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| [[File:Buttergr.jpg|thumb|270px|The frequency response plot from Butterworth's 1930 paper.]]
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| The '''Butterworth filter''' is a type of [[filter (signal processing)|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".<ref>In ''Wireless Engineer'' (also called ''Experimental Wireless and the Wireless Engineer''), vol. 7, 1930, pp. 536–541 - [http://www.gonascent.com/papers/butter.pdf "On the Theory of Filter Amplifiers"-S. Butterworth]</ref>
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| == Original paper ==
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| {{Linear analog electronic filter|filter2=hide|filter3=hide}}
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| 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 [[image parameter filter|theory then in use]]. The filter was not in common use for over 30 years after its publication. Butterworth stated that: {{quote|"An ideal electrical filter should not only completely reject the unwanted frequencies but should also have uniform sensitivity for the wanted frequencies".}}
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| 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 ([[gain]]) was
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| :<math>G(\omega) = \sqrt{\frac{1}{1+{\omega}^{2n}}},</math>
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| where ω is the [[angular frequency]] in radians per second and ''n'' is the number of [[Pole (complex analysis)|pole]]s 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/{{sqrt|2}} ≈ 0.707, which is half power or −3 [[decibel|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.
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| Butterworth solved the equations for two- and four-pole filters, showing how the latter could be cascaded when separated by [[vacuum tube]] [[amplifier]]s and so enabling the construction of higher-order filters despite [[inductor]] losses. In 1930, low-loss core materials such as [[Molypermalloy_Powder_Core|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.
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| 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.
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| Butterworth also showed that his basic low-pass filter could be modified to give [[low-pass filter|low-pass]], [[high-pass filter|high-pass]], [[band-pass filter|band-pass]] and [[band-stop filter|band-stop]] functionality.
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| == Overview ==
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| [[File:Butterworth filter bode plot.svg|thumb|right|350px|The [[Bode plot]] of a first-order Butterworth low-pass filter]]
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| The frequency response of the Butterworth filter is maximally flat (i.e. has no [[ripple (filters)|ripples]]) in the passband and rolls off towards zero in the stopband.<ref name=bianchi>
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| {{cite book
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| | title = Electronic filter simulation & design
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| | author = Giovanni Bianchi and Roberto Sorrentino
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| | publisher = McGraw-Hill Professional
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| | year = 2007
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| | isbn = 978-0-07-149467-0
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| | pages = 17–20
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| | url = http://books.google.com/books?id=5S3LCIxnYCcC&pg=PT32&dq=Butterworth-approximation+maximally-flat#v=onepage&q=Butterworth-approximation%20maximally-flat&f=false
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| }}</ref>
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| 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 (electronics)|octave]] (−20 dB per [[Decade (log scale)|decade]]) (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.
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| Compared with a [[Chebyshev filter|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.
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| == Example ==
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| [[File:LowPass3poleICauer.svg|300px|right|thumb|A third-order low-pass filter ([[Cauer topology]]). The filter becomes a Butterworth filter with [[cutoff frequency]] ω<sub>c</sub>=1 when (for example) C<sub>2</sub>=4/3 farad, R<sub>4</sub>=1 ohm, L<sub>1</sub>=3/2 henry and L<sub>3</sub>=1/2 henry.]] | |
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| A simple example of a Butterworth filter is the third-order low-pass design shown in the figure on the right, with ''C''<sub>2</sub> = 4/3 F, ''R''<sub>4</sub> = 1 Ω, ''L''<sub>1</sub> = 3/2 H, and ''L''<sub>3</sub> = 1/2 H.<ref>Mattaei ''et al.'', p. 107</ref> Taking the [[Electrical impedance|impedance]] of the capacitors ''C'' to be 1/''Cs'' and the impedance of the inductors ''L'' to be ''Ls'', where {{nowrap|''s'' {{=}} σ + ''j''ω}} is the complex frequency, the circuit equations yield the [[transfer function]] for this device:
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| :<math>H(s)=\frac{V_o(s)}{V_i(s)}=\frac{1}{1+2s+2s^2+s^3}.</math>
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| The magnitude of the frequency response (gain) ''G''(ω) is given by
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| :<math>G^2(\omega)=|H(j\omega)|^2=\frac{1}{1+\omega^6},</math>
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| and the [[Phase (waves)|phase]] is given by
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| :<math>\Phi(\omega)=\arg(H(j\omega)).\!</math>
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| [[File:Butterworth3 GainDelay.png|256px|left|thumb|Gain and [[group delay]] of the third-order Butterworth filter with ω<sub>c</sub>=1]]
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| 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.
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| 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. [[File:Butterworth Filter s-Plane Response (3rd Order).svg|250px|right|thumb|Log density plot of the transfer function H(s) in [[complex frequency space]] for the third-order Butterworth filter with ω<sub>c</sub>=1. The three [[Pole (complex analysis)|poles]] lie on a circle of unit radius in the left half-plane.]]
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| These are arranged on a [[Unit circle|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.
