Division of Optimization and Systems Theory
Department of Mathematics
Since cepstrum parameters characterize a spectrum in a similar way as covariances, and can be estimated from data, it makes sense to consider realization based on the cepstrum. By considering a cepstrum interpolation problem, the approaches used in the thesis are described. In particular, an optimization problem is formulated for determining a minimum variance interpolant. Duality theory is then used to obtain a convex optimization problem that can be solved using homotopy methods. Differential geometry is used to study the set of solutions corresponding to different choices of parameter values. The parameters are shown to form local coordinates for the set of interpolants. Since this particular interpolation problem lacks solutions for some parameter combinations, an approximative interpolation problem is introduced and the entropy of the approximative interpolant is studied.
Other interpolation problems studied in the thesis, such as the covariance extension problem with a degree constraint, are described and the main theorems are presented.