Control of Mutual Coupling in Microstrip Antenna Design Based on Bio-inspired Algorithms

Abstract This paper presents a model that uses bio-inspired optimization algorithms to control of the mutual coupling (MC) in Multiple-Input Multiple-Output (MIMO) systems. The method promotes an increase in the channel throughput and a reduction in the distance among the array elements, searching values less than a half wavelength. A microstrip antenna design, at the frequency of 26 GHz, is used as a prototype for MIMO system testing and performance analysis. Genetic Algorithm (GA) is used to optimize, to determine the antenna physical parameters and to improve its performance. The mutual impedance concept is used to model the MC. The particle swarm optimization (PSO) algorithm is used to optimize the channel throughput and to reduce the distance between the elements. Simulation results show that it is possible to obtain a MIMO system performance improvement of 11.1% in channel throughput, the microstrip width is reduced by 14.5% and the distance among the elements can be reduced by 23.7%, considering the MC.


I. INTRODUCTION
In recent years, the number of mobile data network users has grown at a steady pace worldwide, and this has largely increased the number of connections and equipments.According to the Group Speciale Mobile Association (GSMA), there were 5.3 billion mobile users around the world by the end of 2021.The Fourth Generation (4G) network has 58% of connections (excluding licensed cellular technology and Internet of things) and Fifth Generation (5G) network technology is already in use in 70 countries around the world [1].
The MIMO technique, which uses antenna arrays in transmission and reception, increases the transmission rate and is poised to achieve the radio channel transmission limits required in modern wireless networks [2].The concept of MIMO systems brings new challenges for fifth generation (5G) networks, with the massive increase in the number of antennas and the corresponding research associated with its performance [3]- [5].
Some researchers use GA to optimize antennas, in microstrip design, by changing their physical dimensions and the format of the patches array [6], or by the minimization and evaluation of a single rectangular patch [7], [8].In [9], the authors also use GA for linear optimization of an antenna array.

II. THE CHANNEL GAIN MATRIX
Taking advantage of the spatial diversity through antenna arrays in MIMO systems, it assumes that the use of an antenna array increases the performance of the system by ensuring that the signal will reach the receiver with an appropriate Signal-to-Noise Ratio (SNR).MIMO systems exploit the multipath phenomenon to increase the transmission rate in environments that are highly influenced by reflection and diffraction propagation mechanisms.Depending on the number of antennas in the transmitter and receiver, the transmission rate may increase as a result of an increased complexity of the transmitted signal [13].
Considering a MIMO system characterized by a complex-valued matrix H, the output signal is given by [2], [14], [15] Y = Hx + n, (1) in which the input signal and the additive white Gaussian noise are represented by x and n, respectively [15].The transmitter is assumed to be subject to an average power constraint of P across all transmit antennas.Since the noise power is normalized to unity, one commonly refers to the power constraint P as the SNR [14].The perfect channel state information at the transmitter (CSIT) and at the receiver (CSIR) model is justified by the scenario where the channel state can be accurately tracked at the receiver and the statistical channel model at the transmitter is based on information about the channel gain probability distribution fed back from the receiver.Considering perfect CSIT and CSIR, the ergodic capacity of the MIMO channel is given by [14], [16] in which the identity matrix of order N r is represented by I Nr , E[•] is expected value operator, (•) H represents the Hermitian transposition of the channel gain matrix H considering N r and N t the number of receive and transmit antennas, respectively.Finally, P is the transmitting power [14], [16].This research uses the Kronecker channel model to compute the channel gain matrix H [15], [17]- [19].Correlation matrixes in reception and transmission are considered separable and they can be in which the spacial correlation in receiver and transmitter are represented by R Y and R X , respectively, and G H is a Gaussian matrix with elements characterized by identical and independently distributed (i.i.d) Gaussian random variables with zero mean and unit variance [15], [20].
In order to compute the channel gain matrix as a function of the transmitter and receiver impedances it is necessary to adapt Equation (3).This is possible using in which H 1 is the new gain matrix with the MC effect considered, Z r and Z t are the mutual impedances in the receiver and transmitter, respectively [21].

