Improved upper bounds to the causal quadratic rate-distortion function for Gaussian stationary sources

We improve the existing achievable rate regions for causal and for zero-delay source coding of stationary Gaussian sources for mean squared error (MSE) distortion. First, we define the information-theoretic causal rate-distortion function (RDF), R_c^it(D). In order to analyze R_c^it(D), we introduce R_c^it̅(D), the information theoretic causal RDF when reconstruction error is jointly stationary with the source. Based upon R_c^it̅(D), we derive four closed form upper bounds to the gap between R_c^it̅(D) and Shannon´s RDF, two of them strictly smaller than 0.5 bits/sample at all rates. We then show that R_c^it̅(D) can be realized by an AWGN channel surrounded by a unique set of causal pre-, post-, and feedback filters. We show that finding such filters constitutes a convex optimization problem and propose an iterative procedure to solve it. Finally, we build upon R_c^it̅(D) to improve existing bounds on the optimal performance attainable by causal and zero-delay codes.