Optimal weighted least squares based direction finding method for multiple array non-circular signals

Abstract

The application discloses a multi-array non-circular signal positioning method based on optimal weighted least square. Firstly, the spatial information is extended by using the elliptical covariance information of the target signal, so that the virtual array aperture is increased. Secondly, the bearing information of each observation station is obtained by root multiple signal classification (MUSIC) technology, and the non-circular phase is estimated. Finally, the bearing information of each observation station is associated with the signal source by using the non-circular phase, and the information of all observation stations is combined to directly solve the target position according to the optimal weighted least square idea. The multi-array non-circular signal positioning method based on optimal weighted least square can effectively increase the array aperture, has higher spatial freedom and positioning accuracy than the traditional two-step positioning algorithm and the subspace data fusion (SDF) direct positioning method, and can estimate more targets. In addition, the method can ensure the estimation performance while significantly reducing the calculation complexity.

Description

Technical Field

[0001] This invention relates to the field of wireless positioning technology, and in particular to a multi-array non-circular signal positioning method based on optimal weighted least squares. Background Technology

[0002] Traditional multi-array passive localization techniques are mostly designed for unknown signals. From an information theory perspective, the more raw information available, the better the algorithm's performance. Research shows that considering the signal characteristics of the target source when building the algorithm model can further improve localization accuracy. In modern communication systems, amplitude modulation (AM), binary phase shift keying (BPS), pulse amplitude modulation (PAM), and quadrature phase shift keying (QPS) signals are all non-circular (NC) signal types. Therefore, research on multi-array passive localization algorithms for non-circular signals has significant practical application value.

[0003] Existing passive localization algorithms for non-circular signals using multi-array arrays, while expanding the array aperture using the elliptic covariance information of the target signal, also introduce high-dimensional search problems, significantly increasing computational complexity. In practical applications, when signals from the same radiating target impact different observation stations, the received signal-to-noise ratios at different stations often differ and are unstable. Furthermore, traditional two-step localization methods also face challenges in data correlation. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to address the deficiencies involved in the background technology by providing a multi-array non-circular signal localization method based on optimal weighted least squares, which significantly reduces computational complexity while ensuring estimation performance and is easy to process in real time.

[0005] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0006] The multi-array non-circular signal localization method based on optimal weighted least squares includes the following steps:

[0007] Step 1) Construct a multi-array non-circular signal positioning model to obtain the received signal information r. l (t);

[0008] Step 1.1) Construct the array manifold for each observation station.

[0009] M is the number of large-scale uniform linear array elements equipped for each observation station, and K is the number of radiation sources; The guide vector is d, where d represents the element spacing and λ represents the signal wavelength; u l =[x l ,y l ] T Let x represent the position vector of the l-th observation station.l y l These are its x and y coordinates, respectively; the position vector of the k-th radiation source is p. k =[x k ,y k ] T x k y k These are its x and y coordinates; k = 1, 2, ..., K; the total position vector. θ l,k u represents the angle of arrival (°) of the k-th radiation source received by the l-th observation station. l (1) indicates taking vector u l The first element in, p k (1) indicates taking vector p k The first element in the array; ||·|| denotes the 2-norm;

[0010] Step 1.2) uses a multi-array non-circular signal positioning model to obtain the received signal of the l-th observation base station at sampling time t.

[0011] Let diag{·} be the channel loss matrix, and let diag{·} denote a diagonal matrix. To propagate the loss factor, Let be the emission power of the k-th radiation source. Let be the power received by the l-th observation station from the k-th radiation source;

[0012] To consider only strictly second-order noncircular signals with a noncircularity of 1, This indicates the non-circular phase of the k-th radiation source. Let the amplitude of the non-circular signal be represented by the non-circular phase matrix. signal vector It is a real-valued vector;

[0013] n l (t) represents the Gaussian white noise vector;

[0014] Step 2), using the elliptic covariance information of the target signal, the received signal from the observation base station is spread into z-shape. l Calculate the covariance matrix of (t) and perform eigenvalue decomposition.

