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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- G—PHYSICS
- G01—MEASURING; TESTING
- G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
- G01S3/00—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received
- G01S3/02—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received using radio waves
- G01S3/14—Systems for determining direction or deviation from predetermined direction
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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 signal processing technology field, more particularly to the ripple of radar signal, acoustic signal and electromagnetic signal
It is specifically the relatively prime array Wave arrival direction estimating method based on the sparse reconstruction of virtual array covariance matrix up to direction estimation, can
For passive location and target acquisition.
Background technology
Direction of arrival (Direction-of-Arrival, DOA) estimation is one important point of array signal processing field
Branch, it refers to utilize array antenna received spatial domain signal, and passes through modern signal processing technology and the realization pair of all kinds of optimization methods
Receive signal statistics amount to be effectively treated, estimated with the DOA for realizing signal, in the neck such as radar, sonar, voice, radio communication
There is important application value in domain.
The free degree of DOA estimation method refers to that it can be while the number for the incident signal source differentiated, be used as real system
An important measurement index in, decides the overall complexity of system.Existing DOA estimation method is generally using uniform
Linear array carries out the reception and modeling of signal, but the free degree based on uniform linear array method is by actual antennas array element
Number limitation.Specifically, for a uniform linear array for including L bay, its free degree is L-1, i.e., most
L-1 incoming signal can only be differentiated more.Therefore, the number of incident signal source is more than or equal in array in the range of some spatial domain
During the number of bay, the method for existing use uniform linear array will be unable to carry out effective DOA estimations.
In order to increase the free degree, conventional method is needed by increasing physical antenna array element and corresponding radio-frequency module come real
Existing, which results in the increase of system-computed complexity and hardware complexity.Therefore, the DOA estimation sides of existing use uniform array
Method has certain benefit-risk balance between free degree performance and computation complexity.How in physical antenna element number of array
The free degree of Wave arrival direction estimating method is lifted in the case of certain, for economy of the method for improving in real system application
Had great significance with practicality.
The content of the invention
It is an object of the invention to the deficiency existed for above-mentioned prior art, propose a kind of based on virtual array covariance
The relatively prime array Wave arrival direction estimating method of the sparse reconstruction of matrix, derives virtual Domain equivalence by using the characteristic of relatively prime array and connects
The collection of letters number, and the method rebuild by virtual array covariance matrix rebuilds space power spectrum, to lift the freedom of method of estimation
Degree, realizes effective DOA estimations in the case of signal source number is more than or equal to physical antenna element number of array.
The purpose of the present invention is achieved through the following technical solutions：Based on the sparse reconstruction of virtual array covariance matrix
Relatively prime array Wave arrival direction estimating method, the method includes the steps of：
(1) in receiving terminal using the Q relatively prime array of physical antenna array element framework, and believed by the incidence of relatively prime array received
Number；
(2) assume there are K to come from θ1,θ2,…,θKThe far field arrowband incoherent signal source in direction, the then relatively prime array of the dimension of Q × 1
Receiving signal y (t) can be modeled as：
Wherein, sk(t) it is signal waveform, n (t) is noise component(s), a (θ separate with each signal sourcek) it is θkDirection
Steering vector, is expressed as
Wherein, uq, q=1,2 ..., Q represents the physical location of q-th of physical antenna array element in relatively prime array, and u1=0, λ
Represent signal wavelength, []TRepresent transposition operation；T sampling snap is gathered altogether, obtains sample covariance matrix
Here ()HRepresent conjugate transposition；
(3) virtual signal of equal value corresponding to relatively prime array received signal is calculated：The relatively prime array received signal of vectorization
Sample covariance matrixObtain virtual array equivalence and receive signal z：
Wherein,For Q2×
K ties up virtual array guiding matrix, p=[p1,p2,…,pK]TThe power of K incident signal source is included,For noise power, i=
vec(IQ).Here, vec () represent vectorization operation, i.e., each row in matrix are stacked gradually with formed one newly to
Amount, ()*Represent conjugate operation,Represent Kronecker product, IQRepresent Q × Q dimension unit matrixs.The corresponding virtual array of vectorial z
In each Virtual array position be S：
S (i, j)={ ui-uj| i, j=1,2 ..., Q }.
