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Practical 3-D Beam Pattern Based Channel Modeling for Multi-Polarized Massive MIMO Systems^{ †}

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## Abstract

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## 1. Introduction

**First**, nowadays, 3-D channel models show that spherical/plane wave-fronts cannot fulfill the elevation angles with an increased number of antenna elements at a wider space, especially when the user is in the same azimuth angle but at a different elevation. The 3-D beam pattern which is known to provide higher elevation angle is therefore proposed instead of spherical wavefronts. Based on the distance between transmitter (TX) and RX and increased dimension of antenna, spherical waves are emitted in a spherical shape and are considered as a plane wavefront. So, this paper contributes a phenomenon to investigate the beam waves to provide higher elevation angle is therefore proposed instead of spherical/plane wavefronts. Considering this fact, a 3-D massive MIMO in the far-field is assumed such that the phase of each antenna element is determined by the geometrical relationships and EAoAs and EAoDs on the antenna arrays are equal to each antenna element with same power angular spread (PAS). This fact is only possible in the theoretical view if we assumed that the wavefront for each wireless link is spherical. Therefore, in this paper, beam patterns with different PAS were considered to provide various phase excitations for different EAoAs toward different directions of travels (DoTs) on the antenna array with the higher degree of freedom on both the TX and RX.

**Second**, in the massive MIMO antenna, an increase in the dimension of antenna array results in the high correlation between the antenna elements and inaccuracy of channel coefficients. Therefore, for the sake of massive MIMO system design and performance evaluation, it is indispensable to investigate the correlation between antenna elements. This leads us to utilize the spaced rectangular antenna array (SRAA) and spaced uniform linear array (SULA) on received spatial correlation (RSC), considering various elevation angle of arrivals (EAoAs) and azimuth angles of arrivals (AAoAs) for the different antenna elements. This technique provides more accuracy in generating the channel coefficients for a massive MIMO system, where all the antenna elements need to be addressed uniquely at the antenna array to recognize their paths from the elements to the clusters. Antenna element spaces (AESs) technique is an inter-element spacing where the SRAA and SULA are divided into the number of antenna elements in the horizontal and elevation direction of the dipole and omnidirectional antennas. This technique can also be used for any arbitrary choice of the antenna pattern and distribution of azimuth and elevation angles for the polarized massive MIMO antenna with the different configuration of vertical (V), horizontal (H), and Dual (V/H) polarizations as shown in Figure 1.

**Third**, the appearance of clusters on the antenna array is another important characteristic of massive MIMO channel models. In conventional massive MIMO channel models, it is assumed that a cluster is always observable to the entire antennas elements. This model, on the other hand, proposes a non-stationary model based on the movement of the user where the cluster may appear as at least one antenna element and its adjacent elements.

- The impact of beam patterns has been proposed for 3-D massive MIMO channel model for different dipole and omnidirectional antenna elements. Therefore, the beam pattern provide different phase excitation towards different DoTs in the far-field. Given that, it also provides various AoDs and AoAs for each antenna element, contributing different correlation weights for rays related towards/from the clusters. As far as the author’s knowledge is concerned, a practical 3-D channel model for massive MIMO using beam pattern assumption in the far-field has not been considered, yet.
- A closed-form expression for AES has also been studied to reduce the RSC in the horizontal and elevation directions of the antenna array that can be accurately represented as an important aspect of a polarized antenna in 3-D space. Therefore, to design and evaluate a massive MIMO system, the investigation of correlations between antenna elements are necessary. This fact is possible in utilizing the SRAA and SULA, where all the antenna elements need to be addressed uniquely at the antenna array for investigating received spatial correlation. In fact, the model is providing an accurate observation to investigate the received spatial correlation based on the antenna polarizations.
- The movement of the user and clusters make our channel non-stationary which is applied to both time and array axes. It means that the behavior of the clusters varies at different times of EAoAs and AAoAs. Therefore, receiving clusters are observed to at least one antenna element and its adjacent elements depend upon their distance to the clusters at the RX. A novel cluster evolution algorithm in the system level is developed in the antenna pattern.
- The impact of the 3-D beam pattern channel model and elevation angle of the aforementioned channel properties is being investigated by comparing it with those of the 3-D conventional channel model. Statistical properties of the proposed massive MIMO channel model such as ECC and RSC, including signal-to-noise ratio (SNR) of the non-stationary channel model, were investigated. The proposed model has been valid for the far-field effects on the massive MIMO scenarios at the cell edge and the result looks convincing. This might provide a more accurate model for the current LTE-A system. Our good implementation models substantially facilitate the implementation of further techniques for different modeling, especially for massive MIMO antenna where the antenna space will affect the MIMO performance.

