# A Hybrid Optimization Algorithm for the Synthesis of Sparse Array Pattern Diagrams
## 1 Introduction  
Phased array antenna systems have emerged as the core architecture of modern radar and communication systems, due to their unique electronic beam agility and superior low probability of intercept (LPI) characteristics [1, 2]. These systems achieve dynamic beam pattern reconfiguration through precise phase weighting control, not only overcoming the physical limitations of traditional mechanically scanned antennas in terms of response speed but also demonstrating outstanding performance in multi-target tracking, adaptive beamforming, and high-precision direction-finding. In phased array system design, the optimization of array element spacing--as a critical parameter determining radiation characteristics--directly impacts several key performance metrics [3]. Firstly, improper spacing configuration may induce grating lobes, leading to radiation energy dispersion and the significant degradation of main lobe gain. Secondly, inappropriate spacing selection can exacerbate the mutual coupling effects between elements, causing pattern distortion and deterioration of the system noise figure. Moreover, spacing optimization crucially influences essential indicators such as equivalent radiated power (ERP) and angular resolution [4]. Compared to conventional uniform arrays, non-uniform sparse arrays demonstrate remarkable advantages through optimized spatial element distribution [5, 6]. These include: (1) reduced peak sidelobe level (PSLL); (2) narrower 3 dB beamwidth, with identical element count; (3) effective grating lobe suppression through aperiodic arrangement; and(4) significant mitigation of the mutual coupling effects between elements. Current implementations of non-uniform arrays primarily follow two technical approaches: one involves grid-constrained sparse arrays, where element positions are restricted to predefined discrete grid points, while the other involves completely unrestricted thinned arrays, allowing continuous element distribution within a given aperture [7]. In certain specialized applications, thinned arrays exhibit superior performance potential in terms of main-lobe width compression, grating lobe suppression, and pattern synthesis owing to their greater degrees of freedom, making them particularly suitable for advanced phased array system designs with stringent requirements regarding the radar cross-section (RCS) and low observability [8; 9]. The radiation pattern of an antenna is a key indicator for the design of sparse array antennas, affecting the overall design and compensation effect of the antenna array. To solve the complex optimization problem of radiation pattern synthesis, researchers have proposed various intelligent algorithms. In Reference [10], a sparse array optimization algorithm based on a cuckoo search was proposed. This algorithm uses sinusoidal chaotic mapping instead of a fixed step size factor, which improves the global search ability to some extent, but it cannot escape the computational efficiency bottleneck caused by the cuckoo search algorithm. In Reference [11], an improved sparrow search algorithm for sparse linear array synthesis was proposed. Although the use of tent chaotic mapping for antenna element position initialization in this algorithm improves the peak sidelobe level to some extent, its application scope is limited to sparse linear arrays. In Reference [12], a multi-objective array optimization algorithm based on convex programming and particle swarm optimization was proposed, but the test scenarios were too limited to be convincing. Other methods, such as hybrid particle swarm optimization (PSO) [13; 14], the grey wolf optimization algorithm (GWO) [15; 16], and the whale optimization algorithm (WOA) [17; 18], can optimize array performance to some extent, but they generally suffer from inherent defects such as slow convergence speed, limited application scenarios, and easy entrapment in local optima.