Simultaneous Wireless Information and Power Transfer Under Different CSI Acquisition Schemes
Chen-Feng Liu, Marco Maso, Subhash Lakshminarayana, Chia-Han Lee, Tony Q. S. Quek
aa r X i v : . [ c s . I T ] M a y Simultaneous Wireless Information and PowerTransfer Under Different CSI Acquisition Schemes
Chen-Feng Liu, Marco Maso,
Member, IEEE,
Subhash Lakshminarayana,
Member, IEEE,
Chia-Han Lee,
Member, IEEE, and Tony Q. S. Quek,
Senior Member, IEEE
Abstract —In this work, we consider a multiple-input single-output system in which an access point (AP) performs a simul-taneous wireless information and power transfer (SWIPT) toserve a user terminal (UT) that is not equipped with externalpower supply. In order to assess the efficacy of the SWIPT, wetarget a practically relevant scenario characterized by imperfectchannel state information (CSI) at the transmitter, the presence ofpenalties associated to the CSI acquisition procedures, and non-zero power consumption for the operations performed by theUT, such as CSI estimation, uplink signaling and data decoding.We analyze three different cases for the CSI knowledge atthe AP: no CSI, and imperfect CSI in case of time-divisionduplexing and frequency-division duplexing communications.Closed-form representations of the ergodic downlink rate andboth the energy shortage and data outage probability are derivedfor the three cases. Additionally, analytic expressions for theergodically optimal duration of power transfer and channelestimation/feedback phases are provided. Our numerical findingsverify the correctness of our derivations, and also show theimportance and benefits of CSI knowledge at the AP in SWIPTsystems, albeit imperfect and acquired at the expense of the timeavailable for the information transfer.
Index Terms —Simultaneous wireless information and powertransfer (SWIPT), energy harvesting, wireless power transfer,TDD, FDD, analog feedback.
I. I
NTRODUCTION I N conventional wireless systems, the limited battery capac-ity of mobile devices typically affects the overall networklifetime. Increasing the size of the battery might not be afeasible solution to address this problem, due to a consequentreduction of the portability and increase in the cost of theequipment. For these reasons, the study of novel techniques toprolong the lifetime of the battery has triggered an increasedinterest in the wireless communications community. In thiscontext, the study of the so-called wireless power transfer(WPT) has recently gained prominence as means to implementa cable-less power transfer between devices [1], either by
This work was supported in part by the SRG ISTD 2012037, SUTD-MIT International Design Centre under Grant IDSF1200106OH, and theSUTD-ZJU Research Collaboration Grant, and the Ministry of Science andTechnology (MOST), Taiwan under Grant MOST103-2221-E-001-002.C.-F. Liu and C.-H. Lee are with the Research Center for Informa-tion Technology Innovation, Academia Sinica, Taipei 115, Taiwan (email:cfl[email protected]; [email protected]).M. Maso is with the Mathematical and Algorithmic Sciences Lab,Huawei France Research Center, Boulogne-Billancourt 92100, France (email:[email protected]).S. Lakshminarayana is with the Singapore University of Technology andDesign, Singapore 138682 (email: [email protected]).T. Q. S. Quek is with the Singapore University of Technology and Design,Singapore 138682, and also with the Institute for Infocomm Research,A*STAR, Singapore 117685 (email: [email protected]). resonant inductive coupling [2] or by far-field power transfer[3]. The latter approach has seen increasing momentum inrecent years, due to its promising potential for longer rangetransfers. A fundamental breakthrough in this context has beenthe design of rectifying antennas (rectennas) for microwavepower transfer (MPT), key component to achieve an efficientradio frequency to direct current (RF-to-DC) conversion. Thishas brought to several technological advances, e.g., the designof flying vehicles powered solely by microwave [4], whichconfirmed that RF-to-DC conversion can not only be per-formed but also achieve remarkable efficiency. In this regards,commercial products exhibiting an efficiency larger than are already available on the market [5]. Remarkably, perfor-mance of state-of-the-art RF-to-DC converters can be evenhigher, i.e., more larger than , resulting in an impressivepotential DC-to-DC efficiency of [6], [7].
A. Related Works
Recent advances in signal processing and microwave tech-nology have shown that far-field power transfer can offerinteresting perspectives also in the context of traditional wire-less communication systems. For instance, the same electro-magnetic field could be used as a carrier for both energyand information, realizing the so-called simultaneous wirelessinformation and power transfer (SWIPT). The potential of thisapproach has been first highlighted by the studies proposed in[8] and [9]. Therein the trade-off between the two transferswithin SWIPT was investigated, in the case of both flat fad-ing and frequency-selective channels. After these pioneeringworks, several studies have been performed to assess the im-plementability of receivers that can make use of RF signals toboth harvest energy and decode information [10], and analyzethe performance of SWIPT in many scenarios, e.g., multi-antenna systems [11], opportunistic networks [12], wirelesssensor networks [13]. In particular, the aforementioned trade-off is investigated in multiple-input multiple-output (MIMO)systems for two information and power transfer architectures,i.e., time-switching and power-splitting, for both perfect [11]and imperfect [14] channel state information (CSI). A differentapproach is considered in [15], where the trade-off between theinformation and the energy transfer is investigated in a multi-user network under two different constraints, i.e., a constraintexpressed in terms of secrecy rate for the former and amountof harvested energy at the receiver for the latter.A line of work considering orthogonal frequency divisionmultiplexing (OFDM) systems is presented in [16], [17], [18], where the resource allocation policy at the transmitter isstudied for both single and multi-user scenarios, in which thereceiver adopts a power-splitting strategy and considers un-desired interference as an additional energy resource. Finally,ad-hoc scenarios departing from the standard SWIPT paradigmhave been analyzed in works such as [19], [20], where itis assumed that the energy and information transfers areperformed by two different devices, operating in frequency-division duplexing (FDD) mode. More precisely, in [19] thetime allocation policy for the two transmitters is studied, underthe assumption that the efficiency of the energy transfer ismaximized by means of an energy beamformer that exploitsa quantized version of the CSI received in the uplink by theenergy transmitter. Along similar lines, [20] investigates theoptimal time and power allocations strategies such that theamount of harvested energy is maximized, taking into accountthe impact of the CSI accuracy on the latter quantity.
B. Summary of Our Contribution
In general, most existing works for SWIPT rely on idealassumptions: a) availability of perfect CSI at the transmitter,b) no penalty for CSI acquisition, c) no power consumption forsignal decoding operations at the receiver. If these assumptionsare relaxed and the resulting penalties and issues are consid-ered, then the performance of wireless systems can decreasesignificantly. Studies taking into account these aspects havebeen proposed for conventional information transfers, andthe effect of imperfect CSI acquisition, training, feedback aswell as the resource allocation problem have been thoroughlyanalyzed [21]. Departing from these observations, in this workwe aim at investigating the efficacy of SWIPT for practicallyrelevant scenarios, by relaxing the three aforementioned idealassumptions. In particular, we target a multiple-input single-output (MISO) system consisting of a multi-antenna accesspoint (AP) which transfers both information symbols andenergy to a single user terminal (UT) that does not haveaccess to any external power source. Accordingly, in contrastto the previous contributions on this topic, in this work thepower harvested by means of the WPT is used by the UTto perform all the necessary signal processing operationsfor both information decoding and uplink communications.Additionally, we adopt a systematic approach and considerthe three main possible scenarios for an AP that engagesin a downlink transmission in modern networks, to providea more complete characterization of the considered system.Consequently, in this work, the performance and feasibility ofthe SWIPT in a MISO system is studied for the three followingcases: • No CSI available at the AP. • Time-division duplexing (TDD) communications and CSIacquisition at the AP by means of training symbols. • FDD communications and CSI acquisition at the AP bymeans of analog symbols feedback.We compare these three scenarios for three performance met-rics of interest, namely, ergodic downlink rate, energy shortageprobability, and data outage probability. Our contributions inthis work are as follows: • We derive closed-form representations for the three per-formance metrics of interest in all three scenarios andmatch them to the numerical results. • We derive the approximations of the ergodically optimalduration of the WPT phase in all the three scenarios asa portion of the channel coherence time. • Additionally, for the TDD and the FDD scheme, wederive closed-form approximations for the ergodicallyoptimal duration of the channel training/feedback phases,to maximize the downlink rate. • We show that the TDD scheme can outperform the FDDscheme in SWIPT systems in terms of both downlink rateand data outage probability.Our numerical findings verify the correctness of our deriva-tions. More specifically, concerning the downlink rate, weshow that the performance gap between the numerical optimalsolutions and the results obtained by means of our approxima-tions is very small for low to mid signal-to-noise ratio (SNR)values and negligible for high SNR values. Moreover, we showthat both TDD and FDD outperform the non-CSI case at anySNR value in terms of downlink rate. This confirms that CSIknowledge at the AP is always beneficial for the informationtransfer in SWIPT systems, despite both its imperfectness andthe resources devoted to the channel estimation/feedback pro-cedures. The correctness of our derivations is further verifiedwhen numerically evaluating both the energy shortage and dataoutage probability of the considered MISO system adoptingSWIPT, for which a perfect match of analytic and numericalresults is achieved. Finally, it is worth noting that throughoutour study TDD consistently outperforms FDD in terms ofboth downlink rate and data outage probability, confirming thepotential of this duplexing scheme for the future advancementsin modern networks.The rest of the paper is organized as follows. In Sec. II,we specify the system model. In Sec. III-A, III-B, and III-C,we specify the system model and derive the downlink ratefor the non-CSI, the TDD, and the FDD scheme. In Sec. IV,we derive the energy shortage and data outage probability foreach scheme. In Sec. V, we show and discuss the numericalresults. Finally, we conclude in Sec. VI.
