Modeling, Estimation and Optimal Filtration in Signal Processing

Appendix F

The Schur-Cohn Algorithm

In this appendix, our aim is to present the Schur-Cohn algorithm [1] which is often used as a criterion for testing the stability of bounded-input bounded-output systems [2].

To simplify the description of this algorithm, we first take up the analysis of the stability domain of a 2^nd-order transfer function. This particular case leads to a simplification of the stability criteria imposed on the denominator of the transfer function. Unfortunately, it cannot be applied to transfer functions of an order greater than 2. We also present the Schur-Cohn stability algorithm based on the transfer function of an all-pass filter, allowing us to establish equivalence relation between the Schur coefficients and the reflection coefficients.

Let there be a second-order transfer function defined as follows:

The poles of are equal to:

and its zeros are defined as follows:

Depending on the values taken by a₁ and a₂, the poles can be real or complex. For example, when a²₁ < 4a₂, the poles are complex conjugates of each other. Otherwise, they are real. To ensure stability, ...