CN104849694A - Quaternion ESPRIT parameter estimation method for magnetic dipole pair array - Google Patents

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

The hypercomplex number ESPRIT method for parameter estimation of electromagnetic dipole pair array

Technical field

The invention belongs to signal processing technology field, particularly relate to the method for parameter estimation of Electromagnetic Vector Sensor Array.

Background technology

Electromagnetic Vector Sensor Array can not only obtain direction parameter and the polarization parameter of incoming wave signal simultaneously, effectively can also disclose the orthogonal property of each component of electromagnetic wave, once proposition just because its complete electromagnetic wave receiving ability becomes the study hotspot of scholars, and achieve some of great value achievements in research.But the signal processing algorithm of existing Electromagnetic Vector Sensor Array mostly applies mechanically the long vector algorithm of the subspace classes such as classical ESPRIT, MUSIC, long vector algorithm model is that the multi-components output of Electromagnetic Vector Sensor Array in different spatial is stated with a plural long vector, although the correct estimated signal parameter of energy, but destroy the vector structure that each component output data itself have, the Space Time-polarization three-dimensional character of incoming signal can not be embodied completely.

In recent years, people utilize quaternion algebra system theory to expand research to Electromagnetic Vector Sensor Array signal Mutual coupling problem, quaternion model breaches traditional limitation based on complex field long vector model, when carrying out signal transacting to Electromagnetic Vector Sensor Array, the Algebraic Structure of array data is expanded, establish the quaternion model of onlapping number field, thus define the Array Signal Processing algorithm onlapped on number field, thus reach the object that reserved array exports Local Vector characteristic.

Summary of the invention

The object of this invention is to provide a kind of method for parameter estimation that can reduce the Electromagnetic Vector Sensor Array of coupling error.

To achieve these goals, the present invention takes following technical solution:

The hypercomplex number ESPRIT method for parameter estimation of electromagnetic dipole pair array, comprise the following steps: K mutual uncorrelated perfact polarization transverse electromagnetic wave signal incides in Electromagnetic Vector Sensor Array simultaneously, the array element of described array is by an electric dipole and a molecular electromagnetic dipole pair of magnetic dipole

Step one, carry out M sampling obtain first group of sampled data X to the output signal of Electromagnetic Vector Sensor Array, synchronized sampling M time after time delay Δ T, obtains second group of sampled data Y;

Wherein, x _nem () represents the m time sampled data of the electric dipole output signal of the n-th array element, x _nhm () represents the m time sampled data of the magnetic dipole output signal of the n-th array element, y _nethe m time sampled data that m () represents time delay Δ T after, the electric dipole of the n-th array element outputs signal, y _nhthe m time sampled data that m () represents time delay Δ T after, the magnetic dipole of the n-th array element outputs signal;

Step 2, first group of sampled data X and second group sampled data Y is formed first group according to the electric dipole of same array element and the homogeneous snap data investigation of magnetic dipole respectively receive hypercomplex number data matrix Z ₁hypercomplex number data matrix Z is received with first group ₂, construct full array received data matrix Z;

Wherein, represent that the m time sampled data outputed signal by the electric dipole of the n-th array element and magnetic dipole superposes the quaternion algebra certificate formed, represent that the m time sampled data that time delay Δ T is outputed signal by electric dipole and the magnetic dipole of the n-th array element superposes the quaternion algebra certificate formed;

The hypercomplex number data matrix Z that first group of sampled data X is formed ₁=A ₁s+N ₁, wherein, A ₁=[a ₁(θ ₁, φ ₁, γ ₁, η ₁) ..., a ₁(θ _k, φ _k, γ _k, η _k) ..., a ₁(θ _k, φ _k, γ _k, η _k)] be array steering vector corresponding to first group of sampled data, a ₁(θ _k, φ _k, γ _k, η _k)=c _kq (θ _k, φ _k), the hypercomplex number data representation of electromagnetic field in first group of sampled data of a kth incoming signal, and h _kz=sin θ _kcos γ _kbe respectively a kth incoming signal at true origin place electric field component along the z-axis direction and magnetic-field component, q (θ _k, φ _k) be the spatial domain steering vector at full array phase center, θ _kthe angle of pitch of a kth incoming signal, φ _kthe position angle of a kth incoming signal, γ _kthe auxiliary polarization angle of a kth incoming signal, η _kthe polarization phases of a kth incoming signal is poor, N ₁be white Gaussian noise vector, S is the magnitude matrix that incoming signal is formed;

