# GA-Aided Directivity in Volumetric and Planar Massive-Antenna Array Design

Bruno Felipe Costa, Taufik Abrão
Londrina State University (UEL),
Department of Electrical Engineering, Londrina, 86057-970, Parana, Brazil

###### Abstract

The problem of directivity enhancement, leading to the increase in the
directivity gain over a certain desired angle of arrival/departure (AoA/AoD),
is considered in this work. A new formulation of the volumetric array
directivity problem is proposed using the rectangular coordinates to describe
each antenna element, and the desired azimuth and elevation angles with a
general element pattern. Such a directivity problem is formulated to find the
optimal minimum distance between the antenna elements $d_{\text{min}}$ aiming
to achieve as high directivity gains as possible. An expedited implementation
method is developed to place the antenna elements in a distinctive plane
dependent on ($\theta_{0}$; $\phi_{0}$). A novel concept on optimizing
directivity for the uniform planar array (OUPA) is introduced to find a quasi-
optimal solution for the non-convex optimization problem with low complexity.
This solution is reached by deploying the proposed successive evaluation and
validation (SEV) method. Moreover, the genetic algorithm (GA) method was
deployed to find the directivity optimization solution expeditiously. For a
small number of antenna elements , typically $N\in[4,\dots,9]$, the achievable
directivity by GA optimization demonstrates gains of $\sim 3$ dBi compared
with the traditional beamforming technique, using steering vector for uniform
linear arrays (ULA) and uniform circular arrays (UCA), while gains of $\sim
1.5$ dBi is attained when compared with an improved UCA directivity method.
For a larger number of antenna elements , two improved GA procedures, namely
GA-marginal and GA-stall, were proposed and compared with the OUPA method.
OUPA also indicates promising directivity gains surpassing $30$ dBi for
massive MIMO scenarios.

###### keywords:

Directivity , Antenna Array , 5G , Optimization , Omnidirectional , Genetic
Algorithm (GA) , Optimal Uniform Planar Array (OUPA)

††journal: Arxiv (https://arxiv.org/)

##  1 Introduction

Array antennas can provide many advantages in flexibility and scalability
concerning conventional antennas. This is related to the capacity of array
pattern reconfiguration and adaptability. Such adaptability enables various
applications, including radar communications, satellite communications,
wireless communications, radio-astronomy, remote sensing, and direction of
arrival (DoA) estimation. Optimally designed arrays are used to obtain high
directivity patterns, improving the array performance.

Uniform array configurations, such as uniform linear (ULA), planar (UPA),
rectangular (URA), circular or cylindrical (UCA) antenna arrays, have been
deployed to improve the performance of massive antenna-based 5G and B5G
communication systems. Since planar antenna array can deploy a higher number
of antenna elements at a limited physical space and capture the beamspace in
both the horizontal and vertical directions in three-dimensional (3D)
propagation, the development of different planar array configurations, such as
URA and UCA, have become a natural choice among the different possible array
geometries, representing a great interest in the mmWave massive MIMO system
designs.

Array directivity optimization remains an open issue, especially in cases
where the antenna-element position is considered as an optimization variable.
Indeed, the choice of antenna-elements position as a parameter to be optimized
can be justified by the concept of array virtualization, in which antenna-
elements cooperate among each other, forming a virtual antenna array (VAA)
with better directivity performance that the original physical antenna array.

Recent works on array optimization have designed linear sparse array antennas.
For instance, the algorithm proposed in [1] jointly and deterministically
maximizes the aperture efficiency and directivity. Besides, [2] proposes a
planar circular sparse array antenna design, where the positions and
dimensions of the radiators are jointly optimized. In [3] an optimal circular
antenna array for maximum sidelobe levels (SLLs) reduction was investigated
using a metaheuristic approach called player algorithm. An optimization of
sidelobe level and aperture efficiency for aperiodic array antennas was
proposed in [4]. Multiobjective optimization (MOO) is deployed in [5] to
efficiently and effectively design subarrays in linear antenna array; the
optimization was made using a memetic differential evolution (mDE) algorithm.
Following the same research direction, the authors in [6, 7] used MOO and
metaheuristic methods to optimize the beampattern in collaborative array
beamforming design. In [8], the MOO technique is deployed to discuss the
planar antenna array design under different optimization criteria, such as the
side lobe, beam steering quality, minimum power beam width, and maximum
directivity, as a function of element lengths and distances between these
elements. A fundamental question is exploring the Pareto front modeling
optimal trade-offs among multiple conflicting objectives, including the
structure size, its performance, and the minimization of cost. The well-known
heuristic algorithm nondominated sorting genetic algorithm II (NSGA-II) has
been used for this purpose. Many other works are deploying MOO and/or
metaheuristics solutions, such as [9], [10], [11], [12] and [13].

In [14], the optimal radiation pattern is generated using a spherical phased
array. In contrast, in [15], a technique based on changes in the subspace is
deployed to improve the directivity of the omnidirectional UCA. Moreover, the
optimum directive beamformer has been proposed in [16], by deploying the
concept of generalized directivity maximization; the concept of directivity is
based on the mean beam power pattern, which is crucial in assessing and
optimizing the performance of directivity arrays subjected to array
imperfections and sensor mismatch. Recently, by adopting a 3D wireless channel
model, and taking into account both the azimuth and elevation angles of
departure (AoD), the authors of [17] derive the probability density function
for the distances between randomly distributed users and the BS antenna array,
considering three URA, ULA and UCA uniform array topologies. Also, they
derived closed-form expressions for the squared inner product of different
channel vectors, facilitating the interuser interference analysis.

Against this background, the current work proposes a new methodology to
maximize the directivity of omnidirectional volumetric antenna arrays. By
assuming a uniform planar array (UPA) confined on a specific plane with a
minimum distance between the antenna elements optimized, the proposed method
takes in hand the successive evaluation and validation (SEV) procedure. To
demonstrate effectiveness, the proposed method is implemented by deploying
evolutionary heuristic optimization; hence, to implement the directivity
optimization methodology under a large number of antennas in planar arrays, we
suggested two variants of the genetic algorithm: GA-marginal and GA-stall,
both made the GA a promising optimization tool to solve UPA directivity
problem in such large scale antenna scenario. An exciting finding of the
proposed methodology is that a plane space constraints the antenna-element
positioning solutions as a function of the desired elevation and azimuth
angles.

The contribution of this work is fourfold:

  * a)

formulation of optimization problem for directivity maximization; the novelty
of this design is brought by the optimization of each antenna-element position
bounded by a volumetric constraint on the search space, which is given by a
parallelepiped with a central point at the origin and all the points bounded;

  * b)

a geometric interpretation for the solution in case of omnidirectional
scenarios, more specifically, this interpretation results in restraining the
position of each antenna element to a plane equation dependent on the desired
angles ($\theta_{0},\phi_{0}$);

  * c)

using a novel concept based on the UPA geometry, an expedited implementation
method is developed consisting in placing the antenna elements in a
distinctive plane dependent on $\theta_{0}$ and $\phi_{0}$; hence, using the
geometry properties of the UPA, a directivity optimization problem was
formulated to find the optimal minimum distance between the antenna-elements
$d_{\text{min}}$ aiming to achieve as high directivity gains as possible;

  * d)

to corroborate the effectiveness of the proposed method, a genetic algorithm
(GA) is deployed to implement the directivity enhancement method with low
complexity; when compared with a conventional geometric design, the proposed
method has demonstrated a significant improvement in directivity gains on the
desired angle of departure (DoD).

The remainder of the paper is organized as follows. Section 2 describes the
concept of the radiation pattern, directivity optimization via position
antenna-element placement, and the adopted system model. Section 3 formulates
the directivity optimization problem, with particular attention on the
optimization problem for the omnidirectional case; besides, geometric
interpretation is evoked, and then the problem is recast in a simplified way.
For comparison purposes in terms of directivity, different geometric array
structures are considered. A novel directivity optimization method applied to
the UPA geometries is devised in Section 4 as an effective low-complexity
optimization method. Numerical results are developed in Section 5 by deploying
heuristic GA as a practical implementation tool supporting our finding on the
UPA-basis antenna array directivity optimization technique. Concluding remarks
are pointed out in Section 6.

