Coupling signal separation method based on symmetric tensor joint decomposition

Abstract

The invention requests to protect a coupling signal separation method based on symmetric tensor joint decomposition, and provides a new BSS method, namely TenSOFO, for solving instantaneous blind source separation from the perspective of tensor decomposition. According to the method, a novel efficient alternating direction multiplier method framework is provided, and two symmetrical third-order tensors are allowed to be decomposed at the same time in an individual difference scale INDSCAL format. In addition, the basic relation between the BSS based on the second-order statistic-fourth-order statistic SO-FO and the combined INDSCAL decomposition is established, and the provided INDSCAL method can be effectively applied to the instantaneous BSS task. The TenSOFO has the advantages that memory is saved, effective decomposition is achieved, Gaussian noise items disappear when fourth-order FO is used, and source signals can be processed without prior information of other levels. Therefore, the TenSOFO is particularly beneficial to processing long-time sequence signals such as voice and audio sampled at a high rate. Experimental results show that the TenSOFO method provided by the invention is minimum in error measurement and optimal in performance.

Description

Technical Field

[0001] This invention belongs to the field of blind source signal separation technology, specifically to a coupled signal separation method based on symmetric tensor joint decomposition. Background Technology

[0002] In existing technologies, second-order statistics (SO) utilize the non-stationarity or non-whiteness of signals to estimate the mixing matrix, thereby achieving source signal separation. SO is less affected by Gaussian distributions, thus offering stronger noise resistance for blind source separation. Furthermore, since SO primarily involves matrix diagonalization and eigenvalue decomposition, its computational complexity is relatively low. However, SO methods cannot completely resolve the inherent ambiguity caused by the "blindness" of the signal; they can only guarantee the separation of the "equivalent form" of the source signal, and amplitude and order ambiguities still exist.

[0003] The core characteristic of fourth-order statistics (FO) is deeply tied to its statistical basis—non-Gaussianity. It exhibits characteristics of "reliance on non-Gaussianity, resistance to Gaussian noise, and high complexity," complementing second-order statistics. Fourth-order statistics require at least one source signal to be non-Gaussian, because the fourth-order statistic of a Gaussian signal is zero, making it indistinguishable by fourth-order statistics. In contrast, the fourth-order cumulant of a non-Gaussian signal is not zero, and its "degree of non-Gaussianity" can serve as a core criterion for separation. However, the calculation of fourth-order statistics involves the fourth moments of the signal, and the computational complexity increases exponentially with the signal dimension and data length, far exceeding the computational complexity of second-order statistics. Furthermore, fourth-order statistics require more observation data, resulting in poor real-time performance.

[0004] Most traditional methods only study one of the second-order or fourth-order statistics, or only use most of the data. However, the TenSOFO algorithm proposed in this patent combines the second-order and fourth-order statistics. It uses the information on the correlation and linear relationship between observed signals provided by SO to capture the linear structure in the signal and perform preliminary decomposition of the signal. At the same time, it uses FO to capture the non-Gaussian properties of the source signal and the high-order dependencies between data, so as to better identify and separate the source signal with non-Gaussian properties and improve the accuracy of separation.

[0005] After searching, application publication number CN104375976B was found, which discloses a method for identifying a mixture matrix in underdetermined blind source separation based on tensor regularization. This method mainly addresses the problem that existing technologies are limited by specific conditions when estimating the mixture matrix. The implementation steps are as follows: (1) sampling the source signal to obtain observation data; (2) calculating the fourth-order covariance matrix under different time delays using the fourth-order cumulant of the observation data; (3) expanding the fourth-order covariance matrix under different time delays into the form of a third-order tensor; (4) performing tensor regularization on the third-order tensor to obtain the Khatri-Rao product matrix of the mixture matrix to be identified; (5) processing the product matrix using the eigenvalue decomposition method to obtain the estimated value of the mixture matrix. This invention has the advantage of high recognition accuracy and can be used for underdetermined blind source separation of source signals under time-frequency aliasing conditions in the fields of speech, communication, radar and biomedicine.

