Wavelet filtering in the scale domain
Gerald Kaiser · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1996
For a given basic wavelet (psi) (t), two distinct correspondences (called C1 and C2) are established between frequency filters, defined in the frequency domain through multiplication by a transfer function W(f), and scale filters, defined in the wavelet domain through multiplication by a scale transfer function w((sigma) ). W(f) is obtained by performing a scaling convolution of w((sigma) ) with (psi) (f)* (for C1) or its spectral energy density (psi) (f) 2 (for C2). For a large class of transfer functions W(f), this relation can be solved for w((sigma) ) by applying the Mellin transform. We call such frequency filters and their associated time-domain convolution operators C1- or C2-admissible with respect to (psi) . In particular, the identity operator (W(f) equalsV 1) is C2-admissible if and only if the wavelet (psi) is admissible in the conventional sense. The implementation of the correspondence C1 is computationally simpler than C2, but C2 can be generalized to time-dependent filters. Applications are proposed to the analysis of atmospheric turbulence data and wideband Doppler filtering.