# Hybrid Precoding Applied to Multi-Beam Transmitting Reconfigurable Intelligent Surfaces (T-RIS)

^{*}

## Abstract

**:**

## 1. Introduction and Motivations

- (i)
- We derive an analytical propagation model that includes a T-RIS propagation channel and constraints on the resolution of the phase shifters. The obtained model is general and can be easily adapted to most use cases by setting the array geometry, the illumination law, and the pertinent radiation patterns for the focal sources and unit cells.
- (ii)
- (iii)
- We propose and compare two methods to optimize the T-RIS: (i) maximization of the per-user gain (RF-only approach), and (ii) maximization of the per-user rate (hybrid approach). To this end, we fix the array geometry and the set of unit cells (1-bit design) and then we determine the optimal focal distance, i.e., the distance between the lens and the focal sources, with respect to the two aforementioned metrics.
- (iv)
- We provide an evaluation of the two metrics for four served users with imperfect channel knowledge, with an emphasis on the impact of the beam codebook size.

## 2. System Model

#### 2.1. Coordinate System

#### 2.2. Analytical Model: T-RIS

#### 2.3. Analytical Model: Over-the-Air Propagation Channel

#### 2.4. Baseband System Model

## 3. Problem Formulation and Precoder Design

#### 3.1. Problem Formulation

#### 3.2. RF Precoder

#### 3.3. Equivalent Channel and Digital Precoder

#### 3.4. Optimization of T-RIS Structure

- Gain optimization:The T-RIS structure $\mathbf{T}$ is determined by maximizing the antenna array gain, as defined in (19). By doing so, $\mathbf{T}$ is the only function of the RF precoder.$$G\left(\right)open="("\; close=")">{\theta}_{k},{\varphi}_{k}$$The optimization problem can thus be expressed as follows:$${\widehat{d}}_{f}=\underset{{d}_{f}}{\mathrm{maximize}}\phantom{\rule{1.em}{0ex}}{\int}_{{\theta}_{k}=0}^{{\theta}_{\mathrm{max}}}{\int}_{{\varphi}_{k}}^{2\pi}\left(\right)open="["\; close="]">G\left(\right)open="("\; close=")">{d}_{f}d{\theta}_{k}d{\varphi}_{k}$$
- Capacity optimization:Another proposed method is to directly optimize the T-RIS structure $\mathbf{T}$ to maximize the per-user capacity (8). Contrary to the optimization of the gain, the IUI is taken into account here and, therefore, so is the impact of the ZF precoder. The optimization problem can be stated as follows :$${\widehat{d}}_{f}=\underset{{d}_{f}}{\mathrm{maximize}}\phantom{\rule{1.em}{0ex}}R\left(\right)open="("\; close=")">{d}_{f}$$

## 4. Unit-Cell Design and Characterization

#### 4.1. Unit-Cell Design and Frequency Behavior

#### 4.2. Validation of the Proposed Model

## 5. Multi-User Performance Evaluation

#### 5.1. Definitions of Scenarios

#### 5.2. Optimal Focal Distance

#### 5.3. Codebook-Aware Optimization

## 6. Conclusions and Perspectives

#### Perspectives

- (i)
- Power allocation: We observed during the performance evaluation that low, complex ZF can strongly attenuate user streams when beam overlapping occurs. It limits the TX gain and, by extension, the coverage. We believe that a solution with improved power efficiency would greatly improve system performance.
- (ii)
- Source and unit-cell radiation pattern: One challenge for T-RIS is to limit its form factor and investigate systems with reduced focal distances. However, to do so, it is necessary to consider more accurate modeling of sources and unit-cell radiation patterns. In this case, it is possible to use models derived from measurements.
- (iii)
- Wideband transmissions and beam split: With the rise in frequency, signal bandwidth increases and some wideband-induced effects can occur such as beam split. Beam split means that the beam direction moves with the frequency, which induces a power loss and higher IUI. This effect must be taken into account for high-mmWave and sub-THz system design.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Geometry for the analysis of the T-RIS, and (

**b**) Schematics of hybrid digital/analog multi-user beamformer.

**Figure 2.**Representations of the considered codebooks with, respectively, 21, 43, 73, and 111 beams.

**Figure 3.**Proposed T-RIS architecture design and performance: (

**a**) 3D sketch of the 1-bit reconfigurable PIN diode unit cell, (

**b**) unit-cell measurement setup in a standard WR-28 waveguide, (

**c**) simulated and measured transmission coefficients in amplitude with an embedded photograph of the measurement setup, and (

**d**) simulated and measured transmission coefficients in phase.

**Figure 4.**Radiation patterns for a mono-source T-RIS equipped with $20\times 20$ lens ($28.0$ GHz, focal distance 60 mm, focal source = horn antenna with 10 dBi gain).

**Figure 5.**Gains of the broadside T-RIS (mono-source configuration) as a function of frequency: numerical and experimental results. The RF precoder was specified at $28.0$ GHz for a focal distance of 60 mm.

