Discrete Stochastic Processes and Optimal Filtering

5.4. Resolution and calculation of the IIR filter

In order to do this we are going to carry out a pre-whitening of the observation signal.

Firstly let us recall a definition: we say that rational function A(z) represents a minimum phase system if A(z) and are analytical in the set , i.e. if the zeros and poles of A(z) are within the unit disk.

Furthermore the minimum phase system and its inverse are stable.

Paley-Wiener theorem

Given a function S_YY(z) verifying when z = e^iω:

– real function and ≥ 0;

– .

Therefore, there is a causal sequence a_n of the transform in z, A(z) which verifies:

represents the variance of a white noise and A(z) represents a minimum phase system. In addition, the factorization of S_YY(z) is unique.

A(z) being a minimum phase system, is causal and analytical in . Since ...