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Asynchronous Physical-layer Network Coding

Lu Lu, Soung Chang Liew

arXiv:1105.3144v4cs.ITcs.NI

TL;DR

PNC must address signal asynchronies and channel coding for reliable communication. The paper uses joint source/code-node processing and reports reduced asynchrony penalties, including below 1 dB under half-symbol misalignment, while channel-coded performance is not very sensitive to offsets.

  • Problem

    PNC must address asynchronies between transmitted signals, while channel coding is applied for reliable communication.

  • Method

    The approach represents source or code nodes as pairs containing the symbols or code bits transmitted by the two users.

  • Results

    With half-symbol misalignment, the method reduces the penalty to less than 1 dB, while channel-coded performance is not very sensitive to carrier-phase offsets.

  • Takeaways & Limitations

    The results support introducing a half-symbol offset in unchannel-coded PNC and indicate that asynchronies are not major concerns when channel coding is used.

Abstract

from arXiv · show

A key issue in physical-layer network coding (PNC) is how to deal with the asynchrony between signals transmitted by multiple transmitters. That is, symbols transmitted by different transmitters could arrive at the receiver with symbol misalignment as well as relative carrier-phase offset. A second important issue is how to integrate channel coding with PNC to achieve reliable communication. This paper investigates these two issues and makes the following contributions: 1) We propose and investigate a general framework for decoding at the receiver based on belief propagation (BP). The framework can effectively deal with symbol and phase asynchronies while incorporating channel coding at the same time. 2) For unchannel-coded PNC, we show that for BPSK and QPSK modulations, our BP method can significantly reduce the asynchrony penalties compared with prior methods. 3) For unchannel-coded PNC, with half symbol offset between the transmitters, our BP method can drastically reduce the performance penalty due to phase asynchrony, from more than 6 dB to no more than 1 dB. 4) For channel-coded PNC, with our BP method, both symbol and phase asynchronies actually improve the system performance compared with the perfectly synchronous case. Furthermore, the performance spread due to different combinations of symbol and phase offsets between the transmitters in channel-coded PNC is only around 1 dB. The implication of 3) is that if we could control the symbol arrival times at the receiver, it would be advantageous to deliberately introduce a half symbol offset in unchannel-coded PNC. The implication of 4) is that when channel coding is used, symbol and phase asynchronies are not major performance concerns in PNC.

I. INTRODUCTION

PNC must address symbol and carrier-phase asynchronies while integrating channel coding for reliable communication. The paper develops a unified BP-based approach and shows that appropriate decoding can substantially reduce unchannel-coded penalties, while channel-coded systems can benefit from asynchrony.

  • PNC signals may arrive with symbol misalignment and relative carrier-phase offsets, creating a central synchronization challenge.
  • Earlier studies reported worst-case BER penalties of 3 dB for BPSK and up to 6 dB for QPSK under asynchrony.
  • The paper proposes a unified BP-based ML decoding framework that jointly handles symbol and phase asynchronies.
  • For unchannel-coded PNC, the method reduces the BPSK worst-case penalty from 3 dB to less than 0.5 dB.
  • With half-symbol misalignment, QPSK phase-asynchrony penalties fall from more than 6 dB to less than 1 dB, with phase-offset spread no more than 0.5 dB.
  • With channel coding, symbol and phase asynchronies improve BER performance, and all offset combinations have a spread of only around 1 dB.

C. Channel-coded PNC

This section examines channel-coded PNC, emphasizing joint channel decoding and network coding under asynchronous conditions. The paper reports that asynchronies can improve performance and reduce sensitivity to phase offsets, while the proposed joint scheme outperforms disjoint processing.

  • Channel-coded PNC must integrate channel decoding with physical-layer network coding under asynchronous transmissions.
  • The proposed Jt-CNC jointly performs channel decoding and network coding, unlike prior schemes that process them disjointly.
  • The analysis uses Repeat-Accumulate codes and suggests that the general conclusions should also hold for LDPC codes.
  • The paper assumes perfectly estimated channels and leaves detailed channel-estimation procedures outside its treatment.

IV. UNCHANNEL-CODED PNC

Unchannel-coded PNC transmits uncoded packets and uses BP to map asynchronously received signals into network-coded packets. The synchronous case provides the baseline XOR mapping and posterior decision rule.

  • In unchannel-coded PNC, each end node transmits source information directly without channel coding.
  • BP is applied in the PNC mapping process to handle phase and symbol asynchronies.
  • Synchronous unchannel-coded PNC assumes φ = 0 and ∆ = 0, yielding the received packet YR = XA + XB + WR.
  • For BPSK, the relay maps the received signal to an XOR output, with xR = 1 for equal symbols and xR = −1 otherwise.
  • The synchronous MAP decision compares posterior probabilities of equal-symbol and unequal-symbol pairs.

