Embed Size (px)
Citation preview
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 2, FEBRUARY 2015 401
Is Accumulative Information Useful for DesigningEnergy Efficient Transmission?
Chuting Yao, Zhikun Xu, Tingting Liu, and Chenyang Yang
Abstract—Energy efficiency (EE) has become an importantdesign goal for mobile communication systems. Since the EE ofa system is evaluated during a period of time, a transmissionstrategy designed from maximizing instantaneous EE (INEE) maynot achieve the maximal EE of the system. To exploit the accumu-lative nature of the energy, accumulative EE (ACEE) can be usedas the objective function, which is the ratio of the accumulatedamount of data transmitted to the accumulated energy consumeduntil the time for optimization. In this paper, we study whenACEE is beneficial to EE-oriented optimization. By taking a singleuser multi-antenna multi-subcarrier system serving two classesof traffic as an example, we formulate three problems to opti-mize rate allocation among subcarriers and time slots respectivelymaximizing the INEE, ACEE and EE upper-bound, which can beeasily extended to multi-user systems. We proceed to analyze thebehavior and performance of the corresponding solutions. Analyt-ical and simulation results show that using ACEE yields a moreenergy efficient design for the systems serving best effort trafficwith less transmit antennas, serving less users simultaneously orat low signal to noise ratio under time-varying channels. However,the conclusions for real-time traffic are different.
Index Terms—Energy efficiency (EE)-oriented optimization, ac-cumulative EE, instantaneous EE, EE upper-bound.
I. INTRODUCTION
W ITH explosive growth of traffic loads and ever-increasing demands in capacity, the largely ignored
cost in energy is emerging as a major concern, which notonly contributes to the global carbon dioxide emissions butalso incurs rapidly increased operational expenditure [1]. As aresult, green radio has drawn significant attention recently andbecome one of important design goal for next generation highthroughput wireless communication systems [2]–[6].
While a widely accepted definition of green communicationstill remains as an open problem, energy efficient transmissionhas become a synonym for green radio. There are variousdefinitions of energy efficiency (EE) [2], depending on theconcerned system as well as the purpose of the metric. Forinstance, a concerned network can be either coverage-limited
Manuscript received January 9, 2014; revised June 22, 2014, September 12,2014, and November 4, 2014; accepted December 5, 2014. Date of publi-cation December 18, 2014; date of current version February 12, 2015. Thiswork is supported in part by China (NSFC) under Grant 61120106002 andby China 973 Program under Grant 2012CB316003. The associate editorcoordinating the review of this paper and approving it for publication wasB. Clerckx.
C. Yao, T. Liu, and C. Yang are with the School of Electronics andInformation Engineering, Beihang University, Beijing 100191, China (e-mail:[email protected]; [email protected]; [email protected]).
Z. Xu is with China Mobile Research Institute, Beijing 100053, China(e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TCOMM.2014.2382120
or capacity-limited, either single cell or multi-cell, and eitherdownlink or uplink. For the systems deployed in rural areaswhere the coverage is a critical issue, the EE is defined asthe ratio of average power consumption to coverage area ofthe network [7], [8]. For the systems in urban areas with hightraffic load, a commonly used metric to evaluate the EE in bits-per-Joule is the ratio of overall number of bits transmitted tothe energy consumed in a certain period of time [7], or theratio of average throughput to average power [9], or equivalentwith the metric in Joule-per-bit [8], [10]. When consideringdownlink transmission, the amount of data transmitted and theenergy consumed are from one base station (BS) in single cellsystems [3]–[6], [11]–[14], and are aggregated from multipleBSs for multiple users in multi-cell or heterogeneous downlinknetworks [10], [15], [16]. Inheriting the notion of area spectralefficiency, similar metric for EE are also introduced to multi-cell or heterogeneous networks, e.g., [17]–[19]. When consid-ering the EE of uplink systems, the number of bits and energyare from each user [20]–[23].
A performance metric can be used as an objective functionfor optimization, or used for analysis or evaluation. In general,the goal of optimization is to find a solution of good perfor-mance. When a metric is used for optimization, the objectionfunction should be mathematically tractable and the resultingsolution should provide fairly good performance. The goal ofanalysis is to reveal or quantify the dependence of performanceon some key factors. When a metric is used for analysis, themetric should also be tractable in math. The goal of evaluationis to assess the performance or behavior of a system. Whena metric is used for evaluation, the metric to measure perfor-mance should objectively reflect the real performance and bewith good interpretation [24]. On the other hand, both analysisand evaluation can be conducted off-line, such that non-causalinformation (as explained later) can be used. By contrast,transmission strategy should be optimized on-line, since onlycausal information (i.e., the past and current information) ispossibly known. Employing the evaluation metric as an ob-jective function can yield a strategy with best performance,yet this is often not feasible in practice because only causalinformation is available at the moment of optimization in time-varying channels.
To optimize transmission strategies for improving the EE ofhigh throughput single cell systems where causal channel infor-mation should be used, a popular objective function is the ratioof data rate to the power consumed in current time slot, e.g.,[3]–[5], [11]–[13], which is called instantaneous EE (INEE) inthe sequel. If channels are static, maximizing the INEE willalso maximize the evaluation metric, which is however not true
0090-6778 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
402 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 2, FEBRUARY 2015
for time-varying channels. To optimize system configurationssuch as the number of active antennas, the objective function tomaximize EE can be ergodic capacity divided by average powerconsumption [6], [14], [25], such that the designed systems donot need to be configured frequently in time varying channels.In [20], [26], a time-averaged bits-per-Joule metric was used tooptimize transmission strategies towards EE, which is in fact akind of accumulative EE (ACEE), where the accumulative in-formation (i.e., the throughput and energy accumulated withinpast time slots) is exploited. While the optimization problemsfor designing transmission strategies respectively maximizingINEE and ACEE are similar in mathematical tractability, touse ACEE, the accumulative information needs to be stored.Besides, some INEE-oriented designs (e.g., considering userfairness in EE-oriented design [21]) cannot be extended whenconsidering ACEE.
Using different objective functions for optimizing trans-mission strategies indicates exploiting different information.Intuitively, an objective function exploiting more informationshould give rise to better performance but may have a morecomplicated form. For instance, using ACEE as the objectivefunction means exploiting accumulative information, whereonly causal channel information is assumed known and accu-mulative nature of energy is exploited. Yet whether and whenusing ACEE as the objective function to design transmissionstrategies has an EE gain over INEE is not clear, which areseldom addressed in the literature.
In this paper, we consider EE-oriented transmission strategyoptimization problem for downlink high throughput single cellsystems. We strive to analyze when and why using the ac-cumulative information will be beneficial for the EE-orientedtransmission optimization. Our goal is to better understandEE measures for evaluation and optimization, which in turnoffers better guidance for practical optimization problems. Toshow the impact of exploiting different channel information onEE optimization, we model the channel either as static or asblock fading, and we assume that the BS either has perfectknowledge of current channel, or current channel togetherwith accumulative information or even the future channels. Tocapture the essence of the problem with tractable analysis, wetake a single user multi-input-multi-output (MIMO) orthogonalfrequency division multiplexing (OFDM) system as an exam-ple. To reflect the basic principle of EE-oriented optimization,i.e., the EE of a system should be maximized without degradingthe user experience, we consider two typical kinds of services,best effort or real-time traffic. We formulate the optimizationproblems for rate allocation with different objective functionswith and without exploiting the accumulative information in aunified framework, and provide the solutions. By analyzing thebehaviors of the transmission strategies derived from differentobjective functions and the resulting EEs resorting to asymp-totic analysis, we show when using ACEE in the EE-orientedoptimization has gain. While the analytical analysis for the EEof multi-user scenarios are different, the derived solution canbe easily extended to multi-user systems. We use simulationsto show the impact of the number of antennas, the number ofusers, the bandwidth, and the quality of service (QoS) require-ment on the EE comparison between using ACEE and INEE.
The rest of the paper is organized as follows. System modeland EE metrics are introduced in Section II. In Section III,we formulate the optimization problems, provide the solutionsand analyze their behaviors. In Section IV, the EEs achievedby different solutions are compared. Simulation results areprovided in Section V and the paper is concluded in Section VI.
Notations: | · | and ‖ · ‖ denote the magnitude and theEuclidean norm, respectively. (·)H denotes conjugate transpose.E{·} and Var{·} represent expectation and variance respec-tively. (x)+ represents max{x,0}.
II. SYSTEM MODEL AND EE METRICS
A. System and Channel Model
We take a downlink MIMO-OFDM system as an exampleto show the impact of using accumulative information on EE-oriented optimization, where a base station (BS) equipped withNt antennas serves a user with single antenna over K subcarriersin T time slots. The bandwidth of each subcarrier is B and theinterval of each time slot is ΔT .
In current time slot t, 1 ≤ t ≤ T , the received signal at the kthsubcarrier is
ytk =
(√αtht
k
)Hwt
k
√pt
kxtk +nt
k (1)
where xtk is the transmit symbol with E{|xt
k|2} = 1, ptk is the
transmit power, wtk ∈ C
Nt×1 is the beamforming vector with‖wt
k‖2 = 1, αt is the large-scale fading gain including path lossand shadowing, ht
k ∈ CNt×1 is the small scale fading channel
vector, and ntk is a zero-mean additive white Gaussian noise
(AWGN) with variance σ2.Assume that the BS knows perfect channel vectors at current
time slot t (i.e., the moment for optimization and transmission),ht
k,k = 1, · · ·K. Consider maximum ratio transmission (MRT).Then, the beamforming vector at subcarrier k is wt
k = htk/‖ht
k‖,and the achievable rate of the system in time slot t is
Rt = ∑Kk=1 Rt
k = ∑Kk=1 B log2
(1+gt
k ptk
)(2)
where gtk
Δ= αt‖ht
k‖2/σ2 is the channel gain normalized by thenoise power.
