A three-dimensional near-field parameter estimation method based on quadratic correlation calculation

Abstract

The present invention relates to a three-dimensional near-field parameter estimation method based on quadratic correlation operation, including: establishing a cross array reception model, which is composed of a uniform linear array located on the X-axis and a uniform linear array located on the Y-axis, and the element located at the coordinate origin is used as a reference element; in a three-dimensional rectangular coordinate system, it is assumed that there are K near-field targets, and the k-th near-field target S k incides on the cross array at the electrical angles {α k , β k} and the distance r k . By obtaining the estimated values of the electrical angles {α k , β k} from the obtained noise-free covariance matrix, and combining with the MUSIC algorithm, a one-dimensional spectral peak search is performed to obtain the estimated value of the distance r k . The computational complexity of this method is significantly reduced, and it has higher parameter estimation accuracy.

Description

Technical Field

[0001] The invention relates to the technical field of near-field signal positioning, and in particular to a three-dimensional near-field parameter estimation method based on quadratic correlation operation. Background Art

[0002] Near-field signal positioning is used in radar, sonar, wireless communication and other fields. As its application becomes more and more extensive, it has received widespread attention. At first, the near-field positioning algorithm was mainly to improve the far-field positioning algorithm and apply it to the near field. For example, the MUSIC algorithm based on the near field is simple to implement, but it involves multi-dimensional search and huge computational complexity. In recent years, many scholars have used some dimensionality reduction techniques to reduce multi-dimensional search to one-dimensional search, thereby improving computational efficiency, but the inherent high complexity of spectral peak search still exists. On the other hand, the near-field parameter estimation method based on quadratic correlation operation makes full use of the time domain information and spatial domain information of the signal, which can improve the estimation accuracy of the parameters, but the current space-time near-field parameter estimation method in three-dimensional scenes has problems such as high computational complexity and parameter pairing caused by multi-dimensional or multiple searches. Summary of the invention

[0003] The technical problem to be solved by the present invention is to provide a three-dimensional near-field parameter estimation method based on quadratic correlation operation, which can obtain automatically paired near-field two-dimensional electrical angle and distance parameters under the premise of a uniform cross array, and the complexity of the entire parameter estimation process is low.

[0004] The technical solution adopted by the present invention is a three-dimensional near-field parameter estimation method based on quadratic correlation operation, which comprises the following steps:

[0005] S1. Establish a cross array receiving model, and establish a three-dimensional rectangular coordinate system in the cross array receiving model. The cross array receiving model consists of a uniform linear array located on the X axis and a uniform linear array located on the Y axis; the uniform linear array on the X axis includes M x The position of one of the array elements on the X-axis on the XOY plane is represented by (l, 0), where l = -m, ..., -1, 0, 1 ..., m. The uniform linear array on the Y-axis includes M y There are array elements, and the position of one of the array elements on the Y axis on the XOY plane is represented as (0, l), where l = -m, ..., -1, 0, 1 ..., m. The array element at the origin of the coordinate system is used as the reference array element, and the spacing between array elements is set to d;

[0006] S2. In a three-dimensional rectangular coordinate system, assume that there are K near-field targets, and the kth near-field target S k The electrical angle {α k ,βk} and distance r k The incident data is incident on the cross array, where k = 1, 2, ..., K, and the received data model of the array element at (l, 0) on the X-axis is: Among them, s k (t) represents the incident signal of the kth near-field target, n l,0 (t) means the mean is 0 and the variance is σ 2 Independent additive Gaussian noise, τ xl (k) represents the propagation delay between the incident signal of the kth near-field target in space and the reference array element and any array element, τ xl (k)≈ω k l+φ k l 2 , λ represents the signal wavelength, and the received data model of the array element at (0, l) on the Y axis is: Among them, s k (t) represents the incident signal of the kth near-field target, n 0,l (t) means the mean is 0 and the variance is σ 2 Independent additive Gaussian noise, τ yl (k) represents the propagation delay between the incident signal of the kth near-field target in space and the reference array element and any array element, τ yl (k)≈ω k l+φ k l 2 , λ is the wavelength of the incident signal;

