t = -5:0.01:10;

u = double(t>=0);
x = exp(-0.5*t).*u;


x_1 = exp(-0.5*(t-2)) .* double((t-2)>=0); #x_1 = x(t-2)
x_2 = -1*exp(-0.5*(t)) .* double((t)>=0);

plot(t, x_2);
grid on;
title("signal x(t)"); xlabel("t"); ylabel("x")

#------------------------------

t = 0:0.01:10;
x_1 = sin ((2*pi/10)*t);
x_2 = cos((2*pi/10)*t);

T = @(x)  2*x; # x can be any arbitrary input

y_1 = T(x_1)
y_2 = T(x_2)

x_3 = x_1 + x_2

y_3_hat = T(x_3) # the system responds to the linear combination of inputs
y_3 = y_1 + y_2 #the same linear combination of output

threshold = 1e-3;

max_difference = max (abs(y_3 - y_3_hat));

    if max_difference < threshold:
        disp("y=2x is a linear system")
    else:
    disp("y=2x is not a linear system");

end

#------------------------------

t = 0:0.01:10;
x = sin(2*pi*t) + 0.5*cos(pi*t);

T = @(x)  2*x;

t0 = 1;
x_shifted = sin(2*pi*(t-t0)) + 0.5*cos(pi*(t-t0));

y_shifted = T(x_shifted); #the response of the system to x_shifted

y_2 = 2*x_shifted; #y_2 = y(t-t0) = 2tx(t-t0) = 2x_shifted

#check if y2 == y_shifted

max_difference = max( abs(y_2 = y_shifted));


    if max_difference < threshold:
        dis("y=2x is a Time invariant system");
    else:
    disp("y=2x is not a Time variant system");