CN113093093A - Vehicle positioning method based on linear array direction of arrival estimation - Google Patents

Abstract

The invention provides a vehicle positioning method based on linear array direction of arrival estimation, which is characterized in that linear arrays are symmetrically arranged on two sides of a road close to the road with the known road width of b; calculating a covariance matrix R of each array receiving the vehicle signal X; constructing a covariance matching model, and solving by using a non-convex algorithm to obtain a noise-free covariance matrix corresponding to an optimal solution

Recovering the angle by using an angle estimation method, wherein the recovered angle is the incoming direction of the vehicle; the position of array G, J on the road is determined in conjunction with the respective vehicles from array G, J, and the vehicle coordinates are solved using trigonometric functions. The method fully utilizes the rapid convergence of the non-convex algorithm and the high degree of freedom of the sparse linear array, can effectively reduce the computational complexity of the non-grid method and position vehicles with more than array elements, and improves the positioning method pairThe adaptability of the high-speed and high-density driving scene can be used for the high-density and high-speed driving scene.

Description

Vehicle positioning method based on linear array direction of arrival estimation

Technical Field

The invention relates to a vehicle positioning method based on linear array direction of arrival estimation, and belongs to the technical field of Internet of things and signal processing.

Background

The method for vehicle Positioning based on Direction-of-Arrival (DOA) estimation is an important branch of a vehicle Positioning method which is not based on a Global Positioning System (GPS), and is characterized in that an array antenna is used for receiving airspace signals, and the received signals are processed through a statistical signal processing technology and various optimization methods to recover incoming information of the incoming signals, so that the coordinates of a vehicle are given to the incoming information by combining the array position. The vehicle positioning method based on Direction-of-Arrival (DOA) estimation can perform positioning in a scene without GPS signals, and therefore, the method has a wider application range. However, the vehicle positioning method based on DOA estimation not only has high requirements on the direction-finding environment, but also has a limit on the number of vehicles that can be estimated.

Due to the limitations of the traditional DOA estimation method, namely a subspace method, in scenes with low signal-to-noise ratio, high correlation and the like, the compressed sensing direction-finding method is widely concerned. In the compressed sensing direction-finding method, the non-grid division method is widely concerned because grid division and robustness to severe environments such as low signal-to-noise ratio are not needed. However, most of the non-mesh partition methods need to solve the semi-definite planning problem, and have high computational complexity. It is noted that a larger time consumption results in a larger positioning error.

Therefore, in order to adapt to the current high-speed and high-density driving scene, a vehicle positioning method based on grid-free direction finding with low time consumption and high precision needs to be designed.

The above-mentioned problems are problems that should be considered and solved in the vehicle positioning process.

Disclosure of Invention

The present invention aims to provide a vehicle positioning method based on linear array direction of arrival estimation, which solves the problem of large positioning error caused by high computation complexity in the prior art.

The technical solution of the invention is as follows:

a vehicle positioning method based on linear array direction of arrival estimation comprises the following steps,

s1, symmetrically placing linear arrays, namely a left array G and a right array J, close to the two sides of the road on the road with the known road width b, and respectively receiving vehicle signals X by using the arrays on the two sides of the road;

s2 and array G, J receive and process the signals X respectively, and calculate covariance matrix R of each array receiving vehicle signals X;

s3, constructing a covariance matching model, and solving by using a non-convex algorithm to obtain a noise-free covariance matrix corresponding to an optimal solution

S4 corresponding to the optimal solution

Then, the angle theta is recovered by using an angle estimation method₁，…，θ_KThe coming direction of the vehicle is obtained;

s5, and combining the positions of the arrays G, J on the road with the vehicle directions obtained from the arrays G, J, respectively, and solving the coordinates of the vehicle by using trigonometric functions.

Further, in step S1, linear arrays are symmetrically placed on both sides of the road with the known road width b, specifically: the linear array formed by the M array elements is placed on the left side of the road and recorded as an array G, and the linear array formed by the M array elements is placed on the right side of the road and recorded as an array J.

Further, in step S1, the operation mechanism of the array G and the array J for receiving the vehicle signal is the same; wherein the received vehicle signals using array G are: suppose there are K far-field narrow-band signals, i.e., K vehicle signals, at an azimuth angle θ ═ θ₁，…，θ_KIncident on the homogeneous array G at an angle of incidence; after receiving L snapshots, the received signals of array G are respectively represented as: x is AS + N, where X is a reception signal of the array G, S is an incident signal waveform, a is an array manifold matrix of the array G, and a is [ a (θ [) ]₁)，…，a(θ_K)]Is the steering vector of the array G,

is the steering vector of the kth signal on array G,

(·)^Tand N is a noise matrix, and the noises received by different array elements are independent.

