CN108896954A - A kind of direction of arrival estimation method based on joint real value subspace in relatively prime battle array - Google Patents
A kind of direction of arrival estimation method based on joint real value subspace in relatively prime battle array Download PDFInfo
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Abstract
The invention discloses direction of arrival (Direction of arrival, DOA) estimation methods based on joint real value subspace a kind of in relatively prime battle array, have carried out the improvement of array structure to substantially relatively prime battle array first, make symmetrical centered on it；Two sub- covariances are constructed by two submatrixs later, and further obtain the relational matrix between submatrix；Based on the relational matrix, tectonic syntaxis subspace DOA estimate function；It is finally obtained from the intersection of the output of two submatrixs without fuzzy DOA estimation.Due to the presence of relational matrix, the output of two submatrixs is automatic matching, and we have combined two sub-spaces, so estimation performance is guaranteed.Compared to advanced algorithm instantly, it is proposed that method because real-valued is only needed to decompose and the sub- covariance of low dimensional, complexity substantially reduces, but can be kept approximately constant in DOA estimation aspect of performance.Therefore the present invention can substantially reduce complexity in the positioning system in radar, wireless communication, save hardware cost, improve response speed.
Description
Technical Field
The invention relates to a joint real-value subspace-based angle of arrival estimation method in a co-prime matrix, belonging to array signal processing.
Background
Direction of arrival (DOA) estimation is an important research Direction for array signal processing, and is widely used in many fields such as radar and mobile communication. Many classical super-resolution DOA Estimation methods have been proposed, such as a Multiple signal classification (MUSIC) method, an Estimation of parameters by rotation invariant technology (ESPRIT) method, a propagation operator method (PM) and a rotation invariant PM method. However, to avoid the estimation ambiguity problem, these methods all use compact arrays with array element spacing no greater than half a wavelength, limited DOF results in limited DOA estimation resolution, and interference from mutual coupling between antennas is present.
As a non-uniform sparse array, the nested array can obtain degree of freedom (DOF) far greater than the actual physical array element number in a virtual array domain, so that the system capacity and the spatial resolution are increased, however, a compact sub-array still exists in the nested array, the closer antenna spacing is still influenced by the mutual coupling between the antennas, and the DOA estimation performance is reduced. Another non-uniform sparse array that is currently more popular is a co-prime array, which consists of two uniform linear sub-arrays, where the number of array elements and the spacing of the array elements from each other are mutually prime. The co-prime matrix can also obtain a larger DOF in a virtual domain, and compared with a nested matrix, the array elements of the co-prime matrix are more sparsely distributed, so that the co-prime matrix has stronger robustness to the cross coupling problem. In order to solve the problem of DOA estimation ambiguity caused by large array element spacing, a combined MUSIC method in a document 'DOA estimation with combined MUSIC for coprimetry' determines the DOA estimation without ambiguity from the intersection of two sub-array MUSIC spectral peaks based on the co-prime between the sub-arrays of the co-prime array, but the DOA estimation needs two times of global spectral peak search and has high complexity. In order to reduce complexity, a local search MUSIC method in a document 'Partial spectral search-based DOA estimation method for co-prime linear rrarys' utilizes the uniformity of a subarray, converts an angle domain estimation problem into a phase domain, so that all MUSIC spectral peaks can be recovered by local search, and then ambiguity is resolved according to mutual reciprocity. However, both joint-MUSIC and local-search-MUSIC process the sub-arrays separately, so that when the final intersection solution is ambiguous, there is interference between multiple DOA estimates (including true solution and ambiguous solution), and especially error is prone at low snr. In order to avoid this problem, the document "Improved DOA estimation algorithm for co-primary linear estimation using root-MUSIC algorithm" applies an Improved root-finding MUSIC method (Li J, Jiang d. joint elevation and azimuths estimation for L-shaped array) to the co-prime array, first constructs a relationship matrix of two sub-arrays through the signal subspace of the whole array, and then solves DOA estimation by using a root-finding method based on the noise subspace of the whole array. However, the eigen decomposition of the ensemble of arrays and the estimation function based on the ensemble of noise subspaces both involve a high complexity.
Disclosure of Invention
The invention aims to provide a DOA estimation method based on a joint real-valued subspace in a co-prime matrix. Firstly, the manifold of the array of the co-prime matrix is improved, so that the two sub-matrices are centrosymmetric, and the covariance can be changed into a real value through unitary transformation, thereby greatly reducing the complexity. In addition, the covariance of the whole array is not needed, and the relation matrix between two sub-arrays can be obtained only by the sub-covariance of two real values, so that additional pairing is avoided, and the complexity is further reduced. And finally constructing a DOA estimation cost function of a joint subspace based on the noise subspaces of the two subarrays. Compared with the improved root-finding MUSIC method, the complexity of the algorithm is greatly reduced, but the DOA estimation performance can be almost kept unchanged.
