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 :
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.1Also, 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
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.
Example: Second-Order Butterworth Lowpass
Mechanical Equivalent of an Inductor is a Mass