Wireless Information and Power Transfer: Nonlinearity, Waveform Design and Rate-Energy Tradeoff

Transcription

1 1 Wireless Iformatio ad Power Trasfer: Noliearity, Waveform Desig ad Rate-Eergy Tradeoff Bruo Clerckx arxiv: v3 [cs.it 0 Sep 017 Abstract The desig of Wireless Iformatio ad Power Trasfer (WIPT) has so far relied o a oversimplified ad iaccurate liear model of the eergy harvester. I this paper, we depart from this liear model ad desig WIPT cosiderig the rectifier oliearity. We develop a tractable model of the rectifier oliearity that is flexible eough to cope with geeral multicarrier modulated iput waveforms. Leveragig that model, we motivate ad itroduce a ovel WIPT architecture relyig o the superpositio of multi-carrier umodulated ad modulated waveforms at the trasmitter. The superposed WIPT waveforms are optimized as a fuctio of the chael state iformatio so as to characterize the rate-eergy regio of the whole system. Aalysis ad umerical results illustrate the performace of the derived waveforms ad WIPT architecture ad highlight that oliearity radically chages the desig of WIPT. We make key ad refreshig observatios. First, aalysis (cofirmed by circuit simulatios) shows that modulated ad umodulated waveforms are ot equally suitable for wireless power delivery, amely modulatio beig beeficial i sigle-carrier trasmissios but detrimetal i multi-carrier trasmissios. Secod, a multicarrier umodulated waveform (superposed to a multi-carrier modulated waveform) is useful to elarge the rate-eergy regio of WIPT. Third, a combiatio of power splittig ad time sharig is i geeral the best strategy. Fourth, a o-zero mea Gaussia iput distributio outperforms the covetioal capacity-achievig zero-mea Gaussia iput distributio i multi-carrier trasmissios. Fifth, the rectifier oliearity is beeficial to system performace ad is essetial to efficiet WIPT desig. Idex Terms Noliearity, optimizatio, waveform, wireless power, wireless iformatio ad power trasfer I. INTRODUCTION Wireless Iformatio ad Power Trasfer/Trasmissio (WIPT) is a emergig research area that makes use of radiowaves for the joit purpose of wireless commuicatios or Wireless Iformatio Trasfer (WIT) ad Wireless Power Trasfer (WPT). WIPT has recetly attracted sigificat attetio i academia. It was first cosidered i [, where the rateeergy tradeoff was characterized for some discrete chaels, ad a Gaussia chael with a amplitude costrait o the iput. WIPT was the studied i a frequecy-selective AWGN chael uder a average power costrait [3. Sice the, WIPT has attracted sigificat iterests i the commuicatio literature with amog others MIMO broadcastig [4 [6, architecture [7, iterferece chael [8 [10, broadbad Bruo Clerckx is with the EEE departmet at Imperial College Lodo, Lodo SW7 AZ, UK ( b.clerckx@imperial.ac.uk). This work has bee partially supported by the EPSRC of UK, uder grat EP/P003885/1. The material i this paper was preseted i part at the ITG WSA 016 [1. system [11 [13, relayig [14 [16, wireless powered commuicatio [17, [18. Overviews of potetial applicatios ad promisig future research aveues ca be foud i [19, [0. Wireless Power Trasfer (WPT) is a fudametal buildig block of WIPT ad the desig of a efficiet WIPT architecture fudametally relies o the ability to desig efficiet WPT. The major challege with WPT, ad therefore WIPT, is to fid ways to icrease the ed-to-ed power trasfer efficiecy, or equivaletly the DC power level at the output of the rectea for a give trasmit power. To that ed, the traditioal lie of research (ad the vast majority of the research efforts) i the RF literature has bee devoted to the desig of efficiet recteas [1, [ but a ew lie of research o commuicatios ad sigal desig for WPT has emerged recetly i the commuicatio literature [3. A rectea is made of a oliear device followed by a lowpass filter to extract a DC power out of a RF iput sigal. The amout of DC power collected is a fuctio of the iput power level ad the RF-to-DC coversio efficiecy. Iterestigly, the RF-to-DC coversio efficiecy is ot oly a fuctio of the rectea desig but also of its iput waveform (power ad shape) [4 [30. This has for cosequece that the coversio efficiecy is ot a costat but a oliear fuctio of the iput waveform (power ad shape). This observatio has triggered recet iterests o systematic wireless power waveform desig [9. The objective is to uderstad how to make the best use of a give RF spectrum i order to deliver a maximum amout of DC power at the output of a rectea. This problem ca be formulated as a lik optimizatio where trasmit waveforms (across space ad frequecy) are adaptively desiged as a fuctio of the chael state iformatio (CSI) so as to maximize the DC power at the output of the rectifier. I [9, the waveform desig problem for WPT has bee tackled by itroducig a simple ad tractable aalytical model of the diode oliearity through the secod ad higher order terms i the Taylor expasio of the diode characteristics. Comparisos were also made with a liear model of the rectifier, that oly accouts for the secod order term, which has for cosequece that the harvested DC power is modeled as a coversio efficiecy costat (i.e. that does ot reflect the depedece w.r.t. the iput waveform) multiplied by the average power of the iput sigal. Assumig perfect Chael State Iformatio at the Trasmitter (CSIT) ca be attaied, relyig o both the liear ad oliear models, a optimizatio problem was formulated to adaptively chage o each trasmit atea a multisie waveform as a

2 fuctio of the CSI so as to maximize the output DC curret at the eergy harvester. Importat coclusios of [9 are that 1) multisie waveforms desiged accoutig for oliearity are spectrally more efficiet tha those desiged based o a liear model of the rectifier, ) the derived waveforms optimally exploit the combied effect of a beamformig gai, the rectifier oliearity ad the chael frequecy diversity gai, 3) the liear model does ot characterize correctly the rectea behavior ad leads to iefficiet multisie waveform desig, 4) rectifier oliearity is key to desig efficiet wireless powered systems. Followig [9, various works have further ivestigated WPT sigal ad system desig accoutig for the diode oliearity, icludig amog others waveform desig complexity reductio [31 [34, large-scale system desig with may siewaves ad trasmit ateas [3, [33, multi-user setup [3, [33, imperfect/limited feedback setup [35, iformatio trasmissio [36 ad prototypig ad experimetatio [37. Aother type of oliearity leadig to a output DC power saturatio due to the rectifier operatig i the diode breakdow regio, ad its impact o system desig, has also appeared i the literature [30, [38. Iterestigly, the WIPT literature has so far etirely relied o the liear model of the rectifier, e.g. see [ [0. Give the iaccuracy ad iefficiecy of this model ad the potetial of a systematic desig of wireless power waveform as i [9, it is expected that accoutig for the diode oliearity sigificatly chages the desig of WIPT ad is key to efficiet WIPT desig, as cofirmed by iitial results i [1. I this paper, we depart from this liear model ad revisit the desig of WIPT i light of the rectifier oliearity. We address the importat problem of waveform ad