All-pass filter and 2nd order constant delay TF

All-pass filter

An all-pass filter has $|T(j\omega)| = 1$ at all frequencies and changes only the phase. Its zeros are mirror images of its poles about the $j\omega$ axis:

$$T_1(s) = \frac{a - s}{a + s}, \qquad T_2(s) = \frac{s^2 - \dfrac{\omega_o}{Q}s + \omega_o^2}{s^2 + \dfrac{\omega_o}{Q}s + \omega_o^2}$$

The first-order section has phase $\theta = -2\tan^{-1}(\omega/a)$ and delay $\tau = \dfrac{2a}{a^2 + \omega^2}$.

     jw
  x  |  o      x pole, o zero
     |
  x  |  o      |T(jw)| = 1 for all w
 ----+-------> sigma

Importance

• Delay (phase) equalization: added after a filter to make its total group delay flat without changing its magnitude response.
• Phase shifting networks (e.g. 90° phase splitters for SSB), pure time delay of analog signals.
• Correcting phase distortion in communication channels and audio systems.

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 $s\tau_o$.