An Example Using the Fourier
Transform
The following example takes a function and uses the Fourier Transform to figure out the transform of the function. This example also uses Maple 7. Some of the code is left out here as well. You can still follow the example without it.
The notation for the Fourier Transform is indicated below.
The variable t is the variable used to indicate time. The variable w is this case is used to indicate frequency. We can consider the piecewise function below.
The graph of this piecewise function is shown below.
This function has compact support. This means that the function has a finite part that is nonzero. This helps us when we go to integrate the function in order to find the Fourier Transform of it. When a function has compact support we only have to integrate it on its compact support. Therefore, in our case we are going to be integrating our function as follows:
Looking at the above integration, we can see that all we have done is substitute in f(t) and change our integration to the compact support interval of [0,4]. Finally, we can compute the integral to obtain
We now have the Fourier Transform of our function. This allows us to work with frequency instead of time.