An efficient pattern synthesis approach is proposed for the synthesis of a time-modulated sparse linear array (TMSLA) in this paper. Due to the introduction of time modulation, the low/ultralow side lobe level can be obtained with a low amplitude dynamic range ratio. Besides, it helps reduce the difficulty of antenna feeding system effectively. Based on particle swarm optimization (PSO) and convex (CVX) optimization, this paper proposes a hybrid optimization method to suppress the grating lobes of the sparse arrays, peak side lobe level (PSLL), and peak sideband level (PSBL). Firstly, the paper utilizes the CVX optimization as a local optimization algorithm to optimize the elements’ switch-on duration time, which reduces the side lobe of the array. Secondly, with the PSBL as the objective function, the paper adopts the PSO as a global optimization algorithm to optimize the elements’ positions and switch-on time instant, which helps reduce the loss of sideband power caused by time modulation. With respect to the time modulation model, variable aperture sizes (VAS) and more flexible pulse-shifting (PS) schemes are used in this paper. Owing to the introduction of time modulation and CVX optimization, the proposed method is much more feasible and efficient than conventional approaches. Furthermore, it has better array pattern synthesis performance. Numerical examples of the TMSLA and comparisons with the reference are presented to demonstrate the effectiveness of the proposed method.

Because of its low side lobe level and strong radiation directivity, the antenna array is easy to realize beam scanning and shaping and has been widely used in radars, wireless communication, and other fields. However, a large number of elements in the antenna array increase the structural complexity and feeding difficulty of the antenna system. Therefore, the sparse array with random array elements can achieve beam shaping, low mutual coupling, economical efficiency, and high resolution [

A family of position mutated hierarchical particle swarm optimization algorithms with time-varying acceleration coefficients has been presented in [

Evolutionary algorithms are increasingly widely used in the optimization of sparse arrays. Generally, global optimization processes such as genetic algorithm (GA) [

A modified differential evolution (DE) algorithm based on harmony search algorithm was proposed in [

Since the optimal solution of convex optimization is global optimal solution, more scholars use convex optimization in sparse array synthesis. Fuchs used convex optimization to realize beamforming in sparse arrays. The convex problem solved iteratively was transformed into a standard second-order cone programming (SOCP) for calculation, which achieved simultaneous optimization of element position and excitation, thus suppressing the grating lobes and obtaining the desired beam [

The algorithm mentioned above will increase the feed dynamic range ratio (DRR) of the antenna system. In order to reduce it, a time-modulated array with low complexity has been extensively studied in recent years. By optimizing the amplitude excitation of the sparse array in the variable aperture sizes mode, Poli et al. obtained a lower side lobe level and effectively suppressed the sideband level [

According to the analysis of the above research, this paper proposes a joint optimization algorithm based on PSO and time modulation in order to solve the problems of high side lobe and gating lobe caused by the sparse arrays and high dynamic excitation amplitude ratio in the optimization process. The algorithm establishes the mathematical model of the time-modulated sparse array based on convex optimization theory, converts the amplitude and phase weights of the array elements into the optimal switch-on duration time, and reduces the dynamic amplitude ratio of the array. The PSO algorithm is used to optimize the position and switch-on time instant of the array elements; meanwhile, because the solution of convex optimization is the global optimal solution, the global optimal switch-on duration time can be obtained on the condition that the position of array elements and the switch-on time instant are fixed and the ultralow side lobe can be realized ultimately. At the same time, under the constraints of side lobe level and bandwidth, the sideband level caused by time modulation is reduced.

