Question

Exercise 5. Let f(t) denote the 2-periodic function f(t) = 1+2 - <t< . a) Make a representative plot of f(t) showing at least 3 periods. b) Using the numerical integration routine you wrote in your Home work 1 (Exercise 3) compute approx- imately the values of the Fourier series coefficients all of f() for k in the range -20 to 20. c) Make a plot of laul

Answer 1

%Matlab code for Fourier Series clear all close all All time values t=linspace(-pi,pi, 1001); function value zz=1./(1+t. 2); figure(1) Plotting the function plot(t,zz, 'Linewidth', 2) xlabel('t') ylabel('f(t)') title('Plotting of Actual data and Fourier sum') al=t(1); bl=t(end); l=(bl-al)/2; Fourier series of the function for finding a and b coefficients for j=1:50 ssl=zz.*cos(j*pi*t/l); fall a values of the Fourier series aa(i)=(1/1)*trapz(t, ss1); ss2=zz.sin(j*pi*t/1); fall b values of the Fourier series bb (j)=(1/1)*trapz(t, ss2); end 8a0 value of Fourier series aa0=(1/1)*trap(t, 22); t=linspace(-3. *pi, 3. *pi, 6001); s=aa0/2; fprintf('Yes this function is continuous and differentiable.\n\n') fall an and bn terms fprintf( 'Printing few terms for Fourier series\n') for i=1:10 fprintf('\tThe value of ad=lf and båd=f. \n\n',i,aa(i),i,bb(i)) end lgnd{1}='Actual plot'; Fourier series of the function hold on for i=1:50 s=s+bb(i) sin(i*pi*t/1)+aa(i)*cos(i*pi*t/l); if i= 3 plot(t,s) lgnd{2}=sprintf('Fourier sum for $d terms',i); elseif i==5 plot(t,s) lgnd{3}=sprintf('Fourier sum for $d terms', i); elseif i==7 plot(t,s) 1gnd{4}=sprintf('Fourier sum for $d terms', i);

elseif i==20 plot(t,s) lgnd{5}=sprintf('Fourier sum for $d terms', i); elseif i==50 plot(t,s) lgnd{6}=sprintf('Fourier sum for $d terms', i); end end legend (1gnd, 'Location', 'northwest') figure(2) plot(aa) xlabel('count') ylabel('a_k amplitude') title('a_k plot') $$ $$ $$%%%%%% End of Code %%%%%%%%%%%%%%%%% Yes this function is continuous and differentiable. Printing few terms for Fourier series The value of al=0.388891 and bl=-0.000000. The value of a2=0.128226 and b2=0.000000 The value of a3=0.053233 and b3=0.000000. The value of a4=0.016307 and 64--0.000000. The value of a5=0.008047 and b5=0.000000. The value of a6=0.001561 and b6=-0.000000. The value of a 7=0.001591 and b7=0.000000. The value of a8=-0.000186 and b8=0.000000. The value of a 9=0.000537 and b9=0.000000. The value of al0=-0.000290 and bio=-0.000000.

Plotting of Actual data and Fourier sum - Actual plot Fourier sum for 3 terms Fourier sum for 5 terms Fourier sum for 7 terms Fourier sum for 20 terms Fourier sum for 50 terms f(t) -8 6 4 2 0 2 4 6 8 10 a plot a amplitude 0 5 10 15 20 30 35 40 45 50 25 count

%Matlab code for Fourier Series
clear all
close all
%All time values
t=linspace(-pi,pi,1001);
%function value
zz=1./(1+t.^2);

figure(1)
%Plotting the function
plot(t,zz,'Linewidth',2)
xlabel('t')
ylabel('f(t)')
title('Plotting of Actual data and Fourier sum')
a1=t(1); b1=t(end);
l=(b1-a1)/2;
%Fourier series of the function for finding a and b coefficients
for j=1:50
    ss1=zz.*cos(j*pi*t/l);
    %all a values of the Fourier series
    aa(j)=(1/l)*trapz(t,ss1);
    ss2=zz.*sin(j*pi*t/l);
    %all b values of the Fourier series
    bb(j)=(1/l)*trapz(t,ss2);
end

%a0 value of Fourier series
aa0=(1/l)*trapz(t,zz);
t=linspace(-3.*pi,3.*pi,6001);
s=aa0/2;
fprintf('Yes this function is continuous and differentiable.\n\n')
%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
lgnd{1}='Actual plot';
%Fourier series of the function
hold on
for i=1:50
  
    s=s+bb(i)*sin(i*pi*t/l)+aa(i)*cos(i*pi*t/l);
    if i==3
        plot(t,s)
        lgnd{2}=sprintf('Fourier sum for %d terms',i);
    elseif i==5
        plot(t,s)
        lgnd{3}=sprintf('Fourier sum for %d terms',i);
    elseif i==7
        plot(t,s)
        lgnd{4}=sprintf('Fourier sum for %d terms',i);
      
    elseif i==20
        plot(t,s)
        lgnd{5}=sprintf('Fourier sum for %d terms',i);
      
    elseif i==50
        plot(t,s)
        lgnd{6}=sprintf('Fourier sum for %d terms',i);
      
    end
end
legend(lgnd,'Location','northwest')

figure(2)
plot(aa)
xlabel('count')
ylabel('a_k amplitude')
title('a_k plot')
%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%