Numerical Methods Fast Fourier Transform Part: Theoretical Development of Fast Fourier Transform http://numericalmethods.eng.usf.edu

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Numerical MethodsFast Fourier Transform
Part Theoretical Development of Fast Fourier
Transformhttp//numericalmethods.eng.usf.edu
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Chapter 11.06 Theoretical Development of Fast
Fourier Transform
Lecture 16
Major All Engineering Majors Authors Duc
Nguyen http//numericalmethods.eng.usf.edu Numeri
cal Methods for STEM undergraduates
http//numericalmethods.eng.usf.edu
10/26/2020
5
6
Theoretical Development of FFT
Recall Equation (5) from Chapter 11.05 Informal
Development of FFT,
(11.65)
where
(1)
Consider the case . In
this case, we can express and as 2-bit
binary numbers
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(2)
7
Theoretical Development cont.
Eqs. (1) and (2) can also be expressed in
compact forms, as following
(3)
(4)
where
In the new notations, Eq.(11.65) becomes
(5)
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8
Theoretical Development cont.
Consider
9
Theoretical Development cont.
Notice that
Hence Eq. (5) can be simplified to
(7)
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10
Theoretical Development cont.
Define
(8)
Eq. (8) can be modified to
(9)
11
Theoretical Development cont.
Hence
(10)
In matrix notation, Eq.(10) can be written as
(11)
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12
Theoretical Development cont.
It can be seen that Eq. (11) plays the same role
as Eq. (10) from the Informal Development ppt.
Now we can define
(12)
(13)
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13
Theoretical Development cont.
Hence
(14)
Now we have
(15)
14
Theoretical Development cont.
It can be seen that Eq. (15) plays the same role
as Eq. (12) from the Informal Development ppt.
Also, comparing Eq. (5) and Eq. (13), one gets
(16)
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Numerical MethodsFast Fourier Transform
Part Theoretical Development of FFT
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23
Chapter 11.06 Theoretical Development (Contd.)
Lecture 17
The set of Eqs. (8), (12), and (16) represents
the original Cooley-Turkey 14 formulation of
the FFT.
(17)
(18)
where
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24
Theoretical Development cont.
In the new notations, Eq. (5) from the Informal
Development ppt becomes
(19)
Consider
(20)
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25
Theoretical Development cont.
Due to the definitions of (shown in Eq. (4)
of Informal Development), each of the 3 terms
inside the square bracket is equal to 1. Thus,
Eq. (19) can be simplified to
(21)
Define
(22)
(23)
(24)
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26
Theoretical Development cont.
Hence
(25)
Remarks about Eq. (22)
(22a)
companion nodes
(22b)
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27
Theoretical Development cont.
skip computation
28
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29
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  or recommendations expressed in this material are
  those of the author(s) and do not necessarily
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Numerical MethodsFast Fourier Transform
Part Theoretical Development of
FFThttp//numericalmethods.eng.usf.edu
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37
Chapter 11.06 General Case of FFT (Contd.)
Lecture 18
Consider the case where
(26)
(27)
Eq. (5) from Informal Development ppt becomes
(28)
where
(29)
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38
General Case of FFT cont.
The first term of the Eq. (29) can be computed as
Since
Hence all terms inside the brackets are equal to
1. Thus,
(30)
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39
General Case of FFT cont.
Similarly, the second term of Eq. (29) can be
computed as
(31)
(32)
Eq. (28) will eventually become
(33)
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40
General Case of FFT cont.
Let
(34)
(35)
. .
(36)
(37)
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41
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  or recommendations expressed in this material are
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Numerical MethodsFast Fourier Transform
Part Theoretical Development of FFT
http//numericalmethods.eng.usf.edu
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49
Chapter 11.06 FFT Algorithms (Contd.)
Lecture 19
Lets first define
(38)
(39)
where
(40)
(41)
(42)
(43)
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50
FFT Algorithms cont.
Remarks
Using the above notations, Equation (5) from
Informal Development ppt can be expressed as
(44)
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51
FFT Algorithms cont.
Consider
(45)
(46)
Due to the fact that
(47)
Substituting Eq. (46) into (44), one gets
(48)
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52
General Case of FFT cont.
Define
(49)
(50)
Hence
(51)
53
FFT Algorithms cont.
(52)
(53)
Expanding Eq. (52), one obtains
(52a)
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General Case of FFT cont.
Similarly, expanding Eq. (53), one gets
(53a)
54
55
FFT Algorithms cont.
For a typical term corresponding to
, Eq. (52a) gives
(52b)
(52c)
(53b)
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56
General Case of FFT cont.
is presented in Figure 2.
57
Partial/Incomplete Graph of FFT
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Part Theoretical Development of FFT
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66
Chapter 11.06 FFT Algorithms (Contd.)
Lecture 20
In this case, utilizing Eqs. (40-43) into Eqs.
(38) and (39), one obtains
(54)
(55)
where
(56)
(57)
(58)
(59)
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67
FFT Algorithms cont.
Then, Eq. (5) from Informal Development ppt
becomes
(60)
Consider
(61)
Substituting Eq. (61) into Eq. (60), one obtains
(62)
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68
FFT Algorithms cont.
Define
(63)
(64)
Hence
(65)
Expanding (the summation) of Eqs. (63) and (64),
one gets
(66)
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69
FFT Algorithms cont.
Assuming , and then the previous
equation becomes
(67)
Similarly, one has
(68)
(69)
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70
FFT Algorithms cont.
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71
FFT Algorithms cont.
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  or recommendations expressed in this material are
  those of the author(s) and do not necessarily
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Numerical MethodsFast Fourier Transform
Part Theoretical Development of FFT
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77

