A Single-Letter Upper Bound on the Mismatch Capacity via a Multicasting Approach

We introduce a new analysis technique that we use to derive a single-letter upper bound on the mismatch capacity of a stationary point-to-point memoryless channel with decoding metric q. Our bound is obtained by considering a multicast transmission over a two-user broadcast channel with decoding metrics q and ρ at the receivers, referred to as (q, ρ)-surely degraded. This channel has the property that the intersection event of the correct q-decoding of receiver 1 and the erroneous ρ-decoding of receiver 2 has zero probability for any codebook of a certain composition P. Our bound holds in the strong converse sense of exponential decay of the probability of correct decoding at rates above the bound. Several examples that demonstrate the strict improvement of our bound compared to previous results are analyzed. Further, we detect equivalence classes of isomorphic channel-metric pairs (W, q) that share the same mismatch capacity. We prove that if the class contains a matched pair, then our bound is tight and the mismatch capacity of the entire class is fully characterized and is equal to the LM rate, which is the highest rate achievable by random coding, and may be strictly lower than the Shannon (matched) capacity.