A method for estimating the direction of arrival (DOA) of coherent sources in a symmetric array under mutual coupling conditions

Abstract

The application discloses a kind of DOA estimation methods of coherent sources under symmetric array mutual coupling condition, comprising the following steps;With central symmetry MCM is split into several partial matrix difference form;Obtain the reconstruction matrix corresponding to each partial matrix uniquely;Through array flow matrix, the projection matrix of signal subspace is calculated, then the orthogonality of noise subspace and signal subspace is used to construct spectrum function to search spectrum peak and realize preliminary estimation to DOA;The preliminary angle value estimated is used to carry out iterative search to estimate MCM, then the array flow of next iteration is calculated by the MCM estimated, and the projection matrix of signal subspace is constructed to carry out DOA estimation again;Iterative search is carried out multiple times until the termination iteration condition is satisfied.The application not only realizes the purpose of improving the estimation accuracy of DOA and MCM under the condition of low signal-to-noise ratio and snapshot number, but also can achieve good estimation effect when estimating DOA of similar coherent sources.

Description

Technical Field

[0001] The present invention belongs to the technical field of array signal processing, and in particular relates to a DOA estimation method for a coherent signal source under a symmetrical array mutual coupling condition. Background Art

[0002] The most basic problem in the study of Direction of Arrival (DOA) estimation in the field of array signal processing is to use certain methods to distinguish the arrival angles of multiple incident signal sources simultaneously in a certain area of ​​space, so as to obtain the direction and position of the corresponding target. It plays an immeasurable role in positioning, tracking, and navigation.

[0003] When using an antenna array to estimate the angle of incoming signals from space, the positions of the array elements can often be designed to be unequally spaced to increase the array's degrees of freedom. This can effectively improve the performance of the DOA estimation algorithm to a certain extent. Antenna arrays with a centrally symmetric structure are a representative type of non-uniform array. Like uniform arrays, they are also subject to various array errors in practical applications. In particular, mutual coupling error is generated when antenna elements interact with each other. This mutual coupling error effect can be considered as additional noise added to the system, affecting various aspects of the array's performance.

[0004] Existing classic DOA estimation algorithms all rely on highly accurate array received data to accurately represent the array flow pattern. However, in practice, the coupling effects between array elements can have a certain destructive effect on the array flow pattern, causing the array flow pattern in actual applications to deviate significantly from the ideal error-free array flow pattern. This can severely degrade the performance of these DOA estimation algorithms, especially when the signals received by the array are coherent sources. Many DOA estimation algorithms will fail and be unable to achieve the purpose of angle estimation. Furthermore, the actual communication environment is relatively harsh. Non-uniform arrays with mutual coupling effects need to consider the case of coherent signal sources, especially the problem of angle estimation algorithm resolution when the angles of arrival of two coherent signal sources are very close. Angle estimation in low signal-to-noise ratio conditions is also a problem that needs to be solved. Furthermore, to reduce the complexity of receiver data processing, the number of snapshots of sampled data should be as small as possible.

[0005] For the DOA estimation problem of coherent sources under the condition of non-uniform array mutual coupling error, the shortcomings of existing technologies are: the accuracy of DOA estimation depends on the accuracy of the prior condition mutual coupling coefficient matrix (MCM), which often has certain errors; in addition, when the arrival angles of spatially coherent sources are very close or the signal-to-noise ratio is low, the error of MCM will cause more serious performance loss of DOA estimation, and may even cause false peaks, leading to erroneous arrival estimation results. Summary of the Invention

[0006] In order to overcome the shortcomings of the above-mentioned prior art, the purpose of the present invention is to provide a DOA estimation method for coherent signal sources under the condition of symmetric array mutual coupling, which not only achieves the purpose of improving the DOA and MCM estimation accuracy under the conditions of low signal-to-noise ratio and snapshot number, but also can achieve good estimation effect when performing DOA estimation on similar coherent signal sources.

[0007] In order to achieve the above object, the technical solution adopted by the present invention is:

[0008] A DOA estimation method for a coherent signal source under a symmetrical array mutual coupling condition comprises the following steps:

[0009] Step 1: When the number of signal sources is known, the mutual coupling coefficient matrix (MCM) of the centrally symmetric array is split into several matrices with Toeplitz properties.

