%% EE3032, Winter 2019-'20, W4D4, Dr. Durant
% The inputs were given graphically and were 0 outside of [0,2].
% In class we derived the first segment of y(t) using convolution.
% This script completes the example.

%% Get a fresh start...
close all % closes any open windows, etc.
clearvars % removes any existing variables from workspace

%% Setup...
dt = 0.01; % time step for plotting
t = -1 : dt : 5;
r02 = 0 <= t & t <= 2; % Boolean vector matching shape of t, true when 0<=t<=2
r24 = 2 <= t & t <= 4;

%% x and h
x = zeros(size(t));
h = x; % initialize the same way
x(r02) = 1-t(r02)/2;
h(r02) = 4-t(r02);
% Note: In some languages, "h=x" would make the variables refer to the same vector/matrix and would lead to an incorrect result.
% MATLAB is a "copy-on-write" language, which increases efficiency (but makes implementing MATLAB more complex);
% initially "h=x" has the objects share memory, but when you write to h or x,
% MATLAB understands that it needs to make an independent copy.

%% Now let's have MATLAB's Symbolic Toolbox do more of the work for us...
syms tt tau % tt is the symbol corresponding to 't', which we are already using
hh = 4-tau; % h(tau); hh becomes symbolic automatically
xx = tau/2 + 1 - tt/2; % x(t-tau) as derived in class
y02 = int(hh*xx, tau, 0, tt); % integrate d\tau from 0 to tt
y24 = int(hh*xx, tau, tt-2, 2);
disp('y(t), 0<=t<=2'), collect(y02) % collect() is one of several functions that tells MATLAB how to show a symbolic result (factored various ways, expanded, etc.). `doc collect` for more information.
disp('y(t), 2<=t<=4'), collect(y24)

%% y(t) on [0,2] was calculated in class and graphed. Complete the function...
y = zeros(size(t));
y(r02) = t(r02).^3/12 - 3/2*t(r02).^2 + 4 * t(r02); % This matches symbolic result above
% But, let's use MATLAB's symbolic result for 2<=t<=4 to compute numeric results in that range...
y24s = matlabFunction(y24);
y(r24) = y24s(t(r24));
figure, plot(t,x, t,h, t,y)
legend('x(t)','h(t)','y(t)')