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Blind source separation (BSS) recovers source signals from observations without knowing the mixing process or source signals. Underdetermined blind source separation (UBSS) occurs when there are fewer mixes than source signals. Sparse component analysis (SCA) is a general UBSS solution that benefits from sparse source signals which consists of (1) mixing matrix estimation and (2) source recovery estimation. The first stage of SCA is crucial, as it will have an impact on the recovery of the source. Single-source points (SSPs) were detected and clustered during the process of mixing matrix estimation. Adaptive time–frequency thresholding (ATFT) was introduced to increase the accuracy of the mixing matrix estimations. ATFT only used significant TF coefficients to detect the SSPs. After identifying the SSPs, hierarchical clustering approximates the mixing matrix. The second stage of SCA estimated the source recovery using least squares methods. The mixing matrix and source recovery estimations were evaluated using the error rate and mean squared error (MSE) metrics. The experimental results on four bioacoustics signals using ATFT demonstrated that the proposed technique outperformed the baseline method, Zhen’s method, and three state-of-the-art methods over a wide range of signal-to-noise ratio (SNR) ranges while consuming less time.

To address the issues of suboptimal sparsity and the tendency of clustering algorithms to converge to local optima in the estimation of the mixing matrix within underdetermined blind source separation (UBSS) systems, a novel mixing matrix estimation algorithm based on source signal sparsity is proposed. Firstly, the principle of underdetermined mixing matrix estimation leveraging source sparsity is derived. Building upon this foundation, improvements are made from enhancement of signal sparsity and optimization of the clustering algorithm. To overcome the limited sparse representation capability of conventional time–frequency (TF) transformation methods, a source signal sparsity enhancement algorithm based on the Local Maximum Synchroextracting Transform (LMSET) is proposed. This method rearranges the TF coefficients by detecting local maxima in the frequency direction, thereby achieving a more desirable TF resolution and enhanced signal sparsity. Furthermore, to mitigate the sensitivity of the Fuzzy C-Means (FCM) algorithm to initial cluster centers and its propensity for local optima, a robust FCM algorithm optimized by the PID(Proportional-integral-Derivative)-based Search Algorithm (PSA) is adopted. Simulation results demonstrate that the proposed algorithm achieves a superior TF representation and enhances the sparsity of source signals across various environments. Compared to traditional algorithms, the estimation accuracy of the mixing matrix is increased by 19.8%, effectively improving the performance of mixing matrix estimation in underdetermined blind source separation systems.

Underdetermined blind source separation (UBSS) has garnered significant attention in recent years due to its ability to separate source signals without prior knowledge, even when sensors are limited. To accurately estimate the mixed matrix, various clustering algorithms are typically employed to enhance the sparsity of the mixed matrix. Traditional clustering methods require prior knowledge of the number of direct signal sources, while modern artificial intelligence optimization algorithms are sensitive to outliers, which can affect accuracy. To address these challenges, we propose a novel approach called the Genetic Simulated Annealing Optimization (GASA) method with Adaptive Density-Based Spatial Clustering of Applications with Noise (DBSCAN) clustering as initialization, named the CYYM method. This approach incorporates two key components: an Adaptive DBSCAN to discard noise points and identify the number of source signals and GASA optimization for automatic cluster center determination. GASA combines the global spatial search capabilities of a genetic algorithm (GA) with the local search abilities of a simulated annealing algorithm (SA). Signal simulations and experimental analysis of compressor fault signals demonstrate that the CYYM method can accurately calculate the mixing matrix, facilitating successful source signal recovery. Subsequently, we analyze the recovered signals using the Refined Composite Multiscale Fuzzy Entropy (RCMFE), which, in turn, enables effective compressor connecting rod fault diagnosis. This research provides a promising approach for underdetermined source separation and offers practical applications in fault diagnosis and other fields.

The performance of the existing underdetermined blind source separation methods is very sensitive to the initial parameters, meanwhile, the existing setting or selection methods of initial parameters need to be improved. Consequently, an effective underdetermined blind source separation method is proposed in this paper to solve the above engineering problems. Based on the Archimedes optimization algorithm and quantum computing theory, this paper proposes a novel intelligent algorithm named quantum Archimedes optimization algorithm, which solves the objective functions for engineering problems. Then the optimal solution obtained through the quantum Archimedes optimization algorithm is used as the initial clustering centers of the K-means clustering algorithm to achieve mixing matrix estimation. In addition, the original initial estimation signal setting of the source recovery based on radial basis function network is converted into an initial solution in population for quantum Archimedes optimization algorithm. The optimal solution obtained through the quantum Archimedes optimization algorithm is used as the new initial estimation signal setting to achieve source recovery. The simulation results show that the proposed underdetermined blind source separation method has higher accuracy than previous methods. The proposed method that is more robust and applicable makes the setting and selection of initial parameters more reasonable so that the performance is no longer limited to the initial parameters.

The timely discovery of low-frequency oscillations in power systems and accurate identification of their modal parameters is critical in numerous applications. Therefore, we investigated the feasibility of using multi-channel signals and established a relative theory. An algorithm based on the underdetermined blind source separation (UBSS) algorithm was proposed using this theory. First, the energy ratio function was used to determine the fault occurrence time. Then, the Bayesian information criterion was used to estimate the number of fault sources, and the boundary conditions were set to determine the number of fault sources. Next, the UBSS algorithm was used to analyze raw data, extract individual components that characterize faults, and subsequently measure low-frequency oscillation modal parameters through the Hilbert transform. Finally, the fast independent component analysis (FastICA) algorithm was used to separate noise signal from raw data. This separation considerably reduced noise disturbance and ensured the stability of the proposed method. Model simulation was conducted in MATLAB and experimental measurement revealed that the proposed method effectively reduced noise disturbance and could be applied to conditions with considerable disturbance.

