Multiple-relay selection in amplify-and-forward cooperative wireless networks with multiple source nodes

Transcription

1 Wu et al. EURASIP Journal on Wireless Counications and Networing 202, 202:256 RESEARCH Open Access Multiple-relay selection in aplify-and-forward cooperative wireless networs with ultiple source nodes Jinsong Wu *, Yiin D Zhang 2*, Moeness G Ain 2 and Murat Uysal 3 Abstract In this article, we propose ultiple-relay selection schees for ultiple source nodes in aplify-and-forward wireless relay networs based on the su capacity axiization criterion. Both optial and sub-optial relay selection criteria are discussed, considering that sub-optial approaches deonstrate advantages in reduced coputational coplexity. Using sei-definite prograing convex optiization, we present coputationally efficient algoriths for ultiple-source ultiple-relay selection MSMRS with both fixed nuber and varied nuber of relays. Finally, nuerical results are provided to illustrate the coparisons between different relay selection criteria. It is deonstrated that optial varied nuber MSMRS outperfors optial fixed nuber MSMRS under the sae power constraints. Introduction Multihop relaying has eerged as a proising approach to achieve high-rate coverage in wireless counications [,2]. Several aplify-and-forward AF and decode-andforward DF relaying techniques have been introduced such as in [2,3]. Following those pioneer wors, a nuber of cooperative diversity schees have been proposed, including, for exaple, distributed space-tie coding [3-5], adaptive power control for relay networs or relay beaforing [6-9], and relay selection [0-9]. The objective of relay selection is to achieve higher throughput or lower error probability through choosing one or ore relays for transission according to channel conditions. In coparison to relay beaforing, relay selection is attractive due to its deployent of sipler signaling schee and energy saving. Most currently available relay selection approaches assue only a single source node [0-2,4-8], and can be classified into two categories:. A ajority of relay selection rules are restrictive in the sense that they either always use all the available *Correspondence: [email protected]; [email protected] Bell Laboratories, Alcatel-Lucent, Shanghai 20206, P.R. China 2 Center for Advanced Counications, Villanova University, Villanova, PA 9085, USA Full list of author inforation is available at the end of the article relays or always use just a single relay, such as in [0-8,20-29]. In [2], four siple relay selection criteria are described: Two criteria are based on the selection of a single relay according to ean channel gains, while the other two select all available relays. Selecting all available relays are the siplest approach with ultiple relays, and this approach ay not be allowed when the su power liit is less than the suation of the power values of all available relays. A single-relay node is selected based on average channel state inforation CSI, e.g., distance or path loss [20,22,30], and on the instantaneous fading states of the various lins such as in [23]. 2. Multiple-relay selection for a single source has attracted attention as well [3-33]. Jing and Jafarhani proposed sub-optial two-step optiization approaches for single-source ultiple-relay selection in [3,33]: In the first step, phase rotation is perfored at each relay, and thus only power allocation is considered due to signal-to-noise ratio SNR consisting of a suation of purely real ters. In the second step, several sub-optial ethods were introduced [3,33]: a By introducing the idea of relay ordering, several schees with linear coplexity were proposed; 202 Wu et al.; licensee Springer. This is an Open Access article distributed under the ters of the Creative Coons Attribution License which perits unrestricted use, distribution, and reproduction in any ediu, provided the original wor is properly cited.

