A direct array self-positioning method based on particle optimization weighted signal subspace fitting

Abstract

The present invention discloses a method for array direct self-positioning using particle-optimized weighted signal subspace fitting. The method comprises: distributing signal sources and an array for receiving signal sources in a positioning area; acquiring multi-source signals using an array data acquisition model; uniformly selecting grid points in the positioning area and constructing a characteristic matrix of all signal sources at each grid point; calculating a covariance matrix for the acquired data, performing eigenvalue decomposition to obtain a signal subspace, calculating a weight matrix required for weighted signal subspace fitting, fitting a weighted signal subspace function value based on the characteristic matrix, and selecting the point with the maximum function value as an initial estimate of the circular array position; randomly selecting particles near the initial estimated position, weighting each particle position, normalizing the weighted evaluation values, and calculating a weighted average sum of the weights over the particle distribution positions to obtain a circular array position estimate. Compared with traditional methods, the positioning estimate of the present invention minimizes the error variance.

Description

Technical Field

[0001] The present invention belongs to the technical field of passive positioning, and in particular relates to an array direct self-positioning method of particle optimization weighted signal subspace fitting. Background Art

[0002] Positioning technology is a key research area in signal processing, with widespread applications across various industrial sectors. In modern society, determining target location is the foundation of intelligent sensing technology. Precise positioning using wireless sensor nodes has become the core of many technologies, making positioning technology a hot research topic in both academia and industry. After years of development, positioning algorithms have been widely applied in an increasing number of scenarios. However, technological advancements have brought about the emergence of new application scenarios, and these changes have placed higher demands on positioning technology.

[0003] Self-positioning technology has been a topic of widespread concern in recent years. Traditional self-positioning technology uses two-step positioning technologies such as RSS, TDOA, and AOA for auxiliary positioning, and achieves real-time positioning of the array through the least squares positioning algorithm. However, from the perspective of information theory, the two-step positioning method only utilizes the angle information in the data, and therefore the results of the two-step positioning method suffer from information loss.

[0004] The particle filter algorithm is a widely applicable error correction method that works well for both linear and nonlinear systems, making it widely used to correct sensor errors. In direct positioning, the quantization of space by the grid also contributes to perceptual errors, so the particle filter algorithm can effectively reduce this grid quantization error. Therefore, research on array direct positioning algorithms that can use particle optimization to weighted signal subspace fitting is of great significance. Summary of the Invention

[0005] In response to the deficiencies in the prior art, the present invention provides an array direct self-positioning method based on particle optimization weighted signal subspace fitting. Based on the weighted signal subspace fitting method, particle filtering is used to optimize the positioning results to achieve accurate estimation of the source position.

[0006] To achieve the above object, the present invention adopts the following technical solution: a method for direct array self-positioning based on particle optimization weighted signal subspace fitting, which specifically includes the following steps:

[0007] Step 1: Distribute L signal sources and an array for receiving signal sources in the positioning area; the array is a circular array consisting of M array elements;

[0008] Step 2: Collect data from multiple signal sources using a circular array data acquisition model;

[0009] Step 3: Uniformly select grid points in the positioning area, and construct the characteristic matrix of all signal sources at each grid point based on the model of circular array data acquisition;

[0010] Step 4: Calculate the covariance matrix of the collected data and perform eigenvalue decomposition to obtain the signal subspace, noise subspace and diagonal matrix, and calculate the weight matrix required for weighted signal subspace fitting;

[0011] Step 5: Fit the weighted signal subspace function value using the weight matrix constructed at each grid point, the signal subspace, and the characteristic matrix of all signal sources. Compare the weighted signal subspace function values ​​at each grid point and select the point with the largest function value as the initial estimate of the circular array position.

[0012] Step 6: Randomly select particles near the initial estimated position, and the particle distribution satisfies the mean 0 and variance σ 2 The normal distribution relationship of

[0013] Step 7: Perform weight evaluation on each particle position using the particle weighted signal subspace function;

[0014] Step 8: Normalize the weight evaluation value, calculate the weighted average sum of the weight and particle distribution position, and obtain the circular array position estimation result.

