Content uploaded by Shunqing Zhang
Author content
All content in this area was uploaded by Shunqing Zhang
Content may be subject to copyright.
Energy- and Spectral-Efficiency Tradeoff in
Downlink OFDMA Networks∗
Cong Xiong†, Geoffrey Ye Li†, Shunqing Zhang‡, Yan Chen‡, and Shugong Xu‡
†School of ECE, Georgia Institute of Technology, Atlanta, GA, USA
‡GREAT Research Team, Huawei Technologies Co., Ltd., Shanghai, China
Abstract—Conventional design of wireless networks mainly
focuses on system capacity and spectral efficiency (SE). As green
radio (GR) becomes an inevitable trend, energy-efficient design
in wireless networks is becoming more and more important. In
this paper, the fundamental relation between energy efficiency
(EE)andSEindownlinkorthogonal frequency division multiple
access (OFDMA) networks is addressed. We first set up a general
EE-SE tradeoff framework, where the overall EE, SE and per-
user quality-of-service (QoS) are all considered, and prove that
EE is strictly quasiconcave in SE. We also find a tight upper
bound and a tight lower bound on the EE-SE curve for general
scenarios, which reflect the actual EE-SE relation. We then focus
on a special case that priority and fairness are considered and
develop a low-complexity but near-optimal resource allocation
algorithm for practical application of the EE-SE tradeoff. Numer-
ical results corroborate the theoretical findings and demonstrate
the effectiveness of the proposed resource allocation scheme for
achieving a flexible and desirable tradeoff between EE and SE.
Index Terms—Energy efficiency (EE), green radio (GR), or-
thogonal frequency division multiple access (OFDMA), spectral
efficiency (SE)
I. INTRODUCTION
In recent years, the widespread application of high-data-rate
wireless services and requirement of ubiquitous access have
triggered rapidly booming energy consumption. Meanwhile,
the escalation of energy consumption in wireless networks
leads to large amount of greenhouse gas emission and high
operation expenditure. Green radio (GR) [1], which empha-
sizes on energy efficiency (EE) besides spectral efficiency (SE),
has been proposed as an effective solution and becomes an
inevitable trend for future wireless network design. Unfortu-
nately, EE and SE do not always coincides and sometimes
may even conflict [1]. Hence, how to balance EE and SE is
well worth studying.
Orthogonal frequency division multiple access (OFDMA)
has been extensively studied from the SE perspective and
proposed for next generation wireless communication systems,
such as WiMAX and the 3GPP LTE. While OFDMA can
provide high throughput and SE, its EE is previously not much
concerned. To keep pace with GR, it is necessary for OFDMA
to guarantee a certain level of EE at the same time. Recently,
more attention has been paid to energy efficient design in
OFDMA networks. For uplink OFDMA transmission with flat
fading channels, it is shown that EE-oriented design always
∗The work was supported in part by the Research Gift from Huawei
Technologies Co. and the NSF under Grant No. 1017192.
Corresponding author. Email: xiongcong@ece.gatech.edu.
consumes less energy than the traditional fixed power schemes
[2]. Meanwhile, we notice that there is only limited work on
joint design of EE and SE for downlink OFDMA networks.
In this paper, we address the EE-SE relation in downlink
OFDMA networks. We build a general EE-SE tradeoff frame-
work, prove that EE is quasiconcave in SE, then bound the EE-
SE curve for general scenarios by a double-side approximation
process, which relies on a tight upper bound and a lower
bound obtained by Lagrange dual decomposition [3], [4].
When priority and fairness are considered, we propose a
computationally efficient algorithm for resource allocation to
facilitate application of EE-SE tradeoff.
II. SYSTEM MODEL AND PROBLEM FORMULATION
In this section, we introduce the system model of downlink
OFDMA and formulate the problem of EE-SE tradeoff.
A. System Model
We consider the downlink of a single cell OFDMA network
consisting of Kactive users. The total bandwidth Bis equally
divided into Nsubcarriers, each with a bandwidth of W=
B/N.LetK={1,2,··· ,K}and N={1,2,··· ,N}denote
the sets of all users and all subcarriers, respectively.
