Antenna arrays have been widely used in various fields such as radars, communications, navigation, and remote sensing. Compared with a single antenna, an antenna array usually has high gain, beam scanning capability, and flexible pattern shape control ability. Especially in the fifth-generation (5G) and sixth-generation (6G) wireless communications, the design of antenna arrays has been identified as one of the key technologies for systems to support the increasing demand for connectivity and data rates [1].

In the past decades, uniformly spaced arrays have been extensively studied and used due to the convenience of array configurations. However, there are some potential problems in the application of uniform arrays. As is well known, the periodicity of element positions is easy to cause the appearance of grating lobes especially when beam scanning over a relatively large bandwidth is required [2]. To avoid this situation, the element spacing is restricted to not exceed one wavelength for a broadside beam and half a wavelength for a wide-angle scanning beam. In addition, when a narrow beam is required for a high-resolution application, a large number of elements would be required. For an active phased array, each antenna element is usually connected to a single transmit/receive module (TRM). Thus, a large-aperture array with a spacing of half a wavelength would need a large number of TRM, resulting in some challenging problems in the system weight and cost as well as heat dissipation aspects.

To address these issues, various methods for designing sparse antenna array have been developed. The early research on sparse spaced arrays may date back to the 1950s when a matrix relationship between the array elements and the corresponding far-zone pattern was found for linear arrays with arbitrarily distributed elements [3]. Since then, sparse arrays have gained increasing attention, leading to the development of advanced and effective design methods. This paper contributes to the existing literature by categorizing sparse arrays into three distinct types based on their array architecture: Thinned arrays, nonuniformly spaced arrays, and clustered arrays.

A thinned array, achieved by selectively removing elements from a uniformly spaced array, offers a simple configuration. While this reduces the required number of elements, it comes at the cost of a slight increase in beam width for a given aperture. In essence, thinned arrays provide a cost-effective solution, decreasing system cost while maintaining comparable array resolution. Besides, the element positions can be appropriately chosen to produce a pattern with a reduced sidelobe level (SLL). The concept of thinned arrays was firstly reported in the 1960s [4]. Since then, some analytic methods, including density tapering [5,6] and dynamic programming [7,8], have been proposed to design thinned arrays. What is more, the genetic algorithm (GA), which employs binary coded genes to selectively activate elements in a conventionally fully occupied array, was first introduced in the mid-1990s [9]. Subsequently, other stochastic optimization (SO) methods, such as simulated annealing (SA) [10], particle swarm optimization (PSO) [11–13], and some others [14–16] have also been applied to address the sparse array synthesis problem. However, these SO methods would be time-consuming, especially when optimizing a large number of elements. To mitigate the time cost associated with large-scale thinned array synthesis, an iterative Fourier transform (IFT) approach was presented [17–19]. This method successfully exploits an inverse discrete Fourier transform relationship between the array factor (AF) and element excitations, enabling the rapid synthesis of a thinned element distribution. In addition, several improved IFT methods [20–22] have been introduced, in which the number of selected elements gradually decreases as the iterations proceed. Furthermore, the distribution of selected elements can be updated more easily through pattern modification in each iteration. These modifications speed up calculations and promote global convergence. Despite the efficiency of these methods in designing thinned arrays, the obtained elements are usually selected from half-wavelength spaced arrays, significantly limiting the solution space for array geometry optimization.

In nonuniformly spaced arrays, the arrangement of array elements within a given antenna aperture is more flexible, thus offering the potential for enhanced pattern performance compared to thinned arrays. Various advanced methods have been proposed to design these nonuniform arrays. In the early stages, analytical methods [23–26] were developed. For example, a numerical iterative method based on the steepest descent, utilizing element positions and excitations, has proven to be a potent tool for pattern synthesis [25,26]. Besides, SO methods have been employed to synthesize nonuniformly spaced arrays with given element spacing ranges [27–32]. What is more, sparse array reconstruction methods, such as matrix pencil methods (MPM) [33–38] and Bayesian compressive sensing (BCS) [39–42], have been introduced to make the obtained pattern match a reference one. However, these approaches require the predefinition of a reference pattern, which may cause a little bit inconvenience and reduce the solution space to some extent as well. In recent years, some iterative convex optimization (ICO) methods [43–50] have emerged, eliminating the need for reference patterns. In these methods, the sparse array problem is formulated as selecting the element positions from a densely spaced initial array under multiple pattern constraints. Typically, it is solved by executing a sequence of reweighted l₁-norm optimization problems.

