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Low-complexity End-to-End Performance Optimization in MIMO Full-Duplex Relay Systems

Himal A. Suraweera, Ioannis Krikidis, Gan Zheng, Chau Yuen, Peter J. Smith

arXiv:1311.3428v1cs.IT

TL;DR

Full-duplex AF relaying with multiple antennas requires end-to-end design beyond relay-side loopback-interference mitigation. This paper develops joint zero-forcing precoding/decoding and antenna-selection schemes, finding improved end-to-end performance and efficient low-complexity operation under strict computational constraints.

  • Problem

    Prior work emphasizes relay-side loopback-interference mitigation, leaving joint source–relay–destination precoding and decoding for end-to-end performance insufficiently addressed.

  • Method

    The paper jointly designs precoding and decoding with rank-1 zero-forcing self-interference suppression, and studies single-antenna selection with relay power allocation.

  • Results

    Joint zero-forcing precoding improves end-to-end performance, while closed-form outage and diversity analyses characterize the proposed antenna-selection schemes.

  • Takeaways & Limitations

    Antenna selection and relay power allocation provide lower-complexity options for full-duplex AF systems while addressing zero-diversity effects.

Abstract

from arXiv · show

In this paper, we deal with the deployment of full-duplex relaying in amplify-and-forward (AF) cooperative networks with multiple-antenna terminals. In contrast to previous studies, which focus on the spatial mitigation of the loopback interference (LI) at the relay node, a joint precoding/decoding design that maximizes the end-to-end (e2e) performance is investigated. The proposed precoding incorporates rank-1 zero-forcing (ZF) LI suppression at the relay node and is derived in closed-form by solving appropriate optimization problems. In order to further reduce system complexity, the antenna selection (AS) problem for full-duplex AF cooperative systems is discussed. We investigate different AS schemes to select a single transmit antenna at both the source and the relay, as well as a single receive antenna at both the relay and the destination. To facilitate comparison, exact outage probability expressions and asymptotic approximations of the proposed AS schemes are provided. In order to overcome zero-diversity effects associated with the AS operation, a simple power allocation scheme at the relay node is also investigated and its optimal value is analytically derived. Numerical and simulation results show that the joint ZF-based precoding significantly improves e2e performance, while AS schemes are efficient solutions for scenarios with strict computational constraints.

I. INTRODUCTION

The paper addresses full-duplex relaying’s loopback-interference challenge by jointly optimizing precoding and decoding for end-to-end performance, while developing low-complexity antenna-selection and relay power-allocation schemes. It provides closed-form solutions and outage analyses for arbitrary-antenna MIMO terminals.

  • Motivation and challenge: Full-duplex relaying improves spectrum efficiency by simultaneous transmission and reception, but loopback interference creates a large power differential that can saturate the receiver’s analog-to-digital converter.LI mitigation is therefore critical to implementing full-duplex operation.
  • Limitations of prior work: Prior ZF designs can require antenna-number conditions and may ignore non-LI channels and end-to-end performance, producing strictly suboptimal results.The paper identifies a lack of closed-form ZF-based solutions that optimize end-to-end performance.
  • Joint precoding and decoding: The proposed design jointly optimizes source, relay, and destination precoding and decoding for achievable-rate maximization, using single-stream transmission and relay-side ZF LI handling.Closed-form precoder and decoder solutions are given for transmit and receive ZF schemes, together with diversity-order derivations.
  • Antenna selection: Several low-complexity antenna-selection schemes are proposed for MIMO full-duplex relaying, including an optimal scheme that maximizes end-to-end SNR.The paper analytically investigates antenna-selection performance and derives exact and asymptotic outage-probability expressions revealing comparative system performance and diversity effects.
  • Relay power allocation: A simple relay power-allocation method is introduced to address diversity losses associated with antenna selection and full-duplex relaying, with optimal allocation coefficients analytically investigated.The resulting outage expressions expose spatial degrees of freedom through antenna-count-dependent diversity terms.

