%% Difference Equation Solution Characteristics
%
% Copyright 2015 Barry Van Veen
%

clear
close all

set(0,'defaultaxesfontsize',22);
set(0,'defaulttextfontsize',22);
linwidth = 2;

step = [zeros(1,9), ones(1,500)];
n = [-9:499];
b = 1;

%% Individual Terms d^n
%

npos = 0:499;

case1 = [0.95.^npos];
case2 = [ 0.7.^npos];

figure
subplot(211)
g=stem(npos,case1);
xlim([0,80])
title('d = 0.95')
ylabel('d^n')
xlabel('n')
set(g,'LineWidth',linwidth);

subplot(212)
g=stem(npos,case2);
xlim([0,80])
title('d = 0.7')
ylabel('d^n')
xlabel('n')
set(g,'LineWidth',linwidth);

case1n = [(-0.95).^npos];
case2n = [ (-0.7).^npos];

figure
subplot(211)
g=stem(npos,case1n);
xlim([0,80])
title('d = -0.95')
ylabel('d^n')
xlabel('n')
set(g,'LineWidth',linwidth);

subplot(212)
g=stem(npos,case2n);
xlim([0,80])
title('d = -0.7')
ylabel('d^n')
xlabel('n')
set(g,'LineWidth',linwidth);

case3real = [0.95.^npos.*cos(npos*pi/11)];
case3imag = [0.95.^npos.*sin(npos*pi/11)];

figure
subplot(211)
g=stem(npos,case3real);
xlim([0,80])
title('d = 0.95e^{j\pi/11}')
ylabel('Real\{d^n\}')
xlabel('n')
set(g,'LineWidth',linwidth);

subplot(212)
g=stem(npos,case3imag);
xlim([0,80])
ylabel('Imag\{d^n\}')
xlabel('n')
set(g,'LineWidth',linwidth);

case4real = [0.7.^npos.*cos(npos*pi/3)];
case4imag = [0.7.^npos.*sin(npos*pi/3)];

figure
subplot(211)
g=stem(npos,case4real);
xlim([0,40])
title('d = 0.7e^{j\pi/3}')
ylabel('Real\{d^n\}')
xlabel('n')
set(g,'LineWidth',linwidth);

subplot(212)
g=stem(npos,case4imag);
xlim([0,40])
ylabel('Imag\{d^n\}')
xlabel('n')
set(g,'LineWidth',linwidth);

%% Example 1 one real root of char eqn
%
% root at z = 0.95

a = [1,-.95];

y = filter(b,a,step);

ys = (1/sum(a))*step;
yt = y - ys;

figure
subplot(311)
g = stem(n,y);
set(g,'LineWidth',linwidth);
xlim([-9,50])
ylabel('y[n]')
title('Root z = 0.95')

subplot(312)
g = stem(n,ys);
set(g,'LineWidth',linwidth);
xlim([-9,50])
ylabel('y_s[n]')

subplot(313)
g = stem(n,yt);
set(g,'LineWidth',linwidth);
xlim([-9,50])
ylabel('y_t[n]')
xlabel('n')

%% Example 2 one real root of char eqn
%
% root at z = 0.7

a = [1,-.7];

y = filter(b,a,step);

ys = (1/sum(a))*step;
yt = y - ys;

figure
subplot(311)
g = stem(n,y);
set(g,'LineWidth',linwidth);
xlim([-9,50])
ylabel('y[n]')
title('Root z = 0.7')

subplot(312)
g = stem(n,ys);
set(g,'LineWidth',linwidth);
xlim([-9,50])
ylabel('y_s[n]')

subplot(313)
g = stem(n,yt);
set(g,'LineWidth',linwidth);
xlim([-9,50])
ylabel('y_t[n]')
xlabel('n')

%% Example 3 Two complex roots radius 0.95
%
% roots at z = 0.95e^{j*pi/11} and z = 0.95e^{-j*pi/11}

a = [ 1 -2*.95*cos(pi/11) .95^2];

y = filter(b,a,step);

ys = (1/sum(a))*step;
yt = y - ys;

figure
subplot(311)
g = stem(n,y);
set(g,'LineWidth',linwidth);
xlim([-9,80])
ylabel('y[n]')
title('Roots z = 0.95e^{+/-j*\pi/11}')

subplot(312)
g = stem(n,ys);
set(g,'LineWidth',linwidth);
xlim([-9,80])
ylabel('y_s[n]')

subplot(313)
g = stem(n,yt);
set(g,'LineWidth',linwidth);
xlim([-9,80])
ylabel('y_t[n]')
xlabel('n')

%% Example 4 Two complex roots radius 0.95
%
% roots at z = 0.95e^{j*pi/3} and z = 0.95e^{-j*pi/3}

a = [ 1 -2*.95*cos(pi/3) .95^2];

y = filter(b,a,step);

ys = (1/sum(a))*step;
yt = y - ys;

figure
subplot(311)
g = stem(n,y);
set(g,'LineWidth',linwidth);
xlim([-9,80])
ylabel('y[n]')
title('Roots z = 0.95e^{+/-j*\pi/3}')

subplot(312)
g = stem(n,ys);
set(g,'LineWidth',linwidth);
xlim([-9,80])
ylabel('y_s[n]')

subplot(313)
g = stem(n,yt);
set(g,'LineWidth',linwidth);
xlim([-9,80])
ylabel('y_t[n]')
xlabel('n')

%% Example 5 Two complex roots radius 0.7
%
% roots at z = 0.7e^{j*pi/3} and z = 0.7e^{-j*pi/3}

a = [ 1 -2*.7*cos(pi/3) .7^2];

y = filter(b,a,step);

ys = (1/sum(a))*step;
yt = y - ys;

figure
subplot(311)
g = stem(n,y);
set(g,'LineWidth',linwidth);
xlim([-9,80])
ylabel('y[n]')
title('Roots z = 0.7e^{+/-j*\pi/3}')

subplot(312)
g = stem(n,ys);
set(g,'LineWidth',linwidth);
xlim([-9,80])
ylabel('y_s[n]')

subplot(313)
g = stem(n,yt);
set(g,'LineWidth',linwidth);
xlim([-9,80])
ylabel('y_t[n]')
xlabel('n')