- Trending Categories
- Data Structure
- Networking
- RDBMS
- Operating System
- Java
- MS Excel
- iOS
- HTML
- CSS
- Android
- Python
- C Programming
- C++
- C#
- MongoDB
- MySQL
- Javascript
- PHP
- Physics
- Chemistry
- Biology
- Mathematics
- English
- Economics
- Psychology
- Social Studies
- Fashion Studies
- Legal Studies

- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who

# What is Convolution in Signals and Systems?

## What is Convolution?

Convolution is a mathematical tool to combining two signals to form a third signal. Therefore, in signals and systems, the convolution is very important because it relates the input signal and the impulse response of the system to produce the output signal from the system. In other words, the convolution is used to express the input and output relationship of an LTI system.

## Explanation

Consider a continuous-time LTI system which is relaxed at t = 0, i.e., initially,no input is applied to it. Now, if the impulse signal [δ(t)] is input to the system, then output of the system is called the impulse response h(t) of the system and is given by,

$$\mathrm{h(t)=T[\delta(t)]}$$

As any arbitrary signal x(t) can be represented as −

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

Then, the output of the system corresponding to x(t) is given by,

$$\mathrm{y(t)=T[x(t)]}$$

$$\mathrm{\Rightarrow y(t)=T\left [\int_{-\infty}^{\infty}x(\tau) \:\delta (t-\tau )d\tau\right ]}$$

For a continuous-time linear system, the output is given by,

$$\mathrm{y(t)=\int_{-\infty}^{\infty}x(\tau)\:T\left [\delta( t-\tau )\right ]d\tau\:\:\:\:\:\:...(1)}$$

Now, if the response of the system due to impulse signal δ(t) is h(t), then the response of the linear system due to delayed impulse signal is given by,

$$\mathrm{h(t,\tau)=T[\delta(t-\tau)]}$$

By substituting the value of $T[\delta(t-\tau)]$ in the equation (1), we get,

$$\mathrm{y(t)=\int_{-\infty}^{\infty}x(\tau)\:h(t,\tau)d\tau\:\:\:\:\:\:...(2)}$$

Again, for a time-invariant system, the output corresponding to the input delayed by τ units is equal to the output delayed by τ units, i.e.,

$$\mathrm{h(t,\tau)=h(t-\tau)}$$

On putting the value of $h(t,\tau)$ in the equation (2), we have,

$$\mathrm{y(t)=\int_{-\infty}^{\infty}x(\tau)\:h(t-\tau)d\tau\:\:\:\:\:\:...(3)}$$

The expression in equation(3) is called the **convolution integral** or simply **convolution**.

Therefore, the convolution of two continuous-time signals x(t) and h(t) is represented as,

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

The limits of the integration in the convolution integral depends on whether the arbitrary signal x(t) and the impulse response h(t) are causal or not. Therefore,

If both x(t)and h(t) are non-causal, then,

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

If the signal x(t) is non-causal and the impulse response h(t) is causal, then,

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

If the signal x(t) is causal and h(t) is non-causal, then

$$\mathrm{y(t)=\int_{0}^{\infty}x(\tau)\:h(t-\tau)d\tau}$$

If both x(t) and h(t) are causal, then,

$$\mathrm{y(t)=\int_{0}^{t}x(\tau)\:h(t-\tau)d\tau}$$

- Related Articles
- Properties of Convolution in Signals and Systems
- Signals and Systems – Relation between Convolution and Correlation
- What is Correlation in Signals and Systems?
- Signals and Systems – What is Even Symmetry?
- Signals and Systems – What is Odd Symmetry?
- Signals and Systems – What is Inverse Z-Transform?
- Signals and Systems – What is Half Wave Symmetry?
- Signals and Systems – What is Quarter Wave Symmetry?
- Signals and Systems – What is a Linear System?
- Signals & Systems – What is Hilbert Transform?
- Signals and Systems – Classification of Signals
- Signals and Systems: Multiplication of Signals
- Signals and Systems: Even and Odd Signals
- Signals and Systems: Periodic and Aperiodic Signals
- Signals and Systems: Energy and Power Signals