Source number estimation method based on orthogonal matching pursuit and signal subspace matching

Abstract

The application relates to a source number estimation method based on orthogonal matching pursuit and signal subspace matching, and relates to a source number estimation method. The application aims to solve the problem that in an underwater acoustic environment, the signal-to-noise ratio of array snapshot data is usually not high, and is easily disturbed by colored noise and coherent sources, and for a high-speed moving target, the effective snapshot number is small, thus leading to low accuracy of traditional source number estimation. The application uses an orthogonal matching pursuit method to iteratively solve a support vector, and uses a characteristic vector corresponding to a maximum characteristic value of a residual to construct two subspaces, and realizes source number estimation through a signal subspace matching criterion. Simulation results show that compared with existing source number estimation methods, the application method has a lower requirement for a signal-to-noise ratio, performs better under small snapshot conditions, and is not sensitive to coherent sources and colored noise. Lake test data processing results show that the method can effectively estimate the number of fixed and moving underwater acoustic targets.

Description

TECHNICAL FIELD

[0001] The present application relates to a method for estimating the number of sources. BACKGROUND

[0002] The estimation of the number of sources has a wide and key application in underwater acoustic signal processing. When using a subspace-based algorithm to estimate the bearing, the number of sources needs to be known to correctly distinguish the signal subspace from the noise subspace [1] ; when performing blind source separation on underwater acoustic signals, many algorithms take the number of sources as a known condition [2] ; when using a multi-target tracking algorithm based on joint probabilistic data association to track targets, the number of sources is required to be constant and known [3] .

[0003] In a complex and variable underwater acoustic environment, the collected data will be superimposed with strong noise, resulting in a significant decline in the performance of traditional methods for estimating the number of sources, such as Akaike information criterion (AIC) [4] , minimum description length (MDL) criterion [5] , and Gerschgorin radius [6] . Therefore, in order to effectively estimate the number of sources of underwater targets, many studies have been conducted by predecessors to reduce the requirements of the above methods on the signal-to-noise ratio [7-10] . Literature [8] designs a criterion value for representing the contribution of signals and noise in the covariance matrix, and estimates the number of signal sources through the criterion value. When the signal-to-noise ratio is low and there is unequal power, its estimation performance is superior to that of the Gerschgorin radius method. Literature

[10] proposes a spatial focusing method for a rotating uniform circular array, which increases the number of usable fast shots in each frame for static target source number estimation. The above methods rely on a large number of fast shot data to achieve effective source number estimation in a low signal-to-noise ratio underwater acoustic environment, but for high-speed moving targets, the number of fast shots available for estimation in each frame is small

[11] , so the estimation performance of the above methods will be affected.

[0004] In recent years, many scholars have carried out in-depth research on the estimation of the number of sources under the condition of small fast shots [12-15]

[12] improves the AIC method by using large random matrix theory, which improves the estimation performance under small snapshots.

[13] reduces the estimation error of noise eigenvalues under small snapshots by using linear shrinkage method, which improves the estimation performance of MDL method.

[15] combines the ideas of information granule and granular computing with fuzzy set theory to achieve source number estimation under small snapshots and low SNR. The above methods all assume that the noise received by the array is Gaussian white noise, i.e. the noise power of each element is equal and independent. However, the actual marine environment is very complex, and the elements are easily disturbed by various noises, which is difficult to meet the assumption of Gaussian white noise

[16] , which leads to the decline of the estimation performance of these methods.

[0005] In order to achieve effective source number estimation in the scene of small snapshots and colored noise,

[17] first increases the proportion of mutually independent noise by using adaptive diagonal loading method to reduce the influence of correlated noise, and then estimates the source number by using the law that the linear shrinkage coefficient changes with the source number. This method performs well under small snapshots and low degree of colored noise, but has high requirements for SNR. Wax proposed the signal subspace matching (SSM) criterion for source number estimation [18-19] . Initially, Wax used this criterion to match the subspace generated by the original sampling data and the eigenvector of the covariance matrix

[18] , which can achieve good source number estimation under small snapshots and colored noise, but has high requirements for SNR. In order to reduce the requirements for SNR, Wax used two subarrays to generate the signal subspace based on the array rotation invariance

[20] , and used invariant signal subspace matching (ISSM) to achieve source number estimation

[19] , which significantly improves the estimation performance under low SNR. Inspired by the ISSM method,

[21] represents the difference between the two eigenvectors of the two subarrays by calculating the generalized cosine square value, and then uses the difference between the signal eigenvectors and the noise eigenvectors to estimate the source number by combining the sequential ratio. This method requires a large difference between the noise eigenvectors of the two subarrays, so it has high requirements for the number of elements. The above SSM, ISSM and

[21] methods use the matching idea to estimate the source number, which reduces the requirements for snapshots compared to traditional source number estimation methods, but they all estimate the source number based on the eigenvector of the covariance matrix. Under small snapshots, accurate estimation of the covariance matrix requires high SNR. In addition, the above methods generally assume that the sources are independent, but in the underwater acoustic environment, interface reflection may produce a large number of coherent signals

[22] , which leads to the failure of the estimation of eigenvalues and eigenvectors, and further causes the error of the estimation of the number of sources. SUMMARY

[0006] The purpose of the present application is to solve the problem that the signal-to-noise ratio of the array snapshot data is usually not high in the underwater acoustic environment, and is easily disturbed by colored noise and coherent sources, and for high-speed moving targets, the effective snapshots are less, which leads to the low accuracy of the traditional source number estimation, and to propose a source number estimation method based on orthogonal matching pursuit and signal subspace matching.

[0007] The specific process of the source number estimation method based on orthogonal matching pursuit and signal subspace matching is as follows:

[0008] Step one, obtaining an M×L matrix X composed of L snapshots collected by the array, X=[X(1), X(2), …, X(t), …, X(L)];

[0009] Wherein, t=1, 2, …, L; M represents the number of array elements; X(1) represents the snapshot data collected by the array at time 1, X(2) represents the snapshot data collected by the array at time 2, X(t) represents the snapshot data collected by the array at time t, and X(L) represents the snapshot data collected by the array at time L;

[0010] Step two, setting the total number of iterations M-1;

[0011] Step three, initializing the iteration number i=0; initializing r 0,t =X(t), t=1,...,L; initializing

[0012] Wherein, r 0,t represents the residual error of the tth snapshot data collected by the array at the 0th iteration, Λ0 represents the set of support vector indexes obtained by the previous 0 iterations, represents an empty set;

[0013] Step four, let the iteration number i=i+1;

[0014] Step five, taking the inner product of each a(θ v ) in the overcomplete dictionary and the residual error of all snapshot data, and summing the absolute values of all inner products, then selecting a(θ v ) corresponding to the maximum sum value as the support vector obtained by the ith iteration, wherein β i is the index of the selected a(θ v ); v=1, 2, …, J; J represents the total number of observable angles; and the column vector a(θ v ) is the steering vector of the vth far-field narrowband source.

[0015] Step six, based on the support vector obtained in the i-th iteration corresponding index β i and the set of indices Λ corresponding to the support vector obtained in the i-1-th iteration i-1 , obtain the set of indices Λ corresponding to the support vector obtained in the i-th iteration i ;

[0016] based on the set of indices Λ corresponding to the support vector obtained in the i-th iteration i , obtain the matrix composed of the support vector obtained in the i-th iteration

[0017] Step seven, calculate the residual r of the i-1-th iteration i-1 covariance matrix of r, obtain the eigenvector matrix U of the residual corresponding to the i-1-th iteration i-1 ;

[0018] Step eight, based on the matrix composed of the support vector obtained in the i-th iteration and the eigenvector matrix U i-1 , calculate the distance between the subspaces of the i-th iteration using the signal subspace matching criterion

[0019] where (U i-1 )1 represents the first column of the eigenvector matrix U i-1 ;

[0020] represents the subspace generated by the column vectors of the matrix ,

[0021] represents the subspace generated by the column vectors of the matrix and the column vectors of the eigenvector (U i-1 )1,

[0022] <·> represents the subspace generated by the column vectors of the matrix in the brackets;

[0023] represents the distance between the i-th iteration subspace and the subspace ;

[0024] SSM i represents the distance between the subspaces calculated in the i-th iteration;

[0025] Step nine, let

[0026] where Bi denotes a matrix composed of support vectors obtained by the i th iteration;

[0027] Solve z by least square i,t = argmin z ||X(t)-B i z||2,t = 1,...,L;

[0028] Wherein, z denotes the coefficient vector obtained by least square optimization, z = z i,1 ,z i,2 ,...,z i,t ,...,z i,L ; z i,t denotes the coefficient vector of B i ;

[0029] Step ten, based on the array acquisition t th snapshot data X(t), B i and z i,t , calculate the residual r i,t of the i th iteration;

[0030] Step eleven, judge whether i < M-1;

[0031] If yes, repeat steps four to eleven until i = M-1;

[0032] If no, output the source estimation result

[0033] The beneficial effects of the present application are:

[0034] In a complex underwater acoustic environment, the source number estimation faces many challenges. The array snapshot data signal-to-noise ratio is usually not high, and is easily disturbed by colored noise and coherent sources. In addition, for high-speed moving targets, the effective snapshot number is small, and these factors cause the performance of traditional source number estimation methods to decrease significantly. However, the orthogonal matching pursuit method can still estimate the support vector accurately in the above complex scenarios. In order to estimate the source number of underwater acoustic targets stably, the present application uses the orthogonal matching pursuit method to iteratively solve the support vector, and the eigenvector corresponding to the maximum eigenvalue of the residual to construct two subspaces, and realizes the source number estimation through the signal subspace matching criterion. The simulation results show that compared with the existing source number estimation method, the method of the present application has lower requirements for signal-to-noise ratio, performs better under small snapshot conditions, and is not sensitive to coherent sources and colored noise. The lake test data processing results show that the method can effectively estimate the number of fixed and moving underwater acoustic targets.

