Digital Signal Processing Reference
In-Depth Information
xN
( )
x
( )
xn
()
x
(1
x
(0
n
0
N
xN x
() (0)
0
T T
FIGURE 4.2
Periodic digital signal.
Z
1
T
0
xðtÞe
jku
0
t
dt
N
< k <
N
c
k
¼
(4.1)
T
0
where
k
is the number of harmonics corresponding to the harmonic frequency of
kf
0
and
u
0
¼
2
p=T
0
and
f
0
¼
1
=T
0
are the fundamental frequency in radians per second and the fundamental frequency in
Hz, respectively. To apply Equation
(4.1)
, we substitute
T
0
¼ NT
,
u
0
¼
2
p=T
0
and approximate the
integration over one period using a summation by substituting
dt ¼ T
and
t ¼ nT
. We obtain
N
1
n¼
0
xðnÞe
j
1
N
2
pkn
N
c
k
¼
;
N
< k <
N
(4.2)
Since the coefficients
c
k
are obtained from the Fourier series expansion in the complex form, the
resultant spectrum
c
k
will have two sides. There is an important feature of Equation
(4.2)
in which the
Fourier series coefficient
c
k
is periodic of
N
. We can verify this as follows:
N
1
n¼
0
xðnÞe
j
N
1
n¼
0
xðnÞe
j
1
N
1
N
2
pðkþNÞn
N
2
pkn
N
e
j
2
pn
c
kþN
¼
¼
(4.3)
Since
e
j
2
pn
¼
cos
ð
2
pnÞj
sin
ð
2
pnÞ¼
1, it follows that
c
kþN
¼ c
k
(4.4)
Therefore, the two-side line amplitude spectrum
jc
k
j
is periodic, as shown in
Figure 4.3
.
DC component kf
o
= 0xf
o
= 0 Hz
c
Other harmonics ...
1st har
m
o
nic kf
o
= 1xf
o
= f
o
Hz
O
t
her harmonics ...
f
0
-f
s
/ 2
-f
0
f
s
/ 2
f
s
- f
0
f
f
s
f
s
+ f
0
f
0
f
s
= Nf
0
Hz
2nd harmonic kf
o
= 2xf
o
= 2f
o
Hz
FIGURE 4.3
Amplitude spectrum of the periodic digital signal.
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