---
title: "Delay and delay equalization"
subject: "Filter Design"
chapter: 2
key: "semester_5/filter_design:delay-and-delay-equalization"
source: https://www.bctengineer.com/q/semester_5/filter_design/delay-and-delay-equalization
site: BCT Engineering
---

# Delay and delay equalization

Filter Design · Chapter 2: Approximation Methods

- Asked in: 2079 Bhadra, 2+3 marks
- Video: [Equalization Network Synthesis](https://www.youtube.com/watch?v=mE8oA1v8tAA) — Keysight Design Software
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/filter_design/delay-and-delay-equalization
- Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-2

## Question

What is delay and delay equalization? How can delay equalization be performed? Explain with necessary figures.

## Short answer

Group delay τ = -dθ/dω; non-constant delay distorts waveforms. Delay equalization cascades an all-pass network after the filter to sum delays to a constant. The all-pass's delay peak (tuned via ω₀, Q) compensates the filter's rising delay near band edge, flattening overall delay without altering magnitude.

## Answer

#### 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).

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

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

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

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For T(s) = 1/(s+1), the group delay is...
   A. 1/(1+w^2), falling with frequency
   B. 1 for all w
   C. w/(1+w^2)
   D. 1+w^2
   Answer: A. theta = -atan(w), so tau = 1/(1+w^2). (Key point: Delay for 1/(s+1))

2. Which network is used as a delay equalizer?
   A. Low-pass filter
   B. All-pass filter
   C. Notch filter
   D. Band-pass filter
   Answer: B. All-pass leaves magnitude unchanged. (Key point: Equalizer type)

3. In equalization, the desired flat delay tau0 is chosen...
   A. Below the minimum of tau_F
   B. Slightly above the peak of tau_F
   C. Equal to zero
   D. Equal to the cutoff frequency
   Answer: B. All-pass delay can only add delay, so the target must lie above the filter delay peak. (Key point: Choosing target delay)

4. As Q of a second-order all-pass increases, its delay peak...
   A. Gets taller and narrower
   B. Gets lower and wider
   C. Moves to DC
   D. Disappears
   Answer: A. Higher Q gives a sharper delay peak near w0. (Key point: Second-order all-pass Q)

## Close variants set in other exams

- [Delay equalization and how it is done](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Adelay-equalization-how) — Filter Design, chapter 2, asked 3×

## Related in Filter Design

- [All-pass filters in delay equalization; 3rd order Bessel-Thomson](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aallpass-importance-bessel-third-order) — Filter Design, chapter 2, asked 3×
- [All-pass in delay equalization; 4th order Bessel-Thomson](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aallpass-bessel-fourth-order) — Filter Design, chapter 2, asked 3×
- [2nd order constant delay and delay equalization](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aconstant-delay-second-order-equalization) — Filter Design, chapter 2, asked 2×
- [Delay equalization and its importance](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Adelay-equalization-importance) — Filter Design, chapter 2