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 PDF

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CN108680891B
CN108680891B CN201810010456.4A CN201810010456A CN108680891B CN 108680891 B CN108680891 B CN 108680891B CN 201810010456 A CN201810010456 A CN 201810010456A CN 108680891 B CN108680891 B CN 108680891B
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CN108680891A (en
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王洪雁
房云飞
张海坤
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Dalian University
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    • 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
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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

DOA estimation method considering mutual coupling effect under non-uniform noise condition
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 signals
Figure BDA0001540116180000021
From 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,
Figure BDA0001540116180000031
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:
Figure BDA0001540116180000032
wherein,
Figure BDA0001540116180000033
Figure BDA0001540116180000034
representing the noise power superimposed on the m-th array received echo,
Figure BDA0001540116180000035
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,
Figure BDA0001540116180000036
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:
Figure BDA0001540116180000037
wherein, s t ═ s1 t，s2 t，…，sL t]T，
Figure BDA0001540116180000038
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:
Figure BDA0001540116180000041
wherein,
Figure BDA0001540116180000042
ρk，φkrespectively representing mutual coupling coefficients ckAmplitude and phase of;
Figure BDA0001540116180000043
represents a degree of freedom of mutual coupling, and
Figure BDA0001540116180000044
from equation (5), the mutual coupling matrix can be further expressed as:
Figure BDA0001540116180000045
the received signal covariance of equation (4) can be expressed as:
Figure BDA0001540116180000046
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.:
Figure BDA0001540116180000047
wherein | · | purple sweetFRepresents the Frobenius norm,
Figure BDA0001540116180000048
the covariance estimate of the received signal for N snapshots can be obtained, namely:
Figure BDA0001540116180000049
to simplify the solution of the above problem, equation (9) can be rewritten as:
Figure BDA0001540116180000051
wherein B is one
Figure BDA0001540116180000052
The substitution matrix of (a) can be expressed as:
Figure BDA0001540116180000053
wherein,
Figure BDA0001540116180000054
p may be decomposed by Cholesky or features0Is obtained by
Figure BDA0001540116180000055
And 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:
Figure BDA0001540116180000056
wherein,
Figure BDA0001540116180000057
can be obtained by solving the problem (11) in alternating iterations, i.e. by
Figure BDA0001540116180000058
The signal subspace resulting from the sub-iteration is
Figure BDA0001540116180000059
Then it is first
Figure BDA00015401161800000510
Obtained by a sub-iteration
Figure BDA00015401161800000511
Can be expressed as:
Figure BDA00015401161800000512
wherein Dag {. denotes a take diagonal element operator.
Based on (14), the eigenvalue decomposition of equation (13) can be obtained:
Figure BDA00015401161800000513
wherein,
Figure BDA00015401161800000514
is as follows
Figure BDA00015401161800000515
The principal eigenvalue component matrix resulting from the secondary iteration, i.e.
Figure BDA00015401161800000516
Has LdThe number of +1 large eigenvalues,
Figure BDA00015401161800000517
for the purpose of its corresponding feature vector,
Figure BDA00015401161800000518
and
Figure BDA00015401161800000519
respectively representing a noise spatial feature matrix and a feature vector.