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80
Chapter 11.06 General FFT Algorithms (Contd.)
Lecture 21
For the more general case, such as
(70)
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81
General FFT Algorithms
Define (should refer/compare to Eqs. 38-39)
(71)
(72)
With (should refer to Eqs. 40-43)
(73)
(74)
82
General FFT Algorithms cont.
(75)
Hence, Eqs. (71) to (74) will be simplified to
(76)
(77)
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83
General FFT Algorithms cont.
(78)
(79)
84
General FFT Algorithms cont.
Eq. (5) from Informal Development ppt can be
expressed as
(80)
where
(81)
(82)
(83)
84
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General FFT Algorithms cont.
where
(84)
(85)
since
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86
General FFT Algorithms cont.
and with
(86)
(87)
(88)
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87
General FFT Algorithms cont.
Substituting Eqs. (85), (87), and (88) into Eq.
(83), and using Eq. (81), then Eq. (80) will
become
(89)
88
General FFT Algorithms cont.
Define
(90)
(91)
(92)
Then
(93)
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89
General FFT Algorithms cont.
(Connections between
and
FFT Algorithms)
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90
General FFT Algorithms cont.
(94)
(95)
(96)
In fact, Eqs. (94) to (96) are identical to the
earlier derived Eqs.(22) to (24).
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Numerical MethodsFast Fourier Transform
Part Twiddle Factor FFT Algorithmshttp//numer
icalmethods.eng.usf.edu
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99
Chapter 11.06 Twiddle Factor FFT Algorithms
(Contd.)
Lecture 22
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100
Twiddle Factor FFT Algorithms
Eq. (48)
can be re-written as
(97)
101
Twiddle Factor FFT Algorithms cont.
(98)
(99)
(100)
101
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102
Twiddle Factor FFT Algorithms cont.
Remarks
a) Consider the following term in Eq. (98)
(101)
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103
Twiddle Factor FFT Algorithms cont.
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104
Twiddle Factor FFT Algorithms cont.
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105
Table 11.1a Fourier Coefficient Program (in
FORTRAN)
(a) Input Descriptions
.Period say, nterms say, 8 (number of
terms used, for computing
. nsegments 3 (number of segments, to define
the given periodic function)
. integration limits for all 3 segments
function
for the 1st segment
function
for the 2nd segment
function
for the 3rd segment
100
105
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(b) Output Descriptions
The numerical values of the unknown Fourier
coefficients
are printed.
107
Fourier Coefficient Program (in MATLAB)
function f_coeff_final() clc clear close
all clear all inputdata coeff end
function inputdata() global n t1 ft1 ft11
t2 ft2 ft22 t3 ft3 ft33 t4 f4 nterms_ak n
100 nterms_ak 8
101
107
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Fourier Coefficient Program (in MATLAB)
seg 1 t1 -pi(pi/n)(-pi/2) ft1
-pi/2 ft11 (-pi/2. zeros(size(t1))) Seg
2 t2 (-pi/2)(pi/n)(pi/2) ft2 -t2 ft22
(-t2 zeros(size(t2))) Seg 3 t3
(pi/2)(pi/n)pi ft3 -pi/2 ft33 (-pi/2.
zeros(size(t3))) t4 t1 t2 t3 f4 ft11
ft22 ft33 end
109
Fourier Coefficient Program (in MATLAB)
function coeff() global t4 f4
nterms_ak t4_first t4(1) t4_last
t4(length(t4)) p_value ((t4_last -
t4_first)/2) period 2 pi angfreq 2 pi
/ period calculation of a0 a0 trapz(t4,f4)
/ (2pi) angfreq zeros(nterms_ak,1) for i
1nterms_ak angfreq(i) ipi/p_value end
102
109
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Fourier Coefficient Program (in MATLAB)
calculation of ak and bk a
zeros(nterms_ak,1) b zeros(nterms_ak,1) for
i 1nterms_ak theta
ipi.t4./p_value c_value cos(theta)
s_value sin(theta) a_value
c_value.f4 b_value s_value.f4
a(i) (trapz(t4,a_value))/p_value
b(i) (trapz(t4,b_value))/p_value end a
b
103
110
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Fourier Coefficient Program (in MATLAB)
f_ft (a0)ones(size(t4)) for i
1nterms_ak f_ft f_ft
(a(i).cos(ipi.t4./p_value))
(b(i).sin(ipi.t4./p_value)) end
figure plot(t4,f4) hold on
plot(t4,f_ft,'-.') grid on xlabel('t')
ylabel('f(t)') legend('f(t)', 'Fourier
series') end
104
111
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Fourier Coefficient Program (in MATLAB)
f_ft (a0)ones(size(t4)) for i
1nterms_ak f_ft f_ft
(a(i).cos(ipi.t4./p_value))
(b(i).sin(ipi.t4./p_value)) end
113
Fourier Coefficient Program (in MATLAB)
figure plot(t4,f4) hold on
plot(t4,f_ft,'-.') grid on xlabel('t')
ylabel('f(t)') legend('f(t)', 'Fourier
series') end
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113
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Acknowledgement

• This instructional power point brought to you by
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116

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• This material is based upon work supported by the
  National Science Foundation under Grant
  0717624. Any opinions, findings, and conclusions
  or recommendations expressed in this material are
  those of the author(s) and do not necessarily
  reflect the views of the National Science
  Foundation.

117
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