Sparse optimization of planar radio antenna arrays using a genetic algorithm

1.   INTRODUCTION

Radio astronomy is an important branch of astronomical research, using radio waves to study celestial bodies and phenomena, while avoiding some of the limitations of traditional optical telescopes. It is largely unaffected by adverse meteorological conditions, such as fog and clouds, so it can allow astronomical observation at longer distances, with a wider field of view. Radio telescopes and their antenna arrays are essential tools, and as the research field develops, there is increasing requirement for high spatial resolution, high sensitivity, and wide scanning ranges, to facilitate research into the frontiers of radio astronomy, such as the reception of weak signals and high-spatial-resolution detection^[1-3].

A radio antenna array is a system consisting of multiple antenna units, which may be of the same or different types, combined geometrically. In this way, antenna arrays are able to enhance the reception of signals, improve spatial resolution and sensitivity, and achieve highly sensitive directional detections^[4]. In radio astronomy observations, antenna arrays are commonly used to receive radio waves from the Sun, distant galaxies, pulsars, black holes, and other celestial bodies. The use of these arrays has greatly expanded our ability to observe astronomical phenomena.

As technology advances, the design of radio antenna arrays is constantly being optimized and enhanced, improving the performance of radio telescopes, and also advancing the theoretical and technological development of radio astronomy. Using antenna arrays, more complex observations can be carried out, such as using a multi-beam approach for simultaneous observation of multiple targets. For targets such as pulsars, this can allow synchronized observation and timing of multiple sources, (i.e. real-time source-cold-space calibrations)^[5,6], in addition to providing real-time flux calibrations of solar radio bursts. If continuous observation with a wide scanning angle can be carried out under such conditions, with a broad bandwidth, it can be of great significance for continuous solar radio observations and provision of information for space weather early warning and forecasting systems^[7].

Antenna arrays are advantageous and convenient for radio astronomy research. The three parameters of the array—sidelobe level (SLL), beamwidth, and wide-angle scanning—are interrelated, and jointly determine the performance and observing capability of the radio telescope, defining parameters such as spatial resolution, sensitivity, and scanning range of the sky area. To satisfy increasingly complex observational requirements and enhance the observing capability of radio astronomy, the performance of the antenna arrays is also put forward to meet the higher requirements: The arrays should have lower SLL, narrow beamwidth, and wider wide-angle scanning capability^{[8, 9]}.

Considering the relationship between the antenna array full-width-at-half-maximum (FWHM), field of view (FOV), and the number of spatially visible points, the relationship between the beamwidth and SLL with the gain and effective area is

If the beamwidth becomes narrower, the FWHM also becomes narrower and the FOV decreases.

Spatially visible points are defined as

where p refers to the spatially visible points. Since the smaller FOV makes more points, it yields higher resolution for the antenna array.

Gain, G, is defined as

where P_max is the maximum value of the radiation intensity of the antenna, and ${P}_{0}$ is the input power of the antenna. When the beamwidth becomes narrower and the SLL becomes lower, it causes the energy to be more concentrated in the main lobe, increasing the gain.

Effective area is given by

where $\lambda$ is the wavelength, A_e is the effective area. A_e increases with G.

Sensitivity is defined as

expressed in terms of the minimum detectable radio source flow S_min. Here, T_sys is the noise temperature of the entire radio telescope system, k is the Boltzmann constant, T is the observation time, B is the signal reception bandwidth, and n_p is the number of polarization channels. A larger A_e makes the S_min smaller, increasing sensitivity. In addition, the SLL directly affects the antenna's anti-interference ability; if the SLL is high, noise signals from other directions may be received, affecting the accuracy of the observation results.

