Editing Butterworth filter

{{Short description|Type of signal processing filter}}
{{Linear analog electronic filter|filter2=hide|filter3=hide}}
[[File:Buttergr.jpg|thumb|The frequency response plot from Butterworth's 1930 paper.<ref name="Butterworth1930"/>]]

The '''Butterworth filter''' is a type of [[filter (signal processing)|signal processing filter]] designed to have a [[frequency response]] that is as flat 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 name="Butterworth1930"/>

==Original paper==
{{Linear analog electronic filter|filter2=hide|filter3=hide}}

Butterworth had a reputation for solving very complex mathematical problems thought to be 'impossible'. 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".}}

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 [[gain (electronics)|gain]] as a function of frequency (i.e., the magnitude of its [[frequency response]]) is:

:<math>G(\omega) = {\frac{1} \sqrt{1+{\omega}^{2n}}},</math>

where <math>\omega</math> is the [[angular frequency]] in radians per second and <math>n</math> is the number of [[Pole (complex analysis)|pole]]s in the filter—equal to the number of reactive elements in a passive filter. Its [[cutoff frequency]] (the [[half-power point]] of approximately &minus;3 [[decibel|dB]] or a voltage gain of 1/{{sqrt|2}}&nbsp;≈ 0.7071) is normalized to 𝜔 = 1 radian per second. Butterworth only dealt with filters with an even number of poles in his paper, though odd-order filters can be created with the addition of a single-pole filter applied to the output of the even-order filter. 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]] [[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.

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 filter|low-pass]], [[high-pass filter|high-pass]], [[band-pass filter|band-pass]] and [[band-stop filter|band-stop]] functionality.

==Overview==

[[File:Butterworth filter bode plot.svg|thumb|right|350px|The [[Bode plot]] of a first-order low-pass filter]]

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=Bianchi2007/> 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&nbsp;dB per [[Octave (electronics)|octave]] (−20&nbsp;dB per [[Decade (log scale)|decade]]) (all first-order lowpass filters have the same normalized frequency response). A second-order filter decreases at −12&nbsp;dB per octave, a third-order at −18&nbsp;dB and so on. Butterworth filters have a monotonically changing magnitude function with <math>\omega</math>, unlike other filter types that have non-monotonic ripple in the passband and/or the stopband.

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 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:

:<math>\frac{V_o(s)}{V_i(s)}=\frac{R_4}{s^3(L_1 C_2 L_3) + s^2(L_1 C_2 R_4) + s(L_1 + L_3) + R_4}</math>

[[File:LowPass3poleICauer.svg|300px|right|thumb|A third-order low-pass filter ([[Cauer topology]]). The filter becomes a Butterworth filter with [[cutoff frequency]] <math>\omega_c</math>=1 when (for example) <math>C_2</math>=4/3 F, <math>R_4</math>=1 Ω, <math>L_1</math>=3/2 H and <math>L_3</math>=1/2 H.]]

A simple example of a Butterworth filter is the third-order low-pass design shown in the figure on the right, with <math>C_2</math>&nbsp;= 4/3&nbsp;F, <math>R_4</math>&nbsp;= 1&nbsp;Ω, <math>L_1</math>&nbsp;= 3/2&nbsp;H, and <math>L_3</math>&nbsp;= 1/2&nbsp;H.<ref name="MatthaeiYoungJones" /> Taking the [[Electrical impedance|impedance]] of the capacitors <math>C</math> to be <math>1/(Cs)</math> and the impedance of the inductors <math>L</math> to be <math>Ls</math>, where {{nowrap|<math>s= \sigma + j\omega</math>}} is the complex frequency, the circuit equations yield the [[transfer function]] for this device:
:<math>H(s)=\frac{V_o(s)}{V_i(s)}=\frac{1}{1+2s+2s^2+s^3}.</math>

The magnitude of the frequency response (gain) <math>G(\omega)</math> is given by
:<math>G(\omega)=|H(j\omega)|=\frac{1}{\sqrt{1+\omega^6}},</math>

obtained from
:<math>G^2(\omega)=|H(j\omega)|^2=H(j\omega)\cdot H^*(j\omega)=\frac{1}{1+\omega^6},</math>

and the [[Phase (waves)|phase]] is given by

:<math>\Phi(\omega)=\arg(H(j\omega)).\!</math>

[[File:Butterworth3 GainDelay-en.svg|256px|left|thumb|Gain and [[group delay]] of the third-order Butterworth filter with <math>\omega_c=1</math>]]

The [[group delay]] is defined as the negative derivative of the phase shift 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. There are no ripples in the gain curve in either the passband or the stopband.

