On the Performance of Selection Cooperation with Outdated CSI and Channel Estimation Errors

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by  Mehdi Seyfi

IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS,SUBMITTED:SEPT.2010. 1
On the Performance of Selection Cooperation with
Outdated CSI and Channel Estimation Errors
Mehdi Seyfi, Student Member, IEEE, Sami (Hakam) Muhaidat*, Member, IEEE, and Jie Liang, Member, IEEE,
Abstract—Weanalyzetheeffectoffeedbackdelayandchannel In[6],Bletsasetal.proposedafastselectionalgorithmrela-
estimation errors in a decode-and-forward (DF) relay selection tivetothecoherencetimeofthechannel.Foreachrelaynode,
scenario. Amongstalltherelaysthatdecodethesourceinforma-
they assume timers that are inversely proportional to either
tion correctly, only one relay with the best relay-to-destination
1 (R → D) channel quality is selected. More specifically, the the harmonic mean or a max−min function of the back-to-
1 destination terminal first estimates the relay-destination channel back channels. They study the performance of their selection
0 state information (CSI) and then sends the index of the best cooperation scheme in the amplify-and-forwardmode (S-AF)
2 relay to the relay terminals via a delayed feedback link. Due in termsof the outageprobability.The outageprobabilityand
n to the time varying nature of the underlying channel model, the average symbol error rate (ASER) expressions of a relay
a selection is performed based on the old version of the channel
selection with decode-and-forward(S-DF) scenario havebeen
J estimate. In this paper, we investigate the performance of the
underlying selection scheme in terms of outage probability, studied in [8] and [9], respectively. In [10], Beres and Adve
4
averagesymbolerrorrate(ASER)andaveragecapacity.Through analyze relay selection in a network setting, i.e., multi-source

