EMF-Aware Probabilistic Shaping Design for Hardware-Distorted Communication Systems

The fifth-generation cellular network requires dense installation of radio base stations (BS) to support the ever-increasing demands of high throughput and coverage. The ongoing deployment has triggered some health concerns among the community. To address this uncertainty, we propose an EMF-aware probabilistic shaping design for hardware-distorted communication systems. The proposed scheme aims to minimize human exposure to radio frequency (RF) radiations while achieving the target throughput using probabilistic shaping. The joint optimization of the transmit power and nonuniform symbol probabilities is a non–convex optimization problem. Therefore, we employ alternate optimization and successive convex approximation to solve the subsequent problems. Our findings reveal a significant reduction in the users’ exposure to EMF while achieving the requisite quality of service with the help of probabilistic shaping in a hardware-distorted communication system.

1 Introduction

The next-generation wireless communication networks are expected to fulfill the ever-increasing demands of higher data rates, ultra-reliability, minimal latency, high energy efficiency, and massive connectivity for many users/devices (Latva-aho et al., 2020). The fifth-generation (5G) wireless technology is envisioned to support numerous diverse services, such as enhanced mobile broadband (eMBB), ultra-reliable low latency communication (URLLC), and massive machine-type communication (mMTC) (Wan et al., 2018). Some of the new spectra allocated for 5G deployments, for example, millimeter waves, suffer from relatively high path loss, limiting the coverage area. Thus, network densification becomes essential for achieving the promised data rate, which can be realized over space [e.g., dense deployment of base stations (BSs) in small cells] and frequency (large segments of the RF spectrum in diverse bands) (Bhushan et al., 2014).

Installation of 5G cellular technology with extreme node and network densification (with BSs being closer to users) is raising health concerns about the impact of electric and magnetic fields (EMFs) exposure on the population. These worries have sparked several protests against 5G technology and led to some attacks on the 5G BSs (Elzanaty et al., 2021). Recently, anti-5G protests have been held in 30 countries around the world against the threat of 5G wireless technology to public health, the environment, and privacy.

Indeed, the thermal effect is the only proven health impact from RF nonionizing short-term exposure. Therefore, it is necessary to keep the radiation intensity below specific values defined by the exposure regulations and guidelines, such as those of the Federal Communications Commission (FCC) and the International Commission on Non-Ionizing Radiation Protection (ICNIRP) (Buchner and Rivasi, 2020). Nevertheless, there is a debate about severe health impacts due to long-term exposure to EMF (National Toxicology Program, 2018a,b; Vornoli et al., 2019). Hence, the International Agency for Research on Cancer (IARC) classified RF radiation as “possibly carcinogenic to humans” (Vornoli et al., 2019; Group, 2013; Wilbourn et al., 1986).

Recently, a comprehensive study revealed that exposure due to the uplink (UL) from user equipment (UE) is higher than from the BS due to the proximity of the UE to the user (Lou et al., 2021). Another work has proposed an architectural solution to minimize the EMF exposure using reconfigurable intelligent surfaces (Ibraiwish et al., 2022). Another study focused on meticulous cellular network planning to limit exposure from the BSs exploiting MIMO while ensuring coverage and capacity constraints (Matalatala et al., 2018). From the regulatory aspects, some possible risk mitigation strategies are dismission of legacy 2G/3G/4G technologies and reduction of emissions from noncellular sources (Chiaraviglio et al., 2021).

Nevertheless, as mentioned earlier, the EMF exposure–based research does not consider any hardware impairments, which can drastically affect the system performance, such as raising the noise floor. For instance, in order to reach the target data rates in eMBB, hardware distortion (HWD) should be mitigated. Generally, users send a much higher power, escalating their RF exposure to achieve the target rate. The users may still not reach the required throughput due to the saturation for the data rate at a higher SNR as the distortion noise increases with the transmit power.

HWD requires some meticulously designed systems to compensate for its effect and mitigate performance loss. Improper Gaussian signaling (IGS) is an effective compensation signaling scheme that can alleviate the impact of several interference and imperfection sources (Javed et al., 2019). However, realizing the IGS comes with inherent problems of the unbounded peak-to-average power ratio and high detection complexity (Santamaria et al., 2018; Javed et al., 2020). As a consequence, researchers employ finite discrete asymmetric signaling (AS) schemes that can be achieved by geometric shaping (GS), probabilistic shaping (PS), or hybrid shaping (HS) (Elzanaty and Alouini, 2022; Javed et al., 2021). In this work, we propose asymmetric signaling by adopting PS to tackle improper HWD and minimize EMF exposure to the users while maintaining throughput quality of service (QoS). The contributions of this study are summarized as follows:

• We model HWD and EMF exposure in the next-generation wireless cellular network. We present an appropriate receiver and rigorous error probability analysis considering the improper distortion noise.

