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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 Systems
James N. Murdock, Member, IEEE and Theodore S. Rappaport, Fellow, IEEE
Abstract—This paper presents a new theory, called the consumption factor theory, to analyze and compare energy efficient
design choices for wireless communication networks. The approach presented here provides new methods for analyzing and
comparing the power efficiency of communication systems, thus
enabling a quantitative analysis and design approach for “green
engineering” of communication systems. The consumption factor
(CF) theory includes the ability to analyze and compare cascaded
circuits, as well as the impact of propagation path loss on the total
energy used for a wireless link. In this paper, we show several
examples how the consumption factor theory allows engineers to
compare and determine the most energy efficient architectures
or designs of communication systems. One of the key concepts
of the consumption factor theory is the power efficiency factor,
which has implications for selecting network architectures or
particular cascaded components. For example, the question of
whether a relay should be used between a source and sink
depends critically on the ratio of the source transmitter powerefficiency factor to the relay transmitter power-efficiency factor.
The consumption factor theory presented here has implications
for the minimum energy consumption per bit required to
achieve error-free communication, and may be used to extend
Shannon’s fundamental limit theory in a general way. This work
includes compact, extensible expressions for energy and power
consumption per bit of a general communication system, and
many practical examples and applications of this theory.
Index Terms—Power Consumption, Energy Efficiency, Power
Efficiency, Millimeter-wave, Wireless, Cascaded circuits, Capacity, Relay channel.
I. I NTRODUCTION
C
OMMUNICATION systems today, including both wireline and wireless technologies, consume a tremendous
amount of power. For example, the Italian telecom operator
Telecom Italia used nearly 2 Tera-Watt-hours (TWh) in 2006
to operate its network infrastructure, representing 1% of Italy’s
total energy usage [1]. Nearly 10% of the UK’s energy usage
is related to communications and computing technologies [1],
while approximately 2% of the US’s energy expenditure is
dedicated to internet-enabled devices [2]. In Japan, nearly 120
W of power are used per customer in the cellular network
Manuscript received: April 15, 2012, revised: October 12, 2012. Portions
of 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 at
The 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 for
many large infrastructure-based communication systems. [2]
estimates that 1000 homes accessing the Internet at 1 Gigabit-per-second (Gbps) would require 1 Giga-Watt of power.
All of these examples indicate that energy efficiency of
communication systems is an important topic. Given the trend
toward increasing data rates and data traffic, energy efficient
communications will soon be one of the most important challenges for technological development, yet a theory that allows
an engineer to easily compare and analyze, in a quantitative
fashion, the most energy efficient designs has been allusive.
Past researchers have explored analytical and simulation
methods to compare and analyze the power efficiencies
of various wireless networks (see, for example, works in
[3][4][5][6][7][8]). In [3], researchers explored the energy
efficiency in an acoustic submarine channel and illustrated
how the choice of signaling, when matched to the channel,
could approach Shannon’s limit. In [4], researchers considered
a position-based network routing algorithm that could be
optimized locally at each user, in an effort to reduce overall
power consumption of the network, but were unable to derive
convenient and extensible expressions for power efficiency
that could be generalized to any network. In [5], energy
consumption was compared to the obtainable data rate of endusers, and an analysis technique was used to determine energy
efficiency through the use of distributed repeaters. In [6], a
novel bandwidth allocation scheme was devised to optimize
the power consumed in the network while maximizing data
rate, but the analysis was not extensible to a cascaded system
of components, nor could it be easily generalized. [7] illustrates how cumbersome and complicated the field of energy
conservation can be in ad-hoc networks, at both the link and
network layers (e.g. the individual wireless link, as well as the
network topology, where both have a strong impact on energy
utilization). In fact, a recent book, Green Engineering [8],
illustrates the importance, yet immense difficulty, in providing
an easy, generalized, standardized method for analyzing and
comparing power efficiency in a communications network.
Despite the extensive body of literature aimed at energy
efficient communication systems, we believe this paper is the
first to present a generalized analysis that allows engineers
to provide a standard “figure of merit” to compare the power
efficiency (or energy efficiency) of different cascaded circuit
or system implementations over a wide array of problem
domains. The analysis method presented here is general,
in that it may be applied to power efficient circuit design,
c 2014 IEEE
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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 32, NO. 12, DECEMBER 2014
transmitter and receiver design, and also to various network
architectures 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 comparing
the power efficiency of communication systems and network
designs. In particular, as cellular communication networks
evolve, the base station coverage regions will continue to
shrink in size, meaning that there will be a massive increase
in the number of base stations or access points, and relays
are likely to complement the base stations over time [9].
To accommodate the demand for increased data rates to
mobile users, we envisage future millimeter-wave (mm-wave)
communication systems that are much wider in bandwidth
than today’s cellular and Wi-Fi networks. These future systems
will use highly directional steerable antennas and channel
bandwidths of many hundreds of MHz thereby supporting
many Gigabits per second data rates to each mobile device
[9] [10][11][12]. As such systems evolve, small-scale fading
in the channel will become much less of a concern, and more
attention will need to be placed on the power efficient design
of handsets and “light weight” base stations and repeaters that
use wideband channels and multi-element phased arrays with
RF amplifiers. The theory presented here aims to aid in the
design of these wideband wireless networks and devices. As
shown in this paper, the CF framework gives communication
engineers a methodology to analyze, compare and tradeoff
circuit and system design decisions, as well as network architectures (e.g. whether to use relays or small cells, and how
to trade off antenna gain, bandwidth, and power efficiency in
future wireless systems) [9][10][11][12].
In this paper, we provide fundamental insight into the
required power consumption for communication systems, and
create an-easy-to-use theory, which we call the consumption
factor (CF) theory, for analyzing and comparing any cascaded
communication network for power efficiency. In Section II
we present the consumption factor framework for a homodyne transmitter [12]. Section III generalizes the concept
of power-efficiency analysis, which is fundamental to the
consumption factor framework, for any cascaded communication system. Section IV provides numerical examples of the
power-efficiency factor used in the consumption factor theory.
Section V presents a general treatment of the consumption
factor, based on the power-efficiency analysis of the preceding
sections. Section VI demonstrates a key characteristic of the
power-efficiency factor – i.e. that gains of components that
are closest to the sink of a communication system reduce
the impact of the efficiencies of preceding components. In
Section VII, we use the consumption factor framework to
develop fundamental understandings of the energy price of
a bit of information. We use our analysis to demonstrate how
the consumption factor theory may be applied to designing
energy efficient networks, for example by helping to determine
the best route to send a bit of information in a multi-hop
setting to achieve the lowest energy consumption per bit.
Section VIII provides conclusions. The key contribution of this
paper is a powerful and compact representation of the power
consumption and energy consumption per bit of a general
communication system. The representation takes the gains
Fig. 1. Block diagram of a homodyne transmitter used to demonstrate the
power-efficiency factor and consumption factor (CF).
and efficiencies of individual signal-path components (such
as amplifiers and mixers) into account. A second key result is
that, in order to align the goals of lower energy per bit and
higher data rates, it is advantageous to design communication
systems that require as little signal power as possible, so
low, in fact, that ancillary power drain (e.g. for cooling, user
interfaces, etc.) dominates signal power levels. While this may
seem intuitive, the CF theory proves this, and provides a
tangible, objective way of comparing various designs while
showing the degrees to which communication systems must
reduce ancillary power drain, but must also seek means of
reducing required signal levels even more dramatically than
the ancillary power drain. By making every bit as energy
efficient as possible, we show it is possible to greatly expand
the number of bits that can be delivered for a given amount
of energy. Means of achieving this goal include the use very
short link distances (such as femtocells) at millimeter-wave
frequencies for future massively broadband wireless systems.
Earlier, less developed versions of the consumption factor were
presented in [13].