[0007] S3, the receiving data model of the array element located at the (l,0)th position on the X-axis obtained in step S2 is arranged into a matrix form after T sampling, that is: Z x =A x S+N x , where Z x represents the received data of the array element at (l,0) on the X-axis, A x represents the flow matrix of the array element at (l,0) on the X-axis, S represents the incident signal of the near-field target, N x represents additive noise, The received data model of the array element at the (0, l)th position on the Y axis obtained in step S2 is arranged into a matrix form after T sampling, that is: Z y =A y S+N y , where Z y represents the received data of the array element at (l,0) on the Y axis, A y represents the manifold matrix of the array element at (l,0), S represents the incident signal of the near-field target, N y represents additive noise,

[0008] S4, correlating the received data of the two matrix array elements obtained in step S3 with the reference array element as the symmetry center, to obtain two sets of correlated virtual received data;

[0009] S5. According to the received data of the two matrix-form array elements obtained in step S3, the spatial domain information and the time domain information of the received data of the two matrix-form array elements are obtained. According to the obtained spatial domain information and the time domain information, the two correlated sets of virtual received data obtained in step S4 are correlated to obtain a final set of virtual received data. The noise-free covariance matrix is ​​restored for the final set of virtual received data using the covariance matching criterion. The electrical angle {α k ,β k};

[0010] S6, the received data of the two matrix elements obtained in step S3 are cascaded by column, and a one-dimensional spectrum peak search is performed in combination with the MUSIC algorithm to obtain the distance r k The estimated value of .

[0011] The beneficial effects of the present invention are: by using the above-mentioned three-dimensional near-field parameter estimation method based on quadratic correlation operation, the method can effectively reduce the complexity of establishing a cross array receiving model, and under the premise of reducing the complexity, the position parameters of the signal can still be accurately estimated, and the parameters can be automatically paired. At the same time, the second-order statistics are cleverly used in the process of estimating the parameters, so that the calculation complexity is significantly reduced; compared with the existing algorithms, the present invention has higher parameter estimation accuracy and very low complexity, and this conclusion has been proved by simulation experiments.

[0012] Preferably, in step S4, the expression of one set of virtual received data in the two sets of correlated virtual received data is: Another set of expressions for virtual receiving data is: Where E(·) represents the statistical expectation, τ represents the signal delay, (·) * represents the complex conjugate, represents the noise power, δ(·) represents the impulse function,

[0013] Preferably, in step S5, the two sets of correlated virtual received data obtained in step S4 are correlated according to the acquired spatial domain information and time domain information, and the specific process of obtaining a final set of virtual received data includes the following steps:

[0014] S5.01, convert the two sets of virtual received data obtained in step S4 into vector form respectively, and the expression is: in,(·) T Represents a transpose operation;

[0015] S5.02, the r obtained in step S5.01 x (τ) and r y (τ) is sampled at uniform time intervals, then the two sets of virtual received data obtained after N pseudo-snapshot samplings are expressed as: Arrange the two sets of virtual acceptance data into a matrix form, namely: in,

[0016] represents the virtual steering vector, represents the virtual signal on the X-axis, represents additive noise, represents the virtual guide vector on the Y axis, represents the virtual signal on the Y axis, represents additive noise;

[0017] S5.03, the matrix form R obtained in step S5.02 x and R y Do the correlation and get the final set of virtual receiving data as follows: in, represents the covariance matrix of the virtual incident signal, Represents the covariance matrix of the noise.

