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Fourier series example

This MATLAB code provides an example of plotting a truncated Fourier series representation of a square wave signal. It computes the Fourier series in both complex exponential form (yce) and trigonometric form (yt) up to the Nth term, where N is an odd integer. It plots the original square wave, the truncated Fourier series approximations yce and yt, and their amplitude and phase spectra. The code demonstrates how to calculate and visualize truncated Fourier series representations of a periodic signal.

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Fourier Series Example MATLAB Code % ***** MATLAB Code Starts Here ***** % %FOURIER_SERIES_01_MAT % fig_size = [232 84 774 624]; x = [0.1 0.9 0.1]; % 1 period of x(t) x = [x x x x]; % 4 periods of x(t) tx = [-2 -1 0 0 1 2 2 3 4 4 5 6]; % time points for x(t) figure(1),plot(tx,x),grid,xlabel('Time (s)'),ylabel('Amplitude'),... title('Periodic Signal x(t)'),axis([-2 6 0 1]),... set(gcf,'Position',fig_size) % a0 = 0.5; % DC component of Fourier Series ph0 = 0; n = [1 3 5 7 9]; % Values of n to be evaluated an = -3.2 ./ (pi * n).^2; % Fourier Series coefficients mag_an = abs(an); ph_an = -180 * ones(1,length(n)); % n = [0 n]; mag_an = [a0 mag_an]; % Including a0 with a_n ph_an = [ph0 ph_an]; % figure(2),clf,subplot(211),plot(n,mag_an,'o'),grid,xlabel('Harmonic Number'),...
ylabel('Magnitude'),title('Fourier Series Magnitude'),axis([0 10 0 0.6]),... set(gcf,'Position',fig_size) % subplot(212),plot(n,ph_an,'o'),grid,xlabel('Harmonic Number'),... ylabel('Phase (deg)'),title('Fourier Series Phase'),axis([0 10 -200 0]),... set(gcf,'Position',fig_size) % w0 = pi; % Fundamental Frequency t = [-2:0.002:6]; % time vector for approximations % x1 = 0; % approximation with DC + 1 term for i = 1:2 x1 = x1 + mag_an(i)*cos(n(i)*w0*t + ph_an(i)*pi/180); end % x2 = x1; % approximation with DC + 2 terms i = 3; x2 = x2 + mag_an(i)*cos(n(i)*w0*t + ph_an(i)*pi/180); % x3 = x2; % approximation with DC + 3 terms i = 4; x3 = x3 + mag_an(i)*cos(n(i)*w0*t + ph_an(i)*pi/180); % x4 = x3; % approximation with DC + 5 terms for i = 5:6 x4 = x4 + mag_an(i)*cos(n(i)*w0*t + ph_an(i)*pi/180); end
% figure(3),subplot(221),plot(t,x1),grid,xlabel('Time (s)'),... ylabel('Amplitude'),title('DC + 1 Term'),axis([-2 6 0 1]),... subplot(222),plot(t,x2),grid,xlabel('Time (s)'),... ylabel('Amplitude'),title('DC + 2 Terms'),axis([-2 6 0 1]),... subplot(223),plot(t,x3),grid,xlabel('Time (s)'),... ylabel('Amplitude'),title('DC + 3 Terms'),axis([-2 6 0 1]),... subplot(224),plot(t,x4),grid,xlabel('Time (s)'),... ylabel('Amplitude'),title('DC + 5 Terms'),axis([-2 6 0 1]),... set(gcf,'Position',fig_size) % % % ***** MATLAB Code Stops Here ***** Fourier Series Example #2 MATLAB Code % ***** MATLAB Code Starts Here ***** % %FOURIER_SERIES_02_MAT % fig_size = [232 84 774 624]; T0 = 8; w0 = 2*pi/8; t = linspace(-8,16,1001); a0 = 0.25; n = 1:50;
an = (1./(pi*n)) .* sin(n*pi/2); bn = (1./(pi*n)) .* (1 - cos(n*pi/2)); x1 = a0; for i = 1:10 x1 = x1 + an(i)*cos(i*w0*t) + bn(i)*sin(i*w0*t); end x2 = x1; for i = 11:30 x2 = x2 + an(i)*cos(i*w0*t) + bn(i)*sin(i*w0*t); end x3 = x2; for i = 31:50 x3 = x3 + an(i)*cos(i*w0*t) + bn(i)*sin(i*w0*t); end A0 = a0; An = sqrt(an.^2 + bn.^2); thn = atan2(-bn,an)*180/pi; X0 = A0; Xn = An/2; figure(1),clf,plot([-8 -6],[1 1],'b-',[-6 -6],[1 0],'b--',[-6 0],[0 0],'b- ',[0 2],[1 1],'b-',[2 8],[0 0],'b-',... [8 10],[1 1],'b-',[10 16],[0 0],'b-',[0 0],[0 1],'b--',[2 2],[1 0],'b--',[8 8],[0 1],'b--',... [10 10],[1 0],'b--',[16 16],[0 1],'b--'),... axis([-8 16 -.5 1.5]),plotax,xlabel('Time (s)'),ylabel('Amplitude'),title('Periodic Pulse Train x(t)'),... set(gcf,'Position',fig_size),text(5,-0.2,'T_0 = 8 s'),text(5,-0.3,'Pulse width = T_0/4') figure(2),clf,subplot(311),plot(t,x1),subplot(312),plot(t,x2),subplot(313), plot(t,x3),...
