Synthesis of a Radiating Elements for Side Lobe Reduction Using a Hybrid Beamforming Technique Based on Non-Linear Programming and Stochastic Optimization

The geometry of the CAA is seen in Figure 1.

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Figure 1. The circular antenna array geometry

A concentric ring of radius r is analyzed in the x–y plane, with θ and φ indicating the angles of elevation and azimuth, respectively.

$A F(\phi)=\sum_{n=1}^{N_{E}} I_{n} e^{j\left[\beta r \cos \left(\phi-\phi_{n}+\alpha_{n}\right]\right.}$     (1)

With:

$I_{n}$: the amplitude of the excitation current;

$\alpha_{n}$: the phase of the excitation current'; and,

$\varphi_{n}$: the n^th element’ angular position on x-y plane.

$\phi_{n}=\frac{2 \pi(n-1)}{N_{E}}$     (2)

The spacing angle is controlled by the number of antennae components and the spacing angle, the wave number β can be written as:

$\beta \,r=\frac{2\pi \,r}{\lambda }=\sum\limits_{i=1}^{{{N}_{E}}}{{{d}_{i}}={{N}_{E}}d}$      (3)

Maxima of the array factor AF dictate the orientation of the main beam $\left(\varphi_{0}\right)$, which can be adjusted by the moderate adjusting variable $\alpha_{n}$ as follows:

$\alpha_{n}=-\beta r \cos \left(\phi_{0}-\phi_{n}\right)$       (4)

Therefore, the AF may be defined in terms of the intended primary lobe orientation as:

$AF\left( \varphi  \right)=\sum\limits_{n=1}^{{{N}_{E}}}{{{I}_{n}}{{e}^{j\left[ \beta \,r\left( \cos \left( \varphi -{{\varphi }_{n}} \right)-\cos \left( {{\varphi }_{0}}-{{\varphi }_{n}} \right) \right)+\psi  \right]}}}$      (5)

where, $I_{n}$ and $\psi$ are the estimated adjusted amplitude and phase for controlling the figure of merit of the circular antenna array.

The regulated amplitude and phase range vary from [0, 1] and [180°, 180°] when the amplitude is normalized to its greatest value.

The primary goal of this study is to increase directivity and minimize SLL by getting the required major lobe orientation.

The objective may be divided into three parts to obtain the same result:

$O F_{S L L}=\frac{\left|A F\left(\phi_{S L L} \quad , \quad \vec{I}, \quad \vec{\beta}, \quad \phi_{0}\right)\right|}{\left|A F\left(\phi_{\max } \quad , \quad \vec{I}, \quad \vec{\beta}, \quad \phi_{0}\right)\right|}$      (6)

$O F_{D}=\frac{1}{\operatorname{dir}\left(\phi_{0}, \vec{I}, \vec{\beta}\right)}$      (7)

$O F_{M I}=\left|\phi_{0}-\phi_{d e s}\right|$      (8)

The three objectives may be met by decreasing the entire optimization approach for CAA design (OF_CAA).

$O{{F}_{CAA}}=O{{F}_{SLL}}+O{{F}_{D}}+O{{F}_{ML}}$      (9)

The $O F_{S L L}$ is the fitness function aimed towards lowering SLL, where $\varphi_{S L L}$ denotes the greatest side lobe angle. Besides, the directivity at the main lobe direction $\varphi_{S L L}$ should be maximized. The same result may be accomplished by putting this in the desired function's denominator $O F_{D}$. In the third part $\left(O F_{M L}\right)$ of the objective function, the main lobe is aligned with the desired direction $\left(\varphi_{\text {des }}\right)$. As a result, achieving these three figures of merit will be accomplished by minimizing the overall goal function [3].

The inertial mass and acceleration coefficient are neither given a constant value in this section, nor assigned to a time-varying function with linear decay [29]. In Eqns. (12)-(13), the minimization problem's fitness function is described because of the finest available locally (pbest) and global best (gbest) values. (pbesti) average denotes the mean of all the best personal values of this generation.

