Engineering Embedded Systems by Peter Hintenaus

Author:Peter Hintenaus
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Fig. 6.5Step responses of a second, a third, and a fourth order Butterworth filter, all having corner frequency

Fig. 6.6Bode plots of a second, a third, and a fourth order Chebyshev filter, all having a maximum passband ripple of

Fig. 6.7Step responses of a second, a third, and a fourth order Chebyshev filter, all having corner frequency and a maximum passband ripple of

Chebyshev filters (of the first kind) exhibit the sharpest transition for a given order and a given ripple of the gain in their passband. Such a filter introduces a phase shift between its input and its output which is nonlinear with frequency in the filter’s passband, causing considerable overshoot and ringing in the filter’s step response. Chebyshev filters are called for when a design requires a sharp transition from the passband into the stopband, Figs. 6.6 and 6.7.

A Bessel filter, also called a Thomson filter, exhibits the best approximation to a linear dependency in the filter’s passband of the phase shift the filter introduces between its input and its output and frequency. The group delay of this filter, the negative of the derivative of the phase with respect to angular frequency , is almost constant in the filter’s passband, therefore a Bessel filter preserves the temporal relationship between a signal’s spectral components. The step response of a lowpass Bessel filter shows no overshoot and almost no ringing. The transition from the filter’s passband into the filter’s stopband, however, is very gradual. Bessel filters are called for when the temporal relationship between the spectral content of a signal or between signals has to be preserved as much as possible, Figs. 6.8 and 6.9.

Fig. 6.8Bode plots of a second, a third, and a fourth order Bessel filter

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