CN106646344B  A kind of Wave arrival direction estimating method using relatively prime battle array  Google Patents
A kind of Wave arrival direction estimating method using relatively prime battle array Download PDFInfo
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 CN106646344B CN106646344B CN201611163847.7A CN201611163847A CN106646344B CN 106646344 B CN106646344 B CN 106646344B CN 201611163847 A CN201611163847 A CN 201611163847A CN 106646344 B CN106646344 B CN 106646344B
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 G—PHYSICS
 G01—MEASURING; TESTING
 G01S—RADIO DIRECTIONFINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCEDETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
 G01S3/00—Directionfinders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received
Abstract
Description
Technical field
The present invention relates to array signal processing field, especially a kind of Wave arrival direction estimating method.
Background technique
Mutual coupling has and is widely applied very much in radar, sonar and wireless communication, however traditional direction of arrival Estimation method can only solve the case where number of targets is less than array number, therefore how to detect more targets with a small amount of array element is one A good problem to study.In practice, general common N member concentrating rate, can only at most estimate the incoming wave of N1 target Direction.In recent years, propose a kind of linear array of new geometric structure  relatively prime battle array so that estimation target numbers far more than Array element number N, in fact, the target numbers for having the relatively prime battle array of N+M array element that can estimate can achieve O (MN), i.e., and MN The target numbers of same order size, since the ingenious distribution of relatively prime battle array element position can be formed after mathematical operation is handled The bigger virtual array in aperture.Specifically, the relatively prime battle array of N+2M1 physics array element mentioned above can fictionalize aperture and be The concentrating rate of 2MN+1.
For relatively prime battle array, it has been proposed that a kind of smooth MUSIC algorithm in airspace based on relatively prime battle array, for using N+2M1 The relatively prime battle array of a physics array element, MN target is at most estimated that using this method, although the target that this algorithm estimates Number has been much larger than the concentrating rate of equal number physics array element, but due to having used airspace smoothing method, leads Virtual array is caused to be reduced to MN+1, detection performance reduces.Recent years, by D.Donoho, E.Candes and scientist of Chinese origin T.Tao et al. proposes a kind of new acquisition of information guiding theory, i.e. compressed sensing, the theory once proposition, just information theory, The fields such as signal processing, patternrecognition, wireless communication are paid high attention to, which thinks, for a sparse signal or It is that can be carried out after observing on a small quantity to the signal with the signal of rarefaction representation, it is then former using the reconstruct of specific restructing algorithm Beginning signal meets y=Ax, observing matrix in formula for the finite observation y of signal xM < N, if meeting certain item Part can reconstruct original sparse signal x from finite observation y.
Summary of the invention
For overcome the deficiencies in the prior art, the present invention is based on the sparse reconstruct thoughts in compressed sensing, for relatively prime battle array The Wave arrival direction estimating method based on OMP algorithm is constructed, overcoming the smooth MUSIC algorithm in airspace leads to asking for Virtual array reduction Topic so that detection target numbers be more than MN, and emulate proof this method have good estimated accuracy in high s/n ratio.
The technical solution adopted by the present invention to solve the technical problems specifically comprises the following steps:
Step 1: relatively prime battle array structure design
Relatively prime battle array, which is respectively that Md is nested with the concentrating rate of Nd by two spacing, to be formed, and wherein M and N is two matter each other Several constants, d indicate the halfwavelength of reception signal, and a concentrating rate has N number of array element, and array element spacing is Md, another Concentrating rate has 2M array element, and array element spacing is Nd, and first array element of two concentrating rates is overlapped, and shares N+2M1 Array element, since M and N is prime number, so this linear array is referred to as relatively prime battle array, relatively prime battle array is a kind of heterogeneous line of special construction Array；
Step 2: estimating that relatively prime battle array receives the covariance matrix of signal
Relatively prime battle array receives the covariance matrix R of signal_{xx}Estimated by formula (1):
In formula (1),Indicate that relatively prime battle array received signal vector, T indicate array number of snapshots；
Step 3: constructing the received vector y of virtual concentrating rate
Construct the element b that size is the ith row jth column in (N+2M1) × (N+2M1) identity matrix B, B_{ij}=ij, mark Matrix B and covariance matrix R_{xx}It is straightened, obtains by column respectivelyMark vectorThe position that element value isMN to MN is successively found according to sequence from small to large from mark vector b It sets, and is sequentially recorded lower 2MN+1 location information, then successively come out the element extraction of corresponding position from vector z, structure Produce the received vector of virtual concentrating rate
Step 4: angular regions subdivision grid to be detected, constructs array manifold matrix A
By angular regions discretization subdivision grid to be detected, grid vector θ=[θ is formed_{1},θ_{2},…,θ_{D}]^{T}, wherein θ_{k}Table Showing discrete grid angle, k=1 ..., D, D indicates lattice number, and the lattice number D of subdivision is made to be greater than signal number, by Formula (2) and formula (3) construct array manifold matrix A:
A(θ_{1},θ_{2},…,θ_{D})=[a (θ_{1}),a(θ_{2}),…a(θ_{k}),...,a(θ_{D})] (2)
In formula (2), a (θ_{k}) be corresponding array array manifold vector, k=1,2 ..., D, in formula (3), j indicates imaginary number Unit, d indicate to receive the halfwavelength of signal, the wavelength of λ expression reception signal, θ_{k}Indicate discrete grid angle；
Step 5: estimating sparse signal using OMP algorithm
Array manifold matrix A (θ_{1},θ_{2},…,θ_{D}) and the received vector y of virtual concentrating rate meet following equation:
P is corresponding to subdivision grid θ=[θ in formula (4)_{1},θ_{2},…,θ_{D}]^{T}Sparse vector,Indicate noise vector, it is dilute Sparse vector estimated value can be obtained using the OMP algorithm solution in compressed sensing by dredging vector p
Step 6: obtaining signal direction of arrival by sparse vector
If sparse vector estimated valueIn ith it is nonzero, then it represents that corresponding θ_{i}Direction has signal, otherwise indicates do not have Signal.
