Multiple Antennas and Beamforming for SWIPT Systems

Transcription

1 THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF ELECTRICAL ENGINEERING AND TELECOMMUNICATIONS Multiple Ateas ad Beamformig for SWIPT Systems Author: Cheyag She Submissio Date: 27-October-2017 Master of Egieerig (Telecommuicatios)

2 Abstract I covetioal commuicatio systems, the use of radio frequecy (RF) sigals is maily costraied o iformatio trasfer. However, RF sigals also cotai eergy ad this fact leads to the emergece of various ew research areas, like RF-eabled wireless power trasfer (WPT) where wireless receivers could harvest eergy from RF sigals radiated by a eergy trasmitter. There is aother techique that use the same RF sigals for eergy ad iformatio trasfer joitly, this techique is regarded as simultaeous wireless iformatio ad power trasfer (SWIPT). Although a lot of research efforts have made o these ew areas i literature, from practical perspective, further developmet of WPT ad SWIPT still faces some problems like low power trasfer efficiecy, safety cocers etc. These problems slow dow the speed of trasformig SWIPT ad WPT from theory to practice. I this article, we first provide a overview of WPT ad SWIPT which ivolves defiitio of these two paradigms, their possibilities ad costrais as well as relevat techologies that are used to achieve efficiet WPT ad SWIPT. The we focus o a SWIPT system ad we aim to optimize the SWIPT system by maximizig its eergy efficiecy. A detailed discussio of this optimizatio problem is provided i this article.

3 Abbreviatios WPT Wireless power trasfer SWIPT Simultaeous Wireless Iformatio ad Power Trasfer RF Radio Frequecy EH Eergy Harvestig ID Iformatio Decodig MPT Microwave Power Trasfer TDD Time Divisio Duplexig FDD Frequecy Divisio Duplexig CSI Chael State Iformatio DoFs Degree of Freedom MIMO Multiple-iput ad Multiple-output SNR Sigal to Noise Ratio DC Direct Curret QoS Quality of Service AWGN Additive Gaussia Noise 1

4 Cotet Notatios... 3 Chapter 1 Itroductio Wireless Power Trasfer(WPT) Simultaeous wireless iformatio ad power trasfer(swipt)... 6 Chapter 2 System Model System model System optimizatio Chapter 3 Prelimiary Simulatio Chapter 4 Prelimiary Coclusio Chapter Solvig the optimizatio problem mathematically Coclusio Biography... Error! Bookmark ot defied. 2

5 Notatios h Chael vector betwee the trasmitter ad the iformatio receiver. g j Chael vector betwee the trasmitter ad the j th eergy receiver. w Iformatio beamformig vector. σ 2 2 at, σ s Atea oise ad sigal processig oise. ρ Power splittig factor. P mij Miimum required power of the j th eergy receiver. P mi Miimum required power of iformatio receiver. P c Power cosumed for maipulatig trasmitter itself. P 1 Power harvested from o-reewable eergy source. P 2 Power harvested from reewable eergy source 3

6 Chapter 1 Itroductio 1.1 Wireless Power Trasfer(WPT) Wireless power trasfer (WPT) is the trasmissio of electrical eergy from a power source to a electrical load, such as a electrical power grid or appliace, without the use of coductors like wires or cables [1]. I geeral, a wireless power system cosists of a "trasmitter" coected to a source of power such as a power lie, which coverts the power to a time-varyig electromagetic field, ad oe or more "receiver" devices which receive the power ad covert it back to direct curret (DC) or alteratig curret (AC) which is used by a electrical load [2][3]. There are two mai regios of WPT, oe is ear-field regio or o-radiative regio, the other is far-field regio or radiative regio. Near-field regio WPT is maily facilitated by capacitive couplig (electrostatic iductio) betwee metal electrodes, or via magetic fields by iductive couplig (electromagetic iductio) betwee coils of wire [3][4][5][6]. There is o radiative i the ear-field regio WPT, so the use of ear-field regio WPT do ot arise public health cocers caused by strog electromagetic radiatio. However, if there is o receivig device or absorbig material withi their limited rage to "couple" to, earfield regio WPT is ot achievable as o power leaves the trasmitter [7]. Besides, the propagatio rage of ear-field regio WPT is extremely short ad the trasmit power decreases with distace expoetially [8][9][10]. The use of RF sigals for eergy harvestig (EH) eables far-field regio WPT. The propagatio rage i far-field regio WPT is more acceptable tha the ear-field regio WPT ad eergy is radiated by the trasmitter o matter there is a receiver to absorb it or ot. Oe specific example of far-field regio WPT is microwave power trasfer (MPT), MPT takes use of radiative property of microwave to trasmit eergy, it has log propagatio rage due to the support of large-scale atea arrays ad high-power microwave geerators. Although eergy trasmissio i MPT is ot costraied i short distace ad the use of large atea arrays facilitates efficiet MPT, MPT or far-field regio WPT is ot suitable for everyday use because of strog magitude radiatio 4