Thus, first
Figure BDA00015401161800000520
The signal subspace resulting from the sub-iteration can be represented as:
Figure BDA00015401161800000521
as can be seen from the above, the present invention,
Figure BDA00015401161800000522
and
Figure BDA00015401161800000523
the 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:
Figure BDA0001540116180000061
where Γ (θ) may be expressed as:
Figure BDA0001540116180000062
μkcan be expressed as:
Figure BDA0001540116180000063
αkcan be expressed as:
Figure BDA0001540116180000064
wherein,
Figure BDA0001540116180000065
based on equations (18) - (20), equation (17) can be further rewritten as:
Figure BDA0001540116180000066
where T (θ) ═ diag { a (θ) } Θ, Θ can be expressed as:
Figure BDA0001540116180000067
α can be expressed as:
Figure BDA0001540116180000068
wherein alpha is
Figure BDA0001540116180000069
A column vector of
Figure BDA00015401161800000610
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:
Figure BDA0001540116180000071
based on equation (21), then equation (24) may be rewritten as:
Figure BDA0001540116180000072
from formula (25):
αHΥ(θ)α＝0 (26)
wherein,
Figure BDA0001540116180000073
and as is known from the formula (25),
Figure BDA0001540116180000074
can be expressed as:
Figure BDA0001540116180000075
as can be seen from the formula (27), when satisfied
Figure BDA0001540116180000076
Namely, it is
Figure BDA0001540116180000077
Y (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:
Figure BDA0001540116180000078
wherein | is a matrix determinant,
Figure BDA0001540116180000079
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):
Figure BDA00015401161800000710
wherein,
Figure BDA00015401161800000711
based on the formulas (23) and (29):
Figure BDA00015401161800000712
further obtained from formulae (23) and (30):
Figure BDA00015401161800000713
formula (31) can be rewritten as:
Figure BDA0001540116180000081
wherein,
Figure BDA0001540116180000082
the matrix equation of equation (32) can be described as an optimization problem as follows:
Figure BDA0001540116180000083
wherein,
Figure BDA0001540116180000084
to correct the vector, equation (33) can be considered as a general LS optimization problem, whose solution can be expressed as:
Figure BDA0001540116180000085
by the formulae (33) and (34),
Figure BDA0001540116180000086
can be expressed as:
Figure BDA0001540116180000087
based on the formula (35), can be obtained
Figure BDA0001540116180000088
The estimated values, namely:
Figure BDA0001540116180000089
further obtained from formulae (23) and (36):
Figure BDA00015401161800000810
wherein [ ·]kThe k-th element of the matrix vector is represented,
Figure BDA00015401161800000811
the following equations (20) and (37) can be obtained:
Figure BDA00015401161800000812
equation (38) can be further simplified as:
Figure BDA00015401161800000813
wherein,
Figure BDA00015401161800000814
and
Figure BDA00015401161800000815
is composed of
Figure BDA00015401161800000816
Vector, 0kRepresents a 1 xk zero vector; c is a mutual coupling coefficient vector, which can be expressed as:
Figure BDA00015401161800000817
according to the formula (39), when
Figure BDA0001540116180000091
When is at time
Figure BDA0001540116180000092
When in use
Figure BDA0001540116180000093
When is at time
Figure BDA0001540116180000094
Figure BDA0001540116180000095
By
Figure BDA0001540116180000096
Formula (39) can be further converted to:
Figure BDA0001540116180000097
wherein, the vector fkCan be expressed as:
Figure BDA0001540116180000098
variable gkCan be expressed as:
Figure BDA0001540116180000099
based on equation (41), the mutual coupling coefficient vector c can be expressed as:
c＝F-1G (44)
wherein,
Figure BDA00015401161800000910
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 sources
Figure BDA00015401161800000911
Estimating a mutual coupling vector c, wherein
Figure BDA00015401161800000912
From equations (28) to (44), the coefficient matrix is jointly estimated
Figure BDA00015401161800000913
Figure BDA00015401161800000914
Equation (44) may be further expressed as:
Figure BDA00015401161800000915
wherein,
Figure BDA00015401161800000916
representation matrix
Figure BDA00015401161800000917
The generalized inverse matrix of (2).
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:
Figure BDA00015401161800000918
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 invariance
Figure BDA0001540116180000101
Each sub-array including array elements
Figure BDA0001540116180000102
To obtain:
Figure BDA00015401161800001019
and is
Figure BDA0001540116180000103
The covariance matrix of the ith sub-matrix can be expressed as:
Figure BDA0001540116180000104
wherein,
Figure BDA0001540116180000105
array steering vectors and noise power matrixes corresponding to the ith sub-array respectively;
Figure BDA0001540116180000106
in order to be a diagonal rotation matrix,
Figure BDA0001540116180000107
the conventional spatial backward smoothing covariance matrix can be expressed as:
Figure BDA0001540116180000108
wherein
Figure BDA0001540116180000109
Is composed of
Figure BDA00015401161800001010
The second diagonal selection matrix.
Based on equations (47) and (48), the conventional forward and backward spatial smoothing covariance matrix can be expressed as:
Figure BDA00015401161800001011
based on the equations (46) - (49), the noise-free covariance R0The forward and backward smoothed covariance matrix can be further expressed as:
Figure BDA00015401161800001012
wherein,
Figure BDA00015401161800001013
and
Figure BDA00015401161800001014
forward and backward smoothed noiseless signal covariance, respectively.
