Fourier
Series Example - Square Wave
We are going to examine the Fourier Series for a square wave. The
signal we want to work with is given below in Figure 1.
Figure 1
To compute the Fourier
Series we use the integrals for the Fourier coefficients.
We note the following:
-
f(t) = A for 0 < t
< T/2
-
f(t) = -A for T/2 <
t < T
First, note that the square
wave has odd symmetry so that there are no cosine terms. Then using
the expression for the b's above, we have this expression for bk.
Evaluating these integrals
we have:
First, cancel out the
2/T factor everywhere, and we have:
-
bk =
(-A/kp)[cos(kp)
- 1 - 1 + cos(kp)]
-
And, when k is odd, the
two cos(kp)
terms add up to -2.
-
And, when k is even, the
two cos(kp)
terms give +2
-
The net result is:
-
bk =
(4A/kp)
when k is odd,
-
bk =
0 when k is even.