阅读说明：本技术 基于欠定信源克拉美罗界的运动线性稀疏阵列优化方法 (Motion linear sparse array optimization method based on underdetermined information source Cramer-Rao bound ) 是由 秦国栋 刘韦辰 鲍丹 武斌 蔡晶晶 刘高高 李鹏 于 2020-07-22 设计创作，主要内容包括：本发明公开了一种欠定信源下最优运动线性稀疏阵列的设计方法,解决了获取欠定信源下满足孔径和阵元数要求的波达方向估计克拉美罗界(CRB)最小的运动稀疏阵列问题。实现步骤：得到运动线性稀疏阵列的差分阵列D<Sub>c</Sub>及其阵列流型矩阵A<Sub>c</Sub>；计算运动线性稀疏阵列关于波达方向估计的CRB；用改进遗传算法对线性稀疏阵列优化,得到最优阵列结构S<Sub>opt</Sub>。本发明提出了欠定信源下运动线性稀疏阵列关于波达方向估计的克拉美罗界表达式。引入锦标赛选择法,并在交叉变异过程中采用择优策略,避免陷入局部收敛。本发明得到的最优阵列关于波达方向估计的CRB更小,提高了DOA估计性能。信号环境变化时易调整为最优阵列结构。用于高精度的波达方向估计。(The invention discloses a design method of an optimal motion linear sparse array under an underdetermined information source, which solves the problem of obtaining a motion sparse array with the minimum Clalmelo bound (CRB) of estimation of the direction of arrival satisfying the requirements of aperture and array element number under the underdetermined information source. The implementation steps are as follows: differential array D for obtaining motion linear sparse array c And its array flow pattern matrix A c (ii) a Calculating a CRB of the motion linear sparse array with respect to the direction of arrival estimation; optimizing the linear sparse array by using an improved genetic algorithm to obtain an optimal array structure S opt . The invention provides a Cramer-Rao bound expression of a linear sparse array under an underdetermined information source and related to estimation of a direction of arrival. A championship selection method is introduced, and a preference strategy is adopted in the cross variation process, so that the situation of local convergence is avoided. Optimal array obtained by the inventionThe CRB for direction of arrival estimation is smaller, improving DOA estimation performance. And the signal environment is easy to adjust to an optimal array structure when changed. The method is used for high-precision direction-of-arrival estimation.)

1. A motion linear sparse array optimization method based on an underdetermined information source Clarithrome boundary is characterized by comprising the following steps:

1): establishing a mathematical model of a motion linear sparse array and an array flow pattern matrix thereof: the detailed process is as follows

(1.1): parameter setting and initialization: setting an array aperture, an array element number, a direction finding range and an information source distribution condition according to the application requirement of positioning, and initializing parameters of the array aperture, the array element number, the direction finding range and the information source distribution condition;

(1.2): establishing a motion linear sparse array mathematical model: randomly generating N sparse arrays satisfying the requirements of aperture and array element numberWherein Amp represents array aperture, L represents array element number, and C represents combination number operator; for each sparse array, passive aperture synthesis is carried out through motion to obtain a synthetic array S_cDifferential array structure D of_cThe mathematical model of the composite array and its differential array is represented as follows:

D_c＝n₃-n₄n₃,n₄∈S_c

wherein S is_cThe array element coordinate of the array S is expressed by n, and S expresses a sparse array; n is₃、n₄Representing a synthetic array S_cCoordinate of (c) ()_cA parameter index representing a synthetic array;

(1.3): establishing an array flow pattern matrix mathematical model of a moving linear sparse array: in case of Q number of sourcesNext, the array flow type matrix A of the linear sparse array moving under the established underdetermined information source_cIs represented as follows:

wherein a is_c(θ_q) A steering vector of a signal source Q under a synthesis array is shown, wherein Q is 1, … and Q; a is_c(θ_q)＝[a^T(θ_q),b^T(θ_q)]^T＝[1,u₂(θ_q),…,u_L(θ_q),u_d(θ_q),u₂(θ_q)u_d(θ_q),…,u_L(θ_q)u_d(θ_q)]^T，For the synthesis of array element, ()^TRepresents a transpose of a matrix; u. of₍₎Denotes the amount of phase shift, u_l(θ_q)＝exp(-j2πd_lsin(θ_q)/λ)，u_d(θ_q)＝exp(-j2πdsin(θ_q)/λ)，d_lIs the coordinate of the ith array element, belongs to 1,2, … L, and L is the number of the original array element; d is the unit array element distance,

2): calculating a Cramer-Rao bound CRB of the linear sparse array of motion under the underdetermined source with respect to the direction of arrival angle: for each linear sparse array generated in the mathematical model for establishing the moving linear sparse array and the array flow pattern matrix thereof, according to the differential array structure D under the moving condition_cAnd array flow pattern matrix A_cThe cramer-perot boundary CRB (θ) is calculated for its direction of arrival angle.

