the price of self-sustainability for block transmission ... · e.g., opportunistic networks [8],...

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO. 8, AUGUST 2015 1549

The Price of Self-Sustainability for BlockTransmission Systems

Marco Maso, Member, IEEE, Subhash Lakshminarayana, Member, IEEE, Tony Q. S. Quek, Senior Member, IEEE,and H. Vincent Poor, Fellow, IEEE

Abstract—In this work, the self-sustainability of block trans-mission systems is analyzed. In particular, orthogonal frequencydivision multiple access (OFDMA) is taken as a reference, due toits popularity and rather simple signal model. More precisely, ageneralized variant of this scheme in which the transmitted signalis obtained as the sum of an OFDMA and a cognitive interferencealignment (CIA) component, acting as an energy bearer, is consid-ered. In this scenario, the self-sustainability of the transmissionis made possible by the flexibility of the adopted strategy andthe introduction of a novel energy harvesting OFDMA receiver.Both the feasibility conditions for the self-sustainability and theoptimal power allocation to maximize the effectiveness of theenergy transfer performed through the CIA signal are derived.Numerical results show that full self-sustainability can be achievedfor several system configurations and channel statistics. However,this comes at the cost of a rate penalty with respect to a standardclassic OFDMA transmission, which is termed the price of self-sus-tainability. A study of the relationship between the performance ofboth the energy and the information transfer is carried out. A CPsize that minimizes the price of self-sustainability can be found forall the considered configurations.

Index Terms—Energy harvesting, block transmission, self-sustainability, green communications, OFDMA, CIA.

I. INTRODUCTION

W ITH the proliferation of mobile battery-powered devicesin the last decade, energy consumption is emerging as

a key factor in the performance of modern wireless networks.New generations of mobile devices are typically character-ized by superior energy requirements with respect to (w.r.t.)previous generations. Unfortunately, the capacity of current-generation batteries increases at a very slow rate if compared tothe advancements in both signal processing and communicationtechnologies. The energy requirements imposed by the latterare hardly satisfied by the former. In practice, the limited

Manuscript received April 1, 2014; revised September 15, 2014; acceptedDecember 16, 2014. Date of publication January 14, 2015; date of currentversion July 14, 2015. This work was partially supported by the SUTD-MITInternational Design Centre under Grant IDSF1200106OH, the A∗STAR SERCunder Grant 1224104048, the SUTD-ZJU Research Collaboration Grant underSUTD-ZJU/RES/01/2014, and by the U. S. National Science Foundation underGrant ECCS-1343210.

M. Maso is with the Mathematical and Algorithmic Sciences Lab,Huawei France Research Center, 92100 Boulogne-Billancourt, France (e-mail:[email protected]).

S. Lakshminarayana and T. Q. S. Quek are with Singapore University ofTechnology and Design, Singapore 138682 (e-mail: [email protected];[email protected]).

H. V. Poor is with the Department of Electrical Engineering, PrincetonUniversity, Princeton, NJ 08540 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSAC.2015.2391752

energy supply provided by batteries represents an unavoidablebottleneck for the performance of mobile devices, and thussignificantly impacts both the user experience and the networklifetime.

Among the proposed solutions to cope with this problem,scavenging energy from virtually cost-free sources such asambient heat, light or others, is seen as a promising approach toreduce the energy bottlenecks of modern wireless networks. Forthese reasons, energy harvesting techniques have gained promi-nence in the signal processing community, growing from theo-retical concepts into devices for powering mobile electronics[1]. In this context, radio frequency (RF) signals have recentlybecome the subject of increasing interest in the wireless com-munication community as possible energy bearers. Althoughhistorically considered only as information carriers, these sig-nals can be effectively used as means to realize wireless powertransfer (WPT) between devices [2]. This transfer can be real-ized by exploiting various features of the electromagnetic field.The two most prominent solutions to achieve power transferare by means of either near-field resonant inductive coupling[3] or magnetic far-field [4]. Concerning the latter, two mainapproaches have been studied so far: the so-called microwavepower transfer (MPT), in which energy is transmitted fromone device to another wirelessly, and simultaneous wirelessinformation and power transfer (SWIPT), in which the sameelectromagnetic field is used as a carrier for both energy andinformation [5], [6]. The focus of this work is on the latterapproach, due to its potential in terms of both transmissionrange and information transfer capabilities.

A. Related Work

After the pioneering work by Varshney, that first introducedthe idea of SWIPT [5], several studies have characterized thetrade-off between information and the energy transfers [7]. Thestudy of possible solutions to achieve an effective SWIPT hasbeen extended to many scenarios and system configurations,e.g., opportunistic networks [8], multi-antenna systems [9],wireless sensor networks [10] and so on. A common and funda-mental constraint that characterizes all these studies, regardlessof the considered scenario, is the inability of practical circuitsto simultaneously harvest energy and decode the informationfrom the same signal [9]. To overcome this problem, poweror time splitting strategies at the receiver have been proposed.This is true also for the recent contributions that took a stepforward towards the realization of SWIPT in cellular networksand focused on a widely adopted block transmission technique

0733-8716 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

1550 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO. 8, AUGUST 2015

in modern telecommunication standards, such as orthogonalfrequency division multiple access (OFDMA) [11], [12]. Inpractice, as far as SWIPT in OFDMA transmissions is con-cerned, the state-of-the-art solutions require the presence of apower splitting unit at the receiver and possibly adapting theactions of the latter depending on its activity state (idle/active).

B. Summary of Our Contribution

In this work, we present and characterize a method to in-crease the lifetime of a receiver in multi-user systems adoptingblock transmission schemes such as OFDMA, time-divisionmultiple access (TDMA) schemes such as the one proposedin [13], or single carrier frequency-division multiple access(SC-FDMA) [14], to name a few. In particular, we will targetan OFDMA system, owing to both its popularity in many recentwireless communication standards, and its rather simple signalmodel description. We start by focusing on a legacy OFDMAimplementation. We observe that every received OFDMA sym-bol at the receiver includes a redundant part, i.e., the cyclicprefix (CP), prepended to the block to provide an effectivemeans to combat inter-carrier and inter-block interference (ICIand IBI) [15]. This redundant portion of the signal is typicallydiscarded by conventional OFDMA receivers to simplify thefurther decoding operations, inducing a significant waste ofenergy.

To cope with this issue, we propose a novel energy harvestingOFDMA receiver that does remove the CP but instead uses it toscavenge useful energy to power the subsequent digital signalprocessing blocks. This operation can be interpreted in a two-fold sense:

1) It induces a paradigm shift for the OFDMA transmitter,whose transmission could now be seen as a SWIPT inwhich the received signal at the receiver carries bothinformation and energy.

2) Harvesting energy after the RF amplification at the re-ceiver allows to reduce the energy consumption at thereceiver, by recycling resources otherwise wasted in theCP removal operation.

Remarkably, in this context the inability of simultaneous energyand information extraction at the receiver, peculiar to everySWIPT-based system, is naturally addressed. In fact, in ourscheme, the receiver can harvest energy by exploiting the in-herent left-over resources of the OFDMA transmission (whichare intrinsic to the physical structure of any block transmission).Thus, no modification to the receiver actions depending on itsactivity state (idle/active) are required.

To increase the generality and flexibility of the proposedscheme, we consider a composite transmit strategy, in which thetransmitted signal is obtained as the sum of two components,i.e., a standard OFDMA signal and a cognitive interferencealignment (CIA) signal [16]. However, unlike [16], herein weanalyze a multi-user scenario and propose a systematic wayto both derive the linear precoding strategy and study theperformance of the system. Remarkably, this approach providesan evident advantage w.r.t. schemes like the one proposed in[11] and [12], in which the energy transfer/harvesting suf-fers from the same scalability constraints as the information

transfer/decoding. In fact, the amount of energy that eachreceiver can harvest, when the strategies in [11] and [12] areadopted, depends on both the number of receivers and on thescheduling at the transmitter. Conversely, the amount of energythat can be harvested by every receiver in the proposed schemesolely depends on the size of the CP (and does not dependon the number of active receivers). This guarantees its largerflexibility and the full scalability of the transferred energy perreceiver.

