chapter 9 computation of the dft
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123/4/10 1Zhongguo Liu_Biomedical Engineering_Shandong U
niv.
Chapter 9 Computation of the Discrete
Fourier TransformZhongguo Liu
Biomedical Engineering
School of Control Science and Engineering, Shandong University
Biomedical Signal processing
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9.0 Introduction
9.1 Efficient Computation of Discrete Fourier Transform
9.2 The Goertzel Algorithm
9.3 decimation-in-time FFT Algorithms
9.4 decimation-in-frequency FFT Algorithms
9.5 practical considerations ( software realization)
Chapter 9 Computation of the Discrete Fourier Transform
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9.0 Introduction
1. Compute the N-point DFT and of the two sequence and
kX1 kX 2
nx1 nx2
2. Compute for kXkXkX 213 10 Nk
3. Compute as the inverse DFT of kX 3
nxNnxnx 213
Implement a convolution of two sequences by the following procedure:
Why not convolve the two sequences directly?
There are efficient algorithms called Fast Fourier Transform (FFT) that can be orders of magnitude more efficient than others.
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9.1 Efficient Computation of Discrete Fourier Transform
The DFT pair was given as
4
1
0
2 /1[ ]
N
k
j N knx n X k
Ne
1
0
2 /[ ]
N
n
j N knX k x n e
Baseline for computational complexity: Each DFT coefficient requires
N complex multiplications; N-1 complex additions
All N DFT coefficients requireN2 complex multiplications; N(N-1) complex additions
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9.1 Efficient Computation of Discrete Fourier Transform
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Complexity in terms of real operations4N2 real multiplications2N(N-1) real additions (approximate 2N2)
1
0
2 /[ ]
N
n
j N knX k x n e
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9.1 Efficient Computation of Discrete Fourier Transform
Most fast methods are based on Periodicity propertiesPeriodicity in n and k; Conjugate symmetry
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2 / 2 / 2 / 2 /j N k N n j N kN j N k n j N kne e e e 2 / 2 / 2 /j N kn j N k n N j N k N ne e e
Re ]
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X[k] can be viewed as the output of a filter to the input x[n]Impulse response of filter: X[k] is the output of the filter at time n=N
9.2 The Goertzel Algorithm
Makes use of the periodicityMultiply DFT equation with this factor
7
2 / 2 1j N Nk j ke e
1
0
1
0
2 / 2 / 2 /[ ] [ ]
N N
r
k
r
j N kN j N r j N k N rX k x r x re e e
k n NX k y n
2 /[ ]
j N knh n u ne
2 /[ ]k
r
j N k n ry n x r u n re
Define
using x[n]=0 for n<0 and n>N-1
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9.2 The Goertzel AlgorithmGoertzel
Filter:
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Computational complexity4N real multiplications; 4N real additionsSlightly less efficient than the direct method
2 /[ ] [ ] [ ]nk
N
j N knh n u n W u ne
1
1
1k kN
H zW z
[ ] [ 0,1 1,...] [ , ,],kk k Ny n y n W x n n N [ 1] 0ky
0,1,., ..,k n NkX k Ny n
nkNWBut it avoids computation and storage of
1
0
[ ]N
knN
n
X k x n W
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Second Order Goertzel Filter
Goertzel Filter
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2 2
1 1
2 21 21 1
1 12
1 2cos1 1
j k j kN N
kj k j kN N
e z e zH z
kz ze z e z N
Multiply both numerator and denominator
1
2
1
1k
j kN
H z
ze
2[ ] [ 2] 2cos [ 1] 0,1,...,[ ],
ky n y n y n x n n N
N
[ ] [ ] [ 1]kk Ny y WN yN N 0,1,, ...,Nk kX
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Second Order Goertzel Filter
Complexity for one DFT coefficient ( x(n) is complex sequence). Poles: 2N real multiplications and 4N real additions Zeros: Need to be implement only once:
