Sparse array beam reconfigurable design method based on minimum spacing constraint

Abstract

The invention discloses a sparse array beam reconfigurable design method based on minimum spacing constraint. According to the method, firstly, an off-grid group sparse reconstruction model of multiple directional diagrams is established at a half-wavelength interval for an array with a given aperture length, then an inequality constraint condition for minimum adjacent array element interval control is further introduced into the group sparse reconstruction model, and a 10 problem of a joint performance function is converted into a 10 / 11 combined optimization problem. The invention provides an SAI (Sequential Alternating Iteration) algorithm for solving an excitation vector and a position of each array element of a sparse array. Simulation experiments show that the method can effectively realize sparse array antenna pattern reconstruction, and can control the minimum spacing between adjacent array elements, and the designed sparse array is easy for engineering realization.

Description

Technical Field

[0001] The invention belongs to the technical field of array antenna design, and in particular relates to a sparse array beam reconfigurable design method based on minimum spacing constraints. Background Art

[0002] In modern radar and wireless communication electronic systems, array antennas have been widely used to improve antenna gain and beam steering capabilities. Sparse arrays can use fewer array elements to meet the antenna pattern performance when the array is full under certain conditions, which can reduce system complexity, simplify antenna structure, and save costs. Therefore, in recent years, sparse array synthesis has become a widely studied topic in the radar field.

[0003] At present, some sparse array synthesis algorithms have been proposed. Convex optimization, orthogonal matching pursuit and Bayesian algorithms for sparse reconstruction under a single directional graph, as well as sparse array synthesis algorithms based on multi-directional graphs, such as multi-task Bayesian compressed sensing, joint sparse recovery and M-FOCUSS algorithms. In order to improve accuracy, most methods require fine grid division before array synthesis, but this will result in a large amount of calculation. Therefore, off-grid technology is introduced to maintain high accuracy under coarse grids. However, the existing off-grid methods usually integrate the excitation values ​​of different energies in multi-directional graph synthesis, which will lead to large errors. In addition, in actual engineering, in order to reduce the mutual coupling effect between array elements and reduce the mutual interference between elements inside the array, the spacing between array elements should not be too large. And the smaller array element spacing will make the array element components difficult to process, which is not conducive to engineering applications.

[0004] Therefore, the present invention proposes a sparse array beam reconfigurable design method based on minimum spacing constraint. Firstly, a sparse array based on off-grid sparse reconstruction is modeled, and then the minimum array element spacing constraint is introduced into the model to improve the original optimization problem. Finally, the S Al algorithm is proposed to obtain the desired antenna radiation pattern and control the minimum array element spacing by alternately solving the excitation vector and the array element position offset. Summary of the invention

[0005] The technical problem to be solved by the present invention is to provide a sparse array beam reconfigurable design method based on minimum spacing constraints, and introduce the minimum spacing constraint into the sparse array design of multiple directional patterns. Compared with the existing algorithms, the sequential alternating iterative algorithm proposed by the present invention has two advantages: first, it can effectively control the array element spacing, avoiding the phenomenon of too small array element spacing; second, it can improve the sparsity of the sparse array and obtain a high-precision desired directional pattern.

[0006] The present invention adopts the following technical solutions to solve the above technical problems:

[0007] Step 1: Establish a sparse reconstruction model of the off-grid group of multi-directional images, specifically:

[0008] Consider a linear array with an aperture length of D = (M-1)Δd, M uniformly distributed candidate array elements, and a spacing of Δd = 0.1λ between array elements, where λ is the wavelength of the received signal. Consider a sparse array design with L directional patterns, the directional pattern of the lth beam can be expressed as

[0009]

[0010] Among them, θ∈[-90°, 90°] is the observation angle of the pattern, is the excitation of the mth array element of the lth desired pattern, d m =(m-1)Δd(m=1,2,...,M) is the position of the mth array element. Discretizing the above equation, it can be written in the following matrix form:

[0011] F l =AW l

[0012] In the formula, F l =[f l (θ1), f l (θ2),…,f l (θ J )] T , f l (θ j ) is the lth directional pattern at θ j The sampling values ​​at the angle (j=1, 2, ..., J) are sampled, and the directional pattern is sampled at [-90°, 90°] with an interval of 5°, that is, J=37. A is a J×M dimensional complex matrix whose element values ​​are Then the group sparse reconstruction model of the multi-directional graph is

[0013]

[0014] in, represents the number of non-zero rows in W, F = [F 1 , F 2 , …, F L ],W=[W 1 , W 2 , …, W L ], is the reference pattern sampling, and ε represents the error.