Wide-angle scanning capability, i.e., the ability of the telescope to make observations in a wider FOV, is crucial for radio astronomy projects observing a large sky area, and they often need to have a large bandwidth to give a wide scanning sky area range. The main problem faced by ultra-wideband arrays is that they will have grating lobes when scanning a large sky area. The minimum spacing of the array elements is typically unable to satisfy the condition of having no grating lobes in the pattern at the upper frequency, according to the relationship

where d is the array element spacing, $\theta$ is the scanning angle, and λ is the wavelength. At a given wavelength λ, when $\theta$ increases, it will fail to satisfy the constraints of the antenna array pattern grating lobes. To solve this problem, the radio antenna array needs to be designed as a sparse array with unequal spacing of array elements. The sparse array can break the periodic distribution of array radiation energy in space to eliminate grating lobes. In addition, the sparse array also has a series of other advantages, such as increasing the size of the array aperture to improve spatial resolution, without amplitude weighting to achieve low SLL. Based on the advantages of a sparse array, sparse optimization has become a topic of active research in the field of antenna arrays^[10].

This study changes the antenna array layout structure to optimize performance. Previous research on antenna layout has mainly focused on optimizing the SLL or beamwidth, while few studies have been conducted to simultaneously optimize the SLL and beamwidth for a wide-angle observation^{[11, 12]}. Previous studies^[13] have used genetic algorithms to verify the optimization simulation of sparse arrays under specific constraints, and although the optimized SLL and gate flap suppression effects are good, beamwidth has not been optimized. Xie et al.^[14] optimized the array elements using the sparse spacing method of 0–1 planning, focusing on SLL, beamwidth, and the number of concentric circular array units to achieve low SLL and gate suppression. Xiong^[15] used a genetic algorithm together with several improved algorithms to study the sparse spreading of antenna arrays, and optimized the position of the array elements of planar antenna arrays to achieve a reduction in the peak paraflap level. Sun et al.^[16] proposed an ultra-wideband sparse array based on rotationally symmetric distribution, combining it with a covariance matrix adaptive optimization strategy to optimize the distribution of the array elements in a single region only. In this case, with the minimum spacing of the cells being the low-frequency half-wavelength, and with low SLL at different scan angles as optimization targets, to optimize the wide-angle scanning of the array in a wide bandwidth range.

The aim of this study is to investigate the use of genetic algorithms to study and analyze the sparse optimization of the array element spacing of a planar array in wide-angle observations. We carry out a sparse optimization study based on a genetic algorithm, using a 64-element antenna array as the object of the study, with the aim of optimizing SLL and beamwidth of the array pattern.

We employ a linear weighting method to transform the multi-objective problem into a single-objective problem, which is transformed into a fitness function of the genetic algorithm to optimize the layout of the array antenna to achieve a low SLL and narrow beam performance in the wide-angle case. The optimization simulation results show that, compared with the rectangular planar array, the optimized array antenna scanning angle $\theta$ is within ±30°, the SLL is reduced, and the beamwidth becomes narrower.

2.   TECHNIQUES AND METHODS

2.1   Rectangular Planar Array and Simulation Modeling

2.1.1   Antenna arrays

Antenna arrays are divided into two main categories in terms of array element layout: uniformly spaced arrays and non-uniformly spaced arrays. The latter is often broadly referred to as a sparse array^[17]. This refers to an antenna array in which the array elements are arranged at unequal intervals, and can be further subdivided into two types. The first is constructed by removing some of the array elements from a complete uniformly spaced array to form a non-uniform array, with the spacing of the array elements being an integer multiple of some fundamental quantity. The second type allows the spacing of the array elements to be an arbitrary value. However, due to factors such as the physical dimensions of the array elements and the mutual coupling effect between the array elements, the spacing is usually suggested to be not less than half the wavelength, to ensure the performance and efficiency of the antenna array. The sparse array method used in this paper is the latter one.

