阅读说明：本技术 一种基于稀疏重构的无需网格化的近场信号源定位方法 (Near-field signal source positioning method based on sparse reconstruction and without gridding ) 是由 左炜亮 肖同 辛景民 郑南宁 于 2020-05-25 设计创作，主要内容包括：本发明公开了一种基于稀疏重构的无需网格化的近场信号源定位方法,具体为：基于对称均匀线性阵列接收到的信号模型获取协方差矩阵<Image he="67" wi="60" file="DDA0002507419130000011.GIF" imgContent="drawing" imgFormat="GIF" orientation="portrait" inline="no"></Image>提取协方差矩阵<Image he="56" wi="34" file="DDA0002507419130000017.GIF" imgContent="drawing" imgFormat="GIF" orientation="portrait" inline="no"></Image>的反对角线元素构建类似Toeplitz矩阵的相关矩阵<Image he="64" wi="55" file="DDA0002507419130000012.GIF" imgContent="drawing" imgFormat="GIF" orientation="portrait" inline="no"></Image>基于协方差拟合准则构造目标函数,使用半正定规划估计Toeplitz相关矩阵<Image he="67" wi="58" file="DDA0002507419130000013.GIF" imgContent="drawing" imgFormat="GIF" orientation="portrait" inline="no"></Image>基于Toeplitz相关矩阵<Image he="60" wi="35" file="DDA0002507419130000014.GIF" imgContent="drawing" imgFormat="GIF" orientation="portrait" inline="no"></Image>使用Prony方法估计近场信号源的波达方向<Image he="63" wi="57" file="DDA0002507419130000015.GIF" imgContent="drawing" imgFormat="GIF" orientation="portrait" inline="no"></Image>再使用常规的基于子空间的方法有效地估计范围<Image he="59" wi="53" file="DDA0002507419130000016.GIF" imgContent="drawing" imgFormat="GIF" orientation="portrait" inline="no"></Image>可以采用MUSIC方法估计范围<Image he="50" wi="46" file="DDA0002507419130000018.GIF" imgContent="drawing" imgFormat="GIF" orientation="portrait" inline="no"></Image>本发明通过提取信号协方差矩阵反对角线元素构造类似Toeplitz矩阵的相关矩阵,建立针对该相关矩阵重构的凸优化问题,基于恢复的相关矩阵使用Prony方法估计近场信号源的波达方向,避免迭代运算带来的高计算复杂度问题的同时避免传统网格化方法的网格失配问题。(The invention discloses a near field signal source positioning method based on sparse reconstruction and without gridding, which specifically comprises the following steps: obtaining a covariance matrix based on a signal model received by a symmetric uniform linear array Extracting covariance matrices Constructing a correlation matrix similar to the Toeplitz matrix Constructing an objective function based on a covariance fitting criterion, and estimating a Toeplitz correlation matrix by using a semi-definite programming Based on Toeplitz correlation matrix Estimating direction of arrival of near field signal source using Prony method Re-use of conventional subspace-based approaches to efficiently estimate range The range can be estimated using the MUSIC method According to the method, a correlation matrix similar to a Toeplitz matrix is constructed by extracting the anti-diagonal elements of a signal covariance matrix, a convex optimization problem aiming at the reconstruction of the correlation matrix is established, the arrival direction of a near-field signal source is estimated by using a Prony method based on the recovered correlation matrix, the problem of high calculation complexity caused by iterative operation is avoided, and meanwhile the problem of grid mismatch of a traditional gridding method is avoided.)

