CN112611999B - Electromagnetic vector sensor array angle estimation method based on double quaternions - Google Patents
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Abstract
The invention discloses an electromagnetic vector sensor array angle estimation method based on double quaternions, which introduces the double quaternions on the basis of a quaternion model and establishes a time domain tri-orthogonal electric dipole sensor array model. Orthogonal information among array element electric dipoles is effectively reserved, and orthogonality among quaternions is more restrictive than vector orthogonality, so that better angle measurement performance can be realized. Through the dual quaternion multiple signal classification method BQ-MUSIC, the arrival angle estimation is carried out, under the condition of polarization parameter prior, better angle measurement performance than LV-MUSIC is realized through the BQ-MUSIC, the operation amount is reduced, and meanwhile, the robustness of related noise is improved. In addition, the dual-quaternion method not only expands a component on the basis of the quaternion model of the two-component sensor, but also has the potential of solving other problems of the quaternion model.
Description
Technical Field
The invention relates to the field of vector sensor signal processing, in particular to an electromagnetic vector sensor array angle estimation method based on double quaternions.
Background
A complete electromagnetic vector sensor consists of three electric dipoles and three magnetic rings, wherein the electric dipoles and the magnetic rings are mutually orthogonal in space. An electromagnetic vector sensor array (also called a polarization sensitive array) can acquire not only airspace information by a phase difference between array elements like a scalar array, but also direction information contained in a polarization direction vector. The Long Vector (LV) method is a classical vector sensor signal processing method, and the method directly arranges a plurality of output components of each sensor together to form a column vector, and has the advantages of simple implementation, clear meaning and the like. However, this arrangement simply 'piles' the output components together, completely discarding the inherent orthogonal constraint between the individual components of the array element. The quaternion is used as the extension of complex numbers, and has strict orthogonality, so that the quaternion method is naturally introduced into the signal processing of the electromagnetic vector sensor array, and has the advantages of small operand, high noise robustness and the like. Bulow and Sommer originally introduced an supercomplex tool into the signal processing of the array. Bihan and Mirson et al have studied to summarize and illustrate the basic operation of quaternion and applied isomorphic properties to solve the problem of singular value decomposition of quaternion matrix, and have cleared the mathematical barrier of quaternion application in array signal processing. Bihan and Mirson et al also established a uniform linear array frequency domain model of a two-component electromagnetic vector sensor and demonstrated the feasibility of applying quaternions to vector sensor DOA. Based on the model, two persons realize double estimation of airspace and polarization domain by using a Quaternion multiple signal classification method (Q-MUSIC), and the important explanation is that Quaternion orthogonality has stricter constraint conditions than long vector orthogonality, which is also an important reason that Quaternion model has superior estimation performance than long vector model. Bihan and Mirson et al also built a dual quaternion model for a three-component vector sensor array. However, the models created by two persons are built on the frequency domain, and the problem of undefined physical meaning exists. In contrast, quaternion time domain models are widely studied and improved algorithms have evolved greatly due to the relatively clear physical meaning. XIAOFENG and the like derive a double quaternion time domain signal model of the complete electromagnetic vector sensor from the view of Maxwell's equations, but the transplantation is difficult and difficult to popularize because the traditional polarized domain guide vector is not utilized.
Disclosure of Invention
The invention aims to provide an electromagnetic vector sensor array angle estimation method based on double quaternions, which has better angle measurement performance, reduces the operand and improves the robustness of related noise.
In order to achieve the above purpose, the invention is implemented according to the following technical scheme:
an electromagnetic vector sensor array angle estimation method based on double quaternions comprises the following steps:
s1, introducing double quaternions on the basis of a quaternion model, and establishing a time domain tri-orthogonal electric dipole sensor array model;
s2, performing angle-of-arrival estimation through a dual-quaternion multiple signal classification method BQ-MUSIC.
