# [sdiy] Multimode pole-mixing - building notch responses

Donald Tillman don at till.com
Sat Sep 15 20:24:26 CEST 2018

```> On Sep 15, 2018, at 10:46 AM, Tom Wiltshire <tom at electricdruid.net> wrote:
>
> Can anyone explain pole-mixing notch responses to me in a way that I can “get”?
>
> Most of the responses from a multimode pole-mixing filter I understand, but I’m struggling with notches.
>
> The bandpass responses, for example, are the same as the highpass responses, but “shifted right” to effectively make them into a highpass and a lowpass in series - a bandpass. That’s easy to understand. Intuitive, even.

Hey Druid,

That's not really accurate.  Assuming a 2-pole multimode filter, the lowpass will have a 12dB/oct slope down.  The highpass will have a 12dB/oct slope up.  But the bandpass will have a 6dB/oct slope on each side.

So it's not like a highpass and lowpass in series.  It's much more like a lowpass tilted counterclockwise with a 6dB/oct boost throughout.

> I guess the trouble is I don’t really know what a notch filter looks like in terms of the s-domain equations, so it’s hard to see how the pole-mixing leads to that result. Even the Allpass responses are easier, but in that case, I *do* know what the s domain equation is supposed to look like (s-1/s+1 for the one-pole, s^2-s+1 / s^2+s+1 for the two-pole, right?).

The notch function simply adds the lowpass and highpass signals together.  The secret is that, at resonance, the lowpass response shifts the phase 90 degrees behind, the highpass response shifts the phase 90 degrees forward, leaving a 180 degree difference between them, and adding them together cancels completely

For the equation, we add the highpass "s^2" term and the lowpass "1" term, and the "s" term isn't used:

Hnotch(s) = (s^2 + 1) / (s^2 + s + 1)

Since "s" is complex frequency, s^2 has a value of -1 at resonance, and the -1 and +1 cancel for the notch.

-- Don
--
Donald Tillman, Palo Alto, California
http://www.till.com

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