Digital Signal Processing Reference
In-Depth Information
Choosing the duration of one period, N ¼ 4, we have the following sample values:
xð0Þ¼0; xð1Þ¼1; xð2Þ¼0; and xð3Þ¼1
xð0Þþxð1Þþxð2Þþxð3Þ
X
3
c
0
¼
1
4
xðnÞ¼
1
4
¼
1
4
ð0 þ 1 þ 0 1Þ¼0
n ¼0
xð0Þþxð1Þe
jp=2
þ xð2Þe
jp
þ xð3Þe
j3p=2
X
3
c
1
¼
1
4
xðnÞe
j2p1n=4
¼
1
4
n ¼0
xð0Þjxð1Þxð2Þþjxð3Þ¼0 jð1Þ0 þ jð1Þ
¼
1
4
¼j0:5
Similarly, we get
X
3
X
3
c
2
¼
1
4
xðnÞe
j2p2n=4
¼ 0; and c
3
¼
1
4
xðkÞe
j2p3n=4
¼ j0:5
k ¼0
n ¼0
Using periodicity, it follows that
c
1
¼ c
3
¼ j0:5; and c
2
¼ c
2
¼ 0
b. The amplitude spectrum for the digital signal is sketched in
Figure 4.5
.
c
0.
0.
0.
0 .
0 .
0.
2
4
f
Hz
0
5
1
5
4
3
2
1
3
f
/2
2
s
f
4
s
FIGURE 4.5
Two-sided spectrum for the periodic digital signal in Example 4.1.
As we know, the spectrum in the range of 2 to 2 Hz presents the information of the sinusoid with a frequency
of 1 Hz and a peak value of 2j:c
1
j: ¼ 1, which is obtained from converting two sides to one side by doubling the
two-sided spectral value. Note that we do not double the direct-current (DC) component, that is, c
0
.
4.1.2
Discrete Fourier Transform Formulas
Now let us concentrate on development of the DFT.
Figure 4.6
shows one way to obtain the DFT
formula.
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