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Concept
Bessel filter
Summary
In electronics and signal processing, a Bessel filter is a type of analog linear filter with a maximally flat group delay (i.e., maximally linear phase response), which preserves the wave shape of filtered signals in the passband. Bessel filters are often used in audio crossover systems.
The filter's name is a reference to German mathematician Friedrich Bessel (1784–1846), who developed the mathematical theory on which the filter is based. The filters are also called Bessel–Thomson filters in recognition of W. E. Thomson, who worked out how to apply Bessel functions to filter design in 1949.
The Bessel filter is very similar to the Gaussian filter, and tends towards the same shape as filter order increases. While the time-domain step response of the Gaussian filter has zero overshoot, the Bessel filter has a small amount of overshoot, but still much less than other common frequency-domain filters, such as Butterworth filters. It has been noted that the impulse response of Bessel–Thomson filters tends towards a Gaussian as the order of the filter is increased.
Compared to finite-order approximations of the Gaussian filter, the Bessel filter has better shaping factor, flatter phase delay, and flatter group delay than a Gaussian of the same order, although the Gaussian has lower time delay and zero overshoot.
A Bessel low-pass filter is characterized by its transfer function:
where is a reverse Bessel polynomial from which the filter gets its name and is a frequency chosen to give the desired cut-off frequency. The filter has a low-frequency group delay of . Since is indeterminate by the definition of reverse Bessel polynomials, but is a removable singularity, it is defined that .
The transfer function of the Bessel filter is a rational function whose denominator is a reverse Bessel polynomial, such as the following:
The reverse Bessel polynomials are given by:
where
There is no standard set attenuation value for Bessel filters, however, −3.0103 dB is a common choice. Some applications may use a higher or lower attenuation, such as −1 dB or −10 dB.
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