Efficient Relay Beamforming Design with SIC Detection for Dual-Hop MIMO Relay Networks

1 Ef ficient Relay Beamformi ng Design with SIC Detecti on for Dual-Hop MIMO Relay Netw orks Y u Zhang, Hanwen Luo, and W en Chen, Member , IEEE Abstract In this paper, we con sider a dua l-hop Multiple Inpu t M ultiple Outpu t ( MIMO) relay wireless network, in w hich a sour ce-destination pair b oth equipped with multiple ante nnas commun icates t hroug h a large nu mber of half-du plex amplify-and -forward (AF) relay terminals. T wo n ovel linear beam forming schemes based on the m atched filter (MF) and regularized zer o-forcin g (RZF) precod ing techniqu es are propo sed for the MIMO rela y system. W e focu s on the linear p rocess at the relay nodes an d desig n the new relay beamformers by utilizing the chann el state information (CSI) of both bac kward channel and forward chan nel. Th e p roposed b eamform ing designs are based on the QR decomposition ( QRD) filter at the destination no de which pe rforms s uccessiv e interference cancellation (SIC) to achieve the maximum spatial multiplexing gain. Simulation results demonstra te th at the pro posed beam formers that fulfil both the intranod e ar ray gain and distributed ar ray gain ou tperform other relay ing schem es under different system parameters in terms of th e ergodic capacity . Index T erms MIMO relay , beamf orming , suc cessi ve interferen ce cancellatio n, ergodic c apacity . This work is supported by NS F China #60972031 , by SEU SKL project #W200907, by ISN project #ISN11-01, by Huawei Funding #YJCB2009024WL and #YJCB2008 048WL, and by Nati onal 973 project #2009CB8249 00. The authors are wi th the E lectronic E ngineering Department in Shanghai Jiao T ong University , Shanghai, China. Y u Zhang is also w ith the S tate Key Laboratory of Integrated Services Networks, and W en Chen is also with SEU SKL for mobile communications (e-mail: { yuzhang49; hwluo; wenchen } @sjtu.edu.cn). 2 I . I N T R O D U C T I O N Recently relay wireless networks ha ve drawn cons iderable interest from both the academic and industrial communi ties. Due to lo w -complexity a nd low-cost of the re lay elements, the architectures of multiple fixed relay nodes imp lemented in cellular syst ems and many other kinds of networks are considered to be a promi sing technique for future wireless networks [1]. Meanwhile, MIMO technique is well verifi ed to provide si gnificant improve ment in the spectral ef ficiency and link reliability beca use of the m ultiplexing and div ersity gain [2], [3]. Combining the relaying and MIMO techniqu es can m ake use of bo th advantages to increase the data rate in the cell ular edge and extend th e network covera ge. The capacity of MIMO relay networks has b een wel l in vestigated in s e veral papers [4]–[6], in which, [5] deri ves lower bounds on t he capacity of a Gaussian MIMO relay channel under the condition o f transm itting precoding . In order to improve the capacity of relay n etworks, var ious kinds of linear dist ributed MIMO relaying schemes have been inv estigated in [7]–[14]. In [7], the authors analyze the stream signal-to-interference ratio statistic and consider dif ferent relay beamforming based on th e finite-rate feedback of t he channel st ates. Assuming T omlinson - Harashima precodi ng at th e b ase statio n and linear processing at the relay , [8] proposes upper and lower bounds on t he achiev able sum rate for the m ultiuser MIMO sys tem with si ngle relay node. In [9], a linear relaying scheme fulfilling the target SNRs on di f ferent su bstreams is proposed and the power- ef ficient relaying strategy is deri ved in closed form. The optimal relay beamforming scheme a nd po wer control algorithms for a cooperati ve and cogn itiv e radio system are presented i n [12]. In [13], [14], the authors design three relay beamforming schemes b ased on matrix triangul arization which hav e superiority over the con ventional zero-forcing (ZF) and amplify-and-forward (AF) beamformers. Inspired by these heuristic works, this paper proposes two novel relay-beamformer designs for the dual-hop MIMO relay networks, which can achie ve both of the dist ributed array gain and intranode array gain. Intranode array gai n is the gain obtai ned from the introducti on of multiple antennas in each node o f the dual-hop n etworks. Distributed array g ain results from the implementat ion