The hypercomplex number data matrix Z that second group of sampled data Y is formed ₂=A ₂s+N ₂=A ₁Φ S+N ₂, wherein, A ₂=A ₁Φ is array steering vector corresponding to second group of sampled data, and Φ is time delay matrix, N ₂it is white Gaussian noise vector;

Construct full array received data matrix $Z=\left[\begin{array}{c}{Z}_1\\ Z_2\end{array}\right]=\left[\begin{array}{c}{A}_1\\ A_2\end{array}\right]S+N=\mathrm{AS}+N,$ Wherein, $A=\left[\begin{array}{c}{A}_1\\ A_2\end{array}\right]$ The array steering vector matrix that all data is corresponding, $N=\left[\begin{array}{c}{N}_1\\ N_2\end{array}\right]$ It is all data noise matrix;

Step 3, calculate the autocorrelation matrix R of full array received data matrix _z, hypercomplex number feature decomposition is carried out to autocorrelation matrix, obtains the array steering vector estimated value that first group of sampled data is corresponding the array steering vector estimated value that second group of sampled data is corresponding the array steering vector Matrix Estimation value corresponding with all data

$R_z=\frac{1}{M}\left[{\mathrm{ZZ}}^H\right]={\mathrm{AR}}_s{A}^H+{\mathrm{\σ}}^{2}I,$

Wherein, for the autocorrelation function of incoming signal, σ ²for the variance of noise, I is unit matrix, () ^hrepresent transposed complex conjugate operation;

To autocorrelation matrix R _zcarry out Quaternion Matrix feature decomposition and obtain signal subspace E _s, according to subspace principal, there is the nonsingular matrix T of K × K, and E _s=AT, gets E respectively _sthe capable and capable matrix E formed respectively of rear N of front N ₁and E ₂, by the definition of signal subspace, A ₁, A ₂with E ₁, E ₂between meet E ₁=A ₁t, E ₂=A ₂t=A ₁Φ T, then have

To matrix carry out hypercomplex number feature decomposition, K large eigenwert forms delay matrix estimated value eigenwert characteristic of correspondence vector forms nonsingular matrix estimated value according to obtain ${\hat{A}}_1=E_1{T}^{-1},{\hat{A}}_2=E_2{T}^{-1},\hat{A}=E_s{T}^{-1};$

The estimated value of step 4, calculating direction of arrival;

According to calculate the phase difference vector between adjacent two array elements $\stackrel{\‾}{q}({\widehat{\mathrm{\θ}}}_k,{\widehat{\mathrm{\φ}}}_k)=A_1(2:N,k)./A_1(1:N-1,k),$ Wherein, A ₁(2:N, k) represents A ₁kth row the 2 to the N number of element, A ₁(1:N-1, k) represents A ₁kth row the 1 to the N-1 element ./represent correspondence element be divided by;

Calculate phasing matrix phase place is got in arg () expression;

According to $\mathrm{\Ω}=W\·\left[\begin{array}{c}{\widehat{\mathrm{\α}}}_k\\ {\widehat{\mathrm{\β}}}_k\end{array}\right],$ Calculate the direction cosine estimated value in the x-axis direction of a kth incoming signal with the direction cosine estimated value in y-axis direction $\left[\begin{array}{c}{\widehat{\mathrm{\α}}}_k\\ {\widehat{\mathrm{\β}}}_k\end{array}\right]={\left[W\right]}^{\#}\mathrm{\Ω},$ [W] ^#it is the pseudo inverse matrix of location matrix W;

According to the estimated value of direction cosine obtain the estimated value of direction of arrival:

$\left\{\begin{array}{c}{\widehat{\mathrm{\&theta;}}}_k=\arcsin \left(\sqrt{{\widehat{\mathrm{\&alpha;}}}_k^2+{\widehat{\mathrm{\&beta;}}}_k^2}\right)\\ \left\{\begin{array}{c}{\widehat{\mathrm{\&phi;}}}_k=\arctan \left(\frac{{\widehat{\mathrm{\&alpha;}}}_k}{{\widehat{\mathrm{\&beta;}}}_k}\right),{\widehat{\mathrm{\&beta;}}}_k\&GreaterEqual;0\\ {\widehat{\mathrm{\&phi;}}}_k=\mathrm{\&pi;}+\arctan \left(\frac{{\widehat{\mathrm{\&alpha;}}}_k}{{\widehat{\mathrm{\&beta;}}}_k}\right),{\widehat{\mathrm{\&beta;}}}_k<0\end{array}\right.\end{array}\right.;$