The Virtual array repeated in set S in each position is removed, a virtual array S heterogeneous is obtainedn, its is corresponding
Virtual signal of equal valueIt can be obtained by choosing element corresponding in vector z；
(4) virtual array covariance matrix is constructed：Non-homogeneous virtual array S is chosen firstnIn centered on 0 continuous uniform
One section of Virtual array of arrangement, forms a uniform virtual array S for including L Virtual arrayu, its corresponding Virtual array position
It is set to-LvD to LvContinuous position between d, wherein, d be unit interval, be taken into the half for penetrating narrow band signal wavelength, i.e. d=λ/
2,
Correspondingly, the equivalent signal of the uniform virtual arrayInterception can be passed throughIn with corresponding to the L Virtual array
Element on position is obtained, and dimension is L × 1.Then, virtual array covariance matrix RvCan basisBuild following Toeplitz
(the L of structurev+1)×(Lv+ 1) dimension matrix is obtained：
Wherein,It is the reception signal of equal value corresponding to id Virtual array to represent position；
(5) build the sparse reconstruction optimization problem of virtual array covariance matrix and solve：First, by the angle of direction of arrival angle
Degree domain scope is equally spacedly divided intoIndividual mesh pointI.e.Then, according to virtual array association side
Poor matrix RvBuild following with vectorAnd noise powerFor the optimization problem of variable：
Wherein,ForVirtual array guiding matrix is tieed up, its
It is 0 to L corresponding to Virtual array positionvContinuous one section virtual uniform array between d；VectorComprisingIndividual potential incoming wave side
Upward signal power, its corresponding diagonal matrix is∈ is threshold constant, the reconstruction error for constraining covariance matrix；It ensure that the signal power value in all directions is more than or equal to zero；‖·‖0With ‖ ‖F0 norm and F norms are represented respectively,
For (Lv+1)×(Lv+ 1) unit vector is tieed up.Above-mentioned non-convex optimization problem is converted into convex optimization problem, and tries to achieve optimal solution
(6) Mutual coupling result is obtained by spectrum peak search：Using X-axis asIndividual equally distributed space networks lattice point comes
Ripple direction, Y-axis is optimal valueIncluded in element, draw space power spectrum.The peak value in space power spectrum is found, and will
Response corresponding to these peak values is arranged from big to small, and the X-axis angle direction before taking corresponding to K peak value, as ripple reach side
To estimated result.
Further, the relatively prime array structure described in step (1) can be described as：A pair of relatively prime integers M, N are chosen first；So
Afterwards, a pair of sparse homogenous linear subarrays are constructed, wherein first subarray includes the bay that M spacing is Nd, its position
It is set to 0, Nd ..., (M-1) Nd, second subarray includes the bay that N number of spacing is Md, its position is 0, Md ..., (N-
1)Md；Then, two subarrays are subjected to subarray combination according to the overlapping mode of first array element, obtain actual comprising Q=M+
The non-homogeneous relatively prime array architecture of N-1 bay.
Further, the relatively prime array structure described in step (1) can be described as：A pair of relatively prime integers M, N are chosen first, and
M<N；Then, a pair of sparse homogenous linear subarrays are constructed, wherein first subarray includes the antenna array that 2M spacing is Nd
Member, its position is 0, Nd ..., (2M-1) Nd, and second subarray includes the bay that N number of spacing is Md, and its position is 0,
Md,…,(N-1)Md；Then, two subarrays are subjected to subarray combination according to the overlapping mode of first array element, obtain actual
Include the non-homogeneous relatively prime array architecture of Q=2M+N-1 bay.
