經驗模態分解局部性之近場性質的一些證明

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陳駿程(Chun-Cheng Chen) 查詢紙本館藏

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論文名稱

經驗模態分解局部性之近場性質的一些證明
(Some proofs for the nearfield properties of locality of Empirical Mode Decomposition)

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摘要(中)

經驗模態分解(empirical mode decomposition，EMD) 方法是一個自適性分解非穏態訊號拆解方法。它將一個訊號拆解成若干個子訊號。它藉由篩選程序(sifting process)遞迴地將一個時間訊號拆解為幾個在時間頻率平面上分離(well separated)的本質模態函數 (Intrinsic Mode Function，IMF)。由數學角度視之，EMD是一個非線性(nonlinear)和自適應(adaptive)的方法。這兩個性質有別於短時傅立葉 (STFT)與小波 (wavelet)，EMD經常能完整保留波的重要物理參數：相位與波高，因此IMF具有較強之物理或生理意義。在許多應用上已證明EMD在處理非穩態訊號比傅立葉與小波優越，因此已廣泛被應用於許多領域。但EMD仍然存在許多問題，因此許多改進版本陸陸續續被發展。但目前尚無一個演算法可解決所有問題。究其根本原因乃是EMD是一個理論發展遠落後於應用的方法。
之前文獻證明了篩選算子的脈衝響應定理 (Impulse Response Theorem of Sifting Operator，IR theorem)：篩選算子(疊代)等於每個(極值)點所對應之非線性脈衝響應的線性組合，而極值點值恰為其係數。因此可以透過研究脈衝響應的性質來得到篩選運算子的局部性質，進而得到IMF的性質。在訊號的某個時間點給一個振幅為一的脈衝訊號，經由三次木條曲線得到脈衝響應曲線。時間區域可以依據與脈衝的距離區分為遠場與近場。近場是指與脈衝訊號所在節點相鄰的區間，其餘區間為遠場。遠場的局部性是指脈衝響應的振幅隨著與脈衝訊號的距離增加而變小。反之若隨距離變遠而振幅變大，則稱為非局部。遠場的性質已於先前的研究中得到。定義相鄰兩個極大(小)值之間的距離稱為尺度，相鄰兩個尺度的比值稱為尺度比，且其值永遠大於等於一。先前文獻中已證明遠場局部與非局部的充分條件如下，當尺度比小於2.4，遠場的脈衝響應為局部的;當尺度比大於3.1，脈衝響應為非局部的。
相較於遠場，近場數學性質更為複雜，因此目前尚未有近場性質的相關理論，使得局部的性質尚未完整。本論文目的就是要分析近場的性質。在訊號處理中，理想上脈衝響應的振幅應小於脈衝訊號的振幅。稍後的章節將證明在EMD中情況並非如此。因此我們必須重新定義近場附近區域的局部條件，當IR的振幅小於某一閥值時稱為局部，反之則為非局部的。本論文的研究分析發現，近場的局部充分條件與遠場是相同的。
最後我將探討訊號的局部性如何影響分解效能，以檢視EMD的適用性。這是相當重要的問題：給予一個訊號即可知道它是否適合以EMD分析。
當調幅變化(Amplitude Modulation，AM)的變化很小且為局部的，則EMD的分解效果較佳；反之，當AM的變化很大或非局部時，則EMD的分解效果較差。我將舉幾個例子說明此現象。

摘要(英)

