1. The MIMO radar two-dimensional direction of arrival estimation method based on constraint tensor decomposition is characterized by comprising the following steps of:

(1a) constructing a fourth-order tensor model for receiving data by the MIMO radar, wherein the MIMO radar is provided with a plurality of same transmitting sub-arrays;

(1b) reconstructing the fourth-order tensor model and performing singular value decomposition on an obtained reconstruction matrix, wherein a decomposition result comprises a solved left singular matrix, and the left singular matrix comprises a van der mond structure;

(1c) and estimating a column vector of the eigenvalue according to the van der mond structure of the left singular matrix and the correlation of the submatrices of the left singular matrix, and calculating according to the column vector to obtain an estimated value of the two-dimensional direction of arrival.

2. The method of claim 1, wherein the constructing a fourth order tensor model for the MIMO radar to receive data, the MIMO radar having a plurality of identical transmit sub-arrays, is performed by:

(2a) constructing a receiving array and an s-th transmitting sub-array A of the MIMO radar_sReceived data at the q-th pulseWherein S1, 2.. S, Q1, 2.. Q, S I × J, S denotes the total number of transmit sub-arrays, I denotes the number of sub-arrays of the transmit array in the x-axis direction, J denotes the number of sub-arrays of the transmit array in the y-axis direction, Q denotes the total number of pulses, and each transmit sub-array has a transmit sub-arrayThe individual array elements are evenly distributed on the matrix grid,the expression of (a) is:

wherein the content of the first and second substances,is M₀The number of transmitting array elements of the sub-array in the x-axis direction,is M₀The number of transmitting array elements of the sub-array in the x-axis direction, M₀For the number of elements included in each transmit sub-array,echo data representing the q-th pulse of the receiving array and the s-th transmitting sub-array, A_sA transmit steering matrix representing the s-th transmit sub-array, B a steering matrix, Σ, of the receive array_q＝diag(c_q) Is formed by a column vector c_qThe square matrix is opened, and the square matrix is opened,containing doppler shift and radar reflectance information for L targets, represents a target Doppler vector, sigma reflects a target radar reflection coefficient, and L is 1,2_lIndicating the doppler shift of the l-th target, T is the pulse repetition period of the radar,is thatCorresponding gaussian white noise matrix, (.)^TThe transpose of the matrix is represented,meaning defined as;

(2b) analyzing the relation between the transmitting guide vectors of each transmitting sub-array by using the uniform arrangement structure of the transmitting arrays, wherein the s-th transmitting sub-array A_s、A_sCorresponding transverse steering matrix component U_jAnd a longitudinal steering matrix component V_iThe expression of (a) is:

A_s＝U_j⊙V_i

U_j＝U₀Γ_j，V_i＝V₀Γ_i

wherein, s ═ 1I + I denotes the serial number of the s-th transmitting subarray, J ═ 1,2,. J and I ═ 1,2,. I denote the serial numbers of the transmitting subarrays in the transverse and longitudinal directions, respectively, and U ═ 1,. I denotes the serial numbers of the transmitting subarrays in the transverse and longitudinal directions, respectively_jIs the transverse steering matrix component, V, of the transmit sub-array_iIs the longitudinal steering matrix component, U, of the transmit sub-array₀Is a transverse reference matrix, V₀Is a longitudinal reference subarray, gamma_jIs a diagonal matrix containing i column vectors spread apart, Γ_iIs a diagonal matrix spread by j column vectors; an indication of a Khatri-Rao product;

(2c) will be provided withColumn vectorization to obtain A₀＝U₀⊙V₀Is a guide matrix of a reference emission subarray, and a Gaussian white noise matrixColumn vectorization to obtainSubstituting the relation described in step (2b) into the relation, and sequentially assigning all S column vectors in the sequence of sub-array numbersCombined into a new matrix Y^(q)I.e. byExpressed as:

wherein the content of the first and second substances,a set of phase information representing each transmit sub-array in the transverse direction, a set of phase information representing each transmit sub-array in the longitudinal direction, is the noise matrix received by the array at the qth pulse;

(2d) column vectorization matrix Y^(q)To obtain z_qConcrete writing z_q＝[H⊙Δ⊙A₀⊙B]c_q+r_q，r_qIs N^(q)Then splicing the received data of Q pulses according to columns to obtain the MIMO radar multi-pulse received data Z, namelyThe expression is as follows:

Z＝[H⊙Δ⊙A₀⊙B]C^T+R

whereinIs a set of Q pulses of target doppler and RCS information, represents the total noise set received by the array at the Q pulses;

(2e) reconstructing the matrix Z as a 5 th order tensorThe expression is as follows:

wherein the content of the first and second substances,represents the vector outer product, [ [.]]For expressing the set of tensor matrix factors, η_l，δ_l，α_l，β_l，γ_lAre each H, Delta, A₀The l-th column vector of B, C,is the 5 th order noise tensor generated by the same reconstruction of the matrix R.

