The Fading Number of SIMO Fading Channels with Memory

We derive the fading number of a general (not necessarily Gaussian) single-input multiple-output (SIMO) fading channel with memory, where the transmitter and receiver—while fully cognizant of the probability law governing the fading process—have no access to the fading realization. It is demonstrated that the fading number is achieved by IID circularly-symmetric inputs of log squared-magnitude that is uniformly distributed over a signal-to-noise (SNR) dependent interval. The upper limit of the interval is the logarithm of the allowed transmit power, and the lower limit tends to infinity sub-logarithmically in the SNR. Among the new ingredients in the proof is a new theorem regarding input distributions that escape to infinity. Upper and lower bounds on the fading number for SIMO Gaussian fading are also presented. Those are computed explicitly for stationary m-th order autoregressive AR(m) Gaussian fading processes.