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

Richie Burnett rburnett at richieburnett.co.uk
Sun Sep 16 17:28:44 CEST 2018

```> 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!

```