A simple example is presented in this page. Both the functions, h(t) and f(t), are continuous for t > 0.
Given the two waveforms shown in Fig. 16, obtain the convolution integral of h(t) and f(t). Let
y(t) = f(t) * h(t).
SOLUTION :
Differential Equations Approach
Given the impulse response, we can get the differential equation of the system. The response to an impulse input is called as h(t).
When the input is a ramp, the above equation can be expressed as follows, where the response to a ramp input is called as y(t).
Let the initial condition be zero. The response obtained is presented below.
Laplace Transforms Approach
Given h(t) and f(t), the Laplace transform expressions for h(t) and f(t) can be obtained as presented below.
Then the response is obtained as shown below.
Convolution Integral Approach: Analytical Technique
Given h(t) and f(t), the convolution integral of h(t) and f(t) can be obtained as presented below.
Since the convolution integral has commutative property, we can obtain the response as shown below.
Convolution Integral Approach: Graphical Technique
An applet is presented below for illustrating the graphical technique.
This page has presented an example. The subsequent pages contain additional examples.