CN112611999A - Electromagnetic vector sensor array angle estimation method based on double quaternion - Google Patents

Electromagnetic vector sensor array angle estimation method based on double quaternion Download PDF

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CN112611999A
CN112611999A CN202011381512.9A CN202011381512A CN112611999A CN 112611999 A CN112611999 A CN 112611999A CN 202011381512 A CN202011381512 A CN 202011381512A CN 112611999 A CN112611999 A CN 112611999A
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quaternion
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CN112611999B (en
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宋玉伟
胡国平
郑桂妹
主钦宇
陈晨
王鸿帧
彭加强
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Air Force Engineering University of PLA
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    • G—PHYSICS
    • G01—MEASURING; TESTING
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    • G01S3/143—Systems for determining direction or deviation from predetermined direction by vectorial combination of signals derived from differently oriented antennae
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
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    • G01S3/46—Systems for determining direction or deviation from predetermined direction using antennas spaced apart and measuring phase or time difference between signals therefrom, i.e. path-difference systems
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Abstract

The invention discloses an electromagnetic vector sensor array angle estimation method based on a double quaternion. Orthogonal information among array element electric dipoles is effectively reserved, orthogonality among quaternions has stronger constraint than vector orthogonality, and better angle measurement performance can be realized. The arrival angle estimation is carried out by a double-quaternion multiple signal classification method BQ-MUSIC, under the condition that the polarization parameters are prior, the better angle measurement performance than that of LV-MUSIC is realized by the BQ-MUSIC, the operation amount is reduced, and the robustness of related noise is improved. In addition, the dual-quaternion method not only expands one 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

Electromagnetic vector sensor array angle estimation method based on double quaternion
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 a double-quaternion.
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. The electromagnetic vector sensor array (also called a polarization sensitive array) can acquire not only spatial information through phase difference between array elements, but also direction information contained in a polarization guide vector, like a scalar array. 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 method is simple to 'pack' the output components together, and completely discards the inherent orthogonal constraint relationship between the components of the array elements. The quaternion is taken as the extension of the complex number, and because the quaternion has strict orthogonality, the quaternion is naturally introduced into the signal processing of the electromagnetic vector sensor array, and the quaternion method also has the advantages of small operand, strong noise robustness and the like. Bulow and Sommer introduced the hypercomplex number tool into the signal processing of the array at the earliest. Bihan, Mirson and the like research and summarize and expound the basic operation of quaternion and apply isomorphic properties to solve the problem of singular value decomposition of a quaternion matrix and sweep away the mathematical obstacles when the quaternion is applied to array signal processing. Bihan, Mirson and the like also establish a uniform linear array frequency domain model of a two-component electromagnetic vector sensor and demonstrate the feasibility of applying quaternions to carry out vector sensor DOA. On the basis of the model, two people realize dual estimation of an airspace and a polarization domain by using a Quaternion multiple signal classification method (Q-MUSIC), and emphasize that Quaternion orthogonality has a stricter constraint condition than long vector orthogonality, which is also an important reason that the Quaternion model has superior estimation performance than the long vector model. Bihan and Mirson et al also establish a biquad model for a three-component vector sensor array. However, the models created by two people are built on the frequency domain, and the problem of ambiguous physical meaning exists. In contrast, quaternion time domain models are widely studied and improved algorithms have been developed thanks to the relatively clear physical meaning. From the perspective of Maxwell equations, XIAOFENG and the like derive a complete double-quaternion time domain signal model of the electromagnetic vector sensor, but because the traditional polarization domain guide vector is not utilized, the transplantation is difficult, and the popularization is difficult.
Disclosure of Invention
The invention aims to provide an electromagnetic vector sensor array angle estimation method based on a biquaternion, which has better angle measurement performance, reduces the operation amount and improves the robustness of the related noise.
In order to achieve the 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 a double quaternion on the basis of the quaternion model, and establishing a time-domain tri-orthogonal electric dipole sensor array model;
and S2, estimating the arrival angle by a double-quaternion multiple signal classification method BQ-MUSIC.