Assume that each element in htk is subject to Nakagami-m
distribution, which characterizes a wide range of fading envi-ronments via parameter m. For example, m = 1 correspondsto Rayleigh fading. As m increases, the fluctuation of thechannel decreases. When m → ∞, the channel becomes AWGNchannel. With given value of αt , the normalized channel gainfollows Gamma distribution, i.e., gt
k ∼ Gamma(mg,Ω/mg),whose probability distribution function (PDF) is
f(gt
k;mg,Ω)=
mmgg
Ωmg
1Γ(mg)
(gt
k
)mg−1e−
mggtk
Ω (3)
where Ω Δ= E{gt
k} = αtNt/σ2, the shape factor mgΔ=
NtE2{(gt
k)2}/Var{(gt
k)2} = Ntm, and Γ(mg) is Gamma
function.
YAO et al.: IS ACCUMULATIVE INFORMATION USEFUL FOR DESIGNING ENERGY EFFICIENT TRANSMISSION? 403
Fig. 1. Illustration for the system model and EE evaluation.
B. EE Metrics for Evaluation, Optimization and Analysis
While in concept EE is the number of bits transmitted perunit energy, the specific definition has various forms dependingon the concerned system and the purpose of using the metric.
For a downlink system, the power consumption in time slot tcan be modeled as [27],
Pt =
1ξPA(1−ξFE)
∑Kk=1 pt
k +PRF +PBB
(1−ξDC)(1−ξMS)(1−ξCO)
Δ=
1ξ
K
∑k=1
ptk +Pc (4)
where ξPA is the power amplifier (PA) efficiency, ξFE is thefeeder loss, ξDC, ξMS, and ξCO are respectively the loss factorsof direct-current to direct-current power supply, main supplyand cooling, PRF and PBB are the power consumptions of
radio frequency module and baseband processor, ξ Δ= ξPA(1−
ξFE)(1−ξDC)(1−ξMS)(1−ξCO) is an equivalent PA efficiency,
PcΔ= (PRF+PBB)/((1−ξDC)(1−ξMS)(1−ξCO)) is the overall
circuit power consumption.To evaluate the performance in terms of EE of a system
that can be done in an off-line manner, a widely used metricin industry is the total number of bits divided by the energyconsumed during a given period of transmission time [7], whichis proper for measuring the EE at the BS serving mobile users.For the considered system, the EE can be evaluated over T timeslots (shown as Fig. 1), i.e., the metric for evaluation is
η Δ=
∑Tτ=1 RτΔT
∑Tτ=1 PτΔT
=∑T
τ=1 Rτ
∑Tτ=1 Pτ (5)
which is called system EE to differentiate from the objectivefunctions defined later.
To design a transmission strategy to maximize the EE of thesystem, the objective function can be in different forms accord-ing to the available knowledge at the moment of optimization.When only the current channel information gt
k is known foroptimization at time slot t, the INEE, Rt/Pt , is a naturallyresulting objective function. When gt
k and the accumulativeinformation (i.e., the throughput and energy accumulated inthe past time slots 0, · · · , t − 1) are available for optimizationat time slot t, the ACEE, ∑t
τ=1 Rτ/∑tτ=1 Pτ, is intuitively a better
objective function. Note that although energy is accumulative,the EE is not additive, i.e., ∑t
τ=1 Rτ/∑tτ=1 Pτ �= ∑t
τ=1(Rt/Pt).
If we assume that the non-causal channel information at cur-rent time slot t (i.e., the channels in all the T time slots includingfuture information) is available at the BS, the objective functioncan be ∑T
τ=1 Rτ/∑Tτ=1 Pτ, the same as (5). The transmission
strategy derived from such an objective function is not viable
in real-world systems because future channel information is un-known under time-varying channels. Nonetheless, the achievedsystem EE of such a strategy can provide an upper-bound (UB)for all other EE-maximization problems, since the metric usedfor optimization is the same as the metric for evaluation.
These objective functions for optimization can be unified asfollows,
∑T2τ=T1
Rτ
∑T2τ=T1
Pτ(6)
where the values of T1 and T2 lead to different objectivefunctions elaborated as follows.
• T1 = T2 = t: The objective function is the INEE. Werefer to the solution of such a problem as INEE-orienteddesign, and the achieved system EE is denoted asηIN.
• T1 = 1 and T2 = t: The objective function is the ACEE.We refer to the solution of this problem as ACEE-orienteddesign, and the achieved system EE is denoted as ηAC.
• T1 = 1 and T2 = T : The objective function is the evalu-ation metric. We refer to the solution of this problem asUB-oriented design, and the achieved EE, ηUB, is calledEE upper-bound.
To compare the system EEs achieved by the three designsunder fading channels, we consider the following metric foranalysis,
E{Rt}E{Pt} = lim
T→∞η = lim
T→∞
∑Tt=1 Rt
∑Tt=1 Pt
(7)
which is widely used in the literature for analyzing and alsofor evaluating the EE of a system due to the mathematicaltractability [14], [28], where the first equality holds when Rt
and Pt are ergodic [29].
III. EE-ORIENTED RATE ALLOCATION WITH
DIFFERENT OBJECTIVE FUNCTIONS
To show whether ACEE is superior to INEE when servingas an objective function for optimizing the transmission strat-egy, we first provide the corresponding optimized strategies.Because the system EE of the ACEE-oriented design is hard toanalyze, we also find the solution of the UB-oriented design,and show that the two designs behaves similarly. Then, wecan analyze the system EE of the ACEE-oriented design byanalyzing that of the UB-oriented design.
In the sequel, we formulate three EE-oriented optimizationproblems to design the rate allocation among multiple subcarri-ers in current or all time slots according to the assumed channelinformation. We proceed to provide the corresponding solu-tions, and analyze their behaviors owing to exploiting differentinformation.
The design to maximize EE of a system should not compro-mise the QoS perceived by the user. To capture the essenceof this principle and reflect typical QoS constraints imposedby real and non-real time traffic [30], we consider a minimaldata rate requirement R0. Note that the optimimal data rate
404 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 2, FEBRUARY 2015
may exceed R0 to maximize the EE. For best effort traffic,R0 = 0, where the problem is simply to maximize the EEwithout the rate constraint (only with a minimal transmit powerconstraint of zero). We do not consider the maximal transmitpower constraint, since the minimal data rate requirement canusually be satisfied by using admission control in practice. Ifthe transmit power required to support R0 exceeds the maximalvalue when the channel suffers from deep fading, an outage willoccur, which however does not change the final conclusion tobe drawn later.
A. Problem Formulation and Solution
From (2) and (4), the optimization problem maximizingthe objective function in (6) under the QoS constraint and isformulated as
maxR
∑T2τ=T1
∑Kk=1 Rτ
k
∑T2τ=T1
(∑K
k=12
Rτk
B −1ξgτ
k+Pc
) (8a)
s.t. ∑Kk=1 Rτ
k ≥ R0,R ≥ 0
where R =
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
{RT1
1 , . . . ,RT1K , . . . ,RT2
1 , . . . ,RT2K
},
T1 = 1 and T2 = Tfor UB-oriented design
{Rt1, . . . ,R
tK} , otherwise for
ACEE- and INEE-oriented designs.
(8b)
Considering Rτ = ∑Kk=1 Rτ
k in (2), problem (8) can be rewrit-ten as
maxRτ,R
∑T2τ=T1
Rτ
∑T2τ=T1
(∑K
k=12
Rτk
B −1ξgτ
k+Pc
) (9a)
s.t. Rτ ≥ R0,R ≥ 0. (9b)
The objective function is the ratio of a linear function toa convex function of the optimization variables and henceis quasi-concave [3]. Therefore, the problem is quasi-convexand the solution can be found from the Karush-Kuhn-Tucker(KKT) conditions [31]. Similar to the optimal power allocationproblem in [6], the solution of the optimal rate allocated to thekth subcarrier in time slot t follows a water-filling structure,which can be obtained as
Rtk =
⎛⎝B log2
2Rt
BLt
gt −B log21gt
k
⎞⎠
+
(10)
where Lt is the number of allocated non-zero data rates Rtk,k =
1, · · · ,K (i.e., the number of active subcarriers that have data
to transmit) in time slot t, and gt Δ= (∏Lt
l=1 gt[l])
1Lt , in which
gt[l], l = 1, . . . ,Lt are the normalized channel gains sorted in a
descending order.
Upon substituting (10), problem (9) becomes
maxR
∑T2τ=T1
Rτ
∑T2τ=T1
(∑K
k=11ξ
(2
RτBLτ
gτ − 1gτ
k
)+
+Pc
) (11a)
s.t. Rτ ≥ R0, R ≥ 0
where R =
⎧⎪⎪⎨⎪⎪⎩
{RT1 , . . . ,RT2
},T1 = 1 and T2 = T
for UB-oriented designRt , otherwise for ACEE-
and INEE-oriented designs.
(11b)
Problem (11) is quasi-concave, which can be proved by usingsimilar way as in [3]. The structure of this problem is thesame as power allocation problems in [6], [32], and can besolved using standard optimization methods [31]. From theKKT conditions of the problem, it is not hard to obtain theoptimal data rate allocated in time slot t for the ACEE-orienteddesign (T1 = 1 and T2 = t) and INEE-oriented design (T1 = T2 =t) as follows,
Rt∗ =
{R0, if gt < gt
thrBLt
ln2 (W0(ϕt)+1)−∑T2τ=T1,τ �=t Rτ, otherwise
(12)
where gt is defined in (10), and the threshold gtthr is
gtthr
Δ=
(∑T2
τ=T1,τ �=t Rτ+R0
)2
R0BLt ln2−2
R0BLt BLt+ρtBLt
ξB(
∑T2τ=T1,τ �=t Pτ+Pc
) (13)
with ρt Δ= (∑Lt
l=1 gt/gt[l])/Lt , W0(x) is the principle branch of
Lambert-W function [33], and
ϕt Δ=
1Lt
(ξgt
T2
∑τ=T1,τ �=t
Pτ+ξgtPc −ρt
)e
ln2BLt
T2∑
τ=T1τ �=tRτ−1
.
