On the Rate-Distortion Function for Sampled Cyclostationary Gaussian Processes with Memory: Extended Version with Proofs
Zikun Tan, Ron Dabora, H. Vincent Poor
TL;DR
This work derives the rate-distortion function for lossy compression of asynchronously-sampled continuous-time wide-sense cyclostationary Gaussian processes with memory, a setting relevant to communications with clock jitter and relaying. By exploiting the information-spectrum framework, the authors obtain a per-phase RDF $R_{ obreak oldsymbol{eta}}^{oldsymbol{ extphi}_s}(D)= rac{1}{2l} obreak ext{log}ig( rac{ ext{det}(C_{X_{ obreak oldsymbol{eta},l}^{oldsymbol{ extphi}_s}})}{ ext{det}(C_{S_{ obreak oldsymbol{eta},l}^{oldsymbol{ extphi}_s}})}ig)$ and then average over the cyclostationary period to get $R_{ obreak oldsymbol{eta}}(D)= rac{1}{T_c} obreak ext{int}_{0}^{T_c} R_{ obreak oldsymbol{eta}}^{oldsymbol{ extphi}_s}(D) ext{d}oldsymbol{ extphi}_s$, under Gaussian reconstructions with covariance constraints. The main contributions are (i) a rigorous RDF expression for Gaussian processes with memory under asynchronous sampling, (ii) the necessity of initial sampling-phase synchronization at encoder/decoder, and (iii) the connection to the synchronous (information-stable) and dual-capacity formulations. The results support applications in compress-and-forward relaying and storage for temporally varying, cyclostationary signals, while leaving open questions about phase-independence in the asynchronous limit and potential delays between codewords.
Abstract
In this work we study the rate-distortion function (RDF) for lossy compression of asynchronously-sampled continuous-time (CT) wide-sense cyclostationary (WSCS) Gaussian processes with memory. As the case of synchronous sampling, i.e., when the sampling interval is commensurate with the period of the cyclostationary statistics, has already been studied, we focus on discrete-time (DT) processes obtained by asynchronous sampling, i.e., when the sampling interval is incommensurate with the period of the cyclostationary statistics of the CT WSCS source process. It is further assumed that the sampling interval is smaller than the maximal autocorrelation length of the CT source process, which implies that the DT process possesses memory. Thus, the sampled process is a DT wide-sense almost cyclostationary (WSACS) processes with memory. This problem is motivated by the fact that man-made communications signals are modelled as CT WSCS processes; hence, applications of such sampling include, e.g., compress-and-forward relaying and recording systems. The main challenge follows because, with asynchronous sampling, the DT sampled process is not information-stable, and hence the characterization of its RDF should be carried out within the information-spectrum framework instead of using conventional information-theoretic arguments. This work expands upon our previous work which addressed the special case in which the DT process is independent across time. The existence of dependence between the samples requires new tools to obtain the characterization of the RDF.