3. The method according to claim 1, wherein the reconstructing the fourth-order tensor model and performing singular value decomposition on the obtained reconstructed matrix, the decomposition result includes an obtained left singular matrix, and the left singular matrix includes a van der mond structure, and the method includes:

(3a) will tensorConverted into a third order tensor by tensor conversionThe expression is as follows:

wherein G ═ H ∑ Δ is a first matrix factor, and a third matrix factor B | _ C satisfies the column full rank condition;

(3b) according to a tensor reconstruction principle, the third-order tensor is generatedFrom a third dimensionReconstructing the degree to obtain a reconstructed matrix T₍₃₎The expression is:

T₍₃₎＝(G⊙A₀)(B⊙C)^T+N

wherein the matrix N is represented by the noise tensorA noise matrix generated by the same reconstruction method;

(3c) for the reconstructed matrix T₍₃₎Singular value decomposition is carried out, and the expression is as follows:

T₍₃₎＝U∑V^H

wherein, ()^HRepresenting the conjugate transpose of the matrix, the decomposition results being respectively of dimension SM₀A left singular matrix U of x L, a right singular matrix V of dimension nqx L, and a singular value matrix Σ of dimension L x L; since the third factor matrix B ^ C has full rank, there is inevitably one L × L non-singular transformation matrix E, so that UE ═ H ^ Δ ^ A₀And matrices H and Δ are both van der mond matrices.

4. The method according to claim 1, wherein the estimating the column vector of the eigenvalue according to the van der mond structure of the left singular matrix and using the correlation of the submatrices of the left singular matrix and performing the calculation according to the column vector to obtain the estimated value of the two-dimensional direction of arrival comprises:

(4a) first sub-matrix U of two sub-matrices defining left singular matrix₁And a second sub-matrix U₂The expression is:

U₂E＝H⊙Δ⊙A₀

where H is a sub-matrix consisting of the matrix H except the first row,is a sub-matrix composed of a matrix H except the last row, and a matrix E is a non-singular transformation matrix;

(4b) constructing a submatrix U from the matrix U by utilizing row selection according to the operational rule of the Khatri-Rao product₁And U₂The expression is:

wherein, I represents a unit matrix, the dimension size of which is determined by the corresponding subscript, 0 represents a matrix with all elements being 0, and the dimension size is also determined by the corresponding subscript;

(4c) by using the characteristics of van der Mond matrix structure, U₁And U₂The relational expression of (1) is:

U₂E＝U₁EΩ_y

wherein omega_y＝diag(ω_y) Is formed by_yThe square matrix is opened, and the square matrix is opened,is the vector of the generating factor u corresponding to the Van der Mond matrix H₁Is u_l1 st steering vector u_lBeing the first intermediate quantity of the steering vectors of the matrix U,representing the phase stepping of the j +1 th transmitting subarray and the j transmitting subarray in the transverse direction;

(4d) two other sub-matrices defining the left singular matrix: third sub-matrix U₃And a fourth sub-matrix U₄The expression is:

wherein the content of the first and second substances,is a sub-matrix consisting of the matrix delta except the first row,is a sub-matrix composed of the matrix delta except the last row;

(4e) constructing a third sub-matrix U from the matrix U by utilizing row selection according to the operational rule of the Khatri-Rao product₃And a fourth sub-matrix U₄The expression is:

wherein the content of the first and second substances,represents the Kronecker product;

(4f) by using the characteristics of van der Mond matrix structure, U₃And U₄The relational expression of (1) is:

U₄E＝U₃EΩ_x

wherein omega_x＝diag(ω_x) Is formed by_xThe square matrix is opened, and the square matrix is opened,is the vector of the generation factors corresponding to the van der mond matrix delta,the phase of the (i + 1) th transmitting sub-array and the phase of the ith transmitting sub-array in the longitudinal direction are stepped;

(4g) from the correlation of each submatrix, a generation factor vector ω corresponding to the van der mond matrix H is obtained_yVector ω of generation factors corresponding to van der Mond matrix Δ_xThe expression is:

wherein, ()^-1Representing matrix inversion; respectively to the matrixAndperforming characteristic decomposition to obtain a column vector composed of characteristic values, and recording the column vector asAndi.e. the column vector omega_yAnd ω_x(ii) an estimate of (d);

(4h) according toAndthe two-dimensional direction of arrival estimation of the target is realized, and the expression is as follows:

wherein the content of the first and second substances,andare respectively column vectorsAndthe first element of (a) is,andrepresents the median ofIs estimated, andandthe estimation result of the two-dimensional direction of arrival of the pitching angle and the azimuth angle of the ith target is obtained; collecting all L sets of estimatesI.e. to achieve the proposed two-dimensional direction of arrival estimation.

