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Channel Coding and Decoding in a Relay System Operated with Physical layer Network Coding
Shengli Zhang, Soung-Chang Liew
TL;DR
The paper addresses how to design the relay’s Channel-decoding-Network-Coding process for obtaining network-coded packets from superimposed signals. It proposes Arithmetic-sum CNC and reports substantial BER improvements over CNC1 and CNC2 without added decoding complexity, while a capacity conjecture remains unproven.
Problem
The relay’s CNC process must obtain network-coded packets from superimposed signals, while existing CNC designs retain a significant performance gap over straightforward network coding.
Method
The paper proposes Arithmetic-sum CNC, which decodes the arithmetic sum before network coding and provides an implementation based on physical-layer network coding.
Results
ACNC achieves substantial BER improvements over CNC1 and CNC2 without added decoding complexity, with 20 iterations outperforming their 40-iteration decoding.
Takeaways & Limitations
ACNC avoids shortcomings of CNC1 and CNC2 while preserving their advantages without added decoding complexity.
Takeaways & Limitations
The conjecture that ACNC could approach TWRC capacity in both low- and high-SNR regions lacks a rigorous proof.
Abstract
from arXiv · showhide
Physical-layer Network Coding (PNC) can significantly improve the throughput of wireless two way relay channel (TWRC) by allowing the two end nodes to transmit messages to the relay simultaneously. To achieve reliable communication, channel coding could be applied on top of PNC. This paper investigates link-by-link channel-coded PNC, in which a critical process at the relay is to transform the superimposed channel-coded packets received from the two end nodes plus noise, Y3=X1+X2+W3, to the network-coded combination of the source packets, S1 XOR S2 . This is in distinct to the traditional multiple-access problem, in which the goal is to obtain S1 and S2 separately. The transformation from Y3 to (S1 XOR S2) is referred to as the Channel-decoding-Network-Coding process (CNC) in that it involves both channel decoding and network coding operations. A contribution of this paper is the insight that in designing CNC, we should first (i) channel-decode Y3 to the superimposed source symbols S1+S2 before (ii) transforming S1+S2 to the network-coded packets (S1 XOR S2) . Compared with previously proposed strategies for CNC, this strategy reduces the channel-coding network-coding mismatch. It is not obvious, however, that an efficient decoder for step (i) exists. A second contribution of this paper is to provide an explicit construction of such a decoder based on the use of the Repeat Accumulate (RA) code. Specifically, we redesign the belief propagation algorithm of the RA code for traditional point-to-point channel to suit the need of the PNC multiple-access channel. Simulation results show that our new scheme outperforms the previously proposed schemes significantly in terms of BER without added complexity.
A. System model
The system is a half-duplex two-way relay channel in which end nodes transmit simultaneously to a relay, which broadcasts a function of the received signal back to both nodes.
- Channel structure: N1 and N2 exchange information through relay N3 without a direct link between the end nodes.The relay operates in two phases: simultaneous uplink transmission followed by downlink broadcasting.
- Channel structure: All nodes are half-duplex, so a node cannot receive and transmit simultaneously.
- Coding and notation: Channel coding maps each source packet Si to a coded packet Xi before BPSK transmission.The notation Γi denotes the channel-coding scheme adopted by node Ni.
- Uplink model: During the uplink, relay N3 receives a superposition of the two transmitted packets and Gaussian noise.The received packet is treated as a function of X1+X2, with perfect synchronization assumed in the basic model.
- Downlink model: During the downlink, N3 generates X3 from Y3 and broadcasts it to N1 and N2, which use self-information to decode the other packet.
B. Definitions and classification of PNC
Link-by-link coded PNC transforms the relay’s received superposition into S1 XOR S2 through channel decoding and network coding. The paper compares two conventional CNC schemes and motivates ACNC as a way to reduce their mismatch.
- Definitions: Physical-layer network coding transforms simultaneous transmissions into a network-coded packet, here formed as the GF(2) XOR S1 ⊕ S2.Link-by-link coded PNC separately protects uplinks and downlinks with channel coding.
