# Two-Channel Information Fusion Weak Signal Detection Based on Correntropy Method +
## 3 Presentation of the Proposed Method  
Select the effective SCs that are similar to the original signal. It is proposed to use a correntropy induced metric (CIM) to express the similarity between SCs and the original signal.  
For variables \(X\) and \(Y\), define CIM as (7):  
\[\text{CIM}(X,Y)= [k_{\sigma}(0)-V(X,Y)]^{1/2} \tag{7}\]  
When the two variables are close together, the CIM behaves like an L2 norm, which is called the Euclidean zone; outside the Euclidean zone, the CIM behaves like an L1 norm, which is called the transition zone; as the two points move further away, the behavior of CIM is similar to the L0 norm, and this area is called the correction area [23]. Figure 1 gives the contour of \(\text{CIM}(X,0)\) and \(\text{CIM}(Y,0)\) in 2-D, which means the distance from variables \(X\) and \(Y\) to the origin.  
Use CCES to identify the fault characteristics from the reconstructed signal. Let us define \(V_{x}(t,\tau)\) as the correntropy function for a stochastic process \(x(t)\) with a hidden periodicities and for which the time shift is \(\tau\). Then the CE function can be defined as \(V_{x}(t,\tau)=V_{\sigma}(X(t),X(t+\tau))=\mathbb{E}[k_{\sigma}(X(t)-X(t+\tau))]\). Based on the autocorrelation  
Figure 1: The CIM contour map in 2-D.  
function, \(V_{x}(t,\tau)\) for the cyclostationary process of the first and second by Fourier series can be represented as (8):  
\[V_{x}(t+T_{0},\tau)=V_{x}(t,\tau)=\sum_{\alpha}V_{x}^{\alpha}(\tau)\mathrm{e}^{ \mathrm{i}2\pi\alpha t} \tag{8}\]  
where \(t\) is the time of the sequences, \(T_{0}\) is one of the periods of CE, \(\alpha=n/T,\ n\in Z\) is taken as the cyclic frequency, \(T\) is the period of CE.  
Therefore, the CCE function for \(x(t)\) is defined as Fourier coefficients \(V_{x}^{\alpha}\), see (9):  
\[V_{x}^{\alpha}=\frac{1}{T_{0}}{\int_{-T_{0}/2}^{T_{0}/2}V_{x}(t,\tau)\mathrm{e }^{-\mathrm{i}2\gamma\alpha t}dt}=\lim_{T\to\infty}\frac{1}{T}{\int_{-T/2}^{T/2 }k_{\sigma}(x(t)-x(t+\tau))\mathrm{e}^{-\mathrm{i}2\pi\alpha t}dt} \tag{9}\]  
Define CCES as a Fourier transform of CCE see (10):  
\[S_{x}^{\alpha}(f)={\int_{-\infty}^{+\infty}V_{x}^{\alpha}(\tau)\mathrm{e}^{- \mathrm{i}2\gamma f\tau}d\tau} \tag{10}\]  
The specific steps of the method two-SVD-CIM are as follows, Figure 2 shows the flowchart of the SVD-CIM for two-channel information fusion.  
Step 1. Reshape the single-channel signal into Hankel matrix of \(m\)-by-\((N-m+1)\). \(N\) is the length of the signal.  
Step 2. Join the two matrices of two channels into one matrix of \(m\)-by-\((2(N-m+1))\);  
Step 3. PCA for \(m\)-by-\((2(N-m+1))\) matrix;  
Step 4. Calculate the cumulative contribution rate of the first \((N-m+1)\) principal components, if the rate is larger than 95%, reconstruct the signal with these principal components.  
Step 5. SVD for the matrix of principal components.  
Step 6. Calculate CIM between SCs and the original signal, pick out the SCs with smaller values of CIM, and reconstruct the denoised signal with these SCs.  
Step 7. Estimate CCES of the denoising signal to identify weak signal characteristics.  
Figure 2: Flowchart of the two-SVD-CIM.