[0010] Step 2: According to the properties of the Toeplitz matrix, obtain the reconstruction matrix that uniquely corresponds to each partial matrix;

[0011] Step 3: Perform singular value decomposition and dimensionality reduction on the received data matrix to obtain the noise subspace. Calculate the projection matrix of the signal subspace through the array flow matrix. Then, use the orthogonality of the noise subspace and the signal subspace to construct a spectral function to search for the spectral peak and achieve a preliminary estimate of the DOA.

[0012] Step 4: Use the preliminary angle value obtained from the preliminary estimation of the DOA to perform an iterative search to achieve MCM estimation, then calculate the array flow pattern for the next iteration using the obtained MCM, and construct the projection matrix of its signal subspace to perform DOA estimation again;

[0013] Step 5: Repeat steps 3 and 4 and perform iterative search multiple times until the termination condition is met. Such multiple iterations weaken the influence of mutual coupling and coherent sources, thereby further improving the DOA estimation performance.

[0014] In the step 1, a matrix operation is performed on the MCM of the non-uniform array with a central symmetric structure, and the MCM is divided into several parts in the form of a Toeplitz property matrix;

[0015] When the mutual coupling degree of freedom of the array is p=2, C p＝2 =D p＝2 -H p＝2 -Z p＝2

[0016] When the mutual coupling degree of freedom of the array is 3, C p＝3 =Τ p＝3 -Hp＝3 -Z p＝3

[0017] Matrix Z p＝3 Continue with the following decomposition: Z p＝3 =Z' p＝3 -Z" p＝3 ; Where p is the array degree of freedom;

[0018] The mutual coupling coefficient matrix is ​​decomposed into three matrices in the form of addition and subtraction. The matrices T of the first two parts are p＝l and H p＝l (l=2,3) has a band-symmetric Toeplitz property and is uniquely characterized by the first row or first column element. The matrix in the last part has the following characteristics compared to the matrices in the first two parts:

[0019] Z p＝2 and Z" p＝3 The first row, second row, first column, second column, last two rows and last two columns are all 0 elements. The matrix without these rows and columns still has the characteristics of a Toeplitz matrix; and Z' p＝3 Similarly, the first row, first column, last row, and last column all have 0 elements, and the remaining matrix has the characteristics of a Toeplitz matrix.

[0020] The step 2 is specifically as follows: according to the properties of the Toeplitz matrix, a reconstruction matrix uniquely corresponding to each matrix is ​​constructed. When the array degree of freedom p=2,

[0021] C p＝2 a(θ)=(D p＝2 -H p＝2 -Z p＝2 )a(θ)

[0022] =D p＝2 a(θ)-H p＝2 a(θ)-Z p＝2 a(θ)

[0023] =T1 (2) [a(θ)]c-T2 (2) [a(θ)]c-T3 (2) [a(θ)]c

[0024] in, x i Indicates the distance between each array element and the reference array element, c = [c0, c1] H Because D p＝2 The first row or the first column of the column vector consists of non-zero elements, then there is

[0025]

[0026]

[0027]

[0028] Similarly, when the array degree of freedom p = 3, c = [c o ,c1,c2] T , then

[0029] C p＝3 a(θ)=(D p＝3 -H p＝3 -Z p＝3 )a(θ)

[0030] =D p＝3 a(θ)-H p＝3 a(θ)-Z p＝3 a(θ)

[0031] =T1 (3) [a(θ)]c-T2 (3) [a(θ)]c-T3 (3) [a(θ)]c

[0032] in,

[0033]

[0034]

[0035]

[0036] The spectrum peak search function of the multiple signal classification algorithm is expressed as

[0037]

[0038] Wherein, T=T1[a(θ)]-T2[a(θ)]-T3[a(θ)], T1[a(θ)], T2[a(θ)] and T3[a(θ)] are called the reconstruction matrices of D, H and Z respectively.