**Figure 6.**Optimal focal distances for the three scenarios. (

**a**) Gain optimization, (

**b**) Capacity optimization.

Frequency | Reconfigurable Device | Phase Quantization | References |
---|---|---|---|

C-band | Varactors | continuous | [20,21] |

C-band | Varactors and PIN diodes | 16-bit | [22] |

X-band | PIN diodes | 1-bit | [24,25] |

Ku-band | PIN diodes | 1-bit | [26,27,28] |

K-band | Varactors | continuous | [23] |

Ka-band | PIN diodes | 1-bit | [29,30] |

Ka-band | MEMS switches | 2-bit | [31] |

Ka-band | PIN diodes | 2-bit | [32,33,34] |

D-band | PCM-based switches | 1-bit | [35] |

300 GHz | CMOS switches | 8-bit | [36] |

Unit Cell | Insertion Loss (dB) at 28.0 GHz | −2 dB Bandwidth (% @ 27.0 GHz) | −3 dB Bandwidth (% @ 27.0 GHz) |
---|---|---|---|

${0}^{\circ}$ | $1.50$ | $22.9$ | $23.7$ |

${180}^{\circ}$ | $2.10$ | $25.5$ | $26.9$ |

Carrier Frequency | $28.0$ GHz |

Lens | uniform square grid |

$20\times 20$ unit cells | |

size $D\times D$ with $D=20\frac{\lambda}{2}=10.7$ cm | |

1-bit phase quantization | |

omnidirectional antennas | |

Focal Source | |

$2\times 2$ | |

square arrangement with distance $D/2$ | |

horn antennas 10 dBi gain ($p=4$) |

Desk | Office Room | Factory Hanger | |
---|---|---|---|

Max aperture ${\theta}_{\mathrm{max}}$ ${(}^{\circ})$ | 20 | 60 | 80 |

Radius [m] | $0.7$ | $3.5$ | $11.3$ |

Area [m${}^{2}$] | $0.8$ | $18.8$ | $202.1$ |

Scenarios: | Desk | Office Room | Factory Hanger | ||
---|---|---|---|---|---|

Codebook size | Reference | $21.0$ | $20.3$ | $18.2$ | |

21 | ref opt. | $20.0$ | $17.6$ | $13.1$ | |

c-a opt. | $20.1$$(+\mathbf{0.5}\%)$ | $18.2$$(+\mathbf{5.7}\%)$ | $14.0$$(+\mathbf{6.9}\%)$ | ||

43 | ref opt. | $20.4$ | $18.9$ | $13.8$ | |

c-a opt. | $20.5$$(+\mathbf{0.5}\%)$ | $19.1$$(+\mathbf{1.1}\%)$ | $14.8$$(+\mathbf{7.2}\%)$ | ||

73 | ref opt. | $20.4$ | $19.5$ | $14.5$ | |

c-a opt. | $20.5$$(+\mathbf{0.5}\%)$ | $19.6$$(+\mathbf{0.5}\%)$ | $15.3$$(+\mathbf{5.5}\%)$ | ||

111 | ref opt. | $20.6$ | $19.85$ | $15.1$ | |

c-a opt. | $20.6$$(+\mathbf{0.0}\%)$ | $19.8$$(+\mathbf{0.0}\%)$ | $15.8$$(+\mathbf{4.6}\%)$ |

**Table 6.**Optimal focal distances ${\widehat{d}}_{f}/D$ for reference and codebook-aware optimizations.

Scenarios: | Desk | Office Room | Factory Hanger | |
---|---|---|---|---|

Codebook | Reference | $0.26$ | $0.26$ | $0.26$ |

21 | $0.14$$(-\mathbf{23.1}\%)$ | $0.14$$(-\mathbf{46.2}\%)$ | $0.10$$(-\mathbf{61.5}\%)$ | |

43 | $0.22$$(-\mathbf{15.4}\%)$ | $0.21$$(-\mathbf{19.2}\%)$ | $0.12$$(-\mathbf{47.8}\%)$ | |

73 | $0.22$$(-\mathbf{15.4}\%)$ | $0.21$$(-\mathbf{19.2}\%)$ | $0.11$$(-\mathbf{57.7}\%)$ | |

111 | $0.25$$(-\mathbf{3.8}\%)$ | $0.22$$(-\mathbf{15.4}\%)$ | $0.15$$(-\mathbf{42.3}\%)$ |

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## Share and Cite

**MDPI and ACS Style**

Demmer, D.; Foglia Manzillo, F.; Gharbieh, S.; Śmierzchalski, M.; D’Errico, R.; Doré, J.-B.; Clemente, A.
Hybrid Precoding Applied to Multi-Beam Transmitting Reconfigurable Intelligent Surfaces (T-RIS). *Electronics* **2023**, *12*, 1162.
https://doi.org/10.3390/electronics12051162

**AMA Style**

Demmer D, Foglia Manzillo F, Gharbieh S, Śmierzchalski M, D’Errico R, Doré J-B, Clemente A.
Hybrid Precoding Applied to Multi-Beam Transmitting Reconfigurable Intelligent Surfaces (T-RIS). *Electronics*. 2023; 12(5):1162.
https://doi.org/10.3390/electronics12051162

**Chicago/Turabian Style**

Demmer, David, Francesco Foglia Manzillo, Samara Gharbieh, Maciej Śmierzchalski, Raffaele D’Errico, Jean-Baptiste Doré, and Antonio Clemente.
2023. "Hybrid Precoding Applied to Multi-Beam Transmitting Reconfigurable Intelligent Surfaces (T-RIS)" *Electronics* 12, no. 5: 1162.
https://doi.org/10.3390/electronics12051162