An Upper Bound to Zero-Delay Rate Distortion via Kalman Filtering for Vector Gaussian Sources

TL;DR

This paper establishes a tight upper bound on the zero-delay rate distortion function for vector Gaussian AR sources using Kalman filtering, lattice quantization, and feedback schemes, extending to infinite-dimensional cases.

Contribution

It introduces a novel upper bound for zero-delay RDF of vector Gaussian AR sources, leveraging Kalman filtering and feedback realization, and extends results to infinite-dimensional sources.

Findings

The upper bound is tight for finite-order vector Gaussian AR sources.

The NRDF matches the zero-delay RDF for infinite-dimensional sources.

Simulation confirms the theoretical bounds and schemes.

Abstract

We deal with zero-delay source coding of a vector Gaussian autoregressive (AR) source subject to an average mean squared error (MSE) fidelity criterion. Toward this end, we consider the nonanticipative rate distortion function (NRDF) which is a lower bound to the causal and zero-delay rate distortion function (RDF). We use the realization scheme with feedback proposed in [1] to model the corresponding optimal "test-channel" of the NRDF, when considering vector Gaussian AR(1) sources subject to an average MSE distortion. We give conditions on the vector Gaussian AR(1) source to ensure asymptotic stationarity of the realization scheme (bounded performance). Then, we encode the vector innovations due to Kalman filtering via lattice quantization with subtractive dither and memoryless entropy coding. This coding scheme provides a tight upper bound to the zero-delay Gaussian RDF. We extend this result to vector Gaussian AR sources of any finite order. Further, we show that for infinite dimensional vector Gaussian AR sources of any finite order, the NRDF coincides with the zero-delay RDF. Our theoretical framework is corroborated with a simulation example.