The threshold comes from the rate constraint in the rate al-location problems, which has a similar impact as the powerconstraint in the power allocation problems in [6], [32].
To find the optimal solution of problem (8) for the ACEE-and INEE-oriented designs, we need to first find Rt from thecondition in (12) for problem (11), and then compute Rt
k,k =1, · · · ,K with (10), which is the necessary and sufficient con-dition for finding optimal solution of problem (8). Note thatto obtain Rt and Rt
k, the parameter Lt should be known, whichhowever can be known only after Rt
k being known. To find Lt
and Rtk satisfying both (10) and (12), we reduce Lt from K
to 1 until the two conditions are satisfied, which is a standardmethod for solving the problems with water-filling structure in(10). To help understand how to find the optimal solution for theACEE-oriented design and INEE-oriented design in time slot t,i.e., the optimal data rate allocated to multiple subcarriers in
YAO et al.: IS ACCUMULATIVE INFORMATION USEFUL FOR DESIGNING ENERGY EFFICIENT TRANSMISSION? 405
time slot t, R ∗ = {Rt∗1 , . . . ,R
t∗K }, we summarize the procedure
in the following.
Procedure of Finding the Optimal Solutions for the ACEE-and INEE-oriented Designs from Problem (8)
Input: normalized channel gains gtk,k = 1, . . . ,K (already
sorted in descending order) and minimal rate require-ment R0.
Output: optimal rate allocation Rt∗k ,k = 1, . . . ,K.
1: Set Lt as K.2: Compute Rt with (12).3: Compute Rt
k with (10) by substituting Rt , from which anupdated number of active sub-carriers Lt can be obtained.
4: If Lt = Lt , go to next step. Otherwise, set Lt = Lt −1 andgo back to step 2.
5: Obtain Rt∗k ,k = 1, . . . ,K from (10) with Rt and Lt .
The optimal solutions of the ACEE- and INEE-orienteddesigns have the same structure, therefore the computationalcomplexity to find the solutions is the same. Nonetheless,when the two designs are applied for practical systems, thecomplexity to implement the ACEE-oriented design is higherthan the INEE-oriented design. To exploit the accumulativeinformation, the bits transmitted and the energy consumed inthe past time slots need to be stored for the ACEE-orienteddesign and the storage space increases with time.
Different from the ACEE-oriented design and INEE-orienteddesign, with the UB-oriented design the BS allocates the datarate for all T time slots simultaneously, where (12) only pro-vides a relation between the optimal rate allocation Rt in timeslot t and the optimal Rτ,τ �= t,τ= 1,2, . . . ,T in other time slots.Also from the KKT conditions of problem (11), we can obtainthe optimal solution for the UB-oriented design as follows [31],
Rt∗UB =
{R0, if gt < gt
UB
R0 +BLt log2gt
gtUB
, otherwise (14)
where gt is defined in (10) and
gtUB
Δ=
2R0BLt ln2ξB
ηUB. (15)
It follows that the UB-oriented design is a kind of water-fillingrate allocation with multiple water levels gt
UB, t = 1, . . . ,T . With(14) and (10), we can obtain the optimal rate allocation R ∗
UB ={R1∗
1 , . . . ,R1∗K , . . . ,RT∗
1 , . . . ,RT∗K } using the algorithm in [34].
B. Behavior of the Solutions From DifferentObjective Functions
For notational simplicity but without loss of generality, weshow the results for the case of K = 1. Then, Lt = 1, gt is shortfor the normalized channel gain gt
1 in the rest of the paper, andgt defined in (10) reduces to gt .
• ACEE-oriented design (T1 = 1,T2 = t): The BS allocatesthe data rate in the current time slot t, where the accumu-
lated throughput and consumed energy in the interval of(0, t −1] are exploited in the optimization. From (12), theoptimal rate allocation to maximize the ACEE is
RtAC =
⎧⎨⎩
R0 , if gt < gtAC
Bln2
(W0(ϕt
AC)+1)−
t−1∑
τ=1Rτ, otherwise
(16)
where ϕtAC
Δ= (ξgt ∑t−1
τ=1 Pτ +ξgtPc −1)eln2B ∑t−1
τ=1 Rτ−1, and
gtAC
Δ=
(∑t−1
τ=1 Rτ +R0)
2R0B ln2−2
R0B B+B
ξB(∑t−1
τ=1 Pτ +Pc)
• INEE-oriented design (T1 = t,T2 = t): The BS allocatesthe data rate in time slot t only using current channelinformation. From (12), the corresponding optimal rateallocation is
RtIN =
{R0 , if gt < gt
INB
ln2
(W0
(ϕt
IN
)+1
), otherwise (17)
where ϕtIN
Δ= (ξgtPc − 1)e−1, and gt
INΔ= (R02
R0B ln2 −
2R0B B+B)/(ξBPc).
• UB-oriented design (T1 = 1,T2 = T ): The rate allocationto achieve the EE upper-bound can be found from (14) asfollows,
RtUB =
{R0, if gt < gUB
R0 +B log2gt
gUB, otherwise
(18)
where gUBΔ= 2
R0B ln2ηUB/(ξB).
For all the three designs, when the normalized channel gainis lower than a threshold, the optimal data rate is R0, similar to[6], [32]. Otherwise, the rate allocations will be different. In thefollowing, we focus on the case when the normalized channelgain exceeds the threshold.
Proposition 1: For arbitrary time slots t1 and t2 (t1 > t2),when gt ≥ max{gt
IN,gUB}, t = t1, t2, the INEE-oriented andUB-oriented rate allocation respectively satisfy
Rt1IN2
Rt1INB : Rt2
IN2R
t2INB ≈ gt1 : gt2 (19)
2R
t1UBB : 2
Rt2UBB = gt1 : gt2 (20)
and when t1 and t2 are large but t1 − t2 is small, with gt >gt
AC, t = t1, t2, the ACEE-oriented rate allocation satisfies
2R
t1ACB : 2
Rt2ACB ≈ gt1 : gt2 . (21)
Proof: See Appendix A. �It follows that the ACEE-oriented design behaves similar to
the UB-oriented design.Corollary 1: For a normalized channel gain gt ≥ max{gt
IN,gUB}, if there exists gτ
0 > max{gtIN,gUB} that makes Rτ0
IN =
Rτ0UB > R0, then ΔRt Δ
= RtIN −Rt
UB can be approximated as
ΔRt ≈ B log2
(Rτ0
IN/RtIN
)(22)
406 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 2, FEBRUARY 2015
and
{ΔRt < 0, if gt > gτ0
ΔRt ≥ 0, otherwise.(23)
Proof: See Appendix B. �Corollary 1 suggests that the UB-oriented design will trans-
mit with higher data rate than the INEE-oriented design whenthe normalized channel gain exceeds gτ0 and vice versa. In otherwords, the UB-oriented design can exploit “channel diversity”more aggressively with the non-casual channel information: ittransmits more data when the channel condition is good andtransmits less data when the channel becomes worse. Furtherconsidering the behaviors of different solutions reflected inProposition 1, it is the “channel diversity” that gives rise tothe difference in rate allocation between the ACEE- and INEE-oriented design.
IV. ANALYSIS ON EE USING ACCUMULATIVE
OR INSTANTANEOUS INFORMATION
In this section, we study the impact of exploiting accumu-lative information on the EE-oriented transmission strategyoptimization problems under different channels. We considertwo extreme cases of channels. One is static channel where thechannel is constant over the time for evaluation. The other isblock fading where the channel in each time slot is constant andthose among different time slots are independent and identicallydistributed (i.i.d.).
From previous analysis we know that only when the allocateddata rate Rt exceeds the required minimal data rate R0, theACEE- and INEE-oriented designs will be different. Moreover,with the increase of R0, both the thresholds gt
AC and gtIN in-
crease, which indicates that under the same channel condition,the probability of Rt = R0 increases with R0. Therefore, ifthe QoS requirement is stringent, no matter which objectivefunction is applied, the optimal rate allocation in each time slotis equal to R0 with high probability. To highlight the differenceof two designs, we consider best effort traffic in the followinganalysis, i.e., R0 = 0. Again, for notational simplicity and with-out loss of generality, we show the results for the case of K = 1.
A. Comparison Under Static Channels
When the channel is static during evaluation, the normalizedchannel gain gt = Ω for t ∈ (0,T ]. Then from problem (8) it isnot hard to see that the three objective functions are identical,and the three designs will lead to the same system EE.
The optimal rate allocation and the resulting maximal systemEE under static channels can be obtained similarly as theoptimal power allocation and corresponding EE available in theliterature [6]. To facilitate the comparison with time-varyingchannels, we briefly present the result in the sequel with aunified form. From (17), the optimal rate allocation in time slott of the INEE-oriented design when R0 = 0 can be expressed as
RtIN =
Bln2
(W0
((ξPcgt −1)e−1)+1
). (24)
Further from (4) and the definition of Lambert-W function [33],the power consumed by the INEE-oriented design is
PtIN =
2Rt
INB −1ξgt +Pc
=
(W0
((ξPcgt −1)e−1
)+1
)eW0((ξPcgt−1)e−1)+1
ξgt . (25)
From (24) and (25) with W0((ξPcΩ− 1)e−1)Δ= W0, the system
EE achieved by the INEE-oriented design is
ηstaIN =
∑Tt=1 Rt
IN
∑Tt=1 Pt
IN
=Rt
IN
PtIN
∣∣∣∣gt=Ω
=ξBΩln2
e−W0−1 Δ= η0. (26)
By considering the definition of Lambert-W function again,the system EEs achieved by the three designs can be written as
ηstaIN = ηsta
UB = ηstaAC = η0 =
ξBΩ(ξPcΩ−1) ln2
W0. (27)
B. Comparison Under Time-Varying Channels
For block fading channel, the case with m → ∞ is trivial,where the channel reduces to AWGN that can be viewed as aspecial static channel with gt = Ω and ηUB|m→∞ = ηAC|m→∞ =ηIN|m→∞ =η0. Therefore, we analyze the case where m is finite.
