Sparse array design method based on bilinear factor matrix norm

Abstract

The invention discloses a sparse array design method based on a bilinear factor matrix norm, and the method comprises the steps: carrying out the equal-interval sampling of a transducer according to the directivity of an array, obtaining directional sampling points, and enabling the directional sampling points to form a sampling point matrix; and establishing a relation between directional sampling points and the number of transducers in the array in a sampling point matrix by using a matrix beam method. And establishing a relationship between the position of the transducer in the array and the weight and directivity by using a matrix beam method. And the rank of the sampling point matrix to be optimized is minimized through nuclear norm approximation and bilinear decomposition. The rank of the matrix of sampling points is set to the number of transducers in the array. Directive sampling points form a Hankel matrix, the fact that the rank of the matrix is equal to the number of transducers in the array is utilized, the array design problem is converted into the problem of searching for a low-rank matrix meeting the design requirement, the rank of the matrix to be optimized is minimized through nuclear norm approximation and bilinear decomposition, and the array design efficiency is improved. The lowest main lobe width and side lobe level are achieved.

Description

Technical Field

[0001] The invention belongs to the field of processing transducer grating lobes, and in particular relates to a sparse array design method based on a bilinear factor matrix norm. Background Art

[0002] A transducer is a device that converts electrical energy into acoustic energy, including a transmitting transducer that converts electrical energy into acoustic energy, and a receiving transducer that converts acoustic energy into electrical energy. Generally speaking, transmitting and receiving transducers are used separately, but they can also share one. The main performance indicators of a transducer include operating frequency, bandwidth, electroacoustic frequency, impedance at resonant frequency, directivity (transmitting beam width) and sensitivity. A transducer array is an array composed of multiple transducers arranged in a certain form, similar to an antenna array. Each transducer in the transducer array is called a primitive or array element. Large underwater acoustic transducer arrays are several kilometers long and have thousands of primitives. Small ultrasonic transducer arrays, such as those on medical ultrasonic diagnostic instruments, range from a few millimeters to tens of centimeters, and have hundreds of primitives.

[0003] However, when the array element spacing exceeds a certain limit, the radiation field of the phased array antenna will form a regular radiation beam similar to the main beam outside the main lobe. This phenomenon is called grating lobe. Therefore, it can be determined that the array element spacing has a direct impact on the generation of grating lobes. Specifically, when the array element spacing is greater than or equal to half a wavelength, the grating lobe phenomenon will occur. However, in a dual-frequency system using a multiplexed array, when the array is designed with the maximum aperture in the low-frequency mode, switching to the high-frequency mode can obtain a sharper main lobe, and a serious grating lobe problem will occur. Generally speaking, we need to suppress the grating lobe to ensure that the directivity and gain of the array elements in the base array are more stable, so as to improve the overall performance of the sonar system. However, after the grating lobe is suppressed, the weight difference between the array elements in the base array will become larger, resulting in each array element not being able to fully play its role. Therefore, there is an urgent need for a sparse array design method based on the bilinear factor matrix norm that can meet the directivity requirements and have a smaller weight gap. Summary of the invention

[0004] The content of this application is used to introduce concepts in a brief form, which will be described in detail in the detailed implementation section below. The content of this application is not intended to identify the key features or essential features of the technical solution claimed for protection, nor is it intended to limit the scope of the technical solution claimed for protection.

[0005] In view of the problems and shortcomings in the prior art, the present invention aims to provide a sparse array design method based on the bilinear factor matrix norm. The present invention converts the array design problem into a low-rank optimization problem by making the number of array elements in the array equivalent to the rank of the Hankel matrix composed of directional sampling. Under the constraint of directivity requirements, the optimization is completed by the bilinear factor matrix norm minimization algorithm to achieve sparse array design. This solves the problems raised in the above background technology.

[0006] To achieve the above object, the present invention provides the following technical solutions:

[0007] The present invention discloses a sparse array design method based on a bilinear factor matrix norm, comprising the following steps:

[0008] Step 1, obtaining the directivity characteristics of the array elements in the transducer array, and performing equal-space sampling according to the directivity characteristics to obtain directivity sampling points;

[0009] Step 2, using a matrix beam method to establish a connection between the directional sampling points and the number of transducers in the array according to the directional characteristics;

[0010] Step 3, using a matrix beam method to establish a relationship between the transducer position, weight and directivity in the array according to the directivity characteristics;

[0011] Step 4, minimizing the matrix of the directional sampling points to be optimized by nuclear norm approximation and bilinear decomposition;

[0012] Step 5: setting the rank of the directional sampling point matrix to the number of transducers in the array.

