Butterworth Lowpass Design

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 $H(z)=1$ 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 $H(1)=1$ and $H(-1)=0$ . 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 $s$ plane instead of the $z$ plane, where the $s$ plane is the complex plane over which analog filter transfer functions are defined. The analog transfer function $H_a(s)$ is very much like the digital transfer function $H(z)$ , except that it is interpreted relative to the analog frequency axis $s=j\omega_a$ (the ``$j\omega$ axis'') instead of the digital frequency axis $z=e^{j\omega_d T}$ (the ``unit circle''). In particular, analog filter poles are stable if and only if they are all in the left-half of the $s$ 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.