${{\omega }_{i}}=\left( 1.1-\frac{gb}{{{\left( p{{b}_{i}} \right)}_{av}}} \right)$     (12)

${{C}_{i}}=\left( 1+\frac{gb}{\left( p{{b}_{i}} \right)} \right)$     (13)

The circular array antenna was created with the goal of suppressing SLL. In this study, we look at three distinct techniques for handling the beam pattern optimization problem: 8-element CAA, 12-element CAA, and 20-element CAA. Three cases are presented in the simulation-based experimentation. Amplitude-only, angular position-only, and amplitude–angular position synthesis techniques are all examples of synthesis techniques. To find the best solution, each algorithm was run on a 2.30 GHz Intel Core (TM) 2348M processor with 4-GB RAM in MATLAB 2013b.

6.1 Circular antenna array synthesis using amplitude only technique

6.1.1 Synthesis of 8‑elements using amplitude only technique

To deliver the optimal radiation sample for CAA, the BPSO, APSO, and NP-PSO algorithms are applied to synthesize CAA using amplitude only technique.

Table 1 show the best results obtained by using the three algorithms for 8 elements CAA. The present excitation weights, inter-element spacing, and maximum SLL are all listed in the table.

Table 1. Excitement amplitudes for the improved 8-element

Figure 2 depicts the radiation patterns derived by utilizing the amplitude distribution given in Table 1. The efficacy of the suggested strategy may be clearly determined in terms of acquired SLL from these patterns. The statistics show that the suggested NP-PSO has the shortest maximum SLL along with the quickest convergence level, Figure 3.

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Figure 2. Amplitude-only radiation configuration of an adjusted 8-element CAA

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Figure 3. Convergence plot of 8-element synthesis applying amplitude only method

6.1.2 Synthesis of 12‑elements using amplitude only technique

Likewise, with a 12-element CAA, Table 2 shows the amplitude excitation coefficients that have been determined.

Table 2. The optimized 12-element CAA's excitation amplitudes

The radiation pattern graphs of the three methods are shown in Figure 4. BPSO, APSO, and NP-PSO have SLL values of –14.65 dB, –23.97 dB, and –29.13 dB, respectively.

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Figure 4. Amplitude-only radiation pattern of an optimized 12-element CAA

As shown in Figure 5, the NP-PSO algorithm demonstrated fast convergence while being more efficient than both the BPSO and APSO algorithms.

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Figure 5. Convergence diagram of a 12-element CAA synthesis using the Amplitude-Only method

6.1.3 Synthesis of 20‑elements using amplitude only technique

Table 3 shows the amplitudes obtained using BPSO, APSO, and NP-PSO in the example of the 20-element CAA synthesis, with NP-PSO achieving an SLL of -24.57 dB and APSO and BPSO achieving 17.16 dB and -13.26 dB, respectively. Figures 6 and 7 show the radiation pattern as well as the convergence plots.

Table 3. Amplitudes of excitation for the optimized 20-element CAA

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Figure 6. Amplitude-only radiation pattern of an optimized 20-element CAA

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Figure 7. Convergence diagram of amplitude-only synthesis of 20-element CAA

6.2 Circular antenna array synthesis utilizing the position-only method

In this example, as a consequence of maximizing the angular locations of the array's elements, the subsequent separate vector φ_i is supplied.

$\varphi_{i}=\left[\varphi_{i 1}, \varphi_{i 2} \quad \varphi_{i N}\right]$      (18)

and the population matrix is represented by:

$p o p=\left[\begin{array}{c}\varphi_{1} \\ \varphi_{2} \\ \vdots \\ \varphi_{n}\end{array}\right]=\left[\begin{array}{cccc}\varphi_{11} & \varphi_{12} & \cdots & \varphi_{1 N} \\ \varphi_{21} & \varphi_{22} & \cdots & \varphi_{2 N} \\ \vdots & \vdots & \cdots & \varphi_{1} \\ \varphi_{n 1} & \varphi_{n 2} & \cdots & \varphi_{n N}\end{array}\right]$     (19)

Each element of the vector I is chosen so that $\phi_{i N}<360^{\circ}$.

6.2.1 Synthesis of 8‑elements using position only technique

This example considers and synthesizes an 8-element CAA using the angular position-only method, and it is shown in Table 4 how to establish the spatial distribution of the array's components using the NP-PSO, APSO, and BPSO methods. In the radiation pattern depicted in Figure 8, NP-PSO has a greater suppression of maximum SLL, the convergence plots are given in Figure 9.