##  2 Radiation Pattern and Directivity via Element Positioning

For non-isotropic antennas, the directivity is the level of radiated signal
power in a specific direction. This is especially important for applications
such as beamforming in massive MIMO 5G wireless communications. Fig. 1 depicts
the general concept of directivity enhancement via an array of antenna
elements with low directivity; the constructive and destructive interference
can be guided, by changing the position and/or phase of each element, aiming
to achieve a high directivity antenna pattern [18]. In this work, the
omnidirectional antenna-element configuration has been chosen to implement the
array directivity enhancement, which is a more realistic scenario compared to
the isotropic sources, while remaining a suitable configuration to be
analytically analyzed due to its simple element factor, expressed by
$\Upsilon(\theta)=\cos{\theta}$. Furthermore, the applicability of this
configuration is interesting in many fields. Such configuration dramatically
simplifies the analytical directivity expression, while the non-convex
directivity optimization problem remains a special issue.

###  2.1 Radiation Pattern of an Antenna Array

An antenna array is a grouping of $N$ antennas whose geometry can assume many
different forms, such as uniform linear, circular, rectangular, non-uniform
grouping, and so forth; moreover, all elements in the array work jointly to
increase the directivity of the arrangement. The entire field of an antenna
array is obtained by multiplying the field of a single element
$\Upsilon_{e}(\theta,\phi)$ and the array factor $\Upsilon_{a}(\theta,\phi)$;
this is called the radiation pattern of an antenna array and can be written
as:

$\Upsilon(\theta,\phi)=\Upsilon_{e}(\theta,\phi)\Upsilon_{a}(\theta,\phi)$ (1)

The element factor (EF) is the radiation pattern of a single-element antenna,
and the model must resemble the pattern of a real antenna. Using spherical
coordinates, the EF is periodic in $\theta$ and $\phi$ angles. Thus, the
radiated power of an arbitrary antenna can be described using Fourier
analysis, i.e. it can be written using a linear combination of powers of
cosine and sine functions. Hence, the EF can be written:

$\Upsilon_{e}(\theta)=\sin^{u}{(\theta)}\cos^{v}{(\theta)}$ (2)

Figure 1: Directivity representation in a) isotropic source, b)
omnidirectional source, and c) via constructive and destructive interferences
of low-directivity sources.

The array factor (AF) describes a combination of radiating elements in an
array without considering the element radiation pattern. Thus, the array
factor can be interpreted as the radiation pattern by replacing the actual
elements by the isotropic (point) sources, i.e.,
$\Upsilon_{e}(\theta,\phi)=1$. The AF has a dependency in terms of position,
relative phase and relative amplitude of each antenna element and is given by:

$\Upsilon_{a}(\theta,\phi)=\sum\limits_{n=1}^{N}A_{n}e^{j(\alpha_{n}+k\boldsymbol{p}_{n}\cdot\boldsymbol{a}_{p})}$
(3)

where $N$ is the number of antenna elements, $A_{n}$ is the relative amplitude
of $n$-th antenna element, $\alpha_{n}$ is the relative phase of $n$-th
antenna element, $\boldsymbol{p}_{n}$ is the position vector of $n$-th antenna
element; $k$ is the wave number, and $\boldsymbol{a}_{p}$ is the unit vector
of observation point in spherical coordinates, given by:

$\boldsymbol{a}_{p}=\sin(\theta)\cos(\phi)\hat{i}+\sin(\theta)\sin(\phi)\hat{j}+\cos(\theta)\hat{k}$
(4)

The position vector can be written as a combination of the elements in the 3-D
cartesian axis:

$\boldsymbol{p}_{n}={x_{n}}\hat{i}+{y_{n}}\hat{j}+{z_{n}}\hat{k}$ (5)

In this work, we have considered the position of each antenna element of the
array as the variable to be optimized, which the matrix can mathematically
defined:

${\boldsymbol{P}=\left[\begin{array}[]{c}\boldsymbol{p}_{1}\,\,\boldsymbol{p}_{2}\,\,\ldots\,\,\boldsymbol{p}_{N}\end{array}\right]^{T}}$
(6)

where the position vector $\boldsymbol{p}_{n}$ defines the localization of the
$n$-th antenna element in the 3-D space.

###  2.2 Directivity for Arbitrary Volumetric Antenna Arrays

Directivity can be defined as the level of irradiation signal power in a
specific direction to the detriment of others. This is especially important
for some applications, such as beamforming in non-isotropic massive MIMO
antennas in 5G wireless communications. Antenna directivity can be written as
the ratio between the radiation intensity in a desired angle and the sum of
the radiation intensity in all the other directions:

$\mathcal{D}(\theta_{0},\phi_{0})=\frac{|\Upsilon(\theta_{0},\phi_{0})|^{2}}{\frac{1}{4\pi}\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}|\Upsilon(\theta,\phi)|^{2}\sin{(\theta)}d\theta
d\phi}$ (7)

where $|\Upsilon(\theta,\phi)|^{2}$ is the radiation intensity. Besides, for
an arbitrary array with the general element pattern
($\cos^{v}{\theta}\sin^{u}{\theta},\,\,\forall\,\,u,v$), the directivity
expression can be defined as [18]:

$\displaystyle\begin{split}&\mathcal{D}(\theta_{0},\phi_{0},u,v)=\frac{f_{1}}{f_{2}}=\frac{\sin^{2u}{(\theta_{0})}\cos^{2v}{(\theta_{0})}\sum\limits_{\begin{subarray}{c}m,n=1\end{subarray}}^{N}A_{n}A_{m}\xi_{mn}\cos{\left[\Omega_{mn}\right]}}{\mathcal{I}_{n}(u,v)+\mathcal{I}_{m}(u,v)}\end{split}$
(8)

where

$\displaystyle\begin{split}&\mathcal{I}_{n}(u,v)=\sum\limits_{n=1}^{N}A_{n}^{2}\left[\frac{1}{8}((-1)^{2v}+1)\mathcal{B}(u+1,v+\frac{1}{2})\right]\\\
&\mathcal{I}_{m}(u,v)=2\,(-1)^{(v+2u)}\sum\limits_{\begin{subarray}{c}n,m=1\\\
m\neq n\\\
n>m\end{subarray}}^{N}\sum\limits_{\kappa=0}^{u}A_{n}A_{m}\binom{u}{\kappa}\cos{(\alpha_{mn})}\frac{\partial^{2(v+u-\kappa)}}{\partial
z_{mn}^{2(v+u-\kappa)}}\left[\frac{\sin(k\sqrt{\beta^{2}+z_{mn}^{2}})}{k\sqrt{\beta^{2}+z_{mn}^{2}}}\right]\\\
&\Omega_{mn}=\Omega(\boldsymbol{p}_{n},\boldsymbol{p}_{m},\alpha_{n},\alpha_{m},\theta_{0},\phi_{0})=k\,[x_{nm}\sin{\theta_{0}}\cos{\phi_{0}}+y_{nm}\sin{\theta_{0}}\sin{\phi_{0}}+z_{nm}\cos{\theta_{0}}]+\alpha_{nm}\\\
&\boldsymbol{p_{n}}=x_{n}\hat{i}+y_{n}\hat{j}+z_{n}\hat{k},\hskip
14.22636pt\beta=\sqrt{x_{nm}^{2}+y_{nm}^{2}},\hskip
14.22636ptx_{nm}=(x_{n}-x_{m}),\hskip 14.22636pty_{nm}=(y_{n}-y_{m}),\\\
&\hskip 28.45274ptz_{mn}=({z_{n}}-{z_{m}})\\\
&\xi_{mn}=\begin{cases}1\qquad\qquad\,\,m\neq n\\\
\frac{1}{2\cos{\left[\Omega_{mn}\right]}}\quad m=n\end{cases},\hskip
14.22636pt\mathbb{R}e\\{v\\}>-\frac{1}{2},\hskip
14.22636pt{\mathbb{R}e}\\{u\\}>-1,\hskip
14.22636pt\alpha_{mn}=({\alpha_{n}}-{\alpha_{m}}).\end{split}$ (9)

where $\boldsymbol{p_{n}}$ is the position vector of $n$-th antenna element;
$\alpha_{n}$ is the phase of $n$-th antenna element; $A_{n}$ is the amplitude
of $n$-th antenna element; $\theta_{0}$ is the desired elevation angle, where
directivity is evaluated; $\phi_{0}$ represents the desired azimuth angle,
where directivity is evaluated; and $k$ is the wave number of the
transmission.

##  3 Directivity Optimization Problem in Omnidirectional Scenarios

Given a desired angle of departure $(\theta_{0},\phi_{0})$ over an antenna
array composed of $N$ elements, we are looking for the parameter values of an
array configuration that result in the directivity maximization in such an
AoD.

