Ultra-Green Relay Transmission with Wireless Power Transfer for Advanced IoT: Session-Specific Analysis and Optimization

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Wireless Power Transfer for Advanced IoT:

Session-Specific Analysis and Optimization

Item Type Article

Authors Xu, Fang;Yang, Hong Chuan;Alouini, Mohamed-Slim

Citation Xu, F., Yang, H.-C., & Alouini, M.-S. (2023). Ultra-Green Relay Transmission with Wireless Power Transfer for Advanced IoT:

Session-Specific Analysis and Optimization. IEEE Internet of Things Journal, 1–1. https://doi.org/10.1109/jiot.2023.3264654 Eprint version Post-print

DOI 10.1109/jiot.2023.3264654

Publisher Institute of Electrical and Electronics Engineers (IEEE) Journal IEEE Internet of Things Journal

Rights (c) 2023 IEEE. Personal use of this material is permitted.

Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.

Download date 2024-01-08 00:17:41

Link to Item http://hdl.handle.net/10754/690932

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Ultra-Green Relay Transmission with Wireless Power Transfer for Advanced IoT: Session-Specific

Analysis and Optimization

Fang Xu, Member, IEEE,Hong-Chuan Yang,Senior Member, IEEE, and Mohamed-Slim Alouini, Fellow, IEEE

Abstract—Reliable and energy-efficient wireless transmission is of critical importance to the success of future advanced Internet of Things (IoT). Due to the sporadic nature of IoT transmissions, the energy consumption of the individual IoT transmission session varies dramatically with the instantaneous operating environment as well as the quality of service (QoS) requirements. In this paper, we analyze and design the energy- efficient relay transmission systems from an individual data transmission session perspective. Specifically, we consider a dual- hop transmission system with a decode-and-forward relay that is solely powered by wireless power transfer from source node. For both time switching and power splitting modes of simultaneous power and information transmission, we analyze and minimize the total energy consumption of the system when transmitting a fixed amount of data, under a piecewise linear energy harvesting (EH) model. Closed-form expressions for optimal transmission parameters are obtained with and without the consideration of latency constraint. Through selected numerical results, we illustrate various design tradeoffs between energy consumption and latency constraint. We show that with optimal transmission parameters, relay transmission with energy transfer can achieve considerable energy saving compared to direct transmission when the direct link quality is poor and the latency constraint is not stringent. We also show that with optimized parameters, power splitting mode leads to lower energy consumption and smaller transmission duration than time switching mode, at the cost of higher implementation complexity.

Index Terms—Energy efficiency, relay transmission, wireless power transfer, decode-and-forward, time switching, power s- plitting.

I. INTRODUCTION

The carbon emission of wireless communication systems has increased dramatically over the past forty years, as the result of their rapid deployment to support traditional mobile broadband services as well as growing Internet of Things (IoT) applications [1], [2]. With the mass rollout of various IoT applications and the dense deployment of 5G and beyond infrastructures, the carbon footprint of wireless systems will expand even faster. Besides switching to clean or low-carbon energy sources, improving the energy utilization efficiency during system operation is an essential step to scale back wireless emissions. Access networks, which connect wireless

This work was supported in part by a NSERC Discovery Grant.

F. Xu and H.-C. Yang are with the Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC V8W 2Y2, Canada (e-mail:

fangx, [email protected]).

M.-S. Alouini is with the CEMSE Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia (e-mail:

[email protected]).

terminals with the edge/core network, have traditionally been the most energy hungry part of wireless systems. With the ongoing exponential growth of IoT devices, the access networks will be further overwhelmed by connection requirements. To reduce the carbon footprint of wireless systems and contribute to the global carbon-neutral objective, we need to develop highly energy-efficient wireless transmission solutions for the interconnection of numerous IoT devices.

There has been ongoing interest for improving the energy efficiency of wireless transmission systems. Most previous efforts target at traditional broadband communication services.

The energy efficiency of these services was characterized by the ratio of average data rate over corresponding power consumption level, with unit of bps/W [6]. For example, [7]

calculated the average energy consumption for each transmitted bit with the consideration of possible retransmissions.

For fading wireless channel, the channel ergodic capacity is typically used to evaluate the energy efficiency [8], [9].

However, most IoT transmission sessions are very short and sporadic. For example, many IoT devices only send a short measurement update periodically, or upon the reception of a control command. In addition, to support advanced IoT applications, some IoT transmission sessions, e.g. for control commands, may require much higher reliability and lower latency than others, e.g. measurement updates. The resulting energy consumption of IoT transmission sessions will vary dramatically from one session to another. The transmission strategy that is optimal, in the average sense, may perform poorly for a specific transmission session [12]. As such, the average energy efficiency characterization is not suitable for optimizing wireless transmissions to/from IoT devices.

In this work, we maximize the energy efficiency of IoT transmissions from an individual transmission session perspective, with consideration of instantaneous channel realization and particular quality of service (QoS) requirements. Through extending previous works on point-to-point links [14], we analyze and minimize the energy consumption of relay transmission with wireless power transfer in this paper. Particularly, cooperative relay transmission is an effective and economic solution for coverage improvement [3] and wireless power transfer is an attractive approach to power relaying operation.