Further, the S1 specifically includes:
s101, in
Figure BDA0002809511700000021
The representation of the real number field is performed,
Figure BDA0002809511700000022
the representation of the complex field is represented by a complex field,
Figure BDA0002809511700000023
a complex field with an imaginary unit I is represented,
Figure BDA0002809511700000024
a field of a quaternion is represented,
Figure BDA0002809511700000025
denotes a biquad field, Δ denotes a quaternion or biquad, denotes the conjugate of the complex number with respect to the imaginary part I, H denotes the conjugate transpose of the complex matrix,
Figure BDA0002809511700000026
the conjugate of a quaternion is represented,
Figure BDA0002809511700000029
representing a complex conjugate transpose on a quaternion or biquaternion matrix;
double four element number
Figure BDA0002809511700000027
Is defined as:
qΔ＝q0+q1i+q2j+q3k (1)；
wherein i, j, k are three imaginary units of quaternion,
Figure BDA0002809511700000028
quaternion virtual units are interchangeable with complex virtual units, i.e.:
ij＝-ji＝k，jk＝-kj＝i，ki＝-ik＝j，ki＝-ik＝j，iI＝Ii，jI＝Ij，kI＝Ik (2)；
the double quaternion conjugate is defined as:
Figure BDA0002809511700000031
the modulus of the double quaternion is defined as:
Figure BDA0002809511700000032
double four element number matrix
Figure BDA0002809511700000033
Written as B ═ B1+IB2Form (1), wherein B1，B2∈HM×NThen it is accompanied by a quaternion matrix gammaB∈H2M×2NExpressed as:
Figure BDA0002809511700000034
if the matrix B is a Hermitian matrix, then the matrix γBIs also a Hermitian matrix, for gammaBThe characteristic decomposition is carried out as follows:
Figure BDA00028095117000000313
wherein the content of the first and second substances,
Figure BDA0002809511700000035
is gammaBA diagonal matrix formed by the right characteristic values,
Figure BDA0002809511700000036
is a matrix formed by eigenvectors corresponding to eigenvalues in a diagonal matrix D, and B and gamma can be known by utilizing isomorphic relationBWith the same eigenvalues, the eigenvector matrix of the matrix satisfies the relationship:
Figure BDA0002809511700000037
wherein the content of the first and second substances,
Figure BDA0002809511700000038
ψN＝(IN，-IIN)，INis a unit array of nxn;
the characteristic decomposition results D and U of the Hermitian matrix with double four elements are obtainedBVerification is performed by equation (8):
Figure BDA0002809511700000039
s102, assuming that L independent far-field complete polarized waves exist in the space, the polarization domain steering vector of the ith signal source received by the vector sensor is as follows:
Figure BDA00028095117000000310
wherein
Figure BDA00028095117000000311
θl∈[0，π]，
Figure BDA00028095117000000312
ηl∈[-π，π]Respectively an azimuth angle, a pitch angle, a polarization auxiliary angle and a polarization phase difference of the ith signal source;
to fully characterize the orthogonal relationship existing between the components, the three components are arranged in the i, j, k imaginary parts of the quaternion. The polarization domain steering vector can be expressed as a pure biquad number with a real part of 0:
Figure BDA0002809511700000041
combining the spatial phase shift between the array elements, the received data vector of the whole polarization sensitive array is obtained as follows:
Figure BDA0002809511700000042
in the formula (I), the compound is shown in the specification,
Figure BDA0002809511700000043
the space domain guide vector is a vector array, and the origin of coordinates is taken as a phase reference point, so that the method comprises the following steps:
Figure BDA0002809511700000044
wherein d is the array element spacing, and lambda is the electromagnetic wave wavelength;
Figure BDA0002809511700000045
a space-domain steering vector matrix, which is an array, whose values are:
Figure BDA0002809511700000046
in the formula (I), the compound is shown in the specification,
Figure BDA0002809511700000047
is formed by
Figure BDA0002809511700000048
Forming a double-quaternion diagonal array;
Figure BDA0002809511700000049
for the complex envelope of the ith source, the complex envelopes of the L sources form a matrix
Figure BDA00028095117000000410
The noise vector is formed by noise data received by M array elements, and the noise received by the mth array element is expressed by a pure double-quaternion number as follows:
Figure BDA00028095117000000411
wherein n ism，x，nm，y，nm，zNoise data received by electric dipoles in X, Y and Z directions in each array element are respectively pointed, and a matrix A contains all direction information and is called as a guide vector array;
this results in a received data vector x (t) in the form of a double quaternion of the polarization sensitive array, and where the real part of each element is 0.
