An array sparse method and device based on progressive Bayesian compressed sensing and graph fusion

Abstract

The application discloses an array sparse method and device based on progressive Bayesian compressive sensing and graph fusion, which comprises the following steps: taking a candidate array element as input, taking an ideal full-array beam pattern as a full-array reference beam pattern, setting hyperparameters of multi-task Bayesian compressive sensing to control sparsity; solving the multi-task Bayesian compressive sensing problem based on the set hyperparameters to obtain an array distribution and an element weight after preliminary sparsification; taking the element after preliminary sparsification as a vertex, establishing an edge between the vertices with a distance less than or equal to a constraint distance, forming an undirected graph, and dividing the undirected graph into multiple independent subgraphs according to the connectivity of the undirected graph; iteratively merging the vertices in each independent subgraph in parallel to obtain a merged element; and converting the sparse array after element merging into a global convex optimization problem based on a support hyperplane, and solving to obtain a secondary sparse element weight, so that the array element distance constraint is met, and good sparsity and beam performance are realized.

Description

Technical Field

[0001] This invention relates to the field of sparse array design technology, specifically to a method and apparatus for array sparse design based on progressive Bayesian compressed sensing and graph fusion. Background Technology

[0002] Imaging sonar transducer arrays typically employ an evenly spaced, uniformly distributed array configuration, using beamforming for imaging. To improve system resolution, large-aperture transducer arrays are usually required, resulting in a massive number of array elements. Therefore, imaging sonar technology faces two major challenges: hardware system complexity and computational burden. Array sparsity optimization can effectively reduce the number of transducer elements while maintaining system performance, becoming an important method for reducing system complexity and computational load.

[0003] For uniformly distributed arrays, not all elements need to receive acoustic signals; that is, the element coefficients at some locations can be zero. Array sparsity optimization aims to minimize the number of elements by solving for the element weight coefficients under certain constraints on the beam pattern (such as main lobe width and side lobe peak values). Traditional sparse array optimization methods suffer from problems such as low sparsity, inability to process elements with excessively close spacing, and severe attenuation of array beam performance. For example, patent document CN108828603A discloses a sparse optimization method for a cross-shaped three-dimensional imaging sonar array, which does not constrain the element spacing and uses a stochastic optimization algorithm to solve the sparse array, easily generating elements with excessively close positions. Similarly, patent document CN112182953A discloses a modular planar array sparse optimization method and system, which results in a sparse array with weak sidelobe suppression. Summary of the Invention

[0004] In view of the above, the purpose of this invention is to provide an array sparsity method and apparatus based on progressive Bayesian compressed sensing and graph fusion, so as to realize sparse design of array sparsity.

[0005] To achieve the above-mentioned objectives, this invention provides an array sparse method based on progressive Bayesian compressed sensing and graph fusion, comprising the following steps:

[0006] Step 1: Using candidate array elements as input and the ideal full array beammap as the full array reference beammap, set the hyperparameters of multi-task Bayesian compressed sensing to control the sparsity rate.

[0007] Step 2: Solve the multi-task Bayesian compressed sensing problem based on the set hyperparameters to obtain the array distribution and element weights after preliminary sparsity.

[0008] Step 3: Using the initially sparsed array elements as vertices, establish edges between vertices where the spacing between array elements is less than or equal to the constraint distance to form an undirected graph. Divide the undirected graph into multiple independent subgraphs based on its connectivity.

[0009] Step 4: Merge the vertices in each independent subgraph in parallel, that is, prioritize the vertex with the highest weighted merging degree and its nearest connected vertex, update the weight and degree of the vertices in the merged independent subgraph, iterate this step until the distance between all array elements is greater than the distance constraint, and obtain the merged array elements.

[0010] Step 5: Based on the supporting hyperplane, the sparse array after merging array elements is transformed into a global convex optimization problem, and the weights of the quadratic sparse array elements are obtained by solving the problem.

