Robust Beamforming via Matrix Completion

Transcription

1 Robust Beamforming via Matrix Comletion Shunqiao Sun and Athina P. Petroulu Deartment of Electrical and Comuter Engineering Rutgers, the State University of Ne Jersey Piscataay, Ne Jersey {shunq.sun, Abstract Beamforming methods rely on training data to estimate the covariance matrix of the interference ulse noise. Their convergence slos don if the signal of interest is resent in the training data, thus requiring a large numbers of training snashots to maintain good erformance. In a distributed array, in hich the array nodes are connected to a fusion center via a ireless link, the estimation of the covariance matrix ould require the communication of large amounts of data, and thus ould consume significant oer. We roose an aroach that enables good beamforming erformance hile requiring substantially feer data to be transmitted to the fusion center. The main idea is based on the fact that hen the number of signal and interference sources is much smaller than the number of array sensors, the training data matrix is lo rank. Thus, based on matrix comletion theory, under certain conditions, the training data matrix can be recovered from a subset of its elements, i.e., based on sub-nyquist samles of the array sensors. Folloing the recovery of the training data matrix, and to coe ith the errors introduced during the matrix comletion rocess, e roose a robust otimization aroach, hich obtains the beamforming eight vector by otimizing the orst-case erformance. Numerical results sho that combination of matrix comletion and robust otimization is very successful in suressing interference and achieving a near-otimal beamforming erformance ith only artial training data. I. INTRODUCTION Adative beamforming has been idely used in ireless communications, radar and sonar for signal estimation. The adative beamforming method relies on the covariance matrix of the interference-lus noise, hich needs to be estimated based on training data, rior to alying the method for signal estimation. Hoever, in a ractical setting, such as assive source localization alications, the training data alays contain the signal of interest. In that case, the convergence rates of the adative beamforming algorithm are significantly reduced, and can be imroved only by considering very long training data []. If those data ere collected by distributed nodes and they need to be forarded to a fusion enter for the comutation of the covariance matrix, a lot of communication oer ould be required. When the number of signals and interference sources is much smaller than the number of sensors in the array, the training data matrix is lo rank. This means that, under certain conditions, even if some entries of the training data matrix are missing, the full matrix can be recovered via matrix comletion techniques [6][7]. Based on the the above observation, e roose a scheme that significantly reduces the number training This ork as suorted by the Office of Naval Research under Grant ONR-N data needed for estimating the the samle covariance matrix. The idea is that during the training hase, each sensor carries out a uniformly random sub-nyquist samling, and forards the samles to a fusion center. The full training data matrix can then be recovered using matrix comletion. The matrix comletion ste introduces errors hen noisy observation is considered. Assuming that the training data matrix satisfies the restricted isometry roerty [2][3], the relative matrix recovery error is bounded by a number of the order of the observed inverse signal-and-interference-to-noise ratio. Based on that bound, e roose a robust adative beamforming method ith robust otimization [4], hich obtains the beamforming eight vector by otimizing the orst-case erformance. Numerical results sho that combination of matrix comletion and adative beamforming is very successful in suressing interference and achieving a nearotimal SINR outut ith only artial training data. Relation to the literature- This aer follos u on the ork of [8], in