DURING recent decades, multiple-input multiple-output. Multiuser Multi-Hop AF MIMO Relay System Design Based on MMSE-DFE Receiver

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1 1 Mutiuser Muti-op AF MIMO Reay System Design Based on MMSE-DFE Receiver Yang Lv, Student Member, IEEE, Zhiqiang e, Member, IEEE, and Yue Rong, Senior Member, IEEE Abstract To achieve a better ong source-destination distance communication in upin mutiaccess scenarios, we propose a mutiuser muti-hop ampify-and-forward AF mutipe-input mutipe-output MIMO reay technique with noninear minima mean-squared error MMSE-decision feedbac equaization DFE receiver. Under transmission power constraints, this wor focuses on the improvement of reiabiity, meanwhie, doesn t ose the effectiveness or require higher compexity. We demonstrate that the optima structures of reay ampifying matrices ead to a cascading construction for the mean-squared error MSE matrices of respective signa waveform estimations at destination and each reay node. ence, in moderatey high signa-to-noise ratio SNR environment, the intractabe nonconvex optimization probem can be decomposed into easier subprobems for separate optimizations of source precoding and reay ampifying matrices. The source precoding matrix, aong with the decision feedbac matrix, is obtained by an iterative process, which can converge to a Nash point within reasonabe time. As for the reay ampifying matrices, cosed-form water-fiing soutions are derived. Simuation and anaysis resuts show that compared with other existing agorithms, which aso utiize decomposition methods to simpify operations, the proposed agorithms have better MSE and biterror-rate BER performance without increasing the computing time or signaing overhead, thus providing a new step forward in MIMO reay system design. Index Terms MIMO reay, ampify-and-forward, muti-hop reay, mutiuser, MMSE, DFE, mutua information. I. INTRODUCTION DURING recent decades, mutipe-input mutipe-output MIMO wireess communication technique has undergone tremendous deveopment in both academia and industry for its remarabe advantages obtained from mutipexing, diversity, coding or antenna gains 1], 2]. In order to further improve the system reiabiity, reduce the power consumption and expand the networ coverage, extending conventiona singe-hop architecture to more compicated two- or muti-hop ones has aways been regarded as a quite promising strategy, where one or more reay nodes are fixed to forward signas over ong source-destination distance 3], 4]. The ast dozen years has seen a fourishing progress in the fied of MIMO reay system design with considerabe achievements obtained. In genera there are three major types of reay protocos 5], 6]: the simpe and fast ampify-and-forward AF protoco Y. Lv and Z. e are with the Key Laboratory of Universa Wireess Communication, Ministry of Education, Beijing University of Posts and Teecommunications, Beijing , China e-mai: vyangcn@foxmai.com; hezq@bupt.edu.cn. Y. Rong is with the Department of Eectrica and Computer Engineering, Curtin University, Bentey, WA 6102, Austraia e-mai: y.rong@curtin.edu.au /00$00.00 c 2019 IEEE retransmits the received signas in a inear non-regenerative manner 4]; the compicated yet high-performing decodeand-forward DF protoco, contrariwise, regenerates source signas via decoding and recoding operations 7]; and the compress-and-forward CF protoco is in between, offering a worthwhie compexity-performance tradeoff 8]. Thus, for muti-hop systems with constraints on impementation costs and signa processing atency, the AF protoco is a promising soution. Among various AF MIMO reay modes, either fu-dupex FD or haf-dupex D strategy can be appied, depending on whether reay nodes transmit and receive signas simutaneousy or not. The FD strategy, despite maing more efficient use of communication resources, suffers from severe oop-bac interference, and ony under a high canceation quaity, may achieve an advantageous performance 9]. Therefore, we focus on the D strategy in this paper. In a cassic singe-user two-hop system, the direct in can be either negected or considered. For both cases, the system capacity or its upper and ower bounds were discussed in 10]. Given that there may exist a significant mismatch between the true and the estimated channe state information CSI, robust system designs were carried out under such an imperfect channe estimation 11], 12]. Extended from pointto-point reay system, mutipoint-to-mutipoint system enabes a singe MIMO reay node to serve mutipe source-destination pairs concurrenty 13]. According to the capacity scaing aws in 14], instead of singe reay node, mutipe parae reay nodes can be empoyed to expand system capacity. With two source nodes exchanging information bi-directionay through assisting reay nodes, two-way networs were investigated in 15], 16], and the generaized N-way a-cast scenarios aso received attention 17], whie in this paper, we concentrate on one-way upin communication. Inspired from 18], which performed the joint transceiver design for one-hop muticarrier MIMO ins under a unified framewor, the authors in 19] 21] studied two- or mutihop muticarrier AF MIMO reay systems and drew simiar concusions that, for Schur-concave objective functions, the optima source precoding, reay ampifying and minima meansquared error MMSE receiving matrices jointy diagonaize the MIMO channe into a set of parae singe-input singeoutput SISO ones, and the same appies to Schur-convex objectives aong with a rotation of the source matrix. Aiming at 5G Internet of Things IoT, 22] studied the optima routing for muti-hop device-to-device D2D communications. Under

2 2 the interference from Poisson distributed third-party devices, the performance of both D and FD strategies for muti-hop networs was anayzed in 23]. Taing mutiuser issues into consideration under mutiaccess scenarios, with each user having singe antenna, 24] expored the optima cosed-form soution of joint inear fiter design, whie for users equipped with mutipe antennas, iterative agorithms were deveoped in 25] to sove a nonconvex probem. Note that iterative technique requires centraized processing, which invoves a heavy oad of computing time and signaing overhead, especiay for mutiuser muti-hop reay systems. In view of this, using inear MMSE receiver, 26] generaized the decomposabe property, discovered in 27], of the mean-squared error MSE matrix for the signa waveform estimation at the destination node, and proposed simpified agorithms which can be impemented in distributed manners. The approach was further extended to the weighted MMSE WMMSE-based probem in 28], on the basis of the reationship between maxima mutua information MMI and WMMSE objectives. Aiming to improve the quaity of signa reception, the we-nown decision feedbac equaization DFE 29], aso referred to as successive interference canceation SIC 30] or vertica-be aboratories ayered space-time V-BLAST technique 31], successivey cances the inter-symbo interference ISI produced by previousy detected symbos for current decision. Different schemes for impementing the noninear DFE receivers are avaiabe, commony evoving from their inear counterparts, zero-forcing ZF or the aforementioned MMSE. As shown in 32], 33], an MMSE-DFE receiver can provide a significanty better performance than ZF-DFE, particuary under severe channe conditions. Meanwhie, the MMSE-DFE receiver aso has ower compexity than the optima maximum ieihood ML detection 31], and is verified to be information ossess 30], thus preferred in practice. For a basic two-hop mode fitted with this noninear receiver, the wor in 34] designed two cosed-form precoding soutions to be adaptivey seected depending on channe conditions. Using parae MIMO reay nodes, 35] deveoped a joint power oading agorithm. Considering both Schur-convex and Schurconcave objectives, 36] derived