[0015] Step 2.1), according to the following formula, the received signal r of the l-th base station l (t) expands to z l (t):

[0016]

[0017] In the formula, the row commutation matrix

[0018] For the extended direction matrix;

[0019] Extended guide vector

[0020] Step 2.2), calculate z l The covariance matrix of (t) T represents the number of snapshots;

[0021] Step 2.3), for the covariance matrix R l Performing eigenvalue decomposition, we obtain:

[0022]

[0023] In the formula, For the signal subspace, e l,m (m = 1, 2, ..., 2M) represents the eigenvalues ​​λ sorted from largest to smallest. l,m The eigenvectors corresponding to (m = 1, 2, ..., 2M); For the noise subspace, Σ l It is a diagonal matrix composed of eigenvalues;

[0024] Step 3) Apply the root-finding MUSIC technique to obtain azimuth information for each observation station by direction finding and estimate the non-circular phase;

[0025] Step 3.1), define complex variables Construct the guiding vector a(κ) l,k ) and matrix G(κ) l,k ):

[0026]

[0027]

[0028] In the formula, They are matrices of the same dimension and satisfy...

[0029] Step 3.2) Estimate the angle of arrival of the l-th observation station with respect to the k-th radiation source. Non-circular phase

[0030] Then the estimated value of the non-circular phase

[0031] Step 4) Use non-circular phase to associate the azimuth information with the information source, and use the optimal weighted least squares method to associate all observation station data to directly solve for the target position;

[0032] Step 4.1): Pair the azimuth information obtained from each observation station according to the estimated non-circular phase.

[0033] The estimated non-circular phases of each observation station are sorted in ascending order. Since the non-circular phases are unique, the angles of arrival corresponding to the same non-circular phases are automatically aligned.

[0034] Step 4.2), construct the weight matrix In the formula, ε k It is a random error vector with a mean of 0;

[0035] Let be the noise power of the l-th observation station, given by Σ l The average of the 2M-K smaller eigenvalues ​​is obtained; The estimated value of the extended guide vector, i.e.

[0036] Step 4.3) Estimate the location of the k-th radiation source.

[0037] Compared with the prior art, the present invention, employing the above technical solution, has the following technical effects:

[0038] The estimation accuracy of the algorithm proposed in this invention is superior to that of traditional two-step localization and SDF direct localization techniques. Compared with traditional two-step localization and SDF direct localization techniques, the proposed method has more degrees of freedom and can identify more targets. This invention can significantly reduce computational complexity while ensuring estimation performance. Attached Figure Description

[0039] Figure 1 This is a flowchart of the present invention;

[0040] Figure 2 This is a diagram illustrating a multi-array joint positioning scenario.

[0041] Figure 3 This diagram illustrates the computational complexity of the present invention and traditional positioning methods under different numbers of observation stations.

[0042] Figure 4 This is a schematic diagram illustrating the root mean square error performance of the present invention and traditional positioning methods under different signal-to-noise ratios;

[0043] Figure 5 This diagram illustrates the root mean square error performance of the present invention and traditional positioning methods under different snapshot numbers. Detailed Implementation

[0044] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings:

[0045] This invention can be implemented in many different forms and should not be considered limited to the embodiments described herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully express the scope of the invention to those skilled in the art. In the drawings, components are enlarged for clarity.

[0046] This invention provides a multi-array non-circular signal localization method based on optimal weighted least squares, such as... Figure 1 As shown, the specific steps include:

[0047] Step 1), construct as follows Figure 2 The multi-array non-circular signal positioning model is shown; the received signal information r is obtained. l (t);

[0048] Step 1.1) Construct the array manifold for each observation station.

