Block coding for weakly continuous channels

Abstract

Given a discrete stationary channel <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</tex> for which the map <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\mu \rightarrow \mu v</tex> carrying each stationary, ergodic input <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\mu</tex> into the input-output measure <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\mu v</tex> is continuous (with respect to weak convergence) at at least one input, it is shown that every stationary and ergodic source with sufficiently small entropy is block transmissible over the channel. If this weak continuity condition is satisfied at every stationary ergodic input, one obtains the class of weakly continuous channels for which the usual source/channel block coding theorem and converse hold with the usual notion of channel capacity. An example is given to show that the class of weakly continuous channels properly includes the class of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\bar{d}</tex> -continuous channels. It is shown that every stationary channel <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</tex> is "almost" weakly continuous in the sense that every input-output measure <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\mu v</tex> for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</tex> can be obtained by sending <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\mu</tex> over an appropriate weakly continuous channel (depending on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\mu</tex> ). This indicates that weakly continuous channels may be the most general stationary channels for which one would need a coding theorem.