## 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 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.

#### Butterworth Lowpass Poles and Zeros

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

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 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

with

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.

#### Example: Second-Order Butterworth Lowpass

In the second-order case, we have, for the analog prototype,

To convert this to digital form, we apply the bilinear transform

(I.4) | |||

(I.5) | |||

(I.6) | |||

(I.7) |

Note that the numerator is , as predicted earlier. As a check, we can verify that the dc gain is 1:

*i.e.*, that there is a (double) notch at half the sampling rate.

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.

**Next Section:**

Digitizing Analog Filters with the Bilinear Transformation

**Previous Section:**

Lowpass Filter Design