In this monograph, we combine operator techniques with state space
methods to solve factorization, spectral estimation, and interpolation
problems arising in control and signal processing. We present both the
theory and algorithms with some Matlab code to solve these problems. A
classical approach to spectral factorization problems in control theory
is based on Riccati equations arising in linear quadratic control theory
and Kalman ?ltering. One advantage of this approach is that it readily
leads to algorithms in the non-degenerate case. On the other hand, this
approach does not easily generalize to the nonrational case, and it is
not always transparent where the Riccati equations are coming from.
Operator theory has developed some elegant methods to prove the
existence of a solution to some of these factorization and spectral
estimation problems in a very general setting. However, these techniques
are in general not used to develop computational algorithms. In this
monograph, we will use operator theory with state space methods to
derive computational methods to solve factorization, sp- tral
estimation, and interpolation problems. It is emphasized that our
approach is geometric and the algorithms are obtained as a special
application of the theory. We will present two methods for spectral
factorization. One method derives al- rithms based on ?nite sections of
a certain Toeplitz matrix. The other approach uses operator theory to
develop the Riccati factorization method. Finally, we use isometric
extension techniques to solve some interpolation problems.