Further, the virtual array covariance matrix R described in step (4)vOf equal value in the following manner it can obtain：
Further, the virtual array covariance matrix R described in step (4)vOf equal value in the following manner it can obtain：First
By vectorIt is divided into Lv+ 1 dimension is (Lv+ 1) subvector × 1, each subvector includes vectorIn i-th to the i-th+
LvIndividual element, i.e.,：
Then
Thus, RvCan be by seeking Fourth amount in above formulaMatrix extraction of square root obtain.
Further, non-convex optimization problem constructed in step (5) can be by convex relaxing techniques, by optimization problem
0 norm replaces with 1 norm, obtains following with vectorAnd noise powerFor the convex optimization problem of variable：
Wherein ‖ ‖1Represent 1 norm.
Further, non-convex optimization problem constructed in step (5) can be converted into as follows with vectorAnd noise powerDenoising optimization problem is followed the trail of for the base of variable：
Wherein ξ is regularization parameter, for weighing virtual array covariance matrix reconstruction error and vectorIt is openness.
The present invention has advantages below compared with prior art：
(1) present invention takes full advantage of the relatively prime characteristic of relatively prime array, and the derivation of signal is received by virtual array equivalence,
Realize the structure in virtual array covariance matrix.By the Virtual array number that virtual array is included is more than actual physics
The number of bay, the signal transacting based on virtual Domain is laid a good foundation for the lifting of free degree performance；
(2) present invention considers signal and this openness feature is presented in the range of spatial domain, and association is built in virtual Domain
The sparse reconstruction optimization problem of variance matrix, is more than or equal in the case of physical antenna element number of array with realizing in signal source number
Effective DOA estimations；
(3) compared with the method for existing use uniform array, institute's extracting method of the present invention is in terms of the free degree, array aperture
Advantage can effectively reduce physical antenna array element and the number of radio-frequency module unit in real system, embody in actual applications
Go out ideal economy and practicality.
Brief description of the drawings
Fig. 1 is the method overall procedure block diagram of the present invention.
Fig. 2 is a pair of sparse uniform subarray structural representations of the composition relatively prime array of the first kind in the present invention.
Fig. 3 is the structural representation of the relatively prime array of the first kind in the present invention.
Fig. 4 is a pair of sparse uniform subarray structural representations of the composition relatively prime array of Equations of The Second Kind in the present invention.
Fig. 5 is the structural representation of the relatively prime array of Equations of The Second Kind in the present invention.
Fig. 6 is the space power spectrum schematic diagram rebuild using the relatively prime array of the first kind in the present invention.
Fig. 7 is the space power spectrum schematic diagram rebuild using the relatively prime array of Equations of The Second Kind in the present invention.
Embodiment
Referring to the drawings, technical scheme and effect are described in further detail.
For the application of DOA estimation method in systems in practice, we are often desirable to that less antenna equipment can be used
Estimate more incident signal sources, but be constrained to the factors such as antenna array structure, element number of array, existing method is in the two sides
Face can not realize optimization simultaneously, often there is certain benefit-risk balance relation.In order to lift the free degree of DOA estimation method,
The invention provides the relatively prime array Wave arrival direction estimating method based on the sparse reconstruction of virtual array covariance matrix, reference picture 1,
The present invention's realizes that step is as follows：
Step one：The Q relatively prime array of physical antenna array element framework is used in receiving terminal.Relatively prime array structure mainly include with
Lower two classes, put forward DOA estimation method suitable for the present invention.
The relatively prime array structure of the first kind is as follows：A pair of relatively prime integers M, N are chosen first；Then, reference picture 2, are constructed a pair
Sparse homogenous linear subarray, wherein first subarray includes the bay that M spacing is Nd, its position is 0, Nd ...,
(M-1) Nd, second subarray includes the bay that N number of spacing is Md, and its position is 0, Md ..., (N-1) Md；Between unit
The half of incident narrow band signal wavelength, i.e. d=λ/2 are taken as every d；Then, by two subarrays according to the overlapping side of first array element
Formula carries out subarray combination, and reference picture 3 obtains the actual non-homogeneous relatively prime array architecture for including M+N-1 bay.This
When, physical antenna element number of array Q=M+N-1.