Further, in step S2, the covariance matrix R of each array received vehicle signal X is calculated: r ═ XX^H]^T＝ZPZ^H+ σ I, where Z is the eigenvector matrix of the denoised covariance matrix, (. C)^HFor the conjugate transpose operation of the matrix, S is the incident signal waveform, P is the covariance matrix of the incident signal, and P ═ E [ SS ]^H]＝diag(p)，p＝[p₁，…，p_K]^T，E[·]To solve for the desired operation, σ is the noise power, I is the identity matrix;

further, in step S3, a covariance matching model is constructed and solved by a non-convex algorithm to obtain a noise-free covariance matrix corresponding to the optimal solution

In particular to a method for preparing a high-performance nano-silver alloy,

s31, establishing an atom set:

wherein d is an atom in the set of atoms, f is the normalized frequency,

being the phase of an atom, the atomic norm of the sparse signal x on the set of atoms D is defined as: | x | non-conducting phosphor_A＝inf{∑_kc_k:x＝∑_k c_ka_k，c_i>0，a_kE.g. A }, wherein c_kIs the weighting coefficient of the k-th atom, a_kIs the k-th atom selected from the atom set D, c_iAs a weighting coefficient, | · | | non-conducting phosphor_ARepresenting an atomic norm, inf representing an infimum boundary;

s32, according to the definition of the atomic norm, establishing the following atomic norm minimization problem:

wherein, δ is the upper bound of noise, y is the received observation signal, and Φ is a linear mapping matrix; solving the model to obtain the optimal solution

Establishing

Covariance matrix of

Wherein

Is expressed as a vector

The first row of the Tobraz matrix;

s33, the problem of atomic norm minimization is actually to restore the covariance matrix with a semi-positive definite tobutiz matrix structure first, so the problem is transformed into the following covariance matching model:

wherein | · | purple sweet_FFrobenius norm, rank (. cndot.) of the matrix, T (u) of a noise-free covariance matrix, R of a covariance matrix under finite snapshots,

is a sampling covariance matrix, and β is the fitting error; match covarianceThe model is solved by a non-convex algorithm to obtain a noise-free covariance matrix corresponding to the optimal solution

Further, in step S3, the covariance matching model is solved by a non-convex algorithm, specifically,

the covariance matching model constructed in step S331 and step S33 is mathematically transformed as follows:

equivalence is expressed as the following low rank matrix recovery problem:

where T (u) is a noise-free covariance matrix, β is the fitting error, Q and

is an intermediate variable in the model;

s332, representing the low-rank matrix recovery problem obtained in the step S331 as a matrix filling problem according to a non-convex algorithm equivalent as follows:

wherein, two matrix sets R_L、R_HTwo matrix sets are as follows: r_L＝{L∶rank(L)≤K}，

Where T (u) is a noise-free covariance matrix, β is the fitting error, Q and

is an intermediate variable in the model; the matrix L, H is divided intoIs other than R_L、R_HTheir update procedure is as follows:

wherein L is^(l)For iterating the low rank matrix of the L-th time, L^(l+1)A low rank matrix that is iterated the (l + 1) th time; h^(l)For iteration the l-th semi-positive definite Tourette's matrix satisfying the constraints, H^(l+1)A positive semi-definite Tourette's matrix satisfying the constraint for the (l + 1) th iteration; delta₁And delta₂Is a parameter that is a function of,

is to the matrix set R_LThe mapping of (a) to (b) is,

wherein, U_KIs a left eigenvector matrix, Σ_kAs a matrix of eigenvalues, V_KIs a right eigenvector matrix, (.)^HA conjugate transpose operation for the matrix;

is to the matrix set R_HOf a set of matrices R_HIs a satisfying constraint

Of a semi-positive definite Topritz matrix, will therefore

The method comprises the following three steps: ensuring constraint conditions to be satisfied