[0015] Furthermore, the process of using the circular array to collect data from multiple signal sources in step 2 is as follows:

[0016] X=AS+N

[0017] Where X is the signal collected by the circular array, S is the source signal vector, N is the noise signal vector, A is the array flow pattern of the circular array, A=[a1,…,a l ,…,a L ], a l is the steering vector formed by the lth signal source incident on the array azimuth angle, λ is the wavelength of the signal transmitted by the source, r is the radius of the circular array, β l is the elevation angle of the lth signal source measured by the circular array, γ m-1 is the angle between the mth array element and the reference array element relative to the center of the circular array, γ m-1 =2π(m-1) / M, j is the imaginary unit, θ l It represents the incident azimuth angle of the lth radiation source measured by the circular array.

[0018] Furthermore, the characteristic matrix of all sources constructed at each grid point in step 3 is:

[0019] Φ i =[φ 1,i ,…,φ l,i ,…,φL,i ]

[0020]

[0021] Among them, Φ i The characteristic matrix of all sources constructed for the i-th grid point, φ l,i is the characteristic matrix of the lth source constructed in the i-th grid point, λ is the wavelength of the signal transmitted by the source, r is the radius of the circular array, β l is the elevation angle of the lth signal source measured by the circular array, η l,i is the angle between the lth source at the i-th grid point and the north direction, is the angle between the reference element in the circular array and the true north direction, γ m-1 is the angle between the mth array element and the reference array element relative to the center of the circular array, γ m-1 =2π(m-1) / M, j is the imaginary unit, η l,i is the angle between the lth source at the i-th grid point and the north direction, (x i ,y i ) is the position coordinate of the i-th grid point, (p lx ,p ly ) is the position coordinate of the lth source.

[0022] Furthermore, step 4 includes the following sub-steps:

[0023] Step 4.1: Calculate the covariance matrix of the collected data and perform eigenvalue decomposition on the covariance matrix to obtain the signal subspace, noise subspace and diagonal matrix:

[0024]

[0025] Among them, R is the covariance matrix of the collected data, E() is the mathematical expectation, and D S is an L×L dimensional diagonal matrix, D S The diagonal elements of D are composed of the largest L eigenvalues ​​obtained by eigenvalue decomposition; N is a diagonal matrix consisting of ML smallest eigenvalues; U S is the signal subspace, which is composed of the eigenvectors corresponding to the L largest eigenvalues; U N is the noise subspace, which is composed of the eigenvectors corresponding to the ML smallest eigenvalues;

[0026] Step 4.2: Use the diagonal matrix D consisting of the largest L eigenvalues S And the diagonal matrix composed of ML smallest eigenvalues ​​is used to calculate the weight matrix required for weighted signal subspace fitting Where I is the identity matrix, σ 2 is the noise power, D N The mean of the diagonal elements.

[0027] Furthermore, the fitting process of the weighted signal subspace function value in step 5 is:

[0028]

[0029] Among them, f(P i ) is the weighted signal subspace function value of the i-th grid point, tr() is the trace operation of the matrix, I is the identity matrix, U S is the signal subspace, W is the weight matrix, Φ i The characteristic matrix of all sources constructed for the i-th grid point.

[0030] Furthermore, the positions of the J particles randomly selected in step 6 are Pa j =[Pax j ,Pay j ] T , and the position distribution of the particles satisfies the following relationship:

[0031]

[0032]

[0033] Among them, ||||2 is the two-norm operation, N(0,σ 2 ) means that the mean is 0 and the variance is σ 2 The normal distribution relationship, is the initial estimate of the position of the circle.

[0034] Furthermore, the particle weighted signal subspace function in step 7 is:

[0035]

[0036] Among them, C(Pa j ) is the particle Pa j The corresponding weight evaluation value, U S is the signal subspace, W is the weight matrix, Φ′ j The characteristic matrix of all sources constructed for the jth particle, Φ′ j =[φ′ 1,j ,…,φ′ l,j ,…,φ′ L,j ],φ′ l,j is the construction matrix of the lth information source relative to the jth particle:

[0037]

[0038] in, is the angle between the reference element in the circular array and the true north direction, λ is the wavelength of the signal transmitted by the source, r is the radius of the circular array, β l is the elevation angle of the lth signal source measured by the circular array, j is the imaginary unit, γ m-1 is the angle between the mth array element and the reference array element relative to the center of the circular array, γ m-1 =2π(m-1) / M,η′ l,j is the angle between the lth source and the true north direction at the jth particle position, (p lx ,p ly ) is the position coordinate of the lth source.

