Upper Bounds for the Rate Distortion Function of Finite-Length Data Blocks of Gaussian WSS Sources

Abstract

Abstract:In this paper, we present upper bounds for the rate distortion function (RDF) of finite-length data blocks of Gaussian wide sense stationary (WSS) sources and we propose coding strategies to achieve such bounds. In order to obtain those bounds, we previously derive new results on the discrete Fourier transform (DFT) of WSS processes.

Rate-Distortion Function Upper Bounds for Gaussian Vectors and Their Applications in Coding AR Sources

“…From [ 4 ] (Equation ( 10 )) we obtain with and consequently, with being a diagonalization of where the eigenvector matrix is unitary. Therefore, …”

“…In the present paper, we generalize the upper bounds and the two low-complexity coding strategies presented in [ 4 ] to any Gaussian vector. Therefore, in contrast to [ 4 ], here no assumption about the structure of the correlation matrix of the Gaussian vector has been made (observe that since the sources in [ 4 ] were WSS the correlation matrix of the Gaussian vectors there considered was Toeplitz). To obtain such generalization we start our analysis by first proving several new results on the DFT of random vectors.…”

“…In 1973, Pearl [ 3 ] gave an upper bound for the RDF of finite-length data blocks of Gaussian WSS sources, but he did not propose a coding strategy to achieve his bound for a given block length. In [ 4 ], we presented two tighter upper bounds for the RDF of finite-length data blocks of Gaussian WSS sources, and we proposed low-complexity coding strategies, based on the discrete Fourier transform (DFT), to achieve such bounds. Moreover, we proved that those two upper bounds tend to the RDF of the WSS source (computed by Kolmogorov in [ 1 ]) when the size of the data block grows.…”

“…To obtain such generalization we start our analysis by first proving several new results on the DFT of random vectors. Although in [ 4 ] (Theorem 1) another new result on the DFT was presented, it cannot be used here, because such result and its proof rely on the power spectral density (PSD) of a WSS process and its properties.…”

Special Issue on Rate Distortion Theory and Information Theory

“…As was stated in the invitation to the Special Issue, “It is the goal of this Special Issue to reemphasize the critical accomplishments of rate distortion theory and to highlight new directions in rate distortion theoretic and information theoretic research.” Toward this end, we have collected 10 papers that examine source models that are relevant to real-world sources [ 4 , 5 , 6 , 7 ], consider new distortion measures [ 8 ], extend prior work in multiterminal source coding [ 9 , 10 ], examine source model identification and coding [ 11 ], study rate-distortion optimization for 3D video coding [ 12 ] and present a review of the field of information theory and cognition [ 13 ]. These papers not only make significant contributions to the rate distortion theory literature, but also highlight new research directions and indicate that there is much more to be done to harness the full promise of rate distortion theory as defined by Shannon.…”