CN114113808B - DOA-polarization information joint estimation method based on incomplete electric vector sensor - Google Patents
DOA-polarization information joint estimation method based on incomplete electric vector sensor Download PDFInfo
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Abstract
The invention discloses a DOA-polarization information joint estimation method based on a residual electric vector sensor. The performance of the existing parameter estimation algorithm applied to the incomplete electromagnetic vector sensor is seriously reduced under the condition of low signal-to-noise ratio. In addition, most algorithms require spectral searches or parameter matching, which results in a very complex computational process. The invention is as follows: 1. first, two subarrays are divided, and then operation is carried out based on a cross correlation matrix of the two subarrays. 2. The method greatly improves the parameter estimation performance under the condition of low signal-to-noise ratio, and the algorithm does not need spectrum search or parameter matching, so that the calculation speed is further improved, and the simulation experiment result verifies the effectiveness of the algorithm on signal DOA and polarization parameter estimation.
Description
Technical Field
The invention belongs to the technical field of array signal processing, and particularly relates to a DOA (Direction Of Arrival) -polarization information joint estimation method based on a residual electric vector sensor.
Background
Polarization sensitive arrays are an important branch of the array signal processing field, and are widely used in various fields such as communication, radar, military and the like. Compared with a scalar array, the polarization sensitive array has stronger anti-interference capability, more robust detection capability and higher resolution capability.
The number of receiving channels of the complete electromagnetic vector sensor is six times that of the scalar sensor, and the magnetic ring and the electric dipole are arranged on the same array element, so that the polarization sensitive array is more complex to calculate than the scalar array, and the polarization sensitive array has more serious mutual coupling effect, so that the practical application of the polarization sensitive array is limited. The incomplete electric vector sensor consists of complete electromagnetic vector sensor antenna, and is commonly used as a single dipole and a double orthogonal dipole pair. Single dipoles can minimize cross coupling effects, but in some cases can lose polarization information; the mutual coupling effect of the pair of biorthogonal dipoles, although slightly higher, can better acquire and utilize polarization information. Therefore, the two are often used in combination in engineering.
In the last thirty years, students at home and abroad are increasingly active in researching polarization sensitive arrays, and classical parameter estimation algorithms of a plurality of scalar arrays are popularized in the field of polarization sensitive arrays. The most notable of these is the MUSIC algorithm. The rise of such subspace decomposition type algorithms enables a crossing of modern super resolution techniques. With the intensive research of MUSIC algorithm, scholars have proposed a great number of improved algorithms, including feature vector method, weighted MUSIC algorithm, multidimensional MUSIC algorithm, etc. The algorithms can accurately distinguish a plurality of signals in one wave beam, but the algorithms need to perform spectrum searching, so that the calculated amount is increased, and the practical application is influenced. In order to shorten the operation time, scholars have proposed an ESPRIT algorithm, which obtains two identical subarrays through a certain variation, and obtains a phase difference according to the rotation-invariant property between subarrays, so as to estimate the DOA and polarization parameters, unlike a MUSIC algorithm using the characteristic of orthogonality of subspaces. Since the ESPRIT algorithm is small in calculation amount and easy to implement, many derivative algorithms, such as weighted ESPRIT algorithm, real-valued space ESPRIT algorithm, matrix-bundle ESPRIT algorithm, etc., have been proposed by scholars. These ESPRIT-like methods, while not requiring spectral peak searching, require additional pairing of spatial angle and polarization parameters, which on the other hand increases the complexity of the algorithm.
The above mentioned algorithm is not ideal in addition to high computational complexity, and estimation performance in low signal-to-noise ratio situations. This is because these algorithms require eigenvalue decomposition of the autocorrelation matrix of the received signal and then parameter estimation using the decomposed noise subspace. Such noise subspace-based algorithms gradually lose reliability as system noise increases. Therefore, the invention provides a DOA-polarization parameter joint estimation method independent of noise subspace. Compared with the traditional MUSIC algorithm and ESPRIT algorithm, the method has the advantages that the calculated amount is remarkably reduced, and the estimation accuracy is higher when the signal to noise ratio is low.
Disclosure of Invention
The invention aims at least to solve the technical problem of independent noise subspace, and provides a DOA-polarization information joint estimation method based on a residual electric vector sensor.
