The Butterworth filter is a type of
signal processing filter
In signal processing, a filter is a device or process that removes some unwanted components or features from a signal. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspe ...
designed to have a
frequency response
In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of sy ...
that is as flat as possible in the
passband
A passband is the range of frequencies or wavelengths that can pass through a filter. For example, a radio receiver contains a bandpass filter to select the frequency of the desired radio signal out of all the radio waves picked up by its antenn ...
. 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
Filter design is the process of designing a signal processing filter that satisfies a set of requirements, some of which may be conflicting. The purpose is to find a realization of the filter that meets each of the requirements to a sufficient ...
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:
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
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter des ...
could be designed whose cutoff frequency was normalized to 1 radian per second and whose frequency response (
gain
Gain or GAIN may refer to:
Science and technology
* Gain (electronics), an electronics and signal processing term
* Antenna gain
* Gain (laser), the amplification involved in laser emission
* Gain (projection screens)
* Information gain in de ...
) was
:
where
is the
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
in radians per second and
is the number of
pole
Pole may refer to:
Astronomy
*Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets
*Pole star, a visible star that is approximately aligned with the ...
s in the filter—equal to the number of reactive elements in a passive filter. If
= 1, the amplitude response of this type of filter in the passband is 1/ ≈ 0.7071, 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-pole and four-pole filters, showing how the latter could be cascaded when separated by
vacuum tube
A vacuum tube, electron tube, valve (British usage), or tube (North America), is a device that controls electric current flow in a high vacuum between electrodes to which an electric voltage, potential difference has been applied.
The type kn ...
amplifier
An amplifier, electronic amplifier or (informally) amp is an electronic device that can increase the magnitude of a signal (a time-varying voltage or current). It may increase the power significantly, or its main effect may be to boost the v ...
s and so enabling the construction of higher-order filters despite
inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
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 the basic low-pass filter could be modified to give
low-pass
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter des ...
,
high-pass
A high-pass filter (HPF) is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequency d ...
,
band-pass
A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range.
Description
In electronics and signal processing, a filter is usually a two-por ...
and
band-stop
In signal processing, a band-stop filter or band-rejection filter is a filter that passes most frequencies unaltered, but attenuates those in a specific range to very low levels. It is the opposite of a band-pass filter. A notch filter is a b ...
functionality.
Overview
![Butterworth filter bode plot](https://upload.wikimedia.org/wikipedia/commons/a/a8/Butterworth_filter_bode_plot.svg)
The frequency response of the Butterworth filter is maximally flat (i.e. has no
ripples
Ripple may refer to:
Science and technology
* Capillary wave, commonly known as ripple, a wave traveling along the phase boundary of a fluid
** Ripple, more generally a disturbance, for example of spacetime in gravitational waves
* Ripple (electri ...
) 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
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
(−20 dB per
decade
A decade () is a period of ten years. Decades may describe any ten-year period, such as those of a person's life, or refer to specific groupings of calendar years.
Usage
Any period of ten years is a "decade". For example, the statement that "du ...
) (all first-order lowpass filters have the same normalized frequency response). 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
Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics.
Chebyshe ...
Type I/Type II filter or an
elliptic filter
An elliptic filter (also known as a Cauer filter, named after Wilhelm Cauer, or as a Zolotarev filter, after Yegor Zolotarev) is a signal processing filter with equalized ripple (equiripple) behavior in both the passband and the stopband. The amo ...
, the Butterworth filter has a slower
roll-off
Roll-off is the steepness of a Transfer function, transfer function with frequency, particularly in network analysis (electrical circuits), electrical network analysis, and most especially in connection with filter (signal processing), filter cir ...
, and thus will require a higher order to implement a particular
stopband specification, but Butterworth filters have a more linear phase response in the passband than Chebyshev Type I/Type II and elliptic filters can achieve.
Example
A transfer function of a third-order low-pass Butterworth filter design shown in the figure on the right looks like this:
:
![LowPass3poleICauer](https://upload.wikimedia.org/wikipedia/commons/d/dc/LowPass3poleICauer.svg)
A simple example of a Butterworth filter is the third-order low-pass design shown in the figure on the right, with
= 4/3 F,
= 1 Ω,
= 3/2 H, and
= 1/2 H.
Taking the
impedance of the capacitors
to be
and the impedance of the inductors
to be
, where is the complex frequency, the circuit equations yield the
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...
for this device:
:
The magnitude of the frequency response (gain)
is given by
:
obtained from
:
and the
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
is given by
:
![Butterworth3 GainDelay](https://upload.wikimedia.org/wikipedia/commons/8/8d/Butterworth3_GainDelay.png)
The
group delay
In signal processing, group delay and phase delay are delay times experienced by a signal's various frequency components when the signal passes through a system that is linear time-invariant (LTI), such as a microphone, coaxial cable, amplifier, l ...
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
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.
