Lattices for Distributed Source Coding: Jointly Gaussian Sources and Reconstruction of a Linear Function

Consider a pair of correlated Gaussian sources
(X

₁
,
X

₂
). Two separate encoders observe the two components and communicate compressed versions of their observations to a common decoder. The decoder is interested in reconstructing a linear combination of
X

₁
and
X

₂
to within a mean-square distortion of D. We obtain an inner bound to the optimal rate-distortion region for this problem. A portion of this inner bound is achieved by a scheme that reconstructs the linear function directly rather than reconstructing the individual components
X

₁
and
X

₂
first. This results in a better rate region for certain parameter values. Our coding scheme relies on lattice coding techniques in contrast to more prevalent random coding arguments used to demonstrate achievable rate regions in information theory. We then consider the case of linear reconstruction of K sources and provide an inner bound to the optimal rate-distortion region. Some parts of the inner bound are achieved using the following coding structure: lattice vector quantization followed by ldquocorrelatedrdquo lattice-structured binning.