II. C ONSUMPTION FACTOR FOR A H OMODYNE
T RANSMITTER
We define the consumption factor (CF) for a communication
system as the maximum ratio of data rate to power consumed,
or equivalently as the maximum number of bits that may
be transmitted through a communication system for every
Joule of expended energy. A study of the consumption factor
requires a careful analysis of both the power consumption
and data rate capabilities of a communication system. In this
section, we will provide a simple analysis for a homodyne
transmitter as illustrated in Figure 1, to motivate the theory
presented here. We consider a homodyne transmitter because
this topology is attractive for many massively-broadband systems due to its low cost and low complexity [12]. We will
generalize our analysis in Section III to be applicable to a
general cascaded communication system.
The homodyne transmitter in Figure 1 is comprised of
components that directly handle the signal, such as the mixer
and power amplifier, in addition to components that interact
indirectly with the signal, such as the oscillator. Components
that interact directly with the signal are designated “on the
signal path,” while components that are not in the path of the
signal are designated “off the signal path.”
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MURDOCK and RAPPAPORT: CONSUMPTION FACTOR AND POWER-EFFICIENCY FACTOR: A THEORY FOR EVALUATING THE ENERGY EFFICIENCY...
A key component of the consumption factor framework is
understanding that the efficiency of each signal path component may be used to relate the ancillary or “wasted” power
of each component to the total signal power delivered by that
component. For example, the efficiency of the power amplifier
is used to find the partial power required to bias the amplifier
(which is not used, or wasted, in terms of providing signal path
power) as a component of the total signal power delivered to
the load. We define the efficiency of the power amplifier,ηP A ,
and of the mixer, ηMIX ,:
ηP A =
ηMIX =
PA
PRF
PA
PRF
A
+ PNP ON
−RF
(1)
MIX
PRF
MIX + P MIX
PRF
N ON −RF
(2)
PA
is the signal power delivered by the power
where PRF
MIX
amplifier to the matched load, and PRF
is the signal power
A
delivered by the mixer to the power amplifier. PNP ON
−RF and
MIX
PN ON −RF are the power levels used by the power amplifier,
and mixer, respectively, that do not directly contribute to
delivered signal power. Using (1) – (2), we find:
1
PA
PA
PN ON −RF = PRF
−1
(3)
ηP A
PNMIX
ON −RF
=
MIX
PRF
1
ηMIX
−1
(4)
The second key step in the consumption factor analysis results
from the realization that the signal powers delivered by each
PA
MIX
component in the cascade, PRF
, and PRF
, may be related
to the total power delivered by the communication system,
through the gains of each signal path component. As shown
in Section III, this formulation for a cascaded system’s power
efficiency is reminiscent of Frii’s classic noise figure analysis
technique for cascaded systems [24]. Using (1)-(4), we now
RADIO
find the delivered RF power to the matched load, PRF
,
in terms of the signal power from the baseband signal source
and the various gains stages as:
RADIO
BB MIX P A
PRF
= PSIG
G
G
A
PA
PNP ON
−RF = PRF
1
ηP A
MIX
PNMIX
ON −RF =PRF
(5)
1
RADIO
− 1 = PRF
− 1 (6)
ηP A
1
P RADIO
−1 = RFP A
−1 (7)
ηMIX
G
ηMIX
1
BB
where PSIG
is the signal power delivered by the baseband
components to the mixer, and GMIX and GP A , are the power
gains of the mixer and power amplifier, respectively. Equation
(5) simply states that the power delivered to the matched load
is equal to the power delivered by the baseband components
multiplied by the gain of the mixer and of the power amplifier.
Note that we have implicitly assumed an impedance matched
environment. Impedance mismatches may be accounted for by
including a mismatch factor less than one in the gain of each
component.
3
The total power consumption of the homodyne transmitter
may be written as:
PA
=
RADIO
Pconsumed
RADIO
PRF
+ P N ON −RF + PNMIX
ON −RF
+ P BB + P OSC
(8)
where P BB is the power consumed by the baseband components and P OSC is the power consumed by the oscillator.
RADIO
The term PRF
is the total signal power in the homodyne
transmitter delivered to the load. Using equations (5) through
(7) in (8), we re-write the total homodyne power consumption
as:
1
1
1
RADIO
RADIO
Pconsumed
1+
= PRF
−1 + P A
−1
ηP A
G
ηMIX
(9)
+ P BB + P OSC
RADIO
Pconsumed
=
1+
1
ηP A
RADIO
PRF
−1
− 1 + GP1 A ηM1IX − 1
+ P BB + P OSC
(10)
−1
From (10), the factor 1 + ηP1A − 1 + GP1 A ηM1IX − 1
plays a role in the transmitter power consumption analogous
to that of efficiency. In other words, this factor may be
considered the aggregate efficiency of the cascade of the
mixer and power amplifier. In Section III we will generalize
this result and define this factor as the power efficiency factor
for an arbitrary cascaded system (where the cascade may
be either a cascade of components or circuits, or may even
include the propagation channel).
Now that we have formulated a compact representation
of the power consumption of a homodyne transmitter, we
must determine the maximum data rate that the transmitter
can deliver in order to formulate the consumption factor of
the transmitter. To do this, we assume that the transmitter is
communicating through a channel with gain Gchannel to a
receiver of gain GRX having noise figure F with bandwidth
B. We assume also that the transmitter matched load is
T
replaced by an antenna with gain GAN
T X . The signal power
used by the receiver in the detection process,
PRX , is given by:
T
RADIO AN T
GT X Gchannel GAN
PRX = PRF
RX GRX
(11)
T
is the gain of the receiver antenna, and GRX
where GAN
RX
is the gain of the receiver excluding the antenna. We will
assume an AWGN (Additive White Gaussian Noise) channel,
for which the received noise power at the detector, Pnoise , is:
Pnoise = KT F B × GRX
(12)
where K is Boltzmann’s constant (1.38x10−23 J/K) and T is
the system temperature in Kelvin. The SNR at the receiver
detector is therefore:
SN R =
T
RADIO AN T
PRF
GT X Gchannel GAN
RX GRX
KT F B × GRX
(13)
The SNR is related to the minimum acceptable SNR at the
output of the receiver, SN Rmin , as dictated by the modulation
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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 32, NO. 12, DECEMBER 2014
Fig. 2. An example of the use of the power-efficiency factor to find the
consumption factor of two different cascades of a baseband amp, mixer, and
RF amp.
Fig. 3. Higher values of power consumption off of the signal path components result in higher values of SNR needed to maximize the consumption
factor (CF).
and signaling scheme, through a particular operating margin
MSN R :
SN R = MSN R SN Rmin
(14)
where P Go is the close-in free-space path gain (usually a
large negative number in dB) received at a close-in reference
distance do , d is the link distance ( d >do ), and α is the
path loss exponent [10][12][19][20]. Two examples for CF
using equation (18) and (19) are shown in Figures 2 and 3.
Figure 2 shows how the consumption factor of a 60 GHz
wireless communication system varies as the efficiency of the
power amplifier or the mixer are changed, and indicates that
the efficiency of the power amplifier is much more important
in terms of maximizing the overall system efficiency than the
mixer’s efficiency. The key lesson from this example is that the
efficiencies of the devices that handle the highest signal power
levels should be maximized in order to have the most dramatic
effect in maximizing the consumption factor. Figure 3 shows
the impact of changing the minimum required SNR at the
receiver. Note that we have assumed an SNR margin of 0 dB.
The figure indicates that higher levels of power consumption
by non-signal-path devices such as the oscillator result in
higher levels of SNR to maximize the consumption factor. The
figure also indicates an optimum value of SNR to maximize the
consumption factor. This optimum value depends critically on
the amount of power consumed by devices off the signal path.
Note that in these figures, we have assumed the efficiency and
gain of the mixer are equal. This assumption will be explained
in Section III, where we will find that the gain and efficiency
of an attenuating device are equal (similar to Friis’ noise figure
analysis). Note that we have used a logarithmic scale in Figure
3 to allow for easy comparison between the different curves.
