OPTIMUM DISCRETE PHASE-ONLY TRANSMIT BEAMFORMING WITH ANTENNA SELECTION

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1 OPTIMUM DISCRETE PHASE-ONLY TRANSMIT BEAMFORMING WITH ANTENNA SELECTION Özlem Tuğfe Demir, T. Engin Tuncer Elecrical and Elecronics Engineering Deparmen, METU, Ankara, Turkey {ugfe.demir, ABSTRACT Phase-only beamforming is used in radar and communicaion sysems due o is cerain advanages. Anenna selecion becomes an imporan problem as he number of anennas becomes larger han he number of ransmi-receive chains. In his paper, discree single group mulicas ransmi phase-only beamformer design wih anenna subse selecion is considered. The problem is convered ino linear form and solved efficienly by using mixed ineger linear programming o find he opimum subse of anennas and beamformer coefficiens. Several simulaions are done and i is shown ha he proposed approach is an effecive and efficien mehod of subarray ransmi beamformer design. Index Terms Transmi beamformer, discree beamformer, mixed ineger linear programming, anenna selecion. INTRODUCTION Large anenna arrays become cheaper and more feasible wih he advances on fabricaion echniques. In modern radar and communicaion sysems, here are more anennas han he ransmi-receive chains. As a resul, ransmi anenna selecion for he mos appropriae anenna subarray is an imporan problem [], [2]. Pracical radar [3] and communicaion sysems [4], [] hardware is composed of discree phase shifers for ransmireceive beamforming. Phase-only ransmission has cerain advanages [3], [6], [7], [8], [9]. Curren sae of he ar for discree-phase ransmi beamforming discreizes he known beamforming coefficiens or uses ieraive schemes for discree soluions [4], [], [6], [7]. These discree beamformers are far from opimum in erms of imum ransmi power o users. In single group mulicas beamforming, a common signal is ransmied o a pool of N users each equipped wih a single anenna while he ransmier is composed of an anenna array []. The problem is reaed in several previous works [], [] where subopimum soluions are found for coninuous ampliude and phase beamformers. In his paper, he design of opimum discree-phase ransmi beamformer wih anenna selecion is considered. Proposed mehod selecs he bes L ou of M anennas and finds he opimum phase-only ransmi beamformer coefficiens in a join manner using branch and cu echnique. In some previous papers, opimum discree phase-only [2] and opimum discree phase ampliude [3] ransmi beamformer designs are considered. This paper exends he work in [2] by including anenna selecion ino he problem. Max-min fair ransmi beamformer design wih anenna selecion problem is convered o linear form for effecive soluion using mixed ineger linear programming. While he wors case complexiy of his mehod is exponenial when all he possible branches are expanded, i is significanly lower han he brue force search in pracice due o he fac ha soluion is usually found before he full ree expansion. This poin is furher elaboraed inside he paper. To our knowledge, his paper is he firs work which presens he opimum soluion o he aforemenioned problem described above. 2. SYSTEM MODEL In single group mulicas beamforming, i is assumed ha a base saion equipped wih M ransmi anennas ransmis a common signal o N receivers, each having a single anenna. The ransmied signal can be wrien as, x() = s()w () where s() is he source signal and w is he M complex beamformer weigh vecor. The received signal a he k h receiver is given as, y k () = h H k x() + n k () k =,... N (2) where h k is he M complex channel vecor for he k h receiver and n k is addiive noise uncorrelaed wih he source signal and is variance is σk 2. Signal-o-noise raio (SNR) for he k h receiver is, SNR k = σ2 s w H h k 2 σ 2 k where σ 2 s is he source signal variance. σ 2 s = is seleced wihou loss of generaliy hroughou he paper. In phase-only single group mulicas beamforming min problem, beamforming vecor, w, is chosen o imize (3)

