Co-prime array robust DOA estimation method based on few auxiliary array elements

Abstract

The invention discloses a co-prime array robust DOA estimation method based on a small number of auxiliary array elements. According to the scheme, three auxiliary array elements are added on an original co-prime array according to array element spacing, received data of a new co-prime array is decomposed into two sub-array data, and amplitude-phase error estimation is performed on the two sub-arrays by using amplitude-phase characteristics of each element in a covariance matrix; and the amplitude-phase errors of the sub-arrays are sorted and recombined to obtain the amplitude-phase error of the new co-prime array. The method comprises the following steps: firstly, receiving data, then compensating a covariance matrix error of the received data and performing Toeplitz reconstruction, relaxing a non-convex low-rank matrix reconstruction problem through atomic norm optimization, solving by utilizing a convex optimization CVX tool box to obtain a rank recovery matrix, and finally performing angle estimation by utilizing an orientation estimation method. Compared with a traditional method, the method has the advantages that errors of the array are effectively corrected, the number of auxiliary array elements is reduced, resources are effectively saved, and the method has important application value in subsurface buoys or mobile observation platforms for ocean observation.

Description

Technical Field

[0001] The present invention relates to a target detection and processing method for a buoy and a mobile observation platform for ocean observation, and is a robust DOA estimation method for a coprime array based on a small number of auxiliary array elements. Background Art

[0002] In the past decade, coprime arrays have attracted widespread attention in the field of DOA estimation because they can be expanded into virtual uniform linear arrays with higher degrees of freedom and suffer less coupling effects between array elements. In the study of coprime arrays, many methods for DOA estimation have been proposed. However, most methods are only for signals under ideal conditions. Since the actual underwater acoustic environment is very complex, there are many non-ideal factors that cause different amplitude and phase errors in the array. When the array manifold cannot be accurately known, these DOA estimation methods cannot achieve accurate positioning. Therefore, array error estimation and correction are an important part of array signal processing. Without array error correction, any high-resolution spatial spectrum estimation method cannot be used in practice. In order to improve the target detection accuracy of coprime arrays and enable them to be effectively used in real life, the research on amplitude and phase error correction and target detection of coprime arrays is still a hot topic at present, and is closely related to the development and stability of society.

[0003] The existing error correction algorithms are divided into two categories: one is the correction algorithm based on auxiliary information sources, and the other is the correction algorithm based on auxiliary array elements. The classical auxiliary array element method (ISM) requires the setting of multiple accurately calibrated auxiliary array elements, and the number of information sources to be estimated can only be less than the number of auxiliary array elements. The covariance matrix-based auxiliary array element method (R-ISM) does not require the number of auxiliary array elements to be greater than the number of estimated information sources, and is also applicable in the case of multiple information sources, but it is only applicable to uniform linear arrays.

[0004] This patent proposes a robust DOA estimation method for coprime arrays based on a small number of auxiliary array elements. It combines the covariance matrix auxiliary array element method with the coprime array subarray decomposition method to achieve DOA estimation of coprime arrays under amplitude and phase errors. This patent utilizes the array element position characteristics of coprime arrays and only needs to add 3 auxiliary array elements to the coprime array to achieve error correction, reducing the number of auxiliary array elements and effectively saving resources. There is no requirement that the number of auxiliary array elements is greater than the number of estimated signal sources. It is also applicable to the case of multiple signal sources and has important application value in practical engineering. Summary of the invention

[0005] The actual ocean acoustic environment is very complex, and there are many non-ideal factors that cause different amplitude and phase errors in the array, resulting in the inability to accurately know the manifold and the inability to accurately locate the incident target. In order to improve the target detection accuracy of the coprime array and solve the problem of amplitude and phase errors, the present invention proposes a robust DOA estimation method for coprime arrays based on a small number of auxiliary array elements. The method adds a small number of accurately calibrated auxiliary array elements to the coprime array, decomposes the covariance matrix of the coprime array into sub-arrays, and uses the amplitude and phase characteristics of each element in the array covariance matrix to estimate the array error, then compensates the error of the received data covariance matrix, and finally expands the compensated covariance matrix to construct a Toeplitz matrix, optimizes the value of the hole in the Toeplitz matrix using the atomic norm minimization method, and finally combines the MUSIC method to estimate DOA.