The minimum power consumption occurs when MSN R is
RADIO
equal to 0 dB (i.e. MSN R = 1). Solving for PRF
, we
find:
SN Rmin KT F B
RADIO
PRF,min
= AN T
(15)
T
GT X Gchannel GAN
RX
RADIO
RADIO
where we now denote PRF
as PRF,min
to indicate that
this power level corresponds to the minimum acceptable SNR
at the receiver. The minimum power consumption for the
transmitter is found using (10) and (15) as:
RADIO
Pconsumed,min
SN Rmin KT F B
T
AN T
GAN
T X Gchannel GRX
= 1 + ηP1A − 1 +
+ P
BB
+P
1
GP A
1
ηM IX
−1
OSC
−1
(16)
The maximum data rate Rmax at the receiver is given in
terms of the SNR and the bandwidth according to Shannon’s
capacity formula if the modulation and signaling scheme are
not specified. If these are specified, then we find the maximum
data rate in terms of the spectral efficiency of the modulation
and signaling scheme ηsig (bps/Hz):
Rmax = Blog2 (1 + SN R) , General Channel
Rmax = Bηsig , Specif ic M odulation Scheme
(17)
The consumption factor, CF , for the homodyne transmitter is
then found by taking the ratio of (17) to (16):
III. G ENERAL CASCADED COMMUNICATION SYSTEM
We will now generalize the consumption factor to provide
a framework for analyzing a general cascaded communication
(18)
CF = RADIO
system.
Pconsumed,min
The consumption factor is defined [18] as the maximum
Blog2 (1 + SN R)
ratio of data rate to total power consumption for a commu
CF = SN Rmin KT F B
nication system. To determine the consumption factor, we
GAN T Gchannel GAN T
T
X
RX
BB
OSC
−1 + P
+
P
must first determine a compact representation of the power
1+ η 1 −1 + P1 A η 1 −1
G
PA
M IX
consumption of a general cascaded communication system.
(19)
Consider a general cascaded communication system as shown
We will assume a standard log-distance channel gain model:
in Figure 4 in which information is generated at a source, and
do
sent as a signal down a signal path to a sink. Signal path
[dB]
(20)
Gchannel = P Go + 10α × log10
components such as amplifiers and mixers are responsible
d
Rmax
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MURDOCK and RAPPAPORT: CONSUMPTION FACTOR AND POWER-EFFICIENCY FACTOR: A THEORY FOR EVALUATING THE ENERGY EFFICIENCY...
Non-Signal Path Devices
1 ..... k ..... M
1
Source
2
where Gatten is the gain of the attenuator, and is less than
one. Thus, we have shown that ηatten = Gatten for a passive
device or channel.
The total power consumed by the ith stage on the signal
path may be written:
... N
Pconsumedi = Pnon−sigi + Padded−sigi
Sink
Signal Path Devices
Fig. 4. A general communication system composed of components on and
off the signal path.
for transmitting the information signal to the sink. Nonsignal path components include voltage regulation circuitry,
displays or cooling components that do not participate directly
in the signal path, but do consume power. The total power
consumption of the cascaded communication system in Figure
4 (ignoring the source and sink) may be written as:
Pconsumed = Psig +
N
Pnon−sigk +
k=1
M
Pnon−pathk (21)
k=1
where Psig is the sum of all signal powers of each component
in the cascade, Pnon−sigk is the signal power used by the k th
signal path component but not delivered as signal power to
the next signal-path component, and Pnon−pathk is the power
used by the k th component off the signal path. To evaluate
(21), we must consider each component on the signal path
separately. The efficiency of the ith signal path component
may be written as:
ηi =
Psigi
Psigi
+ Pnon−sigi
(22)
Where Psigi is the total signal power delivered by the ith stage
th
to the (i + 1) 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 be
applied to any communication system component. A similar
measure 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 divided
by 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 attenuator
should consume only the signal power delivered to it by
the preceding stage (i.e. the consumption factor theory treats
attenuators as passive components that do not take power from
a power supply). The signal power delivered by an attenuator
to the next stage is a fraction of the signal power delivered to
the attenuator. Therefore, if the ith stage is an attenuator, then
the efficiency of an attenuator, ηatten , as given by (22) is:
ηatten
5
Psigi = Gatten Psigi−1
(23)
Pnon−sigi = (1 − Gatten ) Psigi−1
(24)
Gatten Psigi−1
=
= Gatten
Gatten Psigi−1 + (1 − Gatten ) Psigi−1
(25)
(26)
where Padded−sigi is the total signal power added by the
ith component, which is the difference in the signal power
th
delivered to the (i + 1) component and the signal power
delivered to the ith component. We can sum all the signal
powers added by the components on the signal path (from
left to right in Figure 4) to find:
N
Padded−sigi = PsigN − Psigsource
(27)
i=1
where Psigsource is the signal power provided by the source,
and PsigN is the signal power delivered by the Nth (and last
stage) signal-path component. Adding (27) to the signal power
from the source, we find that the total signal power in the
communication system is equal to the signal power delivered
to the sink (in other words, the signal power delivered by the
last stage is equal to the sum of all signal powers delivered
by each component in the cascade):
Psig = P sigN
(28)
From (22) the total “wasted” power of the k th stage (i.e. power
consumed but not delivered to the next signal path stage) may
be related to the efficiency and total delivered signal power
by that stage:
1
Pnon−sigk = Psigk
−1
(29)
ηk
Also, the signal power delivered by the k th stage may be
related to the total power delivered to the sink by dividing by
the gains of all stages after the k th stage, (i.e. to the right of
the k th ) thus yielding:
N
PsigN = Psigk
(30a)
Gi
i=k+1
Pnon−sigk =
PsigN
N
Gi
1
−1
ηk
(30b)
i=k+1
where Gi is the gain of the ith stage. We can therefore
compute the total power consumed by the communication
system as the power consumed by the source which is assumed
to equal the signal power delivered by the source, and the three
additional terms that represent the power consumed by the inpath cascaded components, and the power dissipated by the
non-signal path components:
Pconsumed = Psigsource +
N
i=1
Pconsumedi +
M
k=1
Pnon−pathk
(31a)
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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 32, NO. 12, DECEMBER 2014
Pconsumed
=
Psigsource +
N
Padded−sigi
i=1
+
N
Pnon−sigk +
k=1
⎛
N
⎜
⎜
= Psig N ⎜1 +
⎝
k=1
1
N
Gi
M
Pnon−pathk (31b)
k=1
⎞
⎟ M
1
⎟
−1 ⎟+
Pnon−pathk
ηk
⎠
k=1
i=k+1
(31c)
In certain circumstances, such as when comparing two different smart phones or other devices that have substantial
power consumed by displays or computer processors, it is
useful to incorporate the impact of the power efficiencies of
the non-path components. To do this, we may simply add these
components to the end of the signal-path cascade in Figure 4
and assume unity gain. For example, write the total power
consumption of the k th non-path component Pnon−pathk
in terms of its usefully dissipated power Puk (power that
directly contributes to its intended functionality) and its power
efficiency ηnon−pathk (the ratio of usefully dissipated power
to its total power consumption):
Pnon−pathk =
Puk
ηnon−pathk
(32)
We may then re-write (31c) as (33). For simplicity, we now
carry on the development of the CF analysis with the power
consumption expression given by (31c) rather than (33), as
we wish to isolate the impact of the efficiencies of non-path
components (noting that such analysis may be done by simply
appending the power efficiencies of non-path components as
described above). We see from (31c) and (35) that the on-path
cascade components may be conveniently represented in the
total power consumption of the cascade as:
PsigN
+ Pnon−path
(34)
H
where Pnon−path is the total power used by devices off the
signal path, and in (34), we introduce the system powerefficiency factor H of all cascaded components defined as:
⎧
⎫−1
⎪
⎪
⎪
⎪
⎪
⎪
N
⎨ ⎬
1
1
H = 1+
−
1
(35)
N
⎪
⎪
ηk
⎪
⎪
k=1
⎪
⎪
G
⎩
⎭
i
Pconsumed =
i=k+1
Where H ranges between 0 and 1, and we call H the powerefficiency factor of the entire signal-path cascade. Note H−1
ranges from 1 to infinity (just like Friis’ Noise Figure). Equation (35) is a very general expression relating the gains and
power efficiencies of the individual components on the signal
path to the signal-path efficiency of the overall communication
system. An implication of this is that the efficiencies of devices
that handle the most power are most important in terms of
the power-efficiency factor of the entire cascade, as these will
be the components in (35) whose efficiencies are divided by
the smallest numbers. As shown subsequently, the presence of
attenuators, such as a wireless channel, makes it such that the
power efficiencies of stages that handle the most power just
prior to the large attenuator, such as a power amplifier, have
the largest impact on overall system power efficiency.
