CN107102291A - The relatively prime array Wave arrival direction estimating method of mesh freeization based on virtual array interpolation - Google Patents

The relatively prime array Wave arrival direction estimating method of mesh freeization based on virtual array interpolation Download PDF

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CN107102291A
CN107102291A CN201710302902.4A CN201710302902A CN107102291A CN 107102291 A CN107102291 A CN 107102291A CN 201710302902 A CN201710302902 A CN 201710302902A CN 107102291 A CN107102291 A CN 107102291A
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CN107102291B (en
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周成伟
樊星
史治国
陈积明
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Zhejiang University ZJU
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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
    • 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
    • G01S3/78—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 electromagnetic waves other than radio waves
    • G01S3/782—Systems for determining direction or deviation from predetermined direction
    • 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
    • G01S3/80—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 ultrasonic, sonic or infrasonic waves
    • G01S3/802—Systems for determining direction or deviation from predetermined direction

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  • Physics & Mathematics (AREA)
  • Engineering & Computer Science (AREA)
  • General Physics & Mathematics (AREA)
  • Radar, Positioning & Navigation (AREA)
  • Remote Sensing (AREA)
  • Electromagnetism (AREA)
  • Variable-Direction Aerials And Aerial Arrays (AREA)
  • Measurement Of Velocity Or Position Using Acoustic Or Ultrasonic Waves (AREA)

Abstract

The invention discloses a kind of relatively prime array Wave arrival direction estimating method of mesh freeization based on virtual array interpolation, the information loss in the prior art caused by the heterogeneity of virtual array is mainly solved the problems, such as.Implementation step is：The relatively prime array of receiving terminal framework；Using relatively prime array received incoming signal and model；Calculate the virtual signal of equal value corresponding to relatively prime array received signal；Construction interpolation virtual array is simultaneously modeled；Construct many sampling snap signals and its sample covariance matrix of interpolation virtual array；Construct projection matrix and define the project related to the projection matrix；Design the optimization problem minimized based on interpolation virtual array signal covariance matrix nuclear norm and solution；Mutual coupling is carried out according to the interpolation virtual array covariance matrix of reconstruction.The present invention improves the free degree and resolution ratio of Mutual coupling, available for passive location and target acquisition.

Description

The relatively prime array Wave arrival direction estimating method of mesh freeization based on virtual array interpolation
Technical field
The invention belongs to signal processing technology field, more particularly to the ripple of radar signal, acoustic signal and electromagnetic signal Up to direction estimation, the relatively prime array Wave arrival direction estimating method of specifically a kind of mesh freeization based on virtual array interpolation can be used In 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 Effective processing of signal statistics amount is received, so that the DOA estimations of signal are realized, 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 the number of its incident signal source that can be estimated.Existing DOA estimation method The reception and modeling of signal are generally carried out using uniform linear array, but the free degree based on uniform linear array method is limited In actual antennas element number of array.Specifically, for a uniform linear array for including L bay, its free degree is L-1.Therefore, it is existing when the number of incident signal source is more than or equal to the number of bay in array in the range of some spatial domain Have and will be unable to carry out effective DOA estimations using the method for uniform linear array.
Relatively prime array can increase the free degree of DOA estimations on the premise of bay number is certain, thus receive The extensive concern of academia.As a classic manifestations of the relatively prime Sampling techniques in spatial domain, relatively prime array is provided The thinned array architectural schemes of one systematization, and the limited bottleneck of the conventional uniform linear array free degree can be broken through, realize The lifting of DOA estimation method free degree performance.The existing DOA estimation method based on relatively prime array is main by using prime number Property, which derives relatively prime array, arrives virtual Domain, and forms virtual uniform linear array reception signal of equal value to realize that DOA estimates.By The Virtual array number included in virtual array is more than actual bay number, and therefore the free degree has obtained effective lifting. But it is due to that the virtual array for deriving and coming from relatively prime array belongs to nonuniform noise, thus it is many existing based on homogenous linear battle array The signal processing method of row can not directly apply to virtual array equivalence and receive signal to realize that effective DOA estimates.Currently adopt It is with a conventional solution of the DOA estimation method of relatively prime array, merely with continuous array element part shape in virtual array Into a virtual uniform linear array to carry out DOA estimations, but which results in the loss of part raw information and correlation estimation performance Reduction.
