A Fast Array Antenna Pattern Synthesis Method Based on a New Optimization Problem

Abstract

The present invention provides a fast array antenna pattern synthesis method based on a new optimization problem, which relates to the field of array antennas. By introducing the phase of the desired radiation field as a design variable, a new optimization problem for the array antenna pattern is constructed; the design variable is initialized; L-BFGS-B is used to find the local optimal solution for the new optimization problem; it is judged whether the optimal solution satisfies the termination condition. If so, the local optimal phase is output to obtain the corresponding array antenna pattern. If not, the iteration continues until a satisfactory array antenna pattern is obtained. The beneficial effects of the present invention are as follows: greatly reducing the dimension of the design variables in the array antenna, reducing the computational complexity, and improving the calculation speed of the array antenna pattern synthesis problem.

Description

Technical Field

[0001] The present invention relates to the field of array antennas, and particularly to a fast array antenna pattern synthesis method based on a new optimization problem. Background Art

[0002] Compared with traditional antennas, array antennas can effectively improve the communication distance, signal quality, coverage area, connection rate, and spectrum efficiency of communication systems, and play an increasingly important role in modern wireless communication systems. By adjusting the number, position, and excitation of the array elements in the antenna array, the desired radiation pattern can be obtained. Array pattern synthesis is a non-linear optimization problem, and many excellent pattern synthesis techniques have been proposed, such as the analytical method, artificial intelligence method, etc.

[0003] Various classical synthesis methods have high computational efficiency, such as the Woodward Lawson method, convex optimization method, matrix pencil method, etc. However, traditional mathematical synthesis methods are not applicable to the optimization of synthesis functions with many local optimal values, and mathematical optimization often requires a large number of function evaluations, which will slow down the computational speed of pattern synthesis.

[0004] Therefore, some artificial intelligence methods, such as the array synthesis method based on artificial neural network (ANN), genetic algorithm (GAs), differential evolution (DE), and particle swarm optimization (PSO), are applied to the design of antenna arrays. Neural networks can handle complex non-linear problems, but the training of neural networks requires a large amount of data and time. These intelligent optimization algorithms have shown their effectiveness and superiority in solving complex optimization problems with certain external conditions. However, with the increase of design variables, the computational efficiency and convergence speed of these artificial intelligence methods will significantly deteriorate.

[0005] In summary, the existing methods mainly have the following defects:

[0006] (1) Many design variables;

[0007] (2) Slow computational speed;

[0008] (3) Poor synthesis effect. Summary of the Invention

[0009] In order to solve the above problems, the present invention provides a fast array antenna pattern synthesis method based on a new optimization problem, which mainly includes the following steps:

[0010] S1: Introduce the phase of the desired radiation field as a design variable to construct a new optimization problem for the array antenna pattern;

[0011] S2: Initialize the design variable;

[0012] S3: Use L - BFGS - B to find the local optimal solution for the new optimization problem;

[0013] S4: Determine whether the optimal solution satisfies the termination condition. If so, output the local optimal phase of the design variables. If not, go back to step S2 and continue the operations of S2 - S4 until the local optimal phase is obtained to get the array antenna pattern that meets the design requirements. The termination condition is to satisfy at least one of the following conditions:

[0014] (1) Less than 2.220446049250313 * 10 -9 ; f(x k ) represents the objective function at the k - th iteration, and f(x k+1 ) represents the objective function at the (k + 1)-th iteration;

[0015] (2) The projection component of the gradient g k is less than 10 -5 ;

[0016] (3) Reach the maximum number of function evaluations 15000;

[0017] (4) Reach the maximum number of iterations 15000.

[0018] Furthermore, the new optimization problem is:

[0019]

[0020] Among them, is the design variable, LSM() represents the optimal solution, p and q are the boundary representations of the shaping region, is the lower bound of the phase, is the upper bound of the phase.

[0021] Furthermore, the process of initializing the design variables includes setting the amplitude of the sidelobe region pattern to 0. At this time, the phase of the radiation field in the sidelobe region does not need to be considered to reduce the number of design variables in the new optimization problem.

