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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 32, NO. 12, DECEMBER 2014 1

Consumption Factor and Power-Efficiency Factor:A Theory for Evaluating the Energy Efficiency of

Cascaded Communication SystemsJames N. Murdock, Member, IEEE and Theodore S. Rappaport, Fellow, IEEE

Abstract—This paper presents a new theory, called the con-sumption factor theory, to analyze and compare energy efficientdesign choices for wireless communication networks. The ap-proach presented here provides new methods for analyzing andcomparing the power efficiency of communication systems, thusenabling a quantitative analysis and design approach for “greenengineering” of communication systems. The consumption factor(CF) theory includes the ability to analyze and compare cascadedcircuits, as well as the impact of propagation path loss on the totalenergy used for a wireless link. In this paper, we show severalexamples how the consumption factor theory allows engineers tocompare and determine the most energy efficient architecturesor designs of communication systems. One of the key conceptsof the consumption factor theory is the power efficiency factor,which has implications for selecting network architectures orparticular cascaded components. For example, the question ofwhether a relay should be used between a source and sinkdepends critically on the ratio of the source transmitter power-efficiency factor to the relay transmitter power-efficiency factor.The consumption factor theory presented here has implicationsfor the minimum energy consumption per bit required toachieve error-free communication, and may be used to extendShannon’s fundamental limit theory in a general way. This workincludes compact, extensible expressions for energy and powerconsumption per bit of a general communication system, andmany practical examples and applications of this theory.

Index Terms—Power Consumption, Energy Efficiency, PowerEfficiency, Millimeter-wave, Wireless, Cascaded circuits, Capac-ity, Relay channel.

I. INTRODUCTION

COMMUNICATION systems today, including both wire-line and wireless technologies, consume a tremendous

amount of power. For example, the Italian telecom operatorTelecom Italia used nearly 2 Tera-Watt-hours (TWh) in 2006to operate its network infrastructure, representing 1% of Italy’stotal energy usage [1]. Nearly 10% of the UK’s energy usageis related to communications and computing technologies [1],while approximately 2% of the US’s energy expenditure isdedicated to internet-enabled devices [2]. In Japan, nearly 120W of power are used per customer in the cellular network

Manuscript received: April 15, 2012, revised: October 12, 2012. Portionsof this work appeared in the 2012 IEEE Global Communications Conference(Globecom).J. N. Murdock is Texas Instruments, Dallas, TX (e-mail:

[email protected]). This work was done while James was a student atThe University of Texas at Austin.T. S. Rappaport is with NYU WIRELESS at New York University and

NYU-Poly,715 Broadway, Room 702, New York, NY 10003 USA (e-mail:[email protected]).Digital Object Identifier 10.1109/JSAC.2014.141204.

[1]. Similar power/customer ratios are expected to hold formany large infrastructure-based communication systems. [2]estimates that 1000 homes accessing the Internet at 1 Giga-bit-per-second (Gbps) would require 1 Giga-Watt of power.All of these examples indicate that energy efficiency ofcommunication systems is an important topic. Given the trendtoward increasing data rates and data traffic, energy efficientcommunications will soon be one of the most important chal-lenges for technological development, yet a theory that allowsan engineer to easily compare and analyze, in a quantitativefashion, the most energy efficient designs has been allusive.Past researchers have explored analytical and simulation

methods to compare and analyze the power efficienciesof various wireless networks (see, for example, works in[3][4][5][6][7][8]). In [3], researchers explored the energyefficiency in an acoustic submarine channel and illustratedhow the choice of signaling, when matched to the channel,could approach Shannon’s limit. In [4], researchers considereda position-based network routing algorithm that could beoptimized locally at each user, in an effort to reduce overallpower consumption of the network, but were unable to deriveconvenient and extensible expressions for power efficiencythat could be generalized to any network. In [5], energyconsumption was compared to the obtainable data rate of end-users, and an analysis technique was used to determine energyefficiency through the use of distributed repeaters. In [6], anovel bandwidth allocation scheme was devised to optimizethe power consumed in the network while maximizing datarate, but the analysis was not extensible to a cascaded systemof components, nor could it be easily generalized. [7] illus-trates how cumbersome and complicated the field of energyconservation can be in ad-hoc networks, at both the link andnetwork layers (e.g. the individual wireless link, as well as thenetwork topology, where both have a strong impact on energyutilization). In fact, a recent book, Green Engineering [8],illustrates the importance, yet immense difficulty, in providingan easy, generalized, standardized method for analyzing andcomparing power efficiency in a communications network.Despite the extensive body of literature aimed at energy

efficient communication systems, we believe this paper is thefirst to present a generalized analysis that allows engineersto provide a standard “figure of merit” to compare the powerefficiency (or energy efficiency) of different cascaded circuitor system implementations over a wide array of problemdomains. The analysis method presented here is general,in that it may be applied to power efficient circuit design,

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2 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 32, NO. 12, DECEMBER 2014

transmitter and receiver design, and also to various networkarchitectures such as relay systems (the relay problem in [5]is validated using the CF theory in this paper).This work has been motivated by the need to have a

compact, repeatable, extensible analysis method for comparingthe power efficiency of communication systems and networkdesigns. In particular, as cellular communication networksevolve, the base station coverage regions will continue toshrink in size, meaning that there will be a massive increasein the number of base stations or access points, and relaysare likely to complement the base stations over time [9].To accommodate the demand for increased data rates tomobile users, we envisage future millimeter-wave (mm-wave)communication systems that are much wider in bandwidththan today’s cellular and Wi-Fi networks. These future systemswill use highly directional steerable antennas and channelbandwidths of many hundreds of MHz thereby supportingmany Gigabits per second data rates to each mobile device[9] [10][11][12]. As such systems evolve, small-scale fadingin the channel will become much less of a concern, and moreattention will need to be placed on the power efficient designof handsets and “light weight” base stations and repeaters thatuse wideband channels and multi-element phased arrays withRF amplifiers. The theory presented here aims to aid in thedesign of these wideband wireless networks and devices. Asshown in this paper, the CF framework gives communicationengineers a methodology to analyze, compare and tradeoffcircuit and system design decisions, as well as network archi-tectures (e.g. whether to use relays or small cells, and howto trade off antenna gain, bandwidth, and power efficiency infuture wireless systems) [9][10][11][12].In this paper, we provide fundamental insight into the

required power consumption for communication systems, andcreate an-easy-to-use theory, which we call the consumptionfactor (CF) theory, for analyzing and comparing any cascadedcommunication network for power efficiency. In Section IIwe present the consumption factor framework for a homo-dyne transmitter [12]. Section III generalizes the conceptof power-efficiency analysis, which is fundamental to theconsumption factor framework, for any cascaded communi-cation system. Section IV provides numerical examples of thepower-efficiency factor used in the consumption factor theory.Section V presents a general treatment of the consumptionfactor, based on the power-efficiency analysis of the precedingsections. Section VI demonstrates a key characteristic of thepower-efficiency factor – i.e. that gains of components thatare closest to the sink of a communication system reducethe impact of the efficiencies of preceding components. InSection VII, we use the consumption factor framework todevelop fundamental understandings of the energy price ofa bit of information. We use our analysis to demonstrate howthe consumption factor theory may be applied to designingenergy efficient networks, for example by helping to determinethe best route to send a bit of information in a multi-hopsetting to achieve the lowest energy consumption per bit.Section VIII provides conclusions. The key contribution of thispaper is a powerful and compact representation of the powerconsumption and energy consumption per bit of a generalcommunication system. The representation takes the gains

Fig. 1. Block diagram of a homodyne transmitter used to demonstrate thepower-efficiency factor and consumption factor (CF).

and efficiencies of individual signal-path components (suchas amplifiers and mixers) into account. A second key result isthat, in order to align the goals of lower energy per bit andhigher data rates, it is advantageous to design communicationsystems that require as little signal power as possible, solow, in fact, that ancillary power drain (e.g. for cooling, userinterfaces, etc.) dominates signal power levels. While this mayseem intuitive, the CF theory proves this, and provides atangible, objective way of comparing various designs whileshowing the degrees to which communication systems mustreduce ancillary power drain, but must also seek means ofreducing required signal levels even more dramatically thanthe ancillary power drain. By making every bit as energyefficient as possible, we show it is possible to greatly expandthe number of bits that can be delivered for a given amountof energy. Means of achieving this goal include the use veryshort link distances (such as femtocells) at millimeter-wavefrequencies for future massively broadband wireless systems.Earlier, less developed versions of the consumption factor werepresented in [13].

