Butterworth Filters

Butterworth filters are a class of all-pole filters, where the poles of the normalized transfer function are equally spaced along the unit circle ($\omega _{0}=1\,{\text{rad/s}}$). This results in a maximally flat pass-band magnitude response, or equivalently:

$\left.{\frac {\partial ^{N}\left|H\left(j\omega \right)\right|}{\partial \omega ^{N}}}\right|_{\omega =0}=0$

(1)

This means that the $N^{\text{th}}$ derivative of the magnitude at DC is zero.

The Low-Pass Butterworth Filter

The low-pass Butterworth filter has the following magnitude response:

$\left|H\left(j\omega \right)\right|^{2}={\frac {1}{1+\left({\frac {\omega }{\omega _{0}}}\right)^{2N}}}$

(2)

Where $N$ is the filter order and $\omega _{0}$ is the $-3\,{\text{dB}}$ frequency. Note that $\left|H\left(j\omega \right)\right|^{2}=\left|H\left(s\right)\right|^{2}=H\left(s\right)\cdot H^{*}\!\left(s\right)$ at $s=j\omega$. Thus:

$H\left(s\right)\cdot H^{*}\left(s\right)={\frac {1}{1+\left({\frac {-s^{2}}{\omega _{0}^{2}}}\right)^{N}}}$

(3)