the optima source and reay matrices together with feed-forward and feedbac matrices for muti-hop systems. This paper investigates mutiuser muti-hop one-way D AF MIMO reay communication with noninear MMSE-DFE receiver at the destination node. To our nowedge, it s the first time to consider both mutiuser context and MMSE- DFE receiver for muti-hop AF MIMO reay research fied. Under this new system architecture, inspired from the optimizing procedures in 26], 28], 34], 36], we propose two distributed transceiver optimization agorithms. To be specific, by utiizing MSE minimization criterion and transmission power constraints, we first formuate a transceiver optimization probem. Next, through further generaizing the aforesaid MSE matrix decomposition method, we obtain the optima structures of reay ampifying matrices and demonstrate that the MSE matrix for the signa waveform estimation at the destination node can recursivey degenerate into those at each reay node. Foowing that, under moderatey high signato-noise ratio SNR environment, the origina probem can be decomposed into separate subprobems. One is to jointy optimize source precoding and decision feedbac matrices, for which an iterative process is deveoped, abe to converge within reasonabe time. The others aim at designing reay ampifying matrices, where two types of cosed-form waterfiing soutions are derived, with the atter being a simpified version of the former. The proposed agorithms are compared with existing approaches in terms of the system MSE, biterror-rate BER and mutua information MI, as we as the computing time and signaing overhead. We show that the proposed agorithms have a better performance than existing approaches. The rest of this paper is organized as foows. In Section II, we describe the system mode and formuate the optimization probem. The agorithms proposed to optimize the source precoding, reay ampifying and decision feedbac matrices are deveoped in Section III. Foowing that, Section IV gives some comments and reevant appications. Numerica simuations and anaysis resuts are presented in Section V. Finay, Section VI draws a concusion. The foowing operators and notations are used throughout the paper: = represents the phrase "is defined as"; C n, C m n respectivey denote the compex vector and matrix space with superscripts n, m n being their dimensions; r + = maxr, 0 for rea number r, and x, x, x stand for the reciproca, compex conjugate, and moduus of scaar x; T, denote the transpose and ermitian transpose of a vector or matrix; X, X, detx, ranx, trx, X F stand for the inverse, pseudo-inverse, determinant, ran, trace, and Frobenius norm of matrix X; represents sequentia mutipication, i.e., for matrices X, = m,, n, n =m X = X m X n ; X], X],, X] m,n indicate the th coumn vector, the th diagona eement, and the mth row and the nth coumn eement of matrix X; x] 1: denotes a subvector of vector x, containing its first eements; X] 1:, X] 1:,1: stand for submatrices of matrix X, containing its eftmost coumns, and its first rows and first coumns, respectivey; UX] represents a matrix with its ower-trianguar part fied by zeros and stricty upper-trianguar part being the same as that of matrix X; bd denotes a boc diagona matrix composed of the entries in parentheses; I is a th-order identity matrix and 0 m n stands for an m n zero matrix; E ] represents statistica expectation; I; denotes the MI between two random vectors; h, h stand for the differentia entropy and the conditiona differentia entropy of a continuous random vector, respectivey. II. SYSTEM MODEL AND PROBLEM FORMULATION In the AF MIMO reay system under consideration, N u users simutaneousy transmit data towards the MMSE-DFE receiver at the destination node with the aid of L 1 sequentia reay nodes as shown in Fig. 1, where each of the rounded rectanges contains a matrix to be optimized. Considering propagation path oss, we assume ony adjacent nodes can estabish direct communication ins. ence the source signas trave through L hops and need to be ampified for L times before reaching

3 LV et a.: MULTIUSER MULTI-OP AF MIMO RELAY SYSTEM DESIGN BASED ON MMSE-DFE RECEIVER 3 s 1 s Nu M 1 B 1 M Nu B Nu v G 1 1 G Nu N 1 y 1 x 2 F 2 y L-1 N L-1 F L x L L v L y L N L W MMSE-DFE Receiver z + s Detector D s Fig. 1. System mode for N u -user L-hop AF MIMO reay communication with MMSE-DFE receiver. the receiver. Moreover, by adopting the D strategy, each reay node utiizes orthogona channes in time and/or frequency domain for separate signa transmission and reception, thus avoiding oop-bac interference. ere proper frequency reuse methods can be appied to tae fu advantage of the imited channe resources. For i = 1,, N u, there are M i independent data streams transmitted from the ith user via M i antennas, so the tota number of independent data streams from a users is N 0 = Nu i=1 M i. At each source node, the moduated signa vector s i C M i is ineary precoded by B i C M i M i, the source precoding matrix, and the precoded signa vector u i = B i s i is then transmitted to the first reay node, where the received signa vector is given by N u y 1 = G i u i + v 1 = 1 x 1 + v 1 1 i=1 with G i C N 1 M i being the MIMO channe matrix between the ith source node and the first reay node, v 1 C N 1 being the independent and identicay distributed i.i.d. additive white Gaussian noise AWGN vector of the first hop, and 1 = G 1,, G Nu ], x1 = u T 1,, ut N u ] T. 2 ere 1 C N 1 N 0 stands for the equivaent first-hop MIMO channe and x 1 C N 0 contains a users precoded symbos. For convenience in anaysis, we specify F 1 = bd B 1,, B Nu 3 as we as s = s T 1,, st N u ] T 4 where F 1 C N 0 N 0 represents the equivaent boc diagona source precoding matrix and s C N 0 invoves a users moduated symbos. Therefore, we obtain x 1 = F 1 s. 5 Besides, a users are assumed to be symbo-synchronous and the mean power of s is normaized, i.e., E ss ] = I N0. For = 1,, L 1, at the th reay node, the reay ampifying matrix F +1 C N N is adopted to fiter the received signa vector y C N in a inear non-regenerative manner of the AF protoco, hence the output signa vector x +1 C N is given by x +1 = F +1 y 6 where y 1 has aready been expressed in 1 and the other inputs can be simiary written as y = x + v, = 2,, L. 7 ere C N N and v C N are respectivey the th-hop MIMO channe matrix and i.i.d. AWGN vector. As commony practiced, a noise vectors are assumed to satisfy circuar symmetry property with zero mean and identity covariance matrix. From 5, 1, 6, 7, we can rewrite the received signa vector at each reay node and the destination node in a more integrated way as foows, y = A s + v, = 1,, L 8 where A C N N 0 stands for the equivaent MIMO channe across the first hops, i.e., from the moduated source signa side to the th reay node or destination node if = L, and iewise, v C N represents the equivaent additive noise. They are given by and v 1 = v 1, v = A = 1 i F i, = 1,, L 9 i= j ] i F i v j + v, = 2,, L. j=2 i= 10 Moreover, we wor out the covariance matrix of v, C = E v v ], for = 1,, L as C 1 = I N1, j C = i F i j=2 i= i=j F i i ] + I N, = 2,, L, 11 which are positive definite matrices. In this paper, we assume a nodes in the reay system are either static or have reativey ow mobiity. So it is reasonabe to assume quasi-static boc fading channes, where, = 1,, L remains constant over a certain time period before changing to other reaizations. Given these, high-precision CSI estimations can be obtained with sma mismatch against the rea CSI. As requested by our agorithms proposed ater, the first reay node shoud acquire 1, i.e., a G i, i = 1,, N u, as we as 2, whie the foowing reay nodes require ony their respective forward channe matrices. To estimate them, either standard training methods 37] or those specific to reayassisted MIMO channes ie the east-squares based agorithm presented in 38] can be appied.

