CN108896954A - A kind of direction of arrival estimation method based on joint real value subspace in relatively prime battle array - Google Patents

Abstract

The invention discloses a method for estimating the angle of arrival (Direction of arrival, DOA) based on a joint real-valued subspace in a coprime array. First, the array structure of the basic coprime array is improved to make it centrosymmetric; Then two sub-covariances are constructed through the two sub-arrays, and the relationship matrix between the sub-arrays is further obtained; based on the relationship matrix, the joint subspace angle-of-arrival estimation function is constructed; finally, the unambiguous DOA estimates. Due to the existence of the relation matrix, the outputs of the two subarrays are automatically paired, and we unite the two subspaces, so the estimation performance is guaranteed. Compared with the state-of-the-art algorithms, our proposed method has greatly reduced complexity because only real-valued eigendecomposition and low-dimensional sub-covariances are required, but the performance of DOA estimation remains almost unchanged. Therefore, the present invention can greatly reduce complexity, save hardware cost and improve response speed in positioning systems in radar and wireless communication.

Description

An Estimation Method of Angle of Arrival Based on Joint Real-valued Subspace in Coprime Array

technical field

The invention relates to a method for estimating the angle of arrival based on a joint real-valued subspace in a coprime array, which belongs to array signal processing.

Background technique

Direction of arrival (DOA) estimation is an important research direction of array signal processing, and it has been widely used in many fields such as radar and mobile communication. At present, many classic super-resolution DOA estimation methods have been proposed, such as the Multiple signal classification (MUSIC) method, the Estimation of signal parameters via rotational invariance technique (ESPRIT) method, the propagation algorithm Sub-method (Propagator method, PM) and rotation invariance PM method and so on. However, in order to avoid the problem of estimation ambiguity, these methods all use compact arrays with element spacing not greater than half a wavelength. The limited DOF leads to limited DOA estimation resolution, and there is mutual coupling interference between antennas.

As a non-uniform sparse array, the nested array can obtain degrees of freedom (DOF) far greater than the number of actual physical array elements in the virtual array domain, thereby increasing the system capacity and spatial resolution. However, the nested array is still There is a compact sub-array, and the closer antenna spacing will still be affected by the mutual coupling between antennas, which will reduce the performance of DOA estimation. Another popular non-uniform sparse array is the coprime array, which is composed of two uniform linear subarrays, and the number of array elements and the distance between array elements are mutually prime. The coprime array can also obtain a larger DOF in the virtual domain, and compared with the nested array, the array element distribution of the coprime array is more sparse, so it has strong robustness to the mutual coupling problem. In order to solve the DOA estimation ambiguity problem caused by the large array element spacing, the joint MUSIC method in the document "DOA estimation with combined MUSIC for coprimarray" is based on the mutual prime property between the subarrays, from the MUSIC peaks of the two subarrays The intersection determines an ambiguity-free DOA estimate, but it requires two global peak searches, which has high complexity. In order to reduce the complexity, the local search MUSIC method in the document "Partial spectral search-based DOA estimation method for co-prime linear arrays" takes advantage of the uniformity of the subarray and transforms the angle domain estimation problem into the phase domain, so that only local search is required All the MUSIC peaks can be recovered, and then defuzzified according to the mutual prime. However, whether it is joint MUSIC or local search MUSIC, the sub-arrays are processed separately, so when the final intersection set is defuzzified, there will be interference between multiple DOA estimates (including the real solution and the fuzzy solution), especially at low It is easy to make mistakes when the signal-to-noise ratio. In order to avoid this problem, the document "Improved DOA estimation algorithm for co-prime linear arrays using root-MUSIC algorithm" applies the improved root-MUSIC method (Li J, Jiang D. Joint elevation and azimuth angles estimation for L-shaped array) to coprime In the array, first construct the relationship matrix of the two subarrays through the signal subspace of the entire array, and then use the root-finding method to solve the DOA estimation based on the noise subspace of the entire array. This method not only realizes the automatic matching of the output results of the two subarrays, It further improves the DOA estimation performance. However, the eigendecomposition of the whole array and the estimation function based on the whole noise subspace involve high complexity.

