% EE3221 W6D1 Dr. Durant 1/2021
% Demonstration cancellation (z-domain and convolution)
% of certain frequencies by an averaging filter.

% z_cancellation

clearvars
close all

% x(n) = sin(Wn) u(n)
% X(z) = (z sin(W))/(z^2 - 2z cos(W) + 1)
% Let W = 2pi/3 radians/sample
% e^jW = -1/2 + j sqrt(3)/2 = cos(W) + j sin(W)
% X(z) = (z sqrt(3)/2)/(z^2 + z + 1)
% Check roots of denominator

% Let h(n) = u(n) - u(n-3)
% H(z) = z/(z-1) - z^-2/(z-1) = (z - z^-2)/(z-1) = (z^3-1)/((z^2)(z-1)))
% check roots and cancel... H(z) = (z^2+z+1)/z^2

% Y(z)/z = H(z)X(z)/z = (sqrt(3)/2) / z^2
% Y(z) = (sqrt(3)/2) / z
% y(n) = (sqrt(3)/2) delta(n-1)

W = 2*pi/3;
n = 0:20;
x = sin(W*n);
h = zeros(size(x));
h(1:3) = 1;
y = conv(x,h);
plot(n,y(1:length(n)),'b*', n,x,'r+', n,h,'g^')
legend('y', 'x', 'h')