trasceiver desig for WIPT ad characterize the rate-eergy tradeoff, accoutig for the rectifier oliearity. I cotrast to the existig WIPT sigal desig literature, our methodology i this paper is based o a bottom-up approach where WIPT sigal desig relies o a soud sciece-drive desig of the uderlyig WPT sigals iitiated i [9. First, we exted the aalytical model of the rectea oliearity itroduced i [9, origially desiged for multi-carrier umodulated (determiistic multisie) waveform, to multicarrier modulated sigals. We ivestigate how a multi-carrier modulated waveform (e.g. OFDM) ad a multi-carrier umodulated (determiistic multisie) waveform compare with each other i terms of harvested eergy. Compariso is also made with the liear model commoly used i the WIPT literature. Scalig laws of the harvested eergy with sigle-carrier ad multi-carrier modulated ad umodulated waveforms are aalytically derived as a fuctio of the umber of carriers ad the propagatio coditios. Those results exted the scalig laws of [9, origially derived for umodulated waveforms, to modulated waveforms. We show that by relyig o the classical liear model, a umodulated waveform ad a modulated waveform are equally suitable for WPT. This explais why the etire WIPT literature has used modulated sigals. O the other had, the oliear model clearly highlights that they are ot equally suitable for wireless power delivery, with modulatio beig beeficial i sigle-carrier trasmissio but detrimetal i multi-carrier trasmissios. The behavior is furthermore validated through circuit simulatios. This is the first paper where the performace of umodulated ad modulated waveforms are derived based o a tractable aalytical model of the rectifier oliearity ad the observatios made from the aalysis are validated through circuit simulatios. Secod, we itroduce a ovel WIPT trasceiver architecture relyig o the superpositio of multi-carrier umodulated ad modulated waveforms at the trasmitter ad a powersplitter receiver equipped with a eergy harvester ad a iformatio decoder. The WIPT superposed waveform ad the power splitter are joitly optimized so as to maximize ad characterize the rate-eergy regio of the whole system. The desig is adaptive to the chael state iformatio ad results from a posyomial maximizatio problem that origiates from the oliearity of the eergy harvester. This is the first paper that studies WIPT ad the characterizatio of the rate-eergy tradeoff cosiderig the diode oliearity. Third, we provide umerical results to illustrate the performace of the derived waveforms ad WIPT architecture. Key observatios are made. First, a multi-carrier umodulated waveform (superposed to a multi-carrier modulated waveform) is useful to elarge the rate-eergy regio of WIPT if the umber of subbads is sufficietly large (typically larger tha 4). Secod, a combiatio of power splittig ad time sharig is i geeral the best strategy. Third, a o-zero mea Gaussia iput distributio outperforms the covetioal capacity-achievig zero-mea Gaussia iput distributio i multi-carrier trasmissios. Fourth, the rectifier oliearity is beeficial to system performace ad is essetial to efficiet WIPT desig. This is the first paper to make those observatios because they are direct cosequeces of the oliearity. Orgaizatio: Sectio II itroduces ad models the WIPT architecture. Sectio III optimizes WIPT waveforms ad characterizes the rate-eergy regio. Sectio IV derives the scalig laws of modulated ad umodulated waveforms. Sectio V evaluates the performace ad sectio VI cocludes the work. Notatios: Bold lower case ad upper case letters stad for vectors ad matrices respectively whereas a symbol ot i bold fot represets a scalar.. F refers to the Frobeius orm a matrix. A{.} refers to the DC compoet of a sigal. E X {.} refers to the expectatio operator take over the distributio of the radom variable X (X may be omitted for readability if the cotext is clear).. refers to the cojugate of a scalar. (.) T ad (.) H represet the traspose ad cojugate traspose of a matrix or vector respectively. The distributio of a circularly symmetric complex Gaussia (CSCG) radom vector with mea µ ad covariace matrix Σ is deoted by CN(µ,Σ) ad stads for distributed as. II. A NOVEL WIPT TRANSCEIVER ARCHITECTURE I this sectio, we itroduce a ovel WIPT trasceiver architecture ad detail the fuctioig of the various buildig blocks. The motivatio behid the use of such a architecture will appear clearer as we progress through the paper. A. Trasmitter ad Receiver We cosider a sigle-user poit-to-poit MISO WIPT system i a geeral multipath eviromet. The trasmitter is

3 3 equipped with M ateas that trasmit iformatio ad power simultaeously to a receiver equipped with a sigle receive atea. We cosider the geeral setup of a multicarrier/bad trasmissio (with sigle-carrier beig a special case) cosistig of N orthogoal subbads where the th subbad has carrier frequecy f ad equal badwidth B s, = 0,...,N 1. The carrier frequecies are evely spaced such that f = f 0 + f with f the iter-carrier frequecy spacig (with B s f ). Uiquely, the WIPT sigal trasmitted o atea m, x m (t), cosists i the superpositio of oe multi-carrier umodulated (determiistic multisie) power waveform x P,m (t) at frequecies f, = 0,...,N 1 for WPT ad oe multicarrier modulated commuicatio waveform x I,m (t) at the same frequecies for WIT 1, as per Fig 1(a). The modulated waveform carries N idepedet iformatio symbols x (t) o subbad = 0,...,. Hece, the trasmit WIPT sigal at time t o atea m = 1,...,M writes as x m (t) = x P,m (t)+x I,m (t), = s P,,m cos(πf t+φ P,,m ) + s I,,m (t)cos(πf t+ φ I,,m (t)), { } = R (w P,,m +x,m (t))e jπft, = R { } (w P,,m +w I,,m x (t))e jπft where we deote the complex-valued basebad sigal trasmitted by atea m at subbad for the umodulated (determiistic multisie) waveform as w P,,m = s P,,m e jφp,,m ad for the modulated waveform as x,m (t) = w I,,m x (t) = s I,,m (t)e j φ I,,m(t). w P,,m is costat across time (for a give chael state) ad x P,m (t) is therefore the weighted summatio of N siewaves iter-separated by f Hz, ad hece occupies zero badwidth. O the other had, x,m (t) has a sigal badwidth o greater tha B s with symbols x (t) assumed i.i.d. CSCG radom variable with zero-mea ad uit variace (power), deoted as x CN(0,1). Deotig the iput symbol x = x e jφ x, we further express the magitude ad phase of x,m as follows s I,,m =s I,,m x with s I,,m = w I,,m ad φ I,,m = φ I,,m +φ x. Hece E { x,m } = s I,,m ad x,m CN(0,s I,,m ). The trasmit WIPT sigal propagates through a multipath chael, characterized by L paths. Let τ l ad α l be the delay ad amplitude gai of the l th path, respectively. Further, deote by ζ,m,l the phase shift of the l th path betwee trasmit atea m ad the receive atea at subbad. Deotigv,m (t) = w P,,m +w I,,m x (t), the sigal received at the sigle-atea receiver due to trasmit atea m ca be expressed as the sum of two cotributios, amely oe origiatig from WPT y P,m (t) ad the other from WIT 1 x I,m (t) ca be implemeted usig e.g. OFDM. followig the capacity achievig iput distributio i a Gaussia chael with average power costrait. (1) y I,m (t), amely y m (t)= y P,m (t)+y I,m (t), =R R { L 1 l=0 { α l v,m (t τ l )e jπf(t τ l)+ζ,m,l }, } h,m (w P,,m +w I,,m x (t))e jπft () where we have assumed max l l τ l τ l << 1/B s so that v,m (t) ad x (t) for each subbad are arrowbad sigals, thus v,m (t τ l ) = v,m (t) ad x (t τ l ) = x (t), l. The quatity h,m =A,m e j ψ,m = L 1 l=0 α le j( πfτ l+ζ,m,l ) is the chael frequecy respose betwee atea m ad the receive atea at frequecy f. Stackig up all trasmit sigals across all ateas, we ca write the trasmit WPT ad WIT sigal vectors as { } x P (t) = R w P, e jπft, (3) x I (t) = R { } w I, x (t)e jπft where w P/I, = [ T w P/I,,1... w P/I,,M. Similarly, we defie the vector chael as h = [ h,1... h,m. The total received sigal comprises the sum of () over all trasmit ateas, amely (4) y(t) = y P (t)+y I (t), { } = R h (w P, +w I, x )e jπft. (5) The magitudes ad phases of the siewaves ca be collected ito N M matrices S P ad Φ P. The (,m) etry of S P ad Φ P write as s P,,m ad φ P,,m, respectively. Similarly, we defie N M matrices such that the (,m) etry of matrix S I ad Φ I write as s I,,m ad φ I,,m, respectively. We defie the average power of the WPT ad WIT waveforms as P P = 1 S P F ad P I = 1 S I F. Due to the superpositio of the two waveforms, the total average trasmit power costrait writes as P P +P I P. Followig Fig 1(b), usig a power splitter with a power splittig ratio ρ ad assumig perfect matchig (as i Sectio II-C1), the iput voltage sigals ρr at y(t) ad (1 ρ)rat y(t) are respectively coveyed to the eergy harvester (EH) ad the iformatio decoder (ID). Remark 1: As it will appear clearer throughout the paper, the beefit of choosig a determiistic multisie power waveform over other types of power waveform (e.g. modulated, pseudo-radom) is twofold: 1) eergy beefit: multisie will be show to be superior to a modulated waveform, ) rate beefit: multisie is determiistic ad therefore does ot iduce ay rate loss at the commuicatio receiver. Remark : It is worth otig the effect of the determiistic multisie waveform o the iput distributio i (1). Recall that x,m = s I,,m e j φ I,,m CN(0,s I,,m ). Hece w P,,m +x,m CN(w P,,m,s I,,m ) ad the effective iput distributio o a give frequecy ad atea is ot zero

4 4 power splitter Power WF x P,1 (t) + x I,1 (t) Comm. WF rectifier x P,M (t) x I,M (t) At. 1 + (a) Trasmitter RF-BB coversio ADC At. M Power WF cacellatio (b) Receiver with waveform cacellatio BB receiver Fig. 1. Trasceiver (Tx ad Rx) architecture for WIPT with superposed commuicatio ad power waveform (WF). mea 3. The magitude w P,,m +x,m is Ricea distributed with a K-factor o frequecy ad atea m give by K,m = s P,,m /s I,,m. Remark 3: The superpositio of iformatio ad power sigals has appeared i other works, but for completely differet purposes; amely for multiuser WIPT i [6, collaborative WIPT i iterferece chael i [43, [44, ad for secrecy reasos i [45, [46. Sice those works relied o the liear model, the superpositio was ot motivated by the rectifier oliearity. Moreover, the properties of the power sigals are completely differet. While the power sigal is a determiistic multisie waveform leadig to o-zero mea Gaussia iput ad the twofold beefit (Remark 1) i this work, it is complex (pseudo-radom) Gaussia CN(0, Σ) i those works. B. Iformatio Decoder Sice x P,m (t) does ot cotai ay iformatio, it is determiistic. This has for cosequece that the differetial etropy of v,m ad x,m are idetical (because traslatio does ot chage the differetial etropy) ad the achievable rate is always equal to I(S I,Φ I,ρ)= log ( ) 1+ (1 ρ) h w I, σ, (6) where σ is the variace of the AWGN from the atea ad the RF-to-basebad dow-coversio o toe. Naturally, I(S I,Φ I,ρ) is larger tha the maximum rate achievable whe ρ = 0, i.e. I(S I,Φ I,0), which is obtaied by performig Maximum Ratio Trasmissio (MRT) o each subbad ad water-fillig power allocatio across subbads. The rate (6) is achievable irrespectively of the receiver architecture, e.g. with ad without waveform cacellatio. I the former case, after dow-coversio from RF-to-basebad (BB) ad ADC, the cotributio of the power waveform is subtracted from the received sigal (as illustrated i Fig 1(b)) 4. I the latter case, the Power WF cacellatio box of Fig 1(b) is removed ad the BB receiver decodes the traslated versio of the codewords. 3 If usig OFDM, x I,m (t) ad x m(t) are OFDM waveforms with CSCG iputs ad o-zero mea Gaussia iputs, respectively. 4 If usig OFDM, covetioal OFDM processig (removig the cyclic prefix ad performig FFT) is the coducted i the BB receiver. v s Fig.. ~ R at v i R i v i ~ v d i d o-liear device v out C i out R L low-pass filter ad load Atea equivalet circuit (left) ad a sigle diode rectifier (right). C. Eergy Harvester I [9, a tractable model of the rectifier oliearity i the presece of multi-carrier umodulated (determiistic multisie) excitatio was derived ad its validity verified through circuit simulatios. I this paper, we reuse the same model ad further expad it to modulated excitatio. The radomess due to iformatio symbols x impacts the amout of harvested eergy ad eeds to be captured i the model. 1) Atea ad Rectifier: The sigal impigig o the atea is y(t) ad has a average power P av = E { y(t) }. A lossless atea is modelled as a voltage sourcev s (t) followed by a series resistace 5 R at (Fig left). Let Z i =R i +jx i deote the iput impedace of the rectifier with the matchig etwork. Assumig perfect matchig (R i = R at, X i = 0), due to the power splitter, a fractio ρ of the available RF power P av is trasferred to the rectifier ad absorbed by R i, so that the actual iput power to the rectifier is P i = ρe { y(t) } = E { v i (t) } /R i ad v i (t)=v s (t)/. Hece, v i (t) ca be formed asv i (t)=y(t) ρr i =y(t) ρr at. We also assume that the atea oise is too small to be harvested. Let us ow look at Fig (right) ad cosider a rectifier composed of a sigle series diode 6 followed by a low-pass filter with load. Deotig the voltage drop across the diode as v d (t) = v i (t) v out (t) where v i (t) is the iput voltage to the diode ad v out (t) is the output voltage across the load resistor, a tractable behavioral diode model is obtaied by Taylor series expasio of the diode characteristic equatio ( v d (t) i d (t) = i s e v t 1 ) (with i s the reverse bias saturatio curret, v t the thermal voltage, the ideality factor assumed equal to 1.05) aroud a quiescet operatig poit v d = a, amelyi d (t) = i=0 k i (v d(t) a) i wherek 0 = i ( a s e v t 1 ) ad k i ev a = i t s, i = 1,...,. Assume a steady-state i!(v t) i respose ad a ideal low pass filter such that v out (t) is at costat DC level. Choosig a = E {v d (t)} = v out, we ca write i d (t) = i=0 k i v i(t) i = i=0 k i ρi/ Raty(t) i/ i. Uder the ideal rectifier assumptio ad a determiistic icomig waveform y(t), the curret delivered to the load i a steady-state respose is costat ad give by i out = A{i d (t)}. I order to make the optimizatio tractable, we trucate the Taylor expasio to the th o order. A oliear model trucates the Taylor expasio to the th o order but retais the fudametal oliear behavior of the diode while a liear model trucates to the secod order term. 5 Assumed real for simplicity. A more geeral model ca be foud i [31. 6 The model holds also for more geeral rectifiers as show i [31.