Background

The MIMO radar is a new system radar with mutually orthogonal transmitting waveforms, and has attracted extensive attention in the last twenty years due to its superior performance in multi-target detection, parameter estimation, and the like, in which a centralized MIMO radar with a relatively small antenna unit spacing is taken as a representative. Numerous researchers have studied and analyzed the advantages and performance improvements of MIMO radar compared to conventional phased array radars, including: the method has the advantages of good anti-interference performance, flexible emission directional diagram, better resolution ratio and higher resolution precision of direction of arrival estimation. The performance improvement mainly comes from the effective utilization of waveform diversity, namely echoes of all receiving and transmitting channels are obtained through matched filtering of a receiving end, so that a virtual array with larger caliber and more array elements is equivalently constructed. As such, past research on the estimation of the direction of arrival of the MIMO radar has mainly focused on analyzing the covariance matrix of the virtual array received data, which can be regarded as a generalization of the estimation method of the direction of arrival of the phased array radar, such as [ z.guo, x.wang, and w.heng ], "Millimeter-wave channel estimation based 2-D beamspace MUSIC method," IEEE trans.wireless communication, vol.16, No.8, pp.5384-5394,2017 ], which generalizes the conventional MUSIC algorithm to the MIMO radar; jinli, g.hong, and s.weimin, "Angle estimation using ESPRIT with outpating in MIMO radar," electron.lett., vol.44, No.24, pp.1422-1423,2008 ], takes into account the application of ESPRIT algorithm in MIMO radar. The algorithms can only utilize single pulse data in MIMO radar multi-pulse receiving data at a time, and need to iterate direction of arrival estimation results among different pulses, so that the algorithms are easily influenced by target fluctuation, and a good estimation effect cannot be kept when the signal-to-noise ratio of a target echo is low.

In view of the above problems, researchers have proposed a Tensor decomposition-based MIMO radar direction-of-arrival estimation algorithm, such as [ d.nion and n.d.si ropoulos, "transducer algebra and multidimensional harmonic in signal processing for MIMO radar," IEEE trans.signal processing, vol.58, No.11, pp.5693-5705, nov.2010 ], and [ n.d.si ropoulos, l.de Lathauwer et al, "transducer composition for signal processing and ma chip learning," IEEE trans.signal processing, vol.65, No.13, pp.3551-3582, jul.2017 ]. The multi-pulse receiving data of the MIMO radar is stored by adopting a tensor model, so that a multi-linear structure between the receiving data of the MIMO radar can be utilized, the directions of arrival of a plurality of targets can be estimated at the same time, and the performance of estimating the directions of arrival is effectively improved. However, the conventional tensor decomposition method, i.e., Alternating Least square method (Alternating Least Squares), has high computational complexity, unstable convergence, and requires information on the number of targets as a priori conditions. These problems are more pronounced when the target tension is higher than third order.

In some application occasions, such as a ground-based radar for multi-target detection and tracking, a transmitting array generally has a large number of array elements, in order to simplify the system structure, measures such as subarray division and the like are adopted, and a corresponding tensor model can reach fourth order or even higher. In this case, to estimate the direction of arrival of a plurality of targets in near real time, an estimation algorithm is required to be used:

therefore, a radar estimation algorithm is needed at present, which is applicable to a high-order tensor model, can still effectively estimate the number of targets under the condition of unknown target number, and has low computational complexity, stable convergence, high accuracy of estimation of the direction of arrival and good resolution.

Disclosure of Invention

In view of this, the invention provides a constraint tensor decomposition-based MIMO radar two-dimensional direction-of-arrival estimation method, which has low computational complexity and stable convergence, can still maintain effective estimation under the condition of unknown target number, can be applied to MIMO radar scenes with a plurality of same sub-arrays, has significantly improved resolution and accuracy of angle estimation compared with other algorithms, and provides a technical approach for MIMO radar engineering application.

In order to achieve the purpose, the technical scheme of the invention is as follows:

the MIMO radar two-dimensional direction of arrival estimation method based on constraint tensor decomposition comprises the following steps:

(1a) and constructing a fourth-order tensor model for receiving data by the MIMO radar, wherein the MIMO radar has a plurality of same transmitting sub-arrays.

