# New Practical Nonlinear Energy-Harvesting Models for Wireless-Powered Communications

Authors: Mohammadreza Babaei; Lütfiye Durak-Ata; Ümit Aygölü
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

As a massive number of nodes are expected to be connected to each other in next-generation wireless networks, the power supply of such multiple-antenna nodes is a challenge and requires new and sustainable solutions. Radio frequency energy harvesting (EH) is among the promising approaches to solve this issue. In this paper, two new mathematically tractable nonlinear EH (NL-EH) models are proposed, for which the system performance is evaluated in terms of average harvested power, throughput, and bit error probability. The system consists of one multiantenna power-constraint source harvesting its power from a dedicated power beacon and transmitting its signal to a destination equipped with multiple antennas. For a comprehensive analysis of the system, closed-form expressions are derived for Nakagami-m fading channels and the special case of Rayleigh channels. For comparison purposes, the performance derivations in Nakagami-m fading channels are provided for linear and piece-wise linear EH models given in the literature. Besides, the simulation results are obtained by applying the Monte-Carlo method for NL-EH models existing in the literature. The results provide a broadening view of EH systems and thoroughly compare the proposed NL-EH models with linear, piece-wise linear, and NL-EH models available in the literature. Hence, these provide better insights towards the analyses and design of EH systems.


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## Body

## 1. Introduction

In next-generation wireless communications, a huge number of nodes with multiple antennas will be connected to each other, and power requirements will be the critical constraint for sustainability. Wireless energy-harvesting (EH) methods are considered among promising solutions to the energy shortage problem of these nodes. Considering the constraint energy resources and green communication, radio frequency (RF) EH provides a small step to solve the energy shortage because it replaces the usage of batteries and electronic power supplies at the sensor nodes. The EH node is equipped with an EH circuit which provides the ability to harvest the incoming signal energy. Wireless power transfer (WPT) and simultaneous wireless information and power transfer (SWIPT) are two EH methods considered in the design of battery-less nodes [1–3]. Based on the EH circuit feature, the power-constraint node can harvest the total incoming signal power or simultaneously harvest energy and process the information [1, 2]. Power splitting (PS) and time switching (TS) are examples of the SWIPT method. In the WPT, a dedicated power beacon (PB) or ambient power nodes are the sources from which the power-constraint nodes harvest their power [3]. In the above-mentioned type of EH circuits and EH sources, the important factors are the amount of harvested power and the characteristic of the EH circuit which directly affect the EH system performance. Linear, piece-wise linear, and nonlinear EH (NL-EH) models characterize the amount of harvested power [1, 4–8]. Most works in the literature consider the linear EH model which has a direct proportion between the input and output powers of the EH circuit [1, 9, 10]. On the other hand, the piece-wise linear EH model is considered in the literature [4, 5] which provides the same amount of energy as in the linear model for the small amount of input power. Different from the linear EH model, it saturates the output power to a fixed maximum value at high input powers.With the measured data from the practical experimental tests given in [11–24], it is figured out that the input and output power relation of the EH circuit is nonlinear and the output power saturates to a maximum power even increasing the input power. Applying the curve fitting over the measurement data, some NL-EH models are proposed in the literature [6–8]. This characteristic of the EH node is different from the linear EH model in which the harvested power is directly proportional to the input power and the harvested power increases with rise in the input power of the EH circuit. Consequently, this results in an overestimation of the amount of harvested power, and this yields an inaccurate understanding of EH system performance. The system performance in terms of outage probability and bit error probability (BEP) is investigated well for the linear EH models [1, 2] and the piece-wise linear EH model [3] given in the literature. However, the system performance of the NL-EH models is mostly studied in terms of the amount of harvested energy and its statistical characteristics [6, 7, 25] since these NL-EH models are not mathematically tractable when investigating their bit error performances. Additionally, a small number of studies investigated the throughput [8]. This problem comes from the fact that for the NL-EH models given in the literature, it is impossible to calculate the probability density function (PDF) or cumulative density function (CDF) of the received signal-to-noise ratio (SNR), which prevents the derivations of BEP or throughput.The ergodic capacity and throughput of dual-hop (DH) amplify-and-forward (AF) relaying system obtained from outage analysis are investigated in [1]. The results are provided for PS, TS, and ideal EH modes where a linear EH model is applied at the power-constraint node. The results are obtained for different parameters, and the authors show the discrepancy of the PS and TS EH modes in terms of the throughput from the ideal EH mode. The closed-form expressions for outage probability and BEP are derived in SWIPT-based DH AF and DH decode-and-forward (DF) relaying systems [2]. The channels are subject to Nakagami-m fading, and the power-constraint relay node applies the linear EH model. The performance is provided for different system parameters, and generally, the PS EH mode outperforms the TS EH mode. System performance of the WPC DH AF relaying system is investigated in [26] where both S and R are power-constraint nodes and harvest their power from a dedicated PB. BER of the system is obtained for different parameters, and theoretical derivations are derived considering the linear EH model.A DH AF relaying system with a power-constraint relay is considered in [4] where the relay applies the TS mode to harvest its power based on the piece-wise linear EH model. The outage capacity of the studied system is derived considering Nakagami-m fading channels. A multirelay system where source (S) and destination (D) communicate with each other with the help of power-constraint relay nodes is investigated under the Rayleigh fading channel assumption [27]. Outage probability and throughput of the system are derived for the piece-wise linear EH model and imperfect channel state information (CSI). In the studied system, both transmit antenna and relay selections are considered for the transmission protocol. In [3], an overlay cognitive radio with a power-constraint secondary transmitter (ST) is investigated. The authors assumed that ST harvests its power from the primary transmitter and applies the piece-wise linear EH model. The closed-form expressions for the outage probabilities and BEPs are derived considering both linear and piece-wise linear EH models which provide a good insight for comparison purposes.The throughput of the DH AF relaying system considering Rayleigh fading is studied in [5]. TS and PS EH modes with a segmented piece-wise linear EH model are applied in the battery-less relay node. The authors show that the PS mode outperforms the TS mode. SWIPT-based cooperative nonorthogonal multiple access (NOMA) system performance is studied in [28] where the near user is the power-constraint and applies a segmented piece-wise linear EH model. The theoretical derivations of outage probability and system throughput are provided for the Rayleigh fading channel.In [7], a NL-EH model based on logistic (sigmoidal) function is proposed. Assuming a nonconvex optimization and simplifying it to a convex optimizing problem, the total harvested power is maximized and the average harvested power is calculated. The rate-energy trade-off region considering both linear and NL-EH models is obtained in [29]. The harvested power versus the number of receive antennas is investigated. In [30], the average harvested power of the point-to-point MIMO WPT system is calculated based on different system parameters. The authors study the maximization of harvested power jointly by optimizing the PS ratio factor and the transmit energy of PB. A MISO NOMA SWIPT system is investigated in [31] where the authors study the physical layer security by adopting the NL-EH model from [7]. The authors simplify the nonconvex optimization problem to a convex one.The quadratic NL-EH model is proposed and studied in a multiuser orthogonal frequency-division multiple access (OFDMA) system [6]. Results are given compared to the linear EH model. Moreover, it is assumed that the parameters of the proposed function in [6] take values of α1<0 and α2>0, and α3<0 which is shown in Table 1. However, a high input power results in a negative value for harvested power. Hence, this proposed model is only valid for small input powers. A heuristic NL-EH model is proposed in [8] applying curve fitting. The throughput of the AF cooperative relaying system using PS and TS modes is investigated where the results are numerically obtained since the proposed model is not mathematically tractable.Table 1
Existing NL-EH and linear EH models.
EH modelsHarvested powerPsLinear [1]Y=ηXPiece-wise linear [4]Y=ηX,X<Pth,ηPth,X>Pthi-segmentPiece-wise linear [5]Y=0,0<X<Pth1,a1X+b1,Pth1<X<Pth2,⋯,Pthi−1<X<Pthi,Pthi,X>PthiNL-EH [6]Y=α1X2+α2X+α3NL-EH [7]Y=M/1+e−aX−b−M/1+eab1−1/1+eabNL-EH [8]Y=p2X3+p1X2+p0Xq3X3+q2X2+q1X+1The NL-EH models given above are based on the curve fitting of experimental measured data. The harvested powers given in [6–8] include the polynomial, exponential, or cubic functions of the input power. Consequently, all these NL-EH models given in the literature are not mathematically tractable, and the harvested power is not provided as PDF and CDF to further calculate the EH system performance. These are the main motivations behind our article.In this paper, we propose two NL-EH models which are mathematically tractable in terms of the PDF, CDF of harvested power, and the system performance at the expense of a small gap in the accuracy between measured data points and those from our models. Apart from the NL-EH models given in the literature, our models depend directly on the system input power rather than the polynomial, exponential, or cubic functions of it. The considered system consists of a PB from where a power-constraint S harvests energy which is then used to transmit data to a D. The channels are assumed subject to Nakagami-m and Rayleigh fading. The former is assumed when the channels between nodes are stronger and the distances are short. In the latter, the link between nodes is subjected to the deleterious effect of the environment in the lack of a line-of-sight component.For both models, PDF and CDF expressions for the harvested power are obtained based on the curve fitting of the measurement data from [11, 15]. Moreover, the average harvested powers are calculated using the PDF of harvested power. Furthermore, the outage probability, throughput, and BEP of the considered system are derived in closed forms for the proposed two NL-EH models. Note that the NL-EH models considered in the literature do not provide closed-form expressions for the system performance since they are not mathematically tractable [6–8]. In the papers considering these models, only the amount of harvested power at the EH receiver is studied without investigating the system BEP or throughput performance.Numerical results provide a comprehensive analysis for all the considered EH models in terms of system performance. The results show that the distance and number of antennas have crucial effects on the system performance which gives an insight into maximizing the system performance.The main contributions of the paper are as follows:(i)
Two mathematically tractable NL-EH models based on the curve fitting of measured data are proposed(ii)
PDF and CDF of the harvested power for the proposed NL-EH models are provided(iii)
Closed-form expressions for the average harvested power, throughput, and BEP are derived, and the results are supported by Monte-Carlo simulations(iv)
Theoretical derivations of the system performance are provided for both Nakagami-m and Rayleigh fading(v)
Results are compared with existing linear, piece-wise linear, and NL-EH models from the literatureThe paper is organized as follows. Section2 describes the considered system model. The proposed NL-EH models are presented in Section 3. Expressions for the average harvested powers are obtained in Section 4. Section 5 deals with the derivation of throughput expressions. Bit error rate (BER) analyses are performed in Section 6. Theoretical and simulation results are presented in Section 7. Section 8 concludes the paper.


