CN104849694A - Quaternion ESPRIT parameter estimation method for magnetic dipole pair array - Google Patents
Quaternion ESPRIT parameter estimation method for magnetic dipole pair array Download PDFInfo
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Abstract
A quaternion ESPRIT parameter estimation method for a magnetic dipole pair array comprises the following steps: sampling output signals of an electromagnetic vector sensor array to obtain a first set of sampling data, and synchronously sampling output signals after a delay of delta T to obtain a second set of sampling data; constructing a first received quaternion data matrix, a second received quaternion data matrix and full-array received data; calculating the autocorrelation matrix of the full-array received data, and performing quaternion feature decomposition on the autocorrelation matrix to obtain array steering vector estimated values corresponding to the first set of sampling data and the second set of sampling data and an array steering vector matrix estimated value corresponding to the full data; obtaining the estimated value of the signal arrival angle based on the array steering vector estimated value corresponding to the first set of sampling data; and reconstructing an array steering vector estimated value of an electric dipole sub-array and an array steering vector estimated value of a magnetic dipole sub-array in the Z-axis direction, and obtaining estimated values of signal polarization parameters according to the rotational invariance relation between sub-array steering vectors.
Description
Technical Field
The invention belongs to the technical field of signal processing, and particularly relates to a parameter estimation method of an electromagnetic vector sensor array.
Background
The electromagnetic vector sensor array not only can simultaneously obtain the azimuth parameter and the polarization parameter of an incoming wave signal, but also can effectively reveal the orthogonal characteristic of each component of an electromagnetic wave, and once the electromagnetic vector sensor array is provided, the electromagnetic vector sensor array becomes a research hotspot of scholars due to the complete electromagnetic wave receiving capability, and obtains some valuable research results. However, the signal processing algorithm of the existing electromagnetic vector sensor array mostly uses the classical long vector algorithms of subspace types such as ESPRIT, MUSIC and the like, and the long vector algorithm model is that the multi-component output of the electromagnetic vector sensor array at different spatial positions is expressed by a complex long vector, although the signal parameters can be correctly estimated, the vector structure of each component output data is destroyed, and the time-space-polarization three-dimensional characteristic of the incident signal cannot be completely reflected.
In recent years, people develop researches on the estimation problem of the wave arrival direction of electromagnetic vector sensor array signals by using a quaternion algebraic system theory, a quaternion model breaks through the limitation of a traditional complex field-based long vector model, the algebraic structure of array data is expanded when the electromagnetic vector sensor array is subjected to signal processing, and a quaternion model of a hypercomplex field is established, so that an array signal processing algorithm on the hypercomplex field is formed, and the aim of keeping the local vector characteristics output by the array is fulfilled.
Disclosure of Invention
The invention aims to provide a parameter estimation method of an electromagnetic vector sensor array, which can reduce coupling errors.
In order to achieve the purpose, the invention adopts the following technical solutions:
the quaternion ESPRIT parameter estimation method of the electromagnetic dipole pair array comprises the following steps of: k mutually independent complete polarization transverse electromagnetic wave signals are simultaneously incident on an electromagnetic vector sensor array, the array element of the array is an electromagnetic dipole pair consisting of an electric dipole and a magnetic dipole,
sampling an output signal of an electromagnetic vector sensor array for M times to obtain a first group of sampling data X, and synchronously sampling for M times after delaying delta T to obtain a second group of sampling data Y;
wherein x isne(m) m-th sampling data, x, of electric dipole output signal of nth array elementnh(m) m-th sampled data, y, representing magnetic dipole output signal of the nth array elementne(m) m-th sampling data of electric dipole output signal of nth array element after time delay of delta T, ynh(m) m-th sampling data of the magnetic dipole output signal of the nth array element after the time delay of delta T is represented;
step two, overlapping the first group of sampling data X and the second group of sampling data Y according to the same snap-shot data of electric dipoles and magnetic dipoles of the same array elements respectivelyForming a first set of received quaternion data matrices Z1And a first set of received quaternion data matrices Z2Constructing a full-array receiving data matrix Z;
