The Fourier analysis I did was a plot of x(angle) versus sin(x) and its sum with successive iterations of (1/n)sin(nx), where n represents the odd order harmonics 1,3,5,... . For example, the first harmonic would be represented by (1/3)sin(3x). There are MANY fourier series. I was simply plotting the most basic mathematical concept of a Fourier series in an effort to enhance my own understanding. Learning the math behind these types of waveforms is very enlightening.

What you are referring to is basically the field of harmonic analysis.

In order to analyze an MP3 file, I presume you would need a method of converting the signals (from a transmission or recording) into data points (e.g., a spreadsheet) for analysis. I currently don’t know how to do this. You would then need to perform harmonic analysis of the data. Although I recently completed the calculus series through differential equations, this is a VERY complex mathematical subject with which I do not currently have a full grasp. Unfortunately, Fourier series were not covered in the Calculus III course I attended, so I am trying to learn more about them on my own.

I am sure there are software programs in existence that can perform harmonic analysis. But this can be complicated by the fact that signals in the real world may also be carrying distortions, interference, over-amplifications, etc. that will make the overall signal very difficult to analyze. So the numerical analysis may end up involving approximations at best. Do you know what fundamental frequency is claimed for the signal you are trying to analyze, or is it of a totally unknown nature? If you knew what fundamental frequency was claimed, then you could approach this from the opposite direction. You could (with a synthesizer for example) attempt to produce a square wave version of that frequency to see how close it matches.

Mathworld.wolfram.com states "the computation and study of Fourier series is known as harmonic analysis and is extremely useful as a way to break up an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical. . . . In particular, since the superposition principle holds for solutions of a linear homogeneous ordinary differential equation, if such an equation can be solved in the case of a single sinusoid, the solution for an arbitrary function is immediately available by expressing the original function as a Fourier series and then plugging in the solution for each sinusoidal component. In some special cases where the Fourier series can be summed in closed form, this technique can even yield analytic solutions" < http://mathworld.wolfram.com/FourierSeries.html>.