[0039] The step 3 is as follows: first, the angle is estimated. Assuming that the initial MCM is the unit matrix, considering that the signal source incident on the array is coherent, the following decorrelation method is used:

[0040] Perform singular value decomposition on the received data matrix;

[0041] X(t)=UDV H

[0042] in, is a left singular matrix, is a right singular matrix, Is a diagonal matrix. Take the diagonal matrix composed of the smaller NM singular values ​​in D as The corresponding NM column of V is recorded as In order to leave only the information of the noise space, the noise space information is Perform dimensionality reduction

[0043]

[0044] So Represents the noise subspace, which is orthogonal to the signal subspace. Let the number of iterations be m, and the first iteration is performed when m = 0. Here, it is assumed that the initial mutual coupling coefficient matrix C0 = I N ,I N is an N-order unit matrix, then let

[0045] W=C m {A(θ)[A H (θ)A(θ)] -1 A H (θ)}

[0046] Here W is the projection matrix of the signal subspace constructed according to the array manifold A(θ), which is still orthogonal to the signal subspace and has nothing to do with the signal source, so decoherence is achieved. The spectral function is

[0047]

[0048] Orthogonal to the projection matrix W of the signal subspace, assuming that the initial MCM is a unit matrix, perform spectrum peak search

[0049]

[0050] Get the incoming signal angle θ m , when there are multiple incoming signals, θ m is a vector.

[0051] The step 5 is specifically as follows: using the preliminary estimate of DOA to obtain θ m Estimate the mutual coupling coefficient and construct the objective function

[0052]

[0053] When using the iterative search method for estimation, when the value of c can make the objective function δ satisfy the constraint condition ε, the iteration is stopped, and the estimated mutual coupling coefficient vector can be obtained. Get the estimated mutual coupling coefficient matrix

[0054] make:

[0055] Calculation of array flow patterns under coupling using the estimated MCM Reuse Perform iterative estimation of DOA and MCM to obtain θ m+1 and Until satisfied

[0056]

[0057] Finally, the termination condition is met The incoming signal angle and MCM are the final estimated results.

[0058] Beneficial effects of the present invention:

[0059] The algorithm proposed in the present invention achieves better DOA estimation results than existing technical solutions under conditions of low signal-to-noise ratio and limited number of snapshots. When two coherent signal sources are very close, the algorithm proposed in the present invention can still clearly distinguish the arrival angles of the two signals, which makes up for the shortcomings of existing technical solutions in this regard.

[0060] This method uses array flow patterns to construct a signal subspace for decorrelation, then performs DOA estimation. Based on the estimated DOA, it uses an iterative search algorithm with Toeplitz matrix reconstruction to perform MCM estimation. The estimated MCM is then used to calculate the array flow pattern for the next iteration, and the previous estimation method is repeated to further enhance DOA estimation. Under low signal-to-noise ratio (SNR) and limited snapshot count conditions, the proposed algorithm achieves lower RMSE for DOA and MCM estimation than existing solutions. Therefore, the proposed algorithm exhibits greater robustness to SNR and effectively improves estimation accuracy.

[0061] When the angles of the incoming signals are very close, the existing technical solutions have large errors in estimating the DOA of the coherent signal source and may even be unable to distinguish the two incoming signal angles. However, the algorithm proposed in the present invention has a good estimation effect and can clearly distinguish two similar coherent signal sources. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] Figure 1 A flowchart of a method for estimating the DOA of a coherent signal source under the mutual coupling condition of a symmetrical non-uniform array provided by an embodiment of the present invention.

[0063] Figure 2 A schematic diagram of a non-uniform array with a centrally symmetrical structure provided by an embodiment of the present invention.

[0064] Figure 3This is a schematic diagram comparing the DOA estimation effects of an embodiment of the present invention with other algorithms when the two coherent sources are at far angles and the signal-to-noise ratio is 5dB.

[0065] Figure 4 This is a schematic diagram comparing the DOA estimation effects of an embodiment of the present invention with other algorithms when the two coherent sources are at far angles and the signal-to-noise ratio is 30dB.

[0066] Figure 5 This is a schematic diagram comparing the DOA estimation effects of an embodiment of the present invention with other algorithms when the angles of two coherent sources are close and the signal-to-noise ratio is 5dB.

[0067] Figure 6 This is a schematic diagram comparing the DOA estimation effects of an embodiment of the present invention with other algorithms when the angles of two coherent sources are close and the signal-to-noise ratio is 30dB.

[0068] Figure 7 A schematic diagram of a simulation showing how the root mean square error of DOA estimation using different methods varies with the signal-to-noise ratio provided by an embodiment of the present invention.

