Note on the FFT

A Note About Fourier Coefficients.

In a Fourier Series, there are two terms in the n^th harmonic - a cosine term and a sine term. Together they give you the components of the signal at that frequency, i.e. the n^th harmonic. Writing them out we have:

a_ncos(nw_ot) + b_nsin(nw_ot) = total component at the n^th harmonic.

However, you should be aware that there are other ways of writing that component, and that you might get more information or insight using those other forms. Actually, you may be forced to deal with this other representation because this is the representation in which results are returned when using FFT algorithms in applications like Mathcad and Matlab.

Consider this diagram:

In this diagram, we have:

a_n = c_n cos(f_n)
b_n = c_n sin(f_n)

Using these relations, we can write the component at the n^th harmonic.

Total component at the n^th harmonic = a_ncos(nw_ot) + b_nsin(nw_ot)
= [c_n cos(f_n)cos(nw_ot)] + [c_n sin(f_n) sin(nw_ot)]
= c_n [cos(f_n)cos(nw_ot) + sin(f_n) sin(nw_ot)]
= c_n cos(nw_ot + f_n)

Now, we can interpret this result. Here are the conclusions.

c_n is actually the size of the component at the n^th harmonic.
You can get a_n and b_n from c_n if you know the phase angle of the harmonic.

Different phase angles will produce different a's and b's, but the size of the harmonic would stay the same.
Ultimately, that all comes down to the choice of t= 0, i.e. where that point is positioned on the waveform. If the waveform is on an oscilloscope that point could be chosen anywhere.