REDUCED RANK SPACE-TIME ADAPTIVE PROCESSING WITH QUADRATIC PATTERN CONSTRAINTS FOR AIRBORNE RADAR

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1 REDUCED RANK SPACE-TIME ADAPTIVE PROCESSING WITH QUADRATIC PATTERN CONSTRAINTS FOR AIRBORNE RADAR Kristine L. Bell Dept. of Appl. & Engr. Statistics George Mason University Fairfax, VA , USA Kathleen E. Wage Dept. of Elec. & Cornp. Engr. George Mason University Fairfax, VA , USA Abstract-Reduced rank (RR) linearly constrained minimum variance (LCMV) adaptive beamforming with quadratic pattern constraints (QPC) is applied to space-time adaptive processing (STAP) for airborne radar. The problem is formulated for general rank reducing transformations and main beam and sidelobe pattern control is achieved by imposing a set of inequality constraints on the mean-square error between the adaptive pattern and a desired beampattern over a set of angle-doppler regions. Both a fixed PRI-staggered post-doppler transformation and a data-dependent principal component transformation are shown to perform at least as well as full-dimension LCMV-QPC STAP in terms of processing gain and sidelobe reduction with significantly reduced computational complexity. 1. INTRODUCTION Space-time adaptive processing (STAP) in airborne radar systems combines signals from N antenna array elements and A f pulses to adaptively suppress clutter and jamming in both the space (angle) and time (Doppler frequency) dimension [I]. The foundation of most STAP techniques is the linearly constrained minimum variance (LCMV) processor [2]. The standard LCMV processor weights are designed to minimize the processor output power subject to a linear distortionless constraint in the angle-doppler steering direction. Full-dimension STAP operates in the full NM-dimensional space and is too computationally costly for practical systems. Partially adaptive techniques use a reduced number of degrees of freedom for interference suppression and can offer computational and performance advantages, particularly under low sample support conditions [I]. Both fully and partially adaptive STAP processors can have unacceptably large sidelobes and mainlobe squinting due to sensor perturbations, pointing error, and low sample support. In radar systems, this behavior can lead to increased false alarms from clutter and unexpected interferers. This rerearch was supported by ONR Gmnt #N To mitigate this problem, a general framework was developed for adaptive and non-adaptive STAP beampattern synthesis for non-linear arrays based on LCMV.beamforming with quadratic beampattern constraints (QPC) [3]. The formulation generalizes the techniques in [4]-[5]. Main beam and sidelobe pattern control is achieved by imposing a set of inequality constraints on the weighted mean-square error (MSE) between the adaptive pattern and a desired beampattern over a set of angle-doppler regions. An important feature of the LCMV-QPC formulation is the specification of multiple quadratic pattern constraints. By proper choice of constraints, the level of pattern control can be traded off against algorithmic complexity. At one extreme, lowcomplexity techniques can be obtained based on one or two constraints similar to the adaptive pattern control method in [4]. At the other extreme, we can achieve.tight pattern control using many constraints, in a manner similar to the technique in [SI. Between the two extremes, an approach using several constraints was shown to achieve good pattern control and maintain a high signal to interference plus noise ratio (SINR) with reasonable complexity [3]. Another key feature is a computationally efficient iterative implementation which can be applied as post-processing to the standard STAP processor. The LCMV-QPC technique was formulated for general rank reducing (RR) transformations in. [6] and applied to beamforming in the spatial dimension using fixed beamspace [2], data dependent principal components (PC) [7], and hybrid PCibeam-space transformations for both linear and nonlinear arrays. Among the reduced rank approaches, the hybrid PCibeamspace technique achieved the best overall performance, very close to full.dimension LCMV-QPC. In this paper, we apply RR-LCMV-QPC processing to the STAP problem and present results from a circular array STAP data set provided by MIT Lincoln Lab [8]. The reduced rank techniques investigated are shown to perform at least as well as full-dimension LCMV-QPC STAP with significantly reduced computational complexity. ' ' /03/$ IEEE 807