Note that with simultaneous wireless information and power transfer (SWIPT) [4], the relay nodes can extract both information and energy from their received signal and use collected energy for data forwarding. For example, [5] maximized the transmission rate for a dual-hop multi-relay IoT system

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with SWIPT, where power and subcarrier allocation were jointly optimized. While there are ongoing interests in SWIPT relaying systems, it is still unclear whether such system can achieve lower energy consumption than direct transmission in terms of the IoT settings, especially under reliability and latency constraints. In this paper, we derive the limit of energy consumption performance for these transmission systems from a session specific transmission perspective, and establish the condition that relay transmission with wireless power transfer can achieve higher energy efficiency.

A. Previous work

There have been continuing interest in wireless transmission systems with energy harvesting (EH) relays [16]–[21]. [16]

reviewed the recent progress of wireless power transfer applications in WSNs and discussed several important performance- enhancing techniques. [17] and [18] applied power and packet size adaptation to optimize the energy efficiency of wireless transmission systems with EH relays. [19] formulated and solved an energy efficiency optimization problem considering relays’ energy harvesting constraints and average throughput constraint. [20] considered a multi-hop transmission system with EH relays, where the energy consumption of source node is minimized. [21] studied the joint transmit power minimization and EH relay selection problem to save the energy of multiple non-energy-harvesting transmitters. While freely available, the energy harvested from the environment is typically unstable and unpredictable, which makes the system design and planning very challenging.

Alternatively, relays can collect energy from the RF signal transmitted by the source node [22]–[31]. [22] considered cooperative sensor network, where energy-constrained relay nodes harvest energy from the transmitter. In [23], the RF energy harvested from source transmission powers a full- duplex amplify-and-forwarding (AF) relay node. [24] considered a RF-powered AF relaying system, where the relay can also harvest energy from its own transmitted signal. The relay node in [25] optimally decides to use its own battery energy or harvest RF energy from the source to enhance system energy efficiency. [26], [27] optimally design precoding scheme to maximize the energy efficiency of a MIMO two- way decode-and-forward (DF) relay system with simultaneous wireless information and power transfer (SWIPT). Generally, relay node can adopt either power splitting or time switching operation mode for SWIPT operation. [28] and [29] considered time-switching operation for single-carrier and multi-carrier transmission systems, respectively, whereas [30] and [31]

employed power splitting mode at the relay. These works apply power allocation, precoding optimization, and relay selection to improve the average energy efficiency of relay transmission.

Most existing works use the average energy-efficiency met- rics to guide the optimal design. The resulting sub-optimal or near-optimal solutions, typically obtained from iterative algorithms, will be applied to all transmission sessions over a wireless link. As mentioned above, such average-based approach is not suitable for IoT transmissions, whose transmission sessions are generally very short and have diverse QoS

requirements. Furthermore, the average-based characterization can only takes into account the statistical QoS constraint, using the concept of effective capacity for example [10], [11], [13]. To best satisfy the QoS requirement of individual IoT transmission sessions while achieving high energy efficiency, the transmission scheme needs to be designed from individual transmission session perspective [12], [14]. To our best knowledge, the only other work that analyzes the energy consumption of individual data transmission session is [15], where the energy consumption of an adaptive modulation and coding system was analyzed when transmitting a large amount of data. However, [12], [14], [15] only focus on point to point transmission scenario. In this work, we extend previous work by considering relay transmission systems with wireless power transfer. To our best knowledge, this is the first work to analyze and optimize relay transmission system from an individual transmission session perspective.

B. Contributions

In this paper, we investigate a relaying transmission system with wireless power transfer from an individual transmission session perspective. Particularly, the relay can harvest RF energy from the transmitted signal of the source and use it to power its relaying operation. We consider a specific transmission session where the source transmits a fixed amount of data to the destination subject to a certain latency requirement. Our goal is to minimize the energy consumption of the transmission session considering the instantaneous channel realization and specific latency requirement. Unlike the average-based design, where the solution can be obtained off-line, our session-specific design need to be carried out on-line in real-time. As such, conventional iterative solution can not apply as they may violate the latency constraint or lead to outdated configuration. In this work, we derive closed-form expressions for the optimal parameters for both power-splitting and time-switching modes of operation under a piecewise-linear EH model, including the transmit/charging powers of the source and the time switching/power splitting factors of the relay. With these analytical results, we can investigate whether the extra energy consumption to charge the relay can enhance the energy efficiency of the transmission system. Note that the system model of relaying transmission with wireless power transfer is much more complex than that of point-to-point transmission system. The derivation process for optimal transmission parameters become much more complex. With closed-form expression for the optimal transmission parameters, the transmitter can perform highly energy-efficient transmission for all sessions in real time, leading to an ultra-green transmission solution for resource constraint IoT applications. We also accurately quantify the effect of the latency constraint on the energy consumption of individual IoT transmission sessions. Selected numerical examples are discussed to illustrate various design tradeoffs.