2 Wu et al. EURASIP Journal on Wireless Counications and Networing 202, 202:256 Page 2 of 3 b Based on recursion, a schee with quadratic coplexity was proposed. Although both single- and ultiple-relay selection approaches for a single source node networ have been investigated, relay selection approaches for ultiple source nodes are rarely addressed in literature. Only the following three existing publications [34-36] have discussed ultiple-source relay selection MSRS approaches. Elzbieta and Raviraj have proposed MSRS for DF relay networs [34]. Xu et al. have presented MSRS approaches in which only a single source is considered as the desired user over each selected relay per transission while other sources or users are considered as interferers during the transission [35]. Guo et al. have analyzed MSRS for opportunistic relays, in other words, only a single source is transitted over each selected relay per transission [36]. Further, there have been several recent research wors on two-way relay selections [37-40]. In this article, we consider AF relay-based cooperative counication systes for siultaneous ultiple-source transission over each selected relay, and ore than one relay is allowed per ultiple-source transission. Each relay is assued to satisfy practical individual short-ter power constraints, that is, each relay has two power levels: zero and its axiu power. This assuption has been used for single-source ultiple-relay selection in, for exaple, [3,33]. The ain contributions of this article can be listed as:. Based on the su capacity criteria, we derive and propose several ultiple-relay selection techniques in AF relay networs with ultiple source nodes. 2. Using sei-definite prograing optiization, we propose coputationally efficient algoriths for ultiple-source ultiple-relay selection MSMRS in the presence of both fixed nuber and varied nuber of relays. The following notations are used: T denotes atrix transpose, conjugate, H atrix conjugate transpose, Hardard product operator, [A] a,b the a, bth entry eleent of atrix A, tr atrix trace operation, Re real part of the object atrix or variable, I iaginary part of the object atrix or variable, E α expectation over rando variable or rando variable set α, diaga denotes a square atrix with all-zeros entries except the ain diagonal filled with the entries of the vector a, φ denotes epty set, and X 0 denotes that X is a positive sei-definite atrix. Syste odel and proble forulation Consider a wireless relay networ with M source nodes transitters, K relay nodes, and one destination node receiver. Each node is equipped with a single antenna. Assue no direct channel path between the source nodes and the destination node. The source nodes and the relay nodes are assued to share the sae transission channel. Based on two-phase half-duplex AF relay assuption, we consider a ultiple-source AF relay selection approach. The period of one two-phase AF relay procedure is defined as one tie channel use. During the tth tie channel use, the two-phase AF protocol is perfored as follows:. In the first phase, the th source node transitter sends source inforation sybol x t using power P S to the relay nodes, where =,..., M, x t E 2 =. the inforation sybols x t, =,..., M, are selected randoly fro M independent codeboos. It is assued that M source nodes siultaneously send uncorrelated signal streas x t, =,..., M, and the corresponding channel sybols are received at relay at the sae tie. 2. Inthesecondphase,L relays with indices,..., L are selected according to soe criteria, which will be elaborated later. Here, L, L K, is an integer, which is referred to as relay selection order inthis article. Then, the i th relay, i =,..., L,scalesits received signal power to unity, and, using power i, aplifies and forwards it to the receiver. Note that, in this two-phase AF protocol, ultiple source nodes share the sae channels. The transission and reception aong the source nodes, the relay nodes and the destination node are assued to be perfectly synchronized. In the tth tie channel use, the channel fro the th source node transitter to the th relay is denoted as and the channel fro the th relay to the receiver is denoted as g t. The channels are odeled as frequency non-selective Rayleigh fading, and are assued to independently vary over different tie channel uses. Denote v t as the noise coponent at the th relay, =,..., K, and denote w t as the noise coponent at the destination node, where v t and w t are assued to be independently and identically distributed i.i.d. coplex Gaussian rando variables with zero ean and unit variance. During the tth tie channel use, the received signal at the th relay is h,t r t = M = P S h,t xt + v t.

