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Physical Layer Network Coding Over Finite And Infinite Fields

Zhang Shengli, Soung chang Liew, Lu Lu

arXiv:0804.2058v1cs.NI

TL;DR

The paper addresses how to define and organize physical-layer network coding in two-way relay networks, where the relay processes combined signals rather than separately decoding both sources. It defines PNC, classifies schemes as finite-field or infinite-field, and evaluates two estimation methods for each class. MMSE schemes outperform non-MMSE schemes, while the better PNC class depends on uplink and downlink quality.

  • Problem

    Physical-layer network coding lacks a precise definition and unified classification for relay processing over finite or infinite network-code structures.

  • Method

    The paper defines PNC, divides it into PNCF and PNCI, and studies two noisy-signal estimation techniques for each class under symbol-level synchronization.

  • Results

    MMSE schemes outperform non-MMSE schemes; PNCF performs better with good uplink and bad downlink channels, whereas PNCI performs better in the reverse condition.

  • Takeaways & Limitations

    PNC performance depends jointly on the network-code domain, estimation method, and relative uplink and downlink channel quality.

Abstract

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Direct application of network coding at the physical layer - physical layer network coding (PNC) - is a promising technique for two-way relay wireless networks. In a two-way relay network, relay nodes are used to relay two-way information flows between pairs of end nodes. This paper proposes a precise definition for PNC. Specifically, in PNC, a relay node does not decode the source information from the two ends separately, but rather directly maps the combined signals received simultaneously to a signal to be relayed. Based on this definition, PNC can be further sub-classed into two categories - PNCF (PNC over finite field) and PNCI (PNC over infinite field) - according to whether the network-code field (or groups, rings) adopted is finite or infinite. For each of PNCF and PNCI, we consider two specific estimation techniques for dealing with noise in the mapping process. The performance of the four schemes is investigated by means of analysis and simulation, assuming symbol-level synchronization only.

I. INTRODUCTION

The paper formalizes physical-layer network coding (PNC), distinguishes it from straightforward network coding, and classifies schemes by whether their network-code structure is finite or infinite. It then separates relay processing into network-code selection and noisy signal estimation, studying four schemes with symbol-level synchronization.

  • I. INTRODUCTION: PNC is precisely defined to distinguish direct physical-layer processing from traditional straightforward network coding.The distinction concerns how the relay handles signals in the wireless two-way relay setting.
  • I. INTRODUCTION: PNC schemes are classified as PNCF or PNCI according to whether the adopted network-code field, group, or ring is finite or infinite.
  • I. INTRODUCTION: A PNC scheme combines network-code determination at the relay with computation of the relayed information from the two received end-node signals.
  • I. INTRODUCTION: The paper studies two signal-estimation techniques for each PNC class, yielding four schemes evaluated under symbol-level synchronization only.

A. System model

The system is a half-duplex two-way relay channel without a direct end-node link, so communication uses simultaneous uplink transmission followed by relay broadcast. The relay forms its transmitted signal from the received signal, and destinations use self-information to decode.

  • A. System model: Half-duplex operation and the absent direct link require separate uplink and downlink phases in the two-way relay channel.During the uplink, both end nodes transmit to the relay simultaneously.
  • A. System model: The relay receives simultaneous transmissions from both end nodes during the uplink phase.
  • A. System model: The model assumes complex Gaussian noise with unit variance, no fading, QPSK modulation, normalized transmitting power of 2, symbol-level synchronization, and perfect channel-coefficient estimation.
  • A. System model: During downlink, the relay generates x3 from y3 and broadcasts it to both end nodes, which decode using their self-information.

B. Definition and classification of PNC

PNC directly maps the relay’s received baseband signal into a network-coded symbol without separately detecting the two source symbols. The resulting schemes are classified as finite-field or infinite-field coding according to the network-code domain.

