Analytical Approach to Uncertainty Evaluation of Exposure Index—Metric for Assessing Human Population Exposure Induced by a Wireless Cellular Network

Abstract

The rapid development of wireless technologies requires reliable assessment of electromagnetic radiation. Under the scope of the project LEXNET, a novel metric named the Exposure Index (EI) was defined to assess the mean exposure of a human population within a geographical area. The EI considers exposure from both base stations/access points and wireless user devices, thereby assessing the total exposure originating from the wireless cellular network. Considering that every measurement result should have an associated uncertainty, this paper proposes an analytical method for evaluating the uncertainty of the EI. Advancing beyond standard GUM evaluations that require scenario-specific derivations of complex partial derivatives, the proposed procedure provides a unified, first-order analytical approximation. Assuming that all input parameters are independent and have small relative standard uncertainties, this framework yields a closed-form relative expression to instantly calculate the combined relative standard uncertainty of the EI. Furthermore, if the determination of expanded uncertainty is required, we propose a method to estimate the probability distribution function of the EI. Assuming that all input parameters of the EI are normally distributed with small uncertainties, we demonstrate that the EI probability distribution can be considered normal. This method provides a highly convenient way to calculate the relative standard uncertainty of the EI and to determine the coverage factor for the relative expanded uncertainty.

1. Introduction

Increasing user demand for accessible, high-quality telecommunication services has led to the tremendous development of wireless communication devices and systems. Due to their intensive deployment, mobile networks (GSM—Global System for Mobile Communications, UMTS—Universal Mobile Telecommunication System, LTE—Long Term Evolution, and 5G NR–5G New Radio) are the most prominent. Additionally, the widespread use of Wireless Local Area Networks (WLANs) is equally significant. With the rapid growth in the number of wireless devices, additional electromagnetic (EM) radiation has been introduced into the modern human environment. Consequently, public concern regarding exposure to EM radiation emitted by these sources has increased [1,2]. Given that wireless communication devices are an integral part of modern life and cannot be easily replaced or removed, an objective assessment of radiofrequency (RF) EM radiation in the human environment is of great importance. Furthermore, human exposure to RF EM fields must be assessed on a regular basis due to the projected intensive growth of wireless traffic [3], the frequent installation of small cell base stations (BS) [4], and the intensive deployment of 5G networks [5].

Authors in [1] have shown that the public is more concerned about BSs than about all other RF EM field sources. Exposure originating from public mobile BSs has been considered in several papers [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. In recent works, authors more often jointly take into account exposure originating from BSs/access points (AP) and exposure originating from user devices [21,22,23,24,25,26]. Moreover, the European Union 7th Framework Programme project LEXNET (Low-EMF Exposure Future Networks) defined a new exposure metric for the evaluation of mean population exposure in the analyzed geographical area, taking into account the exposure originating from BSs and APs as well as exposure induced by wireless user devices [27]. This novel exposure metric, called the Exposure Index (EI) [28], considers many parameters influencing exposure, such as age, posture, usage, technology, and environment. The EI takes into account downlink (DL) exposure (exposure originating from BSs and APs) using average incident power density, and uplink (UL) exposure (exposure originating from wireless user devices) using the average transmitting power of user devices.

According to [29,30], the evaluation of measurement uncertainty is one of the most important components of all metrology applications; the results of measurements must include an uncertainty statement. Without uncertainty, it is not possible to compare measurement results with each other or with reference values specified in standards. Consequently, to evaluate the quality of a measurement result, it is necessary to have a procedure for evaluating and expressing its uncertainty. Measurement uncertainty is a parameter associated with the measurement result, which characterizes the dispersion of the values that could reasonably be attributed to the measurand. Measurement uncertainty is usually determined and expressed as (absolute) standard uncertainty $u\left(y\right)$(the uncertainty of a measurement result $y$ expressed as a standard deviation ${\sigma }_y$) [30]. Relative standard uncertainty $u_r\left(y\right)$ is the standard uncertainty divided by the absolute value of the measured value $y, u_r\left(y\right) =$ $u\left(y\right)/y={\sigma }_y/y$ [31].

The main goal of this paper is to develop a procedure for evaluating the uncertainty of the EI [28], based on the known uncertainties of multiple input parameters. Several approaches for uncertainty evaluation are available:

(a)

The Guide to the expression of Uncertainty in Measurement (GUM) uncertainty framework [30];

(b)

Monte Carlo method (MCM) [32,33];

(c)

Analytic methods.

The GUM uncertainty framework [30], being generally approximate, uses:

• The best estimates $x_i$ of the input quantities $X_i$;
• The standard uncertainties $u\left(x_i\right)$ of the $x_i$;
• The sensitivity coefficients $c_i$,

To determine an estimate $y$ of the final result $Y$ and the corresponding standard uncertainty $u\left(y\right)$:

$$u\left(y\right)=\sqrt{{\displaystyle \sum _{i=1}^N}{c}_i^2{u}^2\left(x_i\right)}$$

(1)

MCM provides a solution with controlled numerical accuracy. Using this method, an approximation of the probability distribution function (PDF) for $Y$ is established numerically by making random draws from the probability distributions of the input quantities and evaluating the model at the resulting values. Furthermore, MCM has fewer operational constraints than the GUM uncertainty framework. Ref. [32] introduces an adaptive MCM procedure in which the number of Monte Carlo trials is determined automatically by utilizing a convergence measure for the overall process.

Analytic methods are based on mathematical analysis used to derive the algebraic form of the probability distribution for the final result $Y$. While these methods do not introduce any approximations, they are applicable only to relatively simple cases. Practical applications of these methods can be found in [34,35].

In this paper, we assume that the measurement uncertainties of the input parameters for the calculation of the EI are known. Given that the analytical expression for the EI is available, utilizing an analytic method is the most appropriate approach to evaluate its uncertainty. The proposed procedure implements a first-order analytical error propagation approach that provides a closed-form algebraic expression to calculate the relative standard uncertainty of the EI using the known relative standard uncertainties of its input parameters. To match the inherent flexibility of the EI metric itself, this procedure is designed to be generic and adaptable, ensuring its applicability across different network and exposure scenarios.

Establishing a metrological framework for the EI requires determining its relative expanded uncertainty, $U_r$, to define an interval around the measurement result that is expected to encompass a large fraction of the distribution of values that could reasonably be attributed to the EI value [30]. A relative expanded uncertainty $U_r$ is conventionally obtained by multiplying the combined relative standard uncertainty $u_r\left(y\right)$ by an appropriately chosen coverage factor k, yielding the expression $U_r=k·u_r\left(y\right)$, which establishes a proportional confidence boundary around the central measurand value. This paper addresses the general methodology of deriving a relative expanded uncertainty $U_{r,p}=k_p·u_r\left(y\right)$ from the combined relative standard uncertainty $u_r\left(y\right)$ of an estimate $y$, thereby establishing a normalized interval around the measurement result that possesses a high, specified coverage probability or level of confidence $p$. Ultimately, this approach serves to determine the precise coverage factor $k_p$ required to produce a relative interval about the measurement result $y$ that encompasses a large, specified fraction $p$ of the distribution of values reasonably attributed to the measurand $Y$ [30,36]. Under a standard normal distribution assumption, coverage factors of $k$ = 2 and $k$ = 3 are universally adopted in compliance testing to correspond to widely recognized confidence levels of approximately 95% and 99%, respectively.

