CN110579737A - Sparse array-based MIMO radar broadband DOA calculation method in clutter environment - Google Patents

Abstract

The invention discloses a sparse array-based MIMO radar broadband DOA calculation method in a clutter environment, and belongs to the field of signal processing. Specifically WCSAB, in the method, clutter interference is suppressed by Capon beam forming, and then target DOA is estimated by a CS method by jointly utilizing different narrowband signal information. Considering that the DOA estimation performance is not only related to the beam forming weight value but also related to the sparse array structure, the invention provides a joint optimization problem of the beam forming weight value and the sparse array, and provides a simple algorithm for solving the optimization problem. The method provided by the invention can improve the performance of target DOA estimation in a clutter environment, including high resolution and low sidelobe, the sparse array reduces the system cost and complexity, the Bayesian Mean Square Error (BMSE) of the target DOA estimation is taken as a performance evaluation index, and the sparse array structure designed by the algorithm is similar to the optimal sparse array performance obtained by an exhaustive method and has better performance than a nested array and a co-prime array.

Description

Sparse array-based MIMO radar broadband DOA calculation method in clutter environment

Technical Field

The invention belongs to the field of signal processing, and particularly relates to a sparse array-based MIMO radar broadband DOA estimation problem in a clutter environment.

Background

An mimo (Multiple Input Multiple output) radar is a novel radar system that synchronously transmits signals by using Multiple transmitting antennas, receives echo signals by using Multiple receiving antennas, and processes the signals in a centralized manner. Compared with the traditional phased array radar, the MIMO radar has obvious advantages such as higher resolution, better target detection, positioning and tracking performance and better target parameter estimation and identification capability. DOA estimation research is an important content in array signal processing, and the application of the DOA estimation research relates to the fields of radar, communication, sonar, radio astronomy, survey, earthquake, biomedicine and the like. At present, there are many classical DOA Estimation methods, such as Multiple Signal Classification (MUSIC), Signal parameter Estimation based on rotation invariant technology (ESPRIT), and so on. In recent years, the theory of Compressive Sensing (CS) has gained wide attention of scholars at home and abroad, and compared with the traditional method, the CS-based MIMO radar DOA estimation has better estimation performance under the conditions of less sampling data and low signal-to-noise ratio.

According to the traditional array signal processing theory, in order to ensure the uniqueness of DOA estimation, the distance between adjacent array elements in the array is less than or equal to half wavelength of an incident signal, and the array meeting the condition is called a full array. The spatial resolution of the array is related to the array aperture, and increasing the resolution requires increasing the array aperture, which means that more antennas are required in a full array. However, due to the practical constraints of software and hardware resources, the number of antennas is usually limited. Sparse arrays have attracted considerable attention in order to increase the array aperture without increasing the number of antennas. When the target is sparse in the observation space, the target DOA can be accurately estimated by the sparse array. However, in a clutter environment, the sparsity of the target in the observation space may be destroyed, thereby causing degradation of DOA estimation performance.

Considering that broadband signals have the advantages of large information amount, strong anti-interference capability, high resolution and the like, relatively representative broadband DOA estimation methods include an Incoherent Signal Subspace (ISSM), a Coherent Signal Subspace (CSSM) and the like. The ISSM divides the broadband signal into a plurality of narrow-band signals on a frequency band, then processes each narrow-band signal respectively, and finally averages the processing results of all the narrow-band signals to obtain a final estimation result. CSSM transforms the covariance matrix of narrowband signals of different frequencies to a reference frequency by focusing and then obtains the final result by using a narrowband estimation method. However, CSSM requires an estimate of the target DOA, and performance is greatly affected by the accuracy of the estimate.

Disclosure of Invention

The invention provides a sparse array-based MIMO radar broadband DOA estimation method in a clutter environment, in particular to WCSAB (wideband compressed sensing after beamforming). Considering that the DOA estimation performance is not only related to the beam forming weight value but also related to the sparse array structure, the invention provides a joint optimization problem of the beam forming weight value and the sparse array, and provides a simple algorithm for solving the optimization problem.

