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A unified framework for optimizing linear nonregenerative multicarrier MIMO relay communication systems
 IEEE TRANS. SIGNAL PROCESS
, 2009
"... In this paper, we develop a unified framework for linear nonregenerative multicarrier multipleinput multipleoutput (MIMO) relay communications in the absence of the direct source–destination link. This unified framework classifies most commonly used design objectives such as the minimal meansqu ..."
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Cited by 94 (50 self)
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In this paper, we develop a unified framework for linear nonregenerative multicarrier multipleinput multipleoutput (MIMO) relay communications in the absence of the direct source–destination link. This unified framework classifies most commonly used design objectives such as the minimal meansquare error and the maximal mutual information into two categories: Schurconcave and Schurconvex functions. We prove that for Schurconcave objective functions, the optimal source precoding matrix and relay amplifying matrix jointly diagonalize the source–relay–destination channel matrix and convert the multicarrier MIMO relay channel into parallel singleinput singleoutput (SISO) relay channels. While for Schurconvex objectives, such joint diagonalization occurs after a specific rotation of the source precoding matrix. After the optimal structure of the source and relay matrices is determined, the linear nonregenerative relay design problem boils down to the issue of power loading among the resulting SISO relay channels. We show that this power loading problem can be efficiently solved by an alternating technique. Numerical examples demonstrate the effectiveness of the proposed framework.
Optimal Beamforming for TwoWay MultiAntenna Relay Channel with Analogue Network Coding
, 2009
"... This paper studies the wireless twoway relay channel (TWRC), where two source nodes, S1 and S2, exchange information through an assisting relay node, R. It is assumed that R receives the sum signal from S1 and S2 in one timeslot, and then amplifies and forwards the received signal to both S1 and S ..."
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Cited by 90 (6 self)
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This paper studies the wireless twoway relay channel (TWRC), where two source nodes, S1 and S2, exchange information through an assisting relay node, R. It is assumed that R receives the sum signal from S1 and S2 in one timeslot, and then amplifies and forwards the received signal to both S1 and S2 in the next timeslot. By applying the principle of analogue network coding (ANC), each of S1 and S2 cancels the socalled “selfinterference ” in the received signal from R and then decodes the desired message. Assuming that S1 and S2 are each equipped with a single antenna and R with multiantennas, this paper analyzes the capacity region of the ANCbased TWRC with linear processing (beamforming) at R. The capacity region contains all the achievable bidirectional ratepairs of S1 and S2 under the given transmit power constraints at S1, S2, and R. We present the optimal relay beamforming structure as well as an efficient algorithm to compute the optimal beamforming matrix based on convex optimization techniques. Lowcomplexity suboptimal relay beamforming schemes are also presented, and their achievable rates are compared against the capacity with the optimal scheme.
Optimality of diagonalization of multihop MIMO relays
 IEEE Trans. Wireless Commun
"... Abstract—For a twohop linear nonregenerative multipleinput multipleoutput (MIMO) relay system where the direct link between source and destination is negligible, the optimal design of the source and relay matrices has been recently established for a broad class of objective functions. The optima ..."
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Cited by 45 (29 self)
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Abstract—For a twohop linear nonregenerative multipleinput multipleoutput (MIMO) relay system where the direct link between source and destination is negligible, the optimal design of the source and relay matrices has been recently established for a broad class of objective functions. The optimal source and relay matrices jointly diagonalize the MIMO relay system into a set of parallel scalar channels. In this paper, we show that this diagonalization is also optimal for a multihop MIMO relay system with any number of hops, which is a further generalization of several previously established results. Specifically, for Schurconcave objective functions, the optimal source precoding matrix, the optimal relay amplifying matrices and the optimal receiving matrix jointly diagonalize the multihop MIMO relay channel. And for Schurconvex objectives, such joint diagonalization along with a rotation of the source precoding matrix is also shown to be optimal. We also analyze the system performance when each node has the same transmission power budget and the same asymptotically large number of antennas. The asymptotic analysis shows a good agreement with numerical results under a finite number of antennas. Index Terms—MIMO relay network, multihop relay, linear nonregenerative relay, majorization.
Grassmannian Beamforming for MIMO AmplifyandForward Relaying
"... Abstract — In this paper, we consider the beamforming codebook design problem for the halfduplex MIMO amplifyandforward relay channel with Rayleigh fading. The analysis is divided into two steps. First, we present the optimal beamforming scheme with full channel state information (CSI) and derive ..."
