Continuous Time Convolution Properties | Continuous Time Signal

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;

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(?)=0 for all integers ?<0 and h(t-?) =0 for all integers t-?<0. Thus the integral on ? in equation (1) may be taken from ?=0 to ?=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 (?=0 to ?=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

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

The order of convolution is not important.                     

Distributive Property

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

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.

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