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Power Allocation for Compute-and-Forward over Fading Channels

Lanwei Zhang, Jamie Evans, Jingge Zhu

TL;DR

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.

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.

Power Allocation for Compute-and-Forward over Fading Channels

TL;DR

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.

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.

Paper Structure

This paper contains 21 sections, 5 theorems, 39 equations, 6 figures, 4 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Paper contributions
  4. Problem formulation
  5. CF computation rate
  6. Computation rate in the fading channel
  7. Discrete case
  8. Theoretical analysis of the optimization problem
  9. General properties of the optimal solution
  10. Guarantee for the symmetric power allocation policy
  11. Further discussion on the asymmetric power allocation policy scenario
  12. Power allocation algorithms for the discrete case
  13. Numerical results for the discrete case
  14. Example 1
  15. Example 2
  16. ...and 6 more sections

Key Result

Lemma 1

If $\mathcal{A}^*$ is an optimal active set of $DP1$ and $\textbf{P}^*$ is the corresponding solution, then all the channel states in $\mathcal{A}^*$ have positive rate, i.e., $R_m(\textbf{p}_m^*) > 0, \forall m \in \mathcal{A}^*$.

Figures (6)

  • Figure 1: Achievable rate in the symmetric power policy case with $\bar{P} > \bar{P}_o$ in Example 1. We observe that Algorithm \ref{['algo_water_filling']} gives the same result as Algorithm \ref{['algo_exhaustive_search']} (hence optimal), as proven in Theorem \ref{['thm_opt_large_p']}.
  • Figure 2: Achievable rate in the symmetric power policy case with $\bar{P} \leq \bar{P}_o$ in Example 1. We observe that Algorithm \ref{['algo_iter_order']} gives the same result as Algorithm \ref{['algo_exhaustive_search']} hence is optimal in this case.
  • Figure 3: Achievable rate in the asymmetric power policy case in Example 1. We observe that Algorithm \ref{['algo_iter_order']} gives the same result as Algorithm \ref{['algo_exhaustive_search']} hence is optimal in this case.
  • Figure 4: Achievable rate against the power constraint in Example 2.
  • Figure 5: Achievable rate against the power constraint in Example 3.
  • ...and 1 more figures

Theorems & Definitions (17)

  • Definition 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 7 more