# Secure Mimo Communication System with Frequency Hopping Aided OFDM-DCSK Modulation
## 2 MIMO-FH-OFDM-DCSK System
### Receiver  
Figure 3 shows the process of FH-OFDM-DCSK demodulation after the receiver receiving the symbols \(\mathbf{R}\). At the legitimate receiver, the user first removes the CP of the received data and performs S/P conversion on the received symbols. Then, use the fast Fourier transform (FFT) to implement OFDM demodulation [15]. The symbols after the FFT operation can be represented by \(\mathbf{r}\). \(\mathbf{r}\) can be expressed as follows:  
\[\mathbf{r}=\frac{1}{\sqrt{N}}\sum_{i=0}^{N-1}\mathbf{R}\exp^{-j\frac{2m}{N}}= \mathbf{H}\cdot\mathbf{F}+\mathbf{n}, \tag{10}\]  
where \(\mathbf{n}\) represents the AWGN, \(\mathbf{H}\) is the channel response matrix, and \(\mathbf{F}=\left[ \begin{array}{c}\mathbf{F}_{1}\\ \mathbf{F}_{2}\end{array} \right]\). For the considered \(2\times 1\) MIMO communication system, we have:  
\[\mathbf{H}=\left[ \begin{array}{cc}h_{1}&h_{2}\end{array} \right], \tag{11}\]  
where \(h_{1}\) and \(h_{2}\) respectively denote the channel impulse response with two channels. After the OFDM demodulation, legitimate users will recover the correct chaotic sequences by the de-hopping matrix that is generated by the FH module which is hidden in chaotic sequences. Note that only legitimate users know how the FH rule for performing the de-hopping operation. Next, the de-hopping matrix is used to de-hop \(\mathbf{r}\), which can be expressed as follows:  
\[\mathbf{r}\cdot\mathbf{U}^{-1}=\mathbf{H}\cdot\mathbf{F}\cdot\mathbf{U}^{-1} +\mathbf{n}\cdot\mathbf{U}^{-1}, \tag{12}\]  
where \((\cdot)^{-1}\) represents the inverse operation of a matrix. Next, the chaotic demodulation is performed with the initial values of the chaotic sequences which are only shared by legitimate users which can be expressed as follows [17]:  
\[yn=\Re\bigg{\{}\sum_{k=0}^{\beta-1}conj(g_{0,k})\cdot g_{n,k}\bigg{\}}, \tag{13}\]where \(yn\) represents the \(n\)th equivalent symbol after chaotic demodulation, \(g_{0,k}\) denotes the \(k\)th chaotic reference symbol, \(g_{n,k}\) denotes the \(k\)th chaotic symbol carrying information used to transmit \(yn\), and \(\Re\bigg{\{}\cdot\bigg{\}}\) represents the real part of the symbol. Let us take \(y1,y2\) for example, the symbols after the chaotic demodulation [14] can be expressed as follows:  
\[ \begin{split} y_{1}&=\Re\bigg{\{}\sum_{k=0}^{\beta -1}(h_{1}a_{1}+h_{2}a_{2})c_{k}^{2}+c_{k}\xi\\ &\qquad+\frac{(h_{1}a_{1}+h_{2}a_{2})c_{k}\xi}{(h_{1}^{*}+h_{2}^{ *})}+\frac{\xi\xi^{*}}{(h_{1}^{*}+h_{2}^{*})}\bigg{\}},\end{split}  \tag{14}\]  
\[ \begin{split} y_{2}&=\Re\bigg{\{}\sum_{k=0}^{\beta -1}(h_{1}(-a_{2}^{*})+h_{2}(a_{1}^{*}))c_{k}^{2}+c_{k}\xi\\ &\qquad+\frac{(h_{1}(-a_{2}^{*})+h_{2}(a_{1}^{*}))c_{k}\xi}{(h_{ 1}^{*}+h_{2}^{*})}+\frac{\xi\xi^{*}}{(h_{1}^{*}+h_{2}^{*})}\bigg{\}},\end{split}  \tag{15}\]  
where \(\xi\) is the complex Gaussian white noise whose real part and imaginary part that followed the Gaussian distribution with a mean value of zero. Meanwhile, the equivalent noise \(N_{k}\) of the \(k\)th equivalent symbol can be expressed as:  
\[N_{1} =\Re\bigg{\{}\sum_{k=0}^{\beta-1}\frac{(h_{1}a_{1}+h_{2}a_{2})c_{k}\xi}{( h_{1}^{*}+h_{2}^{*})}+c_{k}\xi+\frac{\xi^{*}_{\xi}\xi^{*}}{(h_{1}^{*}+h_{2}^{*})} \bigg{\}}, \tag{16}\]  
\[N_{2} =\Re\bigg{\{}\sum_{k=0}^{\beta-1}\frac{(h_{1}(-a_{2}^{*})+h_{2}(a_{1}^{*} ))c_{k}\xi}{(h_{1}^{*}+h_{2}^{*})}+c_{k}\xi^{*}+\frac{\xi\xi^{*}_{\xi}}{(h_{1} ^{*}+h_{2}^{*})}\bigg{\}}. \tag{17}\]  
Finally, after the FH-OFDM-DCSK demodulation, we perform STBC decoding on the demodulated symbols, so that legitimate users can obtain the estimated symbols. The decoding process is expressed as follows:  
\[\mathbf{y}=\bigg{[} \begin{array}{c}y_{1}\\ y_{2}\end{array} \bigg{]}, \tag{18}\]  
\[\bigg{[} \begin{array}{c}\hat{a}_{1}\\ \hat{a}_{2}\end{array} \bigg{]}=(\mathbf{H}^{H}\mathbf{H})^{-1}\mathbf{H}^{H} \cdot\mathbf{y}, \tag{19}\]  
\[\bigg{[} \begin{array}{c}\hat{a}_{1}\\ \hat{a}_{2}\end{array} \bigg{]}=\Bigg{[}