Weighted Sum Rate Maximization for Downlink OFDMA with Subcarrier-pair based Opportunistic DF Relaying

0.41 MB·English

by  Tao Wang

SUBMITTEDTOIEEETRANSACTIONSONSIGNALPROCESSING 1
Weighted Sum Rate Maximization for
Downlink OFDMA with Subcarrier-pair based
Opportunistic DF Relaying
Tao Wang, Senior Member, IEEE, Franc¸ois Glineur, Je´roˆme Louveaux and Luc Vandendorpe, Fellow, IEEE
3 Abstract—This paper addresses a weighted sum rate (WSR) I. INTRODUCTION
1 maximizationproblemfordownlinkOFDMAaidedbyadecode-
0 and-forward (DF) relay under a total power constraint. A novel Orthogonal frequency division multiple access (OFDMA)
2 subcarrier-pair based opportunisticDF relaying protocol is pro- has been widely recognized as one of the dominant wireless
n posed.Specifically,usermessagebitsaretransmittedintwotime technologiesfor high data-rate transmission. One of the main
slots.Asubcarrierinthefirstslotcanbepairedwithasubcarrier
a reasons behind this fact is that spectral efficiency of the
inthesecondslotfortheDFrelay-aidedtransmissiontoauser.In
J particular, the source and the relay can transmit simultaneously OFDM(A) systems can be improved significantly by proper

8 to implement beamforming at the subcarrier in the second slot. resourceallocation(RA)whentransmitterchannelstate infor-
2 Each unpaired subcarrier in either the first or second slot is mation(CSI)isavailable[1]–[3].Theincorporationofdecode-
usedforthesource’sdirecttransmissiontoauser.Abenchmark
and-forward(DF)andamplify-and-forward(AF)relayinginto
] protocol, same as the proposed one except that the transmit
Y OFDM(A) systems through subcarrier-pair based protocols
beamformingisnotusedfortherelay-aidedtransmission,isalso
S considered.Foreachprotocol,apolynomial-complexityalgorithm and associated RA have lately been under intensive investi-
. isdeveloped tofindat least an approximatelyoptimumresource gation [4]–[29]. This class of protocols share the following
s
c allocation(RA),byusingcontinuousrelaxation,thedualmethod, features. User message bits are transmitted during two con-
[ and Hungarian algorithm. Instrumental to the algorithm design secutive equal-duration time slots. In the first slot, the source
is an elegant definition of optimization variables, motivated by
1 theidea of regarding theunpaired subcarriers asvirtual subcarrier broadcasts OFDM symbols, so does the relay in the second
v pairs inthe direct transmission mode. The effectiveness of the RA slot. The source might also emit OFDM symbols during the
0 algorithmandtheimpactofrelaypositionandtotalpoweronthe second slot as will be elaborated later. A subcarrier in the
0 protocols’performanceareillustratedbynumericalexperiments. first slot can be pairedwith a subcarrierin the secondslot for
6 It is shown that for each protocol, it is more likely to pair
transmitting message bits with DF/AF relaying, referred to as
6 subcarriers for relay-aided transmission when the total power
. is low and the relay lies in the middle between the source and the relay-aided transmission mode hereafter.
1
user region. The proposed protocol always leads to a maximum In this paper, we focus on RA for downlink OFDMA with
0
WSR equal to or greater than that for the benchmark one, and subcarrier-pair based DF relaying (there also exist works on
3
the performance gain of using the proposed one is significant
1 RAforOFDMAsystemsusingbidirectionalrelaying[4]).The
especially when the relay is in close proximity to the source
: subcarrier-pairbased AF relayinghas been studied in [5]–[8].
v and the total power is low. Theoretical analysis is presented to
i interpret these observations. Note that the subcarrier-by-subcarrier based pairing may not
X
be sufficientforDF relaying,since the informationfroma set
r IndexTerms—Resourceallocation,decodeandforward,trans- of subcarriers in the first time slot can be decoded and re-
a
mit beamforming, subcarrier pairing, orthogonal frequency di- encoded jointly and then forwarded through a different set of
vision multiple access, convex optimization.
subcarriersin the second time slot [8], [12]. Nevertheless,the
subcarrier-pairbasedDF relayinghas attractedmuchresearch
interest due to simplicity or practical reasons [9]–[29].
When the source-to-destination (S-D) link is unavailable
Copyright (c) 2012 IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be (i.e., the destination lies outside the source’s radio coverage),
obtained fromtheIEEEbysendingarequest [email protected]. RA problems for OFDM systems using subcarrier-pair based
Partofthispaperhasbeenpresentedin2013IEEEWirelessCommunication
DFprotocolshavebeenaddressedin[9]–[12].Intheseworks,
andNetworking Conference, Shanghai, China.
T. Wang is with School of Communication & Information Engineering, everysubcarrierinthefirsttimeslotispairedwithasubcarrier
Shanghai University, 200072 Shanghai, P. R. China. He was with ICTEAM in the second time slot for the relay-aided transmission, as il-
Institute, Universite´ Catholique de Louvain (UCL), 1348 Louvain-la-Neuve,
lustratedinFig.1.a.Tomaximizesumrateunderatotalpower
Belgium (Email:[email protected]).
F. Glineur, J. Louveaux and L. Vandendorpe are with ICTEAM In- constraint,orderedsubcarrierpairinghasbeenproventobethe
stitute, UCL, 1348 Louvain-la-Neuve, Belgium (Email:{francois.glineur, optimum, i.e., the strongest source-to-relay subcarrier should
jerome.louveaux, luc.vandendorpe}@uclouvain.be).
bepairedwiththestrongestrelay-to-destinationsubcarrier,and
This research is supported by The Program for Professor of Special
Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning. so on.
It is also supported by the European Commission in the framework of the The works in [13]–[29] have considered the case where
FP7NetworkofExcellenceinWirelessCOMmunicationsNEWCOM#(Grant
the S-D link is available. When only the relay emits OFDM
agreement no.318306),theIAPproject BESTCOM,andtheARCSCOOP.
symbolsinthesecondtimeslot,opportunisticrelaying(some-
SUBMITTEDTOIEEETRANSACTIONSONSIGNALPROCESSING 2
relay-aidedmode directmode nosource/relaytransmission same subcarrier in the second slot if the relay-aided mode is
used.Theoptimizationofsubcarrierpairingandassignmentto
1sttimeslot 2ndtimeslot usersis addressedin [28] with a graphbased approach.Itis a
subcarrier1 subcarrier1 complicatedRAproblemtojointlyoptimizesubcarrierpairing
subcarrier2 subcarrier2 and mode selection with power allocation and subcarrier
subcarrier3 subcarrier3 assignment to users.
Compared with the above existing works, this paper makes
subcarrier4 subcarrier4
the following contributions:
(a) whentheS-Dlinkisunavailable [9]–[12].
• A novel subcarrier-pair based opportunistic DF protocol
1sttimeslot 2ndtimeslot is proposed for downlink OFDMA aided by a DF relay.
subcarrier1 subcarrier1 Thisprotocolfurthermakesimprovementoverthosepre-
subcarrier2 subcarrier2 viouslystudiedintheliterature[21]–[29],byallowingthe
sourceand the relay to implementtransmit beamforming
subcarrier2
subcarrier3
at a subcarrier in the second time slot for the relay-
subcarrier4 subcarrier4 aided transmission. Note that the protocols studied in
(b) whentheS-Dlinkisavailable butthesourcedoesnottransmit [25], [26] considered the selection of multiple DF relays
inthesecondslot[13]–[20].
