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Encyclopedia > Chebyshev filter
Linear analog electronic filters
Butterworth filter
Chebyshev filter
Elliptic (Cauer) filter
Bessel filter
Gaussian filter
Optimum "L" (Legendre) filter
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Chebyshev filters, are analog or digital filters having a steeper roll-off and more passband ripple than Butterworth filters. Chebyshev filters have the property that they minimize the error between the idealized filter characteristic and the actual over the range of the filter, but with ripples in the passband. This type of filters is named in honor of Pafnuty Chebyshev because their mathematical characteristics are derived from Chebyshev polynomials. Television signal splitter consisting of a hi-pass and a low-pass filter. ... The Butterworth filter is one type of electronic filter design. ... 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. ... An analog or analogue signal is any continuously variable signal. ... A digital system is one that uses discrete values (often electrical voltages), especially those representable as binary numbers, or non-numeric symbols such as letters or icons, for input, processing, transmission, storage, or display, rather than a continuous spectrum of values (ie, as in an analog system). ... Television signal splitter consisting of a hi-pass and a low-pass filter. ... 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. ... In physics, ripples are surface waves on a liquid with wavelengths so short that the liquids motion is governed almost entirely by surface tension forces. ... The Butterworth filter is one type of electronic filter design. ... Pafnuty Lvovich Chebyshev (Russian: ) ( May 26 [O.S. May 14] 1821 – December 8 [O.S. November 26] 1894) was a Russian mathematician. ... In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivres formula and which are easily defined recursively, like Fibonacci or Lucas numbers. ...


Because of the passband ripple inherent in Chebyshev filters, filters which have a smoother response in the passband but a more irregular response in the stopband are preferred for some applications.

Contents

Type I Chebyshev Filters

The frequency response of a fourth-order type I Chebyshev low-pass filter with ε = 1
The frequency response of a fourth-order type I Chebyshev low-pass filter with ε = 1

These are the most common Chebyshev filters. The gain (or amplitude) response as a function of angular frequency ω of the nth order low pass filter is Image File history File links Chebyshev_response. ... Image File history File links Chebyshev_response. ... Amplitude is a nonnegative scalar measure of a waves magnitude of oscillation, that is, magnitude of the maximum disturbance in the medium during one wave cycle. ...

G_n(omega) = left | H_n(j omega) right | = frac{1}{sqrt{1+epsilon^2 T_n^2left(frac{omega}{omega_0}right)}}

where ε is the ripple factor, ω0 is the cutoff frequency and Tn() is a Chebyshev polynomial of the nth order. The Butterworth filters frequency response, with cutoff frequency labeled. ... In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev (Пафнутий Чебышёв), are special polynomials. ...


The passband exhibits equiripple behavior, with the ripple determined by the ripple factor ε. In the passband, the Chebyshev polynomial alternates between 0 and 1 so the filter gain will alternate between maxima at G=1 and minima at G=1/sqrt{1+epsilon^2}. At the cutoff frequency ω0 the gain again has the value 1/sqrt{1+epsilon^2} but continues to drop into the stop band as the frequency increases. This behavior is shown in the diagram on the right. (note: the common definition of the cutoff frequency to −3 dB does not hold for Chebyshev filters!) The decibel (dB) method of calculation , that uses a logarithm to allow very large or very small relations to be represented with a conveniently small number (similar to scientific notation). ...


The order of a Chebyshev filter is equal to the number of reactive components (for example, inductors) needed to realize the filter using analog electronics. It has been suggested that Electric reactance be merged into this article or section. ... An inductor is a passive electrical device employed in electrical circuits for its property of inductance. ... Wikipedia does not yet have an article with this exact name. ...


The ripple is often given in dB: The decibel (dB) method of calculation , that uses a logarithm to allow very large or very small relations to be represented with a conveniently small number (similar to scientific notation). ...

Ripple in dB = 20 log_{10}frac{1}{sqrt{1+epsilon^2}}

so that a ripple of 3 dB results from ε = 1.


An even steeper roll-off can be obtained if we allow for ripple in the stop band, by allowing zeroes on the jω-axis in the complex plane. This will however result in less suppression in the stop band. The result is called an elliptic filter, also known as Cauer 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. ...


Poles and zeroes

Log of the absolute value of the gain of an 8th order Chebyshev type I filter in complex frequency space (s=σ+jω) with ε=0.1 and ω0 = 1. The white spots are poles and are arranged on an ellipse with a semi-axis of 0.3836... in σ and 1.071... in ω. The transfer function poles are those poles in the left half plane. Black corresponds to a gain of 0.05 or less, white corresponds to a gain of 20 or more.
Log of the absolute value of the gain of an 8th order Chebyshev type I filter in complex frequency space (s=σ+jω) with ε=0.1 and ω0 = 1. The white spots are poles and are arranged on an ellipse with a semi-axis of 0.3836... in σ and 1.071... in ω. The transfer function poles are those poles in the left half plane. Black corresponds to a gain of 0.05 or less, white corresponds to a gain of 20 or more.