2 11. REDUCED RANK LCMV STAP WITH QUADRATIC PATTERN CONSTRAINTS We assume a STAP model with N antenna elements and A I pulses. Let v(b, 4, f) denote the NM x 1 space-time array response vector to a signal arriving with elevation angle 8, azimuth angle 4, and Doppler frequency f. In standard full-dimension STAP, the NAf x 1 adaptive weight vector w is designed according to the LCMV criterion, the outpower is minimized subject to a set of d linear constraints. A single distortionless constraint is most commonly used, however additional null or main beam constraints may also he imposed. Let C be the NAf x d constraint matrix, and f be the d x 1 vector of constraint values. The N M x NAf clutter plus interference covariance matrix is estimated from K snapshots of training data from range bins near the target bin. Let x(k) denote the NM x 1 vector of array data at snapshot k. The K-sample covariance matrix estimate is The LCMV optimization problem is The solution has the form min whrw st. &" = f. (2) w = R-'C(CHR-'C)-'f. (3) In partially adaptive methods, the data is transformed by the NAI x L matrix T, i.e. x~(k) = THx(k), (4) and an L x 1 adaptive weight vector WT is designed for the transformed data XT(~). The sample covariance matrix of the transformed data is Fig. 1. Partition of Angle-Doppler space H CT = T c. Partially adaptive techniques can offer computational and performance advantages, particularly under low sample support conditions. However their interference suppression capability is somewhat diminished since they use a reduced number of degrees of freedom. Both fully and partially adaptive LCMV STAP processors can have large. sidelobes and mainlobe squinting due to sensor perturbations, pointing error, and low sample support which can lead to increased false alarms. The LCMV-QPC formulation provides additional pattern control via additional quadratic pattern constraints. As shown in Figure I, we partition angle-doppler space into T sectors ai,..., R,. Let Bd,i(@, h f) be a desired beampattern in the region Qi. The MSE between the beampattern generated by the adaptive weight vector w and the desired beampattern over the region R, is given by (8) If L < NAf, the transformation is rank reducing. This reduces computational complexity but also reduces the adaptive degrees of freedom for interference suppression. Since wfx~(k) = w;thx(k), applying WT to the transformed data XT(~) is equivalent to applying w = TWT to the original data x(k). The RR-LCMV optimization problem can be stated as: min The solution is (5) (TwT)~R(TwT) st. CH(Tw=) = f. (6) Thus the pattern error is a quadratic function ofthe adaptive weight vector. In RR-LCMV-QPC STAP, adaptive weights are designed according to the standard RR-LCMV criterion, while limiting the deviations from the desired pattern using quadratic 808

3 pattern constraints as follows: min St. Defining (TwT)~R(TwT) st. CH(Tw~) = f (13) (TWT)~Q,(TWT) - 2Pe (qf(twt)) +yi I Li,i = 1...r. QT,i = THQiT (14) qt,i = THqi (15) qi = Li -Ti, (16) first order Taylor series approximation of the weight vector for small loading increments is summarized below. One way to achieve fast convergence while ensuring that the small update assumption is valid is to let A? be a fraction of the of the current loading value, i.e. A?) = 01X!p), 01 in the range 0.3 to 1 seems to work well. If the initial loading is small enough, the initial weight vector is essentially the standard RR-LCMV weight vector given in (7). At each iteration, the weights are updated by 1. fori=l,..., T the RR-LCMV-QPC optimization problem can be written in terms of the reduced rank quantities as min wfrtwt st. CFWT = f (17) st. wfqt,~wt - 29e (qgiwt) 5 qi i = I... r. The solution is RQT = RT + CXiQT,i i=l i=l In this processor a weighted sum of reduced rank 'loading' matrices QT,~, i = 1,..., r and a weighted sum of desired beampanern terms qt,,, i = 1,..., T are used to balance the adaptive pattern with the desired pattern. The relative contribution of these terms can be adjusted to achieve pattern control while maintaining high SINR. There are a set of optimum loading levels Xi, i = 1,..., T which satisfy the constraints, however there is no closed form solution for the loading levels, even when r = 1. It can be shown that the mean-square pattern error decreases with increasing Xi, but at the expense of decreased interference suppression. The loading levels must be chosen judiciously to achieve the desired level of performance. In [6], an iterative procedure was developed for computing the optimum loading levels and the corresponding adaptive weight vector in the RR-LCMV-QPC processor. At each iteration, the pattern errors are checked against the constraints. If a constraint is exceeded, the loading for that sector is increased by an incremental factor A?), i.e. A?) = A?-') +A?) and the weight vector is recomputed. A computationally efficient weight update algorithm based on a A,,$' = w$-l) - p$-1) qt (PI (24) (P) - (P-l) 5. PT - P, T - p(p-l)q$)p$-l), 111. REDUCED RANK LCMV-QPC FOR CIRCULAR ARRAY STAP (25) STAP systems have traditionally used a rotating linear array configuration, however a fixed circular array is currently under development under the UHF Electronically Scanned Amy (UESA) program sponsored by the Office of Naval Research (ONR). The array consists of 54 directional antenna elements with suppressed backlobes. Only 20 of the elements will be used at a time to transmit and receive. With this configuration, the antenna can be scanned mechanically in 6.67O increments by choosing the appropriate 20-element sector, and scanned electronically 3~3.33" with the chosen sector of elements. In the MIT Lincoln Lab circular array STAP data set [SI, there are N = 20 elements and AI. = 18 pulses with a 300 Hz pulse repetition frequency. First, the LCMV-QPC technique was used to synthesize a -35 db uniform sidelobe level quiescent pattern steered to c++ = 0" and f = 0 Hz for a range of 50 km, which corresponds to B = -10.5'. Angle-Doppler space was par- titioned into one elevation angle sector B E (-llo,-2"), 11 azimuth angle sectors 4 E (-1Z0,12O), 3Z(12',3O0), +(30", 60"), i(60", loo'), jz(looo, 140 ), 3Z(140, 180 ), and5 Dopplersectorsf E (-30,30), +(30,90), +(go, 150) Hz for a total of 1 x 11 x 5 = 55 sectors. The desired pattern was set to zero outside of the mainlobe region, and the constraint levels were set to -35 db times the volume of the sector. No constraint was used in the mainlobe region. 809

