STRUCTURED COVARIANCE MATRIX ESTIMATION FOR THE RANGE-DEPENDENT PROBLEM IN STAP

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1 STRUCTURED COVARIANCE MATRIX ESTIMATION FOR THE RANGE-DEPENDENT PROBLEM IN STAP Xvier Neyt, Pscl Druyts, nd Mrc Acheroy Dept. of Electricl Engineering Royl Militry Acdemy Brussels, Belgium Jcques G. Verly Dept. of Electricl Engineering nd Computer Science University of Liège Liège, Belgium ABSTRACT We propose method to compute n estimte of the clutterplus-noise covrince mtrix in bisttic rdr configurtions. The estimtion is bsed on the computtion of the clutter scttering coefficients bsed on single dt snpshot t ech rnge using model of the received signl. The covrince mtrix of the dt is modeled s structured covrince mtrix with the scttering coefficients s unknown prmeters. The method is bsed on the computtion of the mximum likelihood. We use the Expecttion- Mximiztion method s estimtion benchmrk. Since the problem is ill-posed, regulriztion is mndtory. This regulriztion is performed by sptil smoothing. The method we propose, unlike the Expecttion Mximiztion, is not itertive nd is thus less computtionlly demnding. The obtined covrince mtrix estimte is used to compute the mtched filter in order to perform trget detection. The performnce of the proposed estimtion method is evluted in terms of signl to interference-plus-noise rtio (SINR) losses nd is found to be lmost indistinguishble from the performnce of the clirvoynt cse. KEY WORDS Bisttic rdr, STAP, mximum likelihood, structured covrince mtrix estimtion, rnge dependence, Expecttion-Mximiztion 1 Introduction Slow-moving trget echoes received by irborne ground moving trget indictor (GMTI) rdrs typiclly compete with very strong clutter return. Spce-Time Adptive Processing (STAP) cn be used to discriminte between trget nd clutter signls [1, 2]. The clutter-plus-noise covrince mtrix (CM) t the rnge of interest is required in order to compute the optimum STAP filter. An estimte of this CM is typiclly obtined from secondry dt, i.e., dt t neighboring rnges round the rnge of interest. To provide n ccurte estimte, the secondry dt needs to be independent nd needs to hve the sme distribution s the clutter-plus-noise signl t the rnge of interest. The two min fctors tht ffect the sttisticl distribution of the clutter signl re ground cover nd rdr geometry (position nd velocity vector of the trnsmitter nd the receiver, ntenn rry geometry nd element rdition pttern, ground relief). Assuming uniform ground cover nd flt terrin, clutter echoes t different rnges re typiclly independent nd identiclly distributed (IID) for monosttic sidelooking configurtions with uniform liner rry (ULA) nd few other prticulr bisttic configurtions [3]. This is however not the cse for most bisttic configurtions [3, 4] nd most multisttic configurtions [5]. If the clutter echoes t different rnges re IID nd Gussin nd if no prticulr structure for the CM other thn hermitin semi-positive definite is ssumed, the mximum likelihood (ML) estimte of the clutter CM cn be obtined by computing the verge of the smple CM t ech rnge. If the clutter echoes t different rnges re not IID, other methods hve to be considered to compenste for the rnge-dependence of the clutter sttistics prior to verging (see for instnce [6, 7, 8]). These methods cn be interpreted s performing n verging in the spectrl domin. However, they do not exploit the fct tht there is direct reltion between the spectrl domin nd the physicl domin nd thus hve difficulties to cope with sptilly inhomogeneous clutter such s in costl res or in res of chnging ground cover. To be ble to cope with inhomogeneous clutter nd be ble to introduce sptil regulrity s constrint on the solution, we wnt to estimte the prmeter of the physicl process t the origin of the scttering, i.e., the scttering coefficients of the clutter ptches on the ground. We show tht the problem cn be cst s the estimtion of structured CM which is clssiclly solved by mximizing the likelihood nd requires the resolution of non-liner eqution. One of the possible method to mximize the likelihood is the Expecttion-Mximiztion lgorithm [9]. However this method is very computtionlly intensive, we thus propose n pproximtion of this method nd show tht the corresponding result re lmost s good. Moreover, given the low-rnk nture of the problem, the inverse problem is illposed nd needs to be regulrized. The pper is orgnized s follows. In Section 2, we introduce the signl model nd the resulting CM model. Bsed on this model, in Section 3, we review the likelihood nd derive the expression to be mximized. In Section 4, we briefly describe the Expecttion-Mximiztion lgorithm pplied to our problem nd show tht it cn be