2.1.2   Establishing a coordinate system

In the design and analysis of rectangular planar arrays, the first step is to establish the coordinate system of the position of the array elements. There are three main coordinate systems: The antenna coordinate system (ACS), radar coordinate system (RCS), and antenna conical angle coordinate system (ACCS)^[18], all of which have the ability to calculate array spatial coordinates. In engineering practice, according to the specific design requirements and application scenarios, an appropriate coordinate system can be flexibly selected. Here, we choose the ACS for the calibration of the position of the array elements and the related calculations, as shown in Fig. 1.

Figure  1.  Schematic diagram of the antenna coordinate system.

DownLoad: Full-Size Img PowerPoint

The orientation of any point in space, R, in an ACS is expressed by two angles $\theta$ and $\phi$. $\theta$ is the angle from the Z-axis to the point R, while $\phi$ is the angle between the X-axis and the line of projection of the point R in the XY plane. This coordinate system is the intuitive definition of a spherical coordinate system. Accordingly, when the magnitude of the point R is set to 1, the coordinate value of the point R is (sin $\theta$ cos $\phi$, sin $\theta$ sin $\phi$, cos $\theta$).

2.1.3   Computational modeling of pattern

In a three-dimensional orthogonal coordinate system, the radiation direction is the positive direction of Z, i.e., the direction of the front hemisphere. The antenna array is located in the XY plane. A two-dimensional case requires two spacings to indicate the spacing of the array elements: The spacing of the X-direction is recorded as d_x, and the spacing of the Y-direction is recorded as d_y. The number of the X-direction of the array element is recorded as M, the number of the Y-direction of the array element is recorded as N, and so the total number of array elements is given by MN, as shown in Fig. 2.

Figure  2.  Schematic of a uniformly arranged planar array.

DownLoad: Full-Size Img PowerPoint

After defining the spacing of the array elements and the number of elements in the X and Y directions, we can find the position coordinates of the array elements in the XY plane, via

where x_m is the position of the m array element in the X-direction, and y_n is the position of the n array element in the Y-direction. Equations (7) and (8) describe a rectangular grid of array element distributions whose centers are located at the coordinate origin (0, 0).

A signal incident along the $(\theta,\ \phi)$ direction onto the array is coherently superimposed to form a synthesized signal after it is received by each array element, and its coherently superimposed voltage is given by

where the array factor ($AF$) is a function of the wavelength $\lambda$, x_l and y_l are the array element spacings, and C_l is the aperture distribution, where the array element spacing is calculated from Equations (7) and (8). $(\theta,\ \phi)$ is the angle of incidence. C_l can be expanded into the form of complex voltages, allowing Equation (9) to be changed into

where $(\theta_0,\ \phi_0)$ is the beam pointing of the array. The synthesized antenna array pattern of the array is obtained by the product of the array element pattern $EP$ and the array factor $AF$. The antenna array pattern synthesis formula is

where $F(\theta,\ \phi)$ is the antenna array pattern. In this paper, for the study of a planar antenna array, only the ideal conditions are considered in the derivation of the antenna array pattern function and the establishment of the optimization model: If all the antenna units are the same, have the same excitation amplitude and are omnidirectional antenna units, then the element factor (EP) of the array element is 1. Substituting Equation (10) into Equation (11), the expression for the planar antenna array pattern is derived as

2.2   Genetic Algorithm-based Antenna Array Sparse Optimization Analysis

2.2.1   Algorithms

A genetic algorithm is a heuristic search algorithm that simulates the mechanism of biological evolution in nature, with a design is inspired by Darwin's principles of natural selection and genetics. As a powerful optimization tool, genetic algorithms are known for efficiency, practicality and robustness, and are especially suitable for solving NP-hard problems as well as complex nonlinear, multimodal, and multi-objective optimization problems. NP-hard problems are a very important class of problems in computer science and mathematics, which are at least as difficult to solve as NP-complete problems (i.e. nondeterministic problems that can be verified in polynomial time). Genetic algorithms have shown excellent application potential in many fields such as engineering, computer science, and economics. Thus, the application of genetic algorithms provides an effective solution for optimal design in the sparse array element placement problem^[19].

