Robustness of the Inner–Outer Factorization and of the Spectral Factorization for FIR Data

The present paper analyzes the construction of outer functions, the factorization of finite-impulse response (FIR) systems into a minimum phase system and an all-pass part, and the spectral factorization. It investigates the behavior of these operations with respect to errors in the given data. It shows that the error in the constructed outer function grows at least and at most proportional with the logarithm of the degree of the given FIR system and proportional to the error in the given data. For the other two operations, it turns out that they show the same behavior with respect to errors in the given FIR data as the construction of outer functions.