## 2 Basic Theory  
### Empirical Mode Decomposition-Based Denoising  
EMD-based denoising consists of two main parts: EMD and threshold denoising. The EMD was pioneered by [PERSON] et al. as an automatic tool for completely data-driven and adaptive multi-component signal processing methods in several IMFs so that the sum of them is equal to the original signal. Each IMF is required to satisfy two basic conditions: the number of the extremum points and the number of zero crossings must be equal or must differ by one at most; the mean of the upper and lower envelopes determined by the local maxima and minima should be zero at any point [PERSON] et al. (2016). Based on the principles of IMF, the process of extracting IMFs can be summarized in the following steps [14]:  
Step 1.1: For an arbitrary signal \(x(t)\), all extrema are searched at first. The upper and the lower envelopes are developed by connecting all the maximal and the minimal points through cubic splines respectively. The average value of the two envelopes is recorded as \(m_{1}(t)\) and the difference between \(x(t)\) and \(m_{1}(t)\) is defined as \(h_{1}(t)\), which can be expressed as follows:  
\[h_{1}(t)=x(t)-m_{1}(t) \tag{1}\]  
If \(h_{1}(t)\) satisfies the two requirements of IMF, then it becomes the first IMF of the original signal, named as \(inf_{1}\). Else, it is treated as the \(x(t)\) and the above operation is repeated. Step 1.2: The reminder \(r_{1}(t)\) is obtained by separating \(inf_{1}\) from \(x(t)\):  
\[r_{1}(t)=x(t)-imf_{1} \tag{2}\]  
Then \(r_{1}(t)\) is regarded as the original signal and the above-mentioned steps are executed again until the \(k\)-th remainder \(r_{k}\) becomes a monotonic function. \(r_{k}\) can be presented as follows:  
\[r_{k}(t)=r_{k-1}(t)-imf_{k} \tag{3}\]  
Step 1.3: \(x(t)\) can be decomposed by the sum of \(k\) IMFs and a residual, which can be shown as follows:  
\[x(t)=\sum_{i=1}^{k}inf_{i}+res \tag{4}\]  
where \(res\) is the residual. Generally, lower-order IMFs have higher frequency components while higher-order IMFs contain lower frequency units. In addition, \(res\) represents the average trend of \(x(t)\). If \(x(t)\) contains some noise, the IMFs and residual will also be polluted by the noise components. In order to eliminate the useless signal, researchers attempt a series of EMD-based denoising methods inspired by wavelet thresholding. Among these methods, the EMD interval thresholding scheme (EMD-IT) is mostly applied, which can be expressed as follows [20]:  
Step 2.1: The denoising thresholds of each IMF are first calculated. In the EMD-IT study, multiples of the IMF dependent universal thresholds are used as follows:  
\[T_{i}=C\sqrt{E_{i}2\ln N} \tag{5}\]  
where \(T_{i}\) is the threshold of \(i\)-th IMF, \(i\) = 1, 2, 3,..., \(k\), \(C\) is a constant, \(N\) is the length of signal and \(E_{i}\) is the estimated energy of the corresponding IMF, which can be calculated as follows:  
\[E_{i}=\frac{E_{1}}{\alpha}\beta^{-i} \tag{6}\]  
where \(E_{1}\) is the variance of the first IMF, \(i\) = 2, 3, 4,..., \(k\), \(\alpha\) and \(\beta\) are parameters that have be estimated by a large number of independent noise realizations and their IMFs. Step 2.2: Unlike the direct application of the hard or soft threshold function in wavelet thresholding, interval thresholding scheme is used in EMD-IT based on the mode cell filter which can reduce the discontinuity. For the arbitrary two adjacent zero-crossings \(z_{i,j}\) = [\(z_{i,j}\), \(z_{i,j+1}\)] in \(i\)-th IMF, the denoising process can be translated into:  
\[\widetilde{h}_{i}(z_{i,j})=\left\{ \begin{array}{ll}h_{i}(z_{i,j})&\left|h_{ i}(r_{i,j})\right|>T_{i}\\ 0&\left|h_{i}(r_{i,j})\right|\leq T_{i}\end{array} \right. \tag{7}\]for hard thresholding and  
\[\widetilde{h}_{i}(\mathbf{z}_{i,j})=\left\{ \begin{array}{ll}\text{sgn}(h_{i}(\mathbf{z }_{i,j}))(h_{i}(\mathbf{z}_{i,j})-T_{i})&\left|h_{i}(r_{ij})\right|>T_{i}\\ 0&\left|h_{i}(r_{ij})\right|\right|\leq T_{i}\end{array} \right.