Behaviour of the spectral factorization for continuous spectral densities

Spectral factorization plays an important role in many applications such as Wiener filter design, prediction, and estimation. It is known that spectral factorization is a non-continuous mapping on the space of all non-negative continuous functions, satisfying the Paley–Wiener condition. Additionally, it will be shown in this paper that every continuous spectrum is a discontinuity point of the spectral factorization. As a consequence, small perturbations in the given data can lead to large errors in the calculated spectral factors. Practical algorithms for the calculation of the spectral factor can only use a finite number of Fourier coefficients of the given spectrum. Consequently, the error in this finite number of coefficients can yield only a finite error in the calculated spectral factor. The paper provides sharp lower and upper bounds for this error. These bounds show that this error grows proportional with the logarithm of the number N of Fourier coefficients which are taken into account.