A Multiobjective Array Beamforming Method for Arrays of Flexible Shape

Abstract

High-performance beamforming incorporating multiple objectives for large-scale antenna arrays becomes increasingly important to improve the capacity and efficiency of wireless communication systems. The speed of synthesizing a desired beam pattern is critical in wireless communications systems to adapt to highly dynamic wireless channels. A modified particle swarm optimization (PSO) algorithm for synthesizing array beam patterns is proposed in this study. The initial positions of particles in PSO are designated following a Taylor distribution instead of being given uniformly distributed random values as in the classical PSO algorithm. The fitness functions are defined to include multiple objectives represented by producing multiple main lobes with customized deep and broadened nulls. Several scenarios have been established to examine the feasibility of the proposed algorithm. Moreover, the performance of the proposed algorithm is compared with those of the ones based on the classical PSO. A significant performance improvement for obtaining beamforming coefficients has been achieved. The robustness of the proposed algorithm is demonstrated further by applying it to a finite array on a curved surface for beamforming.

1. Introduction

Antenna arrays play an essential role in modern wireless communication systems due to their ability to provide high gain and spatial diversity, resulting in high spectral efficiency in wireless communication systems. In addition, a high signal-to-noise (SNR) ratio and low inter-user interference can be achieved by beamforming, offering high performance and flexibility [1]. Phased array antennas are critical technologies for a wide range of applications, including but not limited to radar, communication, remote sensing, navigation, radio astronomy, biomedical imaging, and radiation therapy; hence, they are widely used in many applications nowadays [2]. Multibeam antenna systems become increasingly important in many applications, including communications, imaging, and radio astronomy, in particular, the fifth-generation (5G) and beyond (i.e., 6G) wireless communications [3]. Beamforming and large-scale antenna arrays play a crucial role in many promising communication and sensing techniques towards 5G and beyond communication networks [4,5,6] such as multiple-input multiple-output (MIMO) [7], massive MIMO communications [8,9] and integrated sensing and communication systems [10].

Significant efforts are being put into exploring array antennas with low sidelobe level (SLL), high directivity, high gain, as well as increased beam steering (BS) capability [11]. In order to obtain more agile array beams and increase beam efficiency, SLL must be maintained at a low level. At the same time, placing nulls in certain directions can reject the interfering sources from the angles concerned [12].

The desired beam pattern is usually achieved by controlling the excitation coefficients of the antenna element, namely amplitude-only [13], phase-only [14], array element position-only [15], or any combination of the above parameters. Controlling a set of complex weights (both the amplitude and the phase) is the most efficient way because it has a greater degree of freedom for the solution space [16]; hence, this study aims to design the optimum complex weights in order to achieve the desired array beam pattern. A uniform linear array (ULA) is one of the most commonly used fixed-shape antenna arrays. Consequently, this article focuses on the beamforming design for ULA that suppresses the sidelobe and places nulls simultaneously.

Many algorithms have been developed for beamforming, most notably linearly constrained minimum variance (LCMV) [17], and minimum variance distortionless response (MVDR) [18]. However, these algorithms are derived based on deterministic methods. The achievable goals of these algorithms are very limited. For instance, traditional beamforming methods cannot simultaneously deal with complex scenarios, including multiple targets and specific requirements such as reducing sidelobe levels and generating nulls at specific directions.

Recently, evolutionary algorithms and machine learning techniques have also been applied to several applications of adaptive and reconfigurable antenna arrays, for instance, adaptive nulling, wireless localization, multiple-input, multiple-output (MIMO) communications, element failures, and calibration [19,20]. The evolutionary optimization algorithms are capable of solving multi-objective and nonlinear problems. These algorithms have been successfully used for antenna array pattern synthesis, such as genetic algorithm (GA) [12,20], particle swarm optimization (PSO) [21,22,23], ant colony optimization (ACO) [24], grey wolf optimization (GWO) [25], whale optimization algorithm (WOA) [26].

The PSO approach appears likely to be the most powerful. PSO is acknowledged for its fast convergence speed, fewer parameters, and ease of implementation. According to [27], PSO is more computationally effective (uses a smaller number of function evaluations) than the GA. Hence, PSO is adopted in this paper. However, the initial and convergence conditions are very important in implementing the PSO algorithm. The performance of PSO is primarily influenced by particle initialization as a stochastic search algorithm.

PSO algorithm is used for beamforming in [28] with a restrictive approach to synthesize four radiation patterns for a four-element array. An adaptive array beamformer using PSO is proposed in [29], where an initial DOA estimate is first determined using a PSO estimator to improve the overall beamformer performance. A PSO-based algorithm is utilized in [30] to address the outage problem for the millimeter-wave (mmWave) non-orthogonal multiple access (NOMA) system.

A modified PSO algorithm, which initializes particles based on a Taylor function instead of a uniform distribution, is proposed. This algorithm has a lower fitness at the beginning, and maintains a faster convergence rate in the subsequent iteration process, and is verified in the following design scenarios. The modified algorithm combines the advantages of traditional beamforming methods and intelligent algorithms, making it more suitable for solving array beamforming problems.

The major contributions of this paper are stated as follows: (1) A modified PSO is proposed with a faster convergence speed and validated and compared to the classic PSO with some typical scenarios. (2) A simple fitness function is proposed to flexibly control sidelobe level and null directions, including single null, multiple nulls, and broad nulls. (3) Considering the combination of multiple main beams and multiple nulls in large-scale arrays.

Notation: Throughout the manuscript, the following mathematical notation is used. Bold uppercase letters (e.g., $\mathbf{H}$)

2. Array Beamforming

To illustrate a simple beamforming operation, consider the case shown in Figure 1. The far-field beam pattern of an antenna array can be synthesized by [31]

$$B\left(\theta ,\varphi \right)={\mathbf{w}}^{\mathrm{H}}\mathbf{v}\left(\mathbf{k}\right),$$

(1)

where $\mathbf{w}\in {\mathbb{C}}^{N\times 1}$ is a vector which represents the weights and the components are complex numbers, N is the number of elements in an array, and $\mathbf{v}\in {\mathbb{C}}^{N\times 1}$ is the array manifold vector which is related to the angle of incoming waves, determined by the wavenumber $\mathbf{k}$, and the array element spacing vector $\mathbf{p}$, and the observation angle $\left(\theta ,\varphi \right)$.