[0035] In view of the above deficiencies of the covariance matrix eigenvector-based source number estimation, the application introduces a more robust eigenvector estimation method.

[23] Orthogonal Matching Pursuit (OMP) is a greedy sparse recovery method, which can effectively estimate the source corresponding steering vector under the condition of small snapshot and colored noise, and has lower requirement for signal-to-noise ratio, and is also applicable to coherent sources. When the iteration number is less than or equal to the source number, the support vector estimated by the OMP method in each iteration corresponds to the steering vector of one source. However, when the iteration number exceeds the source number, the support vector corresponds to the noise eigenvector. When the noise intensity is large, these noise eigenvectors may be mistaken for signal steering vectors, so the source number needs to be known.

[0036] In order to perform robust source number estimation on high-speed moving underwater acoustic targets, the application generates two subspaces according to a specific combination mode of the support vector obtained by the OMP method and the eigenvector corresponding to the maximum eigenvalue of the residual, and estimates the source number through the SSM criterion, and the method is simply referred to as OMP-SSM. The simulation results show that OMP-SSM has the advantage of SSM type method that is not sensitive to colored noise, and performs better under the condition of small snapshot, and is also applicable to the scene with coherent sources. The processing results of lake test data further verify the effectiveness of the OMP-SSM method.

[0037] When performing source number estimation in a complex environment, the challenges of low signal-to-noise ratio, small snapshot number, non-white noise and coherent sources are faced, therefore, the application proposes a method for robust source number estimation using orthogonal matching pursuit and signal subspace matching. The method uses the support vector obtained by the orthogonal matching pursuit method and the eigenvector of the residual to construct two subspaces, and calculates the distance between them through the subspace matching criterion, and the iteration number corresponding to the minimum distance is taken as the estimation result of the source number. Theoretical analysis and simulation results show that, compared with the traditional eigenvalue-based source number estimation method, when the noise eigenvalue does not meet the assumption of Gaussian white noise model, the OMP-SSM method still has good estimation performance; compared with the ISSM method which is also based on the signal subspace matching criterion, the OMP-SSM method has lower requirement for signal-to-noise ratio under the condition of small snapshot; in addition, the maximum source number that can be estimated by the OMP-SSM method can reach the theoretical value, and the method is applicable to the scene with coherent sources. The processing results of lake test data show that the method can effectively estimate the number of underwater acoustic targets. In summary, the method has potential for practical application, and can be considered for application in the following scenes: target fast movement leading to small snapshot number; large noise intensity and non-white characteristics in complex water areas; and coherent signal caused by interface reflection. BRIEF DESCRIPTION OF DRAWINGS

[0038] Figure 1 Flow chart of the proposed method;

[0039] Figure 2 Subarray partitioning scheme of the ISSM method, (a) subarray 1 partitioning scheme; (b) subarray 2 partitioning scheme;

[0040] Figure 3 Computational procedure of the ISSM method;

[0041] Figure 4 Computational procedure of the OMP-SSM method in the i-th iteration;

[0042] Figure 5 Estimation performance versus SNR;

[0043] Figure 6 Analysis results of a single Monte Carlo experiment with SNR = -5dB, (a) normalized eigenvalues of the covariance matrix; (b) distance between the subspace spanned by the first k support vectors of OMP and the first k eigenvectors of the covariance matrix and the signal subspace; (c) subspace matching results of ISSM and OMP-SSM;

[0044] Figure 7 Estimation performance versus noise color, (a) SNR = -10dB; (b) SNR = 0dB; (c) SNR = 5dB; (d) SNR = 10dB;

[0045] Figure 8 Estimation performance versus number of sources, (a) Gaussian white noise and SNR = 0dB; (b) colored noise and SNR = 0dB, α = 0.3;

[0046] Figure 9 Estimation performance with coherent sources, (a) estimation performance versus SNR in Gaussian white noise; (b) estimation performance versus noise color with SNR = 5dB in colored noise;

[0047] Figure 10 Estimation performance with coherent sources, (a) estimation performance versus SNR in Gaussian white noise; (b) estimation performance versus noise color with SNR = 5dB in colored noise;

[0048] Figure 11 Estimation performance with coherent sources, (a) estimation performance versus SNR in Gaussian white noise; (b) estimation performance versus noise color with SNR = 5dB in colored noise;

[0049] Figure 12Fig. 4 is a diagram of estimation performance for different incident angle intervals, (a) estimation performance with incident angle interval under Gaussian white noise with SNR of 0dB; (b) estimation performance with incident angle interval under colored noise with SNR of 0dB and a of 0.3;

[0050] Figure 13 Fig. 5 is a diagram of experimental conditions, (a) experimental situation; (b) receiving array; (c) real bearing of the sound source ship; (d) distance between the sound source ship and the receiving ship; (e) sound velocity profile;

[0051] Figure 14 Fig. 6 is a diagram of background noise analysis, (a) time-frequency spectrum of background noise; (b) eigenvalue of normalized covariance matrix;

[0052] Figure 15 Fig. 7 is a diagram of comparison between source number estimation results of each method and the real source number, (a) OMP-SSM method; (b) ISSM method; (c) ISSR method; (d) LS-MDL method; (e) NE-AIC method; (f) ADL-LS method;

[0053] Figure 16 Fig. 8 is a diagram of analysis results of the 41st second lake test data, (a) eigenvalue of normalized covariance matrix; (b) distance between the subspace generated by the first k support vectors of OMP and the first k eigenvectors of the covariance matrix and the signal subspace; (c) subspace matching results of ISSM and OMP-SSM. DETAILED DESCRIPTION

[0054] Embodiment 1: The source number estimation method based on orthogonal matching pursuit and signal subspace matching has the following specific process:

[0055] Step 1: Obtain an MxL matrix X composed of L snapshots collected by the array, X = [X(1), X(2),..., X(t),..., X(L)];

[0056] wherein t = 1, 2,..., L; M represents the number of array elements; X(1) represents the snapshot data collected by the array at time 1, X(2) represents the snapshot data collected by the array at time 2, X(t) represents the snapshot data collected by the array at time t, and X(L) represents the snapshot data collected by the array at time L;

[0057] Step 2: Set the total number of iterations M-1;

[0058] Step 3: Initialize the iteration number i = 0; initialize r 0,t = X(t), t = 1,..., L; initialize

[0059] wherein r 0,tLet represent the residual of the t-th snapshot data collected by the array at the 0th iteration, and Λ0 represent the set of support vector indices obtained from the first 0 iterations. Represents the empty set;

[0060] Step 4: Let the iteration number i = i + 1;

[0061] Step 5: Complete the dictionary Each a(θ) in v ) and the residuals of all snapshot data are used to calculate the inner product, and the absolute values ​​of all inner products are summed. Then, the value corresponding to the largest sum is selected as a(θ). v ) as the support vector obtained in the i-th iteration Where β i For the selected a(θ) v The index of θ; v = 1, 2, ..., J; J represents the total number of observable angles; column vector a(θ) v ) is the steering vector of the v-th far-field narrowband source;

[0062] Step 6: Based on the support vector obtained in the i-th iteration The corresponding index β i The set Λ of indices corresponding to the support vectors obtained in the first i-1 iterations i-1 Obtain the set Λ of indices corresponding to the support vectors obtained in the first i iterations. i ;

[0063] The set Λ of indices corresponding to the support vectors obtained from the previous i iterations i Obtain the matrix composed of the support vectors obtained in the first i iterations.

[0064] Step 7: Calculate the residual r of the (i-1)th iteration. i-1 The covariance matrix is ​​used to obtain the eigenvector matrix U corresponding to the residual at the (i-1)th iteration. i-1 ;r i-1 =[r i-1,1 ,r i-1,2 ,...,r i-1,t ,...,r i-1,L ];

[0065] Step 8: Build a matrix based on the support vectors obtained from the first i iterations. and eigenvector matrix U i-1 The distance between subspaces in the i-th iteration is calculated using the signal subspace matching criterion.

