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Encyclopedia > Butterworth filter
Linear analog electronic filters
Butterworth filter
Chebyshev filter
Elliptic (Cauer) filter
Bessel filter
Gaussian filter
Optimum "L" (Legendre) 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. Television signal splitter consisting of a hi-pass and a low-pass filter. ... The frequency response of a fourth-order type I Chebyshev low-pass filter Chebyshev filters, are analog or digital filters having a steeper roll-off and more passband ripple than Butterworth filters. ... An elliptic filter (also known as a Cauer filter) is an electronic filter with equalized ripple (equiripple) behavior in both the passband and the stopband. ... In electronics and signal processing, a Bessel filter is a variety of linear filter with a maximally flat group delay (linear phase response). ... In electronics and signal processing, A Gaussian filter is designed to give no overshoot to a step function input while maximising the rise and fall time. ... The Optimum L filter (also known as a Legendre filter) was proposed by Athanasios Papoulis in 1958. ... Television signal splitter consisting of a hi-pass and a low-pass filter. ... Frequency response is the measure of any systems response to frequency, but is usually used in connection with electronic amplifiers and similar systems, particularly in relation to audio signals. ... In telecommunications, passband is the portion of spectrum, between limiting frequencies (or, in the optical regime, limiting wavelengths), that is transmitted with minimum relative loss or maximum relative gain. ...

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. For the Technical Symposium of NITK Surathkal Engineer , see Engineer (Technical Fest). ...

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. The Bode plot for a first-order Butterworth filter A Bode plot, named after Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and Bode phase plot: A Bode magnitude plot is a graph of log magnitude against log frequency often used in signal processing to show... The decibel (dB) is a dimensionless unit of ratio which is used to express the relationship between a variable quantity and a known reference quantity. ... In music, an octave (sometimes abbreviated 8ve or 8va) is the interval between one musical note and another with half or double the frequency. ... A decade is a set or a group of ten, commonly a period of 10 years in contemporary English, or a period of 10 days in the French revolutionary calendar. ... In electronics and signal processing, a Bessel filter is a variety of linear filter with a maximally flat group delay (linear phase response). ... The frequency response of a fourth-order type I Chebyshev low-pass filter Chebyshev filters, are analog or digital filters having a steeper roll-off and more passband ripple than Butterworth filters. ... An elliptic filter (also known as a Cauer filter) is an electronic filter with equalized ripple (equiripple) behavior in both the passband and 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. However, Butterworth filter will have a more linear phase response in the passband than the Chebyshev Type I/Type II and elliptic filters. The frequency response of a fourth-order type I Chebyshev low-pass filter Chebyshev filters, are analog or digital filters having a steeper roll-off and more passband ripple than Butterworth filters. ... An elliptic filter (also known as a Cauer filter) is an electronic filter with equalized ripple (equiripple) behavior in both the passband and the stopband. ... In telecommunication, a stopband is a band of frequencies, between specified limits, that a circuit, such as a filter or telephone circuit, does not transmit. ...

## A Simple Example

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 farad, R4=1 ohm, L1=1/2 and L3=3/2 henry.
Log density plot of the transfer function H(s) in complex frequency space for the third order Butterworth filter with ωc=1. Note the three poles which lie on a circle of unit radius in the left half plane.

A simple example of a Butterworth filter is the 3rd order low-pass design shown in the figure on the right, with C2 = 4 / 3 farad, R4 = 1 ohm, L1 = 1 / 2 and L3 = 3 / 2 henry. 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 yields the transfer function for this device: Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links Butterworth_3pole. ... Image File history File links Butterworth_3pole. ... A low-pass filter passes low frequencies fairly well, but attenuates high frequencies. ... Electrical impedance, or simply impedance, is a measure of opposition to a sinusoidal alternating electric current. ... A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ...

$H(s)=frac{V_o(s)}{V_i(s)}=frac{1}{1+2s+2s^2+s^3}$

The gain G(ω) as a function of angular frequency ω is given by: In electronics, gain is usually taken as the mean ratio of the signal output of a system to the signal input of the system. ...

