## 2 System Model
### S-R Link and S-D Link  
The SR-DCSK modulation scheme is adopted at S. The transmitter and receiver system model of SR-DCSK are shown in Figure 2. The frame structure of the transmitted signal at S is shown in Figure 3. We take the nth group of information bit streams as an example. The nth polarity-modulated bit is first mapped to a polarity-modulated symbol \(\mathbf{b_{n}}\), which is carried by the SR-DCSK signal and simultaneously transmitted to R and D in the first time slot. R adopts the DF protocol to recover the polarity-modulated symbol and transmit the code index bits and code-modulated bits to D in the second time slot.  
The chaotic signal \(\mathbf{x}=[x_{1},x_{2},\ldots x_{U}]\) is generated by a second-order Chebyshev polynomial function (i.e., \(x_{k+1}=1-\varphi x_{k}^{2}\)) with initial conditions \(x_{1}=0.9058\) and the parameter \(\varphi=2\). The length of the chaotic signal is U. The length of the information-bearing signal is \(\beta\), where \(\beta=\) J\(\cdot\)U. J is the number of replications. Thus, the transmitted signal of the nth polarity-modulated symbol \(b_{n}\) can be denoted as  
\[\mathbf{e_{n}}=[x,\underbrace{b_{n}x,\ldots,b_{n}x}_{Duplicat  
Figure 3: The frame structure of the transmitted signal at S.  
Figure 2: The system model of SR-DCSK.  
The signals received by R and D can be denoted as  
\[y_{sr,k}=\sum_{l=1}^{L_{sr}}\alpha_{sr,l}\cdot e_{sr,k-\tau_{sr,l}}+n_{sr,k} \tag{2}\]  
\[y_{sd,k}=\sum_{l=1}^{L_{sd}}\alpha_{sd,l}\cdot e_{sd,k-\tau_{sd,l}}+n_{sd,k} \tag{3}\]  
where \(L_{sr(sd)}\) is the number of channel paths. \(\alpha_{sr(sd),l}\) is the _lth_ channel path loss coefficient. \(e_{sr,k-\tau_{sr,l}}\) and \(e_{sd,k-\tau_{sd,l}}\) are the transmitted signals \(e_{sr(sd)}\) after passing through the multipath Rayleigh fading channels. \(n_{sr(sd),k}\) is the AWGN with zero mean and variance \(N_{0,sr(sd)}/2\) for the \(S\to R(D)\) link. \(\tau_{sr(sd),l}\) represents the latency time of the transmitted signal \(e_{sr(sd)}\) after passing through the multipath Rayleigh fading channels. \(k\) represents the _kth_ value of the transmission period.  
The decision variable of the \(S\to R\) link can be denoted as  
\[Z_{sr}=\sum_{j=1}^{J}\sum_{k=1}^{U}y_{sr,k}\cdot y_{sr,k+nU}\]  
\[=\sum_{j=1}^{J}\sum_{k=1}^{U}\left(\sum_{l=1}^{L_{sr}}\alpha_{sr,l}\cdot e_{sr,k-\tau_{sr,l}}+n_{sr,k}\right)\cdot\left(\sum_{l=1}^{L_{sr}}\alpha_{sr,l} \cdot e_{sr,k+nU-\tau_{sr,l}}+n_{sr,k+nU}\right)\]  
\[=\sum_{j=1}^{J}\sum_{k=1}^{U}\left(b_{n}\sum_{l=1}^{L_{sr}}\alpha_{sr,l}^{2} \cdot x_{k-\tau_{sr,l}}^{2}+\sum_{l=1}^{L_{sr}}\alpha_{sr,l}\cdot x_{k-\tau_{ sr,l}}\cdot n_{sr,k+nU}+b_{n}\sum_{l=1}^{L_{sr}}\alpha_{sr,l}\cdot x_{k-\tau_{ sr,l}}\cdot n_{sr,k}+n_{sr,k}\cdot n_{sr,k+nU}\right) \tag{4}\]  
Therefore, if \(Z_{sr}>0\), the polarity-modulated symbol \(b_{n}\) is estimated as \(b_{n}=+1\); otherwise, \(b_{n}\) is estimated as \(b_{n}=-1\).  
The decision variable of the \(S\to D\) link can be denoted as  
\[Z_{sd}=\sum_{j=1}^{J}\sum_{k=1}^{U}y_{sd,k}\cdot y_{sd,k+nU}\]  
\[=\sum_{j=1}^{J}\sum_{k=1}^{U}\left(\sum_{l=1}^{L_{sd}}\alpha_{sd,l}\cdot e_{ sd,k-\tau_{sd,l}}+n_{sd,k}\right)\cdot\left(\sum_{l=1}^{L_{sd}}\alpha_{sd,l} \cdot e_{sd,k+nU-\tau_{sd,l}}+n_{sd,k+nU}\right)\]  
\[=\sum_{j=1}^{J}\sum_{k=1}^{U}\left(b_{n}\sum_{l=1}^{L_{sd}}\alpha_{sd,l}^{2} \cdot x_{k-\tau_{sd,l}}^{2}+\sum_{l=1}^{L_{sd}}\alpha_{sd,l}\cdot x_{k-\tau_{ sd,l}}\cdot n_{sd,k+nU}+b_{n}\sum_{l=1}^{L_{sd}}\alpha_{sd,l}\cdot x_{k-\tau_{ sd,l}}\cdot n_{sd,k}+n_{sd,k}\cdot n_{sd,k+nU}\right) \tag{5}\]