MIMO SAR IMAGING: SIGNAL SYNTHESIS AND RECEIVER DESIGN

Jian Lit Xiayu Zhengt Petre StoicattDept. of Electrical and Computer Engineering tDept. of Information Technology

University of Florida Uppsala UniversityP. 0. Box 116130, Gainesville, FL 32611, USA. P.O. Box 337, SE-75105 Uppsala, Sweden.

Abstract-A multi-input multi-output (MIMO) radar can be used to letters and matrices by boldface uppercase letters. The positive (semi-form a synthetic aperture for high resolution imaging. To successfully )definite Hermitian square root of a positive (semi-)definite matrixutilize the MIMO synthetic aperture radar (SAR) system for practical R is denoted by R1/2. We use (.)T to denote the transpose, ()*imaging applications, constant-modulus transmit signal synthesis and . c

u

optimal receive filter design play critical roles. We present in this for the conjugate transpose, and (.)C for the complex conjugate. Thepaper a computationally attractive cyclic optimization algorithm for the Frobenius matrix norm is denoted by 11 1. The real part of a complex-synthesis of constant-modulus transmit signals with good auto- and cross- valued vector or matrix is denoted by Re(.). The trace of a matrix iscorrelation properties. Then we go on to discuss the use of an instrumental denoted by tr(.). Finally, the operation of stacking the columns of avariables approach to design receive filters that can be used to minimize 'the impact of scatterers in nearby range bins on the received signals matrix on top of each other is denoted as vec(-), and X iS used tofrom the range bin of interest (the so-called range compression problem). denote the Kronecker matrix product.Finally, we present a number of numerical examples to demonstrate theeffectiveness of the proposed approaches. 1I. CYCLIC ALGORITHM FOR CONSTANT-MODULUS SIGNAL

I. INTRODUCTION SYNTHESIS

MIMO (multi-input multi-output) radar is an emerging technology A. Problem Formulationthat has significant potential for advancing the state-of-the-art ofmodern radar. Unlike a standard phased-array radar, which transmits Consider a M olradar e wt N tranmit uand cescaled versions of a single waveform, a MIMO radar system can atna.Ltteclmso LxNb h ouaigcdtransmit viasiits ofantennas multiple proIng sinalysthemay b sequences, referred to as the transmitted signals or waveforms, wheretransmit via its antennas multiple probing signals that may be

L denotes the number of subpulses. We assume that L > N (typicallycompletely different from each other. The MIMO radar's virtual array L » N). The nth transmitted constant-modulus signal has the form:can be used to form a high resolution synthetic aperture radar (SAR)imaging. Xb(l) = do'(l), n = 1, N, I = 1, L (1)

To utilize the MIMO SAR system for practical imaging appli-cations, constant-modulus transmit signal synthesis and receive filter where x, (1) is the (1, n)th-element of X. Letdesign play critical roles. There is an extensive literature on the designof constant-modulus transmit signal waveforms with good auto- and Lcross-correlation properties (see, for example, [1] and the references rnni(P) = Xn(l)4 (l -P) =rnn(-P), p = 0, 1, 2,... (2)therein). In this literature, the design tools used are either analytical l=p+lin nature or they are based on computationally intensive optimization denote the (cross)-correlation of xn (1) and xj (1) at lag p. For least-approaches. However, using such tools or approaches appears to be squares (LS) (i.e., matched filtering) based MIMO radar imagingfeasible only for synthesizing transmitted pulses with rather small applications, an important requirement on {Xn (1)} is that they havesample numbers. In many radar imaging applications, the number of "good correlation properties," or more specifically that:data samples of the transmitted pulses can be rather large (see, e.g.,[2]). For this case, the returned pulses from near and far ranges can N P-1 N N P-1overlap significantly. S S 1rnn4p) 2+ 5 5 2rn(P) ="small",We present in this paper a computationally attractive cyclic op- P+1,p#7 n=1 n=l,i74n p=-P+l