[0018] Preferably, in step S5, the noise-free covariance matrix is ​​restored for the final set of virtual received data using a covariance matching criterion, and the electrical angle {α k ,β k The specific process of estimating the value of} includes the following steps:

[0019] S5.11, the R obtained in step S5.03 xy Vectorizing it, we get: in, ⊙ represents the Khatri-Rao product,

[0020] S5.12. According to the vectorized R xy , the noise-free covariance matrix R is reconstructed using an optimization problem, the optimization problem is: Solve the optimization problem through the optimization toolbox CVX and get the optimal solution In getting the optimized solution After that, the remaining task is to find the parameter ω x and ω y ;

[0021] S5.13. According to in, is the covariance matrix of the signal. Based on the characteristics of the secondary Toeplitz structure of the covariance matrix R, the covariance matrix R is expressed as: Where, the signal power p k >0.;

[0022] S5.14, according to the covariance matrix R obtained in step S5.13, use the Mapp algorithm to convert the optimization solution obtained in step S5.12 Decompose to obtain parameters and According to the obtained parameters and Get the electrical angle and Its expression is:

[0023] Preferably, in step S6, the distance r is obtained. k The specific process of estimating the value includes the following steps:

[0024] S6.1. The electrical angle obtained in step S5 is and Substitute them into the cross array receiving model obtained in step S3 respectively In the equation, we get an unknown parameter r = [r 1 ,...,r K ] T The cross array receiving model is expressed as:

[0025] S6.2, according to the cross array receiving model with an unknown parameter obtained in step S6.1, the maximum likelihood estimate of the covariance matrix of the received signal is obtained as follows:

[0026] S6.3, perform eigendecomposition on the maximum likelihood estimate of the covariance matrix obtained in step S6.2 to obtain eigenvectors and eigenvalues, sort the eigenvalues ​​from large to small, and the first K eigenvalues ​​are large eigenvalues. The eigenvectors corresponding to the large eigenvalues ​​constitute the signal subspace U S , after (M x +M y )-K eigenvalues ​​are small eigenvalues, and the eigenvectors corresponding to the small eigenvalues ​​constitute the noise subspace U N ,get: Among them, Σ S and Σ N Respectively represent the diagonal matrix composed of large eigenvalues ​​and small eigenvalues; then the maximum likelihood estimate of the covariance matrix is ​​expressed as: According to the orthogonality principle of noise subspace and signal subspace, we can get:

[0027] S6.4. Since the matrix R S is a full rank matrix, And the matrix The column vectors in are orthogonal to the noise subspace, resulting in: According to the orthogonal relationship between the noise subspace and the signal vector, the spatial spectrum function of the array is obtained: Let r vary within the Fresnel zone, and estimate the distance r by finding the peak based on the obtained spatial spectrum function of the array. k . BRIEF DESCRIPTION OF THE DRAWINGS

[0028] Figure 1 A schematic diagram of a cross array receiving model established in the present invention;

[0029] Figure 2 A two-dimensional electric angle estimation diagram obtained by using the method of the present invention in an example of an embodiment of the present invention;

[0030] Figure 3 This is a distance estimation graph obtained by using the method of the present invention in an example of an embodiment of the present invention. DETAILED DESCRIPTION

[0031] The invention will be further described below with reference to the accompanying drawings and in combination with specific implementations, so that those skilled in the art can implement the invention with reference to the description. The protection scope of the invention is not limited to the specific implementations.

[0032] The present invention relates to a three-dimensional near-field parameter estimation method based on quadratic correlation operation, the method comprising the following steps:

[0033] S1. Establish a cross array receiving model, and establish a three-dimensional rectangular coordinate system in the cross array receiving model. The cross array receiving model consists of a uniform linear array located on the X axis and a uniform linear array located on the Y axis; the uniform linear array on the X axis includes M x The position of one of the array elements on the X-axis on the XOY plane is represented by (l, 0), where l = -m, ..., -1, 0, 1 ..., m. The uniform linear array on the Y-axis includes M y There are array elements, and the position of one of the array elements on the Y axis on the XOY plane is represented as (0, l), where l = -m, ..., -1, 0, 1 ..., m. The array element at the origin of the coordinate system is used as the reference array element, and the spacing between array elements is set to d;