[0069] Figure 8 A schematic diagram of a simulation of the root mean square error of DOA estimation using different methods as snapshots are digitized is provided in an embodiment of the present invention.

[0070] Figure 9 A schematic diagram of a simulation showing how the RMSE of MCM estimation using different methods varies with the signal-to-noise ratio provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0071] The present invention will be described in further detail below with reference to the accompanying drawings.

[0072] The present invention discloses a DOA estimation method for a coherent signal source under symmetrical array mutual coupling error conditions, which specifically includes the following steps:

[0073] Step 1: Split the centrally symmetric MCM into the difference of several matrices, all of which have the characteristics of a Toeplitz matrix.

[0074] Step 2: According to the properties of the Toeplitz matrix, obtain the reconstruction matrix that uniquely corresponds to each partial matrix;

[0075] Step 3: Perform singular value decomposition and dimensionality reduction on the received data matrix to obtain the noise subspace. Calculate the projection matrix of the signal subspace through the array flow matrix. Then, use the orthogonality of the noise subspace and the signal subspace to construct a spectral function to search for the spectral peak and achieve a preliminary estimate of the DOA.

[0076] Step 4: Use the estimated preliminary angle value to perform iterative search to estimate the MCM, then calculate the array flow pattern for the next iteration through the estimated MCM, and construct the projection matrix of its signal subspace to estimate the DOA again;

[0077] Step 5: Repeat steps 3 and 4, performing multiple iterations until the termination condition is met. This multiple iterations weaken the effects of mutual coupling and coherent sources, further improving the DOA estimation performance.

[0078] like Figure 2 As shown: Step 1 is specifically as follows: suppose there is a non-uniform array symmetrical about the center, with N array elements, assuming that the first array element is used as the reference array element, and the distance between the i-th array element and the reference array element is x i , assume that the unit distance between array elements is d = λ / 2, λ represents the wavelength. Here we only consider the azimuth angle in the incoming wave direction, which can simplify the algorithm derivation. The azimuth angle is θ∈(-90°,90°).

[0079] Assuming that the number of signal sources incident on the array is M, the array flow pattern of the data received by the array can be expressed as

[0080] A=[a(θ1),a(θ2),…,a(θ M )] (1)

[0081] When the mth signal source is incident on the array, the corresponding steering vector a(θ m )(m=1,2,…,M) can be expressed as

[0082]

[0083] Where ζ k is the phase delay of the kth array element relative to the reference array element, which can be expressed as

[0084]

[0085] If the mutual coupling effect between arrays is considered, the received data of the array can be expressed as

[0086] X(t)=CAS(t)+N(t) (4)

[0087] In formula (4), X(t) is the array receiving data matrix; is the mutual coupling coefficient matrix of the array; is the array flow pattern of the non-uniform array; S(t) is the signal vector with the number of snapshots L, which can be expressed as N(t) has a mean of 0 and a variance of Gaussian white noise can be expressed as

[0088] Then the covariance matrix of the array data in formula (4) can be expressed as

[0089]

[0090] in, is the signal covariance matrix, is the signal power, I N Represents the N×N identity matrix. According to the standard subspace decomposition method, R xx Perform eigenvalue decomposition

[0091]

[0092] Where Λ and E=[E s ,E n ] are respectively represented by the covariance matrix R xx The diagonal matrix composed of the eigenvalues ​​of and the eigenvector matrix composed of the eigenvectors corresponding to the eigenvalues, Represents the signal subspace, whose column vectors are matrices composed of the eigenvectors corresponding to the first M larger eigenvalues, Represents the noise subspace, whose column vector is the matrix composed of the eigenvectors corresponding to the last NM smaller eigenvalues;

[0093] Since the signal subspace E s and noise subspace E n mutually orthogonal, a(θ m ) as the mth incoming signal in the signal subspace, we have

[0094]

[0095] Therefore, when the mutual coupling coefficient C is known, the arrival angle of the mth incoming signal can be estimated by searching the spectral function P(θ) for the angle

[0096]

[0097] According to the above description, in order to obtain a more accurate angle value during spectrum peak search, the mutual coupling coefficient matrix must be used as a priori condition. Therefore, the accuracy of angle estimation depends on the accuracy of the mutual coupling coefficient matrix.

