On Tue, 23 Nov 1999, Dana Massie wrote:
>I tried looking empirically at what happens to the poles and zeros of
>filters whose parameters are being interpolated, and they follow arcs on
>the z-plane. I did not try to study analytically what was happening. In
>the case of the parametric eq, amazing weird things happen. I found a
>number of filters where the poles move smoothly from complex poles to
>become real poles. Then the real poles would move along the real axis.
>The movements in the z-plane can be rather complex, actually.
I believe this is something system and control theory people have developed
a lot of intuitive techniques for. That is where the pole-zero methods come
from, too, I think.
>I wonder if the perspective of filter basis sets could help in
>understanding how linear interpolation of direct form coefficients
>works. See above regarding FIR coefficient interpolation.
I think not, since HRTF processing concerns itself with FIR filters. In this
case the math is easy because there is no feedback - the impulse response of
the resulting interpolated filter is the sum of weighted basis impulse
responses. The same does not hold when feedback is added.
>Actually, time varying filters are a huge topic. I wish that I had the
>time (and the skills) to really study the behavior of various classes of
>time varying filters analytically. i actually suspect that the math may
>be intractable for many general cases.
Or maybe not. After all, the filters stay linear no matter how you twiddle
the params. As I said, there are lots of pre-thought out methods to do the
analysis. OTOH, I don't have the skill myself.
>The goal is to design a filter structure that satisfies some design
>constraint for all values of the parameters under interpolation. Is this
>tractable?
Considering how difficult it is to design a digital filter from scratch
(i.e. not going through the analog filter design equations), it probably
will not be.
Sampo Syreeni <decoy@iki.fi>, aka decoy, student/math/Helsinki university
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