What is a pass band

Filter types and parameterization

Parameterization of the Chebyshev filter

properties

The amplitude response of the Chebyshev filter has a configurable ripple in the pass band. For this, however, the amplitude response decreases steeply with a small filter order in the transition area. The following applies: the greater the permissible waviness, the shorter the transition area.

parameter

In addition to the filter order and the cut-off frequency, the transfer function of the Chebyshev filter also contains the ripple parameter “passband ripple” as parameters to be defined. This also affects the definition of the cutoff frequency.

Ripple

The parameter indicates the permissible ripple of the amplitude response in the pass band of the filter. By allowing a ripple, a short transition area between the pass band and the stop band and thus a steep decrease in the amplitude response can be achieved with a significantly lower filter order.

Cutoff frequency

With the Chebyshev filter, the cut-off frequency is defined as the frequency at which the amplitude response penetrates the defined ripple “passband ripple” downwards.

The position of the transition area on the frequency axis is therefore not only associated with the cutoff frequency, but also with the settings for the filter order and ripple.

The following diagram shows three different Chebyshev filters with different filter order and ripple, but the same cutoff frequency.

Graphical representation of the amplitude and phase response of a Chebyshev filter (blue: filter order 4, passband ripple 0.1 dB, red: filter order 2, passband ripple 0.1 dB, cyan: filter order 4, passband ripple 1 dB)

Compare Butterworth and Chebyshev filters

The following graphic shows a direct comparison of the amplitude and phase responses of a Butterworth filter and a Chebyshev filter. Both filters are parameterized in such a way that their amplitude responses intersect at the cut-off frequency of the Butterworth filter with a normalized amplitude of 1 / sqrt (2). Both filters are set up as fifth order filters. The ripple parameter (passband ripple) of the Chebyshev filter is 0.5 dB.

It shows the trade-off between permissible ripple of the amplitude response in the pass band and steepness in the transition area with the same filter order. With the same filter order, the amplitude response of the Chebyshev filter decreases more sharply in the transition area than that of the Butterworth filter. On the other hand, its amplitude response in the passband is not smooth, so that the useful signal is manipulated more strongly here than with the Butterworth filter.

Graphical representation of the amplitude and phase response of a Butterworth filter (blue) and a Chebyshev filter (cyan)