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Joint Source and Relay Precoding Designs for MIMO Two-Way Relaying Based on MSE Criterion

Rui Wang, Meixia Tao

arXiv:1112.3096v1cs.IT

TL;DR

The paper studies joint source-and-relay precoding for MIMO two-way relaying under an MSE criterion. It proposes iterative, channel-parallelization, and antenna-selection designs, with simulations showing effectiveness against conventional schemes.

  • Problem

    The paper addresses joint source and relay precoding design for MIMO two-way relay systems based on the MSE criterion.

  • Method

    The paper decouples the joint design into three sub-problems for iterative optimization, and also develops channel-parallelization and source-antenna-selection precoding designs.

  • Results

    Simulations showed that all proposed precoding designs are effective compared with conventional schemes.

  • Takeaways & Limitations

    Channel parallelization reduces precoding design to power allocation, while antenna selection reduces system feedback overhead and avoids advanced software packages.

Abstract

from arXiv · show

Properly designed precoders can significantly improve the spectral efficiency of multiple-input multiple-output (MIMO) relay systems. In this paper, we investigate joint source and relay precoding design based on the mean-square-error (MSE) criterion in MIMO two-way relay systems, where two multi-antenna source nodes exchange information via a multi-antenna amplify-and-forward relay node. This problem is non-convex and its optimal solution remains unsolved. Aiming to find an efficient way to solve the problem, we first decouple the primal problem into three tractable sub-problems, and then propose an iterative precoding design algorithm based on alternating optimization. The solution to each sub-problem is optimal and unique, thus the convergence of the iterative algorithm is guaranteed. Secondly, we propose a structured precoding design to lower the computational complexity. The proposed precoding structure is able to parallelize the channels in the multiple access (MAC) phase and broadcast (BC) phase. It thus reduces the precoding design to a simple power allocation problem. Lastly, for the special case where only a single data stream is transmitted from each source node, we present a source-antenna-selection (SAS) based precoding design algorithm. This algorithm selects only one antenna for transmission from each source and thus requires lower signalling overhead. Comprehensive simulation is conducted to evaluate the effectiveness of all the proposed precoding designs.

I. INTRODUCTION

The paper studies joint source-and-relay precoding for multi-antenna AF MIMO two-way relaying under a Total-MSE objective. It proposes iterative, channel-parallelization, and source-antenna-selection designs to address the problem's non-convexity, complexity, and signalling overhead.

  • Motivation: MIMO and two-way relaying can improve wireless transmission, but efficient precoding that accounts for the relay is needed to realize their benefits.Two-way relaying completes information exchange in two time slots, while MIMO precoding exploits spatial multiplexing using transmitter CSI.
  • Problem formulation: The paper considers joint source and relay precoding in an AF MIMO two-way relay system where every node has multiple antennas.The study targets AF relaying, whose precoding design is described as more challenging than DF relaying.
  • Iterative precoding design: The authors minimize the two users' Total-MSE with an alternating algorithm that decouples the non-convex joint design into three sub-problems.The relay precoder has a closed-form solution when source precoders and decoders are fixed.
  • Iterative precoding design: Because each sub-problem is solved optimally, the iterative precoding algorithm has guaranteed convergence.The contribution addresses the tractability of joint source-and-relay optimization rather than relay-only precoding.
  • Channel-parallelization design: The channel-parallelization design imposes source-and-relay precoder structures that reduce joint precoding to a simple power-allocation problem.It parallelizes channels in both the MAC and BC phases for certain antenna configurations.
  • Source-antenna selection: For single-data-stream transmission, the SAS design selects one transmitting antenna at each source and can reduce signalling overhead while sometimes outperforming the iterative design.The paper also reports that all proposed designs are effective compared with conventional schemes.

II. SYSTEM MODEL

The system has two multi-antenna sources exchanging messages through a multi-antenna relay in two half-duplex phases. The paper jointly designs source, relay, and receiver processing using global CSI to minimize Total-MSE, while discussing CSI acquisition and feedback overhead.

  • System model: Two source nodes, each with N antennas, exchange messages through an M-antenna relay using two time slots: MAC followed by BC.The sources transmit simultaneously in the MAC phase, and the relay linearly processes and forwards the superimposed signal in the BC phase.
  • Assumptions and implementation: The model normally transmits N data streams per source to use the multiplexing gain, with single-stream transmission treated separately.Under channel reciprocity, the relay can estimate the global CSI during the MAC phase and broadcast the source precoding information.
  • Signal model: Each source transmit vector uses a source precoding matrix whose columns are beamforming vectors for the corresponding data streams.The source and relay transmit powers are constrained by maximum power limits.
  • Signal model: The relay amplifies its received signal with a relay precoding matrix, and each destination subtracts its back-propagated self-interference before decoding.After cancellation, Fi = GiArH¯iA¯i is the equivalent end-to-end MIMO channel for source Si.
  • Design objective: The optimization jointly designs {A1, A2, Ar} from global CSI to minimize the Total-MSE of all data streams for both users.The Total-MSE criterion can yield optimal precoder structures or closed-form solutions in some cases, although it may not be best for overall performance.
  • Assumptions and implementation: Without channel reciprocity, the system requires additional feedback channels and signalling overhead for the relay to obtain G1 and G2.The sources must also estimate effective channels such as G1ArH1 for self-interference cancellation and G1ArH2 for demodulation.

