Synthesis, Analysis, and Processing of Fractal Signals
MASSACHUSETTS INST OF TECH CAMBRIDGE RESEARCH LAB OF ELECTRONICS
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Fractal geometry arises in a truly extraordinary range of natural and man-made phenomena. The 1f 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, 1f 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 1f processes in terms of orthonormal wavelet bases that considerably simplifies their analysis. Wavelet-based representations yield highly convenient synthesis and whitening filters for 1f processes, and allow a number of fundamental detection and estimation problems involving 1f processes to be readily solved. In particular, we obtain robust and computationally efficient algorithms for parameter and signal estimation with 1f signals in noisy backgrounds, coherent detection in 1f backgrounds, and optimal discrimination between 1f signals. Results from a variety of simulations are presented to demonstrate the viability of the algorithms. In contrast to the statistically self-similar 1f processes, homogeneous signals are governed by deterministic self-similarity.
- Numerical Mathematics