[0039] Furthermore, the normalization process of the weights in step 8 is:

[0040]

[0041] Among them, C'(Pa j ) is the normalized weight evaluation value of the j-th particle.

[0042] Furthermore, the circular array position estimation result for:

[0043]

[0044] Compared with the prior art, the present invention has the following beneficial effects: the present invention performs eigenvalue decomposition on the array received signal covariance matrix to obtain a signal subspace, a noise subspace and a corresponding eigenvalue diagonal matrix, and uses a weight matrix to make up for the gap between the theoretical value and the actual value of the signal subspace; and fits the position information of the signal subspace through the characteristic matrix of the signal source. Compared with the multiple parameter estimation calculations of the traditional method, the method of the present invention reduces the cumulative error caused by the multiple parameter calculations, and can effectively improve the accuracy of self-positioning; at the same time, the present invention uses a particle filtering method to optimize the self-positioning result, randomly selects surrounding particle points as samples for the initial estimated position, calculates the particle cost function as an approximation of the probability density function, and uses the mean value instead of the integral operation to minimize the error variance of the positioning estimate value, thereby reducing the grid quantization error. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 It is a flow chart of the array direct self-positioning method of particle optimized weighted signal subspace fitting of the present invention;

[0046] Figure 2 It is a scene diagram of the array direct self-positioning method of particle optimization weighted signal subspace fitting of the present invention;

[0047] Figure 3This is a comparison chart of the simulation results of the array direct self-positioning method based on particle optimization weighted signal subspace fitting of the present invention and the WSSF method;

[0048] Figure 4 This is a comparison chart of array position estimation performance under different signal-to-noise ratios of the direct self-positioning method of the particle optimization weighted signal subspace fitting of the present invention. DETAILED DESCRIPTION

[0049] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings.

[0050] like Figure 1 This is a flow chart of the array direct self-positioning method for particle optimization weighted signal subspace fitting of the present invention. The array direct self-positioning method specifically includes the following steps:

[0051] Step 1: Distribute L signal sources and an array for receiving signal sources in the positioning area. In the present invention, the array is a circular array composed of M array elements. Compared with other array types, the circular array can obtain two-dimensional direction-finding data of the target signal source and uses a relatively small number of array elements.

[0052] Step 2: Collect data from multiple signal sources using a circular array data acquisition model;

[0053] The process of using a circular array to collect data from multiple signal sources in the present invention is as follows:

[0054] X=AS+N

[0055] Where X is the signal collected by the circular array, S is the source signal vector, N is the noise signal vector, A is the array flow pattern of the circular array, A=[a1,…,a l ,…,a L ], a l is the steering vector formed by the lth signal source incident on the array azimuth angle, λ is the wavelength of the signal transmitted by the source, r is the radius of the circular array, β l is the elevation angle of the lth signal source measured by the circular array, γ m-1 is the angle between the mth array element and the reference array element relative to the center of the circular array, γ m-1 =2π(m-1) / M, j is the imaginary unit, θ l It represents the incident azimuth angle of the lth radiation source measured by the circular array.

[0056] Step 3: Uniformly select grid points in the positioning area and construct a characteristic matrix of all signal sources at each grid point based on the circular array data acquisition model. This characteristic matrix can well describe the position constraint relationship between the grid points and the signal sources.

[0057] The characteristic matrix of all signal sources constructed by each grid point in the present invention is:

[0058] Φ i =[φ 1,i ,…,φ l,i ,…,φ L,i ]

[0059]

[0060] Among them, Φ i The characteristic matrix of all sources constructed for the i-th grid point, φ l,i is the characteristic matrix of the lth source constructed in the i-th grid point, λ is the wavelength of the signal transmitted by the source, r is the radius of the circular array, β l is the elevation angle of the lth signal source measured by the circular array, and is β in two-dimensional direction finding. l =90°; is the angle between the reference element in the circular array and the true north direction, γ m-1 is the angle between the mth array element and the reference array element relative to the center of the circular array, γ m-1 =2π(m-1) / M, j is the imaginary unit, η l,i is the angle between the lth source at the i-th grid point and the north direction, (x i ,y i ) is the position coordinate of the i-th grid point, (p lx ,p ly ) is the position coordinate of the lth source.

