[sdiy] Multimode pole-mixing - building notch responses

Donald Tillman don at till.com
Sun Sep 16 03:26:09 CEST 2018

> On Sep 15, 2018, at 3:58 PM, Tom Wiltshire <tom at electricdruid.net> wrote:
> 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?

A 3-pole filter has two bandpass responses; a lower bandpass response with 6dB/oct slope on the left and a 12dB/oct slope on the right, and an upper bandpass, with the mirror image curve.

And you can absolutely add them together for a symmetrical bandpass response.  Though the result has 6dB/octave on both sides like a 2-pole filter, so it's not much of an advantage, but it's certainly nice to have as an option.

But back to the notch...

A notch filter has flat response with a hole in it.  Technically a null.  Varying the Q of the filter varies the width of the notch.  You can do all that with a 2-pole filter.  Any poles beyond that doesn't make it more notchy.

> 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?

Ah, I think I see the confusion...

I believe you're thinking of the notch as summing a lowpass and a highpass, and expecting the slopes of the notch to match the slopes of the those two guys.  That won't work unless the lowpass and highpass are tuned *really* far apart, and then it's not a notch but a bandstop filter.

If they're tuned closer together, the slopes undo the work of each other and you lose your bandstop.

For a notch, the slopes go infinite as it goes to zero, so the number of poles isn't related.  

The notch action comes from the lowpass and highpass phases cancelling.  That's due to the zeroes of the equation being... well, literally being zeroes.

  -- Don
Donald Tillman, Palo Alto, California

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