CN113805139A - Broadband signal sparse representation direction-of-arrival estimation method based on focusing transformation - Google Patents

Broadband signal sparse representation direction-of-arrival estimation method based on focusing transformation Download PDF

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CN113805139A
CN113805139A CN202110975454.0A CN202110975454A CN113805139A CN 113805139 A CN113805139 A CN 113805139A CN 202110975454 A CN202110975454 A CN 202110975454A CN 113805139 A CN113805139 A CN 113805139A
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focusing
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赵永红
辛菁
商桂华
尚治龙
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Xian University of Technology
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S3/00—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received
    • G01S3/02—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received using radio waves
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    • G01S3/143—Systems for determining direction or deviation from predetermined direction by vectorial combination of signals derived from differently oriented antennae

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Abstract

The invention discloses a broadband signal sparse representation direction-of-arrival estimation method based on focusing transformation, which comprises the steps of firstly, calculating echo signals of all sub-bands of a receiving array according to the direction-of-arrival from a far-field broadband signal to the receiving array, the array arrangement structure of all array elements in the receiving array and the sub-band division condition of a receiving filter group; selecting reference frequency points, and calculating a focusing matrix of each sub-band; then focusing the array receiving data of each sub-frequency band to a reference frequency and calculating a focused array covariance matrix; performing characteristic decomposition on the focused array covariance matrix, and establishing a broadband signal sparse representation direction-of-arrival estimation model based on weighted subspace fitting; and finally, converting the model into a second-order cone programming form for solving to obtain an estimated value of the direction of arrival of the broadband signal. The invention realizes the joint estimation of each frequency point of the broadband signal under the condition that the information source estimation angle is unknown, and improves the resolution performance, the estimation performance and the calculation efficiency of the algorithm.

Description

Broadband signal sparse representation direction-of-arrival estimation method based on focusing transformation
Technical Field
The invention belongs to the technical field of radars, and particularly relates to a broadband signal sparse representation direction-of-arrival estimation method based on focusing transformation.
Background
The direction-of-arrival estimation technique is an indispensable technical means for realizing target positioning, and the direction information of a target in an airspace can be obtained from the direction-of-arrival estimation result. The traditional typical method comprises a multiple signal classification method and a rotation invariant subspace method, the two methods break through Rayleigh limit and can realize super-resolution of a target, but the performance is seriously reduced under the environment of low signal-to-noise ratio or few snapshots.
The basic idea of sparse representation theory is to replace a set of basis functions with an atom dictionary composed of a set of redundant functions, so that signals can be represented as a linear combination of a few atom column vectors in the atom dictionary. When a small number of point targets are distributed in the airspace, the targets have sparsity in the whole airspace angle, so that the sparse representation theory can be applied to the estimation of the direction of arrival, and the defects of the traditional method are overcome.
When a broadband signal is incident to an array, envelopes received by each array element are different, and a common processing method is to convert data received by each array element into a frequency domain for frequency division processing, so that an array manifold matrix of the broadband signal is not only a function of an incoming wave direction of a signal source but also a function of frequency, different frequencies correspond to different redundant dictionary matrices, and each frequency point contains angle information of the signal source.
Therefore, for the problem of joint processing of multiband points in a broadband signal sparse representation direction-of-arrival estimation method, the existing methods are mainly divided into two categories, one category is a non-correlation processing method, each frequency point independently estimates the direction of arrival of an information source, then the estimated values of all the frequency points are averaged to obtain a final angle estimation result, the joint sparsity factor among the frequency points is not considered, the other category is a correlation processing method, and although the joint sparsity among the frequency points is considered, the calculation of a focusing matrix needs angle pre-estimation values of the direction of incoming waves of the information source.
Disclosure of Invention
The invention aims to provide a broadband signal sparse representation direction-of-arrival estimation method based on focusing transformation, so as to realize joint estimation of each frequency point of a broadband signal under the condition that an information source estimation angle is unknown, and improve the resolution performance, the estimation performance and the calculation efficiency of an algorithm.
