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Relay Selection for Two-way Relaying with Amplify-and-Forward Protocols

Lingyang Song

arXiv:1111.7076v1cs.IT

TL;DR

The paper addresses limitations in applying one-way-relaying power-allocation algorithms and the unresolved OPA problem in two-way relay systems. It proposes simple Min-Max relay selection, derives and validates an asymptotic SER expression, and reports full diversity with better performance than AP-AF, especially with many relays and OPA.

  • Problem

    Power-allocation algorithms developed for one-way relaying cannot be readily used in relay systems, and the OPA optimization problem in two-way relay systems remains unresolved.

  • Method

    The paper proposes a simple Min-Max relay selection method and derives an asymptotic SER expression for the RS-AF scheme.

  • Results

    The RS-AF scheme achieves full diversity order N, outperforms conventional AP-AF relaying, and its OPA version provides considerable improvement over EPA, particularly with many relays.

  • Takeaways & Limitations

    Min-Max selection achieves almost the same performance as optimal single-relay selection, while RS-AF with OPA improves performance over EPA for large relay networks.

Abstract

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In this paper, we propose a relay selection amplify-and-forward (RS-AF) protocol in general bi-directional relay networks with two sources and $N$ relays. In the proposed scheme, the two sources first transmit to all the relays simultaneously, and then a single relay with a minimum sum symbol error rate (SER) will be selected to broadcast the received signals back to both sources. To facilitate the selection process, we propose a simple sub-optimal Min-Max criterion for relay selection, where a single relay which minimizes the maximum SER of two source nodes will be selected. Simulation results show that the proposed Min-Max selection has almost the same performance as the optimal selection with lower complexity. We also present a simple asymptotic SER expression and make comparison with the conventional all-participate amplify-and-forward (AP-AF) relaying scheme. The analytical results are verified through simulations. To improve the system performance, optimum power allocation (OPA) between the sources and the relay is determined based on the asymptotic SER. Simulation results indicate that the proposed RS-AF scheme with OPA yields considerable performance improvement over an equal power allocation (EPA) scheme, specially with large number of relay nodes.

I. INTRODUCTION

The paper addresses relay selection and SER-oriented power allocation in bi-directional AF networks, proposing RS-AF with a low-complexity Min-Max criterion and OPA.

  • Two-way relay power-allocation methods developed for one-way systems cannot be readily used because the systems operate differently and are more complex.
  • SER-oriented optimal power allocation had not been investigated for two-way relay systems, motivating study of OPA that minimizes system SER.
  • The proposed RS-AF protocol uses two sources and N relays, with one relay selected to forward network-coded signals after simultaneous source transmission.
  • The sub-optimal Min-Max criterion selects the relay minimizing the maximum SER of the two source nodes, reducing selection complexity relative to optimal selection.
  • The paper derives an asymptotic SER expression, compares RS-AF with AP-AF, and determines OPA between the sources and relay from the asymptotic SER.
  • Simulation results verify the proposed RS-AF scheme and report considerable improvement from OPA over EPA, particularly with many relays.

II. SYSTEM MODEL

The system is a two-source, N-relay bi-directional AF network using two transmission phases, relay amplification, and post-self-interference maximum-likelihood detection.

  • Each message exchange uses two phases: both sources simultaneously transmit to all relays, then one selected relay forwards the received signals to both sources.
  • The model assumes single antennas, fading coefficients constant over one frame, independent variation between frames, and perfect channel estimation with all link information available.
  • Source symbols are selected from a unity-power constellation, while source and relay transmit powers are represented by p_s and p_r.
  • The relay receives a superimposed signal from the two sources, amplifies it, and forwards it to both source nodes.
  • After each source subtracts its own information, maximum-likelihood detection recovers the other source's symbol from the received signal.

III. RELAY SELECTION FOR TWO-WAY AF NETWORKS

The RS-AF protocol selects one relay from N candidates for the second transmission phase, using pilot-assisted channel information and leaving the remaining relays idle.

  • Only one best relay is selected from N relays to forward the received analog network-coded signals in the second phase.
  • Pilot symbols transmitted by the two sources assist relay selection at the beginning of each transmission.
  • One source determines the best relay according to a selection criterion and broadcasts its index to all relays.
  • The selected relay is active in the second phase, while all other relays remain idle.

1) Optimal Relay Selection:

Optimal relay selection chooses the relay with the lowest combined destination SER across the two source nodes, conditioned on the relay-to-source channels and instantaneous SNRs.

  • The destination selects the relay minimizing the sum of the two source-node SERs.
  • Each conditional SER is evaluated for a source node given the fading coefficients from both sources to the candidate relay.
  • The SER is conditioned on instantaneous received SNR, whose expression depends on the modulation format.
  • Although optimal single-relay selection is the stated criterion, its performance is very difficult to analyze.

2) Sub-Optimal Relay Selection:

The paper introduces a lower-complexity Min-Max relay-selection criterion that chooses the relay minimizing the worse SER across the two source nodes, aiming for near-optimal performance.