The technical scheme of the invention is a sparse array-based MIMO radar broadband DOA calculation method in a clutter environment, which comprises the following steps:

Step 1: let the position of the transmitting antenna be determined, and the feasible region for placing the receiving antenna be [0, D_r]For simplicity of analysis, the feasible fields are separated by an interval Δ_rDiscretization to N_rA plurality of grid points, and N receiving antennas disposed on some of the grid points, N < N_r；

Step 2: establishing an MIMO radar echo signal model to obtain echo signal time domain sampling data n＝1,...,N_rAnd p 1., L, where p represents a time domain snapshot and L is a snapshot number;

And step 3: for received signalPerforming an L-point discrete Fourier transform to obtain frequency domain data, i.e.

And N is_rThe data of each lattice point is expressed in vector form, i.e. y [ l ]]＝[y₁[l],...,y_Nr[l]]^TWherein p 1, L1;

And 4, step 4: discretizing the target angle observation area into G grid points theta₁,...,θ_GK < G, where K represents the number of targets, the signal model is represented in sparse form:

y[l]＝Φ[l]x+c[l]+u[l]

WhereinWhere a is_r(θ,f_l) Indicating a received steering vector, a_t(θ,f_l) Representing the transmit steering vector, s [ l ]]Representing a frequency domain transmitted signal, x ═ x₁,...,x_G]^TIs K sparse, i.e. x has only K non-zero elements, and the values and positions of the non-zero elements are the target reflection coefficient and DOA, c [ l [ ]]Represents clutter u [ l ]]Representing noise;

And 5: will beam form weight vector w_g,lact on y [ l]The beam forming output result is obtained:

Will r is_g,l(G1., G, L ═ 1., L) is expressed as a vector:

r＝[r_1,1,...,r_G,1,...,r_1,L,...,r_G,L]^T

＝W_rΦx+W_rc+W_ru

wherein the weight matrix W_r＝Diag{W₁,...,W_l,...,W_LIs a block diagonal matrix;

and is provided withΦ＝[Φ^T[1],...,Φ^T[L]]^T，c＝[c^T[1],...,c^T[L]]^TRepresents clutter, u ═ u^T[1],...,u^T[L]]^TRepresenting noise;

Step 6: reconstructing a sparse vector x by basis pursuit desiccation based on the CS theory;

wherein eta is equal to or greater than 0 is a regularization parameter;

And 7: to the solution obtained in step 6The values of the elements in the table are sorted from big to small, and the corresponding lattice point of each sorted element is expressed as { theta [ [ theta ] ]₍₁₎,...,θ_(G)Then the DOA estimation result can be expressed as;

and 8: based on minimum Bayes mean square errorsolving for optimal W_rThe following optimization problem is established

s.t.W_r＝Diag{W₁,...,W_L}

||W||₀＝N

W＝[w_1,1,...,w_1,L,...,w_G,1,...,w_G,L]

Wherein the DOA vector theta of the real target_TIs random in nature and is not only easy to be recognized,Is expressed in the pair theta_Tin the hope of expectation,Denotes theta_TMean square error of DOA estimation, w, when determined_g,lRepresents a weight vector, where G1.., G, L1., L;

And step 9: the problem provided by the step 8 is solved in an optimized way to obtain the optimal W_r。

Further, the specific method of step 9 is as follows:

Step 1: initialization: the number of iterations j is 1,According to the formulaComputing beamforming weightsWhereinR_c(f_l) Is a clutter c [ l ]]The covariance matrix of (a); during each iteration, a set of lattice point selection vectors z is randomly generated₁,...,z_αfor a given z;

Step 2: repeating the iterative process from the step 3 to the step 6:

And step 3: randomly generating a set of lattice point selection vectors z₁,...,z_α}；

And 4, step 4: according to the formula w_g,l＝z⊙ξ_g,lComputingAnd form

According to the formulaCalculating r_g,lR is to_g,lExpressed as a vector r;

Will be provided withSubstituting r into the formulaObtaining x reconstruction resultsAnd target DOA estimation result

According to the formulaobtaining BMSE

And 5: obtaining based on minimum BMSE

Step 6: according toBased onUpdatingGet the corresponding weight valueAnd let j equal j + 1; whereinIs Dc [ l ]]The covariance matrix of (a) is determined,

And 7: when in useThen, iteration stops and the optimal antenna selection is outpute₀Is a threshold value set in advance.

the method provided by the invention can improve the performance of target DOA estimation in a clutter environment, including high resolution and low sidelobe, the sparse array reduces the system cost and complexity, the Bayesian Mean Square Error (BMSE) of the target DOA estimation is taken as a performance evaluation index, and the sparse array structure designed by the algorithm is similar to the optimal sparse array performance obtained by an exhaustive method and has better performance than a nested array and a co-prime array.