In this paper, an array of

Diagram of a linear sparse array with

When the time modulation with a period of

The far-field radiation electric field intensity at the center frequency

The peak side lobe level (PSLL) is defined as

The introduction of time modulation leads to a high sideband level, which reduces the radiation power. Therefore, it is necessary to minimize the sideband level on the premise of constraining the side lobe level. The sideband level can be defined as

Its objective function can be modified to

Because the space of the adjacent element is usually larger than half wavelength in the sparse array, optimizing the array element time sequences alone cannot effectively suppress the gate lobe and reduce the side lobe. Thus, based on the application of time modulation technology, the position of array elements and the switch-on duration time of array elements are taken as joint optimization variables to increase the dimension of optimization variables. However, with the introduction of an element position variable, the optimization problem will become a high-dimensional nonlinear nondeterministic polynomial hard (NP-hard) problem, which is no longer a standard convex optimization problem. Considering that the local optimal solution is the global optimal solution in convex optimization problems, the optimization model of the PSO-CVX hybrid algorithm is given by combining particle swarm optimization and convex optimization. The convex optimization algorithm is used to solve the optimal excitation and switch-on duration time when the element position is determined. The PSO algorithm iteratively solves the optimal element position when the optimal excitation and switch-on duration time are given. Then, the optimization problem is transformed into a local convex optimization problem to optimize the antenna pattern of the sparse array.

Variable aperture size (VAS) modulation makes the aperture size of the uniform linear array change periodically, and the RF switch corresponding to each antenna unit closes simultaneously in the beginning of each timing cycle

Pulse shifting (PS) was proposed by Poli et al., an Italian scholar [

Comparing formulas (

According to the definition of the convex optimization algorithm, formula (

In order to achieve sideband suppression in the time modulation sparse array, the variables needed to be optimized are the normalized switch-on time instant

Scholars Clerc and Kennedy proposed a compression factor method as a new speed update formula for the PSO algorithm [

Usually,

This paper combines the advantages of PSO and convex optimization algorithms. Because of its high efficiency and simplicity, the PSO algorithm is used as a global optimization algorithm. The optimal element position and turn-on time are obtained by particle iterative evolution. Considering that the convex optimization algorithm has the characteristic that the local optimal solution is the global optimal solution, the CVX algorithm is used as the local optimization algorithm to obtain the optimal weight variable

Flow chart of PSO-CVX.

Set the number of array elements, the lobe width of zero power point, and the sampling interval of angle for array model

Initialize the particle population, position, iterations, and speed of each particle

Calculate the value of the steering vector

WHILE the maximum number of cycles has not been reached, DO

Obtain weight coefficients of each array element by the CVX algorithm

Compute fitness values

Update global optimal solution and historical optimal solution

Update positions and velocities of the particles

Increase the loop counter

End cycle and display the best results of pattern synthesis

When the number of array elements is constrained in the limited space platforms such as aircraft and ships, in order to improve the performance of the antenna array and restrain the grating lobe caused by the spacing of array elements larger than

Standard deviation is a measure of the discreteness of the test results, which can reflect the stability of the optimization algorithm. Its formula is as follows:

In order to prove the superiority of the algorithm mentioned above, this section gives several simulation and comparison results of the time modulation sparse linear array optimized by the PSO-CVX algorithm. Firstly, the TMSLA is compared with the time-modulated linear uniform array, traditional linear uniform array, and Chebyshev array to verify the effect of time modulation on side lobe suppression. Then, the sparse array optimized by the PSO-CVX algorithm is compared with the references from different aspects. Secondly, this section analyzes the influence of phase change on the TMSLA. Finally, by optimizing the position, the switch-on duration time, and the switch-on time instant of array elements, the sideband level suppression of the TMSLA is simulated and compared with the algorithms in some references.

The simulation in this section adopts the time modulation mode of VAS. As mentioned in Section

The antenna array consists of 17 elements, and the central frequency is

Figure

TMLA, TMSLA, Chebyshev, and traditional patterns.

Figure

Histogram of normalized switch-on duration time distribution of 17-element TMSLA.

The PSO-CVX algorithm proposed in this paper has been tested 20 times independently for the TMSLA. Figure

Fitness curve of the PSO-CVX algorithm.

