MIMO

Bandwitdh in wireless communications is expensive and limited due to licensing. An easy way to use available bandwidth as efficient as possible is to add antennas on both the sending and transmitting side. This way a simple point to point transmission channel becomes a Multiple-Output-Multiple-Input (MIMO) system. A MIMO system without noise can be described by $$ \begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_N \end{pmatrix} = \begin{pmatrix} h_{y_1 x_1} & h_{y_1 x_2} & \cdots & h_{y_1 x_M} \\ h_{y_2 x_1} & h_{y_2 x_2} & \cdots & h_{y_2 x_M} \\ \vdots & \vdots & \ddots & \vdots \\ h_{y_N x_1} & h_{y_N x_2} & \cdots & h_{y_N x_M} \\ \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_M \end{pmatrix} $$ where

- $x_1, x_2, ..., x_M$ denote the sent data streams
- $y_1, y_2, ..., y_N$ denote the received data streams
- $N$ is the number of receiver antennas
- $M$ is the number of transmitter antennas
- $h_{xy}$ denotes the elements of the impulse response matrix $\mathbf{H}$

The 802.11n mimo model is a Kronecker correlation model which utilizes correlation matrices to describe the MIMO system. Using this method only parameters like angular spread, angle of incidence (arrival/departure) are needed. Contrary to ray tracing an exact description of the environment is not necessary. The channel impulse response matrix $\mathbf{H}$ for each path at a given time is described by $$ \mathbf{H} = \sqrt{P}\left(\sqrt{\frac{K}{K+1}} \begin{bmatrix} \mathrm{e}^{j\phi_{11}} & \mathrm{e}^{j\phi_{12}} & \mathrm{e}^{j\phi_{13}} \\ \mathrm{e}^{j\phi_{21}} & \mathrm{e}^{j\phi_{22}} & \mathrm{e}^{j\phi_{23}} \\ \mathrm{e}^{j\phi_{31}} & \mathrm{e}^{j\phi_{32}} & \mathrm{e}^{j\phi_{33}} \\ \end{bmatrix} + \sqrt{\frac{1}{K+1}} \begin{bmatrix} X_{11} & X_{12} & X_{13} \\ X_{21} & X_{22} & X_{23} \\ X_{31} & X_{32} & X_{33} \\ \end{bmatrix}\right) $$ where

- $X_{ij}$ are correlated, zero-mean, unit variance, complex Gaussian random variables
- $\exp{(j\phi_{ij})}$ are the elements of the fixed LOS matrix
- $K$ denotes the Ricean K-factor
- $P$ denotes the power of each tap

- $R_{\mathrm{TX}}$ denotes the transmit correlation matrix
- $R_{\mathrm{RX}}$ denotes the receive correlation matrix
- $\mathbf{H_{\mathrm{iid}}}$ is a matrix of independent, zero-mean, unit variance, complex Gaussian random variables
- $\otimes$ denotes the Kronecker product operator (an example of an Kronecker product is provided below)

- $\mathbf{A}$ denotes an $m \times n$ matrix
- $\mathbf{B}$ denotes an $x \times y$ matrix