# Study on EEMD-Based KICA and Its Application in Fault-Feature Extraction of Rotating Machinery
## Theory
### Svd  
The signal \(\mathbf{S}(t)=[s_{1}(t),\ldots,s_{n}(t)]^{T}\) is contained with \(n\) source signals. The observed signal \(\mathbf{X}(t)=[x_{1}(t),\ldots,x_{m}(t)]^{T}\) is obtained by \(m\) different sensors. The observed signal is assumed to be a linear superposition of source signals and noise signal \(\mathbf{N}(t)=[n_{1}(t),\ldots,n_{m}(t)]^{T}\) via a hybrid matrix \(\mathbf{A}=\left(a_{ij}\right)_{m\times n}\). The observed signal \(\mathbf{X}(t)\) can be described as follows:  
\[x_{i}(t)=\sum_{j=1}^{n}a_{ij}s_{j}(t)+n_{i}(t),i=1,2,\cdots,m,j=1,2,\cdots,n, \tag{4}\]  
\[\mathbf{X}(t)=\mathbf{A}\mathbf{S}(t)+\mathbf{N}(t). \tag{5}\]  
The matrix \(\mathbf{R}_{X}\) is the covariance matrix of the observed signal \(\mathbf{X}(t)\), and the eigenvalue decomposition of \(\mathbf{R}_{X}\) is performed as follows:  
\[\mathbf{R}_{X}=\mathbf{Q}\mathbf{\Lambda}\mathbf{Q}^{H}, \tag{6}\]  
where \(\mathbf{\Lambda}\) is a diagonal matrix composed of the eigenvalues \(\{\lambda_{1},\lambda_{2},\ldots,\lambda_{m}\}\) of \(\mathbf{R}_{X}\). Every column vector of eigenvector \(\mathbf{Q}\) is a unit eigenvector corresponding to the eigenvalue, and the unit vectors are orthogonal to each other. It can be deduced that the eigenvalues of \(\mathbf{R}_{X}\mathbf{R}_{X}^{H}\) derived by characteristic decomposition are \(\left\{\lambda_{1}^{2},\lambda_{2}^{2},\ldots,\lambda_{m}^{2}\right\}\). That is,  
\[\mathbf{R}_{X}\mathbf{R}_{X}^{H}=\mathbf{Q}\mathbf{\Lambda}\mathbf{Q}^{H}\left(\mathbf{Q}\mathbf{\Lambda} \mathbf{Q}^{H}\right)^{H}=\mathbf{Q}\mathbf{\Lambda}^{2}\mathbf{Q}^{H}. \tag{7}\]  
Therefore, the singular value of \(\mathbf{R}_{X}\) defined by the singular value is \(\{|\lambda_{1}|,|\lambda_{2}|,\ldots,|\lambda_{m}|\}\), because the covariance matrix's singular values are the same as the absolute value of the eigenvalues. So, the number of nonzero eigenvalues is equal to the number of nonzero singular values. It is assumed that \(\widetilde{\mathbf{R}}_{X}\) is the covariance matrix of mixed signal-contained noise. According to the mixed system model \(\mathbf{X}=\mathbf{A}\mathbf{S}+\mathbf{N}\), the following can be calculated:  
\[\begin{array}{ll}\widetilde{\mathbf{R}}_{X}&=\frac{\text{X}\mathbf{X}^{H}}{L}=\frac {(\mathbf{A}\mathbf{S}+\mathbf{N})(\mathbf{A}\mathbf{S}+\mathbf{N})^{H}}{L}\\ &=\frac{\text{A}\mathbf{S}^{H}A^{H}+\text{A}\mathbf{S}\mathbf{N}^{H}+\text{A}\mathbf{N}^{H}A^{ H}+\text{A}\mathbf{N}\mathbf{N}^{H}}{L}\\ &=\mathbf{R}_{X}+\mathbf{R}_{N}+\frac{\text{A}(\mathbf{S}\mathbf{N}^{H})+\text{A}\mathbf{S}^{H}}{ L}\end{array} \tag{8}\]where \(L\) is the number of sampling points, \(\mathbf{R}_{N}\approx\sigma^{2}\mathbf{I}\), \(\frac{\mathbf{S}\mathbf{N}^{H}}{L}\approx 0\), \(\frac{\mathbf{S}\mathbf{N}^{H}}{L}\approx 0\), and then,  
\[\widetilde{\mathbf{R}}_{X}=\mathbf{R}_{X}+\sigma^{2}\mathbf{I}, \tag{9}\]  
where \(\sigma^{2}\) is the power of noise.  
If \(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{k+1}=\lambda_{k+2}=\cdots= \lambda_{m}=0\) is \(m\) eigenvalues of \(\mathbf{R}_{X}\) and \(\mu_{1}\geq\mu_{2}\geq\cdots\geq\mu_{k}\geq\mu_{k+1}\geq\cdots\geq\lambda_{m}\geq 0\) is \(m\) eigenvalues of \(\widetilde{\mathbf{R}}_{X}\),\(\mu_{1}\approx\lambda_{1}+\sigma^{2},\mu_{2}\approx\lambda_{2}+\sigma^{2}, \ldots,\mu_{k}\approx\lambda_{k}+\sigma^{2},\ldots,\)\(\mu_{m}\approx\lambda_{m}+\sigma^{2}\). Therefore, in the case of a high signal-to-noise ratio (SNR), the number of main eigenvalues of the covariance matrix is equal to the number of signals.  
The eigenvalues of \(\widetilde{\mathbf{R}}_{X}\) are in descending order, that is, \(\mu_{1}\geq\mu_{2}\geq\ldots\geq\mu_{m}\geq 0\). Let \(\gamma_{k}=\mu_{k}/\mu_{k+1},k=1,2,\ldots,m-1\); if \(\gamma_{k}\) is the maximum ratio of neighboring singular values, the number of signals will be \(k\).