# Radiation Pattern Synthesis of the Coupled almost Periodic Antenna Arrays Using an Artificial Neural Network Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation: (Radiation Patterns of the Almost Periodic Structures)

- On the metallic part, the electric field can be given by:

- On the discontinuity plane (or on the radiating aperture), the electric field is expressed in function of the guide’s modes as:

- For small and large finite arrays:

- For infinite arrays:

## 3. Concept of Artificial Neural Networks (ANNs)

- 1.
- The structure of the network is first defined. In the network, activation functions are chosen and the network parameters, weights, and biases are initialized.
- 2.
- The parameters associated with the training algorithm—the error goal, the maximum number of epochs (iterations), etc.—are defined.
- 3.
- The training algorithm is called.
- 4.
- Once the neural network is determined, the result is first tested by simulating the output of the neural network with the measured input data. This result is compared to the measured output. The final validation must be performed with independent data.

- Sixty percent are used for training.
- Twenty percent are used to validate that the network is generalizing and to stop training before overfitting.
- The last 20% are used for a completely independent test of network generalization.

## 4. Results and Observations

#### 4.1. Numerical Results

#### 4.2. ANN Application

- 1.
- The training set was used to determine ANN weights.
- 2.
- The validation set was used to check the ANN’s performance and decide when to stop the training process.
- 3.
- The test set was used to assess the performance capabilities of the developed ANN model.

#### 4.3. ANN Performance

## 5. Conclusions

- Reduced computational time and memory usage, especially when adopting the early stopping method, which eliminates the overfitting problem.
- It is suitable for use with a coupled and complex quasi-periodic configuration.
- It is simple and easier to use than other optimization techniques (genetic, LMS, etc.).
- It is adaptable to complex electromagnetic calculations taking into account the effects of mutual coupling.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

MoM-GEC | Method of Moment with Generalized Equivalent Circuits |

FSS | Frequency Selective Surfaces |

EEEE | Electric walls |

EMEM | Electric and Magnetic walls |

EPEP | Electric and Periodic walls |

PPPP | Periodic walls |

1-D | Uni-dimensional |

2-D | Two-dimensional |

ANN | Artificial Neural Network |

GA | Genetic Algorithm |

ES | Early Stopping technique |

MLP | Multilayer Perceptron |

LMA | Levenberg-Marquart Algorithm |

MSE | Mean Squared Error |

CPU | Central Processing Unit |

LMS | Least Mean Squares algorithms |

EM | Electromagnetic Calculation |

HFSS | High-Frequency Structure Simulator, a high frequency electromagnetic simulation software |

## Appendix A

## Appendix B

`1``% Steering angles``2``Phi_s=0;``3``Theta_s=0;% Theta_s=pi/6;``4``5``Phi=0;``6``Theta=0; % Theta=pi/6; (The maximum of radiation change with the steering angle Theta_s)``7``a = 0; b =d_x; c =0; d =d_y;``8``m = 8; n = 12;``9``dx = (b - a)/m; dy = (d - c)/n;``10``i = 1: m; j = 1: n;``11``u = a + (i - 1/2)∗dx;``12``v = c + (j - 1/2)∗dy;``13``[u,v] = meshgrid(u,v);``14``intgralmax=sum( sum(abs(calcul_E_{aperture}(u,v,A,Xs,alfa_moins1,beta_moins1))``15``.∗exp(1i.∗k.∗((u.∗sin(Theta).∗cos(Phi)+v.∗sin(Theta).∗sin(Phi))-``16``(u.∗sin(Theta_s).∗cos(Phi_s)+v.∗sin(Theta_s).∗sin(Phi_s))))∗dx∗dy;``17``radiationmax=intgralmax;``18``Theta=-pi/2:pi/100:pi/2;``19``for j=1:length(Theta)``20``Integral(j)=sum( sum(abs(calcul_E_{aperture}(u,v,A,Xs,alpha,beta))``21``.∗exp(1i.∗k.∗((u.∗sin(Theta).∗cos(Phi)+v.∗sin(Theta).∗sin(Phi))-``22``(u.∗sin(Theta_s).∗cos(Phi_s)+v.∗sin(Theta_s).∗sin(Phi_s))))∗dx∗dy;``23``En(j)=Integral(j)./radiationmax;``24``Edb(j)=20.∗log10(abs(En(j)));``25``deg(j)=Theta(j).∗180./pi;``26``value(j)=Edb(j);``27``end``28``figure,plot(deg,value,’g’);``29``% figure,polar(Theta,abs(En)); % Possible to draw the normalized polar pattern`

