A method for locating hybrid near-field and far-field sources based on sparse reconstruction

Abstract

This invention relates to a method for locating mixed near-field and far-field sources based on sparse reconstruction. It belongs to the field of underwater acoustic detection. This invention addresses the problem that existing methods for locating mixed near-field and far-field sources based on sparse reconstruction cannot achieve both high positioning accuracy and low computational complexity. The process is as follows: 1. Initialize the current iteration count κ = 1; 2. Obtain the covariance matrix based on the received signals from the array elements; 3. Construct a sparse representation model of the covariance vector for mixed near-field and far-field sources and an initial dictionary set of mixed near-field and far-field sources; 4. Construct a Bayesian hierarchical probability model; 5. Update the signal power, accuracy, and perturbation accuracy in the model; 6. Update the dictionary set of mixed near-field and far-field sources; 7. Determine whether the iteration stopping condition is met or the maximum external iteration count is reached; if met, output the far-field spatial spectrum estimation result, the near-field spatial spectrum estimation result, and the corresponding grid point estimation; otherwise, let κ = κ + 1 and return to step 5 until the condition is met.

Description

Technical Field

[0001] This invention relates to a method for locating near-field and far-field hybrid sources based on sparse reconstruction. This invention belongs to the field of underwater acoustic detection. Background Technology

[0002] Source localization is a crucial research area in array signal processing, with wide applications in radar, sonar, and wireless communication. Traditional source localization methods are typically based on pure far-field or pure near-field models. However, in complex real-world scenarios, such as underwater noise source localization using sonar arrays or indoor sound source localization using microphone arrays, near-field and far-field sources coexist. Traditional signal processing methods struggle to effectively separate and locate mixed near-field and far-field sources. Existing subspace-based algorithms for locating mixed near-field and far-field sources often assume the near-field source is located in the Fresnel zone of the array, using Fresnel approximation to separate the near-field source's azimuth and range parameters, introducing model errors. Furthermore, to avoid phase ambiguity, these algorithms require array element spacing to be no greater than 1 / 4 wavelength, resulting in array aperture loss when estimating the far-field source's azimuth. Existing near-field hybrid source localization algorithms based on sparse Bayesian learning (FN-MSBL) construct one-dimensional and two-dimensional overcomplete dictionaries for far-field plane waves and near-field spherical waves, respectively, and then use sparse Bayesian learning to fit and reconstruct the near-field spatial spectrum using matrix snapshot data. While this overcomes the shortcomings of subspace-based algorithms, such as requiring a 1 / 4 wavelength array and low localization accuracy, sparse Bayesian learning requires costly matrix inversion operations in each iteration, resulting in extremely high computational complexity and limiting its real-time applications. In practice, since the true source location is unlikely to be completely on the sampling grid points, it is necessary to correct for off-grid errors. Several one-dimensional off-grid methods have been proposed to improve the grid mismatch problem, but methods for two-dimensional off-grid parameters of near-field sources are relatively few. Summary of the Invention

[0003] The purpose of this invention is to address the problem that existing far-near field hybrid source methods based on sparse reconstruction cannot achieve a balance between high positioning accuracy and low computational complexity, and to propose a far-near field hybrid source positioning method based on sparse reconstruction.

[0004] The specific process of a near-field hybrid source localization method based on sparse reconstruction is as follows:

[0005] Step 1: Initialize the current iteration count κ = 1;

[0006] Step 2: Obtain the covariance matrix based on the received signals of the array elements;

[0007] Step 3: Based on the covariance matrix, construct a sparse representation model of the covariance vector for near and far field mixed sources and an initial dictionary set of near and far field mixed sources;

[0008] Step 4: Based on the sparse representation model of near-field mixed sources using covariance vectors and the initial dictionary set of near-field mixed sources, construct a system containing signal power... Bayesian hierarchical probability model with accuracy α and perturbation accuracy β0;

[0009] Step 5: Update the signal power in the Bayesian hierarchical probability model using GAMP-VBI Precision α, disturbance precision β0;

[0010] Step Six: Update the signal power according to Step Five. Precision α and perturbation precision β0 are used to update the near-field mixed source dictionary set through the gradient off-grid method;

[0011] Step 7: Determine whether the iteration stopping condition is met or the maximum number of external iterations has been reached;

[0012] If the conditions are met, output the far-field spatial spectrum estimation result. Near-field spatial spectrum estimation results and corresponding grid point estimation

[0013] Otherwise, let κ = κ + 1 and return to step five until the iteration stopping condition is met or the maximum number of external iterations is reached, then output the far-field spatial spectrum estimation result. Near-field spatial spectrum estimation results and corresponding grid point estimation

[0014] in, Estimate the angle corresponding to the vector for the far-field grid points; Estimate the angle corresponding to the vector for the near-field grid points; Estimate the distances corresponding to the vectors of near-field grid points.

[0015] The beneficial effects of this invention are as follows:

[0016] To reduce the computational complexity of sparse reconstruction-based far- and near-field hybrid source localization algorithms and correct off-grid errors in far- and near-field hybrid source localization, this invention proposes a fast far- and near-field hybrid source localization algorithm based on a unified sparse representation model of far- and near-field hybrid sources in the covariance domain. This algorithm utilizes variational Bayesian inference and generalized approximate message-passing solutions, and extends the gradient descent off-grid method to near-field source two-dimensional parameter estimation, namely FN-GAMP-OGCVBI (GAMP-based Off-grid Covariance-based VBI for Far and Near-field Sources). This invention effectively improves the computational efficiency and estimation accuracy of the algorithm while reducing computational complexity.

[0017] This invention proposes a fast localization method for mixed near-field and far-field sources based on sparse reconstruction. First, a sparse representation model of the mixed near-field and far-field sources in the covariance domain and a Bayesian hierarchical probability model are constructed. Generalized Approximate Message Passing-Variational Bayesian Inference (GAMP-VBI) is used to avoid matrix inversion operations in the solution process, significantly reducing computational complexity. Then, a gradient descent off-grid method is applied to update grid point parameters, thereby reducing localization errors caused by grid mismatch. Compared with existing sparse Bayesian learning algorithms for mixed near-field and far-field source localization based on element domain data, this method achieves higher localization accuracy and shorter computation time. This invention belongs to the category of fast localization methods for mixed near-field and far-field sources and can be applied to the field of passive sonar signal detection.

[0018] (1) This invention extends the sparse representation model of far-field and near-field mixed sources to the covariance domain and utilizes the higher array signal-to-noise ratio gain of the covariance vector to improve the far-field DOA estimation performance when the number of snapshots is sufficient.

[0019] (2) The present invention compensates for the positioning error caused by mesh mismatch by using the gradient descent off-grid method.

[0020] (3) Compared with the existing near-field hybrid source localization method based on sparse reconstruction, the present invention not only has higher resolution and localization accuracy, but also significantly improved computational efficiency. Attached Figure Description

[0021] Figure 1 This is a flowchart of the present invention;

[0022] Figure 2(a) shows the far-field spatial spectrum of the present invention and other algorithms;

[0023] Figure 2(b) is a magnified view of the far-field spatial spectrum of the present invention and other algorithms;

[0024] Figure 3(a) shows the near-field focusing spatial spectrum of the MVDR algorithm;

[0025] Figure 3(b) shows the near-field spatial spectrum of the FN-OGMSBL algorithm;

[0026] Figure 3(c) shows the near-field spatial spectrum of the FN-OGCVBI algorithm;

[0027] Figure 3(d) is the near-field spatial spectrum of the present invention;

[0028] Figure 4(a) is a comparison of the far-field source DOA estimation performance of the present invention and other algorithms as a function of signal-to-noise ratio;

[0029] Figure 4(b) is a comparison of the near-field source angle estimation performance of the present invention and other algorithms with the change of interference-to-noise ratio;

[0030] Figure 4(c) is a comparison of the near-field source distance estimation performance of the present invention and other algorithms as a function of interference-to-noise ratio;

[0031] Figure 4(d) is a comparison of the computation time of the present invention and other algorithms as a function of signal-to-noise ratio;

[0032] Figure 5(a) is a comparison of the far-field source DOA estimation performance of the present invention and other algorithms with the number of snapshots;

[0033] Figure 5(b) is a comparison of the near-field source angle estimation performance of the present invention and other algorithms with the number of snapshots;

[0034] Figure 5(c) is a comparison of the near-field source distance estimation performance of the present invention and other algorithms with the number of snapshots;

[0035] Figure 5(d) is a comparison of the computation time of the present invention and other algorithms with the number of snapshots;

[0036] Figure 6 This is a comparison chart showing the success probability of far-field source resolution of this invention with other algorithms as a function of angular interval. Detailed Implementation

