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11.06.1
Chapter 11.06
Theoretical Development of FFT
Introduction
The informal development of FFT (presented in the previous chapter 11.05) has captured all
the essential features and characteristics about FFT. In this chapter, however, FFT [1–5] will
first be presented in a more general fashion (for the case where the number of sampled data
points
N
can be expressed as
r
N
2
=
,
with
r
is an integer number) so that it will facilitate
further refined FFT formulation to the cases where the restriction on
r
N
2
=
can be removed.
FFT Algorithms for
r
N
2
=
Recall Equation (5) from Chapter 11.05
∑
−
=
=
1
0
)
(
)
(
~
N
k
nk
W
k
f
n
C
(5, Ch. 11.05)
where
N
i
e
W
π
2
−
=
(4, Ch 11.05)
Considered the case
4
2
2
2
=
=
=
r
N
In this case, we can express
k
and
n
as 2-bit binary numbers
,
3
,
2
,
1
,
0
=
k
(1)
)
,
(
0
1
k
k
=
)
1
,
1
(
),
0
,
1
(
),
1
,
0
(
),
0
,
0
(
=
,
3
,
2
,
1
,
0
=
n
(2)
)
,
(
0
1
n
n
=
)
1
,
1
(
),
0
,
1
(
),
1
,
0
(
),
0
,
0
(
=
Equations (1, 2) can also be expressed in compact forms, as following:
0
1
2
k
k
k
+
=
(3)
0
1
2
n
n
n
+
=
(4)
where
,
0
,
,
,
0
1
0
1
=
n
n
k
k
or 1
In the new notations, Equation (5) from chapter 11.05 becomes
∑∑
=
=
+
+
=
1
0
1
0
)
2
)(
2
(
0
1
0
1
0
1
0
1
0
1
)
,
(
)
,
(
~
k
k
k
k
n
n
W
k
k
f
n
n
C
(5)

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