Gratinglobes Resolving in Sparse ArrayBeamformning

Zhuang Long and Xingzhao LiuDepartment of Electronics Engineering, Shanghai Jiaotong University

Abstract Beamforming in sparse array often results ingratinglobes because of spatial undersample. In thispaper, we investigate the concept of effective apertureto resolve gratinglobes. The central idea of effectiveaperture is to use transmit aperture with anappropriate element number to eliminate undersamplein receive aperture. Quantitative analyze shows thatthe element number of transmit aperture isproportional to the inter-spacing of receive sparsearray. Simulations demonstrate the design method canresolve gratinglobes effectively.

Key words: Gratinglobes, Sparse Array, Beamforming,Effective Aperture

I. INTRODUCTION

Recent years formation flying satellites has arousedgreat interests which can perform multi-missions such asGMTI and SAR imaging [1]. In the cluster each satellitecan be viewed as an element of a sparse array. Bycombining signals received by all the satellites, a newaperture with an especially large baseline is synthesized,which can greatly improve the angular resolution. Whenbeamforming performed, the sparse array can provide witha wealth of independently sampled angle-of-arrivalinformation [2].

But beamforming in array with inter-element spacinglonger than one-half of a wavelength (A/2) , called"sparse array", often results in gratinglobes. Thesegratinglobes aliased in the angle space have the sameresponse amplitude with the mainlobe and with no doubtwould bring in extra energy which is unwanted.

Many methods have been proposed for resolvinggratinglobes in sparse array. Using Techsat2 1 clusterconfiguration H.steyskal et al. [2] [3] proposed a patternsynthesis method which exploits the double periodicitiesof the gratinglobes in the angular domain and of the radarpulses in the frequency domain. However, this technique,instead of eliminating but takes full use of the gratinglobes,would influence the selection of PRF. Other optimizingmethods like genetic algorithm [4] and simulatedannealing [5] can reduce gratinglobes of sparse array byeliminating the periodicity of the thinned elements. ScottD.Berger [6] has given another explanation of gratinglobesin sparse array. He pointed that the formation flyingsatellites can be viewed as a set of nonuniform sampling

points, and gratinglobe problem can be viewed as aproblem of reconstructing a continuous signal fromdiscrete samples. By a nonunform sampling reconstructionformula to synthesize sparse array gratinglobes could bereduced in certain regions.

In this paper, we use the concept of effective apertureproposed by G.Lockwood et al. [7] to design sparse array.We consider satellite cluster here a linear sparse array (i.e.,all the elements (satellites) lie on a line) with uniformsparse distance, much simpler than real case. Antenna oneach satellite is a phased array with controlled elementnumber. Thus the whole cluster can be viewed as a sparsearray with noncontiguous subarray shown in Fig. 1. Byjudicious choice of subarray element number according tosparse distance, gratinglobes can be avoided in the two-way antenna pattern.

Fig. 1. Block diagram of a uniform linear sparse array with N subarrays

In next section, conventional beamforming in sparsearray is analyzed. In section III the design method usingthe concept of effective aperture is described in detail.Simulation results are presented in Section IV and theconclusion is in Section V.

II. SPARSE ARRAY BEAMFORMING

Consider the transmitted signal exp (j2;Tft) with 0from boresight, thus the steering vector is

-T sin 0

P(O)= I e' i - (M-1)dsinO0 (1)e ] e (')

where 2 is the wavelength, d is the inter-elementdistance, M is the element number of the subarray,{ }' denotes transpose. If d> 2 2, gratinglobes would

0-7803-9582-4/06/$20.00 c2006 IEEE

occur in the visible region which is undesirable. In general,we can control the transmit phased array to be A /2element spacing.

When beamforming performed on the whole sparsearray, the steering vector of receiving pattern can bewritten as:

s()ri e -LsinO i (N-1)LsinJ (2S(O) =I0 eie Nl)sn (2)

where L is sparse distance, N is sparse element number(satellite number).