## 2. System Model

The considered half-duplex RF EH system where a power-constraint S harvests its power from a PB is given in Figure1. S and D are equipped with Ns and Nd receive antennas, respectively, while PB is equipped with one antenna. S transmits its signal from one antenna to D using the harvested power. dpbs and dsd denote the distances of links PB→S and S→D, respectively. Additionally, the path loss is represented as Li=1/1+diν where i∈pbs,sd, and ν is the path-loss coefficient. xs is S’s transmitted signal with Exs2=1. h=h1,h2,⋯,hNs and g=g1,g2,⋯,gNd represent the channel gain vectors of links PB→S and S→D, respectively. Furthermore, n=n1,n2,⋯,nNd is the additive white Gaussian noise (AWGN) vector with entries as i.i.d complex Gaussian random variable ni~CN0,N0 for i∈1,2,⋯,Nd. Transmission of the S signal takes T seconds where T/2seconds are allocated for harvesting energy and the remainder for IP. It is assumed that the CSI is perfectly known at node D and channels are assumed to be flat fading and remain constant during each transmission interval and vary independently from an interval to another. Line-of-sight (LoS) and distance are two determinant factors considered in the RF EH communication systems. Since the path loss causes a substantial decrease in the amount of harvested power, the EH communication is possible for short distances and in the presence of a LoS component. Generally, Rician fading is a proper channel model with its LoS component. However, mathematical derivations of the system performance considering the Rician PDF and CDF are not tractable. Meanwhile, Nakagami-m fading is a good approximation of the Rician model [32, 33]. Hence, it is assumed that both links PB→S and S→D are subject to Nakagami-m fading with parameters mh and mg, respectively. Here, we consider that the channel power is identical for each link. In other words, Ωh=Ehi2 and Ωg=Egj2 for i∈1,2,⋯,Ns, and j∈1,2,⋯,Nd, respectively. In the first time slot, the input power of the EH circuit provided in Figure 1 is calculated as
(1)Pin=PTLpbs∑i=1Nshi2,where PT is the energy-bearing signal transmitted from PB. In the second time slot, the received signal at D is given as
(2)y=PsLsdgxs+n,where y=y1,y2,⋯,yNd, g=g1,g2,⋯,gNd, and Ps is the transmit power of S. Considering (1), we have
(3)Ps=fPin,where f· is given in Figure 1. The received SNR after applying the maximum ratio combination (MRC) at node D is given using (2) as
(4)γ=YZ,where Z=Lsd∑i=1Ndgi2/N0. For simplicity, we define X=Pin and Y=Ps in the sequel. The CDF and PDF of r.v.s Z and X are given as [33]
(5)FZz=1−∑i=0mz−1zϕzii!exp−zϕz,(6)fXx=Dxmx−1exp−xϕx,Figure 1
Considered half-duplex wireless-powered system model whereh and g represent channel gain vectors. Pin and Ps are the input and output powers of the EH circuit, respectively. f· is the function converting the input power of the EH circuit to the output power.respectively, whereϕz=mz/z¯, z¯=LsdΩz/N0, Ωz=NdΩg, mz=Ndmg, D=ϕxmx/Γmx, ϕx=mx/x¯, x¯=PTLpbsΩx, Ωx=NsΩh, and mx=Nsmh.


## 3. Linear and NL-EH Models

In the literature, different linear and NL-EH models are provided for the performance evaluation of EH communication systems [1, 4–8]. These EH models are listed in Table 1. For the linear EH model, the harvested power is linearly dependent on the input power of the EH circuit with a constant ratio parameter, namely, the conversion efficiency (η). However, based on the data measurements from designed practical EH circuits, linear EH models overestimate the amount of the harvested energy especially for high input powers which consequently results in a misunderstanding of the system performance. Moreover, the EH circuits have nonlinear properties which can be modeled by a nonlinear function of the amount of harvested power with respect to the input power. Furthermore, the piece-wise linear EH model is provided in the literature where the harvested amount of energy is saturated to a threshold power as Pth while acting the same as the linear EH model for the small input powers as seen from Table 1. In addition to these EH models, NL-EH models are also proposed, applying curve fitting from the measurement data experiments from different EH circuits (Table 1). Please note that the theoretical derivations of system performance are not mathematically tractable considering these known NL-EH models. However, the linear and piece-wise linear EH models are mathematically tractable and provide a general and simple analysis of the system. On the other hand, the proposed NL-EH models in the sequel enable mathematically tractable theoretical derivations of system performance. This provides an advantage on the design and evaluation of EH systems.


### 3.1. Proposed NL-EH Models

In this subsection, two EH models are proposed with their relations between input and output powers. The first proposed NL-EH model (NL-EH-1) is given as(7)Y=fX=MeX−bcX+Mb/e.Here,M denotes the maximum saturation power of the EH circuit, and b, c, and e are constants to be determined by curve fitting. Applying (7), the CDF of the harvested power of NL-EH-1 related to the input power is calculated as
(8)FYy=PraX−bcX+d<y=PrXa−y<yd+bc=FXyd+bca−y,−bcd<y<a,1,y>a,where a=Me, d=Mb/e, and
(9)FXx=1−∑i=0mx−1xϕxii!exp−xϕx.Moreover, the PDF of NL-EH-1 is calculated using (8) as
(10)fYy=da+bca−y2fXyd+bca−y,−bcd<y<a,0,y>a,where fXx is given in (6).The second proposed NL-EH model (NL-EH-2) is given as(11)Y=fX=McXX+b.Similar to NL-EH-1, the CDF of harvested power is obtained as(12)FYy=PraXX+b<y=PrXa−y<yb=FXyba−y,0<y<a,1,y>a,where a=Mc. Taking the derivation of (12), the PDF of NL-EH-2 is calculated as
(13)fYy=baa−y2fXyba−y,0<y<a,0,y>a.Figures2(a) and 2(b) describe the curve fitting of the measured experimental data compared to the linear EH and NL-EH models existing in the literature and the two proposed models in (7) and (11). The results of the curve fitting are obtained based on the measured data from [11, 15]. The measured data are denoted by symbols while the fitted data are represented by lines and dashed lines.Figure 2
Harvested power versus input power comparison of the proposed and existing EH models with experimental results from (a) [11] and (b) [15].
(a)(b)It is seen from both Figures2(a) and 2(b) that both the proposed NL-EH models fit better to data points than the linear and piece-wise linear models [1, 4]. However, as stated, the challenging issue of the existing NL-EH models are the difficulties in mathematical derivations of CDF and PDF of the harvested power and theoretical derivations of throughput as well as BER system performances. In addition, it is observed that apart from the NL-EH model in [6], the other NL-EH models provide similar curve-fitting functions. Moreover, the NL-EH model in [8] provides the best fit compared to all other NL-EH models. However, despite these similarities, these models given in the literature are not mathematically tractable which prevent theoretical investigation of the EH system performance. Meanwhile, the proposed NL-EH models which have an acceptable goodness in fitting are mathematically tractable.Moreover, the goodness of the fits considering the NL-EH models are provided in Table2 for [11, 15], respectively. R2 (R-squared) is the parameter that is statistically calculated from curve-fitting functions and data measurements [34]. The strength of the fit of a model is most commonly evaluated using R2 which always takes values between 0 and 1. This statistical parameter measures how well the data is close to being fit. The closer the value to 1, the fit is better. Considering both Figures 2 and 3 and Table 2, the proposed NL-EH model functions provide similar results as their counterparts given in the literature. However, the important factor in choosing the NL-EH model is having mathematical tractability while being well fit to the data measurements. In other words, a trade-off between accuracy and mathematical tractability is considered. Furthermore, the coefficient parameters of NL-EH-1 and NL-EH-2 considered in (7) and (11) are obtained with curve fitting and provided in Tables 3 and 4, respectively. As provided in both Tables 3 and 4, the parameters of these models are obtained for measurement results of [11–24].Table 2
Goodness comparison of curve fitting for different NL-EH models with data measurements taken from [11, 15].
[11][15]EH modelsR2R2NL-EH-1 (7)0.99970.99994NL-EH-2 (11)0.990130.98066NL-EH [7]0.992770.99126NL-EH [6]0.989790.99328NL-EH [8]0.99990.99901Figure 3
Histogram and theoretical PDF of (a) NL-EH-1 and (b) NL-EH-2 where data measurements are taken from [12], and Ns=1 and Ω=Ω1.
(a)(b)Table 3
Fitting parameters of the proposed NL-EH-1 (7) in different energy-harvesting circuits.
ReferencesMebc[11]244.141.468072.585182003.91475[12]24.646431.302450.50571299.49425[13]9.491.268044.415316.40408[14]994.721.543742.096210.01172[15]7.331.488395.33243.70781[16]87.325.1276951.633262.14287e−4[17]71.421.185172.906846.45064e−4[18]407.732.131497.915120.00272[19]503.952.113276.718260.00122[20]156.8315.38538442.764351.25506e−5[21]225.061.649253.536216.04258e−4[22]594.753.809656.638720.00194[23]924.493.1090613.739950.00653[24]345.3931.64892.862080.03637Table 4
Fitting parameters of the proposed NL-EH-2 (11) in different energy-harvesting circuits.
ReferencesMcb[11]244.141.63202582.15452[12]24.646431.5144826.8605[13]9.491.394470.04939[14]994.721.581951.46469[15]7.331.8059445.66596[16]87.321.18247e142.96048e13[17]71.421.377420.32895[18]407.732.543652.12182[19]503.952.23531.77768[20]156.8333.9674710.38453[21]225.061.770040.56606[22]594.755.557314.38647[23]924.494.08766.04377[24]345.3931.665256.17207Figures3(a) and 3(b) show the histograms of the PDFs of NL-EH-1 and NL-EH-2 given by (10) and (13) for (6), respectively. Moreover, the histograms of both PDFs are provided for different channel powers and parameters where it is assumed that Ns=1, and Ω1=Ω. It is seen from both Figures 3(a) and 3(b) that the histograms overlay the theoretical PDFs for the intended distributions. This verifies the accuracy of the derivations of both PDFs for NL-EH models. Please note that for all cases, the area of shaded regions is always equals to one which verifies the PDF calculations given in (10) and (13). Also, note that both histograms are bounded to the given boundaries defined in (10) and (13).