Further, the step S1 specifically includes:
s101, byRepresenting the real number field, ++>Representing a complex field, ">Then the complex field representing the virtual unit I, < >>Representing a quaternion field>Represents a bi-quaternion domain, delta represents a quaternion or a bi-quaternion, x represents the conjugate of the complex number with respect to the imaginary part I, H represents the conjugate transpose of the complex matrix, +.>Representing quaternion conjugates, ">Representing quaternions or bi-quaternionsComplex conjugate transpose on the number matrix;
double quaternionThe definition is as follows:
q Δ ＝q 0 +q 1 i+q 2 j+q 3 k (1)；
where i, j, k are three imaginary units of a quaternion,
the quaternion dummy units are exchangeable with the plural dummy units, namely:
ij＝-ji＝k，jk＝-kj＝i，ki＝-ik＝j，ki＝-ik＝j，iI＝Ii，jI＝Ij，kI＝Ik (2)；
the dual quaternion conjugate is defined as:
the modulus of the dual quaternion is defined as:
double quaternion matrixWritten as b=b 1 +IB 2 Form (B), wherein B 1 ，B 2 ∈H M×N It accompanies the quaternion matrix gamma B ∈H 2M×2N Expressed as:
if matrix B is a Hermitian matrix, then matrix gamma B Also a Hermitian matrix, for gamma B The characteristic decomposition is carried out:
wherein,is gamma B Diagonal matrix of right eigenvalues, ++>Is a matrix composed of eigenvectors corresponding to eigenvalues in a diagonal matrix D, and B and gamma can be known by using isomorphic relation B With the same eigenvalues, the eigenvector matrix of the matrix satisfies the relationship:
wherein,ψ N ＝(I N ，-II N )，I N is an N x N unit array;
the characteristic decomposition results D and U of the double quaternion Hermitian matrix are obtained B Verification is performed by formula (8):
s102, assuming that L independent far-field complete polarized waves exist in a space, a polarization domain guiding vector of a first information source received by a vector sensor is as follows:
wherein the method comprises the steps ofθ l ∈[0，π]，/>η l ∈[-π，π]The azimuth angle, the pitch angle, the polarization auxiliary angle and the polarization phase difference of the first information source are respectively;
to fully characterize the orthogonal relationship that exists between the components, the three components are arranged in the three imaginary parts i, j, k of the quaternion, respectively. The polarization domain steering vector can be represented as a pure dual quaternion with a real part of 0:
combining the space phase shift among array elements to obtain the received data vector of the whole polarization sensitive array as follows:
in the method, in the process of the invention,is the airspace guide vector of the vector array, takes the origin of coordinates as a phase reference point, and comprises the following steps:
wherein d is the array element spacing, and lambda is the electromagnetic wave wavelength;the airspace guiding vector matrix of the array is as follows:
in the method, in the process of the invention,is composed of->A double quaternion diagonal array is formed; />For the complex envelope of the first source, L source complex envelopes form a matrix +.>The noise vector is composed of noise data received by M array elements, and the noise received by the M array elements is expressed as a pure double quaternion:
wherein n is m，x ，n m，y ，n m，z Noise data received by electric dipoles in X, Y and Z directions are respectively pointed in each array element, and a matrix A contains all direction information and is called a steering vector array;
the resulting received data vector X (t) is in the form of a polarization sensitive array double quaternion, and the real part of each element is 0.