subplot(311),ylabel('Amplitude'),title('Fourier Series Representation of x(t) with 10 Terms'),... subplot(312),ylabel('Amplitude'),title('Fourier Series Representation of x(t) with 30 Terms'),... subplot(313),ylabel('Amplitude'),title('Fourier Series Representation of x(t) with 50 Terms'),xlabel('Time (s)'),... for i = 1:3,subplot(3,1,i),... hold on,plot([0 2],[1 1],'r-',[2 8],[0 0],'r-',[8 10],[1 1],'r-',[10 16],[0 0],'r-',... [0 0],[0 1],'r--',[2 2],[1 0],'r--',[8 8],[0 1],'r--',[10 10],[1 0],'r-- ',[16 16],[0 1],'r--',... [-8 -6],[1 1],'r-',[-6 -6],[1 0],'r--',[-6 0],[0 0],'r-'),hold off,... axis([-8 16 -0.5 1.5]),plotax end set(gcf,'Position',fig_size) figure(3),clf,subplot(211),plot(0,a0,'ro',n,an,'o'),axis([-5 50 -0.2 0.5]),plotax,... hold on,plot([10.5 10.5],[-0.2 0.5],'r--',[30.5 30.5],[-0.2 0.5],'r-- '),hold off,... xlabel('Harmonic Number'),ylabel('Amplitude'),title('Trig Fourier Series Coefficients a_n for x(t)'),... subplot(212),plot(n,bn,'o'),axis([-5 50 -0.05 0.35]),plotax,... hold on,plot([10.5 10.5],[-0.05 0.35],'r--',[30.5 30.5],[-0.05 0.35],'r-- '),hold off,... xlabel('Harmonic Number'),ylabel('Amplitude'),title('Trig Fourier Series Coefficients b_n for x(t)'),... set(gcf,'Position',fig_size) figure(4),clf,subplot(211),plot(0,A0,'ro',n*w0,An,'o'),axis([-2*w0 16 -0.1 0.5]),plotax,... xlabel('Frequency (r/s)'),ylabel('Magnitde'),title('Cosine Fourier Series Magnitudes A_n for x(t)'),... subplot(212),plot(n*w0,thn,'o'),v=axis;axis([-2*w0 16 -200 10]),plotax,... xlabel('Frequency (r/s)'),ylabel('Phase (deg)'),title('Cosine Fourier Series Phases Theta_n for x(t)'),... set(gcf,'Position',fig_size)
figure(5),clf,subplot(211),plot(0,X0,'ro',n*w0,Xn,'o',-n*w0,Xn,'o'),axis([- 16 16 -0.1 0.3]),plotax,... xlabel('Frequency (r/s)'),ylabel('Magnitde'),title('Exponential Fourier Series Magnitudes X_n for x(t)'),... subplot(212),plot(n*w0,thn,'o',-n*w0,-thn,'o'),v=axis;axis([-16 16 v(3:4)]),plotax,... xlabel('Frequency (r/s)'),ylabel('Phase (deg)'),title('Exponential Fourier Series Phases Theta_n for x(t)'),... set(gcf,'Position',fig_size) clear i v % Technical discussion about Matlab and issues related to Digital Signal Processing. Your Email Here Join this Group! Post a new Thread fourier series coefficients - Kurt - Dec 1 12:27:01 2009 hello all, I have a one period square wave on the interval[0,2] defined as: y(t)= 1, 0<=t<1 y(t)= 0, 1<=t<2 I need to find the fourier series coefficients,ck, with k=-10,-9,...,9,10
I heard using a for loop would work but I am completely stuck on how to move through this problem. All help is greatly appreciated, Kurt ______________________________ New Code Sharing Section now Live on DSPRelated.com. Learn about the Reward Program for Contributors here. (You need to be a member of matlab -- send a blank email to matlab-subscribe@yahoogroups.com ) Re: fourier series coefficients - vishwa - Dec 3 7:52:08 2009 you can try for k=-10:1:10 c(k+11) = here you enter the Ck equation; % you cant have negative indexing in MATLAB end Now c gives you the coefficients rgds vishwanath ________________________________ From: Kurt <k...@sbcglobal.net> To: m...@yahoogroups.com Sent: Tue, 1 December, 2009 12:50:19 PM Subject: [matlab] fourier series coefficients  hello all, I have a one period square wave on the interval[0,2] defined as: y(t)= 1, 0<=t<1 y(t)= 0, 1<=t<2 I need to find the fourier series coefficients, ck, with k=-10,-9,... ,9,10 I heard using a for loop would work but I am completely stuck on how to move through this problem. All help is greatly appreciated, Kurt ______________________________ New Code Sharing Section now Live on DSPRelated.com. Learn about the Reward Program for Contributors here. (You need to be a member of matlab -- send a blank email to matlab-subscribe@yahoogroups.com ) Re: fourier series coefficients - Vaibhav Singh - Dec 4 7:44:25 2009
Hey.. For fourier coeff u have to find the fft of the given sequence using matlab. Since u have to find the coeff for kranging from -10:1:10, i.e.21 points u have to define ur function in time domain in 21 samples. Take the fft of these 21 samples. The resultant is your desired fourier coeff . Regards-vaibhav On Thu, Dec 3, 2009 at 5:01 PM, vishwa <v...@yahoo.com> wrote: > you can try > > for k=-10:1:10 > c(k+11) = here you enter the Ck equation; % you cant have negative indexing > in MATLAB > end > > Now > > c gives you the coefficients > > rgds > vishwanath > > ________________________________ > From: Kurt <k...@sbcglobal.net <keg1606%40sbcglobal.net>> > To: m...@yahoogroups.com <matlab%40yahoogroups.com> > Sent: Tue, 1 December, 2009 12:50:19 PM > Subject: [matlab] fourier series coefficients > > hello all, > I have a one period square wave on the interval[0,2] defined as: > y(t)= 1, 0<=t<1 > y(t)= 0, 1<=t<2 > I need to find the fourier series coefficients, ck, with > k=-10,-9,... ,9,10 > I heard using a for loop would work but I am completely stuck on how to > move through this problem. > All help is greatly appreciated, > Kurt > > > -- Vaibhav Singh BE(Hons.) Electronics And Instrumentation BITS-Pilani