The technical scheme adopted by the invention is that the method for estimating the direction of arrival of the broadband signal sparse representation based on the focusing transformation is implemented according to the following steps:
step 1, calculating echo signals of all sub-bands of a receiving array according to the direction of arrival from a far-field broadband signal to the receiving array, the array arrangement structure of all array elements in the receiving array and the sub-band division condition of a receiving filter bank;
step 2, selecting reference frequency points, and calculating a focusing matrix of each sub-band;
step 3, according to the focusing matrix obtained in the step 2, focusing the array receiving data of each sub-frequency band to a reference frequency position and calculating a focused array covariance matrix;
step 4, performing characteristic decomposition on the focused array covariance matrix, and establishing a broadband signal sparse representation direction-of-arrival estimation model based on weighted subspace fitting;
and 5, converting the broadband signal sparse representation direction-of-arrival estimation model based on weighted subspace fitting established in the step 4 into a second-order cone programming form for solving to obtain a broadband signal direction-of-arrival estimation value.
The present invention is also characterized in that,
receiving echo signals y of an array in step 1l(fi) The calculation is as follows:
yl(fi)＝A(fi,θ)sl(fi)+nl(fi),l＝1,…,L,i＝1,…,J，
wherein the content of the first and second substances,
Figure BDA0003227100070000031
indicating the sub-band f on the l-th segment of the received data of the arrayiCorresponding DFT coefficient, M represents array element number,
Figure BDA0003227100070000032
is fiArray manifold matrix corresponding to each frequency point, wherein each column is a target angle thetakCorresponding guide vector a (f)i,θk)，
Figure BDA0003227100070000033
dm,1The distance between the mth array element and the first reference array element is shown, M is 1,2, … M, c is the propagation speed of electromagnetic wave, K is the number of information sources, [ ·]TDenotes a transposition operation, θ ═ θ1,…,θK]Representing the direction of the incoming wave of the source, assuming a target signal vector
Figure BDA0003227100070000034
Obey Gaussian distribution, different sub-bands are independently and identically distributed, and the requirement on the same sub-band is met
Figure BDA0003227100070000035
Figure BDA0003227100070000036
Represents the signal power, δ (l), corresponding to the kth target in the ith sub-band1-l2) Representing an impulse response function, if and only if1＝l2The function value is not zero, the other function values are all 0, E [ ·]Indicating the expected operation, (.)HDenotes a conjugate transpose operation, and diag (·) denotes a diagonal operation. n isl(fi) The complex Gaussian white noise vector is represented and is irrelevant to a target signal, different sub-bands are not relevant, L is the frequency domain fast beat number, and J represents the sub-band division number.
Focusing matrix T (f) of each sub-band in step 2i) The calculation of (a) is specifically as follows:
step 2.1, constructing frequency point fiArray observation data matrix X (f)i)：
X(fi)＝[y1(fi)，y2(fi)，…，yL(fi)]；
Step 2.2, select f0Taking the frequency point as a reference frequency point, and calculating a cross-correlation matrix
Figure BDA0003227100070000037
Sum noise-free covariance matrix
Figure BDA0003227100070000038
Figure BDA0003227100070000039
Figure BDA00032271000700000310
Wherein the content of the first and second substances,
Figure BDA0003227100070000041
is a sample covariance matrix at a reference frequency point, IMIs a matrix of the units,
Figure BDA0003227100070000042
is an estimate of the variance of the noise at the reference frequency point, λjIs a matrix
Figure BDA0003227100070000043
Smaller eigenvalues, K' representing the number of uncorrelated signals;
step 2.3, the cross correlation matrix obtained according to the step 2.2
Figure BDA0003227100070000044
Sum noise-free covariance matrix
Figure BDA0003227100070000045
Calculating a focusing matrix T (f) for each sub-bandi)：
Figure BDA0003227100070000046
