Causal coding of multiple jointly Gaussian sources

We consider causal coding of three jointly Gaussian correlated sources, X₁, X₂, X₃, with a given covariance matrix and determine an analytic closed-form formula for its total rate distortion function subject to Mean Square Error (MSE) distortion constraints when all sources need a positive rate to be represented. It is first shown that the optimal reproduction random variables are jointly Gaussian with the sources. A novel causal coding scheme is then proposed to achieve the total rate distortion function, in which each source is first whitened with respect to all previous original sources and then encoded via encoding a proper linear combination of the residues of the previous sources with respect to all available encoded sources and the residue of the current source with respect to all previous original sources. The more-and-less coding theorem in causal coding of correlated sources - when sources do not form a Markov chain as X₁ → X₂ → X₃, under some conditions on sources and distortion, the more sources need to be encoded, the less total rate is required - is also investigated for Gaussian sources. For the underlying scenario in which all sources need a positive rate to be represented, it is proved that the more-and-less coding is always revealed for non-Markov chain Gaussian sources.