Drawings

FIG. 1 shows the results of BMSE ordering in ascending order for all possible sparse array configurations, and for comparison, FIG. 1 also shows the results for nested arrays (nested arrays) and co-prime arrays (co-prime arrays).

fig. 2(a) shows the optimal sparse array structure based on the minimum BMSE condition, and fig. 2(b) shows the sparse array structure obtained by the proposed algorithm according to the present invention.

Fig. 3 shows DOA estimation results for different sparse array structures when using the WCSAB method.

FIG. 4 shows the DOA estimation results of different array structures using WCSAB and WCT (wideband capacitor technique) methods for single target cases.

FIG. 5 shows the DOA estimation results of different array structures when WCSAB and WCT (wideband capacitor technique) methods are used respectively, considering the dual target case.

Detailed Description

For convenience of description, the following definitions are first made:

Bold capital letters represent matrices, bold lowercase letters represent vectors, (.)^*For conjugation, (.)^TIs a transposition of^HFor the conjugate transposition, | x | | non-conducting phosphor₀And | | x | | non-conducting phosphor₁Respectively representing l of the vector x₀Norm sum l₁Norm, | W | count₀Representing the number of non-zero rows of the matrix W, Diag {. cndot.) representing a block diagonal matrix, Diag_r{. denotes the diagonal matrix after removing zero rows,Denotes the expectation with respect to theta, I_NFor a unit array of order N, 1 is a full 1 vector, and a symbol &indicatesa Hadamard product.

Consider a co-located MIMO radar system with both transmit and receive antennas placed on the horizontal axis of a two-dimensional cartesian coordinate system. Suppose there are M transmit antennas and the position on the horizontal axis is known as d_t,m(M ═ 1.., M). Assume that the feasible domain for placing the receive antenna is [0, D_r]for simplicity of analysis, the feasible fields are separated by an interval Δ_rDiscretization to N_rAnd grid points on which the receiving antennas are placed. Due to the constraint of the number of antennas, the radar system is assumed to have only N (N < N)_r) And available receiving antennas. Order torepresenting a wideband signal transmitted by the mth transmitting antenna, having a frequency range of [ -B ]_m/2,B_m/2]where p denotes a time-domain snapshot, T_sIndicating the sampling period and L the number of fast beats. Let DOA of K far-field point targets be theta_T,k(K1.. K), then the signal received at the nth bin is the signal received at the nth bin

wherein f is_cdenotes the carrier frequency, beta_kThe reflection coefficient of the kth target is represented and assumed to be unknown. Let the first transmitting antenna and the first lattice point doFor reference, then τ_Tt,m,k＝(d_t,m-d_t,1)sinθ_T,kC represents the time delay of the signal from the m-th transmitting antenna to the k-th target relative to the reference array element, tau_Tr,n,k＝(n-1)Δ_rsinθ_T,kRepresenting the time delay of the signal from the kth target to the nth grid point relative to the first grid point. Q denotes the number of clutter scatterers, gamma_q(Q ═ 1., Q) denotes the reflection coefficients of clutter scatterers, and it is assumed that there are independent identically distributed (iid) gaussian random variables between them. Tau is_Ct,m,q＝(d_t,m-d_t,1)sinθ_C,qC represents the time delay of the signal from the m-th transmitting antenna to the q-th clutter-scattering body relative to the reference array element, tau_Cr,n,q＝(n-1)Δ_rsinθ_C,qRepresenting the time delay of the signal from the qth clutter scatterer to the nth grid point, θ_C,qIndicating the orientation of the q-th clutter scatterer relative to the array.Is a variance of σ²White gaussian noise.

by performing an L-point Discrete Fourier Transform (DFT) on the time domain discrete signal, the frequency point f can be obtained_l＝lf_sfrequency domain data of (1...., L), where f_sIs the frequency sampling interval, f_l∈[-B/2,B/2]And isThe signal being at frequency f_lThe DFT result of (a) is

wherein s is_m[l]And u_n[l]Respectively representing the transmitted signalsAnd noiseDFT of (2). Order toAndRespectively expressed at an angle theta and a frequency f_lA receive steering vector and a transmit steering vector. Will N_rThe signal received by each grid point is expressed as a vector

