# Gaussian Approach for the Synthesis of Periodic and Aperiodic Antenna Arrays: Method Review and Design Guidelines

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## Abstract

**:**

## 1. Introduction

## 2. The Problem and the Gaussian Approach

- First, a Gaussian function may be used to approximate a pencil beam with a desired BW by suitably controlling its standard deviation. More precisely, by denoting with $u=kcos\theta $, the desired Gaussian function:$${F}_{\mathrm{d}}\left(u\right)=exp\left(-\frac{{u}^{2}}{2{\sigma}^{2}}\right),$$$$\sigma =k\sqrt{\frac{10}{\mathrm{b}ln10}}cos\left(\frac{\pi +\mathrm{BW}}{2}\right).$$
- Second, a Fourier transform relation holds between a continuous linear infinite source $a\left(z\right)$ and its far-field pattern [40]. Moreover, the Fourier transform of a Gaussian function may be evaluated as another Gaussian function with reciprocal standard deviation (Figure 1).Hence, if ${F}_{\mathrm{d}}\left(u\right)$ in (2) represents the desired far-field pattern, the corresponding continuous source is immediately evaluated as follows:$$a\left(z\right)=\frac{1}{2\pi}{\int}_{-\infty}^{\infty}{F}_{\mathrm{d}}\left(u\right)exp(-jzu)du=\frac{\sigma}{\sqrt{2\pi}}exp\left(-\frac{{\sigma}^{2}{z}^{2}}{2}\right).$$The expression for the continuous infinite source $a\left(z\right)$ in (4) that exactly produces the pattern in (2) can be successively truncated to a finite length L and processed in order to approximate the array factor in (1). Thanks to the Gaussian nature of $a\left(z\right)$, closed-form expressions are obtained in both the PS and the ES problems.

#### 2.1. Position Synthesis—Aperiodic Arrays

#### 2.2. Excitation Synthesis—Periodic Arrays

## 3. Numerical Investigation

#### 3.1. Aperiodic Arrays

#### 3.2. Periodic Arrays

## 4. Parametric Analysis

#### First-Step Design

**Statement.**Design a linear broadside array of length $L/\lambda \le 100$, with HPBW = 1${}^{\circ}$ and maximum SLL not exceeding −20 dB.

**Procedure.**Follow the following steps.

- Step 1: Use Figure 3a for the BW requirement. The design curves suggest that an array with $L/\lambda =30$ is sufficient to satisfy the requirement in terms of HPBW for both ES and PS. Go to step 2.
- Step 2: Use again Figure 3a but now for the SLL requirement. The design curves show that the maximum SLL is considerably lower than the required threshold if ES is adopted, but slightly exceeds the threshold if PS is used. Therefore, if ES is suitable for the specific application, go to Step 3; otherwise, go to Step 4.
- Step 3: Use Figure 4a for b = 3 and $L/\lambda =30$. The identified point reveals that the DRR of the excitations is approximately 7. If this is acceptable, go to Step 6; otherwise, PS must be adopted. Go to Step 4.
- Step 4: Use Figure 3a for the PS with $L/\lambda =30$. The relative curve shows that the maximum SLL is slightly higher than the required −20 dB, but the SLL constraint and the BW one are both satisfied when $L/\lambda =35$. Go to Step 5.
- Step 5: Minimize the number N of elements for the PS with BW = 1${}^{\circ}$, b = 3 and $L/\lambda =35$ by plotting the required curve in Figure 8a, which as outlined at the beginning of this subsection, requires a low computational time (just 16 ms of CPU time using a commercial personal laptop in this case). This novel curve suggests that $N=60$ elements allows one to meet the SLL requirement. Go to Step 7.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) ${F}_{\mathrm{d}}\left(u\right)$ obtained by setting BW = 1${}^{\circ}$ and b = 3 in (3). (

**b**) Corresponding continuous source $a\left(z\right)$. (

**c**) ${F}_{\mathrm{d}}\left(u\right)$ obtained by setting BW = 0.1${}^{\circ}$ and b = 3 in (3). (

**d**) Corresponding continuous source $a\left(z\right)$. As it may be noticed from each pair of figures, the narrower the desired pattern, the wider the continuous source.