Thereafter projecting the result of the previous step onto the Topritz matrix, P_T(L₁)＝T(v)＝L₂Finally, ensuring that the Topriz matrix of the last step is a semi-positive definite matrix, namely P₊(L₂)＝Hdiag(ε₊)H^H＝L₃Wherein P is_Q(. is) a projection onto the matrix Q, L₁Projecting the obtained matrix for the step; p_T(. is a projection onto the Tobraz matrix, L₂Projecting the obtained Tobraz matrix for the step; p₊(. is) a projection onto a semi-positive definite matrix, L₃For the semi-positive definite matrix, epsilon, obtained by projection of this step₊The eigenvalue vector is obtained, wherein the eigenvalue of the solved matrix is reduced to 0 when being less than 0, H is a corresponding eigenvector matrix, beta is a fitting error, and T (v) is a mapped Tobraz matrix;

s333, updating process and of combining the matrix L, H according to the equivalent transformed matrix filling model obtained in the step S332

The optimization problem is solved to obtain a covariance matrix of the sparse signal x

Further, the array G and the array J in step S1 adopt a uniform linear array and a sparse array, wherein the sparse array includes a minimum redundant array, a co-prime array, a nested array or a minimum hole array.

Further, the angle estimation method in step S4 employs the following method: tobraz matrix factorization theorem subspace class method or multiple signal classification subspace class method.

Further, in step S4, the angle θ is restored by the angle estimation method as { θ ═ θ }₁，…，θ_KThe method comprises the following steps:

s41, drawing a virtual domain space spectrum P (u):

where θ is the assumed signal direction of arrival, E_nIs a matrix

A (θ) is the steering vector from the signal, a^H(θ) is a conjugate transpose vector of the steering vector from the signal;

s42, searching for peak values of space spectrum P (u) through spectrum peak search, arranging the peak values in the order from small to large, and taking angle values { theta (theta) { theta) corresponding to the first K peak values₁，…，θ_KAngle θ ═ θ₁，…，θ_KThe term is the coming direction of the vehicle.

Further, in step S6, the vehicle coordinates are solved by using a trigonometric function according to the positions of the array G, J and the directions of the vehicles obtained by the array G, J on the road, specifically:

a rectangular coordinate system is established by taking the uniform linear array G on the left side of the road as an original point, the coordinate of the uniform linear array J on the right side is (b, 0), and the coordinate of the kth vehicle is (x)_k，y_k) Using trigonometric knowledge to set up the equation:

wherein tan θ_{k_1}、tanθ_{k_2}The directions of arrival of the vehicle to the left array and the right array are respectively; and (3) combining the two formulas to obtain the coordinates of the vehicle:

the invention has the beneficial effects that: compared with the prior art, the vehicle positioning method based on the linear array direction of arrival estimation fully utilizes the rapid convergence of a non-convex algorithm, can effectively reduce the computational complexity of a non-grid method, and improves the adaptability of the positioning method to a high-speed driving scene. Experiments prove that the method is less in time consumption, small in error and high in precision. The method of the invention fully utilizes the high degree of freedom characteristic of the sparse linear array, can position vehicles with more than the array number, improves the adaptability of the positioning method to high-density driving scenes, and can be used for high-density and high-speed driving scenes.

Drawings

FIG. 1 is a flowchart illustrating a vehicle positioning method based on linear array direction of arrival estimation according to an embodiment of the present invention.

Fig. 2 is an explanatory diagram of the embodiment in which linear array antennas are symmetrically disposed on both sides of a road.

Fig. 3 is an explanatory diagram of the uniform line array and the sparse line array in the embodiment; wherein, ULA is Uniform linear array of Uniform linear array, SLA: sparse linear array.

Fig. 4 is an explanatory diagram of an embodiment of establishing a rectangular coordinate system with the linear array G on the left side of the road as the origin, where E is the array on the left side located on the left side of the road, F is the array on the right side located on the road, and V is the vehicle.

Fig. 5 is a schematic diagram illustrating the estimation result of a multi-source scene in the embodiment, where the abscissa is the direction of the source, and the ordinate is the power estimation result of the source in each experiment.

Detailed Description

Preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

Examples

Referring to fig. 1 to 4, an embodiment provides a vehicle positioning method based on linear array direction of arrival estimation, including the following steps: wherein the vehicle signal is a narrow-band incident signal,

and S1, symmetrically arranging linear arrays, namely a left array G and a right array J, at two sides of the road with the known road width b, and receiving the vehicle signal X by using the arrays at the two sides of the road.

