Power minimization for OFDM Transmission with Subcarrier-pair based Opportunistic DF Relaying

0.1 MB·English

by  Tao Wang

1
Power minimization for OFDM Transmission with
Subcarrier-pair based Opportunistic DF Relaying
Tao Wang, Senior Member, IEEE, Yong Fang, Senior Member, IEEE and Luc Vandendorpe, Fellow, IEEE
Abstract—This paper develops a sum-power minimized re- a subcarrierin the first slot, the same subcarrierin the second
source allocation (RA) algorithm subject to a sum-rate con- slot is always paired with this subcarrier for DF relaying. In
straintforcooperativeorthogonalfrequencydivisionmodulation
[2]–[4], [10], [11], the optimization of subcarrier pairing was
3 (OFDM) transmission with subcarrier-pair based opportunistic
considered.
1 decode-and-forward (DF) relaying. The improved DF protocol
0 first proposed in [1] is used with optimized subcarrier pairing. Recently, energy-efficient communication is becoming in-
2 InstrumentaltotheRAalgorithmdesignisappropriatedefinition creasingly important [12]. In view of the fact that many
ofvariablestorepresentsource/relaypowerallocation,subcarrier existing works focus on spectral-efficiency maximized RA,
n pairing and transmission-mode selection elegantly, so that after
we develop an RA algorithm for minimizing the sum power
a continuousrelaxation,thedualmethodandtheHungarianalgo-
J subject to a sum-rate constraintfor the improvedDF protocol
rithmcanbeusedtofindan(atleastapproximately)optimumRA

4 with polynomial complexity. Moreover, the bisection method is with optimized subcarrier pairing (OSP). Compared with the
1 usedtospeedupthesearchoftheoptimumLagrangemultiplier algorithms designed in [4], [10], our algorithm uses a new
for the dual method. Numerical results are shown to illustrate methodtodefineindicatorvariablesforrepresentingsubcarrier
] the power-reduction benefit of the improved DF protocol with
T pairing and transmission-mode selection, by regarding the
optimized subcarrier pairing.
I subcarriers for the direct transmission in the two time slots
s. Index Terms—Cooperative communication, resource alloca- as virtual subcarrier pairs. Moreover, the bisection method is
c tion, decode and forward, OFDM. used to find the optimum Lagrange multiplier, which is faster
[
thantheincremental-updatebasedsubgradientmethodusedin

1 I. INTRODUCTION [4], [10].
v Notations: C(x)= 1log (1+x) and [x]+ =max{x,0}.
1 Cooperative orthogonal frequency division modulation 2 2
4 (OFDM) transmission with subcarrier-pair based decode-and-
9 forward(DF)relayingandassociatedresourceallocation(RA) II. SYSTEM ANDPROTOCOL DESCRIPTION
2 were studied in [2]–[5] when the source-to-destination link
. Consider the scenario where a relay assists a source’s
1 exists. In [2]–[4], an “always-relaying” DF protocol was
transmission to a destination. The improved DF protocol in
0 considered.To betterexploitthefrequency-selectivechannels,
[1] is used. Specifically, everydata-transmissionsession takes
3
wehaveproposedanopportunisticDFrelayingprotocolin[5],
1 two consecutiveequal-durationtime slots and OFDM with K
i.e, a subcarrierin the first time slot can either be paired with
: subcarriersis used. To facilitate description, a subcarrierused
v a subcarrier in the second slot for the relay-aided transmis-
inthefirstslotisdenotedbysubcarrierkandoneinthesecond
i
X sion, or used for the direct source-to-destination transmission
slot by subcarrier l hereafter. In the first time slot, the source
without the relay’s assistance. A major drawback for that DF
r radiates OFDM symbols, using P as the transmit power
a protocolisthat,asubcarrierunusedforrelayingin thesecond s,k,1
forsubcarrier k.Thesource-to-relayandsource-to-destination
slotbecomesidle,whichwastesspectrumresource.Toaddress
baseband-channel coefficients for subcarrier k are h and
this issue, we first proposed in [1] an improved DF protocol, sr,k
h , respectively. In the second slot, both the source and
which allows the source to make the direct transmission over sd,k
the relay synchronously radiate OFDM symbols, using P
the subcarriers unused for relaying in the second slot. This s,l,2
andP asthe transmitpowersforsubcarrier l,respectively.
