Direct location method for non-circular signals based on polynomial root finding and weighted fusion

Abstract

The present invention discloses a non-circular signal direct positioning method based on polynomial root finding and weighted fusion, which is characterized by comprising the following steps: first, using the elliptical covariance information of the target signal to expand the spatial information and obtain an enlarged virtual array aperture; second, the high-dimensional spectrum function search problem of each base station is converted into a polynomial root finding problem by a root finding method; then, combining the information of all base stations to construct a spectrum function without complex multiplication to reduce the computational complexity; then, by weighting the received data of each base station, the algorithm estimation performance is improved; finally, the weighted fusion cost function is searched for the spectrum peak, and the one with the largest function value is selected. K The point is the target position. A non-circular source direct positioning method designed by the present invention has higher spatial freedom and positioning accuracy than the traditional direct positioning algorithm, and can estimate more targets. In addition, the method significantly reduces the computational complexity while ensuring the estimation performance.

Description

Technical Field

[0001] The invention belongs to the technical field of wireless positioning, and in particular relates to a non-circular signal direct positioning method based on polynomial root finding and weighted fusion. Background Art

[0002] Traditional direct position determination (DPD) technology is mostly used to study unknown signals. From the perspective of information theory, the more original information that can be used, the better the algorithm performance will be in theory. Studies have shown that considering the signal characteristics of the target source when establishing the algorithm model can further improve the positioning accuracy. In modern communication systems, amplitude modulation signals, binary phase shift keying, pulse amplitude modulation signals, orthogonal phase shift keying signals, etc. are all non-circular (NC) signal types. Therefore, the research on direct positioning algorithms for non-circular signals has important practical application significance.

[0003] The existing non-circular signal direct positioning algorithm ignores the signal loss during the propagation process. In practical applications, when the signal of the same radiating target impacts different observation stations, the received signal-to-noise ratio of different observation stations is often different and unstable. In addition, while expanding the array aperture using the elliptical covariance information of the target signal, it also brings about high-dimensional search, which greatly increases the computational complexity. In order to solve the above problems, the present invention proposes a non-circular source direct positioning method: joint polynomial root finding and multi-array weighted fusion. Summary of the invention

[0004] Purpose of the invention: In order to solve the problems existing in the prior art, the present invention provides a direct positioning method for non-circular signals based on polynomial root finding and weighted fusion, which significantly reduces the computational complexity while ensuring the estimation performance and is easy to process in real time.

[0005] Technical solution: The non-circular signal direct positioning method based on polynomial root finding and weighted fusion described in the present invention comprises the following steps:

[0006] 1) Construct a multi-array non-circular signal direct positioning model, which includes multiple observation base stations and radiation sources, and obtain the received signal r of the lth observation base station l (t);

[0007] 2) Using the non-circular characteristics to expand the received signal r l (t) Get the extended received signal z l (t), calculate z l The covariance matrix R of (t) l , and the covariance matrix R l Perform feature decomposition;

[0008] 3) Estimate the received data weight w of the lth observation base station l , and obtain the weighted cost function

[0009] 4) Construct matrix G l , the high-dimensional spectral function search of each observation station is transformed into polynomial root search, and the weighted fusion cost function f is constructed NC-Root-WDPD ;

[0010] 5) For the simplified cost function f NC-Root-WDPD Perform a peak search to get an accurate estimate of the target location.

[0011] Preferably, the received signal r obtained in step 1) is l (t) is:

[0012]

[0013] in: is the array flow pattern of the lth observation base station; is the steering vector; P k is the signal power of the radiation source at the first observation base station position u l =[x l ,y l ] T The radiation received from the kth source p k =[x k ,y k ] T The signal power is P l，k , k = (1, 2, ..., K), definition is the path propagation loss coefficient, then the propagation loss matrix:

[0014]

[0015] At the same time, consider the non-circular signal with a non-circularity rate of 1 Assumptions represents the phase of the non-circular signal, represents the amplitude of the non-circular signal, then:

[0016]

[0017]

[0018] Where M is the number of array elements, K is the number of targets, t = (1, 2, ..., T) represents the number of snapshots, 1 = (1, 2, ..., L) represents the number of base stations, is the amplitude of the non-circular signal; n l (t) represents the Gaussian white noise vector.

