Power Allocation for Compute-and-Forward over Fading Channels

TL;DR

This paper investigates optimal power allocation strategies for compute-and-forward relaying over fading channels, proposing low-complexity algorithms that effectively maximize the symmetric computation rate in both discrete and continuous channel models.

Contribution

It introduces novel properties of optimal power solutions, develops low-complexity algorithms for both discrete and continuous fading channels, and demonstrates their effectiveness through numerical results.

Findings

Water-filling type solutions for symmetric policies in discrete channels

Asymmetric policies do not follow water-filling form

Proposed algorithms achieve near-optimal or optimal solutions with low complexity

Abstract

Compute-and-forward (CF) is a relaying strategy which allows the relay to decode a linear combination of the transmitted messages. This work studies the optimal power allocation problem for the CF scheme in fast fading channels for maximizing the symmetric computation rate, which is a non-convex optimization problem with no simple analytical or numerical solutions. In the first part of the paper, we investigate the problem when there are finitely many channel states (discrete case). We establish several important properties of the optimal solutions and show that if all users share the same power allocation policy (symmetric policy), the optimal solution takes the form of a water-filling type when the power constraint exceeds a certain threshold. However, if asymmetric policies are allowed, the optimal solution does not take this form for any power constraint. We propose a low-complexity order-based algorithm for both scenarios and compare its performance with baseline algorithms. In the second part of the paper, we state relevant results when the channel coefficients are modelled as continuous random variables (continuous case) and propose a similar low-complexity iterative algorithm for the symmetric policy scenario. Numerical results are provided for both discrete and continuous cases. It is shown that in general our proposed algorithm finds good suboptimal solutions with low complexity, and for some examples considered, finds an exact optimal solution.