Rayleigh Fading Channel Capacity for Coherent Signaling With Asymmetric Constellations

This paper investigates the channel capacity of coherent reception with antenna diversity in Rayleigh fading using asymmetric constellations, such as nonbinary two-sided amplitude-shift keying (ASK) and quadrature amplitude modulation (QAM) with ≥ 8 levels. Due to the asymmetries, the capacity achieving input or a priori probability distributions deviate from uniformity. To perform the investigation, an analytical expression for the mutual information, in terms of an integral over a single variable, between the input (transmitted symbol) and the output (received signal vector comprising signals at receive antennas), is derived. For two-sided ASK, 8-QAM with cross constellation, and 16-QAM, analytical closed form expressions for the high and low signal-to-noise (SNR) approximations, respectively, of the capacity achieving input probability distribution, the capacity, and the relative gain in mutual information (between using the uniform and the capacity achieving input probability distributions) are obtained. Numerical results show that at low SNRs, the use of the capacity achieving nonuniform input probability distribution offers significant advantage over the use of the uniform input probability distribution, while at high SNRs, this advantage is significant only for configurations with one receive antenna. Furthermore, the high and low SNR approximations are shown to be reasonably accurate.