A parameter estimation method for near-field polarimetric MIMO radar based on an accurate model

Abstract

The present invention relates to a near-field polarization MIMO radar parameter estimation method based on an accurate model. The method comprises the following steps: establishing a bistatic MIMO radar system, and estimating a transmission angle, a reception angle, a transmission polarization angle, and a reception polarization angle in the bistatic MIMO radar system using polarization information. The entire process does not require spectral peak searching, and is highly efficient. The estimated parameters can be automatically paired, have no phase ambiguity problem, and have high accuracy. The method is also applicable to non-uniform linear arrays with arbitrary array element spacing and phase uncertainty.

Description

Technical Field

[0001] The present invention relates to the field of radar signal processing, and in particular to a near-field polarization MIMO radar parameter estimation method based on an accurate model. Background Art

[0002] MIMO radar parameter estimation is a current research hotspot and a difficult topic. Compared to phased array radars, MIMO radars can achieve higher angular resolution, greater array degrees of freedom, and flexible transmit waveform design by leveraging mutually orthogonal waveform characteristics. In recent years, numerous excellent algorithms have been proposed to achieve joint parameter estimation of the transmit angle (DOD) and receive angle (DOA) in bistatic MIMO radars. Although these algorithms can accurately locate target parameters in various backgrounds, the transmit and receive arrays used are either scalar uniform or non-uniform arrays, and the target sources are mostly far-field. Compared to scalar arrays, polarimetric arrays can provide not only target angle information but also polarimetric information. Therefore, polarimetric arrays offer higher target resolution and parameter estimation capabilities.

[0003] Furthermore, many existing near-field source localization algorithms are based on approximate models and assume that the spatial amplitudes between the target and the sensor are equal. Furthermore, when the target is located in the near-field radiation of a MIMO radar, the nonlinearity of the signal wavefront representation means that directly applying far-field source direction-finding algorithms will result in significant estimation errors due to model mismatch. Currently, no multidimensional parameter estimation algorithm for MIMO radars has been proposed that utilizes polarization information in near-field scenarios. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a near-field polarization MIMO radar parameter estimation method based on an accurate model, which utilizes polarization information to estimate the transmission angle, reception angle, transmission polarization angle and reception polarization angle, and has high estimation accuracy and high estimation efficiency.

[0005] A near-field polarization MIMO radar parameter estimation method based on an accurate model comprises the following steps:

[0006] S1. Establish a bistatic MIMO radar system, where the transmitting end is composed of M COLD array elements forming a non-uniform linear transmitting array, and the receiving end is composed of N COLD array elements forming a non-uniform linear receiving array;

[0007] S2. Establish a yoz plane rectangular coordinate system in the bistatic MIMO radar system, take the central array elements of the transmitting array and the receiving array as reference array elements, set K near-field targets in the yoz plane rectangular coordinate system, and obtain the distance r between the kth near-field target and the mth array element in the transmitting array.m,tk And the distance r between the kth near-field target and the nth element in the receiving array n,rk , where k = 1, 2, ..., K, m = -M, ..., 0, ..., M, n = -N, ..., 0, ..., N;

[0008] S3, the array elements at the transmitting end simultaneously transmit M orthogonal narrowband waves, which are reflected by K near-field targets and then received by the array elements in the receiving array; according to the r obtained in step S2 m,tk and r n,rk , obtain the spatial amplitude-phase factor a of the kth near-field target relative to the mth array element in the transmit array m,tk (θ tk ,r tk ), and obtain the spatial amplitude-phase factor a of the kth near-field target relative to the nth element in the receiving array n,rk (θ rk ,r rk );

[0009] S4, set in the same transmission signal cycle, the cross-sectional scintillation of a single near-field target is kept constant, the cross-sectional scintillation fluctuations of different targets are independent of each other, and the fluctuation statistics of different targets in different pulse times are also independent of each other. According to the a obtained in step S3 m,tk (θ tk ,r tk ) and a n,rk (θ rk ,r rk ), obtain the output signal of the matched filter at the receiving end at time t;

[0010] S5. Under L snapshots, obtain an output signal matrix of the matched filter at the receiving end based on the output signal of the matched filter at the receiving end obtained in step S4; arrange the output signal matrix of the matched filter at the receiving end into a fifth-order tensor, and then arrange the fifth-order tensor into a third-order parallel factor model Y; obtain a modulo-1 expansion matrix, a modulo-2 expansion matrix, and a modulo-3 expansion matrix of the third-order parallel factor model Y;

[0011] S6. Perform PARAFAC decomposition on the modulo 1 expansion matrix, the modulo 2 expansion matrix, and the modulo 3 expansion matrix of Y obtained in step S5, and use a total least squares method to obtain an estimated value of the transmit array steering vector and an estimated value of the receive array steering vector.

[0012] S7. Construct a transformation matrix C of the emission array t Ignore the error caused by column ambiguity and scale ambiguity in the estimated value of the transmit array steering vector obtained in step S6, and transform the transmit array transformation matrix C tMultiply it with the estimated value of the transmit array steering vector after ignoring the error to obtain the multiplication result, and take the first (M-1) / 2 rows and the last (M-1) / 2 rows of the multiplication result to form the matrix D t1 and matrix D t2 ;

[0013] S8, yes Perform eigendecomposition to obtain its eigenvalues, and derive an estimated value of the transmit polarization angle based on the obtained eigenvalues;

[0014] S9. Obtain an estimated value of the transmitted signal steering vector based on the obtained estimated value of the transmit array polarization angle, that is, obtain an estimated value of the spatial phase factor of the kth target relative to the mth array element in the transmit array. Obtain an estimated value of the transmit angle and an estimated value of the transmit range based on the amplitude relationship of each element in this estimated value relative to the 0th transmit array element.

[0015] S10. Construct another transformation matrix C r , and ignore the error caused by column ambiguity and scale ambiguity in the estimated value of the receiving array steering vector obtained in step S6, and then use the transformation matrix C of the receiving array r Multiply it with the estimated value of the receiving array steering vector after ignoring the error to obtain the multiplication result, and take the first (N-1) / 2 rows and the last (N-1) / 2 rows of the multiplication result to form the matrix D r1 and matrix D r2 ;

[0016] S11, according to The eigenvalue of the eigenvalue is used to get the estimated value of the eigenvalue Using the estimated value To obtain an estimate of the receiving polarization angle;

[0017] S12. Obtain an estimated value of the receiving polarization angle according to step S11, and obtain an estimated value of the received signal steering vector, that is, obtain an estimated value of the spatial phase factor of the kth target relative to the nth array element in the receiving array. Determine an estimated value of the receiving angle and an estimated value of the receiving distance based on the amplitude relationship of each element in the estimated value relative to the 0th receiving array element.