III. MUTUAL COUPLING
Antenna spacing, in the antenna array, is known to influence spatial correlation which limits the channel throughput increase [22].This finite spacing is also responsible for the MC that affects signal transmission and reception due to antenna impedance mismatch.The effect of MC is especially pronounced in arrangements with elements spaced less than λ/2.Due to the considerable demand for compacting the size of mobile systems, the effect of MC cannot be overlooked [15].
The mutual impedance concept is used for evaluating the MC effect on antenna arrays.This paper describes the effects of MC on antenna arrays in reception and transmission, and uses the conventional mutual impedance method (CMIM) [23] and the corresponding receiving mutual impedance method (RMIM) [24] in the design of the transmission and reception antenna arrays.
Conventionally, it is defined the mutual impedance between two elements as shown in Fig. 1 [22], i.e, a transmission antenna connected to one source and a reception antenna is in a open circuit [25].Due to the MC effect, there are two additional sources of excitation in the equivalent circuits.The MC Fig. 1.Equivalent circuit of two receiving antennas considering MC.Adapted from [22].
effect is modeled using the controlled voltage sources V 12 and V 21 in the equivalent circuit.They can be obtained from the currents I 1 and I 2 in both antennas, such as [22] The mutual impedances can be described as [22], [24] and in which Z 12 and Z 21 represent the mutual impedance excited in Antenna 2 and Antenna 1, respectively.Expressions ( 8) and ( 9) describe the total voltage generated by the excitation sources and coupling, and respectively.
For N elements in a transmitter antenna array, the excitation voltages without considering MC, The RMIM method, different from CMIM, considers an impedance Z L terminated in the antenna elements and an external planar wave exciting these elements in receive mode [24], [26].
Fig. 2 shows the equivalent circuit of dipoles in the receive mode.It is possible to observe that the definition of mutual impedance at the reception implies that it depends on the direction of an idealized plane wave that is used as the excitation source.But for omni-directional antennas, for instance, in the dipole or monopole antenna, the mutual impedance at reception is independent of the azimuth angle of the plane wave [25].From this scenario, the following results are obtained: in which Z 12 r and Z 21 r are the mutual impedances in the receive mode in Antenna 1 and Antenna 2. V 12 and V 21 are the voltages coupled of Antenna 1 and Antenna 2m.Finally, I 1 and I 2 are the currents received on Antenna 1 and Antenna 2, respectively [25].
The received voltages across the elements of Antenna 1 and Antenna 2 can be calculated by using ( 12) and (13).Considering U 1 and U 2 , the voltages of the isolated terminal in Antenna 1 and Antenna 2 without coupled voltages V 12 and V 21 , respectively, then [25] Fig. 2. Two dipoles antennas in reception mode.Adapted from [25]. and respectively.Equation ( 14) expresses the relation between V k and U k , considering an antenna array with N elements in the receive mode.V k and U k are the voltage in the antenna terminal and the voltage produced by external source, respectively.
For a general antenna array, the excitation source is not applied directly to the antenna feed ports and the antennas are excited by an external plane wave source [24].In [27] is summarized the basic criteria for the far-field region R for electromagnetic fields from radio frequency sources in the microwave range, including antennas for cell phones, microwave radar and other cases.They are in which D x is the largest linear dimension of the antenna.Only when all the above inequalities are satisfied, one obtains the far-field condition [27].
IV. MICROSTRIP ANTENNA DESIGN One of the characteristics of an antenna array is the irradiation, or the emission of electromagnetic waves in space, efficiently [28], in addition to receiving these waves.As the quality of the transmitted signal increases, the reliability of the system also increases.A system with antenna arrays can contribute to this by increasing the number of users that can be managed and adding the number of services that the system can offer [29].
Usually, the array elements are identical in their geometric form and their most common geometric configuration are linear, circular, rectangular and spherical [30].The MC and the spatial correlation between the elements are factors that have implication on the distance D between the elements.It is recommended that D ≥ λ/2 [30].
Microstrip antennas consist of a thin metallic blade, usually 0.035 mm of thickness, built on a dielectric substrate, with a ground plane on the underside substrate.The rectangular patch is the most used antenna configuration.The radiating elements and power lines are, in general, photographed on the dielectric substrate.The patch can also be square, circular, elliptical, triangular or any shape.The most used feeding methods are the microstrip line, the coaxial probe, opening coupling and proximity coupling [30].The rectangular patch model with a microstrip feed line is considered in this work due to its good performance, easy analysis and wide use.The design of a eight-element microstrip antenna is defined according to Fig. 3. Fig. 3 shows the physical parameters for antenna design, W and L are the width and length of patch, respectively, W g and L g are the width and the length of the ground plane, respectively.They are defined as The width and length of the ground plane is equal in size to the width and length of the substrate, respectively.The width of the feed line is W f , L f is the length of the radiant gap within the patch and G f is the width of the gap.The distance between the elements is represented by D, and the parameters W , L, W f , L f and G f are optimized in this study to minimize the antenna size.
The most suitable substrates for the good performance of the antenna must have a low dielectric constant [30], ϵ r , in the range of 2.2 to 12 and h s , its thickness, corresponds to a small fraction of the wavelength, usually 0.003λ ≤ h s ≤ 0.05λ.The thickest substrates have shown a better performance [30].Based on these recommendations, the substrate chosen was the RO 3003 ® by Rogers Corporation which has a dielectric constant ϵ r = 3, dissipation factor tan(δ) = 0.0010 (for 10 GHz and 23 • C) and h s = 0.5 mm [31].
The frequency of 26 GHz is defined for system evaluation and the parameters of microstrip antenna are optimized according to the goals: The reflection coefficients in decibel (S parameters) are evaluated, which must be less than −10 dB for a resonance frequency the closest possible to 26 GHz.The VSWR, which must be less than two and ensure a bandwidth close to 1 GHz.And the Z parameters, that is, the self-impedance of each element and the mutual impedance between them.The self-impedance must be close to 50 Ω to guarantee a good impedance match.