Meanwhile, current numerous DOA estimation methods are in the design process of optimization problem, it is necessary to which pre-setting signal assumes ripple Up to the space networks lattice point in direction.With the raising to Mutual coupling result required precision, these DOA estimation methods need pre- The space networks lattice point first set will become more and more intensive, and which results in sharply increasing for computation complexity.Moreover, in reality In the situation of border, the direction of arrival of some signals is had unavoidably to be entirely fallen within the mesh point pre-set, so as to cause Intrinsic model mismatch error.
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 interpolation The relatively prime array Wave arrival direction estimating method of mesh freeization, takes full advantage of the full detail that non-homogeneous virtual array is provided, and The Mutual coupling of mesh free is ensure that, so that the free degree and resolution ratio of DOA estimations are improved, and to a certain extent Reduce the computation complexity of DOA estimations.
The purpose of the present invention is achieved through the following technical solutions：A kind of mesh free based on virtual array interpolation Relatively prime array Wave arrival direction estimating method, is comprised the steps of：
(1) receiving terminal carries out framework using M+N-1 antenna, and according to relatively prime array structure；Wherein M and N is relatively prime whole Number；
(2) assume there are K to come from θ1,θ2,…,θKThe far field arrowband incoherent signal source in direction, then the dimension of (M+N-1) × 1 is mutual Matter array received signal x (t) 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, piD, i=1,2 ..., M+N-1 represent the physical location of i-th of physical antenna array element in relatively prime array, and p1 =0；D is the half of incident narrow band signal wavelength X, i.e. d=λ/2,[·]TRepresent transposition operation.Collection T is individual altogether Sampling snap, obtains the sample covariance matrix of relatively prime array received signal
Here, ()HRepresent conjugate transposition operation；
(3) virtual signal of equal value corresponding to relatively prime array received signal is calculated：The relatively prime array received signal of vector quantization Sample covariance matrixObtain virtual array equivalence and receive signal v：
Wherein,For (M+ N-1)2× K ties up virtual array guiding matrix,The power of K incident signal source is included, For noise power, iv=vec (IM+N-1).Here, vec () represents vectoring operations, i.e., each row in matrix are stacked gradually To form a new vector, ()*Represent conjugate operation,Represent Kronecker product, IM+N-1Represent (M+N-1) × (M+N- 1) unit matrix is tieed up.The position of each Virtual array is in the corresponding virtual arrays of vector v
Remove setRepetition Virtual array corresponding to the element of middle repetition on position, obtain one it is heterogeneous virtual ArrayIts corresponding virtual signal v of equal valuecIt can be obtained by choosing the element in vector v on opposite position；
(4) construct interpolation virtual array and its receive signal and model：Firstly for virtual array heterogeneousProtecting On the premise of staying its original Virtual array position constant, some Virtual arrays are inserted in discrete position thereto, so that will be non- Uniform virtual arraySpacing is converted into for d, array aperture be identical with relatively prime array and the increased number of uniform void of Virtual array Matroid is arrangedThe uniform virtual array of the interpolation is included altogetherIndividual Virtual array, wherein | | the gesture of set is represented, its is corresponding Virtual signal v of equal valueIPast vector v can be passed throughcIt is middle insertion 0 obtain, insertion 0 position withThe position of the Virtual array of middle insertion It is corresponding；
(5) sampling snap signal and its sample covariance matrix more than construction interpolation virtual array：WillIt is cut into LIIt is individual long Spend for LIContinuous subarray, wherein
Correspondingly, interpolation virtual arrayMany sampling snap signals can be by intercepting vector vIIn corresponding element obtain , i.e.,：vI,l, l=1,2 ..., LIBy vIIn LI+ 1-l to 2LI- l element compositions. Then, VISample covariance matrix RvIt can be obtained by following manner：
Wherein,<vI>iIt is the reception signal of equal value corresponding to id Virtual array to represent position；
