Characterization of Deletion/Substitution Channel Capacity for Small Deletion and Substitution Probabilities

In this paper, we consider binary input deletion/substitution channels, which model certain channels with synchronization errors encountered in practice. Specifically, we focus on the regimen of small deletion and substitution probabilities, and by extending an approach developed for the deletion-only channel, we obtain an asymptotic characterization of the channel capacity for independent and identically distributed deletion/substitution channels. We first present an upper bound on the capacity for an arbitrary but fixed number of deletions and substitutions, and then we extend the result to the case of random deletions and substitutions. Our final result is as follows: the i.i.d. deletion/substitution channel capacity is approximately $1 - H(p_d) - H(p_s)$, where $p_d$ is the deletion probability, and $p_s$ is the substitution probability.