374 Progress In Electromgnetics Reserch Symposium 2006, Cmbridge, USA, Mrch 26-29 A Subspce-bsed Robust Adptive Cpon Bemforg G. S. Lio, H. Q. Liu, nd J. Li Xidin University, Chin Abstrct Adptive bemforg suffers from performnce degrdtion in the presence of mismtch between the ctul nd presumed rry steering vector of the desired signl. This ide enlightens us, so we propose subspce pproch to dptive bemforg tht is robust to rry errors bsed on imizing MUSIC output power. The proposed method involves two steps, the first step is to estimte the ctul steering vector of the desired signl bsed on subspce technique, nd the second is to obtin optiml weight by utilizing the estimted steering vector. Our method belongs to the clss of digonl loding, but the optiml mount of digonl loding level cn be clculted precisely bsed on the uncertinty set of the steering vector. To obtin noise subspce needs eigen-decomposition tht hs hevy computtion lod nd knows the number of signls priori. In order to overcome this drwbck we utilized the POR (Power of R) technique tht cn obtin noise subspce without eigen-decomposition nd the number of signls priori. It is very interesting tht Li Jin s method is specil cse where m = 1, nd the proposed subspce pproch is the cse where m, so we obtined uniform frmework bsed on POR technique. This is lso n explntion why the performnce of the proposed subspce pproch excels tht of Li Jin s method. The excellent performnce of our lgorithm hs been demonstrted vi number of computer simultions. 1. Introduction Arry signl processing hs wide pplictions in rdr, communictions, sonr, coustics, seismology, nd medicine. One of the importnt tsks of rry processing is bemforg. The stndrd bemformers include the dely-nd-sum pproch, which is known to suffer from poor resolution nd high sidelobe problems. The Cpon bemformer dptively selects the weight vectors to imize the rry output power subject to the liner constrint tht the signl of interest (SOI) does not suffer from ny distortion [1]. The Cpon bemformer hs better resolution nd interference rejection cpbility thn the stndrd bemformer, provided tht the rry steering vector corresponding to the SOI is ccurtely known. In prctice, the knowledge of the SOI steering vector my be imprecise, the cse due to differences between the presumed signl steering vector nd the ctul signl steering vector. When this hppens, the Cpon bemforg my suppress the SOI s n interference, which result in rry performnce drsticlly reduced, especilly rry output signl-to-interference-plus-noise rtio (SINR) [4]. In the pst three decdes mny pproches hve been proposed to improve the robustness of the Cpon bemforg. Additionl liner constrints, including point nd derivtive constrints, hve been imposed to improve the robustness of the Cpon bemforg [2, 3]. However, for every dditionl liner constrints imposed, the bemformer loses one degree of freedom (DOF) for interference suppression. Moreover, these constrints re not explicitly relted to the uncertinty of the rry steering vector. Digonl loding (including its extended versions) hs been populr pproch to improve the robustness of the Cpon bemformer [4]. However, for most of the digonl loding methods, detering the digonl loding res n open problem. Recently there re some methods been proposed (for exmples, [5 7] nd reference therein) to this point. Mismtch between the presumed steering vector of the SOI nd the ctul one result in drsticlly reduced rry SINR, therefore if we cn estimte