Further, the step S2 specifically includes:
the covariance matrix of the dual quaternion received data is:
wherein, R s and R is N The signal covariance matrix and the noise covariance matrix are respectively, and when the noise on each component is uncorrelated, the signal covariance matrix and the noise covariance matrix are:
wherein I is 2M Is a unit matrix with the size of 2M multiplied by 2M, and obviously, the covariance matrix R of X (t) X Is a Hermitian matrix, and is obtained by performing characteristic decomposition by using the formula (5) -formula (8):
wherein, sigma s is a diagonal matrix composed of L large eigenvalues, sigma N Is a diagonal array consisting of 2M-L small eigenvalues, U S Is composed of the eigenvectors corresponding to large eigenvalues, representing the subspace of the signal, U N The noise subspace is represented by a feature vector corresponding to the small feature value; according to subspace theory, signal subspace U S And noise subspace U N Orthogonality, applied in the dual quaternion domain, is expressed as:
covariance matrix R by finite number of snapshot data pairs X Unbiased estimation is performed:
wherein K is the number of shots;
since the direction information of the signal is contained inAnd->Among them, the orthogonalization in the expression (18) can be used to perform +_in relation to noise subspace for the steering vector array A>The expression is:
further, the method further comprises S3, and the specific steps of S3 include:
in the dual quaternion domain, the noise covariance R of the mth array element is calculated by using the formula (14) m，N Expressed as:
in equation (22), the real part retains the autocorrelation of the noise component of the sensor, and the imaginary part is the difference in cross-correlation between the noise components; when the noise received by the electric dipoles on the sensor are not related to each other, the scalar part is reserved in the formula (22), and the covariance matrix of the array is obtained as formula (16); when there is a correlation of the real part and the imaginary part of the correlated noise component or complex noise, the reduction of the three imaginary coefficients reduces the influence of the noise correlation to a certain extent, reducing the estimation performance loss under the correlated noise.
Compared with the prior art, the method introduces double quaternions on the basis of the quaternion model, and establishes a time domain tri-orthogonal electric dipole sensor array model. Orthogonal information among array element electric dipoles is effectively reserved, and orthogonality among quaternions is more restrictive than vector orthogonality, so that better angle measurement performance can be realized. Through the dual quaternion multiple signal classification method BQ-MUSIC, the arrival angle estimation is carried out, under the condition of polarization parameter prior, better angle measurement performance than LV-MUSIC is realized through the BQ-MUSIC, the operation amount is reduced, and meanwhile, the robustness of related noise is improved. In addition, the dual-quaternion method not only expands a component on the basis of the quaternion model of the two-component sensor, but also has the potential of solving other problems of the quaternion model.
Drawings
Fig. 1 is a spectral peak search result of the time domain BQ-MUSIC obtained in the simulation experiment 1 in the simulation example.
Fig. 2 is a spectral peak search result of the time domain LV-MUSIC obtained in simulation experiment 1 in the simulation example.
Fig. 3 is a top view of the BQ-MUSIC spatial spectrum of the correlated noise in simulation experiment 2 in a simulation example.
Fig. 4 is a top view of the spatial spectrum of LV-MUSIC of the correlated noise in simulation experiment 2 in the simulation example.
Fig. 5 is a graph of the relation between the root mean square error and the signal-to-noise ratio of the BQ-MUSIC algorithm and the LV-MUSIC algorithm in the simulation experiment 3 in the simulation example.
Detailed Description
The present invention will be described in further detail with reference to the following examples in order to make the objects, technical solutions and advantages of the present invention more apparent. The specific embodiments described herein are for purposes of illustration only and are not intended to limit the invention.
