On the Capacity of Memoryless Channels with Synchronization Errors

TL;DR

This paper proves that the capacity of memoryless channels with synchronization errors, including deletion channels, is a convex function of the channel parameters, leading to improved upper bounds on capacity.

Contribution

It establishes the convexity of channel capacity with respect to the stochastic channel matrix, providing new bounds and insights for channels with synchronization errors.

Findings

Channel capacity is convex in the stochastic matrix.

i.i.d. deletion channel capacity is convex in deletion probability.

Upper bound on deletion channel capacity as deletion probability approaches 1 is improved.

Abstract

Memoryless channels with synchronization errors as defined by a stochastic channel matrix allowing for symbol insertions and deletions in addition to random errors are considered. Such channels are information stable, hence their Shannon capacity exists. However, computation of the channel capacity is formidable, and only some upper and lower bounds on the capacity (for some special cases) exist. In this short paper, using a simple methodology, we prove that the channel capacity is a convex function of the stochastic channel matrix. Since the more widely studied model of an independent identically distributed (i.i.d.) deletion channel is a particular case, as an immediate corollary to this result we also argue that the i.i.d. deletion channel capacity is a convex function of the deletion probability. We further use this result to improve the existing capacity upper bounds on the deletion channel by a proper "convexification" argument. In particular, we prove that the capacity of the deletion channel, as the deletion probability d --> 1, is upper bounded by $0.4143(1-d)$ (which was also observed by a different (weaker) recent result).