Step 5, by real part and the array steering vector estimated value of electric dipole submatrix of three imaginary parts reconstruct Z-directions with the array steering vector estimated value of magnetic dipole submatrix the estimated value of polarizations parameter is obtained according to the invariable rotary relation between submatrix steering vector;

${\hat{A}}_1={\hat{A}}_{10}+{\hat{A}}_{11}\tilde{i}+{\hat{A}}_{12}\tilde{j}+{\hat{A}}_{13}\tilde{k},$ be real part, be three imaginary parts, according to the formation of Quaternion Matrix in step 2, the steering vector estimated value of the electric dipole submatrix of reconstruct Z-direction with magnetic dipole submatrix steering vector estimated value invariable rotary between two submatrix steering vectors closes be the invariable rotary matrix between two submatrixs, according to calculate the estimated value of polarizations parameter:

${\widehat{\mathrm{\&gamma;}}}_k={\tan }^{-1}\left(\right|\widehat{\mathrm{\&Psi;}}(k,k)\left|\right),$

${\widehat{\mathrm{\&eta;}}}_k=\arg \left(\widehat{\mathrm{\&Psi;}}(k,k)\right),$

Wherein, represent invariable rotary matrix row k kth row element;

N=1 in abovementioned steps ..., N, m=1 ..., M, N are the array number of array, and M is sampling number, for 3 imaginary units of hypercomplex number.

Array of the present invention is circle ring array, and the axis of electric dipole and the axis being parallel of magnetic dipole are in z-axis, and N number of array element is evenly distributed on annulus, and true origin is positioned at the center of circle of annulus.

Because hypercomplex number method can keep hypercomplex number vectorial property better, have performance more better than long vector method, coupling error is less.The inventive method adopts to be estimated based on the multiparameter of ESPRIT algorithm to incoming signal of hypercomplex number, compared with the long vector method of prior art, better can embody the orthogonal property of electromagnetic vector sensor each component amount, improve the precision of parameter estimation.

Accompanying drawing explanation

In order to be illustrated more clearly in the embodiment of the present invention or technical scheme of the prior art, below by need in embodiment or description of the prior art use accompanying drawing do simple introduction, apparently, accompanying drawing in the following describes is only some embodiments of the present invention, for those of ordinary skill in the art, under the prerequisite not paying creative work, other accompanying drawing can also be obtained according to these accompanying drawings.

Fig. 1 is the schematic diagram of embodiment of the present invention Electromagnetic Vector Sensor Array.

Fig. 2 is the process flow diagram of the inventive method.

Fig. 3 is the angle of pitch estimation root-mean-square error of emulation experiment and the graph of a relation of signal to noise ratio (S/N ratio).

Fig. 4 is the position angle estimation root-mean-square error of emulation experiment and the graph of a relation of signal to noise ratio (S/N ratio).

Fig. 5 is the auxiliary polarization angular estimation root-mean-square error of emulation experiment and the graph of a relation of signal to noise ratio (S/N ratio).

Fig. 6 is the polarization phases difference estimation root-mean-square error of emulation experiment and the graph of a relation of signal to noise ratio (S/N ratio).

Fig. 7 is the graph of a relation of the angle-of-arrival estimation probability of success with signal to noise ratio (S/N ratio) of emulation experiment.

Fig. 8 is that the polarizing angle of emulation experiment estimates the graph of a relation of the probability of success with signal to noise ratio (S/N ratio).

Fig. 9 is the inventive method angle-of-arrival estimation scatter diagram.

Figure 10 is long vector algorithm angle-of-arrival estimation scatter diagram.

Figure 11 is that the inventive method polarizing angle estimates scatter diagram.

Figure 12 is that long vector algorithm polarizing angle estimates scatter diagram.

Embodiment

In order to allow above and other objects of the present invention, feature and advantage can be more obvious, the embodiment of the present invention cited below particularly, and coordinate appended diagram, be described below in detail.

Figure 1 shows that the schematic diagram of Electromagnetic Vector Sensor Array of the present invention.As shown in Figure 1, Electromagnetic Vector Sensor Array is circular array, and its array element is that represent electric dipole with arrow in Fig. 1, small circle ring represents magnetic dipole by an electric dipole and a magnetic dipole molecular electromagnetic dipole pair.The axis being parallel of electric dipole is in z-axis, and the axis of magnetic dipole is also parallel to z-axis, and electric dipole and magnetic dipole receive the Electric and magnetic fields in z-axis direction respectively.N number of array element is that the annulus of R is uniformly distributed at radius, and the 1st array element is positioned in x-axis, is counterclockwise circumferentially the 1st respectively ..., N number of array element, true origin is positioned at the center of circle of annulus, the angle of the n-th array element and x-axis forward n=1 ..., N.

