Continuous Time Convolution Properties | Continuous Time Signal

This article discusses the convolution operation in continuous-time linear time-invariant (LTI) systems, highlighting its properties such as commutative, associative, and distributive properties. It also explores the convolution integral and its application in determining the system’s output response.

Continuous Time Convolution

For linear time-invariant (LTI) systems, the convolution is being utilized in order to achieve output response from the knowledge of input and impulse response.

Given two continuous-time signals x(t) and h(t), the convolution is defined as

$y\left( t \right)=\sum\limits_{\tau =-\infty }^{\infty }{x\left( \tau \right)h\left( t-\tau \right)d\tau }~~~~~~~~~~~~~~~~~~~~~~~~\left( 1 \right)$

The integral defined in above equation is an important and fundamental equation in the study of the linear system; it is called the convolution integral. Because it is so often used, it has been given a special shorthand representation;

$Continuous Time Convolution$

It should be noted that convolution integral exists when x(t) and h(t) are both zero for all integers t <0. If x(t) and h(t) are zero for all integers t<0, then x(𝞃)=0 for all integers 𝞃<0 and h(t-𝞃) =0 for all integers t-𝞃<0. Thus the integral on 𝞃 in equation (1) may be taken from 𝞃=0 to 𝞃=t, and the convolution operation is given by;

$x\left( t \right)*h\left( t \right)=\left\{ \begin{matrix} 0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~t=-1,-2,\ldots \\ \underset{0}{\overset{t}{\mathop \int }}\,x\left( \tau \right)h\left( t-\tau\right)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~t=0,1,2,\ldots ~~~~~~~~~~\left( 2 \right)~~~~~~~ \\\end{matrix} \right.$

Since the integral in equation (2) is over the finite range of integers (𝞃=0 to 𝞃=t), the convolution integral exists. Hence, any two signals that are zero for all integers t<0 can be convolved.

Continuous-Time Convolution Properties

The convolution mapping possesses a number of important properties, among those are:

Commutative Property

If x(t) is a signal and h(t) and impulse response, then

$\large x(t)*h(t)=h(t)*x(t)$

An LTI system output with input x(t) and impulse response h (t) is same as an LTI system output with input h(t) and impulse response x(t).

Associative Property

If x(t) is a signal and h₁(t) and h₂(t) are impulse responses, then

$\fn_jvn \large x(t)*(h_1 (t)*h_2 (t))=(x(t)*h_1 (t))*h_2 (t)$

The order of convolution is not important.

Distributive Property

If x(t) is signal and h₁(t) and h₂(t) are impulse responses, then

$\large x(t)*(h_1 (t)+h_2 (t))=x(t)*h_1 (t)+x(t)*h_2 (t)$

LTI systems parallel combination can be substituted with a single LTI system whose unit impulse response is the summation of the separate unit impulse responses in the parallel combination.

You May Also Read: Continuous-Time Graphical Convolution Example

Continuous Time Convolution Properties Key Takeaways

The properties of convolution—commutative, associative, and distributive—play a crucial role in the practical applications of continuous-time linear time-invariant (LTI) systems. These properties simplify the analysis and design of systems, making it easier to predict system behaviors and responses. The convolution integral, which is central to determining output responses from known input and impulse responses, is widely used in fields such as signal processing, control systems, and communication systems.