Cluster Design for Scanned Pattern Interferometric Radar (SP

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c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organizat ion.1 -Conference

A01-39867NM, Aug. 28-30, AIAA 2001-4654CLUSTER DESIGN FOR SCANNED PATTERN INTERFEROMETRICRADAR (SPIR)

Karen MaraisM IT Space Systems Laboratory77 Massachusetts Avenue, Room 37-458Cambridge, M A 02139karenm@mit .edu

Raymond J. SedwictfMIT Space Systems Laboratory77 Massachusetts Avenue, Room 37-439Cambridge, M A 02139sedwick@mit .ed u

ABSTRACTScanned Pattern Interferometric Radar (SPIR) ha s pre-viously been proposed as a processing algorithm fo rspace-based Ground Moving Target Indication (GMTI)radar. Performance of the algorithm is highly depen-dent on cluster design. Aperture placement in the clusterdetermines th e Point Spread Function (PSF). W e showthat certain characteristics of the PSF can be used toidentify ill- or well-conditioned systems. Using theseresults we develop a design algorithm to select apertureplacements that will guarantee good numerical condi-tioning.Keyw ords: radar, interferometry, point spread function ,aperture synthesis, sparse arrays

1 INTRODUCTIONGround Moving Target Indication (GMTI) using sepa-rated spacecraft interferometry at radio wavelengthspromises to be a powerful new all-weather surveillancetool. In previous work, we presented the Scanned Pat-tern Interferometric Radar (SPIR) algorithm as amethod for obtaining high-resolution clutter-tolerantresults1.SPIR uses the high angular variability of a sparse arrayPoint Spread Function (PSF) to extract sufficient infor-mation from the signal return that the clutter and targetscan be separated without an a priori assumption of theclutter statistics. Main lobe clutter can be separated frommoving targets using the deterministic geometric rela-tionship between observation direction an d clutter dop-pler shift.

* Research Assistant, MIT Space Systems Laboratoryt Research Scientist, MIT Space Systems LaboratoryCopyright 2001 by the A merican Institute of Aeronauticsand Astronautics, Inc. A ll rights reserved.

The algorithm synthesises a PSF matrix correspondingto the gain pattern of the cluster as it is swept across thefootprint. By deconvolving the synthesised gain patternfrom the received signals the true ground scene isrevealed.Success of the SPIR algorithm depends on the invert-ibility of the PSF matrix. Using this constraint, a designalgorithm is developed for one-dimensional clusters.This paper discusses the design algorithm and discussessome example arrays.W e begin with a brief review of radar interferometryand the SPIR algorithm. Section 3 discusses perfor-mance parameters for the PSF, and how they are evalu-ated. Section 4 suggests a design algorithm by whichhigh-performance PSFs can be developed. Examples ofsuch PSFs are given in Section 5.

2 RADAR INTERFEROMETRY ANDSPIR2.1 The PSF of an InterferometerThe point spread fun ction (PSF) of an interferometer isa separable function of the individual aperture responseas well as the distribution of the apertures in space. If weassume that the aperture responses are identical, as is thecase for most interferometers, the interferometer PS Fcan be conveniently calculated using Fourier transformtechniques, as shown for a one-dimensional interferom-eter in Figure 1.A n array of identically illuminated apertures is formedby convolving the individual aperture illumination withimpulses placed at the aperture positions. By first takingthe Fourier transforms of the aperture positions and illu-mination, the convolution operation is transformed intomultiplication. The PSF can therefore be written as theproduct of two functions, the array factor and the aper-ture response.

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c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring O rganization.

i t . , it - 'r o . . . . . . . . . .FIG. 1: Calcu lation of the PSF using the FourierTransform

The Fourier transform of the aperture positions isreferred to as the array factor, while the Fourier trans-form of the aperture illumination is just the individualaperture response.Th e array factor for a two-dimensional interferometer isthe norm of the two-dimensional Fourier transform ofthe aperture positions

array(x,y) " dx>dy.S'n=\i>J (1)

where the individual aperture positions are given by( xn,y n) . After transformation to azimuth-elevation coor-dinates (v|/ flz ,v|/e/), the array factor can be expressed as2"M -?2>n = 1 " si n

M / o z + y " si n V e/ (2 )Since the array factor is the sum of complex exponen-tials, it is periodic in v j / fl z and \yel.The PSF for a two-dimensional interferometer is there-fore given by 2

N - -^ ( jc , , s invi /0 .+ y ns inv) / e/ )Z A ,e (3)where E(\\faz , \ \f el ) is the individual aperture response.2.2 A Brief Review of SPIRThe SPIR algorithm relates the true ground scene x,expressed in terms of radar cross section (RCS) andDoppler shift, to the return signal y observed by aninterferometer.

y = Ax (4 )Th e matrix A is referred to as the point spread function(PSF) matrix of the interferometer. In the case of a one-dimensional interferometer, the elements of A aredefined by

A = E ( Q)p (5)which is the PSF of the array, focussed on a cell at angle9^ from boresight.For a complete derivation of the above results, see refer-ence [1],Subsequent discussion is focussed on one-dimensionalclusters, referred to as arrays. Th e discussion extends tothe two-dimensional case in an obvious manner.3 POINT SPREAD FUNCTION DESIGN

FOR SPIRThe main co nstraint on the application of the SPIR algo-rithm is the ability to accurately solve equation (4).Although it is possible to obtain approximate solutionsfor singular systems, such solutions deliver sub-optimalperformance. W e therefore require the PSF matrix to beinvertible. The accuracy of the solution also depends onthe conditioning of the system; m atrices which are well-conditioned (low condition number) are preferred.Some possible measures of PSF performance are resolu-tion, field-of-view, and peak side lobe level.Th e resolution is the smallest distance between twopoints at which they can be resolved as being separate.Points that are closer together than the angular separa-tion of the first nulls of the PSF cannot be identified asseparate points. This angular null separation defines theresolution of the ap erture.The requirement for high-resolution can be m et by usinga large baseline. Unfortunately, this decreases the spa-tial period of the array-factor, as qualitatively discussedin Section 3.2. We shall see that array factors with shortspatial periods may resu lt in PSF matrices which are notfull rank.Th e field-of-view is determined by the width of themain lobe of the aperture response. The intersection ofthe field-of-view with the ground is defined as the inter-ferometer footprint. A wide field-of-view permits cover-age of a large ground area, but has higher processingrequirements. In general, the field-of-view of an aper-ture is inversely proportional to its diameter.

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c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.To prevent "fading" at the edges of the footprint, anaperture response with a sharply defined main lobe ispreferred.The peak side lobe level predicts the clarity of theimage. Information that is located in the side lobes ofthe PSF overlays information in the main lobe, so thatthe reconstructed image is obscured. Low PSF sidelobes result in clearer images.Our goal is thus to obtain high-resolution PSFs withlong spatial periods, which are greater than the desiredfield-of-view.The analysis begins with a discussion of the construc-tion of the PSF matrix, which illustrates the relationshipbetween the PSF and the invertibility of the associatedPSF matrix.3.1 Construction of the PSF MatrixThe PSF matrix is defined by equation (5). It is a squarematrix with the number of rows and columns each equalto the number of cross-range cells.The PSF matrix is constructed by placing the primarygrating lobe of the array factor on successive cross-range cells in the footprint. Th e footprint is defined bythe intersection of the aperture response main lobe withthe ground. The aperture response remains at a constantposition on the ground. Figure 2 illustrates the construc-tion of a 4 x 4 PSF matrix.

Array Factor

FIG. 2: ConstructionMatrix of a 4 x 4 Invertible PSF

In this case the period of the array factor is greater thanthe field-of-view, therefore each row of the resultingPSF matrix is unique. Furthermore, since by construc-tion none of the rows can be obtained by scaling anotherrow, the rows are linearly independent. Therefore thisPS F matrix has full rank and is invertible.

Figure 3 illustrates the effect of an array factor whichrepeats within the field-of-view. The last row of the PSFmatrix is identical to the first row. The resulting PSFmatrix is not full rank and cannot be inverted.

