[sdiy] SVFs with different gains in the integrators

[Andrew Simper]

Hi Guy,

With a perfectly linear filter with no noise, spreading out the integrators
just changes the cutoff, damping, and band pass gain of the filter compared
to a regular SVF. Real world filters won't have perfectly matched cutoffs,
so it's probably useful to understand the impact of this in regular SVF
design as well. If cutoff1 < cutoff2 from my quick working the equations
are:

cutoff_spread = sqrt(cutoff1)*sqrt(cutoff2)
gain_band_spread = gain_band*cutoff_spread / cutoff1
damping_spread = damping*cutoff_spread / cutoff1

This is probably easiest to understand with an example. If you generate a
regular SVF biquad response, where the gain_low, gain_band, and gain_high
are the amounts of gain applied to the low, band, and high outputs of the
SVF, you have:
cutoff = 3000 hz
damping = 2
response = (gain_low*cutoff*cutoff + gain_band*cutoff*s + gain_high*s*s) /
(cutoff*cutoff + cutoff*damping*s + s*s)

then you can match this with the spread out SVF:
cutoff1 = 1000 hz
cutoff2 = 9000 hz
gain_band_spread = gain_band*3
damping_spread = 2*3
response = (gain_low*cutoff1*cutoff2 + gain_band_spread*cutoff1*s +
gain_high*s*s) / (cutoff1*cutoff2 + cutoff1*damping_spread*s + s*s)

In an actual circuit with noise and non-linearities the filter will sound
different to a regular SVF.

Cheers,

Andy

On Sat, 12 Dec 2020 at 18:09, Guy McCusker <guy.mccusker at gmail.com> wrote:

> In Chris McDowell's recent thread I made an off-hand remark about
> setting up an SVF with different gains in the integrators. I don't
> know how well-known this is but I'm wondering what list members know
> about the history and use of this idea in synthesizers.
>
> The theory, if I have it right, is that with different integrator
> gains, the natural frequency is given by the geometric mean of the
> unity gain frequencies, and the Q is enhanced by something like the
> square root of the ratio of the gains. So you can vary Q without
> varying the bandpass feedback.
>
> The only use of this that I know about in synthesizers is the Serge
> Variable Slope filter (VCFS). The claimed varying slope is really
> varying the Q, so that the slope near the natural frequency changes;
> the asymptotic slope is still 12dB/Oct. Are there any other examples?
> Does anyone know any more of the history of this idea?
>
> Incidentally, thinking about this always makes me smile at the
> marketing smarts of Serge in the 1970s. He marketed three filters:
> variable Q filter, variable slope filter, and variable bandwidth
> filter. Since Q and bandwidth are the same thing (one is the
> reciprocal of the other), and since the variable slope filter is
> actually varying the Q, all three of these are in fact variable
> bandwidth filters... but he managed to distinguish them by calling it
> three different things. Smart!
>
> Guy.
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