5 5 ) Liear ad Noliear Models: After trucatio, the output DC curret approximates as o i out = A{i d (t)} k iρ i/ RatA i/ { y(t) i}. (7) i=0 Let us first cosider a multi-carrier umodulated (multisie) waveform, i.e. y(t) = y P (t). Followig [9, we get a approximatio of the DC compoet of the curret at the output of the rectifier (ad the low-pass filter) with a multisie excitatio over a multipath chael as o i out k 0 + i eve,i k i ρi/ R i/ ata { y P (t) i} (8) where A { y P (t) } ad A { y P (t) 4} are detailed i (9) ad (11), respectively (at the top of ext page). The liear model is a special case of the oliear model ad is obtaied by trucatig the Taylor expasio to order ( o = ). Let us the cosider the multi-carrier modulated waveform, i.e. y(t) = y I (t). It ca be viewed as a multisie waveform for a fixed set of iput symbols{ x }. Hece, we ca also write the DC compoet of the curret at the output of the rectifier (ad the low-pass filter) with a multi-carrier modulated excitatio ad fixed set of iput symbols over a multipath chael as k 0 + o i eve,i k i ρi/ R i/ at A{ y I (t) i}. Similar expressios as (9) ad (11) ca be writte fora { y I (t) } ad A { y I (t) 4} for a fixed set of iput symbols { x }. However, cotrary to the multisie waveform, the iput symbols{ x } of the modulated waveform chage radomly at symbol rate 1/B s. For a give chael impulse respose, the proposed model for the DC curret with a modulated waveform is obtaied as i out k 0 + o i eve,i k iρ i/ R i/ ate { x}{ A { yi (t) i}}, (1) by takig the expectatio over the distributio of the iput symbols { x }. For E { A { y I (t) i}} with i eve, the DC compoet is first extracted for a give set of amplitudes { s I,,m } ad phases { φi,,m } ad the expectatio is take over the radomess of the iput symbols x. Due to the i.i.d. CSCG distributio of the iput symbols, x is expoetially distributed with E { x } = 1 ad φ x is uiformly distributed. From the momets of a expoetial distributio, we also have that E { x 4} =. We ca the express (13) ad (14) as a fuctio ofs I,,m adψ I,,m = φ I,,m + ψ,m. Note that this factor of E { x 4} = does ot appear i (11) due to the absece of modulatio, which explais why (11) ad (14) ejoy a multiplicative factor of 3 8 ad 6 8, respectively. Here agai, the liear model is obtaied by trucatig to o =. Let us fially cosider the superposed waveform, i.e. y(t) = y P (t)+y I (t). Both y P (t) ady I (t) waveforms ow cotribute to the DC compoet i out k 0 + o i eve,i k iρ i/ R i/ ate { x}{ A { y(t) i }}. (16) Takig for istace o = 4 ad further expadig the term E { x}{ A { y(t) i }} usig the fact that E {A{y P (t)y I (t)}} = 0, E { A { y P (t) 3 y I (t) }} = 0, E { A { y P (t)y I (t) 3}} = 0 ad E { A { y P (t) y I (t) }} = A { y P (t) } E { A { y I (t) }}, i out ca be writte as i out k 0 +k ρr at A { y P (t) } +k 4ρ R ata { y P (t) 4} +k ρr ate { A { y I (t) }} +k 4 ρ R at E { A { y I (t) 4}} +6k 4ρ R ata { y P (t) } E { A { y I (t) }}. (17) Observatio 1: The liear model highlights that there is o differece i usig a multi-carrier umodulated (multisie) waveform ad a multi-carrier modulated (e.g. OFDM) waveform for WPT, sice accordig to this model the harvested eergy is a fuctio of h w P/I,, as see from (9) ad (13). Hece modulated ad umodulated waveforms are equally suitable. O the other had, the oliear model highlights that there is a clear differece betwee usig a multicarrier umodulated over a multi-carrier modulated waveform i WPT. Ideed, from (13) ad (15) of the modulated waveform, both the secod ad fourth order terms exhibit the same behavior ad same depedecies, amely they are both exclusively fuctio of h w I,. That suggests that for a multi-carrier modulated waveform with CSCG iputs, the liear ad oliear models are equivalet, i.e. there is o eed i modelig the fourth ad higher order term. O the other had, for the umodulated waveform, the secod ad fourth order terms, amely (9) ad (11), exhibit clearly differet behaviors with the secod order term beig liear ad the fourth order beig oliear ad fuctio of terms expressed as the product of cotributios from differet frequecies. Remark 4: The liear model is motivated by its simplicity rather tha its accuracy ad is the popular model used throughout the WIPT literature, e.g. [4. Ideed, it is always assumed that the harvested DC power is modeled as ηp i (y(t)) = ηe { A { y(t) }} where η is the RF-to- DC coversio efficiecy assumed costat. By assumig η costat, those works effectively oly care about maximizig the iput power P i (y(t)) (fuctio of y(t)) to the rectifier, i.e. the secod order term (or liear term) E { A { y(t) }} i the Taylor expasio. Ufortuately this is iaccurate as η is ot a costat ad is itself a fuctio of the iput waveform (power ad shape) to the rectifier, as recetly highlighted i the commuicatio literature [3, [8 [30 but well recogized i the RF literature [1, [. This liear model was show through circuit simulatios i [9 to be iefficiet to desig multisie waveform but also iaccurate to predict the behavior of such waveforms i the practical low-power regime (-30dBm to 0dBm). O the other had, the oliear model, rather tha explicitly expressig the DC output power as η(y(t))p i (y(t)) with η(y(t)) a fuctio of the iput sigal power ad shape, it directly expresses the output DC curret as a fuctio of y(t) (ad therefore as a fuctio of the trasmit sigal ad wireless chael) ad leads to a more tractable formulatio. Such a oliear model with o = 4 has bee validated for the desig of multisie waveform i [8, [9, [31 usig circuit simulators with various rectifier topologies ad iput power ad i [37 through prototypig ad experimetatio. Nevertheless, the use of a liear vs a oliear model for the desig of WPT based o other types of waveforms ad the desig of WIPT has ever bee addressed so far.

