CN116776076A - Underdetermined blind source separation method and system for third-order cumulant and tensor compression - Google Patents
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Abstract
The invention provides an underdetermined blind source separation method for third-order cumulant and tensor compression. And then performing Tack decomposition on the fourth-order tensor to obtain a lower-dimensional nuclear tensor. And then carrying out regular polynary decomposition on the kernel tensor, and decompressing the factor matrix obtained by decomposition to obtain the estimation of the mixed matrix. Finally, a matrix diagonalization method is used for recovering the source signals. Theoretical analysis and simulation results show that the invention can accurately estimate the mixing matrix under the conditions of high interference-to-signal ratio and low signal-to-noise ratio to obtain the separated signals, and can greatly reduce the calculation complexity. Compared with the existing method based on sparsity and accumulation, the method has the advantage that a better separation effect is achieved.
Description
Technical Field
The invention relates to the field of blind source separation, in particular to an underdetermined blind source separation method and system for third-order cumulant and tensor compression.
Background
When the integrated system receives reconnaissance, interference, radar and communication signals, the integrated system has the characteristics of high interference-to-signal ratio and low signal-to-noise ratio. The third-order cumulative amount can better suppress the influence of symmetrically distributed noise such as gaussian noise. The tower decomposition can reduce the dimension of tensors on the premise of guaranteeing the information integrity, so that the calculation complexity is reduced. In summary, the invention provides an underdetermined blind source separation method based on third-order cumulant and tensor compression. The method has strong Gaussian noise resistance and can greatly reduce the operation time. Compared with the classical underdetermined blind source separation method, the method has better identification performance and separation precision under different signal to noise ratios.
Disclosure of Invention
The invention aims to provide an underdetermined blind source separation method and system for third-order cumulant and tensor compression aiming at the defects of the prior art.
In order to achieve the above purpose, the invention adopts the following technical scheme:
the invention provides an underdetermined blind source separation method for third-order cumulant and tensor compression, which comprises the following steps:
s1, centralizing and whitening the observed signal data set to obtain a whitened signal;
s2, solving the third-order accumulation quantity of the whitened signal under different time delays;
s3, stacking a plurality of third-order cumulative tensors into a fourth-order tensor;
s4, performing Tack decomposition on the fourth-order tensor by using singular value decomposition to obtain a low-dimensional kernel tensor;
s5, performing regular multi-component decomposition on the fourth-order tensor by using alternating least square to obtain estimation of a hybrid matrix;
s6, inverting the whitening matrix and multiplying the whitening matrix to obtain an estimated mixing matrix;
s7, performing short-time Fourier transform on the observation signal to obtain an observation signal time-frequency point;
s8, obtaining a time-frequency point of the separated signal by using a matrix diagonalization method;
s9, performing inverse short-time Fourier transform on the time-frequency points of the separated signals to obtain the separated signals.
Further, in the step S1, the observed signal is pre-whitened to a size ofIs:
（1）
wherein ,is a whitening matrix; />Is->A path observation signal matrix; />Is a true mixing matrix; />Is thatA source signal matrix; unitary matrix->Is whitenedThe mixing matrix, herein referred to as whitening mixing matrix.
Further, in S2, the third-order cumulative amount of the zero-mean whitened signal is:
（2）；
wherein ,is time delay; />Is +.>Modulo multiplication; />Is the third order cumulative amount of the source signal;
by varying the delayIs constructed->The three different tensors correspond to the delays, specifically:
（3）。
further, in S3, tensor is calculatedStacked together to form a fourth order tensor->The method comprises the following steps:
（4）；
wherein ,representation vector->Is>An element; /> and />Representation vector->Conjugation +.>；/>An element;representation vector->Is>An element;
fourth order tensorBy vector sumThe matrix can be expressed as:
（5）；
wherein a matrix is defined，/>；/>；/>；/>Is conjugated; />Is a whitening mixing matrix->Is>Individual column vectors>Is a matrix->Conjugation of->Is->Is>Individual column vectors>Is a matrix/>Is>And a column vector.
