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We investigate the communication limits over rapid phase-varying channels and consider the capacity of a discrete- time noncoherent additive white Gaussian noise (NCAWGN) channel under the average power constraint. We obtain necessary and sufficient conditions for the capacity-achieving input distribution and show that this distribution is discrete and possesses an infinite number of mass points. Using this characterization of the capacity-achieving distribution we compute a tight lower bound on the capacity of the channel based on examining suboptimal input distributions. In addition, we provide some easily computable lower and upper bounds on the channel capacity. Finally, we extend some of these results to the partially coherent channel, where it is assumed that a phase-locked loop (PLL) is used to track the carrier phase at the receiver, and that an ideal interleaver and de-interleaver are employed-rendering the Tikhonov distributed residual phase errors statistically independent from one symbol interval to another.