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**Figure 1.**Diagram of periodic planar phased array dipoles: the walls of the environment can be selected from EEEE (waveguide with electric walls), EMEM (waveguide with two electric walls and two magnetic walls), EPEP (waveguide with two electric walls and two periodic walls), and PPPP (waveguide with periodic walls).

**Figure 2.**Section of periodic phased array microstrip lines (planar dipoles): unit cell delimited by periodic walls.

**Figure 5.**Validation of the evaluated input impedance obtained by the MoM-GEC and the HFSS tool for a unit cell of a dipole: ${\alpha}_{0}=0$ rad m${}^{-1}$, ${\beta}_{0}=0$ rad m${}^{-1}$, $w=1$ mm$,\delta =0.75$ mm$\phantom{\rule{3.33333pt}{0ex}}(w\ll {\lambda}_{0},$$\delta \ll {\lambda}_{0}),$${d}_{x}=108$ mm$,{d}_{y}=108$ mm$,L={\lambda}_{0}\approx 54$ mm (at the fixed frequency f = 5.4 GHz$),h=1.25$ mm, and $\u03f5={\u03f5}_{r}=1$ (air).

**Figure 6.**Current density for a unit cell of an almost periodic 2D array (half-wave and full-wave dipoles) defined over the guide aperture, described by the test functions using: f = 5.4 GHz, ${\alpha}_{0}=0$ rad m${}^{-1}$, ${\beta}_{0}=0$ rad m${}^{-1}$, $w=1$ mm, $\delta =0.75$ mm$\phantom{\rule{3.33333pt}{0ex}}(w\ll {\lambda}_{0},\delta \ll {\lambda}_{0})$, ${d}_{x}=108$ mm, and ${d}_{y}=108$ mm. (

**a**) $L=\frac{{\lambda}_{0}}{2}\approx 27$ mm; (

**b**) $L={\lambda}_{0}\approx 54$ mm, $h=1.25$ mm and $\u03f5={\u03f5}_{r}=1$ (air) (see [1] for a comparison).

**Figure 7.**Current density for a unit cell of 2D almost periodic array (half and full wave dipoles) defined by the guide wave’s aperture, described by the basis functions (guide’s modes) using: f = 5.4 GHz, ${\alpha}_{0}=0$ rad m${}^{-1}$, ${\beta}_{0}=0$ rad m${}^{-1}$, w = 1 mm, $\delta $ = 0.75 mm $(w\ll {\lambda}_{0},\delta \ll {\lambda}_{0})$, ${d}_{x}=108$ mm, ${d}_{y}=108$ mm. (

**a**) $L=\frac{{\lambda}_{0}}{2}\approx 27$ mm; (

**b**) $L={\lambda}_{0}\approx $ 54 mm, h = 1.25 mm and $\u03f5={\u03f5}_{r}=1$ (air) (see [1] for a comparison).

**Figure 8.**Electric field of 2D almost periodic array’s unit cell (half and full wave dipoles) defined by the guide wave’s aperture, described by the basis functions (guide’s modes) using: f = 5.4 GHz, ${\alpha}_{0}=0$ rad m${}^{-1}$, ${\beta}_{0}=0$ rad m${}^{-1}$, $w=1$ mm$,\delta =0.75$ mm$\phantom{\rule{3.33333pt}{0ex}}(w\ll {\lambda}_{0},\delta \ll {\lambda}_{0}),{d}_{x}=108$ mm$,{d}_{y}=108$ mm. (

