Sensitivity Analysis of MVDR and MPDR Beamformers

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1 21 IEEE 26-th Covetio of Electrical ad Electroics Egieers i Israel Sesitivity Aalysis of ad Beamformers Livat Ehreberg, Sharo Gaot, Amir Leshem ad Ephraim Zehavi School of Egieerig, Bar-Ila Uiversity, Ramat-Ga, Israel livate1@gmail.com; gaot@eg.biu.ac.il; leshem.amir@gmail.com; ephiz@yahoo.com Abstract A sesitivity aalysis of two distortioless beamformers is preseted i this paper 1. Specifically, two well-kow variats, amely the miimum power distortioless respose ) ad miimum variace distortioless respose ) beamformers, are cosidered. I our sceario, which is typical to may moder commuicatios systems, waves emitted by multiple poit sources are received by a atea array. A aalytical expressio for the sigal to iterferece ad oise ratio SINR) improvemet obtaied by both beamformers uder steerig errors is derived. These expressio are experimetally evaluated ad compared with the robust Capo beamformer ), a robust variat of the beamformer. We show that the beamformer, which uses the oise correlatio matrix i its miimizatio criterio, is more robust to steerig errors tha its couterparts, that use the received sigal correlatio matrix. Furthermore, eve if the oise correlatio matrix is erroeously estimated due to steerig errors i the iterferece directio, the advatage is still maitaied for reasoable rage of steerig errors. These coclusios coform with Cox [1] fidigs. Oly lie of sight propagatio regime is cosidered i the curret cotributio. Ogoig research exteds this work to fadig chaels. Topic: Sigal Processig) I. INTRODUCTION Beamformers are widely used i a variety of sigal processig applicatios, e.g. speech ehacemet, radar, ad wireless commuicatios. I particular, the problem of estimatig sigals of multiple trasmitters usig a atea array is oe of the fudametal problem i array processig. I this cotext, the Capo beamformer [2] gaied popularity due to its high iterferece rejectio capabilities. The Capo beamformer, also referred to by Va Trees as beamformer [3], is however sesitive to steerig errors. Several robust techiques for alleviatig the sesitivity problem were therefore proposed i the literature. Ward et al. [4] provides a compariso of several robust beamformers, based o diagoal loadig, that compesates for model errors. Wax et al. [5] presets a aalysis of the sigal-to-iterferece-plus-oise ratio SINR) for the beamformer, i terms of the sigal-to-oise ratio SNR), iterfereceto-oise ratio INR), sigal-to-iterferece ratio SIR), agular separatio betwee desired sigal ad iterfereces, ad correlatio betwee these sigals. Reddy et al. [6] study the sigal cacellatio ad iterferece rejectio effects of the beamformer i the presece of correlated iterferig sources. Of special iterest to our study, is the aalysis preseted by Cox [1]. I his semial work Cox compares the sesitivity of two robust distortioless beamformers to steerig errors. The two structures differ i the correlatio matrix used. While the beamformer is usig the oise correlatio matrix, the beamformer is usig the received sigals correlatio matrix desired sigal plus oise). The robustess of the two beamformers to steerig errors i the desired sigal directio is aalyzed. It is argued that the beamformer, that icorporates the desired sigal ito the correlatio matrix, teds 1 This research was partially supported by MAGNET ISRC cosortium, Miistry of Idustry, Trade ad Labor, the Govermet of Israel. to suppress the sigal impigig the array from the desired source whe it is erroeously steered. The applicatio of the more robust beamformer ecessitates the availability of a estimate of the oise correlatio matrix. I our work we cosider scearios i which multiple sigals emaatig from poit sources are received by the array. The iterferig sources together with ocoheret sources costitute the oise correlatio matrix. We show that uder ucertaities i the iterferece sigal steerig vectors the beamformer maitais higher robustess as compared with the beamformer. The structure of this paper follows. I Sectio II the problem is formulated ad both the ad beamformers are itroduced. A aalytical expressio for the rovemet of both structures is derived i Sectio III. These cumbersome expressio