The main contributions of this paper can be summarized as follows

• To our best knowledge, this paper is the first work to analyze the energy consumption of relay transmission

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system with wireless power transfer from an individual transmission session perspective. We illustrate various design tradeoff of energy consumption versus latency requirement by analyzing and minimizing the energy consumption when transmitting a fixed amount of data with and without latency constraint.

• We derive the closed-form expressions of all optimal parameters for relaying transmission system, under a piecewise-linear EH model, with and without latency constraint. Also, we establish the condition that no feasible solution exists. With these results, the source and the relay can quickly configure its transmission for each IoT session under a given channel realization and latency requirement, without involving any time-consuming iterative calculation.

• We analyze and optimize both power splitting and time switching operation modes for simultaneous information and power transfer. We show that with optimized parameters in terms of energy consumption minimization, time switching mode leads to higher energy consumption and longer transmission duration than power splitting mode, while enjoying slightly lower implementation complexity.

• We show that relay transmission with wireless energy transfer can achieve considerable energy savings compared to direct transmission when the direct source to destination link has poor quality. Meanwhile, when the transmission session is subject to a stringent latency constraint, the energy consumption of relay transmission increases dramatically, and becomes higher than that of direct transmission.

The rest of the paper is organized as follows. Section II presents the relay transmission system with wireless power transfer under consideration. Section III solves the energy consumption minimization problem for power splitting mode, where as the time switching mode is addressed in Section IV.

After presenting selected numerical examples in Section V, we investigate the effect of latency constraint in Section VI.

Finally, the paper is concluded in Section VII.

II. SYSTEM AND CHANNEL MODEL

We consider a specific IoT transmission session where the source node transmits H bits of data to the destination node. Due to the poor channel quality of the direct link, the transmission is carried out with the help of an intermediate DF relay node. Note that we adopt DF relay as it can achieve higher reliability and capacity than AF relay [32].

To encourage relay cooperation, we assume that the relaying operation will be solely powered by the RF energy harvested from the transmitted signal of the source. Under the energy and information causality constraints, the relay will operate in a half-duplexing fashion. In particular, the relay node first performs data reception and energy harvesting over the first hop and then forward the decoded data to the destination using its harvested energy. Our goal is to minimize the total energy consumption by designing the transmission parameters optimally considering the instantaneous channel condition, reliability requirement, and latency constraint, if applicable.

Fig. 1: Operating modes of relay transmission with wireless power transfer.

For most IoT transmission scenarios, the data amountH is typically small, at most several kbits. As such, such data transmission session will typically complete within one channel coherence time. It follows that the channel gains will remain constant for the whole data transmission duration. Assuming frequency flat fading environment, we denote the complex channel gains of the first hop and the second hop by h_SR and h_RD, respectively. The received signal at the relay can be written as

y_R(t) =h_SRs(t) +n_R(t), (1) wheres(t)is the signal transmitted by the source with power P_S and n_R(t) is the noise at the relay with power σ². The relay performs simultaneous information detection and energy harvesting on y_R(t), following either power splitting or time switching mode of operation. After that, it forwards a copy of its decoded signal, denoted by r(t), to the destination using harvested RF energy. The received signal at the destination given by

y_D(t) =h_RDr(t) +n_D(t), (2) wheren_D(t)is the noise at the destination also with the average power of σ². We will consider the optimal design for power splitting and time switching modes separately in the following two sections, assuming that the instantaneous channel gains have been accurately estimated before the transmission session starts. The effect of imperfect channel knowledge will be investigated in future work. Note that since the optimal designs are carried out based on instantaneous channel gains, the result will be applicable to any fading channel model.

III. POWER SPLITTING MODE

A. Energy consumption analysis

With power splitting operation mode, the total duration of the transmission session,T, is divided into two equal-length slots, one for source transmission and the other for relay transmission, as shown in Fig. 1(a). The relay will divide its received signal y_R(t) into energy signal and information signal, with a power splitting factorβ (0<β<1), for energy harvesting and information decoding, respectively. As most practical energy harvesting circuit is characterized by a linear region and a saturation region, we adopt a piecewise linear EH

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model [33]. In particular, the harvested energy at the relay over the first slot of duration T/2 is given by

E_H^PS=

η βP_Sg_SR^T₂, P_Sg_SRβ <P_sat;

ηP_sat^T₂, P_Sg_SRβ ≥P_sat, (3) where g_SR=|h_SR|² is the channel power gain from source to relay, 0<η<1 denotes the energy conversion efficiency, andP_satis the power saturation threshold of energy harvesting circuit. Meanwhile, the received signal-to-noise ratio (SNR) at the relay for information detection is given by

γ_R^PS=(1−β)P_Sg_SR

σ² . (4)

In general, the energy consumption for information decoding at the relay is negligible compared to that for data forwarding [34], [35]. Accordingly, we assume that the relay uses all of its harvested energy for data forwarding over the second slot. The transmission power of relay is then given by

P_R^PS=E_H^PS

T/2 =ηmin{P_SgSRβ,P_sat}. (5) To establish the performance limit of energy consumption, we assume ideal rate adaptive transmission. Therefore, the effective data rate of the transmission system with power splitting mode can be calculated as

R^PS=B

2log₂(1+ 1

σ²min{P_S(1−β)g_SR, P_R^PSg_RD}), (6) where B is the channel bandwidth and g_RD=|h_RD|² is the channel power gain from relay to destination. It follows that the total transmission duration is equal toT =H/R^PS.

