經驗模態分解(EMD)方法是一個非線性、自適應的時頻分析方法。它藉由幾次的篩選算子作用後將一個複雜時間訊號拆解為若干個本質模態函數(intrinsic mode function、IMF)；這些振盪的IMF往往對應到某些物理或生理現象。由於EMD是建立在極值點所暗示的局部尺度為出發點，因此EMD一直被假設是一個局部性的方法。局部性是指訊號任一個時間點的動態特性經過篩選算子作用後，只會影響其附近的點。而另一方面當極值點分佈不均勻時EMD又會產生所謂的模態混合。它是指同一個IMF中同時摻雜著非穩態的高頻與低頻訊號而模糊了IMF的時頻意義，但如今尚無定量分析；我們將根據極值點分佈距離定義局部不均勻度(DONU)。我們將運用我們過去所證明之篩選算子脈衝響應定理著手去證明對一次篩選算子而言，當DONU超過特定上閥值時，EMD將喪失居部性，而低於特定下閥值時，將能保證其局部性。本研究將找出這上下閥值，並證明其EMD 存在與喪失居部性的關聯。我們也將推論並驗證模態混和發生原因及其與EMD對該訊號喪失局部性的關係，進一步推論與證明脈衝響應定理對多次篩選是否成立，並探討多次篩選算子作用後的局部性。除了數學理論建立，基於本研究所獲得的結論，我們將提出一個EMD改善方法，使其未來應用更具理論與學理基礎。 ;Empirical Mode Decomposition (EMD) is a nonlinear, adaptive time frequency method. A signal is decomposed into a series of intrinsic mode functions (IMFs) by an iterative sifting process. Each IMF often corresponds to a particular physical oscillation. Since EMD is based on the local scale of, it has been hypothesized that EMD is a local method. Locality means that a time point would mainly affect its neighboring points after applying the sifting iterations. On the other hand, EMD would suffer from the mode-mixing phenomenon when the extrema distribution is highly non-uniformly distributed, but there is no a clear quantatively definition till now. We will define the first order of non-uniformity (DFONU) based on the extrema distribution. We will apply the impulse response theorem of the sifting operator to prove that EMD is only a conditionally local method. Yo be more specific, as DFONU is greater then 3.19, EMD will lose its locality, as DFONU is less than 2.45, EMD will be local. We will then explain that the reason for causing mode-mixing is when EMD lose its locality. We will then prove that the impulse response theorem can be extended from single sifting iteration to multiple iterations and study how the locality will be affected by continuous siftings. Finally we will propose a locality-based improved EMD algorithm based on these new finding.