The relatively prime array structure of Equations of The Second Kind is as follows：A pair of relatively prime integers M, N, and M are chosen first<N；Then, reference picture 4, structure
A pair of sparse homogenous linear subarrays are made, wherein first subarray includes the bay that 2M spacing is Nd, its position is
0, Nd ..., (2M-1) Nd, second subarray include the bay that N number of spacing is Md, and its position is 0, Md ..., (N-1)
Md；Then, two subarrays are subjected to subarray combination according to the overlapping mode of first array element, reference picture 5 is actually included
The non-homogeneous relatively prime array architecture of 2M+N-1 bay.Now, physical antenna element number of array Q=2M+N-1.
Step 2：Using relatively prime array received signal and model.Assuming that there is K to come from θ1,θ2,…,θKThe far field in direction is narrow
Band incoherent signal source, using the non-homogeneous relatively prime array received incoming signal of step one framework, obtains Q × 1 and ties up relatively prime array
Signal y (t) is received, can be modeled as：
Wherein, sk(t) it is signal waveform, n (t) is the noise component(s) separate with each signal source, a (θk) it is θkDirection
Steering vector, be expressed as
Wherein, uq, q=1,2 ..., Q represents the physical location of q-th of physical antenna array element in relatively prime array, and u1=0,
[·]TRepresent transposition operation.T sampling snap is gathered altogether, obtains sample covariance matrix
Here ()HRepresent conjugate transposition.
Step 3：Calculate the virtual signal of equal value corresponding to relatively prime array received signal.The relatively prime array received letter of vectorization
Number sample covariance matrixObtain virtual array equivalence and receive signal z：
Wherein,For Q2×
K ties up virtual array guiding matrix, p=[p1,p2,…,pK]TThe power of K incident signal source is included,For noise power, i=
vec(IQ).Here, vec () represent vectorization operation, i.e., each row in matrix are stacked gradually with formed one newly to
Amount, ()*Represent conjugate operation,Represent Kronecker product, IQRepresent Q × Q dimension unit matrixs.The corresponding virtual array of vectorial z
The position of each Virtual array is S in row：
S (i, j)={ ui-uj| i, j=1,2 ..., Q }.
The Virtual array repeated in set S in each position is removed, a virtual array S heterogeneous is obtainedn, its is corresponding
Virtual signal of equal valueIt can be obtained by choosing element corresponding in vector z.
Step 4：Construct virtual array covariance matrix.First, non-homogeneous virtual array S is chosennIn connect centered on 0
Continuous one section of evenly distributed Virtual array, forms a uniform virtual array S for including L Virtual arrayu(due to SuIn void
Matroid member is symmetrical with zero-bit, and L is always odd number), its corresponding Virtual array position is-LvD to LvContinuous position between d
Put, wherein
Correspondingly, the equivalent signal of the uniform virtual arrayInterception can be passed throughIn with corresponding to the L Virtual array
Element on position is obtained, and dimension is L × 1.Then, virtual array covariance matrix RvCan basisBuild following Toeplitz
(the L of structurev+1)×(Lv+ 1) dimension matrix is obtained：
Wherein,It is the reception signal of equal value corresponding to id Virtual array to represent position.Due to uniform virtual array Su
In Virtual array on origin symmetry, the second order equivalence corresponding to its symmetrical array element receives signal statistics amount and is conjugated pass each other
System, therefore RvIt can also be expressed equivalently as：
In addition, RvAlso it can be obtained by Search Space Smoothing, be specially：By vectorIt is divided into Lv+ 1 dimension is (Lv+1)
× 1 subvector, each subvector includes vectorIn i-th to the i-th+LvIndividual element, i.e.,：
Then
Thus, RvCan be by seeking Fourth amount in above formulaMatrix extraction of square root obtain.