ctul steering vector of the SOI, robustness of the rry will be improved. In this pper, from the point of view of the subspce we propose novel robust Cpon bemformer, which involves two steps, the first step is to estimte ctul steering vector of SOI, nd the second is to clculte optiml weight by Cpon method. The rest of this pper is orgnized s follows. Section 2 contins bckground mteril. In section 3, the robust Cpon bemformer is developed. Computer simultion results illustrting the performnce of the robust Cpon bemformer re presented in Section 4. Finlly, Section 5 contins the conclusions. 2. Bckground 2.1. Signl Model We consider the stndrd nrrowbnd bemforg model in which set of M nrrowbnd plne wve signls, impinge on n rry of N sensors with hlf wvelength spcing, where M < N. The N 1 vector of received signls is given by
Progress In Electromgnetics Reserch Symposium 2006, Cmbridge, USA, Mrch 26-29 375 x(t k ) = M 1 m=0 (θ m )s m (t k ) + n(t k ), k = 1,2,...,L (1) where s m (t k ),m = 0,...,M 1; k = 1,2,...,L re the source signls snpshots, (θ m ) = [1,e jπ sin θm,...,e jπ(n 1) sin θm ] T is the steering vector in the direction θ m, nd n(t k ), k = 1,2,...,L re the vectors contining dditive white noise smples, L is the number of the snpshots. Also, in this pper, the sources nd noise re ssumed to be sttisticlly uncorrelted. We ssume tht one of the signls is the desired signl, sy s 0 (t), nd tret the reing signls s interferences. Since s 0 (t) is uncorrelted with the noise nd interferences, the dt covrince mtrix hs the form, M 1 R = σ0(θ 2 0 ) H (θ 0 ) + σk(θ 2 k ) H (θ k ) + R n R s + R i+n (2) k=1 where R s = σ0(θ 2 0 ) H (θ 0 ), σi 2 = E{ s i (t k ) 2} is the power of ith signl, nd R i+n is the interference plus noise covrince mtrix. In prctice, the covrince mtrix R is estimted by ˆR = 1 L x n x H n (3) L where ll received signls hve zero mens nd L smples re independent. n=1 2.2. Cpon Bemforg The Cpon bemforg is s follows. Detere the N 1 vector w 0 tht is the solution to the following linerly constrined qudrtic imiztion problem, w wh Rw s.t.w H ā(θ 0 ) = 1 (4) where ā(θ 0 ) is presumed steering vector of the desired signl. Appling Lgrnge multiplier method results in the following solution, R 1 ā(θ 0 ) w 0 = ā H (θ 0 )R 1 (5) ā(θ 0 ) The rry men output power p 0 is 1 p 0 = ā H (θ 0 )R 1 ā(θ 0 ) The Cpon bemformer hs better resolution nd much better interference rejection cpbility thn the stndrd bemformer, provided tht the presumed rry steering vector of the SOI mtch ctul rry steering vector precisely. In prctice, the exct steering vector of the SOI is unvilble or its mesure/estimtion is imprecise, therefore, we only use the presumed ā(θ 0 ) insted of the ctul (θ 0 ) in the Cpon bemformer, nd the mismtch between the exct steering vector nd the presumed one my drsticlly degrde the performnce of the Cpon bemformer. The rry output SINR cn be written s, SINR = E[ wh 0 s 0 (t) 2 ] w H 0 R i+nw 0 = where σ0 2 = E( s 0 (t) ). Inserting (5) into (7) yields, ā SINR = σ0 2 H (θ 0 )R 1 i+n (θ 0) 2 ā H (θ 0 )R 1 i+nā(θ 0) where (θ 0 ) is the ctul steering vector, then (8) cn be rewritten s: SINR = σ 2 0 H (θ 0 )R 1 i+n (θ 0) w H 0 (6) σ0 w 2 H 0 (θ 0 ) 2 ( M 1 ) (7) σk 2(θ k) H (θ k ) + R n w 0 k=1 H (θ 0 )R 1 i+nā(θ 0) 2 ( H (θ 0 )R 1 i+n (θ 0))(ā H (θ 0 )R 1 i+nā(θ 0)) = SINR m cos 2 ((θ 0 ),ā(θ 0 );R 1 i+n ) (9) (8)