- Definitions: CNC is the relay process that transforms Y3 into S1 ⊕ S2 through both channel decoding and network coding.Once the XOR packet is obtained, it can be channel-encoded for relay transmission.
- Arithmetic-sum CNC Design (ACNC): ACNC is proposed to follow two design principles: avoid decoding extraneous information and preserve useful information for decoding S1 ⊕ S2.The scheme performs channel decoding specifically designed for the network-coding mapping at the relay.
- Conventional schemes: CNC1 separately decodes S1 and S2 before applying XOR, thereby decoding information unrelated to the desired network-coded packet.This extraneous decoding results in an unnecessary power penalty.
- Conventional schemes: CNC2 estimates the symbol-level XOR X1 ⊕ X2 before channel decoding, but this mapping discards useful dependencies among symbols in the complete packet.It uses MMSE estimation and relies on linear channel codes at both end nodes.
Arithmetic-sum CNC Design (ACNC)
ACNC first decodes the received superposition into arithmetic-sum source symbols, then maps that result to the XOR packet. This design uses coded-symbol dependencies while avoiding unnecessary recovery of both individual source packets.
- Arithmetic-sum CNC Design (ACNC): ACNC directs the relay to decode the received packet into the superimposed source symbols before applying symbol-level PNC mapping.The resulting probability mass function for S1+S2 can be transformed into S1 XOR S2.
- Arithmetic-sum CNC Design (ACNC): The relay only needs the sign of the relevant probability difference, rather than all individual probabilities.The individual probabilities are not required for the final symbol-level mapping.
- Arithmetic-sum CNC Design (ACNC): ACNC exploits dependencies among coded symbols that symbol-level PNC mapping in CNC2 neglects.Direct decoding of Y3 makes fuller use of the information and dependency structure within the received packet.
- Arithmetic-sum CNC Design (ACNC): Unlike CNC1, ACNC avoids explicitly obtaining S1 and S2, eliminating extraneous information that constrains their reliable transmission rates.The relay instead obtains the arithmetic-sum distribution needed for network coding.
- Arithmetic-sum CNC Design (ACNC): The paper proposes an ACNC channel-coding scheme based on Repeat Accumulate codes and introduces a new relay decoding algorithm.The decoder is motivated by the previously unstudied practical requirements of joint channel coding and physical-layer network coding.
- Arithmetic-sum CNC Design (ACNC): CNC1 and CNC2 have complementary SNR weaknesses, whereas ACNC has potential for good performance across the full SNR range.CNC1 underperforms at high SNR, CNC2 at low SNR, and both near 0 dB.
A. Encoder at N1 and N2:
The two end nodes use standard RA encoders without transmitter modification. Each source packet is repeated, interleaved, and accumulated by binary summation to form a codeword.
- A. Encoder at N1 and N2:: The end nodes retain the traditional RA encoder, so no transmitter modification is required.Both nodes use the same interleaver pattern and repeat factor q.
- A. Encoder at N1 and N2:: Each input packet is repeated q times, with q ≥ 3, before interleaving and accumulation.Binary summation ⊕ produces the encoded codeword.
B. Decoder at N3:
The relay decoder is built as belief propagation on a virtual RA encoder whose input and output represent the arithmetic sums of the two source packets and codewords.
- B. Decoder at N3:: The relay processes the superposition of the two transmitted RA codewords as decoding of a virtual encoder.The virtual encoder has input Sv = S1 + S2 and output Xv = X1 + X2.
- B. Decoder at N3:: The virtual encoder preserves the RA structure but replaces binary accumulation with a function f designed for arithmetic sums.The function is derived so the virtual encoder matches the source- and codeword-sum specification.
- B. Decoder at N3:: The decoder’s symmetric function makes ambiguity between arithmetic-sum values 0 and 2 harmless for recovering the XOR result.When the sum is not 1, the decoder need not acquire the extraneous distinction between 0 and 2.
