Optimization Design for Sparse Planar Array in Satellite Communications

Abstract

The antenna is one of the key components of satellite communication load. To address the evolving requirements of future satellite communication systems, the sparse planar array has become an important device for transmitting and receiving electromagnetic waves in emerging antenna systems. The advantages of this technology include low cost, low system complexity, and robust anti-interference ability, which have attracted widespread attention within the industry. In this paper, we investigate an optimization design of sparse planar arrays in satellite communication scenarios. Firstly, we introduce the mathematical foundation of the array antennas and establish the optimization design model of the sparse planar array. Secondly, we analyze and compare the impact of different array layout methods on the sparse planar array antenna pattern, and then introduce the latest design trend of array material design. Thirdly, we review some classical optimization methods for optimizing sparse planar arrays and the recent research advancements in promising and novel methods. Lastly, on the basis of the present research status, we propose three future research directions and two critical challenges for optimal design of sparse planar arrays in satellite communication scenarios, which can facilitate the development and realization of array technology under future B5G and 6G wireless networks.

1. Introduction

The antenna is a vital device for satellite communication payloads, as shown in Figure 1; until 19 November 2022, the search volume of the keyword “antenna” demonstrated a growing search volume on IEEE Xplore’s top search. It is currently in third place and also the only device with considerable attention. Antenna technology boasts a rich history spanning over a century; unlike other devices, it enables to effectively transmit RF power signals to the destination and amplify the power density in a certain direction, thereby enhancing the capabilities of wireless systems. In summary, antennas have the characteristics of directionality, spatial directivity, and filtering.

Antenna arrays are radiation devices consisting of multiple antenna units arranged in a specific manner [1], which have found widespread application in various fields such as satellite communications, satellite Internet of things, radar, and astronomy. Planar arrays use the principle of superposition to form regular or random arrays, thereby facilitating beam scanning, multi-beam, shaped beams, and antenna patterns with high gain, strong directionality, narrow beam width, and low sidelobe in satellite communication scenarios [2]. In light of the development requirements for antenna technology in future B5G and 6G networks, investigating the optimal design of spaceborne antenna arrays is imperative.

In the optimal design of planar array antennas, the aperture of the array serves as a key determinant of the upper bounds for system resolution and sensitivity. The actual application range of the antenna array is contingent upon the structure characteristics of the array, as well as the complexity of its software and hardware components. In recent years, array antennas have gained wide usage in satellites due to their narrow beam, which makes them suitable for accurate scanning detection and tracking. Among various antenna types, phased array antennas are the most popular. Unlike common antennas, phased array antennas can use the electric scanning method and the phase difference of the electrical signal sent by the detection unit to control the detection direction. Additionally, they can realize fast scanning by changing the array antenna parameters. Phased array antennas have several characteristics, including long transmission distance, many functions, and good anti-interference performance, making them the most typical planar array antenna in space-borne antennas [3]. At present, mobile satellite communication scenarios predominantly employ intelligent phased array antennas based on digital beamforming technology. These include phased array feeder reflective surface antennas and direct radiation phased antenna arrays. Among them, typical representatives of phased array feeder reflective surface antennas are Inmarsat satellite dishes, Thuraya satellite dishes, Alpha-sat XL satellite dishes, etc. Typical direct radiation phased array antennas include ICO satellite L-band phased array antennas, Harris ADS-B software-defined phased array antennas, Spaceway-3 satellite X-band digital multi-beam phased array antennas, NeLS low-orbit constellation X-band digital multi-beam phased array antennas, SatixFy Ku-band digital multi-beam phased array antennas, and O3b mPower phased array antennas [4]. In phased array antennas, each element has independent amplitude-phase control components and transceiver channels, and the hardware cost is high; moreover, with the expansion of the array aperture, the array requires a considerable number of antenna units and array amplitude-phase control components, making the antenna system very expensive. Additionally, the signal processing complexity of the phased array is high, and an excessive number of receiving channels will make the calculation dimension of the signal processing algorithm too high and reduce the real-time performance of the system. Furthermore, the small element spacing of phased arrays makes it challenging to process T/R components, and there is considerable coupling between elements, especially in the millimeter wave band. Lastly, phased array antennas are heavy and struggle to dissipate heat, which is not conducive for their installation on aircraft and space carriers.

However, phased array antennas exhibit certain limitations, including complex calculations, high costs, and large size and weight, that render them unsuitable for use in satellite transceivers. To address the issue of onboard loading, sparse array antenna technology has emerged as a potential solution. Under specific conditions, sparse planar arrays can achieve the same performance as uniformly spaced full arrays while utilizing a smaller number of elements. Sparse arrays can reduce the complexity and cost of satellite antennas by minimizing the number of elements and mitigating mutual coupling between array elements without significantly altering beam width or aperture size [5]. As shown in Figure 2a, the American PAVE PAWS radar system operates within the frequency range of 420–450 MHz and employs a large double circular array design [6]. Each circular array comprises 2677 elements, with approximately 67% of the elements remaining active during operation, resulting in a detection distance of approximately 4800 km. This system represents the first engineering-level application of sparse planar arrays. In recent years, the research on sparse planar arrays has gained traction, with increasing focus on the optimization design of such arrays, extending from theoretical research to engineering implementation. For instance, the German ARTINO radar system [7] applies sparse plane arrays to drone imaging. Figure 2b (provided by the SKA Program Development Office, https://www.skao.int/, accessed on 30 March 2023) shows the Square Kilometer Array (SKA) project [8] involving 20 countries around the world, which constitutes the largest astronomical telescope array under construction worldwide. At present, the research of sparse arrays on a satellite has become the famous topic. In this paper, the optimization design for sparse arrays we discuss is also based on the general context of satellite communication [9].

Sparse planar arrays possess the necessary attributes of small cost, low software and hardware intricacy, and robust anti-interference capabilities in large-scale array designs. By means of an optimized design, these arrays can attain superior performance metrics, rendering them a vital component for transmitting and receiving electromagnetic waves in emerging antenna systems. The present research topic has garnered significant attention in recent years and encompasses a multitude of investigative directions. Within this context, the current article focuses on the optimization design of sparse planar arrays in satellite communication scenarios. After over five decades of development, theoretical research on the optimal design of sparse planar arrays has attained a relatively advanced level and yielded numerous research findings. As shown in Figure 3, the optimization design of the sparse plane array can be likened to a large tree. wherein the root represents the array arrangement, geometry, mathematics, and physics, while the trunk denotes the material support and production technology levels. The branches and leaves of the tree represent various optimization design methods.

2. Sparse Planar Array Optimization Design Model

The primary objective of optimizing sparse planar arrays is to obtain an antenna pattern that satisfies expectations. The characteristics of the antenna pattern are primarily determined by factors such as amplitude and phase excitation, number of elements, array layout, array geometry, and array material. Optimization parameters include peak sidelobe level, antenna directivity and gain, main lobe width, and zero position. Satellite communication scenarios necessitate the consideration of additional indicators such as array scanning angle and polarization.

This section focuses on the mathematical underpinnings of array antennas, mathematical model of the sparse plane array pattern, optimal design model based on minimum peak sidelobe level, and antenna pattern reconstruction model, as well as key issues to be addressed in optimizing sparse planar arrays.

2.1. Array Antenna Mathematical Underpinnings

The electrical performance of space-borne phased array antennas is primarily evaluated on the basis of several key metrics, including antenna gain, antenna directivity, antenna pattern, beam scanning range, and antenna polarization.

Antenna directivity $D\left(\theta ,\phi \right)$ is a critical parameter that quantitatively quantifies the strength of antenna directivity. The directivity $D\left(\theta ,\phi \right)$ is defined as the ratio of the far-field power density $S\left(\theta ,\phi \right)$ at a distance $r$ from the antenna to the power density $S_0$ of an ideal non-directivity antenna at the same position with the same radiant power $P_r$.

$$D\left(\theta ,\phi \right)=\frac{S\left(\theta ,\phi \right)}{S_0}=\frac{{\left|E\left(\theta ,\phi \right)\right|}^2}{{\left|E_0\right|}^2}.$$

(1)

The physical parameter denoting the ability of an antenna to convert energy effectively is the antenna efficiency. The total antenna efficiency $e_0$ is influenced by the reflection loss at the input of the antenna, conductor loss at the output of the antenna, and medium losses. The expression for the total antenna efficiency is given as

$$e_0=e_r{e}_c{e}_d=\left(1-{\left|\mathsf{\Gamma }\right|}^2\right)e_{cd},$$

(2)

where $e_r$ represents the corresponding efficiency in the circumstance that energy cannot be fully transmitted to the antenna due to the mismatch between the antenna and the transmission line, which is related to the reflection coefficient $\mathsf{\Gamma }$. $e_c$ represents the efficiency attributed to the energy dissipation caused by the heating of the antenna conductor during its operation. $e_d$ represents the efficiency attributed to the energy dissipation caused by the propagation of electromagnetic waves in a dielectric medium. Typically, the product of these two efficiency factors $e_c$ and $e_d$ is utilized to denote the radiation efficiency $e_{cd}$ of the antenna.

