On the Rate-Distortion Function for Sampled Cyclostationary Gaussian Processes with Memory

We study the rate-distortion function (RDF) for lossy compression of discrete-time (DT) processes obtained by sampling continuous-time (CT) wide-sense cyclostationary (WSCS) Gaussian processes with memory. This problem was previously studied for the case in which the sampling interval is commensurate with the period of the cyclostationary statistics (referred to as synchronous sampling), hence we focus on the situation in which these parameters are incommensurate, referred to as asynchronous sampling. The sampling interval is also assumed to be smaller than the maximal autocorrelation length of the CT source process, which results in a DT process with memory, such that the overall DT process is modeled as a Gaussian wide-sense almost cyclostationary (WSACS) process with memory. This problem is motivated by the fact that communications signals are modelled as CT WSCS processes, thus, to facilitate DT processing, e.g., as in compress-and-forward relaying and in recording systems, sampling has to be applied first. The main challenge follows as DT WSACS processes are not information-stable which renders conventional information-theoretic arguments irrelevant, and hence, the characterization of the RDF is carried out within the information-spectrum framework. This work expands upon our previous work which addressed the special case in which the DT process is memoryless. The existence of dependence between the samples requires a new approach for characterizing the RDF.