[0018] The beneficial effects of the present invention are: using the above-mentioned near-field polarization MIMO radar parameter estimation method based on a precise model, the method uses polarization information to estimate the transmission angle, reception angle, transmission polarization angle and reception polarization angle. The entire process does not require spectral peak search and is highly efficient. The estimated parameters can be automatically paired without phase ambiguity problems and have high accuracy. The method is applicable to non-uniform linear arrays with arbitrary array element spacing and phase uncertainty.

[0019] Preferably, in step S2, the distance r between the kth near-field target and the mth array element in the transmitting array is m,tk The expression is: The distance r between the kth near-field target and the nth element in the receiving array n,rk The expression is: Where, m=-M,…,0,…,M, n=-N,…,0,…,N, r 0,tk =r tk , r 0,rk =r rk , d m,t ,d n,r They represent the position of the mth element in the transmitting array and the position of the nth element in the receiving array, θ tk ,θ rk represent the emission angle and acceptance angle of the kth target respectively.

[0020] Preferably, in step S3, the spatial amplitude-phase factor a of the kth near-field target relative to the mth array element in the transmit array is m,tk (θ tk ,r tk ) is: The spatial amplitude-phase factor a of the kth near-field target relative to the nth element in the receiving array n,rk (θ rk ,r rk ) is: Among them, ε m,t ,ε n,r denote the additional phase factor of the mth element in the transmitting array and the additional phase factor of the nth element in the receiving array, respectively, and ε m,t ~U(0,2π),ε n,r ~U(0,2π); They represent the spatial amplitude attenuation of the mth array element in the transmitting array relative to the reference array element and the spatial amplitude attenuation of the nth array element in the receiving array relative to the reference array element; δ m,tk represents the spatial phase caused by the propagation time difference between the transmitted signal from the mth array element to the kth target and the transmitted signal from the reference array element to the kth near-field target; δ n,rk It represents the spatial phase caused by the propagation time difference between the received signal from the kth near-field target to the nth array element in the receiving array and the received signal from the kth near-field target to the reference array element.

[0021] Preferably, in step S4, the expression of the output signal of the matched filter at the receiving end is: y=(Q t ⊙Q r)s(t)+n(t), where s(t) represents the transmitted signal vector, s(t)=[s1(t),s2(t),…,s K (t)] T ∈C K×1 , β k ,f k denote the amplitude and Doppler frequency of the kth near-field target in the near-field space, β1≠β2≠…≠β K , k=1,2,…,K; the noise matrix n(t) represents independent additive Gaussian white noise that is uncorrelated with the transmitted signal at time t, with a mean of 0 and a variance of σ 2 ;Q t =[q t1 ,q t2 ,…,q tK ] represents the steering vector matrix of the transmitting array, Q r =[q r1 ,q r2 ,…,q rK ] represents the steering vector matrix of the receiving array, q tk represents the steering vector of the kth near-field target in the transmitting array, q rk represents the steering vector of the kth target in the receiving array, q tk The expression is: q rk The expression is: a tk represents the kth steering vector of the transmitted signal, a tk =[a -M,tk ,…a 0,tk ,…,a M,tk ] T , a rk represents the kth steering vector of the received signal, a rk =[a -N,rk ,…a 0,rk ,…,a N,rk ] T , v tk represents the kth polarization vector of the transmitted signal, v rk represents the kth polarization vector of the received signal, γ tk ,γ rk Represent the transmit polarization auxiliary angle and receive polarization auxiliary angle respectively, and 0≤γ tk ,γ rk ≤π / 2,η tk ,η rk Represent the transmit polarization phase difference and receive polarization phase difference respectively, and -π≤η tk ,ηrk ≤π.

[0022] Preferably, in step S5, the expression of the output signal matrix of the matched filter at the receiving end is:

[0023] Y=(A t ⊙V t ⊙A r ⊙V r )S+N=(Q t ⊙Q r )S+N, where Y=[y(1),y(2),…,y(L)], A t =[a t1 ,a t2 ,…,a tK ],V t =[v t1 ,v t2 ,…,v tK ] T , A r =[a r1 ,a r2 ,…,a rK ],V r =[v r1 ,v r2 ,…,v rK ] T S=[s(1),s(2),…,s(L)], N=[n(1),n(2),…n(L)], a tk represents the kth steering vector of the transmitted signal, v tk represents the kth polarization vector of the transmitted signal, a rK represents the kth steering vector of the received signal, v rK represents the kth polarization vector of the received signal.

[0024] Preferably, in step S5, the expression of the fifth-order tensor is: in, is the tensor form of N, Γ 5,K×1 is a fifth-order tensor operator, A t×2 is the tensor form of the transmitted signal steering vector, V t×3 is the tensor form of the polarization vector of the transmitted signal, A r×4 is the tensor form of the received signal steering vector, V r×5 is the tensor form of the received signal polarization vector; the expression of the third-order parallel factor model Y is: Among them, Γ 3,K×1 is a third-order tensor operator, Q t×2 is the tensor form of the emission array steering vector, Q r×3is the tensor form of the receiving array steering vector, is the tensor form of the noise matrix; the expressions of the modulo 1 expansion matrix, modulo 2 expansion matrix, and modulo 3 expansion matrix of Y are: Y3=(Q t ⊙Q r )S T , where Q r is the steering vector of the receiving array, Q t is the transmitting array steering vector, and S is the reflection coefficient matrix.

[0025] Preferably, in step S6, the specific process of performing PARAFAC decomposition on the modulo-1 expansion matrix, the modulo-2 expansion matrix, and the modulo-3 expansion matrix of the third-order parallel factor model Y obtained in step S5, and combining the total least squares method to obtain the transmit array steering vector and the receive array steering vector includes the following steps:

[0026] S6.1. Complete the PARAFAC decomposition of the modulo 1 expansion matrix, the modulo 2 expansion matrix, and the modulo 3 expansion matrix of Y through joint optimization. The expression is:

[0027] S6.2, after decomposition, we get Q t ,Q r The estimated values of and S are respectively denoted as and The expressions are:

[0028] Where Π is a permutation matrix, Δ1, Δ2, and Δ3 are the corresponding scale fuzzy matrices, all of which are K×K dimensional real-valued diagonal matrices, and Δ1Δ2Δ3=Ι, N1, N2, and N3 are error matrices.

[0029] Preferably, in step S7, the transformation matrix C of the transmitting array is t The expression is:

[0030] The transformation matrix C of the emission array t Multiply it with the estimated value of the transmit array steering vector after ignoring the error to obtain the multiplication result, and take the first (M-1) / 2 rows and the last (M-1) / 2 rows of the multiplication result to form the matrix D t1 and matrix D t2 The specific process is: the transformation matrix C of the transmitting array t Multiplying by the estimated value of the transmit array steering vector after ignoring the error is expressed as: in, is the estimated value of the transmitting array steering vector, Q tis the transmitting array steering vector, π is a permutation matrix, Δ1 is the scale fuzzy matrix; let Q t =A t ⊙V t , then the expression of the multiplication result is: Among them, Φ t1 =diag(-cosγ t1 ,…,-cosγ tK ), Take the first (M-1) / 2 rows of the multiplication result to form the matrix D t1 , its expression is: D t1 =A t Φ t1 ΠΔ1; take the last (M-1) / 2 rows of the multiplication result to form the matrix D t2 , its expression is: D t2 =A t Φ t2 ΠΔ1.

[0031] Preferably, in step S8, The expression is: in, right Perform eigendecomposition to obtain its eigenvalue and eigenvector. The expression of the obtained eigenvalue is: Ψ=diag(τ t1 ,…τ tK );The expression of the eigenvector is: Ρ=[o t1 ,…,o tK ]; The estimated value of the emission polarization angle obtained based on the obtained characteristic value is:

[0032] Preferably, in step S9, the specific process of obtaining the estimated value of the transmission angle and the estimated value of the transmission distance according to the amplitude relationship of each element in the estimated value of the spatial phase factor of the kth target relative to the mth array element in the transmission array relative to the 0th transmission element is as follows: The estimated value expression of the steering vector of the k-th target transmission signal is obtained: Where unvec(·) represents the vectorized inverse operation. The amplitude relationship of each element in the transmit signal steering vector relative to the 0th transmit array element can be expressed as in, make So we get the expression: in, represents the estimated value of the launch distance, d m,t represents the position of the mth element in the transmit array, Represents the estimated value of the launch angle of the kth target; using all the values of m, construct a and The overdetermined linear equations are:

[0033] Let [v 1k ,v 2k ,v 3k ] represents the right singular vector corresponding to the minimum singular value of the matrix consisting of the coefficients of the overdetermined linear equations, that is:

[0034] Therefore, the overall least squares solution of the system of equations is: in, represents an estimate of the launch distance, Represents an estimate of the launch angle.

[0035] Preferably, in step S10, the transformation matrix C r The expression is:

[0036] The transformation matrix C of the receiving array r Multiply it with the estimated value of the receiving array steering vector after ignoring the error to obtain the multiplication result, and take the (N-1) / 2 rows and the last (N-1) / 2 rows of the multiplication result to form the matrix D r1 and matrix D r2 The specific process is: the transformation matrix C of the receiving array r Multiplying by the estimated value of the receiving array steering vector after ignoring the error is expressed as: Let Q r =A r ⊙V r , get the transformation matrix C of the receiving array r The expression of the multiplication result obtained by multiplying the estimated value of the receiving array steering vector after ignoring the error is: where Φ r1 =diag(-cosγ r1 ,…,-cosγ rK ), Take the last (N-1) / 2 rows of the multiplication result to form the matrix D r1 , its expression is: D r1 =A r Φ r1 ΠΔ2, take the last (N-1) / 2 rows of the multiplication result to form the matrix D r2 , its expression is: D r2 =A r Φ r2 ΠΔ2.

[0037] Preferably, in step S11, The expression is: in, according to The eigenvalue of the eigenvalue is used to get the estimated value of the eigenvalue The expression is: According to the obtained The estimated value of the receiving polarization angle is: η rk =angle(T(k)),γ rk =tan -1 (abs(T(k)), by The estimated value expression of the steering vector of the k-th target received signal is obtained: Where unvec(·) represents the inverse operation of vectorization; the amplitude relationship of each element in the received signal steering vector relative to the 0th receiving array element can be expressed as in, make So we get the expression: in, Indicates the launch distance, d n,r represents the position of the nth element in the receiving array, Represents the estimated value of the launch angle of the kth target; using all the values of n, construct a and The overdetermined linear equations are:

[0038] Let [v 1k ,v 2k ,v 3k ] represents the right singular vector corresponding to the minimum singular value of the matrix consisting of the coefficients of the overdetermined linear equations, that is:

[0039] Therefore, the overall least squares solution of the system of equations is: in, Indicates the estimated value of the receiving distance, Represents an estimate of the acceptance angle. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] Figure 1 A schematic diagram of a bistatic MIMO radar system established in the present invention;

[0041] Figure 2 This is a diagram of the estimated transmission angle and reception angle of a near-field polarization MIMO radar obtained by using the method of the present invention in an example of a specific embodiment of the present invention.

[0042] Figure 3 This is an estimated diagram of the transmission range and receiving range of a near-field polarization MIMO radar obtained by using the method of the present invention in an example of a specific embodiment of the present invention;

[0043] Figure 4 This is an estimated diagram of the transmit polarization angle of a near-field polarization MIMO radar obtained by using the method of the present invention in an example of a specific embodiment of the present invention;

[0044] Figure 5 This is an estimated diagram of the receiving polarization angle of a near-field polarization MIMO radar obtained by using the method of the present invention in an example of a specific embodiment of the present invention;

[0045] Figure 6 This is a phase error estimation diagram of a near-field polarization MIMO radar obtained by using the method of the present invention in an example of a specific embodiment of the present invention. DETAILED DESCRIPTION

[0046] The invention will be further described below with reference to the accompanying drawings and in combination with specific implementations, so that those skilled in the art can implement the invention with reference to the description. The protection scope of the invention is not limited to the specific implementations.

