Non-uniformly spaced linear antenna array design by means of PEEC approach applying Cheetah optimization algorithm

Abstract

This paper presents the application of the Partial Element Equivalent Circuit (PEEC) approach, which is a full wave electromagnetic modelling technique for conductors embedded in arbitrary dielectrics based on equivalent circuits, to the optimal design of antennas with non-uniform spacing between the array elements. The design optimization problem is solved by means of the new nature-inspired Cheetah metaheuristic. The main aim of this paper is to introduce the Cheetah optimization algorithm to the electromagnetics and antenna community. The results are compared to two well-known optimization algorithms and to show the effectiveness of the proposed algorithm on a realistic benchmark problem.

Keywords

Antenna array Cheetah optimization algorithm genetic algorithm partial element equivalent circuit particle swarm optimization algorithm

1. Introduction

An antenna array is a set of similar antenna elements collectively operating as a single radiating element. The process of determining the parameters of an antenna array to obtain the required antenna radiation pattern is known as pattern synthesis [1]. The crucial challenge is to determine the optimum spacing between the elements and their excitations. The design dictates the performance in reducing side lobe levels and placing the nulls at prescribed locations while satisfying the constraints of assured gain and directivity [2]. When the antenna array directs its main lobe with enhanced gain in the desired direction, it forms side lobes and nulls in directions far away from the main lobe.

Antenna array synthesis usually requires an optimization step and constrained numerical optimization techniques have large application in this field. Indeed, the number of successful applications of nature-inspired algorithms to the optimization of linear antenna arrays has increased over the past few years, and includes techniques such as Ant-Lion Algorithm [1,2], Genetic Algorithm (GA) [3–5], Particle Swarm Optimization (PSO) [6–8], Grey Wolf Optimization (GWO) [9], among others [10]. However, most of these methods are quite dependent on the specific tuning of algorithmic control parameters. In this paper, a new swarm-based algorithm, Cheetah [11], is proposed for the optimization of linear antenna arrays. The Cheetah Based Algorithm (CBA) is a paradigm that mimics the social behaviour and hierarchy of African Cheetahs. It has no control parameters to be tuned since all needed ones are based on direct nature observations. Klein et al. (2018) have proposed CBA, and validated it against test functions obtained from the IEEE CEC2014 competition [11]. However, this is the first time that CBA is being applied to the optimization of antenna arrays, to the best of the authors’ knowledge. In particular, in this paper antenna element positions and feeding current amplitudes are optimized in order to obtain peak side lobe level (SLL).

This section has presented a brief introduction to linear antenna array design, the CBA, its applications in optimization problems and the main objective of this paper. The organization of the rest of the paper is as follows: the linear antenna array geometry, discussion about configuration and array equations are presented in Section 2. Section 3 provides a detailed description of CBA along with a flowchart outlining the steps of its implementation. Section 4 presents the test problem formulation and the validation of the obtained results by comparison with other nature-inspired metaheuristic algorithms. Section 5 presents the conclusions.

2. Antenna arrays

2.1. Non-uniformly spaced linear arrays

When a group of similar radiators or antennas is used to produce more than a single radiating source (antenna array), it is possible to obtain an antenna that has a higher gain and a radiation pattern that can be steered in any preferred direction [12]. In a linear antenna array configuration comprising N elements equally spaced along the x-axis, the corresponding array factor AF is governed by the relation given by Eq. ((1)) [15], $\begin{eqnarray}\displaystyle AF({\theta})=2\mathop{\sum }_{i=1}^{N}I_{n}\text{cos}(kx_{n}\cos ({\theta})+{\psi}_{n}) & & \displaystyle\end{eqnarray}$ (1) where I _n, _{$\psi_n$} and x _n are the excitation amplitude, phase and position of nth element in the array and k is the wave number given by 2π∕𝜆 where 𝜆 is the wavelength and θ is the azimuth angle. For a linear array, SLL minimization and null placement can be achieved either by optimizing the excitation amplitude and phase while maintaining uniform spacing as in the case of a conventional array or by optimizing the element spacing while assuming uniform amplitude and phase excitation.

The analysis of non-uniformly spaced linear arrays originated in 1960’s with the work of Unz [13], who developed a matrix formulation to obtain the current distribution necessary to generate a prescribed radiation pattern for a non-uniformly spaced linear array (NUSLA) with specified geometry. Following Unz idea, there is a division of NUSLA in two categories: thinned arrays, which are derived by selectively zeroing some elements of an initial equally spaced linear array (ESLA), and arrays with randomly spaced elements [14]. The second approach starts from a uniform linear array with element spacing of 0.5𝜆, and then the positions are perturbed to reach the optimal positions that produce the lowest side-lobe level, which is the approach applied in this work.