The optimization process of sparse optimization for planar arrays using a genetic algorithm follows the following steps. First, an initial population with real values is generated and the gene sequences of each individual are ordered; in this study the genes of the individuals represent the array element spacing and each individual represents the array element arrangement.

Next, the fitness of the population is evaluated. The value of the fitness is a combination of the SLL and beamwidth of the array, and it is determined by whether the preset termination condition is met; if it is met, the algorithm is terminated, which means that the solution of the array element spacing that meets the optimization condition has been found, and the output array element spacing is the final optimization result. If it is not satisfied, iterative evolution of array element alignment individuals is performed by selection, crossover, mutation, and genetic operations, and the gene sequences are again ordered and mapped to the actual spacing.

For the progeny population produced by each round of iteration, the termination condition judgment is repeated until the termination criteria are satisfied. The flowchart of the spacing algorithm is shown in Fig. 3.

Figure  3.  Flowchart of the sparse array genetic algorithm.

DownLoad: Full-Size Img PowerPoint

While in use, the setting of the fitness function needs to be taken into account. We adopt a method used in the literature [15]. In the analysis based on a three-dimensional antenna array pattern, the sparse optimization is carried out by intercepting multiple cut planes around the main beam at equal angles, and taking the SLL and beamwidth of the antenna array patterns of the cut planes as the optimization targets. Considering the fineness of the sampling interval, when the $\theta$ sampling interval is ≤1° and all the antenna array patterns in the intercepted planes show low SLL and narrow beamwidth characteristics, it can be assumed that the three-dimensional antenna array pattern of the whole rectangular planar array also has low paraflap and narrow beam characteristics. Consequently, we choose to intercept the planes at 1° intervals and construct a mathematical model of the sparse array with the optimization objectives of minimizing the SLL of all the cut planes and guaranteeing that the beamwidth is smaller than the standard beamwidth of the rectangular planar array. In addition, in order for the antenna array to have a wide scanning performance, the maximum sidelobe level (MSLL) of the antenna array pattern under different scanning angles is also an optimization objective. The following is a genetic algorithm analysis for the specific problem of antenna sparse optimization.

2.2.2   Population initialization

First, a planar array is created as the initialization population with an array aperture of L×H; the number of antenna elements is the same as the regular array. The array arrangement of M×N array elements, i.e., the positions of these array elements, is transformed into a two-dimensional complex matrix G with N_x rows and N_y columns. The position matrix of this array element is denoted as dx + jdy, where dx is the distance spacing in the X-direction, and dy is the distance spacing in the Y-direction. The spatial constraints to be satisfied by the array element distance spacing are

where L is the array aperture in the X-direction, H is the array aperture in the Y-direction, and d_c represents the minimum array element spacing constraint, which is taken as half the wavelength here. At this point, let the constrained array element position matrix be f, and the constraint matrix be C, where f(m, n) = x_mn + jy_mn and C(m, n) = (m−1)d_c + j(n−1)d_c. Then, the array element position matrix G is shown by

From the constraints, it can be obtained that the range of the parameter variable boundaries in the X-direction is 0 < x_{q, g} < [L−(N_x−1)d_c], and the range of the parameter variable boundaries in the Y-direction is 0 < y_{q, g} < [H−(N_y−1)d_c], x_{q, g} is the distance interval parallel to the X-axis direction, and y_{q, g} is the distance interval parallel to the Y-axis direction. Thus, only the intermediate individual f needs to be operated during the genetic operation, reducing the search space and speeding up the optimization. When the number of individuals in the population is NP, each intermediate individual is shown by

where q is the ordinal number of the individual in the corresponding population, g is the number of genetic generations.

Since the array aperture is fixed, the boundaries of the populations are constrained as shown by

where dx represents the position of the array element in the X-axis direction and dy represents the position of the array element in the Y-axis direction. This move allows for an array element at each of the four corners of the initialized array, ensuring that the array element does not exceed this boundary during subsequent optimization.