The present invention uses concentrating rate compared to traditional Mutual coupling, this novel due to using relatively prime battle array Nested battle array has so that Mutual coupling performance significantly improves the destination number of estimation in the estimation of array multiple target Estimation performance well；When equally using relatively prime battle array, this method is for the MUSIC algorithm of space smoothing, due to making With the OMP algorithm of compressed sensing, so that the target numbers of estimation are more than NM target, under same array element quantity, further Improve the estimation target numbers of relatively prime battle array；Due to having used OMP algorithm, what can be simple and efficient carries out weight to sparse signal Structure, finally rapidly and efficiently provide Mutual coupling result；In high s/n ratio, the estimated accuracy of the method for the present invention is better than sky Between smooth MUSIC algorithm, in general, performance of the invention is not less than the MUSIC algorithm of space smoothing.
Detailed description of the invention
Fig. 1 is the method flow diagram that the present invention carries out Mutual coupling.
Fig. 2 is the geometry of the relatively prime battle array of the present invention.
Fig. 3 is the Mutual coupling result of the present invention with space smoothing MUSIC algorithm.
Fig. 4 is Mutual coupling result of the present invention to 16 incoming wave signals.
Fig. 5 is the estimated accuracy comparing result of the present invention with the single goal of space smoothing MUSIC algorithm.
The smooth MUSIC algorithm of MUSIC representation space in Fig. 5, OMP indicate that the wave proposed by the present invention based on OMP algorithm reaches Direction determining method.
Specific embodiment
Present invention will be further explained below with reference to the attached drawings and examples.
Step 1: relatively prime battle array structure design
The constant M and N for choosing two prime numbers each other indicate the halfwavelength for receiving signal with constant d, and relatively prime battle array is by between two Away from respectively Md composition nested with the concentrating rate of Nd, one of even linear array has N number of array element, and array element spacing is Md, separately An outer linear array has 2M array element, and array element spacing is Nd, and two linear arrays, first array element is overlapped, a total of N+2M1 battle array Member, since M and N is prime number, so this linear array is referred to as relatively prime battle array, the structure of relatively prime battle array is as shown in Figure 1, relatively prime battle array is one The unequally spaced linear array of kind special construction, therefore can further be derived on the model of linear array；
The linear array comprising N number of array element, element position l are considered first_{i}, it is assumed that K target bearing isAnd And K echo signal is set as narrow band signal, then array received signal indicates are as follows:
1≤t≤T, s in formula_{k}It (t) is kth of incoming wave signal, n (t) is independent identically distributed white noise, It is array manifold vector, and
Ith of element represent kth of signal phase change brought by the time delay of ith of array element.
Step 2: estimating that relatively prime battle array receives the covariance matrix of signal
Relatively prime battle array receives the covariance matrix R of signal_{xx}Estimated by formula (1):
In formula,Indicate that relatively prime battle array received signal vector, T indicate array number of snapshots；
If signal s_{k}(t) obeying variance isIndependent Gaussian distribution, consider that each array element receives the secondorder statistics of data Amount solves the covariance matrix R of array received signal x (t)_{xx}
σ in formula^{2}It is noise power, by matrix R_{xx}It is straightened by column, according to the operation of matrix Padé approximants (also referred to as direct product) Property obtains vector z by formula (7):
Wherein, Indicate the variance of kth of transmitting signal,Expression formula is as follows:
Represent in addition to ith of position be all as 1 remaining element 0 column vector, letter as is regarded vector z according to (8) formula The signal that source vector q is received after matrix Φ observation, matrixIt is middle that there are an array manifold matrixes, and And this matrix is the array manifold matrix of a linear array with more Virtual arrays.
