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Resource Allocation for Multiuser Cooperative OFDM Networks Who


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									  Resource Allocation for Multiuser Cooperative OFDM
  Networks: Who Helps Whom and How to Cooperate
            Zhu Han∗ , Thanongsak Himsoon+ , W. Pam Siriwongpairat+ , and K. J. Ray Liu
                         ∗ Department of Electrical and Computer Engineering

                               University of Houston, Houston, TX 77204
                       + Meteor Communications Corporation, Kent, WA 98032

        Department of Electrical and Computer Engineering and the Institute of Systems Research
                           University of Maryland, College Park, MD 20742

   Abstract— Cooperative transmissions have been shown              in wireless networks. The power control constantly ad-
to be able to greatly improve system performance by                 justs the transmit power so as to maintain the received
exploring the broadcasting nature of wireless channels              link quality, while improving the system performance.
and cooperation among users. We focus, in this paper,
on leveraging cooperation for resource allocation among             In some applications such as wireless sensor networks
users such that the network performance can be improved.            in which each user is powered by batteries, optimizing
Two important questions are answered: who should help               the power management can greatly extend the network
whom among the distributively located users, and how                lifetime. Much work has been done for resource al-
many resources the users should use for cooperation                 location for multiuser wireless networks. In [1], the
to improve the performance. To answer these questions,
a power-optimization, subcarrier-allocation, and relay-             authors gave an overview of radio resource management
selection problem is formulated over a multiuser OFDM               for wireless networks. In [2] and [3], a closed-loop
network, which is applicable to systems such as wireless            power control framework was proposed and proved to
LAN networks. In the multiuser OFDM network, cooper-                converge to a unique optimal point. In [4], a second
ation among different users is conducted by assigning the           order power control algorithm was proposed to improve
subcarriers of the helping users to relay a certain part of
the helped users’ data, while maintaining the desired rates         the convergence speed. In [5], rate adaption and power
of both helping users and helped users by means of power            control were combined to increase system throughput. In
control and rate adaption. In this way, the bandwidth effi-          [6] and [7], power control was combined with antenna
ciency of the multiuser OFDM system with cooperation is             array processing to improve the network performance.
the same as that of the non-cooperative OFDM system. The            In [8], an algorithm considering time, space, and mul-
formulated optimization problem is an assignment problem
for subcarrier usage and corresponding bit loading as               tiuser diversity was proposed for enhancing the system
well as power control. We provide an approximate closed-            efficiency.
form solution for a two-user two-subcarrier case. Then, a              Recently, cooperative transmissions have gained much
suboptimal heuristic algorithm for multiple-user multiple-          attention as an emerging transmit strategy for future
subcarrier case is proposed and implemented in the base             wireless networks. The cooperative transmissions effi-
station to solve the formulated N P hard problem. From
the simulation results, the proposed scheme achieves up             ciently take advantage of the broadcasting nature of wire-
to 50% overall power saving for the two-user system and             less networks, as well as exploit the inherent spatial and
19% ∼ 54% overall power saving for the multiuser case               multiuser diversities. By exploring space diversity and
with random locations, compared to the current multiuser            multiuser diversity, the cooperative transmission scheme
OFDM system without cooperative diversity. The proposed             and implementation algorithms were proposed in [9],
scheme is also compared to a much more complicated
OFDMA system.                                                       assuming full channel state information at the cooperat-
                                                                    ing nodes that utilize beamforming. In [10], transmission
                                                                    protocols of the cooperative transmissions were classi-
                     I. I NTRODUCTION                               fied into different approaches and their performance was
                                                                    analyzed in terms of outage probabilities. The work in
  Resource allocation such as power control has long                [11] analyzed more complicated transmitter cooperative
been regarded as an effective way to dynamically combat             schemes involving dirty paper coding. The authors in
channel fluctuations and reduce co-channel interference              [12] provided rigorous analysis on symbol error rate
  Part of this work was presented in IEEE Wireless Communications   and optimum power allocation for the multi-node decode
and Networking Conference, 2005                                     and forward protocol. Energy-efficient transmission was
considered for broadcast networks in [13]. In [14],                can maintain the desired rates of both helping users and
oversampling was combined with the intrinsic properties            helped users. By doing this, the power of the helped users
of orthogonal frequency division multiplexing (OFDM)               is greatly reduced, because some of its subcarriers have
symbols, in the context of maximal ratio combining                 cooperative diversity and consequent power-reduction
(MRC) and amplify and forward, so that this rate loss              offered by the helping users. On the other hand, the
of cooperative transmissions can be overcome. In [15],             power of the helping users is slightly increased, because
the authors evaluated cooperative-diversity performances           the helping users not only have to relay the information
when the best relay is chosen according to the average             of the helped users, but also utilize the remaining sub-
SNR, and the outage probability of relay selection based           carriers to transmit their own data by higher modulation
on the instantaneous SNR. In [16], the authors proposed            and power. By careful design, the overall system power
a distributed relay selection scheme that requires limited         can be reduced. For such a shared subcarriers cooperative
network knowledge and is based on instantaneous SNRs.              scheme, there is no extra stage purely dedicated for relay.
These ideas are also working their way into standards;             In this way, the bandwidth efficiency of the multiuser
e.g., the IEEE 802.16 (WiMAX) standards body for                   OFDM systems with cooperation is the same as that
future broadband wireless access has established the               of the non-cooperative OFDM systems such as the
802.16j Relay Task Group to incorporate cooperative                IEEE 802.11a/g standard [20]. The optimization for the
relaying mechanisms into this technology.                          system resource allocation is performed by modifying
   Most of the existing cooperative transmission works             the OFDM subcarrier assignment for cooperation and
concentrate on improving the one-to-one link quality,              the corresponding bit loading as well as power control.
while most resource allocation works in the literature
have not considered cooperation transmission yet. Due
to the limited of radio resources and increasing demand               Furthermore, we develop a suboptimal algorithm im-
of variety of services, it is important to consider resource       plemented in the base station to solve the proposed
allocation to fully exploit the cooperative transmission           N P hard problem. We analyze the situation which
paradigm. In this paper, we consider resource allo-                users should be helped or helping, i.e., we answer the
cation among multiple users to optimize the system                 question of “who helps whom”. In addition, we optimize
performance by taking into consideration the cooperative           how many OFDM subcarriers and how much power
transmission strategy. In multiuser wireless networks,             should be used for helping others, i.e., we answer the
there are many open resource allocation questions for              question of “how to cooperate”. An approximate closed-
cooperative transmissions. The most important ones are             form solution for the two-user two-subcarrier case is
who should help whom among the distributively located              derived. Then, a heuristic algorithm for the multiple-
users (i.e. “who helps whom” and relay selection),                 user multiple-subcarrier case is constructed. Performance
and how many resources (like power and subcarrier)                 comparison with a bound using OFDM multiple access
the users should utilize for cooperation to improve the            (OFDMA) system is also studied. From the simulation
system performance (“how to cooperate”). In [17], the              results, the proposed scheme can save up to 50% of
basic problem was formulated for 2-user case and some              the overall transmit power for the two-user system and
preliminary heuristics were constructed. In this paper, we         19% ∼ 54% overall power saving for the multiple-user
aim at answering these two major questions in multiple-            case with random locations, compared with the current
user case and in a more rigorous way.                              OFDM systems without cooperative diversity.
   To answer the questions, we consider the power-
control, subcarrier-allocation, and relay-selection prob-
lem that seeks to minimize the system power over a                    The rest of the paper is organized as follows: In
multiuser OFDM network [18]- [19], which is a key                  Section II, we give the multiuser OFDM system model
element in 4G cellular networks, wireless metropolitan             and provide the traditional non-cooperative transmission
area network (WMAN), and wireless local area networks              solution using the waterfilling method. We construct
(WLAN). In most of current OFDM systems, each time                 the cooperative transmission over multiuser OFDM net-
only one user transmits at a time as in a TDD system,              works and formulate the cooperative optimization as an
and users are scheduled for transmission over different            assignment problem in Section III. We provide algo-
times. The cooperation can be implemented by assigning             rithms to solve the problem in Section IV. Simulation
some subcarriers of the helping users to relay parts of the        results are provided in Section V. Finally, Section VI
helped users’ data, while power control and rate adaption          concludes the paper.

