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.[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]

The Bessel filter is very similar to the Gaussian filter, and tends towards the same shape as filter order increases.[3][4] While the time-domain step response of the Gaussian filter has zero overshoot,[5] the Bessel filter has a small amount of overshoot,[6][7] 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.[3]

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.[8]

The transfer function[edit]

A plot of the gain and group delay for a fourth-order low-pass Bessel filter. Note that the transition from the passband to the stopband is much slower than for other filters, but the group delay is practically constant in the passband. The Bessel filter maximizes the flatness of the group delay curve at zero frequency.

A Bessel low-pass filter is characterized by its transfer function:[9]

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[edit]

The roots of the third-order Bessel polynomial are the poles of the filter transfer function in the plane, here plotted as crosses.

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:[9]



Gain plot of the third-order Bessel filter, versus normalized frequency.
Group delay plot of the third-order Bessel filter, illustrating flat unit delay in the passband.

The transfer function for a third-order (three-pole) Bessel low-pass filter with is

where the numerator has been chosen to give unity gain at zero frequency ().The roots of the denominator polynomial, the filter's poles, include a real pole at , and a complex-conjugate pair of poles at , plotted above.

The gain is then

The −3-dB point, where occurs at . This is conventionally called the cut-off frequency.

The phase is

The group delay is

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 . 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 : the first terms in the series expansion of the group delay will be zero, thus maximizing the flatness of the group delay at .


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,[10][11] which can also be transformed into an allpass filter, to implement fractional delays.[12][13]

See also[edit]


  1. ^ "Bessel Filter". 2013. Archived from the original on 2013-01-24. Retrieved 2022-05-14.
  2. ^ Thomson, W. E. (November 1949). "Delay networks having maximally flat frequency characteristics" (PDF). Proceedings of the IEE - Part III: Radio and Communication Engineering. 96 (44): 487–490. doi:10.1049/pi-3.1949.0101.
  3. ^ a b Roberts, Stephen (2001). "Transient Response and Transforms: 3.1 Bessel-Thomson filters" (PDF).
  4. ^ "comp.dsp | IIR Gaussian Transition filters". Retrieved 2022-05-14. An analog Bessel filter is an approximation to a Gaussian filter, and the approximation improves as the filter order increases.
  5. ^ "Gaussian Filters". Archived from the original on 2020-01-11. Retrieved 2022-05-14.
  6. ^ "How to choose a filter? (Butterworth, Chebyshev, Inverse Chebyshev, Bessel–Thomson)". Retrieved 2022-05-14.
  7. ^ "Free Analog Filter Program". Retrieved 2022-05-14. the Bessel filter has a small overshoot and the Gaussian filter has no overshoot.
  8. ^ Paarmann, L. D. (2001). 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.
  9. ^ a b Bianchi, Giovanni; Sorrentino, Roberto (2007). Electronic filter simulation & design. McGraw–Hill Professional. pp. 31–43. ISBN 978-0-07-149467-0.
  10. ^ Thiran, J.-P. (1971). "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.
  11. ^ Madisetti, Vijay (1997). "Section Classical IIR Filter Types". The Digital Signal Processing Handbook. CRC Press. p. 11-32. ISBN 9780849385728.
  12. ^ Smith III, Julius O. (2015-05-22). "Thiran Allpass Interpolators". W3K Publishing. Retrieved 2022-05-14.
  13. ^ Välimäki, Vesa (1995). Discrete-time modeling of acoustic tubes using fractional delay filters (PDF) (Thesis). Helsinki University of Technology.

External links[edit]