1. Introduction
Due to the capacity of conforming to surfaces, conformal antennas have been successfully applied in many scenarios in recent decades [
1]. A truncated cone, as an aerodynamic structure, has been applied in many streamlined aircrafts such as missiles and rockets. However, for this truncated cone structure, especially in the case of a small cone angle, two issues will come into view. First, it is difficult to install a large amount of electronic equipment such as transceivers and feeding networks in the narrow space of the cone tip, which makes it impossible to deploy a large-scale antenna array. In addition, the small cone angle proposes an arduous challenge to the performance of axial radiation, which has a great impact on the missile’s data transmission. Despite the above difficulties, several studies have been proposed in recent years due to the wide application of truncated cones [
2,
3,
4,
5,
6].
The sparse array can maintain or even improve the radiation characteristics with fewer antenna elements and can reduce installation space for electronic equipment by arranging the antenna elements nonuniformly, which is very applicable in a truncated cone. However, unlike sparse linear and planar arrays, which already have many mature synthesis methods [
7,
8,
9,
10,
11,
12,
13], such as iterative numerical methods and evolutionary algorithms (EAs), sparse conformal array synthesis has not been solved effectively due to its complex structure. Some effort has been made to solve specific problems in sparse conformal array synthesis [
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24], such as analytical techniques [
14], compressive sensing (CS) techniques [
15], distributed aperture synthesis [
16,
17], and evolutionary algorithms [
5,
18,
19,
20,
21,
22]. In [
23], a versatile multi-task Bayesian compressive sensing (MT-BCS) strategy was used to match a given reference pattern based on a nonuniform conformal array. However, the existing research on sparse conformal arrays mainly focuses on array synthesis, rather than reducing the number of antenna elements. Therefore, array synthesis to achieve a sparse array with low PSLL is of great research significance.
The aforementioned works mainly studied suppression of the sidelobes. For the truncated cone array, fixed axial radiation is also of great importance. Some novel antenna elements were designed to enhance the axial radiation in [
25,
26]. A four-element slot array with parasitical dual U-shaped slots in the axial direction is proposed in [
25]; however, this antenna element is not suitable for large antenna arrays due to its undesirable sidelobe level. At the same time, some optimizing algorithms for conformal surfaces have been proposed to make the main beam point in a specified direction [
27,
28,
29]. Concerning both low sidelobe levels and the axial radiation, a brain storm optimization (BSO) method was developed in [
30]. However, the proposed array achieves a good radiation pattern with axial radiation on a spherical surface instead of a truncated cone array. It is an urgent necessity to study the sparse array synthesis based on the truncated cone conformal array that realizes both a low sidelobe pattern and enhanced axial radiation.
Inspired by natural biological evolution, the evolutionary algorithm optimizes the objects by using different operators, such as the crossover operator and mutation operator. Compared with traditional calculus-based methods, the evolutionary algorithm is a mature global optimization method with high robustness and wide applicability. Moreover, owing to the characteristics of self-organization, self-adaptation, and self-learning, it can effectively deal with complex problems that are difficult for traditional optimization algorithms. Evolutionary algorithms have been widely used in antenna array optimization in recent decades, including genetic algorithms (GA) [
30], particle swarm optimization (PSO) algorithms [
31], grey wolf optimization (GWO) [
32], invasive weed optimization (IWO) [
33], ant colony optimization (ACO) [
34], biogeography-based optimization (BBO) [
35], the backtracking search optimization algorithm (BSA) [
30], etc.
In recent years, the application of GA to antenna array synthesis has been widely studied. In [
36], Yan and Lu suppressed the peak SLL of linear antenna arrays by optimizing the excitation current, whereas Tian and Qian [
37] added the positions of elements as a variable and improved the performance by optimizing both variables simultaneously. Furthermore, the fitness function was redesigned to enhance the global searching property of GA. In 2006, Chen et al. [
38] proposed a modified real genetic algorithm (MGA), which improved the optimization efficiency of the algorithm by reducing the search space. As an improvement to the GA, the differential evolution (DE) algorithm is also widely used in the field of array synthesis. S. Yang et al. [
39] applied the DE algorithm to the synthesis of linear arrays, and an element rotation technique was proposed later in [
40]. Moreover, the DE/best/1/bin strategy was adopted to realize the shaping of power patterns, which broaden the application scope of the DE algorithm [
41]. Based on the above research, this paper continues the study of the optimization of conformal sparse arrays.