Step 3: constructing the received vector y of virtual concentrating rate
Construct the element b that size is the ith row jth column in (N+2M1) × (N+2M1) identity matrix B, B_{ij}=ij, mark Matrix B and covariance matrix R_{xx}It is straightened, obtains by column respectivelyMark vector Sequence from mark vector b from small to large successively finds the position that element value isMN to MN, is sequentially recorded lower total 2MN+1 A location information, it is assumed that b=[ 3, 1,1,5,0] successively finds the position that element value is 1 to 1 to big sequence from b with small It sets, the corresponding position information recorded is exactly [2,5,3], then successively the element extraction of corresponding position come out from vector z, Construct the received vector of virtual concentrating rate
Step 4: angular regions subdivision grid to be detected, constructs array manifold matrix A
By angular regions discretization subdivision grid to be detected, grid vector θ=[θ is formed_{1},θ_{2},…,θ_{D}]^{T}, wherein θ_{k}(k =1 ..., D) indicate discrete grid angle, D indicates lattice number, and the lattice number D of subdivision is made to be greater than signal number, Taking D is 10 times or more of signal number, constructs array manifold matrix A by following formula:
A(θ_{1},θ_{2},…,θ_{D})=[a (θ_{1}),a(θ_{2}),…,a(θ_{D})] (2)
In formula (2), a (θ_{k}) be corresponding array array manifold vector, k=1,2 ..., D, in formula (3), j indicates imaginary number Unit, d indicate to receive the halfwavelength of signal, the wavelength of λ expression reception signal, θ_{k}Indicate discrete grid angle；
Step 5: estimating sparse signal using OMP algorithm
Array manifold matrix A (θ_{1},θ_{2},…,θ_{D}) and the received vector y of virtual concentrating rate meet following equation:
P is corresponding to subdivision grid θ=[θ in formula (4)_{1},θ_{2},…,θ_{D}]^{T}Sparse vector,Indicate noise vector, it is dilute Sparse vector estimated value can be obtained using the OMP algorithm solution in compressed sensing by dredging vector p
Step 6: obtaining signal direction of arrival by sparse vector
If sparse vector estimated valueIn ith it is nonzero, then it represents that θ_{i}Direction has signal, otherwise indicates no signal.
In the present invention, more specifically assume that this linear array is relatively prime battle array as shown in Figure 1, be respectively by two spacing Md composition nested with the concentrating rate of Nd, a shared N+2M1 physics array element, one of even linear array have N number of array element, Another linear array has 2M array element, and first array element of two linear arrays is overlapped.Assuming that the covariance matrix R of signal_{xx}It is known that square Battle array size is (N+2M1) × (N+2M1), according to formula (4), by the property of direct product it is recognised that matrix Φ shares (N+2M 1)^{2}A row vector, it has been demonstrated that, if N, M are prime numbers, the subset of these row vectors corresponds to bigger equal in an aperture The array manifold matrix A of even linear array, element position is fromMNd to MNd, and array element spacing is d, at this moment virtual concentrating rate Aperture can achieve (2MN+1) d, below by matrixConstruct the array manifold matrix of concentrating rate
Corresponding row vector is selected from Φ, and these row vectors are ranked up to obtain A, so that array manifold matrix A In (n, k) a element beK=1,2 ..., K, n=MN ..., 0 ..., MN ultimately form size For the matrix A of (2NM+1) × K, A is just the array manifold matrix of the concentrating rate of (2MN+1) d for an aperture at this time, For the vector z of formula (8) left end, according to the corresponding item of method choice that vector y is constructed described in step 3 and sorting obtain to Y is measured, y is the received vector corresponding to this virtual concentrating rate, then A, y meet
Vector in formulaIt is vector 1_{n}Remove the vector that corresponding entry sorts, and NM+1 are 1, remaining is 0.
Mutual coupling is carried out to the sparse reconstruct thought of compressed sensing below to be illustrated:
First to angular regions discretization to be detected: θ=[θ_{1},θ_{2},…,θ_{D}]^{T}, and D > > K, the i.e. grid number of subdivision Mesh is greater than signal number, constructs array manifold matrix A (θ_{1},θ_{2},…,θ_{D}):
A(θ_{1},θ_{2},…,θ_{D})=[a (θ_{1}),a(θ_{2}),…,a(θ_{D})] (2)
In formula It is the array manifold vector for the concentrating rate that an array number is 2NM+1.Therefore formula (4) can be rewritten as
Wherein, y is the received vector for the virtual concentrating rate being calculated according to the reception signal x (t) of relatively prime battle array, such as The arrival bearing of kth of signal of fruit is θ_{i}, then ith of position of vector pAssuming that the lucky position in the position of all targets In on the mesh point of subdivision, then being only nonzero on K arrival bearing, at this point, p is K sparse vector.For formula (9), If given y and A, solves equation by the sparse restructing algorithm in compressed sensing.