[0006] The present invention discloses a coprime array robust DOA estimation method based on a small number of auxiliary array elements, characterized in that: the coprime sensor array is composed as follows: the number of sensors of uniform subarray 1 is N, the sensor spacing is Md, the number of sensors of uniform subarray 2 is M, the sensor spacing is Nd, d=2 / λ, λ is the wavelength, the coprime array is composed of uniform subarray 1 and uniform subarray 2 nested, three corrected auxiliary array elements are added to the two ends of the coprime array, auxiliary array element 1 is placed at the front end of uniform subarray 1, and the distance from the first array element is 1 / 2. The distance between auxiliary array element 2 and the last array element is Md, and auxiliary array element 2 is placed at the rear end of subarray 1, with a distance of Md from the last array element. Auxiliary array element 3 is placed at the front end of subarray 2, with a distance of Nd from the first array element. Since M×[(N-1)+1]=N×[(M-1)+1], the position of auxiliary array element 3 is used in both new uniform subarray 1 and new uniform subarray 2. Then the number of sensors in new uniform subarray 1 becomes N1=N+2, and the number of sensors in new uniform subarray 2 becomes M1=M+2. The sensor spacing remains unchanged. The new coprime array position set is expressed as The number of sensors becomes Z1=M1+N1-2. A robust DOA estimation method for a coprime array based on a small number of auxiliary array elements includes the following steps:

[0007] Step 1: The new coprime array receives data which can be expressed as Perform Hilbert transform on X(t) to obtain the complex signal form of the array receiving signal:

[0008] Step 2: According to the formation and The corresponding relationship is to select the receiving data of auxiliary array element 1, uniform sub-array 1 and auxiliary array element 2 to construct the receiving data of new uniform sub-array 1 Select the receiving data of auxiliary array element 3, uniform sub-array 2 and auxiliary array element 2 to construct the receiving data of new uniform sub-array 2

[0009] Step 3: Use the complex signal of the new coprime array Find the covariance matrix “(·) H ” is the conjugate transpose, where T is the number of snapshots in practical applications;

[0010] Step 4: Use the complex signal of the new uniform subarray 1 Find the covariance matrix Using the complex signal of the new uniform subarray 2 Find the covariance matrix

[0011] Step 5: According to the covariance matrix Estimate the phase error of each element of the new uniform subarray 1, which can be expressed as Estimate the amplitude error of each element of the new uniform subarray 1, which can be expressed as According to the covariance matrix Estimate the phase error of each element of the new uniform subarray 2, which can be expressed as Estimate the amplitude error of each element of the new uniform subarray 2, which can be expressed as

[0012] Step 6: Calculate the correction vector of the new uniform subarray 1 Calculate the correction vector of the new uniform subarray 2

[0013] Step 7: According to the position of each array element in the new uniform sub-array 1 and the new uniform sub-array 2, and The corresponding elements in are sorted and reorganized according to the positions of the new coprime array elements, and the duplicate elements are removed to obtain the amplitude and phase error matrix of the new coprime sensor array.