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| By replacing each inductor with a capacitor and each capacitor with an inductor, a high-pass Butterworth filter is obtained.
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| 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.
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| 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.
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| == Transfer function ==
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| [[File:Butterworth Filter Orders.svg|thumb|350px|left|Plot of the gain of Butterworth low-pass filters of orders 1 through 5, with [[cutoff frequency]] <math>\omega_0 = 1</math>. Note that the slope is 20''n'' dB/decade where ''n'' is the filter order.]]
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| Like all filters, the typical [[prototype filter|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.
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| The gain <math>G(\omega)</math> of an ''n''-order Butterworth low pass filter is given in terms of the transfer function ''H(s)'' as
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| :<math>G^2(\omega)=\left |H(j\omega)\right|^2 = \frac {{G_0}^2}{1+\left(\frac{\omega}{\omega_c}\right)^{2n}}</math>
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| where
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| * n = order of filter
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| * ω<sub>c</sub> = [[cutoff frequency]] (approximately the -3dB frequency)
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| * <math>G_0</math> is the DC gain (gain at zero frequency)
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| It can be seen that as ''n'' approaches infinity, the gain becomes a rectangle function and frequencies below ω<sub>c</sub> will be passed with gain <math>G_0</math>, while frequencies above ω<sub>c</sub> will be suppressed. For smaller values of ''n'', the cutoff will be less sharp.
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| We wish to determine the transfer function ''H(s)'' where <math>s=\sigma+j\omega</math> (from [[Laplace transform]]). Since ''H(s)H(-s)'' evaluated at ''s = jω'' is simply equal to |''H(jω)''|<sup>2</sup>, it follows that:
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| :<math>H(s)H(-s) = \frac {{G_0}^2}{1+\left (\frac{-s^2}{\omega_c^2}\right)^n}.</math>
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| The poles of this expression occur on a circle of radius ω<sub>c</sub> at equally-spaced points along a semi-circle ranging from <math>\frac{\pi}{2}</math> to <math>\frac{3\pi}{2}</math>. 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
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| :<math>-\frac{s_k^2}{\omega_c^2} = (-1)^{\frac{1}{n}} = e^{\frac{j(2k-1)\pi}{n}}
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| \qquad\mathrm{k = 1,2,3, \ldots, n}</math>
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| and hence;
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| :<math>s_k = \omega_c e^{\frac{j(2k+n-1)\pi}{2n}}\qquad\mathrm{k = 1,2,3, \ldots, n}.</math>
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| The transfer function may be written in terms of these poles as
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| :<math>H(s)=\frac{G_0}{\prod_{k=1}^n (s-s_k)/\omega_c}.</math>
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| The denominator is a Butterworth polynomial in ''s''.
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| ===Normalized Butterworth polynomials===
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| 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 <math>s_1</math> and <math>s_n</math>. The polynomials are normalized by setting <math>\omega_c=1</math>. The normalized Butterworth polynomials then have the general form
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| :<math>B_n(s)=\prod_{k=1}^{\frac{n}{2}} \left[s^2-2s\cos\left(\frac{2k+n-1}{2n}\,\pi\right)+1\right]\qquad\mathrm{n = even}</math>
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| :<math>B_n(s)=(s+1)\prod_{k=1}^{\frac{n-1}{2}} \left[s^2-2s\cos\left(\frac{2k+n-1}{2n}\,\pi\right)+1\right]\qquad\mathrm{n = odd}.</math>
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| To four decimal places, they are
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| <center>
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| {|
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| |-----
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| {| class="wikitable" style="text-align: center;"
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| |-
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| !n!!Factors of Polynomial <math>B_n(s)</math>
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| |-
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| !1
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| |<math>(s+1)</math>
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| |-
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| !2
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| |<math>s^2+1.4142s+1</math>
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| |-
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| !3
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| |<math>(s+1)(s^2+s+1)</math>
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| |-
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| !4
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| |<math>(s^2+0.7654s+1)(s^2+1.8478s+1)</math>
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| |-
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| !5
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| |<math>(s+1)(s^2+0.6180s+1)(s^2+1.6180s+1)</math>
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| |-
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| !6
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| |<math>(s^2+0.5176s+1)(s^2+1.4142s+1)(s^2+1.9319s+1)</math>
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| |-
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| !7
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| |<math>(s+1)(s^2+0.4450s+1)(s^2+1.2470s+1)(s^2+1.8019s+1)</math>
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| |-
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| !8
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| |<math>(s^2+0.3902s+1)(s^2+1.1111s+1)(s^2+1.6629s+1)(s^2+1.9616s+1)</math>
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| |}
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| |}</center>
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| The normalized Butterworth polynomials can be used to determine the transfer function for any low-pass filter cut-off frequency <math>\omega_c</math>, as follows
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| :<math>H(s) = \frac{G_0}{B_n(a)}</math> , where <math>a = \frac{s}{\omega_c}.</math>
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| Transformation to other bandforms are also possible, see [[prototype filter]].