**Figure 2.**(

**a**) Position synthesis (PS): the interval extrema ${s}_{n}$ are evaluated by imposing the equi-area requirement of the density tapering approach. The final positions ${z}_{n}$ are the middle points of each interval. (

**b**) Excitation synthesis (ES): each element excitation ${a}_{n}$ is evaluated as the area between the z-axis and the graph of $a\left(z\right)$ in the interval $[{z}_{n}-d/2,{z}_{n}+d/2]$.

**Figure 3.**Beamwidth (BW) and sidelobe levels (SLL) as a function of the array aperture. Solid lines refer to PS, dashed lines refer to ES, lines with markers refer to the case b = 100, and lines without markers refer to the case b = 3: (

**a**) desired BW = 1${}^{\circ}$ and (

**b**) desired BW = 0.1${}^{\circ}$.

**Figure 4.**Dynamic range ratio (DRR) and miminum/maximum interelement distance as a function of the array aperture. Lines with markers refer to the case b = 100, and lines without markers refer to the case b = 3: (

**a**,

**c**) BW = 1${}^{\circ}$, (

**b**,

**d**) BW = 0.1${}^{\circ}$.

**Figure 5.**Obtained array factor with b = 3. Blue dashed lines refers to PS, and red solid lines refer to ES: (

**a**) desired BW = 1${}^{\circ}$ and $L/\lambda $ = 60, (

**b**) desired BW = 0.1${}^{\circ}$ and $L/\lambda $ = 300. The inset in the second subfigure reports a zoom of the main beam and of the first sidelobes.

**Figure 6.**Obtained directivity and SLL as a function of the number of elements. Solid lines refer to PS, and dashed lines refer to ES: (

**a**) desired BW = 1${}^{\circ}$, b = 3, $L/\lambda =60$, (

**b**) desired BW = 0.1${}^{\circ}$, b = 3, $L/\lambda =300$.

**Figure 7.**Array factors of the Gaussian approach obtained with b = 3. Blue dashed lines refer to PS, and red solid lines refer to ES: (

**a**) desired BW = 1${}^{\circ}$, $L/\lambda =60,N=30$, (

**b**) desired BW = 1${}^{\circ}$, $L/\lambda =60,N=300$, (

**c**) desired BW = 0.1${}^{\circ}$, $L/\lambda =300,N=150$, and (

**d**) desired BW = 0.1${}^{\circ}$, $L/\lambda =300,N=1500$. The inset in the last subfigure reports a zoom of the main beam and of the first sidelobes.

**Figure 8.**Application example. (

**a**) SLL as a function of the number of elements for PS with desired BW = 1${}^{\circ}$, b = 3, $L/\lambda =35$. (

**b**) The red solid line represents the array factor obtained by ES, and the blue dashed line represents the array factor obtained by PS.

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**MDPI and ACS Style**

Buttazzoni, G.; Babich, F.; Pastore, S.; Vatta, F.; Comisso, M.
Gaussian Approach for the Synthesis of Periodic and Aperiodic Antenna Arrays: Method Review and Design Guidelines. *Sensors* **2021**, *21*, 2343.
https://doi.org/10.3390/s21072343

**AMA Style**

Buttazzoni G, Babich F, Pastore S, Vatta F, Comisso M.
Gaussian Approach for the Synthesis of Periodic and Aperiodic Antenna Arrays: Method Review and Design Guidelines. *Sensors*. 2021; 21(7):2343.
https://doi.org/10.3390/s21072343

**Chicago/Turabian Style**

Buttazzoni, Giulia, Fulvio Babich, Stefano Pastore, Francesca Vatta, and Massimiliano Comisso.
2021. "Gaussian Approach for the Synthesis of Periodic and Aperiodic Antenna Arrays: Method Review and Design Guidelines" *Sensors* 21, no. 7: 2343.
https://doi.org/10.3390/s21072343