Harvest-and-Opportunistically-Relay: Analyses on Transmission Outage and Covertness
11 Harvest-and-Opportunistically-Relay: Analyseson Transmission Outage and Covertness
Yuanjian Li, Rui Zhao,
Member, IEEE,
Zhiqiao Nie and A.Hamid Aghvami,
Fellow, IEEE
Abstract
For enhancing transmission performance, privacy level and energy manipulating efficiency of wire-less networks, this paper initiates a novel simultaneous wireless information and power transfer (SWIPT)full-duplex (FD) relaying protocol, termed harvest-and-opportunistically-relay (HOR). In the proposedHOR protocol, the relay can work opportunistically in either pure energy harvesting (PEH) or the FDSWIPT mode. Due to the FD characteristics, the dynamic fluctuation of R’s residual energy is difficult toquantify and track. To solve this problem, we apply a novel discrete-state Markov Chain (MC) methodin which the practical finite-capacity energy storage is considered. Furthermore, to improve the privacylevel of the proposed HOR relaying system, covert transmission performance analysis is developed andinvestigated, where closed-form expressions of optimal detection threshold and minimum detection errorprobability are derived. Last but not least, with the aid of stationary distribution of the MC, closed-form expression of transmission outage probability is calculated, based on which transmission outageperformance is analyzed. Numerical results have validated the correctness of analyses on transmissionoutage and covert communication. The impacts of key system parameters on the performance oftransmission outage and covert communication are given and discussed. Based on mathematical analysisand numerical results, it is fair to say that the proposed HOR model is able to not only reliably enhancethe transmission performance via smartly managing residual energy but also efficiently improve theprivacy level of the legitimate transmission party via dynamically adjust the optimal detection threshold.
Yuanjian Li and A. Hamid Aghvami are with Centre for Telecommunications Research (CTR), King’s College London, LondonWC2R 2LS, U.K. (e-mail: [email protected]; [email protected]).Rui Zhao and Zhiqiao Nie are with Xiamen Key Laboratory of Mobile Multimedia Communications, Huaqiao University,Xiamen 361021, China. (e-mail: [email protected]; [email protected]).This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after whichthis version may no longer be accessible. a r X i v : . [ ee ss . SP ] F e b I. I
NTRODUCTION
A. Background
Conventionally, wireless communication systems are basically powered by rechargeable batteryor electrical grid, such as cellular, Bluetooth, Wi-Fi and sensor networks. There are several distin-guish physical or/and economic disadvantages rooted in these traditional wireless communicationpower supply methods, which has been the bottleneck restricting the ubiquitous applicationsof wireless communication [1]. More precisely stated, grid-powered wireless communicationsystems, e.g., cellular networks, require solid support of electrical grid infrastructure, which maynot only need much more construction resources but also lead to enormous energy consumption;The operational lifetime of battery-powered wireless networks is usually limited, for finite batterycapacity in practical applications, leading to periodic battery replacement or recharging. Toprolong the wireless networks lifetime and improve the energy efficiency, the research of energy-aware architectures and transmission strategies has been a hotspot in recent years.Energy harvesting (EH) technique is able to scavenge energy from natural resources (e.g., solarpower, piezoelectric energy, wind and mechanical vibrations), which is known as a promisingcandidate to overcome the aforementioned disadvantages of the traditional power supply strate-gies. Unfortunately, the amount of energy harvested from natural resources highly depends onseveral uncontrollable factors, such as the weather condition, resulting in EH unreliablility. Toaid this, a promising method scavenging energy from man-made radio frequency (RF) radiationhas gained lots of research concentrations [2]. Inspired by the fact that the RF signals can carrythe intended information and energy at the same time, the concept of simultaneous wirelessinformation and power transfer (SWIPT) was coined in [3]. Thereafter, two practical SWIPTstrategies were introduced in [4], i.e., time-switching (TS) and power-splitting (PS) based SWIPT,in which the missions of information decoding (ID) and EH are conducted respectively in timeor power domain. Specifically, the TS-based method allocates part of the time slot to decodeinformation and the remaining to harvest energy, whereas one potion of the received signalpower is utilized for ID and the other potion is used for EH in the PS-based strategy [5].Based on these practical SWIPT strategies, various essential issues about SWIPT were studiedin different wireless transmission systems, e.g., maximizing the ergodic rate for a dynamicSWIPT approach in the cooperative cognitive radio network (CCRN) [6], a non-cooperativegame theoretic approach for the resource optimization in SWIPT enabled heterogeneous small cell network (HetSNet) [7], optimizing the energy efficiency (EE) by delicately designing theprecoders at the transceivers in the multiple-input multiple-output (MIMO) two-way wirelessnetworks [8].Full-duplex (FD) technology which allows transceivers emit and receive information simultane-ously, can potentially achieve efficient utilization of wireless resources (say, time and frequency),and thus it is expected to overcome the shortcomings of half-duplex (HD) counterpart onspectral efficiency (SE) [9]. However, the theoretical performance of FD nodes is significantlylimited by the harmful self-interference (SI) which represents that the emitted signals may bedirectly/indirectly received by their own receivers at the FD nodes [10]. Fortunately, thanks torecent advances in SI cancellation (SIC) techniques (e.g., passive and active SIC approaches), itis possible to suppress the SI to noise level, which makes the FD technology more practical andfeasible in practice [11], [12]. Nevertheless, due to the RF impairments, SI cannot be restrainedperfectly so that the FD networks are still impacted by the so-called residual SI (RSI) [13].Thereafter, FD technique has drawn attention from both academic and industrial communi-ties. Among various wireless FD transmission applications, one popular candidate is the FDrelaying (FDR) technique, which can not only extend the transmission coverage and combat thesevere fading in wireless communications but also enhance the utilization efficiency of wirelessresources [14]. Some of the corresponding works have coped with the performance of variousFDR network backgrounds, in the presence of RSI. For example, two buffer-aided relayingapproaches with adaptive transmission-reception at the FD relay in the absence of direct linkbetween the source and the destination were proposed and studied in [15]; whereas the outageprobability of a amplify-and-forward (AF) FDR network with direct source-destination link wasinvestigated in [16]. Moreover, in the case of decode-and-forward (DF) relaying protocol, [17]researched the ergodic achievable secrecy rate issue in the FDR wiretap channels.With rapid development of the fifth-generation (5G) wireless networks and Internet of Things(IoT), rocketing sorts and amounts of private information (e.g., location data, control orders,social identity information, e-health indexes) are needed to be shared wirelessly among tran-ceivers. Consequently, growing concerns have been posing on security and privacy (low detectionprobability by the third party) of wireless transmissions. In the existing literature, securityissues of wireless communications are much more concerned and investigated than its privacycounterpart. To enhance wireless information security, lots of works have been developed,like, cryptography and information-theoretic physical layer security techniques. However, the inherently public and visible nature of wireless medium (electromagnetic wave) not only leadsto the security vulnerability but also privacy weakness. Recently, increasing research effortshave been pouring into the field of low probability of detection on the existence of wirelesstransmissions, namely, covert communications. For example, communicating covert messagesunder the detection of legitimate party who does not desire the leakage of sensitive informationwhich is supposed to be kept confidential within authorized tranceivers, belongs to covertcommunication research regime. The famous Square Root Law which indicates the fact that O ( √ n ) bits of information can be transmitted reliably and covertly in n channel uses overadditive white Gaussian noise (AWGN) channels as n → + ∞ , was coined in [18]. Afterwards,covert transmissions have been researched and investigated in various wireless communicationscenarios [19]–[21]. In [19], the authors studied covert wireless communications in the presenceof a FD receiver which can generate artificial noise to cause uncertainty at the adversary so thatlow probability of detection can be achievable. Considering both centralized and distributed an-tenna systems (CAS/DAS), multi-antenna-aided covert communications coexisting with randomlylocated wardens and interferers was studied in [20]. Authors of [21] jointly optimized trajectoryand transmit power for covert transmission in unmanned aerial vehicle (UAV) networks, aimingto hide a UAV for communicating critical informations. B. Related Works and Motivation
Hereby, we review the related works, point out the differences and claim our motivation.It has been a promising solution to meet the green communication and the reliable transmissiondemand in the upcoming 5G and IoT era by introducing SWIPT into FDR wireless communi-cations. Particularly, in the scenario which contains power-constrained relay node, the SWIPTFDR has the potential to not only solve power supply problem but also enhance significantlykey wireless transmission performances, like, reliability, SE, valid coverage, quality of service(QoS), etc. Besides, by delicately designing the covert communication detection strategy, it ispromising to improve privacy level of the SWIPT FDR system.To the best of the authors’ knowledge, there already exist inspiring related literature whichinvestigated and studied SWIPT FDR in the context of different wireless network setups. In [22],the characteristics and performance of PS-based two-way SWIPT FDR networks as well as therelay selection issue were researched. In [23], a joint optimization method finding the sourceas well as the relay beamformers was proposed and the numerical results for the mean squared error (MSE) and bit error rate (BER) showed that the proposed method performed well in theMIMO SWIPT FDR systems. In [24], the throughput maximization problem for a FDR wirelesscommunication network with simultaneous down-link energy transfer and up-link informationtransmission was investigated. In [25], outage probability and average throughput performanceswere investigated in a SWIPT FDR wireless network. Unfortunately, the aforementioned worksand the majority of existing literature on SWIPT FDR did not include the consideration on covertcommunications. To bridge this research gap, we investigate covert communication problems ofSWIPT FDR systems in this paper.Regarding the related works of covert communications in the field of wireless relaying net-works, it is still in its infancy stage. In [26], Hu et al. examined the possibility, performancelimits, and associated costs for a power-constrained HD relay transmitting covert informationon top of forwarding the source’s information. Wang et al. [27] investigated how channeluncertainty can influence covert communication performance in wireless relaying networks. Acovert communication scheme under fading channels was proposed and studied in [28] wherethe relay not only forwards source’s information but also plays the role as a cooperative jammer.However, so far, discrete EH technique has not been considered in the existing literature regardingcovert communications, which is a main concern of this paper.Motivated by the aforementioned contents, we propose a novel wireless relaying protocol inwhich discrete-energy-state SWIPT FDR and covert communications are combined and consid-ered, aiming to enhance wireless transmission performance while improving its privacy level.
C. Our Contributions
In this paper, a new transmission protocol termed harvest-and-opportunistically-relay (HOR)is designed and analyzed. Specifically, the FD relay which contains no sustainable power supplybut wireless EH system and rechargeable energy storage is deployed to opportunistically helpthe source and the destination complete their wireless communication. In the proposed HORprotocol, according to the relay’s energy status and channel condition between the source and thedestination, the relay works dynamically in either pure energy harvesting (PEH) or the FD SWIPTmode. Furthermore, to evaluate the detection performance on potential covert communication,i.e., improving the privacy level of the proposed HOR protocol, performance analysis on coverttransmission is developed and investigated. As far as the authors know, we are the first to introduce both discrete EH and covert communications into SWIPT FDR systems. The maincontributions of the paper are concluded in details as follows. • Protocal Design : We systematically establish a novel HOR protocol from listing necessaryhardware facilities to designing feasible transmission stragegy. The proposed HOR schemecan efficiently enhance wireless transmission performance between the source and thedestination, via flexibly managing the relay’s precious stored energy. Besides, through covertcommunication analysis, privacy level of the proposed HOR system can be improved. Itis fair to claim that the proposed HOR scheme is able to not only enchance wirelesstransmission performance but also improve the system’s privacy level. • Hybrid Energy Storage and Markov Chain : We consider the practical energy storage modelunder the limitation of finite capacity at the relay. To enable the relay’s FD functionality,a hybrid energy storage scheme is adopted, which consists of both primary and secondaryenergy storages. To track dynamic fluctuation of residual energy, energy discretization anddiscrete-state MC method is applied to model the complicated energy state transitions. Itis worth noting that all the transition probabilities are calculated in closed-form. Then, thestatinoary distribution of the MC is given. • Covert Commmunication Analysis : Under the cover of forwarded source massages, thereexists potential threat of critical information leakage. To improve the HOR protocol’s privacylevel, we provide covert communication analysis given channel uncertainty in this paper. Theoptimility of radiometer for covert massage detection is proved. Closed-form expressionsof false alarm and missed detection probabilities are derived, based on which we calculateclosed-form expressions of the optimal detection threshold and the coresponding minimumdetection error probability. Furthermore, the impacts of imperfect channle estimation on theminimum detection error probability is also discussed. • Transmission Performance Analysis : Invoking the MC’s stationary distribution, closed-formexpression of transmission outage probability is derived, then we provide transmissionoutage analysis of the proposed HOR scheme. Furthermore, the impacts of key systemparameters on transmission outage performance are investigated via numerical results.
