Synthesis, analysis, and processing of fractal signals

Abstract : Fractal geometry arises in a truly extraordinary range of natural and man-made phenomena. The 1/f family of fractal random processes, in particular, are appealing candidates for data modeling in a wide variety of signal processing scenarios involving such phenomena. In contrast to the well-studied family of ARMA processes, 1/f processes are typically characterized by persistent long-term correlation structure. However, the mathematical intractability of such processes has largely precluded their use in signal processing. We introduce and develop a powerful Karhunen-Loeve-like representation for 1/f processes in terms of orthonormal wavelet bases that considerably simplifies their analysis. Wavelet-based representations yield highly convenient synthesis and whitening filters for 1/f processes, and allow a number of fundamental detection and estimation problems involving 1/f processes to be readily solved. In particular, we obtain robust and computationally efficient algorithms for parameter and signal estimation with 1/f signals in noisy backgrounds, coherent detection in 1/f backgrounds, and optimal discrimination between 1/f signals. Results from a variety of simulations are presented to demonstrate the viability of the algorithms. In contrast to the statistically self-similar 1/f processes, homogeneous signals are governed by deterministic self-similarity.