Butterworth filter
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The Butterworth filter is one type of electronic filter design. It is designed to have a frequency
response which is as flat as mathematically possible in the passband. Another name for them is
'maximally flat magnitude' filters.
Contents
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1 Origin of name
2 Overview
3 Transfer function
o 3.1 Maximally Flat Magnitude Filter
o 3.2 High Frequency Roll Off
4 Filter Realization
5 Normalized Butterworth polynomials
6 Comparison with other linear filters
7 See also
Origin of name
The Butterworth type filter was first described by the British engineer Stephen Butterworth (in
his paper "On the Theory of Filter Amplifiers", Wireless Engineer (also called Experimental
Wireless and the Radio Engineer), vol. 7, 1930, pp. 536-541.
The Bode plot of a first-order Butterworth low-pass filter
Butterworth low-pass filters of orders 1 through 5
Butterworth second-order low-pass active filter example
Overview
The frequency response of the Butterworth filter is maximally flat (has no ripples) in the
passband, and rolls off towards zero in the stopband. When viewed on a logarithmic Bode plot,
the response slopes off linearly towards negative infinity. For a first-order filter, the response
rolls off at −6 dB per octave (−20 dB per decade). (All first-order filters, regardless of name, are
actually identical and so have the same frequency response.) For a second-order Butterworth
filter, the response decreases at −12 dB per octave, a third-order at −18 dB, and so on.
Butterworth filters have a monotonically decreasing magnitude function with ω. The Butterworth
is the only filter that maintains this same shape for higher orders (but with a steeper decline in
the stopband) whereas other varieties of filters (Bessel, Chebyshev, elliptic) have different
shapes at higher orders.
Compared with a Chebyshev Type I/Type II filter or an elliptic filter, the Butterworth filter has a
slower roll-off, and thus will require a higher order to implement a particular stopband
specification. However, Butterworth filter will have a more linear phase response in the passband
than the Chebyshev Type I/Type II and elliptic filters.
Transfer function
Like all filters, the typical prototype is the low-pass filter, which can be modified into a highpass filter, or placed in series with others to form band-pass and band-stop filters, and higher
order versions of these.
The magnitude squared frequency response:
here, n = order of filter
ωc = cutoff frequency = -3dB frequency
ωp = passband edge frequency
1/(1 + ε2) = band edge value of |H(ω)|2.
Since H(s)H(-s) evaluated at s = jω is simply equal to |H(ω)|2, it follows that
The poles occur on a circle of radius ωc at equally spaced points
k = 0, 1, 2, ....., n-1
and hence,
k = 0, 1, 2, ...., n-1
The magnitude of the frequency response of an nth order lowpass filter can be defined
mathematically as:
where G is the gain of the filter, H is the transfer function, j is the imaginary number, n is the
order of the filter, ω is the angular frequency of the signal in radians per second, and ωc is the
cutoff frequency (−3 dB frequency).
Normalizing the expression (setting the cutoff frequency ωc = 1), it becomes:
Bode magnitude plot for Butterworth second-order low-pass filter example
Maximally Flat Magnitude Filter
the first (2n-1)th derivatives of g are zero for ω = 0. This means that
the rate of change of gain with respect to ω is zero i.e., the gain is constant with ω.
Hence Butterworth filters are also called maximally flat filters.
The derivation has been tagged. Comments
High Frequency Roll Off
Hence, the high frequency roll off = 20n dB/decade
Filter Realization
The Butterworth filter having a given transfer function can be realised using Cauer - 1 form:
kth element is given by:
; k = odd
; k = even
Normalized Butterworth polynomials
n
Factors of Polynomial Bn(s)
1
(s + 1)
2
s2 + 1.414s + 1
3
(s + 1)(s2 + s + 1)
4
(s2 + 0.7654s + 1)(s2 + 1.8478s + 1)
5
(s + 1)(s2 + 0.6180s + 1)(s2 + 1.6180s + 1)
6
(s2 + 0.5176s + 1)(s2 + 1.414s + 1)(s2 + 1.9318s + 1)
7
(s + 1)(s2 + 0.4450s + 1)(s2 + 1.247s + 1)(s2 + 1.8022s + 1)
8 (s2 + 0.3986s + 1)(s2 + 1.111s + 1)(s2 + 1.6630s + 1)(s2 + 1.9622s + 1)
Comparison with other linear filters
Here is an image showing the Butterworth filter next to other common kind of filters obtained
with the same number of coefficients:
As is clear from the image, the Butterworth filter rolls off more slowly than all the others but it
shows no ripples.