Bessel Filters

Bessel filters are all-pole filters with maximally flat pass-bands, maximally flat group delays, and poor "out-of-band" or stop-band attenuation compared to Butterworth, Chebyshev, and Elliptic filters of the same order. Since the group delays are relatively constant in the pass-band, Bessel filters exhibit minimal phase distortion, and hence very little overshoot in its step response.

The Low-Pass Bessel Filter

The transfer function of the low-pass Bessel filter is given by:

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

(1)

Where $N$ is the filter order, $\omega _{0}$ is the cut-off frequnecy, and $\theta _{N}\left(s\right)$ is the reverse Bessel polynomial, and is equal to:

$\theta _{N}\left(s\right)=\sum _{k=0}^{N}a_{k}s^{k}$

(2)

Where:

$a_{k}={\frac {\left(2N-k\right)!}{2^{N-k}k!\left(N-k\right)!}}$

(3)

For $k=0,1,\ldots N$. The low-pass Bessel filter characteristics are shown in Figs. 1-4.

Figure 1: The Bessel low-pass filter pole-zero plot for $N=5$.	Figure 2: The Bessel low-pass filter magnitude response for $N=5$.
Figure 3: The Bessel low-pass filter phase response for $N=5$.	Figure 3: The Bessel low-pass filter group delay for $N=5$.