# A Mobility Model for a 3D Non-Stationary Geometry Cluster-Based Channel Model for High Speed Trains in MIMO Wireless Channels

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## Abstract

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## 1. Introduction

- 1.
- A 3D, mobile and non-stationary cluster-based GBSM with scatterers located around the moving MRS is proposed.
- 2.
- The HST’s mobility is described by the enhanced Gauss–Makorv mobility model incorporating acceleration.
- 3.
- The death–birth Markov model is used to model the cluster MPCs.
- 4.
- The channel statistics, i.e., the local space-time correlation function (ST-CF), the root-mean-square Doppler shift spread, and the quasi-stationary intervals, are derived.
- 5.
- The simulated results of the proposed model are compared with the measured results.

## 2. A Mobility Model for a 3D, Non-Stationary Cluster and Geometry-Based Channel Model

#### 2.1. Enhanced Gauss–Markov Mobility Model

#### 2.2. Cluster Process Evolution

#### 2.3. The HST Time-Varying Distances

#### 2.4. Method of Equal Volume and the Proposed Sum of the Sinusoidal Simulation Model

#### 2.5. Cluster Delay Update

#### 2.6. Cluster Power Update

## 3. The Statistical Properties of the Channel Models

#### 3.1. Local Space-Time Correlation Function

#### 3.2. Channel Time-Variant Transfer Function

#### 3.3. The RMS Delay Spread

#### The Stationary Time Interval

## 4. Numerical Simulation and Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**The relationship between the k distributions of intra-cluster paths and time correlations for azimuth angles.

**Figure 5.**The relationship between the k distributions of intra-cluster paths and time correlations for elevation angles.

**Figure 7.**The stationary intervals using the proposed model for time-varying angle, cluster power, and the IMT-A channel.

**Figure 8.**The stationary interval of the proposed model, measured channel, and the IMT-A channel model.

Parameter | Definition |
---|---|

$\phantom{\rule{4pt}{0ex}}D$ | The horizontal distance between the center of the MRS and BS at initial time |

$\phantom{\rule{4pt}{0ex}}{R}_{(n,m)}$ | radius of the sphere around MRS |

$\phantom{\rule{4pt}{0ex}}{f}_{s}\left(t\right)$ | half spacing between the two foci of the ellipse |

${\delta}_{T}$,${\delta}_{R}$ | antenna spacing at the MRS and BS |

${\theta}_{T}$,${\theta}_{R}$ | orientation of the MRS and BS antenna array in the $\phantom{\rule{4pt}{0ex}}x-y$ plane, respectively |

${\phi}^{T}$,${\phi}^{R}$ | angles of elevation of the MRS and BS antenna array relative to the $\phantom{\rule{4pt}{0ex}}x-y$ plane, respectively |

$\phantom{\rule{4pt}{0ex}}{v}_{R}$ | MRS velocity |

${\gamma}_{R}$ | motion direction of the MRS |

${\alpha}_{R}^{Los}$, ${\beta}_{R}^{Los}$ | AAoA and EAoA of the Los path, respectively, |

$\phantom{\rule{4pt}{0ex}}{S}_{(n,m)}$ | The ${m}^{th}$ scatterer in the ${n}^{th}$ cluster |

${\alpha}_{(n,m)}^{R}$ | AAoA of the wave traveling from effective scatterers $\phantom{\rule{4pt}{0ex}}{S}_{(n,m)}$ of the ${n}^{th}$ cluster |

${\alpha}_{(n,m)}^{T}$ | AAoD of the wave that impinges from effective scatterers $\phantom{\rule{4pt}{0ex}}{S}_{(n,m)}$ of the ${n}^{th}$ cluster |

${\beta}_{(n,m)}^{R}$ | EAoA of the wave traveling from effective scatterers $\phantom{\rule{4pt}{0ex}}{S}_{(n,m)}$ of the ${n}^{th}$ cluster |

${\beta}_{(n,m)}^{T}$ | EAoD of the wave that impinges from effective scatterers $\phantom{\rule{4pt}{0ex}}{S}_{(n,m)}$ of the ${n}^{th}$ cluster |

${\gamma}_{S},\varphi $ | Horizontal and elevation moving direction of scatterers, respectively |

$\phantom{\rule{4pt}{0ex}}K$ | Ricean K factors |

$\phantom{\rule{4pt}{0ex}}{V}_{S}$ | Velocity of moving scatterers |

Parameter | Numerical Value |
---|---|

$N\left(t\right)$ | 20 |

${\theta}_{T}={\theta}_{R}$ | ${45}^{\circ}$ |

${\gamma}_{T}={\gamma}_{R}$ | ${0}^{\circ}$ |

${\beta}_{T}={\beta}_{R}$ | ${15}^{\circ}$ |

${\alpha}_{T}={\alpha}_{R}$ | ${25}^{\circ}$ |

${\alpha}_{R}$ | 4 m/s${}^{2}$ |

$\Psi $ | 0.9 |

${\sigma}_{v}$ | 0.01 |

${v}_{R}\left({t}_{0}\right)$ | 250 km/h |

${v}_{S}$ | 0.5 m/s |

${R}_{R}$ | 50 m |

${D}_{min}$ | 50 m |

${H}_{MRS}$ | 30 cm |

${H}_{train}$ | 3.8 cm |

${f}_{c}\left(t\right)$ | 2.6 GHz |

${D}_{s}$ | 1000 m |

${\delta}_{T}$, ${\delta}_{R}$ | $\lambda $/2 |

${\lambda}_{A}$ | 0.8/m |

${\lambda}_{D}$ | 0.04/m |

${P}_{s}$ | 0.3 |

K | 3.8 |

$p=q$ | 2 |

Scenarios | Acceleration | Quasi Stationary Time |
---|---|---|

1 | 0.6 m/s^{s} | 0.21 s |

2 | 0.3 m/s^{s} | 0.3 s |

3 | 0.2 m/s^{s} | 0.451 s |

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**MDPI and ACS Style**

Assiimwe, E.; Marye, Y.W.
A Mobility Model for a 3D Non-Stationary Geometry Cluster-Based Channel Model for High Speed Trains in MIMO Wireless Channels. *Sensors* **2022**, *22*, 10019.
https://doi.org/10.3390/s222410019

**AMA Style**

Assiimwe E, Marye YW.
A Mobility Model for a 3D Non-Stationary Geometry Cluster-Based Channel Model for High Speed Trains in MIMO Wireless Channels. *Sensors*. 2022; 22(24):10019.
https://doi.org/10.3390/s222410019

**Chicago/Turabian Style**

Assiimwe, Eva, and Yihenew Wondie Marye.
2022. "A Mobility Model for a 3D Non-Stationary Geometry Cluster-Based Channel Model for High Speed Trains in MIMO Wireless Channels" *Sensors* 22, no. 24: 10019.
https://doi.org/10.3390/s222410019