The specific steps of the invention are as follows:
firstly, constructing a model of an electric vector of an electromagnetic signal in space;
step two, constructing an array data receiving model;
step three, calculating a cross correlation matrix according to the received data models of the first linear array and the second linear array obtained in the step two;
step four, compressing and smoothing the cross-correlation matrix in sequence;
step five, utilizing a smoothing matrix R m The rotation invariance relation among the row vectors is used for solving a pitch angle estimated value, and the pitch angle estimated value is reversely replaced back to the smoothing matrix R m Then using a known matrix U of substituted pitch angle estimation values m And smoothing matrix R m Further solving an unknown matrix V containing azimuth angle, polarization auxiliary angle and polarization phase difference information m ；
Step six, according to U in step five m and Vm The corresponding relation and the quaternion property are sequentially solved to obtain the azimuth angle estimated valuePolarization auxiliary angle estimation value +.>And polarization phase difference estimation value->
The beneficial effects of the invention are as follows:
1. the invention aims at a polarization sensitive array formed by incomplete electromagnetic vector sensors, divides the array into two subarrays, builds a receiving signal model of the polarization array, and provides a cross-correlation matrix of the two subarrays and carries out operation based on the cross-correlation matrix. Compared with the polarized MUSIC algorithm, the algorithm has smaller operand, does not need parameter matching, and is more flexible compared with the ESPRIT algorithm.
2. Aiming at the problems of large data receiving quantity and serious array element mutual coupling effect of the existing co-point polarization sensitive array, the invention provides a DOA-polarization joint estimation algorithm based on a incomplete electric vector sensor. The complete electromagnetic vector sensor in the array element is replaced by the incomplete electromagnetic vector sensor, so that the mutual coupling effect of the array is reduced while the dimension of the received data is reduced. The algorithm of the invention is verified by simulation, and still keeps higher estimation accuracy under the condition of low signal-to-noise ratio.
Drawings
FIG. 1 is an overall flow chart of the present invention;
FIG. 2 is a block diagram of a defective electrical vector polarization sensitive array of the present invention;
FIGS. 3 (a) and (b) are, respectively, electromagnetic signal propagation and exploded views of the present invention;
FIG. 4 is a specific compression smoothing process for a cross-correlation matrix in accordance with the present invention;
FIG. 5 is a simulation result of the polarized MUSIC method of the present invention;
FIG. 6 is a simulated graph of two-dimensional spatial angle and two-dimensional polarization angle estimated by the method of the present invention; (a) azimuth and pitch coordinates; (b) polarization co-angle and polarization phase difference coordinates;
FIG. 7 is a plot of RMSE versus SNR for DOA parameters in accordance with the present invention;
fig. 8 is a graph showing RMSE of polarization parameters of the present invention as a function of SNR.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
The DOA-polarization information joint estimation method based on the incomplete electric vector sensor shown in fig. 1 comprises the following steps:
step one: model building is carried out on the electric vector of the electromagnetic signal in space:
generating an envelope S (t) of a linear frequency modulation signal according to a preset snapshot number, an information source number, a modulation frequency, a carrier frequency, a bandwidth and a sampling rate; setting a pitch angle theta and an azimuth angle of an incoherent information sourceThe polarization auxiliary angle gamma and the polarization phase difference eta are four electromagnetic signal parameters; setting related parameters of the array, two parallelArray x 1 and x2 And array element spacing d.
As shown in FIG. 3, the pitch angle theta is defined as the angle between the incident electromagnetic signal and the positive direction of the z axis, and 0.ltoreq.theta.ltoreq.pi; azimuth angleFor the angle between the projection of the incident electromagnetic signal on the XY plane and the positive x-axis, +.>The electric field vector E of the incident electromagnetic signal is in two orthogonal unit vectors E θ ,/>Decomposition in the direction V θ and />Two components of the electric field vector E are denoted as:
wherein gamma is E [0, pi/2]And eta epsilon [ -pi, pi]Respectively representing the polarization auxiliary angle and the polarization phase difference, and the tangent value of gamma represents e θ Amplitude of the directional electric fieldThe ratio of the magnitudes of the directional electric fields, η, represents e θ Directional electric field and +.>The phase difference of the directional electric field, j, represents the imaginary part of the complex number.