] the derivation of an ASER expression and asymptotic diversity networksandintroducedaclosed-formformulafortheoutage
T order analysis, we show that the presence of feedback delay probability of a single source single destination network with
I reducestheasymptotic diversityordertoone,whiletheeffect of multiple relays in the DF protocol. In [11], [12] the nearest
. channel estimation errors reduces it to zero. Finally, simulation
cs results are presented to corroborate the analytical results. neighbor selection for the DF mode is proposed. In their
[ work, the best relay is simply the spatially nearest relay to
Index Terms—Selection cooperation, decode-and-forward,
the base station. In [5], [7], the relay whose path introduces
1 feedback delay, channel estimation error.
the maximum SNR is selected, in the amplify-and-forward
v
3 (AF) scheme. In [7], the improvement of S-AF over the all
5 I. INTRODUCTION participate cooperation scheme in the AF mode (AP-AF) is
6 discussed in terms of symbol error rate. The ASER of S-AF
0 IT has been demonstrated that cooperative diversity pro- and AP-AF is discussed based on an approximation of the
.
1 vides an effective means of improving spectral and power cumulativedensityfunctionofthereceivedSNR. In[5],Zhao
0 efficiency of wireless networks as an alternative to MIMO and Adve study the outage probability of S-AF scheme and
1 systems [1], [2]. The main idea behind cooperative diversity analyzetheperformanceimprovementduetoS-AF.Theyalso
1
is that in a wireless environment, the signal transmitted by showdramaticimprovementsofS-AFoverAP-AFintermsof
:
v the source (S) is overheard by other nodes, which can be capacity. In [13] the authors discussed the outage probability,
Xi defined as “relays” [3]. The source and its partners can then ASER,ergodiccapacityandoutagecapacityinS-DFprotocol
jointlyprocessandtransmittheirinformation,therebycreating and compared their results with selection relaying (SR). In
r
a a “virtual antenna array”, although each of them is equipped [14], Ikki and Ahmed derive outage probability, ASER, and
with only one antenna.It is shown in [4], [5] that cooperative capacity expressions for S-AF and illustrate the improvement
diversity networks can achieve a diversity order equal to the of S-AF over AP-AF. It is shown that the relay links in S-AF
number of paths between the source and the destination. as well as AP-AF introduce the same diversity order of M,
However, the need of transmitting the symbols in a time where M is the number of relays.
division multiplexing (TDMA) fashion reduces the maximum Related work and contributions: The effect of feedback
achievablecapacityimprovements.Additionally,duetopower delay on the outage probability of antenna selection in a
allocation constraints, using multiple relay cooperation is not MIMO communications system is discussed in [15]. Relay
economical. To overcome these problems, relay selection is selection in the setups in [6], [8]–[12], assume the perfect
proposed to alleviate the spectral efficiency reduction caused channel state information (CSI) knowelge. However, in prac-
by multiple relay schemes and to moderate the power alloca- tical scenarios, the communication links are not known and
tion constraints [1], [6], [7]. have to be estimated. Since channel estimation is necessary
for the selection procedure, channel estimation error due to
The work of Jie Liang was supported in part by the Natural Sci- imperfection of the estimator is inevitable. The effect of
ences and Engineering Research Council (NSERC) of Canada under grants
channel estimation error in selection cooperation in the AF
RGPIN312262-05, EQPEQ330976-2006,andSTPGP350416-07.
M.Seyfi,S.MuhaidatandJ.LiangarewiththeSchoolofEngineeringSci- modeisstudiedin[16],[17].In[16],weanalyzetheeffectof
ence,SimonFraserUniversity,Burnaby,BC,V5A1S6,Canada.Phone:778- channelestimationerrorontheoutageprobabilityofselection
782-7376. Fax: [phone]. E-mail: [email protected], [email protected],
cooperationintheAFmode.Capacityofselectioncooperation
[email protected].
∗Corresponding author withimperfectchannelestimationisstudiedin[17],[18].The
2 IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS,SUBMITTED:SEPT.2010.
impact of channelestimation error on the ASER performance
Source Destination
of distributed space time block coded (DSTBC) systems is
investigated in [19], assuming the amplify-and-forward pro- Relay Signal
Switching Detection
tocol. Building upon a similar set-up, Gedik and Uysal [20] Center Center
extend the work of [19] to a system with M relays. In [21], (RSC) (RDC)
thesymbolerroranalysisisinvestigatedforthesame scenario M Relays
Feedback RelayIndex Relay Channel
as in [20]. Delay Feedback Selection Estimation
In a practical relay selection scenario, the destination is
responsible for estimating the CSI and performing relay se- Fig. 1. System model for decode-and-forward relay selection with delayed
feedback.
lection. The index of the selected relay is then fed back to
all the relays. Due to the time varying nature of the fading
II. SYSTEMMODEL
channels, which is function of Doppler shift of the moving
Fig.1showstheselectioncooperationsystemmodelstudied
terminals, the CSI correspondingto the selected relay is time
in this paper. We consider a multi-relay scenario with M
varying. In this case, relay selection is performed based on
relays. We assume that the relay R , m = 1,...,M, the
outdated CSI. Hence, the selection scheme may not yield the m
source S, and the destination D are equipped with single
best relay.
transmit and receive antennas, respectively. In our system
Although there have been research efforts on conventional
model, we ignore the direct transmission between the source
MIMO antenna selection with feedback delay and channel
and its destination due to shadowing. h and h represent
estimationerrors(seeforexample[15]andreferencestherein), sm md
the channel fading gains between S R and R D,
only a few isolated results have been reported in the context → m m →
respectively.Assumingahalfduplexconstraint,thedatatrans-
ofcooperativecommunications.In[23],Vicarioetal. analyze
missionisperformedintwotimeslots.Inthefirsttimeslotthe
theoutageprobabilityandtheachievablediversityorderofan
source terminal transmits its data to all potentially available
opportunistic relay selection scenario with feedback delay. In
