On the Beneficial Roles of Fading and Transmit Diversity in Wireless Power Transfer with Nonlinear Energy Harvesting

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(MRT), Transmission Ratio ouae snl-are n ut-are)wvfrsh waveforms multi-carrier) and deter- (single-carrier assumes wave in modulated multi-carrier) commonly unmodulated appeared design (i.e. the further multisine decrea ministic Though have [12]. to complexity [11], not Strategies [9], design does implementation. waveform to which the the easily problem, to itself optimization Due lend non-convex [10]. a resu commonly harvesting from design waveform energy the nonlinearity, nonlinear harvester suit with strategies, [8], acquisition therefore WPT gain and channel CSIT beamforming appropriate on a require trans- rely exploit multi-antenna waveforms additionally Channel-adaptive a to [9]. for as designed so be mitter also can Waveforms oepottenniert fteeeg avse n in- and harvester energy efficiency the conversion of RF-to-DC the nonlinearity crease the exploit to rf A A − rf first second 4.Tesmls emomri bandb Maximum by obtained is beamformer simplest The [4]. taeyi odsg (energy) design to is strategy taeyi odsg (energy) design to is strategy beamforming e rf waveform − e dc rf − 7,[] A [8]. [7], rf × e forms, dfor ed ]. rf oas so oas so egies ently ergy − gies in h ave ap- rm dc nd lts se o d y 1 . 2 also recently appeared, especially in the context of wireless channel. Those fluctuations are shown to boost the RF-to- information and power transfer [13]. DC conversion efficiency and the harvested DC power thanks A third strategy is to design (energy) modulation for single- to the energy harvester nonlinearity. Transmit diversity does carrier transmission so as to exploit the nonlinearity of the not rely on any form of CSIT and its performance analysis energy harvester and increase the RF-to-DC conversion ef- highlights that multiple transmit antennas are useful even in ficiency erf dc [14], [15]. As explained in [13], modulated the absence of CSIT. We also show that transmit diversity can and deterministic− signals do not lead to the same RF-to-DC be combined with the (energy) waveform and modulation to conversion efficiency due to the energy harvester nonlinearity. provide further enhancements. The gain of transmit diversity The design of modulation specifically aimed for WPT has is analyzed using the two aforementioned nonlinear models. recently emerged. It was shown in [14], [15] that a real Third, the analysis and performance benefits are confirmed Gaussian input leads to a higher harvested DC power than a using realistic circuit simulations, prototyping and experimen- circularly symmetric complex Gaussian input. The best known tation. Real-time over-the-air measurements reveal gains of modulation for WPT, developed in [15], exhibits a low proba- about 50% to 100% in harvested DC power with a two- bility of high amplitude signal, reminiscent of flash-signaling. antenna transmit diversity strategy over single-antenna setup, Modulation can also be combined with multi-antenna so as to without any need for CSIT. Transmit diversity is well suited benefit from an additional beamforming gain. for massive IoT for which CSIT acquisition is unpractical. Note that, in contrast with the beamforming strategy, the As a main takeaway observation, results (again) highlight waveform and modulation strategies are deeply rooted in the the crucial role played by the harvester nonlinearity in WPT nonlinearity of the energy harvester. The energy harvester non- signal/system design and demonstrate the previously unknown linearity has now appeared to play a crucial role in WPT signal fact that multiple transmit antennas can be beneficial to WPT and system design but also in various wireless information and even in the absence of CSIT. This opens the door to a re- power transfer signal and system designs [16]. thinking of the role played by multiple antennas in wireless In this paper, in view of the fundamental role played by power-based network design. fading and diversity in wireless communications, and the Organization: Section II introduces the physics-based diode key role played by nonlinearity in WPT, we ask ourselves nonlinear model, Section III analyzes the impact of fading the following questions: Is fading beneficial or detrimental on harvested DC power, Section IV introduces the basic idea to WPT? Can diversity play a role in WPT? Can multiple of transmit diversity for WPT and analyzes its performance, transmit antennas be useful to WPT in the absence of CSIT? Section V shows that transmit diversity can be combined At first, if we consider a naive way of modeling the energy with (energy) waveform and modulation to provide further harvester using the so-called linear model, which assumes a performance benefits, Section VI revisits the impact of fading constant RF-to-DC conversion efficiency independent of the and transmit diversity using a curve fitting-based nonlinear harvester’s input signal (power and shape), maximizing the model, Section VII provides circuit simulation, prototyping output DC power is equivalent to maximizing the average RF and experimentation results, and Section VIII concludes the input power to the energy harvester [4], [8]. Since fading does work and discusses future research directions. not increase the average RF input power, we may be tempted Notations: In this paper, scalars are denoted by italic letters. to answer that fading does not impact the harvested DC power. Boldface lower- and upper-case letters denote vectors and Similarly, since multiple transmit antennas do not increase the matrices, respectively. j denotes the imaginary unit, i.e., j2 = average RF input power in the absence of CSIT, we may 1. E[ ] denotes statistical expectation and represents be tempted to answer that multiple antennas are useless in −the real· part of a complex number. . and ℜ{·}. refer to the the absence of CSIT. Following the same line of thoughts, absolute value of a scalar and the 2-norm| | k ofk a vector. In diversity would be useless either since fading would not have the context of random variables, i.i.d. stands for independent any impact on the harvested DC performance. In this paper, and identically distributed. The distribution of a Circularly we show that, on the contrary, as a consequence of the energy Symmetric Complex Gaussian (CSCG) random variable with harvester nonlinearity, fading is beneficial to the harvested zero-mean and variance σ2 is denoted by (0, σ2); hence DC power, multiple transmit antennas are useful even in the with the real/imaginary part distributed asCN(0, σ2/2). The absence of CSIT and diversity has a role to play in WPT. This distribution of an exponential random variableN with parameter is the first paper to make those observations. Specifically, the λ is denoted by EXPO(λ). stands for “distributed as”. log N ∼ contributions of this paper are summarized as follows. is in base e. ր means approximately equal as N grows large. First, using two different models of the energy harvester ≈ nonlinearity, namely the physics-based diode model of [8] and II. THE NONLINEAR RECTENNA MODEL a curve fitting-based model approach similar to [17]–[19], we We assume the same rectenna model as in [8]. The rectenna study the impact of channel fading on the harvested DC power is made of an antenna and a rectifier. The antenna model and highlight that channel fading is beneficial to WPT. reflects the power transfer from the antenna to the rectifier Second, aside the three aforementioned beamforming, wave- through the matching network. A lossless antenna is modeled form and modulation strategies for WPT, we develop a new 1 as a voltage source vs(t) followed by a series resistance Rant (and therefore a fourth) WPT signal strategy, denoted as (Fig. 1 left hand side). Let Zin = Rin + jXin denote the transmit diversity, that relies on multiple dumb antennas at the transmitter to induce fast fluctuations of the wireless 1Assumed real for simplicity. A more general model can be found in [11]. 3