**a**) $L=\frac{{\lambda}_{0}}{2}\approx 27$ mm; (

**b**) $L={\lambda}_{0}\approx 54$ mm$,h=1.25$ mm, and $\u03f5={\u03f5}_{r}=1$ (air) (see [1] for a comparison).

**Figure 9.**Radiation pattern calculated using the MoM-GEC method for a half-wave and full-wave dipole antenna: f = 5.4 GHz, ${\alpha}_{0}=0$ rad m${}^{-1}$, ${\beta}_{0}=0$ rad m${}^{-1}$, $w=1$ mm, $\delta =0.75$ mm $(w\ll {\lambda}_{0},$$\delta \ll {\lambda}_{0}),$${d}_{x}=108$ mm, ${d}_{y}=108$ mm. (

**Blue color**) $L=\frac{{\lambda}_{0}}{2}\approx 27$ mm; (

**Green color**) $L={\lambda}_{0}\approx 54$ mm, $h=1.25$ mm, and $\u03f5={\u03f5}_{r}=1$ (air).

**Figure 10.**Current density of an example of almost periodic 2D array (half-wave dipoles) defined by the guide wave aperture, described by (

**a**) test functions or (

**b**) basis functions (guide modes) at the operating frequency (f) of 5.4 GHz. The physical parameters used were: ${\alpha}_{0}=0$ rad m${}^{-1}$, ${\beta}_{0}=0$ rad m${}^{-1}$, $w=1$ mm, $\delta =0.75$ mm$\phantom{\rule{3.33333pt}{0ex}}(w\ll {\lambda}_{0},\delta \ll {\lambda}_{0}),{d}_{x}=108$ mm, ${d}_{y}=108$ mm, $L=\frac{{\lambda}_{0}}{2}\approx 27$ mm, $h\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1.25$ mm, and $\u03f5={\u03f5}_{r}=1$ (air) (see [1] for a comparison).

**Figure 11.**Current density of the almost periodic network (half-wave dipoles) defined on the aperture of the waveguide, described by (

**a**) the test functions or (

**b**) the basic functions (guide modes) at the operating frequency (f) of 5.4 GHz. The physical parameters used were: ${\alpha}_{0}=0$ rad m${}^{-1}$, ${\beta}_{0}=0$ rad m${}^{-1}$, $w=1$ mm, $\delta =0.75$ mm$\phantom{\rule{3.33333pt}{0ex}}(w\ll {\lambda}_{0},\delta \ll {\lambda}_{0}),{d}_{x}=108$ mm, ${d}_{y}=108$ mm, $L=\frac{{\lambda}_{0}}{2}\approx 27$ mm, $h\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1.25$ mm, and $\u03f5={\u03f5}_{r}=1$ (air) (see [1] for a comparison).

**Figure 12.**Electric field of the almost periodic array (half-wave dipoles) defined by the waveguide’s aperture, described by the basis functions (guide’s modes) using (

**a**) three cells or (

**b**) seven cells at the operating frequency (f) of 5.4 GHz. The physical parameters were: ${\alpha}_{0}=0$ rad m${}^{-1}$, ${\beta}_{0}=0$ rad m${}^{-1}$, $w=1$ mm, $\delta =0.75$ mm$\phantom{\rule{3.33333pt}{0ex}}(w\ll {\lambda}_{0},\delta \ll \lambda {}_{0}),{d}_{x}=108$ mm, ${d}_{y}=108$ mm, $L=\frac{{\lambda}_{0}}{2}\approx 27$ mm, $h=1.25$ mm, and $\u03f5=\u03f5{}_{r}=1$ (air) (see [1] for a comparison).

**Figure 13.**Current density of 1D aperiodic array’s example (half-wave dipoles) defined by the waveguide’s aperture, described by the test functions at the operating frequency (f) of 5.4 GHz. The physical parameters used were: ${\alpha}_{0}=0$ rad m${}^{-1}$, ${\beta}_{0}=0$ rad m${}^{-1}$, $w=1$ mm, $\delta =0.75$ mm $(w\phantom{\rule{3.33333pt}{0ex}}\ll \phantom{\rule{3.33333pt}{0ex}}{\lambda}_{0},\delta \ll {\lambda}_{0}),{d}_{x}=13.5$ mm, ${d}_{y}=108$ mm, $L=\frac{{\lambda}_{0}}{2}\approx 27$ mm, $h=1.25$ mm, and $\u03f5={\u03f5}_{r}=1$ (air).

