Joint Physical Layer Coding and Network Coding for Bi-Directional Relaying

Abstract: We consider the problem of two transmitters wishing to exchange information through a relay in the middle. The channels between the transmitters and the relay are assumed to be synchronized, average power constrained additive white Gaussian noise channels with a real input with signal-to-noise ratio (SNR) of snr. An upper bound on the capacity is 1/2 log(1+ snr) bits per transmitter per use of the medium-access phase and broadcast phase of the bi-directional relay channel. We show that using lattice codes and lattice decoding, we can obtain a rate of 1/2 log(0.5 + snr) bits per transmitter, which is essentially optimal at high SNRs. The main idea is to decode the sum of the codewords modulo a lattice at the relay followed by a broadcast phase which performs Slepian-Wolf coding with structured codes. For asymptotically low SNR's, jointly decoding the two transmissions at the relay (MAC channel) is shown to be optimal. We also show that if the two transmitters use identical lattices with minimum angle decoding, we can achieve the same rate of 1/2 log(0.5 + snr). The proposed scheme can be thought of as a joint physical layer, network layer code which outperforms other recently proposed analog network coding schemes.

• The paper presents a novel joint coding scheme that integrates lattice decoding with network coding to approach theoretical capacity in AWGN channels.
• The proposed method establishes an upper bound of ½ log(1 + SNR) per transmitter and achieves near-optimal rates at both high and low SNR conditions.
• This work demonstrates the practical advantage of combining physical layer and network coding to enhance efficiency in bi-directional relay communications.

Overview of Joint Physical Layer and Network Coding for Bi-Directional Relaying

This paper investigates the two-way relay problem, where two transmitters exchange information via a central relay. The focus is on additive white Gaussian noise (AWGN) channels, characterized by synchronization and average power constraints. The research presents a nuanced analysis of capacity limits and proposes a scheme that integrates lattice coding and decoding techniques to achieve optimal rates.

The authors establish an upper bound on the exchange capacity, defined in terms of mutual information, to be $\frac{1}{2} \log(1 + \text{SNR})$ per transmitter for both access and broadcast phases. Utilizing lattice codes, the study demonstrates that an achievable rate approaches this bound at high signal-to-noise ratios (SNR). Specifically, a rate of $\frac{1}{2} \log(\frac{2 + \text{SNR}}{2})$ per transmitter is attainable using lattice coding with dither and lattice decoding.

Key to the approach is decoding the sum of codewords modulo a lattice at the relay. This is followed by employing Slepian-Wolf coding with structured codes during the broadcast phase. At low SNR, the proposed joint decoding scheme is shown to be nearly optimal, achieving a rate of $\frac{1}{4} \log(1 + 2P/\sigma^2)$.

The paper references significant prior work on bi-directional relay coding, highlighting the progression from pure network coding schemes to those considering physical layer joint coding advantages. It points out that previous network coding approaches failed to fully exploit physical layer interactions, unlike the proposed joint physical and network coding solutions.

Main Results and Comments

The study achieves several vital results:

• An exchange capacity ceiling of $\frac{1}{2} \log(1 + P/\sigma^2)$ per node is demonstrated.
• High SNR rates closely align with this bound due to lattice techniques, with $\frac{1}{2} \log(\frac{2 + \text{SNR}}{2})$ being shown as achievable.
• At low SNR, a structured joint decoding scheme performs optimally, achieving nearly maximum rates obtainable under the conditions.
• These methods significantly outperform recently suggested analog network coding tactics over varying SNRs.

Implications and Future Directions

This work contributes to the broader understanding of how structured codes, such as lattices, can outperform unstructured random codes in network scenarios, especially under Gaussian noise. It opens avenues for research in optimizing joint physical and network layer coding in distributed systems, potentially influencing multi-user communication systems and cooperative communication networks.

Future developments could include exploring alternative lattice structures or different decoding strategies to further enhance performance, particularly at lower SNR. Further, considering non-orthogonal MAC and broadcast phase capabilities could yield new insights.

Conclusion

The paper presents a rigorous exploration of joint physical layer and network coding strategies, offering a refined method to approach the limits of bi-directional relay channel capacities. The results not only apply foundational information theory principles but also propose tangible coding solutions that extend the capabilities of existing network schemes. The synthesis of lattice-based coding reflects a progressive step in addressing practical communication challenges in relay networks.