The array manifold vector $\mathbf{v}\left(\mathbf{k}\right)\in {\mathbb{C}}^{N\times 1}$ is defined by

$$\mathbf{v}\left(\mathbf{k}\right)=\left[\begin{array}{c}{e}^{-j{\mathbf{k}}^{\mathrm{T}}{\mathbf{p}}_0}\\ e^{-j{\mathbf{k}}^{\mathrm{T}}{\mathbf{p}}_1}\\ ⋮\\ e^{-j{\mathbf{k}}^{\mathrm{T}}{\mathbf{p}}_{N-1}}\end{array}\right],$$

(2)

which incorporates all of the spatial characteristics of the array. The array consists of a set of isotropic antennas located at positions ${\mathbf{p}}_n={\left[p_{x_n},p_{y_n},p_{z_n}\right]}^{\mathrm{T}}$, $n=0,1,\cdots ,N-1$. Moreover, the wavenumber is defined by

$$\mathbf{k}=-\frac{2\pi }{\lambda }\left[\begin{array}{c}sin\theta cos\varphi \\ sin\theta sin\varphi \\ cos\theta \end{array}\right],$$

(3)

where $\lambda$ is the wavelength. The complex weight vector is ${\mathbf{w}}^{\mathrm{H}}=\left[\begin{array}{c}{w}_0^{*},w_1^{*},\cdots ,w_{N-1}^{*}\end{array}\right]$, which can adjust the gain and phase at the output of each antenna to achieve a desirable beam pattern.

A uniform linear array has been chosen as an example to examine the algorithm proposed in this work, as the algorithm proposed can be easily implemented for other array configurations without much modification. The ULA of interest is shown in Figure 2, which consists of N antennas located on the z-axis with uniform spacing equal to d, and the center of the array is placed at the origin of the coordinate system.

The locations of the array elements are:

$$\begin{array}{c}\hfill p_{z_n}=\left(n-\frac{N-1}{2}\right)d,n=0,1,\cdots ,N-1,\end{array}$$

(4)

where, $p_{x_n}=p_{y_n}=0$. The beam pattern of the ULA can be written in the following form:

$$\begin{array}{cc}\hfill B\left(\theta \right)& ={\mathbf{w}}^{\mathrm{H}}\mathbf{v}\left(\theta \right)\hfill \\ \hfill & =e^{-j\left(\frac{N-1}{2}\right)\frac{2\pi d}{\lambda }}\sum _{n=0}^{N-1}{w}_n^{*}{e}^{jn\frac{2\pi d}{\lambda }cos\theta }.\hfill \end{array}$$

(5)

Substituting $cos\theta$ with $\left(cos\theta -cos{\theta }_1\right)$ in (5), and the array vector expression gets modified, the modified ULA beam pattern expression is shown in (6)

$$\begin{array}{cc}\hfill B\left(\theta \right)& ={\mathbf{w}}^{\mathrm{H}}{\mathbf{v}}_1\left(\theta \right)\hfill \\ \hfill & =e^{-j\left(\frac{N-1}{2}\right)\frac{2\pi d}{\lambda }cos\theta }\sum _{n=0}^{N-1}{w}_n^{*}{e}^{jn\frac{2\pi d}{\lambda }\left(cos\theta -cos{\theta }_1\right)}\hfill \end{array}$$

(6)

where ${\theta }_1$ is steering angle and beam pattern $B\left(\theta \right)$ has its maximum when $\theta ={\theta }_1$.

The beam pattern of ULA with multiple beams was synthesized by adding multiple array vectors steering at different directions.

$$\begin{array}{cc}\hfill B\left(\theta \right)& ={\mathbf{w}}^{\mathrm{H}}\sum _{i=1}^K{\mathbf{v}}_i\left(\theta \right)\hfill \\ \hfill & =\sum _{i=1}^K\left(e^{-j\left(\frac{N-1}{2}\right)\frac{2\pi d}{\lambda }cos\theta }\sum _{n=0}^{N-1}{w}_n^{*}{e}^{jn\frac{2\pi d}{\lambda }\left(cos\theta -cos{\theta }_i\right)}\right)\hfill \end{array}$$

(7)

where ${\mathbf{v}}_i\left(\theta \right)$ is the ith array vector corresponding to the ith steering angle ${\theta }_i$, K is the number of main beams. The normalized array beam pattern in dB can be expressed as

$$B\left(\theta \right){\mid }_{\mathrm{dB}}=20{log}_{10}\left|\frac{B\left(\theta \right)}{max\left(B\left(\theta \right)\right)}\right|$$

(8)

In this study, the weights vector, $\mathbf{w}$, consisting of complex values to steer the array beam, needs to be optimized according to (5)–(7) to fulfill the objectives of achieving several peak values for main lobes and desired nulls in the synthesized array pattern, whereas the sidelobe level remains below a specific target value. When the number of array elements increases to a large number, the computation complexity of forming the desired beam pattern becomes significantly high. Hence, it is necessary to develop efficient and low-complexity techniques.

3. Beamforming Algorithms

3.1. Problem Description

Conventionally, array steering vectors (i.e., weights applied for beamforming) are calculated using deterministic methods where the values of interest can be derived from the requirements. For multiobjective problems such as the ones described in this work, these methods become inefficient or even impractical. Hence, intelligent algorithms based on population evolution are introduced for more complex problems. PSO demonstrates its merit, among others, due to its high efficiency in dealing with sophisticated multiobjective problems with limited computational resources.

