A Note About Fourier
Coefficients - 2
In a Fourier Series, there
are two terms in the nth harmonic - a cosine term and a sine
term. Together they give you the components of the signal at that frequency,
i.e. the nth harmonic. Writing them out we have:
ancos(nwot)
+ bnsin(nwot)
= total component at the nth harmonic.
Now, the a's and b's can be
computed starting with the definitions.
 
And, it is possible to compute the
a's and b's by approximating those integrals. However, that isn't necessarily
the way it is done. In most cases a different representation is used. Consider
the following.
In the integral above, the
function, f(t), is multiplied by a complex exponential. However, the complex
exponential can be represented as a complex sum of the cosine and the sine.
ejn2pt/T= cos(n2pt/T)
+ jsin(n2pt/T) Using that representation gives
the two integrals above. (And, note the presence of "j" multiplying the sine in
the integral above, and in the expression for the complex exponential. That is
what makes it complex.)
Now, the neat result from
this is that you can do one integral and get both the a's and b's
simultaneously. In the fft functions in any analysis program that is what
happens most of the time.
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