In electronics and signal processing, a Bessel filter is a type of analog linear filter with a maximally flat group delay (i.e., 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]

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

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

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 ${\displaystyle \theta _{n}(0)=\lim _{x\rightarrow 0}\theta _{n}(x)}$.

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:^[9]

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

where

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

There is no standard set attenuation value for Bessel filters, however, −3.0103 dB is a common choice. Some applications may use a higher or lower attenuation, such as −1 dB or −20 dB. The attenuation is set by frequency scaling the denominator. This can be approximated by interpolating Bessel filter attenuation tables, or calculated precisely with Newton's method or the use of a root finding algorithm.

The denominator may be frequency scaled by using a scaling factor that will be referred to as ${\displaystyle \omega _{c}}$. The frequency-scaled denominator of the Bessel transfer function may be rewritten for the ${\displaystyle n=3}$ case above and the example below as follows:

${\displaystyle H(s)'=H(s)_{{\text{desired dB at }}\omega =1}={\frac {15}{(s\omega _{c})^{3}+6(s\omega _{c})^{2}+15s\omega _{c}+15}}}$

The task for modifying ${\displaystyle H(s)}$ is to find an ${\displaystyle \omega _{c}}$ that results in the desired attenuation at 1 r/s for the normalized transfer function. Newton's method can be summarized to do this by its basic definition:

${\displaystyle \omega _{a}=\omega _{a}+(|H(j\omega _{a})|-{\text{DesiredAttenuation}})/(d|H(j\omega _{a})|/dj\omega _{a})}$

through successive iterations of ${\displaystyle \omega _{a}}$ until ${\displaystyle \omega _{a}}$is the frequency that attenuates ${\displaystyle |H(j\omega _{a})|}$ to the desired attenuation.

While the above summary may be concise and easily understood, the mechanics of obtaining an accurate derivative of the magnitude function along the ${\displaystyle j\omega }$ axis may be problematic. Digital techniques may be used, but it is generally better to apply analytical techniques with continuous functions, if possible, so as to maximize accuracy. ${\displaystyle \omega _{a}}$ may be determined by either of the following methods.

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 give unity gain at zero frequency (${\displaystyle s=0}$).The roots of the denominator polynomial, the filter's poles, include a real pole at ${\displaystyle s=-2.3222}$, and a complex-conjugate pair of poles at ${\displaystyle s=-1.8389\pm j1.7544}$, 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 −3-dB point, where ${\displaystyle |H(j\omega )|={\frac {1}{\sqrt {2}}},\,}$ occurs at ${\displaystyle \omega =1.756}$. This is conventionally called the cut-off frequency.

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 ${\displaystyle \omega ^{2}}$ and ${\displaystyle \omega ^{4}}$ are zero, resulting in a very flat group delay at ${\displaystyle \omega =0}$. 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 ${\displaystyle \omega =0}$ and a second specifies that the gain be zero at ${\displaystyle \omega =\infty }$, 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 ${\displaystyle n}$: the first ${\displaystyle n-1}$ terms in the series expansion of the group delay will be zero, thus maximizing the flatness of the group delay at ${\displaystyle \omega =0}$.

Although the bilinear transform is used to convert continuous-time (analog) filters to discrete-time (digital) infinite impulse response (IIR) filters with comparable frequency response, IIR filters obtained by the bilinear transformation do not have constant group delay.^[10] Since the important characteristic of a Bessel filter is its maximally-flat group delay, the bilinear transform is inappropriate for converting an analog Bessel filter into a digital form.

The digital equivalent is the Thiran filter, also an all-pole low-pass filter with maximally-flat group delay,^[11][12] which can also be transformed into an allpass filter, to implement fractional delays.^[13][14]

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