Random noise attenuation by $f$ -$x$ empirical mode decomposition predictive filtering

Appendix A: Sifting algorithm for empirical mode decomposition

In this appendix, we review the sifting algorithm of empirical mode decomposition (equation 6 in the main paper). For the original signal, we first find the local maxima and minima of the signal. Once identified, fit these local maxima and minima by cubic spline interpolation in turn in order to generate the upper and lower envelopes. Then compute the mean of the upper and lower envelopes $m_{11}$ , the difference between the data and first mean $h_{11}$ .

$$m_{11}=\frac{h^{+}_{10}+h^{-}_{10}}{2},$$ (9)

$$h_{11}=h_{10}-m_{11},$$ (10)

where $h_{ij}$ denotes the remaining signal after $j$ th sifting for generating the $i$ th IMF, $h^+_{ij}$ and $h^-_{ij}$ are corresponding upper and lower envelopes, respectively, and $m_{ij}$ is the mean of upper and lower envelopes after $j$ th sifting for generating the $i$ th IMF. Repeating the sifting procedure (A-2) $k$ times, until $h_{1k}$ reach the prerequisites of IMF, these are:

$$h_{1(k-1)}-m_{1k}=h_{1k}.$$ (11)

The criterion for the sifting process to stop is given by Huang et al. (1998) as:

$$0.2\le SD=\sum_{t=0}^{T}\left[\frac{\vert h_{1(k-1)}(t)-h_{1k}(t)\vert^2}{h^2_{1(k-1)}}\right]\le 0.3,$$ (12)

where $SD$ denotes the standard deviation. When $h_{1k}$ is considered as an IMF, let $c_1=h_{1k}$ , we separate the first IMF from the original data:

$$d-c_1=r_1,$$ (13)

where $d$ is the original signal, $c_n$ denotes the $n$ th IMF, and $r_n$ is the residual after the $n$ th IMF based sifting. Repeating the sifting process from equation A-1 to A-5, changing $h_{1j}$ to $h_{ij}$ , in order to get the following IMFs: $c_2, c_3, \cdots, c_N$ . The sifting process can be stopped when the residual $r_n$ , becomes so small that it is less than a predetermined value of substantial consequence, or when $r_n$ becomes a monotonic function from which no more IMF can be extracted.

Finally, we achieved a decomposition of the original data into N modes, and one residual, as shown in equation 6 in the main context.

Random noise attenuation by $f$ -$x$ empirical mode decomposition predictive filtering