The Fading Number of Multiple-Input Multiple-Output Fading Channels with Memory

The fading number of a general (not necessarily Gaussian) regular multiple-input multiple-output (MIMO) fading channel with arbitrary temporal and spatial memory is derived. The channel is assumed to be non-coherent, i.e., neither receiver nor transmitter have knowledge about the channel state, but they only know the probability law of the fading process. The fading number is the second term in the asymptotic expansion of channel capacity when the signal-to-noise ratio (SNR) tends to infinity. It is shown that the fading number can be achieved by an input that is the product of two independent processes: a stationary and circularly symmetric direction- (or unit-) vector process whose distribution needs to be chosen such that it maximizes the fading number, and a non-negative magnitude process that is independent and identically distributed (IID) and that escapes to infinity. Additionally, in the more general context of an arbitrary stationary channel model satisfying some weak conditions on the channel law, it is shown that the optimal input distribution is stationary apart from some edge effects.