IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 63, NO. 5, JUNE 2014
2139
Energy-Efficient Resource Allocation in Downlink
OFDM Wireless Systems With Proportional
Rate Constraints
Zhanyang Ren, Shanzhi Chen, Senior Member, IEEE, Bo Hu, and Weiguo Ma
Abstract—Orthogonal frequency-division multiplexing (OFDM),
with its own advantages of spectral efficiency (SE) enhancement
and flexible resource allocation, is a promising basic technique
for fulfilling the ever increasing demand for high-data-rate
transmission and various service type support for mobile
multimedia communications. In the meantime, energy efficiency
(EE) has now become a critical metric for green system design.
In this paper, energy-efficient and fair resource allocation is
investigated in a downlink OFDM-based mobile communication
system. Given a subcarrier assignment, the bisection-based
optimal power allocation (BOPA) is proposed at first, which
achieves the maximum EE and guarantees proportional data rates
for users. Then, a two-step subcarrier assignment is designed
to avoid unaffordable computational complexity of exhaustive
search. In the first step, the estimated energy-efficient transmit
power is found via the assumptions on flat fading and subcarrier
sharing. In the second step, the traditional spectral-efficient
subcarrier assignment (SESA) is introduced to complete the
bandwidth resource allocation among users. Although the
two-step subcarrier assignment is suboptimal due to the fact that
the optimization is done independently in two separate steps,
numerical results demonstrate that its performance is very close
to the optimum. This paper also studies the difference between
the energy-efficient solution and the traditional spectral-efficient
policy and observes that they are similar with each other in the
low channel-gain-to-noise ratio (CNR) regime. This observation
is helpful and could enable that the energy-efficient design
can be turned into the relatively simpler spectral-efficient
policy when the CNR is low.
Index Terms—Energy efficiency (EE), mobile multimedia
communications, orthogonal frequency-division multiplexing
(OFDM), proportional rate constraints, spectral efficiency (SE).
Manuscript received July 28, 2013; revised January 18, 2014; accepted
February 27, 2014. Date of publication March 11, 2014; date of current
version June 12, 2014. This work was supported in part by the Major National
Science and Technology Special Project under Grant 2013ZX03001025-001,
by the National High-Technology Program (863 Program) of China under
Grant 2014AA01A701, and by the China Next Generation Internet Project
under Grant CNGI-12-03-003. The review of this paper was coordinated
by the Guest Editors for the Special Section on Green Mobile Multimedia
Communications.
Z. Ren and B. Hu are with State Key Laboratory of Networking and
Switching Technology, Beijing University of Posts and Telecommunications,
Beijing 100876, China (e-mail: [email protected]; [email protected]).
S. Chen and W. Ma are with the State Key Laboratory of Wireless
Mobile Communications, China Academy of Telecommunications Technology (CATT), Beijing 100191, China (e-mail: [email protected];
[email protected]).
Digital Object Identifier 10.1109/TVT.2014.2311235
I. I NTRODUCTION
W
ITH the ever increasing demand for higher reliable
data transmission and various quality-of-service requirements for emerging mobile services and applications such
as content sharing and video streaming [1], mobile multimedia communications are expected to be provided in a more
effective and efficient way. From the wireless communications
perspective, channel variation and interference also challenge
system robustness. In third-generation wireless systems, directsequence code-division multiple access (DS-CDMA) has been
chosen as the basic technique for providing such multimedia
communications [2]. Orthogonal frequency-division multiplexing (OFDM) [3] and CDMA-based technologies, such as multicarrier CDMA (MC-CDMA), are viewed as the evolution path
toward higher network performance [4].
Nowadays, OFDM has been adopted in the next-generation
mobile broadband systems, such as Long Term Evolution
(LTE) and Worldwide Interoperability for Microwave Access
(WiMAX). As a basic broadband technique, OFDM provides
not only a good physical layer with higher and more stable
performance but also benefits service diversity due to its persubcarrier resource-allocation capability with finer granularity.
To this end, in addition to the signal transmission point of view,
the OFDM-based system can be also regarded as a highly potential candidate for mobile multimedia communications from
resource-allocation perspective.
User data rates in mobile multimedia communications vary
and may change dynamically due to the existing and emerging
multimedia applications. To support such dynamic user data
rates, adaptive resource allocation in OFDM systems has been
well investigated in the literature [5]–[9]. Generally speaking,
in addition to user rate consideration, rate-adaptive schemes
[5], [6] aim to maximize system throughput with given transmit power constraints, whereas the marginal-adaptive policies
[7]–[9] try to minimize the power consumption with a target
transmit rate requirement. Both approaches can be viewed as
spectral efficiency (SE) enhancement policies. However, with
the rapidly growing number of communication devices and
the demand for environment protection, energy efficiency (EE)
is adopted to be the key performance in green radio system
design [10]. Different from the traditional SE, EE design aims
to maximize the throughput with each unit of power consumption (bits/Joule), i.e., given the amount of data needed to be
delivered, the objective of EE design is to consume minimal
power resources.
0018-9545 © 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.
2140
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 63, NO. 5, JUNE 2014
It is well known that orthogonal frequency-division multiple access (OFDMA) is optimal for SE design in OFDM
networks [11]; likewise, each subcarrier assigned to only one
user is demonstrated optimal also for EE optimization [12].
In [13] and [14], energy-efficient power allocation for OFDM
systems is studied. Fixed circuit power consumption is considered in [13], whereas in [14], a linear sum rate dependence
is also taken into account in addition to the constant part.
Other than power allocation, more complete resource allocation
with subcarrier assignment included can be found in [15] and
[16] for uplink OFDM, where the flat-fading and frequencyselective fading are considered, respectively. Furthermore,
proportional fairness [17] is also taken into account for the
long-term performance. In the downlink, resources in a base
station, e.g., subcarrier, power and antenna, are allocated among
users to achieve the maximum EE [18]–[22]. In [18], lowcomplexity subcarrier assignment and optimal power allocation
are developed, respectively, to maximize EE and meet user
rate requirements as well. Tradeoff between SE and EE is
investigated in [19]; in addition, tradeoffs between deployment
efficiency and EE, bandwidth and power, and delay and power
in energy-efficient networks are also introduced in [23]. Other
than user rate requirements, a maximum tolerable channel
outage probability is also considered in [20] and [21], where
the base station equipped with a large number of antennas
and secure transmission are addressed, respectively. Given limited backhaul capacity, the work in [22] carried out energyefficient resource allocation and scheduling in a multicell
scenario.
