CN108680891B - DOA estimation method considering mutual coupling effect under non-uniform noise condition - Google Patents
DOA estimation method considering mutual coupling effect under non-uniform noise condition Download PDFInfo
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Abstract
The invention provides a DOA estimation method considering a mutual coupling effect under a non-uniform noise condition. Firstly, based on the least square theory, the algorithm obtains a covariance matrix of a noise-free signal under a cross coupling condition by an alternative iteration method; then, a mutual coupling coefficient matrix is reconstructed by utilizing a signal subspace principle to solve a noise-free signal covariance matrix after mutual coupling compensation; and finally, realizing coherent source decorrelation based on a spatial smoothing method, and realizing DOA parameter estimation by using a traditional MUSIC algorithm. Numerical simulation shows that compared with the traditional MUSIC, non-uniform noise and DOA estimation algorithm under the mutual coupling condition, the provided algorithm can better inhibit the non-uniform noise, obviously relieve the influence of the mutual coupling effect on the spatial smoothing algorithm and obviously improve the DOA estimation performance under the coherent information source condition.
Description
Technical Field
The invention belongs to the field of signal processing, and further relates to a method for improving DOA estimation performance under non-uniform noise. The invention provides a DOA estimation method for jointly optimizing a non-uniform noise covariance matrix and a cross coupling coefficient, aiming at the problems of poor estimation precision and low resolution of a cross coupling array pair coherent information source angle under a non-uniform noise scene. Firstly, based on Least Square (LS) theory, the algorithm obtains a noise-free signal covariance matrix under a cross coupling condition by an alternative iteration method; then, a mutual coupling coefficient matrix is reconstructed by utilizing a signal subspace principle to solve a noise-free signal covariance matrix after mutual coupling compensation; and finally, realizing coherent source decorrelation based on a spatial smoothing method, and realizing DOA parameter estimation by using a traditional MUSIC algorithm. Numerical simulation shows that compared with the traditional MUSIC, non-uniform noise and DOA estimation algorithm under the mutual coupling condition, the provided algorithm can better inhibit the non-uniform noise, obviously relieve the influence of the mutual coupling effect on the spatial smoothing algorithm and obviously improve the DOA estimation performance under the coherent information source condition.
Background
Direction of arrival (DOA) estimation is one of the key issues in array signal processing research, and is widely used in the fields of radar, communication, and the like. Under the assumption of white gaussian noise, a subspace algorithm represented by multiple signal classification (MUSIC) can realize high-resolution DOA estimation. However, in practical applications, the noise superimposed on the echoes received by the sensors is usually uncorrelated non-uniform noise. In this scenario, the DOA estimation performance of the conventional subspace class algorithm may be seriously deteriorated. Furthermore, due to factors such as multipath effect, coherent sources often appear, and the performance of the conventional DOA algorithm is also significantly degraded. For coherent source problems, spatial smoothing is an effective method of decorrelation. However, when the array has a mutual coupling effect such that the flow pattern of the array cannot be known, the spatial smoothing algorithm will fail, thereby causing a significant decrease in the accuracy of the DOA estimation.
Under non-uniform noise, Liao et al propose a subspace iterative algorithm that solves the signal subspace and the noise covariance matrix by Maximum Likelihood (ML) or Least Square (LS) methods to achieve DOA estimation. He et al propose a noise-free covariance sparse reconstruction DOA estimation algorithm. The algorithm removes non-uniform noise from the vectorized signal covariance mainly through a culling operation, so that the influence of the non-uniform noise is eliminated. Wen et al propose a DOA estimation algorithm based on spatial smoothing (Wen algorithm). The algorithm transforms the noise-free covariance matrix reconstruction into an optimization problem based on a coherent signal model to obtain a noise-free signal covariance matrix, thereby eliminating non-uniform noise influence; and then a spatial smoothing method is adopted to reduce the signal coherence. However, the algorithm does not consider the source parameter estimation problem under the mutual coupling condition. Based on this, Liao et al propose a DOA estimation algorithm (Liao algorithm) under the mutual coupling condition, based on the mutual coupling model, signal and noise subspaces are obtained through alternate iteration, and DOA estimation is realized through a parameterized steering vector. However, this algorithm does not take into account the DOA estimation problem under coherent source conditions. Under the condition of a coherent information source, if coherent information source decoherence is not carried out, the performance of the algorithm is seriously reduced.
Aiming at the problems, the invention provides a DOA estimation algorithm for jointly optimizing a non-uniform noise covariance matrix and a cross-coupling coefficient based on a least square method under the condition of non-uniform Gaussian noise so as to improve the DOA estimation performance of an array considering the cross-coupling effect on a coherent signal source. Firstly, the method solves a noise-free signal covariance matrix under a cross coupling condition through alternate iteration to eliminate the influence of non-uniform noise; then according to the signal subspace principle, by utilizing the Toeplitz characteristic of the cross-coupling coefficient matrix, a noise-free signal covariance matrix after cross-coupling compensation can be obtained to eliminate the influence of cross-coupling, and further overcome the influence of the cross-coupling on a space smoothing algorithm; and finally, realizing coherent source decoherence based on a spatial smoothing method to eliminate the influence of the coherent source, and further improving the estimation performance of the array considering the mutual coupling effect on the coherent source DOA under the condition of non-uniform noise. Numerical simulation shows that compared with the traditional MUSIC, non-uniform noise and DOA estimation algorithm under the mutual coupling condition, the algorithm can better inhibit the influence of the non-uniform noise, obviously overcome the influence of the mutual coupling on the spatial smoothing algorithm and obviously improve the DOA estimation performance under the coherent information source condition.
Disclosure of Invention
The invention provides a DOA estimation method for jointly optimizing a non-uniform noise covariance matrix and a cross coupling coefficient, aiming at the problems of poor estimation precision and low resolution of a cross coupling array pair coherent information source angle under a non-uniform noise scene. Firstly, based on Least Square (LS) theory, the algorithm obtains a noise-free signal covariance matrix under a cross coupling condition by an alternative iteration method; then, a mutual coupling coefficient matrix is reconstructed by utilizing a signal subspace principle to solve a noise-free signal covariance matrix after mutual coupling compensation; and finally, realizing coherent source decorrelation based on a spatial smoothing method, and realizing DOA parameter estimation by using a traditional MUSIC algorithm. Numerical simulation shows that compared with the traditional MUSIC, non-uniform noise and DOA estimation algorithm under the mutual coupling condition, the provided algorithm can better inhibit the non-uniform noise, obviously relieve the influence of the mutual coupling effect on the spatial smoothing algorithm and obviously improve the DOA estimation performance under the coherent information source condition. The method comprises the steps of firstly establishing incoherent signal, mutual coupling and coherent signal models, and obtaining a noise-free signal covariance matrix under the mutual coupling condition through an alternative iteration method based on Least Square (LS) theory; then, a mutual coupling coefficient matrix is reconstructed by utilizing a signal subspace principle to solve a noise-free signal covariance matrix after mutual coupling compensation; and finally, realizing coherent source decorrelation based on a spatial smoothing method, and realizing DOA parameter estimation by using a traditional MUSIC algorithm. The method comprises the following specific steps:
1 establishing an incoherent signal model
Considering L uncorrelated far-field narrowband signalsFrom direction thetalWhere L is 1, 2, …, and L is incident on a uniform linear array having M array elements, the array reception vector can be expressed as:
x t=As t+n t (1)
wherein, x t ═ x1 t,x2 t,…,xM t]In order to receive the vector of signals,is an array flow pattern matrix, a (theta) ═ 1, a (theta), …, aM-1(θ)]TIs a steering vector with respect to a direction theta, a (theta)l)=[1,e-jα,…,e-j(M-1)α]TSteering the vector for the array of the ith source, α ═ 2 π d sin (θ)l) The/lambda is the phase difference between adjacent array elements, d and lambda are the array element spacing and the signal wavelength respectively, and d is usually not more than lambda/2; n (t) ═ n1(t),n1(t),…,nM(t)]Non-uniform Gaussian noise that is uncorrelated, and n (t) -CN (0, Q), Q is a non-uniform noise covariance matrix that can be expressed as:
wherein, representing the noise power superimposed on the m-th array received echo,diag {. denotes a diagonalization operator.
Based on equation (1), the received signal covariance can be expressed as:
R=E[x(t)xH(t)]=APAH+Q=R0+Q (3)
wherein,is a signal waveform covariance matrix, R0=APAHRepresenting the noise-free signal covariance.
2 establishing a mutual coupling and coherent signal model
Assume that L incident signals are non-fully coherent sources, including LdAn incoherent signal source (independent source) and LcCoherent source of incident angles thetalAnd thetal', and L ═ Ld+LcThen, the received signal model of equation (1) can be rewritten as:
wherein, s t ═ s1 t,s2 t,…,sL t]T,Representing a matrix of inter-array mutual coupling coefficients.
Because the mutual coupling coefficient between the array elements is inversely proportional to the array element spacing, and the mutual coupling coefficient between the array elements is sharply reduced along with the increase of the array element spacing, when the array element spacing reaches a certain distance, the mutual coupling coefficient can be approximate to 0. And the mutual coupling matrix has a symmetric Toeplitz characteristic, the mutual coupling coefficient can be expressed as:
wherein,ρk,φkrespectively representing mutual coupling coefficients ckAmplitude and phase of;represents a degree of freedom of mutual coupling, and
from equation (5), the mutual coupling matrix can be further expressed as:
the received signal covariance of equation (4) can be expressed as:
wherein, P0=E[s(t)sH(t)]Represents the signal power covariance matrix, and rank (P)0)=Ld+1,CAP0AHCHRepresenting a noise-free covariance matrix under cross-coupling conditions.
As can be seen from the formula (7), due to the influence of non-uniform noise, mutual coupling and coherent information sources, the main feature vector obtained based on R feature decomposition cannot be expanded into the whole signal space, so that the performance of the traditional subspace algorithm is sharply reduced and even fails.
3 LS-based noise covariance matrix Q and signal subspace B joint optimization
From the above analysis, the covariance matrix of the noise-free signal under the cross-coupling condition can be obtained by removing Q in R, that is:
R-Q=CAP0AHCH (8)
the above-mentioned noise-free covariance CAP0AHCHCan be transformed into an LS problem with respect to the noise covariance Q, i.e.:
wherein | · | purple sweetFRepresents the Frobenius norm,the covariance estimate of the received signal for N snapshots can be obtained, namely:
to simplify the solution of the above problem, equation (9) can be rewritten as:
wherein,p may be decomposed by Cholesky or features0Is obtained byAnd B and CA span the same signal subspace, namely span (B) ═ span (CA).