376 Progress In Electromgnetics Reserch Symposium 2006, Cmbridge, USA, Mrch 26-29 where SINR m = σ0 2 H (θ 0 )R 1 i+n (θ 0) nd cos 2 ( ) is defined s, cos 2 H Zb 2 (,b;z) = ( H Z)(b H Zb) Clerly, 0 cos 2 (,b;z) 1. Therefore, rry output SINR is reduced due to mismtch between the presumed steering vector of the SOI nd its true vlue. In recent yers, digonl loding (DL) is populr pproch to improving the robustness of Cpon bemformer to the mismtch bove. In DL methods, the dt covrince ˆR is replced by ˆR+γI, where γ is positive constnt (see reference [4 6] for detils). The DL method proposed in [4] is used in Section 4 for comprisons. In the following section, novel robust bemforg is developed to llevite the effects of the steering vector mismtch on the SINR performnce of Cpon bemformer. 3. Robust Cpon Bemforg The robust bemforg problem we will del with in this pper cn be briefly stted s follows: Extend the Cpon bemformer so s to improve rry output SINR even only n imprecise knowledge of steering vector (θ 0 ) is vilble. To simplify the nottion, in wht follows, we sometimes omit the rgument θ of (θ) nd ā(θ). We ssume tht the only knowledge we hve bout (θ 0 ) is tht it belongs to the following uncertinty [5] [(θ 0 ) ā] H C 1 [(θ 0 ) ā] 1 (11) where C re given positive definite mtrix. As shown bove, rry performnce loses will occur in the presence of mismtch between the presumed nd ctul steering vectors of the SOI. If we estimte the ctul steering vector of the SOI s more precise s we cn, then performnce of the bemformer will be improved. The proposed robust Cpon bemforg is bsed on this ide. From subspce theory we know tht the ctul steering vector of desired signl is orthogonl to noise subspce, our pproch is bsed on the optimizing the projection of signl steering vector onto noise subspce. The steering vector is normed s 2 = H = N. To derive our robust Cpon bemformer, we use following constrined optimiztion H U n U H n (10) s.t.( ā) H C 1 ( ā) 1 (12) 2 = N where ā is known to us in dvnce, but hs error (mismtch to the ctul steering vector of the SOI). U n is the noise subspce, which is obtined by the eigen-decomposition of ˆR. To mke up the noise subspce, we ssume tht the number M, of plne wves impinging on the rry is known priori. We use this ssumption only for derivtions nd cncel it lter. Note tht we cn improve the estimtion ccurcy of the ctul steering vector of the SOI from (12), nd then obtin optiml weight w 0 by Cpon method. Without loss of generlity, we will consider solving (12) for the cse in which C = εi, (ε is user prmeter), then, (12) becomes H U n U H n s.t. ā 2 ε (13) 2 = N We use the Lgrnge multiplier methodology gin, which is bsed on the function L(,λ,µ) = H U n U H n + µ(2n ε ā H H ā) + λ( H N) (14) where µ 0, λ 0 re the Lgrnge multiplier. Hence, the unconstrined imiztion of (14) for fixed µ, λ, is given by Clerly, the optiml solution of is δl(,µ,λ) δ = 2U n U H n 2µā + 2λ = 0 (15) â = µ(u n U H n + λi) 1 ā (16)
Progress In Electromgnetics Reserch Symposium 2006, Cmbridge, USA, Mrch 26-29 377 Inserting â into (14), imizing L(,λ,µ) with respect to µ gives δl(â,µ,λ) = 2N ε ā H â â H ā = 0 δµ (17) Then, we obtin 2N ε ˆµ = 2ā H (U n U H n + λi) 1 ā (18) Inserting ˆµ into (14), imizing Lgrnge function with respect to λ yields nd the following eqution cn be derived, δl(â, ˆµ,λ) δλ = â H â N = 0 (19) ā H (U n U H n + ˆλI) 2 ā [ā H (U n U H n + ˆλI) 1 ā] 2 = N (N ε 2 )2 (20) Then, the solution of ˆλ cn be obtined by some simple mnipultions. Substituting (18) into (16) yields â = (N ε 2 ) (U n U H n + ˆλI) 1 ā ā H (U n U H n + ˆλI) 1 ā To