II. CONSUMPTION FACTOR FOR A HOMODYNETRANSMITTER

We define the consumption factor (CF) for a communicationsystem as the maximum ratio of data rate to power consumed,or equivalently as the maximum number of bits that maybe transmitted through a communication system for everyJoule of expended energy. A study of the consumption factorrequires a careful analysis of both the power consumptionand data rate capabilities of a communication system. In thissection, we will provide a simple analysis for a homodynetransmitter as illustrated in Figure 1, to motivate the theorypresented here. We consider a homodyne transmitter becausethis topology is attractive for many massively-broadband sys-tems due to its low cost and low complexity [12]. We willgeneralize our analysis in Section III to be applicable to ageneral cascaded communication system.The homodyne transmitter in Figure 1 is comprised of

components that directly handle the signal, such as the mixerand power amplifier, in addition to components that interactindirectly with the signal, such as the oscillator. Componentsthat interact directly with the signal are designated “on thesignal path,” while components that are not in the path of thesignal are designated “off the signal path.”


MURDOCK and RAPPAPORT: CONSUMPTION FACTOR AND POWER-EFFICIENCY FACTOR: A THEORY FOR EVALUATING THE ENERGY EFFICIENCY... 3

A key component of the consumption factor framework isunderstanding that the efficiency of each signal path compo-nent may be used to relate the ancillary or “wasted” powerof each component to the total signal power delivered by thatcomponent. For example, the efficiency of the power amplifieris used to find the partial power required to bias the amplifier(which is not used, or wasted, in terms of providing signal pathpower) as a component of the total signal power delivered tothe load. We define the efficiency of the power amplifier,ηPA,and of the mixer, ηMIX ,:

ηPA =PPARF

PPARF + PPA

NON−RF

(1)

ηMIX =PMIXRF

PMIXRF + PMIX

NON−RF

(2)

where PPARF is the signal power delivered by the power

amplifier to the matched load, and PMIXRF is the signal power

delivered by the mixer to the power amplifier. PPANON−RF and

PMIXNON−RF are the power levels used by the power amplifier,and mixer, respectively, that do not directly contribute todelivered signal power. Using (1) – (2), we find:

PPANON−RF = PPA

RF

(1

ηPA− 1

)(3)

PMIXNON−RF = PMIX

RF

(1

ηMIX− 1

)(4)

The second key step in the consumption factor analysis resultsfrom the realization that the signal powers delivered by eachcomponent in the cascade, PPA

RF , and PMIXRF , may be related

to the total power delivered by the communication system,through the gains of each signal path component. As shownin Section III, this formulation for a cascaded system’s powerefficiency is reminiscent of Frii’s classic noise figure analysistechnique for cascaded systems [24]. Using (1)-(4), we nowfind the delivered RF power to the matched load, PRADIO

RF ,in terms of the signal power from the baseband signal sourceand the various gains stages as:

PRADIORF = PBB

SIGGMIXGPA (5)

PPANON−RF =PPA

RF

(1

ηPA−1

)=PRADIO

RF

(1

ηPA−1

)(6)

PMIXNON−RF =PMIX

RF

(1

ηMIX−1)=PRADIORF

GPA

(1

ηMIX−1

)(7)

where PBBSIG is the signal power delivered by the baseband

components to the mixer, and GMIXand GPA, are the powergains of the mixer and power amplifier, respectively. Equation(5) simply states that the power delivered to the matched loadis equal to the power delivered by the baseband componentsmultiplied by the gain of the mixer and of the power amplifier.Note that we have implicitly assumed an impedance matchedenvironment. Impedance mismatches may be accounted for byincluding a mismatch factor less than one in the gain of eachcomponent.

The total power consumption of the homodyne transmittermay be written as:

PRADIOconsumed = PRADIO

RF + PPA

NON−RF + PMIXNON−RF

+ PBB + POSC (8)

where PBB is the power consumed by the baseband com-ponents and POSC is the power consumed by the oscillator.The term PRADIO

RF is the total signal power in the homodynetransmitter delivered to the load. Using equations (5) through(7) in (8), we re-write the total homodyne power consumptionas:

PRADIOconsumed = PRADIO

RF

(1+

(1

ηPA−1

)+

1

GPA

(1

ηMIX− 1

))+ PBB + POSC (9)

PRADIOconsumed =

PRADIORF(

1 +(

1ηPA

− 1)+ 1

GPA

(1

ηMIX− 1

))−1

+ PBB + POSC (10)

From (10), the factor(1 +

(1

ηPA− 1

)+ 1

GPA

(1

ηMIX− 1

))−1

plays a role in the transmitter power consumption analogousto that of efficiency. In other words, this factor may beconsidered the aggregate efficiency of the cascade of themixer and power amplifier. In Section III we will generalizethis result and define this factor as the power efficiency factorfor an arbitrary cascaded system (where the cascade maybe either a cascade of components or circuits, or may eveninclude the propagation channel).Now that we have formulated a compact representation

of the power consumption of a homodyne transmitter, wemust determine the maximum data rate that the transmittercan deliver in order to formulate the consumption factor ofthe transmitter. To do this, we assume that the transmitter iscommunicating through a channel with gain Gchannel to areceiver of gain GRX having noise figure F with bandwidthB. We assume also that the transmitter matched load isreplaced by an antenna with gain GANT

TX . The signal powerused by the receiver in the detection process,PRX , is given by:

PRX = PRADIORF GANT

TX GchannelGANTRX GRX (11)

where GANTRX is the gain of the receiver antenna, and GRX

is the gain of the receiver excluding the antenna. We willassume an AWGN (Additive White Gaussian Noise) channel,for which the received noise power at the detector, Pnoise, is:

Pnoise = KTFB ×GRX (12)

where K is Boltzmann’s constant (1.38x10−23 J/K) and T isthe system temperature in Kelvin. The SNR at the receiverdetector is therefore:

SNR =PRADIORF GANT

TX GchannelGANTRX GRX

KTFB ×GRX(13)

The SNR is related to the minimum acceptable SNR at theoutput of the receiver, SNRmin, as dictated by the modulation



Fig. 2. An example of the use of the power-efficiency factor to find theconsumption factor of two different cascades of a baseband amp, mixer, andRF amp.

and signaling scheme, through a particular operating marginMSNR:

SNR = MSNRSNRmin (14)

The minimum power consumption occurs when MSNR isequal to 0 dB (i.e. MSNR = 1). Solving for PRADIO

RF , wefind:

PRADIORF,min =

SNRminKTFB

GANTTX GchannelG

ANTRX

(15)

where we now denote PRADIORF as PRADIO

RF,min to indicate thatthis power level corresponds to the minimum acceptable SNRat the receiver. The minimum power consumption for thetransmitter is found using (10) and (15) as:

PRADIOconsumed,min =

SNRminKTFBGANT

TX GchannelGANTRX(

1 +(

1ηPA

− 1)+ 1

GPA

(1

ηMIX− 1

))−1

+ PBB + POSC (16)

The maximum data rate Rmax at the receiver is given interms of the SNR and the bandwidth according to Shannon’scapacity formula if the modulation and signaling scheme arenot specified. If these are specified, then we find the maximumdata rate in terms of the spectral efficiency of the modulationand signaling scheme ηsig (bps/Hz):

Rmax = Blog2 (1 + SNR) , General Channel

Rmax = Bηsig , SpecificModulation Scheme (17)

The consumption factor, CF , for the homodyne transmitter isthen found by taking the ratio of (17) to (16):

CF =Rmax

PRADIOconsumed,min

(18)

CF =Blog2 (1 + SNR){

SNRminKTFB

GANTTX

GchannelGANTRX(

1+(

1ηPA

−1)+ 1

GPA

(1

ηMIX−1

))−1 + PBB + POSC

}(19)

We will assume a standard log-distance channel gain model:

Gchannel = PGo + 10α× log10

(dod

)[dB] (20)

