Sparse subarray selection and beamforming design method for partially calibrated arrays

Abstract

The present invention discloses a joint design method for sparse subarray selection and beamforming for a partially calibrated array. Based on the partially calibrated array, the maximization of the output signal to interference noise ratio is taken as the design criterion, and the sparse subarray selection and receiving beamforming design are described as having l 0 norm regularized non-convex constrained optimization problem. This problem selects the optimal subarray while calibrating the errors between subarrays, which can effectively reduce the hardware overhead of some calibration arrays without losing beamforming performance. By using semi-definite relaxation and reweighted l 1 The norm approximation technique is used to transform the optimization problem into a convex problem, and finally the convex optimization method is used to solve it.

Description

Technical Field

[0001] The invention belongs to the field of sparse array design and beam forming, and in particular to a sparse subarray selection and beam forming design method for a partial calibration array. Background Art

[0002] In engineering applications, the geometric structure of the corresponding array cannot be completely ideal due to the limitations of many factors such as the carrier of the equipment itself, the internal layout, and the usage scenario. In addition, there is an error between the given nominal reference position and the actual array element position, and there may be gain and phase mismatch errors between the array elements, which will cause array manifold deviations, thereby sharply reducing the beamforming performance. For example, in the use scenario of airborne array antennas, the aircraft itself is not an absolute rigid body. During the flight, the fuselage will inevitably oscillate and deform, so the airborne antenna will also be displaced, and the corresponding array steering vector will also form a displacement deviation. Only based on the correct array manifold matrix can accurate DOA estimation and good beamforming performance be obtained. Therefore, it is very important and of great practical significance to calibrate the deviation of the array manifold.

[0003] At present, the research on partially calibrated arrays mainly focuses on DOA estimation and adaptive beamforming. By improving the calibration accuracy of subarrays as much as possible under different performance metrics, more robust performance can be obtained. However, the research on sparse subarray selection for partially calibrated arrays is still in its initial stage. Summary of the invention

[0004] The invention is a joint design method for sparse subarray selection and beamforming for a partial calibration array.

[0005] The technical solution to achieve the purpose of the present invention is: a sparse subarray selection and beamforming design method for a partial calibration array, comprising:

[0006] Based on a partially calibrated array, the maximization of the output signal-to-interference-noise ratio is taken as the design criterion, and the sparse subarray selection and receive beamforming design are described as a non-convex constrained optimization problem with l0-norm regularization.

[0007] By using semi-positive definite relaxation and reweighted l1 norm approximation technology, the non-convex constrained optimization problem is converted into a convex optimization problem, and the convex optimization problem is solved by an iterative algorithm to determine the calibration matrix;

[0008] The beamforming weight vector is determined according to the calibration matrix by a method of simultaneous calibration and sparse selection, or sparse selection followed by calibration.

[0009] Compared with the prior art, the present invention has the following significant advantages: the objective function of the optimization problem of the present invention is based on maximizing SINR, the constraint of the number of subarrays is added to the optimization problem, and the sparse subarray selection and receiving beamforming design are described as non-convex optimization problems with l0 norm. The semi-definite relaxation (SDR) technology and the reweighted l1 norm approximation technology are used to convert the original optimization problem into a convex optimization problem, and then the existing convex optimization method is used to solve it; the present invention can obtain robust adaptive beamforming performance and reduce the hardware cost and computational complexity of the system by jointly designing the sparse subarray selection and beamforming of the partially calibrated array. BRIEF DESCRIPTION OF THE DRAWINGS

[0010] Figure 1 Schematic diagram of the changes in signal-to-interference-noise ratio and signal-to-noise ratio under different parameters.

[0011] Figure 2 Schematic diagram of the signal-to-interference-plus-noise ratio and signal-to-noise ratio changes of Algorithm 2.

[0012] Figure 3 Schematic diagram of the sparse sub-array structure obtained by Algorithm 3.

[0013] Figure 4 Schematic diagram of the sparse subarray structure obtained by enumeration method.

[0014] Figure 5 The beam diagram of Algorithm 3 under different parameters.

[0015] Figure 6 Schematic diagram of the signal-to-interference-plus-noise ratio and signal-to-noise ratio changes of Algorithm 3 and Algorithm 2. DETAILED DESCRIPTION

[0016] The following further illustrates a sparse subarray selection and beamforming design method for a partial calibration array of the present invention with reference to the accompanying drawings and examples.