[0014] Step 8. For the covariance matrix in step 3 Perform error compensation to obtain the data covariance matrix of the new coprime sensor array after amplitude and phase error correction

[0015] Step 9: The difference joint array position set of the new coprime sensor array is

[0016] S={±(nMd-mNd)}∪{(n i Md-n j Md)}∪{(m i Nd-m j Nd)},

[0017] Where -1≤n,n i ,n j≤N;-1≤m,m i ,m j ≤M, record d x is an element in the set S, d x The number of occurrences in the set S is defined as the weight coefficient ω(d x ), remove the repeated position differences to obtain the virtual array element position set According to the array element collection Determine the dimension of the virtual array element to be 2N(M+1)+1, expand the corrected data covariance matrix R to form a Toeplitz matrix In the formula

[0018] Step 10: Generate matrix G, which is The zero matrices of the same dimension are the corresponding The non-zero positions of the elements in are set to 1;

[0019] Step 11: Toeplitz matrix in step 9 To optimize the hole filling, the matrix filling problem can be implemented using the following atomic norm optimization model: by solving the corresponding semi-positive definite programming problem To optimize the filling of holes, use After optimization where “tr(·)” represents the trace of the matrix, min represents minimization, μ is the regularization parameter, and The dimensions are the same, represents the Hadamard product, “|| || F " represents the Frobenius norm of the matrix, "≥" represents semi-positive definite, and the semi-positive definite programming problem can be solved by the CVX convex optimization toolbox;

[0020] Step 12: Optimize the matrix Perform feature decomposition and obtain The eigenvalue η i and its corresponding eigenvector, Arrange them from largest to smallest: The K larger eigenvalues ​​correspond to the signal subspace U S , The smaller eigenvalue corresponds to the noise subspace U N , K is the number of incident signals, according to the signal subspace U S Or the noise subspace U N to estimate the direction of the incident signal.

[0021] In step 5 above The estimation method is as follows:

[0022] A1. Extracting the covariance matrix Phase information angle(·) is the phase information of the extraction matrix;

[0023] A2. The upper triangular part of the phase information matrix contains exactly the same information as the lower triangular part of the matrix. Extract the phase information matrix The upper triangle of has:

[0024]

[0025] A3. Sum the first sub-diagonal elements above the main diagonal to get Taking the mean value, we get

[0026] A4. Subtract φ1 from all elements of the first subdiagonal above the main diagonal of matrix T1, and we get α 1n , where n = 1, 2, ..., N1-1, according to this formula Solve for the vector make get

[0027] A5. Transform the covariance matrix in step A1 into Repeat the above steps A1 to A4 to get

[0028] The preferred solution is: The estimation method is as follows:

[0029] B1. Get the covariance matrix The main diagonal elements of Using this vector, we can calculate

[0030] B2. Get the covariance matrix The main diagonal elements of Using this vector, we can calculate

[0031] The preferred processing scheme is: the Toeplitz matrix described in step nine The construction method is as follows: let i = 1, 2, ..., N1 + M1-2, j = 1, 2, ..., N1 + M1-2, when When k=1,2,...,2N(M+1)+1, we can find ω in R k (d x ) elements, of which the lth can be represented as R l (diff k), the data corresponding to the kth virtual array element is The constructed Toeplitz matrix structure is There is a one-to-one correspondence between the zero elements in and the holes in the difference union matrix;

[0032] The preferred processing scheme is: the DOA estimation method in step 12 is: according to the formula The spatial spectrum is calculated and the peak is searched for. The angle corresponding to the extreme value is the estimated value of the incident signal angle.

[0033] The preferred processing scheme is: the DOA estimation method in step 12 is: find the first K large eigenvalues, according to the formula An estimate of the angle of the incident signal can be obtained.

[0034] Compared with the prior art, the technical solution of the present invention has the following technical effects:

[0035] (1) The present invention utilizes a coprime array subarray decomposition method, performs a covariance matrix-assisted array element method on the subarrays of the coprime array, estimates the array error using the amplitude and phase characteristics of each element in the array covariance matrix, and then performs error compensation on the received data covariance matrix, effectively restoring the coprime array DOA estimation performance under amplitude and phase errors, and realizing error correction of the coprime array containing amplitude and phase errors;

