Digital Signal Processing Reference
In-Depth Information
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
x
(0
x
(1
X
(0
X
()
4
W
8
0
1
X
()
2
x
()
2
W
8
2
1
1
X
(6
X
x
(3
x
()
W
8
0
W
8
0
1
(1
4
W
8
0
1
X
(5
x
(5
x
(6
x
()
W
8
0
W
8
1
W
8
2
W
8
3
1
1
X
(3
W
8
0
W
8
2
1
1
1
X
()
7
7
W
8
0
1
1
FIGURE 4.43
The eight-point IFFT using decimation-in-time.
EXAMPLE 4.14
Given a sequence xðnÞ for 0 n 3, where xð0Þ¼1, xð1Þ¼2, xð2Þ¼3, and xð3Þ¼4, evaluate its DFT X ðkÞ
using the decimation-in-time FFT method.
Solution:
Using the block diagram in
Figure 4.42
leads to the result shown in
Figure 4.44
.
4
2
6
2
10
22
x
(01
X
(0
j
x
(23
X
(1
W
4
0
1
2
22
1
x
()12
x
X
()
2
j
W
4
0
1
Wj
4
1
1
(34
X
(3
W
4
0
1
1
1
FIGURE 4.44
The four-point FFT using decimation-in-time.
EXAMPLE 4.15
Given the DFT sequence X ðkÞ for 0 k 3 computed in Example 4.14, evaluate its inverse DFT xðnÞ using the
decimation-in-time FFT method.
Solution:
Using the block diagram in
Figure 4.43
yields
Figure 4.45
.
1
4
1
4
1
4
1
4
8
12
4
j
4
4
8
12
16
X
(0 0
x
(01
X
()
2
2
W
4
0
x
()12
1
1
X
()122
j
x
(23
W
4
0
1
~
Wj
4
1
X
()322
j
x
(34
W
4
0
1
1
1
1
FIGURE 4.45
The four-point IFFT using decimation-in-time.
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