[0061] Step 4: Calculate the covariance matrix of the collected data and perform eigenvalue decomposition to obtain the signal subspace, noise subspace and diagonal matrix, and calculate the weight matrix required for weighted signal subspace fitting. Due to the limited data of the collected signal and the influence of factors such as the noise in the actual environment, the signal subspace calculated based on the data collected by the array is different from the theoretical value. This difference can be compensated by the weight matrix; the specific substeps include the following:

[0062] Step 4.1: Calculate the covariance matrix of the collected data and perform eigenvalue decomposition on the covariance matrix to obtain the signal subspace, noise subspace and diagonal matrix:

[0063]

[0064] Among them, R is the covariance matrix of the collected data, E() is the mathematical expectation, and D S is an L×L dimensional diagonal matrix, D S The diagonal elements of D are composed of the largest L eigenvalues ​​obtained by eigenvalue decomposition;N is a diagonal matrix consisting of ML smallest eigenvalues; U S is the signal subspace, which is composed of the eigenvectors corresponding to the L largest eigenvalues; U N is the noise subspace, which is composed of the eigenvectors corresponding to the ML smallest eigenvalues;

[0065] Step 4.2: Use the diagonal matrix D consisting of the largest L eigenvalues S And the diagonal matrix composed of ML smallest eigenvalues ​​is used to calculate the weight matrix required for weighted signal subspace fitting Where I is the identity matrix, σ 2 is the noise power, D N The mean of the diagonal elements.

[0066] Step 5: Fit the weighted signal subspace function value using the weight matrix constructed at each grid point, the signal subspace, and the characteristic matrix of all signal sources. Compare the weighted signal subspace function value of each grid point. The function value represents the degree of match between the grid point and the true position of the array. The larger the function value, the greater the match. Therefore, the point with the largest function value is selected as the initial estimate of the position of the circular array.

[0067] The fitting process of the weighted signal subspace function value in the present invention is:

[0068] According to array signal processing theory, the space spanned by the signal subspace and the space spanned by the array flow pattern are the same space. It can be seen that:

[0069] span{U s}=span{A}

[0070] At this point there exists a full rank matrix T such that

[0071] U s =AT

[0072] Due to the limited number of sampling times in the actual environment and the interference of the noisy environment, the signal subspace cannot completely coincide with the theoretical one. Therefore, the weight matrix W can be used to make up for the lack of information, so the following relationship can be obtained:

[0073] U s W 1 / 2 =AT

[0074] According to the relationship between the two sides of the above equation, the problem of solving the array position is converted into a least squares fitting problem:

[0075]

[0076] in, represents the estimated value of the initial position of the array, and the coordinates are expressed as represents the estimated value of the full rank matrix T;

[0077] First, the characteristic matrix Φ of the signal source i , for unknown parameters Make the following estimates:

[0078]

[0079] Combining the above two equations, we can get:

[0080]

[0081] Therefore, the grid points can be traversed according to the above formula. i At , the following weighted signal subspace fitting function is obtained:

[0082]

[0083] Where tr() is the trace operation of the matrix, and I is the identity matrix.

[0084] Step 6: Randomly select particles near the initial estimated position, and the particle distribution satisfies the mean 0 and variance σ 2 Compared with random uniform distribution, such normal distribution can ensure that the position information covered by the particles is rich while also constraining the particles to always be distributed near the initial estimated position. Specifically, the positions of the J randomly selected particles are Pa j =[Pax j ,Pay j ] T , and the position distribution of the particles satisfies the following relationship:

[0085]

[0086]

[0087] Among them, ||||2 is the two-norm operation, N(0,σ 2 ) means that the mean is 0 and the variance is σ 2 The normal distribution relationship, is the initial estimate of the position of the circle.