3 Wu et al. EURASIP Journal on Wireless Counications and Networing 202, 202:256 Page 3 of 3 The corresponding scaling factor for the th relay is given by β t = M = P S h,t During the tth tie channel use, the received signal at destination is then obtained as y t = K = + = K = α t α t g t βt rt M g t βt = + w t P S h,t x K = α t s t g t βt vt + w t, q t where α t is the relay selection factor, whose value is equal to 0 or, depending upon different relay selection algoriths. In this article, we choose su capacity per tie channel use as the perforance easure for relay selection [4-43]. The syste su capacity is given by C t = 2 log ρ t, 4 where ρ t is the overall syste effective SNR, and obtained in our case as ρ t = E x t,v t,wt s t q t 2 = E s t 2 x t E v t,wt q t 2. 5 In the above, E x t s t 2 and E v t q t 2 are,wt given by E x t s t 2 = M K P S = = α t gt βt h,t 2 6 and q E t 2 v t =,wt K = Inserting 6 and 7 into 5, we have ρ t = M = P S K = K = α t gt βt α t gt βt h,t α t gt βt Relay selection could be expressed using set partition. Define relay index set =,..., K. There exist L distinct relay indices,..., L, where,..., L K, such that the following hold: α t = 0, /,..., L, K and α t = = =. The optiization proble can be now forulated as arg ax C t. Since log a, a >, is α t L a onotonous function, the proble is equivalent to arg ax ρ t. Relay selection can be ipleented at the destination node receiver. In this case, the receiver is assued to now all instantaneous channel state inforation for source-relay paths and relay-destination paths, which ay be obtained through channel estiation. After one relay selection algorith is perfored at the destination node, α t = = α t L are obtained, and then the destination node feedbacs one-digit relay selection inforation to each relay node. The superscripts t used in this section will be oitted in the rest of the article to siplify the notations whenever no abiguity arises. Fixed nuber ultiple-source ultiple-relay selection When individual relay power constraints are equal or close,thenuberofrelaysaybeusedasaconstraint to stand for su relay power constraints. In this section, for siultaneous transission of ultiple source signals, thenuberofrelaystobeselectedisassuedtobea fixed nuber L, wherel >. This class of approaches are referred to as fixed nuber ultiple-source ultiplerelay selection FN-MSMRS, and the corresponding set partition of relay indices for relay selection can be defined as F M = L >, L is a fixed positive integer. 7 8

4 Wu et al. EURASIP Journal on Wireless Counications and Networing 202, 202:256 Page 4 of 3 Optial FN-MSMRS Using 8 and the values of α t, the output SNR of optial FN-MSMRS OFN-MSMRS can be derived as ρ = M = P S L d= L d= d g d β d h d d g d 2 βd 2 Thus, the proposed selection criterion is arg ax ρ. 0 F M Fixed nuber MSMRS based on sei-definite prograing optiization The coplexity of OFN-MSMRS ay be prohibitive particularly when the diension of the proble becoes larger. Closed-for optiization solutions are unfortunately not possible for 0 due to the involveent of ultiple sources. Based on sei-definite prograing [44], we propose an efficient approach for FN-MSMRS. First, note that 8 can be written in a atrix for as ρ = pt P /2 A s P /2 p p T P /2 A n P /2 p +, where p = [α,..., α K ] T, P = diag M T = P S a s = a s a s,..., PR, A s =, A n = diag a n a n, [ ] β h g,..., β K h K T g K,anda n = [ β g,..., β K g K ] T. Further, note that p is a real integer vector with 0, entries, A s is a Heritian atrix, and A n is a real-valued atrix. It can be readily checed fro 6 that I E x s 2 = I p T P /2 A s P /2 p = 0. Thus, can be further siplified in a real-valued atrix for as ρ = pt P /2 Re A s P /2 p p T P /2 A n P /2 p +. 