summrize, the proposed robust Cpon bemforg consists of following steps. The lgorithm: Step 1: Clculte dt covrince mtrix, i.e., ˆR = 1 L x n x H n L Step 2: Compute the eigen-decomposition of ˆR nd obtin the noise subspce U n. Step 3: Solve ˆλ in (20). Step 4: Use the ˆλ in Step 3 to clculte â = (N ε 2 ) (U n U H n + ˆλI) 1 ā ā H (U n U H n + ˆλI) 1 ā n=1 (21) (22) Step 5: Compute optiml weight by Cpon method, i.e., w 0 = α ˆR 1 1 â, α = (23) â H ˆR 1â The proposed robust bemforg belongs to the clss of digonl loding, but the optiml mount of digonl loding level cn be precisely clculted bsed on the uncertinty set of the steering vector. In the Section 4 computer simultion results demonstrte excellent performnce of the proposed lgorithm. In order to void eigen-decomposition nd knowing the number of signls priori, we use the POR pproch to obtin noise subspce. In [8], R is decomposed by EVD s R = [U s U n ] [ ] [ Λs + σvi 2 0 U H s 0 σvi 2 where Λ s = dig{δ1,...,δ 2 M 2 }, U s denotes the signl subspce. It pproximtes the noise subspce of R bsed on R m (m is positive integer). Accordingly {( σv 2m R m = U n U H σ 2 ) m } v n + U s dig δi 2 + U H σ2 s (25) v Clerly, (σ 2 v/(δ 2 i +σ2 v)) m is less thn 1 nd converge to zero for sufficiently lrge m. Theoreticlly, lim m σ 2m v R m = U n U H n. As result, we modify the criterion (12) nd consider the following POR cost function U H n ] (24) H ˆR m s.t. ā 2 ε (26) 2 = N
378 Progress In Electromgnetics Reserch Symposium 2006, Cmbridge, USA, Mrch 26-29 By contrst, the (26) voids estimting tht dimension directly. Moreover, s m, the proposed the POR bemforg method in (26) converges to the subspce one in (12), nd it cn be shown tht the performnce of the POR method for finite m will converge to the subspce one through computer simultion. We compred our method with previous one in [6], where m = 1 in the section 4. () (b) Figure 1: Output SINR versus different SNR, pointing errors = 3, for () ε = 0.7, for (b) ε = 7. () (b) Figure 2: The Output SINR versus pointing errors for () ε = 0.7, for (b) ε = 7. 4. Computer Results Our motivtion of simultion is to demonstrte the performnce in the presence of some errors in the steering vector. In ll of the exmples considered below, we ssume uniform liner rry (ULA) with N = 20 sensors nd hlf-wvelength spcing is used. The sources emitted mutul independent nrrowbnd wveforms. All the results re chieved vi 50 Monte Crlo trils. In the first exmple, we consider the effect of the pointing error of the SOI on rry output SINR. The exct direction of rrivl of SOI is θ 0, of which ssumed vlue is θ 0 +, i.e., ā(θ 0 ) = (θ 0 + ). We ssume tht (θ 0 ) belongs to the uncertinty set (θ 0 ) ā(θ 0 ) 2 ε (27)
Progress In Electromgnetics Reserch Symposium 2006, Cmbridge, USA, Mrch 26-29 379 where ε is user prmeter. Let ε 0 = (θ 0 ) ā(θ 0 ) 2. Then, choosing ε = ε 0. However, since is unknown in prctice, the ε we choose my be greter or less thn ε 0. To show tht the choice of ε is not criticl issue for our lgorithm, we will present simultion results with severl vlues of ε in eqution (21). In this exmple, the directions of the SOI nd n interference source re θ 0 = 30, θ 1 = 30, respectively. The ssumed direction of the SOI is θ 0 + = 33, which results exct ε 0 = 5.7750. The interference-to-noise rtio (INR) is 40 db. Figure 1 plots rry output SINR versus the SNR of the SOI when the number of snpshots is set to be L = 100. It is observed tht the proposed lgorithm (12) performs