(excluding the source) for transmit beamforming in the
1sttimeslot 2ndtimeslot second slot, while the proposed protocol considers the
subcarrier1 subcarrier1 joint source-relay transmit beamforming. A benchmark
subcarrier2 subcarrier2 protocol, which is the same as the proposed one except
for the relay-aided transmission mode, is also consid-
subcarrier3 subcarrier3
ered. Note that the proposed protocol truly improves the
subcarrier4 subcarrier4 implementation of DF relaying over a subcarrier pair
(c) when the S-D link is available and the source transmits in the with transmitbeamforming,which is notthe case forthe
secondslot[21]–[29]. benchmark protocol.
Fig. 1. Illustration of the subcarrier-pair based DF relaying protocols for • Theweightedsumrate(WSR)maximizedRAproblemis
OFDM(A)-basedsystems,whereeveryarrowindicatesthatthetwoassociated addressedforboththeproposedandbenchmarkprotocols
subcarriers arepairedfortherelay-aided transmission. underatotalpowerconstraintforthewholesystem.First,
itisshownthattheproposedprotocolleadstoamaximum
WSR notsmallerthanthatforthebenchmarkone.Then,
times termed as selection relaying) was studied in [13]–[20].
analgorithmisdevelopedforeachprotocoltofindatleast
Specifically, a subcarrier in the first time slot can either be
an approximately optimum RA with a WSR very close
paired with a subcarrier in the second slot for the relay-aided
to the maximum WSR. Instrumental to the elegance of
transmission,oruseddirectlyfortheS-Dtransmissionwithout
the RA algorithm is a definition of appropriate indicator
the relay’s assistance, referred to as the direct transmission
variables,makingitpossibletocastasubproblemrelated
mode hereafter. It is very important to note that when some
to the joint optimization of transmission-mode selection,
subcarriers in the first slot are used in the direct transmission
subcarrier pairing and assignment to users into an stan-
mode, some subcarriers in the second slot will not be used
dard assignment problem that can be solved efficiently
as illustrated in Fig. 1.b, which leads to a waste of precious
by Hungarian algorithm.
spectrum resource.
The rest of this paper is organized as follows. In the next
Toaddresstheaboveissue,improvedprotocolswhichallow
section, the system and transmission protocols are described.
the source to emit OFDM symbols in the second slot were
The theoretical analysis is made to compare the maximum
proposed and studied in [21]–[29]. The improved protocols
WSRs of the two protocols in Section III. After that, the RA
are the same as those consideredin [13]–[18], exceptthat the
algorithm is developed in Section IV. Numerical experiments
sourcecanalsomakedirectS-Dtransmissionateveryunpaired
are shown to illustrate the effectiveness of the RA algorithm
subcarrier in the second slot, as illustrated in Fig. 1.c. Note
and study the impact of relay position and total power on the
thattheimprovedprotocolsdonotreallyimprovethewaythat
protocols’ performance in Section V. Finally, some conclu-
DF relaying is implemented over a subcarrier pair, but rather
sions are drawn.
letthesourceutilizetheunpairedsubcarriersinthesecondslot
Notations: A letter in bold, e.g. x, represents a set. C(x)=
fordirecttransmissiontoavoidthewasteofspectrumresource. 1log (1+x).
In[24],[27],[29],the subcarrierpairingandpowerallocation 2 2
are jointly optimized for point-to-point OFDM systems. As
II. PROTOCOLS ANDWSR MAXIMIZATION PROBLEM
for OFDMA systems, RA problems considering the joint op-
A. The transmission system and protocols
timization of power allocation, subcarrier assignment to users
and selection of multiple relays for transmit beamforming in ConsiderthedownlinkOFDMAtransmissionfromasource
the second slot are addressed in [25], [26]. In these works, to U users (user u = 1,...,U) aided by a DF relay. The
a prioriand CSI-independentsubcarrierpairing is considered, source, relay and every user are each equipped with a single
i.e., a subcarrier in the first slot is always paired with the antenna, and the channel between every two of them is
SUBMITTEDTOIEEETRANSACTIONSONSIGNALPROCESSING 3
frequencyselective.Thesourceandtherelayaresynchronized r r
sothattheycansimultaneouslyemitOFDMsymbolsusingK
subcarriersandwithsufficientlylongcyclicprefixtoeliminate
s u s u
inter-symbol interference.
The novel transmission protocol is half-duplex, i.e., user transmissionoversubcarrierkinthefirstslot transmissionoversubcarrierlinthesecondslot
(a) (b)
messagebitsaretransmittedintwoconsecutiveequal-duration
time slots, during which all channels are assumed to keep
Fig.2. Therelay-aidedtransmissionmodeoverthesubcarrierpair(k,l)to
unchanged. During the first slot, only the source broadcasts useru.
N OFDM symbols. Both the relay and all users receive these
symbols. After proper processing explained later, the source
of h and h , respectively. At user u, the nth baseband
and relay simultaneously broadcast N OFDM symbols, and su,l ru,l
signal received through subcarrier l is
the users receive them during the second slot.
Due to the OFDMA, each subcarrier is dedicated to trans- y (n)= P |h |+ P |h | θ(n)+z (n),
u,l,2 s,l,2 su,l r,l,2 ru,l u,l,2
mitting a single user’s message exclusively. A subcarrier in (3)
the first slotcan be pairedwith a subcarrierin the secondslot (cid:0)p p (cid:1)
fortherelay-aidedmodetransmissiontoauser.Eachunpaired where zu,l,2(n) is the AWGN with power σ2.
subcarrierineitherthefirstorsecondslotisusedbythesource Finally, user u decodes the message bits from all signals
received during the two slots. These signals can be grouped
for the direct mode transmission to a user.
