CN107015190A - Relatively prime array Wave arrival direction estimating method based on the sparse reconstruction of virtual array covariance matrix - Google Patents
Relatively prime array Wave arrival direction estimating method based on the sparse reconstruction of virtual array covariance matrix Download PDFInfo
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Abstract
The invention discloses a kind of relatively prime array Wave arrival direction estimating method based on the sparse reconstruction of virtual array covariance matrix, the problem of free degree is limited to physical antenna element number of array in the prior art is mainly solved, implementation step is：(1) receiving terminal antenna carries out framework according to relatively prime array structure；(2) use relatively prime array received signal and model；(3) virtual signal of equal value corresponding to relatively prime array received signal is calculated；(4) virtual array covariance matrix is constructed；(5) design virtual array covariance matrix is sparse rebuilds optimization problem and solves；(6) Mutual coupling result is obtained by peak value searching.The present invention, which takes full advantage of relatively prime array, can increase the advantage of free degree performance, and the method for the sparse reconstruction of covariance matrix is introduced in virtual Domain to realize Mutual coupling, the lifting of the free degree is realized, available for passive location and target acquisition.
Description
Technical Field
The invention belongs to the technical field of signal processing, and particularly relates to direction-of-arrival estimation of radar signals, acoustic signals and electromagnetic signals, in particular to a co-prime array direction-of-arrival estimation method based on sparse reconstruction of a virtual array covariance matrix, which can be used for passive positioning and target detection.
Background
Direction-of-Arrival (DOA) estimation is an important branch of the array signal processing field, which means that an array antenna is used for receiving airspace signals, and effective processing is performed on received signal statistics through modern signal processing technology and various optimization methods to achieve DOA estimation of the signals, and the DOA estimation method has important application value in the fields of radar, sonar, voice, wireless communication and the like.
The DOA estimation method has the advantages that the degree of freedom refers to the number of incident signal sources which can be distinguished simultaneously, and the DOA estimation method is used as an important measurement index in practical system application and determines the overall complexity of the system. The existing DOA estimation method generally adopts a uniform linear array to receive and model signals, but the degree of freedom of the uniform linear array-based method is limited by the number of actual antenna elements. Specifically, for a uniform linear array comprising L antenna elements, the degree of freedom is L-1, i.e., only L-1 incident signals can be resolved at most. Therefore, when the number of incident signal sources in a certain airspace range is greater than or equal to the number of antenna array elements in the array, the existing method adopting the uniform linear array cannot carry out effective DOA estimation.
In order to increase the degree of freedom, the conventional method needs to be implemented by adding a physical antenna array element and a corresponding radio frequency module, which results in increasing the system computation complexity and hardware complexity. Therefore, the existing DOA estimation method adopting the uniform array has a certain trade-off problem between the degree of freedom performance and the computational complexity. How to improve the degree of freedom of the direction of arrival estimation method under the condition of a certain number of physical antenna array elements has important significance for improving the economy and the practicability of the method in the practical system application.
Disclosure of Invention
The invention aims to provide a method for estimating the direction of arrival of a co-prime array based on sparse reconstruction of a covariance matrix of a virtual array, aiming at the defects in the prior art.
