Type I Chebyshev Filters
These are the most common Chebyshev filters. The gain (or amplitude) response as a function of angular frequency of the nth-order low-pass filter is
where is the ripple factor, is the cutoff frequency and is a Chebyshev polynomial of the th order.
The passband exhibits equiripple behavior, with the ripple determined by the ripple factor . 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 . At the cutoff frequency the gain again has the value but continues to drop into the stop band as the frequency increases. This behavior is shown in the diagram on the right. The common practice of defining the cutoff frequency at −3 dB is usually not applied to Chebyshev filters; instead the cutoff is taken as the point at which the gain falls to the value of the ripple for the final time.
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 =
so that a ripple amplitude of 3 dB results from
An even steeper roll-off can be obtained if we allow for ripple in the stop band, by allowing zeroes on the -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 filter.
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