The concept of self-sustainability in terms of power con-sumption of the digital signal processing at the receiver isintroduced. Both the achievability of self-sustainability, andthe power allocation strategy that maximizes the effectivenessof the CIA signal as an energy bearer within the SWIPTare characterized analytically. Our numerical findings suggestthat noteworthy levels of self-sustainability can be achievedfor a wide range of realistic system configurations, undercertain reasonable conditions, e.g., indoor scenarios and shortdistances between the devices. However, a price in terms ofrate penalty w.r.t. a standard OFDMA transmission should bepaid due to the presence of the CIA signal. Accordingly, therelationship between the energy that can be harvested by thereceiver and the downlink rate of the transmission is inves-tigated. The performance of the composite OFDMA strategyare evaluated for different channel statistics, i.e., for severalpower delay profiles (PDPs) of the considered multi-pathchannels. Remarkably, a CP size that minimizes the price ofself-sustainability can be found for all the considered configu-rations. This confirms the potential of the proposed approachand provides a constructive way to select suitable system pa-rameters depending on the target performance of the SWIPT,in terms of both downlink (DL) rate and self-sustainability ofthe transmission.

The rest of the paper is organized as follows. In Section II, wedescribe the composite OFDMA signal model. In Section III,we describe and analyze the novel energy harvesting OFDMAreceiver. In Section IV, we discuss the self-sustainability ofthe transmission by means of SWIPT and characterize itsfeasibility. In Section V, we evaluate the performance of theproposed approach numerically, for several system configura-tions. Finally, conclusions and future research directions arediscussed in Section VI.

Notation: Lower and upper case bold characters representvectors and matrices, respectively. All vectors are columnvectors, unless otherwise stated. Given the matrix A ∈ C

M×N ,we denote by A= [a1,a2, . . . ,aN ] its structure, with ai ∈C

M×1,by tr[A] its trace, by [A]m,n its element at the mth row and thenth column and by ker(A) its kernel. 0N×M and IN stand for theall zeros and identity matrix, respectively. For a given vectora = (a1, . . . ,aN), d(a) = diag(a) indicates a diagonal matrixsuch that [d(a)]i,i = ai. The set of all positive real numbers, ex-cluding {0}, is denoted by R

+0 . The expectation of the random

variable x is denoted by E[x], whereas the expectation of thefunction f (x) over its variable x is denoted by Ex[ f (x)]. Givenx ∈ R+, we denote by �x� the smallest integer not less than x.Finally, given the sample outcome b, belonging to the samplespace B, and the event A, we define 1A{b} as the indicatorfunction of A.

MASO et al.: THE PRICE OF SELF-SUSTAINABILITY FOR BLOCK TRANSMISSION SYSTEMS 1551

II. SYSTEM MODEL

Consider a system in which an access point (AP) servicesM user terminals (UEs) in the DL. Let hi = [hi,0, . . . ,hi,l ] bethe frequency-selective, independent and identically distributed(i.i.d.) channels between the AP and the UEs. In particular,let the l + 1 channel taps be independent complex Gaussianrandom variables with zero means and variances given byκ2

j ∈ R, varying depending on the considered PDP, i.e., hi, j ∼CN(0,κ2

j), ∀ j ∈ [0, l]. We assume perfect channel state infor-mation (CSI) at the AP and that each of the devices is equippedwith a single antenna.

OFDMA is a natural choice for such a scenario, as corrobo-rated by its adoption in many modern standards. Nevertheless,the structure of any transmission based on orthogonal frequencydivision multiplexing (OFDM) entails a penalty in terms ofefficiency of both spectral and energy usage, due to the presenceof the CP. One possible way to address this issue is to adopta composite transmit strategy similar to what was proposedin [16], in which a more general signal model w.r.t. existingcontributions on the subject is adopted. In such a scheme, theAP transmits a signal obtained as the linear combination oftwo components: 1) the legacy OFDMA signal and 2) a secondsignal that can harmlessly coexist with the OFDMA signal inthe same band, i.e., a CIA signal. If we let x(o) and x(c) be theOFDMA and CIA signal, respectively, then the overall signaltransmitted by the AP can be expressed as

x = x(o) +ζx(c),

in which ζ ∈ {0,1} is a binary variable that controls the CIAtransmission, with the case ζ = 0 corresponding to the standardOFDMA transmission. Further details on the nature of x, x(o)

and x(c) will be provided in the following.We first focus on the OFDMA component of x. We define

N as the set of available sub-carriers such that |N| = N andL ∈ N as the CP size, such that L ≥ l. The resulting blocklength is N +L. Let Si ⊂ N be the set of sub-carriers allocatedto the ith UE, such that ∪N

i Si = N and Si ∩S j = /0, ∀ j �= i. We

define s(o) = [s(o)1 , . . . ,s(o)N ]T ∼ CN(0,P) as the input vector atthe AP, carrying the information symbols for the UEs, withP = d(p), where p = [pi, . . . , pN ]

T is an N-sized vector such

that pi = E[s(o)i s(o)∗i ] ∈ R+0 . Now, let [F](k+1,l+1) =

1√N

e−i2π klN

for k, l = {0, . . . ,N −1} be a unitary discrete Fourier transform(DFT) matrix and

A =

[0L,N−L IL

IN

]∈ R

(N+L)×N ,

be a CP insertion matrix. Thus, x(o) = [xo1, . . . ,x

oN+L]

T ∈ CN+L

can be written as

x(o) = AF−1s(o).

For the sake of simplicity, let us conclude the description of theOFDMA component of x by considering its contribution to the

received signal at the ith UE, i.e., yi ∈ CN+L. Let

Hi =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

hi,0 0 · · · 0 hi,l · · · hi,1...

. . .. . .

. . .. . .

. . ....

.... . .

. . .. . .

. . .. . . hi,l

hi,l · · · · · · hi,0 0 · · · 0

0. . .

. . .. . .

. . .. . .

......

. . .. . .

. . .. . .

. . . 0

0. . . 0 hi,l · · · · · · hi,0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

∈ C(N+L)×(N+L) be a Toeplitz matrix modeling the convolution

of the channel from the AP to the ith UE, i.e., hi, with x.Assuming perfect time and frequency synchronization and noRF impairments at the receiver, yi can be written as

yi = HiAF−1s(o) +ni,

where ni = [ni,1, . . . ,ni,N+L]T ∼ CN(0,σ2IN+L) is an additive

white Gaussian noise (AWGN) vector, with E[ni, jn∗i, j] = σ2.

Now, let B = [0N×L IN ] ∈ RN×(N+L) be a CP removal matrix

and Di be the mask adopted by the ith UE to retrieve the|Si| sub-carriers of interest, with [Di]m,m = 1, if m ∈ Si and∑M

i=1 Di = IN . At this stage, the first L input samples arediscarded to remove ICI and IBI, yielding

ri = Byi =(

HiAF−1s(o) +ni

), (1)

where Hi ∈ CN×(N+L) and ni ∈ CN are obtained by removingthe first L rows/elements from Hi and ni, respectively. Thesubsequent DFT yields rF

i =DiF(HiAF−1s(o)+ni), an N-sizedfrequency domain representation of the received informationsymbols at the ith UE. After the extraction of the sub-carriersof interest, the standard OFDMA digital signal processingat the receiver typically terminates with a one-tap equal-izer, a parallel-to-serial converter, a demapper and a decisionelement.

We now switch our focus to the CIA component of x. Toderive x(c) we need to consider the channel matrix modelingthe overall equivalent representation of all the links from theAP to the M UEs, i.e., H ∈ C

N×(N+L), obtained as follows. LetrF = ∑M

i=1 rFi ∈ C

N be the vector representing the sum of all thereceived signals at the M UEs, in which the contribution of eachrF

i is preserved thanks to the structure of the mask Di. Thus wecan write

rF =

⎛⎜⎜⎜⎜⎝

M

∑i=1

DiFHi

︸︷︷︸H

⎞⎟⎟⎟⎟⎠AF−1s(o) +

M

∑i=1

DiFni.

We note that H has full rank by construction, and thus for therank-nullity theorem we have that rank(ker(H)) = L,∀h. Weknow from [17] that, in this case, a semi-unitary precoder E ∈C(N+L)×L, such that

DiFHiE = 0N×L, (2)


can always be found. Accordingly, we can let s(c) ∈ RL be

the L-sized random input vector for the CIA precoder, withcovariance matrix Φ =E[s(c)s(c)H] = d([φ1, . . . ,φL]), and definex(c) = Es(c) as the CIA component of x. Further details aboutΦ are deferred until we study the structure of the optimalinput covariance matrix to maximize the transported energy inSection IV-C.

