With the development of radio technology, applications such as mobile communication, radio, television, navigation, space, and radar positioning have progressively permeated every facet of modern defense science and technology as well as the daily lives of people [1]. Array antennas, capable of achieving low sub-lobe, high gain, and assignment of directional maps, are extensively utilized in fields such as radio communication, satellite navigation, and space measurement and control [2,3]. However, although the technology of planar array synthesis has been very mature, the space arrangement restricts its wide-angle scanning characteristics. It cannot realize the simultaneous tracking of multiple targets in the full airspace, therefore, to make up for the defects of planar arrays, and realize the scanning characteristics of the full airspace, the use of conformal array antenna is a preferred solution [4,5]. However, in practical applications, there are still relatively unsolved problems, especially due to the high cost of the transmitter-receiver module, which makes the overall cost of the array quite high. Therefore, reducing the cost of the phased array with a small loss of performance is a considerable challenge [6].

To solve these problems, several scholars have proposed the sparse array approach to reduce the use of transmitting and receiving components by reducing the number of antenna units and optimizing the antenna positions [7–11]. Although this method can reduce the array cost, the aperture of the antenna is wasted, causing an unacceptable decrease in array gain.

With advancements in related algorithms, irregular arrays using subarray tiling have been favored by scholars. In 2009, Mailloux et al. first used an irregular eight-networked lattice to fill an 18×24 grid [12]. Subsequently, in 2012, Rocca and Mailloux used genetic algorithms to solve the problem of accurate delineation of large-scale arrays [13]. However, their approach resulted in a large number of unfilled voids appearing inside the array, failing to maximize caliber efficiency. To fully utilize the array caliber and prevent voids from appearing, the height function method of dominoes [14] was adopted due to its simple form and ease of coding [15–17], and excellent results were obtained. References [18] and [19] used the X algorithm to optimize the subarray structure design, offering an effective perspective on a subarray scheduling scheme. Furthermore, Anselmi et al. introduced an effective subarray structure using the self-replicating method to achieve the effect of low sublobe at wide angles [20]. While these non-regular designs have been implemented in planar arrays, research on the non-regular subarray design for conformal antennas is relatively scarce. Nevertheless, investigating conformal non-regular subarrays is crucial as it allows for the full exploitation of conformal array characteristics, enabling broader scanning angles and reducing the design costs for conformal arrays, which is very important for the development of the arrays. Therefore, based on the design method for the planar irregular subarray, this paper proposes an effective design method of the conformal subarray, which can reduce the design cost while maximizing the use of conformal array characteristics.

The rest of this paper is structured as follows: Section 2 derives the computational patterns for conformal arrays. Section 3 establishes the tiling problem for domino subarrays. Section 4 proposes an optimization method for conformal array subarrays. Section 5 demonstrates the optimization results for the conformal array subarray.

Compared to planar arrays, the pattern synthesis of radiation patterns for conformal arrays is more complex and challenging, mainly because the array elements for conformal arrays are in a three-dimensional coordinate system, which is influenced by the carrier curvature of the conformal surfaces. Consequently, the beam synthesis for conformal arrays needs to be analyzed independently.

To avoid generality, the three-dimensional array of arbitrary positions is discussed. Similar to the planar array, to realize the calculation of the phase compensation amount, the positional coordinates of the array element need to be calculated. As shown in Fig. 1, there is an array unit n in space, and suppose that the radiation direction pattern of this antenna unit in spherical coordinates is ${E_n}\left( {\theta {\mathrm{,}}\left. {{\text{ }}\varphi } \right)} \right.$, with the reference point of the array phase being the origin $O$. For the unit $n$, the phase difference between its received far-field signals and the far-field signals received at the reference point $O$ can be obtained using the dot product method.