We start the analysis assuming non-zero-phase and the non-unitary amplitude in
each antenna; hence, we will solve the proposed problem to obtain a geometric
configuration for each desired angle. For typical applications, the desired
direction changes regularly, which becomes a problem, given that changes in
the positions of antennas are physically impossible. However, in this
formulation, the phase element was not considered, leaving space to adopt
virtualization techniques, which highly depend on the phase elements and will
be the subject of our future research.

Based on the directivity expression provided in eq. (8), we state the
following optimization problem:

$\displaystyle\underset{\boldsymbol{P}}{\text{maximize}}$
$\displaystyle\mathcal{D}(\theta_{0},\phi_{0})=\frac{f_{1}}{f_{2}}$ (10)
subject to
$\displaystyle\boldsymbol{p}_{n}\preceq\boldsymbol{p}_{\max}\quad\text{for}\quad
n=1,2,3,\dots,N$

where the negative semi-definite
$\boldsymbol{p}_{n}\preceq\boldsymbol{p}_{\max}$ defines the 3-D
parallelepiped volume bounded by $[x_{\max},y_{\max},z_{\max}]$. Besides, the
optimization problem in (8) can be rewritten as a minimization problem:

$\displaystyle\underset{\boldsymbol{P}}{\text{minimize}}$
$\displaystyle\frac{f_{2}(\boldsymbol{P},\theta_{0},\phi_{0})}{f_{1}(\boldsymbol{P},\theta_{0},\phi_{0})}$
(11) subject to
$\displaystyle\boldsymbol{p}_{n}\preceq\boldsymbol{p}_{\max}\quad\text{for}\quad
n=1,2,3,\dots,N$

Therefore, the objective of this problem can be reinterpreted as minimizing
$f_{2}(\boldsymbol{P})$. At the same time, simultaneously maximize
$f_{1}(\boldsymbol{P})$ subject to the bounds in the space configuration,
given, for instance, by the 2-norm.

The omnidirectional scenario has been selected due to the concise expressions,
analysis, and manageable complexity while the applicability remains in many
exciting fields. The omnidirectional radiated power is also scattered
symmetrically but not equally in all directions, creating a 3-D torus
radiation pattern, which can be formulated as the expression
$\Upsilon(\theta)=\cos{\theta}$ for all azimuth angles ($\phi$). Hence,
considering the element pattern given in eq. (2), for the omnidirectional
scenario, we set $u=0$ and $v=1$. Such condition simplifies the auxiliary
functions as:

$\displaystyle
f_{1}^{\textsc{o}}(\theta_{0})=\cos^{2}{(\theta_{0})}\sum\limits_{\begin{subarray}{c}m,n=1\end{subarray}}^{N}A_{n}A_{m}\xi_{mn}\cos{\left[\Omega_{mn}\right]}$
(12)

and

$\displaystyle
f_{2}^{\textsc{o}}=\sum\limits_{n=1}^{N}\frac{A_{n}^{2}}{6}-2\sum\limits_{\begin{subarray}{c}m,n=1\\\
m\neq n\\\ n>m\end{subarray}}^{N}A_{n}A_{m}\frac{\partial^{2}}{\partial
z_{mn}^{2}}\left[\frac{\sin(k\sqrt{\beta^{2}+z_{mn}^{2}})}{k\sqrt{\beta^{2}+z_{mn}^{2}}}\right]$
(13) $\displaystyle\text{where}\hskip
85.35826ptd_{mn}=k\sqrt{\beta^{2}+z_{mn}^{2}}$ (14)

Generically, $d_{mn}$ represents the Euclidean distance between the $m$-th and
$n$-th antenna element in the 3-D space, given by
$d_{mn}=k\sqrt{x_{mn}^{2}+y_{mn}^{2}+z_{mn}^{2}}$.

In Eq. (10), we are interested in maximizing $f_{1}^{\textsc{o}}$ while
minimize $f_{2}^{\textsc{o}}$. For this purpose, we can verify in Eq. (12)
that the only way to ensure these conditions consists in maximizing
$\cos{\left[\Omega_{mn}\right]}$, i.e.

$\displaystyle\Omega_{mn}=c_{1}\pi,\quad\text{for
}\,\,\frac{c_{1}}{2}\in\mathbb{Z}\quad c_{1}\,\,\text{even}$ (15)

Using the definitions in (9), and satisfying the condition above, for
$\Omega_{mn}$ we have:

$\displaystyle\begin{split}&x_{nm}\sin{\theta_{0}}\cos{\phi_{0}}+y_{nm}\sin{\theta_{0}}\sin{\phi_{0}}+z_{nm}\cos{\theta_{0}}\,\,=\,\,\frac{c_{1}\pi-(\alpha_{n}-\alpha_{m})}{k}\end{split}$
(16)

Eq. (16) can be interpreted as the plane equation containing all the vectors
of position difference
($\boldsymbol{p}_{mn}=\boldsymbol{p}_{n}-\boldsymbol{p}_{m}$), to fulfill this
condition entirely it is necessary all vectors be restrained in the same plane
of the differences position vectors since the constant $c_{1}$ is not unique.
For each value, we have a different plane, the solution of this condition is
given by infinite parallel planes. However, the combination of position
vectors in these parallel planes does not fulfill the condition, given that
the difference vector for points in different parallel planes will result in a
vector difference in another plane.

Moreover, minimizing $f_{2}^{\textsc{o}}$ results in a more complicated task,
given that the directivity problem in (7) requires minimization of the
summation of all terms in the right-hand side of (13), instead of minimizing
each term. Hence, the $f_{2}^{\textsc{o}}$ minimization problem can be recast
in an alternative version as:

$\displaystyle\underset{\boldsymbol{P}}{\text{maximize}}\quad$
$\displaystyle\sum\limits_{\begin{subarray}{c}m,n=1\\\ m\neq n\\\
n>m\end{subarray}}^{N}A_{n}A_{m}\frac{\partial^{2}}{\partial
z_{mn}^{2}}\left[\frac{\sin(k\sqrt{\beta^{2}+z_{mn}^{2}})}{k\sqrt{\beta^{2}+z_{mn}^{2}}}\right]$
(17)

Besides, the second derivative in (17) can be re-written as:

$\displaystyle\begin{split}\frac{\partial^{2}}{\partial
z_{mn}^{2}}\left[\frac{\sin(\sqrt{\beta^{2}+z_{mn}^{2}})}{\sqrt{\beta^{2}+z_{mn}^{2}}}\right]\,\,=\,\,&\,\,\frac{(\beta^{2}-2z_{mn}^{2})\cos{d_{mn}}}{d_{mn}^{4}}\,\,\\\
&\,\,-\frac{\left[(\beta^{2}-2)z_{mn}^{2}+\beta^{2}+z_{mn}^{4}\right]\sin{d_{mn}}}{d_{mn}^{5}}\end{split}$
(18)

which is function of the $x_{mn}$, $y_{mn}$ and $z_{mn}$, and related to
$\beta$ and $d_{mn}$ through the constraints in eq. (8) and (14),
respectively. Notice that in this formulation, the omnidirectional
configuration has influence only in (18). For another scenarios, i.e.,
different values of $u,v\neq 0$, the only change will be the objective
function (OF) in (17), where the order and number of derivatives dependent on
the values of $u,v\in\mathbb{Z}_{+}$.

