---
title: "Impulse response; conditions for distortionless transmission"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:impulse-response-distortionless-conditions"
source: https://www.bctengineer.com/q/semester_6/communication_systems/impulse-response-distortionless-conditions
site: BCT Engineering
---

# Impulse response; conditions for distortionless transmission

Communication Systems · Chapter 1: Introduction

- Asked in: 2074 Bhadra (CS I), 2+6 marks
- Video: [Distortion Less Transmission of the System](https://www.youtube.com/watch?v=PYZ84ZpMGAI) — TutorialsPoint
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/impulse-response-distortionless-conditions
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

What is impulse response? What are the conditions for distortionless transmission? Explain with necessary diagrams.

## Short answer

Impulse response h(t) is the output of an LTI system to a unit impulse. Distortionless transmission requires constant amplitude response |H(f)|=K, linear phase theta(f)=-2 pi f td (constant delay), and a linear system, so output is K x(t-td).

## Answer

#### Impulse response
The **impulse response** $h(t)$ of a linear time-invariant (LTI) system is its output when the input is a unit impulse $\delta(t)$ (zero initial conditions). It completely describes the system: for any input $x(t)$, the output is the convolution $y(t) = x(t) * h(t)$, and its Fourier transform is the transfer function $H(f)$.

#### Conditions for distortionless transmission
Transmission is distortionless if the output is an exact copy of the input, differing only by a constant gain $K$ and a constant delay $t_d$:

$$
y(t) = K\,x(t - t_d) \;\Rightarrow\; H(f) = K\,e^{-j2\pi f t_d}, \quad h(t) = K\,\delta(t - t_d)
$$

So, over the whole band of the signal:

1. **Flat (constant) amplitude response:** $|H(f)| = K$. Every frequency component gets the same gain. Otherwise **amplitude distortion** occurs, e.g. high frequencies cut more than low ones, making sharp edges rounded.
2. **Linear phase response:** $\theta(f) = -2\pi f t_d$ (or $-2\pi f t_d \pm n\pi$). The phase shift is proportional to frequency, so every component has the same delay $t_d$ (constant group delay $\tau_g = -\frac{1}{2\pi}\frac{d\theta}{df} = t_d$). Otherwise **phase (delay) distortion** occurs, which spreads pulses and causes ISI in data transmission.
3. The system must be **linear**; a non-linear system creates harmonics and intermodulation (non-linear distortion).

```text
 Input x(t)           Output y(t) = K x(t - td)
   |  __                 |        __
   | /  \                |       /  \   (K times)
   |/    \               |      /    \
 --+------\---> t      --+-----/------\---> t
                         |<-td->|
```

```text
 |H(f)|                 theta(f)
   |                       |
 K +---------------        |\
   |                       | \  slope -2*pi*td
   |                     --+--\-----------> f
 --+---------------> f     |   \
  -B       0      B        |    \
```

The required band is only the signal bandwidth $|f| \le B$; outside it, $H(f)$ may be anything. Real channels are not ideal, so **equalizers** are used to make the overall response flat in amplitude and linear in phase.

## Check yourself

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

1. Knowing h(t), the output for any input x(t) is found by:
   A. Convolution y = x * h
   B. Multiplying x(t) h(t)
   C. Adding x and h
   D. Differentiating h
   Answer: A. h(t) completely describes an LTI system via convolution. (Key point: Impulse response meaning)

2. Why must phase be linear in frequency, theta = -2 pi f td?
   A. To remove noise
   B. So amplitude is flat
   C. To raise gain
   D. So every component has the same delay td
   Answer: D. Linear phase gives constant group delay; otherwise phase distortion spreads pulses and causes ISI. (Key point: Linear phase condition)

3. If high frequencies are attenuated more than low frequencies, what results?
   A. Amplitude distortion, e.g. rounded sharp edges
   B. Phase distortion only
   C. No distortion
   D. A time delay only
   Answer: A. Non-flat |H(f)| gives amplitude distortion. (Key point: Amplitude distortion)

4. What is the purpose of an equalizer?
   A. Make the overall response flat in amplitude and linear in phase
   B. Increase carrier frequency
   C. Convert analog to digital
   D. Add noise
   Answer: A. Real channels are non-ideal so equalizers correct them. (Key point: Equalizers)

## Related in Communication Systems

- [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
- [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×
- [Signum signal; distortionless transmission](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asignum-distortionless) — Communication Systems, chapter 1
- [Short note: Distortionless transmission](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashort-note-distortionless-transmission) — Communication Systems, chapter 1, asked 2×
- [Impulse response of matched filter](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amatched-filter-awgn) — Communication Systems, chapter 9, asked 3×
- [Optimum detector; matched filter impulse response](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amatched-filter-impulse-response) — Communication Systems, chapter 9, asked 3×

## 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
- [Deadlock; necessary conditions with examples](https://www.bctengineer.com/question.md?key=semester_6%2Foperating_system%3Adeadlock-necessary-conditions-examples) — Operating System, chapter 7
- [Deadlock conditions and why all necessary](https://www.bctengineer.com/question.md?key=semester_6%2Foperating_system%3Adeadlock-conditions-why-necessary) — Operating System, chapter 7