An Inner SOCP Approximate Algorithm for Robust Adaptive Beamforming for General-Rank Signal Model

1 An Inner SOCP Appr oximate Algorithm for R ob us t Adapti ve Beamforming f or General-Rank Sign al Model Y ongwe i Huan g, Seni or Membe r , IEEE , and Sergiy A. V orobyov F ellow , IEEE Abstract —The worst-ca se r obust adaptiv e beamform ing prob- lem f or general-ra nk signal model is considered. Its fo rmulation is to maximize t he w orst-case si gnal-to- interference-plus-noise ratio (SINR), incorpor ating a positiv e semidefinite constraint on the actual cov ariance matrix of the d esir ed signal. In the literature, semidefinite progra m (SDP) techniques, together with others, hav e been applied to approx imately solve this problem. Herein an inner second-order cone progra m (SOCP) approximate algorithm is p r oposed to so lve it. In particular , a sequence of SOCPs are constructed and solv ed, whil e t he SOCPs ha ve the non decr easing optimal values and conv erge to a locally optimal v alue (it is in f act a globally optimal value through our extensiv e simulations). As a result, our algorithm does not use computationally heav y SDP relax ation t echnique. T o validate our inner approxim ation results, simulation examples are presented, and they demonstrate the impro ve d perf ormance of th e new r obust beamf ormer in ter ms of t he av eraged cpu- time (ind icating how fast the algorithms con verge ) in a high signal-to-noise region. Index T er ms —Robust adaptiv e beamform ing, general-rank signal model, second-order cone progra m, in ner approximatio n. I . I N T RO D U C T I O N Robust ad apti ve be amforming i s a powerful approac h to significantly impr ove the a rray ou tput sign al-to-interfe rence- plus-noise ra tio (SINR) and other p erformanc e metrics such a s mainlob e width , side lobe le vels, etc. [1]–[1 5]. Here the ro b ust- ness typic ally me ans the ability of a method to perf orm well under any im perfect kn o wledge about the sour ce, propaga tion media, a nd sensor array [ 8], [15] –[18]. Th ere exists a numb er of ro b ust ada pti ve beamf orming a pproache s p roposed for the scenario of a rank- one signal mod el (see [ 18] and r eference therein). Howev er, it is also of pr actical i nterest to consi der a genera l-rank signal mod el, as the signal sour ce often can be incohe rently sc attered, so that the design of r obust ad apti ve beamfor ming for gener al-rank sign al models beco mes a mu st. In [ 19], the a uthors have p roposed a n efficient robust adaptive beamf orming metho d f or g eneral-rank so urce m odels, and a closed -form b eamform er has b een de ri ved and go od robustness capa bility has b een dem onstrated, ignor ing h o we ver the positiv e semidefinite (PSD) c onstraint on the worst-case actual covariance for t he d esired signal. T o fix the drawback, the author s of [20] have presen ted a n e w me thod t o th e ro b ust adaptive beam forming, inco rporating the PSD con straint. The resultant r obust beamform ing prob lem has be en formul ated b y introdu cing a m atrix decompo sition (e .g. spectral or Chol esk y Y . Huang is with School of Inf ormation Engineering, Guangdong University of T echnology , Uni ve rsity T own, Guangzhou, Guangdong, 510006, China. Email: ywhuang@gdut.edu.cn. S. A . V orobyov is with Department of Signal Processing and A coustics, School of E lectric al Engineering, Aalto Univ ersity , Konemiehentie 2, 02150 Espoo, Finland. Email: svor@ieee.org. type) of the p resumed signal covariance an d pu tting t he error term in to b oth of the matric es o btained from t he decom posi- tion. I t turns out that the co rrespondi ng robust beamf orming prob lem is a nonco n vex quad ratic program. Howe ver , the author s have proposed an algorith m for solving a semid efi- nite prog ramming (SDP) p roblem in each i terativ e step, that