(2) Source DOA estimation
Equation (50) front-back smooth noiseless covariance matrix
Figure BDA00015401161800001015
The 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 method
Figure BDA00015401161800001016
The characteristic value decomposition is carried out to obtain:
Figure BDA00015401161800001017
as can be seen from equation (51), the signal spatial spectrum estimation can be expressed as:
Figure BDA00015401161800001018
based on the above discussion, the joint optimization DOA estimation algorithm proposed by the present invention can be expressed as follows:
1. solving for
Figure BDA0001540116180000111
2. Initialization
Figure BDA0001540116180000112
Figure BDA0001540116180000113
ε；
3. Feature decomposition
Figure BDA0001540116180000114
Or
Figure BDA00015401161800001122
To obtain
Figure BDA0001540116180000115
Figure BDA0001540116180000116
And
Figure BDA0001540116180000117
4. solving for
Figure BDA0001540116180000118
5. Estimating
Figure BDA0001540116180000119
6. Repeating steps 3, 4 and 5 until the following conditions are met:
Figure BDA00015401161800001110
7. solving for noise subspace by step 3
Figure BDA00015401161800001111
To obtain γ (θ);
8. estimation by equation (28)
Figure BDA00015401161800001112
To calculate
Figure BDA00015401161800001113
And based on the formula (36) to obtain
Figure BDA00015401161800001114
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；
12. Solving by equations (47) and (48), respectively
Figure BDA00015401161800001115
And
Figure BDA00015401161800001116
13. solving by equations (49) and (50)
Figure BDA00015401161800001117
14. To pair
Figure BDA00015401161800001118
Performing eigenvalue decomposition to obtain noise subspace
Figure BDA00015401161800001119
15. Spatial spectrum estimation is achieved by S (θ).
Wherein,
Figure BDA00015401161800001120
is the iteration number;
Figure BDA00015401161800001121
initializing values for the iterations; ε is the iteration stop parameter.
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 signals
Figure BDA0001540116180000121
From 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,
Figure BDA0001540116180000122
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:
Figure BDA0001540116180000123
wherein,
Figure BDA0001540116180000124
Figure BDA0001540116180000125
representing the noise power superimposed on the m-th array received echo,
Figure BDA0001540116180000126
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,
Figure BDA0001540116180000131
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:
Figure BDA0001540116180000132
wherein, s t ═ s1 t，s2 t，…，sL t]T，
Figure BDA0001540116180000133
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:
Figure BDA0001540116180000134
wherein,
Figure BDA0001540116180000135
ρk，φkrespectively representing mutual coupling coefficients ckAmplitude and phase of;
Figure BDA0001540116180000136
represents a degree of freedom of mutual coupling, and
Figure BDA0001540116180000137
from equation (5), the mutual coupling matrix can be further expressed as:
Figure BDA0001540116180000138
the received signal covariance of equation (4) can be expressed as:
Figure BDA0001540116180000139
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.:
Figure BDA0001540116180000141
wherein | · | purple sweetFRepresents the Frobenius norm,
Figure BDA0001540116180000142
the covariance estimate of the received signal for N snapshots can be obtained, namely:
Figure BDA0001540116180000143
to simplify the solution of the above problem, equation (9) can be rewritten as:
Figure BDA0001540116180000144
wherein B is one
Figure BDA0001540116180000145
The substitution matrix of (a) can be expressed as:
Figure BDA0001540116180000146
wherein,
Figure BDA0001540116180000147
p may be decomposed by Cholesky or features0Is obtained by
Figure BDA0001540116180000148
And 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:
Figure BDA0001540116180000149
wherein,
Figure BDA00015401161800001410
can be obtained by solving the problem (11) in alternating iterations, i.e. by
Figure BDA00015401161800001411
The signal subspace resulting from the sub-iteration is
Figure BDA00015401161800001412
Then it is first
Figure BDA00015401161800001413
Obtained by a sub-iteration
Figure BDA00015401161800001414
Can be expressed as:
Figure BDA00015401161800001415
wherein Dag {. denotes a take diagonal element operator.
Based on (14), the eigenvalue decomposition of equation (13) can be obtained:
Figure BDA0001540116180000151
wherein,
Figure BDA0001540116180000152
is as follows
Figure BDA0001540116180000153
The principal eigenvalue component matrix resulting from the secondary iteration, i.e.
