# Ergodic Performance Analysis of Double Intelligent Reflecting Surfaces-Aided NOMA–UAV Systems with Hardware Impairment

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## Abstract

**:**

## 1. Introduction

#### 1.1. Related Works

#### 1.2. Motivations and Our Contributions

- We consider an IRS–NOMA–UAV system without direct links, which consists of a source and several IRSs. We focus on the performance analysis of a group of two users and further determine the impact of hardware impairment.
- We derive closed-form expressions for the achievable rates for two NOMA users under the channel models of Rayleigh and Rician. Compared with recent work [30], our result could be combined with their result to provide complete ergodic performance analysis in a more practical circumstance.
- We employ Monte Carlo simulations to validate the analytical outage probabilities. The achievable rate of each user mainly depends on power allocation factors rather than other main parameters such as the number of IRSs, the number of meta-surface elements and the IRS reflecting coefficient.

## 2. System Model

#### 2.1. The First Scenario

#### 2.2. The Second Scenario (the Benchmark)

## 3. Ergodic Performance Analysis of the Proposed Scheme Using Rayleigh and Rician Fading Channels

#### 3.1. The First Scenario

#### 3.1.1. Upper Bound for the Achievable Rate Using Rayleigh Fading Channels for ${D}_{1}$

**Proposition**

**1.**

**Proof.**

#### 3.1.2. Upper Bound for the Achievable Rate Using Rayleigh Fading Channels for ${D}_{2}$

**Proposition**

**2.**

**Proof.**

#### 3.1.3. Upper Bound for the Achievable Rate Using Rician Fading Channels for ${D}_{1}$

**Proposition**

**3.**

**Proof.**

#### 3.1.4. Upper Bound for the Achievable Rate Using Rician Fading Channels for ${D}_{2}$

**Proposition**

**4.**

**Proof.**

#### 3.2. The Second Scenario (the Benchmark)

#### 3.2.1. Upper Bound for the Achievable Rate Using Rayleigh Fading Channels for ${D}_{1}$

#### 3.2.2. Upper Bound for the Achievable Rate Using Rayleigh Fading Channels for ${D}_{2}$

**Remark**

**1.**

#### 3.2.3. Upper Bound for the Achievable Rate Using Rician Fading Channels for ${D}_{1}$

#### 3.2.4. Upper Bound for the Achievable Rate Using Rician Fading Channels for ${D}_{2}$

**Remark**

**2.**

## 4. Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof.**

## Appendix B

**Proof.**

## Appendix C

**Proof.**

## Appendix D

**Proof.**

## References

- Chu, Z.; Zhu, Z.; Li, X.; Zhou, F.; Zhen, L.; Al-Dhahir, N. Resource Allocation for IRS-Assisted Wireless-Powered FDMA IoT Networks. IEEE Internet Things J.
**2022**, 9, 8774–8785. [Google Scholar] [CrossRef] - Tang, J.; Chen, X.; Zhu, X.; Zhu, F. Dynamic Reallocation Model of Multiple Unmanned Aerial Vehicle Tasks in Emergent Adjustment Scenarios. IEEE Trans. Aerosp. Electron. Syst.
**2022**, 1–43. [Google Scholar] [CrossRef] - Tang, J.; Liu, G.; Pan, Q. A Review on Representative Swarm Intelligence Algorithms for Solving Optimization Problems: Applications and Trends. IEEE/CAA J. Autom. Sin.
**2021**, 8, 1627–1643. [Google Scholar] [CrossRef] - Li, B.; Wu, W.; Li, Y.; Zhao, W. Intelligent Reflecting Surface and Artificial-Noise-Assisted Secure Transmission of MEC System. IEEE Internet Things J.
**2022**, 9, 11477–11488. [Google Scholar] [CrossRef] - Shen, H.; Xu, W.; Gong, S.; He, Z.; Zhao, C. Secrecy rate maximization for intelligent reflecting surface assisted multi-antenna communications. IEEE Commun. Lett.
