On source coding with side information at the decoder

Let \ (Xₖ, Yₖ, Vₖ) \₊=₁^ be a sequence of independent copies of the triple (X, Y, V) of discrete random variables. We consider the following source coding problem with a side information network. This network has three encoders numbered 0, 1, and 2, the inputs of which are the sequences \ Vₖ\, \Xₖ\, and \Yₖ\, respectively. The output of encoder i is a binary sequence of rate Rᵢ, i = 0, 1, 2. There are two decoders, numbered 1 and 2, whose task is to deliver essentially perfect reproductions of the sequences \Xₖ\ and \Yₖ\, respectively, to two distinct destinations. Decoder 1 observes the output of encoders 0 and 1, and decoder 2 observes the output of encoders 0 and 2. The sequence \Vₖ\ and its binary encoding (by encoder 0) play the role of side information, which is available to the decoders only. We study the characterization of the family of rate triples (R₀, R₁, R₂) for which this system can deliver essentially perfect reproductions (in the usual Shannon sense) of \Xₖ\ and \Yₖ\. The principal result is a characterization of this family via an information-theoretic minimization. Two special cases are of interest. In the first, V = (X, Y) so that the encoding of \Vₖ \ contains common information. In the second, Y 0 so that our problem becomes a generalization of the source coding problem with side information studied by Slepian and Wo1f 3.