The rate-distortion function for source coding with side information at the decoder

Let {(X,, Y,J}r= 1 be a sequence of independent drawings of a pair of dependent random variables X, Y. Let us say that X takes values in the finite set 6. It is desired to encode the sequence {X,} in blocks of length n into a binary stream*of rate R, which can in turn be decoded as a sequence { 2k}, where zk E %, the reproduct ion alphabet. The average distorjion level is (l/n) cl= 1 E[D(X,,z&, where D(x,$ 2 0, x E I, 2 E J, is a pre-assigned distortion measure. The special assumption made here is that the decoder has access to the side information {Yk}. In this paper we determine the quantity R*(d). def ined as the infimum of rates R such that (with E > 0 arbitrarily small and with suitably large n) communicat ion is possible in the above setting at an average distortion level (as def ined above) not exceeding d + E. The main result is that R*(d) = inf[Z(X,Z) Z(Y,Z)], where the infimum is with respect to all auxiliary random variables Z (which take values in a finite set 3) that satisfy: i) Y,Z conditiofally independent given X; ii) there exists a functionf: “Y x E + .%, such that E[D(X,f(Y,Z))] 5 d. Let Rx, y(d) be the rate-distortion function which results when the encoder as well as the decoder has access to the side information {Y,}. In nearly all cases it is shown that when d > 0 then R*(d) > Rx, y(d), so that knowledge of the side information at the encoder permits transmission of the {X,} at a given distortion level using a smaller transmission rate. This is in contrast to the situation treated by Slepian and Wolf [5] where, for arbitrarily accurate reproduction of {X,}, i.e., d = E for any E > 0, knowledge of the side information at the encoder does not allow a reduct ion of the transmission rate.