% run this script to plot pseudo-periodic functions as
% y= exp(-t/tau).*sin(omega*t),
% with different values of tau and omega

clear all
close all
clc

% define tau and omega
tau=5;
omega=3;

% define time vector
t_0=0; %initial value
t_f=20; % final value
t_s=0.01; % step time
t=t_0:t_s:t_f; % time vector


% define exp function
y_exp=exp(-t/tau);

% define sin function
y_sin=sin(omega*t);

% define y, a pseudo-periodic function
y=y_exp.*y_sin;

% plot y, y_exp and y_sin in figure 1
figure (1)
hold on
grid on
xlabel('time')
ylabel('y')
plot(t,y,'-r','linewidth',2)
plot(t,y_exp,'--b');
plot(t,y_sin,'--g');

% define exp. modes with different tau values
tau_a=3;
tau_b=6;
y_exp_a=exp(-t/tau_a);
y_exp_b=exp(-t/tau_b);

% define sin functions with different omega values
omega_1=1;
omega_2=2;
y_sin_1=sin(omega_1*t);
y_sin_2=sin(omega_2*t);


figure (2) % plot in figure 2 differents exp, periodic and pseudo periodic funtions
subplot(2,2,1)
% plot exp. modes
hold on
grid on
plot(t, y_exp_a,'-b');
plot(t, y_exp_b,'-r');
ylabel('y')
%xlabel('t')
legend('y_1=e^{-t/3}','y_2=e^{-t/6}')

% plot sin functions (periodic modes)
subplot(2,2,2)
hold on
grid on
plot(t, y_sin_1,'-k');
plot(t, y_sin_2,'-g');
legend('y_1=sin(t)','y_2=sin(2t)')
%xlabel('t')
%ylabel('y')

% plot exp(-t/3), exp(-t/6), sin(t), exp(-t/3).*sin(t) 
% and exp(-t/6).*sin(t)
subplot(2,2,3)
hold on
grid on
y_pseudo_1a=y_exp_a.*y_sin_1;
y_pseudo_1b=y_exp_b.*y_sin_1;

plot(t, y_exp_a,'--b');
plot(t, y_exp_b,'--r');
plot(t, y_sin_1,'--k');

plot(t,y_pseudo_1a,'-b','linewidth',3)
plot(t,y_pseudo_1b,'-r','linewidth',3)

legend('y_1=e^{-t/3}','y_2=e^{-t/6}','y_3=sin(t)','y_4=y_1 y_3','y_5=y_2 y_3');
xlabel('t')
ylabel('y')


% plot exp(-t/3), exp(-t/6), sin(2t), exp(-t/3).*sin(2t)
% and exp(-t/6).*sin(2t)
subplot(2,2,4)
hold on
grid on
y_pseudo_2a=y_exp_a.*y_sin_2;
y_pseudo_2b=y_exp_b.*y_sin_2;

plot(t, y_exp_a,'--b');
plot(t, y_exp_b,'--r');
plot(t, y_sin_2,'--g');

plot(t,y_pseudo_2a,'-c','linewidth',3)
plot(t,y_pseudo_2b,'-m','linewidth',3)
xlabel('t')
%ylabel('y')

legend('y_1=e^{-t/3}','y_2=e^{-t/6}','y_3=sin(2t)','y_4=y_1 y_3','y_5=y_2 y_3');