Contents of the invention

The technical problem to be solved by the present invention is to provide a DOA estimation method based on a joint real-valued subspace in a coprime matrix. First, we improve the array manifold of the coprime matrix, so that the two sub-arrays are centrosymmetric, so that the covariance can be changed to real-valued through unitary transformation, thereby greatly reducing the complexity. In addition, we do not need the covariance of the whole array, but only need the sub-covariance of two real-valued sub-arrays to obtain the relationship matrix between the two sub-arrays, avoiding additional pairing, and further reducing the complexity. Finally, the joint subspace DOA estimation cost function is constructed based on the noise subspace of the two subarrays. Compared with the improved root-finding MUSIC method, the complexity of the proposed algorithm is greatly reduced, but the performance of DOA estimation can remain almost unchanged.

The present invention adopts the following technical solutions for solving the problems of the technologies described above:

The present invention provides a method for estimating the angle of arrival based on a joint real-valued subspace in a coprime array. The structures of the first and second subarrays of the coprime array satisfy centrosymmetry. The specific steps of the method are:

Step 1, constructing the cross-covariance between the first and second sub-arrays and the auto-covariance of the second sub-array;

Step 2, carrying out real-valued transformation to the mutual covariance between the first and second sub-arrays in step 1 and the auto-covariance of the second sub-array by unitary transformation;

Step 3, perform singular value decomposition on the real-valued transformation result of the cross-covariance between the first and second sub-arrays in step 2, and obtain the noise subspace corresponding to the first sub-array and the noise subspace corresponding to the second sub-array ;

Step 4, according to the real-valued transformation result of the mutual covariance between the first and second sub-matrix and the auto-covariance of the second sub-matrix in step 2, construct a relationship matrix;

Step 5, according to the noise subspace corresponding to the first subarray in step 3 and the noise subspace corresponding to the second subarray and the relationship matrix in step 4, construct the angle estimation function of the joint real-valued subspace;

Step 6, using the root-finding method, the uniform distribution of the solution, and the relationship matrix to obtain a group of DOA estimation solutions corresponding to the second sub-array and a group of DOA estimation solutions corresponding to the first sub-array;

Step 7, using the mutual primality of the first and second sub-arrays, the average of the two closest solutions among the two sets of DOA estimation solutions obtained in step 6 is taken as the final DOA estimation.

As a further technical solution of the present invention, in step 1, the mutual covariance between the first and second sub-arrays is The autocovariance of the second subarray is Among them, A ₁ =[a ₁ (α ₁ ),...,a ₁ (α _K )], A ₂ =[a ₂ (β ₁ ),...,a ₂ (β _K )] are respectively The direction matrix of the first and second sub-arrays, K is the number of different directions of the far-field signal existing in the space, α _k ＝Nπsinθ _k , β _k ＝Mπsinθ _k , M and N are the number of elements of the first sub-array and the second sub-array respectively, M and N are relatively prime integers, θ _k is the DOA of the kth signal, k=1,2,...,K, is a diagonal matrix, Represents the energy of the kth signal, σ ² is the noise energy, and I _N is the N×N identity matrix.

As a further technical solution of the present invention, in step 2, the real-valued transformation result of the cross-covariance between the first and second sub-arrays is R _r =H ₁ R _s H ₂ ^H , the auto-covariance of the second sub-array The real-valued transformation result of is R _2r ＝H ₂ R _s H ₂ ^H , where, are the unitary transformation results of A ₁ and A ₂ respectively, and Q _M and Q _N represent unitary matrices of order M and order N respectively.

As a further technical solution of the present invention, the unitary matrix under even and odd dimensions is defined as: and Among them, I _p represents the identity matrix of p×p, Π _p represents the reverse identity matrix of p×p, and p is a natural number.

As a further technical solution of the present invention, step 3 is specifically:

The singular value decomposition of R _r is R _r = UΛV ^T , where U, Λ and V are the left singular vector, singular value and right singular vector respectively;

The noise subspace U corresponding to the first subarray _is formed by the left singular vector corresponding to the minimum MK singular values;

The noise subspace V _n corresponding to the second subarray is composed of right singular vectors corresponding to the smallest NK singular values.

As a further technical solution of the present invention, in step 4, the relationship matrix G=R _r R _2r ⁺ =H ₁ R _s H ₂ ^H (H ₂ R _s H ₂ ^H ) ⁺ =H ₁ H ₂ ⁺ , where (·) ⁺ denotes the generalized inverse of a matrix.