Wherein

Under the CS framework, to estimate DOA θ of K targets_T,k(K1.. K.) the target angle observation region is discretized into G (K < G) grid points θ₁,...,θ_GIt is assumed that the dispersion error is negligible, i.e. the target falls exactly on the grid point. Then the formula (3) can be expressed as

y[l]＝Φ[l]x+c[l]+u[l] (4)

WhereinVector x ═ x₁,...,x_G]^Tis K sparse, i.e. x has only K non-zero elements, and the values and positions of the non-zero elements are the target reflection coefficient and DOA, which can be expressed as

CS theory estimates the target DOA using the sparsity of x, however, this sparsity is destroyed in a cluttered environment, thereby degrading the performance of DOA estimation. Is composed ofThe interference of the spurious wave is suppressed, and a beam forming method is adopted at a receiving end. Order toIs shown in the direction theta_gFrequency f_lAnd the position of the non-zero element represents the lattice point where the antenna is selected to be placed. Since there are only N available receive antennas, the weight vector is required to satisfy | | w_g,l||₀N. The beamformed output is given by

Will r is_g,l(G1., G and L1., L) is expressed as a vector GL × 1

Where phi is [ ]^T[1],...,Φ^T[L]]^T，c＝[c^T[1],...,c^T[L]]^T，u＝[u^T[1],...,u^T[L]]^T，W_r＝Diag{W₁,...,W_L}，according to equation (7), the DOA estimation problem can be converted into a sparse signal reconstruction problem, and based on the CS theory, the K sparse vector x can be reconstructed through base pursuit elimination (BPDN)

where η ≧ 0 is the regularization parameter, for the optimization problem of equation (8), the CVX toolkit can be used to solve. Order toThe solution of the above formula is shown,is the estimation result of the target DOA. Considering a target DOA vector θ_T＝[θ_T,1,...,θ_T,K]^Tis a random case, then the average estimation performance can be given by the Bayesian Mean Square Error (BMSE)

From the equation (9), the DOA estimated performance and the matrix W_rIn connection with, in order to optimize performance, the following optimization problem is given

(10) The last two constraints in the equation are to ensure that w is true for different g and l_g,lThe positions of the non-zero elements in (a) are the same. Due to w_g,lthe position of the non-zero element in the (10) expression indicates that the corresponding lattice point is selected to place the antenna, so that the formula is a joint optimization problem of the weighted value and the sparse array structure.

Considering that equation (10) is an NP-hard problem, a simple algorithm is proposed to solve the optimization problem. The algorithm firstly optimizes the sparse array structure when the weight is given, and then updates the weight for the next iteration. First, how to optimize the sparse array structure when the weight is given is explained. First, a lattice point selection vector is definedWherein z is_nBelongs to {0,1}, only if the element is 1, the corresponding lattice point is selected to place the antenna, and because only N available receiving antennas exist, the requirement of | z | calculation of the virtues₀N. The weight of the first iteration is given by the formation of a full array condition Capon beam

WhereinR_c(f_l) Is a clutter c [ l ]]the covariance matrix of (2). During each iteration, a set of lattice point selection vectors z is randomly generated₁,...,z_αFor a given z, there is

w_g,l＝z⊙ξ_g,l (12)

For different z, different w can be obtained_g,lAnd W_r(W_rFrom w_g,lComposition) of formula (9), BMSE and W_rIn this regard, it can thus be seen that BMSE is also related to z, denoted as e (z). Based on minimum BMSE, the optimal lattice point selection vector z can be obtained_op。

Based on z_opUpdating xi_g,lin the corresponding weight value, orderXi is_g,lIn the formula z_opThe selected element value is updated by

WhereinIs Dc [ l ]]The covariance matrix of (a) is determined,When BMSE e (z)_op) Less than a certain threshold e₀When so, the iteration stops. The detailed algorithm is given in table 1.

TABLE 1 iterative algorithm for solving optimization problem

In order to suppress the interference of the spurious waves, a beam forming method is adopted at the receiving end.is shown in the direction theta_gFrequency f_lThe position of the non-zero element represents the lattice point where the antenna is selected to be placed, and since there are only N available receiving antennas, the weight vector is required to satisfy | | w_g,l||₀n. The array structure of the antenna should be the same for different g and l, i.e. w_g,lThe positions of the non-zero elements in (a) are the same. To represent this constraint, a matrix is constructed

W＝[w_1,1,...,w_1,L,...,w_G,1,...,w_G,L] (14)