protocol and its RA were later intensively investigated, e.g., r,l,2
The relay-to-destination baseband-channel coefficient is h
in[6]–[11].Notethatin[1],[5]–[9],apriorisubcarrierpairing rd,l
for subcarrier l.
was used. i.e., when the relay-aided transmission is used for
A subcarrier in the first slot can either be paired with
one in the second slot for the relay-aided transmission, or
T. Wang is with School of Communication & Information Engineering,
Shanghai University, 200072 Shanghai, P. R. China. He was with ICTEAM be used for the direct transmission without relaying. Every
Institute, Universite´ Catholique de Louvain (UCL), 1348 Louvain-la-Neuve, unpaired subcarrier in the second slot is used for the direct
Belgium. (email:[email protected]).
transmission. In particular, if subcarrier l is used for the
Y. Fang is with School of Communication & Information Engi-
neering, Shanghai University, 200072 Shanghai, P. R. China (email: direct transmission, Ps,l,2 ≥ 0 is used while Pr,l,2 = 0 is
[email protected]). imposed. The maximum average data rates over subcarriers k
L.VandendorpeiswithICTEAMInstitute, UCL,1348Louvain-la-Neuve,
and l used for the direct transmission are C(P G ) and
Belgium (email:[email protected]). s,k,1 sd,k
ResearchsupportedbyTheProgramforProfessorofSpecialAppointment C(Ps,l,2Gsd,l) bits/OFDM-symbol (bpos), respectively, where
(Eastern Scholar) at Shanghai Institutions of Higher Learning. It is also G = |hsd,k|2 and σ2 is the noise variance for each
supportedbytheIAPprojectBESTCOM,theARCSCOOP,theNEWCOM# sd,k σ2
andNSFChina#61271213. subcarrierateverynode’sreceiver.Whensubcarrier kispaired
2
with subcarrier l for the relay-aided transmission, the DF of variables (COV) from P to P = {P ,α ,β |∀k,l},
kl kl kl
relaying is used in which case P = 0 is imposed while whereP ,α andβ satisfyP =tRP ,α =tDα and
s,l,2 kl kl kl kl kl kl kl kl kl
Ps,k,1 ≥0 and Pr,l,2 ≥0 are used (more details are available βkl =tDklβkl, respectively,∀k,l.Aeftercolleectinegallevariables
in [1]). Suppose Ps,k,1+Pr,l,2 =P, it can readily be shown into Xe= {eI,P}, tehe RA proeblem can beerewritten as the
thatthe maximumdatarate is equalto Rk,l =C(GklP) bpos, peroblem (P2):
where e
min P(X)= (P +α +β )
Gsr,kGrd,l if min{G ,G }>G , X kl kl kl
Gkl = Gsr,k−Gsd,k+Grd,l sr,k rd,l sd,k Xk,l
( min{Gsr,k,Gsd,k} if min{Gsr,k,Grd,l}≤Gsd,k, s.t. tR,tD ∈[0,1]e,∀k,le; e
kl kl
Gsr,k = |hsσr,2k|2 and Grd,l = |hrσd2,l|2 [1]. This maximum rate tDkl+tRkl =1,∀k; tDkl+tRkl =1,∀l;
is achieved when Xl (cid:0) (cid:1) Xk (cid:0) (cid:1)
P ≥0,α ≥0,β ≥0,∀k,l;
Grd,l P if min{G ,G }>G , kl kl kl
Ps,k,1 =( PGsr,k−Gsd,k+Grd,l if min{Gssrr,,kk,Grrdd,,ll}≤Gssdd,,kk, e−g(X)≤e −Rreq,e
where g(X) represents the maximum sum rate expressed as
Assumethereexistsacentralcontrolunitwhichknowspre-
cisely {Gsr,k,Gsd,k|∀k} and {Grd,l|∀l}, and determines the g(X)= φ(tRkl,Pkl,Gkl)
optimumRA(i.e.,thesource/relaypowerallocation,subcarrier
k,l
pairingandtransmissionmodeselection)tominimizethesum X(cid:0)+φ(tDe,α ,G )+φ(tD,β ,G ) ,
power subject to the constraint that the sum data rate is not kl kl sd,k kl kl sd,l
smaller than prescribed R bpos. and (cid:1)
req e e
tC(Gx) if t>0,
φ(t,x,G)= t (1)