We can now focus on the received signal at the ith UE,considering both the OFDM and the CIA component of thesignal transmitted by the AP, and write

yi = Hi

(x(o) +ζEs(c)

)+ni. (3)

Thanks to the structure of E, which satisfies (2) by construction,the ith UE can remove the CIA signal from yi by means ofthe combined action of three matrices B, F and Di, associatedwith the CP removal operation, DFT and sub-carriers retrieval,respectively. Accordingly, we obtain

ri = DiFHiAF−1s(o) +DiFHiζEs(c)︸︷︷︸=0, by (2)

+ni.

This guarantees interference-free decoding of the informationsymbols at the ith UE, ∀ζ, and shows both the potential and theflexibility of the composite OFDMA signal model. In practice,ri has the same form as a legacy OFDMA receiver signal, as ifthe CIA signal were not present. In other words, after the CPremoval and DFT, the CIA component of the received signalat the ith UE, i.e., HζEs(c), is aligned to the sub-carriers thatare not assigned to the ith UE and can always be removed bymeans of Di. In this sense, the CP removal operation playsa fundamental role in realizing the alignment. Naturally, theeffectiveness of this approach depends on the accuracy of theCSI at the transmitter. The interested reader may refer to [17]for further details.

Finally, once the structure of the two signals, i.e., x(o) andx(c), has been identified, the AP may choose a suitable powerallocation policy to achieve a target performance. In general, aconstraint that bounds the total power radiated by the transmit-ter is typically imposed in real scenarios; thus we let PM ∈ R

+0

be this upper bound. Accordingly, we can express the powerconstraint to be fulfilled by the AP in the considered compositescheme as

PM ≥ 1Γ

tr[P]+ζ tr[EΦEH] =1Γ

tr[P]+ζ tr[EHE︸︷︷︸IL

Φ]

≥ 1Γ

tr[P]+ζ tr[Φ],

with Γ = NN+L .

III. ENERGY HARVESTING RECEIVER

In OFDMA systems, the CP contains a signal whose contentand nature is neglected by the receiver. Its role is to provide ameans for the OFDM receiver to eliminate IBI and ICI from thereceived signal. However, this entails an unavoidable spectralefficiency penalty for the transmission [15]. Furthermore, a

Fig. 1. Energy harvesting OFDM receiver. The main functions provided bythe CP retrieval block are highlighted.

significant amount of power, e.g., up to 20% in long termevolution (LTE) [14], is wasted at each UE with the CP removaloperation. The proposed approach starts with the aforemen-tioned observation and formulates a strategy to capitalize onthe presence of the CP, transforming it from a redundancy thatinduces the loss of spectral efficiency to a means of powersaving at the receiver.

A fundamental step to achieve this goal is the introduction ofa novel OFDMA receiver architecture that does not discard theCP but uses it as a source of energy to be mined. A schematicrepresentation of such a receiver is depicted in Fig. 1. Therein,the dashed lines connecting the energy harvester to the blocksperforming the OFDMA digital signal processing represent theenergy harvested from the CP. We note that these lines do notsymbolize physical connections between the blocks but ratherare an abstraction to represent the additional source of powerthat the receiver can exploit thanks to the proposed approach,to operate the OFDMA digital signal processing blocks.

Let us consider the operations performed by a receiver suchas the one depicted in Fig. 1. First, no sampling should be per-formed by the RF chain during the RF down-conversion, suchthat the signal fed to the decoding chain is still in analog form.Second, the legacy CP removal element should be replaced bytwo novel blocks, i.e., a CP retrieval element and an energyharvester. We note that a detailed technical description of thestructure and implementation of these two blocks is beyondthe scope of this paper and is deferred for future investigation.However, a brief, and by no means exhaustive, description ofthe structure of both the CP retrieval element and the energyharvester is provided in the following, for the sake of clarity.

In practice, the CP retrieval element serves three purposes:1) timing synchronization to identify the portion of basebandsignal corresponding to the CP, to feed it to the energy harvesterin analog form, 2) sampling of the remainder of the signal, and3) frequency synchronization to yield ri as in (1), for feedingit to the decoding chain of the receiver as in legacy OFDMAreception. Switching the focus to the energy harvester, hereinthe power would be scavenged from a baseband signal, afterthe amplification provided by the RF front-end. This couldsignificantly reduce the impact of path-loss attenuation, and is


in contrast with what is proposed for state-of-the-art solutionsfor SWIPT, in which an RF to direct current conversion (RF-to-DC) is necessary [9], [10]. Naturally, the conversion to DC inour case could still be performed by means of a diode operatingat the appropriate frequency, as for the RF-to-DC operation,given the nature of the signal after the RF amplifier. Thus, theefficiency of the harvesting operation could still be modeled bya coefficient β∈ [0,1], thanks to the law of energy conservation.The energy scavenged from the CP could be used to power theblocks performing the digital signal processing on ri, reducingthe impact of the OFDMA decoding on the battery life of theUE. This significantly modifies the role of the CP within thetransmission. As mentioned in Section I, the interpretation ofthis new role is two-fold and depends on both the consid-ered scenario and adopted perspective. The energy harvestedfrom the CP by the OFDM receiver could reduce the impactof the power consumption for the digital signal processingon the battery life, making the transmission partially or fullyself-sustainable for this aspect.

At this stage, we remark that, thanks to the inherent structureof the OFDMA transmission and unlike the state-of-the-artsolutions, this approach does not require the presence of apower splitting unit in the receiver RF front-end [9], [11],or the modification of either the scheduling at the transmitteror the receiver activity depending on its state [9], [11], [12].In this regard, no impact would be caused by the proposedapproach on surrounding devices or systems adopting differenttechnologies, simplifying the integration of this solution intomodern wireless networks. In practice, the applicability of theproposed approach is not limited to a specific scenario. Anyblock transmission technique could benefit from it, thanks toboth the presence of a redundancy in every block and theflexibility of CIA.1 For instance, this approach could be appliedto OFDMA transmissions, as shown in this contribution, toTDMA schemes such as the one proposed in [13], SC-FDMA[14] and so on.

Finally, we focus on the different trends experienced bythe information and power transfers in the proposed approach.Consider the ergodic achievable rate per UE. As a matterof fact, the number of sub-carriers and channel uses devotedto information transfer in our approach are the same as instandalone OFDMA schemes, regardless of the presence orabsence of the CIA signal (ζ = 1 and ζ = 0, respectively). Thus,as in the standalone cases, herein the ergodic achievable rate peruser is a decreasing function of the number of UEs. Conversely,the amount of energy that can be harvested from the CP by eachreceiver does not depend on the number of UEs, the size ofthe CP being a design parameter of constant size. This ensuresfull scalability of this method for large-scale OFDMA systems,in which the high number of UEs does not alter the amountof energy that the AP can transfer to each one of them. Inparticular, this shows an evident difference w.r.t. schemes likethe one proposed in [12], in which the amount of energy thateach UE can harvest depends on both the number of UEs andon the scheduling at the AP.

1We note that CIA is a sub-space based strategy and its implementability andeffectiveness holds for any block transmission scheme.

IV. SELF-SUSTAINABLE TRANSMISSION

Consider the legacy baseband OFDMA signal model intro-duced in Section II. Therein, the CP size was expressed interms of number of samples, i.e., L. Accordingly, for the sakeof clarity, let us represent the CP retrieval operation describedin the previous section in accordance with that model, i.e., inmatrix form, as

qi = Qyi ∈ CL,

with the matrix Q = [IL 0L,N ] ∈ RL×N+L extracting the CP

from yi, expressed as qi. We remark that this does not explicitlymodel the physical operation performed by the CP retrievalelement but instead provides a useful representation for theprosecution of our analysis.

Now, assume that the digital signal processing of an OFDMblock carrying N information symbols consumes the sameamount of power at each of the M UEs,2 i.e., Pd ∈ R+

0 . Wenote that, in practical implementations, Pd can be on the orderof a few hundreds of mW depending on the adopted hardware[18], [19]. On the other hand, typical values for the total powerradiated by an AP in real system implementations, i.e., PM,range from hundreds of milliwatts to a few watts [20]. Thus,we can safely assume that the energy carried by the CP afterthe RF amplification, and net of the losses experienced in theconversion process, might have the same order of magnitudeas Pd. In this regard, in the remainder of the paper, we willdistinguish between partial and full self-sustainability. In prac-tice, we define as fully-sustainable a transmission for whichthe impact of the OFDMA digital signal processing on theUE’s battery is nulled. Conversely, a partially self-sustainabletransmission is achieved whenever the complete nulling doesnot occur.