[0015] The optimal array element position of the sparse array is not at the M candidate array element position points, and there is a certain offset. Now assume that in a sparse array with an aperture of D, the optimal candidate position of the sparse array element is The first-order Taylor expansion of the direction vector of the mth array element is:

[0016]

[0017] In the formula, b(d m ) is a(d m ) About d m The derivative of . Let A = [a(d1), a(d2), …, a(d M )], B=[b(d1), b(d2),…, b(d M )], then the matrix of the reconfigurable antenna pattern is expressed as F = [A + Bdiag (δ)] W. The mathematical model of off-grid group sparse reconstruction is

[0018]

[0019] Where diag means creating a diagonal matrix, δ = [δ1, δ2, …, δ M ] T , Indicates the offset between the sparse array element position and the candidate array element position.

[0020] Step 2: Introduce the minimum array element spacing constraint into the group sparse reconstruction model, specifically:

[0021] The present invention introduces the minimum array element spacing constraint into the group sparse reconstruction model, and expresses the reconfigurable synthesis problem with spacing constraints as the following l0 norm problem:

[0022]

[0023] Where c = [c1, c2, ..., c M ] T , each element in c is the bi-norm of each row of the L group of excitation solutions W, that is, Q is a positive integer. By changing the value of Q, the minimum spacing between array elements can be adjusted. Q = 5 can be selected to meet the minimum array element spacing d min Greater than or equal to λ / 2.

[0024] Step 3: Improve the l0 optimization problem of the performance function into a l0 / l1 combined optimization problem, specifically:

[0025] The method introduces two weighting coefficients and g, and are alternately selected as optimization variables with L groups of excitation vectors W and the offset δ of the array element spacing. At this time, the synthesis of the reconfigurable multi-directional pattern array with minimum spacing constraints can be expressed as alternately solving the l0 / l1 combination optimization problem:

[0026]

[0027] in, and g are weighted vectors, ⊙ represents the Hadamard product, g = [g1, g2, ..., g M ] T , Represents the XOR logical operation, where 1 is a vector of all 1s with the same dimension as g.

[0028] Problems (a) and (b) are both weighted norm minimization problems, and W and δ are alternately selected as optimization variables. In problem (a), Given by the previous iterative solution, W and δ are given in In problem (b), c is the solution of problem (a), and g is the optimization variable to be solved under given c. The constraints of problem (b) make the weight coefficient g t In each Q connected elements, at most one element has a value of "0", and the rest are all "1". Return to the next iteration. After XOR operation, the weighted coefficient In each Q connected elements, at most one element has a value of "1", and the rest are "0". Make When the value of the element in is "1", the corresponding element in c is retained; When the value of an element in is "0", the corresponding element in c will be heavily penalized. Eventually, in c, at most one element out of every Q connected elements has a value, and the rest are penalized to "0". The element values ​​in c are assigned to L groups of incentives W through the row binorm l The corresponding position makes the minimum spacing constraint applied to W l superior.

[0029] Step 4: The M-FOCUSS algorithm solves the excitation vector, specifically:

[0030] In step 3, W and δ are both optimization variables to be solved. The present invention studies a sequential alternating iterative method to solve the optimization problem. The method alternately iterates between problem (a) and problem (b) to obtain the optimal array element position and excitation.

