Construction of Hilbert transform pairs of wavelet bases and optimal time-frequency localization
We propose a novel method of constructing exact Hilbert transform (HT) pairs of wavelet bases using fractional B- splines and state necessary and sufficient conditions for generating such wavelet pairs. In particular, we demonstrate how HT pairs of biorthogonal wavelet bases of L2 (K) can be constructed using well-localized scaling functions with identical Riesz bounds. Finally, we illustrate this concept by constructing a family of analytic Gabor-like wavelets that exhibit near optimal time-frequency localization.