# Ergodic Rate and Outage Performance of Full-Duplex NOMA Relaying with Channel Estimation Errors and Low-Resolution ADCs

^{*}

## Abstract

**:**

## 1. Introduction

- 1.
- We consider a FD C-NOMA wireless communication system that uses an AF relay to assist in signal transmission between the base station and multiple users. The system performance analysis takes into account non-ideal conditions, such as LI caused by FD operation, channel estimation errors, and quantization errors generated by low-resolution ADCs.
- 2.
- We derive the theoretical expressions of the ergodic rate and outage probability for an FD C-NOMA multi-user system. Specifically, accurate approximate expressions of ergodic rate and closed-form solutions of the outage probability for each user are obtained by using the theories of order statistics and probability statistics.
- 3.
- We verify the mathematical results obtained through Monte Carlo simulations. For the C-NOMA relay network, we discuss the impact of base station transmission power, relay amplification, forwarding power, relay LI level, quantization error level, and channel estimation error level on system performance. Theoretical and simulation results demonstrate the feasibility of the proposed theoretical approach.

## 2. System Model

## 3. Ergodic Rate Analysis

**Theorem**

**1.**

**Proof.**

## 4. Outage Probability Analysis

**Theorem**

**2.**

**Proof.**

## 5. Numerical Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ADCs | Analog-to-digital converters |

AF | Amplify-and-forward |

BER | Bit error rate |

BS | Base station |

cdf | Cumulative (probability measure) distribution function |

CDMA | Code division multiple access |

CSI | Channel state information |

C-NOMA | Cooperative NOMA |

ch.f. | Characteristic function |

CSI | Channel state information |

DF | Decode-and-forward |

ER | Ergodic rate |

FD | Full duplex |

FDMA | Frequency division multiple access |

HD | Half duplex |

i.i.d. | Independent and identically distributed |

LI | Loopback interference |

NOMA | Non-orthogonal multiple access |

OFDMA | Orthogonal frequency division multiple access |

OMA | Orthogonal multiple access |

OP | Outage probability |

Qos | Quality of service |

Probability density function | |

RF | Radio frequency |

r.v. | Random variable |

SER | Symbol error rate |

SIC | Successive interference cancellation |

SINR | Signal-to-interference-plus-noise ratio |

SNR | Signal-to-noise ratio |

TDMA | Time division multiple access |

UE | User equipment |

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**Figure 3.**Outage probability versus SNR in the first phase (${\gamma}_{1}$) with $cee=0.001$, ${\beta}_{\mathrm{LI}}=0.005$, ${\beta}_{\mathrm{SR}}=0.4$, ${\beta}_{\mathrm{RU}}=0.6$, ${\gamma}_{2}=20\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB}$, quantization bits $=6$, power allocation coefficients $=[\frac{1}{2},\frac{1}{3},\frac{1}{6}]$, and ${\gamma}_{\mathrm{th}}=[0.1,0.1,0.1]\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB}$.

**Figure 4.**Ergodic rates versus SNR in the first phase (${\gamma}_{1}$) with $cee=0.001$, ${\beta}_{\mathrm{LI}}=0.005$, ${\beta}_{\mathrm{SR}}=0.4$, ${\beta}_{\mathrm{RU}}=0.6$, ${\gamma}_{2}=20\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB}$, quantization bits $=6$, and power allocation coefficients $=[\frac{1}{2},\frac{1}{3},\frac{1}{6}]$.

**Figure 5.**Outage probability versus SNR in the second phase (${\gamma}_{2}$) with $cee=0.001$, ${\beta}_{\mathrm{LI}}=0.005$, ${\beta}_{\mathrm{SR}}=0.4$, ${\beta}_{\mathrm{RU}}=0.6$, ${\gamma}_{1}=30\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB}$, quantization bits $=6$, power allocation coefficients $=[\frac{1}{2},\frac{1}{3},\frac{1}{6}]$, and ${\gamma}_{\mathrm{th}}=[0.1,0.1,0.1]\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB}$.