7 which results i serious safety cocers. Despite further developmet of WPT faces several challeges, the use of WPT is still attractive. As the use of WPT avoids the costly process of istallig ad displacig power cables i buildigs ad ifrastructures. Besides, it also provides a attractive solutio to prolog lifetime of power-limited etworks ad wireless devices i use, as they ca use eergy harvested from electromagetic radiatio through WPT to charge batteries. WPT has already bee used i simple wireless devices with low eergy cosumptio, for istace, Itel has demostrated the wireless chargig of a temperature ad humidity meter as well as a liquid-crystal display by usig the sigals radiated by a TV statio 4 km away [11]. More importatly, WPT makes it possible to elimiate the last wires coectig mobile devices to charger, thus, WPT has potetial to icrease the mobility of wireless devices sigificatly ad improve user experiece, besides, i a commuicatio etwork with WPT, both trasmitters ad receivers ca be supported without exteral power sources. I additio, WPT is a cotrollable power trasfer method ad it ca be evirometal friedly. I RF-eabled WPT, receivers harvest eergy from ambiet eergy trasmitter. The eergy source is stable, it is also fully cotrolled by the trasmitter. Besides, trasmitter ca also harvest eergy from the surroudig eviromet, like solar eergy or wid eergy, the the harvest eergy ca be used for maipulatig trasmitter itself ad eable wireless eergy trasmissio betwee trasmitter ad receiver. Although WPT provides a lot of possibilities i both academic ad idustrial area, it still faces a lot of challeges. I compariso WPT with traditioal wireless commuicatio, eergy sesitivity i WPT is much more importat tha i traditioal wireless commuicatio. I traditioal wireless commuicatio, the goal is to esure trasmit sigal ca be decoded at receiver side, so the amout of power reachig the receiver is ot so importat as log as it is sufficiet so the sigal to oise ratio is high eough that the iformatio ca be received itelligibly [4] [5] [12]. I cotrast, the amout of eergy received at receiver side ad the eergy efficiecy (fractio of trasmitted eergy that is received) are importat i WPT [4]. 5

8 However, the fact is power trasfer efficiecy i WPT is relatively low as receivers ca oly harvest a small amout of eergy emitted by the trasmitter due to path loss ad chael fadig. The low coversio rate of radio frequecy (RF) sigal to direct curret (DC) also results i iefficiet WPT. Besides, as WPT is iitially used for facilitatig eergy harvestig for critical applicatios with guarateed quality of service (QoS), it requires strog electromagetic radiatio which has raised serious public health cocers. Further developmet of WPT is slow because of these abovemetioed costrais ad problems. Thaks to recet advaces i silico techology ad multiple atea techology that make it possible to solve those above-metioed problems. The breakthrough i silico techology has reduced eergy demad of simple wireless devices sigificatly ad multiple atea techology also cotributes a lot for reducig total trasmit power. Moreover, multiple-atea devices eable eergy beamformig techiques which could shape the trasmit waveform at each atea ad cotrol the collected waveforms at receiver side. The beamformig techiques ca icrease sigal receptio at receiver side so that the eergy efficiecy ca be improved. 1.2 Simultaeous wireless iformatio ad power trasfer(swipt) As RF sigals ca be used ot oly i eergy trasfer but iformatio trasfer, so, recetly, there has bee a icreasig iterest i the itegratio of wireless power trasfer ad iformatio trasmissio. It also motivates the developmet of a ew techology amed simultaeous wireless iformatio ad power trasfer (SWIPT). SWIPT has ability to achieve sigificat gais i spectral efficiecy as it uses the same RF sigal for sigal processig ad eergy harvestig simultaeously, for example, wireless implats could be charged ad calibrated cocurretly with the same sigal, ad wireless sesor odes ca be charged with the cotrol sigal they received from the access poit [13]. As SWIPT has potetial to achieve high spectrum efficiecy, it seems more attractive tha covetioal wireless iformatio trasfer, however, further developmet of SWIPT still faces a lot of challeges ad the implemetatio of efficiet SWIPT has 6