Tosimplifydescription,weusesubcarrierskandltodenote into N vectors, the nth of which is
the kth and lth subcarriers used during the first and second y (n)
y(n)= u,k,1 (4)
slots, respectively (k,l = 1,··· ,K). We define the source y (n)
u,l,2
(cid:20) (cid:21)
transmission powers for subcarrier k in the first slot and
P h
subcarrierlinthesecondslotasP andP ,respectively. = s,k,1 su,k θ(n)+z(n),
s,k,1 s,l,2 P |h |+ P |h |
The relay transmission power for subcarrier l is P . The (cid:20) s,l,2 psu,l r,l,2 ru,l (cid:21)
r,l,2
complex amplitude gains at subcarrier k for the source-to- where z(n) p= [z (n),z p (n)]T. Note that the trans-
u,k,1 u,l,2
relay,source-to-uand relay-to-uchannelsare h , h and mission in effect makes N uses of a discrete memoryless
sr,k su,k
h , respectively.The two transmission modesfor the novel single-input-two-output channel specified by (4), with the
ru,k
protocol are elaborated as follows: nth input and output being θ(n) and y(n), respectively. To
achieve the maximum reliable transmission rate, maximum
1) The relay-aidedtransmission mode: Suppose subcarrier
ratio combining should be used [30], i.e., user u first turns
kispairedwithsubcarrierlfortherelay-aidedmodetransmis-
every y(n) into a decision variable
sion to user u. In such a case, we refer to the two subcarriers
collectively as the subcarrier pair (k,l). A block of message c(n)=( P h )∗y (n)+
s,k,1 su,k u,k,1
bits are first encoded into a code word of complex symbols ∗
{θ(n)|n = 1,··· ,N} with E(|θ(n)|2) = 1, ∀ n. In the first p Ps,l,2|hsu,l|+ Pr,l,2|hru,l| yu,l,2(n),
(5)
slot, the source broadcasts the codeword over subcarrier k (cid:0)p p (cid:1)
as illustrated in Figure 2.a. At the relay and user u, the nth andthendecodesthe messagefrom{c(n)|∀n}.Itcanreadily
baseband signals received through subcarrier k are be derived that the SNR for this decoding is
yr,k(n)= Ps,k,1hsr,kθ(n)+zr,k(n),n=1,··· ,N, (1) γklu(Ps,k,1,Ps,l,2,Pr,l,2)=Gsu,kPs,k,1+
2
and p Gsu,lPs,l,2+ Gru,lPr,l,2 , (6)
y (n)= P h θ(n)+z (n),n=1,··· ,N, where G = |hsu,k|(cid:0)2pand G = |hpru,l|2. (cid:1)
u,k,1 s,k,1 su,k u,k,1 su,k σ2 ru,l σ2
(2) To ensure both the relay and user u can reliably de-
p
code the message bits, the maximum number of mes-
respectively, where z (n) and z (n) are both additive
r,k u,k,1 sage bits that can be transmitted is 2NC(G P )
white Gaussian noise (AWGN) with power σ2. The signal- sr,k s,k,1
and 2NC(γ (P ,P ,P )), respectively. This means
to-noiseratio (SNR) at the relay is P G where G = klu s,k,1 s,l,2 r,l,2
s,k,1 sr,k sr,k that the maximum transmission rate over the subcarrier
|hsr,k|2. At the end of the first time slot, the relay decodes
σ2 pair (k,l) in the relay-aided mode to user u is equal
the message bits from {y (n)|n = 1,··· ,N} and then
r,k toC(min{G P ,γ (P ,P ,P )})bits/OFDM-
sr,k s,k,1 klu s,k,1 s,l,2 r,l,2
reencodes those bits into the same codeword as the source symbol (bpos)1.
did.
2) The direct transmission mode: Suppose subcarrier k
In the second time slot, the source and relay broadcast
(respectively, subcarrier l) is unpaired with any subcarrier in
the codewords {θ(n)e−j∠hsu,l|∀n} and {θ(n)e−j∠hru,l|∀ n} thesecond(respectively,first)slot,andisusedfordirectmode
through subcarrier l, respectively, where ∠h and ∠h
su,l ru,l transmission to user u. The source first encodes message bits
representthephaseofh andh ,respectively.Thismeans
su,l ru,l into a codeword of N symbols, which are then broadcast
that the source and relay implement transmit beamforming to
throughsubcarrier k (respectively,subcarrier l.Insuchacase,
emitthecodewordthroughsubcarrier l asillustratedinFigure
2.b. Note that the source and relay need to know the phase 1Recallthat2N OFDMsymbolsareusedintotalduringthetwotimeslots.
SUBMITTEDTOIEEETRANSACTIONSONSIGNALPROCESSING 4
the relay keeps silent at subcarrier l in the second slot, i.e. A. Rate maximization for the pair in the relay-aided mode
P =0.).User udecodesthemessagebits fromthe signals
r,l,2 1) Analysis for the proposed protocol: To facilitate deriva-
received through subcarrier k (respectively, subcarrier l). The
tion, define ∆ = G −G and G = G +G .
u,k sr,k su,k u,l su,l ru,l
maximumratethroughsubcarrier k (respectively,subcarrier l)
Tomaximizetherate,theoptimumP ,P andP are
s,k,1 s,l,2 r,l,2
inthedirecttransmissionmodeisC(P G )(respectively,
s,k,1 su,k the optimum solution for
C(P G )) bpos.
s,l,2 su,l
A benchmark protocol is also considered. This protocol max min{Gsr,kPs,k,1,γklu(Ps,k,1,Ps,l,2,Pr,l,2)}
Ps,k,1,Ps,l,2,Pr,l,2
is the same as the novel protocol except for the relay-aided
s.t. P +P +P =P, (7)
transmission mode. Specifically, the relay-aided mode is the s,k,1 s,l,2 r,l,2
same as that widely studied in the literature [13]–[18], [21]– Ps,k,1 ≥0,Ps,l,2 ≥0,Pr,l,2 ≥0.
[27],i.e.,thesourcedoesnottransmitatsubcarrierlduringthe
By using the Cauchy-Schwartz inequality, it can be shown
secondslot,ifsubcarrierskandlarepairedfortherelay-aided
that
transmission to user u. In such a case, the maximum rate for
the relay-aided transmission over that subcarrier pair to user γ (P ,P ,P )≤G P +G P , (8)
klu s,k,1 s,l,2 r,l,2 su,k s,k,1 u,l 2
u is equal to C(min{G P ,G P +G P })
sr,k s,k,1 su,k s,k,1 ru,l r,l,2
bpos. It is importantto note that, the benchmarkprotocolis a where P2 = Ps,l,2 +Pr,l,2 and the inequality is tight when
special case of the novelprotocol,since it is equivalentto the Ps,l,2 = GGsuu,,llP2 and Pr,l,2 = GGruu,,llP2. Now, the optimum
novelprotocol with the constraint that P =0 if subcarrier solution for (7) can be found by first solving
s,l,2
l is pairedwith a subcarrierin thefirst slotforthe relay-aided
max min{G P ,G P +G P } (9)
mode transmission. sr,k s,k,1 su,k s,k,1 u,l 2
Ps,k,1,P2
s.t. P +P =P,P ≥0,P ≥0
B. The WSR maximization problem s,k,1 2 s,k,1 2
We assume there exists a central controller which knows for the optimum P and P , and then using that P
s,k,1 2 2
precisely the CSI {G ,G ,G |∀ k}. Before the data to compute the optimum P and P according to the
sr,k su,k ru,k s,l,2 r,l,2
transmission,thecontrollerneedsto findthe optimumsubcar- formulas that tighten the inequality (8). Problem (9) can be
rier and power assignment, i.e., which subcarriers should be solved intuitively as follows. First, the two lines
paired for the relay-aided mode and which should be in the
L ={(x,y (x))|x∈[0,P],y (x)=G x}
direct mode, how these subcarriers should be assigned to the 0 0 0 sr,k
users,aswellasthesource/relaypowerallocationtomaximize L ={(x,y (x))|x∈[0,P],y (x)=G x+G (P −x)}
1 1 1 su,k u,l
the WSR of all users for the adopted transmission protocol
can be plot over the two-dimensional plane of coordinates
(which can be either the novel or benchmark protocol), when
(x,y) in Fig. 3. It can be seen that three different cases are
the total power consumption is not higher than a prescribed
possible, each corresponding to a specific orientation of the
value P . Then, the controller can inform the source and the
t
two lines. The coordinates of points A, B, C and D in the
relay about the optimum subcarrier and power assignment to
figureareshowninTableI.TheoptimumP andobjective
be adopted for data transmission. s,k,1
value for (9) (which are also for(7)) are equal to the x and y
III. THEORETICAL ANALYSIS coordinates of the points A, B and D for the three cases in
It can be shown that the proposed protocol leads to a Fig. 3, respectively. From this fact, it can easily be seen that
maximum WSR greater than or equal to that for the bench- the optimum Ps,k,1, Ps,l,2 and Pr,l,2 for (7) are
mark protocol. To this end, suppose the optimum subcarrier
Gu,l P if min{G ,G }>G ,
assignment and power allocation has been found for the Ps,k,1 = ∆u,k+Gu,l sr,k u,l su,k
benchmarkprotocol. By using the proposed protocolwith the ( P if min{Gsr,k,Gu,l}≤Gsu,k,
same subcarrier assignment and power allocation, the same
WSR can be achieved.Obviously,the maximumWSR forthe
Gsu,l ∆u,k P if min{G ,G }>G ,
proposedprotocolisgreaterthanorequaltothatWSR,namely Ps,l,2 = Gu,l (∆u,k+Gu,l) sr,k u,l su,k
the maximum WSR for the benchmark protocol. ( 0 if min{Gsr,k,Gu,l}≤Gsu,k,
In Section III-A, we assume subcarriers k and l are paired
and
for the relay-aided mode transmission to user u, and a sum
power P is used for this pair. We focus on computing the Gru,l ∆u,k P if min{G ,G }>G ,
maximum rate and optimum power allocation of this pair for Pr,l,2 =( 0Gu,l (∆u,k+Gu,l) if min{Gssrr,,kk,Guu,,ll}≤Gssuu,,kk.