wherein,representing quaternion data formed by overlapping the m-th sampling data of the output signals of the electric dipole and the magnetic dipole of the nth array element,representing quaternion data formed by overlapping the m-th sampling data of the output signals of the electric dipole and the magnetic dipole of the nth array element by the delay delta T;
quaternion data matrix Z formed by first group of sampling data X1＝A1S+N1Wherein A is1＝[a1(θ1，φ1，γ1，η1)，…，a1(θk，φk，γk，ηk)，…，a1(θK，φK，γK，ηK)]Is the array steering vector corresponding to the first set of sample data, a1(θk,φk,γk,ηk)＝ckq(θk,φk)，Is a quaternion data representation of the electromagnetic field in the first set of sampled data of the kth incident signal,and hkz＝sinθkcosγkThe electric field component and the magnetic field component of the kth incident signal along the z-axis direction at the coordinate origin, q (theta)k,φk) Space-domain steering vector, θ, for the full array phase centerkIs the pitch angle, phi, of the kth incident signalkIs the azimuth angle, γ, of the k-th incident signalkIs the auxiliary angle of polarization, η, of the kth incident signalkPolarization phase difference of kth incident signal, N1Is a Gaussian white noise vector, and S is an amplitude matrix formed by incident signals;
quaternion data matrix Z formed by second group of sampling data Y2＝A2S+N2＝A1ΦS+N2Wherein A is2＝A1Phi is an array steering vector corresponding to the second group of sampling data, phi is a time delay matrix, N2Is a gaussian white noise vector;
constructing a full array receive data matrix Wherein, is a matrix of array steering vectors corresponding to the full data, is a full data noise matrix;
step three, calculating the autocorrelation matrix R of the full-array received data matrixzPerforming quaternion characteristic decomposition on the autocorrelation matrix to obtain an array steering vector estimation value corresponding to the first group of sampling dataArray steering vector estimated value corresponding to second group of sampling dataArray steering vector matrix estimation corresponding to full data
Wherein,as a function of the autocorrelation of the incident signal, σ2Is the variance of the noise, I is the identity matrix, (. cndot.)HRepresenting a transposed complex conjugate operation;
for autocorrelation matrix RzPerforming quaternion matrix characteristic decomposition to obtain a signal subspace EsAccording to the subspace principle, there is a K by K non-singular matrix T, and EsTaking E respectively as ATsThe first N rows and the last N rows of the matrix E are respectively formed1And E2By definition of a signal subspace, A1、A2And E1、E2Satisfy E1＝A1T，E2＝A2T＝A1Phi T, then there are
For matrixPerforming quaternion characteristic decomposition, and forming a delay matrix estimation value by K large characteristic valuesThe eigenvectors corresponding to the eigenvalues form nonsingular matrix estimated valuesAccording toTo obtain
Step four, calculating an estimated value of the arrival angle of the signal;
according toCalculating the phase difference vector between two adjacent array elements <math>
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</math> Wherein A is1(2: N, k) represents A12 nd to nth elements, a of the kth column1(1: N-1, k) represents A1The 1 st to the N-1 st elements of the kth column,/denotes the corresponding element division;
calculating a phase matrixarg (·) denotes taking phase;
according to <math>
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</math> Calculating the direction cosine estimated value of the k incident signal in the x-axis directionAnd direction cosine estimate of y-axis direction <math>
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</math> [W]#Is a pseudo-inverse of the position matrix W;
estimation based on directional cosineObtaining an estimated value of the arrival angle of the signal:
step five, byReconstructing array steering vector estimation value of electric dipole sub-array in Z-axis direction by using real part and three imaginary partsArray steering vector estimation value of sum magnetic dipole subarrayObtaining an estimated value of a signal polarization parameter according to a rotation invariant relation among subarray guide vectors;
wherein,representing a rotation invariant matrixThe kth row and the kth column of (1);
in the previous step, N is 1, …, N, M is 1, …, M, N is the array element number of the array, M is the sampling times,is 3 imaginary units of quaternion.
The array is a circular ring array, the axis of the electric dipole and the axis of the magnetic dipole are parallel to the z axis, N array elements are uniformly distributed on the circular ring, and the origin of coordinates is positioned at the center of the circular ring.
The quaternion method can better maintain the characteristics of quaternion vectors, has better performance than a long vector method, and has smaller coupling error. The method adopts an ESPRIT algorithm based on quaternion to estimate the multiple parameters of the incident signal, and compared with the long vector method in the prior art, the method can better reflect the orthogonal characteristics of each component of the electromagnetic vector sensor and improve the precision of parameter estimation.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings needed to be used in the description of the embodiments or the prior art will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art that other drawings can be obtained according to these drawings without creative efforts.