Assuming that the known road distance is b, constructing uniform (sparse) linear arrays on two sides of a road: firstly, placing a uniform linear array formed by M array elements on the right side of a road, and marking as an array G, and then marking as an array J by using a linear array formed by M array elements; since both arrays work identically, only one array will be described. Suppose there are K far-field narrow-band signals, i.e., K vehicle signals, at an azimuth angle θ ═ θ₁，…，θ_KAnd when L snapshots are received, the received signals of the array G can be respectively expressed as:

X＝AS+N

where X is a reception signal of the array G, S is an incident signal waveform, a is an array manifold matrix of the array G, and a ═ a (θ)₁)，…，a(θ_K)]Is a guide of the array GIn the direction of the vector,

is the steering vector of the kth signal over array x,

(·)^Tand N is a noise matrix, and the noises received by different array elements are independent.

S2, calculating a covariance matrix R of each array receiving vehicle signals X: r ═ XX^H]^T＝ZPZ^H+ σ I, where Z is the eigenvector matrix of the denoised covariance matrix, (. C)^HFor the conjugate transpose operation of the matrix, S is the incident signal waveform, P is the covariance matrix of the incident signal, and P ═ E [ SS ]^H]＝diag(p)，p＝[p₁，…，p_K]^T，E[·]For the desired operation, σ is the noise power and I is the identity matrix.

S3, constructing a covariance matching model, and solving by using a non-convex algorithm to obtain a noise-free covariance matrix corresponding to an optimal solution

S31, establishing an atom set:

wherein d is an atom in the set of atoms, f is the normalized frequency,

being the phase of an atom, the atomic norm of the sparse signal x on the set of atoms D is defined as: | x | non-conducting phosphor_A＝inf{∑_kc_k:x＝∑_k c_ka_k，c_i>0，a_kE.g. A }, wherein c_kIs the weighting coefficient of the k-th atom, a_kIs the k-th atom selected from the atom set D, c_iAs a weighting coefficient, | · | | non-conducting phosphor_ARepresenting an atomic norm, inf representing an infimum boundary;

s32, according to the definition of the atomic norm, establishing the following atomic norm minimization problem:

wherein, δ is the upper bound of noise, y is the received observation signal, and Φ is a linear mapping matrix; solving the model to obtain the optimal solution

Establishing

Covariance matrix of

Wherein

Is expressed as a vector

The first row of the Tobraz matrix;

s33, the problem of atomic norm minimization is actually to restore the covariance matrix with a semi-positive definite tobutiz matrix structure first, so the problem is transformed into the following covariance matching model:

wherein | · | purple sweet_FFrobenius norm, rank (. cndot.) of the matrix, T (u) of a noise-free covariance matrix, R of a covariance matrix under finite snapshots,

is a sampling covariance matrix, and β is the fitting error; for covariance matching model, non-convex calculation is usedSolving by the method to obtain a noise-free covariance matrix corresponding to the optimal solution

The covariance matching model constructed in step S331 and step S33 is mathematically transformed as follows:

equivalence is expressed as the following low rank matrix recovery problem:

where T (u) is a noise-free covariance matrix, β is the fitting error, Q and

is an intermediate variable in the model;

s332, representing the low-rank matrix recovery problem obtained in the step S331 as a matrix filling problem according to a non-convex algorithm equivalent as follows:

wherein, two matrix sets R_L、R_HTwo matrix sets are as follows: r_L＝{L∶rank(L)≤K}，

Where T (u) is a noise-free covariance matrix, β is the fitting error, Q and

is an intermediate variable in the model; the matrices L, H belong to R respectively_L、R_HTheir update procedure is as follows:

wherein L is^(l)For iterating the low rank matrix of the L-th time, L^(l+1)For iteration l +1 timesA low rank matrix of (a); h^(l)For iteration the l-th semi-positive definite Tourette's matrix satisfying the constraints, H^(l+1)A positive semi-definite Tourette's matrix satisfying the constraint for the (l + 1) th iteration; delta₁And delta₂Is a parameter that is a function of,

is to the matrix set R_LThe mapping of (a) to (b) is,

wherein, U_KIs a left eigenvector matrix, Σ_kAs a matrix of eigenvalues, V_KIs a right eigenvector matrix, (.)^HA conjugate transpose operation for the matrix;

is to the matrix set R_HOf a set of matrices R_HIs a satisfying constraint

Of a semi-positive definite Topritz matrix, will therefore

The method comprises the following three steps: ensuring constraint conditions to be satisfied

