Enhancing Beamforming Efficiency Utilizing Taguchi Optimization and Neural Network Acceleration

Abstract

This article presents an innovative method for efficiently synthesizing radiation patterns by combining the Taguchi method and neural networks, validating the results on a ten-element antenna array. The Taguchi method aims to minimize product and process variability, while neural networks are used to model the relationship between antenna design parameters and radiation pattern characteristics. This approach utilizes Taguchi parameters as inputs for the neural network, which is then trained on a dataset generated by the Taguchi method. After training, the network is validated using a real ten-element antenna array. Analytical results demonstrate that this method enables efficient synthesis of radiation patterns, with a significant reduction in computation time compared to traditional approaches. Furthermore, validation on the antenna array confirms the accuracy and robustness of the approach, showing a high correlation between the performance predicted by the neural network model and actual measurements on the antenna array. In summary, our article highlights that the combined use of the Taguchi method and neural networks, with validation on a real antenna array, offers a promising approach for efficient synthesis of antenna radiation patterns. This approach combines speed, accuracy, and reliability in antenna system design.

1. Introduction

The synthesis of radiation patterns for antenna arrays stands as a pivotal discipline in both communication systems and radar system design. This sophisticated practice involves crafting and precisely controlling the radiation patterns of interconnected antennas to achieve specific performance goals [1] in signal directionality and spatial coverage [2,3]. By amalgamating antenna theory with advanced signal processing techniques, this practice allows for the precise and adaptable shaping of antenna directional characteristics, offering remarkable flexibility in targeted signal transmission or reception. Sophisticated design methodologies are employed to optimize antenna configurations, yielding high efficiency in diverse applications such as high-performance wireless communication networks and advanced radar systems [4,5]. This introduction aims to explore the fundamental principles, design methodologies, and applications of this intricate yet indispensable discipline in the realm of antenna technologies [5,6]. In radar systems, these arrays revolutionize target detection and tracking by swiftly steering beams electronically, eliminating the need for cumbersome mechanical adjustments [7,8]. They form the backbone of modern communication networks, dynamically adapting to fluctuating environments by employing beamforming techniques to optimize signal strength and coverage. In space communications, their reliability becomes paramount, ensuring seamless connectivity between Earth and satellites to aid in interplanetary exploration and satellite-based services [9]. The military and defense sectors harness their agility for surveillance, electronic warfare, and rapid response capabilities in ever-changing operational scenarios [10]. These arrays are catalysts for the evolution of wireless technologies, refining Wi-Fi, IoT connectivity, and upcoming standards such as 5G by augmenting data rates, minimizing interference, and extending network reach [11]. Additionally, their role in medical imaging, especially ultrasound systems, enables clinicians to achieve detailed high-resolution images, revolutionizing diagnostic capabilities. These diverse and detailed applications underscore the pivotal role of phased antenna arrays across aviation, communications, defense, space exploration, healthcare, and wireless technologies [12,13]. Phased antenna arrays can be paired with neural networks for various applications. Integrating neural networks into phased arrays offers opportunities to optimize and dynamically adapt antenna operations based on environmental conditions and specific requirements. For instance, neural networks can predict and adjust optimal beam parameters in real time to enhance signal quality and range in wireless communication networks. Similarly, in radar systems, neural networks can provide sophisticated analysis of radar return data, aiding in more precise target detection, classification, and tracking by leveraging the learning capabilities of the neural network [14,15]. This fusion of phased antenna arrays with neural networks presents exciting prospects for intelligent adaptation and continual performance optimization across diverse fields such as wireless communications, advanced radar surveillance, and other applications demanding heightened responsiveness and precision [16]. Phased antenna arrays play a pivotal role in the evolution of communication networks such as 5G and the future advancements toward 6 G. These antennas offer essential directionality and adaptability to meet the escalating demands for data throughput and connectivity [17,18] in these next-generation communication networks [19,20]. In 5G, phased arrays are utilized for dynamic beamforming, enabling communications with high speed and low latency and improving coverage in dense environments. Moving towards 6G, these antennas are poised to become even more critical, leveraging higher frequencies and emerging technologies to achieve faster speeds, ultra-low latency, and groundbreaking applications such as extended reality, holographic communications, and more. Phased antenna arrays [21] will be instrumental in maximizing spectral and spatial efficiency to address the technological challenges inherent in these advancements towards next-generation wireless networks [22,23].

The rest of this paper is structured as follows. Section 1 introduces the topic of our study, providing context and outlining its significance. In Section 2, we delve into phased antenna arrays, offering a detailed theoretical formulation and discussing their relevance in the field. Section 3 elaborates on the Taguchi method, providing a thorough explanation of its implementation and showcasing synthesis results for various types of antenna arrays (10, 16, and 24 linear antennas). Section 4 focuses on the synthesis of radiation patterns using neural networks, detailing how the Taguchi method is employed for training and validation in this context. Finally, Section 5 concludes our work by summarizing the findings of our study and outlining potential future research directions.

Table 1. Comparison of antenna array radiation pattern synthesis methods [24,25,26,27].

Method	Efficiency	Complexity	Precision
Taylor	Fast	Moderate	High
PSO	Good	High	Variable
Analytical	Fast	Low	High
Genetic Algorithm	Moderate	Moderate	High
FEM	Fast	Moderate	High
FDTD	Fast	Moderate	High

provides a comprehensive comparative analysis of the various approaches used in synthesizing antenna array radiation patterns. Examining the advantages, disadvantages, and performance of each method allows for a detailed evaluation of the available options. Thus, this comparative study offers valuable insights that can guide the selection of the most appropriate method based on the specific needs of each application. Furthermore, identifying the strengths and limitations of each approach paves the way for future research with the aim of improving existing techniques and exploring new avenues for optimal synthesis of antenna array radiation patterns while considering parameters such as efficiency, complexity, and precision.