**2019**, 23, 1488–1492. [Google Scholar] [CrossRef] [Green Version] - Zhou, G.; Pan, C.; Ren, H.; Wang, K.; Nallanathan, A. A framework of robust transmission design for IRS-aided MISO communications with imperfect cascaded channels. IEEE Trans. Signal Process.
**2020**, 68, 5092–5106. [Google Scholar] [CrossRef] - Pang, X.; Zhao, N.; Tang, J.; Wu, C.; Niyato, D.; Wong, K.K. IRS-Assisted Secure UAV Transmission via Joint Trajectory and Beamforming Design. IEEE Trans. Commun.
**2022**, 70, 1140–1152. [Google Scholar] [CrossRef] - Feng, K.; Wang, Q.; Li, X.; Wen, C.K. Deep Reinforcement Learning Based Intelligent Reflecting Surface Optimization for MISO Communication Systems. IEEE Wirel. Commun. Lett.
**2020**, 9, 745–749. [Google Scholar] [CrossRef] - Wu, Q.; Zhang, R. Towards Smart and Reconfigurable Environment: Intelligent Reflecting Surface Aided Wireless Network. IEEE Commun. Mag.
**2020**, 58, 106–112. [Google Scholar] [CrossRef] [Green Version] - Do, D.-T.; Le, A.-T. NOMA based Cognitive Relaying: Transceiver Hardware Impairments, Relay Selection Policies and Outage Performance Comparison. Comput. Commun.
**2019**, 146, 144–154. [Google Scholar] [CrossRef] - Do, D.-T.; Van Nguyen, M.-S. Device-to-device transmission modes in NOMA network with and without Wireless Power Transfer. Comput. Commun.
**2019**, 139, 67–77. [Google Scholar] [CrossRef] - Dai, L.; Wang, B.; Yuan, Y.; Han, S.; Chih-lin, I.; Wang, Z. Non-orthogonal multiple access for 5G: Solutions, challenges, opportunities, and future research trends. IEEE Commun. Mag.
**2015**, 53, 74–81. [Google Scholar] [CrossRef] - Van Nguyen, M.S.; Do, D.T.; Al-Rubaye, S.; Mumtaz, S.; Al-Dulaimi, A.; Dobre, O. Exploiting Impacts of Antenna Selection and Energy Harvesting for Massive Network Connectivity. IEEE Trans. Commun.
**2021**, 69, 7587–7602. [Google Scholar] [CrossRef] - Do, D.T.; Le, A.T.; Liu, Y.; Jamalipour, A. User Grouping and Energy Harvesting in UAV-NOMA System with AF/DF Relaying. IEEE Trans. Veh. Technol.
**2021**, 70, 11855–11868. [Google Scholar] [CrossRef] - Do, D.T.; Van Nguyen, M.S.; Voznak, M.; Kwasinski, A.; de Souza, J.N. Performance Analysis of Clustering Car-Following V2X System With Wireless Power Transfer and Massive Connections. IEEE Internet Things J.
**2022**, 9, 14610–14628. [Google Scholar] [CrossRef] - Sun, Z.; Jing, Y. On the Performance of Multi-Antenna IRS-Assisted NOMA Networks With Continuous and Discrete IRS Phase Shifting. IEEE Trans. Wirel. Commun.
**2022**, 21, 3012–3023. [Google Scholar] [CrossRef] - Li, Z.; Chen, W.; Wu, Q.; Wang, K.; Li, J. Joint Beamforming Design and Power Splitting Optimization in IRS-Assisted SWIPT NOMA Networks. IEEE Trans. Wirel. Commun.
**2022**, 21, 2019–2033. [Google Scholar] [CrossRef] - Zhang, X.; Zhang, H.; Du, W.; Long, K.; Nallanathan, A. IRS Empowered UAV Wireless Communication with Resource Allocation, Reflecting Design and Trajectory Optimization. IEEE Trans. Wirel. Commun.
**2022**, 21, 7867–7880. [Google Scholar] [CrossRef] - Ding, Z.; Poor, H.V. A simple design of IRS-NOMA transmission. IEEE Commun. Lett.
**2020**, 24, 1119–1123. [Google Scholar] [CrossRef] [Green Version] - Ding, Z.; Schober, R.; Poor, H.V. On the impact of phase shifting designs on IRS-NOMA. IEEE Wirel. Commun. Lett.
**2020**, 9, 1596–1600. [Google Scholar] [CrossRef] - Zhu, J.; Huang, Y.; Wang, J.; Navaie, K.; Ding, Z. Power Efficient IRS-Assisted NOMA. IEEE Trans. Commun.
