Digital Signal Processing Reference
In-Depth Information
4.3
FOURIER SERIES AND FREQUENCY SPECTRA
In this section, we present three examples of Fourier series. These examples lead us
to the important concept of the
frequency spectra
of periodic signals. Then, a table
of Fourier series for some common signals is given.
Fourier series of a square wave
EXAMPLE 4.2
Consider the square wave of Figure 4.4. This signal is common in physical systems. For ex-
ample, this signal appears in many electronic oscillators as an intermediate step in the gener-
ation of a sinusoid.
We now calculate the Fourier coefficients of the square wave. Because
V, 06 t 6 T
0
/2
-V, T
0
/2 6 t 6 T
0
,
x(t) =
b
from (4.23), it follows that
1
T
0
L
T
0
x(t)e
-jkv
0
t
dt
C
k
=
T
0
/2
T
0
V
T
0
L
V
T
0
L
e
-jkv
0
t
dt -
e
-jkv
0
t
dt
=
0
T
0
/2
T
0
/2
T
0
V
T
0
(-jkv
0
)
e
-jkv
0
t
- e
-jkv
0
t
=
B
R
.
0
T
0
/2
The values at the limits are evaluated as
2p
T
0
T
0
v
0
t
t =T
0
/2
=
2
= p; v
0
T
0
= 2p.
Therefore,
jV
2pk
(e
-jkp
- e
-j0
- e
-jk2p
+ e
-jkp
)
C
k
=
x
(
t
)
V
T
0
/2
0
T
0
/2
T
0
3
T
0
/2
t
V
Figure 4.4
Square wave with amplitude
V
.
