## 1 Introduction
However, EMD still have some problems, such as modal aliasing [21] and edge effect [22]. Therefore, the traditional SCBSS method has obvious deficiencies in the analysis of multi-faults. Compared with a one-dimensional space, a multi-dimensional signal always contains more information. As the most natural representation of multi-dimensional data, tensor can preserve the intrinsic structure of the data to the maximum extent. Tensor decomposition method can extract the useful components in the original measured vibration signal. Consequently, the tensor decomposition algorithm has broad application prospects in signal processing and has great practical engineering significance in some aspects such as pattern recognition and big data processing. CANDECOMP/PARAFAC (CP) decomposition [23; 24] is a commonly used tensor decomposition method. If the rank of the tensor is \(R\), the CP decomposition can factorize a tensor into a sum of \(R\)-component rank-one tensors [25]. By the proposed decomposition model, three factor matrixes representing the combination of the vectors can be obtained from the rank-one components. Recently, the tensor-based singular spectrum analysis (TSSA) algorithm, which provides an effective way for solving the above problem of SCBSS, was proposed by [PERSON] et al. [26] and has been applied to the field of EEG signal processing. Firstly, the one-dimensional times series can be segmented as a matrix usinga non-overlapping window. Then, each row of matrix can be expressed as a reconstructed attractor matrix through phase space reconstruction [27]. The obtained every reconstructed attractor matrix formed the corresponding slice of the tensor, thus a 3D tensor was obtained to be decomposed. Then, the above-mentioned CP tensor decomposition method was used here. The key step is performed by the alternating least squares method (ALS) [28] to obtain the three-factor matrix. The TSSA method combines the advantages of the phase space reconstruction, the SSA [29], and the tensor decomposition. However, the TSSA still has some problems when applied to the SCBSS, mainly including unsatisfactory convergence and poor estimating accuracy of the number of the original signals. In this paper, an improved TSSA decomposition method using the weighted optimization CP tensor decomposition model is proposed. The improve method is the so-called CANDECOMP/PARAFAC weighted optimization (CP-WOPT), which is defined as the first-order optimization to solve the least squares objective function over all the factor matrices simultaneously, so as to improve the convergence of this algorithm. Faced with the difficulty in determining the number of original signals, a commonly accepted method is introduced, namely EMD-SVD-BIC [30], which can estimate the number of original signals accurately. Firstly, the intrinsic mode functions (IMFs) of a signal are obtained by using the EMD method. Then, the singular value decomposition (SVD) on the matrix is performed, which consists of the IMFs from the observed signal using SVD. We can obtain the distribution of eigenvalues about the source data. Finally, the BIC is used here to judge the number of source signals. The validity of the proposed method is verified by the numerical simulation signal and the measured vibration signal of the fault test bench in public dataset. The rest of this paper is structured as follows: In Section 2, the basic theory introductions of the TSSA algorithm and blind source separation are briefly described. Then, the proposed single channel blind source separation (SCBSS) method based on the CP-WOPT model is developed. The analysis results of numerical simulation signal and bearing fault signal are, respectively, described in Sections 3 and 4. Section 5 concludes the paper.