# Multilevel Coding for the Full-Duplex Decode-Compress-Forward Relay Channel
## 4 Multilevel Decode-Compress-Forward  
The proposed multilevel coding is shown in Figure 8, over one transmission block. The message at the source, \(W^{(t)}\), is divided into two parts, \(W^{(t)}_{d}\) which is to be transmitted using decode-forward and \(W^{(t)}_{c}\) which is to be transmitted using compress-forward. As shown in the figures, each level at the source and the relay is responsible for either DF or CF.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(H_{12}\) & \(H_{23}\) & \(H_{13}\) & \(R_{\text{No Relay}}\) & \(R_{DF}\) & \(R_{CF}\) & \(R_{DCF}\) & **Upper Bound** \\ \hline \(1\) & \(100\) & \(1\) & \(1.7\) & \(1.7\) & \(1.9\) & \(2.1\) & \(2.19\) \\ \(1\) & \(100\) & \(2\) & \(2.19\) & \(1.7\) & \(2.3\) & \(2.45\) & \(2.47\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The achievable rate of different strategies in the relay channel under different values of channel coefficients.  
Figure 7: The achievable rate of different transmission techniques in the AWGN relay channel under a constrained constellation of 16-PAM at the source and 4-PAM at the relay.  
**Relay transmission:**  
The levels at the relay are divided into two sets. The first set sends the message \(W_{d}^{(t-1)}\). The second set sends the quantized version of \(Y_{2}\) after removing the effect of \(W_{d}\), namely, \(\hat{Y}_{2}\).  
**Source transmission:**  
The levels at the source are divided into three sets. The first set sends the message \(W_{d}^{(t-1)}\) cooperatively with the relay to the destination node. The second set sends the message \(W_{d}^{(t)}\) to be decoded at the relay. The third set of levels sends the message \(W_{c}^{(t)}\).  
The achievable rate of the multilevel coding DCF is given by  
\[\begin{split} R\leq\max_{\hat{B},\hat{B},\hat{C},\tilde{C}}\min \{I(\hat{B};Y_{2}|\hat{C})\},I(\hat{B},\hat{C};Y_{3})\}\\ +I(\hat{B},\hat{B};\hat{Y}_{2},Y_{3}|\hat{C},\hat{C},\hat{B})\end{split} \tag{39}\]  
subject to  
\[I(Y_{2};\hat{Y}_{2}|\hat{C},\tilde{C})\leq I(\hat{C},\tilde{C};Y_{3})-I(\hat{C} ;Y_{3}), \tag{40}\]  
where \(\hat{B}=[B_{1},\ldots,B_{j}]\) are the source levels that are responsible for the decode-forward part. \(\tilde{B}=[B_{j+1}\ldots B_{m}]\) are the source levels responsible for the compress-forward part. Similarly, \(\hat{C}=[C_{1},\ldots C_{l}]\) and \(\tilde{C}=[C_{l+1},\ldots C_{m}]\) are the relay levels responsible for the DF part and CF part, respectively.  
The question now is how to allocate the rate in each level at the source and the relay. This is determined through the following steps.  
1. The optimal function of each level (being a DF or a CF level) at the source and the relay depends on the constellation and the channel conditions. Using an exhaustive search to opti ize (39),  
Figure 8: Multilevel coding for full-duplex decode-compress-forward relay.  
one can find the best set of levels for decode-forward and for compress-forward at the source and the relay.
2. The number of bits of \(W_{d}^{(t)}\) and \(W_{c}^{(t)}\) depends on the rate of the decode-forward and compress-forward components given by \[R_{d}=\min\{I(\hat{B};Y_{2}|\hat{C})\}+I(\hat{B},\hat{C};Y_{3})\}\] (41) and \[R_{c}=I(\hat{B},\hat{B};\hat{Y}_{2},Y_{3}|\hat{C},\hat{C},\hat{B}),\] (42) respectively. Thus, \(W_{d}\) has rate \(nR_{d}\) and \(W_{c}\) has rate \(nR_{c}\).  
**Remark 3**.: _In the proposed multilevel DCF, we assume that each level is limited to either DF or CF. Extensive simulations show that this restriction has a negligible effect on achievable rates._  
**Remark 4**.: _In addition to level-assignment, other labeling variations are also possible, e.g., natural versus Gray labeling. Simulations in the sequel indicate that natural labeling is in general preferable to Gray labeling._