Information-Energy Capacity Region for SLIPT Systems Over Lognormal-Fading Channels

In this paper, we study the fundamental limits of simultaneous lightwave information and power transfer (SLIPT) systems over channels with path loss and lognormal fading conditions. We consider a system with a single transmitter transferring information to a photodiode-based receiver as well as transferring energy to a photovoltaic cell receiver. In particular, we study the information-energy capacity region and the optimal input distribution under (a) peak-power and average-power constraints at the transmitter, and (b) the minimum harvest energy at the energy harvesting receiver. To this end, an expression for the transition probability distribution function of the lognormal channel is derived. By extending Smith's framework and using Hermite polynomial bases, we prove that the optimal input distribution is discrete with a finite number of mass points. Information-energy capacity region for SLIPT over lognormal channel conditions is illustrated and compared with the case of additive white Gaussian noise channel.


I. INTRODUCTION
Optical wireless communication (OWC) has been recognized as a promising technology to achieve high speed, low latency, and highly secure communication.Thanks to the use of the optical spectrum, OWC overcomes the current spectrum limitation of radio frequency (RF) technology, and hence is suitable for many applications in future wireless systems such as 6G networks.OWC has been categorized into many technologies depending on the application environment, such as free-space optical communication, visible light communication, light-fidelity, underwater optical wireless communication, and non-terrestrial satellite communication [1].Light-emitting diodes (LEDs)/laser diodes (LD) are used as transmitters, while photodiodes (PD)/photovoltaic (PV) cells are used for receivers.Moreover, the existing lighting infrastructures can be easily adapted for simultaneous illumination, information transfer, and lightwave power transfer [1].
Recently, simultaneous lightwave information and power transfer (SLIPT) has emerged as a new communication paradigm that exploits optical signals for the dual purpose of information and energy transfer [2], [3].Specifically, SLIPT is promising to charge batteries in energy-deprived devices/sensors efficiently, specifically in applications where other wireless charging methods are challenging, such as underwater, inter-satellite, vehicular, and indoor systems [3].In the foundational analysis of SLIPT in [2], the maximization of energy transfer is considered while satisfying quality of service constraints.In [4], the minimum rate maximization and the transmit power minimization problems have been studied.Various SLIPT policies, receiver designs, and applications have been presented in [5].The rate-power tradeoff in SLIPT has been studied in [6].Moreover, SLIPT has been used in various use cases [7]- [9].In [7], a beamforming design in a multi-cell setup has been investigated to maximize the data rate.In [8], resonant beams have been used to demonstrate highly efficient energy transfer.Finally, SLIPT has also been applied to underwater communication systems in [9].
It is well known that closed-form expressions for the exact capacity of OWC systems are unknown, and thus, several lower and upper bounds have been derived [10], [11].Furthermore, the study of the information-energy capacity region of SLIPT has been limited, with a bound presented in [12].Most of the aforementioned studies on SLIPT systems focus on simple additive white Gaussian noise (AWGN) channels with a static path loss, and therefore the impact of the channel fading on the optimal input distribution has not been investigated.However, SLIPT systems undergo fading effects and several fading models have been used in the literature, i.e., lognormal, gamma-gamma, generalized gamma, and exponential gamma [13].As a widely used fading model, the lognormal distributions have been used to model the fading in underwater weak turbulence conditions, over-the-air systems with fog/weak atmospheric turbulence conditions [13], [14].Hence, it is crucial to study the exact information-energy capacity region and the optimal input distribution of SLIPT systems under lognormal fading conditions for future wireless system designs.
On the other hand, OWC transmitters are subjected to peakpower (PP) and average-power (AP) constraints.To avoid the non-linear distortions in the transmit signals, LED/LD needs to be operated in the linear operation range.This imposes a PP constraint on the transmit signal [10], [11].Moreover, an AP constraint is required due to safety reasons (eye safety) and practical implementation limitations [11].In SLIPT systems, the energy harvesting (EH) receiver is required to harvest a certain average harvest energy level in order to maintain the battery charging or to device operations, which impose an EH constraint [4].Despite the use of the linear EH model in the literature [4], some papers focus on a nonlinear EH model [9], [12] that fit well with hardware requirements.To the best of the authors' knowledge, this is the first work that studies the information-energy capacity region and optimal input distribution of SLIPT systems with PP, AP, and EH constraints and nonlinear EH model over lognormal-fading channels from an information-theoretic standpoint.
This paper studies the information-energy capacity region for SLIPT systems over lognormal-fading channels.Specifically, a system that transmits data to an information receiver and lightwave power to an EH receiver using a single transmitter is considered.We consider a lognormal-fading channel with a path loss and consider PP, AP, and EH constraints.An expression for the transition probability distribution of the lognormal channel is derived.By using Smith's framework [15], a Hermite polynomial base, and the transition probability of the channel, we show that the optimal input distribution is discrete with a finite number of mass points.The informationenergy capacity region of the considered system is illustrated and compared with the conventional AWGN channel.