(6) construct projection matrix and define project：Projection matrix P dimension and RvIt is identical, if matrix RvIn some Element is 0, then the element value of same position is also 0 in projection matrix P；Otherwise the element value of relevant position is in projection matrix P 1.DefinitionFor project, its bracket internal variable is to pass through matrix of variables with P dimension identical matrixes, project In each element and the element in projection matrix P on relevant position be multiplied one by one realization, obtain one it is identical with matrix P dimensions Matrix；
(7) optimization problem of the design based on interpolation virtual array signal covariance matrix nuclear norm minimum and solution.Profit The interpolation virtual array covariance matrix R obtained with (5)vAs reference value, the minimum Toeplitz squares of a nuclear norm are found Battle array and requires itself and R as the covariance matrix of interpolation virtual array signalvDifference be less than a certain threshold value, can build as follows Using vector z as the optimization problem of variable：
Wherein,RepresentNuclear norm,Represent that the hermitian using vector z as first row is symmetrical Toeplitz matrixes；∈ is threshold constant, the reconstruction error for constraining covariance matrix；It ensure that reconstruction Covariance matrix meets positive semi-definite condition；‖·‖FRepresent Frobenius norms.Solve above-mentioned convex optimization problem available most Optimal valueCorrespondingly, the Toeplitz matrixes of reconstructionFor interpolation virtual array covariance matrix；
(8) according to the interpolation virtual array covariance matrix of reconstructionCarry out Mutual coupling.
Further, the relatively prime array structure described in step (1) can be specifically described as：Choose first a pair of relatively prime integer M, N；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, 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 M+N-1 bay.
Further, the V constructed by step (5)ISample covariance matrix RvIt can also be obtained by following method equivalences：
Further, the convex optimization problem in step (7) can be converted into as follows using vector z as the optimization problem of variable：
Wherein μ is regularization parameter, for the trade-off matrix during minimumThe nuclear norm of reconstruction error and z.
Further, the Mutual coupling in step (8), can use following methods：Multiple signal classification method, rotation Invariant subspace method, rooting multiple signal classification method, covariance matrix sparse reconstruction method etc..
Further, in step 8, Mutual coupling is carried out by multiple signal classification method, is specially：Draw virtual Domain space composes PMUSIC(θ)：
Wherein d (θ) is LI× 1 dimension interpolation virtual array steering vector, is by 0 to (L corresponding to positionI- 1) d one section of void Intend uniform array；EnIt is LI×(LI- K) dimension matrix, represent interpolation virtual array covariance matrixNoise subspace；θ The signal direction of arrival assumed that；Spatial spectrum P is found by spectrum peak searchMUSICPeak value on (θ), and by corresponding to these peak values Response arrange from big to small, the angle direction before taking corresponding to K peak value, as Mutual coupling result.
The present invention has advantages below compared with prior art：
(1) present invention introduces the thought of Array interpolation in relatively prime array virtual Domain of equal value, takes full advantage of virtual array The full detail provided is provided.Homogenous linear virtual array is constructed by way of the interpolation Virtual array in non-homogeneous virtual array Row, while the full detail received by original non-homogeneous virtual array is remained so that the virtual Domain signal mode of structure Type meets nyquist sampling law；
(2) present invention is asked based on the thought design optimization that interpolation virtual array signal covariance matrix nuclear norm is minimized Topic, without pre-defined space networks lattice point during optimization problem is designed, realizes the Mutual coupling of mesh free, The resolution ratio and computational efficiency of Mutual coupling are ensure that simultaneously；
(3) optimization problem proposed by the invention rebuild based on interpolation virtual array covariance matrix ensure that optimization is asked Solution result is the symmetrical Toeplitz matrixes of hermitian so that the error between optimal solution and theoretical covariance matrix is smaller.By Toeplitz structures are met in the theoretical covariance matrix of uniform linear array incoherent reception signal, therefore utilize its Toeplitz characteristics carry out the reconstruction of covariance matrix as prior-constrained condition, can cause reconstructed results and actual value difference It is smaller, so as to improve the performance of DOA estimations.