Assume that each subcarrier is exclusively assigned to at
most one user each time to avoid interference among different
users. Denote the transmit power of user kon subcarrier nas
pk,n. Then, the achievable data rate of user kon subcarrier n
is accordingly
rk,n =Wlog21+pk,ngk,n
σ2,
where gk,n =|hk,n|2is the channel power gain of user k
on subcarrier n,hk,n is the corresponding frequency response
and is assumed to be accurately known at the transmitter, and
σ2is the noise power, which is, without loss of generality,
assumed to be the same for the all users on all subcarriers.
B. Problem Formulation
For a downlink OFDMA network, EE and SE are, respec-
tively, defined as
ηEE =R
P+Pc
and ηSE =R
B,
where R,Pand Pcdenote the system overall throughput, total
transmit power, and circuit power, respectively. To obtain a
978-1-61284-231-8/11/$26.00 ©2011 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings
high EE as well as a desirable SE and guarantee quality-of-
service (QoS) for each user, it is reasonable to maximize EE
under a satisfying minimum overall throughput requirement,
ˇ
R, a series of (minimum) rate requirements, ˇ
Rk’s, depending
on the traffic of the corresponding user, and the peak transmit
power, PT. Since the capability of providing differentiate
services is an important feature for future wireless networks, a
heterogeneous traffic model includes both real-time and non-
real-time traffic is considered in our work. For users with
real-time services [5], such as video conferencing and online
gaming, fixed rates ˇ
Ri’s are required. For users with non-
real-time services [5], such as file transfers and online video,
only minimum rate requirements ˇ
Rj’s are demanded. Let
K1={1,2,··· ,K
0−1}and K2={K0,K
0+1,··· ,K}
denote the sets of real-time users and non-real-time users,
respectively. Accordingly, the optimization problem can be
formulated as
max
ρ,PηEE
subject to
K
k=1
ρk,n ≤1,∀n, ρk,n ∈{1,0},∀k∈K,n,
K
k=1
N
n=1
ρk,nrk,n =R≥ˇ
R,
N
n=1
ρk,nrk,n =ˇ
Rk,∀k∈K
1,
N
n=1
ρk,nrk,n ≥ˇ
Rk,∀k∈K
2,
K
k=1
N
n=1
ρk,npk,n =P≤PT,p
k,n ≥0,∀k∈K,n, (1)
where ρ=[ρk,n]K×Nand P=[pk,n]K×Nare the subcarrier
allocation indicator matrix and transmit power matrix, respec-
tively.
For convenience, some other notations used in this paper
are listed in advanced as follows.
Skset of subcarriers assigned to user k,
mknumber of assigned subcarriers of user k,
KEset of users without assigning any subcarrier
(with empty Sk’s)
Pkoverall transmit power for user k, and Pk=
N
n=1 ρk,npk,n,
Rkoverall data rate for user k, and Rk=
N
n=1 ρk,nrk,n,
f(Rk,Sk)power needed by water-filling to fulfill rate Rk
over subcarrier set Sk.
III. EE-SE RELATION
In this section, we will study the EE-SE tradeoff.
A. Fundamentals for EE-SE Tradeoff Relation
The following theorem demonstrates the quasiconcavity of
EE, ηEE, in the SE, ηSE , and is proved in the Appendix.