However, both thinned arrays and nonuniformly spaced arrays, when compared to conventional fully occupied arrays, entail a loss of gain and array aperture efficiency due to the reduction of antenna element. In response to this challenge, the concept of clustered arrays has been proposed, aiming to maintain the same antenna element distributions as uniform arrays while reducing the number of TRMs. Different from conventional array beamforming network (BFN) designs, the clustered arrays place TRMs at a subarray level instead of an element level, leading to significant reduction in the overall number of array channels. To achieve clustered arrays with desired pattern performances, plenty of effective methods have been proposed. The SO methods, as detailed in Refs. [51–55], have been employed to optimize both the sizes of subarrays and excitations at subarray ports, yielding desired patterns for clustered arrays. Besides, the excitation matching (EM) method, introduced in Refs. [56–62], is proposed to synthesize subarray patterns that match a reference pattern by adjusting the subarray excitations based on a set of reference element excitations. Furthermore, by only optimizing the subarray-level excitation, the element excitation vector can be sparsely described, effectively transforming the clustered array synthesis problem into a sparse signal recovery challenge. Consequently, the ICO technique can also be applied to design the layout of clustered arrays, as demonstrated in Refs. [63–65]. More recently, this technique has been extended to facilitate the design of rectangular and circle planar arrays featuring irregular subarray tiling [66–71].

Numerous methods for designing sparse antenna arrays have been developed over the past decades. Consequently, it is necessary to review these methods with a comparative study on their performances. Though some published articles have reviewed synthesis methods for sparse arrays [72,73], the design methodologies have not been systematically and adequately discussed. Moreover, recent years have witnessed the emergence of new technologies [22,45,65], which necessitates their introduction into the discussion. In this paper, we will discuss synthesis methods for thinned arrays, nonuniformly spaced arrays, and clustered arrays, respectively, and give numerical results to analyze the performance of different synthesis methods.

In Section 2, we introduce the definition and formula description of the three types of sparse arrays. Section 3 provides a detailed exploration of various design methods applied to thinned arrays, nonuniformly spaced arrays, and clustered arrays. Moreover, the merits and demerits of these methods across different array types are discussed. In Section 3, we draw conclusions from our findings and discuss potential research directions. Additionally, all presented results were obtained using a personal computer equipped with an AMD Ryzen 7 5800H CPU at 3.20 GHz.

Let us consider an array with N radiating elements located at the positions denoted by $ {{\bf{r}}_n} $ (n = 1, 2, ···, N). The array pattern can be written as follows:

$ \begin{gathered} F(\theta {\mathrm{,}}\;\phi ) = \sum\limits_{n = 1}^N {{I_n}{E_n}(\theta {\mathrm{,}}\;\phi ){e^{{\mathrm{j}}\beta {{\bf{r}}_n} \cdot {{\boldsymbol{\mathbm{α}}}} (\theta {\mathrm{,}}{\text{ }}\phi )}}} { { = }}\sum\limits_{n = 1}^N {{w_n}{E_n}(\theta {\mathrm{,}}\;\phi ){e^{{\mathrm{j}}(\beta {{\bf{r}}_n} \cdot {{\boldsymbol{\mathbm{α}}}} (\theta {\mathrm{,}}{\text{ }}\phi ) + {\varphi _n})}}} \end{gathered} $ (1)

