#### Butterworth Lowpass Poles and Zeros

When the maximally flat optimality criterion is applied to the general (analog) squared amplitude response , a surprisingly simple result is obtained [64]:

where is the desired order (number of poles). This simple result is obtained when the response is taken to be maximally flat at as well as dc (

*i.e.*, when both and are maximally flat at dc).

^{I.1}Also, an arbitrary scale factor for has been set such that the cut-off frequency (-3dB frequency) is rad/sec.

The *analytic continuation*
(§D.2)
of
to the whole
-plane may be obtained by substituting
to obtain

with

A Butterworth lowpass filter additionally has zeros at . Under the bilinear transform , these all map to the point , which determines the numerator of the digital filter as .

Given the poles and zeros of the analog prototype, it is straightforward to convert to digital form by means of the bilinear transformation.

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Example: Second-Order Butterworth Lowpass

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Mechanical Equivalent of an Inductor is a Mass