**Figure 14.**Current densities: examples of 1D aperiodic arrays (half-wave dipoles) that appeared on the guide wave’s aperture, described by the basis functions (Guide’s modes) (without convergence) at the operating frequency (f) of 5.4 GHz: (

**a**) for an aperiodic exitation of type 10101, (

**b**) for an aperiodic exitation type 10001. The physical parameters were: ${\alpha}_{0}=0$ rad m${}^{-1}$, ${\beta}_{0}=0$ rad m${}^{-1}$, $w=1$ mm, $\delta =0.75$ mm$\phantom{\rule{3.33333pt}{0ex}}(w\ll {\lambda}_{0},\delta \ll {\lambda}_{0}),{d}_{x}=27$ mm, ${d}_{y}=108$ mm, $L=\frac{{\lambda}_{0}}{2}\approx 27$ mm, $h=1.25$ mm, and $\u03f5={\u03f5}_{r}=1$ (air).

**Figure 15.**Current densities: 1D and 2D of aperiodic array examples (half-wave dipoles) lying on the guide wave’s aperture and described by the basis functions (guide’s modes) at the operating frequency (f) of 5.4 GHz. The physical parameters were: ${\alpha}_{0}=0$ rad m${}^{-1}$, ${\beta}_{0}=0$ rad m${}^{-1}$, $w=1$ mm, $\delta =0.75$ mm$\phantom{\rule{3.33333pt}{0ex}}(w\ll {\lambda}_{0},\delta \ll {\lambda}_{0}),{d}_{x}=13.5$ mm, ${d}_{y}=108$ mm, $L=\frac{{\lambda}_{0}}{2}\approx 27$ mm, $h=1.25$ mm, and $\u03f5={\u03f5}_{r}=1$ (air).

**Figure 16.**Distribution of the electric density for ($5\times 1$) aperiodic phased half-wavelength planar dipoles (with (1,0,1,0,1) voltage configuration) described with the basis functions (guide’s modes) at f = 5.4 GHz (using EEEE electric walls).

**Figure 17.**Distribution of the electric density for ($5\times 1$) aperiodic phased half-wavelength planar dipoles (with (0,0,1,0,0) voltage configuration) described with the basis functions (guide’s modes) at f = 5.4 GHz (using EEEE electric walls).

**Figure 18.**Radiation pattern computed with a MoM-GEC method against the motifs number at the operating frequency (f) of 5.4 GHz (periodic array).

**Figure 19.**Radiation pattern computed with a MoM-GEC method against the periods at the operating frequency (f) of 5.4 GHz (periodic array).

**Figure 20.**E-and H-plane cuts of the radiation pattern computed with a MoM-GEC method at the operating frequency (f) of 5.4 GHz (periodic array).

**Figure 21.**Radiation patterns for distinct aperiodic configurations and periodic arrays using electromagnetic and analytic formulations (MoM-GEC and analytic formulations) [22].

**Figure 22.**3D Electronically scanned radiation pattern examples for distinct aperiodic array configurations compared to the periodic array obtained using the electromagnetic calculation (MoM-GEC method): ${\varphi}_{s}={0}^{\circ},{\theta}_{s}={45}^{\circ}$ angles of steering: (

**a**) An aperiodic configuration of type (0,0,1,0,0), (

**b**) An aperiodic configuration of type (1,0,1,0,1), (

**c**) A periodic configuration of type (1,1,1,1,1) [22].

**Figure 23.**Electric field at the opening of a waveguide composed of 5 coupled planar dipoles (calculation provided by the MoM method and the HFSS software): ${d}_{x}\approx \frac{{\lambda}_{0}}{2}=27$ mm (period): (

**a**) MoM-GEC with electric walls, (

**b**) MoM-GEC with periodic walls, (

**c**) HFSS Tool.

**Figure 24.**3D radiation patterns proven with the MoM-GEC method (with electric and periodic walls) compared to the HFSS software for a coupled periodic antenna array with 5 elements and ${d}_{x}\approx \frac{{\lambda}_{0}}{2}=27$ mm (period): (

