Sparse beamforming in peer-to-peer relay networks Yunshan Hou a, Zhijuan Qi b, Jianhua Chenc

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1 3rd International Conference on Machinery, Materials and Inforation echnology Applications (ICMMIA 015) Sparse beaforing in peer-to-peer relay networs Yunshan ou a, Zhijuan Qi b, Jianhua Chenc College of Inforation Engineering enan University of Science and echnology Luoyang, enan, 47103, China a houys0134@163.co, b qzj_1978@16.co, cchenjh_16888@163.co eywords: Relay networ, beaforing, transit power, seidefinite relaxation, SINR. Abstract. o reduce active relays in peer-to-peer relay networs, a sparsity prooting penalty ter is introduced into the objective function to obtain beaforing weights through iniization of the total relay transit power while the signal-to-interference-plus-noise ratio (SINR) at the destinations are guaranteed to be above certain thresholds. o deal with the non-convexity of the proble, we use seidefinite relaxation to turn this proble into a seidefinite prograing (SP) proble. hen we can efficiently solve the SP proble using interior point ethod based software tools. Siulation results show that the proposed ethod can efficiently conduct relay selection with a ild increase in iniu total transit power of the relays copared with the traditional full cooperative relay networ. I. Introduction o enhance the utilization of the wireless networ resources, energy efficient designs that enable the counication of ultiple source-destination pairs over a shared channel are deanded. As a result, cooperative relay has attracted uch attention due to its capability of power saving, throughput iproveent and coverage extension. In the ulti-relay aplify-and-forward (AF) networs, the syste perforance can be iproved by the cooperation of relays to assist the source-destination transission[1-5]. Generally, using all relays in the cooperative transission can achieve the best throughput perforance for an AF relay networ. owever, the cost of such full cooperation can be large due to power consuption and signaling overhead. o tacle this proble, another way is partial cooperation by using only a subset of the relays to assist the source-destination transission[6-9]. In this paper, we consider the counication of ultiple soure-destination pairs with the assistance of single-antenna relays. he joint proble of transit beaforing and relay selection is considered. he goal is to obtain sparse beaforing weights through iniization of the total relay transit power while satisfying the SINR requireents at the destinations. o deal with the non-convexity of the proble, we use seidefinite relaxation to turn this proble into a seidefinite prograing (SP) proble. hen we can efficiently solve the SP proble using interior point ethod based software tools. II. Syste odel We consider a wireless networ in which counications between source-destination pairs M ( S, ) tae place in two stages with the aid of M relay nodes { R } =1, as shown in Fig. 1. We assue there is no direct lin between any source and destination. We denote the channel fro S to R as f, and the channel fro R to as g,. A rich-scattering environent is assued so that all the channels undergo independent Rayleigh flat fading he authors - Published by Atlantis Press 396

2 Fig.1 he syste odel of peer-to-peer relay networ In the first stage, all sources transit their essages to the relays siultaneously so that the received signal at relay node R is r = f s + n, for = 1,,, M (1), = 1 where s denotes the data sybol transitted by source node zero-ean coplex Gaussian noise with varianceσ R at For convenience, (1) is rewritten in vector for as R. r = 1 S with { i } E s p = iand n is the r = f s + n () where r = [ r1, r,, r M ], f = [ f1,, f,,, fm, ] and nr = [ n1, n,, nm]. In the second stage, R forwards its received noisy signal weighted by a coplex coefficient w. he transit signal vector fro all the relays can be written as t = Wr (3) where W = diag( w1, w,, w M ) is a diagonal atrix. herefore, the total transit power of the relays is P { M = E t } = w E{ r } where w = [ w, w,, w M ], is a diagonal atrix with diagonal eleents Let s define 1 = 1, = f, p+ σ R = 1 g = [ g,1, g,,, gm, ], then the signal received at d = g t = w w (4) [ ] (5) is s isi r v i= 1 i where v is the zero-ean coplex Gaussian noise with variance can be written as where = g Wf + g W f + g Wn + (6) SINR w Rw + w Sw+ σ = w Q w R g f g f σ at. herefore the SINR at (7) = p( )( ) (8) 397

3 = p ( i)( i) i= 1 i Q g f g f (9) S = σ diag([ gg ],[ gg ],,[ gg ] ) (10) R 11 MM where the notation denotes the eleent-wise adaard product and [ ] l gives the (,) lth eleent of a atrix. III. Proble forulation and proposed ethod In this section, the joint proble of transit beaforing and relay selection is considered. We suppose only the best L relays are selected out of the M relays( L M ). he goal is to find the corresponding beaforing vector w = [ w1, w,, w M ] so that the total relay transission power is iniized, subject to receive-sinr constraints per subscriber. in W w w w Rw w Sw L 0 s.t. w Q w+ +σ γ, = 1,,, w (11) where γ denotes the SINR target at, the l 0 -(quasi)nor is the nuber of nonzero entries of w. Instead of the hard sparsity constraint, an l 0 penalty can be eployed to proote sparsity, leading to in w w + λ w W 0 w Rw w Sw s.t. w Q w+ +σ γ, = 1,,, (1) where λ is a positive real tuning paraeter that controls the sparsity of the solution, and thus the nuber of selected relays. Proble (1) stries a balance between iniizing the transission power and iniizing the nuber of selected relays, where a larger λ iplies a sparser solution. Note that for any λ, there is a corresponding L for which probles (11) and (1) yield the sae sparse solution. Because of the l0-(quasi)nor, solving (1) requires an exhaustive cobinatorial search over all M possible sparsity patterns of w, where the NP-hard proble (1) has to be solved for each of L these patterns. his otivates the pursuit of coputationally efficient solutions. o this objective, we will use the convex l -nor squared as a sparsity inducing regularization to replace the nonconvex l0-nor in (11). Also notice w Q w+ w S w + σ 0, proble (1) is then reforulated as in w w + λ w W ( γ( + )) γσ s.t. w R Q S w, = 1,,, (13) Unfortunately proble (13) is not a convex optiization proble in general. We thus resort to a seidefinite relaxation approach to solve a relaxed version of (13). Let us define X = ww, the optiization proble in (13) is rewritten as in trace(( + λix ) ) X s.t. tr( R γ( Q + S) X ) γσ, = 1,,, ran ( X ) = 1 and X 0 (14) rop the ran constraint(which is not convex), we ai to solve the following optiization proble 398