finally e nables them to ob tain an appro ximate sol ution o f the nonco n vex quadr atic pro gram. I n [21], two beam formers h a ve been built in closed-form fo r the rob ust adaptive beamformin g prob lem studied in [20 ], with the objective to lower the complexity of robust beamforme rs the rein. The authors of [22] have pr oposed a method fo r s olving the r obust beamfor ming prob lem for mulated in [20] u sing t he SDP relaxation technique and bisectio n search . In each ste p of the bisection sear ch, an SDP feasibility pro blem must be solved. In [23] and [24 ], the afo rementioned beamf orming p roblem has b een treated a s a d if ference- of-conv ex (DC) fun ctions optimiza tion prob lem, and a po lynomial time DC (POTDC) approxi mate algo rithm has been pr oposed, wher e a n SDP p roblem had to be solved in every iter ati ve step. Further, th e auth ors have shown t hat under the conditio n th at the erro r norm bou nd for th e a ctual covariance of the desired sig nal is sufficiently small, the locally optimal solu tion outp ut b y the alg orithm i s ind eed globally optimal. In this p aper , we study the robust adaptive beamf orming prob lem, b ut av oid to use hig h complexity SDP pro blem solving-b ase iterations. Instead , we app ly a seco nd-order co ne prog ram (SOCP) to approxi mate the r obust beamf orming pro b- lem. By doing so, only a SOCP is solved at ea ch iterati ve step, which h as less computatio nal c ost. In p articular , a seq uence of SOCPs is con structed and s olved, while th e optimal values of the c orrespondin g SOCPs are gua ranteed to be n ondecreasing and to converge to a lo cally optimal value. M oreover , in our exten si ve simu lations, we find out that our appro ximate algorithm c on verges to a globally optimal s olution . Sinc e every SOCP is a restrict ion of the r ob ust beam forming p roblem, the approx imate algor ithm is said to be a n inne r a pproximati on. T o the best of o ur kn o wledge, this is the first time to pr opose a SOCP appro ximate a lgorithm for the general-r ank ro b ust beamfor ming pro blem unde r cons ideration ( without ut ilizing the SDP relaxatio n tec hnique). I I . S I G N A L M O D E L A N D P RO B L E M F O R M U L A T I O N The a rray ou tput at time ins tance is exp ressed a s where is the complex weight (bea mforming ) vector , is the complex snap shot vector of a rray o bserv a- 2 tions, is the number of sensors in the array , and stands for Hermitian transp ose. T he observation vector is given by (1) where , , an d are the statistically inde pendent compo nents of the d esired s ignal, interf erence, and noise, respectively . The outpu t SINR of the beamf ormer is written as SINR (2) where is the de sired signal covariance matrix and is the interfere nce-plus-no ise covariance matrix. Matrix herein can b e of rank on e or hig her , i.e., Rank . The r ank-one correspo nds to the c ase of the poin t sou rce (see [16], [18]–[ 20]), and we herein are interested in the genera l-rank c ase. In ap plications, the matr ix is typ ically unav ailable. Thus, the samp le covariance o f the da te covariance ma trix is used, and it is expressed as (3) where stands for the number o f tr aining snapsho ts. On the o ther h and, the a ctual covariance m atrix is only known inc ompletely and i mperfectly . Th e b eamvector obtain ed by max imizing th e SINR (2) with and replaced respectively by (a presumed covariance of ) and can howe ver lead to p oor perfo rmance of the sensor array . There- fore, in o rder to imp rov e the beamfo rmer perfo rmance, r obust adaptive beamf orming f or general-ran k signal mo del ha s b een considere d. Se veral works have add ressed the prob lem (see [19]–[ 24] and references there in) in the last tw o decades. Among ro b ust adap ti ve b eamformi ng proble m f ormulations, the following prob lem max imizing