where $ {I_n} = {w_n}{e^{{\mathrm{j}}{\varphi _n}}} $ is the complex excitation of the nth element; w_n and φ_n are the amplitude and phase of $ {I_n} $, respectively. The parameter β = 2π/λ denotes the wave number in free space, $ {{{\boldsymbol{\mathbm{α}}}} _n}(\theta {\mathrm{,}}\;\phi ) = (\sin \theta \cos \phi {\mathrm{,}}\; \sin \theta \sin \phi {\mathrm{,}}\;\cos \theta ) $ is the propagation direction vector, and $ {E_n}(\theta {\mathrm{,}}\;\phi ) $ is the far-field pattern of the nth element. If mutual coupling effect and edge effect can be ignored in a linear or planar array, individual element patterns can be considered identical. Consequently, the array pattern can be expressed as the product of the element pattern and AF. In this situation, the pattern synthesis problem can be transformed as optimizing AF. AF is

$ {\mathrm{AF}}(\theta {\mathrm{,}}\;\phi ) = \sum\limits_{n = 1}^N {{w_n}{e^{{\text{j}}(\beta {{\bf{r}}_n} \cdot {{\boldsymbol{\mathbm{α}}}} (\theta {\mathrm{,}}{\text{ }}\phi ) + {\varphi _n})}}} . $ (2)

Assuming that the array is placed on xoy-plane, the AF can be expressed as

$ \mathrm{AF}(\theta\mathrm{,}\; \phi)=\sum\limits_{n=1}^Nw_ne^{\mathrm{j}[\beta(x_n\sin\theta\cos\phi+y_n\sin\theta\sin\phi)+\varphi_n]}. $ (3)

As mentioned previously, sparse arrays can be classified into three categories: Thinned arrays, nonuniformly spaced arrays, and clustered arrays. Mathematically, these three kinds of sparse arrays can be distinguished based on the parameters of AF.

a) A thinned array means that some of its elements are removed from the corresponding uniformly spaced array. The desired patterns are achieved by turning ON part of elements, while the turned OFF elements are either connected to matching loads or simply removed.

$ I_n=\left\{\begin{gathered}1\mathrm{,}\text{ turned ON element} \\ 0\mathrm{,}\text{ turned OFF element}.\end{gathered}\right. $ (4)

A turned ON element is assigned an amplitude of one, while a turned OFF element is assigned an amplitude of zero. Specifically, this article focused on the selection of element locations, although there are some papers, such as Refs. [74–76], that talk about joint optimization of element positions and excitations.

b) In a nonuniformly spaced array, the array elements can be flexibly placed within the array aperture Ω, providing more degrees of freedom to improve pattern performance compared to thinned arrays. For a general nonuniformly spaced array, the position of the nth element is given by

$ {{\bf{r}}_n} = ({x_n}{\mathrm{,}}\;{y_n}{\mathrm{,}}\;{z_n}) \in \Omega {\mathrm{,}}{\text{ }}n = 1{\mathrm{,}}\;2{\mathrm{,}}\; \cdots {\mathrm{,}}\;N . $ (5)

However, optimizing the element positions of a nonuniformly spaced array is a highly nonlinear problem.

c) When N elements are grouped into M (M$ \ll $N) clusters or subarrays, and the elements in each cluster are equipped with the same TRM—encompassing both amplifier and phase shifter—such an array is referred to as a clustered array:

$ {I_n} = {C_{m{\mathrm{,}}n}}\sum\limits_{m = 1}^M {{w_m}{e^{{\mathrm{j}}{\varphi _n}}}} $ (6)

where

$ C_{m\mathrm{,}\; n}=\left\{\begin{gathered}1\mathrm{,}\text{ if the }n\text{th element belongs to the }m\text{th subarray} \\ 0\mathrm{,}\text{ }\mathrm{othersize.} \\ \end{gathered}\right. $ (7)

Thanks to their architecture modularity and high aperture efficiency, clustered arrays offer a suitable tradeoff, balancing radiation pattern performance and system cost.