Finally, the total energy consumption for transmittingHbits of data with the power splitting mode can be calculated as the product of transmission duration and power consumption for the source, which is given by

E_C^PS= (P_S+P_c)T/2=

H(P_S+P_c)/B

log₂(1+min{P_S(1−β)g_SR, ηmin{P_Sg_SRβ,P_sat}g_RD}/σ²), (7) whereP_cdenotes the circuit power consumption of the source node.

B. Optimal design

We now determine the optimal source transmit powerP_Sand power splitting factor β, based on the instantaneous channel realization, to minimize the total energy consumption, given in Eq. (7). Note that this objective function is non-convex.

As such, we can not directly apply KKT condition to obtain the optimal solution. Note also that we need to determine the optimal parameter values for each data transmission session based on the instantaneous channel gain. In what follows, we derive the closed-form expressions of optimal transmission parameters through mathematical analysis.

We consider the case of P_Sg_SRβ <P_sat and P_Sg_SRβ ≥P_sat separately. WhenPSgSRβ<Psat, the energy consumption minimization problem is formulated as

min

PS,β

H(P_S+P_c)

Blog₂(1+P_Sg_SR(min{(1−β), η βgRD})/σ²), s.t. 0<P_S<min{P_sat/(g_SRβ),P_max}, 0<β <1, where Pmax is the peak transmit power of the source node.

We first observe that objective function is monotonically decreasing with min{1−β, β ηg_RD}, which will achieve its maximum value when 1−β=β ηg_RD. Thus, the optimal value of power splitting factorβ is given by

β^∗= 1

1+g_RDη. (8)

Applying optimal power splitting factor, the optimization problem simplifies to

minP_S

(P_S+Pc)H

Blog₂(1+P_S(g_SR/eg_RD)/σ²),

s.t. 0<P_S<min{P_sat(1+g_RDη)/g_SR,Pmax}, where we define ge_RD =1+ ¹

gRDη for notation conciseness.

It can be verified that, by examining its second derivative with respect to P_S, this objective function is strictly convex with respect to P_S. After taking derivative with respect to P_S and setting the result to zero, the optimal source transmission power, under the the constraint of 0<PS<Psat/(g_SRβ^∗), is determined as

P_S^∗PS=min{ P_c−ge_RDσ²/g_SR W₀[e⁻¹(^P^c^g^SR

eg_RDσ²−1)]−eg_RDσ² g_SR , P_sat(1+g_RDη)

g_SR , P_max},

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whereW0[·] denotes the principle branch of the Lambert W function [38].

For the case of PSgSRβ ≥Psat, the energy consumption minimization problem becomes, assuming thatP_sat/(g_SRβ)is less than P_max ¹

min

PS, β

H(P_S+P_c)

Blog₂(1+min{P_S(1−β)g_SR, ηPsatgRD}/σ²), s.t. P_sat/(g_SRβ)≤P_S≤P_max, 0<β<1,

The objective function will be minimized when PS and β satisfy P_S(1−β)g_SR =ηP_satg_RD. Under this condition, the optimization problem can be simplified to

min

PS

H(P_S+P_c) Blog₂(1+ηPsatgRD/σ²), s.t. P_sat/(g_SRβ)≤P_S≤P_max.

Since the objective function is monotonically increasing with P_S, the optimal P_S should be equal to P_sat/(g_SRβ). Applying this optimal PS, we arrive at the same optimalβ^∗ as given in Eq. (8).

Combining the above two cases, the optimal transmission parameters for power splitting mode in terms of minimizing

1Otherwise, no solution exists for this case.

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total energy consumption is given in Eq. (8) and Eq. (9). The resulting transmission rate of both hops is equal to

R^∗PS=Blog₂(1+ P_S^∗PSηg_SRg_RD

(1+ηg_RD)σ²). (10) Correspondingly, to minimize the energy consumption of the IoT transmission session under consideration, the source will transmit at rate R^∗PS with power P_S^∗PS for a duration of H/R^∗PS/2. The relay will apply power splitting operation with β^∗ and then forward its decoded data with the same rate for the same duration, but with power level ηP_S^∗PSg_SRβ^∗. The resulting minimum energy consumption can be determined as

E_min^PS =











H(P_c−ge_RDσ²/g_SR)ln 2 BW₀[e⁻¹(^PcgSR

egRDσ2−1)] ,

(P_cgSR−egRDσ²)β^∗ W0[e⁻¹(^PcgSR

gRDσe 2−1)]−eg_RDσ²β^∗<P_sat;

H(Psat/g_SR+Pc) Blog₂(1+^η^{Psat gRD}

σ2 ),

Pcg_SR−eg_RDσ²β^∗ W0[e⁻¹(^PcgSR

gRDe σ2−1)]−eg_RDσ²β^∗≥P_sat.

