Chebyshev filter

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Non standard name for "Linear analog electronic filter" template

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 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(\omega) = \left | H_n(j \omega) \right | = \frac\right)}}]

where [|\epsilon| < 1] and [|H(\omega_0)| = \frac}] is the amplification at the cutoff frequency [\omega_0] (note: the common definition of the cutoff frequency to −3 dB does not hold for Chebyshev filters!), and [T_n\left(\frac\right)] is a Chebyshev polynomial of the [n]th order, e.g.:

[T_n\left(\frac\right) = \cos\left(n\cdot\arccos\frac\right) ; 0 \le \omega \le \omega_0]

[T_n\left(\frac\right) = \cosh\left(n\cdot \operatorname\frac\right) ; \omega > \omega_0]

alternatively:

[T_n\left(\frac\right) = a_0 + a_1\frac + a_2\left(\frac\right)^2 +\, \cdots\, + a_n\left(\frac\right)^n; 0 \le \omega \le \omega_0]

[T_n\left(\frac\right) = \frac\sqrt\right)^2 - 1}\right)^n +\left(\frac\sqrt\right)^2 - 1}\right)^} ; \omega > \omega_0]

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.

The ripple is often given in dB:

Ripple in dB = [20 \log_\frac}]

A ripple of 3 dB thus equals a value of [\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\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:

[\left | H( \Omega ) \right | ^2 = \frac }}]

The parameter ε is related to the stopband attenuation γ in decibels by:

[\epsilon = \frac-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 \cosh \left(\frac \cosh^\frac\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:

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.

See also

• Bessel filters
• Butterworth Filters
• Comb filters
• Elliptic filters

 

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