While the proposed first-order analytical approximation efficiently determines the relative standard uncertainty of the EI, proposed frameworks do not inherently provide information on the shape of the output PDF. Where complete knowledge of the output PDF is required, the MCM can be utilized as an alternative; however, numerical simulations remain complex and time-consuming because they must be executed for each individual localized value of the EI [30]. To address this limitation, we extend our analytical framework by proposing a method to estimate the explicit PDF of the EI. We mathematically demonstrate that when all input parameters are normally distributed with small relative uncertainties—which represents the typical case in practical wireless deployments—the resulting overall EI probability distribution can also be characterized as normal. This analytical proof simplifies compliance auditing, as it allows network engineers to instantly determine the coverage factor required to compute the expanded uncertainty without relying on computational simulations.

To summarize, to the authors’ best knowledge, this work represents the first comprehensive framework addressing the uncertainty evaluation of the EI. To match the inherent flexibility of the EI metric itself, the proposed procedure is designed to be generic and adaptable, ensuring its applicability across different network configurations and exposure scenarios. Consequently, this approach offers two primary contributions to the field of electromagnetic dosimetry:

• A Closed-Form Metrological Simplification: While standard GUM evaluations of complex multi-source metrics require the scenario-specific derivation of numerous individual partial derivatives, this work derives a unified, first-order analytical approximation. This replaces matrix-heavy sensitivity coefficients with an elegant, relative closed-form expression, allowing for instantaneous uncertainty budgeting without losing metrological traceability.
• An Analytical Output Distribution Model: We establish the explicit boundary conditions and quantitative thresholds under which the output PDF of the EI converges to a normal distribution. This provides field engineers with a direct, mathematically justified method to determine exact coverage factors ($k_p$) for expanded uncertainty, completely bypassing the need for computationally intensive Monte Carlo simulations.

2. Exposure Index Definition

As previously noted, the EI is a new metric used to evaluate the mean global human population exposure within a geographic area originating from a wireless cellular network as a whole [28]. The EI accounts for DL exposure from BSs and APs using the average incident power density, as well as UL exposure from user devices using their average transmitting power. Additionally, the EI can incorporate exposure originating from nearby devices operated by other users as part of the DL exposure. However, because this surrounding user exposure has a negligible impact on EI values, it is omitted from the analysis in this paper.

Briefly, the EI represents a function used to map a highly complex dataset into a single parameter that is easy to understand, interpret, and use for all stakeholders (from the general public to regulators). For the calculation of EI values, the analytical expression (2) is provided in [28]:

$${EI}^{\mathrm{SAR}}={\displaystyle \frac{1}{\mathrm{T}}}{\displaystyle \sum _t^{N_T}}{\displaystyle \sum _p^{N_P}}{\displaystyle \sum _e^{N_E}}{\displaystyle \sum _r^{N_R}}{\displaystyle \sum _c^{N_C }} {\displaystyle \sum _l^{N_L}}{\displaystyle \sum _{pos}^{N_{pos}}}{f}_{t,p,e,r,l,c,pos}\left[{\displaystyle \sum _u^{N_U}}\left(d^{UL}{ \overline{P}}_{TX}\right)+d^{DL}{ \overline{S}}_{\mathit{inc}}\right] \left[{\displaystyle \frac{\mathrm{W}}{\mathrm{k}\mathrm{g}}}\right]$$

(2)

where

• ${EI}^{\mathrm{SAR}}$ represents the EI value based on the Specific Absorption Rate (SAR), which reflects the mean population exposure for the analyzed geographical area over the analyzed time interval T;
• $N_i$ represents the number of considered cases for: $N_T$—time periods within the T, $N_P$—Population categories, $N_E$—Environments, $N_R$—Radio access technologies (RAT), $N_C$—Cell types, $N_L$—User load profiles, $N_{pos}$—Postures and $N_U$—Usages with devices;
• ${\overline{P}}_{TX}$ represents the average transmit power of all user devices in analyzed area, for the time interval $t$, usage $u$, connected to RAT $r$, in environment $e$;
• ${\overline{S}}_{\mathit{inc}}$ represents the incident power density on the human body average over analyzed area, for the time interval $t$, originating from RAT $r$, in environment $e$;
• $d^{UL} [{\displaystyle \frac{\mathrm{W}\mathrm{s}}{\mathrm{k}\mathrm{g}}}/\mathrm{W}$] and $d^{DL}$ [${\displaystyle \frac{\mathrm{W}\mathrm{s}}{\mathrm{k}\mathrm{g}}}/{\displaystyle \frac{\mathrm{W}}{{\mathrm{m}}^2}}$] represent the normalised values of dose for UL (from user devices) and DL (from BSs and APs), multiplied by the time spent in the configuration;
• $f_{t,p,e,r,l,c,pos}$ represents the fraction of the population that corresponds to category $p$, user load profile $l$, in posture $pos$, connected to RAT $r$, for a cell type $c$, in environment $e,$ during the time period $t$.

Coefficient $d^{UL}$ is associated with the exposure originating from the UL and expressed as an absorbed dose normalized to a transmitted power $P_{TX}^{ref}$ of 1 $\mathrm{W}$:

$$d^{UL}={\displaystyle \frac{{TD}_{t, p, l, e,r, c, u,pos}^{UL}{\left[\mathrm{s}\right]·SAR}_{ p, r, u, pos}^{UL}[\mathrm{W}/\mathrm{k}\mathrm{g}]}{P_{TX}^{ref}[\mathrm{W}]}} \left[{\displaystyle \frac{\mathrm{W}\mathrm{s}}{\mathrm{k}\mathrm{g}}}/\mathrm{W}\right]$$

(3)

where

• $T{D}_{t,p,l,e,r,c,u,pos}^{UL}$ represents the duration of the UL traffic for usage $u$, user profile load $l$, connected to the RAT $r$, in cell type $c$, in environment $e$, for the category of population $p$, in the posture $pos$, for the time period of the day $t$;
• ${\displaystyle \frac{SA{R}_{p,r,u,pos}^{UL}}{P_{TX}^{ref}}}$ represents the $SAR$ value for the usage $u$ and the posture $pos$, in the frequency band of the RAT $r$, and the category of population $p$, calculated and normalized for an emitted power of $P_{TX}^{ref}$.

Coefficient $d^{DL}$ is associated with the exposure originating from the DL and also expressed as an absorbed dose normalized to an incident power density $S_{\mathit{inc}}^{ref}$ of 1 W/m²:

$$d^{DL}={\displaystyle \frac{{TD}_{t,p,e,r,c,pos}^{DL}[\mathrm{s}]·{SAR}_{p,r,pos}^{DL}[\mathrm{W}/\mathrm{k}\mathrm{g}]}{S_{\mathit{inc}}^{ref}[\mathrm{W}/{\mathrm{m}}^2]}} \left[{\displaystyle \frac{\mathrm{W}\mathrm{s}}{\mathrm{k}\mathrm{g}}}/{\displaystyle \frac{\mathrm{W}}{{\mathrm{m}}^2}}\right]$$

(4)

where

• $T{D}_{t,p,e,r,c,pos}^{DL}$ represents the duration in posture $pos$, exposed to the RAT $r$, in cell type $c$, in the environment $e$, for the population $p$, for the time period of the day $t$;
• ${\displaystyle \frac{{SAR}_{p,r,pos}^{DL} }{S_{inc}^{ref}}}$ represents the $SAR$ value originating from the BS or APs of the RAT $r$, in the population $p$, for the posture $pos$, normalized to the received power density $S_{inc}^{ref}$.

