Question

Please answer "b" only.

3.Consider the function { if Tt0 sint if 0t < T 0 and g(t2T) g(t) g(t) (a) Is this function continuous over the interval -r,7? Is this function differentiable over the interval -r, T]? (b) How many terms N do you need to consider so that the truncated Fourier series GN(t) provides a "reasonable" approximation to g(t) for all t (this will be an approximate N, of course)? Plot partial on the interval -T, T], to determine an figure with the actual function sums of the Fourier series in the same appropriate N

%Example code
function plotFS(m);
%m = user provided number of terms desired in the Fourier series;
%this code computes the Fourier series of the function 
%f(x)=0,        for -pi<= x <0, 
%    =cos(x)    for   0<= x <pi


%generate the interval from -pi to pi with step size h;
h = pi/100;
xx1=[-pi:h:0]; 
xx2=[0:h:pi];
xx = [xx1, xx2];
%generate the given function f so that it can be graphed
ff = [zeros(size(xx1)), cos(xx2)];

%compute the first partial sum
%note this is the first column of a larger matrix used to store each
%subsequent partial sum
ps(1,:) = .5*cos(xx); % 

%compute the next partial sum using Matlab's symbolic integrator
syms x %declare the symbolic variable, used in the int command below
ps(2,:) = ps(1,:) + sin(2*xx)*int(cos(x)*sin(2*x), 0, pi)/pi;

%note that many of the coef are zero so are not included nor computed
for n = 4:2:m
    temp = int(cos(x)*sin(n*x), 0, pi)/pi;%integrate this directly
    ps(n,:) = ps(n-2,:) + temp*sin(n*xx);
    %plot(xx,ps(n,:),'k-'); hold on; %plot intermediate partial sums as
    %they're generated
end

figure(1);
%plot the original given function, over three period
plot(xx,ff,'k-', xx+2*pi*ones(size(xx)),ff,'k-',xx-2*pi*ones(size(xx)),ff,'k-'); 
hold on;
%plot the m-th partial sum
plot(xx,ps(m,:),'b-'); 
%extend the graph over two more periods
plot(xx,ps(m,:),'b-', xx+2*pi*ones(size(xx)),ps(m,:),'b-', xx-2*pi*ones(size(xx)),ps(m,:),'b-'); 
legend('Original Fc', 'm-th Partial Sum');
title('Function and its Fourier series');

return
%anything written after "return" will not be executed...

Answer 1

Matlab code for Fourier Series clear all close all All time values t-linspace-pi,pi,1001); creating the function op -1 : 1 if tgth(t, zz (i)-0 for & & t(i)<0 else zz (i)-sin(t(i) ); end end figure (1) %Plotting the function plot (t,zz, 'Linewidth',2) xlabelt) ylabelf(t)' title'Plott ing of Actual data and Fourier sum') al-t (1); bl=t (end); 1l- (b1-al)/2; %Fourier series of the function for finding a and b coefficients for j-1:200 ssl-zz.cos(j*pi*t/1); all a values of the Fourier series aa ((1/1)*trapz(t,ssl); ss2-zz.*sin(j*pi*t / 1); all b values of the Fourier series bb(j (1/1 )* trapz (t, ss2) end %a0 value of Fourier series aa0 (1/1)trapz(t,zz); t-linspace (-pi,pi, 6001); s-aa0/2 fprintfYes this function is continuous and differentiable. \n\n' 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 lgnd1-Actual plot'; Fourier series of the function hold on for i-1:10 s stbb (i)*sin ( i*pi *t /1) +aa (i) *cos (i*pi*t/1); plot (t,s) 1gndi+1sprintf('Fourier sum for %d terms',i); end legend (lgnd, 'Location','northwest') 1

용음몸을몸을 몸을몸용 응을몸을 몸을을 po Jo pu 동용용용용을을을 동물물을동을을동동용 Yes this function is continuous and differentiable Printing few terms for Fourier series The value of al--0.0000 00 and bl=0.500000. The value of a2=-0.212209 and b2=0.000000. The value of a3-0.000000 and b3=-0.000000. The value of a4-0.04 24 43 and b4=0.000000. The value of a5=-0.000000 and b5= -0.000000. The value of a6-0.018191 and b6=0.000000. The value of a7-0.000000 and b7 =0.000000. The value of a8--0.01 0107 and b8=0.000000. The value of a9-0.000000 and b9=-0.000000 The value of a10-0.006433 and b10 0.000000 Plotting of Actual data and Fourier sum 12 Actual plot Fourier sum for 1 terms 1 Fourier sum Fourier sum for 5 terms r 4 terms - Fourier sum for 7 terms -Fourier sum for 8 terms 0.8 r10 terms 0.6 10 0.4 0.2 2

%Matlab code for Fourier Series
clear all
close all
%All time values
t=linspace(-pi,pi,1001);
%Loop for creating the function
for i=1:length(t)
    if t(i)>=-pi && t(i)<0
        zz(i)=0;
    else
        zz(i)=sin(t(i));
    end
end
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=1:200
    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(-pi,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=1:10
  
    s=s+bb(i)*sin(i*pi*t/l)+aa(i)*cos(i*pi*t/l);
    plot(t,s)
    lgnd{i+1}=sprintf('Fourier sum for %d terms',i);  
end
legend(lgnd,'Location','northwest')

%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%