Stability varying with order

Question

I have been doing some research into modelling acoustic spaces, and (with help from this site) I've managed to take the frequency response of fibrous layer model, convert it to a reflection response as a target for an IIR, and then find the polynomial coefficients for it using FDLS. All good so far.

I performed the usual BIBO stability test of finding the magnitude of the IIR denominator roots, and with a 4th order filter everything was fine (<=1.0). But increasing the order, even by just one, then it became unstable.

My question is: What does the order actually represent response-wise? I always assumed it determined the sharpest change in direction of the frequency response i.e. the 'jagged-ness' or complexity of the response curve. If that's true, then my filter is unstable, but the few poles I've calculated just happen to be inside the unit circle.

But is that true?

Answer 1

FDLS is similar to rational polynomial interpolation in that the interpolation quickly becomes unstable if the target polynomial degree is set too high. (e.g. just a tiny amount of variation or noise in the spec at a sub-optimal grid spacing can cause extremely wild excursions in the graph of a high order polynomial). Thus it is usually better if the FDLS order you choose is overdetermined by your specification points to some degree (usually selected by trial-and-error).

The jagged-ness of a frequency response is more related to how close a group of poles or zeros are to the unit circle and each other, rather than to their absolute number.