the worst-case SINR is the most pop ular (4) where t he unce rtainty sets a nd are given by (5) and (6) respectively . Th rougho ut the paper, the matrix norm is assumed to be the Fr obenius norm, whi le a vector no rm is Euclidian norm. Since and are separ able, ( 4) can be reca st int o (7) Observe that for , i t follows that tr and the equality hold s wh en . Theref ore, the solution to t he prob lem in the denom inator o f the ob jectiv e f unction of (7) can b e foun d in closed -form as (8) Thus, (7) can be re expressed a s (9) which i s equ i valent to the fol lo wing optimi zation prob lem (10) in th e sen se that (9) a nd (10 ) sh are the same op timal value and if solves ( 10), the n it i s opt imal for ( 9) too. In or der to incor porate t he PSD co nstraint over into the objective fun ction in (10), th e following robust adaptive be amforming prob lem is con sidered (cf. [20] –[24]) (11) where , , Rank , a nd the nor m of the d istortion is boun ded by a given con stant : (12) Herein we focus on h o w to solve the beamf orming probl em (11), and p resent a new algorithm which we coin as th e inner SOCP app roximate al gorithm. It i s known that [24 ], [ 25] Considering that the objective function o f (11 ) is positive, we rewrite (11) as the following equivalent pr oblem (13) For com parison, in [20], the authors have stud ied the following prob lem: (14) Notice tha t both the inequality con straints in (1 3) and (14 ) can be replace d with equality constraints, without loss o f anything. W e show next that the pr oblems (1 3) and (14 ) ar e eq ui valent to eac h ot her . Proposition II.1 Pr oblems (13 ) and (14 ) a r e eq uivalent to each oth er in the sense that if a nd ar e r espectively an opti mal solution and the optima l value of (14) , then and ar e co rr espond ently an optimal so lution and the optimal va lue of (13) . Convers ely , if and ar e respectively an op timal solution and the optima l v alue of (13) , th en and ar e co rr espond ingly an opt imal solution an d the o ptimal va lue of ( 14) . 3 Pr oof : Sup pose th at and are respectively an optimal so lution an d the optima l value of (14). I t follows that and Suppose that is no t the o ptimal solution of (13), and there is a n opti mal solution suc h that and which i mplies respectively that (15) and This means that is optimal for (14), but with respe ct to (15), it is imp ossible sin ce (15) shall h old with eq uality . Then a con tradiction occ urs, and thereby we conclu de that is o ptimal fo r (13) with th e opt imal value . The p roof for the c onv erse im plication is similar to the above proo f, and thus it is om itted. Giv en th e Propo sition, we can solve (14) in orde r to get an optimal solution of (11 ) . I I I . A N I N N E R S O C P A P P R OX I M AT E P RO C E D U R E T O S O LV E (14) W e desig n h ere an i nner SOCP ap proximate algorit hm fo r nonco n vex prob lem (14 ). T o begin, p roblem (14) is refor mulated e qui valently i nto (16) Indeed , it can b e verified that the optim al solutions of one prob lem are always feasible for th e o ther problem . Note tha t the first con straint in (16 ) is a second -order cone (SOC) constrain t, but the seco nd on e i s a non con vex con straint. Neve rtheless, we can build a c on vex appr oximation for the nonco n vex constra int, and design based on it a n algo rithm to solve (16). T o wards this end , observe that where is an initial po int. There fore, implies that and we establish the following SOCP p roblem instea d o f SDP (17) which has a smaller fe asible set an d its op timal v alue is bigger than tha t of ( 16) . H ere stands for a real par t of a compl e x numbe r . In other words, pr oblem (17) i s a restrictio n of (16 ) . W e wish to desig n a seque nce of prob lems in the form of (17) suc h that the o ptimal value o f e ach con secuti ve pro blem