The invention adopts the following technical scheme for solving the technical problems:
the invention provides a method for estimating an angle of arrival based on a joint real-valued subspace in a co-prime matrix, wherein the structures of a first sub-matrix and a second sub-matrix of the co-prime matrix meet central symmetry, and the method comprises the following specific 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.
As a further technical scheme of the invention, in step 1, the cross covariance between the first and second subarrays isThe second subarray has an autocovariance ofWherein A is1＝[a1(α1),...,a1(αK)]、A2＝[a2(β1),...,a2(βK)]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,αk＝Nπsinθk，βk＝Mπsinθkm, N are the numbers of elements of the first and second sub-arrays, M and N are prime integers, thetakDOA, K-1, 2, …, K,is a diagonal angleThe matrix is a matrix of a plurality of matrices,representing the energy of the k-th signal, σ2As noise energy, INIs an N × N identity matrix.
As a further technical scheme of the invention, in step 2, the real value transformation result of the cross covariance between the first subarray and the second subarray is Rr＝H1RsH2 HThe real value transformation result of the autocovariance of the second subarray is R2r＝H2RsH2 HWhereinare respectively A1、A2Result of unitary transformation of (1), QMAnd QNRepresenting unitary matrices of order M and order N, respectively.
As a further technical solution of the present invention, unitary matrices in even and odd dimensions are defined as:andwherein, IpRepresenting a p x p identity matrix, ΠpDenotes an inverse identity matrix of p × p, p being a natural number.
As a further technical solution of the present invention, step 3 specifically is:
Rris decomposed into Rr＝UΛVTWherein 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 subarraynThe left singular vector corresponding to the minimum M-K singular values is formed;
noise subspace V corresponding to the second subarraynConstructed from right singular vectors corresponding to minimum N-K singular valuesAnd (4) obtaining.
As a further technical solution of the present invention, in step 4, the relationship matrix G ═ RrR2r +＝H1RsH2 H(H2RsH2 H)+＝H1H2 +Wherein, wherein+Representing the generalized inverse of the matrix.
As a further technical scheme of the invention, the angle estimation function of the joint real-valued subspace in the step 5 is
As a further technical solution of the present invention, step 6 specifically is:
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 ejβThe angle estimation function of the joint real-valued subspace is rewritten asa2(z)＝[1,z,…,zN-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 resultk,k＝1,…,K；
Byβk＝MπsinθkObtaining 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 an integer which varies with m and is guaranteedIn the interval [ -1,1 [)]Within;
byAnd h1(αk)＝Gh2(βk) To obtain
ByAnd isTo obtainBy a1(αk) The phase difference between adjacent elements can obtain a DOA estimate corresponding to the first sub-array asWherein a is1,m(αk) And a1,m+1(αk) Respectively represent a1(αk) 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.
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 by a finite number of snapshots:
wherein T represents the number of fast beats, x1(t)、x2And (t) the outputs of the first subarray and the second subarray respectively.
Compared with the prior art, the invention adopting the technical scheme has the following technical effects:
1. the improvement of the coprime array structure leads the coprime array structure to be centrosymmetric;
2. based on a symmetrical array structure, the sub-covariance is converted into real values, so that the operation complexity is reduced;
3. the characteristic decomposition and noise space of the whole array are not needed, the relation matrix can be obtained only through the sub-covariance of the real values, additional pairing is avoided, and the complexity is further reduced;
4. finally, constructing a DOA estimation function based on real-value noise subspaces of the two sub-arrays, obtaining DOA estimation information from the two sub-arrays, and performing ambiguity resolution based on the mutual prime;
5. compared with the current advanced algorithm, the proposed algorithm can greatly reduce the complexity, but the DOA estimation performance is almost unchanged.
Drawings
FIG. 1 is a schematic diagram of the basic co-prime array and the improved co-prime array structure;
FIG. 2 is a schematic flow diagram of the method of the present invention;
FIG. 3 is a schematic diagram of the comparison of the algorithm complexity of the improved root MUSIC method and the method of the present invention;
FIG. 4 is the DOA estimation result of the proposed method in the first 100 experiments;
FIG. 5 is a graph illustrating the comparison between the estimation performance of the method of the present invention and the improved root MUSIC algorithm.