Composition graphs 2, the hypercomplex number method for parameter estimation of electromagnetic dipole pair array of the present invention, comprises the following steps: K mutual uncorrelated perfact polarization transverse electromagnetic wave signal incides in Electromagnetic Vector Sensor Array simultaneously,

Step one, M sampling is carried out to the output signal of Electromagnetic Vector Sensor Array obtain first group of sampled data X, synchronized sampling M time after time delay Δ T, obtain the matrix that second group of sampled data Y, X and Y are 2N × M, K < N-1;

Wherein, x _nem () represents the m time sampled data of the electric dipole output signal of the n-th array element, x _nhm () represents the m time sampled data of the magnetic dipole output signal of the n-th array element, y _nethe m time sampled data that m () represents time delay Δ T after, the electric dipole of the n-th array element outputs signal, y _nhthe m time sampled data that m () represents time delay Δ T after, the magnetic dipole of the n-th array element outputs signal, m=1 ..., M;

Step 2, first group of sampled data X and second group sampled data Y is formed first group according to the electric dipole of same array element and the homogeneous snap data investigation of magnetic dipole respectively receive hypercomplex number data matrix Z ₁hypercomplex number data matrix Z is received with first group ₂:

Wherein, represent that the m time sampled data outputed signal by the electric dipole of the n-th array element and magnetic dipole superposes the quaternion algebra certificate formed, represent that the m time sampled data that time delay Δ T is outputed signal by electric dipole and the magnetic dipole of the n-th array element superposes the quaternion algebra certificate formed;

K perfact polarization and mutual uncorrelated transverse electromagnetic wave signal incide on receiving array simultaneously, then the hypercomplex number data matrix that first group of sampled data X is formed is: Z ₁=A ₁s+N ₁, wherein, A ₁=[a ₁(θ ₁, φ ₁, γ ₁, η ₁) ..., a ₁(θ _k, φ _k, γ _k, η _k) ..., a ₁(θ _k, φ _k, γ _k, η _k)] be array steering vector corresponding to first group of sampled data, a ₁(θ _k, φ _k, γ _k, η _k)=c _kq (θ _k, φ _k), the hypercomplex number data representation of electromagnetic field in first group of sampled data of a kth incoming signal, and h _kz=sin θ _kcos γ _kbe respectively a kth incoming signal at true origin place electric field component along the z-axis direction and magnetic-field component, q (θ _k, φ _k)=[q ₁(θ _k, φ _k) ..., q _n(θ _k, φ _k) ..., q _n(θ _k, φ _k)] be the spatial domain steering vector at full array phase center, the phase differential of the n-th array element relative to true origin, θ _k(0≤θ _k≤ 90 °) be the angle of pitch of a kth incoming signal, φ _k(0≤φ _k≤ 360 °) be the position angle of a kth incoming signal, γ _k(0≤γ _k≤ 90 °) be the auxiliary polarization angle of a kth incoming signal, η _k(-180 °≤η _k≤ 180 °) polarization phases of a kth incoming signal is poor, N ₁be average be zero, variance is σ ²white Gaussian noise vector, S=[s ₁..., s _k] ^tfor K × M magnitude matrix that K mutual uncorrelated signal is formed;

Same, the hypercomplex number data matrix of second group of sampled data Y formation is: Z ₂=A ₂s+N ₂=A ₁Φ S+N ₂, wherein, N ₂be average be zero, variance is σ ²white Gaussian noise vector, A ₂array steering vector corresponding to second group of sampled data, A ₂=A ₁Φ, for time delay matrix, f _kfor the frequency of a kth incoming signal;

Construct full array received data matrix Z: $Z=\left[\begin{array}{c}{Z}_1\\ Z_2\end{array}\right]=\left[\begin{array}{c}{A}_1\\ A_2\end{array}\right]S+N=\mathrm{AS}+N;$ Wherein, $A=\left[\begin{array}{c}{A}_1\\ A_2\end{array}\right]$ The array steering vector matrix that all data is corresponding, $N=\left[\begin{array}{c}{N}_1\\ N_2\end{array}\right]$ It is all data noise matrix;

Step 3, calculate the autocorrelation matrix R of full array received data matrix Z _z, and hypercomplex number feature decomposition is carried out to autocorrelation matrix, thus obtain array steering vector estimated value corresponding to first group of sampled data the array steering vector estimated value that second group of sampled data is corresponding the array steering vector Matrix Estimation value corresponding with all data