Note that we have defined the power efficiency of a signalpath component (22) in terms of the total power it delivers,
i.e. using (26) in (22) we have:
ηi =
Psigi
Psigi −1 + Padded−sigi
=
Psigi + Pnon−sigi Psigi −1 + Padded−sigi + Pnon−sigi
(36a)
Psigi −1 + Padded−sigi
(36b)
ηi =
Psigi −1 + Pconsumedi
This representation of power efficiency is useful as it is
applicable to both passive components, which do not add
signal power, and active components that add signal power.
For an attenuator, Padded−sigi is zero (i.e. for the CF theory
we assume that attenuators can not add additional signal
power), while Pnon−sigi is the signal power removed by the
attenuator. One important application of (36) and (25) is when
the attenuator is a wireless channel. From (25), the powerefficiency factor of a wireless channel Hchannel is given by
its gain Gchannel :
Hchannel = Gchannel
(37)
The power-efficiency factor is a powerful and general means
of determining the power consumption of a communication
system. For example, consider two cascaded sub-systems
whose H’s have already been characterized, where sub-system
#2, with power-efficiency factor Hsub−system 2 and gain
Gsub−system2 follows sub-system #1 with power-efficiency
factor Hsub−system 1 . We can show from (35) that the powerefficiency factor of the entire cascade, Hcascaded system , may
be written much like the classic noise figure theory (Eqn.
38) where the first sub-system is composed of components
1 through M-1, and the second subsystem is composed of
components M through N. Of course, M may be any integer
from 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 for
Hcascaded system . This is easily seen by inverting (38d) and
examining the limiting case in which the first stage has an
optimal power-efficiency factor of 1 as in (39). Consider also
the case of a single component [9]. Using (22) and (35), we
see:
ηi = Hi
(40)
Consider now the case in which a wireless channel exists between a transmitter and receiver. The overall power-efficiency
factor of the entire transmitter-receiver pair, Hlink is given by:
1
1
−1
=
H
+
−
1
H−1
link
RX
GRX Gchannel
−1
1
HT X − 1
(41)
+
GRX Gchannel
where HRX is the power-efficiency factor of the receiver, HT X
is the power-efficiency factor of the transmitter, GRX is the
gain of the receiver, and Gchannel is the channel gain (which
is less than 1) which is equal to the power-efficiency factor of
the channel. Note from (41) that if the receiver gain is much
smaller than the expected channel path loss, the cascaded
power-efficiency factor will be very small and on the order
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MURDOCK and RAPPAPORT: CONSUMPTION FACTOR AND POWER-EFFICIENCY FACTOR: A THEORY FOR EVALUATING THE ENERGY EFFICIENCY...
7
⎞
⎛
M
N
⎜
1 Puk
⎜
Pconsumed = Psig N ⎜1 +
+
⎝
PsigN
ηnon−pathk
k=1
1
N
k=1
Gi
⎟
1
⎟
−1 ⎟
⎠
ηk
(33)
i=k+1
H−1
cascadedsystem
=
+
1
1
1
1
1
1
1
1+
−1 +
− 1 + . . .+
−1 +
−1
ηN
GN ηN −1
GN . . . GM+1 ηM
GN . . . GM ηM−1
⎫−1
⎪
⎪
⎬
1
1
−1
(38a)
... N
⎪
η1
⎪
⎭
Gi
i=1
H−1
sub−system2
1
1
1
1
1
= 1+
−1 +
− 1 + ...+
−1
ηN
GN ηN −1
GN . . . GM+1 ηM
1
1
1
=
1
+
−
1
+
.
.
.
−
1
H−1
sub−system1
M−1
ηM−1
η1
Gi
i=1
−1
H−1
cascadedsystem = Hsub−system2 +
Hcascadedsystem =
Gsub−system2
H−1
sub−system1 −1
Hsub−system1 +
lim
Hsub−system2
Gsub−system2
(39a)
(1 − Hsub−system1 )
Hcascadedsystem =Hsub−system2
of the product of the channel gain with the receiver gain. In
this case, we find that the overall power-efficiency factor is
approximated by:
(42)
This is an important result of this analysis. In particular, it
indicates that in order to achieve a very power-efficient link,
it is desirable to have a high gain receiver and a highly
efficient transmitter. This can be understood by realizing that
a higher gain receiver reduces the output power requirements
at the transmitter. Eqn. (42) indicates the great importance
of the transmitter efficiency. Note, however, that the receiver
efficiency is still important, as from (39b) it is clear that the
receiver’s efficiency is an upper bound on the efficiency of the
overall link.
IV. N UMERICAL E XAMPLES
To better illustrate the use of the consumption factor theory,
and the use of the power-efficiency factor, consider a simple
scenario of a cascade of a baseband amplifier, followed by
a mixer, followed by an RF amplifier. We will consider two
different examples of this cascade scenario, where different
components are used, in order to compare the power efficiencies due to the particular specifications of components.
Assume that for both cascade examples, the RF amplifier is a
commercially available MAX2265 power amplifier by Maxim
technology with 37 % efficiency[15]. In both cases, the mixer
(38c)
(38d)
Hsub−system1 Hsub−system2
Hsub−system1 →1
Hlink ≈ GRX Gchannel HT X (42)
1
(38b)
(39b)
is an ADEX-10L mixer by Mini-Circuits with a maximum
conversion loss of 8.8 dB[16]. In the first case, the baseband
amplifier (the component furthest to the left in Figure 4 if
in a transmitter, and furthest to the right if in a receiver)
is an ERA-1+ by Mini-circuits, and in the second case the
baseband amplifier is an ERA-4+ [17], also by Mini-Circuits.
The maximum efficiencies of these parts are estimated by
taking the ratio of their maximum output signal power to their
dissipated DC power. As the mixer is a passive component,
its gain and efficiency are equal. Table 1 summarizes the
efficiencies and gains of each component in the cascade. Using
(35), the power-efficiency factor of the first scenario is
Hscenario 1 =
1
1
0.37 + 16.17
= 0.2398,
1
0.36 −1
1
1
1
+ 0.36∗16.17
0.1165 − 1
whereas the power-efficiency factor of the second scenario is
Hscenario 2 =
1
1
0.37 + 16.17
= 0.2813.
1
0.36 −1
1
1
1
+ 0.36∗16.17
0.1836 −1
Therefore, we see that the second scenario offers a superior
efficiency compared to the first scenario, due to the better
efficiency of the baseband amplifier, but falls far short of
the ideal power efficiency factor of unity. Using different
components and architectures, it is possible to characterize and
compare, in a quantitative manner, the power-efficiency factor
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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 32, NO. 12, DECEMBER 2014
TABLE I
A N EXAMPLE OF THE USE OF THE POWER - EFFICIENCY FACTOR TO COMPARE TWO CASCADES OF A BASEBAND AMP, MIXER , AND RF AMP.