Further, the S2 specifically includes:
the covariance matrix of the biquad received data is:
Figure BDA0002809511700000057
wherein the content of the first and second substances,R sand RNRespectively, a signal covariance matrix and a noise covariance matrix, and when the noise on each component is uncorrelated, the following are provided:
Figure BDA0002809511700000058
wherein, I2MIs a unit matrix of size 2M × 2M, obviously the covariance matrix R of X (t)XIs a Hermitian matrix, and is obtained by performing characteristic decomposition by using an expression (5) to an expression (8):
Figure BDA0002809511700000051
where, Σ s is a diagonal matrix composed of L large eigenvalues, ΣNIs a diagonal matrix formed by 2M-L small characteristic values, USFormed by eigenvectors corresponding to large eigenvalues, representing a signal subspace, UNThe noise subspace is represented by the eigenvectors corresponding to the small eigenvalues; according to subspace theory, the signal subspace USAnd noise subspace UNOrthogonality, applied in the dual-quad domain, is expressed as:
Figure BDA0002809511700000059
by finite snap-shot data versus covariance matrix RXCarrying out unbiased estimation:
Figure BDA0002809511700000052
wherein K is the number of snapshots;
since the direction information of the signal is contained in
Figure BDA0002809511700000053
And
Figure BDA0002809511700000054
in the method, the noise subspace of the steering vector array A can be determined by using the orthogonal relationship in the formula (18)
Figure BDA0002809511700000055
The expression of the peak search is as follows:
Figure BDA0002809511700000056
further, the method also comprises a step S3, wherein the step S3 specifically comprises:
in the biquad domain, the noise covariance R of the mth array element is expressed by equation (14)m，NExpressed as:
Figure BDA0002809511700000061
in equation (22), the real part retains the autocorrelation of the sensor noise component, and the imaginary part is the cross-correlation difference between the noise components; when the noises received by the electric dipoles on the sensor are not related to each other, only a scalar part is reserved by an equation (22), and the covariance matrix of the array is obtained as an equation (16); when correlated noise components or real part and imaginary part correlation of complex noise exists, the reduction of the three imaginary part coefficients reduces the influence of noise correlation to a certain extent, and the estimation performance loss under the correlated noise is reduced.
Compared with the prior art, the method introduces double quaternion on the basis of the quaternion model, and establishes the time-domain three-orthogonal electric dipole sensor array model. Orthogonal information among array element electric dipoles is effectively reserved, orthogonality among quaternions has stronger constraint than vector orthogonality, and better angle measurement performance can be realized. The arrival angle estimation is carried out by a double-quaternion multiple signal classification method BQ-MUSIC, under the condition that the polarization parameters are prior, the better angle measurement performance than that of LV-MUSIC is realized by the BQ-MUSIC, the operation amount is reduced, and the robustness of related noise is improved. In addition, the dual-quaternion method not only expands one 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 shows the result of the peak search in the time domain BQ-MUSIC obtained in simulation experiment 1 in the simulation example.
FIG. 2 shows the peak search results of the spectrum 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 noise-related data in simulation experiment 2 in the simulation example.
FIG. 4 is a top view of the LV-MUSIC spatial spectrum of noise related signals in simulation experiment 2 in the simulation example.