Detailed Description
The technical scheme of the invention is further explained in detail by combining the attached drawings:
the invention provides a Direction of arrival (DOA) estimation method based on a joint real-valued subspace in a co-prime matrix. Firstly, the basic co-prime matrix is improved to be centrosymmetric, so that the covariance of the matrix can be changed into a real value through unitary transformation, and the operation complexity is greatly reduced. And then constructing two sub-covariances through the two sub-arrays, and further obtaining a relation matrix between the sub-arrays. Based on the relationship matrix, a joint subspace angle of arrival estimation function is constructed. Finally, a DOA estimate is obtained from the intersection of the outputs of the two sub-arrays without ambiguity. Due to the existence of the relation matrix, the output of the two sub-arrays is automatically paired, and the two subspaces are combined, so that the estimation performance is guaranteed. Compared with the current advanced algorithm, the method proposed by the invention greatly reduces the complexity because only real-value feature decomposition and low-dimensional sub-covariance are needed, but can be almost unchanged in DOA estimation performance. Therefore, the invention can greatly reduce the complexity, save the hardware cost and improve the response speed in positioning systems in radar and wireless communication.
The present co-prime array and the improved co-prime array structure shown in fig. 1, the upper half of fig. 1 shows the array structure of the basic co-prime array, which is composed of two sparse Uniform Linear Arrays (ULA), the sub-array 1 has M array elements with an array element spacing Nd, and the sub-array 2 has N array elements with an array element spacing Md, d being a unit spacing, generally set to be a half wavelength, and M and N being co-prime integers. The array structure does not satisfy central symmetry, so real-valued transformation cannot be performed. We therefore show an improved co-prime array structure in the lower half of fig. 1, so that the two sub-arrays are now symmetric about the Z-axis (it should be noted that fig. 1 shows only an example, and it is only necessary to ensure that the two sub-arrays are symmetric about the Z-axis in practical application of the method of the present invention). Assuming that there are far-field signals from K different directions in the space at this time, the outputs of the two sub-arrays are
x1(t)＝A1s(t)+n1(t) (1)
x2(t)＝A2s(t)+n2(t) (2)
Wherein s (t) ═ s1(t),...,sK(t)]TRepresenting K source baseband signals; n is1(t) and n2(t) represents additive white gaussian noise on the respective arrays; a. the1＝[a1(α1),...,a1(αK)]And A2＝[a2(β1),...,a2(βK)]The direction matrices representing sub-array 1 and sub-array 2, respectively, the columns of which are called steering vectors and are represented as
α thereink＝Nπsinθk,βk＝Mπsinθk,θkIndicating the DOA of the k-th signal.
Real value transformation
First, according to (1) - (2), the cross covariance between two sub-arrays is constructed as
WhereinThe diagonal matrix contains K signal energies (Gu J F, Wei P. Joint SVDof Two Cross-Correlation matrix to arrival Automatic Pairing in 2-D angle estimation schemes). Since the two sub-matrices are now centrosymmetric, we can use unitary transform for real-valued transform, thereby reducing complexity. Unitary matrices in odd and even dimensions are defined as follows:
using the steering vector of sub-matrix 1 as an example, the unitary transformation is then transformed into a real steering vector (assuming M is odd)
The corresponding real-valued direction matrix isSimilarly, the directional unitary transformation of sub-matrix 2 becomesSo that the cross-covariance in equation (5) becomes
In practice, to ensure the real value of the cross covariance, we can take the real part of the above equation, i.e. the real part
Based on the cross-covariance in equation (8), its Singular Value Decomposition (SVD) is
Rr＝UΛVT(9)
Wherein U, Λ and V represent the left singular vector, the singular value and the right singular vector, respectively. The left singular vector corresponding to the minimum (M-K) singular values constitutes the noise subspace U of subarray 1nAnd the right singular vector corresponding to the minimum (N-K) singular values constitutes the noise subspace V corresponding to sub-array 2n. The angular information of the two sub-arrays can be obtained from the following cost function
Un Hh1(α)＝0 (10)
Vn Hh2(β)＝0 (11)
However, it should be noted that α and β are obtained separately, which brings about two problems, i.e. 1, α and β are out of order, which requires an additional pairing step and thus increases complexity, and 2, α and β are obtained based on their respective sub-arrays, which is equivalent to using only part of the array aperture, and thus the performance of the estimation needs to be improved.
