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LOGO
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Fast Fourier Transform
(FFT)
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-The Fast Fourier Transform (FFT) is a very efficient algorithm
for performing a discrete Fourier transform.
-The most widely used methods of FFT are based on decomposing
or
breaking the transform into smaller transforms, and combining
them to give
the total transform.
-There are two methods to do this decomposition or
decimation:
(1) Decimation In Frequency domain
(2) Decimation In Time domain
Fast Fourier Transform (FFT)
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4
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5
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2
2
2
log ( )
log ( ) 2
*log ( )
N
NN
N N
stages
complex multiplications
complex additions
2
2
N
N N
DFTFFT
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DFT
1
0
2
( ) ( ) , k=0,1,2,.......,N 1
:
Nkn
N
n
jN
N
X k x n W
where W e
4
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8
(1) Decimation in Time domain
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x(n)={ 1,2,3,4} find DFT for first 4-points
0 1 2 3
2 2
44
j jNW e e j
000000
210011
101102
311113
Example
flip
x(0)=1
x(2)=3
x(1)=2
x(3)=4
Decimation in Time domain
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104 W
jW 14
-1
-1
-1
-1
x(0)=1
x(2)=3
x(1)=2
x(3)=4104 W
104 W
4
-2
6
-2
X(0 )= 10
X(1)= -2+2j
X(2)= -2
X(3)= -2-2j
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x(0)=1
x(2)=3
x(1)=2
x(3)=4
Stage#1
104W
4
-2
104W
6
-2
1
1
0
4 1W
1
4W j
10
-2+2j
-2
-2-2j
Stage#2
X(k)={10,-2+2j,-2,-2-2j}
# stage2increment of
NW
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# stage # stage
2
4
2 2
4for stage#2 1
2
increment of of
increment
NW
W
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x(n)={ 1,1,3,5,0,0,1,3} find DFT for first 8-points
2
8 48
0
8
1
8
2
8
3
8
1(1 )
2
1
1(1 )
2
1( 1 )
2
j j
W e e j
W
W j
W j
W j
100000000
041000011
320100102
161100113
110011004
051011015
530111106
371111117
ExampleDecimation in Time domain
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x(0)=1 X[0]=14
X[1]=0.3-j4.12
X[2]=-3+j7
X[3]=1.7-j0.12
X[4]=-4
0
8W
0
8W
0
8W
0
8W
0
8W
2
8W
0
8W
2
8W
0
8W
1
8W
2
8W
3
8W
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
x(4)=0
x(2)=3
x(6)=1
x(1)=1
x(5)=0
x(3)=5
x(7)=3
1
1
4
2
1
1
8
2
5
1-2j
-3
1+2j
9
1-2j
-7
1+2j
X[5]=1.7+j0.12
X[6]=-3-j7
X[7]=0.3+j4.12
X(k)={14,0.3-j4.12,-3+j7,1.7-j0.121,-4,1.7+j0.121
,-3-j7,0.3+j4.12}
Stage#1 Stage#2 Stage#3
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x(0)=1 X[0]=7
X[1]=0.3-j4.12
X[2]=-3+j7
X[3]=1.7-j0.12
X[4]=-4
0
8W
0
8W
0
8W
0
8W
0
8W
2
8W
0
8W
2
8W
0
8W
1
8W
2
8W
3
8W
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
x(4)=0
x(2)=3
x(6)=1
x(1)=1
x(5)=0
x(3)=5
x(7)=3
1
1
4
2
1
1
8
2
5
1-2j
-3
1+2j
9
1-2j
-7
1+2j
X[5]=1.7+j0.12
X[6]=-3-j7
X[7]=0.3+j4.12
X(k)={14,0.3-j4.12,-3+j7,1.7-j0.121,-4,1.7+j0.121
,-3-j7,0.3+j4.12}
Stage#1 Stage#2 Stage#3
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# stage # stage
2
3
8
2 2
8for stage#2 2
2
8for stage#3 1
2
increment of of
increment
increment
NW
W
W
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IDFT :
Decimation in Time domain
2~
(1)
(2) divided final result by N
jN
NW e
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X(k)={ 1,1+j2,1,1-j2} find x(n) IDFT for 4-points
2 2~4
4
j jNW e e j
000000
210011
101102
311113
ExampleDecimation in Time domain
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X(0)=1
X(2)=1
X(1)=1+j2
X(3)=1-2j
Stage#1
10~4W
2
0
1
2
j4
1
1
4
-4
0
4
Stage#2
Divided by N=4 x(n)={1,-1,0,1}
0~
4W1~
4W j
0~
4W
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(2) Decimation in Frequency domain
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find DFT for first 4-points
0 1 2 3
2 2
44
j jNW e e j
000000
210011
101102
311113
ExampleDecimation in Frequency domain
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x(0)=1
x(1)=2
x(2)=3
x(3)=4
Stage#1
10 = X(0)
-2 = X(2)
-2+2j = X(1)
-2-2j = X(3)
X(k)={10,-2+2j,-2,-2-2j}
1
1
Stage#2 Stage#1
1
0
4 1W
4
6
1
-2
2j
1
4W j
0
4W
0
4W
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x(n)={ 1,1,3,5,0,0,1,3} find DFT for first 8-points
2
8 48
0
8
1
8
2
8
3
8
1(1 )
2
1
1(1 )
2
1( 1 )
2
j j
W e e j
W
W j
W j
W j
ExampleDecimation in Frequency domain
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x(0)=1
-1
-1
-1
-1
x(1)=1
x(2)=3
x(3)=5
x(4)=0
x(5)=0
x(6)=1
x(7)=3
X(k)={14,0.3-j4.12,-3+j7,1.7-j0.121,-4,1.7+j0.121
,-3-j7,0.3+j4.12}
Stage#1 Stage#2 Stage#3
0
8W
1
8W
2
8W
3
8W
1
1
2
2
-1
-1
-1
-1
5
-3
1+2W2
1
1
4
8
9
-7
-1
-1
-1
-1
0
8W
2
8W
0
8W
2
8W
W+2W3
1-2W2
W-2W3
0
8W
0
8W
0
8W
0
8W
X(0)
X(4)
X(2)
X(6)
X(1)
X(5)
X(3)
X(7)
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ExampleDecimation in Frequency domain
X(k)={ 10,-2+j2,-2,-2-j2} find x(n) IDFT for 4-points
2 2~4
4
j jNW e e j
x(0) = 1
x(2) = 3
x(1) = 2
x(3) = 4
X(0) =10
X(2) =-2
X(1) = -2+2j
1~
0
4 W
jW ~
1
41
~0
4 W
1~
0
4 W 12-4
12
j4
8 4
8
16X(3) = -2-2j
-1
-1
1/4
1/4
1/4
1/4
-1
-1
x(n)={ 1,2,3,4}
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For 8-point IDFT
~2
8W
~1
8W
~3
8W
x(0)
x(4)
x(2)
x(6)
X(0)
X(1)
X(2)
X(3)
x(1)
x(5)
x(3)
x(7)
X(4)
X(5)
X(6)
X(7)
~0
8W
~0
8W
~2
8W
~0
8W
~2
8W~
0
8W
~0
8W
~0
8W
~0
8W
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
1/N
1/N
1/N
1/N
1/N
1/N
1/N
1/N
Decimation in Frequency domain