2 K atrix M of size K K, the following relationship always holds, where p T Mp = c T Mc, 4 c = [, c T ] T, 5 and atrix M is related to M by a function f defined as M = f M = [ K T M K K T ] M. 6 4 M K M Now can be re-written as ρ = ct Sc c T Nc +, 7 where S = f P /2 Re A s P /2 and N = f P /2 A n P /2. The optiization proble can be now forulated as axiize ρ, subject to ρ = c T f I K c = L, ct Sc c T Nc +, c T G c =, =,..., K, 8a 8b 8c 8d where G, =,..., K, are all-zero atrices except [G ], =, =,..., K. The fixed relay selection order is quantified in 8c. Note that, it is necessary to include individual relay selection factor constraints, such as 8d, which are actually related to individual relay power constraints, otherwise individual relay selection factors can be arbitrary in the optiization process. Only using vector p, it is hard to quantify individual relay selection factor constraints. However, based on the vector transforation in 3 and 5, individual relay selection factor constraints can be advantageously written as the for shown in 8d. Denote B = cc T.Notethatc T Xc = tr Xcc T = tr XB. Thus, the optiization proble 8 now becoes axiize ρ, 9a Denote c = 2p K, 3 where c is an integer vector with, entries, and K is an all-one colun vector of length K. For an arbitrary subject to: ρ = tr SB tr NB +, tr f I K B = L, 9b 9c

5 Wu et al. EURASIP Journal on Wireless Counications and Networing 202, 202:256 Page 5 of 3 tr G B =, =,..., K, ran B =, 9d 9e Through reoving ran constraint 9e, the proble 9 is now relaxed to axiize tr SW, 26a B 0. 9f The optiization proble is not convex due to ran constraint 9e and fractional constraint 9b. We can perfor sei-definite relaxation through reoving ran constraint 9e [45]. Choosing a positive variable u, trsb where trnb+ u, the above optiization proble can be written as axiize u, 20a subject to tr S un B, u tr f I K B = L, tr G B =, =,..., K, 20b 20c 20d B 0. 20e The optiization proble 20 is still non-convex. However, using the bisection Algorith as shown in Appendix, with the aids of convex prograing tools, such as CVX [46,47] which we have used in the siulations, the proble 20 can be solved iteratively, since it is quasi-convex in each loop within the bisection Algorith, where u acts as a constant. This proble 20 can now be efficiently solved by standard interior point algoriths based on sei-definite prograing SDP [48]. Denote theoptialestiationofb through the proposed bisection Algorith as B. The above SDP procedure requires bisection Algorith, which ight introduce higher coplexity when the nuber of iterative loops is high for convergence. Now we ay consider another approach without the requireent of an bisection algorith. Denote U = N + L f I K. 2 Using 20c, we have tr UB = tr NB Denote w = λc, 23 where λ>0ischosentoaesure w T Uw =. 24 Denote W = ww T,andthus ρ = tr SB tr SW = tr UB tr UW. 25 subject to tr UW =, tr f I K W = Lλ, tr G W = λ, =,..., K, D 0, 26b 26c 26d 26e λ 0. 26f The proble 26 now could be solved using seidefinite prograing without the requireent of a bisection algorith. Note that the above ethod could be considered as the extension of Charnes Cooper algorith [49] fro linear fractional prograing to linear quadratic prograing. Note that the above solutions are obtained through reoving the ran- constraint 9e, which ay lead to an increased proble diension. Thus it is required to convert the sei-definite relaxation solution to soe Boolean solution. In [45,50,5], a randoization ethod has been introduced to achieve this conversion. Note that in those wors, the randoization approach is ipleented without additional constraint. Here, we extend such randoization approach to support extra constraints, such as 20c. Based on the randoization procedure as proposed in the Appendix, the decision of c, ĉ,can be obtained, where ĉ = [ ] T [ĉ],2,..., [ĉ],k+ and [ĉ], is the th entry of ĉ. It should be noted that Steps 9 and 0 of Algorith 2 are introduced to satisfy constraint 20c. In [45,50,5], only c = sign V T u is used in the randoization process. However, it has been further proved that ±sign c = sign V T u holds with probability in Property 2 of [45] a. Thus it ay be eaningful to perfor both + and of sign operations in the randoization process as we have proposed in Steps 9 and 0 of Algorith 2. The above proposed MSMRS based on sei-definite prograing is tered as SDPFN-MSMRS:. In the case of solving Proble 20 and using randoization procedure Algorith 2: SDPFN-MSMRS B, 2. In the case of solving Proble 20 and using randoization procedure Algorith 2 without step 0: SDPFN-MSMRS A, 3. In the case of solving Proble 26 and using randoization procedure Algorith 2: SDPFN-MSMRS B2,

