# Multilevel Coding for the Full-Duplex Decode-Compress-Forward Relay Channel
## 3 Decode-Compress-Forward
### AWGN Full-Duplex Relay  
Assume that all the variables in the Section 3.1 are Gaussian, and the source and relay have an average power constrained by \(P_{1}\) and \(P_{2}\), respectively. Note that Gaussian codebooks are not known to be optimal in this case. A block-Markov encoding of DCF in the AWGN channel is shown in Figure 4 under four transmission blocks. The source and relay signals are given by  
\[X_{1} =U_{1d}+\beta U_{2d}+U_{1c}, \tag{23}\] \[X_{2} =U_{2d}+U_{2c}, \tag{24}\]  
respectively. Each of the codewords \(U_{1d},U_{2d},U_{1c}\) and \(U_{2c}\) are normally distributed. The term \(\beta U_{2d}\) represents the assistance that the source provides for the relay destination transmission. This assistance depends on the correlation between \(U_{1d}+\beta U_{2d}\) and \(U_{2d}\), which is denoted by \(\rho\). **Remark 1**. : _In DCF, normally one can optimize the power allocation of each code at the source and the relay. We fix the power of one of the DF signals \(U_{1d}\) and the CF signal \(U_{1c}\). The power of the remaining signals can be obtained as a function of the power of \(U_{1d}\) and \(U_{1c}\) and the power constraint at the source and relay nodes. Consequently, the design variables of the rate maximization problem become the power of \(U_{1d}\), \(U_{1c}\) and the correlation \(\rho\)._  
Assuming that the power of \(U_{1d}\) is \(P_{1d}\) and the power of \(U_{1c}\) is \(P_{1c}\), the power of \(U_{2d}\) is then given by  
\[P_{2d}=\frac{P_{1}-P_{1d}-P_{1c}}{\beta^{2}}, \tag{25}\]  
and, consequently, the power of \(U_{2c}\) is  
\[P_{2c} =P_{2}-P_{2d}\] \[=P_{2}-\frac{P_{1}-P_{1d}-P_{1c}}{\beta^{2}}, \tag{26}\]  
where  
\[\rho=\frac{E[(U_{1d}+\beta U_{2d})U_{2d}]}{\sqrt{(P_{1d}+\beta^{2}P_{2d})P_{2 d}}}, \tag{27}\]  
\[\beta=\sqrt{\frac{\rho^{2}P_{1d}P_{2d}}{P_{2d}^{2}(1-\rho^{2})}}. \tag{28}\]  
The signals \(Y_{2},\hat{Y}_{2}\) and \(Y_{3}\) are given by  
\[Y_{2} =H_{12}X_{1}+n_{2}, \tag{29}\] \[\hat{Y}_{2} =Y_{2}+\hat{n},\] (30) \[Y_{3} =H_{13}X_{1}+H_{23}X_{2}+n_{3}, \tag{31}\]  
where \(n_{2},n_{3}\) and \(\hat{n}\) are zero mean Gaussian noise with variance \(\sigma_{2}^{2},\sigma_{3}^{2}\) and \(\hat{N}\), respectively. \[X_{2} =U_{2d}+U_{2c}, \tag{24}\]0 Decode-compress-forward transmission for the AWGN full-duplex relay channel over four blocks. Based on the previous characterization for each of the distributions of the variables involved in calculating the transmission rate, the achievable rate for the AWGN DCF relay is as follows:  
\[X_{2} =U_{2d}+U_{2c}, \tag{24}\]1  
\[X_{2} =U_{2d}+U_{2c}, \tag{24}\]2  
\[X_{2} =U_{2d}+U_{2c}, \tag{24}\]3  
where  
\[X_{2} =U_{2d}+U_{2c}, \tag{24}\]4  
To illustrate the performance of DCF as a function of the location of the relay, we adopt the illustration used in [3] and calculate the achievable rate as a function of relay position on a line extending from the source to destination. For simplicity, a path-loss model is considered with the channel coefficient \(H_{ij}=1/d_{ij}^{\alpha}\), where \(d_{ij}\) is the distance between node \(i\) and node \(j\) and \(\alpha\) is the path-loss coefficient, which is usually between 2 and 4. The distance between the source and the destination is fixed to \(d_{13}=1\) while the distances \(d_{12}\) and \(d_{23}\) depend on the relay location where \(d_{23}=1-d_{12}\). In Figure 5, the achievable rate of different transmission techniques are shown as a function of the distance between the source and the relay \(d_{12}\). Figure 5 shows that the achievable rate of DCF in the AWGN relay channel reduces to either DF rate or CF rate. In other words, optimizing DCF results in either DF or CF.