Generalizing the Markov and covariance interpolation problem using input-to-state filters

TL;DR

This paper extends the Markov and covariance interpolation problem by allowing matching of expansion coefficients at multiple points in the complex plane using input-to-state filters, simplifying the solution process.

Contribution

It introduces a generalized interpolation framework utilizing input-to-state filters to match coefficients at various points, solved through Lyapunov and eigenvalue computations.

Findings

Provides a unified approach for multi-point interpolation

Simplifies the solution via Lyapunov and eigenvalue problems

Generalizes previous single-point interpolation methods

Abstract

In the Markov and covariance interpolation problem a transfer function $W$ is sought that match the first coefficients in the expansion of $W$ around zero and the first coefficients of the Laurent expansion of the corresponding spectral density $WW^\star$. Here we solve an interpolation problem where the matched parameters are the coefficients of expansions of $W$ and $WW^\star$ around various points in the disc. The solution is derived using input-to-state filters and is determined by simple calculations such as solving Lyapunov equations and generalized eigenvalue problems.