Array covariance matrix R after focusing in step 30The calculation of (a) is specifically as follows:
step 3.1, obtaining the focusing matrix T (f) according to step 2i) Calculating the array output matrix X of each focused frequency point0(fi)：
X0(fi)＝T(fi)X(fi)；
Step 3.2, according to the array output matrix X of each frequency point obtained in the step 3.10(fi) Calculating the array covariance matrix R after focusing0：
Figure BDA0003227100070000047
The step 4 is as follows:
step 4.1, array covariance matrix R after focusing0Performing characteristic decomposition to obtain
Figure BDA0003227100070000048
Wherein e ismAnd
Figure BDA0003227100070000049
respectively represent covariance matrices R0Eigenvectors and eigenvalues of, diagonal matrices
Figure BDA00032271000700000410
Consisting of K' large eigenvalues, the corresponding eigenvectors constituting the signal subspace Es0Other feature vectors form a noise subspace E orthogonal to the signal subspacen0，
Figure BDA00032271000700000411
Step 4.2, calculating an optimal weight matrix W of the weighted subspace fitting algorithm:
Figure BDA0003227100070000051
4.3, dividing the whole airspace in an angle dimension according to a sparse representation theory to obtain an angle set theta:
Θ＝{θ1,θ2,…,θn,…,θN}，
whereinN represents the number of airspace angle divisions, θnAn angle representing the nth division, N ═ 1,2, …, N;
step 4.4, obtaining an angle set theta according to the optimal weight matrix W obtained in the step 4.2 and the angle set theta obtained in the step 4.3, and establishing a broadband signal sparse representation direction of arrival estimation model based on weighted subspace fitting:
Figure BDA0003227100070000052
wherein the content of the first and second substances,
Figure BDA0003227100070000053
is a coefficient matrix with dimension N x K', each column has the same sparse structure,
Figure BDA0003227100070000054
bnrepresentation matrix
Figure BDA0003227100070000055
The nth row, | · | | non-conducting phosphor2Expressing two-norm operation, min (·) expressing minimum operation, | · | | | luminanceFRepresenting the Frobenius norm of the matrix,
Figure BDA0003227100070000056
represents the regularization parameter, ζ is the threshold value of the chi-squared distribution,
Figure BDA0003227100070000057
representing a reference frequency point f0The overcomplete basis matrix of (a).
Step 5, solving the second-order cone programming form specifically as follows:
Figure BDA0003227100070000058
b is an optimization parameter, and then the above formula is solved by using an optimization program package CVX to obtain a sparse vector q ═ q1,q2,…,qn,…,qN]And forming an angle spectrum, and performing spectrum peak search on the obtained angle spectrum to obtain a broadband signal direction of arrival estimation result.
Compared with a non-relevant processing method, the method for estimating the direction of arrival of the sparse representation of the broadband signal based on the focusing transformation has the advantages that the estimation performance and the operation efficiency of the algorithm can be effectively improved due to the utilization of the joint sparsity of all frequency points; compared with the existing related processing method, the method has the advantages that mutual information and self-information of data are received by using the arrays among different frequency points, the estimation of the information source angle is avoided, the sparse representation theory is introduced into the information source angle estimation, and the algorithm estimation performance is further improved.
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FIG. 1 is a flow chart of an implementation of the present invention;
FIG. 2 is a graph of the simulation results of spatial spectrum of 3 signals in the spatial domain using the method of the present invention;
FIG. 3 is a graph of simulation results of the RMS error versus fast beat number curves for three targets in the airspace using the prior art method and using the method of the present invention;
FIG. 4 is a graph of simulation results of the variation of the runtime of three objects in the airspace with the number of sub-bands using the prior art method and using the method of the present invention.
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.