Fig. 3. Higher values of power consumption off of the signal path com-ponents result in higher values of SNR needed to maximize the consumptionfactor (CF).

where PGo is the close-in free-space path gain (usually alarge negative number in dB) received at a close-in referencedistance do, d is the link distance ( d >do), and α is thepath loss exponent [10][12][19][20]. Two examples for CFusing equation (18) and (19) are shown in Figures 2 and 3.Figure 2 shows how the consumption factor of a 60 GHzwireless communication system varies as the efficiency of thepower amplifier or the mixer are changed, and indicates thatthe efficiency of the power amplifier is much more importantin terms of maximizing the overall system efficiency than themixer’s efficiency. The key lesson from this example is that theefficiencies of the devices that handle the highest signal powerlevels should be maximized in order to have the most dramaticeffect in maximizing the consumption factor. Figure 3 showsthe impact of changing the minimum required SNR at thereceiver. Note that we have assumed an SNR margin of 0 dB.The figure indicates that higher levels of power consumptionby non-signal-path devices such as the oscillator result inhigher levels of SNR to maximize the consumption factor. Thefigure also indicates an optimum value of SNR to maximize theconsumption factor. This optimum value depends critically onthe amount of power consumed by devices off the signal path.Note that in these figures, we have assumed the efficiency andgain of the mixer are equal. This assumption will be explainedin Section III, where we will find that the gain and efficiencyof an attenuating device are equal (similar to Friis’ noise figureanalysis). Note that we have used a logarithmic scale in Figure3 to allow for easy comparison between the different curves.

III. GENERAL CASCADED COMMUNICATION SYSTEM

We will now generalize the consumption factor to providea framework for analyzing a general cascaded communicationsystem.The consumption factor is defined [18] as the maximum

ratio of data rate to total power consumption for a commu-nication system. To determine the consumption factor, wemust first determine a compact representation of the powerconsumption of a general cascaded communication system.Consider a general cascaded communication system as shownin Figure 4 in which information is generated at a source, andsent as a signal down a signal path to a sink. Signal pathcomponents such as amplifiers and mixers are responsible



Source

. . .

Sink

. . . . . . . . . .

Signal Path Devices

Non-Signal Path Devices

1 2 N

1 k M

Fig. 4. A general communication system composed of components on andoff the signal path.

for transmitting the information signal to the sink. Non-signal path components include voltage regulation circuitry,displays or cooling components that do not participate directlyin the signal path, but do consume power. The total powerconsumption of the cascaded communication system in Figure4 (ignoring the source and sink) may be written as:

Pconsumed = Psig +

N∑k=1

Pnon−sigk +

M∑k=1

Pnon−pathk(21)

where Psig is the sum of all signal powers of each componentin the cascade, Pnon−sigk is the signal power used by the k

th

signal path component but not delivered as signal power tothe next signal-path component, and Pnon−pathk

is the powerused by the kth component off the signal path. To evaluate(21), we must consider each component on the signal pathseparately. The efficiency of the ith signal path componentmay be written as:

ηi =Psigi

Psigi + Pnon−sigi

(22)

Where Psigi is the total signal power delivered by the ith stage

to the (i + 1)th stage, and Pnon−sigi is the signal power used

by the ith stage component but not delivered as signal power.This is a very general representation of efficiency that may beapplied to any communication system component. A similarmeasure of efficiency, the PUE (Power Usage Effectiveness),is already used to measure the performance of data centers,and is the total power used for information technology dividedby the total power consumption of a data center[14].Let us consider (22) applied to an attenuating stage, such as

a wireless channel or attenuator. Fundamentally, an attenuatorshould consume only the signal power delivered to it bythe preceding stage (i.e. the consumption factor theory treatsattenuators as passive components that do not take power froma power supply). The signal power delivered by an attenuatorto the next stage is a fraction of the signal power delivered tothe attenuator. Therefore, if the ith stage is an attenuator, thenthe efficiency of an attenuator, ηatten, as given by (22) is:

Psigi = GattenPsigi−1 (23)

Pnon−sigi = (1−Gatten)Psigi−1 (24)

ηatten =GattenPsigi−1

GattenPsigi−1 + (1−Gatten)Psigi−1

= Gatten

(25)

where Gatten is the gain of the attenuator, and is less thanone. Thus, we have shown that ηatten = Gatten for a passivedevice or channel.The total power consumed by the ith stage on the signal

path may be written:

Pconsumedi = Pnon−sigi + Padded−sigi (26)

where Padded−sigi is the total signal power added by theith component, which is the difference in the signal powerdelivered to the (i+ 1)

th component and the signal powerdelivered to the ith component. We can sum all the signalpowers added by the components on the signal path (fromleft to right in Figure 4) to find:

N∑i=1

Padded−sigi = PsigN − Psigsource (27)

where Psigsource is the signal power provided by the source,and PsigN is the signal power delivered by the Nth(and laststage) signal-path component. Adding (27) to the signal powerfrom the source, we find that the total signal power in thecommunication system is equal to the signal power deliveredto the sink (in other words, the signal power delivered by thelast stage is equal to the sum of all signal powers deliveredby each component in the cascade):

Psig = P sigN(28)

From (22) the total “wasted” power of the kth stage (i.e. powerconsumed but not delivered to the next signal path stage) maybe related to the efficiency and total delivered signal powerby that stage:

Pnon−sigk = Psigk

(1

ηk− 1

)(29)

Also, the signal power delivered by the kth stage may berelated to the total power delivered to the sink by dividing bythe gains of all stages after the kth stage, (i.e. to the right ofthe kth) thus yielding:

PsigN= Psigk

N∏i=k+1

Gi (30a)

Pnon−sigk =PsigN

N∏i=k+1

Gi

(1

ηk− 1

)(30b)

where Gi is the gain of the ith stage. We can thereforecompute the total power consumed by the communicationsystem as the power consumed by the source which is assumedto equal the signal power delivered by the source, and the threeadditional terms that represent the power consumed by the in-path cascaded components, and the power dissipated by thenon-signal path components:

Pconsumed = Psigsource +

N∑i=1

Pconsumedi +

M∑k=1

Pnon−pathk

(31a)



Pconsumed = Psigsource +

N∑i=1

Padded−sigi

+

N∑k=1

Pnon−sigk +

M∑k=1

Pnon−pathk(31b)

= PsigN

⎛⎜⎜⎜⎝1 +

N∑k=1

1N∏

i=k+1

Gi

(1

ηk− 1

)⎞⎟⎟⎟⎠+M∑k=1

Pnon−pathk

(31c)In certain circumstances, such as when comparing two dif-ferent smart phones or other devices that have substantialpower consumed by displays or computer processors, it isuseful to incorporate the impact of the power efficiencies ofthe non-path components. To do this, we may simply add thesecomponents to the end of the signal-path cascade in Figure 4and assume unity gain. For example, write the total powerconsumption of the kth non-path component Pnon−pathk

in terms of its usefully dissipated power Puk(power that

directly contributes to its intended functionality) and its powerefficiency ηnon−pathk

(the ratio of usefully dissipated powerto its total power consumption):

Pnon−pathk=

Puk

ηnon−pathk

(32)

We may then re-write (31c) as (33). For simplicity, we nowcarry on the development of the CF analysis with the powerconsumption expression given by (31c) rather than (33), aswe wish to isolate the impact of the efficiencies of non-pathcomponents (noting that such analysis may be done by simplyappending the power efficiencies of non-path components asdescribed above). We see from (31c) and (35) that the on-pathcascade components may be conveniently represented in thetotal power consumption of the cascade as:

Pconsumed =PsigN

H+ Pnon−path (34)

where Pnon−path is the total power used by devices off thesignal path, and in (34), we introduce the system power-efficiency factor H of all cascaded components defined as:

H =

⎧⎪⎪⎪⎨⎪⎪⎪⎩1+

N∑k=1

1N∏

i=k+1

Gi

(1

ηk− 1

)⎫⎪⎪⎪⎬⎪⎪⎪⎭

−1

(35)

