Butterworth filter

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The frequency response plot from Butterworth's 1930 paper. Buttergr.jpg
The frequency response plot from Butterworth's 1930 paper.

The Butterworth filter is a type of signal processing filter designed to have a frequency response 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". [1]

Contents

Original paper

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 theory then in use. The filter was not in common use for over 30 years after its publication. Butterworth stated that:

"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 cutoff frequency was normalized to 1 radian per second and whose frequency response (gain) was

where ω is the angular frequency in radians per second and n is the number of poles 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 1/2 ≈ 0.707, which is half power or 3 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.

Butterworth solved the equations for two- 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 Bode plot of a first-order Butterworth low-pass filter Butterworth filter bode plot.svg
The Bode plot of a first-order Butterworth low-pass filter

The frequency response of the Butterworth filter is maximally flat (i.e. has no ripples) in the passband and rolls off towards zero in the stopband. [2] 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 (−20 dB per 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.

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 pass-band 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:

A third-order low-pass filter (Cauer topology). The filter becomes a Butterworth filter with cutoff frequency oc=1 when (for example) C2=4/3 F, R4=1 O, L1=3/2 H and L3=1/2 H. LowPass3poleICauer.svg
A third-order low-pass filter (Cauer topology). The filter becomes a Butterworth filter with cutoff frequency ωc=1 when (for example) C2=4/3 F, R4=1 Ω, L1=3/2 H and L3=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 C2 = 4/3 F, R4 = 1 Ω, L1 = 3/2 H, and L3 = 1/2 H. [3] Taking the impedance of the capacitors C to be 1/(Cs) and the impedance of the inductors L to be Ls, where s = σ + jω is the complex frequency, the circuit equations yield the transfer function for this device:

The magnitude of the frequency response (gain) G(ω) is given by

obtained from

and the phase is given by

Gain and group delay of the third-order Butterworth filter with oc=1 Butterworth3 GainDelay.png
Gain and group delay of the third-order Butterworth filter with ωc=1

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.

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.

Log density plot of the transfer function H(s) in complex frequency space for the third-order Butterworth filter with oc=1. The three poles lie on a circle of unit radius in the left half-plane. Butterworth Filter s-Plane Response (3rd Order).svg
Log density plot of the transfer function H(s) in complex frequency space for the third-order Butterworth filter with ωc=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 s 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 to be rejected.

Transfer function

Plot of the gain of Butterworth low-pass filters of orders 1 through 5, with cutoff frequency
o
0
=
1
{\displaystyle \omega _{0}=1}
. Note that the slope is 20n dB/decade where n is the filter order. Butterworth Filter Orders.svg
Plot of the gain of Butterworth low-pass filters of orders 1 through 5, with cutoff frequency . Note that the slope is 20n dB/decade where n is the filter order.

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 n-order Butterworth low-pass filter is given in terms of the transfer function H(s) as

where

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

We wish to determine the transfer function H(s) where (from Laplace transform). Because and, as a general property of Laplace transforms at , , if we select H(s) such that:

then, with , we have the frequency response of the Butterworth filter.

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

and hence;

The transfer( or system) 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 s.

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 form

To four decimal places, they are

nFactors of Polynomial
1
2
3
4
5
6
7
8
9
10

The th Butterworth polynomial can also be written as a sum

with its coefficients given by the recursion formula [4] [5]

and by the product formula

where

Further, . The rounded coefficients for the first 10 Butterworth polynomials are:

Butterworth coefficients to four decimals
111
211.41421
31221
412.61313.41422.61311
513.23615.23615.23613.23611
613.86377.46419.14167.46413.86371
714.494010.097814.591814.591810.09784.49401
815.125813.137121.846225.688421.846213.13715.12581
915.758816.581731.163441.986441.986431.163416.58175.75881
1016.392520.431742.802164.882474.233464.882442.802120.43176.39251

The normalized Butterworth polynomials can be used to determine the transfer function for any low-pass filter cut-off frequency , as follows

