a bigger antenna in combination with an adaptedtapering is to be used to provide improved ambiguitysuppression without degrading the resolution.Furthermore, the better the pattern is limited to therelevant Doppler frequencies, the better the emittedpower is used as less power is lost by illuminatingunwanted areas and consequently the NESZ of thesystem is improved. The basic idea is visualized inFig. 11.

Fig. 12. AASRN versus PRF for BD = 7:6 kHz and differenttransmit antenna sizes and patterns. sin(f)=f characteristic for anaperture of 3 m (solid), 4 m antenna with cos(x) excitation

(dashed), 4.6 m antenna with (sin(f)=f)2 characteristic (dotted)and approximately rectangular pattern (dotted dashed) for 11.2 m

antenna.

To demonstrate the potential of pattern taperingon transmit, we consider the system of Section IVand investigate different combinations of transmitantenna dimensions and excitations that can be eitherrealized by a separate transmit antenna or by usingan active array that offers the flexibility to use partsof the receiving antenna for transmit. Fig. 12 showsthe results for the azimuth ambiguity suppressionAASRN in comparison to the 3 m transmit antennawith a uniform taper and hence a sin(f)=f pattern(solid) as used in Section IV. The suppression isalready clearly improved for higher PRF values byapplying a 4 m antenna with a cos(x) excitation(dashed) and becomes even better with a 4.6 mantenna with a simple triangular tapering whichentails a (sin(f)=f)2 characteristic (dotted). However,to provide an improvement in suppression alsofor lower PRF values, a larger transmit antenna isnecessary. An unconventional realization is given bythe quasi-optimum–as it approximates a rectangularpattern ¡sin(x)=x excitation in combination with atransmit antenna of 11.2 m (dotted dashed) whatcorresponds to the antenna length on receive. Itis realized by 35 elements of 0.32 m length each.Especially the sin(x)=x-excitation demonstratesthe potential of tapering to efficiently cancel thespurious spectral components while preserving theresolution. Note that all scenarios show a resolutionbelow 1 m (BD = 7:6 kHz). A full exploitation ofthe benefits of tapering requires a fine adjustment ofantenna dimensions, PRF and BD, taking into accountthe trade-off between resolution and ambiguitysuppression.In the frame of advanced concepts of transmit

antenna architectures in azimuth, a next stepcomprises an antenna that allows for adaptivelyvarying the transmit phase center from pulse to pulseby using only certain parts of the antenna. This wouldenable a sliding phase center on transmit from pulseto pulse that has the potential to compensate for



Fig. 13. “Prebeamshaping on receive” network as analoguerepresentation of preprocessing network. Amplified RF signals ofvarious apertures are adaptively weighted and combined beforeA/D conversion. This yields N optimized output channels that

enter the following DBF network.

a nonoptimum PRF. By choosing the direction ofthe sliding phase center along or against the flightdirection, it is possible to adjust the spatial samplingresulting from too high or too low PRF values,respectively. The PRF range that can be compensatedis determined by the maximum displacement of thephase center, while the fine-tuning ability of theposition, i.e., the “step size” of the phase center frompulse to pulse, depends on the distance between singletransmit elements. Note that a shift of the phase centeris necessary every time it has reached its outermostposition. In the case of a too high PRF this requirescombining or even skipping some samples that mightoverlap while in the case of a too low PRF this resultsin “missing” samples within the synthetic aperture, asa gap occurs when the phase center is switched.

C. Cascaded Beamforming Networks I–AnaloguePrebeamshaping on Receive

In the context of system optimization moresophisticated beamforming on receive approachesare regarded as powerful tools to adaptively improvethe system performance. In a cascaded beamformingnetwork the existing system is extended by asecond network that is used for analogue or digitalpreprocessing of the multi-aperture SAR signalsbefore the reconstruction filter network introducedin Section III is passed. In this section, we stickto an analogue preprocessing technique of themulti-aperture RF signals, a so-called prebeamshapingon receive network, but the results are also valid fora digital preprocessing step introduced in Section D.The system is based on an antenna consisting of anumber of K independent elements. Then a networkfollows, allowing for an individual and reconfigurableweighting and combination of the elements’ signals

Fig. 14. Conventional multi-aperture system with subaperture sized0 equal to the phase center distance (top) and system with

prebeamshaping on receive network (bottom). The multiple use ofcertain elements yields mutually overlapping subapertures ofincreased length d01, but decreased phase center spacing d1.

resulting in N “virtual” output channels j. Theweights may vary from subaperture to subaperture andfrom channel to channel. This allows for “using” thesignal of a certain subaperture in more than one of thevirtual channels resulting in a spatial overlap of thesechannels. The contribution of the signal of element ito the virtual channel j is described by the complexcoefficient wij (cf. Fig. 13). In the case of analoguepreprocessing, each signal is amplified separately byan LNA before weighting, combining, and digitizingthe RF signals.The basic idea behind the network is to modify

the received signal in way to match it to thereconstruction network, primarily by flexibly settingthe number and phase center position of the virtualchannels that are formed. This allows for adjusting thespatial sample positions to the actual system PRF tominimize the nonuniform sampling. In addition, suchnetworks enable to implement pattern tapering onreceive thus offering a powerful tool to e.g. suppressambiguities. The reconfigurable structure allows forflexibly allocating the network resources to emphasizea specific system parameter that finally results inan adjustable trade-off between SNR, AASRN , andresolution.In the following the basic relations regarding

the formation of virtual channels are presentedto demonstrate the idea and potential of theprebeamshaping concept. Consider a multi-apertureantenna of overall length La =N0 ¢ d0, where d0



represents the length of the aperture that formsthe single channel. The spacing between the phasecenters is equal to the length of the subapertures andhence given by d0, which determines the optimumPRF (cf. Fig. 14, top). In Fig. 14, on the bottom, asimple realization of a prebeamshaping network isshown, where some of the receiving elements arepart of multiple virtual channels yielding a numberof N mutually overlapping subapertures of lengthd01 > d0. Due to the overlap of adjacent apertures,the resulting phase center spacing d1 decreases withrespect to the reference case shown on the top, whereno prebeamshaping network is applied, i.e., d1 < d0.Consequently while keeping the overall antennalength constant, the resulting spatial sampling of thesystem is modified as the corresponding optimumPRF is now defined by N and d1. This means thatone obtains a system that allows for flexibly settingthe phase centers appropriate to the operating PRFand improving the SNR. In this context, the modifiedSNR scaling factor of a cascaded beamformingnetworks has to be taken into account as we see inthe following. In addition, a larger subaperture lengthis available enabling pattern tapering of the receivingapertures by adjusting the network’s weightingcoefficients. The enlarged subaperture length d01and the corresponding phase center distance d1 areconnected by (42) and define the overlap dov ofadjacent apertures as given in (43), resulting in a newoptimum PRFuni,c given by (44)

d1 =La¡ d01N ¡1 =

N0 ¢d0¡ d01N ¡ 1 (42)

dov = d01¡ d1 (43)

PRFuni,c =2 ¢ vsN ¢ d1

=N0 ¢ (N ¡ 1)

N ¢ ¡N0¡ d01=d0¢ ¢PRFuni:(44)