[0022] Furthermore, the ratio r of the width of the shaping region to the width of the sidelobe region is:

[0023]

[0024] Among them, W shaped represents the width of the shaping region, and W sidelobe represents the width of the sidelobe region

[0025] Furthermore, if M = 3N, then we get:

[0026]

[0027] Among them, dim(·) represents the dimension of the variable, is the design variable, that is, the phase of the shaping area of the ideal field. N is the number of array elements, and M is the total amount of the direction θ of the array antenna, where i = 1, 2, …, M, and r is the ratio of the width of the shaping area to the width of the sidelobe area; i The preset requirement is that r < 2.

[0028] The beneficial effects brought by the technical solution provided by the present invention are:

[0029] 1. The dimension of the design variable is significantly reduced. By introducing the phase of the shaping area of the ideal radiation field as the design variable, since the shaping area is usually very narrow, the dimension of the design variable can be significantly reduced;

[0030] 2. The computational complexity is low. Compared with other typical mathematical optimizers, the calculation speed is very fast, and the optimal array antenna pattern that meets the conditions can be obtained in a relatively short time.

[0031] BRIEF DESCRIPTION OF THE DRAWINGS The following will further illustrate the present invention in conjunction with the drawings. In the drawings:

[0032]

[0033] Figure 1 is a flowchart of a fast array antenna pattern synthesis method based on a new optimization problem in an embodiment of the present invention.

[0034] Figure 2 is a schematic diagram of a linear array along the x-axis in an embodiment of the present invention.

[0035] Figure 3 are the patterns of two typical linear array antennas in an embodiment of the present invention. (a) is a flat-top fan-shaped pattern, and (b) is a cosecant-squared pattern.

[0036] Figure 4 is the pulse shaping effect diagram when N = 20 in an embodiment of the present invention.

[0037] Figure 5 is the cosecant shaping effect diagram when N = 20 in an embodiment of the present invention.

[0038] Figure 6 is the pulse shaping effect diagram when N = 200 in an embodiment of the present invention.

[0039] Figure 7 is the cosecant shaping effect diagram when N = 200 in an embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0040] In order to have a clearer understanding of the technical features, objectives, and effects of the present invention, the specific implementation manners of the present invention will now be described in detail with reference to the accompanying drawings.

[0041] An embodiment of the present invention provides a fast array antenna pattern synthesis method based on a new optimization problem.

[0042] In the existing array antenna pattern synthesis technology, the increase in the dimension of design variables often slows down the calculation speed of the pattern synthesis problem. To solve this problem, first, the phase of the desired radiation field is introduced as a design variable to construct a new optimization problem. Given the phase, the excitation can be determined by solving a linear least squares problem. When the shaping region is narrow, the dimension of the design variables will be significantly reduced. After taking the phase as a design variable, there are many local optimal solutions for the array pattern synthesis problem, and the local objective values are very close to each other. Then, the L-BFGS-B method is used to find the local optimal phase and approximate the ideal pattern. Experiments prove that the method proposed in this work can obtain a satisfactory pattern in a short time.

[0043] As Figure 1 shown, Figure 1 is a flowchart of a fast array antenna pattern synthesis method based on a new optimization problem in an embodiment of the present invention, which specifically includes the following steps:

[0044] (1) Construction of a new optimization problem

[0045] (1) Principle of array antenna pattern

[0046] As Figure 2 shown, assume that a linear array antenna is composed of N array elements distributed along the x-axis, and each array element can be regarded as an ideal particle radiation source.

[0047] In numerical calculation, [-90°, 90°] is divided into M - 1 parts, with M directions θ i (i = 1, 2,..., M). M is two to three times the number of array elements N.

[0048] The radiation field E i of the antenna array in the θ i direction is

[0049]

[0050] where

[0051]

[0052]

[0053] where, is an n-dimensional direction vector, is the complex excitation vector, θ i is the angle in different directions, θ i ∈[-90°, 90°]; λ represents the wavelength; where d n is the distance between element n and element n + 1; D n is the vector from the origin to the position of the nth element. Since all elements are on a straight line, these vectors can be regarded as one-dimensional vectors.

[0054] The radiation pattern (abbreviation: pattern) is the normalized amplitude of the field. In some applications of pattern synthesis, users only care about the pattern in certain cases, and the phase is not considered at this time.