## 3.1. Proposed NL-EH Models

In this subsection, two EH models are proposed with their relations between input and output powers. The first proposed NL-EH model (NL-EH-1) is given as(7)Y=fX=MeX−bcX+Mb/e.Here,M denotes the maximum saturation power of the EH circuit, and b, c, and e are constants to be determined by curve fitting. Applying (7), the CDF of the harvested power of NL-EH-1 related to the input power is calculated as
(8)FYy=PraX−bcX+d<y=PrXa−y<yd+bc=FXyd+bca−y,−bcd<y<a,1,y>a,where a=Me, d=Mb/e, and
(9)FXx=1−∑i=0mx−1xϕxii!exp−xϕx.Moreover, the PDF of NL-EH-1 is calculated using (8) as
(10)fYy=da+bca−y2fXyd+bca−y,−bcd<y<a,0,y>a,where fXx is given in (6).The second proposed NL-EH model (NL-EH-2) is given as(11)Y=fX=McXX+b.Similar to NL-EH-1, the CDF of harvested power is obtained as(12)FYy=PraXX+b<y=PrXa−y<yb=FXyba−y,0<y<a,1,y>a,where a=Mc. Taking the derivation of (12), the PDF of NL-EH-2 is calculated as
(13)fYy=baa−y2fXyba−y,0<y<a,0,y>a.Figures2(a) and 2(b) describe the curve fitting of the measured experimental data compared to the linear EH and NL-EH models existing in the literature and the two proposed models in (7) and (11). The results of the curve fitting are obtained based on the measured data from [11, 15]. The measured data are denoted by symbols while the fitted data are represented by lines and dashed lines.Figure 2
Harvested power versus input power comparison of the proposed and existing EH models with experimental results from (a) [11] and (b) [15].
(a)(b)It is seen from both Figures2(a) and 2(b) that both the proposed NL-EH models fit better to data points than the linear and piece-wise linear models [1, 4]. However, as stated, the challenging issue of the existing NL-EH models are the difficulties in mathematical derivations of CDF and PDF of the harvested power and theoretical derivations of throughput as well as BER system performances. In addition, it is observed that apart from the NL-EH model in [6], the other NL-EH models provide similar curve-fitting functions. Moreover, the NL-EH model in [8] provides the best fit compared to all other NL-EH models. However, despite these similarities, these models given in the literature are not mathematically tractable which prevent theoretical investigation of the EH system performance. Meanwhile, the proposed NL-EH models which have an acceptable goodness in fitting are mathematically tractable.Moreover, the goodness of the fits considering the NL-EH models are provided in Table2 for [11, 15], respectively. R2 (R-squared) is the parameter that is statistically calculated from curve-fitting functions and data measurements [34]. The strength of the fit of a model is most commonly evaluated using R2 which always takes values between 0 and 1. This statistical parameter measures how well the data is close to being fit. The closer the value to 1, the fit is better. Considering both Figures 2 and 3 and Table 2, the proposed NL-EH model functions provide similar results as their counterparts given in the literature. However, the important factor in choosing the NL-EH model is having mathematical tractability while being well fit to the data measurements. In other words, a trade-off between accuracy and mathematical tractability is considered. Furthermore, the coefficient parameters of NL-EH-1 and NL-EH-2 considered in (7) and (11) are obtained with curve fitting and provided in Tables 3 and 4, respectively. As provided in both Tables 3 and 4, the parameters of these models are obtained for measurement results of [11–24].Table 2
Goodness comparison of curve fitting for different NL-EH models with data measurements taken from [11, 15].
[11][15]EH modelsR2R2NL-EH-1 (7)0.99970.99994NL-EH-2 (11)0.990130.98066NL-EH [7]0.992770.99126NL-EH [6]0.989790.99328NL-EH [8]0.99990.99901Figure 3
Histogram and theoretical PDF of (a) NL-EH-1 and (b) NL-EH-2 where data measurements are taken from [12], and Ns=1 and Ω=Ω1.
(a)(b)Table 3
Fitting parameters of the proposed NL-EH-1 (7) in different energy-harvesting circuits.
ReferencesMebc[11]244.141.468072.585182003.91475[12]24.646431.302450.50571299.49425[13]9.491.268044.415316.40408[14]994.721.543742.096210.01172[15]7.331.488395.33243.70781[16]87.325.1276951.633262.14287e−4[17]71.421.185172.906846.45064e−4[18]407.732.131497.915120.00272[19]503.952.113276.718260.00122[20]156.8315.38538442.764351.25506e−5[21]225.061.649253.536216.04258e−4[22]594.753.809656.638720.00194[23]924.493.1090613.739950.00653[24]345.3931.64892.862080.03637Table 4
Fitting parameters of the proposed NL-EH-2 (11) in different energy-harvesting circuits.
ReferencesMcb[11]244.141.63202582.15452[12]24.646431.5144826.8605[13]9.491.394470.04939[14]994.721.581951.46469[15]7.331.8059445.66596[16]87.321.18247e142.96048e13[17]71.421.377420.32895[18]407.732.543652.12182[19]503.952.23531.77768[20]156.8333.9674710.38453[21]225.061.770040.56606[22]594.755.557314.38647[23]924.494.08766.04377[24]345.3931.665256.17207Figures3(a) and 3(b) show the histograms of the PDFs of NL-EH-1 and NL-EH-2 given by (10) and (13) for (6), respectively. Moreover, the histograms of both PDFs are provided for different channel powers and parameters where it is assumed that Ns=1, and Ω1=Ω. It is seen from both Figures 3(a) and 3(b) that the histograms overlay the theoretical PDFs for the intended distributions. This verifies the accuracy of the derivations of both PDFs for NL-EH models. Please note that for all cases, the area of shaded regions is always equals to one which verifies the PDF calculations given in (10) and (13). Also, note that both histograms are bounded to the given boundaries defined in (10) and (13).


## 4. Average Harvested Power of the Proposed NL-EH Models

In this section, average harvested power of the two proposed NL-EH models are investigated. Additionally, for comparison purposes, the average harvested powers of the linear and piece-wise linear EH models are calculated.


### 4.1. Average Harvested Power of NL-EH-1 Model

The average harvested power of the NL-EH-1 model given in (7) is calculated as
(14)Y¯1=∫0∞aX−bcX+dfXxdx=N1−N2,where
(15)N1=∫0∞aXX+dfXxdx,(16)N2=bc∫0∞1X+dfXxdx.Here,fXx is given in (6). Substituting (6) in (15) and (16) and applying ([35], 3.383-10), we have
(17)N1=DadmxexpdϕxΓmx+1Γ−mx,dϕx,(18)N2=Dbcdmx−1expdϕxΓmxΓ1−mx,dϕx,where Γ·,· is the upper incomplete Gamma function ([35], 8.350-2). Finally, the average harvested power for NL-EH-1 given in (7) is obtained by substituting (17) and (18) in (14).