hich the authors alied matrix comletion to estimate the samle covariance matrix in a distributed ay. Hoever, the estimation of the samle covariance matrix based on a subset of its entries does not reduce the number of samles that each sensor needs to collect and share. Further, as it ill be seen, the robust otimization aroach roosed here is suerior to the diagonal loading scheme used in [8] to combat covariance matrix estimation errors. II. FUNDAMENTALS OF CLASSICAL BEAMFORMING In array signal rocessing, it is often desired to estimate the signal s (k) under the resence of interference and noise ith the hel of an array of M sensors. The array observations x (k) C M can be ritten as [3] x (k) =s (k) a + i (k)+n (k), () here i (k), n (k) are the interference and noise, resectively. Here, k is the time index and a is the signal steering vector. The outut of a narro-band beamformer is y (k) = H x (k), (2) here C M is the comlex vector containing the beamforming eights. The signal-to-interference-lus-noise ratio (SINR) is ritten as [3] SINR = σ2 s H a 2 H R i+n, (3) here { R i+n = E [i (k)+n(k)] [i (k)+n(k)] H} (4) /3/$3. 23 IEEE

2 is the interference-lus-noise covariance matrix and σ 2 s is the signal oer. In the minimum variance distortionless resonse (MVDR) beamformer, the eight vector is obtained by minimizing the outut interference-lus-noise oer hile keeing the signal from a desirable direction distortionless [3]. This leads to the solution and otimal SINR ot = R i+n a a H R (5) i+na. SINR ot = σsa 2 H R i+na. (6) In ractice, the exact interference-lus-noise covariance matrix R i+n is not available. Instead, the samle covariance matrix is used, i.e., ˆR = L XXH (7) here X =[x (),..., x (L)] C M L and L is the number of snashots. Thus, the otimization roblem of MVDR beamformer and the corresonding solution can be is reritten by relacing R ith ˆR hich is knon as Caon method [2], and the solution, referred to as the samle matrix inversion (SMI) beamformer, is SMI = ˆR a a H ˆR a. (8) When there is no signal in the training samles, as L increases the SINR under the eight vector (8) converges very fast to the otimal value defined in (6). If L 2M, the average erformance losses are less than 3dB. Hoever, hen the signals are resent in the training samles, the convergence rate to (6) is much sloer. Usually, in the later case, L M is required []. III. COVARIANCE MATRIX ESTIMATION WITH MATRIX COMPLETION Let us assume the the number of targets and interference sources is K, and the number of sensors in the array is much larger than K, i.e., M K. To achieve high outut SINR, the Caon beamformer requires accurate estimation of the samle covariance matrix, hich in turn requires a number of snashots much larger than the number of sensors, i.e., L M. Thus, for a large array, a large number of training data need to be collected. as The training data matrix in the L snashots can be reritten X = S + Z, (9) here S = [s () a + i (),..., s (L) a + i (L)] is rank K and contains both signal and interference comonents. Here, Z =[n (L),..., n (L)] is the noise matrix. Under the above assumtion, matrix S is lo rank. When the noise level is lo, matrix X and the corresonding covariance matrix, ˆR, are aroximately lo rank. Matrix comletion techniques can thus be alied to recover the training data matrix S based on artial observations by exloiting its lo rank structure. A. Matrix Comletion ith Noise Consider the matrix M R n n2 of rank M is r. The corresonding degree of freedom is (n + n 2 r)r. Suose that the entries are corruted ith noise, i.e., [Y] ij =[M] ij + [E] ij, (i, j) Ω, here, [E] ij reresents noise and Ω is the set of observed entries. The observation can be modeled as P Ω (Y) =P Ω (M) +P Ω (E), here the samling oerator P Ω : R n n2 R n n2 is defined as { Mij, (i, j) Ω [P Ω (M)] ij = (), otherise. The matrix recovery can be done by solving the folloing otimization roblem [7] min B s.t. P Ω (B Y) F δ, () here denotes the nuclear norm and the constant δ satisfies δ 2 ( m + 8m ) σ 2, here m is the number of