For simplicity, assume that the cutoff frequency is equal to unity. The poles pm) of the gain of the Chebyshev filter will be the zeroes of the denominator of the gain. Using the complex frequency s: Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

1+epsilon^2T_n^2(-js)=0

Defining js = cos(θ) and using the trigonometric definition of the Chebyshev polynomials yields:

1+epsilon^2T_n^2(cos(theta))=1+epsilon^2cos^2(ntheta)=0

solving for θ

where the multiple values of the arc cosine function are made explicit using the integer index m. The poles of the Chebyshev gain function are then:

s_{pm}=jcos(theta),
=jcosleft(frac{1}{n}arccosleft(frac{pm j}{epsilon}right)+frac{mpi}{n}right)

Using the properties of the trigonometric and hyperbolic functions, this may be written in explicitly complex form:

s_{pm}^pm= pm sinhleft(frac{1}{n}mathrm{arcsinh}left(frac{1}{epsilon}right)right)sin(theta_m)
+j coshleft(frac{1}{n}mathrm{arcsinh}left(frac{1}{epsilon}right)right)cos(theta_m)

where m=1,2,...n  and

theta_m=frac{pi}{2},frac{2m-1}{n}

This may be viewed as an equation parametric in θn and it demonstrates that the poles lie on an ellipse in s-space centered at s=0 with a real semi-axis of length sinh(arcsinh(1 / ε) / n) and an imaginary semi-axis of length of cosh(arcsinh(1 / ε) / n)


The transfer function

The above expression yields the poles of the gain G. For each complex pole, there is another which is the complex conjugate, and for each conjugate pair there are two more that are the negatives of the pair. The transfer function must be stable, so that its poles will be those of the gain that have negative real parts and therefore lie in the left half plane of complex frequency space. The transfer function is then given by

H(s)=prod_{m=0}^{n-1} frac{1}{(s-s_{pm}^-)}

where s_{pm}^- are only those poles with a negative sign in front of the real term in the above equation for the poles.


In order to obtain a gain of 1 for ω=0 (as shown in the next figure) the transfer function H(s) has to be normalized with a contstant.


The group delay

Gain and group delay of a fifth order type I Chebyshev filter with ε=0.5.
Gain and group delay of a fifth order type I Chebyshev filter with ε=0.5.

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. Image File history File links Size of this preview: 800 × 544 pixel Image in higher resolution (882 × 600 pixel, file size: 28 KB, MIME type: image/png) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Chebyshev filter ... Image File history File links Size of this preview: 800 × 544 pixel Image in higher resolution (882 × 600 pixel, file size: 28 KB, MIME type: image/png) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Chebyshev filter ... In physics, and in particular in optics, the study of waves and digital signal processing, the term group delay has the following meanings: 1. ...

tau_g=-frac{d}{domega}arg(H(jomega))

The gain and the group delay for a fifth order type I Chebyshev filter with ε=0.5 are plotted in the graph on the left. It can be seen that there are ripples in the gain and the group delay in the passband but not in the stop band.


Type II Chebyshev Filters

The frequency response of a fifth-order type II Chebyshev low-pass filter with ε = 0.01
The frequency response of a fifth-order type II Chebyshev low-pass filter with ε = 0.01

Also known as inverse Chebyshev, this type is less common because it does not roll off as fast as type I, and requires more components. It has no ripple in the passband, but does have equiripple in the stopband. The gain is: Image File history File links Size of this preview: 800 × 533 pixel Image in higher resolution (900 × 600 pixel, file size: 27 KB, MIME type: image/png) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Chebyshev filter ... Image File history File links Size of this preview: 800 × 533 pixel Image in higher resolution (900 × 600 pixel, file size: 27 KB, MIME type: image/png) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Chebyshev filter ...

G_n(omega,omega_0) = frac{1}{sqrt{1+ frac{1} {epsilon^2 T_n ^2 left ( omega_0 / omega right )}}}

In the stop band, the Chebyshev polynomial will oscillate between 0 and 1 so that the gain will oscillate between zero and

frac{1}{sqrt{1+ frac{1}{epsilon^2}}}

and the smallest frequency at which this maximum is attained will be the cutoff frequency ω0. The parameter ε is thus related to the stopband attenuation γ in decibels by: 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. ... Attenuation is the reduction in amplitude and intensity of a signal with respect to distance traveled through a medium. ... The decibel (dB) method of calculation , that uses a logarithm to allow very large or very small relations to be represented with a conveniently small number (similar to scientific notation). ...

For a stopband attenuation of 5dB, ε = 0.6801; for an attenuation of 10dB, ε = 0.3333. The frequency fC = ωC/2 π is the cutoff frequency. The 3dB frequency fH is related to fC by:

Poles and zeroes

Log of the absolute value of the gain of an 8th order Chebyshev type II filter in complex frequency space (s=σ+jω) with ε=0.1 and ω0 = 1. The white spots are poles and the black spots are zeroes. All 16 poles are shown. Each zero has multiplicity of two, and 12 zeroes are shown and four are located ouside the picture, two on the positive ω axis, and two on the negative. The poles of the transfer function will be poles on the left half plane and the zeroes of the transfer function will be the zeroes, but with multiplicity 1. Black corresponds to a gain of 0.01 or less, white corresponds to a gain of 3 or more.
Log of the absolute value of the gain of an 8th order Chebyshev type II filter in complex frequency space (s=σ+jω) with ε=0.1 and ω0 = 1. The white spots are poles and the black spots are zeroes. All 16 poles are shown. Each zero has multiplicity of two, and 12 zeroes are shown and four are located ouside the picture, two on the positive ω axis, and two on the negative. The poles of the transfer function will be poles on the left half plane and the zeroes of the transfer function will be the zeroes, but with multiplicity 1. Black corresponds to a gain of 0.01 or less, white corresponds to a gain of 3 or more.