2 he minimum ransmied power o any user. Suppose ha only L ou of M anennas can be ransmiing simulaneously. In his case, he goal is o selec he bes L anennas and find he corresponding beamforming vecor. Considering as he per-anenna power, he problem can be wrien as follows, w C M s.. w H R k w σ 2 k, k =,..., N (4) (ww H ) i,i {, } i =,..., M w H w = L where is he power proporion for he k h receiver and is he -min parameer corresponding o he min{ SNR k }. R k = E{h k h H k } is he correlaion marix of he channel vecor. Solving he above problem requires a combinaorial search over all ( ) M L NP hard problems. In he following par, we consider he discree version of his problem and find he opimum soluion using mixed ineger linear programming wih moderae complexiy wih branch and cu sraegy. 3. DISCRETE PROBLEM In [2], an opimum soluion for he -min syle phaseonly single group mulicas problem wihou anenna selecion is obained when he phase angles of he beamformer are seleced from a discree se. In his paper, opimum soluion is found when he anenna selecion is added o he problem. The problem in (4) can be wrien for he discree phase as, ψ i,α i s.. w H R k w σ 2 k, k =,..., N α i {, } i =,..., M () α i = L ψ i {, θ, 2 θ,..., (2 n ) θ}, θ = 36 2 n where i h elemen of he beamformer vecor w is w i = α i Pan e jψi and ψ i is he discree phase wih n bis. α i is he anenna selecion coefficien. θ is he discree sep size for phase. Since R k is a Hermiian symmeric marix, he inequaliy in () can be expressed as, M p=i+ 2α i α p R k (i, p) cos( R k (i, p) + ψ p ψ i )+ α 2 i R k (i, i) σ 2 k, k =,..., N (6) The inequaliy in (6) can be simplified furher if addiional variables β i,p and µ i,p are defined as, β i,p = ψ i + ψ p µ i,p = α i α p, i =, 2,..., M, p = i +,..., M Using he rigonomeric ideniy, he opimizaion problem can be wrien as, M s.. p=i+ ψ i,α i,β i,p,µ i,p 2µ i,p R k (i, p) [cos( R k (i, p))cosβ i,p sin( R k (i, p))sinβ i,p ] + αi 2 R k (i, i) σ 2 k, k =,..., N (7) ψ i {, θ, 2 θ,...(2 n ) θ}, θ = 36 2 n α i {, }, β i,p = ψ i + ψ p (8) µ i,p = α i α p, (9) i =, 2,..., M, p = i +,..., M α i = L () The above problem seing is no linear. In he following secion, he above problem is convered ino linear form. 4. DISCRETE OPTIMIZATION IN LINEAR FORM Le he firs par of he lef hand side of he inequaliy in (7) be represened as A. A can be expressed in linear form by using some known vecors c and s. c and s are composed of all possible cosβ i,p and sinβ i,p erms and corresponds o a erm relaed o he anenna selecion which nullifies he corresponding elemen in he beamformer weigh vecor w, i.e., c = [, cos( θ), cos( θ),..., cos((2 n ) θ)] T s = [, sin( θ), sin( θ),..., sin((2 n ) θ)] T In order o access each erm in A, indicaor vecors u i,p whose elemens are all zero excep a single elemen are defined. u i,p s are he new variables of he opimizaion. A in (7) can be expressed in erms of u i,p as, A = M p=i+ 2 R k (i, p) [cos( R k (i, p))c T sin( R k (i, p))s T ] u i,p () Noe ha u i,p () and µ i,p are complemens of each oher as binary variables described as in Table. The relaionship beween u i,p vecors should be esablished and used during he opimizaion. Such relaionships can be esablished over binary indicaor vecors, v i. (2 n +) vecor, v i, carries he phase and ampliude informaion of he beamformer vecor. The firs elemen of u i,p and v i, namely u i,p () and v i () are he anenna selecion parameers. v i () is he complemen of α i as given in Table.