[0088] Step 7: Perform weight evaluation on each particle position through the particle weighted signal subspace function, and calculate the probability density function of the event "particle is array position";

[0089] The particle weighted signal subspace function in the present invention is:

[0090]

[0091] Among them, C(Pa j ) is the particle Pa j The corresponding weight evaluation value, U S is the signal subspace, W is the weight matrix, Φ′ j The characteristic matrix of all sources constructed for the jth particle, Φ′ j =[φ′ 1,j ,…,φ′ l,j ,…,φ′ L,j ],φ′ l,j is the construction matrix of the lth information source relative to the jth particle:

[0092]

[0093] in, is the angle between the reference element in the circular array and the true north direction, λ is the wavelength of the signal transmitted by the source, r is the radius of the circular array, β l is the elevation angle of the lth signal source measured by the circular array, j is the imaginary unit, γ m-1 is the angle between the mth array element and the reference array element relative to the center of the circular array, γ m-1 =2π(m-1) / M,η′ l,j is the angle between the lth source and the true north direction at the jth particle position, (p lx ,p ly ) is the position coordinate of the lth source.

[0094] Step 8: Normalize the weight evaluation value Calculate the weighted average of the particle distribution positions to obtain the estimated circular array position. This enables accurate estimation of the source location.

[0095] To verify the effectiveness of the array direct self-positioning method based on particle optimization weighted signal subspace fitting of the present invention, a MATLAB simulation analysis is performed below to prove that the performance estimation indicator is the root mean square error (RMSE), which is defined as:

[0096]

[0097] Where K represents the number of Monte Carlo simulations, q k Represents the estimated value and true value of the kth Monte Carlo experiment.

[0098] like Figure 2This is a scene diagram of the present invention constructed in a simulation experiment. There are L signal sources in the space, and a circular array in the positioning area receives signal data from the signal sources.

[0099] Figure 3 This figure compares the simulation results of the array direct self-positioning method using particle optimization weighted signal subspace fitting of the present invention with the weighted signal subspace fitting (WSSF) method. In this simulation experiment, the circular array element M=7 and the circular array radius r=0.3m were selected; the grid range was 100m long and 100m wide, the grid spacing was 5m, there were three signal sources, and the signal source positions were (25m, 25m), (50m, 75m), and (75m, 25m). The number of snapshots was set to 2048, and the signal-to-noise ratio condition was set to 15dB. It can be seen that the circular array position predicted by the direct self-positioning method of the present invention is closer to the actual circular array position, indicating that the particle optimization method can more accurately estimate the circular array position.

[0100] Figure 4 The figure shows a comparison of the signal source position estimation performance of the method of the present invention under different signal-to-noise ratios. The circular array element M=7 and the circular array radius r=0.3m are selected in the simulation; the grid range is 300m long and 300m wide, and the grid spacing is 2m. In the simulation, the signal source position and the circular array position are random, the number of snapshots is set to 1024, and the number of Monte Carlo experiments is 500. It can be seen from the simulation results that compared with the MUSIC (Multiple Signal Classification), ISF (Initial Signal Fitting), and WSSF methods, the method of the present invention can more accurately estimate the signal source position under different signal-to-noise ratio conditions, and as the signal-to-noise ratio conditions improve, the optimization performance will also be greatly improved.

[0101] The above are merely preferred embodiments of the present invention. The scope of protection of the present invention is not limited to the above embodiments. All technical solutions based on the principles of the present invention are within the scope of protection of the present invention. It should be noted that for those skilled in the art, various improvements and modifications that do not depart from the principles of the present invention should be considered within the scope of protection of the present invention.

Claims

1. A particle-optimized weighted signal subspace fitting array direct self-positioning method, characterized in that: The specific steps include: Step 1: Distribute in the targeted area L A signal source and an array for receiving a signal from the signal source; the array is a circular array composed of M array elements; Step 2: Collect data from multiple signal sources using a circular array data acquisition model; Step 3: Uniformly select grid points in the positioning area, and construct the characteristic matrix of all signal sources at each grid point based on the model of circular array data acquisition; Step 4: Calculate the covariance matrix of the collected data and perform eigenvalue decomposition to obtain the signal subspace, noise subspace and diagonal matrix, and calculate the weight matrix required for weighted signal subspace fitting; Step 5: Fit the weighted signal subspace function value using the weight matrix constructed at each grid point, the signal subspace, and the characteristic matrix of all signal sources. Compare the weighted signal subspace function values ​​at each grid point and select the point with the largest function value as the initial estimate of the circular array position. Step 6: Randomly select particles near the initial estimated position, and the particle distribution satisfies the mean 0 and variance The normal distribution relationship of Step 7: Perform weight evaluation on each particle position using the particle weighted signal subspace function; Step 8: Normalize the weight evaluation value, calculate the weighted average sum of the weight and particle distribution position, and obtain the circular array position estimation result.

