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

(I.1) 
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 cutoff frequency (3dB frequency) is
rad/sec.
The
analytic continuation
(§
D.2)
of
to the whole
plane may be obtained by substituting
to obtain
The
poles of this expression are simply the
roots of unity when
is odd, and the roots of
when
is even. Half of these
poles
are in the lefthalf
plane (
re
) and
thus belong to
(which must be stable). The other half belong
to
. In summary, the poles of an
thorder Butterworth
lowpass prototype are located in the
plane at
, where [
64, p. 168]

(I.2) 
with
for
. These poles may be quickly found graphically
by placing
poles uniformly distributed around the unit circle (in
the
plane, not the
planethis is not a frequency axis) in
such a way that each complex pole has a complexconjugate counterpart.
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.
Next Section: Example: SecondOrder Butterworth LowpassPrevious Section: Mechanical Equivalent of an Inductor is a Mass