Complex-Valued Random Vectors and Channels: Entropy, Divergence, and Capacity

TL;DR

This paper explores how properties like circularity and impropriety of complex signals affect entropy, divergence, and MIMO channel capacity, providing new theorems and insights into optimal signal design and capacity gains.

Contribution

It introduces two maximum entropy theorems for complex vectors, characterizes capacity for various MIMO channels, and analyzes the impact of improper noise on capacity.

Findings

Maximum entropy theorems for complex vectors.

Capacity-achieving input vectors are circular in broad MIMO scenarios.

Improper Gaussian noise can increase capacity if exploited.

Abstract

Recent research has demonstrated significant achievable performance gains by exploiting circularity/non-circularity or propeness/improperness of complex-valued signals. In this paper, we investigate the influence of these properties on important information theoretic quantities such as entropy, divergence, and capacity. We prove two maximum entropy theorems that strengthen previously known results. The proof of the former theorem is based on the so-called circular analog of a given complex-valued random vector. Its introduction is supported by a characterization theorem that employs a minimum Kullback-Leibler divergence criterion. In the proof of latter theorem, on the other hand, results about the second-order structure of complex-valued random vectors are exploited. Furthermore, we address the capacity of multiple-input multiple-output (MIMO) channels. Regardless of the specific distribution of the channel parameters (noise vector and channel matrix, if modeled as random), we show that the capacity-achieving input vector is circular for a broad range of MIMO channels (including coherent and noncoherent scenarios). Finally, we investigate the situation of an improper and Gaussian distributed noise vector. We compute both capacity and capacity-achieving input vector and show that improperness increases capacity, provided that the complementary covariance matrix is exploited. Otherwise, a capacity loss occurs, for which we derive an explicit expression.