## 2 Schemes of SR-ODBR-DCSK
### Receiver  
Figure 5 shows the receiver structure of the SR-ODBR-DCSK. In order to recover the continuous two-bit data of data signal \(b_{k}\) at the same time, it is divided into two branches, one transmitting even bit \(b_{2k}\) and the other transmitting odd bit \(b_{2k+1}\).  
In the \(k\)-code period, when the system restores the information, the receiver first receives the signal \(r_{i,k}\) and its own delay \(R\), and \(r_{i-R,k}\), so as to extract the first \(R\) signal in a data frame, that is, the reference signal. In order to correctly demodulate the even bit value \(b_{2k}\) of data information \(b_{k}\), the information \(r_{i-R,k}\) after receiving information delay \(R\) and the information \(r_{i,k}\) actually received by the demodulator, and finally, the sum of \(P\) independent correlation values is able to be calculated.  
Figure 4: Modulation structure diagram of the SR-ODBR-DCSK.  
The accumulated signal values are compared with the decision threshold. In order to demodulate the odd bit \(b_{2k+1}\) of data information \(b_{k}\), the received reference signal needs to be orthogonally processed with the same Walsh at the transmitter. Then, the chaotic reference signal after orthogonal processing is partially correlated with the reference signal for \(P\) times, and the sum of \(P\) independent correlation values is calculated. The results are compared with the decision threshold to obtain \(b_{2k+1}\), and then the original signal \(b_{k}\) is obtained.  
The decision signals of the two branches can be expressed as  
\[z_{2\text{k}}=\sum\limits_{p=1}^{P}\sum\limits_{i=0}^{R-1}r_{i,p,k}r_{i-R,k} \tag{2}\]  
\[z_{2\text{k+1}}=\sum\limits_{p=1}^{P}\sum\limits_{i=0}^{R-1}r_{i,p,k}r^{\prime }_{i-R,k} \tag{3}\]  
Assuming that the received signal is only affected by the AWGN channel, the received signal is  
\[r_{i-R,k}=s_{i-R}+n_{i-R} \tag{4}\]  
\[r_{i,p,k}=s_{i,p,k}+n_{i,p,k} \tag{5}\]  
\(Z_{2\text{k}}\) can be further expanded to  
\[\begin{array}{c}Z_{2\text{k}}=\sum\limits_{p=1}^{P}\sum\limits_{i=0}^{R-1}r_ {i,p,k}r_{i-R,k}\\ =\sum\limits_{p=1}^{P}\sum\limits_{i=0}^{R-1}\left(s_{i,k}+n_{i,k}\right)\cdot \left(s_{i-R,k}+n_{i-R,k}\right)\\ =A+B+C\end{array} \tag{6}\]  
where \(A\), \(B\), \(C\) are  
\[\begin{array}{c}A=\sum\limits_{p=1}^{P}\sum\limits_{i=0}^{R-1}\frac{1}{\sqrt {2}}b_{2\text{k}}x_{i-R,k}^{2}\\ B=\sum\limits_{p=1}^{P}\sum\limits_{i=1}^{R-1}(\frac{1}{\sqrt{2}}b_{2\text{k +1}}x_{i-R,k}x_{i-R,k}^{2}+\frac{1}{\sqrt{2}}b_{2\text{k}}x_{i-R,k}n_{i-R,k}\\ +\frac{1}{\sqrt{2}}b_{2\text{k+1}}x^{\prime}_{i-R,k}n_{i-R,k}+\alpha_{1}x_{i- R,k}n_{i-R,k})\\ C=\sum\limits_{p=1}^{P}\sum\limits_{i=0}^{R-1}n_{i,k}n_{i-R,k}\end{array} \tag{7}\]  
Figure 5: Receiver structure of the SR-ODBR-DCSK.  
\(A\) is the useful signal component, and the remaining items \(B\) and \(C\) are the interference components with zero mean. These interference components mainly come from the cross-correlation between chaotic sequences in different time periods and the noise interference of correlator input.  
Information bits can be recovered according to decision rules  
\[\hat{b}_{2k}=\left\{\begin{array}{l}+1\ Z_{2k}>0\\ -1\ Z_{2k}\leq 0\end{array}\right. \tag{8}\]  
\[\hat{b}_{2k+1}=\left\{\begin{array}{l}+1\ Z_{2k+1}>0\\ -1\ Z_{2k+1}\leq 0\end{array}\right. \tag{9}\]