9 differet requiremet o the desig of commuicatio system. Like traditioal commuicatio system, system with SWIPT also suffers from chael fadig ad path loss which result i performace loss. Specifically, i practice, efficiecy of power trasfer i SWIPT is low ad the distace of power trasfer i SWIPT is relatively short. Iefficiet SWIPT is ot able to provide guarateed quality of service (QoS). I additio to the covetioal QoS requiremets such as throughput, eergy efficiecy, fairess, ad delay, efficiet trasfer of eergy plays a importat role as a ew QoS requiremet i SWIPT systems. Although eergy efficiecy i SWIPT ca be simply improved by icreasig the power of trasmit sigal, a higher trasmit power leads to a larger susceptibility for iformatio leakage due to broadcast ature of wireless chaels [14]. Therefore, iformatio security is a importat issue i SWIPT system. Besides, desig of receiver side i SWIPT is differet from covetioal commuicatio etwork ad WPT, as receivers i traditioal commuicatio etwork is uable to simultaeously decode iformatio ad harvest eergy from the same RF sigal emitted by the trasmitter due to differet ature of sigal processig ad receiver sesitivities required, it also puts some difficulties i the implemetatio of iformatio ad power trasfer joitly. To improve power trasfer efficiecy i SWIPT, fadig effects eed to be abated. Multiple-ateas techology is a promisig way to ehace performace over fadig chael ad the use of this kid of techology i SWIPT is motivated by the fact that they have the potetial to improve the eergy efficiecy of wireless power trasfer sigificatly [14]. The use of multiple-atea techologies, more specifically, beamformig techologies ca alig the trasmit sigal to a certai power receiver to icrease sigal receptio at receiver side. The implemetatio of multiple-ateas techologies requires chael state iformatio (CSI) is kow to the trasmitter side, geerally, there are two commoly used CSI acquisitio approaches i multiple-ateas systems. Oe is frequecy divisio duplex (FDD) ad the other is time divisio duplex (TDD). 7

10 I FDD, CSI is obtaied by feedback from receivers to trasmitters, specifically, the feedback is determied by a quatizatio codebook ad accuracy of CSI is determied by size of the quatizatio codebook. Although a larger size quatizatio codebook results i more accurate CSI, it also itroduces feedback overheads. Therefore, it is ot efficiet to icrease size of quatizatio codebook to obtai accurate CSI. Istead, for more accurate CSI acquisitio at trasmitter side, the amout of feedback from receivers eed to be icreased. I TDD, CSI ca be obtaied by makig use of chael reciprocity directly. I compariso with FDD, TDD saves feedback resources but trasceiver hardware impairmet may result i performace loss i TDD. As receivers i traditioal commuicatio system may ot be able to facilitate SWIPT, so the implemetatio of SWIPT requires special receiver structures. There are several special receiver structures that ca facilitate SWIPT like separated receiver structure, time switchig receiver structure, power splittig receiver structure, atea switchig receiver structure ad spatial switchig receiver structure, etc. I the followig, we provide a overview of these special receiver structures. Separated Receiver Structure I separated receiver architecture, a eergy harvestig (EH) circuit ad a iformatio decodig (ID) circuit are implemeted as two separated receivers with separated ateas, which are served by a commo multiple atea trasmitter [14]. Both iformatio receiver ad eergy harvestig receiver have ability to sed feedback, for example, iformatio about their eergy demad to the trasmitter side, so that the trasmitter ca take use of the limited system resources efficietly. Besides, the availability of chael state iformatio (CSI) at trasmitter side also lead to efficiet use of limited system resources. Therefore, the use of feedback from receiver side ad CSI ca optimize achievable eergy rate ad iformatio rate. Compared with other special receiver structures, separated receiver structure requires low complexities i hardware implemetatio. It ca be simply implemeted usig off-the-shelf compoets for the two idividual receivers [14]. 8

11 Time Switchig I time switchig receiver structure, each receiver is equipped with a iformatio decoder ad a eergy harvester. The iformatio decoder ad the eergy harvester are cotrolled by a switch. I each time slot, a time switchig receiver ca switch betwee eergy harvestig circuit ad iformatio decodig circuit. I other words, i each time slot, time switchig receiver ca oly use the received sigal for either eergy harvestig or iformatio decodig. Time switchig structure ca be facilitated by simple hardware, but it has high demad o accurate iformatio or eergy schedulig. Power Splittig Receivers equipped with power splittig structure has ability to split the received sigal ito two power streams accordig to a certai power splittig factor. The, these two streams are set to a eergy harvester ad a iformatio decoder. Oe stream is used for harvestig eergy ad the other stream is used for decodig the modulated iformatio. Compared with time switchig techique, power splittig techique has higher demad o hardware, besides, the power splittig factor eed to be optimized accordig to differet requiremet o the eergy demad of iformatio decodig ad eergy harvestig of the receiver. Although complexities of power splittig structure are higher tha time switchig structure, it achieves sigal processig ad eergy harvestig simultaeously i each time slot by usig the same RF sigal. Atea Switchig Atea switchig scheme is motivated by usig atea arrays i geeratig DC powers for reliable device operatio. I geeral, ateas at receiver side are divided ito two groups, oe group is used for ID ad the other group is used for EH [15]. These two groups are cotrolled by a switch for implemetatio of SWIPT i atea domai. The atea switchig techique requires the solutio of a optimizatio problem i each commuicatio frame for optimal assigmet of atea elemets to iformatio decodig ad eergy harvestig [13]. Spatial Switchig 9