Using the same parameters to simulate on the same computer, the PSLL of antenna array is −31.26 dB by optimizing the excitation amplitude and position with the PSO algorithm, which is 5.28 dB higher than that with the PSO-CVX algorithm. Meanwhile, the beam width of 3 dB is 5.4 degree, which is 0.8 degree larger than that optimized by the PSO-CVX algorithm, as shown in Figure

Contrast between the PSO-CVX and PSO algorithms.

Figure

Diagram of optimized array layout of PSO-CVX.

In order to further verify the superiority of time modulation technology, some simulations are made to compare the time modulation sparse array with the 16-element local sparse array by using the exploratory harmony search (EHS) optimization algorithm mentioned in [

PSLL comparison between the array mentioned in [

Comparison object | Optimal PSLL (dB) | Worst PSLL (dB) | SD of PSLL | −3 dB bandwidth (°) | Optimized location (right side) | ||
---|---|---|---|---|---|---|---|

[ |
−13.59 | −13.47 | 0.07 | 10.4 | 3.66 | 4.36 | 5.25 |

PSO-CVX | −17.49 | −17.1 | 0.18 | 5.4 | 2.99 | 3.83 | 4.696 |

The statistical results in Table

The modified Bayesian optimization algorithm (M-BOA) proposed in [

Pattern comparison of PSO-VAS and M-BOA.

Fitness curve of PSO-VAS and M-BOA. (a) Fitness of PSO-VAS. (b) Fitness of M-BOA.

Comparison between the time modulation array and modified Bayesian array.

Comparison object | Optimal PSLL (dB) | Worst PSLL (dB) | Average PSLL |
---|---|---|---|

M-BOA in [ |
−23.9 | −23.15 | −23.38 |

TMSLA | −26.0989 | −26.0989 | −26.0989 |

In [

Performance comparison of the TMSLA and SLA in [

Comparison object | Sparse ratio | PSLL | Iterations | Array aperture ( |
Element spacing ( |
---|---|---|---|---|---|

Array in [ |
31.4% | −35.3 dB | 200 | 18.3 | 0.5∼1 |

PSO-CVX | 31.4% | −39.12 dB | 100 | 18.08 | 0.5∼1 |

PSLL comparison between PSO-CVX and the graph of the optimized excitation amplitude in [

In order to verify the influence of phase change on the peak side lobe level, the peak sideband level, the switch-on duration time, the and sparse ratio of array, some simulations are made in the PS mode for in-phase and out-phase sparse arrays, respectively. Considering a 17-element symmetric sparse array with a peak side lobe level constraint of −30 dB, the optimization objective is to minimize the peak sideband level of the first and second sideband. The normalized turn-on time

Selected from 20 independent experiments, the best normalized antenna pattern is shown in Figure

In-phase excitation pattern of using the PSO-CVX algorithm in the PS mode.

The simulation parameters in this section are consistent with those in the previous section, except that the phase excitation of the array element becomes an adjustable optimization variable which participates in the PSO-CVX algorithm. The best normalized antenna pattern is shown in Figure

Pattern with equal amplitude and out-phase excitation in the PS mode.

As shown in Table

Comparison of robustness between in-phase and out-phase TMSLA.

Excitation mode | Stable iterations | Average PSBL (dB) | Best PSBL (dB) | Worst PSBL (dB) |
---|---|---|---|---|

In-phase | 85 | −21.32 | −22.7 | −20.08 |

Out-phase | 90 | −21.05 | −23.07 | −19.38 |

Comparison of PSBL between in-phase and out-phase TMSLA.

Excitation mode | Expected PSLL (dB) | Optimized PSLL (dB) | First PSBL (dB) | Second PSBL (dB) |
---|---|---|---|---|

In-phase | −30 | −31.05 | −22.7 | −27.1 |

Out-phase | −30 | −30.13 | −23.07 | −23.10 |

The experimental results of the first 30 sideband levels of the time-modulated sparse array with out-phase excitation and in-phase excitation are compared as shown in Figure

Sideband level of in-phase and out-phase excitation.