both protocols. Using these results, theoretical analysis will
be made in Section III-B to show when the maximum WSR The maximum rate associated with the above optimum
for the proposed protocol is strictly greater than that for the solution is equal to C(Gn P) with
klu
benchmark one, and the RA algorithm will be developed in
Section IV. Moreover, this analysis plays an important role Gnklu = ∆Gus,rk,k+GGuu,l,l if min{Gsr,k,Gu,l}>Gsu,k,
to interpret the numerical experiments shown in Section V to ( min{Gsr,k,Gsu,k} if min{Gsr,k,Gu,l}≤Gsu,k.
illustrate the impact of the relay’s position on the benefit of (10)
using the proposed protocol.
SUBMITTEDTOIEEETRANSACTIONSONSIGNALPROCESSING 5
L0:y0(x)=Gsr,kx L1:y1(x)=Gsu,kx+Gu,l(P−x) in Fig. 4. The coordinates of points A and B are the same
as given in Tab. I, and those of points C , C , D and D
A A 1 2 1 2
C aregiveninTab.II.Mostinterestingly,Gn P andGb P are
C D klu klu
B D B B equal to the y-coordinateof D1 and D2, respectively,and C1
A C is above C since G ≥ G . In particular, the following
2 u,l ru,l
points should be noted:
0 x P 0 x P 0 x P • Gnklu >Gbklu holds because D1 is above D2.
(a)Gsr,k≤Gsu,k (b)Gsr,k>Gsu,k≥Gu,l (c)min{Gsr,k,Gu,l}>Gsu,k • when Gsr,k increases (meaningthat point A is elevated),
Gn − Gb increases (since the difference of the y-
klu klu
Fig.3. Illustration ofthetwolines L0 andL1 inthreedifferent cases. coordinate of points D and D is increased).
1 2
TABLEI • whenGru,l increases(meaningthatpointsC1 andC2 are
COORDINATESOFA,B,CANDDINFIGURE3. both elevated), Gn −Gb reduces, because
klu klu
∆ G G
A B C D Gn −Gb = u,k su,l sr,k ,
x P P 0 Gu,l P klu klu (∆u,k+Gsu,l+Gru,l)(∆u,k+Gru,l)
∆u,k+Gu,l
y Gsr,kP Gsu,kP Gu,lP ∆Gus,rk,k+GGuu,l,lP is a decreasing function of Gru,l.
2) Analysis for the benchmark protocol: In this case,
A
Ps,l,2 = 0 and the optimum Ps,k,1 and Pr,l,2 for maximizing y0(x)=Gsr,kx C1 D1
the rate are the optimum solution for
max min{Ps,k,1Gsr,k,Ps,k,1Gsu,k+Pr,l,2Gru,l} y1(x)=Gsu,kx+Gu,l(P−x) C2
Ps,k,1,Pr,l,2 B
(11) y2(x)=Gsu,kx+Gru,l(P−x) D
2
s.t. P +P =P,P ≥0,P ≥0,
s,k,1 r,l,2 s,k,1 r,l,2
0 x P
whichcan also besolvedbythe intuitivemethodasdescribed
above. It can be shown that the optimum Ps,k,1 and Pr,l,2 are Fig.4. Illustration ofGbklu andGnklu whenmin{Gsr,k,Gru,l}>Gsu,k.
Gru,l P if min{G ,G }>G ,
Ps,k,1 =( P∆u,k+Gru,l if min{Gssrr,,kk,Grruu,,ll}≤Gssuu,,kk, TABLEII
COORDINATESOFC1,C2,D1ANDD2INFIG.4.
and
∆u,k P if min{G ,G }>G , C1 C2 D1 D2
Pr,l,2 =( 0∆u,k+Gru,l if min{Gssrr,,kk,Grruu,,ll}≤Gssuu,,kk, xy Gu0,lP Gru0,lP ∆uG,Gknku+l,GulPu,lP ∆u,GGk+rbkulG,ulrPu,lP
and the maximum rate associated with the above optimum
solution is equal to C(Gb P) with The above analysis indicates that Gn ≥ Gb always
klu klu klu
holds, and Gn −Gb increases when either G increases
Gsr,kGru,l if min{G ,G }>G , klu klu sr,k
Gbklu = ∆u,k+Gru,l sr,k ru,l su,k or Gru,l reduces, if min{Gsr,k,Gru,l}>Gsu,k.
