What is the difference equation of a Butterworth filter?

[Elias Kyriakides]

Can anybody provide me with the difference equation of a Butterworth
filter?
e.g. something like y[n] = a1x[n] +a2x[n-1] etc

Thanks,
Elias

[John Doty]

In article <972ge7$4...@smc.vnet.net>, "Elias Kyriakides"
<elias_ky...@hotmail.com> wrote:

No. The Butterworth filter has a frequency response that is a rational
function of finite order. Such a response can be realized by differential
equations, but not by difference equations.

The Buttterworth filter is a classic utilizing the capabilities of analog
electronic components. Attempting to approximate it digitally probably
isn't a particularly useful exercise: digital filters have different
capabilities and limitations.

--
| John Doty "You can't confuse me, that's my job."
| Home: j...@w-d.org
| Work: j...@space.mit.edu

[Bruce Detterich]

You'll have to do a little leg work, but here's the basic approach:

1. Find the desired order of Butterworth filter and frequency scale it
in the S-domain. As a hint to make your life easy, look for
"Butterworth Standard Forms".

2. Once you have the S-domain expression, make the "bilateral"
Z-domain substitution for S. I believe the substitution is something
like s=(z-1)/(z+1).

3. Simplify the result to a Z-domain fraction (polynomials in Z in the
numerator and denominator), from which you can go directly to the
difference equation.

Mathematica should make short work of this. The whole thing should be
covered in any good introductory text on sampled data signal
processing.

BEWARE: Some tables of Butterworth Standard Forms have incorrect
values for orders above the 3rd. This problem persisted in many texts
for many years (case of authors not checking their sources), so you're
best off if you find a text that describes the Butterworth derivation
procedure (it's easy), and capture it in Mathematica.