Note that the preprocessing network has an impacton signal, ambiguities and noise power and thus theexpressions describing these parameters that werederived in Sections IIID—IIIF have to be adapted tothe extended beamforming scenario. The followingparagraph briefly summarizes the differences to theconventional beamforming network and the resultingrelations that are derived in detail in Appendix A.The relevant signal for the output signal power isdetermined by the input signals of the reconstructionnetwork (cf. Section IIID). These signals are givenby the N output signals of the prebeamshapingnetwork, which are defined by the weighted sum ofsignals received by the single elements. The respectiveweighting is given by the coefficients wij(f), thatcan be Doppler frequency dependent in the case ofa digital preprocessing network (cf. Section VD).Further, similar to Section IIIA, the received signalat a single element can be modeled as an equivalent

monostatic signal U(f) filtered with an appropriatefunction Mi(f) that takes into account the offsetbetween transmitter and receiving element. Further,in an analogue implementation the need for powerdividers will result in amplitude imbalances in thesignals that have to be compensated for by adaptingthe weighting functions wij according to (52) (cf.Appendix A). This allows for setting up the followingrelation between input signal power ps,el and outputsignal power before and after focusing, given by ps,cin (45) and ps,c,BD in (46), respectively

ps,c = ps,el ¢N2 ¢G ¢E

24¯¯KXi=1

(Mi(f) ¢wij(f))¯¯235 (45)

ps,c,BD =N2 ¢G ¢E

24A(f)2 ¢ ¯¯KXi=1

Mi(f) ¢wij(f) ¢ rect(f=BD)¯¯235 :(46)

In a similar way, the power of the ambiguities isto be adapted to the prebeamshaping scenario. Thismeans that all expressions derived in Section IIIE canbe adapted to the extended processing network byadding the power gain G of the LNA and replacingU(f) with the resulting equivalent monostatic outputsignal of the prebeamshaping network Uc(f), whichis derived in Appendix A and represents the spectralappearance of the signals before the reconstructionnetwork, but takes into account the prebeamshapingon receive.Finally, let us consider the influence of the

prebeamshaping network on the output noise powerpn,c under the assumption of a lossless network ofweighting elements wij(f). As mentioned before,the receiver noise is assumed to be mutuallyuncorrelated additive white Gaussian noise. Thismeans that the input noise power pn,el is amplifiedby the prebeamshaping network and the followingreconstruction, yielding the output noise power inthe data ps,c and in the image ps,c,BD, according to(47) and (48), respectively. Note that in contrast tothe mere DBF network discussed before, the mutualcorrelations introduced by the prebeamshaping do notallow any more for simplifying the squared sum to asum of squares

pn,c=pn,el ¢G ¢F ¢KXi=1

E

24¯¯NXj=1

Pj(f) ¢wij(f)¯¯235 (47)

pn,c,BD =pn,el ¢G ¢F ¢KXi=1

E

24¯¯NXj=1

Pj(f) ¢wij(f) ¢ rect(f=BD)¯¯235 :

(48)

Combining the expressions for signal power (cf.(45)) and noise power (cf. (47)) and normalizing bythe optimum value F=K yields the SNR scaling factor



Fig. 15. Predicted SNR scaling factor ©bf of the theconventional DBF network (solid) compared with the case whereprebeamshaping network is added (dashed). Simulated values

overlaid in diamonds (BD = 7:6 kHz).

©bf,c of the cascaded network with respect to the data(cf. (49)). As before, the SNR scaling factor withrespect to the image cannot be given explicitly, butcan be determined by calculating the signal powerps,c,BD and the noise power pn,c,BD according to (46)and (48), respectively

©bf,c =pn,c=pn,elps,c=ps,el

¢ KF

=K ¢PK

i=1E·¯PN

j=1Pj(f) ¢wij(f) ¢ rect(f=IS)¯2¸

N2 ¢E·¯PK

i=1Mi(f) ¢wij(f) ¢ rect(f=IS)¯2¸ :

(49)As explained in detail in Appendix A, the

quantization noise power in an analogue realizationpn,q,a is described by (50), while (51) in Section Dwill give the noise resulting from a digitization beforethe preprocessing network. The comparison of (50)with (51) implies that digitizing as late as possiblereduces the scaling of the quantization error

pn,q,a = E

264¯¯ NXj=1

Pj(f) ¢ nqj(f)¯¯2375 : (50)

To demonstrate the potential of theprebeamshaping on receive technique we assume amodified system where each subaperture consists ofmultiple independent elements. With respect to thesystem presented in Section IV, the new system isdesigned to be applicable in a larger PRF range from1150 to 1550 Hz. Hence, the subaperture length isincreased to 1.75 m resulting in an overall length of12.25 m and a decreased optimum PRF of 1234 Hz.Further, a slightly increased aperture antenna ontransmit of 3.15 m is required to ensure a sufficientambiguity suppression as we see later. All othersystem parameters remain unaltered.An analysis of the SNR scaling factor ©bf of

Fig. 16. AASRN versus PRF with prebeamshaping network(dashed) compared with conventional DBF (solid) (BD = 7:6 kHz).

Peak in conventional approach occurs at “singular”PRF = 1440 Hz (cf. Section IIC).

the reconstruction network shows sufficiently lowvalues from 1150 Hz up to approximately 1350 Hz,but yields unacceptably high values for the PRFrange above (cf. Fig. 15, solid line). To providea low SNR scaling over the complete PRF range,a reconfiguration of the prebeamshaping networkbecomes necessary for higher PRF values. In thepresent case, mutually overlapping subapertures of2.625 m length are formed, resulting in a decreasedphase center spacing that corresponds to a newoptimum PRF of 1346 Hz. In order to keep thebeamwidth of the receiving pattern constant and tosuppress its sidelobes, a cosine taper is applied to eachof the subapertures. The respective ©bf,c is given inFig. 15 (dashed line). It is normalised to the optimum©bf of the conventional network and takes intoaccount the impact of the subaperture dimension andtaper on noise power, signal peak power, and azimuthloss Laz that has increased to 3.05 dB due to thepreprocessing network compared with Laz = 2:9 dB inthe conventional case. Fig. 15 shows that an improvedSNR scaling factor is obtained for PRF values above1240 Hz.In terms of the NESZ, we focus on the differences

of the modified system to the system of Section IV.First, one can state a degradation of 0.2 dB due to theincreased Laz. It results from an increased transmitantenna and an increased receive aperture giving riseto an overall Laz of 2.9 dB compared with 2.7 dBof the original system. In contrast, the new antennadimensions result in an increased gain of 0.2 dB ontransmit and 0.4 dB on receive and the SNR scalingfactor of the reconstruction network (cf. Fig. 15) tendsto be better than in the original system (cf. Fig. 9),which might be compensated by the decreasing SNRelwith increasing PRF according to (40). Altogether, theNESZ of the original system (cf. Fig. 10) gives a goodestimation for the new NESZ.Concerning the resolution, the increased

subaperture length of 1.75 m with respect to thelength of 1.6 m in the original system entails a



minor degradation to 1.03 m, both for conventionaland prebeamshaping system operation. Finally, theambiguous energy suppression of the modifiedsystem is investigated. As mentioned above, thetransmit antenna was slightly increased to 3.15 mto guarantee a minimum suppression of ¡21 dB (cf.Fig. 16). When the preprocessing network is applied,the suppression becomes better for PRF valuesabove » 1200 Hz and is clearly improved for higherPRF values due to reduced sidelobes of the receivepattern and minimized amplification of the azimuthambiguous energy caused by the adapted phasecenters (cf. Fig. 16, dashed line). In combinationwith the results for the SNR scaling factor ©bf ,it is hence favourable to use the conventionalconfiguration below a PRF of 1200 Hz and apply theprebeamshaping up from 1240 Hz. In between thefocus can be either turned to the NESZ by choosingthe conventional approach or the preprocessing isapplied to concentrate on the optimization of theAASRN .