However, most existing work does not focus on the system
fairness issue in energy-efficient design for OFDM systems,
except for the uplink long-term fairness consideration in [15]
and [16]. However, downlink fairness is critical for supporting
various multimedia applications in future mobile communication systems. With such a motivation, we propose energyefficient downlink resource allocation and take instantaneous
fairness into account in this paper. To this end, proportional
rate constraints [24] are considered, which can be viewed as
a fairness-level configuration to ensure that each user would
obtain a predetermined proportion of the system throughput
in each resource-allocation determination. The bisection-based
optimal power allocation (BOPA) is first given under a fixed
subcarrier assignment. Then, low-complexity energy-efficient
subcarrier assignment is proposed via two steps: 1) energyefficient transmit power estimation (EETPE), and 2) spectralefficient subcarrier assignment (SESA). The proposed algorithm
enhances EE significantly with the performance approaching
the optimal solution, which can be only obtained by full
enumeration.
The remainder of this paper is organized as follows. In
Section II, the system model and overall description of the
proposed algorithm are given. In Section III, the optimal power
allocation under a fixed subcarrier assignment is proposed. In
Section IV, a suboptimal subcarrier assignment is developed
through two steps: 1) the EETPE, and 2) SESA. The complexity
of the proposed EE solution is further analyzed in Section V.
Numerical results are given and discussed in Section VI, and
finally, this paper is concluded in Section VII.
II. S YSTEM M ODEL AND OVERALL D ESCRIPTION
OF THE P ROPOSED A LGORITHM
A. System Model
In this paper, a downlink multiuser scenario in OFDM wireless systems is considered, and perfect instantaneous channel
information is assumed available at both the base station and the
user side. Based on the channel state information, the resourceallocation algorithm assigns subcarriers set N = {1, 2, . . . , N }
to users set K = {1, 2, . . . , K}, with the total transmit power
constraint Ptrans . In addition, constant circuit power is also
considered to calculate downlink total power consumption, i.e.,
the same as [25]
Ptotal = ζPtrans + PC
(1)
where ζ denotes the reciprocal of drain efficiency of the power
amplifier, and PC is the constant circuit power.
Let B be the system bandwidth and subcarrier bandwidth
can be expressed as B/N accordingly. Let hn, k and N0 be
the channel gain and noise power spectral density of additive
white Gaussian noise, respectively. Further, let μk, n = 1 denote
if subcarrier n is assigned to user k, and μk, n = 0, otherwise;
then, we can express SE in terms of bits/s/Hz based on the
Shannon capacity formula as
μk, n
pk, n h2k, n
Δ
log2 1 +
(2)
SE(μ, p) =
B
N
N0 N
k∈K n∈N
where μ = {μk, n }k∈K, n∈N is the subcarrier assignment indicator set, and p = {pk, n }k∈K, n∈N is the indicator set for power
allocation. For simplicity, gn, k is used to denote the channelgain-to-noise-ratio (CNR) h2n, k /N0 (B/N ) in the following.
Accordingly, EE can be expressed in terms of bits/Hz/Joule as
μk, n
Δ
k∈K
N log2 (1 + pk, n gk, n )
n∈N . (3)
EE(μ, p) =
ζ k∈K n∈N pk, n + PC
Hence, the EE optimization problem can be formulated
mathematically as
EE(μ, p)
subject to
pk,n ≤ Ptrans
maximize
k∈K n∈N
pk,n ≥ 0 for all k, n
μk,n = {0, 1}, for all k, n
μk,n = 1, for all n
k∈K
R 1 : R2 : . . . : RK = α1 : α2 . . . : αK
(4)
where {α1 , . . . , αK } is the proportional rate constraints set for
fairness consideration [24]. It denotes that the data rate among
users would follow a predetermined proportion, or in other
words, it quantizes the user priority. For user k, the proportion
of system throughput distributed would be αk / k∈K αk . The
third and the fourth constraints show that one subcarrier can be
only assigned to one user. User rate Rk can be expressed as
μk, n
log2 (1 + pk, n gk, n ).
Rk =
(5)
N
n∈N
REN et al.: ENERGY-EFFICIENT RESOURCE ALLOCATION IN DOWNLINK OFDM WIRELESS SYSTEMS
2141
Lemma 1: The single-user power consumption function (7)
is strictly convex in Rk , and its first derivative is
1 + p̂k, n gk, n
dPk (Rk |Dk )
= N ln 2 · min
(8)
n∈Dk
dRk
gk, n
Fig. 1.
where {p̂k, n }n∈Dk is the power allocation under rate level of
Rk via (6).
Overall procedure of the proposed algorithm.
B. Overall Description of the Proposed Algorithm
In this paper, the proposed resource-allocation algorithm
is completed through two consecutive steps. The first step is
subcarrier assignment. To reduce the computational complexity,
this step is comprised of two substeps. In the first substep,
we carry out an EETPE algorithm to obtain the estimated
energy-efficient transmit power, followed by another substep
of running the SESA algorithm to assign subcarriers among
users. Then, the second step of the proposed algorithm is power
allocation, which is to execute the BOPA algorithm to complete
the energy-efficient resource allocation. The overall procedure
is shown in Fig. 1.
In the following, we will first introduce the BOPA algorithm
since this power allocation scheme is optimal and we can easily
find a brutal-search-based subcarrier assignment to obtain the
optimal solution. Then, the EETPE and SESA algorithms are
given due to the prohibitive computational complexity of the
brutal search.
III. O PTIMAL P OWER A LLOCATION
Here, we propose the optimal power allocation with a
given subcarrier assignment. Single-user power allocation is
discussed at first, and then the system-level multiuser optimal power allocation under proportional rate constraints is
developed.
B. Multiuser Optimal Power Allocation
Now, we consider all the users in system and the proportional
rate constraints. We introduce a “rate parameter” λ, which is
defined as
Δ
λ=
k∈K
Since Pk (λαk |Dk ) is monotonously increasing in λ, the
upper bound λmax satisfying PT (λmax ) = Ptrans can be found
using the bisection method. Through the conversion, the EE can
be reformed as a function of λ, i.e.,
Δ λ
k∈K αk
(11)
EE(λ) =
ζPT (λ) + PC
and the total transmit power constraint also has a linear form, i.e.,
λ ≤ λmax .
where {p̂k, n }n∈Dk is the power allocation among subcarriers
for user k, and θ is the Lagrangian multiple, which should be
chosen such that user rate Rk is satisfied.
Given subcarriers set Dk , from results (6), we further define
a single-user power consumption function of rate level Rk as
Pk (Rk |Dk ) =
k∈K
(7)
n∈Dk
for which we can have the following lemma. The proof is given
in Appendix A.
R
.