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IV. TIME SWITCHING MODE

A. Energy consumption analysis

With time switching operation mode, the total transmission duration T is divided into three slots, as shown in Fig. 1(b).

Specifically, the source first transfers energy to the relay with power PSC over the first slot of duration αT, where α ∈ (0,1) is the time switching factor. Then, the source transmits information to the relay over the second slot of duration (1−α)T/2. Finally, the relay forwards its decoded information to the destination over the last slot. As such, applying a piecewise linear EH model, the harvested energy at the relay is given by

E_H^{T S}=

αTηP_SCg_SR, P_SCg_SR<P_sat;

αTηP_sat, P_SCg_SR≥P_sat. (12) The received SNR at the relay over the second slot is equal to

γ_R^{T S}=P_Sg_SR

σ² . (13)

Since the relay uses all of its harvested energy for data forwarding over the last slot, the transmit power of the relay is determined as

P_R^{T S}= E_H^{T S}

(1−α)T/2= 2α η

1−αmin{P_SCg_SR,P_sat}. (14) The effective data rate with time switching mode can be calculated, while noting that only 1−α portion of the total duration is used for information transmission, as

R^{T S}=(1−α)B

2 log₂(1+ 1

σ²min{P_SgSR,P_R^{T S}gRD}). (15) It follows that the total duration of the transmission sessionT is equal to H/R^{T S}.

With time switching mode, the source consumes energy at power level P_SC+P_c in the first slot for a duration of αH/R^{T S} and at power level P_S+P_c in the second slot for

(1−α)H/(2R^{T S})duration. As such, we can calculate the total energy consumption as

E_C^{T S}= H(P_S+P_c+αe(P_c+P_SC))/B

log₂(1+min{P_Sg_SR, α ηe min{P_SCg_SR,P_sat}g_RD}/σ²), (16) where we defineαe as 2α/(1−α)for conciseness.

B. Optimal design

We now optimize the charging powerP_SC, information transmission powerP_S, and time switching factorα to minimize the total energy consumption of a specific transmission session, given in Eq. (16). Note that the above objective function is not convex. Again, iterative algorithms are not applicable as the channel state information will soon become outdated.

In the following, we perform some mathematical analysis to derive the closed-form expressions of all optimal transmission parameters.

First of all, we determine the optimal value of charging powerP_SC with the following proposition.

Proposition 1. For time switching mode, the optimal charging power in terms of minimizing the total energy consumption of a data transmission session is equal to P_SC^∗ = min{P_sat/g_SR,P_max}.

Proof. When P_SCg_SR≤P_sat, we can rewrite the total energy consumption as

E_C^{T S}= (P_S+ (1+m/P_SC)P_c+m)H

Blog₂(1+g_SRmin{P_S, mηg_RD}/σ²), (17) where we define an auxiliary variable m=αPe _SC. Note that mcan be adjusted independent of P_SC sinceαe ranges from 0 to+∞. As such, total energy consumption is a monotonically decreasing function of P_SC. So, the optimal P_SC is equal to P_sat/g_SR.

WhenP_SCg_SR≥P_sat, the total energy consumption becomes E_C^{T S}= (P_S+Pc+αe(P_c+P_SC))H

Blog₂(1+min{P_Sg_SR, α ηPe _satg_RD}/σ²), (18) which is monotonically increasing with P_SC. Therefore, the optimalP_SC in this case is also equal toP_sat/g_SR. The proof is completed while noting the peak transmit power of the source P_max.

Applying the optimal value of charging power, i.e.P_SC^∗ , the energy consumption minimization problem can be formulated as

Pmin_S,α

(P_S+P_c+αe(P_c+P_SC^∗ ))H

Blog₂(1+g_SRmin{P_S, α ηPe _SC^∗ g_RD}/σ²), s.t. 0<P_S≤P_max, 0<α<1.

Note that the objective function is minimized when α andP_S satisfy

P_S=α ηPe _SC^∗g_RD. (19) Under this condition, the optimization problem simplifies, after settingαe=P_S/(P_SC^∗ ηg_RD), to

min

PS

(P_SGˆ+P_c)H Blog₂(1+P_Sg_SR/σ²), s.t. 0<P_S≤P_max,

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where we define ˆG=1+ ^P^c^+P

∗ SC

ηP_SC^∗ g_RD for notation conciseness.