Relationship matrix construction
Constructing subarray 2 with an autocovariance of
Wherein sigma2For noise energy, it is also unitary transformed and noise eliminated
Wherein sigma2Can be selected fromAverage of the N-K smaller eigenvalues of ([16 ]]Yunhe C.J. estimation of angle and Doppler frequency for dual MIMO radar) it is noted here that the unitary matrix satisfies orthogonality, i.e. it isThe unitary transform does not have an impact on the noise energy. According to the expressions (8) and (13), a relationship matrix can be constructed as
G＝RrR2r +＝H1RsH2 H(H2RsH2 H)+＝H1H2 +(14)
The G matrix then gives the relation between the two sub-matrix direction matrices, i.e.
H1＝GH2(15)
It should be noted that the construction of the G matrix is based on the sub-covariances in the equations (8) and (13), and the construction complexity is greatly reduced compared with the covariances of the entire array.
Joint subspace-based angle of arrival estimation
Based on the G matrix, h1(α)＝Gh2(β) substituting equation (10), the angle estimation function based on the joint real-valued subspace can be constructed as
(16) The equation can be solved in a root-finding manner becauseWhereinIs a vandermonde vector. Let z be ejβThen, thenWherein a is2(z)＝[1,z,…,zN-1]TThen equation (16) can be written as
WhereinIs the union of two noise subspaces. Based on the polynomial root finding of equation (17), we select the K roots z in the unit circle and closest to the unit circlek,k＝1,…,K。
Because of the fact thatAnd βk＝MπsinθkSo the DOA estimate obtained from subarray 2 (i.e., β) is
From the above equation, it can be seen that the sub-array 2 has a larger array element spacing (M)>1)，zkWill exceed [ -pi, pi]So there is a phase ambiguity, we are based on zkThe solution obtained from equation (18) is only one of the fuzzy solutions, and there are (M-1) solutionsThe M solutions should satisfySo the relationship between them ([11 ]]SUNF, LAN P and GAO B.partial spectral search-based DOA estimation method for co-primary linear arrays)
Where u is an integer which varies with m and is guaranteedIn the interval [ -1,1 [)]Within. Therefore, all M solutions have a uniformly distributed relationship and a distance of 2/M, and according to the relationship and one solution in the formula (18), we can recover all M solutions.
We now obtain a set of solutions from subarray 2Next, we need to consider the DOA estimate from sub-array 1. Because of the fact thatAndand because the relation h is satisfied between the guide vectors1(αk)＝Gh2(βk) Thus obtaining
WhileAnd isThen
ThereinBecause of the scalar quantity, it can be eliminated by normalization. By a1(αk) The phase difference between adjacent elements can obtain the angle estimation of the corresponding subarray 1
Similarly, the above equation is a fuzzy solution, and a set of solutions can be obtained based on the uniform distribution of all N solutions at intervals of 2/N, and one of the solutions obtained by equation (22)
We now obtain two sets of solutions for DOA estimation. These two sets of solutions contain many erroneous solutions but also true DOA estimates. According to ZHOU C, SHI Z, GU Y, et al DECOM: DOA animation with combined music for coprime array, due to the co-prime in the co-prime array, unambiguous DOA can be obtained from the common solution of the two solutions, i.e. the intersection.
However, in practice, the covariance in equations (5) and (12) can only be estimated by a limited number of snapshots, i.e.
Where T represents the number of fast beats. Due to the effect of residual noise, there is often no completely coincident solution in the final two sets of solutions, so we can take the average of the two solutions that are closest in the two sets of solutions as the final unambiguous DOA estimate. Because the solutions obtained for (18) and (22) are both based on zkK, so are auto-paired, i.e., both sets of solutions are for the kth source. This is so that the two closest solutions are solved without being affected by the DOAs from the rest.
The invention relates to a method for estimating DOA (Direction of arrival, DOA) based on a joint real-valued subspace in a co-prime matrix, which comprises the following steps of:
1. according to the formulas (23) - (24), two sub-covariances R are constructedcAnd R2；
2. Based on the formula (8) and the formula (13), the two sub-covariances are subjected to real value transformation to obtain RrAnd R2rAnd according to the formula (9) to RrSingular value decomposition is carried out to obtain a noise subspace U corresponding to the subarray 1nAnd noise subspace V of corresponding subarray 2n；
3. By G ═ RrR2r +Obtaining a relation matrix G;
4. root of herbaceous plantAccording to Un，VnAnd the relation matrix G obtains a joint subspace angle estimation function f (β) in the formula (16);
5. obtaining a DOA estimate for the corresponding subarray 2 using a root-finding approach, and obtaining a set of solutions based on equation (19) using the uniform distribution of the solutions
6. Obtaining a group of DOA estimation solutions corresponding to the subarray 1 by using the relation matrix in the same wayAnd finally, using the reciprocity as the final DOA estimation from the average of the two solutions which are closest in the two groups of solutions.