2.2.3   Selection operations

We use the "roulette" method of selection here, where the probability of an individual f_{q, g} remaining in the population in Equation (15) is determined by measuring its fitness as a proportion of the population. The fitness of each individual determines its probability of being selected; the higher the fitness value, the higher the probability of being selected, and vice versa. The fitness function is given by

where fit_q is the fitness value of corresponding individuals in the population, W₁, W₂, and W₃ are the weighting coefficients, SLL1_max is the MSLL in the antenna array pattern of the cut planes, SLL2_max is the MSLL of the antenna array pattern at all different scan angles, and BW_max is the maximum value of the half-power beamwidth in the antenna array pattern of the cut planes. Since it involves the problem of multi-objective optimization, the linear weighted sum method is used here to set up the weighting coefficients for SLL and beamwidth, to construct the fitness function.

2.2.4   Cross-operation

The selected array element arrangement of odd-numbered individuals and the array element arrangement of even-numbered individuals are paired, and for each array element arrangement, a partial array element spacing between them is exchanged with a crossover probability, P_c. The specific steps are as follows. First, an array element arrangement to be paired is taken out; then, according to the length of a bit string, L, the array element arrangement to be paired is matched, and the integer k in [1, L−1] is randomly selected as a crossover position. Finally, according to P_c, to implement the crossover operation, the paired individuals exchange their respective parts of the array element spacing with each other at the crossover position, forming a new one-array element arrangement.

2.2.5   Variant operations

For each array element arrangement in the population after crossover, some array element intervals are reformed as other equal array element intervals with variant probability, ${P}_{m}$. The specific steps are as follows. In the array element arrangement after crossover, from p = 1~MN and q = 1~NP, a random number r is generated in the interval [0, 1], and the (p, q) array element interval f(p, q) is selected as the variant array element interval if r < P_m. It is then replaced with a randomly generated parameter in the value domain as in the equations

where rand[0, 1] denotes the random generation of a value uniformly distributed within [0, 1]. After the selection of individuals, individual crossover, and mutation operations, a new population is obtained, and the array element intervals in the newly generated population need to be treated again to sort the array element intervals in the population from the smallest to the largest, to obtain the new array element arrangement.

The fitness calculation is performed on the newly generated array element arrangement, and the optimal array element interval is retained in the new generation population for the next genetic operation. After the above steps complete a given number of cycles or satisfy predefined conditions, the genetic algorithm is terminated and the optimal individual is output as the optimization result.

3.   EXAMPLE ANALYSIS

3.1   Array Initialization Information

A standard 8×8 rectangular planar array is sparse optimized and analyzed based on a genetic algorithm. The initial frequency of this array is 300 MHz, the spacing of array elements is $\lambda /2$, the total number of array elements is 64, and the scanning angles $\theta$ = 0° and $\phi$ = 0°.

The arrangement and three-dimensional antenna array pattern of the array element are shown in Fig. 4, where the three-dimensional antenna array pattern is calculated using Equation (12). Since the three-dimensional array pattern of the antenna is symmetric about the u-axis and v-axis, and there are maximum values of SLL and beamwidth in both the u-axis and v-axis directions, the subsequent observations are mainly based on this 3D orientation map sectioned with the sparse optimized comparison. The MSLL is −12.79 dB, and the beamwidth is 13°. Fig. 5 shows the three-dimensional antenna array pattern cross-section diagram for $\phi$ = 0° and $\phi$ = 90°.

Figure  4.  Array element positions in a uniform array (A), and three-dimensional antenna array pattern under uniform distribution scanning angles $\mathit{\theta }$ = 0°, $\mathit{\phi }$ = 0° (B).

DownLoad: Full-Size Img PowerPoint

Figure  5.  The three-dimensional antenna array pattern cross-section at $\mathit{\phi }$ = 0° (A) and at $\mathit{\phi }$ = 90° (B).

