Almost all methods for filter design are optimal in some sense,
and the choice of optimality determines nature of the design.
Butterworth filters are optimal in the sense of having a
maximally flat amplitude response, as measured using a Taylor
series expansion about dc [63, p. 162]. Of course,
the trivial filter has a perfectly flat amplitude response,
but that's an allpass, not a lowpass filter. Therefore, to constrain the
optimization to the space of lowpass filters, we need
constraints on the design, such as and .
That is, we may require the dc gain to be 1, and the gain at half the
sampling rate to be 0.

It turns out Butterworth filters (as well as Chebyshev and Elliptic
Function filter types) are much easier to design as analog
filters which are then converted to digital filters. This means
carrying out the design over the plane instead of the plane,
where the plane is the complex plane over which analog filter
transfer functions are defined. The analog transfer function
is very much like the digital transfer function , except that it
is interpreted relative to the analog frequency axis
(the `` axis'') instead of the digital frequency axis
(the ``unit circle''). In particular, analog filter poles
are stable if and only if they are all in the left-half of the
plane, i.e., their real parts are negative. An
introduction to Laplace transforms is given in Appendix B, and an
introduction to converting analog transfer functions to digital
transfer functions using the bilinear transform appears in
§G.3.