Normally in antenna array synthesis, the array elements are oriented parallel to each other and only one antenna type is used, e.g. linear dipoles with same element length, and it is quite common to apply the approximate Pattern Multiplication (PM) method to calculate the total radiated electric field intensity E _tot of the antenna array structure [15]. Instead in this paper, this problem is addressed by accurately calculating the total field of the antenna array by means of the partial equivalent electric circuit (PEEC) method.

2.2. Partial equivalent electric circuit

The Partial Equivalent Electric Circuit (PEEC) arises from inductance calculations by Ruehli in the 1970’s [16]. The main advantage applying the PEEC method [9] compared ,e.g., to the Finite Element Method (FEM) lies in the fact that the air volume needs not to be discretized into finite domains. In fact, due to the free-space Green’s functions the so-called Electric Field Integral Equation (EFIE) in the frequency domain describes the electric field intensity in the whole problem domain. To obtain the electric field intensity from the calculated current distribution along the wire antenna it is assumed that the one-dimensional stick elements can be interpreted as infinitesimal dipoles. Applying the superimposition principle to all stick elements computed by Eq. ((2)) [15], the total electric field intensity E _tot of the antenna array for far field approximations can be derived, and the directivity D (θ, 𝜙) of the antenna array can be computed by Eq. ((3)) [15], $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle dE_{{\theta}}=j{\eta}\frac{ke^{-jkR}}{4{\pi}R}I_{e}(x^{\prime }, y^{\prime },z^{\prime })\text{sin}{\theta}dz^{\prime }\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle D({\theta}, {\phi})=\frac{\left|\mathbf{E}_{tot}({\theta},{\phi})\right|^{2}2{\pi}r_{FF}^{2}}{{\eta}P_{rad}}.\end{eqnarray}$ (3) In Eq. ((2)) the distance between the source point and the observation point in free space is represented by R, 𝜂 is the intrinsic wave impedance and $I_e(x^{\prime}, y^{\prime}, z^{\prime})$ is the wire current of PEEC stick element with its corresponding source point coordinates. In addition, in Eq. ((3)) P _rad is the radiation power, r _FF is the far field observation sphere radius.

3. Cheetah algorithm

3.1. Cheetah based algorithm

Behavioural ecology studies performed by Durant et al. [17] form the basis of the Cheetah based algorithm (CBA), introduced by Klein et al. [11] in 2018. The formulas of this Section use the same symbology of the cited source, which also provides some further details and more biological context. In the context of optimization the interesting feature of cheetah populations is their arrangement in a social environment where age and gender determine most of the animal behaviour.

The algorithm mimics the social structure of such populations by incorporating demographic data extrapolated from [17]. The population is subdivided in age-groups: (a) from 0 to 1 year cheetas are considered as cubs, (b) adolescents from 1 to 2 years, and (c) adults are older than 2 years. Males and females have distinct living habits: while adult females live alone or with their cubs, males can live in groups with up to three individuals even in an adult phase. Furthermore, males and females have different life expectancies as shown in Table 1, which reports the mean and the variance of the expected life lengths for the two genders. Also the longest recorded living members of the population differ markedly according to gender, with the oldest female having a recorded life of 13.6 years while the oldest male had 7.8 years. For the sake of the algorithm the population can be considered as stationary since the demographic yearly rate of cheetahs is around 1, i.e. the rates of birth and death are almost equivalent. Seasonal parameters, e.g. rainfall, strongly influence the availability of food and thus the gazelle population which is the main food source for cheetahs.

The cornerstone of the CBA algorithm is to represent with sufficiently high fidelity the animal behaviour in order to mimic the successful environmental adaptation and evolutionary success of these predators and therefore the main environmental parameters were taken directly from the reported observed data. Those parameters, shown in Table 2, are the Serengeti plains area, the mean life according to gender and the proportional territory by gender. The only free user selectable parameter is the initial population size, N.