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| ===Maximal flatness===
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| Assuming <math>\omega_c=1</math> and <math>G_0=1</math>, the derivative of the gain with respect to frequency can be shown to be
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| :<math>\frac{dG}{d\omega}=-nG^3\omega^{2n-1}</math>
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| which is [[monotonic]]ally decreasing for all <math>\omega</math> 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
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| :<math>G(\omega)=1 - \frac{1}{2}\omega^{2n}+\frac{3}{8}\omega^{4n}+\ldots</math>
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| In other words, all derivatives of the gain up to but not including the 2''n''-th derivative are zero at <math>\omega=0</math>, 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.
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| ===High-frequency roll-off===
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| Again assuming <math>\omega_c=1</math>, the slope of the log of the gain for large ω is
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| :<math>\lim_{\omega\rightarrow\infty}\frac{d\log(G)}{d\log(\omega)}=-n.</math> | |
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| In [[decibel]]s, the high-frequency roll-off 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]].)
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| ==Filter design==
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| There are a number of different [[electronic filter topology|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.
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| === Cauer topology ===
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| [[File:Cauer lowpass.svg|right|450px|Butterworth filter using [[Cauer topology (electronics)|Cauer topology]]]]
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| The [[Cauer topology (electronics)|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<sup>th</sup> element is given by
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| :<math>C_k = 2 \sin \left [\frac {(2k-1)}{2n} \pi \right ]\qquad\mathrm{k = odd}</math>
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| :<math>L_k = 2 \sin \left [\frac {(2k-1)}{2n} \pi \right ]\qquad\mathrm{k = even}.</math>
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| The filter may start with a series inductor if desired, in which case the <math>\scriptstyle L_k</math> are k odd and the <math>\scriptstyle C_k</math> are k even.
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| These formulae may usefully be combined by making both <math>\scriptstyle L_k</math> and <math>\scriptstyle C_k</math> equal to <math>\scriptstyle g_k</math>. That is, <math>\scriptstyle g_k</math> is the [[immittance]] divided by ''s''.
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| :<math>g_k = 2 \sin \left [\frac {(2k-1)}{2n} \pi \right ]\qquad\mathrm{k = 1,2,3, \ldots, n}.</math> | |
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| These formulae apply to a doubly terminated filter (that is, the source and load impedance are both equal to unity) with <math>\scriptstyle \omega_c=1</math>. This [[prototype filter]] 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<ref>Matthaei, pp. 104-107</ref>
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| :<math>g_j = \frac{a_j a_{j-1}}{c_{j-1} g_{j-1}}\qquad\mathrm{j = 2,3, \ldots, n}</math>
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| where
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| :<math>g_1 = a_1</math>
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| and
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| :<math>a_j = \sin \frac{\pi}{2} \left [\frac {(2j-1)}{n} \right ]\qquad\mathrm{j = 1,2,3, \ldots, n}</math> | |
| :<math>c_j = \cos^2 \left [\frac{\pi j}{2n} \right ]\qquad\mathrm{j = 1,2,3, \ldots, n}.</math>
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| 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]]s and [[multiplexer]]s.<ref>Matthaei, pp. 105,974</ref>
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| === Sallen–Key topology ===
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| [[File:Sallen-Key.svg|right|300px|[[Sallen–Key topology]]]]
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| The [[Sallen–Key topology]] uses active and passive components (noninverting buffers, usually [[op amp]]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 <math>n</math> is odd), this must be implemented separately, usually as an [[RC circuit]], and cascaded with the active stages.
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| For the second-order Sallen–Key circuit shown to the right the transfer function is given by
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| :<math>H(s)=\frac{V_{out}(s)}{V_{in}(s)}=\frac{1}{1+C_2(R_1+R_2)s+C_1C_2R_1R_2s^2}.</math>
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| We wish the denominator to be one of the quadratic terms in a Butterworth polynomial. Assuming that <math>\omega_c=1</math>, this will mean that
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| :<math>C_1C_2R_1R_2=1\,</math>
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| and
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| :<math>C_2(R_1+R_2)=-2\cos\left(\frac{2k+n-1}{2n} \pi\right).</math>
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| This leaves two undefined component values that may be chosen at will.
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| ===Digital implementation===
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| 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 [[Digital biquad filter|biquad sections]], plus one first-order or third-order section for odd orders.
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| ==Comparison with other linear filters==
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| 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.
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| [[File:Electronic linear filters.svg|500px|center]]
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| The Butterworth filter rolls off more slowly around the cutoff frequency than the [[Chebyshev filter]] or the [[Elliptic filter]], but without ripple.
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| ==References==
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| <references/>
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| *Matthaei, George L.; Young, Leo and Jones, E. M. T., ''Microwave Filters, Impedance-Matching Networks, and Coupling Structures'', McGraw-Hill, 1964 {{LCCN|647937}}.
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| [[Category:Linear filters]]
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| [[Category:Network synthesis filters]]
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| [[Category:Electronic design]]
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