[0019] Preferably, the specific steps of step 2) are: according to the non-circular characteristic, the received signal r of the lth observation base station is l (t) is expanded to:

[0020]

[0021] because:

[0022]

[0023] The received signal of each observation base station can be rewritten as:

[0024]

[0025] Among them B l (p) can be regarded as an expanded array flow pattern, and:

[0026]

[0027] Extended Steering Vector for:

[0028]

[0029] in(·) * denotes conjugation;

[0030] Its covariance matrix R l for:

[0031]

[0032] Where T represents the number of snapshots, (·) H represents conjugate transpose;

[0033] The covariance matrix R l Performing feature decomposition, we get:

[0034]

[0035] Assume λ l，m (m=1, 2, ..., 2M) represents the eigenvalues ​​sorted from large to small, and the corresponding eigenvectors are represented by e l，m (m=1,2,…,2M) represents, then the signal subspace is expressed as The noise subspace is expressed as ∑ l is a diagonal matrix of eigenvalues.

[0036] Preferably, the specific steps of step 3) are:

[0037] Rewrite the covariance matrix R l :

[0038]

[0039] Among them, I 2M×2M is a 2M×2M dimensional unit matrix. Assuming that the noise power remains constant during the entire observation process, the signal-to-noise ratio at different observation locations is proportional to Covariance matrix R l Breaks down into:

[0040]

[0041] Where diag{·} represents a diagonal matrix, and the covariance matrix R l The eigenvalues ​​of are expressed as:

[0042]

[0043] in, YesR s The K larger non-zero eigenvalues ​​of represent the received signal power, and the noise power estimate is expressed as:

[0044]

[0045] The estimated power of the received signal at the first observation station is:

[0046]

[0047] The weight of the received data of the first observation station According to the orthogonality of the signal subspace and the noise subspace, we get:

[0048]

[0049] Get the weighted cost function:

[0050]

[0051] Preferably, the specific steps of step 4) are: constructing a matrix G1, transforming the cost function Rewritten as:

[0052]

[0053] Among them, G l is a 2×2 dimensional square matrix:

[0054]

[0055]

[0056] In the formula and are matrices of the same dimension and satisfy the following relationship:

[0057]

[0058]

[0059] Define the complex variable κ:

[0060] The guide vector a l (p) is rewritten as:

[0061] Then the matrix G l Rewritten as:

[0062]

[0063] Let square matrix G l is 0:

[0064]

[0065] By combining the position information of L observation base stations and the weight of received data, a weighted fusion cost function is constructed:

[0066]

[0067] Preferably, a spectrum peak search is performed on the weighted fusion cost function in step 4), and the K points with the largest function values ​​are the target positions.

[0068] Beneficial effects: Compared with the prior art, the beneficial effects of the present invention are: the estimation accuracy of the algorithm proposed in the present invention is better than that of the traditional two-step positioning technology, subspace data fusion technology (SDF), and Capon direct positioning technology; compared with the traditional two-step positioning technology, subspace data fusion algorithm and Capon direct positioning algorithm, the proposed method has more degrees of freedom and can identify more targets; the present invention can significantly reduce the computational complexity while ensuring the estimation performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0069] Figure 1 is a flow chart of the present invention;

[0070] Figure 2 Jointly localize scene graphs for multiple arrays;

[0071] Figure 3 A schematic diagram showing how the algorithm running time of the present invention and the traditional positioning method varies with the number of search grid points;

[0072] Figure 4Schematic diagram of root mean square error performance of the present invention and the traditional positioning method under different signal-to-noise ratios;

[0073] Figure 5 Schematic diagram of the root mean square error performance of the present invention and the traditional positioning method under different snapshot numbers. DETAILED DESCRIPTION

[0074] The present invention is further described in detail below with reference to the accompanying drawings.

[0075] The present invention provides a non-circular source direct positioning method: combining polynomial root finding and multi-array weighted fusion, such as Figure 1 As shown, the specific steps include:

[0076] Step 1: Construct Figure 2 The multi-array non-circular signal direct positioning model shown in the figure is used to obtain the received signal information r l (t).

[0077] The received signal of the lth observation base station is in, For the array flow pattern at each observation station, is the steering vector. Since the signal will be lost when propagating in the air, it is assumed that the signal power of the radiation source is P k , at the observation position u l =[x l ,y l ] T The radiation received from the kth source p k =[x k ,y k ] T The signal power is P l，k ,definition is the path propagation loss coefficient, is the propagation loss matrix. The present invention only considers non-circular signals with a non-circularity of 1, which can be expressed as Assumptions represents a non-circular phase, represents the amplitude of the signal, we can get:

[0078]

[0079]

[0080] in, is a real-valued vector; n l (t) represents the Gaussian white noise vector.

