# Spherical Fourier-Transform-Based Real-TimeNear-Field Shaping and Focusing in Beyond-5G Networks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Problem Statement

#### 2.2. Far-to-Near-Field Transformation

#### 2.3. Phase Center Translation

#### 2.4. Near-Field Intensity Shaping

## 3. Validation

#### 3.1. Numerical Validation

#### 3.2. Full-Wave Validation and Application

#### 3.3. Discussion on Computational Efficiency

- Novel method: The number of operations required is approximately $3{\mathsf{\Lambda}}^{3}$, where $\mathsf{\Lambda}$ is the chosen order of the spherical Fourier transform. For the largest array studied, with N = 200 and $\mathsf{\Lambda}$ = 30, the shaping procedure required around 200 ms and circa 100 MB of pre-computed data.
- O-mt-TR [31]: The amount of operations required is of order ${M}^{L}$, where M are the variables to optimize, and L is the amount of control points used. As the computational time scales exponentially with L, several hours are required to achieve shaping, hence excluding this algorithm from real-time applications. A variation of this technique, being O-mt-LSM [33], does not require any set-up phase, but this results in reduced computational efficiency and resolution.
- Smart skin holography [13]: While a quite good resolution is obtained in near-field shaping, the process required circa 20 min per shape. Again, this algorithm is also not suited to real-time applications.
- Method of Moments (MoM) [30]: A radial slot array is synthesised with an in-house MoM code and tailored for a specific target shape. The amount of time usually required to fill an MoM matrix and solve the matrix system is not compatible with real-time applications.
- Angular spectrum projection method [28]: This method’s efficiency is quantified as largely faster than the O-mt-TR. At the same time, the obtained target shapes exhibit a very poor resolution and are obtained as a discontinuous collection of discrete points.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Green’s Function Multipole Expansion

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**Figure 1.**Application scenario. Users are distributed in the radiative near-field of the antenna array. Each user is targeted by an individual focal spot to ensure the best quality of communication.

**Figure 2.**(

**a**) Conceptual sketch of an array of volume V circumscribed by a sphere of radius R and with its phase center coinciding with the origin of the coordinate system $\mathcal{O}(x,y,z)$. The goal is to shape the array’s emitted field so that it corresponds to a desired pattern on a sphere with radius ${R}_{\mathrm{T}}$. (

**b**) Each $n\mathrm{th}$ array element, fed through the $n\mathrm{th}$ port, is attached to a local phase center, indicated by a vector ${\mathbf{p}}_{n}$, and coinciding with a local coordinate system’s origin ${\mathcal{O}}_{n}({x}_{n},{y}_{n},{z}_{n})$. The circumscribing sphere with regard to the local coordinate system has a radius ${R}_{n}$. A current ${I}_{n}$ is injected into the $n\mathrm{th}$ port (see also Figure 3), generating a current distribution ${\mathbf{j}}_{n}$ on the entire array.

**Figure 3.**Excitation of the $n\mathrm{th}$ active far-field radiation pattern. The $n\mathrm{th}$ port is excited by a Norton equivalent source formed by a current source ${I}_{\mathrm{g},n}$ in parallel with a 50 $\mathsf{\Omega}$ resistance. The source is chosen such that a current ${I}_{n}$ = 1 A is injected into the $n\mathrm{th}$ antenna, represented by its input impedance ${Z}_{n}$. All other ports for i≠n are instead terminated by a 50 $\mathsf{\Omega}$ resistor.

**Figure 4.**Typical sigmoid behavior of the cumulative power spectra ratio ${\mathsf{\Gamma}}_{n}\left({\mathsf{\Lambda}}_{n}\right)/{\mathsf{\Gamma}}_{n}\left({L}_{n}\right)$ for increasing ${\mathsf{\Lambda}}_{n}$. A sufficient value for ${\mathsf{\Lambda}}_{n}$ is reached when this ratio is at least 99%.

**Figure 5.**Results for a synthesized planar array consisting of 197 x-polarized dipoles. The figures show the amplitudes (

**top**) and phases (

**bottom**) of (

**a**) the feeding currents obtained by the novel algorithm, for a (

**b**) circle-like shaped target near-field and (

**c**) the co-polar electric field solution obtained by the novel method. The results in (

**b**,

**c**) are drawn for the upper hemisphere of a target sphere of radius ${R}_{\mathrm{T}}$ = $6\lambda $, where $u=sin\theta cos\varphi $ and $v=sin\theta sin\varphi $.

**Figure 6.**Results for a synthesized planar array consisting of 197 x-polarized dipoles. The figures show the amplitudes (

**top**) and phases (

**bottom**) of (

**a**) the feeding currents obtained by the novel algorithm, for a (

**b**) triangle-like shaped target near-field and (

**c**) the co-polar electric field solution obtained by the novel method. The results in (

**b**,

**c**) are drawn for the upper hemisphere of a target sphere of radius ${R}_{\mathrm{T}}$ = $6\lambda $, where $u=sin\theta cos\varphi $ and $v=sin\theta sin\varphi $.

**Figure 7.**Results for a 7 × 7 planar array consisting of 49 x-polarized dipoles. The figures show the amplitudes (

**top**) and phases (

**bottom**) of (

**a**,

**e**) two sets of feeding currents obtained by the novel algorithm corresponding to (

**b**,

**f**) two differently shaped target near-fields. Further, we have (

**c**,

**g**) the co-polar electric field solutions obtained by the novel method and (

**d**,

**h**) the co-polar near-fields resulting from a full-wave simulation of the 7 × 7 array, excited by the currents in (

**a**) and (

**e**), respectively. The results are drawn for the upper hemisphere of a target sphere of radius ${R}_{\mathrm{T}}$ = $4\lambda $, where $u=sin\theta cos\varphi $ and $v=sin\theta sin\varphi $.

**Figure 8.**Results for a 7 × 7 planar array consisting of 49 x-polarized dipoles. The figures show the amplitudes (

**top**) and phases (

**bottom**) of (

**a**) the feeding currents obtained by the novel algorithm corresponding to (

**b**) four separate focal spots. Further, we have (

**c**) the co-polar electric field solutions obtained by the novel method and (

**d**) the co-polar near-fields resulting from a full-wave simulation of the 7 × 7 array, excited by the currents in (

**a**). The results are drawn for the upper hemisphere of a target sphere of radius ${R}_{\mathrm{T}}$ = $4\lambda $, where $u=sin\theta cos\varphi $ and $v=sin\theta sin\varphi $.

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## Share and Cite

**MDPI and ACS Style**

Felaco, A.; Kapusuz, K.Y.; Rogier, H.; Vande Ginste, D.
Spherical Fourier-Transform-Based Real-TimeNear-Field Shaping and Focusing in Beyond-5G Networks. *Sensors* **2023**, *23*, 3323.
https://doi.org/10.3390/s23063323

**AMA Style**

Felaco A, Kapusuz KY, Rogier H, Vande Ginste D.
Spherical Fourier-Transform-Based Real-TimeNear-Field Shaping and Focusing in Beyond-5G Networks. *Sensors*. 2023; 23(6):3323.
https://doi.org/10.3390/s23063323

**Chicago/Turabian Style**

Felaco, Alessandro, Kamil Yavuz Kapusuz, Hendrik Rogier, and Dries Vande Ginste.
2023. "Spherical Fourier-Transform-Based Real-TimeNear-Field Shaping and Focusing in Beyond-5G Networks" *Sensors* 23, no. 6: 3323.
https://doi.org/10.3390/s23063323