FIG. 5 is a graph of the root mean square error and the signal-to-noise ratio of the BQ-MUSIC algorithm and the LV-MUSIC algorithm in simulation experiment 3 in a simulation example.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail with reference to the following embodiments. The specific embodiments described herein are merely illustrative of the invention and do not limit the invention.
The embodiment specifically discloses an electromagnetic vector sensor array angle estimation method based on a double-quaternion, which comprises the following steps:
introducing a double-quaternion number on the basis of a quaternion model, and establishing a time-domain three-orthogonal electric dipole sensor array model:
to be provided with
Figure BDA0002809511700000071
The representation of the real number field is performed,
Figure BDA0002809511700000072
the representation of the complex field is represented by a complex field,
Figure BDA0002809511700000073
a complex field with an imaginary unit I is represented,
Figure BDA0002809511700000074
a field of a quaternion is represented,
Figure BDA0002809511700000075
denotes a biquad field, Δ denotes a quaternion or biquad, denotes the conjugate of the complex number with respect to the imaginary part I, H denotes the conjugate transpose of the complex matrix,
Figure BDA0002809511700000076
the conjugate of a quaternion is represented,
Figure BDA00028095117000000710
representing a complex conjugate transpose on a quaternion or biquaternion matrix;
double quaternions and quaternions are both found by Hamilton, except that the coefficients of the imaginary and real parts of the double quaternion extend from the real domain to the complex domain, and thus the double quaternion
Figure BDA0002809511700000077
Is defined as:
qΔ＝q0+q1i+q2j+q3k (1)；
wherein i, j, k are three imaginary units of quaternion,
Figure BDA0002809511700000078
in dual quaternion operations, the multiplication between quaternion imaginary units is also not exchangeable, but quaternion imaginary units and complex imaginary units are exchangeable, i.e.:
ij＝-ji＝k，jk＝-kj＝i，ki＝-ik＝j，ki＝-ik＝j，iI＝Ii，jI＝Ij，kI＝Ik (2)；
the double quaternion conjugate is defined as:
Figure BDA0002809511700000079
the modulus of the double quaternion is defined as:
Figure BDA0002809511700000081
eigenvalues and eigenvectors of a quaternion matrix are typically obtained through a isomorphic complex field adjoint matrix, and the decomposition of a biquaternion matrix follows this idea. Double four element number matrix
Figure BDA0002809511700000082
Written as B ═ B1+IB2Form (1), wherein B1，B2∈HM ×NThen it is accompanied by a quaternion matrix gammaB∈H2N×2NExpressed as:
Figure BDA0002809511700000083
if the matrix B is a Hermitian matrix, then the matrix γBIs also a Hermitian matrix, for gammaBThe characteristic decomposition is carried out as follows:
Figure BDA00028095117000000812
wherein the content of the first and second substances,
Figure BDA0002809511700000084
is gammaBA diagonal matrix formed by the right characteristic values,
Figure BDA0002809511700000085
is formed by corresponding characteristic values in a diagonal matrix DThe B and the gamma can be known by utilizing isomorphic relationBWith the same eigenvalues, the eigenvector matrix of the matrix satisfies the relationship:
Figure BDA0002809511700000086
wherein the content of the first and second substances,
Figure BDA0002809511700000087
ψN＝(IN，-IIN)，INis a unit array of nxn;
the characteristic decomposition results D and U of the Hermitian matrix with double four elements are obtainedBVerification is performed by equation (8):
Figure BDA0002809511700000088
assuming that there are L independent far-field fully polarized waves in the space, the polarization domain steering vector of the ith source received by the vector sensor is:
Figure BDA0002809511700000089
wherein
Figure BDA00028095117000000810
θl∈[0，π]，
Figure BDA00028095117000000811
ηl∈[-π，π]Respectively an azimuth angle, a pitch angle, a polarization auxiliary angle and a polarization phase difference of the ith signal source;
to fully characterize the orthogonal relationship existing between the components, the three components are arranged in the i, j, k imaginary parts of the quaternion. The polarization domain steering vector can be expressed as a pure biquad number with a real part of 0:
Figure BDA0002809511700000091
