Digital Signal Processing Reference
In-Depth Information
where
x
(7) represent the intermediate output sequence after the
first iteration, which becomes the input to the second stage.
¢
(0),
x
¢
(1),...,
x
¢
Stage 2
¢
()
+
()
=Æ≤
()
¢
()
+
()
=Æ≤
()
¢
()
-
()
x
0
x
2
2
x
0
x
1
x
3
2
x
1
[
]
=Æ≤
()
x
0
x
2
W
0
0
x
2
[
¢
()
-
()
]
=Æ≤
()
¢
()
+
()
=-Æ≤
()
2
x
1
x
3
W
0
x
3
x
4
x
6
1
j
x
4
¢
()
+
()
=
(
)
+-
(
)
=-
Æ ≤
()
x
5
x
7
0 707
.
-
j
0 707
.
0 707
.
-
j
0 707
.
j
1 41
.
x
5
[
¢
()
-
()
]
=+Æ≤
()
x
4
x
6
W j
0
1
x
6
[
¢
()
-
()
]
Æ ≤
()
x
5
x
7
Wj
2
=-
1 41
.
x
7
The resulting intermediate, second-stage output sequence
x
≤
(0),
x
≤
(1),...,
x
≤
(7)
becomes the input sequence to the third stage.
Stage 3
()
=≤
()
+≤
()
=
()
=≤
()
-≤
()
=
()
=≤
()
+≤
()
=
()
=≤
()
-≤
()
=
()
=≤
()
+≤
()
=-
Xx
0
0
x
1
4
Xx
4
0
x
1
0
Xx
2
2
x
3
0
Xx
6
2
x
3
0
(
)
+-
(
)
=-
Xx
1
4
x
5
1
j
j
141
.
1
j
241
.
()
=≤
(()
-≤
()
=+
()
=≤
()
+≤
()
=+
Xx
5
4
x
510 1
j
.
(
)
+-
(
)
=-
Xx
3
6
x
7
1
j
j
141
.
1
j
041
.
()
=≤
()
-≤
()
=+
Xx
7
6
x
712 1
j
.
We now use the notation of
X
's to represent the final output sequence. The values
X
(0),
X
(1),...,
X
(7) form the scrambled output sequence. These results can be
verified with MATLAB, as described in Appendix D. We show later how to reorder
the output sequence and plot the output magnitude.
Exercise 6.2: Sixteen-Point FFT
Given
x
(0)
0, which represents
a rectangular input sequence. The output sequence can be found using the 16-point
flow graph shown in Figure 6.6. The intermediate output results after each stage
are found in a manner similar to that in Exercise 6.1. Eight twiddle constants
W
0
,
W
1
,...,
W
7
=
x
(1)
=
...
=
x
(7)
=
1, and
x
(8)
=
x
(9)
=
...
x
(15)
=
need to be calculated for
N
=
16.
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