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Stability and Delay of NDMA-MPR Protocol in Rice-Correlated Channels with Co-Channel Interference^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

- Extension of NDMA-MPR to Correlated Rice channels.
- Analysis of backlog retransmission strategies.
- Analysis of NMDA detection protocol with co-channel interference.

## 2. System Model and Assumptions

#### 2.1. Scenario Description and NDMA Protocol Operation

#### 2.2. Backlog Retransmission Schemes

#### 2.3. Epoch-Slot Definition and Feedback Flags

#### 2.4. Examples

## 3. Signal Model

## 4. Receiver Operational Characteristic (ROC) for Terminal Presence Detection

## 5. Persistent Retransmission Strategy

Algorithm 1 Algorithm NDMA with persistent backlog retransmission control. | |

1. | Generate set of colliding terminals $\mathcal{T}$ using traffic model. |

2. | Start super-epoch slot. |

3. | Start of a conventional epoch-slot of NDMA ($\mathit{e}=\mathbf{1}$). |

4. | Detect the presence of contending terminals using ${\mathit{z}}_{\mathit{j}}$ in (4). |

5. | Request retransmissions $\left(\mathit{r}=\u2308\frac{\widehat{\mathit{K}}}{\mathit{M}}\u2309-\mathbf{1}\right)$ to create a virtual MIMO system as in (6) |

6. | Attempt the decoding of the colliding terminals using ZF ($\widehat{\mathbf{S}}={\mathbf{H}}^{-\mathbf{1}}\mathbf{Y}$) or minimum mean square error (MMSE) detection ($\widehat{\mathbf{S}}={(\mathbf{H}+\mathbf{I}{\mathit{\sigma}}_{\mathit{v}}^{\mathbf{2}})}^{-\mathbf{1}}\mathbf{Y}$). |

7. | Is the collision resolved? If Yes, then end of a super-epoch and go back to step 1. If not, the same contending terminals restart one more epoch slot. Go back to step 3. |

## 6. Random Backlog Retransmission Strategy

Algorithm 2 Algorithm NDMA-MPR with random backlog retransmission control. | |

1. | Generate set of colliding terminals $\mathcal{T}$ using traffic model. |

2. | Start of a conventional epoch-slot of NDMA |

3. | Detect the presence of contenting terminals using ${\mathit{z}}_{\mathit{j}}$ in (4). |

4. | Request retransmissions $\left(\mathit{r}=\u2308\frac{\widehat{\mathit{K}}}{\mathit{M}}\u2309-\mathbf{1}\right)$ to create a virtual MIMO system as in (6) |

5. | Attempt the decoding of the colliding terminals using ZF ($\widehat{\mathbf{S}}={\mathbf{H}}^{-\mathbf{1}}\mathbf{Y}$) or minimum mean square error (MMSE) detection ($\widehat{\mathbf{S}}={(\mathbf{H}+\mathbf{I}{\mathit{\sigma}}_{\mathit{v}}^{\mathbf{2}})}^{-\mathbf{1}}\mathbf{Y}$). |

6. | Is the collision resolved? If Yes, then go back to step 1. If not, terminals backlog randomly the lost packet with probability $\mathit{p}$. Go back to step 3. |

## 7. Results

## 8. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 6.**Stable throughput ($\mathit{T}=\mathit{\lambda}\mathit{J}$) vs. transmission probability ($\mathit{p}$) using the first proposed backlog retransmission scheme (persistent) for various values of correlation coefficient $\mathit{\rho}$ and Rice factor ($\mathit{\kappa}$).

**Figure 7.**Stable throughput ($\mathit{T}=\mathit{\lambda}\mathit{J}$) vs. transmission probability ($\mathit{p}$) using the second proposed backlog retransmission scheme for various values of correlation coefficient $\mathit{\rho}$ and Rice factor ($\mathit{\kappa}$).

**Figure 8.**Average Delay ($\mathit{D}$) vs. transmission probability ($\mathit{p}$) using the first proposed backlog retransmission scheme for various values of correlation coefficient $\mathit{\rho}$ and Rice factor ($\mathit{\kappa}$).