Table 1
List of parameters related to cheetahs behaviour

Parameter Value (mean ± variance)

Serengeti plain area [km²] 25,000

Mean female life expectation [years] 7.01 ± 12.27

Mean male life expectation [years] 3.60 ± 1.83

Parameter	Value (mean ± variance)
Serengeti plain area [km²]	25,000
Mean female life expectation [years]	7.01 ± 12.27
Mean male life expectation [years]	3.60 ± 1.83

3.2. Initialization

The initialization of the population in parameter space is based on a random uniform distribution and the gender of each member of the population is attributed prior to the evaluation of the individual fitness. The size of the female group N ⋅ n _f is defined according to the naturally occurring distribution which has 0.65 as mean and a standard deviation of 0.05, while the remaining population, up to N individuals, consists of males. $\begin{eqnarray}\displaystyle n_{f}=0.65+0.05\ast RndUniform(0,1). & & \displaystyle\end{eqnarray}$ (4) Furthermore, all initial population members defined as adults respecting the mean life information of Table I, since the reproductive characteristic is considered to be an interesting characteristic that needs to be preserved on the first iteration of the algorithm. The convergence criteria for the optimization loop are the maximum number of functions evaluations or the reaching of an objective function threshold. Any loop execution represents a one month-life of the population, i.e. the loop is performed 12 times to reach a one year time-interval,

3.3. Population movement

The first operation in the searching loop consists in creating a copy of the current population which is later used in order to determine if the new individuals are improved with respect to their ancestors. First, female members of the population are moved according to: $\begin{eqnarray}\displaystyle x_{f}(k+1)=x_{f}(k)+{\alpha}\ast [x_{best}(k)-x_{bestr}(k)] & & \displaystyle\end{eqnarray}$ (5) where x _f indicates a generic female member of population, x _bext is the optimal individual obtained by the algorithm up to iteration k, x _bestr is a randomly selected best female individual, 𝛼 is a random number generated from a uniform distribution in the range [0, 1]. This movement strategy aims at reproducing the learning capacity that each female could get from other females even if those females live in their own distant home range. Another significant behavioural trait mimicked by the algorithm is the natural repulsion between females, which have been observed to live in isolation (each one will taking care of her own home range), as weaker females are driven away by stronger ones. This behaviour is replicated by ranking females according to their fitness. Euclidean distance calculation among the females is performed to detected if there are any females in one other female home range, in which case the position of the weaker one will be changed according to: $\begin{eqnarray}\displaystyle x_{fi}(k)=x_{fi}(k)+{\beta}\ast fpa & & \displaystyle\end{eqnarray}$ (6) where x _fi is the weaker female population member, fpa is the standard territory area and 𝛽 is a random number generated with uniform distribution in the range [−1, 1]. The distance between all population members is calculated before the movements of the male individuals.

In CBA, two possible rules are applied in order to perform the male movements: (a) if there are free females, the male will move towards the closest free one, (b) otherwise males will move freely, like errants. These two behaviours are realized by implementing following rules: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle x_{mi}(k)=x_{mi}(k)+{\alpha}1\ast [x_{cfi}(k)-x_{mi}(k)+]\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle x_{mi}(k)=x_{mi}(k)+{\alpha}2\ast b_{x}\end{eqnarray}$ (8) where x _mi is the male population member; x _cfi is the closest female and 𝛼1 is a random number taken from a uniform distribution in the range [0, 1]. b _x is a boundary value randomly selected for each problem dimension and 𝛼2 is a random number from the uniform distribution in range [0, 1]. If there are no available females to mate, then all males will follow an errant behaviour, otherwise a newborn will be generated after the males movement. Each newborn consists in the linear combination of the mother’s best position ever and the father’s best position ever. Then the components (problem dimensions) of the newborn are permuted, forming new combinations for the newborns thus mimicking mutation. All new individuals have a group number assigned that is also assigned to their mother so that a family relationship (ancestorship) can be maintained.

Individuals of the two genders which are between 13 and 24 months old, i.e. the previously defined adolescents, tend to move in groups according to: $\begin{eqnarray}\displaystyle x_{ai}(k)=x_{ai}(k)+{\alpha}3\ast [x_{best}(k)-x_{bestgr}(k)] & & \displaystyle\end{eqnarray}$ (9) where x _ai is the adolescent member of the population, x _best is the current optimal solution, x _bestgr is the current best group member position so far and 𝛼3 is a random number generated from the uniform distribution in the range [0,1]. The rule is essentially the same used to update the positions for the female individuals, however the reference in this case is the adolescent group and there is no spacing since no home range has to be guaranteed in this case.

Cubs, i.e. newborn up to the subsequent twelve iterations of the algorithm, follow their mothers’ movement, simulating their initial learning process, thus, their movements are governed by: $\begin{eqnarray}\displaystyle x_{ci}(k)=x_{ci}(k)+{\alpha}4\ast [x_{fm}(k)-x_{ci}(k)] & & \displaystyle\end{eqnarray}$ (10) where x _ci stands for the cub population members, x _fm is the cubs mother’s position (due to ancestorship) and 𝛼4 is a random number generated with uniform distribution in the range [0, 1].