Assuming ${\mathbf{R}}({x_n}{\mathrm{,}}{\text{ }}{y_n}{\mathrm{,}}{\text{ }}{z_n})$ represents the vector from the origin $O$ to the array element $n$, and ${{\mathbf{R}}_{\text{0}}}$ is the unit vector from the origin $O$ to the direction of the far-field target, which is $ \left| {{{\mathbf{R}}_0}} \right| = 1 $. According to the geometric relationship in Fig. 1, the vector ${\mathbf{R}}({x_n}{\mathrm{,}}{\text{ }}{y_n}{\mathrm{,}}{\text{ }}{z_n})$ from the origin $O$ to the array element $n$ can be expressed as: $({r_n}\sin {\theta _n}\cos {\varphi _n}{\mathrm{,}}\;{r_n}\sin {\theta _n}\sin {\varphi _n}{\mathrm{,}}\;{r_n}\cos {\theta _n})$. Likewise, the unit vector $ {{\mathbf{R}}_0} $ from the origin $O$ to the direction of the far-field target can be expressed as $(\sin {\theta _0}\cos {\varphi _0}{\mathrm{,}}{\text{ }}\sin {\theta _0}\sin {\varphi _0}{\mathrm{,}}{\text{ }}\cos {\theta _0})$. By the dot product method, the phase of the array element $n$with respect to the origin $O$ to the far-field target can be derived.

$ {\beta _n} = \frac{{2\pi }}{\lambda }({{\mathbf{R}}_0} \cdot {\mathbf{R}}) = \frac{{2\pi }}{\lambda }{r_n}(\sin {\theta _n}\sin {\theta _0}\cos ({\varphi _0} - {\varphi _n}) + \cos {\theta _{\text{n}}}\cos {\theta _0}) $ (1)

where $ (\cdot) $ denotes the vector dot product, $({r_n}{\mathrm{,}}{\text{ }}{\theta _n}{\mathrm{,}}{\text{ }}{\varphi _n})$ represents polar coordinates of unit $n$, $(\theta {\mathrm{,}}{\text{ }}\varphi )$ is space coordinate, and $\lambda $ is the wavelength.

As a result, the pattern for array element $n$is expressed as

$ {E_n}\left( {\theta {\mathrm{,}} {\;\varphi }} \right) = {g_n}\left( {\theta {\mathrm{,}} {{\text{ }}\varphi } } \right) {A_n}{\text{exp}}({\text{j}}\left(\frac{{2\pi }}{\lambda }{r_n}\left( {\sin {\theta _n}} \sin \theta \cos \left( {\varphi - {{\varphi _n}}} \right) + \left( {\cos {\theta _{\text{n}}}} {\cos \theta } \right) - {{\psi _n}} \right)\right) $ (2)

where ${g_n}\left( {\theta {\mathrm{,}}\left. {{\text{ }}\varphi } \right)} \right.$ is the pattern of unit $n$, ${A_n}$ is the feed amplitude of unit $n$, and $ {\psi _n} $ is the feed phase of the unit $n$.

In the case of conformal arrays, due to the difference in spatial curvature, spatial coordinate transformations are required. He et al. carry out a detailed far-field computation method for conformal arrays, with its core concept being the transformation of the spatial coordinate system[21]. The pattern solution proposed in this paper relies on the analytical expression of the unit pattern; otherwise, the solution would be unattainable. This paper starts from the simulation data based on the antenna unit pattern and directly transforms the data of the pattern. This approach greatly streamlines the computational process and simplifies operational steps, enhancing its applicability in actual engineering scenarios.

For point $p({r_p}{\mathrm{,}}\;{\text{ }}{\theta _p}{\mathrm{,}}\;{\text{ }}{\varphi _p})$, after rotation, the point becomes ${p^\prime }({r_p}{\mathrm{,}}\;{\text{ }}{\theta _p}{\mathrm{,}}\;{\text{ }}{\varphi _p})$ and the test data for that point is the same as the test data for the previous point $p$. Therefore, the first step in the rotation of the pattern is a coordinate rotation in space, that is by three rotations. The rotation formula is as follows

$ {{\mathbf{I}}_x} = \left( {\begin{array}{*{20}{c}} 1&0&0 \\ 0&{\cos \alpha }&{ - \cos \alpha } \\ 0&{\sin \alpha }&{\cos \alpha } \end{array}} \right) $ (3a)

$ {{\mathbf{I}}_y} = \left( {\begin{array}{*{20}{c}} {\cos \alpha }&0&{\sin \alpha } \\ 0&1&0 \\ { - \sin \alpha }&0&{\cos \alpha } \end{array}} \right) $ (3b)

$ {{\mathbf{I}}_z} = \left( {\begin{array}{*{20}{c}} {\cos \alpha }&{ - \sin \alpha }&0 \\ {\sin \alpha }&{\cos \alpha }&0 \\ 0&0&1 \end{array}} \right) $ (3c)

where ${{\mathbf{I}}_x}$ is the rotation matrix rotated along the x-axis, ${{\mathbf{I}}_y}$ is the rotation matrix rotated along the y-axis, ${{\mathbf{I}}_z}$ is the rotation matrix rotated along the z-axis, and $\alpha $ is the rotation angle.