###  3.1 Recasting the Simplified Omnidirectional Problem

One of the most important constraints resulting by the omnidirectional
scenario is given in eq. (16), and where the values of $c_{1}$ can be chosen
arbitrarily; hence, aiming to simplify the analysis we set $c_{1}=0$.
Furthermore, considering all null phases, for the $n$-th and $m$-th antennas,
we obtain:

$\displaystyle\begin{split}&\sin{\theta_{0}}\cos{\phi_{0}}x_{n}+\sin{\theta_{0}}\sin{\phi_{0}}y_{n}+\cos{\theta_{0}}z_{n}=0\\\
&\sin{\theta_{0}}\cos{\phi_{0}}x_{m}+\sin{\theta_{0}}\sin{\phi_{0}}y_{m}+\cos{\theta_{0}}z_{m}=0\\\
\end{split}$ (19)

By subtracting both previous expressions, one can obtain:

$\displaystyle
z_{nm}=\tan{\theta_{0}}(\cos{\phi_{0}}x_{nm}+\sin{\phi_{0}}y_{nm})$ (20)

The restriction (20) can be incorporated into (18), finally resulting in a
simplified optimization problem:

$\displaystyle\underset{\boldsymbol{x},\boldsymbol{y}}{\text{minimize}}$
$\displaystyle\mathcal{G}(\boldsymbol{x},\boldsymbol{y})=-\sum\limits_{\begin{subarray}{c}m,n=1\\\
m\neq n\\\ n>m\end{subarray}}^{N}\mathcal{F}(x_{mn},y_{mn})$ (21) subject to
$\displaystyle|x_{mn}|\,\leq\,x_{\max},\quad|y_{mn}|\,\leq\,y_{\max}\quad\text{for}\quad
n=1,2,3,\dots,N$

with OF defined by (18), and substituting
$\beta^{2}=\frac{d_{mn}^{2}}{k}-z_{mn}^{2}$, results:

$\displaystyle\begin{split}\mathcal{F}(x_{mn},y_{mn})\,\,=\,\,&\frac{\left[(d_{mn}^{2}-3k)z_{mn}^{2}+d_{mn}^{2}\right]\sin{d_{mn}}}{kd_{mn}^{5}}-\frac{(d_{mn}^{2}-3kz_{mn}^{2})\cos{d_{mn}}}{kd_{mn}^{4}}\end{split}$
(22)

The function $\mathcal{G}$ can be rewritten using the definition in (13),
implying the following relation:

$\displaystyle
f_{2}^{\textsc{o}}=\sum\limits_{n=1}^{N}\frac{A_{n}^{2}}{6}+2\mathcal{G}$ (23)

Since $f_{2}^{\textsc{o}}$ represents the denominator integral in (7), for the
omnidirectional scenario this values must be positive; therefore, the function
$\mathcal{G}$ has a lower bound for each value of $N$, given by:

$\displaystyle\mathcal{G}_{\text{bound}}^{N}=-\sum\limits_{n=1}^{N}\frac{A_{n}^{2}}{12}$
(24)

The bound on $z_{mn}$ is implicit, having dependency on $x_{\max}$,
$y_{\max}$, and on the desired angle of departure $(\theta_{0},\phi_{0})$. The
solution of (21) give us the optimal coordinates $x_{n}$ and
$y_{n},\,\,\forall n$ which can be used to find the optimal $z_{n},\,\,\forall
n=1,\ldots,N$ coordinate from (20).

###  3.2 Geometric Planar Arrays Comparison

In this section, different geometric planar array structures are considered
for a directivity comparison purpose. It is shown the importance of setting
the adequate $d_{\text{min}}$ and the associated area occupied by the array,
aiming to exploit the potential of the directivity improvement. All the
selected arrays are planar, aiming to preserve the constraint in (20). Adopted
parameter values and antenna structure include:

Parameter | Adopted Values
---|---
# antennas | $N=16$
Normalized wavelength |  $\lambda=1$ [m]
Desired direction | $\theta_{0}=\phi_{0}=\pi/4$
uniform planar array | UPA
uniform circular array | UCA
uniform hexagonal planar array | UHPA

The selected uniform planar array geometries with $N=16$ elements are depicted
in Fig. 2 with the minimum distance between antenna elements $d_{\text{min}}$
represented by red arrows. Fig. 2(d) exhibits the directivity dependence
regarding the increasing $d_{\text{min}}$ values for each selected uniform
array geometries.

(a) Uniform Planar Array (UPA) (b) Uniform Circular Array (UCA)

(c) Uniform Hexagonal Planar Array (UHPA)

(d) Directivity versus the minimum distance for the three different geometry
arrays: UPA, UCA, and UHPA

Figure 2: Different planar array geometries and respective attained
directivity with $N=16$ elements on the desired plane defined by
$\theta_{0}=\phi_{0}=\pi/4$.

The planar and the hexagon geometry attained superior directivity than the
circular array; the optimal value of $d_{\text{min}}$ for both structures are
close to each other and significantly smaller than the circular array
$d_{\text{min}}$ value. For this application, the UCA has demonstrated do not
be suitable and will be suppressed hereafter in further analyses.

These illustrative results express the importance of setting the adequate
$d_{\text{min}}$ to exploit the directivity enhancement’s potential. However,
the area occupied by the array is also essential, since compact antenna arrays
are more valuable. Although the value of $d_{\text{min}}$ is smaller for the
UPA, the area is not necessarily smaller than the UHPA. To investigate this
aspect, Fig. 3 depicts directivity and the normalized area given by the convex
hull of the arrangement of points for both promising array structures; in (a)
the value of directivity and in (b) the value of area occupied by the convex
hull of the array, selecting the optimal value of $d_{\text{min}}$. It is
apparent in both graphics the remarkable directivity performance proximity for
both structures and respective normalized areas with the increase in the
number of antennas. The area occupied by the convex hull for small values of
antennas are very similar; however, for the maximum directivity configuration
and under a significant number of antenna elements, the UPA demonstrated a
better performance covering a smaller region with the increasing of the
antennas; besides, the points of the UHPA where the performance are
significantly better is given when the hexagon formed by the array is perfect.
Therefore, on can conjecture that both configurations have a very similar
directivity performance. At the same time, for a significant number of
antennas, the UPA surpasses the UHPA directivity vs. area performance tradeoff
unless the hexagon formed by the UHPA is perfect.

(a) Directivity vs. the number of antennas increasing $(N)$ (b) Array area vs.
the number of antennas $(N)$ increasing.

Figure 3: UPA against UHPA in terms of directivity and area, both with optimal
setting of $d_{\text{min}}$.

Given the attained near directivity between UPA and UHPA, both geometries
could be selected without losses in terms of directivity gain and occupied
area. Therefore, the structure selected for the optimization method in the
next section is the UPA, due to the compactness in describing the antenna
elements positioning over a rectangular coordinates system.

##  4 Optimizing the Position of UPA Elements

This section describes the proposed method to achieve high directivity using
the regular planar array structure. This method provides reasonable solutions
for the proposed optimization problem of Eq. (21) with low computation effort
considering the high-complexity operation of searching the geometric locus.

The proposed method consists in assuming a given number of waves ($k$), a
desired direction ($\theta_{0}$,$\phi_{0}$), and a number of antennas ($N$),
finding the optimal UPA on the desired plane defined by (20) aiming to attain
a remarkable improvement in the directivity in a desired direction. The most
critical parameter to be defined to achieve this goal is $d_{\text{min}}$. The
following describes a procedure for finding this parameter for a generic UPA.
Firstly, we can consider the positioning of the antennas on the $xy$ plane,
considering a UPA with $N_{1}\times N_{2}$ antenna elements regularly
distributed, as in Fig. 4.

Figure 4: UPA elements disposition and labeling for $N_{1}\times N_{2}$
antennas.

From this antenna-elements distribution, one can define the distance between
two generic elements, i.e, $d_{mn}$ for $m\neq n$, which is an important
parameter on the use of (22). The distance $d_{mn}$ can be formulated as:

$\displaystyle
d_{mn}=\begin{cases}\frac{|m-n|d_{\text{min}}}{N_{1}}&\text{if}\,\,\psi_{1}=0\\\
|m-n|d_{\text{min}}\quad&\text{if}\,\,\psi_{2}=0\quad\text{and}\quad|m-n|<N_{1}\\\
d_{\text{min}}\sqrt{\psi_{1}^{2}+\psi_{2}^{2}}\quad\,\,&\text{c.c}\end{cases}$
(25)

where:

$\displaystyle\begin{split}&\psi_{1}=\text{mod}(m-1,N_{1})-\text{mod}(n-1,N_{1});\quad\psi_{2}=\left\lfloor\frac{m-1}{N_{1}}\right\rfloor-\left\lfloor\frac{n-1}{N_{1}}\right\rfloor\end{split}$
(26)

To allocate the array on the desired plane without altering the distances, it
is possible to use a rotation matrix considering the desired direction,
defined by:

$\displaystyle\boldsymbol{R}_{\boldsymbol{v}}=\left[\begin{array}[]{ccc}\sin^{2}{\phi_{0}}\mu_{\gamma}+\cos{\gamma}&-\cos{\phi_{0}}\sin{\phi_{0}}\mu_{\gamma}&-\cos{\phi_{0}}\sin{\gamma}\\\
-\sin{\phi_{0}}\cos{\phi_{0}}\mu_{\gamma}&\cos^{2}{\phi_{0}}\mu_{\gamma}+\cos{\gamma}&-\sin{\phi_{0}}\sin{\gamma}\\\
\cos{\phi_{0}}\sin{\gamma}&\sin{\phi_{0}}\sin{\gamma}&\cos{\gamma}\end{array}\right]$
(30)

where $\gamma=\operatorname{acos}{|\cos{\theta_{0}}|}$ and
$\mu_{\gamma}=(1-\cos{\gamma})$; notice that for $\theta_{0}=0$, the
restriction (20) becomes the xy-plane and this rotation matrix becomes an
identity matrix. The position matrix of the array on the desired plane can be
expressed as:

$\displaystyle\widehat{\boldsymbol{P}}=\boldsymbol{P}\boldsymbol{R}_{\boldsymbol{v}}$
(31)

Besides, the variable $z_{mn}$, which consists in a combination of
subtractions on the last column of $\widehat{\boldsymbol{P}}$ can be described
as:

$\displaystyle
z_{mn}=-(\psi_{1}\cos{\phi_{0}}\sin{\gamma}+\psi_{2}\sin{\phi_{0}}\sin{\gamma})$
(32)

Now, with the parameters $d_{mn}$ and $z_{mn}$ available, the OF analytical
expression in (21) can be evaluated; however, this is a non-convex function of
one variable. At this point, the directivity optimization problem can not be
solved using convex optimization tools. On the other hand, notice that finding
the global optimum of this function, either by exhaustive search or by quasi-
optimal evolutionary heuristic methods, the optimal UPA to be placed on the
desired plane can be found, given the advantage of allocating the antenna-
elements on the same plane, while finding the optimal general UPA in terms of
directivity in that plane.

Remark 1: The general methodology is focused on finding the optimal position
of each antenna element that maximizes the antenna array directivity. In this
section, we have developed a solution confined into a specific plane, where
the antenna-elements position on this plane resembles a sequence of
equilateral triangles; hence, the idea of finding the optimal UPA antenna-
elements placement in the such plane is described by the position matrix, Eq.
(31), and solved applying the simplified optimization problem, Eq. (21).

###  4.1 Successive Evaluation and Validation (SEV)

With the $d_{mn}$ and $z_{mn}$ values, the OF of the directivity optimization
problem in (21) for the UPA can be expressed and therefore emerges the
necessity of finding the optimal value for $d_{\text{min}}$ that enhances the
array directivity. Based on numerical results, we concluded that the first
local minimum of the OF in (21) is also the global optimum. Fig. 5 depicts the
value of $\mathcal{G}$ for different values of $N_{1}$ and $N_{2}$.

Figure 5: Values of $\mathcal{G}$ against $d_{\text{min}}$ values for
different configurations of $N_{1}\times N_{2}$

As suspected, for all analyzed $N_{1}\times N_{2}$ configurations, the first
minimum local is also the minimum global. Taking advantage of this aspect, we
propose in Algorithm 1 the successive evaluation and validation (SEV) method,
which basically is a line search procedure, evaluating the function at the
point ($d_{0}=c$), with an increment closely to the origin ($0^{+}$); then, an
increment is computed again for $(d_{1}=d_{0}+c$). Hence, if the increment
brought a decreasing of the OF, i.e., $\mathcal{G}_{n+1}<\mathcal{G}_{n}$,
then the process is repeated; if not the point $d_{n}$ is declared the global
optimum.

Algorithm 1 SEV – Successive Evaluation and Validation

1: Input: $c$

2: $n=0$

3: $d_{n}=(n+1)c$

4: $\mathcal{G}_{n}=-\sum\limits_{\begin{subarray}{c}m,n=1\\\ m\neq n\\\
n>m\end{subarray}}^{N}\mathcal{F}(d_{n})\,\,$: use (21), (25), (32) with
$d_{n}=d_{\text{min}}$

5: while $\mathcal{G}_{n+1}-\mathcal{G}_{n}$ $\leq 0$ do

6: $n=n+1$

7: end while

8: ${d}_{\text{min}}=d_{n}$

9: Output: ${d}_{\text{min}}^{\star}$

A straightforward idea is behind the SEV algorithm: for a minimal input
parameter $c$ value, the number of iterations increases; however, the
precision of the solution improves; on the other hand, for large $c$ values,
the number of iterations and precision can be reduced remarkably, and
depending on how significant is such value, the first local minimum could be
inadvertently skipped, losing the optimum solution. Hence, the step parameter
$c$ must be carefully selected to the SEV achieves a good precision-complexity
tradeoff.

###  4.2 Proposed Optimal Uniform Planar Array (OUPA)

The technique that allocates omnidirectional element-antennas arranged as
uniform planar array (UPA) on a specific $z_{nm}$ plane, given by (20), and
using the SEV method to select the $d_{\text{min}}^{\star}$ is denominated
hereafter optimal uniform planar array (OUPA). We propose this
straightforward, low-computational cost, quasi-optimal method solution that
significantly enhances the directivity in UPA antennas; besides, OUPA design
requires few parameters: the carrier frequency/wave number ($f/\lambda$), the
desired directivity angles ($\theta_{0},\phi_{0}$) and the geometric UPA
structure configuration, i.e., $N_{1}\times N_{2}$. A pseudocode for the OUPA
technique is shown in Algorithm 2.

Algorithm 2 OUPA – Optimal Uniform Planar Array

1: Input: $f/\lambda$, $\theta_{0}$, $\phi_{0}$, $N_{1}$, $N_{2}$

2: Calculate all possible $d_{mn}$ values using (25)

3: Calculate all possible $z_{mn}$ values using (32)

4: Evaluate $\mathcal{G}$ in (21) using the values of $d_{mn}$ and $z_{mn}$

5: Determine $d_{\text{min}}^{\star}$ via SEV method (Algorithm 1)

6: Determine the UPA matrix of position $\boldsymbol{P}$ using
$d_{\text{min}}^{\star}$ via (6)

7: Final array position evaluated by
$\widehat{\boldsymbol{P}}(\theta_{0},\phi_{0})=\boldsymbol{P}\boldsymbol{R}_{v}$,
eq. (31)

8: Output: $\widehat{\boldsymbol{P}}$

Remark 2: The OUPA method exploits the OF features, Eq. (22), in the
simplified optimization formulation, Eq. (21), while for generic $N_{1}\times
N_{2}$ antennas configuration, a practical quasi-optimal solution is developed
in the next section (specifically, subsection 5.3) based on the evolutionary
heuristic optimization approach. The array directivity subject to antenna-
elements positioning has been formulated and solved for various elements in
the range $N\in\\{4;\,36\\}$.

##  5 Numerical Results

Different OUPA performance aspects are numerically evaluated, including a)
directivity performance, b) occupied area, c) comparison with other
directivity optimization methods. The general parameters for the
omnidirectional directivity optimization scenario deployed throughout this
section are summarized in Table 1.

Table 1: General simulation setup Parameter | Adopted Values
---|---
Angle of Departure | ($\theta_{0}$ ,$\phi_{0}$) = ($\frac{\pi}{4}$, $\frac{\pi}{4}$)
Element Amplitude |  $A_{n}=1$ for all $n$.
Element Phase |  $\alpha_{n}$ = 0 for all $n$.
Omni-directional Scenario |  $u=0$ and $v=1$

###  5.1 OUPA Directivity Performance

Numerical simulations have been conducted considering different values of
$N_{1}\times N_{2}$ antenna arrangements. For each configuration, the value of
$d_{\text{min}}^{\star}$ is obtained using the SEV procedure, and then the
relation directivity-area occupied by the array is examined. The goal is
understanding how the occupied area impacts the planar array directivity. The
occupied array area is an essential parameter since compact antenna arrays are
far more helpful. Table 2 depicts the selected parameter values deployed in
the simulations.

Table 2: Simulation setup for OUPA directivity-area evaluation Parameter | Adopted Values
---|---
Frequency / Wave Number |  $5$ GHz / $k\approx 104.8$ m-1
SEV parameter (increment) | $c=10^{-3}$
Number of $N=N_{1}\times N_{2}$ antennas-elements
# Vertical elements | $N_{1}=[2,3,4,\dots 50]$
# Horizontal elements | $N_{2}=[2,3,4,\dots 10]$

Figure 6: a) Directivity and array Area values for different $N=N_{1}\times
N_{2}$ element-antennas configurations; b) correspondent $d_{\min}^{\star}$
values obtained via SEV procedure for different configurations of
$N=N_{1}\times N_{2}$.