To solve optimization problems incorporating a large number of variables, such as designing a sophisticated target beam pattern based on a large-scale antenna array system, PSO can be effectively utilized. The objective of the optimization task is represented by a fitness function, which is to be minimized in the process. Consider a fitness (or cost, error, objective) function: $f\left(\mathbf{x}\right)$, which incorporates and defines the problem that needs to be solved. The general problems in question for PSO algorithms can be described by optimizing $\mathbf{x}$, which satisfies.

$$\begin{array}{cc}\hfill & \underset{\mathbf{x}}{\min }f\left(\mathbf{x}\right),\hfill \\ \hfill & \mathrm{s}.\mathrm{t}.l_d\le x_d\le u_d,d=1,2,\cdots ,D,\hfill \end{array}$$

(9)

where $\mathbf{x}=\left[x_1,x_2,\cdots ,x_D\right]$ is the optimal solution in a vector form and D is the dimension of solution. Each $x_d$ is bounded by lower limits $l_d$ and upper limits $u_d$.

3.2. Classical PSO

3.2.1. Particle Swarm Optimization

PSO algorithm is an evolutionary optimization technique inspired by the behavior of birds in a swarm, which was initially proposed by Eberhart and Kennedy [32], and later modified and further improved by Shi [33].

In PSO, each potential solution to the optimization problem is a particle in the search space. All particles have a fitness function value determined by the problem to be optimized, and each particle has a velocity vector to determine its “flying” direction and distance. The movement of each particle is then determined after searching through the workspace by combining some aspect of the history of its own current and best (best-fitness) locations with those of one or more members of the swarm, adopting some random perturbations. The following iteration takes place after all particles have moved. Eventually, the swarm as a whole, like a flock of birds, is collectively foraging for food.

The position of the ith particle is a D-dimensional vector. The ith particle is represented as ${\mathbf{x}}_i=({\mathit{x}}_{i1},{\mathit{x}}_{i2},\cdots ,{\mathit{x}}_{iD})$. Each particle has its corresponding velocity vector ${\mathbf{v}}_i=({\mathit{v}}_{i1},{\mathit{v}}_{i2},\cdots ,{\mathit{v}}_{iD})$. The best position for the ith particle in the current iteration is ${\mathbf{pbest}}_i=({\mathit{p}}_{i1},{\mathit{p}}_{i2},\cdots ,{\mathit{p}}_{iD})$, and the best position after search for the whole particle swarm is $\mathbf{gbest}=\left({\mathit{g}}_1,{\mathit{g}}_2,\cdots ,{\mathit{g}}_D\right)$. All particles update their position and velocity in every iteration according to the following formula [33]:

$$\begin{array}{cc}\hfill & {\mathit{v}}_{id}=w*{\mathit{v}}_{id}+{\mathrm{c}}_1*r_1\left({\mathit{p}}_{id}-{\mathit{x}}_{id}\right)+{\mathrm{c}}_2*r_2\left({\mathit{g}}_d-{\mathit{x}}_{id}\right)\hfill \\ \hfill & {\mathit{x}}_{id}={\mathit{x}}_{id}+{\mathit{v}}_{id}.\hfill \end{array}$$

(10)

The particles search the whole space and update their values until the fitness function satisfies a target error defined in the optimization problem.

3.2.2. Parameters in PSO

The key parameters used in the PSO algorithm are described as follows:

• Inertia weight, w, which is linearly decreasing with the iteration number, is defined as,

  $$w=w_{\max }-\frac{w_{\max }-w_{\min }}{T_{\max }}t,$$

  (11)

  where $w_{\max }$ is the maximum inertia weight and $w_{\min }$ is the minimum, typically $w_{\max }=0.9$ and $w_{\min }=0.4$; t is current iteration number; $T_{\max }$ denotes the maximum number of iterations designated in the algorithm.
• Two random parameters, $r_1$ and $r_2$ are uniformly distributed values in the interval $\left[0,1\right]$.
• Acceleration constants, $c_1$ and $c_2$ are set equal to 2; they work well for most of the applications. This has been proven by previous work [34]. However, the value of 1.49 was assigned to both parameters in certain cases [35].
• Swarm size, $\mathit{Pop}\_\mathit{size}$ is also named population size or the number of particles, which is usually set between 70 and 100 to be safe. For higher-dimensional problems, PSO often demonstrates a better performance with a larger swarm size [36].

3.2.3. Position and Velocity Initialization

As an iterative algorithm, in the beginning, initial values are given to the positions and velocities of particles, and they play a decisive role in reaching the convergence for the PSO algorithm. The positions of particles are initialized so that they spread among the entire search space [37], ensuring uniform coverage within the search domain and avoiding a local convergence, moreover, leading to a relatively high convergence speed. Another good strategy for initializing the velocities is to set them to zero or random values close to zero instead of giving uniformly distributed random values [38].

3.2.4. Position and Velocity Bounds

Owing to the stochastic nature of updating velocities in iterations, it is possible that the trajectory of a particle expands out of the parameter space, eventually crossing the boundaries of the search space. The method to overcome this problem is to adopt velocity clamping [39], a technique that limits each particle’s velocity; at the same time, the positions of particles are also constrained within a specified range. These boundaries are applied to prevent particles from moving beyond the limits of the search space.

The velocity and position limitations are applied as,

$$v_{i,d}\left(t\right)=\left\{\begin{array}{cc}\hfill \hfill & v_d^{\max },v_{i,d}\left(t\right)>v_d^{\max }\hfill \\ \hfill \hfill & -v_d^{\max },v_{i,d}\left(t\right)<-v_d^{\max }\hfill \\ \hfill \hfill & v_{i,d}\left(t\right),\mathrm{otherwise}\hfill \end{array}\right.$$

(12)

and

$$x_{i,d}\left(t\right)=\left\{\begin{array}{cc}\hfill \hfill & u_d,x_{i,d}\left(t\right)>u_d\hfill \\ \hfill \hfill & l_d,x_{i,d}\left(t\right)<l_d\hfill \\ \hfill \hfill & x_{i,d}\left(t\right),\mathrm{otherwise}\hfill \end{array}\right.,$$

(13)

where $v_d^{\max }$ and $-v_d^{\max }$ are the upper and lower bounds of the velocity of the dth component, $u_d$ and $l_d$ are the upper and lower bounds of the positions of the dth component. The PSO using the above parameter settings and strategies is named “Classical PSO” in this work, and it will be compared in the beamforming application with the “Modified PSO”, which is proposed in the work.