Since the objective function is strictly convex with respect to P_S, we can determine the optimal information transmission power for time switching mode as

P_S^{∗T S}=min{ P_c/G−ˆ σ²/g_SR W₀[e⁻¹(^P^c_ˆ^g^SR

Gσ² −1)]−σ²

g_SR, P_max}. (20) The optimal time switching factor can be obtained, by setting P_StoP_S^{∗T S}, as

α^∗=1− 2g_RDηP_SC^∗

P_S^{∗T S}+2g_RDηP_SC^∗ . (21) With these results, we can calculate the effective information transmission rate as

R^{∗T S}=Blog₂(1+P_S^{∗T S}g_SR

σ² ). (22)

Accordingly, to minimize the energy consumption of the transmission session under consideration, the source first transfers energy to the relay with power P_SC^∗ =min{P_sat/g_SR,P_max} for a duration of α^∗H/R^{∗T S}. Then, it will transmit data at rate R^{∗T S}for duration of(1−α^∗)H/(2R^{∗T S})with powerP_S^{∗T S}. For the remaining (1−α^∗)H/(2R^{∗T S}), the relay will forward its decoded data to the destination at the same rate with power

2α^∗η

1−α^∗P_SC^∗ g_SR. Combining the above descriptions, the resulting minimum energy consumption with time switching mode can be determined as

E_min^{T S} =











H(P_c−Gσˆ ²/g_SR)ln 2 BW₀[e⁻¹(^PcgSR_ˆ

Gσ2 −1)],

Pc/G−σˆ ²/g_SR W0[e⁻¹(^PcgSR_ˆ

Gσ2 −1)]−_g^σ²

SR <P_max;

(Pmax+Pc+_1−α^2α^∗^η∗(Pc+P_SC^∗ ))H Blog₂(1+g_SRmin{Pmax, ^2α_1−α^∗^η∗P_SC^∗ g_RD}/σ²),

Pc/G−σˆ ²/gSR

W0[e⁻¹(^PcgSR

Gσˆ 2 −1)]−_g^σ²

SR ≥Pmax.

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V. NUMERICAL RESULTS

In this section, we present selected numerical examples to illustrate the analytical results in previous sections. Table I summarizes the common parameter values used when gener- ating these results. We set the energy conversion efficiency to 0.9 according to reference [39], which showed in 2017 that energy harvesting rectifier can achieve an energy conversion efficiency of around 0.87. We believe that with the improvement of rectifier technology, an efficiency of 0.9 should be achievable soon, if not already. Note that most existing works on energy efficient design of relay transmissions target the average energy efficiency. It is very difficult, if not impossible to carry out a fair comparison between the proposed session- specific design and conventional average-based design. To verify the effectiveness of the proposed design, we compare the performance of the proposed design with those obtained through exhaustive search and recent session-specific design for point-to-point transmission.

In Fig. 2, we plot the energy consumption of relay transmission system with power splitting mode when transmittingH= 5 kbits of data. In particular, we plot the energy consumption as the function ofg_SRwith different values ofg_RDfor the cases

TABLE I: Common parameter values

Parameters Values

noise power,σ² 1×10⁻⁵ mWatt channel bandwidth,B 100 kHz circuit power,Pc 20 mWatt energy conversion efficiency,η 0.9 maximum transmit power,P_max 300 mWatt power saturation threshold,P_sat 20 mWatt

−300 −28 −26 −24 −22 −20 −18 −16 −14 −12 −10

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Channel power gain g

SR (dB)

Energy consumption (mJ)

PS = 100 mW, β = 0.1, g_RD = −20 dB P_S = 100 mW, β = 0.1, g_RD = −10 dB With P_S^* and β^*, g_RD = −20 dB With P_S^* and β^*, g_RD = −10 dB Exhaustive search, g

RD = −10 dB

Fig. 2: Energy consumption of a transmission session with power splitting energy transfer mode (H=5 kbits).

with and without parameter optimizations. We can see that the energy consumption generally decreases as the channel gains g_RD and g_SR increase, as intuitively expected. We can also see that parameter optimization can dramatically reduce the energy consumption of the session. The perfect match between the analytical results and the results from exhaustive search verifies the validity of our analytical results. Similar behavior can be observed for the energy consumption for time switching mode.

In Fig. 3, we plot the optimal information transmission power of the source P_S^∗ in terms of minimizing the energy consumption for different channel realizations. We can see that when g_SR increases, the optimal power for both modes decreases as intuitively expected. On the other hand, when g_RD increases, while that for power splitting mode decreases, the optimal transmit power for time switching mode increases.

This interesting behavior of time switching mode can be explained as follows. Wheng_RDis relatively small, the second hop will likely be the bottleneck link. The best strategy for the source is to transmit at low rate with low transmit power.

When g_RD increases and relay link can support higher rate, the source will use a higher transmit power level to match the transmission rate of the second hop, which will reduce the overall transmission duration and hence increase the power consumption.

Fig. 4 compares the energy consumption of relay trans-

(8)

−30 −25 −20 −15 −10

−25

−20

−15

−10

−5 0 5 10 15

Channel power gain g RD (dB)

Optimal power level (dBm)

g_SR = −30 dB g_SR = −10 dB

TS mode PS mode

Fig. 3: Optimal transmit power levels of source node (H=5 kbits).

Fig. 4: Energy consumption comparison between relay transmission with time switching mode and direct transmission with optimal parameters (H=5 kbits).