Based on equation (9), the noise-free signal covariance matrix under cross-coupling conditions can be expressed as:
wherein,can be obtained by solving the problem (11) in alternating iterations, i.e. byThe signal subspace resulting from the sub-iteration isThen it is firstObtained by a sub-iterationCan be expressed as:
wherein Dag {. denotes a take diagonal element operator.
Based on (14), the eigenvalue decomposition of equation (13) can be obtained:
wherein,is as followsThe principal eigenvalue component matrix resulting from the secondary iteration, i.e.Has LdThe number of +1 large eigenvalues,for the purpose of its corresponding feature vector,andrespectively representing a noise spatial feature matrix and a feature vector.
as can be seen from the above, the present invention,andthe updating can be repeated by the equations (14) and (16) until the iteration termination condition is satisfied, which is described in detail in the algorithm steps.
Based on equation (13), a noise-free signal covariance matrix under cross-coupling conditions can be obtained to eliminate non-uniform noise effects. However, mutual coupling may cause the DOA estimation algorithm to degrade dramatically, and under coherent source conditions, mutual coupling may also cause the DOA estimation algorithm based on spatial smoothing to degrade or even fail. Therefore, mutual coupling must be overcome to improve DOA estimation performance.
4 mutual coupling coefficient solving
(1) Independent source angle estimation
Based on the mutual coupling model (4), the array steering vector can be rewritten as:
where Γ (θ) may be expressed as:
μkcan be expressed as:
αkcan be expressed as:
based on equations (18) - (20), equation (17) can be further rewritten as:
where T (θ) ═ diag { a (θ) } Θ, Θ can be expressed as:
α can be expressed as:
From the fact that the array steering vector of the incident signal is orthogonal to the noise subspace feature vector, it is further found that:
based on equation (21), then equation (24) may be rewritten as:
from formula (25):
αHΥ(θ)α=0 (26)
as can be seen from the formula (27), when satisfiedNamely, it isY (θ) is a full rank matrix. However, as known from the signal subspace theory, when the incoming wave signal angle θ is an incoherent source angle, i.e., θ ═ θlY (θ) is a non-full rank matrix, i.e., matrix rank deficiency. Therefore, the incoherent source angle theta can be estimated according to the signal space spectrum peaklNamely:
wherein | is a matrix determinant,the incoherent source angles corresponding to the signal space spectrum peaks.
(2) Mutual coupling coefficient estimation
Based on this, the incoherent source angle θ can be obtained by the spatial spectrum peak position estimated by equation (28)lThe estimated value is used to further calculate the mutual coupling sparse vector c, which is further obtained by equations (26) and (27):
based on the formulas (23) and (29):
further obtained from formulae (23) and (30):
formula (31) can be rewritten as:
the matrix equation of equation (32) can be described as an optimization problem as follows:
wherein,to correct the vector, equation (33) can be considered as a general LS optimization problem, whose solution can be expressed as:
further obtained from formulae (23) and (36):
the following equations (20) and (37) can be obtained:
equation (38) can be further simplified as:
wherein,andis composed ofVector, 0kRepresents a 1 xk zero vector; c is a mutual coupling coefficient vector, which can be expressed as:
wherein, the vector fkCan be expressed as:
variable gkCan be expressed as:
based on equation (41), the mutual coupling coefficient vector c can be expressed as:
c=F-1G (44)
as can be seen from equations (28) and (44), the mutual coupling coefficient vector is obtained based on the angle estimation value of some independent source only, and the angle estimation values of other independent sources are not utilized, so that the estimation accuracy is low. In order to improve the accuracy of the estimated value of the cross-coupling vector c, according to the knowledge of the multi-source data fusion theory, L can be comprehensively utilizeddAngular estimation of incoherent sourcesEstimating a mutual coupling vector c, whereinFrom equations (28) to (44), the coefficient matrix is jointly estimated Equation (44) may be further expressed as:
5 noiseless signal covariance solution
As can be seen from equations (13) and (45), the covariance of the noise-free signal after mutual coupling compensation can be expressed as:
based on equation (46), a decoupled noise-free signal covariance matrix R can be obtained0And the mutual coupling influence among array elements is further eliminated. However, under coherent signal conditions, the noise-free signal covariance R0A rank deficit will occur such that R is based0The number of signal subspaces obtained by feature decomposition is smaller than the number of signal sources, and further, the DOA estimation of the signals cannot be effectively realized.