     II. M ULTIUSER OFDM S YSTEM M ODEL AND                                       where y + = max(y, 0) and µi is the water level obtained
             N ON - COOPERATIVE S OLUTION                                         by bisection search of the following expression
   We consider an uplink multiuser OFDM system1 .                                           N                         n
                                                                                                               (µi − Ii )+
Suppose that there are N subcarriers and K users in                                              W log2 1 +         n          = Ti .      (5)
the network. The system is time-multiplexed to serve                                       n=1
all users and each time only one user transmits. We
represent Ti as the transmission rate of the ith user, and                          III. P ROBLEM F ORMULATION FOR C OOPERATIVE
the rate is divided into N subcarriers. We denote ri as                                         R ESOURCE A LLOCATION
                              th            th
the transmission rate of the i user at the n subcarrier,                             Note that, the solution in (4) is based on the assump-
and Pin represents the corresponding transmit power.                              tion that all users do not cooperate with each other. Due
Using adaption techniques such as adaptive modulation,                            to the broadcasting nature of wireless communication
we have [21]                                                                      systems, not only the base station (BS) but also the other
                                                    Pin Gn                        users can hear the transmitted data. If the other users
                  n                                      i
                 ri = W log2 1 +                             ,         (1)        can cooperate and help the transmission, cooperative
                                                     σ2 Λ
                                                                                  diversity can be exploited and the system performance
where W is the subcarrier bandwidth, Λ is a constant                              can be significantly improved. In this section, we em-
for the capacity gap, Gn is the subcarrier gain, and σ 2
                         i                                                        ploy cooperative transmission over the OFDM network
is the thermal noise plus interference power. Without loss                        and then formulate the cooperative resource allocation
of generality, we assume that the noise plus interference                         problem.
power is stable and the same for all subcarriers and all
users. We also assume the channels are constant over
                                                                                  A. Cooperative Transmissions over OFDM Networks
each power control interval.
   The goal of this paper is to minimize the overall                                 In the current OFDM system such as in the IEEE
power consumption, under the sum rate constraint over                             802.11a/g standard, the media access control (MAC)
all subcarriers for each user. If there is no cooperation                         layer provides two different wireless access mechanisms
among users, the overall power minimization problem is                            for wireless medium sharing, namely, the distributed
the same as minimizing each user’s power independently.                           coordination function (DCF) and point coordination
We define Pi = [Pi1 , . . . , PiN ] as a power assignment                          function (PCF). The DCF achieves automatic medium
vector. With the bit loading in (1), the ith user’s power                         sharing among users using carrier sense multiple access
minimization problem can be expressed as:                                         with collision avoidance (CSMA/CA) and request to
                                                                                  send/clear to send (RTS/CTS). The PCF is a more cen-
                              min                Pin ,                 (2)        tralized control mechanism. In both mechanisms, time
                                                                                  division multiple access (TDMA) technology is utilized
                                                                                  for all users to share the channels. Similarly in this paper,
                                             n                                    we assume at each time only one user occupies all the
                           s.t.             ri = Ti .
                                                                                  bandwidth. This is also optimal for TDMA over the
                                                                                  single cell case due to the severe interference.
The above constrained optimization can be solved by the
                                                                                     Most of current systems use ARQ with the
traditional waterfilling method [21]. By representing
                                                                                  ACK/NACK mechanism to ensure the successful packet
                               n            Λσ 2                                  deliver. In cooperative transmission, the reason to use
                              Ii =               ,                     (3)
                                                                                  relay instead of ACK/NACK is due to the wireless
the optimal solution of the waterfilling method is given                           channel. If the source-destination wireless link is not
by                                                                                good, there is possibility that ACK/NACK messages may
                                                                                  not be able to successfully transmit. Moreover, due to
                       +                                         Pin
 Pin = (µi − Ii )                n
                            and ri = W log2 1 +                   n    (4)        the bursty nature of wireless channel, if the source-
                                                                 Ii               destination channel is not good now, in the immediate
   1 Since the relay can improve the performance only if it is located
                                                                                  future time, the channel might not recover yet. How-
close to the source-destination link, we only need to consider the                ever, since the broadcast nature of wireless channel, the
resource allocation within a relatively small region, i.e. within one cell.       relay can also hear the transmission from the source to
For the inter-cell interference that comes from the far away sources, we          destination. Due to the spatial diversity, the relay can
consider the case in which the inter-cell interference has been either
considered as the stable noise-plus-interference or has been solved by            have better channel condition and can help the source to
frequency reuse schemes.                                                          transmit to the destination.

                                                                                                               User 1's           User 2's                             User K's
                User j data                                                                                     data               data                                 data

                                                                                                      v=   1      ...     N   1     ...      N    1    ...   ...   1      ...     N
                                                         Time 1                                     u=
                              j                                                                      1     1
   o e/ e
  P w r at



                                                                        e                                                                                                             Time 1




                                                                                                     1                    1   0


                                                                                                                                                                                      Time 2

                                                                                                     N                                       1
                                                   User i data

                                                                                User j data

                                     wr e


                                   Po e/ at


                                                  Subcarrier                                                                                                       1


                                                                                                                                                                                      Time K
Fig. 1.       OFDM Cooperative Transmission Network
                                                                                                     N                                                                            1