2. The array direct self-positioning method based on particle optimization weighted signal subspace fitting according to claim 1, characterized in that: The process of using the circular array to collect data from multiple signal sources in step 2 is as follows: in, The signal collected by the circular array is is the source signal vector, is the noise signal vector, A is the array flow pattern of the circular array, , For the The steering vector formed by the incident signal source on the array azimuth angle is , is the wavelength of the signal transmitted by the source, is the radius of the circle, The first l The elevation angle of the source, is the angle between the mth array element and the reference array element relative to the center of the circular array, , is the imaginary unit, Indicates the first The incident azimuth of the radiation source.

3. The array direct self-positioning method of particle optimization weighted signal subspace fitting according to claim 1, characterized in that: The characteristic matrix of all sources constructed at each grid point in step 3 is: in, For the The characteristic matrix of all signal sources constructed by grid points is: For the The first grid point constructed l The characteristic matrix of the source, is the wavelength of the signal transmitted by the source, is the radius of the circle, The first l The elevation angle of the source, For the Grid point location The angle between the source and the north direction, is the angle between the reference element in the circular array and the true north direction, is the angle between the mth array element and the reference array element relative to the center of the circular array, , is the imaginary unit, For the Grid point location The angle between the source and the north direction, , , ( , ) is the i The position coordinates of the grid points, For the l The location coordinates of the source.

4. The array direct self-positioning method based on particle optimization weighted signal subspace fitting according to claim 1, characterized in that: Step 4 includes the following sub-steps: Step 4.1: Calculate the covariance matrix of the collected data and perform eigenvalue decomposition on the covariance matrix to obtain the signal subspace, noise subspace and diagonal matrix: in, is the covariance matrix of the collected data, is the mathematical expectation, for dimensional diagonal matrix, The diagonal elements of the largest eigenvalue decomposition The eigenvalues ​​are composed of for the reason The diagonal matrix consisting of the smallest eigenvalues; is the signal subspace, and the signal subspace is composed of The eigenvector corresponding to the largest eigenvalue is composed of is the noise subspace, and the noise subspace is composed of The eigenvector corresponding to the smallest eigenvalue is composed; Step 4.2: By the largest The diagonal matrix consisting of eigenvalues and by The weight matrix required for weighted signal subspace fitting is calculated by using the diagonal matrix composed of the smallest eigenvalues ,in, is the identity matrix, is the noise power, given by The mean of the diagonal elements.

5. The array direct self-positioning method based on particle optimization weighted signal subspace fitting according to claim 1, characterized in that: The fitting process of the weighted signal subspace function value in step 5 is: in, For the i The weighted signal subspace function value of grid points, is the trace operation of the matrix, is the identity matrix, is the signal subspace, is the weight matrix, For the The characteristic matrix of all signal sources constructed by grid points.

6. The array direct self-positioning method based on particle optimization weighted signal subspace fitting according to claim 1, characterized in that: Randomly selected in step 6 The position of a particle is , and the position distribution of the particles satisfies the following relationship: in, is the two-norm operation, It means that the mean is 0 and the variance is The normal distribution relationship, is the initial estimate of the position of the circle.

7. The array direct self-positioning method based on particle optimization weighted signal subspace fitting according to claim 6, characterized in that: The particle weighted signal subspace function in step 7 is: in, For particles The corresponding weight evaluation value, is the signal subspace, is the weight matrix, For the j The characteristic matrix of all information sources constructed by particles, , For the l The source is relative to the The construction matrix of particles: in, is the angle between the reference element in the circular array and the true north direction, is the wavelength of the signal transmitted by the source, is the radius of the circle, The first l The elevation angle of the source, is the imaginary unit, is the angle between the mth array element and the reference array element relative to the center of the circular array, , For the Particle position The angle between the source and the north direction, , , For the l The location coordinates of the source.

8. The array direct self-positioning method based on particle optimization weighted signal subspace fitting according to claim 7, characterized in that: The normalization process of weights in step 8 is: in, For the j The normalized weight evaluation value of each particle.

9. The array direct self-positioning method based on particle optimization weighted signal subspace fitting according to claim 8, characterized in that: The circular array position estimation result for: 。

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