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 that relatively prime array is constituted in the present invention.
Fig. 3 is the structural representation of relatively prime array in the present invention.
Fig. 4 is the structural representation of interpolation virtual array in the present invention.
Fig. 5 is the schematic diagram of interpolation virtual array dividing method in the present invention.
Fig. 6 is the space power spectrum schematic diagram for embodying institute's extracting method free degree performance of the present invention.
Fig. 7 is the normalization spatial spectrum schematic diagram for embodying institute's extracting method resolution ratio performance of the present invention.
Embodiment
Referring to the drawings, technical scheme and effect are described in further detail.
For the application of DOA estimations in systems in practice, relatively prime array can pass through virtual array signal of equal value due to it Calculating and statistic line loss rate, break through physics array element quantity the limitation of the free degree is received much concern.But it is constrained to virtual The heterogeneity of array, at present many methods all can the wherein continuous Virtual array part of Selection utilization carry out DOA estimations so that Cause information loss.Meanwhile, many methods can pre-set the space for assuming that ripple reaches sense before DOA estimations are carried out Mesh point, which results in the contradiction between intrinsic mismatch error and computation complexity and estimated accuracy.It is non-in order to make full use of All information included in uniform virtual array, and avoid estimation resolution ratio caused by predefining space networks lattice point by Limit problem, the invention provides a kind of relatively prime array Wave arrival direction estimating method of mesh freeization based on virtual array interpolation, ginseng According to Fig. 1, step is as follows for of the invention realizing：
Step one：The M+N-1 relatively prime array of bay framework is used in receiving terminal；First, one group of relatively prime integer is chosen M、N；Then, reference picture 2, construct a pair of sparse homogenous linear subarrays, wherein first subarray is Nd's comprising M spacing Bay, its position is 0, Nd ..., (M-1) Nd；Second subarray includes the bay that N number of spacing is Md, its position For 0, Md ..., (N-1) Md；Unit spacing d is taken as the half of incident narrow band signal wavelength, i.e. d=λ/2；Then, by two sons The first bay of array is considered as reference array element, reference picture 3, and the reference array element of two submatrixs is overlapping to realize group of subarrays Close, obtain the actual non-homogeneous relatively prime array architecture for including M+N-1 bay.
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 the dimension of (M+N-1) × 1 mutual Matter array received signal x (t), 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 Relatively prime array steering vector, be expressed as
Wherein, piD, i=1,2 ..., M+N-1 represent the physical location of i-th of physical antenna array element in relatively prime array, and p1 =0；D is the half of incident narrow band signal wavelength X, i.e. d=λ/2,[·]TRepresent transposition operation.Collection T is individual altogether Sampling snap, obtains the sample covariance matrix of relatively prime array received signal
Wherein, ()HRepresent conjugate transposition operation.
Step 3：Calculate the virtual signal of equal value corresponding to relatively prime array received signal.The relatively prime array received letter of vector quantization Number sample covariance matrixObtain virtual array equivalence and receive signal v：
Wherein,For (M+ N-1)2× K ties up virtual array guiding matrix,The power of K incident signal source is included, For noise power, iv=vec (IM+N-1).Here, vec () represents vectoring operations, i.e., each row in matrix are stacked gradually To form a new vector, ()*Represent conjugate operation,Represent Kronecker product, IM+N-1Represent (M+N-1) × (M+N- 1) unit matrix is tieed up.The position of each Virtual array is in the corresponding virtual arrays of vector vWherein
Remove setRepetition Virtual array corresponding to the element of middle repetition on position, obtain one it is heterogeneous virtual ArrayIts corresponding virtual signal v of equal valuecIt can be obtained by choosing the element in vector v on opposite position.