V. BIO-INSPIRED ALGORITHMS
John Holland [32] proposed the method that transforms a population of individuals or chromosomes into a new population that uses the Darwinian principle of reproduction and survival of the fittest, as happens naturally in the genetic operations of crossing and mutation [33].Each individual in a population is associated with a fitness values, which represents a possibility of solving the proposed problem.A search that uses a GA seems to be adequate to find the best solution to the problem based on the genetic operators of selection, crossover and mutation.
Kennedy and Eberhart proposed the PSO algorithm [34] from the particle swarm concept to solve computational intelligence problems [35].The speed and the particle position are attributes of the algorithm's objective function [36].The updating of the speed and particle position show the best solution.
Fig. 4 presents a complete flowchart of the working methodology, using bio-inspired algorithms.The parameters of the microstrip antenna, with a rectangular patch, are optimized using a GA and the PSO to maximize the channel throughput and the distance among the elements.
Initially, the physical and electrical parameters are inserted in the microstrip antenna simulator and defined according to design recommendations for this type of antenna.The simulation and the results of the antenna behavior are performed and evaluated in the CST Studio Suite ® software, it is verified if the expected values are reached and, otherwise, the optimization is carried out with variation of some physical parameters until the goal is reached.
For designing the microstrip antenna, the application area "MW & RF & Optical" for Antennas is chosen, considering patch antennas.The solution used is in time domain, which includes finite integration technique (FIT) and transmission line matrix (TLM) in the same package.E-field, H-field and Farfield monitors are set for performance analysis in 26 GHz.After that, the microstrip antenna is designed with optimization of the size of its physical characteristics.
With the data collected from this first simulation, the MIMO system simulator runs in Matlab ® programming language and is initiated seeking to reduce the distance between the elements that promote the highest channel throughput.The optimization algorithm is executed until it finds the shortest distance or a defined stop point.The MIMO system block receives from the previous procedure the values of the parameters: frequency of operation, number of array elements, internal and load impedances of the antennas, Z parameters and distance between the elements.The SNR values are added to evaluate capacity in function of its variation.After that, an evaluation is made, using the Equations 2, 10 and 14, as to whether the smallest antenna in terms of width has already been found according to commercial smartphones standards.If not, the process is restarted for a new antenna design.

A. Simulations Setup
The values of the algorithms configuration parameters are shown in Tables I and II.The maximum and minimum configuration value of the PSO parameters are shown in Table II, in which c 1 is the cognitive acceleration coefficient and c 2 is the social acceleration coefficients.A uniform distribution is used to obtain the random values r 1 and r 2 .The inertia weight ω is used to control the particle's speed and to avoid its explosions during the process.The most commonly used strategy is to initialize the search process with ω close to one and reduce it linearly during all the process [37].In this paper, the local search space was reduced to distances less than λ/2, values greater than this solution are not important because the MC effect is already considered.After this, the acceleration rate δ is experienced increasing or decreasing the acceleration coefficients , the δ value is obtained from a uniform distribution between 0.05 and 0.10.In strategies in which a slight increase or decrease in the acceleration coefficients is required, a ratio of 0.5δ is used.2.5 and 1.5 c2max and c2min 2.5 and 1.5 r1max and r1min 1.0 and 0.0 r2max and r2min 1.0 and 0.0 ωmax and ωmin 0.9 and 0.4 δmax and δmin 0.1 and 0.05

