Third order Bessel-Thomson constant delay filter

Constant delay filter

A constant delay (Bessel–Thomson) filter is a low-pass filter designed so that its group delay $\tau(\omega) = -d\theta/d\omega$ is maximally flat (nearly constant) over the passband. All frequency components are delayed equally, so the waveform shape is preserved (no phase distortion). The Bessel–Thomson response is the standard example.

Third-order Bessel–Thomson transfer function

The ideal constant-delay function (delay normalized to 1 s) is

$$T(s) = e^{-s} = \frac{1}{\cosh s + \sinh s}$$

Write $M = \cosh s$ (even part) and $N = \sinh s$ (odd part). Their ratio has the continued-fraction expansion

$$\coth s = \frac{M}{N} = \frac{1}{s} + \cfrac{1}{\dfrac{3}{s} + \cfrac{1}{\dfrac{5}{s} + \cfrac{1}{\dfrac{7}{s} + \cdots}}}$$

For an $n$th-order approximation, keep the first $n$ terms, write the result as $M/N$ with polynomials, and use $T(s) = K/(M + N)$, with $K$ chosen so that $T(0) = 1$.

Third order (first three terms):

$$\begin{aligned} \frac{M}{N} &= \frac{1}{s} + \cfrac{1}{\dfrac{3}{s} + \dfrac{s}{5}} = \frac{1}{s} + \frac{5s}{15 + s^2} \\ &= \frac{15 + s^2 + 5s^2}{s(s^2 + 15)} = \frac{6s^2 + 15}{s^3 + 15s} \end{aligned}$$

So $M = 6s^2 + 15$, $N = s^3 + 15s$ and

$$T(s) = \frac{15}{s^3 + 6s^2 + 15s + 15}$$

The same polynomial follows from the Bessel recursion $B_n = (2n-1)B_{n-1} + s^2B_{n-2}$ with $B_0 = 1$, $B_1 = s + 1$: $B_2 = s^2 + 3s + 3$ and $B_3 = 5B_2 + s^2B_1 = s^3 + 6s^2 + 15s + 15$.

Poles: $s = -2.322$ and $s = -1.839 \pm j1.754$. The delay is $\tau(0) = 15/15 = 1$ s and stays at 0.996 s at $\omega$ = 1 rad/s and 0.887 s at 2 rad/s.

For an actual delay $\tau_o$, replace $s$ by $s\tau_o$ (frequency scaling by $k_f = 1/\tau_o$).