## 2 System Model of MC\(-\)ODBR\(-\)DCSK
### Transmitter Structure of MC\(-\)ODBR\(-\)DCSK  
To avoid intra-signal interference in the proposed system, we utilize an orthogonal chaotic signal generator (OCG) to produce two distinct chaotic signals. Figure 2 provides a visual representation of the OCG's block diagram. The OCG generates a pair of orthogonal chaotic carriers that correspond to the two bits of information transmitted in the same  
Figure 1: Time domain diagram. (**a**) Improved Logistic Map (\(x_{0}=0.40\), \(\mu=2\)); (**b**) 2D-Logistic map (\(x_{0}=0.40\), \(y_{0}=0.40\), \(r=1.19\)). information slot. This is achieved through the formulation of specific parameters, as depicted in Figure 2[24]. \[s_{i}^{1}=s_{i-\beta/2}^{1}\,,\text{m}\beta<i\leq\text{m}\beta+\beta/2 \tag{5}\]  
\[s_{i}^{2}=s_{i-\beta/2}^{2}\,,\text{m}\beta<i\leq\text{m}\beta+\beta/2 \tag{6}\]  
where \(s_{i}^{1}\) and \(s_{i}^{2}\) are the output of the OCG, \(\beta\) is the spreading factor. Furthermore, the relationship between \(s_{i}^{1}\) and \(s_{i}^{2}\)\(i\) is given as follows:  
\[\sum_{i=\text{m}\beta+1}^{(\text{m}+1)\beta}s_{i}^{1}s_{i}^{1}=0,\text{m}, \text{n}=0,1,2,\ldots \tag{7}\]  
The experimental parameters are set based on the 3 GPP LTE technical report. The transmission bandwidth is assumed to be a fixed value of \(B\), the MC-ODBR-DCSK system divides the bandwidth \(B\) into \(N_{sub}\) subcarriers and the number of subcarriers is less than that of Fourier transform points when OFDM modulation is performed. The \(N_{sub}\) subcarriers are divided into \(L\) groups, and each group has \(M+2\) subcarriers. The two subcarriers located in the middle are used to transmit the reference signal, and the rest of the subcarriers in the group are used to transmit data information. The number of subcarriers \(N_{sub}\), the number of groups \(L\), and the number of subcarriers \(M+2\) in each group satisfy the relation \(N_{sub}=L\times(M+2)\). The modulator structure of the MC\(-\)ODBR\(-\)DCSK system is shown in Figure 3.  
Figure 3: Transmitter of MC-ODBR-DCSK. Figure 2: Block diagram of OCG. The transmitter generates the data to be sent: \(\mathbf{A}\) and \(\mathbf{B}\). The binary message code elements to be sent are \(\mathbf{A}=[a_{1},a_{2},\dots,a_{M\times L}]\) and \(\mathbf{B}=[b_{1},b_{2},\dots,b_{M\times L}]\). Through serial parallel conversion, it can be converted to \(\mathbf{A_{M\times L}}\), \(\mathbf{B_{M\times L}}\):  
\[\mathbf{A_{M\times L}}= \begin{bmatrix}a_{1}^{1}&\cdots&a_{1}^{L}\\ \vdots&\ddots&\vdots\\ a_{M}^{1}&\cdots&a_{M}^{L}\end{bmatrix} \qquad\mathbf{B_{M\times L}}= \begin{bmatrix} b_{1}^{1}&\cdots&b_{1}^{L}\\ \vdots&\ddots&\vdots\\ b_{M}^{1}&\cdots&b_{M}^{L}\end{bmatrix}  \tag{8}\]  
In the process of \(L\)-group data transmission, the signal structure of the information transmitted in each group is the same, and in order to clearly and intuitively understand the signal transmission process, take one of the groups as an example: \(l\) (\(l\) = 1, 2,..., \(L\)) group. \(a_{k}^{l}\) (\(k\) = 1, 2,..., \(M\)) denotes the kth data bit in the lth group, and \(a_{k}^{l}\) is modulated by the chaotic reference signal \(s_{1}^{l}\). This process can be understood as a spreading process. The binary data bits \(b_{k}^{l}\) (\(k\) = 1, 2,..., \(M\)) are modulated with the chaotic signal \(s_{2}^{l}\) formed by the reference signal passing through the OCG. The two sets of modulated data information are summed up correspondingly, and \(M\) data information and two reference information form the transmitted signal of the \(l\)th set. The \(i\)th spreading sample point of the kth information in the \(l\)th group can be represented as \(a_{k}^{l}s_{1i}^{l}+b_{k}^{l}s_{2i}^{l}\). \(a_{k}^{l}s_{1i}^{l}+b_{k}^{l}s_{2i}^{l}\) (\(i\) = 1, 2,..., \(\beta\)) can be used as the information format of the \(M\)th subcarriers carrying data in the lth group.