Thereafter projecting the result of the previous step onto the Topritz matrix, P_T(X) T (v), and finally ensuring that the Topritz matrix of the previous step is a semi-positive definite matrix, namely P₊(X)＝Hdiag(ε₊)H^HWherein P is_Q(. is a projection onto the matrix Q, P_T(. is a projection onto the Tobraz matrix, P₊(. is a projection onto a semi-positive definite matrix,. epsilon₊The eigenvalue vector is obtained, wherein the eigenvalue of the solved matrix is reduced to 0 when being less than 0, H is a corresponding eigenvector matrix, beta is a fitting error, and T (v) is a mapped Tobraz matrix;

s333, obtained according to step S332, and the likeMatrix fill model for price transformation in conjunction with the above-described updating process and of matrix L, H

The optimization problem is solved to obtain a covariance matrix of the sparse signal x

S4 corresponding to the optimal solution

Then, the angle theta is recovered by using an angle estimation method₁，…，θ_KAnd the direction is the coming direction of the vehicle. The angle estimation method in step S4 employs the following method: tobraz matrix factorization theorem subspace class method or multiple signal classification subspace class method.

In step S4, taking the multi-signal classification as an example, the direction of arrival estimation is performed by the multi-signal classification, which specifically includes:

s41, drawing a virtual domain space spectrum P (u):

where θ is the assumed signal direction of arrival, E_nIs a matrix

A (θ) is the steering vector from the signal, a^H(θ) is a conjugate transpose vector of the steering vector from the signal;

s42, searching for peak values of space spectrum P (u) through spectrum peak search, arranging the peak values in the order from small to large, and taking angle values { theta (theta) { theta) corresponding to the first K peak values₁，…，θ_KAngle θ ═ θ₁，…，θ_KThe term is the coming direction of the vehicle.

S5, and combining the positions of the arrays G, J on the road with the vehicle directions obtained from the arrays G, J, respectively, and solving the coordinates of the vehicle by using trigonometric functions.

When the two roads areThe side arrays are all backward from the vehicle, a rectangular coordinate system is established by taking the linear array G on the left side of the road as the origin, the coordinates of the linear array J on the right side are (b, 0), and the coordinates of the k-th vehicle are (x)_k，y_k) Using trigonometric knowledge to set up the equation:

wherein tan θ_{k_1}、tanθ_{k_2}The directions of arrival of the vehicle to the left array and the right array are respectively; and (3) combining the two formulas to obtain the coordinates of the vehicle:

the vehicle positioning method based on the non-convex algorithm non-grid linear array direction of arrival estimation method fully utilizes the rapid convergence characteristic of the non-convex algorithm, can effectively reduce the calculation complexity, and improves the adaptability of the positioning method to high-speed driving scenes. Meanwhile, the method can also be applied to sparse linear arrays. The method of the embodiment makes full use of the high-degree-of-freedom characteristic of the sparse linear array, can position vehicles with more than the array number, improves the adaptability of the positioning method to high-density driving scenes, and can be used for high-density and high-speed driving scenes.

The effect of the present invention will be further described with reference to the simulation example. In tables 1, 2, 3 and 4 below, spa (sparse and Parametric approach): sparse and parametric methods, which are angle estimation methods, represent methods by which the position of a vehicle is detected; CMRA (Covariance Matrix Recontraction approach): the covariance matrix reconstruction method is an angle estimation method, which represents a method for detecting the position of a vehicle using the covariance matrix reconstruction method; proposed represents the procedure of the examples.

Simulation example 1: the array G and the array J are both 7-array element uniform linear arrays. Assuming that the number of incident far-field narrow-band coherent signals is 1, namely the number of vehicles in a driving scene is 1, and the incident directions to the left and right arrays are theta₁＝50°、θ₂30 ° as shown in fig. 4; two experiments were performed with signal-to-noise ratios set to-5 dB and 0dB, respectively, and a fast sampling beat number of 1000.

The results of comparing the estimation performance of the linear array wave direction estimation method with other existing methods are shown in tables 1 and 2. As can be seen from tables 1 and 2, the estimation performance and the time consumption of the method provided by the invention are superior to those of the existing method based on the covariance matching criterion, and the results in tables 1 and 2 show the effectiveness of the embodiment method.