timization algorithm for the synthesis of constant-modulus transmit where P is an integer. The properties in (3) mean that we want thesignals with good auto- and cross-correlation properties. Then we sidelobes of the auto-correlation functions {rnn(p)} (i.e., for p 7?go on to discuss the use of an instrumental variables (IV) approach 0) to be "small" (note that rnn (0) = L, Vn), and also the cross-to design receive filters that can be used to minimize the impact of correlation functions {rnnI(p)}nA to take on small values at all lagsscatterers in nearby range bins on the received signals from the range p. This is needed because in a LS range-compression operation, thebin of interest (the so-called range compression problem). The reason signals received by the radar are matched filtered by multiplying themfor discussing transmit signal synthesis and receive filter design in with the transmitted signals, appropriately delayed for the range bin ofthe same paper is the interplay that exists between them: a flexible current interest; then signals backscattered to the radar by scatterersreceive filter design approach, such as the IV approach introduced from other range bins (such as range bins adjacent to the one ofin the paper, can be used to compensate for missing features of the interest) will be attenuated in the range-compression stage only iftransmitted signals that could not be easily realized by the synthesis (3) holdsprocedure (such as a higher than desired peak sidelobe level inthe down-range dimension of the SAR image). Finally, we present B. Cyclic Algoithma number of numerical examples to demonstrate the effectivenessof the proposed approaches. The following notations are adopted We begin by reformulating the problem slightly. Let X be thethroughout the paper. Vectors are denoted by boldface lowercase following block-Toeplitz matrix:

978-1-4244-1714-8/07/$25.OO ©007 IEEE 89

The solution to the minimization problem in Step 2 of CA is alsoP+L-1 easily computed. Let LX

, U=U * denote the singular valueXI ... xi(L) 0 decomposition (SVD) of LX, where U is NP x NP, E is NP x

NP, and U is (P + L - 1) x NP. Then the said solution is given

o xi (1) xi(L) by [3]: U (10)(4)

XN XN'(L) 0 If the requirements imposed on down-range and cross-range peaksidelobe levels are not very strict, then we can use the above CAto design xn(l), followed by directly using LS range and cross-

o XN (1) ... XN (L) range compression for radar image formation. However, if strictrequirements (such as the peak sidelobe levels of -50 dB) are imposed

Note that X is NP x (P + L - 1). The auto- and cross-correlations on the down-range and cross-range dimensions for SAR imagethat appear in (3) are the elements of the positive semi-definite matrix formation, then we suggest the use of the IV range compressionXX*. Consequently, we can write (3) more compactly as follows: algorithm discussed in the following.

2XX* - LI "small." (5) III. RECEIVE FILTER DESIGN VIA IV RANGE COMPRESSION

If NP > P + L - 1 for N > 2, and therefore rank(X) < P + Let the M x L received data matrix D* of such a system can beL - 1. In such a case, XX* is singular, and it follows that the written as (see, e.g., [4]):maximum magnitude of its off-diagonal elements must be of the order K P-1 K,0(1) or larger; consequently, the ratio between maxp,n,n 1rni(P)| D* SE akabX* + 5 5 c Jp + E,and rnn(0) = L is of the order O(1/L) but not smaller. To achieve =1 p=-P+1,p#O k=1smaller ratios, we assume that: (11)

+L-1 p L- I where {ak}, {bk}, and {ak} are the receive, transmit steeringNP < P+ L-1 X P < N I1 tpossibly P << N I1 ) vectors, respectively, and the complex amplitudes for the K scatterers

(6) in the range bin of current interest; {apk}, {bpk}, and {apk} alongUnder (6), if we relax any requirement on the elements and the with Kp have the same meaning but for the scatterers in other rangestructure of X, then the class of matrices X that satisfy the equality bins that surround the bin of interest; Jp is a "shift" matrix thatXX* = LI is given by: X L=U, where U is an arbitrary takes into account the fact that the signals reflected by the scattererssemi-unitary matrix, i.e., UU* I. Using this observation, we can in adjacent range bins need different propagation times to reach thereformulate (3) or (5) in the following related (but not equivalent) radar receiver (as shown in (12)); E* is an interference-plus-noiseway: matrix whose columns are assumed to be independent and identically

min X LU 2 (7) distributed (i.i.d.) random vectors with mean zero and covariance{0n(l)},u matrix Q.

