Essential Facts
About Fourier Series
(Otherwise:
Everything you wanted to know about Fourier Series but were afraid to ask)
When Fourier discovered/invented the Series named after him he did two things -
things that you need to remember.
-
Fourier discovered that a
periodic signal could be expressed mathematically as a sum of sines and cosines.
Each sine or cosine is multiplied by a coefficient, and then everything is added
together.
-
Fourier not only discovered
that the signal could be expressed mathematically as a sum of sines and cosines,
he also discovered formulas that let you get the coefficients of the sines and
cosines.
If you know what those two facts mean, then you are well on your way to
understanding Fourier Series.
The Fourier
Sum
Here is the expression that Fourier found for a periodic signal.
In this expression note the
following.
-
This expression can be used to
represent any periodic signal.
-
The sum could have an infinite
number of terms.
-
All terms are at an integral
multiple of a fundamental frequency:
The Fourier
Coefficients
Fourier also figured out a way to compute the coefficients in the series.
  
There is one exception to
the rule:
And that is it. That
is not to say that doing the integrals will be easy. It might not be, and
you might need to do the integration numerically, especially if you have
numerical data. That is another topic.
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