---
title: "LTI system output in time domain"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:lti-output-time-domain"
source: https://www.bctengineer.com/q/semester_6/communication_systems/lti-output-time-domain
site: BCT Engineering
---

# LTI system output in time domain

Communication Systems · Chapter 1: Introduction

- Asked in: 2071 Bhadra (CS I), 2+5 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-output-time-domain
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

Define LTI system. Find the expression (in time domain) for the output of a LTI.

## Short answer

An LTI system satisfies superposition and time-invariance. Its output is the convolution of input and impulse response: y(t) = ∫ x(τ) h(t−τ) dτ = x(t)*h(t).

## Answer

#### LTI system
A system is **linear time-invariant (LTI)** if it satisfies:
- **Linearity (superposition):** if $x_1(t)\to y_1(t)$ and $x_2(t)\to y_2(t)$, then $a x_1(t)+b x_2(t)\to a y_1(t)+b y_2(t)$.
- **Time invariance:** if $x(t)\to y(t)$ then $x(t-t_0)\to y(t-t_0)$; the system behaves the same at all times.

An LTI system is completely described by its **impulse response** $h(t)$, the output when the input is $\delta(t)$.

#### Output in time domain

```text
 x(t) ----->[  LTI system h(t)  ]-----> y(t)
```

**Step 1 – represent the input by impulses.** Using the sifting property of the impulse, any input can be written as a sum (integral) of shifted, weighted impulses:

$$
x(t)=\int_{-\infty}^{\infty}x(\tau)\,\delta(t-\tau)\,d\tau
$$

**Step 2 – time invariance.** By definition $\delta(t)\to h(t)$, so a shifted impulse gives a shifted response: $\delta(t-\tau)\to h(t-\tau)$.

**Step 3 – homogeneity.** A weighted impulse gives a weighted response: $x(\tau)\delta(t-\tau)\to x(\tau)h(t-\tau)$, since $x(\tau)$ is a constant for a fixed $\tau$.

**Step 4 – superposition.** The input is the integral (sum) of such terms, so the output is the integral of the individual responses:

$$
y(t)=\int_{-\infty}^{\infty}x(\tau)\,h(t-\tau)\,d\tau=x(t)*h(t)
$$

This is the **convolution integral**. By changing the variable ($\lambda=t-\tau$) it can also be written as $y(t)=\int_{-\infty}^{\infty}h(\tau)x(t-\tau)d\tau$.

**Special cases:**
- Causal system ($h(t)=0$ for $t<0$) with input starting at $t=0$: $y(t)=\int_0^t x(\tau)h(t-\tau)d\tau$.
- Frequency domain: taking the Fourier transform gives $Y(f)=H(f)X(f)$, where $H(f)$ is the transfer function.

**Example:** for an RC low-pass filter, $h(t)=\frac{1}{RC}e^{-t/RC}u(t)$. With a unit-step input, $y(t)=\int_0^t\frac{1}{RC}e^{-(t-\tau)/RC}d\tau=1-e^{-t/RC}$, the familiar charging curve.

## Check yourself

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

1. An LTI system must satisfy:
   A. Only linearity
   B. Only time invariance
   C. Linearity and time invariance
   D. Causality and stability
   Answer: C. LTI means superposition plus time invariance. (Key point: LTI definition)

2. Why can any input be written as an integral of impulses?
   A. Sifting property of delta
   B. Sampling theorem
   C. Parseval's theorem
   D. Duality
   Answer: A. x(t)=integral x(tau)delta(t-tau)dtau. (Key point: Sifting property)

3. The time-domain output of an LTI system is:
   A. x(t)h(t)
   B. x(t)*h(t)
   C. x(t)+h(t)
   D. x(t)/h(t)
   Answer: B. The convolution integral. (Key point: Output)

4. For RC low-pass with unit step input, the output is:
   A. e^(-t/RC)
   B. t/RC
   C. 1 - e^(-t/RC)
   D. 1 + e^(-t/RC)
   Answer: C. Integrating (1/RC)e^(-(t-tau)/RC) over 0..t gives the charging curve. (Key point: Example)

## Related in Communication Systems

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