, where

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

Maximal flatness

Assuming and , the derivative of the gain with respect to frequency can be shown to be

which is monotonically decreasing for all since the gain G is always positive. The gain function of the Butterworth filter therefore has no ripple. The series expansion of the gain is given by

In other words, all derivatives of the gain up to but not including the 2n-th derivative are zero at , 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 , the slope of the log of the gain for large ω is

In decibels, the high-frequency roll-off is therefore 20n dB/decade, or 6n dB/octave (the factor of 20 is used because the power is proportional to the square of the voltage gain; see 20 log rule.)

Filter implementation and design

There are several different 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.

Cauer topology

Butterworth filter using Cauer topology Cauer lowpass.svg
Butterworth filter using Cauer topology

The 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-th element is given by [6]

The filter may start with a series inductor if desired, in which case the Lk are k odd and the Ck are k even. These formulae may usefully be combined by making both Lk and Ck equal to gk. That is, gk is the immittance divided by s.

These formulae apply to a doubly terminated filter (that is, the source and load impedance are both equal to unity) with ωc = 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 [7]

where

and

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 diplexers and multiplexers. [8]

Sallen–Key topology

Sallen-Key topology Sallen-Key.svg
Sallen–Key topology

The Sallen–Key topology uses active and passive components (noninverting buffers, usually op amps, 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 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

We wish the denominator to be one of the quadratic terms in a Butterworth polynomial. Assuming that , this will mean that

and

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

Butterworth lowpass filters with Sallen-Key topology of 3rd and 4th order, using only one op amp, are described by Huelsman , [9] [10] and further single-amplifier Butterworth filters also of higher order are given by Jurišić et al. . [11]

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 biquad sections, plus one first-order or third-order section for odd orders.

Comparison with other linear filters

Properties of the Butterworth filter are:

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.

Filters order5.svg

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

Related Research Articles

In engineering, a transfer function of an electronic or control system component is a mathematical function which theoretically models the device's output for each possible input. In its simplest form, this function is a two-dimensional graph of an independent scalar input versus the dependent scalar output, called a transfer curve or characteristic curve. Transfer functions for components are used to design and analyze systems assembled from components, particularly using the block diagram technique, in electronics and control theory.

Cutoff frequency frequency response boundary

In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced rather than passing through.

A low-pass filter (LPF) is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter design. The filter is sometimes called a high-cut filter, or treble-cut filter in audio applications. A low-pass filter is the complement of a high-pass filter.

<i>Q</i> factor Parameter describing the longevity of energy in a resonator relative to its resonant frequency

In physics and engineering, the quality factor or Q factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is approximately defined as the ratio of the initial energy stored in the resonator to the energy lost in one radian of the cycle of oscillation. Q factor is alternatively defined as the ratio of a resonator's centre frequency to its bandwidth when subject to an oscillating driving force. These two definitions give numerically similar, but not identical, results. Higher Q indicates a lower rate of energy loss and the oscillations die out more slowly. A pendulum suspended from a high-quality bearing, oscillating in air, has a high Q, while a pendulum immersed in oil has a low one. Resonators with high quality factors have low damping, so that they ring or vibrate longer.

A resistor–capacitor circuit, or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC circuit is composed of one resistor and one capacitor and is the simplest type of RC circuit.

Chebyshev filters are analog or digital filters having a steeper roll-off than Butterworth filters, and have passband ripple or stopband ripple. Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter, but with ripples in the passband. This type of filter is named after Pafnuty Chebyshev because its mathematical characteristics are derived from Chebyshev polynomials. Type I Chebyshev filters are usually referred to as "Chebyshev filters", while type II filters are usually called "inverse Chebyshev filters".

In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response is of finite duration, because it settles to zero in finite time. This is in contrast to infinite impulse response (IIR) filters, which may have internal feedback and may continue to respond indefinitely.