As a further technical solution of the present invention, the angle estimation function of the joint real-valued subspace in step 5 is

As a further technical solution of the present invention, step 6 is specifically:

Use the root-finding method to solve the angle estimation function of the joint real-valued subspace, specifically:

Depend on is a Vandermonde vector, let z=e ^jβ , the angle estimation function of the joint real-valued subspace is rewritten as a ₂ (z)=[1,z,…,z ^N-1 ] ^T , is the union of two noise subspaces;

use the root method to To solve, select K roots z _k , k=1,...,K within the unit circle and closest to the unit circle in the root finding result;

Depend on β _k = Mπsinθ _k , get a DOA estimate from the second sub-array Among them, angle( ) means to take the phase;

According to the uniform distribution of all M solutions with an interval of 2/M, a set of DOA estimation solutions corresponding to the second sub-array is obtained in, u is an integer that varies with m and guarantees Located in the interval [-1,1];

Depend on and h ₁ (α _k ) = Gh ₂ (β _k ), yielding

Depend on and get A DOA estimate corresponding to the first sub-array can be obtained by the phase difference between adjacent elements of a ₁ (α _k ) as Where a _1,m (α _k ) and a _1,m+1 (α _k ) represent the mth and m+1th elements of a ₁ (α _k ), respectively;

According to the uniform distribution of all N solutions with an interval of 2/N, a set of DOA estimation solutions corresponding to the first sub-array is obtained in, v is an integer that varies with n and guarantees that It is within the interval [-1,1].

As a further technical solution of the present invention, the cross-covariance between the first and second sub-arrays and the auto-covariance of the second sub-array are estimated through the finite number of snapshots:

Wherein, T represents the number of snapshots, and x ₁ (t) and x ₂ (t) are outputs of the first sub-array and the second sub-array respectively.

Compared with the prior art, the present invention adopts the above technical scheme and has the following technical effects:

1. Improvements to the coprime array structure, making it centrosymmetric;

2. Based on the symmetrical array structure, we convert the sub-covariances into real values, thus reducing the computational complexity;

3. Without the eigendecomposition and noise space of the entire array, we can obtain the relationship matrix only through the sub-covariance of the real value, avoiding additional pairing, and further reducing the complexity;

4. Finally, we constructed a DOA estimation function based on the real-valued noise subspace of the two sub-arrays, obtained the DOA estimation information from the two sub-arrays, and de-ambiguated based on the mutual prime;

5. Compared with the current advanced algorithms, our proposed algorithm can greatly reduce the complexity, but the DOA estimation performance remains almost unchanged.

Description of drawings

Fig. 1 is a schematic diagram of the structure of the basic coprime array and the improved coprime array;

Fig. 2 is a schematic flow diagram of the method of the present invention;

Fig. 3 is a schematic diagram of the algorithmic complexity comparison between the improved root-finding MUSIC method and the proposed method of the present invention;

Fig. 4 is the DOA estimation result under the first 100 experiments of the proposed method of the present invention;

Fig. 5 is a schematic diagram of the estimation performance comparison between the proposed method of the present invention and the improved root-finding MUSIC algorithm.

Detailed ways

Below in conjunction with accompanying drawing, technical scheme of the present invention is described in further detail:

The invention proposes a method for estimating the angle of arrival (Direction of arrival, DOA) based on joint real-valued subspaces in a coprime matrix. We first improved the array structure of the basic coprime matrix, making it centrosymmetric, so that the array covariance can be transformed into real-valued by unitary transformation, thereby greatly reducing the computational complexity. Then two sub-covariances are constructed through two sub-arrays, and the relationship matrix between sub-arrays is further obtained. Based on the relationship matrix, a joint subspace angle-of-arrival estimation function is constructed. Finally an unambiguous DOA estimate is obtained from the intersection of the outputs of the two subarrays. Due to the existence of the relation matrix, the outputs of the two subarrays are automatically paired, and we unite the two subspaces, so the estimation performance is guaranteed. Compared with the state-of-the-art algorithms, our proposed method has greatly reduced complexity because only real-valued eigendecomposition and low-dimensional sub-covariances are required, but the performance of DOA estimation remains almost unchanged. Therefore, the present invention can greatly reduce complexity, save hardware cost and improve response speed in positioning systems in radar and wireless communication.

