Chebyshev filter

From Academic Kids

Chebyshev filters, named in honor of Pafnuty Chebyshev, are analog or digital filters of a kind related to Butterworth filters, but having a steeper roll-off, but more passband ripple. They have the property that they minimise the error between the idealised filter characteristic and the actual over the range of the filter, but as noted, there can be ripples in the passband. For this reason filters which have a smoother response in the passband but a more irregular response in the rejection bands are preferred for some applications.

The name Chebyshev filter comes from the fact that its mathematical characteristics are derived using Chebyshev polynomials. The frequency (amplitude) characteristic of a Chebyshev filter of the <math>n<math>th order can be described mathematically by:

<math>|H(\omega)| = \frac{1}{\sqrt{1+\epsilon^2 T_n^2\left(\frac{\omega}{\omega_0}\right)}}<math>

where <math>|\epsilon| < 1<math> and <math>|H(\omega_0)| = \frac{1}{\sqrt{1+\epsilon^2}}<math> is the amplification at the cutoff frequency <math>\omega_0<math> (note: the common definition of the cutoff frequency to −3 dB does not hold for Chebyshev filters!), and <math>T_n\left(\frac{\omega}{\omega_0}\right)<math> is a Chebyshev polynomial of the <math>n<math>th order, e.g.:

<math>T_n\left(\frac{\omega}{\omega_0}\right) = \cos\left(n\cdot\arccos\frac{\omega}{\omega_0}\right) ; 0 \le \omega \le \omega_0<math>
<math>T_n\left(\frac{\omega}{\omega_0}\right) = \cosh\left(n\cdot \operatorname{arccosh}\frac{\omega}{\omega_0}\right) ; \omega > \omega_0<math>

alternatively:

<math>T_n\left(\frac{\omega}{\omega_0}\right) = a_0 + a_1\frac{\omega}{\omega_0} + a_2\left(\frac{\omega}{\omega_0}\right)^2 +\, \cdots\, + a_n\left(\frac{\omega}{\omega_0}\right)^n; 0 \le \omega \le \omega_0<math>
<math>T_n\left(\frac{\omega}{\omega_0}\right) = \frac{

\left(\frac{\omega}{\omega_0}\sqrt{\left(\frac{\omega}{\omega_0}\right)^2 - 1}\right)^n + \left(\frac{\omega}{\omega_0}\sqrt{\left(\frac{\omega}{\omega_0}\right)^2 - 1}\right)^{-n} }{2} ; \omega > \omega_0<math>

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 = <math>20 \log_{10} \sqrt{1+\epsilon^2}<math>

A ripple of 3 dB thus equals a value of <math>\epsilon = 1<math>.

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

See also

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