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Physical Network Coding in Two-Way Wireless Relay Channels

Petar Popovski, Hiroyuki Yomo

arXiv:0707.0459v1cs.ITcs.NI

TL;DR

Two-way wireless relay channels offer a setting for physical network coding, but scheme-specific rate limits and operating choices must be characterized. The paper groups schemes into 3-step and 2-step families, derives or bounds their rates, and finds that JDF can match DNF’s upper bound for some source–relay SNR configurations.

  • Problem

    The paper addresses how to maximize two-way rates for physical network coding schemes in two-way relay channels under operational restrictions.

  • Method

    The paper groups schemes into 3-step DF and 2-step AF, JDF, and DNF families, then derives achievable rates or an upper bound for each.

  • Results

    For some source–relay SNR configurations, JDF achieves the same maximal two-way rate as DNF’s upper bound, while no scheme exceeds that bound.

  • Takeaways & Limitations

    DNF has potential for the best two-way rate, but JDF can attain DNF’s upper bound under certain source–relay SNR configurations.

  • Takeaways & Limitations

    The rates are lower bounds because the analysis restricts each round to fresh data and does not explicitly consider full-capacity Gaussian broadcast strategies.

Abstract

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It has recently been recognized that the wireless networks represent a fertile ground for devising communication modes based on network coding. A particularly suitable application of the network coding arises for the two--way relay channels, where two nodes communicate with each other assisted by using a third, relay node. Such a scenario enables application of \emph{physical network coding}, where the network coding is either done (a) jointly with the channel coding or (b) through physical combining of the communication flows over the multiple access channel. In this paper we first group the existing schemes for physical network coding into two generic schemes, termed 3--step and 2--step scheme, respectively. We investigate the conditions for maximization of the two--way rate for each individual scheme: (1) the Decode--and--Forward (DF) 3--step schemes (2) three different schemes with two steps: Amplify--and--Forward (AF), JDF and Denoise--and--Forward (DNF). While the DNF scheme has a potential to offer the best two--way rate, the most interesting result of the paper is that, for some SNR configurations of the source--relay links, JDF yields identical maximal two--way rate as the upper bound on the rate for DNF.

I. INTRODUCTION

Physical network coding applies naturally to two-way wireless relay channels, where two generic architectures combine relay-assisted communication flows in three or two steps.

  • Two-way relay channels connect nodes A and C through relay B, enabling physical network coding for reciprocal communication flows.
  • 3-step scheme: The 3-step scheme has A and C transmit separately, after which B broadcasts a network-coded combination for decoding.In the simple DF version, A and C can recover the other node’s packet using XOR with their own packet.
  • 2-step scheme: The 2-step scheme combines A and C’s transmissions over the multiple access channel before B broadcasts a scheme-dependent signal.Because the nodes transmit simultaneously in Step 1, B receives an interference mixture rather than separately decoded packets.
  • The paper seeks rate-maximizing strategies for both scheme classes, with timing governing 3-step rates and source transmission rates governing 2-step rates.The analyzed schemes are DF for three steps and AF, JDF, and DNF for two steps.

II. NOTATIONS AND DEFINITIONS

The model considers symmetric, time-invariant, half-duplex relay channels carrying only reciprocal A↔C traffic, and evaluates reliable two-way rates under restricted broadcast operation.

  • Only two flows are modeled, A → C and C → A, while relay B is neither a source nor a data sink.
  • All nodes are half-duplex, so each node transmits or receives at a given time, and the links use normalized transmit power.
  • The channels are time-invariant, all nodes know h0, h1, and h2, and the bandwidth is normalized to 1 Hz for SNR-based rate analysis.
  • The source–relay links are assumed better than the direct link, constraining the channel configurations considered.
  • The reported two-way rates are lower bounds because fresh data and potentially suboptimal relay broadcast strategies are imposed.The analysis does not explicitly include broadcast strategies achieving the full Gaussian broadcast-channel capacity region.

III. 3–STEP SCHEME

The 3-step DF scheme uses separate source transmissions followed by a relay broadcast, and its achievable two-way rate depends on coordinating the first two step durations.