[0055] Here, introduce the symbols:

[0056]

[0057]

[0058]

[0059] where is the amplitude of the ideal radiation field, denoted as the ideal pattern; (the phase in which is not given in general pattern synthesis problems) is the ideal radiation field. is the amplitude of the ideal radiation field in the θ i direction, is the ideal radiation field in the θ i direction in Equation (1).

[0060] Then,

[0061]

[0062] where is the amplitude of the designed radiation field, denoted as the actual pattern; is the actual radiation field. P i is the amplitude of the actual radiation field in the θ i direction, E i is the actual radiation field in the θ i direction.

[0063] In pattern synthesis, as Figure 3 shown, the ideal pattern usually includes two parts: the shaping region and the sidelobe region. As Figure 3 shown are two common ideal patterns: (a) is the flat-top fan pattern, and (b) is the cosecant-squared pattern.

[0064] Ideally, the designed radiation pattern is the same as the ideal radiation pattern. However, in most cases, it is impossible to design the radiation pattern as the desired one. Therefore, the goal of pattern synthesis is to minimize the difference between the designed radiation pattern and the desired radiation pattern.

[0065] The distribution of all array elements is known. Pattern synthesis is generally considered an optimization problem of the following form:

[0066]

[0067] where

[0068]

[0069]

[0070] where, is the objective function, is a vector containing excitations. The goal of the optimization problem is to reduce the difference between the designed radiation pattern and the ideal radiation pattern and construct a normalized error in the form of the two-norm.

[0071] It should be noted that for the convenience of solving the optimization problem, the number of sampling points of the radiation pattern should be as small as possible, but for accurate calculation the number of sampling points of the radiation pattern should be as large as possible.

[0072] (2) New Optimization Problem

[0073] In the new optimization problem, the phase of the desired radiation field is taken as the design variable. When the ideal field (including amplitude and phase) is determined, the solution of the optimal excitation is a linear least squares problem. Therefore, the pattern synthesis of the array antenna can be regarded as the phase optimization of the ideal field.

[0074] For the convenience of discussion, the field E in Equation (1) i is rewritten in the form of a real vector:

[0075]

[0076] where

[0077]

[0078]

[0079] where, is the complex excitation, and R(·) and I(·) represent the real part and the imaginary part respectively.

[0080] Let be the phase of the ideal radiation field. According to there is:

[0081]

[0082] Typically, pattern synthesis aims to approximate the ideal radiation pattern (i.e., the shape of the normalized amplitude) without considering the information of the ideal phase. However, in the present invention, the phase of the ideal field should be considered. By introducing the phase of the ideal radiation field, an attempt is made to approximate the ideal radiation field, specifically, to approximate both the amplitude and the phase of the ideal field, rather than only approximating the amplitude of the ideal field, i.e., the radiation pattern, as in the general problem.

[0083] This new idea can be formulated as an optimization problem as follows:

[0084]

[0085] where

[0086]

[0087]

[0088]

[0089]

[0090]

[0091] where, As shown in Equations (2) and (3), are the design variables in Equation (6). It should be noted that the two problem formulations in Equations (4) and (8) are different. In Equation (4), only reflects the error in amplitude between the ideal field and the actual field, while in Equation (8) reflects the errors in both amplitude and phase.

[0092] As described in Equation (8), the design variables consist of the excitation and the phase. When the phase is set to a certain assumed constant, the process of finding the optimal excitation can be regarded as a process of finding the optimal solution in a subspace under constraints.

[0093]

[0094] Taking as a constant, the objective is a linear least-squares problem on Then there is only one optimal solution for the internal minimization, denoted as Therefore, the least-squares solution that minimizes the squared error in Equation (8) can be obtained:

[0095]

[0096] Among them, A + is the Moore-Penrose generalized inverse matrix, is a part of, see Equation (7).

