# Log Likelihood Ratio Based Relay Selection Scheme for Amplify and Forward Relaying with Three State Markov Channel

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. System Model

#### Finite State Markov Channel Modeling

## 4. Results and Discussion

^{−6}is achieved at 16 dB SNR value. In the case of QPSK modulation it achieves 10

^{−6}level at 20 dB SNR for the same parameters.

^{−5}is achieved at 10 dB SNR value. While in the case of QPSK modulation, the best achieved BER is approximately 4 × 10

^{−5}and is achieved at the 12 dB SNR value.

^{−4}. It can be seen that the addition of a source to destination link improves the BER of a proposed scheme as a zero-error point with direct link is found to be at 10 dB SNR, 6 dB better as compared to a system without a direct link, which is found at 16 dB SNR for the considered case with transition matrix presenting equal probability for all considered fading scenarios. Clearly the system with a direct link provides better outage behavior when compared to a system without direct link.

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## Abbreviations

Symbol | Description |

${{\rm Z}}_{S{R}_{N}}$ | received signal at ${N}^{\mathrm{th}}$ relay in first phase |

$\sqrt{{{\rm E}}_{s}}$ | average energy per symbol |

${{\rm H}}_{S{R}_{N}}$ | channel coefficient present between source and ${N}^{\mathrm{th}}$ relay |

${N}_{0}$ | Noise variance |

${b}_{i}$ | best relay pair |

$LL{R}_{N}$ | Log Likelihood ratio magnitude of signal received at ${N}^{\mathrm{th}}$ relay in first phase |

${\eta}_{SR}{}_{N}$ | transmitted symbol from source |

${P}_{b}{}_{i}$ | bit error probability |

${\beta}_{i}$ | amplification factor |

${\mathsf{{\rm Z}}}_{{R}_{i}D}$ | received signal at destination after second phase |

${\delta}_{1}$ and ${\delta}_{2}$ | link weights for MRC |

${\gamma}_{t}$ | Instantaneous SNR |

${p}_{{\gamma}_{t}}$ | probability density function (PDF) for fading channel |

K | ratio between the power in direct path and indirect path in Rican fading |

$\mathsf{\Omega}$ | total available power in Rician Fading |

${I}_{0}(.)$ | Zero order Bessal function |

M | shaping parameter in Nakagami fading |

$\omega $ | controlling spread in Nakagami Fading |

$P$ | transition probability |

$\mathsf{\Pi}$ | steady state vector |

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**Figure 2.**Three-State Markov process with state-A being Rayleigh Fading, State-B represents Rician Fading and State-C represents Nakagami-m fading channel.

**Figure 3.**Bit Error Rate Comparison of a log likelihood ratio (LLR) based relay selection scheme for different values of Transition Probability for (

**a**) Binary Phase Shift Keying (BPSK) Modulation and (

**b**) Quadrature Phase Shift Keying (QPSK) Modulation. Without Source to Destination Link.

**Figure 4.**Comparison of LLR based relay selection scheme for different values of Transition Probability with Source to Destination Link.

Modulation Scheme | BPSK, QPSK |
---|---|

Power Allocation Factor | 0.5 |

SNR Range | 0:2:20 |

Transition Probability $P$ (General case for Example) | [0.1 0.7 0.2; 0.0 0.4 0.6; 0.1 0.2 0.7] |

Fading environment | Three state Markov channel with Rayleigh, Rician, and Nakagami channels |

Frame Length | 256 bit |

Variance S-R, R-D and S-D link | 1 |

Transition probability matrix with equal probability | $P=\left[\begin{array}{ccc}\begin{array}{c}1/3\\ 1/3\\ 1/3\end{array}& \begin{array}{c}1/3\\ 1/3\\ 1/3\end{array}& \begin{array}{c}1/3\\ 1/3\\ 1/3\end{array}\end{array}\right]$ |

Transition probability matrix with Rayleigh only | $P=\left[\begin{array}{ccc}\begin{array}{c}1\\ 1\\ 1\end{array}& \begin{array}{c}0\\ 0\\ 0\end{array}& \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}\right]$ |

Transition probability matrix with Rician only | $P=\left[\begin{array}{ccc}\begin{array}{c}0\\ 0\\ 0\end{array}& \begin{array}{c}1\\ 1\\ 1\end{array}& \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}\right]$ |

Transition probability matrix with Nakagami only | $P=\left[\begin{array}{ccc}\begin{array}{c}0\\ 0\\ 0\end{array}& \begin{array}{c}0\\ 0\\ 0\end{array}& \begin{array}{c}1\\ 1\\ 1\end{array}\end{array}\right]$ |

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**MDPI and ACS Style**

Sahajwani, M.; Jain, A.; Gamad, R.
Log Likelihood Ratio Based Relay Selection Scheme for Amplify and Forward Relaying with Three State Markov Channel. *Future Internet* **2018**, *10*, 87.
https://doi.org/10.3390/fi10090087

**AMA Style**

Sahajwani M, Jain A, Gamad R.
Log Likelihood Ratio Based Relay Selection Scheme for Amplify and Forward Relaying with Three State Markov Channel. *Future Internet*. 2018; 10(9):87.
https://doi.org/10.3390/fi10090087

**Chicago/Turabian Style**

Sahajwani, Manish, Alok Jain, and Radheyshyam Gamad.
2018. "Log Likelihood Ratio Based Relay Selection Scheme for Amplify and Forward Relaying with Three State Markov Channel" *Future Internet* 10, no. 9: 87.
https://doi.org/10.3390/fi10090087