Step 4：Construct interpolation virtual array and its receive signal modeling.Reference picture 4, for virtual array heterogeneousOn the premise of constant in its original Virtual array position of reservation, some Virtual arrays are inserted in the position that there is hole thereto (as shown in the open circles in Fig. 4), so that by non-homogeneous virtual arrayIt is d, array aperture and relatively prime array to be converted into spacing The increased number of uniform virtual array of identical and Virtual arrayInterpolation virtual array is included altogetherIndividual Virtual array, wherein | | represent the gesture of set.The corresponding virtual signal v of equal value of interpolation virtual arrayIPast vector v can be passed throughcThe corresponding positions of Hole Put and insert 0 acquisition.
Step 5：Construct sampling snap signal and its sample covariance matrix more than interpolation virtual array.Reference picture 5, will be interior Insert virtual arrayIt is cut into LIIndividual length is LIContinuous subarray, wherein
Due toIn Virtual array it is symmetrical with zero-bit,It is always odd number, therefore LIFor integer.Correspondingly, interpolation Virtual arrayMany sampling snap signals can be by intercepting vector vIIn corresponding element obtain, i.e.,：VI=[vI,1, vI,2,…,vI,LI], wherein vI,l, l=1,2 ..., LIBy vIIn LI+ 1-l to 2LI- l element compositions.Then, VIAdopt Sample covariance matrix RvIt can be obtained by following manner：
Wherein, ＜ vI〉iIt is the reception signal of equal value corresponding to id Virtual array to represent position.Due to interpolation virtual array Middle Virtual array is symmetrical on zero-bit, therefore equivalence thereon is virtual that to receive signal to correspond to zero-bit be in conjugate relation, institute Equivalence it can also be obtained in the following way with above-mentioned sample covariance matrix：
Step 6：Construction projection matrix simultaneously defines project.Due to the covariance matrix R obtained by step 5vIn include There is the element all 0 on 0 inserted in step 4, therefore its relevant position diagonal.One is defined according to such structure Individual and RvDimension identical projection matrix P, if RvIn element on a certain position be 0, then same position in projection matrix P Element value is also 0；On the contrary then relevant position in projection matrix P element value is 1.DefinitionFor project, wherein including Number internal variable be with P dimension identical matrixes, project is corresponding to projection matrix P by each element of matrix of variables Element on position is multiplied realization one by one, obtains one and matrix P dimension identical matrixes.
Step 7：Design the optimization problem minimized based on interpolation virtual array signal covariance matrix nuclear norm and ask Solution.The interpolation virtual array covariance matrix R obtained using step 5vAs reference value, one nuclear norm minimum of searching Toeplitz matrixes and require itself and R as the covariance matrix of interpolation virtual array signalvDifference be less than a certain threshold value, It can build as follows using vector z as the optimization problem of variable：
Wherein,RepresentNuclear norm,Represent that the hermitian using vector z as first row is symmetrical Toeplitz matrixes；∈ is threshold constant, the reconstruction error for constraining covariance matrix；It ensure that reconstruction Covariance matrix meets positive semi-definite condition；‖·‖FRepresent Frobenius norms.Solve above-mentioned convex optimization problem available most Optimal valueAbove-mentioned convex optimization problem can be converted into the following optimization problem using vector z as variable：
Wherein μ is regularization parameter, for the trade-off matrix during minimumThe nuclear norm of reconstruction error and z. Solve above-mentioned convex optimization problem and can obtain optimum valueCorrespondingly, the Toeplitz matrixes of reconstructionFor interpolation virtual array Row covariance matrix.