[0036] (2) The present invention utilizes the position characteristics of the coprime array and places two auxiliary array elements at the left and right ends of the two sub-arrays respectively. Since the positions of the two auxiliary array elements at the rightmost ends of the two sub-arrays overlap, only three auxiliary array elements need to be placed in the coprime array to achieve amplitude and phase error correction of the coprime array, thereby reducing the number of auxiliary array elements required and effectively saving resource utilization. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 This is a model diagram of the array signal processing method of this patent;

[0038] Figure 2 This is a diagram showing the DOA estimation results of the signal processing method of this patent and the signal processing method when there are amplitude and phase errors;

[0039] Figure 3 This is the relationship curve between the probability of successful resolution and the signal-to-noise ratio of the signal processing method of this patent;

[0040] Figure 4 is the relationship curve between the root mean square error and the signal-to-noise ratio of the signal processing method of this patent;

[0041] Figure 5 The relationship curve between the probability of successful resolution and the number of snapshots of the patented signal processing method is shown in FIG.

[0042] Figure 6 This is the relationship curve between the root mean square error and the number of snapshots of the patented signal processing method. DETAILED DESCRIPTION

[0043] The present invention will now be further described with reference to the embodiments and the accompanying drawings:

[0044] The first embodiment: Figure 1 The coprime array model in the present invention is given. First, we construct a 7-element coprime array. Uniform subarray 1 consists of 5 elements with an element spacing of 3d. Uniform subarray 2 consists of 3 elements with an element spacing of 5d, d=2 / λ, λ is the wavelength. The coprime array is composed of uniform subarray 1 and uniform subarray 2 nested together. Three corrected auxiliary elements are added to the two ends of the coprime array. Auxiliary element 1 is placed at the front end of uniform subarray 1 with a spacing of 3d from the first element. Auxiliary element 2 is placed at the back end of subarray 1 with a spacing of 3d from the last element. Auxiliary element 3 is placed at the front end of subarray 2 with a spacing of 5d from the first element. Since 3×[(5-1)+1]=5×[(3-1)+1], the position of auxiliary element 3 is shared by the new uniform subarray 1 and the new uniform subarray 2. Then the number of sensors in the new uniform subarray 1 becomes N1=7, and the number of sensors in the new uniform subarray 2 is M1=5. The sensor spacing remains unchanged. The new coprime array position set is expressed as The number of sensors becomes Z1 = 10, a signal is incident on the coprime array from the direction of θ1 = 20°, the signal-to-noise ratio is 0dB, and the number of snapshots is T = 500. Using the above conditions for azimuth estimation, the specific implementation process is:

[0045] Step 1: The new coprime array receives data which can be expressed as X(t)=[x1(t),x2(t),…,x 10 (t)] T , perform Hilbert transform on X(t) and get the complex signal form of the array receiving signal:

[0046] Step 2: According to the formation and The corresponding relationship is to select the receiving data of auxiliary array element 1, uniform sub-array 1 and auxiliary array element 2 to construct the receiving data of new uniform sub-array 1 Select the receiving data of auxiliary array element 3, uniform sub-array 2 and auxiliary array element 2 to construct the receiving data of new uniform sub-array 2

[0047] Step 3: Use the complex signal of the new coprime array Find the covariance matrix “(·) H ” is the conjugate transpose;

[0048] Step 4: Use the complex signal of the new uniform subarray 1 Find the covariance matrix Using the complex signal of the new uniform subarray 2 Find the covariance matrix

[0049] Step 5: According to the covariance matrix Estimate the phase error of each element of the new uniform subarray 1, which can be expressed as Estimate the amplitude error of each element of the new uniform subarray 1, which can be expressed as According to the covariance matrix Estimate the phase error of each element of the new uniform subarray 2, which can be expressed as Estimate the amplitude error of each element of the new uniform subarray 2, which can be expressed as

[0050] Step 6: Calculate the correction vector of the new uniform subarray 1 Calculate the correction vector of the new uniform subarray 2

[0051] Step 7: According to the position of each array element in the new uniform sub-array 1 and the new uniform sub-array 2, and The corresponding elements in are sorted and reorganized according to the positions of the new coprime array elements, and the duplicate elements are removed to obtain the amplitude and phase error matrix of the new coprime sensor array.