4 4 On account of the inherent physica property of MIMO channes, a inear non-regenerative MIMO reay system, in order to support N 0 active independent data streams in each reay transmission, shoud satisfy N 0 min{ran 1,, ran L }. 12 Then from the fact that the dimentions of a matrix are greater than its ran, the inequaity N 0 min{n 1,, N L } is further obtained. Besides, noticing that A, = 1,, L represents an equivaent MIMO channe matrix, we shoud aso contro N 0 rana based on the same reason aforesaid, which, together with the restriction of dimensions, rana min{n, N 0 } = N 0, eads to rana = N 0. Foowing that, ranf N 0, = 1,, L is acquired from 9 as we as the right haf of the Syvester inequaity 39]: ranx + rany ranxy 13 min{ranx, rany} where matrices X C m, Y C n. In order to faciitate the anaysis and design of system parameters, we tend to contro ranf = N 0, i.e., to hod the minimum requirement for the transmission of N 0 independent data streams. Lasty, in case any hop aows more than one potentia forward reay nodes, this L-hop communication in shoud be set up by choosing those nodes equipped with enough number of antennas, based on appropriate upper ayer protocos ie the ones in 40], and its specific configurations at physica ayer wi be determined by the agorithms deveoped ater in Section III. To recover source symbos, within the noninear MMSE- DFE receiver at the destination node, the received signa vector first goes through a inear feed-forward fiter, resuting in z = W y L 14 where W C N 0 N L stands for the feed-forward matrix and z = ] T z 1,, z N0 denotes the fitered signa vector. Then for i = 1,, N 0, a weighted inear combination of previous symbo decisions is fed bac and subtracted from z i, foowed by a hard decision detector to successivey determine the current symbo output. ere the order in which the N 0 fitered symbos are detected has a significant infuence on detection performance, and severa ordering agorithms are summarized in 41]. Generay speaing, those symbos received more reiaby shoud be detected earier. In this paper, we detect the N 0 th symbo first and the first symbo ast, based on the assumption of adopting a proper coding scheme whose subcodes at growing symbo indexes are increasingy more powerfu. So we have the N 0 th source symbo estimated as ŝ N0 = z N0, then detected as s N0, foowed by the other source symbos being estimated as ŝ i = z i N 0 j=i+1 d i, j s j, i = 1,, N where d i, j, j = i + 1,, N 0 represents the decision feedbac coefficient empoyed to cance the ISI produced by the jth previousy detected symbo s j from the ith data stream. It ies at the ith row and the jth coumn of the decision feedbac matrix D C N 0 N 0, which is a stricty upper trianguar matrix owing to the detection order set above. Foowing common practice as in 29] 36], 41], etc., we assume the DFE receiver is free of error propagation, i.e., s j = s j. This assumption can be justified by the information theory 42]: there aways exists a coding scheme to achieve an arbitrariy sma probabiity of error for each data stream with any rate beow subchanne capacity, meaning that the infuence of error propagation can be minimized through wedesigned channe coding. ence, via introducing the detected signa vector s = s 1,, s N0 ] T, which is assumed to be equa to s, and the source signa estimator ŝ = ŝ 1,, ŝ N0 ] T, we can reformuate 15 in a matrix form as ŝ = z D s = W y L Ds. 16 Therefore, the estimation error vector e = ŝ s is given by e = W A L D I N0 s + W vl 17 from which we are abe to compute the error covariance, or MSE, matrix E L = E ee ] as E L = W A L D I N0 W AL D I N0 + W CL W. 18 The feed-forward matrix W is schedued to be optimized in the first pace by supposing that the other system parameters, D and F, = 1,, L, have aready been configured. Appying the orthogonaity principe detaied by Theorem 9.1 in 43] for MMSE estimation, we obtain E eyl ] = 0N0 N L, which eads to W = D + I N0 A L A L A L + C L. 19 Now substituting 19 bac into 18 and utiizing the matrix inversion emma 44]: A + XBY = A A X B + Y A X Y A 20 for nonsinguar A and B, we have E L = ] D D + I N0 I N0 A L A L A L + C L AL + IN0 = U I N0 + A L C L A L U 21 where, given for the convenience in mathematica derivations, U = D + I N0 C N 0 N 0 22 is aso caed a decision feedbac matrix, which is an upper trianguar matrix with a its diagona eements equa to 1. To estabish constraints on transmission power, we restrict the power consumed at the ith source node for i = 1,, N u to be no more than budget q i, i.e., tr { E u i ui ]} = tr B i Bi q i, 23 and the power consumed at the 1th reay node for = 2,, L to be no more than budget p, i.e., tr { E x x ]} ] = tr F A A + C F = tr F D F p 24

5 LV et a.: MULTIUSER MULTI-OP AF MIMO RELAY SYSTEM DESIGN BASED ON MMSE-DFE RECEIVER 5 where the matrix D = A A + C C N N, = 1,, L 1 25 is positive definite and essentiay the covariance matrix of y, E y y ]. These hod true as we for = L. At this point, empoying the definitions of U and D L for 19 resuts in W = U A L D L. 26 Utimatey, via minimizing tre L, which is the sum of the MSEs in a data streams for their signa waveform estimations at the destination node, aong with constraints 23 and 24, we can formuate the transceiver optimization probem as ] min tr U I N0 + A L {B i },{F },U C L A L U 27 s.t. tr B i Bi q i, i = 1,, N u 28 tr F D F U] m,n = p, = 2,, L 29 { 0, m > n 1, m = n 30 where sets {B i } = { B 1,, B Nu } and {F } = {F 2,, F L } are made up of source precoding and reay ampifying matrix variabes, respectivey. The probem is a compicated nonconvex probem with matrix variabes, of which the gobay optima soution is intractabe through any practicabe non-exhaustive searching technique. So we tae a step bac and turn to ooing for its ocay optima soutions. ere iterative procedures ie those in 25] may be appied, but their demands for centraized processing impy a high computationa compexity as we as a arge signaing overhead. What inspires us is the MSE matrix decomposition method adopted in 26], where the agorithms presented coud be performed in much more economic distributed manners. Note that 28] extended the resuts in 26] to decompose a weighted MSE matrix. In the foowing, we sha further generaize this decomposition method to the DFEbased MSE matrix E L and accordingy deveop two distributed transceiver design agorithms. III. ALGORITMS DESIGN The proposed transceiver design agorithms have three major steps. Firsty, we show that E L can be decomposed into the cascade of L MSE matrices, which enabes the probem to be decomposed into L subprobems. Then, an