Notations:
In this paper, we denote matrices as boldfaceupper-case letters, vectors as boldface lower-case letters. Ad-ditionally, we let [ · ] † be the conjugate transpose of a vector.All vectors are columns, unless otherwise stated. Furthermore,for a scalar c ∈ C we note by c ∗ its complex conjugate.We use x ⊥ y to express the orthogonality between vec-tors x and y . We denote a circular symmetric Gaussianrandom vector with mean µ and covariance matrix Σ as CN ( µ , Σ ) . The chi-squared distribution with K degrees offreedom is denoted by χ K and its probability density function(PDF) is given by f X ( x ) = ( K ) 2 − K/ x K − e − x/ where Γ( q ) = R ∞ u q − e − u d u is the Gamma function. The non-central chi-squared distribution with K degrees of freedomand non-central parameter ν is denoted by χ ′ K ( ν ) and its PDFis given by f X ( x ) = e − ( x + ν ) / (cid:0) xν (cid:1) K/ − / I K − ( √ νx ) , I n ( · ) , modified Bessel function of the first kind. Γ( q, r ) = R ∞ r u q − e − u d u is the upper incomplete Gamma function. Q M ( q, r ) = R ∞ r ξ M q M − exp (cid:16) − ξ + q (cid:17) I M − ( qξ ) d ξ is thegeneralized Marcum Q-function [22].II. S YSTEM M ODEL
We consider a point-to-point communication system consist-ing of an AP with L antennas and a UT with single antenna.We denote the downlink channel (from the AP to the UT)as h = [ h , · · · , h L ] ⊤ . The channel is assumed to be blockfading, with independent fading from block to block. Theentries of the channel vector are complex Gaussian (Rayleighfading), hence h ∼ CN ( , I L ) . Let T C be the coherencetime length. For simplicity in the notation, we assume thatthe total number of symbols that can be transmitted withinthe coherence time is T C . The AP transmits the symbol x ∈ C L × with a transmit power P , i.e., E (cid:2) k x k (cid:3) = P .The received signal at the UT is given by y = h † x + n, where n ∼ CN (0 , N ) is the thermal noise, modeled as a complexadditive white Gaussian noise (AWGN). We assume the UTdoes not have any external power source (such as the battery)and all the power required for the operations to be performed atthe UT is provided by the AP through the WPT component ofthe SWIPT. Accordingly, the UT is equipped with a circuit thatcan perform two different functions: a) harvest energy from thereceived RF signal, b) information decoding. As considered inprevious literature [11], we assume that the UT cannot harvestenergy and decode information from the same signal, at thesame time. Hence, a time switching strategy is adopted underwhich the AP transmits the signals in two phases: the signalsent during the first phase has WPT purposes and is usedby the UT to harvest energy, whereas the signal sent in thesecond phase has information transfer purposes. Note that,throughout this work, we assume that the energy harvestedin the first phase (power transfer phase) is the sole sourceof power for all the subsequent operations performed by theUT (the exact details of these operations will be specified thefollowing sections).
1) Details of the Power Transfer Phase:
During eachcoherence time interval, the AP first transmits the powerwirelessly to the UT for ǫ < T C time slots. First, the APdivides its power P equally between its L transmit antennas toperform the WPT. Hence the L -sized transmit symbol duringthis phase, denoted by x EH , is given by x EH = q PL s , where s is a random vector with zero mean and covariance matrix E (cid:2) ss † (cid:3) = I L . Thus, the power harvested at the UT is givenby P H = βP k h k L , (1)where β ∈ [0 , is a coefficient that measures the efficiencyof the RF to direct current (RF-to-DC) power conversion [11],[13]. Note that, the words energy and power are used interchangeably in thispaper, for the sake of simplicity, in spite of their conceptual difference. The exact value of ǫ will be specified later depending upon the mode ofoperation.
2) Details of the Information Transmission Phase:
For thesecond phase, namely information transmission phase, weadopt a systematic approach and consider the three scenarios.In the first one, the AP transmits the information symbolswithout the knowledge of CSI (we will refer to this approachas non-CSI scheme). In the second scheme, we considera TDD communication, in which the downlink and uplinkcommunications are performed over the same bandwidth.Accordingly, first the AP acquires the CSI by evaluatinga pilot sequence transmitted by the UT in the uplink, andthen engages in the downlink transmission. In the last case,we consider an FDD communication, in which the downlinkand uplink communications are performed over two separatebandwidths. Consequently, in this case, UT sends an analogfeedback signal in the uplink, carrying the downlink channelestimation, to allow the AP to acquire the CSI and subsequenttransmit information symbols.Under the aforementioned settings, we analyze the perfor-mance of the system for two different metrics of interest,namely the downlink rate and outage probability. We providea detailed analysis of these two metrics in the rest of the paper.III. A
NALYSIS OF THE D OWNLINK R ATE
In this section, we analyze the downlink rate for the threeconsidered schemes.
A. Non-CSI Scheme
We first consider the case where the AP transmits theinformation symbols without the knowledge of CSI. Theschematic diagram of this scenario is shown in Fig. 1. UnderInformation transfer (1 − α N ) T C Power transfer α N T C (a) Operations of the AP. Data decoding (1 − α N ) T C Energy harvesting α N T C (b) Operations of the UT.Figure 1. Operations of the AP and the UT during the coherence time inthe non-CSI scheme. this scheme, the system utilizes ǫ = α N T C symbols to transferpower and the remaining symbols to transmit informationsymbols, where < α N < . The received signal duringthe information transmission phase is given by y = h † x + n, where the x = q PL s and s ∼ CN ( , I L ) . Note that inthe absence of CSI, the AP performs equal power allocationover all its antennas to transmit the information symbol. Forthe information decoding at the UT, we consider that thepower consumption of the circuit components devoted to thedecoding is proportional to the number of received symbols(as typically considered in previous works on the subject [23]).Accordingly, we denote the power consumption per decodedsymbol at the UT as P D .Since the power harvested in the first phase must besufficient to decode all the information symbols, we have that α N T C P H = (1 − α N ) T C P D . Now, if we plug (1) into this equation then, after some manipulations, we have that theminimum fraction of time that should be devoted to the powertransfer, i.e., α N , given by α N = LP D βP k h k + LP D . (2)We now analyze the downlink rate for this scheme. We recallthat the AP can transmit − α N symbols for the informationtransfer. Accordingly, using (2), the downlink rate obtained forthe non-CSI scheme is given by R NC = R NC ( α N ) = (1 − α N ) log (cid:18) P k h k N L (cid:19) = βP k h k βP k h k + LP D log (cid:18) P k h k N L (cid:19) . (3) B. TDD Scheme
We switch our focus to the TDD scheme, whose schematicdiagram is shown in Fig. 2. We recall that, in this case, the APInformation transfer (1 − α T − η T ) T C Channel estimation η T T C Power transfer α T T C (a) Operations of the AP. Data decoding (1 − α T − η T ) T C Pilots transmission η T T C Energy harvesting α T T C (b) Operations of the UT.Figure 2. Operations of the AP and the UT during the coherence time inthe TDD scheme. should provide the UT with sufficient energy for the latter to beable not only to decode the received data but also to perform allthe operations related the uplink signaling inherent to the TDDscheme. Accordingly, the system utilizes ǫ = α T T C symbolsfor power transfer, where < α T < . This is followed by theCSI acquisition phase. Since we assume that all the operationsare performed within the coherence time, the AP can exploitthe reciprocity of the downlink and uplink channels, inherentfeature of the TDD scheme. This way, the channel estimatedin the uplink can be used to design the beamformer for thedownlink transmission. Accordingly, the UT transmits uplinkpilots with power P E for the next η T T C ∈ Z + symbol periods,with < η T < and < α T + η T ≤ . The signal receivedby the AP during the i th symbol period (in the uplink pilottransmission phase) is given by y pT [ i ] = √ P E h ∗ + w [ i ] , where w [ i ] ∼ CN ( , N I L ) is the Gaussian noise at the AP. The APestimates the channel by a minimum variance unbiased (MVU)based estimator [24]. Thus, the channel estimate at the AP isgiven by ˆh = 1 √ P E η T T C η T T C X i =1 (cid:16)p P E h + w ∗ [ i ] (cid:17) = h + ¯ w , (4)where ¯w ∼ CN (cid:16) , N η T T C P E I L (cid:17) denotes the estimation error.This is followed by the information transmission phase.The focus of this work is on the performance of the SWIPTunder non-ideal system assumptions and practical transmitschemes. Therefore, for simplicity we will assume that the AP beamforms the signal carrying the information symbols witha matched filter precoder (MFP) [25], optimal linear filter formaximizing the SNR. Accordingly, the AP exploits the CSIestimate to design the desired beamforming vector, obtainedas m T = ˆh / k ˆh k . Then, the received signal at the UT is givenby y sT = h † ˆh k ˆh k s + n, (5)where s is the information symbol, with E (cid:2) | s | (cid:3) = P .As in the previous case, the UT consumes a power P D to decode every received information symbol. Thus, sincethe harvested power by the UT must be sufficient to sendthe pilot symbols and decode information at the UT, thecondition α T T C P H = η T T C P E + (1 − α T − η T ) T C P D mustbe satisfied for the TDD scheme. Now, if we plug (1) intothis condition then, after some manipulations, we have thatthe minimum fraction of time that should be devoted to thepower transfer, i.e., α T , is given by α T = η T LP E − η T LP D + LP D βP k h k + LP D . (6)Accordingly, we can use (6) to compute the downlink rate forthe TDD scheme as R T = R T ( α T , η T ) = (1 − α T − η T ) log P | h † ˆh | N k ˆh k ! = (1 − η T ) βP k h k − η T LP E βP k h k + LP D log P | h † ˆh | N k ˆh k ! . (7)Note that the channel training time η T T c impacts both theaccuracy of the estimated channel vector and the remainingavailable time for information transfer. As a consequence, let η ⋆T be the duration of the portion of coherence time devotedto the channel training/estimation that maximizes the ergodicdownlink rate, defined as η ⋆T = argmax η T E h , ¯w " (1 − η T ) βP k h k − η T LP E βP k h k + LP D × log P | h † ˆh | N k ˆh k ! . (8)The derivation of the exact value of η ⋆T is very complicated.However, two approximations of this value, valid for highand low SNR, respectively, can be derived as stated in thefollowing result. Lemma 1.