( min{Gsr,k,Gsu,k} if min{Gsr,k,Gru,l}≤Gsu,k. Using the above results, we now show that the proposed
(12) protocol leads to a strictly higher maximum WSR than the
benchmark protocol, if there exist at least two subcarriers
B. Comparison of the two protocols that must be paired for the relay-aided transmission for the
To compare the maximum WSR for the two protocols, benchmark protocol to maximize the WSR. To this end,
it is necessary to first compare Gn and Gb . When collect the subcarrier pairs that must be used by the bench-
klu klu
G ≥ min{G ,G }, Gb = min{G ,G }. If mark protocol to maximize the WSR in the set Φ, and
su,k sr,k ru,l klu sr,k su,k
min{G ,G } ≤ G , Gn = min{G ,G } = ∀ (k,l) ∈ Φ, denote u and P as the user which should
sr,k u,l su,k klu sr,k su,k kl ukl
Gb follows. If min{G ,G } > G , it can be seen use this subcarrier pair and the sum power that should be
klu sr,k u,l su,k
that Gn P and G P correspond to the y-coordinates of assigned to this pair. The rate contributed by this pair must
klu su,k
points D and B in Figure 3.c, respectively, and therefore be equal to C(Gb P ) as shown earlier. In such a case,
klukl ukl
Gn > G since D is higher than B. This means that min{G ,G } > G must be satisfied, because
klu su,k sr,k rukl,l sukl,k
Gn ≥Gb always holds when G ≥min{G ,G }. otherwise simply using subcarriers k and l separately in the
klu klu su,k sr,k ru,l
When min{G ,G } > G , Gn and Gb can be direct mode can lead to a higher sum rate. Suppose the
sr,k ru,l su,k klu klu
compared through a visualization method as follows. Specifi- proposedprotocolis now used with a suboptimum RA which
cally, we plot the lines L , L and adopts the same subcarrier assignment as the optimum RA
0 1
for the benchmarkprotocol.For every subcarrierin the direct
L ={(x,y (x))|x∈[0,P], (13)
2 2 mode, this RA uses the same source power allocation as the
y (x)=G x+G (P −x)}
2 su,k ru,l optimum value for the benchmark protocol, and ∀(k,l)∈Φ,
SUBMITTEDTOIEEETRANSACTIONSONSIGNALPROCESSING 6
this RA uses P as the sum power for the subcarrier pair auniqueuser.(17)and(18)ensurethetotalpowerconstraintis
ukl
(k,l). The maximum rate for this subcarrier pair is equal to satisfied. Theconstraints(19) and(20) are addedto guarantee
C(Gn P ). Since min{G ,G } > G holds, that every S is one-to-one mapped to a new variable for the
klukl ukl sr,k rukl,l sukl,k
Gn > Gb follows from earlier analysis, and therefore change of variable (COV) proposed later to solve the RA
klukl klukl
C(Gn P ) > C(Gb P ) must hold. This means that problem.
klukl ukl klukl ukl
the proposed protocol has a strictly higher maximum WSR Note that an S satisfying (14)-(20) indicates a unique
than the benchmark protocol. feasible RA scheme for the adopted protocol. Viewed from
the other way around, any feasible RA scheme can also be
IV. RA ALGORITHM DESIGN described by an S satisfying those constraints. Interestingly,
A. Formulation of the RA problem the same feasible RA scheme might be described by multi-
ple different S all satisfying these constraints. For instance,
To formulate the WSR maximization problem for the
considerthe scenario where there is only a single user u, and
adoptedprotocol(which can be either the proposedorbench-
the RA scheme requiring messages to be transmitted in the
mark protocol), we define
directmode,respectively,throughsubcarriersk andk during
1 2
G = Gnklu if theproposedprotocolisadopted, the first slot and subcarriers l1 and l2 during the second slot.
klu (cid:26) Gbklu if thebenchmarkprotocolisadopted. This RA scheme can be described by using either an S with
For any configuration of transmission-mode selection, sub- tk1l1uu = tk2l2uu = 1 and tk1l2uu = tk2l1uu = 0, or another
carrier pairing and assignment to users used by the adopted S′ with tk1l2uu =tk2l1uu =1 and tk1l1uu =tk2l2uu =0.
protocol,supposemsubcarrierpairsareassignedtotherelay- Given a feasible S, the maximum WSR for the adopted
aided transmission, then it is always possible to one-to-one protocol is
associate the unpaired subcarriers in the two slots to form
f(S)= t w C(G P )+ (21)
K−mvirtualsubcarrierpairs,eachallocatedtopossiblytwo klu u klu klu
differentusersfordirecttransmissionseparately.Motivatedby k,lX,u,a,b(cid:0)
t w C(G α )+w C(G β ) ,
thisobservation,theRAproblemisformulatedbydefiningthe klab a sa,k klab b sb,l klab
following variables:
where w >0 is t(cid:0)he weight prescribed for user u. The(cid:1)WSR
u
• tklu ∈ {0,1} for any combination of k,l,u. tklu = 1 maximization problem is to solve
indicates that subcarrier k is paired with subcarrier l for
the relay-aided transmission to user u. (P1) max f(S) s.t. (14)−(20)
S
• Pklu ≥0 for any combination of k,l,u. When tklu =1,
for a globally optimum S. We will develop an algorithm in
P is used as the total power for the subcarrier pair
klu
the following subsections to find it, after which the optimum
(k,l).
subcarrier assignment and source/relay power allocation can
• tklab ∈ {0,1} for any combination of k,l and a,b ∈ U.
be computed according to the analysis in Section III-A.
t = 1 indicates that subcarrier k is assigned in the
klab
direct transmission mode to user a during the first slot,
and so is subcarrier l to user b during the second slot. B. The idea behind the RA algorithm design
• αklab ≥0andβklab ≥0foranycombinationofk,l,a,b. Note that (P1) is a nonconvex program consisting of both
When t =1, P and P take the value of α
klab s,k,1 s,l,2 klab continuous and binary variables, thus in general its duality
and β , respectively.
klab gap is not zero. Similar nonconvex optimization problems
Letuscollectallindicatorandpowervariablesin thesets I
for multicarrier systems exist in the literature [31], [32]. A
andP,respectively,anddefineS={I,P}.EveryfeasibleRA
possible approachto tackle them is to show theirdualitygaps
scheme can be described by an S satisfying simultaneously
approachzerowhen a sufficientlylarge numberof subcarriers
t ,t ∈{0,1},∀k,l,u,a,b, (14) is used. This justifies the use of the dual method to find an
klu klab
asymptotically optimum solution.
Here,weuseacontinuous-relaxationbasedapproachtofind
t + t =1,∀k, (15)
klu klab
  atleastanapproximatelyoptimumSfor(P1).Similarmethods
l u a,b
X X X were also used in [33], [34] to compute asymptotic capacity
 
regions.Specifically,all indicatorvariablesare first relaxedto
t + t =1,∀l, (16)
 klu klab becontinuouswithin[0,1],afterwhichwegetanewproblem
k u a,b
X X X
  (P2) max f(S)
(tkluPklu+tklab(αklab+βklab))≤Pt, (17) S
k,lX,u,a,b s.t. tklu,tklab ∈[0,1],∀k,l,u,a,b, (22)
P ≥0,α ≥0,β ≥0,∀k,l,u,a,b, (18)
klu klab klab (15)−(20),
P =0if t =0,∀k,l,u,a,b, (19)
klu klu
as a relaxation of (P1). Define the feasible set of (P2) as FS.
