Coding Theorem and Converse for Abstract Channels with Time Structure and Memory

A coding theorem and converse are proved for abstract channels with time structure that contain continuous-time continuous-valued channels and the result by Kadota and Wyner (1972) as special cases. As main contribution the coding theorem is proved for a significantly weaker condition on the channel output memory and without imposing extra measurability requirements to the channel. These improvements are achieved by introducing a suitable characterization of information rate capacity. It is shown that the previously used ψ-mixing condition is quite restrictive, in particular for the important class of Gaussian channels. In fact, it is proved that for Gaussian (e. g., fading or additive noise) channels the ψ-mixing condition is equivalent to finite output memory. Moreover, a weak converse is derived for all stationary channels with time structure. Intersymbol interference as well as input constraints are taken into account in a general and flexible way, including amplitude and average power constraints as special case. Formulated in rigorous mathematical terms complete, explicit, and transparent proofs are presented. As a side product a gap is closed in the proof of Kadota and Wyner regarding a lemma on the monotonicity of some sequence of normalized mutual informations. An operational perspective is taken and an abstract framework is established, which allows to treat discreteand continuous-time channels with (possibly infinite) memory and arbitrary alphabets simultaneously in a unified way.