The transfer function of a normalized second-order lowpass can be written as
where the normalization maps the desired -3dB frequency to 1, i.e.,
and the ``quality factor'' is defined as
where is a convenient definition for bandwidth in radians per second, given by minus the real part of the complex-conjugate pole locations and in the plane:
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.