經驗模態分解中突波問題的解決架構

Abstract

經驗模態分解是一個被廣為使用的時頻分析工具，然而，訊號中的雜訊干擾，例如突波，可能同時造成模態混合和模態分裂的問題，使得一個物理上有意義的成份被拆解成二個以上的本質模態函數。在此論文中，我們引用近期發展出的經驗模態分解的數學理論，提供突波問題造成模態混合和模態分裂的理論解釋，並且基於此理論基礎，提出了解決突波問題的架構——最小弧長條件。為了更穩健地將突波分離至原先不存在的本質模態函數中，我們加入了以弦波輔助的遮罩方法，而形成了「遮罩—最小弧長—經驗模態分解」。在論文中提供了此方法的數學理論和數值模擬，並且應用至真實世界的訊號，包括電流中的突波干擾、軸承震動訊號、睡眠腦波中的週期性交替模式和核心體溫的生理時鐘。更有甚者，我們將此方法應用在標準十二導心電圖上以分離P波的波形，並且證明由此P波波形所提取的特徵，可以用來偵測受測者是否有潛在的心房顫動。最後，我們將此方法延伸至單位階梯函數上，並且提出一個廣適性的演算法，來處理第N階導數為突波的訊號。

Empirical mode decomposition (EMD) is an extensively utilized tool in time-frequency analysis. However, disturbances such as impulse noise can result in both mode-mixing and mode-splitting effect, in which one physically meaningful component is split in two or more intrinsic mode functions (IMFs). In this work, we provide a mathematical explanation for the cause of mode-mixing and mode-splitting by spikes in EMD, and propose a novel method, the minimum arclength EMD (MA-EMD), to robustly decompose time series data with spikes. To further isolate the spike in a previously non-existed IMF, the masking-aided MA-EMD (MAMA-EMD) is provided. The mathematical foundations and limitations for these two methods are provided. The MAMA-EMD is utilized to deal with four real-world data including electrical current, vibration signals, cyclic alternating pattern in sleep EEG (Electroencephalography), and circadian of core body temperature. In addition, this work developed a tool for P-wave isolation in electrocardiogram (ECG) by the MAMA-EMD method, and showed that the P-wave related features can be used to identify potential atrial fibrillation patients. Finally, we extend our application to the Heaviside step function and propose a general algorithm for signals whose Nth order derivative is a spike function.