A simple example is presented in this page. Both the functions, h(t) and f(t), are pulses.
Given the two waveforms shown in Fig. 17, obtain the convolution integral of h(t) and f(t). Let
y(t) = f(t) * h(t).
SOLUTION :
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. From Fig. 17,
Replace t by λ. Then we get that
Function h(t - λ) is a waveform that is obtained by reflecting h(t) and shifting to the right as the value of t increases from zero onwards. That is,
For t < 0, h(t - λ) lies on the left side of vertical axis and the falling edge of the moving pulse, h(t - λ) would not have reached the rising edge of the stationary pulse f(t) which has a value of unity for values of t varying within the range bounded by zero and one. It can be seen from the equation for h(t - λ) that it has a value of unity when (t - 1) < λ< t. For 0 < t < 1, the falling edge of the moving pulse lies within the time-frame of the stationary pulse, whereas the rising edge of the moving pulse has not yet reached the rising edge of the stationary pulse f(t). For 1 < t < 2, the rising edge of the moving pulse lies within the time-frame of the stationary pulse, whereas the falling edge of the moving pulse has crossed and gone beyond the falling edge of the stationary pulse f(t), which is at t = 1. For t > 2, the rising edge of the moving pulse, h(t - λ) has also gone past the falling edge of the stationary pulse f(t). There is overlap between the moving pulse and the stationary pulse for values of t defined by 0 < t < 2.
The convolution integral is evaluated as shown below.
The plot of the convolution integral is shown in Fig. 17.
Laplace Transforms Approach
The Laplace transform of a time-shifted pulse is presented below.
Hence for this problem,
The inverse transform of Y(s) yields the convolution integral.
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.