Stable adaptive filters: the design domain

Considers a method for the design of stable infinite impulse response (IIR) filters, which are guaranteed to remain stable under adaptation. The reasons for selecting an IIR filter configuration are twofold. In the frequency domain, an IIR filter will generally exhibit a much sharper selectivity than a finite impulse response (FIR) filter of the same complexity. In the time domain, a small size IIR filter can produce a damped ringing response that lasts for a long time. An FIR filter however needs a large number of taps to provide the same response coverage. The damped ringing response can be quite useful for echo cancellation, and so on. In this paper the decomposition of a z-domain polynomial into mirror and antimirror polynomials will be given. Then, the stability of the original polynomial is shown to relate to the placement and interlacing of the roots of these auxiliary polynomials. Next, examples of root sensitivities are explored. Finally, a simple method of forming the mirror and antimirror polynomials is discussed, which is especially useful for verifying stability after an adaptive update of coefficients