## 5 BER Analysis of DCSK and QCSK Systems under the UWA Channel  
In this section, the BER formulas of DCSK and QCSK systems for the UWA model are derived using the noise distribution in Section 3. For the DCSK system, after substituting (15) and (19) in (16) and assuming \(b_{l}=+1\), the output of the correlator represented by \(c_{l}\) can be rewritten as:  
\[\begin{split} c_{l}&=\sum_{k=1}^{\beta}(x_{k}+n_{k} )\cdot(x_{k}+n_{k+\beta})\\ &=\sum_{k=1}^{\beta}x_{k}^{2}+2\sum_{k=1}^{\beta}n_{k}\cdot x_{k} +\sum_{k=1}^{\beta}n_{k}\cdot n_{k+\beta}\end{split} \tag{20}\]  
The mean \(\mu_{1}\) and the variance \(\sigma_{1}^{2}\) values of \(c_{l}\) are calculated using  
Figure 4: UWAS based on differential chaos modulation.  
Figure 3: Proposed QCSK transmitter and receiver system.  
\[\mu_{1} =E(c_{l})=\beta E(x_{k}^{2}) \tag{21}\] \[\sigma_{1}^{2} =V(c_{l}) =2\beta\cdot E(x_{k}^{2})\frac{N_{0}}{2}+\frac{1}{4}\beta\cdot N_{0 }^{2}\] (22) where \(E(\cdot)\) is the mean operator and \(V(\cdot)\) is the variance operator. Since \(E_{b}=2\sum x_{k}^{2}=2\beta E(x_{k}^{2})\), then the mean and variance values in terms of \(E_{b}\) can be expressed as: \[\mu_{1} =E_{b}/2,\] \[\sigma_{1}^{2} =E_{b}\cdot\frac{N_{0}}{2}+\frac{1}{4}\beta\cdot N_{0}^{2} \tag{23}\]  
Similarly, for the QCSK, and by assuming \(a_{l},b_{l}=+1\), the decision variable \(c_{1,l}\) in (18) is rewritten in terms of (17) and (19) as  
\[c_{1,l} =\sum_{k=1}^{\beta}(x_{k}+n_{k})(\frac{1}{\sqrt{2}}(x_{k}+\hat{x }_{k})+n_{k+\beta})\] \[=\frac{1}{\sqrt{2}}\sum_{k=1}^{\beta}x_{k}^{2}+\frac{1}{\sqrt{2}} \sum_{k=1}^{\beta}x_{k}\hat{x}_{k}+\sum_{k=1}^{\beta}x_{k}n_{k+\beta}+\frac{1 }{\sqrt{2}}\sum_{k=1}^{\beta}n_{k}x_{k}+\frac{1}{\sqrt{2}}\sum_{k=1}^{\beta}n _{k}\hat{x}_{k}+\sum_{k=1}^{\beta}n_{k}n_{k+\beta} \tag{24}\]  
By orthogonality, the second term in (24) is zero \((\frac{1}{\sqrt{2}}\sum_{k=1}^{\beta}x_{k}\hat{x}_{k}\approx 0)\). The mean, \(\mu_{2}\), and variance, \(\sigma_{2}^{2}\), values of \(c_{1,l}\) are expressed as:  
\[\mu_{2}=E(c_{1,l})=\frac{1}{\sqrt{2}}\beta E(x_{k}^{2}) \tag{25}\]  
\[\sigma_{2}^{2}=V(c_{1,l})=2\beta\cdot E(x_{k}^{2})\frac{N_{0}}{2}+\frac{1}{4} \beta\cdot N_{0}^{2} \tag{26}\]  
Equations (25) and (26) are rewritten in terms of \(E_{b}=\sum_{k=1}^{\beta}x_{k}^{2}=\beta E(x_{k}^{2})\) as  
\[\mu_{2} =E_{b}/\sqrt{2}\] \[\sigma_{2}^{2} =E_{b}\cdot N_{0}+\frac{1}{4}\beta\cdot N_{0}^{2} \tag{27}\]  
Equation (27) is valid for the decision variable \(c_{2,l}\). From (23) and (14), the BER formula of the DCSK system can be derived for both degrees of freedom such that  
\[\begin{split} P_{e}^{5}&=\int_{0}^{\infty}\frac{0.4 9}{\sigma}\bigg{(}1+\frac{(x+\mu)^{2}}{3\sigma^{2}}\bigg{)}^{-3}dx\\ &=0.5-\frac{0.55125\sigma\mu(\mu^{2}+5\sigma^{2})}{(\mu^{2}+3 \sigma^{2})^{2}}-0.318264\tan^{-1}{(\frac{\mu}{\sqrt{3}\sigma})}\end{split} \tag{28}\]  
\[\begin{split} P_{e}^{6}&=\int_{0}^{\infty}\frac{0.4 6875}{\sigma}\bigg{(}1+\frac{(x+\mu)^{2}}{3\sigma^{2}}\bigg{)}^{-3.5}dx\\ &=0.5-\frac{0.015625\mu(\mu^{4}+10\mu^{2}\sigma^{2}+30\sigma^{4} )}{(0.25\mu^{2}+\sigma^{2})^{2.5}}\end{split} \tag{29}\]  
Similarly, for the QCSK, the BER formula can be written as  
\[\begin{split} P_{e}^{5}&=1-\frac{1.1025\sigma\mu( \mu^{2}+5\sigma^{2})}{(\mu^{2}+3\sigma^{2})^{2}}-0.6365\tan^{-1}{(\frac{\mu}{ \sqrt{3}\sigma})}\\ P_{e}^{6}&=1-\frac{0.03125\mu(\mu^{4}+10\mu^{2} \sigma^{2}+30\sigma^{4})}{(0.25\mu^{2}+\sigma^{2})^{2.5}}\end{split} \tag{30}\]