$R_z=\frac{1}{M}\left[{\mathrm{ZZ}}^H\right]={\mathrm{AR}}_s{A}^H+{\mathrm{\&sigma;}}^{2}I,$

Wherein, for the autocorrelation function of incoming signal, σ ²for the variance of noise, I is unit matrix, () ^hrepresent transposed complex conjugate operation;

To R _zcarry out Quaternion Matrix feature decomposition and obtain signal subspace E _s, according to subspace principal, there is the nonsingular matrix T of K × K, and E _s=AT, gets E respectively _sthe capable and capable matrix E formed respectively of rear N of front N ₁and E ₂, by the definition of signal subspace, A ₁, A ₂with E ₁, E ₂between meet E ₁=A ₁t, E ₂=A ₂t=A ₁Φ T, then have ${\left(E_1^{\#}{E}_2\right)}^H{T}^H=T^H{\mathrm{\&Phi;}}^H;$

To matrix carry out hypercomplex number feature decomposition, K large eigenwert forms delay matrix estimated value eigenwert characteristic of correspondence vector forms nonsingular matrix estimated value according to obtain ${\hat{A}}_1=E_1{T}^{-1},{\hat{A}}_2=E_2{T}^{-1},\hat{A}=E_s{T}^{-1};$

Step 4, by array steering vector estimated value corresponding to first group of sampled data obtain the estimated value of direction of arrival;

According to calculate the phase difference vector between adjacent two array elements $\stackrel{\&OverBar;}{q}({\widehat{\mathrm{\&theta;}}}_k,{\widehat{\mathrm{\&phi;}}}_k)=A_1(2:N,k)./A_1(1:N-1,k),$ Wherein, A ₁(2:N, k) represents A ₁kth row the 2 to the N number of element, A ₁(1:N-1, k) represents A ₁kth row the 1 to the N-1 element ./represent correspondence element be divided by;

Calculate phasing matrix phase place is got in arg () expression;

According to the relation of phasing matrix Ω and location matrix W $\mathrm{\&Omega;}=W\&CenterDot;\left[\begin{array}{c}{\widehat{\mathrm{\&alpha;}}}_k\\ {\widehat{\mathrm{\&beta;}}}_k\end{array}\right],$ Calculate the direction cosine estimated value in the x-axis direction of a kth incoming signal with the direction cosine estimated value in y-axis direction

$\left[\begin{array}{c}{\widehat{\mathrm{\&alpha;}}}_k\\ {\widehat{\mathrm{\&beta;}}}_k\end{array}\right]={\left[W\right]}^{\#}\mathrm{\&Omega;},$

Wherein, [W] ^#the pseudo inverse matrix of location matrix W, [W] ^#=[(W) ^hw] ^-1(W) ^h, location matrix $W=\frac{2\mathrm{\&pi;R}}{{\mathrm{\&lambda;}}_k}\left[\begin{array}{cc}\sin \mathrm{\&Delta;}& \cos \mathrm{\&Delta;}-1\\ \sin 2\mathrm{\&Delta;}-\sin \mathrm{\&Delta;}& \cos 2\mathrm{\&Delta;}-\cos \mathrm{\&Delta;}\\ .& .\\ .& .\\ .& .\\ \sin N-\sin \left[(N-1)\mathrm{\&Delta;}\right]& \cos \mathrm{N\&Delta;}-\cos \left[(N-1)\mathrm{\&Delta;}\right]\end{array}\right],$ $\mathrm{\&Delta;}=\frac{2\mathrm{\&pi;}}{N}$ For the central angle that the circular arc between adjacent two array elements is corresponding, R is the radius of circle ring array, λ _kfor the wavelength of a kth incoming signal;

According to the estimated value of direction cosine obtain the estimated value of direction of arrival further:

$\left\{\begin{array}{c}{\widehat{\mathrm{\&theta;}}}_k=\arcsin \left(\sqrt{{\widehat{\mathrm{\&alpha;}}}_k^2+{\widehat{\mathrm{\&beta;}}}_k^2}\right)\\ \left\{\begin{array}{c}{\widehat{\mathrm{\&phi;}}}_k=\arctan \left(\frac{{\widehat{\mathrm{\&alpha;}}}_k}{{\widehat{\mathrm{\&beta;}}}_k}\right),{\widehat{\mathrm{\&beta;}}}_k\&GreaterEqual;0\\ {\widehat{\mathrm{\&phi;}}}_k=\mathrm{\&pi;}+\arctan \left(\frac{{\widehat{\mathrm{\&alpha;}}}_k}{{\widehat{\mathrm{\&beta;}}}_k}\right),{\widehat{\mathrm{\&beta;}}}_k<0\end{array}\right.\end{array}\right.;$