  • A 3-step round consists of A transmitting, C transmitting, and B broadcasting a function of both packets from that round.
  • The relay must forward uncertainty-resolution packets so each destination can combine the broadcast with its own received information and decode the opposite packet.Random binning makes the required auxiliary packets deterministically known from the corresponding source packet.
  • The source transmission durations are parameterized by θ, with A using N(1−θ) symbols and C using Nθ symbols.
  • The maximal DF two-way rate is given by the theorem’s expression involving C(γ1), C(γ2), and C(γ0).
  • The relay’s third-step packet size comparison determines the relevant inequality for θ and therefore the rate achieved by the 3-step schedule.

1) Case 1:

Case 1 constructs a three-step DF transmission by partitioning the relay’s packet and broadcasting a shared XOR packet. Its total duration and maximal two-way rate are optimized through the parameter θ.

  • Case 1:: The relay partitions DBC into two parts according to the relative packet sizes |DBC| and |DBA|.The region |DBC| < |DBA| determines the partitioning case.
  • Case 1:: The relay forms a bitwise-XOR packet that lets A recover DBA and C recover DBC using their previously received information.A uses the recovered DBA with Step 2 information, while C uses DBC with Step 1 information.
  • Case 1:: The three-step duration is N1,DF(θ) = N(1 − θ) + Nθ + |DBA|/C(γ2).The packet lengths |DBC| and |DBA| depend on θ.
  • Case 1:: R1,DF(θ) increases monotonically with θ and is maximized at the upper limiting value θ = C(γ1) − C(γ0).Substituting this limiting value yields the two-way rate in (12).

2) Case 2:

Case 2 uses zero-padding to equalize packet lengths before XOR transmission. The resulting DF rate decreases with θ and reaches the same maximal two-way rate as Case 1.

  • Case 2:: DBC is padded with zeros so that its transmitted length matches |DBA|.A and C know |DBC| and therefore determine the required padding.
  • Case 2:: The relay broadcasts the XOR packet DpBC ⊕ DBA at rate C(γ1), allowing A to extract DBA.A then combines DBA with its Step 2 information to decode DCA.
  • Case 2:: C removes the padding zeros from DpBC to recover DBC and uses it with Step 1 information to decode DAC.The same broadcast therefore supports decoding in both directions.
  • Case 2:: The two-way rate R2,DF(θ) decreases monotonically with θ and is maximized at the minimal θ in the relevant region.Its maximal value is again given by (12).
  • Case 2:: The DF two-way rate satisfies R*DF < C(γ1) under condition (6).When γ1 = γ2, the capacity expression matches the result obtained from.

IV. 2–STEP SCHEMES

The two-step schemes use simultaneous source transmission followed by a relay broadcast. AF, JDF, and DNF share the same first-step multiple-access transmission but differ in rate selection and relay processing.

  • IV. 2–STEP SCHEMES: In Step 1, nodes A and C transmit simultaneously while relay B receives; in Step 2, B transmits to both destinations.This structure applies to AF, JDF, and DNF.
  • IV. 2–STEP SCHEMES: The source transmission rates RA and RC are scheme-dependent design choices in Step 1.Apart from selecting (RA, RC), the first-step transmission is identical across the three schemes and lasts N symbols.

A. Amplify–and–Forward (AF)

AF amplifies relay B’s received multiple-access signal and broadcasts it in an equal-length second step. The source rates are chosen from the resulting end-to-end SNRs.

  • A. Amplify–and–Forward (AF): Relay B broadcasts xB = βyB after amplifying its received signal by β.Because both steps contain N symbols, the total duration is 2N.
  • A. Amplify–and–Forward (AF): The amplification factor β is selected so relay B’s average per-symbol transmitted energy equals 1.The selection accounts for the received signal power and noise variance N0.
  • A. Amplify–and–Forward (AF): After subtracting its own signal, A observes a Gaussian channel for C’s symbols with an effective SNR.The corresponding SNR determines the rate RC at which C communicates to A.
  • A. Amplify–and–Forward (AF): The AF source-rate pair (RA, RC) is selected from the effective SNRs in the two directions.The paper then evaluates the resulting AF two-way rate.

