| 摘要: | 經驗模態分解（empirical mode decomposition, EMD）及擾動輔助經驗模態分解（disturbanceassisted EMD, D-A EMD）為自適性分解非穩態訊號的拆解方法。它們將一個訊號拆解成若干 個振盪子訊號加上一個低頻的趨勢線。過程是藉由篩選運算子（sifting operator）遞迴地將一 段時間訊號拆解為若干個在時間頻率平面上分離（well separated）的本質模態函數（intrinsic mode functions, IMFs）。由演算法角度視之，它們是自適性（adaptive）和非線性（nonlinear） 的方法。相較於短時傅立葉（STFT）與小波（wavelet）更能完整保留波的重要物理參數：相 位以及波高，尤其可以很好地萃取後者。因此在已被應用的許多領域中，它們已經被證明在處 理非穩態訊號時比傅立葉與小波優越，且其IMF 具有較強之物理或生理意義。
然而，EMD 有時會受到模態混合（mode mixing）的影響，此時可以應用D-A EMD 來解 決此問題，但尚無任何一個改進方法可以解決EMD 所遭遇的所有問題，究其根本乃是因為 EMD 是一個理論遠落後於其應用的方法。
在這項研究中，研究了三項EMD 及D-A EMD 的重要應用。（a）現實世界中的訊號在訊 號的某些區域有時會受到噪音的汙染，當應用EMD 或D-A EMD 對訊號進行拆解時，噪音所 造成的影響會隨著疊代過程擴散到乾淨區域，導致IMF 出現誤差，意即物理意義被扭曲的IMF。 （b）在即時運算的應用中，訊號往往被劃分為一系列相互重疊的時間窗口。窗口外的訊號被 丟棄後將作為誤差源隨著疊代過程擴散到窗口中。這項研究中會證明上述兩項問題在數學上是相同的，它們都是由邊界效應所引起的。（c）三次木條曲線內插是EMD 及D-A EMD 演算 法篩選程序中的重要步驟，它需要在曲線的兩端點代入邊界條件，自然邊界條件（natural BC） 及非節點邊界條件（not-a-knot BC）是最常被採用的兩種邊界條件。這兩種邊界常與鏡像邊界、 線性外插、修正線性外插等方法結合以進一步降低誤差。但是何種方法最適合EMD 目前尚無定論。
由於EMD 篩選疊代的複雜性，從數學上對EMD 邊界效應的理論分析的研究仍舊有限。 先前的研究主要依賴於數值模擬，結果似乎顯示誤差只會侷限在邊界附近的區域。此研究將從 EMD 的IMF 的一次篩選疊代開始，從理論上分析邊界效應，首先證明誤差有可能隨著疊代過 程影響到內部區域，有些情況下誤差甚至會隨著與邊界的距離而被放大。接著將證明分解誤差 呈指數衰減的充分條件，最後透過數值實驗以分析多次篩選後的IMF 和採用不同邊界處理比 較的結果。;Empirical mode decomposition (EMD) and disturbance-assisted EMD (D-A EMD) algorithms are adaptive and nonlinear methods that decompose a nonstationary signal into several intrinsic mode functions (IMFs) through the sifting process. From an algorithmic point of view, they are adaptive and nonlinear methods. Compared with short-time Fourier transform (STFT) and wavelet, the important physical meaning of the wave: phase and amplitude can be completely preserved, especially
the latter can be well extracted. Therefore, in many fields, EMD and D-A EMD algorithms have been applied, they have been proved to be superior to Fourier and wavelet in processing non-stationary signals, and the IMFs have strong physical or physiological significance.
However, EMD sometimes suffers mode-mixing effect. Therefore, D-A EMD algorithms can be applied to solve the problem. But there is no method that can solve all the problems encountered by EMD. Because EMD is a method that its theory is far behind its application. This study will explore three important applications of EMD. (a) When the D-A EMD algorithms are applied to decompose a signal polluted with noise in some regions, the noise will spread into the clean region and introduce errors in the IMFs. (b) During real-time computation of a D-A EMD algorithm, the signal is partitioned into a series of overlapping time windows. Points outside the window are discarded, they act as an error source and the error will spread into the current window. We will prove that the above two problems are mathematically identical. (c) The cubic spline interpolation which is an important step of EMD’s sifting procedure, it requires two boundary conditions at the two ends of the domain. The natural BC and not-a-knot BC are often applied. These two BCs are combined with mirror BC, linear extrapolation, modified linear extrapolations to further reduce errors. But the jury is still out on witch method is best for EMD.
Due to the complex nature of the sifting iteration, a mathematical theory for the error analysis of the boundary effect is still lacking. Previous studies mainly rely on simulations, and the results show that the error seems to be confined locally near the boundary. Beginning with one sifting iteration, which is the kernel of an EMD, we will theoretically analyze the boundary effects completely based on the sifting process without any approximations, prove that the error will propagate into the interior domain and may be amplified and thus obscure the meaning of an IMF. Then, prove a sufficient condition for the exponential decay of the error. When the error is amplified, we propose a method to resolve this problem. Finally, numerical experiments are conducted to analyze multiple siftings and the result of setting different boundary conditions. |
