# A Statistical Estimation of 5G Massive MIMO Networks’ Exposure Using Stochastic Geometry in mmWave Bands

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Current Approaches in Exposure Estiamation

#### 1.2. Our Approach and Contributions

## 2. System Model

_{0}located at the origin without losing generality. To investigate the effect of the network load on the exposure, we assume that the BSs emit with probability $\alpha $, i.e., at a certain time-frame the number of active MTs is less than the number of base stations ${\lambda}_{MT}^{active}\le {\lambda}_{BS}$, which means that the BSs will have an effective density of ${\lambda}_{eff}=\alpha {\lambda}_{BS}$. Since line-of-sight (LOS) paths are the highest contributors to the exposure, we assume that all the paths between the BS and the MT are LOS paths. The BSs will emit with a constant power denoted by ${P}_{tx}$. We assume a 5G massive MIMO network where the BSs are composed of antenna arrays with identical structure, and thus identical radiation patterns. The BS is assumed as single user MIMO, and it emits towards one MT in a single time slot. We assume no downlink power control exist in the network, which seems to be the case for 5G networks [26], so every MT is being allocated the whole power resource for each transmission slot. We implement the channel model NYUSIM, developed by New York University Wireless Group [11] which divides the channel power into clusters and paths between the BS and the MT. The received power ${P}_{rx}^{B{S}_{i}}$ at each MT is dependent on the transmitted power, the propagation distance, the channel gain, and the antenna gain. The global exposure ${P}_{rx}^{t}$ can then be defined as ${P}_{rx}^{t}={{\displaystyle \sum}}_{B{S}_{i}\in {\mathsf{\Psi}}_{\mathrm{BS}}}{P}_{rx}^{B{S}_{i}}$ and it can be rewritten as

#### 2.1. Path Loss Model

**Remark**

**1.**

#### 2.2. Antenna Model

#### 2.2.1. Element Pattern

#### 2.2.2. Array Pattern

#### 2.3. Channel Model

#### 2.3.1. Array Gain

#### 2.3.2. Channel Gain

## 3. Exposure Estimation

**Theorem**

**1.**

## 4. Numerical Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Proof.**

## Appendix B

Abbreviation | Meaning |
---|---|

MIMO | Multiple Input Multiple Output |

SNR | Signal to noise ratio |

MISR | Mean interference to signal ratio |

BS | Base station |

mmWave | Millimeter Wave |

ICNIRP | International Commission on Non-Ionizing Radiation Protection |

PPP | Poisson point process |

CDF | Cumulative distribution function |

MT | Mobile terminal |

URA | Uniformly spaced rectangular array |

AoD | Angle of departure |

UMI | Urban microcell |

Probability distribution function | |

MGF | Moment generating function |

PCE | Polynomial chaos expansion |

PFGL | Probability generating functional |

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**Figure 1.**2D slice of the 3D antenna pattern realization of the 3GPP active antenna array model with 256 antenna elements directed perpendicularly to the antenna array over the steering angles in the azimuth and elevation.

**Figure 2.**Illustration of our mmWave model (

**left**) and the channel model (

**right**). The channel model follows the NYU Wireless Group mmWave channel model [11].

**Figure 3.**Plot of the fit of the antenna array gain into an exponential distribution for a rectangular uniform antenna array with ${N}_{tx}=256$ antenna elements.

**Figure 4.**Channel gain fit into a gamma distribution for MTs uniformly distributed on a 2D plane and having a single element antenna array.

**Figure 5.**Verification of the analytical expression of the exposure versus a Monte-Carlo simulation.

**Figure 6.**90th percentile of the exposure as function of BS Density for different values of the path loss exponent.

**Figure 7.**90th percentile of the exposure as function of system utilization for different values of the BS density.

**Figure 8.**Comparison between the current model developed in this paper, its verification using Monte-Carlo simulations assuming the gain distributions from Section 2.2, and the old model from [25], versus a Monte-Carlo simulation of the exposure using the gain simulated by NYUSIM.

Symbol | Description |
---|---|

${P}_{rx}^{t}$ | Total Power Received at the center of the cell |

${P}_{rx}^{B{S}_{i}}$ | Power received from the ith BS at the center of the cell |

${P}_{tx}$ | Transmitted power from all the BSs |

${H}_{i},{G}_{i}$ | Channel and antenna gain from the ith BS to the MT |

${L}_{i}$ | Path loss experienced by the transmitted signal from the ith BS to the MT |

$\theta ,\varphi $ | Zenith and azimuth angles in the local coordinate system centered at the origin of the array |

${A}_{E},{A}_{E,V},{A}_{E,H}\left(\theta ,\varphi \right)$ | Radiation pattern and its vertical and horizontal components respectively |

${\theta}_{3\mathrm{dB}},{\varphi}_{3\mathrm{dB}}$ | Vertical and horizontal beamwidths of the antennas in degrees |

$\mathrm{min}\left[{x}_{1},{x}_{2}\dots \right]$ | Numerically smallest number of ${x}_{k}$ |