1. A near-field signal source positioning method based on sparse reconstruction and without gridding is characterized by comprising the following steps:

step 1, obtaining a covariance matrix based on a signal model received by a symmetrical uniform linear arrayExtracting covariance matrices

step 2, based on the correlation matrix similar to the Toeplitz matrix obtained in the step 1

Step 3, based on the correlation matrix obtained in step 2Estimating direction of arrival of near field signal source using Prony method

2. The sparse reconstruction-based near-field signal source localization method without gridding according to claim 1, wherein the step 1 is as follows:

it is to set K near-field narrow-band incoherent signal sources to be incident on a symmetrical uniform linear array with 2M +1 omnidirectional sensor array elements, K is less than M, the distance between the array elements is d, and the signal received by the array element M is:

wherein M ═ M, · 0,. 0, M, s_k(n) denotes the kth incident signal, ω_m(n) represents the noise on array element m, where:

the signal received by the symmetric uniform linear array is modeled using a matrix as follows:

in the formula

solving a covariance matrix based on the signal model of expression (3)

Where N represents the number of samples and x (N) represents the data received by the array.

3. The sparse reconstruction-based near-field signal source localization method without gridding according to claim 1, wherein in step 1, the method is based on covariance matrix

the sum of R is theoretical value, the covariance matrix R is 2M +1 dimensional matrix, and the calculation formula can be expressed as

R＝E{x(n)x^H(n)}＝AR_sA^H+σI_2M+1(5)

In the formula

the spatial correlation is derived from the elements on the inverse diagonals of the covariance matrix R:

wherein i ═ 2M, · 1, 0, 1,. 2M, p ═ M + i₂，...，-1，0，-1，M-i₁Where i is₁＝0.5(|i|+i)，i₂＝0.5(|i|-i)，γ_ik＝Ψ_k-iΦ_k；

If i is equal to 0, then i₁＝i₂Constructing an M + 1-dimensional Toeplitz matrix as 0

Substituting formula (6) for formula (7), the Toeplitz matrix can be expressed as:

＝B(θ)R_sB^H(θ)+σI_M+1(8)

in the formula:

4. the sparse reconstruction-based near-field signal source localization method without gridding according to claim 1, wherein the step 2 specifically comprises the following steps:

the following semi-positive definite programming problem (SDP) was constructed from the covariance fitting criterion:

if the sampling number N is more than or equal to 2M +1, the SDP problem is as follows:

if the sampling number N is less than 2M + i, the SDP problem is as follows:

in the formulaT (u) is a Toeplitz matrix, and a correlation matrix can be obtained by solving the semi-definite programming problem

5. The sparse reconstruction-based near-field signal source localization method without gridding according to claim 1, wherein the covariance fitting criterion is:

if the sampling number N is more than or equal to 2M +1:

if the sampling number N is less than 2M +1:

expression (12) and expression (13) wherein θ ═ θ₁，θ₂，...，θ_K]Representing the direction of arrival of K incident signals, { s_k(n)}，σ is the variance of additive noise, and equations (12) and (13) are simplified to obtain:

wherein

6. The sparse reconstruction-based near-field signal source localization method without gridding according to claim 1, wherein in step 3, the method is based on Toeplitz correlation matrixEstimating direction of arrival of near field signal source

301, Toeplitz correlation matrix

＝B(θ)R_sB^H(θ)+σI_M+1

Then there is

in the formula { }^*Represents a conjugate, { }_{{2，...，M+1}}Indicating that the first row of the matrix is removed, and taking the rest elements; the expected direction of arrival is obtained by directly using the Prony method to solve equation (16)

302, get the direction of arrivalThen obtaining the number K of the incident signals, and decomposing the covariance matrix through the eigenvalue under the condition that the number K of the incident signals is known

covariance matrix

estimating distance parameters with known directions of arrival

Technical Field

The invention belongs to the technical field of array signal processing, and particularly relates to a near-field signal source positioning method based on sparse reconstruction and without meshing.

Background

The amount of computation caused by the complexity of the near field problem is usually large, and how to reduce the amount of computation while ensuring the estimation accuracy is a direction of continuous research in the field of near field source positioning. The two-dimensional MUSIC algorithm is to popularize the traditional one-dimensional MUSIC method to two-dimensional parameter estimation, and the method needs to search two parameters of the direction of arrival and the distance, so that the calculation burden is large. The maximum likelihood estimation method has good statistical performance, but generally needs to optimize a highly nonlinear cost function, and needs to iterate continuously, so that the calculation amount is huge. By gridding the surrounding space of the receiving sensor array and estimating the direction of arrival and the position of the signal source by using a grid matching method, the method greatly reduces the calculated amount, but when the actual signal is not on the grid, the error which cannot be overcome by a positioning algorithm can be caused.