Figure BDA0001540116180000154
Has LdThe number of +1 large eigenvalues,
Figure BDA0001540116180000155
for the purpose of its corresponding feature vector,
Figure BDA0001540116180000156
and
Figure BDA0001540116180000157
respectively representing a noise spatial feature matrix and a feature vector.
Thus, first
Figure BDA0001540116180000158
The signal subspace resulting from the sub-iteration can be represented as:
Figure BDA0001540116180000159
as can be seen from the above, the present invention,
Figure BDA00015401161800001510
and
Figure BDA00015401161800001511
the 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:
Figure BDA00015401161800001512
where Γ (θ) may be expressed as:
Figure BDA00015401161800001513
μkcan be expressed as:
Figure BDA00015401161800001514
αkcan be expressed as:
Figure BDA00015401161800001515
wherein,
Figure BDA00015401161800001516
based on equations (18) - (20), equation (17) can be further rewritten as:
Figure BDA0001540116180000161
where T (θ) ═ diag { a (θ) } Θ, Θ can be expressed as:
Figure BDA0001540116180000162
α can be expressed as:
Figure BDA0001540116180000163
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:
Figure BDA0001540116180000164
based on equation (21), then equation (24) may be rewritten as:
Figure BDA0001540116180000165
from formula (25):
αHΥ(θ)α＝0 (26)
wherein,
Figure BDA0001540116180000166
and as is known from the formula (25),
Figure BDA0001540116180000167
can be expressed as:
Figure BDA0001540116180000168
as can be seen from the formula (27), when satisfied
Figure BDA0001540116180000169
Namely, it is
Figure BDA00015401161800001610
Y (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:
Figure BDA0001540116180000171
wherein | is a matrix determinant,
Figure BDA0001540116180000172
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):
Figure BDA0001540116180000173
wherein,
Figure BDA0001540116180000174
based on the formulas (23) and (29):
Figure BDA0001540116180000175
further obtained from formulae (23) and (30):
Figure BDA0001540116180000176
formula (31) can be rewritten as:
Figure BDA0001540116180000177
wherein,
Figure BDA0001540116180000178
the matrix equation of equation (32) can be described as an optimization problem as follows:
Figure BDA0001540116180000179
wherein,
Figure BDA00015401161800001710
to correct the vector, equation (33) can be considered as a general LS optimization problem, whose solution can be expressed as:
Figure BDA00015401161800001711
by the formulae (33) and (34),
Figure BDA00015401161800001712
can be expressed as:
Figure BDA00015401161800001713
based on the formula (35), can be obtained
Figure BDA00015401161800001714
The estimated values, namely:
Figure BDA00015401161800001715
further obtained from formulae (23) and (36):
Figure BDA0001540116180000181
wherein [ ·]kThe k-th element of the matrix vector is represented,
Figure BDA0001540116180000182
the following equations (20) and (37) can be obtained:
Figure BDA0001540116180000183
equation (38) can be further simplified as:
Figure BDA0001540116180000184
wherein,
Figure BDA0001540116180000185
and
Figure BDA0001540116180000186
is composed of
Figure BDA0001540116180000187
Vector, 0kRepresents a 1 xk zero vector; c is a mutual coupling coefficient vector, which can be expressed as:
Figure BDA0001540116180000188
according to the formula (39), when
Figure BDA0001540116180000189
When is at time
Figure BDA00015401161800001810
When in use
Figure BDA00015401161800001811
When is at time
Figure BDA00015401161800001812
Figure BDA00015401161800001813
By
Figure BDA00015401161800001814
Formula (39) can be further converted to:
Figure BDA00015401161800001815
wherein, the vector fkCan be expressed as:
Figure BDA00015401161800001816
variable gkCan be expressed as:
Figure BDA00015401161800001817
based on equation (41), the mutual coupling coefficient vector c can be expressed as:
c＝F-1G (44)
wherein,
Figure BDA00015401161800001818
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 sources
Figure BDA0001540116180000191
Estimating a mutual coupling vector c, wherein
Figure BDA0001540116180000192
From equations (28) to (44), the coefficient matrix is jointly estimated
Figure BDA0001540116180000193
Figure BDA0001540116180000194
Equation (44) may be further expressed as:
Figure BDA0001540116180000195
wherein,
Figure BDA0001540116180000196
representation matrix
Figure BDA0001540116180000197
The generalized inverse matrix of (2).
