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

Tom Wiltshire tom at electricdruid.net
Sat Sep 15 20:44:49 CEST 2018

```Hi Don,

Sorry, I didn’t explain myself well enough. I’m talking about a pole-mixing filter like the Xpander (multimode that way) not a state-variable filter (multimode a different way).

State variable I get well enough, but I hadn’t really considered the notch response there either, so thanks for the explanation. The S domain maths is the same for the pole-mixing case, so I’m looking for  (s^2 + 1) / (s^2 + s + 1) for a 2-pole notch, right?

What does the equation for a 4-pole notch look like? Can you do a 3-pole notch? What does that look like?

Thanks,
Tom

==================
Electric Druid
Synth & Stompbox DIY
==================

> On 15 Sep 2018, at 19:24, Donald Tillman <don at till.com> wrote:
>
>
>> 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:
>
>  (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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