6 information source DOA estimation based on traditional MUSIC algorithm
(1) Spatial smoothing algorithm
The traditional space smoothing algorithm equally divides a uniform linear array into the uniform linear array by utilizing linear array translation invarianceEach sub-array including array elementsTo obtain:and isThe covariance matrix of the ith sub-matrix can be expressed as:
wherein,array steering vectors and noise power matrixes corresponding to the ith sub-array respectively;in order to be a diagonal rotation matrix,
the conventional spatial backward smoothing covariance matrix can be expressed as:
Based on equations (47) and (48), the conventional forward and backward spatial smoothing covariance matrix can be expressed as:
based on the equations (46) - (49), the noise-free covariance R0The forward and backward smoothed covariance matrix can be further expressed as:
(2) Source DOA estimation
Equation (50) front-back smooth noiseless covariance matrixThe DOA estimation can be realized by decomposing the feature space of the subspace type algorithm represented by the MUSIC method, namely, the DOA estimation can be realized by decomposing the feature space of the subspace type algorithm represented by the MUSIC methodThe characteristic value decomposition is carried out to obtain:
as can be seen from equation (51), the signal spatial spectrum estimation can be expressed as:
based on the above discussion, the joint optimization DOA estimation algorithm proposed by the present invention can be expressed as follows:
9. Solving equations (42) and (43) to obtain F and G;
10. solving the formula (45) to obtain a mutual coupling coefficient vector c;
11. estimation of the cross-coupling compensated noise-free signal covariance matrix R by equation (46)0;
15. Spatial spectrum estimation is achieved by S (θ).
Drawings
FIG. 1 is a flow chart of an implementation of the present invention;
FIG. 2 is a comparison graph of spatial spectrum of incoherent signal under non-uniform noise condition (known as mutual coupling);
FIG. 3 is a diagram of spatial spectrum contrast of incoherent signals under non-uniform noise and mutual coupling conditions;
FIG. 4 is a spatial spectrum comparison diagram of coherent signals under non-uniform noise and mutual coupling conditions;
FIG. 5 is a graph comparing RMSE as a function of signal-to-noise ratio (SNR);
FIG. 6 is a graph comparing RMSE with WNPR.
The effects of the present invention can be further illustrated by the following simulations:
compared with the prior art, the invention has the following advantages:
based on the least square theory, a noise-free signal covariance matrix under the cross coupling condition is obtained through alternate iteration to eliminate the influence of non-uniform noise; then according to the signal subspace principle, by utilizing the Toeplitz characteristic of the cross-coupling coefficient matrix, a noise-free signal covariance matrix after cross-coupling compensation can be obtained to eliminate the influence of cross-coupling, and further overcome the influence of the cross-coupling on a space smoothing algorithm; and finally, realizing coherent source decoherence based on a spatial smoothing method to eliminate the influence of the coherent source, and further improving the estimation performance of the array considering the mutual coupling effect on the coherent source DOA under the condition of non-uniform noise. Compared with the traditional MUSIC, non-uniform noise and DOA estimation algorithm under the mutual coupling condition, the algorithm can better inhibit the influence of the non-uniform noise, obviously overcome the influence of the mutual coupling on the spatial smoothing algorithm and obviously improve the DOA estimation performance under the coherent information source condition.
Detailed Description
The implementation steps of the present invention are further described in detail below with reference to fig. 1:
1 establishing an incoherent signal model
Considering L uncorrelated far-field narrowband signalsFrom direction thetalL1, 2, …, L is incident on aA uniform linear array with M array elements, the array receive vector can be expressed as:
x t=As t+n t (1)
wherein, x t ═ x1 t,x2 t,…,xM t]In order to receive the vector of signals,is an array flow pattern matrix, a (theta) ═ 1, a (theta), …, aM-1(θ)]TIs a steering vector with respect to a direction theta, a (theta)l)=[1,e-jα,…,e-j(M-1)α]TSteering the vector for the array of the ith source, α ═ 2 π d sin (θ)l) The/lambda is the phase difference between adjacent array elements, d and lambda are the array element spacing and the signal wavelength respectively, and d is usually not more than lambda/2; n (t) ═ n1(t),n1(t),…,nM(t)]Non-uniform Gaussian noise that is uncorrelated, and n (t) -CN (0, Q), Q is a non-uniform noise covariance matrix that can be expressed as:
wherein, representing the noise power superimposed on the m-th array received echo,diag {. denotes a diagonalization operator.
Based on equation (1), the received signal covariance can be expressed as:
R=E[x(t)xH(t)]=APAH+Q=R0+Q (3)
wherein,is a signal waveform covariance matrix, R0=APAHRepresenting the noise-free signal covariance.
2 establishing a mutual coupling and coherent signal model
Assume that L incident signals are non-fully coherent sources, including LdAn incoherent signal source (independent source) and LcCoherent source of incident angles thetalAnd thetal', and L ═ Ld+LcThen, the received signal model of equation (1) can be rewritten as:
wherein, s t ═ s1 t,s2 t,…,sL t]T,Representing a matrix of inter-array mutual coupling coefficients.
Because the mutual coupling coefficient between the array elements is inversely proportional to the array element spacing, and the mutual coupling coefficient between the array elements is sharply reduced along with the increase of the array element spacing, when the array element spacing reaches a certain distance, the mutual coupling coefficient can be approximate to 0. And the mutual coupling matrix has a symmetric Toeplitz characteristic, the mutual coupling coefficient can be expressed as:
wherein,ρk,φkrespectively representing mutual coupling coefficients ckAmplitude and phase of;represents a degree of freedom of mutual coupling, and
from equation (5), the mutual coupling matrix can be further expressed as:
the received signal covariance of equation (4) can be expressed as:
wherein, P0=E[s(t)sH(t)]Represents the signal power covariance matrix, and rank (P)0)=Ld+1,CAP0AHCHRepresenting a noise-free covariance matrix under cross-coupling conditions.
