So I have been given to understand that the discrete wavelet transform is able to provide both time and frequency resolution in ways that classic Fourier and even short time Fourier cannot. By carrying out discrete convolutions of the wavelet at different scaling factors, one can perform this transform. Now, I have been reading about the implementation and it seems as though this transformation is often carried out by convolving quadrature mirror filters over the signal, downsampling by a factor of two, then performing the same operation on the downsampled version to determine lower frequency components.
I am very confused about the particulars of why this works. It would appear that a particular QMF filter bank, a binomial filter bank, designed by Ali Akansu, is able perform the same transformation as the Daubechies wavelet. https://en.wikipedia.org/wiki/Binomial_QMF
I am wondering if the same can be said for other wavelets. Are there binomial QMF filter banks designed to perform the discrete wavelet transforms for different families of basis wavelets? What are the mathematics behind this alternate approach to calculating discrete wavelet transforms and what are the advantages of using this filter bank technique (computing resources, etc)?
