Lab1

%---------------------------------------
% Fill in your group number.
GroupNumber = 3;
% Fill in your student Name and ID.
Students(1).Name = 'Tyler Young';
Students(1).ID = '260730324';
Students(2).Name = 'Jamie McLeish';
Students(2).ID = '260708573';
%---------------------------------------

%-Documentation-
help dirac
doc abs

%-Arrays-
dt = 0.01; % timestep of discrete signal in seconds
T = 10; % Final value of the time signal in seconds

% This creates an array |t = [0, 0.01, 0.02, 0.03, ..., 9.98, 9.99, 10.0]|.
t = 0:dt:T; % time signal

x1 = sin(2*pi*t);

x2 = zeros(size(t)); % create a zero signal with the same size as time
x2(t < 5) = t(t < 5); % value is equale to time, but only when time < 5
x2(t >= 5) = -t(t >= 5) + 10; % value is equale to -time+10, but only when time >= 5

%-Plotting Signals-
figure;
hold on;
grid on;
plot(t, x1);
plot(t, x2);
title(sprintf('Group %d - Example plot', GroupNumber));
xlabel('Time (seconds)')
ylabel('Signal')
legend('x_1(t)', 'x_2(t)')
hold off;

%-Matrices-
A = [8, 1, 6; 3, 5, 7; 4, 9, 2]
det(A)
eig(A)
inv(A)

%-Saving/Loading Data-
example = sprintf('A variable generated by groupe <%d>', GroupNumber);
save('Lab01.mat', 'example')
clear example
load('Lab01.mat', 'example')
disp(example)

%---Question 1---
t = 0:dt:T; % time signal

y1(t >= 0) = 1;

y2(t < 5) = 0;
y2(t >= 5) = -1;

y3 = zeros(size(t));
y3(t >=0) = 0.1*t(t >= 0);

y4 = exp(-t/10);

figure;
hold on;
% Fill in your plots command
plot(t, y1);
plot(t, y2);
plot(t, y3);
plot(t, y4);
title(sprintf('Group %d - Question 1 - Signals', GroupNumber));
xlabel('Time (seconds)')
ylabel('Signals')
grid on;
legend('y_1(t)', 'y_2(t)', 'y_3(t)', 'y_4(t)', 'location', 'best')

%-Computing Laplace Transforms-
syms t s
f = exp(-t);
F = laplace(f, t, s);
help laplace
simplify(2/(s + 1)-s/(s + 1)^2)
help sym/simplify

%---Question 2---
syms t s
f1 = dirac(t)
f2 = t^2 + 4*t + 1
f3 = t*exp(-t) + 0.5*exp(-t) + exp(-3*t)
f4 = sin(2*t) + cos(t).^2
f5 = exp(-t)*sin(5*t + pi/3) + (t.^2)*exp(-2*t)

%---Question 3---
F1 = simplify(laplace(f1, t, s))
F2 = simplify(laplace(f2, t, s))
F3 = simplify(laplace(f3, t, s))
F4 = simplify(laplace(f4, t, s))
F5 = simplify(laplace(f5, t, s))

%-Computing Inverse Laplace Transforms-
ilaplace(F, s, t);

%---Question 4---
syms s t
G1 = 1/s
G2 = 1/(s*(s+1))
G3 = 5/(s^2 +25)
G4 = (10*s + 1)/(s^2 +2*s +10)
G5 = (2*(s+1)*(s.^2+4))/((s.^2+4*s+1)*(s.^2+9))

g1 = simplify(ilaplace(G1, s, t))
g2 = simplify(ilaplace(G2, s, t))
g3 = simplify(ilaplace(G3, s, t))
g4 = simplify(ilaplace(G4, s, t))
g5 = simplify(ilaplace(G5, s, t))