信道估计模型

Notations

  1. 矩阵拉直或矩阵列化
    设$\boldsymbol{A}=(a_{ij})\in \mathbb{C}^{m\times n}$,记$\boldsymbol{a}_i=(a_{1i},\cdots,a_{mi})^T\ (i=1,\cdots, n)$,令
    \begin{align}
    \text{vec}(\boldsymbol{A})=
    \left[
    \begin{matrix}
    \boldsymbol{a}_1\\
    \vdots\\
    \boldsymbol{a}_n
    \end{matrix}
    \right]
    \end{align}
  2. Kronecker积
    设矩阵$\boldsymbol{A}=(a_{ij})\in \mathbb{C}^{m\times n}$,$\boldsymbol{B}=(b_{ij})\in \mathbb{C}^{p\times q}$,则称如下分块矩阵
    \begin{align}
    \boldsymbol{A}\otimes \boldsymbol{B}=\left[
    \begin{matrix}
    a_{11}\boldsymbol{B} &\cdots &a_{1n}\boldsymbol{B}\\
    \vdots& \ddots&\vdots\\
    a_{m1}\boldsymbol{B} &\cdots &a_{mn}\boldsymbol{B}
    \end{matrix}
    \right]
    \end{align}

System Model

考虑上行单小区多用户大Massive MIMO (multi-input multi-output),其中基站配置有$M$根天线用于同时服务$K$个单天线用户。假设信道是平坦块衰落(flat block fadding),那么基站(base station, BS)的接收信号可以表示为
\begin{align}
\boldsymbol{Y}=\boldsymbol{HX}+\boldsymbol{W}
\end{align}
其中$\boldsymbol{X}\in \mathbb{C}^{K\times L}$是训练信号(training signal)矩阵,该矩阵的每一行对应着每一个用户发动的$L$长导频符号的训练数据。信道矩阵$\boldsymbol{H}\in \mathbb{C}^{M\times K}$表示确定的,待估计的信道参数。$\boldsymbol{W}\in \mathbb{C}^{M\times L}$表示加性高斯白噪声(additive white Gaussian noise,AWGN),其每一个元素均服从均值为0,方差为$2\sigma^2$的循环对称复高斯分布(circularly symmetric complex Gaussian, CSCG)。

该系统模型用实数矩阵表示为
\begin{align}
\tilde{\boldsymbol{Y} }=\tilde{\boldsymbol{A} }\tilde{\boldsymbol{H} }+\tilde{\boldsymbol{W} }
\end{align}
其中
\begin{align}
\tilde{\boldsymbol{Y} }&\overset{\triangle}{=}[\text{Re}(\boldsymbol{Y}),\text{Im}(\boldsymbol{Y})]^T\\
\tilde{\boldsymbol{H} }&\overset{\triangle}{=}[\text{Re}(\boldsymbol{H}),\text{Im}(\boldsymbol{H})]^T\\
\tilde{\boldsymbol{W} }&\overset{\triangle}{=}[\text{Re}(\boldsymbol{W}),\text{Im}(\boldsymbol{W})]
\end{align}
以及
\begin{align}
\tilde{\boldsymbol{A} }\overset{\triangle}{=}\left[
\begin{matrix}
\text{Re}(\boldsymbol{X}) &\text{Im}(\boldsymbol{X})\\
-\text{Im}(\boldsymbol{X}) & \text{Re}(\boldsymbol{X})
\end{matrix}
\right]^T
\end{align}

将矩阵列化(vectorizing),有
\begin{align}
\boldsymbol{y}=\boldsymbol{Ah}+\boldsymbol{w}
\end{align}
其中$\boldsymbol{y}\overset{\triangle}{=}\text{vec}(\tilde{\boldsymbol{Y} })$,$\boldsymbol{h}\overset{\triangle}{=}\text{vec}(\tilde{\boldsymbol{H} })$,$\boldsymbol{w}\overset{\triangle}{=}\text{vec}(\tilde{\boldsymbol{W} })$,以及$\boldsymbol{A}=\boldsymbol{I}_M \otimes \tilde{\boldsymbol{A} }$,其中$\otimes$表示Kronecker积,$\text{vec}$表示矩阵列化操作或矩阵拉直。可以很容易证明,这些参数的维度为:$\boldsymbol{y}\in \mathbb{R}^{2ML}$,$\boldsymbol{A}\in \mathbb{R}^{2ML\times 2MK}$,$\boldsymbol{h}\in \mathbb{R}^{2MK}$。

References:
[1] F. Wang, J. Fang, H. Li, Z. Chen and S. Li, “One-Bit Quantization Design and Channel Estimation for Massive MIMO Systems,” in IEEE Transactions on Vehicular Technology, vol. 67, no. 11, pp. 10921-10934, Nov. 2018.