# FOG De-Noising Algorithm Based on Augmented Nonlinear Differentiator and Singular Spectrum Analysis
## 3 Simulation  
To present the superiority of the proposed SSA-AND de-noising algorithm, simulated noisy signal \(Z(t)=\cos(2\pi t/2700)+\cos(2\pi t/60)+\ \varphi(t)\) is employed for verification, where \(\varphi(t)\) is random white noise. Firstly, the noisy signal \(Z(t)\) is decomposed into components, as shown in Figure 1, we can see that the low frequency signal is extracted mainly in component 1 and high frequency noises are mainly in component 2, component 3, and component 4. Therefore, ANDs for each component should be designed separately. After decomposition, the next step is to calculate the CC of each component, as shown in Table 1, in which we can see that component 1 has the biggest CC. Then AND is applied on component1 to determine the first \(R\) of AND according to Step 3, then the other \(R\)s can be calculated by Equation (10), as shown in Table 1.  
Next step is the application of ANDs on components with determined \(R\)s. It is of note that component 1 will be reserved directly without any processing, which can guarantee there is no signal delay or low frequency component. Figure 2 is the de-noising results of each component by using ANDs with determined \(R\)s. We can see that: The details of component 1 are reserved totally; component 2 is filtered by AND with \(R=0.32\) which makes part of the details is reserved, and signal delay is occurred compared to Figure 1b; and component 3 to component 4 are filtered by AND with bigger \(R\)s which make a set of smooth de-noising results are obtained, and it goes without saying that the signal delays are occurred. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & **Component 1** & **Component 2** & **Component 3** & **Component 4** \\ \hline Correlation Coefficient (\(CC\)) & 0.78 & 0.30 & 0.21 & 0.20 \\ Acceleration Factor (\(R\)) & 0.82 & 0.32 & 0.22 & 0.21 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The correlation coefficients of each component in Figure 1.  
Figure 2: De-noising results of each component by using augmented nonlinear differentiator (AND) with determined \(R\)s: (**a**) component 1 without de-noising; (**b**) component 2 denoising result with \(R=0.32\); (**c**) component 3 denoising result with \(R=0.22\); (**d**) component 2 denoising result with \(R=0.21\). Figure 1: Decomposition results of \(Z(t):\) (**a**) component 1 of decomposition results; (**b**) component 2; (**c**) component 3; (**d**) component 4. The last step is reconstruction. All the filtered components in Figure 2 are added together and then the reconstruction result is obtained, which is shown as Figure 3c. Comparison between Figure 3a-c show that the noise is reduced effectively; comparison between Figure 3a-c show that there is almost no signal delay occurred after de-noising by the proposed SSA-AND algorithm due to the low frequency component1 is reserved totally; from Figure 3d it can be seen that the residual still exists, which means the noise-attenuation ability or signal distortion problem of AND still need to be considered which would be our future work. Furthermore, to verify the robustness of the proposed SSA-AND de-noising algorithm, we added the corresponding noise with different signal-to-noise power ratios (SNRs) for simulation. Additionally, some other advanced de-noising methods, such as Adaptive Robust Kalman Filter (ARKF), Detrended Fluctuation Analysis-VMD (DFA-VMD), empirical mode decomposition-forward linear prediction (EMD-FLP) and traditional AND are employed for comparison. In ARKF algorithm, the measurement noise covariance matrix is adapted using the weighted covariance of the innovation sequence which improves its robustness [13]. In combination of DFA and VMD, the number of band-limited intrinsic mode functions (BLIMF) decomposed by VMD is determined by DFA, aiming to avoid the impact of over-binning or under-binning on the VMD de-noising [24].