(1b) And reconstructing a fourth-order tensor model and performing singular value decomposition on the obtained reconstruction matrix, wherein the decomposition result comprises the solved left singular matrix, and the left singular matrix comprises a Van der Mond structure.

(1c) And estimating a column vector of the eigenvalue according to the van der mond structure of the left singular matrix and the correlation of the submatrices of the left singular matrix, and calculating according to the column vector to obtain an estimated value of the two-dimensional direction of arrival.

Further, a fourth-order tensor model for receiving data of the MIMO radar is constructed, the MIMO radar has a plurality of same transmitting sub-arrays, and the specific method comprises the following steps:

(2a) constructing a receiving array and an s-th transmitting sub-array A of the MIMO radar_sReceived data at the q-th pulseWherein S1, 2.. S, Q1, 2.. Q, S I × J, S denotes the total number of transmit sub-arrays, I denotes the number of sub-arrays of the transmit array in the x-axis direction, J denotes the number of sub-arrays of the transmit array in the y-axis direction, Q denotes the total number of pulses, and each transmit sub-array has a transmit sub-arrayThe individual array elements are evenly distributed on the matrix grid,the expression of (a) is:

wherein the content of the first and second substances,is M₀The number of transmitting array elements of the sub-array in the x-axis direction,is M₀The number of transmitting array elements of the sub-array in the x-axis direction, M₀For the number of elements included in each transmit sub-array,echo data representing the q-th pulse of the receiving array and the s-th transmitting sub-array, A_sA transmit steering matrix representing the s-th transmit sub-array, B a steering matrix, Σ, of the receive array_q＝diag(c_q) Is formed by a column vector c_qThe square matrix is opened, and the square matrix is opened,containing doppler shift and radar reflectance information for L targets, represents a target Doppler vector, sigma reflects a target radar reflection coefficient, and L is 1,2_lIndicating the doppler shift of the l-th target, T is the pulse repetition period of the radar,is thatCorresponding gaussian white noise matrix, (.)^TThe transpose of the matrix is represented,meaning defined as.

(2b) Analyzing the relation between the transmitting guide vectors of each transmitting sub-array by using the uniform arrangement structure of the transmitting arrays, wherein the s-th transmitting sub-array A_s、A_sCorresponding toTransverse steering matrix component U_jAnd a longitudinal steering matrix component V_iThe expression of (a) is:

A_S＝U_j⊙V_i

U_j＝U₀Γ_j,V_i＝V₀Γ_i

wherein, s ═ 1I + I denotes the serial number of the s-th transmitting subarray, J ═ 1,2,. J and I ═ 1,2,. I denote the serial numbers of the transmitting subarrays in the transverse and longitudinal directions, respectively, and U ═ 1,. I denotes the serial numbers of the transmitting subarrays in the transverse and longitudinal directions, respectively_jIs the transverse steering matrix component, V, of the transmit sub-array_iIs the longitudinal steering matrix component, U, of the transmit sub-array₀Is a transverse reference matrix, V₀Is a longitudinal reference subarray, gamma_jIs a diagonal matrix containing i column vectors spread apart, Γ_iIs a diagonal matrix spread by j column vectors; and indicates a Khatri-Rao product.

(2c) Will be provided withColumn vectorization to obtain A₀＝U₀⊙V₀Is a guide matrix of a reference emission subarray, and a Gaussian white noise matrixColumn vectorization to obtainAnd substituting the relational expression described in the step (2b) into the relational expression, and sequentially adding all S column vectors in the order of sub-array numberCombined into a new matrix Y^(q)I.e. byExpressed as:

wherein the content of the first and second substances,a set of phase information representing each transmit sub-array in the transverse direction, a set of phase information representing each transmit sub-array in the longitudinal direction, is the noise matrix received by the array at the q-th pulse.

(2d) Column vectorization matrix Y^(q)To obtain z_qConcrete writing z_q＝[H⊙Δ⊙A₀⊙B]c_q+r_q，r_qIs N^(q)Then splicing the received data of Q pulses according to columns to obtain the MIMO radar multi-pulse received data Z, namelyThe expression is as follows:

Z＝[H⊙Δ⊙A₀⊙B]C^T+R

whereinIs a set of Q pulses of target doppler and RCS information, representing the total noise set received by the array at Q pulses.

(2e) Reconstructing the matrix Z as a 5 th order tensorThe expression is as follows:

wherein the content of the first and second substances,represents the vector outer product, [ [.]]For expressing the set of tensor matrix factors, η_l，δ_l，α_l，β_l，γ_lAre each H, Delta, A₀The l-th column vector of B, C,is the 5 th order noise tensor generated by the same reconstruction of the matrix R.