Thus the total steering vector of the two way antennapattern is:

G (O) = P (O) OS (0)

L P'(O) ejL sPT () e** (N I)Lso PT (O) (3)

where 0 denotes Kronecker product.

Using conventional beamformer, the two-way antennapattern magnitude response is:

y(9d, ) =GH(0d)G(0)12sin2 ;TM d(sin9 - sin Od) sin2zN

L (sin9 - sin (d)

sin2 K (sin -sinfld)I sin2 Z(sin9-sinfd)I

From the above equation we can find the two-wayradiation pattern is the product of transmit pattern andreceive pattern. Gratinglobes would occur in the finalpattern because ofL >>A/2 . Fig 2 shows the two-waybeam pattern withM=l0,d=/22,N=8,L=30 A/2.We use -40dB Chebyshev weights to make thegratinglobes more visible.

02 - - --- ---------- ---- r ----

C

,40 ,*61 i

I, i,

-80 -6.0 40 -20 0 20 40 60 80Angl

Fig.2 Magnitude response of two-way antenna pattern

III. GRATINGLOBES RESOLVING USING THEEFFECTIVE APERTURE CONCEPT

We give a gratinglobe resolving method using the"effective aperture" concept. The effective aperture of anarray is simply the receive aperture that would produce anidentical two-way radiation pattern if the transmit aperturewere a point source [7]. If we want to obtain a two-wayantenna pattern with no gratinglobes, a "desired effectiveaperture" with approximately A /2 element spacing ispreferred. Figure 3(a) gives one example of an effective

aperture with 12 elements and A/2 element spacing.Different ways can be selected to yield this effectiveaperture. We can use a single element transmit array and a12-element receive array with A/ 2 spacing [Fig3.b]. Atwo-element transmit array with A/ 2 spacing and a sparsesix-element receive array with 2(2/2) spacing can alsoyield the effective aperture [Fig.3 (c)]. If using three-element transmit array with A/2 , receive array shouldusing four elements with 3(A/2) spacing [Fig.3 (d)].

The effective aperture concept provides us with auseful strategy for resolving gratinglobes. We can see thatthough the receive aperture is sparse, but convolving witha filled array with appropriate element number couldobtain an aperture with no gratinglobes.

A quantitative analyze is given as follows. Whenreferred to steering vector of an effective aperture, the

phase difference of sequential terms is- dsin 0, we canA

regard the phases distance as /22 . But if the aperture issparse, the phases distance would exceedA/2, which wecall undersample in the spatial domain. It is just theundersample in the spatial domain leads to noncontiguousphase of the steering vector, and in turn the noncontiguousphases bring gratinglobes into the antenna pattern. So wecan resolve gratinglobes by eliminating the noncontiguousphase in the steering vector.

(a)

(b)

(c)

(d)Fig.3. Example of effective aperture with different transmit and receive

aperture, 0 here stands for convolution

Rewrite (3) we can derive:

G(O) = I eJdsinA

2 2j LsinO j (L+d)sinO

e A e A

j (M-1 )dsinO... 2eA

J2;T (M-I)d+(N-I)L]sinO... e A

Attention that the noncontiguous phases in (5) alwayscome up at:

kL+(M-I)d-÷(k+1)L k= 0,1, N-1To eliminate the noncontiguous let

(6)

kL+(M- )d = (k+±)L-212 (7)In practice, L could be viewed as a constant, but the

element number of subarray can be adjusted accordingly.Thus we can obtain the relationship of M and L:

M L- ±12 l (8)d

IV. SIMULATION RESULTS AND DISCUSSION

In this part we give simulation results to illustrate thedesign method with no gratinglobes. According to (8),consider the case with L =20 A/2, so we getM =20.Fig.4 is the response pattern with -40dB Chebyshevweights; we can see that gratinglobes are nonsexistbecause of the consecutive phase of the steering vector.The reason can also be interpreted as Fig.7 depicted:though sparse array has gratinglobes (blue line), but theangles of those gratinglobes are rigorously lie in the nullsof the transmit beam pattern (red line), so energy of thegratinglobes are totally cancelled.