The embodiment particularly discloses an electromagnetic vector sensor array angle estimation method based on double quaternions, which comprises the following steps:
introducing double quaternions on the basis of a quaternion model, and establishing a time domain tri-orthogonal electric dipole sensor array model:
to be used forRepresenting the real number field, ++>Representing a complex field, ">Then the complex field representing the virtual unit I, < >>Representing a quaternion field>Represents a bi-quaternion domain, delta represents a quaternion or a bi-quaternion, x represents the conjugate of the complex number with respect to the imaginary part I, H represents the conjugate transpose of the complex matrix, +.>Representing quaternion conjugates, ">Representing a complex conjugate transpose on a quaternion or a dual quaternion matrix;
the dual quaternion and quaternion are found by Hamilation, except that the coefficients of the imaginary part and the real part of the dual quaternion are expanded from the real number domain to the complex number domain, so the dual quaternionThe definition is as follows:
q Δ ＝q 0 +q 1 i+q 2 j+q 3 k (1)；
where i, j, k are three imaginary units of a quaternion,
in the double quaternion operation, the multiplication between quaternion virtual units is also not exchangeable, but the quaternion virtual units are exchangeable with complex virtual units, namely:
ij＝-ji＝k，jk＝-kj＝i，ki＝-ik＝j，ki＝-ik＝j，iI＝Ii，jI＝Ij，kI＝Ik (2)；
the dual quaternion conjugate is defined as:
the modulus of the dual quaternion is defined as:
the eigenvalues and eigenvectors of the quaternion matrix are typically obtained by an isomorphic complex domain adjoint matrix, and the decomposition of the dual quaternion matrix also follows this idea. Double quaternion matrixWritten as b=b 1 +IB 2 Form (B), wherein B 1 ，B 2 ∈H M ×N It accompanies the quaternion matrix gamma B ∈H 2N×2N Expressed as:
if matrix B is a Hermitian matrix, then matrix gamma B Also a Hermitian matrix, for gamma B The characteristic decomposition is carried out:
wherein,is gamma B Diagonal matrix of right eigenvalues, ++>Is a matrix composed of eigenvectors corresponding to eigenvalues in a diagonal matrix D, and B and gamma can be known by using isomorphic relation B With the same eigenvalues, the eigenvector matrix of the matrix satisfies the relationship:
wherein,ψ N ＝(I N ，-II N )，I N is an N x N unit array;
the characteristic decomposition results D and U of the double quaternion Hermitian matrix are obtained B Verification is performed by formula (8):
assuming that there are L independent far-field completely polarized waves in space, the polarization domain steering vector of the first source received by the vector sensor is:
wherein the method comprises the steps ofθ l ∈[0，π]，/>η l ∈[-π，π]The azimuth angle, the pitch angle, the polarization auxiliary angle and the polarization phase difference of the first information source are respectively;
to fully characterize the orthogonal relationship that exists between the components, the three components are arranged in the three imaginary parts i, j, k of the quaternion, respectively. The polarization domain steering vector can be represented as a pure dual quaternion with a real part of 0:
this representation contains both direction and polarization information while preserving the orthogonal information inherent to the vector sensor that is discarded by the long vector method. Combining the space phase shift among array elements to obtain the received data vector of the whole polarization sensitive array as follows:
in the method, in the process of the invention,is the airspace guide vector of the vector array, takes the origin of coordinates as a phase reference point, and comprises the following steps:
wherein d is the array element spacing, and lambda is the electromagnetic wave wavelength;the airspace guiding vector matrix of the array is as follows:
in the method, in the process of the invention,is composed of->A double quaternion diagonal array is formed; />For the complex envelope of the first source, L source complex envelopes form a matrix +.>The noise vector is composed of noise data received by M array elements, and the noise received by the M array elements is expressed as a pure double quaternion:
wherein n is m，x ，n m，y ，n m，z Noise data received by electric dipoles in X, Y and Z directions are respectively pointed in each array element, and a matrix A contains all direction information and is called a steering vector array;
the resulting received data vector X (t) is in the form of a polarization sensitive array double quaternion, and the real part of each element is 0.