• We propose probability shaping as a form of asymmetric signaling to effectively mitigate improper HWDs and reduce EMF exposure while maintaining QoS in terms of user throughput.

• We employ alternate optimization to jointly design the transmit power of users and nonuniform symbol probabilities. We further use successive convex approximation and the Newton–Raphson method to solve subsequent problems.

The rest of the article is organized such that Section 2 illustrates the system description and adopted models to characterize the radiation exposure and HWD. The error probability analysis is carried out in Section 3, while Section 4 covers the problem formulation and optimization framework. Numerical results are presented in Section 5, followed by the conclusion in Section 6.

2 System Description

In this section, we quantify the EMF exposure of the human population in a next-generation wireless cellular communication system. The network comprises a macro-cell with a macro BS and multiple pico-cells with their serving pico BSs. Apart from the transmitting radiations from the BSs in the downlink (DL), we are interested in the EMF exposure caused by handheld devices. For instance, the RF radiations emitting from cell phones and smartwatches are stronger nearer the users, resulting in near-field exposure, whereas the EMF transmission density from BS towers situated far from the users render far-field exposure. Likewise, increasing applications and use cases including but not limited to autonomous vehicles and smart homes/offices, etc., are extending their radiation footprint near humans. Figure 1 demonstrates a strong radiation pattern near users in the case of voice users when compared with data users. In addition, we account for the performance degradation caused by the additive HWDs accumulating from the various blocks in nonideal RF transceivers. We propose a generalized digital communication system capable of transmitting nonuniformly distributed symbols (from a uniform bitstream using distribution matching) while using an appropriate receiver for optimal detection.

FIGURE 1

FIGURE 1. EMF exposure from 5G network infrastructure.

2.1 Radiation Exposure

The modeling and quantification of human exposure in a wireless cellular network are critical to facilitate the reduction efforts. Numerous studies have identified the essential parameters influencing this end exposure (Kuehn et al., 2019; Lou et al., 2021; Vermeeren et al., 2015a). The EMF exposure, quantified as the exposure index (EI), is primarily dictated by the network topology, environment, radio access technology, user scenarios, and service types (Kuehn et al., 2019). The EI can be written as the weighted sum of all branches in the chain of exposure (Lou et al., 2021).

$\mathrm{E}\mathrm{I}=\sum _g\sum _p\sum _e\sum _r\sum _l\sum _{s}f\left({\mathrm{S}\mathrm{A}\mathrm{R}}^{\text{UL}},\alpha ,{\mathrm{S}\mathrm{A}\mathrm{R}}^{\text{DL}},\varrho \right)(1)$

where the specific absorption rate (SAR), measured in W/kg, is the power absorbed per mass of the exposed tissue for a given period, with SAR^UL and SAR^DL indicating the normalized values of the UL- and DL-induced SARs when the mean transmit power α = 1W and the mean received power density ϱ = 1W/m², respectively. The EI is integrated over different age groups g (e.g., children, young people, adults, and seniors), user posture p (e.g., standing and sitting), environments e (e.g., indoor, outdoor, and commuting), radio access technology r (e.g., GSM, UMTS, Wi-Fi, 5G), layers l (macro, micro, pico, and femto), and service/usage types s (e.g., voice and data) (Tesanovic et al., 2014). We consider an exposure scenario of the next-generation cellular network in an indoor environment where the reference SAR is averaged over different population ages and postures, accommodating both data and voice usage types. The EI for both the UL and DL can be given by EI = EI^UL + EI^DL, combining the DL exposure induced all day long by base stations/access points and the UL exposure incurred by individual wireless communication devices.

Contrary to the general perception, the SAR from the UL renders the dominant part instead of the one from DL with low ϱ (generally $\le 10\mathrm{m}\mathrm{W}/{\mathrm{m}}^2$ according to FCC), given a significant distance between the transmitter and receiver. Thus, the exposure index (EI) metric in the presented scenario is dominated by the UL as $\mathrm{E}\mathrm{I}\left(\alpha \right)={\mathrm{S}\mathrm{A}\mathrm{R}}^{\text{UL}}\alpha$, the reference whole body or localized SAR mainly depends on the required service and the posture (Vermeeren et al., 2015b; Lou et al., 2021).

2.2 Hardware-Impaired Signal Model

The nonlinear transfer functions of various transmitter RF stages, such as the digital-to-analog converter, band-pass filter, and high-power amplifier, result in additive distortion noise η_t, which is distributed as a zero-mean complex Gaussian random variable ${\eta }_{\text{t}}\sim \mathcal{CN}(0,{\kappa }_{\text{t}},{\tilde{\kappa }}_{\text{t}})$ with variance κ_t and pseudo-variance ${\tilde{\kappa }}_{\text{t}}$. The complete statistical characterization requires pseudo-variance, in addition to the variance, accounting for the correlated and/or unequal power distribution among quadrature components of the general complex Gaussian random variable (Javed et al., 2017). Importantly, the value of pseudo-variance is limited by the variance as $|{\tilde{\kappa }}_{\text{t}}|\le {\kappa }_{\text{t}}$ (Björnson et al., 2013; Schenk, 2008).