6 6 A { y P (t) } = 1 A { y P (t) 4} = 3 8 R = 3 8 [ hw P, = 1 s P,,m0 s P,,m1 A,m0 A,m1 cos ( ) ψ P,,m0 ψ P,,m1, (9) m 0,m 1 h 0 w P,0 h 1 w P,1 0, 1,, = + 3 0, 1,, = + 3 m 0,m 1, m,m 3 ( h w P, ) ( h3 w P,3 ), (10) [ 3 s P,j,m j A j,m j cos(ψ P,0,m 0 +ψ P,1,m 1 ψ P,,m ψ P,3,m 3 ). (11) j=0 E { A { y I (t) }} = 1 s I,,m0 s I,,m1 A,m0 A,m1 cos ( [ ) ψ I,,m0 ψ I,,m1 = 1 h w I, m 0,m 1 E { A { [ [ [ y I (t) 4}} = 6 s I,0,m 8 j A 0,m j s I,1,m j A 1,m j cos(ψ I,0,m 0 +ψ I,1,m 1 ψ I,0,m ψ I,1,m 3 ) j=0, j=1,3 = 6 8 0, 1 m 0,m 1, m,m 3 [ hw I, (13) (14) (15) Remark 5: The above model deals with the diode oliearity uder ideal low pass filter ad perfect impedace matchig. However there exist other sources of oliearities i a rectifier, e.g. impedace mismatch, breakdow voltage ad harmoics. Recetly, aother oliear model has emerged i [30. This model accouts for the fact that for a give rectifier desig, the RF-to-DC coversio efficiecy η is a fuctio of the iput power ad sharply decreases oce the iput power has reached the diode breakdow regio. This leads to a saturatio oliearity where the output DC power saturates beyod a certai iput power level. There are multiple differeces betwee those two models. First, our diode oliearity model assumes the rectifier is ot operatig i the diode breakdow regio. Circuit evaluatios i [9, [31 ad i Sectio V-B also cofirm that the rectifier ever reached the diode breakdow voltage uder all ivestigated scearios. We therefore do ot model the saturatio effect. O the other had, [30 assumes the rectifier ca operate i the breakdow regio ad therefore models the saturatio. However, it is to be remided that operatig diodes i the breakdow regio is ot the purpose of a rectifier ad should be avoided. A rectifier is desiged i such a way that curret flows i oly oe directio, ot i both directios as it would occur i the breakdow regio. Hece, [30 models a saturatio oliearity effect that occurs i a operatig regio where oe does ot wish to operate i. I other words, the rectifier is pushed i a iput power rage quite off from the oe it has origially bee desiged for. This may oly occur i applicatios where there is little guaratee to operate the desiged rectifier below that breakdow edge. Secod, the diode oliearity is a fudametal, uavoidable ad itrisic property of ay rectifier, i.e. ay rectifier, irrespectively of its desig, topology or implemetatio, is always made of a oliear device (most commoly Schottky diode) followed by a low pass filter with load. This has for cosequece that the diode oliearity model is geeral ad valid for a wide rage of rectifier desig ad topology (with oe ad multiple diodes) as show i [31. Moreover, sice it is drive by the physics of the rectea, it aalytically liks the output DC metric to the iput sigal through the diode I-V characteristics. O the other had, the saturatio oliearity i [30 is circuit-specific ad modeled via curve fittig based o measured data. Hece chagig the diode or the rectifier topology would lead to a differet behavior. More importatly, the saturatio effect, ad therefore the correspodig oliearity, is actually avoidable by properly desigig the rectifier for the iput power rage of iterest. A commo strategy is to use a adaptive rectifier whose cofiguratio chages as a fuctio of the iput power level, e.g. usig a sigle-diode rectifier at low iput power ad multiple diodes rectifier at higher power, so as to geerate cosistet ad o-vaishig η over a sigificatly exteded operatig iput power rage [39, [40. Third, the diode oliearity model accomodates a wide rage of multi-carrier modulated ad umodulated iput sigals ad is therefore a fuctio of the iput sigal power, shape ad modulatio. The saturatio oliearity model i [30 is restricted to a cotiuous wave iput sigal ad is a fuctio of its power. Hece it does ot reflect the depedece of the output DC power to modulatio ad waveform desigs. Fourth, the diode oliearity is a beeficial feature that is to be exploited as part of the waveform desig to boost the output DC power, as show i [9. The saturatio oliearity is detrimetal to performace ad should therefore be avoided by operatig i the o-breakdow regio ad usig properly desiged rectifier for the iput power rage of iterest. Fifth, the diode oliearity is more meaigful i the lowpower regime (-30dBm to 0dBm with state-of-the-art rectifiers 7 ) while the saturatio oliearity is relevat i the high power regime (beyod 0dBm iput power). 7 At lower power levels, the diode may ot tur o.

7 7 III. WIPT WAVEFORM OPTIMIZATION AND RATE-ENERGY REGION CHARACTERIZATION Leveragig the eergy harvester model, we ow aim at characterizig the rate-eergy regio of the proposed WIPT architecture. We defie the achievable rate-eergy regio as C R IDC (P) { (R,I DC ) : R I, I DC i out, 1 [ SI F + S P F P }. (18) Assumig the CSI (i the form of frequecy respose h,m ) is kow to the trasmitter, we aim at fidig the optimal values of amplitudes, phases ad power splittig ratio, deoted as S P,S I,Φ P,Φ I,ρ, so as to elarge as much as possible the rate-eergy regio. We derive a methodology that is geeral to cope with ay trucatio order o 8. Characterizig such a regio ivolves solvig the problem max i out(s P,S I,Φ P,Φ I,ρ) (19) S P,S I,Φ P,Φ I,ρ subject to 1 [ SI F + S P F P. (0) Followig [9, (19)-(0) ca equivaletly be writte as max z DC(S P,S I,Φ P,Φ I,ρ) (1) S P,S I,Φ P,Φ I,ρ with z DC = o subject to 1 [ SI F + S P F i eve, i P, () k i ρ i/ R i/ at E { x }{ A { y(t) i }} where we defie k i = factor =1.05 ad v t =5.86mV, we get k = ad k 4 = For o =4, similarly to (17), we ca compute z DC as i (3). This eables to re-defie the achievable rate-eergy regio i terms of z DC rather tha i out as follows C R IDC (P) is. Assumig i i!(v t) i s = 5µA, a diode ideality { (R,I DC ) : R I, I DC z DC, 1 [ SI F + S P F P }, (4) This defiitio of rate-eergy regio will be used i the sequel. A. WPT-oly: Eergy Maximizatio I this sectio, we first look at eergy maximizatio-oly (with o cosideratio for rate) ad therefore assume ρ = 1. We study ad compare the desig of multi-carrier umodulated (multisie) waveform (y(t) = y P (t)) ad modulated waveforms (y(t) = y I (t)) uder the liear ad oliear models. Sice a sigle waveform is trasmitted (either umodulated or modulated), the problem simply boils dow to the followig for i {P,I} max z DC (S i,φ i ) subject to 1 S i,φ i S i F P, (5) where z DC (S P,Φ P ) = o i eve,i k ir i/ at A{ y P (t) i} for multi-carrier umodulated (multisie) waveform, ad z DC (S I,Φ I ) = o i eve,i k ir i/ at E { { x }{ A yi (t) i}} for the multi-carrier modulated waveform. 8 We display terms for o 4 but the derived algorithm works for ay o. The problem of multisie waveform desig with a liear ad oliear rectea model has bee addressed i [9. The liear model leads to the equivalet problem max wp, h w P, subject to 1 w P, P whose solutio is the adaptive sigle-siewave (ASS) strategy { wp, P h H = / h, =, (6) 0,. The ASS performs a matched (also called MRT) beamformer o a sigle siewave, amely the oe correspodig to the strogest chael = argmax h. O the other had, the oliear model leads to a posyomial maximizatio problem that ca be formulated as a Reversed Geometric Program ad solved iteratively. Iterestigly, for multisie waveforms, the liear ad oliear models lead to radically differet strategies. The former favours trasmissio o a sigle frequecy while the latter favours trasmissio over multiple frequecies. Desig based o the liear model was show to be iefficiet ad lead to sigificat loss over the oliear model-based desig. The desig of multi-carrier modulated waveform is rather differet. Recall that from (13) ad (15), both the secod ad fourth order terms are exclusively fuctio of h w I,. This shows that both the liear ad oliear model-based desigs of multi-carrier modulated waveforms for WPT lead to the ASS strategy ad the optimum wi, should be desiged accordig to (6). This is i sharp cotrast with the multisie waveform desig ad origiates from the fact that the modulated waveform is subject to CSCG radomess due to the presece of iput symbols x. Note that this ASS strategy has already appeared i the WIPT literature, e.g. i [11, [41 with OFDM trasmissio. B. WIPT: A Geeral Approach We ow aim at characterizig the rate-eergy regio of the proposed WIPT architecture. Lookig at (6) ad (3), it is easy to coclude that matched filterig w.r.t. the phases of the chael is optimal from both rate ad harvested eergy maximizatio perspective. This leads to the same phase decisios as for WPT i [8, [9, amely φ P,,m = φ I,,m = ψ,m (7) ad { { guaratees all argumets of the cosie fuctios i A yp (t) i}} i=,4 ((9), (11)) ad i { E { A { y I (t) i}}} i=,4 ((13), (14)) to be equal to 0. Φ P ad Φ I are obtaied by collectigφ P,,m adφ I,,m,m ito a matrix, respectively. With such phasesφ P ad Φ I, z DC(S P,S I,Φ P,Φ I,ρ) ca be fially writte as (8). Similarly we ca write ( ( I(S I,Φ I,ρ) = log 1+ (1 ρ) ) ) σ C (9) where C = 1 m 0,m 1 j=0 s I,,m j A,mj. Recall from [47 that a moomial is defied as the fuctio g : R N ++ R : g(x) = cx a1 1 xa...xan N where c > 0 ad a i R. A sum of K moomials is called a posyomial ad ca be writte as f(x) = K k=1 g k(x) with g k (x) = c k x a 1k 1 x a k...x a Nk N where c k > 0. As we ca see from (8), z DC (S P,S I,Φ P,Φ I,ρ) is a posyomial.