[0006] The aforementioned patents suffer from two drawbacks: they only utilize the fourth-order cumulant of the observed signal and only address underdetermined blind source separation. Furthermore, the computational complexity of these methods increases exponentially with signal dimension and data length, meaning they require more observation data and suffer from poor real-time performance. Additionally, focusing solely on underdetermined blind source separation ignores the overdetermined case. This invention, while utilizing the fourth-order cumulant of the observed signal, also considers the second-order statistics of the source signal, concatenating the second-order and fourth-order statistical matrices under different time delays into a three-dimensional tensor. This reduces computational complexity while improving separation accuracy. Moreover, this method is applied to the overdetermined case, overcoming the shortcomings of the aforementioned patents that only consider the underdetermined situation. Summary of the Invention

[0007] This invention aims to solve the problems of the prior art. It proposes a method for separating coupled signals based on symmetric tensor joint decomposition. The technical solution of this invention is as follows:

[0008] A method for separating coupled signals based on symmetric tensor joint decomposition includes the following steps:

[0009] The second-order statistics and fourth-order cumulants of the observed signal are calculated, and then the statistics matrix is ​​concatenated into a three-dimensional tensor along three dimensions. Finally, the mixture matrix is ​​obtained by INDSCAL decomposition. and source signal Furthermore, a TenSOFO optimization framework was proposed, which transforms the joint INDSCAL decomposition into a constrained minimization problem through the alternating direction multiplier method. The original variables are alternately minimized and the dual variables are updated to estimate the mixture matrix. The proposed TenSOFO method and previous blind source separation methods are used to perform blind source separation on randomly generated real and complex signals.

[0010] Furthermore, the calculation of the second-order statistics and fourth-order cumulants of the observed signal specifically includes:

[0011] Through calculation and The second-order statistics can be expressed by the following relationship:

[0012]

[0013]

[0014] in, The autocorrelation matrix represents the observed signal. The autocorrelation matrix represents the source signal. It is the autocorrelation function of the r-th source; Indicates time delay; Indicates the observed signal, Indicates the source signal, Represents a mixture matrix, Represents mathematical expectation, Indicates the source signal is delayed The value of time, The autocorrelation function representing the source signal, Indicates the source signal at The value at any given moment.

[0015] The following relationship can be obtained by calculating the fourth-order statistic of x:

[0016]

[0017] in, The weights of the weighted sum are determined by the mixing matrix. It is composed of a combination of elements; The indices represent the source signal indices, which are obtained by summing and traversing the four dimensions of the source signal; i, j, k, and l represent the indices of the observed signal, corresponding to the four dimensions of the observed signal. This represents the fourth-order cumulant of the r-th source signal; , These represent the correlation matrix of the observed signal and the diagonal cumulant matrix of the source signal, respectively.

[0018] Therefore, the fourth-order cumulant of the observed signal is

[0019]

[0020] Represents the mixture matrix The Kronecker product.

[0021] Furthermore, the step of stitching together a statistical matrix into a three-dimensional tensor along three dimensions specifically includes:

[0022] For a second-order statistic, there are N1 observation covariance matrices. By splicing them along the new dimension, we obtain a third-order tensor. For a fourth-order statistic, there are N² observation fourth-order cumulant matrices. Similarly, by splicing along the new dimension, we obtain a third-order tensor. The constructed tensor is shown below:

[0023]

[0024]

[0025] This indicates a tensor splicing operation.

[0026] tensor The first two modes correspond to the expansion dimensions of the fourth-order statistics of the observed signals. The third mode corresponds to the number N² of time delay sets. Each element of these two tensors can be expressed by the following formula:

[0027]

[0028]

[0029] , Each element of the tensor constructed from the second-order statistics and fourth-order cumulants of the observed signal represents a different element.

[0030] In conclusion, and The final representation is as follows:

[0031]

[0032]

[0033] in, Each layer is a weighted sum of the squares of the column elements of the mixing matrix and the variance of the source signal. Each layer is a weighted sum of the squares of the elements of the mixed matrix product and the cumulative amount of the source signal.

[0034] Furthermore, the final step involves performing INDSCAL decomposition to obtain the mixture matrix. and source signal Specifically, it includes:

[0035] Since the statistics of the source signal are a diagonal matrix, using matrix decomposition, the second-order statistics of the source signal in equation (9) can be expressed as follows:

[0036]

[0037] The fourth-order cumulant of the source signal can then be expressed as:

[0038]

[0039] A matrix representing the statistical characteristics of the fourth-order cumulants of a single source signal.

[0040] Substituting equations (18) and (19) into equations (8) and (11), we get:

[0041]

[0042]

[0043] The above analysis shows that the observation statistics at each time lag can be represented as a "factor matrix". The two tensors constructed are in the form of "transpose of factor matrix"; therefore, they can be approximately decomposed as follows:

[0044]

[0045]

[0046] in, The (n, r)th element is .