2. Phased Antenna Arrays

Phased antenna arrays are intricate systems comprising multiple antenna elements, each capable of independent phase and amplitude adjustments in the signals they transmit or receive [28,29,30]. These adjustments enable the array to sculpt the radiation pattern of emitted waves or enhance reception sensitivity. By precisely coordinating these adjustments, the array achieves beamforming, directing a focused beam in specific directions without the need for physical repositioning. This capability proves invaluable in radar systems for swift target tracking and wide-area surveillance [31,32]. In communication networks, phased arrays facilitate adaptive beam steering, optimizing signal reception in dynamic and challenging environments. Despite the complexity inherent in managing numerous elements and ensuring their coordinated actions, ongoing advancements in technology continue to refine these arrays, making them more efficient, adaptable, and pivotal across diverse domains, including telecommunications, space exploration, and defense systems [33,34].

In the world of digital systems, the transition from conventional analog beamforming to digital processes involves intricate signal manipulation. This manipulation occurs at the baseband level, where precise adjustments to signal amplitude and phase are made digitally to align with the desired excitation pattern [35,36]. This departure from traditional analog beamforming involves specialized signal processing components.

Digital systems have the potential to generate a multitude of beams, designated as N, by merging signals from multiple sources [37]. The actual number of formed beams relies on the RF characteristics of M transmission channels and the computational capabilities of the signal processing modules [38].

Critical components such as digital-to-analog converters (DACs), analog-to-digital converters (ADCs), and signal processing modules play pivotal roles in this process. These components require exceptional speed, surpassing that of their analog counterparts, in order to efficiently process digital data and precisely shape analog signals [39]. This rapid processing capability serves as the bedrock of digital beamforming systems, enabling efficient signal manipulation within the digital realm, forming an essential aspect of their operational efficiency [40].

In fully digital beamforming systems, there is a direct relationship between the number of amplifiers, such as Power Amplifiers (PAs) and Low-Noise Amplifiers (LNAs), and the transmission channels. This connection categorizes these systems as active antenna array architectures [41]. Notably, these arrays possess reconfigurable properties, allowing for dynamic modifications during beam formation. This adaptability in altering beams designates these active antenna arrays as reconfigurable antenna arrays, providing flexibility and adaptiveness across diverse communication scenarios [42,43].

In this study, the optimization methodology we introduce was implemented using Matlab scripts, providing a flexible and powerful platform for algorithm development and execution. Moreover, to enhance the user experience and facilitate a deeper understanding of the optimization process, we leveraged Matlab’s Integrated Development Environment (IDE) to provide real-time visual feedback on the convergence of the optimization [44,45]. This interactive feature allows for monitoring the optimization progress while enabling researchers to make informed decisions regarding parameter adjustments or algorithm modifications during runtime. Such detailed insights into the optimization convergence can serve to enrich the methodology and improve its effectiveness in practical applications [46,47,48].

Consider an antenna array consisting of N identical radiating elements. Each element of order n has its own radiation characteristic function $f_n\left({\theta }_n^{\prime }\right)$ and is fed by the current ${\overline{I}}_n=A_{n}exp\left(j{\phi }_n\right)$.

The electric field produced by the $n^{\mathrm{th}}$ element of the antenna array at a point P, located in its far radiation zone, is then written as follows:

$${\overline{E}}_n={\overline{I}}_n{e}^{jk{\mathbf{r}}_{\mathbf{n}}^{\prime }·{\mathbf{u}}_{\mathbf{r}}}{f}_n\left({\theta }_n^{\prime }\right){\mathbf{u}}_{{\mathbf{E}}_{\mathbf{n}}}$$

(1)

with $k=\frac{2\pi }{\lambda }$, ${\mathbf{u}}_{{\mathbf{E}}_{\mathbf{n}}}$ being a unit vector in the direction of the ${\overline{E}}_n$ field produced by the $n^{\mathrm{th}}$ element and ${\theta }_n^{\prime }$ the observation angle in the local coordinate system of the $n^{\mathrm{th}}$ element, while ${\mathbf{r}}_{\mathbf{n}}^{\prime }$ is the position vector of the far point P in the local coordinate system of the $n^{\mathrm{th}}$ element.

The total electric field radiated by the antenna array is the sum of all elementary fields ${\overline{E}}_0$, ${\overline{E}}_1$, …, ${\overline{E}}_{N-1}$. Consequently, when the directions of observation are equal, the distances ${\mathbf{r}}_{\mathbf{n}}^{\prime }$ are considered equal to the distance r between the origin of the global coordinate system and the observation point P. In terms of phase, the difference in path between ${\mathbf{r}}_{\mathbf{n}}^{\prime }$ and r is rewarded by ${\Delta }_{rn}\approx -r_{n}cos\left({\theta }_n\right)$, as shown in Figure 1.

The resulting field is expressed as follows:

$$\overline{E}(r,\theta )=\sum _{n=1}^N\overline{E}(r_n^{\prime },{\theta }_n^{\prime }).$$

(2)

Replacing $\overline{E}(r_n^{\prime },{\theta }_n^{\prime })$ with its expression from (1), the expression of $\overline{E}(r,\theta )$ becomes

$$\overline{E}(r,\theta )=\sum _{n=1}^N{\overline{I}}_n{e}^{-jk{\mathbf{r}}_{\mathbf{n}}^{\prime }·{\mathbf{u}}_{\mathbf{r}}}{f}_n\left({\theta }_n^{\prime }\right){\mathbf{u}}_{{\mathbf{E}}_{\mathbf{n}}},$$

(3)

or ${\mathbf{u}}_{{\mathbf{E}}_{\mathbf{n}}}={\mathbf{u}}_{\mathbf{E}}$ and $f_n\left({\theta }_n^{\prime }\right)=f\left(\theta \right)$ for $\forall n\in 0,..,N$.

Moreover, the relation between the position vectors is $r_n^{\prime }=r-r_{n}cos\left({\theta }_n\right)$; therefore, ${\mathbf{r}}_{\mathbf{n}}^{\prime }·{\mathbf{u}}_{\mathbf{r}}=r-r_{n}cos\left({\theta }_n\right)$.