DownLoad: Full-Size Img PowerPoint

3.2   Optimization Step Using the Genetic Algorithm

In the optimization of the array using the sparse algorithm, the MSLL and beamwidth are made the optimization objective. Since the number of array elements is constant and the array spacing is to be satisfied as much as possible to be greater than $\lambda /2$, the array aperture is set to 5$\lambda$×5$\lambda$, the number of populations is set to be 50, the crossover rate to be 0.8, the variance rate to be 0.08, the maximum number of genetic generations to be 100, and the scanning angles $\theta$ = 0°, $\phi$ = 0°.

Fig. 6 shows the array element layout after optimization by the genetic algorithm, and the evolution of fitness values in genetic iterations calculated with Equation (16), where the fitness evolution curve tends to reach the optimal layout after 50 iterations.

Figure  6.  Post-optimized array element position diagram (A), and variation curve of fitness with number of iterations (B).

DownLoad: Full-Size Img PowerPoint

The element spacing, optimized by the genetic algorithm, is shown in Table 1, where the real numbers are the element spacing in the X-direction and the imaginary numbers are the element spacing in the Y-direction in meters, N_x for rows, N_y for columns.

Table  1.  The element spacing optimized by the genetic algorithm

${N}_{x}/N{}_{y}$	1	2	3	4	5	6	7	8
1	0.00+0.00i	0.02+0.70i	0.20+1.33i	0.12+1.87i	0.06+2.39i	0.06+2.95i	0.20+3.65i	0.00+5.00i
2	0.78+0.00i	0.57+0.51i	0.78+1.06i	0.72+1.85i	0.57+2.51i	0.60+3.12i	0.86+3.75i	0.71+4.94i
3	1.43+0.056i	1.13+0.91i	1.51+1.44i	1.37+2.02i	1.46+2.55i	1.28+3.07i	1.45+3.59i	1.21+4.75i
4	1.98+0.04i	1.73+0.86i	2.10+1.58i	1.89+2.39i	2.26+3.03i	1.88+3.78i	1.96+4.33i	1.82+4.93i
5	2.57+0.05i	2.23+0.64i	2.67+1.15i	2.50+1.80i	2.84+2.60i	2.49+3.15i	2.62+3.77i	2.34+4.64i
6	3.11+0.03i	2.98+0.76i	3.18+1.47i	3.00+2.03i	3.44+2.79i	3.10+3.51i	3.26+4.06i	3.14+4.58i
7	3.76+0.37i	3.50+1.25i	3.72+1.84i	3.71+2.36i	4.20+2.91i	4.36+3.59i	3.92+4.17i	3.71+4.75i
8	5.00+0.00i	4.79+0.97i	4.33+1.63i	4.29+2.62i	4.88+3.19i	4.89+3.75i	4.97+4.30i	5.00+5.00i

 | Show Table

DownLoad: CSV

The optimized three-dimensional antenna array pattern is shown in Fig. 7, where the three-dimensional antenna array pattern is calculated from Equation (12). Its beamwidth at scanning angle $\theta$ = 0° and $\phi$ = 0° is 10° and the MSLL is −14.58 dB.

Figure  7.  Three-dimensional antenna array pattern after dilution (at scanning angles $\mathit{\theta }$ = 0°, $\mathit{\phi }$ = 0°).

DownLoad: Full-Size Img PowerPoint

3.3   Comparison of Performance Improvement after Sparse Optimization

To compare the initial uniform plane antenna array pattern with the sparse optimized pattern more intuitively, the two three-dimensional patterns are given a longitudinal section at 15° intervals. Since the three-dimensional pattern of the initial planar array is symmetric about the u-axis and v-axis, the section antenna array pattern with$\phi$ = 0° of the initial plane array is used here to compare the optimized section antenna array pattern with the optimized section antenna array pattern. The comparative plots are shown in Fig. 8 and the beamwidths and the MSLL values are shown in Table 2.

