Spectral Factorization

Abstract

Spectral factorization is an important tool in the theory of stochastic processes, in information theory, signal processing, control theory and many other fields. This operation factorizes a given function into a causal and an anti-causal part. In signal processing it is necessary for example, for the determination of the causal Wiener filter. Despite a clear and simple derivation of the spectral factorization operator, it shows a quite complicated analytic behavior. The main reason behind this complicated behavior is the non-linearity of the spectral factorization operator. It implies in particular that the boundedness of the spectral factorization operator does not imply its continuity and vice versa. In this chapter, we will investigate the relation between boundedness and continuity of the spectral factorization mapping in detail.