Daubechies Wavelets as Approximate Hilbert-Pairs?

A Hilbert-pair is a pair of wavelets that are Hilbert transforms of each other. A perfect reconstruction multirate filter bank defines a wavelet if the infinite product formula converges. If one chooses two filter banks arbitrarily, in general, the Hilbert transform relationship is not well approximated. This letter reports on an interesting discovery about the celebrated family of orthonormal Daubechies filters with maximum vanishing moments. It is found that if two filters whose lengths differ by four are chosen from this family, a Hilbert-pair of reasonable approximation quality is obtained. An explanation for this discovery is provided, and lessons that can be learned are discussed.