---
title: "Need for all-pass filters; importance of normalization"
subject: "Filter Design"
chapter: 2
key: "semester_5/filter_design:allpass-need-normalization"
source: https://www.bctengineer.com/q/semester_5/filter_design/allpass-need-normalization
site: BCT Engineering
---

# Need for all-pass filters; importance of normalization

Filter Design · Chapter 2: Approximation Methods

- Asked in: 2076 Asoj, 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-need-normalization
- Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-2

## Question

Why do we need all pass filter if it passes all the frequency components provided to it. Explain with practical example. Why normalization and denormalization is important in filter design?

## Short answer

An all-pass filter is used exclusively for its phase/delay characteristics—to equalize group delay distortion caused by sharp filters or channels, preventing pulse overlap (e.g., modem eye diagram), while leaving magnitude untouched. Normalization designs at 1 Ω and 1 rad/s, allowing universal tables and avoiding numerical errors; denormalization scales to desired impedance and frequency, yielding practical component values.

## Answer

#### Why an all-pass filter is needed

An **all-pass filter** is a filter whose magnitude is constant ($|T(j\omega)| = K$) at all frequencies, while its phase changes with frequency. Its zeros are mirror images of its poles about the $j\omega$ axis.

$$
T_1(s) = \frac{s-a}{s+a}, \qquad T_2(s) = \frac{s^2 - \frac{\omega_0}{Q}s + \omega_0^2}{s^2 + \frac{\omega_0}{Q}s + \omega_0^2}
$$

For the first-order section, $|T_1(j\omega)| = 1$, the phase is $\theta = 180^\circ - 2\tan^{-1}(\omega/a)$ (falling from $180^\circ$ at DC to $0^\circ$ at high frequency), and the delay is $\tau = \dfrac{2a}{a^2+\omega^2}$. The sign-flipped form $\dfrac{a-s}{a+s}$ has $\theta = -2\tan^{-1}(\omega/a)$ (0° to −180°) and the same delay.

```text
      jw
      |
  x   |   o      x = pole, o = zero
 -a   |   +a     (mirror images)
 -----+-------> sigma
```

It does not change the magnitude, but it **changes the phase and delay**, and that is exactly why it is used:
- **Delay equalization:** sharp filters (Chebyshev, elliptic) and channels (telephone lines, cables) have non-flat group delay. An all-pass section in cascade adds delay where it is lacking so the total delay becomes flat, removing pulse distortion and inter-symbol interference in data links, video and audio.
- **Phase shifters / phase correction:** e.g. 90° phase-splitting networks in single-sideband (SSB) modulators, phase compensation in control and measurement systems.
- **Time delay circuits:** a cascade of all-pass sections approximates a pure delay $e^{-s\tau}$ for analog delay lines.
- **Building other filters:** a notch or band-pass can be made by adding the input to the all-pass output; used in phasers and audio effects.
- **Hilbert transformers** and phase-locked systems in communication.

**Practical example:** a modem uses a 5th-order Chebyshev low-pass filter for its band limit. Its delay peaks near the cutoff, so pulses near the band edge arrive late and overlap the next symbol. A second-order all-pass section with $\omega_0$ near the cutoff is added; the magnitude is unchanged but the total delay becomes flat, and the eye diagram opens again.

#### Importance of normalization and denormalization

**Normalization** means designing the filter at a reference impedance of $1\ \Omega$ and a reference frequency of $1$ rad/s. **Denormalization** (scaling) converts this prototype into the real filter.

- Design tables (Butterworth, Chebyshev, Bessel poles and element values) are given only for normalized filters; one table serves every application.
- Element values near 1 are easy to compute and avoid numerical errors with very large or small numbers.
- Frequency transformations (LP to HP, BP, BS) are applied to the normalized low-pass prototype.
- Denormalization then gives practical values:

| Scaling | R | L | C |
|---|---|---|---|
| Magnitude ($k_m$) | $k_m R$ | $k_m L$ | $C/k_m$ |
| Frequency ($k_f$) | $R$ | $L/k_f$ | $C/k_f$ |

Example: normalized $C = 1$ F, scaled to $\omega_c = 1000$ rad/s and 10 kΩ gives $C' = 1/(1000\times10^4) = 0.1\ \mu$F.

## Check yourself

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

1. In a modem, why add an all-pass after a Chebyshev low-pass filter?
   A. To flatten the total delay near the cutoff without changing magnitude
   B. To increase the stopband attenuation
   C. To raise the cutoff frequency
   D. To reduce DC gain
   Answer: A. The Chebyshev delay peaks near cutoff; the all-pass compensates, removing symbol overlap. (Key point: Need for all-pass)

2. Normalized filters are designed at...
   A. 1 ohm and 1 rad/s
   B. 50 ohm and 1 kHz
   C. 1 kohm and 1 Hz
   D. 10 ohm and 10 rad/s
   Answer: A. Reference impedance 1 ohm and reference frequency 1 rad/s. (Key point: Normalization definition)

3. Why is normalization important?
   A. Tables for poles/elements exist only for the normalized case, and one table serves all designs
   B. It increases the filter order
   C. It removes the need for denormalization
   D. It improves the Q
   Answer: A. Also keeps numbers near 1 and permits frequency transformations. (Key point: Why normalize)

4. With magnitude scale km and frequency scale kf, how does C scale?
   A. C/(km*kf)
   B. C*km*kf
   C. C*km/kf
   D. C*kf
   Answer: A. C is divided by km and by kf. Example: 1 F -> 1/(1000*10^4) = 0.1 uF. (Key point: Scaling)

## Related in Filter Design

- [Normalization, denormalization and element scaling equations](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Anormalization-element-scaling-equations) — Filter Design, chapter 1, asked 5×
- [Normalization and denormalization with example](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Anormalization-importance-example) — Filter Design, chapter 1
- [Filter definition and denormalization from ωold to ωnew](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Afilter-normalization-frequency-change) — Filter Design, chapter 1
- [Ideal vs practical filters and scaling](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aideal-practical-filters-scaling) — Filter Design, chapter 1, asked 2×
- [Uses of an all-pass filter](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aallpass-filter-uses) — Filter Design, chapter 2
- [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 other subjects

- [RC low and high pass filters](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Afilter-lpf-hpf-transfer) — Basic Electronics Engineering, chapter 1
- [Why do we need DFT](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Awhy-need-dft) — Digital Signal Analysis and Processing, chapter 7, asked 6×
- [Why do we need FFT](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Awhy-need-fft) — Digital Signal Analysis and Processing, chapter 7, asked 2×
- [Ideal low pass and high pass filters](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afrequency-response-lp-hp-filters) — Signals and Systems, chapter 5