[sdiy] Multimode pole-mixing - building notch responses

[Tom Wiltshire]

> On 15 Sep 2018, at 23:03, Donald Tillman <don at till.com> wrote:
> 
> 
>> On Sep 15, 2018, at 11:44 AM, Tom Wiltshire <tom at electricdruid.net> wrote:
>> 
>> 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?
> 
> Well, tell us more...  :-)
> 
> Yes, a 2-pole notch is: (s^2 + 1) / (s^2 + s + 1)
> 
> A 3-pole notch is: (s^3 + s^2 + s + 1) / (s^3 + 2s^2 + 2s + 1)
> 
> But it doesn't buy you anything over a 2-pole notch.
> 
> A 4-pole filter can have a notch, but again, it doesn't buy you anything over a 2-pole notch.
> 
> A 4-pole filter can have a double notch on a Butterworth response, like this:
>     (s^4 + 3.4s^2 + 1) / ((s^2 + 0.848s + 1)(s^2 + 0.765s + 1))
> 
> But that's not very different from two 2-pole filters.

Thanks for bearing with me, Don. I’m going to have to ask another slew of stupid questions.

This is what I mean about I don’t really *get* notches yet. Whereas I can see why you can’t have a 3-pole bandpass (at least, not a symmetrical one) I don’t see why what you say about notches makes sense. Why doesn’t a 3-pole notch “buy you anything over a 2-pole notch”? You can do a 4-pole notch too, but again it’s no different? Ok. Why not?

If I was imagining how this would work, I suppose I’d expect something like 1-pole notches have 6dB slopes down to the notch, whereas 2-pole notches would have 12dB/oct slopes. But I know it doesn’t work like this because you can increase the steepness of the slopes (or equivalently the narrowness of the notch) by increasing the resonance on a 2-pole SVF. So I’m left without any real understanding of what “poles” means in terms of notches. And I’m thinking that focusing on the poles is probably the wrong way to go anyway - what happened to the zeros in all this? Aren’t they basically notches in the response?