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Relay Selection with Network Coding in Two-Way Relay Channels

Yonghui Li, Raymond H. Y. Louie, Branka Vucetic

arXiv:1009.5145v1cs.IT

TL;DR

Two-way relay channels need practical ways to combine relay selection with network coding while controlling BER and relay complexity. This paper proposes single- and dual-relay NC selection schemes, derives and simulates their BER, and finds dual selection superior while gaining over schemes lacking selection or NC.

  • Problem

    The paper studies how to design relay selection for multi-relay two-way channels while improving bidirectional BER and transmission efficiency.

  • Method

    The authors combine relay selection with network coding and propose single- and dual-relay schemes in which selected relays decode, network-code, and forward both users' information.

  • Results

    Dual relay selection with network coding outperforms single selection and other considered schemes, while RS-NC gains over network coding without selection and relay selection without network coding.

  • Takeaways & Limitations

    Combining network coding and relay selection improves system performance and efficiency in two-way relay channels within the studied setting.

Abstract

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In this paper, we consider the design of joint network coding (NC)and relay selection (RS) in two-way relay channels. In the proposed schemes, two users first sequentially broadcast their respective information to all the relays. We propose two RS schemes, a single relay selection with NC and a dual relay selection with NC. For both schemes, the selected relay(s) perform NC on the received signals sent from the two users and forward them to both users. The proposed schemes are analyzed and the exact bit error rate (BER) expressions are derived and verified through Monte Carlo simulations. It is shown that the dual relay selection with NC outperforms other considered relay selection schemes in two-way relay channels. The results also reveal that the proposed NC relay selection schemes provide a selection gain compared to a NC scheme with no relay selection, and a network coding gain relative to a conventional relay selection scheme with no NC.

I. INTRODUCTION

The paper addresses relay selection in multi-relay two-way channels by jointly combining relay selection with network coding. It proposes single- and dual-relay schemes, analyzes them, and finds dual selection superior while gaining over schemes without selection or network coding.

  • The paper proposes practical joint relay-selection and network-coding schemes designed to minimize the average sum BER of the two end users.The schemes use decode-and-forward relaying and target improved spectral efficiency and error performance.
  • The proposed schemes select either one or two relays after both users sequentially broadcast their information, then decode, network-code, and broadcast toward both users.
  • Both proposed RS-NC schemes achieve full diversity order as if all relays were used, while dual relay selection outperforms single selection and other considered schemes.
  • The work derives analytical performance results and verifies them through Monte Carlo simulations.
  • RS-NC provides selection gain over pure network coding without relay selection and network-coding gain over conventional relay selection without network coding.The paper connects these gains to improved system performance and spectral efficiency.

II. SYSTEM MODEL

The system is a two-hop two-way relay network in which both users broadcast to all relays before selected relays network-code and forward the result. Users recover the other user's bit using their own transmitted bit.

  • The model contains two users exchanging information through N relays, with BPSK modulation and a three-step transmission process.The stated extension to other modulation schemes is straightforward.
  • Users 1 and 2 occupy the first two time slots to send their respective symbols to every relay.
  • The channel model assumes independent zero-mean unit-variance circularly symmetric complex Gaussian fading and equal noise variance.
  • Selected relays decode both users' bits, XOR them through network coding, and broadcast the resulting symbol to both users.
  • Each user estimates the network-coded bit and XORs it with its own known bit to recover the other user's information.

III. RELAY SELECTION WITH NETWORK CODING IN TWO-WAY RELAY CHANNELS

The relay-selection analysis uses decode-and-forward relaying and develops BER analysis tools for the proposed network-coded selection schemes. A simplified worst-user criterion is motivated as a practical selection basis.

  • The schemes use decode-and-forward relaying, with relays decoding received signals before forwarding them to destinations.The analysis focuses on a setting where source-to-relay links are much more reliable than relay-to-destination links.
  • The selection analysis derives BER expressions using integration-by-parts identities involving the Q function and the CDF of a random variable.
  • A high-SNR BER approximation is obtained from a first-order expansion of the relevant probability density function.

A. Single Relay Selection with Network Coding (S-RS-NC)

Single relay selection with network coding chooses one relay using a worst-user Min-Max criterion and derives exact and asymptotic BER expressions. The criterion closely matches optimal selection while achieving diversity order N.

  • The worst-user BER is almost identical to the exact two-user BER at medium-to-high SNR, and Min-Max nearly matches optimal selection across all SNRs.
  • Any error in a user's decoded network-coded bit produces an error in the recovered other-user bit, so the relay-to-user BER directly determines the user error probability.
  • The Min-Max criterion selects the relay with the smallest instantaneous BER for the worst user across the two relay-to-user links.
  • The S-RS-NC analysis provides exact and high-SNR approximate average sum BER expressions for the Min-Max criterion.
  • A diversity order of N is achieved by S-RS-NC with the Min-Max selection criterion in a network with N relay nodes.
  • Relay selection can be implemented in a decentralized way using channel-quality-dependent backoff timers, leaving unselected relays idle.

B. Dual Relay Selection With Network Coding (D-RS-NC)

D-RS-NC selects one best relay for each user, combines their network-coded transmissions with Alamouti STBC, and achieves lower BER than single-relay selection with NC.