The Gaussian model for the aggregate residual RF distortions is based on various theoretical investigations, including the central limit theorem and measurement results after applying existing compensation schemes (Wenk, 2010; Zetterberg, 2011; Boulogeorgos et al., 2016; Xia et al., 2015; Suzuki et al., 2008; Studer et al., 2010; Duy et al., 2015; Björnson et al., 2014; and references therein). This can also be motivated analytically by the central limit theorem. The accumulative distortions raise the noise floor of the transmitted signal as x_tx = x_m + η_t, where x_m is the single-carrier band-pass–modulated signal taken from M-ary QAM, M-ary PSK, or M-ary PAM constellation with a probability mass function p_m ≜ p_X (x_m), rendering the transmission probability of symbol x_m, and p ≜ [p₁, p₂, … , p_M]. Let us define the set that includes all the possible symbol distributions as

$\mathbb{S}=\left\{\mathbf{p}:\mathbf{p}=\left[p_1,p_2,\dots ,p_M\right],\sum _{j=1}^M{p}_j=1,p_j\ge 0,\forall j\in \left\{\mathrm{1,2},\dots ,M\right\}\right\}.$

The information-bearing signal is transmitted with an average power α and received under an additive white Gaussian noise (AWGN) condition and receiver distortions ${\eta }_{\text{r}}\sim \mathcal{CN}(0,\alpha {\kappa }_{\text{r}},\alpha \widetilde{{\kappa }_{\text{r}}})$. These distortions result from the nonlinear transfer function of low-noise amplifier, band-pass filters, image rejection low-pass filter, and analog-to-digital converter at the receiver. Thus, the received signal in point-to-point (P2P) communication under improper HWD can be modeled as

$y=\sqrt{\alpha }{x}_m+\sqrt{\alpha }\eta +w;m=\mathrm{1,2},\dots ,M(2)$

where $w\sim \mathcal{CN}(0,{\sigma }_w^2,0)$ is the thermal noise, and the aggregate effect of transceiver distortions is represented by $\eta \sim \mathcal{CN}(0,\kappa ,\tilde{\kappa })$, κ = κ_t + κ_r and $\tilde{\kappa }={\tilde{\kappa }}_{\text{t}}+{\tilde{\kappa }}_{\text{r}}$. Interestingly, the generalized impropriety characterization assists in accurate system modeling, rigorous performance analysis, and appropriate signaling design. Interested readers can study the work by Javed et al. (2020) for the details of statistical impropriety characterization.

2.3 Noise Distribution and Optimal Receiver

Considering the aggregate noise $z=\sqrt{\alpha }\eta +w$ distributed as $z\sim CN\left(0,\alpha \kappa +{\sigma }_w^2,\alpha \tilde{\kappa }\right)$ where in-phase z_I and quadrature-phase z_Q noise components are distributed with the respective variances as

${\sigma }_I^2=\left(\alpha \left(\kappa +\mathfrak{R}\left(\tilde{\kappa }\right)\right)+{\sigma }_w^2\right)/2,(3)$

${\sigma }_Q^2=\left(\alpha \left(\kappa -\mathfrak{R}\left(\tilde{\kappa }\right)\right)+{\sigma }_w^2\right)/2,(4)$

These individual variances are obtained by simultaneously solving the equations of variance, that is, $\mathbb{E}\left\{{\left|z\right|}^2\right\}={\sigma }_I^2+{\sigma }_Q^2$ and pseudo-variance $\mathbb{E}\left\{z^2\right\}={\sigma }_I^2-{\sigma }_Q^2+2i{r}_{z_I{z}_Q}$. Thus, the correlation $r_{z_I{z}_Q}=\alpha \mathfrak{I}\left(\tilde{\kappa }\right)/2$ defines the correlation coefficient ρ_z between z_I and z_Q as

${\rho }_z=\frac{r_{z_I{z}_Q}}{{\sigma }_I{\sigma }_Q}=\frac{\alpha \mathfrak{I}\left(\tilde{\kappa }\right)}{\sqrt{{\left(\alpha \kappa +{\sigma }_w^2\right)}^2-{\left(\alpha \mathfrak{R}\left(\tilde{\kappa }\right)\right)}^2}}.(5)$

Unequal power distribution among the quadrature noise components and nontrivial correlation coefficient marks the improper nature of aggregate additive distortions. Given the nonuniform symbol probabilities and improper noise, we propose a maximum a posterior (MAP) detector for the optimal detection as opposed to the conventional minimum Euclidean or maximum likelihood (ML) detectors (Javed et al., 2021). Thus, the detection criterion is given by

