%% Section 3.2-3.2 Example of signal compression

s = [56 40 8 24 48 48 40 16]'
%% 
% We computed the 3 scale wavelet transform of s and found it is 
%%
v = [35 -3 16 10 8 -8 0 12]'
%% 
% We compress the signal using the threshhold $\varepsilon =4\ldotp$ First 
% we need the wavelet synthesis matrix $W_s$. Here's a calculation of it, starting 
% by calculating the wavelet analysis matrix.
%%
T3a = sym([1/2*eye(4) 1/2*eye(4); 1/2*eye(4) -1/2*eye(4)]);
T3=T3a*downsamp(4);
T2a = sym([1/2*eye(2) 1/2*eye(2); 1/2*eye(2) -1/2*eye(2)])*downsamp(2);
T2 = [T2a zeros(4);zeros(4) eye(4)];
T1a = sym([1/2 1/2; 1/2 -1/2]);
T1 = [T1a zeros(2) zeros(2) zeros(2); zeros(2) eye(2) zeros(2) zeros(2); zeros(4) eye(4)];
Wa = T1*T2*T3
%% 
%  
% 
% I made the entries symbolic with the *sym* command so I would get fractions 
% instead of decimals. The synthesis matrix is the inverse of the analysis matrix,
%%
Ws = Wa^(-1)
%% 
% For the compression, I replace the second component of v by 0 then apply 
% Ws.
%%
v(2,1) = 0; v

%%

y = Ws*v
%% 
% I compress more, using 9 as the threshhold.
%%
v(5,1)=0; v(6,1)=0; v
%%
z = Ws*v
%% 
% I plot the first two vectors. 
%%
plot(s,'b'), hold on
plot(y,'r--'), hold off
%% 
% I see that they are quite close. Now I plot them with the third.
%%
plot(s,'b'), hold on
plot(y,'r--'), 
plot(z,'k:','LineWidth',1.5), hold off