[sdiy] Multimode pole-mixing - building notch responses

[Richie Burnett]

> The equations alone aren’t enough for my brain to get it - not a maths 
> brain in that sense. I need to be able to see it in my mind’s eye, and for 
> that reason your rubber sheet helps a lot!

No worries.  I got tasked with lecturing this to MSc students, so had to 
learn quickly!  The rubber sheet analogy was how it was explained to me as 
an EE, and I found it helpful.  It also works equally well in the z-domain 
with the unit circle as being the "stability line" that you walk along to 
evaluate the frequency response instead of walking up the jw axis.  So once 
you get your head around it, at least you don't have to re-learn it all 
again differently for z-domain digital filters.

I think the thing to remember with notch filters is that it is the "zeros" 
(numerator roots) of the transfer function that actually produce the 
notches.  Their placement determines how deep the notches are, and the 
placement of the accompanying poles determines how broad or narrow the 
notches are.  For example, if you put a pair of complex zeros somewhere on 
the jw axis you will get an infinitely deep notch at that particular 
frequency, but in the absence of any poles it will be quite broad.  If you 
were to move the zeros slightly to the left, off the jw axis, the notch 
won't be as deep, because you won't be walking right over the top of them 
any more, but for now lets assume they're on the jw axis.  If you then place 
a pair of complex poles very close to the zeros, you will get a magnitude 
response that is "propped up by the poles" for all frequencies nearby, 
except for when you walk right over the top of the "zero", where it will 
drop very steeply towards zero.  That's why it is the Q factor (or zeta if 
you prefer) of the poles that determines the width of the notch in a notch 
filter.

Hope this helps,

-Richie,

PS. Try A=1, B=-2, C=2, D=0, E=0 for mixing weights on the Excel sheet I 
sent you to produce a classic 2-zero notch filter response.  If you can work 
through the simplification of the numerator and denominator polynomials it 
spits out with these mixing weights, you should get the classic 2-pole notch 
filter equation that Don posted.  The frequency response certainly looks 
right on my screen, except for right at the very bottom of the notch where 
Excel balks at plotting zero on a log (dB) axis!