Performing angle of arrival estimation by using a dual quaternion multiple signal classification method BQ-MUSIC:
the covariance matrix of the dual quaternion received data is:
wherein R is s And R is N The signal covariance matrix and the noise covariance matrix are respectively, and when the noise on each component is uncorrelated, the signal covariance matrix and the noise covariance matrix are:
wherein I is 2M Is a unit matrix with the size of 2M multiplied by 2M, and obviously, the covariance matrix R of X (t) X Is a Hermitian matrix, and is obtained by performing characteristic decomposition by using the formula (5) -formula (8):
wherein, sigma s is a diagonal matrix composed of L large eigenvalues, sigma N Is a diagonal array consisting of 2M-L small eigenvalues, U S Is composed of the eigenvectors corresponding to large eigenvalues, representing the subspace of the signal, U N The noise subspace is represented by a feature vector corresponding to the small feature value; according to subspace theory, signal subspace U s And noise subspace U N Orthogonality, applied in the dual quaternion domain, is expressed as:
covariance matrix R by finite number of snapshot data pairs X Unbiased estimation is performed:
wherein K is the number of shots;
since the direction information of the signal is contained inAnd->Among them, the orthogonalization in the expression (18) can be used to perform +_in relation to noise subspace for the steering vector array A>The expression is:
it is emphasized that, in comparison with LV - MUSIC, BO-MUSIC reduces the maximum number of sources that can be estimated. For an array containing M array elements, the long vector model can estimate parameters of at most 3M-1 sources, while the dual quaternion can only estimate parameters of at most M-1 sources, because the stronger orthogonality constraint imposed between the signal subspace and the noise subspace comes at the cost of a reduced subspace dimension. This is generally considered worthwhile because the stronger constraints bring additional benefits such as increased robustness of the algorithm to correlated noise, model errors. In addition, because the algorithm is established on the time domain, the feature vector obtained after feature decomposition does not contain polarization information, and the polarization parameter cannot be estimated, so the method is only suitable for the situation that the polarization parameter is known or estimated in advance.
In the long vector algorithm, the received data matrix is expressed as:
X＝[X 1 ，X 2 ，X 3 ] T (21)；
wherein,is the received data vector of the electric dipoles in the X, Y and Z directions. The data covariance matrix size of the M array elements is 3m×3m. Assuming that a memory cell stores a real number or a real number as an imaginary I-coefficient, the memory space required for the covariance matrix in the long vector method is 18M 2 . And the size of the double quaternion covariance matrix is only M 2 Although each dual-quaternion requires 8 memory cells to store, the memory space required by the corresponding dual-quaternion covariance matrix is 8M 2 But the memory occupation is still reduced by nearly half compared with the long vector method, and the memory access time and the running speed are also accelerated.
In the dual quaternion domain, the noise covariance R of the mth array element is calculated by using the formula (14) m，N Expressed as:
in equation (22), the real part retains the autocorrelation of the noise component of the sensor, and the imaginary part is the difference in cross-correlation between the noise components; when the noise received by the electric dipoles on the sensor are not related to each other, the scalar part is reserved in the formula (22), and the covariance matrix of the array is obtained as formula (16); when there is a correlation of the real part and the imaginary part of the correlated noise component or complex noise, the reduction of the three imaginary coefficients reduces the influence of the noise correlation to a certain extent, reducing the estimation performance loss under the correlated noise.
Simulation instance
To verify the feasibility of the invention, the following experiments were performed separately:
simulation experiment 1: in order to estimate azimuth angle and pitch angle simultaneously, an L-shaped polarization sensitive array is adopted in the experiment, the number of array elements is 10, namely 5 tri-orthogonal electric dipole array elements are respectively placed on an x axis and a y axis, and the distance between the array elements is half wavelength. The algorithm time domain BQ-MUSIC algorithm presented herein is compared to the LV-MUSIC algorithm. Assuming that two independent incoherent information sources exist in the space, the information sources are incident on the array from far fields in different directions, and the incident angle isThe two sources are set to the same polarization parameter (γ, η) = (50 °,30 °) for comparison. Under the incoherent noise environment with the signal-to-noise ratio of 5dB, the snapshot number is set to 512, and the spectral peak search results of the time domains BQ-MUSIC and LV-MUSIC obtained under the experimental conditions are shown in the figures 1 and 2 respectively.
Comparing the two spatial spectrums, the spatial spectrum of the BQ-MUSIC has two obvious spectrum peaks, and the peak values of the two main lobes are about 20 dB. In the spatial spectrum of LV-MUSIC, the peak value of the spectral peak 1 does not reach 20dB, and the spectral peak 2 is not obvious enough. From the above comparison, it can be found that the time domain BQ-MUSIC algorithm proposed herein can better distinguish two sources from the spatial domain, and this gap is essentially a result of the well-characterized quadrature information between electric dipoles.