vd as studied in numerous works [2], [8], [11], [13], [17]. The vout nonlinearity of the rectifier is characterized by the fourth and Rant id iout higher order terms [8]. Truncating the expansion to order 4 C (and therefore accounting for the nonlinearity), we can write vs vin Rin vin RL ~ ~ E 2 2 E 4 iout k0′ + k2′ Rant y(t) + k4′ Rant y(t) . Following [8], i≈ , and therefore P , are directly related to the quantity out dc non-linear low-pass E 2 2 E 4 zdc = k2Rant y(t) + k4Rant y(t) (5) device filter and load is 3 where ki = i , i = 2, 4 . Hence, any increase in zdc Fig. 1. Antenna equivalent circuit (left) and a single diode rectifier (right) i!(nvt) [8]. The rectifier comprises a non-linear device (diode) and a low-pass filter would lead to an increase in iout and Pdc. Metric (5) (and (consisting of a capacitor C and a load RL). the truncation to the fourth order term) has been used and validated by circuit simulations in [8], [11], [13]4. input impedance of the rectifier and the matching network. III. EFFECT OF FADING ON HARVESTED ENERGY

Let y(t) also denote the RF signal impinging on the receive In this section, we study the effect of fading on zdc. To that antenna. Assuming perfect matching (Rin = Rant, Xin =0), end, we consider the simplest scenario of a transmitter with a the available RF power Prf is transferred to the rectifier and single antenna whose transmit signal at time t is given by an 2 2 absorbed by Rin, so that Prf = E y(t) = E vin(t) /Rin, unmodulated continuous-wave vin(t)=vs(t)/2, and vin(t)=y(t)√Rin =y(t)√Rant. We also x(t)= √2P cos(w t)= √2P ejw0t , (6) assume that the antenna noise is too small to be harvested. 0 ℜ We consider for simplicity a rectifier composed of a single with w0 the carrier frequency. The average transmit power 2 diode followed by a low-pass filter with a load (RL), as is fixed to P , namely E x(t) = P . Denoting the wireless illustrated in Fig. 1 (right hand side). We consider this setup channel between the transmit and receive antenna as Λ 1/2h − as it is the simplest rectifier configuration. Nevertheless the with Λ the path loss and h the complex fading coefficient model presented in this subsection is not limited to a single (it varies very slowly over time, hence we omit the time series diode but also holds for more general rectifiers with dependency), the received signal can be written as many diodes as shown in [11]. Denote the voltage drop across √ 1/2 jw0t the diode as v (t)= v (t) v (t) where v (t) is the input y(t)= 2P Λ− he . (7) d in − out in ℜ voltage to the diode and vout(t) is the output voltage across the The RF power level at the energyn harvestero input is given by load resistor. A tractable behavioral diode model is obtained 2 1 2 P = E y(t) =Λ− h P. (8) by Taylor series expansion of the diode characteristic function rf | | vd(t) The average RF received power (average RF input power to i (t)= i e nvt 1 , (1) d s − the energy harvester), where the average is taken over the distribution of , is given by with the reverse bias saturation current is, the thermal voltage h , the ideality factor assumed to be equal to , around 1 vt n 1.05 P¯rf = Eh [Prf ]=Λ− P, (9) a quiescent operating point vd(t)= a. We have where we assume that the fading coefficient h is normalized 2 ∞ i such that E h =1. Hence, fading does not change the RF id(t)= k′ (vd(t) a) , (2) | | i − input power to the energy harvester on average. i=0 X Let us plug (7) in the rectenna model (5). We first consider a a nv nv e t the deterministic case where h does not change in time and where k0′ = is e t 1 and ki′ = is i , i =1,..., . − i!(nvt) ∞ is given by 2 , such that 1 ¯ . Recalling Choosing2 a = E[v (t)] = v , we can write h =1 Prf =Λ− P = Prf d − out the analysis| in| [8], we can write ∞ i ∞ i/2 i 3 i (t)= k′v (t) = k′R y(t) . (3) ¯ 2 ¯2 d i in i ant zdc,cw = k2RantPrf + k4RantPrf . (10) i=0 i=0 2 X X We now consider the same unmodulated continuous-wave The DC current delivered to the load and the harvested DC transmission but over a fading channel, with h modeled as power are then given by CSCG with E h 2 = 1. According to [8], after taking the | | i = E[i (t)], P = i2 R , (4) expectation over the fading process, we have out d dc out L E respectively. Note that the operator E[ ] takes the DC com- z¯dc,cw = h [zdc,cw] , ponent of the diode current i (t) but also· averages over the 3 d = k R P¯ E h 2 + k R2 P¯2 E h 4 , potential randomness carried by the input signal y(t) [13]. 2 ant rf | | 2 4 ant rf | | ¯ 2 ¯2 The classical linear model is obtained by truncating to the = k2RantPrf +3 k4R antPrf (11) second order term. This is however inaccurate and inefficient 3 Assuming is =5µA, a diode ideality factor n=1.05 and vt =25.86mV , 2We here assume a steady-state response and an ideal rectification. The low typical values are given by k2 =0.0034 and k4 =0.3829 [8]. 4 pass filter is ideal such that vout(t) is at constant DC level vout (we drop As discussed in Remark 5 of [8], more investigations are needed to assess the dependency on t) and the output current is also at constant DC level iout. the usefulness of higher order terms (beyond 4). 