According to compressive sensing theory, need to acquire the most sparse solution for meeting formula (10)Therefore sparse reconstruction indicates Are as follows:
In formula    _{0}The number of vector nonzero term is indicated, in order to indicate to facilitate A=A (θ_{1},θ_{2},…,θ_{D}), it is assumed that noise Variances sigma^{2}Be it is unknown, in order to solve formula (10), the present invention is solved using OMP algorithm, and the iterative step of OMP algorithm is as follows:
Input: array manifold matrix A, the received vector y of virtual concentrating rate, echo signal quantity K；
Output: the K sparse bayesian learning of pError vector E；
Initialization: surplus E^{0}=y, reconstruction signal p^{0}=0, indexed setThe number of iterations n=0；
Step 1: calculating surplus and construction array manifold matrix A (θ_{1},θ_{2},…,θ_{D}) each column between inner product G^{n}= A^{T}E^{n1}, E^{n1}It is the surplus of nth iteration, when n=1, E^{n1}=E^{0}；
Step 2: finding out G^{n}The corresponding position of the element of middle maximum absolute value, i.e.,
Step 3: updating indexed set Γ^{n}=Γ^{n1}∪ { k } and atom set
Step 4: acquiring approximate solution using least square method
Step 5: updating surplus
Step 6: judging whether iteration meets stop condition, i.e., whether the number of iterations n is greater than K, when the number of iterations is greater than K then Meet stop condition, if meeting stop condition,E=E^{n}, outputOtherwise n ← n+1 is enabled, step is turned Rapid 1, whereinIt isIth.
Method of the invention is further described below: implementing under the premise of the technical scheme of the present invention, provides Detailed embodiment and specific operating process.
Consider the relatively prime battle array comprising 10 physics array element, i.e., N=5, M=3 is taken to the relatively prime battle array in Fig. 2, specifically, the One layer of element position is in [0,3,6,9,12] d, and for second layer element position in [0,5,10,15,20,25] d, taking d is halfwave long value, Several narrow band signals are chosen, arrival bearing is evenly distributed in 60 degree to 60 degree of section, and frequency is taken as f=1000Hz, adopts Sample frequency fs=8192Hz, number of snapshots 500, signaltonoise ratio 0dB.
The narrow band signal that 15 different directions are had chosen in Fig. 3 emulates the detection performance of two kinds of algorithms.In Fig. 4 left figure be by The result obtained according to space smoothing matrix MUSIC algorithm.Right figure illustrates the Mutual coupling based on OMP algorithm in Fig. 3.It is first It first spends to 60 degree with 0.5 degree from 60 for steplength grid division, i.e. then D=240 arrives (3) according to formula (1), calculate structural formula (4) equation equation finally executes OMP algorithm, obtains Mutual coupling result.Dotted line indicates original signal direction in Fig. 4, Solid line indicates estimated result.As can be seen that two kinds of algorithms can obtain good estimation effect, however space smoothing MUSIC is calculated Method can only at most estimate the arrival bearing of MN=15 signal.For 10 yuan of relatively prime battle arrays mentioned above, when there is 16 incoming waves When signal, space smoothing MUSIC algorithm will go wrong, and MUSIC algorithm can only at most estimate NM=15 signal, when 16 When a signal, method is without solution, if still solving according to the input of 15 signals, estimated result differs greatly with true value, at this time Mutual coupling based on OMP algorithm will show advantage.
When Fig. 4 gives 16 incoming wave signals, the result of Mutual coupling is carried out using OMP algorithm, it can be seen that OMP The Mutual coupling result of algorithm can provide accurate estimated result, and this result is that space smoothing MUSIC algorithm cannot Reach.
In the analogous diagram that Fig. 5 is provided, when the present invention considers single target estimation, the estimated accuracy of two kinds of algorithms is with letter It makes an uproar the situation of change of ratio.Different from the emulation of front, specifically in the range of 90 degree are to 90 degree, a signal side is randomly generated To the SNR ranges that horizontal axis indicates are using 2dB as steplength, from 20dB to 10dB；The longitudinal axis indicate error using estimation angle with The absolute value of real angle difference.Each point executes 5000 Monte Carlo Experiments, remaining condition is identical as front emulation.From figure Result in 5 can be seen that the Mutual coupling for single goal, in low signaltonoise ratio, the estimation performance base of two kinds of algorithms This is the same, but in high s/n ratio, the estimated accuracy of OMP algorithm is better than the MUSIC algorithm of space smoothing.
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