TEACHING SELF-AFFINE AND DYNAMICAL SYSTEMS ENGINEERING

Abstract

Teaching digital signal processing at the graduate and undergraduate levels has a long tradition at universities. The signals include time series, still images, videos, and volumetric data such as radar and Doppler radar. The traditional topics such as spectral techniques in single-scale analysis and synthesis are now being expanded to include wavelet bases for multiscale analysis and synthesis [2]. The course described in this paper [1] expands the analysis to polyscale analysis and synthesis as it relates to self-affine processes and dynamical systems [3-4]. This course presents foundations of fractal (polyscale) and chaos theory, with applications to engineering. A unified approach to fractal dimensions provides tools for multiscale and polyscale analysis of time series, images, video, and other objects. Other topics include analysis and synthesis of mono- and multifractal coloured noise for research purposes, as well as stability analysis of dynamical systems, characterization of chaos using Lyapunov exponents, and reconstruction of strange attractors from experimental data. The course also provides a unified description of 19 different fractal dimensions grouped in four classes based on: set- morphology, entropy, spectrum, and variance. Special attention is given to (a) probability and pair-correlation algorithms for E-dimensional images and strange attractors, (b) batch and real-time computation of the variance fractal dimension, and (c) the Rényi dimension spectrum formulation for fractals and multifractals. The objective is to learn how to characterize multifractals through multi- and poly-scale analyses, and how to extract features for their classification. This paper describes the structure of the course, the set of topics covered, the set of course projects, and the lessons learned from the extensive experience with the course.