−40 −35 −30 −25 −20 −15

0.46 0.48 0.5 0.52 0.54 0.56 0.58

Channel power gain g SR (dB)

Minimum energy consumption (mJ)

Direct mode TS mode PS mode

Fig. 5: Energy consumption comparison between power s- plitting, time switching, and direct transmission modes with optimal parameters (g_SD=−50 dB andH=5 kbits).

mission with time switching mode and direct transmission [14] using optimal transmission parameters. In particular, the minimum energy consumption is plotted as function of both g_SR andg_RD. For fair comparison, we assume that the channel attenuation of direct transmission is approximately equal to the sum of the attenuation of both hops, which corresponds to the case that relay is situated on the line connecting the source and the destination. We can see that when the channel quality is poor, i.e. g_SD<−30 dB, relay transmission with energy transfer can save considerable amount of energy compared to direct transmission. On the other hand, when g_SD>−25 dB, direct transmission leads to lower energy consumption. As such, we can conclude that relay transmission with wireless power transfer is more energy efficient than direct transmission when the direct link experience poor channel quality.

Fig. 5 compares the energy consumption of power splitting, time switching, and direct transmission while fixing the direct link channel power gaing_SD to -50 dB and varyingg_SR from -40 dB to -15 dB. Correspondingly, g_SR decreases from -10 dB to -35 dB. The energy consumption of direct transmission is constant asgSDremains the same. The time switching mode leads to lower energy consumption than direct transmission as long as neitherg_SRnorg_RDis too small. On the other hand, the power splitting mode enjoys the lowest energy consumption as long as g_SR is not very small. With the exact analytical expressions for the minimum energy consumption determined above, the source can determine which transmission mode leads to the minimum energy consumption for a given channel realization. The resulting transmission system will achieve the highest possible energy utilization efficiency for each transmission session and save the valuable energy resource

(9)

−305 −28 −26 −24 −22 −20 −18 −16 −14 −12 −10 10

15 20 25 30

SR (dB)

Transmission duration (ms)

TS mode, with P S

* and α^* PS mode, with P

S

* and β^* PS mode, P

S = 200 mW, β = 0.1 TS mode, P

S = 200 mW, α = 0.1

Fig. 6: Transmission duration comparison of power splitting and time switching modes with and without energy consumption minimization (g_RD=−20 dB,H=5 kbits).

of source node.

Fig. 6 plots the time duration required to transmit a fixed amount of data using the relay transmission system under consideration. The results of both power splitting and time switching modes, with and without energy consumption minimization, are presented. We can see that the transmission duration for all cases decreases with increasing g_SR. We also observe that the transmission duration increases for both operating modes after parameter optimization, demonstrating a tradeoff between energy consumption and latency performance. Finally, the optimized power splitting mode leads to smaller transmission duration than the optimized time switching mode. Considering together with the energy consumption comparison in Fig. 5, we can conclude that power splitting is the preferred operating mode as it leads to lower minimum energy consumption and shorter transmission duration, at the cost of slightly higher hardware complexity due to simultaneous energy harvesting and data reception.

VI. EFFECT OFLATENCYCONSTRAINT

The results in previous section show that the optimal transmission parameters that minimizes energy consumption will lead to large transmission duration for transmitting a fixed amount of data. Meanwhile, the QoS requirement of certain IoT applications may mandate a hard latency constraint. In this section, we study the effect of such latency constraint on minimum energy consumption, again from an individual data transmission session perspective. We assume that the source needs to transmit H bits data to the destination under a hard latency constraint, denoted byT_th.

With hard latency constraint, the energy consumption min-

imization problem for power splitting mode becomes min

P_S,β

H(P_S+P_c)/B

log₂(1+^min{P^S^(1−β^)g^SR^, ^ηmin{P^S^g^SR^β^,P^sat^}g^RD^}

σ² )

, s.t. 0<PS≤Pmax, 0<β<1,

2H

Blog₂(1+min{P_S(1−β)g_SR,ηmin{P_Sg_SRβ,Psat}g_RD}/σ²)<T_th. Following a similar process in previous section, while considering the cases ofP_Sg_SRβ ≤P_sat andP_Sg_SRβ>P_sat separately, we can show that the power splitting factor β obtained in Eq. (8) remains optimal under the latency constraint. Accord- ingly, the optimal transmit power is given by

Pe_S^∗PS=max{P_S^∗PS, (2^2H/(BT^th⁾−1)eg_RDσ²/g_SR}, (24) where P_S^∗PS was given in Eq. (9). Essentially, the latency constraint translates to a minimum transmission power requirement. Note that when the minimum power (2^2H/(BT^th⁾− 1)ge_RDσ²/g_SR is greater than P_max, the above optimization problem will have no feasible solutions, i.e. the data transmission can not complete within the latency constraint under the given channel realization. In this case, the source node may decide not to transmit to save its energy.