${A}_{m}$ | Antenna’s front-to-back ratio |

$SL{A}_{V},SL{A}_{H}$ | Vertical and horizontal sidelobe attenuation levels |

${A}_{A}\left(\theta ,\varphi \right)$ | Array pattern of the antenna |

${v}_{m,n}$ | Phase shift due to the $\left[m,n\right]$ antenna element placement |

${w}_{m,n}$ | Weighting factor due to the $\left[m,n\right]$ antenna element |

${N}_{H},{N}_{V}$ | Number of horizontal and vertical antenna elements |

${\theta}_{etilt},{\varphi}_{escan}$ | Zenith and azimuth electrical down-tilt steering angle |

${d}_{v},{d}_{H}$ | Vertical and horizontal antenna element spacing |

$\Gamma \left(b\right)$ | Gamma function $\Gamma \left(b\right)={{\displaystyle \int}}_{0}^{\infty}{x}^{b-1}{e}^{-x}dx$$,\mathfrak{R}\left(b\right)0$ |

${\mathsf{\Psi}}_{BS},{\lambda}_{BS}$ | The Poisson point process and its density describing the BS distribution in the cell |

${\mathbb{E}}_{k}\left[.\right]$ | Expectation with respect to the random variable $k$ |

${\lambda}_{MT}^{active}$ | Density of the active MTs in the cell |

$\alpha $ | Emission probability of the BSs |

$M{T}_{0}$ | Mobile terminal at the center of the cell |

${r}_{i}$ | the distance between BS_{i} and MT_{0} |

$\eta $ | Path loss exponent assumed constant in the whole cell |

${\phi}_{X},{\mathsf{\Phi}}_{X}$ | Moment generating function and characteristic function of the random variable $X$ |

${F}_{\mathrm{X}}\left(x\right)$ | CDF of the random variable $X$ |

$\Gamma \left(a,x\right)$ | Upper incomplete gamma function |

$\gamma \left(a,x\right)$ | Lower incomplete gamma function |

$K$ | Iid random variable describing the channel and antenna gains |

${}_{2}{F}_{2}\left({a}_{1},{a}_{2},{b}_{1},{b}_{2},z\right)$ | Generalized hypergeometric function |

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

Frequency | $28\mathrm{GHz}$ |

Scenario | $UMI$ |

Tx Power | $0\mathrm{dBm}$ |

Array type | $URA$ |

Number of elements | $256$ |

Antenna spacing | $0.5\lambda $ |

Half-Power Beamwidth | $10\xb0$ |

Link Type | $LOS$ |

RF Bandwidth | $800\mathrm{MHz}$ |

User Terminal Height | $1.5\mathrm{m}$ |

Base Station Height | $35\mathrm{m}$ |

**Table 3.**Parameters of the fitted distributions. $a$ for the exponential distribution and $b,c$ for the gamma distribution.

Parameter | Value | Description |
---|---|---|

$a$ | $0.57$ | Gamma distribution shape parameter |

$b$ | $1.45$ | Gamma distribution scale parameter |

$c$ | $966.5$ | Exponential distribution exponent |

**Table 4.**Simulation Parameters used for the verification of the analytical equation with Monte-Carlo simulations

^{1}.

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

$\lambda $ | $2\times {10}^{-5}$ |

$\alpha $ | $0.5$ |

$\eta $ | $4$ |

${P}_{tx}$ | $1\mathrm{mW}$ |

^{1}It should be noted that the parameters in Table 4 are not obtained from realistic 5G mmWave networks. However, obtaining parameters and optimizing analytical models from realistic 5G mmWave networks are of the authors’ interests.

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

$b$ | $1.45$ |

$c$ | $966.5$ |

$a$ | $0.57$ |

$\lambda $ | $2\times {10}^{-5}$ |

$\alpha $ | $0.5$ |

$\eta $ | $3$ |

**Table 6.**Total Sobol indices of the inputs contributing to the 90th percentile of the exposure in the network.

Input Variable | Total Sobol Indices |
---|---|

$\mathsf{\alpha}$ | $0.447$ |

$\mathsf{\eta}$ | $0.932$ |

$\mathsf{\lambda}$ | $0.365$ |

$P$ | $0.254$ |

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**MDPI and ACS Style**

Al Hajj, M.; Wang, S.; Thanh Tu, L.; Azzi, S.; Wiart, J. A Statistical Estimation of 5G Massive MIMO Networks’ Exposure Using Stochastic Geometry in mmWave Bands. *Appl. Sci.* **2020**, *10*, 8753.
https://doi.org/10.3390/app10238753

**AMA Style**

Al Hajj M, Wang S, Thanh Tu L, Azzi S, Wiart J. A Statistical Estimation of 5G Massive MIMO Networks’ Exposure Using Stochastic Geometry in mmWave Bands. *Applied Sciences*. 2020; 10(23):8753.
https://doi.org/10.3390/app10238753

**Chicago/Turabian Style**

Al Hajj, Maarouf, Shanshan Wang, Lam Thanh Tu, Soumaya Azzi, and Joe Wiart. 2020. "A Statistical Estimation of 5G Massive MIMO Networks’ Exposure Using Stochastic Geometry in mmWave Bands" *Applied Sciences* 10, no. 23: 8753.
https://doi.org/10.3390/app10238753