The purpose of the invention is realized by the following technical scheme: a method for estimating the direction of arrival of a co-prime array based on sparse reconstruction of a covariance matrix of a virtual array comprises the following steps:
(1) using Q physical antenna array elements to construct a co-prime array at a receiving end, and receiving an incident signal through the co-prime array;
(2) suppose there are K from θ1,θ2,…,θKFrom a directional far-field narrow-band incoherent signal source, the Q × 1-dimensional co-prime array received signal y (t) can be modeled as:
wherein s isk(t) is a signal waveform, n (t) is a noise component, and a (theta) is independent of each signal sourcek) Is thetakA steering vector of direction, expressed as
Wherein u isqQ1, 2, …, Q representing the actual position of the qth physical antenna element in the co-prime array, and u10, λ represents the signal wavelength, [ ·]TRepresenting a transpose operation; collecting T sampling snapshots to obtain a sampling covariance matrix
This (·)HRepresents a conjugate transpose;
(3) calculating equivalent virtual signals corresponding to the co-prime array receiving signals: sampling covariance matrix vectorizing co-prime array received signalObtaining a virtual array equivalent received signal z:
wherein,is Q2× K-dimensional virtual array steering matrix, p ═ p1,p2,…,pK]TIncluding the power of K incident signal sources,for noise power, I ═ vec (I)Q). Here, vec (·) represents a vectorization operation, i.e., stacking columns in a matrix in order to form a new vector, (·)*It is meant a conjugate operation of the two,denotes the kronecker product, IQAnd representing a Q × Q-dimensional unit matrix, wherein the position of each virtual array element in the virtual array corresponding to the vector z is S:
S(i,j)＝{ui-uj|i,j＝1,2,…,Q}。
removing repeated virtual array elements at each position in the set S to obtain a non-uniform virtual array SnIts corresponding equivalent virtual signalCan be obtained by selecting the corresponding element in the vector z;
(4) constructing a virtual array covariance matrix: firstly, non-uniform virtual array S is selectednA section of virtual array elements which are continuously and uniformly arranged by taking 0 as the center form a uniform virtual array S containing L virtual array elementsuWith a corresponding virtual array element position of-Lvd to Lvd, where d is the unit interval, is taken to be half the wavelength of the incident narrowband signal, i.e. d ═ λ/2,
accordingly, the equivalent signal of the uniform virtual arrayCan be obtained by interceptionThe elements in the positions corresponding to the L virtual array elements are obtained with a dimension L × 1, then the virtual array covariance matrix RvCan be based on(L) of the structure Toeplitzv+1)×(LvThe +1) dimensional matrix yields:
wherein,the equivalent received signal corresponding to the virtual array element with the position id is represented;
(5) constructing a virtual array covariance matrix sparse reconstruction optimization problem and solving: first, the angular range of the direction of arrival angle is divided into equal intervalsA grid pointNamely, it isThen, based on the virtual array covariance matrix RvConstructed as follows as a vectorAnd noise powerFor variable optimization problems:
wherein,is composed ofA dimension virtual array steering matrix corresponding to virtual array element positions of 0 to Lvd is a continuous segment of a virtually uniform array; vector quantityComprisesThe signal power in the potential incoming wave direction corresponds to a diagonal matrix of∈ is a threshold constant used to constrain the reconstruction error of the covariance matrix;the signal power value in each direction is ensured to be larger than or equal to zero; II-0And | · |)FRespectively represent a 0 norm and an F norm,is (L)v+1)×(Lv+1) dimensional unit vector. Converting the non-convex optimization problem into a convex optimization problem and obtaining an optimal solution
(6) Obtaining a direction of arrival estimation result through spectral peak search: about the X axisThe incoming wave direction of the uniformly distributed spatial grid points, and the Y axis is an optimized valueThe elements contained in (1), a spatial power spectrum is drawn. And searching peaks on the spatial power spectrum, arranging the response values corresponding to the peaks from large to small, and taking the X-axis angle direction corresponding to the first K peaks, namely the estimation result of the direction of arrival.
Further, the relatively prime array structure in step (1) can be described as: firstly, selecting a pair of relatively prime integers M, N; then, a pair of sparse uniform linear sub-arrays is constructed, wherein the first sub-array comprises M antenna elements with the distance Nd and the positions of the M antenna elements are 0, Nd, …, (M-1) Nd, and the second sub-array comprises N antenna elements with the distance Md and the positions of the N antenna elements are 0, Md, …, (N-1) Md; and then, performing sub-array combination on the two sub-arrays according to a mode that the first array element is overlapped, and obtaining a non-uniform co-prime array structure actually containing Q + M-1 antenna array elements.
Further, the relatively prime array structure in step (1) can be described as: firstly, selecting a pair of relatively prime integers M, N, wherein M is less than N; then, a pair of sparse uniform linear sub-arrays is constructed, wherein the first sub-array comprises 2M antenna elements with the distance Nd and the positions of the antenna elements are 0, Nd, …, (2M-1) Nd, and the second sub-array comprises N antenna elements with the distance Md and the positions of the antenna elements are 0, Md, …, (N-1) Md; and then, performing sub-array combination on the two sub-arrays according to a mode that the first array element is overlapped, and obtaining a non-uniform co-prime array structure actually containing 2M + N-1 antenna array elements.