The DE algorithm has the advantage of strong applicability [
42]. The addition of a multi-agent system (MAS) can overcome many EA’s drawbacks such as slow iteration speeds and the production of a wide range of applications in many fields [
43,
44]. However, to the best of our knowledge, research on the integration of MAS and DE algorithms has rarely been seen in the field of electromagnetics. Since we expect to obtain an optimal antenna element arrangement and corresponding feeding scheme, so as to generate axial radiation with the lowest PSLL, which is a multi-dimensional nonlinear problem with multi-constraints, the concept of MAS is then applied to the DE algorithm to this end. In addition, mutation–selection strategies and time-varying weighting factors are adopted to accelerate the convergence of the algorithm. The obtained multi-agent composite differential evolution (MCDE) algorithm has proved its superiority with an optimization example of 64 array elements ending with an 80% sparse rate. To test the performance of the proposed algorithm in synthesizing the array on a large scale, an example with 900 antenna elements was simulated and compared with the benchmark algorithm.
The MCDE algorithm is described in
Section 2. The details of the chosen antenna element and the simulation example are shown in
Section 3. The fabrication analysis is analyzed in
Section 4. Finally, the conclusion is summarized in
Section 5.
3. Antenna Element Design and Simulation Results
As shown in
Figure 3, the proposed dual-polarized patch antenna is composed of three substrate layers and one air layer, and Rogers 5880 (
= 2.2) is used as a substrate in this antenna. The patch is etched with a U-shaped groove to achieve miniaturization while broadening the beamwidth. The microstrips with a length of L2 placed on two sides of the patch are connected to the SMP connector to excite the patch, thus radiating dual-polarization electromagnetic waves. Detailed dimensions of the proposed antenna geometry are summarized in
Table 1.
The antenna operates with a voltage standing wave ratio (VSWR) less than 1.7 in the operating bandwidth of 15–17 GHz (
Figure 4a). As shown in
Figure 4b, good similarity is observed in the radiation pattern of two planes with a boresight gain of 6.5 dBi. The half-power beam widths (HPBWs) in the φ = 0° plane and φ = 90° plane are 84° and 86°, respectively, which contributes to axial scanning. Due to the half cone angle of a truncated cone array, the axial radiation can be regarded as a wide-angle scan of the conformal array, which can be achieved by an antenna element with wide HPBW.
The implemented sparse conformal antenna is shown in
Figure 5 for better understanding; 27 aforementioned elements are arranged on the side surface of the cone based on the optimized positions. Since each element has a specific disposing-phase to achieve the axial radiation, phase shifters are needed for each element.
The pattern synthesis of arrays in the previous literature generally treats the element as the ideal omnidirectional antenna and ignores the coupling between elements, which will reduce the optimization time. However, simplification in these two aspects will lead to a large difference between the actual array and the simulated one. In order to solve the problems above, the active element pattern is theoretically verified in [
46]. In the case of an infinite antenna array where each element is excited, the active element pattern contains the coupling effects. Therefore, using the active element pattern can effectively improve the accuracy of the array synthesis. In this research, the antenna element’s active radiation pattern obtained from the full-wave simulation in HFSS is imported into MATLAB to include coupling effects.
MATLAB 2016 on a PC operating at 3.6 GHz with 16 GB of RAM is used for the simulation of MCDE and other benchmark algorithms. For MCDE optimization, the P and Q are taken as 7. Therefore, the size of the population is 49. The generation is set as 200, scaling factor F = 0.85, and crossover rate CR = 0.9. To reduce the computational time for self-learning operator, size sL =3, sGen = 5, and sR = 0.25 are taken.
The benchmark algorithms used in this paper are DE [
44], GA [
31], PSO [
32], GWO [
33], and IWO [
34]. All algorithms have the same population size, which is 49. The parameters CR and F are set as 0.9 and 0.85 in DE. GA has a crossover probability of 0.7 and a mutation probability of 0.1. For PSO, the cognitive component, social component, and inertia weight are set to be 2, 1, and 2, respectively. GWO includes a control parameter that linearly decreases from 2 to 0. The minimum and the maximum number of seed is 0 and 5, respectively, with a modulation index of 2 in IWO. The initial standard deviation and the final standard deviation are set as 0.01 and 0.1, respectively.
3.1. Example 1: Pattern Synthesis of 52-Element Array
An array with
= 52 is established to verify the proposed synthesis method. The parameters of the surface are shown in
Table 2. The DE, GA, PSO, GWO, IWO, and the new MCDE algorithms are applied to synthesize the array and the radiation patterns are compared with the full array result. The geometry of the full array and sparse array obtained by the MCDE algorithm are shown in
Figure 6 for comparison. It can be seen that the elements are unevenly distributed after optimization, thus achieving better performance than uniform arrays in some aspects.
The final arrangement of the 52-element array after MCDE optimization is shown in
Table 3, where one column represents one layer of the array. In each blank, the front number represents the height of each antenna element(λ) in the cone and the latter means the angle of the element in the cylindrical coordinate system (°).