Simulation experiment 2: the experimental conditions in the experiment 1 are kept unchanged, only the internal noise of the vector sensor is changed into coherent noise, meanwhile, the signal to noise ratio is kept unchanged, and the experiment 1 is repeated to obtain a spatial spectrum top view of two algorithms. As shown in fig. 3 and 4, it can be found by comparison that the BQ-MUSIC is still far better than LV-MUSIC in spite of the reduced performance of the two algorithms under the relevant noise. The reason is that the subtraction of the imaginary part in equation (22) reduces the cross-correlation, achieves a noise 'whitening' like effect, and reduces the estimated performance penalty in correlated noise situations.
Simulation experiment 3: because the special double quaternion circuit is not developed yet, a large amount of resources are consumed during simulation, so that the time consumption of the experiment is long, and the simulation experiment is not suitable for carrying out Monte Carlo simulation experiments in a two-dimensional spectrum peak searching mode. Therefore, in experiment 3, a uniform linear array polarization sensitive array with 6 array elements is adopted to search a one-dimensional spectrum peak. The incident angle θ=15° of the signal source, the polarization parameter is the same as that of experiment 1, the snapshot number is set to 512, and through 500 monte carlo experiments, the relation between the root mean square error and the signal to noise ratio of the spatial estimation of the signal by the two algorithms is given as shown in fig. 5. Root mean square error is defined asWherein->Is an estimate of θ.
As can be seen from fig. 5, the time domain BQ-MUSIC algorithm proposed herein always performs better than LV-MUSIC at lower signal-to-noise ratios. With the increase of the signal-to-noise ratio, the estimation errors of the two are smaller and smaller, and finally, the real incidence angle of the signal source is very close.
According to the simulation example, under the condition of polarization parameter priori, better angle measurement performance than LV-MUSIC is realized through BQ-MUSIC, the operation amount is reduced, and meanwhile, the robustness of related noise is improved. I.e. to verify the feasibility of the invention.
The technical scheme of the invention is not limited to the specific embodiment, and all technical modifications made according to the technical scheme of the invention fall within the protection scope of the invention.
Claims (1)
1. The electromagnetic vector sensor array angle estimation method based on the double quaternions is characterized by comprising the following steps of:
s1, introducing a double quaternion on the basis of a quaternion model, and establishing a time domain tri-orthogonal electric dipole sensor array model:
s101, byRepresenting the real number field, ++>Representing a complex field, ">Then the complex field representing the virtual unit I, < >>Representing a quaternion field,represents a dual quaternion domain, delta represents a dual quaternion, and x representsConjugate of complex number with respect to imaginary part I, H representing the conjugate transpose of complex matrix, < >>Representing quaternion conjugates, ">Representing a complex conjugate transpose on a quaternion or a dual quaternion matrix;
double quaternionThe definition is as follows:
q Δ ＝q 0 +q 1 i+q 2 j+q 3 k (1)；
where i, j, k are three imaginary units of a quaternion,
the quaternion dummy units are exchangeable with the plural dummy units, namely:
ij＝-ji＝k，jk＝-kj＝i，ki＝-ik＝j，ki＝-ik＝j，iI＝Ii，jI＝Ij，kI＝Ik (2)；
the dual quaternion conjugate is defined as:
the modulus of the dual quaternion is defined as:
double quaternion matrixWritten as b=b 1 +IB 2 Form (B), wherein B 1 ，B 2 ∈H M×N It accompanies the quaternion matrix gamma B ∈H 2M×2N Expressed as:
if matrix B is a Hermitian matrix, then matrix gamma B Also a Hermitian matrix, for gamma B The characteristic decomposition is carried out:
wherein,is gamma B Diagonal matrix of right eigenvalues, ++>Is a matrix composed of eigenvectors corresponding to eigenvalues in a diagonal matrix D, and B and gamma can be known by using isomorphic relation B With the same eigenvalues, the eigenvector matrix of the matrix satisfies the relationship:
wherein,ψ N ＝(I N ，-II N ),I N is an N x N unit array;
the characteristic decomposition results D and U of the double quaternion Hermitian matrix are obtained B Verification is performed by formula (8):
