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Joint Beamforming Optimization and Power Control for Full-Duplex MIMO Two-way Relay Channel

Gan Zheng

arXiv:1411.6021v1cs.IT

TL;DR

The paper addresses how to improve spectrum efficiency in a full-duplex MIMO two-way relay channel despite residual self-interference. It jointly optimizes relay beamforming and source power for rate-region and sum-rate objectives using iterative algorithms and 1-D search, and simulations show larger rate regions and improved sum rates than conventional half-duplex relaying.

  • Problem

    Full-duplex two-way relaying can reduce communication to one phase, but simultaneous transmission creates residual self-interference that limits end-to-end performance.

  • Method

    The paper jointly optimizes an amplify-and-forward relay’s beamforming matrix and the two sources’ transmit powers using iterative algorithms, 1-D search, and analytical or convex subproblems.

  • Results

    The proposed one-phase full-duplex scheme achieves a significantly larger rate region and higher sum rate than the two-phase full-duplex and conventional half-duplex benchmarks; at 10 dB source SNR, its sum-rate gain is 1.56 over the conventional two-phase HD scheme.

  • Takeaways & Limitations

    Optimizing beamforming and source power makes full-duplex two-way relaying a high-rate alternative to conventional half-duplex two-way relaying in the evaluated settings.

Abstract

from arXiv · show

In this paper we explore the use of full-duplex radio to improve the spectrum efficiency in a two-way relay channel where two sources exchange information through an multi-antenna relay, and all nodes work in the full-duplex mode. The full-duplex operation can reduce the overall communication to only one phase but suffers from the self-interference. Instead of purely suppressing the self-interference, we aim to maximize the end-to-end performance by jointly optimizing the beamforming matrix at the relay which uses the amplify-and-forward protocol as well as the power control at the sources. To be specific, we propose iterative algorithms and 1-D search to solve two problems: finding the achievable rate region and maximizing the sum rate. At each iteration, either the analytical solution or convex formulation is obtained. We compare the proposed full-duplex two-way relaying with the conventional half-duplex two-way relaying, a full-duplex one-way relaying and a performance upper bound. Numerical results show that the proposed full-duplex scheme significantly improves the achievable data rates over the conventional scheme.

I. INTRODUCTION

The paper studies full-duplex two-way relaying with a multi-antenna relay to reduce communication to one phase while jointly managing self-interference, beamforming, and source power. It formulates rate-region and sum-rate problems and develops iterative optimization methods for end-to-end performance.

  • I. INTRODUCTION: Full-duplex operation enables simultaneous information exchange between the two sources in one communication phase, but residual self-interference is the central challenge.The relay’s self-interference can exceed the received signal by over 100 dB and exceed analog-to-digital converter dynamic range.
  • I. INTRODUCTION: The paper jointly optimizes the relay beamforming matrix and source power allocation for a MIMO full-duplex two-way relay channel using amplify-and-forward and physical layer network coding.The sources use single transmit and receive antennas, while the relay has multiple transmit or receive antennas to help suppress residual self-interference.
  • I. INTRODUCTION: The proposed algorithms find the achievable rate region through iterative optimization and 1-D search, with analytical solutions for transmit beamforming and power control at each iteration.A zero-forcing constraint is used to simplify the relay model and problem formulations.
  • I. INTRODUCTION: Sum-rate maximization uses a similar iterative procedure, combining a difference-of-convex approach for transmit beamforming with analytical power allocation.The methods optimize the two end-to-end objectives under source and relay power constraints.
  • I. INTRODUCTION: Simulations compare the proposed full-duplex scheme with three benchmark schemes and report larger achievable rate regions and higher sum rates than conventional half-duplex two-way relaying.The study also models residual self-interference statistically and assumes global channel state information is available at the relay.

B. Signal model

The signal model represents full-duplex relay processing with delayed amplify-and-forward transmission and residual self-interference at the relay and sources. A zero-forcing constraint simplifies the relay recursion and yields SINR-based achievable rates.

  • B. Signal model: The relay receives source signals, residual self-interference, and noise, then transmits a delayed amplified version through matrix W.The relay output includes recursively delayed relay transmission through the residual self-interference channel.
  • B. Signal model: The zero-forcing condition nulls residual self-interference from the relay output to its input, making the relay power and interference expressions more tractable.This constraint is imposed to simplify the signal model and optimization problems.
  • B. Signal model: After each source cancels its delayed transmitted signal, simultaneous transmission leaves residual self-interference from its current transmission.The paper identifies this uncancelled current-symbol interference as the main challenge of full-duplex radio.
  • B. Signal model: The source SINRs account for the desired relayed signal, residual self-interference, and noise, and the achievable rates are log2(1 + γA) and log2(1 + γB).The SINRs are used to evaluate the end-to-end rates at both sources.

C. Problems Statement

The paper formulates two end-to-end optimization objectives: tracing the achievable rate region and maximizing the two-source sum rate. It uses zero-forcing, beamforming decomposition, alternating optimization, and 1-D search to make these problems tractable.