M relays. After, receiving the source signal via independent
this paper, we investigate the impact of feedback delay and
channels, all the relays R , m = 1,2,...M, decode their
channelestimationerrorsontheperformanceofa cooperative m
received signal, and check whether the transmitted signal is
diversity scheme with relay selection. To the best of our
decoded correctly or not. This can be done via some ideal
knowledge, this is the very first paper that investigates relay
cyclic redundancy codes (CRC) [28], which are added to the
selectionwithoutdatedchannelestimates.Assumingimperfect
transmitted information symbols. We define the decoding set
channel estimation and feedback delay, our contributions are
(s) as the set of relays that decode the transmitted signal
summarized as follows: D
correctly. Clearly only those relay nodes with a good source
• We derive exact ASER and outage probability expres- to relay channel can be in the decoding set (s). In the
sionsforrelayselectionwithdecode-and-forward(S-DF) secondtime slot, thebest relaythatsatisfies an inDdexof merit
relaying in the presence of feedback delay and channel participates in the transmission and broadcasts its decoded
estimation errors. symbol towards the destination.
• We derive a lower bound on the average capacity in the In this paper, conditioned on the decoding set (s), the
presenceoffeedbackdelayandchannelestimationerrors. selection function selects the relay with the bestDdownlink
• We demonstrates that the asymptotic diversity order is channel i.e.,
reduced to one with feedback delay and reduced to zero
in the presence of channel estimation error. m⋆ = arg max γmd
m∈D(s){ }
• We present a comprehensive Monte Carlo simulation = arg max P h 2 , (1)
study to confirm the analytical observations and give m∈D(s){ | md| }
insight into system performance.
whereγ isthereceivedSNRfromthemth path.Weassume
md
The rest of the paper is organized as follows: In Sec. II,
that the source has a power constraint of P Joules/symbol
we introduce our system setup and the selection strategy. In
andsimilarlyeachrelaynodein (s)canpotentiallytransmit
thissectionthechannelestimationerroraswellasthedelayed D
its information with P Joules/symbol, and the receiver noise
feedbacklinkmodelsareillustrated.InSec. III,weinvestigate
power is 1.
theoutageprobabilityofthesystemanddevelopaclosed-form
expression for outage probability. In Sec. IV, we study the
A. model of channel estimation error
ASER performance of the system in detail and derive exact
expressionfortheASER.InSec.V,weanalyzetheasymptotic All channelsare assumed to be independentandidentically
order of diversity in S-DF. In particular, we investigate the distributed(i.i.d),withh (0,σ2).Ablockfadingscheme
∼CN h
effect of channel estimation errors and feedback delay on is considered where the channel realizations are assumed to
the diversity of the system. In Sec. VI we introduce a lower be constant over a block and correlated across blocks. The
boundfortheinstantaneouscapacityandderiveaclosed-form correlationcoefficientbetweenthekth andthe(k+i)th block
analytical expression for the average lower bound capacity. isρ =J (2πf Ti),wheref istheDopplerfrequency,J ()
f o d d o ·
Simulation results are presented in Sec. VII, and the paper is is the zeroth order modified Bessel function of first kind and
concluded in Sec. XI. T istheblockduration.Minimummeansquareerror(MMSE)
M.SEYFIetal.:PERFORMANCEANALYSISOFSELECTIONCOOPERATIONWITHFEEDBACKDELAYANDCHANNELESTIMATIONERRORS 3
estimators are employed at the relay nodes. We assume that performed. This might lead to a wrong selection of relay and
the number of training symbols in each frame is sufficiently therefore affect the performance of the system.
less than the data symbols, so that increasing the training In this paper,to simplify the notation, we adoptthe second
symbols’ power would not lead to an increase in the average strategy in relay selection. However, the first strategy obeys
power; hence, we assume that the training symbols’ power is the same rules and formulations.
independently adjustable. Let the true channel be h and the Inthesecondstrategy,sincethefeedbacklinkonlytransmits
estimated channel be hˆ, then h and hˆ are related by [22] the indexof the selected relay,a lower feedbackbandwidthis
required.
hˆ = h+e, (2a) Let the estimated channel be hˆ and the old estimated CSI
h = ρehˆ+u, (2b) based onwhich the selection is performedbe hˆo. Since hˆ and
whereeanduarethechannelestimationerrors.ˆh, eandu,are hˆo arebothzeromeanandjointlyGaussiantheycanberelated
as follows [15]
allzeromeanGaussianrandomvariableswithvariancesσ2,σ2
hˆ e
andσ2.ρ =(σ2/σ2)isthecorrelationcoefficientbetweenthe
u e h hˆ ρ
truechannelandtheestimatedchannel.Thechannelestimation hˆ =σ f ˆh + 1 ρ 2v (4)
error variances can be written as hˆ σhˆo o q − f !
σσu22 == ((11−ρρe))σσh22., ((33ba)) iwshuesreedvfo∼r rNela(y0,s1e)l.ecWtioens,twrehssertehaastˆhˆhoisitshethcehcahnannenlewlhwichhicihs
e − e hˆ used for decoding.
B. Delayed feedback model
III. OUTAGEPROBABILITY OF DFRELAYSELECTION
There are two main selection strategies in the literature. In
In the first time slot, the source terminal communicates
the first scenario, the relay nodes are in charge of monitoring
with the relay terminals by broadcasting the signal x. Let the
of their individual source and destination instantaneous CSI
receivedsignalateachrelaybey ,m=1,...M.Todecode
[6]. In this strategy, CSI is estimated at the relay nodes for sm
the received symbol, each relay estimates the corresponding
furtherdecisionsonwhichrelayisthebestnodetocooperate.
S R channel and then decodes the received signal using
In particular, when the relay nodes overhear a ready-to-send → m
an ML decoder. The old received signal at the mth relay can
(RTS) packet form the source terminal, they estimate the
be written as
source link CSI, i.e., h . Then, upon receiving a clear-to-
sm
send (CTS) packet from the destination, the corresponding xˆ = √Phˆ∗ y
sm,o sm
destination link at the relays, i.e., h is estimated. Right
md = √Phˆ∗ (√Ph x+n ) (5a)
after receiving the CTS packet, the relay nodes ignite a timer sm,o sm,o sm
whichisafunctionoftheinstantaneousrelay-destinationCSI1. = √Phˆ∗ √P ρhˆ +u x+n
sm,o e sm,o sm sm
The timer of the best relay expires first, and a flag packet is
= Pρ hˆ h 2x(cid:16)+ Pˆh∗ u x(cid:17)+√Phˆ∗ in ,(5b)