[0031] When solving problem (a), let The initial values ​​of and g are all 1. Solve W using the M-FOCUSS algorithm. The solution process of the M-FOCUSS algorithm at this time is

[0032]

[0033] Wherein, subscript k represents the number of iterations of the M-FOCUSS algorithm, γ = 0.6 is the sparsity factor, and λ = 0.6 is the regularization parameter. The solution of W is obtained by the M-FOCUSS algorithm, and the excitation of each directional pattern of the corresponding array element is obtained.

[0034] Step 5: Convex optimization is used to solve the position offset of the array element, specifically:

[0035] After obtaining W, we need to solve the offset δ of the array element position. At this time, problem (a) can be equivalent to a least squares problem, that is, solving

[0036] BΛ l δ=E l

[0037] Among them, Λ l =diag(Wl), In order to ensure the real value property of δ, first decompose the complex quantity in the above formula into real and imaginary parts, that is,

[0038]

[0039] In the formula, B R , B I , and B, Λ l and E l The real and imaginary parts of . The above formula becomes For the sparse reconstruction of arrays with L directional patterns, that is, for L groups of different equations All have the same solution δ. The mathematical model for convex optimization to solve δ is

[0040]

[0041] The present invention uses the CVX toolbox to solve, and the δ obtained by the solution is the corresponding offset of the array element. L groups of directional patterns obtain a common array element offset δ, and the position of the array element is d m +δ m Repeat steps 4 and 5 until the integrated pattern reaches the set accuracy error.

[0042] Compared with the prior art, the present invention has the following technical effects:

[0043] 1. Introducing the minimum spacing constraint into the sparse array beam reconfigurable design effectively controls the array element spacing, optimizes the array element arrangement, avoids difficulties in the component processing process, and is conducive to engineering implementation.

[0044] 2. The simulation experimental results show that the sparse array beam reconfigurable design method based on minimum spacing constraints can improve the sparsity of the sparse array and achieve a high-precision desired radiation pattern. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 A schematic diagram of a sparse array

[0046] Figure 2 The sparse array beam reconfigurable design flow chart of the present invention

[0047] Figure 3 Convergence curve of this method

[0048] Figure 4 Antenna pattern synthesized by the two methods

[0049] Figure 5 The array element position diagram synthesized by the two methods DETAILED DESCRIPTION

[0050] The technical solution of the present invention is described in detail below with reference to the accompanying drawings:

[0051] Figure 1 A schematic diagram of sparse array synthesis is shown. Figure 2 The sparse array design process of the present invention is presented, and its main steps are as follows:

[0052] Consider a linear array with an aperture length of D = (M-1)Δd, M uniformly distributed candidate array elements, and a spacing of Δd = 0.1λ between array elements, where λ is the wavelength of the received signal. Consider a sparse array design with L directional patterns, the directional pattern of the lth beam can be expressed as

[0053]

[0054] Among them, θ∈[-90°, 90°] is the observation angle of the pattern, is the excitation of the mth array element of the lth desired pattern, d m =(m-1)Δd(m=1,2,...,M) is the position of the mth array element. Discretizing the above equation, it can be written in the following matrix form:

[0055] Fl=AW l

[0056] In the formula, F l =[f l (θ1), f l (θ2),…,f l (θ J )] T , f l (θ j) is the lth directional pattern at θ j The sampling values ​​at the angle (j=1, 2, ..., J) are sampled, and the directional pattern is sampled at [-90°, 90°] with an interval of 5°, that is, J=37. A is a J×M dimensional complex matrix whose element values ​​are Then the group sparse reconstruction model of the multi-directional graph is

[0057]

[0058] in, represents the number of non-zero rows in W, F = [F 1 , F 2 , …, F L ],W=[W 1 , W 2 , …, W L ], is the reference pattern sampling, and ε represents the error.