4 in trace(( + λix ) ) X s.t. tr( R γ( Q + S) X ) γσ, = 1,,, X 0 (15) We see that the objective function and the first constraints are all affine, and therefore, convex in X. Also, the positive seidefinite constraint on X is convex. As a result, the optiization proble (14) is convex and it can be efficiently solved using software tools such as SeuMi. In general, the atrix X obtained fro (15) is not necessarily of ran one. When the atrix ran is one, then its opt principal eigenvector is the optial solution to the original proble. Otherwise, we have to use randoization technique to obtain a suboptial ran-one solution fro X [10]. opt IV. Siulations In this section, siulation results are presented to evaluate the effectiveness of the proposed approach. In all exaples, the paraeters are set as = and M = 0. For convenience, the target SINRs are set to be the sae, i.e., γ = γ, ( = 1,,, ). We also assue that p = 1 and both channels are in Rayleigh flat fading with E{ f } E{ g } = =.,, 10 We plot the average total transit power of the relays nodes against required target SINR in Fig. for three different penalty paraeter λ = 0, λ = 10 and λ = 50. he case with λ = 0 eans oitting the penalty part in (15), this is the full cooperation case in reference [3] and all relays are active. he perforance coparisons in Fig. show that the average total transit power of the relays nodes of the proposed ethod is only slightly higher than that of the full cooperation ethod in reference [3] when the penalty paraeter λ is set to 10 and 50, which verifies the effectiveness of the proposed ethod. Fig. Average total transit power of the relays nodes versus γ Fig. 3 shows that ore relays are involved in the cooperative transission if the required target SINR γ increases, and larger λ refers to less active relays. Fro Fig.1 and Fig., we can see that copared with the full cooperation case, there is only ild increase of the average total transit power of the relays nodes(less than 1.5 db ) using the proposed ethod. At the sae tie, the proposed ethod involves uch less relay nodes. At γ = 6 db and λ = 10, the proposed ethod yields siilar average total transit power of the relays nodes while using only about 13 relays out of the total 0 relays. At γ = 6 db and λ = 50, the proposed ethod yields siilar average total transit power of the relays nodes while using only about 11 relays out of the total 0 relays. he reason behind is that because different counication lins have different fading characteristics, not all relay nodes contribute to the iproveent of syste perforance. By appropriately selecting relay nodes in beaforing we can produce siilar syste perforance. 399

5 Fig.3 Average nuber of active relays versus γ herefore, we can conclude that the proposed ethod with reasonable λ can achieve siilar iniu average total transit power of the relays copared with the full cooperation case, but just use a subset of relays. V. Conclusion his paper addressed the joint proble of transit beaforing and relay selection in ultiple peer-to-peer networs. he siulation results showed that the proposed ethod can effectively balance the size of active relays and the total relay transit power. he proposed approach can effectively perfor relay selection while satisfying the target SINR constraints and has only a ild increase in the relay transit power as copared to the full cooperation case. Acnowledgent his study was partially supported by the National natural science foundation of China under grant No. U and the natural science foundation of enan province in China under grant No References [1] Y. Zhao, R. Adve, and. Li, Beaforing with liited feedbac in aplify-and-forward cooperative networs, IEEE rans. Wireless Coun., vol. 7, no. 1, pp , ec [] Y. Jing and. Jafarhani, Networ beaforing using relays with perfect channel inforation, IEEE rans. Inf. heory, vol. 55, no. 6, pp , Jun [3] Siavash Fazeli-ehordy, Shahra Shahbazpanahi and Saeed Gazor, Multiple Peer-to-Peer Counications with a networ of relays, IEEE ransactions on Signal Processing, vol. 57, no. 8, pp , August 009. [4] G. Zheng,.-. Wong, A. Paulraj, and B. Ottersten, Robust collaborative-relay beaforing, IEEE rans. Signal Process., vol. 57, no. 8, pp , Aug [5] Y.-W. Liang and R. Schober, Cooperative aplify-and-forward beaforing with ultiple ulti-antenna relays, IEEE rans. Coun., vol. 59, no. 9, pp , Sep [6]. ao and A. Czylwi, Beaforing design and relay selection for ultiple MIMO AF relay systes with liited feedbac, in Proc. IEEE VC Spring, 013, pp

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