Disclosure of Invention

The invention aims to overcome the defects in the prior art, provides a sparse reconstruction-based near-field signal source positioning method for a symmetrical and uniform linear array, which is free from gridding, ensures the precision, avoids huge calculation amount brought by an iterative algorithm, and avoids the problem of grid mismatch.

In order to achieve the purpose, the invention adopts the following technical scheme: a near-field signal source positioning method based on sparse reconstruction and without gridding is characterized by comprising the following steps:

step 1, obtaining a covariance matrix based on a signal model received by a symmetrical uniform linear array

Extracting covariance matricesUsing covariance matrix

Constructing a correlation matrix similar to the Toeplitz matrixThe symmetrical uniform linear array comprises 2M +1 omnidirectional sensor array elements;

step 2, based on the correlation matrix similar to the Toeplitz matrix obtained in the step 1

Constructing an objective function by adopting a covariance fitting criterion, and estimating a correlation matrix by using a semi-positive definite program

Step 3, based on the correlation matrix obtained in step 2Estimating direction of arrival of near field signal source using Prony methodRe-use of subspace-based methods to estimate rangeAngle of rotationAs the information on the orientation of the incident signal,the incident signal is a near-field narrow-band incoherent signal, which is distance information of the incident signal.

The step 1 is as follows:

it is to establish K near-field narrowband incoherent signal sources to incide to the even linear array of symmetry that possess 2M +1 omnidirectional transducer array elements, K < M, and the interval is d between the array element, and the signal that array element M received is:

wherein M ═ M, · 0,. 0, M, s_k(n) represents the kth incident signal, ω m (n) represents the noise on array element m, where:

the signal received by the symmetric uniform linear array is modeled using a matrix as follows:

in the formula The mean value of the additive noise omega (n) is 0, sigma is the variance of the additive noise, and sigma is equal to the incident signal { s }_k(n) uncorrelated, array response matrices

Wherein a is(θ_K，Here (.)^TRepresenting a matrix transposition;

solving a covariance matrix based on the signal model of expression (3)

Where N represents the number of samples and x (N) represents the data received by the array.

In step 1, based on covariance matrixConstruction of a correlation matrix resembling the Toeplitz matrix by the elements of the anti-diagonals

The process is as follows:

the sum of R is theoretical value, the covariance matrix R is 2M +1 dimensional matrix, and the calculation formula can be expressed as

R＝E{x(n)x^H(n)}＝AR_sA^H+σI_2M+1(5)

In the formula I_mIs an m-dimensional unit matrix, E { } represents expectation, and { }^*Represents a conjugate, { }^HRepresents a conjugate transpose;

the spatial correlation is derived from the elements on the inverse diagonals of the covariance matrix R:

wherein i ═ 2M, · 1, 0, 1,. 2M, p ═ M + i₂，...，-1，0，-1，M-i₁Where i is₁＝0.5(|i|+i)，i₂＝0.5(|i|-i)，γ_ik＝Ψ_k-iΦ_k；

If i is equal to 0, then i₁＝i₂Constructing an M + 1-dimensional Toeplitz matrix as 0

Substituting formula (6) for formula (7), the Toeplitz matrix can be expressed as:

＝B(θ)R_sB^H(θ)+σI_M+1(8)

in the formula:

the step 2 specifically comprises the following steps:

the following semi-positive definite programming problem (SDP) was constructed from the covariance fitting criterion:

if the sampling number N is more than or equal to 2M +1, the SDP problem is as follows:

if the sampling number N is less than 2M +1, the SDP problem is as follows:

in the formulaT (u) is a Toeplitz matrix, and a correlation matrix can be obtained by solving the semi-definite programming problem

The covariance fitting criterion is:

if the sampling number N is more than or equal to 2M +1:

if the number of samples N <2M +1:

expression (12) and expression (13) wherein θ ═ θ₁,θ₂,...,θ_K]Indicating the direction of arrival of K incident signals

σ is the variance of additive noise, and equations (12) and (13) are simplified to obtain:

wherein

Extracting covariance matrix for step 1The Toeplitz matrix constructed by the anti-diagonal elements is a theoretically more accurate correlation matrix { } -¹Representing the inverse of the matrix.