8 8 z DC (S P,S I,Φ P,Φ I,ρ) = k ρr ata { y P (t) } +k 4 ρ R at A{ y P (t) 4} +k ρr ate { A { y I (t) }} +k 4 ρ R at E { A { y I (t) 4}} +6k 4 ρ R at A{ y P (t) } E { A { y I (t) }}. (3) [ z DC (S P,S I,Φ P,Φ I,ρ) = k ρ 1 Rat s P,,mj A,mj + 3k 4ρ R at 8 m 0,m 1 j=0 [ + k ρ 1 Rat s I,,mj A,mj + 3k 4ρ R at 4 m 0,m 1 j=0 [ + 3k 4ρ 1 [ 1 R at s P,,mj A,mj j=0 j=0 m 0,m 1 0, 1,, = + 3 m 0,m 1 m 0,m 1 m 0,m 1, m,m 3 [ 1 j=0 [ 3 j=0 s P,j,m j A j,m j s I,,mj A,mj s I,,mj A,mj (8) I order to idetify the achievable rate-eergy regio, we formulate the optimizatio problem as a eergy maximizatio problem subject to trasmit power ad rate costraits max S P,S I,ρ z DC (S P,S I,Φ P,Φ I,ρ) (30) subject to 1 [ SI F + S P F P, (31) I(S I,Φ I,ρ) R. (3) It therefore cosists i maximizig a posyomial subject to costraits. Ufortuately this problem is ot a stadard Geometric Program (GP) but it ca be trasformed to a equivalet problem by itroducig a auxiliary variable t 0 mi S P,S I,ρ,t 0 1/t 0 (33) subject to [ 1 SI F + S P F P, (34) t 0 /z DC (S P,S I,Φ P,Φ I,ρ) 1, (35) [ ( R/ 1+ (1 ρ) ) σ C 1. (36) This is kow as a Reversed Geometric Program [47. A similar problem also appeared i the WPT waveform optimizatio [8. Note that 1/z DC (S P,S I,Φ P,Φ I,ρ) ad 1/ [ ( 1 + (1 ρ) ) σ C are ot posyomials, therefore prevetig the use of stadard GP tools. The idea is to replace the last two iequalities (i a coservative way) by makig use of the arithmetic mea-geometric mea (AM-GM) iequality. Let {g k (S P,S I,Φ P,Φ I,ρ)} be the moomial terms i the posyomial z DC (S P,S I,Φ P,Φ I,ρ) = K k=1 g k(s P,S I,Φ P,Φ I,ρ). Similarly we defie {g k (S I, ρ)} as the set of moomials of the posyomial 1 + ρ σ C = K k=1 g k(s I, ρ) with ρ = 1 ρ. For a give choice of {γ k } ad {γ k } with γ k,γ k 0 ad K k=1 γ k = K k=1 γ k =1, we perform sigle codesatios ad write the stadard GP mi 1/t 0 (37) S P,S I,ρ, ρ,t 0 subject to [ 1 SI F + S P F P, (38) K ( gk (S P,S I,Φ P t,φ I,ρ) ) γk 0 1, (39) k=1 K R k=1 γ k ( ) γk gk (S I, ρ) 1, (40) γ k ρ+ ρ 1, (41) that ca be solved usig existig software, e.g. CVX [48. It is importat to ote that the choice of {γ k,γ k } plays a great role i the tightess of the AM-GM iequality. A iterative procedure ca be used where at each iteratio the stadard GP (37)-(41) is solved for a updated set of{γ k,γ k }. Assumig a feasible set of magitude S (i 1) P ad S (i 1) I ad power splittig ratio ρ (i 1) at iteratio i 1, compute at iteratio i γ k = g k(s (i 1) P,S (i 1) I,Φ P,Φ I,ρ(i 1) ) k = 1,...,K ad γ z DC(S (i 1) P,S (i 1) I,Φ k = P,Φ I,ρ(i 1) ) g k (S (i 1) I, ρ (i 1) )/ ( 1+ ρ(i 1) σ C (S (i 1) I ) ), = 0,...,, k = 1,...,K ad the solve problem (37)-(41) to obtai S (i) P, S(i) I ad ρ (i). Repeat the iteratios till covergece. The whole optimizatio procedure is summarized i Algorithm 1. Algorithm 1 WIPT Waveform ad R-E Regio 1: Iitialize: i 0, R, Φ P ad Φ I, S P, S I, ρ, ρ = 1 ρ, z (0) DC = 0 : repeat 3: i i+1, S P S P, S I S I, ρ ρ, ρ ρ 4: γ k g k ( S P, S I,Φ P,Φ I, ρ)/z DC( S P, S I,Φ P,Φ I, ρ), k = 1,...,K 5: γ k g k ( S I, ρ)/ ( 1+ ρ σc ( S I ) ), = 0,...,, k = 1,...,K 6: S P,S I,ρ, ρ argmi (37) (41) 7: z (i) DC z DC(S P,S I,Φ P,Φ I,ρ) 8: util z (i) < ǫ or i = i max DC z(i 1) DC Note that the successive approximatio method used i the

9 9 Algorithm 1 is also kow as a sequetial covex optimizatio or ier approximatio method [49. It caot guaratee to coverge to the global solutio of the origial problem, but oly to yield a poit fulfillig the KKT coditios [49, [50. C. WIPT: Decouplig Space ad Frequecy Whe M > 1, previous sectio derives a geeral methodology to optimize the superposed waveform weights joitly across space ad frequecy. It is worth woderig whether we ca decouple the desig of the spatial ad frequecy domai weights without impactig performace. The optimal phases i (7) are those of a MRT beamformer. Lookig at (6), (9), (10), (13) ad (15), the optimum weight vectors w P, ad w I, that maximize the d ad 4 th order terms ad the rate, respectively, are MRT beamformers of the form w P, = s P, h H / h, w I, = s I, h H / h, (4) such that, from (5), y P (t) = h s P, cos(w t) = R { h s P, e jwt} ad y I (t) = h s I, x cos(w t) = R { h s I, x e jwt}. Hece, with (4), the multi-atea multisie WIPT weight optimizatio is coverted ito a effective sigle atea multi-carrier WIPT optimizatio with the effective chael gai o frequecy give by h ad the power allocated to the th subbad give by s P, ad s I, for the multi-carrier umodulated (multisie) ad modulated waveform, respectively (subject to s P, + s I, = P ). The optimum magitude s P, ad s I, i (4) ca ow be obtaied by usig the posyomial maximizatio methodology of Sectio III-B. Namely, focusig o o = 4 for simplicity, pluggig (4) ito (9), (10), (13) ad (15), we get (43) ad (44). The DC compoet z DC as defied i (3) simply writes as z DC (s P,s I,ρ), expressig that it is ow oly a fuctio of the N-dimesioal vectors s P/I = [ s P/I,0,...,s P/I,. ( ( Similarly, I(s I,ρ) = log 1+ (1 ρ) σ s I, h )). Followig the posyomial maximizatio methodology, we ca write z DC (s P,s I,ρ) = K k=1 g k(s P,s I,ρ) ad 1 + ρ σ C = K k=1 g k(s I, ρ) with C = s I, h, apply the AM-GM iequality ad write the stadard GP problem mi 1/t 0 (45) s P,s I,ρ, ρ,t 0 subject to 1 [ s P + s I P, (46) K ( ) γk gk (s P,s I,ρ) t 0 1, (47) k=1 K R k=1 γ k ( ) γk gk (s I, ρ) 1, (48) γ k ρ+ ρ 1. (49) Algorithm summarizes the desig methodology with spatial ad frequecy domai decouplig. Such a approach would lead to the same performace as the joit spacefrequecy desig of Algorithm 1 but would sigificatly reduce the computatioal complexity sice oly N-dimesioal vectors s P ad s I are to be optimized umerically, compared to the N M matrices S P ad