Question

3. Consider the function f(t) = ' π2 , with f(t) = f(t +2r). 0<t<π (a) Sketch f(t) by hand for-3r < t < 3T. (b) Determine the general Fourier Series for f(t). (c) Use MATLAB to plot f(t) and the n = 4 Fourier series representation on the same set of axes for -t<T

Answer 1

f(t) = t^2 - pi < t < 0 = pi^2 0 < t < pi sketch f(t) for - 3pi < t < 3 pi f(t) = a_0/2 + Sigma^infinity_n = 1 a_n cos n pi t/i + b_n sin n pi t/L here l = pi - (- pi)/2 = pi Here furies series of f(t) can be written her an f(t) = a_0/2 + Sigma a_n cos notequalto + b_n sin n t Now a_0 = 1/l integral ^pi_ - pi f(t) dt = 1/pi integral ^0_ - pi t^2 dt + 1/pi integral ^pi_0 pi^2 dt = 1/pi [t^3/3]^0_ - pi + 1/pi [pi^2 t]^pi_0 = 1/pi [pi^3/3 + pi^3] = 1 pi^2/3 a_n = 1/l integral ^pi_ - pi cos (notequalto) f(t) dt = 1/pi integral ^0_ - pi t^2 con (notequalto) df + 1/pi integral ^pi_0 pi^2 con (notequalto) dt r\n\ Now integral t^2 cos (notequalto) dt = t^2 sin notequalto/n - integral 2t sin (notequalto)/n dt = t^2 sin notequalto/n - 2 sin t/n^3 + 2t con notequalto/n^2

%Matlab code for Fourier Series
clear all
close all
%All time values
X=linspace(-pi,pi,1001);
%Loop for creating the function
for i=1:length(X)
    if X(i)>=-pi && X(i)<0
        zz(i)=(X(i)).^2;
    else
        zz(i)=pi^2;
    end
end
figure(1)
%Plotting the function
hold on
plot(X,zz)
xlabel('x')
ylabel('f(x)')
title('Plotting of Actual data and Fourier sum for 4 terms')

%fourier series
a0=4*pi^2/3;

an=@(n) (2./(n.^2)).*cos(n*pi);
bn=@(n) pi./n+(2./(n.^3*pi)).*(1-cos(n.*pi));

s=a0/2;
for nn=1:4
    s=s+an(nn).*cos(nn.*X)+bn(nn).*sin(nn.*X);
end

plot(X,s)
legend('Actual data','Fourier sum ')
grid on
%%%%%%%%%%%%%% End of Code %%%%%%%%%%%