The invention discloses a broadband signal sparse representation direction-of-arrival estimation method based on focusing transformation, a flow chart is shown in figure 1, and the method is implemented according to the following steps:
step 1, calculating echo signals of all sub-bands of a receiving array according to the direction of arrival from a far-field broadband signal to the receiving array, the array arrangement structure of all array elements in the receiving array and the sub-band division condition of a receiving filter bank;
receiving echo signals y of an array in step 1l(fi) The calculation is as follows:
yl(fi)＝A(fi,θ)sl(fi)+nl(fi),l＝1,…,L,i＝1,…,J，
wherein the content of the first and second substances,
Figure BDA0003227100070000071
indicating the sub-band f on the l-th segment of the received data of the arrayiCorresponding DFT coefficient, M represents array element number,
Figure BDA0003227100070000072
is fiArray manifold matrix corresponding to each frequency point, wherein each column is a target angle thetakCorresponding guide vector a (f)i,θk)，
Figure BDA0003227100070000073
dm,1The distance between the mth array element and the first reference array element is shown, M is 1,2, … M, c is the propagation speed of electromagnetic wave, K is the number of information sources, [ ·]TDenotes a transposition operation, θ ═ θ1,…,θK]Representing the direction of the incoming wave of the source, assuming a target signal vector
Figure BDA0003227100070000074
Obey Gaussian distribution, different sub-bands are independently and identically distributed, and the requirement on the same sub-band is met
Figure BDA0003227100070000075
Figure BDA0003227100070000076
Represents the signal power, δ (l), corresponding to the kth target in the ith sub-band1-l2) Representing an impulse response function, if and only if1＝l2The function value is not zero, the other function values are all 0, E [ ·]Indicating the expected operation, (.)HDenotes a conjugate transpose operation, and diag (·) denotes a diagonal operation. n isl(fi) The complex Gaussian white noise vector is represented and is irrelevant to a target signal, different sub-bands are not relevant, L is the frequency domain fast beat number, and J represents the sub-band division number.
Step 2, selecting reference frequency points, and calculating a focusing matrix of each sub-band;
focusing matrix T (f) of each sub-band in step 2i) The calculation of (a) is specifically as follows:
step 2.1, constructing frequency point fiArray observation data matrix X (f)i)：
X(fi)＝[y1(fi)，y2(fi)，…，yL(fi)]；
Step 2.2, select f0Taking the frequency point as a reference frequency point, and calculating a cross-correlation matrix
Figure BDA0003227100070000077
Sum noise-free covariance matrix
Figure BDA0003227100070000078
Figure BDA0003227100070000079
Figure BDA00032271000700000710
Wherein the content of the first and second substances,
Figure BDA0003227100070000081
is a sample covariance matrix at a reference frequency point, IMIs a matrix of the units,
Figure BDA0003227100070000082
is an estimate of the variance of the noise at the reference frequency point, λjIs a matrix
Figure BDA0003227100070000083
Smaller eigenvalues, K' representing the number of uncorrelated signals;
step 2.3, the cross correlation matrix obtained according to the step 2.2
Figure BDA0003227100070000084
Sum noise-free covariance matrix
Figure BDA0003227100070000085
Calculating a focusing matrix T (f) for each sub-bandi)：
Figure BDA0003227100070000086
Step 3, according to the focusing matrix obtained in the step 2, focusing the array receiving data of each sub-frequency band to a reference frequency position and calculating a focused array covariance matrix;
array covariance matrix R after focusing in step 30The calculation of (a) is specifically as follows:
step 3.1, obtaining the focusing matrix T (f) according to step 2i) Calculating the array output matrix X of each focused frequency point0(fi)：
X0(fi)＝T(fi)X(fi)；
Step 3.2, according to the array output matrix X of each frequency point obtained in the step 3.10(fi) Calculating the array covariance matrix R after focusing0：
Figure BDA0003227100070000087
Step 4, performing characteristic decomposition on the focused array covariance matrix, and establishing a broadband signal sparse representation direction-of-arrival estimation model based on weighted subspace fitting;
the step 4 is as follows:
step 4.1, array covariance matrix R after focusing0Performing characteristic decomposition to obtain
Figure BDA0003227100070000088
Wherein e ismAnd
Figure BDA0003227100070000089
respectively represent covariance matrices R0Eigenvectors and eigenvalues of, diagonal matrices
Figure BDA0003227100070000091
Consisting of K' large eigenvalues, the corresponding eigenvectors constituting the signal subspace Es0Other feature vectors form a noise subspace E orthogonal to the signal subspacen0，
