## 4 Coded Cooperation with Soft Information Relaying and Relay Selection  
A way to address the problem of error-propagation is SIR, where the relay \(R_{r}\) performs at first a SISO decoding of the received sequence \(\mathbf{y}_{SR_{r}}\). Thereafter, the resulting LLR sequence denoted by \(L(\hat{\mathbf{n}})=(L(\hat{\mathbf{n}}_{1}|\mathbf{y}_{SR_{r}}),\cdots,L(\hat{\mathbf{n}}_{k }|\mathbf{y}_{SR_{r}}))\) is fed into a soft channel encoder of rate \(R_{R}\) to compute the LLRs of the relay code bits denoted by \(L^{(R_{r})}(c_{R,t}|L(\hat{\mathbf{n}}))\), \(1\leq t\leq N_{R}\). In the proposed coded cooperative scheme with SIR and relay selection (CC-SIR-RS), the selected relay is invited to forward the LLRs \(L^{(R_{r})}(c_{R,t}|L(\hat{\mathbf{n}}))\), \(1\leq t\leq N_{R}\), which are referred to as soft estimated symbols. Figure 2a shows the histogram of the LLRs for a given SNR per information bit. Figure 2b shows the normal Q-Q plot of the LLR distribution. A distribution is characterized as Gaussian if the normal Q-Q plot fits the \(y=x\) line. It is obvious that the Q-Q plot does not follow the \(y=x\) line. Hence, the distribution is non-Gaussian. For this reason, we assume that the output LLRs of the SISO encoder are rather affected by quasi-static channel fading coefficients beside the residual additive noise. This can be explained by the fact that channel decoding is able to average out the Gaussian noise but channel fading effect partially persists after decoding. For this reason, a soft estimated symbol is modeled by the following expression:  
\[L^{(R_{r})}(c_{R,t}|L(\hat{\mathbf{n}}))=\frac{2h_{R_{r}}}{\sigma_{R_{ R_{r}}}^{2}}(h_{R_{r}}x_{t}+n_{R_{r},t}),\quad 1\leq t\leq N_{R}, \tag{19}\]  
where \(x_{t}\in\{-1,1\}\) is the BPSK modulated symbol after error-free decoding of the received sequence and hard output encoding, \(h_{R_{r}}\) is a block fading coefficient and is considered as a quasi-static Rayleigh distributed fading coefficient, and \(n_{R_{r},t}\) is a Gaussian distributed random variable with zero-mean and variance \(\sigma_{R_{r}}^{2}\). Hence, the model is referred to as a Rayleigh Gaussian LLR (RGL) model. Let \(\Gamma_{R_{r}}=h_{R_{r}}^{2}/\sigma_{R_{r}}^{2}\) be the obtained SNR at the output of the soft encoder. By means of the moment method (MM), \(\Gamma_{R_{r}}\) can be calculated as follows:  
\[\Gamma_{R_{r}}=\frac{1}{2}(\sqrt{1+E_{2}}-1), \tag{20}\]  
where \(E_{2}\) is the second order moment of the soft estimated symbols \((L^{(R_{r})}(c_{R,1}),\cdots,\)\(L^{(R_{r})}(c_{R,N_{R}}))\). We note that \(\Gamma_{R_{r}}\) is calculated online and forwarded to the destination that is of practical interest. The soft estimated symbols are scaled with a block normalization factor \(\beta_{R_{r}}\) that remains constant during a block transmission and is computed as follows:  
\[\beta_{R_{r}} = \frac{1}{\sqrt{E\{L^{(R_{r})}(c_{R})^{2}\}}} \tag{21}\] \[= \frac{1}{2\sqrt{\Gamma_{R_{r}}^{2}+\Gamma_{R_{r}}}}.\]  
The received symbol at time \(t\) in destination node \(D\) is expressed as  
\[y_{R_{r}D,t}=h_{R_{r}}\sqrt{\frac{G_{SR_{r}}}{d_{R_{r}D}^{n}}\frac{R_{R}}{R_{S} }\rho_{r}\
u|h_{SR_{r}}|^{2}P_{SD}\beta_{R_{r}}L^{(R_{r})}(c_{R,t})+n_{R_{r}D,t}}, \tag{22}\]  
where \(n_{R_{r}D,t}\) is a zero-mean additive white Gaussian noise with variance \(\sigma_{n}^{2}\).