[0098] This paper focuses on the DOA estimation problem for coherent sources with mutual coupling compensation in a non-uniform array symmetric about its center. Analysis reveals that the elements of the MCM for a centrally symmetric array are symmetric about the anti-diagonal line. This allows for decomposition into several matrices with Toeplitz properties, which can be added and subtracted to simplify the mutual coupling coefficient estimation problem.

[0099] Compared with the mutual coupling coefficient matrix of a uniform array, the mutual coupling matrix of the array model with a centrosymmetric structure no longer has the Toeplitz property, but still satisfies the following characteristics:

[0100] (1) According to the reciprocity characteristics of the array antenna, the mutual coupling between element i and element j satisfies

[0101] c i,j =c j,i (9)

[0102] (2) The coupling coefficient of each array element to itself is 1;

[0103] (3) When the distance between two array elements is large enough, the mutual coupling effect disappears, that is, the mutual coupling coefficient is 0;

[0104] (4) The mutual coupling coefficients between array elements with the same spacing are the same.

[0105] In the field of array antennas, the mutual coupling degree of freedom (p) is an inherent property of a given array, describing the number of elements with nonzero mutual coupling coefficients. Due to the influence of the mutual coupling degree of freedom (p) and the element spacing (d), the MCM of a centrosymmetric array antenna is uniquely determined when the antenna spacing and mutual coupling degree of freedom are given.

[0106] In the present invention, a 6-element centrosymmetric antenna array is given as follows Figure 1 As shown, according to the analysis of the literature "Factors affecting the mutual coupling effect of antenna arrays", assuming that the mutual coupling effect between the two array elements can be ignored when the array spacing is greater than 2.5λ, the mutual coupling coefficient matrix of the array when the mutual coupling degrees of freedom are 2 and 3 can be expressed as

[0107]

[0108]

[0109] Although these two matrices no longer have the Toeplitz property, they can be decomposed into the difference between several matrices. For example, when the array mutual coupling degree of freedom is p = 2, Equation (10) can be expressed as:

[0110] C p＝2 =D p＝2 -H p＝2 -Z p＝2 (12)

[0111] in,

[0112]

[0113]

[0114]

[0115] When the mutual coupling degree of freedom of the array is 3, Equation (11) can be expressed as

[0116] C p＝3 =Τ p＝3 -H p＝3 -Z p＝3 (16)

[0117] in

[0118]

[0119]

[0120]

[0121] Matrix Z p＝3 The decomposition can be continued as follows

[0122] Z p＝3 =Z' p＝3 -Z" p＝3 (20)

[0123] In formula (20)

[0124]

[0125]

[0126] From the above matrix decomposition, it can be seen that the mutual coupling coefficient matrix is ​​decomposed into the form of addition and subtraction of three major matrices. The matrices T of the first two parts are p＝l and H p＝l (l=2,3) has the Toeplitz property of band symmetry, so it can be uniquely characterized by its first row or first column elements. The matrix in the last part has the following characteristics compared to the matrices in the first two parts:

[0127] Z p＝2 and Z" p＝3 The first row, second row, first column, second column, last two rows and last two columns are all 0 elements. The matrix without these rows and columns still has the characteristics of a Toeplitz matrix; and Z' p＝3 Similarly, the first row, first column, last row, and last column all have 0 elements, and the remaining matrix has the characteristics of a Toeplitz matrix.

[0128] Then, according to the characteristics of the Toeplitz matrix, when the array degree of freedom p = 2, we have

[0129]

[0130] in, x i (i=1,2,...,5) represents the distance between each array element and the reference array element, c=[c0,c1] H Because D p＝2 The first row or the first column of the column vector consists of non-zero elements, then there is

[0131]

[0132]

[0133]

[0134] Similarly, when the array degree of freedom p = 3, we have

[0135]

[0136] in,

[0137]

[0138]

[0139]

[0140] Decomposing the matrix as in Equation (23) and Equation (27), the peak search function of the multi-signal classification algorithm can be expressed as

[0141]

[0142] Wherein, T=T1[a(θ)]-T2[a(θ)]-T3[a(θ)], T1[a(θ)], T2[a(θ)] and T3[a(θ) are called the reconstruction matrices of D, H and Z respectively.