iterative process is expoited to jointy optimize {B i } and U. Lasty, cosed-form water-fiing soutions are derived for {F }. A. DFE-based MSE Matrix Decomposition By exporing the optima structures of the reay ampifying matrices, we can recursivey decompose the MSE matrix at the receiver into those at respective reay nodes, which forms a cascading construction and consequenty enabes distributed parameter optimizations. Theorem 1: The optima structures of {F } for the probem are given by F = T U A D, = 2,, L 31 where matrices T C N N 0 remain to be soved. Using 31, E L can be decomposed as E L = U I N0 + F1 1 1 F 1 U L + R + T T 32 with =2 R = U A D A U C N 0 N 0, = 2,, L. 33 ence, through substituting {T } = {T 2,,T L } for {F }, the probem can be equivaenty rewritten as tr U I N0 + F1 1 1 F 1 U min ] {B i }, {T },U L 34 + R + T T =2 s.t. tr B i Bi q i, i = 1,, N u 35 tr T R T p, = 2,, L 36 { 0, m > n U] m,n = 37 1, m = n Proof: See Appendix A. ere an interesting, aso reasonabe, point to note about 31 is that, by referring bac to 26, the optima F can be aternativey rewritten as F = T W, = 2,, L 38 where, at the 1th reay node, specific to the received signa vector y = A s + v, matrix W = U A D can be equivaenty viewed as the feed-forward matrix of the MMSE-DFE receiver, with consistent feedbac matrix U for = 2,, L, and the additiona inear weighting matrix T remains to be determined ater. It is worth mentioning that R is exacty the covariance matrix of W y, i.e., R = W E y y] W. Besides, the first term in 32 is equivaent to the DFEbased MSE matrix of the signa waveform estimation at the first reay node. As each user s transmission power goes up, the Frobenius norm of F1 1 1 F 1 tends towards infinity, hence maing the first term in 32 converge to the zero matrix 0 N0 N 0. For = 2,, L, respectivey, the th hop brings about an MSE increment in 32, i.e., R + T. T ere R can be approximated as UU when SNR is moderatey high, which wi be eaborated in the next paragraph. As SNR increases due to the growth of the transmission power at each source and reay node, the Frobenius norm of T T gets cose to infinity, thereby vanishing that MSE increment. Maing use of the matrix inversion emma 20, we can reformuate R for = 2,, L as R = U A C C A I N0 ] + A C A A C A U = U A C A I N0 + A C A U. 39

6 6 In the case of moderatey high SNR environment where A C F IN0 F, 40 A we have the approximation R UU, irreative to variabes {B i } and {T }, which impies that, provided U is a constant, both of the objective function 34 and constraints can be spit in terms of respective {B i } and {T }. ence, assuming reativey good channe state and a given decision feedbac matrix, we are abe to decompose the probem into separate source precoding matrices optimization probem ] min tr U I N0 + F1 1 1 F 1 U 41 {B i },U s.t. tr B i Bi q i, i = 1,, N u 42 { 0, m > n U] m,n = 43 1, m = n and reay ampifying matrix optimization probems, with each handing one T for = 2,, L as foows ] R + T T 44 min tr T s.t. tr T R T p. 45 ere for simpifying agorithm design, we ony consider optimizing U in the probem 41 43, and directy use the ocay optima {B i } and U in the cacuations of reay ampifying matrices. This practice is equivaent to design the decision feedbac matrix merey within the first hop, which maes sense in that, commony the ins between users and the first reay node are more critica to the system performance than those between other reay or destination nodes, so it is reasonabe to assign this extra assistance to the former. In the next two subsections, we sha focus on soving the probem and the probem 44 45, respectivey. B. Joint Source and Feedbac Matrices Optimization It s worth mentioning that each of the agorithms deveoped in 26] and 28] has handed a degraded counterpart of the probem to optimize source precoding matrices. In particuar, for the case of U = I N0 in 26], via introducing a positive semidefinite matrix and using the Schur compement 45, sec. A.5.5], the probem coud be converted into a convex semidefinite programming SDP probem and thus soved through the interior-point method 45, ch. 11]. When U was repaced by a weighting matrix in 28], with its vaue depending on other variabes, the probem, though unabe to be transformed into a convex one, coud be regarded as the WMMSE probem for a mutiuser singe-hop MIMO system with inear receiver. Note that the probem is harder than that in 28], as U is an unnown upper trianguar matrix with unit diagona eements, whie the weighting matrix in 28] does not have such constraint. Interestingy, the probem can be rewritten as the MMSE probem for a mutiuser singe-hop MIMO system with noninear DFE receiver. Namey, the objective function in 41 satisfies ] tr U I N0 + F1 1 1 F 1 U { ]} = min tr E L y 1 Ds s L y 1 Ds s 46 L = min L tr { ]} E L y 1 U s L y 1 U s where, for system across the first hop in Fig. 1, L C N 0 N 1 acts as the feed-forward fiter of the MMSE-DFE receiver, and D is sti the origina feedbac matrix awaiting to be optimized. To see the equivaence of 46 to 41, et s wor out the MMSE probem 46, which, via utiizing y 1 = A 1 s + v 1 = 1 F 1 s + v 1 from 8 10, can be written as min L tr L 1 F 1 F1 1 + I N 1 L L 1 F 1 U UF 1 1 L + UU ]. 47 Apparenty, with positive definite D 1 = 1 F 1 F1 1 + I N 1 as = 1 in 25, the above is an unconstrained convex probem. The probem 47 can be soved by empoying the we-nown optimaity condition 45, eqn. 4.22] to mae its gradient, i.e., 2 1 F 1 F1 1 + I N 1 L 2 1 F 1 U 46], 47], equa to zero. ence we obtain the foowing soution L = 1 F 1 F1 1 + I N 1 1 F 1 U 48 with fu coumn ran. Then substituting 48 bac into 47 resuts in 41. So we are abe to equivaenty reformuate the probem as ] min tr L 1 F 1 U L 1 F 1 U + L L 49 {B i },U, L s.t. tr B i Bi q i, i = 1,, N u 50 { 0, m > n U] m,n = 51 1, m = n which can be soved by an iterative process described beow. Firsty, we initiaize F 1 via setting its diagona submatrices B i equa to q i /M i I Mi for i = 1,, N u, which maes fu use of the power budget q i. Next, with given F 1, et s introduce the QR factorization of the foowing fu coumn ran matrix ] ] 1 F 1 Q = QR = R 52 I N0 Q where Q C N 1+N 0 N 0 is semi-unitary, i.e., Q Q = I N0, its first N 1 and ast N 0 rows respectivey constitute submatrices Q C N 1 N 0 and Q C N 0 N 0, and the other factor R C N 0 N 0 is a nonsinguar upper-trianguar matrix with a its diagona eements being nonzero or even positive if the Gram-Schmidt orthogonaization method 48] is appied. Thus we have 1 F 1 = QR, Q = R. 53 Via utiizing 52 and 53, we are abe to derive the optima U and L as foows.

