---
title: "All-pass in delay equalization; 4th order Bessel-Thomson"
subject: "Filter Design"
chapter: 2
key: "semester_5/filter_design:allpass-bessel-fourth-order"
source: https://www.bctengineer.com/q/semester_5/filter_design/allpass-bessel-fourth-order
site: BCT Engineering
---

# All-pass in delay equalization; 4th order Bessel-Thomson

Filter Design · Chapter 2: Approximation Methods

- Asked in: 2071 Chaitra, 3+3 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/filter_design/allpass-bessel-fourth-order
- Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-2

## Question

Explain the importance of all pass filters in delay equalization. Find the transfer function of fourth order Bessel-Thomson low pass filter.

## Short answer

All-pass filters add delay where the original filter delay is low, flattening total group delay without changing magnitude, crucial for data and video. The fourth-order Bessel-Thomson low-pass transfer function is T(s) = 105/(s^4 + 10s^3 + 45s^2 + 105s + 105).

## Answer

#### Importance of all-pass filters in delay equalization

An **all-pass filter** has $|T(j\omega)| = 1$ at all frequencies but a frequency-dependent phase. A first-order section is $T(s) = \dfrac{s-a}{s+a}$ and a second-order section is $T(s) = \dfrac{s^2 - \frac{\omega_0}{Q}s + \omega_0^2}{s^2 + \frac{\omega_0}{Q}s + \omega_0^2}$: zeros are mirror images of the poles.

- Sharp filters (Chebyshev, elliptic) and transmission channels have **non-constant group delay**, largest near the band edge. This causes phase distortion of pulses and data.
- An all-pass filter cascaded with such a filter **adds delay without changing the magnitude**.
- Its delay is designed to be large where the original delay is small, so that the **total delay is nearly flat**: $\tau_{total}(\omega) = \tau_{filter}(\omega) + \tau_{AP}(\omega) \approx$ constant.
- Used in telephone/modem lines, video and data channels, and after sharp anti-aliasing filters.

```text
 tau    filter delay   +  all-pass delay  = total
  |      _/\              \_    _/         ______
  |   __/   \               \__/
  +--------- w        +-------- w    +-------- w
```

#### Fourth-order Bessel-Thomson low-pass filter

Ideal 1 s delay: $T(s) = e^{-s} = 1/(\cosh s + \sinh s)$. Expand $\coth s$ as a continued fraction and truncate after four terms:

$$
\coth s \approx \frac{1}{s} + \cfrac{1}{\cfrac{3}{s} + \cfrac{1}{\cfrac{5}{s} + \cfrac{1}{\cfrac{7}{s}}}}
$$

Working from the bottom:

$$
\begin{aligned}
\frac{5}{s} + \frac{s}{7} &= \frac{s^2+35}{7s} \\
\frac{3}{s} + \frac{7s}{s^2+35} &= \frac{10s^2 + 105}{s(s^2+35)} \\
\frac{1}{s} + \frac{s(s^2+35)}{10s^2+105} &= \frac{s^4 + 45s^2 + 105}{10s^3 + 105s}
\end{aligned}
$$

So $m(s) = s^4 + 45s^2 + 105$ (approximates $\cosh s$) and $n(s) = 10s^3 + 105s$ (approximates $\sinh s$):

$$
B_4(s) = s^4 + 10s^3 + 45s^2 + 105s + 105
$$

(Check by recursion: $B_4 = 7B_3 + s^2B_2 = 7(s^3+6s^2+15s+15) + s^2(s^2+3s+3)$, which gives the same polynomial.)

$$
T(s) = \frac{105}{s^4 + 10s^3 + 45s^2 + 105s + 105}
$$

This gives unity DC gain and a delay of $105/105 = 1$ s at low frequency; replace $s$ by $s\tau_0$ for a delay of $\tau_0$.

## Check yourself

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

1. What does an all-pass filter do to a signal?
   A. Keeps |T| = 1 at all frequencies but changes phase/delay
   B. Blocks high frequencies
   C. Changes magnitude but not phase
   D. Amplifies near cutoff
   Answer: A. Its zeros mirror the poles, so magnitude is constant while phase varies with frequency. (Key point: All-pass property)

2. How does an all-pass section equalize delay of a sharp filter?
   A. It makes its own delay large where the filter delay is small, so the total is nearly flat
   B. It subtracts the filter magnitude
   C. It lowers the filter cutoff
   D. It adds a fixed delay at all frequencies equally
   Answer: A. tau_total = tau_filter + tau_AP is designed to be nearly constant. (Key point: Delay equalization idea)

3. What is B4(s)?
   A. s^4 + 6s^3 + 15s^2 + 15s + 15
   B. s^4 + 10s^3 + 45s^2 + 105s + 105
   C. s^4 + 10s^3 + 45s^2 + 45s + 105
   D. s^4 + 7s^3 + 21s^2 + 105s + 105
   Answer: B. Truncating coth s after four terms gives m = s^4+45s^2+105 and n = 10s^3+105s. (Key point: Fourth-order polynomial)

4. What numerator K gives unity DC gain for the fourth-order filter?
   A. 10
   B. 45
   C. 15
   D. 105
   Answer: D. K equals the constant term of B4, 105, so T(0) = 1. (Key point: Unity DC gain numerator)

## Close variants set in other exams

- [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×

## Related in Filter Design

- [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×
- [Delay in filter and 3rd order Bessel-Thomson](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Adelay-third-order-bessel-thomson) — Filter Design, chapter 2, asked 8×
- [4th order Butterworth TF; phase and gain equalization](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Abutterworth-fourth-order-tf-equalization) — Filter Design, chapter 2
- [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×
- [Bessel-Thomson characteristics and 3rd order TF](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Abessel-thomson-characteristics-third-order) — Filter Design, chapter 2
- [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×

## Related in other subjects

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- [Transfer function of DT low pass filter](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adt-lowpass-transfer-function) — Signals and Systems, chapter 6
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