A maximum entropy theorem for complex-valued random vectors, with implications on capacity

Recent research has demonstrated significant achievable performance gains by exploiting circularity/non-circularity or properness/improperness of complex-valued signals. In this paper, we investigate the influence of theses properties on important information theoretic quantities such as entropy and capacity. More specifically, we prove a novel maximum entropy theorem that is based on the so-called circular analog of a given (in general, non-Gaussian) complex-valued random vector. Its introduction is supported by a characterization theorem that employs a minimum Kullback-Leibler divergence criterion. As an application of this maximum entropy theorem, we show that the capacity-achieving input random vector is circular for a broad range of multiple-input multiple-output (MIMO) channels including coherent and noncoherent scenarios. This result does not depend on a Gaussian assumption and thus provides a justification for many practical signalling/coding strategies, regardless of the specific distribution of the channel parameters.