[0037] Specific Implementation Method 1: The specific process of this implementation method for locating near-field and far-field hybrid sources based on sparse reconstruction is as follows:

[0038] Step 1: Initialize the current iteration count κ = 1;

[0039] Step 2: Obtain the covariance matrix based on the received signals of the array elements;

[0040] Step 3: Based on the covariance matrix, construct a sparse representation model of the covariance vector for near and far field mixed sources and an initial dictionary set of near and far field mixed sources;

[0041] Step 4: Based on the sparse representation model of near-field mixed sources using covariance vectors and the initial dictionary set of near-field mixed sources, construct a system containing signal power... Bayesian hierarchical probability model with accuracy α and perturbation accuracy β0;

[0042] Step 5: Update the signal power in the Bayesian hierarchical probability model using GAMP-VBI Precision α, disturbance precision β0;

[0043] Step Six: Update the signal power according to Step Five. Precision α and perturbation precision β0 are used to update the near-field mixed source dictionary set through the gradient off-grid method;

[0044] Step 7: Determine whether the iteration stopping condition is met or the maximum number of external iterations has been reached;

[0045] If the conditions are met, output the far-field spatial spectrum estimation result. Near-field spatial spectrum estimation results and corresponding grid point estimation

[0046] Otherwise, let κ = κ + 1 and return to step five until the iteration stopping condition is met or the maximum number of external iterations is reached, then output the far-field spatial spectrum estimation result. Near-field spatial spectrum estimation results and corresponding grid point estimation

[0047] in, Estimate the angle corresponding to the vector for the far-field grid points; Estimate the angle corresponding to the vector for the near-field grid points; Estimate the distances corresponding to the vectors of near-field grid points.

[0048] Specific Implementation Method Two: This implementation method differs from Specific Implementation Method One in that step two involves obtaining the covariance matrix based on the received signals from the array elements; the specific process is as follows:

[0049] Step 21: Consider the existence of K1 narrowband far-field sources and K2 narrowband near-field sources simultaneously incident on a symmetrical uniform linear array composed of M array elements;

[0050] Near-field sources are typically defined as being located in the range [0.62(D). 3 / λ) 1 / 2 2D 2 Within the Fresnel zone of [λ], the far-field source is located at a distance greater than 2D. 2 Outside the Fresnel zone of / λ; D is the array aperture;

[0051] Wherein, the element spacing d = λ / 2, and λ is the wavelength;

[0052] Assuming the number of array elements M is odd, and the central array element is the reference array element, the received signal of the m-th array element at time t is expressed as:

[0053]

[0054] Where, x k (t) represents the signal envelope received by the reference array element from the k-th source at time t, where n m (t) represents the additive background noise received by the m-th array element at time t, τ mk r represents the time delay difference between the arrival of the k-th source at the reference element and the arrival of the m-th element. k r represents the distance from the k-th source to the reference array element. mk This represents the distance from the k-th source to the m-th array element; m = 1, 2, ..., M;

[0055] If the k-th source is a near-field source, the position of the k-th source in polar coordinates is defined as (θ). k ,r k The distance r from the k-th source to the m-th element. mk for:

[0056]

[0057] The time delay difference between the arrival of the k-th source and the m-th array element from the near-field source satisfies

[0058] Where c is the speed of sound, θ k The angle at which the k-th source arrives at the reference array element;

[0059] If the k-th source is a far-field source, assuming the distance r from the k-th source to the reference array element is... k =+∞, the position of the k-th source to the reference element can be determined solely by the angle θ. k It is determined that the time delay difference between the arrival of the k-th source at the reference element and the m-th element is simplified to:

[0060] Step 22: Define the array (a symmetrical uniform linear array composed of M elements) at time t as the received signal vector y(t) = [y1(t),...,y...]. M (t)] T At time t, the reference array element receives the far-field signal vector. The reference array element receives the near-field signal vector at time t. Background noise vector n(t) = [n1(t),...,n M (t)] T ;

[0061] The received signal of the array at time t is:

[0062]

[0063] Where y1(t) is the signal received by the first array element at time t, y M x1(t) represents the signal received by the Mth array element at time t, and x1(t) represents the signal envelope of the first source received by the reference array element at time t. Let K1 be the signal envelope received by the reference array element at time t from the K1th source. Let K1+1 be the signal envelope received by the reference array element at time t. Let n1(t) be the signal envelope received by the reference array element from the (K1+K2)th source at time t, and let n1(t) be the background noise received by the first array element. M (t) represents the background noise received by the Mth array element;

[0064] For the far-field target array manifold, θ f Let a be the set of angles from the k-th source in the far field to the reference array element. f (θ1) is the steering vector of the first source in the far field. is the steering vector of the K1th source in the far field; T′ is the snapshot number;

[0065] For the near-field target array manifold, θ n Let r be the set of angles from the K2 near-field sources to the reference array element. n Let K2 be the set of distances from the near-field sources to the reference array element. For the angle is Distance is The near-field steering vector, For the angle is Distance is The near-field steering vector;

[0066] in, The angle at which the (K1+1)th source arrives at the reference element; The angle at which the K1+K2th source arrives at the reference array element; This is the distance from the (K1+1)th source to the reference array element; This is the distance from the K1+K2th source to the reference array element;

[0067] The expressions for the far-field and near-field steering vectors are:

[0068]

[0069]

[0070] Where j represents the imaginary unit, λ is the signal wavelength, and a f (θ k ( ) is an angle of θ k The far-field steering vector, a n (θ k ,r k ( ) is an angle of θ k The distance is r k The near-field steering vector, r 1k Let r be the distance from the k-th source to the 1st element. Mk θ is the distance from the k-th source to the M-th array element; k r is the angle at which the k-th source arrives at the reference array element; k Let be the distance from the k-th source to the reference array element;

[0071] Steps two and three: Assume that each source signal is a statistically independent stationary Gaussian process, and the reference array element receives signal x at time t. k (t) follows a mean of 0 and a variance of . The complex Gaussian distribution;

[0072] Assume that each array element receives background noise n m (t) follows a function with mean 0 and variance σ. 2 The complex Gaussian distribution;

[0073] signal x k (t) and background noise n m (t) are mutually independent;

[0074] Based on the received signal y(t) of the array at time t, the theoretical covariance matrix R of the array output is obtained. y ; indicates

[0075] R y =E{y(t)y H (t)}

[0076] =A f (θ f )diag(p)A f H (θ f )+A n (θ n ,r n )diag(η)A n H (θ n ,r n )+σ 2 I M ,

[0077] Where p is the far-field signal power vector. The signal power of the first far-field source. The signal power of the K1th far-field source;

[0078] η is the near-field signal power vector. The signal power of the first near-field source. The signal power of the K2th near-field source;

[0079] The superscript H indicates finding the conjugate;

[0080] E{} represents the expected value, I M Let be an M-dimensional identity matrix, and diag() is used to extract the diagonal elements;

[0081] In step two, the covariance of the background noise n(t) is σ. 2 I M ;σ2 Background noise n m The variance of (t);

[0082] Step 24: Considering the sparsity of the source location in space, the far-field angle range of 1 to 180° (inclusive) is uniformly divided into N1 grid points with a grid spacing of I1, resulting in a one-dimensional far-field grid point set.

[0083] in, Let be the angle corresponding to the i-th element in the far-field grid point set, where i is the index of the i-th grid point in the N1 grids;

[0084] "Near-field azimuth" is defined as the angle between the incident azimuth of the near-field source signal and the array, and "near-field distance" is defined as the distance between the near-field source and the reference array element.

[0085] The near-field orientation is uniformly divided into N² grid points with a grid spacing of I², resulting in the near-field orientation grid point set.

[0086] in, The j-th point in the near-field orientation grid set θ The angles corresponding to each element, j θ For the j-th grid in N2 grids θ The index of each grid point;

[0087] The near-field distance is uniformly divided into N3 grid points with a grid spacing of I3, resulting in the near-field distance grid point set.

[0088] in, The j-th point in the near-field distance grid set r The distances corresponding to each element, j r For the j-th grid in N3 grids r Index of points;

[0089] Therefore, the near-field spatial domain is discretized into a set of two-dimensional near-field grid points in polar coordinates. j θ =1,…,N2,j r =1,…N3;

[0090] The set of two-dimensional near-field grid points is uniformly indexed by j.

[0091] Where, N n =N2×N3 is the total number of near-field grid points, and j is N n The index of the j-th point in the grid;

[0092] The theoretical covariance matrix R of the array output yIt can be expressed in the following overcomplete base form:

[0093]

[0094] in, For far-field target array manifold, For the far-field grid point set For near-field target array manifold, Near-field azimuth grid set Near-field range grid point set

[0095] This is the zero-padding extension of p under an overcomplete basis. This represents the spatial spectral power corresponding to the first far-field grid point. This represents the spatial spectral power corresponding to the N1th far-field grid point; the superscript T indicates that the transpose is being calculated.

