  This thesis considers infinite constellations in fading channels, without
power constraint and with perfect channel state information available at the
receiver. Infinite constellations are the framework, proposed by Poltyrev, for
analyzing coded modulation codes. The Poltyrev's capacity, is the highest
achievable normalized log density (NLD) of codewords per unit volume, at
possibly large block length, that guarantees a vanishing error probability. For
a given finite block length and a fixed error probability, there is a gap
between the highest achievable NLD and Poltyrev's capacity. The dispersion
analysis quantifies asymptotically this gap.
  The thesis begins by the dispersion analysis of infinite constellations in
scalar fading channels. Later on, we extend the analysis to the case of
multiple input multiple output fading channels. As in other channels, we show
that the gap between the highest achievable NLD and the Poltyrev's capacity,
vanishes asymptotically as the square root of the channel dispersion over the
block length, multiplied by the inverse Q-function of the allowed error
probability.
  Moreover, exact terms for Poltyrev's capacity and channel dispersion, are
derived in the thesis. The relations to the amplitude and to the power
constrained fading channels are also discussed, especially in terms of
capacity, channel dispersion and error exponents. These relations hint that in
typical cases the unconstrained model can be interpreted as the limit of the
constrained model, when the signal to noise ratio tends to infinity.