The gain of an antenna is determined through the multiplication of its efficiency with its directivity.

$$G\left(\theta ,\varphi \right)=D\left(\theta ,\varphi \right)·e_0.$$

(3)

In spaceborne phased array antennas, a variation in the steering angle ${\theta }_m$ during beam scanning results in a corresponding change in the antenna gain. When ${\theta }_m$ is not significantly large, $G\left({\theta }_m,\phi \right)=G\left({\theta }_0,\phi \right)·\cos {\theta }_m$. $G\left({\theta }_0,{\phi }_0\right)$ represents the gain in the normal direction of the antenna. However, when the phased array antenna is scanned at large angle, the gain of the antenna decreases correspondingly.

The concern of the antenna pattern for spaceborne phased array antennas primarily focuses on parameters such as sidelobe level, beam width, and beam scanning range. The antenna beams can be electrically steered in both azimuth and elevation directions, with the spacing between array elements being a critical factor affecting the antenna’s scanning range. In practical engineering, there exists a constraint relationship between the antenna element spacing and the maximum achievable scanning angle of the antenna, which can be expressed as follows:

$$d\le \frac{\lambda }{1+\sin \left({\theta }_m\right)}.$$

(4)

In satellite communications, the utilization of sparse arrays facilitates the reconfiguration of the array element layout in the antenna pattern as compared to traditional phased arrays. The use of a smaller number of antenna elements can achieve comparable wide-angle scanning performance.

Antenna polarization refers to the polarization of the electromagnetic waves emitted by an antenna. According to the reciprocity theorem, the polarization characteristics of the same antenna acting as a transmitting antenna and a receiving antenna are identical. As the electromagnetic waves radiated by an antenna differ in various directions, different polarizations are observed in different directions. Linear polarization, circular polarization, and elliptical polarization are three commonly used polarization methods that are determined on the basis of the amplitude and phase difference of the electric field vector. For linearly polarized waves, the phase difference between the two components of the electric field vector is an integer multiple of $\pi$. For circularly polarized waves, the amplitude of the two components of the electric field vector is equal, and the phase difference is $\pi /2$. For elliptical polarization, the relationship between the amplitude and phase difference of the electric field vector can be arbitrary. In spaceborne phased arrays, linear polarization typically corresponds to the Ku frequency band, while circular polarization is utilized in the L, S, and Ka frequency bands, among others.

2.2. Sparse Area Array Pattern Mathematical Model

The core of the array antenna is the beamforming technology. The output of each array element is weighted according to specific criteria. This weighting process ensures that the array output is adjusted to align with the desired receiving direction of the array, leading to the concentration of gain in a single direction and the generation of directional beams directed toward the intended user.

The antenna pattern is a critical indicator for evaluating the array’s performance and is generally used to indicate the change in the relative field strength of the radiation field with direction. At the mathematical level, in accordance with the principle of antenna pattern multiplication, it is expressed as the product of the unit factor $T_{mn}\left(\theta ,\phi \right)$ (single element pattern) and the array factor $AF\left(\theta ,\phi \right)$:

$$F\left(\theta ,\phi \right)=T_{mn}\left(\theta ,\phi \right)·AF\left(\theta ,\phi \right).$$

(5)

Assuming that the single-element pattern is wide enough to form an array with isotropic elements, the single-element pattern $T_{mn}\left(\theta ,\phi \right)=1$ in Equation (5) can be rewritten as

$$F\left(\theta ,\phi \right)=AF\left(\theta ,\phi \right).$$

(6)

Assuming that the current distribution of the antenna element can be separated, the planar array with factorable positions is distributed on the xoy plane and contains $M\ast N$ elements within the array aperture; its antenna pattern can be represented as follows [11]:

$$\begin{array}{l}F(\theta ,\phi )={\displaystyle \sum _{m=0}^{M-1}{\displaystyle \sum _{n=0}^{N-1}{w}_{mn}{e}^{jk\left[d_m\left(\sin \theta \cos \phi -\sin {\theta }_0\cos {\phi }_0\right)+d_n\left(\sin \theta \sin \phi -\sin {\theta }_0\sin {\phi }_0\right)\right]}}}\\ \hspace{1em}\hspace{1em}\hspace{1em}=\left|F_1(\theta ,\phi )\right|·\left|F_2(\theta ,\phi )\right|\end{array},$$

(7)

where $w_{mn}$ represents the complex excitation (including amplitude and phase) of the $(m,n)$ element, $k=2\pi /\lambda$ is the number of spatial wavelengths, $\theta$ represents the pitch angle of the observation direction relative to the origin, $\phi$ denotes the azimuth angle of the observation direction relative to the origin, $\left({\theta }_0,{\phi }_0\right)$ is the maximum value of the main beam of the array, $d_m$ and $d_n$ are the coordinates of the $(m,n)$ element, and $\left|F_1(\theta ,\phi )\right|$ and $\left|F_2(\theta ,\phi )\right|$ denote the antenna radiation pattern in the x-direction and y-direction, respectively. The above expression is for the general case of planar arrays.

In order to compare the differences in antenna patterns between a uniform planar array and a sparse planar array, Figure 4a,b show the array layout of a uniform planar array with 169 elements and a sparse planar array with 73 elements, respectively. Figure 5a,b compare the gain patterns of the two arrays on two orthogonal sections ($\phi =0^{\circ }$ and $\phi ={90}^{\circ }$), which are both HFSS full-wave simulation results. It can be seen that, although the two arrays have different numbers of elements, they can still obtain similar antenna pattern shapes and gains. These results are also in line with the theory of array antennas. On the one hand, the similarity in lobe widths between the two arrays can be attributed to their identical aperture sizes. On the other hand, the gain of the distribution of the aperture field is dependent on various factors such as the number of elements, the position of the elements, and the excitation of the elements. Consequently, sparse arrays can achieve the same gain as a full array at the same caliber by optimizing the position and excitation of the elements.

2.3. Optimization Model Based on Minimum Peak Sidelobe Level

Sidelobes, also known as secondary lobes, are lobes in the antenna pattern in addition to the main lobes. An essential performance indicator of the antenna pattern is the peak sidelobe level (PSLL). The lower the peak sidelobe level is, the better the anti-interference performance of satellite communication will be. The peak sidelobe level is a common objective in sparse planar array optimization designs.

When thinning the plane array, under the condition of determining the array aperture, the number of elements can be reduced, the amplitude-phase excitation can be adjusted, and the peak sidelobe level can be reduced by optimizing the array element layout. In sparse planar array optimization design, the peak sidelobe level can generally be utilized as an optimization goal and is defined as

$$PSLL=\underset{(\theta ,\phi )\in S}{\max }\left\{20{\log }_{10}\frac{\left|AF\left(\theta ,\phi \right)\right|}{\left|A{F}_{\max }\right|}\right\},$$

(8)

where $S$ represents the sidelobe range of the antenna pattern, and $A{F}_{\max }$ is the maximum value of the array factor. The basic optimization model based on the minimum peak sidelobe level array design is expressed as

$$\underset{w_{mn}}{\min }PSLL,$$

(9)

where $w_{mn}$ represents the complex excitation of the $(m,n)$ element, where $m=0,1,\dots ,M-1$ and $n=0,1,\dots ,N-1$. The objective of the optimization model in Equation (9) is to obtain a sparse planar array with the lowest PSLL in the visible area. However, since the decrease in the peak sidelobe level often leads to the broadening of the main lobe and the reduction in gain of the main lobe, many studies added control constraints to the main lobe to achieve a reduction in the peak sidelobe level.

2.4. Optimization Model Based on Antenna Pattern Reconstruction

The antenna pattern can be reconstructed to meet the specific demands of satellite communication networks, space remote sensing, and large-angle scanning phased arrays, and it has a wide range of application prospects. In satellite communication scenarios, there are certain special requirements for beam width, main lobes, sidelobes, etc. The goal of the optimization model based on antenna pattern reconstruction is to optimize the excitation and element layout by using fewer elements and to change the antenna pattern according to actual requirements so that the reconstructed antenna pattern is closest to the expected antenna pattern.

Assuming that the number of elements of the sparse planar array is $Q$, and $\tau$ is the upper limit of reconstruction error set by the user, i.e., the acceptable deviation between the reconstruction pattern and the target pattern, the basic optimization model can be expressed as

$$\begin{array}{l}\underset{w_{mn}}{\min }Q\\ s.t.|F_{REF}\left(\theta ,\phi \right)-F\left(\theta ,\phi \right)|\le \tau \end{array}.$$

(10)

The objective of the model presented in Equation (10) is to minimize the number of elements of a sparse planar array when the difference between the reconstructed antenna pattern and the expected antenna pattern is less than the upper limit of reconstruction error set by the user. Here, $F_{REF}\left(\theta ,\phi \right)$ represents the expected antenna pattern, and $\theta$ and $\phi$ are the pitch and azimuth angles of either observation direction in the plane relative to the origin, respectively.