Further, reconstructing a fourth-order tensor model and performing singular value decomposition on an obtained reconstruction matrix, wherein a decomposition result comprises a solved left singular matrix, the left singular matrix comprises a van der mond structure, and the specific method comprises the following steps:

(3a) will tensorConverted into a third order tensor by tensor conversionThe expression is as follows:

wherein G ═ H ∑ Δ is a first matrix factor, and a third matrix factor B | _ C satisfies the column full rank condition.

(3b) According to a tensor reconstruction principle, the third-order tensor is generatedReconstructing from the third dimension to obtain a reconstructed matrix T₍₃₎The expression is:

T₍₃₎＝(G⊙A₀)(B⊙C)^T+N

wherein the matrix N is represented by the noise tensorThe noise matrix generated by the same reconstruction method.

(3c) For the reconstructed matrix T₍₃₎Singular value decomposition is carried out, and the expression is as follows:

T₍₃₎＝U∑V^H

wherein, ()^HRepresenting the conjugate transpose of the matrix, the decomposition results being respectively of dimension SM₀A left singular matrix U of x L, a right singular matrix V of dimension nqx L, and a singular value matrix Σ of dimension L x L; since the third factor matrix B ^ C has full rank, there is inevitably one L × L non-singular transformation matrix E, so that UE ═ H ^ Δ ^ A₀And matrices H and Δ are both van der mond matrices.

Further, according to the van der mond structure of the left singular matrix, by using the correlation of the submatrices of the left singular matrix, estimating a column vector of the eigenvalue and calculating according to the column vector to obtain an estimated value of the two-dimensional direction of arrival, the specific method is as follows:

(4a) first sub-matrix U of two sub-matrices defining left singular matrix₁And a second sub-matrix U₂The expression is:

wherein the content of the first and second substances,is a sub-matrix consisting of the matrix H except the first row,is a sub-matrix composed of the matrix H except the last row, and the matrix E is a non-singular transformation matrix.

(4b) Constructing a submatrix U from the matrix U by utilizing row selection according to the operational rule of the Khatri-Rao product₁And U₂The expression is:

wherein, I represents an identity matrix, the dimension of which is determined by the corresponding subscript, 0 represents a matrix with all elements being 0, and the dimension of which is also determined by the corresponding subscript.

(4c) By using the characteristics of van der Mond matrix structure, U₁And U₂The relational expression of (1) is:

U₂E＝U₁EΩ_y

wherein omega_y＝diag(ω_y) Is formed by_yThe square matrix is opened, and the square matrix is opened,is the vector of the generating factor u corresponding to the Van der Mond matrix H₁Is u_l1 st steering vector u_lA first intermediate quantity, Delta, of steering vectors of the matrix U_my＝m_j+1-m_jIndicating the phase step of the j +1 th transmitting sub-array and the j transmitting sub-array in the transverse direction.

(4d) Two other sub-matrices defining the left singular matrix: third momentArray U₃And a fourth sub-matrix U₄The expression is:

wherein the content of the first and second substances,is a sub-matrix consisting of the matrix delta except the first row,is a sub-matrix consisting of the matrix delta except for the last row.

(4e) Constructing a third sub-matrix U from the matrix U by utilizing row selection according to the operational rule of the Khatri-Rao product₃And a fourth sub-matrix U₄The expression is:

wherein the content of the first and second substances,representing the Kronecker product.

(4f) By using the characteristics of van der Mond matrix structure, U₃And U₄The relational expression of (1) is:

U₄E＝U₃EΩ_x

wherein omega_x＝diag(ω_x) Is formed by_xThe square matrix is opened, and the square matrix is opened,is the vector of the generation factors corresponding to the Van der Mond matrix Delta, Delta_mx＝m_i+1-m_iThe phase of the (i + 1) th transmitting sub-array and the phase of the (i) th transmitting sub-array are stepped in the longitudinal direction.

(4g) From the correlation of each submatrix, a generation factor vector ω corresponding to the van der mond matrix H is obtained_yVector ω of generation factors corresponding to van der Mond matrix Δ_xThe expression is:

wherein, ()^-1Representing matrix inversion; respectively to the matrixAndperforming characteristic decomposition to obtain a column vector composed of characteristic values, and recording the column vector asAndi.e. the column vector omega_yAnd ω_xIs estimated.

(4h) According toAndthe two-dimensional direction of arrival estimation of the target is realized, and the expression is as follows:

wherein the content of the first and second substances,andare respectively column vectorsAndthe first element of (a) is,andrepresents the median ofIs estimated, andandthe estimation result of the two-dimensional direction of arrival of the pitching angle and the azimuth angle of the ith target is obtained; collecting all L sets of estimatesI.e. to achieve the proposed two-dimensional direction of arrival estimation.