3. Arithmetic Propagation of Uncertainty

When a final measurement result depends on multiple input parameters, the law of propagation of uncertainty must be applied. This framework defines the mathematical effect that input variable uncertainties have on the final result, combining them to determine its total uncertainty [30]. Consequently, whenever a calculation requires multiple variables, this propagation procedure is essential for properly determining the combined standard uncertainty.

Given that the analytical expression for calculating EI values relies on sequential multiplication and addition, it is necessary to establish the relationships between the standard uncertainties of the input parameters and the final result for both additive and multiplicative combinations.

According to [37,38], for mutually independent random variables, addition and subtraction lead to absolute standard deviation (combined absolute standard uncertainty), whereas multiplication and division yield relative standard deviations (combined relative standard uncertainty).

It is important to note that the localized arithmetic formulas utilized in this section represent particular algebraic solutions derived directly from the general GUM law of uncertainty propagation specified in Equation (1). By explicitly evaluating the partial derivatives (sensitivity coefficients) for linear sums and multiplicative products of independent variables, the generalized framework naturally simplifies into classical absolute and relative forms. Consequently, this partial derivative approach yields mathematically identical numerical results to the simplified equations, ensuring rigorous theoretical alignment with standard GUM principles while significantly optimizing computational tractability for practical field deployment.

To evaluate how input measurement parameters impact a complex experimental measurement result, consider a scenario where n independent measurement instruments are utilized simultaneously. Let the direct readings from these individual source parameters be random variables $x_1,x_2,\dots ,x_n$. The target parameter of interest, $Y$, is functionally dependent on these inputs such that:

$$Y=f(x_1,x_2,\dots ,x_n)$$

(5)

Each parameter possesses an inherent random deviation relative to its statistical mean. The differential increment $\Delta Y_i$ for the i-th observation of the output variable can be modeled via a first-order Taylor series expansion (linearization) using partial derivatives with respect to each constituent input parameter:

$$\Delta Y_i={\displaystyle \sum _{j=1}^n}\left({\displaystyle \frac{\partial f}{\partial x_j}}\right)\Delta x_{j,i}$$

(6)

where $\Delta x_{j,i}$ represents the deviation of the $j$-th variable during the $i$-th measurement. To establish a functional relationship connecting the variances of these parameters, Equation (6) is squared, and a summation is performed across all observations from $i$ = 1 to $i$ = $N$, where the total sample size $N$ is assumed to be sufficiently large:

$${\displaystyle \sum _{i=1}^N}{\left(\Delta Y_i\right)}^2={\displaystyle \sum _{i=1}^N}\left[{\displaystyle \sum _{j=1}^n}{\left({\displaystyle \frac{\partial f}{\partial x_j}}\right)}^2{\left(\Delta x_{j,i}\right)}^2+{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k\ne j}^n}\left({\displaystyle \frac{\partial f}{\partial x_j}}\right)\left({\displaystyle \frac{\partial f}{\partial x_k}}\right)\Delta x_{j,i}\Delta x_{k,i}\right]$$

(7)

The expansion on the right-hand side of Equation (7) separates cleanly into two distinct algebraic categories:

$$\mathrm{Quadratic}\mathrm{terms}:{\displaystyle \sum _{j=1}^n}{\left({\displaystyle \frac{\partial f}{\partial x_j}}\right)}^2{\left(\Delta x_{j,i}\right)}^2$$

(8)

$$\mathrm{Cross}\text{-}\mathrm{product}\mathrm{terms}:{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k\ne j}^n}\left({\displaystyle \frac{\partial f}{\partial x_j}}\right)\left({\displaystyle \frac{\partial f}{\partial x_k}}\right)\Delta x_{j,i}\Delta x_{k,i}$$

(9)

The quadratic terms represent squared values and are strictly non-negative, meaning they accumulate monotonically. Conversely, the cross-product terms can take either positive or negative values depending on the directional fluctuation of the errors around their respective means. Under the core assumption that the source inputs represent random, uncorrelated, and mutually independent uncertainty distributions, these cross-products will naturally cancel out on average. Consequently, their cumulative summation approaches zero as the sample size $N$ grows large. Given that the input parameters of the EI are treated as mutually independent in this baseline architecture, this cancellation condition is assumed to hold true. However, if the variables are correlated rather than independent, the cross term may not cancel out.

Applying this independence simplification allows Equation (7) to reduce to:

$${\displaystyle \sum _{i=1}^N}{\left(\Delta Y_i\right)}^2={\displaystyle \sum _{j=1}^n}{\left({\displaystyle \frac{\partial f}{\partial x_j}}\right)}^2{\displaystyle \sum _{i=1}^N}{\left(\Delta x_{j,i}\right)}^2$$

(10)

Dividing both sides of this reduced expression by the degrees of freedom ($N$ − 1) yields:

$${\displaystyle \frac{{\displaystyle {\sum }_{i=1}^N}{\left(\Delta Y_i\right)}^2}{N-1}}={\displaystyle \sum _{j=1}^n}{\left({\displaystyle \frac{\partial f}{\partial x_j}}\right)}^2{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^N}{\left(\Delta x_{j,i}\right)}^2}{N-1}}$$

(11)

By standard statistical definition, the left-hand side represents the variance (${{\sigma }_Y}^2$) of the target variable $Y$:

$${\displaystyle \frac{{\displaystyle {\sum }_{i=1}^N}{\left(\Delta Y_i\right)}^2}{N-1}}={\displaystyle \sum _{j=1}^n}{\left({\displaystyle \frac{\partial f}{\partial x_j}}\right)}^2{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^N}{\left(\Delta x_{j,i}\right)}^2}{N-1}}={{\sigma }_Y}^2$$

(12)

Mapping this same definition to the right-hand terms establishes the standard generalized first-order analytical framework for uncertainty propagation:

$${{\sigma }_Y}^2={\displaystyle \sum _{j=1}^n}{\left({\displaystyle \frac{\partial f}{\partial x_j}}\right)}^2{{\sigma }_{x_j}}^2$$

(13)

Equation (13) provides a direct, scalable relationship between independent source variances and the combined output variance. This expression serves as a first-order analytical approximation, operating under the structural constraint that individual relative standard uncertainties remain sufficiently small to justify neglecting higher-order Taylor expansion terms.

Since the structural formulation of the EI relies on sequential combinations of products and sums of these products, specific operational expressions derived from the generalized propagation law in Equation (13) are defined below.

