Bounds on the Capacity of Random Insertion and Deletion-Additive Noise Channels

TL;DR

This paper derives new analytical lower bounds on the capacity of binary insertion and deletion channels, including models with channel noise, providing the first bounds for deletion-AWGN channels and improvements for insertion channels.

Contribution

It introduces the first analytical bounds for deletion-AWGN channels and improves existing bounds for deletion-substitution and small insertion probabilities.

Findings

First analytical bounds for deletion-AWGN channels.

Best available bounds for deletion-substitution channels.

Improved bounds for small insertion probabilities in Gallager's model.

Abstract

We develop several analytical lower bounds on the capacity of binary insertion and deletion channels by considering independent uniformly distributed (i.u.d.) inputs and computing lower bounds on the mutual information between the input and output sequences. For the deletion channel, we consider two different models: independent and identically distributed (i.i.d.) deletion-substitution channel and i.i.d. deletion channel with additive white Gaussian noise (AWGN). These two models are considered to incorporate effects of the channel noise along with the synchronization errors. For the insertion channel case we consider the Gallager's model in which the transmitted bits are replaced with two random bits and uniform over the four possibilities independently of any other insertion events. The general approach taken is similar in all cases, however the specific computations differ. Furthermore, the approach yields a useful lower bound on the capacity for a wide range of deletion probabilities for the deletion channels, while it provides a beneficial bound only for small insertion probabilities (less than 0.25) for the insertion model adopted. We emphasize the importance of these results by noting that 1) our results are the first analytical bounds on the capacity of deletion-AWGN channels, 2) the results developed are the best available analytical lower bounds on the deletion-substitution case, 3) for the Gallager insertion channel model, the new lower bound improves the existing results for small insertion probabilities.