Has the advantages that: according to the method, a high-order tensor model and corresponding matrix expansion thereof are constructed according to a multiple linear structure of multi-pulse receiving data of the MIMO radar, singular value decomposition is carried out on an expansion matrix, four sub-matrixes meeting rotation invariance are generated from a left singular matrix through constraint conditions brought by a transmitting array structure, and then two-dimensional direction of arrival estimation is carried out on a plurality of targets by utilizing an ESPRIT-like algorithm. According to the method, a tensor model is utilized, and a multiple linear relation among multiple pulse receiving data of the MIMO radar is mined; the number of targets can be effectively estimated under the unknown condition without depending on the prior information of the number of the targets, and the direction of arrival estimation is carried out on multiple targets; only matrix operation is involved, iteration is not required, so that the calculation complexity is low, convergence is guaranteed certainly, and the decomposition result is completely consistent with the original input condition under the condition of no noise. Meanwhile, the method has higher resolution and better resolution precision for estimating the direction of arrival.

Drawings

FIG. 1 is a schematic diagram of a system array architecture of the present invention;

FIG. 2 is an algorithmic flow chart of the present invention;

FIG. 3 is a schematic diagram of the accuracy of estimation of the direction of arrival for the simulation synthesis of the present invention;

FIG. 4 is a schematic diagram of the direction of arrival estimation resolution of the simulation synthesis of the present invention.

Detailed Description

The invention is described in detail below by way of example with reference to the accompanying drawings.

The core thought of the invention is as follows: firstly, a high-order tensor model of MIMO radar receiving data with a plurality of same transmitting sub-arrays is deduced, a factor matrix corresponding to the model is proved to have a Vandermonde structure under the constraint of an array structure, then the left singular matrix is ensured to inherit the Vandermonde structure through matrix reconstruction and singular value decomposition of the high-order tensor, finally, phase information of a target in the direction is estimated from the sub-matrix of the left singular matrix by using an ESPRIT algorithm, and further two-dimensional direction of arrival estimation is realized.

The technical principle of the invention is as follows: as shown in fig. 1, consider a MIMO radar with M ═ M_xM_yA plurality of transmitting array elements uniformly distributed on a two-dimensional rectangular grid, wherein M_xAnd M_yThe number of points of the rectangular grid in the x-axis and y-axis directions is represented, respectively. The distance between adjacent array elements isWhere λ is the operating wavelength of the radar system. Without loss of generality, the array is at a pitch angle, provided that electromagnetic waves propagate unattenuated in space and satisfy the far-field conditionThe transmit steering vector in the azimuth theta direction can be expressed as

Wherein the content of the first and second substances, (·)^Tthe transpose of the matrix is represented,the representation defines a variable. Similarly, assuming that the radar has N receiving elements randomly selected from the transmitting elements and having coordinates (xn, yn), where N is 1,2

In general, when there are L objects in space, they are distributed asAndthe transmit steering matrix and the receive steering matrix can be separately defined as

Thus, the multi-target reception data of the MIMO radar can be expressed as

Y＝BΣA^T+N

Wherein Σ is diag (σ),represents a set of L target radar Reflection Coefficients (RCS), and N represents an N × M gaussian white noise matrix. Two-dimensional direction of arrival estimation is to obtain the angle information of all targets from the observation of Y. As shown in fig. 3, the diagram is a schematic diagram of the accuracy of the direction of arrival estimation of the simulation synthesis of the present invention, and fig. 4 is a schematic diagram of the resolution of the direction of arrival estimation of the simulation synthesis of the present invention. As can be seen from the two figures, the method of the invention can improve the estimation rule and the estimation resolution of the direction of arrival.

As shown in fig. 2, the present invention provides a method for estimating a two-dimensional direction of arrival of a MIMO radar based on constrained tensor decomposition, which includes the following steps:

(1a) for a MIMO radar with M transmitting array elements distributed in a rectangular grid and N receiving array elements randomly distributed, constructing a fourth-order tensor model for receiving data of the MIMO radar, wherein the MIMO radar has a plurality of same transmitting sub-arrays:

(2a) constructing a receiving array and an s-th transmitting sub-array A of the MIMO radar_sReceived data at the q-th pulseWhere S is 1, 2.. S, Q is 1, 2.. Q, S is I × J, S denotes the total number of transmission sub-arrays, I denotes the number of sub-arrays of the transmission array in the x-axis direction, J denotes the number of sub-arrays of the transmission array in the y-axis direction, and Q denotes the total number of pulses. The length of the rectangular grid is an x axis, the width of the rectangular grid is a y axis, and the first element of the rectangular grid is an origin. Each transmitting sub-array hasThe individual array elements are evenly distributed on the matrix grid,is expressed as

