# Butterworth filter poles

Hi,

I'm looking at this textbook question and trying to get a better idea of exactly what its asking.

For the processing to be real valued each pole would have to have a complex conjugate right?

So for the two points closest to the real axis, we would call $$e^{j\omega_0}$$ & $$e^{-j\omega_0}$$, then the next two at 45 degree angles $$e^{j\omega_1}$$ & $$e^{-j\omega_1}$$ and the two most vertical points $$e^{j\omega_2}$$ & $$e^{-j\omega_2}$$

And these would be our 6 poles.

But then how does this relate to higher order filters being created with minimal algebra?

• What happens if you add poles right in the middle of each pair? Dec 8 '21 at 20:25

But then how does this relate to higher order filters being created with minimal algebra?

The key phrase -- which should be in the text someplace -- is that the poles of a Butterworth filter are evenly spaced, and hence very predictable.

If the book has already gone into constructing the Butterworth filter transfer function, go back and study up. If this is a book about realizing filters and you're just expected to know how to construct a Butterworth, go study up. Sometimes authors like to do some foreshadowing in their questions, and they'll cover the topic in detail later -- if so, feel free to look ahead.

At any rate, the Butterworth poles are located* at

$$s = \exp \left [j \frac{\pi}{2} \left(1 + \frac{1 + 2k}{n}\right) \right]\ \forall\ k \in [0, n),$$ so you can work that out for higher (or lower) orders.

* from Passive and Active Network Analysis and Synthesis, Aram Budak, Houghton Mifflin, 1974. Oldy, yes, but definitely a goody.

• Wow. Budak is a blast from the past. That's the text I used in undergrad. :-)
– Peter K.
Dec 8 '21 at 20:06
• It was the text for the active filters class I took while getting my Master's degree. It was the only super-easy class that I've taken that I've also used material from throughout my career. Dec 8 '21 at 20:40
• I mean -- what's changed that requires a revision? Resistors and caps are smaller, and GBW bandwidths are bigger, but the basics are the same. Dec 8 '21 at 20:41
• Certainly! I wasn't dissing it, just amazed that someone else knows the book. :-)
– Peter K.
Dec 9 '21 at 1:00

For the processing to be real valued each pole would have to have a complex conjugate right?

Correct

And these would be our 6 poles.

Correct

But then how does this relate to higher order filters being created with minimal algebra?

That depends a bit on your definition of "minimal algebras". The standard way of doing this, is to split this into 3 cascaded section, each of which has one complex conjugate pole pair and dual zero at infinity (in the s-plane) or a dual zero at $$z = -1$$ (in the z-plane).