Since different rotations will make the original observation angle deviate from the observation angle produced after the rotation, this paper adopts the interpolation method to solve the problem of observation angle deviation.

As for any rectangular array, the position of each grid inside it can be represented by a vector ${\mathbf{r}} = (m{\mathrm{,}}{\text{ }}n)$, where $m$ represents the distance from the x-axis and $n$ represents the distance from the y-axis. The vector ${\mathbf{r}}$ corresponding to each position is given in Fig. 2. The blue parts in Fig.2 represent the set $A$ and the green parts in Fig.2 represent the sets ${B_n}$, where the elements of set $A$ represent each position of the array and each element of set ${B_n}$ represents the position of the array occupied by the nth subarray. The set $A$ can be represented as

$ A = \left\{ {(m{\mathrm{,}}{\text{ }}n)|m{\mathrm{,}}{\text{ }}n \in N{\mathrm{,}}{\text{ }}0 \leq m \leq 4{\mathrm{,}}{\text{ }}0 \leq n \leq 4} \right\} $ (4)

where N is set of natural numbers.

The sets ${B_1}$ and ${B_2}$ can be represented as

$ {B_{\mathbf{1}}} = \left\{ {\left( {0{\mathrm{,}}{\text{ }}0} \right){\mathrm{,}}{\text{ }}\left( {0{\mathrm{,}}{\text{ }}1} \right){\mathrm{,}}{\text{ }}\left( {0{\mathrm{,}}{\text{ }}2} \right){\mathrm{,}}{\text{ }}\left( {1{\mathrm{,}}{\text{ }}0} \right)} \right\} $ (5)

$ {B_{\mathbf{2}}} = \left\{ {\left( {2{\mathrm{,}}{\text{ }}1} \right){\mathrm{,}}{\text{ }}\left( {2{\mathrm{,}}{\text{ }}2} \right){\mathrm{,}}{\text{ }}\left( {2{\mathrm{,}}{\text{ }}3} \right){\mathrm{,}}{\text{ }}\left( {3{\mathrm{,}}{\text{ }}3} \right)} \right\}. $ (6)

For this plane there can be several candidate subarrays for splicing. Each element in the set $S$ represents one of the candidates, which can be dented as

$ S = \left\{ {{B_1}{\mathrm{,}}{\text{ }}{B_2}{\mathrm{,}}{\text{ }} \cdots {\mathrm{,}}{\text{ }}{B_K}} \right\} $ (7)

where $K$ is the total number of candidate subarrays.

The subarray splicing problem can be converted into a mathematical problem by the above method, but the coordinate positions of the elements in its set are too complicated. Therefore, in this paper, the set in this method is converted into a sequence of binary vectors. For a given array of size $M \times N$, let the set $B$ convert to a column vector, which represents the arrangement scheme of each subarray. If the subarray occupies a certain position in the array, the value of that position is 1, and vice versa is 0. Based on this, a dictionary matrix ${\mathbf{L}}$ is created, where each row of it represents the arrangement of the array elements in a subarray, with a total of $K$ rows, which can be clearly seen as a binary matrix consisting of 0 and 1.

Thus, the problem of tiling the domino subarray can be established as

$ { {{\boldsymbol{\mathbm{ξ}}}}^{\text{T}}}\;{\mathbf{L}} = {\mathbf{1}}_{1 \times (M \times N)}^{\text{T}} $ (8)

$ {\text{s}}{\text{.t}}{\text{. }}{{\mathbf{1}}_{1 \times K}}{{\boldsymbol{\mathbm{ξ}}}} = t \qquad {{\boldsymbol{\mathbm{ξ}}}} \in \{ 0{\mathrm{,}}\;1\} $ (9)

where $t$ represents the total number of rows needed for complete tiling of the matrix for that multinational network lattice. ${{\boldsymbol{\mathbm{ξ}}}}$ is the selection vector which has $K$ columns, where 1 represents the selection of the sublattice of that row of the dictionary matrix ${\mathbf{L}}$, and vice is 0.

