Butterworth filters are a class of all-pole filters, where the poles of the normalized transfer function are equally spaced along the unit circle (
). This results in a maximally flat pass-band magnitude response, or equivalently:
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![{\displaystyle \left.{\frac {\partial ^{N}\left|H\left(j\omega \right)\right|}{\partial \omega ^{N}}}\right|_{\omega =0}=0}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/cfc9438d456e130a3a298dbec708237c911de7f3)
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(1)
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This means that the
derivative of the magnitude at DC is zero.
The Low-Pass Butterworth Filter
The low-pass Butterworth filter has the following magnitude response:
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![{\displaystyle \left|H\left(j\omega \right)\right|^{2}={\frac {1}{1+\left({\frac {\omega }{\omega _{0}}}\right)^{2N}}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/c504020cc28ea5338716c3ab28b494fc36496ed2)
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(2)
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Where
is the filter order and
is the
frequency. Note that
at
. Thus:
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![{\displaystyle H\left(s\right)\cdot H\left(-s\right)={\frac {1}{1+\left({\frac {-s^{2}}{\omega _{0}^{2}}}\right)^{N}}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/dbb3d00cd3e2e07603706a7412a38498d9b4c6eb)
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(3)
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Thus, the poles are the roots of:
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![{\displaystyle 1+\left({\frac {-s^{2}}{\omega _{0}^{2}}}\right)^{N}=0}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/b4f72394eb693610caa02379fbd91ad13c6af17e)
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(4)
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Or equivalently:
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![{\displaystyle {\frac {-s^{2}}{\omega _{0}^{2}}}=\left(-1\right)^{\frac {1}{N}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/799331eccf812cdade0a5391d2ef10bbc415c733)
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(5)
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Since we can write
, the
roots of
can be written as:
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![{\displaystyle \left(-1\right)^{\frac {1}{N}}=e^{j\pi {\frac {2k-1}{N}}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/9418214996b6f05583059f45f8711086d1a98443)
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(6)
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For
. Thus, we get:
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![{\displaystyle -{\frac {s^{2}}{\omega _{0}^{2}}}=e^{j\pi {\frac {2k-1}{N}}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/b6eeb09d925beeff553688be672c6a5563937ca9)
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(7)
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Solving for
, we get the poles of the low-pass Butterworth filter:
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![{\displaystyle {\begin{aligned}s&=\omega _{0}\left(-1\cdot e^{j\pi {\frac {2k-1}{N}}}\right)^{\frac {1}{2}}\\&=\omega _{0}\left(e^{j\pi }\cdot e^{j\pi {\frac {2k-1}{N}}}\right)^{\frac {1}{2}}\\&=\omega _{0}\cdot e^{j\pi {\frac {2k+N-1}{2N}}}\\&=p_{k}\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/219efa1f8562d6c40c6814bf82d4dd2975cb1774)
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(8)
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We can then write:
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![{\displaystyle H\left(s\right)={\frac {1}{\displaystyle \prod _{k=1}^{N}{\frac {s-p_{k}}{\omega _{0}}}}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/08a300f276589e9178d457207718d172dd52078a)
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(9)
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