Decimation in Frequency Chapter 8 Spring 2012 Ammar AbuHudrouss Islamic University Gaza Decimation in Frequency The discrete Fourier transform can be found using Where N ID: 257305
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Slide1
1
FFTDecimation in Frequency
Chapter 8
Spring
2012
©
Ammar Abu-Hudrouss
-
Islamic University GazaSlide2
Decimation in Frequency
The discrete Fourier transform can be found using
Where
N = 2, 4, 8, 16,… and
X
(k ) can be expressed as Slide3
Decimation in Frequency
But
Then
If k
= 2m or an even number Slide4
Decimation in Frequency
Noting That
If
k
= 2
m+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 forSlide5
Decimation in Frequency
N
/2-point DFT
X
(0)
X
(2)
X
(4)
X
(6)
a
(0)
a
(1)
a
(2)
a(3)
N/2-point
DFT
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)
-1
-1
-1
-1Slide6
Decimation in Frequency
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)
N/4
point
DFT
N/4
point
DFT
N/4
point
DFT
N/4
point
DFT
-1
-1
-1
-1
-1
-1
-1
-1Slide7
Decimation in Frequency
X
(0)X
(4)
X(2)
X(6)
X
(1)
X
(5)
X
(3)
X
(7)
-1
-1
-1
-1
x
(0)
x
(1)
x
(2)
x(3)
x
(4)
x
(5)
x
(6)
x
(7)
-1
-1
-1
-1
-1
-1
-1
-1Slide8
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 generalComplex multiplication of DFT is: N 2
Complex multiplication of FFT is (N/2) log
2(N)
If
N
= 1024
Complex multiplication of DFT is: 1,048,576
Complex multiplication of FFT is: 5,120Slide9
Decimation in Frequency
Index mapping for Fast Fourier Transform
Input Data index n
Index Bits
Reversal Bits
Output data index
k
0
000
000
0
1
001
100
4
2
010
010
2
3
011
110
6
4
100
001
1
5
101
101
5
6
110
011
3
7
111
111
7Slide10
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
-2
6
-2
-2
4
10
-2+2
j
-2-2
j
-1
-1
-1
-1Slide11
Inverse Fourier Transform
The inverse discrete Fourier transform can be found using
Which can be expressed as where
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 factorSlide12
Inverse Fourier Transform
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)
-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/
NSlide13
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
12
-4
12
j
4
8
4
8
16
X
(3) = -2-2
j
-1
-1
1/4
1/4
1/4
1/4
-1
-1