In Figure

Position arrangement of the PS-modulated sparse array with in-phase and out-phase excitation.

From the above comparison, in terms of element timing and element position arrangement, the difference is little between the simulation results of the time-modulated sparse array with out-phase excitation and in-phase excitation. In addition, due to the introduction of phase excitation variables, the optimization variables and computational complexity of the algorithm increase, resulting in a lower optimization speed and an increase in the number of iterations of the algorithmic convergence.

Taking formula (

Sideband level comparison of the time-modulated sparse array with equal amplitude and phase excitation. (a) Comparison of first side band level. (b) Comparison of second side band level.

Sideband level comparison of two time modulation modes.

Algorithm | Expected PSLL (dB) | Optimized PSLL (dB) | First PSBL (dB) | Second PSBL (dB) |
---|---|---|---|---|

PSO-CVX with VAS | NULL | −36.54 | −11.2 | −15.4 |

PSO-CVX with PS | −30 | −31.05 | −22.7 | −27.1 |

Figure

Position arrangement comparison of the sparse array with PS modulation and VAS modulation.

In order to further verify the superiority of the optimization algorithm in this paper, it is compared with [

16-element equal-amplitude in-phase excitation in the PS mode.

Sideband level comparison of the sparse array and array with a spacing of

Parametric comparison of 16-element between TMSLAs in the PS mode and the results in [

Comparison object | Modulation method | PSLL (DB) | PSBL (DB) | −10 dB bandwidth (DB) |
---|---|---|---|---|

[ |
PS | −30 | −19.5 | 13 |

This paper | PS | −30 | −21.75 | 9.4 |

In [

Pattern of 11-element equal-amplitude in-phase excitation in the PS mode.

Parametric comparison of 16-element between TMSLAs in the PS mode and the results in [

Comparison object | Modulation method | PSLL (DB) | PSBL (DB) | −3 dB bandwidth (DB) | Aperture width ( |
---|---|---|---|---|---|

[ |
VAS | −25.0 | −20.08 | 8.5 | 7.836 |

This paper | PS | −25.11 | −21.32 | 7.2 | 6.850 |

As shown in from Figure

Sideband level comparison in the PS and VAS mode.

This paper focuses on the application of the PSO-CVX algorithm in side lobe suppression of the TMSLA. Compared with the traditional uniform array, the introduction of time modulation can replace the amplitude excitation with the timing control of array element to effectively suppress the side lobe level of the antenna array due to the introduction of time modulation. In order to realize the sparse ratio of the array and to avoid the occurrence of gating lobe, the element position variable is introduced, and the optimization model becomes a nonlinear high-dimensional complex optimization, which is transformed into a lower-dimensional particle swarm optimization and a locally convex optimization. The PSO algorithm is used to optimize the position variables of the particles, and the CVX algorithm is used to solve the equivalent complex excitation consisting of switch-on duration time and static phase excitation. The pattern optimized by the PSO-CVX hybrid algorithm shows that the introduction of position variables can effectively reduce the side lobe level of the antenna array, and the TMSLA has more advantages in synthesizing low/ultralow side lobe patterns. At the same time, by optimizing the objective function of variables, the degree of freedom of optimization is increased, and the number of optimization variables is effectively controlled. The algorithm reduces the number of iterations needed to converge to the optimal solution, which shows the effectiveness of the algorithm. In addition, under the same side lobe suppression effect, the PSO-CVX algorithm using the PS mode can synthesize a lower sideband level and narrower main beam. The array antenna system has better robustness and overall radiation characteristics.

Previously reported formula (1), (3), (9), and (12) data were used to support this study and are available at DOI:

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the Fundamental Research Funds for the Central Universities (Grant nos. NS2016043 and kfjj20180406), the National Natural Science Foundation of China (Grant no. 61671239), and the Aeronautics Science Foundation of China (Grant nos. 2017ZC52036 and 20172752019).