Coherent Fading Channels Driven by Arbitrary Inputs: Asymptotic Characterization of the Constrained Capacity and Related Information- and Estimation-Theoretic Quantities

We consider the characterization of the asymptotic behavior of the average minimum mean-squared error (MMSE) and the average mutual information in scalar and vector fading coherent channels, where the receiver knows the exact fading channel state but the transmitter knows only the fading channel distribution, driven by a range of inputs. We construct low-snr and -- at the heart of the novelty of the contribution -- high-snr asymptotic expansions for the average MMSE and the average mutual information for coherent channels subject to Rayleigh fading, Ricean fading or Nakagami fading and driven by discrete inputs (with finite support) or various continuous inputs. We reveal the role that the so-called canonical MMSE in a standard additive white Gaussian noise (AWGN) channel plays in the characterization of the asymptotic behavior of the average MMSE and the average mutual information in a fading coherent channel. We also reveal connections to and generalizations of the MMSE dimension. The most relevant element that enables the construction of these non-trivial expansions is the realization that the integral representation of the estimation- and information- theoretic quantities can be seen as an h-transform of a kernel with a monotonic argument: this enables the use of a novel asymptotic expansion of integrals technique -- the Mellin transform method -- that leads immediately to not only the high-snr but also the low-snr expansions of the average MMSE and -- via the I-MMSE relationship -- to expansions of the average mutual information. We conclude with applications of the results to the characterization and optimization of the constrained capacity of a bank of parallel independent coherent fading channels driven by arbitrary discrete inputs.