International Journal of Antennas and Propagation

Volume 2018, Article ID 6056139, 8 pages

https://doi.org/10.1155/2018/6056139

## Multiobjective Optimization Design of Time-Modulated Concentric Circular Ring Arrays

^{1}Nanjing Research Institute of Electronics Technology, Nanjing 210013, China^{2}College of Physical Science and Technology, Yulin Normal University, Yulin 537000, China^{3}School of Electronic Engineering, Xidian University, Xi’an 710071, China

Correspondence should be addressed to Zhao Wu; moc.361@ytnaik

Received 16 October 2017; Revised 3 January 2018; Accepted 29 January 2018; Published 25 March 2018

Academic Editor: Jaume Anguera

Copyright © 2018 Weilong Liang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A multiobjective approach based on the third evolution step of generalized differential evolution (GDE3) algorithm is proposed for optimizing the time-modulated array (TMA) in this paper. Different from the single-objective optimization, which optimizes a weighted sum of the peak sidelobe level (PSLL) and the peak sideband level (PSBL) of the array, the multiobjective algorithm treats the PSLL and the PSBL as two distinct objectives that are to be optimized simultaneously. Furthermore, not only one outstanding optimization result can be acquired but also a set of solutions known as Pareto front is obtained by using the GDE3 algorithm, which will guide the design of time-modulated array more effectively. Users can choose one appropriate outcome which has a suitable tradeoff between the PSLL and the PSBL. This approach is illustrated through a time-modulated concentric circular ring array (CCRA). The optimal parameters and the corresponding radiation patterns are presented at last. Experimental results reveal that the multiobjective optimization can be an effective approach for the TMA synthesis problems.

#### 1. Introduction

The time-modulated array (TMA) was proposed in 1959 by Shanks and Bickmore [1] and then improved by Kummer et al. in 1963 [2]. Compared with the conventional antenna arrays, an additional high-speed RF switch is connected to each antenna element in the TMA, which introduces the fourth dimension, time, into the design. By controlling the switch-on time interval in a period, the TMA has great flexibility in the control of the aperture excitation which tapers the distribution easily and rapidly. So the realization of low/ultralow sidelobe level (SLL) antenna array becomes much simpler. However, the TMA has an inherent drawback in that there are many sideband signals spaced at multiples of the modulation frequency, which are usually useless. Generally, the sideband level (SBL) needs to be suppressed in order to reduce the energy loss and interference. As the objective function in the TMA pattern synthesis is highly nonlinear and nondifferentiable with different diverse constraint conditions, various algorithms are adopted to optimize the TMA by adjusting the excitation amplitudes or switch-on time sequence. Approaches based on differential evolution (DE) algorithm in [3–5] were proposed to suppress the SBL in the TMA by rearranging the switch-on time. In [6], a linear array with low SLLs, low SBLs, and uniform excitations simultaneously was obtained based on the direct optimization of the switch-on time sequence via the simple genetic algorithm (SGA). Particle swarm optimization (PSO) was used to minimize the power losses in the TMA by properly modifying the modulation sequence [7]. In [8], a novel hybrid algorithm based on the artificial bee colony (ABC) algorithm and DE algorithm called ABC-DE was used to overcome the drawback of the TMA. Furthermore, a hybrid enhanced PSO and DE (hybrid DPSO/DE) was used to optimize the concentric hexagonal antenna array and concentric circular ring array [9]. Also flower pollination algorithm or enhanced flower pollination algorithm is used in the synthesis of circular array antenna [10] and the linear antenna arrays [11].

As mentioned above, the TMA has been optimized by different kinds of optimization algorithms. In these algorithms, single objective is optimized, which means only one best result can be concluded after optimization. However, the electromagnetic optimization objectives are often in conflict with each other, and there may not exist a solution that is the global best one. In fact, there are a set of solutions known as Pareto front or nondominated solutions [12–17]. Therefore, multiobjective evolutionary algorithms have been employed to solve complicated antenna design problems such as the optimal synthesis of linear arrays, planar arrays, or concentric ring arrays [18–22].

In this paper, an approach based on the third evolution step of generalized differential evolution (GDE3) [23] is presented for the TMA optimization. In order to demonstrate the methodology, the time-modulated concentric circular ring array (CCRA) is considered. Two objective functions, the peak sidelobe level (PSLL) and peak sideband level (PSBL) of the time-modulated CCRA, are optimized as a biobjective problem. By optimizing the normalized switch-on time sequence, the number of elements, and the ring spacing, an extensive set of solutions is obtained, and users can choose the most suitable one from it. This method shows the relationship between the PSLL and the PSBL of the TMA, which indicates that the PSLL is inversely proportional to the PSBL.

The rest of this paper is organized as follows. Section 2 describes properties of the time-modulated CCRA and the parameters defined the radiating structure. Then, the numerical results after optimization are presented in Section 3. Finally, the conclusion is given in Section 4.

#### 2. Pattern Synthesis with Time Modulation Technique

The configuration of *N _{r}* ring CCRA with ring

*n*having

*N*equally spaced isotropic elements at a radius of

_{n}*r*is shown in Figure 1. The physical distance between adjacent elements on ring

_{n}*n*is constant. In the time-modulated CCRA, all the antenna elements in the same ring have the same weight and are controlled by a same high-speed RF switch. The far-field array factor of the time-modulated CCRA is expressed as where is the center frequency, is the elevation angle with respect to

*Z*axis,

*β*is the wavenumber, and is the periodic switch-on time sequence function in which antenna elements on ring are switched on for () in each period . is defined as