In this paper we provide a characterization of all positive rational extensions of a given partial covariance sequence. Indeed, motivated by its application to signal processing, speech processing and stochastic realization theory, this characterization is in terms of a complete bianalytic parameterization using familiar objects from systems theory. In particular, this proves a long-standing conjecture by Georgiou. The methodology is based on global analysis of the dynamics of certain fast algorithms for Kalman filtering.