### 4.2. Average Harvested Power of NL-EH-2 Model

The average harvested power of NL-EH-2 model given in (11) is calculated as
(19)Y¯2=∫0∞aXX+bfXxdx,where fXx is given in (6). Applying ([35], 3.383-10), the average harvested power is obtained as
(20)Y¯2=DabmxexpbϕxΓmx+1Γ−mx,bϕx.


### 4.3. Average Harvested Power of Linear and Piece-Wise Linear EH Models

The average harvested powers of linear and piece-wise linear EH models given in Table1 are written as
(21)Y¯l=∫0∞ηxfXxdx,(22)Y¯pw=∫0PthηxfXxdx⏟U1+PM∫Pth∞fXxdx⏟U2,respectively, wherePM=ηPth. Substituting (6) in (21), simplifying and applying ([35], 3.326-2), the average harvested power of the linear EH model for the considered system is obtained as
(23)Y¯l=ηDΓmx+1ϕxmx+1.Moreover, for the piece-wise linear EH model in (22), by substituting (6), we have
(24)U1=ηDγmx+1,ϕxPthϕxmx+1,(25)U2=1−FXPth,where FX· is given in (9). Here, (24) is calculated by ([35], 3.351-1).Based on the obtained expressions given in equations (14), (19), (21), and (22), the average harvested power depends on the PDF of the input power as well as the channel path loss parameter, number of harvesting antennas Ns, and the transmit power of PB, as seen from (6).


## 4.1. Average Harvested Power of NL-EH-1 Model

The average harvested power of the NL-EH-1 model given in (7) is calculated as
(14)Y¯1=∫0∞aX−bcX+dfXxdx=N1−N2,where
(15)N1=∫0∞aXX+dfXxdx,(16)N2=bc∫0∞1X+dfXxdx.Here,fXx is given in (6). Substituting (6) in (15) and (16) and applying ([35], 3.383-10), we have
(17)N1=DadmxexpdϕxΓmx+1Γ−mx,dϕx,(18)N2=Dbcdmx−1expdϕxΓmxΓ1−mx,dϕx,where Γ·,· is the upper incomplete Gamma function ([35], 8.350-2). Finally, the average harvested power for NL-EH-1 given in (7) is obtained by substituting (17) and (18) in (14).


## 4.2. Average Harvested Power of NL-EH-2 Model

The average harvested power of NL-EH-2 model given in (11) is calculated as
(19)Y¯2=∫0∞aXX+bfXxdx,where fXx is given in (6). Applying ([35], 3.383-10), the average harvested power is obtained as
(20)Y¯2=DabmxexpbϕxΓmx+1Γ−mx,bϕx.


## 4.3. Average Harvested Power of Linear and Piece-Wise Linear EH Models

The average harvested powers of linear and piece-wise linear EH models given in Table1 are written as
(21)Y¯l=∫0∞ηxfXxdx,(22)Y¯pw=∫0PthηxfXxdx⏟U1+PM∫Pth∞fXxdx⏟U2,respectively, wherePM=ηPth. Substituting (6) in (21), simplifying and applying ([35], 3.326-2), the average harvested power of the linear EH model for the considered system is obtained as
(23)Y¯l=ηDΓmx+1ϕxmx+1.Moreover, for the piece-wise linear EH model in (22), by substituting (6), we have
(24)U1=ηDγmx+1,ϕxPthϕxmx+1,(25)U2=1−FXPth,where FX· is given in (9). Here, (24) is calculated by ([35], 3.351-1).Based on the obtained expressions given in equations (14), (19), (21), and (22), the average harvested power depends on the PDF of the input power as well as the channel path loss parameter, number of harvesting antennas Ns, and the transmit power of PB, as seen from (6).


## 5. Outage and Throughput Analyses

In this section, closed-form expressions are derived for the outage probability and throughput of the overall system PB→S→D considering both proposed NL-EH models in Nakagami-m fading channels. The throughput of the considered system is calculated as
(26)τ=T21−FγγthRT,where γth=22R−1 and R bits/sec/Hz is the target rate. Moreover, Fγ· is the CDF of the received SNR given in (4).


### 5.1. Outage Probability and Throughput of NL-EH-1 Model

Considering (4) and (7), the conditional CDF of the received SNR is given as
(27)Fγγ=FZγX+daX−bcX,bca<x<∞,1,0<x<bca,where FZz is given in (5). Averaging over X using (6), we have
(28)Fγγ=∫0bc/afXxdx+∫bc/a∞FZγX+daX−bcXfXxdx.(28) is rewritten by substituting (5) and (6) as
(29)Fγγ=1−λi,γ∫bc/a∞x+dax−bcixmx−1exp−xϕxexp−ϕzγx+dax−bcdx,where λi,γ=D∑i=0mz−1γϕzi/i!. By defining ax−bc=t and after some mathematical simplification, (29) is rewritten as
(30)Fγγ=1−λi,γexp−ϕzγaexp−ϕxbca1amx+i×∫0∞t−it+jit+bcmx−1exp−ϕxta−ϕzγJatdt,where J=da+bc. Using ([35], 1.111), we have
(31)Fγγ=1−D∑i=0mz−1γϕzii!exp−ϕzγaexp−ϕxbca1amx+i∑k=0iikJi−k∑l=0mx−1mx−1lbcmx−1−l×∫0∞tk+l−iexp−ϕxta−ϕzγJatdt.Applying ([35], 3.471-9), the CDF of the received SNR at D is calculated as
(32)Fγγ=1−ϖi,k,lγu+i/2exp−ϕzγa×Ku−i2ϕxϕzJa2γ,where u=k+l+1 and
(33)ϖi,k,l=2D1amxexp−ϕxbca∑i=0mz−1ϕz/aii!×∑k=0iikJi−k∑l=0mx−1mx−1lbcmx−1−l×ϕzJϕxk+l−i+1/2,andK·· is the modified Bessel function of the second kind ([35], 8.407-2). The throughput of the considered system for the NL-EH-1 model given in (7) is calculated by substituting Fγγ from (32) into (26).


#### 5.1.1. Special Case of Rayleigh Fading for NL-EH-1 Model

For the case where both links from PB→S and S→D are exposed to Rayleigh fading and both S and D are equipped with one antenna, the CDF and PDF of (5) and (6) are simplified as
(34)FZz=1−exp−zz¯,(35)fXx=1x¯exp−xx¯,respectively, wherex¯=PBLpbsΩx, z¯=LsdΩz/N0, Ωx=Ωh, and Ωz=Ωg. Substituting (34) and (35) into (28), and taking ax−bc=t and simplifying, we have
(36)Fγγ=1−1ax¯exp−γz¯aexp−bcx¯a×∫0∞exp−tx¯aexp−γbc+datz¯adt.Using ([35], 3.471-9), the CDF is calculated as
(37)Fγγ=1−2exp−γz¯aexp−bcx¯aγFK12γF,where F=bc+da/a2x¯z¯. Then, substituting (37) into (26), the throughput for the special case of Rayleigh fading is obtained.


### 5.2. Outage Probability and Throughput of NL-EH-2 Model

Considering (4) and (11), the conditional CDF of the received SNR is given as
(38)Fγγ=FZγX+baXX.Averaging over (6), we have
(39)Fγγ=∫0∞FZγX+baXXfXxdx.Substituting (5) and (6) into (39), and simplifying, we have
(40)Fγγ=1−Θi,γ∫0∞x+baxixmx−1×exp−xϕx−ϕzγbaxdx,where Θi,γ=Dexp−ϕzγ/a∑i=0mz−1γϕzi/i!. Using ([35], 1.111), we have
(41)Fγγ=1−Θi,γ1ai∑k=0iikbi−k×∫0∞xmx−i+k−1exp−xϕx−ϕzγbaxdx.Applying ([35], 3.471-9) and simplifying, we have
(42)Fγγ=1−2Dξi,kγ0.5mx+i+kexp−ϕzγa×Kmx+k−i2ϕxϕzbaγ,where
(43)ξi,k=∑i=0mz−1ϕz/aii!∑k=0iikbi−kϕzbϕxamx+k−i/2.Then, substituting (42) into (26), the system throughput for NL-EH-2 is obtained.


#### 5.2.1. Special Case of Rayleigh Fading for NL-EH-2 Model

The outage probability and throughput of the considered system for the NL-EH-2 model given in (11) is investigated under the assumption of Rayleigh fading for both links PB→S and S→D. Applying (34) and (35) into (39), we have
(44)Fγγ=1−1x¯exp−γaz¯∫0∞exp−xx¯−bγaxz¯dx.Using ([35], 3.471-9), the CDF of received SNR at node D is obtained as
(45)Fγγ=1−2exp−γz¯abγazx¯K12bγazx¯.Finally, the throughput of the considered system for NL-EH-2 is calculated by substituting (45) into (26).


