Bessel filter
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.
[3] Compared to finite-order approximations of the Gaussian filter, the Bessel filter has a slightly better shaping factor (i.e., how well a particular filter approximates the ideal lowpass response), flatter phase delay, and flatter group delay than a Gaussian filter 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] where
is indeterminate by the definition of reverse Bessel polynomials, but is a removable singularity, it is defined that
Setting the cut-off attenuation frequency involves first finding the frequency that achieves the desired attenuation, which will be referred to as
term in each coefficient, as shown in the 3 pole Bessel filter example below.
may be found with Newton's method, or with root finding.
When steps 1) through 4) are complete, the expression involving Newton's method may be written as:
should be limited to prevent it from going negative early in the iterations for increased reliability.
If performed properly, only a handful of iterations are needed to set the attenuation through a wide range of desired attenuation values for both small and very large order filters.
does not contain any phase information, directly factoring the transfer function will not produce usable results.
However, the transfer function may be modified by multiplying it with
A 20-dB cut-off frequency attenuation example using the 3-pole Bessel example below is set as follows.
For even order filters, use the positive real root.
{\displaystyle {\begin{aligned}&H(s)={\frac {15}{s^{3}+6s^{2}+15s+15}}{\text{ (from the example below)}}\\&B_{arith}^{2}=10^{20/10}=0.01{\text{ (the arithmetic gain squared)}}\\&\\&{\text{Find }}H(s)'{\text{ such that }}|H(s)'|=-20{\text{ dB at }}\omega =1{\text{.
}}\\&H(s)H(-s)={\frac {225}{-s^{6}+6s^{4}-45^{2}s+225}}\\&P(s)=225-B_{arith}^{2}(-s^{6}+6s^{4}-45^{2}s+225)=0.01s^{6}-0.06s^{4}+0.45s^{2}+222.75{\text{ (polynomial to be factored)}}\\&R=j5.0771344{\text{ (the positive imaginary root for the above polynomial)}}\\&{\text{For even order filters, use the positive real root.
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 (
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
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
, 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
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.
