On the Capacity and Diversity-Multiplexing Tradeoff of the Two-Way Relay Channel (0810.3900v2)

Abstract: This paper considers a multiple input multiple output (MIMO) two-way relay channel, where two nodes want to exchange data with each other using multiple relays. An iterative algorithm is proposed to achieve the optimal achievable rate region, when each relay employs an amplify and forward (AF) strategy. The iterative algorithm solves a power minimization problem at every step, subject to minimum signal-to-interference-and-noise ratio constraints, which is non-convex, however, for which the Karush Kuhn Tuker conditions are sufficient for optimality. The optimal AF strategy assumes global channel state information (CSI) at each relay. To simplify the CSI requirements, a simple amplify and forward strategy, called dual channel matching, is also proposed, that requires only local channel state information, and whose achievable rate region is close to that of the optimal AF strategy. In the asymptotic regime of large number of relays, we show that the achievable rate region of the dual channel matching and an upper bound differ by only a constant term and establish the capacity scaling law of the two-way relay channel. Relay strategies achieving optimal diversity-multiplexing tradeoff are also considered with a single relay node. A compress and forward strategy is shown to be optimal for achieving diversity multiplexing tradeoff for the full-duplex case, in general, and for the half-duplex case in some cases.

• The paper introduces an iterative algorithm that maximizes achievable rate regions using an amplify-and-forward strategy with optimized power minimization under SINR constraints.
• It establishes a capacity scaling law showing that dual channel matching nearly attains an optimal rate, with throughput scaling as (M/2) log(K) for a large number of relays.
• The paper proposes a compress-and-forward strategy that effectively balances high data rates and robustness, achieving the optimal diversity-multiplexing tradeoff in various relay scenarios.

Capacity and Diversity-Multiplexing Tradeoff in Two-Way Relay Channels

The paper by Rahul Vaze and Robert W. Heath Jr. explores crucial aspects of two-way relay channels (TRC) in multiple input multiple output (MIMO) settings, focusing on both capacity and diversity-multiplexing tradeoff. It offers significant insights into the optimization strategies for achieving optimal relay operations when nodes in a network exchange data through multiple relays.

Optimizing Achievable Rate Regions

The authors propose an iterative algorithm designed to maximize the achievable rate region using the amplify and forward (AF) strategy at relays. They tackle the optimization problem by recasting it as a power minimization problem constrained by signal-to-interference-and-noise ratio (SINR), ensuring that Karush-Kuhn-Tucker (KKT) conditions are met for optimality. This approach necessitates global Channel State Information (CSI) at each relay, which can be resource-intensive. To mitigate this, a less complex alternative called "dual channel matching" is presented, which only requires local CSI and exhibits rate region performance closely approximating that of the optimal AF strategy.

Capacity Scaling Law and Asymptotic Behavior

In asymptotic scenarios with a large number of relays, the paper establishes a capacity scaling law for TRCs, demonstrating that the competitive rate achieved with dual channel matching differs from an upper performance bound by merely a constant term. Specifically, the throughput exponentially scales with the number of relays $K$ as $\frac{M}{2} \log K$, where $M$ denotes the number of antennas at the terminals. This nuanced demonstration is crucial for understanding how TRCs effectively remove the half-duplex rate limitation and improve spectral efficiency significantly compared to uni-directional relay channels.

Diversity-Multiplexing Tradeoff Analysis

Addressing the diversity-multiplexing tradeoff in scenarios with one relay and a direct path between communicating terminals, the authors propose a compress-and-forward (CF) strategy. This method is highlighted as optimal for achieving the desired tradeoff under full-duplex communication conditions universally and in certain cases for half-duplex scenarios. The CF strategy efficiently navigates between high data rates and robustness against errors due to signal fading, which is pivotal in wireless communication environments.

Implications and Future Directions

The findings have practical implications for the design and implementation of relay networks in wireless communications, offering guidelines on choosing relay strategies to optimize either spectral efficiency or robustness to fading, depending on network requirements. These results encourage future research into reducing complexity further for real-time applications and tackling more intricate relay configurations or high-dimensional antenna systems.

Furthermore, advancing this line of investigation into adaptive channel state information techniques could provide more flexible communication strategies, enhancing the applicability of the derived results from this paper to dynamic environments with fluctuating connectivity. Building upon the promising results seen from dual channel matching and compress-and-forward could lead to further innovation in relay-based communication systems.

In conclusion, Vaze and Heath Jr.'s contribution is a vital step forward, providing clarity on operational strategies that leverage the two-way relay channel configurations for optimal communication efficiency within the constraints typical to wireless networking infrastructure.