As can be seen from the formula (7), due to the influence of non-uniform noise, mutual coupling and coherent information sources, the main feature vector obtained based on R feature decomposition cannot be expanded into the whole signal space, so that the performance of the traditional subspace algorithm is sharply reduced and even fails.
3 LS-based noise covariance matrix Q and signal subspace B joint optimization
From the above analysis, the covariance matrix of the noise-free signal under the cross-coupling condition can be obtained by removing Q in R, that is:
R-Q=CAP0AHCH (8)
the above-mentioned noise-free covariance CAP0AHCHCan be transformed into an LS problem with respect to the noise covariance Q, i.e.:
wherein | · | purple sweetFRepresents the Frobenius norm,the covariance estimate of the received signal for N snapshots can be obtained, namely:
to simplify the solution of the above problem, equation (9) can be rewritten as:
wherein,p may be decomposed by Cholesky or features0Is obtained byAnd B and CA span the same signal subspace, namely span (B) ═ span (CA).
Based on equation (9), the noise-free signal covariance matrix under cross-coupling conditions can be expressed as:
wherein,can be obtained by solving the problem (11) in alternating iterations, i.e. byThe signal subspace resulting from the sub-iteration isThen it is firstObtained by a sub-iterationCan be expressed as:
wherein Dag {. denotes a take diagonal element operator.
Based on (14), the eigenvalue decomposition of equation (13) can be obtained:
wherein,is as followsThe principal eigenvalue component matrix resulting from the secondary iteration, i.e.Has LdThe number of +1 large eigenvalues,for the purpose of its corresponding feature vector,andrespectively representing a noise spatial feature matrix and a feature vector.
as can be seen from the above, the present invention,andthe updating can be repeated by the equations (14) and (16) until the iteration termination condition is satisfied, which is described in detail in the algorithm steps.
Based on equation (13), a noise-free signal covariance matrix under cross-coupling conditions can be obtained to eliminate non-uniform noise effects. However, mutual coupling may cause the DOA estimation algorithm to degrade dramatically, and under coherent source conditions, mutual coupling may also cause the DOA estimation algorithm based on spatial smoothing to degrade or even fail. Therefore, mutual coupling must be overcome to improve DOA estimation performance.
4 mutual coupling coefficient solving
(1) Independent source angle estimation
Based on the mutual coupling model (4), the array steering vector can be rewritten as:
where Γ (θ) may be expressed as:
μkcan be expressed as:
αkcan be expressed as:
based on equations (18) - (20), equation (17) can be further rewritten as:
where T (θ) ═ diag { a (θ) } Θ, Θ can be expressed as:
α can be expressed as:
from the fact that the array steering vector of the incident signal is orthogonal to the noise subspace feature vector, it is further found that:
based on equation (21), then equation (24) may be rewritten as:
from formula (25):
αHΥ(θ)α=0 (26)
as can be seen from the formula (27), when satisfiedNamely, it isY (θ) is a full rank matrix. However, as known from the signal subspace theory, when the incoming wave signal angle θ is an incoherent source angle, i.e., θ ═ θlY (θ) is a non-full rank matrix, i.e., matrix rank deficiency. Therefore, the incoherent source angle theta can be estimated according to the signal space spectrum peaklNamely:
wherein | is a matrix determinant,the incoherent source angles corresponding to the signal space spectrum peaks.
(2) Mutual coupling coefficient estimation
Based on this, the incoherent source angle θ can be obtained by the spatial spectrum peak position estimated by equation (28)lThe estimated value is used to further calculate the mutual coupling sparse vector c, which is further obtained by equations (26) and (27):
based on the formulas (23) and (29):
further obtained from formulae (23) and (30):
formula (31) can be rewritten as:
the matrix equation of equation (32) can be described as an optimization problem as follows:
wherein,to correct the vector, equation (33) can be considered as a general LS optimization problem, whose solution can be expressed as:
further obtained from formulae (23) and (36):
the following equations (20) and (37) can be obtained:
equation (38) can be further simplified as:
wherein,andis composed ofVector, 0kRepresents a 1 xk zero vector; c is a mutual coupling coefficient vector, which can be expressed as:
wherein, the vector fkCan be expressed as:
variable gkCan be expressed as:
based on equation (41), the mutual coupling coefficient vector c can be expressed as:
c=F-1G (44)
as can be seen from equations (28) and (44), the mutual coupling coefficient vector is obtained based on the angle estimation value of some independent source only, and the angle estimation values of other independent sources are not utilized, so that the estimation accuracy is low. In order to improve the accuracy of the estimated value of the cross-coupling vector c, according to the knowledge of the multi-source data fusion theory, L can be comprehensively utilizeddAngular estimation of incoherent sourcesEstimating a mutual coupling vector c, whereinFrom equations (28) to (44), the coefficient matrix is jointly estimated Equation (44) may be further expressed as:
5 noiseless signal covariance solution
As can be seen from equations (13) and (45), the covariance of the noise-free signal after mutual coupling compensation can be expressed as:
based on equation (46), a decoupled noise-free signal covariance matrix R can be obtained0And the mutual coupling influence among array elements is further eliminated. However, under coherent signal conditions, the noise-free signal covariance R0A rank deficit will occur such that R is based0The number of signal subspaces obtained by feature decomposition is smaller than the number of signal sources, and further, the DOA estimation of the signals cannot be effectively realized.