Theorem 1. For any given SE, ηSE ≥ˇ
R
B, achieved with
subcarrier allocation matrix, ρ, and power allocation matrix,
P, that satisfy all constraints but not necessarily includ-
ing the peak transmit power one in (1), the maximum EE,
η∗
EE (ηSE) = max
ρ,PηEE (ηSE), is strictly quasiconcave in ηSE
if there is a sufficiently large number of subcarriers. Moreover,
in the region between ηSE =ˇ
R
Band ηSE =ˆ
R
B,η∗
EE (ηSE)is
(i) strictly decreasing with ηSE and maximized at ηSE =ˇ
R
B
if
dη∗
EE (ηSE)
dηSE ηSE =ˇ
R
B
≤0,
(ii) strictly increasing with ηSE and maximized at ηSE =ˆ
R
B
if
dη∗
EE (ηSE)
dηSE ηSE =ˇ
R
B
>0
and dη∗
EE (ηSE)
dηSE ηSE =ˆ
R
B
≥0,
(iii) first strictly increasing and then strictly decreasing with
ηSE and maximized at ηSE =REE,max
Bif
dη∗
EE (ηSE)
dηSE ηSE =ˇ
R
B
>0
and dη∗
EE (ηSE)
dηSE ηSE =ˆ
R
B
<0,
where ˆ
Ris the maximum throughput under all constraints in
(1), which is obviously achieved by transmitting at the peak
transmit power, PT, and REE,max is the throughput which
achieves the global maximum EE, ηmax
EE , under all constraints
except the peak transmit power one in (1).
For any continuous and strictly quasiconcave function, there
is always a unique global maximum over a finite domain
[6, Ch. 8]. Thus, according to Theorem 1, a unique globally
optimal EE of (1) always exits. More importantly, as a result
of this quasiconcavity, (1) can be decomposed into two layers
and solved iteratively,
(i) Inner layer: For a given SE, ηSE, find the maximum EE,
η∗
EE (ηSE), and its derivative, dη∗
EE(ηSE )
dηSE .
(ii) Outer layer: Find the optimal EE, ηopt
EE, by bisection
search like the GABS algorithm in [7].
The corresponding joint inner- and outer-layer optimization
(JIOO) algorithm is listed in Table I. Then the key of the
JIOO algorithm lies in the inner-layer algorithm that finds
η∗
EE (ηSE)and dη∗
EE(ηSE )
dηSE and will be studied in detail in
the following sections.
B. Bounds on the EE-SE Tradeoff
As indicated before, the solution of (1) now relies on finding
η∗
EE (ηSE)and dη∗
EE(ηSE )
dηSE . Since the exact solution is too
complicated to obtain in reality, we will use Lagrange dual
decomposition to approximately solve it, which has been used
in [3], [4] for similar problems and demonstrated to be quite
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings
TAB L E I
JOINT INNER-AND OUTER-LAYER OPTIMIZATION (JIOO) ALGORITHM
Algorithm JIOO
Input: initial value of SE: η(0)
SE =ˇ
R
B
Output: optimal subcarrier and power allocation matrices, ρopt and
Popt, which result in the optimal EE, ηopt
EE
1. η(1)
SE =η(0)
SE,d1←dη∗
EE(ηSE )
dηSE ηSE=η(1)
SE
2. if d1≤0
3. then ηopt
SE ←η(1)
SE
4. else η(2)
SE ←η(1)
SE,η(1)
SE ←αη(1)
SE,P1←P∗(η(1)
SE),
d1←dη∗
EE(ηSE )
dηSE ηSE=η(1)
SE
,whereP∗(η(1)
SE)=
Bη(1)
SE
η∗
EEη(1)
SE−Pcand α>1, i.e., α=2
5. while d1>0&& P1<P
T
6. do η(2)
SE ←η(1)
SE,η(1)
SE ←αη(1)
SE,P1←P∗(η(1)
SE),
d1←dη∗
EE(ηSE )
dηSE ηSE=η(1)
SE
7. while no convergence (∗bisection search ∗)
8. do ηopt
SE ←η(1)
SE+η(2)
SE
2,Popt ←P∗(η(opt)
SE ),
dopt ←dη∗
EE(ηopt
SE )
dηSE ηSE=η(1)
SE
9. if dopt >0&& Popt <P
T
10. then η(2)
SE ←ηopt
SE
11. else η(1)
SE ←ηopt
SE
12. return ρopt ,Popt
accurate with affordable computational complexity. For any
given throughput, R≥ˇ
R, the Lagrange dual problem of the
equivalent problem of (1), which is to minimize the transmit
power, P, can be expressed as
max
λ1,λ2,λ3
min
ρ,PK
k=1
N
n=1
ρk,npk,n
+
N
n=1
λ1,nK
k=1
ρk,n −1
+
K
k=1
λ2,kˇ
Rk−
N
n=1
ρk,nrk,n
+λ3R−
K0−1
k=1
ˇ
Rk−
K
k=K0
N
n=1
ρk,nrk,n
subject to
ρk,n ∈{1,0},p
k,n ≥0,∀k∈K,n,
λ1,n ≥0,∀n, λ2,k ≥0,∀k∈K,λ
3≥0,(2)
where λ1,n,λ2,k , and λ3are the introduced Lagrange
multipliers, and λ1=[λ1,1,λ
1,2,··· ,λ
1,N ]Tand λ2=
[λ2,1,λ
2,2,··· ,λ
2,K ]T, respectively. Note that we drop the
peak power constraint in (1) here since it will be imposed by
the outer-layer processing as shown in the JIOO algorithm.
