[sdiy] Q and Synth Filters

[Magnus Danielson]

From: Glen <mclilith at charter.net>
Subject: [sdiy] Q and Synth Filters
Date: Sat, 26 Mar 2005 21:19:44 -0500
Message-ID: <4.1.20050326211459.00bb4638 at mail.charter.net>

Glen,

> Is the term "Q" always something that makes sense, and can always be
> calculated, within the context of synth filters?

Actually, it applies to linear filter in general, so it can be a usefull thing
to learn more about. It can be measured and calculated by many methods.
However, don't be fooled to beleive it is always easy to measure it at all
times. High-Q values are easy to measure since stick out so much that it
becomes fairly easy to get the characteristics.

> I realize the concept of "Q" applies very well to a bandpass filter, and
> probably to LP and HP as well, but can the concept be applied literally to
> all the synth filter topologies?

Yes, almost all at least depending on how you model things. Delay-based multi-
tap filters (such as digital FIR filters) without feedback is excluded, since
they are only-zero filters.

> Finally, how is Q measured for the various filter types?

Traditionally is the Q-value a measure of how good "quality" a resonant
cuircuit has. A higher Q-value has better "quality" and the quality aspect is
low loss, that is, it looses less energy per cycle than lower-Q resonators do.
Thus, a high Q-value resonator rings longer than low Q-value resonators.
Passive resonators (coils and capacitances, crystals) is safely stable.

One way to measure the Q-value is to measure the frequency on the top of the
resonance, the center frequency fc, and at the -3 dB points on the slopes on
each side of the resonance noting the high and low frequencies fh and fl.
The smaller frequency difference beteween the -3 dB points, the higher Q, so we
got

       fc
Q = -------
    fh - fl

Converting this resonant circuit into a pole-pair using LaPlace analysis gives
us another formula for Q:

     rp
Q = ----
    -2op

where rp is the radius of the polepair (rp² = wp² + op²)
op is the pole-pairs position on the real axis (gain factor per second, is
negative for stable resonators, thus damping)
wp is the pole-pairs position on the imaginary axis (2*Pi*f where f is the
resonance frequency).

Whenever you have a resonant pole-pair and you have the position in the LaPlace
plain, you can calculate the Q-value. Every filter that has a pole-pair (i.e.
all filter except those that only have zeroes or only simple poles) you can
calculate the Q-value for each such pole-pair.

In a typical 4-pole filter for a synth, you have two pole-pairs, which
typically have the same resonance frequency but where one pole-pair is the
resonant pair and the other is far from resonant, so we call them the passive
pair. The passive pair still contributes to the slopes and overall
phase-shiftings.

So, to in some degree come back to your question "Is Q-values generically
usefull?" the answer is yes, you can use that quantity in a generic sense.
There is however some issues when you hit and go pass the stability rules, but
we wouln't do that, now would we? So given the stability rules are followed you
can. It is a handy parameter for synt/audio-work where as when working on it
from a theory point of view, the Q-value is more of a handy analysis measure
rather than a good primary measure.

Cheers,
Magnus