  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.
