# Guaranteeing QoS for NOMA-Enabled URLLC Based on \(\kappa\)-\(\mu\) Shadowed Fading Model
## 2 System Model
In addition, \(|h_{lk}|^{2}\) statistically follows the \(\kappa\)-\(\mu\) shadowed distribution, whose PDF can be expressed approximately by the Gamma distribution, namely, \({|h_{lk}|}^{2}\sim\Gamma(\mathcal{K}_{lk},\theta_{lk})\), where \(\mathcal{K}_{lk}=\frac{m_{lk}\mu_{lk}(1+\kappa_{lk})^{2}}{m_{lk}+\mu_{lk}x_{ lk}^{2}+2m_{lk}x_{lk}}=\frac{\mu_{lk}(1+\kappa_{lk})^{2}}{1+\mu_{lk}x_{lk}^{2}/m_{lk} +2\kappa_{lk}}\); \(\theta_{lk}=\frac{\Omega_{lk}}{\mathcal{K}_{lk}},\Omega_{lk}\) is the mean value of \({|h_{lk}|}^{2}\). The received signal at the BBU under imperfect CSI can be denoted as in the following equation. \[\left[\mathbf{Y}^{(\mathbf{p})}(t),\mathbf{Y}^{(\mathbf{d})}(t)\right]=\sqrt {p_{\mathbf{u}}}\mathbf{G}(t)\Big{[}\mathbf{X}^{(\mathbf{p})}(t),\mathbf{X}^ {(\mathbf{d})}(t)\Big{]}+\Big{[}\mathbf{Z}^{(\mathbf{p})}(t),\mathbf{Z}^{( \mathbf{d})}(t)\Big{]}, \tag{3}\]  
Figure 1: CF mMIMO system under the \(\kappa\)-\(\mu\) shadowed fading model. where \(p_{\rm u}\) is the average transmit power of each user. \(\mathbf{Y}^{(\rm p)}(t)\in\mathbb{C}^{L\times n}\) and \(\mathbf{Y}^{(\rm d)}(t)\in\mathbb{C}^{L\times\hat{n}}\) are received pilot signal and received data signal transmitted by the users at time slot \(t\), respectively. \(n\) and \(\hat{n}\) are the length of the pilots (LoP) and the length of the data signal, respectively, which satisfy \(n+\hat{n}=N=Bt_{f}\). \(\mathbf{X}^{(\rm p)}(t)=\left[\mathbf{x}_{1}^{(\rm p)}(t),\ldots,\mathbf{x}_{K }^{(\rm p)}(t)\right]^{T}\in\mathbb{C}^{K\times n}\) is the transmitted pilot signal at slot \(t\), which satisfies \(\mathbf{X}^{(\rm p)}(t)\Big{(}\mathbf{X}^{(\rm p)}(t)\Big{)}^{H}=n\mathbf{I}_ {K}\). \(\mathbf{X}^{(\rm d)}(t)=\left[\mathbf{x}_{1}^{(\rm d)}(t),\ldots,\mathbf{x}_{K }^{(\rm d)}(t)\right]^{T}\in\mathbb{C}^{K\times\hat{n}}\) is the transmitted data signal at time slot \(t\), obtained by assuming that the elements of \(\mathbf{X}^{(\rm d)}(t)\) are not related to each other, namely \(\mathbb{E}\bigg{(}\left|\mathbf{x}_{k}^{(\rm d)}(t)\right|^{2}\bigg{)}=n\), \(\mathbb{E}\bigg{(}\left(\mathbf{x}_{k}^{(\rm d)}(t)\right)^{T}\mathbf{x}_{j}^ {(\rm d)}(t)\bigg{)}=0\), \(\forall k,j\in[1,K]\) and \(k\
eq j\). \(\mathbf{G}(t)=\left[\mathbf{g}_{1}(t),\ldots\mathbf{g}_{K}(t)\right]\in \mathbb{C}^{L\times K}\) is the CSI of \(K\) users and \(L\) APs, where \(\mathbf{g}_{k}(t)=\left[g_{1k}(t),\ldots g_{Lk}(t)\right]^{T}\), \(\forall k\in[1,K]\). As we mentioned above, the expression for the channel coefficient is \(g_{lk}(t)=\sqrt{\beta_{lk}(t)}h_{lk}\), \(\forall k\in[1,K]\), and \(\forall l\in[1,L]\), which can be simplified as \(g_{lk}=\sqrt{\beta_{lk}}h_{lk}\), \(\mathbf{g}_{k}(t)\equiv\mathbf{g}_{k}\) and \(\mathbf{G}(t)\equiv\mathbf{G}\) because the large-scale fading coefficient \(\beta_{lk}(t)\) and small-scale fading coefficient \(h_{lk}(t)\) are generally assumed to be RVs that are independent of time. \(\mathbf{Z}^{(\rm p)}(t)\) and \(\mathbf{Z}^{(\rm d)}(t)\) represent additive white Gaussian noise (AWGN), whose elements are i.i.d. ; this can be expressed as \(\mathcal{CN}(0,1)\). In a practical communication system, serving users with a large number of APs requires accurate CSI to be available. The estimated channel matrix \(\mathbf{\hat{G}}\) obtained via the least squares estimation method which relies on the received signal with the known pilot sequence is expressed as [52]  
\[\mathbf{\hat{G}}=\mathbf{\Delta G}+\mathbf{\hat{E}}, \tag{4}\]  
where \(\mathbf{\hat{G}}=\left[\mathbf{\hat{g}}_{1},\ldots\mathbf{\hat{g}}_{K}\right] \in\mathbb{C}^{L\times K}\), \(\mathbf{\Delta}=\left[\mathbf{\Lambda}_{1},\ldots,\mathbf{\Lambda}_{K} \right]^{T}\in\mathbb{C}^{K\times L}\), \(\mathbf{\Lambda}_{k}=\left[\Lambda_{1k},\ldots,\Lambda_{LK}\right]^{T}\) and \(\mathbf{\Lambda}_{lk}=\sqrt{\frac{np_{\rm u}\beta_{lk}\Omega_{lk}}{1+np_{\rm u }\beta_{lk}\Omega_{lk}}}\). \(\beta_{lk}\) and \(\Omega_{lk}\) are the large-scale fading coefficient and mean square value of the small-scale fading coefficient, respectively.