- Bessel filter
In
electronics andsignal processing , a Bessel filter is a variety oflinear filter with a maximally flatgroup delay (linearphase response ). Bessel filters are often used inaudio crossover systems. Analog Bessel filters are characterized by almost constant group delay across the entire passband, thus preserving the wave shape of filtered signals in the passband. The filter is named in honor ofFriedrich Bessel , a German mathematician (1784–1846).The transfer function
A Bessel
low-pass filter is characterized by itstransfer 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.
A simple example
The transfer function for a third order Bessel low pass filter is
:
The gain is then
:
The phase is
:
The
group delay is then:
The
Taylor series expansion of the group delay is:
Note that the two terms in and are zero, resulting in a very flat group delay at ω=0. This is the greatest number of terms that can be set to zero, since there are a total of four coefficients in the third order Bessel polynomial, requiring four equations in order to be defined. One equation specifies that the gain be unity at and a second specifies that the gain be zero at , leaving two equations to specify two terms in the series expansion to be zero. This is a general property of the group delay for a Bessel filter of order n: the first "n-1" terms in the series expansion of the group delay will be zero, thus maximizing the flatness of the group delay at .
See also
*
Butterworth filter
*Comb filter
*Chebyshev filter
*Elliptic filter
*Bessel function
*Group delay and phase delay External links
*http://www.filter-solutions.com/bessel.html
*http://www.rane.com/note147.html
*http://www.crbond.com/papers/bsf.pdf
*http://www-k.ext.ti.com/SRVS/Data/ti/KnowledgeBases/analog/document/faqs/bes.htm
*http://ece-www.colorado.edu/~ecen2260/slides/FilterSlides.pdf
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