III. ITERATIVE PRECODING DESIGN

The paper formulates joint source, relay, and decoder precoding as a nonlinear, non-convex Total-MSE minimization and solves it through alternating optimization. Its feasibility depends on the antenna relation: when M ≥ N, Total-MSE can approach zero with increasing source and relay power, whereas M < N imposes a positive lower bound.

  • Alternating optimization: The proposed algorithm alternates updates of one precoder or decoder while fixing the others, using optimal solutions for the resulting sub-problems.The analysis separately considers decoder, relay-precoder, and source-precoder updates.
  • Problem formulation: The joint optimization seeks source, relay, and decoder matrices that minimize the Total-MSE of both users, but the problem is nonlinear and non-convex.The formulation includes the source and relay power constraints and the linear decoding matrices.
  • Feasibility and antenna conditions: When M ≥ N, the Total-MSE J1 + J2 can be made arbitrarily small by increasing power at both source nodes and the relay.This result is established for the considered (N, M, N) two-way relay system.
  • Feasibility and antenna conditions: When M < N, the Total-MSE J1 + J2 is always lower bounded by 2(N −M), regardless of how much power is provided.The bound follows because each direction contributes a lower bound of N −M.

1 P−2G1 ¯ArH2A2E−1

The alternating design decomposes the joint problem into decoder, relay-precoder, and source-precoder sub-problems with tractable structures. Each iteration decreases Total-MSE, and the resulting algorithm converges to a stationary point while retaining a Total-MSE objective’s possible stream-imbalance.

  • Decoder update: With source and relay precoders fixed, decoder optimization is unconstrained and convex in W1 and W2, yielding optimal decoding matrices through KKT conditions.The power constraints do not involve the decoding matrices.
  • Relay update: With source precoders and decoders fixed, relay precoding is a convex Total-MSE minimization problem solvable through KKT conditions.The relay power constraint is included while source power constraints are irrelevant for this sub-problem.
  • Source update: With relay precoding and decoding fixed, source-precoder optimization is convex and can be transformed into a convex quadratically constrained quadratic program.A QCQP solver can be used after accounting for the relay power constraint affected by source precoders.
  • Convergence: Because each sub-problem is solved optimally, Algorithm 1 decreases Total-MSE at every iteration and converges to a stationary point of the original problem.Total-MSE is lower bounded by zero, supporting convergence; the stationarity claim follows at the limit point.
  • Objective limitation: Minimizing Total-MSE can produce unbalanced per-stream MSE, motivating a min-max objective or a rotation that equalizes the MSE-matrix diagonal entries.The paper notes that overall error performance is dominated by the stream with the highest MSE.

IV. LOW-COMPLEXITY PRECODING DESIGN BASED ON CHANNEL PARALLELIZATION

To reduce the high computational complexity of iterative precoding, the paper introduces a structured design that balances performance and complexity. The structure parallelizes MAC and BC channels and reduces precoder optimization to power allocation.

  • Motivation: The iterative precoding design achieves good performance but has high computational complexity, motivating a lower-complexity alternative.The new design is presented as a balance between performance and complexity.
  • Channel parallelization: The structured design aims to simultaneously parallelize bidirectional links in the MAC and BC phases.It uses GSVD for the MAC phase and SVD for the BC phase.
  • Power allocation: The resulting precoder design is reduced to a simple power allocation problem after channel parallelization.This replaces the broader matrix-design task with optimization of diagonal power-allocation factors.

A. Channel Parallelization

The channel-parallelization design jointly decomposes the MAC channel pair with GSVD and the BC phase through a virtual point-to-point channel with SVD. The resulting structure uses unitary factors and diagonal power-allocation matrices, with M = N adopted to avoid unmatched gains between users.

  • MAC phase: The MAC phase jointly decomposes the forward channel pair so that suitable relay and source factors parallelize both forward channels.The relay precoder includes Vh^-1 while source precoders include the corresponding unitary factors.
  • BC phase: The BC phase constructs a virtual point-to-point MIMO channel from the two backward channels and parallelizes it using SVD.The relay precoder incorporates the left singular-vector factor of the virtual channel.
  • Structured precoders: The proposed precoder structure combines channel-decomposition factors with arbitrary unitary matrices and diagonal matrices optimized for power allocation.The diagonal factors correspond to the source and relay precoders and the effective channel gains.
  • Antenna-dimension condition: For M > N, effective gains cannot be matched simultaneously for both sources, so the section focuses on M = N to use all gains for both users.A gain strong for one source may be weak for the corresponding stream of the other source.

B. Joint Power Allocation

The structured precoding design reformulates joint optimization using a tractable MSE upper bound and reduces the problem to a simpler, still non-convex power-allocation formulation.