[0049] M is the number of large-scale uniform linear array elements equipped for each observation station, and K is the number of radiation sources; The guide vector is d, where d represents the element spacing and λ represents the signal wavelength; u l =[x l ,y l ] T Let x represent the position vector of the l-th observation station. l y l These are its x and y coordinates, respectively; the position vector of the k-th radiation source is p. k =[x k ,y k ] T x k y k These are its x and y coordinates; k = 1, 2, ..., K; the total position vector. θ l,k u represents the angle of arrival (°) of the k-th radiation source received by the l-th observation station. l (1) indicates taking vector u l The first element in, p k (1) indicates taking vector p k The first element in the array; ||·|| denotes the 2-norm;

[0050] Step 1.2) uses a multi-array non-circular signal positioning model to obtain the received signal of the l-th observation base station at sampling time t.

[0051] Let diag{·} be the channel loss matrix, and let diag{·} denote a diagonal matrix. To propagate the loss factor, Let be the emission power of the k-th radiation source. Let be the power received by the l-th observation station from the k-th radiation source;

[0052] To consider only strictly second-order noncircular signals with a noncircularity of 1, This indicates the non-circular phase of the k-th radiation source. Let the amplitude of the non-circular signal be represented by the non-circular phase matrix. signal vector It is a real-valued vector;

[0053] n l (t) represents the Gaussian white noise vector;

[0054] Step 2), using the elliptic covariance information of the target signal, the received signal from the observation base station is spread into z-shape. l Calculate the covariance matrix of (t) and perform eigenvalue decomposition.

[0055] Step 2.1), according to the following formula, the received signal r of the l-th base station l (t) expands to z l (t):

[0056]

[0057] In the formula, the row commutation matrix

[0058] For the extended direction matrix;

[0059] Extended guide vector

[0060] Step 2.2), calculate z l The covariance matrix of (t) T represents the number of snapshots;

[0061] Step 2.3), for the covariance matrix R l Performing eigenvalue decomposition, we obtain:

[0062]

[0063] In the formula, For the signal subspace, e l,m (m = 1, 2, ..., 2M) represents the eigenvalues ​​λ sorted from largest to smallest. l,m The eigenvectors corresponding to (m = 1, 2, ..., 2M); For the noise subspace, Σ l It is a diagonal matrix composed of eigenvalues;

[0064] Step 3) Apply the root-finding MUSIC technique to obtain azimuth information for each observation station by direction finding and estimate the non-circular phase;

[0065] Step 3.1), based on the orthogonality of the signal subspace and the noise subspace, we obtain... Where span{·} represents the spanned space, then by minimizing the cost function... The target source location estimate can be obtained. However, the cost function in the above formula includes non-circular phase, and searching for the target location in a two-dimensional plane actually requires a three-dimensional mesh search, which has very high computational complexity. To reduce complexity, we consider removing the non-circular phase and replacing the mesh search with a root-finding method, thus rewriting the cost function.

[0066]

[0067]

[0068] and They are matrices of the same dimension and satisfy...

[0069] This is a quadratic optimization problem, and its minimum value is determined by matrix G. l (p k The smallest eigenvalue of is given. Since the quadratic form is non-negative, the matrix G is given. l (p k The smallest eigenvalue of is also non-negative, when the target position estimate is... When it coincides with the true position p, matrix G l (p k The determinant of ) det{G l (p k The value of )} is 0; in order to use the root-finding method, a complex variable is defined. Then the guiding vector a l (p k (Can be rewritten) Accordingly:

[0070]

[0071] Note G(κ) l,k The determinant of ) is about κ l,k The 4(M-1)th order polynomial has 2(M-1) pairs of conjugate symmetric roots, which means that when the number of physical array elements is M, a maximum of 2(M-1) targets can be estimated; given the number of targets, the estimated l-th observation station is closest to the K roots of the unit circle. It contains the target's location information;

[0072] Step 3.2) Estimate the angle of arrival of the l-th observation station with respect to the k-th radiation source. Non-circular phase

[0073] Then the estimated value of the non-circular phase

[0074] Step 4) Use non-circular phase to associate the azimuth information with the information source, and use the optimal weighted least squares method to associate all observation station data to directly solve for the target position;

[0075] Step 4.1): Pair the azimuth information obtained from each observation station according to the estimated non-circular phase.

