Question

(a) Find the Fourier series for 25 { O if 1 r<O 1-2 if 0<H<1 f(x) defined on the interval -1<rs1. T-2 (b) Using MATLAB, plot the first 20 terms and the first 200 terms of the Fourier series in the interval -3< r<3, In order to do this, the r-interval should be divided into 6001 cqually spaced points by making use of the MATLAB command linspace.

Answer 1

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Matlab code for Fourier Series clear all close all &All time values Xlinspace(-1,1,1001) creating the function for Xi X(i)<0 zz(i)-0 if else 2z (i)-1-(x(i)).^2; end end figure (1) Plotting the function plot (X, zz) xlabelx') ylabelf(x) title Plott ing of Actual data') al x(1 bl=X (end); 1-(bl-al)/2; %Fourier series of the function for finding a and b coefficients for j-1:200 ssl-zz.*cos (j*pi*x/1) all a values of the Fourier series aa (j(1/1)* trapz(x, ssl); ss2-zz.*sin(j*pi *X/1); %all b values of the Fourier series bb (j)(1/1) * trapz(X, ss2); end a0 value of Fourier series aa0 (1/1) trapz(X, zz); x-linspace(-3,3,6001 ); s-aa0/2 all an and bn terms fprintfPrinting few terms for Fourier series \n') for i-1:10 fprintf\tThe value of asd-%f and bsd-%f. \n\n',i,aa(i),i,bb(i)) end %Fourier series of the function for i-1:200 s - s+bb (i) *sin(i*pi*X/1) +aa (i)cos ( i*pi*X/1) . if i-=20 figure (2) plot (X,s) xlabeltime') 1

ylabelf(t) ') titleFourier series of given function for 20 terms ) elseif i--200 figure (3) plot (X,s) XLabel time') i+eFourier series of qiven function for 200 terms') end end s 88%888 % 88888888 % End of Code $%$ 88888888s$888 Printing few terms for Fourier series The value of al-0.203643 and bl-0.447315. The value of a2--0.049661 and b2=0.159153. The value of a3-0.023516 and b3=0.110878. The value of a4-0.011666 and b4=0.0795 73. The value of a5-0.009106 and b5-0.064689. The value of a6-0.00 4630 and b6=0.053045 The value of a7-0.0051 36 and b7=0.045842. The value of a8=-0.002167 and b 8=0.039780 The value of a9-0.003502 and b9=0.035535. The value of a10-0.001 027 and b10=0.031821 2

Plotting of Actual data 0.9 0.8 0.7 0.6 0.5 0 4 03 02 0.1 n6 0.8 02 -0.4 0.6 08 given function for 20 terms Fourier series n8 06 04 02 F 0 05 time 3

Fourier series of given function for 200 terms 1 2 0.8 F 0.6 0.4 02 time Published with MATLAB® R201 8a 4

%Matlab code for Fourier Series
clear all
close all
%All time values
X=linspace(-1,1,1001);
%Loop for creating the function
for i=1:length(X)
    if X(i)>=-1 && X(i)<0
        zz(i)=0;
    else
        zz(i)=1-(X(i)).^2;
    end
end
figure(1)
%Plotting the function
plot(X,zz)
xlabel('x')
ylabel('f(x)')
title('Plotting of Actual data')

a1=X(1); b1=X(end);
l=(b1-a1)/2;
%Fourier series of the function for finding a and b coefficients
for j=1:200
    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);
end

%a0 value of Fourier series
aa0=(1/l)*trapz(X,zz);
X=linspace(-3,3,6001);
s=aa0/2;

%all an and bn terms
fprintf('Printing few terms for Fourier series\n')
for i=1:10
    fprintf('\tThe value of a%d=%f and b%d=%f. \n\n',i,aa(i),i,bb(i))
end
  
%Fourier series of the function
for i=1:200
  
    s=s+bb(i)*sin(i*pi*X/l)+aa(i)*cos(i*pi*X/l);
    if i==20
        figure(2)
        plot(X,s)
        xlabel('time')
        ylabel('f(t)')
        title('Fourier series of given function for 20 terms')
      
      
    elseif i==200
        figure(3)
        plot(X,s)
        xlabel('time')
        ylabel('f(t)')
        title('Fourier series of given function for 200 terms')
      
    end
      
end

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