Bessel filter

A plot of the gain and group delay for a fourth-order low pass Bessel filter. Note that the transition from the pass band to the stop band 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:[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 ${\displaystyle \theta _{n}(0)=\lim _{x\rightarrow 0}\theta _{n}(x)}$.

The roots of the third-order Bessel polynomial are the poles of filter transfer function in the s 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:

${\displaystyle n=1;\quad s+1}$

${\displaystyle n=2;\quad s^{2}+3s+3}$

${\displaystyle n=3;\quad s^{3}+6s^{2}+15s+15}$

${\displaystyle n=4;\quad s^{4}+10s^{3}+45s^{2}+105s+105}$

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

The reverse Bessel polynomials are given by:[11]

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

where

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

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 ${\displaystyle \omega _{0}=1}$ 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 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 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 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]

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

• Bessel and Linear Phase Filters — Nuhertz
• Bessel Filter Constants — C.R. Bond
• Bessel Filters Polynomials, Poles and Circuit Elements — C.R. Bond
• Java source code to compute Bessel filter poles