sent to other relays, informing them to stop their timers as e| sm,o| sm,o sm sm,o sm
the best relay is selected. Then the relays that receive the messagecomponent error component noisecomponent
flag packet, keep silent and the selected relay participates in
where we hav|e su{bzstitut}ed (|2b) {inzto (}5a).|Usin{gz (5b}), the
communication [6].
received effective SNR at each relay is given by
Inthesecondstrategy,therelaynodesestimatetheirindivid-
ualuplinkCSI.Therelaysthatcorrectlydecodetheirreceived Pρ2 hˆ 2
γˆ eff = e | sm,o| =Pγˆ , (6)
symbolsfromthesource,sendaflagpackettothedestination, sm,o 1+Pσ2 sm,o
u
announcing that they are ready to participate in cooperation. sm
The destination terminal, on the other hand, estimates the whereγˆ = ρe2|hˆsm,o|2.Similarly,inthesecondtimeslot,the
downlink CSI, orders the received SNR from each relay in sm,o 1+Pσu2sm
effectiveSNR receivedby thedestinationvia the mth relay is
the decoding set and feedbacks the index of the best relay
that introduces the maximum received SNR via a logM bit Pρ2 hˆ 2
feedbacklink.Theselectedrelaythenoperateswithfullpower γˆ eff = e | md,o| =Pγˆ , (7)
md,o 1+Pσ2 md,o
[24]. umd
Inbothstrategies,averyimportantissue thatmustbetaken where γˆ = ρe2|hˆmd,o|2, given that m (s). Here, we as-
ianmtooncgontseirdmerinatailosnairsethtiemseelveacrtyioinngspweiethd.aCmomacmrousnciocaptiiconraltienkins sumethamtd,γˆo a1+ndPσγˆu2md areexponential∈lyDdistributedwithpa-
sm,o md,o
tthhee ochrdaenrneolfcDoohpeprelenrceshtiifmt,ew[h6i]c.hAisnyinrveelrasyelsyelpercotpioonrtisocnhaelmtoe ramIfettehresrλeslma,oy=Rm(1+isPinσu2tshme)/dρeecoadnidngλmsde,ot,=by(1s+enPdiσnu2gmda)/flρaeg.
mustbeperformednoslowerthanthechannelcoherencetime, packet,it signals its capabilityof participatingin cooperation.
otherwise selection is performed based on the old CSI, while Then, based on the old channel realizations, the destination
the channel conditionsare altered at the time that selection is selects the relay with the best Rm D link
→
rel1aTy-hdeestitmineartioisnaCfSuIncftoiornAoFfmreoladye-dceosotpineartaitoionnC.SIforDFmodeandsource- Rm⋆ =argmm∈Dax(s){γˆmd,o}. (8)
4 IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS,SUBMITTED:SEPT.2010.
A. Outage probability with feedback delay C. Outage probability conditioned on the decoding set (s)
D
Once the best relay is selected by destination, the index of Conditioned on the decoding set with the old channel
R is fed back to all relays via a delayed feedback link. realizations, the outage probability with the new CSI is
m⋆
This means, at the time the relays receive the index, the
system’s CSI has changed, due to the time varying nature
PD(s) =Pr γˆ R ,γˆ max γˆ (s) . (14)
of communication links. Let the current R D channel o  md ≤ o md,o ≥ { id,o} D 
m⋆ → i∈D(s)
realization, the corresponding estimate and the received SNR i6=m (cid:12)
 (cid:12) 
from the selected relay be h , hˆ and γˆ , respectively. Defining
Due to feedback delay, relamy⋆dselme⋆cdtion ism⋆ddone based on χ =∆ max γˆ ,
past channel coefficients instead of current coefficients, i.e., m i∈D(s) { id,o}
i6=m
γˆ =γˆ where m=arg max γˆ .
m⋆d md { id,o} and noting that χ and γˆ are independent, the conditioned
i∈D(s) m md
Assuming that the communication between source and outage probability on the decoding set is
destination targets an end-to-enddata rate R, the system is in
∞
coauptaagceityifptehrebSand→wiRdtmh⋆2C→⋆D= l1inlokgo(b1se+rvPesγˆan)inthsatatnistabneeloouws PoD(s) = Z0 Pr γˆmd ≤Ro D(s),γˆmd,o
the required rate R, i.e., 2 m⋆d × Pr γˆmd,o(cid:2)≥χm D(s(cid:12)(cid:12)),γˆmd,o fγˆm(cid:3)d,o(γˆmd,o) dγˆmd,o
∞
Po = Pr 12log(1+Pγˆm⋆d)≤R = Z0 (cid:2)Fγˆmd(Ro)Fχ(cid:12)(cid:12)m(γˆmd,o)fγˆm(cid:3)d,o(γˆmd,o) dγˆmd,o. (15)
(cid:20) (cid:21)
Conditionedonγˆ andusing(4),γˆ hasanon-centralChi
= Pr[ (s)] md,o md
D squaredistributionwith two degreeof freedomand parameter
DX(s) 2ρ2
η = c γˆ , where c = f b. Therefore we can
× Pr γˆm⋆d ≤Ro,γˆm⋆d,o = max{γˆmd,o} D(s) , (9) m m md,o m (1−ρ2f)σh2ˆmd,o
(cid:20) m∈D(s) (cid:21) write F (x) as [26]
= Pr[ (s)] (cid:12)(cid:12) γˆmd
DX(s) D F (x)= ∞ e−cmλmd2,oγˆmd,o cmλmd,oγˆmd,o k
γˆmd k=0 (cid:18) 2 (cid:19)
× Prγˆmd ≤Ro,γˆmd,o ≥max{γˆid,o} D(s),(10) X γ(k+1,λmdx)
m∈XD(s) ii∈6=Dm(s) (cid:12) × (k!)22σ2 , (16)
 (cid:12) 
where Ro = 22RP−1. where γ(, ) is the incomplete gamma function defined as
· ·
x
γ(s,x)= ts−1e−tdt.
B. Probability of decoding set (s)
D Z0
The relay, R , is in the decoding set (s) if the S R
m m Furthermore, F (x) is given by
D →
link observes an instantaneous capacity per bandwidth C χm
sm,o
that is above the required rate R F (x) = 1 e(−λid,ox)
χm −
1 i∈YD(s) h i
C = log 1+Pγˆ R. (11) i6=m
sm,o 2 2 sm,o ≥ = 1 e−λid,ox+ e−(λid,o+λjd,o)x ...
Noting that γˆ is exponent(cid:0)ially distrib(cid:1)uted, relay R is in − −
sm,o m i∈XD(s) i,jX∈D(s)
the decoding set if [10] i6=m i6=m,j
+ ( 1)|D(s)| e−(λid,o+λjd,o+...λld,o)x. (17)
Pr[R (s)] = Pr γˆ R . −
m ∈D sm,o ≥ o i,j∈XD(s)
= exp λ R . (12) i6=m,j,...,l
(cid:2) − sm,o o(cid:3)
Using (16) and (17) in (15) and noting that f (x) =
Finally, the probability of selecting a(cid:0)specific de(cid:1)coding set is γˆmd,o
[10] λ e−λmd,ox and also the fact that [30]
md,o
∞ k!
Pr[D(s)] = exp −λsm,oRo xkexp(−ax)dx= ak+1.
m∈YD(s) (cid:0) (cid:1) Z0
theoutageprobabilityconditionedonthedecodingsetisgiven
1 exp λ R . (13)
× − − sm,o o by (18) at the top of the next page.
m∈Y/D(s)(cid:2) (cid:0) (cid:1)(cid:3) Finally, substituting (13) and (18) into (10), we obtain (19)
given at the top of the next page.
2Sincethesystemisexposedtochannelestimationerror,theinstantaneous
capacity which is mentioned here, is only a lower bound for the true For the specialcase where all the channelestimation errors
instantaneous capacity; hence, the derived expression for outage probability and delayed feedbacks are the same, i.e., λ=λ =λ =
isalowerboundforthetrueprobabilityofoutageinthepresenceofchannel sm,o md,o
λ , andc =c =...c =c,(19) reducesto(20) given
estimation errorand isexact withthe perfect CSI. Forfurther details please id,o 1 2 |D(s)|
refertoSec.VI. at the top of next page.
M.SEYFIetal.:PERFORMANCEANALYSISOFSELECTIONCOOPERATIONWITHFEEDBACKDELAYANDCHANNELESTIMATIONERRORS 5
PD(s) = ∞ λmkd,+o1γ(k+1,λmdRo/2σ2) cm k ∞γˆk e−(λmd,o+λmd,ocm2 )γˆmd,o e−(λmd,o+λid,o+λmd,ocm2 )γˆmd,o
o (k!)2 2 md,o −
kX=0 (cid:16) (cid:17) Z0 i∈XD(s)