Such a problem as (2) can be solved with dual decomposition
as suggested in [3].
In general, the dual decomposition solution for subcarrier
allocation indicator matrix, transmit power matrix and total
transmit power, (ρ∗
d,P∗
d,P∗
d), to the dual problem (2) yields
a lower bound of total transmit power on the optimal solu-
tion, ρ∗
p,P∗
p,P∗
p, to the primal problem as a result of the
nonconvexity of the primal one [3]. To guarantee the solution
feasible, we find an achievable upper bound on the minimum
transmit power based on the subcarrier allocation strategy
obtained from the dual decomposition. It is easy to verify that
for the subcarrier allocation strategy, ρ=ρ∗
d, the tightest and
achievable upper bound, P∗
ub, on total transmit power can be
achieved in two stages: in the first stage, power is distributed
individually among the subcarriers of each user, Sk, by water-
filling, to merely fulfill its own (minimum) rate requirement,
ˇ
Rk; in the second stage, extra power is then distributed among
all subcarriers of the non-real-time users (each has an initial
water level due to the first phase processing) by water-filling
till the throughput, R, is achieved.
Using dual decomposition, we approximate the minimum
transmit power, P∗
p, tightly from both sides, i.e., P∗
d≤P∗
p≤
P∗
ub, which allows us quite an accurate EE-SE tradeoff relation.
On the other hand, using ηEE =R
P+Pcand ηSE =R
B,the
above bounds on total transmit power, P, correspond to inverse
bounds on EE, η∗
EE (ηSE). Since it was impossible to find out
the closed-form expression for η∗
EE (ηSE),dη∗
EE(ηSE )
dηSE could
only be calculated via the original definition of derivative,
i.e, dη∗
EE(ηSE )
dηSE = limΔηSE→0η∗
EE(ηSE +ΔηSE)−η∗
EE(ηSE )
ΔηSE
.For
practical implementation, we can choose a small positive
ΔηSE =ΔR
B. To further reduce complexity, we do not need
to repeat the whole process for the new SE, ηSE +ΔηSE.
Note that we only need the sign of dη ∗
EE(ηSE )
dηSE but not
necessarily for its value. Since η∗
EE(ηSE +ΔηSE)−η∗
EE(ηSE )
ΔηSE =
BΔR−ΔPR
P+Pc
P+Pc+ΔP, we have that sgn
η∗
EE
(ηSE
+Δ ηSE
)−η∗
EE
(ηSE
)
ΔηSE =
sgn
ΔR
ΔP−η∗
EE (ηSE), where sgn (x)denotes the sign of x.
We can simply distribute ΔRamong all subcarriers of the
non-real-time users by water-filling to approximately find out
ΔPbased on the subcarrier allocation and power allocation
matrix corresponding to η∗
EE (ηSE), and the sign of dη∗
EE(ηSE )
dηSE
is obtained accordingly.
C. Priority, Fairness and Low-Complexity Algorithm
For many practical scenarios, capability to provide different
service priority and fairness among users is important. Let
˜
R.