[0096] This is the zero-padding extension of η under an overcomplete basis. This represents the spatial spectral power corresponding to the first near-field grid point. For the Nth n Spatial spectral power corresponding to each near-field grid point;

[0097] The zero-padding rule is

[0098]

[0099]

[0100] in, Let be the power of the k-th source signal. Let j be the angle corresponding to the j-th element in the near-field grid point set. The distance to the j-th element in the near-field grid set;

[0101] The actual covariance matrix estimate is obtained by using a signal with a finite number of snapshots T′, i.e.

[0102]

[0103] Where, ΔR y The estimation error of the covariance matrix, R is the estimated value of the actual covariance matrix. y This is the theoretical covariance matrix.

[0104] The other steps and parameters are the same as in Specific Implementation Method 1.

[0105] Specific Implementation Method 3: This implementation method differs from Specific Implementation Method 1 or 2 in that, in step 3, a sparse representation model of the covariance vector far-near field mixed source and an initial far-near field mixed source dictionary set are constructed based on the covariance matrix.

[0106] The specific process is as follows:

[0107] Step 31: Estimate the actual covariance matrix Vectorization:

[0108]

[0109] in, The vectorized covariance matrix is ​​vec(·), which is the matrix vectorization operation; Δr is the perturbation vector.

[0110] and These are the far-field and near-field overcomplete bases in the form of the Khatri-Rao product, respectively, with the specific expressions as follows:

[0111]

[0112]

[0113] in, The symbol for the Kronecker product is (·). * Indicates conjugate;

[0114] The angle corresponding to the first element in the far-field grid point set. Let i be the angle corresponding to the i-th element in the far-field grid point set. The angle corresponding to the N1th element in the far-field grid point set;

[0115] For the angle is The far-field steering vector, For the angle is The far-field steering vector, For the angle is The far-field steering vector; For complex numbers;

[0116] The angle corresponding to the first element in the near-field grid point set. The distance is the distance corresponding to the first element in the near-field grid set. Let j be the angle corresponding to the j-th element in the near-field grid point set. Let be the distance to the j-th element in the near-field grid point set. The Nth point in the near-field grid set nThe angle corresponding to each element The Nth point in the near-field grid set n The distance corresponding to each element For the angle is Distance is The near-field steering vector, For the angle is Distance is The near-field steering vector, For the angle is Distance is The near-field steering vector;

[0117] To jointly estimate the noise variance σ 2 ,make

[0118] in To achieve complete basis dimensions, the following unified sparse representation model for near and far fields is obtained.

[0119]

[0120] in, For a complete dictionary; Signal power;

[0121] Step 32: When the incident signal x k When (t) satisfies a complex Gaussian distribution, the perturbation vector Δr follows an asymptotic complex Gaussian distribution Δr~CN(0,W), where the estimated value of W is... Estimated from the covariance matrix, i.e.

[0122] because The non-zero off-diagonal elements indicate that The correlation between the elements can severely impact signal reconstruction performance; therefore, it is necessary to consider... Performing a whitening operation, the resulting sparse representation of the whitened covariance vector is:

[0123]

[0124] in, This is the covariance vector after the whitening operation. Let be the overcomplete dictionary after whitening, ε be the perturbation vector after whitening, and CN() be the complex Gaussian distribution.

[0125] At this point, the covariance vector far- and near-field mixed-source sparse representation model has been constructed. This is the initial dictionary set of near and far field mixed sources.

[0126] Other steps and parameters are the same as in specific implementation method one or two.

[0127] Specific Implementation Method Four: This implementation method differs from Specific Implementation Methods One to Three in that step four involves constructing a signal power... The Bayesian hierarchical probability model with precision α and precision β0 of perturbation vector ε; the specific process is as follows:

[0128] Step 41:

[0129] Assuming the perturbation vector ε follows a complex Gaussian distribution with precision β0, then The likelihood function is

[0130] in, Let CN() be the likelihood function, and let CN() be a complex Gaussian distribution. For dimension M 2 The identity matrix;

[0131] Add a Gamma distribution hyperprior with shape parameter c′ and velocity parameter d′ to the precision β0 of the perturbation vector ε, that is, the prior distribution p(β0;c′,d′)=Gamma(β0|c′,d′);

[0132] Where p(β0;c′,d′) is the prior distribution of β0;

[0133] Assuming signal power It follows a zero-mean complex Gaussian distribution with precision α, i.e. prior distribution

[0134] To α n Add a shape parameter of a′ and a speed parameter of b. n The Gamma distribution of ' is a priori, i.e., the prior distribution of α.

[0135] Where p(α;a′,b′) is the prior distribution of α;

[0136] Step 42: Let the set of latent variables to be estimated be...

[0137] Based on step 41 likelihood function prior distribution Given the prior distribution p(α; a′, b′) of α, the prior distribution p(β0; c′, d′) of β0, and the set of latent variables Θ, calculate the posterior distribution of Θ.

[0138] The expression is:

[0139]

[0140] in, For the posterior distribution, for The prior probability;

[0141] At this point, the Bayesian hierarchical probability model has been constructed (steps 41 and 42 together constitute the Bayesian hierarchical probability model).

[0142] The other steps and parameters are the same as those in one of the specific implementation methods one to three.

[0143] Specific Implementation Method Five: This implementation method differs from Specific Implementation Methods One to Four in that, in step five, the signal power is updated via GAMP-VBI. The accuracy α and the disturbance accuracy β0 are determined; the specific process is as follows:

[0144] Step 51: Update signal power The specific method is as follows:

[0145] 1) Set the tolerance threshold G ξ and maximum number of iterations G l ;

[0146] 2) Initialization: Let

[0147] Among them, |·| 2 This represents the square of the matrix modulo the elements;

[0148] l represents the l-th iteration of the GAMP algorithm. This indicates that the GAMP algorithm is effective for signal power. Expected estimate express The initial value is , and s is the intermediate iteration parameter of the GAMP algorithm. (0) Let represent the initial value of s, where S represents the square of the modulo of the overcomplete dictionary. This represents an overcomplete dictionary after the whitening operation. Represents τ g The initial value, α represents The accuracy parameters;

[0149] 3) Repeat the following steps.

[0150]

[0151]

[0152]

[0153]

[0154]

[0155]

[0156]

[0157]

[0158] Where, μ p ,τ p ,s,τ s ,μ q ,τ q These are intermediate iteration parameters in the GAMP algorithm. For the GAMP algorithm The approximate posterior expectation, τ g For the GAMP algorithm The approximate posterior variance;

[0159] τ represents the l-th iteration p The estimated value, τ represents the (l-1)th iteration p The estimated value, μ represents the l-th iteration. p The estimated value, Let s represent the estimated value of g in the (l-1)th iteration. (l-1) Let s represent the estimated value of s in the (l-1)th iteration. (l) Let λ represent the estimated value of s in the l-th iteration. g Indicates the damping factor. This indicates a scalar output function. Let β0 represent the covariance vector obtained after the whitening operation in step 32, and let β0 represent the perturbation accuracy parameter. τ represents the l-th iteration s The estimated value, express The first derivative, τ represents the l-th iteration q The estimated value, τ represents the l-th iteration s The estimated value, μ represents the l-th iteration. q The estimated value, This indicates that the whitening operation obtained in step 32 has been passed through a complete dictionary. Indicates the l-th iteration The estimated value, This represents a scalar input function. τ represents the l-th iteration g The estimated value, express The first derivative; The product is the Hadamard product, and the damping factor ranges from λ. g ∈(0,1] is used to improve the convergence of the GAMP algorithm;

[0160] Repeat step 3) until the number of iterations l and the tolerance are reached. The condition l≥G is satisfied l or ξ≤G ξ Stop iteration when the time is right;

[0161] || ||2 represents the L2 norm;

[0162] 4) Output the number of iterations κ The approximate posterior distribution estimation results are as follows:

[0163]

[0164]

[0165] in, Indicates the number of iterations κ. The approximate posterior distribution estimation result, where CN() represents the complex Gaussian distribution, and μ represents... The expectation of the approximate posterior distribution, Σ represents The covariance of the approximate posterior distribution, where diag() represents taking the diagonal elements;

[0166] scalar function and derivatives of scalar functions The specific form is:

[0167]

[0168]

[0169]

[0170]

[0171] Where I represents the identity matrix, and α represents... Precision parameters, express The first derivative, express The first derivative;

[0172] This represents Hadamard division. Represents a column vector whose elements are all 1s;

[0173] Step 52: Update the precision parameter α; the specific method is as follows:

[0174]

[0175] Among them, <·> q(·) This indicates that the expectation of the function within the angle brackets is calculated with respect to the probability density q(·), where const represents a constant.

