Zero-Delay Rate Distortion via Filtering for Vector-Valued Gaussian Sources

We deal with zero-delay source coding of a vector-valued Gauss-Markov source subject to a mean-squared error (MSE) fidelity criterion characterized by the operational zero-delay vector-valued Gaussian rate distortion function (RDF). We address this problem by considering the nonanticipative RDF (NRDF) which is a lower bound to the causal optimal performance theoretically attainable (OPTA) function and operational zero-delay RDF. We recall the realization that corresponds to the optimal "test-channel" of the Gaussian NRDF, when considering a vector Gauss-Markov source subject to a MSE distortion in the finite time horizon. Then, we introduce sufficient conditions to show existence of solution for this problem in the infinite time horizon. For the asymptotic regime, we use the asymptotic characterization of the Gaussian NRDF to provide a new equivalent realization scheme with feedback which is characterized by a resource allocation (reverse-waterfilling) problem across the dimension of the vector source. We leverage the new realization to derive a predictive coding scheme via lattice quantization with subtractive dither and joint memoryless entropy coding. This coding scheme offers an upper bound to the operational zero-delay vector-valued Gaussian RDF. When we use scalar quantization, then for "r" active dimensions of the vector Gauss-Markov source the gap between the obtained lower and theoretical upper bounds is less than or equal to 0.254r + 1 bits/vector. We further show that it is possible when we use vector quantization, and assume infinite dimensional Gauss-Markov sources to make the previous gap to be negligible, i.e., Gaussian NRDF approximates the operational zero-delay Gaussian RDF. We also extend our results to vector-valued Gaussian sources of any finite memory under mild conditions. Our theoretical framework is demonstrated with illustrative numerical experiments.Comment: 32 pages, 9 figures, published in IEEE Journal of Selected Topics in Signal Processin

Constellation Optimization in the Presence of Strong Phase Noise

In this paper, we address the problem of optimizing signal constellations for strong phase noise. The problem is investigated by considering three optimization formulations, which provide an analytical framework for constellation design. In the first formulation, we seek to design constellations that minimize the symbol error probability (SEP) for an approximate ML detector in the presence of phase noise. In the second formulation, we optimize constellations in terms of mutual information (MI) for the effective discrete channel consisting of phase noise, additive white Gaussian noise, and the approximate ML detector. To this end, we derive the MI of this discrete channel. Finally, we optimize constellations in terms of the MI for the phase noise channel. We give two analytical characterizations of the MI of this channel, which are shown to be accurate for a wide range of signal-to-noise ratios and phase noise variances. For each formulation, we present a detailed analysis of the optimal constellations and their performance in the presence of strong phase noise. We show that the optimal constellations significantly outperform conventional constellations and those proposed in the literature in terms of SEP, error floors, and MI.Comment: 10 page, 10 figures, Accepted to IEEE Trans. Commu