close all
clear all 
clc

T = 0.01;
N = 1000;
t = (0:N-1)*T;
x = ((t>0)+.5*(t==0)).*(2*exp(-t).*cos(2*pi*t));
X = fftshift(T*fft(x));
om = (-round((N-1)/2):round(N/2)-1) *2*pi/(N*T);          
Xref = 1./(1+1j*(om-2*pi))+1./(1+1j*(om+2*pi));

figure
subplot(2,1,1)
plot(t,x)
grid
xlabel('t')
ylabel('x(t)')
title('signal')
subplot(2,1,2)
semilogy(om,abs(X),om,abs(Xref))
grid
xlabel('\omega')
ylabel('X(\omega)')
legend('MatLab','analytical')
title('Fourier transform')

% printing figure in png format
set(gcf,'PaperUnits','inches','PaperPosition',[0 0 6 4.5])
print -dpng exercise1.png -r100