## 2. 3-D Antenna Configuration

#### 2.1. 3-D AES and Antenna Element’s Positioning

#### 2.1.1. Transmit Antenna Configuration

#### 2.1.2. Receive Antenna Configuration

#### 2.2. 3-D Beam Pattern

## 3. A Practical Non-Stationary of 3-D Massive MIMO Channel Model

#### 3.1. Generation of the Cluster/Channel in System Level

#### 3.1.1. Generating the Clusters at the Transmitter

#### 3.1.2. Toward the Receive Antenna Elements

- if the Cluster${}_{n}$∈$\{Qt(t+\Delta t)\bigcap Qr(t+\Delta t)\}$,$${\mathit{H}}_{n,lk,{l}^{\prime}{k}^{\prime}}(t+\Delta t,\tau )=\left(\right)open="("\; close=")">\begin{array}{c}\delta (n-1)\sqrt{\frac{G}{G+1}}\underset{LOS\phantom{\rule{3.33333pt}{0ex}}TX\phantom{\rule{3.33333pt}{0ex}}distance\phantom{\rule{3.33333pt}{0ex}}vector}{\underbrace{{\mathit{D}}_{LOS,{i}_{1},lk}^{BS}(t+\Delta t)}}\times \underset{LOS\phantom{\rule{3.33333pt}{0ex}}TX\phantom{\rule{3.33333pt}{0ex}}array\phantom{\rule{3.33333pt}{0ex}}response}{\underbrace{{\overrightarrow{\mathit{a}}(\varphi ,\theta )}_{lk}}}\times \underset{LOS\phantom{\rule{3.33333pt}{0ex}}random\phantom{\rule{3.33333pt}{0ex}}phases}{\underbrace{\left[\begin{array}{cc}exp(j{\mathsf{\Phi}}_{LOS}^{v,v})& 0\\ 0& exp(j{\mathsf{\Phi}}_{n}^{h,h})\end{array}\right]}}\times \hfill \\ \underset{LOS\phantom{\rule{3.33333pt}{0ex}}RX\phantom{\rule{3.33333pt}{0ex}}distance\phantom{\rule{3.33333pt}{0ex}}vector}{\underbrace{{\mathit{D}}_{LOS,{i}_{2},{l}^{\prime}{k}^{\prime}}^{UE}(t+\Delta t)}}\times \underset{LOS\phantom{\rule{3.33333pt}{0ex}}RX\phantom{\rule{3.33333pt}{0ex}}array\phantom{\rule{3.33333pt}{0ex}}response}{\underbrace{{\overrightarrow{\mathit{a}}}_{{l}^{\prime}{k}^{\prime}}(\phi ,\vartheta )}}+\sqrt{\frac{{A}_{n}}{G+1}}{\sum}_{n=1}^{N}\underset{NLOS\phantom{\rule{3.33333pt}{0ex}}TX\phantom{\rule{3.33333pt}{0ex}}distance\phantom{\rule{3.33333pt}{0ex}}vector}{\underbrace{{\mathit{D}}_{n,{i}_{1},lk}^{BS}(t+\Delta t)}}\hfill \\ \times \underset{NLOS\phantom{\rule{3.33333pt}{0ex}}TX\phantom{\rule{3.33333pt}{0ex}}array\phantom{\rule{3.33333pt}{0ex}}response}{\underbrace{{\overrightarrow{\mathit{a}}(\varphi ,\theta )}_{lk}}}\times \underset{NLOS\phantom{\rule{3.33333pt}{0ex}}random\phantom{\rule{3.33333pt}{0ex}}phases}{\underbrace{\left[\begin{array}{cc}exp(j{\mathsf{\Phi}}_{n}^{v,v})& \sqrt{{X}^{h}}exp(j{\mathsf{\Phi}}_{n}^{v,h})\\ \sqrt{{X}^{v}}exp(j{\mathsf{\Phi}}_{n}^{h,v})& exp(j{\mathsf{\Phi}}_{n}^{h,h})\end{array}\right]}}\times \hfill \\ \underset{NLOS\phantom{\rule{3.33333pt}{0ex}}RX\phantom{\rule{3.33333pt}{0ex}}distance\phantom{\rule{3.33333pt}{0ex}}vector}{\underbrace{{\mathit{D}}_{n,{i}_{2},{l}^{\prime}{k}^{\prime}}^{UE}(t+\Delta t)}}\times \underset{NLOS\phantom{\rule{3.33333pt}{0ex}}RX\phantom{\rule{3.33333pt}{0ex}}array\phantom{\rule{3.33333pt}{0ex}}response}{\underbrace{{\overrightarrow{\mathit{a}}}_{{l}^{\prime}{k}^{\prime}}(\phi ,\vartheta )}}\hfill \end{array}$$
- Otherwise, if Cluster${}_{n}\notin \{Qt(t+\Delta t)\bigcap Qr(t+\Delta t)\}$,$$\begin{array}{c}\hfill {\mathit{H}}_{n,lk,{l}^{\prime}{k}^{\prime}}(t+\Delta t,\tau )=0\end{array}$$