**2021**, 69, 900–913. [Google Scholar] [CrossRef] - Fu, M.; Zhou, Y.; Shi, Y. Intelligent Reflecting Surface for Downlink Non-Orthogonal Multiple Access Networks. In Proceedings of the IEEE Globecom Workshops (GC Wkshps), Waikoloa, HI, USA, 9–13 December 2019; pp. 1–6. [Google Scholar]
- Yang, G.; Xu, X.; Liang, Y.C. Intelligent Reflecting Surface Assisted Non-Orthogonal Multiple Access. In Proceedings of the IEEE Wireless Communications and Networking Conference (WCNC), Seoul, Korea, 25–28 May 2020; pp. 1–6. [Google Scholar]
- Mu, X.; Liu, Y.; Guo, L.; Lin, J.; Al-Dhahir, N. Exploiting Intelligent Reflecting Surfaces in NOMA Networks: Joint Beamforming Optimization. IEEE Trans. Wirel. Commun.
**2020**, 19, 6884–6898. [Google Scholar] [CrossRef] - Hou, T.; Liu, Y.; Song, Z.; Sun, X.; Chen, Y.; Hanzo, L. Reconfigurable Intelligent Surface Aided NOMA Networks. IEEE J. Sel. Areas Commun.
**2020**, 38, 2575–2588. [Google Scholar] [CrossRef] - Singh, S.K.; Agrawal, K.; Singh, K.; Li, C.P.; Ding, Z. NOMA Enhanced Hybrid RIS-UAV-Assisted Full-Duplex Communication System With Imperfect SIC and CSI. IEEE Trans. Commun.
**2022**, 70, 7609–7627. [Google Scholar] [CrossRef] - Gan, X.; Zhong, C.; Huang, C.; Zhang, Z. RIS-Assisted Multi-User MISO Communications Exploiting Statistical CSI. IEEE Trans. Commun.
**2021**, 69, 6781–6792. [Google Scholar] [CrossRef] - Zhang, Y.; Zhang, J.; Di Renzo, M.; Xiao, H.; Ai, B. Reconfigurable Intelligent Surfaces With Outdated Channel State Information: Centralized vs. Distributed Deployments. IEEE Trans. Commun.
**2022**, 70, 2742–2756. [Google Scholar] [CrossRef] - Selimis, D.; Peppas, K.P.; Alexandropoulos, G.C.; Lazarakis, F.I. On the Performance Analysis of RIS-Empowered Communications Over Nakagami-m Fading. IEEE Commun. Lett.
**2021**, 25, 2191–2195. [Google Scholar] [CrossRef] - Sun, Q.; Qian, P.; Duan, W.; Zhang, J.; Wang, J.; Wong, K.K. Ergodic Rate Analysis and IRS Configuration for Multi-IRS Dual-Hop DF Relaying Systems. IEEE Commun. Lett.
**2021**, 25, 3224–3228. [Google Scholar] [CrossRef] - Ren, H.; Wang, K.; Pan, C. Intelligent Reflecting Surface-Aided URLLC in a Factory Automation Scenario. IEEE Trans. Commun.
**2022**, 70, 707–723. [Google Scholar] [CrossRef] - Xing, Z.; Wang, R.; Wu, J.; Liu, E. Achievable Rate Analysis and Phase Shift Optimization on Intelligent Reflecting Surface With Hardware Impairments. IEEE Trans. Wirel. Commun.
**2021**, 20, 5514–5530. [Google Scholar] [CrossRef] - Tran, P.T.; Nguyen, B.C.; Hoang, T.M.; Nguyen, T.N. On Performance of Low-Power Wide-Area Networks with the Combining of Reconfigurable Intelligent Surfaces and Relay. In IEEE Transactions on Mobile Computing; IEEE: Piscataway, NJ, USA, 2022. [Google Scholar]
- Yue, X.; Liu, Y.; Kang, S.; Nallanathan, A.; Ding, Z. Exploiting Full/Half-Duplex User Relaying in NOMA Systems. IEEE Trans. Commun.
**2018**, 66, 560–575. [Google Scholar] [CrossRef] [Green Version] - Yang, L.; Yang, Y.; Hasna, M.O.; Alouini, M.S. Coverage, Probability of SNR Gain, and DOR Analysis of RIS-Aided Communication Systems. IEEE Wirel. Commun. Lett.
**2020**, 9, 1268–1272. [Google Scholar] [CrossRef] - Abdullah, Z.; Chen, G.; Lambotharan, S.; Chambers, J.A. A Hybrid Relay and Intelligent Reflecting Surface Network and Its Ergodic Performance Analysis. IEEE Wirel. Commun. Lett.
**2020**, 9, 1653–1657. [Google Scholar] [CrossRef]