[0143] First, the angle is estimated. Assuming that the initial MCM is the unit matrix, considering that the signal source incident on the array is coherent, the following decoherence method can be used: First, the received data matrix is ​​singularly decomposed

[0144] X(t)=UDV H (32)

[0145] in, is a left singular matrix, is a right singular matrix, Is a diagonal matrix. Take the diagonal matrix composed of the smaller NM singular values ​​in D as The corresponding NM column of V is recorded as In order to leave only the information of the noise space, the noise space information is Perform dimensionality reduction

[0146]

[0147] So Represents the noise subspace, which is orthogonal to the signal subspace. Let the number of iterations be m, and the first iteration is performed when m = 0. Here we assume that the initial mutual coupling coefficient matrix C0 = I N ,I N is an N-order unit matrix, then let

[0148] W=C m {A(θ)[A H (θ)A(θ)] -1 A H (θ)} (34)

[0149] Here W is the projection matrix of the signal subspace constructed according to the array manifold A(θ), which is still orthogonal to the signal subspace and has nothing to do with the signal source, thus achieving decoherence. Then the spectral function is

[0150]

[0151] Therefore, a peak search is performed

[0152]

[0153] Get the incoming signal angle θ m , when there are multiple incoming signals, θ m is a vector.

[0154] Estimate the angle θ m Then, the objective function is constructed by combining the above formula (31)

[0155]

[0156] Continue to estimate the mutual coupling coefficient. When using the iterative search method for estimation, when the value of c can make the objective function δ satisfy the constraint condition ε, the iteration is stopped, and the estimated mutual coupling coefficient vector can be obtained. Get the estimated mutual coupling coefficient matrix

[0157] make

[0158]

[0159] make According to equations (34)-(36), the m+1th iteration is used to estimate θ m+1, using θ m+1 Repeat the above MCM estimation and iterate until the

[0160]

[0161] Finally, the termination condition is met The incoming signal angle and MCM are the final estimated results.

[0162] The method of the embodiment of the present invention is verified. Assume that the number of array elements of the non-uniform array is 6, the number of signal snapshots is 200, the noise is additive white Gaussian noise, the degree of freedom of the entire array is p=3, and the mutual coupling coefficient vector is

[0163] c=[1,0.9081+0.0256i,-0.1880-0.0582i]

[0164] The termination iteration condition of angle estimation MCM is ε=0.001, and the final termination iteration condition of DOA and MCM estimation is

[0165] The comparison methods used in this paper are: (1) Classical MUSIC-based mutual coupling compensation algorithm: After estimating the MCM using an iterative algorithm, it is substituted into the spectrum function to directly search for spectrum peaks to obtain the DOA estimation result. (2) Smoothing-based mutual coupling compensation algorithm: First, the covariance matrix of the array received data is smoothed, and then the objective function is constructed using the covariance matrix. The DOA and MCM are jointly estimated using an iterative algorithm. (3) Ideal DOA estimation algorithm: DOA estimation is performed when the MCM is known and the source signal is decorrelated.

[0166] Assume that the angles of the two coherent signal sources are 20° and 60°, Figure 3 and Figure 4 The estimation simulation diagrams of the present invention and other algorithms are shown under the conditions of 5dB and 30dB signal-to-noise ratios, respectively. The mutual coupling compensation algorithm based on classic MUSIC lacks the ability to deal with coherence and cannot estimate the angle of the signal arrival wave. The smooth mutual coupling compensation algorithm has a weak spectrum peak and a relatively large estimation error under low signal-to-noise ratio conditions. The algorithm proposed in the present invention has good estimation performance, with a sharp spectrum peak that can clearly distinguish the two signal sources. Its DOA estimation result is very close to that under ideal conditions, and the estimation effect is better.

[0167] When other simulation conditions remain unchanged, only the degree of proximity of the coherent source wave angles is changed. Assuming that the angles of the two wave signals are 20° and 25°, Figure 5 and Figure 6The DOA estimation performance comparison of different algorithms is given when the signal-to-noise ratio is 5dB and 30dB respectively. It can be seen that when the signal-to-noise ratio is low, the smooth mutual coupling compensation algorithm cannot estimate two close coherent signal sources, and even produces pseudo peaks, which has a significant impact on the DOA estimation of the signal. When the signal-to-noise ratio is large, although two angles can be estimated, the peaks are small and not sharp enough, making them difficult to distinguish. However, the algorithm proposed in this paper has better stability as the signal-to-noise ratio changes, and has good results regardless of whether the signal-to-noise ratio is low or high. The spectrum peak for DOA estimation is very close to the DOA estimation under ideal conditions. Therefore, compared with the other algorithms used for comparison, the algorithm in this paper has better performance and estimation accuracy.

