ELEC 302 Butterworth Characteristics 1
BUTTERWORTH FILTERS
• Butterworth filters, often called maximally flat filters, are smooth filters that transition
monotonically from the passband to the stop band. The polynomials that characterize the
Butterworth response are functions of only two parameters: the order of the filter, n, and the
3 dB frequency,
ω
o.
• The response of an nth order low-pass Butterworth filter is an all-pole response characterized
by the nth Butterworth polynomial, Bn(
ω
):
AV
ω
( )
=AV0
Bn
ω
( )
. (9.3-1)
• The magnitude of the nth Butterworth polynomial applied to low-pass filters is given by:
Bn
ω
( )
=1+
ω
ω
o
!
"
#$
%
&
2n
(9.3-2)
• At ω = ωo the magnitude of every Butterworth polynomial is:
Bn
ω
o
( )
=1+1
( )
2n=2
, (9.3-3)
with half-power frequency, here ωo, is also commonly called the 3 dB frequency:
20 log 2
( )
=3.01030 dB ≈3dB
. (9.3-4)
• Since all Butterworth filters have a response that is reduced by 3 dB at the resonant
frequency, it is often common to specify Butterworth filters with γmax = 3 dB.
• The slope of the filter magnitude response at ω = 0 is zero, and at high frequencies, ω 〉〉 ωo,
is -20n dB/decade.