For strictly additive or subtractive linear combinations, where $Y={\sum }_{j=1}^n{c}_j{x}_j$ (with coefficients $c_j=\pm 1$), the resulting absolute standard deviation ${\sigma }_Y$ simplifies to:

$${\sigma }_Y=\sqrt{{\displaystyle \sum _{j=1}^n}{\sigma }_{x_j}^2}$$

(14)

For purely multiplicative or divisive exponential relationships, where Y = ${\displaystyle {\prod }_{j=1}^n}{x}_j^{p_j}$ (with powers $p_j=\pm 1$), evaluating the partial derivatives transforms Equation (13) into a summation of relative terms. The propagated relative standard deviation ${\sigma }_Y/Y$ is formulated as:

$${\displaystyle \frac{{\sigma }_Y}{Y}}=\sqrt{{\displaystyle \sum _{j=1}^n}{\left({\displaystyle \frac{{\sigma }_{x_j}}{x_j}}\right)}^2}$$

(15)

If we assume input parameters of the EI are mutually independent, these propagation rules can be applied to evaluate the EI uncertainty. Although real-world cellular network deployment parameters exhibit operational and architectural dependencies—such as the population fraction scaling simultaneously both uplink (UL) and downlink (DL) terms, or peak network load correlating with traffic duration (TD), transmit power ($P_{TX}$), and incident power density ($S_{inc}$)—this framework treats all input variables as mutually independent. According to the GUM, input quantities can be considered independent if no significant correlation exists between them, or if any existing correlation has a negligible effect on the combined uncertainty. This first-order analytical approximation is robustly justified by three distinct macro-statistical factors:

• Multi-System Operational Decoupling: Parameters are derived from entirely different technical systems, corporate departments, and procedural workflows. Active population fractions are managed using demographic data; traffic durations and load profiles are logged by optimization teams via Operations Support Systems (OSS) telemetry; and SAR coefficients are evaluated by regulatory compliance teams using laboratory dosimetry. These distinct, non-synchronized acquisition pathways naturally isolate measurement uncertainties, satisfying the GUM condition for independent data sources.
• Macro-Statistical Averaging: The input quantities are not high-frequency, instantaneous physical telemetry values subject to rapid environmental fluctuations. Instead, they represent macroscopic data temporally and spatially averaged over specified population fractions. Applying the Central Limit Theorem, this large-scale aggregation smooths out localized transient variations and real-time physical couplings, rendering the averaged multi-source inputs macro-statistically decoupled.
• Two-Stage Hierarchical Mitigation: The uncertainty model operates as a two-stage hierarchical propagation. First, individual parameters are multiplied to yield the intermediate composite exposure terms. Second, these intermediate terms are linearly summed for the final EI. This initial multiplicative step compresses parameter variations into distinct localized terms, significantly mitigating residual cross-correlations before global aggregation.

4. Evaluation of Exposure Index Relative Standard Uncertainty

The expression for the EI calculation, Equation (2), can be written in an alternative form:

$${EI}^{\mathrm{SAR}}={\displaystyle \frac{1}{\mathrm{T}}}{\displaystyle \sum _t^{N_T}}{\displaystyle \sum _p^{N_P}}{\displaystyle \sum _e^{N_E}}{\displaystyle \sum _r^{N_R}}{\displaystyle \sum _c^{N_C }} {\displaystyle \sum _l^{N_L}}{\displaystyle \sum _{pos}^{N_{pos}}}\left[{\displaystyle \sum _u^{N_U}}\left(A\right)+B\right] \left[{\displaystyle \frac{\mathrm{W}}{\mathrm{k}\mathrm{g}}}\right]$$

(16)

where

$$A=f_{\mathrm{t},\mathrm{p},\mathrm{e},\mathrm{r},\mathrm{l},\mathrm{c},\mathrm{p}\mathrm{o}\mathrm{s}}{\displaystyle \frac{{TD}_{t,p,l,e,r,c,u,pos}^{UL}\left[\mathrm{s}\right]·{SAR}_{p,r,u,pos}^{UL} [\mathrm{W}/\mathrm{k}\mathrm{g}]}{P_{TX}^{ref}[\mathrm{W}]}}{ \overline{P}}_{TX}\left[\mathrm{W}\right]$$

(17)

$$B=f_{\mathrm{t},\mathrm{p},\mathrm{e},\mathrm{r},\mathrm{l},\mathrm{c},\mathrm{p}\mathrm{o}\mathrm{s}}{\displaystyle \frac{{TD}_{t,p,e,r,c,pos}^{DL}\left[\mathrm{s}\right]·{SAR}_{p,r,pos}^{DL} [\mathrm{W}/\mathrm{k}\mathrm{g}]}{S_{\mathit{inc}}^{ref}[\mathrm{W}/{\mathrm{m}}^2]}}{ \overline{S}}_{\mathit{inc}}[\mathrm{W}/{\mathrm{m}}^2]$$

(18)

Let us denote the relative standard uncertainties of terms $A$ and $B$ with $u_r\left(A\right)$ and $u_r\left(B\right)$, respectively.

Given that $P_{TX[\mathrm{W}]}^{ref}$ and $S_{\mathit{inc}[\mathrm{W}/{\mathrm{m}}^2]}^{ref}$ are constants, and according to Equation (15), the relative standard uncertainties $u_r\left(A\right)$ and $u_r\left(B\right)$ can be determined as follows:

$$u_r\left(A\right)=\sqrt{{u_r}^2\left(f_{\mathrm{t},\mathrm{p},\mathrm{e},\mathrm{r},\mathrm{l},\mathrm{c},\mathrm{p}\mathrm{o}\mathrm{s}}\right)+{u_r}^2\left({TD}_{t,p,l,e,r,c,u,pos}^{UL}\right){+ u_r}^2\left({SAR}_{ p,r,u,pos}^{UL}\right)+{u_r}^2\left({ \overline{P}}_{TX}\right)}$$

(19)

$$u_r\left(B\right)=\sqrt{{u_r}^2\left(f_{\mathrm{t},\mathrm{p},\mathrm{e},\mathrm{r},\mathrm{l},\mathrm{c},\mathrm{p}\mathrm{o}\mathrm{s}}\right)+{u_r}^2\left({TD}_{t,p,e,r,c,pos}^{DL}\right){+u_r}^2\left({SAR}_{p,r,pos}^{DL}\right)+{u_r}^2\left({ \overline{S}}_{\mathit{inc}}\right)}$$

(20)

Now, according to Equation (14) EI standard uncertainty $u_r\left(\mathrm{E}\mathrm{I}\right)$ can be determined as follows:

$$u_r\left(\mathrm{E}\mathrm{I}\right)={\displaystyle \frac{\sqrt{{\displaystyle {\sum }_t^{N_T}}{\displaystyle {\sum }_p^{N_P}}{\displaystyle {\sum }_e^{N_E}}{\displaystyle {\sum }_r^{N_R}}{\displaystyle {\sum }_c^{N_C }} {\displaystyle {\sum }_l^{N_L}}{\displaystyle {\sum }_{pos}^{N_{pos}}}\left[{\displaystyle {\sum }_u^{N_U}}\left(A^2\ast {u_r}^2\left(A\right)\right)+B^2\ast {u_r}^2\left(B\right)\right] }}{{\displaystyle {\sum }_t^{N_T}}{\displaystyle {\sum }_p^{N_P}}{\displaystyle {\sum }_e^{N_E}}{\displaystyle {\sum }_r^{N_R}}{\displaystyle {\sum }_c^{N_C }} {\displaystyle {\sum }_l^{N_L}}{\displaystyle {\sum }_{pos}^{N_{pos}}}\left[{\displaystyle {\sum }_u^{N_U}}\left(A\right)+\mathrm{B}\right]}}$$

(21)

Equation (21) provides a closed-form algebraic expression to calculate the combined relative standard uncertainty of the EI based on the known relative standard uncertainties of its input parameters.