6 Wu et al. EURASIP Journal on Wireless Counications and Networing 202, 202:256 Page 6 of 3 4. In the case of solving Proble 26 and using randoization procedure Algorith 2 without step 0: SDPFN-MSMRS A2. Note that both the solutions of SDPFN-MSMRS B and SDPFN-MSMRS A require bisection algoriths to solve SDP probles iteratively, while both the solutions of SDPFN-MSMRS B2 and SDPFN-MSMRS A2 do not. Best worse FN-MSMRS and rando FN-MSMRS Note that 9 cannot be further siplified without additional approxiations. Intuitively, it can be questioned whether best-worse single source single-relay selection [33] can be extended to this case. To address this concern, calculate all M a = in = P S h a 2, ga 2 2 βa, where =,..., K. Then,perutatea in descending order such that a σ a σk,whereσ denotes the perutation function. This yields σ,..., σl, and such a selection criterion is tered as best worse FN MSMRS BWFN-MSMRS. For coparison purpose, we also define rando fixed nuber MSMRS RANDFN-MSMRS, which randoly selects L relays, as a baseline benchar FN-MSMRS schee. Varied nuber ultiple-source ultiple-relay selection MSMRS When individual relay power constraints are diverse, su powerconstraintscannolongerbedescribedusingafixed nuber of relays. Unlie the previous section, we assue in this section that the nuber of relays to be selected is not predeterined but rather a varied nuber which is optiized depending on both individual and su relay power constraints. This class of proposed approaches are abbreviated as VN-MSMRS, and the corresponding set partition of relay indices for relay selection can be defined as V M = L K, L is a integer variable, L. P Su a= a For coparison purpose, a baseline benchar VN- MSMRS schee using predeterined relay selection, PVN-MSMRS, is also defined. In this schee, a feasible relay selection is chosen, assuing that this selection satisfies given relay power constraints, and no ore relays can be added, otherwise the given su power constraint is violated. Optial VN-MSMRS OVN-MSMRS For VN-MSMRS, the overall effective syste SNR is still given by 9. However, L is no longer a fixed nuber but a variable to be chosen fro a set L,..., K. The proposed optial selection criterion, OVN-MSMRS, becoes arg ax ρ. 27 V M Varied nuber MSMRS based on sei-definite prograing optiization For VN-MSMRS, the forulation of optiization proble is the sae as 8 except that 8c is replaced by a su power constraint c T f P c P Su. 28 The corresponding sei-definite relaxation forulation is written as axiize u, 29a subject to tr S un B, u tr f P B P Su, tr G B =, =,..., K, 29b 29c 29d B 0 29e The optiization proble 29 can be solved using the bisection procedure siilar to the proposed Algorith as depicted in Appendix. The difference is that Step 4 of the bisection procedure for VN-MSMRS is changed into solve the SDP optiization proble 29. To obtain the estiation of c, ĉ, the randoization procedure Algorith 3 for VN-MSMRS is proposed in Appendix. The above proposed MSMRS based on sei-definite prograing is defined as SDPVN-MSMRS: the SDPVN- MSMRS using randoization procedure Algorith 3 is tered as SDPVN-MSMRS B, while the SDPVN-MSMRS using randoization procedure Algorith 3 without steps 9 and 0 is called as SDPVN-MSMRS A. Nuerical results In this section, we present the perforance of the su capacity per tie channel use for the relay selection approaches under considerations. In all figures, the horizontal axis indicates unit power P, and, =,..., K, and P S, =..., M, are scaled values of P. In this section,thenuberofsourcesissettom = 2. We further assue that channels h,t and g t, =,..., M and =,..., K, are Rayleigh fading channel gains odeled as coplex Gaussian with zero ean and unit variance, and

7 Wu et al. EURASIP Journal on Wireless Counications and Networing 202, 202:256 Page 7 of 3 they change independently over different tie channel uses. FN-MSMRS results In Figures and 2, we assue K = 8, L = 4, = PM, =,..., K, and P S = P, =,..., M. The settings of randoization procedure in SDPFN- MSMRS A, SDPFN-MSMRS A2, SDPFN-MSMRS B, and SDPFN-MSMRS B2 are N c = 2andN l = 4. In Figure, we observe that, to achieve the sae average su capacity per tie channel use,. SDPFN-MSMRS A, SDPFN-MSMRS A2, SDPFN-MSMRS B, and SDPFN-MSMRS B2 use less unit power P than BWFN-MSMRS by.6 and.3 db, respectively; 2. BWFN-MSMRS use less unit power P than RANDFN-MSMRS by only 2.2 db; 3. With the advantage of lower coplexity, SDPFN- MSMRS A, SDPFN-MSMRS A2, SDPFN-MSMRS