is closer to a locally optimal v alue (but in our extensiv e simulations, it converges to the glob ally optima l value). Suppose is an opt imal so lution for (17) an d th e optimal value is . Then we construct the following SOCP prob lem (18) Clearly it is also a restrictio n of ( 16) since t he last constrain t implies that . Solvin g (1 8), we obtain an optimal solution and the o ptimal value . Then th e following prop osition ab out a relationship between optimal values and can be proven. Proposition III.1 I t h olds that . Pr oof : It follows from the secon d const raint of (17) t hat It can b e easily ch ecked that the o ptimal sol ution is feasible for (18). T herefore , we have . Similarly , we can construc t a sequence of SOCP proble ms, such t hat the optim al values compl y with Theref ore, we summ arize the inner SOCP a pproximate algo - rithm as shown in Alg orithm 1. Algorithm 1 I nner SOCP appr oximate a lgorithm fo r ( 14) , , , , ; A sol ution for pro blem ( 14); 1: Suppose that is an initial feasible po int; set and ; 2: 3: solve the fo llo wing SOCP: (19) obtaini ng solu tion and o ptimal value ; 4: ; 5: ; ; 6: until It is worth observing that e very SOCP in Algorith m 1 is a restriction of (16). W e remark t hat th e com putational burden includ es solvin g SOCP (19) in e ach iteratio n, and the worst-case compl e xity o f solv ing the SOCP i s in the or der of (see e.g . [ 26, page 309] ). W e also remar k th at in the approx imate alg orithm pr oposed in [24] , an SDP is solved in 4 eve ry itera tion, which has high er comp utational cost. In fact, therein (14) i s recast into tr tr tr rank (20) Employing the approxi mation and th e con ventional SDP relaxa tion technique, (20 ) is relaxed into th e fo llo wing SDP tr tr tr (21) Then the algori thm searches for the best , and in each itera - tion, sol ves th e corr esponding SDP (2 1) f or finding o ptimal . At the end, 1 for the best giv es a suboptimal solution of (2 0) , which i s p rov en to b e globally optim al un der some co nditions. I nterestingly , the op timal value of t he S DP relaxatio n of (20) (droppin g the rank-one con straint) can serve as a lower bo und of t he robust bea mforming p roblem (1 4) ( a benchm ark, as will be se en i n simu lation exa mples). Finally , we poi nt ou t that (13 ) can b e solved dire ctly , that is, it ca n be appr oximated by the following SOCP pro blem (22) provided that an initial point is a p refix. How ever we choose to solve (14) for the reaso n that it is ea sier to compare this o ur work with t he existing works. I V . S I M U L AT I O N R E S U LT S W e consi der a un iform linear array of 10 om ni-direction al antenn a elements with the inter-elemen t spacing of half wa ve- length (i.e. ). T he p o wer of additive n oise in every antenn a is as sumed to be 0 dB. There is an in terferer w ith the inter ference-to -noise r atio (INR) of 20 d B. Sup pose t hat the des ired signal and the interfer er are locally in coherently scattered with Gaussian and unifor m angu lar power densi- ties with central angles of 30 and 10 , re specti vely . The angula r spreads o f the d esired sign al and the in terferer are assumed to be 4 and 10 , re specti vely . The angu lar power density of the p resumed signa l is Ga ussian with central angle 34 and angular spread 6 . The diagona l lo ading par ameter and th e error norm b ound tr are used, wher e tr denotes t he matr ix trace. T he iteration terminati on thre shold is e qual to . T o find a lower -bou nd on the optimal value o f SDP relax ation of (20), the int erval is d i vided into 100 subsectors. A ll resul ts are a veraged over 100 sim ulation run s. 1 It is easily verified that can be always of rank one, see [27]. More generally , for any gi ven , the SD P (21) admits a r ank-one solution. Fig. 1 shows the req uired CPU-time by o ur A