B. BP-UPNC: A Belief Propagation based Unchannel-coded PNC Scheme

BP-UPNC jointly decodes asynchronous unchannel-coded PNC using a Tanner graph built from oversampled observations and compatibility constraints. Its tree structure enables exact posterior computation with rapid convergence.

  • The asynchronous model allows φ ≠ 0 and/or ∆ ≠ 0 and produces 2N + 1 received samples for relay processing.
  • BP-UPNC is a belief-propagation decoding scheme for jointly handling symbol and phase asynchronies in unchannel-coded PNC.
  • The Tanner graph contains evidence nodes for YR, compatibility nodes Ψ, and source-variable nodes X.
  • Compatibility functions model correlations between adjacent joint symbols by enforcing shared-symbol consistency.
  • Because the Tanner graph is a tree, BP obtains exact a posteriori probabilities after one message-passing iteration.
  • The decoded joint-symbol posterior is converted into a MAP XOR value for network-coded output.

2) Message Update Rules:

The message-update procedure propagates BP messages across the compatibility graph in both directions. It then combines the converged messages to select the maximum-likelihood XOR result.

  • Message Update Rules: BP updates right-bound messages first and then left-bound messages across the compatibility-node graph.
  • Message Update Rules: Messages are represented as Qk and Rk around the k-th compatibility node, while Pk is the associated evidence-node probability vector.
  • Message Update Rules: Each update follows the sum-product principle, keeping node outputs consistent with possible input combinations.
  • Message Update Rules: The compatibility updates use shared symbols between adjacent joint-symbol states and normalize resulting probabilities.
  • Message Update Rules: The procedure repeats directional updates until all successive right-bound and left-bound messages are computed.
  • Message Update Rules: After message convergence, the decoder chooses the a⊕b value corresponding to the largest among four joint-probability elements.

C. Numerical Results

BP-UPNC substantially reduces the BER penalties caused by symbol and phase asynchrony in unchannel-coded BPSK and QPSK PNC. A half-symbol offset particularly improves robustness to phase asynchrony through diversity and certainty propagation.

  • QPSK: 6–7 dB phase-asynchrony penalties with aligned QPSK symbols fall to less than 1 dB under half-symbol misalignment.The comparison is against the perfectly synchronized benchmark.
  • Implication: The paper therefore identifies deliberate half-symbol misalignment as advantageous for controllable unchannel-coded PNC.This implication assumes that symbol arrival timings can be controlled.
  • QPSK: A half-symbol offset makes QPSK performance less sensitive to phase offset, with fixed-BER SNR spread below 0.5 dB.The paper attributes the improvement to diversity and certainty propagation in BP.
  • Mechanism: With a half-symbol offset, good constellation points provide diversity, while BP propagates symbol certainty in both directions and reduces BER.At zero offset, about half of the joint symbols are bad constellation points; at offset 0.5, about N joint symbols are good.

V. CHANNEL-CODED PNC

This section extends belief-propagation decoding to channel-coded PNC, where the relay jointly handles received coded signals and network coding. It considers alternative CNC constructions and highlights the importance of decoder design under asynchronous reception.

  • BP framework: The BP framework incorporates channel coding by transforming relay observations into a network-coded packet through channel decoding and network coding.The relay can either map received symbols directly or detect the source-packet XOR before re-encoding it; the paper studies the latter.
  • Implication: In channel-coded PNC, an appropriate CNC scheme can make symbol and phase asynchronies less performance limiting.The paper reports that symbol offset can produce a positive effect and phase asynchrony can improve performance.
  • Decoder designs: The paper studies Jt-CNC and XOR-CD as two belief-propagation-based channel-coded PNC methods.Jt-CNC jointly integrates channel decoding and network coding, while XOR-CD uses a two-step process.
  • Relay objective: The channel-coded relay operation can decode the source-packet XOR rather than recover the two source packets individually.This network-coded decoding target distinguishes the CNC process from ordinary point-to-point channel decoding.

B. Jt-CNC: A Joint Channel-decoding and Network-Coding Scheme

Jt-CNC jointly performs channel decoding and network coding using a Tanner graph built from asynchronous relay observations. Its joint inference retains more information than XOR-CD, while channel-code-induced loops make BP approximate rather than exact ML decoding.

  • Jt-CNC: Jt-CNC computes joint source-symbol probabilities from asynchronous relay observations and then uses them to construct a Tanner graph.The graph uses oversampled evidence and iteratively updates messages until joint source probabilities are obtained.
  • Graph construction: The Jt-CNC graph represents each source or code node as a pair containing the symbols from both transmitters.This paired representation supports joint channel-decoding and network-coding inference.
  • Decoding property: Because channel coding introduces loops, BP-computed XOR probabilities in Jt-CNC are approximate and require multiple message-update rounds.The unchannel-coded BP graph is tree-structured, whereas the channel-coded graph is not.
  • XOR-CD: XOR-CD first computes soft information for XORed channel-coded symbols and then channel-decodes that information to obtain the source-packet XOR.For a linear code, its channel decoder can match a conventional point-to-point decoder.
  • Comparison: Jt-CNC achieves significantly better BER performance than XOR-CD, by 3 dB on average for both BPSK and QPSK.The comparison is reported for the channel-coded simulations.
  • Comparison: XOR-CD is simpler to implement than Jt-CNC because linear-code properties permit decoding after XORing, but it discards joint-symbol information.The discarded information contributes to its weaker performance relative to Jt-CNC.