At high SNR, η ⋆T can be approximated as η ⋆T ≈ s N L log eB T C P E ( L − , (9) where B = E h h(cid:16) LP E βP k h k (cid:17) log (cid:16) P k h k N (cid:17)i . At low SNR,it can be approximated as η ⋆T ≈ N T C P E − s L − βP T C P E LN ( βP + P E ) − L ! . (10) Proof:
See Appendix-A.
Lemma 1 provides a result whose interpretation is nottrivial. In fact, several parameters are present in (9) and(10), thus understanding their impact on the accuracy of theproposed approximations is rather complex. However, someinteresting insights can drawn from Lemma 1, if we focuson the approximations that are introduced in order to derivethe final results. First, we note that the impact of P D on theaccuracy of the results is likely negligible, due the fact that P ≫ P D by construction. Subsequently, let us focus on thequantity λ = η T T C P E k h k N , introduced in (39). If we fix N atthe denominator of λ then it is straightforward to see that thelatter increases with an increase in P E and L . Now, considerthe low SNR case. In this case, N is very large, thus theapproximation λ ≈ is adopted. In practice, the accuracy ofthis approximation depends on the value of L and P E , i.e., thelower those values are, the more accurate the approximation is.Switching our focus to the high SNR analysis, we observe anopposite behavior. In fact, in this case N is very small, hencethe approximation λ ≫ is introduced. Thus, the accuracyof the approximation is greater when P E and L are large.Concerning the latter parameter, i.e., L number of antennasat the AP, we note that an analysis of its impact on theaccuracy of the results in Lemma 1 is extremely interesting,given its relevance in a MISO systems. Accordingly, a detaileddiscussion on this aspect will be provided in Sec. V. C. FDD Scheme
We now consider the FDD scheme, whose schematic dia-gram is illustrated in Fig. 3. Differently from the TDD case, theInformation transfer (1 − α F − η F − τ F ) T C CSI acquisition τ F T C Pilots transmission η F T C Power transfer α F T C (a) Operations of the AP. Data decoding (1 − α F − η F − τ F ) T C CSI feedback τ F T C Channel estimation η F T C Energy harvesting α F T C (b) Operations of the UT.Figure 3. Operations of the AP and the UT during the coherence time inthe FDD scheme. downlink and uplink channels are in general uncorrelated inthe FDD scheme. Therefore, two separate channel estimationprocedures have to be performed at the UT and the AP, to beable to provide to the latter with the CSI w.r.t. the downlinkchannel. Accordingly, the operations in the FDD scheme areas follows. First, the AP transfers power to the UT for a periodof ǫ = α F T C , with < α F < . As before, we recall thatthe AP should provide the UT with sufficient energy for thelatter to be able not only to decode the received data but also toperform all the operations related the uplink signaling inherentto the FDD scheme. Afterwards, a downlink channel trainingphase takes place, in which the AP sends pilot sequences of η F T C ∈ Z + symbols with power P to the UT for estimatingthe downlink channel, with < η F < . Finally, the UT feedsback in the uplink the estimated CSI in analog form over thesubsequent τ F T C ∈ Z + symbols, where < τ F < and < α F + η F + τ F ≤ . Note that, in this work we adopt asimplified model for the uplink communication, for the sakeof simplicity of the analysis, and matters of space economy.Specifically, we assume that the feedback signal sent by theUT to the AP experiences an AWGN channel. We note that,this follows the typical approach proposed in the literaturefor first studies on CSI acquisition schemes based on analogfeedback signals [21], [26].Now, let us analyze the aforementioned steps in detail.Consider the l th antenna. We denote the pilot sequence sentover it as e l = [ e l [1] · · · , e l [ η F T C ]] ⊤ ∈ C η F T C , l ∈ [1 , L ] .Naturally, the sequences adopted in this phase are knownat both ends of the communications. In particular, withoutloss of generality, we assume orthogonality between pilotsequences sent over different antennas, i.e., e i ⊥ e j , for i = j .Thus, in order to guarantee their orthogonality, and estimate L independent channel coefficients, a lower bound on theminimum sequence size must be satisfied, i.e., η F T C ≥ L .Moreover, the AP equally divides the power P among its L antennas, yielding || e l || = PL and thus k e k = · · · = k e L k = η F T C PL . Then, the signal received by the UT duringthe downlink channel training phase is given by y pUT,F = e h ∗ + · · · + e L h ∗ L + w UT , where w UT ∼ CN ( , N I η F T C ) is the thermal noise at the UT. The UT in turn multipliesthe received signal y pUT,F by e † l / k e l k to estimate the l thchannel coefficient, h l . Similar to the previous section, the L downlink channel coefficients are estimated by an MVUbased estimator. The estimated channel vector at the UT canbe written as ˆh UT = [ˆ h UT, , . . . , ˆ h UT,L ] ⊤ = h + ˆw UT , with ˆw UT ∼ CN (cid:16) , N Lη F T C P I L (cid:17) estimation error vector at the UT.The power consumption at UT to decode a pilot sequence sentfrom one of the L transmit antennas is modeled similarly tothe previous case, i.e., proportional to P D . Accordingly, thetotal power consumed in decoding the pilot symbols is givenby η F T C P D . At this stage, the UT encodes each coefficient by means ofa sequence f l = [ f l [1] , · · · , f l [ τ F T C ]] ⊤ ∈ C τ F T C , ∀ l ∈ [1 , L ] ,such that the L sequences form an orthogonal set, i.e., f i ⊥ f j ,for i = j , and k f k = · · · = k f L k = τ F T C P F L . Asbefore, the adopted sequences are known at both ends of thecommunications. In particular, in order to guarantee their or-thogonality and encode L independent channel coefficients, alower bound on the minimum sequence size must be satisfied,i.e., τ F T C ≥ L .After the encoding, the signal to be fed back by the UT tothe AP is obtained as the sum of all the obtained sequencesat the previous step, i.e., x fF = f ˆ h UT, + · · · + f L ˆ h UT,L .Consequently, its transmission requires a power given by P F L (cid:16) | ˆ h UT, | + · · · + | ˆ h UT,L | (cid:17) = P F k ˆh UT k L . (11)Then, the received signal by the AP is given by y fAP,F = f ˆ h UT, + · · · + f L ˆ h UT,L + w AP , where w AP ∼ CN ( , N I τ F T C ) is the thermal noise at the AP. Now, thelatter multiplies the received sequence by f † k / k f k to estimate h k . Thus, the estimated channel vector at the AP is obtained as ˆh AP = h + ˆw UT + ˆw AP , where ˆw AP ∼ CN (cid:16) , N Lτ F T C P F I L (cid:17) .In particular, we note that ˆw UT and ˆw AP are independent bydefinition.Finally, the AP can exploit the knowledge of ˆh AP to derivethe desired MFP as before, given by ˆh AP / k ˆh AP k , and useit as beamforming vector while transmitting the informationsymbols for the remaining (1 − α F − η F − τ F ) T C symbols.The received information symbol at the UT is given by y sUT,F = h † ˆh AP k ˆh AP k s + n, (12)where s is the information symbol, with E (cid:2) | s | (cid:3) = P .Concerning the energy required to perform all the operationsat the UT, as a matter of fact, since the harvested energymust be sufficient to decode the received pilot sequences,feedback the estimated CSI, and decode the subsequent infor-mation, we have that the condition α F T C P H = η F T C P D + τ F T C P F k ˆh UT k /L + (1 − α F − η F − τ F ) T C P D must besatisfied. Therefore, if we plug (1) into this condition then,after some manipulations, we have that the minimum durationof the energy transfer/harvesting phase for this case, i.e., α F ,should be α F = τ F P F k ˆh UT k − τ F LP D + LP D βP k h k + LP D . (13)Now, we can use (13) to compute the downlink rate for theFDD scheme as R F = R F ( α F , η F , τ F )= (1 − α F − η F − τ F ) log P | h † ˆh AP | N k ˆh AP k ! = (1 − η F − τ F ) βP k h k − τ F P F k ˆh UT k − η F LP D βP k h k + LP D × log P | h † ˆh AP | N k ˆh AP k ! . (14)In this case, two parameters describe the duration of the chan-nel estimation phase, i.e., η F and τ F , related to the channelestimation procedures at the UT and the AP, respectively. Inpractice, these parameters impact both the accuracy of theestimated channel vectors and the remaining available timefor information transfer at the AP. As a consequence, let ( η ⋆F , τ ⋆F ) be the optimal couple of parameters that maximizesthe ergodic downlink rate, defined as ( η ⋆F , τ ⋆F ) = argmax η F ,τ F E h , ˆw " log P | h † ˆh AP | N k ˆh AP k ! × (1 − η F − τ F ) βP k h k − τ F P F k ˆh UT k − η F LP D βP k h k + LP D , (15)where ˆw = ( ˆw AP , ˆw UT ) . As before, the derivation of theexact value of η ⋆F and τ ⋆F is very complicated. Nevertheless,two approximations of this value, valid for high and low SNR,respectively, can be derived as stated in the following result. Lemma 2.