Further, the virtual array covariance matrix R in step (4)vCan be equivalently obtained by the following method:
further, the virtual array covariance matrix R in step (4)vCan be equivalently obtained by the following method: first, vector is calculatedIs divided into Lv+1 dimension of (L)v+1) × 1, each subvector comprising a vectorI th to i + L th ofvThe elements, namely:
then
Whereby RvBy finding the fourth order statistic in the above equationThe matrix of (a) is squared.
Further, the non-convex optimization problem constructed in step (5) can replace the 0 norm in the optimization problem with the 1 norm through a convex relaxation technology, and the following vector is obtainedAnd noise powerConvex optimization problem for variables:
wherein | · |)1Representing a 1 norm.
Further, the non-convex optimization problem constructed in step (5) can be transformed into a vector as followsAnd noise powerThe optimization problem of the basis pursuit denoising for the variable is as follows:
ξ is regularization parameter for weighting the reconstruction error of the covariance matrix of the virtual array and the vectorSparsity of (a).
Compared with the prior art, the invention has the following advantages:
(1) the invention fully utilizes the co-prime characteristic of the co-prime array and realizes the construction of the covariance matrix of the virtual array by deducing the equivalent received signal of the virtual array. Because the number of virtual array elements contained in the virtual array is greater than that of the actual physical antenna array elements, signal processing based on the virtual domain lays a foundation for improving the performance of the degree of freedom;
(2) the method considers the characteristic that the signal presents sparsity in the space domain range, constructs the covariance matrix sparse reconstruction optimization problem on the virtual domain, and realizes effective DOA estimation under the condition that the number of signal sources is more than or equal to the number of physical antenna array elements;
(3) compared with the existing method adopting uniform arrays, the method provided by the invention has the advantages of effectively reducing the number of physical antenna array elements and radio frequency module units in an actual system in the aspects of degree of freedom, array aperture and the like, and has ideal economical efficiency and practicability in practical application.
Drawings
FIG. 1 is a block diagram of the overall flow of the method of the present invention.
Fig. 2 is a schematic structural diagram of a pair of sparse uniform sub-arrays constituting a first kind of co-prime array according to the present invention.
FIG. 3 is a schematic diagram of a first type of relatively prime array according to the present invention.
FIG. 4 is a schematic diagram of a pair of sparse uniform subarrays constituting a second type of co-prime array according to the present invention.
FIG. 5 is a schematic diagram of a second type of relatively prime array according to the present invention.
FIG. 6 is a schematic diagram of a spatial power spectrum reconstructed by using a first kind of co-prime array in the present invention.
FIG. 7 is a schematic diagram of a spatial power spectrum reconstructed by a second kind of co-prime array according to the present invention.
Detailed Description
The technical means and effects of the present invention will be described in further detail below with reference to the accompanying drawings.
For the application of the DOA estimation method in an actual system, it is often desirable to estimate more incident signal sources by using less antenna devices, but the DOA estimation method is limited by factors such as an antenna array structure and the number of array elements, and the existing method cannot achieve optimization in the two aspects at the same time, and a certain advantage-disadvantage trade-off relationship often exists. In order to improve the degree of freedom of the DOA estimation method, the invention provides a co-prime array direction of arrival estimation method based on sparse reconstruction of a covariance matrix of a virtual array, and referring to FIG. 1, the implementation steps of the invention are as follows:
the method comprises the following steps: and a co-prime array is constructed by using Q physical antenna elements at a receiving end. The co-prime array structure mainly includes the following two types, and both are suitable for the DOA estimation method provided by the invention.