From (16), we can find that the ACEE-oriented design de-pends not only on current channel but also on the history infor-mation. As a result, the allocated rate in different time slots isnot a stationary stochastic process, and the corresponding per-formance analysis is hard. Fortunately, Proposition 1 shows thatthe ACEE-oriented design behaves similar to the UB-orienteddesign. In fact, the performance achieved by the two rateallocations are close in block fading channel as will be shownin simulations later. In the sequel, we compare the UB-orienteddesign with the INEE-oriented design, which can implicitlyreflect the EE difference of the ACEE-oriented design with theINEE-oriented design. To investigate the potential of using theaccumulative information, we set the evaluation time T → ∞.
From (17) and (18), it is easy to see that the rate allocationonly depends on the current channel information gt . As a result,the allocated rates by the INEE- and UB-oriented designs arestationary and ergodic stochastic processes when the channelsare ergodic [29]. When T → ∞, the time average is equal to theensemble average, and hence we can use (7) for analysis.
For the INEE-oriented design, considering that the rate RtIN
in (24) is a concave function of gt and the power consumptionPt
IN in (25) is a convex function of gt (see Appendix C), andΩ = E{gt}, from Jensen’s inequality we have,
E{
RtIN
}≤ B
ln2(W0 +1),
E{
PtIN
}≥ (W0 +1)eW0+1
ξΩ.
Then, an upper-bound of the system EE of the INEE-orienteddesign can be obtained as
ηIN =E{
RtIN
}E{
PtIN
} ≤ ξBΩln2
e−W0−1. (28)
YAO et al.: IS ACCUMULATIVE INFORMATION USEFUL FOR DESIGNING ENERGY EFFICIENT TRANSMISSION? 407
where the equality holds if and only if gt = Ω (i.e., staticchannels). Considering (26), we can see that η0 is the upper-bound of the system EE of the INEE-oriented design, i.e.,ηIN ≤ η0.
For the UB-oriented design, from (7) the system EE canbe expressed as (29), shown at the bottom of the page,
where PtUB=(2
RtUBB −1)/(ξgt)+Pc. Based on (18), when gt<gUB,
E{RtUB|gt < gUB} = R0 = 0. When gt ≥ gUB, Rt
UB =
B log2(gt/gUB), from which we have 2
RtUBB /gt = 1/gUB Then,
(29) can be rewritten as
ηUB =
∫ ∞gUB
B log2
(gt
gUB
)f (gt)dgt
1ξ∫ ∞
gUB
(1
gUB− 1
gt
)f (gt)dgt +Pc
(30)
where f (gt) is short for f (gt ;mg,Ω) in (3).
With gUB = 2R0B ln2ηUB/(ξB) and (30), we can obtain a
relation between gUB and gt as
gUB
∫ ∞
gUB
(1
gUB− 1
gt
)f (gt)dgt
−∫ ∞
gUB
ln
(gt
gUB
)f (gt)dgt +gUBξPc = 0. (31)
Further considering (3) and that mg is an integer, (31) can berewritten as (See Appendix D)
v(z,mg)+gUBξPc = 0 (32)
where zΔ= mggUB/Ω,
v(z,mg)=
⎧⎪⎨⎪⎩
e−z − zE1(z)−E1(z), mg=1
e−zmg−1
∑k=0
(mg−1−k
mg−1 −mg−1
∑i=k+1
1i
)zk
k! −E1(z), mg>1
(33)
and E1(·) is the exponential integration function [35].Unfortunately, even when mg is an integer the closed-form
expression of ηUB still can not be derived from (32) due tothe complex integration. In the following, we resort asymptoticanalysis for comparison in high and low signal to noise ratio(SNR) regions by comparing with η0.
1) Comparison Between ηIN and ηUB in High SNR Region(Ω → ∞):
• ηIN vs. η0
In time slot t, the ratio of RtIN and Pt
IN can be derivedfrom (24) and (25) as
RtIN
PtIN
Δ= q(gt) (34)
where q(gt) = ξBgte−W0((ξPcgt−1)e−1)−1/ ln2. Sincelimx→∞ W0(x) → ∞ [33], it is not hard to show thatlimgt→∞ ∂q(gt)/∂gt = 0, which implies that as gt increase,the increasing rate of q(gt) becomes slower. Consequently,if gt is sufficiently large (say when gt > g0), q(gt) can beviewed as a constant approximately. When Ω → ∞, forany given value of g0, we have Ω > g0. Then, the constantcan be expressed as q(Ω). Therefore, when gt ≥ g0, Rt
INcan be approximated as
RtIN ≈ Pt
INq(Ω). (35)
From (7) the asymptotic system EE achieved by the INEE-oriented design is expressed as (36), shown at the bottomof the next page. Since the normalized channel gain fol-lows Gamma distribution, the probability that gt exceedsg0 is Pr(gt ≥ g0) = Γ(mg,mgg0/Ω)/Γ(mg), where Γ(·, ·)is the incomplete Gamma function [36]. When Ω → ∞,we have
limΩ→∞
Pr(gt ≥ g0) =Γ(mg,0)Γ(mg)
= 1. (37)
Substituting (35) into (36) and using (37), the sys-tem EE can be approximated as (38), shown at thebottom of the next page. Further considering q(Ω) =
ξBΩe−W0((ξPcΩ−1)e−1)−1/ ln2= ξBΩe−W0−1/ ln2=η0 de-fined in (26), we have
limΩ→∞
ηIN
η0≈ 1. (39)
• ηUB vs. η0
To find the relationship between ηUB and η0, we firstderive the asymptotic expression of ηUB from (32).
Proposition 2: When the average normalized channel gainΩ → ∞,
limΩ→∞
ηUB = limΩ→∞
BPc ln2
W0
⎛⎝ξPcΩe∑
mg−1k=1
1k −1−ϒ
mg
⎞⎠ (40)
where ϒ is Euler-Mascheroni constant.Proof: See Appendix E. �
ηUB =E{
RtUB
}E{
PtUB
} =E{
RtUB|gt ≥ gUB
}Pr(gt ≥ gUB)+E
{Rt
UB|gt < gUB}
Pr(gt < gUB)
E{
PtUB|gt ≥ gUB
}Pr(gt ≥ gUB)+E
{Pt
UB|gt < gUB}
Pr(gt < gUB)(29)
408 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 2, FEBRUARY 2015
Then, from (40) and (27) with W0 =W0((ξPcΩ−1)e−1) weobtain,
limΩ→∞
ηUB
η0= lim
Ω→∞
ξPcΩ−1ξPcΩ
W0
(ξPcΩe∑
mg−1k=1
1k −1−ϒ
mg
)
W0 ((ξPcΩ−1)e−1). (41)
If there exist x1, x2 and x1 > x2 > 0, then considering themonotonically increasing feature of W0(x) [33], we can obtain
W0(x1)>W0(x2). (42)
With the definition of Lambert-W function [33], we haveW0(x1)eW0(x1)/(W0(x2)eW0(x2)) = x1/x2. Further with (42), wecan derive that eW0(x1)−W0(x2) < x1/x2, from which we have
W0(x1)<W0(x2)+ lnx1
x2. (43)
From (42) and (43), we know that |W0(x1) − W0(x2)|is bounded by | ln(x1/x2)|, ∀x1,x2 > 0. By setting x1 =
ξPcΩe∑mg−1k=1
1k −1−ϒ/mg and x2 = (ξPcΩ−1)e−1, when Ω → ∞,
W0(x1), W0(x2)→ ∞. Furthermore, we have
|W0(x1)−W0(x2)|<∣∣∣∣ln ξPcΩ
ξPcΩ−1+∑mg−1
k=1
1k−ϒ−lnmg
∣∣∣∣ (44)
which is bounded by |∑mg−1k=1
1k −ϒ− lnmg| when Ω → ∞. Thus,
limΩ→∞ W0(x1)/W0(x2) = 1, and (41) becomes
limΩ→∞
ηUB
η0= 1. (45)
From (39) and (45) we see that both the INEE- and UB-orienteddesigns perform close to η0 when Ω → ∞.
2) Comparison in Low SNR Region (Ω → 0): Again, wefirst derive the asymptotic expression of ηUB from (32). Inthis scenario, even the asymptotic expression is hard to derivefor general values of mg. In the sequel, we consider Rayleighfading channel and the number of transmit antennas Nt = 1,which corresponds to an extreme case with most severe channelfluctuation, i.e., mg = 1.
Proposition 3: When the average normalized channel gainΩ → 0, for Rayleigh fading channel we have
limΩ→0
ηUB|mg=1 = limΩ→0
3ξBΩln2
W0
(1
3 3√
ξPcΩ
). (46)
Proof: See Appendix F. �From (28), we know that ηIN ≤ η0. Then, we have
limΩ→0
ηUB|mg=1
ηIN|mg=1≥ lim
Ω→0
ηUB|mg=1
η0. (47)
Considering (46) and (26), we can derive the lower bound ofthe EE gain as
limΩ→0
ηUB|mg=1
η0= lim
Ω→03e1+W0((ξPcΩ−1)e−1)W0
(1
3 3√
ξPcΩ
).
(48)
Since limΩ→0 W0((ξPcΩ− 1)e−1) = −1 and limΩ→0 W0(1/(3 3√
ξPcΩ)) → ∞ [33], the EE gain of the UB-oriented designover the INEE-oriented design is
limΩ→0
ηUB|mg=1
ηIN|mg=1≥ lim
Ω→0
ηUB|mg=1
η0→ ∞. (49)
3) Summary: From these asymptotic analysis we can obtainthe following conclusions.
• The UB-oriented design has no EE gain over the INEE-oriented design in static channels, and in block fadingchannels when the average SNR is very high.
• The UB-oriented design has large EE gain in block fadingchannels when the average SNR is low and the fading issevere.
The analysis in this section implies that ACEE is superior toINEE for the system serving best effort traffic in the low SNRscenarios with severe fading in time-varying channels.