TABLE 1 vehicle coordinates, error, time-consuming at SNR of-5 dB under uniform linear array

Method	Coordinate V/m	Error/cm	Time/s
				SPA	(32.6769，56.3477)	21.98	1.2971
CMRA	(32.6742，56.5317)	4.49	2.8329
				Proposed	(32.6662，56.5227)	3.41	0.014

TABLE 2 vehicle coordinates, error, time consumption at SNR of 0dB under uniform linear array

Method	Coordinate V/m	Error/cm	Time/s
				SPA	(32.6071，56.4647)	8.92	0.9699
CMRA	(32.6068，56.5059)	4.83	2.6759
				Proposed	(32.6089，56.5046)	4.74	0.0095

Simulation example 2: array G and array J are both 4-array element sparse linear arrays, as shown in FIG. 3. Assuming that the number of incident far-field narrow-band coherent signals is 1, namely the number of vehicles in a driving scene is 1, and the incident directions to the left and right arrays are theta₁＝50°、θ₂30 ° as shown in fig. 4; two experiments were performed with signal to noise ratios set to-5 dB and 0dB, respectively, with a snapshot of samplesThe number is 1000.

The results of comparing the estimated performance of the linear array direction of arrival estimation method with other existing methods are shown in tables 3 and 4. As can be seen from tables 3 and 4, the estimation performance and the time consumption of the embodiment method are better than those of the existing method based on the covariance matching criterion, and the results in tables 3 and 4 show the effectiveness of the embodiment method.

TABLE 3 vehicle coordinates, error, time consumption at signal-to-noise ratio of-5 dB for sparse linear arrays

Method	Coordinate V/m	Error/cm	Time/s
				SPA	(32.6559，56.3293)	21.72	0.8157
CMRA	(32.6426，56.5790)	6.06	0.7134
				Proposed	(32.6079，56.5074)	4.57	0.008

TABLE 4 vehicle coordinates, error, time consumption at SNR of 0dB under sparse linear array

Method	Coordinate V/m	Error/cm	Time/s
				SPA	(32.6419，56.5791)	6	0.8835
CMRA	(32.6455，56.6032)	8.68	0.734
				Proposed	(32.6365，56.5665)	4.21	0.06

The calculation complexity of the method of the embodiment can be greatly reduced compared with the existing methods SPA and CMRA based on the covariance matching criterion, because the problems of the SPA and CMRA based on the covariance matching criterion are focused on the convex problem and are solved based on the SDP (simplified programming) problem, and the SDP problem is mostly solved by using a tool kitThe solution, e.g., SDPT 3. However, in the atomic norm, the computational complexity of SDPT3 is as high as O ((N + L)²)²(N+L)^2.5) Wherein N is the number of array elements, and L is the number of fast beats; the embodiment method directly solves the non-convex problem, converts the covariance matching model into the matrix filling problem, and has the calculation complexity of O (PN)³) And P is iteration number, so that the characteristic of low time consumption is achieved.

Simulation example 3: the array G is a 4-array element sparse linear array, as shown in FIG. 3. Assuming that the number of incident far-field narrow-band coherent signals is 6, and the incident directions are [ -50 °, -30 °, -10 °, 10 °, 30 °, 50 ° ]; the signal-to-noise ratio was set to 5dB, the number of fast samples was 1000, and 500 independent experiments were performed.

The result of the linear array direction of arrival estimation method in the multi-source scene is shown in fig. 5. The source arrival estimation values and the power estimation results of each experiment are shown in fig. 5 in the form of scatter diagrams, wherein scatter points of different colors represent estimation values of arrival directions of different incident signals in each experiment, namely, points of six colors of red, green, blue, cyan, violet and yellow correspond to estimation values of-50 degrees, -30 degrees, -10 degrees, -30 degrees and 50 degrees respectively. The blue circles represent the incident signal in the simulation example, i.e., the power to which the blue circles are the true signal. It can be seen that the embodiment method can correctly estimate the incoming directions of these 6 signals. The characteristic is obtained by adapting SLA to a covariance matching model, and the sparse linear array has high-degree-of-freedom characteristic and can detect the number of information sources with more than array elements.

In conclusion, the method provided by the invention fully utilizes the rapid convergence of the non-convex algorithm, can effectively reduce the calculation complexity, and has stronger adaptability to high-speed driving scenes; meanwhile, the method provided by the invention can be suitable for the sparse linear arrays with different structures, and has stronger adaptability to high-density driving scenes.

Claims (10)

1. A vehicle positioning method based on linear array direction of arrival estimation is characterized in that: comprises the following steps of (a) carrying out,

s1, symmetrically placing linear arrays, namely a left array G and a right array J, close to the two sides of the road on the road with the known road width b, and respectively receiving vehicle signals X by using the arrays on the two sides of the road;

s2 and array G, J receive and process the signals X respectively, and calculate covariance matrix R of each array receiving vehicle signals X;

s3, constructing a covariance matching model, and solving by using a non-convex algorithm to obtain a noise-free covariance matrix corresponding to an optimal solution

S4 corresponding to the optimal solution

Then, the angle theta is recovered by using an angle estimation method₁，…，θ_KThe coming direction of the vehicle is obtained;

s5, and combining the positions of the arrays G, J on the road with the vehicle directions obtained from the arrays G, J, respectively, and solving the coordinates of the vehicle by using trigonometric functions.