Where H ranges between 0 and 1, and we call H the power-efficiency factor of the entire signal-path cascade. Note H−1

ranges from 1 to infinity (just like Friis’ Noise Figure). Equa-tion (35) is a very general expression relating the gains andpower efficiencies of the individual components on the signalpath to the signal-path efficiency of the overall communicationsystem. An implication of this is that the efficiencies of devicesthat handle the most power are most important in terms ofthe power-efficiency factor of the entire cascade, as these willbe the components in (35) whose efficiencies are divided bythe smallest numbers. As shown subsequently, the presence ofattenuators, such as a wireless channel, makes it such that thepower efficiencies of stages that handle the most power just

prior to the large attenuator, such as a power amplifier, havethe largest impact on overall system power efficiency.Note that we have defined the power efficiency of a signal-

path component (22) in terms of the total power it delivers,i.e. using (26) in (22) we have:

ηi=Psigi

Psigi + Pnon−sigi

=Psigi−1 + Padded−sigi

Psigi−1 + Padded−sigi + Pnon−sigi(36a)

ηi =Psigi−1 + Padded−sigi

Psigi−1 + Pconsumedi

(36b)

This representation of power efficiency is useful as it isapplicable to both passive components, which do not addsignal power, and active components that add signal power.For an attenuator, Padded−sigi is zero (i.e. for the CF theorywe assume that attenuators can not add additional signalpower), while Pnon−sigi is the signal power removed by theattenuator. One important application of (36) and (25) is whenthe attenuator is a wireless channel. From (25), the power-efficiency factor of a wireless channel Hchannel is given byits gain Gchannel:

Hchannel = Gchannel (37)

The power-efficiency factor is a powerful and general meansof determining the power consumption of a communicationsystem. For example, consider two cascaded sub-systemswhose H’s have already been characterized, where sub-system#2, with power-efficiency factor Hsub−system 2 and gainGsub−system2 follows sub-system #1 with power-efficiencyfactor Hsub−system 1. We can show from (35) that the power-efficiency factor of the entire cascade, Hcascaded system, maybe written much like the classic noise figure theory (Eqn.38) where the first sub-system is composed of components1 through M-1, and the second subsystem is composed ofcomponents M through N. Of course, M may be any integerfrom 1 through M, so (38d) is a completely general result.Note from (38) that the power-efficiency factor of

the second stage Hsub−system 2 is an upper bound forHcascaded system. This is easily seen by inverting (38d) andexamining the limiting case in which the first stage has anoptimal power-efficiency factor of 1 as in (39). Consider alsothe case of a single component [9]. Using (22) and (35), wesee:

ηi = Hi (40)

Consider now the case in which a wireless channel exists be-tween a transmitter and receiver. The overall power-efficiencyfactor of the entire transmitter-receiver pair, Hlink is given by:

H−1link = H−1

RX +1

GRX

(1

Gchannel− 1

)

+1

GRXGchannel

(H−1

TX − 1)

(41)

where HRX is the power-efficiency factor of the receiver,HTX

is the power-efficiency factor of the transmitter, GRX is thegain of the receiver, and Gchannel is the channel gain (whichis less than 1) which is equal to the power-efficiency factor ofthe channel. Note from (41) that if the receiver gain is muchsmaller than the expected channel path loss, the cascadedpower-efficiency factor will be very small and on the order



Pconsumed = PsigN

⎛⎜⎜⎜⎝1 +

1

PsigN

M∑k=1

Puk

ηnon−pathk

+

N∑k=1

1N∏

i=k+1

Gi

(1

ηk− 1

)⎞⎟⎟⎟⎠ (33)

H−1cascadedsystem =

{1+

(1

ηN− 1

)+

1

GN

(1

ηN−1− 1

)+ . . .+

1

GN . . . GM+1

(1

ηM− 1

)+

1

GN . . .GM

(1

ηM−1− 1

)

+ . . .1

N∏i=1

Gi

(1

η1− 1

)⎫⎪⎪⎬⎪⎪⎭

−1

(38a)

H−1sub−system2 = 1+

(1

ηN− 1

)+

1

GN

(1

ηN−1− 1

)+ . . .+

1

GN . . . GM+1

(1

ηM− 1

)(38b)

H−1sub−system1 = 1 +

(1

ηM−1− 1

)+ . . .

1M−1∏i=1

Gi

(1

η1− 1

)(38c)

H−1cascadedsystem = H−1

sub−system2 +1

Gsub−system2

(H−1

sub−system1−1)

(38d)

Hcascadedsystem =Hsub−system1Hsub−system2

Hsub−system1 +Hsub−system2

Gsub−system2(1−Hsub−system1)

(39a)

limHsub−system1→1

Hcascadedsystem =Hsub−system2 (39b)

of the product of the channel gain with the receiver gain. Inthis case, we find that the overall power-efficiency factor isapproximated by:

Hlink ≈ GRXGchannelHTX (42) (42)

This is an important result of this analysis. In particular, itindicates that in order to achieve a very power-efficient link,it is desirable to have a high gain receiver and a highlyefficient transmitter. This can be understood by realizing thata higher gain receiver reduces the output power requirementsat the transmitter. Eqn. (42) indicates the great importanceof the transmitter efficiency. Note, however, that the receiverefficiency is still important, as from (39b) it is clear that thereceiver’s efficiency is an upper bound on the efficiency of theoverall link.

IV. NUMERICAL EXAMPLES

To better illustrate the use of the consumption factor theory,and the use of the power-efficiency factor, consider a simplescenario of a cascade of a baseband amplifier, followed bya mixer, followed by an RF amplifier. We will consider twodifferent examples of this cascade scenario, where differentcomponents are used, in order to compare the power effi-ciencies due to the particular specifications of components.Assume that for both cascade examples, the RF amplifier is acommercially available MAX2265 power amplifier by Maximtechnology with 37 % efficiency[15]. In both cases, the mixer

is an ADEX-10L mixer by Mini-Circuits with a maximumconversion loss of 8.8 dB[16]. In the first case, the basebandamplifier (the component furthest to the left in Figure 4 ifin a transmitter, and furthest to the right if in a receiver)is an ERA-1+ by Mini-circuits, and in the second case thebaseband amplifier is an ERA-4+ [17], also by Mini-Circuits.The maximum efficiencies of these parts are estimated bytaking the ratio of their maximum output signal power to theirdissipated DC power. As the mixer is a passive component,its gain and efficiency are equal. Table 1 summarizes theefficiencies and gains of each component in the cascade. Using(35), the power-efficiency factor of the first scenario is

Hscenario 1 =1

10.37+

116.17

(1

0.36−1)+ 1

0.36∗16.17(

10.1165− 1

)= 0.2398,

whereas the power-efficiency factor of the second scenario is

Hscenario 2 =1

10.37+

116.17

(1

0.36−1)+ 1

0.36∗16.17(

10.1836−1

)= 0.2813.

Therefore, we see that the second scenario offers a superiorefficiency compared to the first scenario, due to the betterefficiency of the baseband amplifier, but falls far short ofthe ideal power efficiency factor of unity. Using differentcomponents and architectures, it is possible to characterize andcompare, in a quantitative manner, the power-efficiency factor



TABLE IAN EXAMPLE OF THE USE OF THE POWER-EFFICIENCY FACTOR TO COMPARE TWO CASCADES OF A BASEBAND AMP, MIXER, AND RF AMP.

Component Gain EfficiencyExample 1MAX2265 RF Amp 24.5 dB (voltage gain of 16.17) 37%ADEX-10L Mixer -8.8 dB 36%ERA-1+BB Amp 10.9 dB 11.65 %Example 2MAX2265 RF Amp 24.5 dB 37%ADEX-10L Mixer -8.8 dB 36%ERA-1+BB Amp 13.4 dB 18.36%

and consumption factor (see subsequent sections) of cascadedcomponents.As a second example, consider the cascade of a transmitter

power amplifier communicating through a free-space channelwith a low-noise amplifier at the receiver. Let us assume thatthe cascade, in the first case, uses the same RF power amplifieras in the previous example (MAX2265), while the LNA isa Maxim Semiconductor MAX2643 with a gain of 16.7 dB(6.68 absolute voltage gain) [18]. We will assume this LNAhas 100% efficiency for purposes of illustrating the impactof the PA’s efficiency and the channel (i.e. here we ignore theLNA’s efficiency, although this can easily be done as explainedabove). For a carrier frequency of 900 MHz, now considerthe cascade for a second case where the MAX2265 RF poweramplifier is replaced with a hypothetical RF amplifier devicehaving 45% power efficiency (a slight improvement). Assumethe link is a 100m free space radio channel with gain of -71.5dB. Since the propagation channel loss greatly exceeds theLNA gain, (42) applies, where HTX is the efficiency of the RFamplifier, so that in the first case using the MAX2265 amplifier(37% efficiency), the power efficiency factor of the cascadedsystem is 173.5e-9, while in the second case (using an RFPower amplifier with 45% efficiency), the power efficiencyfactor is 211.02e-9. The second case has an improved power-efficiency factor commensurate with the power efficiencyimprovement of the RF amplifier stage in the receiver. Thesesimple examples demonstrate how the power-efficiency factormay be used to compare and quantify the power efficienciesof different cascaded systems, and demonstrate the importanceof using higher efficiency RF amplifiers for improved powerefficiency throughout a transmitter-receiver link.