[0059] The optimal array element position of the sparse array is not at the M candidate array element position points, and there is a certain offset. Now assume that in a sparse array with an aperture of D, the optimal candidate position of the sparse array element is The first-order Taylor expansion of the direction vector of the mth array element is:

[0060]

[0061] In the formula, b(d m ) is a(d m ) About d m The derivative of . Let A = [a(d1), a(d2), …, a(d M )], B=[b(d1), b(d2),…, b(d M )], then the matrix of the reconfigurable antenna pattern is expressed as F = [A + Bdiag (δ)] W. The mathematical model of off-grid group sparse reconstruction is

[0062]

[0063] Where diag means creating a diagonal matrix, δ = [δ1, δ2, …, δ M ] T , Indicates the offset between the sparse array element position and the candidate array element position.

[0064] Step 2: Introduce the minimum array element spacing constraint into the group sparse reconstruction model, specifically:

[0065] The present invention introduces the minimum array element spacing constraint into the group sparse reconstruction model, and expresses the reconfigurable synthesis problem with spacing constraints as the following l0 norm problem:

[0066]

[0067] Where c = [c1, c2, ..., c M ] T , each element in c is the bi-norm of each row of the L group of excitation solutions W, that is, Q is a positive integer. By changing the value of Q, the minimum spacing between array elements can be adjusted. Q = 5 can be selected to meet the minimum array element spacing d min Greater than or equal to λ / 2.

[0068] Step 3: Improve the l0 optimization problem of the performance function into a l0 / l1 combined optimization problem, specifically:

[0069] The method introduces two weighting coefficients and g, and are alternately selected as optimization variables with L groups of excitation vectors W and the offset δ of the array element spacing. At this time, the synthesis of the reconfigurable multi-directional pattern array with minimum spacing constraints can be expressed as alternately solving the l0 / l1 combination optimization problem:

[0070]

[0071] in, and g are weighted vectors, ⊙ represents the Hadamard product, g = [g1, g2, ..., g M ] T , Represents the XOR logical operation, where 1 is a vector of all 1s with the same dimension as g.

[0072] Problems (a) and (b) are both weighted norm minimization problems, and W and δ are alternately selected as optimization variables. In problem (a), Given by the previous iterative solution, W and δ are given in In problem (b), c is the solution of problem (a), and g is the optimization variable to be solved under given c. The constraints of problem (b) make the weight coefficient g t In each Q connected elements, at most one element has a value of "0", and the rest are all "1". Return to the next iteration. After XOR operation, the weighted coefficient In each Q connected elements, at most one element has a value of "1", and the rest are "0". Make When the value of the element in is "1", the corresponding element in c is retained; When the value of an element in is "0", the corresponding element in c will be heavily penalized. Eventually, in c, at most one element out of every Q connected elements has a value, and the rest are penalized to "0". The element values ​​in c are assigned to L groups of incentives W through the row binorm l The corresponding position makes the minimum spacing constraint applied to W l superior.

[0073] Step 4: The M-FOCUSS algorithm solves the excitation vector, specifically:

[0074] In step 3, W and δ are both optimization variables to be solved. The present invention studies a sequential alternating iterative method to solve the optimization problem. The method alternately iterates between problem (a) and problem (b) to obtain the optimal array element position and excitation.

[0075] When solving problem (a), let The initial values ​​of and g are all 1. Solve W using the M-FOCUSS algorithm. The solution process of the M-FOCUSS algorithm at this time is

[0076]

[0077] Wherein, subscript k represents the number of iterations of the M-FOCUSS algorithm, γ = 0.6 is the sparsity factor, and λ = 0.6 is the regularization parameter. The solution of W is obtained by the M-FOCUSS algorithm, and the excitation of each directional pattern of the corresponding array element is obtained.

[0078] Step 5: Convex optimization is used to solve the position offset of the array element, specifically:

[0079] After obtaining w, we need to solve the offset δ of the array element position. At this time, problem (a) can be equivalent to a least squares problem, that is, solving

[0080] BΛ l δ=E l

[0081] Among them, Λ l =diag(W l ), In order to ensure the real value property of δ, first decompose the complex quantity in the above formula into real and imaginary parts, that is,

[0082]

[0083] In the formula, B R , B I , and B, Λ l and E l The real and imaginary parts of . The above formula becomes For the sparse reconstruction of arrays with L directional patterns, that is, for L groups of different equations All have the same solution δ. The mathematical model for convex optimization to solve δ is

[0084]

[0085] The present invention uses the CVX toolbox to solve, and the δ obtained by the solution is the corresponding offset of the array element. L groups of directional patterns obtain a common array element offset δ, and the position of the array element is d m +δ m Repeat steps 4 and 5 until the integrated pattern reaches the set accuracy error.