- NLOS: The ${lk}_{th}$ transmit antenna element vector $\overrightarrow{\mathit{a}}{(\varphi ,\theta )}_{lk}$ obtaining by Equation (12) in Section 2 and the vector between ${n}_{th}$ cluster via ${{i}_{1}}_{th}$ path ${\mathit{D}}_{n,{i}_{1},lk}^{BS}(t+\Delta t)$, and the vector between the ${n}_{th}$ cluster and the transmit antenna array ${\mathit{D}}_{n}^{BS}(t)$ at the TX can be given as$$\begin{array}{c}{\mathit{D}}_{n,{i}_{1},lk}^{BS}(t+\Delta t)={\mathit{D}}_{n}^{BS}(t){\left[\begin{array}{c}sin{\varphi}_{n,{i}_{1},lk}^{AAoD}(t+\Delta t),cos{\theta}_{n,{i}_{1},lk}^{EAoD}(t+\Delta t)\\ sin{\varphi}_{n,{i}_{1},lk}^{AAoD}(t+\Delta t),sin{\theta}_{n,{i}_{1},lk}^{EAoD}(t+\Delta t)\\ cos{\theta}_{n,{i}_{1},lk}^{EAoD}(t+\Delta t)\end{array}\right]}^{T}\hfill \end{array}$$Similarly, the ${{l}^{\prime}{k}^{\prime}}_{th}$ receive antenna element vector $\overrightarrow{\mathit{a}}{(\phi ,\vartheta )}_{{l}^{\prime}{k}^{\prime}}$ and the vector between ${n}_{th}$ cluster via ${{i}_{2}}_{th}$ path ${\mathit{D}}_{n,{i}_{2},{l}^{\prime}{k}^{\prime}}^{UE}(t+\Delta t)$, and the vector between the ${n}_{th}$ cluster and the receive antenna array ${\mathit{D}}_{n}^{UE}(t)$ at the RX can be presented as$$\begin{array}{c}{\mathit{D}}_{n,{i}_{2},{l}^{\prime}{k}^{\prime}}^{UE}(t+\Delta t)={\mathit{D}}_{n}^{UE}(t){\left[\begin{array}{c}sin{\phi}_{n,{i}_{2},{l}^{\prime}{k}^{\prime}}^{AAoA}(t+\Delta t),cos{\vartheta}_{n,{i}_{2},{l}^{\prime}{k}^{\prime}}^{EAoA}(t+\Delta t)\\ sin{\phi}_{n,{i}_{2},{l}^{\prime}{k}^{\prime}}^{AAoA}(t+\Delta t),sin{\vartheta}_{n,{i}_{2},{l}^{\prime}{k}^{\prime}}^{EAoA}(t+\Delta t)\\ cos{\vartheta}_{n,{i}_{2},{l}^{\prime}{k}^{\prime}}^{EAoA}(t+\Delta t)\end{array}\right]}^{T}+\mathit{D}\hfill \end{array}$$Then, the four random initial phases for ${n}_{th}$ cluster are derived as$$\begin{array}{c}\hfill \left[\begin{array}{cc}exp(j{\mathsf{\Phi}}_{n}^{(v,v)})& \sqrt{{X}^{h}}exp(j{\mathsf{\Phi}}_{n}^{(v,h)})\\ \sqrt{{X}^{v}}exp(j{\mathsf{\Phi}}_{n}^{(h,v)})& exp(j{\mathsf{\Phi}}_{n}^{(h,h)})\end{array}\right]\end{array}$$