II. SYSTEM MODEL · A. Channel Model

The system is a three-node full-duplex MIMO relay network without a direct source–destination link, modeled with independent Rayleigh block fading, AWGN, and residual loopback interference after cancellation. Channel coefficients remain stationary within each time slot and change independently between slots, representing slow fading with coding over one block.

  • II. SYSTEM MODEL: The network comprises one source S, one full-duplex relay R, and one destination D, with no direct S–D link.The missing direct link may result from heavy path loss and high shadowing.
  • II. SYSTEM MODEL: The source and destination have NT and NR antennas, while the relay has MR receive and MT transmit antennas.The relay uses two antenna groups for full-duplex operation.
  • A. Channel Model: All wireless links experience non-selective independent Rayleigh block fading and additive white Gaussian noise.The modeled channels include HSR for S–R, HRD for R–D, and HRR for loopback interference.
  • A. Channel Model: The relay applies imperfect analog/digital interference cancellation, and residual loopback interference is modeled as a fading feedback channel.This model captures remaining self-interference after cancellation at R.
  • A. Channel Model: Noise at the nodes is modeled as complex AWGN with zero mean and normalized variance.This specifies the noise distribution and normalization used in the channel model.
  • A. Channel Model: The single-input single-output coefficient from transmit antenna j at terminal X to receive antenna i at terminal Y is denoted hi,j.The notation is used for individual antenna-pair channels and their statistics.
  • A. Channel Model: Channel coefficients remain approximately stationary during one time slot, then change independently according to a Rayleigh distribution between slots.The coherence time equals one time slot, corresponding to low-mobility slow fading with coding over one block.

B. System Model … III. JOINT PRECODING/DECODING DESIGN

The paper adopts a low-complexity single-stream AF full-duplex relay model with linear processing and relay power control. It then develops rank-1 ZF-based joint precoding/decoding to eliminate loopback self-interference and maximize end-to-end performance, while treating antenna selection as a lower-complexity special case.

  • B. System Model: The system transmits one data stream and uses unit-norm linear precoding and receive processing at the terminals.Linear processing is motivated as a low-complexity solution for systems with many antennas.
  • B. System Model: AF relaying uses an amplification matrix W constrained to keep relay transmit power below PR, while t, r, and W are jointly optimized for end-to-end performance.The model does not depend on a specific analogue or baseband implementation, but practical implementation issues and frequency-selective environments are outside scope.
  • 1) Joint Precoding/Decoding Design:: The joint precoding/decoding design imposes ZF on W so that relay output produces no loopback self-interference at the relay input.The design combines relay beamforming vectors with source and destination processing under the single-stream model.
  • 1) Joint Precoding/Decoding Design:: The resulting closed-form-oriented design jointly selects processing vectors to realize zero loopback self-interference and maximize the e2e SNR.The ZF constraint is introduced to simplify the otherwise nontrivial MMSE-based optimization.
  • 2) Antenna Selection:: Antenna selection is a special case in which r and t contain one unity element, leaving only one non-zero element of W.Selection can target end-to-end SINR directly or optimize S−R, R−R, and R−D link metrics separately.
  • 2) Antenna Selection:: Sub-optimal antenna-selection schemes provide a better trade-off between implementation complexity and end-to-end performance when many antennas make optimal search costly.The optimal scheme’s search complexity is high, especially as the number of antennas at each terminal increases.
  • III. JOINT PRECODING/DECODING DESIGN: In the detailed joint-design formulation, rank-1 W is appropriate for the single data stream, and its ZF condition is enforced through relay transmit and receive beamforming.The relay input/output recursion, relay power, destination signal, and e2e SINR optimization define the subsequent design problem.

A. Receive ZF with MR > 1

For receive ZF with MR > 1, the design assumes MRT toward the destination and derives closed-form receive and transmit vectors under the ZF criterion. The resulting approach yields an achievable end-to-end SNR expression while simplifying precoder/receiver design, though separate optimization may not be globally optimal because the projection depends on the relay vector.