[0047] Furthermore, the optimization framework of TenSOFO is specifically as follows:

[0048] The core idea is to minimize the objective function and estimate the mixture matrix using the alternating direction multiplier method. Source signal statistics, when , hour, and The joint INDSCAL decomposition is transformed into a constrained minimization problem:

[0049]

[0050] in, Measuring tensors Fitting error to INDSCAL decomposition; Measuring tensors Fitting error to INDSCAL decomposition; Constrain the element-wise quadratic relationship between B and A;

[0051] Due to constraints It is a linear equation, suitable for solving using ADMM; ADMM introduces dual variables. Constructing the augmented Lagrangian function: The core of ADMM is to alternately minimize the original variable and update the dual variable;

[0052]

[0053] in, It is a regularization parameter. It is the scaled dual variable.

[0054] Furthermore, the core of ADMM is to alternately minimize the original variable and update the dual variable; therefore, the iterative steps are divided into the following two steps:

[0055] (1) Update and : It needs to be processed first Decompose the problem into two subproblems with left and right factor matrices and add constraints. :

[0056] (2) Update sum and variance matrix Similar to the process of updating B and K, first construct the ADMM problem, and then... Decomposed into two factor matrices and And add constraints .

[0057] An electronic device includes a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor, when executing the program, implements the coupled signal separation method based on symmetric tensor joint decomposition as described in any one of the claims.

[0058] A non-transitory computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements the coupled signal separation method based on symmetric tensor joint decomposition as described in any one of the claims.

[0059] The advantages and beneficial effects of this invention are as follows:

[0060] Traditional methods apply second-order or fourth-order statistics alone, which can result in either poor noise resistance but poor separation performance or good separation but poor real-time performance and high computational complexity.

[0061] To address this problem, claim 2 calculates the second-order and fourth-order statistics of the observed signals. The second-order statistics provide information on the correlation and linear relationships between the observed signals, capturing the linear structure within the signals and performing preliminary decomposition. Simultaneously, the fourth-order statistics capture the non-Gaussian properties of the source signals and the higher-order dependencies between data, better identifying and separating source signals with non-Gaussian characteristics, thus improving the accuracy of separation. Furthermore, claim 5 proposes a decomposition optimization framework using ADMM, enhancing constraint handling capabilities and flexibility. Attached Figure Description

[0062] Figure 1 This is a flowchart of a preferred embodiment of TenSOFO provided by the present invention;

[0063] Figure 2 These are simulation results of real-valued signals under certain conditions;

[0064] Figure 3 These are simulation results of real-valued signals under underdetermined conditions;

[0065] Figure 4 It is a complex random source signal generated under certain conditions;

[0066] Figure 5 These are simulation results of complex signals under certain conditions;

[0067] Figure 6 It is a complex random source signal generated under underdetermined conditions;

[0068] Figure 7 These are simulation results for complex signals under underdetermined conditions. Detailed Implementation

[0069] The technical solutions of the embodiments of the present invention will be clearly and thoroughly described below with reference to the accompanying drawings. The described embodiments are merely some embodiments of the present invention.

[0070] The technical solution of the present invention to solve the above-mentioned technical problems is:

[0071] BSS based on data statistics:

[0072] Considering a data vector with zero mean, its second-order and fourth-order statistics can be analyzed using the second-order and fourth-order covariance matrices, respectively. The second-order covariance matrix is ​​used to describe the linear correlation and energy distribution of the signal, i.e.

[0073]

[0074] The fourth-order cumulant tensor measures the joint statistical dependency among four signal components, and is particularly sensitive to non-Gaussian signals (the fourth-order cumulant is 0 for Gaussian signals). In BSS, separation can be achieved by utilizing the non-Gaussianity of the source signals, and the fourth-order cumulant can be expressed as...

[0075]

[0076] in, These are elements of a fourth-order tensor, with subscripts corresponding to the four components of the data vector; It represents the cumulative quantity (a more generalized statistic than the mean and variance, which can eliminate the effects of Gaussian noise). Let represent the time delay set, where represents the time difference between the three variables and t; u represents the values ​​of the i, j, k, l-th components of the data vector at different times.