Step 8：According to the interpolation virtual array covariance matrix of reconstructionCarry out Mutual coupling.By introducing Classical method, such as multiple signal classification method, invariable rotary subspace method, rooting multiple signal classification method, covariance The interpolation virtual array covariance matrix to reconstruction such as matrix sparse reconstruction methodOperated, side can be reached in the hope of ripple To estimated result.By taking multiple signal classification method as an example, virtual Domain spatial spectrum P is drawnMUSIC(θ)：
Wherein d (θ) is LI× 1 dimension interpolation virtual array steering vector, is by 0 to (L corresponding to positionI- 1) d one section of void Intend uniform array；EnIt is LI×(LI- K) dimension matrix, represent interpolation virtual array covariance matrixNoise subspace；θ The signal direction of arrival assumed that；Spatial spectrum P is found by spectrum peak searchMUSICOn peak value, and by corresponding to these peak values Response is arranged from big to small, the angle direction before taking corresponding to K peak value, as Mutual coupling result.
One aspect of the present invention introduces the thought of virtual array interpolation, and interior insertion is empty on the basis of the original virtual array of derivation Matroid member, so that original non-homogeneous virtual array is converted into virtual uniform array, while remaining original non-homogeneous virtual All information on array, it is to avoid statistic line loss rate model mismatch caused by the heterogeneity of original virtual array and Information loss problem caused by the virtual uniform submatrix of conventional method interception；On the other hand, introduce based on virtual array signal The thought that covariance matrix nuclear norm is minimized carrys out design optimization problem, to rebuild the covariance matrix of interpolation virtual array, real The mesh free Mutual coupling in virtual Domain is showed.
The effect of the present invention is further described with reference to simulation example.
Simulation example 1：Using relatively prime array received incoming signal, its parameter is chosen for the relatively prime of M=3, N=5, i.e. framework Array is altogether comprising M+N-1=7 physics array element.It is assumed that incident narrow band signal number is 9, and incident direction is uniformly distributed in -50 ° To 50 ° of this space angle domains；Signal to noise ratio is set to 30dB, and sample fast umber of beats T=500；Regularization parameter μ is set to 0.25。
The relatively prime array Wave arrival direction estimating method space of mesh freeization based on virtual array interpolation proposed by the invention Power spectrum is as shown in fig. 6, wherein vertical dotted line represents the actual direction of incident signal source.As can be seen that institute's extracting method of the present invention This 9 incident signal sources can effectively be differentiated.And for the method for conventionally employed uniform linear array, utilize 7 physical antennas Array element can only at most differentiate 6 incoming signals, and result above, which embodies institute's extracting method of the present invention, realizes the increase of the free degree.
Simulation example 2：Using relatively prime array received incoming signal, its parameter is equally chosen for M=3, N=5, i.e. framework Relatively prime array is altogether comprising M+N-1=7 physical antenna array element；It is assumed that incident narrow band signal number is 2, and incident direction for- 0.5 ° to 0.5 °, remaining parameter setting is consistent with simulation example 1.Normalization spatial spectrum as shown in Figure 7 can be seen that this Invention institute extracting method can effectively tell the direction of arrival of the two closely signals, illustrate the resolution of this method well Rate performance.
In summary, institute's extracting method of the present invention takes full advantage of the full detail on non-homogeneous virtual array, can be in letter Number source number realizes the mesh freeization estimation of incoming signal in the case of being more than or equal to physical antenna number, add DOA estimations The free degree and resolution ratio.In addition, compared with the method for conventionally employed uniform linear array, institute's extracting method of the present invention actually should Physical antenna array element and radio-frequency module needed for also can be reduced accordingly, embody economy and high efficiency.