samles. For matrix comletion to be alicable, the singular vectors of matrix M need to satisfy the incoherence roerties. Let the SVD of M be r k= ρ ku k vk H, here ρ k are the singular values. The column and ro saces of M are denoted by U and V, resectively. Define the orthogonal rojection on U as P U = k r u ku H k. Then, the coherence of U is defined as μ (U) = n r max P U e i 2, here e i is the standard basis. i n To assumtions of matrix M are given in [6] as belo A) The coherence obeys max(μ (U),μ(V )) μ for some ositive μ. A) The n n 2 matrix k r u kvk H has a maximum entry bounded by μ r/(n n 2 ) in absolute value for some ositive μ. It is shon in [6] that if u k l μ B /n, v k l μ B /n 2 (2) for some μ B, i.e., the singular vectors are not too siky, then the above incoherence roerties hold. The matrix recovery error is bounded as M ˆM 4 F (2 + )min(n,n 2 )δ +2δ, (3) here = m n n 2 is the fraction of observed entries [7]. B. Training Data Matrix Estimation via Matrix Comletion In [8], the authors alied matrix comletion techniques to estimate the samle covariance matrix ˆR in a distributed ay. Hoever, the estimation of samle covariance matrix directly based on artial observations of covariance matrix entries does not reduce the number of samles at each sensor and the amount of local message assing is also large. This can be seen as follos. In order to recover a matrix, at least one observation er ro and one observation er column are required [6]. Therefore, to recover ˆR directly, each ro and each column of ˆR need to have at least one non-zero element; corresonding to element (i, j), nodes i and j in the array need to do Nyquist samling to obtain L samles and then share

3 those samles. Since ˆR needs to be uniformly oulated, may nodes need to do Nyquist samling and then share data. Instead of estimating ˆR directly, in this aer, e recover the training data matrix S ith only artial samles from each sensor. The recovered Ŝ is then used to construct the samle covariance matrix. It is difficult to sho analytically that the singular vectors of the training matrix meet the conditions A, and A. Hoever, for the case of a linear uniform array and uncorrelated signal and interference, extensive simulation results indicate that the maximal element values in both left and right singular vectors of matrix S is bounded by a small number ith high robability as M and L are large, i.e., the condition defined in (2) holds ith μ B O () (see also [9]). Therefore, matrix S satisfies the incoherence roerties. To recover the data matrix S in (9) ith matrix comletion, during the L snashots, each sensor carries out uniformly random samling and then forardd the sub-nyquist samles to a fusion center. At the fusion center, the observation can be ritten in the vector form b = A (S)+z, (4) here A is a linear transformation maing M L matrices into R m, i.e., A (S) =[S] ij, (i, j) Ω. The adjoint of A is denoted as A and P Ω (X) =A (b). Here, z is a noise vector and its distribution has been described in (9). Considering the corruted observation case, the matrix recovery is done by solving the folloing nuclear norm otimization roblem ith quadratic constraint min W s.t. b A(W) δ. (5) The objective in roblem (5) can be relaced by the aroximate function αw + 2 W F, here 2 W F is a smooth art and the arameter α controls the trade off beteen the accuracy of aroximation and the erformance of the algorithm []. Then, the roblem of (5) is casted as a conic rogramming roblem [] min αw + 2 W F s.t. b A(W) δ. (6) The roblem of (6) can be solved ith the singular value thresholding (SVT) algorithm in an iterative fashion (see []); the iteration converges to that of the original matrix comletion roblem as α []. The recovered training data matrix Ŝ is the otimal solution W ot of roblem (6). Then, the samle covariance matrix is obtained as ˆR mc = LŜŜ H. (7) C. Training Data Matrix Estimation Error Analysis Define the signal-lus-interference-to-noise ratio over the observed data matrix as η = P Ω (S) F /P Ω (Z) F. Define the relative