The Fourier Series is a special case of the Fourier transform
which can be used when the signal is periodic. The basic idea of the
Fourier series is that a periodic
function with period 
could be described by a weighted sum of cosine and
sine functions, i.e.,
where 
.
As shown below the Fourier series can be derived from the Fourier Transform.
The Fourier series is a special case of a more general class of transforms
known as orthogonal transforms. The orthogonal functions used in the Fourier
Series are 
and 
or 
where 
. It can be shown that
and that
Hence, the iEuler functions form an orthonormal set.
Probably the most common form of the iFourier series is,
where 
and 
are,
and 
.
We can also write the Fourier Series as,
where 
and 
are,
where 
and 
are defined above.
An alternate form (known as the exponential Fourier Series) is defined as
where 
and 
is the fundamental period. The
fundamental period is the minimum value of 
for which 
for
all t. 
(which we could write as 
) is
where 
is arbitrary.
and 
and 
are related by the following:
Note that 
.