α =0,β =0if t =0,∀k,l,u,a,b, (20)
klab klab klab Obviously, the feasible set of (P1) is a subset of FS.
where(15)and(16)guaranteetheOFDMA,i.e.,everysubcar- Then, we make the COV from P to P =
rierisusedexclusivelyforthetransmissionofmessagebitsto {P ,α ,β |∀k,l,u,a,b},whereevery P , α and
klu klab klab klu klab
e
e e e e e
SUBMITTEDTOIEEETRANSACTIONSONSIGNALPROCESSING 7
β satisfy, respectively, In practice, it may be difficult to find precisely a global
klab
optimumX⋆ for(P4)ingeneral.Forinstance,existingconvex-
P =t P ,α =t α ,β =t β .
e klu klu klu klab klab klab klab klab klab optimization techniques such as the interior-point method
(23)
or the dual method all search for the global optimum in
e e e an iterative manner, and finally produce an approximately
After the COV, we collect all variables into X={I,P}. It
optimum solution with an objective value very close to the
is important to note that an S ∈FS is one-to-one mapped to
optimum value. Motivated by this fact, suppose a solutionX′
an X∈FX, where FX contains the set of all X’s satisefying
which satisfies
(22), (15)-(16), as well as
1) (26)and(27)andallindicatorvariablesinX′arebinary;
P +α +β ≤P , (24) 2) g⋆−g(X′) is very small;
klu klab klab t
k,lX,u,a,b(cid:16) (cid:17) can be found for (P4), then S(X′) is feasible for (P1) and
P ≥0,eα ≥e0,β e≥0,∀k,l,u,a,b, (25) f⋆−f(S(X′)) is also very small because
klu klab klab
Pklu =0if tklu =0,∀k,l,u,a,b, (26) f⋆−f(S(X′))≤g⋆−g(X′),
e e e
α =0,β =0if t =0,∀k,l,u,a,b. (27)
klab klab klab which means that S(X′) can be taken as an approximately
e
Asea functioneof X∈FX, the WSR can be rewritten as opItinmtuhmefosolllouwtiionngfsourb(sPec1t)i.on,weusethedualmethodtosolve
g(X)=f(S(X)) (P4). Specifically, the ellipsoid method is used to search for
the dual optimum. This ellipsoid method is reduced to the
= w φ(t ,P ,G ) (28)
u klu klu klu
bisection method to update upper and lower bounds for the
k,l,u,a,b
X (cid:0) dual optimum iteratively until convergence. In some cases,
+w φ(t ,α ,eG )+w φ(t ,β ,G ) ,
a klab klab sa,k b klab klab sb,l the global optimum for (P1) can be found, while in other
awnhdere S(X) representes the S corresponding to tehe X ∈ FX(cid:1) cnausmeseriwcael eexxppleariinmebnytsthtehoart,etitchael oapntailmysuims asnodlutiilolunstrfaotre tbhye
Lagrangian relaxation problem (LRP) of (P4) corresponding
tC(Gx) if t>0,
φ(t,x,G)= t (29) to the upper bound produced after convergence can be taken
0 if t=0.
(cid:26) as the X′ described above. Then, S(X′) can be output as an
It can readily be shown that φ(t,x,G) with fixed G is a approximately optimum solution for (P1).
continuousand concavefunctionof t≥0 and x, because it is
a perspective function of C(Gx) which is concave of x (see C. The development of the RA algorithm
pages 89−90 for more details in [35]). Therefore, g(X) is a
Since (P4) is a convex program and it satisfies the Slater
concave function of X∈FX. constraint qualification2, (P4) has zero duality gap (see page
After solving
226of[35]),whichjustifiestheuseofthedualmethodtosolve
(P3) max g(X) (P4).To thisend,µ is introducedasa Lagrangemultiplierfor
X the constraint (24). The LRP for (P4) is
s.t. (22),(15)−(16),(24)−(27),
(P5) max L(µ,X)=g(X)+µ P −P(X)
for its global optimum, the S corresponding to this global X t
(cid:18) (cid:19)
optimum is the optimum solution for (P2). In the following
s.t. (22),(15)−(16),(25),
subsection, we will focus on solving the problem
where L(µ,X) is the Lagrangian of (P4) and P(X) is the
(P4) max g(X)
left-hand side of (24) (i.e., the sum power as a function of
X
s.t. (22),(15)−(16),(24)−(25), X). A global optimum for (P5) is denoted by Xµ. The dual
function is defined as d(µ) = L(µ,X ), which is a convex
µ
which is a relaxation of (P3) by omitting (26) and (27). function of µ. In particular,
Obviously, FX is a subset of the feasible set of (P4). Most
γ(µ)=P −P(X ) (30)
interestingly, (P4) is a convex program, which can be solved t µ
byhighly-efficientconvex-optimizationtechniques.Definethe
is a subgradient of d(µ), i.e., it satisfies
optimum objective value for (P1) and (P4) as f⋆ and g⋆,
respectively. According to the relaxations we made, ∀µ′,d(µ′)≥d(µ)+(µ′−µ)γ(µ), (31)
g⋆ ≥ max g(X)= maxf(S)≥f⋆ and the dual problem is to find the dual optimum
X∈FX S∈FS
µ⋆ =argmind(µ). (32)
follows. Define a global optimum for (P4) as X⋆. If we can
µ≥0
find an X⋆ that satisfies (26) and (27), and contains binary
Since (P4) has zero duality gap, the following properties
indicator variables (i.e., t ,t ∈ {0,1},∀ k,l,u,a,b),
klu klab hold:
then it can readily be shown that S(X⋆) must be a global
optimum for (P1). 2Thereexists atleastanXsatisfyingallinequality constraints strictly.
SUBMITTEDTOIEEETRANSACTIONSONSIGNALPROCESSING 8
• Note that µ⋆ represents the sensitivity3 of the opti- solving
mum objective value for (P4) with respect to P , i.e.,
t
Pg(tPX,t⋆m)e=aniµng⋆.thOabtvµio⋆u>sly0,.g(X⋆) is strictly increasing of I,{tmkl|a∀xk,l} Xk,l uX,a,b(cid:0)tkluAklu+tklabBklab(cid:1)
• µ = µ⋆ and Xµ = X⋆ are true if and only if Xµ s.t. tkl =1,∀k, (37)
is feasible and µγ(µ) = 0 is satisfied according to l
X
Proposition 5.1.5 in [36]. This means that µ⋆γ(µ⋆)=0. t =1,∀l,
kl
Moreover, X =X⋆ if γ(µ)=0.
µ k
X
The idea behind the dual method to solve (P4) is to search tkl = tklu+ tklab,∀k,l.
for µ⋆. Then, the Xµ⋆ that satisfies γ(µ⋆) = 0 can be taken Xu Xa,b
asX⋆.Thekeytothedualmethodconsistsoftwoprocedures t ≥0,t ≥0,∀k,l,u,a,b.
klu klab
to find X for a given µ>0 and µ⋆, respectively, which are
µ
Note that the inequality t A +t B ≤
developed as follows. u,a,b klu klu klab klab
t C holds where C = max{max A ,max B }.
kl kl kl u klu a,b klab
1) FindingX whenµ>0: Thefollowingstrategyisused P (cid:0) (cid:1)
µ Let us call A as the metric for t and B as the
klu klu klab
tofindX for(P5)whenµ>0.First,theoptimumPfor(P5)
µ metric for t . This inequality is tightened when all entries
klab
withfixedIisfoundanddenotedbyPI.DefineXI ={I,PI}. of {t ,t |∀ u,a,b} are assigned to zero, except that the
klu klab
Thenwe find the optimumI to maximizeL(µ,XI)esubjectto one with the metric equal to C is assigned to t .
kl kl
(22),(15)and(16).Finally,XI correespondingtothisoptimeum Therefore, after the problem
I can be taken as X .