Claims (6)

1. the relatively prime array Wave arrival direction estimating method of a kind of mesh freeization based on virtual array interpolation, it is characterised in that include Following steps：
(1) receiving terminal carries out framework using M+N-1 antenna, and according to relatively prime array structure；Wherein M and N is relatively prime integer；
(2) assume there are K to come from θ1,θ2,…,θKThe far field arrowband incoherent signal source in direction, then (M+N-1) × 1 tie up relatively prime battle array Row receive signal x (t) and can be modeled as：
<mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </munderover> <mi>a</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>s</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>n</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
Wherein, sk(t) it is signal waveform, n (t) is the noise component(s) separate with each signal source, a (θk) it is θkDirection is led Draw vector, be expressed as：
<mrow> <mi>a</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <mn>1</mn> <mo>,</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;p</mi> <mn>2</mn> </msub> <mi>d</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>/</mo> <mi>&amp;lambda;</mi> </mrow> </msup> <mo>,</mo> <mo>...</mo> <mo>,</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;p</mi> <mrow> <mi>M</mi> <mo>+</mo> <mi>N</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mi>d</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>/</mo> <mi>&amp;lambda;</mi> </mrow> </msup> <mo>&amp;rsqb;</mo> </mrow> <mi>T</mi> </msup> <mo>,</mo> </mrow>
Wherein, piD, i=1,2 ..., M+N-1 represent the physical location of i-th of physical antenna array element in relatively prime array, and p1=0； D is the half of incident narrow band signal wavelength X, i.e. d=λ/2,[·]TRepresent transposition operation.T sampling is gathered altogether Snap, obtains the sample covariance matrix of relatively prime array received signal
<mrow> <msub> <mover> <mi>R</mi> <mo>^</mo> </mover> <mi>x</mi> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>T</mi> </munderover> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <msup> <mi>x</mi> <mi>H</mi> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
Here, ()HRepresent conjugate transposition operation；
(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 vector quantization Covariance matrixObtain virtual array equivalence and receive signal v：
<mrow> <mi>v</mi> <mo>=</mo> <mi>v</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>R</mi> <mo>^</mo> </mover> <mi>x</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>A</mi> <mi>v</mi> </msub> <msup> <mi>&amp;sigma;</mi> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>&amp;sigma;</mi> <mi>n</mi> <mn>2</mn> </msubsup> <msub> <mi>i</mi> <mi>v</mi> </msub> <mo>,</mo> </mrow>
Wherein,For (M+N-1)2 × K ties up virtual array guiding matrix,The power of K incident signal source is included,For noise Power, iv=vec (IM+N-1).Here, vec () represents vectoring operations, i.e., each row in matrix are stacked gradually to be formed One new vector, ()*Represent conjugate operation,Represent Kronecker product, IM+N-1Represent that (M+N-1) × (M+N-1) dimensions are single Bit matrix.The position of each Virtual array is in the corresponding virtual arrays of vector v
Remove setRepetition Virtual array corresponding to the element of middle repetition on position, obtains a virtual array heterogeneousIts corresponding virtual signal v of equal valuecIt can be obtained by choosing the element in vector v on opposite position；
(4) construct interpolation virtual array and its receive signal and model：Firstly for virtual array heterogeneousRetaining it On the premise of original Virtual array position is constant, some Virtual arrays are inserted in discrete position thereto, so that will be non-homogeneous Virtual arrayBe converted into spacing for d, array aperture be identical with relatively prime array and Virtual array it is increased number of it is uniform virtually ArrayThe uniform virtual array of the interpolation is included altogetherIndividual Virtual array, wherein | | the gesture of set is represented, its is corresponding etc. Valency virtual signal vIPast vector v can be passed throughcIt is middle insertion 0 obtain, insertion 0 position withThe position phase of the Virtual array of middle insertion Correspondence；
(5) sampling snap signal and its sample covariance matrix more than construction interpolation virtual array：WillIt is cut into LIIndividual length is LI Continuous subarray, wherein
Correspondingly, interpolation virtual arrayMany sampling snap signals can be by intercepting vector vIIn corresponding element obtain, i.e.,：vI,l, l=1,2 ..., LIBy vIIn LI+ 1-l to 2LI- l element compositions.Then, VISample covariance matrix RvIt can be obtained by following manner：