3.3. Modified PSO Algorithm

3.3.1. Establishing Initialization Strategy

In implementing “Classical PSO”, the particle initialization step is often overlooked, especially for some applications. In fact, most work gives randomly generated, and uniformly distributed values for the initial particle coordinates [40]. This could lead to redundant iterations and poor performance. A novel particle initialization strategy has been proposed in this work, which can improve the search performance in the case of a large number of parameters for beamforming. The proposed method is named Modified PSO in the following sections.

In PSO, the basic principle of initializing particle positions is to make them cover the entire search space, which can improve the likelihood of achieving the global optimal solution. It is worth mentioning that while ensuring particle diversity, initializing the particle positions with the proposed method yields a better convergence and also significantly reduces the search time for the optimum solution.

One of the key objectives of this study is to obtain the SLL below a specified value rather than simply reducing sidelobes without a goal. Taylor proposed a beamforming technique, also known as Taylor distribution [41], that can put constraints on the maximum sidelobe level and obtain gradually decreasing sidelobes. Using the Taylor distribution to initialize the particle positions, the PSO algorithm can produce a better fitness value because it can satisfy one of the multiple targets at the beginning by suppressing sidelobes. Initializing particle positions with nominal Taylor weights can lead to rapid convergence while ensuring population diversity and avoiding falling into local minima.

The procedure of initializing particle swarm positions is summarized as follows:

• First, work out the weights ${\mathbf{w}}_{taylor}=\left[w_1,w_2,...,w_N\right]$ following a Taylor distribution, the entries required comprise the total number of array elements and the desired peak sidelobe level;
• Obtain the initial particle positions by adding the Taylor weights with random values, and the calculation is described in detail in the following section;
• Assemble the particle swarm using the initial particle positions, ready to carry out the optimizing iterations.

It should be noted that the element in the vector consisting of values generated with the Taylor distribution is real, whereas what is required for beamforming is complex weights. Hence, a transformation was carried out, and the real and imaginary parts of the particles were acquired by incorporating Taylor weights as shown in (14).

The initial position of each particle, represented by ${\mathbf{x}}_i$, with its elements being complex value, are acquired by:

$${\mathit{x}}_{id}=(w_d+r_4)+j(w_d+r_3),$$

(14)

where ${\mathbf{x}}_i=({\mathit{x}}_{i1},{\mathit{x}}_{i2},\cdots ,{\mathit{x}}_{iD})$ and ${\mathit{x}}_{id}$ is its dth dimensional component, $w_d$ is the d-dimensional component of ${\mathbf{w}}_{taylor}$, $d=1,2,\cdots ,N$, $j=\sqrt{-1}$, $r_3$ and $r_4$ are representing random values added to the fixed weights following the Taylor distribution. The range of the values for $r_3$ and $r_4$ was optimized. When the range is too large and close to half of the peak value of Taylor Weights, the benefit of adopting the Taylor distribution tends to disappear; when the range is too small, the difference between different particles is very small, and the algorithm is prone to falling into local convergence; hence the recommended range was taken at $\left[-0.1,0.1\right]$.

The initial particle positions generated for the classic PSO (random values following a uniform distribution for each element in the particle vector) and for the proposed PSO in this study, generated from a Taylor distribution, are given in Figure 3. To visually demonstrate the initial value distribution of particles using two initialization strategies, Figure 3 uses a simplified case study. In Figure 3, the number of antenna units is eight, and the number of particles is ten. A particle as a set of weights has only the real part; the value of particles is limited to [0,1]; the range of random values added to the Taylor weights is $\left[-0.1,0.1\right]$.

The proposed initialization strategy is based on the Taylor distribution, which can also use the Chebyshev distribution since it can specify the maximum level of sidelobes. The initialization based on the Chebyshev distribution follows the same steps given above.

A general beamforming case (where one main beam has a few nulls and low sidelobes) is considered to investigate the impact of different initialization particle strategies on the overall performance of the PSO algorithm. The comparison results are shown in Figure 4, which clearly demonstrates that the algorithm using a Taylor distribution-based initialization strategy provides the most effective performance, yielding a better (i.e., lower) fitness value under the same number of iterations. This advantage applies to more general cases.

When giving values for initializing positions using the Taylor distribution, n is an indicator number of nearly constant-level sidelobes adjacent to the main lobe, specified as a positive integer [41]. The selection of this parameter will also have a slight impact on the algorithm. We also used the same beamforming case to investigate the impact of the value n on different algorithms. The converging process for adopting Taylor weights using different n are given below. As shown in Figure 5, the convergence performances of different values of n are very close. However, the cases with smaller values (2 and 4) perform better in reaching the convergence.

3.3.2. Procedure for the Modified PSO

The proposed algorithm uses a new initialization particle strategy, while other parameters related to the PSO remain unchanged. The main steps of the proposed PSO algorithm with the new initialization strategy are shown in Figure 6. Furthermore, the pseudo-code of the algorithm is given in Algorithm 1.