of m ultiple relay nodes and do es n ot need any cooperation among them. Assuming the same scenario given in [14], the new relay beamformers outperform th e three schemes proposed in [14] under v arious ne twork conditions. The innov ati on points of our relaying 3 schemes are reflected in th e matched filter and regularized zero-forcing beamforming designs implemented at multiple relay nodes while ut ilizing QRD of the effecti ve channel matrix at the destination node. The destination can perform SIC t o decode mul tiple data streams which hav e further enhancement effect on the channel capacity . In this paper , boldface lowerca se letter and bo ldface uppercase letter represent vectors and matrices, respectively . The notations ( A ) i and ( A ) i,j represent t he i th row and ( i, j ) th ent ry of the matrix A . Notations tr ( · ) and ( · ) H denote trace and conjug ate t ranspose operation of a matrix. T erm I N is an N × N identity matrix. and k a k stands for the Euclidean norm of a v ector a . Finally , we denote the expectation operation by E {·} . I I . S Y S T E M M O D E L The consid ered MIMO relay network cons ists of a singl e source and destin ation node both equipped with M antennas, and K N -antenna relay nodes dist ributed betw een the source- destination pair as illustrated in Fig. 1. When the source nod e implements spatial mult iplexing (SM), the re quirement that N ≥ M m ust be satisfied if e very relay node is supposed to support a ll the M in dependent data streams. W e consider half-duplex non-re generativ e relaying throughout this paper where it takes two non-overlapping ti me slots for the data to be t ransmitted from the source to the desti nation node v ia the backward channel (BC) and forward channel (FC). Due to deep large-scale fading effe cts produced by the long distance, we ass ume that t here is no direct link between the source and destination. In this paper , the perfect CSIs of BC and FC are assum ed to be av ailable at relay nodes. In a practical sy stem, each relay uses the traini ng sequences or pilot sent from t he source nod e to acqui re the CSI of all th e backward channels. The acquisition m ethods of FC’ s information would vary with t wo different dup lex forms. If it is a FDD system, the destinati on sh ould estimate the CSI of FC by usin g t he relay-specific pilot s first, and then feedback the CSI to each relay node. As for a TDD system , due to it s intrinsi c reciprocity , relay nodes can u se the CSI of the link from destin ation to relay nodes to acquire the CSI of FC. In the first tim e sl ot, the source no de broadcasts the signal to all the relay nodes through BC. Let M × 1 vector s be the transmit signal vec tor satisfying the power const raint E  ss H  = ( P / M ) I M , wh ere P is defined as the total transmit power at the source node. L et H k ∈ C N × M , ( k = 1 , ..., K ) st and for the BC MIMO channel matrix from the sou rce node t o the k th 4 relay node. All t he relay nodes are s upposed to b e located in a clust er . Then all the backward channels H 1 , · · · , H K can be suppos ed to be ind ependently and identi cally distributed ( i.i.d. ) and e xperience the same Rayleigh flat fading. Then the correspondi ng r ecei ved signal at the k th relay can be written as r k = H k s + n k , (1) where th e term n k is th e sp atio-temporally whit e zero-mean complex additive Gaussian noise vector , independent across k , wi th the covariance m atrix E  n k n H k  = σ 2 1 I N . Therefore, noise var iance σ 2 1 represents the noi se po wer at each relay node. In t he second time slot, fi rstly each relay no de performs linear process by multiply ing r k with an N × N beamforming matrix F k . Cons equently , t he sig nal vector sent f rom the k th relay node is t k = F k r k . (2) From more practical cons ideration, we assum e that each relay no de has it s own power con straint satisfying E  t H k t k  ≤ Q k , which is independent from power P . Hence a power constraint condition of t k can be d eri ved as p ( t k ) = tr  F k  P M H k H H k + σ 2 1 I N  F H k  ≤ Q k . (3) After linear relay beamformin g process, all the relay no des forw ard their data si multaneously to the destination. Thus the signal vector received by the dest ination can be expressed as y = K X k =1 G k t k + n d = K X k =1 G k F k H k s + K X k =1 G k F k n k + n d , (4) where G k , under the s ame assumptio n as H k , is the M × N fo rward channel between the k th relay nod e and the destinati on. n d ∈ C M , s atisfying E  n d n H d  = σ 2 2 I M , deno tes the zero-mean white circularly symm etric complex additive Gaussian n oise at the dest ination node wit h the noise po wer σ 2 2 . I I I . R E L A Y B E A M F O R M I N G D E S I G N In this section, the network er godic capacity with the QR d etector applied at t he destination node for SIC detection is analyzed. And then we will propose two novel relay beamformer schemes based on th e MF and RZF beamforming techniques. 