2 interpreted s hill-climbing method. The regulriztion methods re discussed in Section 5. In Section 6, we describe the pproximtion of the ML solution nd how the scttering coefficients cn be estimted over the whole operting rnge of the rdr. In this section we lso compre the proposed method to other methods found in the literture. Section 7 presents the end-to-end performnce of the method in terms of SINR losses. Finlly, Section 8 concludes the pper. 2 Signl model We consider pulse-doppler rdr with rbitrry trnsmitter nd receiver loctions. An rbitrry ntenn rry is used on the receiver. At ech coherent processing intervl (CPI), the echo signl from the M pulses received on the N ntenn rry elements is smpled t the rnge of interest. The resulting lexiclly-ordered smples form dt snpshot vector of size NM denoted by y. The signl due to the clutter ptches cn be modeled s sum of the contribution over finite number L of clutter ptches [1, 10] L y = i v ci + n, (1) i=1 where i denotes the (complex) reflectivity of clutter ptch i nd is ssumed zero men circulr complex Gussin with unknown vrince r i. v ci is the steering vector corresponding to the clutter ptch i, nd is usully expressed s v ci = c i v i where c i is fctor tht groups the geometric fctors of the rdr eqution (rnge ttenution, element rdition pttern,...), v i is the usul normlized steering vector corresponding to clutter ptch i, nd denotes the Hdmr product (element-wise multipliction). n denotes the therml noise, ssumed Gussin with CM R n = σn 2I. We ssume σ n known, e.g., bsed on direct mesurements [11]. Eqution (1) cn be rewritten in mtrix form s y = V + n, (2) where = { 1,..., L } T nd V = {v c1,..., v cl }. Using this model, the clutter-plus-noise CM R y = E{yy } is R y = V R V + R n, (3) where R = dig{r } with r = {r 1,..., r L } T nd r i = E{ i i }. 3 The likelihood We wnt to find the estimte of R which is the most comptible with the mesurements y. Hence, we wnt to mximize the probbility p(r y) where p(α β) denotes the conditionl probbility of α knowing β. Using the Byes identity we cn write p(r y) = p(y R )p(r ), (4) p(y) where p(r ) is some priori probbility for R, nd p(y) is the probbility of the mesurement y nd is independent of R. Using flt priori probbility for R nd noting tht p(y) is independent of R, the optimum estimte of R cn be found by mximizing p(y R ) which is clssiclly clled the likelihood of R. Since y is ssumed Gussin with CM R y given by (3), one hs p(y R ) 1 R y e y R 1 y y, (5) where R y denotes the determinnt of the mtrix R y. Since y Ry 1 y = tr(ry 1 yy ), the logrithm of the likelihood Λ(R y) is given by Λ(R y) = ln p(y R ) = ln R y tr(r 1 y yy ) + C. (6) where C is known constnt, independent of R. A necessry condition t the mximum of Λ(R y) is dλ(r y) dr i = 0 (7) for ll i. Following [12, 13], we hve ln B / b = tr(b 1 B/ b) nd B 1 / b = B 1 ( B/ b)b 1 nd hence, with {B} d denoting the column vector consisting of the digonl elements of B, we hve nd ln R y r tr(r 1 y yy ) r nd the derivtive of (6) finlly becomes Λ(R y) r = {V R 1 y V } d (8) = {V Ry 1 yy Ry 1 V } d, (9) = {V Ry 1 (yy R y )Ry 1 V } d. (10) Eqution (7) cn thus be rewritten for ll i in vector form s {V Ry 1 (yy R y )Ry 1 V } d = 0, (11) which is equivlent to the trce eqution [14] when ll dmissible vritions of r re considered. This eqution must be solved for R. It is non liner in R nd no closedform solution is known. Noting, s in [13], F = R V R 1 y, (12) nd decomposing R y ccording to (3), (10) cn be rewritten s r Λ(R y) r r = {F yy F } d {F V R V F } d {F R n F (13) } d. Similrly s in [13], defining T = dig{{f V R V F } d }R 1, (14) from (13), the mximum stisfies r = T 1 ({F yy F } d {F R n F } d ). (15) Consequently, the following itertive solution is proposed [13] = r + τ ( ) T 1 ({F yy F } d {F R n F } d ) r r (16).