$G^2(omega)=|H(jomega)|^2=frac{1}{1+omega^6},$

and the phase is given by: To meet Wikipedias quality standards, this article or section may require cleanup. ...

$Phi(omega)=arg(H(jomega)),$

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 log10(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. In physics, and in particular in optics, the study of waves and digital signal processing, the term group delay has the following meanings: 1. ...

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. 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. If we change each capacitor and inductor into a resonant capacitor and inductor in parallel, with the proper choice of component values, a band-pass Butterworth filter is obtained. A high-pass filter is a filter that passes high frequencies well, but attenuates (or reduces) frequencies lower than the cutoff frequency. ... The frequency axis of this symbolic diagram would be logarithmically scaled. ...

log10(gain) and group delay of the third order Butterworth filter with ωc=1

Image File history File links Download high-resolution version (882x512, 20 KB) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Butterworth filter ... Image File history File links Download high-resolution version (882x512, 20 KB) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Butterworth filter ...

## The transfer function

Plot of the gain of Butterworth low-pass filters of orders 1 through 5. 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. Image File history File links Download high resolution version (1229x872, 92 KB) File links The following pages link to this file: Butterworth filter ... Image File history File links Download high resolution version (1229x872, 92 KB) File links The following pages link to this file: Butterworth filter ... A low-pass filter is a filter that passes low frequencies well, but attenuates (or reduces) frequencies higher than the cutoff frequency. ... A high-pass filter is a filter that passes high frequencies well, but attenuates (or reduces) frequencies lower than the cutoff frequency. ... The frequency axis of this symbolic diagram would be logarithmically scaled. ... In electronics, a band-stop filter is a filter that attenuates, usually to very low levels, all frequencies between two non-zero, finite limits and passes all frequencies not within the limits. ...

The gain G(ω) of an n-order Butterworth low pass filter is given in terms of the transfer function H(s) as: A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ...

$G^2(omega)=left |H(jomega)right|^2 = frac {G_0^2}{1+left(frac{omega}{omega_c}right)^{2n}}$

where

• n = order of filter
• ωc = cutoff frequency (approximately the -3dB frequency)
• G0 is the DC gain (gain at zero frequency)

It can be seen that as n approaches infinity, the gain becomes a rectangle function and frequencies below ωc will be passed with gain G0, 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 s = σ + jω. Since H(s)H(-s) evaluated at s = jω is simply equal to |H(ω)|2, it follows that:

$H(s)H(-s) = frac {G_0^2}{1+left (frac{-s^2}{omega_c^2}right)^n}$

The poles of this expression occur on a circle of radius ωc at equally spaced points. The transfer function itself will be specified by just the poles in the negative real half-plane of s. The k-th pole is specified by:

$-frac{s_k^2}{omega_c^2} = (-1)^{frac{1}{n}} = e^{frac{j(2k-1)pi}{n}} qquadmathrm{k = 1,2,3, ldots, n}$

and hence,

$s_k = omega_c e^{frac{j(2k+n-1)pi}{2n}}qquadmathrm{k = 1,2,3, ldots, n}$

The transfer function may be written in terms of these poles as:

$H(s)=frac{G_0}{prod_{k=1}^n (s-s_k)/omega_c}$

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 which are complex conjugates, such as s1 and sn. The polynomials are normalized by setting ωc = 1. The normalized Butterworth polynomials then have the general form:

$B_n(s)=prod_{k=1}^{frac{n}{2}} left[s^2-2scosleft(frac{2k+n-1}{2n},piright)+1right]$ for n even
$B_n(s)=(s+1)prod_{k=1}^{frac{n-1}{2}} left[s^2-2scosleft(frac{2k+n-1}{2n},piright)+1right]$ for n odd