5 A. QR decompos ition an d SIC detectio n Con ventional recei vers such as MF , zero-forcing (l inear decorrelator) and linear minimum mean square error (L-MMSE) decoder hav e been well stud ied in the pre vious works. Matched filter receiv er has bad performance in t he high SNR region while ZF prod uces noise enhancement ef fect. M MSE equ alizer which can be seen as a go od tradeoff of the MF and ZF recei vers, howe ver , achiev es the same order of diversity as ZF does. Hence m uch larger intranode array gain also cannot be obtained from the MMSE receiv er . As analyzed in [15 ], SIC detection based on the QRD has significant adv antage over those con ventional detectors and the performance of the QR detector is asymptotically equiv alent t o that of t he maximum -likelihood detector (ML D). So we wi ll utilize th e QRD detector as the destination receiver W throu ghout thi s paper . From t he above discussion, the final receive d signal at destination can be derived as follows. Let the term P K k =1 G k F k H k = H S D , and P K k =1 G k F k n k + n d = z . Then equati on (4) can be re written as y = H S D s + z , (5) where H S D represents the ef fecti ve channel between th e source and destin ation node, and z is the ef fecti ve noise vector cumulated from the n oise n k at each relay node and the noise vector n d at t he destination. Im plement QR decomp osition of the eff ectiv e channel as H S D = Q S D R S D , (6) where Q S D is an M × M unitary matrix and R S D is an M × M right upper t riangular m atrix. Therefore the QR detector at destination node is cho sen as: W = Q H S D , and the s ignal vector after detection becomes ˜ y = R S D s + Q H S D z . (7) Finally , the optimal relay b eamformer desig n probl em can be formulated mathematically as ˆ F k = a rg max F k C ( F k ) , (8) s.t. p ( t k ) ≤ Q k , (9) where C ( F k ) is t he network ergodic capacity having various specific forms decided by des- tination detector W and relay beamforming m atrix F k that wil l be discussed in detail in the following subsections. 6 Note that the closed-form solution is difficult to obtain when trying to solve the optimization problem (8) directly . In order to g et a specific form of th e relay beamformers, we furt her assume that a power control factor ρ k is set with F k in (2) to guarantee t hat each relay transmit po wer is equal t o Q k . Since H 1 , · · · , H K (and G 1 , · · · , G K ) are i.i.d. dis tributed and exper ience the same Rayleigh fading, all t he relay beamform ers can hav e a uni form design type. Hence th e transmit signal from each relay node after l inear beamforming and power control becomes t k = ρ k F k r k , (10) where the po wer control parameter ρ k can be d eri ved from equation (3) as ρ k =  Q k  tr  F k  P M H k H H k + σ 2 1 I N  F H k  1 2 . (11) B. MF beamfo rming According to the principles of maximum -ratio-transmission (MR T) [16] and maxim um-ratio- combining (MRC) [17], we choos e t he MF as the beamform er for each relay nod e. Therefore we get the beamformin g m atrix as F M F k = G H k H H k , (12) where each relay beamformer can be d ivided into two parts: a recei ve beamform er H H k and a transmit beamformer G H k . The recei ve beamfo rmer H H k is the optim al weight matrix that maximizes recei ved SNR at the relay . Consequently , the recei ved signal at the d estination can be re written from (10) and (12) as y = K X k =1 ρ k G k G H k H H k H k | {z } H M F S D s + K X k =1 ρ k G k G H k H H k n k + n d | {z } z M F , (13) where ρ k is gi ven by substituti ng (12) into equation (11). Performing QRD of th e H M F S D as H M F S D = Q M F S D R M F S D . (14) Then we get the destinati on receiv er as W M F =  Q M F S D  H . (15) Hence the s ignal vec tor after QR detection becomes ˜ y M F = R M F S D s +  Q M F S D  H z M F . (16) 7 Note that the matrix R M F S D has the right u pper triangu lar form as R M F S D =         r 1 , 1 r 1 , 2 . . . r 1 ,M r 2 , 2 . . . . . . 