While this is a non-convex problem, the following cyclic minimiza- P+1tion algorithm, that is conceptually and computationally simple and 1also has good local convergence properties is used to solve (7). , (L x L) (12)The Cyclic Algorithm (CA):

Step 0. Initialize U, or possibly X (in which case the sequence of 1the next steps should be inverted), at some value suggested by 0"prior knowledge;" if no such knowledge is available, then set, (The above definition of Jp is valid for p > 0. For p < 0, Jp shouldfor example, U [ j, (I + jI) J, which is used in our be defined as the transpose of the matrix in (12)).numerical examples. Most MIMO radar techniques for detection or parameter estimation

Step 1. With U equal to its most recent value, minimize (7) with of the scatterers in the range bin of interest rely implicitly on the as-respect to {¢)n (I) }- sumption that the second and third terms in (11) are uncorrelated with

Step 2. With {¢)n(1) } set to their most recent values, minimize (7) X. Note that when this assumption is true, then post-multiplicationwith respect to U. of D* by X (i.e., the matched filtering operation) will have the

Iterate Steps 1 and 2 until a practical convergence criterion is satisfied. beneficial effect of significantly attenuating the second term in (11).The minimization problem in Step 1 has the following generic However, for this to happen, we need that:

form: IIX*JpXl 2 = "small" (ideally, zero),

mil e> 2mi2n{const-2Re[(Szp)e i]} for p= -Pm+1, -1,1 ,P-1. (13)P=J P=J In practice, this often means that IIX*jPX112 is, say -50 dB, relative

P N 1 to 11X*XX 2. However, enforcing orthogonality among the transmitted- mmx cos [arm (5 z-) - (8) waveforms as well as requiring such good auto-correlation properties

\p=1 significantly complicates the waveform synthesis problem. To avoidwhere {zp} are given numbers. The solution to (8) is given by the difficulty indicated above, we just try to do the best we can

/ p\ ~~~~~~regarding both the auto- and cross-correlation properties of ther Ez (9) transmitted waveforms by formulating the signal generation problem

Vp=l J ~~~~~~~asin (3), (5), or (7). Then, we are left with the problem of obtaining

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MF~~~~~~~~~~~~~~VfiIt.,(H.d,..,d,PN)] IVfifIt.(CA)Can equation of the form of (1) from which (i) the residual term MFfilt.,(H.d.md,N MF it' A)]due to signals reflected by scatterers from other range bins has been a

a

attenuated to the desired level, and (ii) the interference-plus-noise =0 .0

term is kept as small as possible.~-30 -40-

Inspired by the method of instrumental variables (IV) used in time 0 0

series analysis, system identification, and array processing (see, e.g.,5

l 60[5], and the references therein), we post-multiply D* by another 50 -60matrix Y in lieu of post-multiplying by X. The result is: -30 -20 -10 0 10 20 30 -30 -20 -10 10 20320

K P-1 K, (a) (b)D*Y S akakbkXY+ 5 5aI: awbkkXJp Y+W Fig. 1. Sidelobe level versus delay p when N 5, L = 256, P = 24,

k=1 p=-P+1,p#O k=1 and the transmitted waveforms are: (a) QPSK Hadamard codes scrambled by(14) a PN sequence, (b) CA codes. The SINR Loss of the IV filter relative to the

where W* = E*Y. We choose the receive "filtering" matrix Y such MF filter for (a) and (b) are 8.8766 dB, 0.0017 dB, respectively.that

X*Y = I. (15)

