On a Source-Coding Problem with Two Channels and Three Receivers

Consider the following communication problem: An encoder is presented with a sequence of source letters {-X*} drawn from alphabet SC. We assume the {X*} are independent and identically distributed with probability mass function p{x) (or a probability density function if 9C is continuous). For each block of N letters (N arbitrary), two discrete encoder outputs /i(X) and /2(X) are produced (X is a vector of N letters). The cardinalities of fx and f2 are limited by ^-log || /i(X) || 1909 1 = 1,2, where the base of the log is arbitrary, but taken to be e in the sequel. Then R t is the maximum rate at which information can be conveyed over the ith channel, in nats per source letter. We assume the existence of three receivers which must estimate X using fi alone, /"2 alone, and both /i and f2. The three estimates, denoted by and X 3 are N vectors in some reproducing alphabets ( SC, SC-i, &3) which in general may not coincide with each other or with SC. Distortions d, d2, and da are incurred at the respective receivers according to X E[8i(Xk,Xik)], i= 1,2,3, iV k-1 where 5,(·,·) is a nonnegative real-valued function defined on 9Cand SCi. This configuration is summarized in Fig. 1. The case of only one receiver is the classical rate-distortion problem. 1 Corresponding to the source statistics and the distortion measure, the rate-distortion function is defined by R(d) =p mfI(X]X), d