V. NUMERICAL AND SIMULATION RESULTS
In this section, we first validate the analytical analysis forsingle user MIMO-OFDM systems under block fading chan-nels, and then show how the analysis translates to multi-usersystems under more realistic time-varying channels.
limΩ→∞
ηIN = limΩ→∞
E{
RtIN|t ≥ g0
}Pr(gt ≥ g0)+E
{Rt
IN|gt < g0}
Pr(gt < g0)
E{
PtIN|gt ≥ g0
}Pr(gt ≥ g0)+E
{Pt
IN|gt < g0}
Pr(gt < g0)(36)
limΩ→∞
ηIN ≈ limΩ→∞
E{
PtINq(Ω)|gt ≥ g0
}Pr(gt ≥ g0)+E
{Rt
IN|gt < g0}
Pr(gt < g0)
E{
PtIN|gt ≥ g0
}Pr(gt ≥ g0)+E
{Pt
IN|gt < g0}
Pr(gt < g0)
= limΩ→∞
q(Ω)E{
PtIN|gt ≥ g0
}Pr(gt ≥ g0)+E
{Rt
IN|gt < g0}(1−Pr(gt ≥ g0))
E{
PtIN|gt ≥ g0
}Pr(gt ≥ g0)+E
{Pt
IN|gt < g0}(1−Pr(gt ≥ g0))
= limΩ→∞
q(Ω) (38)
YAO et al.: IS ACCUMULATIVE INFORMATION USEFUL FOR DESIGNING ENERGY EFFICIENT TRANSMISSION? 409
TABLE IPARAMETERS IN SIMULATION
When there are multiple users in the system, time divi-sion multiple access (TDMA), orthogonal frequency divisionmultiple access (OFDMA) and space division multiple access(SDMA), can be applied. To show the impact of exploitingaccumulative information on different systems, we comparethe system EEs achieved by UB- ACEE- and INEE-orienteddesigns for these three systems. For a fair comparison withTDMA and OFDMA, we employ zero-forcing beamforming(ZFBF) to ensure multi-user interference-free in SDMA. Then,when serving best effort traffic, the optimal solutions for theINEE-, ACEE- and UB-oriented designs derived for single-usersystem can be easily extended to the three kinds of multi-usersystems. When serving real-time traffic, the solutions for singleuser systems can be immediately applied for TDMA system,and can also be applied for OFDMA and SDMA systems byintroducing a strict fairness constraint in data rate among Nu
users, i.e., Rt1 = · · ·= Rt
Nu≥ R0 as in [37].
We consider a micro cell system with cell radius r = 100 m,the main simulation parameters are listed in Table I, which arefrom [27], [38]. The system EE is computed with (5) under onechannel realization in the evaluation duration T . To connect ouranalysis with practical systems, each subcarrier corresponds toa physical resource block (PRB) in LTE. To show the impact ofchannel feature, we consider that the channels between differentsubcarriers are either i.i.d. (i.e., an extreme of frequency selec-tive channel) or remaining constant (i.e., flat fading channel),and channels are spatially uncorrelated. The cell-edge SNR isset as 5 dB, which is the SNR received at distance r when theBS transmits with the maximum power, reflecting both noiseand inter-cell interference. All these settings will be used in thesequel unless otherwise specified.
A. Validation of Analysis and EE Gain in Single User Systems
To validate the analysis, we simulate the system EE for asingle user MIMO-OFDM system under block fading channels,where the location of the user is fixed in a distance d from theBS and shadowing is not considered.
In Fig. 2, we provide the system EEs achieved by the threerate allocation strategies when the evaluation duration variesfrom one to 1000 time slots as well as the correspondingdata rate allocated in each time slot. Fig. 2(a) shows that theACEE-oriented design almost performs the same as the UB-oriented design, even when T is not large, and the system EEof the INEE-oriented design is lower. To observe the differencemore clearly, in Fig. 2(b) we re-order the allocated results asthe value of gt increases. It is shown that for all R0, the rateallocated by the ACEE-oriented design is very close to that bythe UB-oriented design, as implied by Proposition 1. The UB-oriented design transmits with higher data rate than the INEE-
Fig. 2. (a) System EE versus evaluation duration under different minimal raterequirement R0 and (b) allocated data rate Rt versus normalized channel gain gt
and cumulative distribution function (CDF) of gt , Nt = 2, K = 1 and d = 75 m.
oriented design when gt > gτ0 , which verifies Corollary 1. Withthe increase of R0, the probability of Rt = R0 increases. Thisindicates that when the QoS requirement is high, no matterwhich objective function is applied, the allocated rate is almostequal to R0, leading to the same system EE.
In Fig. 2, we compare the simulated system EE evalu-ated over 1000 time slots with the asymptotic analysis underRayleigh fading (i.e., m = 1) at different SNRs. It is shown thatthe asymptotic EE upper-bound for Ω→∞ (computed with (40)by setting m = 1) and Ω → 0 (computed with (46)) are tight inhigh and low SNRs, respectively. The simulated system EE ofthe INEE-oriented design under block fading is always lowerthan the system EE under static channel, i.e., η0 computed with(26). The simulated system EE of the ACEE-oriented design isoverlapped with the simulated EE of the UB-oriented design,both are higher than η0 in low SNR and lower than η0 inhigh SNR. This indicates at high SNR channel fading degradesthe system EE, but at low SNR exploiting channel fading isbeneficial to EE.
410 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 2, FEBRUARY 2015
Fig. 3. (a) Asymptotic and simulated system EE η versus cell-edge SNR whenm = 1 and (b) simulated system EE versus m, Nt = 1, K = 1, d = 100 m, andR0 = 0 Mbps.
In Fig. 3(b), we further show the simulated system EEsversus channel parameter m in high and low SNR levels. Wecan see that the system EEs obtained by the ACEE- and INEE-oriented designs approach η0 and their gap diminishes as mincreases, which again validates previous analysis.
B. Evaluation of EE for Multi-User MIMO-OFDM Systems:Best Effort Traffic
To show how the analytical results for single user systemtranslates to multi-user systems, we simulate the system EEsachieved by MIMO-OFDM systems serving multiple userswith time division (one user is randomly scheduled in eachtime slot), frequency division (multiple users are randomlyscheduled in different frequency bands and equal number ofsubcarriers are allocated to multiple users aimed at maximizingthe sum rate) and space division (multiple users are randomlyscheduled and served with ZFBF in each time slot) multipleaccess. Ten users are distributed uniformly in the cell. Again,in this subsection we consider block fading channels, where the
Fig. 4. System EEs achieved by TDMA, OFDMA and SDMA systems. ForOFDMA and SDMA systems, two users with best effort traffic are served ineach time slot, where Nt = 2, K = 10 PRBs, and cell-edge SNR is 5 dB.
location of the user is fixed in a distance d from the BS but nowshadowing is considered in the simulations.
To observe the impact of using ACEE or INEE for opti-mization in three multi-access systems, in Fig. 4 we providethe system EEs achieved by three multi-access systems un-der frequency selective and flat fading channels. The ACEE-oriented design performs almost the same as the UB-orienteddesign, and both have a remarkable gain over the INEE-orienteddesign, especially under flat fading channels and in TDMA andSDMA systems. This is because large channel fluctuation inTDMA and low SNR in SDMA owing to ZFBF enlarge the gainfrom using the accumulative information as indicated in theanalysis for single user systems, while in frequency selectivefading channels and OFDMA systems the channel fluctuationis small. These results indicate that the same as in single usersystem, in multi-user systems the ACEE-oriented design hasan EE gain over the INEE-oriented design. Yet the EE gainsin different multi-user systems differ: accumulative informationplays a more important role in TDMA and SDMA systems thanthat in OFDMA system.
In Fig. 5, we provide the system EEs achieved by SDMA sys-tem with different antenna and by OFDMA system with differ-ent bandwidth configurations. With more transmit antennas, thegap between the two designs diminishes owing to higher SNRfrom array gain, and the EEs reduce due to the increased circuitpower consumption. With more users served simultaneously,the gap also decreases since channel fluctuations reduce, butthe system EEs increase thanks to the multi-user diversity gain.With more PRBs, both the system EEs and the gap increase.
C. Evaluation of EE for Multi-User MIMO-OFDM Systems:Real-Time Traffic
Finally, we consider a more realistic scenario. In the sim-ulation, whenever a user with real-time traffic is accessed tothe BS, its QoS should be ensured in all time slots during its
YAO et al.: IS ACCUMULATIVE INFORMATION USEFUL FOR DESIGNING ENERGY EFFICIENT TRANSMISSION? 411
Fig. 5. (a) System EEs vs. number of transmit antennas for SDMA systemwith K = 10 PRBs, and (b) number of PRBs for OFDMA system with Nt = 2.Both systems serve two or four users with best effort traffic in each time slotunder flat fading channels, where cell-edge SNR is 5 dB.
connection. In the following, we consider an OFDMA systemserving two users simultaneously (the results for SDMA sys-tem are similar), both have a minimal rate requirement R0 =3 Mbps. To show the impact of user mobility on the perfor-mance, we provide four typical cases. We consider flat fadingchannels that evolve in time following Jakes Model, whichis widely applied in various standardization organizations. Itstemporal correlation function is Rh(τ) = J0(2π fdτ), where J0(·)is the zero-th order Bessel function of the first kind, and fd isthe Doppler spread. The carrier frequency is 2 GHz, one timeslot corresponds to 5 ms and the speed of the users is 3, 60,120 km/h. Since shadowing is a location-dependent randomvariable, its spatial correlation is modeled to be exponentiallydecayed with the distance as in [39] and [40]. When a usermoves with 3 km/h, its location and shadowing almost do notchange. In this simulation set-up, the channels are not blockfading any more.