[0034] S2. In a three-dimensional rectangular coordinate system, assume that there are K near-field targets, and the kth near-field target S k The electrical angle {α k ,β k} and distance r k The incident data is incident on the cross array, where k = 1, 2, ..., K, and the received data model of the array element at (l, 0) on the X-axis is: Among them, s k (t) represents the incident signal of the kth near-field target, n l,0 (t) means the mean is 0 and the variance is σ 2 Independent additive Gaussian noise, τ xl (k) represents the propagation delay between the incident signal of the kth near-field target in space and the reference array element and any array element, τ yl (k)≈ω k l+φ k l 2 , λ represents the signal wavelength, and the received data model of the array element at (0, l) on the Y axis is: Among them, s k (t) represents the incident signal of the kth near-field target, n 0,l (t) means the mean is 0 and the variance is σ 2 Independent additive Gaussian noise, τ yl (k) represents the propagation delay between the incident signal of the kth near-field target in space and the reference array element and any array element, τ yl (k)≈ω k l+φ k l 2 , λ is the wavelength of the incident signal;

[0035] S3, the receiving data model of the array element located at the (l,0)th position on the X-axis obtained in step S2 is arranged into a matrix form after T sampling, that is: Z x =A x S+N x , where Z x represents the received data of the array element at (l,0) on the X-axis, A x represents the flow matrix of the array element at (l,0) on the X-axis, S represents the incident signal of the near-field target, N x represents additive noise, The received data model of the array element at the (0, l)th position on the Y axis obtained in step S2 is arranged into a matrix form after T sampling, that is: Z y =A y S+N y , where Z y represents the received data of the array element at (l,0) on the Y axis, A y represents the manifold matrix of the array element at (l,0), S represents the incident signal of the near-field target, N y represents additive noise,

[0036] S4, correlating the received data of the two matrix array elements obtained in step S3 with the reference array element as the symmetry center, to obtain two sets of correlated virtual received data;

[0037] S5. According to the received data of the two matrix-form array elements obtained in step S3, the spatial domain information and the time domain information of the received data of the two matrix-form array elements are obtained. According to the obtained spatial domain information and the time domain information, the two correlated sets of virtual received data obtained in step S4 are correlated to obtain a final set of virtual received data. The noise-free covariance matrix is ​​restored for the final set of virtual received data using the covariance matching criterion. The electrical angle {α k ,β k};

[0038] S6, the received data of the two matrix elements obtained in step S3 are cascaded by column, and a one-dimensional spectrum peak search is performed in combination with the MUSIC algorithm to obtain the distance r k The estimated value of .

[0039] Preferably, in step S4, the expression of one set of virtual received data in the two sets of correlated virtual received data is: Another set of expressions for virtual receiving data is: Where E(·) represents the statistical expectation, τ represents the signal delay, (·) * represents the complex conjugate, represents the noise power, δ(·) represents the impulse function,

[0040] Preferably, in step S5, the two sets of correlated virtual received data obtained in step S4 are correlated according to the acquired spatial domain information and time domain information, and the specific process of obtaining a final set of virtual received data includes the following steps:

[0041] S5.01, convert the two sets of virtual received data obtained in step S4 into vector form respectively, and the expression is: in,(·) T Represents a transpose operation;

[0042] S5.02, the r obtained in step S5.01 x (τ) and r y (τ) is sampled at uniform time intervals, then the two sets of virtual received data obtained after N pseudo-snapshot samplings are expressed as: Similar to the matrix form of the array receiving data obtained in step S3, the two sets of virtual receiving data are arranged in a matrix form, namely: in, represents the virtual steering vector, which only contains the electrical angle α k Function ω xk related, represents the virtual signal on the X-axis, represents additive noise, which has only one row with value and the rest are zero;

[0043] S5.03, the matrix form R obtained in step S5.02 x and R y Do the correlation and get the final set of virtual receiving data as follows: in, represents the covariance matrix of the virtual incident signal, Represents the covariance matrix of the noise, which has only one value at the symmetric center of the matrix.