There are various ideas for solving the subarray splicing problem, in Refs. [22] and [23], which convert the non-convex problem into a convex optimization problem. The X algorithm is employed in this paper, a method introduced by the eminent computer scientist Donald Knuth, to address the subarray splicing conundrum. The algorithm has been thoroughly detailed in Ref. [24], thus precluding the need for repetition in this paper.

Assuming a spherical conformal array, as shown in Fig. 3 (a), the antenna units are uniformly distributed within θ < 45° according to the spherical latitude and longitude lines. The curvature of the surface at each antenna’s location varies, presenting a complex scenario for subarray design. If two or even more antennas are combined into a subarray according to the optimization method of planar array, multiple subarrays with different curvatures need to be designed. This approach poses significant challenges for the feed network and the antenna structure design.

To facilitate the implementation of the conformal array subarray, this paper proposes a multilevel subarray delineation strategy. Initially, a primary level of subarray delineation is performed on the original spherical surface, and then the small subarray delineation is carried out again on each level of subarray. Due to the characteristics of the sphere, the large subarrays will be spliced with rectangular and trapezoidal shapes. The cells in each large subarray are arranged in a planar grid, and the spacing between cells under different large subarray modules is guaranteed to be equal, as illustrated in Fig. 3 (b). In this way, it is easier to realize the division of the duplex subarray at each level of the subarray, and subsequently complete the subarray layout in the whole sphere. This approach not only reduces the computation time but also makes it easier to be applied in the actual engineering.

To verify the validity of this segmentation, this paper compares the gain and Side Lobe Lever (SLL) of the two arrangements at different scan angles. As shown in Table 1, it can be seen that the gain of the multilevel subarray segmentation approach at different scanning angles decreases by approximately 1.5 dBi, and SLL decreases by 1 dB compared to the conventional spherical array gain. This result shows that the multilevel subarray segmentation approach does not significantly deteriorate the performance of the conformal arrays, and the multilevel subarray segmentation approach is feasible.

Based on the cellular arrangement of each first-level subarray, the dictionary matrix of each first-level subarray is written. The feasible solutions for each one-level subarray are computed using the X algorithm. The overall subarray arrangement of the spherical array can be obtained by subsequently selecting the arrangement scheme among these feasible solutions. If the subarray arrangement is optimized with the subarray amplitude phase, a large amount of computation and a long period are required. Therefore, it is necessary to separate the subarray arrangement and the amplitude phase optimization. Mailloux et al. [25] innovatively proposed the maximum entropy optimization scheduling algorithm. References [16] and [26] used the maximum entropy strategy to obtain satisfactory results. But these researches are based on planar array optimizations. Therefore, a planar grid mapping maximum entropy method is innovatively proposed.

For the set $\zeta = \left\{ {{\zeta _1}{\mathrm{,}}{\text{ }}{\zeta _2}{\mathrm{,}} \; \cdots {\mathrm{,}}{\text{ }}{\zeta _n}} \right\}$ with $n$ variables, define its probability distribution function as $P(\zeta )$. $H(\zeta )$ represents the disorder of this variables. The entropy $H(\zeta )$ of the $\zeta $ set is

$ H\left( \zeta \right) = - \sum\limits_{i = 1}^n {P\left( {{\zeta _i}} \right){{\log }_2}P\left( {{\zeta _i}} \right)} . $ (10)

Compared to planar arrays, conformal arrays have spatial curvature. Their arrangement is not strictly horizontal or vertical throughout the entire space. Therefore, a planar grid is established. Its interval keeps the proper length, generally taking half of the cell spacing in the subarray. Subsequently, the gravity of the subarray is mapped onto the planar grid. As shown in Fig. 4, the gravity of the subarray is obtained within the planar grid. The distribution of the gravity of the subarray in the planar grid can be linked to probability in the entropy $H(\zeta )$.

Specifically, ${r_i}$ and ${c_j}$ represent the number of gravity of subarrays mapping on the $i$ row and $j$ column of the planar grid. The probability of each row is $ {r_p} = {r_i}/2T $. The probability of each row is $ {c_p} = {c_j}/2T $. $T$ is the number of subarrays. The entropy of the irregular subarray is

$ H = - \sum\limits_{i = 1}^M {\frac{{{r_i}}}{{2T}}} {{\mathrm{log}} _2}\frac{{{r_i}}}{{2T}} - \sum\limits_{j = 1}^N {\frac{{{c_j}}}{{2T}}} {{\mathrm{log}} _2}\frac{{{c_j}}}{{2T}}{\mathrm{.}} $ (11)