EE341.01: MATLAB M-FILE FOR PLOTTING TRUNCATED FOURIER SERIES AND ITS SPECTRA MATLAB M-File example6.m: % % Filename: example6.m % % Description: This M-file plots the truncated Fourier Series % representation of a square wave as well as its % amplitude and phase spectrum. clear; % clear all variables clf; % clear all figures N = 11; % summation limit (use N odd) wo = pi; % fundamental frequency (rad/s) c0 = 0; % dc bias t = -3:0.01:3; % declare time values figure(1) % put first two plots on figure 1 % Compute yce, the Fourier Series in complex exponential form yce = c0*ones(size(t)); % initialize yce to c0 for n = -N:2:N, % loop over series index n (odd) cn = 2/(j*n*wo); % Fourier Series Coefficient yce = yce + real(cn*exp(j*n*wo*t)); % Fourier Series computation end subplot(2,1,1) plot([-3 -2 -2 -1 -1 0 0 1 1 2 2 3],... % plot original y(t) [-1 -1 1 1 -1 -1 1 1 -1 -1 1 1], ':'); hold; plot(t,yce); % plot truncated exponential FS xlabel('t (seconds)'); ylabel('y(t)'); ttle = ['EE341.01: Truncated Exponential Fourier Series with N = ',... num2str(N)]; title(ttle); hold; % Compute yt, the Fourier Series in trigonometric form yt = c0*ones(size(t)); % initialize yt to c0 for n = 1:2:N, % loop over series index n (odd) cn = 2/(j*n*wo); % Fourier Series Coefficient yt = yt + 2*abs(cn)*cos(n*wo*t+angle(cn)); % Fourier Series computation end subplot(2,1,2) plot([-3 -2 -2 -1 -1 0 0 1 1 2 2 3],... % plot original y(t) [-1 -1 1 1 -1 -1 1 1 -1 -1 1 1], ':'); hold; % plot truncated trigonometric FS plot(t,yt); xlabel('t (seconds)'); ylabel('y(t)'); ttle = ['EE341.01: Truncated Trigonometric Fourier Series with N = ',... num2str(N)]; title(ttle); hold; % Draw the amplitude spectrum from exponential Fourier Series
figure(2) % put next plots on figure 2 subplot(2,1,1) stem(0,c0); % plot c0 at nwo = 0 hold; for n = -N:2:N, % loop over series index n cn = 2/(j*n*wo); % Fourier Series Coefficient stem(n*wo,abs(cn)) % plot |cn| vs nwo end for n = -N+1:2:N-1, % loop over even series index n cn = 0; % Fourier Series Coefficient stem(n*wo,abs(cn)); % plot |cn| vs nwo end xlabel('w (rad/s)') ylabel('|cn|') ttle = ['EE341.01: Amplitude Spectrum with N = ',num2str(N)]; title(ttle); grid; hold; % Draw the phase spectrum from exponential Fourier Series subplot(2,1,2) stem(0,angle(c0)*180/pi); % plot angle of c0 at nwo = 0 hold; for n = -N:2:N, % loop over odd series index n cn = 2/(j*n*wo); % Fourier Series Coefficient stem(n*wo,angle(cn)*180/pi); % plot |cn| vs nwo end for n = -N+1:2:N-1, % loop over even series index n cn = 0; % Fourier Series Coefficient stem(n*wo,angle(cn)*180/pi); % plot |cn| vs nwo end xlabel('w (rad/s)') ylabel('angle(cn) (degrees)') ttle = ['EE341.01: Phase Spectrum with N = ',num2str(N)]; title(ttle); grid; hold; MATLAB Plots Generated:
Hi, I am trying to write a function to generate Fourier series Coefficients of a given discrete time signal. For instance: x = [1 2 3 4] n = [0 1 2 3] where x holds the values of the signal, and n holds the corresponding time indices. My code for the function is: function a = dtfs(x,n) period = length(x); for k = 1:period a = (1/period)*x(k)*exp(((-j*2*pi)/period)*n(k)); a end i am just not getting the correct values for the fourier series coefficients. Am I setting up the formula wrong? As you can probably tell, i am very very new to MATLAB and I'd also appreciate if someone may guide me in setting up my "for- loop" with a vector to "catch" and store values of "a", so each time the for-loop repeats, the previous value a does not get over- written, instead they are all stored in a vector. Thanks in advance :) Regards Subject: Fourier Series Coefficients From: Andrew Date: 24 Oct, 2008 04:43:01 Message: 2 of 4 Reply to this message Add author to My Watch List View original format Flag as spam I'm guessing the formula, but hopefully the structure of it will help... function a = dtfs(x,n) period = length(x); a = zeros(1, length(x)) for k = 1:period for z = 1:period a(k) = a(k) + x(z) * exp((-j*2*pi)/period * (k-1) * n(z)); end a(k) = a(k) / period; num2str(a(k), '%1.18f'); end
Cheers, Andrew > a = (1/period)*x(k)*exp(((-j*2*pi)/period)*n(k)); >a > end "Raz H" <dilster08@gmail.com> wrote in message <gdrclp$t9b$1@fred.mathworks.com>... > Hi, > > I am trying to write a function to generate Fourier series Coefficients of a given discrete time signal. For instance: > > x = [1 2 3 4] > n = [0 1 2 3] > > where x holds the values of the signal, and n holds the corresponding time indices. > > My code for the function is: > > function a = dtfs(x,n) > period = length(x); > for k = 1:period > a = (1/period)*x(k)*exp(((-j*2*pi)/period)*n(k)); >a > end > > i am just not getting the correct values for the fourier series coefficients. Am I setting up the formula wrong? > > As you can probably tell, i am very very new to MATLAB and I'd also appreciate if someone may guide me in setting up my "for-loop" with a vector to "catch" and store values of "a", so each time the for-loop repeats, the previous value a does not get over-written, instead they are all stored in a vector. > > Thanks in advance :) > > Regards Subject: Fourier Series Coefficients From: Paul Date: 24 Oct, 2008 06:35:05 Message: 3 of 4 Reply to this message Add author to My Watch List View original format Flag as spam
"Andrew" <awbsmith@itee.uq.edu.au> wrote in message <gdrjol$so$1@fred.mathworks.com>... > I'm guessing the formula, but hopefully the structure of it will help... > > function a = dtfs(x,n) > period = length(x); > a = zeros(1, length(x)) > for k = 1:period > for z = 1:period > a(k) = a(k) + x(z) * exp((-j*2*pi)/period * (k-1) * n(z)); > end > a(k) = a(k) / period; > num2str(a(k), '%1.18f'); > end > > Cheers, > Andrew > > > > a = (1/period)*x(k)*exp(((-j*2*pi)/period)*n(k)); >>a > > end > > > > > "Raz H" <dilster08@gmail.com> wrote in message <gdrclp$t9b$1@fred.mathworks.com>... > > Hi, >> > > I am trying to write a function to generate Fourier series Coefficients of a given discrete time signal. For instance: >> > > x = [1 2 3 4] > > n = [0 1 2 3] >> > > where x holds the values of the signal, and n holds the corresponding time indices. >> > > My code for the function is: >> > > function a = dtfs(x,n) > > period = length(x); > > for k = 1:period > > a = (1/period)*x(k)*exp(((-j*2*pi)/period)*n(k)); >>a > > end >> > > i am just not getting the correct values for the fourier series coefficients. Am I setting up the formula wrong? >> > > As you can probably tell, i am very very new to MATLAB and I'd also appreciate if someone may guide me in setting up my "for-loop" with a vector to "catch" and store values of "a", so each time the for-loop repeats, the previous value a does not get over-written, instead they are all stored in a vector. >> > > Thanks in advance :) >> > > Regards
