Mismatched encoding in rate distortion theory

Summary form only given. A length n block code C of size 2^nR over a finite alphabet χˆ₀ is used to encode a memoryless source over a finite alphabet χ. A length n source sequence x is described by the index i of the codeword xˆ₀ (i) that is nearest to x according to the single-letter distortion function d₀(x,xˆ₀). 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 d₁(x,xˆ₁), where d₁ (x, xˆ₁) is in general different from d₀(x,xˆ₀). In fact, the reconstruction alphabets χˆ₀ and χˆ₁ could be different. We study the minimum, over all codebooks C, of the average distortion between the reconstructed sequence xˆ₁(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 d₀(x,xˆ₀), and d₁(x,xˆ₁). 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