III. RAALGORITHM DESIGN 0 if t=0.
(cid:26)
For any subcarrier assignment used by the improved DF
Obviously (P2) is a relaxation of (P1). We will find an
protocol,supposemsubcarrierpairsareassignedtotherelay-
(at least approximately)optimum solution for (P2), and show
aided transmission, then it is always possible to one-to-one that the S corresponding to this solution is still feasible, and
associate the unpaired subcarriers in the two slots to form
hence (at least approximately)optimum for (P1). To this end,
K − m virtual subcarrier pairs for the direct transmission.
note that φ(t,x,G) with fixed G is a continuousand concave
Motivated by this observation, the RA problem is formulated
function of t ≥ 0 and x, because it is a perspective function
by defining: of C(Gx) which is concave of x [13]. As a result, g(X) is a
• tRkl ∈ {0,1} and Pkl ≥ 0, ∀ k,l. tRkl = 1 indicates that concave function of X in its feasible domain for (P2). This
subcarrierk ispairedwithsubcarrierlfortherelay-aided means that (P2) is a convex optimization problem. As can
transmission.WhentRkl =1,Pkl isusedasthetotalpower be checked, it also satisfies the Slater constraint qualification,
for the subcarrier pair (k,l). thereforeithaszerodualitygap,whichjustifiestheapplicabil-
• tDkl ∈ {0,1}, αkl ≥ 0 and βkl ≥ 0, ∀ k,l. tDkl = 1 ity of the dual method to find the globally optimum for (P2),
indicatesthatsubcarriersk andlformavirtualsubcarrier denoted as X⋆ hereafter.
pairforthedirecttransmission.When tDkl =1,Ps,k,1 and To use the dual method, µ is introduced as a Lagrange
Ps,l,2 take the value of αkl and βkl, respectively. multiplier for the rate constraint. The Lagrange relaxation
Letuscollectallindicatorandpowervariablesin thesets I problem for (P2) is the problem (P3):
and P, respectively, and define S={I,P}. The RA problem
can be formulated as the problem (P1): min L(µ,X)=P(X)+µ R −g(X)
req
X
(cid:18) (cid:19)
mSin (tRklPkl+tDklαkl+tDklβkl), s.t. tRkl,tDkl ∈[0,1],∀k,l;
Xk,l tD +tR =1,∀k; tD +tR =1,∀l;
s.t. tR,tD ∈{0,1},∀k,l; kl kl kl kl
kl kl l k
X(cid:0) (cid:1) X(cid:0) (cid:1)
tD +tR =1,∀k; tD +tR =1,∀l; P ≥0,α ≥0,β ≥0,∀k,l;
kl kl kl kl kl kl kl
l k
X(cid:0) (cid:1) X(cid:0) (cid:1) where L(µ,X) is the Lagrangian of (P2). A global optimum
Pkl ≥0,αkl ≥0,βkl ≥0,∀k,l; for (P3) isedenotedeby X aend the dual function is defined as
f(S)≥Rreq, d(µ) = L(µ,X ). Noteµthat d(µ) is concave of µ ≥ 0, and
µ
where f(S) represents the maximum sum rate as Rreq −g(Xµ) is a subgradient of d(µ), i.e., ∀ µ′, d(µ′) ≤
d(µ)+(µ′−µ)(R −g(X )). The dual problem is to find
req µ
f(S)= tRklC(GklPkl)+tDklC(Gsd,kαkl)+tDklC(Gsd,lβkl) . the dual optimum µ⋆ =argminµ≥0d(µ).