  • A. Receive ZF with MR > 1: MRT sets w_t = h_RD, while w_r is optimized subject to the ZF criterion.This reduces the receive-design problem to a constrained optimization involving the ZF condition.
  • A. Receive ZF with MR > 1: The optimal receive direction lies in the null space of E^-1/2H_RRh_RD and aligns with the projected source-relay channel.The projection matrix D defines the feasible ZF subspace, and the active power constraint is met with equality.
  • A. Receive ZF with MR > 1: Substituting the closed-form solutions produces an achievable e2e SNR expression.The SNR is obtained by inserting the optimal receive and transmit solutions into the objective.
  • A. Receive ZF with MR > 1: The proposed design separately optimizes t and r through the terms ∥b_Dh_SR∥^2 and ∥h_RD∥^2.The relay vector r* uniquely maximizes ∥h_RD∥^2, after which t* uniquely maximizes ∥b_Dh_SR∥^2.
  • A. Receive ZF with MR > 1: Because b_D depends on r through h_RD, the separate solutions may not be globally optimal, but they provide simple closed-form precoder/receive-vector design.Their appeal is computational simplicity and direct performance characterization.

B. Transmit ZF with MT > 1

For MT > 1, the relay uses MRC reception while optimizing a transmit zero-forcing vector. The section derives transmit-ZF solutions and expresses the resulting end-to-end SNR.

  • B. Transmit ZF with MT > 1: The relay sets w_r = h_SR, corresponding to an MRC receive beamforming vector.This fixes the receive beamformer before transmit-ZF optimization.
  • B. Transmit ZF with MT > 1: The transmit beamforming vector w_t is optimized after adopting the MRC receive beamformer.The resulting problem is simplified before solving for the transmit-ZF vector.
  • B. Transmit ZF with MT > 1: The solution of problem (25) is obtained by following the procedure used to derive (15).The derivation also invokes monotonicity in an equivalent formulation.
  • B. Transmit ZF with MT > 1: The optimized end-to-end SNR is expressed after obtaining the transmit-ZF solution.The passage introduces the optimized e2e SNR expression without stating its full formula.
  • B. Transmit ZF with MT > 1: Solutions for w_t and w_r are proposed, and substituting w_t* and w_r* into (27) yields the end-to-end SNR expression.The proposed solutions may not satisfy an unstated condition in the supplied passage.

IV. ANTENNA SELECTION … V. OUTAGE PROBABILITY ANALYSIS

The paper presents antenna selection as a lower-complexity alternative to end-to-end optimization for full-duplex MIMO relays, then derives exact and asymptotic outage results for the proposed designs. The schemes trade performance, loop-interference handling, implementation complexity, and diversity-order behavior.

  • IV. ANTENNA SELECTION: Antenna selection targets full-duplex relay systems with stricter computational or energy constraints, where loop interference creates distinctive design challenges.Several selection choices provide different performance/complexity trade-offs, and relay power allocation becomes important under full-duplex operation.
  • IV. ANTENNA SELECTION: The conventional AF amplification factor stabilizes the relay, prevents oscillation, and allows adaptive adjustment of relay transmit power to a constant level.The instantaneous end-to-end SINR is expressed using the relevant link SNRs and loop-interference INR.
  • A. Optimal Antenna Selection: Optimal antenna selection maximizes end-to-end SINR but requires high computation and knowledge of all S−R, R−R, and R−D links.Selected antenna indexes at the relay, source, destination, and relay transmit side are denoted by I, J, K, and L, respectively.
  • B. max-max Antenna Selection: The max-max scheme selects the best S−R and R−D links without considering loop interference, making it strictly sub-optimal for full-duplex relaying.Unlike its SNR-optimal role in conventional half-duplex relaying, this scheme neglects the loop-interference effect.
  • C. Partial Antenna Selection: Partial antenna selection decouples the two relaying hops, reducing the optimal solution’s search set while accounting for loop interference.It offers a good performance/implementation complexity trade-off and does not require destination-to-relay channel feedback for relay transmit-antenna selection.
  • D. Loop Interference Antenna Selection: Loop-interference antenna selection chooses receive and transmit antennas to minimize loop-interference effects while also improving the S−R and R−D channels.Its selection objective is analogous to loop-interference suppression policies using relay precoders.
  • V. OUTAGE PROBABILITY ANALYSIS: Outage analysis derives exact outage-probability expressions for the precoding/decoding designs and antenna-selection schemes, followed by asymptotic analysis.The asymptotic results expose insights including diversity order.