Component
Example 1
Gain
Efficiency
MAX2265 RF Amp
ADEX-10L Mixer
ERA-1+BB Amp
24.5 dB (voltage gain of 16.17)
-8.8 dB
10.9 dB
37%
36%
11.65 %
24.5 dB
-8.8 dB
13.4 dB
37%
36%
18.36%
Example 2
MAX2265 RF Amp
ADEX-10L Mixer
ERA-1+BB Amp
and consumption factor (see subsequent sections) of cascaded
components.
As a second example, consider the cascade of a transmitter
power amplifier communicating through a free-space channel
with a low-noise amplifier at the receiver. Let us assume that
the cascade, in the first case, uses the same RF power amplifier
as in the previous example (MAX2265), while the LNA is
a Maxim Semiconductor MAX2643 with a gain of 16.7 dB
(6.68 absolute voltage gain) [18]. We will assume this LNA
has 100% efficiency for purposes of illustrating the impact
of the PA’s efficiency and the channel (i.e. here we ignore the
LNA’s efficiency, although this can easily be done as explained
above). For a carrier frequency of 900 MHz, now consider
the cascade for a second case where the MAX2265 RF power
amplifier is replaced with a hypothetical RF amplifier device
having 45% power efficiency (a slight improvement). Assume
the link is a 100m free space radio channel with gain of -71.5
dB. Since the propagation channel loss greatly exceeds the
LNA gain, (42) applies, where HT X is the efficiency of the RF
amplifier, so that in the first case using the MAX2265 amplifier
(37% efficiency), the power efficiency factor of the cascaded
system is 173.5e-9, while in the second case (using an RF
Power amplifier with 45% efficiency), the power efficiency
factor is 211.02e-9. The second case has an improved powerefficiency factor commensurate with the power efficiency
improvement of the RF amplifier stage in the receiver. These
simple examples demonstrate how the power-efficiency factor
may be used to compare and quantify the power efficiencies
of different cascaded systems, and demonstrate the importance
of using higher efficiency RF amplifiers for improved power
efficiency throughout a transmitter-receiver link.
V. C ONSUMPTION FACTOR
We now define the consumption factor, CF, and operating
consumption factor (operating CF) for a general communication system such as that in Figure 4, where CF is defined
as:
R
Rmax
CF =
=
(43)
Pconsumed max
Pconsumed,min
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 communication
system. Further analysis based on only maximizing R or
minimizing Pconsumed is also pertinent to system optimization
in terms of consumed power and carried data rate. For a very
general communication system in an AWGN channel, Rmax
may be written using Shannon’s information theory according
to the operational SNR and bandwidth, B:
Rmax = Channel Capacity = Blog2 (1 + SN R)
Or, for frequency selective channels [3]:
B
Pr (f )
Rmax =
df
log2 1 +
N (f )
0
B
2
|H (f )| P t (f )
=
df
log2 1 +
N (f )
0
(45)
(46)
where Pr (f ), Pt (f ), and N (f ) are the power spectral densities of the received power, the transmitted power, and the noise
power at the detector, respectively. H (f ) is the frequency
response of the channel and any blocks that precede the detector. Note that equations (45) and (46) make no assumptions
about the signaling, modulation, or coding schemes used by
the communication system. To support a particular spectral
efficiency ηsig (bps/Hz), there is a minimum SNR required
for the case of an AWGN channel:
SN R
= SN Rmin = 2ηsig − 1
(47)
MSN R
The operating SNR of the system, as well as the operating
margin of the operating SNR ( denoted by MSN R which
represents the operating margin above the minimum SN Rmin )
may be used to find the consumption factor and operating
consumption factor expressed in terms of the system’s powerefficiency factor H:
B log2 (1 + SN R)
R
× Pnoise
Pnon−path + MSN
H
SN R
(48a)
B log2 (1 + M SN R (2ηsig − 1))
,
Pnon−path + (2ηsig − 1 ) × Pnoise
H
(48b)
CF =
CF =
and (49) where we have made use of (34) and the fact that
the signal-power available to the sink, PsigN is related to the
noise power available to the sink, Pnoise and the SNR at the
sink:
PsigN = Pnoise × SN R = KT F BGRX × SN R
(50)
And where the right hand equality in (50) holds for an AWGN
channel. K is Boltzman’s constant (1.38x10−23 J/K), T is the
system 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 future
wireless broadband cellular networks. To see this, consider
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Operating CF =
Operating CF =
Blog2
log2 (1 + SN R)
.
× KT F GRX
SN R
MSN R
(51)
(52)
Equation (52) indicates that for such a link we can increase
data rate by increasing bandwidth, but that unless the signal
path components are made much more efficient (i.e. the system
power-efficiency factor is made closer to 1), then as data rate
increase we will require approximately the same energy per
bit. In other words, if transmission power is the dominant
cause of energy expenditure, then there is little that can
be done to drive down the energy-price per bit through an
increase in bandwidth. There are two problems that arise: A)
efficiency improvements in inexpensive IC components are
becoming harder to achieve due to performance issues when
supply voltages are scaled below 1 volt, which is approximately the supply voltage used by many present-day high
efficiency devices, and B) with the exponential growth in data
traffic that is occurring today, unless the energy cost per bit can
be reduced exponentially, we face an un-tenable requirement
for increased power consumption by communication systems.
The upshot of (52) is that for conventional cellular systems, all
signal-path devices, and particularly the RF power amplifier
the precedes the lossy channel, and other components that
precede lossy attenuators, must be made as power efficient as
possible, thus suggesting that modulation/signaling schemes
should be chosen to support as efficient an RF amplifier as
possible.
Consider the second limiting case of equation (48a), in
which the non-path power dominates the signal power. In
this case, we are assuming that items such as processors,
displays, and other non-signal path components (typical of
smart-phones and tablets) dominate the power drain. We find
(49a)
Pnoise
H
Bηsig
Pnon−path + MSN R (2ηsig − 1) ×
For an AWGN channel, we find that the consumption factor
is relatively insensitive to bandwidth if the signal-path power
dominates the non-path power:
CF ≈ H +1
Pnon−path + SN R ×
two limiting cases illuminated by the consumption factor
theory. In the first case, we assume that the signal path
power 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-station
is communicating to the edge of the macrocell, and the RF
channel requires more power to be used in the RF amplifier
to complete the link than the power used to power other
functions. In this case, the consumption factor equation (48a)
is approximated by:
B log2 (1 + SN R)
CF ≈ H .
SN R
×
P
noise
MSN R
SN R
MSN R
9
(49b)
Pnoise
H
from (48a) that in this case:
CF ≈
B log2 (1 + SN R)
.