  • B. Definition and classification of PNC: Unlike SNC, where the relay separately decodes both source messages before coding, PNC directly transforms the combined received signal into a network-coded symbol.The relay need not identify the individual source information first.
  • B. Definition and classification of PNC: PNCF uses a finite-field network code such as GF(2), while PNCI uses an infinite-field code such as the real or complex field.
  • B. Definition and classification of PNC: With noise, PNCF estimates x1 ⊕ x2 from the relay’s received signal rather than applying a noiseless deterministic transformation.
  • B. Definition and classification of PNC: For QPSK, the finite-field combination is represented through separate addition of the real and imaginary parts.
  • B. Definition and classification of PNC: For PNCI, the network code is matched to the multiple-access channel through a real or complex-field combination, then estimated from the noisy relay observation.

III. PARTICULAR PNC SCHEMES

Given a network code, the relay can use different estimation techniques to compute the relayed value. This section considers three specific estimation functions for particular PNC schemes.

  • III. PARTICULAR PNC SCHEMES: Different relay estimation techniques can produce different PNC schemes even when the network code is fixed.
  • III. PARTICULAR PNC SCHEMES: The section considers three specific estimation functions at the relay node.

A. PNCF schemes

The PNCF schemes estimate the finite-field combination x_1 ⊕ x_2 from the relay observation using MAP or MMSE methods, with symbol-level synchronization only. MAP-based PNCF generalizes the original PNC scheme, while MMSE-based PNCF generalizes an estimate-and-forward scheme and supports both phase-synchronized and unsynchronized cases.

  • A.1 MAP-based PNCF: PNCF estimates x_1 ⊕ x_2 from the relay observation using MAP and MMSE estimation methods.The MAP estimator produces a discrete estimate, whereas the MMSE estimator produces a continuous estimate.
  • A.1 MAP-based PNCF: Under power and carrier-phase synchronization, MAP-based PNCF becomes equivalent to the original PNC scheme.The general MAP formulation therefore extends the original scheme beyond the stronger synchronization condition.
  • A.2 MMSE-based PNCF: MMSE-based PNCF estimates x_1 ⊕ x_2 conditionally on y_3 and applies a coefficient to satisfy the average-power constraint.The packet-level MMSE estimate is formed by estimating each symbol and imposing an overall power constraint.
  • A.2 MMSE-based PNCF: With stronger synchronization, MMSE-based PNCF matches the estimate-and-forward formulation while using the optimal threshold.Unlike the earlier formulation, the MMSE-based scheme does not require carrier-phase synchronization and encompasses both synchronized and unsynchronized cases.
  • PNCI: Linear-MMSE-based PNCI is the ANC scheme of and depends only on the absolute channel coefficients.Consequently, carrier phase offset between the end-node signals does not affect its performance.
  • PNCI: General MMSE-based PNCI is motivated by the non-Gaussian QPSK signal distribution, for which unconstrained MMSE should outperform linear MMSE.The general PNCI estimator is derived from the conditional MMSE estimate of the combined received signal.

IV. PERFORMANCE ANALYSIS AND SIMULATION

The performance analysis uses GSNR to evaluate the four PNC schemes and compares it with BER through numerical simulation. GSNR provides a tractable measure for nonlinear relay functions, and the simulations show correspondence between larger GSNR and smaller BER in these systems.

  • IV. PERFORMANCE ANALYSIS AND SIMULATION: The study analyzes GSNR for all four PNC schemes and compares GSNR and BER through numerical simulation.The analysis follows the scheme definitions and evaluates relay and destination performance.
  • IV. PERFORMANCE ANALYSIS AND SIMULATION: The destination SNR for a linear relay function includes both forwarded relay noise and destination noise.For nonlinear relay functions, the conventional SNR expression is not directly applicable.
  • IV. PERFORMANCE ANALYSIS AND SIMULATION: GSNR extends SNR analysis to nonlinear relay functions by representing the received signal using an uncorrelated error.For nonlinear regeneration, the received signal cannot be decomposed directly into the data signal plus independent noise.
  • IV. PERFORMANCE ANALYSIS AND SIMULATION: GSNR is easier to analyze than BER for general relay functions and is more convenient for theoretical analysis.The metric was originally proposed for one-way relay channels and is adapted here to PNC systems.
  • IV. PERFORMANCE ANALYSIS AND SIMULATION: Numerical simulations show correspondence between GSNR and BER for the PNC systems, with larger GSNR associated with smaller BER.This relationship was previously observed for BPSK and is also reported here for the studied PNC schemes.