A. Joint Precoding/Decoding Designs · 1) Receive ZF: · SR ˘HSR

The section defines rate outage through the e2e SNR cdf and develops exact and asymptotic outage analyses for receive ZF. The receive-ZF design achieves diversity order min(NT(MR −1), MTNR).

  • A. Joint Precoding/Decoding Designs: Rate outage is defined as the probability that instantaneous mutual information I = log2(1 + γ) falls below target rate R0 BPCU.The outage event is Pout = Pr(log2(1 + γ) ≤ R0).
  • A. Joint Precoding/Decoding Designs: The outage probability equals the e2e SNR cdf evaluated at γT = 2^R0 − 1.Specifically, Pout = Fγ(γT).
  • 1) Receive ZF:: Receive ZF transforms the source-relay channel using a unitary matrix Φ, yielding bHSR = ΦHSR and an (MR −1) × NT matrix.The displayed receive-ZF construction includes SRdiag(0, 1, . . ., 1)bHSR.
  • SR ˘HSR: The receive-ZF derivation uses bD = bD†bD, eigen decomposition, and the rank-1 normalized structure of HRRhRD.The resulting norm expression is connected to an eigenvalue of a Wishart matrix.
  • SR ˘HSR: The exact receive-ZF outage is derived from the cdf of the e2e SNR by combining the γSR pdf and γRD cdf, then solving the integral in closed form.The link-average SNRs are γ̄SR = PScSR and γ̄RD = PRcRD; coefficients dl(i, j) can be computed efficiently.
  • SR ˘HSR: The full-duplex receive-ZF design achieves diversity order min(NT(MR −1), MTNR).Here s1 = min(NT, MR −1) and s2 = min(MT, NR) appear in the exact-outage formulation, while t1 = max(NT, MR −1) and t2 = max(MT, NR) support the asymptotic analysis.
  • SR ˘HSR: The asymptotic analysis uses γ ≤ min(γSR, γRD), a bound tight at medium-to-high SNR and asymptotically exact in the high-SNR regime.Simple order statistics and polynomial approximations yield the asymptotic cdf.

2) Transmit ZF: · B. Antenna Selection · 1) Optimal Antenna Selection:

The transmit-ZF design achieves diversity order min (NT MR, (MT −1)NR), while antenna-selection analysis provides exact and asymptotic outage expressions and identifies optimal-selection behavior and relay power allocation.

  • 2) Transmit ZF:: Transmit ZF achieves diversity order min (NT MR, (MT −1)NR).Its asymptotic outage characterization introduces t3 = max (NT , MR) and t4 = max (MT −1, NR).
  • 2) Transmit ZF:: Half-duplex MIMO hop-by-hop MRT/MRC beamforming has diversity order min (NT MR, MT NR), generally exceeding full-duplex transmit ZF.The two approaches offer the same diversity in certain antenna configurations.
  • B. Antenna Selection: The antenna-selection subsection derives exact and approximate outage expressions for the proposed full-duplex AS schemes as PS →∞.The outage distributions of relevant random variables depend on the particular AS scheme.
  • B. Antenna Selection: For optimal, max−max, partial, and loop-interference AS, the notation ⋆ denotes OP, MM, PR, and LI, respectively.These schemes are compared through their end-to-end outage probabilities.
  • 1) Optimal Antenna Selection:: For optimal antenna selection, an analytical outage expression is cumbersome because the maximized SINR variables are dependent.The paper therefore evaluates OP-AS outage performance through simulations, except that OP and MM coincide when MR = MT = 1.
  • 1) Optimal Antenna Selection:: As PS →∞, the OP-AS outage probability admits an asymptotic approximation whose upper and lower bounds have the same diversity order.The proof lower-bounds γOP by γMM and uses positive constants C1 and C2 in the approximation.
  • 1) Optimal Antenna Selection:: The relay’s optimal power-allocation solution is selected by minimizing the dominant asymptotic outage term, and the highest OP-AS diversity order is denoted dmax,OP.The dominant term depends on α; the corresponding expressions decay with powers involving (1−α)NT MR.