The log of the absolute value of the transfer function <math>H(s)</math> 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 <math>H(s)</math> in [[complex frequency space]] for the third-order Butterworth filter with <math>\omega_c</math>=1. The three [[Pole (complex analysis)|poles]] lie on a circle of unit radius in the left half-plane.]]
These are arranged on a [[Unit circle|circle of radius unity]], symmetrical about the real <math>s</math> 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==

[[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_c = 1</math>. Note that the slope is 20<math>n</math>&nbsp;dB/decade where <math>n</math> is the filter order.]]

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.

The gain <math>G(\omega)</math> of an <math>n</math>th-order Butterworth low-pass filter is given in terms of the transfer function <math>H(s)</math> as
:<math>G^2(\omega)=\left |H(j\omega)\right|^2 = \frac {{G_0}^2}{1+\left(\frac{\omega}{\omega_c}\right)^{2n}}</math>

where <math>n</math> is the order of filter, <math>\omega_c</math> is the [[cutoff frequency]] (approximately the −3&nbsp;dB frequency), and <math>G_0</math> is the DC gain (gain at zero frequency).

It can be seen that as <math>n</math> approaches infinity, the gain becomes a rectangle function and frequencies below <math>\omega_c</math> will be passed with gain <math>G_0</math>, while frequencies above <math>\omega_c</math> will be suppressed. For smaller values of <math>n</math>, the cutoff will be less sharp.

We wish to determine the transfer function <math>H(s)</math> where <math>s=\sigma+j\omega</math> (from [[Laplace transform]]). Because <math>\left|H(s)\right|^2 = H(s)\overline{H(s)}</math> and, as a general property of Laplace transforms at <math>s=j\omega</math>, <math>H(-j\omega) = \overline{H(j\omega)}</math>, if we select <math>H(s)</math> such that:

:<math>H(s)H(-s) = \frac {{G_0}^2}{1+\left (\frac{-s^2}{\omega_c^2}\right)^n},</math>

then, with <math>s=j\omega</math>, we have the frequency response of the Butterworth filter.

The <math>n</math> poles of this expression occur on a circle of radius <math>\omega_c</math> at equally-spaced points, and symmetric around the negative real axis. For stability, the transfer function, <math>H(s)</math>, is therefore chosen such that it contains only the poles in the negative real half-plane of <math>s</math>. The <math>k</math>-th pole is specified by

:<math>-\frac{s_k^2}{\omega_c^2} = (-1)^{\frac{1}{n}} = e^{\frac{j(2k-1)\pi}{n}}
\qquad k = 1,2,3,\ldots, n</math>

and hence

:<math>s_k = \omega_c e^{\frac{j(2k+n-1)\pi}{2n}}\qquad k = 1,2,3,\ldots, n.</math>

The transfer (or system) function may be written in terms of these poles as

:<math>H(s)=G_0\prod_{k=1}^n \frac{\omega_c}{s-s_k}=G_0\prod_{k=1}^n \frac{\omega_c}{s-\omega_c e^{\frac{j(2k+n-1)\pi}{2n}}}</math>.

where <math>\textstyle{\prod}</math> is the [[Product (mathematics)#Product of sequences|product of a sequence]] operator. The denominator is a Butterworth polynomial in <math>s</math>.

===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 <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 product form

:<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 n = \text{even}</math>
:<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 n = \text{odd}.</math>

Factors of Butterworth polynomials of order 1 through 10 are shown in the following table (to six decimal places).