[0011] Preferably, in step 1, the ideal full-array beam pattern is a planar array with equally spaced half-wavelength arrays, and the corresponding beamforming result B is represented as:

[0012]

[0013] Where the subscript n is the array element number, N is the number of array elements, and w n For the corresponding array element weights, (x n ,y n Let λ be the position of the array element, λ be the signal wavelength, u = sinθ, v = sinφ, and u, v ∈ [-1, 1], θ and φ be the beam direction angle and elevation angle, respectively. The hyperparameters of multi-task Bayesian compressed sensing include the reciprocal α of the variance of the zero-mean Gaussian distribution of the array element weights and the reciprocal α0 of the variance of the zero-mean Gaussian distribution of the noise ε.

[0014] Preferably, in step 2, when the number of initially sparse candidate array elements is N p When multi-task joint estimation takes the form of:

[0015]

[0016] in Φ represents the real part and the imaginary part, respectively. p Indicates that the number of candidate array elements is N p The corresponding observation matrix,

[0017] Maximum posterior probability As shown in the following formula:

[0018]

[0019]

[0020] Where the subscript l denotes the [l(N] of the matrix. p -1)+1,lN p ] column, B lIndicates the beamforming of column l, C pl and A and Ψ p As an intermediate variable;

[0021] The array distribution and element weights after initial sparseness are obtained by solving the above two formulas.

[0022] Preferably, in step 3, in an undirected graph, the value of the connecting edge between vertices is the ratio of the vertex spacing to the element constraint distance, the value of a vertex is a tuple of its degree value and element weight, and the degree value is the number of edges of the vertex.

[0023] Preferably, in step 4, when merging the vertices corresponding to the array elements, the following merging conditions are followed: 1) Independent subgraphs without connectivity are processed independently; 2) In each round, the vertex with the highest priority degree and its shortest edge connected vertices are merged; 3) The vertex weights and positions are updated by weighted merging points, and the weighted calculation is as follows:

[0024]

[0025] Among them, (x new y new ) represents the updated position, w new To update the element weights, w1 and w2 are the weights of the two vertices being merged, and (x1, y1) and (x2, y2) are the positions of the two vertices being merged.

[0026] Preferably, in step 5, the sparse array after element merging is globally re-optimized. The error between the sparse array beam pattern and the full array reference beam pattern is reduced by constructing and solving a cost function. The cost function is as follows:

[0027]

[0028] in,

[0029] Preferably, in step 5, during the solution process, the cost function is transformed into a convex alternative cost function through the supporting hyperplane, so as to introduce a convex optimization method for global solution. The alternative cost function is as follows:

[0030]

[0031] in, Let f(α) denote any point in α, where the cost function f(α) is convex.

[0032] Preferably, in step 5, the CVX Box convex optimization solver of MATLAB is used to obtain the weights of the quadratic sparse array elements after global re-optimization.

[0033] To achieve the above-mentioned objectives, the embodiments also provide an array sparse device based on progressive Bayesian compressed sensing and graph fusion, including a parameter setting module, a preliminary sparse module, an undirected graph construction module, a vertex merging module, and a secondary sparse module.

[0034] The parameter setting module is used to set the hyperparameters of multi-task Bayesian compressed sensing to control the sparsity rate, with candidate array elements as input and the ideal full array beammap as the full array reference beammap.

[0035] The preliminary sparsity module is used to solve the multi-task Bayesian compressed sensing problem based on the set hyperparameters, and to obtain the array distribution and array element weights after preliminary sparsity.

[0036] The undirected graph construction module is used to form an undirected graph by taking the initially sparsed array elements as vertices and establishing edges between vertices whose array element spacing is less than or equal to the constraint distance. The undirected graph is then divided into multiple independent subgraphs based on its connectivity.

[0037] The vertex merging module is used to merge vertices in each independent subgraph in parallel, that is, to prioritize the vertices with the highest weighted merging degree and their nearest connected vertices, update the weights and degrees of the vertices in the merged independent subgraphs, and iterate this step until the spacing between all array elements is greater than the distance constraint, thus obtaining the merged array elements.

[0038] The quadratic sparse module is used to transform the sparse array after merging array elements into a global convex optimization problem based on the supporting hyperplane, and to solve for the weights of the quadratic sparse array elements.