- LOS: The ${lk}_{th}$ transmit antenna element vector $\overrightarrow{\mathit{a}}{(\varphi ,\theta )}_{lk}$ and the vector between $LO{S}_{th}$ path ${\mathit{D}}_{LOS,{i}_{1},lk}^{BS}(t+\Delta t)$ can be expressed as$$\begin{array}{c}{\mathit{D}}_{LOS,{i}_{1},lk}^{BS}(t+\Delta t)={\left[\begin{array}{c}sin{\varphi}_{LOS,{i}_{1},lk}^{AAoD}(t+\Delta t),cos{\theta}_{LOS,{i}_{1},lk}^{EAoD}(t+\Delta t)\\ sin{\varphi}_{LOS,{i}_{1},lk}^{AAoD}(t+\Delta t),sin{\theta}_{LOS,{i}_{1},lk}^{EAoD}(t+\Delta t)\\ cos{\theta}_{LOS,{i}_{1},lk}^{EAoD}(t+\Delta t)\end{array}\right]}^{T}\hfill \end{array}$$Similarly, the ${{l}^{\prime}{k}^{\prime}}_{th}$ receive antenna element vector $\overrightarrow{\mathit{a}}{(\phi ,\vartheta )}_{{l}^{\prime}{k}^{\prime}}$ and the vector between $LO{S}_{th}$ path ${\mathit{D}}_{LOS,{i}_{1},{l}^{\prime}{k}^{\prime}}^{UE}(t+\Delta t)$ can be presented as$$\begin{array}{c}{\mathit{D}}_{LOS,{i}_{1},{l}^{\prime}{k}^{\prime}}^{UE}(t+\Delta t)={\left[\begin{array}{c}sin{\phi}_{LOS,{i}_{1},{l}^{\prime}{k}^{\prime}}^{AAoA}(t+\Delta t),cos{\vartheta}_{LOS,{i}_{1},{l}^{\prime}{k}^{\prime}}^{EAoA}(t+\Delta t)\\ sin{\phi}_{LOS,{i}_{1},{l}^{\prime}{k}^{\prime}}^{AAoA}(t+\Delta t),sin{\vartheta}_{LOS,{i}_{1},{l}^{\prime}{k}^{\prime}}^{EAoA}(t+\Delta t)\\ cos{\vartheta}_{LOS,{i}_{1},{l}^{\prime}{k}^{\prime}}^{EAoA}(t+\Delta t)\end{array}\right]}^{T}+\mathit{D}\hfill \end{array}$$Then, the two random initial phases for outdoor $LO{S}_{th}$ cluster is derived as$$\begin{array}{c}\hfill \left[\begin{array}{cc}exp(j{\mathsf{\Phi}}_{LOS}^{v,v})& 0\\ 0& exp(j{\mathsf{\Phi}}_{LOS}^{h,h})\end{array}\right]\end{array}$$