To identify the major vibration and radiation noise, a source contribution quantitative estimation method is proposed based on underdetermined blind source separation. First, the single source points (SSPs) are identified by directly searching the identical normalized time-frequency vectors of mixed signals, which can improve the efficiency and accuracy in identifying SSPs. Then, the mixing matrix is obtained by hierarchical clustering, and source signals can also be recovered by the least square method. Second, the optimal combination coefficients between source signals and mixed signals can be calculated based on minimum redundant error energy. Therefore, mixed signals can be optimally linearly combined by source signals via the coefficients. Third, the energy elimination method is used to quantitatively estimate source contributions. Finally, the effectiveness of the proposed method is verified via numerical case studies and experiments with a cylindrical structure, and the results show that source signals can be effectively recovered, and source contributions can be quantitatively estimated by the proposed method.

This paper considers the complex-valued mixing matrix estimation and direction-of-arrival (DOA) estimation of synchronous orthogonal frequency hopping (FH) signals in the underdetermined blind source separation (UBSS). A novel mixing matrix estimation algorithm is proposed by detecting single source points (SSPs) where only one source contributes its power. Firstly, the proposed algorithm distinguishes the SSPs by the comparison of the normalized coefficients of time frequency (TF) points, which is more effective than existing detection algorithms. Then, mixing matrix of FH signals can be estimated by the hierarchical clustering method. To sort synchronous orthogonal FH signals, a modified subspace projection method is presented to obtain the DOAs of FH. One superiority of this paper is that the estimation accuracy of the mixing matrix can be significantly improved by the proposed SSPs detection criteria. Another superiority of this paper is that synchronous orthogonal FH signals can be sorted in underdetermined condition. The experimental results demonstrate the efficiency of the two proposed algorithms.

To solve the separation of multi-source signals and detect their features from a single channel, a signal separation method using multi-constraint non-negative matrix factorization (NMF) is proposed. In view of the existing NMF algorithm not performing well in the underdetermined blind source separation, the β-divergence constraints and determinant constraints are introduced in the NMF algorithm, which can enhance local feature information and reduce redundant components by constraining the objective function. In addition, the Sine-bell window function is selected as the processing method for short-time Fourier transform (STFT), and it can preserve the overall feature distribution of the original signal. The original vibration signal is first transformed into time–frequency domain with the STFT, which describes the local characteristic of the signal from the time–frequency distribution. Then, the multi-constraint NMF is applied to reduce the dimensionality of the data and separate feature components in the low dimensional space. Meanwhile, the parameter WK is constructed to filter the reconstructed signal that recombined with the feature component in the time domain. Ultimately, the separated signals will be subjected to envelope spectrum analysis to detect fault features. The simulated and experimental results indicate the effectiveness of the proposed approach, which can realize the separation of multi-source signals and their fault diagnosis of bearings. In addition, it is also confirmed that the proposed method, juxtaposed with the NMF algorithm of the traditional objective function, is more applicable for compound fault diagnosis of the rotating machinery.

Recovering M sources from N mixtures in underdetermined cases, i.e., M > N, is a great challenge, especially for insufficiently sparse sources in noisy cases. To solve this problem, an improved underdetermined blind source separation (UBSS) method is proposed based on single source points (SSPs) identification and l0-norm. Firstly, we present a mixing matrix estimation method based on SSPs that is identified by directly searching the identical normalized time–frequency (TF) vectors of mixed signals. This method considers the linear representation relations among these TF vectors and therefore could achieve more accurate SSPs identification even in noisy cases. Then, we prove that a non-SSP will be misjudged as SSP with probability zero under some assumptions, which guarantees the stability and effectiveness of the proposed method. Secondly, SSPs are only searched in some optimal frequency bins so that all SSPs in these frequency bins can be identified at one time. Then, the mixing matrix is estimated using hierarchical clustering technique. Thirdly, to recover source signals with real number of active sources, a l0-norm-based source recovery method is proposed which would be transformed to find the submatrix with the least column of the mixing matrix that can linearly represent TF vectors of mixed signals. Therefore, source signals can be recovered with the real number of active sources, which improves the estimation accuracy of source signals. Some experiments are carried out to show the effectiveness of the proposed method. The present research could improve the estimation accuracy of sources for insufficiently sparse sources with noise in underdetermined cases.

To address the issues of poor anti-noise performance of second-order statistics and low estimation accuracy in previous joint underdetermined blind source separation (JUBSS) methods, we propose a novel JUBSS method based on the dependence between different data sets and the advantages of cross third-order cumulant in resisting distributed noise. The method involves several steps. Firstly, we calculate the cross third-order cumulant of multiple whitening data sets with different delays. Then, we stack several third-order cumulants into fourth-order tensors. Next, we decompose the fourth-order tensor using Canonical Polyadic through weight nonlinear least squares, which allows us to estimate the mixed matrix. Finally, depending on the independence of source signals, we propose a matrix diagonalization method to recover the source signal. Experiments demonstrate that the method effectively suppresses the influence of Gaussian noise and performs well in underdetermined, positive and overdetermined cases and produces a better performance than various common approaches. Specifically, for the 3 × 4 mixed model with signal-to-noise ratio of 20 dB, the average relative error is − 14.48 dB, the average similarity coefficient is 0.92 and the signal-to-interference ratio is 24.84 dB.