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/0f2d0c08638c31c0cacad472fcde64f29eb72a86)
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(3)
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