**Figure 9.**Average Delay ($\mathit{D}$) vs. transmission probability ($\mathit{p}$) using the second proposed backlog retransmission scheme for various values of correlation coefficient $\mathit{\rho}$ and Rice factor ($\mathit{\kappa}$).

Variable | Meaning |
---|---|

$\mathit{J}$ | Total number of terminals in the network |

$\mathit{M}$ | Number of antennas at the BS. |

$\mathit{e}$ | Epoch-slot indicator |

$\mathit{l}$ | Length of an epoch-slot |

$\mathit{L}$ | Length of a epoch-slot |

${\mathit{L}}_{\mathit{r}}$ | Length of a relevant super epoch-slot |

${\mathit{L}}_{\mathit{ir}}$ | Length of an irrelevant super epoch-slot |

${\mathit{h}}_{\mathit{j},\mathit{m}}$ | Channel between terminal $\mathit{j}$ and the $\mathit{m}$th antenna of the BS |

${\mathit{\alpha}}_{\mathit{j},\mathit{m}}$ | Random component channel between terminal $\mathit{j}$ and the $\mathit{m}$th antenna of the BS |

${\mathit{\rho}}_{\mathit{m},\tilde{\mathit{m}}}$ | Correlation coefficient between the signal of antenna $\mathit{m}$ with the signal of antenna $\tilde{\mathit{m}}$ |

$\mathit{\gamma}$ | Channel variance between terminals and the BS |

${\mathit{\sigma}}_{\mathit{v}}^{\mathbf{2}}$ | Noise variance |

${\mathit{\sigma}}_{\mathit{g}}^{\mathbf{2}}$ | Interference variance |

$\mathit{\lambda}$ | Packet arrival rate |

$\mathit{p}$ | Terminal transmission probability |

$\mathit{K}$ | Collision multiplicity |

$\widehat{\mathit{K}}$ | Estimated collision multiplicity |

${\mathit{k}}_{\mathit{d}}$ | Number of active terminals correctly detected as active |

${\mathit{k}}_{\mathit{f}}$ | Number of idle terminals incorrectly detected as active |

$\mathcal{T}$ | Set of colliding terminals |

$\widehat{\mathcal{T}}$ | Estimated set of colliding terminals |

${\mathit{z}}_{\mathit{j}}$ | Terminal presence indicator |

$\mathit{\beta}$ | Energy presence detection threshold |

${\mathit{P}}_{\mathit{D}}$ | Terminal presence detection probability |

${\mathit{P}}_{\mathit{F}}$ | Probability of false alarm |

${\mathit{P}}_{\mathit{A}}$ | Total probability of detection |

${\mathit{P}}_{\mathit{c}}$ | Probability of correct resolution |

$\mathsf{\Gamma}$ | Post processing instantaneous Signal-to-Interference-plus-Noise Ratio |

$\mathit{\mu}$ | Line-of-Sight (LOS) component |

${\mathbf{w}}_{\mathit{j}}$ | Orthogonal training sequence for terminal $\mathit{j}$ |

${\mathbf{s}}_{\mathit{j}}$ | Signal transmitted by terminal $\mathit{j}$ |

$\mathbf{H}$ | Mixing channel matrix |

$\mathbf{Y}$ | Signal received by the BS |

$\mathbf{V}$ | Noise vector at the BS |

$\mathbf{G}$ | Interference vector at the BS |

$\mathit{T}$ | Packet throughput |

$\mathit{D}$ | Delay |

$\mathit{\kappa}$ | Rice factor |

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**MDPI and ACS Style**

Sámano-Robles, R. Stability and Delay of NDMA-MPR Protocol in Rice-Correlated Channels with Co-Channel Interference. *Technologies* **2019**, *7*, 22.
https://doi.org/10.3390/technologies7010022

**AMA Style**

Sámano-Robles R. Stability and Delay of NDMA-MPR Protocol in Rice-Correlated Channels with Co-Channel Interference. *Technologies*. 2019; 7(1):22.
https://doi.org/10.3390/technologies7010022

**Chicago/Turabian Style**

Sámano-Robles, Ramiro. 2019. "Stability and Delay of NDMA-MPR Protocol in Rice-Correlated Channels with Co-Channel Interference" *Technologies* 7, no. 1: 22.
https://doi.org/10.3390/technologies7010022