B, and SDPFN-MSMRS B2 require ore unit power P than OFN-MSMRS by 2.2 and 2.5 db, respectively. It is observed that both SDPFN-MSMRS B and SDPFN-MSMRS B2 achieve notably higher average su capacity over both SDPFN-MSMRS A and SDPFN- MSMRS A2 for the sae unit power P. This also verifies the iportance of step 0 of Algorith 2. With very close perforance to SDPFN-MSMRS A and SDPFN-MSMRS B, respectively, SDPFN-MSMRS A2 and SDPFN-MSMRS B2 are quite coputationally effective due to avoiding the needs of additional bisection algoriths. VN-MSMRS results In this section, the settings of randoization procedure in SDPVN-MSMRS B and SDPVNMSMRS A are N c = 2 and N l = 4. In Figures 3 and 4, we assue K = 8, M = 2, P Su = 4PM, = PM, =,..., K, P S = P, =,..., M. Fro Figure 3, we observe that, to achieve the sae average su capacity per tie channel use,. SDPVN-MSMRS B and SDPVN-MSMRS A use less unit power P than PVN-MSMRS by 4.2 and 3.9 db, respectively, 2. With the advantage of lower coplexity, SDPVN-MSMRS B and SDPVN-MSMRS A require ore unit power P than OVN-MSMRS by.4 and.7 db, respectively. 4 Average su capacity per tie channel use bits/hz OFN MSMRS SDPFN MSMRS A with bisection ethod for SDP SDPFN MSMRS B with bisection ethod for SDP SDPFN MSMRS A2 without bisection ethod for SDP SDPFN MSMRS B2 without bisection ethod for SDP BWFN MSMRS RANDFN MSMRS P db Figure Average per tie channel use su capacity versus P for FN-MSMRS, K = 8, M = 2, L = 4, P S = P, =,..., M. = MP, =,..., K,

8 Wu et al. EURASIP Journal on Wireless Counications and Networing 202, 202:256 Page 8 of ProbC >C OFN MSMRS SDPFN MSMRS A with bisection ethod for SDP SDPFN MSMRS B with bisection ethod for SDP SDPFN MSMRS A2 without bisection ethod for SDP SDPFN MSMRS B2 without bisection ethod for SDP BWFN MSMRS RANDFN MSMRS Su capacity per tie channel use bits/hertz Figure 2 Copleentary cuulative distribution function of su capacity per tie channel use at P = 4 db for FN-MSMRS, K = 8, M = 2, L = 4, = MP, =,..., K, P S = P, =,..., M. Average su capacity per tie channel use bits/hz OVN MSMRS SDPVN MSMRS A SDPVN MSMRS B PVN MSMRS P db Figure 3 Average per tie channel use su capacity versus P for VN-MSMRS, K = 8, M = 2, P Su = 4PM, = PM, =,..., K, P S = P, =,..., M.

9 Wu et al. EURASIP Journal on Wireless Counications and Networing 202, 202:256 Page 9 of ProbC > C OVN MSMRS SDPVN MSMRS A SDPVN MSMRS B PVN MSMRS Su capacity per tie channel use bits/hertz Figure 4 Copleentary cuulative distribution function of capacity per tie channel use at P = 4 db for VN-MSMRS, K = 8, M = 2, P Su = 4PM, = PM, =,..., K, P S = P, =,..., M. Unlie in Figures 3 and 4, relay powers in Figures 5 and 6 are not uniforly distributed, and we assue K = 8, M = 2, P Su = 3.62PM, = PM, =, 2, = 0.65PM, = 3, 4, = 0.4PM, = 5,...,8, P S = P, =,..., M. In Figure 5, siilar conclusions can be drawn except for different gains as shown in Figure 3. For exaple, in Figure 5, SDPVN-MSMRS B uses 3.55dB less unit power P than PVN-MSMRS. The Average su capacity per tie channel use bits/hz OVN MSMRS SDPVN MSMRS A SDPVN MSMRS B PVN MSMRS P db Figure 5 Average per tie channel use su capacity versus P for VN-MSMRS, K = 8, M = 2, P Su = 3.62PM, = 0.65PM, = 3, 4, = 0.4PM, = 5,...,8, P S = P, =,..., M. = PM, =, 2,

10 Wu et al. EURASIP Journal on Wireless Counications and Networing 202, 202:256 Page 0 of ProbC > C OVN MSMRS SDPVN MSMRS A SDPVN MSMRS B PVN MSMRS Su capacity per tie channel use bits/hertz Figure 6 Copleentary cuulative distribution function of su capacity per tie channel use at P = 4 db for VN-MSMRS, K = 8, M = 2, P Su = 3.62PM, = PM, =, 2, = 0.65PM, = 3, 4, = 0.4PM, = 5,...,8, P S = P, =,..., M. above results verify the iportance of steps 9 and 0 of Algorith 3. Coparison between OFN-MSMRS and OVN-MSMRS In Figures 7 and 8, we copare OFN-MSMRS with OVN-MSMRS under the sae power constraints, and we assue K = 8, M = 2, P Su = 4PM, = PM, =,..., K, P S = P, =,..., M. Note that, for OFN-MSMRS, P Su = 4PM is equivalent to set L = 4. It is evident that OVN-MSMRS outperfors OFN-MSMRS under the sae power constraints. This Average su capacity per tie channel use bits/hz OFN MSMRS OVN MSMRS P db Figure 7 Average per tie channel use su capacity versus P for OFN-MSMRS and OVN-MSMRS, K = 8, M = 2, P Su = 4PM, = PM, =,..., K, P S = P, =,..., M.