[0047] The present invention provides a near-field polarization MIMO radar parameter estimation method based on an accurate model, the method comprising the following steps:

[0048] S1. Establish a bistatic MIMO radar system, wherein the transmitting end is composed of M (M is an odd number) COLD array elements forming a non-uniform linear transmitting array, and the receiving end is composed of N (N is an odd number) COLD array elements forming a non-uniform linear receiving array;

[0049] S2. Establish a yoz plane rectangular coordinate system in the bistatic MIMO radar system, take the central array elements of the transmitting array and the receiving array as reference array elements, set K near-field targets in the yoz plane rectangular coordinate system, and obtain the distance r between the kth near-field target and the mth array element in the transmitting array. m,tk And the distance r between the kth near-field target and the nth element in the receiving array n,rk , where k = 1, 2, ..., K, m = -M, ..., 0, ..., M, n = -N, ..., 0, ..., N;

[0050] S3, the array elements at the transmitting end simultaneously transmit M orthogonal narrowband waves, which are reflected by K near-field targets and then received by the array elements in the receiving array; according to the r obtained in step S2 m,tk and r n,rk , obtain the spatial amplitude-phase factor a of the kth near-field target relative to the mth array element in the transmit array m,tk (θ tk ,r tk ), and obtain the spatial amplitude-phase factor a of the kth near-field target relative to the nth element in the receiving array n,rk (θ rk ,rrk );

[0051] S4, set in the same transmission signal cycle, the cross-sectional scintillation of a single near-field target is kept constant, the cross-sectional scintillation fluctuations of different targets are independent of each other, and the fluctuation statistics of different targets in different pulse times are also independent of each other. According to the a obtained in step S3 m,tk (θ tk ,r tk ) and a n,rk (θ rk ,r rk ), obtain the output signal of the matched filter at the receiving end at time t;

[0052] S5. Under L snapshots, based on the output signal of the matched filter of the receiving end obtained in step S4, obtain an output signal matrix of the matched filter of the receiving end; arrange the output signal matrix of the matched filter of the receiving end into a fifth-order tensor, and according to the definition of generalized tensorization of the parallel factor model, arrange the fifth-order tensor into a third-order parallel factor model Y; according to the definition of parallel factor modulo n matrix expansion, obtain the modulo 1 expansion matrix, the modulo 2 expansion matrix, and the modulo 3 expansion matrix of the third-order parallel factor model Y;

[0053] S6. Perform PARAFAC decomposition on the modulo 1 expansion matrix, the modulo 2 expansion matrix, and the modulo 3 expansion matrix of Y obtained in step S5, and use a total least squares method to obtain an estimated value of the transmit array steering vector and an estimated value of the receive array steering vector.

[0054] S7. Construct a transformation matrix C of the emission array t Ignore the error caused by column ambiguity and scale ambiguity in the estimated value of the transmit array steering vector obtained in step S6, and transform the transmit array transformation matrix C t Multiply it with the estimated value of the transmit array steering vector after ignoring the error to obtain the multiplication result, and take the first (M-1) / 2 rows and the last (M-1) / 2 rows of the multiplication result to form the matrix D t1 and matrix D t2 ;

[0055] S8, yes Perform eigendecomposition to obtain its eigenvalues, and derive an estimated value of the transmit polarization angle based on the obtained eigenvalues;

[0056] S9. Obtain an estimated value of the transmitted signal steering vector based on the obtained estimated value of the transmit array polarization angle, that is, obtain an estimated value of the spatial phase factor of the kth target relative to the mth array element in the transmit array. Obtain an estimated value of the transmit angle and an estimated value of the transmit range based on the amplitude relationship of each element in this estimated value relative to the 0th transmit array element.

[0057] S10. Construct another transformation matrix C r , and ignore the error caused by column ambiguity and scale ambiguity in the estimated value of the receiving array steering vector obtained in step S6, and then use the transformation matrix C of the receiving array r Multiply it with the estimated value of the receiving array steering vector after ignoring the error to obtain the multiplication result, and take the first (N-1) / 2 rows and the last (N-1) / 2 rows of the multiplication result to form the matrix D r1 and matrix D r2 ;

[0058] S11. According to The eigenvalue of the eigenvalue is used to get the estimated value of the eigenvalue Using the estimated value To obtain an estimate of the receiving polarization angle;

[0059] S12. Obtain an estimated value of the receiving polarization angle according to step S11, and obtain an estimated value of the received signal steering vector, that is, obtain an estimated value of the spatial phase factor of the kth target relative to the nth array element in the receiving array. Determine an estimated value of the receiving angle and an estimated value of the receiving distance based on the amplitude relationship of each element in the estimated value relative to the 0th receiving array element.

[0060] In step S2, the distance r between the kth near-field target and the mth array element in the transmitting array is m,tk The expression is: The distance r between the kth near-field target and the nth element in the receiving array n,rk The expression is: Where, m=-M,…,0,…,M, n=-N,…,0,…,N, r 0,tk =r tk , r 0,rk =r rk , d m,t ,d n,r They represent the position of the mth element in the transmitting array and the position of the nth element in the receiving array, θ tk ,θ rk represent the emission angle and acceptance angle of the kth target respectively.

[0061] In step S3, the spatial amplitude-phase factor a of the kth near-field target relative to the mth array element in the transmit array is m,tk (θ tk ,r tk ) is: The spatial amplitude-phase factor a of the kth near-field target relative to the nth element in the receiving array n,rk (θ rk ,r rk ) is:

[0062] Among them, ε m,t ,ε n,r denote the additional phase factor of the mth element in the transmitting array and the additional phase factor of the nth element in the receiving array, respectively, and ε m,t ~U(0,2π),ε n,r ~U(0,2π); They represent the spatial amplitude attenuation of the mth array element in the transmitting array relative to the reference array element and the spatial amplitude attenuation of the nth array element in the receiving array relative to the reference array element; δ m,tk represents the spatial phase caused by the propagation time difference between the transmitted signal from the mth array element to the kth target and the transmitted signal from the reference array element to the kth near-field target; δ n,rk It represents the spatial phase caused by the propagation time difference between the received signal from the kth near-field target to the nth array element in the receiving array and the received signal from the kth near-field target to the reference array element.