Step 5：Design virtual array covariance matrix is sparse to be rebuild optimization problem and solves.First, according to signal in sky
Between sparse distribution characteristic in the range of domain, the angle domain scope of direction of arrival angle is equally spacedly divided intoIndividual mesh pointI.e.Then, the virtual array covariance matrix R calculated according to step 4vBuild as follows
With vectorAnd noise powerFor the optimization problem of variable：
Wherein,ForVirtual array guiding matrix is tieed up, its
It is 0 to L corresponding to Virtual array positionvContinuous one section virtual uniform array between d；VectorComprisingIndividual potential incoming wave side
Upward signal power, its corresponding diagonal matrix is∈ is threshold constant, the reconstruction error for constraining covariance matrix；It ensure that the signal power value in all directions is more than or equal to zero；‖·‖0With ‖ ‖F0 norm and F norms are represented respectively,
For (Lv+1)×(Lv+ 1) unit vector is tieed up.Because above-mentioned optimization problem includes 0 norm this non-convex, this, which will cause to solve, is stranded
It is difficult；In order to obtain optimization solution, it is contemplated that introducing convex relaxing techniques, 0 norm in above-mentioned optimization problem is replaced with into 1 norm,
Obtain following with vectorAnd noise powerFor the convex optimization problem of variable：
Wherein ‖ ‖1Represent 1 norm.Above-mentioned convex optimization problem can equivalence be written as it is following with vectorAnd noise powerFor
The base of variable follows the trail of denoising optimization problem：
Wherein ξ is regularization parameter, for weighing virtual array covariance matrix reconstruction error and vectorIt is openness.
Solve above-mentioned convex optimization problem and can obtain optimum value
Step 6：Mutual coupling result is obtained by spectrum peak search.Using X-axis asIndividual equally distributed space lattice
Point arrival bearing, Y-axis is optimal valueIncluded in element, draw space power spectrum.The peak value in space power spectrum is found,
And arrange the response corresponding to these peak values from big to small, the X-axis angle direction before taking corresponding to K peak value, as ripple
Up to direction estimation result.
One aspect of the present invention, which takes full advantage of relatively prime array, can increase the advantage of the DOA estimation method free degree, breach
The limited bottleneck of the uniform linear array free degree, is realized in bay number one by the calculating of virtual array equivalent signal
The incident signal source of more numbers is estimated under conditions of fixed；On the other hand the thought of the sparse reconstruction of covariance matrix is introduced, and
The reconstruction and DOA for being applied to virtual Domain to realize space power spectrum are estimated.
The effect of the present invention is further described with reference to simulation example.
Simulation example 1：Using the relatively prime array received incoming signal of the first kind, its parameter is chosen for M=3, N=5, i.e. framework
Relatively prime array altogether comprising Q=M+N-1=7 physical antenna array element.It is assumed that incident narrow band signal number is 7, and incident direction
It is uniformly distributed in the range of -60 ° to 60 ° this space angles；The angle domain scope of direction of arrival angle is [- 90 °, 90 °], space
Domain mesh point uniform sampling spacing is set to 0.1 °；Regularization parameter ξ is set to 0.25；Signal to noise ratio is set to 0dB, snap of sampling
Number K=500.
Relatively prime array Mutual coupling side based on the sparse reconstruction of virtual array covariance matrix proposed by the invention
Method space power spectrum is as shown in fig. 6, wherein vertical dotted line represents the actual direction of incident signal source.In the parameter setting of this example
Under, continuous Virtual array number is L=15, correspondingly, L on virtual arrayv=7.As can be seen that institute's extracting method energy of the present invention
It is enough effectively to differentiate this 7 incident signal sources；In addition, the response of each sense can embody actual signal source on power spectrum
Power information, the direction of arrival information and power information of each signal can be estimated simultaneously.Compared to conventionally employed uniform linear array
Method, can only at most differentiate 6 incoming signals using 7 physical antenna array elements, result above embodies the side of carrying of the invention
Increase of the method in free degree performance.