In step 3, based on Toeplitz correlation matrixEstimating direction of arrival of near field signal sourceThe method comprises the following specific steps:

301, Toeplitz correlation matrix

Is composed of

B(θ)R_sB^H(θ)+σI_M+1

Then there is

Suppose thatThe method is simplified as follows:

in the formula { }^*Represents a conjugate, { }_{2，...，_M+1}Indicating that the first row of the matrix is removed, and taking the rest elements; the expected direction of arrival is obtained by directly using the Prony method to solve equation (16)

302, get the direction of arrivalThen obtaining the number K of the incident signals, and decomposing the covariance matrix through the eigenvalue under the condition that the number K of the incident signals is knownSplitting into a signal subspace and a noise subspace:

covariance matrix

2M +1 eigenvalues are totally arranged, all eigenvalues are sorted according to the size,

is a K-dimensional diagonal matrix formed by the first K larger eigenvalues,

is a subspace spanned by the eigenvectors corresponding to the first K larger eigenvalues, namely a signal subspace;is a 2M +1-K dimensional diagonal matrix formed by the last 2M +1-K small characteristic values,is a subspace spanned by the eigenvectors corresponding to the next 2M +1-K smaller eigenvalues;

estimating distance parameters with known directions of arrival

Compared with the prior art, the invention has the following beneficial technical effects: the invention constructs a correlation matrix similar to a Toeplitz matrix by extracting the anti-diagonal elements of the signal covariance matrix, establishes a convex optimization problem aiming at the reconstruction of the correlation matrix, estimates the arrival direction of a near-field signal source by using a Prony method based on the recovered correlation matrix, the distance is effectively estimated using conventional subspace-based methods in the case of obtaining the direction of arrival, the algorithm has excellent performance on the estimation of the direction of arrival and the distance, particularly under the condition of small sampling number or even single sampling, the accuracy of the estimation result is higher than that of the WLPM and GEMM of the existing algorithms, the algorithm separately estimates the direction of arrival and the distance and automatically associates the directions of arrival and the distance without additional processing, solves the problem in a non-iterative mode, greatly reduces the calculated amount, because the discretization processing is not needed, the performance is improved, and the problem of grid mismatch of the traditional grid matching algorithm is avoided.

Drawings

Figure 1a is a plot of estimated root mean square error with signal to noise ratio for the direction of arrival,

figure 1b is a plot of the estimated root mean square error of the distance as a function of signal to noise ratio.

Figure 2a is a plot of estimated root mean square error in direction of arrival versus number of samples.

Figure 2b is a plot of estimated root mean square error of distance versus number of samples.

Detailed Description

The invention is described in detail below with reference to the figures and examples.

In the present invention, for any variable a, a represents a theoretical value,represents the actual calculated value of the variable a,

representing an estimate of the variable a that is closer to the theoretical value.

Extracting the anti-diagonal elements of the signal covariance matrix, constructing a correlation matrix similar to a Toeplitz matrix, and establishing a convex optimization problem aiming at the reconstruction of the correlation matrix, wherein the correlation matrix which is obtained based on the recovered semi-positive programming estimation and is closer to a theoretical value can estimate the direction of arrival of the near-field signal source by using a Prony method, and the distance between the signal source and the symmetrical uniform linear array can be effectively estimated by using a conventional subspace-based method under the condition of known direction of arrival. The near-field signals are K incoherent narrow-band signals incident on the symmetrical uniform linear arrayThe symmetrical uniform linear array comprises 2M +1 omnidirectional sensor array elements, and the value range of M is M>K，Is azimuth information of incident signal, where_iIndicating the direction of arrival angle of the ith incident signalCounterclockwise angle with respect to the y-axis.

