A Control Theoretical Approach to Spectral Factorization is Unstable

Local stability analysis of a recently proposed recursive feedback-based approach to spectral factorization is performed. The method is found not to give stability guarantees. Interestingly enough, its global behavior often allows one to obtain reasonable approximations of spectral factorizations if a suitable stopping criterion is employed.

On explicit Wiener–Hopf factorization of 2 × 2 matrices in a vicinity of a given matrix

As it is known, the existence of the Wiener–Hopf factorization for a given matrix is a well-studied problem. Severe difficulties arise, however, when one needs to compute the factors approximately and obtain the partial indices. This problem is very important in various engineering applications and, therefore, remains to be subject of intensive investigations. In the present paper, we approximate a given matrix function and then explicitly factorize the approximation regardless of whether it has stable partial indices. For this reason, a technique developed in the Janashia–Lagvilava matrix spectral factorization method is applied. Numerical simulations illustrate our ideas in simple situations that demonstrate the potential of the method.