Let the SVD of X be X = UxExU , where Ux is L x N, Ex isX ~~~~~~~~minllbl2N x N, and Ux is N x N. Let Y = x xUx which guarantees Ithat X*Y = I. The class of matrices Y that satisfy (15) is given by: s.t. 11pb-_ 112 < (N

y = Y~'+ AB, (16) p=-P+ 1, -1,1,... P-1. (23)

where A is a semi-unitary matrix whose columns span the null space The optimization problem in (23) is a convex quadratically con-of X* (which means that UxA - 0 and X*Y - I), and B is strained quadratic program (QCQP), which can be solved efficientlyan arbitrary matrix. Because we assume that the matrix X has full in polynomial time using public-domain software packages (e.g., [6]).column rank, the dimensions of A and B are L x (L - N) and, IV. NUMERICAL ILLUSTRATIONSrespectively, (L - N) x N. We will choose B in (16) to attenuatethe second term in (14). Usually {a}k} and {bpk } are unknown, so We present below several numerical examples. Let N = M = 5.we will select B such that: We consider a MIMO radar system with array configuration of

2 (2.5, 0.5), which indicate the inter-element spacings (in units ofX*JP(Y + AB) <. N, p =-P+ 1,... ,-1,1,... ,P-, wavelengths) of the transmit and receive arrays, respectively. The

(17) array Signal-to-Noise Ratio (SNR) is defined as NPt/72, where Ptwhere ( denotes the desired attenuation level (such as the aforemen- denotes the signal power for each transmit antenna, and o2 is thetioned -50 dB) relative to X*Y

2

N. Let variance of the additive white thermal noise. Hence in the absenceof jamming, Q = u21. Let P = 24 and L = 256. In the examples

-=vec ,) (N2 x 1), (18) below, either we use CA for waveform synthesis or we employ N10p -vec X*JPY (N x 1), (18) QPSK Hadamard codes of length L scrambled by a pseudo-noise

41P =[I x (X*JPA)1, (N2 x [N(L - N)]) (19) (PN) sequence of length L [7].

and A. Signal Synthesis and Receiver Designb = vec(B). (20) For fair comparison purposes, we use MF as: YMF = X(X*X)-1,

which is the original MF scaled by (X*X)-1 so that, like for theThen (17) becomes IV filter, X*YMF = I. To compare the performance of the MF and

_ 2<(N, p -P+.1... -11,... P- 1. (21) IV filters, we define the relative sidelobe level of IV as:

The use of the IV matrix Y to obtain a "data" model with the NTN 2, P -P ......-1,1... Pdesired properties can therefore lead to a more significant attenuation and the corresponding relative sidelobe level of MF is obtained byof the response term corresponding to uninteresting range bins, than replacing the Y above with YMF. We also define the SINR Loss (inthe use of X can. However, this is achieved at the cost of lowering dB) of the IV filter as compared to the MF filter as:the signal-to-interference-plus-noise ratio (SINR). Note that: E LosswhereSINR (INR MF)- 101g10(SINR IV): (25)

={v TcV)1Evec(E* E*)[Y ] where SINR tr([yTy ]Q) = tr(YTYe).tr(Q)defined similarly by replacing the Y above with YMF).

(Tyc)T Q (yTyc + BTBC) 0 Q. (22) Figure 1 shows the relative sidelobe levels of both the MF and IVfilters when SNR = 40 dB. We note that the IV filter can achieve the

Note that if X*X I, then Y*Y I,t t the SINR loss of prescribed sidelobe level of -50 dB, while the MF filter has a muchthe IV filter relative to the matched filter (MF) is due to the additional higher sidelobe level (e.g., Figure 1(a)). To achieve the prescribedterm BTBC. Since Q is unknown, we can try to minimize BTBC sidelobe level, the SINR Loss of the IV filter relative to the MF(or, alternatively, lb l2) to keep (22) small, and therefore minimize filter is 8.8766 dB for QPSK Hadamard codes scrambled by a PNthe SINR loss relative to the matched filter. Hence we can select B sequence, and 0.0017 dB for CA synthesized waveforms which issuch that: much smaller than the former waveforms.