0 r M ,M         , (17) where the d iagonal entries r m,m ( m = 1 , ..., M ) of (17) are real pos itiv e numbers. With the destination node carrying out the SIC detection, the effecti ve SNR for the m th data stream of MF relay beamforming schem e can be deriv ed as S N R M F m = ( P / M ) r 2 m,m  K P k =1     ρ k ( Q M F S D ) H G k G H k H H k  m    2  σ 2 1 + σ 2 2 . (18) C. MF-RZF beamformi ng In th is s ubsection, w e uti lize th e regularized zero-forcing (RZF) precoding [18] as t he transm it beamformer for FC while M F is still kept as the receiv e beamformer matching with the BC condition. So th e MF-RZF b eamformer is const ructed as F M F − RZ F k = G H k  G k G H k + α k I M  − 1 H H k , (19) where α k is an adjustable parameter that control s t he amount of interference among m ultiple data streams in the second hop. One possible metric for c hoosing α k is to maximize the end-to- end ef fectiv e SNR which will be gi ven below . Hence th e corresponding receiv ed signal at the destination is y = K X k =1 G k F k H k s + K X k =1 G k F k n k + n d = K X k =1 ρ k G k G H k  G k G H k + α k I M  − 1 H H k H k s + K X k =1 ρ k G k G H k  G k G H k + α k I M  − 1 H H k n k + n d . (20) The ef fecti ve channel matrix betw een the s ource and the destinatio n is deri ved from (20) as H M F − RZ F S D = K X k =1 ρ k G k G H k  G k G H k + α k I M  − 1 H H k H k . (21) 8 Similarly , after Q RD of H M F − RZ F S D and the SIC detection at the destination node, the effecti ve SNR for the m th data stream of M F-RZF relay beamform ing is obtain ed as S N R M F − RZ F m = ( P / M ) ˜ r 2 m,m  K P k =1     ρ k  Q M F − RZ F S D  H A k  m    2  σ 2 1 + σ 2 2 , (22) where A k = G k F M F − RZ F k . T erm ˜ r m,m is the m th diagon al entry of th e rig ht upper tri angular matrix R M F − RZ F S D deriv ed from QRD operation of H M F − RZ F S D like (14). And ρ k of the MF-RZF relay beamforming is given by substituti ng (19) into equation (11). Finally , the er god ic capacity of a dual-hop MIMO relay network with relay beamforming can be deri ved by summing up the data rate of all the streams as C = E { H k , G k } K k =1 ( 1 2 M X m =1 log 2 (1 + S N R m ) ) , (23) where S N R m refers t o t he effecti ve SNR in (18) or (22). From the cut-set theorem in network information theory [6], th e upper bound capacity of the MIMO relay networks is C upper = E { H k } K k=1 ( 1 2 log det I M + P M σ 2 1 K X k=1 H H k H k !) . (24) D. Compu tational complexity analysis and r emarks In spite of no addi tional signal p rocessing at the destination, referenced schemes in [14] implement QR decompositi on o f matrices at each relay node actually . More precisely , for QR- P-QR scheme i n [14], each backward channel H k and forw ard channel G H k should ha ve a QRD operation. Each relay nod e has twice QRD operations of N × M complex matrix. Therefore, it costs 2 K times of QRD ( N × M complex matrix) for QR-P-QR scheme. For QR-P-ZF scheme, it still needs to imp lement K times of QRD of the N × M matrix. When it com es t o our schemes, for both MF and MF-RZF relay beamforming, the whole signal processin g sp end onl y once QRD at t he destinatio n n ode. Moreover , in our design the QRD is operated on t he effe ctiv e channel matrix H S D between the source and the destinati on. The dimensi on of the complex matrix for QR D is M × M , wh ich is fr ee from the antenna num ber N and the relay num ber K . Obviously , the proposed schemes reduce the computational complexity sharply compared with the referenced methods in [14]. Additionall y , in order to guarantee the effecti ve channel matrix to take the right lower triangular form, the phase cont rol and o rdering m atrix has to be u sed i n the relay beamformers in [14]. Th is 9 results in a performance loss in terms of the network capacity . While the QRD of the com pound ef fecti ve channel at the dest ination proposed in this paper makes the relay b eamformer d esign more flexible, because the ef fecti ve channel matrix i s no t necessary t o be a triangular form. I V . S I M U L A T I O N R E S U L T S In t his section, num erical sim ulations are carried out in order to verify the performance superiority of the propo sed relay b eamforming st rategies. W e compare the ergodic