Delay and delay equalization

Delay

Delay (group delay) is the time taken by a signal component to pass through a filter. If the phase response is $\theta(\omega)$, then

$$\tau(\omega) = -\frac{d\theta(\omega)}{d\omega}$$

If $\tau$ is constant (phase linear in $\omega$), all components are delayed equally and the waveform shape is preserved. If $\tau$ varies with $\omega$, the signal suffers delay (phase) distortion. Example: for $T(s) = \frac{1}{s+1}$, $\theta = -\tan^{-1}\omega$ and $\tau = \frac{1}{1+\omega^2}$, which falls with frequency.

Delay equalization

Delay equalization means adding a network that corrects the delay of a filter or channel so that the total delay is nearly constant over the passband, without changing the magnitude response. The correcting network is an all-pass filter.

How it is performed:

1. Find (or measure) the delay $\tau_F(\omega)$ of the filter; it is usually smallest at low frequency and peaks near the band edge.
2. Choose the desired flat delay $\tau_0$, slightly above the peak of $\tau_F$.
3. Design all-pass sections whose delay fills the gap $\tau_0 - \tau_F(\omega)$:
   • first order: $T(s) = \dfrac{s-a}{s+a}$, $\tau(\omega) = \dfrac{2a}{a^2+\omega^2}$ (largest at DC);
   • second order: $T(s) = \dfrac{s^2 - \frac{\omega_0}{Q}s + \omega_0^2}{s^2 + \frac{\omega_0}{Q}s + \omega_0^2}$, with a delay peak near $\omega_0$ that gets taller and narrower as $Q$ increases.
4. Adjust $a$, $\omega_0$, $Q$ (often by optimization) and cascade the sections with the filter.
5. Realize each all-pass with an op-amp circuit (e.g. a first-order all-pass using one op-amp, two equal resistors, R and C).

  delay
   ^      tau_F (filter)
   |          __
   |        _/  \      tau_F + tau_AP
   |  -----/-----\----------  <- flat
   |  ____/       \
   | tau_AP: large where tau_F small
   +------------------------> w
                 wc

 Vi    +--------+      +-----------+     Vo
 ----->| filter |----->| all-pass  |----->
       +--------+      | equalizer |
                       +-----------+

The magnitude is set only by the filter, while the phase is corrected by the equalizer.