V. CONSUMPTION FACTOR

We now define the consumption factor, CF, and operatingconsumption factor (operating CF) for a general communi-cation system such as that in Figure 4, where CF is definedas:

CF =

(R

Pconsumed

)max

=Rmax

Pconsumed,min(43)

operating CF =R

Pconsumed(44)

where R is the data rate (in bits-per-second or bps), and Rmax

is the maximum data rate supported by the communicationsystem. Further analysis based on only maximizing R orminimizing Pconsumed is also pertinent to system optimizationin terms of consumed power and carried data rate. For a verygeneral communication system in an AWGN channel, Rmax

may be written using Shannon’s information theory accordingto the operational SNR and bandwidth, B:

Rmax = Channel Capacity = Blog2 (1 + SNR) (45)

Or, for frequency selective channels [3]:

Rmax =

∫ B

0

log2

(1 +

Pr (f)

N (f)

)df

=

∫ B

0

log2

(1 +

|H (f)|2 P t (f)

N (f)

)df (46)

where Pr (f), Pt (f), andN (f) are the power spectral densi-ties of the received power, the transmitted power, and the noisepower at the detector, respectively. H (f) is the frequencyresponse of the channel and any blocks that precede the de-tector. Note that equations (45) and (46) make no assumptionsabout the signaling, modulation, or coding schemes used bythe communication system. To support a particular spectralefficiency ηsig (bps/Hz), there is a minimum SNR requiredfor the case of an AWGN channel:

SNR

MSNR= SNRmin = 2ηsig − 1 (47)

The operating SNR of the system, as well as the operatingmargin of the operating SNR ( denoted by MSNR whichrepresents the operating margin above the minimum SNRmin)may be used to find the consumption factor and operatingconsumption factor expressed in terms of the system’s power-efficiency factor H :

CF =B log2 (1 + SNR)

Pnon−path +(

SNRMSNR

)× Pnoise

H

(48a)

CF =B log2 (1 +MSNR(2

ηsig − 1))

Pnon−path + (2ηsig − 1 )× Pnoise

H

, (48b)

and (49) where we have made use of (34) and the fact thatthe signal-power available to the sink, PsigN is related to thenoise power available to the sink, Pnoise and the SNR at thesink:

PsigN = Pnoise × SNR = KTFBGRX × SNR (50)

And where the right hand equality in (50) holds for an AWGNchannel. K is Boltzman’s constant (1.38x10−23 J/K), T is thesystem temperature (degrees K), F is the receiver noise factor,and B is the system bandwidth.There is an important implication of the consumption factor

that relates to the selected cell size and capacity of futurewireless broadband cellular networks. To see this, consider



Operating CF =Blog2

(SNRMSNR

+ 1)

Pnon−path + SNR× Pnoise

H

(49a)

Operating CF =Bηsig

Pnon−path +MSNR (2ηsig − 1)× Pnoise

H

(49b)

two limiting cases illuminated by the consumption factortheory. In the first case, we assume that the signal pathpower consumption is the dominant power drain for a link,as opposed to the non-path power. This may be the case,for example, in a macrocell system in which a base-stationis communicating to the edge of the macrocell, and the RFchannel requires more power to be used in the RF amplifierto complete the link than the power used to power otherfunctions. In this case, the consumption factor equation (48a)is approximated by:

CF ≈ HB log2 (1 + SNR)(

SNRMSNR

)× Pnoise

. (51)

For an AWGN channel, we find that the consumption factoris relatively insensitive to bandwidth if the signal-path powerdominates the non-path power:

CF ≈ Hlog2 (1 + SNR)(

SNRMSNR

)×KTFGRX

. (52)

Equation (52) indicates that for such a link we can increasedata rate by increasing bandwidth, but that unless the signalpath components are made much more efficient (i.e. the systempower-efficiency factor is made closer to 1), then as data rateincrease we will require approximately the same energy perbit. In other words, if transmission power is the dominantcause of energy expenditure, then there is little that canbe done to drive down the energy-price per bit through anincrease in bandwidth. There are two problems that arise: A)efficiency improvements in inexpensive IC components arebecoming harder to achieve due to performance issues whensupply voltages are scaled below 1 volt, which is approx-imately the supply voltage used by many present-day highefficiency devices, and B) with the exponential growth in datatraffic that is occurring today, unless the energy cost per bit canbe reduced exponentially, we face an un-tenable requirementfor increased power consumption by communication systems.The upshot of (52) is that for conventional cellular systems, allsignal-path devices, and particularly the RF power amplifierthe precedes the lossy channel, and other components thatprecede lossy attenuators, must be made as power efficient aspossible, thus suggesting that modulation/signaling schemesshould be chosen to support as efficient an RF amplifier aspossible.Consider the second limiting case of equation (48a), in

which the non-path power dominates the signal power. Inthis case, we are assuming that items such as processors,displays, and other non-signal path components (typical ofsmart-phones and tablets) dominate the power drain. We find

from (48a) that in this case:

CF ≈ B log2 (1 + SNR)

Pnon−path. (53)

Eqn. (53) indicates that wider-band systems are preferableon an energy-per-bit basis provided that signal-power can bemade lower than the total power used by components off thesignal path. This situation is clearly preferable to the first caseas it indicates that by increasing channel bandwidth (say, bymoving to millimeter-wave spectrum bands where there is atremendous amount of spectrum [3][12][13]), we also achievean improvement in the consumption factor, i.e. a reduction inthe energy cost per bit. Interestingly, this indicates that thegoals of massive data rates (through larger) bandwidths andsmaller cell sizes combined together can be used to achievea net reduction in the energy cost per bit. As an increase inbandwidth also enables an increase in data rate, this limitingcase allows us to simultaneously increase both data rate andconsumption factor: i.e. our goals for more data and moreefficient power utilization in delivering this higher speed dataare aligned. This is not to say that we should increase non-path power to the point that equation (53) holds. Rather, wewould desire to decrease the required signal path power tothe point where (53) holds. If the non-path power can bereduced, but the signal power can be reduced even faster, thenwe arrive at the ideal situation of improving power efficiencywith a move to higher bandwidths and greater processingand display capabilities in mobile devices. To achieve thisgoal, it is likely that link distances will need to be reducedas bandwidths are increased. The goal of making the signalpower as low as possible so that the non-path power dominatesmay at first be counter-intuitive. However, realize that inorder to have as many bits as possible flowing through acommunication system it is advantageous to make each bitas cheap as possible. In order for (53) to apply, we require:

Pnon−path >

(SNR

MSNR

)× Pnoise

H(54)

Recall the form of the power-efficiency factor of a wirelesslink given by (41). We will model the channel gain as:

Gchannel =k

dα(55)

Where d is the link distance, α is the path loss exponent,(which equals 2 for free space), and k is a constant. Using(55) in (41) and (54), we find (56). And by isolating distance,we find

dα <PNPMSNR

PnoiseSNRGRXHTXk − kHTX

HRX(GRX −HRX)

≈ PNPMSNR

PnoiseSNRGRXHTXk − kHTX

HRX(GRX) (57)



Pnon−path >

(SNR

MSNR

)× Pnoise

{H−1

RX +1

GRX

(dα

k− 1

)+

dα

GRXk

(H−1

TX − 1)}

(56)

and when further simplifying, we see

dα < GRXHTXk

(PNPMSNR

PnoiseSNR− 1

HRX

)(58)

and, finally, solving for distance, we see that

d <

{GRXHTXk

(PNPMSNR

PnoiseSNR− 1

HRX

)} 1α

(59)