The total radiated field is then

$$\overline{E}(r,\theta )=\sum _{n=1}^N{\overline{I}}_n{e}^{jk(r-r_{n}cos{\theta }_n)}f\left(\theta \right){\mathbf{u}}_{\mathbf{E}}.$$

(4)

We consider ${\overline{E}}_0(r,\theta )=e^{jkr}f\left(\theta \right){\mathbf{u}}_{\mathbf{E}}$ the field radiated by the element at the origin of the global coordinate system. The equation of the total field then becomes

$$\overline{E}(r,\theta )=e^{jkr}f\left(\theta \right)\sum _{n=1}^N{\overline{I}}_n{e}^{-jk{r}_{n}cos{\theta }_n}{\mathbf{u}}_{\mathbf{E}}.$$

(5)

We assume ${\psi }_n$ to be the sum of the excitation phase of the $n^{th}$ element and the decompensation phase of the path difference for ${\Delta }_{rn}$ to be ${\psi }_n={\phi }_n-k{r}_{n}cos{\theta }_n$.

Thus, the total field is written as

$$\overline{E}(r,\theta )=e^{jkr}f\left(\theta \right)\sum _{n=1}^N{A}_n{e}^{j{\psi }_n}{\mathbf{u}}_{\mathbf{E}},$$

(6)

$$AF=\sum _{n=1}^N{A}_n{e}^{j{\psi }_n}.$$

(7)

Equation (6) can be generalized as follows:

$$E=X.A$$

(8)

with $E={[E\left({\theta }_1\right)\cdots E\left({\theta }_K\right)]}^T$ being the vector of the electric field for K directions of radiation, $A={[I_1\cdots I_N]}^T$ the vector of the elementary antenna’s excitations, and X the geometric matrix of the antenna array [48].

Structure of Antenna Arrays

A linear antenna array (Figure 2) has N radiating elements arranged on a line segment with an inter-element distance d, which is not necessarily constant.

These elements are excited by a current of amplitude $a_i$ and phase ${\phi }_i(i=1,2,\dots ,N)$.

The array factor for a very distant point found in the radiation area is written as follows:

$$AF\left(\theta \right)=\sum _{i=1}^{i=N}{a}_i{e}^{j(k{x}_{i}sin\theta +{\phi }_i)}$$

(9)

where ${\phi }_i\in [{\phi }_1,{\phi }_2,\dots ,{\phi }_N]$ represents the excitation phases of the $i^{th}$ element (the first antenna is chosen as the phase reference, ${\phi }_1=0$), $x_i$ is the position of the $i^{th}$ element, $k=2\pi /\lambda$ is the wave number, $\theta$ is the angle of incidence of the desired signal, and $\lambda$ is the wavelength of the signal.

3. Taguchi Method

The Taguchi method, pioneered by Genichi Taguchi, is a statistical approach used to enhance quality and optimize processes in engineering and manufacturing. When applied to antenna arrays, this method involves systematically adjusting various design parameters to improve the array’s performance [48,49].

In the realm of antenna arrays, employing the Taguchi method entails structured experimentation to refine factors influencing performance. These factors encompass elements such as spacing, geometry, materials, feeding techniques, and other design attributes. Engineers conduct experiments based on this approach with the aim of identifying the ideal combination of parameters that enhance metrics such as gain, directivity, radiation pattern, and sidelobe suppression [50,51] (please refer to Appendix B for further details).

Utilizing predefined experimental designs known as orthogonal arrays, the Taguchi method efficiently explores numerous parameter combinations through a limited set of experiments. Statistical analysis of these outcomes aids in pinpointing the most influential parameters and determining the optimal settings for superior antenna performance.

By leveraging the Taguchi method in antenna array design, the goal is to elevate overall performance, reliability, and efficiency by systematically optimizing parameters using a statistically-driven approach [52]. In their quest to improve a phased-array satellite communication antenna, engineers utilized the Taguchi method to optimize its design parameters. They systematically experimented with element spacing, feeding techniques, and substrate materials while varying them across different levels. Analyzing the resulting data on gain, beamwidth, and sidelobe levels using statistical methods, they pinpointed the most impactful factors and identified the ideal combination of element spacing, feeding technique, and substrate material. This optimized configuration maximized gain, narrowed beamwidth, and minimized sidelobes, significantly enhancing the antenna’s performance in satellite communication [53,54].

3.1. Amplitude Synthesis of Antenna Array Radiation Pattern under Constraints

In this type of synthesis, constraints are imposed on the levels of sidelobes and the locations of nulls in the radiation pattern, which the beam should exhibit towards interfering signals. In this initial synthesis type, the Taguchi optimization procedure is applied to three linear antenna arrays, each with a different number of sources (10, 16, and 24), in order to minimize sidelobes.

3.1.1. Ten-Element Antenna Array

The array factor is described by the following equation:

$$AF\left(\theta \right)=2\sum _{n=1}^{5}a\left(n\right)e^{j\varphi \left(n\right)}cos\left[\beta d\left(n\right)sin\left(\theta \right)\right].$$

(10)

To minimize the sidelobes, the fitness function is chosen based on the optimization objective:

$$fitness=min\left(max\left\{20log\left|AF\left(\theta \right)\right|\right\}\right)$$

(11)

subject to constraints $\theta \in \left[0^{\circ },{76}^{\circ }\right],\left[{104}^{\circ },{180}^{\circ }\right]$.

To simplify the task and enhance understanding of this application, we elaborate on the Taguchi optimization procedure in the following subsections.

First Step: Initialization Problem

The first step involves selecting an appropriate experimental design and a well-structured fitness function. There are five parameters requiring optimization. Therefore, the chosen orthogonal experimental design should have five columns ($k=5$) to represent these parameters. To capture nonlinear effects, three levels ($s=3$) for each parameter are deemed sufficient. Typically, an orthogonal experimental design with a strength of 2 ($t=2$) is effective for most problems. In summary, an orthogonal experimental design with five columns, three levels, and a strength of 2 is required.

Second Step: Designating Input Parameters

After accessing the online database OA, an orthogonal array (OA) (27, 5, 3, 2) is available, as indicated in

Table 2. Taguchi experimental design (N = 27, K = 5, S = 3, t = 2).