2.5. The Main Problems to Be Solved in the Optimization Design of the Sparse Area Array

From the perspective of the antenna pattern, three types of problems need to be considered.

The first type of problem is to adjust the excitation amplitude phase weight to determine the array element distribution, such as optimizing peak sidelobe level minimization and main lobe beam width control to obtain the expected antenna pattern.

The second type of problem is to reduce the number of array elements and optimize the weight and array layout according to the expected antenna pattern without changing the array aperture so as to obtain the expected shaped beam or the reconstructed antenna pattern that best matches the desired pattern.

The third type of problem is the comprehensive optimization of arrays, which needs to comprehensively consider the joint optimization of multiple parameters such as array element layout, number of elements, array amplitude phase excitation, and peak sidelobe level so as to design a high-performance antenna pattern that meets expectations.

Satellite communication scenarios are characterized by long transmission distances, complex environmental conditions, numerous influential factors, and strong interference from various sources. Moreover, satellite arrays are constrained by array cost and weight limitations. Consequently, the primary emphasis in such scenarios should be on addressing the third category of issues.

Since the sparse planar array optimization design problem is a multivariate, nonlinear optimization problem that is difficult to deal with, it is necessary to construct a suitable sparse planar array optimization design method, which is also one of the main difficulties in sparse area array optimization design.

3. Sparse Surface Array and Material Design

3.1. Planar Array Geometry

The geometry of the array is an important aspect of the optimized design of the array. Different array geometry configurations result in different antenna patterns. Commonly used planar array geometric configurations mainly include rectangular arrays, circular arrays, polygonal arrays, concentric circle arrays, cross arrays, and spiral arrays. From the perspective of the antenna pattern, the main lobe generally determines the resolution of the array, and the sidelobe determines the anti-interference performance of the array, which is why the peak sidelobe level and control main lobe width should be reduced in array optimization design. Figure 6 shows the four typical arrays that are most frequently used in sparse area array optimization designs.

In order to investigate the influence of array geometry on the antenna pattern, Figure 7, Figure 8 and Figure 9 depict the antenna pattern with equiamplitude distribution and omnidirectional characteristics of the uniform rectangular array (URA), uniform circle array (UCA) and concentric circle array (CCAA). For all arrays, the amplitude weighting $w_{mn}=1$ and the single-element pattern $T_n\left(\theta ,\phi \right)=1$. The arrays considered are uniformly configured with 256 elements and an element spacing of $\lambda /2$. The beam pointing angle is $\left(0^{\circ },0^{\circ }\right)$. Rectangular arrays have a good resolution in the evenly distributed range of elements and are preferred in the radar field, but they exhibit a disadvantage that the other side of the main lobe contains an additional main lobe of the same strength [12]. The uniform circular array arrangement does not have any edge elements, and the main sidelobes of the antenna pattern are evenly distributed, which can decompose the excitation into symmetric spatial components to reduce the coupling; thus, the circular array has lower coupling and better robustness than the rectangular array [1], but the sidelobes are higher than the rectangular array. Circular arrays are often used for wireless communications, especially in smart antenna designs. The concentric circle array and the uniform circle array are significantly lower than the sidelobes, but the concentric circle pattern still shows a high grating lobe close to the level of the main lobe. For instance, Bucci utilized spaceborne isophoric sparse concentric circle arrays to achieve global coverage of GEO orbit while limiting sidelobe effects [13]. Furthermore, another study [14] comparing the antenna pattern of uniform hexagonal arrays with uniform circle arrays revealed that the uniform hexagonal array has superior gain and has lower sidelobes than the uniform circle array [14].

3.2. Sparse Planar Array

When designing sparse planar arrays, the array elements and the layout mode determine the array performance. The array layout involves studying the relationship between array performance and array element distribution. The antenna characteristics can be changed by changing the array layout mode to obtain different antenna patterns. According to the layout mode of the array, the sparse planar arrays can be divided into two basic methods: sparse planar arrays and thinned planar arrays, with the main distinction being periodicity. The former is still periodic, while the latter can be arbitrarily distributed after sparseness and generally does not have periodicity. In the absence of special requirements, the two can also be collectively referred to as sparse plane arrays.

Taking a rectangular array as an example, assuming that the array aperture is $D=D_x\times D_y$ and the planar array contains $Q=M\times N$ elements, the sparse planar array is shown in Figure 10a, and its array elements are arranged equidistant and uniform on the plane, while the thinned planar array is shown in Figure 10b, and its array elements are arranged in the plane with nonuniform and unequal spacing.

In the design of sparse arrays, careful consideration must be given to the space between elements due to the coupling effect and the potential for grating lobe occurrence. As the antenna scanning angle remains fixed, larger element spacings increase the likelihood of grating lobes. Conversely, smaller element spacings result in inter-element coupling effects. Typically, when the spacing between the elements is greater than λ, the coupling effect between the elements is negligible. Therefore, the above two factors need to be balanced in the sparse area array element array to obtain the desired antenna pattern.

3.3. Sparse Planar Array Material Design

In sparse planar array optimization design, the array material is a crucial factor influencing array performance and cost. Maxwell’s equations and constitutive relationships are important physical bases for designing materials in the field of array material design.

Maxwell’s equations and their constitutive relations establish that materials significantly influence the propagation of electromagnetic waves. Antenna array polarization serves as a parameter describing the spatial directivity of the electromagnetic wave vector radiated by an antenna array. As per Maxwell’s theory, alternating electric and magnetic fields intersect perpendicularly. During electromagnetic wave transmission, when the direction of the electric field vector remains fixed or rotates according to specific rules, this wave is referred to as a polarized wave. The polarization direction of an antenna array refers to the direction of the generated electric field. When the receiving and transmitting antennas share identical polarization modes, the highest receiving efficiency and signal level are obtained. In satellite communication scenarios, patch antennas or slot feeding are commonly used to achieve the desired polarization mode. Alternatively, in the design of polarized reconfigurable antennas, the polarization mode can be switched through array material and feed network adjustments. Ultimately, hybrid multi-polarization can be achieved.

When designing spaceborne antenna pattern reconfigurable antennas, two design approaches are generally adopted. One is to design a parasitic unit around the main radiation unit of the antenna and change the beam width and antenna gain of the antenna by increasing the physical size of the antenna aperture. Another is to add parasitic layers and etching gaps or use partial reflective surface antennas to change the beam width and antenna gain of the antenna without changing the physical size of the antenna aperture.

In recent years, metamaterial technology has reached new heights and has become one of the most investigated frontier technologies. Metamaterial technology can break through the traditional thought of material design, enabling artificial composite structures or composite materials with extraordinary physical properties that natural materials do not have. The unique properties of metamaterials make them well-suited for the development of functional devices, including antennas. It is worth noting that atoms serve as the smallest unit of traditional materials, and both the permittivity $\epsilon$ and the permeability $\mu$ are positive, while metamaterials analogize periodic units to atoms, which can achieve negative values of $\epsilon$ and/or $\mu$.

Metamaterial technology has found application in optimizing the design of satellite arrays for use in satellite communication scenarios. For instance, in [15], a high-impedance surface (HIS) structure was utilized to improve the surface wave and wide beam radiation performance of a single antenna element, enhance the wide scanning performance of phased arrays, increase gain while improving the scanning angle range, and facilitate a low-profile array design. In [16], the authors employed an artistic magnetic conductor structure (AMC) to design a broadband planar phased array antenna, broaden bandwidth, enhance phased array beam steering performance, and reduce array size and cost. Similarly, in [17], the electronic bandgap (EBG) structure was used to design a sparse planar array with broadband and wide-angle scanning capabilities, enabling lightweight array design and reduced array fabrication costs. Metamaterial technology offers new possibilities for developing optimization ideas for sparse planar arrays. The unique electromagnetic characteristics and physical properties of metamaterials provide opportunities to realize low-profile array antennas, reduce array volume and weight, regulate electromagnetic wave polarization, broaden bandwidth, and achieve high gain and low element coupling. Additionally, the ease of fabricating metamaterials via methods such as PCB technology, LTCC technology, and 3D printing technology further enhances the potential for metamaterials in sparse array design.

4. Sparse Planar Array Optimization Design Method

The sparse array optimization design method is the primary difficulty of sparse array optimization design. Currently, the optimization design methods of sparse planar arrays can be classified into four main categories: analysis methods, intelligent optimization methods, local optimization methods, and hybrid methods. This section focuses on these four types of methods; for easier understanding, Figure 11 provides a visual representation of the classification of sparse planar array optimization design methods.