Figure  8.  Post-optimized three-dimensional antenna array pattern cross-section diagram.

DownLoad: Full-Size Img PowerPoint

Table  2.  Beamwidth and SLL (at scanning angles $\mathit{\theta }$ = 0°, $\mathit{\phi }$ = 0°)

Cross-section $\phi$/(°)	Beamwidth/(°)	MSLL/dB
Pre-optimization	13	−12.79
Post-optimization 0	10	−14.73
Post-optimization 12	10	−14.58
Post-optimization 15	10	−15.60
Post-optimization 30	10	−20.79
Post-optimization 45	10	−22.86
Post-optimization 60	10	−23.17
Post-optimization 75	10	−16.80
Post-optimization 90	10	−16.22
Post-optimization 105	11	−16.25
Post-optimization 120	11	−18.55
Post-optimization 135	11	−20.28
Post-optimization 150	11	−16.29
Post-optimization 165	10	−18.61

 | Show Table

DownLoad: CSV

Through the scanning angles $\theta$ = 0° and $\phi$ = 0°, the MSLL is −12.79 dB before sparse optimization and −14.58 dB after sparse optimization. The beamwidth is 13° before the sparse optimization, and 10° after the sparse optimization. The MSLL decreases by 1.79 dB and the beamwidth narrows by 3° through the sparse optimization, so the sparse optimization of the array element spacing of the rectangular planar array using the genetic algorithm can reduce the MSLL and narrow the beamwidth.

To verify the scanning characteristics of the array at a wide scanning angle, we analyze and compare the simulation data after changing the scanning angle of the antenna array. Fig. 9 illustrates the main beam cutaway for the sparse optimization with pitch angle, $\theta$, varying from −40° to 40° and azimuth, $\phi$, fixed at 0°. Tables 3 and 4 list the corresponding beamwidth data and the MSLL. The results show that the MSLL decreases significantly and the beamwidth narrows as $\theta$ moves across the ±30° scan width. However, when $\theta$ exceeds 30°, the beamwidth gradually widens with the increasing angle, and the MSLL also shows a rising trend, and when $\theta$ is 35°, the MSLL is higher than before optimization.

Figure  9.  The three-dimensional antenna array pattern cross-section diagram as the scan angle, $\mathit{\theta }$, changes, with the scan angle $\mathit{\phi }$ = 0°.

DownLoad: Full-Size Img PowerPoint

Table  3.  The beamwidth as the scan angle, $\mathit{\theta }$, changes, with the scan angle $\mathit{\phi }$ = 0°

θ/(°)	Pre-optimization beamwidth/(°)	Post-optimization beamwidth/(°)
±5	13	10
±10	13.5	10
±15	14	10.5
±20	14	11
±25	14.5	11
±30	15.5	11.5
±35	16	12.5
±40	17	13.5

 | Show Table

DownLoad: CSV

Table  4.  The MSLL as the scan angle, $\mathit{\theta }$, changes, with the scan angle $\mathit{\phi }$ = 0°

θ/(°)	Pre-optimization MSLL/dB	Post-optimization MSLL/dB
±5	−12.79	−13.75
±10	−12.80	−13.75
±15	−12.80	−13.75
±20	−12.80	−13.48
±25	−12.80	−13.42
±30	−12.80	−13.43
±35	−12.80	−11.95
±40	−12.80	−9.37

 | Show Table

DownLoad: CSV

Here, the rectangular planar array is compared with the antenna array pattern section after sparse optimization with $\theta$ = ±30°, $\phi$ = 30° and $\theta$ = ±30°, $\phi$ = 45°. As an example, Fig. 10 shows the comparison of the planar rectangular array with the antenna array pattern section after sparse optimization when $\theta$ = ±30°, $\phi$ = 30°, and Fig. 11 shows the comparison of the planar rectangular array with the antenna array pattern section after sparse optimization when $\theta$ = ±30°, $\phi$ = 45°.