D. Outline and NotationOrganization : The paper is organized as follows. Section II presents the HOR model and itstransmission strategy. Section III describes the energy discretization, detailed derivation on the
MC and the stationary distribution. Section IV shows covert communication analysis. Section Vgives transmission outage performance analysis. Simulation results are presented in Section VIand conclusions are drawn in Section VII.
Notation : Bold lower case letters denote vectors, e.g., v . Bold upper case letters denotematrices, e.g., M . ( · ) T , ( · ) − , I indicate transpose of matrix, inverse of matrix, unit matrix,while E {·} and | · | mean statistical expectation and modulo operators of a complex number,respectively. CN ( µ, σ ) stands for the complex Gaussian distribution with mean µ and variance σ . The intersection of two sets A and B is denoted by A ∩ B . Pr ( · ) is the operator calculatingprobability of a specific objective. Symbols (cid:80) and (cid:81) represent the summation and productoperations of a sequence of terms, respectively.II. S YSTEM MODEL AND TRANSMISSION STRATEGY
A classical three-node wireless relaying network, which comprises one source (S), one desti-nation (D) and one relay (R), is considered in this paper. Energy-constrained R is equipped withdual antenna so that it can adopt the FD technique, whereas S and D are both single-antennanode. The novel HOR protocol is coined originally to assist wireless communication from S toD, with the ability of managing RF energy smartly, while improving the overall privacy level.
A. Assumptions Regarding Wireless Channels
In this paper, we assume that all wireless channels are modeled as quasi-static Rayleigh fadingchannels, which means that these fading channels remain static within each transmission slot,and vary independently over different transmission slots. The Rayleigh fading distribution thatthe self-interference (SI) channel at R follows is considered because the line-of-sight (LoS)component can be largely eliminated via antenna isolation and the scattering plays the principalrole herein. Note that the aforementioned slot is equivalent to a block of time over which theintended massages are transmitted. Besides, this paper considers the widely used infinite block-length model which means that each transmission slot is composed of n symbols and n → ∞ is assumed. Moreover, the block boundaries in wireless links are predefined to be synchronizedperfectly throughout the whole system. Without loss of generality, the block duration in theconsidered model is normalized to one time unit so that the measures of power and energy areidentical and can be used interchangeably in this paper. Wireless channels S (cid:1) D, S (cid:1)
R, and R (cid:1)
Dare denoted as h SD , h SR , and h RD , respectively. Moreover, h RR indicates the SI link caused by the FD characteristic at R. It is worth noting that the channel coefficients h SD , h SR , h RD and h RR aremanipulated to encompass the gains of transmit and receive antennas as well as the path lossescased by propagation distances among the nodes in this paper. The aforementioned wirelesschannel coefficients follow independently and identically distributed (i.i.d.) complex Gaussiandistribution with zero means and variance E {| h SD | } = Ω SD , E {| h SR | } = Ω SR , E {| h RD | } =Ω RD and E {| h RR | } = Ω RR .Regarding the availability of global CSIs, the instantaneous CSI of channel between S andD are assumed to be available at S via channel estimation, but D can only gain the imperfectinstantaneous CSI estimation of wireless channel between R and D. We note hereby that theavailability of instantaneous S (cid:1) R and R (cid:1)
R CSIs poses no influence on the considered perfor-mance analyses so that we do not rise any assumption on their availabilities.
B. Relay Model
In the considered system model, R is known publicly to be energy-limited, leading to rigorouspower supply problem which is expected to be solved by the promising SWIPT technique.Different from the traditional relay strategy, in this paper, a novel relay protocol named HORis proposed, which allows R to work in either the PEH mode or the FD SWIPT mode oppor-tunistically. In specific, when performing the FD SWIPT, R receives and forwards informationsimultaneously to assist the wireless transmission between S and D, while the PS-based EHsolution is applied to harvest the RF energy. In the case of adopting the PEH mode, R concentrateson capturing wireless energy from the RF signals without any information processing. Apartfrom assisting wireless transmission, R is considered as potential leaker who intends to leakvital information regarding the source signals to the third party, which should keep covert fromthe legitimate party, i.e., S and D. The malicious intention of R keeps secrecy and the legitimateparty cannot make sure whether R is innocent or not, the legitimate party treads R as an innocentand friendly node initially but keeps an eye on R detecting the potential covert communicationgenerated by R.For achieving the proposed HOR functionality, R should equip the following hardwares:1) Three RF chains, enabling the EH, information forwarding and covert massage emitting.2) One rectifier utilized to transform the RF signals into direct currents (DC).3) A battery serving as the principal energy carrier (PEC) with high energy capacity.4) One minor battery (MB) for storing harvested energy temporarily, e.g., a capacitor.
5) A constant energy supply for sending covert massage, whose existence is unaware publicly.In details, the receive antenna at R is permanently bounded with the rectifier via one RF chain.One single battery cannot be charged and discharged simultaneously so that the FD SWIPT modemay not be realized, we herein apply both the PEC and the MB at R to crack this dilemma. Notethat the PEC is directly connected to the rectifier and the transmitting RF chain for absorbing andreleasing energy, respectively. In the PEH mode, the harvested energy are absorbed by the PECdirectly. In the FD SWIPT mode, the PEC releases its residual energy to support the transmittingRF chain. Meanwhile, the MB stores the harvested energy temporarily and delivers all the storedenergy into the PEC when the FD SWIPT mode terminates. The hidden constant energy supplywhich is connected to the other RF chain will release its power only when the relay decides toleak the system’s informations.Alongside assisting signal transmission between S and D, R is a rapacious node whichintends to leak essential information (defined as the covert massage herein) regarding the sourcesignals, when the right opportunity occurs. The legal destination D also plays the role as a warden detecting the potential information leakage. Reducing the probability of being detectedby the legitimate party, R would like to emit the covert massage under some solid covers. Inthis proposed system, the forwarded version of the source signals is the only existing shield.Reasonably, R would consider the worst case (D can gain perfect channel estimation and knowits own noise power) and intends to broadcast the covert massage merely when itself works inthe FD SWIPT mode. Otherwise, the covert communication initiated by R will be detected byD without any hesitation (i.e., probability one), which is definitely an undesired circumstanceR ever expects. This is because, in the case of PEH, R is supposed to focus on EH withoutforwarding, any additional transmit power initiated at R will be detected easily by D.
C. Transmission Protocol
In our proposed HOR protocol, before each transmission block is sent out, S broadcasts pilotsignal to estimate h SD which will be utilized to calculate the received instantaneous signal-to-noise-ratio (SNR) at D, denoted as γ SD = P S | h SD | /σ D where P S represents average transmitpower at S, σ D is the power of additive white Gaussian noise (AWGN) at D. In the case of γ SD ≥ γ th , D feeds back two bits “11” to S through a feedback link, where γ th is a predefinedinstantaneous SNR threshold. Otherwise, D feeds back two bits “00” instead. When S receivesthe feedback bits “11”, S broadcasts two bits “01” to R. Otherwise (i.e., S receives "00"), S sends out bits “10” alternatively. If R receives "01", it means the direct link between S and D is goodenough so that R is not necessarily needed to assist the transmission between S and D, R keepsworking in the PEH mode without forwarding any information (of course, including the possiblecovert massage). If R receives bits “10”, which means the quality of received information at D ispoor, R is expected to help the transmission from S to D. Before participating in transmission, Rhas to estimate its residual energy, checking whether the available energy is sufficient to supportthe transmission. If the energy state of R is greater than a given residual energy threshold E th ,i.e., E i ≥ E th , R feeds back bit “1” to S, otherwise, feeds back bit “0” instead. Once S receivesthe feedback bit “1” from R, S starts to broadcast the intended information signal, and R turnsinto the FD SWIPT mode, i.e., R helps S forward the information signal and harvests energysimultaneously. If S receives the feedback bit “0” from R, S broadcasts energy signal to chargethe battery at R. At this very time, D ceases signal processing, because the energy signal israndomly generated by S and conveys no useful information.The condition γ SD ≥ γ th is referred as the “SNR requirement” which is applied to guaranteethe reliability of communication from S to D. On the other hand, the condition E i ≥ E th isregarded as the “energy requirement”, ensuring that the residual energy at R is sufficient tosupport the relying work.We would like to explain the PEH and the FD SWIPT modes thoroughly in the following:
1) The PEH Mode:
When R works in the PEH mode, R employs the reception antenna forreceiving RF signals. Note that the PEH mode will be enabled in the case of either γ SD ≥ γ th or { γ SD < γ th } ∩ { E i < E th } . By ignoring the negligible energy harvested from the noise at thereceiver, the total amount of energy harvested at R in a transmission slot can be given by E PEH = ηP S | h SR | , (1)where η (0 < η < means the efficiency of energy conversion, and the harvested energy in thisstage will be straight transferred into the PEC.
