The main contents of this chapter include all of the following: Direct computation of the DFT, decimation-in-time FFT algorithms, decimation-in-frequency FFT algorithms, fourier analysis of signals using the DFT.
Trang 1Digital Signal Processing, X, Zheng-Hua Tan, 2005 1
Digital Signal Processing, Fall 2005
EStudy
-Zheng-Hua Tan
Department of Communication Technology Aalborg University, Denmark
zt@kom.aau.dk
Lecture 10: Fast Fourier Transform
Course at a glance
Discrete-time signals and systems
Fourier-domain
representation
DFT/FFT
System analysis
Filter design z-transform
MM1
MM2
MM9, MM10
MM3
MM6 MM4
MM7, MM8
Sampling and reconstruction MM5
System structure System
Trang 2Digital Signal Processing, X, Zheng-Hua Tan, 2005 3
Digital computation of the DFT
additions as a measure of computational complexity
for the efficient and digital computation of the
N-point DFT, rather than a new transform
1 , , 1 , 0 , ] [
1 ]
0
−
=
=
W k X N n
k
kn N
1 , , 1 , 0 , ] [ ]
0
−
=
=
N k
W n x k
n
kn N
Part I: Direct computation of the DFT
Decimation-in-time FFT algorithms
Decimation-in-frequency FFT algorithms
Fourier analysis of signals using the DFT
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N(N-1) complex additions
Compute and store (only over one period)
Compute the DFT using stored and input
1 , , 1 , 0 ), / 2 sin(
) / 2 cos(
) / 2 (
−
= +
=
N k
N k j
N k
e
W k j N k
N
π π
π
Direct computation of the DFT
1 , , 1 , 0 , ] [ ]
0
−
=
=∑−
=
N k
W n x k
n
kn N
]
[n x
k N
W
complex be
may ] [ and
W k
N
1 , , 1 , 0 , ] [ ]
0
−
=
=
N k
W n x k
n
kn N
]
[n
x X [k]
of X[k] requires 4N real multiplications and (4N-2)
real additions
real multiplications and real additions
symmetry and periodicity properties of
) 2 4
N
Direct computation of the DFT
2
4N
kn N
W
1 , , 1 , 0 }), Re{
]}
[ Im{
} Im{
]}
[
(Re{
}) Im{
]}
[ (Im{
} Re{
]}
[ [(Re{
]
0
−
= +
+
−
=∑−
=
N k
W n
x W
n
x
j
W n
x W
n x k
X
kn N
kn N
N
n
kn N kn
N
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Symmetry and periodicity of complex exponential
The number of multiplications is reduced by a factor of
2
} Im{
} Re{
)
]
N
kn N
kn N
kn N n
N
k
n N k N N
n k N
kn
} Re{
]}) [
Re{
]}
[ (Re{
} Re{
]}
[ Re{
} Re{
]}
[
kn N
n N k N
kn N
W n
N x n
x
W n
N x W
n
x
− +
=
−
Part II: Decimation-in-time FFT algorithms
Direct computation of the DFT
Decimation-in-frequency FFT algorithms
Fourier analysis of signals using the DFT
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FFT
the computation of the DFT that is applicable when
N is a composite number, i.e., the product of two or
more integers Later, it resulted in a number of
highly efficient computational algorithms
Fourier transform, FFT
sequence of length N into successively smaller
DFTs
Decimation-in-time FFT algorithms
decomposition is done by decomposing the sequence
into successively smaller subsequences,
and both the symmetry and periodicity of complex
exponential are exploited
(N/2)-point sequences
v
kn N j kn
W = − ( 2 π / )
∑
=
odd even
] [ ]
[ ]
[
n
kn N n
kn
W n x k
X
1 , , 1 , 0 , ] [ ]
0
−
=
=
N k
W n x k
n
kn N
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Decimation-in-time FFT algorithms
y periodicit the
to due ) 1 2 / , , 1 , 0 for compute
only
(
1 , , 1 , 0 ], [ ]
[
] 1 2 [ ]
2
[
) ](
