The transfer function of a *normalized* second-order lowpass can be written as

where the normalization maps the desired -3dB frequency to 1,

and the ``quality factor'' is defined as

where is a convenient definition for

Here we assume , so that the poles have nonzero imaginary parts.

A second-order *Butterworth* lowpass filter is obtained for
. Larger
values give the ``corner resonance''
effect often used in music synthesizers.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University