2) The FD SWIPT Mode:
It is worth noting that the FD SWIPT mode will be invoked whenthe case { γ SD < γ th } ∩ { E i ≥ E th } holds. Only in this circumstance, R gets chance to broadcastcovert massage under the cover of the forwarded version of legitimate signals.When R does not emit covert massages, the received signals at R and D can be expressedrespectively as y R [ ω ] = (cid:112) P S h SR x S [ ω ] + (cid:112) kP R h RR x R [ ω ] + n R [ ω ] , (2) y D [ ω ] = (cid:112) P S h SD x S [ ω ] + (cid:112) P R h RD x R [ ω ] + n D [ ω ] , (3)where P R means average transmit power at R, x S [ ω ] ∼ CN (0 , represents the intended signalemitted from S, ω ∈ { , , ..., n } denotes the symbol index in a transmission block and n measures the block-length, i.e., the total number of channel uses in each specific transmissionslot. x R [ ω ] = x S [ ω − ð ] is the forwarded version of x S [ ω − ð ] after decoding and recodingwhere x R [ ω ] ∼ CN (0 , , and integer ð represents the number of delayed symbols due tosignal processing. The AWGNs received at R and D are respectively marked as n R and n D ,subjected to n R [ ω ] ∼ CN (0 , σ R ) and n D [ ω ] ∼ CN (0 , σ D ) . In the FD SWIPT mode, R suffersfrom the SI which will definitely degrade R’s reception quality. Thanks to the promising SICtechniques, R can debilitate the SI up to a relatively low degree. Practically, constrained bycomputation capacity and impanelment complexity, the perfect SIC cannot be reached. Thus, weconsider a practical scenario where imperfect SIC assumption is adopted, and variable k ∈ (0 , represents the SIC coefficient which implies different SIC levels.When R does decide to broadcast covert massage, the received signals at R and D can beexpressed respectively as y R [ ω ] = (cid:112) P S h SR x S [ ω ] + (cid:112) kP R h RR x R [ ω ] + (cid:112) kP ∆ h RR x c [ ω ] + n R [ ω ] , (4) y D [ ω ] = (cid:112) P S h SD x S [ ω ] + (cid:112) P R h RD x R [ ω ] + (cid:112) P ∆ h RD x c [ ω ] + n D [ ω ] , (5)where P ∆ means average transmit power of covert massage x c subjected to x c [ ω ] ∼ CN (0 , .Note that P ∆ merely comes from the constant energy supply.Enabling the FD SWIPT mode, the PS-based EH protocol is adopted in this paper. Specifically,R splits the power of received signal into ρ : (1 − ρ ) proportions. The ρ portion of the receivedsignal power is used to EH and the remaining (1 − ρ ) portion is allocated to informationprocessing. Therefore, after ignoring the negligible energy harvested form the AWGN, the energyharvested at R in each time slot can be respectively calculated as E FS0 = ηρ (cid:0) P s | h SR | + kP R | h RR | (cid:1) , (6) E FS1 = ηρ (cid:0) P s | h SR | + kP R | h RR | + kP ∆ | h RR | (cid:1) , (7)where the lower suffix “FS0” refers to the circumstance in which the FD SWIPT mode is invokedwithout sending covert massage, another lower suffix “FS1” means that the FD SWIPT modewith covert massage is adopted. Particularly, we constrain the total transmit power at R in the FD SWIPT mode as P FS0 = P R and P FS1 = P R + P ∆ , respectively. Hence, (6) and (7) can bereconstructed uniformly as E FS = ηρ (cid:0) P s | h SR | + kP FS | h RR | (cid:1) , (8)where P FS ∈ { P FS0 , P
FS1 } and E FS ∈ { E FS0 , E
FS1 } . Note that P R = E th holds for each specifictransmission block, the lower suffixes “FS”, “FS0” and “FS1” designed in this paper is forconcise expression. In any specific mathematical expression, “FS” is applied to sorely invoke“FS0” or “FS1”, and no combinations of them will be used. It is worth noting that the harvestedenergy is collected via the MB at the first place, and then transferred into the PEC withinignorable time duration when the FD SWIPT mode completes.III. M ARKOV C HAIN AND S TATIONARY D ISTRIBUTION
Enabling the FD SWIPT mode at R, the hybrid energy container composed of the PECand the MB is considered in our proposed model. This hybrid energy container system makesR possible to absorb and release energy at the same time, which plays the essential role ofhardware foundation in the FD SWIPT mode. However, it leads to highly complex and dynamiccharge-discharge behaviors at R, which poses solid obstacle for tracking energy state changesmathematically. To tackle this problem, the energy capacity of PEC is firstly discretized. Then,the MC is invoked to track the complex transmission procedure among discrete energy states.Via the stationary distribution of the MC, the probability of satisfied energy requirement isdetermined.
A. Energy Discretization
To describe the dynamic charging and discharging behaviors of the PEC, we need to discretizethe battery capacity into discrete energy states delicately. Each energy state implies the availableenergy remained in the PEC, which can be reached by calculating the product of the corre-sponding number of energy levels and the unit energy level. In details, the PEC is quantizedinto L + 1 levels, and each energy level characterizes an energy unit equal to C P /L where C P represents the energy capacity of the PEC. Therefore, the i -th energy state is defined as E i = iC P /L, i ∈ { , , ..., L } . In the case of infinite energy discretization, i.e., L → + ∞ , theproposed discrete battery model can tightly track the behavior of continuous linear battery whichis widely applied in the literature. Note that C P ≥ E th is considered in this paper, otherwise R gets no opportunity working in the FD SWIPT mode. It is also worthy to declare that theanalysis of energy discretization concentrates on arbitrary transmission block.In the PEH mode, the discretized amount of energy absorbed by the PEC can be derived as Ξ PEH (cid:52) = (cid:22) E PEH C P /L (cid:23) C P L = q PEH C P L , (9)where (cid:98)·(cid:99) denotes the floor function and q PEH ∈ { , , ..., L } is defined for notation concision.Here, without loss of generality, we declare that the i -th energy state represents the initial energyamount available in the PEC. After charging in the PEH mode, if E i + Ξ PEH ≥ C P , the PEC willbe charged to the maximal capacity E L = C P and any overflowed energy has to be abandoned.Otherwise, the latest energy state after charging is E i + q PEH = E i + Ξ PEH which is guaranteed tobe fully accommodated by the PEC.In the FD SWIPT mode, the harvested energy should be first stored in the MB and thendelivered into the PEC when the FD SWIPT mode terminates. Because the MB is subjectedto a predefined energy capacity C M , the potential amount of energy transferred into the PECshould be reasonably constrained as min { E FS , C M } where the function min { x, y } outputs thesmaller value between x and y . practically, energy transfer form the MB to the PEC suffers fromcircuitry attenuation. Thus, the actual amount of energy absorbed by the PEC can be given by ˆ E FS = η (cid:48) × min { E FS , C M } , (10)where η (cid:48) denotes the energy transfer coefficient from the MB to the PEC, for circuitry attenuation.Furthermore, the discretized amount of energy absorbed by the PEC should be expressed as Ξ FS (cid:52) = (cid:36) ˆ E FS C P /L (cid:37) C P L = q FS C P L , (11)where q FS ∈ { , , ..., L } is stated for brief expression. While harvesting energy in the FD SWIPTmode, R should decode the source signal and forward the recoded information to D. Invoking theenergy requirement, the consumed energy from the PEC should locates at E CFS ∈ [ E th , E i ] wherewe set E CFS = P R = E th = 0 . C P for each transmission slot for simplicity. After discretization,the amount of energy consumption at the PEC can be given by Ξ CFS = (cid:24) E CFS C P /L (cid:25) C P L = q CFS C P L , (12)where (cid:100)·(cid:101) stands as the ceiling function, and q CFS is defined for notation simplicity. It is worthnoting that R may privately broadcast covert massage under the cover of the legitimate forwardedsignal. The energy supporting convert communication sorely comes from the additional energy supply unknown by the legitimate party and the transmit power of covert massage is fixed as P ∆ . Similarly, if E i − Ξ CFS + Ξ FS ≥ C , the PEC will be fully charged to E L = C P . On thecontrary, the latest energy state after charging is E i − q CFS + q FS = E i − Ξ CFS + Ξ FS . For clarity, we pose the following statement. At the beginning of the ( g + 1) -th block, theinitial energy state E i [ g + 1] is merely determined by the transmission mode and energy variationoccurred in the adjacently former block, i.e., the g -th block. Note that E i [ g + 1] is independentto any transmission block before the g -th block, which implies the Markov property. Specifically, E i [ g + 1] = min { E i [ g ] + Ξ PEH , C P } and E i [ g + 1] = min (cid:8) E i [ g ] − Ξ CFS + Ξ FS , C P (cid:9) correspondrespectively to the PEH mode and the FD SWIPT mode applied at R in the g -th block. Hence,energy state transition among different blocks can be characterized and tracked by the MC. Fromthe aforementioned analysis, we note that energy state transition process in our proposed systemis time-independent, thus the MC is considered as homogeneous in time domain. B. Markov Chain
Following the energy discretization in a specific transmission block and the transition relation-ship between energy states for different blocks, we are able to track the transition procedure ofenergy states at the PEC among multiple transmission blocks as a finite-state time-homogeneousMC. Note that modeling the energy state transition process is indeed not necessary for theMB, because it only plays as a temporary energy storage in the FD SWIPT mode. The transitionprobability p i,j denotes the probability of energy state transition from E i to E j , which is occurredbetween the beginning of a transmission block and that of the next transmission block. The energystate transitions of the PEC can be stated comprehensively in the following six cases:1) From E to E : When initial energy at the PEC is empty, it surly cannot afford the FDSWIPT mode. After a transmission block, the residual energy yet remains empty in the consideredcase. It indicates that the total harvested energy in this PEH block is discretized to zero, namely, Ξ PEH = 0 . Invoking (1) and (9), the transition probability of states E → E can be given by p , = Pr ( q PEH = 0) = Pr (cid:18) | h SR | < C P ηP s L (cid:19) . (13)For h SR is subjected to Rayleigh fading and E {| h SR | } = Ω SR , | h SR | follows the Exponentialdistribution with mean Ω SR . Thus, the cumulative distribution function (CDF) of | h SR | can bederived as F | h SR | ( x ) = 1 − exp ( − x/ Ω SR ) . Furthermore, we get p , = F | h SR | (cid:18) C P ηP s L (cid:19) . (14)
2) From E L to E L : In this case, the initial energy certainly satisfies the energy requirement.Thus, whether R works in the PEH mode or the FD SWIPT mode depends merely on theSNR requirement. If the PEH mode is turned on, the harvested energy in this case can be anypossible value, since the PEC cannot absorb additional energy any more. If the FD SWIPT modeis activated, the consumed energy should be less than or equal to its harvested counterpart. From(8), (11) and (12), the transition probability of states E L → E L can be shown as p L,L = Pr ( γ SD ≥ γ th ) + Pr ( γ SD < γ th ) Pr (cid:0) Ξ CFS ≤ Ξ FS (cid:1) . (15)Similar to the derivation of (14), we can obtain q SD = Pr ( γ SD < γ th ) = F | h SD | ( σ D γ th /P S ) and Pr ( γ SD ≥ γ th ) = 1 − F | h SD | ( σ D γ th /P S ) = 1 − q SD . Regarding Pr (cid:0) Ξ CFS ≤ Ξ FS (cid:1) , we obtain Pr (cid:0) Ξ CFS ≤ Ξ FS (cid:1) = Pr (cid:20)(cid:18) q CFS ≤ η (cid:48) E FS C P /L (cid:19) (cid:92) ( E FS < C M ) (cid:21) + Pr (cid:20)(cid:18) q CFS ≤ η (cid:48) C M C P /L (cid:19) (cid:92) ( E FS ≥ C M ) (cid:21) = Pr (cid:16) E FS ≥ q CFS C P η (cid:48) L (cid:17) , C M ≥ q CFS C P η (cid:48) L , C M < q CFS C P η (cid:48) L . (16)Invoking (8), we can gain Pr (cid:18) E FS ≥ q CFS C P η (cid:48) L (cid:19) = Pr (cid:18) Z ≥ q CFS C P ηρη (cid:48) L (cid:19) , (17)where Z = P s | h SR | + kP FS | h RR | . Via convolution of two Exponential distribution variables,we obtain the CDF of Z as F Z ( x ) = − P S Ω SR P S Ω SR − kP FS Ω RR e − xP S Ω SR + kP FS Ω RR P S Ω SR − kP FS Ω RR e − xkP FS Ω RR , P S Ω SR (cid:54) = kP FS Ω RR γ (cid:16) , xP S Ω SR (cid:17) , P S Ω SR = kP FS Ω RR , (18)where γ ( · , · ) is the lower incomplete Gamma function. Furthermore, we get Pr (cid:18) E FS ≥ q CFS C P η (cid:48) L (cid:19) = 1 − F Z (cid:18) q CFS C P ηρη (cid:48) L (cid:19) . (19)Finally, combining (15), (16) and (19), we can obtain p L,L = − q SD F Z (cid:16) q CFS C P ηρη (cid:48) L (cid:17) , C M ≥ q CFS C P η (cid:48) L − q SD , C M < q CFS C P η (cid:48) L . (20)3) From E i to E j (0 ≤ i < j < L ) : It is easy to find that the energy requirement in this case isnot always met. If the initial energy state cannot satisfy the energy requirement, i.e., E i < E th ,the PEH mode will be selected. Otherwise, we need to evaluate whether the SNR requirementis met or not. When γ SD ≥ γ th , R will choose the PEH mode. On the contrary, R will work inthe FD SWIPT mode. Thus, the transition probability of states E i → E j can be expressed as p i,j = q SD Pr ( E i < E th ) Pr ( q PEH = j − i ) + q SD Pr ( E i ≥ E th ) Pr (cid:0) q FS − q CFS = j − i (cid:1) + (1 − q SD ) Pr ( q PEH = j − i )= Pr ( q PEH = j − i ) , i < ϕ (1 − q SD ) Pr ( q PEH = j − i ) + q SD Pr (cid:0) q FS − q CFS = j − i (cid:1) , i ≥ ϕ , (21)where ϕ = (cid:108) E th C P /L (cid:109) denotes the total number of energy units needed to represent the energyrequirement in the discretized energy regime.Next, we calculate Pr ( q PEH = j − i ) and Pr (cid:0) q FS − q CFS = j − i (cid:1) , shown respectively as (22)and (23) in the following. Pr ( q PEH = j − i ) = Pr (cid:18) ( j − i ) C P ηP S L ≤ | h SR | < ( j − i + 1) C P ηP S L (cid:19) = F | h SR | (cid:18) ( j − i + 1) C P ηP S L (cid:19) − F | h SR | (cid:18) ( j − i ) C P ηP S L (cid:19) . (22) Pr (cid:0) q FS − q CFS = j − i (cid:1) = , C M < ( j − i + q cFS ) C P η (cid:48) L − F Z (cid:18) ( j − i + q cFS ) C P ηρη (cid:48) L (cid:19) , ( j − i + q cFS ) C P η (cid:48) L ≤ C M < ( j − i + q cFS +1 ) C P η (cid:48) L F Z (cid:18) ( j − i + q cFS +1 ) C P ηρη (cid:48) L (cid:19) − F Z (cid:18) ( j − i + q cFS ) C P ηρη (cid:48) L (cid:19) , C M ≥ ( j − i + q cFS +1 ) C P η (cid:48) L . (23)Combining (21), (22) and (23), we get the probability of transition E i → E j as p i,j = F | h SR | (cid:16) ( j − i +1) C P ηP S L (cid:17) − F | h SR | (cid:16) ( j − i ) C P ηP S L (cid:17) , i < ϕ (1 − q SD ) × (cid:104) F | h SR | (cid:16) ( j − i +1) C P ηP S L (cid:17) − F | h SR | (cid:16) ( j − i ) C P ηP S L (cid:17)(cid:105) , i ≥ ϕ & C M < ( j − i + q cFS ) C P η (cid:48) L (1 − q SD ) × (cid:104) F | h SR | (cid:16) ( j − i +1) C P ηP S L (cid:17) − F | h SR | (cid:16) ( j − i ) C P ηP S L (cid:17)(cid:105) + q SD (cid:20) − F Z (cid:18) ( j − i + q cFS ) C P ηρη (cid:48) L (cid:19)(cid:21) , i ≥ ϕ & ( j − i + q cFS ) C P η (cid:48) L ≤ C M < ( j − i + q cFS +1 ) C P η (cid:48) L (1 − q SD ) × (cid:104) F | h SR | (cid:16) ( j − i +1) C P ηP S L (cid:17) − F | h SR | (cid:16) ( j − i ) C P ηP S L (cid:17)(cid:105) + q SD × (cid:20) F Z (cid:18) ( j − i + q cFS +1 ) C P ηρη (cid:48) L (cid:19) − F Z (cid:18) ( j − i + q cFS ) C P ηρη (cid:48) L (cid:19)(cid:21) , i ≥ ϕ & C M ≥ ( j − i + q cFS +1 ) C P η (cid:48) L . (24)
4) From E i to E i (0 < i < L ) : In this case, it is not certain whether the energy requirementis met or not. If E i < E th , the PEH mode will be invoked and the harvested energy should bediscretized as zero. If E i ≥ E th and γ SD ≥ γ th , the PEH mode is enabled and the harvestedenergy should also be discretized as zero, too. If E i ≥ E th and γ SD < γ th , the FD SWIPT modewill be selected, the discretized amount of consumed energy should be equal to that of harvestedenergy. Hence, the transition probability of states E i → E i can be calculated as p i,i = (1 − q SD ) Pr ( q PEH = 0)+ q SD Pr ( E i < E th ) Pr ( q PEH = 0) + q SD Pr ( E i ≥ E th ) Pr (cid:0) q FS − q CFS = 0 (cid:1) = Pr ( q PEH = 0) , i < ϕ (1 − q SD ) Pr ( q PEH = 0) + q SD Pr (cid:0) q FS − q CFS = 0 (cid:1) , i ≥ ϕ , (25)where Pr ( q PEH = 0) and Pr (cid:0) q FS − q CFS = 0 (cid:1) are given respectively by (14) and (26) shown as Pr (cid:0) q FS − q CFS = 0 (cid:1) = Pr (cid:0) q FS = q CFS (cid:1) = , C M < q cFS C P η (cid:48) L − F Z (cid:16) q cFS C P ηρη (cid:48) L (cid:17) , q cFS C P η (cid:48) L ≤ C M < ( q cFS +1 ) C P η (cid:48) L F Z (cid:18) ( q cFS +1 ) C P ηρη (cid:48) L (cid:19) − F Z (cid:16) q cFS C P ηρη (cid:48) L (cid:17) , C M ≥ ( q cFS +1 ) C P η (cid:48) L . (26)Substituting (14) and (26) into (25), we get the transition probability of states E i → E i as p i,i = F | h SR | (cid:16) C P ηP S L (cid:17) , i < ϕ (1 − q SD ) F | h SR | (cid:16) C P ηP S L (cid:17) , i ≥ ϕ & C M < q cFS C P η (cid:48) L (1 − q SD ) F | h SR | (cid:16) C P ηP S L (cid:17) + q SD (cid:104) − F Z (cid:16) q cFS C P ηρη (cid:48) L (cid:17)(cid:105) , i ≥ ϕ & q cFS C P η (cid:48) L ≤ C M < ( q cFS +1 ) C P η (cid:48) L (1 − q SD ) F | h SR | (cid:16) C P ηP S L (cid:17) + q SD (cid:20) F Z (cid:18) ( q cFS +1 ) C P ηρη (cid:48) L (cid:19) − F Z (cid:16) q cFS C P ηρη (cid:48) L (cid:17)(cid:21) , i ≥ ϕ & C M ≥ ( q cFS +1 ) C P η (cid:48) L . (27)5) From E i to E j (0 ≤ j < i ≤ L ) : Obviously, this circumstance can only occur in the FDSWIPT mode because the PEH mode can exclusively lead to energy increasing or energyunchanged. Therefore, the transition probability of states E i → E j can be derived as p i,j = Pr ( γ SD < γ th ) Pr ( E i ≥ E th ) Pr (cid:0) q CFS − q FS = i − j (cid:1) = , i < ϕq SD Pr (cid:0) q CFS − q FS = i − j (cid:1) , i ≥ ϕ . (28) Next, we need to calculate Pr (cid:0) q CFS − q FS = i − j (cid:1) , shown as Pr (cid:0) q CFS − q FS = i − j (cid:1) = , C M < [ q CFS − ( i − j ) ] C P η (cid:48) L − F Z (cid:18) [ q CFS − ( i − j ) ] C P ηρη (cid:48) L (cid:19) , [ q CFS − ( i − j ) ] C P η (cid:48) L ≤ C M < [ q CFS − ( i − j )+1 ] C P η (cid:48) L .F Z (cid:18) [ q CFS − ( i − j )+1 ] C P ηρη (cid:48) L (cid:19) − F Z (cid:18) [ q CFS − ( i − j ) ] C P ηρη (cid:48) L (cid:19) , C M ≥ [ q CFS − ( i − j )+1 ] C P η (cid:48) L (29)Invoking (28) and (29), we can express the transition probability of state E i → E j as p i,j = , i < ϕ (cid:107) (cid:18) j ≥ ϕ & C M < [ q CFS − ( i − j ) ] C P η (cid:48) L (cid:19) q SD (cid:18) − F Z (cid:18) [ q CFS − ( i − j ) ] C P ηρη (cid:48) L (cid:19)(cid:19) , i ≥ ϕ & [ q CFS − ( i − j ) ] C P η (cid:48) L ≤ C M < [ q CFS − ( i − j )+1 ] C P η (cid:48) L q SD (cid:20) F Z (cid:18) [ q CFS − ( i − j )+1 ] C P ηρη (cid:48) L (cid:19) − F Z (cid:18) [ q CFS − ( i − j ) ] C P ηρη (cid:48) L (cid:19)(cid:21) , i ≥ ϕ & C M ≥ [ q CFS − ( i − j )+1 ] C P η (cid:48) L . (30)6) From E i to E L (0 ≤ i < L ) : In this circumstance, we cannot make sure what mode is appliedby R, for whether the initial energy state can satisfy the energy requirement is not determined.When E i < E th , certainly the PEH mode will be activated, and the harvested energy shouldmeet Ξ PEH ≥ E L − E i . Otherwise, if γ SD ≥ γ th , the PEH also will be invoked and the harvestedenergy is supposed to satisfy Ξ PEH ≥ E L − E i . If E i ≥ E th and γ SD < γ th , the FD SWIPTmode will be selected and the relationship between the harvested energy and the released energyshould meet Ξ PEH − Ξ CPEH ≥ E L − E i . Thus, the transition probability of states E i → E L can beexpressed as p i,L = Pr ( γ SD ≥ γ th ) Pr ( q PEH ≥ L − i ) + Pr ( γ SD < γ th ) Pr ( E i < E th ) Pr ( q PEH ≥ L − i )+ Pr ( γ SD < γ th ) Pr ( E i ≥ E th ) Pr (cid:0) q FS − q CFS ≥ L − i (cid:1) = Pr ( q PEH ≥ L − i ) , i < ϕ (1 − q SD ) Pr ( q PEH ≥ L − i ) + q SD Pr (cid:0) q FS − q CFS ≥ L − i (cid:1) , i ≥ ϕ . (31)Next, Pr ( q PEH ≥ L − i ) and Pr (cid:0) q FS − q CFS ≥ L − i (cid:1) can be derived respectively as Pr ( q PEH ≥ L − i ) = 1 − F | h SR | (cid:18) ( L − i ) C P ηP S L (cid:19) , (32) Pr (cid:0) q FS − q CFS ≥ L − i (cid:1) = , C M < ( L − i + q CFS ) C P η (cid:48) L − F Z (cid:18) ( L − i + q CFS ) C P ηρη (cid:48) L (cid:19) , C M ≥ ( L − i + q CFS ) C P η (cid:48) L . (33) E E E p , p , p , p , p , p , p , p , p , M = p , p , p ,L p , p , p ,L p , p , p L,L
Figure 1. The state transition diagram and the corresponding transition probability matrix of the Markov chain, in the case of L = 2 . Invoking (31), (32) and (33), we get the transition probability of states E i → E L , shown as p i,L = − F | h SR | (cid:16) ( L − i ) C P ηP S L (cid:17) , i < ϕ (1 − q SD ) (cid:104) − F | h SR | (cid:16) ( L − i ) C P ηP S L (cid:17)(cid:105) , i ≥ ϕ & C M < ( L − i + q CFS ) C P η (cid:48) L (1 − q SD ) (cid:104) − F | h SR | (cid:16) ( L − i ) C P ηP S L (cid:17)(cid:105) + q SD (cid:20) − F Z (cid:18) ( L − i + q CFS ) C P ηρη (cid:48) L (cid:19)(cid:21) , i ≥ ϕ & C M ≥ ( L − i + q CFS ) C P η (cid:48) L . (34) C. Stationary Distribution
We define M (cid:52) = { p i,j } to denote the ( L + 1) × ( L + 1) state transition matrix. For the con-venience of illustration, we provide an example of the possible energy states and the transitionsamong them in the case of L = 2 . As shown in Figure 1, the state transition diagram and thecorresponding transition probability matrix of the MC are clearly depicted. Theorem In this theorem, we derive the probability that the energy status of arbitrarytransmission slot meets the given energy condition. With the help of stationary distribution ξ ,for arbitrary transmission slot, we have Pr ( E i ≥ E th ) = L (cid:88) i = ϕ ξ i , (35)where ξ i ∈ ξ = ( ξ , ξ , ..., ξ L ) T , which can be gained by applying method in the following proof.Furthermore, we can conclude that Pr ( E i < E th ) = 1 − L (cid:88) i = ϕ ξ i = ϕ − (cid:88) i =0 ξ i . (36) Proof:
Using the similar methods in [29], we can easily verify that the transition matrix M is irreducible and row stochastic , which can be verified via Figure 1 as an example. Thus, thestationary distribution ξ must satisfy the following equation ξ = ( ξ , ξ , ..., ξ L ) T = M T ξ . (37)By solving the above equation, ξ can be derived as ξ = (cid:0) M T − I + B (cid:1) − b , (38)where B i,j = 1 , ∀ i, j, b = (1 , , ..., T and I denotes the unit matrix. Remark In Theorem , ξ i where i ∈ { , , . . . , L } represents the stationary probability of the i -th energy state, on a long-term perspective. The reason why the result Pr ( E i ≥ E th ) = (cid:80) Li = ϕ ξ i in Theorem holds can be straightly explained as that ξ i where i ≥ ϕ describes the probability ofan arbitrary event whose residual energy is higher than the energy threshold and the probabilitysummation of all these events makes up the overall probability of E i ≥ E th . It is worth notingthat Theorem serves as the prerequisite for deriving closed-form expressions of transmissionoutage probability which will be shown in Section V. D. Verification and Discussion
In Figure 2, we illustrate the dynamic charge-discharge behavior of the PEC (subfigure (I))and the comparison of the steady state distribution gained from the analytical framework in thissection against those generated through Monte Carlo simulation (subfigure (II)). Note that forsubfigure (I), (II) and (III), L = 25 , for subfigure (IV), L = 5 the other system parameters areall the same among subfigures in Figure 2. The detailed system parameter setups in this figureis in line with that in Section VI. Remark The initial energy remained in the PEC is set to be empty, and as the proposedHOR system runs with respect to (w.r.t.) block numbers, the complex energy accumulation andconsumption process can be clearly traced as shown in subfigure (I). Observing subfigure (II), In a MC, the transition matrix is said to be irreducible if it is possible to reach any other state form any state in finite numberof steps. In our MC analysis, all possible energy states communicate so that the transition matrix M is irreducible in this paper. In a MC, the transition matrix is said to be row stochastic if the sum of all the elements in a row is one and all elementsare non-negative. In our MC analysis, the transition probabilities from any energy state to all possible energy states sums upto one and the transition probabilities are definitely non-negative, so we say the transition matrix M is row stochastic in thispaper. We also note that M is asymmetric because p i,j (cid:54) = p j,i , ∀ i, j , given the aforementioned analysis. Block Numbers R e s i du a l E n e r gy ( % ) (I) PEC Levels P r ob a b ilit y ( % ) (II) Analytical ValueMonte Carlo Simulation (III)
PEC Levels P r ob a b ilit y ( % ) (IV) PEC Levels P r ob a b ilit y ( % ) Figure 2. Illustration of residual energy fluctuations, validation of the proposed MC analysis and the impact of energydiscretisation levels. it is confirmed that the proposed analytical model matches the actual distribution very well,validating the effectiveness of analysis on the MC in this section.
Remark Comparing subfigures (III) and (IV), one can find that the larger L (i.e. the PEClevels) is, more likely the residual energy in the PEC can satisfy the “energy requirement”which is hereby quantified as that the residual energy in the PEC is greater than or equal to60 % of the PEC’s capacity. This is reasonable for a two-fold reason: 1) the floor function(e.g., formulas (9) and (11)) used to quantify the discretized amount of energy absorbed by thePEC limits that the proposed energy discretization model has to abandon the overflow energyassimilated; 2) the ceil function (e.g., formula (12)) applied to quantify the discretized amountof energy consumed by the PEC restricts that the proposed energy discretization model shouldquantify the underflow amount of discretized energy used up by the PEC as an specific integer,which means the proposed model consumes extra energy than its actual counterpart. Accordingto the aforementioned analysis, we can conclude that the larger L is, i.e., the finer the PECis mathematically discretized, the more efficiently manipulating of RF energy can reach. Asubsequent influence of L on wireless transmission performance can be found in details inSection VI. However, there exists the inherent trade-off between the computation complexityand energy manipulating efficiency of the proposed energy discretization model so that thevalue of L should be chosen carefully and delicately in the practical application scenarios.Based on the MC analysis, we can mathematically track wireless transmission performanceof the proposed HOR protocol, like, the connection outage probability, which will be clearlystated and analyzed in Section V. Besides, the inherent SNR requirement when the FD SWIPT is invoked restricts that γ SD has to be less γ th , which puts congenital influences on the covertperformance analysis in Section IV.IV. C OVERT C OMMUNICATION P ERFORMANCE A NALYSIS
In Section III, we investigated the stationary distribution of energy states discretized at R’sPEC, via energy discretization and finite-state homogeneous MC. In this section, we will analyzethe covert performance of the proposed HOR protocol. Note that R only intends to broadcastcovert massage in the case of working in the FD SWIPT mode, because there exists no solidcover in the PEH mode so that D which also plays the role as warden can detect the arising ofcovert communication easily. Thus, this paper focuses on the circumstance in which D performsdetection regarding covert communication only in the case of FD SWIPT mode. In the PEHmode, R will not broadcast covert massage and D ceases the detection. This consideration isreasonable because the exact work mode R applies is an open consensus among all nodes at thebeginning of each specific transmission block. Note that in this section, the constraint γ SD < γ th holds due to the nature of the FD SWIPT mode. A. Channel Uncertainty Model
To investigate the impact of channel uncertainty on covert detection performance at D, it isassumed that D gets an imperfect estimation of the wireless channel R → D and the imperfectchannel estimation model of D is formulated as h RD = ˆ h RD + ˜ h RD , (39)where ˆ h RD ∼ CN (0 , (1 − β ) Ω RD ) and ˜ h RD ∼ CN (0 , β Ω RD ) are independent complex Gaussianrandom variables (RVs) which represent D’s channel estimation and the corresponding estimationerror, respectively. It is worth noting that β ∈ (0 , measures the degree of channel uncertaintyand the aforementioned assumption of Gaussian estimation error comes from the minimum meansquare error (MMSE) estimation method. Although the instantaneous knowledge of h RD that Dgains is incomplete and contains estimation error, we assume that D does know the fadingdistribution to which h RD is subjected. B. Binary Detection at the Destination
The source node S transmits wireless energy to charge R for gaining assistance helping themain wireless transmission between S and D. As one part of the main party, D performs detection regarding whether R emits illegitimate information, i.e., covert message, under the cover of legalforwarded version of source messages. Hence, apart form receiving desired information formS and R, D also needs to perform simple (binary) hypothesis test in which H means thenull hypothesis indicating that R does not transmit covert information while H represents thealternative hypothesis implicating that R does emit the covert message. In a specific transmissionslot, we define the False Alarm (i.e., type I error) probability by P FA (cid:44) Pr ( D |H ) andthe Missed Detection (i.e., type II error) probability by P MD (cid:44) Pr ( D |H ) , where D and D represent the binary decisions in favor of the occurrence of covert transmission or not,respectively. Besides, the a priori probabilities of hypotheses H and H are assumed to beequal (i.e., both are 0.5) in this paper , which is a widely adopted assumption in the field ofcovert communication. Following this assumption, the detection performance of D is measuredby the detection error probability P E , defined as P E (cid:44) P FA + P MD . (40)For arbitrary (cid:15) > , we define R achieving covert communication if any communicationscheme exists satisfying P E ≥ − (cid:15) . Note that the lower bound on P E characterizes the necessarytrade-off between the false alarms and missed detections in a simple hypothesis test. Specifically, P E ≥ − (cid:15) represents the covert communication constraint and (cid:15) signifies the covert requirement,cause a sufficiently small (cid:15) renders any detector employed at D to be ineffective. C. Derivation and Analytics
When a transmission block is determined to adopt the FD SWIPT mode, D would like tokeep an eye on whether R broadcasts covert massage under the cover of the forwarded versionof source information. In the case of FD SWIPT mode, the received signals at D in the ω -thchannel use within a transmission block can be expressed as y D [ ω ] = √ P S h SD x S [ ω ] + √ P R h RD x R [ ω ] + n D [ ω ] , H √ P S h SD x S [ ω ] + √ P R h RD x R [ ω ] + √ P ∆ h RD x c [ ω ] + n D [ ω ] , H . (41) Lemma A radiometer is utilized by D to perform the detection test monitoring potentialcovert communications launched by R. In the case of availability of noise power at D, it isapproved that radiometer is the optimal detector for covert communication detection. Note that the equal a priori probability assumption corresponds to the circumstance in which D has no a priori knowledgeon whether R emits covert message or not and completely ignores R’s covert transmission possibility. Proof:
See Appendix A.Applying a radiometer as the optimal detection strategy at D, closed-form expressions of falsealarm, missed detection and detection error probabilities for any given threshold τ will be derivedin the following Theorem. Then, closed-form expressions of the optimal detection threshold τ and minimum detection error probability will be derived and given. The impacts of imperfectchannel estimation on minimum detection error probability will be analyzed via discussion ofits monotonicity w.r.t. β . Theorem For arbitrary threshold τ , closed-form expressions of false alarm and misseddetection probabilities can be respectively given by P FA = exp (cid:16) j − τβP R Ω RD (cid:17) , τ ≥ j , otherwise , (42) P MD = − exp (cid:16) j − τβ ( P R + P ∆ )Ω RD (cid:17) , τ ≥ j , otherwise , (43)where j = P S | h SD | + P R | ˆ h RD | + σ D and j = P S | h SD | + ( P R + P ∆ ) | ˆ h RD | + σ D . Furthermore,invoking (40), (42) and (43), we can derive closed-form expression of the detection errorprobability, shown as P E = , τ < j exp (cid:16) j − τβP R Ω RD (cid:17) , j ≤ τ < j (cid:16) j − τβP R Ω RD (cid:17) − exp (cid:16) j − τβ ( P R + P ∆ )Ω RD (cid:17) , τ ≥ j . (44) Proof:
See Appendix B.