2) max −max Antenna Selection: · 3) Partial Antenna Selection:

The max–max and partial antenna-selection sections derive outage expressions and asymptotic behavior for different selection rules. They show that max–max schemes share diversity performance, whereas partial selection balances two decay terms to obtain its highest diversity order.

  • 2) max −max Antenna Selection:: Max–max antenna selection models the selected relay–destination channel as the largest of MTNR exponential random variables.The outage analysis uses auxiliary variables X and Y and requires their cdf and pdf.
  • 2) max −max Antenna Selection:: The exact max–max outage expression lacks a closed-form solution but can be evaluated numerically with standard mathematical software.The asymptotic derivation neglects higher-order product terms in the cdf expansion.
  • 2) max −max Antenna Selection:: Max–max and OP antenna selection achieve the same diversity performance.The comparison is based on the outage expressions and asymptotic forms for the two schemes.
  • 2) max −max Antenna Selection:: The max–max scheme achieves highest diversity order (MTNR)^−1+(NTMR)^−1 with optimal relay power allocation.The OP scheme has higher array gain than max–max antenna selection, as verified numerically.
  • 3) Partial Antenna Selection:: Partial antenna selection uses X = γI,J RR +1 and Y = γK,L SR γI,L RD, whose distributions differ from max–max selection.For each relay receive antenna, selection combines the strongest S–R channel with the weakest R–R channel.
  • 3) Partial Antenna Selection:: In partial selection, A is the largest among NT exponential random variables and B is the smallest among MT exponential random variables.The relay–destination component is separately modeled as the largest among NR exponential random variables.
  • 3) Partial Antenna Selection:: Partial-selection outage minimization occurs when (1 − α)NTMR = αNR, balancing the two asymptotic decay terms.The resulting highest diversity order is denoted dmax,PR and is achieved with optimal relay power allocation.

4) Loop Interference Antenna Selection:

The LI antenna-selection analysis characterizes the selected interference and desired-link gains through order statistics, then derives exact and asymptotic outage expressions. It also identifies the power-allocation condition and the highest achievable diversity order for the LI AS scheme.

  • Order-statistical characterization: The selected relay loop-interference gain RR is the minimum of MRMT exponential random variables, while selected source-relay and relay-destination gains are maxima of NT and NR exponential variables.The exponential parameters are γ̄RR, γ̄SR, and γ̄RD, respectively.
  • Exact outage probability: Combining the cdf of X with the pdf of Y yields the exact outage probability for the LI antenna-selection scheme.The derivation simplifies the integral before combining the distributions.
  • Asymptotic outage probability: For 0 < α < 1, the first asymptotic outage term decays as P^−(1−α)NT.The approximation is developed from the LI AS outage expression.
  • Power allocation: The optimal relay power-allocation parameter α is obtained by equating the exponents, (1−α)NT = αNR.This condition determines the optimum α value for the AS scheme.
  • Diversity order: The LI AS scheme’s highest achievable diversity order is represented by dmax,LI.The section gives an explicit expression for this maximum diversity order.

C. Comparisons of the Schemes · VI. NUMERICAL RESULTS · A. Joint Precoding/Decoding Designs

The paper contrasts ZF precoding and antenna-selection schemes in diversity, complexity, and channel-examination burden, then numerically shows how antenna configurations affect diversity and outage performance. Receive ZF can outperform transmit ZF when both are applicable, while AS schemes offer lower-complexity alternatives.

