---
title: "Short note: LTI system"
subject: "Signals and Systems"
chapter: 5
key: "semester_4/signals_and_systems:short-note-lti-system"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/short-note-lti-system
site: BCT Engineering
---

# Short note: LTI system

Signals and Systems · Chapter 5: Continuous-Time Systems

- Asked in: 2076 Baisakh, 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_4/signals_and_systems/short-note-lti-system
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-5

## Question

Write a short note on LTI system.

## Short answer

An LTI system satisfies linearity and time invariance. It is fully described by impulse response h(t); output is convolution x(t)*h(t). Causal iff h(t)=0 for t<0, BIBO stable iff integral of |h(t)| is finite.

## Answer

A **linear time-invariant (LTI) system** is a system that satisfies both:
- **Linearity (superposition):** $a x_1(t) + b x_2(t) \to a y_1(t) + b y_2(t)$.
- **Time invariance:** $x(t - t_0) \to y(t - t_0)$.

**Key results:**
- An LTI system is completely described by its **impulse response** $h(t)$ (or $h[n]$). The output for any input is the convolution
$$
y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau)\,d\tau, \qquad y[n] = \sum_k x[k]h[n-k]
$$
- In frequency domain, $Y(j\omega) = H(j\omega)X(j\omega)$; complex exponentials are eigenfunctions.
- **Causal** iff $h(t) = 0$ for $t < 0$. **BIBO stable** iff $\int |h(t)|\,dt < \infty$.
- Cascade: $h_1 * h_2$ (order can be swapped). Parallel: $h_1 + h_2$.
- Described by linear constant-coefficient differential/difference equations.

Examples: RC circuit, ideal delay, moving-average filter $y[n] = \frac12(x[n] + x[n-1])$.

## Check yourself

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

1. An LTI system satisfies:
   A. Causality and stability
   B. Linearity and memorylessness
   C. Superposition and time invariance
   D. Time invariance and periodicity
   Answer: C. LTI = linear and time-invariant. (Key point: Definition)

2. What gives the output of an LTI system for any input?
   A. Product of x(t) and h(t)
   B. Derivative of h(t)
   C. Sum x(t)+h(t)
   D. Convolution of input with h(t)
   Answer: D. y = x*h. (Key point: Output)

3. Two LTI systems in cascade have overall impulse response:
   A. h1 * h2
   B. h1 + h2
   C. h1 h2
   D. h1 / h2
   Answer: A. Parallel gives h1+h2; cascade gives convolution. (Key point: Cascade)

4. An LTI system is BIBO stable iff:
   A. h(t)=0 for t<0
   B. The integral of |h(t)| is finite
   C. H(0)=0
   D. h(t) is even
   Answer: B. Absolute integrability of h. (Key point: Stability)

## Related in Signals and Systems

- [Frequency response of a CT LTI system](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Act-frequency-response) — Signals and Systems, chapter 5, asked 3×
- [Describe LTI system properties](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-properties-describe) — Signals and Systems, chapter 5, asked 3×
- [Condition for LTI system to be causal](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-causality-condition) — Signals and Systems, chapter 6, asked 3×
- [LTI system: linearity, stability, TI, causality](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-properties-four) — Signals and Systems, chapter 5, asked 2×
- [Causality condition for CT LTI system](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Acausal-ct-lti-condition) — Signals and Systems, chapter 5
- [Short note: Invertibility of LTI system](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Ashort-note-invertibility) — Signals and Systems, chapter 5

## Related in other subjects

- [Define LTI system and impulse response](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Alti-impulse-response-define) — Data Communication, chapter 2, asked 2×
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- [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