At high SNR, η ⋆F and τ ⋆F can be approximated as η ⋆F ≈ s(cid:18) P F βP (cid:19) P F P × τ ⋆F , (16) τ ⋆F ≈ vuut N L log eB T C ( L − P F (cid:16) P F βP (cid:17) , (17) where B = E h h log (cid:16) P k h k N (cid:17)i . At low SNR, they can beapproximated as τ ⋆F ≈ N L − vuuut βP (cid:18) βPPF +1+ N Lη⋆F TCP (cid:19)(cid:18) N Lη⋆F TC (cid:19) T C (cid:16) βP + P F + P F N Lη ⋆F T C P (cid:17) , (18) η ⋆F ≈ P F N LP T C ( βP + P F ) − s T C P ( βP + P F ) P F N L . (19) Proof:
See Appendix-B.Despite the complexity of (16), (17), (18) and (19), some in-teresting insights can drawn from the approximations adoptedin the derivation in Appendix-B following an approach similarto what has been done for Lemma 1. As before, the impactof P D on the accuracy of the results is likely negligible, duethe fact that P ≫ P D by construction. Now, consider thequantities λ = T C k h k N L (cid:16) ηF P + τF PF (cid:17) and λ = η F T C P k h k N L ,introduced in (49) and (52) respectively. We first focus on thelow SNR case. Therein, the approximations λ , λ ≈ areadopted. In this case, a smaller P F improves the accuracy ofthese approximations, whereas no clear insight can be drawnfor L . Conversely, in the high SNR case, the approximation λ ≫ is adopted. Differently from the previous case, theaccuracy of this approximation increases with P F . A furtherapproximation is introduced in this part of the study, i.e., N L η F P T C ≈ in (50). Accordingly, an additional insight on theimpact of the number of antennas on the accuracy of the resultin Lemma 2 can be drawn, i.e., the smaller L the larger theaccuracy. Interestingly, this is in contrast with the impact of thesame parameter in the TDD case and highlights the expectedlarger penalty for CSI acquisition that FDD pays w.r.t. TDDas the number of antennas grows. A more detailed discussionon its impact on the accuracy of the results in Lemma 2, isdeferred to Sec. V, where a comparative study of the downlinkrate of the three considered schemes is provided.IV. A NALYSIS OF THE O UTAGE P ROBABILITY
In this section, we will study the outage probability for theconsidered system as a function of the parameters introducedso far, and the downlink rate. In the considered practicalSWIPT implementation two possible outage events can occur: • The harvested energy is not sufficient for all the op-erations at the UT (channel estimation, pilot transmis-sion/CSI feedback and information decoding), i.e., theUT experiences an energy shortage . • The harvested energy is sufficient to perform all theoperations at the UT, but the achieved downlink rate issmaller than a target value, i.e., the UT experiences a data outage .We first focus on the case for which energy shortage occurs.Subsequently, we analyze the case for which the harvestedenergy is sufficient for all the operations at the UT, andcompute the data outage probabilities for the three transmitschemes considered in this work. Before we proceed, weremark that, the analytic expressions derived in this section forthe outage probabilities as a function of the system parametersare very complicated, and straightforward inference on theirbehavior is difficult to be drawn. Consequently, as before wedefer the discussion on the outage as a function of the systemparameters for all the cases considered in this work to Sec. V.
A. Energy Shortage Probability1) Non-CSI Scheme:
Referring to (2), for any given valuefor α N , the energy shortage probability for the non-CSI casecan be expressed mathematically as P E,outN ( α N ) = Pr (cid:26) α N βP k h k L < (1 − α N ) P D (cid:27) = γ (cid:16) L, (1 − α N ) LP D α N βP (cid:17) Γ ( L ) , (20)where γ ( q, r ) = R r u q − e − u d u is the lower incompleteGamma function. The closed-form expression of this prob-ability is derived by considering the cumulative distributionfunction (CDF) of χ L if we note that k h k ∼ χ L .
2) TDD Scheme:
Referring to (6), for any given value for α T and η T , the energy shortage probability for the TDD case,denoted by P E,outT ( α T , η T ) , can be expressed mathematicallyas P E,outT ( α T , η T )= Pr (cid:26) α T βP k h k L < (1 − α T − η T ) P D + η T P E (cid:27) = γ (cid:16) L, η T LP E +(1 − α T − η T ) LP D α T βP (cid:17) Γ ( L ) . (21)Using the same approach as for (20), the closed-form expres-sion of the probability in (21) is computed.
3) FDD Scheme:
Consider (13). For any given value for α F , η F and τ F , the energy shortage probability for the FDDcase can be stated mathematically as P E,outF ( α F , η F , τ F ) = Pr (cid:26) α F βP k h k L < τ F P F k ˆh UT k L + (1 − α F − τ F ) P D (cid:27) . (22)The following result provides a closed-form expression of(22). However, for the sake of the simplicity of the rep-resentation of the result, let us denote σ = q N L η F T C P , ρ = √ τ F P F α F βP − τ F P F , ρ = q τ F P F α F βP − τ F P F + τ F P F ( α F βP − τ F P F ) ,and ρ = q − α F − τ F ) LP D α F βP − τ F P F . Lemma 3.
The energy shortage probability for the FDDscheme, as in (22) , can be computed as P E,outF ( α F , η F , τ F )= 1 − Z ∞ θ =0 Q L (cid:16)p ρ σ θ , p ρ σ θ + ρ (cid:17) e θ θ − L L Γ ( L ) d θ . (23) Proof:
The outage probability can be evaluated as follows.First, applying the law of total probability, i.e., given a randomvariable A , Pr ( · ) = E A [Pr ( ·| A )] , we have(22) = E ˆw UT h Pr n k√ h − ρ ˆw UT k < ρ k ˆw UT k + ρ (cid:12)(cid:12) ˆw UT oi . (24)From (24), it can be easily deduced that k√ h − ρ ˆw UT k (cid:12)(cid:12) ˆw UT ∼ χ ′ L (cid:0) ρ k ˆw UT k (cid:1) . Therefore, substitutingthe PDF of k√ h − ρ ˆw UT k (cid:12)(cid:12) ˆw UT into (24), we can rewrite(24) = 1 − E ˆw UT (cid:20) Q L (cid:18) ρ k ˆw UT k , q ρ k ˆw UT k + ρ (cid:19)(cid:21) . (25)Since ˆw UT ∼ CN (cid:0) , σ I L (cid:1) , we have k ˆw UT k = σ Θ ,where Θ ∼ χ L . Substituting the PDF of k ˆw UT k into (25),we derive the RHS of (23), and this concludes the proof.At this stage, if we focus on (20), (21), and (23), we notethat the energy shortage probability in the three consideredcases clearly depends on the values of α N , ( α T , η T ) , and ( α F , η F , τ F ) respectively. However, drawing meaningful in-sights from these results is extremely difficult, due to theircomplexity. Accordingly, we will investigate this aspect inSec. V, by means of suitable numerical analyses. B. Data Outage Probability for the Non-CSI Scheme
We now compute the data outage probability for the non-CSI scheme. Given α N and a specific target downlink rate R NC , the data outage probability can be stated mathematicallyas P D,outN ( α N , R NC ) = Pr (cid:26) α N βP k h k L ≥ (1 − α N ) P D , (1 − α N ) log (cid:18) P k h k N L (cid:19) < R NC (cid:27) , that is the probability that the harvested energy is sufficient forthe decoding operations at the UT, but the achieved downlinkrate is smaller than R NC . Now, let us rewrite P D,outN as Pr (cid:26) (1 − α N ) LP D α N βP ≤ k h k < N LP (cid:18) RNC − αN − (cid:19)(cid:27) . (26)The intersection between the two events in (26) is non-emptywhen (1 − α N ) LP D α N βP < N LP (cid:18) RNC − αN − (cid:19) . (27)If this condition is not satisfied, then (26) in this case is equalto 0. In is worth noting that, assuming R NC = 0 , the dataoutage probability would be 0 only in case of extremely lowvalue of N , given that typically P ≫ P D , as previously discussed. This is in line with what could be expected ina wireless communication system, in which the data outageprobability tends to 0 as the SNR at the receiver increases.If this is not the case, and (27) is satisfied, then (26) can becomputed as P D,outN ( α N , R NC )= γ (cid:18) L, N LP (cid:18) RNC − αN − (cid:19)(cid:19) Γ ( L ) − γ (cid:16) L, (1 − α N ) LP D α N βP (cid:17) Γ ( L ) , (28)where we made use of the CDF of the χ L distribution. C. Data Outage Probability for the TDD Scheme
We switch our focus back to the TDD scheme. For givenvalues of α T , η T , and a target downlink rate R T , the dataoutage probability is expressed as P D,outT ( α T , η T , R T ) = Pr (cid:26) α T βP k h k L ≥ (1 − α T − η T ) P D + η T P E , (1 − α T − η T ) log P | h † ˆh | N k ˆh k ! < R T (cid:27) , (29)that is the probability that the harvested energy is sufficientto engage in the pilots transmission and decode the receiveddata, but the achieved downlink rate is smaller than R T . Thefollowing result provides a closed-form expression for (29)and concludes the study of the TDD case. However, beforeproceeding, let us denote b = N P (cid:18) RT − αT − ηT − (cid:19) , b = η T LP E +(1 − α T − η T ) LP D α T βP , b = N + η T T C P E N and b = N η T T C P E ,for the sake of the simplicity of the representation of the result. Lemma 4.
When N P (cid:18) RT − αT − ηT − (cid:19) < η T LP E +(1 − α T − η T ) LP D α T βP , then the data outage probabilityfor the TDD scheme, as in (29) , can be computed as P D,outT = Z ∞ θ =0 Z b b θ =0 Γ (cid:0) L − , b b − θ (cid:1) θ L − L +1 Γ ( L −
1) Γ ( L ) × I r θ θ b ! e − (cid:16) θ + θ b + θ (cid:17) d θ d θ . (30) Conversely, when N P (cid:18) RT − αT − ηT − (cid:19) ≥ η T LP E +(1 − α T − η T ) LP D α T βP , it can be computed as P D,outT = Z ∞ θ =0 Z b b θ =0 Γ (cid:0) L − , b b − θ (cid:1) I (cid:16)q θ θ b (cid:17) L +1 Γ ( L −
1) Γ ( L ) × θ L − e − (cid:16) θ + θ b + θ (cid:17) d θ d θ + Z ∞ θ =0 e − θ θ L − × (cid:16) Q (cid:16)q θ b , √ b b (cid:17) − Q (cid:16)q θ b , √ b b (cid:17)(cid:17) L Γ ( L ) d θ . (31) Proof:
See Appendix-C.