When considering the design of a sparse antenna array, one must initially consider the array implementation architectures in line with engineering aspects. For different array architectures, the optimization objective, optimization variables as well as design constraints may be much different. To the best of our knowledge, there is no such a method which can universally offer the optimal solution for all synthesis problems across different applications. In the following subsections, we review a variety of sparse array synthesis methods, discussing them separately based on their applicability to synthesizing different types of sparse arrays. Furthermore, we present a series of representative results to illustrate the advantages and disadvantages of these methods.

Over the past sixty years, thinned arrays have found applications in various fields, such as radar system and 5G millimeter-wave (mm-wave) communications. Leveraging turned ON element positions chosen from uniformly spaced grids, a thinned array can achieve desired patterns using fewer elements compared to conventional fully occupied array. Additionally, strategic excitation of elements in specific grids can produce patterns with low SLL. Thinned arrays were firstly reported in the 1960s [4], then numerous improved and advanced methods have been proposed for their design. These methods can be roughly classified into three categories: Analytic methods [5–8,18,77], SO methods [9–16], and IFT techniques [17–22].

At early times, density taper methods were proposed [4–6]. For instance, the positions of turned ON elements are statistically selected by using the amplitude taper as a probability density function in Ref. [5]. However, the results obtained by this method do not exhibit a significant improvement in pattern performance compared with random methods [18,77]. Especially in Ref. [18], a random density taper (RDT) method can easily arrange the random element positions to produce a pattern satisfying the SLL constraint using the MATLAB function randperm Different from the above methods starting from preset spacing, the dynamic programming procedure, acting as an iterative process, builds up one element at a time [7,8]. Though these analytic methods can effectively provide a solution for the distribution of element positions, they cannot guarantee resulting patterns exhibit optimal performance.

As a powerful global optimization tool, SO methods have been applied to design thinned arrays. GA, utilizing binary coded genes to switch on/off the antenna elements, was first applied to achieve optimal thinned arrays in the 1990s [13]. Since then, more and more SO methods such as SA [10], PSO [11–13], and others [14–16] have been used. Moreover, the synergy of SO algorithms with advanced techniques has given rise to hybrid methods, like difference sets of Hadamard type PSO (HSPSO) [12], compact GA (cGA), and modified compact GA (M-cGA) [15], which offer effective solutions with rapid convergence. For example, the HSPSO method is utilized for finding the best compromise solution in terms of both SLL and filling factor (the ratio of turned ON elements to the total number) by combining the noncyclic difference sets of Hadamard type and PSO. Recently, a novel precoded subarray structure strategy has been introduced to transform the thinning of a planar array into the arrangement of predetermined subarrays with specified thinning rates. Additionally, an evolutionary approach has been applied to determine both the quantity and the positions of precoded subarrays with varying element arrangements [16]. However, it is worth noting that these methods are usually time-consuming, especially when optimizing a large number of elements.

The IFT technique proves efficient in realizing thinned arrays, even on a large scale, leveraging the inversediscrete Fourier transform relationship between the AF and the excitations of uniformly spaced elements. This relationship enables the rapid acquisition of thinned element distributions. What is more, the technique has been successively utilized for designing linear, square planar, and circular planar thinned arrays with desired patterns [17–19]. However, the procedure of these methods is easily trapped into local minima. To avoid this situation, some modified IFT techniques, employing a gradual thinning strategy, have been proposed to facilitate global convergence and accelerate computational speed [20,21]. More recently, a modified IFT (MI-FFT) approach is presented to design beam-scanning thinned massive antenna arrays [22].