### 5.3. Outage Probability and Throughput of Linear and Piece-Wise Linear Models

Considering Table1, the conditional CDF of the received SNR at D given in (4) for both linear and piece-wise linear EH models is written as
(46)FγγX=PrZ<γηX=FZγηXX,where FZz is given in (5). For the linear EH model, the CDF is calculated as
(47)Fγlγ=∫0∞FZγηXXfXxdx.Substituting (5) and (6) into (47) and applying ([35], 3.471-9), the CDF in (47) is rewritten as
(48)Fγlγ=1−2D∑i=0mz−1ϕzγ/ηii!ϕzγηϕxmx−i/2×Kmx−i2ϕzγϕx/η.For the piece-wise linear EH model given in Table1, the CDF is expressed as
(49)Fγpwγ=PηXZ<γ,X<Pth+PηPthZ<γ,X>Pth.Considering (5) and (6), and simplifying, we have
(50)Fγpwγ=FXPth−E2γ+FZγPM1−FXPth,where PM=ηPth. Here, E2γ is calculated by substituting (5) and (6) into (49) as
(51)E2γ=D∑i=0mz−1ϕzγ/ηii!∫0Pthxmx−i−1e−xϕxe−ϕzγ/ηxdx.Moreover, applying ([35], 1.211-1), [36, 11], ([35], 9.31-2), and [36, 26], (51) is obtained as
(52)E2γ=D∑i=0mz−1ϕzγ/ηii!∑k=0N−ϕxkk!Pthmx−i+k×G0221ηPthγϕz1,1−mx−i+k−mx−i+k,where N⟶∞ and Gmnpq·apbq is the Meijer-G function ([35], 9.301). Finally, the throughputs of linear and piece-wise linear EH models are obtained by substituting (48) and (50), respectively, into (26).Different from the NL-EH models previously given in the literature, we obtained closed-form outage probability expressions for the Nakagami-m channels in equations (32) and (42) and for the special case of Rayleigh fading channels in equations (37) and (45) for NL-EH-1 and NL-EH-2 models, respectively.


## 5.1. Outage Probability and Throughput of NL-EH-1 Model

Considering (4) and (7), the conditional CDF of the received SNR is given as
(27)Fγγ=FZγX+daX−bcX,bca<x<∞,1,0<x<bca,where FZz is given in (5). Averaging over X using (6), we have
(28)Fγγ=∫0bc/afXxdx+∫bc/a∞FZγX+daX−bcXfXxdx.(28) is rewritten by substituting (5) and (6) as
(29)Fγγ=1−λi,γ∫bc/a∞x+dax−bcixmx−1exp−xϕxexp−ϕzγx+dax−bcdx,where λi,γ=D∑i=0mz−1γϕzi/i!. By defining ax−bc=t and after some mathematical simplification, (29) is rewritten as
(30)Fγγ=1−λi,γexp−ϕzγaexp−ϕxbca1amx+i×∫0∞t−it+jit+bcmx−1exp−ϕxta−ϕzγJatdt,where J=da+bc. Using ([35], 1.111), we have
(31)Fγγ=1−D∑i=0mz−1γϕzii!exp−ϕzγaexp−ϕxbca1amx+i∑k=0iikJi−k∑l=0mx−1mx−1lbcmx−1−l×∫0∞tk+l−iexp−ϕxta−ϕzγJatdt.Applying ([35], 3.471-9), the CDF of the received SNR at D is calculated as
(32)Fγγ=1−ϖi,k,lγu+i/2exp−ϕzγa×Ku−i2ϕxϕzJa2γ,where u=k+l+1 and
(33)ϖi,k,l=2D1amxexp−ϕxbca∑i=0mz−1ϕz/aii!×∑k=0iikJi−k∑l=0mx−1mx−1lbcmx−1−l×ϕzJϕxk+l−i+1/2,andK·· is the modified Bessel function of the second kind ([35], 8.407-2). The throughput of the considered system for the NL-EH-1 model given in (7) is calculated by substituting Fγγ from (32) into (26).


### 5.1.1. Special Case of Rayleigh Fading for NL-EH-1 Model

For the case where both links from PB→S and S→D are exposed to Rayleigh fading and both S and D are equipped with one antenna, the CDF and PDF of (5) and (6) are simplified as
(34)FZz=1−exp−zz¯,(35)fXx=1x¯exp−xx¯,respectively, wherex¯=PBLpbsΩx, z¯=LsdΩz/N0, Ωx=Ωh, and Ωz=Ωg. Substituting (34) and (35) into (28), and taking ax−bc=t and simplifying, we have
(36)Fγγ=1−1ax¯exp−γz¯aexp−bcx¯a×∫0∞exp−tx¯aexp−γbc+datz¯adt.Using ([35], 3.471-9), the CDF is calculated as
(37)Fγγ=1−2exp−γz¯aexp−bcx¯aγFK12γF,where F=bc+da/a2x¯z¯. Then, substituting (37) into (26), the throughput for the special case of Rayleigh fading is obtained.


## 5.1.1. Special Case of Rayleigh Fading for NL-EH-1 Model

For the case where both links from PB→S and S→D are exposed to Rayleigh fading and both S and D are equipped with one antenna, the CDF and PDF of (5) and (6) are simplified as
(34)FZz=1−exp−zz¯,(35)fXx=1x¯exp−xx¯,respectively, wherex¯=PBLpbsΩx, z¯=LsdΩz/N0, Ωx=Ωh, and Ωz=Ωg. Substituting (34) and (35) into (28), and taking ax−bc=t and simplifying, we have
(36)Fγγ=1−1ax¯exp−γz¯aexp−bcx¯a×∫0∞exp−tx¯aexp−γbc+datz¯adt.Using ([35], 3.471-9), the CDF is calculated as
(37)Fγγ=1−2exp−γz¯aexp−bcx¯aγFK12γF,where F=bc+da/a2x¯z¯. Then, substituting (37) into (26), the throughput for the special case of Rayleigh fading is obtained.


## 5.2. Outage Probability and Throughput of NL-EH-2 Model

Considering (4) and (11), the conditional CDF of the received SNR is given as
(38)Fγγ=FZγX+baXX.Averaging over (6), we have
(39)Fγγ=∫0∞FZγX+baXXfXxdx.Substituting (5) and (6) into (39), and simplifying, we have
(40)Fγγ=1−Θi,γ∫0∞x+baxixmx−1×exp−xϕx−ϕzγbaxdx,where Θi,γ=Dexp−ϕzγ/a∑i=0mz−1γϕzi/i!. Using ([35], 1.111), we have
(41)Fγγ=1−Θi,γ1ai∑k=0iikbi−k×∫0∞xmx−i+k−1exp−xϕx−ϕzγbaxdx.Applying ([35], 3.471-9) and simplifying, we have
(42)Fγγ=1−2Dξi,kγ0.5mx+i+kexp−ϕzγa×Kmx+k−i2ϕxϕzbaγ,where
(43)ξi,k=∑i=0mz−1ϕz/aii!∑k=0iikbi−kϕzbϕxamx+k−i/2.Then, substituting (42) into (26), the system throughput for NL-EH-2 is obtained.


### 5.2.1. Special Case of Rayleigh Fading for NL-EH-2 Model

The outage probability and throughput of the considered system for the NL-EH-2 model given in (11) is investigated under the assumption of Rayleigh fading for both links PB→S and S→D. Applying (34) and (35) into (39), we have
(44)Fγγ=1−1x¯exp−γaz¯∫0∞exp−xx¯−bγaxz¯dx.Using ([35], 3.471-9), the CDF of received SNR at node D is obtained as
(45)Fγγ=1−2exp−γz¯abγazx¯K12bγazx¯.Finally, the throughput of the considered system for NL-EH-2 is calculated by substituting (45) into (26).


## 5.2.1. Special Case of Rayleigh Fading for NL-EH-2 Model

The outage probability and throughput of the considered system for the NL-EH-2 model given in (11) is investigated under the assumption of Rayleigh fading for both links PB→S and S→D. Applying (34) and (35) into (39), we have
(44)Fγγ=1−1x¯exp−γaz¯∫0∞exp−xx¯−bγaxz¯dx.Using ([35], 3.471-9), the CDF of received SNR at node D is obtained as
(45)Fγγ=1−2exp−γz¯abγazx¯K12bγazx¯.Finally, the throughput of the considered system for NL-EH-2 is calculated by substituting (45) into (26).


## 5.3. Outage Probability and Throughput of Linear and Piece-Wise Linear Models

Considering Table1, the conditional CDF of the received SNR at D given in (4) for both linear and piece-wise linear EH models is written as
(46)FγγX=PrZ<γηX=FZγηXX,where FZz is given in (5). For the linear EH model, the CDF is calculated as
(47)Fγlγ=∫0∞FZγηXXfXxdx.Substituting (5) and (6) into (47) and applying ([35], 3.471-9), the CDF in (47) is rewritten as
(48)Fγlγ=1−2D∑i=0mz−1ϕzγ/ηii!ϕzγηϕxmx−i/2×Kmx−i2ϕzγϕx/η.For the piece-wise linear EH model given in Table1, the CDF is expressed as
(49)Fγpwγ=PηXZ<γ,X<Pth+PηPthZ<γ,X>Pth.Considering (5) and (6), and simplifying, we have
(50)Fγpwγ=FXPth−E2γ+FZγPM1−FXPth,where PM=ηPth. Here, E2γ is calculated by substituting (5) and (6) into (49) as
(51)E2γ=D∑i=0mz−1ϕzγ/ηii!∫0Pthxmx−i−1e−xϕxe−ϕzγ/ηxdx.Moreover, applying ([35], 1.211-1), [36, 11], ([35], 9.31-2), and [36, 26], (51) is obtained as
(52)E2γ=D∑i=0mz−1ϕzγ/ηii!∑k=0N−ϕxkk!Pthmx−i+k×G0221ηPthγϕz1,1−mx−i+k−mx−i+k,where N⟶∞ and Gmnpq·apbq is the Meijer-G function ([35], 9.301). Finally, the throughputs of linear and piece-wise linear EH models are obtained by substituting (48) and (50), respectively, into (26).Different from the NL-EH models previously given in the literature, we obtained closed-form outage probability expressions for the Nakagami-m channels in equations (32) and (42) and for the special case of Rayleigh fading channels in equations (37) and (45) for NL-EH-1 and NL-EH-2 models, respectively.