+ e−(λmd,o+λid,o+λjd,o+λmd,ocm2 )γˆmd,o −...+ (−1)|D(s)| e−(λmd,o+λid,o+λjd,o+...λld,o+λmd,ocm2 )γˆmd,odγˆmd,o.
i,jX∈D(s) i,j,..X.,l∈D(s)
i6=j i6=j,...,l 

∞ γ(k+1,λ R /2σ2) c k λ k+1 λ k+1
PD(s) = md o m md,o md,o
o (k!) 2 (λ +λ cm)k+1 − (λ +λ + cm)k+1
Xk=0 (cid:16) (cid:17) md,o md,o 2 i∈XD(s) md,o id,o 2

λ k+1 λ k+1
+ md,o ...+( 1)|D(s)| md,o . (18)
(λ +λ +λ +λ cm)k+1 − − (λ +λ +λ +...λ +λ cm)k+1
i,jX∈D(s) md,o id,o jd,o md,o 2 i,j,..X.,l∈D(s) md,o id,o jd,o ld,o md,o 2
i6=j i6=j,...,l 

∞ γ(k+1,λ R /2σ2) c k
P = exp λ R 1 exp λ R md o m
o − sm,o o − − sm,o o (k!) 2
XD(s) mY∈D(s) (cid:0) (cid:1)m∈Y/D(s)(cid:2) (cid:0) (cid:1)(cid:3)mX∈D(s) kX=0 (cid:16) (cid:17)
λ k+1 λ k+1 λ k+1
md,o md,o + md,o ...
×(λ +λ cm)k+1 − (λ +λ + cm)k+1 (λ +λ +λ +λ cm)k+1 −
md,o md,o 2 i∈XD(s) md,o id,o 2 i,jX∈D(s) md,o id,o jd,o md,o 2
i6=j

λ k+1
+( 1)|D(s)| md,o . (19)
− (λ +λ +λ +...λ +λ cm)k+1
i,j,..X.,l∈D(s) md,o id,o jd,o ld,o md,o 2
i6=j,...,l 

l 1
M ∞ γ(k+1,λR /2σ2) c k l m− 1
P = exp( lλR )[1 exp( λR )]M−l l o (cid:18) − (cid:19) . (20)
o − o − − o × × (k!) 2 × (m+ c)k+1
Xl=1 Xk=0 (cid:16) (cid:17) mX=1 2
IV. AVERAGE SYMBOL ERROR RATE where δ(x) is the delta function and Bi =
∞
α Q βPγˆ f dγˆ is given as
0 sio γˆsio sio
In this section we derive a closed form expression for the R (cid:0)p (cid:1) α βP
B = 1 . (23)
ASERoftheDFselectionschemeconsideredinthispaper,in i
2 " −sβP +2λ #
the presence of channel estimation error and feedback delay. sio
LPeγˆtthe. Trehceeniv,ethdeSANSREvRiacathnebseelwecrtiettdenpaatshdenotedbyγˆme⋆ffd = sHigenrea,l fvγiˆma⋆dth|De(ss)e(lxec)teids pthaeth,cwonhdicithioinsalgivPeDnFbyof the received
m⋆d
∂F (x)
f (x)= γˆm⋆d|D(s) . (24)
∞ γˆm⋆d|D(s) ∂x
P¯ = αQ βPγˆ f (γˆ )dγˆ (21)
e Z0 (cid:16)p m⋆d(cid:17) γˆm⋆d m⋆d m⋆d Noting that Fγˆm⋆d|D(s)(x)=PoD(s) Ro=x, (25)
whereαandβ dependonthemodulationscheme,andf is byinserting(18)into(25)andtheresult(cid:12)in(24)theconditional
the probabilitydensity function(PDF) of γˆ , which isγˆmg⋆idven PDF of the received signal is then give(cid:12)n by (26) at the top of
m⋆d
by (see Appendix for details) the next page. Finding a closed form formula for the integral
in(21)isnottractable.However,bysubstituting(22)into(21),
weobtain(27)atthetopofthenextpage,wherewehaveused
M
the approximation [27]
f (x) = B δ(x)
γˆ i
m⋆d iY=1 Q(x) e−x22 na anxn−1, (28)
≈
+  (1−Bi) Bifγˆm⋆d|D(s)(x), where nX=1
XD(s) i∈/YD(s) i∈YD(s) ( 1)n+1(A)n
|D(s)|≥1  (22) an = B√−π(√2)n+1n!, (29)
6 IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS,SUBMITTED:SEPT.2010.
f (x)= ∞ λmde(−λ2mσd2x)(λ2mσd2x)k cm k λmkd,+o1 λmkd,+o1
γˆm⋆d|D(s) 2σ2k! 2 ×(λ +λ cm)k+1 − (λ +λ +λ cm)k+1
Xk=0 (cid:16) (cid:17) md,o md,o 2 i∈XD(s) md,o id,o md,o 2

λ k+1 λ k+1
+ md,o ... +( 1)|D(s)| md,o . (26)
(λ +λ +λ +λ cm)k+1 − − (λ +λ +λ +...λ +λ cm)k+1
i,jX∈D(s) md,o id,o jd,o md,o 2 i,j,..X.,l∈D(s) md,o id,o jd,o ld,o md,o 2
i6=j i6=j,...,l 

P¯ = 1 M B + (1 B ) B α ∞ ∞ anΓ k+ (n+21) /k! cm k λmd (k+1)
e 2iY=1 i XD(s) i∈/YD(s) − ii∈YD(s) i× mX∈D(s)kX=0nX=1 (1+(cid:16)2Pλmβdσ2)k+((cid:17)n+21) (cid:16) 2 (cid:17) (cid:18)2σ2Pβ(cid:19)
|D(s)|≥1
 
λ k+1 λ k+1 λ k+1
md,o md,o + md,o ...
×(λ +λ cm)k+1 − (λ +λ +λ cm)k+1 (λ +λ +λ +λ cm)k+1 −
md,o md,o 2 i∈XD(s) md,o id,o md,o 2 i,jX∈D(s) md,o id,o jd,o md,o 2
 i6=j

λ k+1
+( 1)|D(s)| md,o . (27)
− (λ +λ +λ +...λ +λ cm)k+1
i,j,..X.,l∈D(s) md,o id,o jd,o ld,o md,o 2
i6=j,...,l 

l 1
P¯ = 1BM + M M (1 B)lBM−l αl ∞ na anΓ k+ (n+21) /k! c k λ k+1 l (cid:18)m−−1(cid:19) . (30)
e 2 Xl=1(cid:20)(cid:18) l (cid:19) − (cid:21)× ×kX=0nX=1 (1+(cid:16)2Pλβσ2)k+((cid:17)n+21) (cid:16)2(cid:17) (cid:18)2σ2P(cid:19) ×mX=1(m+ 2c)k+1
with A=1.98 and B =1.135. decoding set, as
In the special case where all the channel estimation errors
and delayed feedbacks are the same, i.e., λ=λsm,o =λmd,o = P¯ =α ∞ na anΓ k+ (n+21) /k! c k
λthied,ot,oapnodfct1he=pacg2e,=w.h.e.rce|DB(s=)| B= c=, (.2.7.)Bred.uces to (30) at e|D(s) Xk=0nX=1 (1+(cid:16)2Pλβσ2)k+((cid:17)n+21) (cid:16)2(cid:17)
1 M
l 1
λ k+1 l m− 1
(cid:18) − (cid:19) ,
× 2σ2Pβ (m+ c)k+1
(cid:18) (cid:19) m=1 2
X
∞ na g(n,k) λ k+1
= ,
V. ASYMPTOTICDIVERSITY ORDER Xk=0nX=1(1+ 2Pλσ2β)k+(n+21) (cid:18)2σ2Pβ(cid:19)
(31)
In the previous section, we have derived an exact ASER where g(n,k) is a constant depending on k and n.
expressionwithfeedbackdelayandchannelestimationerrors, In the following, we consider two different scenarios:
which is valid for the entire SNR range. To gain further
insights into the system’s performance, we focus here on
the high SNR regime and analyze the asymptotic ASER for A. Diversity order with feedback delay and perfect CSI
the selection scheme under consideration conditioned on the
With perfect CSI, we have λ = 1. Therefore, as P ,
decoding set, i.e., P¯e|D(s)3. We assume that all the channel the dominant term in (31) would be the term correspo→nd∞ing
estimationerrorsandfeedbackdelayparametersarethesame,
to k =0 and n=1, i.e.
without loss of generality. Therefore, we have λ = λ =
sm,o
λmd,o =λid,o, and c1 =c2 =...c|D(s)| =c P¯ 1 1,
Using (26), we can write the ASER, conditioned on the e|D(s) ∝ 1+k0/P × P
1
,
∝ P +k
0
= O(P−1). (32)
3IntheDFscheme,selectionrelayingexploitthefulldiversityorder,which
isequaltothenumberofrelaysregardlessofthecardinality ofthedecoding
Thus, in this case, the achievable diversity order in presence
set[8]–[10].Hence,inthiswork,wefocusontheanalysisoftheasymptotic
ASERconditioned onthedecoding set of delayed feedback is 1.
M.SEYFIetal.:PERFORMANCEANALYSISOFSELECTIONCOOPERATIONWITHFEEDBACKDELAYANDCHANNELESTIMATIONERRORS 7
B. Diversity order in the presence of feedback delay and has zero-mean complex Gaussian distribution with the same
imperfect CSI variance,whichistheworsecasedistribution.Inthiscase,we
In the presence of channel estimation error, since λ is can re-write (38) as
proportionalto P, it shouldbe takeninto consideration.From CD(s) arg max (xˆ) (n˜) arg max
(31), we have ≥ fx(x) {H −H }≥ fx(x) {
∞ na λ k+1 H(xˆ)−H(n†)}, (39)
lim P¯ lim P ,
p→∞ e|D(s) =∝ Kp→.∞"Xk=0nX=1(1+(cid:0)kP1λ(cid:1))k+(n+21)#(33) w(3h9e)rewnith† ∼resCpNect(cid:16)t0o,Pfx2(|hˆxm)⋆da|n2dσ2ums⋆idn+g (P36|hˆ)m,⋆ad|2lo(cid:17)w.eMrabxoiumnidzionng
the average capacity, conditioned on the decoding set, can be
Hence, the asymptotic diversity order in the presence of
written as
feedbackdelayandchannelestimationerrorsis0.Thismeans
that the outage probability or the ASER curves are expected CD(s) CD(s)
≥ lb
to be saturated at high SNR with imperfect CSI. 1 q2
= log 1+ ,
2 P2 hˆ 2σ2 +P hˆ 2!
VI. AVERAGE CAPACITY | m⋆d| m⋆d | m⋆d|
1
In this section, we derive an analytical expression for = log(1+Pγˆ ), (40)
2 m⋆d
average capacity in the presence of channel estimation error
and feedback delay. At the destination and after matched where γˆ = ρe2|hˆm⋆d|2. Hence, a lower bound on the S-DF
filtering, the received signal over the selected link is given capacitymc⋆dan be1+oPbtσau2imn⋆edd as
as
C = CD(s)Pr[ (s)]. (41)
xˆ = √Phˆ∗ y , lb lb D
m⋆d m⋆d DX(s)
= √Phˆ∗ (√Ph x+n ),
m⋆d m⋆d m⋆d Therefore,theaveragecapacitybound,i.e.,C¯ ,canbewritten
lb
= √Phˆ∗ √P ρhˆ +u x+n , as
m⋆d e m⋆d m⋆d m⋆d
= Pρe|hˆm⋆d|h2x +(cid:16)Pˆhm∗⋆dum⋆dx +(cid:17)√Pˆhm∗⋆dnmi⋆d .(34) C¯lb = 21 Pr[D(s)]
messagecomponent error component noisecomponent DX(s)
∞
Defining | {z } | {z } | {z } log(1+Pγˆ )f (γˆ )dγˆ . (42)
q =∆ Pρ hˆ 2, (35a) ×Z0 m⋆d γˆm⋆d|D(s) m⋆d m⋆d
e| m⋆d| Substituting (26) into (42) yields (44) given at the top of
n˜ =∆ Pˆh∗ u x+√Pˆh∗ n , (35b) the next page. Note that in our derivation we have used the
m⋆d m⋆d m⋆d m⋆d
integration formula [30]
(34) can be written as
∞ k k!
xˆ=qx+n˜. (36) ln(1+ax)xke−xdx=
(k µ)!
Z0 µ=0 −
The capacity conditioned on the decoding set is defined as X
( 1)k−µ−1e1/a 1 k−µ 1 k−µ−t
CD(s) = arg max (xˆ,x) − Ei + (t 1)! ,
fx(x) I ×" ak−µ (cid:18)−a(cid:19) t=1 − (cid:18)−a(cid:19) #
X
= arg max (xˆ) (xˆx) (43)
fx(x) {H −H | }
= arg max (xˆ) (n˜ x) , (37) where
fx(x) {H −H | } Ei(γ)=∆ ∞ 1 exp( x)dx
where fx(x) is the PDF of the input signal x and H(·) is the Zγ x −
differential entropy function. It can be deduced easily from is the exponential integral function.
(36)thatn˜ andxareuncorrelated,i.e.,E n˜x∗ =0.However, Inthespecialcasewhereallthechannelerrorsandfeedback
{ }
n˜ and x are not independent and therefore (n˜ x) = (n˜). delayparametersarethesame,theaveragecapacityboundcan
H | 6 H
This means that the standard proceduresfor deriving capacity be obtained as in (45) at the top of next page.
expressions are not applicable to (37). Alternatively, noting
that conditioning does not increase the entropy, i.e.,
VII. SIMULATION RESULTS
(n˜ x) (n˜), In the section, we investigate the performance of selection
H | ≤H
cooperation in the presence of channel estimation errors and
we can lower bound the capacity as
feedbackdelaythroughMonte-Carlosimulation.Thetransmit-
CD(s) arg max (xˆ) (n˜) . (38) ted symbolsare drawnfroman antipodalBPSK constellation,
≥ fx(x) {H −H } which means, α = 1 and β = 2. The node-to-node channels
It must be noted that the distribution of n˜ is not Gaussian. areassumedtobezeromeanindependentGaussianprocesses,
However, following [31], Theorem 1, we may assume that n˜ with variance σ2 = σ2 = 1 σ2. In this case, hˆ and
h ho − e
8 IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS,SUBMITTED:SEPT.2010.
C¯lb= ln12  exp −λsm,oRo 1−exp −λsm,oRo × ∞ k (kc2mµk)!(−1)k−µk−−1µe(cid:16)2λσm2dP(cid:17)
XD(s)mY∈D(s) (cid:0) (cid:1)m∈/YD(s)(cid:2) (cid:0) (cid:1)(cid:3) mX∈D(s)Xk=0µX=0 (cid:0) −(cid:1) (cid:16)2λσm2dP(cid:17)
λ k−µ λ k−µ−t λ k+1 λ k+1
Ei md + (t 1)! md md,o md,o
× (cid:18)−2σ2P(cid:19) Xt=1 − (cid:18)−2σ2P(cid:19) #×(λmd,o +λmd,oc2m)k+1 −i∈XD(s)(λmd,o +λid,o +λmd,oc2m)k+1

λ k+1 λ k+1
+ md,o ... +( 1)|D(s)| md,o . (44)
(λ +λ +λ +λ cm)k+1 − − (λ +λ +λ +...λ +λ cm)k+1
i,jX∈D(s) md,o id,o jd,o md,o 2 i,j,..X.,l∈D(s) md,o id,o jd,o ld,o md,o 2
i6=j i6=j,...,l 

