Semi-Orthogonal Wavelets That Behave like Fractional Differentiators

The approximate behavior of wavelets as differential operators is often considered as one of their most fundamental properties. In this paper, we investigate how we can further improve on the wavelet's behavior as differentiator. In particular, we propose semi-orthogonal differential wavelets. The semi-orthogonality condition ensures that wavelet spaces are mutually orthogonal. The operator, hidden within the wavelet, can be chosen as a generalized differential operator $∂ _{ \tau } ^{ \gamma }$ , for a γ-th order derivative with shift τ. Both order of derivation and shift can be chosen fractional. Our design leads us naturally to select the fractional B-splines as scaling functions. By putting the differential wavelet in the perspective of a derivative of a smoothing function, we find that signal singularities are compactly characterized by at most two local extrema of the wavelet coefficients in each subband. This property could be beneficial for signal analysis using wavelet bases. We show that this wavelet transform can be efficiently implemented using FFTs.