## 6 Conclusions  
In this study, we have determined a closed-form analytical representation of the downlink exposure of a 5G massive MIMO network distributed following a PPP with realistic transmission gain and channel representation using statistical distributions instead of approximated value. We have analyzed the distribution of the exposure for different implementation scenarios and we have shown the impact that the network characteristics have on the exposure in the cell. This approach allows the accurate study of the massive MIMO network without the need for costly simulations. We have also shown the significance of using a realistic antenna model as compared to a simple one. We also studied the significance of the key parameters in the network, and we showed the importance the network usage has on the total exposure and the importance of considering it when conducting exposure analysis. This approach can be extended in future work to analyze different deployment scenarios of massive MIMO (e.g., cell-free, IoT... etc.). It can also benefit from realistic data representing the transmission gain after the deployment of the 5G massive MIMO antennas into the 5G network. Conceptualization, M.A.H., S.W. and J.W. ; Methodology, M.A.H., S.W. and L.T.T. ; Software, M.A.H., S.W., and S.A.; Writing--Original Draft Preparation, M.A.H. and S.W. ; Writing--Review and Editing, S.W. and J.W. ; Supervision, S.W. and J.W. ; Project Administration, J.W. All authors have read and agreed to the published version of the manuscript. This research was funded partially by the Agence Nationale des Frequences (ANFR) (CONVENTION 83 ANFR 2016). The authors declare no conflict of interest. The total exposure assuming the emission as shot noise process \(f(K,r)=Kr^{-\eta}\) for \(A<r<B\), and poison distribution of transmitters, can be written as \(P_{rx}^{t}(r)=P_{tx}\sum_{\text{BS}_{i}\in\text{FBS}}K_{i}r_{i}^{-\eta}\) continuing from Equation (1) for \(K_{i}=H_{i}G_{i}\). We let \(K_{i}\) be drawn from a discreet set \(\{K_{i}\}\) so that the shot noise process can be written as sum of independent shot noise processes \(P_{rx}^{i}(r)=P_{tx}\sum_{j}f\big{(}K_{i},r-r_{j}\big{)}\). The MGF of \(P_{rx}^{i}\) can be determined by solving Equation (A1). As per the result in [27], we can obtain the form in Equation (A3) for a continuous set of \(K_{i}\), and using the probability generating functional (PFGL) of a poisson point process. Simplification by integrating by parts gives the form in equation  
\[\varphi(s)=\mathbb{E}\Big{[}e^{-sP_{rx}^{i}(r)}\Big{]}\] (A1)  
\[=\mathbb{E}_{K}\Bigg{[}\exp\Bigg{(}-sP_{tx}\sum_{\text{BS}_{i}\in\Psi_{\text{ NB}}}\sum_{j}K_{i}\big{(}r-r_{j}\big{)}^{-\eta}\Bigg{)}\Bigg{]}\] (A2)  
\[\overset{\text{PGFL}}{=}\exp\Bigg{\{}-\mu\int_{A}^{B}\mathbb{E}_{K}[1-\exp(- sP_{tx}Kr^{-\eta})]\ dr\Bigg{\}}\] (A3)  
The MGF of the exposure then can be represented by the product of the two independent MGFs \(\varphi(s)=\varphi_{\text{in}}(s)\varphi_{\text{out}}(s)\), where \(\varphi_{\text{in}}\) can be obtained by substituting \(A=0\), and \(B=R_{0}=1\) in Equation (13). Likewise, for \(A=1\) and \(B=\infty\), \(\varphi_{\text{out}}\) can be expressed by  
\[\varphi_{out}(s)=\exp\Big{(}\pi\lambda\alpha\left[\mathbb{E}_{K}\big{[}1-e^{-sP _{tx}K}\big{]}-s^{2/\eta}\ \mathbb{E}_{K}\big{[}\big{(}P_{tx}K\big{)}^{2/\eta}\big{]}\Gamma(1-2/\eta)+s^{ 2/\eta}\ \mathbb{E}_{K}\big{[}\big{(}P_{tx}K\big{)}^{2/\eta}\mathbb{T}(1-2/\eta,sP_{ tx}K)\big{]}\right]\Big{)}\] (A4)  
The expression of \(\varphi_{out}\), Equation (A4) can be rewritten into Equation (A5), replacing the expectations with their integral forms. Each component can then be determined as presented below to obtain the closed-form expression  
\[ \begin{split}\varphi_{out}(s)=\exp\Bigg{(}\pi\lambda\alpha\left[1 -\int_{0}^{\infty}e^{-sdP_{tx}}f_{K}(k)dk-\Gamma(1-2/\eta)(sP_{TX})^{2/\eta} \int_{0}^{\infty}k^{2/\eta}f_{K}(k)dk\right.\\ \left.+(sP_{tx})^{2/\eta}\ \int_{0}^{\infty}k^{2/\eta}\Gamma(1-2/ \eta,sdP_{tx})\ f_{K}(k)dk\right]\Bigg{)}\end{split}