The fading number of single-input multiple-output fading channels with memory

We derive the fading number of stationary and ergodic (not necessarily Gaussian) single-input multiple-output (SIMO) fading channels with memory. This is the second term, after the double-logarithmic term, of the high signal-to-noise ratio (SNR) expansion of channel capacity. The transmitter and receiver are assumed to be cognizant of the probability law governing the fading but not of its realization. It is demonstrated that the fading number is achieved by independent and identically distributed (i.i.d.) circularly symmetric inputs of squared magnitude whose logarithm is uniformly distributed over an SNR-dependent interval. The upper limit of the interval is the logarithm of the allowed transmit power, and the lower limit tends to infinity sublogarithmically in the SNR. The converse relies inter alia on a new observation regarding input distributions that escape to infinity. Lower and upper bounds on the fading number for Gaussian fading are also presented. These are related to the mean squared-errors of the one-step predictor and the one-gap interpolator of the fading process respectively. The bounds are computed explicitly for stationary mth-order autoregressive AR(m) Gaussian fading processes.