As shown in Figure 1, the original coprime array and the improved coprime array array structure, the upper part of Figure 1 shows the array structure of the basic coprime array, which consists of two sparse uniform linear arrays (Uniform linear array, ULA) Composition, sub-array 1 has M array elements, the array element spacing is Nd, and sub-array 2 has N array elements, the array element spacing is Md, d is the unit spacing, generally set to half wavelength, M and N are mutually quality integer. The array structure does not satisfy the central symmetry, so real-valued transformation cannot be performed. Therefore, we have given an improved coprime array structure in the lower part of Figure 1, so that the two sub-arrays are centrosymmetric about the Z axis (it should be noted here that Figure 1 is only an example, and in this paper In the practical application of the inventive method, it is only necessary to ensure that the two sub-arrays are centrally symmetrical). Assuming that there are far-field signals from K different directions in the space at this time, the output of the two sub-arrays is

x ₁ (t)＝A ₁ s(t)+n ₁ (t) (1)

x ₂ (t)＝A ₂ s(t)+n ₂ (t) (2)

Where s(t)=[s ₁ (t),...,s _K (t)] ^T represents K source baseband signals; n ₁ (t) and n ₂ (t) represent the additives on the respective arrays Gaussian white noise; A ₁ =[a ₁ (α ₁ ),...,a ₁ (α _K )] and A ₂ =[a ₂ (β ₁ ),...,a ₂ (β _K )] respectively Represents the direction matrix of subarray 1 and subarray 2, whose columns are called steering vectors, denoted as

Among them, α _k =Nπsinθ _k , β _k =Mπsinθ _k , and θ _k represents the DOA of the kth signal.

real-valued transformation

First, according to (1)-(2), the mutual covariance between the two sub-arrays is constructed as

in is a diagonal matrix containing K signal energies (Gu JF, Wei P. Joint SVD of Two Cross-Correlation Matrices to Achieve Automatic Pairing in 2-D AngleEstimation Problems). Because the two sub-arrays at this time are centrosymmetric, we can use unitary transformation for real-valued conversion, thereby reducing complexity. A unitary matrix in odd and even dimensions is defined as follows:

Taking the steering vector of sub-array 1 as an example, its unitary transformation becomes a real-valued steering vector (assuming M is an odd number)

The corresponding real-valued direction matrix is Similarly, after the unitary transformation of the direction matrix of sub-matrix 2, it becomes So the cross-covariance in (5) becomes

In practice, in order to ensure the real value of the cross-covariance, we can take the real part of the above formula, namely

Based on the cross-covariance in (8), its singular value decomposition (Singular value decomposition, SVD) is

R _r = UΛV ^T (9)

where U, Λ and V denote the left singular vector, singular value and right singular vector, respectively. The left singular vectors corresponding to the smallest (MK) singular values constitute the noise subspace U _n of subarray 1, and the right singular vectors corresponding to the smallest (NK) singular values constitute the noise subspace V _n of subarray 2. So the angle information of the two sub-arrays can be obtained from the following cost function

U _n ^H h ₁ (α) = 0 (10)

V _n ^H h ₂ (β) = 0 (11)

However, it should be noted that α and β are obtained separately at this time, so this will bring two problems: 1. α and β are out of order, and an additional pairing step is required at this time, thereby increasing the complexity; 2. α and β are obtained based on their respective sub-arrays, which means that only part of the array aperture is used, so the estimation performance needs to be improved. Therefore, before performing angle estimation, we need to obtain the relationship between the two sub-arrays, so as to jointly use the two sub-arrays.

Relationship matrix construction

The autocovariance of the constructed submatrix 2 is

Where σ ² is the noise energy, which is also subjected to unitary transformation and noise elimination

where ^σ2 can be obtained from The average of the NK smaller eigenvalues ([16]Yunhe C.Jointestimation of angle and Doppler frequency for bistatic MIMO radar), here it is necessary to note that the unitary matrix satisfies the orthogonality, that is So the unitary transformation will not affect the noise energy. According to (8) and (13), the relationship matrix can be constructed as

G＝R _r R _2r ⁺ ＝H ₁ R _s H ₂ ^H (H ₂ R _s H ₂ ^H ) ⁺ ＝H ₁ H ₂ ⁺ (14)

Then the G matrix gives the relationship between the two subarray direction matrices, namely

H ₁ =GH ₂ (15)

It should be noted that the construction of the G matrix is based on the sub-covariance in (8) and (13). Compared with the covariance of the whole array, its construction complexity is greatly reduced. In addition, these two sub-arrays are still real value, the complexity is further reduced.

Angle of Arrival Estimation Based on Joint Subspace

Based on the G matrix, substituting h ₁ (α)=Gh ₂ (β) into Equation (10), then the angle estimation function based on the joint real-valued subspace can be constructed as

Equation (16) can be solved by root-finding method, because in is a Vandermonde vector. Let z=e ^jβ , then where a ₂ (z)=[1,z,…,z ^N-1 ] ^T . Then (16) can be written as

in is the union of two noise subspaces. According to the polynomial root finding result of (17), we select K roots z _k , k=1, . . . ,K within and closest to the unit circle.

because And β _k ＝Mπsinθ _k , so the estimated DOA obtained from sub-matrix 2 (ie β) is

It can be seen from the above formula that because of the large element spacing (M>1) of sub-array 2, the phase value of z _k will exceed [-π, π], so there is phase ambiguity. We use z _k from (18) Only one of the fuzzy solutions is obtained by the formula, and there are (M-1) solutions These M solutions should satisfy So the relationship between them ([11]SUNF,LAN P and GAO B.Partial spectral search-based DOA estimation method for co-prime linear arrays) is

where u is an integer that varies with m, and guarantees It is within the interval [-1,1]. Therefore, there is a uniformly distributed relationship between all M solutions, and the distance is 2/M. According to this relationship and a solution in (18), we can recover all M solutions.

Now we have a set of solutions from subarray 2 Next, we need to consider the DOA estimate from subarray 1. because as well as And because the relationship h ₁ (α _k )=Gh ₂ (β _k ) is satisfied between the steering vectors, we get

and and but

one of them Because it is a scalar, it can be eliminated by normalization. Through the phase difference between adjacent elements of a ₁ (α _k ), the angle of the corresponding sub-array 1 can be estimated as

Similarly, the above formula is a fuzzy solution, according to the uniform distribution of all N solutions with an interval of 2/N, and one of the solutions obtained by formula (22), a set of solutions can also be obtained

We now obtain two sets of solutions for DOA estimation. These two sets of solutions contain many false solutions, but also contain true DOA estimates. According to ZHOU C, SHI Z, GU Y, et al. DECOM: DOA estimation with combinedMUSIC for coprime array, due to the coprime property in the coprime array, the DOA without ambiguity can be obtained from the common solution of the two sets of solutions, namely the intersection.

But in practice, the covariance in (5) and (12) can only be estimated by a limited number of snapshots, namely

Where T represents the number of snapshots. Due to the influence of residual noise, there is often no completely coincident solution in the final two sets of solutions, so we can estimate the final unambiguous DOA from the average of the closest two solutions in the two sets of solutions. Since the solutions obtained in (18) and (22) are based on z _k , k=1,...,K, they are automatically paired, that is, the two sets of solutions correspond to the kth information source. In this way, when solving the closest two solutions, it will not be affected by other DOAs.

The steps of a method for estimating the angle of arrival (Direction of arrival, DOA) based on a joint real-valued subspace in the coprime matrix of the present invention are as shown in Figure 2:

1. According to formula (23)-(24), construct two sub-covariances R _c and R ₂ ;

2. Based on formulas (8) and (13), perform real-valued transformation on the two sub-covariances to obtain R _r and R _2r , and perform singular value decomposition on R _r according to formula (9) to obtain the noise corresponding to sub-array 1 Subspace U _n and noise subspace V _n corresponding to subarray 2;

3. Obtain the relationship matrix G by G=R _r R _2r ⁺ ;

4. Obtain the joint subspace angle estimation function f(β) in formula (16) according to U _n , V _n and relation matrix G;

5. Obtain a DOA estimate corresponding to sub-array 2 by using the root-finding method, and use the uniform distribution of the solution based on (19) to obtain a set of solutions

6. Using the relationship matrix, a set of DOA estimation solutions corresponding to sub-array 1 can be obtained in the same way Finally, the average of the closest two solutions from the two sets of solutions is used as the final DOA estimate by using the coprime property.

Our proposed algorithm does not require additional pairing and spectral peak search, and its computational complexity mainly focuses on the construction of sub-covariances, the calculation of relationship matrices, and the final root-finding. It should be noted that the sub-covariances in the algorithm are all real-valued, so the computational complexity is only 1/4 of the corresponding complex matrix operation. In the end, the total number of multiplications is about O(MNT+N ² T+0.25*(N ³ +2MN ² )+N) times. However, the improved root-finding MUSIC method requires about O((M+N) ² T+(M+N) ³ +MNK+2N ² K+K ³ +N) times of multiplication.

In Fig. 3 we give a comparison of the complexity of the two algorithms under typical parameter configurations, where M=4, N=3, K=2, T=100. It can be found from Figure 3 that the complexity of the proposed algorithm is only 40% of the improved root-finding MUSIC algorithm.

Experimental simulation

In the experiment, we adopted the proposed improved coprime array structure, as shown in Figure 1, where M=4, N=3, and K=2 sources, and their DOAs are θ ₁ =20° and θ ₂ = 35°. Each covariance matrix is estimated by T=100 snapshots respectively. In order to measure the performance of DOA estimation, the root mean square error (RMSE) is defined as

in is the estimated result of DOAθ _k during the l-th Monte Carlo experiment, and we have conducted a total of L=500 experiments.

Figure 4 shows the DOA estimation results of the proposed algorithm under the first 100 experiments. It can be found that the proposed algorithm can always effectively and accurately estimate the DOA of the two sources.

Figure 5 shows the estimated performance comparison between the proposed algorithm and the improved root-finding MUSIC algorithm. The vertical axis is RMSE, and the horizontal axis is SNR. It can be found that the two are almost identical, indicating that the proposed algorithm has almost the same as the improved root-finding MUSIC algorithm. performance, but the complexity of the proposed algorithm is greatly reduced.

This paper proposes a DOA estimation algorithm based on joint real-valued subspaces in coprime matrices. The key work we do can be summarized as follows:

1. Improvements to the coprime array structure, making it centrosymmetric;

2. Based on the symmetrical array structure, we convert the sub-covariances into real values, thus reducing the computational complexity;

3. Without the eigendecomposition and noise space of the entire array, we can obtain the relationship matrix only through the real-valued sub-covariance;

4. Finally, we constructed a DOA estimation function based on the real-valued noise subspace of the two sub-arrays, obtained the DOA estimation information from the two sub-arrays, and de-ambiguated based on the mutual prime;

5. Compared with the current advanced algorithms, our proposed algorithm can greatly reduce the complexity, but the DOA estimation performance remains almost unchanged.

The above is only a specific implementation mode in the present invention, but the scope of protection of the present invention is not limited thereto. Anyone familiar with the technology can understand the conceivable transformation or replacement within the technical scope disclosed in the present invention. All should be covered within the scope of the present invention, therefore, the protection scope of the present invention should be based on the protection scope of the claims.

Claims (9)

1. A method for estimating an angle of arrival based on a combined real-valued subspace in a co-prime matrix is characterized in that the structures of a first sub-matrix and a second sub-matrix of the co-prime matrix respectively meet central symmetry, and the method specifically comprises the following steps:

step 1, constructing a cross covariance between a first subarray and a second subarray and an autocovariance of the second subarray;

step 2, carrying out real-valued transformation on the cross covariance between the first subarray and the second subarray in the step 1 and the auto covariance of the second subarray through unitary transformation;

step 3, carrying out singular value decomposition on the real value transformation result of the cross covariance between the first subarray and the second subarray in the step 2 to obtain a noise subspace corresponding to the first subarray and a noise subspace corresponding to the second subarray;

step 4, constructing a relation matrix according to the real value transformation result of the cross covariance between the first subarray and the second subarray and the auto covariance of the second subarray in the step 2;

step 5, constructing an angle estimation function of the joint real value subspace according to the noise subspace corresponding to the first subarray, the noise subspace corresponding to the second subarray in the step 3 and the relation matrix in the step 4;

step 6, respectively obtaining a group of DOA estimation solutions corresponding to the second subarray and a group of DOA estimation solutions corresponding to the first subarray by utilizing the root solving mode, the solution uniform distribution and the relation matrix;

and 7, taking the average of two solutions closest to each other in the two groups of DOA estimation solutions obtained in the step 6 as a final DOA estimation by utilizing the mutual characteristics of the first subarray and the second subarray.