this representation method contains both directional and polarization information while retaining the orthogonal information inherent to vector sensors that is discarded by long vector methods. Combining the spatial phase shift between the array elements, the received data vector of the whole polarization sensitive array is obtained as follows:
Figure BDA0002809511700000092
in the formula (I), the compound is shown in the specification,
Figure BDA0002809511700000093
the space domain guide vector is a vector array, and the origin of coordinates is taken as a phase reference point, so that the method comprises the following steps:
Figure BDA0002809511700000094
wherein d is the array element spacing, and lambda is the electromagnetic wave wavelength;
Figure BDA0002809511700000095
a space-domain steering vector matrix, which is an array, whose values are:
Figure BDA0002809511700000096
in the formula (I), the compound is shown in the specification,
Figure BDA0002809511700000097
is formed by
Figure BDA0002809511700000098
Forming a double-quaternion diagonal array;
Figure BDA0002809511700000099
for the complex envelope of the ith source, the complex envelopes of the L sources form a matrix
Figure BDA00028095117000000910
The noise vector is formed by noise data received by M array elements, and the noise received by the mth array element is expressed by a pure double-quaternion number as follows:
Figure BDA00028095117000000911
wherein n ism，x，nm，y，nm，zNoise data received by electric dipoles in X, Y and Z directions in each array element are respectively pointed, and a matrix A contains all direction information and is called as a guide vector array;
this results in a received data vector x (t) in the form of a double quaternion of the polarization sensitive array, and where the real part of each element is 0.
The arrival angle estimation is carried out by a double-quaternion multiple signal classification method BQ-MUSIC:
the covariance matrix of the biquad received data is:
Figure BDA0002809511700000107
wherein R issAnd RNRespectively, a signal covariance matrix and a noise covariance matrix, and when the noise on each component is uncorrelated, the following are provided:
Figure BDA0002809511700000108
wherein, I2MIs a unit matrix of size 2M × 2M, obviously the covariance matrix R of X (t)XIs a Hermitian matrix, and is obtained by performing characteristic decomposition by using an expression (5) to an expression (8):
Figure BDA0002809511700000101
where, Σ s is a diagonal matrix composed of L large eigenvalues, ΣNIs a diagonal matrix formed by 2M-L small characteristic values, USFormed by eigenvectors corresponding to large eigenvalues, representing a signal subspace, UNThe noise subspace is represented by the eigenvectors corresponding to the small eigenvalues; according to subspace theory, the signal subspace UsAnd noise subspace UNOrthogonality, applied in the dual-quad domain, is expressed as:
Figure BDA0002809511700000109
by finite snap-shot data versus covariance matrix RXCarrying out unbiased estimation:
Figure BDA0002809511700000102
wherein K is the number of snapshots;
since the direction information of the signal is contained in
Figure BDA0002809511700000103
And
Figure BDA0002809511700000104
in the method, the noise subspace of the steering vector array A can be determined by using the orthogonal relationship in the formula (18)
Figure BDA0002809511700000105
The expression of the peak search is as follows:
Figure BDA0002809511700000106
it is emphasized that compared to LV-MUSIC, BO-MUSIC, reduces the maximum number of estimable sources. For an array containing M array elements, the long vector model can be used at mostTo estimate the parameters of 3M-1 sources, whereas the biquad number can only estimate the parameters of M-1 sources at most, since the stronger orthogonality constraint imposed between the signal subspace and the noise subspace comes at the cost of a reduced subspace dimension. This cost is generally considered to be worthwhile because the stronger constraint also brings other benefits, such as increased robustness of the algorithm to correlated noise, model errors. In addition, the algorithm is established in the time domain, so that the feature vector obtained after feature decomposition does not contain polarization information, and the polarization parameter cannot be estimated, so that the algorithm is only suitable for the condition that the polarization parameter is known or estimated in advance.