[0052] Step 8. For the covariance matrix in step 3 Perform error compensation to obtain the data covariance matrix of the new coprime sensor array after amplitude and phase error correction

[0053] Step 9: The difference joint array position set of the new coprime sensor array is

[0054] S={±(3nd-5md)}∪{(3n i d-3n j d)}∪{(5m i d-5m j d)},

[0055] Where -1≤n,n i ,n j ≤5;-1≤m,m i ,m j ≤3, record d x is an element in the set S, d x The number of occurrences in the set S is defined as the weight coefficient ω(d x ), remove the repeated position differences to get the virtual array element position set: According to the array element collection Determine the dimension of the virtual array element to be 41, expand the corrected data covariance matrix R to form a Toeplitz matrix In the formula

[0056] Step 10: Generate matrix G, which is The zero matrices of the same dimension are the corresponding The non-zero positions of the elements in are set to 1;

[0057] Step 11: Toeplitz matrix in step 9 To optimize the hole filling, the matrix filling problem can be implemented using the following atomic norm optimization model: by solving the corresponding semi-positive definite programming problem To optimize the filling of holes, use After optimization where “tr(·)” represents the trace of the matrix, min represents minimization, μ is the regularization parameter, and The dimensions are the same, represents the Hadamard product, “|| || F " represents the Frobenius norm of the matrix, "≥" represents semi-positive definite, and the semi-positive definite programming problem can be solved by the CVX convex optimization toolbox;

[0058] Step 12: Optimize the matrix Perform eigendecomposition and obtain 21 eigenvalues ​​η i and its corresponding eigenvectors, i = 1, 2, ... 21, and arrange them from large to small: η1>η2>...>η 31 > 0, where a larger eigenvalue corresponds to the signal subspace U S , the 20 smaller eigenvalues ​​correspond to the noise subspace U N , according to the signal subspace U S Or the noise subspace U N to estimate the direction of the incident signal.

[0059] The preferred solution is: The estimation method is as follows:

[0060] A1. Extracting the covariance matrix Phase information angle(·) is the phase information of the extraction matrix;

[0061] A2. The upper triangular part of the phase information matrix contains exactly the same information as the lower triangular part of the matrix. Extract the phase information matrix The upper triangle of has:

[0062]

[0063] A3. Sum the first sub-diagonal elements above the main diagonal to get Taking the mean value, we get

[0064] A4. Subtract φ1 from all elements of the first subdiagonal above the main diagonal of matrix T1, and we get α 1n , where n = 1, 2, ..., 6, according to this formula Solve for the vector make get

[0065] A5. Transform the covariance matrix in step A1 into Repeat the above steps A1 to A4 to get

[0066] The preferred solution is: The estimation method is as follows:

[0067] B1. Get the covariance matrix The main diagonal elements of Using this vector, we can calculate

[0068] B2. Get the covariance matrix The main diagonal elements of Using this vector, we can calculate

[0069] The preferred processing scheme is: the Toeplitz matrix described in step nine The construction method is as follows: Let i = 1, 2, ..., 10, j = 1, 2, ..., 10, when When k = 1, 2, ..., 21, we can find ω in R k (d x ) elements, of which the lth can be represented as R l (diff k ), the data corresponding to the kth virtual array element is The constructed Toeplitz matrix structure is There is a one-to-one correspondence between the zero elements in and the holes in the difference union matrix.

[0070] The preferred processing scheme is: the DOA estimation method in step 12 is: according to the formula The spatial spectrum is calculated and the peak is searched for. The angle corresponding to the extreme value is the estimated value of the incident signal angle.

[0071] The preferred processing scheme is: the DOA estimation method in step 12 is: find the first K large eigenvalues, according to the formula An estimate of the angle of the incident signal can be obtained.