3.4. Fitness comparisons

After the movement stage, which has been performed for all members of the current population, the fitness of the members of the population is evaluated. This step can obviously be performed in parallel. For males which have moved according to an errant natural behaviour the position is always updated even if it is worse than the previous one, while the position of other population members, i.e. cubs, adolescents and females have their position updated only if the new fitness improve over the previous one. These two types of behaviour try to balance the exploratory and exploitative characteristics of the algorithms. Since the observed demographic yearly rate is around 1, a death algorithm will take place in order to ensure that the population remains stable around the initial value, with a male/female composition matching the biologically occurring one. At each iteration the current optimal position is checked against the previously available one and in case of improvement the new optimum will replace the former and will be used for the position updates. The flowchart of CBA is shown in Fig. 1.

Fig. 1.

CBA’s flowchart [11].

4. Problem formulation

The design problem considered here consists in a linear half-wavelength dipole antenna presented in detail in [15]. The E-plane radiation pattern specifications D _main0° = 9 at 𝜙 = 0° and HPWB = 63.64° have to be fulfilled under the assumption that the radiation intensity beyond the main lobe is minimized. The parameter space p is defined by distances between elements d _i, current I _i and by phase 𝜑_i for i = 1, …,4; the excitation of the centre is given by I ₀ = 1A and 𝜑₀ = 0°. The objective function reads: $\begin{eqnarray}\displaystyle f(\boldsymbol{p})=3-{\mu}(D_{Main0^{\circ }}(\boldsymbol{p}))-{\mu}(HPWB_{Main0^{\circ }}(\boldsymbol{p}))-{\mu}(SideInt(\boldsymbol{p})). & & \displaystyle\end{eqnarray}$ (11) Where 𝜇(f ( p )) refers to the nonlinear fuzzy weight function 𝜇 applied to the scalar objective function f of the parameters vector p. It should be noted that other objective functions for the same technical problems are indeed possible and the chosen one, ((11)), stems from a careful investigation of the characteristics of possible objective functions for linear antenna arrays reported in [15] Section 4.

The objective function delivers the fitness value of an antenna array problem, shown in Fig. 2, consisting of five non-uniformly spaced linear half-wavelength dipole antennas. Dipoles are placed along the x-axis and oriented in z-direction. The center element is located at the origin.

Fig. 2.

Antenna array consisting of five parallel oriented linear dipoles along the x-axis.

In this work, we compare the results from CBA with Particle Swarm Optimization algorithm (PSO) and Genetic Algorithm (GA). The parameters adopted for PSO and GA algorithm are presented in Table 2.

Table 2

List of parameters related to PSO and GA algorithm

Algorithm	Parameters	Value
PSO	c ₁, c ₂	2
	w _max	0.9
	w _min	0.2
	Population Size N	30
	Max. Iteration	100
GA	Generation	100
	Population Size N	30
CBA	Max. Iteration	100
	Population Size N	30

Figure 3 shows a comparison between the convergence speed of CBA with respect to the well-known GA and PSO algorithms. The achieved results, considering Eq. (11), lead to the directivity patterns presented in Figs 4 to 6 for GA, PSO and CBA, respectively. Table 3 shows the numerical comparisons between those results.

Fig. 3.

Comparisons GA, PSO and CBA’s result.

Fig. 4.

Radiation pattern from GA result.

Fig. 5.

Radiation pattern from PSO result.

Fig. 6.

Radiation pattern from CBA result.

Table 3

Antenna design result for comparisons

Algorithm	D _Main0°	HPWB _0°	𝜇(D _Main0°(p))	𝜇(HPWB _Main0°(p))	𝜇(SideInt (p))
Cheetah (CBA)	11.648	63.64	1	0.99807	1
GA	10.082	63.64	1	0.99807	0.99641
PSO	11.186	63.64	1	0.99807	0.99998

Although the radiation patterns appear similar at first sight, and all satisfy the specifications D _main0° = 9 at 𝜙 = 0^° and HPWB = 63.64° the results of Table 3 indicate that the solution obtained by CBA has a lower radiation intensity beyond the main lobe with respect to those obtained by PSO and GA.

Furthermore, as shown in Fig. 3, the results obtained by CBA were obtained with approximately 1/3 of the computational effort required by PSO and GA.

5. Conclusions

In this paper the recently developed Cheetah Based Algorithm (CBA) which proved very effective on synthetic benchmark functions is applied to an antenna design problem of engineering interest. Results show that CBA is superior to the well known PSO and GA algorithms both in terms of quality of the final obtained optimal solution as well as the computational effort required, which is about 1/3.

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