capacities of MF and MF-RZF relay beamform ers wit h Q R-P-QR, QR-P-ZF p roposed in [14 ] and t he con ventional AF relaying s cheme in the du al-hop MIMO relay networks. The capacity upper bound is also taken into account as a baseline. All the schem es are compared under th e condition of various system parameters including total n umber of relay nodes and power constraints at source and relay nodes, i.e., different PNR ( P /σ 2 1 , the SNR o f BC), and dif ferent QNR ( Q k /σ 2 2 , the SNR of FC). For simplicity , t he exntries of H k and G k are assum ed to be i.i.d. com plex Gaussian with zero mean and unit var iance. Al l th e relay nodes are suppos ed to ha ve the same power constraint Q k = Q ( k = 1 , ..., K ) , and α k = 1 ( k = 1 , ..., K ) , which, within a li mited range, has no signi ficant im pact on the er godic capacity of the MF-RZF relay beamforming. A. Capacity versus T otal Number of R elay Nodes Like in [13], [14], th e capacity comp arisons are given with the increase o f the total nu mber of relay nodes. In order t o illust rate ho w the SNRs of BC and FC have i mpact on the ergodic capacity with various relay beamforming schemes, three differe nt PNR and QNR are t aken into account. Fig. 2 shows the capacities change with K when N = M = 4 , PNR = QNR = 1 0 dB. Apparently , the proposed M F and MF-RZF relay b eamformers outperform th e QR-P-ZF and QR-P-QR relaying schemes in [14] for K > 1 . F or th is moderate PNR and QNR, the MF-RZF beamformer has the best ergodic capacity performance among the five relaying s chemes and approaches to the capacity upper bound. This can be explained as a result that t he MF recei ve beamformer can maxi mize receiv e SNRs at each relay node while the RZF transmit beamformer pre-cancel inter- stream interference before transmi tting the signal to the d estination node. The relative capacity gain s changing wit h the PNR and QNR is demonstrated in Fig. 3 and Fig. 4. It can be seen from F ig. 3 that MF and MF-RZF keep the superiority over other relaying schemes when t he network has low SNR in BC (PNR = 5 dB) and hig h SNR in FC (QNR = 10 20 dB). This is because that t he MF is used as the receiv e beamformer for the first hop channel, showing the advantage of MF against the low SNR conditi on. Furthermore, Fig. 3 shows th at t he capacity gains of MF-RZF scheme over other beamformers become larger , while the performance superiority of MF decreases when compared to the scenario in Fig. 2. Thi s is because th at the MF performance becom es worse with the in crease o f SNR, while the RZF in FC turns to be better . A l ar ger g ap between QR-P-ZF and QR-P-QR beamforming schemes also con firms the adva ntage of ZF being t he transm it beamform er in the high SNR region. W it h th e kno w ledge of the performance characteristics o f MF in low SNR regions and RZF in hi gh SNR regions, the fact ill ustrated in Fig. 4 that ergodic capacity of M F-RZF becomes a little bit smaller t han M F in lo w QNR en vi ronment is reasonable. Finally , in all the three en vi ronments consi dered above, the con ventional AF relaying keeps as a bad relaying strategy . It can be seen that AF can not obtain the distributed array gain since its er g odic capacity does not increase with the total n umber of relay nodes. Th e reason is that, as for the AF relaying, each relay n ode us es the identity matri x as the beamformer wh ich does not uti lize any CSI of both BC and FC. It is also very im portant to in vestigate t he behaviors of all t he relay beamforming schem es when d istributed array gain is unav ailable, i.e., when there is only a si ngle relay node in the network. From Fig. 2 to Fig. 4 , it can be seen that AF relaying is no longer the worst one and becom es acceptable when K = 1 . M eanwhile, the performance adva ntages of the propos ed methods ov er other con ventional schemes v ary from case to case. Look at the ergodic capacities of all the schemes at the poin t of K = 1 in Fig. 3. At this time, the sin gle relay system has low PNR (PNR = 5 dB)and high QNR (QNR = 10 dB). M F-RZF’ s capacity has about 0 . 1 bps loss than QR-P-ZF beamformi ng while M F has 0 . 03 bps gain over QR-P-QR scheme. Howe ver , if the dual-hop network has moderate PNR and QNR (see Fig . 2) or hi gh QNR (see Fig. 4), the MF and MF-RZF still outperform th e schemes proposed i n [14]. For example, when K = 1 , PNR = QNR = 10 dB, the er godic capacity of MF-RZF b eamforming achie ves 0 . 