6 information source DOA estimation based on traditional MUSIC algorithm
(1) Spatial smoothing algorithm
The traditional space smoothing algorithm equally divides a uniform linear array into the uniform linear array by utilizing linear array translation invarianceEach sub-array including array elementsTo obtain:and isThe covariance matrix of the ith sub-matrix can be expressed as:
wherein,array steering vectors and noise power matrixes corresponding to the ith sub-array respectively;in order to be a diagonal rotation matrix,
the conventional spatial backward smoothing covariance matrix can be expressed as:
Based on equations (47) and (48), the conventional forward and backward spatial smoothing covariance matrix can be expressed as:
based on the equations (46) - (49), the noise-free covariance R0The forward and backward smoothed covariance matrix can be further expressed as:
(2) Source DOA estimation
Equation (50) front-back smooth noiseless covariance matrixThe DOA estimation can be realized by decomposing the feature space of the subspace type algorithm represented by the MUSIC method, namely, the DOA estimation can be realized by decomposing the feature space of the subspace type algorithm represented by the MUSIC methodThe characteristic value decomposition is carried out to obtain:
as can be seen from equation (51), the signal spatial spectrum estimation can be expressed as:
based on the above discussion, the joint optimization DOA estimation algorithm proposed by the present invention can be expressed as follows:
9. Solving equations (42) and (43) to obtain F and G;
10. solving the formula (45) to obtain a mutual coupling coefficient vector c;
11. estimation of the cross-coupling compensated noise-free signal covariance matrix R by equation (46)0;
15. Spatial spectrum estimation is achieved by S (θ).
Wherein,is the iteration number;initializing values for the iterations; ε is the iteration stop parameter.
The effects of the present invention can be further illustrated by the following simulations:
simulation conditions are as follows: uniform linear array, array element number M equal to 8, snapshot number N equal to 500, error parameter epsilon equal to 10-3,Degree of freedom of mutual couplingc10.5exp (j3 pi/5), i.e. simulation experiments only consider mutual coupling between adjacent array elements. The signal-to-noise ratio is defined as:whereinIs a single signal power. The root mean square error is defined as:wherein K is the number of Monte Carlo trials. The non-uniform noise power covariance matrix is assumed to be: q ═ diag {6.0, 12, 0.5, 2.5, 8.0, 1.0, 5.5, 10.0 }.
Simulation content:
simulation 1: considering incoherent signals with three incidence angles of-5 degrees, 7 degrees and 20 degrees respectively under the condition of non-uniform noise, the SNR is 0dB signal space spectrum, and the mutual coupling coefficient among array elements is assumed to be known. As shown in fig. 2, under non-uniform noise, the conventional MUSIC algorithm based on eigen-space decomposition cannot effectively identify three target angles due to rank deficiency of the covariance matrix of the received signal, and the algorithm proposed by the present invention, Wen and Liao algorithm, can correctly distinguish the three target angles. Fig. 2 shows that, compared with Wen and Liao algorithms, the algorithm provided by the invention can also significantly suppress the influence of non-uniform noise, and has better angle estimation accuracy and resolution.
Simulation 2: considering incoherent signals with three incidence angles of-5 degrees, 7 degrees and 20 degrees respectively under the conditions of non-uniform noise and mutual coupling, and the SNR is 0dB signal space spectrum. As can be seen from fig. 3, compared with the conventional MUSIC algorithm under the non-uniform noise condition, the performance of the MUSIC algorithm is further deteriorated due to the mutual coupling effect between the arrays, so that the three target angles cannot be correctly resolved. Although the Wen and Liao algorithms further eliminate the influence of non-uniform noise through alternate iteration based on an LS method, the Wen and Liao algorithms are influenced by array cross coupling, so that the Wen and Liao algorithms cannot accurately estimate three target angles. In addition, as can be seen from fig. 3, the algorithm provided by the present invention has similar DOA estimation performance to the Liao method, which depends on the mutual coupling compensation performed by the two algorithms to further eliminate the mutual coupling effect. FIG. 3 shows that compared with the conventional MUSIC and Wen algorithms, the algorithm provided by the invention has better DOA estimation performance; compared with the Liao algorithm, the algorithm provided by the invention can inhibit mutual coupling influence, and has similar DOA estimation performance.
Simulation 3: considering non-uniform noise and mutual coupling conditions, the three incident angles are respectively-5 °, 7 ° and 20 °, and the SNR is 0dB signal space spectrum, wherein the incident signals of-5 ° and 7 ° are coherent. As can be seen from fig. 4, under the conditions of non-uniform noise, mutual coupling and coherent signal source, the conventional MUSIC algorithm has failed; the mutual coupling compensation is not carried out on the FBSS algorithms proposed by Wen and S.U.Pilai, so that the two algorithms are subjected to spatial smoothing coherent decoupling failure, and the DOA of a signal source cannot be estimated; although the Liao algorithm performs mutual coupling compensation, the Liao algorithm is limited to incoherent signals, and the Liao algorithm fails to estimate coherent sources at-5 degrees and 7 degrees. However, it should be noted that under these three conditions, the proposed algorithm can correctly recognize three target angles with higher main lobe and lower side lobe. Fig. 4 shows that, compared with the conventional MUSIC and other algorithms, the proposed algorithm can not only significantly suppress the non-uniform noise effect, but also achieve mutual coupling compensation to suppress the mutual coupling effect, and has better DOA parameter estimation performance under the condition of coherent signals.