.Aperture Response(FOV)

Row 4 = Row 1FIG. 3: Construction of a 4 x 4 Non-Invertible PSFMatrix

The period of the PSF therefore determines whether thePSF matrix will be invertible. In Section 3.3 we deter-mine the relationship between the array constructionand the PSF period.The next section discusses the parameters by which dif-ferent PSFs can be compared.3.2 PSF Design Param etersFor a given carrier wavelength A , , the aperture responseis a function of the aperture geometry and illumination,while the array factor is a function of the array configu-ration an d baseline. We can therefore isolate the effectsof changing the array from the effects of changing theaperture response.The carrier wavelength determines the scale at whichthe system is sampled, and is usually determined on thebasis of physical size and atmospheric attenuation con-straints. Therefore we assume that it is fixed and thePS F must be designed for best performance at the cho-sen wavelength.Since the aperture diameter is much smaller than thearray baseline, the resolution of the interferometer isdetermined by the baseline, while the the field-of-viewis determined by the individual aperture response.Th e width of the aperture response mainbeam isinversely proportional to the aperture diameter. Increas-ing the aperture diameter therefore decreases the field ofview and vice versa. For a uniformly illuminated rectan-gular aperture, the field-of-view in one dimension is

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FOVrectangular = [-asin Lasin (6)The array configuration determines the shape of the PSFand this problem is examined in detail in Section 4. Thearray baseline determines the resolution, and for a givenconfiguration, the period of the PSF. A s the baseline isincreased the resolution is increased and the period isdecreased. Figure 4 shows the array factor for two iden-tical array and aperture configurations, one with a base-line of 500 m, the other with a baseline of 1000 m.Doubling the baseline halves the period of the array fac-tor, and halves the width of the grating lobes.

Uw*swi}CiwfAntyJS 5 4 4

FIG. 4: Effect of Array Baseline on the PSF3.3 PSF Spatial PeriodAs we have seen the spatial period of the PSF deter-mines whether or no t the PSF matrix will be invertible.In this section we develop an expression for the spatialperiod, based on the aperture positions in a one-dimen-sional array.A one-dimensional aperture array can be uniquelydefined by specifying the relative aperture spacing (con-figuration) and some characteristic length / , such asbaseline or minimum spacing. A t a given wavelength K ,a fully specified array samples a particular set of spatialfrequencies. These spatial frequencies scale inverselywith / and proportionally with A , . Therefore the spatialperiod \\ f p of the array factor scales according to

In particular,.oc^

(7)

(8)

The constant of proportionality can be derived from thearray configuration and is determined as follows:Th e array factor (2) of a one-dimensional array is thenorm of the sum of a sequence of complex exponentialterms, which each have a maximum projection of oneonto the real axis. Therefore the array factor is boundedabove by the total number of apertures, T V

array(\\f) =-2717 sin v j;

n = 1


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c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.The set {kmn} of smallest kmn which will satisfy equa-tions (12) with \\ imn = \\i p is determined by simplifyingeach ratio of positions to irreducible rational form, suchthat the greatest common divisor (GCD) for each pair(kmn, kpq ) is minimised. If at least one kmn does nothave a GC D of one with all the other kpq, then the entireset is reducible and the period of the array is reducedaccording to (12).\y p is found by solving equations (12) with the setI *^mn)

sinx|/p =Recognising thatsetting

(15)*2| 1*1

is the array baseline B, and

the array factor period can be found from= k --max -f t

Fo r smallKnax'ft

(16)

(17)

(18)Alternatively, defining the minimum spacing as xmin ,and setting

kmin =mm[{kmn}]the array factor period can also be found from

(19)

(20)If kmin is greater than one, the behaviour of the arrayfactor is less well defined. Interaction between differentspatial frequencies ma y result in low frequency termswhich decrease the period of the array factor from thedesign value \\ ip . Therefore arrays with kmin = 1 arepreferred.

4 A DESIGN APPROACH TO ENSUREINVERTIBILITY

Armed with equation (17), we can design an array thathas a PSF with a given spatial period and hence ensureinvertibility of the PSF matrix. Th e first step is to deter-mine kmax, based on the system requirements. Thenumber of apertures N is chosen based on high-levelsystem requirements. Given kmax an d N, th e positionsof the remaining apertures (N-2) must be determined.