7 LV et a.: MULTIUSER MULTI-OP AF MIMO RELAY SYSTEM DESIGN BASED ON MMSE-DFE RECEIVER 7 Theorem 2: Based on the QR factorization 52 with given F 1, the optima feedbac matrix U and matrix L can be expressed as U = D R R, L = QD R 54 where D R C N 0 N 0 stands for a diagona matrix extracting a the diagona eements from matrix R. Proof: See Appendix B. Expression 54 is inspired by Theorem 1 in 36], which aims at the MMSE-DFE receiver design of a singe-user mutihop MIMO reay system, whie here, Theorem 2 is specific to that of a mutiuser singe-hop MIMO system, instead. Then, with U and L obtained in the current iteration, {B i } can be optimized by soving ] min tr L 1 F 1 U L 1 F 1 U 55 {B i } s.t. tr B i Bi q i, i = 1,, N u 56 which is derived directy from the probem ere, in order to faciitate anaysis, we define Z = L 1 C N 0 N 0, foowed by introducing submatrices Z i C N 0 M i and U i C N 0 M i for i = 1,, N u, which are respectivey composed of M i coumns in Z and U, from the i j=0 M j + 1 th to the ij=0 M j th one, with M0 set equa to 0. Then, the objective function in 55 can be reformuated as N u tr Z i B i U i Z i B i U i ] 57 i=1 which enabes us to decompose the probem into N u subprobems, each reated to one user s precoding matrix B i for i = 1,, N u as foows min tr Z i B i U i Z i B i U i ] 58 B i s.t. tr B i Bi q i. 59 This convex quadraticay constrained quadratic programming QCQP probem can be soved, with the aid of a Lagrange mutipier λ i R, via Karush-Kuhn-Tucer KKT optimaity conditions 45, sec ], which are necessary and sufficient, appearing here as ] tr B i Bi q i 0, λ i tr B i Bi q i = 0, 60 λ i 0, Zi Z i + λ i I Mi B i = Zi U i. Depending on the vaue of λ i, 60 can be handed under two separate cases. As λ i = 0, we have B i = Zi Z i Z i U i, which is just the optima soution, provided that it aso satisfies the constraint in 59. Otherwise, as λ i > 0, we have B i = Zi Z i + λ i I Mi Z i U i 61 where λ i is determined by substituting 61 into the equation tr B i Bi qi = 0, with the resut abe to be soved through the bisection method 49]. During the above iterative process, each conditiona update of respective U, L and {B i } soves a convex optimization probem, which indicates monotonic convergence towards a Nash point as iterations progress 50]. C. Reay Ampifying Matrices Optimization On the basis of U and {B i } obtained in Subsection III-B, a cosed-form water-fiing soution of each probem with respect to variabe T for = 2,, L can be derived as foows. ere, through introducing T = T R 1/2 C N N 0, we first reformuate the probem as min T tr s.t. tr R 1/2 T T I N0 + T T R 1/2 ] 62 p. 63 Now et us introduce the eigenvaue decomposition EVD of and that of R as = V Λ V, R = P Σ P 64 where V C N N and P C N 0 N 0 are unitary matrices, Λ C N N and Σ C N 0 N 0 are diagona eigenvaue matrices with their diagona eements both sorted in increasing orders. From Lemma 2 in 26], the singuar vaue decomposition SVD structure of the optima T for the probem is given by T = V,1 Ω P 65 where V,1 C N N 0 is a submatrix of V, containing its rightmost N 0 coumns, and Ω C N 0 N 0 is an unnown diagona singuar vaue matrix. Thus we have T = T R /2 = V,1 Ω P P Σ /2 P = V,1 P. 66 ere = Ω Σ /2 C N 0 N 0 remains to be optimized. Further, for j = 1,, N 0, we denote the jth diagona eement of, Σ by δ, j, σ, j, respectivey, and each of the N 0 increasing diagona eements ying at the right-bottom corner of Λ by λ, j. Apparenty, δ, j 0, σ, j > 0, and on account of ran N 0, λ, j > 0. ence by substituting 66 bac into 44 45, this matrix optimization probem can be equivaenty rewritten as the foowing convex power oading probem with respect to the squares of scaar variabes δ, j, min {δ 2, j } s.t. N 0 j=1 N 0 j=1 1 σ, j + δ2, j λ, j 67 δ 2, j σ, j p 68 where { δ, 2 j} = { δ,1 2,, } δ2,n 0. Simiar to Exampe 5.2 in 45], via appying KKT optimaity conditions, the probem has a cosed-form waterfiing soution given by δ 2, j = 1 λ, j λ, j µ σ, j 1 σ, j +, j = 1,, N ere µ > 0 is a Lagrangian mutipier satisfying N + 0 σ, j λ, j 1 = p. 70 λ, j µ σ, j σ, j j=1

8 8 Its eft-hand side can be reformuated as N 0 σ, j λ, j µ λ, j j=1 which is a piecewise-inear non-decreasing function of 1/ µ, with turning points at 1/ λ, j σ, j for j = 1,, N 0. So there is a unique soution meeting 70, abe to be readiy determined. Eventuay, via combining both 31 and 66, we obtain our resutant reay ampifying matrices F = V,1 P U A D, = 2,, L. 72 Moreover, within moderatey high SNR range, inspired from 39 40, we can simpify the optimization process of {F } by approximating R as UU, regardess of the index, which is sti positive definite. Thus the probem for = 2,, L becomes min T tr s.t. tr { UU + T ] T UU T ] } T 73 p. 74 In a simiar way, we first introduce the EVD: UU = PΓP, where P C N 0 N 0 is a unitary matrix, and Γ C N 0 N 0 is a diagona eigenvaue matrix whose diagona eements, denoted by γ j for j = 1,, N 0, are positive and sorted in increasing order. Foowing that, by using Lemma 2 in 26], the SVD structure of the optima T for the probem can be derived as T = V,1 Θ P. 75 ere variabe Θ C N 0 N 0 is a diagona singuar vaue matrix, with its jth nonnegative diagona eement denoted by θ, j for j = 1,, N 0. Substituting 75 bac into the probem and appying KKT optimaity conditions, we obtain the foowing cosed-form water-fiing soution + θ, j = 1 λ, j 1 1/2 λ, j υ γ j γ j, j = 1,, N 0 76 where υ > 0 is the Lagrangian mutipier, determined by N + 0 γ j λ, j 1 = p. 77 λ, j υ γ j γ j j=1 At ast, via combining both 31 and 75, a new design of {F } is given beow, F = V,1 Θ P U A D, = 2,, L. 78 IV. COMMENTS AND APPLICATIONS So far, we have finished the theoretica derivations of our two newy proposed agorithms. For convenience, we denote the agorithm which computes the reay ampifying