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:
Figure BDA0001540116180000198
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 invariance
Figure BDA0001540116180000199
Each sub-array including array elements
Figure BDA00015401161800001910
To obtain:
Figure BDA00015401161800001911
and is
Figure BDA00015401161800001912
The covariance matrix of the ith sub-matrix can be expressed as:
Figure BDA00015401161800001913
wherein,
Figure BDA00015401161800001914
array steering vectors and noise power matrixes corresponding to the ith sub-array respectively;
Figure BDA00015401161800001915
in order to be a diagonal rotation matrix,
Figure BDA00015401161800001916
the conventional spatial backward smoothing covariance matrix can be expressed as:
Figure BDA00015401161800001917
wherein
Figure BDA00015401161800001918
Is composed of
Figure BDA00015401161800001919
The second diagonal selection matrix.
Based on equations (47) and (48), the conventional forward and backward spatial smoothing covariance matrix can be expressed as:
Figure BDA0001540116180000201
based on the equations (46) - (49), the noise-free covariance R0The forward and backward smoothed covariance matrix can be further expressed as:
Figure BDA0001540116180000202
wherein,
Figure BDA0001540116180000203
and
Figure BDA0001540116180000204
forward and backward smoothed noiseless signal covariance, respectively.
(2) Source DOA estimation
Equation (50) front-back smooth noiseless covariance matrix
Figure BDA0001540116180000205
The 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 method
Figure BDA0001540116180000206
The characteristic value decomposition is carried out to obtain:
Figure BDA0001540116180000207
as can be seen from equation (51), the signal spatial spectrum estimation can be expressed as:
Figure BDA0001540116180000208
based on the above discussion, the joint optimization DOA estimation algorithm proposed by the present invention can be expressed as follows:
1. solving for
Figure BDA0001540116180000209
2. Initialization
Figure BDA00015401161800002010
Figure BDA00015401161800002011
ε；
3. Feature decomposition
Figure BDA00015401161800002012
Or
Figure BDA00015401161800002013
To obtain
Figure BDA00015401161800002014
Figure BDA00015401161800002015
And
Figure BDA00015401161800002016
4. solving for
Figure BDA00015401161800002017
5. Estimating
Figure BDA00015401161800002018
6. Repeating steps 3, 4 and 5 until the following conditions are met:
Figure BDA00015401161800002019
7. solving for noise subspace by step 3
Figure BDA00015401161800002020
To obtain γ (θ);
8. estimation by equation (28)
Figure BDA00015401161800002021
To calculate
Figure BDA00015401161800002022
And based on the formula (36) to obtain
Figure BDA00015401161800002023
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；
12. Solving by equations (47) and (48), respectively
Figure BDA0001540116180000211
And
Figure BDA0001540116180000212
13. solving by equations (49) and (50)
