The Butterworth filter is one type of electronic filter design. It is designed to have a frequency response which is as flat as mathematically possible in the passband. Another name for them is 'maximally flat magnitude' filters.

Origin of name

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The Butterworth type filter was first described by the British engineer Stephen Butterworth in his paper "On the Theory of Filter Amplifiers", Wireless Engineer (also called Experimental Wireless and the Radio Engineer), vol. 7, 1930, pp. 536-541.

Overview

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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 plot, the response slopes off linearly towards negative infinity. For a first-order filter, the response rolls off at −6 dB per octave (−20 dB per decade). (All first-order filters, regardless of name, are actually identical and so have the same frequency response.) For a second-order Butterworth filter, the response decreases at −12 dB per octave, a third-order at −18 dB, and so on. Butterworth filters have a monotonically decreasing magnitude function with ω. The Butterworth is the only filter that maintains this same shape for higher orders (but with a steeper decline in the stopband) whereas other varieties of filters (Bessel, Chebyshev, elliptic) have different shapes at higher orders.

Compared with a Chebyshev Type I/Type II filter or an elliptic filter, the Butterworth filter has a slower roll-off, and thus will require a higher order to implement a particular stopband specification. However, Butterworth filter will have a more linear phase response in the passband than the Chebyshev Type I/Type II and elliptic filters.

Transfer function

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Like all filters, the typical prototype is the low-pass filter, which can be modified into a high-pass filter, or placed in series with others to form band-pass and band-stop filters, and higher order versions of these.

The magnitude squared frequency response:

$\backslash left\; |H(\backslash omega)\backslash right|^2\; =\; \backslash frac\; \{1\}\{1+\backslash left(\backslash frac\{\backslash omega\}\{\backslash omega\_c\}\backslash right)^\{2n\}\}\; =\; \backslash frac\{1\}\{1+\backslash epsilon^2\backslash left(\backslash frac\{\backslash omega\}\{\backslash omega\_p\}\backslash right)^\{2n\}\}$

here
n = order of filter
ω_c = cutoff frequency = -3dB frequency
ω_p = passband edge frequency
1/(1 + ε²) = band edge value of |H(ω)|².
Since H(s)H(-s) evaluated at s = jω is simply equal to |H(ω)|², it follows that

$H(s)H(-s)\; =\; \backslash frac\; \{1\}\{1+\backslash left\; (\backslash frac\{-s^2\}\{\backslash omega\_c^2\}\backslash right)^n\}$

The poles occur on a circle of radius ω_c at equally spaced points

$\backslash frac\{-s^2\}\{\backslash omega\_c^2\}\; =\; (-1)^\{\backslash frac\{1\}\{n\}\}\; =\; e^\{\backslash frac\{j(2k+1)\backslash pi\}\{n\}\}$

\qquad\mathrm{k = 0,1,2, \ldots, n-1}

and hence,

$s\_k\; =\; \backslash omega\_ce^\{\backslash frac\{j\backslash pi\}\{2\}\}e^\{\backslash frac\{j(2k+1)\backslash pi\}\{2n\}\}\backslash qquad\backslash mathrm\{k\; =\; 0,1,2,\; \backslash ldots,\; n-1\}$

The magnitude of the frequency response of an nth order lowpass filter can be defined mathematically as:

$G\_n(\backslash omega)\; =\; \backslash left\; |\; H\_n(j\; \backslash omega)\; \backslash right\; |\; =\; \{1\; \backslash over\; \backslash sqrt\{\; 1\; +\; (\backslash omega\; /\; \backslash omega\_\backslash mathrm\{c\})\; ^\; \{2\; n\}\}\; \}$

where G is the gain of the filter, H is the transfer function, j is the imaginary number, n is the order of the filter, ω is the angular frequency of the signal in radians per second, and $\backslash omega\_\backslash mathrm\{c\}$ is the cutoff frequency (−3 dB frequency).

Normalizing the expression (setting the cutoff frequency $\backslash omega\_\backslash mathrm\{c\}\; =\; 1$), it becomes:

$G\_n(\backslash omega)\; =\; \backslash left\; |\; H\_n(j\; \backslash omega)\; \backslash right\; |\; =\; \{1\; \backslash over\; \backslash sqrt\{\; 1\; +\; \backslash omega\; ^\; \{2\; n\}\}\; \}$

Maximally Flat Magnitude Filter

the first (2n-1)^th derivatives of g are zero for ω = 0. This means that

the rate of change of gain with respect to ω is zero i.e., the gain is constant with ω.

Hence Butterworth filters are also called maximally flat filters.
Butterworth filter

High Frequency Roll Off

$=\; \{20n\}\{log\_\{10\}\{\backslash omega\}\}$ Hence, the high frequency roll off = 20n dB/decade

Filter Realization

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The Butterworth filter having a given transfer function can be realised using Cauer - 1 form:
k^th element is given by: $C\_k\; =\; 2\; \backslash sin\; \backslash left\{(2k-1)\}\{2n\}\; \backslash pi\; \backslash right$; k = odd
$L\_k\; =\; 2\; \backslash sin\; \backslash left\{(2k-1)\}\{2n\}\; \backslash pi\; \backslash right$; k = even

Normalized Butterworth polynomials

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Normalized Butterworth polynomials have the general form:

$P(s)=\backslash prod\_\{k=0\}^\{\backslash frac\{n\}\{2\}-1\}\; \backslash left*$ for n even

$P(s)=(s+1)\backslash prod\_\{k=0\}^\{\backslash frac\{n-1\}\{2\}\}\; \backslash left*$ for n odd

To four decimal places, they are:

n Factors of Polynomial $B\_n(s)$ 1 $(s+1)$ 2 $s^2+1.4142s+1$ 3 $(s+1)(s^2+s+1)$ 4 $(s^2+0.7654s+1)(s^2+1.8478s+1)$ 5 $(s+1)(s^2+0.6180s+1)(s^2+1.6180s+1)$ 6 $(s^2+0.5176s+1)(s^2+1.4142s+1)(s^2+1.9319s+1)$ 7 $(s+1)(s^2+0.4450s+1)(s^2+1.2470s+1)(s^2+1.8019s+1)$ 8 $(s^2+0.3902s+1)(s^2+1.1111s+1)(s^2+1.6629s+1)(s^2+1.9616s+1)$

Comparison with other linear filters

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Here is an image showing the Butterworth filter next to other common kind of filters obtained with the same number of coefficients:

As is clear from the image, the Butterworth filter rolls off more slowly than all the others but it shows no ripples.

See also

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• Bessel filters
• Chebyshev filters
• Comb filters
• Elliptic filters

Linear filters

Butterworthfilter | Filtro de Butterworth | Filtre de Butterworth | Filtro Butterworth | Filtr Butterwortha | Filtro Butterworth | 巴特沃斯滤波器