{| style="margin:1em auto;"
|-----
|
{| class="wikitable" style="text-align: center"
|-
!n!!Factors of Butterworth Polynomials <math>B_n(s)</math>
|-
!1
|<math>(s+1)</math>
|-
!2
|<math>(s^2+1.414214s+1)</math>
|-
!3
|<math>(s+1)(s^2+s+1)</math>
|-
!4
|<math>(s^2+0.765367s+1)(s^2+1.847759s+1)</math>
|-
!5
|<math>(s+1)(s^2+0.618034s+1)(s^2+1.618034s+1)</math>
|-
!6
|<math>(s^2+0.517638s+1)(s^2+1.414214s+1)(s^2+1.931852s+1)</math>
|-
!7
|<math>(s+1)(s^2+0.445042s+1)(s^2+1.246980s+1)(s^2+1.801938s+1)</math>
|-
!8
|<math>(s^2+0.390181s+1)(s^2+1.111140s+1)(s^2+1.662939s+1)(s^2+1.961571s+1)</math>
|-
!9
|<math>(s+1)(s^2+0.347296s+1)(s^2+s+1)(s^2+1.532089s+1)(s^2+1.879385s+1)</math>
|-
!10
|<math>(s^2+0.312869s+1)(s^2+0.907981s+1)(s^2+1.414214s+1)(s^2+1.782013s+1)(s^2+1.975377s+1)</math>
|}
|}

Factors of Butterworth polynomials of order 1 through 6 are shown in the following table (Exact).

{| style="margin:1em auto;"
|-----
|
{| class="wikitable" style="text-align: center"
|-
!n!!Factors of Butterworth Polynomials <math>B_n(s)</math>
|-
!1
|<math>(s+1)</math>
|-
!2
|<math>(s^{2}+\sqrt{2} s+1)</math>
|-
!3
|<math>(s+1)(s^2+s+1)</math>
|-
!4
|<math>(s^{2}+\sqrt{2-\sqrt{2} } s+1)(s^{2}+\sqrt{2+\sqrt{2} } s+1)</math>
|-
!5
|<math>(s+1)(s^2+\varphi^{-1} s+1)(s^2+\varphi s+1)</math>
|-
!6
|<math>(s^2+\sqrt{2-\sqrt{3} } s+1)(s^2+\sqrt{2} s+1)(s^2+\sqrt{2+\sqrt{3} } s+1)</math>
|}
|}

where the Greek letter [[Phi (letter)|phi]] ({{nowrap|<math>\varphi</math>}} or <math>\phi</math>) represents the [[golden ratio]]. It is an [[irrational number]] that is a solution to the [[quadratic equation]] <math>x^2 - x - 1 = 0,</math> with a value of<ref name="Weisstein" /><ref name="a001622">{{OEIS2C|id=A001622}}</ref>

{{bi|left=1.6|1=<math>\varphi = \frac{1+\sqrt5}{2} = 1.618033988749...</math>({{OEIS2C|id=A001622}})}}<!-- PLEASE DO NOT add additional digits to the value of φ in this equation; there is long-standing consensus that additional digits do not add to understanding. Thank you.-->

The <math>n</math>th Butterworth polynomial can also be written as a sum

:<math>B_n(s)=\sum_{k=0}^n a_k s^k\,,</math>

with its coefficients <math>a_k</math> given by the recursion formula<ref name=bosse1951/><ref name=weinberg1962/>

:<math>\frac{a_{k+1}}{a_k}=\frac{\cos(k\gamma)}{\sin((k+1)\gamma)}</math>

and by the product formula

:<math>a_k=\prod_{\mu=1}^k\frac{\cos((\mu-1)\gamma)}{\sin(\mu\gamma)}\,,</math>

where

:<math>a_0=1\qquad \text{and}\qquad\gamma=\frac{\pi}{2n}\,.</math>

Further, <math>a_k=a_{n-k}</math>. The rounded coefficients <math>a_k</math> for the first 10 Butterworth polynomials <math>B_n(s)</math> are:

