Joint source-channel coding of a Gaussian mixture source over the Gaussian broadcast channel

Suppose that we want to send a description of a single source to two listeners through a Gaussian broadcast channel, where the channel is used once per source sample. The problem of joint source-channel coding is to design a communication system to minimize the distortion D₁ at receiver 1 and at the same time minimize the distortion D₂ at receiver 2. If the source is Gaussian, the optimal solution is well known, and it is achieved by an uncoded "analog" scheme. We consider a Gaussian mixture source. We derive inner and outer bounds for the distortion region of all (D₁, D₂) pairs that are simultaneously achievable. The outer bound is based on the entropy power inequality, while the inner bound is attained by a digital-over-analog encoding scheme, which we present. We also show that if the modes of the Gaussian mixture are highly separated, our bounds are tight, and hence, our scheme attains the entire distortion region. This optimal region exceeds the region attained by separating source and channel coding, although it does not contain the "ideal" point (D₁ , D₂)=(R^-1(C₁), R^-1(C ₂))