PT (R) + PC
(13)
Apparently, PT (λ) is strictly convex in λ; hence, EE(λ) in (11)
is strictly quasiconcave, and there is a unique globally optimal
solution to obtain the maximum EE.
Therefore, for the reformulated EE problem (11) under linear
constraint (12), we have the following theorem readily. The
proof is given in Appendix B.
Theorem 1: EE(λ) is first strictly increasing and then strictly
decreasing in λ. Furthermore, The unique local optimal solution
λ̂ for EE problem (11) under linear constraint (12) satisfies
λmax , if f (λmax ) ≥ 0
(14)
λ̂ =
otherwise
λ∗ ,
where λ∗ is the global optimum satisfying f (λ∗ ) = 0. f (λ) is
defined as
Δ
f (λ) = ζPT (λ) + PC
p̂k, n ,
(12)
According to [13], we have the following.
Lemma 2: If transmit power PT (R) is strictly convex in rate
vector R, U (R) is strictly quasiconcave, where
U (R) =
Suppose that the given subcarrier assignment is {Dk }k∈K ,
where Dk is the subcarrier set for user k. Obviously, to make
the resource allocation most energy efficient, each user should
consume minimum power resource to reach a target data rate.
It is a water-filling operation and can be solved by Lagrangian
method as follows:
1
1
−
,
0
p̂k, n = max θ ln
2
gk, n
(6)
1
log
(1
+
p
k, n gk, n ) = Rk
2
n∈N
N
(9)
Hence, the user power can be expressed as Pk (λαk |Dk ), and
the total transmit power can be rewritten as
Δ
PT (λ) =
Pk (λαk |Dk ).
(10)
A. Single-User Discussion
Δ
R1
R2
RK
=
= ··· =
.
α1
α2
αK
− λζ ln 2
k∈K
min
n∈Dk
1 + p̂k, n gk, n
gk, n
· αk
(15)
where {p̂k, n }n∈Dk is the power allocation via (6) under the rate
level of λαk .
2142
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 63, NO. 5, JUNE 2014
TABLE I
BOPA A LGORITHM
The essential difference between EE and SE is the transmit
power. For SE issue, the transmitter will consume all the power
resource, whereas for EE aiming to maximize the data rate
per unit of power consumption, there is an optimal transmit
power that may be lower than the maximum in most cases.
We term it the energy-efficient transmit power. Hence, the EE
problem can be viewed as a special SE problem under a special energy-efficient power level. Motivated by such analysis,
energy-efficient subcarrier assignment can be viewed as a twostep operation: 1) find the energy-efficient transmit power; and
2) carry out subcarrier assignment to maximize SE under the
power level obtained in the first step. However, it is generally
very hard to find the optimal solution for both two steps. In
the following, we propose the EETPE and suboptimal SESA
instead, which can reduce the computational complexity and
guarantee a promising performance.
A. Energy-Efficient Transmit Power Estimation
According to Theorem 1, we can obtain the optimal power
allocation easily. The BOPA algorithm is shown in Table I,
where we set the initial points as follows to speed up the
searching of λmax 1 :
⎧
P
Rk ( trans
|Dk )
⎪
K
⎪
⎨ λini, low = min
αk
k∈K
(16)
Ptrans
R
|Dk )
⎪
k(
K
⎪
⎩ λini, high = max
αk
k∈K
where Rk ((Ptrans /K)|Dk ) is the maximum data rate under
power level of (Ptrans /K) for user k, i.e., water-filling with
(Ptrans /K) among subcarrier set Dk .
As shown in Table I, the BOPA algorithm can be divided into
three steps. First, the upper bound of λ related to the maximum
transmit power is initialized by the bisection method. Second,
the derivative of EE at λmax is examined. If the derivative is
positive, then λmax is the optimum according to Theorem 1,
i.e., utilizing all the transmit power would reach the maximum
EE. Otherwise, the optimum λ̂ lies between 0 and λmax , i.e.,
the energy-efficient transmit power is lower than the maximum.
Hence, exploiting the monotonicity of f (λ), λ̂ would be found
by the bisection method in the last step.
IV. L OW-C OMPLEXITY S UBCARRIER A SSIGNMENT
Intuitively, enumerating each possible subcarrier assignment,
the optimum is the one with which applying the BOPA algorithm would obtain the maximum EE. However, the complexity
is too high to be affordable in a real system. Here, we try
to develop a low-complexity policy instead and to achieve a
promising performance meanwhile.
1 Calculate user rate with equal power allocation among users, i.e.,
Rk (Ptrans /K|Dk ). The power level of the user that reaches the highest
proportional rate (Rk /αk ) must be decreased, and meanwhile, the lowest
proportional rate user must increase its power to ensure that all the users
could achieve the same proportional rate level, i.e., λmax . Hence, λmax
must lie between the highest level maxk∈K {(Rk /αk )} and the lowest level
mink∈K {(Rk /αk )}.
The energy-efficient transmit power is estimated via assumptions on flat fading and subcarrier sharing. For flat fading, we
define the average CNR as ḡk = En (gk, n ), ∀ k ∈ K. Since
concave in g, from Jensen’s inequallog2 (1 + pg) is strictly
ity, we know that n∈Dk log2 (1 + pk, n gk, n ) ≤ |Dk | log2 (1 +
pk, n ḡk ), where |Dk | denotes the cardinality of set Dk , i.e., the
number of subcarriers for user k. Furthermore, for subcarrier
sharing, the number of subcarriers assigned to user Nk is
relaxed to a real number on (0, N ]. Hence, the original EE is
upper bounded by the new derived EE. In other words, EE in
frequency-selective fading with subcarrier exclusively taking is
upper bounded by a flat fading assumption with subcarrier sharing. Therefore, the estimated energy-efficient transmit power
would be approximated by the optimal energy-efficient transmit
power of the upper bounded EE here.
From this description, the user rate can be expressed as
Nk
Pk ḡk
log2 1 +
.
(17)
Rk =
N
Nk
Moreover, as the BOPA algorithm in Section II, we also introduce a “rate parameter” η, which is defined as
R2
RK
Δ R1
η=
=
= ··· =
.
(18)
α1
α2
αK
Hence, combining (17) and (18), user power can be formed as
a function of η and Nk , i.e.,
N ηα
k
Nk
− 1 Nk
2
.
(19)
Pk (η, Nk ) =
ḡk
Since the EE optimization problem is converted into a power
consumption minimization one with a given η, we define a
minimum power consumption function of η as2
Δ
P̂ (η) = minimize
Pk (η, Nk ) .
(20)
k∈K
Nk =N
k∈K
2 Apparently, P (η, N ) is strictly decreasing with respect to N . Therek
k
k
fore, the total transmit power is minimized when
N = N rather than
k∈K k
k∈K
Nk < N .
