[[File:Buttergr.jpg|thumb|The frequency response plot from Butterworth's 1930 paper.
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The <math>n</math>th Butterworth polynomial can also be written as a sum
:<math>B_n(s)=\sum_{k=0}^n a_k s^k\,,</math>
with its coefficients <math>a_k</math> given by the recursion formula
:<math display="block">\begin{align}
p_A = p_1 \times (10^{\alpha/10}-1)^
&\qquad \text{For 0} \leq \alpha < \infty
\end{align}
</math>
where:
:<math>p_A</math> is the relocated pole positioned to set the desired cutoff attenuation.
:<math>p_1</math> is a −3.01 dB cutoff pole that lies on the unit circle.
:<math>\alpha</math> is the desired attenuation at the cutoff frequency in dB (1 dB, 10 dB, etc.).
:<math>n</math> is the number of poles, the order of the filter.
Filter implementation and design
There are several different filter topologies available to implement a linear analogue filter. The most often used topology for a passive realisation is the Cauer topology, and the most often used topology for an active realisation is the Sallen–Key topology.
Cauer topology
right|450px|thumb|Butterworth filter using [[Cauer topology (electronics)|Cauer topology ]]
The Cauer topology uses passive components (shunt capacitors and series inductors) to implement a linear analog filter. The Butterworth filter having a given transfer function can be realised using a Cauer 1-form. The k-th element is given by