On the capacity-achieving distribution of the discrete-time non-coherent and partially-coherent AWGN channels

We investigate the ultimate communication limits over rapid phase varying channels and consider the capacity of a discrete-time non-coherent additive white Gaussian noise channel under the average power constraint. We obtain necessary and sufficient conditions for the optimal, capacity achieving, input distribution and show that the optimal distribution is discrete and possesses an infinite number of mass points. In addition, we provide some easily computable lower and upper bounds on the channel capacity. Some of these results are extended to the partially coherent channel, where a phase locked loop (PLL) is used to track the carrier phase at the receiver, and where the residual phase errors are assumed to be Tikhonov distributed and statistically independent.