B. GSNR Analysis

The GSNR analysis derives relay and destination expressions for MAP- and MMSE-based PNCF and PNCI schemes. These expressions use mean square uncorrelated error, with scheme-specific error probabilities or conditional estimation errors.

  • B. GSNR Analysis: The section derives GSNR expressions for all four PNC schemes using the relay’s mean square uncorrelated error.The resulting destination GSNR expressions are obtained from scheme-specific MSUE calculations.
  • B. GSNR Analysis: For MAP-based PNCF, MSUE depends on the probabilities of real-part, imaginary-part, and joint estimation errors.These probabilities are calculated from the MAP estimator and determine the power-constraint coefficient and relay MSUE.
  • B. GSNR Analysis: For MMSE-based PNCF, the relay MSUE is obtained from the conditional MMSE estimate of the finite-field combination.The destination GSNR follows by substituting this MSUE into the destination expression.
  • B. GSNR Analysis: For linear-MMSE PNCI, the relay’s uncorrelated error is the normalized Gaussian noise, simplifying the MSUE analysis.The destination GSNR is then expressed in terms of the resulting MSUE and channel coefficients.

MMSE-based PNCI

The paper proves that MMSE-based estimation is optimal within both PNCF and PNCI: it minimizes MSUE at the relay and destination nodes and therefore maximizes GSNR. The proof links relay MSUE minimization to destination GSNR maximization.

  • MMSE-based PNCI: MMSE-based PNCF minimizes MSUE and maximizes GSNR at the relay and both destination nodes among all PNCF schemes.This is stated as Theorem 1 for the considered system.
  • MMSE-based PNCI: MMSE-based PNCI minimizes MSUE and maximizes GSNR at the relay and both destination nodes among all PNCI schemes.This is stated as Theorem 2 for the considered system.
  • MMSE-based PNCI: The optimality argument has two steps: MMSE minimizes relay MSUE, and relay MSUE minimization is equivalent to destination GSNR maximization.The equivalence is established using the destination analyses in the appendices.

C. Simulation Result

Simulations compare MMSE and non-MMSE estimators across PNCF and PNCI under symmetric-channel conditions, showing that relative class performance depends on uplink and downlink quality.

  • MMSE estimation outperforms non-MMSE estimation for both PNCF and PNCI in GSNR and MSUE.The reported BER behavior generally corresponds to GSNR: larger GSNR usually leads to smaller BER with QPSK modulation.
  • PNCF performs better than PNCI when the uplink channel is good, whereas PNCI performs better when the uplink channel is bad.With a good uplink, estimation errors are negligible, while the PNCI signal requires substantially more transmit power than the PNCF signal.
  • PNCF performs better than PNCI when the downlink channel is bad, whereas PNCI performs better when the downlink channel is good.When the downlink is very good, PNCF loses information by transforming the PNCI signal into the finite-field network-coded signal.

V. CONCLUSION

The paper defines PNC as direct mapping of simultaneous end-node signals into a network-coded relay symbol and distinguishes finite- and infinite-field subclasses. It reports MMSE advantages and channel-dependent performance differences, while focusing on memoryless relaying without relay-side channel coding.

  • PNC maps simultaneous signals from two end nodes directly into a network-coded symbol instead of separately decoding their information.Different mapping functions produce different PNC schemes.
  • PNCF uses finite-field network codes, whereas PNCI uses infinite-field network codes.
  • MMSE schemes outperform non-MMSE schemes, while PNCF is better for good uplink and bad downlink channels, and PNCI is better in the reverse conditions.
  • The study focuses on memoryless relay protocols without channel coding at the relay, and PNCI lacks a channel code supporting decoding over its complex-field symbol.For PNCF, linear LDPC or Turbo codes over GF(2) can support decoding of the finite-field network-coded symbol.
  • The analysis relates destination GSNR maximization to relay MSUE minimization for both PNCF and PNCI schemes.This equivalence is stated for the destination GSNR expressions in both network-code classes.
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