## 5 Application to MIMO Secure Communications  
A simple communication system was developed in [28] where system (30) was used as both the master and slave. In our case, we will consider the same master-slave pair from the previous section, i.e., (29) and (30) yielding the pair  
Figure 4: Time evolution of the synchronization error for the 3D case. Figure 5: Time evolution of the synchronization error for the 2D case. \[\left\{ \begin{array}{l}D_{t}^{p}X(t)=f(X(t)),\\ D_{t}^{q}Y(t)=BY(t)+g(Y(t))+U.\end{array} \right.\]  
The basic idea here is to use the master-slave combination to form a multiple input multiple output (MIMO) secure communications system. MIMO uses multiple antennas at both ends of the transmission in order to send/receive multiple messages simultaneously, thereby increasing the throughput (data rate) of the system. Several MIMO communication schemes have been proposed in the literature such as [30; 31], but none of these schemes have attracted much attention. In this section, we assume an **F-M** synchronization strategy with a dimension \(d=3\). Recall from the last section that we chose  
\[\textbf{F}=\left( \begin{array}{c}y_{1}+y_{4}^{2}\\ y_{2}+y_{4}\\ y_{3}+1\end{array} \right)\text{ and }\textbf{M}\left(t\right)=\left( \begin{array}{ccc}1&\sin t&0\\ 2&0&\frac{1}{1+\ln\left(t+1\right)}\\ 0&\frac{1}{1+\sqrt{t}}&3\end{array} \right).\]  
Assuming that we want to transmit three different streams in the form \(S\left(t\right)=\left(s_{1}\left(t\right),s_{2}\left(t\right),s_{3}\left(t \right)\right)^{T}\), we can define the transmitted signals by  
\[\widetilde{S}\left(t\right)=\textbf{M}\left(t\right)X(t)+S\left(t\right).\]  
Note that this proposed scheme can be considered a masking algorithm as each message is hidden inside a chaotic signal. Normally, the message signals are transmitted through a common medium such as air, a copper wire, or an optical fiber. The effect of the medium on the signals is two fold. First, the signals are coupled, meaning that each antenna at the receiver is pinged by a combination of all transmitted streams. Second, consecutive symbols transmitted from the same antenna undergo a phenomenon referred to as inter-symbol interference (ISI), which is a result of the low-pass and dynamic nature of the transmission medium. For simplicity, let us assume a narrowband channel **H**, whereby ISI is ignored. The received signals are given by  
\[R\left(t\right)=\textbf{H}\widetilde{S}\left(t\right)+V\left(t\right),\]  
with \(V\left(t\right)\) being a vector of additive white Gaussian noise (AWGN) streams. At the receiver, a process known as equalization is usually used to reverse the effect of the channel. A wide variety of equalization schemes can be found in the literature, most of which require some form of training in order to estimate the channel matrix **H**. These include zero-forcing, minimum mean square error (MMSE), decision feedback (DF), precoding, and many more. For the purpose of our study, we assume an MMSE equalizer of the form  
\[\textbf{W}=\left(\textbf{H}^{H}\textbf{H}+\sigma^{-1}\textbf{I}\right)^{-1} \textbf{H}^{H},\]  
where \(\sigma\) is the signal-to-noise ratio (SNR) per stream and \(\left(\cdot\right)^{-1}\) denotes the inverse of a square nonsingular matrix. It is easy to see that for \(\sigma=0\), this is simply the pseudo-inverse of \(\textbf{H}^{H}\), which is known as the zero-forcing equalizer. The inclusion of \(\sigma\) aims to minimize the noise amplification by the equalizer. After equalization, we have  
\[\widetilde{R}\left(t\right) = \textbf{W}R\left(t\right)\] \[= \textbf{W}\left[\textbf{H}\widetilde{S}\left(t\right)+V\left(t \right)\right]\] \[\approx \widetilde{S}\left(t\right)+\textbf{W}V\left(t\right)\] \[\approx \widetilde{S}\left(t\right).\]We now go back to the chaotic slave, which is synchronized to the master by means of a training sequence. The synchronization process is based on Theorem 2 as we saw in Case 2 of the previous section where the control parameters were chosen as \(\left(c_{1},c_{2},c_{3}\right)=\left(1,2,3\right)\).