3 bps and 1 . 01 bps gains over QR-P-QR and QR-P-ZF schemes respective ly . As for the MF beamformer , these gains become 0 . 05 bps and 0 . 77 bps . From the above d iscussion, it can be concluded that our proposed relaying schemes are sti ll ef ficient w hen the relay network has no distributed array condit ion and only intranode array gain is a v ailable. It should be noticed that si mplest AF relayi ng has desirable capacity p erformance in this case. Therefore, the AF scheme m ight be re garded as an alternative solution, especially when the network has only on e 11 relay node and moderate SNRs of two-hop channels. B. Capacity versus PNR The er go dic capacity versus t he PNR and QNR is another important aspect to measure the performance of t he prop osed schemes. The performances of MF and MF-RZF linear relaying schemes are shown i n Fig. 5 and Fig. 6. W e set QNR = PNR in Fig. 5, which i s the same as d one in [14]. The er godic capa cities of both MF-RZF and MF relaying strategies grow approximately linearly with th e PNR (and QNR) like t he upper bound and outperform other schemes. In Fig. 6, we ev aluate how t he capacities change with t he PNR by keeping QNR = 10 dB. The two proposed relay beamformers can still achieve much better performance than the con ventional schemes. Ho wev er , the er godic capacities of all the relay beamforming schemes become saturated as the PNR i ncreases. Note that AF scheme can e ven outperform the QR-P-ZF beamforming in the high PNR region in this case. An d capacity upper bo und ke eps growing linearly with PNR since it is determi ned only by the BC conditions as can be verified in equation (24). The result in Fig. 6 illus trates that if t he SNR o f FC keeps u nder certain va lues, s imply increasing t he source transmit power has lim ited impact on the network capacity . V . C O N C L U S I O N A N D F U T U R E W O R K In this paper , two novel relay beamformer desi gn schemes based on MF and RZF tech- niques hav e been derived for a dual-hop MIMO relay network with Amplify-and-Forward (AF ) relaying protocol. The proposed MF and MF-RZF beamformers are constructed jointly with the QR d ecomposition filer at the destinati on node which transforms the ef fectiv e compound channel into a righ t up per triangular form. Consequentl y m ultiple d ata streams can be decoded with the destination SIC detector . Simulation results demonstrate that our proposed schemes outperform the con ventional relay beamformi ng st rategies in the sens e o f the ergodic capacity under various network parameters. Furthermore, the two proposed relay beamforming schemes still ha ve desirable performance when the distri buted array gain is unav ailabl e in the network. Although the proposed relay beamforming strategies ha ve per formance gain over the conv en- tional schemes, the original optimizati on prob lem (8) and (9), the imperfect CSIs of BC and FC, the overhead of the feedback traffi c, and th e optimal α k values of the MF-RZF beamformer are still challenging prob lems that need further research ef fort. 12 R E F E R E N C E S [1] R. Pabst, B. H. W alke, D. C. S chultz, H. Y aniko meroglu, S. Mukherjee, H. V iswanathan, M. L ett, W . Zirwas, M. Dohler , H. Aghv ami, D. D. Falconer , and G. P . Fettweis, “Relay-Based Deployment Concepts for Wireless and Mobile Broadband Radio, ” IEEE Commun. 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Ergodic capacity comparisons versus K ( N = M = 4 , P N R = QN R = 10 dB ) . 15 5 10 15 20 25 30 1 0 5 10 15 20 Total Number of Relay Nodes, K Ergodic Capacity (bit/s/Hz) Upper Bound MF−RZF MF QR−P−ZF QR−P−QR AF Fig. 3. Ergodic capacity comparisons versus K ( N = M = 4 , P N R = 5 dB , Q N R = 20 dB ). 16 5 10 15 20 25 30 1 0 5 10 15 20 25 30 35 Total Number of Relay Nodes, K Ergodic Capacity (bit/s/Hz) Upper Bound MF−RZF MF QR−P−ZF QR−P−QR AF Fig. 4. Ergodic capacity comparisons versus K ( N = M = 4 , P N R = 20 dB , QN R = 5 dB ). 17 0 5 10 15 20 25 30 0 10 20 30 40 50 60 Ergodic Capacity (bit/s/Hz) PNR(QNR) (dB) Upper Bound MF−RZF MF QR−P−ZF QR−P−QR AF Fig. 5. Ergodic capacity comparisons versus PNR (QNR) ( N = M = 8 , K = 10 ). 18 0 10 20 30 40 0 10 20 30 40 50 60 70 Ergodic Capacity (bit/s/Hz) PNR (dB) Upper Bound MF−RZF MF QR−P−ZF QR−P−QR AF Fig. 6. Ergodic capacity comparisons versus PNR ( N = M = 8 , QN R = 10 dB , K = 10 ).