Was this a HW problem? It looks like one to me! Subject: Fourier Series Coefficients From: Raz H Date: 24 Oct, 2008 06:54:02 Message: 4 of 4 Reply to this message Add author to My Watch List View original format Flag as spam "Paul" <par@ceri.memphis.edu> wrote in message <gdrqap$sl5$1@fred.mathworks.com>... > "Andrew" <awbsmith@itee.uq.edu.au> wrote in message <gdrjol$so$1@fred.mathworks.com>... > > I'm guessing the formula, but hopefully the structure of it will help... >> > > function a = dtfs(x,n) > > period = length(x); > > a = zeros(1, length(x)) > > for k = 1:period > > for z = 1:period > > a(k) = a(k) + x(z) * exp((-j*2*pi)/period * (k-1) * n(z)); > > end > > a(k) = a(k) / period; > > num2str(a(k), '%1.18f'); > > end >> > > Cheers, > > Andrew >> >> > > > a = (1/period)*x(k)*exp(((-j*2*pi)/period)*n(k)); >>>a > > > end >> >> >> >> > > "Raz H" <dilster08@gmail.com> wrote in message <gdrclp$t9b$1@fred.mathworks.com>... > > > Hi, >>> > > > I am trying to write a function to generate Fourier series Coefficients of a given discrete time signal. For instance: >>> > > > x = [1 2 3 4] > > > n = [0 1 2 3] >>> > > > where x holds the values of the signal, and n holds the corresponding time indices. >>> > > > My code for the function is:
>>> > > > function a = dtfs(x,n) > > > period = length(x); > > > for k = 1:period > > > a = (1/period)*x(k)*exp(((-j*2*pi)/period)*n(k)); >>>a > > > end >>> > > > i am just not getting the correct values for the fourier series coefficients. Am I setting up the formula wrong? >>> > > > As you can probably tell, i am very very new to MATLAB and I'd also appreciate if someone may guide me in setting up my "for-loop" with a vector to "catch" and store values of "a", so each time the for-loop repeats, the previous value a does not get over-written, instead they are all stored in a vector. >>> > > > Thanks in advance :) >>> > > > Regards > > Was this a HW problem? It looks like one to me! @Andrew Thank you very much! I guess my loop was not set up correctly, plus I was not signifying the time indices correctly. @Paul This was not a homework problem, though I'll be taking Signals soon, so I am trying to become familiar with MATLAB. Thanks to all who replied! :) EE341.01: MATLAB M-FILE FOR PLOTTING TRUNCATED FOURIER SERIES This example shows a MATLAB M-file for plotting a truncated Fourier Series. Various numbers of terms are used. MATLAB M-File example5.m: % % Filename: example5.m % % Description: Example to show how the truncated Fourier series in % complex exponential form approximates the real % signal. More and more terms are taken showing a % better and better representation of the original signal. % clear; % clear all variables clf; % clear all figures
% Define parameters to plot original sawtooth tr = [-1 0 0 1 1 2 2]; yr = [0 1 0 1 0 1 0]; % Plot Truncated Fourier Series Approximation (N = 1) N = 1; % define number of terms to use (n = -N..N) c0 = 0.5; % define dc bias coefficient t = -1:0.001:2; % define time values for y(t) y = c0 * ones(size(t)); % let initial y = c0 (dc bias) for all times for n = -N:-1, % compute y for negative n cn = exp(j*pi/2)/(2*pi*n); y = y + real(cn * exp(j*n*2*pi*t)); end; % compute y for positive n and add to y for n = 1:N, % found using negative n cn = exp(j*pi/2)/(2*pi*n); y = y + real(cn * exp(j*n*2*pi*t)); end; subplot(2,2,1); % plot approximation plot(t,y); hold; plot(tr,yr,':'); hold; xlabel('time (seconds)'); ylabel('y(t) approximation'); title('EE341.01: Truncated FS, -1<=n<=1'); % Plot Truncated Fourier Series Approximation (N = 2) clear; % clear all variables N = 2; % define number of terms to use (n = -N..N) c0 = 0.5; % define dc bias coefficient t = -1:0.001:2; % define time values for y(t) y = c0 * ones(size(t)); % let initial y = c0 (dc bias) for all times for n = -N:-1, % compute y for negative n cn = exp(j*pi/2)/(2*pi*n); y = y + real(cn * exp(j*n*2*pi*t)); end; % compute y for positive n and add to y for n = 1:N, % found using negative n cn = exp(j*pi/2)/(2*pi*n); y = y + real(cn * exp(j*n*2*pi*t)); end; subplot(2,2,2); % plot approximation plot(t,y); hold; plot(tr,yr,':'); hold; xlabel('time (seconds)'); ylabel('y(t) approximation'); title('EE341.01: Truncated FS, -2<=n<=2'); % Plot Truncated Fourier Series Approximation (N = 3) clear; % clear all variables
N = 3; % define number of terms to use (n = -N..N) c0 = 0.5; % define dc bias coefficient t = -1:0.001:2; % define time values for y(t) y = c0 * ones(size(t)); % let initial y = c0 (dc bias) for all times for n = -N:-1, % compute y for negative n cn = exp(j*pi/2)/(2*pi*n); y = y + real(cn * exp(j*n*2*pi*t)); end; % compute y for positive n and add to y for n = 1:N, % found using negative n cn = exp(j*pi/2)/(2*pi*n); y = y + real(cn * exp(j*n*2*pi*t)); end; subplot(2,2,3); % plot approximation plot(t,y); hold; plot(tr,yr,':'); hold; xlabel('time (seconds)'); ylabel('y(t) approximation'); title('EE341.01: Truncated FS, -3<=n<=3'); % Plot Truncated Fourier Series Approximation (N = 10) clear; % clear all variables N = 10; % define number of terms to use (n = -N..N) c0 = 0.5; % define dc bias coefficient t = -1:0.001:2; % define time values for y(t) y = c0 * ones(size(t)); % let initial y = c0 (dc bias) for all times for n = -N:-1, % compute y for negative n cn = exp(j*pi/2)/(2*pi*n); y = y + real(cn * exp(j*n*2*pi*t)); end; % compute y for positive n and add to y for n = 1:N, % found using negative n cn = exp(j*pi/2)/(2*pi*n); y = y + real(cn * exp(j*n*2*pi*t)); end; subplot(2,2,4); % plot approximation plot(t,y); hold; plot(tr,yr,':'); hold; xlabel('time (seconds)'); ylabel('y(t) approximation'); title('EE341.01: Truncated FS, -10<=n<=10'); MATLAB Plot Generated:
Fourier series example

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Fourier series example

  • 1.
