Butterworth filter

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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.

Contents

1 Origin of name
2 Overview
3 Transfer function

3.1 Maximally Flat Magnitude Filter
3.2 High Frequency Roll Off

4 Filter Realization
5 Normalized Butterworth polynomials
6 Comparison with other linear filters
7 See also

Origin of name

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.

The Bode plot of a first-order Butterworth low-pass filter

Butterworth low-pass filters of orders 1 through 5

Butterworth second-order low-pass active filter example

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 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

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:

[\left |H(\omega)\right|^2 = \frac \right)^} = \frac\right)^}]

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) = \frac \right)^n}]

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

[\frac = (-1)^} = e^}\qquad\mathrm]

and hence,

[s_k = \omega_ce^}e^}\qquad\mathrm]

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

[ G_n(\omega) = \left | H_n(j \omega) \right | = ) ^ } } ]

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 [\omega_\mathrm] is the cutoff frequency (−3 dB frequency).

Normalizing the expression (setting the cutoff frequency [\omega_\mathrm = 1]), it becomes:

[ G_n(\omega) = \left | H_n(j \omega) \right | = } } ]

Bode magnitude plot for Butterworth second-order low-pass filter example

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.

High Frequency Roll Off

[ = }] Hence, the high frequency roll off = 20n dB/decade

Filter Realization

The Butterworth filter having a given transfer function can be realised using Cauer - 1 form:
k^th element is given by: [C_k = 2 \sin \left [frac pi right ]]; k = odd
[L_k = 2 \sin \left [frac pi right ]]; k = even

Normalized Butterworth polynomials

Normalized Butterworth polynomials have the general form:

[P(s)=\prod_^-1} \left[s^2+2scosleft(frac,piright)+1right]] for n even

[P(s)=(s+1)\prod_^} \left[s^2+2scosleft(frac,piright)+1right]] 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

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.