![Butterworth Filter s-Plane Response (3rd Order)](https://upload.wikimedia.org/wikipedia/commons/9/9c/Butterworth_Filter_s-Plane_Response_%283rd_Order%29.svg)
These are arranged on a
circle of radius unity, symmetrical about the real
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 that is to be rejected.
Transfer function
![Butterworth Filter Orders](https://upload.wikimedia.org/wikipedia/commons/c/cd/Butterworth_Filter_Orders.svg)
Like all filters, the typical
prototype
A prototype is an early sample, model, or release of a product built to test a concept or process. It is a term used in a variety of contexts, including semantics, design, electronics, and Software prototyping, software programming. A prototyp ...
is the low-pass filter, which can be modified into a high-pass filter, or placed in series with others to form
band-pass
A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range.
Description
In electronics and signal processing, a filter is usually a two-por ...
and
band-stop
In signal processing, a band-stop filter or band-rejection filter is a filter that passes most frequencies unaltered, but attenuates those in a specific range to very low levels. It is the opposite of a band-pass filter. A notch filter is a b ...
filters, and higher order versions of these.
The gain
of an
th-order Butterworth low-pass filter is given in terms of the transfer function
as
:
where
is the order of filter,
is the
cutoff frequency
In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced ( attenuated or reflected) rather than ...
(approximately the −3 dB frequency), and
is the DC gain (gain at zero frequency).
It can be seen that as
approaches infinity, the gain becomes a rectangle function and frequencies below
will be passed with gain
, while frequencies above
will be suppressed. For smaller values of
, the cutoff will be less sharp.
We wish to determine the transfer function
where
(from
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
). Because
and, as a general property of Laplace transforms at
,
, if we select
such that:
:
then, with
, we have the frequency response of the Butterworth filter.
The
poles of this expression occur on a circle of radius
at equally-spaced points, and symmetric around the negative real axis. For stability, the transfer function,
, is therefore chosen such that it contains only the poles in the negative real half-plane of
. The
-th pole is specified by
:
and hence
:
The transfer (or system) function may be written in terms of these poles as
:
.
where
is the
product of a sequence
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition ...
operator. The denominator is a Butterworth polynomial in
.
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 that are complex conjugates, such as
and
. The polynomials are normalized by setting
. The normalized Butterworth polynomials then have the general product form
:
:
Factors of Butterworth polynomials of order 1 through 10 are shown in the following table (to six decimal places).
Factors of Butterworth polynomials of order 1 through 6 are shown in the following table (Exact).
where the Greek letter
phi
Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek: ''fi'' ) is the 21st letter of the Greek alphabet.
In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voicele ...
( or
) represents the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
. It is an
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
that is a solution to the
quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
with a value of
The
th Butterworth polynomial can also be written as a sum
:
with its coefficients
given by the recursion formula
[
:
and by the product formula
:
where
:
Further, . The rounded coefficients for the first 10 Butterworth polynomials 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
Prototype filters are electronic filter designs that are used as a template to produce a modified filter design for a particular application. They are an example of a nondimensionalised design from which the desired filter can be scaled or tra ...
.
Maximal flatness
Assuming and , the derivative of the gain with respect to frequency can be shown to be
:
which is monotonic
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
ally decreasing for all since the gain is always positive. The gain function of the Butterworth filter therefore has no ripple. The series expansion of the gain is given by
:
In other words, all derivatives of the gain up to but not including the 2-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 is
:
In decibel
The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a po ...
s, the high-frequency roll-off is therefore 20 dB/decade, or 6 dB/octave (the factor of 20 is used because the power is proportional to the square of the voltage gain; see 20 log rule
The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a p ...
.)
Filter implementation and design
There are several different filter topologies available to implement a linear analogue filter. The most often used topology for a passive realisation is the Cauer topology, and the most often used topology for an active realisation is the Sallen–Key topology.
Cauer topology
The Cauer topology
Electronic filter topology defines electronic filter circuits without taking note of the values of the components used but only the manner in which those components are connected.
Filter design characterises filter circuits primarily by their ...
uses passive components (shunt capacitors and series inductors) to implement a linear analog filter. The Butterworth filter having a given transfer function can be realised using a Cauer 1-form. The ''k''-th element is given by
:
:
The filter may start with a series inductor if desired, in which case the ''Lk'' are ''k'' odd and the ''Ck'' are ''k'' even. These formulae may usefully be combined by making both ''Lk'' and ''Ck'' equal to ''gk''. That is, ''gk'' is the immittance In electrical engineering and acoustics, immittance is a concept combining the impedance and admittance of a system or circuit. The term ''immittance'' was invented by H. W. Bode.
It is sometimes convenient to use ''immittance'' to re ...
divided by ''s''.