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15 ig

20~~~~~~~~~ ~ 20 20 01 0X

-30 -20 -10 0 10 2030-30-20 -10 0 10 20 30 -30 -20 -10 0 10 20 30-30 -20 -10 0 10 20 30

AMgl(dAg) Angle (dAg) Angle (dAg) Angle (dAg)

(c) (d) (c) (d)Fig. 2. MIMO SAR images formed via transmitting the QPSK Hadamard Fig. 3. MIMO SAR images formed via transmitting the CA optimizedcodes scrambled by a PN sequence as probing waveforms and using the probing waveforms and using the IV filter for range compression. (a) DAS,matched filter for range compression. (a) DAS, (b) Capon, (c) GLRT, and (b)Capon, (c) GLRT, and (d) refined CAPES.(d) refined CAPES.

V. CONCLUSIONB. MIMO SAR Imaging Constant-modulus transmit signal synthesis and receive filter de-

We now consider using MIMO SAR to form angle-range images. sign play critical roles in many MIMO radar applications includingThe array is the same as before. The target scatterers are of equal MIMO SAR imaging. We have presented a computationally attractiveunit strength and are arranged to form "UFL" in 24 range bins. cyclic optimization algorithm for the synthesis of constant-modulusA strong uncorrelated QPSK jammer signal impinges at 100, with transmit signals with good auto- and cross-correlations. When strictthe Array Interference-to-Noise Ratio (AINR) being equal to 100 requirements on the peak sidelobe levels (e.g., -50 dB relative todB. The MIMO SAR data matrix obtained after IV filtering (range sidelobe level) are imposed, we have taken an alternative route andcompression) has the following form: proposed a "filtering" operation based on the use of instrumental

variables (IV) matrix, which can be effectively designed via solvingDIV= [ D1Y D2Y . DNY ] 7 (26) a QCQP problem in polynomial time to achieve the desired goals at

the cost of a slight loss of the signal-to-interference-and-noise ratio.where DX is the data matrix collected at the position i, i -1, 2, ... N, with N denoting the total number of data collection VI. ACKNOWLEDGEMENTpositions used to form the synthetic aperture. The dimension of DIV This work was supported in part by the Army Research Officeis M x MN. We assume that L = 256 and N = 10 in our numerical under Grant No. W91 INF-07-1-0450, the Office of Naval Researchexamples. The MIMO SAR data matrix DMF obtained after the MF under Grant No. N000140710293, and the National Science Founda-filtering is similar except that the Y in (26) is replaced with YMF. tion under Grant No. CCF-0634786.

Consider first using the QPSK Hadamard codes scrambled bya PN sequence as the transmitted waveforms and the MF filter REFERENCESfor range compression. The MIMO SAR images formed by using [1] H. Deng, "Polyphase code design for orthogonal netted radar systems,"the data-independent Delay-And-Sum (DAS) (i.e., least-squares (LS) IEEE Transactions on Signal Processing, vol. 52, pp. 3126-3135, Novem-[7]) approach and three data-adaptive methods: the Capon [7], the ber 2004.Generalized Likelihood-Ratio Test (GLRT) [7], the refined combined [2] D. R. Fuhrmann and G. San Antonio, "Transmit beamforming for MIMO

Capon and APES (CAPS)7 pprochearshradar systems using partial signal correlations," 38th Asilomar ConferenceCapon and APES (CAPES) [7] approaches are shown in Figure on Signals, Systems and Computers, Pacific Grove, CA, vol. 1, pp. 295-2(a)-(d). The data-adaptive methods perform better than the data- 299, November 2004.independent DAS and the refined CAPES method provides the [3] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.:best image for this example. Consider next using the waveforms Cambridge University Press, 1985.optimized via the CA algorithm and the IV filter for MIMO SAR [4] P. Stoica, J. Li, and Y. Xie, "On probing signal design for MIMO radar,"

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' ~~~~~oversymmetric cones," Optimization Methods and Software Online,SAR images can be obtained for all methods that we have considered. vol. 11-12, pp. 625-653, Oct. 1999.In particular, the refined CAPES image shown in Figure 3(d) contains [7] L. Xu, J. Li, and P. Stoica, "Radar imaging via adaptive MIMO tech-no false scatterers and is the best among all images presented in this niques," 14th European Signal Processing Conference, (invited), Florence,paper. Italy, September 2006.

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