12 Spatial switchig (SS) techique ca be applied i multiple-iput ad multipleoutput (MIMO) cofiguratios ad achieves SWIPT i the spatial domai by exploitig multiple degrees of freedom (DoFs) of the iterferece chael [16]. As MIMO chael ca be decomposed ito parallel eigechaels ad each eigechael is orthogoal to other eigechaels, so each chael ca be exploited for either iformatio decodig or eergy harvestig. At the output of each eigechael, there is a switch that drives the chael output to either covetioal decodig or the rectificatio circuit [16]. I this article, we focus o a SWIPT system which is powered by hybrid eergy source icludig o-reewable eergy source ad reewable eergy source. This system cosists of a multiple-atea trasmitter ad several sigle-atea receivers. I the system, separated receiver structure is equipped at receiver side. I other words, there are two types of receivers i the system, oe is iformatio receiver, the other oe is eergy harvestig receiver. The iformatio receiver has power splittig structure, so it ca harvest eergy ad decode iformatio from the same RF sigal emitted by the trasmitter simultaeously while the eergy harvestig receivers ca oly harvest eergy from the trasmit sigal. Our aim i this article is to maximize eergy efficiecy betwee trasmitter ad iformatio receiver. This ca be implemeted by takig use of total harvest power from hybrid eergy source properly, besides, the use of multiple atea techology, more specifically, beamformig techology ad cotrol of power splittig factor of the iformatio receiver ca also result i sigificat icrease i eergy efficiecy. 10

13 Chapter 2 System Model 2.1 System model I this article, we cosider a SWIPT system which is powered by hybrid eergy source, see i figure 2.1. Figure 2.1 The use of reewable eergy source like solar or wid make the system gree. Stability of the system is esured by usig o-reewable eergy source. I the dow lik of this system, there are J umbers eergy harvestig receivers ad oe iformatio receiver. Both eergy harvestig receiver ad iformatio receiver are sigle atea devices. The eergy harvestig receiver ca oly harvest eergy from RF sigal emitted by the trasmitter while the iformatio receiver has power splittig structure which eables it harvest eergy ad decode iformatio from the same RF sigal emitted by the trasmitter joitly. At trasmitter side, the trasmitter is equipped with multiple ateas. I the SWIPT system, the trasmissio process is divided ito time slot, i each time slot, the trasmitter first harvest eergy from both reewable eergy source ad oreewable eergy source. The it uses the harvested power to maipulate itself 11

14 properly ad trasmit RF sigal to the receiver side. I covetioal commuicatio system, we maily focus o whether the trasmit sigal could be decoded at receiver side, as high sigal to oise ratio (SNR) of the receive sigal lead to better result i iformatio decodig, so the trasmitter always emit RF sigal that the amout of eergy cotaied by the RF sigal is far beyod the eergy demad for iformatio decodig at receiver side. I other words, large amout of eergy is wasted which lead to iefficiet use of limited system resources as trasmitter lacks iformatio of eergy demad of receivers. However, decode iformatio is ot the oly goal i SWIPT system. Both iformatio rate ad eergy efficiecy are importat criteria i SWIPT. It is obvious to see that iefficiet use of limited system resources leads to low eergy efficiecy. I order to improve eergy efficiecy i SWIPT system, separated receiver structure is equipped at the receiver side, as both iformatio receiver ad eergy harvestig receiver have ability to sed iformatio about their eergy demad to the trasmitter, so the trasmitter ca avoid waste i power trasmissio ad assig the limited system resources appropriately. The receive sigals at iformatio receiver ad the j th eergy receiver are show as below. y = h H x + z a (2.1) y j = g H j x + z j j {1 J} (2.2) where x deotes the trasmit sigal vector, h H is the chael vector betwee trasmitter ad iformatio receiver ad g H j is the chael vector betwee H trasmitter ad the j th eergy receiver. Both h ad g H j has fadig ad path loss. z a ad z j are additive Gaussia oises (AWGNs) of iformatio receiver ad the 2 j th eergy harvestig receiver respectively, with zero mea ad variace σ at ad 2 σ atj, respectively. As the trasmitter is equipped with multiple ateas, so, we use beamformig 12