2. The linear array direction of arrival estimation based vehicle localization method of claim 1, wherein: in step S1, linear arrays are symmetrically placed on both sides of the road with the known road width b, specifically: the linear array formed by the M array elements is placed on the left side of the road and recorded as an array G, and the linear array formed by the M array elements is placed on the right side of the road and recorded as an array J.

3. The linear array direction of arrival estimation based vehicle localization method of claim 1, wherein: in step S1, the operation mechanism of the array G and the array J for receiving the vehicle signal is the same; wherein the received vehicle signals using array G are: suppose there are K far-field narrow-band signals, i.e., K vehicle signals, at an azimuth angle θ ═ θ₁，…，θ_KIncident on the homogeneous array G at an angle of incidence; after receiving L snapshots, the received signals of array G are respectively represented as:x is AS + N, where X is a reception signal of the array G, S is an incident signal waveform, a is an array manifold matrix of the array G, and a is [ a (θ [) ]₁)，…，a(θ_K)]Is the steering vector of the array G,

is the steering vector of the kth signal on array G,

(·)^Tand N is a noise matrix, and the noises received by different array elements are independent.

4. A method for vehicle localization based on linear array direction of arrival estimation according to any of claims 1-3, characterized by: in step S2, the covariance matrix R of each array received vehicle signal X is calculated: r ═ XX^H]^T＝ZPZ^H+ σ I, where Z is the eigenvector matrix of the denoised covariance matrix, (. C)^HFor the conjugate transpose operation of the matrix, S is the incident signal waveform, P is the covariance matrix of the incident signal, and P ═ E [ SS ]^H]＝diag(p)，p＝[p₁，…，p_K]^T，E[·]For the desired operation, σ is the noise power and I is the identity matrix.

5. A method for vehicle localization based on linear array direction of arrival estimation according to any of claims 1-3, characterized by: in step S3, a covariance matching model is constructed and solved by a non-convex algorithm to obtain a noise-free covariance matrix corresponding to an optimal solution

In particular to a method for preparing a high-performance nano-silver alloy,

s31, establishing an atom set:

wherein d is an atom in the set of atoms, f is the normalized frequency,

being the phase of an atom, the atomic norm of the sparse signal x on the set of atoms D is defined as: | x | non-conducting phosphor_A＝inf{∑_kc_k：x＝∑_kc_ka_k，c_i＞0，a_kE.g. A }, wherein c_kIs the weighting coefficient of the k-th atom, a_kIs the k-th atom selected from the atom set D, c_iAs a weighting coefficient, | · | | non-conducting phosphor_ARepresenting an atomic norm, inf representing an infimum boundary;

s32, according to the definition of the atomic norm, establishing the following atomic norm minimization problem:

s.t.||y-Φx||₂delta is less than or equal to delta, wherein delta is the upper bound of noise, y is the received observation signal, and phi is a linear mapping matrix; solving the model to obtain the optimal solution

Establishing

Covariance matrix of

Wherein

Is expressed as a vector

The first row of the Tobraz matrix;

s33, the atomic norm minimization problem is actually the first recovery of the positive half-definite Tourette' S momentCovariance matrix of the array structure, therefore the problem translates to a covariance matching model as follows: min_urank[T(u)]，

T (u) is greater than or equal to 0, wherein | · | | non-woven phosphor_FFrobenius norm, rank (. cndot.) of the matrix, T (u) of a noise-free covariance matrix, R of a covariance matrix under finite snapshots,

is a sampling covariance matrix, and β is the fitting error; solving the covariance matching model by using a non-convex algorithm to obtain a noise-free covariance matrix corresponding to the optimal solution

6. The linear array direction of arrival estimation based vehicle localization method of claim 5, wherein: in step S3, the covariance matching model is solved by a non-convex algorithm, specifically,

the covariance matching model constructed in step S331 and step S33 is mathematically transformed as follows:

equivalence is expressed as the following low rank matrix recovery problem: min_urank[T(u)]，

