# Performance Analysis of Multihop Full-Duplex NOMA Systems with Imperfect Interference Cancellation and Near-Field Path-Loss

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## Abstract

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## 1. Introduction

- We take into account the impact of the imperfect interference cancellation (IC) at all receivers. We consider the near-field path-loss at relays to better capture the short transmission distance from the transmit and receive antennae at the relay.
- We take into account the interhop interference and self-interference at all relays due to the full-duplex protocol. It, as a consequence, makes the mathematical framework more complicated compared with half-duplex relaying where the orthogonal transmission between hops is employed.
- We derive closed-form expressions of the OP and potential throughput (PT) of the considered systems.
- We unveil the impact of the total transmit power on the performance of both OP and PT by employing rigorously mathematical frameworks instead of numerical computations.
- We provide remarks to highlight the influence of elements in the OP framework.
- We also derive the mathematical framework of the baseline system to highlight the advantage of the proposed system.
- We supply numerical results via the Monte Carlo method to verify the accuracy of the derived mathematical framework.

## 2. System Model

#### 2.1. Channel Modelling

#### 2.1.1. Small-Scale fading

#### 2.1.2. Path-Loss

#### Far-Field Path-Loss

#### Near-Field Path-Loss

#### 2.2. Signal-to-Interference-Plus-Noise Ratios (SINRs)

#### 2.2.1. Perfect Interference Cancellation (PIC)

#### 2.2.2. Imperfect Interference Cancellation (IIC)

## 3. Performance Analysis and Trends

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

**Theorem**

**3.**

**Proof.**

#### 3.1. Performance Trends

**Proposition**

**1.**

**Proof.**

#### 3.2. Performance of Baseline System

## 4. Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Proof of Proposition 1

## Appendix B. Proof of Theorem 2

## Appendix C. Proof of Theorem 3

## Appendix D. Proof of Proposition 1

## References

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**Figure 3.**Potential throughput vs. R under all schemes. Solid lines are plotted by employing (13) while markers are Monte Carlo simulation.

**Figure 5.**Potential throughput vs. ${P}_{\mathrm{tot}}$ under all schemes. Solid lines are plotted by employing (13) while markers are Monte Carlo simulation.

**Figure 7.**Potential throughput vs. ${d}_{m,m}$ under all schemes. Solid lines are plotted by employing (13) while markers are Monte Carlo simulation.

**Figure 9.**Potential throughput vs. ${\alpha}_{1}$ under all schemes. Solid lines are plotted by employing (13) while markers are Monte Carlo simulation.

**Figure 11.**Potential throughput vs. $\mathcal{M}$ under all schemes. Solid lines are plotted by employing (13) while markers are Monte Carlo simulation.

**Figure 12.**Outage probability vs. R under the impact of imperfect interference cancellation, hardware impairment (HI), and imperfect channel state information (ICSI). Solid lines are plotted by employing (16) while markers are Monte Carlo simulation.

Symbol | Definition |
---|---|

$\mathbb{E}\left\{.\right\}$, $Pr\left(.\right)$ | Expectation and probability operators |

${h}_{u,v}$ | Channel coefficient between transmitter u and receiver v |

${\varsigma}_{u,v}^{F}$ | Far-field path-loss between transmitter u and receiver v |

${\varsigma}_{m}^{N}$ | Near-field path-loss between of the ${R}_{m}$ relay |

${K}_{0}$, c | Path-loss constant, speed of light |

v, ${f}_{c}$, $\eta $ | Wavelength, carrier frequency, path-loss exponent |

${d}_{u,v}$ | Transmission distance from node u to node v |

${P}_{\mathrm{tot}}$ | Total transmit power of the whole networks |

${P}_{m}$, ${P}_{0}$ | Transmit power of the ${R}_{m}$ relay and source node |

${\alpha}_{1}$, ${\alpha}_{2}$ | Coefficients of power allocation for ${D}_{1}$ and ${D}_{2}$ |

L, $\mathcal{M}$ | Maximum size of the received antenna & number of relays |

${d}_{RD}$, ${G}_{T}$, ${G}_{R}$ | Rayleigh distance, transmit and receive antennae gain |

${R}_{a}$, $a\in \left\{1,2\right\}$ | Targeted rate of ${D}_{a}$ destination |

${\epsilon}_{v}$ | Residue of the imperfect SIC at v receiver |

${r}_{m}$ | Residue of the self-interference cancellation at ${R}_{m}$ relay |

${x}_{a}$, ${\tilde{x}}_{a}$, ${\tilde{\tilde{x}}}_{a}$ | Intended signals for ${D}_{a}$ sent by ${R}_{m-1}$, ${R}_{z}$ and ${R}_{t}$ relays |

${y}_{{R}_{m}}$, ${y}_{{D}_{a}}$ | Received signals at the ${R}_{m}$ relay and ${D}_{a}$ destination |

${n}_{m}$, ${n}_{{D}_{a}}$ | AWGN noise at the ${R}_{m}$ relay and ${D}_{a}$ destination |

${\sigma}^{2}$ | Noise variance at all the receiver |

NF, Bw | Noise figure, transmission bandwidth |

$\mathsf{\Phi}$, $H\left(.\right)$ | Average transmit-power-to-noise-ratio and Heaviside function |

${\omega}_{u,v}$ | Variance of small-scale fading from transmitter u to receiver v |

$exp\left(.\right)$, $log\left(.\right)$ | Exponential and logarithm functions |

$max\left(.\right)$, $min\left(.\right)$ | Maximum and minimum functions |

${F}_{X}\left(x\right)$ | Cumulative distribution function (CDF) of RV X |

${\overline{F}}_{X}\left(x\right)$ | Complementary Cumulative distribution function (CCDF) of RV X |

${M}_{X}\left(x\right)$ | Moment generating function (MGF) of RV X |

${f}_{X}\left(x\right)$ | Probability density function (PDF) of RV X |

OP${}_{a}^{w}$ | Outage probability of the ${D}_{a}$ destination under w scheme |

PT${}^{w}$ | Potential throughput of the whole networks under w scheme |

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## Share and Cite

**MDPI and ACS Style**

Tu, L.-T.; Phan, V.-D.; Nguyen, T.N.; Tran, P.T.; Duy, T.T.; Nguyen, Q.-S.; Nguyen, N.-T.; Voznak, M.
Performance Analysis of Multihop Full-Duplex NOMA Systems with Imperfect Interference Cancellation and Near-Field Path-Loss. *Sensors* **2023**, *23*, 524.
https://doi.org/10.3390/s23010524

**AMA Style**

Tu L-T, Phan V-D, Nguyen TN, Tran PT, Duy TT, Nguyen Q-S, Nguyen N-T, Voznak M.
Performance Analysis of Multihop Full-Duplex NOMA Systems with Imperfect Interference Cancellation and Near-Field Path-Loss. *Sensors*. 2023; 23(1):524.
https://doi.org/10.3390/s23010524

**Chicago/Turabian Style**

Tu, Lam-Thanh, Van-Duc Phan, Tan N. Nguyen, Phuong T. Tran, Tran Trung Duy, Quang-Sang Nguyen, Nhat-Tien Nguyen, and Miroslav Voznak.
2023. "Performance Analysis of Multihop Full-Duplex NOMA Systems with Imperfect Interference Cancellation and Near-Field Path-Loss" *Sensors* 23, no. 1: 524.
https://doi.org/10.3390/s23010524