[sdiy] Different pole notches and phasers

[Richie Burnett]

I think one of the things that is confusing is that mathematically "notches" 
don't come from poles.  They come from zeros.  And zeros in a transfer 
function often arise from a signal taking two parallel paths through a 
circuit.  Like in the classic cascaded all-pass phaser, or those 
combinations of the Expander filter outputs that result in responses with 
notches.  You don't get notches until you start to combine the input with a 
phase-shifted version,  or combine outputs that have different phase 
relationships.

In the simplest instance, if you take a signal, delay it and mix it back in 
with the original, you will get notches.  This is the classic comb filter. 
And the notches result from destructive interference between the original 
and the delayed (phase shifted) version.

If you actually wanted to design a digital notch filter to remove something 
like an annoying 1kHz tone, you would do this by placing a pair of zeros 
right on the unit circle at the angle that corresponds to 1kHz.  And this 
would annihilate anything at 1kHz (to the limits of numerical precision.) 
You can alter the width of the notches by placing poles close to them but 
just in from the unit circle.  The closer you put the poles to the zeros the 
narrower the notch width will be.  (A narrow digital 1kHz notch filter at 
8kHz sample rate would have a pair of complex zeros at 0.7071+j0.7071 and 
0.7071-j0.7071, and a pair of poles just inward from the unit circle at 
something like 0.69+j0.69 and 0.69-j0.69.  If you move the poles closer to 
the origin the notches will get wider.)

Stay healthy everyone!

-Richie,



-----Original Message----- 
From: Tom Wiltshire
Sent: Monday, April 13, 2020 11:58 AM
To: SDIY List
Subject: [sdiy] Different pole notches and phasers

Hi all,

Could someone explain something I don’t understand clearly to me?

What does it mean to talk about the number of poles when discussing a notch 
filter?

Additionally, what’s the difference between a notch filter and a notch 
created with an allpass filter and the input added together? Is there a 
difference?

It seems to me there is, since multiple stages of allpass filters with the 
input added together (a phaser) creates multiple notches, whereas a 
multi-pole notch would just be a deeper/narrower notch. Is that right? Or 
could a 4-pole notch filter have two notches?

There’s something I don’t quite get here, and I’m struggling to find exactly 
what it is, hence the rather confused questions. Any clarification anyone 
can offer would be greatly appreciated.

Thanks,
Tom

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