Extensions of the stability criterion for ARMA filters

Using the lattice representation of an ARMA filter, it is well known that the necessary and sufficient condition so that the poles are inside the unit circle is where the ki\´s are the reflection coefficients. The filter is said to be wide sense stable if no pole is located outside the unit circle, and it is interesting to characterize this stability by an appropriate necessary and sufficient condition. To establish this condition, the concept of canonical reflection coefficient is introduced, which eliminates the problems appearing when the Levinson recursion is not inversible. Some examples are discussed and a simple and practical test for wide sense stability is given.