A. Price of Self-Sustainability

In traditional OFDM transmissions, the amount of powercarried by the CP of each received block cannot be arbitrarilyaltered by a standard OFDM transmitter. Conversely, in thecomposite scheme studied in this work, the entirety of the CIAsignal is contained within the CP of each received block, asshown in (3), irrespective of the channel. From a practical pointof view, the portion of the total power budget allocated to theCIA signal, i.e., PC, can be flexibly varied by the OFDMAcomposite transmitter, choosing among a wider set of powerallocation policies as compared to its legacy counterpart. On theone hand, this provides a high level of flexibility that allows theAP to decide on how to privilege either one of the two transfers,depending on the system target rate and self-sustainability level.On the other hand, this may also entail a penalty in terms of DLrate, paid by the AP w.r.t. the legacy standalone OFDMA caseto increase the effectiveness of the energy transfer. Hereafter,we will refer to this penalty as the price of self-sustainability ofthe transmission, formally defined as follows.

Definition 1 (Price of Self-Sustainability): Let R1 ∈ R+0 and

R0 ∈ R+0 be the DL rate achieved for ζ = 1 (composite

2This assumption is made to simplify the notation in the following analysisand does not affect the generality of the approach.


OFDMA) and ζ = 0 (legacy standalone OFDMA), respectively.Accordingly, the price of self-sustainability of the transmissionis defined as Π = 1− R1

R0∈ [0,1].

In other words, the price of self-sustainability expresses therate loss experienced by the AP if a composite OFDMA trans-mission is adopted instead of a standalone OFDMA scheme.

B. Feasibility Conditions

In practical scenarios, the UEs typically expect an adequateservice from the AP in terms of DL rate. In this context,an AP usually operates such that the DL rate is maximizedunder a given power constraint, possibly aiming at transmittinginformation at the optimum rate. Thus, to increase the practicalrelevance of our study, we adhere to this realistic paradigm andassume that the main purpose of the transmission from the APto the UEs does not change w.r.t. to typical system implemen-tations. Naturally, as previously discussed, in our approach theUEs make a different, and arguably more profitable use, of theCP as compared to a legacy OFDM receiver.

Now, let us assume N fixed by a certain standard, e.g.,LTE, and consider the standard rate maximization problem forOFDMA transmissions (i.e., without WPT). In this case, the DLrate is maximized if each sub-carrier is assigned to the UE withthe best fading condition and a successive water-filling (WF)power loading over the selected sub-carriers [21] is performed.Switching our focus to the CP size, i.e., L, we note that thisparameter affects many aspects of the transmission, and thusis typically subject to some well defined criteria. First of all,L must always be larger than the delay spread of the channel,to guarantee IBI and ICI free reception. Subsequently, on theone hand the DL rate is maximized if L is minimized (assumingthat perfect synchronization at each UE is achievable). On theother hand, a reduction of the CP size may also decrease theself-sustainability of the transmission, due to a lower amountof energy that can be scavenged by the receiver. Accordingly,if we aim at minimizing L to maximize the DL rate, whileensuring the self-sustainability of the OFDMA transmission,the following problem needs to be solved:

minL∈N

L

s.t. β∑Mj=1 tr

[QH j(AFHPFAH + . . .

+ ζEΦEH)HHj QH]−δMPd ≥ 0 (a)

1Γ

tr[P]+ζtr[Φ]−PM ≤ 0 (b)

pi ≥ 0 (c)

φn ≥ 0, (d)

L ≥ l, (e) (4)

∀ i ∈ [1,N] and ∀n ∈ [1,L]. Unfortunately, a solution to theproblem in (4) cannot be found in closed form. In this regard,we note that the integer structure of the problem is intrinsicto the considered scenario and would arise even if we wereto express the CP as a period in time (rather than an integervalue). In fact, L would still have to be chosen from a finite

set of real values (instead of a finite number of integer values),depending on the sampling period at the receiver. Nevertheless,if we consider the constraints in (4) a study of its feasibilitycan still be performed as follows: The constraint denoted by(a) enforces the self-sustainability of the transmission, witha scaling factor δ ∈ R

+0 that models different levels of self-

sustainability, i.e., full (δ = 1) or partial (0 ≤ δ < 1). Now, letGi = QHiAFH = [gi,1,gi,2, . . . ,gi,N ] ∈ C

L×N and Vi = QHE =[vi,1,vi,2, . . . ,vi,L]∈C

L×L. Then, the N+L+3 constraints in (4)can be rewritten as⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∑Ni=1 pi

(∑M

j=1 ‖g j,i‖2)+ . . .

+ζ∑Ln=1 φn

(∑M

j=1 ‖v j,n‖2)− δ

βMPd ≥ 0

− 1Γ ∑N

i=1 pi −ζ∑Ln=1 φ j +PM ≥ 0

pi ≥ 0

φn ≥ 0

L− l ≥ 0,

(5)

∀ i ∈ [1,N], ∀n ∈ [1,L]. The feasibility of (4) can be assessedby demonstrating the non-emptiness of the set of solutions to(5). By construction, (5) is an overdetermined linear system ofinequalities, hence the following holds.

Lemma 1: Let ξ = PMPd

∈ R+0 be the ratio between the upper

bound on the total radiated power by the transmitter and thepower consumed by the receiver for the digital signal process-ing, ϑL = max

n∈[1,L]{∑M

j=1 ‖v j,n‖2} and υN = maxi∈[1,N]

{∑Mj=1 ‖g j,i‖2}.

The feasibility of (4), and thus a solution to (5), exists if andonly if ⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

l ≤ N(υNξβδM −1), i f ϑL ≤ υN

l ≥N(

δβ M−ϑLξ

)ξ(ϑL−υN)− δ

β M, i f ϑL >

δβ M+υN ξ

ξ

l <N(

δβ M−ϑLξ


β M, otherwise.

(6)

Proof: See Appendix A. �Lemma 1 guarantees that if (6) is fulfilled, then (4) is feasible.

Nevertheless, this does not imply that the WF power allocationobtained by the AP as a solution to the legacy OFDMA ratemaximization problem belongs to the solution to (5). In otherwords, assume that a given portion (or the entirety) of the totalpower budget is allocated to the OFDMA transmission. Then ifthe AP aims at maximizing the DL rate as a prior task, the fullself-sustainability may not always be feasible. The same is nottrue if we consider partial self-sustainability, i.e., 0 ≤ δ < 1 in(4), as can be trivially verified by letting ζ = 0 and δ = 0 in (a)in (4). In this case, we see that a suitable value for δ such that(a) is verified can always be found, regardless of the OFDMApower allocation vector p. This implies that a partially self-sustainable OFDMA transmission is always feasible and thusfurther motivates the presence of the scaling factor δ in (4). Inpractice, the AP can use Lemma 1 either to verify the feasibilityof a level of self-sustainability, or to identify the maximumachievable self-sustainability, given the system parameters and


channel realization. At this stage, a suitable value for L can beset by the AP according to two policies:

1) Standard DL rate maximization: the AP sets the lowestpossible value for L, i.e., L = l, implicitly assumingthe feasibility of perfect time synchronization at eachUE, and accept the risk of performance detriment if asynchronization error occurs. No special focus is put onthe self-sustainability of the transmission.

2) Self-sustainable DL rate maximization: the AP sets anyvalue for L according to Lemma 1, increasing the self-sustainability of the transmission at the expense of aDL rate penalty w.r.t. the first policy. In practice, thiscould be achieved by performing an exhaustive searchover N − l possible values, to determine the suitable CPsize. This policy would also increase the tolerance of thetransmission to synchronization errors at the receivers.