REN et al.: ENERGY-EFFICIENT RESOURCE ALLOCATION IN DOWNLINK OFDM WIRELESS SYSTEMS
2143
TABLE II
TPM A LGORITHM
For P̂ (η), we have the following property and give the proof
in Appendix C.
Lemma 3: The minimum power consumption function P̂ (η)
is strictly convex in η, and its first derivative is
N ηαk
αk 2 N̂k
dP̂ (η)
= N ln 2
dη
ḡk
(21)
k∈K
where {N̂k }k∈K is the solution to (20) at η.
Based on this description, the upper bounded EE under flat
fading and subcarrier sharing can be formed as a function of
η, i.e.,
Δ η
k∈K αk
.
(22)
EEub (η) =
ζ P̂ (η) + PC
According to Lemma 1, EEub (η) is also strictly quasiconcave in η, and there is a unique solution to achieve the
maximum EE. Since (dP̂ (η)/dη) > 0, P̂ (η) is monotonously
increasing with respect to η; hence, the maximum transmit
power constraint can be converted into a linear constraint as
η ≤ ηmax
(23)
where P̂ (ηmax ) = Ptrans .
As Theorem 1, we readily have the following properties. The
proof is similar with Appendix B and is thus omitted.
Theorem 2: The upper bounded EE under flat fading and
subcarrier sharing EEub (η) is first strictly increasing and then
strictly decreasing in η, and EE optimization problem (22)
under linear constraint (23) has a unique optimal solution as
∗
if η ∗ ≤ ηmax
η ,
(24)
η̂ =
ηmax , otherwise
where P̂ (ηmax ) = Ptrans , and η ∗ is the global optimum satisfying δ(η ∗ ) = 0. δ(η) is defined as
Δ
δ(η) = ζ P̂ (η) + PC − ζN ln 2
αk 2
k∈K
N ηαk
N̂k
ḡk
(25)
where {N̂k }k∈K is the solution to (19) at η.
Based on Theorem 2, the estimated energy-efficient transmit
power can be obtained readily as
∗
∗
P̃ = P̂ (η ), if P̂ (η ) ≤ Ptrans
(26)
Ptrans , otherwise.
Before describing the whole algorithm, we should propose
the solution to power minimization problem (20) at first.
Obviously, (20) is a standard convex optimization. Karush–
Kuhn–Tucker conditions can be listed as follows:
N ηαk
1 − N ηαNkk ln 2 2 Nk − 1 + ḡk γ = 0
(27)
N − k∈K Nk = 0
where γ is the Lagrangian multiplier. Define γk (Nk ) as
N ηα
k
N ηαk ln 2
Δ
2 Nk
γk (Nk ) = 1 − 1 −
Nk
(28)
and we can have
dγk (Nk )
N ηαk ln 2 NNηαk
=−
·2 k
dNk
N2
k
N ηα
k
N ηαk ln 2
N ηαk ln 2
+ 1−
·
· 2 Nk
Nk
Nk2
N ηαk
(N ηαk ln 2)2
=−
· 2 Nk < 0.
(29)
2
Nk
Hence, with a given γ, we can use the bisection method to get
{Nk }k∈K . Furthermore, we can also
use the bisection method to
get the optimum γ ∗ to satisfy N = k∈K Nk . The transmission
power minimization (TPM) solution can be regarded as a twolayer bisection method, and the detail is shown in Table II.
Generally, the number of subcarriers assigned to users satisfies
1 ≤ Nk ≤ N ; hence, we set the initial points for searching
γ ∗ as
⎧
ηα
⎨ γlow = min 1−(1−ηαḡk ln 2)2 k
k
k∈K (30)
⎩ γhigh = max 1−(1−N ηαk ln 2)2N ηαk .
ḡk
k∈K
By combining it with the TPM algorithm, we can propose
the EETPE algorithm as in Table III. The initial points can be
set as3
ηlow = 0
ḡk
(31)
ηhigh = max log2 1 + Ptrans
αk .
KN
k∈K
In step 1, we first calculate δ(ηhigh ) with {Nk }k∈K , which is
obtained by the TPM algorithm. If δ(ηhigh ) is nonnegative, then
δ(ηmax ) would be positive since δ(η) is strictly decreasing with
respect to η and ηhigh > ηmax . Hence, the maximum transmit
power would be the estimated energy-efficient transmit power.
Otherwise, if δ(ηhigh ) is negative, the global optimum η ∗ would
lie between ηlow and ηhigh . Then, in step 2, η ∗ is found by
the bisection method. Finally, the estimated energy-efficient
transmit power is determined according to (26).
3 Suppose that all N subcarriers are assigned to each user and then allocate
the total power Ptrans among users equally. The maximum proportional
rate level maxk∈K {(Rk /αk )} is much higher than ηmax that is obtained
from P̂ (ηmax ) = Ptrans under any possible subcarrier assignment satisfying
N = N.
k∈K k
2144
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 63, NO. 5, JUNE 2014
TABLE III
EETPE A LGORITHM
the subcarrier is assigned according to the max–min policy as
aforementioned.
V. C OMPLEXITY S TUDY
TABLE IV
SESA A LGORITHM
Based on our previous analysis, the proposed EE-suboptimal
algorithm can be implemented in three steps: 1) EETPE;
2) SESA; and 3) BOPA algorithms.
In the EETPE algorithm, suppose that the number of iterations of bisection for finding η ∗ is TOL , and the number
of iterations for finding γ ∗ and that for calculating {Nk }k∈K
in the TPM algorithm are TML and TIL , respectively;4 thus,
the complexity of the EETPE algorithm is O(TOL TML TIL K).
For the SESA algorithm, the complexity is mainly caused by
channel sorting and is thus O(KN log2 N ). Let TWF be the
number of iterations of a practical algorithm for water-filling
(6), such as 1-D search; the complexity of BOPA algorithm is
thus O(TOL TWF K).
On the other hand, the optimal solution EE-optimal can be
obtained by exhaustive search of any subcarrier assignment
combining the BOPA algorithm. Obviously, there are K N kinds
of subcarrier assignment; hence, the complexity of the optimal
solution is O(TOL TWF K N +1 ).
The complexity of both the optimal and proposed suboptimal
solutions is summarized in Table V.