   In OFDM networks, each user has flexibility to assign                                           Fig. 2. Assignment Matrix A Example: User 2’s Subcarrier 1 Helps
                                                                                                  User 1’s Subcarrier N
the transmission over different subcarriers. This flexi-
bility gives the possibility of cooperation among users.
In this paper, we consider cooperative OFDM system
over TDMA by using this flexibility. At each time, still                                           answer the question of “how to cooperate”. Moreover,
only one user transmits with positive power. However,                                             because of the users’ different locations and channel
this user can select the number of subcarriers for its                                            conditions, some users are more effective to help others’
own data while keeping the same rate by using adaptive                                            transmissions. Hence, it is essential to find the optimal
modulation and power control. At the same time, this                                              cooperative groups, i.e., to answer the question of “who
user can determine the number of subcarriers for relaying                                         helps whom”.
parts of others’ data. Notice that, unlike most of current                                           We will answer the above two questions by first defin-
cooperative transmission schemes in which extra relay                                             ing an assignment matrix AKN ×KN , given in Figure 2,
stages are necessary, the overall bandwidth efficiency for                                         whose element Au,v ∈ {0, 1} where u = 1, · · · , KN
relay and self-transmission of the proposed system is                                             and v = 1, · · · , KN . The value of Au,v represents
the same as the current OFDM system such as IEEE                                                  the subcarrier indexes of both helping user and helped
802.11a/g standard. To fully understand the proposed                                              user. For notation convenience, we denote (i, n) = (i −
scheme, an example is given as follows.                                                           1)N + n. We use (i, n) to represent the helping user
   In Figure 1, user i relays user j’s data to the BS.                                            i at subcarrier n, and (j, n ) as the helped user j at
At time one, user j transmits data, while all other users                                         subcarrier n . The value of each element of A has the
including the BS can listen. In the next time period, user                                        following interpretation:
i transmits its own data, while at the same time user i                                             1) A(i,n),(i,n) = 1 means the ith user transmits its
can help transmit user j’s data if user i’s location is close                                          own data at the nth subcarrier to the BS.
to the BS and the channel is good. Specifically, user i                                              2) A(i,n),(j,n ) = 1, for i = j, means the ith user at
can relay some parts of user j’s data in some of the N                                                 the nth subcarrier relays the data for the j th user
subcarriers, so as to reduce user j’s transmit power. In                                               at the n th subcarrier .
doing so, user i has to transmit its own data in the rest                                            Since each subcarrier contains only data from one user
of the available subcarriers2 . Consequently the power for                                                                KN
                                                                                                  at a time, we have v=1 Au,v = 1, ∀u = 1, . . . , KN .
user i is increased to maintain its own data transmission.                                        Note that, in case of A = IKN ×KN , the solutions of
However, the overall system power can be reduced. From                                            the proposed scheme are the same as those obtained
the system optimization point of view, the overall power                                          from the traditional non-cooperative method in Section
of both user i and user j can be minimized by selecting                                           II. We also show an example in Figure 2 where user
the proper number of subcarriers for cooperation, i.e., to                                        2 uses its subcarrier 1 to relay the data for user 1 at
  2 In practice, this can be implemented by adaptive modulation and                               the N th subcarrier, i.e. A(2,1),(1,N ) = 1. As shown in
by sending zero over the relaying subcarriers.                                                    Figure 2, each set of N rows represents data transmitted

at a specific time and each set of N columns represents             where without loss of generality ω2 ∼ N (0, σ 2 ) and
whose data are being transmitted at that time.                                                           Rj,i
                                                                                         Xi,BS =                                          (9)
B. Problem Formulation for Resource Allocation over                                                     |Rj,i |
Cooperative Transmissions                                          is the transmitted signal from user i to the BS that is
   In this subsection, we formulate the resource alloca-           normalized to have unit energy.
tion optimization with cooperative transmissions as an                Substituting (7) into (9), then we can rewrite (8) as
assignment problem. We define PK×N = [P1 , . . . , PN ]                                                                   Ri,BS =
as the power allocation matrix whose components are all
nonnegative, and GKN ×KN as the channel gain matrix                  Pin G(i,n)(i,n) (   Pjn   G(j,n    )(i,n) Xj,i   + ω1 )
whose elements obey the following rules:                                                                                           + ω2 . (10)
   1) G(j,n ),(i,n) , for i = j, denotes the channel gain                         Pjn G(j,n    )(i,n)   + σ2
       from the j th user at the n -th subcarrier to the ith          Using (10), the relayed SNR for the n -th subcarrier
       user at the nth subcarrier.                                 of the j th user, which is helped by the nth subcarrier of
   2) G(i,n),(i,n) represents the channel gain from the            the ith user, is given by:
       ith user at the nth subcarrier to the BS.
   3) In order to prevent A(i,n),(i,n ) = 1, for n = n              j,n
                                                                             A(i,n),(j,n ) Pin Pjn G(i,n)(i,n) G(j,n           )(i,n)
       (each user will not relay its own data), we define           Γr =                                                                 . (11)
       G(i,n),(i,n ) = 0, ∀n = n , and ∀i.                                  σ 2 Pin G(i,n)(i,n) + Pjn G(j,n           )(i,n)   + σ2
   In this paper, we use the amplified-and-forward (AF)             Here A(i,n),(j,n ) has a value of 0 or 1, depending on the
cooperative protocol [10] which is simple to be im-                helping and helped relation. Therefore, by (6) and (11),
plemented in relays and can be more mathematically                 we have the rate at the output of MRC as
tractable. Other cooperative protocols such as decode-
and-forward can be employed in a similar way. The                             n                          Γj,n + Γj,n
                                                                                                          d      r
                                                                             rj = W log2 1 +                                   .         (12)
helping user helps the helped user by relaying the data on                                                    Λ
the selected subcarriers using AF. The receiver at the BS          If user i helps with subcarrier n, user i transmit no
combines together the directly received signal from the            information for its own data with this subcarrier, i.e.,
helped user and the relayed signal from the helping user,            n
                                                                   ri = 0. The overall occupied bandwidth for user j’s
using MRC. In what follows, we will derive ri in (1)                                                                 1
                                                                   information is thus 2W . So there is no factor of 2 in (12).
that incorporates cooperative transmissions. Notice that           Each helping user (like user i) utilizes less bandwidth for
helping users can select multiple subcarriers to assist the        its own transmission while using the rest of its available
helped users. Suppose the ith user at the nth subcarrier           bandwidth (like subcarrier n) for helping others. There
helps the j th user at the n th subcarrier. We express the         is no stage purely dedicated to the cooperative trans-
SNR that results from the direct transmission from the             mission. This fact leads to the same average bandwidth
j th user at the n th subcarrier to the BS by                      efficiency as the non-cooperative case.
                   A(j,n ),(j,n ) Pjn G(j,n ),(j,n )                   In this paper, we determine the assignment matrix A
         Γj,n =
          d                                          . (6)         with combinatorial components, and the corresponding
Here A(j,n ),(j,n ) has a value of 0 or 1, depending on if         power allocation matrix P with nonnegative real com-
user j transmits its own information on subcarrier n .             ponents, for an objective to minimize the overall power
   Next, we consider the SNR at the BS that results from           and satisfy all the constraints. The optimization problem
user i relaying user j’s data to the BS. By assuming that          can be formulated as:
Xj,i is the transmitted signal from user j to user i, the                                        K      N
received signal at user i is                                                             min                Pin                          (13)
                                                                                                i=1 n=1
           Rj,i =    Pjn G(j,n   )(i,n) Xj,i   + ω1 ,    (7)                                  N    n
                                                                         Transmission Rate:
                                                                                              n=1 ri (A, P) = Ti , ∀i;
where ω1 ∼ N (0, σ 2 ) and σ 2 is the noise plus inter-                                 KN
                                                                          Assignment:    v=1 Au,v = 1, ∀u = 1, . . . , KN,
ference variance. The noise values at different users are                             and Au,v ∈ {0, 1}, ∀u, v,
assumed to be the same. User i amplifies Rj,i and relays                                       N     n      max
                                                                          Power Constraint:   n=1 Pi ≤ Pi       , ∀i,
it to the BS in which the received signal is
                                                                   where Pimax is the maximal power constraint due to
         Ri,BS =      Pin G(i,n)(i,n) Xi,BS     + ω2 ,   (8)       the hardware limitation. Here the subscript i represents