4 because E h 4 = 2. We note that the second order term has not changed| | between (10) and (11), while the fourth order term in the fading case is twice as large as in the deterministic case5. In other words, fading is beneficial in WPT if the rectifier nonlinearity is modeled. This benefit originates from the rectifier nonlinearity that leads to the convexity of the expression of the diode current (3) with respect to y(t). According to Jensen’s inequality, this con- E 4 E 2 2 ¯2 vexity leads to y(t) ( y(t) ) = Prf in (5), and E h 4 (E h 2 )2 in (11),≥ and implies that fluctuations of the| signal| ≥ due to| | fading at the input of the rectifier boost the harvested energy. This contrasts with communication where Fig. 2. General architecture of transmit diversity for WPT with M dumb fading is known to be detrimental [1]. Indeed, the ergodic antennas. Signal s(t) is an energy modulation/waveform (s(t) = 1 in (12)). capacity of a fast fading channel is lower than the AWGN capacity due to the concavity of the log in the rate expression. where Expression (11) was obtained under the assumption of a M jψm(t) continuous wave (CW) as per (6). Nevertheless, if the CW is h(t)= hme (14) m=1 modulated using energy modulation or replaced by an energy X waveform, the factor of 2 in the fourth order term originating is the effective time-varying channel, whose rate of change is from fading would still appear on top of an additional gain determined by the rate of change of the phases ψm(t) m. originating from the use of energy modulation/waveform. This Fig. 2 illustrates the general architecture of transmit{ diversity.}∀ will be discussed in more details in Section V. In order to get some insights into the performance benefit IV. BASIC TRANSMIT DIVERSITY of the transmit diversity strategy, we assume the phases ψ (t) are uniformly distributed over 2π and are inde- Motivated by the above observation that fading is help- m m pendent.{ }∀ Moreover, we assume7 h =1 m. We then write ful to the harvested energy, we design a transmit diversity m ∀ scheme using dumb antennas to boost the harvested energy of E h(t) 2 3 E h(t) 4 z¯ = k R P¯ | | + k R2 P¯2 | | rectennas that are exposed to slow fading channels. We aim dc,td cw 2 ant rf 4 ant rf 2 − M 2 M at inducing channel fluctuations and creating an effective fast 3 M 1 = k R P¯ + k R2 P¯2 1+ − . (15) fading channel by using multiple transmit antennas. During a 2 ant rf 2 4 ant rf M given harvesting period, the rectenna will therefore see a fast M 1 8 varying channel that will lead to a higher harvested energy. The quantity Gtd = 1+ M− is obtained from computing E h(t) 4 /M 2 over the uniform distribution of the phases A. Phase Sweeping Transmit Diversity | | ψm(t) m. Gtd can be viewed as the diversity gain offered In its simplest form, the proposed transmit diversity for {by the transmit}∀ diversity strategy. It ranges from 1 for M =1 WPT induces fast fading by transmitting a continuous wave on (i.e. no gain) to 2 in the limit of large M (which corresponds to each antenna with an antenna dependent time-varying phase6 the gain offered by CSCG fading in Section III)9, as illustrated ψm(t). Considering M transmit antennas and one receive in Fig. 3. For M = 2 and M = 4, the scheme boosts the antenna, we can write the transmit signal on antenna m as fourth order term by a factor of 1.5 and 1.75, respectively. This is remarkable as it shows that multiple transmit antennas 2P xm(t)= cos(w0t + ψm(t)), are beneficial to WPT even in the absence of CSIT. r M In a multi-user setup, transmit diversity benefits all users by 2P = ej(w0t+ψm(t)) . (12) providing a diversity gain to all of them simultaneously, i.e., r M ℜ adding more energy receivers comes for free and does not The average total transmit powern is kept fixedo to P . Denoting compromise each user’s performance. This behavior was also the wireless channel between antenna m and the receive observed with energy waveforms in frequency flat channels[8]. 1/2 antenna as Λ− hm (it varies slowly over time, hence we It is important to recall that the transmit diversity gain is omit the time dependency), the received signal is written as deeply rooted in the rectifier nonlinearity. If the nonlinearity modeled by the fourth order term is ignored, the transmit 2P 1/2 jw t y(t)= Λ− h(t)e 0 , (13) diversity gain disappears. With the linear model, there is no M ℜ r n o benefit in using transmit diversity since the second order 5Higher order terms in the Taylor expansion would lead to terms propor- 7 tional to E |h|i, i = 6, 8,.... Since the moments of Gaussian distribution This assumption could be experienced in strong LoS situations for instance are unbounded as the exponent goes to infinity, the harvested energy would and is made for simplicity in the analysis and circuit simulations to highlight scale to infinity if we account for all terms in the Taylor expansion. However, the role of the antenna dependent time-varying phases of transmit diversity recall that the assumption behind the rectenna model is that the diode operates to induce fast fluctuations of a slow-varying channel. in the nonlinear region, as detailed in Section III-B and Remark 4 of [8]. 8The derivations are pretty straightforward and similar to the ones used to 6We here assume for simplicity that the transmit power is uniformly spread compute the scaling laws in [8]. They are therefore not provided here. across antennas and the phases only are time-varying. We could also imagine 9The diversity gain is larger when the channels on each antenna have similar a strategy where both the amplitude and the phase on each antenna vary over strengths (e.g. hm = 1 ∀m). If there is a severe imbalance between the time. Note also that ψ1(t) can be set to 0 without loss of generality. channel magnitudes |hm|, transmit diversity will have less benefit. 5

