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 [64, 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 D, and an introduction to converting analog transfer functions to digital transfer functions using the bilinear transform appears in §I.3.
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
In the second-order case, we have, for the analog prototype,
To convert this to digital form, we apply the bilinear transform
Note that the numerator is , as predicted earlier. As a check, we can verify that the dc gain is 1:
In the analog prototype, the cut-off frequency is rad/sec, where, from Eq.(I.1), the amplitude response is . Since we mapped the cut-off frequency precisely under the bilinear transform, we expect the digital filter to have precisely this gain. The digital frequency response at one-fourth the sampling rate is
and dB as expected.
Note from Eq.(I.8) that the phase at cut-off is exactly -90 degrees in the digital filter. This can be verified against the pole-zero diagram in the plane, which has two zeros at , each contributing +45 degrees, and two poles at , each contributing -90 degrees. Thus, the calculated phase-response at the cut-off frequency agrees with what we expect from the digital pole-zero diagram.
In the plane, it is not as easy to use the pole-zero diagram to calculate the phase at , but using Eq.(I.3), we quickly obtain
A related example appears in §9.2.4.
Digitizing Analog Filters with the Bilinear Transformation
Lowpass Filter Design