Experiment	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{a}}_4$	${\mathit{a}}_5$
1	1	1	1	1	1
2	2	1	2	2	2
3	3	1	3	3	3
4	1	2	1	2	2
5	2	2	2	3	3
6	3	2	3	1	1
7	1	3	1	3	3
8	2	3	2	1	1
9	3	3	3	2	2
10	1	1	2	1	2
11	2	1	3	2	3
12	3	1	1	3	1
13	1	2	2	2	3
14	2	2	3	3	1
15	3	2	1	1	2
16	1	3	2	3	1
17	2	3	3	1	2
18	3	3	1	2	3
19	1	1	3	1	3
20	2	1	1	2	1
21	3	1	2	3	2
22	1	2	3	2	1
23	2	2	1	3	2
24	3	2	2	1	3
25	1	3	3	3	2
26	2	3	1	1	3
27	3	3	2	2	1

. The fitness function is chosen based on the optimization objective. In this optimization example, the function is selected to achieve a low sidelobe level. The input parameters need to be selected before conducting the experiments. When the orthogonal experimental design is chosen, numerical values corresponding to the three levels of each input parameter must be determined in the first iteration. The value for level 2 is selected at the midpoint of the optimization range. The values for levels 1 and 3 are calculated using the following equations:

$$D{N}_1=\frac{A_{max}+A_{min}}{S+1},$$

(12)

$$A{\left(n\right)}_2^1=\frac{A_{max}+A_{min}}{2},$$

(13)

$$A{\left(n\right)}_1^1=A{\left(n\right)}_1^2-D{N}_1,$$

(14)

$$A{\left(n\right)}_3^1=A{\left(n\right)}_1^2+D{N}_1.$$

(15)

Numerical Application:

$$D{N}_1=\frac{1+0}{3+1}=0.25,$$

(16)

$$A{\left(n\right)}_2^1=\frac{1+0}{2}=0.5,$$

(17)

$$A{\left(n\right)}_1^1=0.5-0.25=0.25,$$

(18)

$$A{\left(n\right)}_3^1=0.5+0.25=0.75.$$

(19)

Using these equations, Table 2 can be converted into numerical values, as shown in

Table 3. Numerical values of the levels in the first iteration.

Experiment	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{a}}_4$	${\mathit{a}}_5$
1	0.25	0.25	0.25	0.25	0.25
2	0.5	0.25	0.5	0.5	0.5
3	0.75	0.25	0.75	0.75	0.75
4	0.25	0.5	0.25	0.5	0.5
5	0.5	0.5	0.5	0.75	0.75
6	0.75	0.5	0.75	0.25	0.25
7	0.25	0.75	0.25	0.75	0.75
8	0.5	0.75	0.5	0.25	0.25
9	0.75	0.75	0.75	0.5	0.5
10	0.25	0.25	0.5	0.25	0.5
11	0.5	0.25	0.75	0.5	0.75
12	0.75	0.25	0.25	0.75	0.25
13	0.25	0.5	0.5	0.5	0.75
14	0.5	0.5	0.75	0.75	0.25
15	0.75	0.5	0.25	0.25	0.5
16	0.25	0.75	0.5	0.75	0.25
17	0.5	0.75	0.75	0.25	0.5
18	0.75	0.75	0.25	0.5	0.75
19	0.25	0.25	0.75	0.25	0.75
20	0.5	0.25	0.25	0.5	0.25
21	0.75	0.25	0.5	0.75	0.5
22	0.25	0.5	0.75	0.5	0.25
23	0.5	0.5	0.25	0.75	0.5
24	0.75	0.5	0.5	0.25	0.75
25	0.25	0.75	0.75	0.75	0.5
26	0.5	0.75	0.25	0.25	0.75
27	0.75	0.75	0.5	0.5	0.25

: $a{\left(n\right)}_1^2=0.5;a{\left(n\right)}_1^3=0.75;a{\left(n\right)}_1^1=0.25$.

Third Step: Conducting Experiments and Building a Response Table

After determining the input parameters, the fitness function for each experiment can be calculated. For instance, the fitness value for experiment 1 (i.e., the first row of Table 3) is computed using Equation (14), resulting in 12.97. Then, the fitness value in the Taguchi method is converted to a signal-to-noise ratio (S/N), denoted as ($\eta$), using Formula (14). The corresponding fitness values and S/N ratios are listed in

Table 4. Experiments and fitness values.

Experiment	${\mathit{a}}_1$	${\mathit{a}}_{21}$	${\mathit{a}}_{31}$	${\mathit{a}}_4$	${\mathit{a}}_5$	Fitness R	(S/N)
1	0.25	0.25	0.25	0.25	0.25	12.97	−22.26
2	0.5	0.25	0.5	0.5	0.5	11.19	−20.98
3	0.75	0.25	0.75	0.75	0.75	10.56	−20.47
4	0.25	0.5	0.25	0.5	0.5	9.91	−19.92
5	0.5	0.5	0.5	0.75	0.75	9.70	−19.73
6	0.75	0.5	0.75	0.25	0.25	13.86	−22.83
7	0.25	0.75	0.25	0.75	0.75	8.67	−18.76
8	0.5	0.75	0.5	0.25	0.25	15.53	−23.82
9	0.75	0.75	0.75	0.5	0.5	16.81	−24.51
10	0.25	0.25	0.5	0.25	0.5	9.32	−19.39
11	0.5	0.25	0.75	0.5	0.75	9.31	−19.38
12	0.75	0.25	0.25	0.75	0.25	7.61	−17.63
13	0.25	0.5	0.5	0.5	0.75	8.28	−18.36
14	0.5	0.5	0.75	0.75	0.25	9.88	−19.90
15	0.75	0.5	0.25	0.25	0.5	10.99	−20.82
16	0.25	0.75	0.5	0.75	0.25	9.03	−19.12
17	0.5	0.75	0.75	0.25	0.5	13.93	−22.88
18	0.75	0.75	0.25	0.5	0.75	11.27	−21.04
19	0.25	0.25	0.75	0.25	0.75	6.84	−16.70
20	0.5	0.25	0.25	0.5	0.25	10.13	−20.11
21	0.75	0.25	0.5	0.75	0.5	9.70	−19.73
22	0.25	0.5	0.75	0.5	0.25	8.26	−18.34
23	0.5	0.5	0.25	0.75	0.5	10.97	−20.81
24	0.75	0.5	0.5	0.25	0.75	10.95	−20.78
25	0.25	0.75	0.75	0.75	0.5	8.28	−18.36
26	0.5	0.75	0.25	0.25	0.75	7.90	−17.96
27	0.75	0.75	0.5	0.5	0.25	21.51	−26.65

. These results are then used to construct a response table (Table 4) for the first iteration by averaging the $S/N$ ratios for each parameter. The corresponding signal-to-noise ratio values are listed in

Table 5. Response table (in decibels) after the first iteration, i.e., element levels (dB).