lgorithm 1 herein (“New be amformer ”) and th e algo rithm pr oposed in [24]) (“K-V be amformer” ). It c an b e seen that the averaged CPU-time o f our beam former is less than that by [24 ], espe- cially a t the high S NR region (o ur co mputer has a p rocessor of In ter Core i7- 6650U and a 16 GB RAM). Fig. 2 de picts the appr oximate o ptimal values by our a lgorithm and th e o ne in [24], as well as the o ptimal value of (1 4), or eq ui valently (20) (term ed as “Lower- Bound ”). It c an be seen that the th ree curves coinc ide, mean ing that both our ap proximate algo rithm and the appr oach in [24] achieve the optimal value of the origina l pro blem ( 14). 30 35 40 45 50 55 60 SNR (dB) 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 Averaged cpu-time (S ec.) New beamformer K-V beamformer Fig. 1. A veraged cpu-time versus SNR, w ith INR=20 dB and 30 35 40 45 50 55 60 SNR (dB) 0 0.5 1 1.5 2 2.5 3 3.5 4 Objective value 10 6 New beamformer K-V beamformer Lower-Bound Fig. 2. Objecti ve function value of the problem (14) (or equiv alently (20) ) versus S NR, with INR=20 dB and V . C O N C L U S I O N W e h a ve co nsidered the robust ada pti ve bea mforming pr ob- lem for g eneral-ra nk signa l models inc orporatin g one P SD constrain t over the actual covariance m atrix of the d esired signal. Unlike solving the pr oblem by the SDP relaxation in every iterative step, we h a ve developed the inner SOCP approx imate algo rithm. In particula r , a sequence o f SOCPs have been co nstructed and solved, wh ere the optimal values of the SOCPs are non decreasing and conver ge t o an o ptimal value of the problem considerd. Sinc e on ly solving an SOCP is require d in iter ati ve step s, th e total compu tational com plexity is lower tha n t hat of existing a lgorithms inv o lving solv ing SDPs. Th e impr oved perf ormance of the prop osed robust beamfor mer has bee n dem onstrated by s imulations in terms of the av erage d CPU-time requir ed fo r the algor ithms, which indicate how fast t he algori thms ar e. 5 R E F E R E N C E S [1] A. B. Gershm an, “Robust adapti ve beamform ing in sensor arrays, ” International Journal of Electr onics and Comm unicat ions , vol. 53, pp. 305C314, Dec. 1999. [2] Y . I. Abramovich, “Controlled method for adaptiv e optimization of filters using the criterion of maximum SNR, ” Radio Engineering and Electr oni c Physics , vol. 26, pp. 87C 95, Mar. 1981. [3] B.D. Carlson, “Covar iance matrix estimation errors and diagonal loading in adaptiv e arrays, ” IEE E T ransactions on Aerospace and Electr oni c Systems , vol. 24 no. 7, pp. 397C401, July 1988. [4] B. Widro w , K. M. Duv all, R .P . Gooch, and W .C. Newman, “Signal cancellati on phenomena i n adapti ve antennas: causes and cures, ” IEE E T ransactions on A ntennas and Pr opagation , vol. 30, pp. 469C478, 1982. [5] H. Cox, R. M. Zeskind, and M. H. Owen, “Robust adapti ve beamform- ing, ” IEEE T ransacti ons on A coustics, Speech, and Signal Processing , vol. 35, pp. 1365C1376, Oct. 1987. [6] K. L. Bell, Y . Ephraim, and H. L. V an Trees, “ A Bayesian approach to robust adaptiv e beamforming, ” IEEE T ransactions on Signal Processing , vol. 48, pp. 386C398, Feb . 2000. [7] L. Chang and C. C. Y eh, “Performance of D MI and eigenspace- based beamform ers, ” IEE E Tr ansactions on A ntennas and Propa gation , vol. 40, pp. 1336C1347, Nov . 1992. [8] S.A . V orobyov , A.B. Gershman, and Z.- Q. L uo, “Robust adaptiv e beamforming us ing worst-case performance optimization: a solution to the s ignal mismatch problem, ” IEEE Tr ansactions on Signal Pr ocess ing , vol. 51, pp. 313C324, Feb . 2003. [9] R.G. Lorenz and S .P . Boyd, “Robust minimum varianc e beamforming, ” IEEE T ransactions on Signal Processing , vol. 53, pp. 1684C1696, May 2005. [10] A. B. Gershman, Z.