VI. S IMULATIONS
A. Simulation Environment
B. Suboptimal Spectral-Efficient Subcarrier Assignment
After finding the estimated energy-efficient transmit power,
the subcarrier assignment can be carried out with the objective
of maximizing the SE. This is a spectral-efficient resourceallocation problem, which has been investigated thoroughly
in traditional OFDM systems [24], [26], OFDM-based relay
systems, [27], and cognitive radio networks as well [28]. The
discrete stochastic optimization policy proposed in [28] is much
more complex since power allocation is evolved in each subcarrier assignment attempt; although [26] could be more effective
via water-filling, it is also more complex compared with that of
[24]. To this end, the equal-power-distribution-based method in
[24] is adopted to carry out subcarrier assignment in this paper.
It is of low complexity and could reach average performance.
The key idea is maximizing the minimum proportional rate
(Rk /αk ). That is, the user that has the minimum (Rk /αk )
would be assigned with the corresponding best subcarrier, and
its data rate would be updated with equal power distribution
(P̃ /|Dk |). The SESA is shown in Table IV. In step 1, each user
chooses its best subcarrier to initialize user rate, and in step 2,
In simulations, we assume that each user’s subcarrier signal
undergoes identical Rayleigh fading independently. The total
transmit power Ptrans and the circuit power PC are 1 and
0.2 W, respectively. The reciprocal of drain efficiency of power
amplifier ζ is 0.38. The average CNR is defined as En (gk, n )
for all k, where En (gk, n ) denotes statistical expectation of
gk, n with respect to n. All the experiment data are obtained
by Monte Carlo simulations over 1000 channel realizations.
The EE-optimal and SE methods in [24] are considered the
comparison schemes with our proposed EE-suboptimal solution.
B. Computational Complexity
Figs. 2 and 3 show the average running time of EEsuboptimal. The simulation was run on a personal computer,
and the software and hardware conditions are: Windows XP
Home Edition, Pentium-M 1.73-GHz processor, and 2-GB
memory. The average CNR is 5 dB and each user has the same
rate requirement.
Fig. 2 gives the comparison between EE-optimal and EEsuboptimal. Since EE-optimal is a brutal-force-search-based
scheme, we fix the number of users as 2, and run simulation
with a small number of subcarriers. It is clear that the EEoptimal is much more complex than EE-suboptimal. The running time of EE-optimal grows exponentially with the number
of subcarriers, whereas that of EE-suboptimal only increases
linearly.
4 These three iterations can be viewed as outer layer, middle layer, and inner
layer search of the EETPE algorithm.
REN et al.: ENERGY-EFFICIENT RESOURCE ALLOCATION IN DOWNLINK OFDM WIRELESS SYSTEMS
2145
TABLE V
C OMPUTATIONAL C OMPLEXITY C OMPARISON
Fig. 2. Average running time versus the number of subcarriers. The number
of users is two, and the average CNR is 5 dB for both two users.
Fig. 3. Average running time versus the number of subcarriers. Average CNR
is 5 dB for all users.
The average running times of EE-suboptimal under different
scenarios are given in Fig. 3. We can observe that even in
condition of 256 subcarriers and 32 users, EE-suboptimal takes
about 0.35 s to complete one resource allocation. It is still much
more efficient than EE-optimal with only two users and ten
subcarriers, which takes 20 s according to Fig. 2.
C. EE and SE Performance
Figs. 4 and 5 show the EE and SE, respectively. To reduce
the simulation time, we set ten subcarriers and two users. Proportional rate constraints are set to 1:1 and 1:10, respectively, to
evaluate the effect of the system fairness requirement. As shown
Fig. 4. EE versus average CNR. The number of subcarriers is ten and the
number of users is two. All users have the same channel condition.
Fig. 5. SE versus average CNR. The number of subcarriers is ten and the
number of users is two. All users have the same channel condition.
in Fig. 4, EE methods are more effective in terms of EE than
the SE method in the high-SNR regime. While in the low-SNR
regime, the SE and EE policies could reach a similar EE level.
In Fig. 5, we can also see a similar trend in terms of SE, i.e.,
the SE method is more spectral efficient than the EE method in
the high-SNR regime, whereas in the low-SNR regime the two
schemes are similar with each other.
It can be explained by the reason that the rate function of
power is concave. Hence, the rate increment of each power
unit is higher in the low-CNR regime than that in the highCNR regime, and the energy-efficient transmit power would
be the maximum transmit power with high possibility. In fact,
2146
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 63, NO. 5, JUNE 2014
Fig. 6. EE versus average CNR of user 2. The number of subcarriers is ten
and the number of users is two. User 2’s average CNR is 5 dB more than that
of user 1.
Fig. 8. EE versus average CNR. The number of subcarriers is 64 and the
number of users is four. All users have the same channel condition.
Fig. 7. SE versus average CNR of user 2. The number of subcarriers is ten
and the number of users is two. User 2’s average CNR is 5 dB more than that
of user 1.
Fig. 9. SE versus average CNR. The number of subcarriers is 64 and the
number of users is four. All users have the same channel condition.
when the energy-efficient transmit power is the same as the
maximum transmit power, the SE scheme and the EE scheme
are equivalent. On the other hand, in the high-CNR regime,
the energy-efficient transmit power would be lower than the
maximum transmit power since the power would be less efficient as rate increases slowly. Hence, the discrepancy of the two
designs is significant, i.e., the SE method would achieve more
throughputs, and the EE method could be more efficient with
each unit of power consumption.
Mathematically, as gk, n is small, i.e., in the low-CNR
regime, f (λmax ) and δ(ηmax ) are more likely to be positive,
which can be observed from (15) and (25). This means that the
maximum transmit power is more likely to be the most energy
efficient, which verifies our earlier discussion.
In Figs. 4 and 5, we can also see that either EE or SE under
proportion 1:1 achieves a higher performance than that of 1:10.
It is reasonable since the channel conditions are similar between
two users and neither has priority over the other.
To show the connection between proportion and channel
condition more clearly, we set different average CNRs for two
users in Figs. 6 and 7. For simplicity, average CNR of user 2 is
5 dB more than that of user 1. The other parameters are the same
as in Figs. 4 and 5. Apparently, when the proportion is 1:10,
system performance reaches a higher level in terms of either
EE or SE. In short, when the user with better channel condition
requires more data rate, the system performance is better.
Figs. 8–11 show a scenario of 64 subcarriers and four users.
Proportional rate constraints are set to 1:1:1:1 and 1:1:1:10 in
this case. In Figs. 8 and 9, all users have the same average CNR,
and in Figs. 10 and 11, user 4’s average CNR is 5 dB higher than
that of the others.