II. SYSTEM MODEL
We consider a SLIPT system over a lognormal channel, wherein the transmitter is comprised of a single, narrow-beam LED/LD.A PD receiver is used for information reception and a PV cell receiver is used for lightwave EH, as detailed in previous works [2], [3], [5].The transmitter uses intensity modulation (IM), and hence, the instantaneous output optical power is proportional to the driving current signal.At the receiver, direct detection (DD) is used, and hence, a current is generated through the PD/PV cell that is proportional to the incident optical power.The generated current signals from the PD and the PV cell are separately sent for information decoding and EH.The harvested energy from the PV cell receiver can be used to charge a battery while the received signal from the PD is first sent through a trans-impedance amplifier to convert it to a voltage signal and then goes through the decoding stage [4].A system diagram of this model is depicted in Fig. 1.

A. Channel Model
The channel gain from the LED/LD to the PD can be modeled using h 1 = h 1,l h T , where h 1,l is the path loss, and h T is the lognormal distributed turbulence-induced fading.h 1,l is deterministic and depends on the transmitter-receiver geometry.It can be expressed as [16] where η t and η r are the optical efficiencies of the transmitter and receiver, respectively, c(λ) is the extinction coefficient of the channel, λ is the optical wavelength, l is the perpendicular distance between the transmitter plane and the receiver plane, θ 0 is the transmitter beam divergence angle, θ P D is the angle between the perpendicular axis to the receiver plane and the transmitter-PD trajectory, and A P D is the PD aperture area.
The lognormal distribution is used to model the fading coefficient h T in weak turbulence conditions in OWC, (i.e., weak oceanic turbulence and weak atmospheric turbulence).Its probability density function (pdf) is given by [13], [17] The fading coefficient in the above takes the form h T = e 2X l , where X l is the fading log-amplitude, which is Gaussian distributed with mean µ X l and variance σ 2 X l .Normalizing the fading coefficient to ensure that fading does not affect the average power leads to µ X l = −σ 2 X l .We assume that the channel gain between the LED/LD to the PV is static, and hence, it can be modeled as h 2 = h 2,l , where h 2,l is the geometric path loss from the LED/PD transmitter to the PV cell receiver.It can be obtained by replacing A P D and θ P V in (1) with A P V and θ P V , respectively, where A P V is the effective area of the PV cell and θ P V is the angle between the perpendicular axis to the receiver plane and the transmitter-PV cell trajectory.

B. Information Transfer
We consider a memoryless discrete-time lognormal channel [18].Let x ∈ R + be the transmit signal.For a given A > 0, and ε > 0, its PP and AP constraints are 0 ≤ x ≤ A, and E{x} ≤ ε [11].The received electrical signal at the information receiver is expressed as where a is the electrical-to-optical conversion efficiency at the LED/LD, R P is the responsivity of the PD, and n is the real AWGN with zero mean and variance σ 2 g , i.e., n ∼ N (0, σ 2 g ).

C. Power Transfer
To perform EH, the PV receiver generates a current from incident optical power and stores charges in a battery.The received current signal at the PV receiver is expressed as where R E is the responsivity of the PV cell.The effect of the noise can be neglected in PV cell receivers 1 .The harvested energy at the EH receiver for the time duration T is expressed as [2], [9] where f E is the fill factor of the PV cell, I s is the shunt current, and V oc is the open circuit voltage.Within our mathematical framework, I s is expressed as aR E h 2 x, i.e., the received current signal at the PV receiver, and V oc is , where v t is the thermal voltage, and I 0 is the dark saturation current [2].Combining these expressions and taking the expectation, the average harvest energy can be expressed as To satisfy the EH requirement at the receiver, ĒH needs to be greater than the given threshold value, E th , i.e., E th ≤ ĒH .

D. Problem Formulation
We form an optimization problem to maximize the mutual information between the channel input X and the channel output Y subject to PP and AP constraints at the transmitter, and a minimum harvested energy constraint at the EH receiver.The optimization problem is expressed as where b = f E v t T aR E h 2 and c = aR E h 2 /I 0 are constants, F A is the set of all input distributions that satisfy the PP constraint, i.e., F A = F ∈ F, A 0 dF (x) = 1 , and F is the set of all possible input distributions.The mutual information between random variables X and Y of input and output, respectively, can be written as a function of the input distribution F , where i(x; F ) = y p(y|x) log 2 p(y|x) p(y;F ) dy is the marginal information density.Denote by g j : F → R with j ∈ {1, 2} the following functions and, let Ω be the set of all input distributions, such that Hence, the optimization problem in ( 7) could be written in the compact form as C = sup