${\widehat{m}}_{\text{PS}}=\underset{1\le m\le M}{\mathrm{argmax}}{p}_X\left(x_m\right)f_{Y_{\text{I}},Y_{\text{Q}}|X}\left(y_{\text{I}},y_{\text{Q}}|x_m\right),(6)$

where $f_{Y_{\text{I}},Y_{\text{Q}}|X}\left(y_{\text{I}},y_{\text{Q}}|x_m\right)$ is the conditional Gaussian probability density function (PDF) of y given that x_m is derived using Javed et al. (2020, eq. 43)

$f_{Y_{\text{I}},Y_{\text{Q}}|X}\left(y_{\text{I}},y_{\text{Q}}|x_m\right)=\frac{1}{2\pi {\sigma }_{\text{I}}{\sigma }_{\text{Q}}\sqrt{1-{\rho }_z^2}}\exp \left\{\frac{-1}{2\left(1-{\rho }_z^2\right)}\left[\begin{array}{c}\hfill \frac{{\left(y_{\text{I}}-\sqrt{\alpha }\mathfrak{R}\left(x_m\right)\right)}^2}{{\sigma }_{\text{I}}^2}+\frac{{\left(y_{\text{Q}}-\sqrt{\alpha }\mathfrak{I}\left(x_m\right)\right)}^2}{{\sigma }_{\text{Q}}^2}+\hfill \\ \hfill -\frac{2{\rho }_z\left(y_{\text{I}}-\sqrt{\alpha }\mathfrak{R}\left(x_m\right)\right)\left(y_{\text{Q}}-\sqrt{\alpha }\mathfrak{I}\left(x_m\right)\right)}{{\sigma }_{\text{I}}{\sigma }_{\text{Q}}}\hfill \end{array}\right]\right\}.(7)$

3 Error Probability Analysis

In this section, we derive the symbol error probability P_s for a system transmitting M-ary–modulated symbols with prior probabilities p_m and subjected to HWD using the union-bound and pairwise error probability as follows:

$P_s\le \sum _{m=0}^{M-1}\sum _{n\ne m}{P}_m\mathrm{Pr}\left(x_m\to x_n|x_m\right).(8)$

The pairwise error probability, for receiving an erroneous symbol x_m given x_n was transmitted, can be derived using the following MAP rule:

$\mathrm{Pr}\left(x_m\to x_n|x_m\right)=\mathrm{Pr}\left\{P_m{f}_{Y_I,Y_Q}\left(y_I,y_Q|x_m\right)\le P_n{f}_{Y_I,Y_Q}\left(y_I,y_Q|x_n\right)\right\}.(9)$

Using (7) and few simplifications, we get the following bit error probability as

$P_{\text{b}}\le {\mathrm{P}}_{\text{b}}^{\text{UB}}\left(M,\alpha ,\mathbf{p}\right)\triangleq \frac{1}{{\log }_2\left(M\right)}\sum _{m=1}^M\sum _{n=1n\ne m}^M{p}_m\mathcal{Q}\left({\beta }_{mn}\left(\alpha \right)\ln \left(\frac{p_m}{p_n}\right)+\frac{1}{2{\beta }_{mn}\left(\alpha \right)}\right),(10)$

where β_mn is defined as

${\beta }_{mn}\left(\alpha \right)\triangleq \sqrt{\frac{{\alpha }^2\left({\kappa }^2-\mathfrak{R}{\left(\tilde{\kappa }\right)}^2-\mathfrak{I}{\left(\tilde{\kappa }\right)}^2\right)+2\alpha \kappa {\sigma }_w^2+{\sigma }_w^4}{{\alpha }^2{\wp }_{\text{mn}}+\alpha \left({{\xi }_{mn}}_I^2+{{\xi }_{mn}}_Q^2\right)2{\sigma }_w^2}},(11)$

with ${\wp }_{\text{mn}}=2{{\xi }_{mn}}_I^2\left(\kappa -\mathfrak{R}\left(\tilde{\kappa }\right)\right)+2{{\xi }_{mn}}_Q^2\left(\kappa +\mathfrak{R}\left(\tilde{\kappa }\right)\right)-4\mathfrak{I}\left(\tilde{\kappa }\right){{\xi }_{mn}}_I{{\xi }_{mn}}_Q$, while ξ_mn = d_mn = x_m − x_n represents the distance between the m^th and n^th symbols.