Simulation example 2：Using the relatively prime array received incoming signal of Equations of The Second Kind, its parameter is chosen for M=3, N=5, i.e. framework
Relatively prime array altogether comprising Q=2M+N-1=10 physical antenna array element；It is assumed that incident narrow band signal number is 15, remaining parameter
Setting is consistent with simulation example 1.Now, continuous Virtual array number is L=35, correspondingly, L on virtual arrayv=
17.Space power spectrum as shown in Figure 7 can be seen that institute's extracting method of the present invention can just have only with 10 physical antenna array elements
The direction of arrival and angle information of 15 incident signal sources are differentiated in effect ground, embody the advantage in free degree performance.
In summary, institute's extracting method of the present invention can be real in the case where signal source number is more than or equal to physical antenna number
Effective estimation of existing incoming signal, adds the free degree and computational efficiency.In addition, the method with conventionally employed uniform linear array
Compare, also can be corresponding in the physical antenna array element and radio-frequency module needed for real application systems using institute's extracting method of the present invention
Reduce, embody economy and high efficiency.
Claims (7)
1. a kind of relatively prime array Wave arrival direction estimating method based on the sparse reconstruction of virtual array covariance matrix, its feature exists
In comprising the steps of：
(1) in receiving terminal using the Q relatively prime array of physical antenna array element framework, and relatively prime array received incoming signal is passed through.
(2) assume there are K to come from θ1,θ2,…,θKThe far field arrowband incoherent signal source in direction, the then relatively prime array received of the dimension of Q × 1
Signal y (t) can be modeled as：
Wherein, sk(t) it is signal waveform, n (t) is noise component(s), a (θ separate with each signal sourcek) it is θkThe guiding in direction
Vector, is expressed as
Wherein, uq, q=1,2 ..., Q represents the physical location of q-th of physical antenna array element in relatively prime array, and u1=0, λ are represented
Signal wavelength, []TRepresent transposition operation；T sampling snap is gathered altogether, obtains sample covariance matrix
Here ()HRepresent conjugate transposition.
(3) virtual signal of equal value corresponding to relatively prime array received signal is calculated：The sampling of the relatively prime array received signal of vectorization
Covariance matrixObtain virtual array equivalence and receive signal z：
Wherein,For Q2× K is tieed up
Virtual array guiding matrix, p=[p1,p2,…,pK]TThe power of K incident signal source is included,For noise power, i=vec
(IQ).Here, vec () represents vectorization operation, i.e., each row in matrix are stacked gradually to form a new vector,
(·)*Represent conjugate operation,Represent Kronecker product, IQRepresent Q × Q dimension unit matrixs.In the corresponding virtual array of vectorial z
The position of each Virtual array is
Remove setThe Virtual array repeated in middle each position, obtains a virtual array heterogeneousIts corresponding equivalence
Virtual signalIt can be obtained by choosing element corresponding in vector z.
(4) virtual array covariance matrix is constructed：Non-homogeneous virtual array is chosen firstIn centered on 0 continuous uniform arrange
One section of Virtual array, form a uniform virtual array comprising L Virtual arrayIts corresponding Virtual array position
For-LvD to LvContinuous position between d, wherein, d is unit interval, is taken into the half for penetrating narrow band signal wavelength, i.e. d=λ/2,
Correspondingly, the equivalent signal of the uniform virtual arrayInterception can be passed throughIn with the position corresponding to the L Virtual array
On element obtain, dimension be L × 1.Then, virtual array covariance matrix RvCan basisBuild following Toeplitz structures
(Lv+1)×(Lv+ 1) dimension matrix is obtained：
Wherein,It is the reception signal of equal value corresponding to id Virtual array to represent position.
