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.
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