[0066] Among them, (U) i-1 )1 represents the eigenvector matrix U i-1 The first column (the first column is U) i-1the eigenvector corresponding to the largest eigenvalue of the matrix

[0067] denotes the subspace generated by the column vectors of the matrix

[0068] denotes the subspace generated by the column vectors of the matrix i-1 and the column vectors of the eigenvector (U

[0069] denotes the subspace generated by the column vectors of the matrix in the bracket;

[0070] denotes the distance between the ith iteration subspace and the subspace

[0071] SSM i denotes the distance between the subspaces calculated in the ith iteration;

[0072] Step nine, let

[0073] where B i denotes the matrix composed of the support vectors obtained in the first i iterations;

[0074] Solve z i,t by least squares z = argmin i ||X(t)-B i,1 z||2, t = 1,..., L;

[0075] X(1) corresponds to z i,2 ; X(2) corresponds to z i,L ; X(L) corresponds to z i,1 ;

[0076] where z denotes the coefficient vector obtained by least squares optimization, z = z i,2 , z i,t ,..., z i,L ; z i,t denotes the coefficient vector of B i ;

[0077] Step ten, based on the tth snapshot data X(t) collected by the array, B i and z i,t , calculate the residual r i,t of the ith iteration;

[0078] Step eleven, judge whether i < M-1;​​​

[0079] If yes, repeat steps four to eleven until i=M-1;

[0080] If no, output the source estimation result

[0081] Specific implementation two: the difference between this embodiment and the specific implementation one is that the step one specific process is:

[0082] Suppose that K far-field narrow-band sources are incident to a M-element uniform linear array, and the incident direction θ is the included angle between the signal direction and the normal direction of the array;

[0083] Let X=[X(1), X(2), …, X(t), …, X(L)] be a M×L matrix composed of L snapshots collected by the array, X(1) represents the snapshot data collected by the array at time 1, X(2) represents the snapshot data collected by the array at time 2, X(t) represents the snapshot data collected by the array at time t, and X(L) represents the snapshot data collected by the array at time L;

[0084] Let S=[S(1), S(2), …, S(t), …, S(L)] be a K×L matrix composed of K sources in L snapshots, S(1) represents a vector composed of the amplitudes of the K sources at time 1, S(2) represents a vector composed of the amplitudes of the K sources at time 2, S(t) represents a vector composed of the amplitudes of the K sources at time t, and S(L) represents a vector composed of the amplitudes of the K sources at time L;

[0085] Let N=[N(1), N(2), …, N(t), …, N(L)] be a M×L matrix composed of M noises in L snapshots, N(1) represents a vector composed of the amplitudes of the M array elements at time 1, N(2) represents a vector composed of the amplitudes of the M array elements at time 2, N(t) represents a vector composed of the amplitudes of the M array elements at time t, and N(L) represents a vector composed of the amplitudes of the M array elements at time L;

[0086] The array observation model is represented as

[0087] X=AS+N (1)

[0088] A=[a(θ1),a(θ2),...,a(θ k ),...,a(θ K )] (2)

[0089]

[0090] where A is the array manifold matrix, column vector a(θ1) is the steering vector of the 1st far-field narrowband source, column vector a(θ2) is the steering vector of the 2nd far-field narrowband source, column vector a(θ k ) is the steering vector of the kth far-field narrowband source, and column vector a(θ K ) is the steering vector of the Kth far-field narrowband source.

[0091] λ represents the signal wavelength, d represents the inter-element distance, is an intermediate variable, θ k represents the angle between the direction of the kth far-field narrowband source and the normal direction of the array, j represents the imaginary unit, j 2 =-1; the superscript T represents transposition;

[0092] In order to sparsely represent the array signal model, an overcomplete dictionary of the array manifold matrix is introduced

[0093]

[0094] where J represents the total number of observable angles, which is much larger than the actual number of sources K.

[0095] Column vector a(θ1) is the steering vector of the 1st far-field narrowband source, column vector a(θ2) is the steering vector of the 2nd far-field narrowband source, column vector a(θ v ) is the steering vector of the kth far-field narrowband source, and column vector a(θ J ) is the steering vector of the Kth far-field narrowband source.

[0096] {θ1, θ2,..., θ J} represents the set of observable angles of the linear array, which is obtained by quantizing the far-field space [-90°, 90°] at certain angle intervals.

[0097] The source corresponding to L snapshots is represented as a JxL matrix Each snapshot corresponds to a J-dimensional column vector, which has only K non-zero elements, corresponding to the K angle positions {θ1, θ2,..., θ v ,..., θ J} in the set {θ1, θ2,..., θ k ,..., θ K}.

[0098] represents the vector composed of the amplitudes of all incident sources within the observable angle at time 1, represents the vector composed of the amplitudes of all incident sources within the observable angle at time 2, represents the vector composed of the amplitudes of all incident sources within the observable angle at time L;

[0099] In the case of {θ1,θ2,...,θ} k ,...,θ K The corresponding column vector a(θ) v ) is called the support vector of X;

[0100] The sparse representation of the array signal model is as follows

[0101]

[0102] The other steps and parameters are the same as in Specific Implementation Method 1.

[0103] Specific Implementation Method Three: This implementation method differs from Specific Implementation Method One or Two in that the specific process of step five is as follows:

[0104] Where, β i Denotes the support vector obtained in the i-th iteration. The corresponding index;

[0105] a(θ v ) represents the steering vector of the v-th far-field narrowband source, where v = 1, 2, ..., J;

[0106] θ v Let v represent the angle between the direction of the v-th far-field narrowband source and the direction of the array normal, where v represents the v-th source.

[0107] The superscript H indicates finding the conjugate;

[0108] r i-1 Let r represent the residual obtained in the (i-1)th iteration. i-1 =[r i-1,1 ,r i-1,2 ,...,r i-1,t ,...,r i-1,L ], r i-1,1 Indicates r i-1 The residual corresponding to X(1); r i-1,2 Indicates r i-1 The residual corresponding to X(2); r i-1,t Indicates r i-1 The residual corresponding to X(t), r i-1,L Indicates r i-1 The residual corresponding to X(L) in the middle.

[0109] Other steps and parameters are the same as in specific implementation method one or two.

[0110] Specific Implementation Method Four: This implementation method differs from Specific Implementation Methods One to Three in that the specific process of step six is ​​as follows:

[0111] 1) Support vectors obtained based on the i-th iteration corresponding to the support vector obtained in the i-th iteration i corresponding to the support vector obtained in the i-th iteration i-1 corresponding to the support vector obtained in the i-th iteration i corresponding to the support vector obtained in the i-th iteration

[0112] corresponding to the support vector obtained in the i-th iteration i corresponding to the support vector obtained in the i-th iteration i-1 corresponding to the support vector obtained in the i-th iteration i corresponding to the support vector obtained in the i-th iteration

[0113] corresponding to the support vector obtained in the i-th iteration i-1 corresponding to the support vector obtained in the i-th iteration i-1 corresponding to the support vector obtained in the i-th iteration i corresponding to the support vector obtained in the i-th iteration i corresponding to the support vector obtained in the i-th iteration

[0114] corresponding to the support vector obtained in the i-th iteration i-1 corresponding to the support vector obtained in the i-th iteration i corresponding to the support vector obtained in the i-th iteration

[0115] 2) obtaining a matrix composed of support vectors obtained in the i-th iteration based on the index set i corresponding to the support vector obtained in the i-th iteration

[0116]

[0117] The other steps and parameters are the same as one of the first to third embodiments.

[0118] The fifth embodiment is different from one of the first to fourth embodiments in that the step eight has the following specific process:

[0119] 1) calculating the projection matrix of the subspace

[0120] 2) calculating the projection matrix of the subspace

[0121] 3) calculating the difference between the projection matrix and the projection matrix , and obtaining the distance between the subspace and

[0122] The other steps and parameters are the same as one of the first to fourth embodiments.

[0123] Specific Implementation Method Six: This implementation method differs from Specific Implementation Methods One to Five in that, in step 1), the computational subspace... projection matrix Represented as:

[0124]

[0125] The other steps and parameters are the same as those in one of the specific implementation methods one to five.

[0126] Specific Implementation Method Seven: This implementation method differs from Specific Implementation Methods One through Six in that the computational subspace in step 2) is... projection matrix Represented as:

[0127]

[0128] The other steps and parameters are the same as those in one of the specific implementation methods one to six.

[0129] Specific Implementation Method Eight: This implementation method differs from Specific Implementation Methods One to Seven in that, in step 3), the projection matrix is ​​calculated... and projection matrix The difference is used to obtain the subspace by taking the norm of the difference. and Distance between Represented as:

[0130]

[0131] Where <·> denotes the subspace generated by the column vectors of the matrix within the parentheses; This represents the square of the Frobenius norm.

[0132] The other steps and parameters are the same as those in any of the specific implementation methods one to seven.

[0133] Specific Implementation Method Nine: This implementation method differs from Specific Implementation Methods One to Eight in that, in step ten, the data X(t) and B, acquired by the array, are based on the t-th snapshot data X(t) and B. i and z i,t Calculate the residual r of the i-th iteration. i,t ;

[0134] r i,t =X(t)-B i z i,t ,t=1,...,L.

[0135] The other steps and parameters are the same as those in one of the specific implementation methods one to eight.

[0136] 2. Derivation of the OMP-SSM method and introduction of related theories:

[0137] 2.1 Signal Subspace Matching Criterion

[0138] The key to estimating the number of sources using the signal subspace matching criterion lies in constructing two sets of vectors, where the first K vectors are signal feature vectors and the remainder are noise feature vectors, and the signal feature vectors in the two sets of vectors must not be completely identical. When both sets select the first i vectors to generate subspaces, the subspaces generated by the two sets are equal only when i = K, i.e., both are signal subspaces.

[0139] Next, the signal subspace matching criterion is further introduced using the ISSM method. A uniform linear array of M elements is divided into two subarrays, each composed of elements numbered 1:MI and I+1:M, where 1≤I≤M-1. Figure 2 This demonstrates the subarray partitioning methods when I=1 and I=2. Based on the array rotation invariance, it can be concluded that…

[20] When the noise is 0, the signal eigenvectors of the covariance matrix of these two subarrays can generate the same signal subspace. The ISSM method uses the eigenvectors of the covariance matrix of these two subarrays to perform signal subspace matching and realize source number estimation. In order to improve the robustness of the method, the reference

[19] uses the two sets of subarrays obtained when I is 1 and 2 for matching, which is called the multiple ISSM method.

