On multiscale wavelet analysis for step estimation

We consider step detection and estimation using a multiscale wavelet analysis, based on the ability of a certain discrete wavelet transform (DWT) to characterize signal steps and edges. This DWT, developed by Mallat and Zhong (1992), estimates the gradient at various smoothing levels without downsampling in time. As first proposed by Rosenfeld (1970) for edge sharpening, multiple scales are combined by forming the pointwise product across scales. We show that this approach is a non-linear whitening transformation, and characterize the non-Gaussian PDF of the output. Detection curves are shown for parameterized sigmoidal step change signals. Step location estimation performance is also shown, with comparison to the Cramer-Rao bound in additive white Gaussian noise