[0176] q opt (α) denotes the best approximate posterior distribution of α. express Let p(α; a′, b′) represent the prior distribution of α. express The approximate posterior distribution;

[0177] Based on variational Bayesian inference, an approximate posterior is obtained.

[0178] Where, α n Represents the nth element of α. Indicates shape parameters as The rate parameter is The Gamma distribution, This represents the estimate of the shape parameter a′. a′ represents the shape parameter set in step 4.1. The rate parameter b n The estimate of ′ b n ′ represents the rate parameter set in step four-one. express The nth element, Represents the square of the L2 norm; Indicates intermediate variables. μ n Let μ be the nth element of μ, where μ represents the value obtained in step 51. The expected value of the approximate posterior distribution, superscript * Indicates conjugation, Σ n,n The element in the nth row and nth column;

[0179] The elements α of the precision parameter α under the number of iterations κ n The estimation result is

[0180] Step 53: Update the perturbation accuracy parameter β0; the specific update method is as follows:

[0181]

[0182] Among them, <·> q(·) This indicates that the expectation of the function within the angle brackets is calculated with respect to the probability density q(·), where const represents a constant.

[0183] q opt (β0) represents the best approximate posterior distribution of β0. Let p(β0; c′, d′) represent the likelihood function, and let c′, d′ represent the prior distribution of β0.

[0184] Based on variational Bayesian inference, an approximate posterior is obtained.

[0185] in, tr(·) represents finding the trace of a matrix;

[0186] The estimation result of β0 under the number of iterations κ is as follows

[0187] The other steps and parameters are the same as those in one of the specific implementation methods one to four.

[0188] Specific Implementation Method Six: This implementation method differs from Specific Implementation Methods One to Five in that, in step six, the signal power updated according to step five... Precision α and perturbation precision β0 are used to update the near-field mixed source dictionary set through an off-grid gradient method; the specific process is as follows:

[0189] Step Sixty-One: Order For the current far-field signal power estimation, For the current near-field signal power estimation, the signal power estimate μ is divided into: and

[0190] in, Let μ be a vector consisting of the 1st to N1st elements. For μ, the N1+2 to N1+Nth n A vector consisting of n elements; Background noise power;

[0191] Similarly, Σ is divided into and

[0192] Where Σ is the signal covariance;

[0193] Σ p For far-field signal covariance, Σ is a matrix consisting of elements located in rows 1 to N1 and columns 1 to N1;

[0194] Σ η For near-field signal covariance, For Σ located between the N1+2th and N1+Nth , n Rows located between the (N1+2)th and (N1+N)th rows n A matrix composed of the elements of each column; Let Σ be the variance of the noise power.

[0195] Step 62: Let the current far-field one-dimensional and near-field two-dimensional grid point estimates be respectively

[0196] Current far-field one-dimensional grid point estimation Near-field two-dimensional grid point estimation The corresponding expression for the grid point estimation vector is:

[0197]

[0198]

[0199]

[0200] in,

[0201] Estimate the angle corresponding to the vector for the far-field grid points. To estimate the angle corresponding to the first element in the vector for far-field grid points, To estimate the angle corresponding to the i-th element in the vector for far-field grid points. The angle corresponding to the N1th element in the estimation vector of the far-field grid points;

[0202] Estimate the angle corresponding to the vector for the near-field grid points. To estimate the angle corresponding to the first element in the vector for near-field grid points, To estimate the angle corresponding to the j-th element in the near-field grid point estimation vector, The Nth point in the near-field grid point estimation vector n The angle corresponding to each element

[0203] Estimate the distance corresponding to the vector for near-field grid points. To estimate the distance corresponding to the first element in the vector for near-field grid points, To estimate the distance corresponding to the j-th element in the vector for near-field grid points, The Nth point in the near-field grid point estimation vector n The distance corresponding to each element;

[0204] Step 63: Based on the log-likelihood function To each and Taking the partial derivative, we get partial derivatives partial derivatives partial derivatives

[0205] Step 64: Based on the near and far field grid points in the κth iteration as well as partial derivatives partial derivatives partial derivatives Update the near and far field grid points in the (κ+1)th iteration.

[0206] Step 65: Based on the near and far field grid points in the (κ+1)th iteration Update the near-far mixed source dictionary set in the (κ+1)th iteration.

[0207] The other steps and parameters are the same as those in one of the specific implementation methods one to five.

[0208] Specific Implementation Method Seven: This implementation method differs from Specific Implementation Methods One through Six in that step six-three is based on the log-likelihood function. To each and Taking the partial derivative, we get partial derivatives partial derivatives partial derivatives The expression is:

[0209]

[0210]

[0211]

[0212] in, for The partial derivatives, for The partial derivatives, for The partial derivatives, where real{} is the real part;

[0213] Q1 is a far-field overcomplete base pair. The partial derivatives, Q2 is a near-field overcomplete base pair. The partial derivatives, Q3 is a near-field overcomplete base pair. The partial derivatives,

[0214] To estimate using the new far-field grid points The obtained far-field overcomplete basis, To estimate using the new near-field grid points The obtained near-field overcomplete basis;

[0215] Σ p For the far-field signal covariance, Σ p For near-field signal covariance;

[0216] diag() retrieves the diagonal elements.

[0217] The other steps and parameters are the same as those in one of the specific implementation methods one to six.

[0218] Specific Implementation Method Eight: This implementation method differs from Specific Implementation Methods One to Seven in that, in step six-four, the near and far field grid points are based on the κ-th iteration. as well as partial derivatives partial derivatives partial derivatives Update the near and far field grid points in the (κ+1)th iteration.

[0219] The updated formula is:

[0220]

[0221]

[0222]

[0223] in, This is the far-field one-dimensional grid point estimation vector in the (κ+1)th iteration. This is the estimated vector of the near-field angular dimension grid points in the (κ+1)th iteration. This is the estimated vector of the near-field range grid points in the (κ+1)th iteration. Let be the far-field one-dimensional grid point estimation vector in the κ-th iteration. Let be the near-field angular dimension grid point estimation vector in the κ-th iteration. Let be the near-field distance dimension grid point estimation vector in the κ-th iteration. For the κ-th iteration The partial derivatives, For the κ-th iteration The partial derivatives, For the κ-th iteration The partial derivatives;

[0224] sign(·) is the sign function, ζ f , ζ n|θ and ζ n|r They are respectively and Fixed step size for gradient descent.

[0225] The other steps and parameters are the same as those in any of the specific implementation methods one to seven.

[0226] Specific Implementation Method Nine: This implementation method differs from Specific Implementation Methods One through Eight in that, in step six-five, the near and far field grid points are based on the (k+1)th iteration. Update the near-far mixed source dictionary set in the (κ+1)th iteration.

[0227] The expression is:

[0228]

[0229]

[0230]

[0231] in, For the far-field overcomplete basis in the form of the Khatri-Rao product estimated in the (κ+1)th iteration, For the near-field overcomplete basis in the form of the Khatri-Rao product estimated in the (κ+1)th iteration, I M Let be an M-dimensional identity matrix, and vec{} be a vectorization operator;

[0232] The far-field one-dimensional grid point estimation vector in the (κ+1)th iteration The angle corresponding to the first element in the diagram. The far-field one-dimensional grid point estimation vector in the (κ+1)th iteration The angle corresponding to the i-th element in the equation. The far-field one-dimensional grid point estimation vector in the (κ+1)th iteration The angle corresponding to the N1th element in the array;

[0233] The near-field angular dimension grid point estimation vector under the (κ+1)th iteration The angle corresponding to the first element in the diagram. The near-field angular dimension grid point estimation vector in the (κ+1)th iteration. The angle corresponding to the j-th element in the equation. The near-field angular dimension grid point estimation vector under the (κ+1)th iteration The Nth n The angle corresponding to each element; The near-field distance dimension grid point estimation vector in the (κ+1)th iteration The distance corresponding to the first element in the middle. The near-field distance dimension grid point estimation vector in the (κ+1)th iteration The distance corresponding to the j-th element in the array. The near-field distance dimension grid point estimation vector in the (κ+1)th iteration The Nth n The distance corresponding to each element;

[0234] For the angle is The far-field steering vector, For the angle is The far-field steering vector, For the angle is The far-field steering vector;

[0235] For the angle is Distance is The near-field steering vector, For the angle is Distance is The near-field steering vector, For the angle is Distance is The near-field steering vector;

[0236] This is the estimated value of the covariance W of the disturbance vector Δr in step 3.2.

