基於 EMD 的廣義多層次立方雲小波轉換;EMD-based Generalized Multilevel Cubic Spline Wavelet Transform

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Please use this identifier to cite or link to this item: https://ir.lib.ncu.edu.tw/handle/987654321/98639

Title:	基於 EMD 的廣義多層次立方雲小波轉換;EMD-based Generalized Multilevel Cubic Spline Wavelet Transform
Authors:	羅宇傑;LUO, YU-JIE
Contributors:	機械工程學系
Keywords:	經驗模態分解;立方雲小波;訊號處理;理論分析;改良式的鏡像拓延;EMD;cubic spline wavelet;signal processing;theoretical analysis;improved mirror extension
Date:	2025-07-07
Issue Date:	2025-10-17 13:02:18 (UTC+8)
Publisher:	國立中央大學
Abstract:	傅立葉（Fourier）、小波（Wavelets）與經驗模態分解（Empirical Mode Decomposition, EMD）是三種常見的時頻分析方法。小波與傅立葉之間的關係在學術界早已被充分探討，但由於 EMD 的非線性特性，其與小波之間的關係仍不明確。有鑑於 EMD 在構造上下包絡線時使用了立方雲曲線（cubic spline curves），因此學術界一直推測 EMD 應該與立方雲小波（cubic spline wavelets）之間存在某種潛在關聯。若這種關係能被建立，將有助於我們對時頻分析的理解更進一步。在本研究中，我們將證明改良版的均勻相位 EMD（Uniform Phase EMD, UPEMD），一種基於 EMD 的演算法，其實是一種廣義多層次立方雲小波轉換（Generalized Multilevel Cubic Spline Wavelet Transform, GMWT）。當其篩選迭代次數設定為 1 時，它實際上就可簡化為一個多層次立方雲小波（Multilevel cubic spline wavelet）。這種方法具有以下優勢：(1) 它能指定遮罩頻率（mask frequency）來設定每個本質模態函數（IMF）的頻率範圍，並且透過調整篩選疊代次數，還可以控制欲萃取 IMF 的自適應頻寬。眾所皆知，EMD 要將頻率非常接近的兩個訊號分離是非常困難的。而利用此特性，我們將能分離頻率極為相近之訊號成分。(2) 由於本研究所提出的 GMWT 是一種廣義的小波轉換，每個被萃取的 IMF 都可以用卷積積分（convolution integral）的形式表示，這使得我們可以使用更有效率的方法來實作 GMWT。Wang已在2014年證明EMD 的計算複雜度與傅立葉轉換相同，然而在實際應用中，傅立葉轉換的運算速度仍快上數倍。藉由利用第 (2) 點的特性，我們將能大幅提升 GMWT 的計算速度。此外，我們將證明GMWT的頻寬與收斂性以及發展改良式的鏡像拓延（mirror extension）來解決GMWT因週期性邊界條件以及進行即時運算時所造成的邊界效應。此方法雖看似簡單，但在(嵌入式系統)運算實務上，卻大大縮減了計算延遲(例如: 心源性呼吸訊號)或對資料長度的要求(例如: 一天的長度訊號)。最後，我們將比較GMWT與其他文獻上方法(如：快速疊代濾波、經驗小波轉換)在人造訊號、生醫訊號、地球科學資料與物理系統等應用的分解效能。 ;Fourier, Wavelets, and Empirical Mode Decomposition (EMD) are the three widely used methods for time-frequency analysis. The relationship between wavelets and Fourier has long been thoroughly investigated in academia；however, the nonlinear challenges of EMD still makes its relationship with wavelets unclear. Since EMD applies cubic spline curves to construct the upper/lower envelope, the academic community has speculated that there may be a connection between EMD and cubic spline wavelets. If this relationship can be established, it would advance our understanding of time-frequency analysis. In this study, we will demonstrate that the improved version of the uniform phase EMD, when the number of sifting iteration is set to 1, is equivalent to a multilevel cubic spline wavelet. Therefore, we refer to this new algorithm as the Generalized Multilevel Cubic Spline Wavelet Transform (GMWT). This method has the following advantages : (1) It allows the specification of mask frequency values to set the frequency limit for each Intrinsic Mode Function (IMF). By adjusting the number of sifting iteration, the adaptive bandwidth of the IMF to be extracted can be controlled. For EMD, it is quite challenging to decompose a signal into two sub-signals with very similar frequencies.; however, this property enables GMWT to separate such closely spaced components. (2) Since the proposed GMWT is a generalized wavelet transform, each extracted IMF can be expressed in the form of a convolution integral, which allows for more efficient methods for implementing GMWT. Wang have demonstrated in 2014 that the computational complexity of EMD is comparable to Fourier transforms, though in practice, Fourier calculations are still several times faster. By leveraging property (2), we will significantly improve the computational speed of GMWT. Furthermore, we theoretically establish the bandwidth properties and convergence of GMWT, and propose an improved mirror extension technique to address the boundary effects caused by the periodic boundary conditions of GMWT and those arising during real-time processing. Although this method appears simple, it significantly reduces computational latency (e.g., EDR signal) and alleviates constraints on signal length (e.g., LOD signal) in practical implementations such as embedded systems. Finally, we compare the decomposition performance of GMWT with other existing methods, such as Fast Iterative Filtering (FIF) and Empirical Wavelet Transform (EWT), across a range of signals, including synthetic signals, biomedical data, geophysical measurements, and nonlinear physical systems.
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