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 $i^{th}$ channel tap during time slot $t$ is uniformly distributed over an interval of centre $c_i$ and radius $ r_{i}$. The array of channel taps is independent along both $t$ and $i$. The channel state information is unavailable at both the transmitter and the receiver. Lower and upper bounds are derived on the White-Gaussian-Input (WGI) capacity $C_{\scriptscriptstyle{WGI}}$ for arbitrary values of the radii $ r_i$. It is shown that $C_{\scriptscriptstyle{WGI}}$ does not scale with the average input power. The proposed lower bound is achieved by a joint-typicality decoder that is tuned to a set of candidates for the channel matrix. This set forms a net that covers the range of the random channel matrix and its resolution is optimized in order to yield the largest achievable rate. Tools in matrix analysis such as Weyl's inequality on perturbation of eigenvalues of symmetric matrices are used in order to analyze the probability of error.