Algorithm 1: Modified PSO

Input: array elements N, $\mathit{Pop}\_\mathit{size}$, $T_{\max }$, fitness function $fit$, position upper bound $\left[u_1,u_2,\cdots ,u_D\right]$, position lower bound $\left[l_1,l_2,\cdots ,l_D\right]$, Output: Optimized $\mathbf{gbest}$ 1: $w=0.9$;  $c_1=c_2=2$; 2: calculate $w\_taylor$; 3: for i = 1:$\mathrm{Pop}\_\mathrm{size}$do                                ▹ Initialize particle 4:     for d = 1:D do 5:         ${\mathit{x}}_{id}=(w_d+r_4)+j(w_d+r_3)$; 6:         ${\mathit{v}}_{id}=0$; 7:         $pbes{t}_{id}=x_{id}$; 8:     end for 9: end for 10: choose $\mathbf{gbest}$; 11: $t=0$; 12: repeat                                       ▹ Main loop 13:     $t=t+1$; 14:     $w=0.9-(t-1)*(0.9-0.4)/\left(T_{\max }\right)$; 15:     for i = 1:$\mathrm{Pop}\_\mathrm{size}$ do 16:         for d = 1:D do 17:            ${\mathit{v}}_{id}=w*v_{id}+c_1*r_1*(pbes{t}_{id}-x_{id})+c_2*r_2*(gbes{t}_d-x_{id})$; 18:            ${\mathit{x}}_{id}={\mathit{x}}_{id}+{\mathit{v}}_{id}$; 19:         end for 20:         if $fit\left({\mathit{x}}_i\right)<fit\left({\mathit{pbest}}_i\right)$ then 21:            ${\mathit{pbest}}_i={\mathit{x}}_i$; 22:            if $fit\left({\mathit{pbest}}_i\right)<fit\left({\mathit{gbest}}_i\right)$ then 23:                ${\mathit{gbest}}_i={\mathit{pbest}}_i$; 24:            end if 25:         end if 26:     end for 27: until $t=T_{\max }$ or $fit\left(\mathbf{gbest}\right)=0$ 28: return  $\mathbf{gbest}$

3.4. Fitness Function Definition

The fitness function defines the convergence requirement of satisfying the objective such that the positions of all particles are optimized. For beamforming problems, we are generally interested in achieving a desired beam pattern, including a low sidelobe level, several beams, and nulls at specified angles in the synthesized far-field radiation patterns. Meanwhile, the depth and width of nulls may also need to be managed. The fitness function f, which incorporates multiple goals to achieve, is defined as

$$\begin{array}{cc}\hfill & f=0.8*f_1+0.2*f_2,\mathrm{WHERE}\hfill \\ \hfill & f_1=\left\{\begin{array}{cc}\hfill \mathrm{MSLL}-\mathrm{DSLL}\hfill & \hfill \mathrm{if}\mathrm{MSLL}>\mathrm{DSLL}\hfill \\ 0& \hfill \mathrm{if}\mathrm{MSLL}\le \mathrm{DSLL}\hfill \end{array}\right.,\hfill \\ \hfill & f_2=\left\{\begin{array}{cc}\hfill \mathrm{MNDL}-\mathrm{DNDL}\hfill & \hfill \mathrm{if}\mathrm{MNDL}>\mathrm{DNDL}\hfill \\ 0& \hfill \mathrm{if}\mathrm{MNDL}\le \mathrm{DNDL}\hfill \end{array}\right.,\hfill \end{array}$$

(15)

where $\mathrm{MSLL}$ denotes the peak sidelobe level of the array pattern that is calculated from particles in each iteration (the particle vector is assigned with a set of weights) using (4), $\mathrm{DSLL}$ denotes the desired peak sidelobe level, $\mathrm{MNDL}$ denotes the calculated maximum value of radiated powers in the directions where nulls are expected, and $\mathrm{DNDL}$ denotes the desired power level for nulls. One or multiple nulls can be synthesized using this fitness function.

The optimized values for $\mathrm{MSLL}$ and $\mathrm{MNDL}$ are obtained once they are smaller than the target values, $\mathrm{DSLL}$ and $\mathrm{DNDL}$. The iterations for optimization stop when the value of the fitness function converges to 0, which is determined by a condition for convergence.

4. Numerical Analysis

In order to demonstrate the performance of the proposed PSO technique, beamforming design and synthesis problems with sophisticated requirements have been considered. In simulations, we have used MEEP for electromagnetic simulations and Python to run PSO-based optimizations and system simulations. Five typical scenarios of array beamforming are investigated. The specific requirements and settings of the five scenarios are summarized in

Table 1. Definition of the scenarios and objectives.

Scenario	Array Elements	Null Number or Type	Main Lobe Number	DNDL	DSLL
1	16	1	1	−70 dB	−20 dB
2	16	8	1	−60 dB	−20 dB
3	16	range	1	−50 dB	−20 dB
4	16	3	2	−60 dB	−20 dB
5	100	12	13	−50 dB	−20 dB

. These five scenarios are grouped into two categories, described and studied in Section 4.1 and Section 4.2. Section 4.3 compares the results from the proposed method to the ones obtained based on the classic PSO. In addition, the influence of different SLL and null requirements on beamwidth was studied in Section 4.4.

4.1. One Main Lobe

The beam patterns with one main lobe were considered in this category. The main lobe was assumed in the direction where $\theta ={90}^{\circ }$, and several extra requirements on nulls were investigated by incorporating them in the objective functions.

4.1.1. Single Null

In scenario 1, one main lobe in the radiation pattern with a single null is studied. The direction for the null can be arbitrarily set at any angle (except for the main lobe direction). In the first trial, it is set with $\theta ={150}^{\circ }$ and $\theta ={101}^{\circ }$. A null near the main lobe is set to demonstrate the algorithm’s agility to place the null at an arbitrary angle. The target sidelobe level (SLL) and null depth level (NDL) are −20 dB and −70 dB, respectively.

After implementing the algorithm for an array of 16 elements as defined for Scenario 1, Figure 7 and Figure 8 show that the null in the specified direction is obtained. Meanwhile, SLL is maintained at a relatively low level. The NDL of the prescribed angle and peak SLL are −71.65 dB and −20.04 dB in Figure 7, and in Figure 8 they are −75.57 dB, −20.26 dB, respectively. The Beam Width between First Nulls (FNBW) in Figure 7 and Figure 8 are 17.2 degrees and 19.4 degrees. The difference in FNBWs for the two null configurations is 2.2 degrees.

