A Bessel filter is a type of analogue linear filter used within RF and other electronic applications that has a maximally flat group or phase delay. This preserves a wave shape of signals within the pass-band.

There are some RF and in particular audio applications where the preservation of the wave shape and phase of components within a signal is important. Audio crossover units is just one example, although many others exist. It is in these applications where the Bessel filter is an ideal solution.

As might be expected, the Bessel filter provides a slower transition from pass-band to stop-band than for other forms of filter of the same order.

## Bessel filter development

The Bessel filter takes its name from a German mathematician and astronomer named Friedrich Bessel who lived between1784 and 1846. Bessel developed the mathematical theory on which this form of filter is based.

Occasionally the filters may also be referred to as Bessel-Thomson filters. This is because W. E. Thomson developed the methodology of using Bessel functions within the design of this form of filter.

## Bessel filter basics

Some of the key features of the Bessel filter can be summarised as below:

The maximally flat group delay of the Bessel filter means that it equally exhibits a maximally linear phase response.*Maximally flat group delay:*A direct result of the maximally flat group delay of the Bessel filter it gives an output for a square wave input with no overshoot because all the frequencies are delayed by the same amount.*Overshoot:*The transition from the pass band to the stop band for the Bessel filter is much slower or shallower than for other filters.*Slow cut-off:*

## Bessel filter calculations

A Bessel low-pass filter has a transfer function of the form:

$H\left(s\right)=\frac{{\theta}_{n}\left(0\right)}{{\theta}_{n}(\frac{s}{{\omega}_{0}})}$

The transfer function of the Bessel filter is a rational function whose denominator is a reverse Bessel polynomial, such as the following:

Polynomial for Bessel Filter | |
---|---|

Order | Polynomial |

1 | s + 1 |

2 | s^{2} + 3s + 3 |

3 | s^{3} + 6s^{2} + 15 s + 15 |

4 | s^{4} + 10s^{3} + 45s^{2} 105s + 105 |

5 | s^{5} + 15s^{4} + 105s^{3} + 420s^{2} 945s + 945 |