=R−K
k=1 ˇ
Rk, then the two rate constraints in (1) can
be rewritten as
N
n=1
ρk,nrk,n =ˇ
Rk+ωk˜
R,
K
k=1
ωk=1,ω
k≥0,∀k,
where ωkis the weight factor for user k. We can determine the
weights according to user traffic types, fairness and priority
requirements. For users with real-time services, ωican be
simply set to zero. For users with non-real time services,
which have only minimum rate requirements ˇ
Rj’s, we can
prioritize them and enforce certain notations of fairness by
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings
adjusting ωj. For example, we can determine ωjby making
user rates proportional to a set of predetermined values to
ensure proportional fairness among non-real-time users [8].
With certain predetermined weight vector, ω, the equivalent
problem of (1) given the throughput Rcan be regarded as the
conventional margin adaptation (MA) optimization problem
[3], [9]–[11] as follows.
min
ρ,P
K
k=1
N
n=1
ρk,npk,n
subject to
K
k=1
ρk,n ≤1,∀n, ρk,n ∈{1,0},∀k, n,
rk,n ≥0,∀k, n
N
n=1
ρk,nrk,n =ˇ
Rk+ωk˜
R, ∀k, (3)
Although dual decomposition can be still employed simi-
larly as before, it typically needs O(NK3)times of water-
filling to converge [3] and its complexity is still very high.
Motivated by the BABS algorithm [10] for finding distribution
of subcarriers in flat-fading channels, we suggest the following
maximum-power-decrease-first (MPDF) algorithm. In initial-
ization of this approach, each user is only virtually assigned its
worst subcarrier and its transmit power needed in this situation
will be used as a benchmark to measure how much power can
be saved if the user is actually assigned a subcarrier when he
has none. Then, in each iteration, each user finds its optimal
subcarrier among the unassigned ones and calculates its power
decrease with the additional subcarrier. Only the user with the
maximum power decrease (power saving) will be assigned its
favorite subcarrier in this iteration. The above iteration process
will proceed until all the subcarriers have been assigned or no
user needs additional subcarriers. Note that to guarantee each
user is assigned at least one subcarrier at last, when the the
number of unassigned subcarriers equals the number of users
without any subcarriers, the subcarrier assignment should be
implemented among such empty users.
TAB L E II
MAXIMUM-POWER-DECREASE-FIRST (MPDF) ALGORITHM
1) Initialize: KE←K;Sk←∅,m
k←0,∀k∈K.
2) Calculate benchmarks: ˇnk←arg min
n∈N gk,n,Pk=f(Rk,ˇnk),
∀k∈K.
3) ˆnk←arg max
n∈N gk,n,ΔPk←Pk−f(Rk,Sk∪{ˆnk}),∀k∈K.
ˆ
k←arg max
k∈K 1+δN−K
k=1 mk−|KE|δ(mk)ΔPk.
Assign and update: Sˆ
k←S
ˆ
k∪{ˆnˆ
k},N←N\{ˆnˆ
k},mˆ
k←
mˆ
k+1,Pˆ
k←Pˆ
k−ΔPˆ
k,KE←K
E\{
ˆ
k}.
4) Repeat Step 3) until K
k=1 mk=Nor max
k∈K ΔPk=0.
Theorem 2. The power required by water-filling with a fixed
target rate is convex and non-increasing with the number of
subcarriers assigned if the subcarriers are added in descend-
ing order of the power gains of subcarriers. (The proof is
omitted.)
From Theorem 2, the MPDF algorithm naturally prevents
one user getting too many subcarriers since benefit of acquir-
ing subcarriers is decreasing. Meanwhile, the MPDF algorithm
needs O(NK)times of water-filling in total.
On the other hand, to obtain the sign of dη∗
EE(ηSE )
dηSE to
implement the JIOO algorithm, we can follow the same way
as for the general case indicated before.
