Exponent Function for Source Coding with Side Information at the Decoder at Rates below the Rate Distortion Function

We consider the rate distortion problem with side information at the decoder posed and investigated by Wyner and Ziv. The rate distortion function indicating the trade-off between the rate on the data compression and the quality of data obtained at the decoder was determined by Wyner and Ziv. In this paper, we study the error probability of decoding at rates below the rate distortion function. We evaluate the probability of decoding such that the estimation of source outputs by the decoder has a distortion not exceeding a prescribed distortion level. We prove that when the rate on the data compression is below the rate distortion function this probability goes to zero exponentially and derive an explicit lower bound of this exponent function. On the Wyner-Ziv source coding problem the strong converse coding theorem has not been established yet. We prove this as a simple corollary of our result. I. SOURCE CODING WITH SIDE INFORMATION AT THE DECODER Let X and Y be finite sets and {(Xt, Yt)}∞t=1 be a stationary discrete memoryless source. For each t = 1, 2, · · ·, the random pair (Xt, Yt) takes values in X × Y , and has a probability distribution pXY = {pXY (x, y)}(x,y)∈X×Y We write n independent copies of {Xt}∞t=1 and {Yt} ∞ t=1, respectively as X = X1, X2, · · · , Xn and Y n = Y1, Y2, · · · , Yn. We consider a communication system depicted in Fig. 1. Data sequences X is separately encoded to φ(X) and is sent to the information processing center. At the center the decoder function ψ observes φ(X) and Y n to output the estimation Z of X. The encoder function φ is defined by φ : Xn → Mn = { 1, 2, · · · ,Mn } , (1) where ‖φ(n)‖ (= Mn) stands for the range of cardinality of φ. Let Z be a reproduction alphabet. The decoder function ψ is defined by ψ : Mn × Y → Z. (2) Let d : X × Z → [0,∞) be an arbitrary distortion measure on X × Z . The distortion between x ∈ Xn and z ∈ Zn is defined by d(x, z) △ = n