[0072] analyze Figure 2 It can be seen that the angle estimated by the patented method is basically consistent with the true angle, and the angle cannot be correctly estimated without amplitude and phase error correction, so the patented method can effectively correct the amplitude and phase errors.

[0073] Second embodiment: The relationship between the root mean square error (RMSE) and the signal-to-noise ratio and the relationship between the probability of successful resolution and the signal-to-noise ratio of the signal processing method of the present patent are studied respectively, and the effect diagram is as follows: Figure 3 , Figure 4 The application conditions of the algorithm in the present invention are as follows:

[0074] We used a 7-element coprime array. Uniform subarray 1 consists of 5 elements with an element spacing of 3d, uniform subarray 2 consists of 3 elements with an element spacing of 5d, and the two uniform subarrays share the first element, where d is half of the signal wavelength 0.5m. After adding auxiliary elements, uniform subarray 1 consists of 7 elements with an element spacing of 3d, uniform subarray 2 consists of 5 elements with an element spacing of 5d, and the two subarrays share the third and last elements. The number of elements in the new coprime array is 10. A signal θ1 = 20° is incident on the new coprime array, the number of snapshots T = 500, and the signal-to-noise ratio is changed from -5dB to 20dB in steps of 5dB. 200 independent Monte Carlo experiments are performed and simulated by the MATLAB software system.

[0075] analyze Figure 3 It can be seen that when the snapshot number T=200, as the signal-to-noise ratio increases, the success probability of resolution of the method of the present invention and the method of placing 4 auxiliary array elements at the front end of the coprime array increases. When the signal-to-noise ratio is 5dB, the success probability of resolution of the two methods reaches 100%. This shows that the method of the present invention can effectively distinguish the true angle.

[0076] analyze Figure 4It can be seen that when the snapshot number T = 500, as the signal-to-noise ratio increases, the root mean square error of the two methods is decreasing. When the signal-to-noise ratio is -5dB, the root mean square error of the patented method is about 0.834, and the root mean square error of the method of placing 4 auxiliary array elements at the front end of the coprime array is about 0.913. When the signal-to-noise ratio is 15dB, the root mean square error of the patented method is about 0.064, and the root mean square error of the method of placing 4 auxiliary array elements at the front end of the coprime array is about 0.051. It can be seen that the performance of the patented method is basically the same as that of the method of placing 4 auxiliary array elements at the front end of the coprime array, but this method only uses 3 auxiliary array elements, reducing the number of auxiliary array elements.

[0077] The third embodiment: The relationship between the root mean square error (RMSE) and the number of snapshots and the relationship between the probability of successful resolution and the number of snapshots of the signal processing method of the present patent are studied respectively, and the effect diagram is as follows: Figure 5 , Figure 6 The application conditions of the algorithm in the present invention are as follows:

[0078] We used a 7-element coprime array. Uniform subarray 1 consists of 5 elements with an element spacing of 3d, uniform subarray 2 consists of 3 elements with an element spacing of 5d, and the two uniform subarrays share the first element, where d is half of the signal wavelength 0.5m. After adding auxiliary elements, uniform subarray 1 consists of 7 elements with an element spacing of 3d, uniform subarray 2 consists of 5 elements with an element spacing of 5d, and the two subarrays share the third and last elements. The number of elements in the new coprime array is 10. A signal θ1 = 20° is incident on the new coprime array, with a signal-to-noise ratio snr = 10dB. The number of snapshots is changed from 50 to 400 in steps of 50. 200 independent Monte Carlo experiments are performed and simulated using the MATLAB software system.

[0079] analyze Figure 5 It can be seen that when the signal-to-noise ratio snr=10, as the number of snapshots increases, the success probability of resolution of the method of the present patent and the method of placing 4 auxiliary array elements at the front end of the coprime array increases. When the number of snapshots is 150, the success probability of resolution of the two methods reaches 100%, which shows that the method of the present patent can effectively distinguish the true angle.