[0063] In step S4, the output signal of the matched filter at the receiving end is expressed as:

[0064] y=(Q t ⊙Q r )s(t)+n(t), where s(t) represents the transmitted signal vector, s(t)=[s1(t),s2(t),…,s K (t)] T ∈C K×1 , β k ,f k denote the amplitude and Doppler frequency of the kth near-field target in the near-field space, β1≠β2≠…≠β K , k=1,2,…,K; the noise matrix n(t) represents independent additive Gaussian white noise that is uncorrelated with the transmitted signal at time t, with a mean of 0 and a variance of σ 2 ;Q t =[q t1 ,q t2 ,…,q tK ] represents the steering vector matrix of the transmitting array, Q r =[q r1 ,q r2 ,…,q rK ] represents the steering vector matrix of the receiving array, q tk represents the steering vector of the kth near-field target in the transmitting array, q rk represents the steering vector of the kth target in the receiving array, q tk The expression is: q rk The expression is: a tk represents the kth steering vector of the transmitted signal, a tk =[a -M,tk ,…a 0,tk ,…,a M,tk ] T , a rk represents the kth steering vector of the received signal, a rk =[a -N,rk ,…a 0,rk ,…,a N,rk ] T , v tk represents the kth polarization vector of the transmitted signal, v rk represents the kth polarization vector of the received signal, γ tk ,γ rk Represent the transmit polarization auxiliary angle and receive polarization auxiliary angle respectively, and 0≤γ tk ,γ rk ≤π / 2,η tk ,η rk Represent the transmit polarization phase difference and receive polarization phase difference respectively, and -π≤η tk ,η rk ≤π.

[0065] In step S5, the output signal matrix of the matched filter at the receiving end is expressed as:

[0066] Y=(A t ⊙V t ⊙A r ⊙V r )S+N=(Q t ⊙Q r )S+N, where Y=[y(1),y(2),…,y(L)], A t =[a t1 ,a t2 ,…,a tK ],V t =[v t1 ,v t2 ,…,v tK ] T , A r =[a r1 ,a r2 ,…,a rK ],V r =[v r1 ,v r2 ,…,v rK ] TS=[s(1),s(2),…,s(L)], N=[n(1),n(2),…n(L)], a tk represents the kth steering vector of the transmitted signal, v tk represents the kth polarization vector of the transmitted signal, a rK represents the kth steering vector of the received signal, v rK represents the kth polarization vector of the received signal.

[0067] In step S5, the expression of the fifth-order tensor is: in, is the tensor form of N, Γ 5,K×1 is a fifth-order tensor operator, A t×2 is the tensor form of the transmitted signal steering vector, V t×3 is the tensor form of the polarization vector of the transmitted signal, A r×4 is the tensor form of the received signal steering vector, V r×5 is the tensor form of the received signal polarization vector; the expression of the third-order parallel factor model Y is: Among them, Γ 3,K×1 is a third-order tensor operator, Q t×2 is the tensor form of the emission array steering vector, Q r×3 is the tensor form of the receiving array steering vector, is the tensor form of the noise matrix; the expressions of the modulo 1 expansion matrix, modulo 2 expansion matrix, and modulo 3 expansion matrix of Y are: Y3=(Q t ⊙Q r )S T , where Q r is the steering vector of the receiving array, Q t is the transmitting array steering vector, and S is the reflection coefficient matrix.

[0068] In step S6, the modulo-1 expansion matrix, the modulo-2 expansion matrix, and the modulo-3 expansion matrix of the third-order parallel factor model Y obtained in step S5 are subjected to PARAFAC decomposition, and the transmit array steering vector and the receive array steering vector are obtained by combining the total least squares method. The specific process includes the following steps:

[0069] S6.1. Complete the PARAFAC decomposition of the modulo 1 expansion matrix, the modulo 2 expansion matrix, and the modulo 3 expansion matrix of Y through joint optimization. The expression is:

[0070] S6.2, after decomposition, we get Q t ,Q r The estimated values of and S are respectively denoted as and The expressions are:

[0071] Where Π is a permutation matrix, Δ1, Δ2, and Δ3 are the corresponding scale fuzzy matrices, all of which are K×K dimensional real-valued diagonal matrices, and Δ1Δ2Δ3=Ι, N1, N2, and N3 are error matrices.

[0072] In step S7, the transformation matrix C of the transmit array is t The expression is:

[0073] The transformation matrix C of the emission array t Multiply it with the estimated value of the transmit array steering vector after ignoring the error to obtain the multiplication result, and take the first (M-1) / 2 rows and the last (M-1) / 2 rows of the multiplication result to form the matrix D t1 and matrix D t2 The specific process is: the transformation matrix C of the transmitting array t Multiplying by the estimated value of the transmit array steering vector after ignoring the error is expressed as: in, is the estimated value of the transmitting array steering vector, Q t is the transmitting array steering vector, π is a permutation matrix, Δ1 is the scale fuzzy matrix; let Q t =A t ⊙V t , then the expression of the multiplication result is: Among them, Φ t1 =diag(-cosγ t1 ,…,-cosγ tK ), Take the first (M-1) / 2 rows of the multiplication result to form the matrix D t1 , its expression is: D t1 =A t Φ t1 ΠΔ1; take the last (M-1) / 2 rows of the multiplication result to form the matrix D t2 , its expression is: D t2 =A t Φ t2 ΠΔ1.

[0074] In step S8, The expression is: in, right Perform eigendecomposition to obtain its eigenvalue and eigenvector. The expression of the obtained eigenvalue is: Ψ=diag(τ t1 ,…τ tK );The expression of the eigenvector is: Ρ=[ot1 ,…,o tK ]; The estimated value of the emission polarization angle obtained based on the obtained characteristic value is:

[0075] In step S9, the specific process of obtaining the estimated value of the transmission angle and the estimated value of the transmission distance is as follows: The estimated value expression of the steering vector of the k-th target transmission signal is obtained: Where unvec(·) represents the vectorized inverse operation. The amplitude relationship of each element in the transmit signal steering vector relative to the 0th transmit array element can be expressed as in, make So we get the expression: in, represents the estimated value of the launch distance, d m,t represents the position of the mth element in the transmit array, Represents the estimated value of the launch angle of the kth target; using all the values of m, construct a and The overdetermined linear equations are: Let [v 1k ,v 2k ,v 3k ] represents the right singular vector corresponding to the minimum singular value of the matrix consisting of the coefficients of the overdetermined linear equations, that is: Therefore, the overall least squares solution of the system of equations is: in, represents an estimate of the launch distance, Represents an estimate of the launch angle.

