# Multilevel Coding for the Full-Duplex Decode-Compress-Forward Relay Channel
## 3 Decode-Compress-Forward
### Constellation-Constrained Full-Duplex Relay  
For the discrete input AWGN relay channel, the rate expressions in Theorem 1 can be calculated via numerical integrations. The optimizing distribution may require an exhaustive search. As shown in many point-to-point and multi-user scenarios [33; 34], when the constellation becomes large enough, the achievable rate under a constrained constellation becomes very close to the Gaussian input rate.  
Figure 5: The achievable rate of different transmission techniques in the AWGN full-duplex relay channel.  
However, the main difficulty comes from the restriction in Equation (21), which is now even harder to satisfy since the mutual information \(I(X_{2};Y_{3})\) is no longer equal to  
\[\frac{1}{2}\log\left(1+\frac{|H_{23}|^{2}P_{2}}{\sigma_{3}}\right)\]  
and is limited by the cardinality of the input size \(|X_{2}|\). The exact value of the constellation constrained capacity [35], \(I(X_{2};Y_{3})\), can be obtained from  
\[I(X_{2};Y_{3})=\max_{P_{X_{2}}(x_{2})}\sum_{\tilde{Y}_{2}}P_{X_{2}}(x_{2})\int _{y_{3}}P_{Y_{3}|X_{2}}(y_{3}|x_{2})\log\left(\frac{P_{Y_{3}|X_{2}}(y_{3}|x_{2 })}{P_{Y_{3}}(y_{3})}\right)dy_{3}. \tag{34}\]  
An accurate approximation for \(I(X_{2};Y_{3})\) under constrained constellation can be obtained using the Blahut-Arimoto algorithm [36; 37]. Constellation constrained point-to-point channel capacity for various constellations is shown in Figure 6.  
By calculating the value of \(I(X_{2};Y_{3})\), one can find the achievable rate of compress-forward and DCF. In the following, an upper bound is provided on the achievable rate of DCF under discrete relay-destination input \(X_{2}\). The upper bound is based on bounding \(I(X_{2};Y_{3})\) by either the cardinality of \(X_{2}\) or the Gaussian capacity  
\[I(X_{2};Y_{3})\leq\min\Bigg{\{}|X_{2}|,\frac{1}{2}\log\left(1+\frac{|H_{23}|^{ 2}P_{2}}{\sigma_{3}^{2}}\right)\Bigg{\}}. \tag{35}\]  
Therefore, the constraint in Equation (21) becomes  
\[I(Y_{2};\hat{Y}_{2}|X_{2},U_{1d})\leq\min\Bigg{\{}|X_{2}|,\frac{1}{2}\log\left( 1+\frac{|H_{23}|^{2}P_{2}}{\sigma_{3}^{2}}\right)\Bigg{\}}-I(U_{2d};Y_{3}). \tag{36}\]  
By using the bound in Equation (36), one can find an upper bound on the DCF as follows:  
\[R\leq\min\Big{\{}I(U_{1d};Y_{2}|U_{2d}),I(U_{1d},U_{2d};Y_{3})\Big{\}}+I(X_{1} ;\hat{Y}_{2},Y_{3}|X_{2},U_{1d}) \tag{37}\]  
Figure 6: The constellation constrained capacity of several constellations in the point-to-point AWGN channel compared with the Gaussian input capacity.  
subject to  
\[I(Y_{2};\hat{Y}_{2}|X_{2},U_{1d})\leq\min\left\{\frac{1}{2}\log\left(1+\frac{|H_{2 3}|^{2}P_{2}}{\sigma_{3}^{2}}\right),|X_{2}|\right\}-I(U_{2d};Y_{3}). \tag{38}\]  
The capacity of the decode-compress-forward technique can be obtained by evaluating these expressions using numerical integrations and an exhaustive search for the optimal distribution.  
In a similar manner to Section 3.2, the achievable rate for different strategies is shown in Figure 7, showing that when the relay is close to the destination, CF suffers a rate penalty while DCF does not.  
**Remark 2**.: _Compress-forward works well when the source-relay channel is weak, but the relay destination channel is strong so that an estimate of \(Y_{2}\) can be communicated with the destination. If the relay-destination channel supports insufficient rate (either because of channel gain or limitation of constellation) communicating an estimate of \(Y_{2}\) is not an attractive strategy for the relay. This can in part explain the effect of the constellation constraint on the performance of compress-forward._  
Clearly, the relay channel model in Figure 7 does not represent every possible scenario. We show two more results in Table 1 that cannot be observed in Figure 7.