[0039] Compared with the prior art, the beneficial effects of the present invention include at least the following:

[0040] This invention provides a method and apparatus for sparse array synthesis based on progressive Bayesian compressed sensing and graph fusion. It proposes a novel processing architecture for sparse array synthesis, including fast gridded sparsification, undirected graph-based element weighted merging, and global BCS element weight re-optimization. The undirected graph-based element weighted merging method ensures that array elements satisfy the minimum spacing constraint, and a novel merging strategy reduces the number of merging operations. A globally optimized element weight re-estimation method improves the beammap performance of the sparse array. Therefore, the proposed method achieves good sparsity and beammap performance while satisfying array element spacing constraints, making it suitable for sparse array design. Attached Figure Description

[0041] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0042] Figure 1 A flowchart illustrating the array sparsity method based on progressive Bayesian compressed sensing and graph fusion provided for an embodiment;

[0043] Figure 2 for Figure 2 A schematic diagram of a planar array signal model provided for an embodiment;

[0044] Figure 3 For reference, a top-down comparison of the beam diagrams of a full array and a sparse array is provided for the implementation example.

[0045] Figure 4 For the example, a horizontal slice comparison of the beam patterns of full array and sparse array is shown.

[0046] Figure 5 The array element distribution and normalized weight diagram of the sparse array are shown in the example.

[0047] Figure 6 This is a schematic diagram of the array sparse device based on progressive Bayesian compressed sensing and graph fusion provided in the embodiment. Detailed Implementation

[0048] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and do not limit the scope of protection of this invention.

[0049] To address the issues of low sparsity and excessively close element spacing in traditional array sparsity methods, this embodiment provides an array sparsity method and apparatus based on progressive Bayesian compressed sensing and graph fusion, which achieves high beam performance and high sparsity sparse array generation while satisfying element spacing constraints.

[0050] like Figure 1 As shown in the embodiment, the array sparse method based on progressive Bayesian compressed sensing and graph fusion includes the following steps:

[0051] S1 takes candidate array elements as input and the ideal full array beammap as the full array reference beammap to set the hyperparameters of multi-task Bayesian compressed sensing to control the sparsity rate.

[0052] The embodiment provides an ideal full array, taking a planar array with equal spacing and half-wavelength arrangement as an example. The beamforming uses the following azimuth and elevation angle models: Figure 2 As shown. For an array containing N elements, with beam azimuth and elevation angles of θ and φ respectively, the corresponding beamforming result can be expressed as:

[0053]

[0054] Where the subscript n is the array element number, N is the number of array elements, and w n For the corresponding array element weights, (x n ,y n Let λ be the position of the array element, λ be the signal wavelength, u = sinθ, v = sinφ, and u, v ∈ [-1, 1], θ and φ be the beam direction angle and elevation angle, respectively. The hyperparameters of multi-task Bayesian compressed sensing include the reciprocal α of the variance of the zero-mean Gaussian distribution of the array element weights and the reciprocal α0 of the variance of the zero-mean Gaussian distribution of the noise ε.

[0055] S2 solves the multi-task Bayesian compressed sensing problem based on the set hyperparameters, and obtains the array distribution and array element weights after preliminary sparsity.

[0056] In the embodiment, the array sparsity is controlled by hyperparameters α and α0, which are the element weights w, respectively. p The reciprocal of the variance of the zero-mean Gaussian distribution of the noise ε, i.e.

[0057] When the initial sparse candidate array element number is N p When multi-task joint estimation takes the form of:

[0058]

[0059] in Φ represents the real part and the imaginary part, respectively. p Indicates that the number of candidate array elements is N p The corresponding observation matrix;

[0060] Maximum posterior probability As shown in the following formula:

[0061]

[0062]

[0063] Where the subscript l denotes the [l(N] of the matrix. p -1)+1,lN p ] column, B l Indicates the beamforming of column l, C pl and A and Ψ p As an intermediate variable;

[0064] The array distribution and element weights after initial sparseness are obtained by solving the above two formulas.

[0065] S3. Using the initially sparse array elements as vertices, and establishing edges between vertices where the spacing between array elements is less than or equal to the constraint distance, an undirected graph is formed. Based on the connectivity of the undirected graph, the undirected graph is divided into multiple independent subgraphs.

[0066] In this embodiment, a minimum element spacing constraint is imposed on the sparse array to facilitate actual fabrication and subsequent maintenance. An element weighting merging method based on an undirected graph is used to ensure that the element spacing meets the constraint.

