MAE591 Random Data
FFT
It is convenient to start with
n
= 0 to define the finite Fourier transform of sampled data, i.e.
(
)
∑
−
=
−
=
1
0
)
2
exp
)
(
)
,
(
N
n
fnh
j
n
x
h
T
f
X
π
At the discrete frequencies of
Nh
k
T
k
f
k
=
=
,
1
,
......
,
3
,
2
,
1
,
0
−
=
N
k
, the discrete Fourier
transform (DFT) is often defined as
1
∑
−
=
=
=
1
0
)
(
)
(
)
,
(
N
n
k
k
kn
W
n
x
h
T
f
X
X
,
1
,
......
,
3
,
2
,
1
,
0
−
=
N
k
.
and the inverse discrete Fourier transform (IDFT) as
∑
=
−
=
∆
=
N
k
k
n
kn
W
X
N
t
n
x
x
1
)
(
1
)
(
,
1
,
......
,
3
,
2
,
1
,
0
−
=
N
n
where
h
has been included with
to have a scale factor of unity before the summation.
Here
)
(
k
f
X
π
−
=
N
kn
j
kn
2
exp
)
(
W
, having the properties such as
i
integer
for
iN
W
u
W
N
u
W
v
W
u
W
v
u
W
1
)
(
),
(
)
(
),
(
)
(
)
(
=
=
+
=
+
.
Note that
N
2
multiplyadds operations are needed for computation of
N
point discrete Fourier
transforms. Fast Fourier Transform (FFT) can reduce the computational effort to 2
pN
multiplyadds operations, where
N
= 2
p
. Thus the speed ratio between the FFT and the
converntional DFT becomes
p
N
4
.
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 Spring '09
 .
 0 W, Fourier T Nh, 00 01 W

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