T (u) ≧ 0, where T (u) is a noise-free covariance matrix, β is a fitting error, Q and

is an intermediate variable in the model;

s332, a stepS331, the low-rank matrix recovery problem is expressed as the following matrix filling problem according to the non-convex algorithm equivalence:

wherein, two matrix sets R_L、R_HTwo matrix sets are as follows: r_L＝{L：rank(L)≤K}，

Where T (u) is a noise-free covariance matrix, β is the fitting error, Q and

is an intermediate variable in the model; the matrices L, H belong to R respectively_L、R_HTheir update procedure is as follows:

juqueshi, L^(l)For iterating the low rank matrix of the L-th time, L^(l+1)A low rank matrix that is iterated the (l + 1) th time; h^(l)For iteration the l-th semi-positive definite Tourette's matrix satisfying the constraints, H^(l+1)A positive semi-definite Tourette's matrix satisfying the constraint for the (l + 1) th iteration; delta₁And delta₂Is a parameter that is a function of,

is to the matrix set R_LThe mapping of (a) to (b) is,

wherein, U_kIs a left eigenvector matrix, Σ_kAs a matrix of eigenvalues, V_KIs a right eigenvector matrix, (.)^HA conjugate transpose operation for the matrix;

is to the matrix set R_HOf a set of matrices R_HIs a matrix satisfying the equationBundle of

Of a semi-positive definite Topritz matrix, will therefore

The method comprises the following three steps: ensuring constraint conditions to be satisfied

Thereafter projecting the result of the previous step onto the Topritz matrix, P_T(L₁)＝T(v)＝L₂Finally, ensuring that the Topriz matrix of the last step is a semi-positive definite matrix, namely P₊(L₂)＝Hdiag(ε₊)H^H＝L₃Wherein P is_Q(. is) a projection onto the matrix Q, L₁Projecting the obtained matrix for the step; p_T(. is a projection onto the Tobraz matrix, L₂Projecting the obtained Tobraz matrix for the step; p₊(. is) a projection onto a semi-positive definite matrix, L₃For the semi-positive definite matrix, epsilon, obtained by projection of this step₊The eigenvalue vector is obtained, wherein the eigenvalue of the solved matrix is reduced to 0 when being less than 0, H is a corresponding eigenvector matrix, beta is a fitting error, and T (v) is a mapped Tobraz matrix;

s333, updating process and of combining the matrix L, H according to the equivalent transformed matrix filling model obtained in the step S332

The optimization problem is solved to obtain a covariance matrix of the sparse signal x

7. A method for vehicle localization based on linear array direction of arrival estimation according to any of claims 1-3, characterized by: in the step S1, the array G and the array J adopt a uniform linear array and a sparse array, wherein the sparse array includes a minimum redundant array, a co-prime array, a nested array or a minimum hole array.

8. A method for vehicle localization based on linear array direction of arrival estimation according to any of claims 1-3, characterized by: the angle estimation method in step S4 employs the following method: tobraz matrix factorization theorem subspace class method or multiple signal classification subspace class method.

9. A method for vehicle localization based on linear array direction of arrival estimation according to any of claims 1-3, characterized by: in step S4, the angle θ is restored by the angle estimation method as { θ ═ θ }₁，…，θ_KThe method comprises the following steps:

s41, drawing a virtual domain space spectrum P (u):

where θ is the assumed signal direction of arrival, E_nIs a matrix

A (θ) is the steering vector from the signal, a^H(θ) is a conjugate transpose vector of the steering vector from the signal;

s42, searching for peak values of space spectrum P (u) through spectrum peak search, arranging the peak values in the order from small to large, and taking angle values { theta (theta) { theta) corresponding to the first K peak values₁，…，θ_KAngle θ ═ θ₁，…，θ_KThe term is the coming direction of the vehicle.

10. A method for vehicle localization based on linear array direction of arrival estimation according to any of claims 1-3, characterized by: in step S6, the vehicle directions and the positions of the array G, J on the road are obtained by combining the vehicles obtained by the array G, J, and the vehicle coordinates are solved by using a trigonometric function, specifically:

the uniform linear array G on the left side of the road isA rectangular coordinate system is established at the origin, the coordinates of the uniform linear array J on the right side are (b, 0), and the coordinate of the k-th vehicle is (x)_k，y_k) Using trigonometric knowledge to set up the equation:

wherein tan θ_{k_1}、tanθ_{k_2}The directions of arrival of the vehicle to the left array and the right array are respectively; and (3) combining the two formulas to obtain the coordinates of the vehicle:

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