Bessel filter

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 in 1949. (In fact, a paper by Kiyasu of Japan predates this by several years.)

The Bessel filter is very similar to the Gaussian filter, and tends towards the same shape as filter order increases. While the time-domain step response of the Gaussian filter has zero overshoot, the Bessel filter has a small amount of overshoot, but still much less than common frequency domain filters.

Compared to finite-order approximations of the Gaussian filter, 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 and zero overshoot.

A Bessel low-pass filter is characterized by its transfer function:

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 ${\displaystyle \theta _{n}(0)=\lim _{x\rightarrow 0}\theta _{n}(x)}$.

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