matrices as 72 by the MMSE-DFE agorithm, whie its simpified version based on 78 is named as the SMMSE-DFE agorithm. Their compete procedures of transceiver design are summarized in Tabe I, where we initiaize the feedbac matrix U as I N0 to indicate the origina inear receiver, and the precoding matrices TABLE I PROCEDURES OF SMMSE-DFE ALGORITMS 1 Initiaize U 0 = I N0 and B 0 i = q i /M i I Mi, i = 1,, N u. Construct F 0 1 as 3. Specify toerance ɛ > 0. Set counter n = 0. a Let n = n + 1. Compute U n, L n as 54 based on fixed F n 1. b For i = 1,, N u : Sove probem to obtain B n i based on fixed U n, L n ; Construct F n 1 as 3. c Continue to Step 1a unti U n U n F + F n 1 F n 1 / 2N0 F < ɛ. 2 For = 2,, L, with fixed U n, F n 1 : MMSE-DFE: Acquire via 69 70; Compute F as 72. SMMSE-DFE: Acquire Θ via 76 77; Compute F as Compute W as 26 and D = U n I N0. B i as q i /M i I Mi to eep the source signas unchanged with maximum permissibe transmission power q i. For the oop of joint U and F 1 optimization, L acts as an auxiiary variabe, the variabe with superscript n denotes that it is at the nth iteration, and toerance ɛ taes a sma enough positive vaue, ony ower than which the convergence error, expressed as U n U n F + F n 1 F n / 1 2N0 F, 79 is acceptabe. 79 is essentiay an average of the differences of a entries in both U and F 1 over two consecutive iterations, which wors we through testing. Simiar to the agorithms in 26], the proposed agorithms can aso be impemented in distributed manners, with most system parameters optimized ocay. For detais about the MMSE-DFE agorithm, at the first reay node, we carry out the iterative process of joint source and feedbac matrices design, with the resuting B i for i = 1,, N u sent bac to each respective user. Next, from A 1, D 1 and U, R 2 is computed so as to obtain the first reay ampifying matrix F 2, after which we prepare A 2, D 2 and deiver them aong with U to the second reay node. Then for = 2,, L 1, based on A, D and U forwarded by the 1th reay node, the th reay node, in turns, computes R +1, obtains F +1, prepares A +1, D +1 and deivers them as we as U to the next node in need. During the agorithm execution, preparations for both A and D are faciitated by their recursive reationships. In particuar, from 9 and 11, we have A = F A, 80 C = F C F + I N 81 for = 2,, L, and by further considering definition 25, we obtain D = F D F + I N, = 2,, L. 82 For the SMMSE-DFE agorithm, R is simpified and fixed to UU despite the index, which, compared with 33, eaves out one matrix inversion and severa matrix mutipications. Particuary, we may aso perform the computation of R = UU and its EVD for ony one time in this case, with the resuts, Γ and P, achieved at the first reay node and forwarded to subsequent ones, together with U and other parameters, to

9 LV et a.: MULTIUSER MULTI-OP AF MIMO RELAY SYSTEM DESIGN BASED ON MMSE-DFE RECEIVER 9 be directy used there. ence the SMMSE-DFE agorithm may significanty reduce the computationa compexity. Noteworthiy, in view of notationa convenience, this paper focuses on narrow-band singe-carrier systems, whie the generaized resuts for broadband muticarrier reay systems can be directy obtained, foowing either subcarrier-independent or subcarrier-cooperative approaches as anayzed in 19]. In this paper, we mae comparisons among four agorithms, incuding those two proposed here, the first agorithm designed in 26], denoted by the LMMSE agorithm since it uses inear receiver, and the first agorithm in 28], named as the WMMSE agorithm for its constructed objective function. The reasons for seecting them are that they a target at mutiuser mutihop AF MIMO reay systems and utiize certain MSE matrix decomposition methods. In the LMMSE agorithm, {B i } are obtained via soving a convex SDP probem through the wedeveoped interior-point method, and simiar to our treatments, {F } are optimized hop-by-hop in a distributed manner, except no need to optimize and deiver U. The criterion empoyed by the WMMSE agorithm is to maximize the MI between source and destination nodes, which can be transformed into a weighted MSE minimization probem with definite weighting matrix. Two nested iterative oops constitute its optimization procedures, with the inner oop optimizing {B i } and the outer oop focusing on {F } as we as the weighting matrix, which require centraized processing. Therefore, the WMMSE agorithm has a higher compexity than the other three distributed approaches, among which the LMMSE agorithm owns the owest compexity as it ony needs to optimize the source and reay matrices. Compared with the LMMSE and WMMSE agorithms, the utiization of DFE receivers may further reduce the system MSE, with the SMMSE-DFE agorithm sighty inferior to the MMSE-DFE agorithm. Section V wi eaborate their performance comparisons in more detai via numerica simuations. We anticipate that the newy proposed agorithms coud be appicabe in a few practica scenarios, where fixed reay nodes hep to deiver mutipe users aggregate data streams over ong distances. Typica appication exampes incude: the muti-hop wireess bachau networ for mutimedia broadband access 51], the muti-hop wireess sensor networ for mine tunne monitoring 52], the muti-hop underwater acoustic networ for reiabe communication 53], etc. Besides, severa industry standards tae into consideration the reay issues as we, e.g., IEEE j ] specifying both physica and medium access contro MAC ayer enhancements to enabe reay operations for high-speed wireess access systems. Aso worth mentioning, the miimeter-wave distribution networ MDN 55] may benefit from our wor to estabish dense gigabitspeed bachau. Note that the IEEE P Tas Group AY has recenty made MDN a use case and been adding features to support it 56]. V. NUMERICAL SIMULATIONS