C. Power Allocation Strategy

Before starting, we note that in this section the case inwhich the CIA signal is present, i.e., ζ = 1, is considered.From a practical point of view, the power allocation strategyat the AP in case of composite transmission requires a two-step procedure. This process is significantly different fromthe approach adopted for legacy OFDMA transmissions, inwhich only information transfer is performed. In fact, first theavailable power budget at the AP needs to be shared amongthe OFDMA and CIA signals. Subsequently, the AP shouldidentify the optimal power loading strategy for each of the twosignals, according to the goal of each transfer. Let us start fromthe latter aspect and assume that the AP may deterministicallychoose a suitable power sharing among the two signals. LetPO = tr[P] ∈ [0,PM] and PC = tr[Φ] = PM −PO be the portionsof power allocated to the OFDMA and CIA signal, respectively.Once the two portions have been set, the AP proceeds to theoptimization of the two power allocation strategies, separately.

We first focus on the OFDMA transmission. Assuming thatthe AP’s goal is throughput maximization for information trans-fer, as is typically done in the literature on the subject, then theoptimal power loading strategy is obtained as described in [21].Accordingly, each [P]i,i, ∀ i ∈ [1,N] is determined by meansof the WF power loading strategy derived therein, and brieflydiscussed in Section IV-B.

The identification of the optimal power allocation strategyfor the CIA signal requires a different and less conventionalapproach. In fact, the sole purpose of this signal is the realiza-tion of an energy transfer from the AP to the UEs. Let H̃ =Q∑M

i Hi ∈ CL×(N+L), for the sake of compactness of notation.

To maximize the efficiency of the energy transfer the AP mustsolve

maxΦ

E = tr[H̃EΦEHH̃H]

s.t. tr[Φ]≤ PC, (7)

where E ∈R is the overall amount of energy received within theCP by the M UEs. The solution to (7) is given by the followingproposition.

Proposition 1: Let W = H̃E and W = UΛ12W TH be its singu-

lar value decomposition (SVD), with U ∈ CL×L and T ∈ C

L×L

unitary matrices and Λ12W = d([

√λW,1, . . . ,

√λW,L]) a diagonal

matrix carrying the singular values of W (obtained as the squareroot of the eigenvalues of WHW, i.e., λW,i, ∀ i∈ [1,L]), such that√

λW,1 ≥√

λW,2 ≥ . . .≥√

λW,L. The solution to (7) is given by

Φ = PCt1tH1 .

Proof: See Appendix B. �We now come back to the first step of the power loading strat-

egy, i.e., the identification of the optimal values for PM and PC.Unfortunately, an analytical optimization of the power sharingstrategy between the OFDM and CIA signals is not feasible.The interested reader may refer for further details to [22], wherea similar signal model (even though arising from a differentapproach) characterizes this issue. Thus, numerical iterativeapproaches are the only viable option to identify suitable valuesfor PM and PC. To better understand the implications of thisaspect for practical implementations of the proposed scheme,let us focus on the time constraints the AP should complywith in this case. Consider the CSI, a key element needed tocompute the CIA precoder and adapt the power loading strategyto current channel conditions. The validity and reliability ofthis information can be guaranteed only within the coherencetime of the channel of interest. This imposes rather stringenttime constraints on the AP to perform its optimizations. Inpractice, the latter may not be able to identify the optimal powersharing for realistic scenarios by mean of iterative numericalalgorithms, due to their inherent computational burden. Consid-ering all the aforementioned limitations and constraints, hereinwe do not aim at identifying the optimal power sharing rule forthe AP. Conversely, we study this aspect of the transmissionby means of an extensive numerical analysis, whose resultsare presented in Section V. This way, we can assess the per-formance of the considered scheme for both different budgetsharing proportions and system configurations, providing anoverall and practically relevant picture for the reader.

V. NUMERICAL ANALYSIS

The numerical findings in this section are obtained by meansof a set of Monte-Carlo simulations. A practically relevantrange of AP, without the Saxon genitive transmit power val-ues is considered, i.e., PM ∈ [0.9,2.7] W [20]. Similarly, inaccordance with the characteristics of realistic OFDMA re-ceiver implementations, we assume Pd = 600 mW [18], [19].We note that, all the obtained results are depicted as a func-tion of ξ = PM

Pd∈ [1.5,4.5], for the sake of simplicity. Fur-

thermore, we consider β = 0.5 as is typically done in theliterature [9]. Finally, we let the number of UEs and sub-carriers be M = 4 and N = 64, respectively. The CP sizeis modeled according to [14], i.e., L ∈ L =

[⌈9N128

⌉, N

4

]. In

this regard, we note that M and N have been chosen for thesake of computational tractability. A study of the impact ofthe number of both UEs and sub-carriers on the performanceof the composite scheme will be the subject of future research.

We assume that the AP and UEs are deployed in an in-door scenario, e.g., commercial/office/residential, with the UEsuniformly distributed around the AP, at a distance ranging


between 5 m and 10 m, possibly separated by 0–2 thin walls.Furthermore, we assume a carrier frequency of 1800 MHz andthat each UE disposes of a variable gain RF amplifier, whosemaximum delivered gain ranges between 25 dB and 30 dB [23].In practice, in such a setting, the distance-dependent path-lossexperienced by the transmitted RF signal, modeled accord-ing to [24], can be effectively compensated by the variablegain of the RF amplifier. The rationale for the choice ofthis setting is two-fold. On the one hand, we aim at assess-ing the merit of the proposed idea in a real-life setting. Onthe other hand, we want to preserve the meaningfulness ofboth interpretations of the proposed approach, as discussed inSection I and Section III, i.e., a) OFDM-based SWIPT andb) recycling of otherwise wasted energy. It is worth noting thatthis paper is a first step towards a complete characterizationand realization of self-sustainable block transmissions. Thus,parameters such as antenna gains, frequency response of RF/IFhardware components, e.g., mixers, to name a few, will notbe taken into account in the considered scenarios. In fact, aninappropriate choice for these parameters could significantlyundermine the relevance of the obtained results. However, wenote that the proposed parametric model could be suitablyextended to account for a detailed link budget model. This willbe the subject of a future investigation.

For the sake of simplicity in the representation of the results,we assume thermal noise at the receiver such that the signal-to-noise ratio (SNR) is 10 dB.3 In this regard, we note that theimpact of the adoption of the composite strategy, i.e., ζ = 1and PO < 1, on the performance of the system is two-fold: 1)the effectiveness of the energy transfer from the AP to the UEsis increased w.r.t. when ζ = 0, and 2) the performance of theinformation transfer is reduced w.r.t. the legacy OFDMA case.This is due to the power sharing strategy adopted in the compos-ite case, which results in an SNR reduction at each UE. Theseaspects need to be correctly modeled to obtain consistent andreliable outcomes during the simulations. In practice, both thestandalone and the composite OFDMA transmissions shouldexperience the same average noise power at the receiver, i.e.,σ2, regardless of the adopted power sharing strategy. Thus, inour tests, the AWGN at the UEs is generated as if the AP wereperforming a standalone OFDMA transmission, i.e., ζ = 0,PO = 1 and PC = 0.

The channels between the AP and the UEs are modeledas frequency-selective Rayleigh fading channels with l + 1taps, characterized by exponentially decreasing PDPs. In other

words, we let κ2j = exp

{− jT

τrms

}with T the sampling period

at each UE, τrms the root mean square delay spread of thechannel and j =

√−1. In particular, we let l = min{L}= � 9N

128�and restrict our study to two families of exponential PDPs:1) slowly decaying, and 2) rapidly decaying. For matters ofspace economy, we will consider two specific values for κ2

j ,one representative for each of the two considered families. In

particular, we let κ2j = exp

{− j

3

}and κ2

j = exp{− j

1.3

}for the

slowly decaying and rapidly decaying PDP, respectively.

3We note that this is a likely occurring value in real scenarios to be able tomeet the target performance for modern network deployments.

Fig. 2. δ∗, as L changes, N = 64 and ζ = 0. (a) Slow decay. (b) Fast decay.

A. Average Level of Self-Sustainability

With this analysis, we study the level of self-sustainabilityδ that the composite OFDMA transmission can achieve forthe considered scenarios. A natural way to characterize theperformance of the considered system is the computation ofδ∗ = Eh[δh], the average level of self-sustainability of thetransmission, with δh defined as the instantaneous achievablelevel of self-sustainability for a given channel h. This choicenot only yields a statistically relevant metric but also simplifiesthe presentation of our results. We start our analysis from thecase ζ = 0, i.e., standard OFDMA transmission, and computeδ∗ for the aforementioned values of ξ, as L changes, in Fig. 2.

