A Single-Letter Upper Bound on the Mismatch Capacity via Multicast Transmission

TL;DR

This paper develops a new single-letter upper bound on the mismatch capacity of stationary, memoryless channels using a multicast transmission approach over a two-user broadcast channel, improving previous bounds and characterizing capacity classes.

Contribution

It introduces a novel multicast-based analysis technique to derive tighter bounds on mismatch capacity and identifies equivalence classes of channel-metric pairs with shared capacities.

Findings

The new bound is at least as tight as previous bounds.

The bound is tight if the class contains a matched pair.

The mismatch capacity can be characterized by the LM rate in certain classes.

Abstract

We introduce a new analysis technique to derive a single-letter upper bound on the mismatch capacity of a stationary, single-user, memoryless channel with a decoding metric $q$. Our bound is obtained by considering a multicast transmission over a two-user broadcast channel with decoding metrics $q$ and $\rho$ at the receivers, referred to as $(q,\rho)$-surely degraded. This channel has the property that the intersection event of correct $q$-decoding of receiver $1$ and erroneous $\rho$-decoding of receiver $2$ has zero probability for any fixed-composition codebook of a certain composition $P$. Our bound holds in the strong converse sense of an exponential decay of the probability of correct decoding at rates above the bound. Further, we refine the proof and present a bound that is at least as tight as that of any choice of $\rho$. Several examples that demonstrate the strict improvement of our bound compared to previous results are analyzed. Finally, 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 achievable by random coding, and may be strictly lower that the matched capacity.