3 Tking (13) into ccount, this lst eqution cn be rewritten s ( = r + τ T 1 r Λ(R y). r r ) r (17) This shows tht the itertive scheme proposed in [13] is equivlent to modified grdient descent where the grdient is multiplied by mtrix. Moreover, tking τ = 1 in (17) results in = ( T 1 ({F yy F } d {F R n F } d ) ) r. (18) The fixed points of this eqution stisfy (15) nd this might be interpreted s n itertive solution to (15). Notice tht we do not clim convergence of (18). 4 Expecttion-Mximiztion The Expecttion-Mximiztion (EM) lgorithm cn be used to find the vlue of R tht mximizes (6) [9]. In the EM lgorithm, the coefficients re clled the complete dt while y re the incomplete dt nd (2) provides mpping between them. The EM lgorithm is n itertive lgorithm tht consists in succession of n expecttion (E) step nd mximiztion (M) step. In the E-step, the expecttion of the log-likelihood of the complete dt Λ cd (R ) conditioned on n estimte of R nd on the incomplete dt y is computed, Q(R R ) = E{Λ cd(r ) R, y}. (19) In the M-step, the vlue of R tht mximizes Q(R R ) is computed R (k+1) = rgmx Q(R R ). (20) R These two steps re repeted until convergence. Since is complex Gussin distributed with CM R, we hve Λ cd (R ) = ln R tr(r 1 ) + C (21) where C is known constnt independent of R nd (19) cn be evluted s Q(R R ) = ln R tr(r 1 E{ R, y}) + C. (22) Since the term E{ R, y} does not depend on R, tking the derivtive of Q(R R ) with respect to r yields Q(R R ) r = {R 1 } d +{R 1 R 1 E{ R, y}} d (23) nd setting this derivtive equl to zero to find the mximum of Q(R R ) yields i = {E{ R, y}} d, (24) which is set of L decoupled equtions. Eqution i of this set is = E{ i 2 R, y}. (25) The term E{ R, y} ppering in (24) cn be expressed s function of the conditionl men ā = E{ R, y} nd the conditionl covrince of E{ R, y} = āā + cov{ R, y} (26) where the conditionl men nd the conditionl covrince re well-known results from estimtion theory [15, 16] nd, with R y = R y (R ) R, re respectively given by nd by ā = E{ R cov{ R, y} = R Hence, (24) cn be rewritten s, y} = R R = {R V Ry 1 yy Ry 1 V R } d {R V R 1 +{R Λ(R y) r V R 1 y y (27) V Ry 1 V R. (28) y R } d V R } d (29) or, using the expression for Λ(R y) r obtined in the previous section, we finlly cn rewrite one itertion of the EM lgorithm s = r + r r. (30) This shows tht t lest in the context in which we use it the EM lgorithm cn be interpreted s modified grdient descent method to find the vlue of R tht mximizes the log-likelihood Λ(R y). 5 Regulriztion In the intended ppliction, V R V is not full rnk, which implies tht there is no unique solution to the inverse problem. This kind of problem is clled ill conditioned nd regulriztion is necessry to obtin solution [17]. Regulriztion is obtined by introducing priori knowledge bout the solution R. This cn be done, e.g., by directly imposing n priori probbility density of R in the likelihood to be minimized [13, 18] or by restricting the spce in which the solution is looked for, e.g., by decomposing the CM R in fmily of CM [12, 19]. A review of vrious regulriztion methods pplied to the EM lgorithm cn be found in [20]. A prticulr regulriztion is one tht consists in dding smoothing step fter computtion of the M-step of the EM lgorithm. In [13, 20], it is shown tht, in some prticulr cses, there is n equivlence between the imposition of some prior probbility density p(r ) nd the ddition of sptil smoothing step. The Tikhonov regulriztion introduced in [13] results in the modifiction of the mtrix T in (17), which introduces sptil smoothing. This motivtes the introduction of sptil smoothing step in (18) s nd in (30) s = W ( T 1 ({F yy F } d {F R n F } d ) ) R (31)

