Design of Linear Precoders for Correlated Sources in MIMO Multiple Access Channels

TL;DR

The paper addresses transmitting correlated sources over fading MIMO MACs by designing distributed linear precoders that minimize the sum-MSE $\xi_{\text{sum}}$, a non-convex objective when sources are correlated.It introduces multiple approaches, including a projected gradient method, a low-complexity aligned MRT strategy, a gain–direction factorization for the precoders, and SDP/SDR formulations for special cases, providing closed-form results for the two-user SIMO MAC. The results show substantial gains from exploiting source correlation, especially at low SNR and when the number of receive antennas is small relative to the number of users, with a range of complexity-performance tradeoffs. The work provides practical design tools for scalable, distributed precoding in sensor networks and similar setups, and benchmarks against source–channel separation bounds to quantify distortions under different operating regimes.

Abstract

This work focuses on distributed linear precoding when users transmit correlated information over a fading Multiple-Input and Multiple-Output Multiple Access Channel. Precoders are optimized in order to minimize the sum-Mean Square Error (MSE) between the source and the estimated symbols. When sources are correlated, minimizing the sum-MSE results in a non-convex optimization problem. Precoders for an arbitrary number of users and transmit and receive antennas are thus obtained via a projected steepest-descent algorithm and a low-complexity heuristic approach. For the more restrictive case of two single-antenna users, a closed-form expression for the minimum sum-MSE precoders is derived. Moreover, for the scenario with a single receive antenna and any number of users, a solution is obtained by means of a semidefinite relaxation. Finally, we also consider precoding schemes where the precoders are decomposed into complex scalars and unit norm vectors. Simulation results show a significant improvement when source correlation is exploited at precoding, especially for low SNRs and when the number of receive antennas is lower than the number of transmitting nodes.

Figures (8)

• Figure 1: Block-diagram of a fading MIMO MAC system with $K$ users, linear precoders, and a linear receiver.
• Figure 2: Example of the three feasible regions defined according to the sign of the gradient vectors for the two-user SIMO MAC with a correlation factor $\rho = 0.95$ (top) and $\rho = 0.99$ (bottom), for $\mathbf{h}_1=(1, 1)^T$, $\mathbf{h}_2 = (1, 0.5)^T$ and $\sigma_n^2=1$.
• Figure 3: Performance with and without precoding for a two-user SIMO MAC with $N_R = 2$ receive antennas and equal power constraints. Two different correlation factors are considered: $\rho = 0.8$ and $\rho = 0.95$.
• Figure 4: Average power allocated to each user after precoding the source symbols for a two-user SIMO MAC with $N_R = 2$ antennas when the power constraints are set such that $\text{SNR}_1=30$ dB and $\text{SNR}_2=5$ dB (top). Four-user SISO MAC with $N_R = 1$ antennas and power constraints $\text{SNR}=[35, 25, 15, 5]$ (bottom).
• Figure 5: Performance of the optimal linear precoder depending on the number of receive antennas for the two-user SIMO MAC, for SNR=5 dB and SNR=20 dB, and different correlation factors.

Theorems & Definitions (6)

• Lemma IV.1
• proof
• Lemma V.1
• proof
• Lemma V.2
• proof