And satisfy | | W | count of the north₀By this constraint, w can be satisfied for different g and l, i.e. the number of non-zero rows of the matrix is N_g,lthe positions of the non-zero elements in (a) are the same. Weight vector w_g,lacting on received signal y [ l]Obtaining a beam forming output r according to the formula (6)_g,lIt is expressed as a G × 1 vector

From the above formula, it can be found that for different frequencies f_lThe vectors x are equally sparse. In order to jointly utilize signal information of different frequencies, r is_l(L ═ 1., L) is expressed as a vector of GL × 1, i.e., expression (7). By converting the DOA estimation problem into the sparse signal reconstruction problem, the reconstruction result of the sparse vector x can be obtained according to the formula (8)ThenThe position of the maximum K elements in the target DOA is the estimation result and is expressed asTo pairthe values of the elements in the table are sorted from big to small, and the corresponding lattice point of each sorted element is expressed as { theta [ [ theta ] ]₍₁₎,...,θ_(G)Then the DOA estimation result can be expressed as

Full array beam forming weight vector xi_g,lCan make the direction theta_gFrequency f_lThe signal passes through without distortion while suppressing interference and noise in other directions, and is expressed as

The optimal solution of the above equation is equation (11).

Two simulation examples are given for sparse array-based MIMO radar broadband DOA estimation in a clutter environment, and the parameters are set as follows: suppose D_r11 λ/2, where λ represents the wavelength corresponding to the highest frequency of the signal. The feasible region in which the receiving antenna can be placed is divided by delta_rλ/2 is the interval dispersion of 12 grid points. The number of transmitting and receiving antennas available for the MIMO radar system is assumed to be M-N-6, and the transmitting end array structure is determined to be known.

to simplify the analysis, it is assumed that the transmitted signal bandwidths are the same, i.e. B_m200MHz (M1.., M), the carrier frequency is 1 GHz.

the target angular field of view is discretized into 41 grid points-20 °, -19 °,20 °.

The clutter consists of 250 scatterers, distributed at angles of-90 °, -90 ° +180 °/250 °,90 °.

Defining a signal-to-noise ratioAnd signal to noise ratioWithout loss of generality, assuming a target reflection coefficient of 1, SNR and SCR are set to-5 dB and-30 dB, respectively.

in simulation 1, it is assumed that the targets are uniformly and randomly distributed on the grid points after the angular observation domain is discretized. To ensure that the aperture of the array does not change, letThen shareDifferent sparse array structures. Fig. 1 shows the results of the BMSE arrangement in ascending order for all possible sparse array structures, diamonds represent the optimal sparse array structure under the minimum BMSE condition, squares represent the sparse array structures obtained according to the algorithm given in table 1, and it can be seen that the two structures have similar performance, and the specific structures of the two sparse arrays are given in fig. 2. For comparison, fig. 1 also shows the results of a nested array (nested array) and a co-prime array (co-prime array), which are marked by asterisks and circles, respectively, and it can be seen that the performance of various sparse array structures is better than that of the nested array and the co-prime array. Assuming that there is only one target, DOA is-14 °, fig. 3 shows the DOA estimation results of the above four sparse array structures when using the WCSAB method. As can be seen from the figure, the optimal sparse array and the sparse array obtained by the algorithm can accurately estimate the target DOA, and the nested array and the co-prime array have errors in estimation.

In simulation 2, the performance of two wideband DOA estimation methods, WCSAB and wct (wideband Capon technique), were compared. The WCT method belongs to one of ISSM, and obtains a corresponding DOA estimation result by using a Capon method for each narrow-band signal, and then averages all the results to obtain a final estimation result. Fig. 4 considers the single target case with a target DOA of 10 °, and fig. 5 considers the dual target case with target DOAs of 6 ° and 10 °. In the figure, the solid line represents a full array structure at intervals of λ/2, the dotted line represents an optimal sparse array structure, and the dotted line represents a sparse array structure obtained according to an algorithm. As can be seen from fig. 4 and 5, compared to the full array structure, the two sparse arrays have narrower main lobe widths, i.e. the sparse arrays have higher resolution. But sparse arrays result in higher sidelobes and this problem is more severe in the dual target case. By comparison, it can be found that the side lobe of the WCSAB method is lower than the WCT. It can also be seen from fig. 5 that the WCSAB method can accurately estimate the target DOA in both sparse arrays, while the WCT estimates have errors. By contrast, the WCSAB method performs better.