[0044] Preferably, in step S5, the noise-free covariance matrix is ​​restored for the final set of virtual received data using a covariance matching criterion, and the electrical angle {α k ,β k The specific process of estimating the value of} includes the following steps:

[0045] S5.11, the R obtained in step S5.03 xy Vectorizing it, we get: in, ⊙ represents the Khatri-Rao product, It can be seen that r xy It can be regarded as a virtual output in a far-field scenario;

[0046] S5.12. According to the vectorized R xy , the noise-free covariance matrix R is reconstructed using an optimization problem, the optimization problem is: Solve the optimization problem through the optimization toolbox CVX and get the optimal solution In getting the optimized solution After that, the remaining task is to find the parameter ω x and ω y ;

[0047] S5.13. According to in, is the covariance matrix of the signal. Based on the characteristics of the secondary Toeplitz structure of the covariance matrix R, the covariance matrix R is expressed as: Where, the signal power p k >0.;

[0048] S5.14, combined with the relationship in S5.13, according to the Mapp algorithm, the optimized solution obtained in S5.12 Decompose to obtain parameters and Get parameters and Then, the electrical angle can be obtained and Its expression is:

[0049] Preferably, in step S6, the distance r is obtained. k The specific process of estimating the value includes the following steps:

[0050] S6.1. The electrical angle obtained in step S5 is and Substitute them into the cross array receiving model obtained in step S3 respectively In the equation, we get an unknown parameter r = [r 1 ,...,r K ] T The cross array receiving model is expressed as:

[0051] S6.2, according to the cross array receiving model with an unknown parameter obtained in step S6.1, the maximum likelihood estimate of the covariance matrix of the received signal is obtained as follows:

[0052] S6.3, perform eigendecomposition on the maximum likelihood estimate of the covariance matrix obtained in step S6.2 to obtain eigenvectors and eigenvalues, sort the eigenvalues ​​from large to small, and the first K eigenvalues ​​are large eigenvalues. The eigenvectors corresponding to the large eigenvalues ​​constitute the signal subspace U s , after (M x +M y )-K eigenvalues ​​are small eigenvalues, and the eigenvectors corresponding to the small eigenvalues ​​constitute the noise subspace U N ,get: Among them, Σ S and Σ N Respectively represent the diagonal matrix composed of large eigenvalues ​​and small eigenvalues; According to the relevant knowledge of spatial spectrum estimation theory, the maximum likelihood estimate of the covariance matrix can be expressed as: Combined with the above analysis, multiply both sides of the equation by the noise subspace U N : And further according to the orthogonality principle of noise subspace and signal subspace, we can get:

[0053] S6.4. Since the matrix R S is a full rank matrix, And the matrix The column vectors in are orthogonal to the noise subspace, and we get: According to the orthogonal relationship between the noise subspace and the signal vector, the spatial spectrum function of the array is obtained: Let r vary within the Fresnel zone and estimate the distance r by finding the peak according to the spatial spectrum function of the array. k .

[0054] The effectiveness of the near-field polarization MIMO radar parameter estimation method based on an accurate model proposed in the present invention is demonstrated by the following examples:

[0055] Assume that there are four independent near-field signals incident on the single-sided array element M at four directions: {126°, 24°, 0.22λ}, {75°, 39°, 0.27λ}, {34°, 86°, 0.32λ} and {141°, 130°, 0.41λ}. x =1,M y = 2, that is, the total number of array elements in the two uniform linear arrays is 7. The signal-to-noise ratio and the number of snapshots are set to 30 dB and 2000 respectively, and pseudo snapshots are used. The results are as follows Figure 1 and Figure 2 As shown. Figure 1 and Figure 2 We can see that the estimated parameters of the four near-field sources can be correctly estimated and paired; therefore, the method proposed in the present invention is effective.