    Fourier Series Example MATLABCode % ***** MATLAB Code Starts Here ***** % %FOURIER_SERIES_01_MAT % fig_size = [232 84 774 624]; x = [0.1 0.9 0.1]; % 1 period of x(t) x = [x x x x]; % 4 periods of x(t) tx = [-2 -1 0 0 1 2 2 3 4 4 5 6]; % time points for x(t) figure(1),plot(tx,x),grid,xlabel('Time (s)'),ylabel('Amplitude'),... title('Periodic Signal x(t)'),axis([-2 6 0 1]),... set(gcf,'Position',fig_size) % a0 = 0.5; % DC component of Fourier Series ph0 = 0; n = [1 3 5 7 9]; % Values of n to be evaluated an = -3.2 ./ (pi * n).^2; % Fourier Series coefficients mag_an = abs(an); ph_an = -180 * ones(1,length(n)); % n = [0 n]; mag_an = [a0 mag_an]; % Including a0 with a_n ph_an = [ph0 ph_an]; % figure(2),clf,subplot(211),plot(n,mag_an,'o'),grid,xlabel('Harmonic Number'),...
  • 2.
    ylabel('Magnitude'),title('Fourier Series Magnitude'),axis([010 0 0.6]),... set(gcf,'Position',fig_size) % subplot(212),plot(n,ph_an,'o'),grid,xlabel('Harmonic Number'),... ylabel('Phase (deg)'),title('Fourier Series Phase'),axis([0 10 -200 0]),... set(gcf,'Position',fig_size) % w0 = pi; % Fundamental Frequency t = [-2:0.002:6]; % time vector for approximations % x1 = 0; % approximation with DC + 1 term for i = 1:2 x1 = x1 + mag_an(i)*cos(n(i)*w0*t + ph_an(i)*pi/180); end % x2 = x1; % approximation with DC + 2 terms i = 3; x2 = x2 + mag_an(i)*cos(n(i)*w0*t + ph_an(i)*pi/180); % x3 = x2; % approximation with DC + 3 terms i = 4; x3 = x3 + mag_an(i)*cos(n(i)*w0*t + ph_an(i)*pi/180); % x4 = x3; % approximation with DC + 5 terms for i = 5:6 x4 = x4 + mag_an(i)*cos(n(i)*w0*t + ph_an(i)*pi/180); end
  • 3.
    % figure(3),subplot(221),plot(t,x1),grid,xlabel('Time (s)'),... ylabel('Amplitude'),title('DC +1 Term'),axis([-2 6 0 1]),... subplot(222),plot(t,x2),grid,xlabel('Time (s)'),... ylabel('Amplitude'),title('DC + 2 Terms'),axis([-2 6 0 1]),... subplot(223),plot(t,x3),grid,xlabel('Time (s)'),... ylabel('Amplitude'),title('DC + 3 Terms'),axis([-2 6 0 1]),... subplot(224),plot(t,x4),grid,xlabel('Time (s)'),... ylabel('Amplitude'),title('DC + 5 Terms'),axis([-2 6 0 1]),... set(gcf,'Position',fig_size) % % % ***** MATLAB Code Stops Here ***** Fourier Series Example #2 MATLAB Code % ***** MATLAB Code Starts Here ***** % %FOURIER_SERIES_02_MAT % fig_size = [232 84 774 624]; T0 = 8; w0 = 2*pi/8; t = linspace(-8,16,1001); a0 = 0.25; n = 1:50;
  • 4.
    an = (1./(pi*n)).* sin(n*pi/2); bn = (1./(pi*n)) .* (1 - cos(n*pi/2)); x1 = a0; for i = 1:10 x1 = x1 + an(i)*cos(i*w0*t) + bn(i)*sin(i*w0*t); end x2 = x1; for i = 11:30 x2 = x2 + an(i)*cos(i*w0*t) + bn(i)*sin(i*w0*t); end x3 = x2; for i = 31:50 x3 = x3 + an(i)*cos(i*w0*t) + bn(i)*sin(i*w0*t); end A0 = a0; An = sqrt(an.^2 + bn.^2); thn = atan2(-bn,an)*180/pi; X0 = A0; Xn = An/2; figure(1),clf,plot([-8 -6],[1 1],'b-',[-6 -6],[1 0],'b--',[-6 0],[0 0],'b- ',[0 2],[1 1],'b-',[2 8],[0 0],'b-',... [8 10],[1 1],'b-',[10 16],[0 0],'b-',[0 0],[0 1],'b--',[2 2],[1 0],'b--',[8 8],[0 1],'b--',... [10 10],[1 0],'b--',[16 16],[0 1],'b--'),... axis([-8 16 -.5 1.5]),plotax,xlabel('Time (s)'),ylabel('Amplitude'),title('Periodic Pulse Train x(t)'),... set(gcf,'Position',fig_size),text(5,-0.2,'T_0 = 8 s'),text(5,-0.3,'Pulse width = T_0/4') figure(2),clf,subplot(311),plot(t,x1),subplot(312),plot(t,x2),subplot(313), plot(t,x3),...
  • 5.
    subplot(311),ylabel('Amplitude'),title('Fourier Series Representationof x(t) with 10 Terms'),... subplot(312),ylabel('Amplitude'),title('Fourier Series Representation of x(t) with 30 Terms'),... subplot(313),ylabel('Amplitude'),title('Fourier Series Representation of x(t) with 50 Terms'),xlabel('Time (s)'),... for i = 1:3,subplot(3,1,i),... hold on,plot([0 2],[1 1],'r-',[2 8],[0 0],'r-',[8 10],[1 1],'r-',[10 16],[0 0],'r-',... [0 0],[0 1],'r--',[2 2],[1 0],'r--',[8 8],[0 1],'r--',[10 10],[1 0],'r-- ',[16 16],[0 1],'r--',... [-8 -6],[1 1],'r-',[-6 -6],[1 0],'r--',[-6 0],[0 0],'r-'),hold off,... axis([-8 16 -0.5 1.5]),plotax end set(gcf,'Position',fig_size) figure(3),clf,subplot(211),plot(0,a0,'ro',n,an,'o'),axis([-5 50 -0.2 0.5]),plotax,... hold on,plot([10.5 10.5],[-0.2 0.5],'r--',[30.5 30.5],[-0.2 0.5],'r-- '),hold off,... xlabel('Harmonic Number'),ylabel('Amplitude'),title('Trig Fourier Series Coefficients a_n for x(t)'),... subplot(212),plot(n,bn,'o'),axis([-5 50 -0.05 0.35]),plotax,... hold on,plot([10.5 10.5],[-0.05 0.35],'r--',[30.5 30.5],[-0.05 0.35],'r-- '),hold off,... xlabel('Harmonic Number'),ylabel('Amplitude'),title('Trig Fourier Series Coefficients b_n for x(t)'),... set(gcf,'Position',fig_size) figure(4),clf,subplot(211),plot(0,A0,'ro',n*w0,An,'o'),axis([-2*w0 16 -0.1 0.5]),plotax,... xlabel('Frequency (r/s)'),ylabel('Magnitde'),title('Cosine Fourier Series Magnitudes A_n for x(t)'),... subplot(212),plot(n*w0,thn,'o'),v=axis;axis([-2*w0 16 -200 10]),plotax,... xlabel('Frequency (r/s)'),ylabel('Phase (deg)'),title('Cosine Fourier Series Phases Theta_n for x(t)'),... set(gcf,'Position',fig_size)
  • 6.