## 6. BEP Analyses

In this section, closed-form expressions for the BEP of the considered system for NL-EH-1, NL-EH-2, linear, and piece-wise linear EH models are derived. The BEP is calculated as(53)Pb=AB2ρπ∫0∞e−BγγFγγdγ,where Fγγ is the CDF obtained in (32), (37), (42), and (45) for NL-EH-1 and NL-EH-2 in Nakagami-m and Rayleigh fading channels, respectively. Here, constants A and B are modulation-specific parameters [33] and ρ=log2M, M being the modulation order. Under the assumption of Gray mapping, (53) provides a tight upper bound at high and medium SNR regions. Substituting (32), (37), (42), and (45) into (53), (53) is obtained in the general form of
(54)Pb=AB2ρπΞ−Iζ,where ζ∈1,2,3,4 and
(55)Ξ=∫0∞e−Bγγdγ=πB.(55) is calculated using ([35], 3.381-4). In (54), I1, I2, I3, and I4 are obtained by substituting (32), (37), (42), and (45), respectively, as explained in the following subsections.


### 6.1. BEP Analysis for NL-EH-1 Model

The BEP for the NL-EH-1 model given in (7) is calculated for the Nakagami-m fading channel, from (54), by taking for ζ=1(56)I1=ϖi,k,l∫0∞e−Bγγγ0.5u+iexp−ϕzγa2×Ku−i2ϕxϕzJa2γdγ.Furthermore, assumingγ=z2 in (56), we have
(57)I1=2ϖi,k,l∫0∞zu+iexp−z2κKu−iδzdz,where κ=B+ϕz/a and δ=2ϕxϕzJ/a2. Using ([35], 6.631-3) in (57) and simplifying, we have
(58)I1=ϖi,k,l1κς+iΓ0.5+ςΓ0.5+i×expδ2/8κW−0.5ς+i,0.5ς−iδ2/4κδ,where ς=k+l+1 and W·,·· is the Whittaker function ([35], 9.22). Finally, the closed-form BEP for the NL-EH-1 model is calculated by substituting (58) into (54).


#### 6.1.1. Special Case of Rayleigh Fading for NL-EH-1

The BEP for the Rayleigh fading channel is calculated replacing forζ=2(59)I2=2Fexp−bcx¯a∫0∞exp−γz¯a−BγK12γFdγ,in (54). Moreover, using ([35], 6.614-4) in (59), we have
(60)I2=exp−bc/x¯a+F/2αΓ1.5Γ0.5W−0.5,0.5F/αα,where α=B+1/z¯a. Thus, the closed-form BEP expression for the NL-EH-1 model in Rayleigh fading is calculated by substituting (60) into (54).


### 6.2. BEP Analysis for NL-EH-2 Model

The BEP for the NL-EH-2 model given in (11) is calculated for the Nakagami-m fading channel from (54) by replacing for ζ=3(61)I3=2Dξi,k∫0∞γ0.5mx+k+i−1exp−γϕza+B×Kmx+k−i2ϕxϕzbaγdγ.Assumingγ=z2 in (61), we have
(62)I3=4Dξi,k∫0∞zmx+k+iexp−z2κKmx+k−iσzdz,where κ=B+ϕz/a and σ=2ϕxϕzb/a. Using ([35], 6.631-3) and simplifying, (62) is rewritten as
(63)I3=2Dξi,k1κmx+k+iΓ0.5+mx+kΓ0.5+i×expσ2/8κW−0.5mx+k+i,0.5mx+k−iσ2/4κσ.Finally, substituting (63) into (54), the closed-form BEP expression for NL-EH-2 is obtained.


#### 6.2.1. Special Case of Rayleigh Fading for NL-EH-2

The BEP is calculated from (54) by taking ζ=4 and replacing
(64)I4=2baz¯x¯∫0∞exp−γB+1az¯K12baz¯x¯γdγ.Using ([31], 6.614-4), we have
(65)I4=expβ/2αΓ1.5Γ0.5W−0.5,0.5β/αα,where α=B+1/z¯a and β=b/az¯x¯. Hence, the BEP expression for the NL-EH-2 model under the Rayleigh fading channel is derived substituting (65) into (54).


### 6.3. BEP Analysis for Linear and Piece-Wise Linear EH Models

The conditional BEP of the system considering both linear and piece-wise linear EH models can be expressed as(66)PbeX=AB2ρπ∫0∞e−BγγFγγXdγ=AB2ρπΞ−M2x,where Fγγx=FZγ/ηXX is given in (46) and (5). Moreover, Ξ is obtained in (55), and
(67)M2x=∑i=0mz−1ϕz/ηxii!∫0∞γi−0.5exp−γB+ϕzηxdγ.Applying ([35], 3.381-4) into (67) and after some mathematical simplifications, we have
(68)M2x=ηϕz∑i=0mz−1Γi+0.5i!x1+xGi+0.5,where G=ηB/ϕz. For the linear EH model, the unconditional BEP of the considered system is calculated as
(69)Pbl=∫0∞PbexfXxdx=AB2ρπM11−M21,where
(70)M11=Ξ∫0∞fXxdx=Ξ,(71)M21=∫0∞M2xfXxdx,where M2x and fXx are given in (67) and (6), respectively. Substituting, simplifying, and using Equation (10) of [36], we have
(72)M21=Dηϕz∑i=0mz−11i!∫0∞xmx−0.5exp−ϕxx×G1111xG0.5−i0dx.Using ([35], 7.813-1), we have
(73)M21=Dηϕz∑i=0mz−11i!ϕx−mx+0.5×G1221Gϕx−mx+0.5,0.5−i0.Finally, the BEP expression for the linear model is obtained substituting (70) and (73) into (69).Now, considering the piece-wise linear EH model, the BEP of the considered system is written as(74)Pbpw=∫0PthPbexfXxdx⏟T1+PbeX=Pth⏟T2∫Pth∞fXxdx⏟T3,where T2 is calculated by substituting Pth into (66), and T3=1−FXPth is calculated using (9). Moreover, we have
(75)T1=AB2ρπN11−N21,where
(76)N11=Ξ∫0PthfXxdx=ΞFXPth,(77)N21=∫0PthM2xfXxdx,where M2x and fXx are given in (67) and (6), respectively. Substituting into (77), simplifying, and applying y=xG, the upper bound of integration is changed from Pth to GPth, and (77) is rewritten as
(78)N21=ΔiR1+R2,GPth≥1,ΔiR3,GPth<1,where
Δi=Dηϕz∑i=0mz−1Γi+0.5i!Gmx+0.5,(79)Rι=∫ymx−0.51+yi+0.5exp−ϕxyGdy,where 0,1, 1,GPth, and 0,GPth are the integration boundaries for ι=1, ι=2, and ι=3, respectively. Considering ι=1 and ι=3 into (79) and applying [3, 36], we have
(80)Rι=∑k=0N−εk−1kk!∫0υymx−0.5+kexp−ϕxyGdy,where ε=−i+0.5 and υ=1 and GPth for R1 and R3, respectively. Applying ([35], 07.20.07.0001.01) into (80), we have
(81)R1=∑k=0N−εk−1kk!Γmx+k+0.5Γmx+k+1.5×1F1mx+k+0.5;mx+k+1.5;−ϕxG,(82)R3=∑k=0N−εk−1kk!Γmx+k+0.5Γmx+k+1.5PthGmx+k+0.5×1F1mx+k+0.5;mx+k+1.5;−ϕxPth,where 1F1a;b;· is the Kummer confluent hypergeometric function given in ([35], 07.20.02.0001.01). Considering ι=2 in (79) and applying Equation (3) in [36], we have
(83)R2=∑k=0N−εk−1kk!R21−R22,where
(84)R21=∫1∞ymx−i−k−1exp−ϕxyGdy=ϕxG−mx−i−kΓmx−i−k,ϕxG,(85)R22=∫PthG∞ymx−i−k−1exp−ϕxyGdy=ϕxG−mx−i−kΓmx−i−k,ϕxPth.Here, (84) and (85) are calculated using ([35], 3.351-2). Then, (83) is calculated by substituting (84) and (85). Moreover, (78) is calculated using (81), (82), and (83). Finally, substituting (75), T2, and T3 into (66) and (9), respectively, the BEP expression for the piece-wise linear EH model is obtained.Thanks to the mathematical tractability of our NL-EH models, we obtained closed-form expressions for the average BEP of the considered power-constraint system. Moreover, to gain a better insight through the new NL-EH models, we calculated BEP of the reference systems applying linear and piece-wise linear EH models. The detailed effects of the system parameters on the obtained BEP expressions are evaluated in the next section.


