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Compute-Forward Multiple Access for Gaussian MIMO Channels

Lanwei Zhang, Jamie Evans, Jingge Zhu

TL;DR

This paper extends the CFMA scheme to a two-user Gaussian multiple-input multiple-output (MIMO) multiple access channel (MAC) and proposes the CFMA Serial Coding Scheme (SCS) and the CFMA Parallel Coding Scheme (PCS) with nested lattice codes.

Abstract

Compute-forward multiple access (CFMA) is a multiple access transmission scheme based on Compute-and-Forward (CF) which allows the receiver to first decode linear combinations of the transmitted signals and then solve for individual messages. This paper extends the CFMA scheme to a two-user Gaussian multiple-input multiple-output (MIMO) multiple access channel (MAC). We propose the CFMA serial coding scheme (SCS) and the CFMA parallel coding scheme (PCS) with nested lattice codes. We first derive the expression of the achievable rate pair for MIMO MAC with CFMA-SCS. We prove a general condition under which CFMA-SCS can achieve the sum capacity of the channel. Furthermore, this result is specialized to single-input multiple-output (SIMO) and $2$-by-$2$ diagonal MIMO multiple access channels, for which more explicit sum capacity-achieving conditions on power and channel matrices are derived. We construct an equivalent SIMO model for CFMA-PCS and also derive the achievable rates. Its sum capacity achieving conditions are then analysed. Numerical results are provided for the performance of CFMA-SCS and CFMA-PCS in different channel conditions. In general, CFMA-PCS has better sum capacity achievability with higher coding complexity.

Compute-Forward Multiple Access for Gaussian MIMO Channels

TL;DR

This paper extends the CFMA scheme to a two-user Gaussian multiple-input multiple-output (MIMO) multiple access channel (MAC) and proposes the CFMA Serial Coding Scheme (SCS) and the CFMA Parallel Coding Scheme (PCS) with nested lattice codes.

Abstract

Compute-forward multiple access (CFMA) is a multiple access transmission scheme based on Compute-and-Forward (CF) which allows the receiver to first decode linear combinations of the transmitted signals and then solve for individual messages. This paper extends the CFMA scheme to a two-user Gaussian multiple-input multiple-output (MIMO) multiple access channel (MAC). We propose the CFMA serial coding scheme (SCS) and the CFMA parallel coding scheme (PCS) with nested lattice codes. We first derive the expression of the achievable rate pair for MIMO MAC with CFMA-SCS. We prove a general condition under which CFMA-SCS can achieve the sum capacity of the channel. Furthermore, this result is specialized to single-input multiple-output (SIMO) and -by- diagonal MIMO multiple access channels, for which more explicit sum capacity-achieving conditions on power and channel matrices are derived. We construct an equivalent SIMO model for CFMA-PCS and also derive the achievable rates. Its sum capacity achieving conditions are then analysed. Numerical results are provided for the performance of CFMA-SCS and CFMA-PCS in different channel conditions. In general, CFMA-PCS has better sum capacity achievability with higher coding complexity.
Paper Structure (21 sections, 8 theorems, 95 equations, 3 figures, 1 table)

This paper contains 21 sections, 8 theorems, 95 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. Paper contributions
  4. System Model
  5. The CFMA serial coding scheme (SCS)
  6. Transmission scheme for CFMA-SCS
  7. Achievable Rates with CFMA-SCS
  8. Sum capacity achievability with CFMA-SCS
  9. Case study for CFMA-SCS
  10. SIMO MAC
  11. Diagonal $2$-by-$2$ MIMO MAC
  12. Channels with a similar SVD structure
  13. The CFMA parallel coding scheme (PCS)
  14. Transmission scheme for CFMA-PCS
  15. Achievable rates with CFMA-PCS
  16. ...and 6 more sections

Key Result

Theorem 1

For a two-user MIMO Gaussian MAC given the channel matrices $\textbf{H}_1,\textbf{H}_2$ and the precoding matrices $\textbf{B}_1,\textbf{B}_2$, with CFMA-SCS, the following rate pair is achievable for any linearly independent $\textbf{a},\textbf{b}\in\mathbb{Z}^2$ and $\boldsymbol{\beta}\in\mathbb{R}^2$ if $r_l(\textbf{a},\boldsymbol{\beta})\geq 0$ and $r_l(\textbf{b}|\textbf{a},\boldsymbol{\beta

Figures (3)

  • Figure 1: Simulation results for SIMO MAC and $2$-by-$2$ MIMO MAC with CFMA-SCS where the channel coefficients are generated from Uniform $[0,1]$ and Uniform $[1,2]$. It is harder for CFMA-SCS to achieve the sum capacity in the MIMO case compared to the SIMO case. When the channel matrices are diagonal in the MIMO case, CFMA-SCS has a very poor sum capacity achievability. In the SIMO MAC case where the channel coefficients are generated from Uniform $[1,2]$, CFMA-SCS can achieve the sum capacity when the SNR is larger than $0$ dB.
  • Figure 2: Simulation results for a two-user generic $2$-by-$2$ MIMO MAC with CFMA-SCS, where the channel coefficients are randomly distributed over the interval $[0,1]$. There is a nearly constant improvement (around $0.01$) of $R_A$ with permutation when the power constraint $P$ is larger than $20$ dB.
  • Figure 3: Simulation results for a two-user generic $2$-by-$2$ MIMO MAC, where the channel coefficients are randomly distributed over the interval $[0,1]$ and $[1,2]$. It can be observed that CFMA-PCS has larger $R_A$'s in all cases. However, CFMA-PCS has a larger increase in $R_A$ when the channel coefficients are generated from Uniform $[1,2]$ compared to when they are from Uniform $[0,1]$.

Theorems & Definitions (16)

  • Definition 1: Ambiguity lattice decoder
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Remark 2
  • Lemma 1
  • Remark 3
  • Corollary 1
  • Lemma 2
  • Remark 4
  • ...and 6 more