3): linear sparse array optimization using improved genetic algorithm: minimizing the CRB described above, the following expression embodies the optimization of a linear sparse array:

wherein Tr () represents the trace of the matrix;

selecting a linear sparse array S generated in a mathematical model for establishing a motion linear sparse array and an array flow pattern matrix thereof as an individual in a population, and taking a corresponding CRB as the fitness of the individual; when the initial generation population is selected, only the linear sparse array with the CRB smaller than the set threshold value of the genetic algorithm can become individuals in the initial generation population; repeatedly executing crossing, variation and selection operations according to a genetic algorithm flow, wherein a tournament selection method is adopted as a selection mechanism; finally obtaining the optimal solution S of the motion linear sparse array under the underdetermined information source_opt。

2. The method for optimizing a linear sparse array in motion based on an underdetermined source cramer-circle as claimed in claim 1, wherein said calculating cramer-circle CRB (θ) of the under-determined source sparse array with respect to the direction of arrival angle in step 2) comprises the following steps:

(2.1): calculating a differential array flow type matrix and an augmentation matrix of the motion linear sparse array: let theta_qThe Q direction of arrival angle of the qth signal source, Q is 1, …, Q, and the differential array steering vector of the qth signal source is

A differential array flow pattern matrix of

Corresponding augmentation matrix is

<>_γRepresenting the gamma-th element of the vector, m being the difference array D_cArray element coordinates of (1);

(2.2): computing array signal covariance matrices

then the covariance matrix of the array signal is

J_:,z＝vec(K(z))，z∈D

D is a differential array of the linear sparse array, z represents the array element coordinates of the differential array of the linear sparse array, vec () represents vectorization of the matrix, namely the matrix is arranged into column vectors according to the column weight; k (z) is as defined

Wherein<>_i,jRepresenting the ith row and jth column element of the matrix, r1 and r2 representing the array element coordinates of the sparse array S, ()_:,wRepresents the w-th column of the matrix;

(2.3): computing a matching matrixQuadrature complement ofSum signal power matrix G₀：

Wherein (C)^HRepresents a conjugate transpose of the matrix;

(2.4): calculating a Cramer-Rao bound CRB of the linear sparse array of motion under the underdetermined source with respect to the direction of arrival angle:

wherein L is_SIndicating a fast beat number.

3. The method for optimizing a linear sparse array based on motion of an underdetermined source cramer-perot as claimed in claim 1, wherein said optimizing the linear sparse array by using an improved genetic algorithm in step 3) comprises the following steps:

let N_pIndicating the size of the population, i.e. the total number of individuals in the population, p_sIndicates the selection probability, N_cDenotes the number of crossovers, p_mRepresenting the mutation probability; n is a radical of_iDenotes the total number of iterations, μ denotes the initial CRB value, p_cExpressing the preferential probability; the linear sparse array optimization based on the improved genetic algorithm comprises the following specific steps:

(3.1): initializing the first generation population P₀: randomly generating N_pA CRB is smaller than mu, and the current iteration number i is made to be 0;

(3.2): entering an algorithm iteration process;

(3.3): finding the optimal solution of the contemporary population: calculating the CRB of each linear sparse array, finding the minimum CRB in the current generation population, and recording the corresponding linear sparse array;

(3.4): generation of next generation population P using tournament selection_k: for each selection ∈ N_pFrom P to P_kSelecting N in generation_i＝round(N_p×p_s) Individual, using N_iUpdating P for the individual with the least CRB among individuals_k；

(3.5): for updated P_kAnd (3) executing a cross operation: for each operation ∈ N_cFrom P_kTwo individuals are selected, and the array element positions of the two arrays are crossed randomly to select a preferred probability p_cUpdating P_k；

(3.6): performing mutation operation: to P_kEach individual in the group generates a random number gamma of between 0 and1, if gamma is less than or equal to p_mRandomly selecting one of the 2 nd to the L-1 th array elements, randomly selecting a bit at the position xi, and obtaining xi after negation_mWith a preferred probability p_cUpdating P_k；

(3.7): judging whether to jump out of iteration: if i < N_iIf so, making i equal to i +1, returning to (3.2), otherwise, jumping out of iteration, and executing the step (3.8);

(3.8): finding a global optimal solution: after iteration is finished, the minimum CRB in all generations is found, and the corresponding individual is the optimal solution S_opt；

(3.9): obtaining the optimal array structure S_optThe optimal linear sparse array structure under the underdetermined information source;

finally, the output S of the genetic algorithm is improved_optNamely the optimal linear sparse array under the underdetermined information source of the multi-sensor motion platform.