The convergence plot of all optimization algorithms is shown in
Figure 7. It can be seen that the MCDE algorithm attains a PSLL of about −20.84 dB within 50 iterations, whereas the final PSLL obtained by the benchmark algorithm is above −16 dB after 100 iterations, showing the capacities of fast convergence, stability, and the small required population.
The proposed array is installed on a truncated cone surface instead of a flat plane, so the main beam is not directed to the array axis (0°, 0°) when the elements are all fed in equal-amplitude and in-phase. Therefore, axial radiation is achieved thorough phase disposing. The normalized radiation patterns (0° < θ < 180°) of the full array and sparse array synthesized by MCDE and other benchmark algorithms are shown in
Figure 7. It can be seen that the main lobe synthesized by any algorithm can accurately scan to (0°, 0°) with high symmetry along θ = 0°.
The performances of the sparse array after different optimizations and the full array are detailed in
Table 4, including PSLL, directivity, gain, and beamwidth. Compared with the PSLL = −9.97 dB of the full array, a certain reduction is achieved by different optimizations. The comparison are shown in
Figure 8. Among them, MCDE achieves −20.84 dB and −21.53 dB in the φ = 0° and 90° planes, respectively, which are better than the PSO result of −16.30 dB. For directivity, it is worth emphasizing that the gain of the array is theoretically reduced to 21.55 dBi due to the number of elements being reduced to 80%. Therefore, only the IWO and the proposed MCDE algorithm achieve gains larger than the theoretical one of 0.02 dBi and 0.32 dBi, which verify the enhancement of the axial gain. The gain of the array is calculated by:
where
D is the directivity obtained by array optimization, and
Ea is the efficiency of the proposed dual-polarized patch antenna at the corresponding frequency, which is 85% at 16 GHz. Since the beam width has a negative correlation with the aperture of the array, the HPBW of the array will remain unchanged if the aperture is maintained after optimization. Different from the IWO algorithm that narrows the beam, the MCDE algorithm can keep the HPBW similar to that of the full array.
In conclusion, the MCDE algorithm shows certain competitiveness in terms of PSLL and gain compared with other optimization algorithms. The 3D radiation pattern of the sparse array synthesized by the MCDE algorithm is shown in
Figure 9 to reveal more details of the array’s radiation.
3.2. Example 2: Pattern Synthesis of 512-Element Array
The large-scale array optimization problem will challenge the stability and the efficiency of the optimization algorithm. To further investigate the optimization capability of the MCDE algorithm for large-scale arrays, a 512-element sparse array based on the truncated cone surface was built, in which the sparsity was set to 56.8%, i.e., 900 antenna elements existed in the array before sparsity. The specific metrics of the array are shown in
Table 5. Similar to Example 1, a dual-polarized patch antenna is used as the array element, and its active radiation pattern obtained from full-wave simulation is imported for synthesis. Again, the objective of the array optimization is to achieve a fixed main beam to (0°, 0°) and the lowest PSLL. Likewise, the present large-scale array is regarded as a phased array for beam steering. The sparse array layout obtained from the MCDE optimization is shown in
Figure 10.
The radiation pattern of the full array and the sparse array obtained by the benchmark algorithms are shown in
Figure 11. It can be seen that the arrays after different optimizations can still achieve axial and symmetrical radiation in large-scale cases.
The array performances obtained by full arrays and sparse arrays synthesized by different algorithms are shown in
Table 6, including PSLL, directivity, gain, and HPBW. Compared with the PSLL of the full array, all the sparse arrays achieved reductions of more than 4 dB, especially for the GWO and MCDE, which are 6.38 dB and 12.23 dB, respectively. As a reference for comparison, the theoretical gain of the array after elements reduction is calculated to be 29.98 dBi, while the full array achieves a directivity of 32.44 dBi. The gains of arrays synthesized by the IWO and MCDE algorithms are 0.04 and 0.16 dB higher than the theoretical one, respectively. Other algorithms can hardly guarantee the HPBW, while the array optimized by the MCDE algorithm maintains the HPBW the same as the full array. It can be seen that in the case of nearly 1000 elements before synthesis with a sparsity of 56.8%, the algorithm still shows stable performance, achieving the goals of both a large number of elements and large sparsity. The three-dimensional radiation pattern of the array is shown in
Figure 12.
3.3. Comparison with Existing Research
In this part, some studies based on sparse conformal arrays are compared with the proposed method, including PSO-SOCP, hybrid IWO/PSO, and the Compressed-Sensing Inspired Deterministic Algorithm. Commonly, these researches achieve good PSLL with a larger sparse rate by optimizing the excitation and phase of elements. Compared to the above-mentioned arrays, the proposed array can achieve radiation on a surface with a greater slope (
Table 7). However, since the method is based on evolutionary algorithms, the performance is relatively lower than compressed-sensing optimization methods.