IV. NUMERICAL RESULTS AND CONCLUSION
In this section, we present some simulation results to verify
the theoretical analysis and the effectiveness of the proposed
approaches. In our simulation, the channel is modeled as a
frequency-selective fading channel consisting of six indepen-
dent Rayleigh multipaths with power delay profile, e−2l,l=
0,1,··· ,5. The total bandwidth, 1.28 MHz, is equally divided
into 64 non-overlapping subcarriers. The circuit power is
2.5W. There are two real-time users each has a fixed rate
requirement of 125 kbps, and two non-real-time users each
has a minimum rate requirement of 25 kbps. The fairness
notion employed here for non-real-time users is the partial
proportional constraint (PPC) modified from the propotional
constraint in [8], where ω1:ω2:··· :ωK=α1:α2:··· :
αK. For simplicity, we let αk=ˇ
Rklog2¯gk,n. Consequently
ωk=ˇ
Rklog2¯gk,n/K
k=1 ˇ
Rklog2¯gk,n.
Figure 1 shows the EE-SE relation in the case that all the
four users have a same average channel-gain-to-noise ratio
(CNR), ¯gk,n
σ2, of 20 dB and no specific overall throughput
requirement is imposed here, i.e, ˇ
R=0. From the figure, the
EE-SE relation has a bell shape curve and is also quasiconcave,
since the upper bound and the lower bound derived from
Lagrange dual decomposition almost perfectly match together
and they are in a bell shape.
Figure 2 demonstrates the relation between EE, η∗
EE, and
the minimum prescribed SE, ˇηSE =ˇ
R
B. In this case, the two
real-time users have the same average CNR of 20 dB. One
of the two non-real-time users has an average CNR of 20
dB, while the other has an average CNR of 17 dB. Here no
peak power constraint is imposed to investigate performance
limit. From the figure, the performance of the MPDF-based
method is quite close to that of the dual method, where the
performance loss is within 3%; and it is slightly better than
the DPRA [11] -based method in the high power regime.
The complexity (average computational time) is evaluated
using the profile in Matlab, which is shown in Fig.
3. Obviously, our MPDF-based method has relatively lower
complexity and offers an attractive performance to complexity
good tradeoff.
APPENDIX
Proof: Let R∗
1,R∗
2and R∗
3denote the optimal rate
vectors corresponding to the overall throughput R1,R2, and
R3, respectively, and they also satisfy all constraints but not
necessarily including the peak power constraint in (1). Without
loss of generality, assume that R1<R
2<R
3.LetR2denote
the rate vector as follows.
R2=R3−R2
R3−R1R∗
1+R2−R1
R3−R1R∗
3
=λR∗
1+(1−λ)R∗
3,
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings
00.5 11.5 22.5 3
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
0.28
0.3
SE (bits/s/Hz)
EE (Mbits/Joule)
UB (dual)
LB (dual−based)
Fig. 1. EE-SE relation
00.5 11.5 22.5 3
0.12
0.14
0.16
0.18
0.2
0.22
SEmin (bits/s/Hz)
EE (Mbits/Joule)
UB (dual)
LB (dual−based)
DPRA
MPDF
Fig. 2. Performance comparison of the MPDF-based, DPRA-based and dual-
based JIOO algorithms
2 3 4 5 6 7 8
100
101
102
103
User number (K)
Normalized CPU Time
UB (dual)
DPRA
MPDF (proposed)
Fig. 3. Complexity comparison in the case ˇηSE =1bit/s/Hz,CNRk=
20 dB,ˇ
Rk= 25 kbps and with PPC
where λ=(R3−R2)/(R3−R1) and 0<λ<1.
Obviously, R2is also in the feasible region of (1) and its
sum rate is R2. From [3], it is known that P(R)is strictly
convex in R, given a sufficiently large number of subcarriers.
Thus, P(R2)<λP(R∗
1)+(1−λ)P(R∗
3). Since P(R∗
2)
is the optimal rate vector among all the rate vectors with a
summation of R2,wehaveP(R∗
2)≤P(R2). Consequently,
we have that P(R∗
2)<λP(R∗
1)+(1−λ)P(R∗
3). Thus,
the minimum transmit power needed given the throughput
R(= BηSE),P(R)=P(R∗), is strictly convex in R(and
ηSE).