[0017] The present invention assumes that multiple uniform linear subarrays form a large array, in which each subarray is precisely calibrated, so the steering vectors of all subarrays are accurately known. However, the gain or phase deviation between subarrays is unknown, that is, only error calibration is required between subarrays. In order to achieve sparse subarray selection and beamforming design, the present invention takes maximizing SINR as the optimization goal, and selects the optimal subarray while calibrating the error between subarrays. Through this joint design method, it is possible to obtain robust adaptive beamforming performance and reduce the hardware cost and computational complexity of the system.

[0018] A sparse subarray selection and beamforming design method for partially calibrated arrays performs weighted combination of signals at the receiving end. Based on the maximum SINR criterion, the sparse subarray selection and receive beamforming design are described as a non-convex optimization problem with l0 norm. This optimization problem can not only calibrate the errors between subarrays, but also select the optimal sparse subarray. The semi-positive relaxation technique and the reweighted l1 norm approximation technique are used to convert the original optimization problem into a convex optimization problem, and then the existing convex optimization method is used to solve it. The specific implementation steps are as follows:

[0019] Any K non-overlapping uniform linear subarrays form a complete array of M omnidirectional sensors. Assume that the kth subarray contains M k sensors, namely Each subarray is precisely calibrated internally, and there are disturbance errors between subarrays. The array steering vector with dimension M×1 is expressed as:

[0020] a=Vh(1)

[0021] Among them, the full array structure matrix V of the combination of K uniform linear subarrays, with dimension M×K, is defined as:

[0022]

[0023] Among them, v k (k=1,...,K) is a column vector describing the known geometry of the subarray, and the corresponding dimension is M k ×1; 0 represents a vector whose elements are all 0. h=[1,h2,...,h K ] T is a vector of dimension K×1, whose h k (k=2,...,K) represents the unknown disturbance between sub-arrays. The nominal array response is expressed as:

[0024]

[0025] In formula (3), h0 = [1, 1, ..., 1] T is the nominal (undisturbed) value of h, which corresponds to the case where there is no interference between the subarrays. Taking h1=1 means that the first subarray is used as the reference array, then the error of any k-th array can be described as g k and φ k represent the unknown gain and phase mismatch between the first sub-array and the kth sub-array respectively.

[0026] Although the diagonal loading (DL) method can significantly improve the robustness of the MVDR beamformer in the case of array response mismatch and low sample size, the most prominent disadvantage of the traditional diagonal loading MVDR beamforming technology is that the optimal value of γ is heavily dependent on the usage scenario. Although some scholars have proposed several methods for automatically selecting diagonal loading parameters, these methods cannot utilize the potential structure in the uncertainty model. Moreover, it may lack sufficient robustness when the displacement between sub-arrays is completely unknown or the interference error between sub-arrays is large. In order to improve the robustness of the beamforming algorithm, scholars then studied a beamforming design method based on steering vector estimation.

[0027] Assuming that there is arbitrary interference between sub-arrays, the vector h can be estimated first, and then the real steering vector can be obtained according to a=Vh, so as to achieve calibration between sub-arrays.

[0028] Step 1: Sparsely select the subarray. The present invention takes maximizing the signal-to-interference-noise ratio as the optimization criterion and describes the sparse subarray selection and receive beamforming design as a non-convex optimization problem with a l0 norm:

[0029]

[0030] Among them, taking the l0 norm of the calibration vector h can realize the sparse selection of subarrays. The non-zero element position in the optimization variable h corresponds to the selected subarray, and P represents the preset number of selected subarrays. The constraint equation sets h1 = 1 because the first subarray is usually selected as the reference array, and it can avoid the trivial solution of h = 0, which can be equivalently expressed as h H e1=1,where R i+n =E{(x i (t)+x n (t))(x i (t)+x n (t)) H} represents the covariance matrix of interference plus noise at time t, where The angles are θ1,...,θ q q interference signals, a(θ i ) represents θ i Angle of the steering vector, s i (t) is the interference signal at time t, Indicates that the mean is 0 and the variance is δ 2 Additive Gaussian white noise, represents a set of complex numbers. Substituting equation (1) into equation (4), the optimization problem is transformed into:

[0031]

[0032] Step 2: In order to replace the strict sparsity constraint, the l0 norm can be put into the objective function as a penalty term, and a regularization factor μ can be introduced. The value of μ can control the sparsity of the solution, thereby controlling the number of selected subarrays. The optimization problem is further expressed as:

[0033]

[0034] Since the objective function of equation (6) contains a non-convex l0 norm, this combinatorial optimization problem is NP-hard and cannot be solved in polynomial time. Next, the reweighted l1 norm is used to convexly relax the l0 norm in the objective function, and squaring the l1 norm does not change the original sparsity of the norm. The optimization problem is restated as:

[0035]

[0036] in, represents the regularization weight vector at the rth iteration, which can promote the sparsity of the solution. Symbol Represents vector z r Hadamard product with the vector h.

