Depending on the arrangement and spacing of antenna elements, the propagation channels between different antenna elements may be correlated. To include this correlation into channel matrix, a popular Kronecker model is introduced assuming that the transmit correlation $\mathbf{R}_{t}$ is independent of the receive correlation $\mathbf{R}_{r}$. Mathematically, the channel matrix can be written as $$ \mathbf{H} = \mathbf{R}_{r}^{1/2} \mathbf{H}_{w} \mathbf{R}_{t}^{1/2} $$ where $\mathbf{H}_{w}$ denotes an identical independent distributed (i.i.d.) channel without any transmit and receive correlation. By using this model the covariance matrix of $\rm{vec}\left(\mathbf{H}\right)$ is given by $\mathbf{R}_{r} \otimes \mathbf{R}_{t}$.

In this webdemo, we consider an approximate receive and transmit correlation model based on linear antenna arrays with equidistant spacing. The corresponding correlation matrices can be written as: $$ \mathbf{R}_{t}=\begin{bmatrix} 1 & \rho_{t} & \rho_{t}^{4} & \cdots & \rho_{t}^{\left(N_{t}-1\right)^2} \\ \rho_{t} & 1 & \rho_{t} &\cdots & \vdots\\ \vdots & \ddots & &\ddots & \vdots \\ \rho_{t}^{\left(N_{t}-1\right)^2} & \cdots & \rho_{t}^{4} & \rho_{t} &1 \end{bmatrix} \, \mathbf{R}_{r}=\begin{bmatrix} 1 & \rho_{r} & \rho_{r}^{4} & \cdots & \rho_{r}^{\left(N_{r}-1\right)^2} \\ \rho_{r} & 1 & \rho_{r} &\cdots & \vdots\\ \vdots & \ddots & &\ddots & \vdots \\ \rho_{r}^{\left(N_{r}-1\right)^2} & \cdots & \rho_{r}^{4} & \rho_{r} &1 \end{bmatrix} $$ Thus the correlation can be characterized by two parameters, i.e., $\rho_{t} \in [0,1)$ and $\rho_{r} \in [0,1)$ which depends on the angular spread and distance of antenna elements at transmitter and receiver respectively.