## 2 System Model of MC\(-\)ODBR\(-\)DCSK
### Receiver Structure of MC\(-\)ODBR\(-\)DCSK  
The transmitter block diagram and signal structure of the transmitter are described in the previous section, and this subsection describes the structure of the receiver of the MC\(-\)ODBR\(-\)DCSK system. Because of the combination of OFDM key technology, the system does not need a delay circuit, channel estimation, and equalizer when setting up the receiver, and the structure has changed compared to the traditional DCSK technology. The structure of the receiver is the opposite process of the transmitter, and similar to the transmitter. The demodulation process is the same for all groups of signals at the receiver. Again, the analysis of the received signal of group \(l\) is used as an example to be able to make the process clearly understood. The block diagram of the receiving end of the MC\(-\)ODBR\(-\)DCSK system is shown in Figure 6.  
Figure 5: Format of the signal transmitted in the MC\(-\)ODBR\(-\)DCSK system. Figure 6: Receiver of MC\(-\)ODBR\(-\)DCSK. The received signal is converted by series-parallel and FFT transform, and the matrix is obtained by FFT transform. The two subcarriers located in the middle of each group are summed and normalized to obtain the reference signal \(r_{ref1}^{l}\) of one of the branches, which is deposited into the reference matrix \(R_{1}^{l}\). The reference signal sequence \(r_{ref1}^{l}\) is stored in matrix \(R_{2}^{l}\) through \(r_{ref2}^{l}\) generated by the orthogonalizer, and other \(M\) information \(r_{k}^{l}\) (\(k=1,2,\ldots,M\)) in each group is stored in matrix \(W_{l}\). According to the above analysis, the expressions of reference matrix \(R_{1}^{l}\), \(R_{2}^{l}\) and information matrix \(W_{l}\) are:  
\[R_{1}^{l}=s_{1}^{l}+n_{1}^{l} \tag{9}\]  
\[R_{2}^{l}=s_{2}^{l}+n_{2}^{l} \tag{10}\]  
\[W_{l}=\left[ \begin{array}{c}a_{1}^{l}s_{1}^{l}+b_{1}^{l}s_{2}^{l}+n_{1,1}^{l }\\ a_{2}^{l}s_{1}^{l}+b_{2}^{l}s_{2}^{l}+n_{2,1}^{l}\\ \vdots\\ a_{M}^{l}s_{1}^{l}+b_{M}^{l}s_{2}^{l}+n_{M,1}^{l}\end{array} \right] \tag{11}\]  
\(n_{i}\) represents the noise impact received during transmission. The received data information and reference information are calculated by the matrix product \(\hat{d}_{1,l}=\text{sign}\left(R_{1}^{l}\times W_{l}^{l}\right)\), \(\hat{d}_{2,l}=\text{sign}\left(R_{2}^{l}\times W_{l}^{l}\right)\) (\"\(\times\)\" denotes the multiplication operation of two matrices and \"\(\times\)\" denotes the transpose operation of the matrix). The original binary information is recovered through the positive and negative results. The demodulation process is as follows:  
\[z_{k}^{l}|a_{k}^{l}=\sum_{l=0}^{\beta-1}\left(a_{k}^{l}s_{1,i}^{l}+b_{k}^{l}s_ {2,i}^{l}+n_{k,i}^{l}\right)\times\left(s_{1,i}^{l}+n_{i}^{l}\right) \tag{12}\]  
\[z_{k}^{l}|b_{k}^{l}=\sum_{i=0}^{\beta-1}\left(a_{k}^{l}s_{1,i}^{l}+b_{k}^{l}s_ {2,i}^{l}+n_{k,i}^{l}\right)\times\left(s_{2,i}^{l}+n_{i}^{l}\right) \tag{13}\]  
\(n_{k,i}^{l}\) and \(n_{i}^{l}\) are two independent Gaussian noise sequences from information slots and reference slots, respectively. The structures of the above two equations are similar, so one group is selected for analysis. By deformation of Equation (12), we can get:  
\[z_{k}^{l}|a_{k}^{l}=\sum_{i=0}^{\beta-1}\left[a_{k}^{l}(s_{1,i}^{l})^{2} \right]+\sum_{i=0}^{\beta-1}b_{k}^{l}s_{1,i}^{l}s_{2,i}^{l}+\sum_{i=0}^{\beta -1}(s_{1,i}^{l}n_{k,i}^{l}+a_{k}^{l}s_{1,i}^{l}n_{i}^{l}+b_{k}^{l}s_{2,i}^{l}n_ {i}^{l}+n_{i}^{l}n_{k,i}^{l}) \tag{14}\]  
where:  
\[z_{1}=\sum_{i=0}^{\beta-1}\left[a_{k}^{l}(s_{1,i}^{l})^{2}\right] \tag{15}\]  
\[z_{2}=\sum_{i=0}^{\beta-1}b_{k}^{l}s_{1,i}^{l}s_{2,i}^{l} \tag{16}\]  
\[z_{3}=\sum_{i=0}^{\beta-1}(s_{1,i}^{l}n_{k,i}^{l}+a_{k}^{l}s_{1,i}^{l}n_{i}^{ l}+b_{k}^{l}s_{2,i}^{l}n_{i}^{l}+n_{i}^{l}n_{k,i}^{l}) \tag{17}\]  
The Equation (14) can be divided into three parts, which can solve the useful part \(z_{1}\) of the original signal. The chaotic signals generate the correlation part \(z_{2}\), which generates the internal interference between signals.