[0080] analyze Figure 6It can be seen that when the signal-to-noise ratio snr=10, as the number of snapshots increases, the root mean square error of the two methods decreases. When the number of snapshots is 50, the root mean square error of the patented method is about 0.298, and the root mean square error of the method of placing 4 auxiliary array elements at the front end of the mutually prime array is about 0.312. When the number of snapshots is 250, the root mean square error of the MUSIC algorithm under the patented method is about 0.137, and the root mean square error of the method of placing 4 auxiliary array elements at the front end of the mutually prime array is about 0.133. It can be seen that the performance of the patented method is basically the same as that of the method of placing 4 auxiliary array elements at the front end of the mutually prime array, but this method only uses 3 auxiliary array elements, which reduces the number of auxiliary array elements and effectively saves resources.

[0081] The specific examples described herein are merely illustrative of the present invention. A person skilled in the art may make various modifications, supplements or substitute similar methods to the specific examples described, but this will not deviate from the present invention or exceed the scope defined by the appended claims.

Claims

1. A robust DOA estimation method for coprime arrays based on a small number of auxiliary array elements, characterized by: The coprime sensor array is constructed as follows: the number of sensors in uniform subarray 1 is N, and the sensor spacing is Md; the number of sensors in uniform subarray 2 is M, and the sensor spacing is Nd, d = 2 / λ, λ is the wavelength. The coprime array is composed of uniform subarray 1 and uniform subarray 2 nested together. Three corrected auxiliary array elements are added to the two ends of the coprime array. Auxiliary array element 1 is placed at the front end of uniform subarray 1, with a spacing of Md from the first array element. Auxiliary array element 2 is placed at the rear end of subarray 1, with a spacing of Md from the last array element. Auxiliary array element 3 is placed at the front end of subarray 2, with a spacing of Nd from the first array element. Since M×[(N-1)+1]=N×[(M-1)+1], the position of auxiliary array element 3 is shared by the new uniform subarray 1 and the new uniform subarray 2. Then the number of sensors in the new uniform subarray 1 becomes N1=N+2, and the number of sensors in the new uniform subarray 2 is M1=M+2. The sensor spacing remains unchanged. The new coprime array position set is expressed as The number of sensors becomes Z1=M1+N1-2. A robust DOA estimation method for a coprime array based on a small number of auxiliary array elements includes the following steps: Step 1: The new coprime array receives data which can be expressed as Perform Hilbert transform on X(t) to obtain the complex signal form of the array receiving signal: Step 2: According to the formation and The corresponding relationship is to select the receiving data of auxiliary array element 1, uniform sub-array 1 and auxiliary array element 2 to construct the receiving data of new uniform sub-array 1 Select the receiving data of auxiliary array element 3, uniform sub-array 2 and auxiliary array element 2 to construct the receiving data of new uniform sub-array 2 Step 3: Use the complex signal of the new coprime array Find the covariance matrix "(·) H ” is the conjugate transpose, where T is the number of snapshots in practical applications; Step 4: Use the complex signal of the new uniform subarray 1 Find the covariance matrix Using the complex signal of the new uniform subarray 2 Find the covariance matrix Step 5: According to the covariance matrix Estimate the phase error of each element of the new uniform subarray 1, which can be expressed as Estimate the amplitude error of each element of the new uniform subarray 1, which can be expressed as According to the covariance matrix Estimate the phase error of each element of the new uniform subarray 2, which can be expressed as Estimate the amplitude error of each element of the new uniform subarray 2, which can be expressed as Step 6: Calculate the correction matrix of the new uniform subarray 1 Calculate the correction matrix of the new uniform subarray 2 Step 7: According to the position of each array element in the new uniform sub-array 1 and the new uniform sub-array 2, and The corresponding elements in are sorted and reorganized according to the positions of the new coprime array elements, and the duplicate elements are removed to obtain the amplitude and phase error matrix of the new coprime sensor array. Step 8. For the covariance matrix in step 3 Perform error compensation to obtain the data covariance matrix of the new coprime sensor array after amplitude and phase error correction Step 9: The difference joint array position set of the new coprime sensor array is S={±(nMd-mNd)}∪{(n i Md-n j Md)}∪{(m i Nd-m j Nd)}, Where -1≤n,n i ,n j ≤N;-1≤m,m i ,m j ≤M; denoted by d x is an element in the set S, d x The number of occurrences in the set S is defined as the weight coefficient ω(d x ), remove the repeated position differences to get the virtual array element position set: According to the array element collection Determine the dimension of the virtual array element to be 2N(M+1)+1, expand the corrected data covariance matrix R to form a Toeplitz matrix In the formula Step 10: Generate matrix G, which is The zero matrices of the same dimension are the corresponding The non-zero positions of the elements in are set to 1; Step 11: Toeplitz matrix in step 9 To optimize the hole filling, the matrix filling problem can be implemented using the following atomic norm optimization model: by solving the corresponding semi-positive definite programming problem To optimize the filling of holes, use After optimization where "tr(·)" represents the trace of the matrix, min represents minimization, and μ is the regularization parameter. and The dimensions are the same, represents the Hadamard product, "|| || F " represents the Frobenius norm of the matrix, "≥" represents semi-positive definite, and the semi-positive definite programming problem can be solved by the CVX convex optimization toolbox; Step 12: Optimize the matrix Perform feature decomposition and obtain The eigenvalue η i and its corresponding eigenvector, Arrange them from largest to smallest: The K larger eigenvalues ​​correspond to the signal subspace U S , The smaller eigenvalue corresponds to the noise subspace U N , K is the number of incident signals, according to the signal subspace U S Or the noise subspace U N to estimate the direction of the incident signal.