D. Data Outage Probability for the FDD Scheme
We conclude our study on the data outage probability byconsidering the FDD case. For given α F , η F , τ F , and aspecific target downlink rate R F , the data outage probabilitycan be stated mathematically as P D,outF ( α F , η F , τ F , R F )= Pr ( α F βP k h k L ≥ τ F P F k ˆh UT k L + (1 − α F − τ F ) P D , (1 − α F − η F − τ F ) log P | h † ˆh AP | N k ˆh AP k ! < R F ) , (32)that is the probability that the harvested energy is sufficient toestimate the downlink channel, feed back its estimated versionin the uplink, and decode the received data, but the achieveddownlink rate is smaller than R F . The following result pro-vides a closed-form expression of (32) and concludes the studyof the FDD case. However, as before, let us introduce somenew notation to further simplify representation of the results.Accordingly, we let σ = N L + η F P T C N L , σ = N Lη F P T C , σ = N Lτ F P F T C , σ = (1+ σ ) σ σ + σ , b = N P (cid:18) RF − αF − ηF − τF − (cid:19) , b = (1 − α F − τ F ) LP D α F βP , and b = τ F P F α F βP . Lemma 5.
The data outage probability for the FDD scheme,as in (32) , can be computed as P D,outF = Z ∞ θ =0 Z θ + θ > b − b b σ Z σ b θ =0 θ L − θ L − Γ ( L ) Γ ( L − × Q L − s θ σ σ σ , s σ (cid:18) b + b ( θ + θ ) σ (cid:19) − θ ! × I (cid:16)q θ θ σ σ σ (cid:17) I (cid:18)q θ θ (1+ σ ) σ (cid:19) L +1 × e (cid:18) θ σ σ σ + θ θ θ θ + (1+ σ z σ (cid:19) d θ d θ d θ d θ (33) + Z ∞ θ =0 Z θ + θ ≤ b − b b σ Z σ (cid:16) b + b θ θ σ (cid:17) θ =0 Q L − s θ σ σ σ , s σ (cid:18) b + b ( θ + θ ) σ (cid:19) − θ ! × I s θ θ σ σ σ ! I s θ θ (1 + σ ) σ θ L − θ L − × e − (cid:18) θ σ σ σ + θ θ θ θ + (1+ σ θ σ (cid:19) Γ ( L −
1) Γ ( L ) 2 L +1 d θ d θ d θ d θ (34) + Z ∞ θ =0 Z θ + θ ≤ b − b b σ I (cid:18)q θ θ (1+ σ ) σ (cid:19) L × e (cid:16) (1+ σ θ σ + θ θ θ (cid:17) × " Q s θ σ σ σ , s σ (cid:18) b + b ( θ + θ ) σ (cid:19)! − Q s θ σ σ σ , p σ b ! × θ L − θ L − Γ ( L −
1) Γ ( L ) d θ d θ d θ . (35) Proof:
See Appendix-D.V. N
UMERICAL R ESULTS
In this section we evaluate the performance of SWIPT forMISO systems, to assess its merit under the transmit schemesconsidered in this work. The parameters used in our numericalresults are as follows. We consider β = 0 . , which is a goodapproximation of the performance delivered by state-of-the-art commercial products [5], and typically adopted value inthe literature on this subject [11], [12], [27]. We consider P = 1 and T C = 1000 for simplicity, and L ∈ { , } . Weassume that the system operates in the industrial, scientificand medical (ISM) band, i.e., carrier frequency of 2.4 GHz.Accordingly, we set a distance between the AP and the UTin the order of meters such that we can ensure that thelatter is situated in the far-field region of the radiating AP.Furthermore, we assume that the signals transmitted by boththe AP and the UT experience a generic path loss attenuation,with a path loss exponent equal to 3. As a consequence, wecan safely let PP D = 1000 . The rationale for this is that byincorporating the propagation losses in PP D , we frame a morerealistic scenario. Finally, we model the ratio between thepower budgets available at the AP and the UT following thesame logic. We let PP E = PP F = 100 , in accordance with thetypical ratio between the available power budgets at both sidesof the communication in modern networks, which is roughly dB [28]. A. Downlink Rate
Now, we focus on the ergodic downlink rate. First, wecompute the optimal numerical performance of the system bynumerically solving the problems in (8) and (15), by means ofan exhaustive search whose complexity and time requirementsare not suitable for realistic implementations. Subsequently,we evaluate the accuracy of our theoretical results by compar-ing them to the numerical performance results. Throughout thissection, we will refer to the derived approximated parametersin Lemma 1 and Lemma 2 as analytic results, for the sake ofclarity. Now, for the TDD scheme, let R ⋆T and η ⋆T be the op-timal downlink rate and the optimal duration of the portion ofthe coherence time devoted to the channel training/estimation,computed by extensive Monte-Carlo simulations. For the sakeof clarity, with a little abuse of notation, we denote ˆ η ⋆T as theoptimal parameter of interest for the TDD scheme, computedaccording to Lemma 1. For the FDD scheme, a similar notationis defined. Now, we define ζ T = R T ( α T , ˆ η ⋆T ) R ⋆T ∈ [0 , , for TDD,and ζ F = R F ( α F , ˆ η ⋆F , ˆ τ ⋆F ) R ⋆F ∈ [0 , , for FDD, as the ratio betweenthe downlink rate obtained with the analytic and optimalnumerical results. We let SNR ∈ [0 , dB and compute ζ T and ζ F for both L = 3 and L = 6 in Fig. 4, Fig. 5, Fig. 6, andFig. 7. Quantitatively, if we focus on the best performer for Note that, α T and α F are computed according to (6) and (13) respectively. SNR [dB] ζ T High SNR analyticLow SNR analytic
Figure 4. ζ T for analytic and numerical parameters, TDD and L = 3 antennas. SNR [dB] ζ T High SNR analyticLow SNR analytic
Figure 5. ζ T for analytic and numerical parameters, TDD and L = 6 antennas. each of the considered SNR values, the gap between ζ T ( ζ F inthe FDD case) and 1 is remarkably small. Thus, the accuracyof our derivations is confirmed. If we focus on the impactof L on the two analytic results, we note that they confirmthe intuitions provided in Sec. III-B and Sec. III-C. However,the difference in terms of the best ζ T (and ζ F ) between thetwo antenna configurations is rather small. This shows that theimpact of the number of antennas at the AP on the accuracyof the analytic results is not very significant. Furthermore, wesee that ζ F ≤ ζ T , ∀ SNR ∈ [0 , dB and ∀ L ∈ { , } . This isdue to the two-step channel estimation process that is neededin the FDD scheme for the CSI acquisition at the AP. As aconsequence, a greater number of approximations is necessary.This reduces the accuracy of our closed-form representation SNR [dB] ζ F High SNR analyticLow SNR analytic
Figure 6. ζ F for analytic and numerical parameters, FDD and L = 3 antennas. SNR [dB] ζ F High SNR analyticLow SNR analytic
Figure 7. ζ F for analytic and numerical parameters, FDD and L = 6 antennas. of η ⋆F and τ ⋆F . Focusing on the practical implementation, we note that thepresence of the analytic results provides a twofold alternativefor the AP, depending on the system intrinsic constraints.When the time available for the optimization of the trans-mit parameters is small, the analytic results could be usedto achieve a performance which is reasonably close to theoptimal, without resorting to an exhaustive search. Conversely,if more time is available for the AP, the analytic results canbe used to improve the efficiency of the search for the optimalparameters. In this regard, we note that the downlink rate is aconcave function of ( η, τ ) . Accordingly, in this case, the local The interested reader may refer to Appendix-B for further details. optimum coincides with the global optimum. Now, assumethat the results of Lemma 1 and Lemma 2 were adopted as astarting point for finding the numerically optimal parameters,by means of an exhaustive search inside a smaller set. Then,the necessary time to identify the global optimum could besignificantly reduced w.r.t. a “blind” exhaustive search, dueto the proximity of the analytic results and the actual globaloptimum.To conclude our analysis on the downlink rate, we inves-tigate the advantages, if any, that the two duplexing schemesdiscussed so far can bring w.r.t. the non-CSI case in termsof the downlink rate. We remark that our goal is to char-acterize the performance of the system under the realisticassumptions made in Sec. I. Thus, in the following study,all the system parameters discussed so far are set accordingto the analytic results derived in Sec. III. Moreover, forsimplicity in the representation, we let R = R T ( α T , ˆ η ⋆T ) and R = R F ( α F , ˆ η ⋆F , ˆ τ ⋆F ) be the downlink rate for TDD and FDD,respectively, when the analytic results are adopted. The ratiobetween these rates and their counterpart for the non-CSI case(i.e., RR NC , with R NC as in (3)) is represented in Fig. 8, forSNR ∈ [0 , dB. Remarkably, both duplexing schemes clearly SNR [dB] RR N C TDD for L = 6FDD for L = 6TDD for L = 3FDD for L = 3 Figure 8. Ratio between the ergodic downlink rate for the CSI acquisitionschemes and the non-CSI case. outperform the non-CSI approach in terms of downlink rate.This shows that, despite the penalties incurred to acquire theCSI, evident downlink rate enhancements are experienced bythe AP, thanks to presence of the CSI, however imperfect thelatter might be. The result in Fig. 8 is even more remarkable,considering that therein the two duplexing schemes alwaysoutperform the non-CSI approach, regardless of the antennaconfiguration and the SNR