For coding applications it would be interesting to retain the downsampled coefficients of the lowpass analysis filter and neglect the high pass coefficients(Assume them to be zeros). In this exersise, we will investigate whether such as system would retain basic polynomials. Consider the Daubechies 2 filter bank (You can get the filter coefficients with the "wfilters" command in Matlab). As the input signal, consider the samples of
a) A constant function
b) A linear polynomial - (ax+b)
b) A quadratic polynomial - (ax^2+bx+c)
Perform the analysis operation and retain only the lowpass coefficients (The lowpass filter is the one which have 2 zeros at z=-1). Now perform the reconstruction and and plot the outputs.
Now we will compare the results with the filter bank(with 2 zeros at z=1) you have designed in problem 3(Homework 4). Retain only the coefficents of the filter with with 2 zeros at z=1 and perform the reconstruction. Compare the results with the previous case. What do you think is the reason??
Can you find out the functions which the new system will preserve? Show experimentally.
For practical coding applications, which of the two systems would you prefer ? Why??