Source Precoding Research Articles

Joint Source and Relay Optimization for Non-Regenerative MIMO Two-Way Relay Systems with Imperfect CSI

In this paper, we consider a non-regenerative MIMO two-way relay system with imperfect channel state information (CSI). We employ a stochastic approach to model the channel uncertainties and address the robust joint source and relay optimization problem based on the minimum mean squared error (MMSE) criterion. With imperfect CSI, the self-interference (SI) cannot be completely canceled at destination nodes. Hence, both channel uncertainties and residual self-interference should be considered. We develop an optimization framework that unifies both frequency-division duplex (FDD) and time-division duplex (TDD) systems despite their different channel statistical properties. Two robust algorithms are proposed to minimize the sum mean squared error (MSE) averaged over channel uncertainties. The first algorithm adopts alternating optimization to update the source precoders, relay precoder and destination receivers iteratively with guaranteed convergence. In the second algorithm, only the relay precoder with certain structure is considered. Then the relay precoder design is reduced to the simple power allocation problem. Simulation results show that the proposed algorithms provide robustness against channel uncertainties, especially when the signal-to-noise ratio is high.

Joint source and relay optimization for parallel MIMO relay networks

Abstract In this article, we study the optimal structure of the source precoding matrix and the relay amplifying matrices for multiple-input multiple-output (MIMO) relay communication systems with parallel relay nodes. Two types of receivers are considered at the destination node: (1) The linear minimal mean-squared error (MMSE) receiver; (2) The nonlinear decision feedback equalizer based on the minimal MSE criterion. We show that for both receiver schemes, the optimal source precoding matrix and the optimal relay amplifying matrices have a beamforming structure. Using such optimal structure, joint source and relay power loading algorithms are developed to minimize the MSE of the signal waveform estimation at the destination. Compared with existing algorithms for parallel MIMO relay networks, the proposed joint source and relay beamforming algorithms have significant improvement in the system bit-error-rate performance.

Joint Source and Relay Design for Multiuser MIMO Nonregenerative Relay Networks With Direct Links

In this paper, we investigate joint source precoding matrices and relay processing matrix design for multiuser multiple-input-multiple-output (MIMO) nonregenerative relay networks in the presence of the direct source-destination links. We consider both capacity and mean-square-error (MSE) criteria subject to the distributed power constraints, which are nonconvex and apparently have no simple solutions. Therefore, we propose an optimal source precoding matrix structure based on the point-to-point MIMO channel technique and a new relay processing matrix structure under the modified power constraint at relay node, based on which a nested iterative algorithm of jointly optimizing sources precoding and relay processing is established. We show that the capacity-based optimal source precoding matrices share the same structure with the MSE-based matrices and so does the optimal relay processing matrix. Simulation results demonstrate that the proposed algorithm outperforms the existing results.

Channel Estimation of Dual-Hop MIMO Relay System via Parallel Factor Analysis

The optimal source precoding matrix and relay amplifying matrix have been developed in recent works on multiple-input multiple-output (MIMO) relay communication systems assuming that the instantaneous channel state information (CSI) is available. However, in practical relay communication systems, the instantaneous CSI is unknown, and therefore, has to be estimated at the destination node. In this paper, we develop a novel channel estimation algorithm for two-hop MIMO relay systems using the parallel factor (PARAFAC) analysis. The proposed algorithm provides the destination node with full knowledge of all channel matrices involved in the communication. Compared with existing approaches, the proposed algorithm requires less number of training data blocks, yields smaller channel estimation error, and is applicable for both one-way and two-way MIMO relay systems with single or multiple relay nodes. Numerical examples demonstrate the effectiveness of the PARAFAC-based channel estimation algorithm.

Precoder Design for MIMO Relay Networks with Direct Link and Decision Feedback Equalisation

In this letter we derive the source and relay precoders for a multiple-input multiple-output (MIMO) relay system with zero-forcing (ZF) decision feedback equalisation (DFE) when a direct link exists between the source and destination. Assuming full channel state information (CSI) is available at each node in the network, the processors are optimised for a large class of objective functions subject to the ZF condition and average power constraints at both the source and relay terminals. Simulation results show that the proposed design outperforms existing techniques in terms of bit error ratio (BER).