recovery error of the samle data matrix Ŝ as φŝ = Ŝ S F /S F. Then, e have the folloing lemmas. Lemma 3.: The relative recovery error bound of the training data matrix is on the order of bound of η, i.e., bound ( ) ( ) φŝ =(± ε) bound, here ε is a small number. η The roof is given in Aendix A. Lemma 3.2: The error bound of the recovered data matrix Ŝ to the original noisy training data matrix X is on the order of noise level, i.e., Ŝ X F (C +) MLσ ith a numerical constant C, here σ is the standard deviation of the hite noise sequence. The roof is given in Aendix B. IV. ROBUST LOW RANK BEAMFORMING The recovered training data matrix Ŝ contains errors hich result in degradation of the erformance of the traditional beamforming methods. Therefore, robust beamforming methods are required. In this aer, ith the robust otimization techniques [4], e develo a beamformer that is robust against training data matrix mismatch, and it is based on orstcase erformance otimization, along the lines of [4]. The training data matrix X can be modeled as X = Ŝ + Δ. (8) Based on Lemma 3.2, the error matrix Δ is bounded, i.e., Δ F (C +) MLσ β. In the Caon beamformer, the objective is to minimize H ˆR = L H XX H = ( X H ) H ( X H ) L = X H 2. (9) L Therefore, minimizing H ˆR is equivalent to minimizing X H. We ant to obtain the eight vector by solving the folloing otimization roblem min max Δ F β X H s.t. H a. (2) As in [4], e ant to minimize the orst-case outut oer corresonding to the orst-case error matrix Δ subject to the distortionless resonse for the desired signal. The roblem (2) can be equivalently ritten as min max ŜH + Δ H s.t. Δ F β H a. (2) Let f () = max ΔF β ŜH + Δ H, then, it as shon in [4] that f () = ŜH + β. Thus, e can rerite the roblem of (2) as min ŜH + β s.t. H a. (22) The absolute oeration in the constraint of roblem (22) makes it nonconvex. Fortunately, hen the the eight vector undergoes any hase rotation, the cost function in (22) is

4 unchanged. Therefore, the eight vector can be chosen to satisfy Re { H a }, Im { H a } =. (23) Thus, the roblem of (22) can be equivalently ritten as min ŜH + β s.t. H a. (24) If the constraint of roblem (24) is true, then the conditions in (23) ould be satisfied. Also, the inequality constraint in (24) can be relaced by the equality H a =. Alternative robust beamforming methods are the diagonal loading (DL) [5] and the eigensace-based beamformer []. The idea of DL is to relace the samle covariance matrix ˆR mc ith ˆR DL = ˆR mc + ξi. The main difficulty in the DL method is in choosing the arameter ξ; ifξ is too large, the beamformer fails to suress the interference since most of the effort is used for hite noise suression. In ractice, ξ is usually chosen as σ 2 [3]. In the eigensace-based beamformer, the basic idea is to use a rojection of the steering vector a onto the samle signal-lus-interference subsace. The eigensace beamformer is knon to be oerful but its erformance degrades a lot at lo SNR [3]. V. NUMERICAL RESULTS In the simulations, e use a ULA ith M =4sensors. The intersace distance beteen each sensor is set to λ/2. Assume there is one signal source and its direction of arrival (DOA) is 2 and suosed to be knon. To interference sources are in the lane sace ith DOA as 5 and, resectively. Thus, K =3. The frequencies of both signal and interference sources are set to f = 9 Hz. The observation noise sequence is assumed to be hite Guassian ith zero mean and standard deviation as σ. The samle data matrix recovery is done using the SVT algorithm []. The arameter α is set to α =2 ML. Each sensor carries out a uniformly random samling and forards the sub-nyquist samles to the fusion center. The DL and and eigensace beamformers ere also imlemented. In DL, the diagonal loading factor is chosen as ξ =σ 2. Define the interference-to-noise ratio (INR) as INR = S int F /Z F, here S int is the samled interference data matrix. Fig. shos the relative recovery error φŝ of the training data matrix versus the number of samles er degree of freedom. The simulation arameters