Step 5, by real part and the array steering vector estimated value of electric dipole submatrix of three imaginary parts reconstruct Z-directions with the array steering vector estimated value of magnetic dipole submatrix the estimated value of polarizations parameter is obtained according to the invariable rotary relation between submatrix steering vector;

${\hat{A}}_1={\hat{A}}_{10}+{\hat{A}}_{11}\tilde{i}+{\hat{A}}_{12}\tilde{j}+{\hat{A}}_{13}\tilde{k},$ be real part, be three imaginary parts, 3 imaginary units of hypercomplex number, according to the formation of Quaternion Matrix in step 2, the array steering vector estimated value that first group of sampled data is corresponding can be expressed as the steering vector estimated value of the electric dipole submatrix of reconstruct Z-direction with magnetic dipole submatrix steering vector estimated value invariable rotary between two submatrix steering vectors closes be the invariable rotary matrix between two submatrixs, the diagonal matrix that it is diagonal element with the element in bracket that diag [] represents;

According to calculate the estimated value of polarizations parameter:

${\widehat{\mathrm{\&gamma;}}}_k={\tan }^{-1}\left(\right|\widehat{\mathrm{\&Psi;}}(k,k)\left|\right),$

${\widehat{\mathrm{\&eta;}}}_k=\arg \left(\widehat{\mathrm{\&Psi;}}(k,k)\right),$

Wherein, represent invariable rotary matrix row k kth row element.

The present invention utilizes two groups of synchronously sampled datas to construct full array received data and autocorrelation matrix thereof, hypercomplex number feature decomposition is carried out to autocorrelation matrix, and the estimation of array steering vector is obtained according to subspace theory, the direction cosine in x-axis direction and y-axis direction are obtained by spatial domain steering vector piecemeal computing, thus obtain the estimation of signal two dimensional arrival angles, reconstruct the electric dipole submatrix steering vector in x-axis direction and y-axis direction according to array steering vector, utilize the relation between two submatrix steering vectors to obtain the estimation of polarization parameter.

Effect of the present invention can be further illustrated by following simulation result:

Emulation experiment condition is as follows: adopt radius be the Homogeneous Circular array of R=0.5 λ as receiving array, N=14 electromagnetic dipole is to being uniformly distributed in circumferentially.The parameter of two incoming signals is respectively (θ ₂, φ ₂, γ ₂, η ₂)=(30 °, 43 °, 67 °, 80 °), carry out 1024 snaps, 500 independent Monte Carlo experiments.

Emulation experiment adopts the long vector algorithm of prior art and the Quaternion Algorithm of the inventive method to contrast.As shown in Figures 3 to 6, in signal to noise ratio (S/N ratio) interval, the root-mean-square error of the inventive method is all lower than the root-mean-square error of long vector algorithm, when signal to noise ratio (S/N ratio) is 0dB, the inventive method the angle of pitch, position angle, auxiliary polarization angle, polarization phases difference estimate root-mean-square error with long vector method compare respectively little 0.8 °, 0.15 °, 0.2 ° and 0.5 °.

As seen from Figure 7, be 0.95 when 0dB based on the angle-of-arrival estimation probability of success of the inventive method, and based on the angle-of-arrival estimation probability of success of long vector less than 0.55.As seen from Figure 8, when 0dB, the polarization parameter probability of success of the inventive method is close to 1, and is only 0.65 based on the probability of success of the polarization parameter of long vector.

Can find out from the contrast of Fig. 9 and Figure 10, when signal to noise ratio (S/N ratio) is 10dB, the angle-of-arrival estimation value of the inventive method near true value more among a small circle in disturbance, the error of estimated value is less, and then deviation true value is more for long vector algorithm estimated value, and evaluated error is larger.

Can find out from the contrast of Figure 11 and Figure 12, when signal to noise ratio (S/N ratio) is 10dB, the polarization estimation value of the inventive method is tightly centered around near true value, and evaluated error is less.And long vector algorithm estimated value to depart from actual value more, evaluated error is larger.

Therefore can find out that the inventive method has higher parameter estimation performance than long vector method from the root-mean-square error of parameter estimation, the probability of success and scatter diagram.