[0067] Specifically, using the initially sparsed array elements as vertices, connected edges are established between all vertices where the distance between array elements is less than the constraint distance. The value of the connected edge is the ratio of the vertex distance to the array element constraint distance, and the value of a vertex is a tuple of its degree (number of edges in the vertex) and weight. This constructs an undirected graph of the array elements. After obtaining the undirected graph, it is divided into multiple independent subgraphs based on its connectivity, with each independent subgraph having connected vertices.

[0068] S4. Merge the vertices in each independent subgraph in parallel, that is, prioritize the vertices with the highest weighted merging degree and their nearest connected vertices, update the weights and degrees of the vertices in the merged independent subgraphs, iterate this step until the spacing between all array elements is greater than the distance constraint, and obtain the merged array elements.

[0069] In this embodiment, when merging vertices corresponding to array elements, the following merging conditions are followed: 1) Independent subgraphs without connectivity are processed independently; 2) In each round, the vertex with the highest priority degree and its shortest edge connected vertices are merged; 3) Vertex weights and positions are updated by weighted merging of the merging points, and the weighted calculation is as follows:

[0070]

[0071] Among them, (x new y new ) represents the updated position, w new To update the element weights, w1 and w2 are the weights of the two vertices being merged, and (x1, y1) and (x2, y2) are the positions of the two vertices being merged.

[0072] During the iterative merging process, until all array elements are no longer connected, i.e., all elements satisfy the distance constraint between them being less than the distance distance, we obtain N. s The merged array elements.

[0073] S5 transforms the sparse array after merging array elements into a global convex optimization problem based on the supporting hyperplane, and obtains the weights of the quadratic sparse array elements by solving the problem.

[0074] In this embodiment, the sparse array after element merging is globally re-optimized to improve array beam performance. The error between the sparse array beam pattern and the full array reference beam pattern is reduced by constructing and solving a cost function. The cost function is as follows:

[0075]

[0076] in,

[0077] During the solution process, the cost function is transformed into a convex alternative cost function through the supporting hyperplane, thereby introducing a convex optimization method for global solution. The alternative cost function is as follows:

[0078]

[0079] in, Let f(α) denote any point in α, where the cost function f(α) is convex. In this embodiment, the CVX Box convex optimization solver in MATLAB is used to obtain the globally re-optimized sparse element weights.

[0080] To demonstrate the performance of this method in terms of array sparsity and beammap, the embodiment uses a 40×40 sonar element planar array with equal spacing at half-wavelengths for comparison with other array sparsity methods. The sidelobe suppression index is set to -35dB. The comparison methods include BCS and the MT-BCS (S-MT-BCS) method considering element spacing constraints. The reference beammap of the standard full array and the beammap of the sparse array obtained by this method are shown below. Figure 3 The corresponding beam slice comparison is shown in [link to comparison]. Figure 4 The positions and weights of the array elements after sparsening are shown in [link to relevant documentation]. Figure 5 As can be seen from the results of the embodiment, the sparse array obtained by this method has beam performance that is basically consistent with the ideal effect, the sidelobes are lower than the constraint value, and the array elements are significantly sparser than the original array.

[0081] The key indicators of sparse arrays, such as the number of array elements, sparsity, mean square error (MSE) compared with the standard beammap, peak sidelobe suppression ratio (PSR), and minimum element spacing, were statistically compared. The comparison results are shown in the table below.

[0082] Table 1

[0083] method Reference to the whole formation BCS S-MT-BCS This method Number of array elements 1600 1024 734 683 sparsity 1 0.64 0.46 0.43 MSE / <![CDATA[1.7×10 -2 ]]> <![CDATA[3.0×10 -2 ]]> <![CDATA[5.5×10 -4 ]]> PSR(dB) -35 -35 -34.7 -35.7 Minimum element spacing (λ) 0.5 0.246 0.500 0.500

[0084] As can be seen from the table, the main indicators of this method are all better than those of the comparison method, and the array sparsity, manufacturability and array beammap performance are significantly improved.