4.1. Sparse Area Array Optimization Based on Analysis Methods

The primary way to solve the problem of sparse array optimization design in the early stage is to use analysis methods, which are mainly used for array synthesis under known array layout conditions; most of these methods are aimed at uniform arrays. For example, the Dolph–Chebyshev synthesis method first proposed in 1946 obtained an equal secondary lobe level. It achieved the narrowest main lobe for a uniform array [18], and the Taylor synthesis method obtained a lower sidelobe level by changing the zero position of the sidelobe. The pattern decreased with the secondary lobe interval [19]. Both of these methods are optimized for single targeting. Bayliss synthesis is a two-parameter differential beam synthesis method that minimizes peak sidelobe levels and maximizes angular accuracy [20]. The Elliott synthesis method is based on the Taylor and Bayliss distribution to achieve different sidelobe structures on the left and right [21]. The above analytic methods only optimize the peak sidelobe level and main lobe width, but do not optimize the position of the array elements. The almost difference set (ADS) synthesis method was proposed in [22,23], which can determine the position of the array elements, predict the upper bound of the sidelobes a priori, and reduce the peak sidelobe level. The above sparse planar array optimization method based on the analysis method has simple ideas, but strong pertinence and a small scope of application. The Woodward–Lawson method is aimed at the beamforming problem and is suitable for the antenna pattern synthesis problem of arbitrary array beamforming [24], which can achieve a good fit of the expected pattern, but the peak sidelobe level cannot be controlled in the non-fitting area.

The analysis methods based on statistics mainly include the dynamic programming method (DP) [25], density cone method (DT) [26], and fractional order Legendre transform method (FLT) [27]. Statistical-based optimization methods are feasible when dealing with the design of small-scale sparse arrays; however, as the scale of the array increases, the processing level of these algorithms decreases significantly. In order to improve the calculation speed, some scholars proposed a matrix decomposition-based method, which can solve the problem of sparse area array pattern reconstruction and beamforming; the most commonly used are the matrix pencil method (MPM) [28,29] and the augmented matrix beam (MEMP) method [30], which can minimize the number of array elements and do not need iteration, thus exhibiting low computational complexity. The authors of [31] proposed a 0–1 integer programming method (ILP), which can optimize the element excitation, reduce the peak sidelobe level, and improve the computational efficiency without iteration. The representative analysis methods for large-scale sparse area matrix synthesis problems are based on Fourier transform methods, such as fast Fourier transform technology [32] and iterative Fourier transform (IFT) [33]. The IFT method is an analysis method that has been used more frequently in sparse area array optimization designs in recent years.

IFT was proposed by Dutch scholar Keiser and applied to large sparse planar matrix synthesis [34]. IFT requires a known sparsity rate and uses the Fourier variation relationship between the excitation and array functions to iterate the excitation coefficient. Its advantage is that the expected peak sidelobe level can be obtained, and the excitation can be optimized. It has a fast convergence speed in processing large planar arrays, but the disadvantage of this method is that it is easy to fall into the local optimum.

Assume $M\times N$ array elements are arranged in a square grid. Let $u=sin\theta \cos \phi$, $v=\sin \theta \sin \phi$, and $\Delta d_x$ and $\Delta d_y$ represent the element space on the $x$- and $y$-axes. Under the conditions of an ideal omnidirectional element, the antenna pattern expression for a planar array can be expressed as follows [35]:

$$F(u,v)={\displaystyle \sum _{m=0}^{M-1}{\displaystyle \sum _{n=0}^{N-1}{w}_{mn}{\mathrm{e}}^{jk(n\Delta d_{x}u+m\Delta d_{y}v)}}}.$$

(11)

Applying the inverse Fourier transform on Equation (11) yields

$$f(x,y)=\frac{{\displaystyle \sum _{m=0}^{M-1}{\displaystyle \sum _{n=0}^{N-1}F\left(m,n\right){\mathrm{e}}^{j2\pi (mx/M+ny/N)}}}}{MN}.$$

(12)

It is not difficult to find that there is a two-dimensional Fourier transform relationship between the array element excitation $w_{mn}$ and the antenna pattern, which can be understood as the antenna pattern representing the time-domain signal and the array element excitation denoting the spectrum. The flowchart of the iterative Fourier algorithm is shown in Figure 12.

Following the proposal of the IFT algorithm, some scholars also proposed many improved algorithms. For example, for the large sparse planar array, the IFT method easily falls into the local optimum. The authors of [36] proposed an improved iterative Fourier transform (MIFT), which introduces a variable filling factor to obtain a lower peak sidelobe level than the IFT algorithm. In [36], circular arrays with diameters ranging from $25\lambda$ to $100\lambda$ were thinned. Results indicated that, when the array element number was 828, and the fill factor was 42.95%, an MSLL value 1.05 dB lower than that obtained using the IFT method and a directivity 0.12 dB higher than that obtained using the IFT method were achieved. Similarly, when the array element number was 3127 and the fill factor was 40.15%, an MSLL value 1.60 dB lower than that obtained using the IFT method and a directivity of 0.13 dB higher than that obtained using the IFT method were observed. Lastly, for an array element number of 12,514 and a fill factor of 40.00%, the MSLL value was 6.18 dB lower than that obtained using the IFT method, while the directivity was 0.06 dB higher than that obtained using the IFT method.

The authors of [37] proposed a sparse area matrix synthesis algorithm based on adaptive probability model optimization by constructing a probability estimation model to characterize the distribution of sparse arrays and combined it with the iterative Fourier method; the probability model was used to determine the position of the smallest sidelobe level element in the sparse area matrix, highlighting the global search ability and robustness. In the experimental results presented in [37], a circular array aperture with a diameter of $50\lambda$ was utilized with 3116 turned ON elements and a filling factor of 40%. Results indicated that the PLIFT method yielded a PSLL value that was 1.5 dB lower than that obtained using the IFT method. Aiming at the problem of beamforming, the authors of [38] proposes a fast pencil beamforming method for large-scale 4D thin cloth arrays by combining the least squares interpolation method with iterative fast IFT, which greatly improves the calculation speed. In the proposed method for nonuniform array synthesis, the array is first divided into “conventional arrays” based on excitation. Subsequently, a uniform virtual array structure is obtained from interpolation of the nonuniform array structure via the least squares interpolation method. Lastly, beam synthesis is accomplished using the IFT method. Aiming at the sparse area array synthesis problem, the authors of [39] used amplitude monotonicity conditions to propose an IFT method with high gain and smaller amplitude dynamics. This study presented nine experimental simulation results, covering array element numbers ranging from nine to 9999. The directivity of the proposed method was shown to improve by 0.42–0.51 dB, while the beam width was reduced by ${0.1}^{\circ }$–${3.2}^{\circ }$, and the amplitude was dynamically decreased by 2.56–7.72, when compared to the IFT method. Furthermore, the proposed method exhibited a high level of efficiency, with a simulation time of 4.3 s, which can facilitate the acquisition of low sidelobe levels.

4.2. Intelligent Optimization Methods

With the development of computer level, intelligent optimization methods have been proposed one after another, being widely used in various disciplines due to their unique advantages of strong universality and efficient solution of complex problems. At present, intelligent optimization methods can be divided into two categories: traditional intelligent methods and artificial intelligence methods.

4.2.1. Sparse Area Array Optimization Based on Traditional Intelligent Methods

The traditional intelligent method is the earliest intelligent method applied to the optimization design of sparse area arrays; these methods are adapted functions of the target optimization. In the optimization design of sparse plane arrays, we generally define the fitness function as some array parameters, where the goal is to solve the optimization of fitness. The traditional intelligent optimization method is suitable for optimizing planar arrays with small array scale, and it can achieve good results. However, as the array’s scale increases, this method needs to break through the bottleneck of computational efficiency. When the amount of computation is large, this type of algorithm faces convergence deterioration and easily falls into the problem of local optimization. As the scale of the array expands, the CPU consumption increases, which reduces the efficiency of the algorithm.

Among the many traditional intelligent optimization methods, the genetic algorithm, simulated annealing method, particle swarm algorithm, and ant colony algorithm are most frequently used. For the sake of easier understanding, we summarize the flowchart of these four types of methods for designing sparse area arrays in Figure 13.

Traditional intelligent algorithms generally take the adaptation function as the optimization goal; the most typical approach is to take the peak sidelobe level as the fitness function of the population, where the output function is generally the antenna pattern, and the goal is to obtain the smallest peak sidelobe level. The number of elements and their arrangement also affect the output of the function.

In 1994, Haupt [40] and Neil [41] applied genetic algorithms to sparse array optimization design problems for the first time. In order to reduce the peak sidelobe level, Haupt used a genetic algorithm to optimize an array of 20 × 10 sparse planes, obtaining a lower peak sidelobe level than the statistical method with a filling rate of 58%. However, due to the above document element position limitations, it can only be located in an equidistant grid, which can neither optimize the excitation weight nor minimize the number of elements. The authors of [42] divided the planar array into several subarrays, and used genetic algorithms to weight the subarrays by amplitude and phase excitation. Another study [35] was aimed at large sparse planar arrays, where the excitation amplitude phase weights of the array elements were optimized, the weak excitation was removed, and a comprehensive method of planar antenna array cosecant square radiation pattern based on genetic algorithm was proposed to optimize the peak sidelobe level and small ripple deviation. The genetic algorithm is suitable for processing sparse planar arrays of small and medium scale, whereby it can obtain the global optimal solution. However, as the aperture of the array becomes larger, the calculation complexity is high, the convergence is slow, it easily falls into local optimization, and the number of elements cannot be minimized.