[0081] Step 2: Using the elliptical covariance information of the target signal, the received signal of each observation base station is expanded to:

[0082]

[0083] because

[0084]

[0085] The received signal of each observation base station can be rewritten as

[0086]

[0087] Among them B l (p) can be regarded as an expanded array flow pattern, and

[0088]

[0089] Extended Steering Vector for

[0090]

[0091] in(·) * Indicates conjugation.

[0092] Then the covariance matrix of the lth observation station can be calculated as Where T represents the number of snapshots, (·) H represents the conjugate transpose; perform eigendecomposition on the covariance matrix and obtain:

[0093]

[0094] Assume λ l，m (m=1, 2, ..., 2M) represents the eigenvalues ​​sorted from large to small, and the corresponding eigenvectors are represented by e l，m (m=1,2,…,2M) represents, then the signal subspace is expressed as The noise subspace is expressed as ∑ l is a diagonal matrix of eigenvalues.

[0095] Step 3: Estimate the weight of the data received by each observation station.

[0096] Rewrite the covariance matrix R l :

[0097]

[0098] Among them, I 2M×2M is a 2M×2M dimensional unit matrix. Assuming that the noise power remains constant during the entire observation process, the signal-to-noise ratio at different observation locations is proportional to Covariance matrix R l can be broken down into:

[0099]

[0100] Where diag{·} represents a diagonal matrix. The eigenvalue of the covariance matrix is ​​expressed as:

[0101]

[0102] in, YesR s The K larger non-zero eigenvalues ​​of represent the received signal power, and the noise power estimate is expressed as:

[0103]

[0104] The estimated value of the received signal power at the lth observation station is:

[0105]

[0106] Then the receiving data weight of the lth observation station can be obtained

[0107] Step 4: Construct the matrix G l , the high-dimensional spectral function search of each observation station is transformed into polynomial root search, and the weighted fusion cost function f is constructed NC-Root-WDPD .

[0108] According to the orthogonality of the signal subspace and the noise subspace, we can get

[0109]

[0110] That is, by minimizing the cost function

[0111]

[0112] The target source position can be estimated. However, it should be noted that the cost function of the above formula contains non-circular phase. To search for the target position in a two-dimensional plane, a three-dimensional spectrum peak search is actually required, and the computational complexity is very high. In order to reduce the complexity, consider stripping the non-circular phase and using the root-finding method instead of the spectrum search to define the reversible matrix. Rewrite the cost function of the above formula as

[0113]

[0114] Among them, G l is a 2×2 dimensional square matrix

[0115]

[0116]

[0117] In the formula and are matrices of the same dimension and satisfy the following relationship

[0118]

[0119]

[0120] because It is a quadratic optimization problem, whose minimum value is given by the matrix G l The minimum eigenvalue of is given. Since the quadratic form is non-negative, the matrix G l The minimum eigenvalue of is also non-negative, when the target position estimate When it coincides with the true position p, the matrix G l The determinant det{G l} is 0. In order to apply the root-finding method, define a complex variable κ

[0121]

[0122] Then the steering vector a l (p) can be rewritten as

[0123]

[0124] Correspondingly, note that the matrix G l There are conjugate terms in G. In order to make the root-finding process as simple as possible, l can be rewritten as

[0125]

[0126] It can be observed that the matrix G l The determinant det{G l} is a 4(M-1)th order polynomial in κ, let it be equal to zero

[0127]

[0128] The above equation has 2(M-1) pairs of conjugate symmetric solutions, which means that when the number of physical array elements is M, at most 2(M-1) targets can be estimated. When the number of targets is known, the estimated lth base station is closest to the K roots of the unit circle. The target's position information is included in . By combining the position information of L base stations with the received data weight, the following spectrum function can be constructed:

[0129]

[0130] Step 5: Get an accurate estimate of the target location by searching for the peak.

[0131] For the weighted fusion cost function f in step 4 NC-Root-WDPD Perform spectrum peak search, and the K points with the largest function value are the target positions.