users which can be helped users, helping users, or                               Note that the problem in (13) can be viewed as a
non-cooperative users. Notice that A indicates both                           generalized assignment problem, which is an N P hard
“who help whom” and the number of subcarriers for                             problem [22]. When A is fixed, the problem in (13) can
cooperation, and P illustrates the level of power for                         be viewed as a nonlinear continuous optimization over
cooperation. From (6), (11) and (12), ri is a function of                     P. So we divide the problem into two subproblems in the
both assignment matrix A and power allocation matrix                          next section. The first subproblem is to find the optimal
P. The optimal choice of A also depends on the channel                        P with a fixed A. Then in the second subproblem, we
conditions from the helped users to the helping users as                      try to find A that generates the optimal solution by using
well as those from the helping users to the BS. In case of                    the results of the first subproblem.
A = IKN ×KN , the problem in (13) reduces to the non-
cooperative problem in (2) and the waterfilling method                            IV. P ROPOSED R ESOURCE A LLOCATION S CHEME
can be used to find the optimal solution.                                                OVER C OOPERATIVE T RANSMISSIONS
   It is worth mentioning that the problem formulation                           In this Section, we first provide an analytical approx-
in (13) is from a network point of view based on the                          imation of optimum power allocation with fixed A for
assumptions that all the users will follow the protocol                       a two-user two-subcarrier case. This analysis provides
and there is no greedy or malicious user. In [27], the                        some insight for the formulated problem. Second, we
behaviors and incentives of greedy users are studied for                      prove the unique optimality of (13) with fixed A. Then,
cooperative transmissions.                                                    based on the insight obtained from the two-user two-
   Next, we study the maximum transmit power con-                             subcarrier case, we develop a greedy suboptimal algo-
straint in (13). To cope with this constraint, one common                     rithm to optimize A and solve the problem in (13) for the
approach is to add the barrier functions which can be                         multiple-user case. Finally, a performance comparison
written as:                                                                   with OFDMA for multi-user multi-subcarrier case is also
       N                                 n      maxN                          provided.
                             if 0, n=1 Pi ≤ Pi      ,
Ii (       Pin , Pimax ) =
                             otherwise.                                       A. Analytical Approximation for Two-User Two-
                                                  (14)                        Subcarrier System With Fixed A
We can modify the objective function in (13) as
                                                                                 In what follows, we will show the analytical evaluation
             K    N            K            N
                                                                              of the optimum power allocation for the cooperative
       min             Pin +         Ii (         Pin , Pimax ),   (15)       transmission system with two users, as shown in Figure
             i=1 n=1           i=1          n=1
                                                                              1. Since the users’ locations are random and mostly
The optimization goal is equal to the original goal in                        asymmetric, it is effective for one user to help the other
(13) if each user’s power is less than or equal to the                        instead of both users helping each other3 . Without loss
maximum power limit. Otherwise, the optimization goal                         of generality, we consider the case in which user 2
will achieve infinity. One good approximation for Ii is                        helps user 1. For simplicity of the exposition, we assume
the log function which is widely utilized in nonlinear nu-                    that an OFDM modulator for each user utilizes two
merical optimization. By adding these barrier functions,                      subcarriers (N = 2), and we consider the case that user
the power constraint in (13) can be removed. In addition,                     2 allocates subcarrier 1 to relay the data of user 1 at
for the single cell case, the users located close to the BS                   subcarrier 24 . Based on the system under consideration,
have better channels and need lower transmitted power.                        we know that the power of user 2 at subcarrier 2 is
These users can help other faraway users with higher                          not used for relay transmission. Therefore, the optimum
transmitted power. The cooperation transmission reduces                       power allocation can be determined by
the helped users’ power a lot and increases the helping                                                           2
users’ power slightly. The underlying reason for this                                                        min P2 ,                            (16)
unequal power reduction and power increase is because                                                            2
                                                                                                         s.t.   r2   = T2 ,
the propagation loss factor in the wireless networks is
usually greater than 1. Since the helping users’ power                        Thus, we have
is usually much smaller than the helped users’ power, if                                                   G(2,2)(2,2) P2          T2
                                                                                            log2 1 +                           =      .          (17)
there exists a feasible non-cooperative solution in (13),                                                      Λσ 2                W
the cooperative solutions still satisfy the maximal power
                                                                                 3 Nevertheless, the two users helping each other case has been studied
constraint. This fact will also be shown in the simulation
                                                                              in [23].
results. For mathematical simplicity, we do not consider                         4 This can also be generalized to the N -subcarrier case with helping
the maximal power constraint in the following analysis.                       percentage equal to 50% by using similar techniques.

                                     2                                                             2      1             2
Hence, the optimum power allocation P2 is given by                  To find the optimum power P1 and P2 in terms of ν1 ,
                         Λσ 2                                       we apply the Lagrange multiplier method to (23). After
             P2 =                 2T2 /W − 1 .           (18)       some manipulations, we find that the optimum power is
                                                                    the solution of a quadratic equation:
   For the relay transmission link, r2 = 0 since user
2 uses subcarrier 1 to help user 1. So the bandwidth
                                                                                                                       1    2
                                                                         Y 2 G(1,2)(1,2) + G(1,2)(2,1) G(2,1)(2,1) Y (P2 − P1 ) +
                                                                                             2 1
efficiency is the same as the noncooperative scheme. The             G(1,2)(2,1) G(2,1)(2,1) P1 P2 (G(1,2)(2,1) − G(2,1)(2,1) ) = 0, (25)
optimum power allocation can be obtained by solving the
following optimization problem:                                     where
                         1    2    1                                                            2                1
                    min P1 + P1 + P2 ,                   (19)                  Y = G(1,2)(2,1) P1 + G(2,1)(2,1) P2 .         (26)
                           1    2
                     s.t. r1 + r1 = T1 .                            By substituting (26) into (25), we have the quadratic
Lagrange multiplier method can be applied to obtain an
analytical solution. However, it is difficult, if possible, to
                                                                                   2         2 1       1
                                                                              A(P1 )2 + BP1 P2 + C(P2 )2 = 0,       (27)
get a closed-form solution by directly applying Lagrange            Where A, B, and C are functions of channel gains only.
multiplier technique to (19). In the sequel, we provide             Assuming that the channel gains are known, we can
an alternative approach that allows us to obtain a closed-                                       2
                                                                    express the transmit power P1 in terms of the relay
form solution for the optimization problem in (19).                         1
                                                                    power P2 as
   First, we use a tight approximation on the SNR                                          2      1
                                                                                         P1 = ηP2 .                   (28)
expression in (11) as proved in [24]
                                           2 1
                  G(1,2)(2,1) G(2,1)(2,1) P1 P2                     By substituting (28) into (25) and solving the quadratic
    Γ1,2 ≈
     r                         2                1
                                                  .      (20)       equation for η, if G(1,2)(2,1) = G(2,1)(2,1) , we obtain
             σ2   G(1,2)(2,1) P1 + G(2,1)(2,1) P2
We divide W in both sides of the rate constraint in (19)                 −G(2,1)(2,1) G(1,2)(1,2) + G(2,1)(2,1)    G(1,2)(2,1) G(2,1)(2,1) − G(1,2)(
and denote ν1 = T1 . Then we use the approximation in
                  W                                                 η=
(20) to the constraint as:                                                                                    G(1,2)(2,1) G(2,1)(2,1) − G(1,2)(1,2)
                           1                               2                                     2 1
            G(1,1)(1,1) P1                   G(1,2)(1,2) P1           If (1,2)(2,1) G(2,1)(2,1) P1 P2, then we solve for P1 and
                                                                       GG                                                    2
log2 1 +                       + log2 1 +                     +             (1,2)(2,1) = G(2,1)(2,1)           = ν1 ,
                  Λσ 2                            Λσ 2                                 2                  1
                                                                   P 1 from the P1 + G(2,1)(2,1) P2
                                                                 Λσ 22 G(1,2)(2,1)optimization problem in (23) by applying
                                                                   the 2Lagrange multiplier method in a similar way. After
                                                           (21) some manipulations, we can find that
with ν1 corresponding to the first logarithmic term                                            1                 1                      1
                         1          2                                    −(2G(1,2)(1,2) P2 + G(1,2)(2,1) )P2 + [(2G(1,2)(1,2) P2 + G(1,2)(2,1)
that involves only P1 , and ν1 relating to the second η =
                                            2          1                                                                             1
                                                                                                     2(G(1,2)(1,2) − G(1,2)(2,1) )(P2 )2
logarithmic term that contains both P1 and P2 . Both
  1       2                                                                                                                    (30)
ν1 and ν1 have non-negative values. Therefore, we can
separate (19) into two subproblems:                                where Ω            4(G(1,2)(1,2) − G(1,2)(2,1) )[(G(1,2)(1,2) +
                                                                   G(1,2)(2,1) )P2 − G(1,2)(2,1) ](P2 )3 .
                                                                                    1                   1
                            min P1 ,                       (22)       If η is not a positive real number or not a number
                             G(1,1)(1,1) P1        1
                                                                   caused by dividing by zero in (29) and (30), because of
           s.t. log2 1 +                      = ν1                 polynomial nature of (27), the optimal solution of (23)
                                  Λσ 2
                                                                   happens on the boundary, i.e. P2 = 0. So under this
                                    2     1                        condition, user 1 and user 2 do not cooperate with each
                        min P1 + P2                        (23)
                    P12 ≥0,P 1 ≥0
                                                                   other. Otherwise if η is a positive real number, using
                                                                   (28) and (29) to solve the optimization problem in (23),
                                                                   2 1
                 G(1,2)(1,2) P1           G(1,2)(2,1) G(2,1)(2,1) P1 P2 can express 2 optimum P 2 and P 1 in terms of ν 2
                                                                   we                   the
s.t. log2 1 +                     +                      2                  1
                                                                                  = ν1 .                1         2               1
                      Λσ 2           Λσ 2 G(1,2)(2,1) P1 + G(2,1)(2,1) P2
                                                                                           1     1       2
The optimum power P1 in (22) can be easily obtained                                      P2 =         2ν1 − 1 ,                (31)
in terms of ν1 as
                  1        Λσ 2         1
                                       ν1                                                        1      2
                P1 =                  2 −1 .               (24)                             2
                                                                                         P1 =        2ν1 − 1 ,                 (32)
                       G(1,1)(1,1)                                                              β1