11 Wu et al. EURASIP Journal on Wireless Counications and Networing 202, 202:256 Page of ProbC > C OFN MSMRS OVN MSMRS Su capacity per tie channel use bits/hertz Figure 8 Copleentary cuulative distribution function of su capacity per tie channel use at P = 4 db for OFN-MSMRS and OVN-MSMRS, K = 8, M = 2, P Su = 4PM, = PM, =,..., K, P S = P, =,..., M. iplies that the best selection solution for soe channel realizations ay not necessarily always reach full su power constraints. Note that the coplexity of optial MSMRS significantly increases when K becoes larger. In siulations, we choose a sall nuber of K = 8, for reduced siulation tie. For such low K values, the coplexity advantage for the proposed approaches ay not be that significant. However, with the increase of K, coplexity advantage for proposed approaches in Sections Fixed nuber ultiple-source ultiple-relay selection and Varied nuber ultiple-source ultiple-relay selection will becoe ore pronounced. Conclusion Based on the su capacity axiization criterion, we have proposed a nuber of ultiple-relay selection approaches for siultaneously transitting ultiple source nodes with fixed power relays in an aplifyand-forward cooperative relay networ. We propose coputationally efficient algoriths based on seidefinite prograing for MSMRS with both fixed nuber and varied nuber of relays. We have deonstrated that optial varied nuber MSMRS outperfors fixed nuber MSMRS under the sae su power constraints. Although we have discussed the convex relaxation approaches in this article, as the future research directions, it ay be deserved to investigate other nonconvex-relaxation approaches with better perforance, such as in [52-54]. Endnote a In [45], the authors express sign operation using notation σ instead of sign Appendix Algoriths Algorith Bisection procedure:. Initialize the upper and lower liits of u, u U and u L ; 2. If u U u L <ε, go to step 7, otherwise go to step 3; 3. u := 2 u U + u L ; 4. Perfor the SDP optiization procedure for proble 20; 5. If the optiization proble 20 is infeasible or unbounded, u U := u; else u L := u; B = B; 6. Go to step 2; 7. The optiization procedure ends. Algorith 2 Randoization procedure for FN-MSMRS:. Copute V such that B = V T V,where V = [v,..., v K ]; 2. Set a c = 0, a s = 0,andρ ax = 0; 3. If a s = 0, go to step 3, otherwise go to step 4; 4. If a c N c,

12 Wu et al. EURASIP Journal on Wireless Counications and Networing 202, 202:256 Page 2 of 3 choose ĉ using BWFN-MSMRS such that L entries of ĉ equal to and the rest equal to,thengoto step 3; else go to step 5; 5. Set a l = 0; 6. Choose rando vector u fro the unifor distribution on the unit sphere; 7. Copute c = sign V T u, and thus obtain c as in 5; 8. Copute p = 2 c + ; K 9. If [p], == L, = Copute ρ based on 8b; If ρ>ρ ax, ρ ax = ρ, ĉ = c; a s = ; K 0. If K [p], == L, = Copute c = c,[c], =, and thus obtain c; Copute ρ based on 8b; If ρ>ρ ax, ρ ax = ρ,ĉ = c; a s = ;. a l = a l + ; 2. If a l N l, a c = a c + ; go to 3; else go to 6; 3. The randoization procedure ends. Algorith 3 Randoization procedure for VN-MSMRS:. Copute V such that B = V T V,where V = [v,..., v K ], v is the th colun