[0013] Furthermore, in step 3, the relationship between the position, weight and directivity of the transducer in the array is established by using a matrix beam method, which specifically includes the following steps:

[0014] Step 3.1, writing the directional sampling points obtained by equally spaced sampling into the form of a Hankel matrix;

[0015] Step 3.2, removing the first row and the last row of the Hankel matrix;

[0016] Step 3.3, solving the parameters including the array element position and converting them into conventional eigenvalues, and calculating the transducer array element position;

[0017] Step 3.4, finding the minimum product solution to obtain the weight corresponding to the transducer array element position.

[0018] Furthermore, in step 4, the rank of the sampling point matrix is ​​minimized by nuclear norm approximation, which specifically includes the following steps:

[0019] Step 4.1.1, performing Vandermonde decomposition on the Hankel matrix to obtain the product of the position matrix and the weight matrix of the array element;

[0020] Step 4.1.2, after relaxing the decomposed Hankel matrix using the nuclear norm, convert it into an augmented Lagrangian function;

[0021] Step 4.1.3, recursively optimize each variable of the Hankel matrix using the variable direction multiplier method;

[0022] Step 4.1.4, use the singular value threshold to optimize the Hankel matrix in the augmented Lagrangian form.

[0023] Furthermore, in step 4, the rank of the sampling point matrix is ​​optimized by bilinear decomposition, which specifically includes the following steps:

[0024] Step 4.2.1, using the augmented Lagrangian function to solve its minimum value by the variable direction multiplier method;

[0025] Step 4.2.2, updating the sampling point matrix with the Lagrangian multipliers in the augmented Lagrangian function after the minimum value;

[0026] Step 4.2.3, converting the sampling point matrix into a corresponding directional sampling vector by using the pseudo-inverse of the Hankel operator;

[0027] Step 4.2.4, using the matrix beam to infer the weights and positions of the transducers in the array from the directional sampling vector.

[0028] Furthermore, the directional sampling points obtained by equally spaced sampling in step 3.1 are written in the form of a Hankel matrix, which is expressed as:

[0029]

[0030] in, is the Hankel operator, and k is the bundle parameter.

[0031] Furthermore, in step 4.1.2, the decomposed Hankel matrix is ​​relaxed using the nuclear norm and then converted into an augmented Lagrangian form, which is expressed as:

[0032]

[0033] Among them, |||| * is the nuclear norm of the matrix, Λ is the Lagrange multiplier, and k is the bundle parameter.

[0034] Furthermore, the updating formula of the directional sampling vector in step 4.2.3 is expressed as:

[0035]

[0036] in, is represented as the projection operator, Λ is represented as the Lagrange multiplier, Expressed as the pseudo-inverse of the Hankel operator.

[0037] Compared with the prior art, the beneficial effects of the present invention are as follows: the present invention provides a sparse array design method based on the bilinear factor matrix norm. The proposed method mainly includes arranging the array elements in the array according to the maximum aperture at low frequency. At this time, the array has a higher angular resolution at low frequency, and the grating lobe in the high-frequency directivity of the array is suppressed by the natural directivity of the transmitting transducer. By optimizing the position and weight of the array elements in the array, the main lobe width is further reduced while maintaining the sidelobe level and the high-frequency grating lobe spacing, thereby enhancing the angular resolution of the array. The proposed method uses the matrix beam method to establish a connection between the array directivity sampling and the transducer position and weight in the array. By forming the directivity sampling points into a Hankel matrix, the array design problem is transformed into a problem of finding a low-rank matrix that meets the design requirements by using the fact that the rank of the matrix is ​​equal to the number of transducers in the array. On this basis, the rank of the matrix to be optimized is minimized by nuclear norm approximation and bilinear decomposition. In view of the specific design problem to be solved in the present invention, the proposed method achieves the lowest main lobe width and sidelobe level using the same number of array elements. In addition, compared with other methods, the weight difference between array elements obtained by the method proposed in the present invention is smaller, and the role of each array element in the array is fully utilized. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] The drawings constituting a part of this application are used to provide a further understanding of this application, so that other features, purposes and advantages of this application become more obvious. The schematic embodiment drawings and their descriptions of this application are used to explain this application and do not constitute an improper limitation on this application. In the drawings:

[0039] Figure 1 : is a block diagram of the main steps in the embodiment of the present invention;

[0040] Figure 2 : is a performance diagram of the array directivity in an embodiment of the present invention, (a) is the field of view of the system, (b) is the main lobe width, field of view and side lobe level;

[0041] Figure 3 : is a performance diagram of the array design result of the present invention in an embodiment of the present invention, (a) is the array directivity, and (b) is the position and weight of the array element in the array. DETAILED DESCRIPTION

[0042] Embodiments of the present disclosure will be described in more detail below with reference to the accompanying drawings. Although certain embodiments of the present disclosure are shown in the accompanying drawings, it should be understood that the present disclosure can be implemented in various forms and should not be construed as being limited to the embodiments set forth herein. On the contrary, these embodiments are provided to provide a more thorough and complete understanding of the present disclosure. It should be understood that the drawings and embodiments of the present disclosure are only for exemplary purposes and are not intended to limit the scope of protection of the present disclosure.

[0043] It should also be noted that, for ease of description, only the parts related to the invention are shown in the drawings. In the absence of conflict, the embodiments and features in the embodiments of the present disclosure can be combined with each other.

[0044] In a dual-frequency system using a multiplexed array, when the array is designed with the maximum aperture in the low-frequency mode, a sharper main lobe can be obtained by switching to the high-frequency mode, but the grating lobe problem will occur at this time. If the grating lobes in the directivity of the receiving array are to be eliminated, on the one hand, the angular resolution can be reduced and the array aperture can be reduced, and on the other hand, more high-frequency array elements can be added to the array at the cost of increasing system complexity. Different from the above two solutions, the present invention retains the grating lobes in the directivity of the receiving array and suppresses them by using the directivity of the transmitting transducer.

[0045] In imaging applications, the transducer array needs to have a thinner main lobe width so that the system has a higher angular resolution. Usually, the directivity of the transducer array will directly affect the performance of the imaging system. Therefore, the array can obtain a specific directivity by adjusting the position and weight of the array elements in the transducer array. Specifically, for a linear array containing N array elements, the position of these array elements is d = [d 1 , …, d N ] T , and the corresponding weight is w=[w 1 ,…,w N ] T , then the corresponding directivity y(u) can be expressed as,

[0046]

[0047] Where u = cosθ, θ represents the corresponding angle direction, and λ represents the array design frequency f c Take a linear array composed of 32 array elements as an example. When each array element uses the same weight and the array element spacing is 0.5λ c When c is the underwater sound speed. However, we can further reduce the main lobe width of the array by increasing the frequency at which the transducer array operates. Still taking a uniform linear array composed of 32 array elements as an example. When the array operates at a higher frequency, such as 3fc , at this time the spacing between array elements becomes 1.5λ h , where λ h =c / 3f c , c is still the underwater sound speed. It can be seen that increasing the operating frequency significantly reduces the width of the main lobe, but the appearance of the grating lobe still significantly affects the imaging quality of the system.

[0048] The present invention discloses a sparse array design method based on a bilinear factor matrix norm, which will be described in detail below with reference to the accompanying drawings and in combination with embodiments. Specifically, the following steps are included:

[0049] Step 1, obtaining the directivity characteristics of the array elements in the transducer array, and performing equal-space sampling according to the directivity characteristics to obtain directivity sampling points;

[0050] Step 2, using a matrix beam method to establish a relationship between the directional sampling points and the number of transducers in the array according to the directional characteristics;

[0051] Step 3, using the matrix beam method to establish the relationship between the transducer position, weight and directivity in the array according to the directivity characteristics;

[0052] Step 4, minimizing the matrix of the directivity sampling points to be optimized by nuclear norm approximation and bilinear decomposition;

[0053] Step 5, setting the rank of the directional sampling point matrix to the number of transducers in the array.

[0054] Specifically, we first sample the transducers at equal intervals according to the directivity of the array to obtain directional sampling points, and then establish a relationship between the directional sampling points and the number of transducers in the array. We sample the directivity y(u) shown in Formula 1 at equal intervals, and the corresponding sampling points are v m =mT s , m=0,…,M, then the corresponding sampling result can be written as,

[0055]

[0056] in, N is the number of elements in the array. Based on the sampling results, the problem of minimizing the number of elements can be solved by optimizing the array weight vector w = [w 1 , w 2 ,…,w N ], and the position vector z = [z 1 , z 2 , …, z N] is achieved. The number of array elements is limited by minimizing the weights and the dimensions of the position vector, while the array directivity requirement is determined by the upper and lower bounds of the directivity amplitude sampling. Since the designed array is symmetrical, the directivity amplitude will not appear complex.