  • C. Comparisons of the Schemes: ZF precoding outperforms AS schemes in diversity because using all antenna elements mitigates LI, with diversity dominated by the weakest relaying branch.One antenna element is reserved for spatial cancellation at the relay’s input/output under received/transmitted ZF operation.
  • C. Comparisons of the Schemes: OP AS and MM AS achieve similar diversity and significantly outperform PR AS and LI AS, whose diversity orders are respectively independent of MT and the number of relay antennas.The passage attributes these comparisons to Table 1’s diversity-order results.
  • C. Comparisons of the Schemes: ZF designs have general complexity O(n^3), whereas AS schemes provide different complexity levels suited to networks with different computational capabilities.ZF requires RF chains for each antenna and beamforming operations including matrix multiplication, matrix inversion, and eigen-decomposition.
  • C. Comparisons of the Schemes: OP AS examines NT MRMT NR channels, MM AS examines (NT MR + MT NR) channels, PR AS examines (NT MRMT + NR) channels, and LI AS examines (NT + MRMT + NR) channels.These channel counts define the schemes’ comparative selection complexity.
  • VI. NUMERICAL RESULTS: The numerical results evaluate outage probability for the proposed precoding and AS schemes under a symmetric setup, with the AS observations also applying to asymmetric setups.The stated setup uses R0 = 2 BPCU and cSR = cRD = 1; the asymmetric condition is cSR̸ = cRD.
  • A. Joint Precoding/Decoding Designs: For receive ZF, the considered antenna configurations achieve diversity orders 1, 2, and 3 according to min (NT (MR −1), MT NR).Even with one receive antenna at D, selecting suitable source and relay design parameters improves performance.
  • A. Joint Precoding/Decoding Designs: For transmit ZF, the considered configurations again achieve diversity orders 1, 2, and 3 according to min (NT MR, (MT −1)NR).The reported receive-ZF diversity order for configuration (2, 3, 2, 3) is four.
  • A. Joint Precoding/Decoding Designs: For (2, 3, 2, 3), receive ZF achieves fourth-order diversity and superior performance, while transmit ZF achieves diversity order three; designers must choose configurations carefully.The passage notes that only one ZF form may be deployable when MT = 1 or MR = 1, whereas both can apply in other configurations.

B. Antenna Selection · VII. CONCLUSION

Antenna selection exhibits zero diversity without relay power control, but optimized power allocation restores diversity; the conclusion highlights joint rank-1 ZF designs, closed-form outage analysis, and complexity-reducing selection schemes.

  • B. Antenna Selection: Without relay power control, all full-duplex antenna-selection schemes suffer zero-diversity behavior, with OP AS best, PR AS next, and MM AS better than PR AS and LI AS at low PS.The setup uses α = 1, and PR AS converges to OP AS’s error floor.
  • B. Antenna Selection: Full-duplex antenna-selection schemes perform favorably at low-to-medium PS, whereas half-duplex transmission is superior at high PS because it avoids LI and benefits from diversity.Half-duplex selection maximizes the SNRs of the S−R and R−D links.
  • B. Antenna Selection: With optimized α, outage decays with PS and the diversity orders of OP, MM, PR, and LI AS are respectively 2, 2, 1.33 and 1.The OP AS scheme provides the best performance among the considered schemes.
  • B. Antenna Selection: At cRR = 0.5, OP AS outperforms MM AS through higher array gain, while their performance difference is almost negligible at cRR = 0.1.Both schemes provide the same diversity.
  • B. Antenna Selection: For PR AS, αopt = 0.667 is best asymptotically, while α = 0.99 and 0.9 outperform it for PS < 30 dB before low-diversity effects emerge.Values close to one can approach an error floor; α should therefore depend on the operating region.
  • VII. CONCLUSION: The study introduces joint precoding/decoding designs with rank-1 ZF self-interference suppression and derives closed-form results revealing diversity order and array gain.The results were verified through simulations.
  • VII. CONCLUSION: Outage probability depends on antenna counts and receive-ZF or transmit-ZF precoding, while several antenna-selection schemes reduce system complexity and receive exact and asymptotic outage analyses.These schemes are evaluated through outage probability expressions and approximations.
  • VII. CONCLUSION: A single-parameter relay power-allocation scheme overcomes antenna-selection zero diversity, supports desired outage performance, and has optimum values for diversity maximization.The parameter can be selected according to the desired outage performance.
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