## 4 Performance Analysis
### Outage Probability
#### 4.2.1 Co-Carrier-Based Direct Combination  
Let \(\Gamma_{k}\) denote the demodulation threshold of UT \(k\). The outage probability of UT \(k\) for the co-carrier-based DC is given below.  
**Theorem 3**: _The outage probability of the \(k\)-th UT in the co-carrier-based DC scheme is given by_  
\[\Pr_{k}=1-(1/|\beta_{k}|^{2})^{M}(\frac{\alpha\Gamma_{k}}{M|\beta_{k}|^{2}}+ \frac{1}{|\beta_{k}|^{2}})^{-M}e^{-\frac{\Gamma_{k}}{\alpha M\beta_{k}|^{2}}}. \tag{28}\]  
Figure 3: Co-carrier-based SC method.  
Proof.: The outage probability of UT \(k\) can be defined as \(\Pr_{k}=\mathbb{P}(SINR_{k}<\Gamma_{k})\), which can be further calculated as  
\[\Pr_{k} =\mathbb{P}(\frac{|\sum_{m=1}^{M}g_{k,m}|^{2}P_{k}}{\sum_{m=1}^{M}| g_{k,m}|^{2}P+\sigma^{2}}<\Gamma_{k})\] \[=\mathbb{P}(\frac{YP_{k}}{ZP+\sigma^{2}}<\Gamma_{k})\] \[=F_{X}(\Gamma_{k}), \tag{29}\]  
where \(X=\frac{YR_{k}}{ZP+\sigma^{2}}\).  
Following (19), (29) can be derived as  
\[F_{X}(\Gamma_{k})=1-(1/|\beta_{k}|^{2})^{M}(\frac{\alpha\Gamma_{k}}{M|\beta_{ k}|^{2}}+\frac{1}{|\beta_{k}|^{2}})^{-M}e^{-\frac{\Gamma_{k}}{\delta M \beta_{k}|^{2}}}. \tag{30}\]  
The theorem is proved.  
Next, we analyze the outage probability of the beam-center UT. Assume that the demodulation threshold is \(\Gamma_{m}\). The outage occurs when \(SINR_{m,k}<\Gamma_{k}\) or \(SNR_{m,m}<\Gamma_{m}\) since the beam-center UT should decode \(s_{k}\) and \(x_{m}\) successively. Then, we derive the following theorem.  
Theorem 4.: _The outage probability of the beam-center UT in the co-carrier-based DC scheme is given by_  
\[\Pr_{m}=\left\{\begin{array}{ll}1-e^{-\frac{\Gamma_{m}}{\alpha \delta|\beta_{m}|^{2}}},&\frac{\Gamma_{k}}{1-\alpha\Gamma_{k}}\leq\frac{ \Gamma_{m}}{\alpha}\\ 1-e^{-\frac{\Gamma_{k}}{\delta|\beta_{m}|^{2}(1-\alpha\Gamma_{k})}},&\frac{ \Gamma_{k}}{1-\alpha\Gamma_{k}}>\frac{\Gamma_{m}}{\alpha}\end{array}\right. \tag{31}\]  
Proof.: The outage probability of the beam-center UT can be expressed as  
\[\Pr_{m} =\mathbb{P}(SINR_{m,k}<\Gamma_{k}\text{ or }SNR_{m,m}<\Gamma_{m})\] \[=1-\mathbb{P}(\frac{|g_{m,m}|^{2}P_{k}}{|g_{m,m}|^{2}P_{m}+ \sigma^{2}}\geq\Gamma_{k},\frac{|g_{m,m}|^{2}P_{m}}{\sigma^{2}}\geq\Gamma_{m})\] \[=1-\mathbb{P}(|g_{m,m}|^{2}\geq\max\{\frac{\Gamma_{k}}{\delta- \Gamma_{k}\alpha\delta},\frac{\Gamma_{m}}{\alpha\delta}\}), \tag{32}\]  
which can be further discussed as  
\[\Pr_{m} =1-\mathbb{P}(|g_{m,m}|^{2}\geq\max\{\frac{\Gamma_{k}}{\delta- \Gamma_{k}\alpha\delta},\frac{\Gamma_{m}}{\alpha\delta}\})\] \[=\left\{\begin{array}{ll}1-\mathbb{P}(|g_{m,m}|^{2}\geq\frac{ \Gamma_{m}}{\alpha\delta}),&\frac{\Gamma_{k}}{1-\alpha\Gamma_{k}}\leq\frac{ \Gamma_{m}}{\alpha}\\ 1-\mathbb{P}(|g_{m,m}|^{2}\geq\frac{\Gamma_{k}}{\delta-\Gamma_{k}\alpha\delta }),&\frac{\Gamma_{k}}{1-\alpha\Gamma_{k}}>\frac{\Gamma_{m}}{\alpha}\end{array}\right. \tag{33}\]  
When \(\frac{\Gamma_{k}}{1-\alpha\Gamma_{k}}\leq\frac{\Gamma_{m}}{\alpha}\), we have  
\[\mathbb{P}(|g_{m,m}|^{2}\geq\frac{\Gamma_{m}}{\alpha\delta})=1-\mathbb{P}(|g_{ m,m}|^{2}<\frac{\Gamma_{m}}{\alpha\delta})=e^{-\frac{\Gamma_{m}}{\alpha \delta|\beta_{m}|^{2}}}. \tag{34}\]  
Similarly, we can derive  
\[\mathbb{P}(|g_{m,m}|^{2}\geq\frac{\Gamma_{k}}{\delta-\Gamma_{k}\alpha\delta})= 1-\mathbb{P}(|g_{m,m}|^{2}<\frac{\Gamma_{k}}{\delta-\Gamma_{k}\alpha\delta})= e^{-\frac{\Gamma_{k}}{\delta|\beta_{m}|^{2}(1-\alpha\Gamma_{k})}}. \tag{35}\]  
for \(\frac{\Gamma_{k}}{1-\alpha\Gamma_{k}}>\frac{\Gamma_{m}}{\alpha}\).  
The theorem is proved.