15 techique at trasmitter side to icrease sigal receptio at receiver side ad achieve efficiet SWIPT. The trasmit sigal vector ow is give by x = w s (2.3) where s is the trasmit sigal ad w is the correspodig beamformig vector. We assume without loss of geerality so that the eergy of trasmit sigal is ormalized to 1. At receiver side, desig of differet beamformig vector ca result i sigificat differece o efficiecy of eergy harvestig ad iformatio decodig. Besides, as the iformatio receiver has power splittig structure, we eed to take the effect of power splittig factor ito cosideratio. We cosider the iformatio receiver splits the received sigal ito two power streams accordig to power splittig factor ρ ad 1 ρ for iformatio decodig ad eergy harvestig respectively. See figure 2.2. Figure 2.2 We assume the power splittig uit is a perfect passive aalog device ad the use of it does ot itroduce extra oise to the iformatio receiver. The rage of power splittig factor is from 0 to 1 which idicates that the power splittig uit does ot itroduce ay extra power. The harvested power ca be stored or used as a power supply for sigal processig through chargig the rechargeable battery whe the eergy provided by trasmitter to iformatio receiver is ot eough for iformatio decodig. 2.2 System optimizatio I this sectio, we formulate our optimizatio problem for the SWIPT system. We aim to maximize eergy efficiecy betwee iformatio receiver ad trasmitter. Defiitio of eergy efficiecy is give by 13

16 log 2 (1 + SNR) (2.4) P where SNR is the sigal to oise ratio of iformatio receiver, P is the total cost at the trasmitter side. P cosists of two parts eergies. Oe part is eergy cosumed for maipulatig trasmitter itself. The other part is power harvested by trasmitter from other eergy resources. It is show that icrease value of SNR at receiver side ad decrease amout of either total trasmit power or power cosumed by trasmitter itself ca improve eergy efficiecy betwee iformatio receiver ad trasmitter. The trasmissio is divided ito time slot, we use i to idicate the i th time slot. SNR i idicates SNR of iformatio receiver at the i th time slot which is give by SNR i = ρ h H i w i 2 2 ρσ at + σ2 s i {1 } (2.5) where ρ is the power splittig factor for iformatio decodig, h i H is the Hermitia chael betwee iformatio receiver ad trasmitter at i th time slot ad w i is the 2 correspodig beamformig vector at i th time slot. σ at is atea oise power ad σ s is sigal processig oise power. Both σ at ad σ s are costat. It is obvious to see that maximizatio of SNR i requires high receive sigal power ad low oise power, this ca be implemeted by cotrol the power splittig factor ρ ad beamformig vector w i. It is feasible to maximize receive sigal power of iformatio receiver by the desig of beamformig vector. However, icreasig of power splittig factor results i icreasig i both receive sigal power ad atea oise power, i order to get high SNR, the icreasig rate of receive sigal power is required to exceed the icreasig rate of atea oise power. At trasmitter side, the sum of power cosumed by trasmitter itself ad eergy harvested by trasmitter from other eergy resources is give by P = P c + P 1,i + P 2,i (2.6) P c represets the power cosumed by trasmitter itself, here, P c is a costat. P 1,i represets harvest power from o-reewable eergy source at the i th time slot ad P 2,i represets harvest power from reewable eergy source at the i th time slot. The use of eergy harvested from reewable eergy source is free of charge as we ca use 14

17 the harvested reewable eergy source directly without extra pay. The use of harvested eergy from o-reewable eergy source at trasmitter side requires extra cost, for example, if the harvested eergy by the trasmitter is produced by fossil fuels, at least, the use of this harvested eergy eed to pay for the cosumptio of fossil fuels. We iitially aim to reduce the cost at trasmitter side, icreasig use of reewable eergy source is a promisig way to achieve this goal, however, this will result i decrease of system stability. As i some occasios, the amout of eergy provided by reewable eergy source is ot able to support the trasmitter, sometimes, the total harvest eergy from reewable eergy source could eve be zero, for example, it is impractical to harvest solar eergy whe the weather is cloudy. I order to esure stability of the SWIPT system ad maximize eergy efficiecy, there is a trade-off betwee the use of reewable eergy source ad o-reewable eergy source. The objective fuctio of the optimizatio problem is give by maximize i=1 log 2(1 + SNR i ) P c + (P 1,i + P 2,i ) i=1 (2.7) s. t. η j g i H w i 2 P jmii, η(1 ρ) h i H w i 2 P mii i {1 } i {1 } 0 ρ w 2 P 2, + P 1, + [( P 2,i + P 1,i ) ( w 2 )] i=1 i=1 The amout of power harvested by iformatio receiver ad eergy harvestig receiver should larger or equal to the miimum required power of these two kids of receivers. P jmii represets the miimum required power of the j th eergy harvestig receiver at i th time slot, P mii represets the miimum required power of iformatio receiver at the i th time slot. I this article, these two values are kow to the trasmitter side, as both iformatio receiver ad eergy harvestig receiver ca sed feedback which cotais their demad of eergy to the trasmitter. η j ad η is 15

