The capacity of a random-phase additive white Gaussian noise (AWGN) channel, referred to as noncoherent channel, is investigated in the case of a transmission of N information symbols. The non-Gaussianity of the capacity-achieving distribution is shown and a lower bound on the channel capacity is derived. For increasing values of the number of transmitted symbols N, the capacity of a noncoherent channel is shown to asymptotically approach that of a coherent channel, i.e., a known-phase AWGN channel. The asymptotical Gaussianity of the capacity-achieving distribution is also shown. Based on the derived lower bound, the inherent capacity loss of a noncoherent channel, as compared to a coherent one, may be considered very limited for all but very small values of N. This result may be viewed as the information theoretic counterpart of a similar conclusion derived by many authors with reference to the probability of detection error.
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