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|Linear analog electronic filters|
In electronics and signal processing, a Bessel filter is a type of analog linear filter with a maximally flat group/phase delay (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.
The Bessel filter is very similar to the Gaussian filter, and tends towards the same shape as filter order increases. The Bessel filter has better shaping factor, flatter phase delay, and flatter group delay than a Gaussian of the same order, though the Gaussian has lower time delay.
The 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 reverse Bessel polynomials are given by:
The transfer function for a third-order (three-pole) Bessel low-pass filter, normalized to have unit group delay, is
The roots of the denominator polynomial, the filter's poles, include a real pole at s = −2.3222, and a complex-conjugate pair of poles at s = −1.8389 ± j1.7544, plotted above. The numerator 15 is chosen to give a gain of 1 at DC (at s = 0).
The gain is then
The phase is
The group delay is
The Taylor series expansion of the group delay is
Note that the two terms in ω2 and ω4 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 ω = 0 and a second specifies that the gain be zero at ω = ∞, 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 ω = 0.
As the important characteristic of a Bessel filter is its maximally-flat group delay, and not the amplitude response, it is inappropriate to use the bilinear transform to convert the analog Bessel filter into a digital form (since this preserves the amplitude response and not the group delay). The digital all-pole filter with maximally-flat group delay is derived using a Gaussian hypergeometric function connected with the Legendre functions.
- Butterworth filter
- Comb filter
- Chebyshev filter
- Elliptic filter
- Bessel function
- Group delay and phase delay
- Thomson, W.E., "Delay Networks having Maximally Flat Frequency Characteristics", Proceedings of the Institution of Electrical Engineers, Part III, November 1949, Vol. 96, No. 44, pp. 487–490.
- Design and Analysis of Analog Filters: A Signal Processing Perspective By Larry D. Paarmann p 238 http://books.google.com/books?id=l7oC-LJwyegC
- Giovanni Bianchi and Roberto Sorrentino (2007). Electronic filter simulation & design. McGraw–Hill Professional. pp. 31–43. ISBN 978-0-07-149467-0.
- Thiran - Recursive Digital Filters with Maximally Flat Group Delay
- Madisetti - Digital Signal Processing Handbook, section 184.108.40.206 Classical IIR Filter Types