Suppose we have a linear two-pole low-pass (=all-pole in continuous-time) with a transfer function of 1/(s^2+s/Q+1).Marvinh wrote: Sun Aug 18, 2024 6:32 am edit: it resonates a little to loudly i am wondering if this is a reason for a saturator? I know we can add smoothers to the g parameter.
Now, let's look at the response at three frequencies of interest:
At DC we have w=0. Substituting s=i*w=i*0=0, we have 1/(0^2+0/Q+1). So clearly this filter has unity gain at DC.
Since this filter has "normalized" cutoff at w=1 (we could set another cutoff by substituting s/wc for each s where wc is the desired cutoff, but working with normalized responses is often easier).
So at cutoff we have 1/(i^2 + i/Q + 1) where i^2=-1 (by definition), so this simplifies to 1/(i/Q) which is the same thing as Q/i. So at cutoff we have an amplitude gain of Q and 90 degrees phase-shift (from the imaginary unit in the denominator). This is the most important part: Q is essentially acting as a "gain control at cutoff" in terms of the amplitude response. The situation is a bit more complicated if our filter has zeroes as well as poles, since the "zeroes" also can have a "Q" value that's acting the opposite and giving inverse gain, but things kinda still follow the same idea (eg. a peaking EQ is literally poles and zeroes at the same frequency and the gain at cutoff is Qp/Qz where Qp and Qz are the Q-values of poles and zeroes respectively).
If we take w into the limit at infinity, then 1/(s^2+s/Q+1) = 1/inf = 0, which is why this is a lowpass.
The situation with higher order filters is a little bit more complicated, because while we can factor each filters into second order sections, the poles might be at different frequencies and how Q is actually defined can be chosen in a few ways.. but such filters can still be factored into a cascade of second order sections (each with their own cutoff and Q) and then the above applies to each such section again. If there's a "dominant" pole-pair that gives the resonance (eg. moog-style cascades) then it is mostly the Q value of that pole-pair that's most significant in terms of gain at cutoff. If we build a 4th order filter by stacking two 2nd order sections with the same cutoff and Q that we control directly, then the effective gain at cutoff becomes Q^2.
The thing with saturation in "musical" filters is that this linear behaviour is only allowed at rather low amplitudes, because we want to use a high resonance (=high Q), but we don't want it to get stupidly loud if the cutoff frequency is strongly excited. As soon as the resonance amplitude tries to grow higher, the non-linearities act (in a well-behaved filter) to dynamically lower the Q until the amplitude has fallen back to sane levels, at which point the tail of the ringing is allowed to continue at high-Q again. One can make this less obvious by driving the filter at much lower gain (at which point non-linear filters usually sound more like their linear counter-parts with annoyingly loud resonance), but the point is, this "dynamic Q" is a thing in many musical filters even if you don't drive them so hard that you'd actually start getting significant amounts of actual distortion (which.. we obviously often do, because it can also sound nice, but that's a different thing).
Note that "self-oscillation" is "infinite Q" which translates to "infinite amplitude at cutoff" yet often the self-oscillation amplitude of a musical filter might be less than the input level and that should give you an idea of how strongly the resonance is typically limited by non-linearities even when the filter still sounds like it has high resonance that's fairly "pure" in terms of distortion.
