[sdiy] Feedback in filters [was: Re: From a commercial standpoint...]

[Magnus Danielson]

Hi,

On 05/04/2016 11:08 AM, Lanterman, Aaron wrote:
>
>> On May 4, 2016, at 4:28 AM, Mattias Rickardsson <mr at analogue.org
>> <mailto:mr at analogue.org>> wrote:
>>
>> Den 3 maj 2016 6:46 em skrev "Quincas Moreira" <quincas at gmail.com
>> <mailto:quincas at gmail.com>>:
>> >
>> > You can always patch VC control of resonance.  Just send the filter
>> > output to an inverter, then to a VCA, then back into another filter
>> > input.  Voila, control the VCA = control filter resonance.
>>
>> Nope, not on all filters. The usual state-variable topology has its Q
>> feedback from the bandpass output and with opposite direction:
>> Less feedback -> higher resonance
>> More feedback -> lower resonance
>>
>> So there you need to do some odd tricks *and* have the bandpass.
>>
> There’s also a slight quirk in the way that the negative feedback works
> depending on the number of poles, in the case of a filter where the core
> is a series of cascaded 1st-order sections.
>
> If you have four sections in your cascade, like a Moog ladder (or
> Prophet 5, etc.), then inverting the output and feeding it to the input
> sharpens the curve makes the usual resonance, the poles expand in an X
> pattern, and at a feedback multiplier of 4 you get self-resonance.
>
> But if you have just two sections, and you apply negative feedback from
> the output to the input, the poles split apart and spread, but
> vertically along the imaginary coordinate but. They never actually hit
> the imaginary axis itself, so it won’t self-resonate. I seem to recall
> that the Elka Synthex in 2-pole lowpass mode behaves like that, or
> something like that.

It is illustrative to consider the three-pole filter also, there you 
need to have only two integrators between the tap and the feedback 
point. Look at the Formant filter!

In general, one has to realize that it is the modification of the root 
polynomial and that is what you so with the integrators and the feedback 
terms. A state-variable has a very simple relationship, where as 
feedback-hacked filters takes a little more analysis.

If you have a line of integrators in series and provide a feedback to 
the input, you will get as many poles as you have integrators. Depending 
on the sign of the feedback, you get two different arrangements of the 
poles, but essentially they will form a circle.
With the wrong polarity, one of the poles will go towards 0 Hz and 
create a strong integrator, base-lift. This often saturates in practice 
and is quite useless, shifting the polarity rotates the arrangement of 
poles with 180/n (where n is the number of poles) and two of the poles 
becomes dominant as in close to the jw axis and represent the resonance 
as we are (musically) used to it.

The state-variable filter actually has the polynomial feedback and 
consider the classic s^2+p*s+q=0 being the polynomial for the roots,
then the solution with complex roots have s=-p/2+/-sqrt(p/4-q) and
thus p is the main control of the real part, thus the distance to the 
imaginary axis and thus in relationship to the resonance. There is more 
equations to be written, but I think this is enough for illustration.

Ah, all the fun.

Cheers,
Magnus