[0237] The other steps and parameters are the same as those in one of the specific implementation methods one to eight.

[0238] Specific Implementation Method 10: This implementation method differs from one of the specific implementation methods 1 to 9 in that step 7 determines whether the iteration stopping condition is met or the maximum number of external iterations is reached;

[0239] If the conditions are met, output the far-field spatial spectrum estimation result. Near-field spatial spectrum estimation results and corresponding grid point estimation

[0240] Otherwise, let κ = κ + 1 and return to step five, where the near and far field mixing source dictionary set is used. Updated to the result obtained in step six The far-field spatial spectrum estimation result is output until the iteration stopping condition is met or the maximum number of external iterations is reached. Near-field spatial spectrum estimation results and corresponding grid point estimation

[0241] in, Estimate the angle corresponding to the vector for the far-field grid points; Estimate the angle corresponding to the vector for the near-field grid points; Estimate the distance corresponding to the vector for the near-field grid points;

[0242] The specific process is as follows:

[0243] Determine if the iteration stopping condition is met. Or, the maximum number of external iterations κ≥G is reached. κ ;

[0244] If the iteration stopping condition is met Or, the maximum number of external iterations κ≥G is reached. κ If the iteration stops, output the far-field spatial spectrum estimation result. Near-field spatial spectrum estimation results and corresponding grid point estimates;

[0245] Among them G κ To initially set the maximum number of external iterations, G τ α is the initially set tolerance threshold. (κ) Let α be the precision corresponding to the κ-th iteration. (κ-1) This represents the precision corresponding to the (κ+1)th iteration;

[0246] Otherwise, let κ = κ + 1 and return to step five until the iteration stopping condition is met or the maximum number of external iterations is reached, then output the far-field spatial spectrum estimation result. Near-field spatial spectrum estimation results And the corresponding grid point estimation.

[0247] The other steps and parameters are the same as those in any of the specific implementation methods one to nine.

[0248] The beneficial effects of the present invention are verified using the following embodiments:

[0249] Performance simulations were performed to compare the following methods: Minimum Variance Distortionless Response (MVDR), FN-OGMSBL (Element Domain SBL-based far-field and near-field hybrid source off-grid localization), FN-OGCVBI (Covariance Domain Variational Bayesian Inference (VBI)-based far-field and near-field hybrid source localization), and FN-GAMP-OGCVBI (Covariance GAMP-VBI-based far-field and near-field hybrid source fast localization method, the algorithm of this invention). A uniform symmetrical linear array with 11 elements and an element spacing of 0.5m (designed received signal frequency 1.5kHz) was considered. The algorithm parameters were set as follows: the far-field angle range [1~180] degrees was uniformly divided into N1=90 grid points with a grid spacing I1=2°; the near-field angle was uniformly divided into N2=36 grid points with a grid spacing I2=4°; the near-field distance was uniformly divided into N3=10 grid points with a grid spacing I3=1m; and the maximum number of external EM iterations G was set. q =200, tolerance τ=10 -5 The maximum number of iterations G in the GAMP algorithm l =5, tolerance ξ=10 -3All methods assume that the number of sources is known.

[0250] Simulation 1 compares the far-field and near-field spatial spectrum estimation results of various algorithms. It is assumed that two far-field sources and two near-field sources are simultaneously incident on a uniform linear array. The azimuths of the far-field sources are 60.4° and 70.8°, respectively, with a signal-to-noise ratio (SNR) of 0 dB for both. The positions of the near-field sources in the polar coordinate system are (43.8°, 3.5 m) and (135.5°, 7.8 m), with an SNR of 15 dB for both, and the number of snapshots is 200. The far-field spatial spectra of each algorithm are shown in Figure 2. All algorithms can resolve two far-field targets. Among them, the MVDR far-field spatial spectrum has a wider spectral peak and a peak at the near-field azimuth. Compared with FN-OGMSBL, FN-GAMP-OGCVBI has the highest accuracy and the sharpest spectral peak, indicating that the present invention has higher resolution and far-field direction finding accuracy. Figures 3(a), 3(b), 3(c), and 3(d) show the near-field spatial spectra of MVDR focused beamforming, FN-OGMSBL, FN-OGCVBI, and FN-GAMP-OGCVBI, respectively. All algorithms can distinguish two near-field targets, with MVDR having the highest background level. The near-field spatial spectrum of FN-OGMSBL contains spurious peaks due to inaccurate distance estimation. The near-field spatial spectrum of the present invention, FN-GAMP-OGCVBI, shows the clearest target with no obvious spurious peaks, exhibiting the highest near-field positioning accuracy.

[0251] Simulation 2 compares the statistical performance of each algorithm under different signal-to-noise ratios (SNRs). It is assumed that two far-field sources and two near-field sources are simultaneously incident on a uniform linear array in space. The azimuths of the far-field sources are 92.2° and 100.6°, respectively, with the SNR varying from -15dB to 10dB. The near-field sources are located at (66.4°, 7.2m) and (125.2°, 5.6m) in polar coordinates, treated as interference signals, with a fixed SNR of -15dB. The number of snapshots is fixed at 200, and 200 Monte Carlo simulations are performed. Figure 4(a) shows the variation of the far-field source DOA estimation RMSE for each algorithm with SNR; Figure 4(b) shows the variation of the near-field source azimuth estimation RMSE for different algorithms with SNR; Figure 4(c) shows the variation of the near-field source distance estimation RMSE for different algorithms with SNR; Figure 4(d) shows the variation of the average computation time for different algorithms with SNR. As shown in the figure, due to the first-order Taylor approximation error, the near-field and far-field positioning accuracy of FN-OGMSBL is unsatisfactory. FN-GAMP-OGCVBI, while maintaining a similar high positioning accuracy to FN-OGCVBI, significantly reduces computation time. The FN-GAMP-OGCVBI of this invention exhibits the highest near-field and far-field hybrid source positioning accuracy, demonstrating significant advantages across various signal-to-noise ratios, and its average computation time is only 19.02% of FN-OGMSBL and 8.79% of FN-OGCVBI. These results demonstrate the advantages of this invention in both positioning accuracy and computational efficiency.

[0252] Simulations were performed to compare the statistical performance of different algorithms under different snapshot numbers. It was assumed that two far-field sources and two near-field sources were simultaneously incident on a uniform linear array. The azimuths of the far-field sources were 92.2° and 100.6°, respectively, with a fixed signal-to-noise ratio (SNR) of 0 dB. The near-field sources were located at (66.4°, 7.2 m) and (125.2°, 5.6 m) in polar coordinates, treated as interference signals, with a fixed SNR of -15 dB. The snapshot number varied from 50 to 400, and 200 Monte Carlo simulations were performed. Figure 5(a) shows the variation of the RMSE for far-field source DOA estimation with the number of snapshots for different algorithms; Figure 5(b) shows the variation of the RMSE for near-field source angle estimation with the number of snapshots for different algorithms; Figure 5(c) shows the variation of the RMSE for near-field source distance estimation with the number of snapshots for different algorithms; Figure 5(d) shows the variation of the average computation time for different algorithms with the number of snapshots. As shown in the figure, the accuracy advantage of the FN-GAMP-OGCVBI algorithm for near-field source localization compared to the FN-OGMSBL algorithm increases with the number of snapshots. This is because the covariance domain data improves the array output signal-to-noise ratio as the number of snapshots increases. The FN-GAMP-OGCVBI algorithm exhibits higher near-field source localization accuracy and lower computation time when the number of snapshots is sufficient, with an average computation time of only 26.8% of that of FN-OGMSBL for each number of snapshots. While maintaining a similar high localization accuracy to FN-OGCVBI, the computation time is significantly reduced to 9.92% of that of FN-OGCVBI. These results further demonstrate the advantages of this invention in terms of localization accuracy and computational efficiency.

[0253] The simulation compares the resolution performance of various algorithms for the azimuth of two far-field targets with different angular intervals under the presence of strong near-field interference sources. The two far-field sources are located at 60.3° and 60.3°+Δθ, respectively, where Δθ is the angular interval between the two far-field sources, and the signal-to-noise ratio (SNR) is 0 dB. One near-field interference source is located at (50.4°, 6.7 m) in polar coordinates, with a fixed SNR of 15 dB, a fixed number of snapshots of 200, and a fixed angular interval. The angle was changed from 2° to 8°, and the Monte Carlo simulation was performed 200 times. Figure 6 The effects of different algorithms on the resolution probability of two targets with far-field sources under strong near-field interference are presented, and the resolution probability is defined as the probability estimated in the simulation that satisfies the equation. The ratio of the number of simulations to the total number of simulations. Figure 6 The results show that the FN-GAMP-OGCVBI of the present invention has the smallest resolvable far-field angular spacing. It can resolve far-field dual-target signals with an angular spacing of 4 degrees with a probability of more than 90% at a signal-to-noise ratio of 0dB. This proves that the present invention effectively suppresses the influence of strong near-field interference on the accuracy of far-field source azimuth estimation and achieves high-resolution far-field azimuth estimation. It also proves the effectiveness of the off-grid method proposed in the present invention.