[0077] INDSCAL:

[0078] Individual Difference Scaling (INDSCAL) is a special variant of canonical polynomial factorization (CP factorization) / parallel factorization, its core function being to achieve factorization of symmetric tensors (transforming high-dimensional tensors into low-dimensional ones). Under the INDSCAL model, the following conditions are met: The third-order tensor It can be decomposed into two factor matrices and (where R is the tensor rank), the decomposition form can be expressed as:

[0079]

[0080] in and They are matrices and The r-th column has a tensor size of The computation of INDSCAL typically follows the same iterative process as the traditional CP decomposition (i.e., the CP-ALS algorithm). Two The matrices are treated as independent factor matrices, denoted as follows: and The two factors are independent of each other during the update process, and there are no explicit constraints forcing them to be equal. (Reuse factor) To capture the symmetry of the first two dimensions of the tensor, combined with factors By addressing the third dimension, a low-rank decomposition of the symmetric third-order tensor is ultimately achieved. Despite differing initial estimates, the inherent symmetry of the data and the essential uniqueness of the CP decomposition will ultimately allow the two... Convergence (allowing for scaling factor differences). As a special case of CP decomposition, the uniqueness of INDSCAL can also be guaranteed under mild conditions.

[0081] ADMM:

[0082] The Alternating Direction Method of Multipliers (ADMM) is suitable for solving convex-constrained optimization problems of the following form, whose cost function can be expressed as:

[0083]

[0084] The corresponding augmented Lagrangian function is:

[0085]

[0086] in, It is a regularization parameter. These are dual variables. ADMM is based on the duality theory of convex optimization, and its core idea is to fix the dual variables. In the case of x, minimize the augmented Lagrangian function with respect to the original variables x and y; conversely, maximize the dual function. Regarding dual variables The value of .

[0087] The method described in this article:

[0088] Calculate the statistic:

[0089] In many applications, the fundamental sources are stationary, non-Gaussian, statistically independent (a core premise of BSS), and individually correlated at different lags. In these cases, by calculating... and The second-order statistics can be expressed by the following relationship:

[0090]

[0091]

[0092] in, The autocorrelation matrix represents the observed signal. The autocorrelation matrix represents the source signal. It is the autocorrelation function of the r-th source. Since the source signals are statistically independent, the covariance between different sources is 0, and the source signals are stationary, the autocorrelation is only related to the time delay. Relevant. Therefore It is a diagonal matrix.

[0093] The following relationship can be obtained by calculating the fourth-order statistic of x:

[0094]

[0095] in, The weights of the weighted sum are determined by the mixing matrix. It is composed of a combination of elements.

[0096] Therefore, the fourth-order cumulant of the observed signal is

[0097]

[0098] Constructing tensors:

[0099] For a second-order statistic, there are N1 observation covariance matrices. By splicing them along the new dimension, we obtain a third-order tensor. For a fourth-order statistic, there are N² observation fourth-order cumulant matrices. Similarly, by splicing along the new dimension, we obtain a third-order tensor. The constructed tensor is shown below:

[0100]

[0101]

[0102] tensor The first two modes correspond to the expansion dimensions of the fourth-order statistics of the observed signals. The third mode corresponds to the number N² of time delay sets. Each element of these two tensors can be expressed by the following formula:

[0103]

[0104]

[0105] In conclusion, and The final representation is as follows:

[0106]

[0107]

[0108] in, Each layer is a weighted sum of the squares of the column elements of the mixing matrix and the variance of the source signal. Each layer is a weighted sum of the squares of the elements of the mixed matrix product and the cumulative amount of the source signal.

[0109] The introduction of N1 observation covariance matrices and N2 observation fourth-order cumulant matrices is to utilize the redundant information provided by "multiple sets of statistics," satisfying the uniqueness condition of tensor decomposition, ensuring that the mixture matrix can be uniquely estimated, and improving the robustness of the mixture matrix estimation. In BSS, the second-order and fourth-order statistics of the source signal are usually related to time delays. For the second-order covariance matrix, different time delays will correspond to different matrices—because the source signal may have different autocorrelation at different time differences. For the fourth-order cumulant matrix, different sets of time delays will also correspond to different matrices—because the non-Gaussianity of higher-order statistical properties may manifest differently at different time delays. Therefore, by selecting N1 different time delays, N1 different second-order covariance matrices can be obtained, and the same applies to the fourth-order.