4 Expecttion-Mximiztion Single itertion Crude mtched filter Figure 1. Comprison of the scttering coefficient mps obtined using different estimtion methods (ech column correspond to one method) for different scenrios (ech line corresponds to different scenrio). The grphs shre the sme color scle. = W ( r + r Λ(R y) r R r (32) where the squre mtrix W is circulnt smoothing mtrix. The mtrix W is circulnt becuse the scttering coefficients re locted long closed isornge on the ground. The effect of W is to perform weighted verging of r i with the neighboring scttering coefficients. 6 Appliction to scttering coefficient mps estimtion Equtions (2) nd (3) cn be written down for ech rngegte. Notice tht for bisttic scenrios, V typiclly depends on the rnge. If the mesurements y t different rnges re uncorrelted, i.e., if the rnge sidelobes re negligible, the estimtion of the unregulrized R is decoupled in rnge, wht simplifies the computtions. However, the physics clerly imposes sptil constrint on the scttering coefficients r i in the sense tht smooth evolution of the scttering coefficients should be fvored, both in zimuth (cross-rnge) nd in rnge. Hence the smoothing step discussed in the previous section will be extended to smooth the estimte of r i ), lso cross different rnges. Equtions (31) nd (32), while ble to provide solution to the problem, result in n extremely computtionlly intensive lgorithm. We thus propose to consider single itertion of (31) s estimtor. The motivtion is tht, if the initil estimte R (0) is close enough to the true vlue, single itertion might be sufficient. Moreover, if the clutterto-noise rtio is high enough, the contribution of the second term in (31) my be neglected. At one rnge, n estimte of R is thus obtined by computing r = W ( T 1 {āā } d ) R (0), (33) where ā is the ML estimte of conditioned on R (0) nd y is given by (27). The lgorithm we propose thus consists in two steps. The first step estimtes ā t ech rnge nd computing r = T 1 {āā } d t ech rnge independently. The second step performs locl sptil verging of the estimted scttering coefficients. As comprison, we consider the estimtor used in [8], which roughly results from tking F = V. In order to void the introduction of trivil bis on tht crude estimtor, normliztion fctor [13] is used to correct the men vlue. We will denote this estimtor s the crude mtched filter (MF). We compred the performnces of the different estimtion methods for different scenrios. In ll cses, we considered bisttic setup, where the trnsmitter is locted t the origin nd the receiver is locted t (0, 100). The trnsmitter pltform is flying est while the receiver is flying north. Unless otherwise specified, the clutter-to-noise rtio is 20dB. The initil vlue of the covrince mtrix

5 R (0) is tken equl to ri where r is +30dB offset from the true verge vlue. The resulting scttering coefficient mps re illustrted in Fig. 1. Ech column corresponds to different method nd ech row to different scenrio. The first column is the EM lgorithm (32) where the smoothing step is extended in rnge s discussed bove. The middle column is the single-itertion lgorithm described bove nd the third column is the crude MF. # of coefficients ML (sing. it.) Crude MF EM smoothed σ (db (normlized)) # of coefficients ML (sing. it.) Crude MF EM smoothed σ (db (normlized)) () (b) Figure 2. Scttering coefficient histogrm for () uniform scttering coefficients nd (b) the checkerbord ptterned scttering coefficients. The first scenrio (first row) corresponds to uniform scttering coefficients. The fct tht the estimte obtined using the crude MF exhibits sptil rtifcts indictes tht constnt correction fctor does not completely compenste the bis. It should lso be noted tht the