2nd order constant delay and delay equalization

Constant delay filter

A constant delay filter (Bessel-Thomson filter) is a low-pass filter whose group delay $\tau(\omega) = -d\theta/d\omega$ is (nearly) constant over the passband, i.e. its phase is linear, $\theta(\omega) \approx -\omega\tau_0$. The ideal is $T(s) = e^{-s\tau_0}$: every frequency is delayed by the same $\tau_0$, so the output is an undistorted, delayed copy of the input. Since $e^{-s\tau_0}$ is not rational, it is approximated by an all-pole function whose delay is maximally flat at $\omega = 0$.

Properties: maximally flat delay, almost no overshoot in the step response, but a gradual magnitude roll-off.

Transfer function of the second-order constant delay filter

Take a second-order all-pole low-pass function normalized for a delay of 1 s at DC:

$$T(s) = \frac{b_0}{s^2 + b_1 s + b_0}$$

Phase and delay. With $s = j\omega$:

$$\theta(\omega) = -\tan^{-1}\frac{b_1\omega}{b_0 - \omega^2}, \qquad \tau(\omega) = -\frac{d\theta}{d\omega} = \frac{b_1(b_0 + \omega^2)}{(b_0-\omega^2)^2 + b_1^2\omega^2}$$

Expanding the denominator:

$$\tau(\omega) = \frac{b_1 b_0 + b_1\omega^2}{b_0^2 + (b_1^2 - 2b_0)\omega^2 + \omega^4}$$

Conditions for maximally flat delay equal to 1. $\tau(\omega)$ stays at 1 for as many powers of $\omega$ as possible if the numerator matches the denominator term by term:

$$\begin{aligned} \omega^0:&\quad b_1 b_0 = b_0^2 \;\Rightarrow\; b_1 = b_0 \\ \omega^2:&\quad b_1 = b_1^2 - 2b_0 \;\Rightarrow\; b_0 = b_0^2 - 2b_0 \;\Rightarrow\; b_0 = 3 \end{aligned}$$

So $b_0 = b_1 = 3$ and

$$T(s) = \frac{3}{s^2 + 3s + 3}, \qquad \tau(\omega) = \frac{9 + 3\omega^2}{9 + 3\omega^2 + \omega^4}$$

The delay is $1$ s at $\omega = 0$ and deviates only by the $\omega^4$ term, i.e. it is maximally flat. The poles are $s = -1.5 \pm j0.866$ ($\omega_0 = \sqrt{3} = 1.732$, $Q = 1/\sqrt{3} = 0.577$). The denominator $s^2+3s+3$ is the second-order Bessel polynomial. For a delay of $\tau_0$ seconds, replace $s$ by $s\tau_0$:

$$T(s) = \frac{3/\tau_0^2}{s^2 + (3/\tau_0)s + 3/\tau_0^2}$$

Importance of 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