We note that, full self-sustainability (δ∗ ≥ 1) is neverachieved in either of the two cases, regardless of the consideredvalue of ξ. However, non-negligible values of partial self-sustainability are achievable for the considered scenarios. Theimpact of the CP size on δ∗ is evident. A quantitative differ-ence in favor of the slowly decaying PDP case is noticeableas well. This is due to the larger IBI affecting the receivedsignal at each UE in this case, which in turn increases theaverage energy carried by the CP of each received OFDMA


block. Interestingly, the slope of the performance curve in-creases with ξ. This can be explained by considering thatthe OFDMA symbol is characterized by a significant peak-to-average power ratio (PAPR), which induces a highly non-uniform power profile of the OFDMA symbol, CP included.Accordingly, increasing the length of the CP increases itsPAPR, whose effects become more evident as ξ grows. Thisinduces a non-linear relationship between δ∗ and ξ, enhancedby the adoption of the non-uniform power loading strategy,i.e., WF, which translates in different slopes of the obtainedcurves.

We now consider the case ζ = 1, i.e., the AP does not devoteits power budget entirely to the OFDMA signal, but insteadassigns a portion of it to the CIA signal. In practice, the levelof self-sustainability achieved in this study will depend on thechosen value for PC. In what follows, similarly to the notationadopted for the standalone OFDMA transmission, we will referto δh(PC) to explicitly show this dependency. We start by notingthat if the average power carried by the CP of the each receivedOFDMA block at the UE is already sufficient to ensure full self-sustainability, then the AP would not be interested in allocatingfurther power to the CIA signal. Accordingly, we resort to aslight abuse of notation and redefine δ∗ as in (8), shown at thebottom of the page. The results obtained for ζ = 1 are shown inFig. 3.

As opposed to standard OFDMA transmission, herein a valueof L for which δ∗ ≥ 1 can be found for almost all the consideredconfigurations, thanks to the presence of the CIA signal. Morespecifically, let us focus on the results for the slowly decayingPDP, depicted in Fig. 3(a). In this case, a CP size lower than themaximum allowed in modern standards, i.e., L = N

4 , is alreadysufficient to perform a fully self-sustainable transmission forN = 64, ∀ξ ∈ [1.5,4.5]. Similarly, in case of rapidly decayingPDP, δ∗ ≥ 1 for all the considered configurations except one,i.e., ξ = 1.5, as can be seen in Fig. 3(b). We note that, the per-formance difference between the two families of PDPs followsa different trend as compared to what we observed in Fig. 2. Infact, herein δ∗ is larger in case of rapidly decaying PDP whenthe CP size is shorter, whereas the opposite behavior is observedfor larger CP sizes. This difference is small but noticeable and isdue to the structure of the CIA signal. In fact, as shown in [25],the latter may suffer from very high PAPR. This issue is moreevident when the CIA precoder is built upon a channel charac-terized by uniform or slowly decaying exponentially decreasingPDPs. This entails a lower effectiveness of the CIA signal as anenergy bearer in the case of slowly decaying PDP and smallCP size. However, the impact of the PAPR on the effectivenessof the transfer is mitigated when the CP is larger, and, as aresult, the slowly decaying PDP case yields better performance.This is due to the larger channel diversity in the latter case,if compared to the rapidly decaying one, which increases the

Fig. 3. δ∗, as L changes, N = 64 and ζ = 1. (a) Slow decay. (b) Fast decay.

probability of finding a large√

λW,1 (i.e., the largest singularvalue of W, the equivalent channel matrix representation forthe CIA transmission, as defined in Section IV-C). To concludewe observe that whenever the average power harvested by eachUE is larger than Pd, i.e., δ∗ > 1 for ζ = 1, the resulting energysurplus may be stored by each UE for future usage. This aspecthas not been considered in this work but certainly representsone of the most interesting subjects for future research.

B. Price of Self-Sustainability

The presence of the CIA signal increases δ∗, however thiscomes at a cost for the AP, i.e., the price of self-sustainabilityof the transmission, denoted by Π, as discussed in Section IV-A.The focus of this section is specifically on this parameter. Now,

δ∗ =

⎧⎨⎩

maxPC

{Eh[δh(PC)]} if maxPC

{Eh[δh(PC)]} ≤ 1

minPC

{Eh[δh(PC)]1{Eh[δh(PC)]>1}{Eh[δh(PC)]}} otherwise(8)


Fig. 4. Portion of the power budget dedicated to the OFDM signal to achieveδ∗, as L changes, N = 64. (a) Slow decay. (b) Fast decay.

let us take a step back and consider the value of PC for which theδ∗ in (8) is obtained, denoted by P∗

C. For the sake of simplicity,we define η = 1− PC

PMas the portion of the total power budget

devoted to the OFDMA signal and η∗ = 1 − P∗C

PMas its value

when δ∗ is achieved. The behavior of η∗, as both ξ and L vary,is depicted in Fig. 4.

As could be expected, η∗ increases with the CP size, forboth PDP families. In fact, the amount of energy carried bythe OFDMA signal in the CP depends on the size of the latter.If L is small then a significant contribution from the CIAsignal is required to achieve high levels of self-sustainability.Consequently, when ζ = 1 and L is large, δ∗ can be achieved forsmaller values of PC. As a matter of fact, the results in Fig. 4(a)and Fig. 4(b) are not dissimilar quantitatively. Nevertheless,a non-negligible difference is present in the rate at which η∗

grows with L. If we consider small CP sizes, η∗ is larger in thecase of rapidly decaying PDP, whereas an opposite behavior isobserved when the CP size is large. This is due to the greatereffectiveness of the CIA signal as an energy bearer, in the caseof small CP size and channel characterized by rapidly decaying

Fig. 5. Price of self-sustainability, as L changes, N = 64. (a) Slow decay.(b) Fast decay.

PDP, discussed in the previous section. Conversely, a slowlydecaying PDP is more advantageous than a rapidly decayingone for large CP sizes, and sufficient values of δ∗ are achievedfor lower values of PC.

We now switch our focus to the price of self-sustainability ofthe transmission, i.e., Π, as defined in Section IV-A. As seen inFig. 4, when L is small, PC ≥ PO for a large number of systemconfigurations. The price of self-sustainability for these casescould be severe. In this sense, a CP size increase induces botha larger η∗, i.e., an SNR gain at the UEs, and a smaller pre-logfactor for the DL rate. A trade-off between these two effects isexpected, and a value for L such that Π is minimized is likelyto be found for all ξ. To illustrate this aspect, we compute Π forthe two considered families of PDPs, as L changes, and depictit in Fig. 5.

In general, a larger ξ reduces the price of self-sustainability,as could have been intuitively expected. Interestingly, a valueof L for which Π is minimized can be found for all thetested configurations. This is the result of the aforementionedtrade-off between the SNR gain at the receiver and the lower


pre-log factor of the DL rate, a consequence of the larger L. Inparticular, the value of L for which Π is minimized decreases asξ increases, for both considered families of PDPs. The reasonfor this behavior can be intuitively explained by consideringthat the rate increases logarithmically with the SNR. Accord-ingly, the impact of an increase of η∗ is larger when its valueis small. In practice, the SNR gain induced by an increase ofη∗ is less beneficial for the DL rate in case of larger ξ, andhence Π is minimized by a smaller L. Naturally, the range of Πin Fig. 5 depends on the nature of δ∗, i.e., either the highestpossible level of self-sustainability or the lowest level largerthan 1. If we changed the definition of δ∗, that is if the levels ofself-sustainability of interest were defined otherwise, a differentrange for Π would be obtained. However, this would not changethe validity of our previous insights, as can be inferred byrecalling the behavior of η∗ in Fig. 4(b) and (b) as L increases.In fact, from a quantitative point of view, the results in Fig. 5are a direct consequence of the aforementioned behavior of η∗.In practice, for small CP sizes the price of self-sustainabilityis lower for the rapidly decaying PDP than for the slowlydecaying PDP case, whereas the converse behavior is observedfor large CP sizes. The price paid by the AP when ζ = 1, fora target level of self-sustainability given by δ∗, ranges fromreasonably acceptable (rate loss of 19% w.r.t. legacy OFDMA)to highly penalizing (rate loss of 76%) values. This shows that,in practical scenarios, the power sharing strategy to allocate PO

and PC should be selected by the AP depending on the targetperformance of the SWIPT, in terms of both DL rate and self-sustainability of the transmission.