**Figure 6.**Ergodic rates versus SNR in the second phase (${\gamma}_{2}$) with $cee=0.001$, ${\beta}_{\mathrm{LI}}=0.005$, ${\beta}_{\mathrm{SR}}=0.4$, ${\beta}_{\mathrm{RU}}=0.6$, ${\gamma}_{1}=30\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB}$, quantization bits $=6$, and power allocation coefficients $=[\frac{1}{2},\frac{1}{3},\frac{1}{6}]$.

**Figure 7.**Outage probability versus different loopback interference levels (${\beta}_{\mathrm{LI}}$) with $cee=0.001$, ${\beta}_{\mathrm{SR}}=0.4$, ${\beta}_{\mathrm{RU}}=0.6$, ${\gamma}_{1}=30\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB}$, ${\gamma}_{2}=20\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB}$, quantization bits $=6$, power allocation coefficients $=[\frac{1}{2},\frac{1}{3},\frac{1}{6}]$, and ${\gamma}_{\mathrm{th}}=[0.1,0.1,0.1]\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB}$.

**Figure 8.**Ergodic rates versus different loopback interference levels (${\beta}_{\mathrm{LI}}$) with $cee=0.001$, ${\beta}_{\mathrm{SR}}=0.4$, ${\beta}_{\mathrm{RU}}=0.6$, ${\gamma}_{1}=30\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB}$, ${\gamma}_{2}=20\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB}$, quantization bits $=6$, and power allocation coefficients $=[\frac{1}{2},\frac{1}{3},\frac{1}{6}]$.

**Figure 9.**Outage probability versus channel estimation error level and quantization bits with ${\beta}_{\mathrm{LI}}=0.005$, ${\beta}_{\mathrm{SR}}=0.4$, ${\beta}_{\mathrm{RU}}=0.6$, ${\gamma}_{1}=30\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB}$, ${\gamma}_{2}=20\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB}$, power allocation coefficients $=[\frac{1}{2},\frac{1}{3},\frac{1}{6}]$, and ${\gamma}_{\mathrm{th}}=[0.1,0.1,0.1]\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB}$.

**Figure 10.**Ergodic rates versus different channel estimation error levels and quantization bits with ${\beta}_{\mathrm{LI}}=0.005$, ${\beta}_{\mathrm{SR}}=0.4$, ${\beta}_{\mathrm{RU}}=0.6$, ${\gamma}_{1}=30\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB}$, ${\gamma}_{2}=20\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB}$, and power allocation coefficients $=[\frac{1}{2},\frac{1}{3},\frac{1}{6}]$.

Quantization Bits | $\mathit{\alpha}$ |
---|---|

1 | 0.6366 |

2 | 0.8825 |

3 | 0.96546 |

4 | 0.990503 |

5 | 0.997501 |

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**MDPI and ACS Style**

Wang, S.; Huang, Y.; Yang, Y.
Ergodic Rate and Outage Performance of Full-Duplex NOMA Relaying with Channel Estimation Errors and Low-Resolution ADCs. *Appl. Sci.* **2023**, *13*, 6950.
https://doi.org/10.3390/app13126950

**AMA Style**

Wang S, Huang Y, Yang Y.
Ergodic Rate and Outage Performance of Full-Duplex NOMA Relaying with Channel Estimation Errors and Low-Resolution ADCs. *Applied Sciences*. 2023; 13(12):6950.
https://doi.org/10.3390/app13126950

**Chicago/Turabian Style**

Wang, Siye, Yeqin Huang, and Yong Yang.
2023. "Ergodic Rate and Outage Performance of Full-Duplex NOMA Relaying with Channel Estimation Errors and Low-Resolution ADCs" *Applied Sciences* 13, no. 12: 6950.
https://doi.org/10.3390/app13126950