A sparse reconstruction method applied to near-field signal source positioning based on a symmetrical uniform linear array is specifically realized by the following steps:

1) covariance matrix calculation based on signal model received by symmetric uniform linear array

Wherein N represents the number of samples, x (N) represents the array received data, { }^HRepresents a conjugate transpose;

2) based on covariance matrixConstruction of a correlation matrix resembling the Toeplitz matrix by the elements of the anti-diagonals

3) Correlation matrix based on Toeplitz-like matrixConstructing an objective function using covariance fitting criteria, and recovering a correlation matrix using a semi-positive plan

4) Based on a correlation matrixEstimating direction of arrival of near field signal source using Prony method

5) Efficient estimation of range using conventional subspace-based methods (e.g., MUSIC)

The following is a detailed description.

K near-field narrow-band incoherent signal sources are incident to a symmetrical uniform linear array with 2M +1 omnidirectional sensor array elements, K is required to be less than M, the distance between the array elements is d, and signals received by the array elements M are as follows:

wherein M ═ M, · 0,. 0, M, s_k(n) denotes the kth incident signal, ω_m(n) represents the noise on array element m, where:

where λ is the signal wavelength, θ_kIs the direction of arrival, r, of the kth incident signal_kIs the distance of the kth incident signal from the uniform symmetric linear array.

The signal received by the symmetric uniform linear array is modeled using a matrix as follows:

in the formula While the additive noise ω (n) has a mean value of 0, a variance of σ and is related to the incident signal { s_k(n) } not relevant; response matrix of arrayWherein Here (.)^TRepresenting a matrix transposition.

Step 1) solving a covariance matrix based on a signal model received by a symmetric uniform linear array

Covariance matrix calculation based on signal model (4)

Where N represents the number of samples and x (N) represents the data received by the array.

Step 2) based on covariance matrix

Construction of a correlation matrix resembling the Toeplitz matrix by the elements of the anti-diagonals

Hereinafter, R and R are theoretical values, and a correlation matrix similar to a Toeplitz matrix is constructed by using the method

For actual values with large errors:

a. the covariance matrix R is a 2M +1 dimensional matrix, and the calculation formula can be expressed as

R＝E{x(n)x^H(n)}＝AR_sA^H+σI_2M+1(6)

In the formulaI_mFor an m-dimensional unit matrix, E { } denotes expectation, { } denotes conjugation^HRepresenting a conjugate transpose.

b. The spatial correlation can be derived from the elements on the inverse diagonals of the covariance matrix R:

wherein i ═ 2M, · 1, 0, 1,. 2M, p ═ M + i₂，...，-1，0，-1，M-i₁Here i₁＝0.5(|i|+i)，i₂＝0.5(|i|-i)，γ_ik＝Ψ_k-iΦ_k。

c. If i is equal to 0, then i₁＝i₂Construct M +1 dimensional Toeplitz matrix Г at 0

d. Substituting formula (7) for formula (8), the Toeplitz matrix can be expressed as:

＝B(θ)R_sB^H(θ)+σI_M+1(9) in the formula:

step 3) based on the correlation matrix similar to the Toeplitz matrix obtained in step 2)Constructing an objective function using covariance fitting criteria, and recovering a correlation matrix using a semi-positive plan

The following semi-positive definite programming problem (SDP) was constructed from the covariance fitting criterion:

if the sampling number N is more than or equal to 2M +1, the SDP problem is as follows:

if the number of samples N <2M +1, the SDP problem is:

in the formulaT (u) is a Toeplitz matrix.

