Per Enqvist

Laurent Baratchart, Per Enqvist, Andrea Gombani et al.

We consider the symmetric Darlington synthesis of a p x p rational symmetric Schur function S with the constraint that the extension is of size 2p x 2p. Under the assumption that S is strictly contractive in at least one point of the imaginary axis, we determine the minimal McMillan degree of the extension. In particular, we show that it is generically given by the number of zeros of odd multiplicity of I-SS*. A constructive characterization of all such extensions is provided in terms of a symmetric realization of S and of the outer spectral factor of I-SS*. The authors's motivation for the problem stems from Surface Acoustic Wave filters where physical constraints on the electro-acoustic scattering matrix naturally raise this mathematical issue.

Per Enqvist

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.

Per Enqvist

When methods of moments are used for identification of power spectral densities, a model is matched to estimated second order statistics such as, e.g., covariance estimates. If the estimates are good there is an infinite family of power spectra consistent with such an estimate and in applications, such as identification, we want to single out the most representative spectrum. We choose a prior spectral density to represent a priori information, and the spectrum closest to it in a given quasi-distance is determined. However, if the estimates are based on few data, or the model class considered is not consistent with the process considered, it may be necessary to use an approximative covariance interpolation. Two different types of regularizations are considered in this paper that can be applied on many covariance interpolation based estimation methods.