The Sallen–Key topology is an electronic filter topology used to implement second-order active filters that is particularly valued for its simplicity. It is a degenerate form of a voltage-controlled voltage-source (VCVS) filter topology.

A resistor–inductor circuit, or RL filter or RL network, is an electric circuit composed of resistors and inductors driven by a voltage or current source. A first-order RL circuit is composed of one resistor and one inductor and is the simplest type of RL circuit.

An elliptic filter is a signal processing filter with equalized ripple (equiripple) behavior in both the passband and the stopband. The amount of ripple in each band is independently adjustable, and no other filter of equal order can have a faster transition in gain between the passband and the stopband, for the given values of ripple. Alternatively, one may give up the ability to adjust independently the passband and stopband ripple, and instead design a filter which is maximally insensitive to component variations.

In electronics and signal processing, a Bessel filter is a type of analog linear filter with a maximally flat group/phase delay, which preserves the wave shape of filtered signals in the passband. Bessel filters are often used in audio crossover systems.

Stub (electronics) Short electrical transmission line

In microwave and radio-frequency engineering, a stub or resonant stub is a length of transmission line or waveguide that is connected at one end only. The free end of the stub is either left open-circuit, or short-circuited. Neglecting transmission line losses, the input impedance of the stub is purely reactive; either capacitive or inductive, depending on the electrical length of the stub, and on whether it is open or short circuit. Stubs may thus function as capacitors, inductors and resonant circuits at radio frequencies.

Ripple in electronics is the residual periodic variation of the DC voltage within a power supply which has been derived from an alternating current (AC) source. This ripple is due to incomplete suppression of the alternating waveform after rectification. Ripple voltage originates as the output of a rectifier or from generation and commutation of DC power.

Electronic filter topology Electronic filter circuits without taking note of the values of the components used but only the manner in which those components are connected

Electronic filter topology defines electronic filter circuits without taking note of the values of the components used but only the manner in which those components are connected.

m-derived filters or m-type filters are a type of electronic filter designed using the image method. They were invented by Otto Zobel in the early 1920s. This filter type was originally intended for use with telephone multiplexing and was an improvement on the existing constant k type filter. The main problem being addressed was the need to achieve a better match of the filter into the terminating impedances. In general, all filters designed by the image method fail to give an exact match, but the m-type filter is a big improvement with suitable choice of the parameter m. The m-type filter section has a further advantage in that there is a rapid transition from the cut-off frequency of the pass band to a pole of attenuation just inside the stop band. Despite these advantages, there is a drawback with m-type filters; at frequencies past the pole of attenuation, the response starts to rise again, and m-types have poor stop band rejection. For this reason, filters designed using m-type sections are often designed as composite filters with a mixture of k-type and m-type sections and different values of m at different points to get the optimum performance from both types.

Prototype filter

Prototype filters are electronic filter designs that are used as a template to produce a modified filter design for a particular application. They are an example of a nondimensionalised design from which the desired filter can be scaled or transformed. They are most often seen in regard to electronic filters and especially linear analogue passive filters. However, in principle, the method can be applied to any kind of linear filter or signal processing, including mechanical, acoustic and optical filters.

mm'-type filters, also called double-m-derived filters, are a type of electronic filter designed using the image method. They were patented by Otto Zobel in 1932. Like the m-type filter from which it is derived, the mm'-type filter type was intended to provide an improved impedance match into the filter termination impedances and originally arose in connection with telephone frequency division multiplexing. The filter has a similar transfer function to the m-type, having the same advantage of rapid cut-off, but the input impedance remains much more nearly constant if suitable parameters are chosen. In fact, the cut-off performance is better for the mm'-type if like-for-like impedance matching are compared rather than like-for-like transfer function. It also has the same drawback of a rising response in the stopband as the m-type. However, its main disadvantage is its much increased complexity which is the chief reason its use never became widespread. It was only ever intended to be used as the end sections of composite filters, the rest of the filter being made up of other sections such as k-type and m-type sections.