In a space rectangular coordinate system, the electric field vector E can be decomposed into:
wherein ex 、e y 、e z The unit vectors in the x, y, and z directions are indicated, respectively.
Writing the electric field vector E in a matrix form:
the electric field vector of the electric dipole includes both spatial information and polarization information.
And step two, constructing an array data receiving model.
As shown in fig. 2, it is assumed that K far-field narrowband completely polarized electromagnetic wave signals are incident to a polarization sensitive array in the spatial far field, wherein the polarization sensitive array is composed of incomplete electric vector sensors, and the polarization sensitive array comprises two parallel first linear arrays and second linear arrays; the first linear array is composed of M electric dipole pairs pointing to the XZ direction, and can only receive electromagnetic waves parallel to the x axis and the z axis; the second linear array is composed of M electric dipoles pointing in the Y direction, and can only receive electromagnetic waves parallel to the Y axis. The polarization sensitive array is assumed to be placed on the YOZ plane, and the array element spacing d is half-wavelength lambda/2.
When the kth signal source is incident to the polarization sensitive array, according to the quaternion model, the received data of the first linear array in the polarization sensitive array is expressed as:
wherein sk (t) represents an incident signal vector matrix of the kth signal source, S (t) = [ S ] 1 (t) s 2 (t)…s k (t)]Which represents all incident signals; n (N) 1 (t) is a space-time-polarization domain white noise matrix of the first linear array; i represents the imaginary part of the complex number;an electric field guiding matrix for the first linear array, in which diagonal matrix +.>Represents the kth electric field steering vector, +.>A quaternion representation of the kth signal along the x-axis and z-axis electric field directions;
wherein θ k ，γ k ，η k Respectively representing the azimuth angle, pitch angle, polarization auxiliary angle and polarization phase difference of the kth signal source;
the received data for the second linear array in the polarization-sensitive array is expressed as:
wherein N2 (t) represents a space-time-polarization domain white noise matrix of the second linear array;an electric field guiding matrix being a second linear array, in which diagonal matrix the +.>Represents the kth electric field steering vector, +.>A quaternion representation of the kth signal along the y-axis electric field direction;
in the formulas (4) and (7), a (θ) = [ a (θ) 1 ) a(θ 2 )…a(θ K )]Is an array space domain epidemic matrix, wherein the spatial phase shift vector of the kth signal can be defined as:
wherein
Thus, the received data model of the entire polarization-sensitive array can be written as:
wherein
And thirdly, calculating a cross correlation matrix according to the received data models of the first linear array and the second linear array obtained in the second step.
The cross-correlation matrix R is calculated from equations (4) and (7) as follows:
in the formula (11), the amino acid sequence of the compound,represents x 1 (t) and->The superscript H represents the conjugate transpose, N 1(t) and N2 (t) independent of each other in the space-polarization domain; incident electromagnetic wave signal vector matrixThe elements in S (t) are independent of each other, so that the autocorrelation matrix R of the incident electromagnetic wave signal S Is a diagonal matrix:
p in formula (12) k Representing the energy of the kth incident electromagnetic wave signal.
Taking into account that and />Also a diagonal matrix, then equation (11) is further reduced to:
R＝A(θ)ZA H (θ) (13)
wherein ：
wherein superscript denotes conjugate operation, and M×M-dimensional cross-correlation matrix shown in formula (13) contains polarization-spatial domain thought information of the signal.
And step four, sequentially compressing and smoothing the cross-correlation matrix, as shown in fig. 4.