Considering a larger input case, we also give a scenario of
256 subcarriers and 16 users in Figs. 12–15. Four users have
higher rate requirements and the proportional rate constraints
are set to 1:· · ·1:1:1:1 and 1:· · ·:10:10:10:10. In Figs. 12 and
13, all users have the same average CNRs, whereas in Figs. 14
REN et al.: ENERGY-EFFICIENT RESOURCE ALLOCATION IN DOWNLINK OFDM WIRELESS SYSTEMS
2147
Fig. 10. EE versus average CNR of user 4. The number of subcarriers is 64
and the number of users is four. User 4’s average CNR is 5 dB more than that
of the others.
Fig. 12. EE versus average CNR. The number of subcarriers is 256 and the
number of users is 16. All users have the same channel condition.
Fig. 11. SE versus average CNR of user 4. The number of subcarriers is 64
and the number of users is 4. User 4’s average CNR is 5 dB more than that of
the others.
Fig. 13. SE versus average CNR. The number of subcarriers is 256 and the
number of users is 16. All users have the same channel condition.
and 15, the last four users’ average CNRs would be 5 dB higher
than that of the others.
Similar with Figs. 4–7, we can also see the relation between
SE design and EE design in these two scenarios. In the highCNR regime, the two schemes are distinct from each other,
whereas in the low-CNR regime, the discrepancy would not be
significant. Moreover, the connection between system performance and proportion requirements is illustrated as well. When
more data rate is distributed to the user that has a higher channel
gain, the system performance would be enhanced in terms of
both EE and SE.
portional rate constraints are incorporated into the proposed
resource-allocation approaches. First, the optimal power allocation with a given subcarrier assignment is carried out.
Then, a two-step suboptimal method for assigning subcarriers
is proposed to avoid the expense and complexity of exhaustive
search. In the first step, the energy-efficient transmit power
is estimated via an upper bounded EE with assumptions on
flat fading and subcarrier sharing. In the second step, the
traditional spectral-efficient scheme is introduced to complete
the subcarrier assignment. The numerical results show that the
proposed algorithm could approach the optimum and is more
energy efficient than the traditional spectral-efficient method in
the high-CNR regime. In the low-CNR regime, the SE design
and the EE design are similar to the result of the concavity of
Shannon capacity formulation. This observation simplifies the
EE design to a certain extent since we can turn to a relatively
less complex SE policy in this situation.
VII. C ONCLUSION
In this paper, we have investigated downlink energy-efficient
and fair resource allocation in OFDM wireless systems to
support mobile multimedia communications. Both EE and pro-
2148
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 63, NO. 5, JUNE 2014
1
2
3
{r̂k,
n }n∈Dk , {r̂k, n }n∈Dk , and {r̂k, n }n∈Dk be the rate distribution accordingly; thus, we can have
Pk Rk1 |Dk + (1 − )Pk Rk2 |Dk
=
p̂1k, n + (1 − )
p̂2k, n
n∈Dk
n∈Dk
1
2
2N r̂k,
2N r̂k,
n − 1
n − 1
=
+ (1 − )
gk, n
gk, n
n∈Dk
n∈Dk
1
2
N r̂
N r̂
2 k, n − 1 + (1 − ) 2 k, n − 1
=
gk, n
n∈Dk
1
2
3
+(1−)N r̂k,
N r̂k,
2N r̂k,
n
n − 1 b 2
n − 1
≥
≥
gk, n
gk, n
n∈Dk
n∈Dk
(33)
= Pk Rk1 + (1 − )Rk2 |Dk
a
Fig. 14. EE versus average CNR of the last four users. The number of
subcarriers is 256 and the number of users is 16. The last four users’ average
CNR is 5 dB more than that of the others.
where a utilizes the convexity of 2x − 1, and b utilizes the
definition of Pk (Rk |Dk ). This shows the convexity.
Let {r̂k, n + Δrk, n }n∈Dk be the rate
distribution under user
rate level Rk + ΔRk , where ΔRk = n∈Dk Δpk, n ; then, we
can have (32), shown at the bottom of the page.
This gives the derivative.
A PPENDIX B
P ROOF OF T HEOREM 1
Proof: Based on the derivative of Pk (λαk |Dk ) given in
Lemma 1, the derivative of PT (λ) can be obtained as
dPT (λ) dPk (λαk |Dk )
=
· αk
PT (λ) =
dλ
d(λαk )
k∈K
1 + p̂k, n gk, n
= N ln 2
min
· αk (36)
n∈Dk
gk, n
k∈K
Fig. 15. SE versus average CNR of the last four users. The number of
subcarriers is 256 and the number of users is 16. The last four users’ average
CNR is 5 dB more than that of the others.
A PPENDIX A
P ROOF OF L EMMA 1
Proof: Let {p̂1k, n }n∈Dk , {p̂2k, n }n∈Dk , and {p̂3k, n }n∈Dk
be the power allocation under user rate Rk1 , Rk2 , and Rk1 +
(1 − )Rk2 based on (6), respectively, where 0 ≤ ≤ 1. Let
where {p̂k, n }n∈Dk is the single-user power allocation under
rate level of λαk .
Obviously, PT (λ) is monotonously increasing in λ; hence,
let PT (λ) be the second derivative of PT (λ), we have PT (λ) >
Δ 0. For simplicity, define α = K
k=1 αk ; we can also have
dEE(λ) α (ζPT (λ) + PC ) − (λα)ζPT (λ)
=
dλ
(ζPT (λ) + PC )2
(ζPT (λ) − λζPT (λ) + PC ) · α
=
(ζPT (λ) + PC )2
f (λ) · α
=
(ζPT (λ) + PC )2
− n∈Dk
Pk (Rk + ΔRk |Dk ) − Pk (Rk |Dk )
lim
= lim
ΔRk →0
ΔRk
ΔRk
ΔRk →0
2N r̂k, n (2N Δrk, n −1)
2N r̂k, n +N Δrk, n −2N r̂k, n
min
min
n∈Dk
n∈Dk
gk, n
gk, n
= lim
= lim
ΔRk →0
ΔRk →0
ΔR
ΔR
k
k
N r̂ k, n
· n∈Dk Δrk, n · N ln 2
min 2 gk, n
1 + p̂k, n gk, n
n∈Dk
= lim
= N ln 2 · min
ΔRk →0
n∈Dk
ΔRk
gk, n
dPk (Rk |Dk )
=
dRk
min
n∈Dk
2N r̂k, n +N Δrk, n −1
gk, n
(37)
2N r̂k, n −1
gk, n
(32)
REN et al.: ENERGY-EFFICIENT RESOURCE ALLOCATION IN DOWNLINK OFDM WIRELESS SYSTEMS
Δ
where f (λ) = ζPT (λ) − λζPT (λ) + PC . The derivative of
f (λ) can be given as
Since the Hessian matrix is negative semidefinite, Pk (η, Nk ) is
convex. We further proof the convexity of P̂ (η) in the following.