VI. RESULTS
After the whole design optimization process for the microstrip antenna and the MIMO system, and after the evaluation and rounding, the final values of the physical parameters for the microstrip antenna are described in Table III.
It is possible to observe a change in the values of each parameter and a reduction of 14.5% in W g , which is one of the goals established in the limit of 67.3 mm.This happens, mainly, due to the reduction of W and D values.Table IV shows the response of the design objectives of the microstrip antenna after optimization.It is possible to observe that the optimization method achieved five of the six design objectives with efficiency.The resonance frequency, the reflection coefficients (S parameters), the VSWR, the bandwidth and the substrate width achieved better than expected values.The line reference impedance used in this research is 50 Ω.The algorithm didn't achieve optimal values of internal impedance that would promote a perfect impedance match with the transmission line, but this is a non-complex problem to solve with an adequate design of transmission line with stub and will be the subject of future works.
The far-field measurement graphs were evaluated to analyze the gain, directivity and efficiency of   the antenna, and are presented in Fig. 7(a) and 7(b).
The antenna presents good directivity with a gain of 7.24 dBi in the frequency of 26 GHz, and also has good efficiency.In this paper, the distance between the elements of the transmitting antenna array was fixed at a wavelength (λ) and this distance in the receiving antenna elements was varied from 0.001λ to 1λ.Table V details the values of throughput optimization versus distance reduction for a MIMO 8 × 8 system with a signal-noise-ratio (SNR) of 20 dB.
The displacement behavior of the elements can be observed, with and without optimization, showing that the method is valid to achieve higher channel throughputs for a distance smaller than λ/2.There   was a reduction in the distance between the elements by 23.7%, which represents, at a frequency of 26 GHz, a reduction of 10.5 mm in the total width of the microstrip.Also, at a distance between the elements of 0.401λ, there is an increase of 11.1% in channel throughput.The response of the throughput optimization of the MIMO systems implemented as a function of the SNR can be seen in Fig. 8.
To calculate the throughput, the smallest distance between the elements found by the algorithm is considered, in this case, 0.401λ, for the transmitter and receiver mode, simulating two mobile stations with the microstrip antennas designed.It is possible to observe that the increase in SNR also increases the capacity optimization, that is, the higher the SNR, the better the application of this method.And   once again the increase in channel capacity of 11.1% for a 20 dB SNR is confirmed.
The results confirm the efficiency of the algorithm in solving the throughput optimization problem with reduction of the distance between the elements considering the MC.They also show that a well founded methodology combined with artificial intelligence methods can generate satisfactory results and reduce computational complexity.

VII. CONCLUSION
This paper presentes a methodology to increase the channel throughput of a MIMO system, considering the mutual coupling effect, with the use of bio-inspired algorithms in the optimization process.Another objective of this study was to minimize the size of the microstrip antenna, based on considerations about the distance between the elements and on the physical dimensions of the antenna.The rectangular patch with transmission line was chosen as a template.The physical parameters of the antenna and some performance measures were evaluated, such as: operating frequency, parameters S and Z, VSWR, internal resistance, directivity, gain and efficiency.
The GA and PSO algorithms form a simple and powerful tool to minimize the size of the antenna and optimize the channel throughput for MIMO systems.The best results show an increase of 11.1% in the throughput of MIMO channels 8 × 8 with the PSO algorithm, including a proposal to modify the search strategy of the algorithm and a reduction in the distance between the antenna elements by 23.7%.
For future works, the authors plan to extend the methodology to other frequency bands, including millimeter waves, such as 28 and 30 GHz.The study of mutual coupling in the reception mode must be further compared with other methods from the literature, as well as, the effect of this methodology for non-uniform distances between the antenna elements.Impedance matching solutions will also be studied, for the proposed antenna.
in which Z in and Z g are the antennas input impedance and the source impedance.The coupled voltage of the Antenna 1 and Antenna 2 are represented by V 12 and V 21 , considering the terminal with open circuit.The excitation current of the Antenna 1 and Antenna 2 are represented by I 1 and I 2 considering the terminal with short-circuit, respectively.

Fig. 5 (
a) and 5(b) show the values of the S parameters between each element and in itself and, due to software limitations, only the first 25 curves are shown.Fig. 6(a) and 6(b) show the values of the S parameters in the resonance frequency.It is possible to observe from the graphs the efficiency of the GA algorithm in the search for the values of the S parameters and the resonance frequency for all elements.

Fig. 5 .
Fig. 5. Results of the S parameters S with all the antenna elements.
(a) S Parameters at the resonance frequency in transmission mode.(b) S Parameters at the resonance frequency in reception mode.

Fig. 6 .
Fig. 6. Results of the S parameters S with all the antenna elements.

Fig. 8 .
Fig. 8.Comparison graph of throughput as a function of SNR in receiver mode with and without optimization.

TABLE II .
PSO OPTIMIZATION PARAMETERS.

TABLE III .
RESULTS OF THE PARAMETERS FOR MICROSTRIP ANTENNA.

TABLE IV .
RESULTS OF THE DESIGN GOALS FOR THE MICROSTRIP ANTENNA.GHz 27.3 GHz 26 ± 0.028 GHz 26 ± 0.017 GHz

TABLE V .
RESULT OF THE CHANNEL THROUGHPUT OPTIMIZATION IN MIMO 8 × 8 SYSTEM AS A FUNCTION OF DISTANCE.