[0057] Next, the relationship between the position, weight and directivity of the transducer in the array is established by using the matrix beam method. The present invention uses the matrix beam method (MPM) to achieve the association. Specifically, the following steps are included:

[0058] Step 3.1, writing the directional sampling points obtained by equally spaced sampling into the form of a Hankel matrix;

[0059] Step 3.2, removing the first row and the last row of the Hankel matrix;

[0060] Step 3.3, solving the parameters including the array element position and converting them into conventional eigenvalues, and calculating the transducer array element position;

[0061] Step 3.4, finding the minimum product solution to obtain the weight corresponding to the transducer array element position.

[0062] Specifically, the array directivity is sampled at equal intervals, and the sampling results are written in the form of a Hankel matrix as follows:

[0063]

[0064] in is the Hankel operator, and k is the bundle parameter. By removing the last row of matrix Y, we get matrix Y 0 as follows,

[0065]

[0066] Similarly, removing the first row of matrix Y yields matrix Y 1 as follows,

[0067]

[0068] If k=N and M-1≤2N is satisfied, it can be found that for the matrix bundle Y 1 -zY 0 , z takes the value of {z 0 , …, z N-1} is not full rank. Therefore, the solution to the parameter z containing the array element position can be obtained by solving the generalized characteristic, that is, (Y 1-z Y 0 )q=0. Convert the above formula into a conventional eigenvalue and express it as In the formula Yes0 The pseudo-inverse of . After obtaining the corresponding eigenvalues, the position of the transducer can be obtained as,

[0069]

[0070] By finding the minimum product solution of s=Zw, we can get the weight corresponding to the array element.

[0071]

[0072] Therefore, the MPM links the Hankel matrix consisting of directional sampling to the positions and weights of the transducers in the array.

[0073] In step 4, the rank of the Hankel matrix is ​​optimized by approximating the nuclear norm, which specifically includes the following steps:

[0074] Step 4.1.1, perform Vandermonde decomposition on the Hankel matrix to obtain the product of the position matrix and the weight matrix of the array element;

[0075] Step 4.1.2, after relaxing the decomposed Hankel matrix using the nuclear norm, convert it into an augmented Lagrangian function;

[0076] Step 4.1.3, recursively optimize each variable of the Hankel matrix using the variable direction multiplier method;

[0077] Step 4.1.4, use the singular value threshold to optimize the Hankel matrix in the augmented Lagrangian form.

[0078] Specifically, we perform Vandermonde decomposition on the Hankel matrix and obtain:

[0079]

[0080] It can be found that the rank of the Hankel matrix is ​​equal to the number of transducers in the array. Based on this feature, it can be obtained that

[0081]

[0082] in, is the Hankel operator of the Hankel matrix, k is the beam parameter, and M is the number of sampling points. However, since directly constraining the rank of the matrix is ​​non-convex and difficult to solve, we use the nuclear norm to relax the above equation and transform it into,

[0083]

[0084] Where ||||* is the nuclear norm of the matrix, defined as Where σi is the singular value of the matrix X. Rewriting the above formula into the corresponding augmented Lagrangian form can be expressed as,

[0085]

[0086] Then, the Alternating Direction Method of Multipliers (ADMM) is used to recursively optimize each variable. The singular value threshold (SVT) is used to optimize the matrix Y in the augmented Lagrangian form, and the obtained When Y k+1 Required to obtain the minimum value If defined Where ∑ 0 is the part with singular values ​​greater than 1 / β. Accordingly, ∑ 1 are the remaining singular values. If the matrix Y is optimally chosen as, Combining the above two equations, we can get

[0087] Combined with the definition of nuclear norm, the gradient can be expressed as Then W can be expressed as Because 1 All elements in are less than 1 / β, then the loop for solving the optimal Hankel matrix under the nuclear norm constraint can be expressed as,

[0088]

[0089]

[0090]

[0091] In the formula, is the Hankel operator The inverse operator of is a mapping operator, defined as,

[0092]

[0093] Through the update process of the matrix Y, it can be found that there are still small singular values ​​in the Hankel matrix that are retained, and these small singular values ​​will still affect the rank of the matrix. To this end, the present invention performs bilinear decomposition (Matrix Factorization, MF) through the matrix, that is, artificially limits the maximum rank of the Hankel matrix, that is, the number of array elements in the array.

