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.

翻译：计算转发（CF）是一种中继策略，允许中继节点解码发送消息的线性组合。本文研究了快衰落信道下CF方案中最大化对称计算速率的最优功率分配问题，该问题是一个非凸优化问题，缺乏简单的解析解或数值解。论文第一部分研究了信道状态数量有限（离散情形）时的问题。我们建立了最优解的几个重要性质，并证明当所有用户采用相同的功率分配策略（对称策略）且功率约束超过特定阈值时，最优解具有注水式结构。然而，若允许非对称策略，则任何功率约束下最优解均不呈现此形式。针对两种场景，我们提出了一种低复杂度的基于排序的算法，并与基线算法进行性能比较。论文第二部分阐述了当信道系数建模为连续随机变量（连续情形）时的相关结论，并为对称策略场景提出了类似的低复杂度迭代算法。研究提供了离散与连续情形的数值结果，表明所提算法能以较低复杂度获得良好的次优解，且在部分算例中能找到精确最优解。