Elements	1	2	3	4	5
Level 1	−19.02	−19.63	−19.92	−20.83	−21.18
Level 2	−20.62	−20.17	−20.95	−21.03	−20.82
Level 3	−21.61	−21.46	−20.38	−19.39	−19.24

. These results are then used to construct a response table (Table 5) by averaging the signal-to-noise ratios ($S/N$) for each parameter using the following equation:

$$\overline{\eta }(m,n)=\frac{1}{N}{\int }_{i,OA(i,n)=m}\eta \left(i\right)$$

(20)

where n is the number of parameters, m is the number of levels (1, 2, 3), N is the number of level combinations, and i stands for the i-th iteration.

For example, the average of the ratios (S/N) for $A\left(4\right){\mid }_1^2$ and $A\left(5\right){\mid }_1^3$ is as follows:

$$\overline{\eta }(2,4)=\frac{1}{9}{\int }_{i,OA(1,4)=2}\eta \left(1\right)=-20.03\mathrm{dB}$$

(21)

$$\overline{\eta }(3,5)=\frac{1}{9}{\int }_{i,OA(1,5)=3}\eta \left(1\right)=-19.24\mathrm{dB}.$$

(22)

To identify the optimal level value for each parameter, it is necessary to find the highest signal-to-noise (S/N) ratio within each column of Table 4. This is highlighted in Table 5. When these optimal levels have been identified, a confirmation test is conducted using the corresponding numerical values of these optimal levels in the response table, (

Table 6. Optimized values of $a_i$ and maximum SLL for networks.

Optimized Values of ${\mathit{a}}_{\mathit{i}}$	Maximum SLL
1.0000	−13.1526
0.8413	−22.8982
0.9322
0.7675
0.6049
0.5715
0.4746
0.4877

).

If the results from the current iteration do not meet the termination criteria, which are examined using Equation (23), and after determining the optimal level values from the current iteration that are used as central values for the next iteration (24) (in our case, the central values for the next iteration are $a{\left(1\right)}_1^1$; $a{\left(2\right)}_1^1$; $a{\left(3\right)}_1^1$; $a{\left(4\right)}_1^1$; $a{\left(5\right)}_1^1$), the optimization process is repeated in the subsequent iteration while reducing the optimization interval using Equation (25).

$$\frac{D{N}_{i+1}}{D{N}_1}<0.01$$

(23)

$$a_n{\mid }_{i+1}^2=a_n{\mid }_i^{opt}$$

(24)

$$D{N}_{i+1}=RR\left(i\right)\times D{N}_1=r{r}^i\times D{N}_1$$

(25)

In this optimization example, the reduced function ($rr$) is set to $0.9$.

This first type of synthesis (amplitude synthesis under constraint) aims to minimize the secondary lobes as much as possible using our optimization technique. In Figure 3a, the obtained results demonstrate successful minimization of the secondary lobe level (SLL), also knows as the sidelobe level, down to $-25.2722$ dB, where the gain is approximately $12.3$ dB higher compared to the secondary lobe level of the uniform linear antenna array (SLL = $-12.9651$ dB). The optimization objective is achieved after 80 iterations, as indicated in Figure 3b.

By comparing the results obtained with those of PSO (Particle Swarm Optimization) (see Figure 4), we observe the following:

• A gain of 0.6 dB in terms of secondary lobe minimization.
• A convergence speed of 80 iterations for the Taguchi method.
• The actual time required for our digital optimization tool is approximately 10 s.

With this method, we aim to minimize the error criterion; this amounts to minimizing the difference between the desired pattern (template) and the pattern obtained through synthesis.

3.1.2. Sixteen-Element Antenna Array

In this example, the array factor is described by Equation (26).

$$AF\left(\theta \right)=2\sum _{n=1}^{8}a\left(n\right)cos\left[\beta d\left(n\right)cos\theta \right]$$

(26)

The fitness function is chosen based on the following optimization objective:

$$fitness=min\left(max\right\{20log\left|AF\right(\theta \left)\right|\left\}\right)$$

(27)

under constraint $\theta \in \left[0^{\circ },{80}^{\circ }\right],\left[{100}^{\circ },{180}^{\circ }\right]$.

To use the Taguchi method, the following steps [3.3]–[3.1] must be followed:

• Step 1: Determine the number of parameters ($k=8$).
• Step 2: Determine the number of levels ($s=3$).
• Step 3: Determine the strength ($t=2$).
• Step 4: Determine the OA experimental design ($27,8,3,2$).
• Step 5: Determine the reduced function ($rr=0.9$).
• Step 6: Determine the convergence value $=0.01$.

Figure 5a shows that the level of sidelobes (SLL) is minimized down to SLL = −31.3151 dB. The gain is approximately 18.17 dB compared to the level of side lobes of the uniform array (SLL = −13.148 dB). As shown in Figure 5b, the optimization objective is achieved after 66 iterations. The results obtained compared to those of PSO (see Figure 5) show a gain of 0.8 dB in terms of minimizing the side lobes, a convergence speed of 66 iterations for the Taguchi method, and a time requirement of 9 s for our digital optimization tool.

3.1.3. Antenna Array with 24 Elements

In this example, the network factor is described by the equation

$$AF\left(\theta \right)=2\sum _{n=1}^{12}a\left(n\right)cos\left[\beta d\left(n\right)cos\theta \right].$$

(28)

The fitness function is chosen based on the following optimization objective:

$$fitness=min\left(max\right\{20log\left|AF\right(\theta \left)\right|\left\}\right)$$

(29)

Under constraint $\theta \in \left[0^{\circ },{82}^{\circ }\right],\left[{102}^{\circ },{180}^{\circ }\right].$

To use the Taguchi method, we need to follow these steps:

• Step 1: Determine the number of parameters ($k=12$).
• Step 2: Determine the number of levels ($s=3$)
• Step 3: Determine the strength ($t=2$).
• Step 4: Determine the OA experimental design ($27,12,3,2$).
• Step 5: Determine the reduced function ($rr=0.8$).
• Step 6: Determine the convergence value $=0.001$.