[0086] The effectiveness of the method of the present invention is verified by computer simulation. The Chebyshev pattern with a sidelobe of -20dB is selected as the reference pattern for multi-pattern synthesis, and the method without the minimum spacing constraint is used as the experimental comparison algorithm. In this simulation experiment, an array consisting of 20 array elements with an interval of λ / 2 is considered, and the array aperture is D=9.5λ. The array is used for beam scanning, and the scanning angle is [-10°, 0°, 10°]. The array of this aperture is now gridded, and the spacing between the virtual array elements is Δd=0.1λ. The reference pattern is spatially sampled at intervals of 5°, and the sampling range is [-90°, 90°], that is, J=37. The minimum array element spacing is set to d min =0.5λ, that is, Q = 5. In order to measure the accuracy of this method, the standard mean square error NMSE error and sparsity are used as the measurement criteria:

[0087] (1) The standard mean square error NMSE error is defined as

[0088]

[0089] (2) The number of array elements in a uniform full array is N, and the number of array elements after sparse array is P. Then the sparsity is defined as

[0090]

[0091] Figure 3 The convergence curve of the accuracy of the SAI algorithm as the number of iterations changes. As can be seen from the figure, the error of the algorithm is reduced to a very low level after only one iteration. And as the number of iterations increases, the NMSE obtained by the SAI method tends to be stable and unchanged. The SAI method converges quickly and has good convergence performance. Figure 4The antenna patterns synthesized by the two algorithms are shown in Figure 2. It can be seen that the pattern synthesized by the comparison algorithm is poor, the pattern is distorted and the sidelobe level is increased, which will have a greater impact on the array performance. The pattern obtained by the SAI method has a better fit with the reference pattern, the main lobe area is basically the same, and the sidelobe level is not significantly increased.

[0092] Figure 5 The following is a comparison of the positions of the array elements obtained by the two methods. It can be seen that some of the array elements formed by the comparison algorithm are too dense, and the number of sparse array elements synthesized is 18, with a sparsity of 10%. The spacing between some array elements is less than half a wavelength, and the minimum array element spacing is 0.0854λ. Such a small array element spacing poses a challenge to practical applications in engineering. The sparse array cannot meet the minimum spacing constraint, resulting in difficulty in processing array element components and increased coupling between array elements. The elements formed by the SAI method are more evenly distributed, with 14 array elements after sparseness, and the array sparsity is increased to 30%. The position arrangement of the array elements is also optimized, with a minimum array element spacing of 0.7059λ. The results show that the sparse array synthesized by the SAI method has a constraint of minimum unit spacing and can be used in practical engineering applications.