The spectrum of a chirp pulse describes its characteristics in terms of its frequency components. This frequency-domain representation is an alternative to the more familiar time-domain waveform, and the two versions are mathematically related by the Fourier transform.
The spectrum is of particular interest when pulses are subject to signal processing. For example, when a chirp pulse is compressed by its matched filter, the resulting waveform contains not only a main narrow pulse but, also, a variety of unwanted artifacts many of which are directly attributable to features in the chirp's spectral characteristics.
The simplest way to derive the spectrum of a chirp, now that computers are widely available, is to sample the time-domain waveform at a frequency well above the Nyquist limit and call up an FFT algorithm to obtain the desired result. As this approach was not an option for the early designers, they resorted to analytic analysis, where possible, or to graphical or approximation methods, otherwise. These early methods still remain helpful, however, as they give additional insight into the behavior and properties of chirps.

Frequency selective surface

A frequency-selective surface (FSS) is any thin, repetitive surface designed to reflect, transmit or absorb electromagnetic fields based on the frequency of the field. In this sense, an FSS is a type of optical filter or metal-mesh optical filters in which the filtering is accomplished by virtue of the regular, periodic pattern on the surface of the FSS. Though not explicitly mentioned in the name, FSS's also have properties which vary with incidence angle and polarization as well - these are unavoidable consequences of the way in which FSS's are constructed. Frequency-selective surfaces have been most commonly used in the radio frequency region of the electromagnetic spectrum and find use in applications as diverse as the aforementioned microwave oven, antenna radomes and modern metamaterials. Sometimes frequency selective surfaces are referred to simply as periodic surfaces and are a 2-dimensional analog of the new periodic volumes known as photonic crystals.

Lattice delay network

Lattice delay networks are an important subgroup of lattice networks. They are all-pass filters, so they have a flat amplitude response, but a phase response which varies linearly with frequency. All lattice circuits, regardless of their complexity, are based on the schematic shown below, which contains two series impedances, Za, and two shunt impedances, Zb. Although there is duplication of impedances in this arrangement, it offers great flexibility to the circuit designer so that, in addition to its use as delay network it can be configured to be a phase corrector, a dispersive network, an amplitude equalizer, or a low pass filter, according to the choice of components for the lattice elements.

References

  1. 1 2 In Wireless Engineer (also called Experimental Wireless and the Wireless Engineer), vol. 7, 1930, pp. 536–541 – "On the Theory of Filter Amplifiers", S. Butterworth (PDF)
  2. Giovanni Bianchi and Roberto Sorrentino (2007). Electronic filter simulation & design. McGraw-Hill Professional. pp. 17–20. ISBN   978-0-07-149467-0.
  3. Matthaei et al., p. 107
  4. Bosse, G. (October 1951). "Siebketten ohne Dämpfungsschwankungen im Durchlaßbereich (Potenzketten)". Frequenz. 5 (10): 279–284.
  5. Louis Weinberg (1962). Network analysis and synthesis. Malabar, Florida: Robert E. Krieger Publishing Company, Inc. (published 1975). pp. 494–496. ISBN   0-88275-321-5.
  6. US 1849656,William R. Bennett,"Transmission Network",published March 15, 1932
  7. Matthaei, pp. 104-107
  8. Matthaei, pp. 105,974
  9. Huelsman, L. P. (May 1971). "Equal-valued-capacitor active--network realisation of a 3rd-order lowpass Butterworth characteristic". Electronics Letters. 7 (10): 271–272.
  10. Huelsman, L. P. (December 1974). "An equal-valued capacitor active network realization of a fourth-order low-pass Butterworth characteristic". Proceedings of the IEEE. 62 (12): 1709–1709.
  11. Jurišić, Dražen; Moschytz, George S.; Mijat, Neven (2008). "Low-sensitivity, single-amplifier, active- allpole filters using tables". Automatika. 49 (3–4): 159–173.