To assess the synthesis capabilities of some above-mentioned methods for designing thinned arrays, three square arrays with different sizes (N = 144, 576, 900) are considered in this comparison. To ensure fairness, a consistent filling factor is maintained, and the synthesized SLLs obtained by five methods are presented in Table 1. The methods include RDT [18] as an analytic method, HSPSO [12], cGA, and M-cGA [15] as SO methods, and IFT density taper (IFTDT) [18] and MI-FFT [22] as modified IFT methods. Particularly, most of the results in Table 1 are drawn from published articles [12,15], with the exception of the MI-FFT technique. As can be seen, SLLs obtained by HSPSO are almost worse than those of other methods. This is because the HSPSO method aims to find a compromise between SLL and filling factor, while the other methods can preset the filling factor value to effectively suppress SLL. More importantly, the MI-FFT technique, incorporating a gradual array thinning strategy, consistently generates patterns with the lowest SLL in this example. With the increase of the array size, SLLs obtained by MI-FFT are −19.2 dB, −23.5 dB, and −26.4 dB, respectively. As a representative case, the MI-FFT technique is utilized for synthesizing a 24×24 square array with a 44% filling factor, and the final obtained 254-element layout is showed in Fig. 1. Fig. 2 depicts the 3D pattern in (u, v)-plane along with two orthogonal cuts of the pattern.

Nonuniformly spaced arrays offer increased spatial freedom compared to thinned arrays. In these arrays, elements can be flexibly positioned within the aperture and unconstrained by a preset grid. This flexibility allows for achieving desired patterns with a reduced number of elements.

As is well known, the synthesis of a nonuniformly spaced array with optimized element positions poses a strongly non-linear and non-convex optimization challenge. Over the past several decades, numerous efficient methods have been proposed to tackle this intricate problem. The early research on sparse spaced arrays can date back to 1950s when a matrix relationship between the array elements and corresponding far-zone pattern was found in a linear array with arbitrarily distributed elements [3]. In this subsection, we discuss various methods of designing nonuniformly spaced arrays. In addition to analytic methods and the SO methods commonly used in thinned array synthesis, we also introduce sparse array reconstruction methods and ICO approaches.

For analytic methods, some early attempts at designing nonuniform arrays used spatial tapering of array elements to suppress SLL [23,24]. However, the optimality of the realizing arrays is doubtful [78]. Besides, the numerical iterative methods, including the steepest descent method [25] and the iterative least-squares technique [26], have been exploited to find suitable element positions and excitations. While these methods allow the incorporation of linear and non-linear constraints, the design process requires a predefined number of elements, limiting the optimized arrays from achieving a minimum number of elements. To address this limitation, a search algorithm based on l_p quasi-norm [79] and a recursive inversion algorithm [80] have been proposed, aiming to obtain a sparse configuration that matches a reference pattern.

Due to the ability of solving non-linear and multi-constrained problems, SO methods such as differential evolution (DE) [27] and PSO [28] find applications in optimizing nonuniformly spaced arrays. Different from selecting array element positions on a uniform grid when synthesizing thinned arrays, the distance range between any two adjacent elements is generally preset in the design of nonuniform linear arrays. This ensures achieving constraints on the resulting maximum and minimum spacings of adjacent elements can be achieved. However, these methods face challenges when directly controlling the element spacing of resulting planar arrays. To address this issue, the minimum element spacing constraint is converted to the Chebyshev distance constraint, providing more manageable control [29–31]. In particular, modified real GA is proposed to optimize element positions within a rectangular boundary [29]. Another study in Ref. [30] exploits an asymmetric mapping method to search for optimal solutions. In addition, this mapping method is integrated into the joint optimization problem to offer spacing-constrained sparse array layout candidates, particularly when considering rotated dipole antennas in Ref. [30]. Moreover, a DE algorithm with a new encoding mechanism and Cauchy mutation (DE-NEM-CM) is presented in Ref. [32] for optimizing large unequally spaced planar array layouts while adhering to the constraint of minimum element spacing. Despite these innovative approaches, the time constraints inherent in large-scale array optimization with SO algorithms remain an unavoidable challenge.