Pnon−path
(53)
Eqn. (53) indicates that wider-band systems are preferable
on an energy-per-bit basis provided that signal-power can be
made lower than the total power used by components off the
signal path. This situation is clearly preferable to the first case
as it indicates that by increasing channel bandwidth (say, by
moving to millimeter-wave spectrum bands where there is a
tremendous amount of spectrum [3][12][13]), we also achieve
an improvement in the consumption factor, i.e. a reduction in
the energy cost per bit. Interestingly, this indicates that the
goals of massive data rates (through larger) bandwidths and
smaller cell sizes combined together can be used to achieve
a net reduction in the energy cost per bit. As an increase in
bandwidth also enables an increase in data rate, this limiting
case allows us to simultaneously increase both data rate and
consumption factor: i.e. our goals for more data and more
efficient power utilization in delivering this higher speed data
are aligned. This is not to say that we should increase nonpath power to the point that equation (53) holds. Rather, we
would desire to decrease the required signal path power to
the point where (53) holds. If the non-path power can be
reduced, but the signal power can be reduced even faster, then
we arrive at the ideal situation of improving power efficiency
with a move to higher bandwidths and greater processing
and display capabilities in mobile devices. To achieve this
goal, it is likely that link distances will need to be reduced
as bandwidths are increased. The goal of making the signal
power as low as possible so that the non-path power dominates
may at first be counter-intuitive. However, realize that in
order to have as many bits as possible flowing through a
communication system it is advantageous to make each bit
as cheap as possible. In order for (53) to apply, we require:
Pnoise
SN R
Pnon−path >
(54)
×
MSN R
H
Recall the form of the power-efficiency factor of a wireless
link 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
<
≈
PN P MSN R
GRX HT X k −
Pnoise SN R
PN P MSN R
GRX HT X k −
Pnoise SN R
kHT X
(GRX − HRX )
HRX
kHT X
(GRX )
(57)
HRX
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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 32, NO. 12, DECEMBER 2014
Pnon−path >
SN R
MSN R
× Pnoise H−1
RX +
and when further simplifying, we see
1
PN P MSN R
−
dα < GRX HT X k
Pnoise SN R HRX
and, finally, solving for distance, we see that
α1
1
PN P MSN R
−
d < GRX HT X k
Pnoise SN R HRX
(58)
(59)
If (59) is satisfied, then increasing bandwidth will have the
double benefit of enabling both increased data rates and
higher consumption factors –i.e. lower energy consumption
per bit. One caveat to (59) is that an increase in bandwidth
clearly also requires a smaller radio link distance in order for
(59) to apply. To ensure that this is the case, we may introduce
a scaling factor β > 1 to ensure that by increasing bandwidth
within a given bound, we do not violate (59):
α1
1
1
PN P MSN R
d<
GRX HT X k
−
(60)
β
Pnoise SN R HRX
Re-writing (60) in terms of bit rate, we find that for a given
operating SNR:
SN R
R = Blog2
+1
(61)
MSN R
where
1
MSN R
= R
(62)
SN R
2B − 1
hence both data rates and consumption factors increase when
α1
1
PN P
1
1
GRX HT X k
(63)
−
d<
R
β
KT F B 2 B
− 1 HRX
In addition to characterizing the power consumption, powerefficiency factor, and CF of a transmitter-receiver pair, the
consumption factor framework may be applied to an individual
transmitter or receiver. To do this for a transmitter, simply
replace the transmitter antenna and channel with a matched
dummy load as was done in Section II for the homodyne
receiver. Similarly, to analyze the case of a receiver, simply
apply 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 as
a 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 MHz
at millimeter-wave bands, recent propagation measurements
show that small scale fading is almost neglible, and large-scale
fading is less variable with directional antennas that “find” the
best pointing directions at both the transmitter and receiver
[10][19][20] thus validating (20) as a reasonable first-order
assumption.
An interesting extension of the theory presented here, which
is beyond the scope of this paper, would consider more
sophisticated channel models that include fading or variability
due to transients in beam switching, or the power efficiencies
1
GRX
dα
−1
k
+
dα
GRX k
H−1
TX − 1
(56)
and power consumption tradeoffs for various antenna array
hardware or beam steering processing needed to implement
future mm-wave cellular networks. For example, antennas that
exploit multipath or beam combining, and can be beamsteered
towards the strongest reflections will be used in future wireless
networks [11][21][22][23]. In [22][23], it was shown that
the direction of arrival of multipath energy for a steerable
antenna can be found by measuring the cross correlation of
narrowband (e.g. CW) fading signals, thus suggesting that future broadband millimeter-wave devices might simultaneously
use narrow band pilot tones that can be detected by closely
spaced low gain omni-directional antennas on the handset or
base station, while the communication traffic is simultaneously
carried using high gain (narrow beam) steerable directional
antennas[11][10]. The CF theory can be easily extended to
analyze the power tradeoff for this additional antenna complexity (and many others). This is readily seen by considering
the homodyne transmitter example, where equations (1), (3)
and (11) may be used to represent the power consumption and
power efficiency of a transmitting antenna that is actually a
combined phased array antenna with multiple RF power amplifier stages. The power drain caused by signal processing would
be represented in the efficiency and power consumption of the
off-path components (e.g., the signal processing components)
as represented in equations (21) and (22), or (33) and (34) in
the general result. As should be clear, by quantifying the additional power consumption and power efficiencies of different
types of processing requirements and hardware requirements,
the CF theory allows for a quantitative comparison of a wide
range of circuit and system implementations. We now illustrate
some numerical examples to highlight the use of this analysis
method, and show how to apply the CF analysis to network
architectures (e.g. relay systems) subsequently in the paper.
VI. CF AND P OWER -E FFICIENCY FACTOR E XAMPLE
To illustrate some pertinent effects of the Consumption
Factor and power-efficiency factor of a communication system,
first consider how the efficiencies of the individual blocks in a
communication system impact the power-efficiency factor H .
For simplicity, we will consider HT X , the power-efficiency
factor of a transmitter. First assume the transmitter is composed of a cascade of N stages, each with an absolute power
gain of G. Further, assume that we may vary the efficiency of
the ith stage in the transmitter from 0 to 100%. The rest of
the stages are assumed 100% efficient. For this case, HTX is
given by:
1
HTX =
(64)
1
i−N
1+G
−
1
T
X
η
i
(64) is plotted in Figure 5 for a seven-stage transmitter, with
each block having an absolute power gain of 2. The figure
makes it clear that stages closer to the output of the transmitter,
which would be closer to the sink if the transmitter were
used with a receiver, have the largest impact on HTX . Further,
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11
bandwidth. We may therefore apply Shannon’s theorem [24]
to find:
1
1
Eb
×
=
(69)
Ebc,min
No Tb ln (2) PN P + TEbH
b
No ln(2)
PN P
ln (2) No
ln (2) No PN P
+
=
+
Eb
C
H
C
H
(70)
where we have made the substitution C = T1b , i.e. that in
the limit, the bit-rate approaches the channel capacity C.
Eqn. (70) should be interpreted as the minimum energy that
must be expended/consumed by the communication system
per bit (as opposed to the energy per bit in the signal) over
the noise spectral density in order to obtain arbitrarily low
error rate. This interpretation should not be confused with
the interpretation of the original Shannon limit, which relates
to the bit energy per noise spectral density within the signal
that flows through the communication system. Note that if the
system is 100 % efficient on the signal-path, and no power is
used off the signal path, then (70) degenerates to Shannon’s
limit, indicating that in effect the communication system and
the signal it carriers have become identical. Equation (70)
indicates the importance of the power-efficiency factor of
a communication system in determining the true, practical
energy cost of a single bit. Note also from (34) that the total
power consumption to send a single bit is given by Pbc,bit :
Ebc,min =
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 most
impact on the overall systems power-efficiency factor. In other words, it is
most important to maximize the efficiencies of components that handle the
highest power levels.
equations (48) – (49) make clear that as HTX increases from
0 to 1, the Consumption Factor also increases.
VII. E NERGY P ER B IT
Shannon’s limit describes the minimum energy-per-bit-pernoise spectral density required to achieve arbitrarily low probability of bit error through proper coding scheme selection:
Eb
= ln (2)
(65)
No
This limit is generally found by using Shannon’s capacity
theorem, and allowing the code used to occupy an infinite
bandwidth [24].
As shown in (48) the CF is given as the maximum ratio of
data rate to power for a communication system, and may be
written as:
Blog2 (1 + SN R)
CF =
(66)
PN P + SN Rmin × Pnoise
H
Let us take the limit of (66) as bandwidth approaches infinity,
assuming AWGN. This is equivalent to allowing our system’s
coding scheme to spread out infinitely in bandwidth, driving
our SNR down to the minimum acceptable to still achieve arbitrarily low error. First, recall that the SNR may be written in
terms of the energy-per-bit Eb , the time required to transmit a
single bit Tb , the noise spectral density No , and the bandwidth
of the system B [24]:
SN R =
Eb
Tb
(67)
No B
In the limit as B approaches infinity, the SNR approaches the
minimum acceptable SNR. Therefore we have (68). Where
Ebc,min is the minimum energy per bit that must be consumed
by the communication system, and Eb is the minimum energyper-bit that must be present in the signal carried by the
communication system and delivered to the receiver’s detector.