The practical operational boundary for this first-order analytical uncertainty propagation approximation is defined by an upper threshold where individual input parameters satisfy $u_r\left(x_i\right) \le$ 20%. Within a typical high-accuracy zone where $u_r\left(x_i\right)\le$ 10%. (as demonstrated in our Section 6 case study), omitting the higher-order Taylor series cross-product variance terms introduces a completely negligible error of only 0.50% to the total uncertainty budget. Even when expanding the model’s applicability to its maximum 20% boundary to handle higher real-world network deployment variations, a strict error-bound analysis reveals that the neglected higher-order terms account for a mere 1.96% of the total variance budget. Because an analytical approximation error under 2% is widely accepted in complex population-scale dosimetry, this closed-form relative expression remains robustly valid up to this 20% limit, beyond which the linearization breaks down and numerical Monte Carlo methods must be deployed instead.

5. Exposure Index Probability Distribution Function Assessment

The proposed analytical method for the EI relative standard uncertainty calculation does not provide information on the type of PDF. To obtain the relative expanded uncertainty $U_r$ of the EI or the appropriate coverage factor k, the PDF of the EI must be known [30,36]. This section addresses the estimation of the PDF for cases where the input parameters are normally distributed with large mean values and small standard deviations (i.e., with small relative standard uncertainties).

As demonstrated in the following section, assuming that all input parameters of the EI have small relative uncertainties allows the PDF of the EI to be approximated as normal. It should be emphasized that this assumption is typically the case in practical applications.

Within this framework, a relative standard uncertainty is mathematically classified as “small” satisfying a practical metrological upper threshold of $u_r\left(x_i\right)\le$ 10% (or 0.1) for high-quality network metrics, with an extended framework validity limit up to $u_r\left(x_i\right)\le$ 20% (or 0.2). At these low variance levels, the skewness of a log-normal distribution approaches zero, forcing its asymmetric profile to tightly converge to a symmetric normal distribution.

To prove this claim, we begin with the analytical expression for the EI in Equation (2). Structurally, the EI consists of sums of products of input parameters. It is well-established that the product of N mutually independent log-normal random variables also follows a log-normal distribution. This represents a specific application of a broader mathematical property where the logarithm of a product is equivalent to the sum of the logarithms of its factors. Furthermore, for random variables characterized by large mean values and small standard deviations (i.e., small uncertainties), a log-normal distribution asymptotically converges to a normal distribution. Consequently, the product of N mutually independent normal random variables with small uncertainties can also be approximated as normal. To verify this behavior analytically, we begin by examining the PDF of a log-normal distribution.

The PDF of a log-normal distribution is given by:

$$f\left(x\right)={\displaystyle \frac{1}{x\sigma \sqrt{2\pi }}}{e}^{-{\displaystyle \frac{{\left(\ln x-\mu \right)}^2}{2{\sigma }^2}}}, x>0$$

(22)

For log-normal distribution, the mean $\overline{x }$ and variance ${\sigma }_x^2$ are:

$$\overline{x }=e^{\mu +{\sigma }^2/2}$$

(23)

$${\sigma }_x^2=\left(e^{{\sigma }^2}-1\right)e^{2\mu +{\sigma }^2}$$

(24)

Let us divide ${\sigma }_x$ by $\overline{x }$:

$${\displaystyle \frac{{\sigma }_x}{\overline{x }}} = \sqrt{e^{{\sigma }^2}-1}$$

(25)

By assumption of small uncertainty: ${\displaystyle \frac{{\sigma }_x}{\overline{x }}} = \sqrt{e^{{\sigma }^2}-1}\ll 1$. Further, it means that $1<e^{{\sigma }^2}\ll 2$. Applying Taylor series expansion, expression (25) becomes:

$${\displaystyle \frac{{\sigma }_x}{\overline{x }}} \approx \sigma$$

(26)

Knowing that ${\displaystyle \frac{{\sigma }_x}{\overline{x }}} \approx \sigma \ll 1$, expression (23) becomes:

$$\overline{x }\approx e^{\mu }$$

(27)

On the other side, substituting $\mu$ with $\mu =\ln \overline{x }$ in expression (22):

$$f\left(x\right)={\displaystyle \frac{1}{x\sigma \sqrt{2\pi }}}{e}^{-{\displaystyle \frac{{\left(\ln x-\ln \overline{x }\right)}^2}{2{\sigma }^2}}}$$

(28)

$$f\left(x\right)={\displaystyle \frac{1}{x\sigma \sqrt{2\pi }}}{e}^{-{\displaystyle \frac{{\left(\ln \frac{x}{\overline{x }}\right)}^2}{2{\sigma }^2}}}$$

(29)

Knowing that ${\displaystyle \frac{x}{\overline{x }}}\approx 1$:

$$f\left(x\right)={\displaystyle \frac{1}{x\sigma \sqrt{2\pi }}}{e}^{-{\displaystyle \frac{{\left(\ln [1+\left(\frac{x}{\overline{x }}-1\right)]\right)}^2}{2{\sigma }^2}}}$$

(30)

$\left|{\displaystyle \frac{x}{\overline{x }}}-1\right|\ll 1$ and applying Taylor series expansion:

$$f\left(x\right)={\displaystyle \frac{1}{x\sigma \sqrt{2\pi }}}{e}^{-{\displaystyle \frac{{(\frac{x}{\overline{x }}-1)}^2}{2{\sigma }^2}}}$$

(31)

$$f\left(x\right)={\displaystyle \frac{1}{x\sigma \sqrt{2\pi }}}{e}^{-{\displaystyle \frac{{(x-\overline{x })}^2}{2{\sigma }^2{\overline{x }}^2}}}$$

(32)

$$f\left(x\right)={\displaystyle \frac{1}{\frac{x}{\overline{x }}\sigma \overline{x }\sqrt{2\pi }}}{e}^{-{\displaystyle \frac{{(x-\overline{x })}^2}{2{\sigma }^2{\overline{x }}^2}}}$$

(33)

Finally, using expression (26), expression (33) becomes:

$$f\left(x\right)={\displaystyle \frac{1}{\frac{x}{\overline{x }}{\sigma }_x\sqrt{2\pi }}}{e}^{-{\displaystyle \frac{{(x-\overline{x })}^2}{2{\sigma }_x^2}}}$$

(34)

And finally, knowing that ${\displaystyle \frac{x}{\overline{x }}}\approx 1$, expression (34) becomes:

$$f\left(x\right)={\displaystyle \frac{1}{{\sigma }_x\sqrt{2\pi }}}{e}^{-{\displaystyle \frac{{(x-\overline{x })}^2}{2{\sigma }_x^2}}}$$

(35)

which represents normal distribution.