Further, the method comprises the steps of,the tower decomposition matrix form of (2) is:
（6）；
wherein ,is +.>；/>For nuclear tensor>，/> and />Factor matrices of Take decomposition respectively; />Is a super diagonal tensor; i.e. < ->，/>，/>The rank of (2) is expressed as:
(7)；
record->，/>Then->Is a rank +.>Tensors;
i.e.Is called Tak-1 decomposition, i.e. +.>,/>,/>，/>Is a unitary matrix, so
（8）；
Thereby willTensor->Compression to->Nuclear tensor->；/>Is +.>；
（9）；
wherein ,is split into principal components, so +.>Is composed of->Before->A matrix consisting of left singular vectors, consisting of +.>Singular value decomposition of (2) to obtain +.>Then, the kernel tensor is obtained from the formula (10)>：
（10）。
Further, in S5, the kernel tensorPerforming regular multi-component decomposition:
（11）；
wherein ,，/> and />Are nuclear tensors->Factor matrix of regular polynary decomposition> and />Is of the size of；/>Is +.>The standard decomposition factor matrix can be solved by alternating least squares iterations>，/> and />From the formulae (6) and (11):
（12）；
namely:
（13）；
so for the kernel tensorAfter regular polynary decomposition, a +.>And two->For->Conjugation is taken and marked as +.>The method comprises the steps of carrying out a first treatment on the surface of the Three matrices are fused->The estimation of the mixing matrix is realized by singular value decomposition, and is as follows:
（14）；
respectively is a matrix->Is>A plurality of column vectors; matrix->Singular value decomposition is carried out, and the formula is as follows:
（15）；
wherein ,is a left singular matrix>Is a singular value matrix, & lt & gt>For right singular matrix, the estimated matrix is, wherein ,/>For matrix->Column 1 on the left.
Further, in the step S6, the whitening matrix is obtainedTo obtain an estimate of the mixing matrix +.>：
（16）。
Further, in S8, considering the influence of noise, the mixing matrix corresponding to the source signal is estimated by equation (17), which is:
（17）；
wherein ,for mixingMatrix->Is>A set of sub-matrices, together comprising->An element; />Is made use of->The detection amount of the diagonalization degree of the covariance matrix of the source signal obtained by estimation is as follows:
（18）；
wherein ,is->A matrix of non-diagonal elements; />Is->F-norm of (c); />Inverting the matrix;
the smaller the ∈>The better the diagonalization degree of (c), the more independent the signal.
Further, the underdetermined blind source separation system for third-order cumulant and tensor compression is realized by the underdetermined blind source separation method for third-order cumulant and tensor compression, and further comprises the following steps:
the data matrix construction module is used for generating a source signal and an observation signal;
the data whitening module is used for constructing a whitening matrix according to the sampling covariance matrix and whitening the observed signal by using the whitening matrix;
the third-order cumulative quantity solving module is used for solving the third-order cumulative quantity under different time delays for the whitened signal;
the tower decomposition module is used for performing tower decomposition on the fourth-order tensor to obtain a compressed kernel tensor;
the regular polynary decomposition module is used for carrying out regular polynary decomposition on the kernel tensor to obtain estimation of the mixing matrix;
the signal recovery module is used for obtaining a separated signal according to a matrix diagonalization method by using the estimated mixed matrix;
and the performance evaluation module is used for evaluating the estimation effect of the mixing matrix by adopting the average relative error and evaluating the separation performance by using the average similarity coefficient.
The beneficial effects of the invention are as follows: since the third-order cumulative amount has a strong capability of resisting symmetrically distributed noise, the suppression capability of suppressing gaussian white noise having symmetrically distributed characteristics is strong. Therefore, the method based on the third-order cumulant can effectively inhibit Gaussian white noise, so that the estimation performance of the hybrid matrix under the low signal-to-noise ratio is improved;
a high-dimensional fourth-order tensor is compressed into a low-dimensional kernel tensor by Tak decomposition. The estimation accuracy of the hybrid matrix is guaranteed, the calculation complexity is greatly reduced, the estimation accuracy of the hybrid matrix is further improved, and the tensor decomposition time is saved.