In Fig. 6, the values of directivity, array area and correspondent
$d_{\text{min}}^{\star}$ are depicted for a wide range of $N$ element-antennas
values. As expected, increasing $N$, the directivity and array area are
increased proportionally. Since the array area increment is not desired, the
directivity-area tradeoff must be carefully evaluated. The correspondent
$d_{\text{min}}^{\star}$ values found by simulations using SEV procedure are
illustrated in Fig. 6.b. The most important feature of this plot is that the
optimal value of $d_{\text{min}}$ has a discrete distribution regarding
$N_{1}$ and $N_{2}$; besides, the difference between the levels of optimum
values is constant, which is very interesting and can be used to reduce
significantly the search space of the optimal value. Another essential
characteristic is that the increase in $N$ not necessarily increases
$d_{\text{min}}^{\star}$ value. Finally, the $N_{1}$ and $N_{2}$ values impact
the optimal $d_{\text{min}}$ value; hence, this effect is analyzed in more
details in the next section.

To gain a deep understanding of the Area and Directivity dependence, Fig. 7
depicts directivity values against array area for different configurations of
$N=N_{1}\times N_{2}$. Observing the Pareto frontier among conflicting
directivity vs occupied area by the UPA array is possible. As expected, it is
necessary to increase the array area to achieve higher directivity values. The
Pareto front establishes the optimal trade-off between both conflicting
parameters; possible solutions (points) above this frontier are unfeasible for
such an array structure, and the points below are sub-optimal in the
directivity-area tradeoff.

Figure 7: Pareto front for directivity vs occupied area of UPA arrays.

###  5.2 Directivity vs Area under Different $N_{1}\times N_{2}$ Arrangements

Due to constructive aspects and integer combination of $N_{1}$ and $N_{2}$,
the structure realization for a given area is constrained, given that the
total number of antennas is a multiplication of two integers, i.e.,
$N_{1}\times N_{2}=N$. Moreover, most of the analyses available in the
literature focus on the UPA structures, with squared $N_{1}=N_{2}$ antenna
arrangements. In this work, we also have analyzed rectangular $N_{1}\neq
N_{2}$ antenna arrangements.

To investigate the impact on the directivity, four different UPA structures
with $N=[36;48;60;72]$ antennas were compared. Fig. 8.a) depicts possible
achievable directivity values and respective UPA normalized area (assuming
$k=1$) for possible $N_{1}$ and $N_{2}$ combinations. The directivity value
and area occupied by $N$ antennas are compared considering different UPA
configurations, each column representing the same number of antennas; the
markers on the same column indicate specific configurations of $N_{1}\times
N_{2}$. As indicated in Fig. 8.b), decreasing the difference between $N_{1}$
and $N_{2}$ results in better values of maximum attainable directivity111In
Fig. 8.b), the variation of $d_{\text{min}}$ causes an impact on the value of
directivity. It is possible to notice that better directivity values are
achieved when $N_{1}=N_{2}$. Indeed, these numerical results indicate that the
smaller difference between $N_{1}$ and $N_{2}$ better is the array
directivity. as a function of $d_{\min}$. However, the area occupied by the
UPA also increases. On the other hand, choosing configurations with a great
disparity among $N_{1}$ and $N_{2}$ implies a substantial directivity
reduction, being possible to attain a similar performance, or even be
surpassed, when compared with the same UPA structure but with a smaller number
of antennas, for instance, $36\times 2$ against $6\times 8$, in which, despite
the great difference of $24$ antennas, the directivity are quite similar in
both array arrangements.

(a) (b)

Figure 8: a) Directivity and Area vs. $N$ for different configurations with
the same number of antennas $N=[36;48;60;72]$. b) Directivity vs $d_{\min}$ fo
four possible $N_{1}\times N_{2}$ arrangements in an UPA with $N=36$ antennas.

As the focus of this work is the directivity enhancement and aiming at
exploiting the UPA structure, the difference between $N_{1}$ and $N_{2}$ must
be reduced; therefore, the choice for $N_{1}$ and $N_{2}$ should respect the
following condition:

$\displaystyle|N_{1}-N_{2}|\,\,=\,\,\left\\{{\begin{array}[]{*{20}{ll}}0&\text{squared
(UPA)}\\\ 1&\text{rectangular (UPA)}\\\ \end{array}}\right.$ (35)

###  5.3 Genetic Algorithm solving Non-Convex Directivity Problem

Since the original directivity optimization is a non-convex problem, in this
subsection, the performance of the genetic algorithm (GA) in solving the
omnidirectional uniform array directivity problem is evaluated. The pseudocode
for the GA deployed to find the enhanced directivity solutions is presented in
Algorithm 3. Besides, the GA parameter values deployed in this context are
summarized in Table 3.

Algorithm 3 GA-UPA – GA Optimizer for Omnidirectional Uniform Antenna Array

1: Input: $n_{\text{vars}}$, $c_{f}$, $g_{\text{max}}$, $M_{\text{Gaussian}}$,
$p_{\text{size}}$

2: Population $\leftarrow$
InitializePopulation($p_{\text{size}}$,$n_{\text{vars}}$)

3: $\boldsymbol{S}=\mathcal{G}(\textit{Population})$

4: $\boldsymbol{s}_{\text{best}}$ $\leftarrow\boldsymbol{s}$ which results in
the minimum value of $\mathcal{G}$($\boldsymbol{s}$)

5: while $\\#\text{{ number of Generation}}<g_{\text{max}}$ do

6: Parents $\leftarrow$ SelectParents(Population,$p_{\text{size}}$)

7: Children $\leftarrow$ $\emptyset$

8: for [Parent1,Parent2] $\in$ Parents do

9: [Children1,Children2] $\leftarrow$ Crossover(Parent1,Parent2,$c_{f}$)

10: Children $\leftarrow$ $M_{\text{Gaussian}}$(Child1)

11: Children $\leftarrow$ $M_{\text{Gaussian}}$(Child2)

12: end for

13: $\boldsymbol{S}=\mathcal{G}(\textit{Children})$

14: $\boldsymbol{s}_{\text{best}}$ $\leftarrow\boldsymbol{s}$ which results in
the minimum value of $\mathcal{G}$($\boldsymbol{s}$)

15: Population $\leftarrow$ Replace(Population, Children)

16: end while

17: Output: ${\boldsymbol{s}}_{\text{best}}$

Table 3: Simulation Setup used in the GA-UPA directivity Parameter | Adopted Values
---|---
Wave Number / Wavelength |  $k=1$ $[m^{-1}]$ / $\lambda=2\pi$ $[m]$
# antenna-elements | $N\in\\{4,5,6,7,8,9\\}$
Elements placement bounds | $[x_{\max},\,y_{\max},\,z_{\max}]=[5,\,5,\,z_{\max}]$
Objective function (OF) | eq. (21)
Number of Variables, ($n_{\text{vars}}$) | $2N$
Crossover Fraction ($c_{f}$) | 0.7
Max. # Generations ($g_{\text{max}}$) | $40$
Mutation ($M_{\text{Gaussian}}$) | Gaussian
Initial Population | Feasible and Random
Population Size ($p_{\text{size}}$) | $200N$
Population Type | Double Vector
Search space |  $[0,\,x_{\max}]$; $[0,\,y_{\max}]$

Remark 3: GA has been selected as a powerful, well-established evolutionary
technique to solve non-convex optimization problems, among other also well-
known techniques, such as particle swarm optimization (PSO), ant colony
optimization (ACO), artificial bee colony algorithm (BCA), and other
evolutionary algorithms (EA) optimization. GA or EA applies the natural
evolution principles to find an optimal local solution. In GA, the problem is
encoded in a series of bit strings manipulated by the algorithm; the decision
variables and problem functions are deployed directly. The main drawbacks of
an EA are: a) it is much slower222often by factors of a hundred times. than
alternatives such as the gradient-based and Simplex methods; b) as problem
size scales up, e.g., ten to a hundred or a thousand decision variables, an EA
is often overwhelmed by the dimensionality of the problem, being unable to
find a solution close to a locally optimal solution; c) a solution is
acceptable only in comparison to other, previously discovered solutions;
indeed, an EA has no concept of an optimal solution, or any way to test
whether a given solution is optimal, even locally optimal; d) An EA never
really knows when to stop, since it does not know whether a given solution is
optimal. Finally, EAs usually finish running manually by the user, or by a
predetermined limit on the number of iterations.

Fig. 9 depicts the solution of problem eq. (21) found by the GA for $N=6,7$
and $8$ antennas. When the solution is visualized in the 3-D plots, it is
possible to observe the plane that restrains the solution and how the
solutions found strive to attain a significant number of equal Euclidean
distances. However, when analyzing the projection of the solution in the
$xy$-plane, it is possible to verify the distortion introduced by the term
$z_{nm}$, eq. (20).

