Digital Signal Processing Reference
In-Depth Information
Fig. 4.1 Periodic function
a, its Fourier series
coefficients b and absolute
values of the complex Fourier
series coefficients c
(a)
x
(
t
)
A
t
2
T
0
τ
T
0
(b)
a
n
0.2
0
-0.2
0
1
2
3
4
5
6
n
b
0.2
0
-0.2
0
1
2
3
4
5
6
n
(c)
0.4
c
n
0
n
-10
-5
0
5
10
where
Z
T
0
c
n
¼
1
2
Þ ¼
1
T
0
ð
a
n
jb
n
x
ð
t
Þ
exp
ð
jnx
0
t
Þ
dt
:
0
Example 4.1 Determine the coefficients of Fourier series as well as complex
Fourier series for a periodic time function shown in Fig.
4.1
a.
Solution From Eq.
4.1
one obtains:
;
Z
s
a
n
¼
2
T
0
A cos
ð
nx
0
t
Þ
dt
¼
A
s
T
0
np
sin
2pn
0
;
Z
s
b
n
¼
2
T
0
A sin
ð
nx
0
t
Þ
dt
¼
A
np
s
T
0
1
cos
2pn
0
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