The independence among the source signals is fully utilized, the mixed vector corresponding to the source signals which are not zero in any adjacent domain is estimated through measuring the diagonalization degree of the covariance matrix, the underdetermined equation is converted into the overdetermined equation, and then matrix inversion is utilized to finish the estimation of the source signals. Compared with the prior method, the method relaxes the sparsity condition of the signal in the time-frequency domain.
Drawings
FIG. 1 is a schematic flow chart of an underdetermined blind source separation method of third-order cumulant and tensor compression in the invention;
FIG. 2 is an exploded view of a tower;
FIG. 3 is a regular multivariate decomposition schematic;
FIG. 4 is a time domain waveform of a source signal;
FIG. 5 is a time domain waveform of an observed signal;
FIG. 6 is a time domain waveform diagram of a split signal;
fig. 7 is a graph of hybrid matrix estimation error versus different signal to noise ratios.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention.
Because the array is adopted to receive far-field narrowband signals, the invention uses a time delay mixing model and the process is as follows
wherein ,representation->The observed signal received by the respective receive channel,representation->Individual source signal->Is->A hybrid matrix of dimensions, also known as an array streaming matrix, is used in the processing of array signals. Because of the underdetermined situation +.>。/>Representing additive white gaussian noise during signal acquisition. Currently underdetermined blind source separation is generally solved by a two-step method, i.e. the mixing matrix +.>The source signal is then recovered from the estimated mixing matrix.
It should be noted that when the time-frequency aliasing detection, interference, radar and communication signals are simultaneously received by the array, the problems of low estimation accuracy and poor separation performance of the hybrid matrix under the severe environments of high interference-to-signal ratio and low signal-to-noise ratio are necessarily faced. And if the dimension of the tensor is too high, the decomposition time increases substantially.
To achieve the above object, referring to fig. 1,
an underdetermined blind source separation method for third-order cumulant and tensor compression comprises the following steps:
s1, centralizing and whitening the observed signal data set to obtain a whitened signal;
s2, solving the third-order accumulation quantity of the whitened signal under different time delays;
s3, stacking a plurality of third-order cumulative tensors into a fourth-order tensor;
s4, performing Tack decomposition on the fourth-order tensor by using singular value decomposition to obtain a low-dimensional kernel tensor;
s5, performing regular multi-component decomposition on the fourth-order tensor by using alternating least square to obtain estimation of a hybrid matrix;
s6, inverting the whitening matrix and multiplying the whitening matrix to obtain an estimated mixing matrix;
s7, performing short-time Fourier transform on the observation signal to obtain an observation signal time-frequency point;
s8, obtaining a time-frequency point of the separated signal by using a matrix diagonalization method;
s9, performing inverse short-time Fourier transform on the time-frequency points of the separated signals to obtain the separated signals.
In the S1, the observed signal is pre-whitened with the size ofIs:
（1）
wherein ,is a whitening matrix; />Is->A path observation signal matrix; />Is a true mixing matrix; />Is thatA source signal matrix; unitary matrix->Is a whitened mixing matrix, herein referred to as a whitened mixing matrix.
In S2, the third-order cumulative amount of the zero-mean whitened signal is:
（2）；
wherein ,is time delay; />Is +.>Modulo multiplication; />Is the third order cumulative amount of the source signal;
by varying the delayIs constructed->The three different tensors correspond to the delays, specifically:
（3）。
in the S3, tensor is toStacked together to form a fourth order tensor->The method comprises the following steps:
（4）；
wherein ,representation vector->Is>An element; /> and />Representation vector->Conjugation +.>；/>An element;representation vector->Is>An element;
fourth order tensorThe vector sum matrix can be expressed as:
（5）；
wherein a matrix is defined，/>；/>；/>；/>Is conjugated; />Is a whitening mixing matrix->Is>Individual column vectors>Is a matrix->Conjugation of->Is->Is>Individual column vectors>Is a matrix->Is>And a column vector.