value. Furthermore, the largestadvantage over the non-CSI performance is obtained in thelow-to-mid SNR regime. In this regard, we first focus on theTDD case. In both cases, i.e., L = 3 and L = 6 , RR NC isa monotonically decreasing function of the SNR, confirmingthat the MFP performs better for low than for high SNR values[25], [29]. In particular, this shows that the availability of the CSI at the AP, albeit imperfect, is sufficient to achieve a muchlarger downlink rate as compared to the non-CSI approach.Furthermore, the performance for L = 6 is strictly largerthan for L = 3 , showing that, as in the case of traditionalwireless communications, the SWIPT can effectively exploitthe transmit diversity gain delivered by a MISO system as L grows. Interestingly, the same is true for the FDD scheme. TheCSI acquisition procedure in this case is more complex andprone to a higher uncertainty, especially at low SNR. Thisimpacts the behavior of RR NC that presents a maximum atSNR = 10 dB, for both the considered antenna configurations.On one hand, the gain brought by the FDD scheme over thenon-CSI approach is dominated by the power gain at the UT,brought by a more accurate beamformer design at the AP,for SNR ≤ dB. On the other hand, the reduction of themultiplexing gain due to the increasing impact that both thechannel estimation and feedback phases have on available timefor information transfer, as the quality of the CSI increases,determines the decreasing behavior of RR NC for SNR > dB.Finally, we note that the difference at high SNR between thevalues of RR NC for TDD and FDD is very low, but increaseswith the L . In fact, when the SNR is high, the channelestimation/feedback phases are very short, thus the differencein the amount of time available for the information transfer inboth cases is small. Nevertheless, a bigger L entails a larger τ F (thus α F ) and, in turn, increases the difference betweenthe values of RR NC for TDD and FDD at high SNR as well. B. Outage Probability
We switch our focus to the analysis of the energy short-age and the data outage probability. A set of Monte-Carlosimulations is performed to obtain the numerically computedprobabilities. Subsequently, we set the values of η T , η F , and τ F according to Lemma 1 and Lemma 2 and compute theexact value of both metrics by means of the analytic results inSec. IV. At this stage, we only consider the case L = 3 owingto space economy. In the previous subsection, we verifiedthat the impact of change in the number of antennas onthe accuracy of the analytic results on the downlink rate israther small. Accordingly, a robustness of the accuracy ofour results to a change in the number of antennas could beconjectured. For the sake of clarity we let p E,out and p D,out bethe energy shortage probability and the data outage probabilitywhen no energy shortage occurs, respectively. Furthermore,we let R NC = R T = R F = 6 (bit/s/Hz) be the targetrate for the considered system. Finally, we depict p E,out forSNR ∈ [0 , dB in Fig. 9, Fig. 10, and Fig. 11 and p D,out for SNR ∈ [0 , dB in Fig. 12. As shown in these figures, thenumerical results perfectly match the analytic results derivedin Sec. IV for all the three schemes. This perfect match verifiesthe correctness of our derivations.We start by noting that the energy shortage probabil-ity strongly depends on the considered parameters. Thus, acomparison between schemes could have limited interestedw.r.t. a comparison between the results obtained for each Referring to Sec. IV-B, IV-C, and IV-D, we note that R NC , R T , and R F are specified values. −4 −3 −2 −1 SNR [dB] P E , o u t Analytic for α N = 0 . α N = 0 . α N = 0 . α N = 0 . α N = 0 . α N = 0 . Figure 9. Energy shortage probability, non-CSI and L = 3 antennas. −4 −3 −2 −1 SNR [dB] P E , o u t Analytic for α T = 0 . α T = 0 . α T = 0 . α T = 0 . α T = 0 . α T = 0 . Figure 10. Energy shortage probability, TDD and L = 3 antennas. scheme, as the duration of the energy transfer phase varies.Accordingly, we restrain our focus to the latter aspect. Asexpected, the energy shortage probability is independent of theSNR, regardless of the value of α N . However, for both theTDD and the FDD scheme, the energy shortage probabilitydecreases with the SNR, regardless of the value of α T and α F . In these cases, a larger SNR reduces the optimal timefor both devices to perform the operations intrinsic to theCSI acquisition and achieve accurate channel estimations. Inother words, the channel estimation accuracy increases withthe SNR value, thus the CSI acquisition requires less time.Therefore, the energy consumption at the UT is lower whenthe SNR is large. Now, if the duration of the energy transferphase is doubled or tenfold, a reduction of the energy shortageprobability from almost one to three orders of magnitude is −4 −3 −2 −1 SNR [dB] P E , o u t Analytic for α F = 0 . α F = 0 . α F = 0 . α F = 0 . α F = 0 . α F = 0 . Figure 11. Energy shortage probability, FDD and L = 3 antennas. −6 −5 −4 −3 −2 −1 SNR [dB] P D , o u t Analytic for non-CSINumerical for non-CSIAnalytic for FDDNumerical for FDDAnalytic for TDDNumerical for TDD
Figure 12. Data outage probability when no energy shortage occurs, L = 3 antennas. observed, depending on the considered scheme. In practice, ifthe coherence time is long enough, even a rather small increaseof the duration of the energy transfer phase can positivelyimpact the energy shortage probability.We now switch our focus to the data outage probabilityillustrated in Fig. 12. We start by noting that, to compute thenumerical data outage probability in this case, the duration ofthe energy transfer phase for the three considered schemes, i.e., α N , α T , and α F , is chosen at each iteration of the simulationssuch that the harvested energy at the UT is sufficient toperform the receiver operations intrinsic to each scheme. Asa matter of fact, the obtained quantitative results for a studyof this kind are not extremely relevant, in fact they clearlydepend on the selected target rate. In practice, their qualitative behavior is definitely more interesting. In this regard, thelowest data outage probability is experienced by the consideredsystem in the case of the TDD scheme. This could havebeen expected after our findings on the downlink rate in theprevious section, in which the TDD scheme resulted as thebest performer out of the three considered cases.VI. C ONCLUSION
In this work, we have examined the efficacy of SWIPT ina MISO system consisting of an AP and a single UT. Inparticular, the latter is not equipped with any local powersource, but instead harvests the necessary energy for itsoperations from the received RF signals. The performanceof the considered system has been analyzed under realisticand practically relevant system assumptions. Three practicalcases have been considered: a) absence of CSI at the AP,b) imperfect CSI at the AP acquired by means of pilotsestimation (TDD), c) imperfect CSI at both the UT andthe AP acquired by means of analog CSI feedback in theuplink (FDD). We have compared the considered scenarios bymeans of three performance metrics of interest, i.e, the ergodicdownlink rate, the energy shortage probability, and the dataoutage probability. Accordingly, we have derived closed-formexpressions for each metric, and for the ergodically optimalduration of both the WPT and the channel training/feedbackphases, to maximize the downlink rate in all the three sce-narios. The accuracy of our derivations has been verifiedby an extensive numerical analysis. First, it is worth notingthat TDD has consistently been the best performer for eachconsidered metric, confirming the potential of this duplexingscheme for the future advancements in modern networks. Morespecifically, concerning the downlink rate, our findings showthat CSI knowledge at the AP is always beneficial for theinformation transfer in SWIPT systems, despite the resourcesdevoted to the channel estimation/feedback procedures and thepresence of estimation errors. In a follow-up of this work, wewill study both strategies to maximize the efficiency of theWPT, in the case of the availability of CSI knowledge at theAP prior to the WPT phase (or part of it), and their impact onthe energy shortage probability. Additional subject of futureinvestigation will be the extension of the considered set-up toa multi-user scenario. A
PPENDIX AP ROOF OF L EMMA E h , ¯w [ · ] = E h [ E ¯w [ ·| h ]] . Furthermore, we neglect LP D in (8)since, in practice, P D ≪ P generally [28]. We proceed byfirst computing the following expression: (cid:18) − η T − η T LP E βP k h k (cid:19) E ¯w " log P | h † ˆh | N k ˆh k ! , (36)for a given channel realization h . In order to compute (36),the following straightforward results can be derived: | h † ˆ h | = N k h k η T T C P E Ψ and k ˆh k = N η T T C P E Ψ , (37) where Ψ ∼ χ ′ (cid:16) η T T C P E k h k N (cid:17) and Ψ ∼ χ ′ L (cid:16) η T T C P E k h k N (cid:17) . We break up the subsequent analysisinto two cases, namely, the high SNR and low SNR cases.First, we consider the analysis at high SNR. In this case,applying the approximation, log (1 + SN R ) ≈ log SN R when