The first type of coprime array structure is as follows: firstly, selecting a pair of relatively prime integers M, N; then, referring to fig. 2, a pair of sparse uniform linear sub-arrays is constructed, wherein the first sub-array comprises M Nd-spaced antenna elements at positions 0, Nd, …, (M-1) Nd, and the second sub-array comprises N Md-spaced antenna elements at positions 0, Md, …, (N-1) Md; the unit interval d is half of the wavelength of the incident narrow-band signal, namely d is lambda/2; and then, performing sub-array combination on the two sub-arrays according to a mode that the first array element is overlapped, and referring to fig. 3, obtaining a non-uniform co-prime array structure actually containing M + N-1 antenna array elements. In this case, the number Q of physical antenna elements is M + N-1.
The second type of relatively prime array structure is as follows: firstly, selecting a pair of relatively prime integers M, N, wherein M is less than N; then, referring to fig. 4, a pair of sparse uniform linear sub-arrays is constructed, wherein the first sub-array comprises 2M Nd-spaced antenna elements at positions 0, Nd, …, (2M-1) Nd, and the second sub-array comprises N Md-spaced antenna elements at positions 0, Md, …, (N-1) Md; and then, performing sub-array combination on the two sub-arrays according to a mode that the first array element is overlapped, and referring to fig. 5, obtaining a non-uniform co-prime array structure actually containing 2M + N-1 antenna array elements. In this case, the number Q of physical antenna elements is 2M + N-1.
Step two: and receiving signals by adopting a relatively prime array and modeling. Suppose there are K from θ1,θ2,…,θKThe far-field narrow-band incoherent signal source in the direction receives an incident signal by adopting the nonuniform co-prime array constructed in the step one to obtain a Q × 1 dimensional co-prime array received signal y (t), and can be modeled as follows:
wherein s isk(t) is a signal waveform, n (t) is a noise component independent of each signal source, and a (theta)k) Is thetakA steering vector of direction, expressed as
Wherein u isqQ1, 2, …, Q representing the actual position of the qth physical antenna element in the co-prime array, and u1＝0，[·]TRepresenting a transpose operation. Collecting T sampling snapshots to obtain a sampling covariance matrix
This (·)HRepresenting a conjugate transpose.
Step three: and calculating equivalent virtual signals corresponding to the co-prime array receiving signals. Sampling covariance matrix vectorizing co-prime array received signalObtaining a virtual array equivalent received signal z:
wherein,is Q2× K-dimensional virtual array steering matrix, p ═ p1,p2,…,pK]TIncluding the power of K incident signal sources,for noise power, I ═ vec (I)Q). Here, vec (·) represents a vectorization operation, i.e., stacking columns in a matrix in order to form a new vector, (·)*It is meant a conjugate operation of the two,denotes the kronecker product, IQAnd representing a Q × Q-dimensional unit matrix, wherein the position of each virtual array element in the virtual array corresponding to the vector z is S:
S(i,j)＝{ui-uj|i,j＝1,2,…,Q}。
removing repeated virtual array elements at each position in the set S to obtain a non-uniform virtual array SnIts corresponding equivalent virtual signalThis can be obtained by selecting the corresponding element in the vector z.
Step four: a virtual array covariance matrix is constructed. Firstly, choose the non-uniform virtual array SnA section of virtual array elements which are continuously and uniformly arranged by taking 0 as the center form a uniform virtual array S containing L virtual array elementsu(due to S)uThe virtual array elements in the array are symmetrically distributed at zero position, L is always an odd number), and the corresponding virtual array element position is-Lvd to Lvd successive positions therebetween, wherein
Accordingly, the equivalent signal of the uniform virtual arrayCan be obtained by interceptionThe elements in the positions corresponding to the L virtual array elements are obtained with a dimension L × 1, then the virtual array covariance matrix RvCan be based on(L) of the structure Toeplitzv+1)×(LvThe +1) dimensional matrix yields:
wherein,and the equivalent received signals corresponding to the virtual array elements with the positions id are represented. Due to the uniform virtual array SuThe virtual array element is symmetrical about the origin, and the statistics of the second-order equivalent received signals corresponding to the symmetrical array element are conjugate, so that RvCan also be equivalently expressed as:
furthermore, RvOr can be obtained by a spatial smoothing technique, specifically: will vectorIs divided into Lv+1 dimension of (L)v+1) × 1, each subvector comprising a vectorI th to i + L th ofvThe elements, namely:
then
Whereby RvBy finding the fourth order statistic in the above equationThe matrix of (a) is squared.