[0076] In step S10, the transformation matrix C r The expression is: The transformation matrix C of the receiving array r Multiply it with the estimated value of the receiving array steering vector after ignoring the error to obtain the multiplication result, and take the (N-1) / 2 rows and the last (N-1) / 2 rows of the multiplication result to form the matrix D r1 and matrix D r2 The specific process is: the transformation matrix C of the receiving array r Multiplying by the estimated value of the receiving array steering vector after ignoring the error is expressed as: Let Q r =A r ⊙V r , get the transformation matrix C of the receiving arrayr The expression of the multiplication result obtained by multiplying the estimated value of the receiving array steering vector after ignoring the error is: where Φ r1 =diag(-cosγ r1 ,…,-cosγ rK ), Take the last (N-1) / 2 rows of the multiplication result to form the matrix D r1 , its expression is: D r1 =A r Φ r1 ΠΔ2, take the last (N-1) / 2 rows of the multiplication result to form the matrix D r2 , its expression is: D r2 =A r Φ r2 ΠΔ2.

[0077] In step S11, The expression is: in, according to The eigenvalue of the eigenvalue is used to get the estimated value of the eigenvalue The expression is: According to the obtained The estimated value of the receiving polarization angle is: η rk =angle(T(k)),γ rk =tan -1 (abs(T(k)), by The estimated value expression of the steering vector of the k-th target received signal is obtained: Where unvec(·) represents the inverse operation of vectorization; the amplitude relationship of each element in the received signal steering vector relative to the 0th receiving array element can be expressed as in, make So we get the expression: in, Indicates the launch distance, d n,r represents the position of the nth element in the receiving array, Represents the estimated value of the launch angle of the kth target; using all the values of n, construct a and The overdetermined linear equations are: Let [v 1k ,v 2k ,v 3k ] represents the right singular vector corresponding to the minimum singular value of the matrix consisting of the coefficients of the overdetermined linear equations, that is: Therefore, the overall least squares solution of the system of equations is: in, Indicates the estimated value of the receiving distance, Represents an estimate of the acceptance angle.

[0078] The above-mentioned near-field polarization MIMO radar parameter estimation method based on an accurate model is adopted. This method uses polarization information to estimate the transmission angle, reception angle, transmission polarization angle, and reception polarization angle. The entire process does not require spectral peak search and is highly efficient. The estimated parameters can be automatically paired without phase ambiguity problems, with high accuracy, and are applicable to non-uniform linear arrays with arbitrary array element spacing and phase uncertainty.

[0079] The accuracy of the near-field polarization MIMO radar parameter estimation method based on an accurate model proposed in the present invention is demonstrated by the following examples:

[0080] Assume M = N = 2, d t =d r =2×λL=512; Assuming that there are three near-field targets in the near-field space of the MIMO radar, the three near-field targets are obtained by the above method (θ tk ,θ rk , γ tk , γ rk , η tk , η rk , r tk , r rk ) are (30°, 80°, 10°, 42°, 50°, 17°, 1, 2), (45°, 45°, 22°, 33°, 55°, 28°, 3.5, 3.6), (20°, 10°, 45°, 60°, 80°, 56°, 2.5, 2.0838); Assuming that the signal-to-noise ratio (SNR) of the three near-field targets is 20 dB, after six simulation experiments, the parameter estimation results of the near-field targets are as follows: Figures 2 to 6 The experimental results show that the proposed method can correctly identify the three target sources, and the eight-dimensional parameters can be automatically paired with high estimation accuracy.

Claims

1. A near-field polarization MIMO radar parameter estimation method based on an accurate model, characterized by: The method comprises the following steps: S1. Establish a bistatic MIMO radar system, where the transmitting end is composed of M COLD array elements forming a non-uniform linear transmitting array, and the receiving end is composed of N COLD array elements forming a non-uniform linear receiving array; S2. Establish a yoz plane rectangular coordinate system in the bistatic MIMO radar system, take the central array elements of the transmitting array and the receiving array as reference array elements, set K near-field targets in the yoz plane rectangular coordinate system, and obtain the distance r between the kth near-field target and the mth array element in the transmitting array. m,tk And the distance r between the kth near-field target and the nth element in the receiving array n,rk , where k = 1, 2, ..., K, m = -M, ..., 0, ..., M, n = -N, ..., 0, ..., N; S3, the array elements at the transmitting end simultaneously transmit M orthogonal narrowband waves, which are reflected by K near-field targets and then received by the array elements in the receiving array; according to the r obtained in step S2 m,tk and r n,rk , obtain the spatial amplitude-phase factor a of the kth near-field target relative to the mth array element in the transmit array m,tk (θ tk ,r tk ), and obtain the spatial amplitude-phase factor a of the kth near-field target relative to the nth element in the receiving array n,rk (θ rk ,r rk ); the spatial amplitude-phase factor a m,tk (θ tk ,r tk ) is: The spatial amplitude-phase factor a n,rk (θ rk ,r rk ) is: Among them, ε m,t ,ε n,r denote the additional phase factor of the mth element in the transmitting array and the additional phase factor of the nth element in the receiving array, respectively, and ε m,t ~U(0,2π),ε n,r ~U(0,2π); They represent the spatial amplitude attenuation of the mth array element in the transmitting array relative to the reference array element and the spatial amplitude attenuation of the nth array element in the receiving array relative to the reference array element; δ m,tk represents the spatial phase caused by the propagation time difference between the transmitted signal from the mth array element to the kth target and the transmitted signal from the reference array element to the kth near-field target; δ n,rk It represents the spatial phase caused by the propagation time difference between the received signal from the kth near-field target to the nth array element in the receiving array and the received signal from the kth near-field target to the reference array element; S4, set in the same transmission signal cycle, the cross-sectional scintillation of a single near-field target is kept constant, the cross-sectional scintillation fluctuations of different targets are independent of each other, and the fluctuation statistics of different targets in different pulse times are also independent of each other. According to the a obtained in step S3 m,tk (θ tk ,r tk ) and a n,rk (θ rk ,r rk ), obtain the output signal of the matched filter at the receiving end at time t; S5. Under L snapshots, obtain an output signal matrix of the matched filter at the receiving end based on the output signal of the matched filter at the receiving end obtained in step S4; arrange the output signal matrix of the matched filter at the receiving end into a fifth-order tensor, and then arrange the fifth-order tensor into a third-order parallel factor model Y; obtain a modulo-1 expansion matrix, a modulo-2 expansion matrix, and a modulo-3 expansion matrix of the third-order parallel factor model Y; S6. Perform PARAFAC decomposition on the modulo 1 expansion matrix, the modulo 2 expansion matrix, and the modulo 3 expansion matrix of Y obtained in step S5, and use a total least squares method to obtain an estimated value of the transmit array steering vector and an estimated value of the receive array steering vector. S7. Construct a transformation matrix C of the emission array t , the transformation matrix C of the emission array t The expression is: Ignoring the errors in the estimated value of the transmit array steering vector obtained in step S6 due to column ambiguity and scale ambiguity, the transformation matrix C of the transmit array is converted to t Multiply it with the estimated value of the transmit array steering vector after ignoring the error to obtain the multiplication result, and take the first (M-1) / 2 rows and the last (M-1) / 2 rows of the multiplication result to form the matrix D t1 and matrix D t2 ; S8, yes Perform eigendecomposition to obtain its eigenvalues, and derive an estimated value of the transmit polarization angle based on the obtained eigenvalues; S9. Obtain an estimated value of the transmitted signal steering vector based on the obtained estimated value of the transmit array polarization angle, that is, obtain an estimated value of the spatial phase factor of the kth target relative to the mth array element in the transmit array. Obtain an estimated value of the transmit angle and an estimated value of the transmit range based on the amplitude relationship of each element in this estimated value relative to the 0th transmit array element. S10. Construct another transformation matrix C r : Ignore the errors in the estimated value of the receiving array steering vector obtained in step S6 due to column ambiguity and scale ambiguity, and then transform the receiving array matrix C r Multiply it with the estimated value of the receiving array steering vector after ignoring the error to obtain the multiplication result, and take the first (N-1) / 2 rows and the last (N-1) / 2 rows of the multiplication result to form the matrix D r1 and matrix D r2 ; S11, according to The eigenvalue of the eigenvalue is used to get the estimated value of the eigenvalue Using the estimated value To obtain an estimate of the receiving polarization angle; S12. Obtain an estimated value of the receiving polarization angle according to step S11, and obtain an estimated value of the received signal steering vector, that is, obtain an estimated value of the spatial phase factor of the kth target relative to the nth array element in the receiving array. Determine an estimated value of the receiving angle and an estimated value of the receiving distance based on the amplitude relationship of each element in the estimated value relative to the 0th receiving array element.

