﻿ 一种3D Massive MIMO Kronecker信道模型
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 应用科技  2018, Vol. 45 Issue (6): 37-41  DOI: 10.11991/yykj.201801003 0

### 引用本文

DUAN Jingjing, ZHANG Wei. A 3D Massive MIMO kronecker channel model[J]. Applied Science and Technology, 2018, 45(6), 37-41. DOI: 10.11991/yykj.201801003.

### 文章历史

A 3D Massive MIMO kronecker channel model
DUAN Jingjing, ZHANG Wei
College of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: To establish a rational channel model for performance analysis and evaluation suitable for massive multiple-input multiple-output (MIMO) system, this paper proposes a 3D Massive MIMO Kronecker channel model. On basis of the study of traditional 3D MIMO Kronecker channel model, the non-stationary characteristics of Massive MIMO channel are characterized by using the birth-death process to model the non-stationary evolution of scattering clusters on the array axis. The birth-death process is abstracted as the survival probability matrix. Simulation results show that the proposed channel model can not only capture the spatial correlation of large-scale antenna arrays, but also describe the evolution of scattering clusters on the large-scale antenna array axis.
Keywords: Massive MIMO    channel model    Kronecker model    non-stationary property    scattering cluster    birth-death process    survival probability matrix    spatial correlation

Kronecker信道模型是最为广泛使用的信道模型之一，由于该模型复杂度较低且实现简单，常用于研究具有空间相关性的MIMO系统容量和性能仿真分析[5-6]。为了更好地描述3D信道传播特性，近来有学者提出了一种3D MIMO Kronecker信道模型用于任意结构天线阵列的3D MIMO信道容量和天线间相关性的分析研究[7]，但该Kronecker模型缺乏对Massive MIMO信道非平稳特性的描述，适用于Massive MIMO系统的3D Kronecker模型还需要进一步研究。

1 3D MIMO Kronecker信道建模 1.1 3D MIMO信道模型

3D MIMO信道模型中散射传输路径可以抽象为散射簇，由方位角、俯仰角和散射簇功率表示，如图1所示。

 $\begin{array}{l}{h_{u,s,n}}(t) = \sqrt {{{{P_n}} / M}} {\sum\limits_{m = 1}^M {\left[ {\begin{array}{*{20}{c}}{{F_{{\rm{RX}},u,\theta '}}\left( {\theta ',\phi '} \right)}\\{{F_{{\rm{RX}},u,\phi '}}\left( {\theta ',\phi '} \right)}\end{array}} \right]} ^{\rm{T}}} \times \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left[ {\begin{array}{*{20}{c}}{\exp \left( {j\varPhi _{n,m}^{\theta \theta }} \right)} & {\sqrt {{\kappa ^{ - 1}}} \exp \left( {{\rm{j}}\varPhi _{n,m}^{\theta \phi }} \right)}\\{\sqrt {{\kappa ^{ - 1}}} \exp \left( {j\varPhi _{n,m}^{\phi \theta }} \right)} & {\exp \left( {{\rm{j}}\varPhi _{n,m}^{\phi \phi }} \right)}\end{array}} \right] \times \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left[ {\begin{array}{*{20}{c}}{{F_{{\rm{TX}},s,\theta }}\left( {\theta ,\phi } \right)}\\{{F_{{\rm{TX}},s,\phi }}\left( {\theta ,\phi } \right)}\end{array}} \right] \times \exp \left( {{\rm{j}}2{\rm{{\text {π}} }}\lambda _0^{ - 1}\left( {{{{{\Omega}} '}_{n,m}} \cdot {{{d}}_{{\rm{RX}},u}}} \right)} \right) \times \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\exp \left( {{\rm{j2{\text {π}} }}\lambda _0^{ - 1}\left( {{{{\Omega }}_{n,m}} \cdot {{{d}}_{{\rm{TX}},s}}} \right)} \right) \times \exp \left( {{\rm{j2{\text {π}} }}{v_{n,m}}t} \right)\end{array}$ (1)

 ${h_{u,s}}(t){\rm{ = }}\sum\limits_{n = 1}^N {{h_{u,s,n}}(t)}$ (2)
1.2 3D MIMO Kronecker模型分析

Kronecker模型假设发送端相关矩阵和接收端相关矩阵是可分离的，最终通过获得的信道相关矩阵来计算信道系数矩阵。下面以式(2)为基础给出3D MIMO Kronecker信道模型。一个MIMO系统由 $S$ 根发射天线和 $U$ 根接收天线组成，信道矩阵H