In order to solve the problem of the minimum number of elements, Trucco [43] applied the simulated annealing method to the synthesis of sparse planar arrays, where the position and weight were optimized. The method was applied to a small planar array composed of 112 array elements and a large planar array composed of 3228 elements under the full array condition, while the small array was optimized with a filling rate of 55% and 60%, and the large array was thinned by 88.9%. All results obtained the peak sidelobe level equivalent to the full array situation, but the effect of this method on the reduction in peak sidelobe level was not obvious. The authors of [44] adopted a mirror operation, took the first lobe level as the optimization function, introduced the control coefficient of the number of elements, used the simulated annealing method to obtain a sparse array with an excellent low sidelobe pattern, and reduced the calculation amount of the algorithm. The advantage of using simulated annealing is that the position and weight of the array elements can be optimized at the same time. Although this method cannot guarantee that the global optimal solution will be obtained, it can greatly increase the probability of searching for the optimal solution; it is necessary to debug and compare the initial temperature, decay rate, and other parameter combinations to find the optimal solution suitable for specific scenarios. The disadvantage is that the searchability is not enough, whereby it is sometimes impossible to obtain a good solution, and it is not easy to reduce the peak sidelobe level.

The particle swarm algorithm is a swarm intelligence algorithm obtained by simulating the behavior of flocks of birds foraging for food in coordination with each other. Aiming at the problem of reducing peak sidelobe level, the improved binary Boolean particle swarm optimization (BPSO) is applied to the design of sparse planar arrays [45], which obtains lower sidelobes and better reliability. Although the principle of the BPSO algorithm is simpler, it only considers itself and does not consider the characteristics of sparse area array, which can cause huge computational problems in practical applications. The authors of [46] proposed a hybrid synthesis particle swarm optimization (HSPSO) algorithm based on the Hadamard set, which is more in line with the design characteristics of a sparse area array, and reduces the peak sidelobes while limiting the number of elements. This study presents a comparison between the proposed HSPSO method and the SPSO method. The PSLL results obtained using the HSPSO and SPSO methods were evaluated for different scenarios with varying array element numbers and filling factors. Specifically, for an array element turned ON number of 21 and a filling factor of 0.58, the PSLL obtained using the HSPSO method was found to be 0.51 dB lower than that achieved using the SPSO method. Similarly, for an array element turned ON number of 75 and a filling factor of 0.52, the PSLL obtained using the HSPSO method was 1.27 dB lower than that obtained using the SPSO method. Lastly, for an array element turned ON number of 323 and a filling factor of 0.56, the PSLL obtained using the HSPSO method was 3.25 dB lower than that obtained using the SPSO method. In order to solve the problem that particle swarm optimization converges quickly and easily falls into the local optimal solution, the authors of [47] proposed an improved PSO algorithm, which adopts the analytic initial value in the initialization process and analyzes the element weight through matrix operation, specifies the weight as the random initial value of any particle, and uses the PSO algorithm to iterate. This algorithm can improve the convergence characteristics of the algorithm. The advantage of using this algorithm to optimize the sparse area array is that the algorithm complexity is low and the parameters are few, but it faces the problem of fast convergence and falling into local extremes.

Ant colony algorithms benefit from the natural phenomenon of ants looking for food. The ant colony algorithm was applied to the sparse area array design for the first time in [48], and a low sidelobe was obtained, but other factors affecting the antenna pattern, such as amplitude phase, were not limited. Aiming at the problem of peak sidelobe level reduction, the authors of [49] used the ant colony optimization method to adjust the position of the elements in the uniform sparse circle array, obtaining a lower peak sidelobe level and providing the maximum directivity in the opposite direction. The authors of [49] reported a comparative evaluation of the proposed ACO method with the GA and IWO methods for an array of 10 elements and ${\phi }_0=0^{\circ }$. The results indicated that the PSLL achieved using the ACO method was 3.4 dB lower than that obtained using the GA method. Additionally, the directivity achieved using the ACO method was 1.25 dB higher than that obtained using the GA method. Moreover, the PSLL achieved using the ACO method was 5.4 dB lower than that obtained using the IWO method, and the directivity achieved using the ACO method was 1.88 dB higher than that obtained using the GA method. For large arrays, in [50], the ant migration path was used as the subarray division scheme, the peak sidelobe level was taken as the optimization goal, an adaptive beamforming method based on the ant colony algorithm was proposed, and the peak sidelobe level was effectively lowered. The advantages of the ant colony algorithm are its simple design, good robustness, easy parallel implementation, and the characteristics of implicit local search, but the disadvantage are its slow convergence and easy settling into a local optimum.

In recent years, with the continuous development and improvement of intelligent algorithms, the main research trends have begun to gravitate toward the mixing and improvement of traditional intelligent algorithms and the proposal of emerging intelligent optimization methods, such as the use of biogeography-based optimization (BBO) to optimize the amplitude excitation of rectangular arrays, concentric circle arrays, and hexagonal arrays, which obtain a lower peak sidelobe level, but lead to a decrease in the directionality of the array shape [51]. In order to solve this problem, the authors of [52] used the teaching–learning-based optimization algorithm (TLBO) to reduce the peak sidelobe level of planar array, which does not need to tune parameters, shortens the algorithm processing time, and highlights the superior array pattern characteristics while effectively reducing the number of elements. The symbiotic organism search (SOS) algorithm [53] is used to design concentric circle arrays with the best peak sidelobe level, which significantly reduces the number of elements compared with the BBO algorithm and the TLBO algorithm, but can obtain lower sidelobes and has better robustness without tuning parameters. The study presented in [53] conducted a comparison of the thinning techniques of CCAA using SOS, BBO, and TLBO methods for an array of 440 elements. The results revealed that the BBO method employed 211 array elements to achieve a PSLL of −26.58 dB. The TLBO method achieved a PSLL of −28.82 dB by utilizing 215 array elements. However, the SOS method achieved a lower PSLL of −31.18 dB with the use of 208 array elements. Aiming at the problem that the traditional intelligent method is slow to converge and easily falls into a local optimum, the authors of [54] used the whale optimization algorithm (WOA) to optimize the concentric circle array, which could obtain a low sidelobe level, while only a small number of iterations could obtain the global optimality, and the results could be prevented from falling into local optimum. The authors of [54] presented a comparative analysis of the PSO and WOA methods for thinning CCAA under identical array size conditions. The evaluation was conducted on a CCAA with 168 array elements, and the results indicated that the WOA method achieved a PSLL of −27.98 dB by employing a reduced set of 87 array elements. In contrast, the PSO method required 97 array elements to achieve a slightly lower PSLL of −27.81 dB. In order to optimize multiple targets at the same time, moth-flame optimization (MFO) was used to improve the far-field radiation characteristics of the antenna [55], resulting in a narrower zero beamwidth and a lower peak sidelobe level. In addition to the above methods, many emerging intelligent algorithms have been applied to the optimization design of sparse planar arrays, such as invasive weed optimization (IWO) [56], atomic search optimization (ASO) [57], brainstorming optimization (BSO) [11], and firefly algorithm (FA) [58]. More and more new methods have been introduced in this field, and it is still an important research direction to deal with sparse planar array optimization problems based on traditional intelligent algorithms.

4.2.2. Sparse Area Array Optimization Based on Artificial Intelligence Methods

Artificial intelligence methods have attracted great attention in recent years [59], and they are widely used in all walks of life, opening up new ways for intelligent optimization methods. Especially in mobile communication scenarios, the artificial intelligence method adaptively adjusts the weight coefficients of each element in the antenna array on the basis of the dynamic perception of the electromagnetic environment in space, and then changes the pattern shape, polarization, and other performance parameters of the smart antenna while in orbit. The ultimate goal is to enable the satellite communication load to respond effectively to the electromagnetic environment it faces, thereby achieving optimized communication performance. Some scholars have begun applying artificial intelligence methods such as deep learning (DL), artificial neural network (ANN), support vector machine (SVM), and Gaussian regression (GP) to the optimization design of intelligent antenna arrays (SAA).

The advantages of deep learning (DL) are its strong learning ability and good adaptability, and it is generally used to solve the adaptive beamforming problem of sparse area arrays. For example, the authors of [60] proposed a beamforming method based on deep learning, constructed a convolutional neural network (CNN) model to calculate the optimal weight vector, and calculated the excitation weight in real time. This method could still show good beamforming performance when the array had amplitude-phase error and mutual coupling.

The artificial neural network has the function of self-learning and high-speed optimization, and it is good at finding the relationship between certain variables and results from complex processes, which is often used to improve the complexity of array optimization calculation and deal with the coupling effect of sparse area arrays. For example, the authors of [61] used the artificial neural network (ANN) to construct a multilayer feedforward neural network with supervised learning by introducing the numerical electromagnetic radiation map of the coupled periodic array as a database, which effectively solved the sparse area array synthesis problem, including the coupling effect. This method considers the coupling between elements, has good directionality, and has significant advantages in speed and memory consumption, but the peak sidelobe level of this method is high, and the suppression effect is not obvious. In order to solve this problem, the authors of [62] used a neural network to carry out adaptive optimization design of array element radiation performance and array element spacing, used the convex optimization method to optimize the array comprehensively, and proposed an antenna array optimization design method based on an artificial neural network (ANN) and convex optimization. This method considers the coupling between array elements and synergistically optimizes the array factor, which can control the main lobe and obtain a lower peak sidelobe.

