Estimation of fractal signals from noisy measurements using wavelets

The role of the wavelet transformation as a whitening filter for 1/f processes is exploited to address problems of parameter and signal estimations for 1/f processes embedded in white background noise. Robust, computationally efficient, and consistent iterative parameter estimation algorithms are derived based on the method of maximum likelihood, and Cramer-Rao bounds are obtained. Included among these algorithms are optimal fractal dimension estimators for noisy data. Algorithms for obtaining Bayesian minimum-mean-square signal estimates are also derived together with an explicit formula for the resulting error. These smoothing algorithms find application in signal enhancement and restoration. The parameter estimation algorithms find application in signal enhancement and restoration. The parameter estimation algorithms, in addition to solving the spectrum estimation problem and to providing parameters for the smoothing process, are useful in problems of signal detection and classification. Results from simulations are presented to demonstrated the viability of the algorithms