If you haven’t been able to visualize a sum of sine waves, the video will be a real eye-opener. The second part shows how those circles moving together describe a square wave with ripple. The first part of the video makes the point (visually) that a sine wave is really just a point on a rotating circle moving through time. If you don’t have MATLAB yourself, you can always check out the video (see below). But can you visualize why the transform works the way it does? If you can’t (or even if you can), you should check out MATLAB visualization of harmonic circles. Like a lot of abstract concepts, it is easy to understand the basic premise and you could look up any of the mathematical algorithms that can take a signal and perform a Fourier transform on it. Of course, to get a perfect square wave, you need an infinite number of odd harmonics, but in practice only a few will do the job. A square wave of frequency F can be made with a sine wave of frequency F along with all of its odd harmonics (that is, 3F, 5F, 7F, etc.). Conversely, you can generate any signal by adding up a bunch of sine waves. If you do any electronics work–especially digital signal processing–you probably know that any signal can be decomposed into a bunch of sine waves.
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