Bessel filter

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.[1] 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.[2] (In fact, a paper by Kiyasu of Japan predates this by several years.[3][4])

The Bessel filter is very similar to the Gaussian filter, and tends towards the same shape as filter order increases.[5][6] While the time-domain step response of the Gaussian filter has zero overshoot,[7] the Bessel filter has a small amount of overshoot,[8][9] but still much less than common frequency domain filters.

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, though the Gaussian has lower time delay and zero overshoot.[10]

The transfer function

A Bessel low-pass filter is characterized by its transfer function:[11]

${\displaystyle H(s)={\frac {\theta _{n}(0)}{\theta _{n}(s/\omega _{0})}}\,}$

where ${\displaystyle \theta _{n}(s)}$ is a reverse Bessel polynomial from which the filter gets its name and ${\displaystyle \omega _{0}}$ is a frequency chosen to give the desired cut-off frequency. The filter has a low-frequency group delay of ${\displaystyle 1/\omega _{0}}$ . Since ${\displaystyle \theta _{n}(0)}$ is indeterminate by the definition of reverse Bessel polynomials, but is a removable singularity, it is defined that ${\displaystyle \theta _{n}(0)=\lim _{x\rightarrow 0}\theta _{n}(x)}$ .

Bessel polynomials

The transfer function of the Bessel filter is a rational function whose denominator is a reverse Bessel polynomial, such as the following:

${\displaystyle n=1;\quad s+1}$
${\displaystyle n=2;\quad s^{2}+3s+3}$
${\displaystyle n=3;\quad s^{3}+6s^{2}+15s+15}$
${\displaystyle n=4;\quad s^{4}+10s^{3}+45s^{2}+105s+105}$
${\displaystyle n=5;\quad s^{5}+15s^{4}+105s^{3}+420s^{2}+945s+945}$

The reverse Bessel polynomials are given by:[11]

${\displaystyle \theta _{n}(s)=\sum _{k=0}^{n}a_{k}s^{k},}$

where

${\displaystyle a_{k}={\frac {(2n-k)!}{2^{n-k}k!(n-k)!}}\quad k=0,1,\ldots ,n.}$

Example

The transfer function for a third-order (three-pole) Bessel low-pass filter with ${\displaystyle \omega _{0}=1}$ is

${\displaystyle H(s)={\frac {15}{s^{3}+6s^{2}+15s+15}},}$

where the numerator has been chosen to given unity gain at zero frequency (s = 0).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 gain is then

${\displaystyle G(\omega )=|H(j\omega )|={\frac {15}{\sqrt {\omega ^{6}+6\omega ^{4}+45\omega ^{2}+225}}}.\,}$

The phase is

${\displaystyle \phi (\omega )=-\arg(H(j\omega ))=\arctan \left({\frac {15\omega -\omega ^{3}}{15-6\omega ^{2}}}\right).\,}$

The group delay is

${\displaystyle D(\omega )=-{\frac {d\phi }{d\omega }}={\frac {6\omega ^{4}+45\omega ^{2}+225}{\omega ^{6}+6\omega ^{4}+45\omega ^{2}+225}}.\,}$

The Taylor series expansion of the group delay is

${\displaystyle D(\omega )=1-{\frac {\omega ^{6}}{225}}+{\frac {\omega ^{8}}{1125}}+\cdots .}$

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.

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 but not the group delay).

The digital equivalent is the Thiran filter, also an all-pole low-pass filter with maximally-flat group delay,[12][13] which can also be transformed into an allpass filter, to implement fractional delays.[14][15]

References

1. "Bessel Filter". 2013-01-24. Archived from the original on January 24, 2013. Retrieved 2016-01-06.
2. 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.
3. Kiyasu, Z (August 1943). "On A Design Method of Delay Networks". J. Inst. Electr. Commun. Eng. Japan. 26: 598–610.
4. Bohn, Dennis; Miller, Ray (1998). "RaneNote 147: A Bessel Filter Crossover, and Its Relation to Others". www.rane.com. Retrieved 2016-01-06.
5. Roberts, Stephen. "SIGNAL PROCESSING & FILTER DESIGN: 3.1 Bessel-Thomson filters" (PDF). The impulse response of Bessel-Thomson filters tends towards a Gaussian as the filter order is increased
6. "comp.dsp | IIR Gaussian Transition filters". www.dsprelated.com. Retrieved 2016-01-06. An analog Bessel filter is an approximation to a Gaussian filter, and the approximation improves as the filter order increases.
7. "Gaussian Filters". www.nuhertz.com. Retrieved 2016-03-29. The most significant characteristic of the Gaussian filter is that the step response contains no overshoot.
8. "How to choose a filter? (Butterworth, Chebyshev, Inverse Chebyshev, Bessel or Thomson)". www.etc.tuiasi.ro. Retrieved 2016-03-29. Bessel ... Advantages: Best step response-very little overshoot or ringing.
9. "Free Analog Filter Program". www.kecktaylor.com. Retrieved 2016-03-29. the Bessel filter has a small overshoot and the Gaussian filter has no overshoot.
10. Paarmann, L. D. (2001-06-30). Design and Analysis of Analog Filters: A Signal Processing Perspective. Springer Science & Business Media. ISBN 9780792373735. the Bessel filter has slightly better Shaping Factor, flatter phase delay, and flatter group delay than that of a Gaussian filter of equal order. However, the Gaussian filter has less time delay, as noted by the unit impulse response peaks occurring sooner than they do for Bessel filters of equal order.
11. Giovanni Bianchi and Roberto Sorrentino (2007). Electronic filter simulation & design. McGraw–Hill Professional. pp. 31–43. ISBN 978-0-07-149467-0.
12. Thiran, J. P. (1971-11-01). "Recursive digital filters with maximally flat group delay". IEEE Transactions on Circuit Theory. 18 (6): 659–664. doi:10.1109/TCT.1971.1083363. ISSN 0018-9324.
13. Madisetti, Vijay (1997-12-29). "Section 11.3.2.2 Classical IIR Filter Types". The Digital Signal Processing Handbook. CRC Press. p. 282. ISBN 9780849385728. A fifth IIR filter ... is the all-pole filter that possesses a maximally flat group delay .... this filter is not obtained directly from the analog equivalent, the Bessel filter ... Instead, it can be derived directly in the digital domain [Thiran]
14. Smith III, Julius O. (2015-05-22). "Thiran Allpass Interpolators". W3K Publishing. Retrieved 2016-04-29.
15. Välimäki, Vesa (1995-01-01). "Discrete-time modeling of acoustic tubes using fractional delay filters" (PDF). Otaniemi: Helsinki University of Technology. Thiran (1971) proposed an analytic solution for the coefficients of an all-pole low-pass filter with a maximally flat group delay ... it seems that the result of Thiran is better suited to the design of allpass than all-pole filters. Cite journal requires |journal= (help)