2 fading in communications and in WPT, as well as contrast 1.9 the proposed transmit diversity for WPT with various forms 1.8 of diversity strategies used in communications. 1.7 1) Fading: Channel fading in point-to-point (resp. multi- 1.6 user) scenario is commonly viewed as a source of unreliability td 1.5 G (resp. randomization) in communications that has to be mit- 1.4 igated (resp. exploited) [20]. In contrast, channel fading for 1.3 both point-to-point and multi-user scenarios can be viewed as 1.2 a source of randomization in WPT that has to be exploited. 1.1

1 2) Time Diversity: In [21], a phase-rotated/sweeping trans- 100 101 102 mit diversity strategy was proposed for point-to-point trans- Number of transmit antennas M mission so as to convert spatial diversity into time diversity.

Fig. 3. Transmit diversity gain Gtd as a function of M. The effective SISO channel resulting from the addition of multiple branches seen by the receiver fades over time, and term in (15) is constant irrespective of M in the absence of this selective fading can be exploited by channel coding. In the CSIT. In other words, the second order term can only by proposed transmit diversity for WPT, there is no information increased by the use of beamforming, which requires CSIT. transmission and therefore no channel coding. Time-selective The fourth order term, on the other hand, can be enhanced by fading is also induced by a phase rotation/sweeping, but it is beamforming, waveform, modulation and transmit diversity, exploited by the rectifier rather than channel coding. with the last three schemes applicable in the absence of CSIT10. 3) Multi-User Diversity: In [20], [22], multi-user diversity Transmit diversity has a number of practical benefits leading was introduced as a new form of diversity at the system level to low cost deployments. First, it relies on dumb antennas fed that leverages the independent time-varying channels across with a low Peak-to-Average Power Ratio (PAPR) continuous the different users. By tracking the channel fluctuations using wave at a single carrier frequency w0. Second, it does not a limited channel feedback and selecting the best user at a time require any form of synchronization among transmit antennas (typically when its instantaneous channel is near its peak), since a time-varying phase is applied to each of them. Third, a multi-user diversity gain is exploited that translates into it can be applied to co-located and distributed antenna deploy- a sum-rate scaling double logarithmically with the number ments. Fourth, it is transparent to the energy receivers, which of users at high SNR. By increasing the number of users, eases the system implementation. Fifth, for deployments with the sum-rate increases, though each user gets access to a a massive number of devices, it provides a transmit diversity smaller portion of the resources. Such a multi-user diversity gain simultaneously to all users without any need for CSIT. strategy can also be applied in multi-user WPT in the same Remark 1: It is worth contrasting with (energy) beamform- way as communications, as investigated in [23]. The proposed ing, waveform and modulation. Note that the transmit diversity transmit diversity however differs from multi-user diversity as gain Gtd is obtained without any CSIT. This contrasts with a it does not rely on independent channels across users and does (energy) beamforming approach for WPT that always relies not require any form of tracking and scheduling. Moreover, on CSIT in order to beam energy in the channel direction. all users do not have access to a portion of the resources but The philosophy behind transmit diversity is also different from rather benefit simultaneously from the channel fluctuations. energy waveform [8] and energy modulation [15] that transmit Therefore adding more users comes for free and does not specific signal with suitable characteristics. Energy waveform compromise each user’s harvested DC power. in [8] relies on deterministic multisine at the transmitter whose The proposed transmit diversity for WPT also has some amplitudes and phases are carefully selected (as a function similarities with multi-user diversity. Indeed, the multi-user of the wireless channel in general settings) so as to excite diversity gain is known to increase with the dynamic range the rectifier in a periodic manner by sending high energy of the fluctuations. This may naturally occur in fast fading pulses. Energy modulation in [13]–[15] relies on single-carrier conditions. However, in deployments subject to Line-of-Sight transmission but whose input follows a probability distribution (LoS) and/or slow fading, multiple transmit antennas with that boosts the fourth order term in the Taylor expansion of time-varying phases and magnitudes can be used to induce the diode. One of such distribution, proposed in [15] and large and fast channel fluctuations, which are then exploited reminiscent of flash signaling, exhibits a low probability of by channel tracking and scheduling [20]. Similarly, in the pro- high amplitude transmit signals. In contrast, transmit diversity posed transmit diversity for WPT, large channel fluctuations is designed to generate amplitude fluctuations of the wireless boost the harvested DC power and multiple transmit antennas channel, not amplitude fluctuations of the transmit signal. with time-varying phases (and potentially magnitudes) are used to generate those channel fluctuations. B. Comparisons with Fading/Diversity in Communications The notion of fading and diversity has a long history in V. TRANSMIT DIVERSITY WITH ENERGY communications. It is interesting to compare the effect of MODULATION/WAVEFORM

10Extra performance benefits are nevertheless obtained with CSIT based- Energy modulation and energy waveform are two different waveform design in frequency selective channels, as detailed in [8]. techniques to also boost the amount of DC power at the 6 output of the rectenna. Similarly to transmit diversity, they both exploit the nonlinearity of the energy harvester. However, in contrast to transmit diversity, both create large amplitude fluctuations of the transmit signal. Energy waveform in [7], [8] relies on deterministic multisine at the transmitter so as to excite the rectifier in a periodic manner by sending high energy pulses. Energy modulation in [13]–[15] relies on single-carrier transmission with inputs following a probability distribution that boosts the fourth order term in the diode Taylor expansion. Fig. 4. Rectenna used for circuit simulations. In this section, we show that transmit diversity can be 2 combined with energy modulation/waveform so as to jointly (0, 1) and Gmod = l for the flash signaling distribution N jθ generate amplitude fluctuations of the wireless channel and of [15] characterized by s = re with the phase θ uniformly amplitude fluctuations of the transmit signal. This combined distributed over [0, 2π) and the amplitude distributed according to the following probability mass function (with l 1) effect leads to a joint diversity and modulation/waveform ≥ gain and can be leveraged in practice to balance the cost 1 1 l2 , r =0, of the power amplifiers (PA) versus the number of antennas. pr(r)= − (19) 1 , r = l. Indeed, one extreme consists in using one single transmit ( l2 antenna to generate energy waveform/modulation. That would Hence, transmit diversity with energy modulation brings a require an expensive PA to cope with the high PAPR transmit total gain in the fourth order term equal to the product of the signal and would not benefit from a transmit diversity gain. diversity gain Gtd and the modulation gain Gmod. Another extreme would consist in using transmit diversity with multiple dumb antennas with cheap PAs and low PAPR B. Transmit Diversity with Energy Waveform continuous wave signal. That would not benefit from a modulation/waveform gain. In between those two extremes, we Instead of using an energy modulated single-carrier signal, can use of mixture of transmit diversity and energy modula- we can use a deterministic in-phase multisine waveform with 11 tion/waveform and partially trade the transmit diversity gain uniform power allocation on each antenna . In such a case, with the modulation/waveform gain so as to balance the cost (16) still holds but s(t) is replaced by the baseband multisine N 1 1 waveform s(t) = − ejn∆wt with ∆ = 2π∆ the of the PAs and the number of transmit antennas. n=0 √N w f inter-carrier frequency spacing. Assuming a frequency-flat channel, the receivedP signal writes as in (17). Following [8], A. Transmit Diversity with Energy Modulation we can then compute Rather than transmitting an unmodulated continuous wave ¯ 3 2 ¯2 with a time-varying phase on each antenna, we can modulate z¯dc,td wf = k2RantPrf + k4RantPrf GtdGwf (20) the carrier using a suitable input distribution. The transmit − 2 2 N signal on antenna m can be written as 2N +1 ր 2N where Gwf = 3N 3 . Hence, transmit diversity≈ with energy waveform brings a 2P j(w0t+ψm(t)) xm(t)= s(t)e (16) total gain in the fourth order term equal to the product of the r M ℜ n o diversity gain Gtd and the waveform gain Gwf . where s(t) is the complex energy symbol drawn from a given distribution with E s(t) 2 = 1 and transmitted on VI. PERFORMANCE ANALYSIS WITH CURVE all antennas. s(t) can be drawn| | for instance from a CSCG FITTING-BASED NONLINEAR RECTENNA MODEL distribution, real distribution or any other distribution with high energy content [15]. The received signal becomes The previous sections were based on a diode nonlinear model driven by the physics of the rectenna. We here study 2P 1/2 jw t another rectenna nonlinear model that is based on fitting data y(t)= Λ− h(t)s(t)e 0 , (17) M ℜ using a polynomial. We use this model to assess the impact r n o with h(t) given in (14). In order to analyze the performance of fading on the harvested DC power and the potential gain of transmit diversity. Observations made in this section confirm gain, let us again assume hm = 1. Following [13], we can then compute those made in previous sections. Hence, despite the numerous differences between the physics-based model and the curve 2 2 z¯dc,td mod = k2RantP¯rf E h(t) E s(t) fitting-based model [13], [16], both approaches lead to the − | | | | 3 same conclusion that fading and transmit diversity enhance + k R2 P¯2 E h(t) 4 E s(t) 4 2 4 ant rf | | | | the RF-to-DC conversion efficiency. This confirms that the 3 benefits of fading and transmit diversity for WPT originate = k R P¯ + k R2 P¯2 G G . (18) 2 ant rf 2 4 ant rf td mod from the nonlinearity of the harvester and are not an artefact M 1 4 of the underlying nonlinear model. where G = 1+ − and G = E s(t) . G td M mod | | mod varies depending on the input distribution, namely G =2 11 mod Recall from [8] that allocating power uniformly across all sinewaves leads for a CSCG input s(t) (0, 1), G = 3 for s(t) to performance very close to the optimum in frequency flat channels. ∼ CN mod ∼ 7

