ESTIMATION OF FRACTAL SIGNALS FROM NOISY MEASUREMENTS USING WAVELETS

The 1/f family of fractal processes are increasingly appealing candidates for data modeling in a variety of signal processing applications in light of the fact that such a wide range of phenomena are inherently well suited to these models. In contrast to the well-studied family of ARMA processes, 1/f processes are characterized by an inherent scale invariance and persistent long-term correlation structure. Despite their apparent applicability in many scenarios, they have received relatively little attention in the traditional signal processing literature. This has been due, at least in part, to the mathematical intractability of fractal processes. However, fractal signal representations in terms of orthonormal wavelet bases have recently been described that considerably simplify the analysis of these processes. We exploit the role of the wavelet transformation as a whitening filter for 1/f processes to address problems of parameter and signal estimation 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 error 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, 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. A variety of results from simulations are presented to demonstrate the viability of the algorithms.