s102, assuming that L independent far-field complete polarized waves exist in a space, a polarization domain guiding vector of a first information source received by a vector sensor is as follows:
wherein the method comprises the steps ofθ l ∈[0，π]，/>η l ∈[-π，π]The azimuth angle, the pitch angle, the polarization auxiliary angle and the polarization phase difference of the first information source are respectively;
to fully characterize the orthogonal relationship existing between the components, three components are respectively arranged in three imaginary parts of i, j and k of the quaternion; the polarization domain steering vector is represented by a pure double quaternion with a real part of 0:
combining the space phase shift among array elements to obtain the received data vector of the whole polarization sensitive array as follows:
in the method, in the process of the invention,is the airspace guide vector of the vector array, takes the origin of coordinates as a phase reference point, and comprises the following steps:
wherein d is the array element spacing, and lambda is the electromagnetic wave wavelength;is the empty of the arrayDomain oriented vector matrix with values:
in the method, in the process of the invention,is composed of->A double quaternion diagonal array is formed; />For the complex envelope of the first source, L source complex envelopes form a matrix +.> The noise vector is composed of noise data received by M array elements, and the noise received by the M array elements is expressed as a pure double quaternion:
wherein n is m，x ，n m，y ，n m，z Noise data received by electric dipoles in X, Y and Z directions are respectively pointed in each array element, and a matrix A contains all direction information and is called a steering vector array;
thereby obtaining a received data vector X (t) in the form of a dual quaternion of the polarization sensitive array, wherein the real part of each element is 0;
s2, estimating the angle of arrival through a dual-quaternion multiple signal classification method BQ-MUSIC:
the covariance matrix of the dual quaternion received data is:
wherein R is S And R is N The signal covariance matrix and the noise covariance matrix are respectively, and when the noise on each component is uncorrelated, the signal covariance matrix and the noise covariance matrix are:
wherein I is 2M Is a unit matrix with the size of 2M multiplied by 2M, and obviously, the covariance matrix R of X (t) X Is a Hermitian matrix, and is obtained by performing characteristic decomposition by using the formula (5) -formula (8):
wherein, sigma S Is a diagonal matrix composed of L large eigenvalues, sigma N Is a diagonal array consisting of 2M-L small eigenvalues, U S Is composed of the eigenvectors corresponding to large eigenvalues, representing the subspace of the signal, U N The noise subspace is represented by a feature vector corresponding to the small feature value; according to subspace theory, signal subspace U S And noise subspace U N Orthogonality, applied in the dual quaternion domain, is expressed as:
covariance matrix R by finite number of snapshot data pairs X Unbiased estimation is performed:
wherein K is the number of shots;
since the direction information of the signal is contained inAnd->Among them, the orthogonalization in the expression (18) is used to perform the noise subspace for the steering vector array A>The expression is:
the method also comprises S3, wherein the specific steps of S3 include:
in the dual quaternion domain, the noise covariance R of the mth array element is calculated by using the formula (14) m，N Expressed as:
in equation (22), the real part retains the autocorrelation of the noise component of the sensor, and the imaginary part is the difference in cross-correlation between the noise components; when the noise received by the electric dipoles on the sensor are not related to each other, the scalar part is reserved in the formula (22), and the covariance matrix of the array is obtained as formula (16); when there is a correlation of the real part and the imaginary part of the correlated noise component or complex noise, the reduction of the three imaginary coefficients reduces the influence of the noise correlation to a certain extent, reducing the estimation performance loss under the correlated noise.
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