4 real multiplications and 4 real additionsComplexity for all DFT coefficients
Each pole is used for two DFT coefficients Approximately N2 real multiplications and 2N2 real
additions10
2[ ] [ 2] 2cos [ 1] 0,1,...,[ ],
ky n y n y n x n n N
N
[ ] [ ] [ 1]kk Ny y WN yN N 0,1,, ...,Nk kX
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Second Order Goertzel Filter
If do not need to evaluate all N DFT coefficientsGoertzel Algorithm is more efficient than FFT if
less than M DFT coefficients are needed,M < log2N
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2[ ] [ 2] 2cos [ 1] 0,1,...,[ ],
ky n y n y n x n n N
N
[ ] [ ] [ 1]kk Ny y WN yN N 0,1,, ...,Nk kX
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9.3 decimation-in-time FFT Algorithms
Makes use of both periodicity and symmetryConsider special case of N an integer power of 2Separate x[n] into two sequence of length N/2
Even indexed samples in the first sequenceOdd indexed samples in the other sequence
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1
0
n even n odd
2 /
2 / 2 /
[ ]
[ ] [ ]
N
n
j N kn
j N kn j N kn
X k x n
x n x n
e
e e
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9.3 decimation-in-time FFT Algorithms
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/2 1 /2 1
2 12
r 0 r 0
[2 ] [2 1]N N
r krkN NX k x r W x r W
Substitute variables n=2r for n even and n=2r+1 for odd
G[k] and H[k] are the N/2-point DFT’s of each subsequence
n even n odd
2 / 2 /[ ] [ ]
j N kn j N knX k x n x ne e
kNG k W H k
/2 1 /2 1
/2 /2r 0 r 0
[2 ] [2 1]N N
rk k rkN N Nx r W W x r W
2
/2
2/2
22
N NN
jjNW We e
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9.3 decimation-in-time FFT Algorithms
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G[k] and H[k] are the N/2-point DFT’s of each subsequence
kNG k W H k
/2 1 /2 1
/2 /2r 0 r 0
[2 ] [2 1]N N
rk k rkN N NX k x r W W x r W
/2
22 2/2 rk
N
rrN N
j kj kWe e
2
NG k G k
2
NH k H k
10,1,...,
2
Nk
0,1,...,k N
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8-point DFT using decimation-in-time
Figure 9.3
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computational complexityTwo 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
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9.3 decimation-in-time FFT Algorithms
Repeat same process , Divide N/2-point DFTs into Two N/4-point DFTsCombine outputs
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N=8
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9.3 decimation-in-time FFT Algorithms
After two steps of decimation in time
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Repeat until we’re left with two-point DFT’s
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9.3 decimation-in-time FFT Algorithms
flow graph for 8-point decimation in time
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Complexity:Nlog2N complex multiplications and additions
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Butterfly Computation
Flow graph constitutes of butterflies
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We can implement each butterfly with one multiplication
Final complexity for decimation-in-time FFT(N/2)log2N complex multiplications and additions
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9.3 decimation-in-time FFT Algorithms
Final flow graph for 8-point decimation in time
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Complexity:(Nlog2N)/2 complex multiplications and Nlog2N additions
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9.3.1 In-Place Computation同址运算
Decimation-in-time flow graphs require two sets of registersInput and output for each stage
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0
0
0
0
0
0
0
0
0 0
1 4
2 2
3 6
4 1
5 5
6 3
7 7
X x
X x
X x
X x
X x
X x
X x
X x
0
4
2
6
1
5
3
7
x
x
