Filter types and parameterization
 | This description is limited to low-pass filters. However, the concepts can be applied to other filter types (high-pass, band-pass and stop-band filters). |
The Butterworth filter and the Chebyshev filter are common implementations of a digital filter.
The difference between the two implementations essentially consists of the balance between the permissible ripple of the amplitude response in the passband and the slope of the amplitude response in the transition between the passband and the stopband. While the Butterworth filter has a maximally flat amplitude response in the passband, for the Chebyshev filter the permissible ripple of the amplitude response in the passband is specified as a parameter. The advantage of the Chebyshev filter is a steeper decrease of the amplitude response in the transition range from the passband to the stopband.
The filter types are compared and described in more detail below. First, some basic terms are explained briefly.
Transfer function in the amplitude/frequency diagram
The filter is described mathematically by the transfer function (see Digital filters). The transfer function can be displayed in the form of an amplitude and a phase response.
Passband
The passband (blue zone) allows spectral components of a signal to pass through. Modification of the signal in this frequency range should be avoided.
Stopband
In the stopband (red zone), the filter attenuates the corresponding frequency components of the signal.
Transition
The transition (yellow zone) separates the passband and the stopband. It should normally be as small as possible. The design of the transition phase is a defining criterion for the selection of the filter type and its parameterization.
Passband ripple
The ripple in the passband describes the waviness of the amplitude response in the passband.
Parameterization of the Butterworth filter
Properties
The amplitude response of the Butterworth filter is maximally flat in the passband, so that the wanted signal in this range is only minimally manipulated. In addition, the entire course of the amplitude response is monotonous, i.e. without passband ripple. This filter type is therefore one of the most frequently used filter types.
Parameter
The transfer function of the Butterworth filter contains only two parameters that have to be defined: the cut-off frequency and the filter order.
Filter order
The filter order determines how steeply the amplitude response decreases in the transition range. The higher the filter order, the steeper the amplitude response decreases and the smaller the transition range.
The following applies for the slope of the amplitude response of a Butterworth filter: -n x 20 dB/decade, with n = order, i.e. -20 dB/decade for filter order 1, -40 dB/decade for filter order 2, etc.
Cut-off frequency
The cut-off frequency of the Butterworth filter is defined according to its transfer function as the frequency at which the normalized amplitude response assumes the value 1/sqrt(2) ≈ -3 dB. This applies to all filter orders. Accordingly, when designing the filter, care must be taken to ensure that the spectral components of a signal are already attenuated by 3 dB at the cut-off frequency. This parameter causes a parallel shift of the amplitude response along the frequency axis (distortion due to the logarithmic frequency axis).
Parameterization of the Chebyshev filter
Properties
The amplitude response of the Chebyshev filter has a parameterizable passband ripple. However, the amplitude response decreases steeply in the transition range when the filter order is small. The following applies: The greater the permissible passband ripple, the shorter the transition range.
Parameter
In addition to the filter order and the cut-off frequency as parameters to be defined, the transfer function of the Chebyshev filter contains a "passband ripple" parameter. This also affects the definition of the cut-off frequency.
Passband ripple
The parameter specifies the permissible ripple of the amplitude response in the passband of the filter. By allowing a passband ripple, a short transition range between passband and stopband, and thus a steep decrease of the amplitude response, can be achieved with a significantly lower filter order.
Cut-off frequency
The cut-off frequency of the Chebyshev filter is defined as the frequency at which the amplitude response passes downwards through the defined "passband ripple".
The position of the transition range on the frequency axis is thus associated not only with the cut-off 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 cut-off frequency.
Comparison of Butterworth and Chebyshev filters
The following diagram shows a direct comparison of the amplitude and phase response of a Butterworth filter and a Chebyshev filter. Both filters are parameterized so that their amplitude responses intersect at the cut-off frequency of the Butterworth filter at a normalized amplitude of 1/sqrt(2). Both filters are defined as fifth order filters. The passband ripple parameter of the Chebyshev filter is 0.5 dB.
The weighing up referred to above between the permissible passband ripple of the amplitude response and the slope in the transition with the same filter order becomes apparent. With the same filter order, the amplitude response of the Chebyshev filter decreases more sharply in the transition than that of the Butterworth filter. On the other hand, its amplitude response is not smooth in the passband, so that the wanted signal is manipulated more strongly here than with the Butterworth filter.