Fig. 6. (a) System EEs and (b) CDF of the transmit power for an OFDMAsystem with two users served simultaneously (from which we can see that theoutage probability due to the constraint on Pmax is less than 3%), R0 = 3 Mbps,Nt = 4 and K = 20 PRBs. Case 1: both users move towards BS from d = 100 min a speed of 60 km/h and are connected in 1000 time slots. Case 2: both usersmove away from BS to 100 m in 60 km/h and are connected in 1000 timeslots. Case 3: user 1 moves to BS in 60 km/h from 100 m and is connected in1000 time slots, while user 2 in 3 km/h is connected with 100 time slots. Toensure there always exist two active users during evaluation, user 2 is randomlyscheduled from the user pool after each 100 time slots. Case 4: user 1 movestowards BS from d = 100 m in a speed of 120 km/h and are connected in 1000time slots. Another user is the same as Case 3.
In Fig. 6(a), we show the system EEs for the four cases.In Cases 1 and 2, the ACEE-oriented design do not performclose to the UB-oriented design as in previous results. Tounderstand this result, we provide the corresponding CDF ofthe allocated overall transmit power pt in Fig. 6(b). In Case 1the accumulative information used in ACEE provides an op-timistic “prediction” on future channel condition that leads toan aggressive transmission strategy, and in Case 2 just on theopposite where the pessimistic expectation leads to a conserva-tive transmission strategy. In both cases, the channels amongmultiple time slots during evaluation are no longer ergodic,and therefore the accumulative information exploited by the
412 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 2, FEBRUARY 2015
ACEE-oriented design can not reflect future channel condition.In Case 3, the ACEE-oriented design still outperforms theINEE-oriented design remarkably and performs closely to theUB-oriented design. This is because the second user simultane-ously served with the first user is randomly scheduled, whosechannel fluctuation plays a dominant role in the accumulativeinformation. In Case 4 where user 1 is the same as the firstuser in Case 1 except that the moving speed is much higher, theACEE-oriented is inferior to the INEE-oriented design, owingto the same reason as explained for Case 1.
VI. CONCLUSION
In this paper, we answered the question when accumulativeinformation is useful for EE-oriented transmit strategy opti-mization. By taking a downlink single user MIMO-OFDMsystem as an example and considering best effort and real-timetraffic, we formulated the INEE-, ACEE- and UB-maximizationproblems in a unified framework and provided the correspond-ing solutions, which can be extended into multi-user scenariosusing TDMA, OFDMA and SDMA with ZFBF. We analyzedthe behavior of the optimal rate allocation from the three ob-jective functions, which shows that the ACEE-oriented designbehaves similar to the UB-oriented design when channels areergodic. We derived the closed-form expressions of the systemEEs achieved by the INEE- and UB-oriented designs for singleuser system under various channels in high and low SNR levels,resorting to the asymptotic analysis. The analysis shows thatthe EE gain provided by using accumulative information comesfrom exploiting a kind of “channel diversity” more aggres-sively, which is more remarkable in time-varying channels withsevere fluctuation or at low SNR. Simulation results validatedanalytical analysis and shown that when serving best efforttraffic, for the systems with less transmit antennas, servingless users in the same time-frequency resource simultaneouslyor using wider bandwidth for multiple users, exploiting theaccumulative information can provide large gain. When servingreal-time traffic, accumulative information is not always benefi-cial for improving system EE, especially when the minimal raterequirement is high and when future channels of the users areneither predicted nor ergodic.
APPENDIX APROOF OF PROPOSITION 1
From (17), we have W0((ξgtPc − 1)e−1) = RtIN ln2/B − 1
when gt ≥ gtIN, t = t1, t2. Then, considering the definition of
Lambert-W function [35] (i.e., x = W0(x)eW0(x)), we have
(ξgtPc − 1)e−1 = (RtIN ln2/B − 1)e
RtIN ln2
B −1, from which weobtain
gt =
(Rt
IN ln2−B)
2Rt
INB +B
ξBPc. (A.1)
When the normalized channel gain gt is high enough suchthat (ξgtPc − 1)e−1 0, from the monotonic feature of theLambert-W function [33], we know W0 ((ξgtPc − 1)e−1) =
RtIN ln2/B− 1 0, i.e., Rt
IN ln2 B. Then, (RtIN ln2)2
RtINB
B and hence (RtIN ln2−B)2
RtINB +B ≈ (Rt
IN ln2)2Rt
INB . Conse-
quently, (A.1) be approximated as gt ≈ (RtIN ln2)2
RtINB /(ξBPc).
Then, the relation between Rt1IN and Rt2
IN can be approximated as
(Rt1
IN ln2)
2R
t1INB
ξBPc:
(Rt2
IN ln2)
2R
t2INB
ξBPc≈ gt1 : gt2 (A.2)
which is exactly (19).
From (18), we have 2Rt
UBB = 2
R0B gt/gUB when gt ≥ gUB, t =
t1, t2. As a result, the relation between Rt1UB and Rt2
UB satisfies
2R
t1UBB : 2
Rt2UBB =
2R0B gt1
gUB:
2R0B gt2
gUB= gt1 : gt2 (A.3)
which is (20).From (16) we have
W0(ϕt
AC
)=(
RtAC +∑t−1
τ=1 RτAC
) ln2B
−1 (A.4)
when gt ≥ gtAC, t = t1, t2, where ϕt
AC = (ξgt ∑t−1τ=1 Pτ
AC +ξgtPc −1)e
ln2B ∑t−1
τ=1 RτAC−1.
From the definition of Lambert-W function and PtAC =
(2Rt
ACB − 1)/(ξgt) + Pc obtained from the power consumption
in dominator of (11a) with Lt = 1, (A.4) can be rewritten
as ∑tτ=1 Pτ
AC − 2Rt
ACB ln2ξBgt ∑t
τ=1 RτAC = 0, from which we obtain
gt = 2Rt
ACB ln2ξB
∑tτ=1 Rτ
AC∑t
τ=1 PτAC
. As a result, the relation between Rt1AC and
Rt2AC satisfies
2R
t1ACB
∑t1τ=1 Rτ
AC
∑t1τ=1 Pτ
AC
: 2R
t2ACB
∑t2τ=1 Rτ
AC
∑t2τ=1 Pτ
AC
= gt1 : gt2 . (A.5)
Considering t1 > t2, we have ∑t1τ=1 Rτ
AC
∑t1τ=1 Pτ
AC
=∑
t2τ=1 Rτ
AC+∑t1−t2τ=1 R
τ+t2AC
∑t2τ=1 Pτ
AC+∑t1−t2τ=1 P
τ+t2AC
.
When t2 is very large but t1 − t2 is small, we have∑t2
τ=1 RτAC ∑t1−t2
τ=1 Rτ+t2AC and ∑t2
τ=1 PτAC ∑t1−t2
τ=1 Pτ+t2AC . Con-
sequently, ∑t1τ=1 Rτ
AC/∑t1τ=1 Pτ
AC ≈ ∑t2τ=1 Rτ
AC/∑t2τ=1 Pτ
AC. Uponsubstituting into (A.5), we obtain (21).
APPENDIX BPROOF OF COROLLARY 1
When gt ≥ max{gtIN,gUB}, to find ΔRt , we first employ
the approximation expressions in Proposition 1 to find therelationship between Rt
IN and RtUB. From (17) and (18) we can
see that RtIN and Rt
UB are respectively Lambert-W function andlogarithm function of gt when gt exceeds the threshold. Thissuggests that there exists a specific gτ0 such that Rτ0
IN = Rτ0UB.
Let t1 = t and t2 = τ0, from (19) and (20) in Proposition 1,we have
RtIN2
RtINB : Rτ0
IN2R
τ0INB ≈ 2
RtUBB : 2
Rτ0UBB . (B.1)
YAO et al.: IS ACCUMULATIVE INFORMATION USEFUL FOR DESIGNING ENERGY EFFICIENT TRANSMISSION? 413
Considering Rτ0IN = Rτ0
UB, (B.1) reduces to 2Rt
INB : 2
RtUBB ≈ Rτ0
IN :Rt
IN, which can be rewritten as
RtIN
B− Rt
UB
B≈ log2 Rτ0
IN − log2 RtIN. (B.2)
Based on the definition of ΔRt in Corollary 1, we obtain (22).Since W0(x) is a monotonic increasing function of x [33],
from (17) we know that RtIN is also an increasing function of
gt . Therefore, if gt > gτ0 , we have RtIN > Rτ0
IN, and vice versa.Substituting into (22), we obtain (23).
APPENDIX CPROOF OF THE CONCAVITY OF Rt
INAND THE CONVEXITY OF Pt
IN
Since the Lambert-W function is a concave function [33],from (17) it is easy to see that Rt
IN is a concave function of gt .To prove that Pt
IN in (25) is a concave function of gt , we showthat ∂2Pt
IN/∂(gt)2 > 0.Considering that the first order derivative of W0(x) is
∂W0(x)/∂x= e−W0(x)/(1+W0(x)) [33], the first order derivativeof Pt
IN can be derived as
∂PtIN
∂gt =1
ξ(gt)2
2+W0(ϕt
IN
)− e1+W0(ϕt
IN)
1+W0(ϕt
IN
) (C.1)
where ϕtIN = (ξPcgt − 1)e−1. Considering that W0(ϕt
IN) > −1[33] and the numerator in (C.1) is a decreasing function ofW0(ϕt
IN) because its derivative with respect to W0(ϕtIN)
is 1 − e1+W0(ϕtIN) < 0, the numerator 2 + W0(ϕt
IN) −e1+W0(ϕt
IN) < limW0(ϕtIN)→−1(2 + W0(ϕt
IN) − e1+W0(ϕtIN)) = 0.
Therefore, ∂PtIN/∂gt < 0. The second order derivative of Pt
INcan be derived as
∂2PtIN
∂(gt)2 =G(W0(ϕt
IN))
ξ(gt)3(1+W0(ϕt
IN))3 (C.2)
where G(x) = (x2 +4x+2)ex+1 −2(1+ x)3 −2(1+ x)2 −2x−e−x−1,∀x >−1.
Because ∂4G/∂x4 = (x2 + 12x + 30)e1+x − e−1−x > 0 and∂3G/∂x3|x→−1 = 0, we have ∂3G/∂x3 > ∂3G/∂x3|x→−1 = 0.Because ∂2G/∂x2|x→−1 = ∂G/∂x|x→−1 = 0, we have G(x) >limx→−1 G(x) = 0,∀x > −1. Therefore, the numerator of (C.2)exceeds zero. Further considering that the denominator of (C.2)is positive, we have ∂2Pt
IN/∂(gt)2 > 0,∀W0(ϕtIN)>−1.