By following the same procedure as for examples 1 and 2, the results obtained in this example can be summarized as follows:

• The optimized maximum SLL found by the Taguchi method is −39.2263 dB (Figure 6a).
• The gain is 25.2718 dB, as the SLL of the uniform array is −13.9545 dB.
• The number of iterations is 73 (Figure 6b).
• The excitation weights of the antenna array optimized with the Taguchi method are listed in

  Table 7. Taguchi-optimized weights for the 24 elements.

  Elements	Weights
  1	1.0000	7	0.5292
  2	0.9717	8	0.4203
  3	0.9171	9	0.3182
  4	0.8399	10	0.2275
  5	0.7454	11	0.1512
  6	0.6397	12	0.1262

  .
• The comparative study between the results obtained by the Taguchi method and those obtained by PSO indicates a considerable gain of approximately 3.7 dB.

In summary, the results of these three examples lead us to conclude that there is proportionality between the number of elements on one hand and the convergence speed on the other, such that the convergence speed will be faster with a higher the number of elements.

3.2. Phase Synthesis of the Antenna Array Radiation Pattern

Phase synthesis allows for the realization of directed lobes with a moderately controllable level of sidelobes. Using this technique, it is possible to control the received level in the direction of useful and interfering radiation. The linear antenna array has N equally spaced elements along the z-axis. The spacing between elements is half the wavelength ($\lambda /2$), and the excitations of the array elements are symmetric and uniform with respect to the x-axis. Hence, the excitation phases of the antenna array are optimized within the interval $[-\pi ,\pi ]$.

The array factor of this linear array is defined by

$$AF\left(\theta \right)=2\sum _{n=1}^{N}a\left(n\right)e^{j\varphi \left(n\right)}cos\left[\beta d\left(n\right)sin\theta \right],$$

(30)

where:

• $d\left(n\right)$: distance between sources $=\frac{\lambda }{2}$
• $a\left(n\right)$: amplitude $=1$
• $\varphi \left(n\right)$: phase $\in [-\pi ,\pi ]$
• $\beta =\frac{2\pi }{2}$

Ten-Element Antenna Array

The Taguchi method used in this type of synthesis aims to minimize the sidelobes as much as possible. The array factor is described by the (31):

$$AF\left(\theta \right)=2\sum _{n=1}^{5}cos\left[\beta d\left(n\right)sin\left(\theta \right)+\varphi \left(n\right)\right]$$

(31)

The array factor in decibels is

$$A{F}_{dB}\left(\theta \right)=20.log\left|\frac{AF\left(\theta \right)}{AF\left({\theta }_0\right)}\right|,$$

(32)

with ${\theta }_0$ being the main beam direction = $0^{\circ }$.

The objective (fitness) function is chosen based on the optimization objective:

$$fitness=k_1\times f_{SL}\left(\theta \right)+k_2\times f_{NS}\left(\theta \right).$$

(33)

$$\begin{array}{cccc}\hfill f_{SL}& :\mathrm{Region}\mathrm{of}\mathrm{SLL}\mathrm{minimization};\hfill & \hfill f_{SL}\left(\theta \right)& =max\left\{A{F}_{dB}\left(\theta \right)\right\}\hfill \\ \hfill {\theta }_{NS}& :\mathrm{Region}\mathrm{of}\mathrm{interference};\hfill & \hfill f_{NS}\left(\theta \right)& =\sum _{k}A{F}_{dB}\left({\theta }_{null}^k\right)\hfill \end{array}$$

To use the Taguchi method, the following steps are required:

• Step 1: Determine the number of parameters ($k=10$).
• Step 2: Determine the number of levels ($s=3$).
• Step 3: Determine the strength ($t=2$).
• Step 4: Determine the OA experimental design ($27,10,3,2$).
• Step 5: Determine the reduced function ($rr=0.9$).
• Step 6: Determine the convergence value $=0.001$.

We succeeded in minimizing the secondary lobes by approximately 3 dB compared to those of the uniform array. To achieve beamforming in a desired direction, phase synthesis aims to determine the relative phases applied to each individual antenna element in the array. Let us start with the case of a uniform amplitude array where the sources are fed with the same amplitude; then, the array factor is as follows.

After optimization with the Taguchi algorithm, we obtain directive radiation patterns with moderately controllable levels of secondary lobes sweeping across the entire useful area [−70°, 70°] (Figure 7).

Table A1. Optimization of linear antenna array radiation patterns using taguchi method (ten elements) with $rr=0.9$ and $SLL=-20$ dB.

Angles (Degrees)
−70	−60	−50	−40	−30	−20	−10	0
84.1463	77.4561	68.4900	58.3565	45.5440	30.3652	15.2609	0
−106.7147	−126.6421	−153.5394	173.1063	134.6896	92.9003	46.6549	0
62.5153	29.3031	−15.5709	−70.2386	−134.2522	154.4626	78.0024	0
−128.3519	−174.4535	122.4385	45.5405	−44.1573	−144.8638	109.4055	0
40.8783	−17.9783	−99.6457	160.2969	45.0250	−83.2515	140.8011	0
− 40.8783	17.9783	99.6457	−160.2969	−45.0250	83.2515	−140.8011	0
−128.3519	174.4535	−122.4385	−45.5405	44.1573	144.8638	−109.4055	0
−62.5153	29.3031	15.5709	70.2386	134.2522	−154.4626	−78.0024	0
106.7147	126.6421	153.5394	−173.1063	−134.6896	−92.9003	−46.6549	0
−84.1463	−77.4561	−68.4900	−58.3565	−45.5440	−30.3652	−15.2609	0
Angles (Degrees)
10	20	30	40	50	60	70
−15.1075	−31.1920	−45.4145	−58.3105	−69.3280	−78.4109	−84.1271
−46.2167	−91.7536	−135.2755	−173.9426	153.8277	126.7011	106.7805
−77.2720	−154.1821	135.8536	70.4664	16.0056	−30.0311	−63.2958
−108.3717	144.2690	45.0192	−45.1731	−121.7617	−180.7680	127.5157
−140.4088	83.7520	−43.9049	−160.8637	100.5142	18.2899	−41.5726
140.4088	−83.7520	43.9049	160.8637	−100.5142	−18.2899	41.5726
108.3717	−144.2690	−45.0192	45.1731	121.7617	180.7680	−127.5157
77.2720	154.1821	−135.8536	−70.4664	−16.0056	30.0311	63.2958
46.2167	91.7536	135.2755	173.9426	−153.8277	−126.7011	−106.7805
15.1075	31.1920	45.4145	58.3105	69.3280	78.4109	84.1271

in Appendix A.1 contains all the synthesized weights at different pointing angles. The figures below show a reduction of the secondary lobes to −20 dB (Figure 7b), −29 dB (Figure 7c), and −38 dB (Figure 7d) (please refer to Appendix A.2 for further details).