Figure BDA0001540116180000213
14. To pair
Figure BDA0001540116180000214
Performing eigenvalue decomposition to obtain noise subspace
Figure BDA0001540116180000215
15. Spatial spectrum estimation is achieved by S (θ).
Wherein,
Figure BDA0001540116180000216
is the iteration number;
Figure BDA0001540116180000217
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，
Figure BDA0001540116180000218
Degree of freedom of mutual coupling
Figure BDA0001540116180000219
c10.5exp (j3 pi/5), i.e. simulation experiments only consider mutual coupling between adjacent array elements. The signal-to-noise ratio is defined as:
Figure BDA00015401161800002110
wherein
Figure BDA00015401161800002111
Is a single signal power. The root mean square error is defined as:
Figure BDA00015401161800002112
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},
Figure BDA0001540116180000221
wherein WNPR is the ratio of the maximum noise power to the minimum power of non-uniform Gaussian noise (WNPR),
Figure BDA0001540116180000222
Figure BDA0001540116180000223
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 signals
Figure FDA0003302849760000011
From 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,
Figure FDA0003302849760000012
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:
Figure FDA0003302849760000013
wherein,
Figure FDA0003302849760000014
Figure FDA0003302849760000015
representing the noise power superimposed on the m-th array received echo,
Figure FDA0003302849760000016
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)
wherein,
Figure FDA0003302849760000017
is a signal waveform covariance matrix, R0＝APAHRepresenting a noise-free signal covariance;
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:
Figure FDA0003302849760000018
wherein s (t) ═ s1(t)，s2(t)，…，sL(t)]T，
Figure FDA0003302849760000019
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:
Figure FDA0003302849760000021
wherein,
Figure FDA0003302849760000022
ρk,φkrespectively representing mutual coupling coefficients ckAmplitude and phase of;
Figure FDA0003302849760000023
represents a degree of freedom of mutual coupling, and
Figure FDA0003302849760000024
the mutual coupling matrix is further represented by equation (5):
Figure FDA0003302849760000025
the received signal covariance of equation (4) is expressed as:
Figure FDA0003302849760000026
wherein,
Figure FDA00033028497600000212
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:
Figure FDA0003302849760000027
wherein | · | purple sweetFRepresents the Frobenius norm,
Figure FDA0003302849760000028
obtained from the covariance estimates of the received signal for N snapshots, i.e.:
Figure FDA0003302849760000029
rewriting formula (10) as:
Figure FDA00033028497600000210
wherein B is one
Figure FDA00033028497600000211
Is represented as:
Figure FDA0003302849760000031
wherein,
Figure FDA0003302849760000032
resolving P by Cholesky or feature0Is obtained by
Figure FDA0003302849760000033
And 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:
Figure FDA0003302849760000034
wherein,
Figure FDA0003302849760000035
obtained by solving for alternative iterations of equation (12), i.e. let
Figure FDA0003302849760000036
The signal subspace resulting from the sub-iteration is
Figure FDA0003302849760000037
Then it is first
Figure FDA0003302849760000038
Obtained by a sub-iteration
Figure FDA0003302849760000039
Expressed as:
Figure FDA00033028497600000310
wherein Dag {. denotes a diagonal element operator;
based on (15), the eigenvalues of equation (14) are resolved:
Figure FDA00033028497600000311
wherein,
Figure FDA00033028497600000312
is as follows
Figure FDA00033028497600000313
The principal eigenvalue component matrix resulting from the secondary iteration, i.e.
Figure FDA00033028497600000314
Has LdThe number of +1 large eigenvalues,
Figure FDA00033028497600000315
for the purpose of its corresponding feature vector,
Figure FDA00033028497600000316
and
Figure FDA00033028497600000317
respectively representing a noise space characteristic matrix and a characteristic vector;
then it is first
Figure FDA00033028497600000318
The signal subspace resulting from the sub-iteration is represented as:
Figure FDA00033028497600000319
as can be seen from the above, the present invention,
Figure FDA00033028497600000320
and
Figure FDA00033028497600000321
the update can be repeated by equations (15) and (17) until the iteration end condition is satisfied:
Figure FDA00033028497600000322
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:
Figure FDA00033028497600000323
where Γ (θ) is expressed as:
Figure FDA00033028497600000324
μkexpressed as:
Figure FDA0003302849760000041
αkexpressed as:
Figure FDA0003302849760000042
wherein,
Figure FDA0003302849760000043
based on the expressions (19) to (21), the expression (18) is further rewritten as:
Figure FDA0003302849760000044