[0168] Assuming the incoming wave angle is 20° and the signal-to-noise ratio changes from 0dB to 20dB, Figure 7 The results show how the root mean square error (RMSE) of DOA estimation using the present invention and the smoothing-based mutual coupling compensation algorithm varies with the signal-to-noise ratio. As can be seen, the DOA estimation algorithm proposed in the present invention has a lower RMSE than the smoothing-based mutual coupling estimation algorithm as the signal-to-noise ratio changes, demonstrating that the algorithm proposed in the present invention has better estimation performance.

[0169] Figure 8 The root mean square error (RMSE) of DOA estimation using the present invention and a smoothing-based mutual coupling compensation algorithm as the number of snapshots changes is shown. As can be seen, the present invention achieves a lower RMSE when performing DOA estimation than the smoothing-based mutual coupling estimation algorithm as the number of snapshots changes, demonstrating its superior estimation performance.

[0170] In this paper, the performance of MCM estimation is characterized by the relative RMSE of MCM. Assuming that the number of cycle simulations is k, the following definition is given:

[0171]

[0172] Among them, ||·|| F represents the Frobenius norm. Figure 9 The RMSE of the mutual coupling coefficient estimation for 100 Monte Carlo cycles as a function of the signal-to-noise ratio is shown. It can be seen that the proposed algorithm has a lower RMSE when performing MCM estimation than the smoothing-based MCM estimation algorithm, thus achieving better estimation performance.

[0173] The present invention discloses a DOA estimation method for coherent signal sources under symmetric array mutual coupling conditions, comprising the following steps: when the number of signal sources is known, decomposing an array mutual coupling matrix (MCM) with a central symmetric structure into several matrices with Toeplitz properties; secondly, constructing a reconstruction matrix uniquely corresponding to each Toeplitz matrix according to the characteristics of the Toeplitz matrix; then performing singular value decomposition and dimensionality reduction processing on a received data matrix to obtain a noise subspace, calculating a projection matrix of a signal subspace using an array flow matrix, constructing a spectral function using the orthogonality of the noise subspace and the signal subspace, searching for a spectral peak to achieve DOA estimation; performing an iterative search using an estimated angle value to estimate the MCM; multiplying the estimated MCM by the array flow pattern to obtain a new array flow pattern, constructing a projection matrix of the signal subspace using the new array flow pattern to estimate the DOA again, and weakening the influence of mutual coupling and coherent sources through iterative estimation of the MCM and the DOA, thereby further improving the accuracy of the DOA estimation. Under the conditions of low signal-to-noise ratio and limited number of snapshots, the present invention can obtain better DOA estimation performance and realize the resolution estimation of coherent signal sources.

Claims

1. A DOA estimation method for coherent sources under symmetrical array mutual coupling conditions, characterized in that: The following steps are included: Step 1: When the number of signal sources is known, the MCM of the non-uniform array with central symmetry is split into several parts with the Toeplitz property matrix; Step 2: According to the properties of the Toeplitz matrix, obtain the reconstruction matrix that uniquely corresponds to each partial matrix; Step 3: Perform singular value decomposition and dimensionality reduction on the received data matrix to obtain the noise subspace. Calculate the projection matrix of the signal subspace through the array flow matrix. Then, use the orthogonality of the noise subspace and the signal subspace to construct a spectral function to search for the spectral peak and achieve a preliminary estimate of the DOA. Step 4: Use the preliminary angle value obtained from the preliminary estimation of the DOA to perform an iterative search to achieve MCM estimation, then calculate the array flow pattern for the next iteration using the obtained MCM, and construct the projection matrix of its signal subspace to perform DOA estimation again; Step 5: Repeat steps 3 and 4, and perform iterative search multiple times until the termination condition is met.