APPENDIX DSIMPLIFYING (31) WHEN mg IS AN INTEGER
Considering (3) and by defining I1Δ=
∫ ∞gUB
(gt)mg−1e−mggt
Ω dgt ,
I2Δ= gUB
∫ ∞gUB
(gt)mg−2e−mggt
Ω dgt , and I3Δ=
∫ ∞gUB
ln( gt
gUB)(gt)mg−1e−
mggt
Ω dgt , (31) can be rewritten as
mmgg
Ωmg
1Γ(mg)
(I1 − I2 − I3)+gUBξPc = 0. (D.1)
From the definition of the incomplete gamma function [36],we can obtain I1 and I2 as
I1 =(mg
Ω
)−mgΓ(
mg,mggUB
Ω
),
I2 =gUB
(mg
Ω
)−mg+1Γ(
mg −1,mggUB
Ω
)(D.2)
where Γ(·, ·) is the incomplete Gamma function.Considering that mg is an integer and letting
x = mggt/Ω, then I3 can be rewritten as I3 =
( Ωmg
)mg∫ ∞
mggUBΩ
ln( xmggUB
Ω)xmg−1e−xdx, which can be derived
from integration by parts [35] as
I3 = e−mggUB
Ω
mg
∑i=2
mg−i
∑k=0
(mggUB
Ω
)k 1(mg − i+1)k!
+ E1
(mggUB
Ω
)(D.3)
where E1(·) is the exponential integration function [35]. Substi-tuting (D.2) and (D.3) into (D.1), we have v(z,mg)+gUBξPc =
0, where zΔ= mggUB/Ω and
v(z,mg) =1
Γ(mg)Γ(mg,z)−
1Γ(mg)
zΓ(mg −1,z)
− e−zmg
∑i=2
mg−i
∑k=0
zk 1(mg − i+1)k!
−E1(z). (D.4)
When mg is an integer, from [35] the incomplete gammafunction Γ(mg,z) can be written as
Γ(mg,z) =
{E1(z), mg = 0
(mg −1)!e−z ∑mg−1k=0
zk
k! , mg > 0.(D.5)
Considering Γ(mg) = (mg − 1)! and substituting (D.5) into(D.4), we obtain (33).
APPENDIX EPROOF OF PROPOSITION 2
Considering gUB = ln2ηUB/(ξB) when R0 = 0, (40) inProposition 2 can be rewritten as
limΩ→∞
gUB = limΩ→∞
1ξPc
W0
⎛⎝ξPcΩe∑
mg−1k=1
1k −1−ϒ
mg
⎞⎠ . (E.1)
With the definition of Lambert-W function, (E.1) can be
rewritten as limΩ→∞−e∑mg−1k=1
1k −1−ϒ +
mggUBΩ egUBξPc = 0, which
is equivalent to
limΩ→∞
(1+ϒ−
mg−1
∑k=1
1k+ ln
mggUB
Ω+gUBξPc
)= 0. (E.2)
From (32), we know that v(z,mg) ∈ (−∞,0) since gUBξPc ∈(0,∞). It is not hard to find ∂v(z,mg)/∂z > 0,∀z,mg > 0from (33), which indicates that v(z,mg) is monotonically in-creasing with z. Then for a fixed mg, (32) can be rewrittenas z = v−1(−ξPcgUB), where v−1(·) is the inverse functionof v(·). Considering the monotonic feature of v−1(·), we
414 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 2, FEBRUARY 2015
know limx→−∞ v−1(x) = 0. Therefore, with limΩ→∞ gUB → ∞,we obtain limΩ→∞ z = v−1(−ξPc limΩ→∞ gUB) = 0. In a sim-ilar way, we can prove that limΩ→0 z → ∞. Then, (E.2) isequivalent to
limz→0
(1+ϒ−∑mg−1
k=1
1k+ lnz+gUBξPc
)= 0. (E.3)
In the following, we prove the proposition by showing that(E.3) holds.
From (33), v(z,mg) is a summation of finite series andexponential integration function. Since z = mggUB/Ω is a realnumber, E1(z) can be expressed by convergent series [41],
which is E1(z) = −ϒ − lnz + ∑∞k=1
(−1)k+1zk
kk! , from which wehave
limz→0
E1(z) = limz→0
(−ϒ− lnz). (E.4)
Substituting (E.4) into (33), we obtain
limz→0
v(z,mg)
=
{limz→0(1+ϒ+ lnz), mg = 1
limz→0
(1−∑
mg−1k=1
1k +ϒ+ lnz
), mg > 1
(E.5)
i.e., limz→0 v(z,mg) = limz→0(1−∑mg−1k=1
1k +ϒ+ lnz) holds for
mg ≥ 1.From (E.5) and (32), it is easy to see that (E.3) holds, i.e.,
Proposition 2 holds.
APPENDIX FPROOF OF PROPOSITION 3
Considering gUB = ln2ηUB/(ξB) when R0 = 0, (46) can berewritten as
limΩ→0
gUB = limΩ→0
3ΩW0
(1
3 3√
ξPcΩ
). (F.1)
By using the definition of Lambert-W function, (F.1) can berewritten as
limΩ→0
egUB3Ω
(gUB
Ω
)3√
ξPcΩ−1 = 0 (F.2)
which is equivalent to
limΩ→0
egUB
Ω(gUB
Ω
)3ΩξPc −1
= limΩ→0
egUB
Ω(gUB
Ω
)2gUBξPc −1 = 0. (F.3)
From the analysis on (32), we have limΩ→0 z → ∞. Whenmg = 1, z = mggUB/Ω = gUB/Ω. Then (F.3) can be rewritten as
limz→∞
z2ezgUBξPc −1 = 0. (F.4)
In the following, we prove the proposition by showing that (F.4)holds.
The expansion of E1(z) used in Appendix E cannot help here,because the expansion is unlimited when z → ∞. Fortunately,
we can obtain the limit of E1(z) when z → ∞ from the results of[42] as follows,
limz→∞
E1(z) = limz→∞
e−z
z
∞
∑k=0
k!(−z)k . (F.5)
Since z > 0, we rewrite (32) by multiplying z2ez andsubstituting (F.5) as limz→∞(z2 − z2ez(1 + z) e−z
z ∑∞k=0
k!(−z)k +
z2ezgUBξPc) = limz→∞(−1−∑∞k=1
(k+1)(k+1)!(−z)k + z2ezgUBξPc) =
0. Considering limz→∞ ∑∞k=1(k+ 1)(k+ 1)!/(−z)k = 0, we ob-
tain (F.4), i.e., the proposition holds.
REFERENCES
[1] S. Tombaz, A. Vastberg, and J. Zander, “Energy- and cost-efficient ultra-high-capacity wireless access,” IEEE Wireless Commun., vol. 18, no. 5,pp. 18–24, Oct. 2011.
[2] Z. Hasan, H. Boostanimehr, and V. K. Bhargava, “Green cellular net-works: A survey, some research issues and challenges,” IEEE Commun.Surveys Tuts., vol. 13, no. 4, pp. 524–540, 2011.
[3] G. Miao and J. Zhang, “On optimal energy-efficient multi-user MIMO,”in Proc. IEEE GLOBECOM, 2011, pp. 1–6.
[4] C. Xiong, G. Y. Li, S. Zhang, Y. Chen, and S. Xu, “Energy- andspectral-efficiency tradeoff in downlink OFDMA networks,” IEEE Trans.Commun., vol. 10, no. 11, pp. 3874–3886, Nov. 2011.
[5] D. Ng, E. Lo, and R. Schober, “Energy-efficient resource allocation inOFDMA systems with large numbers of base station antennas,” IEEETrans. Wireless Commun., vol. 11, no. 9, pp. 3292–3304, Sep. 2012.
[6] C. Isheden, Z. Chong, E. Jorswieck, and G. Fettweis, “Frameworkfor link-level energy efficiency optimization with informed transmit-ter,” IEEE Trans. Wireless Commun., vol. 11, no. 8, pp. 2946–2957,Aug. 2012.
[7] “Environmental Engineering (EE), Energy Efficiency Of Wireless AccessNetwork Equipment,” ETSI TS 102.706, Aug. 2009.
[8] G. Auer et al., “D 2.3: Energy efficiency analysis of the refer-ence systems, areas of improvements and target breakdown,” EARTH,Nov. 2010. [Online]. Available: https://www.ict-earth.eu/publications/deliverables/deliverables.html
[9] L. Xiang, X. Ge, C.-X. Wang, F. Y. Li, and F. Reichert, “Energy efficiencyevaluation of cellular networks based on spatial distributions of traffic loadand power consumption,” IEEE Trans. Wireless Commun., vol. 12, no. 3,pp. 961–973, Mar. 2013.
[10] F. Heliot, M. Imran, and R. Tafazolli, “Low-complexity energy-efficientresource allocation for the downlink of cellular systems,” IEEE Trans.Commun., vol. 61, no. 6, pp. 2271–2281, Jun. 2013.
[11] A. Akbari, F. Héliot, M. A. Imran, and R. Tafazolli, “Energy efficiencycontours for broadcast channels using realistic power models,” IEEETrans. Wireless Commun., vol. 11, no. 11, pp. 4017–4025, Nov. 2012.
[12] J. Xu and L. Qiu, “Energy efficiency optimization for MIMO broadcastchannels,” IEEE Trans. Wireless Commun., vol. 12, no. 2, pp. 690–701,Feb. 2013.
[13] C. He, B. Sheng, P. Zhu, X. You, and G. Y. Li, “Energy-and spectral-efficiency tradeoff for distributed antenna systems with proportionalfairness,” IEEE J. Sel. Areas Commun., vol. 31, no. 5, pp. 894–902,May 2013.