Theorem The optimal detection threshold of D’s radiometer, which is supposed to minimize P E , is given by τ ∗ = j , j ≥ τ k =0 τ k =0 , j < τ k =0 , (45)where τ k =0 = − βP R ( P R + P ∆ ) Ω RD P ∆ ln P R P R + P ∆ + P S | h SD | + σ D . (46) Proof:
See Appendix C. -9 D e t ec ti on E rr o r P r ob a b ilit y ( % ) (I) SimulationAnalytical, eq. (44)Analytical, eq. (45) -15 -10 -5 P S (dBm) P r ob a b ilit y ( % ) (II) eq. (44)eq. (47) P r ob a b ilit y ( % ) (III) eq. (47) Figure 3. Validation of the derived closed-from expressions of detection error probability and the optimal detection threshold,illustration of performance superiority of the proposed minimum detection error probability and its monotonicity w.r.t. β . Corollary To achieve the best detection performance, D will always select the optimal detec-tion threshold as per (45). Thus, closed-form expression of minimum detection error probabilitycan be calculated as P ∗ E = exp (cid:16) j − j βP R Ω RD (cid:17) , j ≥ τ k =0 (cid:16) j − τ k βP R Ω RD (cid:17) − exp (cid:16) j − τ k β ( P R + P ∆ )Ω RD (cid:17) , j < τ k =0 . (47) Remark According to
Theorem , Theorem and Corellary , it is confirmed that P E , τ ∗ and P ∗ E are independent to parameters k , L , γ th , C M , η , η (cid:48) , σ R , h RR and h SR . This isbecause, concisely speaking, covert communication is constrained to be possible only withinthe FD SWIPT mode, and parameters C P and E th can affect covert metrics in the manner ofthe aforementioned P R = E th = 0 . C P . Moreover, P ∗ E is not subjected to P S and h SD either,because of subtractions of j − j , j − τ k and j − τ k . This finding can guide designers tounderstand clearly what parameters are valid to pose impacts on covert communication detectionperformance.With the help of Corollary , it is mathematically guaranteed that the detection error proba-bility at D is minimized on the perspective of imperfect channel estimation. However, how doesthe factor β influence the performance of minimum detection error probability? This questionmotivates us to provide the following Corollary. Corollary Minimum detection error probability P ∗ E is monotonically increasing functionw.r.t. β . Proof:
See Appendix D. Remark Based on
Corollary , the imperfect channel estimation is proved to be an importantfactor posing significant impacts on P ∗ E . A smaller β , i.e., the better channel estimation method,is desired to enhance the covert communication detection performance at D.To better show the covert communication performance analysis and verify the correctness ofthe corresponding analytical expressions, Figure 3 is illustrated in which β = 0 . stands unlessotherwise specified and the other system parameters are set in line with those in Section VI.Note that in Figure 3, we evaluate covert metrics for arbitrarily selected transmission block pairin which h SD = − . − . j and h RD = 0 . . j stand. From subfigure (I), theMonte Carlo simulation nodes mach perfectly with the analytical curve of (44) and the dashline generated from (45) coincides tightly with the simulated optimal τ ’s coordinate, validatingthe correctness of the derived analytical expressions in Theorem , Theorem . Subfigure (II)depicts clearly that applying Corellary can significantly reduce the detection error probability,compared with its counterpart without the optimal detection threshold. It can also be observedfrom subfigure (II) that the curve of P ∗ E holds instant w.r.t. P S , the reason was explained in Remark . Last but not least, subfigure (III) shows that P ∗ E is a monotonically increasing function w.r.t. β , justifying the effectiveness of Corollary and Remark .In this section, we analyzed the covert communication performance by proving the optimalityof radiometer on detection of potential covert communication and deriving closed-form expres-sions of detection error probability. Based on the mathematical analysis, we calculated andstated closed-form expressions of the optimal detection threshold and minimum detection errorprobability. Note that in this section, we focused on the situation where the FD SWIPT modeis invoked at R. For each particular transmission block which is located in the domain of FDSWIPT, we provided closed-form expressions of the optimal detection threshold and minimumdetection probability from the perspective of imperfect channel estimation of instantaneouswireless channel between S and D, which means the values of (45) and (47) stand for thisspecific transmission block and vary among different transmission blocks when the FD SWIPTmode is activated. Besides, we would like to emphasize hereby that the optimality of our analysisin this section is valid for any particular wireless channel applications when the FD SWIPT modeis active. It is also worth noting that the proposed HOR model inherently limits γ SD < γ th forthe analysis in this section, due to the SNR requirement. V. T
RANSMISSION O UTAGE P ERFORMANCE A NALYSIS
In this section, a typical transmission performance metrics, namely, transmission outage prob-ability (TOP) is derived and analyzed in details. In this paper, we consider the circumstance inwhich D applies Maximum Ratio Combination (MRC) protocol to combine the received signalsfrom S and R, when the FD SWIPT mode stands.In the FD SWIPT mode, invoking (3) and (5), the received SINR at D can be given by γ D = γ SD + Y H , H γ SD + Y H , H , (48)where Y H = min (cid:26) (1 − ρ ) P S | h SR | (1 − ρ ) kP R | h RR | + σ R , P R | h RD | σ D (cid:27) , (49) Y H = min (cid:26) (1 − ρ ) P S | h SR | (1 − ρ ) k ( P R + P ∆ ) | h RR | + σ R , P R | h RD | P ∆ | h RD | + σ D (cid:27) . (50)Note that the term min {· , ·} in (49) and (50) is introduced by the fixed DF relaying policy appliedat R [30]. Knowing | h SR | ∼ E (Ω SR ) , | h RR | ∼ E (Ω RR ) and | h RD | ∼ E (Ω RD ) , closed-formCDF expressions of Y H and Y H can be calculated as F Y H φ ( x ) = − P S Ω SR exp (cid:18) − (cid:18) σ R (1 − ρ ) P S Ω SR + σ D P R Ω RD (cid:19) x (cid:19) P S Ω SR + kP R Ω RR x , φ = 01 − P S Ω SR exp (cid:18) − (cid:18) σ R (1 − ρ ) P S Ω SR + σ D ( P R − P ∆ x ) Ω RD (cid:19) x (cid:19) P S Ω SR + k ( P R + P ∆ )Ω RR x , φ = 1&& x < P R P ∆ , φ = 1&& x ≥ P R P ∆ . (51) Lemma Closed-form expression of CDF of γ D |H can be derived as F γ D |H ( x ) = q SD − v [ Ei ( v ) − Ei ( v )] × exp P S Ω SR (cid:16) σ R (1 − ρ ) P S Ω SR + σ D P R Ω RD (cid:17) − σ D P S Ω SD ( P S Ω SR + kP R Ω RR x ) kP R Ω RR , (52)where Ei ( · ) represents the one-argument Exponential Integral function. For concise expression,we define the following variables in (52) as v = σ D Ω SR kP R Ω RR Ω SD , (53) v = σ D P S Ω SD − σ R (1 − ρ ) P S Ω SR − σ D P R Ω RD kP R Ω RR , (54) v = v ( P S Ω SR + kP R Ω RR x ) , (55) v = v ( P S Ω SR + kP R Ω RR ( x − γ th )) . (56) Proof:
See Appendix E.
Lemma Closed-form CDF expression of γ D |H in the case of FD SWIPT mode can bederived approximately as F γ D |H ( x ) ≈ quadgk ( fun ( y ) , , γ th ) , (57)where the definitions of quadgk ( · , · , · ) and fun ( y ) can be found in the following proof. Proof:
See Appendix F.
Remark In Lemma , the approximation of F γ D |H is achieved by converting infinite integralto finite summation. The accuracy of this approximation is mainly affected by the amount ofnodes used within the finite summation, the more nodes is applied, the more complex thesummation is, though the preciser approximation it can achieve. Theorem Closed-form expression of the TOP in the FD SWIPT mode can be given by
T OP FS = 12 L (cid:88) i = ϕ ξ i (cid:2) F γ D |H (cid:0) R th − (cid:1) + F γ D |H (cid:0) R th − (cid:1)(cid:3) . (58) Proof:
See Appendix G.
Theorem Closed-form expression of the TOP in the PEH mode can be given as
T OP
PEH = q SD ϕ − (cid:88) i =0 ξ i + F γ SD | γ SD ≥ γ th (cid:0) R th − (cid:1) , (59)where the concept of F γ SD | γ SD ≥ γ th ( x ) can be found in the following proof. Proof:
See Appendix H.
Corollary Finally, invoking (58) and (59), closed-form expression of overall TOP for ourproposed HOR model can be derived as
T OP = q SD ϕ − (cid:88) i =0 ξ i + F γ SD | γ SD ≥ γ th (cid:0) R th − (cid:1) + 12 L (cid:88) i = ϕ ξ i (cid:2) F γ D |H (cid:0) R th − (cid:1) + F γ D |H (cid:0) R th − (cid:1)(cid:3) . (60)In this section, the developed closed-form expression of the TOP is indeed in a form ofcomplicated composition, from which the impacts of system parameters on the TOP performanceare impossible to be unveiled and discussed thoroughly. To analyse the TOP performance of the proposed HOR system and thus highlight the superiority of the HOR scheme as well as theimpacts of various system parameters on the TOP performance, we pose detailed investigationvia showing numerical results in Section VI.VI. N UMERICAL R ESULTS
In this section, applying the analytical expressions derived in the previous contents, numericalresults will be performed and the impact of key parameters on the performance will also beinvestigated. The simulation is deployed in a 2-dimensional (2D) topology where all the nodesare placed with the same altitude, i.e., terrestrial relaying scenario. Unless otherwise specified,the simulation results are based on the following parameter setups. The distances among nodesare allocated as d SD = 15 m, d SR = 8 m, d RR = 0 . m and d RD = 8 m, where it is reasonableto consider that the distance between R’s dual antennas is relatively near. We set the averagewireless channel gains as Ω ij = 1 / (1 + d αij ) , { i.j } ∈ { S, R, D } where the path loss exponentis predefined as α = 3 , the AWGN powers σ R = σ D = − dBm, the target transmission rate R th = 1 bps/Hz, the SNR threshold γ th = 1 , the energy threshold E th = 0 . C P , the transmitpower of S P S = − dBm, the PS factor ρ = 0 . and the covert transmitting power P ∆ = 0 . P R .Regarding parameters of the hybrid energy storage, we set C P = C M = 10 − Joule, the energyconversion efficiency η = 0 . , the energy transfer coefficient η (cid:48) = 0 . and the discretisation level L = 25 . -15 -10 -5 0 P S (dBm)00.020.040.060.080.1 T r a n s m i ss i on O u t a g e P r ob a b ilit y Simulation L = L = 25 L No Relay
Figure 4. Transmission outage probability versus P S with various L values. A. Validation of The Proposed Energy Discretization Method
In this part, we validate the feasibility and accuracy of the proposed discrete energy modeldescribed in Section III, by showing curves generated from the MC based TOP analysis andthe corresponding Monte Carlo simulation points. Figure 4 depicts curves of the TOP versus P S with different energy discretisation levels. Note that L → ∞ serves as upper bound of the TOPperformance, in the case of a massive energy discretisation. It can be observed from Figure 4that even a small energy discretisation level ( L = 5 ) is enough to provide considerable TOPperformance gain for majority of the simulated P S regime, compared to the circumstance in whichno relay assists wireless communication between S and D. Comparing the TOP performancecurves of various L values, one can conclude that the TOP performance approaches the upperbound gradually as the value of L increases. The reason why L can affect the HOR system hasbeen explained in details in Remark . Specifically, the TOP performance curve when L ’s valueis not so large, i.e., L = 25 can coincide with the upper bound in the most region of simulated P S . The aforementioned observations validates the effectiveness of the proposed HOR system onhelping wireless transmissions between devices, even with practical energy discretisation levels( L = 5 , L = 25 ). -9 -8 -7 -6 -5 -4 -3 C P (Joule)0.020.0240.0280.0320.034 T r a n s m i ss i on O u t a g e P r ob a b ilit y Simulation L = 5 L = 25 L No Relay
Figure 5. Transmission outage probability versus C P withvarious L values. T r a n s m i ss i on O u t a g e P r ob a b ilit y Simulation L = 5 L = 25 L No RelayOptimality
Figure 6. Transmission outage probability versus ρ withvarious L values. B. The Impact of Capacity of The PEC
In this subsection, we examine that how C P influences the TOP performance. Figure 5 showsthe TOP curves versus C P with various L values. It is straightforward to find that for specificHOR system parameter setup, there exists optimal value of C P to minimise the TOP performance.The existence of the optimal C P is because, briefly speaking, it influences the values of P R and E th by the means of P R = E th = 0 . C P . Under the system parameter setup of this example, thevalues of L does not pose any impact on value of the optimal C P . It can be observed that L = 25 can almost act as a feasible alternative of the TOP performance’s upper bound, revealing theefficiency of the proposed energy discretisation model. The observation of this example allowsthe system designer to determine an optimal C P while reducing computation by selecting a smallbut sufficient L , for various system parameter setups. C. The Impact of The PS Factor
In this part, we investigate the impact of ρ on the TOP performance. Figure 6 demonstrates theTOP curves versus ρ with various L values. Alongside all the possible values of ρ towards ρ = 1 ,we can find that the TOP curves first decreases, reach the optimality and then rapidly rocketto the worst case at which performance gain offered by the proposed HOR protocol does notexist any more. The existence of the optimality is because the inherent trade-off at R betweenharvesting more energy and gaining stronger received SNR of the signals from S. Also, onecan find that the energy discretisation levels does pose impact on the value of the optimality.Specifically, a larger L leads to a smaller value of the optimality. It does make sense becausea larger L can reduce the energy loss in the proposed energy discretisation model based on thediscussion in Remark so that R has the space to pose more efforts on information processing. D. The Impact of R’s AWGN Power
In this subsection, we show the influence of σ R on the TOP performance. Figure 7 depicts theTOP curves versus σ R with various values of ρ . From the figure, it is straightforward to concludethat the TOP performance is getting worse with the increasing of σ R . Specifically, when R isless or equally “noisy” than D, i.e., in the case of σ R ≤ σ D , the TOP performance remains staticat the minimum value. On the contrary, a “noisier” R will lead to the loss of performance gainoffered by the proposed HOR system. This is because, in short, the min function introduced -90 -70 -50 -30 -10 R2 (dBm)0.020.0220.0260.030.034 T r a n s m i ss i on O u t a g e P r ob a b ilit y Simulation = 0.1 = 0.5 = 0.84 = 0.9No Relay -600.02080.0210.02120.02140.0216
Figure 7. Transmission outage probability versus σ R withvarious ρ values. k -3 -2 -1 T r a n s m i ss i on O u t a g e P r ob a b ilit y SimulationNo Relay P S = -10 dBm P S = -5 dBm P S = Figure 8. Transmission outage probability versus k withvarious P S values. by the DF relaying strategy in formulas (49) and (50) forces the overall received SNR γ D tobehave the segmentation feature. Besides, with the increasing of σ R , the impact of ρ on theTOP performance gradually becomes negligible, e.g., in the case of ρ ∈ [ − , dBm. This isbecause, at this moment, Y H i , i ∈ { , } is way too small compared with γ SD . Moreover, wegive the detailed illustration in the case of σ R = − dBm. At this point, the TOP performanceof ρ = 0 . (the empirical optimal PS factor from Figure 6) is superior to that of ρ = 0 . ,validating the existing of the optimal ρ which was found and discussed in the aforementionedSubsection C . E. The Impact of SIC Strength
In this part, we examine how k can affect the TOP performance. Figure 8 shows the TOP curvesversus k with various P S values. It is direct to find from this figure that the TOP performanceis becoming worse with the increasing of k , for all simulated P S setups. The reason is that alarger k means a stronger SI which suppresses the received SNR of R more. Although a larger k can lead R to harvest more energy from the loop SI channel, from Figure 8, it is still better topursue a good SIC efficiency, i.e., a smaller value of k , when implementing the proposed HORsystem. Besides, with a higher P S , the impact of k becomes less obvious. This is because thestrengths of both energy harvested from the loop SI channel and the interference caused by the SI link become minor, in the front of a high value of P S , which is determined by formulas (8),(49) and (50). d SR (m)0.020.0240.0280.0320.034 T r a n s m i ss i on O u t a g e P r ob a b ilit y Simulation L = 5 L = 25 L No Relay
Figure 9. Transmission outage probability versus d SR withvarious L values. th T r a n s m i ss i on O u t a g e P r ob a b ilit y Simulation k = 0.1 k = 0.5 k = 0.9No Relay Figure 10. Transmission outage probability versus γ th withvarious k values. F. The Impact of The Distance Between S and R
In this subsection, we discuss the impact of d SR on the TOP performance. Figure 9 illustratesthe TOP curves versus d SR with various values of L . Under the subjective of the Triangle SideLength Rule, the possible length of d SR should locates in d SR ∈ (7 , m. From Figure 9, itis straightforward to find that no matter what value L is, a reasonable shorter distance betweenS and R is always preferred for achieving more TOP performance gain. The reason is simplybecause the amount of harvested energy is very sensitive to d SR , which can be found in theassumption of Ω SR = 1 / (1 + d SR ) . From this figure, the approaching speed of the TOP curvesto the “No Relay” line is much slower for a larger L , validating the discussion in Remark . G. The Impact of The SNR Threshold
In this part, we analyse how the value of γ th affects the TOP performance. Figure 10 depictsthe TOP curves versus γ th with different k values. From this figure, one can observe that thereexists an optimal value of γ th which can minimise the TOP curves. This is because, conciselyspeaking, the value of γ th directly influences the occurrence frequency of the FD SWIPT mode,which is determined by the activation condition as { γ SD < γ th } ∩ { E i ≥ E th } . The dilemma of “never or less frequently using R” and “using R too much” makes the optimal γ th standing.Besides, the optimal value of γ th is independent to k . However, a more solid SIC degree, i.e., asmaller k , is still preferable, which is consistent with the discussion in Subsection E . E th -7 T r a n s m i ss i on O u t a g e P r ob a b ilit y Simulation k = 0.1 k = 0.5 k = 0.9No Relay Figure 11. Transmission outage probability versus E th withvarious k values. P R (Watt) 10 -7 T r a n s m i ss i on O u t a g e P r ob a b ilit y Simulation k = 0.1 k = 0.5 k = 0.9No Relay Figure 12. Transmission outage probability versus P R withvarious k values. H. The Impact of The Energy Threshold
In this subsection, we discuss the impact of E th on the TOP performance. Figure 11 illustratesthe TOP curves versus E th with various values of k . It is easy to conclude that an optimal choiceof E th which can minimise the TOP performance does exist for a specific k . The reason is similarto that discussed in Subsection G , which can be explained by the activation condition of theFD SWIPT mode, i.e., { γ SD < γ th } ∩ { E i ≥ E th } . This observation can help the designer todetermine a feasible setup of E th in practical applications. I. The Impact of R’s Transmit Power
In this part, we discuss the impact of P R on the TOP performance. Figure 12 depicts the TOPcurves versus P R with various values of k . The overall appearance of this figure is similar tothat of Figure 11, however the subtle differences can be found by comparing these two figures,illustrating the different influence strengths of E th and P R on the TOP performance. The existenceof the optimal P R is due to the following two trade-offs: 1) a larger P R will consume more storedenergy at the PEC but also lead the PEC to absorb more energy from the SI channel. 2) the min function introduced by the DF relaying strategy limits that γ D is not always increasing withthe increasing of P R . This two kinds of dilemma cause that simply enlarging P R does not leadto a better TOP performance, and also make the optimal value of P R existing. This finding isbeneficial for designer to choose a feasible value of P R in implement of the proposed HORsystem. VII. C ONCLUSION
In this paper, we initiated a novel wireless relaying transmission scheme termed HOR, viaboth listing the necessary hardware devices and designing the essential transmission protocol. Torealise the SWIPT and true FD functionalities of the proposed HOR system, a practical finite-capacity hybrid energy storage model is applied, which is composed of two independent energycontainers. The relay can work opportunistically in either PEH or FD SWIPT mode accordingto the proposed HOR scheme, not only providing a better way to manipulate available wirelessenergy but also improving the overall wireless transmission performance within the end-to-endwireless communication scenario. To track the dynamic charge-discharge behaviour of the PEC,a discrete-state MC method is adopted, based on which the long-term stationary distribution ofenergy states is quantified. Furthermore, covert communication and transmission performancesof the proposed HOR system were analysed via deriving closed-form expressions of minimumdetection error probability and transmission outage probability. Numerical results validated thecorrectness of the aforementioned analyses and the impacts of key system parameters wereinvestigated. The proposed HOR scheme can enhance wireless energy manipulating efficiency,wireless transmission performance and privacy level of the end-to-end wireless transmissionsystem, which has been proved throughout this paper.A