The tower decomposition matrix form of (2) is:
（6）；
wherein ,is +.>；/>For nuclear tensor>，/> and />Factor matrices of Take decomposition respectively; />Is a super diagonal tensor; i.e. < ->，/>，/>The rank of (2) is expressed as:
(7)；
record->，/>Then->Is a rank +.>Tensors;
i.e.Is called Tak-1 decomposition, i.e. +.>,/>,/>，/>Is a unitary matrix, so
（8）；
Thereby willTensor->Compression to->Nuclear tensor->；/>Is +.>；
（9）；
wherein ,is split into principal components, so +.>Is composed of->Before->A matrix consisting of left singular vectors, consisting of +.>Singular value decomposition of (2) to obtain +.>Then, the kernel tensor is obtained from the formula (10)>：
（10）。
In the S5, the kernel tensorPerforming regular multi-component decomposition:
（11）；
wherein ,，/> and />Are nuclear tensors->Factor matrix of regular polynary decomposition> and />Is of the size of；/>Is +.>The standard decomposition factor matrix can be solved by alternating least squares iterations>，/> and />From the formulae (6) and (11):
（12）；
namely:
（13）；
so for the kernel tensorAfter regular polynary decomposition, a +.>And two->For->Conjugation is taken and marked as +.>The method comprises the steps of carrying out a first treatment on the surface of the Three matrices are fused->The estimation of the mixing matrix is realized by singular value decomposition, and is as follows:
（14）；
respectively is a matrix->Is>A plurality of column vectors; matrix->Singular value decomposition is carried out, and the formula is as follows:
（15）；
wherein ,is a left singular matrix>Is a singular value matrix, & lt & gt>For right singular matrix, the estimated matrix is, wherein ,/>For matrix->Column 1 on the left.
In the step S6, the whitening matrix is obtainedTo obtain an estimate of the mixing matrix +.>：
（16）。
In S8, considering the influence of noise, the mixing matrix corresponding to the source signal is estimated by equation (17), which is:
（17）；
wherein ,for a mixed matrix->Is>A set of sub-matrices, together comprising->An element; />Is made use of->The detection amount of the diagonalization degree of the covariance matrix of the source signal obtained by estimation is as follows:
（18）；
wherein ,is->A matrix of non-diagonal elements; />Is->F-norm of (c); />Inverting the matrix;
the smaller the ∈>The better the diagonalization degree of (c), the more independent the signal.
An underdetermined blind source separation system for third-order cumulant and tensor compression is realized by an underdetermined blind source separation method for third-order cumulant and tensor compression, and further comprises:
the data matrix construction module is used for generating a source signal and an observation signal;
the data whitening module is used for constructing a whitening matrix according to the sampling covariance matrix and whitening the observed signal by using the whitening matrix;
the third-order cumulative quantity solving module is used for solving the third-order cumulative quantity under different time delays for the whitened signal;
the tower decomposition module is used for performing tower decomposition on the fourth-order tensor to obtain a compressed kernel tensor;
the regular polynary decomposition module is used for carrying out regular polynary decomposition on the kernel tensor to obtain estimation of the mixing matrix;
the signal recovery module is used for obtaining a separated signal according to a matrix diagonalization method by using the estimated mixed matrix;
and the performance evaluation module is used for evaluating the estimation effect of the mixing matrix by adopting the average relative error and evaluating the separation performance by using the average similarity coefficient.
The effects of the present invention will be further described with reference to simulation experiments.
