# Optimization of Effective Throughput in NOMA-Based Cognitive UAV Short-Packet Communication
## 2 System Model and Problem Formulation
### Transmission Model  
When the GBS is idle, the UAV will transmit \(N-N_{s}\) symbols to the secondary users using NOMA. In contrast to long-packet communication, the decoding error probability in short-packet communication is not negligible. According to [22], for a given transmission rate \(R\), the decoding error probability of the secondary users can be approximated as  
\[\varepsilon=\mathcal{Q}(f(\gamma,N_{s},R)), \tag{2}\]  
where \(f(\gamma,N_{s},R)=\ln 2\sqrt{\frac{N-N_{s}}{V}}(\log_{2}(1+\gamma)-R)\), \(\gamma\) denotes the SNR at the secondary users, and \(V\) denotes the channel dispersion, \(V=1-(1+\gamma)^{-2}\). Considering the effect of the non-zero error decoding probability, a perfect SIC with the NOMA transmission scheme cannot be guaranteed. Thus, we introduce the effective decoding error probability, which is defined as \(\varepsilon_{i}\), \(i\in(1,2)\), to represent the actual error probability at \(u_{i}\). In the following, we derive the expressions for the effective decoding error probability for \(u_{1}\) and \(u_{2}\), respectively. (1) Transmission to user \(u_{1}\): The received signal at \(u_{1}\) can be expressed as  
\[y_{1}=\sqrt{p_{1}h_{1}}x_{1}+\sqrt{p_{2}h_{1}}x_{2}+n_{1}, \tag{3}\]  
where \(n_{1}\sim CN(0,\delta_{1}^{2})\) denotes the AWGN with zero mean and variance \(\delta_{1}^{2}\) at \(u_{1}\). \(x_{1}\) and \(x_{2}\) denote the signals sent by the UAV to \(u_{1}\) and \(u_{2}\), respectively. \(p_{1}\) and \(p_{2}\) are the transmit power assigned to \(u_{1}\) and \(u_{2}\), respectively. For the received signal \(y_{1}\), the secondary user \(u_{1}\) needs to eliminate \(x_{2}\) before decoding \(x_{1}\). To this end, \(x_{2}\) will be firstly decoded by \(u_{1}\) and \(x_{1}\) is regarded as interference. According to (2), for a given transmission rate \(R_{2}\), the decoding error probability of \(x_{2}\) at \(u_{1}\) is  
\[\varepsilon_{2,1}=\mathcal{Q}(f(\gamma_{2,1},N_{s},R_{2})), \tag{4}\]where \(\gamma_{2,1}=\frac{p_{2}h_{1}}{p_{1}h_{1}+\delta_{1}^{2}}\) is the signal-to-interference-plus-noise ratio (SINR) at \(u_{1}\). Then, if the secondary user \(u_{1}\) successfully decodes \(x_{2}\), \(u_{1}\) can use SIC to remove the interference caused by \(x_{2}\). In this case, for a given transmission rate \(R_{1}\), the decoding error probability of \(x_{1}\) at the secondary user \(u_{1}\) is  
\[\varepsilon_{1}=\mathcal{Q}(f(\gamma_{1},N_{s},R_{1})), \tag{5}\]  
where \(\gamma_{1}=\frac{p_{1}h_{1}}{\delta_{1}^{2}}\) is the SNR at the secondary user \(u_{1}\). If the secondary user \(u_{1}\) fails to decode \(x_{2}\), \(u_{1}\) can only decode \(x_{1}\) and \(x_{2}\) is regarded as interference. According to (2), for a given transmission rate \(R_{1}\), the decoding error probability of \(x_{1}\) at \(u_{1}\) is  
\[\varepsilon_{1}^{\prime}=\mathcal{Q}\big{(}f(\gamma_{1}^{\prime},N_{s},R_{1}) \big{)}, \tag{6}\]  
where \(\gamma_{1}^{\prime}=\frac{p_{1}h_{1}}{p_{2}h_{1}+\delta_{1}^{2}}\) is the SINR at the secondary user \(u_{1}\). Based on the above analysis, the effective decoding error probability of \(x_{1}\) at \(u_{1}\) is given by  
\[\bar{\varepsilon}_{1}=(1-\varepsilon_{2,1})\varepsilon_{1}+\varepsilon_{2,1} \varepsilon_{1}^{\prime} \tag{7}\]  
(2) Transmission to user \(u_{2}\): The received signal at \(u_{2}\) can be expressed as  
\[y_{2}=\sqrt{p_{1}h_{2}}x_{1}+\sqrt{p_{2}h_{2}}x_{2}+n_{2}, \tag{8}\]  
where \(n_{2}\sim CN(0,\delta_{2}^{2})\) denotes the AWGN with zero mean and variance \(\delta_{2}^{2}\) at \(u_{2}\). According to the NOMA scheme, we decode \(x_{2}\) at \(u_{2}\) directly and \(x_{1}\) is considered as interference. According to (2), for a given transmission rate \(R_{2}\), the decoding error probability of \(x_{2}\) at \(u_{2}\) is  
\[\varepsilon_{2}=\mathcal{Q}(f(\gamma_{2},N_{s},R_{2})), \tag{9}\]  
where \(\gamma_{2}=\frac{p_{2}h_{2}}{p_{1}h_{2}+\delta_{2}^{2}}\) is the SINR at \(u_{2}\).