A correlation matrix may be obtained by solving the semi-positive definite programming problem

Step 4) based on the correlation matrix obtained in step 3

Estimating direction of arrival of near field signal source using Prony method

Correlation matrixIs a Toeplitz matrix and can be expressed in the form shown in formula (8), then there are

It can be assumed from equation (9)It can also be simply expressed as:

in the formula { }^*Represents a conjugate, { }_{{2，...，M+1}}Representing the removal of the first row of the matrix, taking the remaining elements, the equation in equation (13) can be solved directly using the Prony methodTo obtain a desired direction of arrival

Step 5) efficient estimation of distance using conventional subspace-based methodsThe invention preferably adopts MUSIC subspaces

Inter-method estimation of distance

a. In the case where the number K of incident signals is known, the covariance matrix is decomposed by eigenvalues

Splitting into a signal subspace and a noise subspace:

covariance matrix2M +1 eigenvalues are totally arranged, all eigenvalues are sorted according to the size,

is a K-dimensional diagonal matrix formed by the first K larger eigenvalues,is a subspace spanned by eigenvectors corresponding to K larger eigenvalues, namely a signal subspace;

is a 2M +1-K dimensional diagonal matrix formed by the next 2M +1-K smaller eigenvalues,is a subspace spanned by the eigenvectors corresponding to the next 2M +1-K small eigenvalues.

b. Distance parameters can be efficiently estimated using conventional subspace-based methods with known directions of arrivalIf the MUSIC algorithm can be solved by the following formula

The effect of the above method is explained by the following different situations:

there are two incoherent near-field signals whose directions of arrival and distances are unknown in space, and their incident directions and distances are (12 °, 1.5 λ), (31 °, 1.7 λ), respectively. The symmetric uniform linear array comprises 7 array elements with 2M +1, the interval of the array elements is d lambda/4, a near-field source positioning Weighted Linear Prediction Method (WLPM) and a near-field source positioning (GEMM) based on the symmetric subarray are added in simulation for comparison, and a CRB boundary is given. Each simulation result was obtained through 1000 independent replicates.

Referring to fig. 1a and 1b, all methods estimate the direction of arrival and the distance of the near-field signal relatively accurately, and the estimated Root Mean Square Error (RMSE) of the direction of arrival and the distance decreases significantly with increasing signal-to-noise ratio; the sampling number N is 200, and the solid line is obtained by the method; the long dotted line is obtained by the GEMM algorithm; dotted and dashed lines are obtained by a WLPM algorithm; the short dashed line is the lower CrB; the method provided by the invention is completely superior to the WLPM method and the GEMM method under the condition of low signal-to-noise ratio, and the result in the graph b shows that the method provided by the invention has excellent performance on distance estimation and is far superior to the WLPM method and the GEMM method.

Referring to fig. 2a and 2b, the signal-to-noise ratio is 10dB, and the solid line is obtained by the method of the present invention; the long dotted line is obtained by the GEMM algorithm; dotted and dashed lines are obtained by a WLPM algorithm; the short dashed line is the lower CrB; all the methods relatively accurately estimate the direction of arrival and the distance of the near-field signal, the Root Mean Square Error (RMSE) of the direction of arrival and the distance is obviously reduced along with the increase of the number of samples, the effect of the method provided by the invention is completely superior to that of a WLPM method and a GEMM method under the condition of low number of samples and even single sample, and the result of fig. 2b shows that the method provided by the invention has excellent performance on distance estimation and is far superior to that of the WLPM method and the GEMM method.

According to the method, a correlation matrix similar to a Toeplitz matrix is constructed by extracting the anti-diagonal elements of the signal covariance matrix, a convex optimization problem aiming at the reconstruction of the correlation matrix is established, the arrival direction of a near-field signal source is estimated by using a Prony method based on the recovered correlation matrix, and the distance can be effectively estimated by using a conventional subspace-based method under the condition of acquiring the arrival direction. The sparse reconstruction method without gridding is used, the problem of high calculation complexity caused by iterative operation is avoided, and the problem of grid mismatch of the traditional gridding method is solved.