Constant delay filter significance; 2nd order TF

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).

Significance

• A filter with phase $\theta(\omega) = -\omega\tau_o$ delays every frequency by the same $\tau_o$, so the output is an exact delayed copy of the input within the passband.
• Needed where waveform shape matters: pulse and digital data transmission, video, radar, oscilloscope and ECG signals, and audio crossovers.
• Bessel filters have little overshoot and ringing in their step response, unlike Butterworth or Chebyshev filters.
• The cost is a poor magnitude roll-off (wider transition band).

Second-order constant delay filter

Let

$$T(s) = \frac{a_0}{s^2 + a_1 s + a_0}$$

The phase is $\theta(\omega) = -\tan^{-1}\dfrac{a_1\omega}{a_0 - \omega^2}$, and the group delay is

$$\tau(\omega) = -\frac{d\theta}{d\omega} = \frac{a_1(a_0 + \omega^2)}{a_0^2 + (a_1^2 - 2a_0)\omega^2 + \omega^4}$$

Condition 1 – normalized delay of 1 s at dc: $\tau(0) = a_1a_0/a_0^2 = a_1/a_0 = 1 \Rightarrow a_1 = a_0$.

Condition 2 – maximally flat delay: $\tau(\omega)$ is flattest when the $\omega^2$ terms of numerator and denominator are in the same ratio as the constant terms:

$$\frac{a_1}{a_1^2 - 2a_0} = \frac{a_1 a_0}{a_0^2} \;\Rightarrow\; a_1^2 = 3a_0$$

With $a_1 = a_0$: $a_0^2 = 3a_0 \Rightarrow a_0 = 3,\ a_1 = 3$.

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

Poles: $s = -1.5 \pm j0.866$ ($\omega_o = \sqrt{3}$, $Q = 0.577$). The delay is $\tau(\omega) = \dfrac{9 + 3\omega^2}{9 + 3\omega^2 + \omega^4}$: exactly 1 s at dc and 0.923 s at 1 rad/s.

The same result comes from truncating $\coth s = \dfrac{1}{s} + \dfrac{1}{3/s} = \dfrac{s^2 + 3}{3s}$, giving $M + N = s^2 + 3s + 3$. For a delay $\tau_o$, replace $s$ by $s\tau_o$.