The simulation experiment adopts a three-array element uniform linear array to receive four paths of narrow-band far-field signals. Array element spacing isThe incidence angles are respectively->The observation time is 100us, and the snapshot number is 10000./>The LFM pulse signal has a starting frequency of 4MHz, a cut-off frequency of 9MHz, a pulse width of 60us and an arrival time of 20us. />Is a pair of->The interference signal is intermittently sampled, the interference sampling frequency is 5, the sampling period is 20us, and the sampling pulse width is 16us. />The bandwidth of the LFM echo signal is 10MHz, and the pulse width is 100us. />Is 16QAM signal, carrier frequency is 5MHz, and the signal is formed by basebandAnd (5) shape filtering, wherein the transmission rate is 5M/s. The sampling frequency of the signals is 100MHz. The signal and mixing matrix are complex. Fig. 2 is a schematic illustration of a tower decomposition, and fig. 3 is a schematic illustration of a canonical multiple decomposition. Fig. 4 is a source signal waveform. Fig. 5 is an observed signal waveform at an interference-to-signal ratio of 15dB and a signal-to-noise ratio of 20 dB. The observed signal waveform is relatively similar to the interfering signal because the interfering signal is relatively powerful and the remaining signals are submerged. Fig. 6 is a waveform of a split signal that is distorted due to noise and interference, but still more similar to the source signal. The calculated similarity coefficient is 0.91. Fig. 7 shows the relative error of hybrid matrix estimation for different methods at different signal-to-noise ratios. The algorithm has better performance under different signal-to-noise ratios, and compared with other methods, the method has the best estimation effect.
The foregoing examples merely illustrate embodiments of the invention and are described in more detail and are not to be construed as limiting the scope of the invention. It should be noted that it will be apparent to those skilled in the art that several variations and modifications can be made without departing from the spirit of the invention, which are all within the scope of the invention. Accordingly, the scope of protection of the present patent is to be determined by the appended claims.
Claims (9)
1. The underdetermined blind source separation method for third-order cumulant and tensor compression is characterized by comprising the following steps of:
s1, centralizing and whitening the observed signal data set to obtain a whitened signal;
s2, solving the third-order accumulation quantity of the whitened signal under different time delays;
s3, stacking a plurality of third-order cumulative tensors into a fourth-order tensor;
s4, performing Tack decomposition on the fourth-order tensor by using singular value decomposition to obtain a low-dimensional kernel tensor;
s5, performing regular multi-component decomposition on the fourth-order tensor by using alternating least square to obtain estimation of a hybrid matrix;
s6, inverting the whitening matrix and multiplying the whitening matrix to obtain an estimated mixing matrix;
s7, performing short-time Fourier transform on the observation signal to obtain an observation signal time-frequency point;
s8, obtaining a time-frequency point of the separated signal by using a matrix diagonalization method;
s9, performing inverse short-time Fourier transform on the time-frequency points of the separated signals to obtain the separated signals.
2. The underdetermined blind source separation method of third-order cumulant and tensor compression according to claim 1, wherein: in the S1, the observed signal is pre-whitened with the size ofIs:
（1）；
wherein ,is a whitening matrix; />Is->A path observation signal matrix; />Is a true mixing matrix; />Is->A source signal matrix; unitary matrix->Is a whitened mixing matrix, herein referred to as a whitened mixing matrix.
3. The underdetermined blind source separation method of third-order cumulant and tensor compression of claim 2, wherein in S2, the third-order cumulant of the zero-mean whitened signal is:
（2）；
wherein ,is time delay; />Is +.>Modulo multiplication; />Is the third order cumulative amount of the source signal;
by varying the delayIs constructed->The three different tensors correspond to the delays, specifically:
（3）。
4. a method according to claim 3An underdetermined blind source separation method of third-order cumulant and tensor compression is characterized in that in S3, tensor is carried outStacked together to form a fourth order tensor->The method comprises the following steps:
（4）；
wherein ,representation vector->Is>An element; /> and />Representation vector->Conjugation +.>；/>An element; />Representation vector->Is>An element;
fourth order tensorThe vector sum matrix can be expressed as:
（5）；
wherein a matrix is defined，/>；/>；/>；/>Is conjugated;is a whitening mixing matrix->Is>Individual column vectors>Is a matrix->Conjugation of->Is->Is>The number of column vectors is a function of,is a matrix->Is>And a column vector.