  • The Double-Max criterion selects relay S(j) with the maximum relay-to-user-j link quality for each user.
  • At BER 10^-4, Double-Max selection is about 1 dB worse than optimal selection but is much simpler and decentralized.
  • The selected relays use Alamouti STBC to transmit simultaneously while avoiding mutual interference.
  • The D-RS-NC average sum BER has exact and high-SNR asymptotic expressions, with the asymptotic comparison showing BER reduction relative to S-RS-NC.
  • D-RS-NC can be implemented through two decentralized selection steps, one best relay for each user.
  • Dual-relay selection with NC is always superior to single-relay selection with NC in two-way relay networks.

C. Network Coding without Relay Selection (NC -No-RS)

NC-No-RS lets all relays transmit on orthogonal channels with equal total relay power, providing a baseline for evaluating relay-selection gains.

  • NC-No-RS uses all relays on orthogonal channels, allowing each receiver to separate and combine their signals.
  • The comparison fixes total relay transmission power across NC-No-RS and RS-NC schemes, with NC-No-RS power equally distributed among relays.
  • Relay selection is always superior to NC-No-RS, and its performance gain increases as the number of relays grows.

D. Conventional Relay Selection With No Network Coding (RS-No-NC)

The conventional RS-No-NC scheme selects one relay for each destination user and forwards only the other user’s information without network coding. Its BER is compared analytically with the proposed RS-NC schemes, showing that network coding provides an additional gain.

  • D. Conventional Relay Selection With No Network Coding (RS-No-NC): RS-No-NC selects one best relay for each destination and forwards each other user’s information without network coding.The two selected relays use orthogonal channels to avoid interference.
  • D. Conventional Relay Selection With No Network Coding (RS-No-NC): The S-RS-NC error rate is always half of conventional RS-No-NC, demonstrating a network-coding gain.
  • D. Conventional Relay Selection With No Network Coding (RS-No-NC): D-RS-NC achieves the maximum BER reduction among the relay-selection schemes, while S-RS-NC achieves the second-best performance.RS-No-NC is better than NC without relay selection when more than six relays are available.

IV. SIMULATION RESULTS

Simulations compare relay-selection and network-coding schemes across relay counts and validate the analytical BER expressions. D-RS-NC performs best and is near-optimal, while its advantage over S-RS-NC requires a synchronization trade-off.

  • IV. SIMULATION RESULTS: D-RS-NC performs almost as well as exhaustive optimal relay-set selection while using much lower complexity.The exhaustive scheme may select any number of relays, whereas D-RS-NC selects up to two.
  • IV. SIMULATION RESULTS: D-RS-NC outperforms the other considered schemes across the simulated relay counts.The comparisons include S-RS-NC, RS-No-NC, and NC without relay selection.
  • IV. SIMULATION RESULTS: D-RS-NC provides selection gains over NC without relay selection, while combined relay selection and network coding gain over conventional RS-No-NC.The latter gains are attributed to network coding.
  • IV. SIMULATION RESULTS: The analytical BER exactly matches simulation results for D-RS-NC, RS-No-NC, and NC without relay selection across all SNR ranges.S-RS-NC shows slight low-SNR deviation but matches simulations at high SNR.
  • IV. SIMULATION RESULTS: D-RS-NC requires strict synchronization between two selected relays, whereas S-RS-NC avoids this requirement.The schemes therefore involve a practical implementation-complexity and performance trade-off.

V. CONCLUSIONS

The paper combines network coding with relay selection for two-way relay channels, proposing single- and dual-relay schemes with simplified criteria. Dual relay selection is near-optimal and outperforms single selection, while the combined designs provide selection and network-coding gains.

  • Single relay selection uses a Min-Max criterion, whereas dual relay selection uses a Double-Max criterion selecting one best relay for each user.
  • Dual relay selection outperforms single relay selection and is near-optimal in two-way relay channels.
  • The proposed schemes provide selection gains over network coding without relay selection and network-coding gains over relay selection without network coding.
  • Double-Max dual relay selection is about 1dB from the optimal criterion, while single relay selection is easier to implement because dual selection requires relay-transmission synchronization.

A. Proof of Theorem 1

The proof derives the average BER for single relay selection with network coding by modeling the selected relay’s minimum user SNR and averaging the conditional BER.

  • The selected relay is characterized through the minimum instantaneous SNR to the two users, whose PDF is derived from the selection statistics.
  • The conditional overall BER is approximated and then averaged over the minimum selected-relay SNR to obtain the average sum BER.
  • A high-SNR first-order expansion and PDF manipulation yield the asymptotic BER expression for single relay selection with network coding.

B. Proof of Theorem 2

The proof derives BER expressions for dual relay selection with network coding and for network coding without relay selection using ordered SNR statistics, MGFs, and high-SNR expansions.

  • The derivation decomposes the BER into terms, evaluates one through the single-relay procedure, and evaluates the other using the MGF of a sum of selected SNRs.
  • Partial-fraction expansion and subsequent substitutions produce the desired closed-form BER expression for the dual-relay scheme.
  • A Taylor expansion and Laplace-transform argument establish p = N −1, yielding the high-SNR asymptotic BER expression.
  • The average BER for network coding without relay selection is obtained from the CDF of the summed SNR.
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