Table 4. Angular positions for the optimized 8-element CAA

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Figure 8. Position-only radiation pattern of an optimized 8-element CAA

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Figure 9. Convergence plot of an 8-element CAA synthesis by means of a position-only approach

6.2.2 Synthesis of 12 elements using angular position only

Figure 10 illustrates the radiation patterns produced using both techniques in 12-element CAA synthesis using an angular position-only technique and indicate a considerable improvement in the SLL. NP-PSO, APSO, and BPSO are used to determine the angular positions of the elements, as shown in Table 5. The convergence plots are shown in Figure 11, which allows us to observe the fast convergence of NP-PSO.

Table 5. Angular positions for the optimized 12-element CAA

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Figure 10. Radiation pattern of an improved 12-element CAA using a position-only approach

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Figure 11. Convergence diagram for a 12-element CAA synthesis using the position-only method

6.2.3 Synthesis of a 20-element system based solely on angular position

Figure 12 exhibits an evaluation of the radiation patterns of the 20-element CAA obtained using various techniques. The SLL values for 20 elements in the BPSO, APSO, and NP-PSO algorithm–based results are 14.59 dB, 17.46 dB, and 23.91 dB, respectively, and the improved values required to plot Figure 10 with both techniques are provided in Table 6. The convergence graph for the specified objective function for 20 antenna components is shown in Figure 13.

Table 6. Angular locations for the improved 20-element CAA

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Figure 12. Viewpoint radiation pattern of an optimized 20-element CAA

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Figure 13. The convergence curve of a 20-element CAA synthesis using the position alone approach

6.3 Synthesis of a circular antenna array combining the amplitude and position approach

Among the two previous examples, only one attribute was utilized in the optimization process. Furthermore, in multi attribute optimization methods, multiple variables can be considered such as an array's angular locations and amplitudes.

6.3.1 Synthesis of a 20-element system based solely on angular position

This approach can manage SLL by manipulating both the amplitudes and placements of the components. Both algorithms consistently created patterns with substantial performance in minimizing the maximum SLL in the first example of 8-element CAA synthesis using the Amp-Pos approach.

The magnitude and location distributions predicted by the BPSO, APSO, and NP-PSO are shown in Table 7. The SLL obtained using NP-PSO and APSO is substantially lower than the BPSO, as seen by the radiation pattern plots in Figure 14. Figure 15 depicts the consistency of the algorithms' performance in terms of convergence.

6.3.2 Synthesis of 12 Elements Using the Approach of Amplitude and Angular Position

As compared to the previous instance, the SLL of the 12-element CAA has improved significantly. Table 8 shows the outcomes of the three methods, which also includes the maximum SLL determined by the algorithms with the aim of suppressing SLL. In this case, Figures 16 and 17 show the radiation pattern and convergence plots for a 12-element CAA synthesis.

Table 7. The improved 8-element CAA's excitation amplitude and angular locations

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Figure 14. Amplitude and angular position approach was used to create a radiation pattern for an optimized 8-element CAA

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Figure 15. Convergence diagram of an 8-element CAA synthesis utilizing the amplitude and position approach

Table 8. The improved 12-element CAA's excitation amplitude and angular locations

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Figure 16. Amplitude and angular position approach was used to create a radiation pattern for an optimized 12-element CAA

6.3.3 The amplitude and angular position approach used to synthesize 20 components

A 20 elements CAA is synthesized using the Amp-Pos approach with BPSO, APSO, and NP-PSO in this example. Consequently, the amplitudes and angular coordinates are determined using the methods shown in Table 9 as well as the estimated SLL maximum. As depicted in Figure 18, the NP-PSO performs better in terms of convergence. The radiation pattern chart in Figure 19 shows the same thing.

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Figure 17. The convergence curve of a 12-element CAA synthesis combining the amplitude and position approach

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Figure 18. Convergence diagram of amplitude and position approach for 20-element CAA synthesis

Table 9. The improved 20-element CAA's excitation amplitude and angular coordinates

6.4 Comparative study

Table 10 compares the results achieved with NP-PSO to the results obtained with other methods for circular antenna array synthesis. Table 10 confirms that the SLL value achieved by using NP-PSO is the good quality when compared to the published findings obtained by utilizing previous algorithms such as CS [35] and HS [36]. Figure 20 illustrates the effect of the maximum SLL adopting various techniques.

Table 10. Maximum side lobes levels obtained by different algorithms for 20-element CAA

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Figure 19. Amplitude and angular position approach used to improve the radiation pattern of a 20-element CAA

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Figure 20. Radiation pattern obtained by different algorithms for 20-element CAA