In recent years, sparse array reconstruction methods, notably MPM and BCS, have gained much attention due to their efficiency in achieving desired patterns rapidly. The initial application of MPM for synthesizing nonuniformly spaced arrays is introduced in Ref. [33]. By performing singular value decomposition (SVD) of desired pattern samples, this method rearranges the element position and excitation distribution, resulting in a lower-rank approximation corresponding to fewer elements. Building on this concept, the forward-backward MPM [34] and the extended MPM [35] are developed to achieve accurate pattern shapes and multiple patterns for linear arrays, respectively. To synthesize arbitrary sparse planar arrays withthe MPM method, an innovative approach is utilized [36]. This approach incorporates the matrix decomposition method for SVD into matrix enhancement and matrix pencil (MEMP) [81], and then compresses the decomposed matrices with SVD in parallel to accelerate the synthesis procedure. What is more, a unitary matrix pencil (UMP) method based on the unitary transformation is presented to enhance matching accuracy for shaped patterns in the synthesis of linear arrays [37] and planar arrays [38]. Importantly, these MPM-based methods can prior estimate the optimal number of elements, which is not possible with traditional sparse methods. As for BCS, utilizing a probabilistic formulation of array synthesis and the fast relevance vector machine, the technique efficiently determines element excitations for matching reference patterns in a nonuniform layout. Over the past decade, this technique has been developed to design linear arrays [39], planar arrays [40], and conformal arrays [41] with an ultrasparse position distribution, and a comprehensive review of the technique is available in Ref. [42]. However, both MPM and BCS methods face common challenges. One such challenge is the prerequisite for providing realizable reference patterns in advance. Another issue is the complexity of achieving minimum element spacing control, which may result in synthesized arrays that cannot be practically realized due to the excessively close element spacing.

In contrast to sparse array reconstruction methods, ICO approaches may not require a reference pattern in the optimization process. By means of formulating the sparse array synthesis problem as a reweighted l₁-norm problem, these approaches can be applied to efficiently synthesize focused or shaped patterns [43,44]. However, the weighting vector in the reweighted l₁-norm optimization is solely determined by the excitation vector during the iteration process, limiting the control over the minimum element spacing. To overcome this challenge, an alternating convex optimization (ACO) approach based on the reweighted l₁-norm optimization has been presented in Ref. [45]. In this novel approach, both the element excitation vector and the weighting vector are alternately chosen as optimization variables. This allows the minimum element spacing constraint to be easily integrated into the optimization process by optimizing the distribution of the weighting vector. Inspired by the alternating strategy, another technique called alternating linear programming optimization [46] has been presented for synthesizing a sparse linear array with sum and difference patterns based on a typical two-section BFN frame. Considering real antenna array structures and mutual coupling effect, a refined extended ACO [47] technique is provided to design multibeam sparse circular-arc antenna arrays with minimum element spacing control. Furthermore, effective algorithms are presented in Refs. [49] and [50] for synthesizing large-scale sparse planar arrays. Specifically, an iterative approach involving polygonal expansion and contraction of the sources is employed in Ref. [49], while layout optimization is addressed in Ref. [50] by framing it as a constrained least squares problem and iteratively refining the solution.

To demonstrate the effectiveness of various methods for synthesizing nonuniformly spaced arrays, we consider the Chebyshev pattern with SLL of −30 dB as a reference pattern for a linear array with length L = 14.5λ. Table 2 shows the synthesis results of five methods, including SLL, directivity, realized minimum element spacing, the number of optimized elements, and time cost. Since the performance of patterns obtained by sparse array reconstruction methods is described by the matching error, SLLs obtained by MPM and BCS methods may slightly exceed the reference value. With respect to the directivity, the results obtained by different methods are relatively close since the synthesized array apertures hardly change compared with those before optimization. Besides, it can be found that the realized minimum element spacing with reweighted l₁-norm optimization may be too small, at just 0.05λ, making it difficult to realize in practice. Although the minimum element spacing obtained by MPM and BCS methods exceeds half a wavelength in this example, both methods generally cannot control the distance of inter-element spacing, as constraints on element positions cannot be added. On the contrary, both PSO in Ref. [28] and ACO in Ref. [45] methods are able to control the minimum element spacing. Particularly, the former constrains the range of adjacent element spacing in the initial array setting, and the latter realizes the control by optimizing the distribution of the weighting vector in the alternating optimization process. In terms of the sparsity effect, these methods, including PSO, MPM, BCS, reweighted l₁-norm optimization, and ACO, use 20, 18, 20, 21, and 19 elements to reconstruct the Chebyshev pattern, respectively. Notably, the MPM method only requires 18 elements, saving 40% channel resources compared to the Chebyshev array, which is also fewer than those obtained by other methods. What is more, sparse array reconstruction methods show great advantage in computing efficiency. Especially for the MPM method, it only takes 0.08 s to obtain the element distribution. Figs. 3 (a) and (b) show the patterns and excitation distributions optimized by these methods, respectively.

