Shaping gain of the partially coherent additive white Gaussian noise channel

We derive necessary conditions on the channel input and output probability density functions (PDFs) to achieve channel capacity for the additive white Gaussian noise (AWGN) channel with Tikhonov distributed phase error. We show that the Gaussian input does not achieve capacity, and we obtain a lower bound on the shaping gain. The shaping gain can be as large as 2.4 dB at rates as low as 0.5 b/symbol/Hz. This contrasts with the well-known shaping gain for the AWGN channel, which is small at low transmission rates and 1.53 dB at higher rates.