### Doctoral Thesis Defense, Optimization and Systems Theory

Friday, April 6, 2001, 10.00, Kollegiesalen, Administration building,
Valhallavägen 79, KTH

**Per Enqvist**

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Spectral estimation by geometric, topological and optimization methods

*
Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan
framlägges till offentlig granskning för avläggande av teknologie
doktorsexamen fredagen den 6:e april 2001 kl 10.00 i Kollegiesalen,
Administrationsbyggnaden, Kungliga Tekniska Högskolan,
Valhallavägen 79.
*
This thesis consists of four papers dealing with various aspects of
spectral estimation and the stochastic realization problem.

In Paper A a robust algorithm for solving the Rational Covariance
Extension Problem with degree constraint (RCEP) is presented. This
algorithm improves on the current state of art that is based on convex
optimization. The new algorithm is based on a continuation method,
and uses a change of variables to avoid spectral factorizations and
the numerical ill-conditioning occuring in the original formulation
for some parameter values.

In Paper B a parameterization of the RCEP is described in the context
of cepstral analysis and homomorphic filtering. Further, it is shown
that there is a natural extension of the optimization problem
mentioned above to incorporate cepstral parameters as a
parameterization of zeros. The extended optimization problem is also
convex and, in fact, it is shown that a window of covariances and
cepstral lags form local coordinates for ARMA models of order
*n*.

In Paper C the geometry of shaping filters is analyzed by considering
parameterizations using various combinations of poles, zeros,
covariance lags, cepstral lags and Markov parameters. In particular,
the covariance and cepstral interpolation problem is studied using
differential geometry and duality theory. Assuming there is an
underlying system that is stable and minimum phase, it is shown in
this paper that there is a one-to-one correspondence between Markov
parameters and cepstral coefficients. An approach based on
simultaneous Markov and covariance parameter interpolation has been
studied by Skelton et. al. In this paper it is studied from a global
analysis point of view.

Paper D deals with a regularization of two filter design methods,
namely the covariance and cepstral matching ARMA design method and
covariance matching for MA filters. Both methods are posed as
optimization problems, and a barrier term is introduced to achieve a
strictly minimum phase solution. As a result of the regularization,
exact interpolation is traded for a gain in entropy, and the map from
data to filter defined by the optimization problems is turned into a
diffeomorphism.

Calendar of seminars

*Last update: March 22, 2001 by
Anders Forsgren,
anders.forsgren@math.kth.se.
*