Claims

1. A three-dimensional near-field parameter estimation method based on quadratic correlation operation, Features: The method comprises the following steps: S1. Establish a cross array receiving model, and establish a three-dimensional rectangular coordinate system in the cross array receiving model. The cross array receiving model consists of a uniform linear array located on the X axis and a uniform linear array located on the Y axis; the uniform linear array on the X axis includes M x The position of one of the array elements on the X-axis on the XOY plane is represented by (l, 0), where l = -m, ..., -1, 0, 1 ..., m. The uniform linear array on the Y-axis includes M y There are array elements, and the position of one of the array elements on the Y axis on the XOY plane is represented as (0, l), where l = -m, ..., -1, 0, 1 ..., m. The array element at the origin of the coordinate system is used as the reference array element, and the spacing between array elements is set to d; S2. In a three-dimensional rectangular coordinate system, assume that there are K near-field targets, and the kth near-field target S k The electrical angle {α k ,β k } and distance r k The incident data is incident on the cross array, where k = 1, 2, ..., K, and the received data model of the array element at (l, 0) on the X-axis is: Among them, s k (t) represents the incident signal of the kth near-field target, n l,0 (t) means the mean is 0 and the variance is σ 2 Independent additive Gaussian noise, τ xl (k) represents the propagation delay between the incident signal of the kth near-field target in space and the reference array element and any array element, τ xl (k)≈ω k l+φ k l 2 , λ represents the signal wavelength, and the received data model of the array element at (0, l) on the Y axis is: Among them, s k (t) represents the incident signal of the kth near-field target, n 0,l (t) means the mean is 0 and the variance is σ 2 Independent additive Gaussian noise, τ yl (k) represents the propagation delay between the incident signal of the kth near-field target in space and the reference array element and any array element, τ yl (k)≈ω k l+φ k l 2 , λ is the wavelength of the incident signal; S3, the receiving data model of the array element located at the (l,0)th position on the X-axis obtained in step S2 is arranged into a matrix form after T sampling, that is: Z x =A x S+N x , where Z x represents the received data of the array element at (l,0) on the X-axis, A x represents the flow matrix of the array element at (l,0) on the X-axis, S represents the incident signal of the near-field target, N x represents additive noise, The received data model of the array element at the (0, l)th position on the Y axis obtained in step S2 is arranged into a matrix form after T sampling, that is: Z y =A y S+N y , where Z y represents the received data of the array element at (l,0) on the Y axis, A y represents the flow matrix of the array element at (l,0) on the Y axis, S represents the incident signal of the near-field target, N y represents additive noise, S4, correlating the received data of the two matrix array elements obtained in step S3 with the reference array element as the symmetry center, to obtain two sets of correlated virtual received data; S5. According to the received data of the two matrix-form array elements obtained in step S3, the spatial domain information and the time domain information of the received data of the two matrix-form array elements are obtained. According to the obtained spatial domain information and the time domain information, the two correlated sets of virtual received data obtained in step S4 are correlated to obtain a final set of virtual received data. The noise-free covariance matrix is ​​restored for the final set of virtual received data using the covariance matching criterion. The electrical angle {α k ,β k }'s estimated value; S6, the received data of the two matrix elements obtained in step S3 are cascaded by column, and a one-dimensional spectrum peak search is performed in combination with the MUSIC algorithm to obtain the distance r k The estimated value of .