4.1.2. Multiple Nulls

In Scenario 2, one main beam with multiple nulls is considered. Initially, the directions for eight nulls were set for the array with 16 elements at ${25}^{\circ }$, ${40}^{\circ }$, ${55}^{\circ }$, ${70}^{\circ }$, ${110}^{\circ }$, ${125}^{\circ }$, ${140}^{\circ }$, and ${155}^{\circ }$.

After performing the algorithm with the new fitness function, as shown in Figure 9, the synthesized beam pattern with a low SLL and eight nulls at predefined directions has been obtained. The peak SLL was −20.44 dB, the FNBW was 15.0 degrees, and the depths of all eight nulls at their specified angles were summarized in

Table 2. The depths of obtained nulls.

Angle (Degree)	Depth (dB)	Angle (Degree)	Depth (dB)
25	−63.24	110	−60.03
40	−60.26	125	−61.42
55	−60.91	140	−60.43
70	−60.05	155	−61.01

. The target SLL and NDL are −20 dB and −60 dB, respectively. Moreover, all NDLs were below −60 dB as desired.

4.1.3. Broad Nulls

A broad null is sometimes required if the interfering signal’s direction of arrival (DOA) cannot be estimated exactly. To demonstrate the capability of the algorithm to suppress the interference from a range of angles, scenario 3 is considered; the range of angles for the interfering source is set in $\left[{120}^{\circ },{130}^{\circ }\right]$, the target SLL and NDL is −20 dB and −50 dB, respectively. After implementing the algorithm, the synthesized beam pattern with a broad null as deep as −50 dB, an FNBW of 15.0 degrees and a peak SLL of −20.08 dB have been demonstrated, which was illustrated in Figure 10.

4.2. Multiple Beams

Multiple beams are required in some applications such as radio astronomy; more than one beam pointing to different angles simultaneously is expected to accelerate the survey speed; meanwhile, a number of nulls are produced to reject interfering sources. In scenarios 4 and 5, the proposed algorithm was implemented to synthesize multiple main lobes, whereas more than one null was anticipated.

4.2.1. Two Main Lobes

An array beam pattern with the main lobes pointing to ${90}^{\circ }$ and ${120}^{\circ }$ can be calculated using (4). In scenario 4, the angles for nulls are designated at ${30}^{\circ }$, ${65}^{\circ }$, ${135}^{\circ }$, and the target SLL and NDL were set as −20 dB and −60 dB. As shown in Figure 11, the beam pattern with three nulls and two beams at their predefined directions has been obtained. The peak SLL is −19.97 dB, the FNBW for the first main lobe is 20.6 degrees, the FNBW for the second main lobe is 27.3 degrees and the depth for the nulls at the directions of ${30}^{\circ }$, ${65}^{\circ }$ and ${135}^{\circ }$ are −61.13 dB, −62.02 dB and −60.04 dB, respectively.

4.2.2. Thirteen Main Lobes

In order to evaluate the full potential of the proposed algorithm for beamforming, in Scenario 5, an array with 100 elements was studied, where 13 main lobes and 12 nulls were expected to be formed. As the requirement is much more stringent than the previous ones, −50 dB was assigned for the target level of nulls. However, the SLL maintained 20 dB lower than the main lobe.

Due to the increase in the dimensions of the problem to be optimized, the $\mathit{Pop}\_\mathit{size}$ and $T_{\max }$ were set to 1000 and 1000, respectively, when using the proposed PSO. After implementing the algorithm with the new fitness function, the synthesized pattern with the 13 main lobes and 12 nulls is achieved, and this was illustrated in Figure 12, the peak SLL was −20.27 dB, and the power levels for the 12 specified angles of nulls were summarized in

Table 3. The null depths of the synthesized beam.

Angle (Degree)	Depth (dB)	Angle (Degree)	Depth (dB)	Angle (Degree)	Depth (dB)
23	−50.09	72	−50.49	120	−50.35
36	−49.98	84	−50.04	132	−50.94
48	−58.07	96	−53.02	144	−50.00
60	−55.53	108	−52.61	157	−49.99

.

4.3. Modified PSO versus Classic PSO

Finally, the effectiveness of the proposed modified PSO was evaluated by comparing it with the Classic PSO. The evaluation was performed for all scenarios except scenario 5, as it is too complicated for the classic PSO. The shared values of parameters for implementing the Modified PSO and Classic PSO are given in

Table 4. The parameter settings, average execution time, and final average fitness values obtained after ten repeated operations using two algorithms on each scenario.

Scenario	$\mathit{Pop}\_\mathit{size}$	Average Fitness Value		Average Execution Time (s)
		Classic PSO	Modified PSO	Classic PSO	Modified PSO
1	100	2.089	0	67.97	3.97
2	200	2.729	0.093	129.83	85.28
3	100	3.239	0.027	64.97	39.12
4	100	2.009	0.016	46.82	33.89

for 4 scenarios investigated, and the performance of searching for both methods was obtained and then compared by adopting the values given in Table 4.

The gradual change of fitness function values during the optimization iterations for scenarios 1 to 4 is illustrated in Figure 13. They were calculated by averaging over 10 repeating runs for each curve independently. The final average fitness value is shown in Table 4, where $T_{\max }$ is the number of iterations per run. From Figure 13 and Table 4, it can be seen that the proposed algorithm yielded lower fitness values in the early stage of iterations and reached convergence more efficiently under all scenarios of investigation.

The optimal complex weights meeting the desired objectives for the four scenarios were obtained by running the two algorithms, and they are given in

Table 5. The weights of each element optimized by the classical and modified PSO in four designed scenarios.

