| 摘要: | 經驗模態分解（Empirical Mode Decomposition, EMD）是一種用於對非穩態(non-stationary)訊號進行時頻分析的非線性（Nonlinear）、自適性（Adaptive）方法。在許多應用上已證明它比傳統傅立葉轉換優越。EMD透過篩選運算子(Sifting)將訊號拆解成多組不同頻帶的本質模態函數（Intrinsic Mode Functions, IMFs）。然而EMD 的一個主要缺點是模態混合效應(Mode Mixing)。擾動輔助經驗模態分解（Disturbance-Assisted EMD, D-A EMD）被提出用來解決這個問題。例如使用白噪音的總體經驗模態分解（Ensemble EMD, EEMD）、CEEMD，但也產生了殘餘噪音(Residue Noise)和模態分裂(Mode Splitting)的問題，D-A EMD中使用正/餘弦波的均勻相位經驗模態分解(Uniform Phase EMD, UPEMD)最小化了模態混合效應，同時避免了EEMD所產生的兩個副作用。這些擾動輔助方法使 EMD 在分析真實世界數據時具有更好的性能。
EMD在分解時，表現出太多的模態，有些IMF難以解釋其現象，缺乏嚴謹的數學及物理意義，且當訊號的頻帶太過於接近時，EMD容易將其視為單一波型的調幅訊號，例如: Beat effect，此時需要更多的篩選才有可能分解出有意義的IMF，Wang在2014年已經證明篩選的次數與計算時間成正比，不論是否發生模態混合效應多次篩選的計算時間依舊太長，故本篇論文提出一種以三次樣條小波(Cubic Spline Wavelet)基於EMD的非自適性算法，將EMD的篩選過程以頻域的內積方法所取代，使計算時間快，不受多次篩選疊代的影響，並在處理頻率相近的數據時能更容易拆解，且由於演算法為線性在數學上可分析。
我們透過實驗比較新方法在生醫、地球科學、類聲音訊號、物理等領域與其他不同基於EMD算法(如:EMD、UPEMD、EEMD)的差別，利用人造合成訊號與真實世界例子驗證新方法之性能。
;Empirical Mode Decomposition (EMD) is a nonlinear and adaptive method used for time-frequency analysis of non-stationary signals. It has been proven superior to traditional Fourier transforms in many applications. EMD decomposes the signal into a set of intrinsic mode functions (IMFs) with different frequency bands through the sifting operation. However, a major drawback of EMD is the mode mixing effect. Disturbance-Assisted EMD (D-A EMD) has been proposed to address this issue. For example, Ensemble EMD (EEMD) and CEEMD utilize white noise, but they introduce residual noise and mode splitting problems. D-A EMD with Uniform Phase EMD (UPEMD) using sinusoids minimizes the mode mixing effect and avoids the two side effects caused by EEMD. These disturbance-assisted methods improve the performance of EMD in analyzing real-world data.
During the decomposition, EMD often generates numerous modes, some of which are difficult to interpret and lack rigorous mathematical and physical meanings. When the frequency bands of the signal are too close, EMD tends to treat them as amplitude-modulated signals of a single waveform, such as the beat effect. In such cases, more sifting is required to obtain meaningful IMFs. Wang demonstrated in 2014 that the number of siftings is proportional to the computation time. Regardless of whether mode mixing occurs or not, the computation time for multiple siftings remains too long. Therefore, this paper proposes a non-adaptive algorithm based on EMD using cubic spline wavelets, replacing the sifting process of EMD with a frequency-domain inner product method. This approach speeds up the computation time and is not affected by multiple iterations of siftings. It also facilitates the decomposition of data with close frequencies. Additionally, the algorithm is linear and can be mathematically analyzed.
We conducted experiments to compare the new method with other EMD-based algorithms (such as EMD, UPEMD, and EEMD) in various fields, including biomedical, Earth sciences, audio signals, and physics. Synthetic and real-world examples were used to validate the performance of the new method. |
