小波轉換及其在混沌信號上之應用

Abstract

本論文的主要目的在討論當小波轉換 (Wavelet transform) 時，不同的 基礎小波函數 (basic wavelet function) 在信號處理上所產生的效應以 及將小波轉換應用於混沌 (chaotic) 信號上。一般來說，在應用小波轉 換時，會遭遇到較困難的問題，乃是如何選取或設計一個基礎小波函數 ，和在不同信號處理時如何決定其內部的大小參數 (scale factor)。因 此，我們提出一種時間-頻率視窗(time-frequency window) 法則來決定 如何選取基礎小波函數，同時對於大小參數選取的問題提出一套有系統的 準則。再者，至今小波轉換應用於混沌信號的仍不常見。傳統上，我們是 從動態系統的里亞譜諾指數(Lyapunov exponents) 和分叉圖 (Bifurcation diagram) 去判斷混沌現象。然而，此兩種方法皆是以時域 的觀點去看待混沌行為。在此論文中，我們將小波轉換應用於混沌信號上 ，且成功的以頻域的觀點來判斷動態系統中的混沌現象。 The purposes of this thesis are to discuss the effect for erent basic wavelet functions based on the wavelet transform (WT) in signal processing and to apply the $WT$ to chaotic signals. In general, the difficult issues in $WT$ are how to design or search a adequate basic wavelet function and how to decide the scale factor according to different signal analysis. So some basic wavelet functions are studied and the time-frequency window criterion is proposed to the choice of basic wavelet functions, and give a rule to select scale factor in $WT$ systematically. Moreover, little attentions have been paid to the application of chaotic signals for $WT$. Traditionally, to identify a chaotic system is by means of its bifurcation diagram or the Lyapunov exponent. While these two methods are to see the chaotic phenomena in the viewpoint of time domain, we apply the $WT$ to chaotic signals and change the viewpoint of time domain into frequency domain to indentify chaotic behaviors of some chaotic systems successfully.