---
title: "Short note: Distortionless transmission"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:short-note-distortionless-transmission"
source: https://www.bctengineer.com/q/semester_6/communication_systems/short-note-distortionless-transmission
site: BCT Engineering
---

# Short note: Distortionless transmission

Communication Systems · Chapter 1: Introduction

- Asked in: 2070 Magh (CS I), 5 marks; 2070 Bhadra (CS I), 5 marks (asked 2 times)
- Video: [Distortionless LTI System](https://www.youtube.com/watch?v=9VzN_CohuNs) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/short-note-distortionless-transmission
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

Write a short note on distortionless transmission.

## Short answer

Distortionless transmission requires the output to be a scaled, delayed copy of the input, y(t)=K x(t−t_d). The channel must provide constant amplitude response |H(f)|=K and linear phase θ(f)=−2πf t_d, so all frequency components are equally amplified and have the same time delay t_d.

## Answer

**Distortionless transmission** means the output of a system (channel, filter or amplifier) has exactly the same **shape** as the input; only its amplitude may be scaled by a constant and it may be delayed by a fixed time:

$$
y(t) = K\,x(t - t_d)
$$

where $K$ is a constant gain and $t_d$ a constant delay.

**Transfer function:** taking the Fourier transform,

$$
Y(f) = K X(f)\,e^{-j2\pi f t_d} \;\Rightarrow\; H(f) = K\,e^{-j2\pi f t_d}
$$

and the impulse response is $h(t) = K\,\delta(t - t_d)$.

**Conditions** (over the band occupied by the signal):
1. **Constant amplitude response:** $|H(f)| = K$. All frequency components are amplified or attenuated equally.
2. **Linear phase response:** $\theta(f) = -2\pi f t_d$ (a straight line through the origin, or through $\pm n\pi$). All components are delayed by the same time $t_d$; the group delay $-\frac{1}{2\pi}\frac{d\theta}{df} = t_d$ is constant.

```text
 |H(f)|              theta(f)
   |                   |\
 K +-----------        | \  slope = -2*pi*td
   |                   |  \
   +-----------> f   --+---\------> f
                       |    \
```

**Types of distortion when conditions fail:**
- **Amplitude distortion:** $|H(f)|$ not constant, so some frequencies are boosted or cut more than others.
- **Phase (delay) distortion:** phase not linear, so components arrive at different times; serious for video and data pulses (causes ISI).
- **Non-linear distortion:** the system is non-linear, creating harmonics and intermodulation products.

In practice exact distortionless transmission is impossible over all frequencies, but it is enough for $|H(f)|$ to be flat and the phase linear over the **signal bandwidth**. Equalizers are used to correct channel amplitude and phase response.

## Check yourself

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

1. A distortionless system has output y(t) equal to:
   A. K x(t) + t_d
   B. K x(t) with K a function of frequency
   C. x(t) filtered to half the bandwidth
   D. K x(t - t_d), a scaled and delayed copy of the input
   Answer: D. Only a constant gain K and a constant delay t_d are allowed; the shape must be unchanged. (Key point: Distortionless output form)

2. Which pair of conditions on H(f) gives distortionless transmission over the signal band?
   A. Constant |H(f)| and linear phase through the origin
   B. Constant phase and linearly rising |H(f)|
   C. Linear |H(f)| and constant phase
   D. Zero phase and |H(f)| that falls with frequency
   Answer: A. H(f) = K exp(-j 2 pi f t_d) means flat amplitude response and phase -2 pi f t_d, a straight line through the origin. (Key point: Conditions on H(f))

3. If the amplitude response is flat but the phase response is not linear, which distortion occurs and why?
   A. Amplitude distortion, since some frequencies are boosted more
   B. Non-linear distortion, since harmonics are created
   C. Phase (delay) distortion, since components are delayed by different times
   D. No distortion, since only the amplitude matters
   Answer: C. Non-linear phase means group delay varies with frequency, so components arrive at different times, which is serious for video and data pulses. (Key point: Phase distortion)

## Related in Communication Systems

- [Hilbert transform properties; distortionless transmission](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahilbert-properties-distortionless) — Communication Systems, chapter 2, asked 3×
- [Impulse response, transfer function and distortionless transmission](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adistortionless-transfer-function) — Communication Systems, chapter 1
- [Impulse response; conditions for distortionless transmission](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aimpulse-response-distortionless-conditions) — Communication Systems, chapter 1
- [Signum signal; distortionless transmission](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asignum-distortionless) — Communication Systems, chapter 1

## Related in other subjects

- [Distortionless transmission conditions](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adistortionless-transmission) — Signals and Systems, chapter 5
- [Distortionless transmission; ideal LPF step response](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adistortionless-ideal-lpf-step) — Signals and Systems, chapter 5
- [Distortionless line and telephone lines](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Adistortionless-line-telephone) — Electromagnetism, chapter 5