18 coversio rate of RF sigal to direct curret (DC) of the j th eergy harvestig receiver ad iformatio receiver respectively. w 2 is the total power trasmit to the receiver side, the value of it should be small or equal to the amout of power cotaied i the trasmitter. As the trasmissio process is divide ito time slot, so the amout of power cotaied i trasmitter at the th time slot is comprised by two parts. Oe part is harvest power from both reewable eergy source ad o-reewable eergy source at th time slot which is represeted as P 2 + P 1, the other part is the remaied power from the 1 st time slot to the ( 1) th time slot. This remaied 1 power is represeted as ( i=1 P 2i + P 1i ) ( i=1 w 2 ), where P 2i + P 1i 1 1 i=1 is the remaied power from the 1 st time slot to the ( 1) th time slot at trasmitter 1 i=1 side ad w 2 is the trasmitted power from the 1 st time slot to the ( 1) th time slot. Chapter 3 Prelimiary Simulatio I this sectio, we oly cosider trasmissio process at the 1 st time slot. We first focus o the effects of icreasig power splittig factor ad weight of beamformig vector o SNR of iformatio receiver. SNR of iformatio receiver at the 1 st time slot is give by SNR 1 = ρ h H 1 w ρσ at + σ2 s (3.1) I prelimiary simulatio, we assume chael betwee trasmitter ad iformatio receiver is time ivariat over the 1 st time slot ad the eergy beam aligs with the directio of the strogest eigemode of the chael matrix. We set chael gai of h 1 equal to 1, the rage of power splittig factor is from 0 to 1 ad the rage of beamformig vector is from 0 to 10. Atea ad sigal processig 2 2 oise power are costat. I the prelimiary simulatio, we set value of σ at ad σ s equal to ad respectively. The relatioship betwee power splittig factor ad beamformig vector with SNR 1 is show i figure 3.1 ad figure 3.2 respectively. 16

19 Figure 3.1 Figure 3.2 Ituitively, the result shows that SNR 1 as a fuctio of power splittig factor ρad beamformig vector w 1 is a mootoe icreasig fuctio. Although icreasig power splittig factor lead to icrease i both receive sigal power ad atea oise power, icrease of beamformig vector weight make the icreasig rate of receive sigal power is far beyod the icreasig rate of atea oise power. As SNR 1 is a mootoe icreasig fuctio, so log 2 (1 + SNR 1 ) is also a mootoe icreasig fuctio, but the relatioship betwee power splittig factor ad beamformig vector with the efficiecy of iformatio trasmissio betwee trasmitter ad iformatio receiver is ot mootoe icreasig. We ow focus o effect of beamformig ad power splittig o efficiecy of iformatio trasmissio 17

20 ad we also cosider effect of differet use of both reewable eergy source ad oreewable eergy source o efficiecy of iformatio trasmissio. s. t. We still cosider the 1 st slot, ad the objective fuctio is give by, maximize log 2(1 + SNR 1 ) P 1,1 + P 2,1 + P c (3.2) C1: η j g 1 H w 1 2 P jmi1 C2: η(1 ρ) h H 1 w 1 2 P mi1 C3: 0 ρ 1 C4: w 1 P 2,1 + P 1,1 For the calculatio of SNR 1, value of chael gai, rage of power splittig factor ad beamformig vector remai same as the previous calculatio of SNR 1. P c represets power cosumed for maipulatig trasmitter itself, it is a costat ad i prelimiary simulatio, we set the value of it equal to 1W. We assume all settigs of parameters i the objective fuctio satisfy costraits C 1 to C 4. From previous aalysis, the use of reewable eergy source ca icrease efficiecy of iformatio trasmissio rate sigificatly, i the prelimiary simulatio, we first set the amout of harvest power from reewable eergy source equal to zero ad chage the value of harvest power from o-reewable eergy source. The, we cosider a situatio that the harvest power from reewable eergy source is eough to support sigal trasmissio from trasmitter side to receiver side ad maipulatio of trasmitter. The relatioship betwee weight of beamformig vector ad efficiecy of iformatio trasmissio is show i figure

21 Figure 3.3 It is obvious to see the efficiecy of iformatio trasmissio does ot icrease with the icrease of beamformig vector weight. With icreasig weight of beamformig vector, the efficiecy of iformatio trasmissio first keeps icreasig, the, it starts to decrease after reachig its maximum value. I occasio 1, 2 ad 3, we cosider the trasmitter do ot harvest power from reewable eergy source, so, the value of P2 is equal to 0. P1 is harvest power from o-reewable eergy source, we assume at the 1 st time slot, eergy harvest from o-reewable eergy source is used up by the trasmitter, so there is o remaied power ad the value of P1 is submitted ito the objective fuctio directly. From figure 5, we ca see that icrease use of o-reewable eergy source lead to efficiecy of iformatio trasmissio decrease. I occasio 4, we cosider the trasmitter harvest 1W eergy from reewable eergy source, ad we assume 1W is eough to support the trasmitter work properly ad trasmit sigal to receiver side so we do ot eed to use eergy harvest from oreewable eergy source. Besides, eergy harvest from reewable eergy source is used up by trasmitter. Compare occasio 4 with occasio 2, it is easy to see uder the same level of eergy demad, the use of reewable eergy source ca icrease the efficiecy of iformatio trasmissio sigificatly ad lead to decrease of beamformig vector weight. 19