    figure(5),clf,subplot(211),plot(0,X0,'ro',n*w0,Xn,'o',-n*w0,Xn,'o'),axis([- 16 16 -0.10.3]),plotax,... xlabel('Frequency (r/s)'),ylabel('Magnitde'),title('Exponential Fourier Series Magnitudes X_n for x(t)'),... subplot(212),plot(n*w0,thn,'o',-n*w0,-thn,'o'),v=axis;axis([-16 16 v(3:4)]),plotax,... xlabel('Frequency (r/s)'),ylabel('Phase (deg)'),title('Exponential Fourier Series Phases Theta_n for x(t)'),... set(gcf,'Position',fig_size) clear i v % Technical discussion about Matlab and issues related to Digital Signal Processing. Your Email Here Join this Group! Post a new Thread fourier series coefficients - Kurt - Dec 1 12:27:01 2009 hello all, I have a one period square wave on the interval[0,2] defined as: y(t)= 1, 0<=t<1 y(t)= 0, 1<=t<2 I need to find the fourier series coefficients,ck, with k=-10,-9,...,9,10
  • 7.
    I heard usinga for loop would work but I am completely stuck on how to move through this problem. All help is greatly appreciated, Kurt ______________________________ New Code Sharing Section now Live on DSPRelated.com. Learn about the Reward Program for Contributors here. (You need to be a member of matlab -- send a blank email to matlab-subscribe@yahoogroups.com ) Re: fourier series coefficients - vishwa - Dec 3 7:52:08 2009 you can try for k=-10:1:10 c(k+11) = here you enter the Ck equation; % you cant have negative indexing in MATLAB end Now c gives you the coefficients rgds vishwanath ________________________________ From: Kurt <k...@sbcglobal.net> To: m...@yahoogroups.com Sent: Tue, 1 December, 2009 12:50:19 PM Subject: [matlab] fourier series coefficients  hello all, I have a one period square wave on the interval[0,2] defined as: y(t)= 1, 0<=t<1 y(t)= 0, 1<=t<2 I need to find the fourier series coefficients, ck, with k=-10,-9,... ,9,10 I heard using a for loop would work but I am completely stuck on how to move through this problem. All help is greatly appreciated, Kurt ______________________________ New Code Sharing Section now Live on DSPRelated.com. Learn about the Reward Program for Contributors here. (You need to be a member of matlab -- send a blank email to matlab-subscribe@yahoogroups.com ) Re: fourier series coefficients - Vaibhav Singh - Dec 4 7:44:25 2009
  • 8.
    Hey.. For fourier coeffu have to find the fft of the given sequence using matlab. Since u have to find the coeff for kranging from -10:1:10, i.e.21 points u have to define ur function in time domain in 21 samples. Take the fft of these 21 samples. The resultant is your desired fourier coeff . Regards-vaibhav On Thu, Dec 3, 2009 at 5:01 PM, vishwa <v...@yahoo.com> wrote: > you can try > > for k=-10:1:10 > c(k+11) = here you enter the Ck equation; % you cant have negative indexing > in MATLAB > end > > Now > > c gives you the coefficients > > rgds > vishwanath > > ________________________________ > From: Kurt <k...@sbcglobal.net <keg1606%40sbcglobal.net>> > To: m...@yahoogroups.com <matlab%40yahoogroups.com> > Sent: Tue, 1 December, 2009 12:50:19 PM > Subject: [matlab] fourier series coefficients > > hello all, > I have a one period square wave on the interval[0,2] defined as: > y(t)= 1, 0<=t<1 > y(t)= 0, 1<=t<2 > I need to find the fourier series coefficients, ck, with > k=-10,-9,... ,9,10 > I heard using a for loop would work but I am completely stuck on how to > move through this problem. > All help is greatly appreciated, > Kurt > > > -- Vaibhav Singh BE(Hons.) Electronics And Instrumentation BITS-Pilani
  • 9.
    EE341.01: MATLAB M-FILEFOR PLOTTING TRUNCATED FOURIER SERIES AND ITS SPECTRA MATLAB M-File example6.m: % % Filename: example6.m % % Description: This M-file plots the truncated Fourier Series % representation of a square wave as well as its % amplitude and phase spectrum. clear; % clear all variables clf; % clear all figures N = 11; % summation limit (use N odd) wo = pi; % fundamental frequency (rad/s) c0 = 0; % dc bias t = -3:0.01:3; % declare time values figure(1) % put first two plots on figure 1 % Compute yce, the Fourier Series in complex exponential form yce = c0*ones(size(t)); % initialize yce to c0 for n = -N:2:N, % loop over series index n (odd) cn = 2/(j*n*wo); % Fourier Series Coefficient yce = yce + real(cn*exp(j*n*wo*t)); % Fourier Series computation end subplot(2,1,1) plot([-3 -2 -2 -1 -1 0 0 1 1 2 2 3],... % plot original y(t) [-1 -1 1 1 -1 -1 1 1 -1 -1 1 1], ':'); hold; plot(t,yce); % plot truncated exponential FS xlabel('t (seconds)'); ylabel('y(t)'); ttle = ['EE341.01: Truncated Exponential Fourier Series with N = ',... num2str(N)]; title(ttle); hold; % Compute yt, the Fourier Series in trigonometric form yt = c0*ones(size(t)); % initialize yt to c0 for n = 1:2:N, % loop over series index n (odd) cn = 2/(j*n*wo); % Fourier Series Coefficient yt = yt + 2*abs(cn)*cos(n*wo*t+angle(cn)); % Fourier Series computation end subplot(2,1,2) plot([-3 -2 -2 -1 -1 0 0 1 1 2 2 3],... % plot original y(t) [-1 -1 1 1 -1 -1 1 1 -1 -1 1 1], ':'); hold; % plot truncated trigonometric FS plot(t,yt); xlabel('t (seconds)'); ylabel('y(t)'); ttle = ['EE341.01: Truncated Trigonometric Fourier Series with N = ',... num2str(N)]; title(ttle); hold; % Draw the amplitude spectrum from exponential Fourier Series
  • 10.
