The Trigonometric Fourier Series

This infinite series separates a periodic waveform into its DC and sine wave components. The constants a_o, a_n and b_n tells us how much of each component exists in f(t) and they are calculated by using a type of a correlation function. Click here to see these functions used to calculate the Fourier coefficients.

Here is an example:  Lets say I have a square wave, square(t), with a fundamental frequency of   f  and I want to "check to see how much of sin(f*t) is in square(t)"  From the above equation we know that the coefficient b₁ must be solved to answer this question. In this case b₁= 4/(pi*f) which simply tells us that the magnitude of sin(f*t) in square(t) is 4/(pi*f).   And if we decide to "check to see how much of sin(2f*t) is in square(t)" we will find that there is no correlation (i.e.b₂ = 0).   

Visually, this is much easier to see. The graphs below show square(t), sin(f*t) and sin(2f*t) next to each other.  If the average of square(t)*sin(f*t) is taken one can see from inspection that the result is non-zero, likewise it is just as easy to see that the average of square(t)*sin(2f*t) is indeed zero.