---
title: "Steps in designing a constant delay filter"
subject: "Filter Design"
chapter: 2
key: "semester_5/filter_design:constant-delay-design-steps"
source: https://www.bctengineer.com/q/semester_5/filter_design/constant-delay-design-steps
site: BCT Engineering
---

# Steps in designing a constant delay filter

Filter Design · Chapter 2: Approximation Methods

- Asked in: 2078 Bhadra, 1+5 marks; 2070 Chaitra, 6 marks (asked 2 times)
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/filter_design/constant-delay-design-steps
- Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-2

## Question

What is constant delay filter? What are the steps involved in designing a constant delay filter? Explain with suitable example.

## Short answer

A constant delay filter has maximally flat group delay. Design steps: specify delay tolerance, normalize frequency, choose order from delay tables, write normalized Bessel polynomial, denormalize by replacing s with sτ₀. Example: τ₀=0.5 ms, flat delay up to 2000 rad/s gives order 3 with T(s)=1.2×10¹¹/(s³+12000s²+6×10⁷s+1.2×10¹¹).

## Answer

#### Constant delay filter
A **constant delay (Bessel–Thomson) filter** is a low-pass filter designed so that its group delay $\tau(\omega) = -d\theta/d\omega$ is maximally flat (nearly constant) over the passband. All frequency components are delayed equally, so the waveform shape is preserved (no phase distortion). Its transfer function is $T(s) = K/B_n(s)$, where $B_n(s)$ is a Bessel polynomial: $B_1 = s + 1$, $B_2 = s^2 + 3s + 3$, $B_3 = s^3 + 6s^2 + 15s + 15$, and $B_n = (2n-1)B_{n-1} + s^2B_{n-2}$.

#### Design steps
1. **Specifications:** dc delay $\tau_o$, allowed delay error (e.g. within 1%) up to frequency $\omega_1$, and any attenuation limits.
2. **Normalize:** take $\tau_o$ = 1 s, so the frequency axis becomes $\omega\tau_o$; the normalized band edge is $\omega_1\tau_o$.
3. **Choose the order:** from Bessel delay curves/tables, pick the lowest $n$ whose normalized delay stays within the tolerance up to $\omega_1\tau_o$ (also check the loss there).
4. **Write $T_n(s) = B_n(0)/B_n(s)$.**
5. **Denormalize:** replace $s$ by $s\tau_o$.
6. **Realize** the function with a passive ladder or active sections (Sallen–Key/MFB), then apply magnitude scaling for practical values.

#### Example
Design a filter with dc delay $\tau_o$ = 0.5 ms whose delay is within 1% up to $\omega_1$ = 2000 rad/s.

Normalized frequency: $\omega_1\tau_o = 2000 \times 0.5\times10^{-3} = 1$.

Normalized delay at $\omega$ = 1 rad/s (computed from $\tau = -d\theta/d\omega$):

| Order $n$ | $\tau(1)$ (s) | Error |
|---|---|---|
| 2 | 0.923 | 7.7% (fails) |
| 3 | 0.996 | 0.36% (meets) |

So $n = 3$: $T(s) = \dfrac{15}{s^3 + 6s^2 + 15s + 15}$. Replacing $s$ by $s\tau_o = s/2000$:
$$
\begin{aligned}
T(s) &= \frac{15}{(s/2000)^3 + 6(s/2000)^2 + 15(s/2000) + 15} \\
&= \frac{1.2\times10^{11}}{s^3 + 12000s^2 + 6\times10^{7}s + 1.2\times10^{11}}
\end{aligned}
$$
The magnitude loss at 2000 rad/s is only about 0.9 dB. The function can be realized as a first-order RC section (pole at $-2.322 \times 2000 = -4644$ rad/s) cascaded with a Sallen–Key biquad (poles at $(-1.839 \pm j1.754)\times2000$ rad/s).

## Check yourself

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

1. What is a constant delay (Bessel-Thomson) filter?
   A. A filter with equiripple passband and the sharpest cut-off
   B. A low-pass filter with maximally flat group delay in the passband, so waveform shape is preserved
   C. A filter with |T| = 1 at all frequencies
   D. A filter with zero delay at all frequencies
   Answer: B. Equal delay for all frequency components gives no phase distortion. (Key point: Constant delay filter)

2. In the design procedure, what is the normalized band edge for dc delay τo and band edge ω1?
   A. ω1 / τo
   B. ω1 τo
   C. ω1 + τo
   D. 1/(ω1 τo)
   Answer: B. Taking τo = 1 s makes frequency ω·τo. (Key point: Normalize step)

3. How is the order chosen?
   A. Lowest n whose normalized delay stays within tolerance up to ω1 τo
   B. Always n = 2
   C. Highest n available
   D. From the stopband attenuation only
   Answer: A. In the example n = 2 gives 7.7% error, n = 3 gives 0.36%. (Key point: Order choice)

4. In the example (τo = 0.5 ms), how is the final T(s) obtained from 15/B3(s)?
   A. Replace s by 2000 s
   B. Replace s by s/0.5
   C. Replace s by s τo = s/2000
   D. Multiply the numerator by 2000
   Answer: C. Frequency scaling kf = 1/τo = 2000 rad/s. (Key point: Denormalize)

## Related in Filter Design

- [Designing a constant delay filter (2nd order example)](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aconstant-delay-design-second-order-example) — Filter Design, chapter 2
- [Third order Bessel-Thomson constant delay filter](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aconstant-delay-third-order-bessel) — Filter Design, chapter 2, asked 9×
- [Importance of constant delay filter; 3rd order TF](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aconstant-delay-importance-third-order) — Filter Design, chapter 2, asked 8×
- [Second order constant delay filter](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aconstant-delay-second-order) — Filter Design, chapter 2, asked 4×
- [Constant delay filter significance; 2nd order TF](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aconstant-delay-significance-second-order) — Filter Design, chapter 2, asked 3×
- [All-pass filter and 2nd order constant delay TF](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aallpass-second-order-constant-delay) — Filter Design, chapter 2, asked 2×

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