2. The DOA estimation method for a coherent signal source under symmetrical array mutual coupling conditions according to claim 1, characterized in that: In the step 1, a matrix operation is performed on the MCM of the symmetric non-uniform array to divide it into several parts in the form of a Toeplitz property matrix; When the mutual coupling degree of freedom of the array is p=2, C p＝2 =D p＝2 -Η p＝2 -Z p＝2 When the mutual coupling degree of freedom of the array is 3, C p＝3 =D p＝3 -H p＝3 -Z p＝3 Matrix Z p＝3 Continue with the following decomposition: Z p＝3 =Z' p＝3 -Z" p＝3 ; Where p is the array degree of freedom; The mutual coupling coefficient matrix is ​​decomposed into three matrices in the form of addition and subtraction. The matrices T of the first two parts are p＝l and Η p＝l (l=2,3) has a band-symmetric Toeplitz property and is uniquely characterized by the first row or first column element. The matrix in the last part has the following characteristics compared to the matrices in the first two parts: Z p＝2 and Z" p＝3 The first row, second row, first column, second column, last two rows and last two columns are all 0 elements. The matrix without these rows and columns still has the characteristics of a Toeplitz matrix; and Z' p＝3 Similarly, the first row, first column, last row, and last column all have 0 elements, and the remaining matrix has the characteristics of a Toeplitz matrix.

3. The DOA estimation method for a coherent signal source under symmetrical array mutual coupling conditions according to claim 1, characterized in that: The step 2 is specifically as follows: according to the properties of the Toeplitz matrix, a reconstruction matrix uniquely corresponding to each matrix is ​​constructed. When the array degree of freedom p=2, in, x i Indicates the distance between each array element and the reference array element, c = [c0, c1] H Because D p＝2 The first row or the first column of the column vector consists of non-zero elements, then there is Similarly, when the array degree of freedom p = 3, c = [c o ,c1,c2] T , then in, The spectrum peak search function of the multiple signal classification algorithm is expressed as Where T = T1[a(θ)]-T2[a(θ)]-T3[a(θ)], T1[a(θ)], T2[a(θ)] and T3[a(θ)] are called the reconstruction matrices of D, H and Z respectively; represents the noise subspace.

4. The DOA estimation method for a coherent signal source under symmetrical array mutual coupling conditions according to claim 1, characterized in that: The step 3 is as follows: first, the angle is estimated. Assuming that the initial MCM is the unit matrix, considering that the signal source incident on the array is coherent, the following decorrelation method is used: Perform singular value decomposition on the received data matrix; X(t)=UDV H in, is a left singular matrix, is a right singular matrix, Is a diagonal matrix; take the diagonal matrix composed of the smaller NM singular values ​​in D as The corresponding NM column of V is recorded as In order to leave only the information of the noise space, the noise space information is Perform dimensionality reduction So Represents the noise subspace, which is orthogonal to the signal subspace. Let the number of iterations be m, and the first iteration is performed when m = 0. Here, it is assumed that the initial mutual coupling coefficient matrix C0 = I N ,I N is an N-order unit matrix, then let W=C m {A(θ)[A H (θ)A(θ)] -1 A H (i)} Here W is the projection matrix of the signal subspace constructed according to the array manifold A(θ), which is still orthogonal to the signal subspace and has nothing to do with the signal source, so decoherence is achieved. The spectral function is Orthogonal to the projection matrix W of the signal subspace, assuming that the initial MCM is a unit matrix, perform spectrum peak search Get the incoming signal angle θ m , when there are multiple incoming signals, θ m is a vector.

5. The DOA estimation method for a coherent signal source under symmetrical array mutual coupling conditions according to claim 1, characterized in that: The step 5 is specifically as follows: using the initial estimate of DOA to θ m Estimate the mutual coupling coefficient and construct the objective function When using the iterative search method for estimation, when the value of c can make the objective function δ satisfy the constraint condition ε, the iteration is stopped, and the estimated mutual coupling coefficient vector can be obtained. Get the estimated mutual coupling coefficient matrix make: Calculation of coupled array flow patterns using estimated DOA and MCM Reuse Perform iterative estimation of DOA and MCM to obtain θ m+1 and Until satisfied Finally, the termination condition is met The incoming signal angle and MCM are the final estimated results.

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