---
title: "LTI system and its significance"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:lti-significance"
source: https://www.bctengineer.com/q/semester_6/communication_systems/lti-significance
site: BCT Engineering
---

# LTI system and its significance

Communication Systems · Chapter 1: Introduction

- Asked in: 2070 Bhadra (CS I), 2+3 marks
- Video: [Linear Time-Invariant (LTI) Systems](https://www.youtube.com/watch?v=Y8iFJVmSQIk) — 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/lti-significance
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

Define linear time invariant system. What is the significance of such system in communication engineering?

## Short answer

An LTI system obeys superposition and time-invariance, fully described by h(t) or H(f). In communications, it simplifies analysis via convolution and multiplication, models channels, enables filter design, sets distortionless conditions (flat |H(f)|, linear phase), and permits signal superposition without intermodulation.

## Answer

#### Linear time-invariant system
A system is **LTI** if it obeys:
- **Linearity (superposition):** $a x_1(t)+b x_2(t)\to a y_1(t)+b y_2(t)$.
- **Time invariance:** a time shift of the input gives the same shift of the output: $x(t-t_0)\to y(t-t_0)$.

Such a system is fully described by its impulse response $h(t)$; the output is $y(t)=x(t)*h(t)$ and $Y(f)=H(f)X(f)$.

#### Significance in communication engineering
1. **Simple analysis:** knowing only $h(t)$ or $H(f)$, the output for any input can be found by convolution or multiplication. This makes channel and filter analysis easy.
2. **Frequency-domain design:** sinusoids are eigenfunctions of LTI systems; a sinusoid passes with only a change of amplitude $\lvert H(f)\rvert$ and phase $\angle H(f)$. Filters, equalizers and channels are therefore designed using $H(f)$.
3. **Model of channels:** cables, free space (over short time) and most filters are well modelled as LTI, giving tools for distortion, bandwidth and delay calculations.
4. **Distortionless transmission:** the condition $H(f)=Ke^{-j2\pi ft_d}$ (flat amplitude, linear phase) is stated for LTI systems.
5. **Noise analysis:** for a random input, $S_y(f)=\lvert H(f)\rvert^2S_x(f)$, used to find output noise power of receivers.
6. **Superposition of signals:** in FDM many signals pass through the same channel without creating new frequencies; non-linear systems would create intermodulation.

**Example:** a telephone channel modelled as an LTI band-pass filter (300–3400 Hz) lets the designer predict exactly which speech frequencies pass and how much delay each suffers.

## Check yourself

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

1. An LTI system is fully described by:
   A. Its input only
   B. Its impulse response h(t) or H(f)
   C. Its noise power
   D. Its carrier
   Answer: B. Output is x*h, or X H in frequency. (Key point: Description)

2. Why is frequency-domain design convenient for LTI systems?
   A. Sinusoids pass with only amplitude and phase change
   B. Sinusoids become square waves
   C. They produce harmonics
   D. Only DC passes
   Answer: A. Sinusoids are eigenfunctions. (Key point: Eigenfunction)

3. Output PSD of an LTI filter for input PSD S_x is:
   A. S_x |H|
   B. |H(f)|^2 S_x(f)
   C. S_x / |H|^2
   D. S_x + |H|^2
   Answer: B. S_y=|H|^2 S_x, used for output noise power. (Key point: Noise analysis)

4. Why do FDM signals pass through an LTI channel without mixing?
   A. It adds new frequencies
   B. It squares the signals
   C. It creates no new frequencies, unlike a non-linear system which causes intermodulation
   D. It removes all carriers
   Answer: C. Superposition holds, so no intermodulation. (Key point: FDM)

## Related in Communication Systems

- [LTI system output in time domain](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Alti-output-time-domain) — Communication Systems, chapter 1
- [LTI output of a WSS input is WSS](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Alti-wssp-output) — Communication Systems, chapter 9
- [Parseval's theorem; output PSD of LTI system](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aparseval-power-lti-psd) — Communication Systems, chapter 1
- [Hilbert transformation; properties of LTI system](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahilbert-define-lti-properties) — Communication Systems, chapter 1
- [Periodic and non-periodic signals; LTI output is convolution](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aperiodic-nonperiodic-convolution) — Communication Systems, chapter 1
- [Analog and digital system block diagrams; linear and non-linear distortion](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aanalog-digital-block-linear-nonlinear-distortion) — Communication Systems, chapter 1

## Related in other subjects

- [Is y[n] = nx[n] − x[n−1] time invariant](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Atime-invariance-check) — Signals and Systems, chapter 6, asked 3×
- [Control system and its significance in engineering](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Awhat-is-cs-significance) — Control Systems, chapter 1
- [LTI: causality and time variance](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-causality-time-variance) — Signals and Systems, chapter 5, asked 2×
- [LTI system and closed loop in modern systems](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Alti-define-cl-pronounced) — Control Systems, chapter 1