[0110] The uniqueness of tensor decomposition is a core prerequisite for BSS; only when the decomposition is unique can the mixture matrix be accurately recovered. Concatenating these into a new tensor introduces an additional "pattern dimension," providing more constraints for tensor decomposition. Real-world data contains noise, and a single statistic may be severely affected by noise. However, N1 and N2 statistics provide redundant information. Through the "joint fitting" process of tensor decomposition, the impact of noise can be averaged, improving performance. Robustness and accuracy of the estimation.

[0111] INDSCAL decomposition:

[0112] Given that the second-order statistics of Gaussian noise are weak, its fourth-order cumulant is 0, and the statistics of the source signal are diagonal matrices, the second-order statistics of the source signal can be expressed as follows:

[0113]

[0114] The fourth-order cumulant of the source signal can then be expressed as:

[0115]

[0116] Substituting equations (18) and (19) into equations (8) and (11), we get:

[0117]

[0118]

[0119] The above analysis shows that the observation statistics at each time lag can be represented as a "factor matrix". The form is "transpose of factor matrix". Therefore, the two tensors constructed can be approximately decomposed as:

[0120]

[0121]

[0122] in, The (n, r)th element is .

[0123] The process of TenSOFO is shown in the following figure:

[0124] Depend on Figure 1 It can be seen that the mixing matrix of R source signals Generate M observation signals First, statistical calculations are performed on the observed signals. SO is a second-order statistic, used to calculate the covariance matrix of the observed signals, reflecting their temporal correlation. FO is a fourth-order statistic, used to calculate the fourth-order cumulant matrix of the observed signals, reflecting their non-Gaussianity. Then, a tensor is constructed using these statistical calculations, and the statistical matrix is... and They are spliced ​​together along three dimensions to form a three-dimensional tensor. and Finally, joint scaling analysis (INDSCAL) is performed to estimate the mixture matrix. and source signal .

[0125] Optimize the framework:

[0126] The core idea is to minimize the objective function and estimate the mixture matrix using the alternating direction multiplier method. Source signal statistics. When , hour, and The joint INDSCAL decomposition is transformed into a constrained minimization problem:

[0127]

[0128] in, Measuring tensors Fitting error to INDSCAL decomposition; Measuring tensors Fitting error to INDSCAL decomposition; Constrain the element-order square relation between B and A.

[0129] Due to constraints It is a linear equation, suitable for solving using ADMM. ADMM introduces dual variables. Construct the augmented Lagrangian function:

[0130]

[0131] in, It is a regularization parameter. It is the scaled dual variable.

[0132] The core of ADMM is to alternately minimize the original variable and update the dual variable; therefore, the iterative steps consist of the following two steps:

[0133] (1) Update and : It needs to be processed first Decompose the problem into two subproblems with left and right factor matrices and add constraints. :

[0134]

[0135] To address this constraint problem, we again use ADMM to introduce dual variables (cumulative difference matrix). Construct the augmented Lagrangian function, track the degree of constraint violation, and guide alignment. Iterate and update using the following formula: , , , :

[0136]

[0137] Then, the optimal solution is obtained through iterative loops using the least squares closed-form solution. The Lagrange term is combined with matrix inversion to update the value. and The update rule for the i-th step within the k-th iteration is as follows:

[0138]

[0139]

[0140]

[0141]

[0142] in, , In the initial step of the k-th iteration (i=0), set... , .

[0143] After the iteration ends, due to numerical errors or permutation-scaling uncertainties in the decomposition, , It may not be perfectly aligned, therefore a permutation matrix is ​​required. and diagonal matrix Scaling is performed to achieve precise alignment, i.e.

[0144]

[0145] Final decision .

[0146] (2) Update sum and variance matrix Similar to the process of updating B and K, first construct the ADMM problem, and then... Decomposed into two factor matrices and And add constraints :

[0147]

[0148] Starting from j=0, initialize Then, according to the following formula, we get , :

[0149]

[0150] in, These are newly added Lagrange multipliers from the previous iteration, used for constraint relaxation.

[0151] Depend on We can obtain:

[0152]

[0153] By reshaping the matrix (that is, to put) The rth column is reshaped into (matrix), and perform singular value decomposition, taking its most important left and right singular vectors, we can obtain respectively. and The rth column.

[0154] The update formula for the j-th step within the k-th iteration is as follows:

[0155]

[0156]

[0157]

[0158]

[0159] in:

[0160]

[0161]

[0162] These are two regularization matrices, used for two... The update incorporates intermediate variables and regularization terms from the previous step, ensuring the stability and convergence of the update process.