k,l Since (P2) has zero duality gap, two important properties
X(cid:0) (cid:1)
Obviously,(P1)isanonconvexmixed-integernonlinearpro- should be noted. One is that µ⋆ > 0. This is because µ⋆
gram.To find the optimum S, we first relax all indicatorvari- represents the sensitivity of the optimum objective value for
ablestobecontinuouswithin[0,1].Then,wemakethechange (P2) with respect to Rreq, i.e., ∂∂PR(Xreq⋆) =µ⋆ [13]. Obviously,
3
P(X⋆) is strictly increasing of R , meaning that µ⋆ > 0. 1 [15]. After knowing {t⋆ |∀ k,l}, the optimum I can be
req kl
The other is that µ=µ⋆ and X =X⋆, if and only if X is constructed according to the way mentioned earlier. Finally,
µ µ
feasible and µ(g(Xµ)−Rreq) = 0 according to Proposition the corresponding XI ={I,PI} is assigned to Xµ. Note that
5.1.5 in [14]. Based on the above property, the µ > 0 and the Hungarian algorithm to solve (P5) has a complexity of
Xµ that satisfies g(Xµ) = Rreq can be found as µ⋆ and O(K3) [15]. e
X⋆. Therefore, the key to using the dual method consists of 2) To find µ⋆: an incremental-update based subgradient
two procedures to finding X and µ⋆, respectively. We first method can be used as in [4], [10]. However, this method
µ
introduce the one to finding X as follows. convergesvery slowly. To develop a faster algorithm, we first
µ
1) To find X when µ>0: the following strategy is used. showthatg(X )isanon-decreasingfunctionofµ≥0.Tothis
µ µ
First,theoptimumPfor(P3)withfixedIisfoundanddenoted end, suppose µ ≥ µ . Since R −g(X ) is a subgradient
1 2 req µ
by PI. Define XI ={I,PI}. Then we find the optimum I to of d(µ) at µ, d(µ1) ≤ d(µ2)+(µ1 −µ2)(Rreq −g(Xµ2))
maximizeL(µ,XI)esubjecttotheconstraintsonIin(P3).XI and d(µ2)≤d(µ1)+(µ2−µ1)(Rreq−g(Xµ1)) follow. As a
correesponding to this opteimum I can be taken as Xµ. result,
SupposeIisfixed,itcanreadilybeshownthattheoptimum
(µ −µ )(R −g(X ))≤d(µ )−d(µ )
Pkl, αkl and βkl for (P3) are 1 2 req µ1 1 2
≤(µ −µ )(R −g(X ))
P =tRΛ(µ,G );α =tDΛ(µ,G );β =tDΛ(µ,G ) 1 2 req µ2
ekl e kl e kl kl kl sd,k kl kl sd,l holds, and thus g(X )) ≥ g(X ), meaning that g(X ) is
+ µ1 µ2 µ
wehere Λ(µ,G) = elog22eµ− G1 . Usineg these formulas, indeed non-decreasing with µ. Based on the above property,
XI ={I,PI} can behfound. It cani readily be shown that the bisection method can be used to the µ > 0 satisfying
g(X )=R as µ⋆.
µ req
Le(µ,XI)=µRreq+ tRklAkl+tDklBkl (2) The overall procedure to solving (P2) for X⋆ is shown in
k,l Algorithm 1, where ǫ>0 is small and prescribed. As can be
X(cid:0) (cid:1) shown in a similar way as in [16], the finally produced X
where µ
is either equal to (if g(X ) = R is satisfied), or a close
µ req
Akl =Λ(µ,Gkl)−µ·C(GklΛ(µ,Gkl)) approximation (if Rreq < g(Xµ) ≤ Rreq+ǫ is satisfied) for
Bkl =Λ(µ,Gsd,k)−µ·C(Gsd,kΛ(µ,Gsd,k)) X⋆. Moreover, the indicator variables in that Xµ are either 0
or 1, and therefore the corresponding S is either optimum or
+Λ(µ,G )−µ·C(G Λ(µ,G )).
sd,l sd,l sd,l approximatelyoptimumfor(P1). It can readily be shown that
Now, it can be readily shown that the optimum I for (P3) Algorithm 1 has a polynomial complexity with respect to K.
is the solution to the problem (P4),
Algorithm 1 The algorithm to solve (P1).
