Steps in designing a 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). Its transfer function is $T(s) = K/B_n(s)$, where $B_n(s)$ is a Bessel polynomial: $B_1 = s + 1$, $B_2 = s^2 + 3s + 3$, $B_3 = s^3 + 6s^2 + 15s + 15$, and $B_n = (2n-1)B_{n-1} + s^2B_{n-2}$.

Design steps

1. Specifications: dc delay $\tau_o$, allowed delay error (e.g. within 1%) up to frequency $\omega_1$, and any attenuation limits.
2. Normalize: take $\tau_o$ = 1 s, so the frequency axis becomes $\omega\tau_o$; the normalized band edge is $\omega_1\tau_o$.
3. Choose the order: from Bessel delay curves/tables, pick the lowest $n$ whose normalized delay stays within the tolerance up to $\omega_1\tau_o$ (also check the loss there).
4. Write $T_n(s) = B_n(0)/B_n(s)$.
5. Denormalize: replace $s$ by $s\tau_o$.
6. Realize the function with a passive ladder or active sections (Sallen–Key/MFB), then apply magnitude scaling for practical values.

Example

Design a filter with dc delay $\tau_o$ = 0.5 ms whose delay is within 1% up to $\omega_1$ = 2000 rad/s.

Normalized frequency: $\omega_1\tau_o = 2000 \times 0.5\times10^{-3} = 1$.

Normalized delay at $\omega$ = 1 rad/s (computed from $\tau = -d\theta/d\omega$):

So $n = 3$: $T(s) = \dfrac{15}{s^3 + 6s^2 + 15s + 15}$. Replacing $s$ by $s\tau_o = s/2000$:

$$\begin{aligned} T(s) &= \frac{15}{(s/2000)^3 + 6(s/2000)^2 + 15(s/2000) + 15} \\ &= \frac{1.2\times10^{11}}{s^3 + 12000s^2 + 6\times10^{7}s + 1.2\times10^{11}} \end{aligned}$$

The magnitude loss at 2000 rad/s is only about 0.9 dB. The function can be realized as a first-order RC section (pole at $-2.322 \times 2000 = -4644$ rad/s) cascaded with a Sallen–Key biquad (poles at $(-1.839 \pm j1.754)\times2000$ rad/s).