CE 40763 Digital Signal Processing Fall 1992 Fast Fourier transform (FFT). Hossein Sameti Department of Computer Engineering Sharif University of Technology. Motivation.
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CE 40763Digital Signal ProcessingFall 1992Fast Fourier transform (FFT)
HosseinSameti
Department of Computer Engineering
Sharif University of Technology
N pt. DFT of x(n)
Q: For each k:
How many adds and how many mults?
A: (N-1) complex adds and N complex mults.
How many k values do we have?
N
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Direct computation:
Ideal case:
FFT:
Example:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
FFT
Decimation in time
Decimation in frequency
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
N: power of 2
n: even
n: odd
n: even
n=2r
n:0N-2
r:0N/2-1
n: odd
n=2r+1
n:1N-1
r:0N/2-1
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Suppose:
Solution: use the relationship between DFS and DFT.
We thus need to replicate G(k) and H(k) “once”, to get X(k).
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
pt. DFT
g(r)
After replication
+
pt. DFT
h(r)
(twiddle factor)
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
pt. DFT
g(r)
pt. DFT
h(r)
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
N/2 pt. DFT block
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
r(0)
r(1)
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
r(0)
r(1)
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Flow graph of a the 2-pt. DFT
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
How many stages do we have?
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
2 mults+ 2 adds
1 mult+ 2 adds
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
In-place computation (only N storage locations are needed)
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Total number of operations:
adds
mults
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Output indexing is in order.
input indexing is shuffled.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
1
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
N/2 pt. DFT of g(n)
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
-1
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
N/2 pt. DFT of h(n)
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology