# Comparative Analytical Study of SCMA Detection Methods for PA Nonlinearity Mitigation

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## Abstract

**:**

## 1. Introduction

## 2. System Model

## 3. Bussgang Decomposition-Based MPA

- Notably, (12) quantifies the gap between the BER of the proposed approach and that of a universally optimal MPA (the RFF-based MPA in [12]). As mentioned before, this quantification helps when trading off computational complexity with BER performance subject to achieving a given BER-based level of QoS.
- It is further noted that the above deviation is independent of the fading distribution. In this context, it is indeed worth mentioning that the ideal BER, ${\psi}_{p\left(h\right)}\left({\mathrm{SNR}}^{*}\right)$, is mostly an integral of a Q-function over the concerned PDF $p\left(h\right)$ [2]. However, when ${\psi}_{p\left(h\right)}\left({\mathrm{SNR}}^{*}\right)$ (and hence its derivative ${\psi}_{p\left(h\right)}^{\prime}$) are known, the optimality gap is found to be independent of the underlying distribution.
- It is possible to further improve the error approximation in (12) as follows:$${\mathrm{BER}}_{\mathrm{Bussgang}}={\displaystyle \sum _{l=0}^{\infty}}\frac{{\psi}_{p\left(h\right)}^{\left(l\right)}\left({\mathrm{SNR}}^{*}\right)}{l!}\mathbb{E}\left[\Delta {m}_{jk}^{l}\right],$$$$\mathbb{E}\left[\Delta {m}_{jk}^{l}\right]={\displaystyle \sum _{s=0}^{l}}\left(\genfrac{}{}{0pt}{}{l}{s}\right)\mathbb{E}\left[{\mathcal{P}}^{s}{\mathcal{Q}}^{l-s}\right].$$From ([22] p. 546), this is simplified as:$$\mathbb{E}\left[\Delta {m}_{jk}^{l}\right]={\displaystyle \sum _{s=0}^{l}}{\displaystyle \sum _{u=0}^{\mathrm{min}\left[s,(l-s)\right]}}\left(\genfrac{}{}{0pt}{}{l}{s}\right)\frac{2s!\left[2\left(l-s\right)!\right]{\left(\alpha {\sigma}_{h}^{2}{\sigma}_{x}^{2}\right)}^{2u}}{{2}^{l}\left[(s-u)!\right]\left[(l-s-u)!\right]2u!},$$$${\mathrm{BER}}_{\mathrm{Bussgang}}={\displaystyle \sum _{l=0}^{\infty}}\frac{{\psi}_{p\left(h\right)}^{\left(l\right)}\left({\mathrm{SNR}}^{*}\right)}{l!}\mathbb{E}\left[\Delta {m}_{jk}^{l}\right].$$

Algorithm 1 Bussgang based MPA. |

1: Initialization: ${I}_{kj}=p\left({\mathbf{x}}_{j}\right)$ according to a uniform distribution. 2: Initialization: ${I}_{jk}:=\frac{1}{2\pi {\sigma}_{n}^{2}}exp\left[-\frac{{\left|y\left[k\right]-\alpha h\left[k\right]{\displaystyle \sum _{\forall j\in {B}_{k}}}x\left[k\right]\right|}^{2}}{{\sigma}_{v}^{2}+{\sigma}_{n}^{2}}\right]$ $\alpha =\frac{\mathbb{E}\left[{\mathbf{y}}^{T}\mathrm{diag}\left(\mathbf{h}\right)\mathbf{x}\right]}{\mathbb{E}\left[{\u2225\mathrm{diag}\left(\mathbf{h}\right)\mathbf{x}\u2225}^{2}\right]}$, ${\sigma}_{v}^{2}={(1-\alpha )}^{2}\mathbb{E}\left[{\u2225\mathrm{diag}\left(\mathbf{h}\right)\mathbf{x}\u2225}^{2}\right]$. 3: Initialize the maximum number of iterations, $\mathit{ITER}$. 4: while c < ITER do ${I}_{jk}:=\mathrm{log}\left(p\left({\mathbf{x}}_{j}\right)\right)+{\displaystyle \sum _{j\in {\mathcal{B}}_{k}}}{I}_{kj}.$ ${I}_{kj}:=\underset{\forall {\mathbf{x}}_{j}\in {\mathcal{C}}_{j},k\in {\mathcal{B}}_{j}}{max}log\left(p\left(y\left[k\right]\right|\mathbf{x})\right)+{\displaystyle \sum _{k\in {\mathcal{B}}_{j}}}{I}_{jk}$ $c:=c+1$ end while5: Detect user-symbols as per ([2] Equation (12.12)) using the steady-state message-values ${I}_{jk}$ and codebook ${\mathcal{C}}_{j}$ |

## 4. Simulations

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

NOMA | Non-orthogonal multiple access |

SCMA | Sparse code multiple access |

MPA | Message passing algorithm |

BER | Bit error rate |

RKHS | Reproducing kernel Hilbert space |

RFF | Random Fourier features |

IIoT | Industrial internet of things |

PD-NOMA | Power domain NOMA |

SIC | Successive interference cancellation |

PA | Power amplifier |

QoS | Quality of service |

Probability density function | |

AWGN | Additive white Gaussian noise |

GSNR | Generalized signal-to-noise ratio |

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**Figure 4.**BER vs. SNR comparison of the Bussgang-based detector with RFF-based detector for a Rayleigh Channel by varying the number of pilots.

**Figure 6.**BER vs. SNR validation for the Bussgang-based detector for a Nakagami-m channel with $m=0.5$.

Codebook | Section II.A [23] |

Modulation | OOK |

Value of p | 1 |

Kernel-width assignment | Silverman’s rule [24] |

Number of MPA iterations | 15 |

Number of transmitted bits | ${10}^{7}$ |

Parameter values for Rayleigh distribution | ${\sigma}_{h}^{2}=1$ |

Parameter values for the Nakagami-m distribution | Shape: $m=0.5$, |

Spread parameter: 1 | |

${n}_{G}$ | 110 |

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**MDPI and ACS Style**

Sfeir, E.; Mitra, R.; Kaddoum, G.; Bhatia, V.
Comparative Analytical Study of SCMA Detection Methods for PA Nonlinearity Mitigation. *Sensors* **2021**, *21*, 8408.
https://doi.org/10.3390/s21248408

**AMA Style**

Sfeir E, Mitra R, Kaddoum G, Bhatia V.
Comparative Analytical Study of SCMA Detection Methods for PA Nonlinearity Mitigation. *Sensors*. 2021; 21(24):8408.
https://doi.org/10.3390/s21248408

**Chicago/Turabian Style**

Sfeir, Elie, Rangeet Mitra, Georges Kaddoum, and Vimal Bhatia.
2021. "Comparative Analytical Study of SCMA Detection Methods for PA Nonlinearity Mitigation" *Sensors* 21, no. 24: 8408.
https://doi.org/10.3390/s21248408