On information rates for mismatched decoders

Reliable transmission over a discrete-time memoryless channel with a decoding metric that is not necessarily matched to the channel (mismatched decoding) is considered. It is assumed that the encoder knows both the true channel and the decoding metric. The lower bound on the highest achievable rate found by Csiszar and Korner (1981) and by Hui (1983) for DMC´s, hereafter denoted C_LM, is shown to bear some interesting information-theoretic meanings. The bound C_LM turns out to be the highest achievable rate in the random coding sense, namely, the random coding capacity for mismatched decoding. It is also demonstrated that the ε-capacity associated with mismatched decoding cannot exceed C_LM. New bounds and some properties of C_LM are established and used to find relations to the generalized mutual information and to the generalized cutoff rate. The expression for C_LM is extended to a certain class of memoryless channels with continuous input and output alphabets, and is used to calculate C_LM explicitly for several examples of theoretical and practical interest. Finally, it is demonstrated that in contrast to the classical matched decoding case, here, under the mismatched decoding regime, the highest achievable rate depends on whether the performance criterion is the bit error rate or the message error probability and whether the coding strategy is deterministic or randomized