Denote the superlevel set of ηEE (ηSE)as
Sβ={R≥ˇ
R|ηEE (ηSE)≥β, β ∈R}.
When β≥0,Sβis equivalent to Sβ={R≥ˇ
R|βP (ηSE)+
βPc−BηSE ≤0}, where P(ηSE)is the minimum total
transmit power needed for any SE ηSE ≥ˇ
R/B.Asa
result of the convexity of P(ηSE)(i.e., P(R)) proved above,
Sβis strictly convex in ηSE. Hence, η∗
EE (ηSE)is strictly
quasiconcave and has a unique global maximum.
One the other hand, lim
ηSE→∞ η∗
EE (ηSE) = lim
ηSE→∞
BηSE
P+Pc=
lim
P→∞
BηSE
P= lim
P→∞
O(P)
P=0. Thus, starting from ηSE =
ˇ
R/B,η∗
EE (ηSE)is either strictly decreasing with ηSE if
dη∗
EE(ηSE )
dηSE ηSE=ˇ
R/B ≤0, or first strictly increasing and then
strictly decreasing with ηSE if dη∗
EE(ηSE )
dηSE ηSE=ˇ
R/B >0.
And the maximum EE within the SE region, ˇ
R
B,RPT
B,is
straightforward as indicated in Theorem 1.
REFERENCES
[1] Y. Chen, S. Zhang, S. Xu, and G. Y. Li, “Fundamental tradeoffs on
green wireless networks,” to appear in IEEE Commun. Mag.
[2] G. Miao, N. Himayat, G. Y. Li, and D. Bormann, “Energy-efficient de-
sign in wireless OFDMA,” in Proc. IEEE Int. Conf. Commun. (ICC’08),
Beijing, China, May 2008.
[3] K. Seong, M. Mohseni, and J. M. Cioffi, “Optimal resource allocation
for OFDMA downlink systems,” in Proc. IEEE Int. Symp. Inf. Theory
(ISIT’06), Seattle, USA, July 2006.
[4] W. Yu and R. Lui, “Dual methods for nonconvex spectrum optimization
of multicarrier systems,” IEEE Trans. Commun., vol. 54, no. 7, pp. 1310–
1322, July 2006.
[5] X. Wang, G. B. Giannakis, and A. G. Marques, “A unified approach
to QoS-guaranteed scheduling for channel-adaptive wireless networks,”
Proc. IEEE, vol. 95, no. 12, pp. 2410–2431, Dec. 2007.
[6] R. K. Sundaram, A First Course in Optimization. Cambridge University
Press, 1996.
[7] G. Miao, N. Himayat, and G. Y. Li, “Energy-efficient link adaptation
in frequency-selective channels,” IEEE Trans. Commun., vol. 58, no. 2,
pp. 545–554, 2010.
[8] Z. Shen, J. G. Andrews, and B. L. Evans, “Adaptive resource allocation
in multiuser OFDM systems with proportional rate constraints,” IEEE
Trans. Wireless Commun., vol. 4, no. 6, pp. 2726–2737, Nov. 2005.
[9] C. Y. Wong, R. S. Cheng, K. B. Letaief, and R. D. Murch, “Multiuser
OFDM with adaptive subcarrier, bit, and power allocation,” IEEE J. Sel.
Areas. Commun., vol. 17, no. 10, pp. 1747–1758, Oct. 1999.
[10] D. Kivanc, G. Li, and H. Liu, “Computationally efficient bandwidth
allocation and power control for OFDMA,” IEEE Trans. Wireless
Commun., vol. 2, no. 6, pp. 1150–1158, Nov. 2003.
[11] Y. Lin, T. Chiu, and Y. T. Su, “Optimal and near-optimal resource
allocation algorithms for OFDMA networks,” IEEE Trans. Wireless
Commun., vol. 8, no. 8, pp. 4066–4077, Aug. 2009.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings
