[sdiy] Ladder filters and gain drop, that old chestnut

[Richie Burnett]

> This is very interesting. I was aware of the effect, but I'd never heard 
> of the "crossover frequency" before. Why don't we use it more?

It's more of an EE term.  We talk about the "loop cross-over frequency" in 
discussions about stability in systems with negative feedback.  Ironically 
most EEs aim to tame systems with negative feedback and generally keep away 
from regions of self-oscillation.  (You don't want the turret of a rocket 
launcher, or the cruise-control in your car breaking into self-oscillation!) 
...but musicians love resonance and the sound of almost self-oscillation.

> I mean, musically, for filters, it's the frequent where the resonance 
> occurs that's usually the significant thing about it. When people tune a 
> filter to 1V/Oct, they usually do it with the resonance right up so you 
> can hear the filter oscillating, so what they're tuning is the crossover 
> frequency, not the cutoff, right?

Yes.  Musicians probably just don't realise that the two are different. 
Another reason is that the self-oscillating frequency is well defined:  You 
push up the resonance until it howls and you can measure the frequency of 
the resulting oscillation right there with a frequency counter, and 
calibrate it for 1v/oct or whatever.  The cutoff frequency however, is a 
kind of more woolly thing, and harder to measure casually.  Firstly you 
could define it as the -3dB point if you were an EE, or might choose 
the -12dB point if you were Robert Moog designing a 4-pole ladder.  If you 
define the cutoff frequency as the -12dB point then it just happens to give 
you the same answer as the cross-over frequency at self-oscillation for the 
moog ladder.  My point is that a musician can hear the self-oscillation and 
is well equipped (with good ears) and trained to characterise its frequency 
(pitch).  But they probably have less idea of what a filter with a -3dB 
cutoff frequency of f Hz actually sounds like when the resonance is turned 
right down.

> The -3dB definition of the cutoff always seemed very arbitrary to me, but 
> I could see that you've got a smooth response curve and you need to define 
> some point on it as the "corner", so -3dB is as good as any.

It's an EE thing again, and just a commonly used convention.  It comes from 
the "half power" definition.  If the voltage amplitude is down to -3dB, then 
the power is halved.  It doesn't work for everything though.  For instance 
you could have a Chebyshev filter with 5dB of passband ripple - It's 
passband response would cross the -3dB line several times before finally 
disappearing into the stopband.  So clearly -3dB is not a good definition 
for this filter.

I encourage EE students to define the cutoff frequency of their filters by 
where the passband and stopband asymptote lines cross each other.  We have a 
lab session where they generate bode-plots for a range of different 1-pole 
and 2-pole filters, lowpass, high-pass, resonant, non-resonant, etc.  They 
find the asymptote method much more intuitive once you start considering 
higher order filters with resonant peaks, and it still gives the same cutoff 
frequency answer as the -3dB point for first order filters.

> Crossover frequency offers a much better defined point (nothing arbitrary 
> about 180 degrees) and it makes much more sense in resonant filters.

I think so too.  If you want some other useful terms to describe filters 
borrowed from EE control theory, search for "phase margin" and "gain 
margin".  These can be summarised as how far a given filter is away from 
self oscillation in terms of its phase shift, and it's loop gain 
respectively.

-Richie,

PS. Apologies for the long post to anyone not finding this discussion 
interesting.