EM-bsed estimtor nd the single-itertion estimtor provide very similr scttering coefficients mp. This is lso visible in Fig. 2() showing the histogrm of the scttering coefficients obtined using the different methods. In the scenrio of the second row, sinc-shped trnsmit ntenn without bcklobe pttern ws considered. The scttering coefficients in the bcklobe of the ntenn hve no influence on the mesured dt nd, thus, cnnot be estimted. The vlues for those coefficients thus results from the regulriztion. Agin, clerly, the crude MF fils to provide useful estimte. The crude MF is directly ffected by the ntenn digrm through c i. Following similr resoning s in the previous sections, similr expression to the crude MF hs been used [8] to obtin n estimte of E{ i c i 2 }, but in tht cse, the prior knowledge is difficult to express due to the mixture of the geometric effects c i with knowledge bout i wht mkes regulriztion difficult whenever c i is not constnt. Finlly, in the lst scenrio, non-uniform scttering coefficients were considered. A difference of 10dB ws considered between the high nd low scttering coefficients. This is the typicl difference tht is observed between (monosttic) ground bckscttering nd monosttic bckscttering by the se-surfce in C-bnd [21], demnding scenrio in STAP. As cn be seen, despite the nonuniformity, the pttern is still esily recognizble. Moreover, the histogrm of Fig. 2(b) is clerly bimodl with peks round -5dB nd +5dB, i.e. the exct vlues. While the pttern is still present in the crude MF estimte, the corresponding histogrm does not exhibit the bimodlity, thus denoting lrge estimtion error. 7 End-to-end results Figure 3 presents comprison of the performnce of vr- SINR loss for ν s = Clirvoynt SCM+DL ML (single itertion) ν d 0.5 Figure 3. Comprison of the SINR loss for different covrince mtrix estimtion methods. ious CM estimtion methods in terms of SINR loss. The scenrio considered is the sme s the first scenrio of Fig. 1. In ll cses, the filter considered hs the sme expression, w = Ry 1 v (34) where v is the trget steering vector nd the CM R y is replced with n estimte or the clirvoynt CM. The clssicl smple CM [22], complemented with digonl loding to cope with the limited number of trining smples, is denoted by SCM+DL [10]. The estimte obtined using the proposed method, where the clutter reflectivity is estimted using single itertion of the ML itertive solution, is denoted by ML (single itertion). As cn be seen, the SCM+DL method fils due to the rnge-dependence of the clutter distribution nd due to the very limited number of vilble trining snpshots. The proposed method results in SINR loss tht is lmost identicl to tht of the clirvoynt cse. 8 Summry nd conclusion We propose new method to estimte the clutter covrince mtrix (CM) in the presence of rnge-dependence effects (due both the ground cover vrition nd to geometric effects). The method relies on the knowledge of the structure of the CM to obtin n pproximtion of the ML estimte of the CM of the dt. An estimte of the true ML solution is be obtined s the results of n itertive hill climbing method, using, e.g., the Expecttion-Mximiztion lgorithm. We pproch this using single itertion. Due to the ill-posed nture of the problem, the estimtion needs to be regulrized. We chieve this by performing sptil verging. The estimted clutter scttering coefficients re very close to the estimted ML solution. Moreover, significnt reduction of the computtionl burden is chieved by using single itertion. The method relies on model nd will of course fil if the model is inccurte. This is the cse in the presence of trget nd the corresponding rnges should be removed before proceeding to the sptil verging step.