D. Cascaded Beamforming Networks II–AdaptedDigital Beamforming

Similar to the prebeamshaping network presentedbefore, this approach is based on subsequentprocessing networks, but in contrast to the analogueprebeamshaping, all signals are sampled, digitized,and stored before processing. Hence, the amount ofdata to be handled limits in this case the maximumnumber of individual elements but a completea posteriori processing becomes possible enablingwide flexibility and reconfigurability in terms oforder and structure of the networks. As the basicprinciple is again a cascaded structure of networks,the expressions for the SNR scaling factor ©bf in(45)—(49) are valid.Recall the “classical” reconstruction algorithm

presented in Section III that aims at recovering theunambiguous Doppler spectrum by suppressing theambiguous frequency components. In terms of thegroup antenna characteristic that results from theweighting coefficients of the beamforming algorithm,this corresponds to placing nulls at angles whereambiguous Doppler frequencies are situated. Further,one observes that only for uniform sampling, themaximum in the group characteristic is steered atthe Doppler frequency that is to be recovered. Forincreasing nonuniformity, the maximum drifts awayfrom that angle resulting in a decreased gain. Hence,the degradation of the SNR is caused by a loss ofthe signal energy that is compensated for by a laterscaling of the signal that lifts up the noise floor. Abasic idea of the “adapted beamforming” by cascadednetworks is now to introduce a trade-off betweenambiguity suppression and optimized signal energy

to improve the SNR. This means that a number ofdegrees of freedom is used for placing nulls in thegroup pattern while other weights are chosen tooptimize the signal power. Basically this approachresults in the cascaded structure of two subsequentnetworks where one effectuates the steering ofthe mainlobe while the other cancels the azimuthambiguities. The order of the networks and thecombinations of the subapertures are in principlearbitrary as long as the structure allows for themultiplication of the respective patterns of the twostages. This means that the weighting functions are thesame for each of the formed channels. Such networksshow flexibility with respect to the receiving patternand the effective sampling ratio given by the numberof different reconstructed channels. Further, the spatialsample positions can be adapted, but only with certainflexibility as this is directly linked to the number ofreconstructed channels. Regarding SNR, AASRN ,and resolution this results in a trade-off between theeffective sampling ratio, the receiving pattern, and aneventual amplification by the reconstruction algorithmdepending on the spatial sampling.Concerning the scaling of the quantization noise,

the position of the A/D converter with respect tothe weighting networks has to be considered, asonly processing steps after the digitization affectthe quantization error. This means that in contrastto (50) the preprocessing is taken into account fordetermining the power of the quantization noise pn,q,d

pn,q,d =KXi=1

E

264¯¯ NXj=1

Pj(f) ¢wij(f) ¢nqi(f)¯¯2375 : (51)

One possible realization of a cascaded networkstructure is given by Fig. 17, where the multi-aperturereconstruction is followed by a beam-steering withweights wi(f). Consider an antenna of N0 receivingelements, where N channels are processed by themulti-aperture reconstruction algorithm, respectively,to form N2 output channels that may mutually overlapor not. The resulting N2 channels are then weightedand combined where the weighting functions wi(f)are chosen to steer the maximum of the joint antennacharacteristic for every f 2 [¡BD=2,BD=2] in thedirection corresponding to that frequency. As themulti-aperture reconstruction processing does notaffect the signal envelope, the beamwidth of theoutput signal is defined by the single element length,while the gain is determined by the number ofchannels that are combined in the second stage. Thismeans that the signal amplitude gain rises by a factorof N2 at the cost of an effective sampling ratio thatis decreased by a factor of N0=N and an increasednoise floor. The latter is caused by mutual correlationsbetween the noise components introduced by partlyoverlapping output channels. Equations (45)—(49)



Fig. 17. Cascaded networks for ambiguity suppression followedby beam-steering with mutual overlap of adjacent filter groups of

first stage.

Fig. 18. Cascaded networks: Beam-steering and pattern taperingfollowed by reconstruction of N channels with mutual overlap ofadjacent beam-steering groups. This representation is similar toprebeamshaping network but with less elements and A/D

conversion carried out before processing.

describe the resulting SNR scaling factor of thenetwork.The commutativity of the system allows for

changing the different stages as long as the resultingprocessing of each element’s branch remainsunaltered. Consequently, the system of Fig. 17can be equivalently represented by inverselyordered networks. Fig. 18 shows an example ofsuch a structure where the steering of the beam incombination with a possible tapering is carried outbefore multi-aperture reconstruction.One recognizes that the way of combining the two

stages offers a wide range of flexibility, for examplea closer spacing of the resulting phase centers can beachieved by subapertures that are no longer formedby adjacent but distributed elements. The minimumpossible phase center spacing is given by the distancebetween adjacent elements. It can be achieved ifsubapertures made up of N2 elements overlap byN2¡ 1 elements with their neighbored subaperturesor if distributed elements form the subapertures. Themaximum spacing is obtained in the conventionalcase where adjacent subapertures do not overlap.In combination with the number of channels thatare formed, this offers a wide range of adaptation

Fig. 19. Prediction (dashed) and simulation (diamond symbols)of SNR scaling factors ©bf for adapted beamforming compared

with prediction of conventional DBF (solid).

regarding the PRF. Note that all processing is applieda posteriori and consequently the system setting canbe reconfigured arbitrarily to focus on the respectiveperformance parameter of interest, be it ambiguitysuppression, resolution, or NESZ.In the following, the potential of the adapted

beamforming approach is demonstrated with a systemexample. Consider again the system of Section Cwith 7 subapertures of length 1.75 m, each, and aPRF range of operation from 1150 Hz to 1550 Hz.As derived in Section C, the SNR scaling factor ofthe reconstruction network shows sufficiently lowvalues up from 1150 Hz, but yields unacceptably highvalues for PRF values above 1400 Hz (cf. Fig. 19,solid line). To operate the system in these PRF ranges,the gathered data are processed in a different way:The signals of six adjacent subapertures, respectively,are processed by the multi-aperture reconstructionalgorithm, yielding two output channels that areweighted and combined to maximize the signal power.Changing the order of the processing networks, thisis equivalent to combining two adjacent receivingelements by the beam-steering, respectively, yieldingsix channels that are reconstructed unambiguously inthe second stage. A further interpretation arises, if thebeam-steering is combined with the multi-aperturereconstruction to a single filter function for eachchannel. In this case two systems of six elementseach are obtained, where the respective processingis equivalent to the multi-aperture reconstructionif the systems are considered separately. Thereconstructed signals from these systems are simplyadded afterwards.Due to the adapted processing the number of

phase centers is reduced and consequently theoptimum PRF is shifted to 1440 Hz. Further,mutual correlations cause the noise to rise. Thisis compensated by the increased signal power dueto the doubled area of each subaperture, finallyleading to a SNR scaling factor ©bf that is improvedby 0.35 dB when comparing the values at therespective optimum PRFs (cf. Fig. 19, dashedline). Due to the reduced effective sampling, thiscurve starts only at PRF = BD=6. As the azimuth



Fig. 20. AASRN for adapted beamforming (dashed) comparedwith conventional DBF (solid) (BD = 7:6 kHz). Peak in

conventional approach occurs at “singular” PRF at 1440 Hz(cf. Section IIC).

loss remains the same due to the beam-steering,the NESZ of this system can be deduced from theNESZ derived in Section C in combination withthe modified ©bf of Fig. 19. Additionally, a “loss”factor of 6=7 has to be considered, as the number ofeffective channels decreased from N = 7 to N = 6 (cf.(40)).Next, the AASRN of the system is investigated

and presented in Fig. 20. The solid line shows thesuppression of the original system that is improvedby the adapted beamforming above 1340 Hz(dashed line). In combination with Fig. 19, one canconclude that the conventional approach is suitablebelow 1340 Hz, while the adapted beamforming isfavourable for PRF values above.Regarding the resolution, the adapted beamforming

yields a value of 1.04 m similar to the conventionalcase.

