# Bessel filter

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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 Friedrich Bessel, a German mathematician (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.^{[1]}

The Bessel filter is very similar to the Gaussian filter, and tends towards the same shape as filter order increases.^{[2]}^{[3]} 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.^{[4]}

The time-domain step response of the Bessel filter has some overshoot, but less than common frequency domain filters.

## The transfer function

A Bessel low-pass filter is characterized by its transfer function:^{[5]}

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 .

## Bessel polynomials

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:^{[5]}

where

## Example

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 ± *j*1.7544, 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 *ω*^{2} and *ω*^{4} 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.

## Digital

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.^{[6]}^{[7]}

## See also

- Butterworth filter
- Comb filter
- Chebyshev filter
- Elliptic filter
- Bessel function
- Group delay and phase delay

## References

- ↑ 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. - ↑ http://www.robots.ox.ac.uk/~sjrob/Teaching/SP/l3.pdf
- ↑ http://www.dsprelated.com/showmessage/130958/1.php
- ↑ Design and Analysis of Analog Filters: A Signal Processing Perspective By Larry D. Paarmann p 238 http://books.google.com/books?id=l7oC-LJwyegC
- ↑
^{5.0}^{5.1}{{#invoke:citation/CS1|citation |CitationClass=book }} - ↑ Thiran - Recursive Digital Filters with Maximally Flat Group Delay
- ↑ Madisetti - Digital Signal Processing Handbook, section 11.3.2.2 Classical IIR Filter Types