Since the cross correlation matrix of equation (13) is too complex, according to the matrix compression concept, the array cross correlation matrix is compressed into a matrix with lower dimension under the condition of not losing signal information, firstly, a switching matrix J is defined, and a column adding matrix 1 is defined * The following are provided:
1 * ＝[1 1…1] T ∈K×1 (16)
thus, the cross-correlation compression matrix R z The expression is as follows:
first row of matrix A (θ) in equation (17) and matrix JA * The last line of (θ) is the same, and one is reserved, then the (2M-1) x 1-dimensional cross-correlation compression matrix expression is as follows:
to (18) R z Performing space smoothing on the mapped polarization-airspace, and defining a space smoothed matrix as R m Selecting a cross-correlation compression matrix R z (n: 2M-K+n) rows as a smoothing matrix R m Then the matrix R is smoothed m The expression is as follows:
wherein ,
further to smooth matrix R m Is decomposed into a matrix U containing pitch angle information only m And a matrix V containing azimuth angle, polarization auxiliary angle, polarization phase difference information m ；
R m ＝U m ×V m (20)
Step five, utilizing a smoothing matrix R m The rotation invariance relation among the row vectors is used for solving a pitch angle estimated value, and the pitch angle estimated value is reversely replaced back to the smoothing matrixR m Then using a known matrix U of substituted pitch angle estimation values m And smoothing matrix R m Further solving an unknown matrix V containing azimuth angle, polarization auxiliary angle and polarization phase difference information m The method comprises the steps of carrying out a first treatment on the surface of the The method comprises the following steps:
r is selected m Line 1 to (2M-K-1) of (B) to form subarrays R 1 R is selected m Line 2 to (2M-K) of the array to form a subarray R 2 Analysis shows that there is a fixed rotation invariant relationship between the two matrices:
R 2 ＝R 1 V m -1 ΛV m (22)
where Λ is the diagonal matrix diag { ζ ] 1 ,ξ 2 ,…,ξ K }。
Matrix is arrangedWherein the superscript->Generalized inverse operation of the representation matrix, R p Similar to Λ, for R p Decomposing characteristic value, R p The K eigenvalues of Λ correspond to the K diagonal elements of Λ, and the Λ matrix contains pitch angle θ information of the signal. Therefore, the calculation formula of the pitch angle estimation value is obtained as follows:
in equation (23), Λ (k) is the kth diagonal element of Λ, and function angle represents the phase angle of the complex number.
Bringing k pitch angle estimates into a matrix U m Transforming equation (20) and solving unknown matrix V containing azimuth angle, polarization auxiliary angle and polarization phase difference information m ：
In (24)) Equation left V m Is unknown, the right of the equation is known.
Step six, according to U in step five m and Vm The corresponding relation and the quaternion property are sequentially solved to obtain the azimuth angle estimated valuePolarization auxiliary angle estimation value +.>And polarization phase difference estimation value->The method specifically comprises the following steps:
for equation (24), the right two matrices of the equation are known, and the pitch angle estimate of the kth signalAnd V is equal to m Corresponds to row k, thus selecting V m Is estimated in the first column of (a); v (V) m First column and k row elements v k The expression is:
in (16)In the form of quaternion>Is a conjugated form of quaternion;
for two quaternions, if there are the following forms:
the conjugate operation of Q is:
Q * ＝q 0 -q 1 i-q 2 j-q 3 l (27)
the multiplication of Q and P is defined as:
v in formula (25) based on the property relationship of quaternions of formulas (26), (27), (28) k Unfolding and writing:
wherein l represents the imaginary part of the complex number;
imaginary part v of j in comparison (29) k ] j And l imaginary part v k ] l The following formula is obtained:
further, the azimuth estimate of the kth signalThe calculation formula is as follows:
from the formula (29), v is known k The real part is:
estimating pitch angleAnd azimuth estimate +.>Substitution of the real part [ v ] k ] r Obtaining the estimated value of the polarization auxiliary angle +.>The calculation formula of (2) is as follows:
estimating pitch angleAnd polarization auxiliary angle estimation value +.>Substitution of the imaginary part of j v k ] j Obtaining the estimated value of the polarization phase differenceThe calculation formula of (2) is as follows:
the simulation analysis of the present invention was performed as follows:
simulation the selected array is shown in fig. 2, wherein the number of array elements of two subarrays is m=4, and the array element spacing is lambda/2.