[0140] When I = 1, the subarray element indices are 1:M-1 and 2:M. The calculation process is as follows: Figure 3 As shown. First, the covariance matrix of the two subarrays is estimated using snapshot data, and then eigenvalues ​​are obtained by eigenvalue decomposition, denoted as U = [u1, u2, ..., u...]. M-1 ] and V = [v1, v2, ..., v M-1 The first K vectors are signal feature vectors, and the last MK-1 are noise feature vectors. Then, the first i feature vectors from U and V are selected to generate a subspace, i∈{1,2,...,M-1}. Finally, the distance between subspaces is calculated using a subspace matching criterion; the i corresponding to the minimum distance is the estimated number of sources. To characterize the distance between subspaces, the projection matrix of the subspace generated by the first i feature vectors is first calculated.

[0141] P(U 1:i )=U 1:i (U 1:i H U 1:i ) -1 U 1:i H (7)

[0142] P(V 1:i) = V 1:i (V 1:i H V 1:i ) -1 V 1:i H , (8)

[0143] U 1:i and V 1:i are matrices composed of the first i column vectors of U and V respectively. Then the difference of the projection matrices of the two subspaces is calculated, and its Frobenius norm is computed

[0144]

[0145] where "<·>" denotes the subspace generated by the column vectors of the matrix in the bracket. The value of this norm characterizes the distance between the subspaces generated by U 1:i and V 1:i .

[0146] The distance between <U 1:i > and <V 1:i > is analyzed as follows when i takes different values.

[0147] (1) When i = K, <U 1:i > and <V 1:i > are generated by all the estimated signal eigenvectors. In this case, the distance between the subspaces only depends on the estimation accuracy of the signal eigenvectors. If the signal eigenvectors can be accurately estimated, the distance is 0.

[0148] (2) When i < K, <U 1:i > and <V 1:i > are generated by partial estimated signal eigenvectors. Since U and V are obtained by different subarrays, the subspaces generated by partial signal eigenvectors are not equal. If the signal eigenvectors can be accurately estimated, the distance between <U 1:i > and <V 1:i > will be larger than the case of i = K. It is worth noting that the eigenvectors corresponding to the largest eigenvalues in the two sets of eigenvectors are more similar, and after being disturbed by noise, the distance between the subspaces when i = 1 is smaller than that when i = K. This is because when i = 1, only one estimated signal eigenvector is selected to generate the subspace, and the noise disturbance is smaller than that when i = K. At the same time, since the two vectors are similar, the distance between the subspaces generated by them is also smaller, so this phenomenon is prone to occur.

[0149] (3) When i > K, <U 1:i > and <V 1:i > are generated by all the estimated signal eigenvectors U 1:K and V 1:Kand noise feature vector U K+1:i and V K+1:i Generate. Due to U K+1:i and V K+1:i It has randomness, therefore, when the signal feature vector can be accurately estimated,

[0150] 1:i >and <V 1:i The distance > will be greater than when i = K. When i is slightly greater than K, fewer noise feature vectors are selected.

[0151] K+1:i >and <V K+1:i The similarity is low, so add U 1:K and V 1:K back, 1:i >and <V 1:i The distance between them increases with the increase of i, reaching its maximum at a certain value. Because u i and v i Both are M-1 dimensional feature vectors. When i approaches M-1 and continues to increase, 1:i >and <V 1:i By gradually approximating the same M-1 dimensional linear space, the similarity between the two gradually increases. When i = M-1, 1:i >and <V 1:i If the two sources are in the same M-1 dimensional linear space, then the distance between them is 0. This also indicates that the maximum number of sources that this method can theoretically detect is M-2.

[0152] When I=2, the subarray element numbers are 1:M-2 and 3:M. The calculation and analysis process is similar to that of I=1. At this time, the maximum number of sources that can be estimated is reduced to M-3. The multiple ISSM method adds the calculation results of equation (9) when I=1 and I=2, and takes the iteration number i corresponding to the minimum value after addition as the source number estimation result.

[0153] Currently, many source number estimation methods have high requirements for the number of snapshots. [4-10] and sensitivity to colored noise [12-15] As the above analysis shows, the signal subspace matching criterion for source number estimation does not assume that the noise model is Gaussian white noise. When the level of colored noise is not very high and its impact on the signal eigenvector estimation performance is small, the signal subspace matching criterion can still maintain good estimation performance. Furthermore, this criterion is based on the idea of ​​matching and has low requirements for the number of snapshots. [18-19] Existing methods utilize signal eigenvector matching to estimate the number of information sources. [18-19,21] ​​​​​The eigenvectors are obtained through eigenvalue decomposition of the covariance matrix. However, obtaining a relatively accurate covariance matrix requires high signal-to-noise ratio (SNR) and a high snapshot number. Furthermore, the eigenvector estimation using these methods fails when coherent sources are present. Therefore, a eigenvector estimation method is needed that is less demanding on SNR under small snapshot conditions and is applicable to coherent sources. Based on this method, two sets of eigenvectors for matching are constructed, requiring low similarity between the two eigenvectors used for matching when i=1. Finally, the source number is estimated using a signal subspace matching criterion.

[0154] 2.2 Orthogonal Matching Pursuit Method

[0155] OMP is a greedy sparse recovery method. When the number of sources K is known, it can be used based on snapshot data X and an overcomplete dictionary. The steering vector corresponding to the signal source is obtained iteratively. The standard OMP method is only applicable to sparse recovery of single-shot data. When the data consists of multiple shots, the array signal model satisfies the multiple measurement vector model in compressed sensing theory, allowing the OMP method to be extended to achieve joint sparse recovery of multi-shot data.

[24] .

[0156] 2.2.1 Calculation process of OMP method

[0157] The calculation process of the OMP method is shown in Table 1. β i Λ represents the index of the support vector obtained in the i-th iteration; i Λ represents the set of indices of the first i support vectors. i ∈{β1,β2,...,β i}; r is a matrix composed of the first i support vectors; i,t Remove X(t) The residual after the relevant part, r i =[r i,1 ,r i,2 ,...,r i,L After initialization is complete, in step (b), the overcomplete dictionary will be... Each atom in the sequence is multiplied by the inner product of all snapshot residuals, and the absolute values ​​of all snapshot inner products are summed. The atom with the largest sum is then selected as the support vector obtained in this iteration. In steps (c) to (e), the support vector obtained in the current iteration is incorporated into... The residuals are updated by solving a least-squares problem. This iterative process is repeated until K support vectors are selected, which correspond to the estimated steering vectors of the K sources.

[0158] Table 1. Calculation steps of the orthogonal matching pursuit method

[0159]

[0160] If the number of sources K is known, the DOA estimation can be realized by OMP method, and the angle corresponding to the estimated steering vector is the incident angle of the source. When the number of sources is unknown, if the iteration number exceeds the number of sources K, the support vector obtained will be determined by the residual of the noise. If the noise energy is large, the amplitude of the support vector corresponding to it may be close to the support vector corresponding to the source, which is easy to lead to misjudgment.

[0161] The focus of the present application is not to use the OMP method for DOA estimation, but to generate two subspaces by using the support vector estimated by the OMP method and the eigenvector corresponding to the maximum eigenvalue of the residual, and to realize the source number estimation by the SSM criterion.

[0162] 2.2.2 Performance comparison and analysis of OMP and eigenvalue decomposition method under small snapshot condition

[0163] The following further analyzes that under the condition of small snapshot, the error of using OMP to estimate the support vector is smaller compared with estimating the eigenvector by the covariance matrix. In order to simplify the analysis, it is assumed that when the support vector is solved in the i th iteration of OMP, the residual r i-1 contains only noise N and support vector When the signal model is single snapshot

[0164]

[0165] In step (b), each atom is multiplied by r i-1 to select the support vector, and when the atom is , the inner product result corresponding to it is the largest

[0166]

[0167] In formula (11) When the array element noise is independent Gaussian white noise, Since the norm of is 1, it can be obtained that

[0168] It can be seen that the variance σ 2 of the noise part decreases with the increase of the number of array elements. Therefore, even under the condition of single snapshot, the influence of noise on the estimation of support vector will be weakened with the increase of the number of array elements. When the array element noise is colored noise, the noise intensity of each array element is not equal and may be correlated, at this time the corresponding to each array element is no longer independent and identically distributed. But when the degree of color is not very high, The variance of the estimation error of the covariance matrix can be reduced by increasing the number of array elements. In the case of multiple snapshots, the inner product operation is performed for each snapshot, and the results are summed. The support vector is then selected based on the sum. This reduces the influence of large errors in individual snapshot data on the selection of the support vector, thereby improving the estimation performance.

[0169] The array covariance matrix is R X = E(X(t)X(t) H ), which is approximated by computing the time average when using snapshot data

[0170]

[0171] According to the literature

[26] , the estimation error of the covariance matrix obeys the following multivariate complex Gaussian distribution

[0172]

[0173] where "vec" represents the conversion of a matrix to a vector; represents the computation of the Kronecker product of two matrices. It can be seen that the variance of the estimation error of the covariance matrix increases as the number of snapshots L decreases. Therefore, the error of the eigenvector obtained by performing eigenvalue decomposition on is large in the case of small snapshots.