[0097] The next step is to optimize In this step, only use to calculate the error between the ideal radiation pattern and the actual radiation pattern. The objective of the new optimization problem can be constructed as the squared error of the amplitude, which is the same as the general radiation pattern synthesis optimization problem. Assign the calculated by Equation (11) to the excitation Then replace the position of in Equation (8) with Then there is:

[0098]

[0099] In the ideal case, the antenna does not radiate energy to the sidelobe region, so the required radiation pattern in the sidelobe region is set to zero. In this way, the phase in the sidelobe region does not need to be considered, and the design variable can be set as the phase of the shaping region of the ideal field, denoted as So the new least squares method-assisted optimization problem is:

[0100]

[0101] Among them, is the design variable, p and q are the boundaries of the shaping region, is the lower bound of the phase, is the upper bound of the phase.

[0102] (3) Reduction of design variables

[0103] The second part is to convert the design variable of the nonlinear optimization problem from to And this part will explain the reduction of the dimension of the variable vector. According to Equation (13), formula (14) is obtained:

[0104]

[0105] Among them

[0106]

[0107]

[0108] The number of sampling directions of the sample in the shaping region is the size of. W shaped represents the width of the shaping region, W sidelobeDenote the width of the sidelobe region as \(r\), and use \(r\) to represent the ratio of the width of the shaped region to the width of the sidelobe region. Then

[0109]

[0110] If \(M = 3N\), then

[0111]

[0112] Then

[0113]

[0114] According to Equation (17), if \(r\lt2\) is satisfied, the number of design variables of the new optimization problem will be less than that of the general optimization problem. In practical engineering, the shaped region is usually narrow and sharp, and the ratio of the shaped region to the sidelobe region is close to zero. The reduction of design variables is very significant. Moreover, the smaller \(r\) is, the more significant the reduction of optimization variables is.

[0115] (II) Fast Optimization of the Ideal Field Phase

[0116] After using the phase as the design variable, the problem has many local optimal solutions, and the local objective values are very close to each other. Based on the full analysis and utilization of the problem characteristics, the present invention uses L - BFGS - B to find the local optimal phase and gradually approach the ideal pattern.

[0117] The input and output of the L - BFGS - B algorithm are as follows:

[0118] Input: The objective function \(f(x)\), The accuracy requirement \(\epsilon\);

[0119] Output: The minimum point \(x\) of \(f(x)\) * ;

[0120] The specific steps are as follows:

[0121] (1) Select the parameters \(x_0\), \(B_0\) in the algorithm, and set the parameter \(k = 0\); \(x_0\) is the initial value, and its value can be randomly selected. \(B_0\) is the initial approximate inverse Hessian matrix, which is an identity matrix; \(k\) is the number of iterations, \(k\leq15000\);

[0122] (2) Calculate the gradient \(g\) of the \(k\) - th iteration k \(= g(x\) (k) ), if the termination condition is satisfied, stop the iteration and obtain the local optimal solution;

[0123] (3) From \(B\) k P k \(= - g\) k , find \(p\) k ;

[0124] (4) Ball

[0125] (5) Set x (k+1) = x (k) + λ k P k ;

[0126] (6) Calculate g k+1 = g(x (k+1) ), if the termination condition is satisfied, stop the iteration and obtain the local optimal solution. If the termination condition cannot be satisfied, calculate the approximate inverse Hessian matrix B of the (k + 1)-th iteration k+1 ;

[0127] (7) k = k + 1, go back to step (3);

[0128] Wherein, represents the calculation of the gradient, and B k represents the approximate inverse Hessian matrix of the k-th iteration.

[0129] (III) The technical key points of the present invention are as follows:

[0130] 1. Introduce the phase of the shaping region of the ideal radiation field as a design variable, establish a new optimization problem, and greatly reduce the dimension of the design variables;

[0131] 2. Make full use of the characteristic that there are many very close local optimal values in the new optimization problem, and propose to use the L-BFGS-B algorithm to find the local optimal phase, reducing the computational complexity of the antenna synthesis problem.

[0132] The present invention constructs a new optimization problem, transforms the original optimization problem with the excitation as a variable into an optimization problem with the phase as a variable, and uses L-BFGS-B to find the local optimal solution. To illustrate the advantages of the new optimization problem and the effectiveness of the algorithm, in this embodiment, a set of widely used fan-shaped and cosecant-squared patterns in a linear array are used as test problems. The shaping region of the fan-shaped pattern is selected as [-20°, 20°], and the shaping region of the cosecant-squared pattern is selected as [3°, 40°]. The sampling points M in the radiation region of the LSM should be selected as 2 to 3 times the array size N. In this embodiment, N = 20, N = 200, and M = 3N are adopted. The array antenna is an equally spaced linear array with a spacing of 0.5λ.

