Mismatched encoding in rate distortion theory

Summary form only given. A length n block code C of size 2/sup nR/ over a finite alphabet /spl chi//spl circ//sub 0/ is used to encode a memoryless source over a finite alphabet /spl chi/. A length n source sequence x is described by the index i of the codeword x/spl circ//sub 0/(i) that is nearest to x according to the single-letter distortion function d/sub 0/(x,x/spl circ//sub 0/). Based on the description i and the knowledge of the codebook C, we wish to reconstruct the source sequence so as to minimize the average distortion defined by the distortion function d/sub 1/(x,x/spl circ//sub 1/), where d/sub 1/(x, x/spl circ//sub 1/) is in general different from d/sub 0/(x,x/spl circ//sub 0/). In fact, the reconstruction alphabets /spl chi//spl circ//sub 0/ and /spl chi//spl circ//sub 1/ could be different. We study the minimum, over all codebooks C, of the average distortion between the reconstructed sequence x/spl circ//sub 1/(i) and the source sequence x as the blocklength n tends to infinity. This limit is a function of the code rate R, the source's probability law, and the two distortion measures d/sub 0/(x,x/spl circ//sub 0/), and d/sub 1/(x,x/spl circ//sub 1/). This problem is the rate-distortion dual of the problem of determining the capacity of a memoryless channel under a possibly suboptimal decoding rule. The performance of a random i.i.d. codebook is found, and it is shown that the performance of the "average" codebook is in general suboptimal.