For time switching mode, the energy consumption minimization problem is updated to

Pmin_S, α

(P_S+P_c+α(Pe _c+P_SC))H

Blog₂(1+min{P_Sg_SR, α ηe min{P_SCg_SR,P_sat}g_RD}/σ²), s.t. 0<P_S≤P_max, 0<α<1,

2H

B(1−α)log₂(1+min{P_SgSR,α ηe min{P_SCgSR,Psat}g_RD}/σ²)<T_th. Note that increasing effective transmission rate will reduce the energy consumption as well as the transmission duration. Therefore, the optimal charging power is still equal to min{P_sat/g_SR,Pmax}under the latency constraint. And the op- timalα andP_Sstill need to satisfy Eq. (19). After eliminating α using Eq. (19), the optimization problem simplifies to

minP_S

(P_SGˆ+P_c)H Blog₂(1+P_Sg_SR/σ²), s.t. 0<P_S≤P_max,

g_RDηP_SC^∗

PS+2gRDηP_SC^∗ Blog₂(1+P_Sg_SR/σ²)≥H/T_th. The optimal power determined in Eq. (20) should also satisfy the latency contraint. After some manipulations, we can equivalently transform the latency constraint to

( U

g_SRg_RD− U

σ²g_RDP_S)exp( U

g_SRg_RD− U

σ²g_RDP_S)≤

− U

g_SRg_RDexp(− U

g_SRg_RD+2Hln 2 BT_th ).

(25)

whereU denotes _ηP^Hσ∗²^{ln 2}

SCBT_th. Applying the definition of Lambert W function, the value range of transmit power that satisfies the latency constraint is determined as

P₀^{T S}≤P_S≤P₋₁^{T S}, (26) where

P₀^{T S}=−σ²( 1 gSR

+g_RD

U W₀[− U gSRgRD

exp(− U gSRgRD

+2Hln 2 BTth

)]), (27)

(10)

0.012 0.013 0.014 0.015 0.016 0.017 0.018 0.019 0.02 0.021 0.2

0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4

Latency constraint (s)

TS mode, g_SR = −30 dB, g_RD = −10 dB TS mode, g_SR = −10 dB, g_RD = −30 dB PS mode, g_SR = −30 dB, g_RD = −10 dB PS mode, g_SR = −10 dB, g_RD = −30 dB

Fig. 7: Effect of latency constraint on the energy consumption of the relay transmission system with optimal parameter values (H=5 kbits).

and

P₋₁^{T S}=−σ²( 1 g_SR+g_RD

U W−1[− U g_SRgRD

exp(− U g_SRgRD

+2Hln 2 BT_th )]).

(28) Here, W−1[·] denotes the negative branch of Lambert W function. It can be verified using the properties of the W function that P₀^{T S} is greater than zero. As such, the optimal information transmission power for the time switching mode under the latency constraint is determined as

PeS

T S=max{P₀^{T S}, min{P₋₁^{T S}, P_S^{∗T S}}}, (29) whereP_S^{∗T S} was given in Eq. (20). The corresponding optimal time switching factor for this case becomes

αe=1− 2g_RDηP_SC^∗ PeS

T S+2g_RDηP_SC^∗

. (30)

The energy consumption minimization problem for time switching mode may also have no solution due to the latency constraint. Specifically, for a particular channel realization, if the right hand side of Eq. (25), i.e. −_g ^U

SRg_RDexp(−_g ^U

SRg_RD+

2Hln 2

BT_th ), is less than−1/e or P₀^{T S} in Eq. (27) is greater than Pmax, the latency constraint can not be satisfied. In this case, the source node may again decide not to transmit to save its energy.

In Fig. 7, we plot the minimum energy consumption of the relay transmission system, with optimal transmission parameters, as the function of the latency constraint threshold T_th. As we can see, the minimum energy consumption of relay transmission system under different channel realization follows a similar decreasing trend as latency threshold increases.

This confirms again the tradeoff between energy consumption and transmission latency. We also observe that the energy

−40 −35 −30 −25 −20 −15

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

SR (dB)

TS mode PS mode Direct mode

Fig. 8: Minimum energy consumption comparison for power splitting, time switching, and direct modes under latency constraint (g_SD=−50 dB,H=5 kbits,T_th=0.015 s).

consumption of time switching mode is slightly higher than that with power splitting mode. Finally, better source-to-relay channel quality leads to lower energy consumption as less energy is consumed during wireless power transfer to the relay.

In Fig. 8, we compare the minimum energy consumption of relay transmission system with direct transmission under a hard latency constraint ofT_th=0.015 s. For fair comparison, we again assume that the relay is situated on the line between the source and destination and fix the channel power gain of direct link to -50 dB. The minimum energy consumption is plotted as function of g_SR, while the corresponding g_RD is decreasing from -10 dB to -35 dB. We can see that under the given latency constraint, relay transmission with power transfer requires much higher energy consumption than direct transmission. This is because that a much high transmission power is required for relay transmission system to satisfy the latency requirement. We also note that power splitting mode leads to lower energy consumption than time switching mode, especially wheng_SR is large, which again justifies that power splitting mode is preferred in terms of energy consumption minimization.

VII. CONCLUSION

In this paper, we investigated the problem of energy consumption minimization for relaying transmission with wirelessly-powered relay from an individual data transmission session perspective. For both time switching and power splitting energy transfer modes, we analyzed and minimized the energy consumption of the system when transmitting a fixed amount of data, with and without latency constraint. Selected numerical results were presented to illustrate the mathematical formulation and various design tradeoffs. We show that there