A. A Curve Fitting-based Nonlinear Rectenna Model B. Effect of Fading on Harvested Energy Since a curve fitting-based model requires actual data, we Similarly to Section III, we here assess the impact of fading designed, optimized and simulated the rectenna circuit of Fig. on harvested DC power. We assume that the fading coefficient 4. We used a conventional single series rectifier circuit that h is modeled as CSCG with E h 2 = 1. With a fading | | consists of a rectifying diode, impedance matching circuit, channel, Prf fluctuates as a function of h around its mean ¯ and low pass filter. The Schottky diode Skyworks SMS7630 Prf . As a consequence, Pdc also fluctuates as a function of h. [24] is chosen for the rectifying diode because it requires For a given realization of h, Prf relates to Pdc through (21). ¯ E low biasing voltage level, which is suitable for low power Defining the average DC power as Pdc = h [Pdc], with the rectifier. The impedance matching and low pass filter circuits expectation taken over the channel distribution, and making 2 ¯ are designed for an in-phase 4-tone ( ) multisine input use of the nonlinear model (21) with Prf = h Prf , we write N = 4 | | signal centered around 2.45GHz with an RF input power of 2 ¯ 2 2 ¯ Pdc,dB a(log( h Prf )) +b log( h Prf )+c P¯ = E e = E e | | | | , -20dBm and with 2.5MHz inter-carrier frequency spacing. dc h h The load impedance R2 is chosen as 10KΩ in order to = Pdc,no fading efadingh (25)i reach maximum RF-to-DC conversion efficiency with the 4- where tone multisine waveform. The matching network capacitor C1, 2 aP¯ +bP¯rf,dB+c inductor L1 and output capacitor C2 values are optimized Pdc,no fading = e rf,dB , (26) (using an iterative process) to maximize the output DC power 2 2 ¯ 2 E a(log( h )) +(2aPrf,dB+b) log( h ) under a given load impedance and for the given multisine input efading = h e | | | | . (27) waveform at -20dBm RF input power. The chosen values are h i According to (25), the average harvested DC power P¯dc with given by 0.4pF for C1, 8.8nH for L1, and 1nF for C2. The fading can be written as the product of the harvested DC power antenna impedance is set as R1 = R = 50Ω and the voltage ant P without fading and a fading gain e that source is expressed as √ . dc,no fading fading V1 V1 = vs(t)=2y(t) Rant characterizes the impact of channel fading on the harvested Using PSpice circuit simulations, we evaluate in Fig. 5 and DC power. Interestingly, (25) can be equivalently written as Fig. 6 the DC power Pdc harvested at the load as a function of Pdc,no fading the RF input power Prf for a continuous wave and multisine (N =8) input RF signals, respectively. Based on the collected P¯dc = P¯rf erf dc efading, data, we can then choose a model that fits the data. We fit the − = Pz¯rf }|erf dc{ efading, (28) data (in a least-squares sense) using a polynomial as follows − e¯rf−dc 2 Pdc,dB = aP + bPrf,dB + c (21) rf,dB where erf dc = Pdc,no fading| /P¯rf{zis the} conventional RF- to-DC conversion− efficiency (with e.g. continuous wave or with multisine excitation) in the absence of fading and e¯rf dc = (22) − Pdc,dB = log (Pdc) , P¯dc/P¯rf is the RF-to-DC conversion efficiency in the presence P = log (P ) = log h 2 + P¯ , (23) of fading. Interestingly, the RF-to-DC conversion efficiency in rf,dB rf | | rf,dB the presence of fading e¯rf dc can be split into two contri- P¯rf,dB = log P¯rf . (24) butions: the conventional RF-to-DC− conversion efficiency in The coefficients a, b and c of the polynomials are given for the absence of fading erf dc and a fading gain efading that − two different RF input power ranges in the legends of Fig. accounts for the impact of fading on the harvester nonlinearity. ¯ 2 5 and Fig. 6, for continuous wave (N = 1) and multisine Defining d = 2aPrf,dB + b and X = h , and noting that | | (N =8) excitations, respectively. Note that, similarly to [19], X EXPO(1), the fading gain efading can be expressed as ∼ the fitting is performed in the log-log scale instead of using 2 E a(log(X)) +d log(X) a linear scale. Using a sigmoid or a polynomial to fit data in efading = X e , the linear scale, as in [17] and [18] respectively, may lead to h d+a log(X) i = EX X , severe inaccuracies in the low power regime as highlighted in [16]. We note that a second degree polynomial fits the data ∞h d+a log(x) ix = x e− dx. (29) relatively well, especially whenever fitted to the low input 0 12 Z power regime -40 dBm to -5 dBm, as per subfigures (b). Numerical integration can be used to compute the last integral. Subfigures (c) illustrate the RF-to-DC conversion efficiency. The fading gain efading is illustrated in Fig. 7 for continuous Second degree fitting in the low power regime (-40 dBm to -5 wave (’Continuous wave (N=1)’) and multisine (’Multisine dBm) appears relatively accurate, though RF-to-DC conversion (N=8)’) excitations using the fitting parameters taken from efficiency is in general sensitive to any discrepancy in the Fig. 5 and Fig. 6, respectively. It appears that efading is larger curve fitting since it describes the ratio between Pdc and Prf . than one for most input power, especially in the low input We also note that multisine excitation leads to higher harvested power regime, suggesting that fading is beneficial to energy DC power and RF-to-DC conversion efficiency compared to harvesting. This confirms observations made in Section III. continuous wave [7], [8], [11]. The gain can be explained again by the convexity of the Pdc- 12We are interested in the non-diode breakdown region, i.e. below -5dBm Prf relationship in the low power regime. The fading gain leads for the considered circuit, for the reasons discussed in Remark 5 of [13]. to an increase of the RF-to-DC conversion efficiency in the low 8