VI. DISCUSSION

The paper summarized the state-of-the-art for SARsystems and principles that enable HRWS imagingand recalled the basic properties of multi-apertureSAR signals regarding the spatial sampling. Then,an algorithm for DBF on receive in multi-apertureSAR systems was presented that allows for theunambiguous recovery of the azimuth signal even incase of a nonuniformly sampled synthetic aperture,thereby avoiding any stringent PRF restriction. Thismulti-aperture reconstruction algorithm combinesthe individual receiver signals in a linear space-timeprocessing and can be interpreted as solving a linearsystem of equations in a way that ambiguous partsin the azimuth signal are canceled. In a next step, thealgorithm was investigated in detail with respect tothe influence of the processing in the DBF networkon signal power, ambiguities, and noise, yieldinganalytical expressions for the estimation of residualambiguities and the determination of the SNR scalingfactor ©bf of a beamforming network. This analysisshowed that especially strong nonuniform samplingof the azimuth signal may cause a degraded SNR dueto an increased network SNR scaling factor and an

amplification of residual azimuth ambiguities. Thetheory was verified in simulations when evaluatingan example system that was designed to image aswath of 100 km with a geometric resolution of 1 m.Concerning the performance parameters, simulationswere carried out including a comparison of thereconstruction algorithm to alternative methods ofprocessing the azimuth signal. The results showedthat only the reconstruction algorithm and the verysimilar null-steering technique provide high resolutionin combination with efficient ambiguity suppression.In a next step, the main error sources that may disturbthe output signal were identified to be aliased partsin the received signal and the amplification of errorsand noise in the input signal due to the processing inthe case of nonuniformly sampled data. Based on thisanalysis, optimization strategies as pattern tapering ontransmit (Section VB), prebeamshaping on receive(Section VC) and adapted beamforming networks(Section VD) were introduced and the resultingperformance improvement demonstrated. Patterntapering on transmit enables an efficient suppressionof ambiguous energy by adapting the pattern tothe processed Doppler bandwidth and thus limitingthe aliased part in the signal. In an extension of anadvanced transmit architecture, an adaptive adjustmentof the transmit phase center to the PRF is possible byletting the phase center “slide” over the whole antennafrom pulse to pulse. Prebeamshaping on receivenetworks can enforce the limitation of the Dopplerspectrum by tapering the receive pattern. Furthermore,such networks allow for adaptively adjusting thephase center positions to the PRF thus optimizing thespatial sampling. As a consequence, the system can beoperated in a wider PRF range. Finally, the adaptedbeamforming method established the idea of pushingthe processing from a pure suppression of ambiguousenergy to a trade-off between SNR and ambiguitysuppression by combining a beam-steering approachwith the reconstruction algorithm. In the case ofprebeamshaping on receive networks and the adaptedbeamforming approach, the theoretical examination ofresidual errors, ambiguities and the SNR scaling factorwas extended to the class of cascaded beamformingnetworks. The extensions consider mutual correlationsbetween different channels in contrast to conventionalbeamforming systems.The focus of future work is turned to distributed

SAR systems with multiple receiver satellitesforming a sparse array SAR system. In principlethe reconstruction algorithm is also applicable tothose systems, but new aspects of importance arisethat have to be considered. Firstly, the increasedalong-track separations of the receiving apertureshave to be taken into account as they entail a highsensitivity of the sampling against PRF variationsdue to modulo influence on the effective sampleposition. On the one hand, this might cause problems,as small PRF changes have a huge impact on the



sampling scenario, but on the other hand this offers abig opportunity for optimizing the spatial sampling byadapting the PRF as was mentioned in Section VA andAppendix D. Further, large along-track displacementsmight require a steering of the antenna footprintsto illuminate a joint area on ground resulting in adifferent squint of the satellites and thus the receptionof mutually shifted Doppler spectra in the receivers.This might lead to a subsampling and require newprocessing approaches, e.g. the combination ofmulti-aperture reconstruction and superresolutiontechniques [37]. In this context, the investigation ofthe impact of different antenna gains versus Dopplerfrequency on the processing will also be of interest.Secondly, any cross-track separation of the receiverswill introduce an additional phase in the receivedsignals, which has to be compensated, e.g., via thesimultaneous acquisition of a digital elevation modelin case of multiple satellites. Further, a sparse arrayconfiguration has to be carefully investigated withrespect to a two-dimensional processing approach asdifferent sensors might receive the signal of the sametarget with different ranges and range cell migrationsas introduced in [28].Further potential fields of application for DBF

are e.g. the application of the ScanSAR or theTOPS-SAR principle in multi-aperture systems,enabling ultrawide swaths. In this context the impactof the multi-aperture reconstruction algorithm whenoperated in combination with burst modes is to beinvestigated with respect to the particular properties ofsuch modes. As the scaling of the ambiguous energyis strongly dependent on the Doppler sub-band, theuse of different subspectra for different targets in aScanSAR or TOPS-SAR mode will result in a largevariation of the residual azimuth ambiguities of thesetargets. So, in ScanSAR operation and even in theTOPS-SAR mode, this can result in a scalloping-likeeffect for the azimuth ambiguities, which is currentlyinvestigated in detail.

APPENDIX A. CASCADED BEAMFORMINGNETWORKS

In the context of system optimization moresophisticated beamforming approaches are regardedas powerful tools to adaptively improve the systemperformance. Sections VC and VD establishedthe idea of cascaded beamforming networks andinvestigated the new concept with respect to thenetwork capability to adaptively position the systems’phase centers. In the following, the impact on signal,ambiguities and noise power that were only mentionedbriefly in Section VC are derived in more detail.

A. Extended System Model

Based on the system model presented inSection IIIF and on the block diagram given inFig. 13, we obtain new system models that are

Fig. 21. Multi-aperture system model taking into accountnetwork to perform prebeamshaping on receive before signals are

sampled and pass through reconstruction filter network.

Fig. 22. Multi-aperture system model including cascadednetworks to perform preprocessing after the signals are sampledand before they pass through reconstruction filter network.

extended by a block representing the additionalprocessing steps, either a prebeamshaping (PBS)network situated before the analog-to-digital converter(ADC) (cf. Fig. 21) or a network processingthe already digitized data. In the case where thepreprocessing network is located behind the ADC,the order of the applied filters can of course bechanged as long as the overall transfer functionof each branch remains the same. The weightfunctions are denominated by wij that (althoughnot necessary) may be frequency dependent inthe case of a filtering after the ADC (cf. Fig. 22).We start with a linear system of K channels i withan input signal ui(t), a receiver noise componentni,B(f), and the corresponding signal-to-noise ratioSNRel,i, respectively. As before, receiver noise isconsidered to be the dominating thermal noise sourceand is assumed to be additive, white, and mutuallyuncorrelated between the channels. After reception,the signal is amplified before the K input signalsare weighted and combined to form N intermediateoutput signals, each representing a virtual channelfor the reconstruction filter network. These channelsare then (either before or after the preprocessing)subsampled in azimuth with a ratio of PRF inducinga quantization error that is modelled as additivenoise sources nqj(f) and nqi(f), respectively. Thenfiltering by the reconstruction filters Pj(f) takes placefollowed by coherent combination of all N outputbranches yielding the signal uout[k] (cf. Fig. 21,Fig. 22).