Claims (2)

1. A MIMO radar broadband DOA calculation method based on sparse array in clutter environment comprises the following steps:

Step 1: let the position of the transmitting antenna be determined, and the feasible region for placing the receiving antenna be [0, D_r]For simplicity of analysis, the feasible fields are separated by an interval Δ_rDiscretization to N_rA plurality of grid points, and N receiving antennas disposed on some of the grid points, N < N_r；

Step 2: establishing an MIMO radar echo signal model to obtain echo signal time domain sampling datan＝1,...,N_rAnd p 1., L, where p represents a time domain snapshot and L is a snapshot number;

And step 3: for received signalPerforming an L-point discrete Fourier transform to obtain frequency domain data, i.e.

And N is_rThe data of the grid points being represented in vector form, i.e.Wherein p 1, L1;

and 4, step 4: discretizing the target angle observation area intoG grid points theta₁,...,θ_GK < G, where K represents the number of targets, the signal model is represented in sparse form:

y[l]＝Φ[l]x+c[l]+u[l]

WhereinWhere a is_r(θ,f_l) Indicating a received steering vector, a_t(θ,f_l) Representing the transmit steering vector, s [ l ]]Representing a frequency domain transmitted signal, x ═ x₁,...,x_G]^TIs K sparse, i.e. x has only K non-zero elements, and the values and positions of the non-zero elements are the target reflection coefficient and DOA, c [ l [ ]]Represents clutter u [ l ]]Representing noise;

And 5: will beam form weight vector w_g,lAct on y [ l]the beam forming output result is obtained:

Will r is_g,l(G1., G, L ═ 1., L) is expressed as a vector:

r＝[r_1,1,...,r_G,1,...,r_1,L,...,r_G,L]^T

＝W_rΦx+W_rc+W_ru

Wherein the weight matrix W_r＝Diag{W₁,...,W_l,...,W_LIs a block diagonal matrix;

And is provided withΦ＝[Φ^T[1],...,Φ^T[L]]^T，c＝[c^T[1],...,c^T[L]]^TRepresents clutter, u ═ u^T[1],...,u^T[L]]^TRepresenting noise;

Step 6: reconstructing a sparse vector x by basis pursuit desiccation based on the CS theory;

Wherein eta is equal to or greater than 0 is a regularization parameter;

And 7: to the solution obtained in step 6The values of the elements in the table are sorted from big to small, and the corresponding lattice point of each sorted element is expressed as { theta [ [ theta ] ]₍₁₎,...,θ_(G)Then the DOA estimation result can be expressed as;

and 8: based on minimum Bayes mean square errorSolving for optimal W_rThe following optimization problem is established

s.t.W_r＝Diag{W₁,...,W_L}

||W||₀＝N

W＝[w_1,1,...,w_1,L,...,w_G,1,...,w_G,L]

Wherein the DOA vector theta of the real target_Tis random in nature and is not only easy to be recognized,Is expressed in the pair theta_TIn the hope of expectation,Denotes theta_TMean square error of DOA estimation, w, when determined_g,lRepresents a weight vector, where G1.., G, L1., L;

And step 9: the problem provided by the step 8 is solved in an optimized way to obtain the optimal W_r。

2. The method according to claim 1, wherein the method for calculating the MIMO radar wideband DOA based on the sparse array in the clutter environment comprises the following specific steps:

Step 9.1: initialization: the number of iterations j is 1,According to the formulaComputing beamforming weightswhereinR_c(f_l) Is a clutter c [ l ]]The covariance matrix of (a); during each iteration, a set of lattice point selection vectors z is randomly generated₁,...,z_αFor a given z;

Step 9.2: repeating the iterative process from step 9.3 to step 9.6:

Step 9.3: randomly generating a set of lattice point selection vectors z₁,...,z_α}；

step 9.4: according to the formula w_g,l＝z⊙ξ_g,lComputingAnd form

According to the formulaCalculating r_g,lR is to_g,lExpressed as a vector r;

Will be provided withSubstituting r into the formulaObtaining x reconstruction resultsAnd target DOA estimation result

According to the formulaTo obtain

Step 9.5: obtaining based on minimum BMSE

Step 9.6: according toBased onUpdatingGet the corresponding weight valueAnd let j equal j + 1; wherein Is Dc [ l ]]The covariance matrix of (a) is determined,

Step 9.7: when in useThen, iteration stops and the optimal antenna selection is outpute₀Is a threshold value set in advance.