# FOG De-Noising Algorithm Based on Augmented Nonlinear Differentiator and Singular Spectrum Analysis
## 2 Algorithm
### Innovation Solutions  
Inspired by the above problem analysis of AND, the question is focused on how to decompose the noisy signal into different components and filter the components with different appropriate acceleration factors. It is well known that singular spectrum analysis (SSA) is one of the representations of multi-scale transformation methods with good time-frequency analysis ability, which can adaptively decompose arbitrary signals into a set of components with different frequencies [8; 9]. Moreover, experimental results have also shown that the SSA outperforms empirical mode decomposition (EMD) and WT in noise robustness. Therefore, SSA is selected for decomposing noisy signal before AND filtering. Another important issue needing to be resolved is how to determine the \(R\) of AND for each component obtained by SSA decomposition. If \(R\) is just determined by experience, the final de-noising result cannot be guaranteed. The correlation coefficient (\(CC\)) can be used to reflect the correlation degree between decomposed component and original signal. Therefore, \(CC\) is used here as a determination criterion for each \(R\) of AND. The steps of SSA combining AND is shown as follows:  
Step 1: Decomposition. The signal \(\mathbf{X}\) is decomposed into components by using SSA, which are \(u_{1}\), \(u_{2}\),..., \(u_{n}\), respectively.  
Step 2: Calculate \(CC\) for each component.  
\[CC_{j}=\frac{\sum_{i=1}^{m}(x_{i}-\overline{x})\big{(}u_{ji}-\overline{u_{j}}\big{)} }{\sqrt{\sum_{i=1}^{m}(x_{i}-\overline{x})^{2}\sum_{i=1}^{m}\big{(}u_{ji}- \overline{u_{j}}\big{)}^{2}}}\qquad i=1,2,\ldots m;\ j=1,2,\ldots n \tag{8}\]  
Step 3: Judge the biggest \(CC_{j}\) and select the corresponding \(u_{j}\), where \(u_{j}\) is the \(j\)th decomposed component by using SSA. Set an origin value of \(R\) which is the acceleration factor of AND, then AND is used to track \(u_{j}\) and \(d\_u_{j}\) is obtained, where \(d\_u_{j}\) is the tracking result output by AND. Calculate the difference between \(u_{j}\) and \(d\_u_{j}\), if the difference is smaller than the defined threshold \(\theta\), then \(R\) is selected as \(R_{j}\), where \(R_{j}\) is acceleration factor of the \(j\)th AND for \(u_{j}\). If the difference is bigger than \(\theta\), adjusting the value of \(R\) until \(u_{j}-d\_u_{j}\leq\theta\) is reached. The threshold \(\theta\) is defined as Equation (9) by experience, where \(q\) is the length of \(u_{j}\).  
\[\theta=\frac{\text{mean}\Big{\{}\sum_{1}^{q}\big{|}u_{j}(i)\big{|}\Big{\}}}{q \times 100} \tag{9}\]  
After \(R_{j}\) is determined, the other \(R_{i}\) can be calculated by Equation (10), where \(R_{i}\) is acceleration factor of the \(i\)th AND for \(u_{i}\), \(i=1,2,\ldots,n,\ i\
eq j\).  
\[R_{i}=\frac{CC_{i}}{CC_{j}}R_{j} \tag{10}\]  
Step 4: Design \(n\) ANDs for all the decomposed components by using the calculated \(R\) in Equation (10). Then each component is de-noised by the corresponding AND.  
Step 5: Reconstruction. The last step is to get the final de-noising result by reconstruction. Note that \(u_{j}\) with the biggest \(CC_{j}\) is the most useful component of original signal according to the definition of correlation coefficient, therefore \(u_{j}\) will be reserved and this step can make sure that the time-delay would be avoid. The \(d\_u_{1}\), \(d\_u_{2}\),..., \(u_{j}\),..., \(d\_u_{n}\) are added together and the final de-noising result is obtained.  
Since the proposed de-noising algorithm is the combination of AND and SSA, it is named as the SSA-AND algorithm.