We could also perform experiments on the computer where we caused the weights to vary with frequency, thus, allowing us to design and placedigital filters on the seismometers to optimize the response of the array to earthquake signals while suppressing seismic noise associated with nearbycities and other sources of seismic noise.
A general purpose mathematical transform
I mention all of this simply to illustrate the general nature of the Fourier transform. Once again, the Fourier transform is simply amathematical process that can be used to transform a set of complex values in one domain into a set of complex values in a different domain.
Before getting into the details of this discussion, I want to refer you to a couple of excellent references on the FFT. Of course, you can find manymore by performing a Google search for the keyword FFT.
Fourier transform images
Many of the images that you will see in this module were produced using an applet named FftLab that I originally downloaded from a website named sepwww.stanford.edu/oldsep/hale/FftLab.html. (As of October 2015, that website no longer exists. However, you can now downloadthe applet from (External Link) . Also, as of October 2015, you can learn more about the applet's author, DaveHale here .)
In order to use the applet to create the illustrations for this document, I changed the name of the applet class to cause it to fit into my file-naming scheme. I alsomade a couple of minor modifications to the code to force the applet's output to fit into this narrow publication format. Otherwise, I used the applet in its originalform. This applet is extremely useful in performing FFT experiments very quickly and easily. I strongly recommend that you become familiar with it.
As an alternative to downloading the applet from the web page given above, click here to download a zip file containing the modified source code for the applet along with a Windowsbatch file that will compile and execute the applet. ((Assuming that you have the Java Development Kit (JDK) and Oracle's appletviewer program installed on your computer, you should be able to simply extract thecontents of the zip file into an empty folder and double-click on the batch file to run the applet.)
Will discuss underlying concepts
As mentioned earlier, the FFT algorithm is very complicated. I won't discuss the mechanics of the algorithm in this module. Rather, I willexplain the underlying concepts that make the FFT algorithm possible.
Hopefully after reading my explanation of the basic concepts, you will be able to understand the explanation of the mechanics of the algorithmprovided by others.
A linear transform
The FFT algorithm is an algorithm that takes advantage of several reasonably well-know facts along with some less well-known facts.
One of those facts is that the Fourier transform is a linear transform. By this, I mean that the transform of the sum of two or more input series isequal to the sum of the transforms of the individual input series. I will attempt to illustrate this in Figure 1 , Figure 2 , and Figure 3 .