Note that Ebc,min and Eb are not equal, as Ebc,min is the
amount of energy consumed/expended by the communication
system per bit (including the operation of ancillatory functions
such as powering non-signal path components like oscillators),
while Eb is the amount of energy per bit in the signal
itself. Note that the denominator is no longer a function of
Eb × C
= C×E bc,min .
(71)
H
Bits delivered to the edge of a wireless cell (greater propagation distance) are expected to be the most costly from an
energy perspective. It is instructive to estimate the required
power consumption per bit as a function of a cell radius for
a 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 the
link channel gain, HRX is the power-efficiency factor of the
receiver, HT X is the power-efficiency factor of the transmitter,
and GRX is the gain of the receiver. Using (72) in (70), we
find (73).
If we factor out the inverse of the channel gain, we find
(74). In the limit of very small channel gains (see (42)), this
yields:
Hlink → HT X GRX Gchannel
(75)
Pbc,bit = PN P +
Ebc,min =
ln (2) No
PN P
+
C
HT X GRX Gchannel
(76)
The interpretation of (76) is that for cases in which
GRX Gchannel is much smaller than unity (i.e. a highly
attenuating wireless channel), the stage immediately after
the attenuation should have high gain, so that the stage
immediately before does not need to have an extremely high
output power, resulting in increased loss. Secondly, we see
the importance of power amplifier efficiency and the need to
overcome 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. Several
examples using equation (76) are shown in Figure 6 through
Figure 9. Figures 6 and 7 show an example of a 20 GHz carrier
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12
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 32, NO. 12, DECEMBER 2014
lim CF =
B→∞
1
Ebc,min
=
=
(Blog2 (1 + SN Rmin ))
limB→∞
PN P + SN Rmin × Pnoise
H⎫
⎧
Eb
Tb
⎪
⎪
⎪
⎪
⎨ Blog2 1 + No B
⎬
limB→∞
Eb
⎪
⎪
⎪
⎩ PN P + NTbB × NHo B ⎪
⎭
(68)
o
Hlink =
Ebc,min =
HRX HT X GRX Gchannel
HT X GRX Gchannel + HRX HT X (1 − Gchannel ) + HRX (1 − HT X )
−1
1
PN P
1
1
+ ln (2) No H−1
H
+
−
1
+
−
1
.
RX
TX
C
GRX Gchannel
GRX Gchannel
Ebc,min =
ln (2) No PN P
−1
+
GRX Gchannel H−1
RX + HT X − Gchannel
C
GRX Gchannel
system with path loss modeled according to a log-distance
break-point model.
Contrasting Figures 6 and 7 shows that highly efficient
systems can afford to use longer link distances while systems
with less efficient signal path components should use shorter
distances. The decrease in efficiency is reflected in the change
in HT X 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 higher
carrier frequencies (k is the value of the path loss at a close-in
reference measurement distance).
With equation (74), we can determine the maximum wireless transmission distance d for which non-path power dominates the power expenditure per bit, and hence the maximum
distance before each bit becomes progressively more energyexpensive:
PN P
ln (2) No
GRX
>
(77)
−1 Gchannel +H−1
TX
C
GRX Gchannel
HRX
Gchannel >
ln (2) No C
RX
HT X PN P GRX +No Cln (2) 1− G
HRX
(78)
If we model the channel gain as (55), we find:
k
ln (2) No C
>
dα
HT X PN P GRX + No Cln (2) 1 −
GRX
HRX
(79)
and (80). If PN P < NoHCln(2)
, then (80) is unlikely to have
RX
a positive solution due to the small value of ln(2)No C. Our
interest in keeping the non-path energy per bit larger than
the signal energy per bit stems from interest in making every
bit as cheap from an energy perspective as possible while
simultaneously achieving the goal of a higher capacity. As
discussed in Section V, by forcing non-path energy to be larger
than signal energy per bit, we can achieve the simultaneous
(72)
(73)
(74)
Fig. 6. For a system with high signal path efficiency and high non-path power
consumption, we see that the energy expenditure per bit is dominated by nonpath power, indicating little advantage to shortening transmission distances.
goals of lower energy per bit and higher capacities through an
increase in bandwidth.
Note that the maximum value of d that ensures that nonpath power exceeds signal power increases as the amount of
non-path power increases. Also, as the gain of the receiver increases, the link distance may be extended while still achieving
a lower price-per-bit through an increase in bandwidth versus
a lower receiver gain system. As expected, as the path loss
exponent α increases, the maximum value of d decreases, as
indicated by (80).
Required energy consumption per bit can be used to evaluate the energy requirements of multi-hop versus singlehop communications. For example, consider the situation
illustrated in Figure 10 in which the source and sink may
communicate directly or through a relay. A similar analysis,
[4], showed the importance of path loss exponent and how
it 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 circumstances
it is advantageous to use a relay from an energy perspective
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MURDOCK and RAPPAPORT: CONSUMPTION FACTOR AND POWER-EFFICIENCY FACTOR: A THEORY FOR EVALUATING THE ENERGY EFFICIENCY...
d<
HT X k
ln (2) N o C
α1
GRX
PN P GRX + No Cln (2) 1 −
HRX
Fig. 7. When signal-path components are less efficient, as illustrated here,
then shorter transmission distances start to become advantageous, as signalpath power starts to represent a larger portion of the power expenditure per
bit.
13
(80)
Fig. 9. Lower efficiencies of signal-path components motivates the use of
shorter transmission distances.
power efficiency factor of the direct link, Hsource is the powerefficiency factor of the source transmitter, GRX,sink is the
gain of the sink receiver, and Gchannel,d3 is the channel gain
through the direct link. For the link through the relay, we find
the minimum energy consumption per bit Ebc,min,relay :
Ebc,min,relay =
Fig. 8. A higher carrier frequency system that provides a much higher bit
rate capacity (e.g. bandwidth) without substantially increasing non-path power
consumption may result in a net reduction in the energy price per bit.
based on the placement of the relay. But, to our knowledge
this paper presents the first such treatment in terms of the
gain of the sink and power-efficiency of the source transmitter.
Consider a three node network as shown in Figure 10. If the
path loss exponent for the network is α, then the required
power to transmit over a given distance d is proportional to dα .
Here we extend the analysis of [4] to account for the gain and
efficiencies of the devices participating in the network. The
results indicate when, on a per bit basis, it is advantageous to
use 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 hop
as illustrated in Figure 10, we simply compare the energy
consumptions for each link. For the single link through
distance d3 , we find the minimum energy consumption per
bit Ebc,min,direct :
PN Pd3
N0 ln (2)
+
Cd3
Hd3
(81)
Hd3 ≈ GRX,sink Gchannel,d3 Hsource
(82)
Ebc,min,direct =
where Cd3 is the capacity of the direct link, PN Pd3 is the
non-path power associated with the direct link, Hd3 is the
PN Pd1 PN Pd2 N0 ln (2) N0 ln (2)
+
+
+
(83)
Cd1
Cd2
Hd1
Hd2
Hd1 ≈ GRX,relay Gchannel,d1 Hsource
(84)
Hd2 ≈ GRX,sink Gchannel,d2 Hrelay
(85)
where Cd1 and Cd2 are the capacities of the links through
distances d1 and d2 , Hd1 and Hd3 are the power-efficiency
factors associated with the links through d1 and d2 , PN Pd1
and PN Pd2 are 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 through
link distances d1 and d2 , respectively. Assume that all nonpath powers are equal PN Pd1 = PN Pd2 = PN Pd3 = PN P , and
that the capacities through each link are equal to C. The ratio
of the minimum energy consumption per bit in both cases
determines when it is advantageous to transmit through the
relay:
PN P
1
1
2
+
N
ln
(2)
+
0
C
Hd1
Hd2
Ebc,min,relay
=
PN P
1
Ebc,min,direct
+ N0 ln (2)
C
=
Hd3
2Hd1 Hd2 Hd3 PN P + N0 CHd3 ln (2) (Hd1 + Hd2 )
Hd1 Hd2 Hd3 PN P + N0 Cln (2) Hd1 Hd2
(86)
Whenever (86) evaluates to be less than one, then it is
advantageous 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 <
No Cln (2) Hd1 Hd2
Hd1 Hd2 PN P + No Cln (2) (Hd1 + Hd2 )
(87)
If we model the channel gain according to equation (55), we
find (90). As the value of k depends only on the path loss
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14
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 32, NO. 12, DECEMBER 2014
No Cln (2) Hd1 Hd2
Hd1 Hd2 PN P + No Cln (2) (Hd1 + Hd2 )
No Cln (2) Hd1 Hd2
1
<
GRX,sink Hsource Hd1 Hd2 PN P + No Cln (2) (Hd1 + Hd2 )
GRX,sink Gchannel,d3 Hsource <
Gchannel,d3
k
<
dα
3
1
GRX,sink Hsource
(88)
(89)
No Cln (2) Hd1 Hd2
Hd1 Hd2 PN P + No Cln (2) (Hd1 + Hd2 )
(90)
or
dα
3 >GRX,sink Hsource
Hd1 Hd2 PN P +No Cln (2) (Hd1 +Hd2 )
No Cln (2) Hd1 Hd2
Fig. 10.