Consequently, the linear Taylor series approximation and normal PDF convergence remain highly robust up to our maximum defined input uncertainty boundary of $u_r\left(x_i\right)$= 20%. A strict mathematical error-bound analysis reveals that at this maximum 20% limit, omitting the higher-order cross-product variance terms introduces a minor variance error of only 1.96% to the total uncertainty budget. However, if input parameters or log-normal shadowing dispersions exceed this 20% to 25% threshold, the mounting skewness and non-linearities compromise the model boundaries, causing the normal PDF approximation to break down. In such high-dispersion environments, the analytical approach must be substituted with numerical methods, such as adaptive Monte Carlo simulations, to capture the skewed distribution profile.

This implies that for variables with a mean value much larger than their standard deviation (small relative standard uncertainty), a log-normal distribution asymptotically converges to a normal distribution. Therefore, the product of normal random variables with relatively large mean values and small standard deviations can also be approximated as a normal distribution.

The second step in the EI calculation involves adding the products of the input parameters. It is well-established that the sum of N mutually independent normal random variables also follows a normal distribution. Given that all input parameters of the EI, as well as their individual products, are mutually independent, and assuming they are normal variables with small relative standard uncertainties, it can be concluded that the PDF of the EI is approximately normal.

6. Practical Example of Exposure Index Uncertainty Evaluation

To validate the theoretical framework and demonstrate the practical application of the proposed EI uncertainty evaluation model, this section presents an example based on data obtained from a live network deployment [39]. The evaluation considers an indoor scenario within the building of the School of Electrical Engineering, University of Belgrade (ETF), where two micro base stations are deployed to provide dedicated indoor coverage.

To systematically map this live network deployment to the mathematical formulations described in previous sections, a specific operational scenario was defined over a total observation time period of 8 h (T = 8 h). The environmental, demographic, and network configuration parameters are mapped into the model framework using the following conditions:

• Time periods within the observation time T ($N_T$ = 1): Evaluated across one time interval corresponding to the active part of the day.
• Population categories ($N_P$ = 1): Restricted to the adult population category.
• Environments ($N_E$ = 1): Limited to a single indoor environment.
• Radio access technologies (RAT) ($N_R$ = 2): Two micro base stations for implementation of GSM 900 MHz and UMTS 2100 MHz networks.
• Cell types ($N_C$ = 1): Focused solely on a microcell deployment topology.
• User load profiles ($N_L$ = 1): Considering heavy user profile.
• Postures ($N_{pos}$ = 1): Evaluated for users in a static sitting position.
• Usages with devices ($N_U$ = 1): Limited to active voice calls using standard mobile phone devices.

For the specific operational scenario analysed, the general EI formulation from Equation (3) can be explicitly expanded as follows:

$${EI}^{\mathrm{SAR}}={\displaystyle \frac{1}{\mathrm{T}}}{\displaystyle \sum _t^{Day}}{\displaystyle \sum _p^{Adults}}{\displaystyle \sum _e^{Indoor}}{\displaystyle \sum _r^{GSM, UMTS}}{\displaystyle \sum _c^{Microcell }} {\displaystyle \sum _l^{Heavy user}}{\displaystyle \sum _{pos}^{Sitting}}{f}_{t,p,e,r,l,c,pos}\left[{\displaystyle \sum _u^{Voice call}}\left(d^{UL}{ \overline{P}}_{TX}\right)+d^{DL}{ \overline{S}}_{\mathit{inc}}\right] \left[{\displaystyle \frac{\mathrm{W}}{\mathrm{k}\mathrm{g}}}\right]$$

(36)

Equation (36) simplifies considerably by focusing on the static parameters of the defined scenario. By fixing the indices for a single day, an adult demographic, an indoor environment, a heavy user profile, microcell coverage, a sitting position, and a voice call service, the multi-dimensional summations collapse. This reduction yields the simplified expression shown in Equation (37):

$${EI}^{\mathrm{SAR}}={\displaystyle \frac{1}{\mathrm{T}}}{\displaystyle \sum _r^{GSM, UMTS}}{f}_{r, Day,Adults,Indoor,Heavy user,Microcell,Sitting}\left[d^{UL}{ \overline{P}}_{TX}+d^{DL}{ \overline{S}}_{\mathit{inc}}\right]\left[{\displaystyle \frac{\mathrm{W}}{\mathrm{k}\mathrm{g}}}\right]$$

(37)

To enhance readability, the scenario-specific probability function can be abbreviated by dropping the constant fixed indices. By substituting the full subscript profile with the compact notation $f_r$, the equation reduces to its final operational form:

$${EI}^{\mathrm{SAR}}={\displaystyle \frac{1}{\mathrm{T}}}{\displaystyle \sum _r^{GSM, UMTS}}{f}_r·\left[d^{UL}{ \overline{P}}_{TX}+d^{DL}{ \overline{S}}_{\mathit{inc}}\right] \left[{\displaystyle \frac{\mathrm{W}}{\mathrm{k}\mathrm{g}}}\right]$$

(38)

The fully expanded uplink normalized dose coefficient, $d^{UL}$, is defined by the fixed parameters of the scenario as:

$$d^{UL}={\displaystyle \frac{{TD}_{r,Day,Adults,Heavy user,Indoor,Microcell,Voice call,Sitting}^{UL}\left[\mathrm{s}\right]·{SAR}_{ r,Adults,Voice call,Sitting}^{UL}[\mathrm{W}/\mathrm{k}\mathrm{g}]}{P_{TX}^{ref}[\mathrm{W}]}} \left[{\displaystyle \frac{\mathrm{W}\mathrm{s}}{\mathrm{k}\mathrm{g}}}/\mathrm{W}\right]$$

(39)

Using the same compact notation by omitting the fixed scenario baselines, the abbreviated uplink coefficient is written as:

$$d^{UL}={\displaystyle \frac{{TD}_r^{UL}[\mathrm{s}]·{SAR}_{ r,}^{UL}[\mathrm{W}/\mathrm{k}\mathrm{g}]}{P_{TX}^{ref}[\mathrm{W}]}} \left[{\displaystyle \frac{\mathrm{W}\mathrm{s}}{\mathrm{k}\mathrm{g}}}/\mathrm{W}\right]$$

(40)

Similarly, the detailed downlink dose coefficient, $d^{DL}$, is mapped to the scenario as:

$$d^{DL}={\displaystyle \frac{{TD}_{r, Day,Adults,Indoor,Microcell,Sitting}^{DL}[\mathrm{s}]·{SAR}_{r,Adults,Sitting}^{DL}[\mathrm{W}/\mathrm{k}\mathrm{g}]}{S_{\mathit{inc}}^{ref}[\mathrm{W}/{\mathrm{m}}^2]}} \left[{\displaystyle \frac{\mathrm{W}\mathrm{s}}{\mathrm{k}\mathrm{g}}}/{\displaystyle \frac{\mathrm{W}}{{\mathrm{m}}^2}}\right]$$

(41)

Dropping the static subscripts yields the abbreviated operational version of the downlink coefficient:

$$d^{DL}={\displaystyle \frac{{TD}_r^{DL}[\mathrm{s}]·{SAR}_r^{DL}[\mathrm{W}/\mathrm{k}\mathrm{g}]}{S_{\mathit{inc}}^{ref}[\mathrm{W}/{\mathrm{m}}^2]}} \left[{\displaystyle \frac{\mathrm{W}\mathrm{s}}{\mathrm{k}\mathrm{g}}}/{\displaystyle \frac{\mathrm{W}}{{\mathrm{m}}^2}} \right]$$