If is is desirable to switch between various beam pointing directions, either active feeding systems (variable amplifiers for amplitudes and phase shifters) or passive circuits can be used. In the latter case, for each desired lobe direction, a different distribution circuit would theoretically be required. In practice, circuits are used that apply the desired phases to the antennas depending on the chosen input. These are known as passive beamformers.

Here, we present the results of beam synthesis using the Taguchi method, crossing at −3 dB and minimizing a lobe to −20 dB, −29 dB, and −38 dB, corresponding to the maximum of adjacent lobes. The goal is to cover a given angle by scanning a beam of high gain instead of a wide beam with low gain. For optimal coverage, it is necessary for the beams to intersect at −3 dB. As a possible example, we mention application to a communication system covering −90° to +90° in steps of 20°. The simulations developed here were based on secondary lobe level synthesis, phase synthesis, and template synthesis achieved by adjusting the geometric and electrical parameters of the array. The obtained results indicate a remarkable reduction in the level of secondary lobes and almost perfect adherence to the radiation pattern against the imposed functions.

It is worth mentioning that the Taguchi method is effective in addressing the problem of antenna array synthesis as well, and yields excellent results, paving the way for targeted synthesis of shaped antenna arrays.

4. Neural Networks for Synthesis and Optimization of Antenna Arrays

Neural networks offer a promising approach for the synthesis and optimization of antenna arrays. Antenna arrays are crucial in wireless communication systems for controlling radiation characteristics such as directionality, beamforming, and sidelobe suppression [55,56,57,58]. Traditional methods involve complex mathematical models and iterative optimization algorithms, which are often slow and computationally demanding. Neural networks provide an alternative, potentially speeding up the design process and improving antenna array performance. They can be used for pattern synthesis by learning from beam specifications and sidelobe levels, or for optimization by adjusting array parameters to maximize performance metrics such as gain or sidelobe suppression. By using datasets of existing antennas, neural networks can also learn patterns to guide design. As surrogate models, they can accelerate prototypes by avoiding costly electromagnetic simulations [59]. Additionally, neural networks enable adaptive antenna arrays, dynamically adjusting their configuration based on environmental conditions or system requirements. Overall, while neural networks offer prospects for faster designs, improved performance, and increased adaptability in wireless communication systems, they require careful design and training to ensure robustness and generalization [60]. Neural networks have been increasingly utilized for the synthesis and optimization of antenna arrays due to their ability to learn complex relationships between antenna parameters and radiation patterns. The process typically involves the following steps [61]:

Neural networks offer several advantages for antenna array synthesis and optimization, including their ability to handle nonlinear relationships, adapt to complex antenna configurations, and provide fast and efficient solutions.

Combining neural networks with the Taguchi method offers a robust approach for optimizing and designing antenna arrays and various engineering systems. The Taguchi method, developed by Genichi Taguchi, provides a statistical framework for design optimization, aiming to enhance product or process quality while minimizing sensitivity to external factors. Initially, the Taguchi method guides the design of experiments to collect data on system performance. These data are then utilized to train a neural network model, which learns the intricate relationships between input parameters and performance metrics, such as signal strength or interference levels in the case of antenna arrays. Subsequently, employing Taguchi’s optimization principles, such as the signal-to-noise ratio (SNR) and optimization criteria (smaller-the-better, larger-the-better, or nominal-the-best), the trained neural network predicts the system’s performance for different parameter combinations, aiding in optimization efforts. Moreover, the Taguchi method’s focus on robust design ensures sensitivity to variations is minimized, with the neural network facilitating sensitivity analysis to identify critical factors affecting performance. This integrated approach enables iterative improvement, fostering more efficient and effective design optimization processes in antenna arrays and beyond [62,63,64].

Taguchi–Neural Network Architectures

When integrating neural networks with the Taguchi method for antenna array optimization, a comprehensive understanding of the underlying mathematical framework is crucial [65]. The Taguchi method typically employs a loss function, often derived from the signal-to-noise ratio (SNR), aimed at optimizing antenna array performance. One common formulation is the smaller-the-better loss function, shown below.

$$L=-10{log}_{10}\left(\frac{1}{n}\sum _{i=1}^n{\left(\frac{1}{{\varphi }_i}\right)}^2\right)$$

(34)

Here, ${\varphi }_i$ represents the observed performance from individual experiments. This loss function encapsulates the overall quality of the antenna array, with lower values indicating better performance.

Neural networks are then utilized to predict the output performance $\hat{\varphi }$ based on th einput parameters X and network parameters $\theta$. Mathematically, this relationship is expressed as $\hat{\varphi }=f(X;\theta )$, where the function f encapsulates the mapping learned by the neural network.

The optimization objective is to minimize the loss function $L\left(X\right)$ subject to any parameter constraints. During the training phase of the neural network, optimization algorithms such as gradient descent are employed to iteratively update the network parameters $\theta$. Specifically, the parameters are updated according to

$$\theta :=\theta -\alpha \nabla J\left(\theta \right),$$

(35)

where $\alpha$ represents the learning rate and $\nabla J\left(\theta \right)$ denotes the gradient of the loss function with respect to the network parameters.

These equations form the backbone of the mathematical framework when combining neural networks with the Taguchi method for antenna array optimization. By leveraging neural networks, engineers can effectively design and optimize antenna arrays, leading to enhanced performance and efficiency in various applications [66,67,68] (

Table 8. Performance comparison of training algorithms for neural networks [69].