where T (θ) [ diag { a (θ) } ] Θ, Θ is expressed as:
Figure FDA0003302849760000045
α is represented as:
Figure FDA0003302849760000046
wherein alpha is
Figure FDA0003302849760000047
A column vector of
Figure FDA0003302849760000048
According to the fact that the array guide vector of the incident signal is orthogonal to the noise subspace characteristic vector, further obtaining:
Figure FDA0003302849760000049
based on equation (22), equation (25) is rewritten as:
Figure FDA00033028497600000410
from formula (26):
αHΥ(θ)α＝0 (27)
wherein,
Figure FDA0003302849760000051
and as is known from the formula (26),
Figure FDA0003302849760000052
expressed as:
Figure FDA0003302849760000053
as can be seen from the formula (27), when satisfied
Figure FDA0003302849760000054
Namely, it is
Figure FDA0003302849760000055
Y (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:
Figure FDA0003302849760000056
wherein | is a matrix determinant,
Figure FDA0003302849760000057
the incoherent source angles corresponding to the signal space spectrum peaks;
(2) mutual coupling coefficient estimation
Further obtained by equations (27) and (28):
Figure FDA0003302849760000058
based on the formulae (24) and (30):
Figure FDA0003302849760000059
further obtained by equations (24) and (31):
Figure FDA00033028497600000510
equation (32) is rewritten as:
Figure FDA00033028497600000511
wherein,
Figure FDA00033028497600000512
the matrix equation of equation (33) is expressed as an optimization problem as follows:
Figure FDA00033028497600000513
wherein,
Figure FDA00033028497600000514
to correct the vector, equation (34) is considered as a general LS optimization problem whose solution is expressed as:
Figure FDA00033028497600000515
by the formulae (34) and (35),
Figure FDA00033028497600000516
expressed as:
Figure FDA00033028497600000517
based on formula (36), obtaining
Figure FDA0003302849760000061
The estimated values, namely:
Figure FDA0003302849760000062
further obtained by equations (24) and (37):
Figure FDA0003302849760000063
wherein [ ·]kThe k-th element of the matrix vector is represented,
Figure FDA0003302849760000064
obtained from equations (21) and (38):
Figure FDA0003302849760000065
equation (39) is further simplified as:
Figure FDA0003302849760000066
wherein,
Figure FDA0003302849760000067
and
Figure FDA0003302849760000068
is composed of
Figure FDA0003302849760000069
Vector, 0kRepresents a 1 xk zero vector; c is a vector of cross-coupling coefficients, expressed as:
Figure FDA00033028497600000610
from the formula (40), when
Figure FDA00033028497600000611
When is at time
Figure FDA00033028497600000612
When in use
Figure FDA00033028497600000613
When is at time
Figure FDA00033028497600000614
Figure FDA00033028497600000615
By
Figure FDA00033028497600000616
Then formula (40) further translates to:
Figure FDA00033028497600000617
wherein, the vector fkCan be expressed as:
Figure FDA00033028497600000618
variable gkExpressed as:
Figure FDA00033028497600000619
based on equation (42), the mutual coupling coefficient vector c is expressed as:
c＝F-1G (45)
wherein,
Figure FDA0003302849760000071
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 sources
Figure FDA0003302849760000072
Estimating a mutual coupling vector c, wherein
Figure FDA0003302849760000073
As can be seen from equations (29) to (45), the coefficient matrix is jointly estimated
Figure FDA0003302849760000074
Equation (45) is further expressed as:
Figure FDA0003302849760000075
wherein,
Figure FDA0003302849760000076
representation matrix
Figure FDA0003302849760000077
The generalized inverse matrix of (2);
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:
Figure FDA0003302849760000078
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 invariance
Figure FDA0003302849760000079
Each sub-array including array elements
Figure FDA00033028497600000710
To obtain
Figure FDA00033028497600000711
And is
Figure FDA00033028497600000712
The covariance matrix of the ith sub-array is then expressed as:
Figure FDA00033028497600000713
wherein,
Figure FDA00033028497600000714
array steering vectors and noise power matrixes corresponding to the ith sub-array respectively;
Figure FDA00033028497600000715
in order to be a diagonal rotation matrix,
Figure FDA00033028497600000716
the conventional spatial backward smoothing covariance matrix is represented as:
Figure FDA00033028497600000717
wherein,
Figure FDA00033028497600000718
is composed of
Figure FDA00033028497600000719
A secondary diagonal selection matrix;
based on equations (48) and (49), the conventional forward and backward spatial smoothing covariance matrix is expressed as:
Figure FDA00033028497600000720
based on the equations (47) - (50), the noise-free covariance R0The forward and backward smoothed covariance matrix is further expressed as:
Figure FDA00033028497600000721