(42)

To isolate the separate contributions for the subsequent uncertainty evaluation, let $A$ and $B$ represent the total scaled uplink and downlink components per technology $r$, respectively:

$$A=f_r{\displaystyle \frac{{TD}_r^{UL}[\mathrm{s}]·{SAR}_{ r}^{UL}[\mathrm{W}/\mathrm{k}\mathrm{g}]}{P_{TX}^{ref}[\mathrm{W}]}}{ \overline{P}}_{TX}[\mathrm{W}]$$

(43)

$$B=f_r{\displaystyle \frac{{TD}_r^{DL}[\mathrm{s}]·{SAR}_r^{DL}[\mathrm{W}/\mathrm{k}\mathrm{g}]}{S_{\mathit{inc}}^{ref}[\mathrm{W}/{\mathrm{m}}^2]}}{ \overline{S}}_{\mathit{inc}}[\mathrm{W}/{\mathrm{m}}^2]$$

(44)

Applying Equations (19) and (20), the relative standard uncertainties $u_r\left(A\right)$ and $u_r\left(B\right)$ for each radio access technology are determined by:

$$u_r\left(A\right)=\sqrt{{u_r}^2\left(f_r\right)+{u_r}^2\left({TD}_r^{UL}\right){+ u_r}^2\left({SAR}_{ r}^{UL}\right)+{u_r}^2\left({ \overline{P}}_{TX}\right)}$$

(45)

$$u_r\left(B\right)=\sqrt{{u_r}^2\left(f_r\right)+{u_r}^2\left({TD}_r^{DL}\right){+u_r}^2\left({SAR}_r^{DL}\right)+{u_r}^2\left({ \overline{S}}_{\mathit{inc}}\right)}$$

(46)

Finally, the combined relative standard uncertainty of the total exposure index, $u_r(\mathrm{E}\mathrm{I})$, simplifies to a weighted summation over the two radio access technologies (GSM and UMTS):

$$u_r(\mathrm{E}\mathrm{I})={\displaystyle \frac{\sqrt{{\displaystyle {\sum }_r^{GSM,UMTS}}\left[A^2\ast {u_r}^2\left(A\right)+B^2\ast {u_r}^2\left(B\right)\right] }}{{\displaystyle {\sum }_r^{GSM, UMTS}}\left[A+B\right]}}$$

(47)

The fraction of users per GSM and UMTS microcell is obtained via core network signaling data, specifically through Location Update messages. These data reflect the actual number of users connected to the observed cells during the specified measurement period.

Cell-level uplink transmitted power statistics for the user equipment (UE) are gathered directly from network elements, such as the Radio Network Controller (RNC) and Base Station Controller (BSC), using Layer 3 (L3) messages containing UE transmitter power data. Per-cell network reports provide the network measurements utilized to determine the UE transmit power distribution for both 2G and 3G voice services. The average uplink transmit power values are extracted from the network using these cell statistics, while the uplink duty factor (DF) for voice services is determined experimentally using a Rohde & Schwarz ESPI Test Receiver managed by R&S ROMES software version 4.11. The DF, or activity factor, represents the actual operational activity of the transmitters based on signal presence on the radio interface. This factor is critical for the time-averaging of uplink transmitting power. Time-averaging is required because power samples extracted from network-generated UE measurement reports or drive-test data are recorded exclusively when the transmitter is actively transmitting. Consequently, to calculate the average UE power necessary for EI evaluations, the active transmission power must be combined with the DF.

Frequency-selective electric field strength measurements were conducted within the School of Electrical Engineering, University of Belgrade. The measurements were carried out using a calibrated Rohde & Schwarz portable measurement system, consisting of an R&S FSH6 spectrum analyzer and an R&S TS-EMF isotropic radiator antenna. Band-selective measurements targeted the GSM 900 MHz and UMTS 2100 MHz operator bands across multiple uniformly distributed measurement positions throughout the building. Finally, statistical analyses were performed on the values obtained from these measurement locations to calculate the corresponding average incident power density parameters.

The usage time durations are established using user profiles derived from Customer Analytics System (SAS) and cell statistics. To accurately capture high-load operational periods, the dataset isolates daytime hours within an urban environment, aligning with the temporal window of the field measurements. Average voice call durations for both GSM and UMTS, are computed based on these user profiles and network cell statistics.

Reference Specific Absorption Rate (SAR) data are computed using 3D electromagnetic simulation platforms based on the Finite Difference Time Domain (FDTD) and Finite Integration Technique (FIT) methods. Near-field and far-field exposure models are evaluated utilizing the Virtual Family “Duke” anatomical model, representing a 34-year-old male. The simulation configuration evaluates a sitting posture with a mobile phone positioned adjacent to the head to replicate realistic user exposure conditions.

Parameters, input values, and relative standard uncertainties utilized for the calculation, as well as determined values of EI values, relative standard uncertainties and relative expanded uncertainties for confidence level of approximately 95% and 99%, for the presented example are shown in Table 1.

Table 1. Parameters, input values, and relative standard uncertainties utilized for the calculation of uncertainty of Exposure Index.

Parameter	Value	Relative Standard Uncertainty ${\mathit{u}}_{\mathit{r}}$	Source
$f_{GSM}$	28.17%	3.4%	Fraction of the population connected to GSM obtained via core network signaling data
$f_{UMTS}$	71.83%	3.4%	Fraction of the population connected to UMTS obtained via core network signaling data
${ \overline{P}}_{TX GSM}$	12.1 mW	7%	Average transmit power of all user devices connected to GSM obtained using L3 messages and uplink duty factor for voice services
${ \overline{P}}_{TX UMTS}$	9.03 µW	2.5%	Average transmit power of all user devices connected to UMTS obtained using L3 messages and uplink duty factor for voice services
${ \overline{S}}_{\mathit{inc} \mathrm{GSM}}$	571.08 $\mu \mathrm{W}/{\mathrm{m}}^2$	6.5%	Incident power density averaged over area originating from GSM and determined by band selective measurements
${ \overline{S}}_{\mathit{inc} \mathrm{UMTS}}$	182.08 $\mu \mathrm{W}/{\mathrm{m}}^2$	8.4%	Incident power density averaged over area originating from UMTS and determined by band selective measurements
${TD}_{GSM}^{UL}$	1372 s	7.5%	Duration of the UL traffic for voice calls for heavy users in GSM technology determined by analysis of user data from SAS and cell statistics.
${TD}_{UMTS}^{UL}$	1347 s	9%	Duration of the UL traffic for voice calls for heavy users in UMTS technology determined by analysis of user data from SAS and cell statistics.
${TD}_{GSM}^{DL}$	8 h	0%	Duration of the DL exposure to the GSM technology
${TD}_{UMTS}^{DL}$	8 h	0%	Duration of the DL exposure to the UMTS technology
${SAR}_{GSM}^{UL}$	0.012 ${\displaystyle \frac{\mathrm{W}\mathrm{s}}{\mathrm{k}\mathrm{g}}}/\mathrm{W}$	7.8%	SAR value for the voice calls sitting posture, in the frequency band of GSM calculated by simulation and normalized for emitted power of 1 W
${SAR}_{ UMTS}^{UL}$	0.0052 ${\displaystyle \frac{\mathrm{W}\mathrm{s}}{\mathrm{k}\mathrm{g}}}/\mathrm{W}$	8.7%	SAR value for the voice calls sitting posture, in the frequency band of UMTS calculated by simulation and normalized for emitted power of 1 W
${SAR}_{GSM}^{DL}$	0.0056 ${\displaystyle \frac{\mathrm{W}\mathrm{s}}{\mathrm{k}\mathrm{g}}}/{\displaystyle \frac{\mathrm{W}}{{\mathrm{m}}^2}}$	5.4%	SAR value for the sitting posture, in the frequency band of GSM calculated by simulation and normalized for Incident power density of 1 ${\displaystyle \frac{\mathrm{W}}{{\mathrm{m}}^2}}$
${SAR}_{UMTS}^{DL}$	0.0043 ${\displaystyle \frac{\mathrm{W}\mathrm{s}}{\mathrm{k}\mathrm{g}}}/{\displaystyle \frac{\mathrm{W}}{{\mathrm{m}}^2}}$	5.7%	SAR value for the sitting posture, in the frequency band of UMTS calculated by simulation and normalized for Incident power density of 1 ${\displaystyle \frac{\mathrm{W}}{{\mathrm{m}}^2}}$
${\mathit{E}\mathit{I}}^{\mathit{S}\mathit{A}\mathit{R}}$	3.413${\displaystyle \frac{\mathbf{\mu }\mathbf{W}}{\mathbf{k}\mathbf{g}}}$	8.172%	Calculated EI which reflects the mean population exposure for the analyzed geographical area over the analyzed time interval T
${\mathit{U}}_{\mathit{r}\mathbf{,}\mathbf{95}}$	16.344%		Calculated relative expanded uncertainty (k = 2, p ≈ 95%)
${\mathit{U}}_{\mathit{r}\mathbf{,}\mathbf{99}}$	24.516%		Calculated relative expanded uncertainty (k = 3, p ≈ 99%)