FIG. 1 is a schematic diagram of an electromagnetic vector sensor array according to an embodiment of the present invention.
FIG. 2 is a flow chart of the method of the present invention.
FIG. 3 is a diagram of root mean square error of pitch angle estimation versus signal-to-noise ratio for a simulation experiment.
FIG. 4 is a plot of root mean square error of azimuth angle estimation versus signal-to-noise ratio for simulation experiments.
FIG. 5 is a graph of the root mean square error of the auxiliary polarization angle estimation versus the signal-to-noise ratio of the simulation experiment.
FIG. 6 is a diagram of root mean square error of polarization phase difference estimation versus signal-to-noise ratio in a simulation experiment.
Fig. 7 is a diagram of arrival angle estimation success probability with signal-to-noise ratio in a simulation experiment.
Fig. 8 is a diagram of the polarization angle estimation success probability with the signal-to-noise ratio of the simulation experiment.
FIG. 9 is a scatter plot of angle-of-arrival estimates for the method of the present invention.
Fig. 10 is a scatter diagram of the angle-of-arrival estimates for the long vector algorithm.
FIG. 11 is a scatter plot of the estimation of the polarization angle according to the method of the present invention.
FIG. 12 is a scatter plot of long vector algorithm polarization angle estimates.
Detailed Description
In order to make the aforementioned and other objects, features and advantages of the present invention more apparent, embodiments of the present invention will be described in detail below with reference to the accompanying drawings.
FIG. 1 is a schematic diagram of an electromagnetic vector sensor array of the present invention. As shown in fig. 1, the electromagnetic vector sensor array is a circular array, and its array elements are electromagnetic dipole pairs consisting of an electric dipole and a magnetic dipole, the electric dipole is represented by an arrow in fig. 1, and the magnetic dipole is represented by a small circular ring. The axis of the electric dipole is parallel to the z-axis, the axis of the magnetic dipole is also parallel to the z-axis, and the electric dipole and the magnetic dipole respectively receive the electric field and the magnetic field in the z-axis direction. N array elements are evenly distributed on a ring with the radius of R, the 1 st array element is positioned on an x axis, the 1 st, …, N array elements and the origin of coordinates are respectively positioned at the circle center of the ring along the anticlockwise direction of the circumference, and the N th array element and the x axis are at a positive included anglen＝1,…,N。
With reference to fig. 2, the method for estimating quaternion parameters of an array of electromagnetic dipole pairs of the present invention comprises the following steps: k mutually independent complete polarization transverse electromagnetic wave signals are simultaneously incident on the electromagnetic vector sensor array,
sampling output signals of an electromagnetic vector sensor array for M times to obtain a first group of sampling data X, synchronously sampling for M times after delaying delta T to obtain a second group of sampling data Y, wherein X and Y are both 2 NxM matrixes, and K is less than N-1;
wherein x isne(m) m-th sampling data, x, of electric dipole output signal of nth array elementnh(m) m-th sampled data, y, representing magnetic dipole output signal of the nth array elementne(m) m-th sampling data of electric dipole output signal of nth array element after time delay of delta T, ynh(M) M-th sampling data of the magnetic dipole output signal of the nth array element after the time delay of delta T, wherein M is 1, … and M;
step two, overlapping the first group of sampling data X and the second group of sampling data Y according to the same snap-shot data of electric dipoles and magnetic dipoles of the same array elements to form a first group of receiving quaternion data matrix Z1And a first set of received quaternion data matrices Z2：
Wherein,representing quaternion data formed by superposition of m-th sampling data of electric dipole and magnetic dipole output signals of the nth array element，Representing quaternion data formed by overlapping the m-th sampling data of the output signals of the electric dipole and the magnetic dipole of the nth array element by the delay delta T;