are set as SNR = db, INR = 3dB and L = 2. The degree of freedom of training data matrix S is df = (M + L K) K = 6. In total, iterations are run and the relative errors are averaged. It can be seen that hen m/df increases from 2 to 4, the relative error φŝ dros sharly to the recirocal of the observed signal-lus-interference-to-noise ratio level, i.e., the hase transition haens. In our folloing simulations, e set samling ratio as = m ML =.5, i.e., m/df 6. Fig. 2 comares the the recirocal of the observed signallus-interference-to-noise ratio η and the relative error φŝ in Ŝ as ell as the relative error in ˆR mc. Set SNR = db and L = 2. iterations are run and relative errors are averaged. It can be seen that in the entire INR range, the φŝ Values Recirocal of η Relative error of the training data matrix m/df Fig. : The relative recovery error. The x-axis is the number of samles er degree of freedom (df). Value Recirocal of η Relative error of recovered data matrix Relative error of estimated covariance matrix INR (db) Fig. 2: The relative errors of Ŝ and ˆR mc versus INR. are much smaller than the recirocal of η, hich confirms the conclusion in Lemma 3.. In addition, the relative error in ˆR mc is even smaller. Fig. 3 comares the erformances of several beamformers in terms of the number of snashot L, i.e., the robust otimization, DL as ell as eigensace-based beamformers, alied on the covariance matrix estimate obtained via matrix comletion (MC). The SNR and INR are set as db and 3dB, resectively. iterations have been run to calculate the average outut SINR. The otimal SINR (6) is also lotted for reference. It can be seen from Fig. 3 that as L increases, the SINR outut under both robust otimization and eigensacebased lo rank beamformers aroaches the otimal SINR ithin about 3dB. The results also confirms that to achieve a high SINR outut, large snashots are required. Therefore, our matrix comletion based lo rank beamforming method is imortant. The SINR erformance of these beamformers versus the SNR is shon in Fig. 4 for INR = 3dB and L = 2.

5 28 26 MC: DL MC: Robust otimization Outut SINR (db) Otimal SINR 4 Full data: Robust otimization MC: Eigensace 2 MC: DL MC: Robust otimization Number of snashots Fig. 3: The outut SINR versus the number of snashots. Beamattern (db) DOA angle (degree) Fig. 5: Beamatterns comarison. Outut SINR (db) Otimal SINR Full data: Robust otimization MC: Eigensace MC: DL MC: Robust otimization SNR (db) Fig. 4: The outut SINR versus SNR. iterations ere run and the relative errors ere averaged. It should be noted that at lo SNR, the signal of interest is buried in the background noise and the training data matrix X is aroximately of rank 2, hich is the number of interference sources. Via matrix comletion, the recovered training data matrix Ŝ is of rank 2, and contains the interference signal information, based on hich, the suression of interference can be achieved ith the beamformers. It can be seen in Fig. 4 that among all matrix comletion based robust beamformers, the robust otimization lo rank adative beamformer has the best SINR outut in the entire SNR range, and its SINR is identical to that of the robust otimization beamformer ith full data. The DL beamformer loses some erformance in the high SNR region, hile the eigensace-based beamformer does not ork ell at lo SNR. Last, the beamattern comarison is dislayed in Fig. 5. The arameters are set as L = 2, SNR = db and INR = 3dB. Comared ith the DL beamformer, the roosed robust otimization lo rank adative beamformer gives the best suression to the interference sources at DOA 5 and hile keeing the signal source at DOA 2 distortionless. VI. CONCLUSION In this aer, beamforming roblem in the large size sensor array ith signal of interest resent has been studied, in hich huge training data needs to be collected to generate a comarable SINR outut. When the training data matrix is lo rank, each sensor only needs to carries out a uniformly random samling and forards the sub-nyquist samles to the fusion center. The full training