are experimetally evaluated i Sectio IV ad compared with the algorithm, which is a robust variat of the beamformer. The experimetal results are discussed ad possible explaatios for the beamformers behavior are give i the same sectio. The paper is cocluded i Sectio IV-C. II. PROBLEM FORMULATION Cosider L elemets atea array ad deote the complex evelope of the received arrow-bad sigals as y[t] C L 1. The received sigals are related to the trasmitted sigal by yt) =dt)st)+gt)s It)+vt) 1) where, the desired sigal is deoted s, its power is σ d 2, ad d is the correspodig steerig vector. s I =[s I,1, s I,K] T is a vector of K ucorrelated iterferece sigals, Λ=diag [ σ 2 I,1,,σ 2 I,K]) is its correspodig correlatio matrix, ad G =[g 1,, g K ] is a matrix cosistig of the correspodig steerig vectors. v are spatially-white sesor oise sigals with correlatio matrix σ 2 vi. For the sake of brevity, we have omitted the time idex. We will adopt this covetio wheever o cofusio ca arise. The correlatio matrix of the received sigal y is give by R y = R s + R 2) where R s = σ 2 d dd H is the desired sigal compoet, ad R = GΛG H +σ 2 v I is the iterferece sigals compoet. Defiig B = GΛ 1 2 we ca restate the iterferece sigals correlatio matrix as R = BB H + σ 2 v I. The received sigal is processed by a beamformer, i order to ehace the desired sigal while mitigatig all iterferece sigals: x = w H y. 3) There are several alteratives for desigig the beamformer, such as the miimum mea square error MMSE) criterio [3]. I this paper we are cocered with distortioless array respose. I particular, two alterative beamformer criteria are cosidered, amely the /1/$26. c 21 IEEE 416

2 ad the. A robust versio of the beamformer, the well-kow [7], will be detailed ad compared i Sec. IV. The beamformer seeks for the best filter set w 1 such that the overall iterferece sigals power is suppressed while the desired sigal is maitaied: w 1 = argmi w H R w s.t. w H d =1. 4) The well-kow solutio to this criterio is give by: w 1 = R d d H R d. 5) The criterio is closely related to the : w 2 = argmi w H R yw s.t. w H d =1 6) with a solutio give by: w 2 = R y d d H R y d. 7) Ideally, without model-errors, both criteria coicide [3]. I the followig sectios we aalyze the beamformers performace uder model errors i both the iterfereces ad desired chael steerig vectors. III. THEORETICAL ERROR ANALYSIS I practical applicatios, accurate steerig vectors or oise correlatio matrix are rarely available. This may lead to both poor iterferece reductio ad desired sigal distortio, ad hece cause performace degradatio. Cox [1] compared ad evaluated the sesitivity of the ad beamformers ad showed that the former, which is directly usig the oise correlatio matrix, is more robust to desired sigal steerig errors. I both beamformers, the aalysis was doe with o model errors o the correlatio matrices. I our cotributio, a specific model for the oise correlatio matrix is cosidered. I this sceario, the oise correlatio matrix is based o the estimated chaels ad cosists of both iterferig poit sources ad ocoheret oise sources. We derive ow a sesitivity aalysis to iterferece sigals steerig errors. Deote the estimatio of the desired chael vector by, ad the estimated iterferece chael as G. σ 2 v ad Λ are assumed to be kow. Hece, the estimated oise correlatio matrix is give by: = B B H + σ 2 v I. 8) Note, that while the oise correlatio matrix R, used by the beamformer, is assumed to be erroeously estimated, the received sigal correlatio matrix R y, used by the, is assumed to be accurately estimated. A. Figure-of-Merit The output power of both beamformers is give by: σ 2 x = w H R yw = w H R sw + w H R w 9) where w is either the filters w 1 or filters w 2. Followig Cox [1], we use the rovemet, amely, = SINRout SINR i, as a performace measure comparig both beamformers. Sice the iput SINR is idepedet of the beamformer criterio, the output SINR will be used for the compariso. Defie, SINR out = wh R sw w H R = σ2 d w H d 2 1) w w H R w Note, that w is calculated usig iaccurate estimates of d ad R. Hece, the costrait w H d =1is ot met. Therefore, the umerator ca be regarded as a distortio term ad the deomiator is related to the oise level at the output of the beamformer. B. Error