Computing Reflection Coefficients to
Check Filter Stability

Since we know that a recursive filter is stable if and only if all its poles have magnitude less than 1, an obvious method for checking stability is to find the roots of the denominator polynomial $A(z)$ in the filter transfer function Eq. (7.3). If the moduli of all roots are less than 1, the filter is stable. This test works fine for low-order filters (e.g., on the order of 100 poles or less), but it may fail numerically at higher orders because the roots of a polynomial are very sensitive to round-off error in the polynomial coefficients [61]. It is therefore of interest to use a stability test that is faster and more reliable numerically than polynomial root-finding. Fortunately, such a test exists based on the filter reflection coefficients.

It is a mathematical fact [48] that all poles of a recursive filter are inside the unit circle if and only if all reflection coefficients (which are always real) are strictly between -1 and 1. The full theory associated with reflection coefficients is beyond the scope of this book, but can be found in most modern treatments of linear prediction [48,47] or speech modeling [92,20,66]. An online derivation appears in [86].^9.3Here, we will settle for a simple recipe for computing the reflection coefficients from the transfer-function denominator polynomial $A(z)$. This recipe is called the step-down procedure, Schur-Cohn stability test, or Durbin recursion [48], and it is essentially the same thing as the Schur recursion (for allpass filters) or Levinson algorithm (for autocorrelation functions of autoregressive stochastic processes) [38].