# Performance Analysis of MIMO System with Single RF Link Based on Switched Parasitic Antenna
## 2 Theory of MIMO System
### Channel Design  
In the MIMO communication systems, signals transmitted from the antennas to the received antennas needs to pass through the wireless channels. In the transmission, it is impossible for the electromagnetic wave to realize the direct transmission from the transmitting antenna to the receiving antenna. The buildings, trees, cars, and pedestrians during the transmission will cause reflection, scattering, and attenuation, which forms the multipath effect [30]. For a point from the receiving end, the final vibration presentation is the synthesis triggered by the multiple paths vibration, but the power required for each path vibration trigger is not necessarily the same. Besides, the vibration starting time can be different. In Equation (10), \(\alpha\) is the power attenuation factor, and \(x(t)\) and \(\tau\) present the emission signal and the delay, respectively. The whole vibration can be described as:  
\[\sum_{i}\alpha_{i}x(t-\tau_{i}) \tag{10}\]  
The closing formula represents the superposition of vibrations triggered by all paths. Multipath effects can be divided into two kinds: the objective multipath effect and the multipath effect with time resolution [31]. For an objective multipath, the received signal can be represented as:  
\[\sum_{i}\alpha_{i}x(t-\tau)=\sum_{i}\alpha_{i}x(t-\tau)=Ax(t-\tau) \tag{11}\]  
It can be seen from Equation (11) that for the inherent multipath effect, the final rendering effect is consistent with the effect of single path transmission. For a multipath effect with time resolution, \(\tau_{i}\) denotes different delays in each path. Normally, the multipath effect is time dependent, which can be described as a stochastic process of time and space, which satisfies certain statistical characteristics after a large number of field tests. The Rayleigh distribution is often used to describe the fading range of multipath signals in analog buildings with dense urban communication environments. The probability density function is:  
\[f(\mathrm{x})=\frac{x}{\sigma^{2}}\exp\left(-\frac{x^{2}}{2\sigma^{2}}\right) z\geq 0 \tag{12}\]  
It represents a stationary narrow-band Gauss random process with a mean value of 0 and a variance of \(\sigma^{2}\). When the ambient environment is simple and the transmitting antenna and the receiving antenna can realize direct path transmission, the Rician distribution is used to describe the fading range of the multipath signal. The Rician distribution can be regarded as the sum of the multipath signal components of the dominant signal and the Rayleigh distribution. The probability density function can be expressed as:  
\[f(\mathrm{x})=\frac{x}{\sigma^{2}}\exp{(-\frac{x^{2}+A^{2}}{2\sigma^{2}})}I_{ 0}(\frac{xA}{\sigma^{2}}) \tag{13}\]  
\(A\) denotes the peak signal amplitude and \(I_{0}\) is the zero order Bessel function of the first order. Both the Rayleigh channel and Rician channel belong to fast fading channels. The specific channel  
\begin{table}
\begin{tabular}{c c c} \hline \(\{\mathbf{b_{1}},\mathbf{b_{2}}\}^{T}\) & \(\{\mathbf{s_{1}},\mathbf{s_{2}}\}^{T}\) & \(\mathcal{S}\) \\ \hline \([1,1]^{T}\) & \([1,1]^{T}\) & 2 \\ \([1,0]^{T}\) & \([1,-1]^{T}\) & 1 \\ \([0,1]^{T}\) & \([-1,1]^{T}\) & 1 \\ \([0,0]^{T}\) & \([-1,-1]^{T}\) & 2 \\ \hline \end{tabular}
\end{table}
Table 1: Synthesis of two channel binary signals.  
fading factor at a specific time of the transmitter is unknown, but it is always changing and satisfies a certain probability distribution. In this paper, the Rayleigh channel model is adopted because the default communication environment is densely populated in urban areas.