Channel Capacity of an Asymmetric Constellation in Rayleigh Fading with Noncoherent Energy Detection

The channel capacity of noncoherent reception of multi-level one-sided amplitude-shift keying (ASK), which is an asymmetric constellation, in Rayleigh fading with receive diversity and energy detection is considered. The asymmetries result in capacity achieving input probability distributions, that is, a priori probability distributions, that deviate from uniformity. An analytical expression for the mutual information in terms of a single integral is derived, and from it the set of equations, which can be solved to obtain the optimum or capacity achieving input probabilities, is obtained. High and low signal-to-noise ratio (SNR) approximations of the optimum input probabilities and the capacity are derived next. Furthermore, a logarithmic upper bound on the mutual information is obtained. Numerical results confirm that the uniform distribution of input probabilities is not capacity achieving. For example, with average SNR per symbol per branch of 6 dB, the relative deviation of the mutual information (with uniform input distribution) from the capacity is nearly 20% for 4-level ASK with one transmit diversity branch and two receive diversity branches. Furthermore, the derived high and low SNR approximations to the capacity are shown to be reasonably accurate.