22 The relatioship betwee power splittig factor ad efficiecy of iformatio trasmissio is show i figure 3.4. Figure 3.4 The efficiecy of iformatio trasmissio first icreases with the icrease of power splittig factor, after reachig its highest poit, the efficiecy of iformatio trasmissio starts to decrease. Chapter 4 Prelimiary Coclusio The result of prelimiary simulatio idicates that it is feasible to maximize efficiecy of iformatio trasmissio betwee trasmitter ad iformatio receiver by appropriate desig of power splittig factor ad beamformig vector. Besides, the use of o-reewable eergy source i the SWIPT system icrease the efficiecy of iformatio trasmissio sigificatly. Oce we get the maximum value of efficiecy of iformatio trasmissio, we ca set the correspodig value of beamformig vector, power splittig factor to reach perpetual maximizatio of efficiecy of iformatio trasmissio. 20

23 Chapter Solvig the optimizatio problem mathematically From the result of prelimiary simulatio, we ca see the objective fuctio is a cocave fuctio. More specifically, the umerator is cocave ad the deomiator is affie. As a costraied cocave maximizatio problem is equivalet to a costraied covex miimizatio problem (through a sig chage i the objective), so, alteratively, we may view the problem i (9) as the miimizatio problem subject to all the costraits. Moreover, costraits C1 ad C2 are o-covex, which does ot facilitate the desig of a computatioally efficiet beamformer, so, we adopt trace operator ad rewrite (2.7) i the followig form. For simplicity, we assume ρ is give. miimize i=1 log 2(1 + SNR i ) Pc + (P 1,i + P 2,i ) i=1 (5.1) s. t. where SNR i = ρtr(h iw i ) 2 ρσ at + σ2 s i {1 } C1: Tr(G i W i ) P jmi i η j i {1 } C2: Tr(H i W i ) P mi i η(1 ρ) i {1 } C3:( i=1 Tr(W i )) i=1 (P 2,i + P 1,i ) 0 C4: W 0 C5: Rak(W) 1 Where W = w i w i H, G = g i g i H ad H = h i h i H (h i C N T 1, g i C N T 1 ). Solvig problem (5.1) is still ot easy due to the presece of iequality costraits, the o-covexity of costrait 5 as well as fractioal patter of the objective fuctio. Accordig to Werer Dikelbach theorem [17]: For the followig problems: 21

24 max { N(x) D(x) x S} (5.2) max{n(x) q 0 D(x) x S} for q E 1 (5.3) Where E is the Euclidea space of dimesio 1 ad S be a compact ad coected subset of E. if ad oly if q 0 = N(x 0) D(x 0 ) = max { N(x) D(x) x S} F(q 0 ) = F(q 0, x 0 ) = max{n(x) q 0 D(x) x S} = 0 Where q 0 is maximum of problem (5.2) ad x 0 is a solutio vector of problem (5.2) ad this theorem is still valid if we replace max by mi [17] Hece, we further rewrite the objective fuctio, s. t. C1 C5 miimize i=1 log 2 (1 + SNR i ) q (P 1,i + P 2,i ) Pc (5.4) i=1 + For computatioal efficiet, we let τ i = Tr(W i G i ) ad add aother costrait C6, where C6 is Evetually, the optimizatio problem is C6: τ i Tr(W i G i ) miimize log 2 (1 + ρτ i i=1 q i=1 (P 1,i + P 2,i ) + Pc (5.5) s. t C1 C6 ρσ at +σ s ) ow the oly difficulty of solvig the optimizatio problem is due to o-covexity of costrait 5. Costraits 5 is a combiatorial costrai [18-26]. This costrait is required for fidig a global optimal solutio [18]. I order to elimiate the effect of the ocovexity, we adopt a semidefiite programmig(sdp) relaxatio to (5.3) by relaxig costrait 5: Rak(W) = 1 [18]. The we get miimize log 2 (1 + ρτ i i=1 q i=1 (P 1,i + P 2,i ) + Pc (5.6) s. t C1 C4, C6 ρσ at +σ s ) C5: Rak(W) = 1 22

