Strange values for approximating coefficients in wavelet decompsition in Matlab

Question

I'm trying to get wavelet decomposition of arcsin(x) using, say, Haar wavelets When using both Matlab's dwt or wavedec functions, I get strange values for approximating coefficients. Since applying low-pass Haar wavelets's filter equals to performing half-sum and the maximum of arcsin is pi/2, I assume that approximating coefficients can't surpass pi/2, yet this code:

x = linspace(0,1,128);
y = asin(x);
[cA, cD] = dwt(y, 'haar'); %//cA for approximating coefficients

returns values more than pi/2 in cA. Why is that?

Answer 1

I believe what makes you confused here is thinking that Haar's filter just averages two adjacent numbers when computing 1-level approximation coefficients. Due to the energy preservation feature of the scaling function, each pair of numbers gets divided by sqrt(2) instead of 2. In fact, you could see what a particular wavelet filter does by typing in the following command (for the Haar filter in this case):

[F1,F2] = wfilters('haar','d')
F1 =
    0.7071    0.7071
F2 =
   -0.7071    0.7071

You can then check the validity of what you have gotten above by constructing a simple loop:

CA_compare = zeros(1,64);
for k = 1 : 64
CA_compare(k) = dot( y(2*k-1 : 2*k), F1 );
end

You will then see that "CA_compare" contains exactly the same values as your "cA" does.

Hope this helps.