Re: stability of time-varying biquad

On Oct 29, 9:00 am, robert bristow-johnson <r...@xxxxxxxxxxxxxxxxxxxx>
wrote:
On Oct 29, 10:14 am, Vladimir Vassilevsky <antispam_bo...@xxxxxxxxxxx>
wrote:



Jerry Avins wrote:
HardySpicer wrote:

On 29 Oct, 10:00, SYL <sya...@xxxxxxxxx> wrote:

It is known that time-varying may make an IIR unstable even it is
stable on its own. Imposing limits on the amount of varying is able to
eliminate this possibility of becoming unstable.

I am wondering, is there a cookbook formula I can obtain these limits
on fc, Q or gain?

Not sure what you mean. Just work out its poles and make sure they are
in the left hand plane (or within the unit circle).
Time-varying or not the poles must lie there for it to be stable.

Most time-varying filters are in transition, with the coefficients
linearly interpolated between initial and final values. The end states
being stable doesn't guarantee that intermediate states will also be
stable.

For the IIR filter of the second order, the linear transition of the
coefficients guarantees that the intermediate states are always stable
if the end points are stable.

sure, for 2nd-order filters with complex conjugate poles, the sqrt of
the coef for the 2nd-order term (i call it "a2" but some old texts
called it "b2") in the denominator is the distance the poles are from
the origin (and -a1/2 is the real part of the two poles).  as long as |
a2| < 1, then supposedly the filter is stable.

but you can have modulated parameters (e.g. a vibrato applied to the
resonant frequency) where all of the instantaneous pole locations are
all inside the unit circle, yet the filter blows up (if the modulation

============

is large enough, fast enough, and the Q is high enough).  that is what
=======================================

these papers cited (by me and Martin) are all about.

r b-j

This is what I was asking. Jean Laroche's 2007 paper mentions that one
could impose a limit of the rate or speed or amount of the change. If
these limits can not be easily calculated, what would be the typical
values? I will do some experiments.

As I understand, the papers Martin cited were more about interpolating
poles/zeros or [a]/[b]. The filter can blow up even without
interpolating.

Syl
.