The above, it is only preferred embodiment of the present invention, not any pro forma restriction is done to the present invention, although the present invention discloses as above with preferred embodiment, but and be not used to limit the present invention, any those skilled in the art, do not departing within the scope of technical solution of the present invention, make a little change when the technology contents of above-mentioned announcement can be utilized or be modified to the Equivalent embodiments of equivalent variations, in every case be the content not departing from technical solution of the present invention, according to any simple modification that technical spirit of the present invention is done above embodiment, equivalent variations and modification, all still belong in the scope of technical solution of the present invention.

Claims (2)

1. the hypercomplex number ESPRIT method for parameter estimation of electromagnetic dipole pair array, it is characterized in that, comprise the following steps: K mutual uncorrelated perfact polarization transverse electromagnetic wave signal incides in Electromagnetic Vector Sensor Array simultaneously, described array elements is by an electric dipole and a molecular electromagnetic dipole pair of magnetic dipole

Step one, carry out M sampling obtain first group of sampled data X to the output signal of Electromagnetic Vector Sensor Array, synchronized sampling M time after time delay Δ T, obtains second group of sampled data Y;

Wherein, x _nem () represents the m time sampled data of the electric dipole output signal of the n-th array element, x _nhm () represents the m time sampled data of the magnetic dipole output signal of the n-th array element, y _nethe m time sampled data that m () represents time delay Δ T after, the electric dipole of the n-th array element outputs signal, y _nhthe m time sampled data that m () represents time delay Δ T after, the magnetic dipole of the n-th array element outputs signal;

Step 2, first group of sampled data X and second group sampled data Y is formed first group according to the electric dipole of same array element and the homogeneous snap data investigation of magnetic dipole respectively receive hypercomplex number data matrix Z ₁hypercomplex number data matrix Z is received with second group ₂, construct full array received data matrix Z;

Wherein, represent that the m time sampled data outputed signal by the electric dipole of the n-th array element and magnetic dipole superposes the quaternion algebra certificate formed, represent that the m time sampled data that time delay Δ T is outputed signal by electric dipole and the magnetic dipole of the n-th array element superposes the quaternion algebra certificate formed;

The hypercomplex number data matrix Z that first group of sampled data X is formed ₁=A ₁s+N ₁, wherein,

A ₁=[a ₁(θ ₁, φ ₁, γ ₁, η ₁) ..., a ₁(θ _k, φ _k, γ _k, η _k) ..., a ₁(θ _k, φ _k, γ _k, η _k)] be array steering vector corresponding to first group of sampled data, a ₁(θ _k, φ _k, γ _k, η _k)=c _kq (θ _k, φ _k), the hypercomplex number data representation of electromagnetic field in first group of sampled data of a kth incoming signal, and h _kz=sin θ _kcos γ _kbe respectively a kth incoming signal at true origin place electric field component along the z-axis direction and magnetic-field component, q (θ _k, φ _k) be the spatial domain steering vector at full array phase center, θ _kthe angle of pitch of a kth incoming signal, φ _kthe position angle of a kth incoming signal, γ _kthe auxiliary polarization angle of a kth incoming signal, η _kthe polarization phases of a kth incoming signal is poor, N ₁be white Gaussian noise vector, S is the magnitude matrix that incoming signal is formed;

The hypercomplex number data matrix Z that second group of sampled data Y is formed ₂=A ₂s+N ₂=A ₁Φ S+N ₂, wherein, A ₂=A ₁Φ is array steering vector corresponding to second group of sampled data, and Φ is time delay matrix, N ₂it is white Gaussian noise vector;

Construct full array received data matrix $Z=\left[\begin{array}{c}{Z}_1\\ Z_2\end{array}\right]=\left[\begin{array}{c}{A}_1\\ A_2\end{array}\right]S+N=\mathrm{AS}+N,$ Wherein, $A=\left[\begin{array}{c}{A}_1\\ A_2\end{array}\right]$ The array steering vector matrix that all data is corresponding, $N=\left[\begin{array}{c}{N}_1\\ N_2\end{array}\right]$ It is all data noise matrix;

Step 3, calculate the autocorrelation matrix R of full array received data matrix _z, hypercomplex number feature decomposition is carried out to autocorrelation matrix, obtains the array steering vector estimated value that first group of sampled data is corresponding the array steering vector estimated value that second group of sampled data is corresponding the array steering vector Matrix Estimation value corresponding with all data

$R_z=\frac{1}{M}\left[{\mathrm{ZZ}}^H\right]={\mathrm{AR}}_s{A}^H+{\mathrm{\&sigma;}}^{2}I,$

Wherein, for the autocorrelation function of incoming signal, σ ²for the variance of noise, I is unit matrix, () ^hrepresent transposed complex conjugate operation;