Support vector machines (SVMs) are good at dealing with problems that require classification and decision making in sparse area array design, such as antenna array element failure problems [63], which can improve the robustness of the array. At the same time, SVM is a convex optimization problem, which can also be used to assist the optimization design of sparse area arrays [64].

In summary, the sparse area array optimization method based on artificial intelligence algorithms is more comprehensive and more robust, while having a shorter calculation time and higher calculation accuracy, which is very suitable for scenarios such as wireless communication that require real-time and adaptive optimization and adjustment of arrays. At present, in the optimization design of sparse area arrays, the application of this type of method is relatively rare. Nevertheless, in the past 1 or 2 years, the combination of artificial intelligence methods and antenna-related fields has strengthened. This type of method will become the mainstream method of intelligent reconfigurable antenna design in the future, with good application prospects.

4.2.3. Local Optimization Methods

The essence of the local optimization method is based on convex optimization (CO) [65]. In 1997, Lebret and Boyd first used CO technology to solve the problem of minimizing peak sidelobe levels [66]. In array synthesis, the optimization problem can be described as a CO problem; then, the problem is essentially solved.

Before introducing CO technology, convex sets and convex functions are first introduced. A convex set can be defined as a convex set for any $x_1,x_2\in C$ and $0\le \theta \le 1$, if it satisfies

$$\theta x_1+\left(1-\theta \right)x_2\in C.$$

(13)

If the function $f$ is a convex set in the domain, using $domf$ to describe the domain, and for any $x_1,x_2\in domf$ and any $\theta \in R$, $0\le \theta \le 1$, there is

$$f\left(\theta x_1+\left(1-\theta \right)x_2\right)\le \theta f\left(x_1\right)+\left(1-\theta \right)f\left(x_2\right).$$

(14)

When the inequality in the above equation holds strictly at $x_1\ne x_2$ and $0\le \theta \le 1$, then the function $f$ can be called strictly convex.

According to the above basic concepts, we can describe the convex optimization problem as

$$\begin{array}{l}\min f\left(x\right)\\ s.t.g_i\left(x\right)\le 0,i=1,2,\dots ,m\\ \hspace{1em}\hspace{1em}{h}_i\left(x\right)=0,i=1,2,\dots ,p,\end{array}$$

(15)

where $f$ and inequality $g_i$ are convex functions, and $h_i$ is a radiative function.

There are three main standard forms of convex optimization, as shown in Figure 14. Sparse area array synthesis problems can be converted into these standard forms and then solved using toolkits such as CVX [67], and SeDuMi [68]. At present, the second-order cone program (SOCP) is the most widely used in sparse area matrix synthesis [69]. For example, the authors of [70] proposed to transform the array pattern synthesis problem into an SOCP problem, and then synthesize the pattern with very low peak sidelobe level using the cyclic iteration method under the condition of satisfying the expected main lobe, which is suitable for any array.

At present, local optimization methods have been widely used in sparse array optimization design. Aiming at the beamforming problem, an iterative beam synthesis method based on sequence convex optimization was proposed in [71], which realizes the synthesis of arbitrary-shaped beams and the minimization of peak sidelobe levels. Aiming at the array synthesis problem, the authors of [72] proposed an array synthesis method based on convex optimization, which uses conjugate symmetric beamforming technology to form weights, so that the upper and lower bounds of the antenna pattern are constrained as convex. The problem is transformed into a second-order cone planning problem. A coupling matrix is introduced to reduce the coupling effect between the elements, the main lobe of any beam width and response ripple can be obtained by using a smaller number of elements, and the designed element spacing can be greater than half wavelength, reducing the coupling effect without the presence of the grating lobes. This method has good robustness. Aiming at the synthesis of ultrasparse array patterns in large aperture scenes, a fast algorithm using the alternating direction multiplier method was proposed to optimize the position and weight of the array element and effectively reduce the peak sidelobe level [73]. An iterative scheme was used to propose a sparse area array synthesis method based on sequence convex optimization, reducing CPU time consumption while reducing the number of elements and peak sidelobe level [74]. In a follow-up study, the author proposed a reconfigurable sparse array synthesis method based on sequence convex optimization [75], which simultaneously optimizes the number of elements and excitation weights and can achieve an arbitrary beam width and secondary lobe level, with good robustness. The authors of [76] proposed a joint (JCO) integrated approach. By introducing beamforming, a suitable SPA with a lower lobe and grating lobe can be reconstructed during the scanning process based on the JCO method, and a more accurate reconstruction map can be obtained. In addition, the alternating projection method mentioned in [77], the semi-fixed relaxation method mentioned in [78], and the heavy weighted $l_1$ mode minimization method mentioned in [79] can all be used to deal with antenna pattern reconstruction problems.

Some methods can also be classified as convex optimizations. In 2004, Candes, Romberg, Tao, and Donoho proposed compressed sensing (CS) [80,81], providing new ideas for the design of sparse planar arrays, such as the basis purity (BP) reconstruction algorithm [82], orthogonal matching tracking method (OMP) [83], underdetermined system local method (FOCUSS) algorithm [84], and Bayesian compressed sensing (BCS) reconstruction algorithm [85], which have become popular in recent years.

Compressed sensing theory can reconstruct the original signal with a sampling rate far lower than the Nyquist sampling theorem, and its basic idea can be summarized as follows: if the signal is sparse in a transformation domain, a linear measurement matrix can be selected to project the high-dimensional signal in the sparse domain into the low-dimensional space, and a small number of projections on the low-dimensional space can be used to solve an optimization problem. Finally, the original signal is reconstructed with a high probability. The sparse array construction process is shown in Figure 15.

The construction of sparse area arrays according to compressed sensing theory is mainly divided into three basic steps: sparse representation of signal, linear measurement of signal, and signal reconstruction.

Sparse representation of signal: when there is at most one nonzero value in the signal, the signal is said to be sparse; signals can also be considered sparse if they are not sparse themselves but are expressed sparingly in some basis matrix.

Linear measurement of the signal: After determining that the original signal is sparse in a transformation domain, a sublinear measurement of the signal is performed to obtain a measured value.

Signal reconstruction: Signal reconstruction, also known as signal reconstruction or signal recovery, relies on the measurement matrix and low-dimensional measurement vectors to reconstruct the high-dimensional original signal. When the signal satisfies the restricted isometry property (RIP) condition, the signal satisfies the uniqueness of signal reconstruction.

According to the principle of compressed sensing, the antenna pattern reconstruction problem of sparse planar arrays can be transformed into a solution:

$$\begin{array}{l}\underset{w}{\min }||w|{|}_{{}_0}\\ s.t.\left|\right|f-f_{REF}|{|}_2\le \tau \end{array},$$

(16)

where $f_{REF}$ represents the target antenna pattern, and $\tau$ is the upper limit of the reconstruction error set by the user.

According to the convex optimization technique, the pattern reconstruction problem established according to the principle of compressed sensing can be transformed into

$$\begin{array}{l}\underset{w}{\min }||w|{|}_1\\ s.t.\left|\right|f-f_{REF}|{|}_2\le \tau \end{array}.$$

(17)

Aiming at the sparse area array synthesis problem, the authors of [86] combined compressed sensing theory to propose an iterative convex optimization scannable sparse array synthesis method, which optimizes the excitation weight and element position, significantly reduces the number of elements, and obtains a low peak sidelobe level. Compressed sensing and low-rank recovery techniques were applied to sparse array pattern synthesis [87], and an improved compressed sensing reconstruction algorithm orthogonal multi-match tracking (OMMP) algorithm was proposed, which has better robustness, faster atomic selection speed, and lower sidelobes while saving 20–30% of the array elements.

Aiming at the problem of pattern reconstruction, a pattern reconstruction method based on compressed sensing theory was proposed in [88], which realizes multi-beam reconstruction with an appropriate excitation coefficient and minimizes the number of elements. Aiming at large arrays, a pattern reconstruction method based on multi-task Bayesian compressed sensing was proposed according to the stimulus and merging of small-pitch elements [89]; on the basis of ensuring the accuracy of the reconstruction pattern, a sparse solution with a small number of elements is obtained in a short time.

Aiming at the beamforming problem, a rectangular sparse area array synthesis method based on compressed sensing theory was proposed in [90], which can quickly and effectively shape the beam map of interest and obtain a smaller number of elements. The authors of [91] proposed a beamforming method based on convex optimization and MT-BCS to realize controllable secondary lobe-level reference shape beamforming, and optimized the number of elements and layout.