The basic consumption factor analysis determines when it is
advantageous to use a relay.
value at a reference distance from the transmitter, we shall
here let k = 1, as it gives little insight by retaining it in the
equations.
1
1
PN P
= GRX,sink Hsource
(91)
+
+
No Cln (2) Hd1
Hd2
See also (92) next page. To gain intuition, assume that the
non-path power is zero, in which case:
GRX,sink
Hsource
α
α
d1 +
dα
d3 >
(93)
2
GRX,relay
Hrelay
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 sink
and the relay receivers (gain here is the ratio of the power that
is retransmitted – as for the relay - or processed/demodulated
– as for the sink - to the power that is received from the
transmitter). Antenna gain is included in this gain because
we compute path loss assuming that the path loss has been
normalized for antenna gain. Therefore, if we have a relay
with a very low gain compared to the gain of the sink receiver,
the high output power required to communicate with the relay
may outweigh the benefit derived from communicating over
a shorter distance. The second key point is that the link
over which the relay acts as a transmitter (i.e. d2 ) is scaled
according to the ratio of the power-efficiency factors of the
source transmitter and relay transmitter. Therefore, if the relay
has a very low efficiency transmitter, any savings derived from
the shorter link distance between the relay and the sink may
be outweighed by the loss incurred due to the inefficiencies of
the relay. For the case of free-space channels, we may re-write
Fig. 11. From an energy perspective, it is advantageous to use a relay
provided the relay link distances are contained within the ellipse defined by
equation (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 illustrated
by Figure 11. It is advantageous to use the relay rather than
a direct path from source to sink if the distances d1 , d2 , and
d3 are such that they are in the interior of the ellipse in Figure
11.
Figures 12 through 14 illustrate the use of equation (93) and
how the region in which it is advantageous to place a relay
changes in size as certain parameters are varied. In Figure 12,
see that this region increases in size as the relay’s transmitter
becomes more efficient. In Figure 13, the same impact occurs
as the relay’s gain increases. Figure 14 indicates that higher
path loss exponents result in a larger area in which it is
beneficial to use the relay from an energy perspective. This
is because an increase in path loss incentivizes a means of
shortening link distances.
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MURDOCK and RAPPAPORT: CONSUMPTION FACTOR AND POWER-EFFICIENCY FACTOR: A THEORY FOR EVALUATING THE ENERGY EFFICIENCY...
dα
3 >
GRX,sink Hsource PN P
+
No Cln (2)
Fig. 12. The interior regions of each ellipse indicate where it is advantageous
to 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 are
where 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. C ONCLUSIONS
We have presented a basic theory of consumption factor
(CF) and have used this framework to develop useful concepts in designing future energy efficient wireless broadband
networks. Additionally, we have developed understanding of
how the efficiency of a communication system impacts the
minimum required energy expenditure per bit. We demonstrated the use of the consumption factor by showing when
it is advantageous to use a relay in a multi-hop setting.
Our analysis indicates the importance of the receiver gain
and transmitter efficiency for wireless communications. The
theory presented here allows engineers to compare cascades
of various components, and show quantitatively that when
the receiver gain is high, the transmitter may use lower
power, often resulting in a net energy savings. The continued
expansion of world-wide communication systems and the
exponential increase in data traffic necessitates reducing the
energy costs per bit. A key realization from the consumption
factor analysis is that, in order to align the goals of higher
data rates and lower energy expenditure per bit, it is necessary
to reduce the signal powers used in communication systems
to a point where ancillary power consumption (e.g. power
GRX,sink
GRX,relay
dα
1 +
Hsource
Hrelay
15
dα
2
(92)
Fig. 14.
Higher values of path loss result in a larger area where it is
advantageous to use a relay. The regions inside the ellipses are where it is
advantageous to use a relay.
consumed by oscillators and cooling equipment) is higher than
on-path signal power. Ideally, such ancillary forms of power
consumption will be decreased rapidly, but on-path signal
powers should be decreased even more quickly. To achieve this
goal for wireless systems, very short link distances, such as
those in a femtocells, become advantageous, or alternatively,
much more efficient RF power amplifiers if longer distances
are to be used. Our theory shows that shorter link distances
combined with massive bandwidths (e.g. at millimeter-wave
carrier frequencies) and highly directional antennas will enable
unprecedented data rates and lower energy consumption per
bit. This in turn enables continued exponential growth in total
data traffic while mitigating the dramatic increase in energy
consumption. In fact, as future mm-wave wireless systems
evolve using untapped spectrum above 5GHz [11][4][12], the
power consumption factor theory presented here may give
insight into proper beam forming and minimum power configurations for future wireless devices that use high gain adaptive
antennas that sense from where most multipath energy arrives.
ACKNOWLEDGMENTS
The authors would like to acknowledge F. Gutierrez, E. BenDor, Y. Qiao, J. I. Tamir, and K. Shabaik for their useful
thoughts, 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, West
Lafayette, IN, in 1982, 1984, and 1987, respectively, and is an Outstanding
Electrical Engineering Alumnus from his alma mater. Currently, he holds
the David Lee/Ernst Weber Chair in Electrical Engineering at NYU-Poly,
and is also center director of the NSF I/UCRC WICAT center at NYUPoly, and Professor of Computer Science and Professor of Radiology at New
York University. Earlier in his career, he founded the Wireless Networking
and Communications Group (WNCG) at the University of Texas at Austin
(UT), where he also was founding NSF I/UCRC site director for the Wireless
Internet Center for Advanced Technology (WICAT). Prior to UT, he was on
the electrical and computer engineering faculty of Virginia Tech where he
founded the Mobile and Portable Radio Research Group (MPRG), one of the
world’s first university research and teaching centers dedicated to the wireless
communications field. In 1989, he founded TSR Technologies, Inc., a cellular
radio/PCS software radio manufacturer that he sold in 1993 to what is now
CommScope, Inc. In 1995, he founded Wireless Valley Communications Inc.,
a site-specific wireless network design and management firm that he sold in
2005 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 national
committees pertaining to communications research and technology policy. He
is a highly sought-after consultant and technical expert, and serves on various
boards of several high-tech companies. He has authored or coauthored over
200 technical papers, over 100 U.S. and international patents, and several
prize papers and bestselling books. In 2006, he was elected to serve on the
Board of Governors of the IEEE communications Society (ComSoc), and was
elected 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 from
the University of Texas at Austin, Austin, in 2008, where he specialized
in communication systems and signal processing. He obtained a Master of
Engineering degree from The University of Texas at Austin in 2011, for
which he focused on sub-THz and electromagnetic engineering, in addition
to channel modeling and scientific data archiving. Currently, he is working at
Texas Instruments, where he focuses on low power radio systems and sub-THz
radar applications. James has co-authored over ten conference and technical
magazine papers and two journal papers.