min tRA +tDB
I,{tkl|∀ks.,lt}. tXRkkl,l,t(cid:0)Dklk,ltklk∈l [0,k1l],∀kl(cid:1)k,l; 123::: cµwomhmiinlpeu=gte(0XG;µµkmlm,aax∀x)k=≤,l1R;;recqomdopute g(Xµmax);
tkl =tRkl+tDkl,∀k,l; 4: µmax =2µmax; compute g(Xµmax);
t =1,∀k; t =1,∀l; 5: end while
kl kl
6: while 1 do
wtRhAere +exttrDaBvari≥abtleXslC{tklh|o∀ldks,wl}hearreeXCkintro=dumceind{.ANo,teBth}at. 78:: µif=Rreµqma≤x+2gµ(mXinµ;)s≤olvRere(qP3+)ǫfotrhXenµ;
kl kl kl kl kl kl kl kl kl 9: go to line 15;
tLDet.uTshliasbienleAquklalaitsythisemtigehtrtiecnfeodrwtRkhleanndthBekelnatsrythienm{etRtri,ctDfo}r 10: else if g(Xµ)>Rreq+ǫ then
kl kl kl 11: µmax =µ;
withthesmallermetricisassignedtot ,whiletheotherentry
kl 12: else
assigned to 0. This means that after the problem (P5):
13: µmin =µ;
min t C 14: end if
kl kl
{tkl|∀k,l} k,l 15: end while
s.t. tX∈[0,1],∀k,l; 16: compute the S corresponding to Xµ as an (at least
kl approximately) optimum solution for (P1).
t =1,∀k; t =1,∀l;
kl kl
l k
X X
is solved for its optimum solution {t⋆ |∀k,l}, an optimum I
kl IV. NUMERICAL EXPERIMENTS
for(P4) canbe constructedas follows.Foreverycombination
of k and l, the entry in {tR,tD} with the smaller metric is Consider the scenario where the relay is located in the
kl kl
assigned with t⋆ , while the other entry with 0. straight line between the source and the destination. The
kl
Most interestingly, (P5) is a standard assignment problem, source-to-destination and source-to-relay distances are 1 km
hence {t⋆ |∀ k,l} can be found efficiently by the Hungarian and d km (d ∈ [0,1]), respectively. The parameters are set
kl
algorithm, and every entry in {t⋆ |∀ k,l} is either 0 or as σ2 = −50 dBm, R = 100 bpos and ǫ = 1. When K
kl req
4
and d are fixed, every channel impulse response is randomly using the intuitive method explained in the Appendix of [8].
generated in the same way as in [17]. When the relay lies in the middle between the source and the
Toillustratethepower-reductionbenefitoftheimprovedDF relay, it is more likely to have G and G both be much
sr,k rd,l
protocol with OSP, two benchmark protocols are considered. greaterthan G , and thus G is more likely to take a high
sd,k kl
The first one is the improved DF protocol with a priori value. This explains the observation.
subcarrierpairingasstudiedin[1].Thesecondoneisthenon- When d is fixed and K increases, it can be observed that
cooperative transmission, i.e., the direct transmission is used the average P and P reduce while the average Nsp and
sp fsp K
at every subcarrier. Define P , P and P as the minimum Nfsp increase. Moreover, the average P and P are much
sp fsp D K sp fsp
sum power needed for the improved DF protocol with OSP, smaller than P , and the average P is much smaller than
D sp
the firstand secondbenchmarkprotocols,respectively.Define the average P , especially when K takes a high value. This
fsp
N andN astheoptimumnumberofsubcarrierpairsused is because using more subcarriers leads to more flexibility
sp fsp
for the relay-aidedtransmission by the improvedDF protocol of subcarrier pairing and transmission-mode selection for the
with OSP and the first benchmark protocol, respectively. P sum-power reduction.
sp
and N can be computed with Algorithm 1. It can readily
sp + V. CONCLUSION
be shown that P = 2 λ− 1 , where λ satisfies
D k Gsd,k We have developed a sum-power minimized RA algo-
that kC([λGsd,k−1]+P)=hRr2eq. Moreiover, Pfsp is equal to rithm subject to a sum-rate constraint for cooperative OFDM
the optimum objective value of (P1) imposed with the extra transmission using the improved DF protocol with optimized
consPtraint tR = tD = 0, ∀ k,l : k 6= l. An algorithm similar subcarrierpairing.Thepower-reductionbenefitofthisprotocol
kl kl
as Algorithm 1 can be designed to find P and N , which has been illustrated by numerical results.
fsp fsp
is omitted here due to space limitation.
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