wherein,
Figure FDA00033028497600000722
and
Figure FDA00033028497600000723
forward and backward smooth noiseless signal covariance, respectively;
(2) source DOA estimation
Equation (51) front-back smooth noiseless covariance matrix
Figure FDA0003302849760000081
The DOA estimation is realized by decomposing the feature space of the subspace type algorithm represented by the MUSIC method, namely, the DOA estimation is realized
Figure FDA0003302849760000082
The characteristic value decomposition is carried out to obtain:
Figure FDA0003302849760000083
as can be seen from equation (52), the signal spatial spectrum estimate is expressed as:
Figure FDA0003302849760000084
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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
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CN114325561A (en) * 2021-11-16 2022-04-12 西安电子科技大学 Improved coherent signal DOA estimation method based on subarray selection strategy
CN114624665B (en) * 2022-03-24 2023-11-07 电子科技大学 Mutual coupling error DOA self-correction method based on dynamic parameter iterative optimization
CN114966640B (en) * 2022-07-29 2022-12-09 宁波博海深衡科技有限公司 Orientation estimation method and system based on array background noise statistics covariance estimation

Citations (8)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
DE69933807D1 (en) * 1998-03-24 2006-12-14 Nec Corp Direction finding device for processing measured data
CN105068041A (en) * 2015-08-28 2015-11-18 哈尔滨工程大学 Single-base MIMO radar angle estimation method based on covariance vector sparse representation under cross coupling condition
CN105974366A (en) * 2016-04-29 2016-09-28 哈尔滨工程大学 Four-order cumulant sparse representation-based MIMO (multiple-input-multiple-output) radar direction of arrival estimation method under mutual coupling condition
CN106291449A (en) * 2016-08-04 2017-01-04 大连大学 Direction of arrival angular estimation new method under symmetric-stable distribution noise
CN106980106A (en) * 2017-04-21 2017-07-25 天津大学 Sparse DOA estimation method under array element mutual coupling
CN107037397A (en) * 2017-06-21 2017-08-11 哈尔滨工业大学 A kind of method that a variety of array errors are corrected in Mutual coupling
CN107290730A (en) * 2017-05-10 2017-10-24 长江大学 Bistatic MIMO radar angle evaluation method under array mutual-coupling condition
CN107315162A (en) * 2017-07-25 2017-11-03 西安交通大学 Far field DOA Estimation in Coherent Signal method with Wave beam forming is converted based on interpolation

Family Cites Families (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US6812889B2 (en) * 2002-01-24 2004-11-02 Motorola, Inc. Methods and apparatus for determining a direction of arrival in a wireless communication system
FR2940461B1 (en) * 2008-12-23 2011-01-21 Thales Sa METHOD FOR DETERMINING ANGLES OF ARRIVAL INTO AZIMUT AND ELEVATION OF COHERENT SOURCES

Patent Citations (8)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
DE69933807D1 (en) * 1998-03-24 2006-12-14 Nec Corp Direction finding device for processing measured data
CN105068041A (en) * 2015-08-28 2015-11-18 哈尔滨工程大学 Single-base MIMO radar angle estimation method based on covariance vector sparse representation under cross coupling condition
CN105974366A (en) * 2016-04-29 2016-09-28 哈尔滨工程大学 Four-order cumulant sparse representation-based MIMO (multiple-input-multiple-output) radar direction of arrival estimation method under mutual coupling condition
CN106291449A (en) * 2016-08-04 2017-01-04 大连大学 Direction of arrival angular estimation new method under symmetric-stable distribution noise
CN106980106A (en) * 2017-04-21 2017-07-25 天津大学 Sparse DOA estimation method under array element mutual coupling
CN107290730A (en) * 2017-05-10 2017-10-24 长江大学 Bistatic MIMO radar angle evaluation method under array mutual-coupling condition
CN107037397A (en) * 2017-06-21 2017-08-11 哈尔滨工业大学 A kind of method that a variety of array errors are corrected in Mutual coupling
CN107315162A (en) * 2017-07-25 2017-11-03 西安交通大学 Far field DOA Estimation in Coherent Signal method with Wave beam forming is converted based on interpolation

Non-Patent Citations (5)

* Cited by examiner, † Cited by third party
Title
Comparison of circular,Uniform-and non Uniform Y-Shaped Array Antenna for DOA Estimation using Music Algorithm;M.Hajian等;《2006 European Conference on Wireless Technology》;20070115;第143-146页 *
Mutual coupling effect and compensation in non-uniform arrays for direction-of-arrival estimation;Elie BouDaher等;《Digital Signal Processing》;20170228;第61卷(第2期);第3-14页 *
一种互耦和相干源条件下的DOA估计方法;廖斌;《雷达科学与技术》;20080430;第6卷(第2期);第151-154页 *
均匀线阵混合信源DOA估计与互耦误差自校正;景小荣等;《***工程与电子技术》;20140930;第36卷(第9期);第1696-1701页 *
非线性阵列DOA估计算法研究;潘捷;《中国优秀博硕士学位论文全文数据库(博士)信息科技辑(月刊)》;20141115(第11期);第I136-11页 *

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