Fourier series example

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Fourier series example

  1. 1. Fourier Series ExampleMATLAB 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 Seriesph0 = 0;n = [1 3 5 7 9]; % Values of n to be evaluatedan = -3.2 ./ (pi * n).^2; % Fourier Series coefficientsmag_an = abs(an);ph_an = -180 * ones(1,length(n));%n = [0 n];mag_an = [a0 mag_an]; % Including a0 with a_nph_an = [ph0 ph_an];%figure(2),clf,subplot(211),plot(n,mag_an,o),grid,xlabel(HarmonicNumber),...
  2. 2. ylabel(Magnitude),title(Fourier Series Magnitude),axis([0 10 00.6]),...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 Frequencyt = [-2:0.002:6]; % time vector for approximations%x1 = 0; % approximation with DC + 1 termfor i = 1:2x1 = x1 + mag_an(i)*cos(n(i)*w0*t + ph_an(i)*pi/180);end%x2 = x1; % approximation with DC + 2 termsi = 3;x2 = x2 + mag_an(i)*cos(n(i)*w0*t + ph_an(i)*pi/180);%x3 = x2; % approximation with DC + 3 termsi = 4;x3 = x3 + mag_an(i)*cos(n(i)*w0*t + ph_an(i)*pi/180);%x4 = x3; % approximation with DC + 5 termsfor i = 5:6x4 = x4 + mag_an(i)*cos(n(i)*w0*t + ph_an(i)*pi/180);end
  3. 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 #2MATLAB 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. 4. an = (1./(pi*n)) .* sin(n*pi/2);bn = (1./(pi*n)) .* (1 - cos(n*pi/2));x1 = a0;for i = 1:10x1 = x1 + an(i)*cos(i*w0*t) + bn(i)*sin(i*w0*t);endx2 = x1;for i = 11:30x2 = x2 + an(i)*cos(i*w0*t) + bn(i)*sin(i*w0*t);endx3 = x2;for i = 31:50x3 = x3 + an(i)*cos(i*w0*t) + bn(i)*sin(i*w0*t);endA0 = 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--,[88],[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,Pulsewidth = T_0/4)figure(2),clf,subplot(311),plot(t,x1),subplot(312),plot(t,x2),subplot(313),plot(t,x3),...
  5. 5. subplot(311),ylabel(Amplitude),title(Fourier Series Representation ofx(t) with 10 Terms),...subplot(312),ylabel(Amplitude),title(Fourier Series Representation ofx(t) with 30 Terms),...subplot(313),ylabel(Amplitude),title(Fourier Series Representation ofx(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],[00],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]),plotaxendset(gcf,Position,fig_size)figure(3),clf,subplot(211),plot(0,a0,ro,n,an,o),axis([-5 50 -0.20.5]),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 SeriesCoefficients 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 SeriesCoefficients 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 -0.10.5]),plotax,...xlabel(Frequency (r/s)),ylabel(Magnitde),title(Cosine Fourier SeriesMagnitudes 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 FourierSeries Phases Theta_n for x(t)),...set(gcf,Position,fig_size)
  6. 6. 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 FourierSeries Magnitudes X_n for x(t)),...subplot(212),plot(n*w0,thn,o,-n*w0,-thn,o),v=axis;axis([-16 16v(3:4)]),plotax,...xlabel(Frequency (r/s)),ylabel(Phase (deg)),title(Exponential FourierSeries 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 Threadfourier series coefficients - Kurt - Dec 1 12:27:01 2009hello all,I have a one period square wave on the interval[0,2] defined as:y(t)= 1, 0<=t<1y(t)= 0, 1<=t<2I need to find the fourier series coefficients,ck, withk=-10,-9,...,9,10
  7. 7. I heard using a for loop would work but I am completely stuck on how tomovethrough this problem.All help is greatly appreciated,Kurt______________________________New Code Sharing Section now Live on DSPRelated.com. Learn about the Reward Program forContributors 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 2009you can tryfor k=-10:1:10c(k+11) = here you enter the Ck equation; % you cant have negative indexinginMATLABendNowc gives you the coefficientsrgdsvishwanath________________________________From: Kurt <k...@sbcglobal.net>To: m...@yahoogroups.comSent: Tue, 1 December, 2009 12:50:19 PMSubject: [matlab] fourier series coefficientsÂhello all,I have a one period square wave on the interval[0,2] defined as:y(t)= 1, 0<=t<1y(t)= 0, 1<=t<2I need to find the fourier series coefficients, ck, withk=-10,-9,... ,9,10I heard using a for loop would work but I am completely stuck on how tomovethrough this problem.All help is greatly appreciated,Kurt______________________________New Code Sharing Section now Live on DSPRelated.com. Learn about the Reward Program forContributors 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. 8. Hey..For fourier coeff u have to find the fft of the given sequence usingmatlab.Since u have to find the coeff for kranging from -10:1:10, i.e.21 points uhave to define ur function in time domain in 21 samples. Take the fft ofthese 21 samples. The resultant is your desired fourier coeff .Regards-vaibhavOn 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 negativeindexing> 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 SinghBE(Hons.) Electronics And InstrumentationBITS-Pilani
  9. 9. EE341.01: MATLAB M-FILE FOR PLOTTING TRUNCATED FOURIER SERIES AND ITS SPECTRAMATLAB 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 variablesclf; % clear all figuresN = 11; % summation limit (use N odd)wo = pi; % fundamental frequency (rad/s)c0 = 0; % dc biast = -3:0.01:3; % declare time valuesfigure(1) % put first two plots on figure 1% Compute yce, the Fourier Series in complex exponential formyce = c0*ones(size(t)); % initialize yce to c0for 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 computationendsubplot(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 FSxlabel(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 formyt = c0*ones(size(t)); % initialize yt to c0for 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 computationendsubplot(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 FSplot(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. 10. figure(2) % put next plots on figure 2subplot(2,1,1)stem(0,c0); % plot c0 at nwo = 0hold;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 nwoendfor 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 nwoendxlabel(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 Seriessubplot(2,1,2)stem(0,angle(c0)*180/pi); % plot angle of c0 at nwo = 0hold;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 nwoendfor 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 nwoendxlabel(w (rad/s))ylabel(angle(cn) (degrees))ttle = [EE341.01: Phase Spectrum with N = ,num2str(N)];title(ttle);grid;hold;MATLAB Plots Generated:
  11. 11. 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 Id 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 Im 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
  12. 12. 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$1@fred.mathworks.com>... > Hi, > > I am trying to write a function to generate Fourier series Coefficients of a given discrete time signal. For instance: > > x = [1 2 3 4] > n = [0 1 2 3] > > where x holds the values of the signal, and n holds the corresponding time indices. > > My code for the function is: > > function a = dtfs(x,n) > period = length(x); > for k = 1:period > a = (1/period)*x(k)*exp(((-j*2*pi)/period)*n(k)); >a > end > > i am just not getting the correct values for the fourier series coefficients. Am I setting up the formula wrong? >> As you can probably tell, i am very very new to MATLAB and Id also appreciate if someone may guide me in setting up my"for-loop" with a vector to "catch" and store values of "a", so each time the for-loop repeats, the previous value a does not get over-written, instead they are all stored in a vector. > > Thanks in advance :) > > Regards Subject: Fourier Series Coefficients From: Paul Date: 24 Oct, 2008 06:35:05 Message: 3 of 4 Reply to this message Add author to My Watch List View original format Flag as spam
  13. 13. "Andrew" <awbsmith@itee.uq.edu.au> wrote in message <gdrjol$so$1@fred.mathworks.com>... > Im guessing the formula, but hopefully the structure of it will help... > > function a = dtfs(x,n) > period = length(x); > a = zeros(1, length(x)) > for k = 1:period > for z = 1:period > a(k) = a(k) + x(z) * exp((-j*2*pi)/period * (k-1) * n(z)); > end > a(k) = a(k) / period; > num2str(a(k), %1.18f); > end > > Cheers, > Andrew > > > > a = (1/period)*x(k)*exp(((-j*2*pi)/period)*n(k)); >>a > > end > > > > > "Raz H" <dilster08@gmail.com> wrote in message <gdrclp$t9b$1@fred.mathworks.com>... > > Hi, >> > > I am trying to write a function to generate Fourier series Coefficients of a given discrete time signal. For instance: >> > > x = [1 2 3 4] > > n = [0 1 2 3] >> > > where x holds the values of the signal, and n holds the corresponding time indices. >> > > My code for the function is: >> > > function a = dtfs(x,n) > > period = length(x); > > for k = 1:period > > a = (1/period)*x(k)*exp(((-j*2*pi)/period)*n(k)); >>a > > end >> > > i am just not getting the correct values for the fourier series coefficients. Am I setting up the formula wrong? >>> > As you can probably tell, i am very very new to MATLAB and Id also appreciate if someone may guide me in setting up my"for-loop" with a vector to "catch" and store values of "a", so each time the for-loop repeats, the previous value a does not get over-written, instead they are all stored in a vector. >> > > Thanks in advance :) >> > > Regards
  14. 14. Was this a HW problem? It looks like one to me! Subject: Fourier Series Coefficients From: Raz H Date: 24 Oct, 2008 06:54:02 Message: 4 of 4 Reply to this message Add author to My Watch List View original format Flag as spam "Paul" <par@ceri.memphis.edu> wrote in message <gdrqap$sl5$1@fred.mathworks.com>... > "Andrew" <awbsmith@itee.uq.edu.au> wrote in message <gdrjol$so$1@fred.mathworks.com>... > > Im guessing the formula, but hopefully the structure of it will help... >> > > function a = dtfs(x,n) > > period = length(x); > > a = zeros(1, length(x)) > > for k = 1:period > > for z = 1:period > > a(k) = a(k) + x(z) * exp((-j*2*pi)/period * (k-1) * n(z)); > > end > > a(k) = a(k) / period; > > num2str(a(k), %1.18f); > > end >> > > Cheers, > > Andrew >> >> > > > a = (1/period)*x(k)*exp(((-j*2*pi)/period)*n(k)); >>>a > > > end >> >> >> >> > > "Raz H" <dilster08@gmail.com> wrote in message <gdrclp$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:
  15. 15. >>> > > > 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 Id also appreciate if someone may guide me in setting upmy "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 Ill 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 SERIESThis example shows a MATLAB M-file for plotting a truncated Fourier Series. Variousnumbers 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 variablesclf; % clear all figures
  16. 16. % Define parameters to plot original sawtoothtr = [-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 coefficientt = -1:0.001:2; % define time values for y(t)y = c0 * ones(size(t)); % let initial y = c0 (dc bias) for all timesfor 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 yfor 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 approximationplot(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 variablesN = 2; % define number of terms to use (n = -N..N)c0 = 0.5; % define dc bias coefficientt = -1:0.001:2; % define time values for y(t)y = c0 * ones(size(t)); % let initial y = c0 (dc bias) for all timesfor 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 yfor 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 approximationplot(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
  17. 17. N = 3; % define number of terms to use (n = -N..N)c0 = 0.5; % define dc bias coefficientt = -1:0.001:2; % define time values for y(t)y = c0 * ones(size(t)); % let initial y = c0 (dc bias) for all timesfor 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 yfor 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 approximationplot(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 variablesN = 10; % define number of terms to use (n = -N..N)c0 = 0.5; % define dc bias coefficientt = -1:0.001:2; % define time values for y(t)y = c0 * ones(size(t)); % let initial y = c0 (dc bias) for all timesfor 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 yfor 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 approximationplot(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:

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