In the long vector algorithm, the received data matrix is represented as:
X＝[X1，X2，X3]T (21)；
wherein the content of the first and second substances,
Figure BDA0002809511700000111
is the received data vector of the electric dipole in the X, Y, Z direction. The data covariance matrix size of the M array elements is 3 mx 3M. Assuming that one memory cell stores one 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 18M2. And the size of the biquad covariance matrix is only M2Although each biquad requires 8 memory cells to store, the memory space required for the corresponding biquad covariance matrix is 8M2However, the memory occupation is still reduced by nearly half compared with the long vector method, and the memory access time and the operation speed are also increased.
In the biquad domain, the noise covariance R of the mth array element is expressed by equation (14)m，NExpressed as:
Figure BDA0002809511700000112
in equation (22), the real part retains the autocorrelation of the sensor noise component, and the imaginary part is the cross-correlation difference between the noise components; when the noises received by the electric dipoles on the sensor are not related to each other, only a scalar part is reserved by an equation (22), and the covariance matrix of the array is obtained as an equation (16); when correlated noise components or real part and imaginary part correlation of complex noise exists, the reduction of the three imaginary part coefficients reduces the influence of noise correlation to a certain extent, and the estimation performance loss under the correlated noise is reduced.
Simulation example
To verify the feasibility of the invention, the following experiments were performed, respectively:
simulation experiment 1: in order to estimate the azimuth angle and the pitch angle simultaneously, an L-shaped polarization sensitive array is adopted in the experiment, the number of array elements is 10, namely 5 three 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 there are two independent incoherent sources in space, incident on the array from the far field in different directions at angles of incidence
Figure BDA0002809511700000121
For comparison, the two sources set the same polarization parameters (γ, η) to (50 °, 30 °). Under the incoherent noise environment with the signal-to-noise ratio of 5dB, the fast beat number is set to 512, and fig. 1 and fig. 2 are respectively the spectral peak search results of the time domains BQ-MUSIC and LV-MUSIC obtained under the experimental conditions.
Comparing the spatial spectra of the two, the spatial spectrum of BQ-MUSIC has two obvious spectral peaks, and the peak values of the two main lobes are both about 20 dB. In the spatial spectrum of LV-MUSIC, peak 1 did not reach 20dB and peak 2 was not sufficiently distinct. From the above comparison, it can be seen that the time domain BQ-MUSIC algorithm proposed herein is able to better resolve the two sources from the spatial domain, and this difference is essentially the result of the quadrature information between the electric dipoles being well characterized.
Simulation experiment 2: and (3) keeping the experimental conditions in the experiment 1 unchanged, only changing the internal noise of the vector sensor into coherent noise, keeping the signal-to-noise ratio unchanged, and repeating the experiment 1 to obtain a spatial spectrum top view of the two algorithms. As shown in fig. 3 and fig. 4, it can be found by comparison that, although the performance of the two algorithms is degraded under correlated noise, the effect of BQ-MUSIC is still much better than that of LV-MUSIC. The reason is that the subtraction of the imaginary part in equation (22) reduces the cross-correlation, achieving a noise-like 'whitening' effect, reducing the estimation performance loss in the case of correlated noise.
Simulation experiment 3: because a special double-quaternion circuit is not developed yet, a large amount of resources are consumed during simulation, so that the experiment time is long, and the Monte Carlo simulation experiment is not suitable for being carried out in a two-dimensional spectral peak searching mode. Therefore, experiment 3 adopts a 6-array uniform linear array polarization sensitive array to perform one-dimensional spectral peak search. The source incidence angle θ is 15 °, the polarization parameter is the same as that in experiment 1, the fast beat number is set to 512, and the relationship between the root mean square error and the signal-to-noise ratio of the estimation of the signal space domain by the two algorithms is shown in fig. 5 through 500 monte carlo experiments. Root mean square error is defined as
Figure BDA0002809511700000131
Wherein
Figure BDA0002809511700000132
Is an estimate of theta.