2. The robust DOA estimation method for a coprime array based on a small number of auxiliary array elements as claimed in claim 1, characterized in that: Step 5 The estimation method is as follows: A1. Extracting the covariance matrix Phase information angle(·) is the phase information of the extraction matrix; A2. The upper triangular part of the phase information matrix contains exactly the same information as the lower triangular part of the matrix. Extract the phase information matrix The upper triangle of has: A3. Sum the first sub-diagonal elements above the main diagonal to get Taking the mean value, we get A4. Subtract φ1 from all elements of the first subdiagonal above the main diagonal of matrix T1, and we get Where n1=1,2,...,N1-1, according to this formula Solve for the vector make get A5. Transform the covariance matrix in step A1 into Repeat the above steps A1 to A4 to get 3. The robust DOA estimation method for a coprime array based on a small number of auxiliary array elements as claimed in claim 1, characterized in that: Step 5 The estimation method is as follows: B1. Get the covariance matrix The main diagonal elements of Using this vector, we can calculate B2. Get the covariance matrix The main diagonal elements of Using this vector, we can calculate 4. The robust DOA estimation method for a coprime array based on a small number of auxiliary array elements as claimed in claim 1, characterized in that: The Toeplitz matrix described in step 9 The construction method is as follows: let i = 1, 2, ..., N1 + M1-2, j = 1, 2, ..., N1 + M1-2, when When k = 1, 2, ..., 2N(M+1)+1, we can find ω in R k (d x ) elements, of which the lth can be represented by R l (diff k ), the data corresponding to the kth virtual array element is The constructed Toeplitz matrix structure is There is a one-to-one correspondence between the zero elements in and the holes in the difference union matrix.

5. The robust DOA estimation method for a coprime array based on a small number of auxiliary array elements as claimed in claim 1, characterized in that: The DOA estimation method in step 12 is: The spatial spectrum is calculated and the peak is searched for. The angle corresponding to the extreme value is the estimated value of the incident signal angle.

6. The robust DOA estimation method for a coprime array based on a small number of auxiliary array elements as claimed in claim 1, characterized in that: The DOA estimation method in step 12 is: find the first K large eigenvalues, according to the formula An estimate of the angle of the incident signal can be obtained.

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