[0174] 2.2.3 Performance comparison between OMP and eigenvalue decomposition method in the presence of coherent sources

[0175] There are K sources S(t) = [s1(t), s2(t),..., s K (t)] T , A = [a(θ1), a(θ2),..., a(θ K )], and it can be seen from equation (1) that

[0176] X(t) = AS(t) + N(t) (14)

[0177] Assuming that s1(t) and s2(t) are coherent, we have s2(t) = γs1(t), where γ is a constant. Let We can obtain

[0178]

[0179] According to the literature

[27] , if eigenvalue decomposition is performed on the covariance matrix, the steering vectors a(θ1) and a(θ2) will be combined into a(θ1) + γa(θ2). Therefore, the source number estimation method based on eigenvalue decomposition of the covariance matrix estimates the result as K-1.

[0180] When there is a coherent source, the residual r i-1,t For

[0181] r i-1,t = s1(t)a(θ1) + γs1(t)a(θ2) + C (16)

[0182] where C represents the noise and the product of the non-coherent source and its corresponding steering vector. The inner product operation in step (b) of the OMP method in Table 1 is

[0183]

[0184] where, when performing the inner product operation, s2(t) = γs1(t) and s1(t) as a scalar only plays a role in scaling the result. Therefore, when there is a coherent source, it does not affect the OMP method using the inner product to solve the support vector.

[0185] 2.3 Estimation of the number of sources using orthogonal matching pursuit and signal subspace matching

[0186] 2.3.1 Theoretical derivation

[0187] Table 2 Calculation steps of the OMP-SSM method

[0188]

[0189] According to the analysis in Section 2.2, compared with the eigenvalue decomposition method, the OMP method has higher estimation accuracy of the steering vector under small snapshot conditions, and is not sensitive to colored noise and coherent sources. At the same time, when estimating the support vector, the OMP method first obtains the source steering vector, and then obtains the noise eigenvector, which meets the requirements for signal subspace matching in Section 2.1. Therefore, the present application uses the OMP method and the signal subspace matching criterion to estimate the number of sources. In order to minimize the use of eigenvectors of the covariance matrix, and to make the method applicable to coherent sources, in the i-th iteration, one subspace for matching is generated by the first i support vectors of the OMP method, and the other subspace is generated by the first i-1 support vectors and the eigenvector corresponding to the largest eigenvalue of the residual.

[0190] The calculation process of the OMP-SSM method is shown in Table 2. In steps (a) to (c), the calculation process of OMP-SSM is the same as that of the OMP method, and the support vector is found by inner product and the matrix composed of the support vectors is obtained In step (d), the residual r i-1 is calculated, and the covariance matrix is decomposed to obtain the corresponding eigenvector matrix Ui-1 Next, step (e) uses the SSM criterion in equation (9) to calculate the... and The distance of the generated subspace, where (U i-1 )1 represents the eigenvector matrix U i-1 The first column (the first column is U) i-1 The eigenvector corresponding to the largest eigenvalue in the subspace is used. Steps (f) and (g) update the residuals by solving a least-squares problem. Finally, the number of iterations corresponding to the minimum subspace distance is selected as the estimate of the number of sources.

[0191] The calculation process for the i-th iteration is as follows: Figure 4 As shown. For the sake of the following discussion, it is assumed that the incident angle spacing of the signal sources is large and the correlation of each steering vector is small, so that the OMP method can obtain the correct sparse recovery result under noise-free conditions.

[0192] (1) When i = K, Figure 4 In for It contains exactly the entire estimated steering vector. r i-1 For r K-1 ,express exist The residual of the upper projection. Due to the low correlation between the guiding vectors, the residual r K-1 Still retained Most of the information, therefore (U K-1 )1 is The normalized vector that constitutes the majority component. At this point... and The distance is mainly affected by noise. When the noise intensity is close to 0, (U K-1 )1 is very close Make and The distance is close to 0.

[0193] (2) When i > K, assuming i = K + 1, at this time Figure 4 In for It contains noise support vectors r i-1 For r K , indicating noise at The residual of the projection upwards, therefore (U K )1 is also a noise vector. Because the noise vector has a high degree of randomness, it makes... and The distance is greater than and The distance.

[0194] (3) When i < K, assume i = K-1, at this time Figure 4 In for It does not include all estimated steering vectors. i-1 For r K-2 ,express and exist The residual of the projection. Similarly, due to the low correlation between the steering vectors, the residual r K-2 Still retained and Most of the information. But at this time (U K-2 )1 is not or The normalized vector that constitutes the majority component, therefore and The distance is greater than and The distance between them. It is worth noting that when i = 1, the two subspaces used for matching are... and <[(U0)1]>. Among them, (U0)1 represents the first support vector estimated by the OMP method, corresponding to the steering vector of a certain source; (U0)1 represents the eigenvector corresponding to the largest eigenvalue obtained by eigenvalue decomposition of the covariance matrix. At this time, the similarity between the two is low, thus reducing the probability that the distance between the subspaces is greater than that when i=1 is greater than that when i=K.

[0195] The above discussion does not take the independence between information sources as a prerequisite, so this method is also applicable to the case where there are coherent information sources.

[0196] 2.3.2 Computational Complexity Analysis

[0197] In the i-th iteration of the OMP-SSM method, the computational complexity of the inner product of the residuals of a single snapshot and a single atom of an overcomplete dictionary in step (b) is O(M). Therefore, the computational complexity of the inner product of L snapshots and J atoms is O(LJM). The computational complexity of the eigenvalue decomposition in step (d) is O(M). 3 Step (e) calculates the subspace distance, and the computational complexity of calculating the projection matrix using equation (7) is O(iM). 2 The computational complexity of Equation (9) for finding the norm of the two projection matrices is O(M). 2 Therefore, the computational complexity of this step is O(iM). 2 The computational complexity of updating the single snapshot residual using least squares in steps (f) and (g) is O(iM).

[23] Then the computational complexity of L snapshots is O(iLM).

[0198] Under the condition of small snapshots, the number of snapshots L is close to the number of array elements M, and the complete dictionary is passed. The total number of atoms J is greater than M. Therefore, the computational complexity of the OMP-SSM method in the i-th iteration is O(LJM). The method performs M-1 iterations in total, with a computational complexity of O(LJM). 2 ).

[0199] 2.3.3 Advantages and limitations of the method

[0200] Based on the analysis in Section 2.2, under small snapshot conditions, the OMP method has higher estimation accuracy than the eigenvalue decomposition method. Furthermore, the matching subspace of the OMP-SSM method is mainly generated from the support vectors of OMP; therefore, compared to methods that only utilize the eigenvectors of the covariance matrix to estimate the number of sources, OMP-SSM performs better under small snapshot conditions. Moreover, under colored noise conditions, when the degree of coloration of the noise is not high, the OMP method still maintains good estimation performance. Finally, this method is also applicable to the presence of coherent sources.

[0201] The limitations of the SSM-OMP method mainly stem from the inherent limitations of the OMP method itself. As an overcomplete dictionary, when using the OMP method to estimate the steering vector of an incident signal, the estimation accuracy is affected not only by noise but also by the distance between the incident angles of the signal sources. [28-29] When the incident angle of the signal source is close and the aperture is small, the support vector corresponding to the signal source will have a high correlation, which may cause the support vector estimated in step (b) to fail. Furthermore, when calculating the residuals in steps (f) and (g), if the steering vector has a high correlation, then when i = K-1, the residual r K-1 Only A small portion of the information. At this time (U K-1 )1 will deviate make and As the distance increases, it can easily lead to errors in estimating the number of sources. Therefore, when there are nearby incident sources, it is necessary to increase the array aperture to reduce the correlation of the steering vector. [30-31] .

[0202] Example:

[0203] The following simulation analysis examines the impact of noise, snapshot number, target source number, coherent sources, and angular spacing between adjacent sources on the estimation performance of the OMP-SSM method. Simultaneously, it introduces the existing ISSM...

[19] ISSR

[21] NE-AIC

[12] LS-MDL

[13] and ADL-LS

[17] The methods were compared in performance. Specifically, when simulating the ISSM method, to improve the robustness of the estimation results, the multiple ISSM method recommended in reference 19 was used for source number estimation. The simulation used an isotropic uniform linear array with 16 elements M, an element spacing equal to half the wavelength of the narrowband source, and an azimuth grid spacing of 0.5°. The signal-to-noise ratio (SNR) was defined as 10lg(1 / 2) * ( ... 10 (p / σ n 2 ), where p represents the source power, σ n 2 The variance (power) of the Gaussian white noise is represented. 1000 Monte Carlo experiments are conducted under each condition, and the estimation performance of each method is characterized by the estimation accuracy, defined as the ratio of the number of experiments that accurately estimate the number of sources to the total number of experiments.

[0204] 3.1 Analysis of the impact of noise on source number estimation:

[0205] Assume there are four independent narrowband random-phase sources with incident angles of [-24°, 2°, 10°, 25°], and a snapshot count of 20. Consider both white noise and colored noise. The colored noise model is a banded colored noise.

[19] As shown in equations (18) and (19)

[0206]

[0207] Where, n i (t) represents the white noise of the i-th element, cn m (t) represents the colored noise obtained by the m-th array element after being affected by neighboring array elements. The parameter α is used to characterize the degree of color of the noise. When α is 0, cn m (t) then becomes white noise.

