Post on 18-Aug-2018
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FFTFFTDecimation in FrequencyDecimation in Frequency
Spring
2009© Ammar Abu-Hudrouss -Islamic
University Gaza
Slide ٢Digital Signal Processing
Decimation in Frequency
1
01,.....,2,1,0)()(
N
n
knN NkWnxkX
12/
0
1
2/)()()(
N
n
N
Nn
knN
knN WnxWnxkX
12/
0
12/
0
)2/()2/()()(N
n
N
n
NnkN
knN WNnxWnxkX
NjN eW /2
The discrete Fourier transform can be found using
Where N = 2, 4, 8, 16,… and
X (k ) can be expressed as
12/
0
12/
0
2/ )2/()()(N
n
N
n
knN
kNN
knN WNnxWWnxkX
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Slide ٣Digital Signal Processing
Decimation in Frequency
kkNNkkNN eeW )1(/)2/(2)2/(
12/
0
2)2/()()2(N
n
mnNWNnxnxmX
12/
0
12/
0
)2/()1()()(N
n
N
n
knN
kknN WNnxWnxkX
But
Then
If k = 2m or an even number
12/
02/)()2(
N
n
mnNWnamX
Slide ٤Digital Signal Processing
Decimation in Frequency
mnN
NmnjNmnjmnN WeeW 2/
)2//(2/222
Noting That
)2/()()( Nnxnxna
12/
0
2)2/()()12(N
n
mnN
nNWWNnxnxmX
)2/()()( Nnxnxnb
12/
02/)()12(
N
n
mnN
nN WWnbmX
If k = 2m+1 (odd number) and using the same method
Then X(2m) is N/2-point DFT for a(n)
X(2m+1) is N/2-point DFT for nNWnb )(
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Slide ٥Digital Signal Processing
Decimation in Frequency
28W
18W
38W
N/2-pointDFT
X(0)
X(2)
X(4)
X(6)
a(0)
a(1)
a(2)
a(3)
N/2-pointDFT
X(1)
X(3)
X(5)
X(7)
b(0)
b(1)
b(2)
b(3)
x(0)
x(1)
x(2)
x(3)
x(4)
x(5)
x(6)
x(7)
08W
-1
-1
-1
-1
Slide ٦Digital Signal Processing
Decimation in Frequency
28W
18W
38W
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)
08W
N/4 point DFT
N/4 point DFT
N/4 point DFT
N/4 point DFT
28
14 WW
08
04 WW
-1
-1
-1
-1
-1
-1-1
-1
28
14 WW
08
04 WW
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Slide ٧Digital Signal Processing
Decimation in Frequency
X(0)
X(4)
X(2)
X(6)
X(1)
X(5)
X(3)
X(7)0
8W
08W
08W
08W
-1
-1
-1
-1
28W
18W
38W
x(0)
x(1)
x(2)
x(3)
x(4)
x(5)
x(6)
x(7)
08W
08W
28W
08W
28W
-1
-1
-1
-1
-1
-1-1
-1
Slide ٨Digital Signal Processing
Decimation in Frequency
Using the previous algorithm , the complex multiplications needed is only 12. While using the normal DFT would require 64 complex multiplications
In general
Complex multiplication of DFT is: N2
Complex multiplication of FFT is (N/2) log2(N)
If N = 1024
Complex multiplication of DFT is: 1,048,576
Complex multiplication of FFT is: 5,120
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Slide ٩Digital Signal Processing
Decimation in Frequency
Index mapping for Fast Fourier Transform
Input Data index n
Index Bits Reversal Bits
Output data indexk
0 000 000 01 001 100 42 010 010 23 011 110 64 100 001 15 101 101 56 110 011 37 111 111 7
Slide ١٠Digital Signal Processing
Decimation in Frequency
Example Given a sequence x(n) where x(0) = 1, x(1) = 2, x(2) = 3, x(3) = 4 and x(n) = 0 elsewhere ,find DFT for the first four points
solution
X(0)
X(2)
X(1)
X(3)
x(0) =1
x(1) =2
x(2) =3
x(3) =4
104 W
jW 14 10
4 W
104 W -26
-2
-2
4 10
-2+2j
-2-2j-1
-1
-1
-1
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Slide ١١Digital Signal Processing
Inverse Fourier Transform
1
01,.....,2,1,0)(1)(
N
k
knN NnWkX
Nnx
1
0
~
1,.....,2,1,0)(1)(N
k
knN NnWkX
Nnx
The inverse discrete Fourier transform can be found using
Which can be expressed as
where knN
knN WW
~
We can see that the difference between the inverse discrete Fourier and forward Fourier transform is the twiddled factor and the division by 1/N
is called the twiddled factor
Slide ١٢Digital Signal Processing
Inverse Fourier Transform
~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~
08W
~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
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Slide ١٣Digital Signal Processing
Inverse Fourier Transform
Example Given a sequence X(n) given in the previous example. Find the IFFT using decimation in frequency method
solution
x(0) = 1
x(2) = 3
x(1) = 2
x(3) = 4
X(0) =10
X(2) =-2
X(1) = -2+2j
1~
04 W
jW ~
14 1
~0
4 W
1~
04 W 12-4
12
j4
8 4
8
16X(3) = -2-2j
-1
-1
1/4
1/4
1/4
1/4-1
-1