[0254] This invention may have other embodiments. Without departing from the spirit and essence of this invention, those skilled in the art can make various corresponding changes and modifications according to this invention, but these corresponding changes and modifications should all fall within the protection scope of the appended claims.

Claims

1. A method for locating hybrid near-field and far-field sources based on sparse reconstruction, characterized in that: The specific process of the method is as follows: Step 1: Initialize the current iteration count κ = 1; Step 2: Obtain the covariance matrix based on the received signals of the array elements; Step 3: Based on the covariance matrix, construct a sparse representation model of the covariance vector for near and far field mixed sources and an initial dictionary set of near and far field mixed sources; Step 4: Based on the sparse representation model of near-field mixed sources using covariance vectors and the initial dictionary set of near-field mixed sources, construct a system containing signal power... Bayesian hierarchical probability model with accuracy α and perturbation accuracy β0; Step 5: Update the signal power in the Bayesian hierarchical probability model using GAMP-VBI Precision α, disturbance precision β0; Step Six: Update the signal power according to Step Five. Precision α and perturbation precision β0 are used to update the near-field mixed source dictionary set through the gradient off-grid method; Step 7: Determine whether the iteration stopping condition is met or the maximum number of external iterations has been reached; If the conditions are met, output the far-field spatial spectrum estimation result. Near-field spatial spectrum estimation results and corresponding grid point estimation Otherwise, let κ = κ + 1 and return to step five until the iteration stopping condition is met or the maximum number of external iterations is reached, then output the far-field spatial spectrum estimation result. Near-field spatial spectrum estimation results and corresponding grid point estimation in, Estimate the angle corresponding to the vector for the far-field grid points; Estimate the angle corresponding to the vector for the near-field grid points; Estimate the distances corresponding to the vectors for near-field grid points; In step three, based on the covariance matrix, a sparse representation model of the covariance vector for near-field and far-field mixed sources and an initial dictionary set of near-field and far-field mixed sources are constructed; the specific process is as follows: Step 31: Estimate the actual covariance matrix Vectorization: in, The vectorized covariance matrix is ​​vec(·), which is the matrix vectorization operation; Δr is the perturbation vector. and Let be the far-field overcomplete basis and the near-field overcomplete basis in the form of the Khatri-Rao product, respectively, where σ is the power of the source signal, and I... M It is the identity matrix; Step 32: When the incident signal x k When (t) satisfies a complex Gaussian distribution, the perturbation vector Δr follows an asymptotic complex Gaussian distribution Δr~CN(0,W), where the estimated value of W is... Estimated from the covariance matrix, i.e. right Performing a whitening operation, the resulting sparse representation of the whitened covariance vector is: in, This is the covariance vector after the whitening operation. ε is the overcomplete dictionary after whitening, CN() is the perturbation vector after whitening, T' is the number of finite snapshots.

2. The near-field and far-field hybrid source localization method based on sparse reconstruction according to claim 1, characterized in that: In step two, the covariance matrix is ​​obtained based on the received signals from the array elements; the specific process is as follows: Step 21: K1 narrowband far-field sources and K2 narrowband near-field sources are simultaneously incident on a symmetrical uniform linear array composed of M array elements; Wherein, the element spacing d = λ / 2, and λ is the wavelength; Assuming the number of array elements M is odd, and the central array element is the reference array element, the received signal of the m-th array element at time t is expressed as: Where, x k (t) represents the signal envelope received by the reference array element from the k-th source at time t, where n m (t) represents the additive background noise received by the m-th array element at time t, τ mk r represents the time delay difference between the arrival of the k-th source at the reference element and the arrival of the m-th element. k r represents the distance from the k-th source to the reference array element. mk This represents the distance from the k-th source to the m-th array element; m = 1, 2, ..., M; If the k-th source is a near-field source, the position of the k-th source in polar coordinates is defined as (θ). k ,r k The distance r from the k-th source to the m-th element. mk for: The time delay difference between the arrival of the k-th source and the m-th array element from the near-field source satisfies Where c is the speed of sound, θ k The angle at which the k-th source arrives at the reference array element; If the k-th source is a far-field source, assuming the distance r from the k-th source to the reference array element is... k =+∞, the position of the k-th source to the reference element can be determined solely by the angle θ. k It is determined that the time delay difference between the arrival of the k-th source at the reference element and the m-th element is simplified to: Step 22: Define the received signal vector y(t) of the array at time t as y(t) = [y1(t),...,y M (t)] T At time t, the reference array element receives the far-field signal vector. The reference array element receives the near-field signal vector at time t. Background noise vector n(t) = [n1(t),...,n M (t)] T ; The received signal of the array at time t is: Where y1(t) is the signal received by the first array element at time t, y M x1(t) represents the signal received by the Mth array element at time t, and x1(t) represents the signal envelope of the first source received by the reference array element at time t. Let K1 be the signal envelope received by the reference array element at time t from the K1th source. Let K1+1 be the signal envelope received by the reference array element at time t. Let n1(t) be the signal envelope received by the reference array element from the (K1+K2)th source at time t, and let n1(t) be the background noise received by the first array element. M (t) represents the background noise received by the Mth array element; For the far-field target array manifold, θ f Let a be the set of angles from the k-th source in the far field to the reference array element. f (θ1) is the steering vector of the first source in the far field. is the steering vector of the K1th source in the far field; T′ is the snapshot number; For the near-field target array manifold, θ n Let r be the set of angles from the K2 near-field sources to the reference array element. n Let K2 be the set of distances from the near-field sources to the reference array element. For the angle is Distance is The near-field steering vector, For the angle is Distance is The near-field steering vector; in, The angle at which the (K1+1)th source arrives at the reference element; The angle at which the K1+K2th source arrives at the reference array element; This is the distance from the (K1+1)th source to the reference array element; This is the distance from the K1+K2th source to the reference array element; The expressions for the far-field and near-field steering vectors are: Where j represents the imaginary unit, λ is the signal wavelength, and a f (θ k ( ) is an angle of θ k The far-field steering vector, a n (θ k ,r k ( ) is an angle of θ k The distance is r k The near-field steering vector, r 1k Let r be the distance from the k-th source to the 1st element. Mk θ is the distance from the k-th source to the M-th array element; k r is the angle at which the k-th source arrives at the reference array element; k Let be the distance from the k-th source to the reference array element; Steps two and three: Assume that the reference array element receives signal x at time t. k (t) follows a mean of 0 and a variance of . The complex Gaussian distribution; Assume that each array element receives background noise n m (t) follows a mean of 0 and a variance of σ. 2 The complex Gaussian distribution; signal x k (t) and background noise n m (t) are mutually independent; Based on the received signal y(t) of the array at time t, the theoretical covariance matrix R of the array output is obtained. y ; indicates R y =E{y(t)y H (t)} =A f (i f )diag(p)A f H (i f )+A n (i n ,r n )diag(η)A n H (i n ,r n )+s 2 I M , Where p is the far-field signal power vector. The signal power of the first far-field source. The signal power of the K1th far-field source; η is the near-field signal power vector. The signal power of the first near-field source. The signal power of the K2th near-field source; The superscript H indicates finding the conjugate; E{} represents the expected value, I M Let be an M-dimensional identity matrix, and diag() is used to extract the diagonal elements; In step two, the covariance of the background noise n(t) is σ. 2 I M ;σ 2 Background noise n m The variance of (t); Step 24: Divide the far-field angle range of 1 to 180° into N1 grid points with a grid spacing of I1 to obtain a one-dimensional far-field grid point set. in, Let be the angle corresponding to the i-th element in the far-field grid point set, where i is the index of the i-th grid point in the N1 grids; The near-field azimuth is defined as the angle between the incident azimuth of the near-field source signal and the array; The near-field distance is defined as the distance between the near-field source and the reference array element; The near-field orientation is uniformly divided into N² grid points with a grid spacing of I², resulting in the near-field orientation grid point set. in, For the j-th point in the near-field orientation grid set θ The angles corresponding to each element, j θ For the j-th grid in N2 grids θ The index of each grid point; The near-field distance is uniformly divided into N3 grid points with a grid spacing of I3, resulting in the near-field distance grid point set. in, The j-th point in the near-field distance grid set r The distances corresponding to each element, j r For the j-th grid in N3 grids r Index of points; Therefore, the near-field spatial domain is discretized into a set of two-dimensional near-field grid points in polar coordinates. The set of two-dimensional near-field grid points is uniformly indexed by j. Where, N n =N2×N3 is the total number of near-field grid points, and j is N n The index of the j-th point in the grid; The theoretical covariance matrix R of the array output y It can be expressed in the following overcomplete base form: in, For far-field target array manifold, For the far-field grid point set For near-field target array manifold, Near-field azimuth grid set Near-field range grid point set This is the zero-padding extension of p under an overcomplete basis. This represents the spatial spectral power corresponding to the first far-field grid point. This represents the spatial spectral power corresponding to the N1th far-field grid point; the superscript T indicates that the transpose is being calculated. This is the zero-padding extension of η under an overcomplete basis. This represents the spatial spectral power corresponding to the first near-field grid point. For the Nth n Spatial spectral power corresponding to each near-field grid point; The zero-padding rule is in, Let be the power of the k-th source signal. Let j be the angle corresponding to the j-th element in the near-field grid point set. The distance to the j-th element in the near-field grid set; The actual covariance matrix estimate is obtained by using a signal with a finite number of snapshots T′, i.e. Where, ΔR y The estimation error of the covariance matrix, R is the estimated value of the actual covariance matrix. y This is the theoretical covariance matrix.

