## 2 Theoretical Descriptions
### Source Number Estimation Based on Bayesian Information Criterion  
Generally, BSS belongs to underdetermined blind source separation in the field of mechanical fault diagnosis. Basically, this involves the number of multi-channel observation signals being less than the number of source signals, especially in the situation of a single-channel observation signal. Since RPSEMD has the ability to adaptively decompose signals into a series of linear and stationary IMFs, it is hypothesized that the IMFs obtained from RPSEMD can solve the problem of underdetermined BSS. Nevertheless, the number of source signals of the system should be first estimated. The single channel observation signal \(x(t)\) is decomposed by RPSEMD to obtain sub-band components \(c_{i}(t)(i=1,2,...d)\) and the remainder \(r_{d}(t)\). Following this, \(x(t)\) and the sub-band components of the decomposition are composed of a multi-dimensional observation signal \(x_{imf}(t)=\left(x(t),c_{1}(t),...,c_{d}(t),r_{d}(t)\right)^{\mathrm{T}}\). In addition, the correlation matrix of \(x_{imf}(t)\) is defined as  
\[R_{x}=E[x_{imf}(t)x_{imf}^{\mathrm{H}}(t)] \tag{5}\]  
where the operator of H is the complex conjugate transformation. When the noise belongs to Gaussian distribution and the corresponding IMFs are not relevant, the correlation matrix of \(x_{imf}(t)=\left(x(t),c_{1}(t),...,c_{d}(t),r_{d}(t)\right)^{\mathrm{T}}\) is expressed as  
\[R_{x}=E[s(t)s^{\mathrm{H}}(t)]+\sigma^{2}\mathrm{I}_{p-d} \tag{6}\]  
where \(v\) is the dimension of new observed signal \(x_{imf}(t)\), \(\mathrm{I}_{p-d}\) is unit matrix and \(\sigma^{2}\) is the noise power.  
Performing singular value decomposition on \(R_{x}\) gives the following expression:  
\[R_{x}=V_{s}\Lambda_{s}V_{s}^{\mathrm{T}}+V_{z}\Lambda_{z}V_{z}^{\mathrm{T}} \tag{7}\]  
where \(\Lambda_{s}=diag\{\lambda_{1},...,\lambda_{d}\}\) are the principal eigenvalues in descending order, while \(\Lambda_{z}=diag\{\lambda_{d+1},...,\lambda_{v}\}\) is the characteristic values for noise. Commonly, there is often clear distinction of the characteristic value between the noise components and useful signals. Therefore, for an accurate estimation of the covariance matrix and under the premise that the noise variance is relatively small, the dimensions of noise subspace of \(R_{x}\) can be determined through judging the feature eigenvalue. Thus, the number of source signals can be obtained theoretically. However, it is difficult to determine the threshold value of the main characteristic value and the noise characteristic value. Therefore, the dimension of the noise subspace cannot be judged accurately. In order to solve the problem of threshold setting, the Bayesian information criterion (BIC) is employed to estimate the dimensions of the source signal and the noise subspace. BIC establishes the method of source number estimation based on the Bayesian Minaka selection model [41]. Thus, the BIC model can be approximately expressed as  
\[\mathrm{BIC}(k)=\left(\prod\limits_{j=1}^{k}\lambda_{j}\right)^{-N/2}\widetilde {\sigma}_{k}^{\mathrm{\ -}N(l-k)/2}\ \ N^{-(d_{k}+k)/2} \tag{8}\]  
where \(\widetilde{\sigma}_{k}^{\mathrm{\ \ 2}}=\left(\sum\limits_{j=k+1}^{l}\lambda_{j} \right)/(l-k)\), \(d_{k}=lk-k(k+1)/2\), \(1\leq k\leq l\), \(l\) is the number of non-zero eigenvalues. The goal of BIC is to find the maximum value of the cost function, which corresponds to the estimated number of source signals.