Step five: and designing and solving a virtual array covariance matrix sparse reconstruction optimization problem. Firstly, according to the sparsity distribution characteristic of the signal in the space domain range, the angular domain range of the direction of arrival angle is divided into equal intervalsA grid pointNamely, it isThen, the virtual array covariance matrix R is calculated according to the step fourvConstructed as follows as a vectorAnd noise powerFor variable optimization problems:
wherein,is composed ofA dimension virtual array steering matrix corresponding to virtual array element positions of 0 to Lvd is a continuous segment of a virtually uniform array; vector quantityComprisesThe signal power in the potential incoming wave direction corresponds to a diagonal matrix of∈ is a threshold constant used to constrain the reconstruction error of the covariance matrix;the signal power value in each direction is ensured to be larger than or equal to zero; II-0And | · |)FRespectively represent a 0 norm and an F norm,is (L)v+1)×(Lv+1) dimensional unit vector. Since the above optimization problem contains a non-convex term of 0 norm, it will cause difficulty in solving; to obtain an optimized solution, we consider introducing a convex relaxation technique to replace the 0 norm in the above optimization problem with a 1 norm, resulting in the following vectorAnd noise powerConvex optimization problem for variables:
wherein | · |)1Representing a 1 norm. The convex optimization problem described above can be written equivalently as the following vectorAnd noise powerThe optimization problem of the basis pursuit denoising for the variable is as follows:
ξ is regularization parameter for weighting the reconstruction error of the covariance matrix of the virtual array and the vectorSparsity of (a). Solving the convex optimization problem can obtain the optimized value
Step six: and obtaining a wave arrival direction estimation result through spectral peak searching. About the X axisThe incoming wave direction of the uniformly distributed spatial grid points, and the Y axis is an optimized valueThe elements contained in (1), a spatial power spectrum is drawn. And searching peaks on the spatial power spectrum, arranging the response values corresponding to the peaks from large to small, and taking the X-axis angle direction corresponding to the first K peaks, namely the estimation result of the direction of arrival.
On one hand, the method fully utilizes the advantage that the co-prime array can increase the DOA estimation method freedom degree, breaks through the bottleneck that the uniform linear array freedom degree is limited, and estimates more incident signal sources under the condition of certain number of antenna array elements by calculating the equivalent signal of the virtual array; on the other hand, the idea of sparse reconstruction of the covariance matrix is introduced and applied to a virtual domain to realize reconstruction and DOA estimation of a spatial power spectrum.
The effect of the present invention will be further described with reference to the simulation example.
Simulation example 1: the first kind of co-prime array is used to receive the incident signal, and its parameters are selected as M-3 and N-5, i.e. the co-prime array contains Q-M + N-1-7 physical antenna elements. The number of incident narrow-band signals is assumed to be 7, and the incident directions are uniformly distributed in a space angle range of-60 degrees to 60 degrees; the angular domain range of the direction of arrival angle is [ -90 degrees, 90 degrees ], and the uniform sampling interval of spatial domain grid points is set to be 0.1 degree; the regularization parameter ξ is set to 0.25; the signal-to-noise ratio is set to 0dB, and the sampling fast beat number K is 500.
The spatial power spectrum of the co-prime array direction of arrival estimation method based on the sparse reconstruction of the virtual array covariance matrix is shown in fig. 6, wherein a vertical dotted line represents the actual direction of an incident signal source. Under the parameter setting of the present example, the number of consecutive virtual array elements on the virtual array is L-15, and accordingly, L is equal to Lv7. It can be seen that the method provided by the invention can effectively distinguish the 7 incident signal sources; in addition, the response value of each signal direction on the power spectrum can reflect the power information of the actual signal source, and the direction-of-arrival information and the power information of each signal can be estimated simultaneously. Compared with the traditional method adopting a uniform linear array, the method can only distinguish 6 incident signals at most by utilizing 7 physical antenna array elements, and the result shows that the method provided by the invention is improved in the degree of freedom performance.