[0076] The estimated non-circular phases of each observation station are sorted in ascending order. Since the non-circular phases are unique, the angles of arrival corresponding to the same non-circular phases are automatically aligned.

[0077] Step 4.2), when the distance between the observation station and the target is far enough, p k (n) represents the vector p k The nth element, u l (n) represents the vector u l The nth element is transformed into matrix multiplication: [1 - tan(θ)] l,k )]p k =[1 -tan(θ)] l,k )]u l Since the non-circular phase of the signal is unique, the azimuth information obtained from each observation station can be paired based on the estimated non-circular phase. Then, by combining the information from all observation stations, F can be obtained. 1,k p k =F 2,k G+ε k ;

[0078] ε k It is a random error vector with a mean of 0;

[0079]

[0080] As the number of snapshots approaches infinity, the error vector ε k The covariance has the following form:

[0081]

[0082] Step 4.3) Estimate the location of the k-th radiation source.

[0083] The spatial degrees of freedom obtained by the method of this invention are 2M-2, while the spatial degrees of freedom of a traditional uniform linear array with the same number of array elements are M-1, thus increasing the degree of freedom. Figure 3 This diagram illustrates the computational complexity (number of complex multiplications) of the method described in this invention compared to traditional positioning methods, as a function of the number of observation stations. Simulation conditions are: 3 targets, each observation station equipped with a uniform linear array of 8 elements, 100 snapshots, a global search range of 1000 meters, and a search step size of 0.5 meters. Figure 3 As can be seen, the method described in this invention does not require spectral peak search, and the computational complexity is significantly reduced.

[0084] The performance estimation standard of this invention is the root mean square error (RMSE), defined as follows:

[0085]

[0086] Where Mon represents the number of Monte Carlo trials, and K represents the number of targets. Let x represent the estimated position of the k-th target in the mn-th experiment, (x) k ,y k ) represents the actual value of the k-th target position.

[0087] Figure 4 This is a performance curve showing the root mean square error (RMSE) of the method described in this invention compared to the traditional two-step localization method and the SDF direct localization algorithm, as a function of signal-to-noise ratio. The simulation conditions are as follows: there are three targets with non-circular phases and positions of (10 radians, 30 radians, 70 radians) and [(-500 m, 50 m), (50 m, 220 m), (800 m, 110 m)], respectively; four observation base stations [(-900 m, -900 m), (-300 m, -700 m), (300 m, -1000 m), (900 m, -1200 m)], each base station equipped with a uniform linear array of 5 elements; 200 snapshots; and 500 simulations. Figure 4 As can be seen, this invention achieves higher positioning accuracy.

[0088] Figure 5This is a performance curve showing the root mean square error (RMSE) of the method described in this invention compared to the traditional two-step positioning method and the SDF direct positioning algorithm, as a function of the number of snapshots. The simulation conditions were as follows: three targets with non-circular phases and positions of (10 radians, 30 radians, 70 radians) and [(-500 m, 50 m), (50 m, 220 m), (800 m, 110 m)], respectively; four observation base stations [(-900 m, -900 m), (-300 m, -700 m), (300 m, -1000 m), (900 m, -1200 m)], each equipped with a uniform linear array of 5 elements; a signal-to-noise ratio of 10 dB; and 500 simulations. Figure 5 It can be seen that the method proposed in this invention has better estimation performance than the traditional two-step localization method and the SDF direct localization algorithm.

[0089] In summary, analysis of the simulation results shows that the multi-array non-circular signal localization method based on optimal weighted least squares proposed in this invention can effectively increase the array aperture, achieve higher spatial degrees of freedom and localization accuracy, and estimate more targets. Furthermore, this method eliminates the need for spectral peak search, significantly reducing computational complexity.