4 Next, a scenario with two 30 db interference-to-noise ratio (INR) jammers at 60 and -20, in addition to clutter, was considered. The pointing direction was chosen to be (Oa,bS,fs) = (-10.5,00,60Hz). An8 kmtrainingwindow (200 snapshots) was used to estimate the covariance matrix. Tapered adaptive STAP processor weights were computed by replacing the distortionless constraint with the quiescent -35 db sidelobe level beampanern steered to the pointing direction. Diagonal loading was added at a level of 0 db to allow the covariance matrix to be inverted and for sidelobe control. The resulting space-time beampattern, and beampattern cuts are shown in Figure 2. The beamformer has put nulls on the clutter ridge and the two jammen, but the sidelobes have risen above the constraint level. The fully adaptive LCMV-QPC STAP processor was then used to reduce the sidelobes. The initial loading levels were set to Xo = , which roughly corresponds to 0 db diagonal loading, and then iteratively increased using a = 0.8. Processing was limited to five iterations. The fully adaptive LCMV-QPC processor is able to reduce most of the sidelobes below the -35 db level while improving SlNR slightly and maintaining a well behaved main-beam, and deep clutter and jammer nulls. The final beampattern is shown in Figure 3. Next, a fixed pulse repetition interval (PR1)-staggered post-doppler rank reducing transformation was applied. Let fs denote the Doppler steering direction, and AT denote the number oftaps in the staggered Doppler filter. The transformation matrix has the form [I]: TPRI = Fa@IN, (26) F, = Toeplitz([G; Onf-~,], [f8(0) O;,-,W]) (27) [ - f, = 1 e J2xfs _ e-j 2r(Af -1)fs 1. (28) Excellent SlNR performance was achieved with as few as 54 degrees of freedom (M = 16), however the QPC technique had no flexibility to reduce sidelobe levels. The rank had to be increased to 162 (AT = 10) to provide enough degrees of freedom for sidelobe control. This technique provided a further increase in SlNR while controlling sidelobes with less than half the dimension of the fully adaptive processor. The final beampattern is shown in Figure 4. Finally, we applied data dependent principle component rank reduction using the first (L- 1) principal eigenvectors of the clutter plus interference subspace, augmented by the array response vector for the pointing direction, T~~ = iv(e,,b,,fs) U~I, (29) U, is the N x (L - 1) matrix of clutter plus interference eigenvectors obtained from the eigendecomposition of R. The principle component processor gave performance close to the fully adaptive processor with 11 1 degrees of freedom, but the QPC technique again had difficulty reducing sidelobe levels. Increasing the rank by one by augmenting the transformation matrix with the tapered quiescent weight vector provided the additional flexibility needed to control the sidelobes. The SINR was about the same as the fully adaptive LCMV-QPC processor using less than onethird of the full dimensionality. The final beampattern is shown in Figure 5. Both of the reduced rank techniques performed at least as well as full-dimension LCMV-QPC STAP with significantly reduced computational complexity. The PRI-staggered post-doppler processor provided a higher SlNR but required more degrees of freedom. The principle components processor had better sidelobe suppression capability with fewer degrees of freedom, but with a lower SINR. IV. REFERENCES [I] J. Ward, Space-Time Adaptive Processing for Airborne Radar, MIT Lincoln Laboratory Technical Report 1015, Dec [2] H. L. Van Trees, Optiniun, Array Processing: Detection, Estiniation, and Modulation Theoy, Part IV, New York, NY John Wiley and Sons, [3] K. L. Bell, H. L. Van Trees, and L. J. Griffiths, Adaptive Beampattern Control Using Quadratic Constraints for Circular Array STAP, ASAP 2000, MIT Lincoln Lab, Lexington, MA, pp , March [4] D. T. Hughes and J. G. McWhirter, Using the Penalty Function Method to Cope with Mainbeam Jammers, Third Inil. Conf: on Sip. Process. (ICSP 96), Beijing, China, Oct [5] P. Y. Zhou and M. A. Ingram, Pattern Synthesis for Arbitrary Arrays Using an Adaptive Array Method, IEEE Trans. Antennas Propagot., vol. 47, no. 5, pp , May [6] K. L. Bell and K. E. Wage, Partially Adaptive LCMV Beamforming with Quadratic Pattern Constraints, 4th. World Multiconf: on Systemics. Cybernetics and lnformaiics (SCI 2000), Orlando, FL, vol. VI, pp , July [71 Kirsteins and D. Detection Using Low Rank Approximation to a Data Matrix, IEEE Trans. Aerospace and Electronic Syst., vol. 30, no. 1, pp , Jan [8] M. Zahnan and B. Freburger, Circular STAP Data Package, May 17,

5 .. -a*. SINR I Fig. 2. Full-dimension Tapered Adaptive LCMV Fig. 4. PRI-Staggered Post-Doppler RR-LCMV-QPC with DOF = lw tm 1% Omplsr F muw IW Fig. 3. Full-dimension LCMV-QPC. Fig. 5. PC RR-LCMV-QPC with DOF =