The practical evaluation results compiled in Table 1 demonstrate the proposed first-order analytical framework, yielding a global ${EI}^{\mathrm{SAR}}$ of 3.413 µW/kg with a combined relative standard uncertainty $u_r\left(\mathrm{E}\mathrm{I}\right)$ of 8.172%. Based on the practical evaluation data, the model establishes reliable relative expanded uncertainties of $U_{r,95}$ = 16.344% (for k = 2) and $U_{r,99}$= 24.516% (for k = 3) under a standard normal assumption, providing precise confidence intervals for compliance auditing without requiring computational simulations.

7. Discussion

The EI is a metric defined to evaluate the mean global human population exposure within a geographic area originating from a wireless cellular network as a whole. It maps a highly complex dataset into a single parameter that represents real-life exposure, making it easy to understand, interpret, and use for all stakeholders. Furthermore, the EI aims to help telecommunication operators optimize network architecture, operations, radio technologies, and management to reduce overall human exposure without compromising user quality of service (QoS). Ultimately, minimization of the EI is intended to serve as a new key performance indicator (KPI) for developing new wireless cellular networks. By treating the EI as an average value over different contributing parameters, the metric enables the evaluation of each parameter’s contribution to derive targeted measures for lowering EM field exposure in an area.

The EI accounts for DL exposure from BSs and APs based on the average incident power density, as well as UL exposure from user equipment UE using their average transmitting power. The metric incorporates a multitude of parameters influencing exposure, including age, posture, usage, technology, and environmental factors. Methodologically, the framework of this novel metric is complex and structured around an exposure chain that comprehensively maps all relevant exposure configurations [40]. Individual exposure is integrated across diverse radio access technologies (e.g., GSM, UMTS, LTE, 5G), network layers (macro-, micro-, and femtocells), device types (e.g., smartphones, tablets), application modalities (voice calls or data sessions), and user profiles (ranging from heavy users to non-users). Ultimately, the final EI is computed as a weighted sum encompassing all operational branches of this exposure chain.

The EI calculation relies on input parameters determined through distinct methodologies [41]. Reference SAR data are pre-calculated and integrated into the uplink $d^{UL}$ and downlink $d^{DL}$ dose components. Demographic life-segmentation data are primarily sourced from established literature, while information communication technology (ICT) usage metrics are extracted via Key Performance Indicator (KPI) measurements from network-side sensors. The variables ${\overline{P}}_{TX}$ and ${ \overline{S}}_{\mathit{inc}}$ are derived from deterministic simulations, empirical measurements, and live network data. These raw measurements and network datasets often require translation or calibration into the specific input exposure metrics defined by the EI governing equation. Given that the input parameters originate from distinct sources and are determined through varying methodologies, a standard uncertainty must be quantified for each individual parameter according to its respective source and methodology.

Bearing in mind the inherent variability of the input parameters, quantifying and expressing the standard uncertainty for each determined EI value is essential for comparative and regulatory analyses. Based on the standard uncertainties of the input parameters for the EI calculation in a specific scenario and utilizing the analytical expression provided in Equation (21), the standard uncertainty of the total EI for that scenario is straightforward to calculate. This approach represents a highly convenient and computationally efficient alternative to numerical simulations, ensuring its applicability across different network configurations and exposure scenarios to match the inherent flexibility of the EI metric itself.

For analytical applications involving compliance comparisons against established reference levels, the determination of expanded uncertainty is indispensable [30,36]. The expanded uncertainty defines a confidence interval bounding the measurement result, which encompasses a significant proportion of the distribution of values reasonably attributable to the measurand. This expanded uncertainty is derived by scaling the combined standard uncertainty by a numerical coverage factor (k), the magnitude of which is fundamentally governed by the PDF of the measurement result. Consequently, characterizing the PDF is a prerequisite for robust uncertainty estimation. The selection of k, typically bounded between 2 and 3, dictates the target level of statistical confidence [36].

For typical practical scenarios, all input parameters of the EI have small uncertainties, which allows the PDF of the EI to be approximated as a normal distribution. In the case of a normally distributed EI, empirical practice conventionally uses a coverage factor of k = 2 to yield an approximate 95.45% confidence interval, whereas a factor of k = 3 expands the interval to encompass a confidence threshold exceeding 99%.

8. Conclusions

An objective assessment of human population exposure to RF EM fields is crucial. For this reason, a new exposure metric was defined under the EU FP7 project LEXNET to evaluate the mean global population exposure within an analyzed geographical area [28]. This metric simultaneously accounts for exposure originating from BSs and AP as well as exposure from wireless user devices.

Since evaluating measurement uncertainty is a critical component of all metrology applications, this paper proposes a procedure for evaluating EI uncertainty. The proposed procedure provides a unified, first-order analytical approximation to calculate EI relative standard uncertainty based on the known relative standard uncertainties of multiple independent input parameters, bypassing the calculation of dense sensitivity derivative matrices. Furthermore, we propose a method to estimate the output PDF of the EI. Mathematically demonstrating that normal input parameters with small standard deviations asymptotically converge to a normal distribution, we establish the explicit validity limits up to an upper bound of $u_r\left(x_i\right)$= 20% where the final output distribution can safely be considered normal. This boundary model provides strict metrological traceability, allowing network compliance engineers to instantly determine exact coverage factors for relative expanded uncertainty calculations without relying on heavy Monte Carlo simulations.