where                                                                       Proof: All users are divided into two groups. The
        G(1,2)(1,2)         G(1,2)(2,1) G(2,1)(2,1)                   first group of users does not cooperate with others, i.e.,
 β1 =               +                                   .             {i : A(i,n),(j,n ) = 0, ∀i = j or n = n }. Therefore, the
           Λσ 2       Λσ 2 ηG(1,2)(2,1) + G(2,1)(2,1)
                                                      (33)            problem can be considered in the same way as the single
                                          1         2                 user case and the waterfilling method can be used to find
From (24), (31), (32), and the relation ν1 = ν1 − ν1 , the
              1     2    1                                            the only local optimum which is the global optimum for
total power P1 + P1 + P2 that satisfies the constraint in
                                      2                               this kind of users.
(19) can be expressed in terms of ν1 , i.e.,
                                                                         In the second group, users cooperate with each other,
   1    2    1              Λσ 2             2
  P1 + P1 + P2 =                    2(ν1 −ν1 ) − 1         (34)       i.e., {i : ∃A(i,n),(j,n ) = 1, ∀i = j or n = n }. For
                      G(1,1)(1,1)                                     a fixed A, the optimization problem for this group of
          1    2         1        2
                                                                      users can be expressed as
      +      2ν1   −1 +       2ν1 − 1             2
                                              f (ν1 ).
          β1            ηβ1
                                                                                                       K    N
Finally, we can find the optimum value of ν1 by solving                                        min                Pjn                     (39)
the unconstrained optimization problem                                                         P
                                                                                                      j=1 n =1
                          min f (ν1 ),                     (35)                                     N     n
                                                                                                    n =1 rj   = Tj , ∀j.
                           ν1                                                          s.t.
which results in
                                                                        We express the constraint by the use of (12) as:
  2                                  Λσ 2 ηβ1
 ν1   = 0.5 ν1 + 0.5 log2                                . (36)                    N
                                G(1,1)(1,1) (η + β1 )                                               Γj,n + Γj,n
                                                                                                     d      r
                                                                                              1+                         = 2W .          (40)
Consequently, ν1 can be found from (21) as                                                               Λ
                                                                                  n =1

  1                                  Λσ 2 ηβ1                         Since each user can be helped by at most one other
 ν1 = 0.5 ν1 − 0.5 log2                                  . (37)
                                G(1,1)(1,1) (η + β1 )                 user, each product for each subcarrier n in (40) is not
                               1        2                             coupled with each other. Suppose the cooperative rate for
Substituting the obtained ν1 and ν1 into (24), (31)
                                                                                                                            Γj,n +Γj,n
                                          1   2
and (32) gives the optimum power P1 , P1 , and P2 ,     1             user j and subcarrier n is 2τ , i.e.,            1+    d
respectively.                                                         2τ . For a fixed A and using (6) and (11), all power
   From the above derivations, we know that in order to               components result in a quadratic form of Pjn , because
minimize the overall power, the ratio η in (28) (i.e. the             of the uncoupling fact. If the two roots of the polynomial
helped user’s transmit power over the helping user’s re-              function have opposite signs, the problem in (39) has
lay power) is only determined by the channel conditions               a unique positive solution for power P. Suppose each
between the two users and those to the BS. Then for                   cooperative subcarrier has the rate larger than W if τ >
each helped user, it balances the power allocation used               1, and the helping user i helps by subcarrier n. From (6)
between the subcarriers that do not get help from others              and approximation of (11), we can rewrite (40) as
and the subcarriers whose information is relayed by
others, so as to minimize the overall power. We will show                G(j,n )(j,n ) (G(j,n )(j,n ) + G(i,n)(i,n) )(Pjn )2             (41)
by computer simulation that the above analysis with the                          (n )           2
                                                                           +C   Pj       + Λσ       G(i,n)(i,n) Pin (1   − τ ) = 0.
SNR approximation in (20) provides the solutions that
are very close to the optimal solutions.                              where C is a constant of channel gain and Pin . It is
                                                                      obvious that in order to obtain unique positive Pjn , τ
B. Power Minimization Optimality for Multiple-user                    should be greater than 1. As a result, the global optimum
Multiple-Subcarrier System with Fixed A                               can always be achieved.
   In this subsection, we assume that the assignment ma-                 The above theorem proves that there is a unique
trix A is known and fixed. We show the characteristics                 optimal solution for the case with fixed A under the high
of the solution by the following theorem.                             rate assumption. So with the fixed A, any nonlinear or
   Theorem 1: For a fixed A, as long as each cooperative               convex optimization methods [25], [26] can be employed
subcarrier has the rate rj in (12) that is larger than W ,            to solve (39). In the case of low rates, the optimization
there is only one local optimum which is also the global              problem will have multiple local optima. Under this
optimum for (13). In other words, the corresponding                   condition, there are generally two possible approaches.
SNR satisfies                                                          The first one is simulated annealing, and the other is to
                    Γj,n + Γj,n > Λ.
                      d       r                       (38)            find good heuristic initialization.