[0094] Then, the rank of the sampling point matrix is ​​optimized by bilinear decomposition, which specifically includes the following steps:

[0095] Step 4.2.1, use the augmented Lagrangian function to solve its minimum value by the variable direction multiplier method;

[0096] Step 4.2.2, update the sampling point matrix with the Lagrangian multipliers in the augmented Lagrangian function after the minimum value;

[0097] Step 4.2.3, convert the sampling point matrix into the corresponding directional sampling vector through the pseudo-inverse of the Hankel operator;

[0098] Step 4.2.4, use the matrix beam to infer the weights and positions of the transducers in the array from the directional sampling vector.

[0099] Specifically, for matrix Y, if it is transformed into the product of two matrices through matrix decomposition, then its maximum rank is completely determined by these two matrices. For example, if There exists a matrix decomposition Y=UV T , and U and V satisfy Then the maximum rank of Y can only be r. Based on this idea, the nuclear norm is expressed as,

[0100]

[0101] In the formula ||·|| F is the Frobenius norm. Substituting another representation of the nuclear norm in formula 8 into formula 5, we can find that the problem is transformed into

[0102]

[0103] The augmented Lagrangian function corresponding to Formula 9 can be expressed as:

[0104]

[0105] Similarly, using the ADMM method to solve the minimum value of formula 10 requires looping to solve the following problems:

[0106]

[0107]

[0108]

[0109] The definition of Ω is consistent with that in Formula 8, which represents the set of all directional sampling that meets the design requirements. At the same time, the update method of the Lagrange multiplier Λ is,

[0110]

[0111] Taking the derivative of the right side of formula 12 with respect to U and setting the derivative to zero, we can get the corresponding update formula:

[0112]

[0113] Similarly, taking the derivative of the right side of Formula 13 with respect to V and setting the derivative to zero, we can obtain the update formula of the matrix V as follows:

[0114]

[0115] In order to obtain the update formula of the sampling vector y, an intermediate result can be obtained by taking the derivative of the right side of Formula 14 and making the result zero

[0116]

[0117] The next step is to use the pseudo-inverse of the Hankel operator The Hankel matrix Converted to the corresponding directional sampling vector y (i+1) , the pseudo-inverse of the Hankel operator Defined as In the formula is the Hankel operator The adjoint operator. According to the definition of the adjoint operator Then for the matrix It can be expressed as,

[0118]

[0119] Furthermore, in order to ensure middle in is the unit mapping operator, It can be expressed as where c m , m = 0, ..., M is equal to the number of elements on the mth anti-diagonal line of matrix B. Based on the above description, the directional sampling vector y (i+1) The update formula can be expressed as In the formula is the projection operator defined by Formula 8. By recursively solving the sub-problems shown in Formulas 12 to 15, the directional sampling that meets the design requirements can be obtained, and then the matrix beam method MPM can be used to infer the weights and positions of the transducers in the array from the sampling results. Compared with directly minimizing the nuclear norm of the Hankel matrix composed of directional sampling, the bilinear decomposition directly limits the number of transducers in the array and avoids the problem of small singular values ​​in the direct minimization of the nuclear norm method.

[0120] Experimental Results

[0121] The present invention adopts the proposed array directivity-based method to design the required array. The vector y from the nuclear norm consists of 129 samples sampled from the grating lobe-free area (cosθ from 0 to 0.6441), and the position and amplitude restrictions of the samples are the same as other design methods. Let the dimensions of the matrices U and V in formula 10 be 65*32, and design an array consisting of 32 transducers. The results are shown in the figure, and the final directivity parameters are as follows: HPBW is 1.07°; PSL = -19.94dB; FOV = 37.75. For the convenience of comparison, this paper summarizes the design results of all comparison methods in the following Table 1.

[0122]

[0123] Table 1

[0124] It can be found from the experimental results that when the number of array elements is the same, the design result of the proposed method has the optimal main lobe width and the highest side lobe level. When designing using matrix bundles and nuclear norm minimization, the design result is completely dependent on the preset reference directivity, and the corresponding result will also be limited by the reference design. In the method based on reweighted minimization, in order to give the array element position enough degrees of freedom, a large number of virtual array elements need to be set, which not only increases the difficulty of obtaining sparse results, but also introduces greater errors when discarding a large number of virtual array elements with small weights, resulting in a higher highest side lobe level in the result. Similar problems also occur in the method based on EMaC. Since the rank of the matrix is ​​limited by the singular value threshold, some small singular values ​​will still be retained, affecting the design results. Based on EMaC, this method uses bilinear decomposition to add a hard threshold to the rank of the matrix, which fundamentally solves the influence of small singular values. In addition, this method does not require specific reference directivity and preset array element positions, and can better meet the needs of array design.