Let {N̂k1 }k∈K , {N̂k2 }k∈K , and {N̂k3 }k∈K be the subcarrier
assignments to reach the minimum power consumption under
rate parameter η1 , η2 , and η1 + (1 − )η2 , respectively, where
0 ≤ ≤ 1. Therefore, we can have
f (λ) = (ζPT (λ) − λζPT (λ) + PC )
= ζPT (λ) − ζPT (λ) − λζPT (λ)
= −λζPT (λ) < 0.
(38)
P̂ (η1 ) + (1 −)P̂ (η2 )
Pk η1 , N̂k1 + (1 − )
Pk η2 , N̂k1
=
Hence, f (λ) is strictly decreasing in λ. Since f (0) = PC > 0,
EE(λ) is either strictly increasing, or first strictly increasing and
then strictly decreasing in λ. On the other hand, we know that
EE(0) = 0, and
lim EE(λ) = lim
λ→∞
λ→∞
k∈K
a
≥
Therefore, from the given discussion, it can be derived that
EE(λ) is first strictly increasing and then strictly decreasing
in λ. From (37), it can be further derived that the unique
global optimum λ∗ satisfies f (λ∗ ) = 0. Local optimum λ̂ for
the constrained EE can be also obtained readily, as expressed
in (14).
This completes the proof.
≥
N ηαk
N̂k
N̂k
2
2 N̂k +ΔN̂k ·
−1
N̂k
N̂k
−
ḡk
−1
N̂k
=
2
2 N̂k +ΔN̂k −1
ḡk
Δη
k∈K
N (η+Δη)αk
N ηαk
N ηαk
2
N̂k
k∈K
−1
N̂k
ḡk
2
N (Δη)αk
N̂k
N̂k
· lim
ḡk
k∈K
k∈K
2
Accordingly, we can further have (34) and (35), shown at the
bottom of the page.
Δη
Δη
Δη→0
= N ln 2
(42)
k∈K
2
dP̂ (η)
P̂ (η + Δη) − P̂ (η)
k∈K
= lim
≥ lim
Δη→0
Δη→0
dη
Δη
N (Δη)αk N ηαk
= lim
Pk η1 + (1 − )η2 , N̂k3
where we use the convexity of Pk (η, Nk ) at a and the definition
of P̂ (η) at b.
Now, we calculate the first derivative of P̂ (η). Let {N̂k }k∈K
and {N̂k + ΔN̂k }k∈K be the subcarrier assignments under rate
parameter η and η + Δη, respectively. From the definition of
P̂ (η), we can have
N ηαk
N̂k+ ΔN̂k
N̂
−
1
+
Δ
N̂
2
k
k
P̂ (η) ≤
(43)
ḡk
k∈K
N (η+Δη)α
k
N̂k
N̂k
−
1
2
P̂ (η + Δη) ≤
.
(44)
ḡk
k∈K
ḡk
Δη→0
k∈K
N (Δη)αk
·
k∈K
= lim
Pk η1 + (1 − )η2 , N̂k1 + (1 − )N̂k2
k∈K
and then the Hessian matrix can be derived as
Nk
2 −1
N
α
ln
2
k
∇2 Pk (η, Nk ) = η
ḡk . (41)
·η
−1 Nηk
Nk
2
= P̂ (η1 + (1 − )η2 )
Proof: First, we prove that Pk (η, Nk ) is convex. The
Jacobian matrix can be obtained as
N ηαk
Nk
ln
2
·
2
N
α
k
N ηαk
∇Pk (η, Nk ) = ḡk (40)
1 − N ηαNkk ln 2 2 Nk − 1
k∈K
Pk η1 , N̂k1 + (1 − )Pk η2 , N̂k2
k∈K
b
A PPENDIX C
P ROOF OF L EMMA 3
P̂ (η + Δη) − P̂ (η)
dP̂ (η)
= lim
≤ lim
Δη→0
Δη→0
dη
Δη =
λα
α
= 0.
= lim
λ→∞ ζPT (λ)
ζPT (λ) + PC
(39)
2149
Δη→0
N (η+Δη)αk
N̂k +ΔN̂k
−1
2
N̂k
Δη
= N ln 2
(N̂k +ΔN̂k )
−
ḡk
−1
2
k∈K
N ηαk
k∈K
N̂k
ḡk
2 N̂k +ΔN̂k −1
N ηαk
αk
(34)
(N̂k +ΔN̂k )
ḡk
Δη
N̂k
N ηαk
=
2 N̂k +ΔN̂k (N̂k + ΔN̂k )
k∈K
ḡk
N (Δη)αk
2 N̂k +ΔN̂k − 1
· lim
Δη→0
Δη
N ηαk
N̂k
ḡk
αk
(35)
2150
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 63, NO. 5, JUNE 2014
Therefore, according to the squeeze theorem, we can have
N ηαk
2 N̂k αk
dP̂ (η)
= N ln 2
.
dη
ḡk
(45)
k∈K
This completes the proof.
R EFERENCES
[1] H. Moustafa, N. Marechal, and S. Zeadally, “Mobile multimedia applications: Delivery technologies,” IEEE IT Prof., vol. 14, no. 5, pp. 12–21,
Sep./Oct. 2012.
[2] F. Adachi, M. Sawahashi, and H. Suda, “Wideband DS-CDMA for
next-generation mobile communications systems,” IEEE Commun. Mag.,
vol. 36, no. 9, pp. 56–69, Sep. 1998.
[3] T. Hwang, C. Yang, G. Wu, S. Li, and G. Ye Li, “OFDM and its wireless applications: A survey,” IEEE Trans. Veh. Technol., vol. 58, no. 4,
pp. 1673–1694, May 2009.
[4] F. Adachi, “Evolution towards broadband wireless systems,” in Proc. Int.
Symp. Wireless Pers. Multimedia Commun., 2002, vol. 1, pp. 19–26.
[5] H. Yin and H. Liu, “An efficient multiuser loading algorithm for OFDMbased broadband wireless systems,” in Proc. IEEE Global Telecommun.,
2000, vol. 1, pp. 103–107.
[6] Z. Mao and X. Wang, “Efficient optimal and suboptimal radio resource
allocation in OFDMA system,” IEEE Trans. Wireless Commun., vol. 7,
no. 2, pp. 440–445, Feb. 2008.
[7] C. Wong, R. Cheng, K. Lataief, and R. Murch, “Multiuser OFDM with
adaptive subcarrier, bit, and power allocation,” IEEE J. Sel. Areas Commun., vol. 17, no. 10, pp. 1747–1758, Oct. 1999.
[8] 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.
