## 3 Coded Cooperation with Hard Information Relaying and Relay Selection
### Outage Probability Analyses of the CC-HIR-RS Scheme
#### 3.1.1 CC-HIR-RS Scheme with Constant PSR (CC-HIR-RS-CPS)  
In this subsection, we discuss the CC-HIR-RS-CPS scheme and perform the derivation of its exact closed-form outage probability expression. The PSR is assumed to be constant for all relays. In order to avoid error-propagation, only relays that correctly decode the received message from the source node are considered in the second time-slot of the cooperative transmission. This can be verified by means of a cyclic redundancy check (CRC) code, which allows one to check the correctness of the received message. To ensure that the messages are decoded correctly at the decoder of relay \(R_{r}\), the channel capacity of the \(S-R_{r}\) link should be no less than the information rate \(R\). Hence, the set of candidate relays for forwarding the message can be given by  
\[\Psi = \{r:C_{SR_{r}}=\frac{1}{2}\log_{2}(1+\Gamma_{SR_{r}})>R\}, \tag{8}\]  
where \(\Gamma_{SR_{r}}\) is given in (2). In the proposed CC-HIR-RS-CPS scheme, the best relay is selected based on maximizing \(\Gamma_{R,D}\) through all relays that correctly decode the message from the source node. The best relay of the CC-HIR-RS-CPS scheme is obtained by the following expression:  
\[R_{\text{opt}}^{\text{CC-HIR-RS-CPS}} = \arg\max_{K_{r},r\in\Upsilon}\{\Gamma_{R_{r}D}\} \tag{9}\] \[= \arg\max_{K_{r},r\in\Upsilon}\{\bar{\Gamma}_{R_{r}D}|h_{SR_{r}}| ^{2}|h_{R_{r}D}|^{2}\}.\]  
For simple notation, let \(\gamma_{SR_{r}}=|h_{SR_{r}}|^{2}\) and \(\gamma_{R_{r}D}=|h_{R_{r}D}|^{2}\); \(\gamma_{SR_{r}}\) and \(\gamma_{R,D}\) are exponential distributed with means \(\bar{\gamma}_{SR_{r}}\) and \(\bar{\gamma}_{R,D}\), respectively. The outage probability of the CC-HIR-RS-CPS occurs in the two cases described below:  
* Case 1: All source-relay links are in outage, i.e., \(\Psi=\emptyset\). The resulting outage probability is denoted by \(P_{1}\).
* Case 2: \(\Psi\
eq\emptyset\), but the selected relay-destination link is in outage. The resulting outage probability is denoted by \(P_{2}\). Hence, the outage probability of CC-HIR-RS-CPS system is expressed as \[P_{\text{out}}^{\text{CC-HIR-RS-CPS}}=P_{1}+P_{2},\] (10) where\[P_{1} = \prod_{r=1}^{n_{R}}P(\Gamma_{SR_{r}}<2^{2R}-1) \tag{11}\] \[= \prod_{r=1}^{n_{R}}(1-\exp^{-\frac{2^{2R}-1}{T_{SR_{r}}}})\]  
and  
\[P_{2} = \sum_{k=1}^{n_{R}}\binom{n_{R}}{k}\prod_{r\in\mathbb{Y}_{k}}P( \Gamma_{SR_{r}}<2^{2R}-1)\prod_{r\in\mathbb{Y}_{k}}P(\Gamma_{SR_{r}}>2^{2R}-1)\cdot \tag{12}\] \[P\bigg{(}\max_{r\in\mathbb{Y}_{k}}\{\bar{\Gamma}_{R_{r}D}\gamma_ {SR_{r}}\gamma_{R_{r}D}\}<2^{2R}-1\bigg{)}\] \[= \sum_{k=1}^{n_{R}}\binom{n_{R}}{k}\prod_{r\
otin\mathbb{Y}_{k}}(1 -\exp^{-\frac{2^{2R}-1}{T_{SR_{r}}}})\prod_{r\in\mathbb{Y}_{k}}(\exp^{-\frac{ 2^{2R}-1}{T_{SR_{r}}}})\prod_{r\in\mathbb{Y}_{k}}P(\gamma_{SR_{r}}\gamma_{R_{r} D}<\frac{2^{2R}-1}{\bar{\Gamma}_{R_{r}D}})\] \[= \sum_{k=1}^{n_{R}}\binom{n_{R}}{k}\prod_{r\
otin\mathbb{Y}_{k}}( 1-\exp^{-\frac{2^{2R}-1}{T_{SR_{r}}}})\prod_{r\in\mathbb{Y}_{k}}(\exp^{-\frac{ 2^{2R}-1}{T_{SR_{r}}}})\prod_{r\in\mathbb{Y}_{k}}F_{\gamma_{SR_{r}}\gamma_{R_{ r}D}}(\frac{2^{2R}-1}{\bar{\Gamma}_{R_{r}D}}).\]  
Here, \(F_{\gamma_{SR_{r}}\gamma_{R_{r}D}}\) is the CDF of the random variable \(\gamma_{SR_{r}}\gamma_{R_{r}D}\), which is expressed as  
\[F_{\gamma_{SR_{r}}\gamma_{R_{r}D}}(\gamma) = 1-P(\gamma_{SR_{r}}\gamma_{R_{r}D}>\gamma) \tag{13}\] \[= 1-\int_{0}^{\infty}P(\gamma_{SR_{r}}>\gamma/x)f_{\gamma_{R_{r}D} }(x)dx\] \[= 1-\int_{0}^{\infty}\exp^{-\gamma/\gamma_{SR_{r}}x}\frac{1}{\bar {\gamma}_{R_{r}D}}\exp^{-x/\bar{\gamma}_{R_{r}D}}dx\] \[= 1-2\sqrt{\frac{\gamma}{\gamma_{SR_{r}}\gamma_{R_{r}D}}}K_{1}(2 \sqrt{\frac{\gamma}{\gamma_{SR_{r}}\gamma_{R_{r}D}}}),\]  
as \(\int_{0}^{\infty}\exp^{-ax-b/x}dx=2\sqrt{b/a}K_{1}(2\sqrt{ab})\)[29], where \(K_{1}(\cdot)\) is the first-order modified Besse function.