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We consider the characterization of the asymptotic behavior of the averageminimum mean-squared error (MMSE) and the average mutual information in scalarand vector fading coherent channels, where the receiver knows the exact fadingchannel 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 thenovelty of the contribution -- high-snr asymptotic expansions for the averageMMSE and the average mutual information for coherent channels subject toRayleigh fading, Ricean fading or Nakagami fading and driven by discrete inputs(with finite support) or various continuous inputs. We reveal the role that theso-called canonical MMSE in a standard additive white Gaussian noise (AWGN)channel plays in the characterization of the asymptotic behavior of the averageMMSE and the average mutual information in a fading coherent channel. We alsoreveal connections to and generalizations of the MMSE dimension. The mostrelevant element that enables the construction of these non-trivial expansionsis the realization that the integral representation of the estimation- andinformation- theoretic quantities can be seen as an h-transform of a kernelwith a monotonic argument: this enables the use of a novel asymptotic expansionof integrals technique -- the Mellin transform method -- that leads immediatelyto not only the high-snr but also the low-snr expansions of the average MMSEand -- via the I-MMSE relationship -- to expansions of the average mutualinformation. We conclude with applications of the results to thecharacterization and optimization of the constrained capacity of a bank ofparallel independent coherent fading channels driven by arbitrary discreteinputs.

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