To four decimal places, they are:

n Factors of Polynomial Bn(s)
1 (s + 1)
2 s2 + 1.4142s + 1
3 (s + 1)(s2 + s + 1)
4 (s2 + 0.7654s + 1)(s2 + 1.8478s + 1)
5 (s + 1)(s2 + 0.6180s + 1)(s2 + 1.6180s + 1)
6 (s2 + 0.5176s + 1)(s2 + 1.4142s + 1)(s2 + 1.9319s + 1)
7 (s + 1)(s2 + 0.4450s + 1)(s2 + 1.2470s + 1)(s2 + 1.8019s + 1)
8 (s2 + 0.3902s + 1)(s2 + 1.1111s + 1)(s2 + 1.6629s + 1)(s2 + 1.9616s + 1)

### Maximal flatness

Assuming ωc = 1 and G0 = 1, the derivative of the gain with respect to frequency can be shown to be:

$frac{dG}{domega}=-nG^3omega^{2n-1}$

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

$G(omega)=1 - frac{1}{2}omega^{2n}+frac{3}{8}omega^{4n}+ldots$

In other words, all derivatives of the gain up to but not including the 2n-th derivative are zero, resulting in "maximal flatness".

### High Frequency Roll Off

Again assuming ωc = 1, the slope of the log of the gain for large ω is:

$lim_{omegarightarrowinfty}frac{dlog(G)}{dlog(omega)}=-n$

In decibels, the high frequency roll off is therefore 20n dB/decade. (The factor of 20 is used because the power is proportional to the square of the voltage gain.) The decibel (dB) is a dimensionless unit of ratio which is used to express the relationship between a variable quantity and a known reference quantity. ...

## Filter Design

There are a number of different filter topologies available to implement a linear analog filter. These circuits differ only in the values of the components, but not in their connections.

### 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 kth element is given by: Image File history File links No higher resolution available. ... Wilhelm Cauer proposed a number of electronic filter designs or circuit topologies for realization of a driving point impedance. ...

$C_k = 2 sin left [frac {(2k-1)}{2n} pi right ]$; k = odd
$L_k = 2 sin left [frac {(2k-1)}{2n} pi right ]$; k = even

### Sallen-Key topology

The Sallen-Key topology uses active and passive components (op amps 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 n is odd), this must be implemented separately, usually as an RC circuit, and cascaded with the op-amp stages. A Sallen and Key filter is a type of active filter, particularly valued for its simplicity. ... An operational amplifier or op-amp is an electronic circuit module (normally built as an integrated circuit, but occasionally with discrete transistors or vacuum tubes) which has a non-inverting input (+), an inverting input (-) and one output. ... A resistor-capacitor circuit (RC circuit), or RC filter or RC network, is one of the simplest analogue electronic filters. ...

The Sallen-Key transfer function is given by

$H(s)=frac{1}{1+C_2(R_1+R_2)s+C_1C_2R_1R_2s^2}$

We wish the denominator to be one of the quadratic terms in a Butterworth polynomial. Assuming that ωc = 1, this will mean that

$C_1C_2R_1R_2=1,$

and

$C_2(R_1+R_2)=2cosleft(frac{2k+n-1}{2n}right)$

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

## Comparison with other linear filters

Here is an image showing the gain of a discrete-time Butterworth filter next to other common filters types. All filters are fifth-order.

All filters are of the same order, in this case five, which means that all filters roll off by 5 times 20 dB per decade, or 100 dB per decade. The Butterworth filter rolls off more slowly around the cutoff frequency than the others, but shows no ripples. No file by this name exists; you can upload it. ...

Results from FactBites:

 A BUTTERWORTH-FILTER COOKBOOK (450 words) The appropriate bandpass filter is one whose time decay can be chosen to be reasonable (in combination with a reasonable necessary compromise on the shape of the rectangle). Figure 14 Spectra of Butterworth filters of various-order n. Conceptually, the easiest form of Butterworth filtering is to take data to the frequency domain and multiply by equation (21), where you have selected some value of n to compromise between the demands of the frequency domain (sharp cutoff) and the time domain (rapid decay).
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