B. Compensation for Power Loss

In the analogue representation of a preprocessingnetwork, amplitude imbalances might occur asnot all elements’ signals are to be split equally.Hence, depending on the necessary branching of therespective signal, the gain loss is different and has to



be compensated to ensure uniform amplitude of allsignals when entering the first processing network.This can be achieved by including the compensationin the weighting functions of this first stage byreplacing the amplitude gain jwij j by the new factorjw0ij j according to (52), where Nch,i represents thenumber of signals in which the original signal ofelement i is split. By this, the amplitude of the signalafter being split is recovered as given in (53), whereUi,split(f) gives the signal after splitting while Ui(f)denotes the original signal of channel i. With theseadapted weighting functions all following equationsremain valid

jw0ij j= jwij j ¢qNch,i (52)

Ui,split(f) ¢w0ij(f) =Ui(f)pNch,i

¢w0ij(f) =Ui(f) ¢wij(f):

(53)

C. Signal Power

The desired signal that is relevant for the outputsignal power is determined by the input signal of thereconstruction network (cf. Section IIID) and doesnot depend on the location of the ADC. In contrastto the case where no prebeamshaping is applied, thissignal is given by Uc,j(f) that forms the input of thereconstruction filter network and is determined by theweighted sum of signals Ui(f) received by the singleelements and respective weighting coefficients

Uc,j(f) =KXi=1

Gi ¢Ui(f) ¢wij(f): (54)

As derived in Section IIIA, Ui(f) can be modelledas an equivalent monostatic function U(f) filteredwith an appropriate function that corresponds basicallyto a time-shift and the addition of a constant phase.To avoid confusion with the reference functionsHj(f) that characterize the virtual channels, wedenominate these functions Mi(f). We obtain (againassuming identical LNAs in all channels and takinginto account (52) if analogue preprocessing is applied)the following, where U(f) represents the signalreceived by a single element at the same position asthe transmit aperture

Uc,j(f) =KXi=1

Gi ¢Ui(f) ¢wij(f)

=G ¢U(f) ¢KXi=1

Mi(f) ¢wij(f): (55)

In a next step we assume the wij(f) being thesame for every virtual channel j which meansthat all resulting virtual channels j are identical interms of their envelope and only shifted relative toeach other with respect to their phase centers. This

interpretation allows for expressing the signals by aequivalent monostatic signal Uc(f) and functions Hj(f)characterizing the virtual channels j as done in (12).We can simplify (55)—(56), where Uc(f) correspondsto the signal that arises by the weighted sum of thesingle elements, if the resulting phase center wassituated at the same position as the transmitter. TheLNA’s power gain G is mentioned separately as it willcancel later

Uc,j(f) =G ¢Uc(f) ¢Hj(f): (56)

Only the prebeamshaping changes thecharacteristic of Uc(f) while the DBF network thatreconstructs the signal only introduces a scalingfactor of N without modifying the pattern of Uc(f).Hence, the signal power is determined by the spectralappearance of Uc(f) given by the resulting patterncharacteristic of the combined subapertures and thenumber of virtual channels N. Using (57), we obtainthe following output signal after reconstruction in thedata (cf. (58)) and in the image after focusing (cf.(59)) that correspond to (45) and (46), respectively

jUc(f)j=¯¯U(f) ¢

KXi=1

Mi(f) ¢wij(f)¯¯

= A(f) ¢¯¯KXi=1

Mi(f) ¢wij(f)¯¯ (57)

ps,c =N2 ¢G ¢E[jUc(f) ¢ rect(f=IS)j2]

= ps,el ¢N2 ¢G ¢E

24¯¯KXi=1

Mi(f) ¢wij(f)¯¯235 (58)

ps,c,BD =N2 ¢G ¢E

24¯¯U(f) ¢KXi=1

Mi(f) ¢wij(f) ¢ rect(f=BD)¯¯235 :

(59)

D. Residual Reconstruction Error

If a preprocessing network is used beforereconstruction is applied, the power of the ambiguitiesis to be adapted to the prebeamshaping scenario. Thismeans that all expressions derived in Section IIIEcan be adapted to the extended processing networkby adding the gain factor G and replacing U(f) bythe resulting output signal of the prebeamshapingnetwork Uc(f) defined by (56), that representsthe spectral appearance of the signal before thereconstruction network, but takes into account thepreprocessing (either prebeamshaping on receiveor adapted beamforming). Assuming that all virtualchannels that enter the reconstruction filter networkare identical in terms of weighting of the elements thatform the channel, Uc(f) is determined by the weighted



sum given in (56) but does not include the gain G ofthe LNA.

E. SNR Scaling of Cascaded BeamformingNetworks–©bf,c

Let us consider the influence of theprebeamshaping network on the noise power underthe assumption of a lossless network of weightingelements wij(f). As mentioned above, the receivernoise is assumed to be mutually uncorrelated additivewhite Gaussian noise ni,B(f) with a power spectraldensity of N0,B with respect to the system bandwidthB. Further, quantization noise is assumed to beuniformly distributed and spectrally white as proposedin [40]. Again identical elements i, implying the samecharacteristics for Gi, Fi, gi, fi for all i are assumed,yielding the following output noise power pn,c of thesystem

pn,c = pn,el ¢G ¢F ¢KXi=1

E

264¯¯ NXj=1

Pj(f) ¢wij(f)¯¯2375

| {z }Receiver noise pn,c,rx

+ pn,q|{z}Quantization noise

= pn,c,rx+pn,q: (60)

This means that the input noise power isamplified by the prebeamshaping and the followingreconstruction and weighted quantization noise isadded. Further, in contrast to a mere DBF network asdiscussed before, the mutual correlations introducedby the prebeamshaping do not allow any morefor simplifying the squared sum that gives thethermal output noise power to a sum of squares. Thisamplification of the receiver noise is independentof the system, while the quantization error dependson the location of the ADC. If the preprocessing isapplied before digitization, the quantization noise atthe output is given by (61) while (62) is valid if thequantization error is introduced in the signal beforethe preprocessing network is passed

pn,q,a = E

264¯¯ NXj=1

Pj(f) ¢ nqj(f)¯¯2375 (61)

pn,q,d =KXi=1

E

264¯¯ NXj=1

Pj(f) ¢wij(f) ¢ nqi(f)¯¯2375 : (62)

Again, the number of bits is assumed to be chosensuch that the quantization error in the output signalwill be negligible compared with the (thermal)receiver noise and consequently the overall noise is

approximated by pn,c,rx. Analogously to (34) that givesthe SNR scaling factor of conventional beamformingnetworks, the SNR scaling factor of the cascadednetwork with respect to the data ©bf,c (cf. (63))is determined by expression (58) (that gives theoutput signal power) and the first term of (60) thatapproximates the noise power. As before, the SNRscaling factor for the focused image cannot be givenexplicitly, as the relation between the input and outputsignal power depends on the shape of U(f) and otherparameters as defined in (59). Of course ©bf,c canbe calculated using (59) for the signal power andexpression (64) giving the remaining the noise powerafter focusing

