Fourier Series Example - Pulse
        We are going to examine the Fourier Series for repetitive.  The signal we want to work with is given below in Figure 1.

        We will compute the Fourier Series of a general pulse that repeats.  The pulse sequence is shown below.  The pulse signal varies between zero volts and one volt.
 
 

Figure 1

Now, to evaluate the coefficients, we do the integrations indicated above.  We have the following.

or:

an = 2Asin(nwoTp)/(nTwo)

an = Asin(nwoTp)/(np)

Similarly,

or:
bn = 2A[-cos(nwoTp) + 1]/(nTwo)

bn = A[-cos(nwoTp) + 1]/(np)

and,

ao = (Tp/T)

Now, we can compute some of the coefficients for a particular case.  We will examine the situation where the pulse is high for one-fourth of the period, i.e. when Tp = T/4.  In that situation we have:

nwoTp = (n2p/T)Tp = np/2

Note that the a's (the cosine coefficients) will all be zero for even n's, while the b's (the sine coefficients) will be zero for every fourth n.  That being said, the coefficients we have computed are given in the table below.  For this table we have assumed a period of 4 seconds.
 
 

n
an
bn
0
.25
-
1
.31831
.31831
2
0
.31831
3
-.10610
.10610
4
0
0
5
.06366
.06366
6
0
.10610
7
-.04547
.04547
8
0
0
9
.03537
  .03537
10
0
 .06366