2. The joint real-valued subspace-based angle of arrival estimation method of claim 1 in the co-prime matrix, wherein in step 1, the cross-covariance between the first and second sub-matrices isThe second subarray has an autocovariance ofWherein A is₁＝[a₁(α₁),...,a₁(α_K)]、A₂＝[a₂(β₁),...,a₂(β_K)]Respectively, the direction matrixes of the first and second sub-matrixes, K is the number of different directions of the far-field signals existing in the space, α_k＝Nπsinθ_k，β_k＝Mπsinθ_km, N are the numbers of elements of the first and second sub-arrays, M and N are prime integers, theta_kDOA, K-1, 2, …, K,in the form of a diagonal matrix,representing the energy of the k-th signal, σ²As noise energy, I_NIs an N × N identity matrix.

3. The joint real-valued subspace-based angle of arrival estimation method of claim 2, wherein in step 2, the real-valued transform result of the cross-covariance between the first and second sub-arrays is R_r＝H₁R_sH₂ ^HThe real value transformation result of the autocovariance of the second subarray is R_2r＝H₂R_sH₂ ^HWherein are respectively A₁、A₂Result of unitary transformation of (1), Q_MAnd Q_NRepresenting unitary matrices of order M and order N, respectively.

4. The method of claim 3, wherein the unitary matrices in even and odd dimensions are defined as:andwherein, I_pRepresenting a p x p identity matrix, Π_pDenotes an inverse identity matrix of p × p, p being a natural number.

5. The method for estimating an angle of arrival based on a joint real-valued subspace in a co-prime matrix according to claim 3, wherein step 3 specifically comprises:

R_ris decomposed into R_r＝UΛV^TWherein U, Lambda and V are respectively a left singular vector, a singular value and a right singular vector;

noise subspace U corresponding to the first subarray_nThe left singular vector corresponding to the minimum M-K singular values is formed;

noise subspace V corresponding to the second subarray_nConsisting of right singular vectors corresponding to the smallest N-K singular values.

6. The method of claim 5, wherein the relationship matrix G-R in step 4 is a co-prime matrix_rR_2r ⁺＝H₁R_sH₂ ^H(H₂R_sH₂ ^H)⁺＝H₁H₂ ⁺Wherein, wherein⁺Representing the generalized inverse of the matrix.

7. The method of claim 6, wherein the angle of arrival estimation method based on the joint real-valued subspace in the co-prime matrix is characterized in that the angle estimation function of the joint real-valued subspace in step 5 is

8. The method for estimating an angle of arrival based on a joint real-valued subspace in a co-prime matrix according to claim 7, wherein step 6 specifically comprises:

solving an angle estimation function of the joint real-valued subspace by using a root solving method, specifically comprising the following steps:

byIs a vandermonde vector, let z equal e^jβThe angle estimation function of the joint real-valued subspace is rewritten asa₂(z)＝[1,z,…,z^N-1]^T，Is the union of two noise subspaces;

by means of root findingSolving is carried out, and K roots z which are in the unit circle and are closest to the unit circle are selected from the root finding result_k,k＝1,…,K；

Byβ_k＝Mπsinθ_kObtaining a DOA estimate from the second sub-arrayWherein, angle () represents the phase;

obtaining a group of DOA estimation solutions corresponding to the second subarray according to the uniform distribution of all M solutions at intervals of 2/MWherein,u is integer varying with mNumber and guaranteeIn the interval [ -1,1 [)]Within;

byAnd h₁(α_k)＝Gh₂(β_k) To obtain

ByAnd isTo obtainBy a₁(α_k) The phase difference between adjacent elements can obtain a DOA estimate corresponding to the first sub-array asWherein a is_1,m(α_k) And a_1,m+1(α_k) Respectively represent a₁(α_k) The m-th and m + 1-th elements of (a);

obtaining a group of DOA estimation solutions corresponding to the first subarray according to the uniform distribution of all N solutions at intervals of 2/NWherein,v is an integer which varies with n and is guaranteedIn the interval [ -1,1 [)]Within.

9. The joint real-valued subspace-based angle of arrival estimation method in a co-prime matrix of claim 2, wherein the cross-covariance between the first and second sub-matrices and the auto-covariance of the second sub-matrix are estimated by finite snapshot numbers:

wherein T represents the number of fast beats, x₁(t)、x₂And (t) the outputs of the first subarray and the second subarray respectively.