# Bessel filter

Linear analog electronic filters
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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 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.[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 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:[5]

$H(s) = \frac{\theta_n(0)}{\theta_n(s/\omega_0)}\,$

where $\theta_n(s)$ is a reverse Bessel polynomial from which the filter gets its name and $\omega_0$ is a frequency chosen to give the desired cut-off frequency. The filter has a low-frequency group delay of $1 / \omega_0$. Since $\theta_n (0)$ is indeterminate by the definition of reverse Bessel polynomials, but is a removable singularity, it is defined that $\theta_n (0) = \lim_{x \rightarrow 0} \theta_n (x)$.

## Bessel polynomials

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:

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

The reverse Bessel polynomials are given by:[5]

$\theta_n(s)=\sum_{k=0}^n a_ks^k,$

where

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

## Example

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, normalized to have unit group delay, is

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

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 numerator 15 is chosen to give a gain of 1 at DC (at s = 0).

The gain is then

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

The phase is

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

The group delay is

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

The Taylor series expansion of the group delay is

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

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 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]