Technical Field

The invention belongs to the technical field of direction of arrival estimation, mainly relates to the design of a direction of arrival estimation sparse array, and particularly relates to a motion linear sparse array optimization method based on an underdetermined information source Clarmero bound. The method is suitable for the design problem of the optimal motion linear sparse array under the underdetermined information source.

Background

The direction finding technology based on the sparse array structure is widely applied to the fields of communication, radar, sonar, satellite navigation, radio telescopes and the like. Compared to a uniform array, a sparse array has a higher degree of freedom and a larger array aperture for the same number of sensors. Common sparse arrays include co-prime arrays, Minimum Redundant Arrays (MRA), minimum aperture arrays (MHA), and Nested Arrays (NA). In designing nested and co-prime arrays, it is generally considered to use O (N) sensors to estimate O (N)²) Uncorrelated far-field narrow-band sources, where O (N) denotes order N, O (N)²) Represents N²And (4) carrying out step.

However, sparse arrays tend to have some limitations, such as the presence of some holes in differential arrays of co-prime arrays and minimum hole arrays, which greatly limits the direction-finding performance. In order to solve the problem, a sparse array is carried on a motion platform, time gain is converted into space gain by using a passive synthetic aperture technology, and the original hole positions are filled by using the motion characteristic of motion, so that the array freedom degree is improved, the direction-finding performance of the system is improved, and support is provided for high-precision positioning.

In the estimation problem, a quantitative comprehensive evaluation index is needed to measure the estimation performance. The Cramer-Rao Bound (CRB) is the lower limit of variance for any unbiased estimator. That is, it is impossible to obtain an unbiased estimate with a variance less than the lower limit; the Cramer-Lo boundary provides a standard for comparing the performance of unbiased estimates, and is a commonly used performance evaluation index.

The CRB is directly related to the inverse of the Fisher Information Matrix (FIM), which contains information for all unknown parameters. For typical applications, the number of sources under a uniform line array is generally less than the number of sensors. For the case of sparse arrays, Liu et al, in the references [ C. -L.Liu, P.Vaidyanathan, Cramer-Rao bases for coprime and others for space arrays, while white find more than one sources sensors, Digital Signal processing61.doi:10.1016/j.dsp.2016.04.011 ], gave specific expressions of CRBs under conditions where the number of sources is greater than the number of sensors, and demonstrated that CRBs are present in the presence of an augmented array flow matrix.

Sparse arrays (such as co-prime arrays, nested arrays, etc.) mainly studied and used at present can obtain higher degree of freedom by utilizing the characteristics of motion, so as to obtain better super-resolution direction finding performance. But the structure of these sparse arrays is generally fixed when the aperture and array element number are determined. Although their degrees of freedom can be improved by the motion of the arrays, this does not mean that the arrays are optimal for estimating performance under an underdetermined source. That is, CRBs of these arrays with respect to the direction of arrival angle under an underdetermined source are not the smallest of all arrays satisfying the aperture and array element number constraints, so that the direction of arrival estimation performance of the arrays is not excellent enough.

Disclosure of Invention

The invention aims to provide a motion linear sparse array optimization method which meets the requirements of aperture and array element number and has the minimum CRB (minimum cross correlation) under an underdetermined information source aiming at a motion linear sparse array.

The invention relates to a motion linear sparse array optimization method based on an underdetermined information source Cramer Row bound, which is characterized by comprising the following steps of:

1): establishing a mathematical model of a motion linear sparse array and an array flow pattern matrix thereof: the detailed process is as follows

(1.1): parameter setting and initialization: setting an array aperture, an array element number, a direction finding range and an information source distribution condition according to the application requirement of positioning, and initializing parameters of the array aperture, the array element number, the direction finding range and the information source distribution condition;

(1.2): establishing a motion linear sparse array mathematical model: randomly generating N sparse arrays satisfying the requirements of aperture and array element number

Wherein Amp represents array aperture, L represents array element number, and C represents combination number operator; for each sparse array, passive aperture synthesis is carried out through motion to obtain a synthetic array S_cDifferential array structure D of_cThe mathematical model of the composite array and its differential array is represented as follows:

D_c＝n₃-n₄n₃,n₄∈S_c

wherein S is_cThe array element coordinate of the array S is expressed by n, and S expresses a sparse array; n is₃、n₄Representing a synthetic array S_cCoordinate of (c) ()_cA parameter index representing a synthetic array;

(1.3): establishing an array flow pattern matrix mathematical model of a moving linear sparse array: under the condition that the number of the information sources is Q, the established array flow type matrix A of the sparse array under the underdetermined information sources_cIs represented as follows:

wherein a is_c(θ_q) A steering vector of a signal source Q under a synthesis array is shown, wherein Q is 1, … and Q; a is_c(θ_q)＝[a^T(θ_q),b^T(θ_q)]^T＝[1,u₂(θ_q),…,u_L(θ_q),u_d(θ_q),u₂(θ_q)u_d(θ_q),…,u_L(θ_q)u_d(θ_q)]^T，For the synthesis of array element, ()^TRepresents a transpose of a matrix; u. of₍₎Denotes the amount of phase shift, u_l(θ_q)＝exp(-j2πd_lsin(θ_q)/λ)，u_d(θ_q)＝exp(-j2πdsin(θ_q)/λ)，d_lIs the coordinate of the ith array element, belongs to 1,2, … L, and L is the number of the original array element; d is the unit array element distance,lambda is the signal wavelength and pi is the circumference ratio; theta_qThe direction of arrival angle of the qth signal source is Q1, …, Q.

2): calculating a Cramer-Rao bound CRB of the linear sparse array of motion under the underdetermined source with respect to the direction of arrival angle: for each linear sparse array generated in the mathematical model for establishing the moving linear sparse array and the array flow pattern matrix thereof, according to the differential array structure D under the moving condition_cAnd array flow pattern matrix A_cThe cramer-perot boundary CRB (θ) is calculated for its direction of arrival angle.

3): sparse arrays were optimized using improved genetic algorithms: minimizing the CRB described above, the following expression embodies the optimization of sparse arrays:

wherein Tr () represents the trace of the matrix;

selecting a sparse array S generated in a mathematical model for establishing a motion linear sparse array and an array flow pattern matrix thereof as an individual in a population, and taking a corresponding CRB as the fitness of the individual; when the initial generation population is selected, only the linear sparse array with the CRB smaller than the set threshold value of the genetic algorithm can become individuals in the initial generation population; repeatedly executing crossing, variation and selection operations according to a genetic algorithm flow, wherein a tournament selection method is adopted as a selection mechanism; finally obtaining the optimal solution S of the motion linear sparse array under the underdetermined information source_opt。

The method solves the problem that the fixed structure sparse array estimation performance is not excellent enough under the condition of underdetermined information sources.

Compared with the prior art, the invention has the following advantages:

obtaining a sparse array with better estimation performance under the condition that a signal source is underdetermined: compared with sparse arrays with fixed structures such as co-prime arrays and nested arrays, the method provided by the invention is implemented under the condition that the aperture and the array element number are the same, so that the array with better estimation performance under the underdetermined information source can be obtained, and support is provided for high-precision positioning under the underdetermined information source.

The array structure is easy to adjust when the signal environment changes: when the signal environment changes, the motion linear sparse array of the fixed structure can only maintain the original estimation performance by increasing the array aperture or array element number; the method does not need to increase the aperture and the array element number of the array, and only needs to train the optimal sparse array under all possible signal environments in advance according to the method and store the structure of the optimal sparse array; when the signal environment changes, the original sparse array is changed into the corresponding optimal sparse array by changing the arrangement condition of the array elements, so that the original excellent estimation performance can be maintained, and the method is relatively more flexible in practical application.

Description of the drawings:

FIG. 1 is a flow chart of an implementation of the present invention.

FIG. 2 is a CRB plot of the optimal array obtained by the present invention versus the conventional sparse array for direction of arrival angle at different snapshot numbers.

FIG. 3 is a diagram of the position comparison of the optimal array and its differential array with the conventional sparse array elements obtained by the present invention under different snapshot numbers.

FIG. 4 is a graph of the CRB versus the Direction of arrival angle for different signal-to-noise ratios for the optimal array obtained by the present invention versus a conventional sparse array.

FIG. 5 is a diagram of the position comparison of the optimal array and its differential array with the conventional sparse array element under different signal-to-noise ratios obtained by the present invention.

The specific implementation mode is as follows:

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