µ
SupposeIisfixed,wefindPIasfollows.Specifically,every max tklCkl
Pklu in PI is equal to 0 when tklu = 0. When tklu > 0, the {tkl|∀k,l} Xk,l uX,a,b
optimum Pklu can be foundeby using the KKT conditions s.t. tkl =1,∀k, (38)
reelated toePklu. In summary, the optimum Pklu can be shown Xl
to be e t =1,∀l,
kl
e e
k
X
P =t Λ(w ,µ,G ), (33) t ≥0,∀k,l,
klu klu u klu kl
is solved for its optimum solution {t⋆ |∀k,l}, an optimum I
where Λ(w ,µ,eG) is defined as Λ(w ,µ,G) = kl
u u for(37)canbeconstructedbyassigningforeverycombination
+
wu2loµg2e − G1 . In a similar way, the optimum αklab of k and l, all entries in {tklu,tklab|∀ u,a,b} ⊂ I to zero,
except for the one with the metric equal to C to t⋆ .
ahnd β canibe shown to be kl kl
klab
Most interestingly, (38) is a standard assignment prob-
e
lem, hence every entry in {t⋆ |∀ k,l} is either 0 or 1
e αklab =tklabΛ(wa,µ,Gsa,k), (34) and {t⋆ |∀ k,l} can be foundkelfficiently by the Hungarian
kl
β =t Λ(w ,µ,G ), (35) algorithm[37].Afterknowing{t⋆ |∀k,l},theoptimumIcan
klab klab b sb,l kl
e beconstructedaccordingtothewaymentionedearlier.Finally,
respectively. Usieng these formulas, XI = {I,PI} can be the corresponding XI = {I,PI} is assigned to Xµ. Note
that to compute X , {A ,B |∀ k,l,u,a,b} containing
found. It can readily be shown that µ klu klab
K2(U+U2)entrieshastobeceomputedfirst, whichimpliesa
e
complexity of O(K2U2). Moreover,the Hungarian algorithm
L(µ,XI)=µPt+ tkluAklu+tklabBklab (36) to solve (38) has a complexity of O(K3) [37]. This means
k,lX,u,a,b(cid:0) (cid:1) that the complexity of finding Xµ is O(K2U2+K3).
2) Finding µ⋆: To find µ⋆, an incremental-update based
where subgradient method which updates µ with µ = [µ−δ(P −
t
P(X ))]+ can be used, where δ > 0 is a prescribed step
µ
A =w C(G Λ(w ,µ,G ))−µ·Λ(w ,µ,G ) size [36]. However, this method converges very slowly, since
klu u klu u klu u klu
δ has to be very small to guarantee convergence. To speed
B =w C(G Λ(w ,µ,G ))−µ·Λ(w ,µ,G )+
klab a sa,k a sa,k a sa,k up the search for µ⋆, we use the ellipsoid method. The idea
w C(G Λ(w ,µ,G ))−µ·Λ(w ,µ,G ).
b sb,l b sb,l b sb,l behind the ellipsoid method is to find a series of contracting
ellipsoids that always contain µ⋆ [35]. The ellipsoid method
Finally, we find the optimum I for maximizing L(µ,XI) can be reduced to the bisection method as follows.
subject to (22), (15) and (16). This problem is equivalent to First, a lower bound µ and an upper bound µ for µ⋆
l u
are initialized. As said earlier, µ⋆ > 0 holds, thus µ can
l
be initialized with 0. As shown in the Appendix, µ can
u
for3aNcootenvthexatmthienismenizsaittiivointyparnoabllyesmis.wItascainntrboedupcreodveinnpthaagtesg2(X49⋆-)25=3oµf⋆[3b5y] be initialized with KwmaPxtlog2e. Then, µl and µu are up-
casting the problem (P4) into an equivalent convex minimizPattion problem. dated iteratively as follows. In every iteration, Xµm where
Theproofisstraightforward andomitted hereduetospacelimitation. µm = µl+2µu is computed. If γ(µm) > 0, then ∀ µ > µm,
SUBMITTEDTOIEEETRANSACTIONSONSIGNALPROCESSING 9
d(µ) ≥ d(µ )+(µ−µ )γ(µ ) > d(µ ). This means that Algorithm 1 The RA algorithm to find an approximately
m m m m
µ⋆ mustbeconfinedin[µ,µ ],soµ shouldbeupdatedwith optimum S for (P1)
l m u
µm.Ifγ(µm)<0,itcanbeshownsimilarlythatµl shouldbe 1: compute Gklu, ∀k,l,u.
updatedwithµm.Theiterationisterminatedwhenγ(µm)=0 2: µl =0; µu = KwmaPxtlog2e;
or µu−µl ≤ǫ where ǫ>0 is a prescribed small value. 3: while µu−µl >ǫ do
When the iteration is terminated with γ(µm) = 0 being 4: µm = µu+2µl;
satisfied, X⋆ = Xµm must hold as said earlier. Note that 5: solve (P5) with µ=µm for Xµm; compute γ(µm);
S(Xµm) must be a global optimum for (P1) since Xµm 6: if γ(µm)=0 then
satisfies (26) and (27), and containsbinaryindicatorvariables 7: compute S(Xµm) and output it as an optimum solu-
as said in Section IV.B. tion for (P1);
We now consider the case where the iteration is terminated 8: exit the algorithm;
with µu − µl ≤ ǫ being satisfied. In such a case, we find 9: else if γ(µm)>0 then
thatXµu isan approximatelyoptimumsolutionfor(P4).This 10: µu =µm;
findingwillbeillustratedbynumericalexperimentsinSection 11: else
V. Itcan be explainedby theoreticalanalysisas follows. Note 12: µl =µm;
that 13: end if
14: end while
g⋆−g(Xµu)≤d(µu)−g(Xµu)=µuγ(µu) (39) 15: solve (P5) with µ=µu for Xµu;
holds since ∀ µ ≥ 0, g⋆ ≤ d(µ). In addition, we present the 16: compute S(Xµu) and output it as an approximately opti-
mum solution for (P1).
following lemma:
Lemma 1: γ(µ) is an increasing function of µ≥0.
Proof: Suppose µ ≥µ . According to (31),
1 2 1 2
d(µ1)≥d(µ2)+(µ1−µ2)γ(µ2) s r 5 50 m
d(µ2)≥d(µ1)+(µ2−µ1)γ(µ1) d km 4 3
follow. As a result,
1 km
(µ −µ )γ(µ )≥d(µ )−d(µ )≥(µ −µ )γ(µ )
1 2 1 1 2 1 2 2 Fig.5. Therelay-aided downlinkOFDMAsystemconsidered innumerical
experiments.
holds, and thus γ(µ )≥γ(µ ). This completes the proof.
1 2
According to Lemma 1, γ(µ ) ≥ γ(µ⋆) = 0 because
u
Mµuor≥eovµer⋆,,µmγe(aµnin)gretdhuactesXaµsuthies iatelwraatiyosnfperaoscibeeledsfoanrd(Pit4i)s. leadsto at mostlog2(KwmǫaPxtlog2e)≈21+log2(PKt) iterations
u u for a given combination of K and P .
t
very small after convergence,since µ decreases to approach
u The channels are independent of each other and generated
µ⋆ whichsatisfies µ⋆γ(µ⋆)=0.Thismeansthatg⋆−g(X )
µu in the same way as in [1], [3]. For every user u, the impulse
is very small according to (39). Moreover, X also satisfies
µu response of the source-to-u channel is modeled as a delay
(26) and (27) and all indicator variables in X are binary.