[0208] 3.1.1 Analysis of the impact of Gaussian white noise on the estimation of the number of information sources

[0209] Figure 5 This demonstrates how the estimation accuracy of different source number estimation methods varies with SNR under Gaussian white noise background. From... Figure 5 It can be observed that when SNR ≤ -5dB, the OMP-SSM method outperforms other comparative methods in estimation accuracy. When SNR ≥ -4dB, the NE-AIC and LS-MDL methods slightly outperform OMP-SSM. Furthermore, the overall performance of the OMP-SSM method is significantly better than that of the ISSM and ISSR methods. In particular, when SNR = -3dB, the estimation accuracy of the OMP-SSM method is close to 0.9, while the estimation accuracy of ISSM and ISSR is less than 0.5.

[0210] Figure 6The analysis results of a single Monte Carlo experiment with a signal-to-noise ratio of -5dB are presented. Figure 6 Figure (a) shows the eigenvalues ​​of the normalized covariance matrix, ordered from largest to smallest. The simulation uses four incident sources; therefore, the first four eigenvalues ​​correspond to signal eigenvalues, and the remaining twelve are noise eigenvalues. Theoretically, under Gaussian white noise, the noise eigenvalues ​​should tend to be equal, reflecting the uniformity and unbiasedness of the noise. However, when the number of snapshots is small, the estimation error of the sample covariance is large, resulting in significant differences between the noise eigenvalues, deviating from the theoretically flat distribution. Nevertheless, from... Figure 6 In (a), it can still be observed that the changes in noise characteristic values ​​are relatively gradual, and the overall trend is linearly decreasing.

[0211] Figure 6 Figure (b) shows the subspace generated by the first k support vectors of OMP. and the subspace <U generated by the first k eigenvectors of the covariance matrix 1,...,k >, the distance to the signal subspace . It can be observed that since the number of sources is 4, when k=4, and<U 1,...,k The distance between > and is the smallest. However, due to the low signal-to-noise ratio and the small number of snapshots, the eigenvectors estimated by the covariance matrix have a large error, resulting in when k=4. 1,...,k The distance between > and is relatively large, close to 1. Based on the analysis in Section 2.2, the OMP method has good noise suppression capability when the number of snapshots is small; therefore, when k=4, The distance from is small, close to 0.

[0212] Figure 6 Figure (c) shows the distance changes between the subspaces used for matching in the ISSM and OMP-SSM methods at different iteration numbers i. It can be seen that the OMP-SSM method achieves the minimum distance between subspaces at i=4, successfully estimating the number of sources. For the ISSM method, according to the analysis in Section 2.1, the first eigenvector in the two sets of eigenvectors has high similarity. When the noise intensity is high (SNR=-5dB), the following may occur: Figure 6 The matching result shown in (c) indicates that the distance between subspaces is less when i=1 than when i=4. Furthermore, since multiple ISSMs are used for source number estimation, the theoretically maximum estimable number of sources is 13. From... Figure 6 As can be seen in (c), when i > 4, the distance between subspaces of the ISSM method first increases and then decreases with the increase of i, which is consistent with the analysis in section 2.1. And according to... Figure 4It can be seen that, in the OMP-SSM method, only the last vector is different in the two sets of vectors used to generate the matching subspaces, so the distance between the subspaces mainly depends on the difference of the last vector. This also makes Figure 6 In Fig. 8(c), the distance between the subspaces fluctuates greatly when i > 4 in the OMP-SSM method.

[0213] 3.1.2 Analysis of the influence of colored noise on the estimation of the number of sources

[0214] Figure 7 Fig. 9 shows the estimation accuracy of each method for estimating the number of sources as a function of a when the SNR is 5 dB in the presence of colored noise. From Figure 7 It can be seen from Fig. 9 that, as a increases, the performance of the OMP-SSM and ADL-LS methods is affected the least. The estimation accuracy of the OMP-SSM method remains above 0.9 at all times, while the estimation accuracy of the ADL-LS method decreases from 0.85 to 0.72. The robustness of the ISSM and LS-MDL methods is poorer, and they show a significant jump when a is 0.7 and 0.5, respectively. The NE-AIC and ISSR methods are easily affected by colored noise, and the estimation accuracy of both methods is close to 0 when a is 0.5.

[0215] Figure 8 Fig. 10 shows the analysis results of a single Monte Carlo experiment when the SNR is 5 dB and a is 0.6 in the presence of colored noise. By comparing with Figure 6 , the influence of the change in noise type (from Gaussian white noise to colored noise) on the eigenvalues and the influence of the improvement in SNR (from -5 dB to 5 dB) on the distance between the subspaces generated by the eigenvectors of the covariance matrix and the signal subspace can be analyzed. From Figure 8 (a) in Fig. 10, it can be seen that, in the presence of colored noise, the noise eigenvalues no longer show a linear change. The first few noise eigenvalues decrease rapidly, and the 10th eigenvalue is close to 0. In this case, if the number of sources is estimated only by relying on the eigenvalues, the rapidly decreasing part of the noise eigenvalues may be mistaken for signal eigenvalues, resulting in an overestimation of the number of sources. From Figure 8 (b) in Fig. 10, it can be seen that, as the SNR improves from -5 dB to 5 dB, the interference of the noise decreases, reducing the distance between the signal subspace generated by the covariance matrix and the true signal subspace. In addition, Figure 8 (c) in Fig. 10 shows that, in the presence of colored noise, both the OMP-SSM and ISSM methods can accurately estimate the target number of sources, verifying their effectiveness in the colored noise environment.

[0216] 3.2 Analysis of the influence of the number of snapshots on the estimation of the number of sources

[0217] Assume that there are four independent sources with narrowband random phase, and the incident angles are [-24°, 2°, 10°, 25°]. Figure 9 Fig. 9 Figure 9 Fig. 9 (d) shows the estimation accuracy of different source number estimation methods with the number of snapshots varying under the Gaussian white noise background with the signal-to-noise ratio of -10dB, 0dB, 5dB and 10dB. From Figure 9 As can be seen from Fig. 9 (a), when the signal-to-noise ratio is -10dB and the number of snapshots is 80, the estimation accuracy of the OMP-SSM method is 0.8, while that of the ADL-LS method is 0.6, and the estimation accuracy of other methods is less than 0.5. Figure 9 Fig. 9 (b) shows that when the signal-to-noise ratio is 0dB, the probability curve of the OMP-SSM method is relatively stable, and the estimation accuracy can reach 0.8 when the number of snapshots is 7, while the estimation accuracy of other methods is less than 0.5. Figure 9 Fig. 9 (c) and Fig. 9 (d) show that when the signal-to-noise ratio is 5dB and 10dB respectively, the estimation accuracy of the OMP-SSM method can reach 0.9 when the number of snapshots is 5 and 4 respectively.

[0218] 3.3 Analysis of the influence of the target source number on source number estimation

[0219] In order to analyze the influence of the target source number on the performance of each method, first, 15 independent sources with narrowband random phase are constructed, and the incident angles are [0°, 8°, 15°, 23°, 31°, 40°, 51°, 63°, -8°, -15°, -23°, -31°, -40°, -51°, -63°]. Then, the first k sources are selected from the constructed source group to generate the signals for simulation, k ∈ {1, 2, …, 15}, and the number of snapshots is 20. Figure 10 Fig. 9 (a) and Fig. 10 (b) show the estimation accuracy varying with the source number under the Gaussian white noise background with the signal-to-noise ratio of 0dB. The α of the colored noise is 0.3. As can be seen, the estimation accuracy of the OMP-SSM method always remains above 0.9, and the maximum target source number that can be estimated also reaches the theoretical value. In contrast, the estimation accuracy of other methods gradually decreases with the increase of the target source number. It is worth noting that according to the analysis in sections 2.1 and 3.1.1, the distance between the two matched subspaces of the ISSM method is small at iteration i = 1 and i = 13. Therefore, when the true source number is 1 and 13, the source estimation accuracy of this method is very high. In Figure 10 In Fig. 10 (b), the performance of the NE-AIC, LS-MDL and ISSR methods has decreased significantly due to the addition of the colored noise, and the estimation accuracy of the ISSM method fluctuates greatly due to its poor robustness, while the performance of only the OMP-SSM and ADL-LS methods remains almost unchanged.

[0220] 3.4 Analysis of the influence of coherent sources on the source number estimation

[0221] Assume that there are four narrowband sources with the incident angles [-24°, 2°, 10°, 25°], corresponding to s1, s2, s3 and s4, and the number of snapshots is 20. Among them, s1 and s4 are coherent, and s1(s4), s2 and s3 are independent of each other. Figure 11 Fig. 11(a) and 11(b) respectively show the estimation accuracy of different source number estimation methods under Gaussian white noise and colored noise with varying SNR and a. Compared with Fig. 10(a) and 10(b), it can be seen that the estimation accuracy of the OMP-SSM method is almost unchanged, while the other methods have failed. Figure 5 and Figure 7 It can be seen that the estimation performance of the OMP-SSM method remains almost unchanged in the presence of coherent sources, while the other methods have failed. This is because when the snapshot data containing coherent sources is decomposed into eigenvalues and eigenvectors, the eigenvalues and eigenvectors of coherent sources will be combined. Therefore, the source number estimation methods based on eigenvectors (such as ISSM and SR-ISSM) and the methods based on eigenvalues (such as LS-MDL, NE-AIC and ADL-LS) cannot accurately estimate the source number in this case.

