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Digital Filters Design for Signal and Image Processing by Mohamed Najim

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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:

images

images

and

images

with:

images

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:

images

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

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