# Performance Analysis of MIMO System with Single RF Link Based on Switched Parasitic Antenna
## 2 Theory of MIMO System
### Receiver Design  
The algorithm introduced at the receiver is mainly for correcting the distortion introduced by the channel. Three receiver algorithms are presented, including the zero forcing algorithm, the maximum merge ratio algorithm, and the minimum mean square error algorithm. The core idea of the zero forcing algorithm is to use linear transformation matrix to multiply the receiver matrix, and it will completely or partially eliminate the interference of other antennas except the Gauss white noise. For the \(t\)\(\times\)\(r\) MIMO system, the relation between the received signal \(Y\) of the receiving antenna, the transmitting signal \(X\) of the transmitting antenna, the channel matrix \(H\), the interference \(W\), and the Gauss white noise \(N\) can be described as:  
\[Y=HX+W+N \tag{14}\]  
It can be rewritten into the matrix format, as is shown in Equation (15). \[\left[ \begin{array}{c}y_{0}\\ y_{1}\\ y_{2}\\ \vdots\\ y_{r-1}\end{array} \right]=\left[ \begin{array}{ccc}h_{11}&\cdots&h_{1t}\\ \vdots&\ddots&\vdots\\ h_{r1}&\cdots&h_{rt}\end{array} \right]\left[ \begin{array}{c}x_{0}\\ x_{1}\\ x_{2}\\ \vdots\\ x_{r-1}\end{array} \right]+\left[ \begin{array}{c}w_{0}\\ w_{1}\\ w_{2}\\ \vdots\\ w_{r-1}\end{array} \right]+\left[ \begin{array}{c}n_{0}\\ n_{1}\\ n_{2}\\ \vdots\\ n_{r-1}\end{array} \right] \tag{15}\]  
A linear merging method can be found to eliminate interference \(W\), and the merging coefficients \(\left[ \begin{array}{ccc}c_{0}&\ldots&c_{r-1}\end{array} \right]\) needs to be found to satisfy:  
\[\left[ \begin{array}{ccc}c_{0}&\ldots&c_{r-1}\end{array} \right]\left[ \begin{array}{c}y_{0}\\ y_{1}\\ y_{2}\\ \vdots\\ y_{r-1}\end{array} \right]=\left[ \begin{array}{ccc}c_{0}&\ldots&c_{r-1} \end{array} \right]\left[ \begin{array}{ccc}h_{11}&\cdots&h_{1t}\\ \vdots&\ddots&\vdots\\ h_{r1}&\cdots&h_{rt}\end{array} \right]\left[ \begin{array}{c}x_{0}\\ x_{1}\\ x_{2}\\ \vdots\\ x_{r-1}\end{array} \right]+0+\left[ \begin{array}{ccc}c_{0}&\ldots&c_{r-1} \end{array} \right]\left[ \begin{array}{c}n_{0}\\ n_{1}\\ n_{2}\\ \vdots\\ n_{r-1}\end{array} \right] \tag{16}\]  
Therefore, the middle product term should be zero. \[\left[ \begin{array}{ccc}c_{0}&\ldots&c_{r-1}\end{array} \right]\left[ \begin{array}{c}w_{0}\\ w_{1}\\ w_{2}\\ \vdots\\ w_{r-1}\end{array} \right]=0 \tag{17}\]  
Therefore, we need to find a vector orthogonal to \(\left[ \begin{array}{ccc}w_{0}&\ldots&w_{r-1}\end{array} \right]^{T}\) in the \(r\) dimensional linear space. Based on the matrix theory, there are many solutions that are orthogonal to the \(w\) matrix. From the linear space point of view, the zero forcing algorithm is to project the received signal to the orthogonal direction of interference, so that the interference becomes zero or approaches zero, which is the maximum signal to interference ratio (SIR). The principle of the maximum combination ratio algorithm is the projection of the received signal to a certain direction to ensure the maximum received signal-to-noise power ratio. For the \(t\)\(\times\)\(r\) MIMO system, the maximum combination ratio algorithm uses the weighted factor \(\hat{h}_{ij}\) to linearly combine the reception signal:\[\sum\hat{h}_{ij}y_{i}=\left\langle\mathbf{y},\hat{\mathbf{h}}^{*}\right\rangle=\left\langle \mathbf{h},\hat{\mathbf{h}}^{*}\right\rangle x+\left\langle\mathbf{n},\hat{\mathbf{h}}^{*}\right\rangle \tag{18}\]  
The \(h\) matrix can be expressed as:  
\[\mathbf{h}=\left[ \begin{array}{ccc}h_{11}&\cdots&h_{1t}\\ \vdots&\ddots&\vdots\\ h_{r1}&\cdots&h_{rt}\end{array} \right],\hat{\mathbf{h}}=\left[ \begin{array}{ cccc}h_{11}^{*}&\cdots&\frac{h_{1t}^{*}}{\left|\mathbf{h}\right|}\\ \vdots&\ddots&\vdots\\ \frac{h_{r1}^{*}}{\left|\mathbf{h}\right|}&\cdots&\frac{h_{rt}^{*}}{\left|\mathbf{h} \right|}\end{array} \right] \tag{19}\]  
The independent high four noise does not change the projection length \(\left\langle\mathbf{n},\hat{\mathbf{h}}^{*}\right\rangle\) in any direction, and it owns the isotropic character.