Converse Theorems for the DMC With Mismatched Decoding

The problem of mismatched decoding with an additive bounded metric q for a discrete memoryless channel W is addressed. We study two kinds of decoders. The δ-margin mismatched decoder outputs a message whose metric with the channel output exceeds that of all the other codewords by at least δ. The τ-threshold decoder outputs a single message whose metric with the channel output exceeds a threshold τ. It is proved that the mismatch capacity with a constant margin decoder is equal to the “product-space” improvement of the random coding lower bound on the mismatch capacity, C
_q
^(∞)
(W), which was introduced by Csiszár and Narayan. We next consider sequences of P-constant composition codebooks. Using the Central Limit Theorem, it is shown that for such sequences of codebooks the supremum of achievable rates with constant threshold decoding is upper bounded by the supremum of the achievable rates with a constant margin decoder, and therefore also by C
_q
^(∞)
(W). Further, a soft converse is proved stating that if the average probability of error of a sequence of codebooks with ordinary mismatched decoding converges to zero sufficiently fast, the rate of the code sequence is upper bounded by C
_q
^(∞)
(W). In particular, if q is a bounded rational metric, and the average probability of error converges to zero faster than O(n
^-1
), then R ≤ C
_q
^(∞)
(W). Finally, a max-min multi-letter upper bound on the mismatch capacity that bears some resemblance to C
_q
^(∞)
(W) is presented.