# Secure Mimo Communication System with Frequency Hopping Aided OFDM-DCSK Modulation
## 3 Performance Analysis
### Ber Performance  
Firstly, according to the FH-OFDM-DCSK demodulation and STBC decoding process in Section 2.2, we are able to derive the energy per bit \(E_{b}\) as:  
\[E_{b}=\Re\bigg{\{}\sum_{k=0}^{\beta-1}c_{k}^{2}\bigg{\}}. \tag{21}\]  
Then, substitute \(E_{b}\) into Equation (20), and the estimates of received symbols can be expressed as:  
\[\left[ \begin{array}{c}\hat{a}_{1}\\ \hat{a}_{2}\end{array} \right]=\left[ \begin{array}{c}E_{b}a_{1}\\ E_{b}a_{2}\end{array} \right]+\left[ \begin{array}{cc}h_{1}^{*}&h_{2}\\ h_{2}^{*}&-h_{1}\end{array} \right].\left[ \begin{array}{c}N_{1}\\ N_{2}^{*}\end{array} \right]. \tag{22}\]  
Let \(\mathrm{E}\{\cdot\}\) represent the mean and \(\mathrm{var}\{\cdot\}\) represents the variances. Next, we use the method of numerical analysis to deduce the numerical characteristics of the estimated symbol, and the derivation process is as follows:  
\[\mathrm{E}\{\hat{a}_{1}\}=E_{b}\mathrm{E}\{a_{1}\}, \tag{23}\]  
\[\mathrm{var}\{\hat{a}_{1}\}=\mathrm{E}\{a_{1}^{2}\}-\mathrm{E}\{a_{1}\}^{2}= \mathrm{var}\{h_{1}^{*}N_{1}\}+\mathrm{var}\{h_{2}N_{2}^{*}\}, \tag{24}\]  
\[\mathrm{var}\{h_{1}^{*}N_{1}\} = \frac{\beta\mathrm{E}\{\xi^{*}\xi\}\mathrm{E}\{(h_{1}^{*})^{2}\} }{\mathrm{E}\{(h_{1}^{*}+h_{2}^{*})^{2}\}}+E_{b}\mathrm{\times}\mathrm{E}\{ \xi^{2}\}\mathrm{\times}\mathrm{E}\{(h_{1}^{*})^{2}\} \tag{25}\] \[\times\mathrm{E}\{(\frac{(h_{1}a_{1}+h_{2}a_{2})^{2}}{(h_{1}^{*}+ h_{2}^{*})^{2}}+1)\}\] \[= \frac{\beta N_{0}^{2}\mathrm{E}\{(h_{1}^{*})^{2}\}}{2\mathrm{E}\{ (h_{1}^{*}+h_{2}^{*})^{2}\}}+\frac{E_{b}N_{0}\mathrm{E}\{(h_{1}^{*})^{2}\}}{2 }\times\mathrm{E}\{(\frac{(h_{1}a_{1}+h_{2}a_{2})^{2}}{(h_{1}^{*}+h_{2}^{*})^{ 2}}+1)\},\]  
Figure 3: Structure of DE-FH-OFDM-DCSK. \[ \begin{split}\text{var}\{h_{2}N_{2}^{*}\}&=\frac{\beta \text{E}\{(\tilde{s}^{*}\xi\}\text{E}\{(h_{2})^{2}\}}{\text{E}\{(h_{1}+h_{2})^{2} \}}+E_{b}\times\text{E}\{\xi^{2}\}\times\text{E}\{(h_{2})^{2}\}\\ &\times\text{E}\{(\frac{h_{1}^{*}(-a_{2})+h_{2}^{*}a_{1})^{2}}{( h_{1}+h_{2})^{2}}+1)\}\\ &=\frac{\beta N_{0}^{2}\text{E}\{(h_{2})^{2}\}}{2\text{E}\{(h_{1} +h_{2})^{2}\}}+\frac{E_{b}N_{0}\text{E}\{(h_{2})^{2}\}}{2}\times\text{E}\{( \frac{(h_{1}^{*}(-a_{2})+h_{2}^{*}a_{1})^{2}}{(h_{1}+h_{2})^{2}}+1)\},\end{split}  \tag{26}\]  
where \(N_{0}\) represents the noise power. Furthermore, \(\Gamma\) represents the signal-to-noise ratio (SNR) can be determined as:  
\[\Gamma=\frac{(\text{E}\{\hat{a}_{1}\})^{2}}{\text{var}\{\hat{a}_{1}\}}. \tag{27}\]  
Based on the derived numerical characteristics, we are able to derive the theoretical BER expressions over the selective Rayleigh fading channel, flat Rayleigh fading channel, and AWGN channel. For the selective Rayleigh fading channel, the theoretical BER can be derived as:  
\[ \begin{split} BER_{Selective}&=\frac{1}{2}\text{erfc}( \sqrt{\frac{\Gamma}{2}})=\frac{1}{2}\text{erfc}((\frac{\text{E}\{\hat{a}_{1}\} ^{2}}{2\text{var}\{\hat{a}_{1}\}})^{\frac{1}{2}})\\ &=\frac{1}{2}\text{erfc}(((\frac{\beta N_{0}^{2}}{2\text{E}\{(h_ {1}^{*}+h_{2}^{*})^{2}\}}+\frac{(\frac{\text{E}_{b}N_{0}}{2})\times\text{E}\{ (h_{1}^{*}+h_{2}^{*})^{2}\}+1)}{E_{b}^{2}}\\ &\quad+\frac{\frac{E_{b}N_{0}}{2}\times(\frac{\text{E}\{(-h_{1} ^{*}+h_{2}^{*})^{2}\}}{\text{E}\{(h_{1}^{*}+h_{2}^{*})^{2}\}}+1)}{E_{b}^{2}}\\ &\quad+\frac{\frac{\beta N_{0}^{2}}{2\text{E}\{(h_{1}^{*}+h_{2}^ {*})^{2}\}}}{E_{b}^{2}})\times\frac{2}{(|h_{1}|^{2}+|h_{2}|^{2})^{2}})^{-\frac {1}{2}}).\end{split}  \tag{28}\]  
where \(\text{erfc}(\cdot)\) represents complementary error function, and \(|\cdot|\) denotes complex number modulo. For the flat Rayleigh fading channel, let channel impulse response \(h_{k}=h\), and the theoretical BER can be derived as:  
\[ \begin{split} BER_{flat}&=\frac{1}{2}\text{erfc}((( \frac{\beta}{8\text{E}\{(h^{*})^{2}\}(\frac{\text{E}_{b}}{N_{0}})^{2}}\\ &\quad+\frac{\frac{\text{E}\{(h^{*})^{2}\}}{\text{E}\{(h^{*})^{ 2}\}}+2}{2\frac{E_{b}}{N_{0}}}+\frac{\beta}{8\text{E}\{(h^{*})^{2}\}(\frac{ \text{E}_{b}}{N_{0}})^{2}})\times\frac{1}{2|h|^{2})^{-\frac{1}{2}}).\end{split}