Formulatio for The received sigal correlatio matrix R y is assumed to be accurately estimated. Hece the expressio for the output SINR of the beamformer, give by Cox [1] remais ualtered: SINR w γ 2 SINR 2 max out = 1+ 11) 2SINR max + SINRmax) 2 1 γ2 ) where γ 2 <, ; R <, d; R > 2 >< d, d; R > with γ 2 1. The weighted ier product is defied as < a, b; C > a H Cb 12) ad the maximum SINR obtaied i the error-free estimatio case is defied as SINR max = σ 2 d < d, d; R >. C. Error formulatio for I our sceario the oise correlatio matrix is comprised of all iterferece steerig vectors. We derive i the sequel a expressio for the output SINR of the beamformer. The power of the sigal compoet at the output of the beamformer is give by σd 2 w H 1 d 2 = σd 2 H 2 H d 13) <, d; = σd 2 > 2 <, ; > 2 ad the power of the oise compoet is give by H H w H 1 R w 1 = R <, ; = <, ; H R > > 2 14) Collectig terms we get SINR w <, d; 1 out = σd 2 > 2 15) <, ; R > Usig Woodbury idetity [8], the iverse of the estimated oise correlatio matrix [defied by 8)] is give by = 1 I σ B σ 2 vi 2 + B ) H B BH). 16) v Usig the last expressio ad 8) we have R 1 σ 2 v = 1 σ 2 σv 2 v I + BB H) 17) σvi 2 + BB H) B σ 2 v I + B B) H BH. 417

3 The last expressio is comprised of two terms. Substitutig each of the expressios ito the deomiator of 15) we get after cumbersome maipulatios): α 1 σ 2 v <, ; H σ 2 vi + BB H) = 18) > + 1 σ 2 v K <, b i; i=1 >< b i, ; I > β 1 2 H σvi 2 + BB H) B 19) σ v σvi 2 + B B) H BH [ K K = t i,j <, b i; >< b j, ; I > i= K < σ, b j; >< b j, b i; I > v ] K t i,j < b j, ; I > 2) where the elemets t i,j are the compoets of the matrix T defied as T = σvi 2 + B B) H. 21) Collectig terms usig 15), 18) ad 19), we fially get the expressio for the SINR at the output of the beamformer: SINR w 1 out = σ2 d <, d; > 2. 22) α β IV. PERFORMANCE EVALUATION This sectio is dedicated to performace evaluatio of the ad beamformers. The [7], which is a variat of the beamformer, robust agaist erroeous desired sigal steerig vector, is used for compariso. A. Test procedure We begi the evaluatio by examiig the ratio of the SINR improvemet [defied by 1)] betwee the 11) ad the 15). We the compare the SINR gai of the ad beamformers with that of the, proposed by Stoica et al. [7]. The is a extesio of the stadard Capo beamformer i.e. ) to the case of ucertaity i the steerig vectors. Leshem ad Gaot [9] used the for successive iterferece cacelatio i the MIMO commuicatios framework. The miimizes the followig criterio: ˆd = argmi d H R y d s.t. d 2 = ɛ 23) d where is the iitial estimate of the steerig vector. The solutio for ˆd is give by: ˆd = I + λr y). 24) The Lagrage multiplier λ is obtaied by solvig gλ) = I + λr y) 2 = ɛ. 25) Solvig this equatio ecessitates kowig the ucertaity parameter ɛ, which is uavailable i may cases. Although uavailable, we assume, for simplicity that ɛ is kow. Fially, we obtai the robust )/ )[db] SNR = db SNR = db 5 SNR = 4dB SNR = 8dB SNR = 12dB SNR = 16dB SNR = 2dB meaδθ [degrees] Fig. 1. rovemet ratio of ad beamformers for various values of SNR. Steerig error mea i the rage o 8 o. beamformer by substitutig i the stadard beamformer the iitial steerig vector with the updated estimatio: w = R y ˆd ˆd H R y. 26) ˆd Simulatios were carried out usig 4 sigals impigig o a array of 5 atea elemets from directios θ = o, 4 o, 8 o, 11 o. All sigals are equi-power ad the spatial white oise power σ 2 v is set i the rage db to 2dB. The erroeous steerig agle of all sigals either desired or iterferece) was set to θ i = θ i +Δθ, i = 1, 2, 3, 4. The steerig error is assumed to be Gaussia distributed Δθ Nη,.2η) 2 ), where η>. We repeated each experimet 1 times ad averaged the results. B. Results Fig. 1 depicts the ratio of the rovemets of the ad whe the oise correlatio matrix is erroeously estimated, due to steerig errors. It ca be see that i spatial white oise regime SNR = db), the differeces betwee ad are egligible. However, i the spatial iterferece regime SNR > db), the rovemet ratio betwee the beamformers is sigificatly higher. To gai some isight o the of the beamformers we proceed by evaluatig the actual rovemet for the, beamformer i compariso with. The improvemet as a fuctio of Δθ is depicted i Fig. 2a) for SNR = db ad i Fig.2b) for SNR =2dB. Table I provides a compariso of the rovemet for various values of SNR for η =6 o. As stated above, i the spatial iterferece regime, there are sigificat differeces betwee ad. It is also evidet that the outperforms the, ad that the differeces gets larger as the estimatio error i the steerig agle icreases. C. Discussio As preseted above, the performace differeces betwee the, ad are isigificat i the white oise regime. This ca be attributed to the tedecy of all beamformers to coverge to the delay ad sum beamformer whe the white oise level is high. I the spatial iterferece regime, the experieces 418

