A Note on the Deletion Channel Capacity

Memoryless channels with deletion errors as defined by a stochastic channel matrix allowing for bit drop outs are considered in which transmitted bits are either independently deleted with probability d or unchanged with probability 1 − d. 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 exist. In this paper, we first show a simple result that the parallel concatenation of two different independent deletion channels with deletion probabilities d1 and d2, in which every input bit is either transmitted over the first channel with probability of λ or over the second one with probability of 1−λ, is nothing but another deletion channel with deletion probability of d = λd1+(1−λ)d2. We then provide an upper bound on the concatenated deletion channel capacity C(d) in terms of the weighted average of C(d1), C(d2) and the parameters of the three channels. An interesting consequence of this bound is that C(λd1 + (1 − λ)) ≤ λC(d1) which enables us to provide an improved upper bound on the capacity of the i.i.d. deletion channels, i.e., C(d) ≤ 0.4143(1 − d) for d ≥ 0.65. This generalizes the asymptotic result by Dalai [1] as it remains valid for all d ≥ 0.65. Using the same approach we are also able to improve upon existing upper bounds on the capacity of the deletion/substitution channel. Index Terms Deletion channel, deletion/substitution channel, channel capacity, capacity upper bounds.