If (59) is satisfied, then increasing bandwidth will have thedouble benefit of enabling both increased data rates andhigher consumption factors –i.e. lower energy consumptionper bit. One caveat to (59) is that an increase in bandwidthclearly also requires a smaller radio link distance in order for(59) to apply. To ensure that this is the case, we may introducea scaling factor β > 1 to ensure that by increasing bandwidthwithin a given bound, we do not violate (59):

d <1

β

{GRXHTXk

(PNPMSNR

PnoiseSNR− 1

HRX

)} 1α

(60)

Re-writing (60) in terms of bit rate, we find that for a givenoperating SNR:

R = Blog2

(SNR

MSNR+ 1

)(61)

whereMSNR

SNR=

1

2RB − 1

(62)

hence both data rates and consumption factors increase when

d <1

β

{GRXHTXk

(PNP

KTFB

1

2RB − 1

− 1

HRX

)} 1α

(63)

In addition to characterizing the power consumption, power-efficiency factor, and CF of a transmitter-receiver pair, theconsumption factor framework may be applied to an individualtransmitter or receiver. To do this for a transmitter, simplyreplace the transmitter antenna and channel with a matcheddummy load as was done in Section II for the homodynereceiver. Similarly, to analyze the case of a receiver, simplyapply a passive matched source to the receiver input.The preceding analysis uses a distant-dependent large-scale

spatial channel model that represents the channel path loss asa function of distance between the transmitter and receiver,as expressed by the path loss exponent (see equation (20)).As channel bandwidths increase to several hundreds of MHzat millimeter-wave bands, recent propagation measurementsshow that small scale fading is almost neglible, and large-scalefading is less variable with directional antennas that “find” thebest pointing directions at both the transmitter and receiver[10][19][20] thus validating (20) as a reasonable first-orderassumption.An interesting extension of the theory presented here, which

is beyond the scope of this paper, would consider moresophisticated channel models that include fading or variabilitydue to transients in beam switching, or the power efficiencies

and power consumption tradeoffs for various antenna arrayhardware or beam steering processing needed to implementfuture mm-wave cellular networks. For example, antennas thatexploit multipath or beam combining, and can be beamsteeredtowards the strongest reflections will be used in future wirelessnetworks [11][21][22][23]. In [22][23], it was shown thatthe direction of arrival of multipath energy for a steerableantenna can be found by measuring the cross correlation ofnarrowband (e.g. CW) fading signals, thus suggesting that fu-ture broadband millimeter-wave devices might simultaneouslyuse narrow band pilot tones that can be detected by closelyspaced low gain omni-directional antennas on the handset orbase station, while the communication traffic is simultaneouslycarried using high gain (narrow beam) steerable directionalantennas[11][10]. The CF theory can be easily extended toanalyze the power tradeoff for this additional antenna com-plexity (and many others). This is readily seen by consideringthe homodyne transmitter example, where equations (1), (3)and (11) may be used to represent the power consumption andpower efficiency of a transmitting antenna that is actually acombined phased array antenna with multiple RF power ampli-fier stages. The power drain caused by signal processing wouldbe represented in the efficiency and power consumption of theoff-path components (e.g., the signal processing components)as represented in equations (21) and (22), or (33) and (34) inthe general result. As should be clear, by quantifying the addi-tional power consumption and power efficiencies of differenttypes of processing requirements and hardware requirements,the CF theory allows for a quantitative comparison of a widerange of circuit and system implementations. We now illustratesome numerical examples to highlight the use of this analysismethod, and show how to apply the CF analysis to networkarchitectures (e.g. relay systems) subsequently in the paper.

VI. CF AND POWER-EFFICIENCY FACTOR EXAMPLE

To illustrate some pertinent effects of the ConsumptionFactor and power-efficiency factor of a communication system,first consider how the efficiencies of the individual blocks in acommunication system impact the power-efficiency factor H .For simplicity, we will consider HTX , the power-efficiencyfactor of a transmitter. First assume the transmitter is com-posed of a cascade of N stages, each with an absolute powergain of G. Further, assume that we may vary the efficiency ofthe ith stage in the transmitter from 0 to 100%. The rest ofthe stages are assumed 100% efficient. For this case, HTX isgiven by:

HTX =1

1 +Gi−N(

1ηTXi

− 1) (64)

(64) is plotted in Figure 5 for a seven-stage transmitter, witheach block having an absolute power gain of 2. The figuremakes it clear that stages closer to the output of the transmitter,which would be closer to the sink if the transmitter wereused with a receiver, have the largest impact on HTX. Further,



Fig. 5. If all the blocks in the communication system have positive gain,then the efficiencies of the blocks closest to the sink will have the mostimpact on the overall systems power-efficiency factor. In other words, it ismost important to maximize the efficiencies of components that handle thehighest power levels.

equations (48) – (49) make clear that as HTX increases from0 to 1, the Consumption Factor also increases.

VII. ENERGY PER BIT

Shannon’s limit describes the minimum energy-per-bit-per-noise spectral density required to achieve arbitrarily low prob-ability of bit error through proper coding scheme selection:

Eb

No= ln (2) (65)

This limit is generally found by using Shannon’s capacitytheorem, and allowing the code used to occupy an infinitebandwidth [24].As shown in (48) the CF is given as the maximum ratio of

data rate to power for a communication system, and may bewritten as:

CF =Blog2 (1 + SNR)

PNP + SNRmin × Pnoise

H

(66)

Let us take the limit of (66) as bandwidth approaches infinity,assuming AWGN. This is equivalent to allowing our system’scoding scheme to spread out infinitely in bandwidth, drivingour SNR down to the minimum acceptable to still achieve ar-bitrarily low error. First, recall that the SNR may be written interms of the energy-per-bit Eb, the time required to transmit asingle bit Tb, the noise spectral density No, and the bandwidthof the system B [24]:

SNR =

Eb

Tb

NoB(67)

In the limit as B approaches infinity, the SNR approaches theminimum acceptable SNR. Therefore we have (68). WhereEbc,min is the minimum energy per bit that must be consumedby the communication system, and Eb is the minimum energy-per-bit that must be present in the signal carried by thecommunication system and delivered to the receiver’s detector.Note that Ebc,min and Eb are not equal, asEbc,min is theamount of energy consumed/expended by the communicationsystem per bit (including the operation of ancillatory functionssuch as powering non-signal path components like oscillators),while Eb is the amount of energy per bit in the signalitself. Note that the denominator is no longer a function of

bandwidth. We may therefore apply Shannon’s theorem [24]to find:

1

Ebc,min=

Eb

NoTb ln (2)× 1

PNP + Eb

TbH

(69)

Ebc,min =ln (2)No

Eb

PNP

C+

Noln(2)

H=

PNP

C+

ln (2)No

H(70)

where we have made the substitution C = 1Tb, i.e. that in

the limit, the bit-rate approaches the channel capacity C.Eqn. (70) should be interpreted as the minimum energy thatmust be expended/consumed by the communication systemper bit (as opposed to the energy per bit in the signal) overthe noise spectral density in order to obtain arbitrarily lowerror rate. This interpretation should not be confused withthe interpretation of the original Shannon limit, which relatesto the bit energy per noise spectral density within the signalthat flows through the communication system. Note that if thesystem is 100 % efficient on the signal-path, and no power isused off the signal path, then (70) degenerates to Shannon’slimit, indicating that in effect the communication system andthe signal it carriers have become identical. Equation (70)indicates the importance of the power-efficiency factor ofa communication system in determining the true, practicalenergy cost of a single bit. Note also from (34) that the totalpower consumption to send a single bit is given by Pbc,bit:

Pbc,bit = PNP +Eb × C

H= C×Ebc,min. (71)

Bits delivered to the edge of a wireless cell (greater propa-gation distance) are expected to be the most costly from anenergy perspective. It is instructive to estimate the requiredpower consumption per bit as a function of a cell radius fora single user at the edge of the cell.Recall first that the power-efficiency-factor over a wireless

link may be written with (41) as (72), where Gchannel is thelink channel gain, HRX is the power-efficiency factor of thereceiver, HTX is the power-efficiency factor of the transmitter,and GRX is the gain of the receiver. Using (72) in (70), wefind (73).If we factor out the inverse of the channel gain, we find