5. The underdetermined blind source separation method of third-order cumulant and tensor compression of claim 4,the tower decomposition matrix form of (2) is:
（6）；
wherein ,is +.>；/>For nuclear tensor>，/> and />Factor matrices of Take decomposition respectively; />Is a super diagonal tensor; i.e. < ->，/>，/>The rank of (2) is expressed as:
(7)；
record->，/>Then->Is a rank +.>Tensors;
i.e.Is called Tak-1 decomposition, i.e. +.>,/>,/>，/>Is a unitary matrix, so
（8）；
Thereby willTensor->Compression to->Nuclear tensor->；/>Is of the size of；
（9）；
wherein ,is split into principal components, so +.>Is composed of->Before->A matrix of left singular vectors consisting ofSingular value decomposition of (2) to obtain +.>Then, the kernel tensor is obtained from the formula (10)>：
（10）。
6. The underdetermined blind source separation method of third-order cumulant and tensor compression according to claim 5, wherein in S5, the kernel tensor isPerforming regular multi-component decomposition:
（11）；
wherein ,，/> and />Are nuclear tensors->Factor matrix of regular polynary decomposition> and />Is +.>；Is +.>The standard decomposition factor matrix can be solved by alternating least squares iterations>，/> and />From the formulae (6) and (11):
（12）；
namely:
（13）；
so for the kernel tensorAfter regular polynary decomposition, a +.>And two->For->Conjugation is taken and marked as +.>The method comprises the steps of carrying out a first treatment on the surface of the Three matrices are fused->The estimation of the mixing matrix is realized by singular value decomposition, and is as follows:
（14）；
respectively is a matrix->Is>A plurality of column vectors; matrix->Singular value decomposition is carried out, and the formula is as follows:
（15）；
wherein ,is a left singular matrix>Is a singular value matrix, & lt & gt>For right singular matrix, the estimated matrix is, wherein ,/>For matrix->Column 1 on the left.
7. The underdetermined blind source separation method of third-order cumulant and tensor compression according to claim 6, wherein in S6, the whitening matrix is obtained by solving forTo obtain an estimate of the mixing matrix +.>：
（16）。
8. The underdetermined blind source separation method of third-order cumulant and tensor compression according to claim 7, wherein in S8, the mixing matrix corresponding to the source signal is estimated by equation (17) in consideration of the influence of noise, which is:
（17）；
wherein ,for a mixed matrix->Is>A set of sub-matrices, together comprising->An element; />Is made use of->The detection amount of the diagonalization degree of the covariance matrix of the source signal obtained by estimation is as follows:
（18）；
wherein ,is->A matrix of non-diagonal elements; />Is->F-norm of (c); />Inverting the matrix;
the smaller the ∈>The better the diagonalization degree of (c), the more independent the signal.
9. An underdetermined blind source separation system for third-order cumulant and tensor compression, characterized by being implemented by the underdetermined blind source separation method for third-order cumulant and tensor compression as claimed in any one of claims 1 to 8, further comprising:
the data matrix construction module is used for generating a source signal and an observation signal;
the data whitening module is used for constructing a whitening matrix according to the sampling covariance matrix and whitening the observed signal by using the whitening matrix;
the third-order cumulative quantity solving module is used for solving the third-order cumulative quantity under different time delays for the whitened signal;
the tower decomposition module is used for performing tower decomposition on the fourth-order tensor to obtain a compressed kernel tensor;
the regular polynary decomposition module is used for carrying out regular polynary decomposition on the kernel tensor to obtain estimation of the mixing matrix;
the signal recovery module is used for obtaining a separated signal according to a matrix diagonalization method by using the estimated mixed matrix;
and the performance evaluation module is used for evaluating the estimation effect of the mixing matrix by adopting the average relative error and evaluating the separation performance by using the average similarity coefficient.
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