S I of Algorithm. Algorithm WIPT Waveform with Decouplig 1: Iitialize: i 0, R, wp/i, ad i (4), s P, s I, ρ, ρ = 1 ρ, z (0) DC = 0 : repeat 3: i i+1, s P s P, s I s I, ρ ρ, ρ ρ 4: γ k g k ( s P, s I, ρ)/z DC ( s P, s I, ρ), k = 1,...,K 5: γ k g k ( s I, ρ)/ ( 1+ ρ σc ( s I ) ), = 0,...,, k = 1,...,K 6: s P,s I,ρ, ρ argmi (45) (49) 7: z (i) DC z DC(s P,s I,ρ) 8: util z (i) < ǫ or i = i max DC z(i 1) DC D. WIPT: Characterizig the Twofold Beefit of usig Determiistic Power Waveforms Superposig a power waveform to a commuicatio waveform may boost the eergy performace but may lead to the drawback that the power waveform iterferes with the commuicatio waveform at the iformatio receiver. Uiquely, with the proposed architecture, this issue does ot occur sice the power waveform is determiistic, which leads to this twofold eergy ad rate beefit highlighted i Remark 1. Nevertheless, i order to get some isights ito the cotributios of the determiistic power waveform to the twofold beefit, we compare i Sectio V the R-E regio of Sectio III-B to that of two baselies. The first baselie is the simplified architecture where the determiistic multisie waveform is abset ad for which the R-E regio ca still be computed usig Algorithm 1 by forcig S P = 0. I this setup, the twofold beefit disappears. The secod baselie is the hypothetical system where the power waveform is a determiistic multisie from a eergy perspective but CSCG distributed from a rate perspective. The eergy beefit is retaied but the rate beefit disappears sice the power waveform ow creates a iterferece term 1 ρh w P, (with power level give by (1 ρ) h w P, ) at the iformatio receiver ad therefore a rate loss. The achievable rate of that system is give by I LB (S P,S I,Φ P,Φ I,ρ) ) (1 ρ) h w I, = log (1+ σ +(1 ρ) h w P, (50) ad the correspodig R-E regio is a lower boud o that achieved i Sectio III-B. Comparig those two R-E regios, we get a sese of the rate beefit of usig a determiistic power waveform over a modulated oe. The lower boud o the R-E regio is obtaied by replacig I by I LB i (4). Because of the iterferece term i the SINR expressio, decouplig the space frequecy desig by choosig the weights vectors as i (4) is ot guarateed to be optimal. To characterize the R-E regio of the hypothetical system, we therefore resort to the more geeral approach where space ad frequecy domai weights are joitly desiged, similarly to the oe used i Sectio III-B. Let us assume the same phases Φ P ad Φ I as i (7). Such a choice is optimal for M = 1 (eve though there is o

10 10 A { y P (t) } = 1 E { A { y I (t) }} = 1 [ [ h s P, h s I,,A { y P (t) 4} = 3 8,E { A { y I (t) 4}} = 6 8 0, 1,, = + 3 [ [ 3 j=0 s hj P,j, (43) h s I,. (44) claim of optimality for M > 1). With such a choice of phases, z DC (S P,S I,Φ P,Φ I,ρ) also writes as (8), while the lower boud o the achievable rate is writte as ( ( I LB (S I,Φ I,ρ) = log 1+ (1 ρ)c ) ) σ+(1 ρ)d (51) where C = 1 m 0,m 1 j=0 s I,,m j A,mj ad D = 1 m 0,m 1 j=0 s P,,m j A,mj. The optimizatio problem ca ow be writte as (33)-(36) with (36) replaced by R ( ) 1+ ρ σ D ( 1+ ρ σ (D +C ) ) 1. (5) Defie the set of moomials{f j (S P,S I,ρ)} of the posyomial 1+ ρ σ (D +C )= J j=1 f j(s P,S I,ρ). For a give choice of {γ j } with γ j 0 ad J j=1 γ j=1, we perform sigle codesatios ad write the stadard GP as mi 1/t 0 (53) S P,S I,ρ, ρ,t 0 subject to 1 SI [ F + S P F P, (54) K ( gk (S P,S I,Φ P t,φ I,ρ) ) γk 0 1, (55) k=1 R J ( 1+ ρ j=1 σ γ k ) D (S P ) ( ) γj fj (S P,S I,ρ) 1, (56) γ j ρ+ ρ 1. (57) The whole optimizatio procedure is summarized i Algorithm 3. Algorithm 3 boils dow to Algorithm 1 for D = 0. IV. SCALING LAWS I order to get some isights ad assess the performace gai/loss of a umodulated waveform over a modulated waveform for WPT, we quatify how z DC scales as a fuctio of N for M = 1. For simplicity we trucate the Taylor expasio to the fourth order ( o = 4). We cosider frequecy-flat (FF) ad frequecy-selective (FS) chaels. We assume that the complex chael gais α l e jξ l are modeled as idepedet CSCG radom variables. α l are therefore idepedet Rayleigh distributed such that α l EXPO(λ l ) with 1/λ l = β l = E { α l}. The impulse resposes have a costat average received power ormalized to 1 such that L 1 l=0 β l = 1. I the frequecy flat chael, ψ = ψ ad Algorithm 3 Lower-Boud o R-E Regio 1: Iitialize: i 0, R, Φ P ad Φ I, S P, S I, ρ, ρ = 1 ρ, z (0) DC = 0 : repeat 3: i i+1, S P S P, S I S I, ρ ρ, ρ ρ 4: γ k g k ( S P, S I,Φ P,Φ I, ρ)/z DC( S P, S I,Φ P,Φ I, ρ), k = 1,...,K 5: γ j f j ( S P, S I, ρ)/ ( 1+ ρ σ(d ( S P )+C ( S I )) ), = 0,...,N 1, j = 1,...,J 6: S P,S I,ρ, ρ argmi (53) (57) 7: z (i) DC z DC(S P,S I,Φ P,Φ I,ρ) 8: util z (i) < ǫ or i = i max DC z(i 1) DC A = A. This is met whe() f is much smaller tha the chael coherece badwidth. I the frequecy selective chael, we assume that L >> 1 ad frequecies f are far apart from each other such that the frequecy domai CSCG radom chael gais h,m fade idepedetly (phase ad amplitude-wise) across frequecies ad ateas. Takig the expectatio over the distributio of the chael, we deote z DC = E {z DC }. Leveragig the derivatio of the scalig laws for multisie waveforms i [9, we ca compute the scalig laws of multi-carrier modulated waveforms desiged usig the ASS strategy (relyig o CSIT) ad the uiform power (UP) allocatio strategy (ot relyig o CSIT). UP is characterized by S i = P/ N1 N ad Φ i = 0 N, i {P,I} [9. Table I summarizes the scalig laws for both modulated ad umodulated waveforms for N = 1 (sigle-carrier SC) ad for N > 1 (multi-carrier) with ASS ad UP strategies. It also compares with the UPMF strategy for multi-carrier umodulated (multisie) waveforms (itroduced i [9) that cosists i