A General Formula for the Mismatch Capacity

TL;DR

This paper derives a comprehensive formula for the mismatch capacity of channels with mismatched decoding, unifying existing theories, providing bounds, and exploring threshold decoding and capacity conditions.

Contribution

It introduces a general mismatch capacity formula, proves a conjecture for bounded metrics, and links threshold capacity with the product-space lower bound.

Findings

Established a general mismatch capacity formula.

Proved Csiszár and Narayan's conjecture for bounded metrics.

Derived bounds and conditions for capacity and strong converse.

Abstract

The fundamental limits of channels with mismatched decoding are addressed. A general formula is established for the mismatch capacity of a general channel, defined as a sequence of conditional distributions with a general decoding metrics sequence. We deduce an identity between the Verd\'{u}-Han general channel capacity formula, and the mismatch capacity formula applied to Maximum Likelihood decoding metric. Further, several upper bounds on the capacity are provided, and a simpler expression for a lower bound is derived for the case of a non-negative decoding metric. The general formula is specialized to the case of finite input and output alphabet channels with a type-dependent metric. The closely related problem of threshold mismatched decoding is also studied, and a general expression for the threshold mismatch capacity is obtained. As an example of threshold mismatch capacity, we state a general expression for the erasures-only capacity of the finite input and output alphabet channel. We observe that for every channel there exists a (matched) threshold decoder which is capacity achieving. Additionally, necessary and sufficient conditions are stated for a channel to have a strong converse. Csisz\'{a}r and Narayan's conjecture is proved for bounded metrics, providing a positive answer to the open problem introduced in [1], i.e., that the "product-space" improvement of the lower random coding bound, $C_q^{(\infty)}(W)$, is indeed the mismatch capacity of the discrete memoryless channel $W$. We conclude by presenting an identity between the threshold capacity and $C_q^{(\infty)}(W)$ in the DMC case.