Clustered array techniques have played a significant role in 5G cellular systems and satellite communications. Their ability to simplify array design and reduce system costs while maintaining high-resolution performance has been widely acknowledged [2]. In general, a clustered array involves partitioning an array with N elements into M subarrays, where M is considerably smaller than N. In this configuration, only M array channels are required, as the array is excited at the subarray level instead of the element level. Unlike thinned and nonuniformly spaced arrays which achieve sparsity through element position optimization, the clustered arrays attain sparsity by reducing the number of array channels while maintaining a uniform layout. In particular, when each subarray of an array is of the same size, resulting patterns may exhibit grating lobes, especially when the array requires a wide-angle beam scanning capability [51]. This phenomenon arises from the large average spacing of array subarrays. Thus, various effective and innovative methods have been proposed to design clustered arrays with desired patterns by optimizing both subarray size and subarray excitation. In this subsection, we mainly discuss three types of methods: SO, EM, and compress sensing (CS).

As a representative SO algorithm, GA has been adopted to generate focused and scanning beams in clustered linear and planar arrays [52,53]. In the context of linear array optimization, SLL is minimized by dividing the array into several subarrays of unequal sizes and applying amplitude weighting at the subarray ports [52]. Additionally, the optimization process allows for control over the minimum number of elements (N_min) in each subarray. This is achieved by initially distributing N_min elements in each subarray and suitably placing the remaining elements among the subarrays. In the synthesis of planar clustered arrays, an optimal solution has been found by utilizing polyomino-shaped subarrays and optimizing subarray orientations and positions [53]. However, this solution cannot guarantee full coverage of the optimized arrays, potentially leading to some existing holes that could reduce array aperture efficiency and directivity. To overcome this problem, an irregular two-size square tiling method [55] exploiting innovative integer-code GA is presented to fully cover rectangular array apertures. This tiling method not only increases the directivity of patterns but also reduces optimization complexity, albeit with a constraint on the angular scanning ability of the synthesized arrays.

The EM approach is proposed to make the optimized pattern match to a reference one as closely as possible. In this approach, the subarray excitations are determined based on a set of reference element excitations. Initially proposed in the 1980s [57], the primary objective was to obtain an optimum sum and a best compromise difference pattern by independently optimizing element excitations and subarray weighting coefficients. Building upon this concept, various methods have been developed for design clustered arrays. For instance, by utilizing the relationship between the independently optimal sum and difference excitations, the synthesis problem is defined as searching for the optimal path in a non-complete binary tree, and a fast resolution algorithm is introduced [58,59]. However, this algorithm lacks element continuity in each clustered array, leading to complex BFNs. To overcome the drawback, contiguous partition methods (CPM) have been presented to design contiguously clustered geometries [60–62]. For instance, the clustering problem is addressed by reformulating it into the search for the best path in a non-complete binary tree in Ref. [63]. Simultaneously, the grouping of array elements into contiguous subarrays and the computation of subarray weights are performed. However, these methods typically require setting the number of subarrays in advance. To address this limitation, a sparseness-regularized solver is utilized in Ref. [41]. This solver, combined with elementary modules from a user-defined dictionary, enables obtaining the final array with the optimized number of subarrays.

