Bessel filter

A Bessel filter is a frequency filter whose design aims at the following (equivalent) properties:

optimal "rectangular transmission behavior", d. H. a waveform whose frequency components lie within the passband of the filter appears almost unchanged at the output (except for a delay);
constant group delay in the passband;
linear phase response in the pass band.

It is accepted that the amplitude curve does not bend as sharply as with the Butterworth filter or Chebyshev filter .

The filter was developed by WE Thomson in 1949 as the optimal passive delay network - in terms of group delay - and named after the German mathematician Friedrich Wilhelm Bessel (1784–1846).

In digital signal processing , Bessel filters can be implemented by selecting appropriate filter coefficients in IIR filters (recursive filter structure).

Transfer function

Note: The cutoff frequency of the Butterworth filter and the low-pass cascade have been matched to the Bessel filter

The transfer function is optimized to make the group delay independent of the frequency.

With the transfer function for an nth order filter

{\ displaystyle {\ underline {A}} = {\ frac {A_ {0}} {1+ \ sum _ {i = 1} ^ {n} c_ {i} ^ {\ prime} P ^ {i}} }}

With

{\ displaystyle A_ {0}}

DC voltage gain

{\ displaystyle P = {\ frac {p} {\ omega _ {g}}} \ Leftrightarrow j \ Omega = j {\ frac {\ omega} {\ omega _ {g}}}}

and cutoff frequency

{\ displaystyle \ omega _ {g}}

the recursion formula can be used for the coefficients ${\ displaystyle c_ {i} ^ {\ prime}}$

i = 1:	${\ displaystyle c_ {1} ^ {\ prime} = 1}$
i = 2 ... n:	${\ displaystyle c_ {i} ^ {\ prime} = {\ frac {2 (n-i + 1)} {i (2n-i + 1)}} \ cdot c_ {i-1} ^ {\ prime} }$

determine.

However, the coefficients are not normalized to the limit frequency , but to the group delay, i.e. H. at the amplitude has not decreased by 3 dB. ${\ displaystyle \ Omega = 1}$

properties

The Bessel filter has the following properties:

smooth frequency response in the pass band
low steepness of the amplitude response (less than with the Butterworth filter) in the range of the cutoff frequency
slight overshoot in the step response , decreases with the order
constant group delay in the passband

Normalized Bessel polynomials

n	Bessel polynomial
1	${\ displaystyle 1 + P}$
2	${\ displaystyle 1 + P + {\ frac {1} {3}} P ^ {2}}$
3	${\ displaystyle 1 + P + {\ frac {2} {5}} P ^ {2} + {\ frac {1} {15}} P ^ {3}}$
4th	${\ displaystyle 1 + P + {\ frac {3} {7}} P ^ {2} + {\ frac {2} {21}} P ^ {3} + {\ frac {1} {105}} P ^ {4}}$
5	${\ displaystyle 1 + P + {\ frac {4} {9}} P ^ {2} + {\ frac {1} {9}} P ^ {3} + {\ frac {1} {63}} P ^ {4} + {\ frac {1} {945}} P ^ {5}}$