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

Richie Burnett rburnett at richieburnett.co.uk
Sun Aug 30 18:09:45 CEST 2015

```> 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.

```