[0163] and The functions are respectively with and The same applies when the stopping criterion is met, using and estimating... and Estimate in the same way and Then set Finally, it was concluded that :

[0164]

[0165] Stopping criteria, parameter selection, and complexity:

[0166] Stopping criteria and parameter selection:

[0167] The proposed method consists of an outer loop (for the entire iteration process) and two inner loops (for updates). , ;renew , The maximum number of iterations for three loops is preset to a fixed value: K. stop =100, I stop =10, J stop =10. A stopping criterion based on the original residual and the dual residual is adopted:

[0168]

[0169] Where, cur and dual represent the current estimate and the previous estimate, respectively, and are calculated using the following formula:

[0170]

[0171] In the formula, and Both are greater than zero, representing absolute tolerance and relative tolerance, respectively. The original variables are... and express. include , include dual variables include , Represents the regularization parameter .

[0172] Each iteration uses the following adaptive rule to select the value of the regularization parameter:

[0173]

[0174] in, , Represents the original and target variables for the current iteration; This represents the target variable of the previous iteration; in practical applications, it is taken as... =10, =2; It is the Frobenius norm of the matrix, used to measure the overall difference between the elements.

[0175] when If the value is too large, it indicates that the difference between the original variable and the target variable is too great, and the constraint is not effectively satisfied; therefore, it should be increased. Increase the weight of the penalty term to force the variable closer to the constraint; when If the value is too large, it indicates that the update range of the target variable is too large, and the optimization is too aggressive. Therefore, it should be reduced. Reduce the weight of the penalty term to make the update smoother; otherwise, If it remains unchanged, it means that the current parameters balance convergence and accuracy.

[0176] Complexity:

[0177] Computational complexity is used to measure the operational efficiency of an algorithm (i.e., the number of basic operations required to complete one iteration or the entire process). Used to describe the gradual trend of the number of algorithm operations as the input size increases, indicating a growth rate that does not exceed a certain function.

[0178] Assuming N1=N2=N (i.e., the time delays / time delay sets of the second and fourth-order statistics are the same), the inner first-level ADMM loop is used for updating. and The second layer ADMM is updated cyclically. and Processing tensors Its complexity consists of the following parts:

[0179] Total complexity of inner iteration: number of iterations is Update in time loop and Time processing tensors The total complexity is . Update cyclically and Time processing tensors The number of iterations is J stop When the total complexity is .

[0180] Matrix alignment steps: involve permutation and scaling operations, with a complexity of O(n log n). The complexity at this point is much smaller than that of the iteration part, and can be ignored or merged.

[0181] Total complexity: TenSOFO's outer loop iterations are... Therefore, the overall computational complexity is .

[0182] Numerical Experiment:

[0183] TenSOFO simulates real signals:

[0184] Each source signal is configured to be a length of... Filter and a length of random coefficient vector Convolutional, i.e. The number of sensors J is set to 7; the number of source signals R is set to 3 and 9 respectively to discuss overdetermined and underdetermined cases; the number of samples N is set to... The signal-to-noise ratio (SNR) was set to a range of 0 dB to 20 dB, and the delay parameter K was set to 5 to construct the tensor. In the experiment, R source signals were first generated, each processed by a random signal and a filter, which introduced some correlation between the source signals. Then, a random matrix and observation data were generated. In the loop, white noise of appropriate intensity was added to the observation data to generate the final noisy observation result x.

[0185] When the number of source signals R=3, the over-limit case is as follows: Figure 2 As shown, with the gradual increase of the signal-to-noise ratio (SNR), the estimation errors of both SOBIUM (using only second-order statistics for blind source separation) and TenSOFO (using a combination of second-order statistics and fourth-order cumulants proposed in this paper) generally show a decreasing trend, and TenSOFO's estimation error is smaller than that of SOBIUM. The estimation error of FOOBI (using only fourth-order cumulants for blind source separation) does not change significantly. Therefore, it can be seen that, under certain conditions, TenSOFO performs best in blind source separation of real signals, with the smallest estimation error.

[0186] When the number of source signals R=9, the underdetermined condition is as follows: Figure 3 As shown, with the signal-to-noise ratio (SNR) gradually increasing, the estimation errors of SOBIUM and TenSOFO generally show a decreasing trend, while the estimation error of TenSOFO is smaller than that of SOBIUM in most cases. The estimation error of FOOBI, however, does not change significantly. Therefore, it can be seen that, under underdetermined conditions, TenSOFO performs best in blind source separation of real signals, exhibiting the smallest estimation error.

