Gaussian distortion-rate function under sub-nyquist nonuniform sampling

A bound on the amount of distortion in the reconstruction of a stationary Gaussian process from its rate-limited samples is derived. The bound is based on a combined sampling and source coding problem in which a Gaussian stationary process is described from a compressed version of its values on an infinite discrete set. We show that the distortion in reconstruction cannot be lower than the distortion-rate function based on optimal uniform filter-bank sampling using a sufficient number of sampling branches. This can be seen as an extension of Landau's theorem on a necessary condition for optimal recovery of a signal from its samples, in the sense that it describes both the error as a result of sub-sampling and the error incurred due to lossy compression of the samples.