[0125] The above descriptions are only some preferred embodiments of the present disclosure and an explanation of the technical principles used. Those skilled in the art should understand that the scope of the invention involved in the embodiments of the present disclosure is not limited to the technical solutions formed by a specific combination of the above-mentioned technical features, but should also cover other technical solutions formed by any combination of the above-mentioned technical features or their equivalent features without departing from the above-mentioned inventive concept. For example, the above-mentioned features are replaced with the technical features with similar functions disclosed in the embodiments of the present disclosure (but not limited to) and the technical solutions formed.

Claims

1. A sparse array design method based on bilinear factor matrix norm, characterized in that: The steps include: Step 1, obtaining the directivity characteristics of the array elements in the transducer array, and performing equal-space sampling according to the directivity characteristics to obtain directivity sampling points; Step 2, using a matrix beam method to establish a connection between the directional sampling points and the number of transducers in the array according to the directional characteristics; Step 3, using a matrix beam method to establish a relationship between the transducer position, weight and directivity in the array according to the directivity characteristics; Step 4, minimizing the matrix of the directional sampling points to be optimized by nuclear norm approximation and bilinear decomposition; Step 5: setting the rank of the directional sampling point matrix to the number of transducers in the array.

2. A sparse array design method based on bilinear factor matrix norm according to claim 1, characterized in that: In step 3, a relationship between the position, weight and directivity of the transducer in the array is established using a matrix beam method, which specifically includes the following steps: Step 3.1, writing the directional sampling points obtained by equally spaced sampling into the form of a Hankel matrix; Step 3.2, removing the first row and the last row of the Hankel matrix; Step 3.3, solving the parameters including the array element position and converting them into conventional eigenvalues, and calculating the transducer array element position; Step 3.4, finding the minimum product solution to obtain the weight corresponding to the transducer array element position.

3. A sparse array design method based on bilinear factor matrix norm according to claim 2, characterized in that: In step 4, the rank of the sampling point matrix is ​​minimized by nuclear norm approximation, which specifically includes the following steps: Step 4.1.1, performing Vandermonde decomposition on the Hankel matrix to obtain the product of the position matrix and the weight matrix of the array element; Step 4.1.2, using the nuclear norm to relax the decomposed Hankel matrix, and then converting it into an augmented Lagrangian function; Step 4.1.3, recursively optimizing each variable of the Hankel matrix using the variable direction multiplier method; Step 4.1.4, use the singular value threshold to optimize the Hankel matrix in the augmented Lagrangian form.

4. A sparse array design method based on bilinear factor matrix norm according to claim 3, characterized in that: In step 4, the rank of the sampling point matrix is ​​optimized by bilinear decomposition, which specifically includes the following steps: Step 4.2.1, using the augmented Lagrangian function to solve its minimum value by the variable direction multiplier method; Step 4.2.2, updating the sampling point matrix with the Lagrangian multipliers in the augmented Lagrangian function after the minimum value; Step 4.2.3, converting the sampling point matrix into a corresponding directional sampling vector by using the pseudo-inverse of the Hankel operator; Step 4.2.4, using the matrix beam to infer the weights and positions of the transducers in the array from the directional sampling vector.

5. The sparse array design method based on bilinear factor matrix norm according to claim 4, characterized in that: In step 3.1, the directional sampling points obtained by equally spaced sampling are written in the form of a Hankel matrix, which is expressed as: in, is the Hankel operator, and k is the bundle parameter.

6. The sparse array design method based on bilinear factor matrix norm according to claim 4, characterized in that: In step 4.1.2, the decomposed Hankel matrix is ​​relaxed using the nuclear norm and then converted into an augmented Lagrangian form, which is expressed as: Among them, |||| * is the nuclear norm of the matrix, Λ is the Lagrange multiplier, and k is the bundle parameter.

7. The sparse array design method based on bilinear factor matrix norm according to claim 4, characterized in that: The updating formula of the directional sampling vector in step 4.2.3 is expressed as: in, is represented as the projection operator, Λ is represented as the Lagrange multiplier, Expressed as the pseudo-inverse of the Hankel operator.

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