The entropy $H(\zeta )$ is maximized when the variable is uniformly distributed. The performance of the array depends on the periodicity of the subarray arrangement. A larger $H(\zeta )$ represents a more disordered subarray arrangement and weaker periodicity. Therefore, a more capable subarray arrangement implies lager $H(\zeta )$. The irregular subarray arrangements optimization problem can be transformed to the calculation of $H(\zeta )$. Consequently, the problem can be described as

$ {\text{min}}: - \left( { - \sum\limits_{i = 1}^M {\frac{{{r_i}}}{{2T}}} {{{\mathrm{log}} }_2}\frac{{{r_i}}}{{2T}} - \sum\limits_{j = 1}^N {\frac{{{c_j}}}{{2T}}} {{{\mathrm{log}} }_2}\frac{{{c_j}}}{{2T}}} \right){\mathrm{.}} $ (12)

While the enumeration algorithm can provide the optimal arrangement for subarray configurations, it inevitably results in a substantial computational burden. To address this challenge, the particle swarm optimal algorithm (PSO) will be used to encode each arrangement, using the maximum entropy as the fitness function.

The global search ability of PSO has been verified in many literature. In this paper, multiple swarms are introduced to ensure the effectiveness of PSO and to enhance its global search capability. Specifically, the individuals within PSO are divided into four clusters. Each cluster has different ${c_1}$ and ${c_2}$ acceleration parameters. The functions of ${c_1}$ and ${c_2}$ for each cluster are shown in Table 2.

The Cauchy variation is introduced to enhance the population diversity and prevent PSO from prematurely converging into local optimal solutions in the early stage. A speed change between 0 and 1 is produced according to the Cauchy probability density distribution of (13). The speed, independent of the particle’s inherent inertia, consciousness, and group consciousness, is completely random. Consequently, individuals within the swarm will proceed in the variable direction to transcend of the local optimal solution and to explore in a larger range. The diversity of the particle population and global search ability will be enhanced.

$ f\left( \chi \right) = \left( {1/\pi } \right)t/\left( {{t^2} + {\chi ^2}} \right) $ (13)

It is necessary to optimize the amplitude phase of the antenna unit to obtain different beams after obtaining the subarray arrangement. Therefore, the optimization method of convex optimization is used to optimize the amplitude and phase of the subarray. This method has been proposed and verified in Ref. [27]. The MOSEK solver in the CVX toolkit will be used to solve the problem in this paper.

Given a sphere of radius $6\lambda $, a first-level subarray division is performed with θ < 45°. The first-level subarray is divided into two parts: The top is arranged in a 6×6 rectangle and the sides are arranged with the same six trapezoidal shaped subarray distributions. As illustrated in Fig. 5, the final arrangements of the distribution of cells in each first-level subarray are given.

To evaluate the effectiveness of mapping for maximum entropy, this paper compares the performance of arrays with different entropy. As shown in Fig. 6, the distribution in the planar mapping grid and the subarray arrangement with different entropy can be demonstrated. The performance of arrays with different entropy can be observed in Table 3. The subarray arrangement with different entropy obtains the same performance in terms of gain. However, when comparing the arrangement with maximum entropy to minimum entropy, the arrangement with minimum entropy declines by 2 dB in terms of SLL at each scan angle. The results illustrate that arrays possessing higher entropy perform better and that the method, of mapping the maximum entropy, is feasible. This approach can obtain the optimized subarray arrangement from several arrangements without optimizing the antenna excitation.

As shown in Fig. 7, patterns of the different scanning angle with maximum entropy are presented. The array has a gain of 25.61 dBi and an SLL is = 23.03 dB at θ = 45°, and a gain of 21.21 dBi and an SLL is = 19.19 dB at θ = 60°. The gain drops by 4.4 dB at θ = 60°, but the SLL does not decline. The result proves that the antenna has good scanning performance.

In this paper, an optimization algorithm based on the maximum entropy of plane mapping is proposed for the conformal array subarray. Initially, the method divides the conformal array into several first-level subarrays and uses X algorithm to determine feasible solutions for first-level subarray tiling. Subsequently, PSO is utilized to optimize the conformal array subarray arrangement using maximum entropy as the fitness function. After that, convex optimization is harnessed to optimize the subarray excitation. The results prove that the method can effectively find the optimal conformal array subarray arrangement.

The authors declare no conflicts of interest.