[0187] TenSOFO simulates complex signals:

[0188] In the experiment, R random complex source signals are first generated. Each source signal is obtained by passing a random signal and a filter, which gives the source signals a certain correlation. Then, a random matrix and observation data are generated. In the loop, complex Gaussian white noise of appropriate intensity is added to the observation data to generate the final noisy complex signal observation result x. When the number of source signals R=3, the generated complex random source signals under certain conditions are as follows: Figure 4 As shown.

[0189] The result of blind source separation of complex signals under certain conditions is as follows: Figure 5 As shown, with the signal-to-noise ratio (SNR) gradually increasing, the estimation errors of SOBIUM and TenSOFO generally show a decreasing trend, while the estimation error of TenSOFO is smaller than that of SOBIUM. The estimation error of FOOBI, however, does not change significantly. Therefore, it can be seen that, under underdetermined conditions, TenSOFO performs best in blind source separation of complex signals, with the smallest estimation error. When the number of source signals R=9, the complex random source signals generated under underdetermined conditions are as follows: Figure 6 As shown.

[0190] The result of blind source separation of complex signals under underdetermined conditions is as follows: Figure 7As shown, with the signal-to-noise ratio (SNR) gradually increasing, the estimation errors of both SOBIUM and TenSOFO generally show a decreasing trend, while the estimation error of TenSOFO is smaller than that of SOBIUM. The estimation error of FOOBI, however, does not change significantly. Therefore, it can be seen that, under underdetermined conditions, TenSOFO exhibits the best performance in blind source separation of complex signals, with the smallest estimation error.

[0191] in conclusion:

[0192] This patent studies the problem of instantaneous blind source separation from the perspective of tensor decomposition. Addressing the ambiguity and high computational cost of traditional methods, it proposes a method called TenSOFO that simultaneously utilizes second-order statistics and fourth-order cumulants. This method combines the advantages of using second-order statistics and fourth-order cumulants individually, offering benefits such as memory savings and the ability to process source signals without requiring additional prior information. The method first calculates the second-order statistics and fourth-order cumulants of the observed signal, then concatenates the statistics matrices along three dimensions to form a three-dimensional tensor, and finally performs INDSCAL decomposition to obtain the mixing matrix. and source signal Building upon this method, a TenSOFO optimization framework is proposed. This framework transforms the joint INDSCAL decomposition into a constrained minimization problem using the alternating direction multiplier method. The original variables are alternately minimized while the dual variables are updated, thereby estimating the mixture matrix. The proposed TenSOFO method and previous blind source separation methods were used to perform blind source separation on randomly generated real and complex signals, and the estimation errors of different methods were observed under underdetermined and overdetermined conditions. Experimental results show that the proposed TenSOFO method has the smallest estimation error and the best performance, regardless of whether the signal is real or complex, or whether it is underdetermined or overdetermined.

[0193] The systems, devices, modules, or units described in the above embodiments can be implemented by computer chips or entities, or by products with certain functions.

[0194] It should also be noted that the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitation, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0195] The above embodiments should be understood as illustrative only and not as limiting the scope of protection of the present invention. After reading the description of the present invention, those skilled in the art can make various alterations or modifications to the present invention, and these equivalent changes and modifications also fall within the scope defined by the claims of the present invention.

Claims

1. A method for separating coupled signals based on symmetric tensor joint decomposition, characterized in that, Includes the following steps: The second-order statistics and fourth-order cumulants of the observed signal are calculated, and then the statistics matrix is ​​concatenated into a three-dimensional tensor along three dimensions. Finally, the mixture matrix is ​​obtained by INDSCAL decomposition. and source signal Furthermore, a TenSOFO optimization framework was proposed, which transforms the joint INDSCAL decomposition into a constrained minimization problem through the alternating direction multiplier method. The original variables are alternately minimized and the dual variables are updated to estimate the mixture matrix. The proposed TenSOFO method and previous blind source separation methods are used to perform blind source separation on randomly generated real and complex signals.