2. The near-field polarization MIMO radar parameter estimation method based on an accurate model according to claim 1, characterized in that: In step S2, the distance r between the kth near-field target and the mth array element in the transmitting array is m,tk The expression is: The distance r between the kth near-field target and the nth element in the receiving array n,rk The expression is: Where, m=-M,…,0,…,M, n=-N,…,0,…,N, r 0,tk =r tk , r 0,rk =r rk , d m,t ,d n,r denote the position of the mth element in the transmitting array and the nth element in the receiving array, θ tk ,θ rk denote the transmitting angle and receiving angle of the kth target respectively; in step S3, the spatial amplitude-phase factor a of the kth near-field target relative to the mth array element in the transmitting array is m,tk (θ tk ,r tk ) is: The spatial amplitude-phase factor a of the kth near-field target relative to the nth element in the receiving array n,rk (θ rk ,r rk ) is: , where ε m,t ,ε n,r denote the additional phase factor of the mth element in the transmitting array and the additional phase factor of the nth element in the receiving array, respectively, and ε m,t ~U(0,2π),ε n,r ~U(0,2π); They represent the spatial amplitude attenuation of the mth array element in the transmitting array relative to the reference array element and the spatial amplitude attenuation of the nth array element in the receiving array relative to the reference array element; δ m,tk represents the spatial phase caused by the propagation time difference between the transmitted signal from the mth array element to the kth target and the transmitted signal from the reference array element to the kth near-field target; δ n,rk It represents the spatial phase caused by the propagation time difference between the received signal from the kth near-field target to the nth array element in the receiving array and the received signal from the kth near-field target to the reference array element.

3. The near-field polarization MIMO radar parameter estimation method based on an accurate model according to claim 2, characterized in that: In step S4, the expression of the output signal of the matched filter at the receiving end is: y = (Q t ⊙Q r )s(t)+n(t), where s(t) represents the transmitted signal vector, s(t)=[s1(t),s2(t),…,s K (t)] T ∈C K×1 , β k ,f k denote the amplitude and Doppler frequency of the kth near-field target in the near-field space, β1≠β2≠…≠β K , k=1,2,…,K; the noise matrix n(t) represents independent additive Gaussian white noise that is uncorrelated with the transmitted signal at time t, with a mean of 0 and a variance of σ 2 ;Q t =[q t1 ,q t2 ,…,q tK ] represents the steering vector matrix of the transmitting array, Q r =[q r1 ,q r2 ,…,q rK ] represents the steering vector matrix of the receiving array, q tk represents the steering vector of the kth near-field target in the transmitting array, q rk represents the steering vector of the kth target in the receiving array, q tk The expression is: q rk The expression is: a tk represents the kth steering vector of the transmitted signal, a tk =[a -M,tk ,…a 0,tk ,…,a M,tk ] T , a rk represents the kth steering vector of the received signal, a rk =[a -N,rk ,…a 0,rk ,…,a N,rk ] T , v tk represents the kth polarization vector of the transmitted signal, v rk represents the kth polarization vector of the received signal, γ tk ,γ rk Represent the transmit polarization auxiliary angle and receive polarization auxiliary angle respectively, and 0≤γ tk ,γ rk ≤π / 2,η tk ,η rk Represent the transmit polarization phase difference and receive polarization phase difference respectively, and -π≤η tk ,η rk ≤π.

4. The near-field polarization MIMO radar parameter estimation method based on an accurate model according to claim 3, characterized in that: In step S5, the output signal matrix of the matched filter at the receiving end is expressed as: Among them, Y=[y(1),y(2),…,y(L)], A t =[a t1 ,a t2 ,…,a tK ], V t =[v t1 ,v t2 ,…,v tK ] T , A r =[a r1 ,a r2 ,…,a rK ],V r =[v r1 ,v r2 ,…,v rK ] T , S=[s(1),s(2),…,s(L)], N=[n(1),n(2),…n(L)], a tk represents the kth steering vector of the transmitted signal, v tk represents the kth polarization vector of the transmitted signal, a rK represents the kth steering vector of the received signal, v rK represents the kth polarization vector of the received signal; the expression of the fifth-order tensor is: in, is the tensor form of N, Γ 5,K×1 is a fifth-order tensor operator, A t×2 is the tensor form of the transmitted signal steering vector, V t×3 is the tensor form of the polarization vector of the transmitted signal, A r×4 is the tensor form of the received signal steering vector, V r×5 is the tensor form of the received signal polarization vector; the expression of the third-order parallel factor model Y is: Among them, Γ 3,K×1 is a third-order tensor operator, Q t×2 is the tensor form of the emission array steering vector, Q r×3 is the tensor form of the receiving array steering vector, is the tensor form of the noise matrix; the expressions of the modulo 1 expansion matrix, modulo 2 expansion matrix, and modulo 3 expansion matrix of Y are: Y3=(Q t ⊙Q r )S T , where Q r is the steering vector of the receiving array, Q t is the transmitting array steering vector, and S is the reflection coefficient matrix.

