%% Example 3.8 Comparison of Haar, CDF(2,2) and Daub transforms
% I sample the analog signal f(t) at 64 points where 
% 
% $f\left(t\right)=16t^2 \text{  }for\text{ }0\text{ }\le t\le \frac{1}{4}$, 
% 1 for $\frac{1}{4\text{ }}\le t\text{ }\le \frac{1}{2},\text{ }\frac{1}{2}for\text{ 
% }\frac{1}{2}\le t\le \frac{3}{4},and\text{ }2-2t\text{ }for\text{ }\frac{3}{4}\le 
% t\le 1$

x = [16*((0:15)/64).^2 ones(1,16) ones(1,16)/2 (2*ones(1,16)-2*(49:64)/64)];
plot(x)
axis([0 64 -0.2 1.2])
xticks([0:16:64])
title('Signal of length 64 (quadratic, step, linear)')
%% 
% Next I compute the one-scale Haar transform using the command dwthaar 
% from Computer Exercise 3.7.4 and plot it.
%%
y = dwthaar(x);
plot(y)
axis([0 64 -0.2 1.2])
xticks([0:8:64])
annotation('doublearrow','x',[.13,.51],'y',[.3,.3])
text(12,.2,'trend')
annotation('doublearrow','x',[.51,.9],'y',[.3,.3])
text(45,.2,'detail')
title('One-scale Haar transform')
%% 
% The trend vector is from 0 to 31 and the detail is from 32 to 63. Notice 
% that the trend looks quite similar to the original signal.
% 
% Next I compute the CDF(2,2) transform using the command cdfamat from Computer 
% Exercise 3.7.2.
%%
z = cdfamat(64)*x';
plot(z')
axis([0 64 -0.2 1.6])
xticks([0:8:64])
annotation('doublearrow','x',[.13,.51],'y',[.3,.3])
text(12,.3,'trend')
annotation('doublearrow','x',[.51,.9],'y',[.3,.3])
text(45,.3,'detail')
title('One-scale CDF(2,2) transform')
%% 
% Notice that the trend vector is distorted around n=16. This reflects the 
% jump discontinuity of $f$ at $t=\frac{1}{2}$. The derivative of $f$ has a jump 
% at$\text{ }t=\frac{3}{4}$ so n=48. This causes the blip you see in the detail 
% vector at 48. You don't see this in the Haar transform. 
% 
% Finally, I compute the Daub4 transform, using the command daub4mat from 
% Computer Exercise 3.7.3.
%%
w = daub4mat(64)*x';
plot(w')
axis([0 64 -0.2 1.6])
xticks([0:8:64])
annotation('doublearrow','x',[.13,.51],'y',[.3,.3])
text(12,.3,'trend')
annotation('doublearrow','x',[.51,.9],'y',[.3,.3])
text(45,.3,'detail')
title('One-scale Daub4 transform')
%% 
% Again there's some distortion in the trend vector around n=16. This jump 
% discontinuity also appears in the detail vector of the Daub4 transform, at about 
% n=40. Again, the slope discontinuity appears at in the detail vector at n=48.
% 
% Notice that all three transforms give information about the location in 
% time of the singullaities. The CDF(2,2) transform has more information than 
% the Haar transform and the Daub4 transform has the most information.