This section presents the performance of the MMSE-DFE and SMMSE-DFE agorithms in optimizing mutiuser mutihop AF MIMO reay systems through numerica simuations. TABLE II EXAMPLES OF SYSTEM SETTINGS Ex. N u {M i } L {N } Q db 1 2 {3, 2} 2 {8, 7} {2, 3, 2} 3 {8, 10, 9} {2, 3, 2} 5 {8, 10, 9, 11, 8} 25 ere we compare them with the aforementioned LMMSE and WMMSE agorithms over severa criteria, incuding MSE, BER, MI and computationa compexity. Inte R Core TM 2 Duo Processor E ] is empoyed to run MATLAB R R2015b under 32-bit Windows 10 operating system. When programming to sove the SDP probem in the LMMSE agorithm, we utiize CVX 58], a MATLAB-based software for convex optimization. Foowing Monte Caro method, the simuation resuts shown beow are set to be averaged over 1000 independent channe reaizations, where a fat Rayeigh fading environment is assumed, with a the entries in channe matrices having zero means. Besides, to normaize the impact from the number of transmitting antennas at each source and reay node upon the received SNR at the destination node, we configure the variances of entries in G i as 1/M i, i = 1,, N u, and those in as 1/N, = 2,, L. The distribution of noise v for = 1,, L is subject to CN 0, I N, denoting a circuary symmetric compex Gaussian CSCG random vector with zero mean and identity covariance matrix I N. Without oss of generaity, we assume unified transmission power budgets for both source and reay nodes, i.e., q i = Q, i = 1,, N u and p = P, = 2,, L. Note that the propagation path oss is impicity considered through P and Q, as we normaize the noise power to be unit. Based on severa trias, 10 3 may act as a suitabe vaue of the toerance ɛ to stop the iterations in Tabe I. For respective performance indicators versus variabe P, comparisons among the four agorithms are impemented over three different exampes of system settings as isted in Tabe II, where "Ex." is short for the word "Exampe", {M i } = { M 1,, M Nu } and {N } = {N 1,, N L }. A. MSE Performance For a data streams, the arithmetic average of their MSE is computed as tre L /N 0. Fig. 2 shows the MSE comparisons among a four agorithms under different exampes, with P ranging from 0 db to 60 db. It s verified by a three exampes that, when the vaue of P is reativey ow, signifying a poor SNR environment where to decompose the probem into subprobems and is not suitabe, our two agorithms perform worse than the others, whie under moderatey high SNR environment, in contrast to the LMMSE and WMMSE agorithms, we observe much enhanced performance of both MMSE-DFE and SMMSE-DFE agorithms. Moreover, the MSE curves of the two proposed agorithms amost overap with each other so that we can substitute the atter for the former with ony a negigibe performance degradation. Remaraby, in each curve, there s

10 Ex. 1: LMMSE Ex. 1: MMSE-DFE Ex. 1: SMMSE-DFE Ex. 1: WMMSE Ex. 2: LMMSE Ex. 2: MMSE-DFE Ex. 2: SMMSE-DFE Ex. 2: WMMSE 10 0 Ex. 3: LMMSE Ex. 3: MMSE-DFE Ex. 3: SMMSE-DFE Ex. 3: WMMSE MSE 10-1 MSE P db a Fig. 2. MSE versus P performance comparisons for a Ex. 1-2 and b Ex P db b N u 2 3 Users TABLE III TE AMMING CODES IN USE LMMSE/WMMSE agorithms SMMSE-DFE agorithms 1 7, 4 15, , 4 7, 4 1 7, 4 31, , 4 15, , 4 7, 4 aways a rapid fa of MSE as P varies in medium range, which is mainy attributed to the growth of SNR, as we as the high spatia diversity order obtained by the underoaded utiization of avaiabe data streams, since we have N 0 = N u i=1 M i < N for = 1,, L in Tabe II. Besides, a the simuations are executed over fixed Q, which restricts the performance to be further improved once P becomes arge enough. This saturation effect can be readiy noticed in Fig. 2a for P greater than 40 db. Regarding each agorithm, Ex. 2 aways underperforms Ex. 1 owing to the increased number of users, which adds system oad, and the increased number of hops, which brings in more noise. Meanwhie, Ex. 3 has the maximum numbers of both users and hops, eading to the highest MSE among the three exampes for P ower than approximatey 30 db. owever, its performance turns out to be the best after P exceeds 40 db, thans to the raising of Q from 20 db to 25 db. B. BER Performance ere we iustrate the BER performance of the proposed agorithms under two different simuation schemes in Fig. 3 and Fig. 4, respectivey. The first scheme foows the assumption of no error propagation in DFE receivers to see the idea agorithm performance without the aid of channe coding. For each channe reaization, ten thousand QPSK symbos per data stream are transmitted through the channe, then are demoduated with the resuting error bits counted up for computing the average BER. Remaraby, in Fig. 3, it can be seen that the proposed agorithms have a better BER performance than existing approaches, e.g., for Ex. 2 at P = 30 db, the SMMSE- DFE agorithms outperform the others by even more than three orders of magnitude. Moreover, via observing both Fig. 2 and Fig. 3, one can recognize that an agorithm having a better MSE aso has a ower system BER, indicating that MSE is a sensibe design criterion. We can aso observe that the SMMSE-DFE agorithm has ony a negigibe higher BER than the MMSE-DFE agorithm. The second simuation scheme, which is coser to a practica communication system, adopts amming codes 59] and considers the error propagation in DFE receivers, i.e., to feed bac the QPSK symbos regenerated from previousy decoded information bits for each data stream. The simuation resuts in Fig. 4 are obtained by transmitting twenty thousand information bits per data stream through the system for each channe reaization. As shown in Tabe III, for the SMMSE-DFE agorithms, in order to minimize the error propagation from earier detected symbos to ater detected ones, we resort to the idea of unequa error protection UEP 41], 60] and empoy increasingy more powerfu amming codes from the first to the ast user. Besides, for a the users in the LMMSE and WMMSE agorithms, we empoy the 7, 4 amming code, which is the most powerfu code used for the SMMSE-DFE agorithms. Namey, the channe coding for the LMMSE and WMMSE agorithms is more advantageous. Even so, within