Wherein the content of the first and second substances,is M₀The number of transmitting array elements of the sub-array in the x-axis direction,is M₀The number of transmitting array elements of the sub-array in the x-axis direction, M₀For the number of elements included in each transmit sub-array,echo data representing the q-th pulse of the receiving array and the s-th transmitting sub-array, A_sA transmit steering matrix representing the s-th transmit sub-array, B a steering matrix of the receive array, sigma_q＝diag(c_q) Is formed by a column vector c_qThe square matrix is opened, and the square matrix is opened,containing doppler shift and radar reflectance information for L targets, represents a target Doppler vector, sigma reflects a target radar reflection coefficient, and L is 1,2_lIndicating the doppler shift of the l-th target, T is the pulse repetition period of the radar,is thatCorresponding gaussian white noise matrix, (.)^TThe transpose of the matrix is represented,meaning defined as.

(2b) Analyzing the relation between the transmitting guide vectors of each transmitting sub-array by using the uniform arrangement structure of the transmitting arrays, wherein the s-th transmitting sub-array A_s、A_sCorresponding transverse steering matrix component U_jAnd a longitudinal steering matrix component V_iThe expression of (a) is:

A_s＝U_j⊙V_i

U_j＝U₀Γ_j，V_i＝V₀Γ_i

wherein, s ═ 1I + I denotes the serial number of the s-th transmitting subarray, J ═ 1,2,. J and I ═ 1,2,. I denote the serial numbers of the transmitting subarrays in the transverse and longitudinal directions, respectively, and U ═ 1,. I denotes the serial numbers of the transmitting subarrays in the transverse and longitudinal directions, respectively_jIs the transverse steering matrix component, V, of the transmit sub-array_iIs the longitudinal steering matrix component, U, of the transmit sub-array₀Is a transverse reference matrix, V₀Is a longitudinal reference subarray, gamma_jIs a diagonal matrix of i elements of which the column vector is open, Γ_iIs a diagonal matrix spread by a column vector containing j elements. In the invention, a first transmitting subarray is selected as a reference matrix. And indicates a Khatri-Rao product.

Further, the expression of U, V is:

collecting all L groups of guide vectors generated by the transmitting array, and writing the guide vectors of the right singular matrix V:

the steering vectors for the left singular matrix U are written as:

first intermediate amountThe first steering vector of the matrix U, the second intermediate quantity Is the ith steering vector of matrix V. Both containing azimuth information theta of the target_lAnd pitch angle informationΓ_j＝diag(h_j) Is a column vector h_jOpen diagonal matrix, Γ_i＝diag(d_i) Is a column vector d_iOpen diagonal matrix, the set of column vectors representing phase information between subarrays, for the phase center coordinate (m) of the s-th subarray_i，m_j) The column vectors of the two diagonal arrays are specifically represented as

(2c) Will be provided withColumn vectorization to obtain A₀＝U₀⊙V₀Is a guide matrix of a reference emission subarray, and a Gaussian white noise matrixColumn vectorization to obtainAnd substituting the relational expression described in the step (2b) into the relational expression, and sequentially adding all S column vectors in the order of sub-array numberCombined into a new matrix Y^(q)I.e. byExpressed as:

wherein the content of the first and second substances,a set of phase information representing each transmit sub-array in the transverse direction, a set of phase information representing each transmit sub-array in the longitudinal direction, is received by the array at the q-th pulseTo the noise matrix.

(2d) Column vectorization matrix Y^(q)To obtain z_qConcrete writing z_q＝[H⊙Δ⊙A₀⊙B]c_q+r_q，r_qIs N^(q)Then splicing the received data of Q pulses according to columns to obtain the MIMO radar multi-pulse received data Z, namelyThe expression is:

Z＝[H⊙Δ⊙A₀⊙B]C^T+R

whereinIs a set of Q pulses of target doppler and RCS information, representing the total noise set received by the array at Q pulses.

(2e) Reconstructing the matrix Z as a 5 th order tensorThe expression is:

wherein the content of the first and second substances,represents the vector outer product, [ [.]]For representing a set of tensor factor matrices, η_l，δ_l，α_l，β_l，γ_lAre each H, Delta, A₀The l-th column vector of B, C,is the 5 th order noise tensor generated by the same reconstruction of the matrix R.

