Bounds on the Information Rate for Sparse Channels with Long Memory and i.u.d. Inputs

In this paper we propose new bounds on the achievable information rate for discrete-time Gaussian channels with intersymbol interference (ISI) and independent and uniformly distributed (i.u.d.) channel input symbols drawn from finite-order modulation alphabets. Specifically, we are interested in developing new bounds on the achievable rates for sparse channels with long memory. We obtain a lower bound which can be achieved by practical receivers, based on MMSE channel shortening and suboptimal symbol detection for a reduced-state channel. An upper bound is given in the form of a semi-analytical solution derived using basic information theoretic inequalities, by a grouping of the channel taps into several clusters resulting in a newly defined single-input multiple-output (SIMO) channel. We show that the so obtained time-dispersive SIMO channel can be represented by an equivalent single-input single-output (SISO) channel with a significantly shorter channel memory. The reduced computational complexity allows the use of the BCJR algorithm for the newly defined channel. The proposed bounds are illustrated through several sparse channel examples and i.u.d. input symbols, showing that the upper bound significantly outperforms existing bounds. Performance of our lower bound strongly depends on the channel structure, showing best results for minimum-phase and maximum-phase systems.