- Q. Luo, and S. Shahbazpanahi, “ Robust adaptiv e beamforming based on worst-case performance optimization, ” in: P . Stoica, J. L i (Eds .), R obust A daptive Beamforming , Wile y, Hoboken, NJ, 2006, pp. 49C89. [11] J. L i, P . Stoica, and Z . W ang, “On robust Capon beamforming and diagonal loading, ” I EE E T ransactions on Signal Processing , vol. 51, pp. 1702C1715, July 2003. [12] S.A . V orobyov , A.B. G ershman, Z.-Q. Luo, N. Ma, “ A dapti ve beamform- ing with joint robustness against mismatched signal steering vect or and interference nons tati onarity , ” IE EE Signal P r ocess ing Letters , vol. 11, pp. 108C111, Feb . 2004. [13] Y . Gu, A. Leshem , “Robust adapti ve beam forming based on interference cov ariance m atrix reconstruction and steering vect or estimation, ” IEE E T ransactions on Signal Pr ocessing , vol . 60, pp. 3881C3885, July 2012. [14] S.A . V orobyo v , H . Chen, A.B. Gershman, “On the relationship between robust minimum v ariance beamformers with probabilistic and worst- case distrortionless response constraints, ” IEE E T ransactions on Signal Proc essing , vol. 56, pp. 5719C5724, Nov . 2008. [15] A. Khabbazibasmenj, S .A. V orobyov , A. Hassanien, “Robust adaptiv e beamforming based on steering vector estimation with as little as possible prior information, ” IEE E T ransactions on Signal P r ocessing , vol. 60, pp. 2974C2987, June 2012. [16] J. Li and P . Stoica, Robust Adaptive Beamform ing , John Wi ley Sons, Hoboken, NJ, 2006. [17] A. B. Gershm an, N . D. Sidiropoulos, S. S hahbazpana hi, M. Bengts- son, and B. Ottersten, “Con ve x optimization-based beamform ing: from recei ve to transmit and network designs, ” IE EE Signal P r ocess ing Magazine , vol. 27, no. 3, pp. 62-75, M ay 2010. [18] S. A. V orobyov , “Principles of m inimum variance r ob ust adaptiv e beamforming design, ” Signal Processing , vol.93, pp. 3264-3277, 2013. [19] S. Shahbazpanahi, A. B. Gershm an, Z.-Q. L uo, and K. M. W ong, “Robust adapti ve beamform ing for general-rank signal models, ” IE EE T ransactions on Signal Processing , vol. 51, no. 9, pp. 2257-2269, September 2003. [20] H. Chen and A. B. Gershman, “Robust adpati ve beamforming for general-rank signal m odels using positiv e semidefinite cov ariance con- straints, ” Proceedings of IEE E ICASSP , pp. 2341-2344, 2008. [21] C. X ing, S. Ma, and Y .-C. Wu, “On lo w complexi ty robust beamforming with positiv e semi-definite constraints, ” IEE E T ransactions on Signal Proc essing , vol. 57, no. 12, pp. 4942-4945, December 2009. [22] H. Chen and A. B. G ershman, “W orst-case based robust adaptiv e beamforming f or general-rank signal m odels using positiv e semi-definite cov ariance constraint, ” Proceedi ngs of IEE E ICASSP , pp. 2628- 2631, 2011. [23] A. K habbazibasmenj and S. V orobyov , “ A computationall y efficient robust adapti ve beamforming for general-r ank signal models with posi- ti ve semi-definitness constraint, ” Pr oceedings of 4th IEE E Internati onal W orkshop on Computational A dvances in Multi-Sensor Adaptive Pro- cessing (CAMSAP ’11) , pp. 185-188, December 2011. [24] A. Khabbazibasmenj and S.A. V orobyov , ”Robust adaptiv e beamfor ming for general-rank signal model with positiv e sem i-definite constraint via PO TDC, ” IEEE T ransactions on Signal Processing , vol. 61, no. 23, pp. 6103-6117, December 2013. [25] Y . Huang, Q . Li, W .-K. Ma, and S. Zhang, ”Robust m ultica st beamform- ing for spectrum sharing-based cognitiv e r adios, ” IEEE T ransaction on Signal Pr ocessing , vol. 60, no. 1, pp. 527-533, January 2012. [26] A. Ben-T al and A. Nemirovski, “L ectur es on Modern Con vex Optimiza- tion, ” Class Notes, Georgia Institute of T echnology , Fall 2005. O nline : http://www2.isye.gatech.edu/ nemirovs/Lect ModCon vOpt.pdf. [27] Y . H uang and S. Zhang, “Complex m atrix decomposition and quadratic programming, ” Mathematics of Operations Research , vol. 32, no. 3, pp. 758-768, 2007.