(1b) Reconstructing the fourth-order tensor model and performing singular value decomposition on the obtained reconstruction matrix, wherein the decomposition result comprises a solved left singular matrix which comprises a van der mond structure:

(3a) will tensorConverted into a third order tensor by tensor conversionThe expression is as follows:

wherein G ═ H ^ Δ is a first matrix factor, A₀Can be used as a second matrix factor, and a third factor matrix B | _ C satisfies the column full rank condition.

(3b) According to a tensor reconstruction principle, the third-order tensor is generatedReconstructing from the third dimension to obtain a reconstructed matrix T₍₃₎The expression is:

T₍₃₎＝(G⊙A0)(B⊙C)^T+N

wherein the matrix N is represented by the noise tensorThe noise matrix generated by the same reconstruction method.

(3c) For the reconstructed matrix T₍₃₎Singular value decomposition is carried out, and the expression is as follows:

T₍₃₎＝U∑V^H

wherein, ()^HRepresenting the conjugate transpose of the matrix, the decomposition results being respectively of dimension SM₀A left singular matrix U of x L, a right singular matrix V of dimension NQ x L,and a singular value matrix sigma with dimension L x L. Since the third factor matrix B ^ C has full rank, there is inevitably one L × L non-singular transformation matrix E, so that UE ═ H ^ Δ ^ A₀And matrices H and Δ are both van der mond matrices. The above equation shows that there are special van der mond structures for the left singular matrix constrained by the array structure.

(1c) And estimating a column vector of the eigenvalue according to the van der mond structure of the left singular matrix and the correlation of the submatrices of the left singular matrix, and calculating according to the column vector to obtain an estimated value of the two-dimensional direction of arrival.

(4a) First sub-matrix U of two sub-matrices defining left singular matrix₁And a second sub-matrix U₂The expression is:

wherein the content of the first and second substances,is a sub-matrix consisting of the matrix H except the first row,is a sub-matrix composed of the matrix H except the last row, and the matrix E is a non-singular transformation matrix.

(4b) Constructing a submatrix U from the matrix U by utilizing row selection according to the operational rule of the Khatri-Rao product₁And U₂The expression is:

wherein, I represents an identity matrix, the dimension of which is determined by the corresponding subscript, 0 represents a matrix with all elements being 0, and the dimension of which is also determined by the corresponding subscript.

(4c) By using the characteristics of van der Mond matrix structure, U₁And U₂The relational expression of (1) is:

U₂E＝U₁EΩ_y

wherein omega_y＝diag(ω_y) Is formed by_yThe square matrix is opened, and the square matrix is opened,is the vector of the generating factor u corresponding to the Van der Mond matrix H₁Is u_l1 st steering vector u_lA first intermediate quantity, Delta, of steering vectors of the matrix U_my＝m_j+1-m_jIndicating the phase step of the j +1 th transmitting sub-array and the j transmitting sub-array in the transverse direction.

(4d) Two other sub-matrices U defining the left singular matrix₃And U₄The expression is:

wherein the content of the first and second substances,is a sub-matrix consisting of the matrix delta except the first row,is a sub-matrix consisting of the matrix delta except for the last row.

(4e) Constructing a submatrix U from the matrix U by utilizing row selection according to the operational rule of the Khatri-Rao product₃And U₄The expression is:

wherein the content of the first and second substances,representing the Kronecker product.

(4f) By using the characteristics of van der Mond matrix structure, U₃And U₄The relational expression of (1) is:

U₄E＝U₃EΩ_x

wherein omega_x＝diag(ω_x) Is formed by_xThe square matrix is opened, and the square matrix is opened,is the vector of the generation factors corresponding to the van der mond matrix delta,the phase stepping of the i +1 th transmitting sub-array and the ith transmitting sub-array in the longitudinal direction is shown.

(4g) From the correlation of each submatrix, a generation factor vector ω corresponding to the van der mond matrix H is obtained_yVector ω of generation factors corresponding to van der Mond matrix Δ_xThe expression is:

wherein, ()^-1Representing matrix inversion; respectively to the matrixAndperforming characteristic decomposition to obtain a column vector composed of characteristic values, and recording the column vector asAndi.e. the column vector omega_yAnd ω_xIs estimated.

(4h) According toAndthe two-dimensional direction of arrival estimation of the target is realized, and the expression is as follows:

wherein the content of the first and second substances,andare respectively column vectorsAndthe first element of (a) is,andrepresents the median ofIs estimated, andandnamely the two-dimensional direction of arrival estimation result of the pitch angle and the azimuth angle of the ith target. Collecting all L sets of estimatesThe proposed two-dimensional direction of arrival estimation can be achieved.

In summary, the above description is only a preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.