Claims

1. A sparse array beam reconfigurable design method based on minimum spacing constraints, characterized by The steps include: Step 1: Establish a sparse reconstruction model of the off-grid group of multi-directional images, specifically: Consider a linear array with an aperture length of D = (M-1)Δd, M uniformly distributed candidate array elements, and a spacing of Δd = 0.1λ between array elements, where λ is the wavelength of the received signal. Consider a sparse array design with L directional patterns, the directional pattern of the lth beam can be expressed as Among them, θ∈[-90°, 90°] is the observation angle of the pattern, is the excitation of the mth array element of the lth desired pattern, d m =(m-1)Δd(m=1,2,...,M) is the position of the mth array element. Discretizing the above equation, it can be written in the following matrix form: F l =AW l In the formula, F l =[f l (θ1), f l (θ2),…,f l (θ J )] T , f l (θ j ) is the lth directional pattern at θ j The sampling values ​​at the angle (j=1, 2, ..., J) are sampled, and the directional pattern is sampled at [-90°, 90°] with an interval of 5°, that is, J=37. A is a J×M dimensional complex matrix whose element values ​​are Then the group sparse reconstruction model of the multi-directional graph is in, represents the number of non-zero rows in W, F = [F 1 , F 2 , …, F L ],W=[W 1 , W 2 , …, W L ], is the reference pattern sampling, and ε represents the error. The best array element position of the sparse array is not at the M candidate array element position points, and there is a certain offset. Now assume that in a sparse array with an aperture of D, the optimal candidate position of the sparse array element is The first-order Taylor expansion of the direction vector of the mth array element is: In the formula, b(d m ) is a(d m ) About d m The derivative of . Let A = [a(d1), a(d2), …, a(d M )], B=[b(d1), b(d2),…, b(d M )], then the matrix of the reconfigurable antenna pattern is expressed as F = [A + Bdiag (δ)] W. The mathematical model of off-grid group sparse reconstruction is in, diag means to create a diagonal matrix, δ = [δ1, δ2, …, δ M ] T , Indicates the offset between the sparse array element position and the candidate array element position. Step 2: Introduce the minimum array element spacing constraint into the group sparse reconstruction model, specifically: The present invention introduces the minimum array element spacing constraint into the group sparse reconstruction model, and expresses the reconfigurable synthesis problem with spacing constraints as the following l0 norm problem: Where c = [c1, c2, ..., c M ] T , each element in c is the bi-norm of each row of the L group of excitation solutions W, that is, Q is a positive integer. By changing the value of Q, the minimum spacing between array elements can be adjusted. Q = 5 can be selected to meet the minimum array element spacing d min Greater than or equal to λ / 2. Step 3: Improve the l0 optimization problem of the performance function into a l0 / l1 combined optimization problem, specifically: The method introduces two weighting coefficients and g, and are alternately selected as optimization variables with L groups of excitation vectors W and the offset δ of the array element spacing. At this time, the synthesis of the reconfigurable multi-directional pattern array with minimum spacing constraints can be expressed as alternately solving the l0 / l1 combination optimization problem: in, and g are weighted vectors, ⊙ represents the Hadamard product, g = [g1, g2, ..., g M ] T , Represents the XOR logical operation, where 1 is a vector of all 1s with the same dimension as g. Step 4: The M-FOCUSS algorithm solves the excitation vector, specifically: In step 3, W and δ are both optimization variables to be solved. The present invention studies a sequential alternating iterative method to solve the optimization problem. The method alternately iterates between problem (a) and problem (b) to obtain the optimal array element position and excitation. When solving problem (a), let The initial values ​​of and g are all 1. Solve W using the M-FOCUSS algorithm. The solution process of the M-FOCUSS algorithm at this time is Wherein, subscript k represents the number of iterations of the M-FOCUSS algorithm, γ = 0.6 is the sparsity factor, and λ = 0.6 is the regularization parameter. The solution of W is obtained by the M-FOCUSS algorithm, and the excitation of each directional pattern of the corresponding array element is obtained. Step 5: Convex optimization is used to solve the position offset of the array element, specifically: After obtaining W, we need to solve the offset δ of the array element position. At this time, problem (a) can be equivalent to a least squares problem, that is, solving BL l δ=E l Among them, Λ l =diag(W l ), In order to ensure the real value property of δ, first decompose the complex quantity in the above formula into real and imaginary parts, that is, In the formula, B R , B I , and B, Λ l and E l The real and imaginary parts of . The above formula becomes For the sparse reconstruction of arrays with L directional patterns, that is, for L groups of different equations All have the same solution δ. The mathematical model for convex optimization to solve δ is s.t.-Δd / 2<δ m <Δd / 2 The present invention uses the CVX toolbox to solve, and the δ obtained by the solution is the corresponding offset of the array element. L groups of directional patterns obtain a common array element offset δ, and the position of the array element is d m +δ m Repeat steps 4 and 5 until the integrated pattern reaches the set accuracy error.

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