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Cited by 44 (2 self)
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Abstract — In this paper, we consider the beamforming codebook design problem for the halfduplex MIMO amplifyandforward relay channel with Rayleigh fading. The analysis is divided into two steps. First, we present the optimal beamforming scheme with full channel state information (CSI) and derive the optimal source and relay beamforming vectors. Next, we consider the beamforming problem with receiver CSI only and provide a beamforming vector quantization scheme. Based on the statistics of the optimal beamforming vectors, we show that Grassmannian codebooks minimize the upper bound for SNR loss caused by quantization, and therefore these codebooks are appropriate choices for quantizing the optimal beamforming vectors. The efficiency of the Grassmannian codebooks is verified by simulation results.
Robust Joint Design of Linear Relay Precoder and Destination Equalizer for DualHop AmplifyandForward MIMO Relay Systems
"... Abstract—This paper addresses the problem of robust linear relay precoder and destination equalizer design for a dualhop amplifyandforward (AF) multipleinput multipleoutput (MIMO) relay system, with Gaussian random channel uncertainties in both hops. By taking the channel uncertainties into acc ..."
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Cited by 25 (5 self)
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Abstract—This paper addresses the problem of robust linear relay precoder and destination equalizer design for a dualhop amplifyandforward (AF) multipleinput multipleoutput (MIMO) relay system, with Gaussian random channel uncertainties in both hops. By taking the channel uncertainties into account, two robust design algorithms are proposed to minimize the meansquare error (MSE) of the output signal at the destination. One is an iterative algorithm with its convergence proved analytically. The other is an approximated closedform solution with much lower complexity than the iterative algorithm. Although the closedform solution involves a minor relaxation for the general case, when the column covariance matrix of the channel estimation error at the second hop is proportional to identity matrix, no relaxation is needed and the proposed closedform solution is the optimal solution. Simulation results show that the proposed algorithms reduce the sensitivity of the AF MIMO relay systems to channel estimation errors, and perform better than the algorithm using estimated channels only. Furthermore, the closedform solution provides a comparable performance to that of the iterative algorithm. Index Terms—Amplifyandforward (AF), equalizer, minimum meansquareerror (MMSE), multipleinput multipleoutput
Optimal linear nonregenerative multihop MIMO relays with MMSEDFE receiver at the destination
 IEEE Trans. Wireless Commun
, 2010
"... Abstract—In this paper, we study multihop nonregenerative multipleinput multipleoutput (MIMO) relay communications with any number of hops. We design the optimal source precoding matrix and the optimal relay amplifying matrices for such relay network where a nonlinear minimal meansquared error ..."
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Cited by 23 (17 self)
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Abstract—In this paper, we study multihop nonregenerative multipleinput multipleoutput (MIMO) relay communications with any number of hops. We design the optimal source precoding matrix and the optimal relay amplifying matrices for such relay network where a nonlinear minimal meansquared error (MMSE)decision feedback equalizer (DFE) is used at the destination node. We first derive the structure of the optimal source and relay matrices. Then based on the link between most commonly used MIMO system design objectives and the diagonal elements of the MSE matrix, we classify the objective functions into two categories: Schurconvex and Schurconcave composite objective functions. We show that when the composite objective function is Schurconvex, the MMSEDFE receiver together with the optimal source and relay matrices enable an arbitrary number of source symbols to be transmitted at one time, and yield a significantly improved BER performance compared with nonregenerative MIMO relay systems using linear receivers at the destination. We also show that for Schurconcave composite objective functions, the optimal source and relay matrices, and the optimal feedforward matrix at the destination node jointly diagonalize the multihop MIMO relay channel, and thus in such case, the nonlinear MMSEDFE receiver is essentially equivalent to a linear MMSE receiver. Index Terms—MIMO relay network, multihop relay, MMSE, DFE, nonregenerative relay, majorization.
NonRegenerative Multicarrier MIMO Relay Communications Based on Minimization of MeanSquared Error
"... Abstract—In this paper we propose nonregenerative multicarrier multipleinput multipleoutput (MIMO) relay techniques that minimize the meansquared error (MSE) of the signal waveform estimation. We establish the closedform optimal precoding matrices at the source and relay nodes in the absence of ..."