**Figure 3.**The performance of overall achievable rates by varying the number of IRSs (N = ${N}_{1}$= ${N}_{2}$= ${N}_{3}$) when changing $\gamma $ with Rayleigh fading.

**Figure 4.**Impact of power allocation factors ${\chi}_{1}$ on overall achievable rates by varying $\gamma $ with Rayleigh fading.

**Figure 5.**Impact of hardware impairments ${\mathrm{{\rm Y}}}_{S}={\mathrm{{\rm Y}}}_{{D}_{1}}={\mathrm{{\rm Y}}}_{{D}_{2}}$ on overall achievable rates by varying $\gamma $ with Rayleigh fading.

**Figure 6.**Impact on the performance of overall achievable rates by varying ${x}_{{I}_{1}}={x}_{{I}_{2}}={x}_{{I}_{3}}$ when changing $\alpha $ with Rayleigh fading.

**Figure 7.**The achievable rates versus N = ${N}_{1}$= ${N}_{2}$= ${N}_{3}$ when changing ${\chi}_{1}$ with Rician fading.

**Figure 8.**Impact on the performance of overall achievable rates by varying $\gamma $ when changing the Rician-K factor.

**Figure 9.**Impact on the performance of overall achievable rates by varying $\gamma $ when changing $\eta $ with Rician fading.

**Figure 10.**Impact on the performance of overall achievable rates of Rayleigh fading and Rician fading by varying $\gamma $.

**Figure 11.**Impact on the performance of overall achievable rates of Rayleigh fading and Rician fading between NOMA and OMA with two IRSs by varying $\gamma $.

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**MDPI and ACS Style**

Van Nguyen, M.-S.; Do, D.-T.; Phan, V.-D.; Ullah Khan, W.; Imoize, A.L.; Fouda, M.M.
Ergodic Performance Analysis of Double Intelligent Reflecting Surfaces-Aided NOMA–UAV Systems with Hardware Impairment. *Drones* **2022**, *6*, 408.
https://doi.org/10.3390/drones6120408

**AMA Style**

Van Nguyen M-S, Do D-T, Phan V-D, Ullah Khan W, Imoize AL, Fouda MM.
Ergodic Performance Analysis of Double Intelligent Reflecting Surfaces-Aided NOMA–UAV Systems with Hardware Impairment. *Drones*. 2022; 6(12):408.
https://doi.org/10.3390/drones6120408

**Chicago/Turabian Style**

Van Nguyen, Minh-Sang, Dinh-Thuan Do, Van-Duc Phan, Wali Ullah Khan, Agbotiname Lucky Imoize, and Mostafa M. Fouda.
2022. "Ergodic Performance Analysis of Double Intelligent Reflecting Surfaces-Aided NOMA–UAV Systems with Hardware Impairment" *Drones* 6, no. 12: 408.
https://doi.org/10.3390/drones6120408