©bf,c =pn,c=pn,elps,c=ps,el

¢ KF

=

K ¢PK

i=1E

·¯PN

j=1Pj(f) ¢wij(f) ¢ rect(f=IS)¯2¸

N2 ¢E·¯PK

i=1Mi(f) ¢wij(f) ¢ rect(f=IS)¯2¸

(63)

pn,c,BD= pn,el ¢G ¢F ¢

KXi=1

E

24¯¯NXj=1

Pj (f) ¢wij(f) ¢ rect(f=BD)¯¯235 :(64)

APPENDIX B. MULTI-APERTURE RECONSTRUCTIONIN THE FRAME OF CLASSICAL STAP

Consider, according to [47], a multi-channelbeamforming system consisting of N channels. Eachchannel i is described by its input signal si, noisecomponent ni, and weighting function ai that aresummarized by s= [s1, : : : ,sN]

T, n= [n1, : : : ,nN]T

and a= [a1, : : : ,aN]T, respectively. The output y of

the system can then be described by (65), where theoperator ± denotes the scalar product

y = a ± s+ a ± n: (65)

Further, if we assume an incident plane waveand use quasi-parallel approximation, each si can berelated to s1 by a phase term that accounts for theazimuth angle-dependent path difference betweenreceivers i and 1. Hence, s simplifies to a steeringvector, where μ gives the azimuth angle and ¢xidenotes the distance of receiver i to receiver 1.Without restriction of generality we assume ¢x1 = 0and finally obtain

s=

·1,exp

·j2¼¸sin(£)¢xi

¸: : : ,exp

·j2¼¸sin(£)¢xN

¸¸T:

(66)



If coloured noise is assumed, the optimumbeamforming weights a to minimize the noise powerare defined by

a¤ =R¡1 ± sk

(67)

where * denotes the complex conjugate, k representsa normalization factor and R describes the noisecovariance matrix that consists of the covariancematrix of the spatially structured noise Rc and theunity matrix I representing the white noise of power¾2:

R= E[n ± n¤] =Rc+¾2I: (68)

In the special case when interpreting theambiguous Doppler frequencies as a directiveinterference arriving from the respective angles andfurther assuming that no white noise is present,i.e., ¾ = 0, one can show that (69) holds true, if thedependency on the azimuth angle μ is replaced byDoppler frequency f

a¤(f) =R(f)¡1 ± s(f)

k=Rc(f)

¡1 ± s(f)k

= P¤(f):

(69)

The matrix P represents the reconstruction filterfunctions derived in Section IIIB and contains in itsith line the weighting function ai. Hence P offers analternative way of deriving the optimum beamformerto suppress azimuth ambiguities that are interpreted asdirective intereferences in the STAP context.

APPENDIX C. ANALOGY OF NULL-STEERING ANDMULTI-APERTURE RECONSTRUCTION

Consider a linear array of N receiving apertureswhere the distance of aperture i to aperture 0 isdenoted by ¢xi and the look direction is designatedby μ (cf. Fig. 23). By definition we set ¢x1 = 0. Afterreceiving the incident signal sin(μ), an individualcomplex weighting coefficient ai is applied to eachof the channels’ signals si(μ). Finally these signals arecombined coherently, what is indicated by the “§,”yielding the output signal s(μ).Applying the quasi-parallel approximation, the

well-known array manifold vector v that characterizesthe signal at each receiver i with respect to the signalat receiver 1 looks as follows [46]:

v(£) =·1,exp

μ¡j 2 ¢¼

¸¢¢x2 ¢ sin£

¶, : : : ,

expμ¡j 2 ¢¼

¸¢¢xN ¢ sin£

¶¸T= [v1, : : : ,vi, : : : ,vN]

T: (70)

Taking into account the weighting coefficientsgiven by the vector a= [a1, : : : ,aN]

T and thesubsequent coherent summation, the resulting

Fig. 23. Incident signal sin(μ), arriving under angle μ on lineararray of N receiving apertures. After reception, each receiver’ssignal si(μ) is weighted by ai and summed up yielding output

signal s(μ).

output signal s(μ) is determined by (71), where s1(μ)represents the signal at receiver 1

s(£) =NXi=1

si(£) ¢ ai

= s1(£) ¢NXi=1

(vi ¢ ai) = s1(£) ¢ v(£) ± a: (71)

Consider now N output signals for input signalsarriving from N different angles μj , j = f1,2, : : : ,Ng.The respective output signals are given by the vectors that can be expressed by the matrix-vector operationof (72), where V represents the manifold vectors forthe different angles as given in (73)

s1(£) ¢V ¢ a= s (72)

V= [v1, : : : ,vN] =

264 v1(£1) ¢ ¢ ¢ vN(£1)

¢ ¢ ¢ vi(£j) ¢ ¢ ¢v1(£N) ¢ ¢ ¢ vN(£N)

375 :(73)

The weighting coefficients which are necessary tosteer nulls in N ¡ 1 directions μj in order to “extract”the signal from the Nth direction are then determinedby the inversion of V. One obtains a matrix A=V¡1

that contains in its kth column the respective weightsto steer the nulls in the pattern to all directions μj ,j 6= k.In the next step, consider a frequency f1 and

its N ¡ 1 ambiguous frequencies within the systemazimuth bandwidth N ¢PRF, that are separated byinteger multiples of PRF (cf. (74)). Using the relationbetween angle μ and Doppler frequency f of (75), thevariable μj in (73) can be replaced by the respectivefj , what yields V(f) in (76). Then the matrix A(f) =V¡1(f) contains in its kth column the respectiveweights to steer the nulls in the pattern to allambiguous frequencies fj , j 6= k, while recovering fk.



Hence, calculating and applying A(f) for all f1 withinthe interval I1 = [¡N ¢PRF=2,¡(N ¡ 2) ¢PRF=2] andthe respective ambiguous Doppler frequency sets in(74) allows for unambiguously recovering the originalspectrum of bandwidth N ¢PRF

f1 + [0,PRF, : : : , (N ¡ 1) ¢PRF] =: [f1, : : : ,fj , : : : ,fN ] with f1 2h¡N ¢PRF

2,¡ (N ¡ 2) ¢PRF

2

i(74)

f(£) =2 ¢ v¸¢ sin£ (75)

V(f) =

26666666664

exph¡j ¢ ¼ ¢¢x1

v¢fi1

¢ ¢ ¢ exp

·¡j ¢ ¼ ¢¢xN

v¢f1¸

¢ ¢ ¢ exp

·¡j ¢ ¼ ¢¢xi

v¢ (f1 + (j¡ 1) ¢PRF)

¸¢ ¢ ¢

exph¡j ¢ ¼ ¢¢x1

v¢ (f1 + (N ¡ 1) ¢PRF)

i¢ ¢ ¢ exp

·¡j ¢ ¼ ¢¢xN

v¢ (f1 + (N ¡ 1) ¢PRF)

¸

37777777775:

(76)

Let us now compare above shown matrix V(f)to the system matrix H(f) derived in Section III. Weobserve that under the assumed approximations, V(f)is nearly identical to H(f). The only difference is theconstant phase term relating the bi- and monostaticsample position, that is accounted for in H(f) whileit is neglected in V(f). Hence the multi-aperturereconstruction algorithm can be regarded as anextension of the classical null-steering approach thatalso accounts for large receiver spacing. In the caseof a single-platform system, the distances between thereceiving apertures are small, and consequently theresulting phase differences are negligible.