4 Problem Formulation and Optimization

We propose a PS scheme, as a possible form of AS schemes, to effectively mitigate the drastic effects of additive distortions while transmitting with minimum power to minimize the EMF exposure. To this end, we employ a higher-order M_nu probabilistically shaped quadrature amplitude modulation (QAM) offering more degrees of freedom and adaptive rates. The optimization problem targets the joint design of the transmit power and symbol probabilities which minimize the exposure index while maintaining a throughput quality constraint. Assuming that the set includes all possible values of α as $\mathbb{A}=\{\alpha :0\le \alpha \le {\alpha }_{\text{max}}\}$, we can formulate the optimization problem as

$\mathbf{P}\mathbf{1}:\underset{\alpha \in \mathbb{A},\mathbf{p}\in \mathbb{S}}{\text{minimize\hspace{0.17em}}}\mathrm{E}\mathrm{I}\left(\alpha \right)(\mathrm{12a})$

$\text{subject\hspace{0.17em}to}\sum _{m=1}^{M_{\text{nu}}}{\left|x_m\right|}^2{p}_m\le 1,(\mathrm{12b})$

$\left(1-{\mathrm{P}}_{\text{b}}^{\text{UB}}\left(M_{\text{nu}},\alpha ,\mathbf{p}\right)\right)\mathrm{H}\left(\mathbf{p}\right)\ge {\mathcal{T}}_{\text{u}},(\mathrm{12c})$

where ${\mathcal{T}}_{\text{u}}=\left(1-{\mathrm{P}}_{\text{b}}^{\text{UB}}\left(M_{\text{u}},{\alpha }_{\text{max}},{\mathbf{p}}_{\text{u}}\right)\right){\log }_2\left(M_{\text{u}}\right)$ is the throughput of the constellation with maximum power transmission. Moreover, (12b) and (12c) represent the average power and throughput QoS constraints, respectively. In addition, H(p) is the source entropy, which represents the transmitted rate in terms of bits per symbol per channel use and is defined as

$\mathrm{H}\left(\mathbf{p}\right)\triangleq \sum _{m=1}^{M_{\text{nu}}}-p_m{\log }_2\left(p_m\right).(13)$

The joint optimization problem is challenging due to the non-convex constraints. Therefore, we adopt an alternate optimization approach to iteratively solve the transmit power and symbol probabilities using the subproblems (14) and (21), respectively. The alternate optimization Algorithm 1 begins with some initial feasible points p^(j) and α^(j) and evaluates $\mathrm{E}\mathrm{I}\left({\alpha }^{(j)}\right)$ for a reference benchmark. Then, it finds a feasible solution of P1(a), that is, p^(j*⁾ that satisfies (12b) and (12c). Given a probability mass function (PMF) p^(j*⁾, we optimize P1(b) to minimize the exposure index obtaining optimal α^(j*⁾, which is updated to attain initial points for the next iteration. This iterative process continues until reaching an acceptable tolerance δ. Consequently, the solution parameters yield a suitable PMF and transmission power, which render a minimum exposure index while maintaining a throughput QoS.

Algorithm 1. Alternate Optimization

$\mathbf{P}\mathbf{1}\left(\mathbf{a}\right):\underset{\mathbf{p}\in \mathbb{S}}{\text{find\hspace{0.17em}}}\mathbf{p}(\mathrm{14a})$

$\text{which\hspace{0.17em}satisfies}(12b)\text{\hspace{0.17em}and}(\mathrm{14b})$

$1-{\mathrm{P}}_{\text{b}}^{\text{UB}}\left(M_{\text{nu}},\alpha ,\mathbf{p}\right)\ge {\mathcal{T}}_{\text{u}}/\mathrm{H}\left(\mathbf{p}\right),(\mathrm{14c})$

The problem P1(a) is a non–convex optimization problem due to the constraint (14c). Interestingly, 1/H(p) is convex, as the second derivative is always positive (see Appendix). However, the bit error probability is a non–convex function in p. Therefore, we tackle this challenge using the successive convex approximation approach based on the Taylor series approximation of the bit error probability. The first-order Taylor series approximation of a function $f\left(x\right)$ around a point x^(k) is given as

$\tilde{f}\left(x,x^{\left(k\right)}\right)\approx f\left(x^{\left(k\right)}\right)+{\nabla }_{x}f\left(x^{\left(k\right)}\right)\left(x-x^{\left(k\right)}\right).(15)$

Thus, we need to compute ${\nabla }_{\mathbf{p}}{\mathrm{P}}_{\text{b}}^{\text{UB}}\left(M_{\text{nu}},\alpha ,\mathbf{p}\right)$ and evaluate it as ${\mathbf{p}}^{\left(k\right)}$.