3. The near-field and far-field hybrid source localization method based on sparse reconstruction according to claim 2, characterized in that: In step three, based on the covariance matrix, a sparse representation model of the covariance vector for near-field and far-field mixed sources and an initial dictionary set of near-field and far-field mixed sources are constructed; the specific process is as follows: Step 31: Estimate the actual covariance matrix Vectorization: in, The vectorized covariance matrix is ​​vec(·), which is the matrix vectorization operation; Δr is the perturbation vector. and These are the far-field and near-field overcomplete bases in the form of the Khatri-Rao product, respectively, with the specific expressions as follows: in, The symbol for the Kronecker product is (·). * Indicates conjugate; The angle corresponding to the first element in the far-field grid point set. Let i be the angle corresponding to the i-th element in the far-field grid point set. The angle corresponding to the N1th element in the far-field grid point set; For the angle is The far-field steering vector, For the angle is The far-field steering vector, For the angle is The far-field steering vector; For complex numbers; The angle corresponding to the first element in the near-field grid point set. The distance is the distance corresponding to the first element in the near-field grid set. Let j be the angle corresponding to the j-th element in the near-field grid point set. Let be the distance to the j-th element in the near-field grid point set. The Nth point in the near-field grid set n The angle corresponding to each element The Nth point in the near-field grid set n The distance corresponding to each element For the angle is Distance is The near-field steering vector, For the angle is Distance is The near-field steering vector, For the angle is Distance is The near-field steering vector; make in To achieve complete basis dimensions, the following unified sparse representation model for near and far fields is obtained. in, For a complete dictionary; Signal power; Step 32: When the incident signal x k When (t) satisfies a complex Gaussian distribution, the perturbation vector Δr follows an asymptotic complex Gaussian distribution Δr~CN(0,W), where the estimated value of W is... Estimated from the covariance matrix, i.e. right Performing a whitening operation, the resulting sparse representation of the whitened covariance vector is: in, This is the covariance vector after the whitening operation. Let be the overcomplete dictionary after whitening, ε be the perturbation vector after whitening, and CN() be the complex Gaussian distribution. At this point, the covariance vector far- and near-field mixed-source sparse representation model has been constructed. This is the initial dictionary set of near and far field mixed sources.

4. The near-field and far-field hybrid source localization method based on sparse reconstruction according to claim 3, characterized in that: Step four involves constructing a signal power... The Bayesian hierarchical probability model with precision α and precision β0 of perturbation vector ε; the specific process is as follows: Step 41: Assuming the perturbation vector ε follows a complex Gaussian distribution with precision β0, then The likelihood function is in, Let CN() be the likelihood function, and let CN() be a complex Gaussian distribution. For dimension M 2 The identity matrix; Add a Gamma distribution hyperprior with shape parameter c′ and velocity parameter d′ to the precision β0 of the perturbation vector ε, that is, the prior distribution p(β0;c′,d′)=Gamma(β0|c′,d′); Where p(β0;c′,d′) is the prior distribution of β0; Assuming signal power It follows a zero-mean complex Gaussian distribution with precision α, i.e. prior distribution To α n Add a shape parameter of a′ and a speed parameter of b. n The Gamma distribution of ' is a priori, i.e., the prior distribution of α. Where p(α;a′,b′) is the prior distribution of α; Step 42: Let the set of latent variables to be estimated be... Based on step 41 likelihood function prior distribution Given the prior distribution p(α; a′, b′) of α, the prior distribution p(β0; c′, d′) of β0, and the set of latent variables Θ, calculate the posterior distribution of Θ. The expression is: in, For the posterior distribution, for The prior probability; At this point, the Bayesian hierarchical probability model has been constructed.

5. The near-field and far-field hybrid source localization method based on sparse reconstruction according to claim 4, characterized in that: In step five, the signal power is updated via GAMP-VBI. The accuracy α and the perturbation accuracy β0 are determined; the specific process is as follows: Step 51: Update signal power The specific method is as follows: 1) Set the tolerance threshold G ξ and maximum number of iterations G l ; 2) Initialization: Let l = 1, s (0) =0, Among them, |·| 2 This represents the square of the matrix modulo the element; l represents the l-th iteration of the GAMP algorithm. This indicates the GAMP algorithm's effect on signal power. Expected estimate express The initial value is , and s is the intermediate iteration parameter of the GAMP algorithm. (0) Let represent the initial value of s, where S represents the square of the modulo of the overcomplete dictionary. This represents an overcomplete dictionary after the whitening operation. Represents τ g The initial value, α represents The accuracy parameters; 3) Repeat the following steps. Where, μ p ,τ p ,s,τ s ,μ q ,τ q These are intermediate iteration parameters in the GAMP algorithm. For the GAMP algorithm The approximate posterior expectation, τ g For the GAMP algorithm The approximate posterior variance; τ represents the l-th iteration p The estimated value, τ represents the (l-1)th iteration p The estimated value, μ represents the l-th iteration. p The estimated value, Let s represent the estimated value of g in the (l-1)th iteration. (l-1) Let s represent the estimated value of s in the (l-1)th iteration. (l) Let λ represent the estimated value of s in the l-th iteration. g Indicates the damping factor. This indicates a scalar output function. Let β0 represent the covariance vector obtained after the whitening operation in step 32, and let β0 represent the perturbation accuracy parameter. τ represents the l-th iteration s The estimated value, express The first derivative, τ represents the l-th iteration q The estimated value, τ represents the l-th iteration s The estimated value, μ represents the l-th iteration. q The estimated value, This indicates that the whitening operation obtained in step 32 has been passed through a complete dictionary. Indicates the l-th iteration The estimated value, This represents a scalar input function. τ represents the l-th iteration g The estimated value, express The first derivative; The product is the Hadamard product, and the damping factor ranges from λ. g ∈(0,1]; Repeat step 3) until the number of iterations l and the tolerance are reached. The condition l≥G is satisfied l or ξ≤G ξ Stop iteration when the time is right; || ||2 represents the L2 norm; 4) Output the number of iterations κ The approximate posterior distribution estimation results are as follows: in, Indicates the number of iterations κ. The approximate posterior distribution estimation result, where CN() represents the complex Gaussian distribution, and μ represents... The expected value of the approximate posterior distribution, ∑ represents The covariance of the approximate posterior distribution, where diag() represents taking the diagonal elements; scalar function and derivatives of scalar functions The specific form is: Where I represents the identity matrix, and α represents... Precision parameters, express The first derivative, express The first derivative; This represents Hadamard division. Represents a column vector whose elements are all 1s; Step 52: Update the precision parameter α; the specific method is as follows: Among them, <·> q(·) This indicates that the expectation of the function within the angle brackets is calculated with respect to the probability density q(·), where const represents a constant. q opt (α) denotes the best approximate posterior distribution of α. express Let p(α; a′, b′) represent the prior distribution of α. express The approximate posterior distribution; Obtaining an approximate posterior Where, α n Represents the nth element of α. Indicates shape parameters as The rate parameter is The Gamma distribution, This represents the estimate of the shape parameter a′. a′ represents the shape parameter set in step 4.