vector of V; 2. Set a c = 0, a s = 0,andρ ax = 0; 3. If a s = 0, go to step 3, otherwise go to step 4; 4. If a c N c, Choose ĉ using optial ultiple-source single-relay selection such that the su power constraint 28 is satisfied, then go to step 3; else go to step 5; 5. Set a l = 0; 6. Choose rando vector u fro the unifor distribution on the unit sphere; 7. Copute c = sign V T u, and thus obtain c; 8. If the su power constraint 28 is satisfied, Copute ρ based on 8b; If ρ>ρ ax, ρ ax = ρ, ĉ = c; a s = ; 9. Copute c = c, [c], =, and thus obtain c; 0. If the su power constraint 28 is satisfied, Using c,coputeρ based on 8b; If ρ>ρ ax, ρ ax = ρ, ĉ = c; a s = ;. a l = a l + ; 2. if a l N l, a c = a c + ; go to 3; else go to 6; 3. The randoization procedure ends. Copeting interest The authors declare that they have no copeting interests Acnowledgeents The study was perfored when J. Wu was with the Center for Advanced Counications, Villanova University, Villanova, PA 9085, USA Author details Bell Laboratories, Alcatel-Lucent, Shanghai 20206, P.R. China. 2 Center for Advanced Counications, Villanova University, Villanova, PA 9085, USA. 3 Faculty of Engineering, Ozyegin University, 34794, Istanbul, Turey. Received: 28 June 20 Accepted: 3 July 202 Published: 6 August 202 References. A Sendonaris, E Erip, B Aazhang, User cooperation diversity part I: syste description. IEEE Trans. Coun. 5, JN Lanean, DNC Tse, GW Wornell, Cooperative diversity in wireless networs: efficient protocols and outage behavior. IEEE Trans. Inf. Theory. 502, PA Anghel, M Kaveh, On the perforance of distributed space-tie coding systes with one and two non-regenerative relays. IEEE Trans. Wirel. Coun. 53, J Lanean, G Wornell, Distributed space-tie-coded protocols for exploiting cooperative diversity in wireless networs. IEEE Trans. Inf. Theory. 490, Y Jing, B Hassibi, Distributed space-tie coding in wireless relay networs. IEEE Trans. Wirel. Coun. 52, A Rezni, SR Kularni, S Verdu, Degraded Gaussian ultirelay channels: Capacity and optial power allocation. IEEE Trans. Inf. Theory. 502, X Tang, Y Hua, Optial design of non-regenerative MIMO wireless relays. IEEE Trans. Wirel. Coun, N Khajehnouri, AH Sayed, Distributed MMSE relay strategies for wireless sensor networs. IEEE Trans. Signal Process. 55, X Li, Y Zhang, M Ain, Joint optiization of source power allocation and relay beaforing in ultiuser cooperative wireless networs. Mobile Netw. Appl. 65, V Sreng, H Yanioeroglu, DD Falconer, Relayer selection strategies in cellular networs with peer-to-peer relaying. in Proc. IEEE Vehicular Tech. Conf. vol , pp A Ribeiro, X Cai, GB Giannais, Sybol error probabilities for general cooperative lins. IEEE Trans. Wirel. Coun. 4, AK Sade, Z Han, KJR Liu, A distributed relay-assignent algorith for cooperative counications in wireless networs. in Proc. IEEE Int. Conf. Coun. Istanbul, Turey, 2006, pp Y Zhao, R Adve, TJ Li, Sybol error rate of selection aplify-and-forward relay systes. IEEE Coun. Lett. 0, Z Lin, E Erip, A Stefanov, Cooperative regions and partner choice in coded cooperative systes. IEEE Trans. Coun. 54,

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