  • The optimal single-relay selection is very difficult to analyze.
  • The criterion is motivated by the sum SER being typically dominated by the worst user’s SER.
  • The Min-Max criterion selects the relay that minimizes the maximum SER of the two source users.It is formulated by comparing each relay’s two source-node SERs, including their effective SNRs.
  • The selected relay is reported to achieve near-optimal SER performance with lower selection complexity.

A. Asymptotic SER of the RS-AF scheme

This section derives an analytical average SER for the proposed RS-AF scheme and develops a high-SNR asymptotic expression under Rayleigh-fading analysis.

  • The section derives the analytical average SER of the proposed RS-AF schemes.
  • The Min-Max selection scheme has almost the same performance as the optimal selection scheme.
  • The derivation assumes independent effective SNRs γ1,rk and γ2,rk when obtaining the PDF of the relay-related variable.
  • High-SNR asymptotic analysis uses the limiting behavior of modified Bessel functions, including K1(z) approaching 1/z near zero.
  • The average SER is obtained by averaging over Rayleigh fading channels and rewriting the calculation with a standard Normal random variable.
  • A diversity order of N is achieved for the proposed RS-AF scheme.

B. SER Comparison with the AP-AF Scheme

The paper compares RS-AF with conventional AP-AF using asymptotic SER analysis under a common transmit-energy and relay-power-division assumption.

  • The section derives the SER of the conventional all-participate amplify-and-forward scheme for comparison with RS-AF.
  • The AP-AF analysis models effective SNR at each source and assumes equal power division among relay nodes with equal total transmit energy.
  • For N > 1, RS-AF achieves better SER than AP-AF, and the gain increases with the number of relay nodes.
  • The comparison attributes the difference to RS-AF concentrating transmit power in the best relay, whereas AP-AF uses all relay nodes.

V. TRANSMIT POWER ALLOCATION

The paper formulates source–relay power allocation as asymptotic-SER minimization under a fixed total-power constraint and derives the resulting allocation behavior.

  • The allocation problem minimizes the RS-AF asymptotic SER over source and relay powers subject to 2ps + pr = p.The optimization also imposes positive-power bounds on ps and pr.
  • The optimum is characterized by setting Lagrangian derivatives with respect to ps and pr equal to zero.
  • The relay power should equal the total transmit power used at both sources, regardless of the number of relays.
  • The SER improvement from optimum power allocation over equal power allocation increases exponentially with the number of relay nodes.

VI. SIMULATION RESULTS

Simulations evaluate RS-AF under BPSK over Rayleigh fading with equal noise variance and compare it against AP-AF. The setup examines one source node because symmetry gives both sources the same SER.

  • Simulation setup: The AP-AF scheme is included as a comparison baseline.The simulations compare the proposed RS-AF scheme with conventional all-participate amplify-and-forward relaying.
  • Simulation setup: Simulations use BPSK modulation over Rayleigh fading channels.Both source nodes have the same SER by symmetry, so only one source node is examined.
  • Simulation setup: The total energy is p = 3, and S1, S2, and Rk use the same noise variance N0.The relay index is k = 1, . . . , N.
  • Simulation setup: The source SNR is defined as ψs = ps/N0.

A. Simulated Results

Simulations show that Min-Max relay selection closely matches optimal selection while RS-AF outperforms AP-AF. The RS-AF SER advantage over AP-AF increases with the number of relays.

  • Relay selection comparison: Min-Max selection achieves almost the same SER as optimal relay selection, especially as the relay count increases.The two methods become nearly indistinguishable with more relays, indicating near-optimal single-relay selection performance.
  • Relay selection comparison: RS-AF performance improves as the number of relays increases.
  • Comparison with AP-AF: For N = 2, 3, 4, RS-AF uses ps = pr = 1, whereas AP-AF uses ps = 1 and pr = 1/N.
  • Comparison with AP-AF: At SNR = 20 dB, RS-AF has an SER factor of about 0.6, 0.5, and 0.3 over AP-AF for N = 2, 3, and 4, respectively.The SER gain increases with the number of relay nodes.
  • Analytical verification: The asymptotic analytical SER converges to the simulated SER for optimal relay selection at high SNR.This agreement verifies the derived analytical expressions.

C. Power Allocation

The paper evaluates optimum power allocation for RS-AF and finds better high-SNR performance than equal allocation, with the improvement increasing with relay count. The selected allocation uses ps = p/4 and pr = p/2.

  • Power-allocation comparison: The OPA experiment sets ps = p/4 and pr = p/2 under the total power constraint, while EPA uses ps = pr = p/3.
  • Power-allocation comparison: With OPA, RS-AF achieves better high-SNR performance than equal power allocation.The improvement increases exponentially with the number of relay nodes, satisfying (34).
  • Optimal power ratio: The best performance occurs at λ = ps/pr = 0.5.Equivalently, ps = p/4 and pr = p/2.
  • Overall conclusions: The proposed RS-AF scheme achieves full diversity order N and better performance than conventional AP-AF.
  • Power-allocation method: The paper determines OPA between the sources and relay based on the asymptotic SER.
  • Overall conclusions: RS-AF with OPA yields considerable performance improvement over EPA, particularly with large numbers of relays.
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