The local optimization method is suitable for solving the problem of additional constraints, and this kind of method has strong applicability and can maintain high computational efficiency. In recent years, the method based on compressed sensing theory has shown many excellent properties, such as effectively alleviating sampling pressure, reducing storage, processing, and transmission costs, and exhibiting high matching accuracy; however, due to the grid mismatch problem [92], it still has certain limitations in sparsity rate. If there is a breakthrough in this aspect, improvements can be made in the optimization design method of sparse area arrays.

4.2.4. Sparse Array Optimization Design Based on Hybrid Algorithm

In the field of sparse planar array design, there are often certain drawbacks and limitations due to the use of a single method. Therefore, in order to improve the efficiency of the algorithm and overcome the shortcomings of a single method in solving the optimization design problem of sparse arrays, a major research trend focuses on the design of sparse planar arrays using comprehensive hybrid methods. Commonly used hybrid methods can be divided into three categories, namely, the mixing of intelligent optimization methods and local optimization methods, the mixing of intelligent optimization methods and analysis methods, and the mixing of local optimization methods and analysis methods.

In order to solve the problem that only the intelligent optimization method is used to search for a large sparse area matrix synthesis and easily falls into the local solution, combining the global optimization method and the analysis method can combine the advantages of the two, which not only ensures a fast calculation speed, but also prevents the algorithm from easily falling into local optimization. In order to prevent the IFT method from falling into the local minimum [93], an algorithm (IWO-IFT) was proposed to mix the invasive weed algorithm with the iterative Fourier transform method, which uses the IFT method to generate weeds, and performs IWO disturbance on the weeds by changing the order of the excitation amplitude of the array elements to obtain better weeds and lower peak sidelobe levels. The authors investigated the synthesized outcomes of circular arrays in [93], where each array possessed different array apertures with varying diameters $25\lambda$, $50\lambda$, and $100\lambda$. The results of their inquiry proved to be nothing short of intriguing. Specifically, when the filling factor of the array was set at an alluring 40%, the PSLL obtained through the IWO-IFT methodology exhibited remarkable discrepancies compared to the PSLL accomplished using the IFT approach. To be exact, the PSLL manifested through the IWO-IFT method was 0.73 dB lower for the $25\lambda$ diameter aperture, 0.66 dB lower for the $50\lambda$ diameter aperture, and 2.60 dB lower for the $100\lambda$ diameter aperture in comparison to the PSLL acquired from the IFT methodology. Nevertheless, despite these substantial variations, the directivity of the synthesized consequences via both the IWO-IFT strategy and the IFT approach was not found to possess any significant differences. The authors of [94] used the IFT method to form a sparse plane array first; then, the DE algorithm was used to optimize the position, and a two-step IFT-DE method was proposed by combining the iterative Fourier transform method and the differential evolution method to reduce the number of elements and the peak sidelobe level and accelerate the convergence speed of the algorithm. The authors explored the synthesized outcomes of rectangular arrays in [94]. Each array possessed diverse aperture dimensions with 36, 64, and 100 array elements, and the fill factor of the array was set at 50%. Upon rigorous analysis, the PSLL manifested through the IFT-DE technique was lower than the PSLL acquired from the IFT approach, exhibiting noteworthy differences. To be exact, the PSLL portrayed via the IFT-DE method was 2.87 dB lower for the 36 elements, 2.91 dB lower for the 64 elements, and 2.28 dB lower for the 100 elements in comparison to the PSLL achieved through the IFT methodology. Aiming at the optimization problem of rotationally symmetric sparse circular arrays [95], a VM-HSDE method combining the differential evolution method (HSDE) and vector mapping (VM) method based on the harmonic search was proposed to reduce the peak sidelobe level and the number of elements, as well as obtain a faster convergence speed. The authors of [96] integrated the FFT method into the PSO algorithm to improve the generation efficiency of the antenna pattern, and proposed a particle swarm algorithm based on CUDA, which is a parallel particle swarm method that integrates the FFT algorithm and intelligence method to realize the synthesis of large-scale reflection arrays.

The combination of intelligent optimization methods and analysis methods can realize multiparameter joint optimization with strong adaptability. The hybrid method proposed in [97] combines CO technology and the SA method to successfully realize the joint optimization of counter-element excitation and position. This hybrid optimization approach is highly adaptable to various array synthesis problems. A hybrid method using the density cone (DT) method to initialize genetic algorithms was applied to synthesize thinning arrays [98], enhancing the ability of global optimization methods to synthesize large aperture sparse areal arrays. The authors of [99] proposed a hybrid method combining convex programming (CP) and the genetic algorithm (GA) to minimize peak sidelobes of sparse surface arrays and optimize the position and excitation of arrays. The authors explored the sparse division of an array with a size of $16\lambda \times 13\lambda$ and 891 elements into a sparse array consisting of only 265 elements. The fill factor was set at a mere 29.7%. The results were nothing short of astonishing. The PSLL obtained from this novel approach proved to be significantly lower than the expected value of −20 dB.

Combining the analysis method with the local optimization method can realize multiparameter joint optimization, taking into account the method’s calculation speed and strong pertinence. The Fourier transform of the model designed by the Elliott–Stern method is used to obtain the initial element excitation [100], and IFT is applied to iteratively optimize the element excitation, which synthesizes a large torus sparse array, improving the algorithm’s speed and robustness. The authors of [34] combined the density cone (DT) method and iterative Fourier method; the IFTDT algorithm was proposed to perform ring partition of the array aperture and determine the number of zonal array elements through the Taylor aperture distribution model, which significantly improved the convergence ability and convergence stability of the IFTDT algorithm. Aiming at beamforming problems, a hybrid method combining IFT and quadratic programming was proposed in [101], which could obtain better radiation performance while reducing the number of elements. The authors of [102] combined the fast Fourier transform method and generalized the analytic equation, proposing a uniformly distributed circular array synthesis method with quantitative weights to reduce the peak sidelobe level and the narrowest half-power beamwidth.

Ultimately, to facilitate a more intuitive comparison of the synthesis results yielded by the four optimization techniques, for the benefit of our readers,

Table 1. Quantitative indications obtained from the literature.

Subject	Method	Number of Referenced Array Elements	Number of Optimized Array Elements	Reference PSLL (dB)	Optimized PSLL (dB)	Filling Factor	Directivity (dB)
An efficient approach for the synthesis of large sparse planar array [30]	MEPM	196	100 121	−30.0 dB	−25.11 dB −29.96 dB	0.51 0.62	20.05 dB 22.39 dB
Synthesis of Thinned Planar Arrays Using 0–1 Integer Linear Programming Method [31]	0–1 ILP	144 144 144 144 256 256 256	49 69 76 88 128 128 128	−17.60 dB −19.40 dB −23.07 dB −18.65 dB −26.72 dB −25.30 dB −25.84 dB	−19.54 dB −23.20 dB −24.26 dB −23.77 dB −31.04 dB −30.95 dB −29.74 dB	0.34 0.48 0.53 0.61 0.50 0.50 0.50	/ / / / 25.80 dB 25.42 dB 25.15 dB
Research and application of large aperture ultra-sparse array synthesis algorithm [73]	ADMM	400	129	−20.0 dB	−20.2 dB	0.32	/
Synthesis of maximally-sparse square or rectangular arrays through compressive sensing [90]	CS	162	94	−30.3 dB	−28.9 dB	0.58	/
Application of compressed sensing theory to sparse array orientation graph synthesis [87]	OMMP	27	19	−30.0 dB	−29.9 dB	0.70	/
Synthesis of pattern reconfigurable sparse arrays with multiple measurement vectors FOCUSS method [84]	FOCUSS	20	16	−19.50 dB	−19.80 dB	0.8	/
Near-field pattern synthesis for sparse focusing antenna arrays based on Bayesian compressive sensing and convex optimization [91]	BCS	121	52	−20.00 dB	−20.30 dB	0.43	/
Synthesis of linear and planar arrays via sequential convex optimizations [74]	SCO	49 121 121 121 121	35 77 89 102 105	−17.6 dB −24.33 dB −24.33 dB −24.33 dB −24.33 dB	−17.64 dB −24.35 dB −24.37 dB −24.32 dB −24.34 dB	0.71 0.64 0.74 0.84 0.87	/
Synthesis of Concentric Circular Antenna Array Using Whale Optimization Algorithm [54]	WOA	60 90 126 168	33 58 73 87	−18.09 dB −21.89 dB −20.88 dB −27.81 dB	−24.15 dB −26.71 dB −25.46 dB −27.98 dB	0.55 0.64 0.58 0.52	/
Pattern design of 2D antenna arrays using biogeography-based optimization [51]	BBO	49 100	49 100	−12.65 dB −12.96 dB	−20.96 dB −20.82 dB	1	17.14 dB 20.27 dB
Effective minimization of side lobe level of sparse thinned planar array antenna in multiple planes with constraints [52]	TLBO	108	84	−26.35 dB	−22.27 dB	0.78	/
Planar Thinned Antenna Array Synthesis Using Modified Brain Storm Optimization [103]	BSO	200	100	−20.0 dB	−19.9 dB	0.50	/
Concentric circular antenna array synthesis for side lobe suppression using moth flame optimization [55]	MFO	18 30	18 30	−11.23 dB −9.55 dB	−36.84 dB −27.92 dB	1	12.15 dB 10.76 dB
Design of planar concentric circular antenna arrays with reduced side lobe level using symbiotic organism’s search [53]	SOS	105	51	−18.26 dB	−18.19 dB	0.48	/
A hybrid approach based on PSO and Hadamard difference sets for the synthesis of square thinned arrays [46]	HSPSO	36 144 576	21 75 323	−13.06 dB −16.74 dB −18.97 dB	−13.57 dB −18.01 dB −22.22 dB	0.58 0.52 0.56	/
Adaptive Learning of Probability Density Taper for Large Planar Array Thinning [37]	PLIFT	7790	3116	−31.80 dB	−33.30 dB	0.40	/
Synthesis of large planar thinned arrays using IWO-IFT algorithm [93]	IWO-IFT	1928	772 578	−23.0 dB −22.4 dB	−27.13 dB −25.93 dB	0.40 0.30	32.92 dB 31.79 dB
A two-step method for the low-sidelobe synthesis of uniform amplitude planar sparse arrays [94]	IFT-DE	144	70	−17.26 dB	−20.91 dB	0.48	/
Multiple-constraint synthesis of rotationally symmetric sparse circular arrays using a hybrid algorithm [95]	VM-HSDE	231	201	−23.74 dB	−24.21 dB	0.87	/
Low-sidelobe pattern synthesis for sparse conformal arrays based on PSO-SOCP optimization [104]	PSO-SOCP	62	50 40 30	−21.53 dB	−22.12 dB −21.65 dB −21.20 dB	0.81 0.65 0.48	/
A Hybrid Approach for the Synthesis of Planar Thinned Arrays with Sidelobes Reduction [99]	CP-GA	891	265	−20.00 dB	−22.50 dB	0.30	/