#### 3.2. Delay of the Clusters

#### 3.3. Energy Transferring

## 4. Received Spatial Correlation

## 5. Experimental Results and Discussions

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Detailed of antenna configuration X${}_{Pol}$, V, H, and V/H polarizations of an antenna elements in the horizontal and vertical angles.

**Figure 2.**A more detailed of field experiments of the beam pattern using spherical coordinate system for non-stationary massive MIMO channel.

**Figure 3.**Phase difference between antenna elements for the varying (

**a**) DoT = ${90}^{\circ}$ and (

**b**) DoT = ${0}^{\circ}$.

**Figure 4.**Overview of the measurement area at the campus of Shanghai Jiao Tong University, China. A spaced rectangular antenna array with 16 cross polarized patch antenna elements at the TX was placed on the rooftop of the Biomedical building during their respective measurement campaigns and a spaced uniform linear array as a user was moved around the international buildings acting as multiple-antenna user.

**Figure 5.**Performance comparison between E3-D and C3-D models based on different configurations, (

**a**). V/H, (

**b**). V, (

**c**). H, receiving signals from TX with cross antenna elements polarization toward RX, in case there are polarization matching between TX and RX. And (

**d**), in case there are polarization mismatching between TX and RX, the channel variations are based on the user movement with different DoT = ${0}^{\circ}$ and DoT = ${90}^{\circ}$. It is illustrated that the channel has been changed when the distance is increasing and also, the signal orientations do not match with the antenna polarization at the RX. Therefore, variation in polarization causes changes in the received signal level due to the inability of the antenna to receive such polarization changes.

Parameters | Values |
---|---|

Frequency range | 2.620–2.630 (GHz) |

Duplexing | TDD and FDD |

Channel coding | Turbo code |

Channel bandwidth | 10 (MHz) |

FFT size | 1024 sub-carriers |

CP length | 80, 72 normal |

Total symbols | 140 s |

TX block configuration | 50 Resource blocks |

Modulation schemes | QPSK and QAM |

Multiple access schemes | OFDM |

Parameters | Values |
---|---|

$(\theta ,\varphi )$, $(\vartheta ,\phi )$ | elevation and azimuth angles of the departure and arrival, respectively |

${\mathit{D}}_{n,{i}_{1},lk}^{BS}(t+\Delta t)$ | distance vector between ${n}_{th}$ cluster and ${lk}_{th}$ transmit antenna element via ${{i}_{1}}_{th}$ path |

${\mathit{D}}_{n}^{BS}(t)$ | distance vector between ${n}_{th}$ cluster and transmit antenna |

${\mathit{D}}_{n,{i}_{2},{l}^{\prime}{k}^{\prime}}^{UE}(t+\Delta t)$ | distance vector between ${n}_{th}$ cluster and ${{l}^{\prime}{k}^{\prime}}_{th}$ receive antenna element via ${{i}_{2}}_{th}$ path |

${\mathit{D}}_{n}^{UE}(t)$ | distance vector between ${n}_{th}$ cluster and receive antenna |

${\mathit{D}}_{LOS,{i}_{1},lk}^{BS}(t+\Delta t)$ | distance vector between $LO{S}_{th}$ path and ${lk}_{th}$ transmit antenna element |

${\mathit{D}}_{LOS,{i}_{1},{l}^{\prime}{k}^{\prime}}^{UE}(t+\Delta t)$ | distance vector between $LO{S}_{th}$ path and ${{l}^{\prime}{k}^{\prime}}_{th}$ receive antenna element |

$\overrightarrow{\mathit{a}}{(\varphi ,\theta )}_{lk},\overrightarrow{\mathit{a}}{(\phi ,\vartheta )}_{{l}^{\prime}{k}^{\prime}}$ | array response of the TX and RX, respectively |

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**MDPI and ACS Style**

Aghaeinezhadfirouzja, S.; Liu, H.; Balador, A.
Practical 3-D Beam Pattern Based Channel Modeling for Multi-Polarized Massive MIMO Systems. *Sensors* **2018**, *18*, 1186.
https://doi.org/10.3390/s18041186

**AMA Style**

Aghaeinezhadfirouzja S, Liu H, Balador A.
Practical 3-D Beam Pattern Based Channel Modeling for Multi-Polarized Massive MIMO Systems. *Sensors*. 2018; 18(4):1186.
https://doi.org/10.3390/s18041186

**Chicago/Turabian Style**

Aghaeinezhadfirouzja, Saeid, Hui Liu, and Ali Balador.
2018. "Practical 3-D Beam Pattern Based Channel Modeling for Multi-Polarized Massive MIMO Systems" *Sensors* 18, no. 4: 1186.
https://doi.org/10.3390/s18041186