    figure(2) % put next plots on figure 2 subplot(2,1,1) stem(0,c0); % plot c0 at nwo = 0 hold; for n = -N:2:N, % loop over series index n cn = 2/(j*n*wo); % Fourier Series Coefficient stem(n*wo,abs(cn)) % plot |cn| vs nwo end for n = -N+1:2:N-1, % loop over even series index n cn = 0; % Fourier Series Coefficient stem(n*wo,abs(cn)); % plot |cn| vs nwo end xlabel('w (rad/s)') ylabel('|cn|') ttle = ['EE341.01: Amplitude Spectrum with N = ',num2str(N)]; title(ttle); grid; hold; % Draw the phase spectrum from exponential Fourier Series subplot(2,1,2) stem(0,angle(c0)*180/pi); % plot angle of c0 at nwo = 0 hold; for n = -N:2:N, % loop over odd series index n cn = 2/(j*n*wo); % Fourier Series Coefficient stem(n*wo,angle(cn)*180/pi); % plot |cn| vs nwo end for n = -N+1:2:N-1, % loop over even series index n cn = 0; % Fourier Series Coefficient stem(n*wo,angle(cn)*180/pi); % plot |cn| vs nwo end xlabel('w (rad/s)') ylabel('angle(cn) (degrees)') ttle = ['EE341.01: Phase Spectrum with N = ',num2str(N)]; title(ttle); grid; hold; MATLAB Plots Generated:
  • 12.
    Hi, I am trying to write a function to generate Fourier series Coefficients of a given discrete time signal. For instance: x = [1 2 3 4] n = [0 1 2 3] where x holds the values of the signal, and n holds the corresponding time indices. My code for the function is: function a = dtfs(x,n) period = length(x); for k = 1:period a = (1/period)*x(k)*exp(((-j*2*pi)/period)*n(k)); a end i am just not getting the correct values for the fourier series coefficients. Am I setting up the formula wrong? As you can probably tell, i am very very new to MATLAB and I'd also appreciate if someone may guide me in setting up my "for- loop" with a vector to "catch" and store values of "a", so each time the for-loop repeats, the previous value a does not get over- written, instead they are all stored in a vector. Thanks in advance :) Regards Subject: Fourier Series Coefficients From: Andrew Date: 24 Oct, 2008 04:43:01 Message: 2 of 4 Reply to this message Add author to My Watch List View original format Flag as spam I'm guessing the formula, but hopefully the structure of it will help... function a = dtfs(x,n) period = length(x); a = zeros(1, length(x)) for k = 1:period for z = 1:period a(k) = a(k) + x(z) * exp((-j*2*pi)/period * (k-1) * n(z)); end a(k) = a(k) / period; num2str(a(k), '%1.18f'); end
  • 13.
    Cheers, Andrew > a = (1/period)*x(k)*exp(((-j*2*pi)/period)*n(k)); >a > end "Raz H" <dilster08@gmail.com> wrote in message <gdrclp$t9b$1@fred.mathworks.com>... > Hi, > > I am trying to write a function to generate Fourier series Coefficients of a given discrete time signal. For instance: > > x = [1 2 3 4] > n = [0 1 2 3] > > where x holds the values of the signal, and n holds the corresponding time indices. > > My code for the function is: > > function a = dtfs(x,n) > period = length(x); > for k = 1:period > a = (1/period)*x(k)*exp(((-j*2*pi)/period)*n(k)); >a > end > > i am just not getting the correct values for the fourier series coefficients. Am I setting up the formula wrong? > > As you can probably tell, i am very very new to MATLAB and I'd also appreciate if someone may guide me in setting up my "for-loop" with a vector to "catch" and store values of "a", so each time the for-loop repeats, the previous value a does not get over-written, instead they are all stored in a vector. > > Thanks in advance :) > > Regards Subject: Fourier Series Coefficients From: Paul Date: 24 Oct, 2008 06:35:05 Message: 3 of 4 Reply to this message Add author to My Watch List View original format Flag as spam
  • 14.
    "Andrew" <awbsmith@itee.uq.edu.au> wrotein message <gdrjol$so$1@fred.mathworks.com>... > I'm guessing the formula, but hopefully the structure of it will help... > > function a = dtfs(x,n) > period = length(x); > a = zeros(1, length(x)) > for k = 1:period > for z = 1:period > a(k) = a(k) + x(z) * exp((-j*2*pi)/period * (k-1) * n(z)); > end > a(k) = a(k) / period; > num2str(a(k), '%1.18f'); > end > > Cheers, > Andrew > > > > a = (1/period)*x(k)*exp(((-j*2*pi)/period)*n(k)); >>a > > end > > > > > "Raz H" <dilster08@gmail.com> wrote in message <gdrclp$t9b$1@fred.mathworks.com>... > > Hi, >> > > I am trying to write a function to generate Fourier series Coefficients of a given discrete time signal. For instance: >> > > x = [1 2 3 4] > > n = [0 1 2 3] >> > > where x holds the values of the signal, and n holds the corresponding time indices. >> > > My code for the function is: >> > > function a = dtfs(x,n) > > period = length(x); > > for k = 1:period > > a = (1/period)*x(k)*exp(((-j*2*pi)/period)*n(k)); >>a > > end >> > > i am just not getting the correct values for the fourier series coefficients. Am I setting up the formula wrong? >> > > As you can probably tell, i am very very new to MATLAB and I'd also appreciate if someone may guide me in setting up my "for-loop" with a vector to "catch" and store values of "a", so each time the for-loop repeats, the previous value a does not get over-written, instead they are all stored in a vector. >> > > Thanks in advance :) >> > > Regards
  • 15.