(5) build the sparse reconstruction optimization problem of virtual array covariance matrix and solve：First, by the angle domain of direction of arrival angle
Scope is equally spacedly divided intoIndividual mesh pointI.e.Then, according to virtual array covariance square
Battle array RvBuild following with vectorAnd noise powerFor the optimization problem of variable：
Wherein,ForVirtual array guiding matrix is tieed up, its correspondence
In Virtual array position L is arrived for 0vContinuous one section virtual uniform array between d；VectorComprisingOn individual potential arrival bearing
Signal power, its corresponding diagonal matrix is∈ is threshold constant, the reconstruction error for constraining covariance matrix；It ensure that the signal power value in all directions is more than or equal to zero；‖·‖0With ‖ ‖F0 norm and F norms are represented respectively,
For (Lv+1)×(Lv+ 1) unit vector is tieed up.Above-mentioned non-convex optimization problem is converted into convex optimization problem, and tries to achieve optimal solution
(6) Mutual coupling result is obtained by spectrum peak search：Using X-axis asIndividual equally distributed space networks lattice point incoming wave side
To Y-axis is optimal valueIncluded in element, draw space power spectrum.Find space power spectrum on peak value, and by these
Response corresponding to peak value is arranged from big to small, and the X-axis angle direction before taking corresponding to K peak value, as direction of arrival is estimated
Count result.
2. the relatively prime array Mutual coupling according to claim 1 based on the sparse reconstruction of virtual array covariance matrix
Method, it is characterised in that：Relatively prime array structure described in step 1 can be described as：A pair of relatively prime integers M, N are chosen first；Then,
A pair of sparse homogenous linear subarrays are constructed, wherein first subarray includes the bay that M spacing is Nd, its position is
0, Nd ..., (M-1) Nd, second subarray include the bay that N number of spacing is Md, and its position is 0, Md ..., (N-1)
Md；Then, two subarrays are subjected to subarray combination according to the overlapping mode of first array element, obtain actual comprising Q=M+N-1
The non-homogeneous relatively prime array architecture of individual bay.
3. the relatively prime array Mutual coupling according to claim 1 based on the sparse reconstruction of virtual array covariance matrix
Method, it is characterised in that：Relatively prime array structure described in step 1 can be described as：A pair of relatively prime integers M, N, and M are chosen first<
N；Then, a pair of sparse homogenous linear subarrays are constructed, wherein first subarray includes the bay that 2M spacing is Nd,
Its position is 0, Nd ..., (2M-1) Nd, and second subarray includes the bay that N number of spacing is Md, and its position is 0,
Md,…,(N-1)Md；Then, two subarrays are subjected to subarray combination according to the overlapping mode of first array element, obtain actual
Include the non-homogeneous relatively prime array architecture of Q=2M+N-1 bay.
4. the relatively prime array Mutual coupling according to claim 1 based on the sparse reconstruction of virtual array covariance matrix
Method, it is characterised in that：Virtual array covariance matrix R described in step 4vOf equal value in the following manner it can obtain：
5. the relatively prime array Mutual coupling according to claim 1 based on the sparse reconstruction of virtual array covariance matrix
Method, it is characterised in that：Virtual array covariance matrix R described in step 4vOf equal value in the following manner it can obtain：First will
VectorIt is divided into Lv+ 1 dimension is (Lv+ 1) subvector × 1, each subvector includes vectorIn i-th to the i-th+Lv
Individual element, i.e.,：
Then
Thus, RvCan be by seeking Fourth amount in above formulaMatrix extraction of square root obtain.
6. the relatively prime array Mutual coupling according to claim 1 based on the sparse reconstruction of virtual array covariance matrix
Method, it is characterised in that：Constructed non-convex optimization problem can be by convex relaxing techniques, by 0 model in optimization problem in step 5
Number replaces with 1 norm, obtains following with vectorAnd noise powerFor the convex optimization problem of variable：
Wherein ‖ ‖1Represent 1 norm.
7. the relatively prime array Mutual coupling according to claim 1 based on the sparse reconstruction of virtual array covariance matrix
Method, it is characterised in that：Constructed non-convex optimization problem can be converted into as follows with vector in step 5And noise powerFor
The base of variable follows the trail of denoising optimization problem：
Wherein ξ is regularization parameter, for weighing virtual array covariance matrix reconstruction error and vectorIt is openness.
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