better thn other two lgorithms t ll input SNR. Also, since the error in steering vector of SOI is reltively lrge nd cnnot be negligible, the stndrd Cpon bemformer nd its digonl loding version suffer from severe performnce degrdtion when SNR increses. However, the proposed bemformer hs SINR loss of 5 db when SNR = 20 db. The proposed the POR method for different m over vrious input SNRs is lso illustrted in Figure 1. Obviously, the Output SINR for m = 2 nd m = 3 ll converge to subspce pproch (12), the counterprt for m = 1 [6] hs the lrge output SINR loss. Figure 2 shows the rry output SINR curve versus the pointing errors, in which SNR = 0 db, INR = 20 db, L = 100. In this figure, the excellent performnce chieved by the proposed lgorithm is observed, which shows the robustness to the pointing errors. It is noted tht, similr to other robust pproches, our method will worsen if there is/re strong interference sptilly closed to the SOI. The reson is tht for given uncertinty region (11), the solution of in optimiztion (12) is converge to the strong interference source. Also, it cn be seen tht the Output SINR of the proposed POR method increses s m increses, with m = 3 hs sme performnce with subspce one (12). 5. Conclusion In this pper, we discuss the performnce degrdtion due to the presence of steering vector uncertinty of the SOI, such s, direction of rrivl estimtion error, finite number of snpshots, nd rry response error, etc. A robust Cpon bemformer is developed by utilizing the orthogonlity between signl nd noise subspce. A more ccurte estimte of the ctul steering vector of the SOI is obtined vi constrined optimiztion, by which the optiml weight is computed ccording to Cpon bemforg. We hve shown tht the proposed lgorithm belongs to the clss of digonl loding pproches, nd the optiml mounts of digonl loding cn be precisely clculted. In order to void eigen-decomposition nd knowing the number of signls priori, we hve proposed POR-bsed robust bemforg scheme. It significntly outperforms the method proposed in [6] nd converge to the subspce one (12). The excellent performnce of our lgorithm hs been demonstrted vi number of computer simultions. Acknowledgment The work described in this pper ws supported by Ntionl Science Fund under grnt NFS60472097. REFERENCES 1. Cpon, J., High resolution frequency-wvenumber spectrum nlysis, Proc. IEEE, Vol. 57, 1408 1418, Aug. 1969. 2. Er, M. H. nd A. Cntoni, Derivtive constrints for brod-bnd element spce ntenn rry processor, IEEE Trns. Acoust., Speech, Signl Processing, Vol. ASSP-31, 1378 1393, Dec. 1983. 3. Buckley, K. M. nd L. J. Griffiths, An dptive generlized sidelobe cnceller with derivtive constrints, IEEE Trns. Antenns Propgt., Vol. AP-34, 311 319, Mr. 1986. 4. Crlson, B. D., Covrince mtrix estimtion errors nd digonl loding in dptive rrys, IEEE Trns. Aerospce nd Electronic System., Vol. 24, 397 401, Jul. 1988. 5. Stoic, P., Z.-S. Wng, nd J. Li, Robust cpon bemforg, IEEE Signl Processing Letters, Vol. 10, No. 6, 172 175, June 2003. 6. Li, J., P. Stoic, nd Z.-S. Wng, Doubly constrined robust cpon bemforg, IEEE Trns. Signl Processing, Vol. 52, No. 9, 2407 2423, Sep. 2004. 7. Shhbzpnhi, S., A. B. Gershmn, Z.-Q. Luo, nd K. M. Wong, Robust dptive bemforg for generl-rnk signl models, IEEE Trns. Signl Processing, Vol. 51, No. 9, 2257 2269, Sep. 2003. 8. Xu, Z., P. Liu, nd X. Wng, Blind multiuser detection: from moe to subspce methods, IEEE Trns. Signl Processing, Vol. 52, No. 2, 510 524, Feb. 2004.