Signal Processing Framework for the Analysis and Application of Chaotic Systems.

摘要

One contribution of chaos theory to the engineering research community is the notion that complex, erratic behavior in physical systems need not be the result of stochastic phenomena-such phenomena may result from deterministic mechanisms. This idea has been used in the analyses of several engineering systems. Perhaps more interesting are the several proposed engineering applications that take advantage of the structure of signals generated by chaotic systems. In order to take full advantage of the unique properties of chaotic signals in future applications, this structure must be well characterized. This thesis explores two aspects of this issue-the statistical structure of chaotic signals and the to linear distortion of chaotic signals. In the first portion of the thesis, we concentrate on the time- average behavior of signals generated by chaotic systems with one state variable. Using an analogy between such signals and stationary stochastic processes, we present a framework for analyzing the statistical properties of these chaotic signals. In particular, we provide readily computable analytic expressions for a broad class of statistics of a large class of chaotic signals. We also present a technique for approximating the statistics of certain chaotic signals for which exact results are unavailable. As an example of the utility of these results, we use them to determine the power spectra of chaotic signals and to analyze a model of a switching DC-DC power converter operating in a chaotic regime. In the second portion of the thesis, we concentrate on chaotic signals that have been linearly filtered. Such signals may arise, for example, when chaotic phenomena are measured through sensors with linear dynamics. We present results relating certain parameters of the original and distorted signals.