25 Accordig to the basic priciples of optimizatio theory, if solutio of W is a rak-oe matrix, the it is the optimal solutio of the origial problem. However, rak of W may larger tha oe ad this lead to the costrait relaxatio might ot be tight. I the followig, we proof the tightess of the origial problem after SDP relaxatio via dual problem ad KKT coditios of problem (5.6). We also eed the Lagragia fuctio of problem (5.6), which is ρτ i ) ρσ 2 at +σ 2 s D(W i, α, β, γ, δ, Y) = i=1 log 2 (1 + q (t) ( i=1 (P 1,i + P 2,i ) + Pc) + α( P jmi i η j Tr(G i W i )) + β( P mi i Tr(H η(1 ρ) iw i )) + γ i (( i=1 Tr(W i )) i=1(p 2,i + P 1,i ))+δ(τ i Tr(W i H i )) Tr(WY) (5.7) Where α 0 is the dual variable for the miimum required power of eergy receiver i C1. β 0 is the dual variable for the miimum required power of iformatio receiver i C2. γ i 0, i (1 ) is the dual variable vector associated with the maximum trasmit power i costrait C3. The matrix Y 0 is the dual variable for the semi-defiiteess costrait o matrix W. Now, we do the first order differetiatio to W. By KKT coditio, we the get: Y = α G i β H i + I NT γ i δ H i i=1 = α G i + I NT γ i (β + δ )H i i=1 = A (β + δ ) H i (5.8) where α, β, δ, γ i, are optimal dual variables ad we let A = α G i + I NT Besides, by complemetary slackess coditio, we get Y W i = 0 (5.9) i=1 γ i. (5.9) is satisfied whe the colums of W i lay i the ull space of Y [19]. Therefore, if Rak(Y ) = N T 1, the the rak of optimal W 0 must be oe ad the optimal w i ca be obtaied through eigevalue decompositio o W [19]. Now, we prove the tightess of relaxed optimizatio problem by cotradictio that 23

26 A is a full rak matrix with rak N T [19]. We assume that A is a rak deficiet matrix with at least oe zero eigevalue ad we deote the associated eigevector as u. Without loss of geerality, we create a matrix U = u u H from the eigevector. We the multiply U at both side of (5.8) ad apply trace operator, we get Tr(Y U) = Tr(AU) (β + δ )Tr(H i U) (5.10) = (β + δ )Tr(H i U) (5.11) (5.11) is obtaied because u is geerated from the ull space of A ad Tr(AU) = Tr(u H Au ) = 0. We the examie the sigs of both side of the equality i (5.11). We first cosider the right-had side. Recall costrait 6 is C6: τ i Tr(W i G i ) ad the optimal coditio of costrait 6 is τ i = Tr(W i G i ), accordig to KKT coditio, uder the optimal coditio, δ > 0, hece β + δ > 0. Next, we approve Tr(H i U) > 0. Sice h i ad g i are statistically idepedet. As a cosequece, the probability that H i ad G i share the same ull space is zero which yields Tr(H i U) 0. Moreover, h i h H i is a positive semidefiite matrix thus Tr(H i U) must be positive ad the right-had side of (5.11) must be egative. For the left-had side of (5.11), Y is a positive semidefiite matrix ad the lefthad side of (5.11) is o-egative, which cotradicts the sig of the right-had side of (5.11). Therefore, matrix A must be a full rak matrix with rak N T [19]. We the have Rak(Y ) + Rak((β + α )H i ) Rak(Y + (β + α )H i ) = Rak(A) = N T Rak(Y ) N T 1 (21) (21) idicates that Rak(Y ) is either N T or N T 1. I order to satisfy miimum required power at receiver side, W should ot equal to zero. Hece, Rak(Y ) = N T 1 ad Rak(W ) = 1 24

27 25

28 5.2 Coclusio System parameters Item value ρ: power splittig factor of iformatio receiver 0.5 η: coversio rate of iformatio receiver 0.5 η j : coversio rate of j th eergy receiver 0.5 P 1i : Harvested power from o-reewable eergy resources P 2i : Harvested power from reewable eergy resources P c : Power cosumed by trasmitter 3dB 3dB 1.5e-4 For simulatio part, Dikelbach method is applied. Importat assumptio 1. From the 1 st time slot to the th time slot, for ay give, chael h i ad chael g i are time ivariat. 2. Chael h i ad chael g i are statistically idepedet. 3. P 1,i ad P 2,i is give ad for ay ith time slot, value of P 1,i ad P 2,i are fixed. 4. For the simulatio part, we set the umber of iformatio receiver ad eergy receiver i the system model all equal to 1 Figure 5.1 Figure 5.1 shows that whe icreasig the miimum required power at receiver 26

29 side, eergy efficiecy will decrease. Moreover, icreasig umber of ateas at trasmitter side ca icrease the eergy efficiecy sigificatly. The gap betwee N T = 2 ad N T = 4 is larger tha the gap betwee N T = 4 ad N T = 6 due to chael hardeig. Figure 5.2 Figure 5.2 illustrates that with icreasig of miimum required power, outage probability of chael is icreasig. The icreasig rate of chael outage probability is decreasig because the proposed optimizatio which utilities the resources efficietly. 27

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