${\nabla }_{\mathbf{p}}{\mathrm{P}}_{\text{b}}^{\text{UB}}\left(M_{\text{nu}},\alpha ,\mathbf{p}\right)=\left[\frac{\partial {\mathrm{P}}_{\text{b}}^{\text{UB}}}{\partial p_1}\frac{\partial {\mathrm{P}}_{\text{b}}^{\text{UB}}}{\partial p_2}\dots \frac{\partial {\mathrm{P}}_{\text{b}}^{\text{UB}}}{\partial p_{M_{\text{nu}}}}\right].(16)$

In order to compute $\partial {\mathrm{P}}_{\text{b}}^{\text{UB}}/\partial p_t$, we rewrite (10) as

${\mathrm{P}}_{\text{b}}^{\text{UB}}\left(M_{\text{nu}},\alpha ,\mathbf{p}\right)=\frac{1}{{\log }_2\left(M_{\text{nu}}\right)}\sum _{m=1}^{M_{\text{nu}}}\sum _{n=1n\ne m}^{M_{\text{nu}}}{p}_m\underset{{\mathrm{\Omega }}_{mn}}{\overset{\infty }{\int }}\frac{e^{-\frac{u^2}{2}}}{\sqrt{2\pi }}du,(17)$

where Ω_mn is defined as

${\mathrm{\Omega }}_{mn}={\beta }_{mn}\ln \left(\frac{p_m}{p_n}\right)+\frac{1}{2{\beta }_{mn}}.(18)$

Applying the Leibniz integral rule on (17) yields the following partial derivative:

$\frac{\partial {\mathrm{P}}_{\text{b}}^{\text{UB}}}{\partial p_t}\le \frac{1}{{\log }_2\left(M_{\text{nu}}\right)}\sum _{n=1,n\ne t,m=t}^{M_{\text{nu}}}\left(\mathcal{Q}\left({\mathrm{\Omega }}_{mn}\right)-\frac{{\beta }_{mn}}{\sqrt{2\pi }}{e}^{-\frac{{\mathrm{\Omega }}_{mn}^2}{2}}\right)+\frac{1}{{\log }_2\left(M_{\text{nu}}\right)}\sum _{m=1,m\ne t,n=t}^{M_{\text{nu}}}\frac{{\beta }_{mn}{p}_m}{\sqrt{2\pi }{p}_n}{e}^{-\frac{{\mathrm{\Omega }}_{mn}^2}{2}}.(19)$

Now, ${\mathrm{P}}_{\text{b}}^{\text{UB}}\left(M_{\text{nu}},\alpha ,\mathbf{p}\right)$ can be approximated from Eqs. 15, 16, 19 using first-order Taylor series expansion around an initial probability vector p^(k) as

${\tilde{\mathrm{P}}}_{\text{b}}^{\text{UB}}\left(M_{\text{nu}},\alpha ,\mathbf{p},{\mathbf{p}}^{\left(k\right)}\right)\triangleq {\mathrm{P}}_{\text{b}}^{\text{UB}}\left(M_{\text{nu}},\alpha ,{\mathbf{p}}^{\left(k\right)}\right)+{\nabla }_{\mathbf{p}}{\mathrm{P}}_{\text{b}}^{\text{UB}}\left(M_{\text{nu}},\alpha ,{\mathbf{p}}^{\left(k\right)}\right)\left(\mathbf{p}-{\mathbf{p}}^{\left(k\right)}\right).(20)$

Conclusively, we can solve P1 (a) by replacing ${\mathrm{P}}_{\text{b}}^{\text{UB}}\left(M_{\text{nu}},\alpha ,\mathbf{p}\right)$ in (14c) with its Taylor series approximation ${\tilde{\mathrm{P}}}_{\text{b}}^{\text{UB}}\left(M_{\text{nu}},\alpha ,\mathbf{p},{\mathbf{p}}^{(k)}\right)$ and solving the resultant convex feasibility problem iteratively using the well-known successive convex approximation approach (Liu et al., 2019). On the other hand, we solve P1 (b) for a given PMF to obtain an optimal α* which minimizes the EMF exposure.

$\mathbf{P}\mathbf{1}\left(\mathbf{b}\right):\underset{\alpha \in \mathbb{A}}{\text{minimize\hspace{0.17em}}}\mathrm{E}\mathrm{I}\left(\alpha \right)(\mathrm{21a})$

$\text{subject\hspace{0.17em}to}(12c)(\mathrm{21b})$

Intuitively, the throughput of probabilistically shaped M_nu-ary QAM is an increasing function of α because ${\mathrm{P}}_{\text{b}}^{\text{UB}}\left(M_{\text{nu}},\alpha ,\mathbf{p}\right)$ is a decreasing function of α. Therefore, the solution to the problem P1 (b) is simply obtained by solving the inequality constraint with equality as

${\mathrm{P}}_{\text{b}}^{\text{UB}}\left(M_{\text{nu}},\alpha ,\mathbf{p}\right)=1-{\mathcal{T}}_{\text{u}}/\mathrm{H}\left(\mathbf{p}\right)(22)$

The root of the nonlinear equation $\mathrm{\Upsilon }(\alpha )={\mathrm{P}}_{\text{b}}^{\text{UB}}\left(M_{\text{nu}},\alpha ,\mathbf{p}\right)+{\mathcal{T}}_{\text{u}}/\mathrm{H}(\mathbf{p})-1$ can be obtained using the Newton–Raphson method, which begins with an initial guess α^(j) and updates it with every iteration as explained by Kelley (2003):