APPENDIX D. ADAPTIVE PRF MANAGEMENT INSPARSE ARRAY SYSTEMS

As mentioned in Section VA, the conditionsfor the multi-aperture processing can be improvedby harmonizing the phase centers with the PRF toobtain a sampling as uniform as possible. In sparsearray systems, an adaptive management of the PRFshows great potential to adjust the virtual samplepositions. Recalling the results derived in Section II,we can express the distance between adjacent samplesoriginating from transmit pulse k at channel j andtransmit pulse l at channel i by (77), where ¢xjdenotes the distance of the receiver to the transmitter

¢xj,i = xj,k ¡ xi,l =¢xj2¡ ¢xi

2+ (k¡ l) ¢ vs

PRF,

k, l 2 Z: (77)

In a conventionally operated single platformsystem, adjacent samples originate from the same orsubsequent pulses, i.e., jk¡ lj · 1, and consequently

the inter-sample distance is smaller than theinter-pulse distance. Hence in such a system, thepotential is limited as small changes in the PRFdo not allow for large variations in the sampling.In contrast, in sparse arrays, the spatial position of

adjacent samples is not bound to the same transmittedpulse as in single platform systems. One recognizesthat variations in the PRF result in variations of theposition ¢xj,i amplified by a factor (k¡ l) and henceonly small changes in the PRF might already have alarge impact on the spatial distribution of the samples.This offers wide flexibility to adapt the sampling evenif only a small variation of the PRF is possible andallows thus for compensating for the high sensitivityof the systems’ SNR and AASRN regarding PRFvariations.

REFERENCES

[1] Curlander, J., and McDonough, R.Synthetic Aperture Radar–Systems and Signal Processing.New York: Wiley, 1991.

[2] Carrara, W., Goodman, R., and Majewski, R.Spotlight Synthetic Aperture Radar: Signal ProcessingAlgorithm.Boston: Artech House, 1995.

[3] Tomiyasu, K.Conceptual performance of a satellite borne, wide swathsynthetic aperture radar.IEEE Transactions on Geoscience and Remote Sensing,GRS-19 (1981), 108—116.

[4] Moore, R. K., Claassen, J. P., and Lin, Y. H.Scanning spaceborne synthetic aperture radar withintegrated radiometer.IEEE Transactions on Aerospace and Electronic Systems,AES-17 (1981), 410—421.

[5] De Zan, F., and Monti Guarnieri, A. M.TOPSAR: Terrain observation by progressive scans.IEEE Transactions on Geoscience and Remote Sensing, 44(2006), 2352—2360.

[6] Claassen, J. P., and Eckerman, J.A system for wide swath constant incident anglecoverage.In Proceedings of Synthetic Aperture Radar TechnologyConference, Las Cruces, NM, 1978.



[7] Jain, A.Multibeam synthetic aperture radar for globaloceanography.IEEE Transactions on Antennas and Propagation, AP-27(1979), 535—538.

[8] Jean, B. R., and Rouse, J. W.A multiple beam synthetic aperture radar design conceptfor geoscience applications.IEEE Transactions on Geoscience and Remote Sensing,GRS-21 (1983), 201—207.

[9] Griffiths, H., and Mancini, P.Ambiguity suppression in SARs using adaptive arraytechniques.In Proceedings of IEEE International Geoscience andRemote Sensing Symposium (IGARSS), Espoo, Finland,1991, 1015—1018.

[10] Currie, A., and Brown, M. A.Wide-swath SAR.IEE Proceedings–Radar Sonar and Navigation, 139, 2(1992), 122—135.

[11] Callaghan, G. D., and Longstaff, I. D.Wide swath spaceborne SAR using a quad element array.IEE Proceedings–Radar Sonar and Navigation, 146, 3(1999), 159—165.

[12] Goodman, N., Rajakrishna, D., and Stiles, J.Wide swath, high resolution SAR using multiple receiveapertures.In Proceedings of IEEE Geoscience and Remote SensingSymposium (IGARSS), vol. 3, Hamburg, Germany, 1999,1767—1769.

[13] Younis, M., and Wiesbeck, W.SAR with digital beamforming on receive only.In Proceedings of IEEE Geoscience and Remote SensingSymposium (IGARSS), vol. 3, Hamburg, Germany, 1999,1773—1775.

[14] Suess, M., Grafmuller, B., and Zahn, R.A novel high resolution, wide swath SAR system.In Proceedings of IEEE Geoscience and RemoteSensing Symposium (IGARSS), Sydney, Australia, 2001,1013—1015.

[15] Suess, M., and Wiesbeck, W.Side-looking synthetic aperture system.EP 1 241 487 A 1, 2001.

[16] Wiesbeck, W.SDRS: Software-defined radar sensors.In Proceedings of IEEE Geoscience and Remote SensingSymposium (IGARSS), Sydney, Australia, 2001,3259—3261.

[17] Suess, M., Zubler, M., and Zahn, R.Performance investigation on the high resolution, wideswath SAR system.In Proceedings of European Conference on SyntheticAperture Radar (EUSAR), Ulm, Germany, 2002,187—191.

[18] Brule, L., and Baeggli, H.Radarsat-2 program update.In Proceedings of IEEE Geoscience and Remote SensingSymposium (IGARSS), Toronto, Canada, 2002.

[19] Goodman, N., Lin, S., Rajakrishna, D., and Stiles, J.Processing of multiple-receiver spaceborne arrays forwide area SAR.IEEE Transactions on Geoscience and Remote Sensing, 40,4 (Apr. 2002), 841—852.

[20] Krieger, G., Fiedler, H., Rodriguez-Cassola, M., Hounam,D., and Moreira, A.System concepts for bi- and multistatic SAR missions.In Proceedings of the ASAR Workshop 2003, Saint-Hubert,Quebec, Canada, June 2003.

[21] Krieger, G., and Moreira, A.Potentials of digital beamforming in bi- and multistaticSAR.In Proceedings of IEEE International Geoscience andRemote Sensing Symposium (IGARSS), Toulouse, France,2003, 527—529.

[22] Heer, C., Soualle, F., Zahn, R., and Reber, R.Investigations on a new high resolution wide swath SARconcept.In Proceedings of IEEE International Geoscience andRemote Sensing Symposium (IGARSS), Toulouse, France,2003, 521—523, 2003.

[23] Younis, M., Fischer, C., and Wiesbeck, W.Digital beamforming in SAR systems.IEEE Transactions on Geoscience and Remote Sensing, 41,7 (July 2003), 1735—1739.

[24] Younis, M., Venot, Y., and Wiesbeck, W.A simulator for digital beam forming SAR.In Proceedings of International Radar Symposium (IRS),Dresden, Germany, 2003.

[25] Mittermayer, J., and Runge, H.Conceptual studies for exploiting the TerraSAR-X dualreceiving antenna.In Proceedings of Geoscience and Remote SensingSymposium (IGARSS), Toulouse, France, 2003.

[26] Aguttes, J. P.The SAR train concept: Required antenna area distributedover N smaller satellites, increase of performance by N.In Proceedings of Geoscience and Remote SensingSymposium (IGARSS), Toulouse, France, 2003.

[27] Krieger, G., Gebert, N., and Moreira, A.Unambiguous SAR signal reconstruction fromnonuniform displaced phase center sampling.IEEE Geoscience and Remote Sensing Letters, 1 (2004),260—264.

[28] Krieger, G., Gebert, N., and Moreira, A.SAR signal reconstruction from non-uniform displacedphase center sampling.In Proceedings of Geoscience and Remote SensingSymposium (IGARSS), Anchorage, AK, Sept. 2004.