[0037] definition and Represent the real and imaginary parts of the complex variable respectively. Similarly, the following definitions can be made:

[0038]

[0039] According to these definitions, the optimization problem of formula (7) can be transformed from the complex domain to the real domain to facilitate further solution. The transformed optimization problem is:

[0040]

[0041] in is a vector The first element of .

[0042] Step 3: It is very difficult to directly solve the optimization problem (10), but according to the form of the objective function, the semi-positive definite relaxation method can be used to convexly relax the optimization problem. Therefore, the quadratic function in the objective function is rewritten as:

[0043]

[0044] In formula (11), we define The operator Tr{·} represents taking the trace of the matrix. Similarly, can be converted to:

[0045]

[0046] in,

[0047] Then, after a series of transformations and convex relaxation operations, the original optimization problem (5) can be relaxed into a convex problem, which can be solved by standard solvers in Matlab, such as CVX. Therefore, the problem is restated as:

[0048]

[0049] Among them, the operator [·] 11 Represents the value of the first diagonal element in the square matrix. Constraint This means that the matrix is a semi-positive definite matrix. In the above optimization problem, the rank-one constraint is non-convex, so it is discarded in the process of convex relaxation of the problem. The superscript r represents the number of iterations, and the regularized weight matrix The iterative update formula is:

[0050]

[0051] Z is a weight matrix, which is usually initialized to a matrix with all elements equal to 1. Each element in the matrix Z can be updated iteratively in turn, and the update expression for the (m,n)th element is:

[0052]

[0053] in, And ε>0 is a very small number, which is to prevent the denominator from being 0 and to avoid the final solution converging to the local minimum.

[0054] By iteratively solving the convex optimization problem (13), the required calibration matrix is ​​finally obtained Next, we need to extract the matrix Extract the main eigenvector from Finally, the original calibration vector is restored:

[0055]

[0056] Among them, I K represents the K×K dimensional identity matrix, j is the imaginary unit, and h is the sparse calibration vector.

[0057] In summary, the original problem is relaxed to a convex problem using a semi-positive relaxation algorithm, and then solved using the convex optimization toolbox in Matlab. In order to enhance sparsity, the sparse coefficient μ is optimized using the bisection method. The specific solution steps are summarized in Algorithm 1:

[0058] 1) Input parameters: P, K, M k ,ε,μ min ,μ max

[0059] 2) Initialization settings:

[0060] Z is a K×K matrix of all 1s, that is, is a 2K×2K matrix of all ones.

[0061] 3) Start iteration:

[0062] i. By formula (13), we can obtain:

[0063] ii. Calculate h using formula (16): r

[0064] iii. Using formulas (15) and (14), we can obtain:

[0065] iv. Update the value of μ by binary search

[0066] vr=r+1

[0067] 4) If ||h||0=P, the loop stops when the desired number of subarrays is selected; otherwise, the loop returns to 3).

[0068] 5) Output: sparse calibration vector

[0069] Through Algorithm 1, the optimization problem (5) can be efficiently solved and the sparse calibration vector h can be obtained. Next, it is necessary to obtain the weight vector of the corresponding beamformer and examine the performance of the beamformer.

[0070] Under the minimum variance distortion-free response criterion, if no sparsity constraint is imposed on the calibration vector h, the calibration vector h and the beamforming weight vector w can be directly obtained as follows:

[0071]

[0072] If the obtained sparse calibration vector h is substituted into equation (18), the corresponding weight vector w is obtained. However, due to the matrix is full rank, and the weight vector w actually obtained is non-sparse. In order to truly realize the selection of subarrays and obtain the calibrated beamforming weight vector, the present invention proposes two implementation strategies: simultaneous calibration and sparse selection, and sparse selection first and then calibration.