Figure BDA0003227100070000092
Step 4.2, calculating an optimal weight matrix W of the weighted subspace fitting algorithm:
Figure BDA0003227100070000093
4.3, dividing the whole airspace in an angle dimension according to a sparse representation theory to obtain an angle set theta:
Θ＝{θ1,θ2,…,θn,…,θN}，
wherein N represents the number of airspace angle divisions, thetanAn angle representing the nth division, N ═ 1,2, …, N;
step 4.4, obtaining an angle set theta according to the optimal weight matrix W obtained in the step 4.2 and the angle set theta obtained in the step 4.3, and establishing a broadband signal sparse representation direction of arrival estimation model based on weighted subspace fitting:
Figure BDA0003227100070000094
wherein the content of the first and second substances,
Figure BDA0003227100070000095
is a coefficient matrix with dimension N x K', each column has the same sparse structure,
Figure BDA0003227100070000096
bnrepresentation matrix
Figure BDA0003227100070000097
The nth row, | · | | non-conducting phosphor2Expressing two-norm operation, min (·) expressing minimum operation, | · | | | luminanceFRepresenting the Frobenius norm of the matrix,
Figure BDA0003227100070000098
represents the regularization parameter, ζ is the threshold value of the chi-squared distribution,
Figure BDA0003227100070000099
representing a reference frequency point f0The overcomplete basis matrix of (a).
Step 5, converting the broadband signal sparse representation direction-of-arrival estimation model based on weighted subspace fitting established in the step 4 into a second-order cone planning form solution to obtain a broadband signal direction-of-arrival estimation value, wherein the second-order cone planning form solution in the step 5 is as follows:
Figure BDA0003227100070000101
b is an optimization parameter, and then the above formula is solved by using an optimization program package CVX to obtain a sparse vector q ═ q1,q2,…,qn,…,qN]And forming an angle spectrum, and performing spectrum peak search on the obtained angle spectrum to obtain a broadband signal direction of arrival estimation result.
The estimation performance of the invention on the target angle information can be further verified by the following simulation.
1. Experimental parameters:
the total number M of array elements is 16, the array arrangement mode is a uniform linear array, the beam width is 6.4 degrees, the frequency range of the broadband signal is 80 Hz-120 Hz, and the signal-to-noise ratio is 0 dB.
Experiment one: the broadband signal has an angle of incidence of 0, 3, 14, where the first two signals are correlated with a correlation coefficient of 0.98, and are uncorrelated with the third signal.
Experiment two: the broadband signal had an angle of incidence of-27.6 °, 3.2 °, 40.5 °, the first two signals were correlated with a correlation coefficient of 0.98, both uncorrelated with the third signal, fast beat numbers varied from 100 to 400 at 25 intervals, and monte carlo times of 200.
Experiment three: the incidence angle of the broadband signal is-27.6 degrees, 3.2 degrees and 40.5 degrees, the first two signals are related, the correlation coefficient is 0.98, the two signals are not related to the third signal, the number of sub-bands is changed from 10 to 50 at intervals of 5, and the Monte Carlo number is 200.
2. Content and analysis of experiments
Experiment one:
the method of the invention is used for estimating the direction of arrival of 3 equi-power broadband signals of a far field in space to obtain a space spectrum of angle estimation, as shown in figure 2, wherein 'o' represents the direction of arrival of a real target.
As can be seen from fig. 2, the method of the present invention can successfully resolve the equal power signals from 3 different directions, wherein the angle interval between the first two signals is smaller than the beam width, which indicates that the method of the present invention can achieve super resolution.
Experiment two:
changing the fast beat number, carrying out 200 times of simulation experiments by using the existing method and the method of the invention for each fast beat number, and respectively counting the variation curve of the root mean square error of the existing method and the method of the invention for 3 equal-power broadband signals in the airspace along with the fast beat number, as shown in 3.
As can be seen from FIG. 3, as the number of snapshots increases, the root mean square error of the method of the present invention decreases gradually, and the root mean square error of the method of the present invention at each number of snapshots is smaller than that of the existing method, which indicates that the estimation performance of the method of the present invention is superior to that of the existing method.