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Cited by 20 (10 self)
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Abstract—In this paper we propose nonregenerative multicarrier multipleinput multipleoutput (MIMO) relay techniques that minimize the meansquared error (MSE) of the signal waveform estimation. We establish the closedform optimal precoding matrices at the source and relay nodes in the absence of the direct sourcedestination link. Interestingly, we show that the proposed precoding matrices jointly convert the multicarrier MIMO relay channel into parallel singleinput singleoutput (SISO) relay channels. In order to reduce the computational complexity of the optimal algorithm, a suboptimal precoding approach based on an upperbound of the MSE expression is developed. Numerical examples illustrate a significant performance improvement of the proposed algorithms over the existing techniques. I.
Power allocation for a MIMO relay system with multipleantenna users
 IEEE Trans. Signal Process
, 2010
"... Abstract—A power allocation or scheduling problem is studied for a multiuser multipleinput multipleoutput (MIMO) wireless relay system where there is a nonregenerative relay between one access point and multiple users. Each node in the system is equipped with multiple antennas. The purpose of thi ..."
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Cited by 20 (4 self)
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Abstract—A power allocation or scheduling problem is studied for a multiuser multipleinput multipleoutput (MIMO) wireless relay system where there is a nonregenerative relay between one access point and multiple users. Each node in the system is equipped with multiple antennas. The purpose of this study is to develop fast algorithms to compute the source covariance matrix (or matrices) and the relay transformation matrix to optimize a system performance. We consider the minimization of power consumption subject to rate constraint and also the maximization of system throughput subject to power constraint. These problems are nonconvex and apparently have no simple solutions. In this paper, a number of computational strategies are presented and their performances are investigated. Both uplink and downlink cases are considered. The use of multiple carriers is also discussed. Moreover, a generalized waterfilling (GWF) algorithm is developed to solve a special class of convex optimization problems. The GWF algorithm is used for two of the strategies shown in this paper. Index Terms—Convex optimization, generalized water filling, medium access control, multiuser MIMO relays, network of MIMO links, nonconvex optimization, space–time power allocation, space–time power scheduling. I.
Linear precoding designs for amplifyandforward multiuser twoway relay systems
 in Proc. IEEE GLOBECOM
, 2011
"... AbstractWe investigate the linear precoding designs for multiuser twoway relay system (MUTWRS) where a multiantenna basestation (BS) communicates with multiple singleantenna mobile stations (MSs) via a multiantenna relay station (RS). The amplifyandforward (AF) relay protocol is employed. Th ..."
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Cited by 20 (6 self)
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AbstractWe investigate the linear precoding designs for multiuser twoway relay system (MUTWRS) where a multiantenna basestation (BS) communicates with multiple singleantenna mobile stations (MSs) via a multiantenna relay station (RS). The amplifyandforward (AF) relay protocol is employed. The design goal is to optimize the precodings at BS, RS or both so as to minimize the total meansquare error (MSE) of the uplink messages while maintaining the individual signaltointerferenceplusnoise ratio (SINR) requirement for each downlink signal. We show that the BS precoding design problem can be converted to a standard second order cone programming (SOCP), while the RS precoding is nonconvex for which a local optimal solution is obtained using an iterative algorithm. A joint BSRS precoding is also obtained by alternating optimization of BS precoding and RS precoding with guaranteed convergence. Numerical results show that RSprecoding is superior to BSprecoding. Furthermore, the joint BSRS precoding can significantly outperform the two individual precoding schemes. The implementation issues including complexity and feedback overhead are also discussed.
Optimal design of source and relay pilots for MIMO relay channel estimation
 IEEE Trans. Signal Process
, 2011
"... Abstract—In this paper, we consider a channel estimation scheme for a twohop nonregenerative MIMO relay system without the direct link between source and destination. This scheme has two phases. In the first phase, the source does not transmit while the relay transmits and the destination receives. ..."
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Cited by 20 (1 self)
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Abstract—In this paper, we consider a channel estimation scheme for a twohop nonregenerative MIMO relay system without the direct link between source and destination. This scheme has two phases. In the first phase, the source does not transmit while the relay transmits and the destination receives. In the second phase, the source transmits, the relay amplifies and forwards, and the destination receives. At the destination, the data received in the first phase are used to estimate the relaytodestination channel, and the data received in the second phase are used to estimate the sourcetorelay channel. The linear minimum meansquare error estimation (LMMSE) is used for channel estimation, which allows the use of prior knowledge of channel correlations. For phase 1, an algorithm is developed to compute the optimal source pilot matrix for use at the relay. For phase 2, an algorithm is developed to compute the optimal source pilot matrix for use at the source and the optimal relay pilot matrix for use at the relay. Index Terms—Convex optimization, MIMO wireless relays, nonconvex optimization, pilot waveform design, relay channel estimation. I.