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.