The fundamental & the higher frequencies(harmonics) generate periodic signals from the original wave. And every periodic signal can be written as a sum of the variuos harmonics using the Fourier series.

Hence, to find the various harmonics using the fourier series, we can use...

nth harmonic : (a_ncosx+b_nsinx)

where,

&

where p is the number of unique values of the function y. The following example will make things a bit more clear...

Example : y is a function of x periodic with period 2pi. Some experimental values of y are given below calculated for certain values of x. Expand y to 2 harmonics.

Solution :

Clearly, in the above, p=6,

& We simply need to find:

1st harmonic + 2nd harmonic = (a₁cosx+b₁sinx) + (a₂cos2x+b₂sin2x)

So, all we need is a₁, b₁, a₂ &
b₂

for which we use the formula mentioned above:

&

where x_i=0, 60, 120... & so on.