  • C. Problems Statement: Full-duplex physical layer network coding reduces two-way communication to one phase but requires careful source power selection because higher power increases residual self-interference.The proposed design optimizes beamforming and source powers rather than simply using maximum transmit power.
  • C. Problems Statement: The achievable rate region is obtained by maximizing one source’s rate while constraining the other source’s rate to exceed a target, then enumerating that target.This is formulated as problem P1 under source and relay power constraints.
  • C. Problems Statement: The sum-rate objective maximizes RA + RB subject to transmit-power constraints on the two sources and the relay.This is formulated as problem P2.
  • C. Problems Statement: The relay beamforming matrix is decomposed into transmit and receive vectors, reducing the zero-forcing condition to w_r†H_RRw_t = 0 with ||w_r|| = 1.The decomposition uses the single-stream structure and physical layer network coding’s signal mixing.
  • C. Problems Statement: Alternating optimization updates coupled variables separately while using 1-D search for the receive beamforming vector.At least two relay transmit or receive antennas are required to make the zero-forcing constraint feasible.

A. Parameterization of the receive beamforming vector wr

The receive beamforming vector w_r is parameterized by a scalar α to balance source signals, while accounting for the relay’s zero-forcing constraint. This enables alternating optimization of w_t and source powers followed by one-dimensional search over α.

  • Receive-beamformer parameterization: w_r balances the signals received from the two sources through a parameterization indexed by 0 ≤ α ≤ 1.The parameterization is motivated by w_r’s role in the received-signal terms.
  • Receive-beamformer parameterization: The α-based parameterization is incomplete because w_r must also satisfy the relay zero-forcing constraint.Despite this limitation, the parameterization makes the optimization more tractable.
  • Overall optimization procedure: For fixed α, the algorithm optimizes w_t and (p_A, p_B), then performs one-dimensional search to select α*.Each variable optimization admits an analytical solution without iterative or numerical methods.
  • Transmit-beamforming subproblem: The transmit-beamforming subproblem can be reformulated using W_t = w_t w_t† and semidefinite relaxation, although its special structure permits an analytical solution.The zero-forcing constraint is removed by expressing w_t through the null space of w_r†H_RR.
  • Source-power subproblem: After w_t is obtained, the procedure proceeds to optimize the source power allocation using the resulting beamforming quantities.The source-power stage uses terms involving the relay-to-source channels and self-interference.

C. Optimization of the source power (pA, pB)

With both relay beamformers fixed, source power control is optimized under self-interference and relay-power constraints. The resulting linear-fractional problem is converted to a linear program and solved analytically through feasibility and active-constraint cases.

  • Problem formulation: Full-duplex sources cannot generally use their maximum powers because single-antenna nodes cannot spatially suppress residual self-interference.The relay can eliminate self-interference with multiple antennas and transmit at full relay power.
  • Problem formulation: The source-power problem is a linear-fractional program that can be converted into a linear program.Its special structure enables analytical solutions rather than generic numerical optimization.
  • Analytical solution: The procedure first checks feasibility, then solves the power allocation while initially ignoring the relay-power constraint.At the unconstrained optimum, at least one source reaches its maximum power.
  • Analytical solution: If the obtained allocation violates the relay-power constraint, that constraint becomes active and the final powers are determined from the resulting equation set.Otherwise, the checked allocation is optimal.

D. The overall algorithm

The achievable-rate-region algorithm alternates beamforming and source-power optimization for a fixed receive-beamformer parameter, while sum-rate maximization uses DC programming for transmit beamforming. One-dimensional search selects the receive-beamformer parameter in both problems.

  • Achievable-rate-region algorithm: For a fixed α or w_r, the rate-region algorithm alternates optimization of w_t and (p_A, p_B) until convergence.The objective increases monotonically across iterations, yielding a local optimum.
  • Achievable-rate-region algorithm: One-dimensional search over 0 ≤ α ≤ 1 selects α*, and enumerating source B’s required rate traces the achievable-rate-region boundary.The boundary is obtained numerically after the alternating optimization.
  • Sum-rate algorithm: For sum-rate maximization, w_r is selected by one-dimensional search and w_t and source powers are optimized alternately.The sum-rate formulation uses the same receive-beamformer characterization.
  • Sum-rate algorithm: The transmit-beamforming objective is handled with DC programming by linearizing its convex component and solving sequential convex problems.The method seeks a local optimum rather than directly solving the nonconcave formulation.
  • Sum-rate algorithm: When M_t > 2, the relaxed semidefinite solution satisfies the rank-one requirement; for M_t = 2 or M_t = 1, scalar optimization handles the special cases.For M_t = 2, the zero-forcing constraint fixes the beamforming direction.

B. Optimization of source power (pA, pB)

With beamformers fixed, sum-rate power allocation is solved by distinguishing whether the relay-power constraint is active. The active case reduces to a one-variable problem whose stationary points satisfy a cubic equation.

  • Problem formulation: The fixed-beamformer sum-rate problem optimizes source powers under relay and individual source-power constraints.The formulation includes the source self-interference terms in the rate expressions.
  • Inactive relay-power constraint: If the relay-power constraint is inactive, the allocation reduces to conventional power allocation over two interference links and has a binary optimum.The paper identifies this as the known solution for that case.
  • Active relay-power constraint: When the relay-power constraint is active, it satisfies p_A C_rA + p_B C_rB + 1 = P_R and reduces the objective to a function of p_B.The resulting problem is one-dimensional after expressing p_A through the active constraint.
  • Active relay-power constraint: The stationary-point condition becomes a cubic equation, whose candidate roots and boundary point are compared to select the maximum objective value.The candidate root set may contain 0, 1, or 3 elements.
  • Complexity: The overall sum-rate algorithm is dominated by solving an SDP with one M_t × M_t matrix variable and two constraints, with worst-case complexity O(M^4.5/ε).Here ε denotes the desired solution accuracy.

V. BENCHMARK SCHEMES

The proposed full-duplex network is evaluated against three benchmark schemes: conventional two-phase half-duplex relaying, two-phase one-way full-duplex relaying, and an upper-bound scheme that ignores residual relay self-interference.

  • The conventional two-phase HD TWRC uses analog network coding and is known to outperform three- and four-phase HD schemes.
  • The two-phase one-way FD benchmark operates the relay in full-duplex mode while the two sources remain half-duplex.
  • The upper-bound benchmark ignores residual self-interference at the relay while retaining self-interference at the two sources.

A. Two-phase HD relaying using analog network coding

The conventional two-phase HD relay uses analog network coding: both sources transmit first, then the relay beamforms and broadcasts, with no self-interference but a two-phase throughput factor.

  • Both sources transmit to the relay in the first phase, and the relay broadcasts a beamformed signal in the second phase.
  • Because HD relaying has no self-interference, every node can use full power and only the relay beamforming matrix W requires optimization.
  • The relay power consumption combines amplified source signals and relay noise as pR = ∥WhAR∥2PA + ∥WhBR∥2PB + trace(WW†).
  • A factor of 1/2 appears because information exchange requires two transmission phases.
  • The achievable-rate-region and maximum-sum-rate problems are solved using the resulting rate and relay-power expressions.

B. Two-phase one-way FD

The two-phase one-way FD benchmark lets the relay operate full-duplex while sources transmit at maximum power, using zero-forcing beamforming and time sharing to characterize rates. The supplied evaluation passages also define an idealized no-relay-self-interference upper bound and the simulation comparisons.

  • Two-phase one-way FD: In two-phase one-way FD relaying, the relay operates full-duplex while the two sources operate half-duplex, allowing both sources to transmit at maximum power.
  • Two-phase one-way FD: Closed-form source-A rates are derived under receive zero forcing and transmit zero forcing, then combined through RA = log2(1+max(γRZF, γTZF)).
  • Two-phase one-way FD: The two directional rates cannot be achieved simultaneously because each source would need the entire transmission time.
  • Two-phase one-way FD: The rate-region boundary is obtained by time sharing with t ∈ [0, 1], yielding (tRA, (1 − t)RB).
  • FD-Upper Bound: The no-relay-self-interference benchmark assumes HRR = 0 and provides an unrealistic performance upper bound for evaluating the proposed algorithms.
  • Numerical comparison: The simulations compare the proposed scheme with Two-phase HD, Two-phase FD, and the Proposed one-phase FD upper bound using 100 independent channel realizations.

A. Achievable rate region

The proposed one-phase full-duplex scheme enlarges the achievable rate region and improves sum-rate performance over conventional half-duplex and two-phase full-duplex benchmarks across several operating conditions. Gains depend on residual self-interference, relay power, antenna count, channel asymmetry, and available channel state information.

  • Achievable rate region: The proposed one-phase FD scheme achieves a significantly larger rate region than the two-phase FD and conventional HD schemes.
  • Sum rate performance: At source SNR 10 dB, the proposed and two-phase FD schemes achieve sum-rate gains of 1.56 and 1.22, respectively, over two-phase HD.
  • Relay transmit SNR: At relay transmit SNRs above 5 dB, the proposed one-phase FD scheme outperforms two-phase FD, with especially large gains at high relay SNR.At low relay SNR, it can underperform two-phase FD at low transmit SNR.
  • Residual self-interference: When source residual SI exceeds -5 dB, two-phase FD outperforms one-phase FD, although both remain better than two-phase HD even at 5 dB SI gain.
  • Relay antennas: Increasing relay antennas from 2 to 6 steadily increases sum rate, while the rate gain remains about 1.55 beyond two antennas.The increase is attributed to array gain.
  • Asymmetric channel gain: In the asymmetric case, weaker R–B channels reduce both sources’ rates, with source B suffering more loss and two-phase FD remaining close to proposed FD.The end-to-end performance is restricted by the weaker R–B channel, limiting the benefit of simultaneous source transmission.
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