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Fractional Programming for Stochastic Precoding over Generalized Fading Channels

Wenyu Wang, Kaiming Shen

TL;DR

Simulations show that the proposed stochastic precoding method outperforms the benchmark methods in both Gaussian and non-Gaussian fading channel cases and improves the efficiency of the proposed algorithm by eliminating the large matrix inverse.

Abstract

This paper seeks an efficient algorithm for stochastic precoding to maximize the long-term average weighted sum rates throughout a multiple-input multiple-output (MIMO) network. Unlike many existing works that assume a particular probability distribution model for fading channels (which is typically Gaussian), our approach merely relies on the first and second moments of fading channels. For the stochastic precoding problem, a naive idea is to directly apply the fractional programming (FP) method to the data rate inside the expectation; it does not work well because the auxiliary variables introduced by FP are then difficult to decide. To address the above issue, we propose using a lower bound to approximate the expectation of data rate. This lower bound stems from a nontrivial use of the matrix FP, and outperforms the existing lower bounds in that it accounts for generalized fading channels whose first and second moments are known. The resulting approximate problem can be efficiently solved in closed form in an iterative fashion. Furthermore, for large-scale MIMO, we improve the efficiency of the proposed algorithm by eliminating the large matrix inverse. Simulations show that the proposed stochastic precoding method outperforms the benchmark methods in both Gaussian and non-Gaussian fading channel cases.

Fractional Programming for Stochastic Precoding over Generalized Fading Channels

TL;DR

Simulations show that the proposed stochastic precoding method outperforms the benchmark methods in both Gaussian and non-Gaussian fading channel cases and improves the efficiency of the proposed algorithm by eliminating the large matrix inverse.

Abstract

This paper seeks an efficient algorithm for stochastic precoding to maximize the long-term average weighted sum rates throughout a multiple-input multiple-output (MIMO) network. Unlike many existing works that assume a particular probability distribution model for fading channels (which is typically Gaussian), our approach merely relies on the first and second moments of fading channels. For the stochastic precoding problem, a naive idea is to directly apply the fractional programming (FP) method to the data rate inside the expectation; it does not work well because the auxiliary variables introduced by FP are then difficult to decide. To address the above issue, we propose using a lower bound to approximate the expectation of data rate. This lower bound stems from a nontrivial use of the matrix FP, and outperforms the existing lower bounds in that it accounts for generalized fading channels whose first and second moments are known. The resulting approximate problem can be efficiently solved in closed form in an iterative fashion. Furthermore, for large-scale MIMO, we improve the efficiency of the proposed algorithm by eliminating the large matrix inverse. Simulations show that the proposed stochastic precoding method outperforms the benchmark methods in both Gaussian and non-Gaussian fading channel cases.

Paper Structure

This paper contains 11 sections, 6 theorems, 50 equations, 9 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. System Model and Problem Formulation
  3. Stochastic Matrix FP Methods
  4. Stochastic Extension of Matrix FP
  5. Proposed Lower Bound
  6. FP-Based Stochastic Precoding
  7. Acceleration for Large-Scale MIMO
  8. Simulations
  9. Settings
  10. Test Results
  11. Conclusion

Key Result

Theorem 1

Problem log prob is equivalent to in the sense that $x$ is a global solution (or a stationary point solution) to log prob iff it is a global solution (or a stationary point solution) to prob:sqt, where a positive semidefinite auxiliary variable $\bm\Gamma_n\in\mathbb S^{M\times M}_+$ is introduced for each matrix ratio.

Figures (9)

  • Figure 1: Approximation error between the lower bounds and original objective for different noise power levels and different values of the temporal correlation coefficient $\rho_{jk}$.
  • Figure 2: The long-term average sum rates achieved by the different algorithms under the different fading models and under the different noise power levels when $v=60$ and $90$ km/h.
  • Figure 3: Convergence of different algorithms versus number of iterations under various configurations when $v=60$ km/h.
  • Figure 4: Convergence of different algorithms versus number of iterations under various configurations when $v=90$ km/h.
  • Figure 5: Convergence of different algorithms versus elapsed time under various configurations when $v=60$ km/h.
  • ...and 4 more figures

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • Remark 4
  • Proposition 2
  • ...and 1 more