At this stage, a question related to the possible impact of dif-ferent average SNR values at the receiver on the outcome of ourtests would be legitimate. This parameter plays an importantrole in the WF algorithm, especially in the low SNR regime. AnSNR variation may have an impact on both the optimal L, theachievable δ∗, η∗ and Π. Unfortunately, an extensive study ofthe impact of the SNR on these parameters could not be carriedout due to space limitations. Thus, we will only study thisimpact on the price of self-sustainability, due to the prominenceof this parameter in this contribution. This allows us to bettercharacterize the performance of the information transfer withinthe SWIPT in the considered scenarios, as a function of theSNR at the receiver. Accordingly, in the next test, the price ofself-sustainability will depend not only on L, as previously seen,but also on the SNR. For the sake of simplicity, and similarly towhat was previously done, we will use Π(L) to explicitly showthe dependence on the former parameter. Furthermore, we letL∗ = min

L∈L{Π(L)} be the CP size inducing the minimum price

of self-sustainability for the transmission, i.e., Π∗ = Π(L∗). Inthis regard, considering the previous results, we note that thisdefinition of L∗ implicitly entails the values of δ∗ and η∗ whichminimize the price of self-sustainability for a given SNR value.Finally, Π∗ is depicted in Fig. 6 as a function of the SNR.

The price of self-sustainability consistently decreases withthe SNR for all the values of ξ. Let us consider the realizationof the WF solution for different SNR values. It is a well-knownfact that optimal WF approximates equal power allocation(EPA) for high SNR values. More precisely, EPA approaches

Fig. 6. Price of self-sustainability, as the SNR changes, N = 64. (a) Slowdecay. (b) Fast decay.

the optimal solution for the DL rate maximization problemin OFDMA systems as the SNR grows [21]. In practice, anddifferently from the WF case for low and mid SNR, all sub-carriers positively contribute to the signal carried by the CPwhen EPA is adopted, increasing its diversity. Accordingly, thecontribution of the OFDMA signal to the scavenged energy atthe UEs is significant and a lower contribution from the CIAsignal is required to achieve δ∗ in this case. Thus, smallervalues for both L and PC are necessary. Concerning the latterparameter, we could conjecture that η∗ may follow a similartrend to Π∗, as the SNR grows. From a quantitative pointof view, the largest difference is observed for ξ = 1.5, i.e.,around 32%, and it decreases as ξ increases. In practice, thehigher PM the smaller is the performance loss of the informationtransfer w.r.t. legacy OFDMA. This is due to the previouslyanalyzed behavior of η∗, and its impact on the DL rate of thetransmission, the latter increasing only logarithmically with theSNR. In fact, η∗ and Π follow opposite trends as ξ grows, i.e.,when η∗ increases the price of self-sustainability decreases.


VI. CONCLUSION

In this paper, the self-sustainability of block transmissionsin terms of power consumption of the digital signal process-ing at the receiver, has been studied. A very popular blocktransmission scheme, i.e., OFDMA, has been specifically tar-geted. In particular, a composite variant of this scheme inwhich the transmitter transmits a signal obtained as the sumof an OFDMA and a CIA component is analyzed. In thisscheme, the OFDMA receiver can harvest energy from the CPof each received OFDMA block, without either modificationsto the scheduling at the transmitter or the need to modifythe activities of the receiver depending on its activity state(active/idle). We have derived both a condition for which theself-sustainability of the transmission is feasible, depending onseveral system parameters, and the optimal power allocationstrategy to maximize the effectiveness of the CIA signal asan energy bearer. The performance of the composite OFDMAstrategy has been evaluated for different system configurationsand channel statistics. The presence of the CIA signal has beenshown to be highly beneficial for the effectiveness of the energytransfer, yielding significant levels of self-sustainability.

The obtained results show a noteworthy consistency as dif-ferent system parameters are tested. The self-sustainability interms of power consumption of the OFDMA digital signalprocessing can be achieved for a large range of configurations.Remarkably, the average power harvested by each UE in somecases may also generate a surplus that could be stored forfuture use. However, the AP pays a price of self-sustainabilityto increase the effectiveness of the energy transfer, expressedas a rate penalty w.r.t. a legacy OFDMA transmission. The re-lationship between the performance of the two transfers withinthe SWIPT has been investigated. The obtained results showthat a CP size that minimizes the price of self-sustainability canbe found for all the considered configurations.

These findings open interesting research directions, includ-ing but not limited to: 1) the extension of the adopted para-metric signal model such that a comprehensive link budget isaccounted for, 2) the study of strategies to capitalize on thepossible energy surplus at the receiver, depending on both thenumber of adopted sub-carriers and UEs serviced by the AP,and 3) the practical design and implementation of an energyharvesting OFDMA receiver.

APPENDIX APROOF OF LEMMA 1

Proof: The feasibility of (4), hence the existence of asolution to (5), can be proved by means of the so-calledFourier-Motzkin elimination (FME) algorithm [26]. The FMEiteratively eliminates one variable from the system at each step,by pairing the inequalities in which the variable to eliminateappears with opposite sign. This yields a new system whosesolution coincides with the solution of the previous system overthe common variables. The key idea of this method is thatthe solvability of the system of constant inequalities obtainedat the last step is a sufficient and necessary condition for thesolvability of the original system [26].

For simplicity, we start by letting ∑Mj=1 ‖g j,i‖2 = υi and

∑Mn=1 ‖v j,n‖2 =ϑn. In particular, we assume υ1 ≤ υ2 ≤ ·· · ≤ υN

and ϑ1 ≤ϑ2 ≤ ·· · ≤ϑL. We note that this comes without loss ofgenerality. In fact, any ordering would be completely equivalentby means of an appropriate variable substitution. Thus, thesystem obtained at the first step of the algorithm is given by

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∑Ni=2 pi(υi −υ1)+ . . .

+ζ∑Ln=1 φn(ϑn −υ1)≥−Γυ1PM +

δβ

MPd

− 1Γ ∑N

i=2 pi −ζ∑Ln=1 φn ≥−PM

pi ≥ 0, ∀ i ∈ [2,N]

φn ≥ 0, ∀n ∈ [1,L]

L− l ≥ 0.

Now, let υk be the variable eliminated by the FME at the kthstep. We define υ0 = 0 and let ϒk = {υi|(υi −υk) �= 0}, ∀k ∈[0,N]. The general form of the system obtained at the kth stepcan be written as⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∑i∈ϒkpi(υi −υk)+ . . .

+ζ∑Ln=1 φn(ϑn −υk)≥−ΓυkPM +

δβ

MPd

− 1Γ ∑N

i=k+1 pi −ζ∑Ln=1 φn ≥−PM

pi ≥ 0, ∀i ∈ [k+1,N]

φn ≥ 0, ∀n ∈ [1,L]

L− l ≥ 0.

(9)

Specifically, we can write the Nth step as

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

ζ∑Ln=1 φn(ϑn −υN)≥ −ΓυNPM +

δβ

MPd

−ζ∑Ln=1 φn ≥ −PM

φn ≥0, ∀n ∈ [1,L]

L− l ≥0,

(10)

irrespective of whether υN = υi,∀ i ∈ [1,N] or not, as can beverified by looking at (9). At this stage, if ϑL ≤ υN then (ϑn −υN) ≤ 0, ∀n ∈ [1,L]. As a consequence, each following stepof the algorithm will just eliminate the corresponding variable,yielding the system of constant inequalities given by⎧⎪⎨

⎪⎩N

N +LυNPM − δ

βMPd ≥ 0

L− l ≥ 0.

(11)

Therefore, in this case, (4) is feasible if and only if (11) is veri-

fied, that is if l≤N

(max

i∈[1,N]{∑M

j=1 ‖g j,i‖2} ξβδM −1

), with ξ= PM

Pd.


Consider now the case ϑL > υN . Let G = {υN ,ϑn s.t. ϑn −υN > 0} be the set given by υN and all the ϑn > υN . We start bynoting that if ϑL = ϑn,∀n ∈ [1,L], then the solution is still l ≤N

(max

i∈[1,N]

{∑M

j=1 ‖g j,i‖2}

ξβδM −1

), as can be trivially verified

by looking at (10). Alternatively, if there exists an n such thatϑL > ϑn, then the system obtained after the (N +L−1)th stepof the algorithm is given by

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

φL(ϑL−max{G\{ϑL}})≥−PM[max{G\{ϑL}}− . . .