11 LV et a.: MULTIUSER MULTI-OP AF MIMO RELAY SYSTEM DESIGN BASED ON MMSE-DFE RECEIVER BER 10-3 BER Ex. 1: LMMSE Ex. 1: MMSE-DFE Ex. 1: SMMSE-DFE Ex. 1: WMMSE Ex. 2: LMMSE Ex. 2: MMSE-DFE Ex. 2: SMMSE-DFE Ex. 2: WMMSE P db a Ex. 3: LMMSE Ex. 3: MMSE-DFE Ex. 3: SMMSE-DFE Ex. 3: WMMSE P db Fig. 3. BER versus P performance comparisons for a Ex. 1-2 and b Ex. 3 with no error propagation in DFE receivers and no channe coding. b BER 10-3 BER Ex. 1: LMMSE Ex. 1: MMSE-DFE Ex. 1: SMMSE-DFE Ex. 1: WMMSE Ex. 2: LMMSE Ex. 2: MMSE-DFE Ex. 2: SMMSE-DFE Ex. 2: WMMSE P db a Ex. 3: LMMSE Ex. 3: MMSE-DFE Ex. 3: SMMSE-DFE Ex. 3: WMMSE P db Fig. 4. BER versus P performance comparisons for a Ex. 1-2 and b Ex. 3 with error propagation in DFE receivers and amming codes. b reativey high range of P in Fig. 4, the BER performance of our two proposed agorithms is sti better than that of the LMMSE and WMMSE agorithms. It can be aso observed that as the vaue of P is ow, e.g., as P < 19 db for Ex. 2, our agorithms perform sighty worse than the others, but the disadvantage is not obvious. Thus overa, the SMMSE-DFE agorithms, despite suffering from the error propagation, have an outstanding BER performance. C. MI Performance The MI performance is aso an important indicator used for evauating the effectiveness of data transmission. Note that the capacity of a muti-hop MIMO reay channe with arbitrary signa processing methods adopted at each reay node is sti a chaenging probem under research. Nevertheess, aiming at the specific system mode as shown in 8 with = L, i.e., y L = A L s+ v L, which is estabished on the basis of AF protoco and D strategy, we can yet derive the upper boundary MI between

12 MI b/s/z Ex. 1: LMMSE Ex. 1: MMSE-DFE Ex. 1: SMMSE-DFE Ex. 1: WMMSE Ex. 2: LMMSE Ex. 2: MMSE-DFE Ex. 2: SMMSE-DFE Ex. 2: WMMSE P db a Fig. 5. MI versus P performance comparisons for a Ex. 1-2 and b Ex. 3. MI b/s/z Ex. 3: LMMSE Ex. 3: MMSE-DFE Ex. 3: SMMSE-DFE Ex. 3: WMMSE P db b source and destination nodes by assuming CSCG input, I y L ; s = og det I N0 + A L C L A L. 83 This resut can be directy obtained through a simiar approach as that in 61], for competeness, detais of which are shown in Appendix C. Observed from Fig. 5, the four agorithms have simiar MI performance, but within the cose-up views, we are sti abe to see the sma differences. ere the WMMSE agorithm and the LMMSE agorithm respectivey own the best and the worst behavior, whie the curves of the SMMSE-DFE agorithms ying in the midde and approximating to that of the WMMSE agorithm, so the DFE-based agorithms aso ensure reativey good MI performance. Note that we compute the tota MI for a data streams. ence, from Ex. 1 to Ex. 2, despite the increased number of hops, which woud resut in the decine of MI, we eventuay obtain a higher MI due to the increase of N 0 from 5 to 7. Besides, compared with Ex. 2, the system in Ex. 3 has the same number of data streams. owever, the atter system has more hops as we as higher transmission power Q for each user, which have opposite effects on the system MI. Simiar to the case in Fig. 2, ony at about 35 db for the vaue of P, the curves of Ex. 3 start to exceed those owned by Ex. 2, i.e., the second setting pays a dominant roe herefrom. D. Computing Time and Signaing Overhead To refect the computationa compexity of each agorithm, which depends on the number of iterations that is hard to be predicted beforehand, we record the execution time instead as shown in Tabe IV. Note that the unit of time adopted by its fourth coumn is miisecond ms, and our simuation patform can ony be accurate to 1 microsecond µs, hence the digits at the ast decima pace of this coumn are not significant, but the outcomes of averaging operations. ere the first three distributed agorithms optimize {B i } through iterative oops, which are irreevant to variabe P, then directy wor out F for = 2,, L one after another, whie the WMMSE agorithm adopts nested inner and outer oops to jointy optimize {B i } and {F }, with both of their time costs strongy depending on the vaue of P. Consequenty, for the WMMSE agorithm, the entries in Tabe IV are presented in ranges, rather than averages. Among those three agorithms whose computing time is insensitive to P, the LMMSE agorithm is the fastest, and as the numbers of users and hops increase, its high efficiency over the other agorithms becomes even more apparent. For our two proposed agorithms, their respective oops to obtain both {B i } and U are exacty the same and thus consume equa time, whie the distinctions are in the procedures to obtain {F }, where, since the SMMSE-DFE agorithm ony needs to compute the EVD of R = UU for one time, it has ess matrix operations and, therefore, is faster than the MMSE-DFE agorithm. It s worth mentioning that, towards a three distributed agorithms, the first reay node undertaes the heaviest amount of computations, whie the tass of the other reay nodes are much easier. This shoud be considered in resource aocation. As to the issue on signaing overhead, detaied anaysis has aready been presented in Section IV, so here we intend to mae a brief summary. The WMMSE agorithm requires centraized processing, which has a high signaing overhead to coect and deiver system parameters, whie the other three agorithms are abe to be impemented in distributed manners, hence having ighter signaing overhead. Among them, the LMMSE agorithm is the most economic, as the DFE-based