D. Numerical Results

Channel-coded PNC with Jt-CNC turns the adverse effects of asynchrony into performance gains and remains relatively insensitive to phase offsets. Jt-CNC also substantially outperforms the simpler XOR-CD scheme.

  • Jt-CNC results: The QPSK channel-coded BER spread across phase offsets is no more than 1 dB, with or without symbol misalignment.This indicates limited sensitivity across the tested phase-offset combinations.
  • Jt-CNC versus XOR-CD: Jt-CNC achieves 3 dB better BER performance on average than XOR-CD for both BPSK and QPSK.XOR-CD is less complex but has significantly worse performance and exhibits phase penalty rather than phase reward.
  • Jt-CNC results: Symbol misalignment further desensitizes Jt-CNC performance to phase asynchrony.With a 0.5-symbol offset, the required SNR for a target QPSK BER varies little with phase offset.
  • Conclusion: With an appropriate CNC algorithm such as Jt-CNC, symbol and phase asynchronies are not performance-limiting factors in channel-coded PNC.This is the paper’s general conclusion for the channel-coded setting.
  • Jt-CNC results: Phase asynchrony improves Jt-CNC performance instead of imposing a penalty in channel-coded PNC.For QPSK, channel coding changes the aligned-symbol phase penalty from 6–7 dB in the unchannel-coded case to a reward of around 0.5 dB.

E. Diversity and Certainty Propagation in Jt-CNC

Jt-CNC uses channel coding and joint BP-based processing to create diversity and certainty-propagation effects that improve BER performance and reduce sensitivity to phase and symbol asynchrony. Unlike XOR-CD, whose disjoint processing weakens certainty propagation, Jt-CNC remains robust with or without symbol misalignment.

  • Diversity and certainty propagation: Jt-CNC exhibits these positive effects with or without symbol misalignment, unlike unchannel-coded PNC, where they occur only when symbols are misaligned.
  • Phase asynchrony: Even with perfectly aligned symbols, Jt-CNC can remove phase penalties through diversity and certainty propagation, making phase asynchrony less critical.
  • XOR-CD comparison: XOR-CD retains significant phase-asynchrony penalties because it uses XOR probabilities and loses individual-symbol certainties needed for propagation.
  • Unchannel-coded PNC: For unchannel-coded PNC, BP decoding reduces phase-asynchrony penalties under symbol misalignment through diversity and certainty propagation.
  • Unchannel-coded PNC: With half-symbol misalignment, phase-offset sensitivity falls to around 0.5 dB instead of more than 6 dB for perfectly aligned symbols, motivating deliberate misalignment.
  • Processing schemes: Jt-CNC jointly performs network coding and channel decoding, whereas XOR-CD separates these operations before decoding.
  • Performance under asynchrony: Across channel-coded PNC, symbol and phase asynchronies can produce performance rewards, with the spread across offset combinations only slightly more than 1 dB.
  • Diversity and certainty propagation: Channel coding embeds each source symbol across multiple coded symbols, allowing Jt-CNC to exploit correlations for diversity and certainty propagation.

MESSAGE UPDATE STEPS OF JT-CNC

Jt-CNC updates belief-propagation messages through code, check, and source nodes in alternating upward and downward directions. The updates incorporate additional incoming messages and QPSK XOR mappings before passing normalized probabilities onward.

  • Message-update sequence: Jt-CNC updates messages below code nodes X, then propagates information upward into check nodes C and source nodes S.The procedure is organized into successive message-update steps across the Tanner graph.
  • Graph operations: Messages emanating from compatibility nodes are updated as in the earlier BP-UPNC procedure, while PNC mapping is performed at triangle nodes.The Tanner-graph figures identify the compatibility-node updates and the triangle-node PNC mapping.
  • Message-update sequence: Even code nodes receive two additional incoming messages, U↓ and V↓, from check nodes above before right-bound message updates.These messages must be incorporated when updating the right-bound message below X.
  • QPSK message mapping: QPSK source-node updates combine 256 input combinations into 16 output possibilities using the sum-product principle.Each XOR check-node input has 16 possibilities, and each output probability sums 16 input probabilities.
  • QPSK message mapping: The QPSK XOR combines corresponding real and imaginary XOR components to form the output symbol.For example, the component-wise mapping produces a = −1 + j from the specified input pair.
  • Downward propagation: After upward updates, messages flow downward into check nodes and code nodes, with normalization factors maintaining probability sums.The downward procedure includes updates from source nodes to check nodes and from check nodes to code nodes.
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