Fourier Analysis

In the early 19th century, the French mathematician Jean-Baptiste Fourier proved that any reasonably behaved periodic function, g(t) with period T, can be constructed as the sum of a (possibly infinite) number of sines and cosines:

where f = 1/T is the fundamental frequency, an and bn are the sine and cosine amplitudes  of the nth harmonics (terms), and c is a constant. Such a decomposition is called a Fourier series.From the Fourier series, the function can be reconstructed. That is, if the period, T, is known and the amplitudes are given, the original function of time can be found by performing the sums of First Eq.

A data signal that has a finite duration, which all of them do, can be handled by just imagining that it repeats the entire pattern over and over forever (i.e., the interval from T to 2T is the same as from 0 to T, etc.).

The an amplitudes can be computed for any given g(t) by multiplying both sides of First Eq. by sin(2πkft) and then integrating from 0 to T. Since

only one term of the summation survives: an. The bn summation vanishes completely. Similarly, by multiplying First Eq. by cos(2πkft) and integrating between 0 and T, we can derive bn. By just integrating both sides of the equation as it stands, we can find c. The results of performing these operations are as follows: