Numerical Methods Fast Fourier Transform Part: Theoretical Development of Fast Fourier Transform

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Major: All Engineering Majors Authors: Duc Nguyen Numerical Methods for STEM undergraduates 6/3/20165 Chapter 11.06: Theoretical Development of Fast Fourier Transform Lecture # 16

Theoretical Development of FFT For the case of Recall Equation (5) from Chapter Informal Development of FFT, where 6/3/20166 (11.65) Consider the case. In this case, we can express and as 2-bit binary numbers: (1) (2)

Theoretical Development cont. Eqs. (1) and (2) can also be expressed in compact forms, as following where (3) (4) In the new notations, Eq.(11.65) becomes (5) 6/3/20167

Consider Theoretical Development cont.

Notice that Hence Eq. (5) can be simplified to (7) 6/3/20169

Define (8) Eq. (8) can be modified to (9) Theoretical Development cont.

Hence (10) In matrix notation, Eq.(10) can be written as (11) 6/3/201611

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) 6/3/201612

Hence (14) Now we have (15) 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) Thus, Eq. (16) implies that unscrambling (or bit-reversed operations) the results of will give us the desired results for. 6/3/201614

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Numerical Methods Fast Fourier Transform Part: Theoretical Development of FFT

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Chapter 11.06: Theoretical Development (Contd.) The set of Eqs. (8), (12), and (16) represents the original Cooley-Turkey [1–4] formulation of the FFT. Consider the case In this case, and can be expressed in compact forms (using 3-bit binary numbers) as (18) (17) where 6/3/ Lecture # 17

Theoretical Development cont. In the new notations, Eq. (5) from the Informal Development ppt becomes (19) Consider (20) 6/3/201624

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) 6/3/201625

Theoretical Development cont. Hence (25) Remarks about Eq. (22) (22a) companion nodes (22b) 6/3/201626

skip computation Theoretical Development cont.

Figure 1. Graphical Form of FFT (for the case ). 6/3/201628

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Numerical Methods Fast Fourier Transform Part: Theoretical Development of FFT

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Chapter 11.06: General Case of FFT (Contd.) Consider the case where Eqs. (17) and (18) can be generalized to, where can be any integer number, as following (26) (27) Eq. (5) from Informal Development ppt becomes where (28) (29) 6/3/ Lecture # 18

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) 6/3/201638

General Case of FFT cont. Similarly, the second term of Eq. (29) can be computed as (31) (32) Eq. (28) will eventually become (33) 6/3/201639

General Case of FFT cont. …. Let (34) (35) (36) (37) 6/3/201640

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Numerical Methods Fast Fourier Transform Part: Theoretical Development of FFT

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Chapter 11.06: FFT Algorithms (Contd.) Consider when (where are integers) Let’s first define (38) (39) where (40) (41) (42) (43) 6/3/ Lecture # 19

FFT Algorithms cont. Remarks: a) The smallest value for (when ). b) The largest value for “ “ will occur when and. Hence, Eq. (38) gives Thus, if then 6/3/ Using the above notations, Equation (5) from Informal Development ppt can be expressed as (44)

FFT Algorithms cont. Consider (45) (46) Due to the fact that (47) Substituting Eq. (46) into (44), one gets (48) 6/3/201651

Define (49) (50) Hence (51) General Case of FFT cont.

FFT Algorithms cont. For,then Eqs. (49) and (50) give (52) (53) Expanding Eq. (52), one obtains (52a) 6/3/2016

Similarly, expanding Eq. (53), one gets 54 (53a) General Case of FFT cont.

FFT Algorithms cont. For a typical term corresponding to, Eq. (52a) gives (52b) (52c) For a typical term, such as, Eq. (53a) gives 6/3/ (53b)

A partial/incomplete graph of FFT (based on Eqs. (52b), (52c) and (53b)), for the case is presented in Figure 2. General Case of FFT cont.

Partial/”Incomplete” Graph of FFT Figure 2. An “Incomplete” FFT for. 6/3/201657

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Numerical Methods Fast Fourier Transform Part: Theoretical Development of FFT

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Chapter 11.06: FFT Algorithms (Contd.) Consider the case In this case, utilizing Eqs. (40-43) into Eqs. (38) and (39), one obtains (54) (55) where (56) (57) (58) (59) 6/3/ Lecture # 20

FFT Algorithms cont. Then, Eq. (5) from Informal Development ppt becomes (60) Consider (61) Substituting Eq. (61) into Eq. (60), one obtains (62) 6/3/

FFT Algorithms cont. Define Hence (63) (64) (65) Expanding (the summation) of Eqs. (63) and (64), one gets (66) 6/3/201668

FFT Algorithms cont. Assuming, and then the previous equation becomes (67) Similarly, one has (68) Assuming, and, then (69) 6/3/201669

FFT Algorithms cont. Thus, computation of each term for arrays and (see Eqs. (67) and (69)) will require the “previous” 4 terms and 2 terms, respectively. The partial (or incomplete) graphical display for FFT (with ;, ) based on Eqs. (67) and (69), is shown in Figure 3. 6/3/201670

FFT Algorithms cont. Figure 3. An “Incomplete” FFT for 6/3/201671

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Numerical Methods Fast Fourier Transform Part: Theoretical Development of FFT

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Chapter 11.06: General FFT Algorithms (Contd.) The relationship between FFT algorithms for versus is important to notice. For the more general case, such as where are any ‘integer” numbers. (70) 6/3/ Lecture # 21

With (should refer to Eqs ) (73) (74) Define (should refer/compare to Eqs ) (71) (72) General FFT Algorithms

General FFT Algorithms cont. In order to simplify the “arithmetic efforts”, and to easily identify the “patterns” of the general FFT algorithms/formulas, we assume, and therefore the following case will be considered (75) Hence, Eqs. (71) to (74) will be simplified to 6/3/ (76) (77)

(78) (79) General FFT Algorithms cont.

84 General FFT Algorithms cont. Eq. (5) from Informal Development ppt can be expressed as (80) where (81) (82) (83) 6/3/2016

General FFT Algorithms cont. where (84) (85) since 6/3/201685

General FFT Algorithms cont. and with (88) (87) (86) 6/3/201686

Substituting Eqs. (85), (87), and (88) into Eq. (83), and using Eq. (81), then Eq. (80) will become (89) General FFT Algorithms cont.

Define (90) (91) (92) Then (93) 6/3/201688

General FFT Algorithms cont. 6/3/ (Connections betweenandFFT Algorithms) In order to see the “connections” between the general FFT algorithms (such as ) and the base 2 FFT algorithms (such as ), we now consider the special case where

Hence we have that In this case, Eqs. (90) to (92) can be simplified to (94) (95) (96) In fact, Eqs. (94) to (96) are “identical” to the earlier derived Eqs.(22) to (24). General FFT Algorithms cont.

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Numerical Methods Fast Fourier Transform Part: Twiddle Factor FFT Algorithms

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Chapter 11.06: Twiddle Factor FFT Algorithms (Contd.) Consider the case To facilitate the discussions for better understanding about the “improved FFT” algorithms by using the “twiddle factor” [1-4], a specific case for will be explained in the following paragraphs. 6/3/ Lecture # 22

Eq. (48) (97) can be re-written as Twiddle Factor FFT Algorithms

101 Twiddle Factor FFT Algorithms cont. The factor can be included either in the “inner summation” or in the “outer summation” (as shown in Eq. (50). In this section, however, the factor is included in the “inner summation”. Thus, one defines (98) (99) (100) 6/3/2016

Twiddle Factor FFT Algorithms cont. Remarks: a) Consider the following term in Eq. (98) (101) Consider the following few possibilities for, in Table 1. 6/3/

Twiddle Factor FFT Algorithms cont i3 1 -i Table 1. Values of twiddle factor in FFT. Thus depending on the numerical (integer) value of, the value of can only be. Hence there is “no multiplication” involved in computing. 6/3/

Twiddle Factor FFT Algorithms cont. (b) Since the twiddle factor is included in the first vector, as shown in Eq (98), the remaining factor can also be either Thus there is also “no multiplication” involved in calculating the second vector as shown in Eq. (99). c) The twiddle factor can also be applied for the case, or for the more general case 6/3/

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 1 st segment function =for the 2 nd segment function =for the 3 rd segment /3/2016

(b) Output Descriptions The numerical values of the unknown Fourier coefficients are printed.

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; /3/2016

% 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 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) / (2*pi) angfreq = zeros(nterms_ak,1); for i = 1:nterms_ak angfreq(i) = i*pi/p_value; end /3/2016 Fourier Coefficient Program (in MATLAB)

% calculation of ak and bk a = zeros(nterms_ak,1); b = zeros(nterms_ak,1); for i = 1:nterms_ak theta = i*pi.*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 /3/2016 Fourier Coefficient Program (in MATLAB)

f_ft = (a0)*ones(size(t4)); for i = 1:nterms_ak f_ft = f_ft + (a(i).*cos(i*pi.*t4./p_value)) + (b(i).*sin(i*pi.*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 /3/2016 Fourier Coefficient Program (in MATLAB)

f_ft = (a0)*ones(size(t4)); for i = 1:nterms_ak f_ft = f_ft + (a(i).*cos(i*pi.*t4./p_value)) + (b(i).*sin(i*pi.*t4./p_value)); end 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 /3/2016 Fourier Coefficient Program (in MATLAB)

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