[0132] The spatial degree of freedom obtained by the method of the present invention is DOF=2(M-1), while the spatial degree of freedom of the traditional uniform linear array is DOF=M-1 under the condition of the same number of array elements, which increases the degree of freedom to a certain extent. Figure 3 The following is a schematic diagram showing the running time of the method of the present invention and the traditional positioning method as a function of the number of search grid points. The simulation conditions are: 3 targets, 6 observation base stations, each base station is equipped with a uniform linear array with 6 elements, the number of snapshots is 200, and the signal-to-noise ratio is 15 decibels. Figure 3 It can be seen that the method described in the present invention (weighted root-finding direct positioning) transforms the high-dimensional spectral function search of each observation station into polynomial root-finding and constructs a cost function that does not require complex multiplication. Compared with the high-dimensional spectral search method before the transformation, the computational complexity is significantly reduced.

[0133] The performance estimation standard of the present invention is the root mean square error (RMSE) which is defined as:

[0134]

[0135] Among them, Mon is the number of Monte Carlo experiments, K is the number of targets, represents the estimated value of the kth target position in the mnth experiment, (x k ,y k ) represents the actual value of the kth target position.

[0136] Figure 4 The performance curve of the root mean square error of the method of the present invention, the traditional two-step positioning method, the subspace data fusion algorithm and the Capon direct positioning algorithm as the signal-to-noise ratio changes. The simulation conditions are: there are 3 targets, their non-circular phases and positions are (10 radians, 20 radians, 30 radians) and [(100 meters, 900 meters), (500 meters, -200 meters), (900 meters, 200 meters)], 6 observation base stations [(-800 meters, -900 meters), (-480 meters, -700 meters), (-160 meters, -1000 meters), (160 meters, -800 meters), (480 meters, -1100 meters), (800 meters, -1200 meters)], each base station is equipped with a uniform linear array with 6 elements, the number of snapshots is 100, and the simulation is repeated 1000 times. From Figure 4 It can be seen that the present invention achieves higher positioning accuracy.

[0137] Figure 5The performance curve of the root mean square error of the method of the present invention, the traditional two-step positioning method, the subspace data fusion algorithm and the Capon direct positioning algorithm as the number of snapshots changes. The simulation conditions are: there are 3 targets, their non-circular phases and positions are (10 radians, 20 radians, 30 radians) and [(100 meters, 900 meters), (500 meters, -200 meters), (900 meters, 200 meters)], 6 observation base stations [(-800 meters, -900 meters), (-480 meters, -700 meters), (-160 meters, -1000 meters), (160 meters, -800 meters), (480 meters, -1100 meters), (800 meters, -1200 meters)], each base station is equipped with a uniform linear array with 6 elements, the signal-to-noise ratio is 20 decibels, and the simulation is repeated 1000 times. From Figure 5 It can be seen that the method proposed in the present invention has better estimation performance than the traditional two-step positioning method, the subspace data fusion algorithm and the Capon direct positioning algorithm.

[0138] In summary, from the analysis of the simulation effect diagram, it can be seen that the non-circular source direct positioning method proposed in the present invention: the combined polynomial root finding and multi-array weighted fusion can effectively increase the array aperture, have higher spatial degrees of freedom and positioning accuracy, and can estimate more targets. In addition, while ensuring the estimation performance, this method significantly reduces the computational complexity by converting the high-dimensional spectral function search problem of each observation station into a polynomial root finding problem and constructing a cost function that does not require complex multiplication.

[0139] The embodiments of the present invention are described in detail above with reference to the accompanying drawings, but the present invention is not limited to the above embodiments, and various changes can be made within the knowledge scope of ordinary technicians in this field without departing from the purpose of the present invention.

