On a Class of Time-Varying Gaussian ISI Channels

This paper studies a class of stochastic and time-varying Gaussian intersymbol interference (ISI) channels. The probability law for the
$i^{th}$
channel tap during time slot
$t$
is supported over an interval of centre
$c_{i}$
and radius
$r_{i}$
. The transmitter and the receiver only know the centres
$c_{i}$
and the radii
$r_{i}$
. The joint distribution for the array of channel taps and their realizations are unknown to both the transmitter and the receiver. A lower bound (achievability result) is presented on the channel capacity which results in an upper bound on the capacity loss compared to when all radii are zeros. The lower bound on the channel capacity saturates at a positive value as the maximum average input power
$P$
increases beyond what is referred to as the saturation power
$P_{sat}$
. Roughly speaking,
$P_{sat}$
is inversely proportional to the sum of the squares of the radii
$r_{i}$
. It is also verified that in the presence of channel state information at the receiver, if the so-called central channel frequency response is everywhere nonzero, then the aforementioned capacity loss is bounded from above by a constant that does not depend on
$P$
. A partial converse result is provided in a scenario where different channel taps vary independently of each other and each channel tap process is stationary with a finite differential entropy rate. It is shown that for every sequence of codebooks with vanishing probability of error, if the size of each symbol in every codeword is bounded away from zero by a constant that is proportional to
$\sqrt {P}$
, then the rate of that sequence of codebooks does not scale with
$P$
. Tools in matrix analysis such as matrix norms and Weyl’s inequality on perturbation of eigenvalues of symmetric matrices are used in order to analyze the probability of error. A result in the paper that may find other applications is a new and tight upper bound on the size of a Gaussian typical set.