x
x
x
x
x
x
0
1
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X
X
X
X
X
X
X
X
2
2
2
2
2
2
2
2
0
1
2
3
4
5
6
7
X
X
X
X
X
X
X
X
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9.3.1 In-Place Computation同址运算
Note the arrangement of the input indicesBit reversed indexing(码位倒置)
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0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0 000 000
1 4 001 100
2 2 010 010
3 6 011 110
4 1 100 001
5 5 101 101
6 3 110 011
7 7 111 111
X x X x
X x X x
X x X x
X x X x
X x X x
X x X x
X x X x
X x X x
0
4
2
6
1
5
3
7
x
x
x
x
x
x
x
x
0
1
2
3
4
5
6
7
X
X
X
X
X
X
X
X
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Figure 9.13
cause of bit-reversed order
binary coding for position :000
001
010
011
100
101
110
111
MN 2
must padding 0 to
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9.3.2 Alternative forms
Note the arrangement of the input indicesBit reversed indexing(码位倒置)
25
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0 000 000
1 4 001 100
2 2 010 010
3 6 011 110
4 1 100 001
5 5 101 101
6 3 110 011
7 7 111 111
X x X x
X x X x
X x X x
X x X x
X x X x
X x X x
X x X x
X x X x
0
4
2
6
1
5
3
7
x
x
x
x
x
x
x
x
0
1
2
3
4
5
6
7
X
X
X
X
X
X
X
X
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Figure 9.14
9.3.2 Alternative forms
strongpoint : in-place computations
shortcoming : non-sequential access of data
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Figure 9.15
shortcoming : not in-place computation
non-sequential access of data
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Figure 9.16
shortcoming : not in-place computation
strongpoint: sequential access of data
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9.3 decimation-in-time FFT Algorithms
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/2 1 /2 1
2 12
r 0 r 0
[2 ] [2 1]N N
r krkN NX k x r W x r W
Substitute variables n=2r for n even and n=2r+1 for odd
G[k] and H[k] are the N/2-point DFT’s of each subsequence
n even n odd
2 / 2 /[ ] [ ]
j N kn j N knX k x n x ne e
kNG k W H k
/2 1 /2 1
/2 /2r 0 r 0
[2 ] [2 1]N N
rk k rkN N Nx r W W x r W
2
/2
2/2
22
N NN
jjNW We e
Review
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9.3.1 In-Place Computation同址运算Bit reversed indexing(码位倒置)
30
0
0
0
0
0
0
0
0
000 000
001 100
010 010
011 110
100 001
101 101
110 011
111 111
X x
X x
X x
X x
X x
X x
X x
X x
0
4
2
6
1
5
3
7
x
x
x
x
x
x
x
x
0
1
2
3
4
5
6
7
X
X
X
X
X
X
X
X
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Figure 9.14
9.3.2 Alternative forms
strongpoint : in-place computations
shortcoming : non-sequential access of data
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9.4 Decimation-In-Frequency FFT Algorithm
The DFT equation
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1
0
[ ]N
nkN
n
X k x n W
1 / 2 1 1
2 2 2
0 0 / 2
2 [ ] [ ] [ ]N N N
n r n r n rN N N
n n n N
X r x n W x n W x n W
/2 1
/20
[ ] [ / 2]N
nrN
n
x n x n N W
Split the DFT equation into even and odd frequency indexes
Substitute variables
/2 1 /2 1
/2 22
0 0
[ ] [ / 2]N N
n N rn rN N
n n
x n W x n N W
/2 1
0/2( )
Nrn
nNg n W
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9.4 Decimation-In-Frequency FFT Algorithm
The DFT equation
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1
0
[ ]N
nkN
n
X k x n W
1 /2 1 1
(2 1) (2 1) (2 1)
0 0 /2
2 1 [ ] [ ] [ ]N N N
n r n r n rN N N
n n n N
X r x n W x n W x n W
/2 1
(2 1)
0
[ ] [ / 2]N
n rN
n
x n x n N W
/2 1 /2 1
/2 (2 1)(2 1)
0 0
[ ] [ / 2]N N
n N rn rN N
n n
x n W x n N W
(2 1) /22 1N
r Nr NN N NW W W
2
/22 1n r rn n rn n
N NN N NW W W W W
/
2
2 1
0/[ ] [ / 2]
Nn rnN
nNx n x n N W W
/2
/2 1
0
( )N
n rN N
n
n
h n W W
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decimation-in-frequency decomposition of an N-point DFT to N/2-point DFT