C. Finding Suboptimal A for Multiple-user Multiple-        we go back to determine the helping and helped user
Subcarrier Case                                            pair again and continue the iteration. Notice that there
   In previous sections, we analyze the optimization might be more than one subcarrier of a helping user for
problem with a fixed A. In this subsection, we develop relaying the information of the helped users. If the user
a suboptimal solution to find A, which represents “who with the minimal power cannot help others anymore,
helps whom” and how many subcarriers should be used the random pair of helping user and helped user is
for cooperation (i.e. “how to cooperate”). Because of formed to further explore the possibility that the power
the combinatorial nature, the problem in (39) is also an can be reduced. If the power cannot be reduced, this
N P hard problem [25], [26], since any element of A random pair of users is not applicable and the original
has a value of either 0 or 1 and the search dimension A and P are restored. Otherwise, the new A and its
of A is 2KN ×KN . For any specific A, we calculate the corresponding P are updated. This process stops when
transmit power vector P(A), and then select the one that the power cannot be further reduced for a period of
generates the minimal overall power. The problem can iterations. The resulting A is the bandwidth assignment
be formulated as                                           for the helping users to the helped users, which is the
                        K N
                                                           answer to the questions of “who helps whom” and “how
                  min              n
                                 Pi (A)             (42) many subcarriers to cooperate”. The resulting matrix P
                       i=1 n=1
                                                           represents power allocation. So ”how to cooperate” is
                                                           also answered. Notice that in each iteration, the sum rate
                                                           for each user is kept unchanged. The detailed algorithm
s.t.    Au,v = 1, ∀u = 1, . . . , KN, and Au,v ∈ {0, 1}, ∀u, v.
                                                           is shown as follows:

However, the complexity is high especially when a large               Multiuser Suboptimal Algorithm for A
number of subcarriers are utilized, and when there are a             Initialization: A = IKN ×KN , and calculate (39) to
substantial number of users in the OFDM network. This              obtain P.
prohibits the full search method in practice.                        Iteration:
   Next, we propose a suboptimal greedy algorithm to
                                                                     1) select user i with the minimal power and user j
find the assignment matrix A. The basic idea is to let
                                                                         with maximal power under the condition:
the user with the least transmit power (likely close to
the BS) to help the user with the most transmit power                      • user i has at least one subcarrier for transmit-

(likely far away from the BS). Consequently, the user                         ting its own data.
with the least power has to increase its power to help the
user with the most power in order to reduce the overall               2) Hypotheses:
system power. Then we determine the users’ selection                       • Only the subcarrier that transmits the user’s

again, and repeat the above steps. The iteration stops                       own data is eligible for the hypotheses.
when no overall power can be reduced. The suboptimal                       • If user i’s subcarrier n helps user j’s subcarrier

greedy algorithm is implemented in the base station.                         n:
   Initially, A is assigned as an identity matrix, i.e., the                 Set [A](i−1)∗N +n,(i−1)∗N +n = 0 and
initial scheme is the non-cooperative scheme. Second,                        [A](i−1)∗N +n,(j−1)∗N +n = 1;
we sort according to users’ transmit power. We select                      • Solve (39) for P.

the users with the maximal and minimal transmit power                    Among all hypotheses, find the maximal power
as helped user and helping user, respectively, according                 reduction.
to the following condition: Each user has at least one
subcarrier to send its own data. The above condition                  3) If all hypotheses are not effective for power reduc-
makes sure that helping user’s data must be transmitted.                 tion, random helping and helped pairs are formed
Among N subcarriers of the helping user, we make                         to see if the power can be further reduced or not.
N hypotheses that the nth subcarrier is assigned to                   4) Update A, go to step 1. End if no power reduction
assist the helped user and the remaining subcarriers are                  for a period of iterations, return A and P.
unchanged. The overall power for this hypothesis is
                                                                      Since the system power is non-increasing in each
obtained by solving (39) with some numerical method.
                                                                   iteration 5 and is lower bounded by the full search result,
From all these hypotheses, the algorithm selects the one
that maximally reduces the overall power and keeps the                5 The results from the random disturbance are applicable only when
rest N − 1 subcarriers the same in each iteration. Then            there is power reduction.

the iteration always converges. Note that the complexity          αi is no more than 2. αi is equal to 2 only if all the
for each iteration of the proposed algorithm is O(N 3 )           subcarriers of the helped user get helped by others. The
and the algorithm is suboptimal because of the greedy             problem formulation for the performance comparison is
local search. But the local optimum problem is alleviated         to minimize the overall power under the constraints of
by the random disturbance in step 3 of the algorithm.             total bandwidth and individual rates:
Moreover, the helping users cannot be helped again in
the future. As we have mentioned in the two-user case,
                                                                                          min            αi N Pi                 (43)
due to the asymmetry of channel conditions for different                                 0<αi ≤2
users, it is not effective for users to help each other or
form helping loop. So in the proposed algorithm, we
implicitly exclude the helping loop and users cannot be                                 i=1   αi = K;
helping users and helped users at the same time.                            s.t.                               ¯
                                                                                                            Pi Gi
                                                                                     αi N W log2 1 +        Λσ 2    = Ti ,
   The channel information required to perform the al-
gorithm can be obtained in the following way: Since the                   ¯
                                                                  where Gi is the average channel gain 6 over the band-
users take turns to transmit information to the BS, the           width occupied by user i. The objective function is the
channel information from the source to the destination            overall transmitted power since here Pi is the power
and from the relay to the destination can be easily               per bandwidth W , the first constraint is the overall
estimated. When one user transmits to the BS, the                 bandwidth, and the second constraint is the required rate.
possible relays of this user can estimate the channels            The problem in (43) is different from the problem in
(from sources to relays). All this information can be sent        (13) for two reasons. First, the bandwidth assignment
to the BS for the optimization. Since the average number          is relaxed to continuous functions αi instead of the
of users is small and number of subcarriers are around 48         discrete function A 7 . So the solution to (43) is an upper
for WLAN, the amount of information is limited. The BS            bound for OFDMA. Second, the received SNR of the
needs to send control bits to the potential helping/helped        helped user is obtained by the assumption that all the
users to indicate relay-selection, power-optimization, and        bandwidth is occupied by the helped user directly. The
subcarrier-allocation. This requires some overheard for           purpose is to have a fair comparison from the bandwidth
signaling. Here we also assume the mobility of the users          point of view. Moreover, this simplifies the analysis but
is low, so that the channel conditions are stable for             removes the multipath diversity provided by cooperative
sufficiently long time and the frequency and overhead              transmission. Since each users’ transmission can be
to update the channel information are also low.                   allocated to not only the different time slots but also dif-
                                                                  ferent frequency in (43), the multiuser, time, frequency
D. Performance Comparison                                         diversity of OFDMA can be achieved. The problem in
   For the multiuser case, it is very difficult to obtain          (43) can be solved by some numerical methods [25],
the analytical closed-form solution of (13), because of           [26]. The solutions of (43) give us some insights on
the non-convexity and integer nature of the N P hard              the performance comparisons of the proposed suboptimal
problem. In order to better understand the performance            solution, which are shown in the next Section.
of the proposed suboptimal solution, we also compare
the performances with OFDMA. Since the helped user’s
information is relayed by the helping user in some                       V. S IMULATION R ESULTS AND A NALYSIS
subcarriers, it can be viewed as the helped user occupied
                                                                     We perform computer simulations for multiuser
more bandwidth (subcarriers) by the help of the helping
                                                                  OFDM systems to evaluate the system performance and
users. The basic idea is to view these helping subcarriers
                                                                  answer the questions like “who helps whom” and “how
as a part of the bandwidth of the helped users. Suppose
                                                                  to cooperate”. In what follows, the simulation results for
the helping user i helps the helped user j with N
                                                                  the two-user system are presented in the first part, and
subcarriers. So the bandwidth occupied by the helped
                                                                  those for the multiuser system are shown in the second
user is W N (1 + N ) and the bandwidth for the helping
                    N                                             part.
user is W N (1 − N ). We define each user to have a
bandwidth of αi N W , where αi ∈ (0, 2]. Note that αi is
                                                                     6 Notice that flat fading is assumed for OFDMA system to reduce the
greater than 0 since any user must have some bandwidth
                                                                  analysis complexity, since OFDMA resource allocation in frequency
to transmit its own data. Since we only consider the case         selective fading is still an open issue in the literature.
in which a user can only get helped by one other user,               7 This is typical relaxation in the OFDMA literature [18].