{| class="wikitable" style="margin:1em auto; text-align: center"
|+ Butterworth Coefficients <math>a_k</math> to Four Decimal Places
|-
| <math>n</math>
|| <math>a_0</math>
|| <math>a_1</math>
|| <math>a_2</math>
|| <math>a_3</math>
|| <math>a_4</math>
|| <math>a_5</math>
|| <math>a_6</math>
|| <math>a_7</math>
|| <math>a_8</math>
|| <math>a_9</math>
|| <math>a_{10}</math>
|-
|<math>1</math>||<math>1</math>||<math>1</math>
|-
|<math>2</math>||<math>1</math>||<math>1.4142</math>||<math>1</math>
|-
|<math>3</math>||<math>1</math>||<math>2</math>||<math>2</math>||<math>1</math>
|-
|<math>4</math>||<math>1</math>||<math>2.6131</math>||<math>3.4142</math>||<math>2.6131</math>||<math>1</math>
|-
|<math>5</math>||<math>1</math>||<math>3.2361</math>||<math>5.2361</math>||<math>5.2361</math>||<math>3.2361</math>||<math>1</math>
|-
|<math>6</math>||<math>1</math>||<math>3.8637</math>||<math>7.4641</math>||<math>9.1416</math>||<math>7.4641</math>||<math>3.8637</math>||<math>1</math>
|-
|<math>7</math>||<math>1</math>||<math>4.4940</math>||<math>10.0978</math>||<math>14.5918</math>||<math>14.5918</math>||<math>10.0978</math>||<math>4.4940</math>||<math>1</math>
|-
|<math>8</math>||<math>1</math>||<math>5.1258</math>||<math>13.1371</math>||<math>21.8462</math>||<math>25.6884</math>||<math>21.8462</math>||<math>13.1371</math>||<math>5.1258</math>||<math>1</math>
|-
|<math>9</math>||<math>1</math>||<math>5.7588</math>||<math>16.5817</math>||<math>31.1634</math>||<math>41.9864</math>||<math>41.9864</math>||<math>31.1634</math>||<math>16.5817</math>||<math>5.7588</math>||<math>1</math>
|-
|<math>10</math>||<math>1</math>||<math>6.3925</math>||<math>20.4317</math>||<math>42.8021</math>||<math>64.8824</math>||<math>74.2334</math>||<math>64.8824</math>||<math>42.8021</math>||<math>20.4317</math>||<math>6.3925</math>||<math>1</math>
|}

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

:<math>H(s) = \frac{G_0}{B_n(a)}</math> , where <math>a = \frac{s}{\omega_c}.</math>

Transformation to other bandforms are also possible, see [[prototype filter]].

===Maximal flatness===

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

:<math>\frac{dG}{d\omega}=-nG^3\omega^{2n-1}</math>

which is [[monotonic]]ally decreasing for all <math>\omega</math> since the gain <math>G</math> is always positive. The gain function of the Butterworth filter therefore has no ripple. The series expansion of the gain is given by

:<math>G(\omega)=1 - \frac{1}{2}\omega^{2n}+\frac{3}{8}\omega^{4n}+\ldots</math>

In other words, all derivatives of the gain up to but not including the 2<math>n</math>-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.

===High-frequency roll-off===

Again assuming <math>\omega_c=1</math>, the slope of the log of the gain for large <math>\omega</math> is

:<math>\lim_{\omega\rightarrow\infty}\frac{d\log(G)}{d\log(\omega)}=-n.</math>

In [[decibel]]s, the high-frequency roll-off is therefore 20<math>n</math>&nbsp;dB/decade, or 6<math>n</math>&nbsp;dB/octave (the factor of 20 is used because the power is proportional to the square of the voltage gain; see [[20 log rule]].)

=== Minimum order ===
To design a Butterworth filter using the minimum required number of elements, the minimum order of the Butterworth filter may be calculated as follows.<ref name="Paarmann2001" />

:<math>n = \left\lceil\frac{\log{\bigr(\frac{10^{\alpha_s/10}-1}{10^{\alpha_p/10}-1}}\bigr)}{2\log{(\omega_s /\omega_p)}}\right\rceil</math>

where:

:<math>\omega_p</math> and <math>\alpha_p</math> are the pass band frequency and attenuation at that frequency in dB.

:<math>\omega_s</math> and <math>\alpha_s</math> are the stop band frequency and attenuation at that frequency in dB.

:<math>n</math> is the minimum number of poles, the order of the filter.

:<math>\lceil \cdot \rceil</math> denotes the [[Floor and ceiling functions|ceiling function]].

