Lossy source coding for a cascade communication system with side-informations

We investigate source coding in a cascade communication system consisting of an encoder, a relay and an end terminal, where both the relay and the end terminal wish to reconstruct source $X$ with certain fidelities. Additionally, side-informations $Z$ and $Y$ are available at the relay and the end terminal, respectively. The side-information $Z$ at the relay is a physically degraded version of side-information $Y$ at the end terminal. Inner and outer bounds for the rate distortion region are provided in this work for general discrete memoryless sources. The rate distortion region is characterized when the source and side-informations are jointly Gaussian and physically degraded. The doubly symmetric binary source is also investigated and the inner and outer bounds are shown to coincide in certain distortion regimes. A complete equivalence of the rate-distortion region is established between the problem being considered and the side-information scalable source coding problem, when there is no side-information at the relay. As a byproduct, the same equivalence can be established between the well-known successive refinement problem and Yamamoto's cascade communication system, without relying on their rate-distortion characterization.