The
Butterworth filter is one type of
signal processing filterIn signal processing, a filter is a device or process that removes from a signal some unwanted component or feature. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal...
design. It is designed to have a
frequency responseFrequency response is the measure of any system's output spectrum in response to an input signal. In the audible range it is usually referred to in connection with electronic amplifiers, microphones and loudspeakers. Radio spectrum frequency response can refer to measurements of coaxial cables,...
which is as flat as mathematically possible in the
passbandIn brief, the passband is the range of frequencies or wavelengths that can pass through a filter without being attenuated.- Passband in terms of filters :...
. Another name for it is
maximally flat magnitude filter.
The Butterworth type filter was first described by the
BritishThe United Kingdom of Great Britain and Northern Ireland is a sovereign state located off the northwestern coast of continental Europe. It is an island country, spanning an archipelago including Great Britain, the northeastern part of Ireland, and many small islands...
engineerEngineers are concerned with developing economical and safe solutions to practical problems, by applying mathematics and scientific knowledge while considering technical constraints. The term is derived from the Latin root "ingenium," meaning "cleverness"...
Stephen ButterworthStephen Butterworth was a British physicist who invented the Butterworth filter, a class of electrical circuits that are used to filter electrical signals....
in his paper "On the Theory of Filter Amplifiers",
Wireless Engineer (also called
Experimental Wireless and the Wireless Engineer), vol. 7, 1930, pp. 536–541.
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 plotA Bode plot is a graph of the logarithm of the transfer function of a linear, time-invariant system versus frequency, plotted with a log-frequency axis, to show the system's frequency response...
, the response slopes off linearly towards negative infinity. For a first-order filter, the response rolls off at −6
dBThe decibel is a logarithmic unit of measurement that expresses the magnitude of a physical quantity relative to a specified or implied reference level. Since it expresses a ratio of two quantities with the same unit, it is a dimensionless unit...
per
octaveIn music, an octave , is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon which has been referred to as the "basic miracle of music," the use of which is "common in most musical systems." It may be derived from the...
(−20 dB per
decadeA decade is a period of ten years. The word is derived from the late Latin decas, from Greek decas, from deca. The other words for spans of years also come from Latin: lustrum , century , millennium . The term usually refers to a period of ten years starting at a multiple of ten...
) (all first-order lowpass filters have the same normalized frequency response). For a second-order lowpass filter, the response ultimately 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
ChebyshevChebyshev filters are analog or digital filters having a steeper roll-off and more passband ripple or stopband ripple than Butterworth filters...
Type I/Type II filter or an
elliptic filterAn elliptic filter is a signal processing filter with equalized ripple behavior in both the passband and the stopband...
, the Butterworth filter has a slower
roll-offRoll-off is a term commonly used to describe the steepness of a transmission function with frequency, particularly in electrical network analysis, and most especially in connection with filter circuits in the transition between a passband and a stopband...
, and thus will require a higher order to implement a particular
stopbandA stopband is a band of frequencies, between specified limits, in which a circuit, such as a filter or telephone circuit, does not let signals through, or the attenuation is above the required stopband attenuation level...
specification. However, Butterworth filters have a more linear phase response in the passband than the Chebyshev Type I/Type II and elliptic filters.
A simple example
A simple example of a Butterworth filter is the 3rd order low-pass design shown in the figure on the right, with farad, ohm, and henry. Taking the
impedanceElectrical impedance, or simply impedance, describes a measure of opposition to a sinusoidal alternating current . Electrical impedance extends the concept of resistance to AC circuits, describing not only the relative amplitudes of the voltage and current, but also the relative phases...
of the capacitors
C to be
1/Cs and the impedance of the inductors
L to be
Ls, where is the complex frequency, the circuit equations yields the
transfer functionA transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...
for this device:
The magnitude of the frequency response (gain) is given by:
and the
phaseThe phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0. Phase is a frequency domain or Fourier transform domain concept, and as such, can be readily understood in terms of simple harmonic...
is given by:
The
group delayGroup delay is a measure of the transit time of a signal through a device under test , versus frequency. Group delay is a useful measure of phase distortion, and is calculated by differentiating the insertion phase response of the DUT versus frequency. Another way to say this is that group delay is...