And (4) simulation: considering signals at three angles of incidence of-5 °, 7 °, and 20 °, respectively, the signal-to-noise ratio SNR [ -2: 2: 18], 100 independent Monte Carlo replicates were performed in which-5 ° and 7 ° incident signals were coherent. As can be seen from fig. 5, as the SNR increases, the RMSE of the proposed algorithm decreases continuously, and after repeated alternate iterations, the RMSE performance of the proposed algorithm is similar to the MUSIC algorithm under ideal conditions. Fig. 5 shows that the proposed algorithm still has good DOA estimation performance under non-uniform noise, mutual coupling and coherent source conditions.
And (5) simulation: considering signals at three incident angles of-5 °, 7 ° and 20 °, respectively, with a signal-to-noise ratio SNR of 5dB, 100 monte carlo independent repeat experiments were performed, where the incident signals at-5 ° and 7 ° were coherent. The non-uniform noise power is assumed to be Q ═ diag {2.0, 2.0, 1.0, 1.0, 2.0, 1.0, 1.0, 2.0},wherein WNPR is the ratio of the maximum noise power to the minimum power of non-uniform Gaussian noise (WNPR), namely WNPR ═ 2: 22]. As can be seen from fig. 6, under the conditions of non-uniform noise, mutual coupling and coherent source, the proposed algorithm still has a relatively small RMSE fluctuation range, and as the number of repeated alternating iterations increases, the RMSE of the proposed algorithm approximates to the RMSE of the conventional MUSIC under ideal conditions, thereby indicating that the proposed algorithm has better noise robustness.
In summary, the invention provides a DOA estimation method for jointly optimizing a non-uniform noise covariance matrix and a cross-coupling coefficient based on the least square theory. Firstly, based on the least square theory, the algorithm obtains a noise-free signal covariance matrix under a cross coupling condition through an alternating iteration method so as to reduce the influence of non-uniform noise; then, deriving a cross coupling coefficient matrix by utilizing the orthogonal characteristic of the signal and noise subspace to solve a noise-free signal covariance matrix after cross coupling compensation so as to eliminate cross coupling influence and overcome the influence of the cross coupling on a spatial smoothing algorithm; and finally, realizing coherent source decorrelation based on a spatial smoothing method, and realizing DOA parameter estimation by using a traditional MUSIC algorithm. Numerical simulation shows that compared with the traditional MUSIC, non-uniform noise and DOA estimation algorithm under the mutual coupling condition, the provided algorithm can better inhibit the non-uniform noise, obviously relieve the influence of the mutual coupling effect on the spatial smoothing algorithm and obviously improve the DOA estimation performance under the coherent information source condition. Therefore, the algorithm provided by the invention can provide a solid theory and implementation basis for the research of DOA estimation performance in the field of array signal processing in engineering application.
Claims (1)
1. A DOA estimation method considering mutual coupling effect under the condition of non-uniform noise is characterized by comprising the following steps:
the first step is as follows: establishing an incoherent signal model:
considering L uncorrelated far-field narrowband signalsFrom direction thetalAnd L is 1, 2, …, and L is incident on a uniform linear array with M array elements, the received signal vector is expressed as:
x(t)=As(t)+n(t) (1)
wherein x (t) ═ x1(t),x2(t),…,xM(t)]In order to receive the vector of signals,is an array flow pattern matrix, a (theta) ═ 1, a (theta), …, aM-1(θ)]TIs a steering vector with respect to a direction theta, a (theta)l)=[1,e-jα,…,e-j(M-1)α]TSteering the vector for the array of the ith source, α ═ 2 π dsin (θ)l) The/lambda is the phase difference between adjacent array elements, d and lambda are the array element spacing and the signal wavelength respectively, and d is less than or equal to lambda/2; n (t) ═ n1(t),n1(t),…,nM(t)]Non-uniform Gaussian noise that is uncorrelated, and n (t) CN (0, Q), Q is a non-uniform noise covariance matrix expressed as:
wherein, representing the noise power superimposed on the m-th array received echo,diag {. } represents a diagonalization operator;
based on equation (1), the received signal covariance is expressed as:
R=E[x(t)xH(t)]=APAH+Q=R0+Q (3)
the second step is that: establishing a mutual coupling and coherent signal model
Assume that L incident signals are non-fully coherent sources, including LdAn incoherent signal source (independent source) and LcCoherent source of incident angles thetalAnd θ'lAnd L ═ Ld+LcThe received signal model of equation (1) is rewritten as:
wherein s (t) ═ s1(t),s2(t),…,sL(t)]T,Representing a matrix of mutual coupling coefficients between the arrays;
according to the Toeplitz characteristic of the mutual coupling coefficient matrix, the mutual coupling coefficient can be expressed as:
wherein,ρk,φkrespectively representing mutual coupling coefficients ckAmplitude and phase of;represents a degree of freedom of mutual coupling, and
the mutual coupling matrix is further represented by equation (5):
the received signal covariance of equation (4) is expressed as:
wherein,represents the signal power covariance matrix, and rank (P)0)=Ld+1,CAP0AHCHRepresenting a noise-free covariance matrix under a cross-coupling condition;
