# Derivation of Bessel filters

I was reading Winder's Analog and Digital Filter Design and the section on Bessel filter.

I was hoping to see a complete derivation of the Bessel filter theory, but Winder's book gives only

\begin{align} H(s)&=e^{-sT} \tag{$\scriptstyle\text{pure delay}$}\\ H(s)&=e^{-s}=\frac{1}{\sinh(s)+\cosh(s)} \tag{$\scriptstyle\text{normalized pure delay, T=1}$} \end{align}

and then using series expansion of $\sinh$ and $\cosh$ we're left with

$$H(s)=\frac{a_0}{B_n(s)}$$

where $B_n(s)$ is a Bessel polynomial.

Anyone know where can I get a more exhaustive derivation of the Bessel filters?

• I don't know of any explicit derivation, but the Wikipedia page Example section suggests what is happening: The idea is to derive the group delay $D(\omega)$ and set as many of the Taylor series terms to zero.
– Peter K.
Jun 27 '16 at 12:13

The optimality criterion for Bessel (low pass) filters is that they have a maximally flat group delay at $\omega=0$, as correctly pointed out in a comment by Peter K.. The derivation is slightly tortuous, but you can find it for the continuous-time case in these lecture notes, which are taken from the book Introduction to Circuit Synthesis and Design by Temes and La Patra. I can recommend this book to anybody who wants to gain a deep understanding of electrical network theory.