- B. Decoder at N3:: The virtual code uses a Tanner graph with information, code, evidence, and check nodes connected by local constraints.Each check node enforces the corresponding f-function relation among neighboring variable nodes.
- B. Decoder at N3:: With noise, the decoder passes posterior probabilities rather than exact symbol values through repeated right-to-left and left-to-right message updates.After several iterations, the probabilities are used to decode the arithmetic-sum information symbols.
- B. Decoder at N3:: The received symbol is rewritten as a function of the summed coded symbols and noise, supporting evidence-node probability initialization.The construction also extends to general modulation when a q-ary signal decomposes into 2 log q bits.
Update Equations for Output Messages Going Out of a Variable Node
The decoder updates probability messages across variable and check nodes using normalized belief-propagation rules, then iterates these updates until a stopping criterion is met.
- Update Equations for Output Messages Going Out of a Variable Node: Variable-node updates combine incoming probability vectors and normalize them to produce outgoing messages.The normalization factor ensures the three symbol probabilities sum to one.
- Update Equations for Output Messages Going Out of a Variable Node: The message-passing independence assumption is valid for iterations within a cycle-free neighborhood.The probability of this cycle-free condition approaches one as code length increases, allowing larger iteration depths.
- Update Equations for Output Messages Going Out of a Variable Node: Check-node updates apply the ACNC function f to incoming probability vectors to compute outgoing symbol probabilities.The resulting expression combines products of input probabilities for the possible arithmetic-sum configurations.
- Update Equations for Output Messages Going Out of a Variable Node: The proposed update rules require four real-number multiplications, matching the traditional RA decoder under the same message format.Other terms can be obtained with simple addition.
- Update Equations for Output Messages Going Out of a Variable Node: The complete algorithm initializes all messages, performs the four update classes iteratively, and outputs an information-node message when stopping criteria are satisfied.The final output is formed by combining the relevant incoming messages through variable-node operations.
V. Numerical Simulation
The simulations compare ACNC with CNC1 and CNC2 under varying iterations and packet lengths. ACNC achieves lower BER than both conventional schemes without added decoding complexity.
- BER versus SNR and iterations: BER decreases as SNR and iteration number increase for all three schemes.
- BER versus SNR and iterations: ACNC outperforms CNC2 by about 0.5 dB at BER near 10^-4 and exceeds CNC1 by an even larger gap.
- BER versus SNR and iterations: ACNC with 20 iterations outperforms both CNC1 and CNC2 with 40 iterations.
- BER versus packet length: Larger packet lengths lead to smaller BER for all schemes, while ACNC retains its advantage over CNC1 and CNC2.For all packet lengths, ACNC exceeds CNC2 by about 0.5 dB at BER 10^-4 and exceeds CNC1 by a larger gap.
- Simulation setup: The ACNC implementation uses an RA-code belief-propagation decoder tailored to the PNC multiple-access channel.The simulations use BPSK, equal power allocation, AWGN, and a repeat factor q of 3.
- Conclusion: ACNC avoids the shortcomings of CNC1 and CNC2 while preserving their advantages without added decoding complexity.The conclusion reports substantial BER improvements over both conventional schemes.
Appendix I: Decoding algorithm with non-perfect synchronization
The appendix extends the joint decoding algorithm to unequal-power, non-perfectly synchronized transmissions. It notes that synchronization imperfections can impose power penalties and describes modified decoding functions and update rules.
- Synchronization assumption: The proposed joint decoding algorithm is based on the assumption of perfect synchronization between the two end nodes.
- Synchronization assumption: In practice, perfect synchronization is difficult, especially in fading channels, and non-perfect synchronization results in power penalties.
- Unequal power allocation: For unequal power allocation, the received relay signal is represented using fading coefficients P_1 and P_2.
- Modified decoding: One extension keeps the virtual encoder unchanged and modifies the channel-related initialization for the decoding algorithm.
- Modified decoding: A second extension constructs a new function g whose virtual-encoder output supports analogous updating rules for the new decoding algorithm.
- Modified decoding: The function g satisfies the appendix's two stated design properties through its specified output cases.