10-3

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diode breakdown region diode breakdown region 10-6 10-6

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Circuit simulations -9 Circuit simulations Curve fitting polynomial degree 1 10 Curve fitting polynomial degree 1 Curve fitting polynomial degree 2 Curve fitting polynomial degree 2 -10 10 10-10 -7 -6 -5 -4 -3 -2 -1 10 10 10 10 10 10 10 10-7 10-6 10-5 10-4 10-3 10-2 10-1 RF input power P [W] RF input power P [W] rf rf (a) Curve ﬁtting for input RF power range -40dBm to 20dBm. Second (a) Curve ﬁtting for input RF power range -40dBm to 20dBm. Second degree polynomial: a = −0.0977, b = −0.9151, c = −11.1684. First degree polynomial: a = −0.1089, b = −1.1660, c = −12.0041. First degree polynomial character: a = 0, b = 0.8848, c = −4.6926. degree polynomial: a = 0, b = 0.8396, c = −4.7881.

10-3 10-3

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-9 Circuit simulations -9 Circuit simulations 10 Curve fitting polynomial degree 1 10 Curve fitting polynomial degree 1 Curve fitting polynomial degree 2 Curve fitting polynomial degree 2 10-10 10-10 10-7 10-6 10-5 10-4 10-3 10-7 10-6 10-5 10-4 10-3 RF input power P [W] RF input power P [W] rf rf (b) Curve ﬁtting for input RF power range -40dBm to -5dBm. Second (b) Curve ﬁtting for input RF power range -40dBm to -5dBm. Second degree polynomial: a = −0.0669, b = −0.1317, c = −6.3801. First degree polynomial: a = −0.1105, b = −1.1468, c = −11.4342. First degree polynomial: a = 0, b = 1.4848, c = 2.9250. degree polynomial: a = 0, b = 1.5239, c = 3.9400.

0.3 0.6 Circuit simulations Circuit simulations Curve fitting (degree 2) [-40 dBm to -5 dBm] Curve fitting (degree 2) [-40 dBm to -5 dBm] 0.25 Curve fitting (degree 2) [-40 dBm to 20 dBm] 0.5 Curve fitting (degree 2) [-40 dBm to 20 dBm]

0.2 0.4

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0 0 10-7 10-6 10-5 10-4 10-3 10-2 10-1 10-7 10-6 10-5 10-4 10-3 10-2 10-1 RF input power P [W] RF input power P [W] rf rf

Fig. 5. Circuit evaluations of harvested DC power Pdc versus RF input power Fig. 6. Circuit evaluations of harvested DC power Pdc versus RF input Prf for a continuous wave (N = 1) excitation and corresponding curve fitting power Prf for a multisine (N = 8) excitation and corresponding curve fitting model for the rectenna in Fig. 4. model for the rectenna in Fig. 4. input power regime as illustrated in Fig. 8. We also note in Remark 2: Circuit simulations have not been conducted Fig. 7 that the fading gain with a multisine is smaller than with below -40dBm because such a power level is likely too low a continuous wave for high RF input power levels and larger to be harvested due to the limited sensitivity of the rectifier. for low RF input power levels. The behavior is again observed Indeed, we note in Fig. 5(c) and 6(c) that the RF-to-DC because of the convexity at low input power and concavity at conversion efficiency is close to 0 at around -40dBm. Though high input powers of the Pdc-Prf relationship. As it appears the polynomial-based fitting (21) was done using the available from Fig. 8, erf dc without fading increases more rapidly with data in the range -40dBm to 20dBm, we assumed that the same a multisine than− with a continuous wave at low input power, polynomial can be used for power levels below -40dBm to but decreases more rapidly at high input power levels. compute the fading gain in (29) (and transmit diversity gain 9

2.6 1.8 Continuous wave (N=1) Continuous wave (N=1) 2.4 1.7 Continuous wave (N=1) with sensitivity (P =10-7) Multisine (N=8) rf,min 2.2 Continuous wave (N=1) with sensitivity (P =3*10-7) 1.6 rf,min 2 Multisine (N=8) 1.5 Multisine (N=8) with sensitivity (P =10-7) rf,min 1.8 1.4 Multisine (N=8) with sensitivity (P =3*10-7) rf,min

1.6 td 1.3 e fading e 1.4 1.2

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Fig. 7. Modeled fading gain efading (with and without rectifier sensitivity) Fig. 9. Modeled transmit diversity gain etd with a continuous wave (N = 1) with a continuous wave (N = 1) and multisine (N = 8) excitation for an and multisine (N = 8) excitation for an average RF input power P¯rf ranging average RF input power P¯rf ranging from -40dBm to -5dBm. Second degree from -40dBm to -5dBm. Second degree polynomial fitting parameters taken polynomial fitting parameters taken from Fig. 5(b), 6(b). from Fig. 5(b), 6(b).