[0133] To illustrate the reduction of the dimension of the design variables, according to the above tests, Table 1 shows the comparison of the dimensions of the design variables in the synthesis problems of the fan-shaped and cosecant-squared patterns of the array antennas with N = 20 and N = 200

[0134] Table 1

[0135] Shaped radiation pattern and number of array elements General optimization problem New optimization problem Pulse shaping (N = 20) 40 11 Cosecant shaping (N = 20) 40 9 Pulse shaping (N = 200) 400 110 Cosecant shaping (N = 200) 400 90

[0136] As can be seen from Table 1, the variable dimension of the new optimization problem with the phase of the shaping region as the design variable is much smaller than that of the ordinary optimization problem with the excitation as the design variable. The search space often grows exponentially with the number of decision variables. With the reduction of the variable dimension, the computational cost of solving pattern synthesis is greatly reduced.

[0137] The method disclosed in the present invention is compared with all other unconstrained algorithms and bound-constrained algorithms in scipy.optimize.minimize. The experiment is repeated 25 times, and 25 different initial phases are set to find different local optimal solutions to compare the performance of different algorithms. In the 25 experiments, the objective function value and the number of function evaluations are counted to evaluate the performance of the algorithms. The objective function value can reflect the effectiveness of the algorithm, and the number of function evaluations can reflect the computational complexity of the algorithm.

[0138] Table 2 shows the comparison of the mean of the target value, the standard deviation of the target value, the mean of the number of function evaluations, and the standard deviation of the number of function evaluations for sector shaping when N = 20. Table 3 shows the comparison of the mean of the target value, the standard deviation of the target value, the mean of the number of function evaluations, and the standard deviation of the number of function evaluations for cosecant shaping when N = 20.

[0139] Table 2 Comparison of the errors and the average number of function evaluations for 25 experiments of sector pattern synthesis of array antennas when N = 20 ("-" indicates that the algorithm is not effective in all experiments)

[0140]

[0141]

[0142] Table 3 Comparison of the errors and the average number of function evaluations for 25 experiments of cosecant pattern synthesis of array antennas when N = 20 ("-" indicates that the algorithm is not effective in all experiments)

[0143]

[0144] Then, in order to verify the effectiveness of the proposed method when the number of elements is large, the performance of the proposed method is compared with that of CG, BFGS, L-BFGSB, and Powell when N = 200 because they have better performance in previous experiments.

[0145] Table 4 shows the comparison of the mean target value, mean number of function evaluations, and mean time consumption for sector shaping when N = 200. Table 5 shows the comparison of the mean target value, mean number of function evaluations, and mean time consumption for cosecant shaping when N = 200. Since Tables 2 and 3 have already given the general rules of the mean and standard deviation, the standard deviation is not given in Tables 4 and 5.

[0146] Table 4 Comparison of the errors and average number of functions in 25 experiments of array antenna sector pattern synthesis when N = 200

[0147]

[0148] Table 5 Comparison of the errors and average number of functions in 25 experiments of array antenna cosecant pattern synthesis when N = 200

[0149]

[0150] It can be seen from Tables 2, 3, 4, and 5 that the target values obtained by different methods are relatively close. In the case of different initial phases, the optimal target values are different but very close. This indicates that there are many local optimal values for this problem, and the local optimal values are very close to each other. The number of function evaluations of L - BFGS - B is significantly less than that of other methods. Since other methods all need to calculate the gradient, Hessian, or approximate inverse Hessian of the objective function, this requires a large number of objective function evaluations.

[0151] Figure 4 is the pulse shaping effect diagram when N = 20, Figure 5 is the cosecant shaping effect diagram when N = 20, Figure 6 is the pulse shaping effect diagram when N = 200, Figure 7 is the cosecant shaping effect diagram when N = 200. Through Figures 4 - 7 it can be known that the method proposed by the present invention can obtain a satisfactory pattern in a linear array, with a very low ripple level, and most sidelobes are below - 30 dB without constraints.