1 2 [ )
](
2
[
] 1 2 [ ]
2
[
] [ ]
[ ]
[
1 ) 2 / ( 0
2 /
1
)
2
/
(
0
2 /
1 ) 2 / ( 0
2 1
)
2
/
(
0
2
1 ) 2 / ( 0
) 1 2 ( 1
)
2
/
(
0
2
odd even
−
=
−
= +
=
+ +
=
+ +
=
+ +
=
+
=
∑
∑
∑
∑
∑
∑
∑
∑
−
=
−
=
−
=
−
=
−
=
+
−
=
N k
N k
k H W
k
G
W r x W
W r
x
W r
x W
W r
x
W r x W
r
x
W n x W
n x k
X
k N
N r
rk N
k N N
r
rk N
N r
rk N
k N N
r
rk N
N r
k r N N
r
rk N
n
kn N n
kn N
Flow graph of the decimation-in-time
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Decimation-in-time FFT
∑
∑
∑
∑
∑
∑
∑
−
=
−
=
−
=
−
=
−
=
+
−
=
−
=
+ +
=
+ +
=
+ +
=
=
1 ) 4 / (
0
4 / 2
/
1 ) 4 / (
0
4 /
1 ) 4 / (
0
4 / 2
/
1
)
4
/
(
0
4 /
1 ) 4 / (
0
) 1 2 ( 2 /
1 ) 4 / (
0
2 2 /
1 ) 2 / (
0
2 /
] 1 2 [ ]
2 [ ]
[
] 1 2 [ ]
2 [
] 1 2 [ ]
2 [ ]
[ ]
[
N l
lk N
k N N
l
lk N
N l
lk N
k N N
l
lk N
N l
k l N N
l
lk N N
r
rk N
W l h W
W l h k
H
W l g W
W l g
W l g W
l g W
r g k
G
Combination of Fig 9.3 and 9.4
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2-point DFT
Flow graph
N
2
log stages and each stage has N complex multiplications and N complex
additions
N
Nlog 2
In total, complex multiplications and additions.
240 , 10 log
576 , 048 , 1
1024 2
e.g.
2 2 10
=
=
=
=
N N N N
A reduction of 2 orders!
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Flow graph of butterfly computation
Flow graph
by a factor of 2 over the number in Fig 9.7
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Part III: Decimation-in-frequency FFT algorithms
Direct computation of the DFT
Decimation-in-time FFT algorithms
Fourier analysis of signals using the DFT
Decimation-in-frequency FFT algorithms
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Part IV: Fourier analysis of signals using the DFT
Direct computation of the DFT
Decimation-in-time FFT algorithms
Decimation-in-frequency FFT algorithms
Fourier analysis of signals using the DFT
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Fourier analysis of signals using the DFT
Not ideal
Low-pass filtered and modified
∑∞
−∞
=
+
=
r c
j
T
r j T j X T e
Fourier analysis of signals using the DFT
∫ −
−
= π
π
θ ω θ
π X e V e d
e
2
1
)
Windowing
j
N
n
kn N
e
n
v
k
0
) / 2
( , 0 , 1 , , 1
]
[
)
(
ω
π
−
=
−
=
−
=
=∑
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Effect of Windowing on Fourier analysis
Effect of Windowing on Fourier analysis
A rectangular window of length 64.
) ( 2 ) ( 2
) ( 2 ) ( 2 ) (
) cos(
] [ ) cos(
] [ ] [
) cos(
) cos(
] [
) cos(
) cos(
) (
) ( 1
) ( 1
) ( 0
) ( 0
1 1 1
0 0 0
1 1 1 0 0 0
1 1 1 0 0 0
1 1
1 1
0 0
0 0
ω ω θ ω
ω θ
ω ω θ ω
ω θ ω
θ ω θ
ω
θ ω θ
ω
θ θ
+
−
−
+
−
−
+ +
+
=
+ +
+
=
+ +
+
=
+ Ω +
+ Ω
=
j j j
j
j j j
j j
c
e W e A e
W e A
e W e A e
W e A e V
n n w A n
n w A n v
n A n
A n x
t A t
A t s
Effect of Windowing on Fourier analysis
The DTFT of a sinusoidal signal
is a pair of impulses
Windowing broadens the impulses and reduces the distinction of signals that are close in frequency
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Summary
Course at a glance
Discrete-time signals and systems
Fourier-domain
representation
DFT/FFT
System analysis
Filter design z-transform
MM1
MM2
MM6 MM4
MM7, MM8
Sampling and reconstruction MM5
System structure System
Trang 15Digital Signal Processing, X, Zheng-Hua Tan, 2005 29
The end.
Thanks for your attention!