As can be seen from fig. 5, the time-domain BQ-MUSIC algorithm proposed herein consistently performs better than LV-MUSIC at lower signal-to-noise ratios. Along with the increase of the signal-to-noise ratio, the estimation error of the two is smaller and smaller, and finally the estimation error is very close to the true incident angle of the information source.
According to the simulation example, under the condition that the polarization parameters are prior, the BQ-MUSIC is used for realizing better angle measurement performance than the LV-MUSIC, the operation amount is reduced, and the robustness of the related noise is improved. I.e. the feasibility of the invention was verified.
The technical solution of the present invention is not limited to the limitations of the above specific embodiments, and all technical modifications made according to the technical solution of the present invention fall within the protection scope of the present invention.

Claims (4)

1. An electromagnetic vector sensor array angle estimation method based on double quaternions is characterized by comprising the following steps:
s1, introducing a double quaternion on the basis of the quaternion model, and establishing a time-domain tri-orthogonal electric dipole sensor array model;
and S2, estimating the arrival angle by a double-quaternion multiple signal classification method BQ-MUSIC.
2. The method according to claim 1, wherein the S1 specifically includes:
s101, in
Figure FDA0002809511690000011
The representation of the real number field is performed,
Figure FDA0002809511690000012
the representation of the complex field is represented by a complex field,
Figure FDA0002809511690000013
a complex field with an imaginary unit I is represented,
Figure FDA0002809511690000014
a field of a quaternion is represented,
Figure FDA0002809511690000015
denotes a biquad field, Δ denotes a quaternion or biquad, denotes the conjugate of the complex number with respect to the imaginary part I, H denotes the conjugate transpose of the complex matrix,
Figure FDA0002809511690000016
the conjugate of a quaternion is represented,
Figure FDA0002809511690000017
representing a complex conjugate transpose on a quaternion or biquaternion matrix;
double four element number
Figure FDA0002809511690000018
Definition ofComprises the following steps:
qΔ＝q0+q1i+q2j+q3k (1)；
wherein i, j, k are three imaginary units of quaternion,
Figure FDA00028095116900000114
quaternion virtual units are interchangeable with complex virtual units, i.e.:
ij＝-ji＝k，jk＝-kj＝i，ki＝-ik＝j，ki＝-ik＝j，iI＝Ii，jI＝Ij，kI＝Ik (2)；
the double quaternion conjugate is defined as:
Figure FDA00028095116900000110
the modulus of the double quaternion is defined as:
Figure FDA00028095116900000111
double four element number matrix
Figure FDA00028095116900000112
Written as B ═ B1+IB2Form (1), wherein B1，B2∈HM×NThen it is accompanied by a quaternion matrix gammaB∈H2M×2NExpressed as:
Figure FDA00028095116900000113
if the matrix B is a Hermitian matrix, then the matrix γBIs also a Hermitian matrix, for gammaBThe characteristic decomposition is carried out as follows:
Figure FDA0002809511690000021
wherein the content of the first and second substances,
Figure FDA0002809511690000022
is gammaBA diagonal matrix formed by the right characteristic values,
Figure FDA0002809511690000023
is a matrix formed by eigenvectors corresponding to eigenvalues in a diagonal matrix D, and B and gamma can be known by utilizing isomorphic relationBWith the same eigenvalues, the eigenvector matrix of the matrix satisfies the relationship:
Figure FDA0002809511690000024
wherein the content of the first and second substances,
Figure FDA0002809511690000025
ψN＝(IN，-IIN)，INis a unit array of nxn;
the characteristic decomposition results D and U of the Hermitian matrix with double four elements are obtainedBVerification is performed by equation (8):
Figure FDA0002809511690000026
s102, assuming that L independent far-field complete polarized waves exist in the space, the polarization domain steering vector of the ith signal source received by the vector sensor is as follows:
Figure FDA0002809511690000027
wherein
Figure FDA00028095116900000210
θl∈[0，π]，
Figure FDA0002809511690000028
ηl∈[-π，π]Respectively an azimuth angle, a pitch angle, a polarization auxiliary angle and a polarization phase difference of the ith signal source;