## 6.1. BEP Analysis for NL-EH-1 Model

The BEP for the NL-EH-1 model given in (7) is calculated for the Nakagami-m fading channel, from (54), by taking for ζ=1(56)I1=ϖi,k,l∫0∞e−Bγγγ0.5u+iexp−ϕzγa2×Ku−i2ϕxϕzJa2γdγ.Furthermore, assumingγ=z2 in (56), we have
(57)I1=2ϖi,k,l∫0∞zu+iexp−z2κKu−iδzdz,where κ=B+ϕz/a and δ=2ϕxϕzJ/a2. Using ([35], 6.631-3) in (57) and simplifying, we have
(58)I1=ϖi,k,l1κς+iΓ0.5+ςΓ0.5+i×expδ2/8κW−0.5ς+i,0.5ς−iδ2/4κδ,where ς=k+l+1 and W·,·· is the Whittaker function ([35], 9.22). Finally, the closed-form BEP for the NL-EH-1 model is calculated by substituting (58) into (54).


### 6.1.1. Special Case of Rayleigh Fading for NL-EH-1

The BEP for the Rayleigh fading channel is calculated replacing forζ=2(59)I2=2Fexp−bcx¯a∫0∞exp−γz¯a−BγK12γFdγ,in (54). Moreover, using ([35], 6.614-4) in (59), we have
(60)I2=exp−bc/x¯a+F/2αΓ1.5Γ0.5W−0.5,0.5F/αα,where α=B+1/z¯a. Thus, the closed-form BEP expression for the NL-EH-1 model in Rayleigh fading is calculated by substituting (60) into (54).


## 6.1.1. Special Case of Rayleigh Fading for NL-EH-1

The BEP for the Rayleigh fading channel is calculated replacing forζ=2(59)I2=2Fexp−bcx¯a∫0∞exp−γz¯a−BγK12γFdγ,in (54). Moreover, using ([35], 6.614-4) in (59), we have
(60)I2=exp−bc/x¯a+F/2αΓ1.5Γ0.5W−0.5,0.5F/αα,where α=B+1/z¯a. Thus, the closed-form BEP expression for the NL-EH-1 model in Rayleigh fading is calculated by substituting (60) into (54).


## 6.2. BEP Analysis for NL-EH-2 Model

The BEP for the NL-EH-2 model given in (11) is calculated for the Nakagami-m fading channel from (54) by replacing for ζ=3(61)I3=2Dξi,k∫0∞γ0.5mx+k+i−1exp−γϕza+B×Kmx+k−i2ϕxϕzbaγdγ.Assumingγ=z2 in (61), we have
(62)I3=4Dξi,k∫0∞zmx+k+iexp−z2κKmx+k−iσzdz,where κ=B+ϕz/a and σ=2ϕxϕzb/a. Using ([35], 6.631-3) and simplifying, (62) is rewritten as
(63)I3=2Dξi,k1κmx+k+iΓ0.5+mx+kΓ0.5+i×expσ2/8κW−0.5mx+k+i,0.5mx+k−iσ2/4κσ.Finally, substituting (63) into (54), the closed-form BEP expression for NL-EH-2 is obtained.


### 6.2.1. Special Case of Rayleigh Fading for NL-EH-2

The BEP is calculated from (54) by taking ζ=4 and replacing
(64)I4=2baz¯x¯∫0∞exp−γB+1az¯K12baz¯x¯γdγ.Using ([31], 6.614-4), we have
(65)I4=expβ/2αΓ1.5Γ0.5W−0.5,0.5β/αα,where α=B+1/z¯a and β=b/az¯x¯. Hence, the BEP expression for the NL-EH-2 model under the Rayleigh fading channel is derived substituting (65) into (54).


## 6.2.1. Special Case of Rayleigh Fading for NL-EH-2

The BEP is calculated from (54) by taking ζ=4 and replacing
(64)I4=2baz¯x¯∫0∞exp−γB+1az¯K12baz¯x¯γdγ.Using ([31], 6.614-4), we have
(65)I4=expβ/2αΓ1.5Γ0.5W−0.5,0.5β/αα,where α=B+1/z¯a and β=b/az¯x¯. Hence, the BEP expression for the NL-EH-2 model under the Rayleigh fading channel is derived substituting (65) into (54).


## 6.3. BEP Analysis for Linear and Piece-Wise Linear EH Models

The conditional BEP of the system considering both linear and piece-wise linear EH models can be expressed as(66)PbeX=AB2ρπ∫0∞e−BγγFγγXdγ=AB2ρπΞ−M2x,where Fγγx=FZγ/ηXX is given in (46) and (5). Moreover, Ξ is obtained in (55), and
(67)M2x=∑i=0mz−1ϕz/ηxii!∫0∞γi−0.5exp−γB+ϕzηxdγ.Applying ([35], 3.381-4) into (67) and after some mathematical simplifications, we have
(68)M2x=ηϕz∑i=0mz−1Γi+0.5i!x1+xGi+0.5,where G=ηB/ϕz. For the linear EH model, the unconditional BEP of the considered system is calculated as
(69)Pbl=∫0∞PbexfXxdx=AB2ρπM11−M21,where
(70)M11=Ξ∫0∞fXxdx=Ξ,(71)M21=∫0∞M2xfXxdx,where M2x and fXx are given in (67) and (6), respectively. Substituting, simplifying, and using Equation (10) of [36], we have
(72)M21=Dηϕz∑i=0mz−11i!∫0∞xmx−0.5exp−ϕxx×G1111xG0.5−i0dx.Using ([35], 7.813-1), we have
(73)M21=Dηϕz∑i=0mz−11i!ϕx−mx+0.5×G1221Gϕx−mx+0.5,0.5−i0.Finally, the BEP expression for the linear model is obtained substituting (70) and (73) into (69).Now, considering the piece-wise linear EH model, the BEP of the considered system is written as(74)Pbpw=∫0PthPbexfXxdx⏟T1+PbeX=Pth⏟T2∫Pth∞fXxdx⏟T3,where T2 is calculated by substituting Pth into (66), and T3=1−FXPth is calculated using (9). Moreover, we have
(75)T1=AB2ρπN11−N21,where
(76)N11=Ξ∫0PthfXxdx=ΞFXPth,(77)N21=∫0PthM2xfXxdx,where M2x and fXx are given in (67) and (6), respectively. Substituting into (77), simplifying, and applying y=xG, the upper bound of integration is changed from Pth to GPth, and (77) is rewritten as
(78)N21=ΔiR1+R2,GPth≥1,ΔiR3,GPth<1,where
Δi=Dηϕz∑i=0mz−1Γi+0.5i!Gmx+0.5,(79)Rι=∫ymx−0.51+yi+0.5exp−ϕxyGdy,where 0,1, 1,GPth, and 0,GPth are the integration boundaries for ι=1, ι=2, and ι=3, respectively. Considering ι=1 and ι=3 into (79) and applying [3, 36], we have
(80)Rι=∑k=0N−εk−1kk!∫0υymx−0.5+kexp−ϕxyGdy,where ε=−i+0.5 and υ=1 and GPth for R1 and R3, respectively. Applying ([35], 07.20.07.0001.01) into (80), we have
(81)R1=∑k=0N−εk−1kk!Γmx+k+0.5Γmx+k+1.5×1F1mx+k+0.5;mx+k+1.5;−ϕxG,(82)R3=∑k=0N−εk−1kk!Γmx+k+0.5Γmx+k+1.5PthGmx+k+0.5×1F1mx+k+0.5;mx+k+1.5;−ϕxPth,where 1F1a;b;· is the Kummer confluent hypergeometric function given in ([35], 07.20.02.0001.01). Considering ι=2 in (79) and applying Equation (3) in [36], we have
(83)R2=∑k=0N−εk−1kk!R21−R22,where
(84)R21=∫1∞ymx−i−k−1exp−ϕxyGdy=ϕxG−mx−i−kΓmx−i−k,ϕxG,(85)R22=∫PthG∞ymx−i−k−1exp−ϕxyGdy=ϕxG−mx−i−kΓmx−i−k,ϕxPth.Here, (84) and (85) are calculated using ([35], 3.351-2). Then, (83) is calculated by substituting (84) and (85). Moreover, (78) is calculated using (81), (82), and (83). Finally, substituting (75), T2, and T3 into (66) and (9), respectively, the BEP expression for the piece-wise linear EH model is obtained.Thanks to the mathematical tractability of our NL-EH models, we obtained closed-form expressions for the average BEP of the considered power-constraint system. Moreover, to gain a better insight through the new NL-EH models, we calculated BEP of the reference systems applying linear and piece-wise linear EH models. The detailed effects of the system parameters on the obtained BEP expressions are evaluated in the next section.


