Wavelet transform analysis directly from sinusoid crossings

We show that the wavelet transforms of an analog signal ${\mathit{f}}_{\mathit{a}}$(t) can be computed directly and efficiently from a data representation consisting only of locations {${\mathit{t}}_{\mathit{i}}$}, where ${\mathit{f}}_{\mathit{a}}$(t) intersects with the reference sinusoid r(t) of frequency W and amplitude A. To achieve a measurement bandwidth of W, one crossing must occur within each interval Δ=1/2W. This is satisfied when A≥‖${\mathit{f}}_{\mathit{a}}$(t)‖ for all ${\mathit{f}}_{\mathit{a}}$(t) values within the sampling period T. A total of 2M sinusoid crossings occur within T. Crossing-based wavelet analysis is demonstrated with respect to the sombrero-shaped wavelet, using 256 crossings of an interferogram signal. © 1996 The American Physical Society.