**a**) MoM-GEC with electric walls, (

**b**) MoM-GEC with periodic walls, (

**c**) HFSS Tool.

**Figure 25.**Normalized polar radiation patterns proven with the MoM-GEC method (with electric and periodic walls) compared to the HFSS software for a coupled periodic antenna array with 5 element and ${d}_{x}\approx \frac{{\lambda}_{0}}{2}=27$ mm (period): (

**a**) $\varphi =0$ deg, (

**b**) $\varphi =90$ deg.

**Figure 26.**Validation of the radiation pattern obtained using the MoM-GEC and HFSS tools for a coupled periodic antenna array with 5 elements and ${d}_{x}\approx \frac{{\lambda}_{0}}{2}=27$ mm (period).

**Figure 27.**Flow diagram illustrating the numerical radiation pattern optimization using an artificial neural network algorithm.

**Figure 28.**Radiation field data were divided into three subsets: a training set, validation set, and testing set. The parameters chosen to simulate the suggested almost periodic array antennas were: $\varphi =0$ deg, ${N}_{x}=3$ elements, and ${d}_{x}=2\lambda $ = 108 mm.

**Figure 29.**Radiation pattern function approximation with early stopping (ES): improving generalization with early stopping (ES). The parameters chosen to simulate the suggested almost periodic array antennas were: $\varphi =0$ deg, ${N}_{x}=3$ elements, and ${d}_{x}=2\lambda $ = 108 mm (For periodic array example).

**Figure 30.**The ANN output as a radiation pattern with early stopping (ES) (improving generalization with early stopping): for ($5\times 1$) aperiodic phased half-wavelength planar dipoles with (10101) voltage configuration [22].

Number of input neurons | 100 |

Number of hidden layers | 2 |

Number of output neurons | 1 |

Algorithm | lm |

Learning rate | 0.01 |

Momentum | 0.95 |

MSE goal | $1\times {10}^{-3}$ |

Minimum performance gradient | $1\times {10}^{-5}$ |

Initial mu | 0.001 |

mu decrease factor | 0.1 |

mu increase factor | 10 |

Maximum mu | $1\times {10}^{10}$ |

Epochs between displays | 25 |

Generate command-line output | false |

Show training GUI | true |

Maximum time to train in seconds | inf |

Maximum number of epochs | 300 |

Regularization parameter | 0.8 |

Transfer function in hidden layer | tan-sigmoid (“tansig”) |

Transfer function in output layer | linear(“purelin”) |

**Table 2.**Time consumption of learning for artificial neural networks with and without early stopping.

Training | Time Consumed by the Algorithm (s) |
---|---|

Default training (with levenberg algorithm) | $126.863$ |

Training with early stopping method (using levenberg algorithm) | $19.276$ |

**Table 3.**Total CPU time (in seconds) used by electromagnetic calculation optimized through an ANN algorithm.

EM Calculation Using ANN Optimization | ANN with Spatial MoM Coding | ANN with Floquet MoM Coding |
---|---|---|

Elapsed CPU Time (in seconds) | $2704.502344$ | $1750.368348$ |

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## Share and Cite

**MDPI and ACS Style**

Bilel, H.; Taoufik, A.
Radiation Pattern Synthesis of the Coupled almost Periodic Antenna Arrays Using an Artificial Neural Network Model. *Electronics* **2022**, *11*, 703.
https://doi.org/10.3390/electronics11050703

**AMA Style**

Bilel H, Taoufik A.
Radiation Pattern Synthesis of the Coupled almost Periodic Antenna Arrays Using an Artificial Neural Network Model. *Electronics*. 2022; 11(5):703.
https://doi.org/10.3390/electronics11050703

**Chicago/Turabian Style**

Bilel, Hamdi, and Aguili Taoufik.
2022. "Radiation Pattern Synthesis of the Coupled almost Periodic Antenna Arrays Using an Artificial Neural Network Model" *Electronics* 11, no. 5: 703.
https://doi.org/10.3390/electronics11050703