## 7. Performance Evaluation

In this section, simulation and theoretical results are obtained for different values of the system parameters. In all figures of this section, lines denote the theoretical results while symbols or symbols with dashed lines present the simulation results. In addition to the two proposed NL-EH models, the simulation results for linear, piece-wise linear, and existing NL-EH models are provided which gives an extra insight in analyzing the system performance. In all cases, the theoretical results are in perfect match with the Monte-Carlo simulation results. Theoretical curves for average harvested power are obtained from (14), (20), (21), and (22). Theoretical throughput curves are depicted by replacing (32), (42), (48), and (50) into (26). For the BER analysis, theoretical results are obtained by applying (54), (69), and (74). The fitting parameters for [11, 15] are listed in Tables 3 and 4, for the NL-EH-1 and NL-EH-2 models, respectively, while the coefficient of other models are provided in Table 5.Table 5
Fitting parameters of different EH models given in Table1 with data measurements taken from energy-harvesting circuits of [11, 15].
[11][15]EH modelsParametersParametersLinear [1]η=0.6η=0.25Piece-wise linear [4]η=0.6,Pth=410η=0.25,Pth=29.5NL-EH [7]a=0.00409,b=−63.88202,M=244a=0.08271,b=8.34322,M=7.3NL-EH [8]p0=406.5963,p1=−46878.9296,p2=3389602.655,q1=570.147,q2=−115371.7,q3=1.0291e7p0=10.24347,p1=−25.14551,p2=16.37787,q1=21.21917,q2=−7.88436,q3=138.76844Unless stated differently, the fading parameters for each link are taken asmh=mg=2, and the nodes S and D are both equipped with Ns=Nd=2 receive antennas. It is assumed that the modulation is binary phase-shift keying (BPSK) and the system throughput is obtained for R=1 bits/sec/Hz. Furthermore, it is assumed that dpbs=dsd=1 and the path-loss coefficient is taken equal to ν=2.7 [38]. Moreover, the channel power gains for both links are chosen as Ωh=Ωg=1, and the noise power at node D is fixed to N0=1. Note that the optimization of the time slots dedicated for EH and IP is a difficult task due to the dependent parameters PT, dpbs, dsd, ν, Ns, Nd, mh, mg, Ωh, Ωg, and R.Figure4 describes the average harvested power at node S versus PT for data measurements taken from (14). The resulting curves are depicted for dpbs=1 and dpbs=10. It is shown that the amount of average harvested power for dpbs=1 is 0 dBm which is approximately 25 dBm more compared to the case of dpbs=10. This verifies that the distance dpbs has a significant effect on the amount of harvested power. Moreover, it is seen that the average harvested powers of the NL-EH models for dpbs=1 and dpbs=10 are saturated at PT=45 dBm and PT=70 dBm, respectively. Please note that all NL-EH models are saturated at a predefined Pth value. Furthermore, compared to dpbs=1,25 dBm, more power is needed to saturate for dpbs=10.Figure 4
Average harvested power versusPT where data measurements are taken from [15].The behavior of average harvested power versusPT is shown in Figure 5 considering the same system parameters for two different EH circuits given in [11, 15] for dpbs=1. It is observed that the EH circuit in [11] provides approximately 2 dBm more average harvested power compared to that in [15] considering the linear EH model. Moreover, it is shown that the average harvested power saturates at PT=40 dBm and PT=55 dBm for the EH circuits given in [11, 15], respectively, and results from the design of the EH system. This means that the EH circuit design has an important impact on the performance of the EH system since more power is harvested at the power-constraint node S. However, despite the EH circuit design, NL-EH models saturate at a predefined threshold value. Please note that all the NL-EH models along with the proposed NL-EH models have the same behavior on the amount of average harvested power.Figure 5
Average harvested power versusPT where dpbs=1; data measurements are taken from [11, 15].In Figure6, average harvested power is depicted versus Ns. The results are provided for dpbs=1 and PT=50 dBm. It is concluded from Figure 6 that more antennas at node S result in higher average harvested power at the EH node which significantly affects the proposed system performance. However, it is shown that by increasing the number of receive antennas at node S, the average harvested power approximately saturates to the values between 38 dBm and 41 dBm for different NL-EH models when Ns=6. In other words, node S with Ns=3 antennas provides an almost equal amount of harvested power compared to node S with Ns=6 antennas. Please note that the differences among NL-EH models are related to the curve fitting of the experimental measured data in a specified interval of input values. Moreover, it is seen from Figure 6 that the amount of average harvested power for the linear EH model is overestimated compared to the NL-EH and piece-wise linear EH models where for Ns=6, the average harvested power for the linear EH model is 48.5 dBm. This means that the linear EH model overestimates approximately 8.5 dBm more average harvested power compared to NL-EH models.Figure 6
Average harvested power versusNs for PT=50 dBm and dpbs=1.In Figure7, the system throughput is depicted versus R. The results are provided for different numbers of receive antennas Ns and Nd. It is seen that by increasing the number of antennas at both nodes S and D, the throughput is increased. Besides, it is observed from Figure 7 that all of the throughput curves start from approximately τ=0.2, reach their maximum values, and then get decreased by increasing the value of R until they get toward zero. Moreover, it is observed that the linear EH model overestimates the system performance for all cases. Additionally, the more receive antennas at nodes S and D, the more overestimation of throughput is resulted. In other words, the differences between NL-EH models and linear EH model throughput curves are increased by increasing Ns and Nd. This is due to the fact that no saturation power is defined in the linear EH model.Figure 7
Throughput of considered system versusR (bits/sec/Hz) for PT=50 dBm and dpbs=dsd=1.Figure8 presents BER versus PT when dpbs=dsd=1. The curves are obtained for Ns=Nd=2. It is shown that at high SNR values, the system performance considering the NL-EH and piece-wise linear EH models go to the error floor while the linear EH model system performance is increased at high SNR values due to the overestimation of harvested power which causes an unreasonable expectation of the system performance. However, at low SNR values, the BER performances of linear/piece-wise linear/NL-EH models are equivalent. Additionally, based on the obtained parameters from curve fitting of [15], piece-wise linear and logistic EH models provide approximately the same performance. Similarly, the heuristic model, fraction model, and proposed NL-EH-1 and NL-EH-2 models provide similar system performances. This comes from the fact that the difference in these results is caused by curve-fitting functions which are calculated based on a specific interval of data measurements.Figure 8
BER versusPT for Ns=Nd=2 and dpbs=dsd=1.The BER performance curves versusNd are shown in Figure 9. Here, the results are provided for PT=50 dBm, dpbs=dsd=1, Ns=1, and Ns=2. It is seen that by increasing Nd, the BER performance gets better for both Ns values. Also, the linear EH model outperforms the piece-wise linear and NL-EH models for Ns=1 and Ns=2 due to more harvested power. It comes from the fact that more receive antennas at node S provide more harvested energy which results in higher transmit power at node S and consequently high received SNR at node D. Hence, it ensures a better system performance.Figure 9
BER versusNd for PT=50 dBm and dpbs=dsd=1 where the results are provided for Ns=1 and Ns=2.In Figure10, the BER performance versus Ns is given for PT=50 dBm, and dpbs=dsd=1. It is observed that by increasing the number of antennas from Nd=1 to Nd=2, the system performance gets significantly better. Additionally, by increasing the number of antennas at S, more power is harvested which results in better system performance for the linear EH model. However, for the NL-EH and piece-wise linear EH models, at high input power, the harvested power is saturated at a threshold power value, and then, even by increasing the number of antennas at S, namely, Ns, the system performance goes to the error floor. Obviously, for both Nd=1 and Nd=2, the same performances are obtained compared to Ns=2 and Ns=10, respectively. In other words, the S node with only Ns=1 antenna provides approximately equivalent performance compared to Ns=10. Considering the results from both Figures 9 and 10, it is concluded that the performances of piece-wise linear and NL-EH models are mostly dependent on the value of Nd rather than Ns since Ns provides only an increase in the harvested power which is saturated at some value. However, even with small harvested power values but more antennas at node D, the system performance is improved. However, for the linear EH model, since the system performance is directly related to Ns, increasing Ns results in better system performance.Figure 10
BER versusNs for PT=50 dBm and dpbs=dsd=1 where the results are provided for Nd=1 to Nd=2.


## 8. Conclusion

The performance of a wireless-powered communication system equipped with multiple antennas at both nodes S and D has been investigated based on the practical NL-EH models. Two new mathematically tractable NL-EH models have been proposed for which the PDFs and CDFs of the harvested power and their histograms have been provided considering the data measurements taken from the literature. Different from the NL-EH models given in the literature which are not analytically tractable, these new NL-EH models make the performance analyses analytically tractable. Furthermore, considering the Nakagami-m fading and as a special case the Rayleigh fading, closed-form expressions for the average harvested powers, outage probabilities, throughputs, and BER performances have been analytically derived. Moreover, for comparison purposes, the same performance measures have been analytically derived for linear and piece-wise linear EH models. In each case, theoretical derivations have been supported by Monte-Carlo simulation results and have been compared with the existent EH models given in the literature such as linear, piece-wise linear, and other NL-EH models. It has been shown that apart from the linear EH model which overestimates the realistic system performance, the NL-EH models and piece-wise linear EH model provide practical system performances. Finally, the results obtained from different NL-EH models provide a comprehensive insight into the performance of the considered system. We have considered the case of one transmit antenna in this paper; however, it is possible to have more antennas for transmission at the source and to even analyze the system performance based on transmit antenna selection (TAS). However, these assumptions are out of scope of our paper and left as a future work.



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*Source: 6534754-2022-06-25.xml*