[0222] 3.5 Analysis of the influence of the incident angle spacing between adjacent sources on the source number estimation

[0223] Assume that there are four narrowband random phase independent sources with the incident angles [-24°, 2°, 10°, 10°+Δ], Δ∈{1°, 2°,..., 15°} and the number of snapshots is 20. The estimation accuracy of the OMP-SSM and ISSM methods based on signal subspace matching is compared under different incident angle spacings. Figure 12 Fig. 12(a) and 12(b) respectively show the estimation accuracy under Gaussian white noise and colored noise with SNR of 0 dB and varying incident angle spacing, where the a of the colored noise is 0.3. From Figure 12 It can be seen from Fig. 12(a) that the estimation accuracy of the OMP-SSM and ISSM methods reaches 0.8 when the incident angle spacing is 6° and 7° respectively. In addition, the curve of the OMP-SSM method changes more smoothly, showing better robustness. From Figure 12 It can be observed from Fig. 12(b) that under the condition of colored noise, the estimation accuracy curves of the OMP-SSM and ISSM methods are very close, but the curve of the OMP-SSM method changes more smoothly.

[0224] 4 Verification by lake trial data

[0225] 4.1 Lake trial conditions

[0226] The effectiveness of the method is verified by using lake trial data. The experimental scene is as follows Figure 13As shown, there are 3 sound source ships and 1 receiving ship in the experiment, and the AD sampling rate of the receiving ship is 200 kHz. The sound source ships include fixed sound source ship 1, fixed sound source ship 2 and mobile sound source ship. The fixed sound source ships 1 and 2 are dropped into water by about 8 m through lead blocks, and the mobile sound source ship is fixedly connected by a steel pipe and enters water by about 5 m deep. Among them, the fixed sound source 1 emits a single frequency signal of 25 kHz per second, the fixed sound source 2 emits a 14-30 kHz broadband interference noise for 6 seconds every 6 seconds, and the mobile sound source emits a single frequency signal of 25 kHz per second along the direction of the arrow. Figure 13 Fig. 24(b) shows a regular hexagonal receiving array of the receiving ship, each side of which is a uniform linear array of 16 array elements, and the array element spacing is 0.03 meters. The data of the best receiving side are selected for source number estimation. Figure 13 Fig. 24(c) shows the real bearing change of the 3 sound source ships over time. Figure 13 Fig. 24(d) shows the distance change between each sound source ship and the receiving ship over time. Figure 13 Fig. 24(e) is a sound velocity profile of the lake area, and the sound velocity is set to 1460 m / s when calculating the steering vector.

[0227] 4.2 Background noise analysis

[0228] When the sound source is turned off, the collected data of a single array element are taken out, and the background noise time-frequency spectrum is calculated by short-time Fourier transform, the window length is set to 0.05 seconds, and the background noise time-frequency spectrum is obtained as shown in Fig. 24(a). Figure 14 From Fig. 24(a), it can be observed that the power spectrum density of the background noise at different frequencies is not equal, for example, the power spectrum density near 50 kHz is obviously larger, which does not conform to the characteristics of white noise. Figure 14 In order to further analyze the eigenvalue distribution of the array covariance matrix in the frequency band of interest, FIR digital filtering is used to filter the array sampling data, the order is 128, and the passband range is 15 kHz-30 kHz. After filtering, the data are processed in segments, and the frequency domain amplitude of the 24.5 kHz-25.5 kHz frequency band is extracted as the frequency domain snapshot. When performing FFT, the sampling rate is set to 200 kHz, so that there are 10000 sampling points in 0.05 seconds, the frequency spectrum interval is 20 Hz, and there are 51 frequency domain snapshots in the 24.5 kHz-25.5 kHz frequency band. Therefore, there are 1020 frequency domain snapshots in 1 second for calculating the covariance matrix, and the eigenvalues obtained are shown in Fig. 24(b).

[0229] As can be seen from Fig. 24(b), the downward trend of the noise eigenvalues does not present a slow and linear change trend, which does not conform to the expected characteristics of white noise model. Figure 14 4.3 Source number estimation

[0230]

[0231] ​When the sound sources are turned on, the data collected by the array is preprocessed as in Section 4.2, filtered and processed in segments. The sampling rate is 200 kHz, the passband range is 15 kHz ~ 30 kHz, and the FFT is performed every 0.05 seconds, and then the frequency domain amplitude of the 24.5 kHz ~ 25.5 kHz frequency band is extracted as the frequency domain snapshot. 1020 frequency domain snapshots of 1 second are accumulated to form a frame of data for estimating the number of sources. 80 seconds of data are intercepted for source number estimation, and the estimation results of each method are compared with the true source number, as shown in Figure 15 . Figure 15 The "6+" in the middle of the ordinate scale indicates that the number of sources is ≥ 6.

[0232] Figure 15 Figure (a) shows the comparison of the source number estimation results of the OMP-SSM method per second and the true source number. The angular quantization interval of the overcomplete dictionary constructed by the OMP-SSM method is 0.5°. Since the fixed sound source ship 2 does not emit signals every second, the true source number will fluctuate periodically between 2 and 3. During the first 15 seconds, due to the close azimuth angle of the moving sound source ship and the fixed sound source ship 1, the source number cannot be accurately estimated. When the time reaches 16 seconds, according to Figure 13 Figure (c), the azimuth angle of the moving sound source ship and the fixed sound source ship 1 is close to 7 degrees at this time, and the OMP-SSM method can accurately estimate the true source number. As can be seen, the performance of OMP-SSM in the actual environment is mainly limited by the angular interval of adjacent sources (array aperture), which is consistent with the discussion in Section 2.3.3 and the simulation in Section 3.5. In addition, the fixed sound source ship 2 emits a broadband signal, and only the data of the 24.5 kHz to 25.5 kHz frequency band is used in this data processing, resulting in the failure to detect 3 sources during the time when the sound source ship 2 emits signals. Therefore, from Figure 15 Figure (a), it can be seen that since the 20th second, the OMP-SSM method has underestimation in the first second of each emission period of the sound source ship 2.

[0233] Observing Figure 15 Figures (b) to (f) from 15 to 41, it can be found that almost all the remaining source number estimation methods fail. In order to further explore the failure reason, the data of the 41st second are selected for in-depth analysis. Similar to Figure 6 and Figure 8 , Figure 16 Figure (a) shows the analysis results of the 41st second. According to Figure 13 Figure (c), there are 3 incident sources at the 41st second, so the 4th to 16th eigenvalues correspond to noise eigenvalues. From Figure 16As can be seen in (a), the decreasing trend of these noise eigenvalues ​​does not exhibit a slow, linear change. In particular, the decrease in the 4th and 5th eigenvalues ​​is significantly greater than the decrease in subsequent eigenvalues, which is inconsistent with the expected characteristics of the Gaussian white noise model. According to the analysis in Section 3.1.2, the source number estimation method based on eigenvalues ​​is prone to overestimating the value in this case. Therefore, in Figure 15 As can be observed in Tables (d) to (f), the estimation results of the LS-MDL, NE-AIC, and ADL-LS methods are all greater than the actual number of sources. It is worth noting that... Figure 7 The results show that the LS-MDL and ADL-LS methods have some resistance to colored noise, but their estimation almost fails when faced with real-world data. This is mainly because, in real-world environments, the noise eigenvalue distribution of acoustic signals after passing through complex underwater acoustic channels does not conform to the standard Gaussian white noise or colored noise under simulation conditions. Therefore, the performance of methods that rely solely on eigenvalues ​​for source number estimation degrades, leading to the estimation failure of the LS-MDL and ADL-LS methods.

[0234] Depend on Figure 16 As shown in (b), the distance between the signal subspace generated by the support vectors of the OMP method and the true signal subspace is slightly smaller than the distance between the subspace generated by the eigenvectors of the covariance matrix and the true signal subspace. However, according to the analysis in Sections 2.1 and 3.1.1, the high similarity of the first eigenvector in the two sets of eigenvectors used for matching by the ISSM method leads to... Figure 16 The matching result shown in (c) indicates that when the iteration number i = 1, the distance between subspaces is less than the distance when i = K (K is the number of real sources). Therefore, in Figure 15 As can be seen in (b), the source number estimation result of the ISSM method is almost always 1. For Figure 15 The ISSR method in (c) almost fails to estimate due to the small number of array elements and the interference of colored noise.

[0235] 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.