−υN(1−Γ)]+δβ

MPd

−φL ≥ −PM

φL ≥ 0

L− l ≥ 0.

Finally, the system of constant inequalities obtained at the(N +L)th step of the FME is

⎧⎪⎨⎪⎩

ξ[ϑL −υN(1−Γ)]− δβ

M ≥ 0

L− l ≥ 0.

Therefore, when ϑL > υN , the necessary and sufficient condi-tion for which (4) admits a solution can be written as

⎧⎪⎪⎪⎨⎪⎪⎪⎩

l ≥N(

δβ M−ϑLξ


β M, if ϑL >

δβ M+υN ξ

ξ

l <N(

δβ M−ϑLξ


β M, otherwise,

and this concludes the proof. �

APPENDIX BPROOF OF PROPOSITION 1

Proof: Let Φ = UDUH be the spectral decomposition ofthe covariance matrix of s(c), where U ∈ C

L×L is a unitarymatrix and D = d(φ1, . . . ,φL), with tr[D] ≤ PC. Let us assumethat φ1 ≥ φ2 ≥ . . .≥ φL, without loss of generality. Accordingly,we can rewrite E in (7) as

E = tr

⎡⎣WU︸︷︷︸

W̃D

DUHWH︸︷︷︸W̃H

⎤⎦= tr[W̃DW̃H] =

L

∑i=1

φi‖w̃i‖2. (12)

If we look closely at (7), we see that the power constraintin (7) is satisfied as long as ∑L

i=i φi ≤ PC. Thus, from (12),we can write E ≤ PC‖w̃1‖2, with the equality holding if andonly if ‖w̃1‖2 = max

i∈[1,L]‖w̃i‖2, φ1 = 0 and φi = 0, ∀ i ∈ [2,L].

It is straightforward to see that this is verified if and only ifu1 = t1, with t1 the right-singular vector of W associated withthe largest singular value of W, i.e.,

√λW,1. As a consequence,

the covariance matrix for the CIA input signal that maximizesE in (7) is

Φ = PCt1tH1 ,

and this concludes the proof. �

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Marco Maso (S’08–M’14) received the Bachelor’sdegree in 2005 and the M.Sc. degree in telecom-munications engineering in 2008, both from Uni-versity of Padova, Padova, Italy. He received boththe Ph.D. degree in information engineering (I.C.T.)from University of Padova and the Ph.D. degree intelecommunications (ICST) from Supélec, France,in 2013.

He has been engaged in research on practicalimplementations of OFDM packet synchronizationin 2005/06, DVB-T2 systems implementation in

2008/09, signal processing algorithms for high speed coherent optical commu-nications in 2012/13 and dynamic spectrum management strategies for mobilead-hoc networks in 2013/2014. He held a position as a research engineerat Supélec in 2012/13 and as a post-doctoral research fellow at SingaporeUniversity of Technology and Design in 2013/14. Since September 2014 is aResearcher with the Mathematical and Algorithmic Sciences Lab of HuaweiFrance Research Center.

Dr. Maso has been actively involved in organizing and chairing sessions,and has served as a member of the Technical Program Committee in a numberof international conferences, including IEEE flagship conferences. He wasincluded in the “Best of Globecom,” 50 best papers in IEEE Globecom 2014.His research interests broadly span the areas of wireless communicationsand signal processing for the physical layer, with focus on heterogeneousnetworks, self-organizing networks, MIMO systems, interference managementand suppression techniques, wireless power transfer and cognitive radios.

Subhash Lakshminarayana (S’07–M’13) receivedthe M.S. degree in electrical and computer engi-neering from The Ohio State University, Columbus,OH, USA, in 2009, and the Ph.D. degree from theAlcatel Lucent Chair on Flexible Radio and the De-partment of Telecommunications at École Supérieured’Électricité (Supélec), France, in 2012. Currently heis with Singapore University of Technology and De-sign (SUTD). He has held visiting research appoint-ments at Princeton University from Aug.–Dec. 2013and May–Nov. 2014. He has also been has been

a student researcher at the Indian Institute of Science, Bangalore, India,during 2007.

Dr. Lakshminarayana was included in the “Best of Globecom,” 50 bestpapers in IEEE Globecom 2014. His research interests broadly spans wirelesscommunication and signal processing with emphasis on small cell networks(SCNs), cross-layer design wireless networks, MIMO systems, stochastic net-work optimization, energy harvesting and smart grid systems.

Tony Q. S. Quek (S’98–M’08–SM’12) received theB.E. and M.E. degrees in electrical and electron-ics engineering from Tokyo Institute of Technology,Tokyo, Japan, respectively. He received the Ph.D.degree in electrical engineering and computer sci-ence from Massachusetts Institute of Technology,Cambridge, MA, USA. Currently, he is an AssistantProfessor with the Information Systems Technologyand Design Pillar at Singapore University of Tech-nology and Design (SUTD). He is also a Scientistwith the Institute for Infocomm Research. His main

research interests are the application of mathematical, optimization, and statis-tical theories to communication, networking, signal processing, and resourceallocation problems. Specific current research topics include sensor networks,heterogeneous networks, green communications, smart grid, wireless security,compressed sensing, big data processing, and cognitive radio.

He has been actively involved in organizing and chairing sessions, andhas served as a member of the Technical Program Committee as well assymposium chairs in a number of international conferences. He is serving asthe Co-chair for the PHY & Fundamentals Track for IEEE WCNC in 2015,the Communication Theory Symposium for IEEE ICC in 2015, the PHY &Fundamentals Track for IEEE EuCNC in 2015, and the Communication andControl Theory Symposium for IEEE ICCC in 2015. He is currently an Editorfor the IEEE TRANSACTIONS ON COMMUNICATIONS, the IEEE WIRELESS

COMMUNICATIONS LETTERS, and an Executive Editorial Committee Memberfor the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS. He wasGuest Editor for the IEEE COMMUNICATIONS MAGAZINE (Special Issueon Heterogeneous and Small Cell Networks) in 2013, the IEEE SIGNAL

PROCESSING MAGAZINE (Special Issue on Signal Processing for the 5G Rev-olution) in 2014, and the IEEE WIRELESS COMMUNICATIONS MAGAZINE

(Special Issue on Heterogeneous Cloud Radio Access Networks) in 2015.Dr. Quek was honored with the 2008 Philip Yeo Prize for Outstanding

Achievement in Research, the IEEE Globecom 2010 Best Paper Award, the2011 JSPS Invited Fellow for Research in Japan, the CAS Fellowship for YoungInternational Scientists in 2011, the 2012 IEEE William R. Bennett Prize, andthe IEEE SPAWC 2013 Best Student Paper Award.

H. Vincent Poor (S’72–M’77–SM’82–F’87) re-ceived the Ph.D. degree in EECS from PrincetonUniversity, Princeton, NJ, USA, in 1977. From 1977until 1990, he was on the faculty of the Universityof Illinois at Urbana-Champaign. Since 1990 hehas been on the faculty at Princeton, where he isthe Michael Henry Strater University Professor ofElectrical Engineering and Dean of the School ofEngineering and Applied Science. His research inter-ests are in the areas of information theory, statisticalsignal processing and stochastic analysis, and their

applications in wireless networks and related fields including social networksand smart grid. Among his publications in these areas are the recent booksPrinciples of Cognitive Radio (Cambridge University Press, 2013) and Mech-anisms and Games for Dynamic Spectrum Allocation (Cambridge UniversityPress, 2014).

Dr. Poor is a member of the National Academy of Engineering, the NationalAcademy of Sciences, and is a foreign member of Academia Europaea andthe Royal Society. He is also a fellow of the American Academy of Arts andSciences, the Royal Academy of Engineering (U.K), and the Royal Societyof Edinburgh. He received the Marconi and Armstrong Awards of the IEEECommunications Society in 2007 and 2009, respectively. Recent recognition ofhis work includes the 2014 URSI Booker Gold Medal, and honorary doctoratesfrom Aalborg University, Aalto University, the Hong Kong University ofScience and Technology, and the University of Edinburgh.

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