Experiment three:
the number of sub-bands is changed, and for each sub-band, 200 simulation experiments are performed by using the existing method and the method of the present invention, and the variation curves of the running time of the existing method and the method of the present invention along with the number of sub-bands are respectively counted, as shown in fig. 4.
As can be seen from fig. 4, the operation time of the method of the present invention is smaller than that of the conventional method for all the number of sub-bands, and the operation time is much smaller than that of the conventional method for most of the number of sub-bands, which indicates that the calculation efficiency of the method of the present invention is better than that of the conventional method.
In conclusion, the method can effectively estimate the direction of arrival of the air-domain broadband signal, and improves the estimation performance and the calculation efficiency of the broadband signal.

Claims (6)

1. The method for estimating the direction of arrival of the sparse representation of the broadband signal based on the focusing transformation is characterized by comprising the following steps:
step 1, calculating echo signals of all sub-bands of a receiving array according to the direction of arrival from a far-field broadband signal to the receiving array, the array arrangement structure of all array elements in the receiving array and the sub-band division condition of a receiving filter bank;
step 2, selecting reference frequency points, and calculating a focusing matrix of each sub-band;
step 3, according to the focusing matrix obtained in the step 2, focusing the array receiving data of each sub-frequency band to a reference frequency position and calculating a focused array covariance matrix;
step 4, performing characteristic decomposition on the focused array covariance matrix, and establishing a broadband signal sparse representation direction-of-arrival estimation model based on weighted subspace fitting;
and 5, converting the broadband signal sparse representation direction-of-arrival estimation model based on weighted subspace fitting established in the step 4 into a second-order cone programming form for solving to obtain a broadband signal direction-of-arrival estimation value.
2. The method for estimating direction of arrival of sparse representation of broadband signals based on focusing transformation as claimed in claim 1, wherein the echo signals y of the receiving array in step 1l(fi) The calculation is as follows:
yl(fi)＝A(fi,θ)sl(fi)+nl(fi),l＝1,…,L,i＝1,…,J，
wherein the content of the first and second substances,
Figure FDA0003227100060000011
indicating the sub-band f on the l-th segment of the received data of the arrayiCorresponding DFT coefficient, M represents array element number,
Figure FDA0003227100060000012
is fiArray manifold matrix corresponding to each frequency point, wherein each column is a target angle thetakCorresponding guide vector a (f)i,θk)，
Figure FDA0003227100060000013
dm,1The distance between the mth array element and the first reference array element is shown, M is 1,2, … M, c is the propagation speed of electromagnetic wave, K is the number of information sources, [ ·]TDenotes a transposition operation, θ ═ θ1,…,θK]Representing the direction of the incoming wave of the source, assuming a target signal vector
Figure FDA0003227100060000021
Obey Gaussian distribution, different sub-bands are independently and identically distributed, and the requirement on the same sub-band is met
Figure FDA0003227100060000022
Figure FDA0003227100060000023
Represents the signal power, δ (l), corresponding to the kth target in the ith sub-band1-l2) Representing an impulse response function, if and only if1＝l2The function value is not zero, the other function values are all 0, E [ ·]Indicating the expected operation, (.)HRepresenting a conjugate transpose operation, diag (·) representing a diagonal operation, nl(fi) The complex Gaussian white noise vector is represented and is irrelevant to a target signal, different sub-bands are not relevant, L is the frequency domain fast beat number, and J represents the sub-band division number.