The empirical mode decomposition (EMD) method is an adaptive decomposition method for non-stationary signals. It can decompose a signal into several sub-signals. Due to the recursively sifting process it can decompose a time signal into several well separated Intrinsic Mode Functions (IMF) on the time-frequency plane. From a mathematical point of view, EMD is a non-linear and adaptive method. Finally, these two properties are different from Fourier transform and wavelet methods. Compared with short-time Fourier transform (STFT) and wavelet, EMD can usually completely retain the important physical parameters of wave: phase and wave height, so IMF has strong physical or physiological significance. In many applications, it has been proved that EMD is superior to Fourier transform and wavelet in processing non-stationary signals, so EMD has been widely used in many fields such as mechanical engineering and so on. But EMD still has many problems, so many improved versions have been developed one after another. But currently there is no single algorithm that can solve all problems. The root cause is that EMD is a method whose theoretical development lags far behind its application.
Early study has proved the Impulse Response Theorem of Sifting Operator: the sifting operator (iteration) is equal to the linear combination of the nonlinear impulse response corresponding to each (extreme) point, and the extreme value is exactly its coefficient. Therefore, the properties of the sifting operator can be obtained by the properties of impulse response, and then obtain the properties of the IMF. Give an impulse signal with an amplitude of one at a certain point in time of the signal, and obtain the impulse response curve through the cubic spline. The time domain can be divided into far-field and near-field according to the distance from the impulse signal. The near-field refers to the interval adjacent to the node where the impulse signal is located, and the rest of intervals are the far-field. Locality of the far field means that the amplitude of the impulse response decreases as the distance from the impulse signal increases. Otherwise, if the amplitude becomes larger as the distance increase called non-local. The properties of far-field have been obtained by early study. The distance between two adjacent maximum (minimum) values is defined as the scale, and the ratio of the two adjacent scales is called the scale ratio, its value is always greater than or equal to one. Early study has proved that local and non-local sufficient condition for the far-field are as follows. When the scale ratio is less than 2.4, the far-field impulse response is local; The scale ratio is greater than 3.1, the far-field impulse response is nonlocal.
The mathematical properties of Near-field are more complicated rather than far-field, so there is no relevant theory of the near-field properties at present, make local properties are not yet complete. The purpose of this paper is to analyze the properties of the near-field. In signal processing, ideally amplitude of the impulse response should be smaller than the amplitude of the impulse signal. Later chapters will prove that is not true in EMD. Therefore, we must redefine the local conditions in the vicinity of near-field. When the amplitude of IR is less than certain threshold called local, and vice versa, it called nonlocal. The research and analysis of this study found that the local sufficient conditions of the near-field are the same as those of the far-field.
Finally, I will explore how locality of the signal affects the decomposition performance to examine the applicability of EMD. It is a very important question: give a signal to know whether it is suitable for EMD analysis.
When the variety in amplitude modulation (AM) is small and local, the decomposition efficiency of EMD is better; Conversely, when the change in AM is large or nonlocal, the decomposition efficiency of EMD is poor. At last, I will give a few examples to illustrate this phenomenon.

關鍵字(中)

★ 經驗模態分解
★ 本質模態函數
★ 局部性
★ 脈衝響應定理
★ 近場
★ 分解效能

關鍵字(英)

★ Empirical Mode Decomposition (EMD)
★ Intrinsic mode function (IMF)
★ Locality
★ Impulse response (IR) Theorem
★ Near-field
★ Efficiency

論文目次

摘要 i
Abstract iii
誌謝 vi
目錄 vii
圖目錄 ix
表目錄 x
符號說明 xi
一、緒論 1
1-1 研究動機 1
1-2 文獻探討 1
1-3 研究目的 3
1-4 論文章節 3
二、EMD的基礎 4
2-1 經驗模態分解(Empirical Mode Decomposition, EMD) 4
2-2 三次木條曲線(Cubic Spline) 5
2-3 篩選運算子的脈衝響應定理(Sifting Operator of IR Theorem) 5
2-4 脈衝響應介紹(Introduce of IR) 6
2-4-1 脈衝響應定義 6
2-4-2 脈衝響應遠場y_k性質 9
三、脈衝響應的近場局部性 10
3-1 近場yk性質與脈衝響應(IR) 10
3-2 脈衝響應(IR)的近場局部定義 11
3-3 脈衝響應(IR)的近場局部性證明 12
3-3-1 y、u_(e,k)對於近場變數的單調性 12
3-3-2 脈衝響應近場的局部與非局部條件 18
3-3-3 脈衝響應的局部與非局部條件 20
四、討論 22
五、結論與未來展望 24
5-1 結論 24
5-2 未來展望 24
參考文獻 25
附錄一、引理二證明 29
附錄二、引理三(2.、3.)證明 31
附錄三、引理四證明 34
附錄四、引理五證明 45
附錄五、定理六證明 53
附錄六、引理八證明 54
附錄七、引理九證明 56

參考文獻

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指導教授

王淵弘

審核日期

2021-9-29

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