DSP_FOEHU - Lec 09 - Fast Fourier Transform

1. ‫ر‬َ‫ـد‬ْ‫ق‬‫ِـ‬‫ن‬،،،‫لما‬‫اننا‬ ‫نصدق‬ْْ‫ق‬ِ‫ن‬‫ر‬َ‫د‬
LECTURE (09)
Fast Fourier Transform
Assist. Prof. Amr E. Mohamed

2. FAST FOURIER TRANSFORM ALGORITHMS
 The basic idea behind the fast Fourier transform (FFT)
 Decompose the N-point DFT computation into computations of smaller-size DFTs
 Take advantage of the periodicity and symmetry of the complex number .
 Assuming that N is an even number. Divide the data sample into two groups by
decimation
2
1
2
,...1,0'
)1'2()'(
)'2()'(





 N
nfor
nxnv
nxnu
)...7()3();5()2();3()1();1()0(
)...6()3();4()2();2()1();0()0(
xvxvxvxv
xuxuxuxu



3. FAST FOURIER TRANSFORM ALGORITHMS
 Consider special case of N an integer power of 2
 Separate x[n] into two sequence of length N/2
 Even indexed samples in the first sequence
 Odd indexed samples in the other sequence
 Substitute variables n=2r for n even and n=2r+1 for odd
3
       









1
oddn
/2
1
evenn
/2
1
0
/2
][][][
N
knNj
N
knNj
N
n
knNj
enxenxenxkX 
 






















1
2
]12[]2[
1
2
0]12[]2[
12/
0r
)2/(
2/
12/
0r
)2/(
2/
12/
0r
2/
12/
0r
2/
Nk
N
WrxWWrx
N
kWrxWWrx
kX
N
Nkr
N
k
N
N
Nkr
N
N
rk
N
k
N
N
rk
N

4. Computation Complexity
 To calculate the DFT of N-Points discrete time signal, we need:
 (N-1)2 Complex Multiplications
 N(N-1) Complex Additions.
 To calculate the FFT of N-Points discrete time signal, we need:
 Complex Multiplications
 Complex Additions.
4
NlogN 2
Nlog
2
N
2

5. Decimation In Time
 8-point DFT example using decimation-in-time
 Two N/2-point DFTs
 2(N/2)2 complex multiplications
 2(N/2)2 complex additions
 Combining the DFT outputs
 N complex multiplications
 N complex additions
 Total complexity
 N2/2+N complex multiplications
 N2/2+N complex additions
 More efficient than direct DFT
 Repeat same process
 Divide N/2-point DFTs into
 Two N/4-point DFTs
 Combine outputs
5

6. 6
 After two steps of decimation in time
On
the whiteboard
 Repeat until we’re left with two-point DFT’s
On
the whiteboard
Decimation In Time

7. 7
Decimation-In-Time FFT Algorithm
 Final flow graph for 8-point decimation in time
On
the whiteboard

8. 8