Optimum Linear Design of Two-Hop MIMO Relay Networks With QoS Requirements

This paper investigates the optimal design issue of general two-hop amplify-and-forward (AF) MIMO relay networks with one source, one destination and multiple relays, each having multiple antennas. Beginning with the study of the mean-square error (MSE) matrix of the system and the use of Wiener filter for the destination processing, a constrained optimization problem with respect to the unknown relay precoder matrix is formulated with a goal of minimizing the total relay transmit power subject to the MSE-based quality of service (QoS) requirement on each data stream. As the original optimization problem is very difficult to solve, a lower bound of the cost function is pursued to achieve an approximate optimization problem. The modified problem is then solved under the perfect channel state information (CSI) condition using the diagonalization method along with the majorization theory to obtain optimal relay precoder with either nonoptimal or optimal source precoding. It is shown that a convex relay precoding solution is available in the case of nonoptimal source precoding whereas a joint optimal source and relay precoder can be obtained through a multilevel waterfilling algorithm. The optimal design of the relay network under imperfect CSI condition is also considered by modifying the optimization problem with the channel mean and covariance matrix, leading to a robust optimization design of the relay precoder. The proposed technique is validated by Monto Carlo simulations and is also compared with one existing method.

Joint Tomlinson–Harashima Source and Linear Relay Precoder Design in Amplify-and-Forward MIMO Relay Systems via MMSE Criterion

Existing minimum-mean-square-error (MMSE) transceiver designs in amplify-and-forward (AF) multiple-input-multiple-output (MIMO) relay systems all assume a linear precoder at the source. Nonlinear precoders in such a system have yet to be considered. In this paper, we study a nonlinear transceiver in AF MIMO relay systems in which a Tomlinson-Harashima (TH) precoder is used at the source, a linear precoder is used at the relay, and an MMSE receiver is used at the destination. Since two precoders and three links are involved, the transceiver design, which is formulated as an optimization problem, is difficult to solve. We first propose an iterative method to overcome the problem. In the method, the two precoders are separately optimized in an iterative step. To further improve the performance, we then propose a non-iterative method that can yield closed-form solutions for the precoders. This method uses the primal decomposition technique in which the original optimization can first be decomposed into a master and a subproblem optimization. In the subproblem, the optimum source precoder is solved as a function of the relay precoder. In the master problem, the optimization is then transferred to a relay-precoder-only problem. However, the optimization is not convex, and the primal decomposition cannot be directly applied. We then propose cascading a unitary precoder after the TH precoder so that the optimization in the subproblem and the master problem can be conducted. Furthermore, using a relay precoder structure, we can transfer the master problem to a convex optimization problem and obtain a closed-form solution by the Karuch-Kuhn-Tucker (KKT) conditions. Simulations show that the proposed transceivers can significantly outperform existing linear transceivers.

Nonlinear Transceiver Designs in MIMO Amplify-and-Forward Relay Systems

Various linear transceiver design methods have been developed in three-node amplify-and-forward (AF) multiple-input-multiple-output (MIMO) relay systems. Nonlinear designs in such systems, however, have yet to be investigated. In this paper, we propose nonlinear transceiver designs for a linear source and relay precoded system with the QR successive-interference-cancellation (SIC) receiver and another linear source and relay precoded system with the minimum-mean-squared-error (MMSE) SIC receiver. Our designs minimize the criterion of the block error rate, which is a complicated function of the source and relay precoders. Solving the two precoders simultaneously is not feasible. To overcome the difficulties, we first resort to the primal decomposition approach, i.e., transferring the original optimization to a subproblem and a master problem and solving the two precoders individually. However, since two power constraints are mutually coupled, the decomposition cannot actually be conducted. We then propose a unitary structure for the source precoder and show that the power constraints can be decoupled. As a result, the source precoder can be solved as a function of the relay precoder in the subproblem. With a proposed relay precoder structure, the master problem can further be transferred to a scalar-valued concave optimization problem. A closed-form solution can finally be derived by the Karuch-Kuhn-Tucker (KKT) conditions. Simulations show that the proposed transceivers can significantly outperform the existing linear transceivers.

Multihop Nonregenerative MIMO Relays— QoS Considerations

For nonregenerative multihop multiple-input multiple-output (MIMO) relay communication systems, the optimal source precoding matrix and the optimal relay amplifying matrices have been recently established for a broad class of objective functions subjecting to the transmission power constraint at each node. However, existing works do not consider any quality-of-service (QoS) constraints, which are important in practical communication systems. In this paper, we derive the optimal source and relay matrices of a multihop MIMO relay system that guarantee the predetermined QoS criteria be attained with the minimal total transmission power. In particular, we consider two types of receivers at the destination node: the linear minimal mean-squared error (MMSE) receiver and the nonlinear decision feedback equalizer (DFE) based on the MMSE criterion. We show that for both types of receivers, the solution to the original optimization problem can be upper-bounded by using a successive geometric programming (GP) approach and lower-bounded by utilizing a dual decomposition technique. Simulation results show that both bounds are tight, and to obtain the same QoS, the MIMO relay system using the nonlinear MMSE-DFE receiver requires substantially less total transmission power than the linear MMSE receiver-based system.