SN R ≫ , and (37) to (36), we can derive E ¯w " log P | h † ˆh | N k ˆh k ! ≈ E ¯w (cid:20) log (cid:18) P k h k Ψ N Ψ (cid:19)(cid:21) . (38)Subsequently, using the Taylor series expansion of log Ψ and log Ψ at their respectively mean values (i.e. λ and L + λ respectively, where λ = η T T C P E k h k N ), we have(38) = log P k h k N + log e × E Ψ , Ψ " ln (2 + λ ) + Ψ − − λ λ − (Ψ − − λ ) λ ) + · · ·− ln (2 L + λ ) − Ψ − L − λ L + λ + (Ψ − L − λ ) L + λ ) − · · · (39) = log P k h k N + log e ln (2 + λ ) − (2 + 2 λ )(2 + λ ) + · · ·− ln (2 L + λ ) + (2 L + 2 λ )(2 L + λ ) − · · · ! (40) ( a ) ≈ log P k h k λN (2 L + λ ) = log P k h k N − log (cid:18) Lλ (cid:19) , (41)where ( a ) in (41) is derived by noting that λ is large (at highSNR), and hence we can neglect the higher order fractionalterms in (40). Moreover, in the ln(2 + λ ) term is neglectedowing to λ ≫ . Further, we use (41) in (36) and take theexpectation over h . Moreover, since L/λ is small at high SNR,we use the approximation, log (1 + x ) ≈ x log e when x ≈ , in the derivation. By some straightforward computations,we can rewrite (8) as η ⋆T ≈ argmax η T B − η T B − N L log eη T T C P E ( L − , (42)where B = E h h(cid:16) LP E βP k h k (cid:17) log (cid:16) P k h k N (cid:17)i , and B is aconstant value. In order to derive η ⋆T , we differentiate (42)with respect η T and set it equal to . By some straightforwardcomputations, we can derive the result (9).We now move to the analysis at low SNR. Before deriving,we first note that E [ X ] = E [exp(ln X )] . Subsequently, usingthe approximation, log (1 + SN R ) ≈ SN R log e when SN R ≈ , and Jensen’s inequality, i.e., E [exp(ln X )] ≥ exp( E [ln X ]) , we have(36) ≥ (cid:18) − η T − η T LP E βP k h k (cid:19) P log eN × exp E ¯w " ln | h † ˆh | k ˆh k ! . (43) Once again, we apply Taylor series expansion to E ¯w h ln (cid:16) | h † ˆh | k ˆh k (cid:17)i , i.e., steps (38), (39), (40), and (41).In this case, since λ is small at low SNR, we approximatethe higher order terms, such as λ (2+ λ ) in (40), by a constantvalue κ . Using this, we can rewrite (43) as κ (cid:16) − η T − η T LP E βP k h k (cid:17) P k h k log eN × λ L + λ . (44)Finally, we take the expectation over h . By using Jensen’sinequality (similar approaches in (43)) and applying Taylorseries expansion for the logarithm term at the mean value k h k (similar approaches in (39), (40), and (41)), we rewrite (8) as η ⋆T ≈ argmax η T κ (cid:16) η T T C P E LN (cid:17) (cid:16) L − η T L − η T LP E βP (cid:17) η T T C P E N where κ is a constant. In order to derive η ⋆T , we differentiatethe above formula with respect η T and set it equal to . Bysome straightforward computations, we can derive the result(10). A PPENDIX BP ROOF OF L EMMA E ˆw " − τ F − τ F P F k ˆh UT k βP k h k − η F ! × log P | h † ˆh AP | N k ˆh AP k ! . (45)To compute (45), the following results can be derived (detailsomitted for lack of space): | h † ˆh AP | = N L k h k T C (cid:18) η F P + 1 τ F P F (cid:19) Φ , (46) k ˆh AP k = N L T C (cid:18) η F P + 1 τ F P F (cid:19) Φ , (47) k ˆh UT k = N L η F T C P Φ , (48)where Φ ∼ χ ′ (cid:18) T C k h k N L (cid:16) ηF P + τF PF (cid:17) (cid:19) , Φ ∼ χ ′ L (cid:18) T C k h k N L (cid:16) ηF P + τF PF (cid:17) (cid:19) , and Φ ∼ χ ′ L (cid:16) η F T C P k h k N L (cid:17) .Before proceeding, we note that the pre-log term and theterm inside the logarithm in (45) are correlated, due to thepresence of ˆw UT in both terms. However, at high SNR, thevariance of ˆw UT will be small. Thus, we assume that thepre-log term and term inside the logarithm are approximatelyindependent (and hence we can take the expectations ofthese two terms in (45) separately). For the term inside thelogarithm, using (46), (47), and (48) and by Taylor seriesexpansion (similar approaches in (39), (40), and (41)), weobtain E ˆw " log P | h † ˆh AP | N k ˆh AP k ! ≈ log P k h k λ N (2 L + λ ) , (49) where λ = T C k h k N L (cid:16) ηF P + τF PF (cid:17) . For the pre-log term, we have E ˆw " − τ F − τ F P F k ˆh UT k βP k h k − η F ≈ − τ F − τ F P F βP − η F . (50)The approximation in (50) is derived using E ˆw [ k ˆh UT k ] = k h k + N L η F T C P ≈ k h k (since at high SNR, N L η F P T C issmall enough to be neglected). Finally, we substitute (49) and(50) into (45) and take the expectation over h . Using theapproximation log (1 + x ) ≈ x log e when x ≈ and bysome straightforward computations, we can rewrite (15) as ( η ⋆F , τ ⋆F ) ≈ argmax η F ,τ F (cid:18) − τ F − τ F P F βP − η F (cid:19) × (cid:18) B − N L log eT C ( L − (cid:18) η F P + 1 τ F P F (cid:19)(cid:19) , (51)where B = E h h log (cid:16) P k h k N (cid:17)i . In order to find η ⋆F and τ ⋆F ,we differentiate (51) with respect η F and τ F and equate themto . By some straightforward computations, we can derivethe results (16) and (17).We now turn to the analysis at low SNR. Herein, we followthe similar derivations as in Appendix-A. First we use the sameapproaches as in (43). Further, we apply (46), (47), and (48)and use the Taylor series expansion (similar to the approachesin (44)). Thus, we can derive(45) ≥ (cid:18) (1 − η F − τ F ) βP k h k − τ F P F N L η F T C P (2 L + λ ) (cid:19) × κ log e (2 + λ ) N β (2 L + λ ) , (52)where λ = η F T C P k h k N L , and κ is a constant value. Finally,as before we take the expectation over h . By using Jensen’sinequality (similar approaches in (43)) and applying Taylorseries expansion of the logarithm term at the mean value k h k (similar approaches in (39), (40), and (41)), we can rewrite theexpected value of (52) over h as κ (cid:16) − η F − τ F − τ F P F βP − τ F P F N Lη F T C βP (cid:17)(cid:18) L + T C N (cid:16) ηF P + τF PF (cid:17) (cid:19) (cid:18) T C N (cid:16) ηF P + τF PF (cid:17) (cid:19) − , (53)where κ is a constant value. Now, we differentiate (53)with respect η F and τ F and equate them to . Using somestraightforward computations, we obtain the result (18). When x ≈ , √ x ≈ x . By utilizing this approximation in (18)since N is large at low SNR, we can derive τ ⋆F ≈ ( η ⋆F ) T C βP P F N L .Substituting the latter approximated τ ⋆F into the differentiatedequation ∂ (53) ∂η F = 0 , we can also derive the result (19).A PPENDIX CP ROOF OF L EMMA (cid:12)(cid:12)(cid:12) h † ˆh k ˆh k (cid:12)(cid:12)(cid:12) and k h k . To evaluate(29), we first express k h k as the sum of (cid:12)(cid:12)(cid:12) h † ˆh k ˆh k (cid:12)(cid:12)(cid:12) and another independent random variable (see the steps below). This stepsimplifies the evaluation as shown in the following. In order todo so, first we project the row vector h † onto an orthonormalset of vectors Ω = n ˆh † k ˆh k , g † , · · · , g † L o , where g † , · · · , g † L arechosen arbitrarily such that the vectors in Ω span the complex L dimensional space. Recall that h † | ˆh ∼ CN (cid:16) b ˆh † , b I L (cid:17) .Since the distribution of a Complex Gaussian random vector(with distribution CN ( , I ) ) projected onto an orthonormalset remains unchanged [29], we can conclude that h † ˆh k ˆh k (cid:12)(cid:12)(cid:12) ˆh ∼CN (cid:16) k ˆh k (1+ b ) , b − (cid:17) and h † g l (cid:12)(cid:12) ˆh ∼ CN (cid:0) , b − (cid:1) , l = 2 , . . . , L. Additionally, the h † g l (cid:12)(cid:12) ˆh l = 2 , . . . , L are independent randomvariables. Thus we can conclude the following: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h † ˆh k ˆh k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆh = 12 b Θ , (54) (cid:16)(cid:12)(cid:12) h † g (cid:12)(cid:12) + · · · + (cid:12)(cid:12) h † g L (cid:12)(cid:12) (cid:17) (cid:12)(cid:12)(cid:12) ˆh = 12 b Θ , (55)where Θ ∼ χ ′ (cid:16) b b k ˆh k (cid:17) , and Θ ∼ χ L − . Furthermore, Θ and Θ are independent and Θ + Θ = 2 b k h k .Now, applying the same approach as in (24) to (29) andusing (54) and (55), we can derive the following: P D,outT = E ˆh [Pr { Θ < b b , Θ + Θ ≥ b b } ] . Let us focus on therelationship between b and b and consider two possiblecases, i.e., b < b and b ≥ b . We start from the former. Inthis case, denoting the PDF of Θ i as f Θ i ( θ i ) for i ∈ { , } ,we have, P D,outT = E ˆh Z b b θ =0 Γ (cid:0) L − , b b − θ (cid:1) I (cid:18)r k ˆh k θ b b (cid:19)