µu linewithL=6taps,whichareindependentlygeneratedfrom
This means that S(X ) can be output as an approximately
µu circularlysymmetriccomplexGaussiandistributionswith zero
optimum solution for (P1) as said in Section IV.B. mean and variance equal to 1 dsu −2.5, where d = 1
The overall procedure to find an approximately optimum L dref ref
km and d represents the source-to-u distance. The source-
su
solutionfor(P1)issummarizedinAlgorithm1.Itscomplexity (cid:0) (cid:1)
to-relay and relay-to-u channels are generated in the same
can be studied as follows. First, {Gklu|∀ k,l,u} needs to be way, with each tap having the variance as 1 d −2.5 and
computed, which needs K2U operations. Then, finding µ⋆ L dref
1 dru −2.5, respectively, where d represents the relay-
with the bisection method requires at most a number of iter- L dref ru (cid:0) (cid:1)
to-u distance. The CSI {h |∀ k}, {h |∀ k,u} and
aXtionhsasinathcoemoprdleexriotyfolofgO2((KK)2.UF2o+r eKac3h).itTehraetrieofno,rec,omthpeuttoitnagl {h(cid:0)ru,k|(cid:1)∀ k,u} are computedsr,bky making Ksu-,pkoint FFT over
µ
complexity of Algorithm 1 is O(log (K)(K2U2+K3)). the impulse response of the associated channels.
2
Inordertoillustratethebenefitofoptimizedsubcarrierpair-
ing and opportunistic DF relaying, we also consider another
V. NUMERICAL EXPERIMENTS benchmark protocol (BP-2) in addition to the already studied
In numerical experiments, we consider the relay-aided benchmark mark protocol (BP-1). BP-2 is the one studied in
downlink OFDMA system illustrated in Figure 5. The relay [25] using a single relay, i.e., subcarrier k in the first slot
is located in the line between the source and the center of and subcarrier k in the second slot are allocated to a user for
the user region, and the source-to-relay distance is d km. either the relay-aided transmission or the direct transmission
U =5 users are served and they are randomly and uniformly separately. The RA algorithm proposed in [25] is used for
distributed in a circular region of radius 50 m. Their weights BP-2.
are randomly chosen between 0.8 and 1.2 for every system AccordingtotheanalysisinSectionIV.C,S(X )isfinally
µu
realizationsimulated.ForAlgorithm1,ǫissetas10−6,which output as an approximately optimum solution if the iteration
SUBMITTEDTOIEEETRANSACTIONSONSIGNALPROCESSING 10
is terminatedwith µ −µ ≤ǫ being satisfied. In such a case, Average WSR
u l 10.5
f⋆−f(S(X ))≤µ γ(µ ) after convergence,and the proposed protocol
µu u u 10 BBPP−−12
δ(µu)= fµ(Su(γX(µµuu))) (40) M−symbol) 9.95
can be computed to evaluate the relative difference between OFD 8.5
the WSR finally achieved and the maximum WSR for (P1). R (bits/ 8
S
To illustrate the effectiveness of Algorithm 1, we have W
7.5
executed Algorithm 1 for both the proposed protocol and
7
BP-1 over 104 random system realizations. Specifically, the
system realizations are generated by randomly choosing a 6.50 .1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
d (km)
combination of d ∈ [0.1,0.9] km, K ∈ {8,16,32,64,128},
P /σ2 ∈ [0,45] dB, then generating the channels as said (a)
t
earlier. It can readily be shown that at most 28 iterations are Nsp/K
1
executedforAlgorithm1foreveryrandomchannelrealization the proposed protocol
BP−1
generated.The δ(µ )isevaluatedandcollectedforallsystem 0.9 BP−2
u
realizationswhenthe iterationof Algorithm1 terminateswith 0.8
µu−µl ≤ ǫ being satisfied. The probability density function 0.7
(PDF) of these δ(µ ) in dB scale (i.e., 10*log (δ(µ ))) is
u 10 u 0.6
showninFigure6.Itcanbeseenthatδ(µ )isalwayssmaller
u
0.5
than 3%, which indicates that the finally produced S(X )
µu
is indeed an approximately optimum solution with a WSR 0.4
very close to the maximum WSR for (P1) if the iteration is 0.3
terminated with µ −µ ≤ǫ being satisfied.
u l 0.2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
d (km)
pdf of 10log (δ(µ ))
10 m (b)
0.07
the proposed protocol
0.06 BP−1 Fig.7. TheaverageoptimumWSRsand Nsp astherelaypositionchanges
K
whenPt/σ2=20dBandK=32.
0.05
ensity0.04
d
ability 0.03 It is interesting to observe that for every protocol,the opti-
b
o
pr mum WSR is higher and it is more likely to pair subcarriers
0.02
for the relay-aided transmission to maximize the WSR when
0.01
the relay moves toward the middle between the source and
0 the user-region center. This behavior is interpreted for the
−140 −120 −100 −80 −60 −40 −20 0
10*log10(δ(µm)) proposed protocol as follows (those for BP-1 and BP-2 can
be interpreted in a similar way and thus omitted due to space
Fig. 6. ThePDFof10∗log10(δ(µu)simulated over104 random system limitation). It is important to note that the optimum WSR for
realizations.
theproposedprotocol,astheoptimumobjectivevalueof(P1),
To show the impact of relay position on the protocols’ depends on {Gsu,k,Gnklu|∀ k,l,u}. If ∀ k,l,u, Gnklu is more
performance, we choose P /σ2 = 20 dB and K = 32, likely to take a high value, the subcarriers are more likely
t
then evaluated the average optimum WSRs and Nsp for to be paired for the relay-aided transmission to maximize
K
everyprotocolover 1000randomchannelrealizationswhen d the WSR, and the average optimum WSR for the proposed
increases from 0.1 to 0.9 km. Here, Nsp denotes the average protocol increases. As can be seen from Fig. 3, Gnklu is high
numberofthesubcarrierpairsthatshouldbeusedintherelay- if both Gsr,k and Gu,l are much greater than Gsu,k. When
aidedmodetomaximizetheWSR.Itcanreadilybecomputed the relay lies in the middle between the source and the user-
thatatmost20iterationsisexecutedforAlgorithm1forevery regioncenter,bothGsr,kandGu,larelikelytobemuchgreater
channelrealizationgenerated.TheresultsareshowninFigure thanGsu,k, meaningthatGnklu is likely to be high.Therefore,
7. the optimum WSR is higher and it is more likely to pair
When d is fixed, the proposed protocol leads to a greater subcarriersfortherelay-aidedtransmissionwhentherelaylies
average optimum WSR than BP-1, which illustrates the the- in the middle between the source and the user-region center.
oretical analysis in Section III-B. Moreover, the proposed When d is small, the optimum WSR for the proposed
protocol and BP-1 both have greater average optimum WSRs protocol is much greater than that for BP-1, and it is more
than BP-2. This is because they can betterexploitthe degrees likely to pair subcarriers for the relay-aided transmission to
offreedomforsubcarrierpairingandassignmenttousersthan maximize the WSR for the proposed protocol than for BP-1.
BP-2 to improve the spectrum efficiency. This can be explained as follows. Note that if Gn −Gb
klu klu