(74). In the limit of very small channel gains (see (42)), thisyields:

Hlink → HTXGRXGchannel (75)

Ebc,min =PNP

C+

ln (2)No

HTXGRXGchannel(76)

The interpretation of (76) is that for cases in whichGRXGchannel is much smaller than unity (i.e. a highlyattenuating wireless channel), the stage immediately afterthe attenuation should have high gain, so that the stageimmediately before does not need to have an extremely highoutput power, resulting in increased loss. Secondly, we seethe importance of power amplifier efficiency and the need toovercome the loss incurred in the channel.Equation (76) confirms that the energy cost of a single bit

does indeed increase as the channel gain decreases. Severalexamples using equation (76) are shown in Figure 6 throughFigure 9. Figures 6 and 7 show an example of a 20 GHz carrier



limB→∞

CF =1

Ebc,min= limB→∞

{(Blog2 (1 + SNRmin))

PNP + SNRmin × Pnoise

H

}

= limB→∞

⎧⎪⎪⎨⎪⎪⎩

(Blog2

(1 +

EbTb

NoB

))

PNP +EbTb

NoB× NoB

H

⎫⎪⎪⎬⎪⎪⎭ (68)

Hlink =HRXHTXGRXGchannel

HTXGRXGchannel +HRXHTX (1−Gchannel) + HRX (1−HTX)(72)

Ebc,min =PNP

C+ ln (2)No

{H−1

RX +1

GRX

(1

Gchannel− 1

)+

1

GRXGchannel

(H−1

TX − 1)}

. (73)

Ebc,min =PNP

C+

ln (2)No

GRXGchannel

{GRXGchannelH

−1RX +H−1

TX −Gchannel

}(74)

system with path loss modeled according to a log-distancebreak-point model.Contrasting Figures 6 and 7 shows that highly efficient

systems can afford to use longer link distances while systemswith less efficient signal path components should use shorterdistances. The decrease in efficiency is reflected in the changein HTX and HRX between the two plots. Figures 8 and 9,where the carrier frequency has been increased to 180 GHz,for which k is higher due to atmospheric absorbtion [12],indicate that shorter link distances should be used for highercarrier frequencies (k is the value of the path loss at a close-inreference measurement distance).With equation (74), we can determine the maximum wire-

less transmission distance d for which non-path power domi-nates the power expenditure per bit, and hence the maximumdistance before each bit becomes progressively more energy-expensive:

PNP

C>

ln (2)No

GRXGchannel

((GRX

HRX−1

)Gchannel+H−1

TX

)(77)

Gchannel>ln (2)NoC

HTX

(PNPGRX+NoCln (2)

(1−GRX

HRX

)) (78)

If we model the channel gain as (55), we find:

k

dα>

ln (2)NoC

HTX

(PNPGRX +NoCln (2)

(1− GRX

HRX

)) (79)

and (80). If PNP < NoCln(2)HRX

, then (80) is unlikely to havea positive solution due to the small value of ln(2)NoC. Ourinterest in keeping the non-path energy per bit larger thanthe signal energy per bit stems from interest in making everybit as cheap from an energy perspective as possible whilesimultaneously achieving the goal of a higher capacity. Asdiscussed in Section V, by forcing non-path energy to be largerthan signal energy per bit, we can achieve the simultaneous

Fig. 6. For a system with high signal path efficiency and high non-path powerconsumption, we see that the energy expenditure per bit is dominated by non-path power, indicating little advantage to shortening transmission distances.

goals of lower energy per bit and higher capacities through anincrease in bandwidth.

Note that the maximum value of d that ensures that non-path power exceeds signal power increases as the amount ofnon-path power increases. Also, as the gain of the receiver in-creases, the link distance may be extended while still achievinga lower price-per-bit through an increase in bandwidth versusa lower receiver gain system. As expected, as the path lossexponent α increases, the maximum value of d decreases, asindicated by (80).

Required energy consumption per bit can be used to eval-uate the energy requirements of multi-hop versus single-hop communications. For example, consider the situationillustrated in Figure 10 in which the source and sink maycommunicate directly or through a relay. A similar analysis,[4], showed the importance of path loss exponent and howit can often be beneficial to transmit through relay nodes,especially with high path loss exponents. Work such as that by[5] similarly attempts to determine under what circumstancesit is advantageous to use a relay from an energy perspective



d <

{HTXk

ln (2)NoC

(PNPGRX +NoCln (2)

(1− GRX

HRX

))} 1α

(80)

Fig. 7. When signal-path components are less efficient, as illustrated here,then shorter transmission distances start to become advantageous, as signal-path power starts to represent a larger portion of the power expenditure perbit.

Fig. 8. A higher carrier frequency system that provides a much higher bitrate capacity (e.g. bandwidth) without substantially increasing non-path powerconsumption may result in a net reduction in the energy price per bit.

based on the placement of the relay. But, to our knowledgethis paper presents the first such treatment in terms of thegain of the sink and power-efficiency of the source transmitter.Consider a three node network as shown in Figure 10. If thepath loss exponent for the network is α, then the requiredpower to transmit over a given distance d is proportional to dα.Here we extend the analysis of [4] to account for the gain andefficiencies of the devices participating in the network. Theresults indicate when, on a per bit basis, it is advantageous touse the relay or the direct path from source to sink.To determine the difference in energy consumption that

would result from using a double-hop versus a single hopas illustrated in Figure 10, we simply compare the energyconsumptions for each link. For the single link throughdistance d3, we find the minimum energy consumption perbit Ebc,min,direct:

Ebc,min,direct =PNPd3

Cd3+

N0 ln (2)

Hd3(81)

Hd3 ≈ GRX,sinkGchannel,d3Hsource (82)

where Cd3 is the capacity of the direct link, PNPd3is the

non-path power associated with the direct link, Hd3 is the

Fig. 9. Lower efficiencies of signal-path components motivates the use ofshorter transmission distances.

power efficiency factor of the direct link,Hsource is the power-efficiency factor of the source transmitter, GRX,sink is thegain of the sink receiver, and Gchannel,d3 is the channel gainthrough the direct link. For the link through the relay, we findthe minimum energy consumption per bit Ebc,min,relay :

Ebc,min,relay=PNPd1

Cd1+PNPd2

Cd2+N0 ln (2)

Hd1+

N0 ln (2)

Hd2(83)

Hd1 ≈ GRX,relayGchannel,d1Hsource (84)

Hd2 ≈ GRX,sinkGchannel,d2Hrelay (85)

where Cd1 and Cd2 are the capacities of the links throughdistances d1 and d2, Hd1 and Hd3 are the power-efficiencyfactors associated with the links through d1 and d2, PNPd1

and PNPd2are the non-path powers used by the links through

distance d1 and d2. GRX,relay is the gain of the relay receiver,Hrelay is the power-efficiency factor of the relay transmitter,and Gchanneld1 and Gchanneld2 are the channel gains throughlink distances d1 and d2, respectively. Assume that all non-path powers are equal PNPd1

= PNPd2= PNPd3

= PNP , andthat the capacities through each link are equal to C. The ratioof the minimum energy consumption per bit in both casesdetermines when it is advantageous to transmit through therelay:

Ebc,min,relay

Ebc,min,direct=

2PNP

C +N0 ln (2)(

1Hd1

+ 1Hd2

)PNP

C +N0 ln (2)(

1Hd3

)

=2Hd1Hd2Hd3PNP +N0CHd3 ln (2) (Hd1 + Hd2)

Hd1Hd2Hd3PNP +N0Cln (2)Hd1Hd2(86)

Whenever (86) evaluates to be less than one, then it isadvantageous from an energy perspective to use the relay.Solving for the power-efficiency factor of the direct link

needed to satisfy this condition, we find (87) through (89).