2. The coupled signal separation method based on symmetric tensor joint decomposition according to claim 1, characterized in that, The calculation of the second-order statistics and fourth-order cumulants of the observed signal specifically includes: Through calculation and The second-order statistics can be expressed by the following relationship: in, The autocorrelation matrix represents the observed signal. The autocorrelation matrix represents the source signal. It is the autocorrelation function of the r-th source; Indicates time delay; Indicates the observed signal, Indicates the source signal, Represents a mixture matrix, Represents mathematical expectation, Indicates the source signal is delayed The value of time, The autocorrelation function representing the source signal, Indicates the source signal at The value at time; The following relationship can be obtained by calculating the fourth-order statistic of x: in, The weights of the weighted sum are determined by the mixing matrix. It is composed of a combination of elements; The index of the source signal is represented by summing and traversing the four dimensions of the source signal; i, j, k, l represent the indices of the observed signal, corresponding to the four dimensions of the observed signal. This represents the fourth-order cumulant of the r-th source signal; , These represent the correlation matrix of the observed signal and the diagonal cumulant matrix of the source signal, respectively. Therefore, the fourth-order cumulant of the observed signal is Represents the mixture matrix The Kronecker product.

3. The coupled signal separation method based on symmetric tensor joint decomposition according to claim 2, characterized in that, The process of stitching together a statistical matrix into a three-dimensional tensor in three dimensions specifically includes: For a second-order statistic, there are N1 observation covariance matrices. By splicing them along the new dimension, we obtain a third-order tensor. For a fourth-order statistic, there are N² observation fourth-order cumulant matrices. Similarly, by splicing along the new dimension, we obtain a third-order tensor. The constructed tensor is shown below: This indicates a tensor concatenation operation; tensor The first two modes correspond to the expansion dimensions of the fourth-order statistics of the observed signals. The third mode corresponds to the number N2 of time-delay sets; each element of these two tensors can be expressed by the following formula: , Each element of the tensor constructed from the second-order statistics and fourth-order cumulants of the observed signal is represented, respectively. In conclusion, and The final representation is as follows: in, Each layer is a weighted sum of the squares of the column elements of the mixing matrix and the variance of the source signal. Each layer is a weighted sum of the squares of the elements of the mixed matrix product and the cumulative amount of the source signal.

4. The coupled signal separation method based on symmetric tensor joint decomposition according to claim 3, characterized in that, Finally, INDSCAL decomposition is performed to obtain the mixture matrix. and source signal Specifically, it includes: Since the statistics of the source signal are a diagonal matrix, using matrix decomposition, the second-order statistics of the source signal in equation (9) can be expressed as follows: The fourth-order cumulant of the source signal can then be expressed as: A matrix representing the statistical characteristics of the fourth-order cumulants of a single source signal; Substituting equations (18) and (19) into equations (8) and (11), we get: The above analysis shows that the observation statistics at each time lag can be represented as a "factor matrix". The two tensors constructed are in the form of "transpose of factor matrix"; therefore, they can be approximately decomposed as follows: in, The (n, r)th element is .

5. The coupled signal separation method based on symmetric tensor joint decomposition according to claim 4, characterized in that, The optimization framework for TenSOFO is as follows: The core idea is to minimize the objective function and estimate the mixture matrix using the alternating direction multiplier method. Source signal statistics, when , hour, and The joint INDSCAL decomposition is transformed into a constrained minimization problem: in, Measuring tensors Fitting error to INDSCAL decomposition; Measuring tensors Fitting error to INDSCAL decomposition; Constrain the element-wise quadratic relationship between B and A; Due to constraints It is a linear equation, suitable for solving using ADMM; ADMM introduces dual variables. Constructing the augmented Lagrangian function: The core of ADMM is to alternately minimize the original variable and update the dual variable; in, It is a regularization parameter. It is the scaled dual variable.

6. The coupled signal separation method based on symmetric tensor joint decomposition according to claim 5, characterized in that, The core of ADMM is to alternately minimize the original variable and update the dual variable; therefore, the iterative steps are divided into the following two steps: (1) Update and : It needs to be processed first Decompose the problem into two subproblems with left and right factor matrices and add constraints. : (2) Update sum and variance matrix Similar to the process of updating B and K, first construct the ADMM problem, and then... Decomposed into two factor matrices and And add constraints .

7. An electronic device, characterized in that, It includes a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor, when executing the program, implements the coupled signal separation method based on symmetric tensor joint decomposition as described in any one of claims 1 to 6.

8. A non-transitory computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the coupled signal separation method based on symmetric tensor joint decomposition as described in any one of claims 1 to 6.

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