双极化（天线）ULA的信道模型

双极化（天线）均匀线性阵列

Uniform Linear Array of $M$ dual-polarized antenna elements (如图所示)

基站有$M$个双极化天线阵元，终端UE为1根单极化天线，那么上行信道可以表征为：
$${\mathbf{h}}_{ul}=\left[\begin{array}{c}{\mathbf{h}}_{ul,H}\\ {\mathbf{h}}_{ul,V}\end{array}\right]\in {\mathbb{C}}^{2M\times 1}$$

其中，

• ${\mathbf{h}}_{ul,H}\in {\mathbb{C}}^{M\times 1}$ is the channel vector corresponding to the $M$ H-polarzied antenna ports
• ${\mathbf{h}}_{ul,V}\in {\mathbb{C}}^{vM\times 1}$ is the channel vector corresponding to the $M$ V-polarzied antenna ports

回顾ULA阵列的阵列响应向量为(array)
$${\mathbf{a}}_{ul}(\xi )=[1,e^{j\pi \xi },\cdots ,e^{j\pi (M-1)\xi }]\in {\mathbb{C}}^{M\times 1}$$

其中$\xi =\sin (\theta )$,.$\theta \in [-\pi /2,\pi /2]$是到达角（Angle of Arrival, AOA），注意这里我们假设$d={\lambda }_{ul}/2$。因此 H- & V- channel vector可以表征为
$$\begin{array}{rl}{\mathbf{h}}_{ul,H}& ={\int }_{-1}^{+1}{W}_H(\xi ){\mathbf{a}}_{ul}(\xi )d\xi \\ {\mathbf{h}}_{ul,V}& ={\int }_{-1}^{+1}{W}_V(\xi ){\mathbf{a}}_{ul}(\xi )d\xi \end{array}$$

其中$W_H(\xi ),W_V(\xi )$是表征角度增益(angular gain)的随机过程。我们假设$W_H,W_V$是零均值的复高斯过程，具有下面的自相关关系
$$\begin{array}{rl}\mathbb{E}\left[W_H(\xi )W_H^{\ast }({\xi }^{\prime })\right]& ={\gamma }_H(\xi )\delta (\xi -{\xi }^{\prime })\\ \mathbb{E}\left[W_V(\xi )W_V^{\ast }({\xi }^{\prime })\right]& ={\gamma }_V(\xi )\delta (\xi -{\xi }^{\prime })\end{array}$$

where we have adopted the wide-sense stationary uncorrelated scattering (WSSUS) model, which assumes stationary second order channel statistics (over resonably short time intervals) and uncorrelated angular scattering gains.

${\gamma }_H,{\gamma }_V\in {\mathbb{R}}_{+}$分别表示H- & V-的角度功率(along each AOA)。We call these horizontal and vertical angular spread functions (ASFs)。

实际中，H & V的关系一般不是孤立的，两者之间存在信道能量的泄露。因此对于每个AoA，$W_H(\xi )$和$W_V(\xi )$是相关的，满足
$$\mathbb{E}[W_H(\xi )W_V({\xi }^{\prime })]=\rho (\xi )\delta (\xi -{\xi }^{\prime })$$

其中$\rho \in \mathbb{C}$。

我们常把信道向量显式地分为两个部分，一部分是LOS成分(component)，另一部分是NLOS成分，构成为
$${\mathbf{h}}_{ul,(H/V)}=\sqrt{\alpha }{\mathbf{h}}_{ul,(H/V)}^{LOS}+\sqrt{1-\alpha }{\mathbf{h}}_{ul,(H/V)}^{NLOS}$$

其中$\alpha \in [0,1]$是一个能量归一化系数。

LOS和NLOS成分分别为：
$$\begin{array}{rl}{\mathbf{h}}_{ul,(H/V)}^{LOS}& ={\beta }_{ul,(H/V)}^{LOS}{\mathbf{a}}_{ul}({\xi }_{LOS})\\ {\mathbf{h}}_{ul,(H/V)}^{NLOS}& =\sum _{i=1}^{p-1}{\beta }_{ul,(H/V),i}^{NLOS}{\mathbf{a}}_{ul}({\xi }_{NLOS,i})\end{array}$$

其中${\xi }_{LOS},\{{\xi }_{NLOS,i}{\}}_i$是到达角AOAs，${\beta }_{ul,(H/V)}^{LOS},\{{\beta }_{ul,(H/V),i}^{NLOS}{\}}_i$是复增益(complex-valued gains)。

那么双极化的上行信道可以被表征为：
$$\begin{array}{rl}{\mathbf{h}}_{ul}& ={\int }_{-1}^1\left[\begin{array}{cc}{\mathbf{a}}_{ul}\left(\xi \right)& 0\\ 0& {\mathbf{a}}_{ul}\left(\xi \right)\end{array}\right]\left[\begin{array}{c}{W}_H\left(\xi \right)\\ W_V\left(\xi \right)\end{array}\right]d\xi \\ & ={\int }_{-1}^1\left({\mathbf{I}}_2\otimes {\mathbf{a}}_{ul}(\xi )\right)\mathbf{w}(\xi )d\xi \end{array}$$

其中$\mathbf{w}(\xi ):=[W_H\left(\xi \right),W_V\left(\xi \right){]}^T$，那么信道的协方差矩阵可以表示为
$${\mathbf{\Sigma }}_h^{ul}=\mathbb{E}\left[{\mathbf{h}}_{ul}{\mathbf{h}}_{ul}^H\right]={\int }_{-1}^1\mathbf{\Gamma }(\xi )\otimes {\mathbf{A}}_{ul}(\xi )d\xi$$

其中${\mathbf{A}}_{ul}(\xi )={\mathbf{a}}_{ul}(\xi ){\mathbf{a}}_{ul}(\xi {)}^H$，$\mathbf{\Gamma }(\xi )$定义为
$$\mathbf{\Gamma }(\xi )=\mathbb{E}\left[\mathbf{w}(\xi )\mathbf{w}(\xi {)}^H\right]=\left[\begin{array}{cc}{\gamma }_H\left(\xi \right)& \rho \left(\xi \right)\\ \rho {\left(\xi \right)}^{\ast }& {\gamma }_V\left(\xi \right)\end{array}\right]\in {\mathbb{C}}^{2\times 2}$$

$\mathbf{\Gamma }(\xi )$是一个半正定矩阵，我们把$\mathbf{\Gamma }(\xi )$称为dual-polarized angular spread function (DP-ASF). DP-ASF captures the angular spectral properties of the channel, i.e. the power density along H and V links and the power leakage density between the two.

[1] M. B. Khalilsarai, T. Yang, S. Haghighatshoar, X. Yi and G. Caire, “Dual-Polarized FDD Massive MIMO: A Comprehensive Framework,” in IEEE Transactions on Wireless Communications, vol. 21, no. 2, pp. 840-854, Feb. 2022, doi: 10.1109/TWC.2021.3099727.