Hd3 <NoCln (2)Hd1Hd2

Hd1Hd2PNP +NoCln (2) (Hd1 +Hd2)(87)

If we model the channel gain according to equation (55), wefind (90). As the value of k depends only on the path loss



GRX,sinkGchannel,d3Hsource <NoCln (2)Hd1Hd2


Gchannel,d3 <

(1

GRX,sinkHsource

)NoCln (2)Hd1Hd2


k

dα3<

(1

GRX,sinkHsource

)NoCln (2)Hd1Hd2

Hd1Hd2PNP +NoCln (2) (Hd1 +Hd2)

(90)

or

dα3 >GRX,sinkHsourceHd1Hd2PNP+NoCln (2) (Hd1+Hd2)

NoCln (2)Hd1Hd2

Fig. 10. The basic consumption factor analysis determines when it isadvantageous to use a relay.

value at a reference distance from the transmitter, we shallhere let k = 1, as it gives little insight by retaining it in theequations.

= GRX,sinkHsource

(PNP

NoCln (2)+

1

Hd1+

1

Hd2

)(91)

See also (92) next page. To gain intuition, assume that thenon-path power is zero, in which case:

dα3 >

(GRX,sink

GRX,relay

)dα1 +

(Hsource

Hrelay

)dα2 (93)

From (93), the link over which the relay is receiving (i.e. d1)is scaled according to the ratio of the overall gains of the sinkand the relay receivers (gain here is the ratio of the power thatis retransmitted – as for the relay - or processed/demodulated– as for the sink - to the power that is received from thetransmitter). Antenna gain is included in this gain becausewe compute path loss assuming that the path loss has beennormalized for antenna gain. Therefore, if we have a relaywith a very low gain compared to the gain of the sink receiver,the high output power required to communicate with the relaymay outweigh the benefit derived from communicating overa shorter distance. The second key point is that the linkover which the relay acts as a transmitter (i.e. d2) is scaledaccording to the ratio of the power-efficiency factors of thesource transmitter and relay transmitter. Therefore, if the relayhas a very low efficiency transmitter, any savings derived fromthe shorter link distance between the relay and the sink maybe outweighed by the loss incurred due to the inefficiencies ofthe relay. For the case of free-space channels, we may re-write

Fig. 11. From an energy perspective, it is advantageous to use a relayprovided the relay link distances are contained within the ellipse defined byequation (94). This assumes a free space path loss exponent of 2.

(93) as:

1 >

(d1

d3

)2

(GRX,relay

GRX,sink

) +

(d2

d3

)2

(Hrelay

Hsource

) (94)

which is the equation for the interior of an ellipse, as illustratedby Figure 11. It is advantageous to use the relay rather thana direct path from source to sink if the distances d1, d2, andd3 are such that they are in the interior of the ellipse in Figure11.

Figures 12 through 14 illustrate the use of equation (93) andhow the region in which it is advantageous to place a relaychanges in size as certain parameters are varied. In Figure 12,see that this region increases in size as the relay’s transmitterbecomes more efficient. In Figure 13, the same impact occursas the relay’s gain increases. Figure 14 indicates that higherpath loss exponents result in a larger area in which it isbeneficial to use the relay from an energy perspective. Thisis because an increase in path loss incentivizes a means ofshortening link distances.



dα3 >

(GRX,sinkHsourcePNP

NoCln (2)+

(GRX,sink

GRX,relay

)dα1 +

(Hsource

Hrelay

)dα2

)(92)

Fig. 12. The interior regions of each ellipse indicate where it is advantageousto place a relay. This figure shows that as the relay becomes more efficient,that it is advantageous to use the relay over a wider area.

Fig. 13. Impact of changing the relay gain. The areas inside the ellipses arewhere it is advantageous to use a relay. We see that as relay gain increases,the regions over which the relay is advantageous also increases.

VIII. CONCLUSIONS

We have presented a basic theory of consumption factor(CF) and have used this framework to develop useful con-cepts in designing future energy efficient wireless broadbandnetworks. Additionally, we have developed understanding ofhow the efficiency of a communication system impacts theminimum required energy expenditure per bit. We demon-strated the use of the consumption factor by showing whenit is advantageous to use a relay in a multi-hop setting.Our analysis indicates the importance of the receiver gainand transmitter efficiency for wireless communications. Thetheory presented here allows engineers to compare cascadesof various components, and show quantitatively that whenthe receiver gain is high, the transmitter may use lowerpower, often resulting in a net energy savings. The continuedexpansion of world-wide communication systems and theexponential increase in data traffic necessitates reducing theenergy costs per bit. A key realization from the consumptionfactor analysis is that, in order to align the goals of higherdata rates and lower energy expenditure per bit, it is necessaryto reduce the signal powers used in communication systemsto a point where ancillary power consumption (e.g. power

Fig. 14. Higher values of path loss result in a larger area where it isadvantageous to use a relay. The regions inside the ellipses are where it isadvantageous to use a relay.

consumed by oscillators and cooling equipment) is higher thanon-path signal power. Ideally, such ancillary forms of powerconsumption will be decreased rapidly, but on-path signalpowers should be decreased even more quickly. To achieve thisgoal for wireless systems, very short link distances, such asthose in a femtocells, become advantageous, or alternatively,much more efficient RF power amplifiers if longer distancesare to be used. Our theory shows that shorter link distancescombined with massive bandwidths (e.g. at millimeter-wavecarrier frequencies) and highly directional antennas will enableunprecedented data rates and lower energy consumption perbit. This in turn enables continued exponential growth in totaldata traffic while mitigating the dramatic increase in energyconsumption. In fact, as future mm-wave wireless systemsevolve using untapped spectrum above 5GHz [11][4][12], thepower consumption factor theory presented here may giveinsight into proper beam forming and minimum power config-urations for future wireless devices that use high gain adaptiveantennas that sense from where most multipath energy arrives.

ACKNOWLEDGMENTS

The authors would like to acknowledge F. Gutierrez, E. Ben-Dor, Y. Qiao, J. I. Tamir, and K. Shabaik for their usefulthoughts, discussions, and careful reviews.

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Theodore (Ted) S. Rappaport (Fellow, IEEE) received the B.S., M.S.,and Ph.D. degrees in electrical engineering from Purdue University, WestLafayette, IN, in 1982, 1984, and 1987, respectively, and is an OutstandingElectrical Engineering Alumnus from his alma mater. Currently, he holdsthe David Lee/Ernst Weber Chair in Electrical Engineering at NYU-Poly,and is also center director of the NSF I/UCRC WICAT center at NYU-Poly, and Professor of Computer Science and Professor of Radiology at NewYork University. Earlier in his career, he founded the Wireless Networkingand Communications Group (WNCG) at the University of Texas at Austin(UT), where he also was founding NSF I/UCRC site director for the WirelessInternet Center for Advanced Technology (WICAT). Prior to UT, he was onthe electrical and computer engineering faculty of Virginia Tech where hefounded the Mobile and Portable Radio Research Group (MPRG), one of theworld’s first university research and teaching centers dedicated to the wirelesscommunications field. In 1989, he founded TSR Technologies, Inc., a cellularradio/PCS software radio manufacturer that he sold in 1993 to what is nowCommScope, Inc. In 1995, he founded Wireless Valley Communications Inc.,a site-specific wireless network design and management firm that he sold in2005 to Motorola, Inc. (NYSE: MOT). Rappaport has testified before the U.S.Congress, has served as an international consultant for the ITU, has consulted

for over 30 major telecommunications firms, and works on many nationalcommittees pertaining to communications research and technology policy. Heis a highly sought-after consultant and technical expert, and serves on variousboards of several high-tech companies. He has authored or coauthored over200 technical papers, over 100 U.S. and international patents, and severalprize papers and bestselling books. In 2006, he was elected to serve on theBoard of Governors of the IEEE communications Society (ComSoc), and waselected to the Board of Governors of the IEEE Vehicular Technology Society(VTS) in 2008 and 2011.

James N. Murdock (Member, IEEE) received the B.S.E.E. degree fromthe University of Texas at Austin, Austin, in 2008, where he specializedin communication systems and signal processing. He obtained a Master ofEngineering degree from The University of Texas at Austin in 2011, forwhich he focused on sub-THz and electromagnetic engineering, in additionto channel modeling and scientific data archiving. Currently, he is working atTexas Instruments, where he focuses on low power radio systems and sub-THzradar applications. James has co-authored over ten conference and technicalmagazine papers and two journal papers.


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