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Chebyshev filter

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Image:Chebyshev_response.png thumb|350px|The frequency response of a fourth-order type I Chebyshev low-pass filter '''Chebyshev filters''', are analog (signal) analog or digital electronic filter filters having a steeper roll-off and more passband ripple than Butterworth filters. 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.

Description


Type I Chebyshev Filters
These are the most common Chebyshev filters. The frequency (amplitude) characteristic of the $n$th order filter can be described mathematically by: :$G\_n(\backslash omega)\; =\; \backslash left\; |\; H\_n(j\; \backslash omega)\; \backslash right\; |\; =\; \backslash frac\{1\}\{\backslash sqrt\{1+\backslash epsilon^2\; T\_n^2\backslash left(\backslash frac\{\backslash omega\}\{\backslash omega\_0\}\backslash right)\}\}$ where and $|H(\backslash omega\_0)|\; =\; \backslash frac\{1\}\{\backslash sqrt\{1+\backslash epsilon^2\}\}$ is the amplification at the cutoff frequency $\backslash omega\_0$ (''note'': the common definition of the cutoff frequency to −3 decibel dB does ''not'' hold for Chebyshev filters!), and $T\_n\backslash left(\backslash frac\{\backslash omega\}\{\backslash omega\_0\}\backslash right)$ is a Chebyshev polynomial of the $n$th order, e.g.: :$T\_n\backslash left(\backslash frac\{\backslash omega\}\{\backslash omega\_0\}\backslash right)\; =\; \backslash cos\backslash left(n\backslash cdot\backslash arccos\backslash frac\{\backslash omega\}\{\backslash omega\_0\}\backslash right)\; ;\; 0\; \backslash le\; \backslash omega\; \backslash le\; \backslash omega\_0$ :$T\_n\backslash left(\backslash frac\{\backslash omega\}\{\backslash omega\_0\}\backslash right)\; =\; \backslash cosh\backslash left(n\backslash cdot\; \backslash operatorname\{arccosh\}\backslash frac\{\backslash omega\}\{\backslash omega\_0\}\backslash right)\; ;\; \backslash omega\; >\; \backslash omega\_0$ alternatively: :$T\_n\backslash left(\backslash frac\{\backslash omega\}\{\backslash omega\_0\}\backslash right)\; =\; a\_0\; +\; a\_1\backslash frac\{\backslash omega\}\{\backslash omega\_0\}\; +\; a\_2\backslash left(\backslash frac\{\backslash omega\}\{\backslash omega\_0\}\backslash right)^2\; +\backslash ,\; \backslash cdots\backslash ,\; +\; a\_n\backslash left(\backslash frac\{\backslash omega\}\{\backslash omega\_0\}\backslash right)^n;\; 0\; \backslash le\; \backslash omega\; \backslash le\; \backslash omega\_0$ :$T\_n\backslash left(\backslash frac\{\backslash omega\}\{\backslash omega\_0\}\backslash right)\; =\; \backslash frac\{\; \backslash left(\backslash frac\{\backslash omega\}\{\backslash omega\_0\}\backslash sqrt\{\backslash left(\backslash frac\{\backslash omega\}\{\backslash omega\_0\}\backslash right)^2\; -\; 1\}\backslash right)^n\; +\; \backslash left(\backslash frac\{\backslash omega\}\{\backslash omega\_0\}\backslash sqrt\{\backslash left(\backslash frac\{\backslash omega\}\{\backslash omega\_0\}\backslash right)^2\; -\; 1\}\backslash right)^\{-n\}\; \}\{2\}\; ;\; \backslash omega\; >\; \backslash omega\_0$ The order of a Chebyshev filter is equal to the number of reactance reactive components (for example, inductors) needed to realize the filter using analog electronics. The ripple is often given in decibel dB: :Ripple in dB = $20\; \backslash log\_\{10\}\; \backslash sqrt\{1+\backslash epsilon^2\}$ A ripple of 3 dB thus equals a value of $\backslash epsilon\; =\; 1$. An even steeper roll-off can be obtained if we allow for ripple in the pass band, by allowing zeroes on the $j\backslash omega$-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.

Type II Chebyshev Filters
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 ripple in the stopband. The transfer function is: :$\backslash left\; |\; H(\; \backslash Omega\; )\; \backslash right\; |\; ^2\; =\; \backslash frac\{1\}\{\backslash sqrt\{1+\; \backslash frac\{1\}\; \{\backslash epsilon^2\; T\_n\; ^2\; \backslash left\; (\; \backslash omega\_0\; /\; \backslash omega\; \backslash right\; )\}\}\}$ The parameter ε is related to the stopband attenuation γ in decibels by: :$\backslash epsilon\; =\; \backslash frac\{1\}\{\backslash sqrt\{10^\{0.1\backslash gamma\}-1\}\}$ For a stopband attenuation of 5dB, ε = 0.6801; for an attenuation of 10dB, ε = 0.3333. The frequency ''f_C = ω_C/2 π'' is the cutoff frequency. The 3dB frequency f_H is related to f_C by: :$f\_H\; =\; f\_C\; \backslash cosh\; \backslash left(\backslash frac\{1\}\{n\}\; \backslash cosh^\{-1\}\backslash frac\{1\}\{\backslash epsilon\}\backslash right)$

Applicability
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.

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: Image:Electronic_linear_filters.svg 500px|center As is clear from the image, Chebyshev filters are sharper than the Butterworth filter; they are not as sharp as the Elliptic filter elliptic one, but they show fewer ripples over the bandwidth.

See also
* Bessel filters * Butterworth filter Butterworth Filters * Comb filters * Elliptic filters Category:Linear filters de:Tschebyschefffilter es:Filtro de Chevyshev pt:Filtro Chebyshev zh:切比雪夫滤波器

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