k completely polarized and uncorrelated transverse electromagnetic wave signals are simultaneously incident on the receiving array, and the quaternion data matrix formed by the first group of sampling data X is: z1＝A1S+N1Wherein A is1＝[a1(θ1,φ1,γ1,η1),…,a1(θk,φk,γk,ηk),…,a1(θK,φK,γK,ηK)]Is the array steering vector corresponding to the first set of sample data, a1(θk,φk,γk,ηk)＝ckq(θk,φk)，Is a quaternion data representation of the electromagnetic field in the first set of sampled data of the kth incident signal,and hkz＝sinθkcosγkThe electric field component and the magnetic field component of the kth incident signal along the z-axis direction at the coordinate origin, q (theta)k,φk)＝[q1(θk,φk),…,qn(θk,φk),…,qN(θk,φk)]Is a space-domain steering vector at the center of the full array phase,is the phase difference of the nth array element relative to the origin of coordinates, thetak(0≤θkNot more than 90 deg. is the pitch angle of the k-th incident signal, phik(0≤φk360 DEG or less is the azimuth angle of the k-th incident signal, gammak(0≤γkLess than or equal to 90 deg.) is the auxiliary polarization angle, eta, of the kth incident signalk(-180°≤ηkPolarization phase difference of kth incident signal of not more than 180 DEG, N1Is that the mean is zero and the variance is sigma2Gaussian white noise vector of (S) [ < S >1,…,sK]TA K multiplied by M amplitude matrix formed by K mutually uncorrelated signals;
similarly, the quaternion data matrix formed by the second group of sampling data Y is: z2＝A2S+N2＝A1ΦS+N2Wherein N is2Is that the mean is zero and the variance is sigma2Gaussian white noise vector of, A2Is the array steering vector corresponding to the second set of sample data, A2＝A1Φ，As a delay matrix, fkThe frequency of the kth incident signal;
constructing a full-array received data matrix Z: wherein, is a matrix of array steering vectors corresponding to the full data, is a full data noise matrix;
step three, calculating an autocorrelation matrix R of the full-array received data matrix ZzAnd performing quaternion characteristic decomposition on the autocorrelation matrix to obtain an array steering vector estimation value corresponding to the first group of sampling dataArray steering vector estimated value corresponding to second group of sampling dataArray steering vector matrix estimation corresponding to full data
Wherein,as a function of the autocorrelation of the incident signal, σ2Is the variance of the noise, I is the identity matrix, (. cndot.)HRepresenting a transposed complex conjugate operation;
to RzPerforming quaternion matrix characteristic decomposition to obtain a signal subspace EsAccording to the subspace principle, there is a K by K non-singular matrix T, and EsTaking E respectively as ATsThe first N rows and the last N rows of the matrix E are respectively formed1And E2By definition of a signal subspace, A1、A2And E1、E2Satisfy E1＝A1T，E2＝A2T＝A1Phi T, then there are <math>
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For matrixPerforming quaternion characteristic decomposition, and forming a delay matrix estimation value by K large characteristic valuesThe eigenvectors corresponding to the eigenvalues form nonsingular matrix estimated valuesAccording toTo obtain
Fourthly, array steering vector estimation values corresponding to the first group of sampling dataObtaining an estimated value of a signal arrival angle;
according toCalculating the phase difference vector between two adjacent array elements <math>
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calculating a phase matrixarg (·) denotes taking phase;
according to the relation between the phase matrix omega and the position matrix W <math>
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Wherein [ W ]]#Is a pseudo-inverse of the position matrix W, [ W ]]#＝[(W)HW]-1(W)HPosition matrix <math>
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</math> The central angle corresponding to the circular arc between two adjacent array elements, R is the radius of the circular ring array, lambdakIs the wavelength of the kth incident signal;
estimation based on directional cosineFurther obtaining an estimated value of the angle of arrival of the signal:
step five, byReconstructing array steering vector estimation value of electric dipole sub-array in Z-axis direction by using real part and three imaginary partsArray steering vector estimation value of sum magnetic dipole subarrayObtaining an estimated value of a signal polarization parameter according to a rotation invariant relation among subarray guide vectors;
according toCalculating an estimated value of a signal polarization parameter:
wherein,representing a rotation invariant matrixRow k and column k.
The method comprises the steps of constructing full-array received data and an autocorrelation matrix thereof by using two groups of synchronous sampling data, carrying out quaternion characteristic decomposition on the autocorrelation matrix, obtaining estimation of an array guide vector according to a subspace theory, obtaining direction cosines in the x-axis direction and the y-axis direction through airspace guide vector block operation, thus obtaining estimation of a two-dimensional arrival angle of a signal, reconstructing electric dipole subarray guide vectors in the x-axis direction and the y-axis direction according to the array guide vector, and obtaining estimation of a polarization parameter by using the relation between the two subarray guide vectors.
