Chapter 9 Computation of the Discrete Fourier Transform
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104/19/2023 1Zhongguo Liu_Biomedical Engineering_Shandong
Univ.
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
DFTcomputational 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
5
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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2 / ( ), ( )* j N k N k kN N N
n nN
ne W W W W
6
9.1 Efficient Computation of Discrete Fourier Transform
Most fast methods are based on Periodicity propertiesPeriodicity in n and k; Conjugate symmetry
6
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
( ) ( ) nk k N N kN N N
n nW W W
( )( )* ( )* n nk N k kN N N
nW W W
( )( )* ( )* n nk N k kN N N
nW W W
Re ]
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X[k] can be viewed as output of a filter to 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:
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
9
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 [ 0,1,...,1] [ ],
y n y n y n
N
kn nx N
[ ] [ ] [ 1] kk Ny y W yN N N 0,1,, ...,X kk N
2 ( )[ ] [ 2] 2 0,cos [ 1] [ ], 1,...,
y n y n y n x n
Nn
N kN
, X k X N k
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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
11
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
13
/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
14
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
/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
k
NG k W H k
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computational complexityTwo N/2-point DFTs
2(N/2)2 complex multiplications2(N/2)2 complex additions
Combining the DFT outputsN complex multiplicationsN complex additions
Total complexityN2/2+N complex multiplicationsN2/2+N complex additionsMore 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
19
Complexity:Nlog2N complex multiplications and additions
N=2m
m=log2N
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Butterfly Computation
Flow graph of basic butterfly computation
20
We can implement each butterfly with one multiplication
Final complexity for decimation-in-time FFT
(N/2)log2N complex multiplications and Nlog2N additions
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Nlog2N complex multiplications and additions
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
N=2m
m=log2N
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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
22
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
2
3
4
5
6
7
X
X
X
X
X
X
X
X
1
1
1
1
1
1
1
1
0
1
2
3
4
5
6
7
X
X
X
X
X
X
X
X
one set of registers
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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)
23
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 000
1 001
2 010
3 0
0 000
4 100
2 010
6 110
1 001
5 101
3 011
7 111
11
4 100
5 101
6 110
7 111
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
normal order binary coding for position :
000
001
010
011
100
101
110
111
0
4
2
6
1
5
3
7
x
x
x
x
x
x
x
x
2 1 0x n n n
000
001
010
011
100
101
110
111
x
x
x
x
x
x
x
x
2 1 0n n n2 1 0n n n
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Figure 9.13
cause of bit-reversed order binary coding for position :
000
001
010
011
100
101
110
111
0
4
2
6
1
5
3
7
x
x
x
x
x
x
x
x
2 1 0x n n n
000
100
010
110
001
101
011
111
x
x
x
x
x
x
x
x
2 1 0n n n 0 1 2n n n
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9.3.2 Alternative forms
Note the arrangement of the input indicesBit reversed indexing (码位倒置)
26
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
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 000
1 001
2 010
3 0
0 000
4 100
2 010
6 110
1 001
5 101
3 011
7 111
11
4 100
5 101
6 110
7 111
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
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Figure 9.14
9.3.2 Alternative forms
strongpoint : in-place computationsshortcoming : non-sequential access of data
input in normal order and output in bit-reversed order
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Figure 9.15
shortcoming : not in-place computation non-sequential access of data
Input, output same normal order
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Figure 9.16
shortcoming : not in-place computation strongpoint: sequential access of data
Same structure, Input bit-reversed order , output normal order
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9.4 Decimation-In-Frequency FFT Algorithm
The DFT equation
30
1
0
[ ]N
nkN
n
X k x n W
1
/2
1 /2 2 2
2 1
0 0
[ ] [ ] [ ]2
r rN N
n n nN N N
N
nn n N
rX x n W x n W x n Wr
/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
0
/2 12
0
2 2/[ ] [ / 2]
N
rn rN N
n
NN
n
nx n xW N Wn
/2 1
0/2( )
Nrn
nNg n W
/ 2 n Nn n
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9.4 Decimation-In-Frequency FFT Algorithm
The DFT equation
31
1
0
[ ]N
nkN
n
X k x n W
2 1 21 /2 1 1
( ) ( )1 2( )
0 0
1
/2
[ ] [ ] [2 ]1 r rN N N
n n nN N N
n n
r
n N
X x n W x n W x n Wr
/2 1
(2 1)
0
[ ] [ / 2]N
n rN
n
x n x n N W
/2 1 /2 1
(2 1)(2 1) /2
0 0
2[ /[ ] ]N N
rn rN N
n n
n Nx n W x n WN
(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 1
0/2[ ] [ / 2] n
Nrn
Nn
Nx n x n N W W
/2
/2 1
0
( )N
rnN N
n
n
Wh n W
/ 2 n Nn n
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decimation-in-frequency decomposition of an N-point DFT to N/2-point DFT
32
/2 1
0/22 [ ] [ / 2]
N
n
nrNr x n x n NX W
/2 1
0/22 1 [ ] [ / 2]
Nrn
nN
nNr x n x n N WX W
/2
/2 1
0
( )N
rnN N
n
n
Wh n W
/2 1
0/2( )
NrnN
n
Wg n
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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
33
/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
34
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
39
N=8
4-point DFT 2-point DFT
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9.4.1 In-Place Computation同址运算
40
DIF FFT
DIT FFT
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41
9.4.1 In-Place Computation同址运算
41
DIF FFT
DIT FFT
transpose
![Page 42: Chapter 9 Computation of the Discrete Fourier Transform](https://reader035.fdocuments.net/reader035/viewer/2022062217/56812d3c550346895d92402b/html5/thumbnails/42.jpg)
42
9.4.2 Alternative forms
42
decimation-in-time Butterfly Computation
decimation-in-frequecy Butterfly Computation
transpose
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43
The DIF FFT is the transpose of the DIT FFT
43
DIF FFT
DIT FFT
transpose
![Page 44: Chapter 9 Computation of the Discrete Fourier Transform](https://reader035.fdocuments.net/reader035/viewer/2022062217/56812d3c550346895d92402b/html5/thumbnails/44.jpg)
44
9.4.2 Alternative forms
DIF FFT
DIT FFT
transpose
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45
9.4.2 Alternative forms
DIF FFT
DIT FFT
transpose
![Page 46: Chapter 9 Computation of the Discrete Fourier Transform](https://reader035.fdocuments.net/reader035/viewer/2022062217/56812d3c550346895d92402b/html5/thumbnails/46.jpg)
46
Figure 9.24 erratum
0
4
2
6
1
5
3
7
x
x
x
x
x
x
x
x
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47
9.4.2 Alternative forms
DIF FFT
DIT FFT
transpose
![Page 48: Chapter 9 Computation of the Discrete Fourier Transform](https://reader035.fdocuments.net/reader035/viewer/2022062217/56812d3c550346895d92402b/html5/thumbnails/48.jpg)
48 04/19/202348Zhongguo Liu_Biomedical
Engineering_Shandong Univ.
Chapter 8 HW9.1, 9.2, 9.3,
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