Claims

1. A non-circular signal direct positioning method based on polynomial root finding and weighted fusion, characterized in that: The method comprises the following steps: 1) Construct a multi-array non-circular signal direct positioning model, which includes multiple observation base stations and radiation sources, and obtain the received signal r of the lth observation base station l (t); 2) Using the non-circular characteristics to expand the received signal r l (t) Get the extended received signal z l (t), calculate z l The covariance matrix R of (t) l , and the covariance matrix R l Perform feature decomposition; 3) Estimate the received data weight w of the lth observation base station l , and obtain the weighted cost function 4) Construct matrix G l , the cost function of each observation base station The search is transformed into a polynomial root and a weighted fusion cost function f is constructed. NC-Root-WDPD ; 5) Weighted fusion cost function f NC-Root-WDPD Perform a peak search to obtain an accurate estimate of the target location; The specific steps of step 4) are: construct the matrix G l , the cost function Rewritten as: in, is the phase of the non-circular signal, L is the number of base stations, w l is the received data weight of the lth (l=1, 2, ..., L) observation base station, and the vector G l is a 2×2 dimensional square matrix: Among them, a l (p) is the steering vector of the th observation base station, (-) H represents the conjugate transpose, [] T represents the transpose of the matrix, (·)* represents the conjugate, and are matrices of the same dimension and satisfy the following relationship: in, represents the noise subspace of the,th observed base station; Define the complex variable κ: In the formula, d represents the array element spacing, ||·|| represents the 2-norm, u(1) represents the first element of vector u, and p(1) represents the first element of vector p; The guide vector a l (p) is rewritten as: Where, M is the number of array elements; Then the matrix G l Rewritten as: Let square matrix G l is 0: By combining the position information of L observation base stations and the weight of received data, a weighted fusion cost function is constructed: in is the kth root of the observation base station closest to the unit circle, K is the number of targets, is the estimated value of the received signal power of the lth observed base station, is the noise power estimate, ang() is the phase of the complex number, and |·| is the absolute value.

2. The non-circular signal direct positioning method based on polynomial root finding and weighted fusion as claimed in claim 1, characterized in that: Step 1) The received signal r l (t) is: in: is the array flow pattern of the lth observation base station; is the steering vector of the kth radiation source of the kth observation base station, d represents the array element spacing; P k is the signal power of the kth radiation source at the position u of the observation base station l =[x l ,y l ] T The radiation received from the kth source p k =[x k ,y k ] T The signal power is P l，k ,k=(1,2,...,K),[ ] T represents the transpose of the matrix, and defines is the path propagation loss coefficient, then the propagation loss matrix: At the same time, consider the non-circular signal with a non-circularity rate of 1 Assumptions represents the phase of the non-circular signal of the kth radiation source, represents the amplitude of the non-circular signal, then: represents the amplitude of the non-circular signal of the first K radiation sources, where M is the number of array elements, K is the number of targets, t = (1, 2, ..., T) represents the number of snapshots, l = (1, 2, ..., L) is the number of base stations, is the amplitude of the non-circular signal; n l (t) represents the Gaussian white noise vector.

3. The non-circular signal direct positioning method based on polynomial root finding and weighted fusion as claimed in claim 2, characterized in that: The specific steps of step 2) are: according to the non-circular characteristic, the received signal r of the lth observation base station is l (t) is expanded to: because: The received signal of each observation base station can be rewritten as: Among them B l (p) can be regarded as an expanded array flow pattern, and: Extended Steering Vector for: in(·) * denotes conjugation; Its covariance matrix R l for: Where T represents the number of snapshots, (·) H represents conjugate transpose; The covariance matrix R l Performing feature decomposition, we get: Assume λ l，m (m=1, 2, ..., 2M) represents the eigenvalues ​​sorted from large to small, and the corresponding eigenvectors are represented by e l，m (m=1,2,…,2M) represents, then the signal subspace is expressed as The noise subspace is expressed as, ∑ l is a diagonal matrix of eigenvalues.

4. The non-circular signal direct positioning method based on polynomial root finding and weighted fusion as claimed in claim 3, characterized in that: The specific steps of step 3) are: Rewrite the covariance matrix R l : Among them, I 2M×2M is a 2M×2M dimensional unit matrix. Assuming that the noise power remains constant during the entire observation process, the signal-to-noise ratio at different observation locations is proportional to Covariance matrix R l Breaks down into: Where diag{·} represents a diagonal matrix, and the covariance matrix R l The eigenvalues ​​of are expressed as: in, YesR s The K larger non-zero eigenvalues ​​of represent the received signal power, and the noise power estimate is expressed as: The estimated value of the received signal power at the observation station is: The weight of the received data of the observation station According to the orthogonality of the signal subspace and the noise subspace, we get: Get the weighted cost function:

5. The non-circular signal direct positioning method based on polynomial root finding and weighted fusion as claimed in claim 1, characterized in that: The weighted fusion cost function in step 4) is searched for spectrum peaks, and the K points with the largest function values ​​are the target positions.

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