%% Example 3.13 - CDF(2,2) transform, N=8
% We put the normalization in the synthesis matrix. 

format rational
Ta = 4*sqrt(2)*cdfamat(8);
Ts = cdfsmat(8)/(4*sqrt(2));

%% 
% I plot the basis for the trend subspace.
%%
u0t = Ts(:,1);
S = shift(8);
plot(u0t), hold on
plot(S^2*u0t)
plot(S^4*u0t)
plot(S^6*u0t)
hold off
title('Basis for CDF(2,2) trend subspace, N=8')
xticklabels({'0','1','2','3','4','5','6','7'})
%% 
% I plot the basis for the detail subspace.
%%
v0t = Ts(:,5);
S = shift(8);
plot(v0t), hold on
plot(S^2*v0t)
plot(S^4*v0t)
plot(S^6*v0t)
hold off
title('Basis for CDF(2,2) detail subspace, N=8')
xticklabels({'0','1','2','3','4','5','6','7'})
%% 
% 
% 
% I decompose the signal $x={\left\lbrack 0\text{ }1\text{ }2\text{ }3\text{ 
% }4\text{ }5\text{ }6\text{ }7\right\rbrack }^T$ into its projections onto the 
% trend and detail subspaces. First we look at the basis vectors for the trend 
% and detail subspaces.
%%
x = [0 1 2 3 4 5 6 7]';
y = Ta*x
%% 
% The trend is 
%%
s = y(1:4)
%% 
% and the detail is 

d = y(5:8)
%% 
% The projection of $x$  onto the trend subspace is 
%%
xs = Ts*[s;zeros(4,1)]
%% 
% The projection onto the detail subspace is 
%%
xd = Ts*[zeros(4,1);d]
%% 
% You see that $x=xs+xd$.
% 
% I plot the signal as a solid line and its trend projection as a dashed 
% line.
%%
plot(x), hold on
plot(xs,'--')
xticklabels({'0','1','2','3','4','5','6','7'})
%% 
% The relative compression error in approximating $x$ by $xs$ is
%%
format short
norm(x-xs)^2/norm(x)^2