In recent years, ICO techniques have been utilized for efficiently synthesizing the clustered arrays with desired patterns. With respect to clustered linear arrays, the array synthesis problem can be recast as a reweighted l₁-norm problem by introducing the first order difference or total-variation norm (TV-norm) of the excitation vector [63,64], and subsequently solved. Besides, a lower triangular 0-1 matrix is constructed in Ref. [65] to connect the sparse vector and the element excitation vector. Through the iterative reweighted l₁-norm algorithm, this construction allows finding the minimum number of subarrays with desired patterns. However, these methods in Refs. [63–65] are not directly applicable to clustered planar array synthesis because planar arrays are usually covered with only one or a few irregular sub-arrays, thereby facilitating mass production. To address this issue, several representative irregular subarrays such as domino-shape, L-tetromino, and polyhex-shape are considered, and the planar antenna aperture could be covered by employing rotating and flipping operations [82,83]. More interestingly, some Tetris-like subarrays are explored in Ref. [84]. Inspired by the sparseness-regularized model [85], the concept of a user-defined binary dictionary, containing the membership between antenna elements and subarrays, has gained widespread adoption. In particular, the synthesis problem can be recast to a dictionary-based sparse and linear retrieval problem. To solve this problem, a heuristic iterative convex relaxation programming method is proposed to synthesize modular subarrayed planar phased arrays [66,67]. Besides, utilizing a user-defined dictionary matrix, a binary sparse vector, which possesses the same structure as the resulting sparse excitation vector, is introduced to describe the pattern performance and exact tile of the aperture [68–71]. By the minimization of the reweighted l₁-norm, a focused beam with low SLL or a shaped beam can be obtained. Recently, the concept of 3D space entropy is introduced in Ref. [86], which transforms the tiling problem for spherical conformal phased array antennas into a subarray partition for the full array. Then, the tiling topology, formed by two-element tiles, is binary coded and optimized using convex integer optimization programming.

To compare the performance of different methods for synthesizing clustered arrays, a linear array of N = 128 elements with an inter-element spacing d = λ/2 is considered. Table 3 lists the synthesis results of four methods in terms of SLL, first null beamwidth (FNBW), directivity (D), the number of subarrays (Q), realized minimum element number in each subarray (N_min) as well as the time cost. The adopted methods include the hybrid GA [52] as an improved SO method, hybrid iterative contiguous partition method (I-CPM) [61] as EM, and TV-norm [64] and CS [65] as ICO methods. As can be seen from the table, the pattern performance (including SLL and D) and the subarray geometry (including Q and N_min) obtained by the hybrid GA and the hybrid I-CPM are comparable. However, FNBWs obtained by the two ICO methods with SLL of −36.5 dB are narrower than those obtained by the hybrid methods, while requiring a higher number of subarrays. Moreover, due to the absence of a constraint on the element count in each subarray, the ICO methods lack direct control over the minimum element count in each subarray, a capability achieved by the two hybrid methods. Specifically, when the SLL constraint changes from −36.5 dB to −40.0 dB for CS, the number of required subarrays grows from 17 to 25. Notably, when SLL is constrained not to exceed −40.0 dB, a subarray with a single excitation even appears, contradicting the initial goal of subarray design. In terms of the computational efficiency, the ICO methods only take approximately 10 s, whereas the hybrid I-CPM requires more than 100 s, indicating that the ICO methods are more efficient in subarray synthesis. As an illustration, the synthesized patterns and excitation distributions are shown in Figs. 4 (a) and (b), respectively (due to the symmetry of these excitations, only half of them are provided).

This review has summarized numerous methods for designing sparse arrays, each with its own set of advantages and disadvantages. These methods are tailored to different array architectures, and we have discussed various approaches for designing thinned arrays, nonuniformly spaced arrays, and clustered arrays. While all three array architectures aim to reduce system costs, they achieve this goal through different mechanisms. Thinned and nonuniformly spaced arrays directly decrease the number of antenna elements, thereby reducing the number of TRMs. On the other hand, clustered arrays achieve sparsity by sharing TRMs among the array elements within subarrays. Future research in this field could explore the synergies between nonuniform array layouts and subarray synthesizing methods. Combining these approaches has the potential to further reduce the overall cost and weight of the system, providing innovative solutions for practical applications.

The authors declare no conflicts of interest.