[0073] For the design idea of ​​simultaneous calibration and sparse selection, the specific mathematical transformation can be written as:

[0074]

[0075] Among them, the symbol represents the Hadamard product; symbol represents the Kronecker product. The dimension is M k ×1, a vector whose elements are all 1. Select vector It is composed of elements 0 and 1. Set the positions of non-zero elements in vector h to 1, and keep the positions of elements 0 unchanged to obtain

[0076] Finally, under the strategy of simultaneous calibration and sparse selection, the complete solution process of beamforming weights is summarized as Algorithm 2.

[0077] 1) Input parameters: P, K, M k ,ε,μ min ,μ max

[0078] i. Use Algorithm 1 to solve the optimization problem (5) and obtain the corresponding sparse calibration vector

[0079] ii. Based on sparse calibration vector Construct the corresponding selection vector

[0080] iii. Obtain the sparse weight vector through formulas (18) and (19)

[0081] 2) Output: beamformer weight vector

[0082] For the design idea of ​​sparse selection followed by calibration, we first iteratively solve the optimization problem (5) to obtain a sparse calibration vector Then, the positions of non-zero elements in the sparse calibration vector h are set to 1, and the positions of elements 0 remain unchanged, and the selection vector is obtained. So, The position where the element is 1 corresponds to the selected subarray, and the position where the element is 0 corresponds to the discarded subarray. Finally, for the selected subarray, the inter-array error calibration is performed using formula (17) to obtain Then Substituting into equation (18), we can obtain the corresponding beamforming weight vector The mathematical expression is:

[0083]

[0084] Finally, under the strategy of sparse selection followed by calibration, the complete solution process of the beamforming weights is summarized as Algorithm 3.

[0085] 1) Input parameters: P, K, M k ,ε,μ min ,μ max

[0086] i. Use Algorithm 1 to solve the optimization problem (5) and obtain the corresponding sparse calibration vector

[0087] ii. Select vector by construction Determine the selected subarray;

[0088] iii. For the selected optimal sparse subarray, use formula (17) to obtain the new calibration vector

[0089] iv. Obtain the corresponding weight vector through formula (20)

[0090] 2) Output: beamformer weight vector

[0091] Example

[0092] The specific implementation scheme of the sparse subarray selection and beamforming design for a partial calibration array of the present invention is further illustrated by Matlab simulation.

[0093] 1) Simulation system parameter settings

[0094] In the simulation experiment, it is assumed that there are K sub-arrays, each of which contains M k=6 (k=1,...,K) antenna array elements, then there are a total of M=6×K antennas forming a complete array. For all subarrays, the internal element spacing is set to λ / 2, where λ represents the wavelength of the signal. Because the present invention takes the first subarray as a reference, we set the displacement vectors between the remaining K-1 subarrays and the first subarray to: [5λ,0], [(5×2)λ,0],…, [(5×(K-1))λ,0]. In order to make the simulation experiment more concise and efficient, we assume that there is only a phase disturbance error between the subarrays. In other words, this phase error is caused by the inability to accurately know the displacement between the subarrays. Assume that the incident angle of the desired signal is θ0=0 ° , there are four interference signals at the same time, and their corresponding incident angles are θ1 = -30°, θ2 = -11°, θ3 = 40°, and θ4 = 70°. The four interference signals are of equal power, and INR = 20dB.

[0095] 2) Beam diagram drawing

[0096] In order to intuitively display the beamforming effect after sparse selection of subarrays, the weight vector solved by Algorithm 3 is used to weighted combine the received signals, and finally the beam patterns under different parameters are drawn. The horizontal axis of the beam pattern represents the angle range of [-90°, 90°]. The amplitude value after weighted combination of the signal is modulo and normalized to the maximum value, so the unit of the vertical axis of the beam pattern is dB.

[0097] 3) Metrics

[0098] In the present invention, the effect of the final beamforming needs to be measured. In addition to drawing the beam diagram, the output signal to interference and noise ratio (SINR) is also required. After using Algorithm 2 and Algorithm 3 to obtain the corresponding beamforming weight vector, substituting it into the solution formula, the output signal to interference and noise ratio under different parameters can be obtained:

[0099]

[0100] In the formula, is the covariance matrix of the expected signal at time t, x s (t) = a(θ0)s0(t) represents the desired signal at an angle of θ0. At the same time, in an ideal case, the signal-to-interference-noise ratio can be expressed as:

[0101]

[0102] λ max {·} means taking the maximum eigenvalue of the matrix. Under the same initial conditions, the larger the output signal-to-interference-noise ratio, the better the beamforming effect.

