小波轉換在波散射之應用

Abstract

本論文探討小波轉換的基本原理及特性並將其應用於求解電磁積分方程式。傳統動差解法(Method of Moments, MoM)的主要瓶頸在於造成滿矩陣(Full Matrix)及巨額的計算量，由於計算時間取決於非零元素的多寡且不同的小波其稀疏化的效果差異甚大，本論文旨在設計不同的小波來將阻抗矩陣最疏化(sparsified)而不犧牲解的精確度。同時我們也提出可見能量的觀念來選取最合適的小波，這一能量指標利於設計出積分方程數值解的最佳小波，因為最小可見能量的小波其相對應的阻抗矩陣最為稀疏。 基於論文的完整性，我們首先簡述小波的數學基礎及傳統的動差法。隨後介紹可見能量的概念並用於Daubechies 小波的設計，一般的Daubechies小波是最小相位型(minimum-phase)，經由數值實驗發現最小可見能量的小波是混合相位型(mix-phase)，這種改良的小波更利於電磁數值解。為了得到更佳的稀疏度及驗證可見能量的正確性，我們使用最佳化來設計另一種型式的小波，格式結構的正交濾波器（Lattice-structure Quadrature Mirror Filter, QMF)。配合小波封包最佳基底選擇法(Best basis selection for wavelet packet)，我們得到更好的稀疏度。對於圓柱型散射體，其阻抗矩陣的稀疏度幾乎接近對角化。這意味著可見能量指標可以選取各個不同領域內的小波來有效的解散射積分方程，同時這也拓展了求解的自由度。 論文最後，我們指出本研究在電磁領域內幾個可以更深入研究的方向及應用。

The fundamental principles and characteristics of wavelet transform are discussed and applied to solve electromagnetic integral equations (IE). The major drawback of conventional method of moments (MoM) is the full matrix generation and huge computation time. Since different wavelets will results in diverse sparsity and computation time depends on the number of non-zero elements. The purposes of this dissertation are focus on the design of different wavelets to sparsify MoM impedance matrix without sacrificing much accuracy. We also present the concept of visible energy (VE) as a criterion to select a more suitable wavelet. The VE index is useful in designing new wavelets for electromagnetic IE. The lower VE index is, the more sparsity is the impedance matrix. In this dissertation, we first introduce the mathematical preliminaries of wavelets and conventional MoM for the sake of completeness. The VE concept is introduced thereafter and applied to Daubechies wavelet design. Daubechies wavelet is a minimum-phase one. However, we found that mix-phase wavelets are more useful in numerical solution of IE for their lower VE index. To acquire more sparsity and verify the validity of VE, we design another type of wavelet by optimization technique. That is the lattice-structure quadrature mirror filter (QMF) which is widely used in digital signal processing community. Much more sparsity is arrived by the QMF and best basis selection algorithm of wavelet packet. For example, the impedance matrix of a cylindrical scatterer is almost diagonalized. This result suggested that more wavelts from different fields could contribute to solve scattering IE efficiently by VE selection. This approach also extends the degree of freedom. Last but not least, we pinpointed several possible extensions and applications for further studied in electromagnetic field.