The Butterworth filter is a type of signal processing filter designed to have as flat
a frequency response as possible in the passband. It is also referred to as a maximally flat
magnitude filter. It was first described in 1930 by the British engineer and physicist
Stephen Butterworth in his paper entitled "On the Theory of Filter Amplifiers".
Original paper Butterworth had a reputation for solving "impossible"
mathematical problems. At the time, filter design required a considerable amount of designer
experience due to limitations of the theory then in use. The filter was not in common
use for over 30 years after its publication. Butterworth stated that:
"An ideal electrical filter should not only completely reject the unwanted frequencies
but should also have uniform sensitivity for the wanted frequencies".
Such an ideal filter cannot be achieved but Butterworth showed that successively closer
approximations were obtained with increasing numbers of filter elements of the right values.
At the time, filters generated substantial ripple in the passband, and the choice of
component values was highly interactive. Butterworth showed that a low pass filter could be designed
whose cutoff frequency was normalized to 1 radian per second and whose frequency response
was
where ω is the angular frequency in radians per second and n is the number of poles in
the filter—equal to the number of reactive elements in a passive filter. If ω = 1,
the amplitude response of this type of filter in the passband is 1/√2 ≈ 0.707, which
is half power or −3 dB. Butterworth only dealt with filters with an even number of
poles in his paper. He may have been unaware that such filters could be designed with an
odd number of poles. He built his higher order filters from 2-pole filters separated by vacuum
tube amplifiers. His plot of the frequency response of 2, 4, 6, 8, and 10 pole filters
is shown as A, B, C, D, and E in his original graph.
Butterworth solved the equations for two- and four-pole filters, showing how the latter
could be cascaded when separated by vacuum tube amplifiers and so enabling the construction
of higher-order filters despite inductor losses. In 1930, low-loss core materials such as molypermalloy
had not been discovered and air-cored audio inductors were rather lossy. Butterworth discovered
that it was possible to adjust the component values of the filter to compensate for the
winding resistance of the inductors. He used coil forms of 1.25″ diameter and
3″ length with plug in terminals. Associated capacitors and resistors were contained inside
the wound coil form. The coil formed part of the plate load resistor. Two poles were
used per vacuum tube and RC coupling was used to the grid of the following tube.
Butterworth also showed that his basic low-pass filter could be modified to give low-pass,
high-pass, band-pass and band-stop functionality. Overview
The frequency response of the Butterworth filter is maximally flat 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. A first-order filter's response rolls off
at −6 dB per octave. A second-order filter decreases at −12 dB per octave, a third-order
at −18 dB and so on. Butterworth filters have a monotonically changing magnitude function
with ω, unlike other filter types that have non-monotonic ripple in the passband and/or
the stopband. 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, but Butterworth filters have
a more linear phase response in the pass-band than Chebyshev Type I/Type II and elliptic
filters can achieve. Example
A simple example of a Butterworth filter is the third-order low-pass design shown in the
figure on the right, with C2 = 4/3 F, R4 = 1 Ω, L1 = 3/2 H, and L3 = 1/2 H. Taking
the impedance of the capacitors C to be 1/Cs and the impedance of the inductors L to be
Ls, where s = σ + jω is the complex frequency, the circuit equations yield the transfer function
for this device:
The magnitude of the frequency response G(ω) is given by
and the phase is given by
The group delay is defined as the derivative of the phase with respect to angular frequency
and is a measure of the distortion in the signal introduced by phase differences for
different frequencies. The gain and the delay for this filter are plotted in the graph on
the left. It can be seen that there are no ripples in the gain curve in either the passband
or the stop band. The log of the absolute value of the transfer
function H(s) is plotted in complex frequency space in the second graph on the right. The
function is defined by the three poles in the left half of the complex frequency plane.
These are arranged on a circle of radius unity, symmetrical about the real s axis. The gain
function will have three more poles on the right half plane to complete the circle.
By replacing each inductor with a capacitor and each capacitor with an inductor, a high-pass
Butterworth filter is obtained. A band-pass Butterworth filter is obtained
by placing a capacitor in series with each inductor and an inductor in parallel with
each capacitor to form resonant circuits. The value of each new component must be selected
to resonate with the old component at the frequency of interest.
A band-stop Butterworth filter is obtained by placing a capacitor in parallel with each
inductor and an inductor in series with each capacitor to form resonant circuits. The value
of each new component must be selected to resonate with the old component at the frequency
to be rejected. Transfer function
Like all filters, the typical prototype is the low-pass filter, which can be modified
into a high-pass filter, or placed in series with others to form band-pass and band-stop
filters, and higher order versions of these. The gain of an n-order Butterworth low pass
filter is given in terms of the transfer function H(s) as
where n = order of filter
ωc = cutoff frequency is the DC gain
It can be seen that as n approaches infinity, the gain becomes a rectangle function and
frequencies below ωc will be passed with gain , while frequencies above ωc will be
suppressed. For smaller values of n, the cutoff will be less sharp.