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-passA high-pass filter is an LTI filter that passes high frequencies well but attenuates frequencies lower than the cutoff frequency. The actual amount of attenuation for each frequency is a design parameter of the filter...
Butterworth filter is obtained. If we change each capacitor and inductor into a resonant capacitor and inductor in parallel, with the proper choice of component values, a
band-passA band-pass filter is a device that passes frequencies within a certain range and rejects frequencies outside that range. An example of an analogue electronic band-pass filter is an RLC circuit...
Butterworth filter is obtained.
The transfer function
Like all filters, the typical
prototypePrototype 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 transformed. They are most often seen with regard to...
is the
low-pass filterA low-pass filter is a filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter when used in...
, which can be modified into a
high-pass filterA high-pass filter is an LTI filter that passes high frequencies well but attenuates frequencies lower than the cutoff frequency. The actual amount of attenuation for each frequency is a design parameter of the 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 functionA transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...
H(s) as:
where
- n = order of filter
- ωc = 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 rather than passing through.Typically in electronic systems such as filters and...
(approximately the -3dB frequency)
- is the DC gain (gain at zero frequency)
It can be seen that as
n approaches infinity, the gain becomes a rectangle function and frequencies below ω
c will be passed with gain , while frequencies above ω
c 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
H(s)H(-s) evaluated at
s = jω is simply equal to |
H(jω)|
2, it follows that:
The poles of this expression occur on a circle of radius ω
c at equally spaced points. The transfer function itself will be specified by just 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:
for n even for n odd
To four decimal places, they are:
| n | Factors of Polynomial |
|---|
| 1 |
|
| 2 |
|
| 3 |
|
| 4 |
|
| 5 |
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| 6 |
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| 7 |
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| 8 |
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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 2
n-th derivative are zero, 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 that is flatter in the passband than the "maximally flat" Butterworth. Such a filter is the inverse Chebyshev filter.
High-frequency roll-off
Again assuming , the slope of the log of the gain for large ω is:
In
decibelThe decibel is a logarithmic unit of measurement that expresses the magnitude of a physical quantity relative to a specified or implied reference level. Since it expresses a ratio of two quantities with the same unit, it is a dimensionless unit...
s, the high-frequency
roll-offRoll-off is a term commonly used to describe the steepness of a transmission function with frequency, particularly in electrical network analysis, and most especially in connection with filter circuits in the transition between a passband and a stopband...
is therefore 20
n dB/decade, or 6
n dB/octave (The factor of 20 is used because the power is proportional to the square of the voltage gain; see 20 log rule.)
Filter design
There are a number of different
filter topologiesAn electronic filter topology is an electronic analog filter circuit in which the values of the components remain undefined. A particular topology is then characterized entirely by the manner in which the components are connected, and not by their values....
available to implement a linear analogue filter. These circuits differ only in the values of the components, but not in their connections.
Cauer topology
The Cauer topology 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:
- k = od
- k = eve
The filter may start with a series inductor if desired, in which case the are k odd and the are k even.
Sallen-Key topology
The
Sallen-Key topologyThe Sallen–Key topology is an electronic filter topology used to implement second-order active filters that is particularly valued for its simplicity.. It is a degenerate form of a voltage-controlled voltage-source filter topology...
uses active and passive components (noninverting buffers, usually op amps, 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 circuitA resistor–capacitor circuit ', or RC filter or RC network, is an electric circuit composed of resistors and capacitors driven by a voltage or current source...
, and cascaded with the active stages.
The Sallen-Key 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 component values undefined, which may be chosen at will.
Digital implementation
Digital implementations of Butterworth filters often use
bilinear transformThe bilinear transform is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa...
or
matched z-transformIn mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation.It can be considered as a discrete equivalent of the Laplace transform...
to discretize an analog filter. For higher orders, they are sensitive to quantization errors. For this reason, they are often calculated as cascaded
biquad sectionsFor the analog implementation of a biquad filter, check biquad filter.In signal processing, a digital biquad filter is a second-order recursive linear filter, containing two poles and two zeros...
and a cascaded first order filter, 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 others, but shows no ripples.