## 2 System Model
### R-D Link  
The MCIM-MDCSK modulation scheme is adopted at R. The block diagram of the MCIM-MDCSK modulation scheme is shown in Figure 4. The transmitted signal at R can be expressed as  
\[e_{rd}=W_{R}\bigotimes x+c_{n}\cdot W_{x_{n}}\bigotimes x+d_{n}\cdot W_{y_{n} }\bigotimes x+b_{n}\cdot W_{k}\bigotimes x \tag{6}\]In Equation (6), \(W_{R}\) is the reference Walsh code. \(W_{k}\) is utilized for the polarity-modulated bits. \(W_{x_{n}}\) and \(W_{y_{n}}\) are the outputs of the Walsh code selector, where \(x_{n}=2S_{n}+1\) and \(y_{n}=2S_{n}+2\). Here, \(S_{n}\) represents the code index symbols. The length of Walsh codes is P. \(c_{n}\) and \(d_{n}\) are the in-phase and quadrature components of the \(n^{th}\) M-ary constellation symbol \(M_{n}=c_{n}+jd_{n}\).  
Figure 4: The block diagram of the MCIM-MDCSK modulation scheme.  
The reference signal received at D can be expressed as  
\[y_{R,k}=\frac{1}{P}\sum\limits_{p=0}^{P-1}W_{R,p+1}\left[\sum\limits_{l=1}^{L_{ ml}}\alpha_{rd,l}\cdot W_{R,p+1}\cdot x_{k-\tau_{rd,l}}+n_{rd,k}\right]=\sum \limits_{l=1}^{L_{ml}}\alpha_{rd,l}\cdot x_{k-\tau_{rd,l}}+n_{R,k} \tag{7}\]  
where  
\[n_{R,k}=\frac{1}{P}\sum\limits_{p=0}^{P-1}W_{R,p+1}\cdot n_{rd,k} \tag{8}\]  
The \(n\)th information signal received at D can be expressed as  
\[y_{rd,k}^{nq}=\frac{1}{P}\sum\limits_{p=0}^{P-1}W_{nq,p+1}\left[\sum\limits_{l =1}^{L_{ml}}\alpha_{rd,l}\cdot(c_{n}W_{x_{n},p+1}\cdot x_{k-\tau_{rd,l}}+d_{n}W _{y_{n},p+1}\cdot x_{k-\tau_{rd,l}}+b_{n}W_{k,p+1}\cdot x_{k-\tau_{rd,l}})+n_{rd,k}\right]\]  
\[=\left\{\begin{array}{ll}\sum\limits_{l=1}^{L_{dl}}\alpha_{rd,l}\cdot c_{n} \cdot x_{k-\tau_{rd,l}}+n_{x,k},&q=x_{n}\\ \sum\limits_{l=1}^{L_{ml}}\alpha_{rd,l}\cdot d_{n}\cdot x_{k-\tau_{rd,l}}+n_{y,k},&q=y_{n}\\ n_{q,k},&q\
eq x_{n}\text{ and }\ q\
eq y_{n}\\ \sum\limits_{l=1}^{L_{ml}}\alpha_{rd,l}\cdot b_{n}\cdot x_{k-\tau_{rd,l}}+n_{W _{k}},&W_{nq}=W_{k}\end{array}\right. \tag{9}\]  
where  
\[n_{x,k}=\frac{1}{P}\sum\limits_{p=0}^{P-1}W_{nq,p+1}\cdot n_{rd,k},\ \ \ \ q=x_{n} \tag{10}\]  
\[n_{y,k}=\frac{1}{P}\sum\limits_{p=0}^{P-1}W_{nq,p+1}\cdot n_{rd,k},\ \ \ \ q=y_{n} \tag{11}\]  
\[n_{q,k}=\frac{1}{P}\sum\limits_{p=0}^{P-1}W_{nq,p+1}\cdot n_{rd,k},\ \ \ q\
eq x_{n}\text{ and }q\
eq y_{n} \tag{12}\]  
\[n_{W_{k}}=\frac{1}{P}\sum\limits_{p=0}^{P-1}W_{k,p+1}\cdot n_{rd,k},\ \ \ \ W_{nq}=W_{k} \tag{13}\]  
The decision variable of the nth signal in the \(R\to D\) link can be denoted as  
\[Z_{nq}=\sum\limits_{k=1}^{U}y_{R,k}\cdot y_{rd,k}\]  
\[=\left\{\begin{array}{ll}\sum\limits_{k=1}^{U}\left(\sum\limits_{l=1}^{L_{ ml}}\alpha_{rd,l}\cdot x_{k-\tau_{rd,l}}+n_{R,k}\right)\cdot\left(\sum\limits_{l=1}^{L_{ ml}}\alpha_{rd,l}\cdot c_{n}\cdot x_{k-\tau_{rd,l}}+n_{x,k}\right),&q=x_{n}\\ \sum\limits_{k=1}^{U}\left(\sum\limits_{l=1}^{L_{dl}}\alpha_{rd,l}\cdot x_{k- \tau_{rd,l}}+n_{R,k}\right)\cdot\left(\sum\limits_{l=1}^{L_{ml}}\alpha_{rd,l} \cdot d_{n}\cdot x_{k-\tau_{rd,l}}+n_{y,k}\right),&q=y_{n}\\ \sum\limits_{k=1}^{U}\left(\sum\limits_{l=1}^{L_{ml}}\alpha_{rd,l}\cdot x_{k- \tau_{rd,l}}+n_{R,k}\right)\cdot n_{q,k},&q\
eq x_{n}\text{ and }\ q\
eq y_{n}\\ \sum\limits_{k=1}^{U}\left(\sum\limits_{l=1}^{L_{ml}}\alpha_{rd,l}\cdot x_{k- \tau_{rd,l}}+n_{R,k}\right)\cdot\left(\sum\limits_{l=1}^{L_{ml}}\alpha_{rd,l} \cdot b_{n}\cdot x_{k-\tau_{rd,l}}+n_{W_{k}}\right)=Z_{W_{k}},&W_{nq}=W_{k} \end{array}\right. \tag{14}\]  
Finally, by utilizing an equal-gain combiner (EGC) to process the decision metrics of the \(S\to D\) and \(R\to D\) links, the polarity-modulated symbol can be estimated as  
\[\hat{b_{n}}=\left\{\begin{array}{ll}+1,&\left(Z_{sd}+Z_{W_{k}}\right)/2>0\\ -1,&\text{others}\end{array}\right. \tag{15}\]The code index bits can be estimated as  
\[\xi_{n}=\underset{q=1,\ldots Q}{\text{max}}\big{(}\big{|}Z_{mq}\big{|}\big{)} \tag{16}\]  
By using the Euclidean distance decision method [22], the M-ary constellation symbols corresponding to the maximum energy in each group can be estimated. Then the code-modulated bits can be recovered by the symbol-to-bit converters.