The Rate Distortion Function of Asynchronously Sampled Memoryless Cyclostationary Gaussian Processes
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Man-made communications signals are typically modelled as continuous-time (CT) wide-sense cyclostationary (WSCS) processes. As modern processing is digital, it operates on sampled versions of the CT signals. When sampling is applied to a CT WSCS process, the statistics of the resulting discrete-time (DT) process depends on the relationship between the sampling interval and the period of the statistics of the CT process: When these two parameters have a common integer factor, then the DT process is WSCS. This situation is referred to as synchronous sampling. When this is not the case, which is referred to as asynchronous sampling, the resulting DT process is wide-sense almost cyclostationary (WSACS). In this work, we study the fundamental tradeoff of sources codes applied to sampled CT WSCS processes, namely, their rate-distortion function (RDF). We note that RDF characterization for the case of synchronous sampling directly follows from classic information-theoretic tools utilizing ergodicity and the law of large numbers; however, when sampling is asynchronous, the resulting process is not information stable. In such cases, commonly used information-theoretic tools are inapplicable to RDF analysis, which poses a major challenge. Using the information spectrum framework, we show that the RDF for asynchronous sampling in the low distortion regime can be expressed as the limit superior of a sequence of RDFs in which each element corresponds to the RDF of a synchronously sampled WSCS process (but their limit is not guaranteed to exist). The resulting characterization allows us to introduce novel insights on the relationship between sampling synchronization and RDF. For example, we demonstrate that, differently from stationary processes, small differences in the sampling rate and the sampling time offset can notably affect the RDF of sampled CT WSCS processes.
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