When Fourier discovered/invented the Series named after him he did two things - things that you need to remember.

• Fourier discovered that a periodic signal could be expressed mathematically as a sum of sines and cosines.  Each sine or cosine is multiplied by a coefficient, and then everything is added together.
• Fourier not only discovered that the signal could be expressed mathematically as a sum of sines and cosines, he also discovered formulas that let you get the coefficients of the sines and cosines.

        Here is the expression that Fourier found for a periodic signal.

• This expression can be used to represent any periodic signal.
• A periodic signal repeats.  Say the time for a repetition is T seconds.  Then, if the periodic signal is f(t), we would have:
• f(t+T) = f(t)
• The sum could have an infinite number of terms.
• All terms are at an integral multiple of a fundamental frequency:
• f_o = 1/T = fundamental frequency (Hertz)
• w_o = 2pf_o = fundamental angular frequency (radians/second)
• The multiples of the fundamental frequency are call the harmonics.

        Fourier also figured out a way to compute the coefficients in the series.

There is one exception to the rule:

And that is it.  That is not to say that doing the integrals will be easy.  It might not be, and you might need to do the integration numerically, especially if you have numerical data.  That is another topic.