${\alpha }^{\left(j+1\right)}={\alpha }^{\left(j\right)}-\frac{\mathrm{\Upsilon }\left({\alpha }^{\left(j\right)}\right)}{{\mathrm{\Upsilon }}^{\prime }\left({\alpha }^{\left(j\right)}\right)}.(23)$

It is worthy to note that ϒ′(α^(j)) ≠ 0 and is computed from (17) as

${\mathrm{\Upsilon }}^{\prime }\left(\alpha \right)=\frac{1}{{\log }_2\left(M_{\text{nu}}\right)}\sum _{m=1}^{M_{\text{nu}}}\sum _{n=1n\ne m}^{M_{\text{nu}}}{p}_m\frac{e^{-{\mathrm{\Omega }}_{mn}^2/2}}{\sqrt{2\pi }}\left(\frac{1}{2{\beta }_{mn}^2}-\ln \left(\frac{p_m}{p_n}\right)\right)\frac{\partial {\beta }_{mn}}{\partial \alpha },(24)$

where $\frac{\partial {\beta }_{mn}}{\partial \alpha }$ is expressed as

$\frac{\partial {\beta }_{mn}}{\partial \alpha }=\frac{1}{2}\sqrt{\frac{{\alpha }^2{\wp }_{\text{mn}}+\alpha \left({{\xi }_{mn}}_I^2+{{\xi }_{mn}}_Q^2\right)2{\sigma }_w^2}{{\alpha }^2\left({\kappa }^2-\mathfrak{R}{\left(\tilde{\kappa }\right)}^2-\mathfrak{I}{\left(\tilde{\kappa }\right)}^2\right)+2\alpha \kappa {\sigma }_w^2+{\sigma }_w^4}}\mathrm{x}(25)$

$\left\{\frac{2\alpha \left({\kappa }^2-\mathfrak{R}{\left(\tilde{\kappa }\right)}^2-\mathfrak{I}{\left(\tilde{\kappa }\right)}^2\right)+2\kappa {\sigma }_w^2}{{\alpha }^2{\wp }_{\text{mn}}+\alpha \left({{\xi }_{mn}}_I^2+{{\xi }_{mn}}_Q^2\right)2{\sigma }_w^2}-\frac{\psi \left(2\alpha {\wp }_{\text{mn}}+\left({{\xi }_{mn}}_I^2+{{\xi }_{mn}}_Q^2\right)2{\sigma }_w^2\right)}{{\left({\alpha }^2{\wp }_{\text{mn}}+\alpha \left({{\xi }_{mn}}_I^2+{{\xi }_{mn}}_Q^2\right)2{\sigma }_w^2\right)}^2}\right\}(26)$

with $\psi ={\alpha }^2\left({\kappa }^2-\mathfrak{R}{\left(\tilde{\kappa }\right)}^2-\mathfrak{I}{\left(\tilde{\kappa }\right)}^2\right)+2\alpha \kappa {\sigma }_w^2+{\sigma }_w^4$. The process is repeated until the desired criterion is met in terms of precision, that is, ϒ(α) becomes acceptably small and the change α is lesser than the predefined limit or maximum number of iterations.

5 Numerical Results

In this section, we present some numerical results to quantify the EMF exposure caused by the proposed PS scheme instead of conventional no-shaping (NS). We consider an indoor active user in the next-generation cellular network suffering from improper HWDs. We assume κ ∈ {0, 1}, ρ = 0.9, ${\sigma }_w^2=1$, M_u = 8 − QAM, and M_nu = 16 − QAM unless specified otherwise. The reference whole-body SAR values are taken from the study by Vermeeren et al. (2015a, Table 27), which were computed using 3D EM-simulation platforms based on the finite difference time domain and finite integration technique method. For instance, the whole-body SAR^ref for a standing adult is 0.0053 and while sitting is 0.0047. On the other hand, for a child it is 0.015 and 0.014, respectively, for the uplink voice communication at the 2.6-GHz frequency band. In number results, we take the average reference SAR weighted by the number of users in each category along with their respective postures. In particular, we consider SAR^ref = 41 × 10⁻⁴ W/Kg for data users and SAR^ref = 63 × 10⁻⁴ W/Kg for voice users per unit power (Ibraiwish et al., 2022). Evidently, the SAR reference value of voice users is higher than that of data users because the voice users are expected to keep their mobile phones near their brains.

