Capacity-Achieving Input Phase Distributions for Noncoherent Rayleigh-Fading Channels with Memory

The present paper considers single-input single-output non-coherent Rayleigh fading channels in which the memory is modelled by a Gauss-Markov process, and in which the magnitude of the input signal is constrained in an arbitrary manner. A contribution is made to the understanding of such channels by proving that, for any given input magnitude distribution, it is optimum from a capacity perspective to choose the phase of the input independent and identically distributed, with a distribution which is uniform over the interval [-pi, pi). In particular, if the capacity of such a channel can be achieved with an input distribution f, then it is also achievable by an input distribution f' having the same magnitude distribution and a phase distribution with the above property.