    Was this aHW problem? It looks like one to me! Subject: Fourier Series Coefficients From: Raz H Date: 24 Oct, 2008 06:54:02 Message: 4 of 4 Reply to this message Add author to My Watch List View original format Flag as spam "Paul" <par@ceri.memphis.edu> wrote in message <gdrqap$sl5$1@fred.mathworks.com>... > "Andrew" <awbsmith@itee.uq.edu.au> wrote in message <gdrjol$so$1@fred.mathworks.com>... > > I'm guessing the formula, but hopefully the structure of it will help... >> > > function a = dtfs(x,n) > > period = length(x); > > a = zeros(1, length(x)) > > for k = 1:period > > for z = 1:period > > a(k) = a(k) + x(z) * exp((-j*2*pi)/period * (k-1) * n(z)); > > end > > a(k) = a(k) / period; > > num2str(a(k), '%1.18f'); > > end >> > > Cheers, > > Andrew >> >> > > > a = (1/period)*x(k)*exp(((-j*2*pi)/period)*n(k)); >>>a > > > end >> >> >> >> > > "Raz H" <dilster08@gmail.com> wrote in message <gdrclp$t9b$1@fred.mathworks.com>... > > > Hi, >>> > > > I am trying to write a function to generate Fourier series Coefficients of a given discrete time signal. For instance: >>> > > > x = [1 2 3 4] > > > n = [0 1 2 3] >>> > > > where x holds the values of the signal, and n holds the corresponding time indices. >>> > > > My code for the function is:
  • 16.
    >>> > > > function a = dtfs(x,n) > > > period = length(x); > > > for k = 1:period > > > a = (1/period)*x(k)*exp(((-j*2*pi)/period)*n(k)); >>>a > > > end >>> > > > i am just not getting the correct values for the fourier series coefficients. Am I setting up the formula wrong? >>> > > > As you can probably tell, i am very very new to MATLAB and I'd also appreciate if someone may guide me in setting up my "for-loop" with a vector to "catch" and store values of "a", so each time the for-loop repeats, the previous value a does not get over-written, instead they are all stored in a vector. >>> > > > Thanks in advance :) >>> > > > Regards > > Was this a HW problem? It looks like one to me! @Andrew Thank you very much! I guess my loop was not set up correctly, plus I was not signifying the time indices correctly. @Paul This was not a homework problem, though I'll be taking Signals soon, so I am trying to become familiar with MATLAB. Thanks to all who replied! :) EE341.01: MATLAB M-FILE FOR PLOTTING TRUNCATED FOURIER SERIES This example shows a MATLAB M-file for plotting a truncated Fourier Series. Various numbers of terms are used. MATLAB M-File example5.m: % % Filename: example5.m % % Description: Example to show how the truncated Fourier series in % complex exponential form approximates the real % signal. More and more terms are taken showing a % better and better representation of the original signal. % clear; % clear all variables clf; % clear all figures
  • 17.
    % Define parametersto plot original sawtooth tr = [-1 0 0 1 1 2 2]; yr = [0 1 0 1 0 1 0]; % Plot Truncated Fourier Series Approximation (N = 1) N = 1; % define number of terms to use (n = -N..N) c0 = 0.5; % define dc bias coefficient t = -1:0.001:2; % define time values for y(t) y = c0 * ones(size(t)); % let initial y = c0 (dc bias) for all times for n = -N:-1, % compute y for negative n cn = exp(j*pi/2)/(2*pi*n); y = y + real(cn * exp(j*n*2*pi*t)); end; % compute y for positive n and add to y for n = 1:N, % found using negative n cn = exp(j*pi/2)/(2*pi*n); y = y + real(cn * exp(j*n*2*pi*t)); end; subplot(2,2,1); % plot approximation plot(t,y); hold; plot(tr,yr,':'); hold; xlabel('time (seconds)'); ylabel('y(t) approximation'); title('EE341.01: Truncated FS, -1<=n<=1'); % Plot Truncated Fourier Series Approximation (N = 2) clear; % clear all variables N = 2; % define number of terms to use (n = -N..N) c0 = 0.5; % define dc bias coefficient t = -1:0.001:2; % define time values for y(t) y = c0 * ones(size(t)); % let initial y = c0 (dc bias) for all times for n = -N:-1, % compute y for negative n cn = exp(j*pi/2)/(2*pi*n); y = y + real(cn * exp(j*n*2*pi*t)); end; % compute y for positive n and add to y for n = 1:N, % found using negative n cn = exp(j*pi/2)/(2*pi*n); y = y + real(cn * exp(j*n*2*pi*t)); end; subplot(2,2,2); % plot approximation plot(t,y); hold; plot(tr,yr,':'); hold; xlabel('time (seconds)'); ylabel('y(t) approximation'); title('EE341.01: Truncated FS, -2<=n<=2'); % Plot Truncated Fourier Series Approximation (N = 3) clear; % clear all variables
  • 18.
    N = 3; % define number of terms to use (n = -N..N) c0 = 0.5; % define dc bias coefficient t = -1:0.001:2; % define time values for y(t) y = c0 * ones(size(t)); % let initial y = c0 (dc bias) for all times for n = -N:-1, % compute y for negative n cn = exp(j*pi/2)/(2*pi*n); y = y + real(cn * exp(j*n*2*pi*t)); end; % compute y for positive n and add to y for n = 1:N, % found using negative n cn = exp(j*pi/2)/(2*pi*n); y = y + real(cn * exp(j*n*2*pi*t)); end; subplot(2,2,3); % plot approximation plot(t,y); hold; plot(tr,yr,':'); hold; xlabel('time (seconds)'); ylabel('y(t) approximation'); title('EE341.01: Truncated FS, -3<=n<=3'); % Plot Truncated Fourier Series Approximation (N = 10) clear; % clear all variables N = 10; % define number of terms to use (n = -N..N) c0 = 0.5; % define dc bias coefficient t = -1:0.001:2; % define time values for y(t) y = c0 * ones(size(t)); % let initial y = c0 (dc bias) for all times for n = -N:-1, % compute y for negative n cn = exp(j*pi/2)/(2*pi*n); y = y + real(cn * exp(j*n*2*pi*t)); end; % compute y for positive n and add to y for n = 1:N, % found using negative n cn = exp(j*pi/2)/(2*pi*n); y = y + real(cn * exp(j*n*2*pi*t)); end; subplot(2,2,4); % plot approximation plot(t,y); hold; plot(tr,yr,':'); hold; xlabel('time (seconds)'); ylabel('y(t) approximation'); title('EE341.01: Truncated FS, -10<=n<=10'); MATLAB Plot Generated: