Question

Consider the function -1.0<xso x, (a) Find the Fourier series of the function f(x (b) Use any graphing software to plot the function f (x) (c) Graph the partial sums S (x) for N 1,2,3,5,10,20,50 in ONE graph (d) Analyse your results. END

Answer 1

%Matlab code for Fourier Series clear all close all %All x-linspace (-1,3,3001); %Loop for creating the function for i= 1:length (x) time values elseif x(i)>0 && x(i)<-2 else end ePlotting the function hold on plot (x,zz, 'Linewidth,4) al-x (1)i bl-x(end) 1-(bl-al)/2; %Fourier series of the function for finding a and b coefficients fprintf( First 10 terms of Fourier series\n') for j-l:50 ssl-zz.*cos j*pi*x/l) %a11 a values of the Fourier series aa(j)-(1/1)*trapz(x, ss1) ss2-zz.*sin j*pi*x/l) %all b values of the Fourier series bb (j)-(1/1) trapz (x, ss2); if j<-10 (1/1)*trapz(x, ssl); fprint f ( '\tFourier fprint f ( "\tFourier a(2d) b(%d ) is ís %f\n', j ,aa(j) ) %f\n\n', j,bb(j)) term term end end N-[1 2 3 5 10 20 50] a0 value of Fourier series aa0-(1/1) trapz(x, zz); s-aa0/2; k-1; %Fourier series of the function for i=1:50 s-stbb i)*sin(i*pi*x/l)+aa (i)*cos(i*pi*x/l); if i-N ( k ) k-kt1; plot(x,s) lgnd (k)-sprintf fourier term-8d',i) end

lgnd(1Actual data; %plotting Fourier series of the function xlabel("time) ylabel ('f(t) title('Fourier series of given function') legend (lgnd, 'Location, northwest) 8888868888888888888888 End of Code 웅888888888888888888 First 10 terms of Fourier series Fourier term a(1) is -0.752 954 Fourier term b(1) is 1.354287 Fourier term a(2) is -0.102988 Fourier term b(2) is 0.795774 Fourier term a(3) is 0.340158 Fourier term b(3) is 0.209802 Fourier term a(4) is -0.050328 Fourier term b(4) is -0.079577 Fourier term a(5) is -0.183545 Fourier term b(5) is 0.138557 Fourier term a(6) is -0.012925 Fourier term b(6) is 0.265255 Fourier term a(7) is 0.139885 Fourier term b(7) is 0.089313 Fourier term a(8) Fourier term b(8) is -0.039788 is -0.012332 Fourier term a(9) is-0.104265 Fourier term b(9) is 0.072943 Fourier term a(10) is -0.005720 Fourier term b(10) is 0.159149

Fourier series of given function 3.5 Actual data tourier term-1 3 fourier term-2 fourier term-3 fourier term. fourier term-10 fourier term 20 fourier term:50 1.5 0.5 -0.5 1.5 0.5 0.5 1.5 2.5 time Published with MATLAB R2018a

%Matlab code for Fourier Series
clear all
close all
%All time values
x=linspace(-1,3,3001);
%Loop for creating the function
for i=1:length(x)
    if x(i)>-1 && x(i)<=0
        zz(i)=x(i);
    elseif x(i)>0 && x(i)<=2
        zz(i)=2+2.*x(i)-(x(i)).^2;
    else
        zz(i)=2;
    end
end
%Plotting the function
hold on
plot(x,zz,'Linewidth',4)

a1=x(1); b1=x(end);
l=(b1-a1)/2;
%Fourier series of the function for finding a and b coefficients
fprintf('First 10 terms of Fourier series\n')
for j=1:50
    ss1=zz.*cos(j*pi*x/l);
    %all a values of the Fourier series
    aa(j)=(1/l)*trapz(x,ss1);
    ss2=zz.*sin(j*pi*x/l);
    %all b values of the Fourier series
    bb(j)=(1/l)*trapz(x,ss2);
    if j<=10
        fprintf('\tFourier term a(%d) is %f\n',j,aa(j))
        fprintf('\tFourier term b(%d) is %f\n\n',j,bb(j))
    end
end

N=[1 2 3 5 10 20 50];
%a0 value of Fourier series
aa0=(1/l)*trapz(x,zz);
s=aa0/2; k=1;
%Fourier series of the function
for i=1:50
  
    s=s+bb(i)*sin(i*pi*x/l)+aa(i)*cos(i*pi*x/l);
  
    if i==N(k)
      
        k=k+1;
        plot(x,s)
        lgnd{k}=sprintf('fourier term=%d',i);
      
    end
      

end
lgnd{1}='Actual data';
%Plotting Fourier series of the function
xlabel('time')
ylabel('f(t)')
title('Fourier series of given function')
legend(lgnd,'Location','northwest')
  
%%%%%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%%%%