uiformly allocatig power to all siewaves ad matchig the waveform phase to the chael phase (hece it requires CSIT). Such a UPMF strategy is suboptimal for multisie excitatio with the oliear model-based desig ad its scalig law is therefore a lower boud o what ca be achieved with the optimal multisie strategy of (5) [9. Let us first discuss multi-carrier trasmissio (N >> 1) with CSIT. Recall that, i the presece of CSIT, the ASS strategy for multi-carrier modulated waveforms is optimal for the maximizatio of z DC with the liear ad oliear model-based desigs. The ASS for umodulated (multisie) waveform is oly optimal for the liear model-based desig. As it ca be see from the scalig laws 9, umodulated (multisie) waveform with UPMF strategy leads to a liear 9 Takig N to ifiity does ot imply that the harvested eergy reaches ifiity as explaied i Remark 4 of [9.

11 11 TABLE I SCALING LAWS OF MULTI-CARRIER UNMODULATED VS MODULATED WAVEFORMS. Waveform Strategy N, M Frequecy-Flat (FF) Frequecy-Selective (FS) No CSIT Modulated z DC,SC N=1,M=1 k R atp +6k 4 R at P Modulated z DC,UP N>1,M=1 k R atp +6k 4 R at P k R atp +6k 4 R at P Umodulated z DC,SC N=1,M=1 k R atp +3k 4 R at P Umodulated z DC,UP N>>1,M=1 k R atp +k 4 R at P N k R atp +3k 4 R at P CSIT Modulated z DC,ASS N>>1,M=1 k R atp +6k 4 R at P k R atp logn +3k 4 R at P log N Umodulated z DC,ASS N>>1,M=1 k R atp +3k 4 R at P k R atp logn + 3 k 4R at P log N Umodulated z DC,UPMF N>>1,M=1 k R atp +k 4 R at P N k R atp + π /16k 4 R at P N k R atp +k 4 R at P N icrease of z DC with N i FF ad FS chaels while the ASS strategy for both modulated ad umodulated oly lead to at most a logarithmic icrease with N (achieved i FS chaels). Hece, despite beig suboptimal for the oliear model-based desig, a umodulated (multisie) waveform with UPMF provides a better scalig law tha those achieved by modulated/umodulated waveforms with ASS. I other words, eve a suboptimal multi-carrier umodulated waveform outperforms the optimal desig of the multi-carrier modulated (with CSCG iputs) waveform. Let us ow look at multi-carrier trasmissio i the absece of CSIT. A modulated waveform does ot eable a liear icrease of z DC with N i FF chaels, cotrary to the umodulated waveform. This is due to the CSCG distributios of the iformatio symbols that create radom fluctuatios of the amplitudes ad phases across frequecies. This cotrasts with the periodic behavior of the umodulated multisie waveform that has the ability to excite the rectifier i a periodic maer by sedig higher eergy pulses every 1/ f. Fially, for the sigle-carrier trasmissio i the absece of CSIT case (N = 1), a opposite behavior is observed with z DC of the modulated waveform outperformig that of the umodulated waveform thaks to the fourth order term beig twice as large. This gai origiates from the presece of the fourth order momet of x, amely E { x 4} =, i the fourth order term of (1). Hece modulatio through the CSCG distributio of the iput symbols is actually beeficial to WPT i sigle-carrier trasmissios. Actually, this also suggests that a modulatio with a iput distributio leadig to a large fourth order momet is beeficial to WPT 10. The scalig laws also highlight that the effect of chael frequecy selectivity o the performace depeds o the type of waveform. Ideed, without CSIT, frequecy selectivity is detrimetal to umodulated waveform performace (as already cofirmed i [9) but has o impact o modulated waveform performace. O the other had, with CSIT, frequecy selectivity leads to a frequecy diversity gai that is helpful to the performace of both types of waveforms 11. Observatio : The scalig laws highlight that, due to the diode oliearity of the fourth order term, there is a clear differece betwee usig a umodulated (multisie) waveform ad a modulated waveform i WPT. For sigle-carrier 10 Asymmetric Gaussia sigalig (asymmetric power allocatios to the real ad imagiary dimesios) may be a better optio tha CSCG for WPT [ This behavior was already cofirmed i [9 for multisie waveforms. Fig. 3. Frequecy respose MHz 1 MHz Frequecy [MHz Frequecy respose of the wireless chael. trasmissios (N = 1), the scalig law of the modulated waveform outperforms that of the umodulated waveform due to the beeficial effect of the fourth order momet of the CSCG distributio to boost the fourth order order term. O the other had, for multi-carrier trasmissios (N >> 1), the scalig law of multisie sigificatly outperforms that of the modulated waveform. While z DC scales liearly with N with a multisie waveform thaks to all carriers beig i-phase, it scales at most logarithmically with N with the modulated waveform due to the idepedet CSCG radomess of the iformatio symbols across subbads. This also shows that the diode oliearity ca be beeficial to WPT performace i two differet ways: first, by ivolvig higher order momets of the iput distributio, ad secod by eablig costructive cotributios from various frequecies. Observatio further motivates the WIPT architecture of Sectio II that is based o a superpositio of multi-carrier umodulated (determiistic multisie) waveform for efficiet WPT ad multi-carrier modulated waveform for efficiet WIT. V. PERFORMANCE EVALUATIONS We cosider two types of performace evaluatios, the first oe is based o the simplified oliear model of Sectio II-C with o = 4, while the secod oe relies o a actual ad accurate desig of the rectea i PSpice. A. Noliear Model-Based Performace Evaluatios We ow illustrate the performace of the optimized WIPT architecture. Parameters are take as k = , k 4 = ad R at = 50Ω. We assume a WiFi-like eviromet at a ceter frequecy of 5.18GHz with a 36dBm EIRP, dbi receive atea gai ad 58dB path loss. This leads to a