CEN352 Digital Signal Processing
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CEN352 Digital Signal Processing Lecture No. 12 to 14 Department of Computer Engineering, College of Computer and Information Sciences, King Saud University, Riyadh, Kingdom of Saudi Arabia October, 2012 BY Dr. Anwar M. Mirza Office No. 2185 Phone: 4697362 [email protected] or [email protected] / ا رز مي د ي ج م وز ن ا وز ت ك الد
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By Dr . Anwar M. Mirza Office No. 2185 Phone: 4697362 [email protected] or [email protected]. CEN352 Digital Signal Processing. الدكتور / انور مجيد ميرزا. Lecture No. 12 to 14. Department of Computer Engineering, College of Computer and Information Sciences, - PowerPoint PPT Presentation
Transcript of CEN352 Digital Signal Processing
CEN352Digital Signal
Processing
Lecture No. 12 to 14Department of Computer Engineering,
College of Computer and Information Sciences,King Saud University, Riyadh, Kingdom of Saudi Arabia
October, 2012
BY
Dr. Anwar M. Mirza
Office No. 2185Phone: 4697362
[email protected] or [email protected]
ميرزا / مجيد انور الدكتور
Fast Fourier Transform
It is a very efficient algorithm for computing the DFT coefficients.
x
y
z=x+y x
yz=x - y
-1
xw
z=wx
Definitions of the graphical operations
(N/2)-point DFT
(N/2)-point DFT
x(4)x(5)x(6)x(7)
x(0)x(1)x(2)x(3)
b(0)b(1)b(2)b(3)
a(0)a(1)a(2)a3)
-1-1-1-1
WN0
WN1
WN2
WN3
X(1)X(3)X(5)X(7)
X(0)X(2)X(4)X(6)
The first iteration of the 8-point FFT
Method of Decimation-in-Frequency
The second iteration of the 8-point FFT
x(4)x(5)x(6)x(7)
x(0)x(1)x(2)x(3)
-1-1-1-1
X(1)X(3)X(5)X(7)
X(0)X(2)X(4)X(6)
WN0
WN1
WN3
WN2
WN0
WN2
WN2
WN0
-1-1
-1-1
(N/4)-point DFT
(N/4)-point DFT
(N/4)-point DFT
(N/4)-point DFT
Method of Decimation-in-Frequency
Block diagram for the 8-point FFT (total twelve multiplications)
X(1)X(3)X(5)X(7)
X(0)X(2)X(4)X(6)
x(4)x(5)x(6)x(7)
x(0)x(1)x(2)x(3)
-1-1-1-1
WN0
WN1
WN3
WN2
WN2
WN0
-1-1
-1-1
WN0
WN0
WN0
WN0
-1
-1
-1
-1
Method of Decimation-in-Frequency
Example 4.12
Solution
First Iteration Second Iteration
x(0)=1
x(1)=2
x(2)=3
x(3)=4
4
6
-2
-2 W40 = 1
W41 = -j
-2
2j
10
-2
-2+2j
-2-2j
W40 = 1
W40 = 1
10
-2
-2+2j
-2-2j
00
01
10
11
Bit Index00
10
01
11
Bit Reversal
= X(0)
= X(2)
= X(1)
= X(3)
First Iteration Second Iteration
X(0)=10
X(1)=-2+j2
X(2)=-2
X(3)=-2-j2
8
-4
12
j4
12
-4
10
-2
-2+2j
-2-2j
4
12
8
16
00
01
10
11
Bit Index00
10
01
11
Bit Reversal
= x(0)
= x(2)
= x(1)
= x(3)
Solution
Example 4.13
The first iteration of the 8-point FFT using Decimation-in-Time
Method of Decimation-in-Time
The second iteration of the 8-point FFT using Decimation-in-Time
Method of Decimation-in-Time
The 8-point FFT using Decimation-in-Time (12 complex multiplications)
Method of Decimation-in-Time
The 8-point IFFT using Decimation-in-Time
Method of Decimation-in-Time
The 4-point FFT using Decimation-in-Time
The 4-point FFT using Decimation-in-Time
That’s it for today !!
Example 4.12
Solution
First Iteration Second Iteration
x(0)=6
x(1)=4
x(2)=2
x(3)=0
8
4
4
4 W40 = 1
W41 = -j
4
-4j
12
4
4-4j
4+4j
W40 = 1
W40 = 1
12
4
4-4j
4+4j
00
01
10
11
Bit Index00
10
01
11
Bit Reversal
= X(0)
= X(2)
= X(1)
= X(3)
Example 4.12
Solution
First Iteration Second Iteration
x(0)=1
x(1)=2
x(2)=3
x(3)=4
4
6
-2
-2 W40 = 1
W41 = -j
-2
2j
10
-2
-2+2j
-2-2j
W40 = 1
W40 = 1
10
-2
-2+2j
-2-2j
00
01
10
11
Bit Index00
10
01
11
Bit Reversal
= X(0)
= X(2)
= X(1)
= X(3)