The algorithm does not need additional pairing and spectral peak search, and the computational complexity of the algorithm mainly focuses on the construction of the sub-covariance and the solving of the relation matrix and the final root solving. It should be noted that the sub-covariances in the algorithm are all real values, so the computational complexity is only 1/4 for the corresponding complex matrix operation. Eventually, the total number of complex multiplications is required to be about O (MNT + N)2T+0.25*(N3+2MN2) + N) times. The improved root-finding MUSIC method requires the multiplication times of O ((M + N)2T+(M+N)3+MNK+2N2K+K3+N)。
In fig. 3, we show a comparison of two algorithm complexities under a typical parameter configuration, where M is 4, N is 3, K is 2, and T is 100. From fig. 3, it can be seen that the complexity of the algorithm we propose is only 40% of the improved root-finding MUSIC algorithm.
Simulation of experiment
In the experiment, we adopted the proposed improved co-prime array structure, as shown in fig. 1, where M is 4, N is 3, and K is 2 sources, whose DOAs are θ respectively120 ° and θ235 ° is set. Each covariance matrix is estimated by T-100 snapshots. To measure the DOA estimation performance, a Root Mean Square Error (RMSE) is defined as
WhereinIs DOA theta at the first Monte Carlo experimentkIn total, we performed 500 experiments.
Fig. 4 shows the DOA estimation result of the proposed algorithm in the previous 100 experiments, and it can be found that the proposed algorithm can always effectively and accurately estimate the DOA of two information sources.
Fig. 5 shows a comparison between the estimated performance of the proposed algorithm and the improved root MUSIC algorithm, where the vertical axis is RMSE and the horizontal axis is SNR, which can be found to be almost coincident, which shows that the proposed algorithm has almost the same performance as the improved root MUSIC algorithm, but the complexity of the proposed algorithm is greatly reduced.
A DOA estimation algorithm based on a joint real-valued subspace in a co-prime matrix is proposed. The key work we do can be summarized as follows:
1. the improvement of the coprime array structure leads the coprime array structure to be centrosymmetric;
2. based on a symmetrical array structure, the sub-covariance is converted into real values, so that the operation complexity is reduced;
3. the characteristic decomposition and noise space of the whole array are not needed, and a relation matrix can be obtained only through the sub-covariance of the real values;
4. finally, constructing a DOA estimation function based on real-value noise subspaces of the two sub-arrays, obtaining DOA estimation information from the two sub-arrays, and performing ambiguity resolution based on the mutual prime;
5. compared with the current advanced algorithm, the proposed algorithm can greatly reduce the complexity, but the DOA estimation performance is almost unchanged.
The above description is only an embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art can understand that the modifications or substitutions within the technical scope of the present invention are included in the scope of the present invention, and therefore, the scope of the present invention should be subject to 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(αK)]、A2＝[a2(β1),...,a2(βK)]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, αk＝Nπsinθk，βk＝Mπsinθkm, N are the numbers of elements of the first and second sub-arrays, M and N are prime integers, thetakDOA, K-1, 2, …, K,in the form of a diagonal matrix,representing the energy of the k-th signal, σ2As noise energy, INIs 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 Rr＝H1RsH2 HThe real value transformation result of the autocovariance of the second subarray is R2r＝H2RsH2 HWherein are respectively A1、A2Result of unitary transformation of (1), QMAnd QNRepresenting 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, IpRepresenting a p x p identity matrix, ΠpDenotes 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:
Rris decomposed into Rr＝UΛVTWherein 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 subarraynThe left singular vector corresponding to the minimum M-K singular values is formed;
noise subspace V corresponding to the second subarraynConsisting 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 matrixrR2r +＝H1RsH2 H(H2RsH2 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 ejβThe angle estimation function of the joint real-valued subspace is rewritten asa2(z)＝[1,z,…,zN-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 resultk,k＝1,…,K；
Byβk＝MπsinθkObtaining 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(αk)＝Gh2(βk) To obtain
ByAnd isTo obtainBy a1(αk) The phase difference between adjacent elements can obtain a DOA estimate corresponding to the first sub-array asWherein a is1,m(αk) And a1,m+1(αk) Respectively represent a1(αk) 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.
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