Simulation example 2: adopting a second type of co-prime array to receive an incident signal, wherein the parameters are selected to be M-3 and N-5, namely the co-prime array comprises 10 physical antenna elements Q-2M + N-1; assuming that the number of incident narrowband signals is 15, the remaining parameter settings are consistent with simulation example 1. In this case, the number of successive virtual array elements on the virtual array is L35, and accordingly, Lv17. As can be seen from the spatial power spectrum shown in fig. 7, the method provided by the present invention can effectively resolve the direction of arrival and angle information of 15 incident signal sources by using only 10 physical antenna elements, and thus, the advantage of the performance of the degree of freedom is embodied.
In summary, the method provided by the present invention can realize effective estimation of the incident signal under the condition that the number of signal sources is greater than or equal to the number of physical antennas, and increases the degree of freedom and the calculation efficiency. In addition, compared with the traditional method adopting a uniform linear array, the method provided by the invention can correspondingly reduce the physical antenna array elements and radio frequency modules required in the practical application system, thereby reflecting the economy and high efficiency.
Claims (7)
1. A method for estimating the direction of arrival of a co-prime array based on sparse reconstruction of a covariance matrix of a virtual array is characterized by comprising the following steps:
(1) and a receiving end uses Q physical antenna array elements to construct a co-prime array, and receives an incident signal through the co-prime array.
(2) Suppose there are K from θ1,θ2,…,θKFrom a directional far-field narrow-band incoherent signal source, the Q × 1-dimensional co-prime array received signal y (t) can be modeled as:
wherein s isk(t) is a signal waveform, n (t) is a noise component, and a (theta) is independent of each signal sourcek) Is thetakA steering vector of direction, expressed as
Wherein u isqQ1, 2, …, Q representing the actual position of the qth physical antenna element in the co-prime array, and u10, λ represents the signal wavelength, [ ·]TRepresenting a transpose operation; collecting T sampling snapshots to obtain a sampling covariance matrix
This (·)HRepresenting a conjugate transpose.
(3) Calculating equivalent virtual signals corresponding to the co-prime array receiving signals: sampling covariance matrix vectorizing co-prime array received signalObtaining a virtual array equivalent received signal z:
wherein,is Q2× K-dimensional virtual array steering matrix, p ═ p1,p2,…,pK]TIncluding the power of K incident signal sources,for noise power, I ═ vec (I)Q). Here, vec (·) represents a vectorization operation, i.e., stacking columns in a matrix in order to form a new vector, (·)*It is meant a conjugate operation of the two,denotes the kronecker product, IQRepresenting a Q × Q-dimensional identity matrix, the position of each virtual array element in the virtual array corresponding to vector z is
Removing collectionsRepeating virtual array elements at each position to obtain a non-uniform virtual arrayIts corresponding equivalent virtual signalThis can be obtained by selecting the corresponding element in the vector z.
(4) Constructing a virtual array covariance matrix: firstly, non-uniform virtual array is selectedA section of virtual array elements which are continuously and uniformly arranged by taking 0 as the center form a uniform virtual array comprising L virtual array elementsIts corresponding virtual array element position is-Lvd to Lvd are connected with each otherA continuous position, where d is a unit interval, and is taken as half the wavelength of the incident narrowband signal, i.e. d ═ λ/2,
accordingly, the equivalent signal of the uniform virtual arrayCan be obtained by interceptionThe elements in the positions corresponding to the L virtual array elements are obtained with a dimension L × 1, then the virtual array covariance matrix RvCan be based on(L) of the structure Toeplitzv+1)×(LvThe +1) dimensional matrix yields:
wherein,and the equivalent received signals corresponding to the virtual array elements with the positions id are represented.