At first, we investigate the EMF exposure of a single active user for a range of HWD levels. We present the comparison between the traditional NS scheme employing M_u − QAM and the proposed PS scheme employing M_nu − QAM. For a fair comparison, we minimize the exposure index of M_u − QAM with uniform distribution p_u (NS) scheme to ensure a throughput threshold of ${\mathcal{T}}_{\text{0}}=2.997$ bits/sec, that is,

$\mathbf{P}\mathbf{2}\left(\mathbf{a}\right):\underset{{\alpha }_{\text{u}}\in \mathbb{A}}{\text{minimize\hspace{0.17em}}}\mathrm{E}\mathrm{I}\left({\alpha }_{\text{u}}\right)(\mathrm{27a})$

$\text{subject\hspace{0.17em}to}\left(1-{\mathrm{P}}_{\text{b}}^{\text{UB}}\left(M_{\text{u}},\alpha ,{\mathbf{p}}_u\right)\right){\log }_2\left(M_{\text{u}}\right)\ge {\mathcal{T}}_{\text{0}},(\mathrm{27b})$

The simulation results provide insights into the exposure index of non-shaping versus probabilistic shaping for a range of HWDs to achieve a target throughput, as shown in Figure 2. Evidently, they reveal the superiority of employing PS to successfully limit the EMF exposure while maintaining the QoS in terms of throughput. The advantage of PS over NS is particularly prominent for higher distortion levels. Moreover, we move a step further to investigate the performance of the HS scheme with M_nu − QAM employing an aggregate of GS and PS as detailed in the study by Javed et al. (2021). Interestingly, HS reduces the EMF exposure for higher distortion levels, but the gain is insignificant given the added complexity in designing GS and PS parameters. Conclusively, PS is the preferred choice as it can reduce EMF exposure up to 98% with affordable complexity.

FIGURE 2

FIGURE 2. Exposure index [W/kg] versus HWD power when NS transmits with α_max.

Next, we study the impact of varying the modulation order for three different distortion levels on the EMF exposure as shown in Figure 3. We assumed a modulation order M_nu ranging from 8-QAM to 64-QAM for the proposed PS. Noticeably, the increasing modulation order increases the EMF exposure for all HWD levels, that is, least, medium, and high distortion levels, however, at different paces. Intuitively, more transmit power is required to meet the QoS constraint in a highly distorted system instead of the least distorted system, which advocates a considerable EMF exposure of higher distortion levels.

FIGURE 3

FIGURE 3. Exposure index [W/kg] versus modulation order.

Similarly, we analyzed the benefits of PS over NS for a range of noise variance (−98dBm to −89dBm) for two different target throughputs, that is, ${\mathcal{T}}_{\text{0}}=2.7$ bits/sec and ${\mathcal{T}}_{\text{0}}=2.85$ bits/sec. We can observe a 55 and 81.82% reduction in EMF exposure at −98dBm for the target threshold rates of 2.7 and 2.85 bits per sec per channel, respectively, with PS over NS as shown in Figure 4. Intuitively, PS outperforms NS for the entire range of noise variance in decreasing the EMF exposure on a user.

FIGURE 4

FIGURE 4. Exposure index [W/kg] versus noise variance.

6 Conclusion

Throughout this study, we highlight the significance of employing probabilistic shaping (PS) to mitigate the drastic effects of improper hardware distortions and effectively reduce the user’s EMF exposure while maintaining a target threshold. The numerical results reveal up to a 98% reduction in the exposure index with the help of PS as compared to the conventional NS. Further investigation demonstrates a minor gain with HS over PS with significant added complexity. Thus, we conclude that mere PS is the preferred choice to reduce the exposure index, given a trade-off between lowering the EI and increasing computational complexity.

Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

Author Contributions

The study was conducted as a collaboration among all authors. AE conceived the work. The manuscript was mainly drafted by SJ and was revised and approved by all co-authors.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Appendix

Appendix A: Proof of Convexity

The first- and second-order derivatives of 1/H(p) are given as

${\nabla }_{\mathbf{p}}\left(1/\mathrm{H}\left(\mathbf{p}\right)\right)=-{\mathrm{H}}^{\prime }\left(\mathbf{p}\right)/{\left[\mathrm{H}\left(\mathbf{p}\right)\right]}^2(28)$

${\nabla }_{\mathbf{p}}^2\left(1/\mathrm{H}\left(\mathbf{p}\right)\right)=-\frac{{\left[\mathrm{H}\left(\mathbf{p}\right)\right]}^2{\mathrm{H}}^{″}\left(\mathbf{p}\right)-2{\left[{\mathrm{H}}^{\prime }\left(\mathbf{p}\right)\right]}^2\mathrm{H}\left(\mathbf{p}\right)}{{\left[\mathrm{H}\left(\mathbf{p}\right)\right]}^4}\ge 0\because \mathrm{H}\left(\mathbf{p}\right)\ge 0\text{and}{\mathrm{H}}^{″}\left(\mathbf{p}\right)\le 0(29)$

The second-order derivative is always positive given the nonnegative and concave nature of information entropy. Hence, we can safely conclude that 1/H(p) is a convex function in p.