3 Once u i,p and v i indicaor vecors are defined, i is possible o wrie (8) in erms of hese vecors. Firs (8) is normalized by θ, hen β i,p θ = ψ i θ + ψ p (2) θ The above expression can be wrien as, d T u i,p = d T ( v i + v p ) (mod 2 n ) (3) where d = [,,, 2,..., (2 n ) ] T. Noe ha he firs zero in d is for he anenna selecion and he second one is for zero phase ( θ). d T u i,p is always nonnegaive and modulo operaion on he righ hand side ensures he equaliy. Modulo operaion can be removed if an addiional binary variable a i,p is used, i.e., d T u i,p = d T ( v i + v p ) + a i,p 2 n (4) Noe ha (4) is defined when i h and p h anennas are used simulaneously. For he oher cases of anenna exisences, (ex: i h anenna does no exiss), equaion (4) can be sill made valid if an addiional variable b i,p is added o he lef hand side of equaion (4). In his case, he righ hand side of (4) is always posiive and d T u i,p = requiring a posiive b i,p o saisfy he equaliy, i.e., d T u i,p + b i,p = d T ( v i + v p ) + a i,p 2 n () Noe ha () is more comprehensive form of linear expression in (8). 4.. Final Form of he Discree Problem The final variable of opimizaion,, corresponds o he min{ SNR k } and hence i is real and posiive, i.e., R +. Furhermore all he expressions in (6-22) are linear. Therefore he problem can be solved using mixed ineger linear programming [4]. The new opimizaion problem can be wrien as, M s.. p=i+ v i,u i,p,a i,p,b i,p sin( R k (i, p))s T ] u i,p + 2 R k (i, p) [cos( R k (i, p))c T ( v i ())R k (i, i) σ 2 k k =,..., N (6) d T u i,p + b i,p = d T ( v i + v p ) + a i,p 2 n (7) a i,p {, } (8) b i,p (9) b i,p + 2 n ( u i,p ()) 2 n (2) v i () + v p () 2u i,p () (2) Table. α i α p µ i,p v i () v p () u i,p () b i,p v i () = M L (22) i =, 2,..., M, p = i +,..., M 2 n + 2 n + v i (k), u i,p (k) {, }, v i (k) =, u i,p (k) =. k= k= The expression in (6) sands for (7) in he previous secion. The expressions (7), (8), (9) and (2) are for (8) where a i,p and b i,p are he addiional variables o make (7) valid for all cases including he case when any one of he anennas is no seleced. (9) and (2) guaranees ha b i,p is when i h and p h anennas are seleced in accordance wih Table. (2) is used for he anenna selecion as in (9) and i realizes he Table. Noe ha (2) is an inequaliy over binary variables o realize (9). (22) sands for () and i shows he number of seleced anennas. The problem in (6-22) is always feasible and he global opimum can be found using mixed ineger linear programming [], [6], [7] wih branch and cu echnique. Once he soluion for v i s are found, he phase angles and he anenna selecion coefficiens of he beamformer vecor are obained as, ψ i = f T ψ v i, α i = v i () i =,..., M (23) where f ψ = [,, θ,..., (2 n ) θ ] T = θd. The following example shows he srucure of u i,p and v i vecors as well as heir inerrelaions. The consrucion of (7) is also elaboraed in his example. Example: Le M = 3, n = 2 ( θ = 9 ) and he beamformer weigh vecor be w = [ e j8 e j9 ] T, i.e., he firs anenna is no used. In his case, α i s are given as α =, α 2 =, and α 3 = respecively. ψ 2 = 8 and ψ 3 = 9. Then v i vecors are; v = [ v 2 = [ ] T = α ] T = α 2 9 ψ 2=8 27 v 3 = [ ] T = α 3 ψ 3= Since he firs anenna is no used µ i,p s are given as µ,2 = µ,3 = and µ 2,3 =. β 2,3 is β 2,3 = ψ 2 + ψ 3 = 9 =