III. OPTIMAL INPUT DISTRIBUTION
We study the properties of the capacity-achieving distribution of the SLIPT system, the solution to the optimization problem in (7).To this end, the mathematical framework proposed in [15] is extended to the lognormal channel with a EH receiver.Firstly, Theorem 1 establishes the existence and the uniqueness of the optimal input distribution.Secondly, by using the Lagrangian Theorem, the dual equivalent problem is given by Corollary 1. Thirdly, we provide necessary and sufficient conditions for the optimal input distribution in Theorem 2 and extend to a more useful set in Corollary 2. Finally, we show that the capacity-achieving input distribution is discrete in Theorem 3.
Theorem 1.The capacity C is achieved by a unique input distribution F * , i.e., C = sup Proof.The proof follows a similar approach as in [18].
Corollary 1.The strong duality holds for the optimization problem in (12).i.e., there are constants, λ j ≥ 0, for j ∈ {1, 2} such that Proof.The proof follows a similar approach as in [18].
Theorem 2. F * is the capacity-achieving input distribution, if and only if, ∀F ∈ Ω there exist λ j > 0, for j ∈ {1, 2} such that Proof.The proof follows a similar approach as in [19].
Corollary 2. Let Supp(F * ) be the points of support of a distribution F * .Then, F * is the optimal input distribution, if there exist λ 1 ≥ 0, and λ 2 ≥ 0, such that for all x, with equality if x ∈ Supp(F * ).
Proof.The proof follows a similar approach as in [19].Due to the space limitations, we do not provide it here.
In the following, we will show that the equality in Corollary 2 can not be satisfied in a set that has an accumulation point, hence the support of F * must be discrete.The discretness property of the optimal input distribution is given by the following theorem, Theorem 3. The optimal input distribution that achieves the capacity in (7) is discrete with a finite number of mass points.
Proof.The proof is presented in Appendix A.

IV. NUMERICAL RESULTS AND DISCUSSION
We characterized the optimal input distribution for a lognormal channel with AP, PP, and EH constraints.Now, we numerically evaluate the information-energy capacity region by using a numerical solver such as CVX [20].Unless stated otherwise, the parameter values are set to, a = 20 W/A, , and σ 2 Xt = 0.1 [2], [4], [9].Fig. 2 shows the information-energy capacity region for different PP constraints.The region corresponding to this scenario is determined through the resolution of the optimization problem presented in (7).A notable trade-off emerges between the information rate sent to the information decoder and the energy rate to the EH receiver.This trade-off becomes apparent as higher EH constraints lead the transmitter to choose a symbol with greater amplitude, consequently reducing the performance of information transfer.It is noteworthy that, for lower amplitude constraints, there is no trade-off between the two objectives.In this regime, the optimal input distribution is binary, maximizing both information and energy transfer simultaneously.
Fig. 3 highlights the impact of the channel on the information-energy capacity region.The comparison reveals a noticeable gap between the two regions, and this discrepancy is primarily attributed to the adverse effects introduced by the lognormal channel.Specifically, we observed that the information energy capacity region of the Gaussian channel outperforms that of the lognormal channel.This gap in performance can be attributed to the inherent characteristics of the lognormal channel, which introduces challenges such as signal attenuation and fluctuations.

V. CONCLUSION
In this paper, we studied the information-energy capacity region of a SLIPT system over a lognormal-fading channel with nonlinear EH, where the PP, AP, and EH constraints were considered.An expression for the transition probability distribution of the lognormal channel was derived.By using Smith's framework and introducing appropriate Hermite polynomial bases, we proved that the optimal input distribution is discrete with a finite number of mass points.Numerical results of the information-energy capacity region were presented and compared it with AWGN channel conditions.Results show that under low PP constraints the optimal distribution is binary and no trade-off between the information rate and the EH.Further, lognormal fading results in degradation of the informationenergy capacity region as compared with AWGN channel conditions.
APPENDIX A PROOF OF THEOREM 3 We show that the equality in (15) can not be satisfied on a set of points that has an accumulation point, which indicates that the set Supp(F * ) must be discrete and the optimal input X must be a discrete random variable.We start with the necessary and sufficient conditions for the optimality of F * .Extending the necessary and sufficient conditions of Corollary 2 to the complex domain, the left-hand side (LHS) of ( 15 where the domain D is defined by ℜ(z) > 0. The function s(z) is analytic over the complex domain, since the linear function, and the logarithmic function are all analytic.The necessary condition for the optimal input distribution F * is that s(z) must be zero ∀z ∈ Supp(F * ).However, from the identity theorem, if the set Supp(F * ) has an accumulation point and the analytic function s(z) = 0, ∀z ∈ D, then s(z) is necessarily zero over the whole D, and hence, from

A. Conditional Probability of the Channel
The conditional probability p Y |X (y|x) of the lognormal channel is not present in the literature.Hence, we derive p Y |X (y|x), which will be used to derive the properties of

Fig. 1 .
Fig. 1.A SLIPT system over a lognormal-fading channel with an LED/LD transmitter, a PD-based information decoder, and a PV cell EH receiver.