furnishes the quantitative indications for element quantity, fill factor, PSLL, and directionality drawn from key references pertaining to the four optimization methods expounded upon in this chapter.

5. Summary and Outlook

In this paper, the optimization design of sparse planar arrays in satellite communication scenarios was studied, the mathematical foundation of antenna arrays was introduced, and the optimization design model of sparse planar arrays was established. Then, the influence of different array layout methods on the sparse planar array antenna pattern was analyzed and compared, and the recent new design trend of array material design was also introduced. Then, four types of methods (analysis methods, local optimization methods, intelligent optimization methods, and hybrid methods) were reviewed. On this basis, the frequently used iterative Fourier analysis method, the four traditional intelligence methods and emerging artificial intelligence methods, the methods based on convex optimization technology and compressed sensing theory, and the three-type hybrid method based on the combination of the above were introduced, while the advantages and disadvantages of each method were summarized.

5.1. Future Research Directions

On the basis of the above research content, three future research directions and two important challenges are proposed for optimizing sparse planar arrays in satellite communication scenarios.

5.1.1. The New Method Brings a New Dawn to the Optimization Design of Sparse Area Arrays

At present, in the satellite communication scenario, with the continuous expansion of the scale of the array, for sparse area arrays, the efficiency of some methods, such as the traditional intelligence method, will continue to decline with the expansion of the array scale, and relying on a single method is obviously no longer up to the standard at the engineering application level, while the accuracy requirements of the method have also been increased. Therefore, the use of hybrid methods becomes even more important. In future research, artificial intelligence algorithms such as deep learning, reinforcement learning, and various neural networks (ANN, LMS, etc.) have better generalization ability and better robustness for complex arrays, providing more degrees of freedom to break through the performance boundary of existing technologies and create new functions for antenna design, which will become a hotspot in future research for sparse array optimization design or auxiliary sparse array optimization design. The optimization design method based on compressed sensing shows many excellent properties, such as high matching accuracy, but its own mesh mismatch problem limits the performance of the algorithm in a sparse environment. Thus, how to effectively solve this problem is a research direction that needs addressing. It is worth mentioning that, in order to reduce the software and hardware complexity of the system, increase the computing speed, accelerate convergence, and improve the performance of the array, the combination of sub-array division technology and various methods can also become a new research direction in the future, such as in the distributed constellation. The array of satellites in the constellation can be regarded as different sub-arrays, forming a multi-star joint array, which has important research significance in the optimization design of sparse arrays in the future.

5.1.2. Intelligent Reconfigurable Antennas Have Become the Focus of Research

The reconfigurability and intelligence of the communication load antenna are effective ways to improve the flexibility and anti-interference ability of the satellite communication system. To achieve beamforming under different requirements and realize flexible arrays, the optimization design of intelligent reconfigurable antennas and digital antennas will become an important direction in this research field. In recent years, with the rapid development of satellite technology, the location of sparse arrays has continuously shown a trend of diversification. Spaceborne sparse area arrays require arrays to have higher flexibility, lower system complexity and cost loss, and higher reliability and anti-interference solid performance; hence, the array is required to have adaptive, reconfigurable, anti-interference performance and real-time dynamic adjustment of beams. The intelligent reconfigurable antenna can flexibly use beams, embodying the characteristics of self-selection, self-optimization, and adaptation, which can exponentially expand the communication capacity and has broad application prospects in the field of satellite communication. In satellite communication, intelligent reconfigurable antennas cover frequency, polarization, and pattern reconstruction from the reconstruction domain. The implementation methods include on-orbit RF reconstruction and on-orbit autonomous digital reconstruction. At the level of RF reconstruction, the smart antenna system can change the working frequency band, polarization characteristics, and pattern characteristics of the spaceborne smart antenna in orbit by changing the antenna unit’s radiation structure and array arrangement according to the electromagnetic space situation perception results. In terms of autonomous digital reconstruction, the generalized array response of the smart antenna transceiver array can adaptively adjust the weight coefficient of each array element in the antenna array according to the dynamic spatial electromagnetic environment perception in the broadband range of a specific frequency band, as well as change the shape and polarization of the smart antenna in orbit, such that the satellite communication payload can adapt to the electromagnetic environment faced by the payload and achieve better communication performance in the electromagnetic environment. In future satellite communication systems, smart antennas will mainly depend on two aspects. First, when designing future satellite communication systems, it is necessary to fully consider the characteristics of smart antennas and continuously improve and upgrade them to ensure the compatibility of smart antenna technology, as well as better undertake the development of wireless communication technology. Second, the actual performance of the smart antenna needs to be evaluated according to the key parameters of the future satellite communication system to determine the actual performance of the smart antenna. In conclusion, the development and production of an antenna array for upcoming satellite communication systems ought to prioritize not only the optimization of particular mode performance indicators (e.g., peak sidelobe level, antenna gain, and main lobe width), but also the consideration of propagation scenarios, in order to attain a superior equilibrium between capacity performance and system architecture complexity [105].

5.1.3. Array Material Design Will Be Closely Integrated with Metamaterial Technology

At the engineering level, in order to adapt to the new development situation of emerging fields and expand the new uses of sparse area arrays in satellite communication scenarios, the material design of the array component part is a key factor influencing the performance and cost of the array in optimization design. Currently, the design of feed networks, substrates, and other materials in the optimization design of sparse area arrays is one of the hot research fields in recent years. Metamaterial technology has gradually developed from theory and trial to application. Many devices in the world are often used in complex environments, having high requirements for performance indicators; thus, they are often combined with metamaterial technology. In satellite communication scenarios, sparse surface arrays are required to have the characteristics of ultra-wideband, low profile, low cost, low array element coupling, light weight, and radiation resistance. Therefore, in the future, combining metamaterial technology with array material design will become an important research direction for sparse area array material design.

5.2. Key Challenges

5.2.1. The Array Can Be Located on Any Undulating Carrier and Terrain

In practical engineering applications, the actual environment deviates from the ideal situation, and the antenna array may be located on any undulating and shaped carrier or terrain; for example, carrying the array on the satellite load cannot achieve the ideal area array. This also makes the actual array layout of the array elements different from theory, and the antenna pattern will inevitably cause deviations. This deviation will increase with the expansion of the array scale. To avoid losing generality, two methods can be considered to solve this problem; the first method is to use a three-dimensional layout space surface array in the pattern design. The second method is to equalize the three-dimensional layout array antenna to a sparse planar array, but the actual error needs to be considered, and some parameters to correct the error can be introduced to reduce the error and improve the design accuracy.

5.2.2. The Element Spacing Leads to a Contradiction between the Gate Lobe and the Coupling

In the optimization design of sparse planar arrays, it is imperative to address the challenge posed by the impact of array element spacing on the antenna pattern. Excessive spacing between elements can introduce grating lobes into the antenna pattern, while inadequate spacing leads to strong coupling effects that adversely affect the pattern. In particular, the optimization design of sparse cloth plane arrays is related to the sparsity of the array, the array element layout, and the antenna pattern. Consequently, achieving a balance between suppressing grating lobes and reducing coupling effects represents a key engineering challenge.