On remote source coding for signal estimation via Poisson sampling

Remote source coding for signal estimation via random sampling is analyzed. We consider a stationary Gaussian source to be estimated from samples randomly scattered according to a homogeneous Poisson point process, under a constraint on the number of bits per space unit. Building on the fact that homogeneous Poisson sampling introduces a white noise in the estimated signal spectrum and on results for indirect source coding by Dobrushin and Tsybakov, we derive an expression for the overall distortion rate as a function of the average sample intensity and of the bit rate constraint. The results enable the understanding of separation principle, between the distortion component due to sampling and that due to source coding, for Poisson random sampling. Finally, closed-form expressions for band-limited and non band-limited signals are provided.