Increasing the element number of transmit aperture candecrease the gain of the gratinglobes because of thenarrow main beam. We present the relationship ofmaximum magnitude response (using -40dB Chebyshevweights) with subarray element number increasing (Fig.5).We can see that when subarray element number satisfies

(8), i.e.M =20 at M=20 and M=30 atL =30- , the2

maximum response can be obtained. But with the numberincreased, magnitude response reduced inversely because(8) is not satisfied and the nulls are not matched to thegrating lobes. Attention should be paid when elementnumber equals 2M, though locations of certain nulls arealso the same with those of gratinglobes, the widths of thenulls are adversely decreased because of the aperturelength increased, so perfect cancels can not be obtained,and remained energy can be seen in Fig.6.

V. CONCLUSION

Beamforming in sparse array would lead togratinglobes in the two-way radiation pattern. The reasonlies in the undersample in spatial domain. We find that theundersample would lead to the phase distance in thesteering vector larger than A/2 We use the concept of"effective aperture" to eliminate the noncontiguous phaseby judicious choosing the element number of transmitaperture according to the inter-spacing of receive sparsearray. Simulation results demonstrated that the two-wayradiation pattern is sensitive to the inter-spacing of sparsearray L . If L is not accurate, then a wrong subarrayelement number may leads to remained energy ofgratinglobes.

REFERENCES[1] Gerhard Krieger, Hauke Fiedler, Alberto Moreira, "Bi- and

multistatic SAR: potentials and chanllenges", EUSAR2004, Ulm,May 2004, pp.365-369.

[3] H.Steyskal, and I.K.Schindler, "Separable Space-Time PatternSynthesis for Moving Target Detection with TechSat 21 --ADistributed Space-Based Radar System," Proceeding of the IEEEAerospace Conference, March 2003.

[4] R. L. Haupt, "Thinned arrays using genetic algorithms," IEEETRANS. Antennas Propagat., vol.42, pp.993-999, July 1994.

[5] C.A.Meijer, "Simulated annealing in the design of thinned arrayshaving low sidelobe levels," Proc. South African Symp.Communications and Signal Processing, pp.361-366, 1998

[6] Scott. Berger, "Nonuniform sampling reconstruction applied to sparsearray beamforming", IEEE radar conference 2002, pp98-103.

[7] G. R. Lookwood, et al., F.S." Optimizing the radiation pattern ofsparse periodic linear arrays" IEEE Transactions on Ultrasonics,Ferroelectrics and Frequency Control, vol.43, issuel, pp: 15 - 19,Jan. 1996

).20- -~--------------~-----20

&-40

-80-

-80 -60 .40 -20 0 20 40 60 80An gLe

Fig.4 Magnitude response withM = 20,M = 20, -40dB Chebyshevweights

m -3 0rt:

. 1 ~~~~L=20AJ2

------------ - 0 1

- |~~~~~~~~~~~~~~~- - -L

----- - - - - - - - - - - - - - - -- - - - - - - - - - -- - - - - -

0 10 20 30 40Su barray E leTnent

£0 60Iumber

70

Fig.5 Relationship of maximum magnitude response with subarrayelement number, -40dB Chebyshev weights

0 -I ---- T - - - - r r - - - - - - - - n- - - - n - - - - T

I~0C l l

n~~~~~~~~~~~~~j J1

IL40

.80 60 -40 .20 0

Ang.le20 40 60 80

Fig.6 Magnitude response withM = 40, L = 20-2

[2] H.Steyskal, I.K.Schindler, P.Franchi and R. J.Mailloux, "PatternSynthesis for Techsat2l -A Distributed Space-Based RadarSystem," IEEE Antennas and Propagation Magazine, August 2003.

,-20 L plH0o i ~------ - - - -- Ij -1 5.'- - - --

*-40 * ..-

ITm-60' ¶1I

.4.0 E-0 -40 20 40 60 80

Fig.7 Grating lobes cancel using nulls of transmit beam pattern