4 Ideal [db] Gai[dB] meaδθ [degrees] θ[degrees] 4 a) SNR = db Fig. 3. Beampatter for ideal beamformer with exact steerig vector,, ad beamformers i high SNR. Steerig error mea η =1 o. [db] Fig meaδθ [degrees] b) SNR =2dB rovemet as a fuctio of η for, ad. SNR TABLE I SINR IMPROVEMENT FOR VARIOUS VALUES OF SNR FOR, AND. STEERING ERROR MEAN η =6 o.all VALUES ARE IN DB. a cosiderable performace degradatio, which icreases with the steerig error. This ca be explaied by examiig the beampatter depicted i Fig. 3 i the rage ±6 o ) for the case of η =1 o ad SNR =2dB. Both beamformers are erroeously steered towards the desired source. The, which utilizes the received sigal correlatio matrix i the miimizatio criterio, cosiders the actual desired source directio as a iterferece, ad therefore aims at directig ull i its agle. This pheomeo dramatically decreases the output SINR. The partially compesates for the steerig errors by broadeig the mai lobe), resultig i less desired sigal suppressio. The which oly uses the oise compoet i the miimizatio criterio does ot direct a ull towards the desired source ad therefore exhibits a icreased robustess to steerig errors. V. CONCLUSION I this paper, a sesitivity aalysis of distortioless beamformer was performed. The case of multiple iterferece sigals with erroeous steerig directios was aalyzed. A expressio for the output SINR obtaied by the ad beamformers was derived ad experimetally evaluated. Furthermore, the SINR improvemets of the two desigated beamformers was compared with rovemet of the, a well-kow robust variat of the beamformer. It is show that the which utilizes the oise correlatio matrix R i its miimizatio criterio, outperforms the ad, both usig the received sigal correlatio matrix R y, although the latter exhibits a improved robustess to steerig errors. A fudametal coclusio is that i scearios i which R is available, it is advisable to use the beamformer, which shows sigificat robustess to errors i steerig vectors estimatio. I moder commuicatio systems this sceario is ofte ecoutered. We stress, that i applicatios for which R is uavailable, the should be used istead of the beamformer. A ogoig research exteds the results of this paper to fadig rather tha lie of sight chaels. REFERENCES [1] H. Cox, Resolvig power ad sesitivity to mismatch of optimum array processors, The joural of the acoustical society of America, vol. 54, pp , Feb [2] J. Capo, High-resolutio frequecy-waveumber spectrum aalysis, Proceedigs of the IEEE, vol. 57, o. 8, pp , aug [3] H. L. V. Trees, Optimum Array Processig, Detectio, Estimatio, ad Modulatio Theory, Part IV,. New York: Wiley, 22. [4] J. Ward, H. Cox, ad S. Kogo, A compariso of robust adaptive beamformig algorithms, i Sigals, Systems ad Computers, 23. Coferece Record of the Thirty-Seveth Asilomar Coferece o, vol. 2, , pp Vol

5 [5] M. Wax ad Y. Au, Performace aalysis of the miimum variace beamformer, Sigal Processig, IEEE Trasactios o, vol. 44, o. 4, pp , apr [6] V. Reddy, A. Paulraj, ad T. Kailath, Performace aalysis of the optimum beamformer i the presece of correlated sources ad its behavior uder spatial smoothig, Acoustics, Speech ad Sigal Processig, IEEE Trasactios o, vol. 35, o. 7, pp , jul [7] J. Li, P. Stoica, ad Z. Wag, O robust capo beamformig ad diagoal loadig, Sigal Processig, IEEE Trasactios o, vol. 51, o. 7, pp , july 23. [8] K. B. Peterse ad M. S. Pederse, The matrix cookbook, imm.dtu.dk/pubdb/p.php?3274, Nov. 28. [9] A. Leshem ad S. Gaot, Robust sequetial iterferece cacellatio for space divisio multiple access commuicatios, i Europea Sigal Processig coferece. Poza, Polad: EURASIP, Sep