Again, assuming that the cutoff frequency is equal to unity, the poles pm) of the gain of the Chebyshev filter will be the zeroes of the denominator of the gain: Image File history File links Size of this preview: 599 × 600 pixel Image in higher resolution (600 × 601 pixel, file size: 20 KB, MIME type: image/png) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Chebyshev filter ... Image File history File links Size of this preview: 599 × 600 pixel Image in higher resolution (600 × 601 pixel, file size: 20 KB, MIME type: image/png) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Chebyshev filter ...

1+epsilon^2T_n^2(-1/js_{pm})=0

The poles of gain of the type II Chebyshev filter will be the inverse of the poles of the type I filter:

frac{1}{s_{pm}^pm}= pm sinhleft(frac{1}{n}mathrm{arcsinh}left(frac{1}{epsilon}right)right)sin(theta_m)
qquad+j coshleft(frac{1}{n}mathrm{arcsinh}left(frac{1}{epsilon}right)right)cos(theta_m)

where m=1,2,...,n . The zeroes zm) of the type II Chebyshev filter will be the zeroes of the numerator of the gain:

The zeroes of the type II Chebyshev filter will thus be the inverse of the zeroes of the Chebyshev polynomial.

where m=1,2,...,n .


The transfer function

The transfer function will be given by the poles in the left half plane of the gain function, and will have the same zeroes but these zeroes will be single rather than double zeroes.


The group delay

Gain and group delay of a fifth order type II Chebyshev filter with ε=0.1.
Gain and group delay of a fifth order type II Chebyshev filter with ε=0.1.

The gain and the group delay for a fifth order type II Chebyshev filter with ε=0.1 are plotted in the graph on the left. It can be seen that there are ripples in the gain in the stop band but not in the pass band.
Image File history File links Size of this preview: 800 × 533 pixel Image in higher resolution (900 × 600 pixel, file size: 27 KB, MIME type: image/png) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Chebyshev filter ... Image File history File links Size of this preview: 800 × 533 pixel Image in higher resolution (900 × 600 pixel, file size: 27 KB, MIME type: image/png) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Chebyshev filter ...


Digital Implemention

As with most analog filters, the Chebyshev may be converted to a digital (discrete-time) recursive form via the bilinear transform. However, as digital filters have a finite bandwidth, the response shape of the transformed Chebyshev will be warped. Alternatively, the Matched Z-transform may be used, which does not warp the response. Infinite impulse response (IIR) filters have an impulse response function which is non-zero over an infinite length of time. ... In digital signal processing, the bilinear transform is a conformal mapping, often used to convert a transfer function of a linear, time-invariant (LTI) filter in the continuous-time domain (often called an analog filter) to a transfer function of a linear, shift-invariant filter in the discrete-time domain... An FIR filter In electronics,nirali a digital filter is any electronic filter that works by performing digital mathematical operations on an intermediate form of a signal. ... This article or section does not cite its references or sources. ... In digital signal processing, the bilinear transform is a conformal mapping, often used to convert a transfer function of a linear, time-invariant (LTI) filter in the continuous-time domain (often called an analog filter) to a transfer function of a linear, shift-invariant filter in the discrete-time domain...


Comparison with other linear filters

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

As is clear from the image, Chebyshev filters are sharper than the Butterworth filter; they are not as sharp as the elliptic one, but they show fewer ripples over the bandwidth. No file by this name exists; you can upload it. ... The Butterworth filter is one type of electronic filter design. ... 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. ...


See also

In electronics and signal processing, a Bessel filter is a variety of linear filter with a maximally flat group delay (linear phase response). ... The Butterworth filter is one type of electronic filter design. ... In signal processing, a comb filter adds a delayed version of a signal to itself, causing constructive and destructive interference. ... 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. ...

References

  • Daniels, Richard W. (1974). Approximation Methods for Electronic Filter Design. New York: McGraw-Hill. ISBN 0-07-015308-6. 

  Results from FactBites:
 
Chebyshev filter - Wikipedia, the free encyclopedia (404 words)
Chebyshev filters have the property that they minimise the error between the idealised filter characteristic and the actual over the range of the filter, but with ripples in the passband.
This type of filters is named in honor of Pafnuty Chebyshev because their mathematical characteristics are derived from Chebyshev polynomials.
As is clear from the image, Chebyshev filters are sharper than the Butterworth filter; they are not as sharp as the elliptic one, but they show fewer ripples over the bandwidth.
  More results at FactBites »

 
 

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