[0103] 4) Result analysis

[0104] The present invention has carried out three instance simulations in total. Figure 1 This is the first simulation experiment, the purpose is to explore the impact of the number of subarrays K on the performance of the beamformer. The beamforming weight vector under the condition of no sparsity constraint can be obtained by equations (17) and (18). According to the form of equation (22), the ideal signal-to-interference-noise ratio at the output of the beamformer can be obtained, that is, there is no error in the complete array. This simulation sets four different numbers of subarrays: K = 6, K = 5, K = 4, and K = 3. The simulation experiment is repeated 100 times and the average value is taken to obtain the final result figure. Figure 2 This is the second simulation experiment, the purpose is to study the beamforming performance of Algorithm 2. In this simulation experiment, P = 3, P = 4, and P = 5 subarrays are selected from K = 6 subarrays. The corresponding sparse weight vectors can be obtained through Algorithm 2, and then substituted into formula (21) to analyze the change of the output signal-to-noise ratio. In order to obtain the optimal signal-to-noise ratio in the sparse subarray scenario, the present invention adopts the enumeration method to conduct an exhaustive search for all possible combinations to obtain the optimal signal-to-noise ratio under the corresponding parameters. Repeat the simulation experiment 100 times and take the average value to obtain the final result diagram. Figure 3 to Figure 6 This is the third simulation experiment, the purpose is to study the beamforming performance of Algorithm 3. This simulation selects P=3, P=4, and P=5 subarrays from K=6 subarrays, and then performs error calibration and beamforming on the selected sparse subarrays. When SNR=0dB and INR=20dB are obtained, the corresponding beam diagram and sparse subarray structure diagram are drawn. Similarly, the enumeration method is used to select the sparse array with the largest output signal-to-interference-noise ratio, and the effect is compared with that of Algorithm 3. Figure 3 , Figure 4 Schematic diagrams of the sparse sub-matrix structures obtained by Algorithm 3 and the enumeration method respectively. Figure 5 Represents the beam diagram of Algorithm 3 under different parameters. Figure 6 Schematic diagram showing the change of SINR and SNR of Algorithm 3 and Algorithm 2.

[0105] Depend on Figure 1 It can be seen that the more subarrays are used, the more array elements there are, that is, the higher the degree of freedom, so the ideal output signal to noise ratio is higher and the beamforming performance is better. However, the more subarrays there are, the more error terms need to be calibrated, and the greater the signal to noise ratio loss is. If the number of subarrays used is smaller, then the signal to noise ratio loss after subarray calibration is smaller than in the ideal scenario.

[0106] Depend on Figure 2 It can be clearly seen that when the subarrays are calibrated and sparsely selected at the same time, the more subarrays are selected, that is, the vector The more weight values ​​retained in , the smaller the output SINR loss of the beamformer. However, compared with the beamforming effect in the optimal scenario, Algorithm 2 will cause a large performance loss. On the other hand, the enumeration method can select the optimal sparse subarray combinations of P = 3, P = 4, and P = 5, respectively, and their corresponding output SINR loss is very small.

[0107] Depend on Figure 5 It can be found that the optimal subarray selected by Algorithm 3 can form a deep null for the interference signal after error calibration, and has good spatial filtering performance.

[0108] Depend on Figure 6 It can be seen that Algorithm 3 can select the best sparse subarray, and its output signal to interference noise ratio is very close to the optimal signal to interference noise ratio, which fully proves the effectiveness and accuracy of Algorithm 3. And it can be clearly seen that under the same experimental parameters, the performance of the beamformer in Algorithm 3 is far better than the beamformer designed by Algorithm 2.