	Scenario 1		Scenario 2		Scenario 3		Scenario 4
	Modified PSO	Classic PSO	Modified PSO	Classic PSO	Modified PSO	Classic PSO	Modified PSO	Classic PSO
E1	0.64 + j0.60	0.57 + j0.68	0.26 + j0.37	0.16 + j0.01	0.29 + j0.42	0.35 + j0.35	0.26 + j0.15	0.39 + j0.98
E2	0.41 + j0.23	0.35 + j0.59	0.31 + j0.26	0.25 + j0.19	0.43 + j0.01	0.27 + j0.45	0.12 + j0.32	0.63 + j0.51
E3	0.61 + j0.36	0.48 + j0.39	0.38 + j0.31	0.43 + j0.47	0.43 + j0.46	0.76 + j0.36	0.11 + j0.12	0.32 + j0.96
E4	0.69 + j0.58	0.61 + j0.49	0.59 + j0.63	0.53 + j0.49	0.45 + j0.54	0.27 + j0.51	0.35 + j0.49	0.79 + j0.96
E5	0.61 + j0.59	0.59 + j0.63	0.63 + j0.62	0.38 + j0.51	0.80 + j0.77	0.75 + j0.58	0.48 + j0.70	0.92 + j0.79
E6	0.79 + j0.73	0.36 + j0.92	0.78 + j0.77	0.66 + j0.74	0.91 + j0.84	0.84 + j0.70	0.82 + j0.80	1.00 + j0.77
E7	0.79 + j0.82	0.54 + j0.64	0.83 + j0.88	0.80 + j0.79	0.87 + j0.72	0.78 + j0.72	0.90 + j0.66	1.00 + j0.51
E8	0.74 + j0.87	0.45 + j0.63	0.70 + j0.73	0.60 + j0.68	0.96 + j0.86	0.72 + j0.72	0.75 + j0.87	0.74 + j1.00
E9	0.77 + j0.84	0.68 + j0.89	0.71 + j0.75	0.53 + j0.65	0.86 + j0.81	0.61 + j0.87	0.84 + j0.89	0.90 + j0.83
E10	0.84 + j0.80	0.75 + j0.63	0.84 + j0.89	0.67 + j0.64	0.86 + j0.80	0.72 + j0.55	0.76 + j0.74	0.79 + j0.00
E11	0.76 + j0.73	0.37 + j0.86	0.77 + j0.75	0.73 + j0.67	0.84 + j0.87	0.62 + j0.46	0.45 + j0.86	0.80 + j0.91
E12	0.69 + j0.71	0.67 + j0.73	0.60 + j0.59	0.72 + j0.68	0.71 + j0.73	0.58 + j0.52	0.65 + j0.68	0.77 + j1.00
E13	0.55 + j0.47	0.49 + j0.36	0.59 + j0.69	0.58 + j0.48	0.43 + j0.72	0.68 + j0.43	0.71 + j0.37	0.58 + j0.29
E14	0.57 + j0.43	0.39 + j0.56	0.40 + j0.42	0.33 + j0.29	0.49 + j0.63	0.36 + j0.41	0.58 + j0.01	0.75 + j0.00
E15	0.39 + j0.40	0.12 + j0.27	0.30 + j0.33	0.41 + j0.37	0.04 + j0.49	0.48 + j0.41	0.26 + j0.37	0.51 + j0.76
E16	0.87 + j0.63	0.56 + j0.54	0.24 + j0.34	0.40 + j0.18	0.35 + j0.35	0.21 + j0.00	0.21 + j0.21	0.31 + j0.05

.

4.4. Beamwidth

The beamwidth is another important factor to consider for beam pattern synthesis. It includes Half-Power Beam Width (HPBW) and FNBW. The characteristics of both of them were studied for the cases with various null requirements. After implementing the modified PSO on the array of 16 elements, the FNBW and HPBW produced with different SLLs and null requirements are shown in

Table 6. The FNBW and HPBW of the beam patterns generated under various SLLs and nulls requirements.

Element Number	FNBW (Degree)	HPBW (Degree)	Target SLL (dB)	Target Null Angle (Degree)	Obtained SLL (dB)	Obtained NDL (dB)
16	15.1	6.4	−15	135	−15.02	−64.25
16	15.8	6.6	−15	85	−15.21	−68.45
16	15.5	6.8	−15	20	−15.67	−67.02
16	18.1	7.2	−20	135	−20.03	−68.66
16	17.3	7.1	−20	85	−20.07	−72.50
16	17.5	7.1	−20	20	−21.00	−65.05
16	19.9	7.7	−25	135	−25.19	−64.25
16	19.6	7.6	−25	85	−25.02	−62.37
16	20.2	7.7	−25	20	−25.21	−60.01

. Only one null was considered here, and the desired null depth is below −60 dB. From Table 6, it can be seen that lower nulls resulted in wider FNBWs. For the HPBW parameter, it stayed nearly constant for the three cases where the null was formed at ${135}^{\circ }$, ${85}^{\circ }$, and ${20}^{\circ }$, respectively.

5. Results and Discussion

To demonstrate the effectiveness of the proposed PSO-based method for realistic beamforming problems, a curved array is considered for beam steering. Full-wave electromagnetic (EM) simulations require a long running time to determine the optimized weights, as each iteration of the full-wave simulation requires computationally expensive calculation with the whole array structure. However, with the proposed algorithm, the optimized weights can be efficiently computed by employing the known radiation pattern of each element, which can be extracted from full-wave simulations with each element being excited independently. The structure of the curved array used is illustrated in Figure 14.

The purpose of this example is to demonstrate the multiobjective beamforming capability of the proposed method. This example demands manipulating the positions of desired main beams in addition to maintaining the sidelobes at low levels. The requirement for beams synthesized from the curved array is stringent, which includes 60 degrees of scanning capability from the broadside with a sidelobe level 20 dB lower than the main lobe. It is challenging to achieve this goal using the analytical method where the phase shift for each array element is calculated by their relative locations with respect to the scan angle, which is particularly difficult when a curved array is considered. Excitation amplitude and phase associated with each array element must be optimized to achieve this goal. Therefore, the modified PSO is used for the beam pattern optimization, and the results are accordingly validated through EM simulations.

