Chebyshev filter

Chebyshev filters are continuous frequency filter adapted to a possible bending of the sharp frequency response at the cut-off frequency ωg. But the gain in the passband or stopband extends not monotonic but has a definable ripple (ripple ). Within an order of the drop is steeper, the larger the ripple is approved. They are named after Pafnuti Lvovitch Chebyshev (formerly transliterated as Chebychev ).

A distinction is made between Chebyshev filters of type I and type II. Chebyshev type I have in the passband of an oscillating shape of the transfer function. Chebyshev Type II have the ripple of the transfer function in the stop band and are referred to in the literature as inverse Chebyshev filter.

Transfer function

For the area, the Chebyshev polynomials have the desired properties. For the Chebyshev polynomials grow monotonous.

To prepare with the help of Chebyshev polynomials a low-pass, is given to

With selected so that x = 0 is a measure of waviness.

Coefficients

Bring to the transfer function in the form

Are obtained for the coefficients and the following relations:

Order n of the filter just:

Order n of the filter odd:

These coefficients are chosen such that the cut-off frequency is normalized to the last frequency on which the selected gain is assumed the last time.

Properties

The Chebyshev filter has the following properties:

• Wave frequency response depending on the type in the passband or stopband.
• Very steep kinks at the cutoff frequency improves with the order and the ripple.
• Significant overshoot in the step response deteriorates with the order and ripple.
• Allowed to go to 0, the ripple, the Chebyshev filter is transferred to a Butterworth filter.
• Not a constant group delay in the passband.

Digital realization

For a digital implementation of the Chebychev filter is first transformed by the different biquads and bilinear transform cascaded these with the corresponding coefficients, and. In the following, this has been carried out on a low-pass filter with even order n.

The Z- transform of biquads looks generally as follows:

This equation is transformed into the time domain as follows:

The coefficients and are calculated from the coefficients and as follows:

Is a measure of the overshoot:

The coefficients then are calculated as follows:

In order to implement higher order filters, one needs only to cascade several biquad sections. The implementation of digital Tschebyscheffilter done in IIR filter structures ( recursive filter structure ).