  • The joint optimization of source and relay precoder parameters minimizes the two users’ Total-MSE.
  • A tractable upper bound is introduced because the MSE covariance matrices remain non-diagonal after simplification.
  • The upper-bound MSE matrix is diagonal, enabling a simpler optimization sub-problem.
  • The resulting precoder-design problem is more analytically tractable but remains non-convex.
  • An iterative approach converts the structured problem into two convex sub-problems.

1) Sub-problem 1:

The paper solves coupled precoding sub-problems iteratively, while also developing channel-parallelized and source-antenna-selection designs with lower implementation complexity.

  • 1) Sub-problem 1:: The power-allocation sub-problem is convex and admits a water-filling solution derived from the KKT conditions.
  • 1) Sub-problem 1:: Because source and relay powers are tightly coupled, the algorithm alternates updates until convergence.
  • Channel parallelization: The channel-parallelization design decomposes channel pairs and updates relay and source power allocations iteratively.
  • Source antenna selection: For single-data-stream transmission, source antenna selection chooses the antenna pair and associated vectors and relay precoder minimizing Total-MSE.
  • Source antenna selection: Antenna selection is a special case of beamforming that generally lowers computational complexity and feedback overhead.
  • Source antenna selection: The SAS algorithm uses two update steps per iteration, closed-form solutions, and no advanced software package.

VI. SIMULATION RESULTS AND DISCUSSIONS

Simulations use Rayleigh fading and evaluate convergence under varying SNR and initialization conditions. The iterative algorithm converges within a bounded number of iterations and is reported as robust to initialization.

  • The simulations use Rayleigh-fading channel matrices with complex Gaussian entries of zero mean and unit variance.
  • The evaluation simulates average BER under QPSK modulation using defined MAC- and BC-phase SNRs.
  • Different initialization points produce minimal BER performance differences, leading the authors to describe the algorithm as robust and near optimal.
  • At N = M = 2, convergence occurs within 10 iterations at low SNR, about 30 at medium SNR, and 50 at high SNR.
  • The iterative algorithm finds only a local optimum because the primal problem is non-convex, so initialization points can yield different convergent solutions.

B. Performance Comparison for Multi-data-stream Transmission

The proposed designs improve BER relative to non-precoding and relay-only baselines, with unrestricted iterative precoding performing best. Channel parallelization reduces complexity but can degrade performance, while SAS lowers feedback overhead for single-stream transmission.

  • Multi-data-stream transmission: Uniform channel-parallelized precoding provides only marginal gain over non-precoding because uniform power can disadvantage weaker sub-channels.
  • Multi-data-stream transmission: The unrestricted iterative precoding design achieves the best performance among the proposed designs because it imposes no precoder structure.
  • Multi-data-stream transmission: Increasing relay antennas improves BER through increased diversity gain, and the proposed scheme’s gain over non-precoding grows with relay-antenna count.
  • Multi-data-stream transmission: Joint source/relay precoding significantly outperforms relay-only precoding with either MMSE or ZF receivers.
  • Single-data-stream transmission: For single-stream transmission, SAS can outperform iterative precoding with limited initialization, while more random initializations allow iterative precoding to approach the optimal solution.
  • Single-data-stream transmission: SAS can match stronger iterative configurations while requiring lower feedback overhead, but the best iterative configurations have substantially higher computational complexity.
  • Conclusions: The conclusion reports that channel parallelization reduces computational complexity despite degraded performance, whereas all proposed designs outperform conventional schemes.

APPENDIX A PROOF OF LEMMA 1

The proof establishes convexity of the relay-precoding subproblem by showing that its objective and feasible set are convex. It also derives properties used to characterize the optimal solution.

  • Positive semidefiniteness of Kronecker products supports the convexity arguments for the objective and constraints.
  • The objective function J_r1 + J_r2 is convex because both component objectives have positive semidefinite Hessian structures.
  • The feasible set in problem (15) is convex after verifying the associated constraint functions and block matrices are positive semidefinite.
  • Because both the objective and feasible set are convex, optimization problem (15) is a convex problem.
  • The proof further uses monotonicity with respect to λ and positivity conditions to establish the relevant upper-bound relations.
  • Trace products of positive semidefinite matrices are nonnegative, completing the required sign arguments in the proof.

APPENDIX C PROOF OF LEMMA 3

The proof transforms the optimization into a convex quadratically constrained quadratic program and analyzes its KKT conditions. It then selects the unique positive solution required by the constraints.

  • The optimization in (23) is transformed into a convex QCQP because its objective and constraint matrices are positive semidefinite.
  • Positive semidefinite matrices establish the convexity required for the transformed optimization problem.
  • The proof rewrites the MSE using the matrix inversion lemma before deriving the Lagrangian and KKT conditions.
  • The KKT analysis separates cases according to the sign of A_r and identifies the relevant equation for the optimal variable.
  • When A_r is positive, the solution is chosen as the only positive root of the associated monotonic equation.
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