Wherein,<vI>iIt is the reception signal of equal value corresponding to id Virtual array to represent position；
(6) construct projection matrix and define project：Projection matrix P dimension and RvIt is identical, if matrix RvIn some element For 0, then the element value of same position is also 0 in projection matrix P；Otherwise the element value of relevant position is 1 in projection matrix P.It is fixed JusticeFor project, its bracket internal variable is to pass through every in matrix of variables with P dimension identical matrixes, project One element and the element in projection matrix P on relevant position are multiplied realization one by one, obtain one and matrix P dimension identical squares Battle array；
(7) optimization problem of the design based on interpolation virtual array signal covariance matrix nuclear norm minimum and solution.Utilize (5) Obtained interpolation virtual array covariance matrix RvAs reference value, the minimum Toeplitz matrix conducts of a nuclear norm are found The covariance matrix of interpolation virtual array signal, and require itself and RvDifference be less than a certain threshold value, can build as follows with vector z For the optimization problem of variable：
Wherein,RepresentNuclear norm,Represent that the hermitian using vector z as first row is symmetrical Toeplitz matrixes；∈ is threshold constant, the reconstruction error for constraining covariance matrix；It ensure that reconstruction Covariance matrix meets positive semi-definite condition；‖·‖FRepresent Frobenius norms.Solve above-mentioned convex optimization problem available most Optimal valueCorrespondingly, the Toeplitz matrixes of reconstructionFor interpolation virtual array covariance matrix；
(8) according to the interpolation virtual array covariance matrix of reconstructionCarry out Mutual coupling.
2. the relatively prime array Wave arrival direction estimating method of the mesh freeization according to claim 1 based on virtual array interpolation, It is characterized in that：Relatively prime array structure described in step (1) can be specifically 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 M+N-1 The non-homogeneous relatively prime array architecture of individual bay.
3. the relatively prime array Wave arrival direction estimating method of the mesh freeization according to claim 1 based on virtual array interpolation, It is characterized in that：V constructed by step (5)ISample covariance matrix RvIt can also be obtained by following method equivalences：
4. the relatively prime array Wave arrival direction estimating method of the mesh freeization according to claim 1 based on virtual array interpolation, It is characterized in that：Convex optimization problem in step (7) can be converted into as follows using vector z as the optimization problem of variable：
Wherein μ is regularization parameter, for the trade-off matrix during minimumThe nuclear norm of reconstruction error and z.
5. the relatively prime array Wave arrival direction estimating method of the mesh freeization according to claim 1 based on virtual array interpolation, It is characterized in that：Mutual coupling in step (8), can use following methods：Multiple signal classification method, invariable rotary Space-wise, rooting multiple signal classification method, covariance matrix sparse reconstruction method etc..
6. the relatively prime array Wave arrival direction estimating method of the mesh freeization according to claim 1 based on virtual array interpolation, It is characterized in that：In step 8, Mutual coupling is carried out by multiple signal classification method, is specially：Draw virtual domain space Compose PMUSIC(θ)：
<mrow> <msub> <mi>P</mi> <mrow> <mi>M</mi> <mi>U</mi> <mi>S</mi> <mi>I</mi> <mi>C</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msup> <mi>d</mi> <mi>H</mi> </msup> <mrow> <mo>(</mo> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> <msub> <mi>E</mi> <mi>n</mi> </msub> <msubsup> <mi>E</mi> <mi>n</mi> <mi>H</mi> </msubsup> <mi>d</mi> <mrow> <mo>(</mo> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow>
Wherein d (θ) is LI× 1 dimension interpolation virtual array steering vector, is by 0 to (L corresponding to positionI- 1) one section of d is virtual Even array；EnIt is LI×(LI- K) dimension matrix, represent interpolation virtual array covariance matrixNoise subspace；θ is false Fixed signal direction of arrival；Spatial spectrum P is found by spectrum peak searchMUSICPeak value on (θ), and by the sound corresponding to these peak values It should be worth and arrange from big to small, the angle direction before taking corresponding to K peak value, as Mutual coupling result.
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