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Claims

1. A method for estimating the number of sources based on orthogonal matching pursuit and signal subspace matching, characterized in that: The method specifically comprises the following steps: Step one, An M*L matrix X=[X(1), X(2),..., X(t),..., X(L)] composed of L snapshots collected by the array is obtained; Wherein, t=1, 2,..., L; M represents the number of array elements; X(1) represents the snapshot data collected by the array at time 1, X(2) represents the snapshot data collected by the array at time 2, X(t) represents the snapshot data collected by the array at time t, and X(L) represents the snapshot data collected by the array at time L; Step two, the total number of iterations M-1 is set; Step three, initialize iteration number i = 0; initialize r 0,t = X(t), t = 1,..., L; initialize ; wherein r 0,t represents the residual of the tth snapshot data collected by the array at the 0th iteration, Λ0represents the set of support vector indexes obtained by the previous 0 iterations, represents an empty set; Step four, let the iteration number i=i+1; Step five, complete the overcomplete dictionary Step four, calculate the inner product of each a(θ v ) and the residual of all snapshot data, sum the absolute value of all inner products, and then select the a(θ v ) corresponding to the maximum sum value as the support vector obtained in the i-th iteration , where β i is the index of the selected a(θ v ); v = 1, 2, …, J; J represents the total number of observable angles; and the column vector a(θ v ) is the steering vector of the v-th far-field narrowband source. Step six, obtaining a set of indices corresponding to support vectors obtained in the i-th iteration corresponding indices β i corresponding to support vectors obtained in the i-th iteration i-1 corresponding to support vectors obtained in the i-th iteration i ; a set of indices corresponding to the support vectors obtained in the previous i iterations i obtaining a matrix composed of the support vectors obtained in the previous i iterations ; Step seven, calculate the residual r of the i-1th iteration i-1 covariance matrix of the residual r of the i-1th iteration, to obtain the eigenvector matrix U of the residual r at the i-1th iteration i-1 ; Step eight, based on the matrix composed of the support vectors obtained in the previous i iterations and the eigenvector matrix U i-1 , the distance between the subspaces of the i-th iteration is calculated using the signal subspace matching criterion ; Wherein, (U i-1 )1 represents the first column of the eigenvector matrix U i-1 in the middle represents a subspace generated by the column vectors of the matrix ;​ represents a subspace generated by the column vectors of the matrix i-1 )1, and ;​ <·> represents the subspace generated by the column vectors of the matrix in the bracket; represents the distance between the ith iteration subspace and subspace ; SSM i denotes the distance between the subspaces calculated at the i-th iteration; Step nine, let ; where B i represents the matrix composed of the support vectors obtained by the previous i iterations; Solve for z using least squares i,t = argmin z ||X(t) - B i z||2, t = 1,..., L; where z represents a coefficient vector obtained by least square optimization, z = z i,1 ,z i,2 ,…,z i,t ,…,z i,L ; z i,t represents a coefficient vector of B i ; Step ten, compute the residual r i and z i,t for the i-th iteration based on the array-acquired t-th snapshot data X(t), B i,t ; Step eleven, whether i If yes, steps four to eleven are repeatedly executed until i=M-1; If not, output the source estimation result .

2. The method of claim 1, wherein: The M*L matrix X=[X(1), X(2),..., X(t),..., X(L)] composed of L snapshots collected by the array in step one is obtained; Wherein, t=1, 2,..., L; M represents the number of linear arrays; X(1) represents the snapshot data collected by the array at time 1, X(2) represents the snapshot data collected by the array at time 2, X(t) represents the snapshot data collected by the array at time t, and X(L) represents the snapshot data collected by the array at time L; The specific process is as follows: Suppose that K far-field narrow-band sources are incident to a M-element uniform linear array, and the incident direction θ is the included angle between the signal direction and the normal direction of the array; Let X=[X(1), X(2),..., X(t),..., X(L)] be an M*L matrix composed of L snapshots collected by the array, X(1) represents the snapshot data collected by the array at time 1, X(2) represents the snapshot data collected by the array at time 2, X(t) represents the snapshot data collected by the array at time t, and X(L) represents the snapshot data collected by the array at time L; Let S=[S(1), S(2),..., S(t),..., S(L)] be a K*L matrix composed of K sources in L snapshots, S(1) represents a vector composed of the amplitude values of the K sources at time 1, S(2) represents a vector composed of the amplitude values of the K sources at time 2, S(t) represents a vector composed of the amplitude values of the K sources at time t, and S(L) represents a vector composed of the amplitude values of the K sources at time L; Let N=[N(1), N(2),..., N(t),..., N(L)] be an M*L matrix composed of M noises in L snapshots, N(1) represents a vector composed of the amplitude values of the M array elements at time 1, N(2) represents a vector composed of the amplitude values of the M array elements at time 2, N(t) represents a vector composed of the amplitude values of the M array elements at time t, and N(L) represents a vector composed of the amplitude values of the M array elements at time L; The array observation model is represented as X=AS+N (1) A=[a(θ1),a(θ2),...,a(θ k ),...,a(θ K )] (2) Wherein, A is an array manifold matrix, column vector a(θ1) is a steering vector of the first far-field narrowband signal source, column vector a(θ2) is a steering vector of the second far-field narrowband signal source, column vector a(θ k ) is a steering vector of the kth far-field narrowband signal source, and column vector a(θ K ) is a steering vector of the Kth far-field narrowband signal source. λ denotes the signal wavelength, d denotes the array element spacing, denotes an intermediate variable, θ k denotes the angle between the kth far-field narrowband source direction and the array normal direction, j denotes the imaginary unit, j 2 = -1; the upper index T denotes the transpose; Introducing an array manifold matrix for overcomplete dictionaries ; Wherein, J represents the total number of observable angles, which is much larger than the actual source number K; column vector a(θ v ) is the steering vector of the vth far-field narrowband source, and column vector a(θ J ) is the steering vector of the Jth far-field narrowband source. {θ1,θ2,...,θ J} represents the set of observable angles for the linear array; The source corresponding to L snapshots is expressed as a J x L matrix There are only K non-zero elements in the J-dimensional column vector corresponding to each snapshot, which correspond to the K angle positions {θ1, θ2,..., θ v ,...,θ J} of the actual sources in the set {θ1, θ2,..., θ k ,...,θ K} represents a vector of all the incident source amplitudes within the observable angle at time 1, represents a vector of all the incident source amplitudes within the observable angle at time 2, represents a vector of all the incident source amplitudes within the observable angle at time L; The column vector a(θ) corresponding to {θ1, θ2,..., θN} is called the support vector of X. k ,...,θ K} is called the support vector of X. v The column vector a(θ) corresponding to {θ1, θ2,..., θN} is called the support vector of X. The sparse representation of the array signal model is 3. The method of claim 2, wherein: The step five will make inner product of each a(θ ) in the overcomplete dictionary v and the residual of all snapshot data, sum the absolute value of all inner products, then select the a(θ v ) corresponding to the maximum sum value as the support vector of the i-th iteration , where β i is the index of the selected a(θ v ); v = 1, 2, …, J; J represents the total number of observable angles; the column vector a(θ) is the steering vector of the v-th far-field narrowband source; the specific process is as follows: Wherein, β i represents the support vector obtained at the i-th iteration corresponding index; a(θ v ) denotes the steering vector of the vth far-field narrowband source, v = 1, 2,..., J; θ v denotes the angle between the direction of the v-th far-field narrowband source and the array normal direction, v denotes the v-th; The upper subscript H represents the conjugate; r i-1 denotes the residual obtained at the i-1th iteration, r i-1 = [r i-1,1 , r i-1,2 ,..., r i-1,t ,..., r i-1,L ], r i-1,1 denotes the residual corresponding to X(1) in r i-1 ; r i-1,2 denotes the residual corresponding to X(2) in r i-1 ; r i-1,t denotes the residual corresponding to X(t) in r i-1 ; r i-1,L denotes the residual corresponding to X(L) in r i-1 .

4. The method of claim 3, wherein: the support vector obtained based on the i-th iteration in step six the corresponding index β i the set Λ of the corresponding indices of the support vectors obtained in the first i-1 iterations i-1 obtaining the set Λ of the corresponding indices of the support vectors obtained in the first i iterations i ; a set of indices corresponding to the support vectors obtained at the previous i iterations i obtaining a matrix composed of the support vectors obtained at the previous i iterations The specific process is as follows: 1) the support vector obtained based on the i-th iteration the corresponding index β i the set Λ of the corresponding indices of the support vectors obtained based on the first i-1 iterations i-1 obtaining the set Λ of the corresponding indices of the support vectors obtained based on the first i iterations i is denoted as: Λ i = Λ i-1 ∪ β i Λ i-1 = [β1, β2,..., β i-1 ], Λ i = [β1, β2,..., β i ]; Wherein, β1represents an index corresponding to a support vector obtained in the first iteration, β2represents an index corresponding to a support vector obtained in the second iteration, β i-1 β1represents an index corresponding to a support vector obtained in the first iteration, β2represents an index corresponding to a support vector obtained in the second iteration, β i β1represents an index corresponding to a support vector obtained in the first iteration, β2represents an index corresponding to a support vector obtained in the second iteration, β 2) the index set Λ corresponding to the support vector obtained in the i-th iteration i obtain the matrix composed of the support vectors obtained in the i-th iteration ; 5. The method of claim 4, wherein: The step eight matrix composed of the support vectors obtained based on the previous i iterations and the eigenvector matrix U i-1 The distance between the subspaces of the i-th iteration is calculated using the signal subspace matching criterion Wherein, (U i-1 )1 represents the first column of the eigenvector matrix U i-1 in the middle represents a subspace generated by the column vectors of the matrix ​ represents a subspace generated by the column vectors of the matrix i-1 )1, and ​ <·> represents the subspace generated by the column vectors of the matrix in the bracket; represents the distance between the ith iteration subspace and subspace ; SSM i represents the distance between the subspaces calculated at the i-th iteration; The specific process is as follows: 1) Computing the projection matrix of a subspace ​ 2) Computing the projection matrix of a subspace ​ 3) Compute projection matrix and projection matrix Difference, norm of difference gives distance between subspaces and ​ 6. The method of claim 5, wherein: the projection matrix of the 1) calculated subspace is represented as:​ 7. The method of claim 6, wherein: the projection matrix of the 2) calculated subspace the projection matrix of the 2) calculated subspace is represented as:

8. The method of claim 7, wherein: The projection matrix in 3) is calculated and the projection matrix The difference between the two, and the norm of the difference gives the subspace and The distance between is represented as: where <·> denotes the subspace generated by the column vectors of the matrix within the brackets; denotes the square of the Frobenius norm.

9. The method of claim 8, wherein: The step ten calculates the residual r i and z i,t of the i-th iteration based on the t-th snapshot data X(t), B i,t arrayed r i,t = X(t) - B i z i,t t = 1,..., L.

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