The effect of the present invention can be further illustrated by the following simulation results:
the simulation experiment conditions are as follows: a uniform circular array with the radius of 0.5 lambda is used as a receiving array, and N14 electromagnetic dipole pairs are uniformly distributed on the circumference. The parameters of the two incident signals are respectively(θ2,φ2,γ2,η2) 1024 snapshots were taken, 500 independent monte carlo tests (30 °,43 °,67 °,80 °).
The simulation experiment adopts the long vector algorithm of the prior art to compare with the quaternion algorithm of the method of the invention. As shown in fig. 3 to fig. 6, the root mean square error of the method of the present invention is lower than that of the long vector algorithm in the signal-to-noise ratio interval, and when the signal-to-noise ratio is 0dB, the root mean square error of the method of the present invention is respectively 0.8 °, 0.15 °, 0.2 ° and 0.5 ° smaller in the pitch angle, the azimuth angle, the auxiliary polarization angle and the polarization phase difference estimation than that of the long vector method.
As can be seen from fig. 7, the success probability of the angle of arrival estimation based on the method of the present invention is 0.95 at 0dB, whereas the success probability of the angle of arrival estimation based on the long vector is less than 0.55. It can be seen from fig. 8 that the probability of success of the polarization parameter of the method of the present invention is close to 1 at 0dB, whereas the probability of success of the polarization parameter based on long vectors is only 0.65.
It can be seen from the comparison between fig. 9 and fig. 10 that, when the signal-to-noise ratio is 10dB, the estimated value of the arrival angle of the method of the present invention is disturbed in a smaller range near the true value, the error of the estimated value is smaller, while the estimated value of the long vector algorithm deviates more from the true value, and the estimated error is larger.
As can be seen from the comparison between fig. 11 and fig. 12, the estimated value of the polarization parameter of the method of the present invention is close to the true value at the signal-to-noise ratio of 10dB, and the estimation error is small. And the estimated value of the long vector algorithm deviates more from the true value, and the estimation error is larger.
Therefore, the method has higher parameter estimation performance than the long vector method from the root mean square error, success probability and scatter diagram of parameter estimation.
Although the present invention has been described with reference to a preferred embodiment, it should be understood that various changes, substitutions and alterations can be made herein without departing from the spirit and scope of the invention as defined by the appended claims.
Claims (2)
1. The quaternion ESPRIT parameter estimation method of the electromagnetic dipole pair array is characterized by comprising the following steps of: k mutually independent complete polarization transverse electromagnetic wave signals are simultaneously incident on an electromagnetic vector sensor array, the array element is an electromagnetic dipole pair consisting of an electric dipole and a magnetic dipole,
sampling an output signal of an electromagnetic vector sensor array for M times to obtain a first group of sampling data X, and synchronously sampling for M times after delaying delta T to obtain a second group of sampling data Y;
wherein x isne(m) m-th sampling data, x, of electric dipole output signal of nth array elementnh(m) m-th sampled data, y, representing magnetic dipole output signal of the nth array elementne(m) m-th sampling data of electric dipole output signal of nth array element after time delay of delta T, ynh(m) m-th sampling data of the magnetic dipole output signal of the nth array element after the time delay of delta T is represented;
step two, overlapping the first group of sampling data X and the second group of sampling data Y according to the same snap-shot data of electric dipoles and magnetic dipoles of the same array elements to form a first group of receiving quaternion data matrix Z1And a second set of received quaternion data matrices Z2Constructing a full-array receiving data matrix Z;
wherein,representing quaternion data formed by overlapping the m-th sampling data of the output signals of the electric dipole and the magnetic dipole of the nth array element,representing quaternion data formed by overlapping the m-th sampling data of the output signals of the electric dipole and the magnetic dipole of the nth array element by the delay delta T;
quaternion data moments formed by the first set of sample data XArray Z1＝A1S+N1Wherein