Figure  10.  The antenna array pattern cross-section diagram, before and after optimization, at the scanning angles of $\mathit{\theta }$ = ±30°, $\mathit{\phi }$ = 30°.

DownLoad: Full-Size Img PowerPoint

Figure  11.  The three-dimensional antenna array pattern cross-section diagram, before and after optimization, at scanning angles of $\mathit{\theta }$ = ±30°, $\mathit{\phi }$ = 45°.

DownLoad: Full-Size Img PowerPoint

As shown in Fig. 10 and Fig. 11, the sparse optimization produces a significant improvement in the performance of the antenna array under the configurations of scanning angles of $\theta$ = ±30°, $\phi$ = 30° and $\theta$ = ±30°, $\phi$ = 45°. The specific data are shown in Table 5. After the sparse optimization, the MSLL is reduced by 0.65 dB and the beamwidth is narrowed by 4.5° when the scanning angle is $\theta$ = ±30°, $\phi$ = 30°; the MSLL is reduced by 2.24 dB and the beamwidth is narrowed by 4° when the scanning angle is $\theta$ = ±30°, $\phi$ = 45°. This result shows that the sparse optimized scanning angles of $\theta$ = ±30°, $\phi$ = 30° and $\theta$ = ±30°, $\phi$ = 45° are both effective in reducing the MSLL and narrowing the beamwidth.

Table  5.  The beamwidth and MSLL at scanning angles of $\mathit{\theta }$ = ±30°, $\mathit{\phi }$ = 30°, and $\mathit{\theta }$ = ±30°, $\mathit{\phi }$ = 45°

	Pre-optimization $\theta$ = ±30°, $\phi$ = 30°	Post-optimization $\theta$ = ±30°, $\phi$ = 30°	Pre-optimization $\theta$ = ±30°, $\phi$ = 45°	Post-optimization $\theta$ = ±30°, $\phi$ = 45°
Beamwidth/(°)	15.0	11.5	15.5	11.5
MSLL/dB	−12.80	−13.43	−12.80	−15.02

 | Show Table

DownLoad: CSV

4.   CONCLUSIONS

In this study, we primarily explore the optimal design of genetic algorithms for the sparse optimization of element spacing in rectangular planar arrays. Our core strategy is to employ a linear weighting method to transform the multi-objective optimization problem involving MSLL and beamwidth into a single-objective problem. Subsequently, we use the genetic algorithm to optimize the layout of the antenna array, aiming to achieve low MSLL and narrow beamwidth characteristics for the antenna array under wide-angle scanning conditions.

Specifically, we select the MSLL and beamwidths of multiple tangent antenna array patterns passing through the main beam by slicing the three-dimensional antenna array patterns. The MSLL of the antenna array patterns at different scan angles are the optimization objectives. To transform the multi-objective optimization problem into a single-objective problem, we use the linear weighting method, and this single objective is transformed into a fitness function in the genetic algorithm. By iteratively solving the genetic algorithm, we obtain the optimal array arrangement scheme.

The simulation results show that after the sparse optimization, the MSLL decreases by 1.79 dB, and the beam width narrows by 3°. In addition, to verify the performance of the sparse optimization under wide-angle scanning conditions, we select specific scanning angles ($\theta$ = ±30°, $\phi$ = 30°; $\theta$ = ±30°, $\phi$ = 45°) for detailed performance verification of the tangent antenna array pattern of the overall beam. The results show that sparse optimization of the antenna array exhibits optimization in both SLL and beamwidth for $\theta$ within a scan width of ±30°.

Here, sparse optimization is only performed for a single frequency point, but as radio astronomy develops, to carry out a wide range of sky scanning, the wide bandwidth required may cause grating lobes. To effectively address this problem, the periodic pattern of the radiated energy from the array can be disrupted by using an unequal spacing array element layout, which, in turn, achieves the purpose of reducing or eliminating the grating lobe effect. Future work will focus on sparse optimization for ultra-wideband arrays.