Digital Filters Design for Signal and Image Processing

11.6. Appendix B: optimum 2-D stability criteria

Let us consider a recursive filter of transfer function H(z) = B(z)/A(z) that is causal or semi-causal and without non-essential singularity of the second kind in the unit bi-circle T₂. We write:

and

with:

In this section, we ask these question: “how do we choose a criterion to test the stability of H(z) Is there a criterion that is better adapted than the others for computer implementation?”

To answer these questions, we will introduce the complexity degree of a 2-D stability criterion and show that certain criteria have a minimum degree of complexity.

We saw at the end of Chapter 10 that the resultant R(z₁) obtained by eliminating z₂ between the two polynomial equations in z₂:

is a complex polynomial in the variables z₁ and whose coefficients ...