to fully characterize the orthogonal relationship existing between the components, the three components are arranged in the i, j, k imaginary parts of the quaternion. The polarization domain steering vector can be expressed as a pure biquad number with a real part of 0:
Figure FDA0002809511690000029
combining the spatial phase shift between the array elements, the received data vector of the whole polarization sensitive array is obtained as follows:
Figure FDA0002809511690000031
in the formula (I), the compound is shown in the specification,
Figure FDA0002809511690000032
the space domain guide vector is a vector array, and the origin of coordinates is taken as a phase reference point, so that the method comprises the following steps:
Figure FDA0002809511690000033
wherein d is the array element spacing, and lambda is the electromagnetic wave wavelength;
Figure FDA0002809511690000034
a space-domain steering vector matrix, which is an array, whose values are:
Figure FDA0002809511690000035
in the formula (I), the compound is shown in the specification,
Figure FDA0002809511690000036
is formed by
Figure FDA0002809511690000037
Forming a double-quaternion diagonal array;
Figure FDA0002809511690000038
for the complex envelope of the ith source, the complex envelopes of the L sources form a matrix
Figure FDA0002809511690000039
Figure FDA00028095116900000310
The noise vector is formed by noise data received by M array elements, and the noise received by the mth array element is expressed by a pure double-quaternion number as follows:
Figure FDA00028095116900000311
wherein n ism，x，nm，y，nm，zNoise data received by electric dipoles in X, Y and Z directions in each array element are respectively pointed, and a matrix A contains all direction information and is called as a guide vector array;
this results in a received data vector x (t) in the form of a double quaternion of the polarization sensitive array, and where the real part of each element is 0.
3. The method according to claim 2, wherein the S2 specifically includes:
the covariance matrix of the biquad received data is:
Figure FDA00028095116900000312
wherein R issAnd RNRespectively, a signal covariance matrix and a noise covariance matrix, and when the noise on each component is uncorrelated, the following are provided:
Figure FDA00028095116900000313
wherein, I2MIs a unit matrix of size 2M × 2M, obviously the covariance matrix R of X (t)xIs a Hermitian matrix, and is obtained by performing characteristic decomposition by using an expression (5) to an expression (8):
Figure FDA0002809511690000041
where, Σ s is a diagonal matrix composed of L large eigenvalues, ΣNIs a diagonal matrix formed by 2M-L small characteristic values, USFormed by eigenvectors corresponding to large eigenvalues, representing a signal subspace, UNThe noise subspace is represented by the eigenvectors corresponding to the small eigenvalues; according to subspace theory, the signal subspace USAnd noise subspace UNOrthogonality, applied in the dual-quad domain, is expressed as:
Figure FDA0002809511690000048
by finite snap-shot data versus covariance matrix RXCarrying out unbiased estimation:
Figure FDA0002809511690000042
wherein K is the number of snapshots;
since the direction information of the signal is contained in
Figure FDA0002809511690000043
And
Figure FDA0002809511690000044
in the method, the noise subspace of the steering vector array A can be determined by using the orthogonal relationship in the formula (18)
Figure FDA0002809511690000047
The expression of the peak search is as follows:
Figure FDA0002809511690000046
4. the method of claim 3, further comprising S3, wherein the S3 specifically comprises:
in the biquad domain, the noise covariance R of the mth array element is expressed by equation (14)m，NExpressed as:
Figure FDA0002809511690000051
in equation (22), the real part retains the autocorrelation of the sensor noise component, and the imaginary part is the cross-correlation difference between the noise components; when the noises received by the electric dipoles on the sensor are not related to each other, only a scalar part is reserved by an equation (22), and the covariance matrix of the array is obtained as an equation (16); when correlated noise components or real part and imaginary part correlation of complex noise exists, the reduction of the three imaginary part coefficients reduces the influence of noise correlation to a certain extent, and the estimation performance loss under the correlated noise is reduced.
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