0.4 0.4 without fading - CW (N=1) 1 antenna - CW (N=1) 0.35 with fading - CW (N=1) 0.35 2 antennas - TD - CW (N=1) without fading - multisine (N=8) 1 antenna - multisine (N=8) with fading - multisine (N=8) 2 antennas - TD - multisine (N=8) 0.3 0.3

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Fig. 8. Modeled RF-to-DC conversion efficiency without fading erf−dc and Fig. 10. Modeled RF-to-DC conversion efficiency with one transmit antenna with fading e¯rf−dc for a continuous wave (N = 1) and multisine (N = 8) without fading erf−dc and with two antenna-transmit diversity erf−dc,td for excitation in the average RF input power P¯rf range of -40dBm to -5dBm. a continuous wave (N = 1) and multisine (N = 8) in the average RF input Second degree polynomial fitting parameters taken from Fig. 5(b), 6(b). power P¯rf range of -40dBm to -5dBm. Second degree polynomial fitting parameters taken from Fig. 5(b), 6(b). in (33)). As in [19], [25], we can improve the analysis by jψ (t) accounting for the limited sensitivity of the rectifier in the 2π. The effective channel can be written as h(t)= e 1 + jψ2(t) 2 very low power regime by stating e and h(t) = 2(1+cos(ψ1(t) ψ2(t))). Defining the random| variable| U = ψ (t) ψ −(t), also uniformly 1 − 2 (21), Prf Prf,min, distributed over 2π, and equally splitting the transmit power P Pdc,dB = ≥ (30) (0, otherwise, among the two antennas (P/2 on each antenna), the effect of transmit diversity on the harvested DC power can be computed where refers to the sensitivity threshold below which Prf,min following the same steps as in Subsection VI-B by replacing the rectifier does not turn on. Doing so, the fading gain in (29) h 2 with 1+cos(U). Following (28), we therefore write can be adjusted as follows | | Pdc,no fading ∞ d+a log(x) x efading = x e− dx, (31) P¯dc,td = P¯rf erf dc etd, Zxmin − = Pz¯rf }|erf dc{ etd (32) where xmin = Prf,min/P¯rf . As illustrated in Fig. 7, when − the rectifier sensitivity is taken into account, the fading gain erf−dc,td decreases in the very low power regime as Prf,min increases. where the transmit diversity gain| etd{zis obtained} by replacing X in (29) by 1+cos(U), such that

C. Effect of Transmit Diversity on Harvested Energy 2 e = E ea(log(1+cos(U))) +d log(1+cos(U)) , Similarly to Subsection IV-A, we here assess the beneﬁts of td U h d+a log(1+cos(U)) i transmit diversity on the harvested DC power with the curve = EU (1+cos(U)) , ﬁtting-based nonlinear model. Let us consider the simplest 2π h d+a log(1+cos(u)) i1 case of two transmit antennas (M =2) with hm =1 and the = (1+cos(u)) du. (33) 2π phases ψ1 and ψ2 independent and uniformly distributed over Z0 10

1.35 0.55 Continuous wave (N=1) 1 antenna - CW (N=1) 0.5 1.3 Multisine (N=8) 2 antennas - TD - CW (N=1) 0.45 1 antenna - multisine (N=8) 1.25 2 antennas - TD - multisine (N=8) 0.4 1.2 0.35

1.15 0.3 td e 1.1 0.25

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0.9 0 10-7 10-6 10-5 10-4 10-3 10-2 10-1 10-7 10-6 10-5 10-4 10-3 10-2 10-1 Average RF input power [W] Average RF input power [W]

Fig. 11. Simulated transmit diversity gain etd with a continuous wave (N = Fig. 12. Simulated RF-to-DC conversion efﬁciency with one transmit antenna 1) and multisine (N = 8) for an average RF input power P¯rf ranging from without fading erf−dc and with two antenna-transmit diversity erf−dc,td for -40dBm to 20dBm. a continuous wave (N = 1) and multisine (N = 8) in the average RF input power P¯rf range of -40dBm to 20dBm.

Numerical integration can be used to compute the last integral. than 150 times in order to average out over a sufficient number The transmit diversity gain etd is illustrated in Fig. 9 for continuous wave and multisine excitations using the fitting pa- of realizations of the phase. rameters taken from Fig. 5 and Fig. 6, respectively. It appears We chose a transmit diversity phase change rate of 2.5 MHz though we also considered several phase change rates between that etd is larger than one for most input power, especially in the low input power regime, suggesting that transmit diversity 1 MHz and 10 MHz. The periods of those phase changes range is beneficial to energy harvesting. This is also inline with from 0.1 µs to 1 µs. As stated in Subsection VI-A, the rectenna observations made in Subsection IV-A and Section V. The has been optimized for a 4-tone multisine input waveform with an inter-carrier frequency spacing MHz. Such a transmit diversity gain etd leads to an RF-to-DC conversion ∆f = 2.5 waveform generates large peaks with a periodicity of efficiency with transmit diversity erf dc,td larger than the RF- 1/∆f = to-DC conversion efficiency achieved− with a single transmit 0.4 µs. The RC constant of the low pass filter is 10KΩ 1nF = . The period of the phase change is therefore smaller× than antenna erf dc, as illustrated in Fig. 10. Following Remark 2, 10µs accounting− for the limited rectifier sensitivity would lead to a the rectenna’s RC constant. We did not observe significant performance gap between different phase change rates. We decrease of etd in the very low power regime. set the phase change rate to 2.5 MHz as mentioned above because it is applicable for the simulations and the prototype. VII. PERFORMANCE EVALUATIONS Fig. 11 illustrates the simulated gain of transmit diversity Previous sections highlighted the benefit of transmit diver- with two transmit antennas (M = 2). Hence, in Fig. 9 and sity in WPT with both continuous wave or more complicated Fig. 11, etd refers to the transmit diversity gain obtained modulation/waveform. The gains were predicted from the from analysis/modeling and simulations, respectively. Fig. 12 analysis using two different nonlinear models of the energy displays the simulated RF-to-DC conversion efficiency (in harvester. We here evaluate the performance benefits of trans- contrast to the modeled one in Fig. 10) with and without mit diversity using circuit simulations and real-time over-the- transmit diversity. A performance gain of transmit diversity air experimentation, and contrast the results with the analysis. with a continuous wave input is observed in circuit simulations, therefore validating the prediction from the analysis. Though a transmit diversity gain with a continuous wave is A. Circuit Simulations observed in both analysis (Fig. 9) and simulations (Fig. 11), We consider a two-antenna (M = 2) transmit diversity the general behavior is not the same in both figures. Indeed, system over a deterministic channel with hm = 1 (similarly while the analysis highlights a decreasing gain as the input to the analysis). Using Matlab, we generate the input voltage power level increases, the simulations display larger gains source V1 to the realistic rectenna of Fig. 4. This is done by in the practical RF input power level range of -20dBm to - generating the transmit signal s(t), using either a continuous 10dBm. Recall that the rectenna of Fig. 4 was designed for an wave (CW) or multisine (N =8), passing it through the two- RF input power of -20dBm and that the limited sensitivity antenna transmit diversity strategy whose phase is randomly of the rectifier was not modeled in the very low power swept at a 2.5 MHz rate, and converting the received signal regime in Fig. 9, which partly explain why the simulated y(t) (assuming hm = 1) to the input voltage source V1. We transmit diversity gain is higher (than the modeled one) at then feed the input voltage source V1 into the PSpice circuit -20dBm RF input power level and lower at -40dBm. The simulator. The duration of the generated signal was determined circuit simulations suggest that the limited rectifier sensitivity to be long enough (about 70 µs) for the rectifier to be in a is indeed observed at very low power levels since the simulated steady state response mode. For each input power level ranging transmit diversity gain displays a severe drop similar to the 7 from -40dBm to 20dBm, we repeat the same process for more one illustrated in Fig. 7 (for Prf,min = 3e− ). On the other 11