[0090] It will be understood by those skilled in the art that, unless otherwise defined, all terms used herein (including technical and scientific terms) have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. It should also be understood that terms such as those defined in general dictionaries should be understood to have the same meaning as in the context of the prior art, and should not be interpreted in an idealized or overly formal sense unless defined as herein.

[0091] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A multi-array non-circular signal localization method based on optimal weighted least squares, characterized by the following steps: Step 1) Construct a multi-array non-circular signal positioning model to obtain the received signal information r. l (t); Step 1.1) Construct the array manifold for each observation station. M is the number of large-scale uniform linear array elements equipped for each observation station, and K is the number of radiation sources; The guide vector is d, where d represents the element spacing and λ represents the signal wavelength; u l =[x l ,y l ] T Let x represent the position vector of the l-th observation station. l y l These are its x and y coordinates, respectively; the position vector of the k-th radiation source is p. k =[x k ,y k ] T x k y k These are its x and y coordinates; k = 1, 2, ..., K; the total position vector. θ l,k u represents the angle of arrival (°) of the k-th radiation source received by the l-th observation station. l (1) indicates taking vector u l The first element in, p k (1) indicates taking vector p k The first element in the array; ||·|| denotes the 2-norm; Step 1.2) uses a multi-array non-circular signal positioning model to obtain the received signal of the l-th observation base station at sampling time t. Let diag{·} be the channel loss matrix, and let diag{·} denote a diagonal matrix. To propagate the loss factor, Let be the emission power of the k-th radiation source. Let be the power received by the l-th observation station from the k-th radiation source; To consider only strictly second-order noncircular signals with a noncircularity of 1, This indicates the non-circular phase of the k-th radiation source. Let the amplitude of the non-circular signal be represented by the non-circular phase matrix. signal vector It is a real-valued vector; n l (t) represents the Gaussian white noise vector; Step 2), using the elliptic covariance information of the target signal, the received signal from the observation base station is spread into z-shape. l , calculate its covariance matrix, and then... Eigenvalue decomposition; Step 2.1), according to the following formula, the received signal r of the l-th base station l (t) expands to z l (t): In the formula, the row commutation matrix For the extended direction matrix; Extended guide vector Step 2.2), calculate z l The covariance matrix of (t) T represents the number of snapshots; Step 2.3), for the covariance matrix R l Performing eigenvalue decomposition, we obtain: In the formula, For the signal subspace, e l,m (m = 1, 2, ..., 2M) represents the eigenvalues ​​λ sorted from largest to smallest. l,m The eigenvectors corresponding to (m = 1, 2, ..., 2M); For the noise subspace, Σ l It is a diagonal matrix composed of eigenvalues; Step 3) Apply the root-finding MUSIC technique to obtain azimuth information for each observation station by direction finding and estimate the non-circular phase; Step 3.1), define complex variables Construct the guiding vector a(κ) l,k ) and matrix G(κ) l,k ): In the formula, and They are matrices of the same dimension and satisfy... Step 3.2) Estimate the angle of arrival of the l-th observation station with respect to the k-th radiation source. Non-circular phase Then the estimated value of the non-circular phase Step 4) Use non-circular phase to associate the azimuth information with the information source, and use the optimal weighted least squares method to associate all observation station data to directly solve for the target position; Step 4.1): Pair the azimuth information obtained from each observation station according to the estimated non-circular phase. The estimated non-circular phases of each observation station are sorted in ascending order. Since the non-circular phases are unique, the angles of arrival corresponding to the same non-circular phases are automatically aligned. Step 4.2), construct the weight matrix In the formula, ε k It is a random error vector with a mean of 0; Let Σ be the noise power of the l-th observation station. l The average of the 2M-K smaller eigenvalues ​​is obtained; The estimated value of the extended guide vector, i.e. Step 4.3) Estimate the location of the k-th radiation source.

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