(5) Constructing a virtual array covariance matrix sparse reconstruction optimization problem and solving: first, the angular range of the direction of arrival angle is divided into equal intervalsA grid pointNamely, it isThen, based on the virtual array covariance matrix RvConstructed as follows as a vectorAnd noise powerFor variable optimization problems:
wherein,is composed ofA dimension virtual array steering matrix corresponding to virtual array element positions of 0 to Lvd is a continuous segment of a virtually uniform array; vector quantityComprisesThe signal power in the potential incoming wave direction corresponds to a diagonal matrix of∈ is a threshold constant used to constrain the reconstruction error of the covariance matrix;the signal power value in each direction is ensured to be larger than or equal to zero; II-0And | · |)FRespectively represent a 0 norm and an F norm,is (L)v+1)×(Lv+1) dimensional unit vector. Converting the non-convex optimization problem into a convex optimization problem and obtaining an optimal solution
(6) Obtaining a direction of arrival estimation result through spectral peak search: about the X axisThe incoming wave direction of the uniformly distributed spatial grid points, and the Y axis is an optimized valueThe elements contained in (1), a spatial power spectrum is drawn. And searching peaks on the spatial power spectrum, arranging the response values corresponding to the peaks from large to small, and taking the X-axis angle direction corresponding to the first K peaks, namely the estimation result of the direction of arrival.
2. The method for estimating the direction of arrival of a co-prime array based on sparse reconstruction of a covariance matrix of a virtual array according to claim 1, wherein: the coprime array structure described in step 1 can be described as follows: firstly, selecting a pair of relatively prime integers M, N; then, a pair of sparse uniform linear sub-arrays is constructed, wherein the first sub-array comprises M antenna elements with the distance Nd and the positions of the M antenna elements are 0, Nd, …, (M-1) Nd, and the second sub-array comprises N antenna elements with the distance Md and the positions of the N antenna elements are 0, Md, …, (N-1) Md; and then, performing sub-array combination on the two sub-arrays according to a mode that the first array element is overlapped, and obtaining a non-uniform co-prime array structure actually containing Q + M-1 antenna array elements.
3. The method for estimating the direction of arrival of a co-prime array based on sparse reconstruction of a covariance matrix of a virtual array according to claim 1, wherein: the coprime array structure described in step 1 can be described as follows: firstly, selecting a pair of relatively prime integers M, N, wherein M is less than N; then, a pair of sparse uniform linear sub-arrays is constructed, wherein the first sub-array comprises 2M antenna elements with the distance Nd and the positions of the antenna elements are 0, Nd, …, (2M-1) Nd, and the second sub-array comprises N antenna elements with the distance Md and the positions of the antenna elements are 0, Md, …, (N-1) Md; and then, performing sub-array combination on the two sub-arrays according to a mode that the first array element is overlapped, and obtaining a non-uniform co-prime array structure actually containing 2M + N-1 antenna array elements.
4. The method for estimating the direction of arrival of a co-prime array based on sparse reconstruction of a covariance matrix of a virtual array according to claim 1, wherein: step 4, the virtual array covariance matrix RvCan be equivalently obtained by the following method:
5. the method for estimating the direction of arrival of a co-prime array based on sparse reconstruction of a covariance matrix of a virtual array according to claim 1, wherein: step 4, the virtual array covariance matrix RvCan be equivalently obtained by the following method: first, vector is calculatedIs divided into Lv+1 dimension of (L)v+1) × 1, each subvector comprising a vectorI th to i + L th ofvThe elements, namely:
then
Whereby RvBy finding the fourth order statistic in the above equationThe matrix of (a) is squared.
6. The method for estimating the direction of arrival of a co-prime array based on sparse reconstruction of a covariance matrix of a virtual array according to claim 1, wherein: the non-convex optimization problem constructed in the step 5 can replace 0 norm in the optimization problem with 1 norm through a convex relaxation technology to obtain the following vectorAnd noise powerConvex optimization problem for variables:
wherein | · |)1Representing a 1 norm.
7. The method for estimating the direction of arrival of a co-prime array based on sparse reconstruction of a covariance matrix of a virtual array according to claim 1, wherein: the non-convex optimization problem constructed in step 5 can be transformed into a vector as followsAnd noise powerThe optimization problem of the basis pursuit denoising for the variable is as follows:
ξ is regularization parameter for weighting the reconstruction error of the covariance matrix of the virtual array and the vectorSparsity of (a).
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