Simulation 1 target DOA and polarization parameter estimation
Considering that two incoherent signals are incident to the array, the pitch angle of the signals is theta= [5 degrees, 20 degrees ]]The method comprises the steps of carrying out a first treatment on the surface of the Azimuth anglePolarization auxiliary angle gamma= [30 °,45 ]]The method comprises the steps of carrying out a first treatment on the surface of the Polarization phase difference η= [60 °,80 ]]. Snap number snap=1000, signal to noise ratio snr= -10dB. Fig. 5 shows simulation results of the polarized MUSIC algorithm, from which it can be seen that one of the spectral peaks appears slightly blurred, there are several next highest side peaks around the main peak,this is because the MUSIC algorithm needs to perform eigenvalue decomposition on the autocorrelation matrix to find the noise subspace, and when the signal-to-noise ratio is low, the algorithm cannot clearly decompose the signal subspace and the noise subspace. Fig. 6 (a) and fig. 6 (b) show graphs of the two-dimensional space angle and the two-dimensional polarization angle estimated by the algorithm of the present invention, respectively, and it can be seen from the graphs that the algorithm can accurately estimate the space domain parameter and the polarization parameter at a low signal-to-noise ratio.
Simulation 2 algorithm performance and signal to noise ratio relationship
The simulation compares the space domain and polarization domain parameter estimation performance of the polarized MUSIC algorithm with that of the method. Monte Carlo number L is 500, the signal to noise ratio is changed from-10 dB to 20dB, the airspace and polarization domain parameter estimation performance of the algorithm is estimated by using root mean square error (root mean squareerror, RMSE), and the RMSE calculation formula in the experiment is shown as (1)
in the formula Respectively representing the estimated values in the kth source first Monte Carlo simulation. Fig. 7 and 8 show graphs comparing RMSE of spatial and polarization parameters with system SNR, respectively.
As can be seen from fig. 7 and 8, the error of the estimation of the three algorithms is acceptable in the case of a large signal-to-noise ratio; in the case of low signal-to-noise ratio, RMSE of MUSIC algorithm and ESPRIT algorithm increases drastically, and when the signal-to-noise ratio decreases to-20 dB, RMSE of both algorithms is already close to 5 °, which means that the algorithm is completely disabled. In contrast, the RMSE of the algorithm in the condition of low signal-to-noise ratio does not exceed 0.15 degrees, the estimation accuracy of the algorithm is more accurate than that of the traditional four-dimensional MUSIC algorithm, and the estimation error of the algorithm is in a convergence trend. For estimation of polarization parameters, the estimation performance of the two algorithms is basically the same, and the performance of the algorithm is slightly better when the signal to noise ratio is small.
Comparison of the operating efficiency of the simulation 3 algorithm
The simulation verifies the operation efficiency of the polarized MUSIC algorithm and the method. Table 1 shows the comparison of the running time of the polarized MUSIC algorithm and the algorithm of the present invention with the change of the number of array elements when the number of sources k=3.
Table 1 two algorithm run times
As can be seen from table 1, the polarized MUSIC algorithm consumes more time in operation, the operation time increases significantly as the number of array elements increases, while the invention text algorithm replaces the spectral peak search with mathematical operation, and the operation time decreases greatly.
Claims (5)
1. The DOA-polarization information joint estimation method based on the incomplete electric vector sensor is characterized by comprising the following steps of:
firstly, constructing a model of an electric vector of an electromagnetic signal in space;
step two, constructing an array data receiving model;
assuming that K far-field narrowband completely polarized electromagnetic wave signals are incident to a polarization sensitive array in a space far-field, wherein the polarization sensitive array consists of a residual electric vector sensor and comprises two parallel first linear arrays and second linear arrays; the first linear array is composed of M electric dipole pairs pointing to the XZ direction, and can only receive electromagnetic waves parallel to the x axis and the z axis; the second linear array is composed of M electric dipoles pointing to the Y direction, and can only receive electromagnetic waves parallel to the Y axis; assuming that the polarization sensitive array is placed on a YOZ plane, and the array element intervals d are half-wavelength lambda/2;
when the kth signal source is incident to the polarization sensitive array, according to the quaternion model, the received data of the first linear array in the polarization sensitive array is expressed as:
wherein sk (t) represents an incident signal vector matrix of the kth signal source, S (t) = [ S ] 1 (t) s 2 (t) … s k (t)]Which represents all incident signals; n (N) 1 (t) is a space-time-polarization domain white noise matrix of the first linear array; i represents the imaginary part of the complex number;an electric field guiding matrix for the first linear array, in which diagonal matrix +.>Represents the kth electric field steering vector, +.>A quaternion representation of the kth signal along the x-axis and z-axis electric field directions;
wherein θ k ，γ k ，η k Respectively representing the azimuth angle, pitch angle, polarization auxiliary angle and polarization phase difference of the kth signal source;
the received data for the second linear array in the polarization-sensitive array is expressed as:
wherein N2 (t) represents the space-time-pole of the second linear arrayA white noise matrix of the chemical domain;an electric field guiding matrix being a second linear array, in which diagonal matrix the +.>Represents the kth electric field steering vector, +.>A quaternion representation of the kth signal along the y-axis electric field direction;
in the formulas (4) and (7), a (θ) = [ a (θ) 1 ) a(θ 2 ) … a(θ K )]Is an array space domain epidemic matrix, wherein the spatial phase shift vector of the kth signal can be defined as:
wherein
Thus, the received data model for the entire polarization sensitive array is:
wherein
Step three, calculating a cross correlation matrix according to the received data models of the first linear array and the second linear array obtained in the step two; specifically, the cross-correlation matrix R is calculated from the formulas (4) and (7) as follows:
in the formula (11), the amino acid sequence of the compound,represents x 1 (t) and->The superscript H represents the conjugate transpose, N 1(t) and N2 (t) independent of each other in the space-polarization domain; the elements in the vector matrix S (t) of the incident electromagnetic wave signal are independent of each other, so that the autocorrelation matrix R of the incident electromagnetic wave signal S Is a diagonal matrix:
p in formula (12) k Representing the energy of the kth incident electromagnetic wave signal;
taking into account that and />Also a diagonal matrix, then equation (11) is further reduced to:
R＝A(θ)ZA H (θ) (13)
wherein ：
wherein the superscript indicates conjugate operation, and the M x M-dimensional cross-correlation matrix shown in formula (13) contains polarization-airspace thinking information of the signal;
step four, compressing and smoothing the cross-correlation matrix in sequence;
step five, utilizing a smoothing matrix R m The rotation invariance relation among the row vectors is used for solving a pitch angle estimated value, and the pitch angle estimated value is reversely replaced back to the smoothing matrix R m Then using a known matrix U of substituted pitch angle estimation values m And smoothing matrix R m Further solving an unknown matrix V containing azimuth angle, polarization auxiliary angle and polarization phase difference information m ；
Step six, according to U in step five m and Vm The corresponding relation and the quaternion property are sequentially solved to obtain the azimuth angle estimated valuePolarization auxiliary angle estimation value +.>And polarization phase difference estimation value->
2. The DOA-polarization information joint estimation method based on the incomplete electric vector sensor according to claim 1, wherein the step one is specifically to define a pitch angle theta as an included angle between an incident electromagnetic signal and the positive direction of a z-axis, wherein θ is more than or equal to 0 and less than or equal to pi; azimuth angleFor the angle between the projection of the incident electromagnetic signal on the XY plane and the positive x-axis, +.>The electric field vector E of the incident electromagnetic signal is in two orthogonal unit vectors E θ ,/>Decomposition in the direction V θ and />Two components of the electric field vector E are denoted as:
wherein gamma is E [0, pi/2]And eta epsilon [ -pi, pi]Respectively representing the polarization auxiliary angle and the polarization phase difference, and the tangent value of gamma represents e θ Amplitude of the directional electric fieldThe ratio of the magnitudes of the directional electric fields, η, represents e θ Directional electric field and +.>The phase difference of the directional electric field, j, represents the imaginary part of the complex number;
in a space rectangular coordinate system, the electric field vector E is decomposed into:
wherein ex 、e y 、e z The unit vectors in the x, y and z directions are respectively represented;
writing the electric field vector E in a matrix form:
the electric field vector of the electric dipole includes both spatial information and polarization information.