We wish to determine the transfer function H(s) where . Since and as a general property
of Laplace transforms at , then if we select H(s) such that:
then for imaginary inputs, , we have the frequency response of the Butterworth filter.
The poles of this expression occur on a circle of radius ωc at equally-spaced points, and
symmetric around the imaginary axis. For stability, the transfer function, H(s), is therefore
be chosen such that it contains only the poles in the negative real half-plane of s. The
k-th pole is specified by
and hence;
The transfer function may be written in terms of these poles as
The denominator is a Butterworth polynomial in s.
Normalized Butterworth polynomials The Butterworth polynomials may be written
in complex form as above, but are usually written with real coefficients by multiplying
pole pairs which are complex conjugates, such as and . The polynomials are normalized by
setting . The normalized Butterworth polynomials then have the general form
To four decimal places, they are
The normalized Butterworth polynomials can be used to determine the transfer function
for any low-pass filter cut-off frequency , as follows
, where Transformation to other bandforms are also
possible, see prototype filter. Maximal flatness
Assuming and , the derivative of the gain with respect to frequency can be shown to
be
which is monotonically decreasing for all since the gain G is always positive. The gain
function of the Butterworth filter therefore has no ripple. Furthermore, the series expansion
of the gain is given by
In other words, all derivatives of the gain up to but not including the 2n-th derivative
are zero at , resulting in "maximal flatness". If the requirement to be monotonic is limited
to the passband only and ripples are allowed in the stopband, then it is possible to design
a filter of the same order, such as the inverse Chebyshev filter, that is flatter in the passband
than the "maximally flat" Butterworth. High-frequency roll-off
Again assuming , the slope of the log of the gain for large ω is
In decibels, the high-frequency roll-off is therefore 20n dB/decade, or 6n dB/octave
Filter design There are a number of different filter topologies
available to implement a linear analogue filter. The most often used topology for a passive
realisation is Cauer topology and the most often used topology for an active realisation
is Sallen–Key topology. Cauer topology
The Cauer topology uses passive components to implement a linear analog filter. The Butterworth
filter having a given transfer function can be realised using a Cauer 1-form. The kth
element is given by
The filter may start with a series inductor if desired, in which case the are k odd and
the are k even. These formulae may usefully be combined by
making both and equal to . That is, is the immittance divided by s.
These formulae apply to a doubly terminated filter with . This prototype filter can be
scaled for other values of impedance and frequency. For a singly terminated filter the element
values are given by
where
and
Voltage driven filters must start with a series element and current driven filters must start
with a shunt element. These forms are useful in the design of diplexers and multiplexers.
Sallen–Key topology
The Sallen–Key topology uses active and passive components to implement a linear analog
filter. Each Sallen–Key stage implements a conjugate pair of poles; the overall filter
is implemented by cascading all stages in series. If there is a real pole, this must
be implemented separately, usually as an RC circuit, and cascaded with the active stages.
For the second-order Sallen–Key circuit shown to the right the transfer function is
given by
We wish the denominator to be one of the quadratic terms in a Butterworth polynomial. Assuming
that , this will mean that
and
This leaves two undefined component values that may be chosen at will.
Digital implementation Digital implementations of Butterworth and
other filters are often based on the bilinear transform method or the matched Z-transform
method, two different methods to discretize an analog filter design. In the case of all-pole
filters such as the Butterworth, the matched Z-transform method is equivalent to the impulse
invariance method. For higher orders, digital filters are sensitive to quantization errors,
so they are often calculated as cascaded biquad sections, plus one first-order or third-order
section for odd orders. Comparison with other linear filters
Here is an image showing the gain of a discrete-time Butterworth filter next to other common filter
types. All of these filters are fifth-order.
The Butterworth filter rolls off more slowly around the cutoff frequency than the Chebyshev
filter or the Elliptic filter, but without ripple.
References ^ In Wireless Engineer, vol. 7, 1930, pp.
536–541 - "On the Theory of Filter Amplifiers"-S. Butterworth
^ Giovanni Bianchi and Roberto Sorrentino. Electronic filter simulation & design. McGraw-Hill
Professional. pp. 17–20. ISBN 978-0-07-149467-0. ^ Matthaei et al., p. 107
^ Matthaei, pp. 104-107 ^ Matthaei, pp. 105,974
Matthaei, George L.; Young, Leo and Jones, E. M. T., Microwave Filters, Impedance-Matching
Networks, and Coupling Structures, McGraw-Hill, 1964 LCCN 64-7937.