0.3pF SMS7630

2.4nH 1nF 10㏀

Fig. 13. Rectenna used in the experiment. hand, transmit diversity appears less beneficial when used Fig. 14. Experimental setup with the SDR (PXIe-7966R and NI 5791R) implementing the transmit diversity strategy, the two PAs linked to Tx antenna in conjunction with multisine waveform. Recall from the 1 and 2, and the Rx antenna connected to the rectifier. analysis in subsection V-B and from Figs. 9 and 10 that transmit diversity was expected to provide an additional gain on top of the multisine waveform gain. It seems from circuit but different Tx-Rx distances of 2.5m and 5m. The received simulations that this additional gain is rather negligible. This DC voltage measurement is carried out five times, and each is partially due to the rectifier design. Since the rectifier used measurement, non-continuous in time, is one minute long. in the simulations has been optimized for a 4-tone input Final results are averaged value of 5 minutes total for each signal, its has a limited ability to exploit the large fluctuations case. Fig. 14 is a picture of the overall experimental setup. induced by transmit diversity with an 8-tone waveform. If Fig. 15 illustrates the harvested DC power measured at the we could adjust the rectifier (and its load) as a function load of the rectifier for various transmit strategies, namely of the input signal, we could see more gain of combining a single antenna with continuous wave (CW), a two-antenna multisine and transmit diversity. Nevertheless, it is somewhat transmit diversity (TD) with continuous wave, a single antenna intuitive that, in practice, we observe a diminishing return of multisine (N = 8) and a two-antenna transmit diversity with combining transmit diversity and multisine waveform. Indeed, multisine (N = 8). Two different distances of 2.5 m and 5 the multisine waveform already significantly boosts the RF-to- m from the transmitter are considered, corresponding to a DC conversion efficiency as evidenced by Fig. 12, therefore relatively high and low input power to the energy harvester, leaving less room for improvement for transmit diversity. respectively. We observe that measurements are inline with theory and circuit simulations, with transmit diversity provid- B. Prototyping and Experimentation ing gain with both continuous wave and multisine waveforms, Leveraging the early WPT prototype reported in [26], we though larger with continuous wave and at a moderate distance have implemented a two-antenna transmit diversity prototype of 2.5 m. In Fig. 16, we plot the same results but in a different to verify that transmit diversity provides in real world condi- way to highlight the relative gain over the DC power harvested tions the performance benefits predicted from the analysis and with the single antenna continuous wave. TD with CW offers simulation results. The prototype is divided into two parts, a gain over CW of about 8% and 45%, at 5 m and 2.5 m, Tx and Rx, respectively. Rx part is a simple rectenna. The respectively. By combining TD with multisine, a combined rectifier printed circuit board (PCB) is fabricated with a λ/4 transmit diversity and multisine (N = 8) waveform gain is length of microstrip, 1005 (1.0 mm x 0.5 mm lumped element observed of about 58% and 105% over a single antenna CW. package size) passive elements pads for π-matching network, Recall that those gains are obtained without any need for and followed by a Schottky diode rectifier circuit. The same CSIT and that, in a multi-user deployment, each device would diode and low pass filter circuit as the rectenna used for the benefit from this performance enhancement, irrespectively of circuit simulations in Fig. 4 are implemented in the prototype. the number of devices. It is also worth reminding that the The values of the matching network components are modified prototype relies on two transmit antennas. The gain grows with to fit the real fabricated PCB. We use universal 2.4 GHz band the number of antennas, as discussed in Subsection IV-A. WiFi antennas with 3dBi gain. Fig. 13 illustrates the circuit diagram of the prototype rectenna used in the experiment. VIII. CONCLUSIONS AND FUTURE WORKS Tx part is an RF signal generator with 2.45GHz center frequency implemented using a software defined radio (SDR) This work sheds some new light on the role of fading and relying on NI FlexRIO PXIe-7966R and NI 5791R transceiver diversity in WPT. Thanks to the energy harvester nonlinearity, module. Tx signal generator continuously generates the se- fading and diversity are found beneficial to boost the RF-to- lected signal type such as CW (continuous wave), multisine DC conversion efficiency. We developed a new form of signal (N = 8), TD-CW (transmit diversity with CW), and TD- design for WPT relying on phase sweeping transmit diversity. multisine (transmit diversity with multisine). The experiment is Transmit diversity relies on multiple antennas to induce fast carried out at two different locations. The Rx antenna locations fluctuations of the wireless channel and does not rely on any are fixed and Tx antennas are moved to the test locations. form of CSIT. The effect of fading and transmit diversity have Both locations exhibit line-of-sight (LoS) between Tx and Rx, been analyzed using two different types of energy harvester 12

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ACKNOWLEDGMENTS ergy harvesting models in wireless information and power transfer,” We thank B. Lavasani and National Instruments for provid- https://arxiv.org/pdf/1802.09994 [26] J. Kim, B. Clerckx, and P. D. Mitcheson, “Prototyping and Experi- ing some of the equipments needed to conduct the experiment. mentation of a Closed-Loop Wireless Power Transmission with Channel REFERENCES Acquisition and Waveform Optimization,” IEEE WPTC 2017. [1] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge, U.K.: Cambridge Univ. Press, 2005. 13

Bruno Clerckx is a Reader in the Electrical and Electronic Engineering Department at Imperial Col- lege London. He received his PhD degrees from Uni- versité catholique de Louvain (Louvain-la-Neuve, PLACE Belgium) in 2005, and worked for Samsung Elec- PHOTO tronics (Suwon, Korea) from 2006 to 2011. His HERE research area is communication theory and signal processing for wireless networks.

Junghoon Kim received his BSc in electronic and electrical engineering from Hongik University, Seoul, Korea, in 2008 and MSc in Telecommunica- tions from University College London, UK, in 2015. PLACE From 2008 to 2014, He worked for Samsung Elec- PHOTO tronics, Suwon, Korea. He is currently a PhD student HERE at Imperial College London. His research interests are RF energy harvesting and wireless information and power transfer.