Signals and Systems with MATLAB
Computing and Simulink
Modeling, Fourth Edition
7
−
37
Copyright
©
Orchard Publications
Line Spectra
Figure 7.35.
Line spectrum for half
−
wave rectifier, Page 7
−
17
The line spectra of other waveforms can be easily constructed from the Fourier series.
Example 7.4
Compute the exponential Fourier series for the waveform of Figure 7.36, and plot its line spectra.
Assume
.
Solution:
This recurrent rectangular pulse is used extensively in digital communications systems. To deter-
mine how faithfully such pulses will be transmitted, it is necessary to know the frequency compo-
nents.
Figure 7.36.
Waveform for Example 7.4
As shown in Figure 7.36, the pulse duration is
. Thus, the recurrence interval (period)
, is
times the pulse duration. In other words,
is the ratio of the pulse repetition time to the dura-
tion of each pulse. For this example, the components of the exponential Fourier series are found
from
(7.110)
The value of the average (
component) is found by letting
. Then, from (7.110) we
obtain
or
n
ω
t
0
1
2
4
6
8
A
/
π
A
/
2
DC
ω
1
=
0
−π/κ
2
π
T
ω
t
A
π
π/κ
T
/
κ
−π
−
2
π
T
k
⁄
T
k
k
C
n
1
2
π
------
Ae
jnt
–
t
d
π
–
π
∫
A
2
π
------
e
jnt
–
t
d
π
k
⁄
–
π
k
⁄
∫
=
=
DC
n
0
=
C
0
A
2
π
------t
π
k
⁄
–
π
k
⁄
A
2
π
------
π
k
--
π
k
--
+
=
=

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Chapter 7
Fourier Series
7
−
38
Signals and Systems with MATLAB
Computing and Simulink
Modeling, Fourth Edition
Copyright
©
Orchard Publications
(7.111)
For the values for
, integration of (7.110) yields
or
(7.112)
and thus,
(7.113)
The relation of (7.113) has the
form, and the line spectra are shown in Figures 7.37
through 7.39, for
,
and
respectively by using the MATLAB scripts below.
fplot('sin(2.*x)./(2.*x)',[
−
4
4
−
0.4
1.2])
fplot('sin(5.*x)./(5.*x)',[
−
4
4
−
0.4
1.2])
fplot('sin(10.*x)./(10.*x)',[
−
4
4
−
0.4
1.2])
Figure 7.37.
Line spectrum of (7.113) for
C
0
A
k
---
=
n
0
≠
C
n
A
jn2
–
π
--------------e
jnt
–
π
k
⁄
–
π
k
⁄
A
n
π
------
e
jn
π
k
⁄
e
jn
π
k
⁄
–
–
j2
-------------------------------------
⋅
A
n
π
------
n
π
k
------
sin
⋅
A
n
π
k
⁄
(
)
sin
n
π
-------------------------
=
=
=
=
C
n
A
k
---
n
π
k
⁄
(
)
sin
n
π
k
⁄
-------------------------
⋅
=
f t
( )
A
k
---
n
π
k
⁄
(
)
sin
n
π
k
⁄
-------------------------
⋅
n
∞
–
=
∞
∑
=
x
sin
x
⁄
k
2
=
k
5
=
k
10
=
-4
-3
-2
-1
0
1
2
3
4
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
K=2
k
2
=