                                 Comparison of Three Methods for Overall Power                                                                         Cooperation Region for 50% Helping
                                                                      water filing
                                                                      user 1 helps user 2
                                                                      user 2 helps user 1                                                                                                       User 1 help User 2
                                                                      analysis : user 1 helps user 2                                       20
                                                                      analysis : user 2 helps user 1
  Overall Power (dBm)

                                                                                                                                                       Waterfilling Area

                                                                                                                      User 2 Location Y
                                                                                                                                            0                              User 2 help User 1


                        −10                                                                                                               −20

                                                                                                                                           −30   −20          −10             0            10              20        30

                                                                                                                                                                     User 2 Location X
                         −30   −20       −10          0          10                20                  30

                                               User 2 Location
                                                                                                                 Fig. 4. Cooperative Region For Two-User System with User 1 Located
                                                                                                                 at (10m, 0m)
Fig. 3. Comparison of Overall Power of Three Different Schemes
and Analytical Results

                                                                                                                 when user 2 is located far away from the BS, even in
                                                                                                                 the opposite direction to user 1 such as (-30m,0), the 1-
A. Two-User System
                                                                                                                 H-2 scheme can reduce the overall transmit power. This
   We set simulations of the first part as follows: There                                                         can be explained by the same reason as above. In the
are a total of K = 2 users in the OFDM network. A BS                                                             extreme case, when user 2 is located very far away from
is located at coordinate (0,0), user 1 is fixed at coordinate                                                     the BS compared with the location of user 1 to the BS,
(10m,0), and user 2 is randomly located within the range                                                         both user 1 and the BS can be considered as multiple
of [-30m 30m] in both x-axis and y-axis. The propaga-                                                            sinks for user 2’s signal. This provides so called “virtual
tion loss factor is set to 3. The noise plus interference                                                        multiple antenna diversity”. In addition, the analytical
level is σ 2 = −60dBm and we select the capacity gap                                                             results obtained by derivation closely match the optimal
as Λ = 1. An OFDM modulator for each user utilizes                                                               results by the numerical algorithm.
N = 32 subcarriers and each subcarrier occupying a                                                                  In Figure 4, we show the region where different
bandwidth of W = 1. Without loss of generality, we                                                               schemes should be applied based on user 2’s location.
assume flat fading in simulations so that the results such                                                        With the same simulation setup as in the previous case,
as regions can be clearly related to distances and can be                                                        we can see that when user 2’s location is close to
compared with OFDMA results obtained from Section                                                                the BS or lies in between user 1 and the BS, the 2-
IV-D. The system under frequency selective fading can                                                            H-1 scheme is preferred. When user 2 is located far
be studied in a similar way. The performance can be                                                              away from the BS, the 1-H-2 scheme produces minimal
further improved by exploring frequency diversity.                                                               overall power. When user 2’s location is in between the
   In Figure 3, we show a comparison of the overall                                                              above two cases, the waterfilling scheme is the optimum
power in dB of the waterfilling scheme, user 1 helps                                                              choice. This figure presents the answers to the question
user 2 (1-H-2) scheme, user 2 helps user 1 (2-H-1)                                                               of “who helps whom”. It is worth mentioning that both
scheme, and the analyzed results from Section IV-B. In                                                           x axis and y axis represent users’ channel conditions
this simulation, user 2 moves from location (-30m,0) to                                                          by using locations, so that the answer of “who helps
(30m,0). The transmission rate for each user is Ti =                                                             whom” can be clearly illustrated. If channel effects such
2N W and half of the subcarriers are used for helping                                                            as shadowing and fading are considered, both x axis and
the other. We observe that when user 2 is located close to                                                       y axis can be channel conditions and are not related to
the BS, the 2-H-1 scheme can reduce the overall power                                                            the distances.
up to 50%. The reason is that user 1 can use user 2 as                                                              In Figure 5, we answer the question of how the users
a relay node to transmit user 1’s data such that user 1’s                                                        should cooperate with each other. The simulation setup
power can be reduced. Since user 2 is close to the BS,                                                           is the same as in the previous case except that the
even with only half of the subcarriers to carry its own                                                          number of helped subcarriers is not fixed. Instead, we
data, the increase of power for user 2 is still smaller                                                          find the optimal percentage of subcarriers that are used
than the power reduction for user 1. On the other hand,                                                          for helping others. We show the helping percentage as a

                                                                                                                                                                                                Helping Percentage vs. Rate and Location

                                                Optimal Helping Percentage For Different Location of User 2

                                                                                                                                                  Helping Percentage

      Helping Percentage


                             −60                                                                                                                                                2
                             −30                                                                                                     30                                                                                                                 −30
                                      −20                                                                                                                                           4                                                             −20
                                                 −10                                                                 10                                                                     6                                              −10
                                                           0                                                 0                                                                                      8                      10
                             User 2 Location X                                                                                                                                  Rate (NW)                                       User 2 Location
                                                                 10                                   −10   User 2 Location Y                                                                                       20
                                                                                                                                                                                                         10   30
                                                                        20                      −20
                                                                                 30       −30

                                                                                                                                               Fig. 7.                        Helping Percentage versus User’s Rate
Fig. 5.                       Cooperative Percentage For Two-User System

                                                           Helping Range vs. Rate and User 2 Location
                                                                                                                                               to (30m, 0). As we can see from Figure 6, the helping
                                                                                                                                               regions are changed with increase of the transmission
                                                               User 1 help User 2                                                              rate. The waterfilling regions increase to maxima around
                                                                                                                                               the rate equal to 3.4N W , then the waterfilling regions
                                 10                               Waterfilling
                                                                                                                                               are shrunk until all regions are occupied by 1-H-2 or
           User 2 Location