A1＝[a1(θ1,φ1,γ1,η1),…,a1(θk,φk,γk,ηk),…,a1(θK,φK,γK,ηK)]is the array steering vector corresponding to the first set of sample data, a1(θk,φk,γk,ηk)＝ckq(θk,φk)，Is a quaternion data representation of the electromagnetic field in the first set of sampled data of the kth incident signal,and hkz＝sinθkcosγkThe electric field component and the magnetic field component of the kth incident signal along the z-axis direction at the coordinate origin, q (theta)k,φk) Space-domain steering vector, θ, for the full array phase centerkIs the pitch angle, phi, of the kth incident signalkIs the azimuth angle, γ, of the k-th incident signalkIs the auxiliary angle of polarization, η, of the kth incident signalkPolarization phase difference of kth incident signal, N1Is a Gaussian white noise vector, and S is an amplitude matrix formed by incident signals;
quaternion data matrix Z formed by second group of sampling data Y2＝A2S+N2＝A1ΦS+N2Wherein A is2＝A1Phi is an array steering vector corresponding to the second group of sampling data, phi is a time delay matrix, N2Is a gaussian white noise vector;
constructing a full array receive data matrix Wherein, is a matrix of array steering vectors corresponding to the full data, is a full data noise matrix;
step three, calculating the autocorrelation matrix R of the full-array received data matrixzPerforming quaternion feature decomposition on the autocorrelation matrix to obtain array steering vector estimation corresponding to the first group of sampling dataValue ofArray steering vector estimated value corresponding to second group of sampling dataArray steering vector matrix estimation corresponding to full data
Wherein,as a function of the autocorrelation of the incident signal, σ2Is the variance of the noise, I is the identity matrix, (. cndot.)HRepresenting a transposed complex conjugate operation;
for autocorrelation matrix RzPerforming quaternion matrix characteristic decomposition to obtain a signal subspace EsAccording to the subspace principle, there is a K by K non-singular matrix T, and EsTaking E respectively as ATsThe first N rows and the last N rows of the matrix E are respectively formed1And E2By definition of a signal subspace, A1、A2And E1、E2Satisfy E1＝A1T，E2＝A2T＝A1Phi T, then there are
For matrixPerforming quaternion characteristic decomposition, and forming a delay matrix estimation value by K large characteristic valuesThe eigenvectors corresponding to the eigenvalues form nonsingular matrix estimated valuesAccording toTo obtain
Step four, calculating an estimated value of the arrival angle of the signal;
according toCalculating the phase difference between two adjacent array elementsVectorWherein A is1(2: N, k) represents A12 nd to nth elements, a of the kth column1(1: N-1, k) represents A1The 1 st to the N-1 st elements of the kth column,/denotes the corresponding element division;
calculating a phase matrixarg (·) denotes taking phase;
according to <math>
<mrow>
<mi>Ω</mi>
<mo>=</mo>
<mi>W</mi>
<mo>·</mo>
<mfenced open='[' close=']'>
<mtable>
<mtr>
<mtd>
<msub>
<mover>
<mi>α</mi>
<mo>^</mo>
</mover>
<mi>k</mi>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mover>
<mi>β</mi>
<mo>^</mo>
</mover>
<mi>k</mi>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>,</mo>
</mrow>
</math> Calculating the direction cosine estimated value of the k incident signal in the x-axis directionAnd direction cosine estimate of y-axis direction <math>
<mrow>
<mfenced open='[' close=']'>
<mtable>
<mtr>
<mtd>
<msub>
<mover>
<mi>α</mi>
<mo>^</mo>
</mover>
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</msub>
</mtd>
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<mi>β</mi>
<mo>^</mo>
</mover>
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</msub>
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</mtable>
</mfenced>
<mo>=</mo>
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<mrow>
<mo>[</mo>
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</mrow>
<mo>#</mo>
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</mrow>
</math> [W]#Is a pseudo-inverse of the position matrix W;
estimation based on directional cosineObtaining an estimated value of the arrival angle of the signal:
step five, byReconstructing array steering vector estimation value of electric dipole sub-array in Z-axis direction by using real part and three imaginary partsArray steering vector estimation value of sum magnetic dipole subarrayObtaining an estimated value of a signal polarization parameter according to a rotation invariant relation among subarray guide vectors;
wherein,representing a rotation invariant matrixRow k and column k elements;
in the previous step, N is 1, …, N, M is 1, …, M, N is the array element number of the array, M is the sampling times,is 3 imaginary units of quaternion.
2. The method of quaternion ESPRIT parameter estimation of an array of electromagnetic dipole pairs as recited in claim 1, wherein: the array is a circular ring array, the axis of the electric dipole and the axis of the magnetic dipole are parallel to the z axis, the N array elements are uniformly distributed on the circular ring, and the origin of coordinates is positioned at the center of the circular ring.
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