###### Abstract  
Sparse arrays with low cost and engineering complexity are widely applied in many fields. However, the high peak sidelobe level (PSLL) of a sparse array causes the degradation of weak target detection performance. Particularly for the large size of underwater low-frequency sensors, the design problem requires a minimum spacing constraint, which further increases the difficulty of PSLL suppression. In this paper, a novel swarm-intelligence-based approach for sparse sensor array design is proposed to reduce PSLL under spacing constrains. First, a global enhancement whale optimization algorithm (GEWOA) is introduced to improve the global search capability for optimal arrays. A three-step enhanced strategy is used to enhance the ergodicity of element positions over the aperture. In order to solve the adaptation problem for discrete array design, a position decomposition method and a V-shaped transfer function are introduced into off-grid and on-grid arrays, respectively. The effectiveness and superiority of the proposed approach is validated using experiments for designing large-scale low-frequency arrays in the marine environment. The PSLL of the off-grid array obtained by GEWOA was nearly 3.8 dB lower than that of WOA. In addition, compared with other intelligent algorithms, the on-grid array designed using GEWOA had the lowest PSLL. sparse array design; underwater acoustic sensor array; peak sidelobe level; intelligence optimization algorithm +
Footnote †: journal: Applied Sciences  
elements deviate from the designed positions. For the seabed array mounted at the bottom of sea, the uncertainties of the element positions and the entire array configuration are induced by ship drift and subsurface currents when the array deploys to the seafloor from the layout vessel. We focus on the first type problem under the marine environment to design a random sparse array with low PSLL to enhance the detection of weak targets [8]. The existing methods to solve the first problem are mainly divided into two categories: deterministic optimization methods and stochastic optimization methods. Deterministic optimization methods, including minimum redundancy array [9], coprime array [10], and difference coarray [11], are easy to implement; however, the designed arrays are generally suboptimal. Stochastic optimization methods use intelligent algorithms to find the element positions that satisfy the design requirements. Genetic algorithm (GA) [12], particle swarm optimization (PSO) algorithm [13], and simulated annealing algorithm [14] have been successfully applied to sparse sensor array design. However, these intelligent algorithms converge slowly and easily fall into local optima. Some improved algorithms [15] have been proposed to solve the problem for array design--for example, a series of swarm intelligence optimization algorithms mimicking swarm behavior, including the wolf pack algorithm (WPA) [16], bat algorithm (BA) [17], and cuckoo search (CS) algorithm [18], which show good search accuracy and quick convergence speed. Although the stochastic optimization method suffers from global performance degradation during the design process of large-scale arrays, this method outperforms the deterministic optimization method in terms of the array performance as the calculation capacity increases. The whale optimization algorithm (WOA) [19]--a swarm intelligence algorithm--has been applied successfully to sparse sensor array design for both fixed aperture and non-fixed aperture and has shown its advantages in sidelobe suppression and null steering for small- and medium-scale arrays [20,21]. Combining the salp swarm algorithm (SSA) and WOA, a novel swarm algorithm was proposed to improve its convergence accuracy for conformal antenna array design [22] and for dual-band aperture-coupled antenna design [23]. Nevertheless, the global search capability of WOA decreases with the expansion of the search range in large-scale array design. Researchers have attempted to improve the algorithm from several aspects in successive steps.