(a) $N=6$ antennas (b) $N=6$, $xy$-plane

(c) $N=8$ (d) $N=8$, $xy$-plane

Figure 9: GA solution for $N=6,\,8$ UPA antennas: (left) final element-
antennas position; (right) projection onto the $xy$-plane.

The evolution of antenna-element position through the GA-UPA generations is
depicted in Fig. 10; the left plots show the placement of the coordinates
($\boldsymbol{x},\boldsymbol{y}$) found by the GA solution and the right plots
indicate the distribution of the difference-coordinate points
($x_{mn}$,$y_{mn}$), i.e., all possible differences between the correspondent
coordinates ($\boldsymbol{x},\boldsymbol{y}$), together with the contour plots
of the objective function $\mathcal{F}$ defined in (22). Notice that in the
right plots, the number of points on each plot is given by the combinations of
all possible differences $(x_{mn}$,$y_{mn})$, which result in $N(N-1)$ points.
Another interpretation for these coordinates points is that they represent the
terms of each summation in $\mathcal{G}$, eq. (21).

(a) $1^{\text{th}}$ Generation (b) $1^{\text{th}}$ Generation (c)
$35^{\text{th}}$ Generation (d) $35^{\text{th}}$ Generation

Figure 10: Evolution of GA-UPA solutions for $N=8$ antennas: (left) antenna-
element position; (right) distribution of ($x_{mn}$,$y_{mn})$ and contour
lines for the objective function $\mathcal{F}$ of eq. (22).

The evolution of the antenna-element positions through the GA generations is
given in the xy-plane projection. Moreover, it is possible to observe the
evolution of the coordinates points ($x_{mn}$, $y_{mn}$); this evolution leads
to an arrangement where all allocated points are concentrated in sites of
small values of $\mathcal{F}$, regarding the range of all the possible values.

The selection of the difference-coordinate points ($x_{mn},y_{mn}$) can not be
performed freely. Indeed, these points result in Euclidean distances
constraints, which in some cases did not physically feasible; therefore, the
obvious solution consisting in allocating all points to the same minimum of
$\mathcal{F}$ is not possible due to the implication that all Euclidean
distances must be the same. Indeed, the modification of position in one
element of the antenna implies in the allocation of $N-1$ points; hence, the
allocation of these points must be resolved simultaneously.

Remark 4: The GA solution (Fig. 10) converges to a geometric figure similar to
the points on a regular triangular tiling (RTT); the only reason for the
convergence is not exactly an RTT geometry is found examining plane constrain
$z_{mn}$, eq. (20); hence, for $\theta_{0}=\phi_{0}=0$, the solution will be
given by a set of points allocated on an RTT. This arrangement presents
symmetry which imposes that all possible distances can be reduced remarkably
by repetition of points regularity. As a result, the difference-coordinate
points ($x_{mn}$,$y_{mn}$) can be distributed exploiting such standard,
selecting the feasible points corresponding to small values of $\mathcal{F}$
and using the repetition to allocate more points.

Comparison with the UCA and ULA: The values of directivity found with GA-UPA
is compared with the two classical steering vector beamforming: a) the uniform
circular array (UCA), and b) the uniform linear array (ULA). The directivity
using steering vector consists of changing each antenna element’s phase,
aiming to minimize the array factor in (3), depending on the position of each
element in a well-defined geometric structure. For the ULA and UCA the
steering vector is given, respectively, by:

$\displaystyle\alpha_{n}^{\textsc{ula}}=-dn\cos{\theta_{0}}\,\,\hat{k}$ (36)
$\displaystyle\alpha_{n}^{\textsc{uca}}=-r\sin{\gamma_{n}}\sin{\theta_{0}}\cos{\phi_{0}}\,\,\hat{i}-r\cos{\gamma_{n}}\sin{\theta_{0}}\sin{\phi_{0}}\,\,\hat{j}$

where: $\gamma_{n}=\frac{(n-1)2\pi}{N}$, and $d$ is the regular distance
between adjacent element. In this formulation, ${r}$ is the ratio of the
circle that contains all the points of the UCA. This ratio was considered so
that the minimal distance between the antenna’s element was defined as half
carrier wavelength, $\frac{\lambda}{2}$.

The values of directivity $\mathcal{D}$ through the generations of the GA-UPA
is compared in Fig. 11.a with the regular ULA design; it is noticeable the
excellent performance increase in the early generations, being capable of
surpassing the conventional ULA design after the first generation. The
convergence of the proposed algorithm is observable with the stationary
directivity performance over the generations, which occurs after $g=21,20$ and
$35$ generations for $N=6,7$ and $8$ antenna elements, respectively.

(a) Directivity (b) Objective function $\mathcal{G}$ values

Figure 11: Directivity and OF values $\mathcal{G}$ through the GA-UPA
generations for $N=6,7$ and $8$. ULA directivity is included in (a); a lower
bound for $\mathcal{G}^{N}$ following (24) is defined by dashed lines in (b).

The sum of all the difference-coordinates points ($x_{mn}$,$y_{mn}$)
corresponding to the OF in eq. (21) typically evolves through generations, as
depicted in Fig. 11.b for $N=6,7$ and $8$; also, lower bound values can be
calculated using eq. (24) and considering $A_{n}=1,\,\forall\,n$. The bound
values $\mathcal{G}_{\text{bound}}^{6}=-\frac{1}{2}$,
$\mathcal{G}_{\text{bound}}^{7}=-\frac{7}{12}$ and
$\mathcal{G}_{\text{bound}}^{8}=-\frac{2}{3}$ are included for comparison
purpose. Hence, after algorithm convergence, the GA-UPA performance gap can be
calculated as:

$\displaystyle\Delta\mathcal{G}^{N}=\mathcal{G}_{\text{bound}}^{N}-\mathcal{G}_{N}$
(37)

where $\mathcal{G}_{N}$ is the OF value attained with the GA for $N$ antenna-
elements after convergence. Applying (37) on the values found in Fig. 11.b,
the gaps can be established:

$\Delta\mathcal{G}^{6}=0.0418;\qquad\Delta\mathcal{G}^{7}=0.0318;\qquad\Delta\mathcal{G}^{8}=0.0423$

The gap indicates the proximity with the best solution possible; hence, for
all three simulation scenarios, the GA-UPA method achieves almost the best
solution in terms of the performance gap. Moreover, by increasing $N=6$ to 8,
it is possible to verify a decrease in $\mathcal{G}$ values for the same
number of generations. Indeed, a decreasing in $\mathcal{G}$ is responsible
for the gain in the directivity, Fig. 11.a.

###  5.4 Improved GA-aided UPA Directivity Methods

The GA-aided optimization directivity approach of the previous section
demonstrated certain difficulty in finding the optimal solution when the
number of antennas increases. Indeed, considering exhaustive combinations in
the numerical simulations, the GA-ULA could not outperform the proposed OUPA
directivity method, in a manageable time, for $N>12$, with the constrain of
searching the solution only on the desired plane, which significantly reduces
the GA-ULA complexity.

Aiming at using the GA heuristic optimization approach in scenarios with more
antennas, we propose changing the GA initialization parameters. Hence, to
solve the UPA element-antenna position problem in a manageable time, the first
approach named GA-marginal includes an initial population based on a near-
local solution. This approach’s main changes are summarized in Table 4.

Table 4: GA-marginal input parameters Parameter | Alteration
---|---
Initial Population |  Near-local solution
Mutation ($M_{\text{Uniform}}$) | Uniform Mutation
Population Size ($p_{\text{size}}$) | $8N^{2}$
Elements placement bounds | $[x_{\max},\,y_{\max}]=[2p_{\max},\,2p_{\max}]$
Max. # Generations ($g_{\text{max}}$) | Unlimited
Stopping Criterion | Outperform OUPA
Mutation rate | Determined via simulation
Crossover Fraction ($c_{f}$) | Determined via simulation

The near-local solution strategy consists in adding the OUPA solution to the
initial population, but with a small perturbation, preventing the GA search
from being restricted to the local solutions. Besides, the mutation function
was changed to be uniform, increasing the diversity to escape local optima.
The population size was considerably increased to expand the chances of
finding the better solution at the cost of increasing algorithm complexity.
Moreover, the bounds of element placement were altered considering the OUPA
solution, the new values are dependent on $p_{\max}$, which is the max value
of the coordinate $x$ or $y$ of the OUPA solution. The stopping criterion was
modified in a way that the $g_{\text{max}}$ became unlimited, and the new
criterion consisted in incrementing the generations until GA-marginal
outperforms the OUPA method. Furthermore, the mutation rate and the crossover
fraction is now determined by a simulation wherein a combination of both
values are evaluated jointly for every 100 generations, and then, the
combination that achieves the best value of directivity is selected; the range
of both values are set equal and multiple percentages of 10$\%$.