4 27. Then u i,p vecors are; u,2 = u,3 = [ u 2,3 = [ ] T = µ i,p ] T = µ 2,3 9 8 β 2,3=27 d is given as d = [,,, 2, 3 ] T. Then (7) for i = 2, p = 3 is, d T u 2,3 + b 2,3 = d T ( v 2 + v 3 ) + a 2,3 2 2 (24) In he above expression d T ( v 2 + v 3 ) is negaive and a 2,3 should be in order o have he equaliy wih he posiive righ hand side. Noe ha b 2,3 = in accordance wih Table. (7) for i =, p = 2 is given as, d T u,2 + b,2 = d T ( v + v 2 ) + a,2 2 2 (2) a,2 and b,2 are seleced appropriaely by he opimizaion program in order o saisfy he equaliy.. SIMULATIONS In his paper, Gurobi [4] which is an efficien mixed ineger linear programming solver is used by employing branch and cu sraegy. The evaluaion of he proposed mehod is done for a uniform linear array (ULA) wih M anennas where L elemens are chosen a each case. For simpliciy, = W is seleced. In he firs experimen, M = 8 and L = 4 anennas are seleced as in LTE sandard [8]. Line of sigh condiion is assumed and here are N = 4 users a azimuh angles, 9, 2 and respecively. The power proporions for each user are seleced as γ =, γ 2 = 4, γ 3 =, and γ 4 = 2 respecively. n = bis are used for he discree phase-only beamformer. Fig. shows he beampaerns for he proposed discree phase-only beamformer (DPOB) and he fixed ULA wih 4 elemens. Boh of he DPOB s in Fig. are opimum [2] bu DPOB wih anenna selecion chooses he bes 4- elemen subarray ou of M = 8. The ransmi power for each user is significanly improved as seen Fig. for he anenna selecion case. In he second experimen, he azimuh angles for N = 4 users are seleced randomly a each rial. In his case, L = 4 bu M is changed as 8, 2 and 6 respecively. Brue force resul is shown only for M = 8 due o compuaional complexiy. As i is seen from he Fig. 2, proposed mehod and brue force are exacly he same since he mehod is opimum. I is also seen ha L = 4 ou of M = 6 anenna case gives he bes resul for = min{ wh h k 2 } L2 γ = 6 4 = 4 and i is almos equal o he upper bound = 4. In he hird experimen, Rayleigh channel model is assumed and he channels are seleced randomly a each rial. Resuls of boh DPOB wih fixed array ( M = L = 4 ) and anenna selecion ( M = 8, L = 4 ) are shown for differen Table 2. Compuaional ime of DPOB and brue force search L = 4 M = 8 M = 2 M = 6 DPOB 7.96 s 2.8 s 46.9 s BRUTE FORCE 88 s 43 s 2 s number of bi values, n = 3, 4,. Fig. 3 shows he minimum SNR for differen channel realizaions. Minimum SNR obained wih anenna selecion is higher han ha of fixed array even wih small number of bis. Table 2 shows he compuaional complexiy of he brue force and he proposed mehod where he average of rials are repored. As i is seen from his able, he proposed opimum mehod has significanly lower complexiy hanks o he efficiency of he mixed ineger linear programming wih branch and cu echnique [4]. ARRAY BEAMPATTERN (db) 2 2 DPOB wih anenna selecion M=8, L=4 DPOB wih fixed array M=L= AZIMUTH ANGLE (DEGREES) Fig.. Beampaerns of DPOB wih anenna selecion and fixed array (The doed lines show he azimuh angles of he users). (MINIMUM SNR) DPOB wih anenna selecion M=8, L=4 DPOB wih anenna selecion M=2, L=4 DPOB wih anenna selecion M=6, L=4 DPOB wih fixed array M=L=4 Brue Force M=8, L= TRIAL FOR CHANNEL REALIZATION Fig. 2. Minimum SNR for channel realizaions for DPOB wih anenna selecion for differen number of anennas and fixed array in line of sigh condiion wih ULA.

