The Rate-Distortion Function for Sampled Cyclostationary Gaussian Processes with Memory and with Bounded Processing Delay: Extended Version with Proofs

We study the rate-distortion function (RDF) for the lossy compression of discrete-time (DT) wide-sense almost cyclostationary (WSACS) Gaussian processes with memory, arising from sampling continuous-time (CT) wide-sense cyclostationary (WSCS) Gaussian source processes. The importance of this problem arises as such CT processes represent communications signals, and sampling must be applied to facilitate the DT processing associated with their compression. Moreover, the physical characteristics of oscillators imply that the sampling interval is incommensurate with the period of the autocorrelation function (AF) of the physical process, giving rise to the DT WSACS model considered. In addition, to reduce the loss, the sampling interval is generally shorter than the correlation length, and thus, the DT process is correlated as well. The difficulty in the RDF characterization follows from the information-instability of WSACS processes, which renders the traditional information-theoretic tools inapplicable. In this work we utilize the information-spectrum framework to characterize the RDF when a finite and bounded delay is allowed between processing of subsequent source sequences. This scenario extends our previous works which studied settings without processing delays or without memory. Numerical evaluations reveal the impact of scenario parameters on the RDF with asynchronous sampling.