3. The method according to claim 2, wherein the focusing matrix T (f) of each sub-band in step 2 is a focusing matrix T (f)i) The calculation of (a) is specifically as follows:
step 2.1, constructing frequency point fiArray observation data matrix X (f)i)：
X(fi)＝[y1(fi)，y2(fi)，…，yL(fi)]；
Step 2.2, select f0Taking the frequency point as a reference frequency point, and calculating a cross-correlation matrix
Figure FDA0003227100060000024
Sum noise-free covariance matrix
Figure FDA0003227100060000025
Figure FDA0003227100060000026
Figure FDA0003227100060000027
Wherein the content of the first and second substances,
Figure FDA0003227100060000028
is a sample covariance matrix at a reference frequency point, IMIs a matrix of the units,
Figure FDA0003227100060000029
is an estimate of the variance of the noise at the reference frequency point, λjIs a matrix
Figure FDA00032271000600000210
Smaller eigenvalues, K' representing the number of uncorrelated signals;
step 2.3, the cross correlation matrix obtained according to the step 2.2
Figure FDA00032271000600000211
Sum noise-free covariance matrix
Figure FDA00032271000600000212
Calculating a focusing matrix T (f) for each sub-bandi)：
Figure FDA0003227100060000031
4. The method according to claim 3, wherein the array covariance matrix R after focusing in step 3 is0The calculation of (a) is specifically as follows:
step 3.1, obtaining the focusing matrix T (f) according to step 2i) Calculating the array output matrix X of each focused frequency point0(fi)：
X0(fi)＝T(fi)X(fi)；
Step 3.2, according to the array output matrix X of each frequency point obtained in the step 3.10(fi) Calculating the array covariance matrix R after focusing0：
Figure FDA0003227100060000032
5. The method for estimating direction of arrival of sparse representation of broadband signals based on focusing transformation as claimed in claim 4, wherein said step 4 is specifically as follows:
step 4.1, array covariance matrix R after focusing0Performing characteristic decomposition to obtain
Figure FDA0003227100060000033
Wherein e ismAnd
Figure FDA0003227100060000034
respectively represent covariance matrices R0Eigenvectors and eigenvalues of, diagonal matrices
Figure FDA0003227100060000035
Consisting of K' large eigenvalues, the corresponding eigenvectors constituting the signal subspace Es0Other feature vectors form a noise subspace E orthogonal to the signal subspacen0，
Figure FDA0003227100060000036
Step 4.2, calculating an optimal weight matrix W of the weighted subspace fitting algorithm:
Figure FDA0003227100060000037
4.3, dividing the whole airspace in an angle dimension according to a sparse representation theory to obtain an angle set theta:
Θ＝{θ1,θ2,…,θn,…,θN}，
wherein N represents the number of airspace angle divisions, thetanAn angle representing the nth division, N ═ 1,2, …, N;
step 4.4, obtaining an angle set theta according to the optimal weight matrix W obtained in the step 4.2 and the angle set theta obtained in the step 4.3, and establishing a broadband signal sparse representation direction of arrival estimation model based on weighted subspace fitting:
Figure FDA0003227100060000041
wherein the content of the first and second substances,
Figure FDA0003227100060000042
is a coefficient matrix with dimension N x K', each column has the same sparse structure,
Figure FDA0003227100060000043
bnrepresentation matrix
Figure FDA0003227100060000044
The nth row, | · | | non-conducting phosphor2Expressing two-norm operation, min (·) expressing minimum operation, | · | | | luminanceFRepresenting the Frobenius norm of the matrix,
Figure FDA0003227100060000045
represents the regularization parameter, ζ is the threshold value of the chi-squared distribution,
Figure FDA0003227100060000046
representing a reference frequency point f0The overcomplete basis matrix of (a).
6. The method for estimating the direction of arrival of a sparse representation of a broadband signal based on focusing transformation according to claim 5, wherein the solving in the form of the step 5 second-order cone programming is as follows:
Figure FDA0003227100060000047
b is an optimization parameter, and then the above formula is solved by using an optimization program package CVX to obtain a sparse vector q ═ q1,q2,…,qn,…,qN]And forming an angle spectrum, and performing spectrum peak search on the obtained angle spectrum to obtain a broadband signal direction of arrival estimation result.
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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116626591A (en) * 2023-05-26 2023-08-22 长安大学 Broadband sound source positioning method, system, equipment and medium based on unmanned aerial vehicle harmonic focusing
CN116626591B (en) * 2023-05-26 2024-02-20 长安大学 Broadband sound source positioning method, system, equipment and medium based on unmanned aerial vehicle harmonic focusing

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