5. The near-field polarization MIMO radar parameter estimation method based on an accurate model according to claim 4, characterized in that: In step S6, the modulo-1 expansion matrix, the modulo-2 expansion matrix, and the modulo-3 expansion matrix of the third-order parallel factor model Y obtained in step S5 are subjected to PARAFAC decomposition, and the transmit array steering vector and the receive array steering vector are obtained by combining the total least squares method. The specific process includes the following steps: S6.

1. Complete the PARAFAC decomposition of the modulo 1 expansion matrix, the modulo 2 expansion matrix, and the modulo 3 expansion matrix of Y through joint optimization. The expression is: S6.2, after decomposition, we get Q t ,Q r The estimated values of and S are respectively denoted as and The expressions are: Where Π is a permutation matrix, Δ1, Δ2, and Δ3 are the corresponding scale fuzzy matrices, all of which are K×K dimensional real-valued diagonal matrices, and Δ1Δ2Δ3=Ι, N1, N2, and N3 are error matrices.

6. The near-field polarization MIMO radar parameter estimation method based on an accurate model according to claim 5, characterized in that: In step S7, the transformation matrix C of the transmit array is t Multiply it with the estimated value of the transmit array steering vector after ignoring the error to obtain the multiplication result, and take the first (M-1) / 2 rows and the last (M-1) / 2 rows of the multiplication result to form the matrix D t1 and matrix D t2 The specific process is: the transformation matrix C of the transmitting array t Multiplying by the estimated value of the transmit array steering vector after ignoring the error is expressed as: in, is the estimated value of the transmitting array steering vector, Q t is the transmitting array steering vector, π is a permutation matrix, Δ1 is the scale fuzzy matrix; let Q t =A t ⊙V t , then the expression of the multiplication result is: Among them, Φ t1 =diag(-cosγ t1 ,…,-cosγ tK ), Take the first (M-1) / 2 rows of the multiplication result to form the matrix D t1 , its expression is: D t1 =A t Φ t1 ΠΔ1; take the last (M-1) / 2 rows of the multiplication result to form the matrix D t2 , its expression is: D t2 =A t Φ t2 ΠΔ1.

7. The near-field polarization MIMO radar parameter estimation method based on an accurate model according to claim 6, characterized in that: In step S8, The expression is: in, right Perform eigendecomposition to obtain its eigenvalue and eigenvector. The expression of the obtained eigenvalue is: Ψ=diag(τ t1 ,…τ tK );The expression of the eigenvector is: Ρ=[o t1 ,…,o tK ]; The estimated value of the emission polarization angle obtained based on the obtained characteristic value is:

8. The near-field polarization MIMO radar parameter estimation method based on an accurate model according to claim 7, characterized in that: In step S9, the specific process of obtaining the estimated value of the transmission angle and the estimated value of the transmission distance is as follows: The estimated value expression of the steering vector of the k-th target transmission signal is obtained: Where unvec(·) represents the inverse operation of vectorization; the amplitude relationship of each element in the transmit signal steering vector relative to the 0th transmit array element is expressed as: in, make So we get the expression: in, represents the estimated value of the launch distance, d m,t represents the position of the mth element in the transmit array, Represents the estimated value of the launch angle of the kth target; using all the values of m, construct a and The overdetermined linear equations are: Let [v 1k ,v 2k ,v 3k ] represents the right singular vector corresponding to the minimum singular value of the matrix consisting of the coefficients of the overdetermined linear equations, that is: Therefore, the total least squares solution of the system of equations is: in, represents an estimate of the launch distance, Represents an estimate of the launch angle.

9. The near-field polarization MIMO radar parameter estimation method based on an accurate model according to claim 8, characterized in that: In step S10, the transformation matrix C of the receiving array is received. r Multiply it with the estimated value of the receiving array steering vector after ignoring the error to obtain the multiplication result, and take the (N-1) / 2 rows and the last (N-1) / 2 rows of the multiplication result to form the matrix D r1 and matrix D r2 The specific process is: the transformation matrix C of the receiving array r Multiplying by the estimated value of the receiving array steering vector after ignoring the error is expressed as: Let Q r =A r ⊙V r , get the transformation matrix C of the receiving array r The expression of the multiplication result obtained by multiplying the estimated value of the receiving array steering vector after ignoring the error is: where Φ r1 =diag(-cosγ r1 ,…,-cosγ rK ), Take the last (N-1) / 2 rows of the multiplication result to form the matrix D r1 , its expression is: D r1 =A r Φ r1 ΠΔ2, take the last (N-1) / 2 rows of the multiplication result to form the matrix D r2 , its expression is: D r2 =A r Φ r2 ΠΔ2.

10. The near-field polarization MIMO radar parameter estimation method based on an accurate model according to claim 9, characterized in that: In step S11, The expression is: in, according to The eigenvalue of the eigenvalue is used to get the estimated value of the eigenvalue The expression is: According to the obtained The estimated value of the receiving polarization angle is: η rk =angle(T(k)),γ rk =tan -1 (abs(T(k)), by The estimated value expression of the steering vector of the k-th target received signal is obtained: where unvec(·) represents the inverse operation of vectorization; the amplitude relationship of each element in the received signal steering vector relative to the 0th receiving array element is expressed as: in, make So we get the expression: in, Indicates the launch distance, d n,r represents the position of the nth element in the receiving array, Represents the estimated value of the launch angle of the kth target; using all the values of n, construct a and The overdetermined linear equations are: Let [v 1k ,v 2k ,v 3k ] represents the right singular vector corresponding to the minimum singular value of the matrix consisting of the coefficients of the overdetermined linear equations, that is: The total least squares solution of the system of equations is: in, Indicates the estimated value of the receiving distance, Represents an estimate of the acceptance angle.

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