## 3 Exposure Estimation  
In this section, we derive the analytical framework based on the system model of the 5G network given in Section 2. Since the distribution of \(P_{rx}^{t}\) is unknown, we adopt the Gil-Pelaez inversion theorem [30] to compute the CDF defined in Equation (2), which is based on characteristic function of random variable \(P_{rx}^{t}\). The CDF \(P_{rx}^{t}\) is given by  
\[F_{P_{rx}^{t}}(x)=\frac{1}{2}-\frac{1}{\pi}\int_{0}^{\infty}\frac{1}{t}Im\Big{[} e^{-j\pi t}\Phi_{P_{rx}^{t}}(t)\Big{]}dt \tag{12}\]  
where \(\Phi_{P_{rx}^{t}}(t)\) represents the characteristic function of \(P_{rx}^{t}\), which can be derived from moment-generating function as \(\Phi_{P_{rx}^{t}}(t)~{}=~{}\varphi_{P_{rx}^{t}}(jt)\). The moment generating function (MGF) of a random variable, defined as \(\varphi_{X}(x):=\mathbb{E}\Big{[}e^{iX}\Big{]},~{}\forall~{}t\in\mathbb{R}\), is an alternative representation of the random variable other than its probability distribution. It can be used to determine the distribution's moments as \(\mathbb{E}[X^{n}]=\frac{d^{n}\varphi_{X}(x)}{dx^{n}}\Big{|}_{0},~{}\forall~{}n \in\mathbb{R}\). Since the expected value of the distribution is determined by derivation of the MGF in comparison to integrating the probability density, using the MGF becomes more attractive for complicated random variables thanks  
Figure 4: Channel gain fit into a gamma distribution for MTs uniformly distributed on a 2D plane and having a single element antenna array. Figure 3: Plot of the fit of the antenna array gain into an exponential distribution for a rectangular uniform antenna array with \(N_{tx}=256\) antenna elements. to the simpler operation. Following the path loss model defined in Equation (3), the total power at the \(MT_{0}\) can then be represented by: \(P_{rx}^{t}(t)=\sum_{i\in\Pi}\mathcal{B}_{i}f(K_{i},r_{i})\). Where \(\mathcal{B}_{i}\) is a set of random variables such that \(P(\mathcal{B}_{i}=1)=1-P(\mathcal{B}_{i}=0)=\alpha\). Since we divided the space into two regions, the total power received in the whole region can be written as \(P_{rx}^{t}=P_{rx}^{in}+P_{rx}^{out}\) where \(P_{rx}^{in}\) is the total power received from transmitters within distance \(R_{0}\) from the MT, and \(P_{rx}^{out}\) is the total received power from transmitters at distances greater than \(R_{0}\). To account for transmission gain, we assume that \(K_{i}\) is drawn from the distribution having the PDF: \(f_{K}(k)=f_{G}(g).f_{H}(h)\). Where, G and H are the antenna gain and the channel gain distributions, respectively. As mentioned in 2.2, we fit the antenna gain into an exponential distribution with PDF \(f_{G}(g)=ae^{-qg}\), and the channel gain into a gamma distribution with PDF \(f_{H}(h)=\frac{1}{\Gamma(k)\theta}t^{k-1}e^{-h/c}\). The MGF of this model is presented in Theorem 1. **Theorem 1**. : _The MGF of the exposure in the network modeled in Section 2, given the array gain and channel gain in (10) and (11) is formulated as:_  
\[\varphi(s)=\exp(\pi\lambda\alpha\Theta), \tag{13}\]  
_where_  
\[ \begin{array}{ll}\Theta=1-\left(\frac{a}{P}\right)^{b}e^{\frac{a}{P}}\Gamma& \left(1-b,\frac{a}{P}\right)-\pi\frac{2}{\eta}\left(\frac{\widetilde{P}}{a} \right)^{\frac{2}{\eta}}\frac{\csc(\pi\frac{a}{P})\Gamma(b+\frac{2}{\eta})}{ \Gamma(b)}\\ &+(-1)^{b}\frac{\pi nb}{\Gamma(b)}\left(-\frac{c}{a}\right)^{\frac{2}{\eta}} \csc(\pi b)\gamma\left(b+\frac{2}{\eta},-\frac{a}{P}\right)\\ &+\frac{\eta}{2\Gamma(b)}\left[-\left(\frac{c}{x}\right)^{\frac{1}{2}}\left( (-1)^{b}\frac{2}{\eta}\left(-\frac{x}{a}\right)^{\frac{2}{\eta}}\pi\csc(\pi b )\gamma\left(1+b+\frac{2}{\eta},-\frac{a}{P}\right)\right.\right.\\ &\left.\left.-\left(\frac{xc}{aP}\right)^{\frac{2}{\eta}}\Omega(b+1,)\right)+ \Omega(b)\right]\\ \end{array}