[29] Krieger, G., Gebert, N., and Moreira, A.Digital beamforming and non-uniform displaced phasecentre sampling in bi- and multistatic SAR.In Proceedings of European Conference on SyntheticAperture Radar (EUSAR), Ulm, Germany, 2004.

[30] Younis, M.Digital beam-forming for high-resolution wide swath realand synthetic aperture radar.Karlsruhe, Germany, 2004.

[31] Gebert, N., Krieger, G., and Moreira, A.SAR signal reconstruction from non-uniform displacedphase centre sampling in the presence of perturbations.In Proceedings of Geoscience and Remote SensingSymposium (IGARSS), Seoul, South Korea, 2005.

[32] Gebert, N., Krieger, G., and Moreira, A.High resolution wide swath SAR imaging–Systemperformance and influence of perturbations.In Proceedings of International Radar Symposium (IRS),Berlin, Germany, 2005.

[33] Li, Z., Wang, H., Su, T., and Bao, Z.Generation of wide-swath and high-resolution SARimages from multichannel small spaceborne SAR system.IEEE Geoscience and Remote Sensing Letters, 2, 1 (Jan.2005), 82—86.

[34] Li, Z., Bao, Z., Wang, H., and Liao, G.Performance improvement for constellation SAR usingsignal processing techniques.IEEE Transactions on Aerospace and Electronic Systems,42, 2 (2006), 436—452.



[35] Fischer, C., Heer, C., Krieger, G., and Werninghaus, R.A high resolution wide swath SAR.In Proceedings of European Conference on SyntheticAperture Radar (EUSAR), Dresden, Germany, 2006.

[36] Gebert, N., Krieger, G., and Moreira, A.High resolution wide swath SAR imaging with digitalbeamforming–Performance analysis, optimization andsystem design.In Proceedings of European Conference on SyntheticAperture Radar (EUSAR), Dresden, Germany, 2006.

[37] Krieger, G., and Moreira, A.Spaceborne bi- and multistatic SAR: Potential andchallenges.IEE Proceedings–Radar, Sonar and Navigation, 153(2006), 184—198.

[38] Papoulis, A.Generalized sampling expansion.IEEE Transactions on Circuits and Systems, CAS-24, 11(1977), 652—654.

[39] Brown, J.Multi-channel sampling of low-pass signals.IEEE Transactions on Circuits and Systems, CAS-28, 2(1981), 101—106.

[40] Oppenheim, A.Discrete-Time Processing.Upper Saddle River, NJ: Prentice-Hall, 1999.

[41] Wang, T., and Bao, Z.Improving the image quality of spacebornemulitple-aperture SAR under minimization of sidelobeclutter and noise.IEEE Geoscience and Remote Sensing Letters, 3, 3 (July2006), 297—301.

Nicolas Gebert received the Dipl.-Ing (M.S.) from the Munich University ofTechnology (TU Munchen), Munich, Germany, in 2003.He is currently as a Ph.D. student with the German Aerospace Center (DLR)

in Oberpfaffenhofen, Germany. His major research areas are digital beamformingconcepts with focus on signal processing, system design, and optimizationof multi-channel SAR systems for high performance imaging applications.Further interests are the development of innovative and advanced concepts forultra-wide-swath imaging with SAR systems.

[42] Marquez-Martínez, J., Mittermayer, J., andRodríguez-Cassola, M.Radiometric resolution optimization for future SARsystems.In Proceedings of Geoscience and Remote SensingSymposium (IGARSS), Anchorage, AK, Sept. 2004.

[43] Kraus, J.Antennas.New York: McGraw-Hill, 1950.

[44] Krieger, G., Gebert, N., and Moreira, A.Multidimensional waveform encoding: A new digitalbeamforming technique for synthetic aperture radarremote sensing.IEEE Transactions on Geoscience and Remote Sensing, 46,1 (2008), 31—46.

[45] Krieger, G., et al.TanDEM-X: A satellite formation for high-resolutionSAR interferometry.IEEE Transactions on Geoscience and Remote Sensing, 45,11, 1 (2007), 3317—3341.

[46] van Trees, H. L.Optimum Array Processing–Part IV of Detection,Estimation, and Modulation Theory.New York: Wiley, 2002.

[47] Guerci, J. R.Space-Time Adaptive Processing.Boston: Artech House, 2003.



Gerhard Krieger (M’03) received the Dipl.-Ing. (M.S.) and Dr.-Ing. (Ph.D.)degrees (with honors) in electrical and communication engineering from theTechnical University of Munich, Germany, in 1992 and 1999, respectively.From 1992 to 1999, he was with the Ludwig-Maximilians University, Munich,

where he worked as an interdisciplinary research scientist on the modeling ofbiological and technical vision systems. In 1999, he joined the Microwaves andRadar Institute (HR) of the German Aerospace Center (DLR), Oberpfaffenhofen,Germany, where he developed signal processing algorithms for an innovativeforward looking radar system. From 2001 to 2007 he led the New SARMissions Group which devised new bistatic and multistatic radar systems likethe forthcoming TanDEM-X mission as well as innovative digital beamformingtechniques for advanced multi-channel SAR. Since 2008, he has been Head of thenew Radar Concepts Department of the Microwaves and Radar Institute, DLR,Oberpfaffenhofen, Germany. His research interests include innovative remotesensing systems and applications based on radar interferometry, tomography, bi-and multistatic satellite formations, digital beamforming, and the development ofadvanced signal and image processing algorithms.

Alberto Moreira (M’92–SM’96–F’04) was born in Soo Jose dos Campos,Brazil, in 1962. He received the B.S.E.E. and the M.S.E.E. degrees, in 1984 and1986, respectively, from the Aeronautical Technological Institute ITA, Brazil andthe Eng. Dr. degree (Honors) from the Technical University of Munich, Germany,1993.In 2003, he received a full professorship from the University of Karlsruhe,

Germany, in the field of microwave remote sensing. As its chief scientist andengineer, he managed from 1996 to 2001 the SAR Technology Department of theMicrowaves and Radar Institute at the German Aerospace Center (DLR). Underhis leadership, the DLR airborne SAR system, E-SAR, has been upgraded tooperate in innovative imaging modes like polarimetric SAR interferometry andSAR tomography. Since 2001, he is the director of the Microwaves and RadarInstitute at DLR. The Institute contributes to several scientific programs andspace projects for actual and future air- and space-borne SAR missions. Recently,the mission proposal TanDEM-X lead by his Institute has been approved forthe realization phase. He is the principal investigator for this mission. Hisprofessional interests and research areas encompass radar end-to-end systemdesign and analysis, innovative microwave techniques and system concepts, signalprocessing, and remote sensing applications.Dr. Moreira is serving as a member of the IEEE GRSS Administrative

Committee (1999—2001, 2004—2006), is the chair of the German Chapterof the GRSS since 2003, and is actively serving as an associate editor forthe Geoscience and Remote Sensing Letters. Since 2003 he is also serving asa member of the Board of Directors of the ITG (Information TechnologySociety) of VDE (German Association for Electrical, Electronic and InformationTechnologies). In 1995, he was the recipient of the DLR Science Award. He andhis colleagues received the GRSS Transactions Prize Paper Awards in 1997 and2001, respectively. He is also the recipient of the IEEE Nathanson Award (1999)and the IEEE Kiyo Tomiyasu Award (2007). He has contributed to the successfulseries of the European SAR conferences (EUSAR) since 1996 as member of theTechnical Program Committee, Technical Chairman (2000), Awards Chairman(2002—2004) and General Chairman (2006).



digital beamforming on receive: techniques and …digital beamforming on receive: techniques and...

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