:
These formulae apply to a doubly terminated filter (that is, the source and load impedance are both equal to unity) with ωc = 1. This prototype filter
Prototype filters are electronic filter designs that are used as a template to produce a modified filter design for a particular application. They are an example of a nondimensionalised design from which the desired filter can be scaled or tra ...
can be scaled for other values of impedance and frequency. For a singly terminated filter (that is, one driven by an ideal voltage or current source) 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 diplexer
A diplexer is a passive device that implements frequency-domain multiplexing. Two ports (e.g., L and H) are multiplexed onto a third port (e.g., S). The signals on ports L and H occupy disjoint frequency bands. Consequently, the signals on L and ...
s and multiplexer
In electronics, a multiplexer (or mux; spelled sometimes as multiplexor), also known as a data selector, is a device that selects between several analog or digital input signals and forwards the selected input to a single output line. The sel ...
s.
Sallen–Key topology
The Sallen–Key topology
The Sallen–Key topology is an electronic filter topology used to implement second-order active filters that is particularly valued for its simplicity. uses active and passive components (noninverting buffers, usually op amp
An operational amplifier (often op amp or opamp) is a DC-coupled high- gain electronic voltage amplifier with a differential input and, usually, a single-ended output. In this configuration, an op amp produces an output potential (relative to ...
s, resistors, and capacitors) 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 (in the case where is odd), this must be implemented separately, usually as an RC circuit
A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC c ...
, 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.
Butterworth lowpass filters with Sallen–Key topology of 3rd and 4th order, using only one op amp
An operational amplifier (often op amp or opamp) is a DC-coupled high- gain electronic voltage amplifier with a differential input and, usually, a single-ended output. In this configuration, an op amp produces an output potential (relative to ...
, are described by Huelsman,[
][
] and further single-amplifier Butterworth filters also of higher order are given by Jurišić et al.[
]
Digital implementation
Digital implementations of Butterworth and other filters are often based on the bilinear transform
The bilinear transform (also known as Tustin's method, after Arnold Tustin) is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa.
The bilinear t ...
method or the matched Z-transform method
The matched Z-transform method, also called the pole–zero mapping or pole–zero matching method, and abbreviated MPZ or MZT, is a technique for converting a continuous-time filter design to a discrete-time filter (digital filter) design.
The ...
, 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
Impulse invariance is a technique for designing discrete-time infinite-impulse-response (IIR) filters from continuous-time filters in which the impulse response of the continuous-time system is sampled to produce the impulse response of the discre ...
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
Properties of the Butterworth filter are:
* Monotonic
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
amplitude response in both passband and stopband
* Quick roll-off
Roll-off is the steepness of a Transfer function, transfer function with frequency, particularly in network analysis (electrical circuits), electrical network analysis, and most especially in connection with filter (signal processing), filter cir ...
around the cutoff frequency, which improves with increasing order
* Considerable overshoot and ringing
Ringing may mean:
Vibrations
* Ringing (signal), unwanted oscillation of a signal, leading to ringing artifacts
* Vibration of a harmonic oscillator
** Bell ringing
* Ringing (telephony), the sound of a telephone bell
* Ringing (medicine), a ri ...
in step response
The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the out ...
, which worsens with increasing order
* Slightly non-linear phase response
In signal processing, phase response is the relationship between the phase of a sinusoidal input and the output signal passing through any device that accepts input and produces an output signal, such as an amplifier or a filter.
Amplifiers, filt ...
* Group delay
In signal processing, group delay and phase delay are delay times experienced by a signal's various frequency components when the signal passes through a system that is linear time-invariant (LTI), such as a microphone, coaxial cable, amplifier, l ...
largely frequency-dependent
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
Chebyshev filters are analog or digital filters that have a steeper roll-off than Butterworth filters, and have either passband ripple (type I) or stopband ripple (type II). Chebyshev filters have the property that they minimize the error betwee ...
or the Elliptic filter
An elliptic filter (also known as a Cauer filter, named after Wilhelm Cauer, or as a Zolotarev filter, after Yegor Zolotarev) is a signal processing filter with equalized ripple (equiripple) behavior in both the passband and the stopband. The amo ...
, but without ripple.
References
{{Reflist, refs=
[ {{cite book , last1=Matthaei , first1=George L. , last2=Young , first2=Leo , last3=Jones , first3=E. M. T. , title=Microwave Filters, Impedance-Matching Networks, and Coupling Structures , publisher=McGraw-Hill , year=1964 , LCCN=64007937 , pages=104-107, 105, and 974 ]
[
{{cite journal
, last1 = Bosse
, first1 = G.
, year = 1951
, title = Siebketten ohne Dämpfungsschwankungen im Durchlaßbereich (Potenzketten)
, journal = Frequenz
, volume = 5
, issue = 10
, pages = 279–284
, doi = 10.1515/FREQ.1951.5.10.279
]
[
{{cite book
, title = Network analysis and synthesis
, first = Louis
, last = Weinberg
, publisher = Robert E. Krieger Publishing Company, Inc.
, date = 1962
, publication-date = 1975
, publication-place = Malabar, Florida
, isbn = 0-88275-321-5
, pages = 494–496
, url = https://hdl.handle.net/2027/mdp.39015000986086
, access-date = 2022-06-18
]
Linear filters
Network synthesis filters
Electronic design