Abbreviation	Algorithm	Performance
LM	Levenberg-Marquardt	High convergence rate
BFG	BFGS Quasi-Newton	Fast convergence
RP	Resilient Backpropagation	Robust to noise
BR	Bayesian Regularization	Effective for small data
SCG	Scaled Conjugate Gradient	Memory efficient
CGB	Conjugate Gradient with Powell/Beale Restarts	Balanced performance
CGF	Fletcher-Powell Conjugate Gradient	Stable convergence
CGP	Polak-Ribiére Conjugate Gradient	Good for sparse data
OSS	One-Step Secant	Fast convergence
GDX	Variable Learning Rate Backpropagation	Adaptive learning rate
GD	Basic Gradient Descent	Simple, easy to implement
GDM	Gradient Descent with Momentum	Accelerated convergence

).

We employed the backpropagation algorithm, a widely-used method in neural network training, to train our model. The neural network architecture consisted of three layers: an input layer with 15 neurons, a hidden layer with 50 neurons, and an output layer with 10 neurons (see Figure 8).

To measure the performance of the network during training, we utilized the Mean Squared Error (MSE) as the performance criterion. Our goal was to minimize the MSE to below 0.0001, indicating a high level of accuracy in the model’s predictions (see Figure 9).

During the training process (see Figure 10), we set parameters to monitor the progress and control the learning dynamics. We configured the network to display progress updates every 50 training epochs, allowing us to track the improvement of the model over time. Additionally, we set the learning rate to 0.03 to regulate the size of weight updates during training.

To ensure effective convergence of the model, we specified the maximum number of training epochs as 30,000. This parameter defines the number of iterations the training algorithm undergoes to optimize the network parameters. Furthermore, we employed a momentum coefficient of 0.9, which helps accelerate convergence by considering past weight updates during parameter optimization [70].

We chose the backpropagation algorithm for several reasons. First, backpropagation is widely recognized for its effectiveness in training artificial neural networks, with proven successes in domains such as computer vision and natural language processing. Additionally, its implementation is relatively straightforward, and is supported by numerous online resources and well-established code libraries. The flexibility of backpropagation allows it to adapt to a wide variety of neural network structures and to work with different data types and architectures(see Figure 11). Furthermore, it provides efficient optimization of network weights through techniques such as gradient descent, and is widely supported by popular libraries such as TensorFlow and PyTorch. Backpropagation is often chosen for its simplicity, effectiveness, and versatility, making it a preferred choice for machine learning and neural network training.

With a fifteen-neuron input layer and a ten-neuron output layer in a Self-Organizing Map (SOM), each neuron in the output layer represents a cluster or grouping of similar data from the input layer. The SOM (see Figure 12) reduces the dimensionality of the input space into a ten-dimensional output space, preserving spatial relationships between similar data points. Each output neuron acts as a centroid or representative of a cluster of similar data, grouping input data points that are closest in terms of distance. This interpolation of data into the output space allows for information aggregation and helps mitigate overfitting issues. Having only ten neurons in the output layer facilitates visualization and interpretation of the resulting map, allowing for an understanding of how different clusters are organized spatially. In summary, this configuration provides a compact representation of input data, easing visualization, interpretation, and data analysis while preserving the relationships between similar data points.

The SOM neighbor weight distances and sample hits (see Figure 13) are crucial metrics for comprehending Self-Organizing Maps (SOMs). The neighbor weight distances measure the similarity or dissimilarity between adjacent neurons’ weights. Smaller distances indicate a smooth transition in representing input space, suggesting similar responses to similar input patterns among neighboring neurons. Conversely, larger distances imply a more abrupt transitions, indicating differing responses to similar inputs. The sample hits metric tracks how often each neuron is selected as the best-matching unit (BMU) for input samples during training. Neurons with higher hits represent densely populated regions of the input space, while lower-hit neurons represent sparser regions. Analyzing the neighbor weight distances helps evaluate the map’s ability to capture the underlying structure of input data (see Figure 14), while the sample hits reveal the distribution and density of the data, aiding in identifying clusters or patterns. Together, these metrics provide insight into how the SOM organizes and represents input data, facilitating interpretation and understanding of its performance.

One possible concrete application of the Taguchi method with smart antennas and neural networks could be the design of an adaptive wireless communication system. By using the Taguchi method to optimize antenna parameters in combination with neural networks for decision-making and dynamic adaptation, a system could be created that optimizes its performance in real time based on changing communication channel conditions. This approach would enable more efficient spectrum utilization, improved quality of service, and greater reliability in wireless communications.

5. Conclusions

The synthesis of antenna arrays employs a spectrum of techniques, ranging from intricate analytical methods to iterative numerical methods driven by optimization algorithms. However, a common limitation of these techniques is their predominant focus on the array factor, while often neglecting the intricate interactions among array elements and real-time operational constraints. This oversight can introduce inaccuracies in the resulting radiation pattern, prompting the need to incorporate the physical relationships between array feeding parameters and the resulting radiation patterns in order to enhance precision. Due to the inherently nonlinear behavior of antenna arrays, attempting to address these complexities using traditional methodologies poses significant challenges and is frequently disregarded. In contrast, leveraging neural network-based solutions offers a promising avenue by establishing connections between desired radiation patterns and key feeding parameters, such as voltage and spacing, within the real antenna array. This approach facilitates the transition from a conventional antenna array to a smart array, while bypassing the complexities associated with conventional methods. This paper provides an overview of various neural network applications in smart antenna array synthesis, underscoring their potential to address the limitations of traditional techniques and improve the accuracy and efficiency of antenna array synthesis processes.

In conclusion, our research demonstrates the effectiveness of combining the Taguchi method with neural networks for accurate synthesis of antenna radiation patterns. Validation against real-world data confirms the reliability and practical applicability of our approach, opening avenues for further refinement and exploration.

In our future work, in addition to exploring phase and amplitude synthesis using this Taguchi-neural network method, we plan to delve deeper into experimentation with various learning algorithms. We aim to compare the effectiveness of these algorithms against our current approach in order to identify the one that provides the best performance in terms of accuracy and generalization of antenna radiation pattern models. This approach will help us to gain a better understanding of the strengths and limitations of each algorithm and select the most suitable one for our specific application.