Question

A periodic signal, x(t) is shown below. A = 10, T-4 sec. -T Write a MATLAB script to plot the signal, using enough points to get a smooth curve. Compute the Fourier series coefficients for the signal (if you can find them in the text, that is ok). Plot the single-sided or double-sided spectra for each signal. Include enough frequencies in the plots to adequately represent the frequency content of the signals. Plot partial sums of the Fourier series for the signal x(t) (terms 1 through N in the infinite series). Decide what N should be to get a get a good reproduction of the corresponding x(t).

Answer 1

MATLAB code is given below in bold letters.

clc;
close all;
clear all;

% define the signal parameters
A = 10;
T = 4;
w0 = 2*pi/T;

% define the symbolic variables
syms n t;

% define the signal
f = -A/T*t+A;

% evaluate the fourier series integral
an1 = 2/T*int(f*cos(n*w0*t),0,T);
bn1 = 2/T*int(f*sin(n*w0*t),0,T);
a0 = 1/T*int(f,0,T);

% obtain C by substituting n in c[n]
nmax = 100;
n = 1:nmax;
a = subs(an1);
b = subs(bn1);

% define the time vector
ts = 1e-1; % ts is sampling the
t = 0:ts:4*T-ts;

% directly plot the signal x(t)
t1 = 0:ts:4-ts;
x1 = -A/T*t1+A;
x = repmat(x1,1,4);

figure;
plot(t,x,'r','linewidth',2);grid on;
xlabel('time');ylabel('x(t)');
ylim([-5 15]);

% Now fourier series reconstruction
N = [5 50 100];

for p = 1:length(N)
for i = 1:length(t)
for k = 1:N(p)   
x(k,i) = a(k)*cos(k*w0*t(i)) + b(k)*sin(k*w0*t(i));
end
y(i) = a0+sum(x(:,i)); % Add DC term

end
figure;
plot(t,y,'color',rand(1,3),'linewidth',2); % Plot results
xlabel('Time (sec)','FontSize',14); grid on;
ylabel('x(t)','FontSize',14);ylim([-5 15]);
STRING = ['Approximation with Number of terms : ',num2str(N(p))];
title(STRING);
end

15 10 16 14 12 10 time

Approximation with Number of terms: 5 10 E.. 10 12 14 16 Time (sec)

Approximation with Number of terms 50 10 10 12 14 16 Time (sec)

Approximation with Number of terms 100 10E.... 10 12 14 16 Time (sec)