                                                                                                                                               2-H-1 regions. This is because when the power grows
                                  0                                                                   User 2 help User 1
                                                                                                                                               exponentially with increase of rate, the helping with little
                                                                                                                                               percentage can reduce a lot of power in the proposed
                                                                                                                                               OFDM cooperative network. In Figure 7, however, we
                                                                                                                                               observe that the helping percentages are reduced as the
                             −20                                                                       User 1 help User 2                      users’ rates keep increasing. This is because the users
                                                                                                                                               need more subcarriers to send their own data, so the
                                                                                                                                               number of subcarriers to help others is reduced. Note
                               0.2          1          2         3         4          5         6       7        8          9        10
                                                                                 Rate (NW)                                                     also from the figure that when the rate is high, the
                                                                                                                                               original waterfilling area becomes the 1-H-2 or 2-H-1
Fig. 6.                       Cooperative Region versus User’s Rate                                                                            area. But the percentage of helping is very small.

function of different user 2’s locations. For pure notation                                                                                    B. Multiuser System
purpose, when the 2-H-1 scheme is used, the percentage                                                                                            We consider, in this subsection, the performance of
is illustrated to be positive; when the waterfilling scheme                                                                                     the proposed scheme for the multiuser scenario. In the
is chosen, the percentage is zero; when the 1-H-2 scheme                                                                                       simulations, all distributed users are located within a
is applied, the percentage is shown to be negative. By                                                                                         circle of radius 50m. The BS is located at the center
doing this, with the same percentages, we can distinguish                                                                                      of the circle, i.e. (0m, 0m) and the closest distance from
who helps whom. From Figure 5, the closer user 2 is                                                                                            a user to the BS is 1m. The other settings are the same
to the BS, the larger percentage user 2 will help user                                                                                         as those for the two-user case.
1. This observation follows the fact that user 2’s own                                                                                            In Figure 8, we show simulation results for a 4-
transmission virtually costs nothing in terms of power                                                                                         user case. Over different random locations, we select
usage. On the other hand, when user 2 is far away from                                                                                         a typical snapshot of user 1, 2, 3, and 4 with location
the BS, user 1 helps user 2 more and more. Therefore, the                                                                                      (3.2331m, 7.2071m), (1.7768m, -7.8635m), (35.7636m,
figure gives us insight as to how the level of cooperation                                                                                      5.8430m), and (-32.7265m, 21.5281m), respectively. We
and partner assignments affect the system performance.                                                                                         show the power changes in the proposed cooperative
   Figure 6 and Figure 7 show the effects of user’s                                                                                            scheme versus the iteration number. Because user 1 and
transmission rate Ti on who should help whom and how                                                                                           user 2 are close to the BS, they help user 3 and user
users should cooperate. We modify Ti from 0.5N W to                                                                                            4’s transmission to reduce their power. The subcarriers
10N W for all users. User 2 is located from (-30m,0)                                                                                           of user 1 and user 2 are assigned to help user 3 and user

                                          Power Reduction vs. Iteration                                                            Power vs. Rate

                                                                                  User 1
                                                                                  User 2
                                                                                  User 3
                                                                                  User 4                                −10

                                                                                                           Power (dB)
       Power (dB)


                                                                                                                        −20                    Waterfilling, No. of User=4
                                                                                                                                               Cooperation, No. of User =4
                                                                                                                                               Waterfilling, No. of User=8
                                                                                                                                               Cooperation, No. of User=8
                                                                                                                                               OFDMA, No. of User=4
                                                                                                                                               OFDMA, No. of User=8

                     −70                                                                                                −30
                      0          10        20             30           40       50          60                             1   2   3            4            5               6
                                                       Iteration                                                                       Rate

Fig. 8.              Power Convergence                                                                Fig. 10. Overall Power versus Each User’s Rate in the Unit of N W

                                          Power vs. Number of Users                                                                TABLE I
                                                                                                      E FFECT OF S HADOW FADING ON P OWER S AVING (P ERCENTAGE OF
                                                                                                       P OWER R EDUCTION USING C OOPERATION OVER THE P OWER OF
                                                                                                                           N ONCOOPERATION )
      Power (dBm)

                                                                                                       Shadow Fading Standard Deviation         Rate=2NW            Rate=4NW
                                                                                                                    1dB                           23.5%               37.3%
                                                                                                                    3dB                           41.0%               53.5%
                                                                                                                    5dB                           59.5%               78.9%

                                                                       Waterfilling Rate=2NW
                                                                       Cooperation Rate=2NW
                                                                       Waterfilling Rate=4NW
                                                                                                      the waterfilling scheme will produce the minimal overall
                                                                       Cooperation Rate=4NW           power. Under this occasion, the proposed cooperative
                         2   3        4         5         6
                                                    Number of Users
                                                                   7        8        9      10
                                                                                                      scheme has the same performance as the non-cooperative
Fig. 9.              Overall Power versus Number of Users                                                In Figure 10, we show the overall system power versus
                                                                                                      the rate requirements with 4-user and 8-user cases.
                                                                                                      Obviously the overall power increases almost exponen-
4. Consequently, the remaining numbers of subcarriers                                                 tially with the rate requirement. This corresponds to the
for user 1 and user 2 are reduced, so that their transmit                                             relation between rate and power given in (1) and (12).
power is increased. But the overall system power is                                                   The proposed cooperative scheme can reduce 37% to
reduced. Notice that after iteration 37, user 4 still has                                             54% of overall power under 4-user condition and 8-user
the largest power, but no user can further help him/her,                                              condition. The proposed cooperative scheme can reduce
since to help user 4 will increase the power of the helping                                           more power when the rate is high. This is because the
users more than the power reduction of user 4. From                                                   power reduction is large if the user is helped by others
iteration 50, user 2 tries to help user 3 which has the                                               when the rate is higher, even though the percentages
second largest power. This is due to the third step of the                                            of subcarriers that other users would like to help are
algorithm, in which the random pair of users is formed.                                               reduced according to Figure 7.
   In Figure 9, we show the system overall power in dB                                                   In Table I, we investigate the shadow fading effect
versus the number of users with different rate require-                                               on the proposed scheme. Specifically, we study the
ments. We can see that the proposed cooperative scheme                                                lognormal shadow fading model with different variance
can save up to 19% of overall power for rate equal to                                                 in dB. As shown in the table, the proposed scheme can
2N W and 42% for rate equal to 4N W , respectively. The                                               achieve more power saving (defined as the percentage
power reduction increases when the number of users is                                                 of power reduction using cooperation over the power
large, which is due to the increase of the choices for                                                of noncooperation using waterfilling), when the shadow
cooperation is more. In the simulation, we find some                                                   fading is more severe. The intuitive reason is because the
occasions where the users are located in the areas where                                              variance for the channel gains between different users

increases. As a result, the proposed scheme can have                           [4] R. Jantti and S. L. Kim, “Second-order power control with
more opportunities to improve the system performances.                             asymptotically fast convergence,” IEEE Journal on Selected
                                                                                   Areas in Communications: Wireless Communication Series, vol.
   In Figure 10, we also show the performance com-                                 SAC-18 (3), pp. 447-457, 2000.
parisons using (43) which is an upper bound for the                            [5] X. Qiu and K. Chawla, “On the performance of adaptive modula-
OFDMA system. The proposed suboptimal scheme has                                   tion in cellular systems,” IEEE Transactions on Communications,
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system, at each time, more than one user is transmitting                           IEEE Journal on Selected Areas in Communication, vol. 16, no.
by using different subcarriers. The synchronization of                             8, pp. 1437-1449, October 1998.
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