Transcript Chapter 9
Chapter 9 Computation of the Discrete
Fourier Transform
9.1
9.2
9.3
9.4
9.5
decimation-in-time FFT Algorithms
decimation-in-frequency FFT Algorithms
IFFT Algorithm
FFT Algorithm of real sequence
practical considerations（software realization)
Direct computation:
N 1
X [k ]
x[ n]W
n 0
N 1
x[n]
1
N
kn
0 k N 1
N
X [k ]W
k 0
kn
0 n N 1
N
Complex multiplication:
N2
Complex addition:
N ( N 1)
Real multiplication:
4N 2
Real addition:
4N 2
9.1 decimation-in-time FFT Algorithms
G[0]
x[0]
x[2]
x[4]
N/2 POINT
G[1]
DFT
G[2]
X[0]
X[1]
X[2]
G[3]
x[6]
x[1]
X[3]
H[0]
WN0
H[1]
WN1
Figure 9.3
X[4]
-1
x[3]
N/2 POINT
-1
H[2]
x[5]
x[7]
WN2
-1
DFT
H[3]
WN3
-1
X[5]
X[6]
X[7]
Figure 9.9
complex multiplica tions :
2
N N2
N
, ( N 1)
2
2
2
2
x[0]
N/4point
DFT
x[4]
x[2]
N/4point
G[0]
G1[1]
G[1]
G2[0]
WN0
DFT
DFT
x[6]
x[1]
G1[0]
X[0]
X[1]
G[2]
X[2]
-1
G2[1]
WN2
G[3]
X[3]
-1
N/4point
DFT
DFT
x[5]
H1[0]
H[0]
WN0
-1
H1[0]
H[1]
WN1
-1
H2[0]
x[3]
WN0
N/4point
DFT
H2[0]
WN2
H[3]
-1
DFT
WN2
-1
DFT
x[7]
H[2]
Figure 9.5
X[4]
-1
WN3
-1
X[5]
X[6]
X[7]
Figure 9.10
times of complex multiplica tion ：
1 time / butterfly computatio n * N / 2 butterfly computatio ns / stage * log 2N stage
times of complex addition ：
2 times / butterfly computatio n * N / 2butterfly computatio ns / stage * log 2N stage
strongpoint：in-place computations
shortcoming：non-sequential access of data
compare the operation quantity
alternative forms：
Figure 9.14
strongpoint：in-place computations
shortcoming：non-sequential access of data
Figure 9.15
shortcoming：not in-place computation
non-sequential access of data
Figure 9.16
shortcoming：not in-place computation
strongpoint: sequential access of data
9.2 decimation-in-frequency FFT Algorithms
Figure 9.17
Figure 9.19
Figure 9.18
Figure 9.20
the two kinds
of butterfly
computation
are transpose
of each other.
alternative forms：
Figure 9.22
Figure 9.23
Figure 9.24
9.3 IFFT Algorithm
N 1
kn
n 0
N
X [ k ] x[ n]W
1
x[n]
N
0 k N 1
N 1
kn
k 0
N
X [k ]W
0 n N 1
method1 : x X
1。
kn
N
W
kn
N
W
1/ N
attention：the difference to transpose
DIT-FFTDIF-IFFT
X[0]
X[4]
1/2
1/2WN-0
1/2
1/2
1/2
1/2
x[0]
x[1]
-1
X[2]
X[6]
1/2WN-0
1/2
X[5]
1/2WN-2
1/2WN-0
1/2
1/2
X[7]
x[3]
-1
1/2WN-0
1/2
-1
1/2WN-0
1/2WN-1
1/2
-1
X[3]
x[2]
-1
-1
X[1]
1/2
1/2
-1
1/2WN-0
x[5]
-1
x[6]
1/2WN-3
1/2WN-2
-1
-1
1/2WN-2
1/2WN-0
x[4]
-1
-1
x[7]
DIF-FFTDIT-IFFT
X[0]
X[1]
1/2
1/2
1/2
1/2
1/2
1/2WN0
-1
1/2
X[2]
1/2WN0
1/2
1/2WN2
1/2WN0
x[0]
x[4]
x[2]
-1
1/2
X[3]
X[4]
X[5]
X[6]
X[7]
-1
-1
1/2WN0
1/2
1/2
1/2WN1
1/2
1/2WN0
-1
-1
-1
1/2WN2
-1
1/2
1/2WN2
1/2WN0
-1
1/2WN3
-1
1/2WN0
-1
-1
x[6]
x[1]
x[5]
x[3]
x[7]
2，
method 2 :
1
x[n]
N
kn
X [k ]W
N
k 0
N 1
*
0 n N 1
kn
1
1
*
*
X
[
k
]
W
FFT
X
[k ]
N
N k 0
N
N 1
step:
（1）X*[k]
（2）FFT{ }
（3）（）*/N
can transfer FFT subprogram directly
3。Method 3( exercise 9.1)
*
9.4 FFT Algorithm of real sequence
calculate two N po int real FFT by
one N po int complex FFT
(1) y[n] x1[n] jx2[n], n 0,...N 1
(2)Y [k ] FFT { y[n]}, k 0,...N 1
Y [k ] Y *[ N k ]
(3) X1[k ]
2
Y [k ] Y *[ N k ]
X 2[k ]
, k 0,...N 1
2j
j (Y [k ] Y *[ N k ])
2
calculate one N po int real FFT by
one N / 2 po int complex FFT
(1) y[n] x1[n] jx2[n] x[2n] jx[2n 1], n 0,...N / 2 1
(2)Y [k ] FFT { y[n]}, k 0,...N / 2 1
Y [k ] Y *[ N / 2 k ]
(3) X1[k ]
2
Y [k ] Y *[ N / 2 k ]
X 2[k ]
, k 0,...N / 2 1
2j
(4) X [k ] X1[k ] WNk X 2[k ], k 0,....N / 2 1
X [k N / 2] X1[k ] WNk X 2[k ], k 0,....N / 2 1
9.5 practical considerations
in-place
computati
on
Figure 9.11
cause of bit-reversed order
binary coding for position：
000
001
010
011
100
101
110
111
must padding 0 to
N 2M
Figure 9.13
Coefficient:
WNr r=0,1,…N/2-1
summary:
9.1
9.2
9.3
9.4
9.5
decimation-in-time FFT Algorithms
decimation-in-frequency FFT Algorithms
IFFT Algorithm
FFT Algorithm of real sequence
practical considerations（software realization)
requirements：
1. derivation of decimation-in-time and decimation-in-frequency
FFT algorithms, and draw the graphs ;
2. draw flow graph of IFFT;
3. special arithmetic of real-sequence FFT;
4. concepts of in-place computation and bit-reversed order.