=== Nonstandard cutoff attenuation ===
The cutoff attenuation for Butterworth filters is usually defined to be &minus;3.01 dB. If it is desired to use a different attenuation at the cutoff frequency, then the following factor may be applied to each pole, whereupon the poles will continue to lie on a circle, but the radius will no longer be unity.<ref name="Paarmann2001" /> The cutoff attenuation equation may be derived through algebraic manipulation of the Butterworth defining equation stated at the top of the page.<ref>[[Butterworth filter#Original paper]]</ref>

:<math display="block">\begin{align}
p_A = p_1 \times (10^{\alpha/10}-1)^{{-1}/{2n}}  
&\qquad \text{For 0}    \leq  \alpha  <  \infty  
\end{align}
</math>

where:

:<math>p_A</math> is the relocated pole positioned to set the desired cutoff attenuation.

:<math>p_1</math> is a &minus;3.01 dB cutoff pole that lies on the unit circle.

:<math>\alpha</math> is the desired attenuation at the cutoff frequency in dB (1 dB, 10 dB, etc.).

:<math>n</math> is the number of poles, the order of the filter.

==Filter implementation and design==

There are several different [[electronic filter topology|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===

[[File:Cauer lowpass.svg|right|450px|thumb|Butterworth filter using [[Cauer topology (electronics)|Cauer topology]] ]]

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&nbsp;1-form. The ''k''-th element is given by<ref name="Bennett1929" />

:<math>C_k = 2 \sin \left [\frac {(2k-1)}{2n} \pi \right ]\qquad k = \text{odd}</math>

:<math>L_k = 2 \sin \left [\frac {(2k-1)}{2n} \pi \right ]\qquad k = \text{even}.</math>

The filter may start with a series inductor if desired, in which case the ''L<sub>k</sub>'' are ''k'' odd and the ''C<sub>k</sub>'' are ''k'' even. These formulae may usefully be combined by making both ''L<sub>k</sub>'' and ''C<sub>k</sub>'' equal to ''g<sub>k</sub>''. That is, ''g<sub>k</sub>'' is the [[immittance]] divided by ''s''.

:<math>g_k = 2 \sin \left [\frac {(2k-1)}{2n} \pi \right ]\qquad k = 1,2,3, \ldots, n.</math>

These formulae apply to a doubly terminated filter (that is, the source and load impedance are both equal to unity) with &omega;<sub>c</sub> = 1. 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 name="MatthaeiYoungJones" />

:<math>g_j = \frac{a_j a_{j-1}}{c_{j-1} g_{j-1}}\qquad j = 2,3, \ldots, n</math>

where

:<math>g_1 = a_1</math>

and

:<math>a_j = \sin \left [\frac {(2j-1)}{2n} \pi \right ]\qquad j = 1,2,3, \ldots, n</math>
:<math>c_j = \cos^2 \left [\frac{j}{2n} \pi \right ]\qquad j = 1,2,3, \ldots, n.</math>

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 name="MatthaeiYoungJones" />

===Sallen–Key topology===

[[File:Sallen-Key.svg|right|300px|thumb|[[Sallen–Key topology]] ]]

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.

For the second-order Sallen–Key circuit shown to the right the transfer function is given by

:<math>H(s) = \frac{V_\text{out}(s)}{V_\text{in}(s)} = \frac{1}{1 + C_2(R_1+R_2)s + C_1 C_2 R_1 R_2 s^2}.</math>

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

:<math>C_1C_2R_1R_2=1\,</math>

and

:<math>C_2(R_1+R_2)=-2\cos\left(\frac{2k+n-1}{2n} \pi\right).</math>

This leaves two undefined component values that may be chosen at will.

Butterworth lowpass filters with Sallen–Key topology of third and fourth order, using only one [[op amp]], are described by Huelsman,<ref name="huelsman1971"/><ref name="huelsman1974"/> and further single-amplifier Butterworth filters also of higher order are given by Jurišić et al.<ref name="jurisic2008"/>

===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 [[Digital biquad filter|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 function|Monotonic]] [[frequency response|amplitude response]] in both passband and stopband
* Quick [[roll-off]] around the cutoff frequency, which improves with increasing order
* Considerable [[overshoot (signal)|overshoot]] and [[ringing (signal)|ringing]] in [[step response]], which worsens with increasing order
* Slightly non-linear [[phase response]]
* [[Group delay]] 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.

[[File:Filters order5.svg|500px|center]]

The Butterworth filter rolls off more slowly around the cutoff frequency than the [[Chebyshev filter]] or the [[Elliptic filter]], but without ripple.

==See also==

* [[Bessel filter]]
* [[Chebyshev filter]]
* [[Comb filter]]
* [[Elliptic filter]]
* [[Filter design]]

==References==

{{Reflist|refs=

<ref name="Butterworth1930">{{cite journal |first=S. |last=Butterworth |journal=Experimental Wireless and the Wireless Engineer |volume=7 |year=1930 |pages=536–541 |url=https://www.changpuak.ch/electronics/downloads/On_the_Theory_of_Filter_Amplifiers.pdf |title=On the Theory of Filter Amplifiers }}</ref>

<ref name="MatthaeiYoungJones">{{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 }}</ref>

<ref name=bosse1951>
{{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
| bibcode = 1951Freq....5..279B
 | s2cid = 124123311
 }}</ref>

<ref name=weinberg1962>
{{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
 | hdl = 2027/mdp.39015000986086
 | url = https://hdl.handle.net/2027/mdp.39015000986086
 | access-date = 2022-06-18
 }}</ref>
 
 <ref name="huelsman1971">
 {{cite journal
 | last1      = Huelsman
 | first1     = L. P.
 | date       = May 1971
 | title      = Equal-valued-capacitor active-''RC''-network realisation of a 3rd-order lowpass Butterworth characteristic
 | journal    = Electronics Letters
 | volume     = 7
 | issue      = 10
 | pages      = 271–272
 | doi = 10.1049/el:19710185
 | bibcode = 1971ElL.....7..271H
 }}</ref>
 
 <ref name="huelsman1974">
 {{cite journal
 | last1      = Huelsman
 | first1     = L. P.
 | date       = December 1974
 | title      = An equal-valued capacitor active ''RC'' network realization of a fourth-order low-pass Butterworth characteristic
 | journal    = Proceedings of the IEEE
 | volume     = 62
 | issue      = 12
 | pages      = 1709
 | doi = 10.1109/PROC.1974.9689
 }}</ref>

<ref name="jurisic2008">
 {{cite journal
 | last1      = Jurišić
 | first1     = Dražen
 | last2      = Moschytz
 | first2     = George S.
 | last3      = Mijat
 | first3     = Neven
 | date       = 2008
 | title      = Low-sensitivity, single-amplifier, active-''RC'' allpole filters using tables
 | journal    = Automatika
 | volume     = 49
 | issue      = 3–4
 | pages      = 159–173
 }}</ref>
 
<ref name=Bianchi2007>
{{cite book
 | title = Electronic filter simulation & design
 | first1 = Giovanni
 | last1  = Bianchi
 | first2 = Roberto
 | last2  = Sorrentino
 | publisher = McGraw-Hill Professional
 | year = 2007
 | isbn = 978-0-07-149467-0
 | pages = 17–20
 | url = https://books.google.com/books?id=5S3LCIxnYCcC&q=Butterworth-approximation+maximally-flat&pg=PT32
 }}</ref>
 
<ref name="Bennett1929">{{cite patent |country=US |number=1849656 |inventor=William R. Bennett |title=Transmission Network |fdate=June 29, 1929 |pubdate=March 15, 1932}}</ref>

<ref name="Paarmann2001">{{cite book |last=Paarmann |first=Larry D. |title=Design and Analysis of Analog Filters: A Signal Processing Perspective |date=2001 |publisher=Kluwer Academic Publishers |isbn=0-7923-7373-1 |location=Norwell, Massachusetts, US |publication-date=2001 |pages=117, 118 |language=EN |url=https://archive.org/details/designanalysisof0000paar }}</ref>

<ref name="Weisstein">{{cite web |last=Weisstein |first=Eric W. |title=Golden Ratio |url=https://mathworld.wolfram.com/GoldenRatio.html |access-date=2020-08-10 |website=mathworld.wolfram.com |language=en}}</ref>

}}

[[Category:Linear filters]]
[[Category:Network synthesis filters]]
[[Category:Electronic design]]