Transceiver Design for Dual-Hop Nonregenerative MIMO-OFDM Relay Systems Under Channel Uncertainties

In this paper, linear transceiver design for dual-hop nonregenerative [amplify-and-forward (AF)] MIMO-OFDM systems under channel estimation errors is investigated. Second order moments of channel estimation errors in the two hops are first deduced. Then based on the Bayesian framework, joint design of linear forwarding matrix at the relay and equalizer at the destination under channel estimation errors is proposed to minimize the total mean-square-error (MSE) of the output signal at the destination. The optimal designs for both correlated and uncorrelated channel estimation errors are considered. The relationship with existing algorithms is also disclosed. Moreover, this design is extended to the joint design involving source precoder design. Simulation results show that the proposed design outperforms the design based on estimated channel state information only.

Joint Linear Processing for an Amplify-and-Forward MIMO Relay Channel with Imperfect Channel State Information

The problem of jointly optimizing the source precoder, relay transceiver, and destination equalizer has been considered in this paper for a multiple-input-multiple-output (MIMO) amplify-and-forward (AF) relay channel, where the channel estimates of all links are assumed to be imperfect. The considered joint optimization problem is nonconvex and does not offer closed-form solutions. However, it has been shown that the optimization of one variable when others are fixed is a convex optimization problem which can be efficiently solved using interior-point algorithms. In this context, an iterative technique with the guaranteed convergence has been proposed for the AF MIMO relay channel that includes the direct link. It has been also shown that, for the double-hop relay case without the receive-side antenna correlations in each hop, the global optimality can be confirmed since the structures of the source precoder, relay transceiver, and destination equalizer have closed forms and the remaining joint power allocation can be solved using Geometric Programming (GP) technique under high signal-to-noise ratio (SNR) approximation. In the latter case, the performance of the iterative technique and the GP method has been compared with simulations to ensure that the iterative approach gives reasonably good solutions with an acceptable complexity. Moreover, simulation results verify the robustness of the proposed design when compared to the nonrobust design that assumes estimated channels as actual channels.

Optimal Linear Non-Regenerative Multi-Hop MIMO Relays with MMSE-DFE Receiver at the Destination

In this paper, we study multi-hop non-regenerative multiple-input multiple-output (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 mean-squared 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: Schur-convex and Schur-concave composite objective functions. We show that when the composite objective function is Schur-convex, the MMSE-DFE 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 non-regenerative MIMO relay systems using linear receivers at the destination. We also show that for Schur-concave composite objective functions, the optimal source and relay matrices, and the optimal feed-forward matrix at the destination node jointly diagonalize the multi-hop MIMO relay channel, and thus in such case, the nonlinear MMSE-DFE receiver is essentially equivalent to a linear MMSE receiver.

Optimal joint source and relay beamforming for MIMO relays with direct link

In this letter, we investigate the optimal structure of the source precoding matrix and the relay amplifying matrix for non-regenerative multiple-input multiple-output (MIMO) relay communication systems with the direct source-destination link. We show that both the optimal source precoding matrix and the optimal relay amplifying matrix have a beamforming structure. Based on this structure, an iterative joint source and relay beamforming algorithm is developed to minimize the mean-squared error (MSE) of the signal waveform estimation. Numerical example demonstrates an improved performance of the proposed algorithm.

Optimality of diagonalization of multi-hop MIMO relays

For a two-hop linear non-regenerative multiple-input multiple-output (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 Schur-concave 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 Schur-convex 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.

A Unified Framework for Optimizing Linear Nonregenerative Multicarrier MIMO Relay Communication Systems

In this paper, we develop a unified framework for linear nonregenerative multicarrier multiple-input multiple-output (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 mean-square error and the maximal mutual information into two categories: Schur-concave and Schur-convex functions. We prove that for Schur-concave 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 single-input single-output (SISO) relay channels. While for Schur-convex 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.

Joint source/relay precoder design in nonregenerative cooperative systems using an MMSE criterion

This paper considers transmitter precoding in an amplify-and-forward cooperative system where multiple antennas are equipped at the source, the relay, and the destination. Existing methods for the problem only consider the design of the relay precoder. To further improve the performance, we include the source precoder into the design. Using a minimum-meansquare- error (MMSE) criterion, we propose a joint source/relay precoder design method, taking both the direct and relay links into account. It is shown that the MMSE is a highly nonlinear function of the precoding matrices, and a direct minimization is not feasible. To facilitate analysis, we propose to design the precoders toward first diagonalizing the MSE matrix of the relay link. This imposes certain structural constraints on both precoders that allow us to derive an analytically tractable MSE upper bound. By conducting minimization with respect to this bound, the solution can be obtained by an iterative water-filling technique.