PPENDIX AP ROOF OF L EMMA Because each symbol of the received message vector y D in a specific transmission slot followsi.i.d. complex Gaussian distribution, y D [ ω ] is ruled by the distribution shown as CN (cid:16) , P S | h SD | + P R | ˆ h RD | + P R | ˜ h RD | + σ D (cid:17) , H CN (cid:16) , P S | h SD | + ( P R + P ∆ ) | ˆ h RD | + ( P R + P ∆ ) | ˜ h RD | + σ D (cid:17) , H . (61)Denote the observation conditioned on ψ by y D ( ψ ) = [ y D [1] ( ψ ) , y D [2] ( ψ ) , . . . , y D [ n ] ( ψ )] in which y D [ ω ] ( ψ ) ∼ CN (0 , σ D + ψ ) . Note that ψ represents the sum variance of D’s receivedsignals from S and R. To distinguish the null hypothesis H from the alternative hypothesis H ,we here introduce a couple of non-negative and real-value RVs Ψ and Ψ , whose PDFs areintegratedly given by f Ψ q ( ψ ) = exp (cid:16) − ψ − φ βP R Ω RD (cid:17) βP R Ω RD , x > φ , q = 0 exp (cid:18) − ψ − φ β ( P R + P ∆ ) Ω RD (cid:19) β ( P R + P ∆ )Ω RD , x > φ , q = 10 , otherwise , (62)where φ = P S Ω SD + (1 − β ) P R Ω RD and φ = P S Ω SD + (1 − β ) ( P R + P ∆ ) Ω RD .Furthermore, the PDF of vector y D given ψ can be calculated as f y D ( ψ ) ( y ) = n (cid:89) ω =1 exp (cid:16) − | y D [ ω ]( ψ ) | σ D + ψ (cid:17) π ( σ D + ψ ) = (cid:18) π ( σ D + ψ ) (cid:19) n exp (cid:18) − (cid:80) nω =1 | y D [ ω ] ( ψ ) | σ D + ψ (cid:19) . (63)Here, invoking the Fisher-Neyman Fctorization Theorem, the total received power in a transmis-sion slot (cid:80) nω =1 | y D [ ω ] ( ψ ) | is a sufficient statistic for D’s hypothesis test. It is worth notingthat (cid:80) nω =1 | y D [ ω ] ( ψ ) | = ( σ D + ψ ) X n where X n denotes chi-squared RV with n degreesof freedom. Because D performs testing between two simple hypotheses and he knows thestatistical knowledge of his received signals when either hypothesis stands, with help of theNeyman-Pearson Lemma, the best testing rule for D to decide which hypothesis stands is thelikelihood ratio test (LRT), given by Λ ( y D ) = f y D |H ( y ) f y D |H ( y ) D ≷ D Γ , (64)where Γ = Pr ( H ) / Pr ( H ) = 1 due to the application of equal a priori assumption. D doesnot have instantaneous knowledge of either Ψ or Ψ , so he modifies his LRT as Λ ( y D ) = E Ψ (cid:2) f y D ( ψ ) ( y ) (cid:3) E Ψ (cid:2) f y D ( ψ ) ( y ) (cid:3) D ≷ D Γ . (65)We introduce here that RV X is smaller than RV Y in the likelihood ratio order, i.e., X ≤ lr Y ,when f Y ( x ) /f X ( x ) is an non-decreasing function over the union of their supports. Invoking (62), we have f Ψ ( ψ ) f Ψ ( ψ ) = P R P R + P ∆ exp (cid:18) P ∆ ψ − ( P R + P ∆ ) φ + P R φ βP R ( P R + P ∆ ) Ω RD (cid:19) . (66)It is straightforward to find that (66) is non-decreasing over the union of supports of Ψ and Ψ , hence Ψ ≤ lr Ψ . From the statistical nature of chi-squared RVs, for any ψ ≤ ψ , we have y D ( ψ ) ≤ lr y D ( ψ ) . Then, according to Theorem 1, Chapter 11 in [32], the monotonicity of Λ ( y D ) is ruled by Stochastic Ordering and Λ ( y D ) is non-decreasing w.r.t. (cid:80) nω =1 | y D [ ω ] ( ψ ) | .Hence, the LRT (65) is equivalent to a received power threshold test. Since any one-to-onetransformation of a sufficient statistic remains the sufficiency, the term (cid:80) nω =1 | y D [ ω ] | /n isalso a sufficient statistic. From the strong law of large numbers, we have X n /n → wheninfinite blocklength ( n → ∞ ) assumption is considered. Invoking the Lebesgue’s DominatedConvergence Theorem, it is allowed to replace X n /n by 1, when n → ∞ . Thus, we get T = lim n →∞ n n (cid:88) ω =1 | y D [ ω ] | = P S | h SD | + P R | ˆ h RD | + P R | ˜ h RD | + σ D , H P S | h SD | + ( P R + P ∆ ) | ˆ h RD | + ( P R + P ∆ ) | ˜ h RD | + σ D , H . (67)Then, the optimal decision rule at D can be expressed as T D ≷ D τ, where τ denotes the threshold which will be optimized to minimize P E .After all, a radiometer which is able to detect the total power of received messages at D isproved to be optimal. Besides, radiometers are also beneficial for D due to its low complexity andease of implementation. So, it is sufficient and optimal for D to apply a radiometer to performhypothesis test regarding covert communication detection.A PPENDIX BP ROOF OF T HEOREM P FA = Pr ( T > τ |H ) = Pr (cid:16) P R | ˜ h RD | + j > τ (cid:17) = Pr (cid:16) | ˜ h RD | > τ − j P R (cid:17) , τ ≥ j , otherwise , (68) P MD = Pr ( T < τ |H )= Pr (cid:16) ( P R + P ∆ ) | ˜ h RD | + j < τ (cid:17) = Pr (cid:16) | ˜ h RD | < τ − j P R + P ∆ (cid:17) , τ ≥ j , otherwise . (69)Because the uncertain part of channel R → D follows the distribution ˜ h RD ∼ CN (0 , β Ω RD ) , itis straightforward to know that | ˜ h RD | obeys the Exponential distribution. Thus, the CDF of | ˜ h RD | can be gained as F | ˜ h RD | ( x ) = 1 − exp ( − x/ ( β Ω RD )) . Then, after some simple algebracalculation, we gain closed-form expressions of false alarm and missed detection probabilities,expressed respectively as (42) and (43). Invoking (40), (42) and (43), closed-form expression ofdetection error probability can be gained after simple derivation as (44).A PPENDIX CP ROOF OF T HEOREM τ ∗ = arg min τ P E . (70)In the case of τ < j , the detection error probability at D remains 1. This is the worst casefor D and D will never choose any value satisfying τ < j . Thus, the optimization problem didnot stand in this case.In the case of j ≤ τ < j , it is easy to find that P E monotonically decreases w.r.t. τ . Besides,the piecewise function P E is a continuous function along side the whole feasible domain of τ .Thus, D will choose j to minimize P E , leading to P E = exp (( j − j ) / ( βP R Ω RD )) .In the case of τ ≥ j , to determine the optimal value of τ , the first derivative of function P E w.r.t. τ is calculated as ∂ P E ∂τ = kβP R ( P R + P ∆ ) Ω RD , (71)where k = P R exp [( j − τ ) / ( β ( P R + P ∆ ) Ω RD )] − ( P R + P ∆ ) exp [( j − τ ) / ( βP R Ω RD )] . It iseasy to find that whether (71) is positive or not depends only on the value of k . After simplemanipulations, we can modify k as k = exp (cid:18) ln P R + j − τβ ( P R + P ∆ ) Ω RD (cid:19) − exp (cid:18) ln ( P R + P ∆ ) + j − τβP R Ω RD (cid:19) . (72) Besides, the Exponential function exp is monotonically increasing w.r.t. the feasible independentvariable region. Thus, we can determine whether k is positive or not by k = ln P R P R + P ∆ + P R ( j − τ ) − ( P R + P ∆ ) ( j − τ ) βP R ( P R + P ∆ ) Ω RD = ln P R P R + P ∆ + P ∆ ( τ − P S | h SD | − σ D ) βP R ( P R + P ∆ ) Ω RD . (73)Because τ ≥ j stands in this considered case, the right hand of (73) is absolutely positive.However, the left hand of (73) is negative due to P R < P R + P ∆ . Most importantly, from (73),we can find that k is a monotonically increasing function w.r.t. τ . Let k = 0 , we can getthe solution as (46). From (46), we can conclude that k ≥ in the case of τ ≥ τ k =0 and k < otherwise. If j ≥ τ k =0 holds, in the case of τ ≥ j , we can determine that k > andfurthermore ∂ P E /∂τ > which means P E monotonically increases w.r.t. τ when τ ≥ j . Here,it is the optimal choice for D to choose j as the optimal threshold which is able to minimize P E . If j < τ k =0 , we know that for τ ∈ ( j , τ k =0 ) , ∂ P E /∂τ < and for τ ∈ ( τ k =0 , + ∞ ) , ∂ P E /∂τ > . Thus, the optimal detection threshold for D is τ k =0 in this case.A PPENDIX DP ROOF OF C OROLLARY j ≥ τ k =0 , i.e., β ≥ − P ∆ | ˆ h RD | / (cid:16) P R Ω RD ln P R P R + P ∆ (cid:17) , the first derivative of P ∗ E w.r.t. β can be calculated as ∂ P ∗ E ∂β | j ≥ τ k = − j − j β P R Ω RD exp (cid:18) j − j βP R Ω RD (cid:19) , (74)whose value is positive due to j < j . For j < τ k =0 , i.e., β < − P ∆ | ˆ h RD | / (cid:16) P R Ω RD ln P R P R + P ∆ (cid:17) ,the first derivative of P ∗ E w.r.t. β can be calculated as ∂ P ∗ E ∂β | j <τ k = | ˆ h RD | β Ω RD × (cid:34) exp (cid:32) | ˆ h RD | β Ω RD + P R P ∆ ln P R P R + P ∆ (cid:33) − exp (cid:32) | ˆ h RD | β Ω RD + P R + P ∆ P ∆ ln P R P R + P ∆ (cid:33)(cid:35) , (75)whose value is also positive due to the truth of P R > P ∆ > . Thus, we can conclude that P ∗ E monotonically increases as β increases. A PPENDIX EP ROOF OF L EMMA γ D |H can be constructed as F γ D |H ( x ) = Pr (cid:16) γ SD + Y H < x (cid:92) γ SD < γ th (cid:17) . (76)Note that the limitation of variable γ SD should be constrained as γ SD < γ th due to the nature ofFD SWIPT mode. Invoking (51) and after some simple mathematical computation, we can earnclosed-form expression of (76) as (52). A PPENDIX FP ROOF OF L EMMA γ D |H should be calculated in the way similar to the derivationof (52). However, we found that it is mathematically intractable. To tackle this problem, we resortto Gauss-Kronrod Quadrature (GKQ) method to approximately solve it, shown as F γ D |H ( x ) = Pr (cid:104) γ SD + Y H < x (cid:92) γ SD < γ th (cid:105) = (cid:90) γ th σ D P S Ω SD F Y H ( x − y ) exp (cid:18) − σ D yP S Ω SD (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) dy fun ≈ n (cid:88) i =1 (cid:37) i fun ( y i ) , (77)where (cid:37) i and y i denote the weights and points which are essential to evaluate the function fun ( y ) .Note that the GKQ formula is an adaptive method for numerical integration, which is a variant ofGaussian quadrature. In this paper, we use the built-in function of Matlab named quadgk ( · , · , · ) to calculate (77), which implements adaptive quadrature based on a Gauss-Kronrod pair ( th and th order formulas). Applying quadgk ( · , · , · ) , we can derive closed-form approximate CDFexpression of γ D |H as (57). A PPENDIX GP ROOF OF T HEOREM
T OP FS = Pr (cid:104) log (1 + γ D ) < R th (cid:92) H (cid:92) FS (cid:105) + Pr (cid:104) log (1 + γ D ) < R th (cid:92) H (cid:92) FS (cid:105) a = Pr (cid:104) log (1 + γ D ) < R th (cid:92) H (cid:92) γ SD < γ th (cid:105) L (cid:88) i = ϕ ξ i +Pr (cid:104) log (1 + γ D ) < R th (cid:92) H (cid:92) γ SD < γ th (cid:105) L (cid:88) i = ϕ ξ i = 12 L (cid:88) i = ϕ ξ i × Pr (cid:104) γ D |H < R th − (cid:92) γ SD < γ th (cid:105)(cid:124) (cid:123)(cid:122) (cid:125) f + Pr (cid:104) γ D |H < R th − (cid:92) γ SD < γ th (cid:105)(cid:124) (cid:123)(cid:122) (cid:125) f , (78)where the factor / is due to the assumption of equal a priori , R th is the target rate underwhich the transmission outage occurs. Note that step (a) in (78) holds, because of the fact thatthe energy requirement is independent with other factors. With the help of Lemma and Lemma , we are able to derive closed-form expressions of f and f , which can be achieved by simplyreplacing variable x in (52) and (57) with factor R th − . Substituting f and f into (78), wecan derive closed-form expression of the TOP in the FD SWIPT mode as (58) and this completesthe proof. A PPENDIX HP ROOF OF T HEOREM
T OP
PEH = Pr (cid:104) log (1 + γ D ) < R th (cid:92) PEH (cid:105) = Pr (cid:0) γ SD < R th − ∩ γ SD < γ th (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) f ϕ − (cid:88) i =0 ξ i + Pr (cid:0) γ SD < R th − ∩ γ SD ≥ γ th (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) f . (79)In the case of γ SD < γ th ∩ E i < E th , we have Pr (cid:0) γ SD < R th − (cid:1) = 1 . It is worth notingthat hereby Pr (cid:0) γ SD < R th − (cid:1) and Pr ( γ SD < γ th ) are independent with each other, because D cases signal-processing and forces Pr (cid:0) γ SD < R th − (cid:1) = 1 , leading f = Pr ( γ SD < γ th ) = q SD .In the case of γ SD ≥ γ th , the main wireless channel is good enough, closed-form expression ofCDF of γ SD | γ SD ≥ γ th can be derive as F γ SD | γ SD ≥ γ th ( x ) = exp (cid:16) − σ D γ th P S Ω SD (cid:17) − exp (cid:16) − σ D xP S Ω SD (cid:17) , x > γ th , x ≤ γ th . (80)Hence, closed-form expression of f can be given by f = F γ SD | γ SD ≥ γ th (cid:0) R th − (cid:1) . Substituting f and f into (79), we can derive closed-form expression of the TOP in the PEH mode as (59)and this completes the proof. R EFERENCES [1] Y. Bi and H. Chen, “Accumulate and jam: Towards secure communication via a wireless-powered full-duplex jammer,”
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