1. The rate parameter b n The estimate of ′ b n ′ represents the rate parameter set in step four-one. express The nth element, Represents the square of the L2 norm; Indicates intermediate variables. μ n Let μ be the nth element of μ, where μ represents the value obtained in step 51. The expected value of the approximate posterior distribution, with the superscript * indicating conjugate, ∑ n,n For the elements in the nth row and nth column; The elements α of the precision parameter α under the number of iterations κ n The estimation result is Step 53: Update the disturbance accuracy parameter β0; the specific update method is as follows: Among them, <·> q(·) This indicates that the expectation of the function within the angle brackets is calculated with respect to the probability density q(·), where const represents a constant. q opt (β0) represents the best approximate posterior distribution of β0. Let p(β0; c′, d′) represent the likelihood function, and let c′, d′ represent the prior distribution of β0. Obtaining an approximate posterior in, tr(·) represents finding the trace of a matrix; The estimation result of β0 under the number of iterations κ is as follows 6. The near-field and far-field hybrid source localization method based on sparse reconstruction according to claim 5, characterized in that: In step six, the signal power is updated according to step five. Precision α and perturbation precision β0 are used to update the near-field mixed source dictionary set through an off-grid gradient method; the specific process is as follows: Step Sixty-One: Order For the current far-field signal power estimation, For the current near-field signal power estimation, the signal power estimate μ is divided into: and in, Let μ be a vector consisting of the 1st to N1st elements. For μ, the N1+2 to N1+Nth n A vector consisting of n elements; Background noise power; Similarly, ∑ is divided into and Where ∑ is the signal covariance; ∑ p For far-field signal covariance, Let ∑ be the matrix consisting of elements located in rows 1 to N1 and columns 1 to N1; ∑ η For near-field signal covariance, For ∑ located at N1+2 to N1+N n Rows located between the (N1+2)th and (N1+N)th rows n A matrix composed of the elements of each column; Let ∑ be the variance of the noise power. Step 62: Let the current far-field one-dimensional and near-field two-dimensional grid point estimates be respectively and Current far-field one-dimensional grid point estimation Near-field two-dimensional grid point estimation The corresponding expression for the grid point estimation vector is: in, Estimate the angle corresponding to the vector for the far-field grid points. To estimate the angle corresponding to the first element in the vector for far-field grid points, To estimate the angle corresponding to the i-th element in the vector for far-field grid points. The angle corresponding to the N1th element in the estimation vector of the far-field grid points; Estimate the angle corresponding to the vector for the near-field grid points. To estimate the angle corresponding to the first element in the vector for near-field grid points, To estimate the angle corresponding to the j-th element in the near-field grid point estimation vector, The Nth point in the near-field grid point estimation vector n The angle corresponding to each element Estimate the distance corresponding to the vector for near-field grid points. To estimate the distance corresponding to the first element in the vector for near-field grid points, To estimate the distance corresponding to the j-th element in the vector for near-field grid points, The Nth point in the near-field grid point estimation vector n The distance corresponding to each element; Step 63: Based on the log-likelihood function To each and Taking the partial derivative, we get partial derivatives partial derivatives partial derivatives Step 64: Based on the near and far field grid points in the κth iteration as well as partial derivatives partial derivatives partial derivatives Update the near and far field grid points in the (κ+1)th iteration. Step 65: Based on the near and far field grid points in the (κ+1)th iteration Update the near-far mixed source dictionary set in the (κ+1)th iteration.

7. The near-field and far-field hybrid source localization method based on sparse reconstruction according to claim 6, characterized in that: In step six-three, the log-likelihood function is used as a basis. To each and Taking the partial derivative, we get partial derivatives partial derivatives partial derivatives The expression is: in, for The partial derivatives, for The partial derivatives, for The partial derivatives, where real{} is the real part; Q1 is a far-field overcomplete base pair. The partial derivatives, Q2 is a near-field overcomplete base pair. The partial derivatives, Q3 is a near-field overcomplete base pair. The partial derivatives, To estimate using new far-field grid points The obtained far-field overcomplete basis, To estimate using the new near-field grid points The obtained near-field overcomplete basis; ∑ p For the far-field signal covariance, ∑ p For near-field signal covariance; diag() retrieves the diagonal elements.

8. The near-field and far-field hybrid source localization method based on sparse reconstruction according to claim 7, characterized in that: In step six-four, the near and far field grid points are based on the κ-th iteration. as well as partial derivatives partial derivatives partial derivatives Update the near and far field grid points in the (κ+1)th iteration. The updated formula is: in, This represents the far-field one-dimensional grid point estimation vector in the (κ+1)th iteration. This is the estimated vector of the near-field angular dimension grid points in the (κ+1)th iteration. This is the estimated vector of the near-field range grid points in the (κ+1)th iteration. Let be the far-field one-dimensional grid point estimation vector in the κ-th iteration. Let be the near-field angular dimension grid point estimation vector in the κ-th iteration. Let be the near-field distance dimension grid point estimation vector in the κ-th iteration. For the κ-th iteration The partial derivatives, For the κ-th iteration The partial derivatives, For the κ-th iteration The partial derivatives; sign(·) is the sign function, ζ f , ζ n|θ and ζ n|r They are respectively and Fixed step size for gradient descent.

9. The near-field and far-field hybrid source localization method based on sparse reconstruction according to claim 8, characterized in that: In step six-five, the near and far field grid points are based on the (κ+1)th iteration. Update the near-far mixed source dictionary set in the (κ+1)th iteration. The expression is: in, For the far-field overcomplete basis in the form of the Khatri-Rao product estimated in the (κ+1)th iteration, For the near-field overcomplete basis in the form of the Khatri-Rao product estimated in the (κ+1)th iteration, I M Let be an M-dimensional identity matrix, and vec{} be a vectorization operator; The far-field one-dimensional grid point estimation vector in the (κ+1)th iteration The angle corresponding to the first element in the diagram. The far-field one-dimensional grid point estimation vector in the (κ+1)th iteration The angle corresponding to the i-th element in the equation. The far-field one-dimensional grid point estimation vector in the (κ+1)th iteration The angle corresponding to the N1th element in the array; The near-field angular dimension grid point estimation vector in the (κ+1)th iteration. The angle corresponding to the first element in the diagram. The near-field angular dimension grid point estimation vector in the (κ+1)th iteration. The angle corresponding to the j-th element in the equation. The near-field angular dimension grid point estimation vector in the (κ+1)th iteration. The Nth n The angle corresponding to each element; The near-field distance dimension grid point estimation vector in the (κ+1)th iteration The distance corresponding to the first element in the array. The near-field distance dimension grid point estimation vector in the (κ+1)th iteration The distance corresponding to the j-th element in the array. The near-field distance dimension grid point estimation vector in the (κ+1)th iteration The Nth n The distance corresponding to each element; For the angle is The far-field steering vector, For the angle is The far-field steering vector, For the angle is The far-field steering vector; For the angle is Distance is The near-field steering vector, For the angle is Distance is The near-field steering vector, For the angle is Distance is The near-field steering vector; This is the estimated value of the covariance W of the disturbance vector Δr in step 3.

2.

10. The near-field and far-field hybrid source localization method based on sparse reconstruction according to claim 9, characterized in that: In step seven, it is determined whether the iteration stopping condition is met or the maximum number of external iterations is reached. If the conditions are met, output the far-field spatial spectrum estimation result. Near-field spatial spectrum estimation results and corresponding grid point estimation Otherwise, let κ = κ + 1 and return to step five, where the near and far field mixing source dictionary set is used. Updated to the result obtained in step six The far-field spatial spectrum estimation result is output until the iteration stopping condition is met or the maximum number of external iterations is reached. Near-field spatial spectrum estimation results and corresponding grid point estimation in, Estimate the angle corresponding to the vector for the far-field grid points; Estimate the angle corresponding to the vector for the near-field grid points; Estimate the distances corresponding to the vectors for near-field grid points; The specific process is as follows: Determine if the iteration stopping condition ||α is met. (κ) -α (κ-1) ||2 / ||α (κ-1) ||2≤G τ Or, the maximum number of external iterations κ≥G is reached. κ ; If the iteration stopping condition ||α is satisfied (κ) -α (k-1) ||2 / ‖α (k-1) ||2≤G τ Or, the maximum number of external iterations κ≥G is reached. κ If the iteration stops, output the far-field spatial spectrum estimation result. Near-field spatial spectrum estimation results and corresponding grid point estimates; Among them G κ To initially set the maximum number of external iterations, G τ α is the initially set tolerance threshold. (κ) Let α be the precision corresponding to the κ-th iteration. (κ-1) This represents the precision corresponding to the (κ+1)th iteration; Otherwise, let κ = κ + 1 and return to step five until the iteration stopping condition is met or the maximum number of external iterations is reached, then output the far-field spatial spectrum estimation result. Near-field spatial spectrum estimation results And the corresponding grid point estimation.

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