3. The DOA-polarization information joint estimation method based on the incomplete electric vector sensor according to claim 2, wherein the step four is specifically as follows:
since the cross correlation matrix of equation (13) is too complex, according to the matrix compression concept, the array cross correlation matrix is compressed into a matrix with lower dimension under the condition of not losing signal information, firstly, a switching matrix J is defined, and a column adding matrix 1 is defined * The following are provided:
1 * ＝[1 1 … 1] T ∈K×1 (16)
thus, the cross-correlation compression matrix R z The expression is as follows:
first row of matrix A (θ) in equation (17) and matrix JA * The last line of (θ) is the same, and one is reserved, then the (2M-1) x 1-dimensional cross-correlation compression matrix expression is as follows:
to (18) R z Performing space smoothing on the mapped polarization-airspace, and defining a space smoothed matrix as R m Selecting a cross-correlation compression matrix R z (n: 2M-K+n) rows as a smoothing matrix R m Then the matrix R is smoothed m The expression is as follows:
wherein ,
further to smooth matrix R m Is decomposed into a matrix U containing pitch angle information only m And a matrix V containing azimuth angle, polarization auxiliary angle, polarization phase difference information m ；
R m ＝U m ×V m (20)
4. The DOA-polarization information joint estimation method based on the incomplete electric vector sensor according to claim 3, wherein the fifth step is specifically as follows:
r is selected m Line 1 to (2M-K-1) of (B) to form subarrays R 1 R is selected m Line 2 to (2M-K) of the array to form a subarray R 2 Analysis shows that there is a fixed rotation invariant relationship between the two matrices:
R 2 ＝R 1 V m -1 ΛV m (22)
where Λ is the diagonal matrix diag { ζ ] 1 ,ξ 2 ,…,ξ K }；
Matrix is arrangedWherein the superscript->Generalized inverse operation of the representation matrix, R p Similar to Λ, for R p Decomposing characteristic value, R p The K eigenvalues of the (a) correspond to the K diagonal elements of the Λ, and the pitch angle theta information of the signals is contained in the Λ matrix; therefore, the calculation formula of the pitch angle estimation value is obtained as follows:
in the formula (23), Λ (k) is a kth diagonal element of Λ, and the function angle represents a complex phase angle;
bringing k pitch angle estimates into a matrix U m Transforming equation (20) and solving unknown matrix V containing azimuth angle, polarization auxiliary angle and polarization phase difference information m ：
In the formula (24), the left side of the equation V m Is unknown, the right of the equation is known.
5. The DOA-polarization information joint estimation method based on the incomplete electric vector sensor according to claim 4, wherein the step six is specifically:
for equation (24), the right two matrices of the equation are known, and the pitch angle estimate of the kth signalAnd V is equal to m Corresponds to row k, thus selecting V m Is estimated in the first column of (a); v (V) m First column and k row elements v k The expression is:
in (16)In the form of quaternion>Is a conjugated form of quaternion;
v in equation (25) based on the property relationship of quaternions k Unfolding and writing:
wherein l represents the imaginary part of the complex number;
imaginary part v of j in comparison (29) k ] j And l imaginary part v k ] l The following formula is obtained:
further, the azimuth estimate of the kth signalThe calculation formula is as follows:
from the formula (29), v is known k The real part is:
estimating pitch angleAnd azimuth estimate +.>Substitution of the real part [ v ] k ] r Obtaining the estimated value of the polarization auxiliary angle +.>The calculation formula of (2) is as follows:
estimating pitch angleAnd polarization auxiliary angle estimation value +.>Substitution of the imaginary part of j v k ] j Obtaining estimated value of polarization phase difference +.>The calculation formula of (2) is as follows:
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| CN104122533A (en) * | 2014-07-29 | 2014-10-29 | 电子科技大学 | Joint parameter estimation method based on distributed polarization sensitive array |
| CN109375152A (en) * | 2018-09-05 | 2019-02-22 | 南京航空航天大学 | The DOA and polarization combined estimation method of L gusts of electromagnetic vector nesting lower low complex degrees |
| CN109959891A (en) * | 2019-04-11 | 2019-07-02 | 南京航空航天大学 | The dimensionality reduction spectrum peak search method of Space Angle and polarization parameter in L gusts of electromagnetic vector |
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| CN104122533A (en) * | 2014-07-29 | 2014-10-29 | 电子科技大学 | Joint parameter estimation method based on distributed polarization sensitive array |
| CN109375152A (en) * | 2018-09-05 | 2019-02-22 | 南京航空航天大学 | The DOA and polarization combined estimation method of L gusts of electromagnetic vector nesting lower low complex degrees |
| CN109959891A (en) * | 2019-04-11 | 2019-07-02 | 南京航空航天大学 | The dimensionality reduction spectrum peak search method of Space Angle and polarization parameter in L gusts of electromagnetic vector |
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