I'm very new in a research topic about wavelet filtering and I received an observation by a professor, but I actually cannot understand it, so I ask you to help me. Suppose to perform a stationary wavelet packet transform of a signal (or a wavelet packet transform without decimation, but it is irrelevant for this question). I can see it from a filter bank point of view: I can recursively convolve the signal x[t] with the impulsive responses g0 and h0 of the high-pass and low-pass filters to obtain the wavelet coefficients of a new level. Suppose to compute the coefficients until level 2. So, I will obtain four sets of coefficients, which are also obtainable through four equivalent filters, build by convolving g0 and h0 in different ways. If I want to reconstruct the signal, I can build four equivalent reconstruction filters, convolve the wavelet coefficients with them and sum up the results. It works fine since I tried in Matlab, but now the problem arises. Before summing up all the signals obtained, I can express them as: yi[t] = (x[t] * di[t]) * ri[t] and because of the properties of convolution, (x[t] * di[t]) * ri[t] = x[t] * (di[t] * ri[t]) = x[t] * bi[t], where bi[t] is the convolution between the i-th decomposition and reconstruction filter. I also tried it in matlab and obviously it works. Now, a professor says:
"the convolution between orthonormal basis for the wavelet filter for the same order must be a constant value (e.g. the convolution result is not a time varying function like b_i[t], but a constant value)"
How can he say it? As far as I know, bi[t] is not a constant! Otherwise yi[t] would be a scaled version of x[t]. Please help: how would you interpret the professor's sentence?