[9] Y.-B. Lin, T.-H. Chiu, and Y. Su, “Optimal and near-optimal resource
allocation algorithms for OFDMA networks,” IEEE Trans. Wireless Commun., vol. 8, no. 8, pp. 4066–4077, Aug. 2009.
[10] G. Y. Li, Z. Xu, C. Xiong, C. Yang, S. Zhang, Y. Chen, and S. Xu,
“Energy-efficient wireless communications: Tutorial, survey, and open
issues,” IEEE Wireless Commun., vol. 18, no. 6, pp. 28–35, Dec. 2011.
[11] G. Li and H. Liu, “On the optimality of the OFDMA network,” IEEE
Commun. Lett., vol. 9, no. 5, pp. 438–440, May 2005.
[12] Q. Shi, W. Xu, D. Li, Y. Wang, X. Gu, and W. Li, “On the energy
efficiency optimality of OFDMA for SISO-OFDM downlink system,”
IEEE Commun. Lett., vol. 17, no. 3, pp. 541–544, Mar. 2013.
[13] 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, Feb. 2010.
[14] C. Isheden and G. P. Fettweis, “Energy-efficient multi-carrier link adaptation with sum rate-dependent circuit power,” in Proc. IEEE Global
Telecommun., 2010, pp. 1–6.
[15] G. Miao, N. Himayat, Y. Li, and D. Bormann, “Energy efficient
design in wireless OFDMA,” in Proc. IEEE Int. Conf. Commun., 2008,
pp. 3307–3312.
[16] G. Miao, N. Himayat, G. Y. Li, and S. Talwar, “Low-complexity energyefficient scheduling for uplink OFDMA,” IEEE Trans. Commun., vol. 60,
no. 1, pp. 112–120, Jan. 2012.
[17] R. Mazumdar, L. G. Mason, and C. Douligeris, “Fairness in network optimal flow control: Optimality of product forms,” IEEE Trans. Commun.,
vol. 39, no. 5, pp. 775–782, May 1991.
[18] C. Xiong, G. Y. Li, S. Zhang, Y. Chen, and S. Xu, “Energy-efficient
resource allocation in OFDMA networks,” IEEE Trans. Commun., vol. 60,
no. 12, pp. 3767–3778, Dec. 2012.
[19] C. Xiong, G. Y. Li, S. Zhang, Y. Chen, and S. Xu, “Energy-and spectralefficiency tradeoff in downlink OFDMA networks,” IEEE Trans. Wireless
Commun., vol. 10, no. 11, pp. 3874–3886, Nov. 2011.
[20] D. W. K. Ng and R. Schober, “Energy-efficient resource allocation in
OFDMA systems with large numbers of base station antennas,” in Proc.
IEEE Int. Conf. Commun., 2012, pp. 5916–5920.
[21] D. W. K. Ng, E. S. Lo, and R. Schober, “Energy-efficient resource allocation for secure OFDMA systems,” IEEE Trans. Veh. Technol., vol. 61,
no. 6, pp. 2572–2585, Jul. 2012.
[22] D. W. K. Ng, E. S. Lo, and R. Schober, “Energy-efficient resource allocation in multi-cell OFDMA systems with limited backhaul capacity,” in
Proc. IEEE Wireless Commun. Netw. Conf., 2012, pp. 1146–1151.
[23] Y. Chen, S. Zhang, S. Xu, and G. Y. Li, “Fundamental trade-offs on
green wireless networks,” IEEE Commun. Mag., vol. 49, no. 6, pp. 30–37,
Jun. 2011.
[24] Z. Shen, J. Andrews, and B. 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.
[25] S. Cui, A. J. Goldsmith, and A. Bahai, “Energy-constrained modulation
optimization,” IEEE Trans. Wireless. Commun., vol. 4, no. 5, pp. 2349–
2360, Sep. 2005.
[26] C. Mohanram and S. Bhashyam, “A sub-optimal joint subcarrier and
power allocation algorithm for multiuser OFDM,” IEEE Commun. Lett.,
vol. 9, no. 8, pp. 685–687, Aug. 2005.
[27] Y. Pan, A. Nix, and M. Beach, “Distributed resource allocation for
OFDMA-based relay networks,” IEEE Trans. Veh. Technol., vol. 60, no. 3,
pp. 919–931, Mar. 2011.
[28] R. Xie, F. R. Yu, and H. Ji, “Dynamic resource allocation for heterogeneous services in cognitive radio networks with imperfect channel sensing,” IEEE Trans. Veh. Technol., vol. 61, no. 2, pp. 770–780, Feb. 2012.
Zhanyang Ren received the B.E. and Ph.D. degrees from Beijing University of Posts and Telecommunications, Beijing, China, in 2008 and 2013,
respectively.
He then joined the Wireless Network Research
Department, Huawei Technologies Company Ltd.,
Beijing, China. His current research interests include
wireless communications theory and wireless communications systems.
Shanzhi Chen (SM’04) received the Ph.D. degree
from Beijing University of Posts and Telecommunications, Beijing, China, in 1997.
In 1994, he joined Datang Telecom Technology &
Industry Group, where he has served as the Chief
Technology Officer since 2008. From 1999 to 2011,
he was a member of the Steering Expert Group on
Information Technology of the 863 Program of
China. He is currently the Director of the State Key
Laboratory of Wireless Mobile Communications,
China Academy of Telecommunications Technology, Beijing, China, and a board member of Semiconductor Manufacturing International Corporation, Shanghai, China. He has considerable contributions to
time-division (TD) synchronous code-division multiple-access third-generation
industrialization and TD Long-Term Evolution Advanced fourth-generation
standardization. His current research interests include wireless mobile communications, Internet of Things, and emergency communications.
Dr. Chen received the State Science and Technology Progress Award in 2001
and 2012.
Bo Hu received the Ph.D. degree in communications
and information systems from Beijing University
of Posts and Telecommunications (BUPT), Beijing,
China, in 2006.
He is currently an Associate Professor with the
State Key Laboratory of Networking and Switching
Technology, BUPT. His current research interests include future wireless mobile communication systems
and mobile-centric networking.
Weiguo Ma received the B.Sc. degree in electrical
engineering and the Ph.D. degree in signal and information systems from Beijing Institute of Technology,
Beijing, China, in 1993 and 1998, respectively.
He is currently a Professor with the State Key Laboratory of Wireless Telecommunications, China Academy of Telecommunications Technology, Beijing.
He also served as the Chair to several research
projects, including the National High-Technology
R&D 863 Program of China as well as industrybased projects. His current research interests include
prototype verification of key evolution technology of time-division Long-Term
Evolution and beyond.