Bessel Filter

Problem:
What is a Bessel filter?

Solution:
The term Bessel refers to a type of filter response, not a type of filter. It features flat group delay in the passband:



This is the characteristic of Bessel filters that makes them valuable to digital designers. Very few filters are designed with square waves in mind. Most of the time, the signals filtered are sine waves, or close enough that the effect of harmonics can be ignored. If a waveform with high harmonic content is filtered, such as a square wave, the harmonics can be delayed with respect to the fundamental frequency if a Butterworth or Chebyshev response is used. The Fourier Series of a square wave is:



This means that a square wave is an infinite series of odd harmonics, or sinewaves, summed together to create the square shape. Obviously, if a square wave is to be transmitted without distortion, all of the harmonics - out to infinity - must be transmitted. This means that the square wave can be high pass filtered without distortion, if the 3 dB point of the filter is significantly lower than the fundamental. If the square wave is low pass filtered, however, the situation changes dramatically. Harmonics will be eliminated, producing distortion in the square wave. It is the job of the designer to decide just how many harmonics must be passed and what can be eliminated. Suppose that the designer wants to keep five harmonics. The resulting waveform looks something like this:



This may be acceptable to the designer - it depends on the timing of the leading and trailing edge of the waveform. The elimination of harmonics will result in rounding of the edges, and therefore delay in the leading and trailing edges of the digital signal. Of more importance, the harmonics that are passed will not be delayed.

The Bessel approximation has a smooth passband and stopband response, like the Butterworth. For the same filter order, the stopband attenuation of the Bessel approximation is much lower that that of the Butterworth approximation.

The following figures are representative of a low pass filter. The response characteristics are mirror imaged for high pass filters.



The designer can see that there is no ripple in the passband of a Bessel filter, and that it does not have as much rejection in the stop band as a Butterworth filter.

The phase response of the three filter types is shown below. The Bessel response has the slowest rate of change of phase.