2Γ ( L − e (cid:18) θ + k ˆh k b b (cid:19) d θ . (56)Since ˆh ∼ CN ( , (1 + b ) I L ) , it follows that k ˆh k = (1+ b )2 Θ , where Θ ∼ χ L . Substituting the PDF of k ˆh k into (56), we obtain our result (30). In the second case b ≥ b ,following the same approaches as in (56), we obtain our result(31) and conclude the proof.A PPENDIX DP ROOF OF L EMMA ˆh UT given ˆh AP as f (cid:16) ˆh UT (cid:12)(cid:12) ˆh AP (cid:17) , the analytic expression for the outageprobability, i.e., P D,outF in (32), can be written as E ˆh AP (cid:20) Z Pr ( (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h † ˆh AP k ˆh AP k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < b , k h k ≥ b + b k ˆh UT k (cid:12)(cid:12)(cid:12) ˆh AP , ˆh UT o f (cid:16) ˆh UT (cid:12)(cid:12) ˆh AP (cid:17) d ˆh UT i . (57)To compute (57), we use the projection approach that wehave used in Appendix-C. First, we project the row vector h † onto an orthonormal set of vector n ˆh † AP k ˆh AP k , g † , · · · , g † L o , (such that these vectors span the L dimensional complex space).Since h | ˆh AP , ˆh UT ∼ CN (cid:16) σ ˆh UT , σ I L (cid:17) , following thesame approach as in Appendix-C, we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h † ˆh AP k ˆh AP k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆh AP , ˆh UT = 12 σ Θ , (58) (cid:16)(cid:12)(cid:12) h † g (cid:12)(cid:12) + · · · (cid:12)(cid:12) h † g L (cid:12)(cid:12) (cid:17) (cid:12)(cid:12)(cid:12) ˆh AP , ˆh UT = 12 σ Θ , (59)where Θ ∼ χ ′ (cid:18) σ σ (cid:12)(cid:12)(cid:12) ˆh † UT ˆh AP k ˆh AP k (cid:12)(cid:12)(cid:12) (cid:19) and Θ ∼ χ ′ L − (cid:18) σ σ (cid:18) k ˆh UT k − (cid:12)(cid:12)(cid:12) ˆh † UT ˆh AP k ˆh AP k (cid:12)(cid:12)(cid:12) (cid:19)(cid:19) . Moreover, Θ and Θ are independent and Θ + Θ = 2 σ k h k . Using (58)and (59), we rewrite (57) as E ˆh AP (cid:20) Z Pr n Θ + Θ ≥ σ (cid:16) b + b k ˆh UT k (cid:17) , Θ < σ b o f (cid:16) ˆh UT (cid:12)(cid:12) ˆh AP (cid:17) d ˆh UT i . (60)To compute the double integration over θ and θ in (60),we have two two cases, i.e., b < b + b k ˆh UT k and b ≥ b + b k ˆh UT k . In the first case, the probability term in (60)can be computed as Z σ b θ =0 Q L − vuuut σ σ k ˆh UT k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆh † UT ˆh AP k ˆh AP k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , r σ (cid:16) b + b k ˆh UT k (cid:17) − θ ! I vuut θ σ σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆh † UT ˆh AP k ˆh AP k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × e − θ + σ σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆh † UT ˆh AP k ˆh AP k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ! d θ . (61)Since ˆh UT | ˆh AP ∼ CN (cid:16) σ σ ˆh AP , σ I L (cid:17) , using the projectionapproach once again, we can derive (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆh † UT ˆh AP k h AP k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆh AP = σ , (62) k ˆh UT k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆh † UT ˆh AP k h AP k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆh AP = σ , (63)where Θ ∼ χ ′ (cid:16) σ σ k ˆh AP k (cid:17) and Θ ∼ χ L − . Further-more, Θ and Θ are independent and Θ + Θ = σ k ˆh UT k .Lastly, we note that since b < b + b k ˆh UT k , we have b < ( b + b ) (Θ +Θ ) σ and hence Θ + Θ > b − b ) b σ . Now,applying (61), (62), and (63) to (60), the integral of (60) can be computed as Z θ + θ > b − b b σ Z σ b θ =0 I s σ θ k ˆh AP k σ θ L − L +1 × Q L − s θ σ σ σ , s σ (cid:18) b + b ( θ + θ ) σ (cid:19) − θ ! × I (cid:16)q θ θ σ σ σ (cid:17) Γ ( L − e (cid:18) θ θ θ + θ σ σ σ + σ k ˆh AP k σ (cid:19) d θ d θ d θ . (64)We note that, since ˆh AP ∼ CN ( , (1 + σ + σ ) I L ) , wehave that k ˆh AP k = σ + σ Θ , where Θ is χ L . Usingthis fact and (64) in (60), we obtain (33) in Lemma 5.We now consider the second aforementioned case, i.e., b ≥ b + b k ˆh UT k . Following similar steps as in the first case,we obtain (34) and (35) in Lemma 5 (the detailed steps havebeen omitted for matters of space economy). At this stage, theoutage probability in (32) is obtained as the sum of (33), (34),and (35), and this concludes the proof.R EFERENCES[1] L. Xie, Y. Shi, Y. Hou, and A. Lou, “Wireless power transfer and ap-plications to sensor networks,”
IEEE Trans. Wireless Commun. , vol. 20,no. 4, pp. 140–145, Aug. 2013.[2] A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, andM. Soljaˇci´c, “Wireless power transfer via strongly coupled magneticresonances,”
Science , vol. 317, no. 5834, pp. 83–86, Jul. 2007.[3] W. C. Brown, “The history of power transmission by radio waves,”
IEEETrans. Microw. Theory Techn. , vol. 32, no. 9, pp. 1230–1242, Sep. 1984.[4] J. Schlesak, A. Alden, and T. Ohno, “A microwave powered high altitudeplatform,” in
Proc. IEEE MTT-S Int. Microw. Symp. Dig. , May 1988, pp.283–286 vol.1.[5] P. Corp., “P2110-915MHz RF powerharvester receiver,”
ProductDatasheet , pp. 1–12, 2010.[6] T. Le, K. Mayaram, and T. Fiez, “Efficient far-field radio frequencyenergy harvesting for passively powered sensor networks,”
IEEE J.Solid-State Circuits , vol. 43, no. 5, pp. 1287–1302, May 2008.[7] J. McSpadden and J. Mankins, “Space solar power programs andmicrowave wireless power transmission technology,”
IEEE Microw.Mag. , vol. 3, no. 4, pp. 46–57, Dec. 2002.[8] L. R. Varshney, “Transporting information and energy simultaneously,”in
Proc. IEEE Int. Symp. Inf. Theory , Jul. 2008, pp. 1612–1616.[9] P. Grover and A. Sahai, “Shannon meets Tesla: wireless informationand power transfer,” in
Proc. IEEE Int. Symp. Inf. Theory , Jun. 2010,pp. 2363–2367.[10] S. Y. R. Hui, W. Zhong, and C. K. Lee, “A critical review of recentprogress in mid-range wireless power transfer,”
IEEE Trans. PowerElectron. , vol. 29, no. 9, pp. 4500–4511, Sep. 2014.[11] R. Zhang and C. K. Ho, “MIMO broadcasting for simultaneous wire-less information and power transfer,”
IEEE Trans. Wireless Commun. ,vol. 12, no. 5, pp. 1989–2001, May 2013.[12] S. Lee, R. Zhang, and K. Huang, “Opportunistic wireless energyharvesting in cognitive radio networks,”
IEEE Trans. Wireless Commun. ,vol. 12, no. 9, pp. 4788–4799, Sep. 2013.[13] H. J. Visser and R. J. M. Vullers, “RF energy harvesting and transportfor wireless sensor network applications: principles and requirements,”
Proc. IEEE , vol. 101, no. 6, pp. 1410–1423, Jun. 2013.[14] Z. Xiang and M. Tao, “Robust beamforming for wireless informationand power transmission,”
IEEE Wireless Commun. Lett. , vol. 1, no. 4,pp. 372–375, Aug. 2012.[15] D. W. K. Ng, E. S. Lo, and R. Schober, “Robust beamforming forsecure communication in systems with wireless information and powertransfer,”
IEEE Trans. Wireless Commun. , vol. 13, no. 8, pp. 4599–4615,Aug. 2014.[16] D. W. K. Ng and R. Schober, “Spectral efficient optimization in OFDMsystems with wireless information and power transfer,” in
Proc. 21stEuropean Signal Process. Conf. , Sep. 2013, pp. 1–5. [17] D. W. K. Ng, E. S. Lo, and R. Schober, “Energy-efficient resourceallocation in multiuser OFDM systems with wireless information andpower transfer,” in Proc. IEEE Wireless Commun. and Netw. Conf. , Apr.2013, pp. 3823–3828.[18] ——, “Wireless information and power transfer: energy efficiency opti-mization in OFDMA systems,”
IEEE Trans. Wireless Commun. , vol. 12,no. 12, pp. 6352–6370, Dec. 2013.[19] X. Chen, C. Yuen, and Z. Zhang, “Wireless energy and informationtransfer tradeoff for limited feedback multi-antenna systems with energybeamforming,”
IEEE Trans. Veh. Technol. , vol. 63, no. 1, pp. 407–412,Jan. 2014.[20] G. Yang, C. K. Ho, and Y. L. Guan, “Dynamic resource allocation formultiple-antenna wireless power transfer,”
IEEE Trans. Signal Process. ,vol. 62, no. 14, pp. 3565–3577, Jul. 2014.[21] G. Caire, N. Jindal, M. Kobayashi, and N. Ravindran, “Multiuser MIMOachievable rates with downlink training and channel state feedback,”
IEEE Trans. Inf. Theory , vol. 56, no. 6, pp. 2845–2866, Jun. 2010.[22] J. I. Marcum, “Table of Q functions,”
Project RAND Research Memoran-dum M-339, ASTIA Document AD 1165451, Rand Corporation, SantaMonica, CA , vol. abs/1302.0585, Jan. 1950.[23] W. R. Heinzelman, A. Chandrakasan, and H. Balakrishnan, “Energy-efficient communication protocol for wireless microsensor networks,”in
Proc. 33rd Annu. Hawaii Int. Conf. Syst. Sci. , Jan. 2000, pp. 1–10.[24] S. M. Kay,
Fundamentals of Statistical Signal Processing: EstimationTheory . Prentice Hall PTR, 1993, ch. 3.4, pp. 30–35.[25] R. de Miguel and R. R. Muller, “Vector precoding for a single-userMIMO channel: matched filter vs. distributed antenna detection,” in
Proc. 1st Int. Symp. Appl. Sci. Biomed., and Commun. Tech. , Oct. 2008,pp. 1–4.[26] D. Samardzija and N. Mandayam, “Unquantized and uncoded channelstate information feedback in multiple-antenna multiuser systems,”
IEEETrans. Commun. , vol. 54, no. 7, pp. 1335–1345, Jul. 2006.[27] L. Liu, R. Zhang, and K.-C. Chua, “Wireless information and powertransfer: a dynamic power splitting approach,”
IEEE Trans. Commun. ,vol. 61, no. 9, pp. 3990–4001, Sep. 2013.[28] 3GPP, “TR 36.814, further advancements for E-UTRA physical layeraspects, v.9.0.0,” 3GPP, Tech. Rep., Mar. 2010.[29] D. Tse and P. Viswanath,