5 (MINIMUM SNR) DPOB wih fixed array M=L=4, n=3 DPOB wih fixed array M=L=4, n=4 DPOB wih fixed array M=L=4, n= DPOB wih anenna selecion M=8, L=4, n=3 DPOB wih anenna selecion M=8, L=4, n=4 DPOB wih anenna selecion M=8, L=4, n= TRIAL FOR CHANNEL REALIZATION Fig. 3. Minimum SNR for realizaions of Rayleigh fading channels for DPOB wih anenna selecion and fixed array for differen number of bis. 6. CONCLUSION Single group mulicas ransmi beamformer design wih anenna subarray selecion problem is considered. A mehod for opimum discree phase-only beamformer wih anenna selecion is presened. I is shown ha he proposed mehod performs significanly beer compared o opimum fixed array. Compuaional complexiy is much beer han he exhausive search hanks o he efficiency of he seleced algorihm for opimizaion. REFERENCES [] O. Mehanna, N. D. Sidiropoulos, and G. B. Giannakis, Join mulicas beamforming and anenna selecion, IEEE Trans. Signal Processing, vol. 6, no., pp , May 23. [2] A. Dua, K. Medepalli, and A. Paulraj, Receive anenna selecion in MIMO sysems using convex opimizaion, IEEE Trans. Wireless Commun., vol., no. 9, pp , Sep. 26. [3] H. Kamoda, J. Tsumochi, F. Suginoshia, Reducion in quanizaion lobes due o digial phase shifers for phased array radars, in Asia-Pacific Microwave Conference 2, -8 Dec. 2. [4] Y. Sao, K. Fujia, H. Sawada, and S. Kao, Design and performance of beam-forming anenna wih discree phase shifer for pracical millimeer-wave communicaions sysems, in Microwave Conference Proceedings (APMC), 2 Asia-Pacific, 7- Dec. 2. [] V. Venkaeswaran, and A.-J. van der Veen Analog beamforming in MIMO communicaions wih phase shif neworks and online channel esimaion, IEEE Trans. Signal Processing, vol. 8, no. 8, pp , Aug. 2. [6] T. K. Sinhamahapara, A. Ahmed, G. K. Mahani, N. Pahak, and A. Chakrabary, Design of discree phaseonly dual-beam array anennas wih minimum dynamic range raio, in Applied Elecromagneics Conference, 27. AEMC 27. IEEE, 9-2 Dec. 27. [7] T. H. Ismail and Z. M. Hamici, Array paern synhesis using digial phase conrol by quanized paricle swarm opimizaion, IEEE Trans. Anennas Propag., vol. 8, no. 6, pp , June 2. [8] S.-Ho Tsai, Equal gain ransmission wih anenna selecion in MIMO communicaions, IEEE Trans. Wireless Commun., vol., no., pp , May 2. [9] X. Zhang, A. F. Molisch, and S.-Y. Kung, Variablephase-shif-based RF-baseband codedesign for MIMO anenna selecion, IEEE Trans. Signal Processing, vol. 3, no., pp , Nov. 2. [] N. D. Sidiropoulos, T. N. Davidson, and Z.-Q. Luo, Transmi beamforming for physical layer mulicasing, IEEE Trans. Signal Processing, vol. 4, no. 6, pp , June 26. [] N. D. Sidiropoulos and T. N. Davidson, Broadcasing wih channel sae informaion, in Proc. IEEE Sensor Array and Mulichannel (SAM) Workshop, Siges, Barcelona, Spain, Jul. 82, 24, vol., pp [2] O. T. Demir and T. E. Tuncer, Opimum design of discree ransmi phase only beamformer, in 2h European Signal Processing Conference (EUSIPCO), 9-3 Sep. 23. [3] O. T. Demir and T. E. Tuncer, Opimum discree single group mulicas beamforming, in IEEE In. Conf. Acousics, Speech and Signal Processing (ICASSP 4), 4-9 May 24. [4] Gurobi. [Online]. Available: hp:// [] M. Elf, C. Guwenger, M. Jünger, and G. Rinaldi, Branch-and-cu algorihms for combinaorial opimizaion and heir implemenaion in ABACUS, in Compuaional Combinaorial Opimizaion, M. Jünger, D. Naddef Ed. Berlin, Germany: Springer- Verlag, 2 [6] D. Wei and A. V. Oppenheim, A branch-and-bound algorihm for quadraically-consrained sparse filer design, IEEE Trans. Signal Processing, vol. 6, no. 4, pp. 6-8, Feb. 23. [7] E. Amaldi, A. Capone, S. Coniglio, and L. G. Gianoli, Nework opimizaion problems subjec o -min fair flow allocaion, IEEE Signal Process. Le., vol. 7, no. 7, pp , July 23. [8] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzea, O. Edfors, and F. Tufvesson, Scaling up MIMO: Opporuniies and challenges wih very large arrays, IEEE Signal Processing Mag., vol. 3, no., pp. 4-6, Jan. 23.