# Guaranteeing QoS for NOMA-Enabled URLLC Based on \(\kappa\)-\(\mu\) Shadowed Fading Model
## 2 System Model
\(\mathbf{\hat{E}}=\left[\mathbf{\hat{e}}_{1},\ldots,\mathbf{\hat{e}}_{K}\right] \in\mathbb{C}^{L\times K}\) is the channel estimation error matrix, and its elements \(\mathbf{\hat{e}}_{k}=\left[\mathbf{\hat{e}}_{1k},\ldots,\mathbf{\hat{e}}_{Lk }\right]^{T}\) follow \(\mathbf{\hat{e}}_{lk}\sim\mathcal{CN}\Big{(}0,\sigma_{\mathbf{\hat{e}}_{lk}}^ {2}\Big{)}\), where \(\sigma_{\mathbf{\hat{e}}_{lk}}^{2}=\Lambda_{lk}^{2}\sigma_{\mathbf{\hat{e}}_{ lk}}^{2}=\frac{\Lambda_{lk}^{2}}{np_{\rm u}}=\frac{\beta_{lk}\Omega_{lk}}{1+np_{\rm u }\beta_{lk}\Omega_{lk}}\). The elements of \(\mathbf{\hat{G}}\) can be expressed as \(\mathbf{\hat{g}}_{lk}=\mathbf{\Lambda}_{lk}\mathbf{\hat{g}}_{lk}+\mathbf{\hat{ e}}_{lk}=\sqrt{\beta_{lk}}h_{lk}\); thus, \(\mathbf{\hat{h}}_{lk}=\mathbf{\Lambda}_{lk}h_{lk}+\frac{\mathbf{\hat{e}}_{lk}}{ \sqrt{\beta_{lk}}}\). Adopting the standard linear zero-forcing detector \(\mathbf{\hat{A}}=\mathbf{\hat{G}}(\mathbf{\hat{G}}^{H}\mathbf{\hat{G}})^{-1} \in\mathbb{C}^{L\times K}\), the received signal of the \(k\)th user is  
\[\mathbf{R}(t)=\sqrt{p_{\rm u}}\mathbf{\hat{A}}^{H}\mathbf{G}(t)\mathbf{X}^{( \rm d)}(t)+\mathbf{\hat{A}}^{H}\mathbf{Z}^{(\rm d)}(t). \tag{5}\]  
After completing zero-forcing detection, the PPSNR of the \(k\)th user is [53]  
\[\hat{\gamma}_{k} =\frac{p_{\rm u}}{p_{\rm u}\sum\limits_{l=1}^{L}\left|\mathbf{ \hat{a}}_{lk}\right|^{2}\sum\limits_{n=1}^{K}\sigma_{\mathbf{\hat{a}}_{lk}}^{2}+ \left\|\mathbf{\hat{a}}_{k}\right\|^{2}}\] \[\stackrel{{(a)}}{{\approx}}\frac{\frac{p_{\rm u}}{ \sum\limits_{\rm\frac{K}{n}}\left\|\mathbf{\hat{a}}_{k}\right\|^{2}+\left\| \mathbf{\hat{a}}_{k}\right\|^{2}}}{\frac{\mathbf{\hat{e}}}{n}\left\|\mathbf{ \hat  
where \(\mathcal{L}_{k}=\mathcal{L}/\mathcal{B}_{k}=\left\{\hat{l}\big{|}\forall\hat{l}=1,..., \hat{L}\right\}\), \(\mathcal{L}=\left\{l|\forall l=1,...,L\right\}\), \(\mathcal{B}_{k}\) is defined as \(\mathcal{B}_{k}\stackrel{{\Delta}}{{=}}\mathrm{Unique}\bigg{(} \bigg{\{}l_{n}^{*}=\arg\max\limits_{l}\beta_{lk}|\forall n\
eq k\bigg{\}} \bigg{)}\) and \(\mathrm{Unique}(\mathcal{T})\) returns the same values as those in the set \(\mathcal{T}\) but with no repetitions. \(\hat{a}_{lk}\) is the \(l\)th entry of \(\hat{\mathbf{a}}_{k}\), and \(\hat{\mathbf{a}}_{k}\) is the \(k\)th column of \(\hat{\mathbf{A}}\). \(\eta_{lk}\) is the channel gain of the \(l\)th AP and the \(k\)th user and is an RV with both large-scale fading coefficient and a small-scale fading coefficient. (\(a\)) is based on the consideration that when \(p_{\mathrm{u}}\) is high enough, \(p_{\mathrm{u}}\sigma_{\hat{a}_{lk}}^{2}\approx\frac{1}{n}\). (\(b\)) occurs because \(\frac{1}{\|\hat{\mathbf{a}}_{k}\|^{2}}\approx\sum\limits_{l\in\mathcal{L}_{k}} \big{|}\delta_{lk}\big{|}^{2}\)[53]. Because the gain of small-scale fading \(|\eta_{lk}|^{2}\) statistically follows the \(\kappa-\mu\) shadowed distribution, the distribution of \(\eta_{lk}\) is obtained in Lemma 1 as follows.  
**Lemma 1**.: _Based on a \(\kappa-\mu\) shadowed fading model, the channel gain \(\eta_{lk}\) follows the Gamma distribution, which is_  
\[\eta_{lk}\sim\Gamma\big{(}\mathcal{K}_{\hat{lk}},\mathcal{Q}_{\hat{lk}}\big{)}, \tag{7}\]  
_where \(\mathcal{K}_{\hat{lk}}=\frac{m_{\hat{lk}}\mu_{lk}\left(1+\kappa_{\hat{lk}} \right)^{2}}{m_{\hat{lk}}^{2}+\mu_{\hat{lk}}\kappa_{\hat{lk}}^{2}+2m_{\hat{lk }}\kappa_{\hat{lk}}}=\frac{\mu_{\hat{lk}}\left(1+\kappa_{\hat{lk}}\right)^{2} }{1+\mu_{\hat{lk}}\kappa_{\hat{lk}}^{2}/m_{\hat{lk}}^{2}+2\kappa_{\hat{lk}}}\), and \(\mathcal{Q}_{\hat{lk}}=\beta_{\hat{lk}}\theta_{\hat{lk}}=\frac{\beta_{\hat{lk }}\Omega_{\hat{lk}}}{\mathcal{K}_{\hat{lk}}}=\frac{\Omega_{\hat{lk}}}{\mathcal{ K}_{\hat{lk}}}\). \(\beta_{\hat{lk}}\Omega_{\hat{lk}}\) is the mean value of \(\eta_{\hat{lk}}\), as well as the mean value of the channel gain._  
Proof. : Please refer to Appendix A. Based on the \(\kappa-\mu\) shadowed fading model, \(\zeta_{k}=\sum\limits_{\hat{l}\in\mathcal{L}_{k}}\eta_{\hat{lk}}\) can be regarded to the sum of i.n.i.d. Gamma RVs.