On Duality between Filtering and Interpolation*

Abstra ct. Positive real rational functions play a central role in both deterministic and stochastic linear systems theory, arising in circuit synthesis, filtering, interpolation, spectral analysis, speech processing, stability theory, stochastic realization theory and systems identification – to name just a few. For this reason, results about positive real transfer functions and their realizations typically have many applications and manifestations. In this paper, we survey a recent study of certain spaces of positive real transfer functions, describing a fundamental geometric duality between filtering and interpolation. Not surprisingly, then, this duality, while interesting in its own right, has several corollaries which provide solutions and insight into some very interesting and intensely researched problems. One of these is the rational covariance extension problem, which was formulated by Kalman and for which the duality theorem yields a complete solution. In this paper, we shall describe the duality theorem, which we motivate in terms of both the rational covariance extension problem, viewed as an interpolation problem, and a fast algorithm for Kalman filtering, viewed as a nonlinear dynamical system on the space of positive real transfer functions. We also outline a new proof of the recent solution to the rational covariance extension problem, using a global inverse function theorem due to Hadamard. We conclude by describing some additional corollaries which relate to the minimal stochastic partial realization problem.