Butterworth filter
The Butterworth filter is a type of 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".
Original paper
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 theory then in use. The filter was not in common use for over 30 years after its publication. Butterworth stated that: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 as a function of frequency is:
where is the angular frequency in radians per second and is the number of poles in the filter—equal to the number of reactive elements in a passive filter. Its cutoff frequency is normalized to ? = 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 amplifiers and so enabling the construction of higher-order filters despite inductor losses. In 1930, low-loss core materials such as 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, high-pass, band-pass and band-stop functionality.
Overview
The frequency response of the Butterworth filter is maximally flat 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. A first-order filter's response rolls off at −6 dB per octave . 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, unlike other filter types that have non-monotonic ripple in the passband and/or the stopband.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, 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:File:LowPass3poleICauer.svg|300px|right|thumb|A third-order low-pass filter. The filter becomes a Butterworth filter with cutoff frequency =1 when =4/3 F, =1 Ω, =3/2 H and =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 = 4/3 F, = 1 Ω, = 3/2 H, and = 1/2 H. Taking the impedance of the capacitors to be and the impedance of the inductors to be, where is the complex frequency, the circuit equations yield the transfer function for this device:
The magnitude of the frequency response is given by
obtained from
and the phase is given by
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 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.svg|250px|right|thumb|Log density plot of the transfer function in complex frequency space for the third-order Butterworth filter with =1. The three poles lie on a circle of unit radius in the left half-plane.
These are arranged on a circle of radius unity, symmetrical about the real 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
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 gain of an th-order Butterworth low-pass filter is given in terms of the transfer function as
where is the order of filter, is the cutoff frequency, and is the DC gain.
It can be seen that as approaches infinity, the gain becomes a rectangle function and frequencies below will be passed with gain, while frequencies above will be suppressed. For smaller values of, the cutoff will be less sharp.
We wish to determine the transfer function where . Because and, as a general property of Laplace transforms at,, if we select such that:
then, with, we have the frequency response of the Butterworth filter.
The poles of this expression occur on a circle of radius at equally-spaced points, and symmetric around the negative real axis. For stability, the transfer function,, is therefore chosen such that it contains only the poles in the negative real half-plane of. The -th pole is specified by
and hence
The transfer function may be written in terms of these poles as
where is the product of a sequence operator. The denominator is a Butterworth polynomial in.
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 and. The polynomials are normalized by setting . The normalized Butterworth polynomials then have the general product formFactors of Butterworth polynomials of order 1 through 10 are shown in the following table.
Factors of Butterworth polynomials of order 1 through 6 are shown in the following table.
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