As shown in Figure 15, before implementing the proposed algorithm for coefficient determination, the steering vectors were derived directly from (1) while a uniform amplitude was assumed, resulting in a beam pattern with a peak SLL of −8.26 dB with the main lobe direction at ${54}^{\circ }$. After implementing the proposed method for pattern synthesis, the peak SLL acquired was as low as −13.95 dB with the main beam pointing at ${53}^{\circ }$. The obtained weights from the algorithm were then used to control the excitation signals for the array elements in the EM simulation. The beam pattern obtained through the EM simulation has a peak SLL of −12.9 dB and the main lobe direction of $52.{25}^{\circ }$. The good agreement between the result based on the algorithm and that of the full-wave simulation for verification indicated the robustness of the algorithm where multiple objectives of the beam pattern can be achieved without requiring complex EM calculation for each set of parameters during the optimizing process.

The excitation amplitude and phase obtained using different methods are shown in the

Table 7. The comparison of the excitation schemes on the finite array between the direct analytic determination with a Taylor tapering and the Modified PSO.

		E1	E2	E3	E4	E5	E6	E7	E8
Analytic	Amplitude	0.76	0.85	1.11	1.27	1.27	1.11	0.85	0.76
Determination	Phase (rad)	8.94	6.46	3.89	1.25	−1.47	−4.26	−7.12	−10.03
Modified PSO	Amplitude	0.63	0.69	0.75	0.75	0.68	0.60	0.42	0.42
	Phase (rad)	8.98	6.46	4.02	1.34	−1.36	−3.81	−6.80	−9.87

. The acquisition time for optimal array coefficients can be significantly reduced by adopting the proposed method. In the full-wave simulations, the time to have the beam pattern produced with a planar array is approximately 5 min, and it tripled when the array is on the curved surface. It takes hundreds of simulations to seek the coefficients for a simple objective. However, it took only few minutes by the modified PSO algorithm once the embedded elements patterns were known. Additionally, it can incorporate more comprehensive requirements on the beam patterns without a marked increase for computation resources and time.

Table 8. Comparison of optimization techniques of synthesizing beam patterns.

Ref.	Algorithm	Design	Function	Performance
[25]	GWO	Uniform linear array.	Optimal antenna positions and amplitudes for achieving minimum SLL along with null placement and suppression of the first side lobe.	Lower SLL was obtained; however, GWO is more computationally time-consuming than PSO.
[26]	WOA	Linear aperiodic array.	WOA was applied to the synthesis of uniformly excited broadside linear aperiodic arrays with sidelobe suppression and null steering under constraints on beamwidth requirements.	The average fitness result of WOA is slightly higher than that of CLPSO [42]. WOA has fewer iterations than CLPASO [42] and higher iterations than IWO-WDO [15].
[11]	Mayfly Algorithm (MA)	Uniform and sparse linear array.	MA was applied for sidelobe suppression and null placement in the following two ways: by optimizing amplitudes while maintaining uniform spacing and by optimizing the antenna positions while assuming a uniform amplitude excitation.	MA is able to obtain a considerable improvement in peak SLL suppression and null control. However, MA requires a longer computation time and more parameters than PSO.
[15]	Hybrid invasive weed optimization and wind-driven optimization (IWO/WDO)	Linear sparse array.	The proposed algorithm is implemented to synthesize the uniformly excited linear sparse array pattern having an SLL and null control with a constraint on beamwidth by optimizing the element position only.	The Hybrid algorithm has improved performance in terms of null control, beamwidth control, and the rate of convergence. The proposed algorithm requires fewer iterations for convergence compared to PSO.
[43]	Cuckoo Search (CS)	Large scale concentric circular antenna array (CCAA).	A hybrid approach to suppress the SLL of large-scale CCAA is proposed, which includes an improved discrete cuckoo search algorithm (IDCSA) for thinning the CCAA and a cuckoo search-invasive weed algorithm (CSIWA) for further optimizing the thinned CCAA.	The proposed IDCSA and CSIWA can obtain a lower maximum SLL but do not have advantages in terms of processing time.
This work	Modified PSO	Uniform linear array.	A modified PSO algorithm is proposed for optimizing multiple objectives in radiation pattern, including SLL, null, and main lobe direction, by optimizing the excitation phase and amplitude of the elements.	The proposed algorithm has shown good performance compared to classical PSO in multiple design scenarios, and its effectiveness was verified with a curved array.

provides a comprehensive overview of algorithms for synthesizing beam patterns for antenna arrays. These optimization technologies have specific application scenarios, and each technology has its own advantages and limitations. The algorithm proposed in this work can be improved by taking into account the mutual coupling effect between antennas in the arrays, and the proposed algorithm still needs to be tested for various antenna array configurations.

6. Conclusions

The particle swarm optimization algorithm is utilized for synthesizing array beams with multiple objectives, including multiple main lobes and several nulls in the radiation pattern. Based on scrutinizing classic PSO in the beamforming application, a modified PSO is proposed where the initial positions of particles are chosen based on the Taylor distribution instead of being randomly assigned. In addition, the fitness function incorporating multiple objectives is established. The combination of the new particle initialization and introduction of the new convergence condition enables the modified PSO to calculate beamforming weights for arrays with a large number of elements. The effectiveness of the proposed method is demonstrated with simulations and numerical results on four different scenarios, and its performance is compared with that of the classical PSO. The studied scenarios confirmed that the proposed algorithm is more efficient than the classical PSO in reaching convergence. It is potentially advantageous for solving beamforming problems with multiple objectives and sophisticated beamforming requirements. Finally, the proposed algorithm has also been used to optimize the directional pattern of a curved antenna array, and the results have verified the ability of the proposed algorithm to solve practical beamforming problems.