B. Joint Decode–and–Forward (JDF)

JDF maximizes its two-way rate by selecting source transmission rates within the multiple-access channel’s achievable region and optimizing their balance across the two steps.

  • Rate-pair selection: JDF chooses (RA, RC) inside the convex region of rate pairs that node B can jointly decode in Step 1.The sum-rate is maximized on segment LALC, with time-sharing parameter λ governing the selected pair.
  • Rate-pair selection: The decoding order and padding procedure depend on whether RC exceeds RA or RA exceeds RC.When RC > RA, B pads DAC before forming the network-coded packet; the reverse-rate case uses analogous transmission logic.
  • Maximization: When γ2 exceeds γ1 + γ2, the two-way rate increases with λ and is maximized at λ = 1.Other rate pairs can also achieve the maximal rate, including pairs on segment LALE; LE satisfies RA = RC = C(γ1).

C. Denoise–and–forward (DNF)

DNF lets the relay denoise a multiple-access observation into a codeword that both end nodes use to recover each other’s packets. Its rate analysis provides an upper bound, with achievability conditional on a stated conjecture.

  • Scheme operation: DNF does not require node B to decode xA and xC individually in Step 1.Instead, B maps its received sequence yB to a denoising codeword broadcast in Step 2.
  • Scheme operation: The denoising mapping must let each end node retrieve the other packet when its own codeword is known.This property is stated for typical received sequences and jointly selected codebooks.
  • Rate bound: The DNF upper bound is achievable if the paper’s denoising-codeword conjecture is valid.The stated choice is guaranteed to provide an upper bound regardless, and equality with the achievable DNF rate is conditional.
  • Rate bound: The upper bound uses RA = C(γ1) and RC = C(γ), with γ1 ≤ γ ≤ γ2, and broadcasts the denoising codeword at rate C(γ1).The weaker source-relay link determines the common broadcast rate in the described construction.

V. NUMERICAL ILLUSTRATION

The numerical comparisons show how SNR configurations and the direct link affect the two-way rates of DF, AF, JDF, and DNF. DNF's upper bound remains highest in one configuration, while increasing the source-relay SNR improves AF and DF.

  • SNR configurations: The DNF upper bound is highest across all γ1 values when γ2 = γ1.In this configuration, RAF is below RJDF at low SNR, but AF surpasses JDF at high SNR as noise amplification becomes less significant.
  • Direct-link effects: Improving the direct-link SNR produces a significant increase in DF's two-way rate.DF is evaluated with γ0 = 0 and γ0 = γ1.
  • SNR configurations: JDF reaches the DNF upper bound at the lowest source-relay-link SNR configuration identified in the numerical comparison.The supplied figure discussion identifies this configuration through the threshold for γ2 at which JDF becomes equal to the DNF upper bound.
  • SNR configurations: Increasing γ2 improves the two-way rates of AF and DF while leaving the DNF curve unchanged.The improvement is larger for AF, which slightly outperforms DF with γ0 = γ1 at higher SNRs.

VI. CONCLUSION

The paper organizes physical network-coding methods for two-way relay channels into 3-step and 2-step schemes and derives achievable rates or an upper bound for them. No scheme exceeds DNF's upper bound, but JDF matches it under certain source-relay SNR configurations.

  • Scheme classification: The paper groups the schemes into 3-step DF and 2-step AF, JDF, and DNF categories.This provides the framework for comparing the physical network-coding strategies.
  • Rate analysis: The authors derive achievable rates for DF, AF, and JDF, plus an upper bound on DNF's achievable rate.The analysis targets two-way-rate maximization under the considered operational restrictions.
  • Main result: No evaluated scheme achieves a higher two-way rate than DNF's upper bound.This is the paper's numerical comparison across the considered schemes.
  • Main result: Under certain source-relay SNR configurations, JDF's maximal two-way rate equals DNF's upper bound.The paper identifies this equality as its most interesting result.
  • Future work: The authors leave proving DNF-bound achievability and studying efficient broadcasting effects as future work.They also propose investigating 3-step designs when the direct link exceeds one source-relay link.
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