[0085] like Figure 6As shown, the embodiment also provides an array sparsity device 60 based on progressive Bayesian compressed sensing and graph fusion, including a parameter setting module 61, a preliminary sparsity module 62, an undirected graph construction module 63, a vertex merging module 64, and a secondary sparsity module 65; wherein, the parameter setting module 61 is used to set the hyperparameters of multi-task Bayesian compressed sensing to control the sparsity rate by taking candidate array elements as input and an ideal full array beammap as the full array reference beammap; the preliminary sparsity module 62 is used to solve the multi-task Bayesian compressed sensing problem based on the set hyperparameters to obtain the array distribution and element weights after preliminary sparsity; the undirected graph construction module 63... The module 64 is used to form an undirected graph by using the initially sparse array elements as vertices and establishing edges between vertices whose array element spacing is less than or equal to the constraint distance. Based on the connectivity of the undirected graph, it divides the undirected graph into multiple independent subgraphs. The vertex merging module 64 is used to merge the vertices in each independent subgraph in parallel, that is, to prioritize merging the vertex with the highest weighted merging degree and its nearest connected vertex, update the weights and degrees of the vertices in the merged independent subgraph, and iterate this step until the spacing between all array elements is greater than the distance constraint, thus obtaining the merged array elements. The quadratic sparse module 65 is used to transform the sparse array after array element merging into a global convex optimization problem based on the supporting hyperplane, and solve for the weights of the quadratic sparse array elements.

[0086] It should be noted that the array sparse device based on progressive Bayesian compressed sensing and graph fusion provided in the above embodiments should be illustrated using the above-described division of functional modules as an example when performing array sparse. The functions can be assigned to different functional modules as needed, that is, the internal structure of the terminal or server can be divided into different functional modules to complete all or part of the functions described above. Furthermore, the array sparse device based on progressive Bayesian compressed sensing and graph fusion provided in the above embodiments and the array sparse method embodiment based on progressive Bayesian compressed sensing and graph fusion belong to the same concept. For details of its implementation process, please refer to the array sparse method embodiment based on progressive Bayesian compressed sensing and graph fusion, which will not be repeated here.

[0087] The specific embodiments described above illustrate the technical solution and beneficial effects of the present invention in detail. It should be understood that the above description is only the most preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, additions, and equivalent substitutions made within the scope of the principles of the present invention should be included within the protection scope of the present invention.

Claims

1. An array sparse method based on progressive Bayesian compressed sensing and graph fusion, characterized in that, Includes the following steps: Step 1: Using candidate array elements as input and the ideal full array beammap as the full array reference beammap, set the hyperparameters of multi-task Bayesian compressed sensing to control the sparsity rate. Step 2: Solve the multi-task Bayesian compressed sensing problem based on the set hyperparameters to obtain the initially sparse array distribution and element weights. When the number of candidate elements in the initial sparse distribution is... When multi-task joint estimation takes the form of: in These represent the real part and the imaginary part, respectively. Indicates the number of candidate array elements. The corresponding observation matrix, For noise, As the array element weight; Step 3: Using the initially sparsed array elements as vertices, establish edges between vertices where the spacing between array elements is less than or equal to the constraint distance to form an undirected graph. Divide the undirected graph into multiple independent subgraphs based on its connectivity. Step 4: Merge vertices in each independent subgraph in parallel, that is, prioritize merging the vertex with the highest weighted degree and its nearest connected vertex, update the weights and degrees of the vertices in the merged independent subgraph, and iterate this step until the distance between all array elements is greater than the distance constraint, thus obtaining the merged array elements. When merging the vertices corresponding to the array elements, the following merging conditions are followed: 1) Independent subgraphs without connections are processed independently; 2) In each round, prioritize merging the vertex with the highest degree and its shortest edge connected vertex; 3) Vertex weights and positions are updated by weighting the merged points, and the weighting calculation is as follows: in,( , () indicates the updated position. To update the array element weights, and The weights of the two vertices being merged, ( , )and( , ) represents the positions of the two vertices to be merged; Step 5: Based on the supporting hyperplane, the sparse array after merging array elements is transformed into a global convex optimization problem, and the weights of the quadratic sparse array elements are obtained by solving the problem. During the solution process, the cost function is transformed into a convex alternative cost function through the supporting hyperplane, so as to introduce a convex optimization method for global solution. The alternative cost function is as follows: in, ; , express At any point in the interval, the cost function is... It is convex. , These are hyperparameters that control the sparsity of the array. Represents column Beamforming, subscript Representing a matrix List.