Remark 5: the idea of introducing modifications on the initial population
generation in the GA-aided directivity method aims to facilitate the guided
search to attain incremental improvements over the OUPA solution, in a
manageable time, given the non-convexity nature of the problem. Indeed, the
computational difficulty of outperforming our proposed OUPA method in such
burden computational conditions is a clear indication of the expedited and
advantageous solution given by the proposed OUPA method.

However, limiting the GA technique to surpass the proposed OUPA marginally
will not give us a promising perspective on how much gain one can attain in
the UPA directivity context using heuristic evolutionary techniques, and how
much the cost to achieve such improvement. Hence, to address this issue,
another GA modification is proposed herein, using almost the same set of
parameters described for the GA-marginal, but changing only the stopping
criterion that now consists of achieving hundred consecutive stalled
generations, without directivity gain. This method henceforth will be called
GA-stall.

###  5.5 Array Directivity Methods: Numerical Comparison

Considering a small number of element-antennas yet, Table 5 exposes values of
array directivity found by the OUPA method, by GA-aided UPA improved
solutions, and the two classical steering vector beamforming, the UCA, and the
conventional ULA, as well as an improved directivity method for the
omnidirectional UCA proposed in [15].

Table 5: Directivity Methods Comparison for small $N$’s $N$ | 6 | 8 | 9
---|---|---|---
OUPA |  $11.70$ dBi |  $12.91$ dBi |  $14.12$ dBi
GA-marginal |  $11.81$ dBi |  $13.19$ dBi |  $14.12$ dBi
GA-stall |  $12.35$ dBi |  $13.49$ dBi |  $14.5$ dBi
UCA |  $7.96$ dBi |  $8.73$ dBi |  $9.17$ dBi
ULA |  $9.17$ dBi |  $10.38$ dBi |  $10.88$ dBi
UCA [15] | n.a |  $12.00$ dBi | n.a.

The last row in Table 5 depicts the directivity found in [15], which consists
in a technique based on the subspace changes for the omnidirectional UCA
directivity. For instance, considering $N=8$, the maximum directivity value
found is close to $\mathcal{D}=12$ dBi. Such value remains remarkably reduced
compared with the optimization methods proposed herein, i.e., $\sim 1.5$ dBi
less compared with the GA-stall approach and $\sim 0.9$ dBi less when compared
with our OUPA approach.

For computational complexity analysis, Fig. 12 exhibits the simulation time
vs. # antennas $N\in[4;\,36]$, considering the OUPA improved GA-marginal and
GA-stall methods proposed, the GA-marginal and GA-stall. Aiming to maximize
the directivity, the quasi-squared UPA condition in eq. (35) is applied;
hence, possible combinations of $N_{1}$ and $N_{2}$ was bounded by
$\max(N_{1},N_{2})=6,$ implying in the following possible antenna-elements
arrangements

$\displaystyle N=N_{1}\times N_{2}\rightarrow\begin{array}[]{ccc}2\times
2&2\times 3&3\times 3\\\ 3\times 4&4\times 4&4\times 5\\\ 5\times 5&5\times
6&6\times 6\\\ \end{array},$ (41)

which are identified by the respective markers in Fig. 12.

The OUPA technique proposed herein, even with far less complexity, can follow
the performance of both GA’s techniques very closely. As expected, the GA-
marginal was able to surpass the OUPA method marginally, proving its sub-
optimality. However, the time spent by both GA heuristic methods was far
superior, configuring a far inferior performance-complexity tradeoffs. The GA-
stall aided UPA directivity optimization method was able to achieve a
considerable and consistent improvement when compared with the OUPA, the
amount of time consumed to find such a solution is extensively large.

(a) Directivity (b) Computational Time

Figure 12: Directivity and run-time comparison between OUPA, GA-marginal and
GA with the increasing in the number of antennas $N=N_{1}\times N_{2}$.

Remark 6: One can conjecture that among the three proposed directivity
methods, the improvement beyond the OUPA solution provided by the GA-stall
aided UPA algorithm comes with a much high computational cost. Moreover, both
GA-stall and GA-marginal optimization schemes have resulted in less
effectiveness than the proposed OUPA method in terms of performance-complexity
tradeoff.

Finally, Table 6 shows the crossover fraction ($c_{f}$) and the mutation rate
($m_{r}$) values as a function of dimensionality problem ($N$), required in
the GA-aided methods that maximize the directivity in the simulation of Fig.
12.

Table 6: Input parameters configuration for both GA-marginal and GA-stall algorithms N | 4 | 6 | 9 | 12 | 16 | 20 | 25 | 30 | 36
---|---|---|---|---|---|---|---|---|---
$c_{f}$ ($\%$) | 80 | 40 | 10 | 90 | 40 | 80 | 80 | 60 | 80
$m_{r}$ ($\%$) | 100 | 20 | 10 | 10 | 10 | 70 | 10 | 10 | 20

For most configurations, the crossover fraction percentage has high values,
while the mutation rate presents a low percentage. Such a combination lead to
a population that has several changes to escape local optima between the
generations, nonetheless, the value of $m_{r}$ remaining low can be
interpreted as a mean of guaranteeing an improvement around the solution.
Notice that in the case $N=20$, both GA-stall directivity optimization
algorithms selected a high $m_{r}$ percentual value; as a result, the
simulation time resulted significantly smaller; however, the performance was
incremental better when compared to the OUPA and GA-marginal. The discrepant
values found for $N=4$ antennas can be explained by the simplicity of the
geometric configuration, therefore, given a population, a volatile evolution
did not prevent the method from finding a remarkable solution.

##  6 Conclusions and Final Remarks

This work proposes a new approach to maximize the directivity of an
omnidirectional volumetric antenna array. This technique assumes a uniform
planar array (UPA) confined on a specific plane with a minimum distance
between the antenna elements optimized by the successive evaluation and
validation (SEV) procedure, namely optimal uniform planar array (OUPA).
Moreover, evolutionary heuristic GA optimization is employed to validate the
proposed methodology; hence, for a large number of antennas, two modifications
on the initial parameters are suggested, denoted GA-marginal and GA-stall,
both made the GA a promising optimization tool to solve UPA directivity
problem in such large scale antenna scenario.

An exciting finding is that a plane space constrains the element positioning
solutions as a function of the desired elevation ($\theta_{0}$) and azimuth
($\phi_{0}$) angles. A new method based on the UPA structure is introduced to
address this constraint to solve the directivity optimization problem with
low-complexity effort. Then an evolutionary heuristic technique is selected to
implement the directivity optimization analysis devised herein. Indeed, the
genetic algorithm (GA) was selected as an expedited optimization tool.

The proposed methodology is focused on finding the optimal position of each
antenna element that maximizes the antenna array directivity. The solution was
confined to a specific plane, where the antenna elements position on this
plane resembling a sequence of equilateral triangles; hence, the idea of
finding the optimal UPA in such a plane is introduced. The OUPA method
exploits the OF features, while in the evolutionary heuristic optimization
approach, the directivity based on antenna-elements position has been
formulated and solved for a different number of antenna-elements in the range
$N\in\\{4;\,36\\}$. Indeed, numerical results deploying the proposed GA-
marginal and GA-stall heuristic evolutionary techniques against the also
proposed OUPA method for a different number of antennas has demonstrated
higher directivity gains when compared with the well-known regular UCA and ULA
arrangements, as well as when compared with recent literature design for the
specific case of $N=8$ antennas.

The computational complexity reveals a superior performance of the OUPA
method, implying in an excellent performance-complexity tradeoff. Moreover,
the OUPA design method has achieved impressive performance in massive MIMO
scenarios ($N\geq 50$), achieving a directivity of $30$ dBi in an extensive
quasi-squared planar configuration, i.e., $N=15\times 16$ antennas.

## Acknowledgement

This work was partly supported by The National Council for Scientific and
Technological Development (CNPq) of Brazil under Grants 310681/2019-7, partly
by the CAPES- Brazil - Finance Code 001, and the Londrina State University -
Parana State Government (UEL).

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