[14] J. Jiang, M. Dianati, M. Imran, R. Tafazolli, and Y. Chen, “On the relationbetween energy efficiency and spectral efficiency of multiple-antennasystems,” IEEE Trans. Veh. Technol., vol. 62, no. 7, pp. 3463–3469,Sep. 2013.
[15] D. Ng, E. Lo, and R. Schober, “Energy-efficient resource allocation inmulti-cell OFDMA systems with limited backhaul capacity,” IEEE Trans.Wireless Commun., vol. 11, no. 10, pp. 3618–3631, Oct. 2012.
[16] S. He, Y. Huang, S. Jin, and L. Yang, “Coordinated beamforming forenergy efficient transmission in multicell multiuser systems,” IEEE Trans.Commun., vol. 61, no. 12, pp. 4961–4971, Dec. 2013.
[17] L. Zhang, H.-C. Yang, and M. Hasna, “Generalized area spectral effi-ciency: An effective performance metric for green wireless communica-tions,” IEEE Trans. Commun., vol. 62, no. 2, pp. 747–757, Feb. 2014.
[18] Y. S. Soh, T. Q. Quek, M. Kountouris, and H. Shin, “Energy efficientheterogeneous cellular networks,” IEEE J. Sel. Areas Commun., vol. 31,no. 5, pp. 840–850, May 2013.
YAO et al.: IS ACCUMULATIVE INFORMATION USEFUL FOR DESIGNING ENERGY EFFICIENT TRANSMISSION? 415
[19] H. Tabassum, M. Z. Shakir, and M. Alouini, “Area Green Efficiency(AGE) of two tier heterogeneous cellular networks,” in Proc. IEEEGLOBECOM Workshops, 2012, pp. 529–534.
[20] G. Miao, N. Himayat, G. Y. Li, and S. Talwar, “Low-complexity energy-efficient scheduling for uplink OFDMA,” IEEE Trans. Commun., vol. 60,no. 1, pp. 112–120, Jan. 2012.
[21] L. Liu, G. Miao, and J. Zhang, “Energy-efficient scheduling for downlinkmulti-user MIMO,” in Proc. IEEE ICC, 2012, pp. 4390–4394.
[22] A. Zappone, P. Cao, and E. Jorswieck, “Energy efficiency optimizationin relay-assisted MIMO systems with perfect and statistical CSI,” IEEETrans. Signal Process., vol. 62, no. 2, pp. 443–457, Jan. 2014.
[23] Y. Rui, Q. Zhang, L. Deng, P. Cheng, and M. Li, “Mode selectionand power optimization for energy efficiency in uplink virtual MIMOsystems,” IEEE J. Sel. Areas Commun., vol. 31, no. 5, pp. 926–936,May 2013.
[24] X. R. Li and Z. Zhao, “Evaluation of estimation algorithms part I: In-comprehensive measures of performance,” IEEE Trans. Aerosp. Electron.Syst., vol. 42, no. 4, pp. 1340–1358, Oct. 2006.
[25] Z. Xu et al., “Energy-efficient configuration of spatial and frequencyresources in MIMO-OFDMA systems,” IEEE Trans. Commun., vol. 61,no. 2, pp. 564–575, Feb. 2013.
[26] Y. Wang, J. Zhang, and P. Zhang, “Low-complexity energy-efficient powerand subcarrier allocation in cooperative networks,” IEEE Commun. Lett.,vol. 17, no. 10, pp. 1944–1947, Oct. 2013.
[27] G. Auer, V. Giannini, C. Desset, and I. Godor, “How much energy isneeded to run a wireless network?” IEEE Wireless Commun., vol. 18,no. 5, pp. 40–49, Oct. 2011.
[28] O. Onireti, F. Heliot, and M. Imran, “On the energy efficiency-spectral ef-ficiency trade-off of distributed MIMO systems,” IEEE Trans. Commun.,vol. 60, no. 5, pp. 3741–3753, Sep. 2013.
[29] A. Papoulis and S. U. Pillai, Probability, Random Variables and StochasticProcesses With Errata Sheet. New York, NY, USA: McGraw-Hill, 2002.
[30] C.-F. Tsai, C.-J. Chang, F.-C. Ren, and C.-M. Yen, “Adaptive radio re-source allocation for downlink OFDMA/SDMA systems with multimediatraffic,” IEEE Trans. Wireless Commun., vol. 7, no. 5, pp. 1734–1743,May 2008.
[31] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.:Cambridge Univ. Press, 2004.
[32] C. Xiong, G. Li, S. Zhang, Y. Chen, and S. Xu, “Energy-efficient resourceallocation in OFDMA networks,” IEEE Trans. Commun., vol. 60, no. 12,pp. 3767–3778, Dec. 2012.
[33] R. Corless, G. Gonnet, D. Hare, D. Jeffrey, and D. Knuth, “On the LambertW function,” Adv. Comput. Math., vol. 5, no. 1, pp. 329–359, 1996.
[34] D. P. Palomar and J. R. Fonollosa, “Practical algorithms for a familyof waterfilling solutions,” IEEE Trans. Signal Process., vol. 53, no. 2,pp. 686–695, Feb. 2005.
[35] A. Jeffrey and D. Zwillinger, Table of Integrals, Series, Products.Amsterdam, The Netherlands: Elsevier, 2007.
[36] E. W. Weisstein, “Incomplete gamma function,” MathWorld, A WolframWeb Resource, 2006. [Online]. Available: http://mathworld.wolfram.com/IncompleteGammaFunction.html
[37] Z. Ren, S. Chen, B. Hu, and W. Ma, “Energy-efficient resource allocationin downlink OFDM wireless systems with proportional rate constraints,”IEEE Trans. Veh. Technol., vol. 63, no. 5, pp. 2139–2150, Jun. 2014.
[38] “Further advancements for E-UTRA physical layer aspects (Release 9),”Sophia-Antipolis, France, TR 36.814 V1.2.0, Jun. 2009.
[39] M. Gudmundson, “Correlation model for shadow fading in mobile radiosystems,” Electron. Lett., vol. 27, no. 23, pp. 2145–2146, Nov. 1991.
[40] K. J. Yang and Y. R. Tsai, “Location tracking in mobile networks undercorrelated shadowing effects,” in Proc. IEEE WCNC, 2009, pp. 1–5.
[41] S. Orszag and C. Bender, Advanced Mathematical Methods for Scientistsand Engineers. New York, NY, USA: McGraw-Hill, 1978.
[42] S. J. Malham, “An Introduction to Asymptotic Analysis.” Lecture Notes,Department of Mathematics, Heriot-Watt University, U.K., 2005. [On-line]. Available: www.macs.hw.ac.uk/~simonm/ae.pdf
Chuting Yao received the B. Eng. degree from theSchool of Advanced Engineering, Beihang Univer-sity (BUAA), Beijing, China, in 2011. She is nowpursuing the Ph.D. degree from the School of Elec-tronics and Information Engineering, BUAA. FromOctober 2014 to January 2015, she is a VisitingStudent with the Department of Electrical and Com-puter Engineering, Texas A&M University, CollegeStation, TX, USA. Her research interests lie in thearea of energy efficient.
Zhikun Xu received the B.S.E. degree in elec-tronics engineering and the Ph.D. degree in signaland information processing from Beihang Univer-sity (BUAA), Beijing, China, in 2007 and 2013,respectively. He had been a Visiting Researcher inthe School of Electrical and Computer Engineering,Georgia Institute of Technology, USA, from 2009 to2010. After graduation, he joined the Green Commu-nication Research Center (GCRC) of China MobileResearch Institute (CMRI) as a project manager. Hiscurrent research interests include green radio, cross-
layer resource allocation in cellular networks, and advanced signal processingand transmission techniques. In these areas, he has published more than 20papers in IEEE TRANSACTIONS and IEEE flagship conferences like ICC,GLOBECOM, PIMRC, VTC, etc.
Tingting Liu received the B.S. and Ph.D. degreesin signal and information processing from BeihangUniversity, Beijing, China, in 2004 and 2011, respec-tively. From December 2008 to January 2010, shewas a Visiting Student with the School of Electronicsand Computer Science, University of Southampton,Southampton, U.K. She is currently a Lecturer withthe School of Electronics and Information Engi-neering, Beihang University. Her research interestsinclude wireless communications and signal process-ing, the degrees of freedom (DoF) analysis and inter-
ference alignment transceiver design in interference channels, energy efficienttransmission strategy design and joint transceiver design for multicarrier andmultiple-input multiple-output communications. She received the awards of2012 Excellent Doctoral Thesis in Beijing and 2012 Excellent Doctoral Thesisin Beihang University.
Chenyang Yang received the Ph.D. degrees inelectrical engineering from Beihang University (for-merly Beijing University of Aeronautics and Astro-nautics, BUAA), China, in 1997. She has been aFull Professor with the School of Electronics andInformation Engineering, BUAA since 1999. She haspublished over 200 international journal articles andconference papers and filed over 60 patents in thefields of energy efficient transmission, CoMP, inter-ference management, cognitive radio, relay, etc. Shewas nominated as an Outstanding Young Professor
of Beijing in 1995 and was supported by the 1st Teaching and Research AwardProgram for Outstanding Young Teachers of Higher Education Institutionsby Ministry of the Education during 1999–2004. She was the Chair of theBeijing chapter of IEEE Communications Society during 2008–2012, and theMDC Chair of APB of the IEEE Communications Society during 2011–2013.She has served as Technical Program Committee Member for a numerousIEEE conferences, and was the publication chair of IEEE ICCC 2012 andSpecial Session Chair of IEEE ChinaSIP 2013. She has served as an AssociateEditor for IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS during2009–2014, and a Guest Editor for IEEE JOURNAL OF SELECTED TOPICS IN
SIGNAL PROCESSING to be published in February 2015. She is now a GuestEditor for IEEE JOURNAL OF SELECTED AREAS IN COMMUNICATIONS, anAssociate Editor-in-Chief of Chinese Journal of Communications and ChineseJournal of Signal Processing. Her recent research interests include greencommunications and interference control for 5G wireless systems.