6 Finlly, the proposed method is shown to provide CM estimte tht performs nerly s well s the clirvoynt CM in terms of SINR loss, which indictes tht the resulting filter will hve nerly-optiml detection performnce. References [1] J. Wrd. Spce-time dptive processing for irborne rdr. Technicl Report 1015, MIT Lincoln Lbortory, Lexington, MA, December [2] Richrd Klemm. Principles of spce-time dptive processing. The Institution of Electricl Engineers (IEE), UK, [3] X. Neyt, Ph. Ries, J. G. Verly, nd F. D. Lpierre. Registrtion-bsed rnge-dependence compenstion method for conforml-rry STAP. In Proc. Adptive Sensor Arry Processing (ASAP) Workshop, MIT Lincoln Lbortory, Lexington, MA, June [4] Y. Zhng nd B. Himed. Effects of geometry on clutter chrcteristics of bisttic rdrs. In Proceedings of the IEEE Rdr Conference 2003, pges , Huntsville, AL, My [5] X. Neyt, J. G. Verly, nd M. Acheroy. Rngedependence issues in multisttic STAP-bsed rdr. In Proceedings of the Fourth IEEE Workshop on Sensor Arry nd Multi-chnnel Processing (SAM 06), Wlthm, MA, July [6] B. Himed, Y. Zhng, nd A. Hjjri. STAP with ngle-doppler compenstion for bisttic irborne rdrs. In Proceedings of the IEEE Rdr Conference 2002, pges , Long Bech, CA, April [7] W. L. Melvin, B. Himed, nd M. E. Dvis. Doubly dptive bisttic clutter filtering. In Proceedings of the IEEE Rdr Conference 2003, pges , Huntsville, AL, My [8] F. D. Lpierre, Ph. Ries, nd J. G. Verly. Computtionlly-efficient rnge-dependence compenstion methods for bisttic rdr STAP. In Proceedings of the IEEE Rdr Conference, pges , Arlington, VA, My [9] A. P. Dempster, N. M. Lird, nd D. B. Rubin. Mximum likelihood from incomplete dt vi de EM lgorithm. Journl of the Royl Sttisticl Society, B39:1 37, [10] J. R. Guerci. Spce-Time Adptive Processing for Rdr. Artech House, Norwood, MA, [11] P. Lecomte. The ERS sctterometer instrument nd the on-ground processing of its dt. In Proceedings of Joint ESA-Eumetst Workshop on Emerging Sctterometer Applictions From Reserch to Opertions, pges , The Netherlnds, November ESTEC. [12] J. P. Burg, D. G. Luenberger, nd D. L. Wenger. Estimtion of structured covrince mtrices. Proceedings of the IEEE, 70(9): , September [13] Y. V. Shkvrko. Estimtion of wvefield power distribution in the remotely sensed environment: Byesin mximum entropy pproch. IEEE Trnsctions on Signl Processing, 50(9): , September [14] D. L. Snyder, J. A. O Sullivn, nd M. I. Miller. The use of mximum likelihood estimtion for forming imges of diffuse rdr trgets from dely-doppler dt. IEEE Trnsctions on Informtion Theory, 35(3): , My [15] M. Brkt. Signl detection nd estimtion. Artech house, [16] D. R. Fuhrmnn. Structured covrince estimtion: Theory, ppliction nd recent results. In IEEE Sensor Arry nd Multichnnel Signl Processing Workshop, Wlthm, MA, July [17] A. N. Tikhonov nd V. Y Arsenin. Solutions of illposed problems. John Wiley & Sons, [18] Y. V. Shkvrko. Unifying regulriztion nd byesin estimtion methods for enhnced imging with remotely sensed dt Prt I: Theory. IEEE Trnsctions on Geoscience nd Remote Sensing, 42(5): , My [19] P. Moulin, J. A. O Sullivn, nd D. L. Snyder. A method of sieves for multiresolution spectrum estimtion nd rdr imging. IEEE Trnsctions on Informtion Theory, 38(2): , Mrch [20] A. D. Lntermn. Sttisticl imging in rdio stronomy vi n expecttion-mximiztion lgorithm for structured covrince estimtion. In J. A. O Sullivn, editor, Sttisticl Methods in Imging: In Medicine, Optics, nd Communiction, festschrift in honor of Donld L. Snyder s 65th birthdy. Springer-Verlg, Jnury [21] P. Pettiux, X. Neyt, nd M. Acheroy. Vlidtion of the ERS-2 sctterometer ground processor upgrde. In Proceedings of SPIE Remote Sensing of the Ocen, Se Ice nd Lrge Wter Regions 2002, volume 4880, Crete, Greece, September [22] I. S. Reed, J. D. Mllett, nd L. E. Brennn. Rpid convergence rte in dptive rrys. IEEE Trnsctions on Aerospce nd Electronic Systems, 10(6): , November 1974.