 ${{H}}{\rm{ = }}{\left[ {{h_{u,s}}} \right]_{U \times S}}$

3D MIMO Kronecker信道模型中，信道相关矩阵 ${{{R}}_H}$ 表示为发送端相关矩阵 ${{{R}}_{{\rm{TX}}}}$ 和接收端相关矩阵 ${{{R}}_{{\rm{RX}}}}$ 的Kronecker积，即

 ${{{R}}_H} = {{{R}}_{{\rm{TX}}}} \otimes {{{R}}_{{\rm{RX}}}}$

 ${{{R}}_{{\rm{TX}}}} = {\rm{E}}\left\{ {\left[ {\begin{array}{*{20}{c}} {\sum\limits_{u = 1}^U {h_{{u_1}}^ * {h_{{u_1}}}} }&{...}&{\sum\limits_{u = 1}^U {h_{{u_1}}^ * {h_{{u_s}}}} } \\ \vdots &{...}& \vdots \\ {\sum\limits_{u = 1}^U {h_{{u_s}}^ * {h_{{u_1}}}} }&{...}&{\sum\limits_{u = 1}^U {h_{{u_s}}^ * {h_{{u_s}}}} } \end{array}} \right]} \right\}$

${{ R}_{{\rm{TX}}}}$ 矩阵中任意一个元素 ${R_{{\rm{TX}}}}\left( {k,l} \right)$ ，其中 $k,l \leqslant U$ ，可得：

 ${R_{{\rm{TX}}}}(k,l) = {\rm{E}}\left\{ {\sum\limits_{u = 1}^U {h_{{u_k}}^ * {h_{{u_l}}}} } \right\}$ (3)

 ${R_{{\rm{RX}}}}(k,l) = {\rm{E}}\left\{ {\sum\limits_{s = 1}^S {h_{{k_s}}^ * {h_{{l_s}}}} } \right\}$

 ${{H}}{\rm{ = }}{{R}}_{{\rm{RX}}}^{1/2}{{GR}}_{{\rm{TX}}}^{1/2}$ (4)

2 3D Massive MIMO信道建模

3D MIMO Kronecker模型虽然可以很好地描述传统MIMO信道的三维传播场景，但缺少对Massive MIMO信道非平稳特性的描述，本文提出的3D Massive MIMO Kronecker模型中采用散射簇在阵列轴上生灭过程演变来建模Massive MIMO信道的非平稳特性。

2.1 信道的非平稳特性

2.2 生灭过程

 ${P_{{\rm{TX}}}} = {{\rm{e}}^{ - \beta {d_T}}}$

 $\begin{array}{l}{P_{{\rm{TX}}}}(k,l) = {P_{{\rm{TX}}}}\left\{ {k \to s} \right\}{P_{{\rm{TX}}}}\left\{ {s \to l} \right\} = \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{{\rm{e}}^{ - \beta {d_T}(s{\rm{ - }}k)}}{{\rm{e}}^{ - \beta {d_T}(l - s)}} = {{\rm{e}}^{ - \beta {d_T}\left( {l - k} \right)}}\end{array}$
2.3 3D Massive MIMO Kronecker模型

 ${R}_{{\rm{TX}}}{^{\rm{'}}}\left( {k,l} \right) = {P_{{\rm{TX}}}}(k,l){R_{{\rm{TX}}}}\left( {k,l} \right)$

 ${{ R}}_{{\rm{TX}}}{^{\rm{'}}}{\rm{ = }}{{{R}}_{{\rm{TX}}}} \circ {{{P}}_{{\rm{TX}}}}$

3 仿真分析

 $C = {\log _2}\det [{{{I}}_S} + \frac{\rho }{S}{{H}}{{{H}}^{\rm{H}}}]$ (5)

 $C = {\log _2}\det [{{{I}}_S} + \frac{\rho }{S}{{R}}_{{\rm{RX}}}^{1/2}{{G}}{{{R}}_{{\rm{TX}}}}{{{G}}^{\rm{H}}}{{R}}_{{\rm{RX}}}^{{\rm{H}}/2}]$

4 结论

1）与传统3D MIMO Kronecker信道模型相比，所提模型更加准确地反映了Massive MIMO的信道特性；

3）通过计算机仿真实验可以看出，散射簇在天线阵列轴的非平稳演变可以进一步降低Massive MIMO信道的空间相关性。

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