% EE3032 W6D2 F19 Example - Dr. Durant

% Key point: All (reasonable) periodic signals can be represented a
% (possibly infinite) sum of sinusoids. Since we know how an LTI system
% affects each sinusoid, we can apply superposition to get the overall
% response.

% TRY: ADD/REMOVE components to k and X to get a higher/lower order approximation

% Today, we just show that a square wave can be approximated arbitrarily
% well (technically, in an energy sense) by a series of sinusoids.

% Let x(t) be an even, 1 V peak (with x(0) = 1), square wave with fundamental...
T0 = 1; % s
k = [ 1  3    5   7]; % multipliers of the fundamental frequency
X = [1 -1/3 +1/5 -1/7]; % we will learn how to derive this series given the square wave later.
 
omega = 2*pi*k/T0; % radians/s for each of the components. 1/T0 is Hz, convert to radians, and calculate frequency multiple/harmonic

t = linspace(-3, 3, 1000);

x = zeros(size(t)); % initialize x(t) to be the correct size
for idx = 1:length(omega) % for each of the frequencies
   part = abs(X(idx)) * cos (omega(idx)*t + angle(X(idx))); % calculate the contribution using the corresponding X phasor
   plot(t,part), hold on % plot that component, and "hold" previous plots as more are added
   x = x + part;
end

plot(t,x) % and plot the final signal