## 3 Polar Coded SDF Cooperation
### Proposed PCSDF Cooperation  
In original SDF cooperation, OP performance is limited by the channel quality of the S-R\({}_{i}\) link. Whether the relay nodes can decode the source information correctly or not has a crucial impact on the system performance. To reduce error propagation and increase BER performance, an improved PCSDF relay cooperation is proposed in this section. After the relay receives messages from the source in Phase I, the procedure of decoding and forwarding begins. To ensure that the messages are correctly decoded, the channel capacity of link S-R\({}_{i}\) is required to be no less than the information transmission rate \(V\). Therefore, the set of candidate relay nodes is expressed as  
\[\Omega_{k}=\{\mathbb{R}_{i}:C_{sr_{i}}\geq V\}, \tag{15}\]where \(C_{sr_{j}}\) is the channel capacity of link S-R\({}_{i}\). If the candidate relay set \(\Omega_{k}\) is empty, an attempt at direct transmission through link S-D is performed. In this case, the channel capacity of the transmission system is expressed as  
\[C_{\text{DRT}}{=}\frac{1}{2}\log_{2}(1+2\gamma_{sd}), \tag{16}\]  
where \(C_{\text{DRT}}\) is the channel capacity of direct link S-D, and \(\gamma_{sd}=\frac{P_{c}|h_{rd}|^{2}}{N_{0}}\) is the instantaneous SNR when the destination node D receives the signals from the source node S directly. Furthermore, a relationship \(E_{b}/N_{0}=\gamma_{sd}\cdot\log_{2}(MI)\), exists, where \(MI\) is the modulation index, e.g., 2 for binary phase shift keying (BPSK), 4 for quadrature phase shift keying (QPSK), etc. In another situation, if the candidate relay set \(\Omega_{k}\) is not empty, a relay node with different instantaneous channel quality from all relay nodes is selected as the \(i\)-th candidate relay node R\({}_{c,i}\) in set \(\Omega_{k}\). The SNR of the received signals from the relay node R\({}_{c,i}\) is expressed as  
\[\gamma_{rd} =\frac{P_{rd}|h|^{2}}{N_{0}}\quad, \tag{17}\] \[Subject\quad to\quad R_{c,i}\in\Omega_{k}\]  
where \(\gamma_{rd}\) is the SNR when destination node D receives signals from the relay node R\({}_{i}\), and mainly depends on the transmission power of the relay node and the channel fading coefficient of link \(R_{i}\)-D. When the transmission power of the source node S and each relay node R\({}_{i}\) is equal, \(R_{c,i}\) is only related to the channel fading coefficient \(h_{rd}\). After the relay node is successful, the ratio of the cooperative capability of R\({}_{c,i}\), i.e., \(\beta_{r_{c,i}}=|h_{r_{c,i}d}|^{2}\) with instantaneous channel gain of link S-D, i.e., \(\beta_{sd}=|h_{sd}|^{2}\) is obtained at the destination. With an efficient information rate, or code rate \(\tau\), used in the DF cooperation, the OP \(P_{DF}\) can be expressed as  
\[ \begin{array}{l}P_{\text{DF}}(\gamma_{sr_{j}},\tau)=P(I_{\text{DF}}<\tau)\\ =P[|h_{sr}|^{2}<(2^{2\tau}-1)/\gamma_{sr_{j}}]+P[|h_{sr}|^{2}\geq(2^{2\tau}-1)/ \gamma_{sr_{j}}]\\ =P[|h_{sd}|^{2}+|h_{rd}|^{2}<(2^{2\tau}-1)/\gamma_{sr_{j}}]\\ \sim(2^{2\tau}-1)/(\sigma_{sr_{i}}^{2}\cdot\gamma_{sr_{j}})\end{array} , \tag{18}\]  
where \(I_{\text{DF}}\) is the channel capacity of the DF cooperation system and is represented as \(I_{\text{DF}}=\min\limits_{i}\{I_{sr},I_{sr_{j}d}\}\). Thus the cooperation threshold \(\theta\) is in proportion to the reverse of the SNR \(\gamma_{sr_{j}}\); it can be derived from (18) and expressed as \(\theta=(2^{2\tau}-1)/\gamma_{sr_{j}}\). Subsequently, the ratio is compared with the cooperation threshold \(\theta\). If \(\beta_{r_{c,i}}/\beta_{sd}>\theta\), the quality of the link including the selected relay is optimal. Thus, \(R_{c,i}\) can be used to forward messages. The received signals in the destination D are then expressed as  
\[y_{r_{c,i}d}=h_{r_{c,i}d}\sqrt{P_{r_{c,i}}}x_{r_{c,i}}+n_{rd_{c,i}}. \tag{19}\]  
If \(\beta_{r_{opt}}/\beta_{sd}\leq\theta\), the channel quality of the chosen link R\({}_{opt}\)-D is rather poor.