Delay equalization and its importance

Delay equalization

Delay equalization is the process of making the overall group delay of a system (filter or channel) constant over the band of interest by cascading it with an all-pass network (delay equalizer), which changes only the phase, not the magnitude.

Why it is needed:

• Sharp magnitude filters (Chebyshev, elliptic, even Butterworth) have group delay that rises steeply near the band edge. Transmission lines and channels also have non-uniform delay.
• Different frequency components then arrive at different times, causing phase (delay) distortion: pulses spread, square waves ring, and in data links this causes inter-symbol interference and higher bit error rate.
• In video, the eye notices delay distortion as smearing and ghost edges; in audio it changes transients.
• It lets a designer use a sharp, economical magnitude filter and fix the phase separately, instead of using a high-order Bessel filter with poor selectivity.

How it is done: the all-pass section $T_{AP}(s) = \dfrac{s^2 - \frac{\omega_0}{Q}s + \omega_0^2}{s^2 + \frac{\omega_0}{Q}s + \omega_0^2}$ has $|T_{AP}| = 1$ and a delay peak near $\omega_0$. Its $\omega_0$ and $Q$ are chosen so that the delay is large where the filter delay is small, giving $\tau_{filter} + \tau_{AP} \approx$ constant.

 in    +----------+     +-----------+    out
 ----->| filter   |---->| all-pass  |---->
       | |T| sharp|     | |T| = 1   |
       +----------+     +-----------+
 delay:  _/\_      +     \_  _/    =  _____
                           \/        flat

Example: a 4th-order Chebyshev low-pass filter used before an A/D converter in a modem has a delay peak near its cutoff. A second-order all-pass section with $\omega_0$ a little below the cutoff is added after it; the combined delay becomes nearly flat across the data band, and the received pulses no longer overlap.