To autocorrelation matrix R _zcarry out Quaternion Matrix feature decomposition and obtain signal subspace E _s, according to subspace principal, there is the nonsingular matrix T of K × K, and E _s=AT, gets E respectively _sthe capable and capable matrix E formed respectively of rear N of front N ₁and E ₂, by the definition of signal subspace, A ₁, A ₂with E ₁, E ₂between meet E ₁=A ₁t, E ₂=A ₂t=A ₁Φ T, then have

To matrix carry out hypercomplex number feature decomposition, K large eigenwert forms delay matrix estimated value eigenwert characteristic of correspondence vector forms nonsingular matrix estimated value according to obtain ${\hat{A}}_1=E_1{T}^{-1},{\hat{A}}_2=E_2{T}^{-1},\hat{A}=E_s{T}^{-1};$

The estimated value of step 4, calculating direction of arrival;

According to calculate the phase difference vector between adjacent two array elements wherein, A ₁(2:N, k) represents A ₁kth row the 2 to the N number of element, A ₁(1:N-1, k) represents A ₁kth row the 1 to the N-1 element ./represent correspondence element be divided by;

Calculate phasing matrix phase place is got in arg () expression;

According to $\mathrm{\&Omega;}=W\&CenterDot;\left[\begin{array}{c}{\widehat{\mathrm{\&alpha;}}}_k\\ {\widehat{\mathrm{\&beta;}}}_k\end{array}\right],$ Calculate the direction cosine estimated value in the x-axis direction of a kth incoming signal with the direction cosine estimated value in y-axis direction $\left[\begin{array}{c}{\widehat{\mathrm{\&alpha;}}}_k\\ {\widehat{\mathrm{\&beta;}}}_k\end{array}\right]={\left[W\right]}^{\#}\mathrm{\&Omega;},$ [W] ^#it is the pseudo inverse matrix of location matrix W;

According to the estimated value of direction cosine obtain the estimated value of direction of arrival:

$\left\{\begin{array}{c}{\widehat{\mathrm{\&theta;}}}_k=\arcsin \left(\sqrt{{\widehat{\mathrm{\&alpha;}}}_k^2+{\widehat{\mathrm{\&beta;}}}_k^2}\right)\\ \left\{\begin{array}{c}{\widehat{\mathrm{\&phi;}}}_k=\arctan \left(\frac{{\widehat{\mathrm{\&alpha;}}}_k}{{\widehat{\mathrm{\&beta;}}}_k}\right),{\widehat{\mathrm{\&beta;}}}_k\&GreaterEqual;0\\ {\widehat{\mathrm{\&phi;}}}_k=\mathrm{\&pi;+arctan}\left(\frac{{\widehat{\mathrm{\&alpha;}}}_k}{{\widehat{\mathrm{\&beta;}}}_k}\right),{\widehat{\mathrm{\&beta;}}}_k<0\end{array}\right.\end{array}\right.;$

Step 5, by real part and the array steering vector estimated value of electric dipole submatrix of three imaginary parts reconstruct Z-directions with the array steering vector estimated value of magnetic dipole submatrix the estimated value of polarizations parameter is obtained according to the invariable rotary relation between submatrix steering vector;

${\hat{A}}_1={\hat{A}}_{10}+{\hat{A}}_{11}\tilde{i}+{\hat{A}}_{12}\tilde{j}+{\hat{A}}_{13}\tilde{k},$ be real part, be three imaginary parts, according to the formation of Quaternion Matrix in step 2, the steering vector estimated value of the electric dipole submatrix of reconstruct Z-direction with magnetic dipole submatrix steering vector estimated value invariable rotary between two submatrix steering vectors closes be the invariable rotary matrix between two submatrixs, according to calculate the estimated value of polarizations parameter:

${\widehat{\mathrm{\&gamma;}}}_k={\tan }^{-1}\left(\right|\widehat{\mathrm{\&Psi;}}(k,k)\left|\right),$

${\widehat{\mathrm{\&eta;}}}_k=\arg \left(\right|\widehat{\mathrm{\&Psi;}}(k,k)\left|\right),$

Wherein, represent invariable rotary matrix row k kth column element;

N=1 in abovementioned steps ..., N, m=1 ..., M, N are the array number of array, and M is sampling number, for 3 imaginary units of hypercomplex number.

2. the hypercomplex number ESPRIT method for parameter estimation of electromagnetic dipole pair array according to claim 1, it is characterized in that: described array is circular array, the axis of electric dipole and the axis being parallel of magnetic dipole are in z-axis, N number of array element is evenly distributed on annulus, and true origin is positioned at the center of circle of annulus.