[0152] The beneficial effects of the present invention are:

[0153] 1. The dimension of the design variables is significantly reduced. By introducing the phase of the ideal radiation field shaping region as the design variable, the shaping region is usually very narrow, and the dimension of the design variables can be significantly reduced;

[0154] 2. The computational complexity is low. Compared with other typical mathematical optimizers, the calculation speed is very fast, and a qualified pattern can be obtained in a relatively short time.

[0155] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A fast array antenna pattern synthesis method based on a new optimization problem, characterized in that: It includes the following steps: S1: Introduce the phase of the desired radiation field as a design variable and construct a new optimization problem for the array antenna pattern; the new optimization problem is: Among them, is a design variable, LSM() represents the optimal solution obtained by the least squares method, p and q are the boundary representations of the shaping region, is the lower bound of the phase, is the upper bound of the phase; S2: Initialize the design variable; S3: Use L-BFGS-B to find the local optimal solution for the new optimization problem; The input and output of the L-BFGS-B algorithm are as follows: Input: objective function f(x), precision requirement ε; Output: The minimum point x of f(x) * ; The specific steps are as follows: (1) Select the parameters x0 and B0 in the L-BFGS-B algorithm and set the parameter k = 0; x0 is the initial value, B0 is the initial approximate inverse Hessian matrix, and k is the number of iterations; (2) Calculate the gradient g for the k-th iteration k = g(x (k) ). If the termination condition is satisfied, stop the iteration and obtain the local optimal solution; (3) From B k p k = -g k , find p k ; (4) Find (5) Set x (k+1) = x (k) + λ k p k ; (6) Calculate g k+1 = g(x (k+1) ). If the termination condition is satisfied, stop the iteration and obtain the local optimal solution. If the termination condition cannot be satisfied, calculate the approximate inverse Hessian matrix B for the (k + 1)-th iteration k+1 ; (7) Let k = k + 1 and return to step (3); Among them, represents the calculation of the gradient, and B k represents the approximate inverse Hessian matrix of the k-th iteration; S4: Determine whether the optimal solution satisfies the termination condition. If so, output the local optimal phase of the design variable. If not, return to step S2 and continue the operations of S2 - S4 until the local optimal phase is obtained to obtain an array antenna pattern that meets the design requirements.

2. The fast array antenna pattern synthesis method based on a new optimization problem according to claim 1, characterized in that: In step S2, the process of initializing the design variable includes setting the amplitude of the pattern in the sidelobe region to 0. At this time, the phase of the radiation field in the sidelobe region does not need to be considered to reduce the number of design variables in the new optimization problem.

3. A fast array antenna pattern synthesis method based on a new optimization problem according to claim 1, characterized in that: In step S2, the ratio r of the width of the shaping region to the width of the sidelobe region is: Among them, W shaped represents the width of the shaped region, and W sidelobe represents the width of the sidelobe region.

4. A fast array antenna pattern synthesis method based on a new optimization problem according to claim 2, characterized in that: In step S2, if M = 3N, then: where dim(·) represents the dimension of the variable, is the design variable, i.e., the phase of the shaping region of the ideal field, N is the number of array elements, and M is the total amount of the direction θ of the array antenna, i = 1, 2, …, M, and r is the ratio of the width of the shaping region to the width of the sidelobe region; i i.e., the total amount of the direction θ of the array antenna, i = 1, 2, …, M, and r is the ratio of the width of the shaping region to the width of the sidelobe region; The preset requirement is that r < 2.

5. A fast array antenna pattern synthesis method based on a new optimization problem according to claim 1, characterized in that: In step S4, the termination condition is to satisfy at least one of the following conditions: (1) Less than 2.220446049250313*10 -9 ; f(x (k) ) represents the objective function at the k-th iteration, and f(s (k+1) ) represents the objective function at the (k + 1)-th iteration; (2) The projection component of the gradient g k is less than 10 -5 ; (3) Reach the maximum number of function evaluations of 15000; (4) Reach the maximum number of iterations of 15000.

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