Genetic Algorithm for Optimizing Sidelobe Levels of Rectangular Planar Array Antennas

This paper uses genetic algorithm to optimize the maximum relative sidelobe level of unequal amplitude and unequal distance rectangular planar arrays. By proposing a new adaptive mutation operator, the convergence performance of the algorithm is improved. Good calculation results show that genetic algorithm is an effective method for solving such problems.
Keywords: array antenna; sidelobe level; genetic algorithm; optimization

Sidelobe Reduction of Plane Array Using Genetic Algorithm

HU Xing-hang
(Dept.of Physics,Yueyang Teachers College,Yueyang 414000,China)
LIN De-yun
(Dept.of Electronic Engineering,Tsinghua Univ.,Beijing 100084,China)

Abstract:In this paper, the maximum relative sidelobe level of a large plane array is optimized by genetic algorithm. An adaptive mutation operator is presented. Good results show that the genetic algorithm is an effective method to solve antenna array optimization problems.
Key words: antenna array; sidelobe level; genetic algorithm; optimization

1. Introduction
The maximum relative sidelobe level of an antenna is an important parameter for evaluating antenna performance. Given the antenna shape and the number of array elements, how to minimize the sidelobe level by properly selecting the spacing, feed current amplitude and phase of each array element is an important topic in array antenna synthesis. For large array antennas with complex shapes, traditional analytical methods (such as the Dolph-Chebyshev synthesis method, etc.) are difficult to calculate, and numerical analysis methods are more appropriate. Since the objective functions or constraints in antenna optimization problems are mostly multi-parameter, nonlinear, non-differentiable or even discontinuous, traditional numerical optimization methods based on gradient optimization technology cannot effectively obtain satisfactory engineering solutions; In recent years, a genetic algorithm that simulates natural evolution has been applied to the field of computational electromagnetics [1, 2]. This algorithm only requires that the problem to be solved is computable and has no other restrictions such as differentiability. At the same time, because the algorithm uses an optimized random search technique, it can obtain the global optimal solution with a higher probability and a faster rate. This paper uses a genetic algorithm to optimize the array element spacing and feed current amplitude of a rectangular planar array with 1024 array elements, so that the maximum relative sidelobe level of the array is reduced from -13.27dB of a uniform array to -34.56dB. The results show that genetic algorithms have broad application prospects in solving a large number of complex optimization problems in antenna systems.

2. Optimization of array antenna sidelobe level
1. Array antenna pattern function
Consider a rectangular planar array consisting of 2Nx rows and 2Ny columns of array elements. The phases of the array elements are the same, and the spacing dxi, dyj and normalized current amplitudes Ixm, Iyn can be different, but they are symmetrical about the x-axis and y-axis, as shown in Figure 1. Assume that the pattern function of the array element is cosθ, then the pattern function of this array is [4]:

(1)

Here, k = 2π/λ, λ is the wavelength. The current amplitude of the m-th row and n-th column array element is calculated by the following formula:

I _mn =I _xm I _yn I ₀ (2)

In the formula, I0 is the current amplitude base.

Figure 1 Schematic diagram of rectangular planar array structure

2. Using genetic algorithm to optimize the sidelobe level of array antenna
(1) Coding scheme and calculation process
When applying genetic algorithm, the solution parameters should be encoded first. The encoding scheme of the normalized current and spacing of the array element is given by the following two equations:

(3)
(4)

In the formula, t is the adjustment parameter, which is taken as t=12.5 in the example of this paper; b is the binary code vector, whose element value is 0 or 1; the calculation flow chart of the genetic algorithm is shown in Figure 2.

Figure 2 Genetic algorithm calculation flow chart

(2) Adaptive mutation operator
In order to avoid evolutionary stagnation and premature convergence caused by closed competition in the late iteration, this paper proposes an adaptive mutation operator that dynamically adjusts the mutation ratio according to the evolution of the best individual in each generation. Let Pm be the mutation probability, and its value in each generation is determined by the following formula:

P _m =P _m0 (R ^S +1) (5)

Where Pm0 is the initial mutation ratio; R is the number of generations of the best individual that have not evolved continuously; S is the mutation parameter, which is taken as S=2 in the calculation example of this paper.
Comparative studies show that the use of adaptive mutation operators can better improve the evolution rate and optimization results.

3. Calculation Example
A rectangular planar array with 32 rows and 32 columns was selected as a research example, and its maximum relative sidelobe level was optimized. Selecting dxi=dyi and Ixm=Iym, according to the principle of pattern multiplication and the symmetry of this problem, it is only necessary to calculate the optimal distribution of 16 array elements on one side of the y axis in a row. The scale of the trial solutions of this array reached (32×32)16=1.4615×1048, but the genetic algorithm used its unique selection optimization strategy and gave the optimal value of the maximum relative sidelobe level: -34.56dB after evaluating only 28,200 trial solutions. This result is 21.29dB lower than the -13.27dB of the uniform array［4］, and the optimization effect is obvious. Figure 3 shows the directional pattern of the optimized non-uniform array in the XZ main plane. The corresponding array element spacing and normalized current are shown in Table 1.

Figure 3 Optimized non-uniform array XZ main surface pattern

Table 1 Optimized array element current amplitude and spacing

Figure 4 depicts the three-dimensional radiation pattern of the optimized planar array in this example. It can be clearly seen from the figure that, except for a series of sidelobe peaks below -34.56dB at the X-axis (Φ=0°) and the Y-axis (Φ=90°), there are no sidelobe peaks with a level higher than -35dB in other areas; and, while the sidelobe is greatly reduced, the main lobe is not significantly widened. Calculation shows that the main lobe width of a uniform array with the same number of array elements is 3.23°, while the main lobe width of the optimized array is 4.15°, which is only 0.92° more than before optimization, indicating that the array still has good directivity.

Figure 4 Optimized planar array stereogram

IV. Conclusion
The genetic algorithm is used to solve the optimization problem of large array antennas. It has the characteristics of simple algorithm structure, high search efficiency and good optimization results. The adaptive mutation operator proposed in this paper can avoid the evolutionary sluggishness and immature convergence caused by closed competition, and has made a useful discussion on enriching the application of genetic algorithms.