On Zero-delay Source Coding of LTI Gauss-Markov Systems with Covariance Matrix Distortion Constraints

We introduce a new class of zero-delay source coding problems where a vector-valued Gauss-Markov source is conveyed subject to covariance matrix distortion constraints. We address this problem by defining an information theoretic measure where we minimize mutual information subject to causality constraints and covariance matrix distortion constraints. The resulting measure serves as a lower bound to the zero-delay rate distortion function (RDF). We solve this problem by showing that it is semidefinite representable and, thus, can be computed numerically. We also show that for this new class of information measures, it is possible to have achievable rates up to a constant space-filling loss due to a vector lattice quantizer and a constant loss due to entropy coding. We corroborate our framework with illustrative simulation examples.