2. A three-dimensional near-field parameter estimation method based on quadratic correlation operation according to claim 1, Features: In step S4, the expression of one set of virtual received data in the two sets of correlated virtual received data is: Another set of expressions for virtual receiving data is: Where E(·) represents the statistical expectation, τ represents the signal delay, (·) * represents the complex conjugate, represents the noise power, δ(·) represents the impulse function, 3. The three-dimensional near-field parameter estimation method based on quadratic correlation operation according to claim 2, Features: In step S5, the two sets of correlated virtual received data obtained in step S4 are correlated according to the acquired spatial domain information and temporal domain information. The specific process of obtaining a final set of virtual received data includes the following steps: S5.01, convert the two sets of virtual received data obtained in step S4 into vector form respectively, and the expression is: in,(·) T Represents a transpose operation; S5.02, the r obtained in step S5.01 x (τ) and r y (τ) is sampled at uniform time intervals, then the two sets of virtual received data obtained after N pseudo-snapshot samplings are expressed as: Arrange the two sets of virtual acceptance data into a matrix form, namely: in, represents the guidance vector on the virtual X-axis, represents the virtual signal on the X-axis, represents additive noise, represents the virtual guide vector on the Y axis, represents the virtual signal on the Y axis, represents additive noise; S5.03, the matrix form R obtained in step S5.02 x and R y Do the correlation and get the final set of virtual receiving data as follows: in, represents the covariance matrix of the virtual incident signal, Represents the covariance matrix of the noise.

4. The three-dimensional near-field parameter estimation method based on quadratic correlation operation according to claim 3, Features: In step S5, the noise-free covariance matrix is ​​restored for the final set of virtual received data using the covariance matching criterion, and the electrical angle {α k ,β k The specific process of estimating the value of} includes the following steps: S5.11, the R obtained in step S5.03 xy Vectorizing it, we get: in, ⊙ represents the Khatri-Rao product, S5.

12. According to the vectorized R xy , the noise-free covariance matrix R is reconstructed using an optimization problem, and the optimization is: Solve the optimization problem through the optimization toolbox CVX and get the optimal solution In getting the optimized solution After that, the remaining task is to find the parameter ω x and ω y ; S5.

13. According to in, is the covariance matrix of the signal. Based on the characteristics of the secondary Toeplitz structure of the covariance matrix R, the covariance matrix R is expressed as: Where, the signal power p k >0.; S5.14, according to the covariance matrix R obtained in step S5.13, use the Mapp algorithm to convert the optimization solution obtained in step S5.12 Decompose to obtain parameters and According to the obtained parameters and Get the electrical angle and Its expression is:

5. The three-dimensional near-field parameter estimation method based on quadratic correlation operation according to claim 4, Features: In step S6, the distance r is obtained. k The specific process of estimating the value includes the following steps: S6.

1. The electrical angle obtained in step S5 is and Substitute them into the cross array receiving model obtained in step S3 respectively In the equation, we get an unknown parameter r = [r 1 ,...,r K ] T The cross array receiving model is expressed as: S6.2, according to the cross array receiving model with an unknown parameter obtained in step S6.1, the maximum likelihood estimate of the covariance matrix of the received signal is obtained as follows: S6.3, perform eigendecomposition on the maximum likelihood estimate of the covariance matrix obtained in step S6.2 to obtain eigenvectors and eigenvalues, sort the eigenvalues ​​from large to small, and the first K eigenvalues ​​are large eigenvalues. The eigenvectors corresponding to the large eigenvalues ​​constitute the signal subspace U S , after (M x +M y )-K eigenvalues ​​are small eigenvalues, and the eigenvectors corresponding to the small eigenvalues ​​constitute the noise subspace U N ,get: Among them, Σ S and Σ N Respectively represent the diagonal matrix composed of large eigenvalues ​​and small eigenvalues; then the maximum likelihood estimate of the covariance matrix is ​​expressed as: According to the orthogonality principle of noise subspace and signal subspace, we can get: S6.

4. Since the matrix R S is a full rank matrix, And the matrix The column vectors in are orthogonal to the noise subspace, resulting in: According to the orthogonal relationship between the noise subspace and the signal vector, the spatial spectrum function of the array is obtained: Let r vary within the Fresnel zone, and estimate the distance r by finding the peak based on the obtained spatial spectrum function of the array. k .

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