4.1 Determining the Maximum SpatialFrequency NumberTh e field-of-view of an aperture can be exp ressed as

FOV =2 a - _ (21)where D is the aperture diameter and a is a shape fac-tor such that a rectangular aperture of width

Deff = (22)has the same field-of-view as the actual aperture. Foruniformly illuminated rectangular apertures a = 1 andfor uniformly illuminated circular apertures a = 1.22 .The PSF matrix will be invertible if the period of thearray factor is greater than the field-of-view of the aper-ture

yp>FOV (23)Substituting equations (18), (21) and (22) into this rela-tion gives

7 - - B (24)The effective aperture diameter De^ is selected to sat-isfy the field-of-view requirement.Th e baseline is chosen to satisfy the cross-range resolu-tion requirement ^cross_range, according to

B = 2K - rground (25)cross_range

where rground is the range to the ground from the array.The limiting case is taken by setting

B (26)In fact, the limiting case will no t provide an invertiblePSF m atrix, since the last row of the PSF matrix will bethe same as the first row. However, we can analyse thelimiting case with the caveat that in the actual system byeither decreasing B or increasing Dejy .Since kmax is an integer, the baseline must be a multipleof half the effective aperture diameter. Likewise, from(15), all internal separations between apertures mustalso be multiples of this spacing. Therefore De 2 pro-vides a discretisation of the design space, which can bestudied non-dim ensionally in terms of the minimal spa-tial frequency numbers kmn. The requirement thatkmin = 1 means that the two most closely spaced aper-

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c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.tures physically overlap. This limitation can be over-come by using phased-array antennas.Given kmax and the number of apertures, the next sec-tion shows how the remaining (N-2) apertures shouldbe placed.4.2 Optimally Irreducible ArraysTh e period of the array factor is independent of the posi-tion of the array in space, therefore we place the firstaperture at k} = 0, and the N * aperture at kN = kmax,with no loss of generality. From the earlier discussionon the reducibility of the kmn, the only way to ensureirreducibility is to place the remaining (N-2) aper-tures such that

TABLE 1: System Parameters

= 1The resulting array configuration, {k l',k 2;...:>kN} , willhave an array factor with period \\i p given by (17). W ewill refer to such an array as being (kmax, N ) optimallyirreducible. A (kmax, N ) array which does not satisfy(27) will have a smaller period than the desired designvalue \\f p.A rrays which have the same PSF are referred to as beingsimilar; this is indicated by arrayl ~ array2. Arrayswhich are mirror images of each other{0 2;...^_1^WflJC}~{0^WflJC-V,;--^m-^max} (28)

are similar and are discarded when determining the con-figuration solution space.With N = 2 apertures, there is only on e difference pairku = kmax and the irreducibility condition requireskmax = kmin = 1. Therefore there is one unique solu-tion, {0;!}. The array factor has the same shaperegardless of A , and the baseline. The period is given by

V \ N = 2 --B WWith N = 3 apertures, on e unique solution exists fo reach kmax, namely {0; l ; fc w f l j c} . It is always possible todesign an array with the required period, but the shapeof the array factor is defined by kmax.Fo r 4


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c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.S&ttvi* O p t i m a l l y Four y a p s r t u r * Optimally Irreducible Any JD;1:38;1200]

FIG. 6: {0;1;2;1200} Optimally Irreducible Array

0.02 0 002Aigle from BoresigNt [radians]FIG. 7 : {0;1 ;14;1200} Optimally Irreducible Array

By carefully choosing the position of the third aperturevarious characteristics of the PSF, such as the width ofthe envelope main lobe, can be changed.6 CONCLUSIONS AND FUTURE WORKThe success of the SPIR processing algorithm is highlydependent on the design of the satellite cluster. In thispaper we have developed an algorithm for the design of

0.02 0 002

FIG. 8: {0;1;38;1200} Optimally Irreducible Arrayoptimally irreducible one-dimensional arrays, whichresult in numerically well-conditioned systems.Future work will develop additional criteria with whichto select particular irreducible arrays that provide thebest system-level performance. Using such arrays theperformance of SPIR in the presence of clutter and noisecan be accurately estimated.

ACKNOWLEDGMENTThis work wa s made possible in part by the support ofthe Air Force Office of Scientific Research Contract #F49620-99-1-0217 under the technical supervision ofDr Robert Herklotz.

REFERENCES1 Marais, K., Sedwick, R. J., Space Based GMTI UsingScanned Pattern Interferometric Radar (SPIR), IEEEAerospace Conference, Proceedings of, March 2001,Montana, 0-7803-6599-2/01.2 Kong, E. M., Optimal Trajectories and Orbit Designfor Separated Spacecraft Interferometry, MIT S.M.Thesis in Aeronautical and Astronautical Engineering,November 1998.

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Cluster Design for Scanned Pattern Interferometric Radar (SP