the third step: LS-based noise covariance matrix Q and signal subspace B joint optimization
From equation (8), the noise-free signal covariance matrix under the cross-coupling condition can be obtained by removing Q in R, that is:
R-Q=CAP0AHCH (9)
the above noise-free covariance CAP0AHCHThe solution problem of (a) translates into an LS problem with respect to the noise covariance Q, namely:
wherein | · | purple sweetFRepresents the Frobenius norm,obtained from the covariance estimates of the received signal for N snapshots, i.e.:
rewriting formula (10) as:
wherein,resolving P by Cholesky or feature0Is obtained byAnd B and CA span the same signal subspace, namely span (B) ═ span (CA);
based on equation (10), the noise-free signal covariance matrix under cross-coupling conditions is represented as:
wherein,obtained by solving for alternative iterations of equation (12), i.e. letThe signal subspace resulting from the sub-iteration isThen it is firstObtained by a sub-iterationExpressed as:
wherein Dag {. denotes a diagonal element operator;
based on (15), the eigenvalues of equation (14) are resolved:
wherein,is as followsThe principal eigenvalue component matrix resulting from the secondary iteration, i.e.Has LdThe number of +1 large eigenvalues,for the purpose of its corresponding feature vector,andrespectively representing a noise space characteristic matrix and a characteristic vector;
as can be seen from the above, the present invention,andthe update can be repeated by equations (15) and (17) until the iteration end condition is satisfied:wherein epsilon is an iteration termination parameter;
the fourth step: mutual coupling coefficient solving
(1) Independent source angle estimation
Based on the cross-coupling model equation (4), the array steering vector is rewritten as:
where Γ (θ) is expressed as:
μkexpressed as:
αkexpressed as:
based on the expressions (19) to (21), the expression (18) is further rewritten as:
where T (θ) [ diag { a (θ) } ] Θ, Θ is expressed as:
α is represented as:
According to the fact that the array guide vector of the incident signal is orthogonal to the noise subspace characteristic vector, further obtaining:
based on equation (22), equation (25) is rewritten as:
from formula (26):
αHΥ(θ)α=0 (27)
as can be seen from the formula (27), when satisfiedNamely, it isY (θ) is a full rank matrix; however, as known from the signal subspace theory, when the incoming wave signal angle θ is an incoherent source angle, i.e., θ ═ θlY (θ) is a non-full rank matrix, i.e., matrix rank deficiency; therefore, the incoherent source angle theta is estimated according to the signal space spectrum peaklNamely:
wherein | is a matrix determinant,the incoherent source angles corresponding to the signal space spectrum peaks;
(2) mutual coupling coefficient estimation
Further obtained by equations (27) and (28):
based on the formulae (24) and (30):
further obtained by equations (24) and (31):
equation (32) is rewritten as:
the matrix equation of equation (33) is expressed as an optimization problem as follows:
wherein,to correct the vector, equation (34) is considered as a general LS optimization problem whose solution is expressed as:
further obtained by equations (24) and (37):
obtained from equations (21) and (38):
equation (39) is further simplified as:
wherein,andis composed ofVector, 0kRepresents a 1 xk zero vector; c is a vector of cross-coupling coefficients, expressed as:
wherein, the vector fkCan be expressed as:
variable gkExpressed as:
based on equation (42), the mutual coupling coefficient vector c is expressed as:
c=F-1G (45)
in order to improve the accuracy of the estimated value of the cross-coupling vector c, L is comprehensively utilized according to the multi-source data fusion theorydAngular estimation of incoherent sourcesEstimating a mutual coupling vector c, whereinAs can be seen from equations (29) to (45), the coefficient matrix is jointly estimatedEquation (45) is further expressed as:
the fifth step: noise-free signal covariance solution
As can be seen from equations (14) and (46), the covariance of the noise-free signal after mutual coupling compensation is expressed as:
and a sixth step: source DOA estimation based on traditional MUSIC algorithm
(1) Spatial smoothing algorithm
The traditional space smoothing algorithm equally divides a uniform linear array into the uniform linear array by utilizing linear array translation invarianceEach sub-array including array elementsTo obtainAnd isThe covariance matrix of the ith sub-array is then expressed as:
wherein,array steering vectors and noise power matrixes corresponding to the ith sub-array respectively;in order to be a diagonal rotation matrix,
the conventional spatial backward smoothing covariance matrix is represented as:
based on equations (48) and (49), the conventional forward and backward spatial smoothing covariance matrix is expressed as:
based on the equations (47) - (50), the noise-free covariance R0The forward and backward smoothed covariance matrix is further expressed as:
(2) source DOA estimation
Equation (51) front-back smooth noiseless covariance matrixThe DOA estimation is realized by decomposing the feature space of the subspace type algorithm represented by the MUSIC method, namely, the DOA estimation is realizedThe characteristic value decomposition is carried out to obtain:
as can be seen from equation (52), the signal spatial spectrum estimate is expressed as:
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