Claims

1. A sparse subarray selection and beamforming design method for a partially calibrated array, characterized in that: include: Based on the partially calibrated array, with the maximization of the output signal-to-interference-noise ratio as the design criterion, the sparse subarray selection and receive beamforming design are described as a non-convex constrained optimization problem with l0 norm regularization, which is specifically expressed as: sth1=1 ||h||0≤P Where a is the array steering vector, h = [1,h2,...,h K ] T is a calibration vector of dimension K×1, where h k represents the disturbance between unknown subarrays, k = 2, ..., K, P represents the number of pre-set selected subarrays, h1 = 1 means that the first subarray is used as the reference array, R i+n =E{(x i (t)+x n (t))(x i (t)+x n (t)) H } represents the covariance matrix of interference plus noise at time t, where The angles are θ1,…,θ q q interference signals, a(θ i ) represents θ i Angle of the steering vector, s i (t) is the interference signal at time t, Indicates that the mean is 0 and the variance is δ 2 Additive Gaussian white noise, represents a complex set; The array steering vector is specifically: In the formula, v k Describes the known geometric structure of the subarray, the corresponding dimension is M k ×1, 0 represents a vector whose elements are all 0, k=1,...,K; Substituting the array steering vector into the non-convex constrained optimization problem with l0 norm regularization, we obtain the transformed non-convex constrained optimization problem with l0 norm regularization: s.t.h H e1=1 ||h||0≤P In the formula, h H e1=1 is equivalent to h1=1 because the first subarray is usually selected as the reference array, and it can also avoid the trivial solution of h=0. represents the set of real numbers; By using semi-positive definite relaxation and reweighted l1 norm approximation technology, the non-convex constrained optimization problem is converted into a convex optimization problem. The convex optimization problem is solved by an iterative algorithm to determine the calibration matrix. The specific method is: The l0 norm is added as a penalty term to the objective function of the transformed non-convex constrained optimization problem, and a regularization factor μ is introduced. The optimization problem is expressed as: s.t.h H e1=1; The reweighted l1 norm is used to convexly relax the l0 norm in the above objective function, and the optimization problem is reformulated as: s.t.h H e1=1 in, Represents the regularization weight vector at the rth iteration, symbol Represents vector z r Hadamard product with vector h; definition and denote the real and imaginary parts of the complex variable respectively, and define The optimization problem after convex relaxation is converted from the complex number domain to the real number domain. The transformed optimization problem is: in is a vector The first element of The semi-positive definite relaxation method is used to perform convex relaxation on the optimization problem, and the quadratic function in the transformed objective function is rewritten as: In the formula, The operator Tr{·} represents taking the trace of the matrix; The objective function Translates to: in, The optimization problem is reformulated as: Among them, the operator [·] 11 Represents the value of the first diagonal element in the square matrix, constraining This means that the matrix is a semi-positive definite matrix, and the problem is solved directly using the standard convex optimization toolbox; In the above optimization problem, the rank-one constraint is non-convex, so it is discarded in the process of convex optimization of the problem, where the matrix The superscript r represents the number of iterations, and the regularized weight matrix The iterative update formula is: Z is the weight matrix, which is initialized to a matrix with all elements set to 1; Iteratively update the (m,n)th element in the matrix Z: in, And ε>0 is a very small value; The beamforming weight vector is determined according to the calibration matrix by a method of simultaneous calibration and sparse selection, or sparse selection followed by calibration.

2. The sparse subarray selection and beamforming design method for a partially calibrated array according to claim 1, characterized in that: The specific method of determining the beamforming weight vector according to the calibration matrix by simultaneous calibration and sparse selection is: Based on sparse calibration vector Contains information about subarray selection and constructs the corresponding selection vector Under the minimum variance distortion-free response criterion, according to the sparse calibration vector determining a beamforming weight vector without imposing a sparsity constraint; A final beamforming weight vector is determined according to the constructed selection vector and the beamforming weight vector without applying the sparsity constraint.

3. The sparse subarray selection and beamforming design method for a partially calibrated array according to claim 2, characterized in that: Under the minimum variance distortion-free response criterion, according to the sparse calibration vector The beamforming weight vector without applying sparsity constraint is determined as follows:

4. The sparse subarray selection and beamforming design method for a partially calibrated array according to claim 2, characterized in that: The final beamforming weight vector is determined according to the constructed selection vector and the beamforming weight vector without applying the sparsity constraint: Among them, the symbol represents the Kronecker product, The dimension is M k ×1, a vector whose elements are all 1, and w is the beamforming weight vector without sparsity constraint.

5. The sparse subarray selection and beamforming design method for a partially calibrated array according to claim 1, characterized in that: The specific method of determining the beamforming weight vector by sparse selection followed by calibration according to the calibration matrix is: The positions of non-zero elements in the sparse calibration vector h are set to 1, and the positions of elements 0 remain unchanged, and it is converted into a selection vector The position where the element is 1 corresponds to the selected subarray, and the position where the element is 0 corresponds to the discarded subarray; For the selected optimal sparse subarray, the inter-array error calibration is performed to obtain The corresponding expression is: And obtain the corresponding weight vector It is expressed as:

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