data matrix is reconstructed by matrix comletion. In the noisy observation case, the recovery is not exact. To deal ith the errors in the reconstructed training data matrix introduced by the matrix comletion, robust beamforming method as then roosed based on the orst-case erformance otimization. Numerical results sho that the rosed beamformer achieves near-otimal erformance ith half observations. APPENDIX A PROOF OF THE LEMMA 3. Proof: The roof of Lemma 3. is based on the assumtion that matrix S satisfies the restricted isometry roerty (RIP) [2]. The RIP asserts that the samling oerator obeys ( ε) S 2 F P Ω (S) 2 F ( + ε) S2 F, (25) rovided that S is lo rank. Here, ε is a smaller constant. Since the training data matrix S satisfies the incoherence roerties defined in Section III-A, then ith high robability, on the observation set Ω hose elements are uniformly random samled entries, the restricted isometry roerty holds []. The RIP roerty means that the energy of S on the observation set Ω is about roortional to the size of Ω. Based on RIP, the results in [3] ould yield that the recovered matrix Ŝ by solving the convex otimization roblem (5) satisfies Ŝ S F C δ, (26)

6 here C is a numerical constant [7]. This means that the ( right δ hand of (26) is on the order of, i.e., Ŝ S F δ O ). Thus, e have ( ) Ŝ S F O δ φŝ =. (27) S F S F From the RIP (25), e have P Ω (S) F S F, here means there is a smaller number ε in (25). Since P Ω (Z) F δ, ehave η = P Ω (Z) F P Ω (S) F δ SF. (28) By comaring the right hands of inequalities (27) and (28), the conclusion in Lemma 3. is roved. [] J. F. Cai, E. J. Candès and Z. Shen, A singular value thresholding algorithm for matrix comletion, SIAM Journal on Otimization, vol. 2, no. 2, , 2. [] S. Becker, E. J. Candès, and M. Grant, Temlates for convex cone roblems ith alications to sarse signal recovery, Math. Prog. Com. vol. 3, no. 3, , 2. [2] E. J. Candès and T. Tao, Decoding by linear rogramming, IEEE Trans. on Information Theory, vol. 5, no. 2, , 25. [3] M. Fazel, E. Candès, B. Recht, and P. Parrilo, Comressed sensing and robust recovery of lo rank matrices, in Proc. 42nd Asilomar Conf. Signals, Syst. Comut., Pacific Grove, CA, Oct. 28. [4] A. Ben-Tal, L. E. Ghaoui, and A. Nemirovski, Robust Otimization. Princeton University Press, 29. Proof: We have Ŝ X F = APPENDIX B PROOF OF THE LEMMA 3.2 Ŝ S + S X F (29) Ŝ S F + S X F (3) C δ + MLσ (3) m + 8m C σ + MLσ (32) (C +) MLσ. (33) Here, in (3) the triangle inequality of matrix norm is alied. In (3), the bound (26) is alied. REFERENCES [] D. D. Feldman and L. J. Griffiths, A rojection aroach to robust adative beamforming, IEEE Trans. Signal Processing, vol. 42, , 994. [2] J. Caon, High-resolution frequency-avenumber sectrum analysis, Proceedings of the IEEE, vol. 57, no. 8, , 969. [3] S. A. Vorobyov, A. B. Gershman and Z. Q. Luo, Robust adative beamforming using orst-case erformance otimization: A solution to the signal mismatch roblem, IEEE Trans. on Signal Processing, vol. 5, no.2, , 23. [4] S. A. Vorobyov, A. B. Gershman, Z. Q. Luo and N. Ma, Adative beamforming ith joint robustness against mismatched signal steering vector and interference nonstationarity, IEEE Signal Processing Letters, vol., no. 2,. 8-, 24. [5] J. Li, P. Stocia and Z. Wang, On robust Caon beamforming and diagonal loading, IEEE Trans. on Signal Processing, vol. 5, no. 7, , 23. [6] E. J. Candès and B. Recht, Exact matrix comletion via convex otimization, Foundations of Comutational Mathematics, vol. 9, no. 6, , 29. [7] E. J. Candès and Y. Plan, Matrix comletion ith noise, Proceedings of the IEEE, vol. 98, no. 6, , 2. [8] A. Waters and V. Cevher, Distributed bearing estimation via matrix comletion, in Proc. of IEEE International Conference on Acoustic, Seech and Signal Processing (ICASSP), Dallas, TX, Mar. 2. [9] S. Sun, A. P. Petroulu and W. U. Baja, Target estimation in colocated MIMO radar via matrix comletion, submitted in Proc. of IEEE International Conference on Acoustic, Seech and Signal Processing (ICASSP), Vancouver, Canada, May 23.