TABLE OF CONTENTS 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 T 2. We write:
| (11.45) |  |
and
with:
| (11.46) |  |
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 (z 1 ) obtained by eliminating z 2 between the two polynomial equations in z 2 :
is a complex polynomial 
in the variables z 1 and 
whose coefficients depend on the coefficients of A(z 1 , z 2 ). Moreover, whatever the complex value of z 1 , the resultant R (z 1 ) only takes real values. Hence, it is also a real polynomial R(x, y)
of the variables x and y associated with the real and imaginary parts of z 1 respectively:
By dividing the polynomial R(x, y) by y 2 + x 2 ?1 in the space of real polynomials in x and y, we obtain the equation:
| (11.47) |  |
Here T(x)