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/2 1
0/22 [ ] [ / 2]
N
n
nrNX r x n x n N W
/2 1
0/22 1 [ ] [ / 2]
Nn rnN
nNX r x n x n N W W
/2
/2 1
0
( )N
n rN N
n
n
h n W W
/2 1
0/2( )
Nrn
nNg n W
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/4
/4 12
0
( )N
n sN N
n
n
q n W W
/
2
0/4
4 1
2*(2 1) [ ( ) ( / 4)] N N
Nn sn
n
X s g n g n N W W
decimation-in-frequency decomposition of an 8-point DFT to four 2-point DFT
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/4 1
0/4( )
Nsn
nNp n W
/
0/4
4 1
2*2 [ ( ) ( / 4)]N
sn
nNX s g n g n N W
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2-point DFT
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081 1( ) ( ) ( )v v vX q X p X q W 8when N
1 1( ) ( ) ( )v v vX p X p X q
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/2 1
0/22 [ ] [ / 2]
N
n
nrNX r x n x n N W
/2 1
0/2( )
Nrn
nNg n W
/4 1 /2 1
2 2
0 /4
/4 1 /4 12 2 ( /4)
0 0
/4 1 /4
/2 /2
/2 /2
/4 /4
/
1
0 0
04
/4 1
2*2 ( ) ( )
( ) ( / 4)
( ) ( / 4)
[ ( ) ( / 4)]
N Nsn sn
n n N
N Nsn s n N
n n
N Nsn sn
N N
N N
N N
N
n n
Nsn
n
X s g n W g n W
g n W g n N W
g n W g n N W
g n g n N W
/4 1
0/4( )
Nsn
nNp n W
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/2 1
0/22 [ ] [ / 2]
N
n
nrNX r x n x n N W
/2 1
0/2( )
Nrn
nNg n W
/2 1
(2 1)
0
/4 1 /2 1(2 1)
/2
/2 /2
/4 /2
(2 1)
0 /
/2
/4 /4 /
4
/4 1 /4 1(2 1)( /4)
0 0
/4 12 2 (2 1)
02
2*(2 1) ( )
( ) ( )
( ) ( / 4)
( ) ( / 4)
Ns n
n
N Ns n s n
n n N
N
N
N N
N N N
N N N N
Nsn n s n N
n n
Nsn n s
Nn n s N
n
X s g n W
g n W g n W
g n W W g n N W
g n W W g n N W W W
/4 1
/4
0
/4 12
0/4[ ( ) ( / 4)] N N
N
n
Nn sn
n
g n g n N W W
/2 /2
(2 1) /4 /2 /4/2 1s N sN N
N N NW W W
/4
/4 12
0
( )N
n sN N
n
n
q n W W
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/4
/4 1
0
2*2 ( )N
sn
nNX s p n W
/4
/4
/4 12
0
/8 1 /4 1
/2
/4
2
0 8
2*2*2 ( )
( ) ( )
Ntn
n
N Nt
N
n tn
n n NN N
X t p n W
p n W p n W
/8 1 /8 1
2 2 ( /8)
04
0/4 /( ) ( / 8)
N Ntn t n N
Nn n
Np n W p n N W
/8 1
0/8[ ( ) ( / 8)]
Ntn
nNp n p n N W
( ) ( 1)p n p n 8when N
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/4
/4 1
0
2*2 ( )N
sn
nNX s p n W
/4
/4
/4 1(2 1)
0
/8 1 /4 1(2 1) (2 1)
04
//
8
2*2*(2 1) ( )
( ) ( )
Nt n
n
N Nt n t n
N
Nn n N
N
X t p n W
p n W p n W
/8 1 /8 1
(2 1) (2 1)( /8)
0 0/4 /4( ) ( / 8)
N Nt n t n N
nN
nNp n W p n N W
/8 1
0
4/8[ ( ) ( / 8)] N
Ntn n
Nn
p n p n N W W
0
8[ ( ) ( 1)]p n p n W 8when N
/8 1 /8 12 2 (2
/4 /4 /4 /4 /41) /8
0 0
( ) ( / 8)N N
tn nN N N
tn n t N
n nN Np n W W p n N W W W
/4 /4(2 1) /8 /4 /8
/4 1t N tN NN N NW W W
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Final flow graph for 8-point DFT decimation in frequency
41
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9.4.1 In-Place Computation同址运算
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DIF FFT
DIT FFT
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9.4.1 In-Place Computation同址运算
43
DIF FFT
DIT FFT
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9.4.2 Alternative forms
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decimation-in-time Butterfly Computation
decimation-in-frequecy Butterfly Computation
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The DIF FFT is the transpose of the DIT FFT
45
DIF FFT
DIT FFT
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9.4.2 Alternative forms
DIF FFT
DIT FFT
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9.4.2 Alternative forms
DIF FFT
DIT FFT
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Figure 9.24 erratum
0
4
2
6
1
5
3
7
x
x
x
x
x
x
x
x
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9.4.2 Alternative forms
DIF FFT
DIT FFT
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23/4/1050Zhongguo Liu_Biomedical Engineering_Shandong U
niv.
Chapter 9 HW9.1, 9.2, 9.3,
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