2. The array sparse method based on progressive Bayesian compressed sensing and graph fusion according to claim 1, characterized in that, In step 1, the ideal full-array beam pattern is a planar array with equally spaced half-wavelength arrays, and the corresponding beamforming result is... B Represented as: Among them, subscript Number the array elements. N The number of array elements. For the corresponding array element weights, For the position of the array element, For the signal wavelength, ,and , and These are the beam direction angle and elevation angle, respectively. The hyperparameters of multi-task Bayesian compressed sensing include the reciprocal of the variance of the zero-mean Gaussian distribution of the element weights. ,noise The reciprocal of the variance of a zero-mean Gaussian distribution .

3. The array sparse method based on progressive Bayesian compressed sensing and graph fusion according to claim 2, characterized in that, In step 2, the maximum posterior probability As shown in the following formula: in, and as well as As an intermediate variable; Solve to obtain the array distribution and element weights after the initial sparse distribution.

4. The array sparse method based on progressive Bayesian compressed sensing and graph fusion according to claim 1, characterized in that, In step 3, in an undirected graph, the value of the connecting edge between vertices is the ratio of the vertex spacing to the element constraint distance, the value of a vertex is a pair of its degree value and element weight, and the degree value is the number of edges of the vertex.

5. The array sparse method based on progressive Bayesian compressed sensing and graph fusion according to claim 2, characterized in that, In step 5, the sparse array after element merging is globally re-optimized. The error between the sparse array beam pattern and the full array reference beam pattern is reduced by constructing and solving a cost function. The cost function is as follows: in, .

6. The array sparse method based on progressive Bayesian compressed sensing and graph fusion according to claim 1, characterized in that, In step 5, the CVX Box convex optimization solver in MATLAB is used to obtain the weights of the quadratic sparse array elements after global re-optimization.

7. An array sparse device based on progressive Bayesian compressed sensing and graph fusion, characterized in that, It includes a parameter setting module, a preliminary sparsity module, an undirected graph construction module, a vertex merging module, and a secondary sparsity module; The parameter setting module is used to set the hyperparameters of multi-task Bayesian compressed sensing to control the sparsity rate, with candidate array elements as input and the ideal full array beammap as the full array reference beammap. The preliminary sparsity module is used to solve the multi-task Bayesian compressed sensing problem based on set hyperparameters, obtaining the array distribution and element weights after preliminary sparsity. When the number of candidate elements in the preliminary sparsity is... When multi-task joint estimation takes the form of: in These represent the real part and the imaginary part, respectively. Indicates the number of candidate array elements. The corresponding observation matrix, For noise, As the array element weight; The undirected graph construction module is used to form an undirected graph by taking the initially sparsed array elements as vertices and establishing edges between vertices whose array element spacing is less than or equal to the constraint distance. The undirected graph is then divided into multiple independent subgraphs based on its connectivity. The vertex merging module is used to merge vertices in each independent subgraph in parallel. Specifically, it prioritizes merging the vertex with the highest weighted degree and its nearest connected vertex, updates the weights and degrees of the vertices in the merged independent subgraph, and iterates this step until the distance between all elements is greater than the distance constraint, resulting in the merged element. When merging vertices corresponding to element elements, the following merging conditions are followed: 1) Independent subgraphs without connections are processed independently; 2) In each round, the vertex with the highest degree and its shortest edge connected vertex are merged first; 3) Vertex weights and positions are updated by weighting the merged points, and the weighting calculation is as follows: in,( , () indicates the updated position. To update the array element weights, and The weights of the two vertices being merged, ( , )and( , ) represents the positions of the two vertices to be merged; The quadratic sparse module is used to transform the sparse array after element merging into a global convex optimization problem based on the supporting hyperplane, and to obtain the weights of the quadratic sparse array elements. During the solution process, the cost function is transformed into a convex alternative cost function through the supporting hyperplane, so as to introduce a convex optimization method for global solution. The alternative cost function is as follows: in, ; , express At any point in the interval, the cost function is... It is convex. , These are hyperparameters that control the sparsity of the array. Represents column Beamforming, subscript Representing a matrix List.

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