C¯lb = ln12XlM=1exp(−lλRo)[1−exp(−λRo)]M−l×l×Xk∞=0µX=k0λk+1(cid:0)(2ck(cid:1)−k µ)! ×"(−2σ1λ2)Pk−kµ−−µ1e(2σλ2P)Ei(cid:18)−2σλ2P(cid:19)
l 1
k−µ λ k(cid:0)−µ−t(cid:1) c k l m− 1
+ (t 1)! (cid:18) − (cid:19) . (45)
Xt=1 − (cid:18)−2σ2P(cid:19) #(cid:16)2(cid:17) ×mX=1(m+ 2c)k+1
hˆ are zero mean Gaussian processes with unit variance and knowledge, unless otherwise indicated.
o
ρ = 1 σ2. The variance of noise components is set to Fig. 7 shows the diversity performance of the system in
e − e
N =1 and R =1 bps/Hz. presence of feedback delay link for different values of ρ .
0 o
f
Outage Probability: Fig. 2 shows the performance of the We assume perfect CSI knowledge. It is observed that the
systemwithM =4relayswithfeedbackdelay(FD)forρ = asymptoticaldiversityorderofthesystemtendsto1forallρ
f f
0.6,0.7,0.8,0.9and 1. In Fig. 2, we assume perfectCSI, i.e., values.Thisillustratesthedestructiveeffectoffeedbackdelay
ρ = 1. We observe a perfect match between the analytical anddemonstratesthatrelayselectioninthiscaseisannihilated.
e
and simulation results. It can also be deduced from the slope Fig. 8 illustrates the asymptotical diversity for different
of the curves that, for M =4 and ρ <1, the diversity order number of relays i.e., M = 2,3,4. It is obvious that at high
f
of the selection scheme for is 1. In the case of ideal feedback SNRthediversityorderisindependentofthenumberofrelays
link, i.e., ρ =1, the diversity order of 4 is observed. in the system.
f
Fig. 3 illustrates the performance of the system for M = Fig. 9 depicts the asymptotical diversity order in presence
2,3,4, with ρ = 0.9,1. In Fig. 3, the same slope for ofchannelestimationerrors.Itisnoticedthattheasymptotical
f
differentnumberofrelaysisnoticed,confirmingouranalytical diversity order in this case is reduced to zero, confirming our
observations. For the case of ρ =1 and choosing M =3,4, earlier observations in Figs. 4 and 6.
f
the diversity orders of 3 and 4 are observed, respectively. Average Capacity: Fig. 10 shows the lower bound average
In Fig. 4, we study the effect of both feedback delay capacity in bits per second per Hz per bandwidth versus
and channel estimation errors. An error floor, in presence of SNR. It is observed that the presence of channel estimation
channel estimation error, is noticed, as predicted by (36). errors result in capacity ceilings in the average capacity
ASERperformance:Fig.5showstheASERinthepresence curves. It can be also seen that feedback delay aggravatesthe
of feedback delay for M = 3. We assume perfect CSI averagecapacityperformanceofthesystem.However,channel
knowledge.Assuminganidealfeedbacklink,thefulldiversity estimationerrorshaveagreatereffectinworseningtheaverage
order of 3 is achieved. However, with feedback delay, the capacity performance of the system.
diversity order is reduced to 1, as predicted earlier. (cid:4)
Fig. 6 illustrates the performance of the ASER in presence
of channelestimation error and feedback delay for M =2. It
is clear from Fig. 6 that channel estimation errors reduce the
diversity order of the system to zero, confirming our earlier XI. CONCLUSION
analysis.
In this paper, we discuss a relay selection scheme in DF
Diversityorder:Notingthattheasymptoticaldiversityorder
networks. We show that the presence of channel estimation
d is given by the magnitude of the slope of ASER against
errors and feedback delay degrades the performance and also
average SNR in a log-log scale [2]:
reduces the diversity order of S-DF. We derive an exact ana-
log P¯ lytical expressions for the outage probability, average symbol
e
d= lim . (46)
SNR→∞− logSNR error rate and average capacity bound.
(cid:0) (cid:1)
in this subsection we analyze the asymptoticaldiversity order
of the underlying selection scheme. We assume perfect CSI APPENDIX
M.SEYFIetal.:PERFORMANCEANALYSISOFSELECTIONCOOPERATIONWITHFEEDBACKDELAYANDCHANNELESTIMATIONERRORS 9
100 100
S−DF, with FD, simulation, M=3
S−DF, with FD, analytical, M=3
S−DF, ideal feedback link, M=3
10−1 ρf=0.6,...0.9 10−1
delayed feedback link
10−2 10−2
Pout10−3 ideal feedback link ASER10−3
10−4 10−4
10−5 10−5 ideal feedback link
S−DF with FD, M=4, simulation
S−DF with FD, M=4, analytical
10−6 10−6
0 5 10 15 20 25 30 0 5 10 15 20 25 30
SNR(dB) SNR(dB)
Fig. 2.Outage Probability forM =4and perfect CSIin presence ofdelay Fig.5.ASERforM =3andperfectCSIinpresenceofdelayinthefeedback
inthefeedback link.InthisfigurePe =1andρf =0.6,0.7,...1. link.Inthisfigureρe =1andρf =0.6,0.7,...,1.
100 100
S−DF, with DF and CE error, simulation, M=2
S−DF with FD
S−DF, with DF and CE error, analytical, M=2
M=2,3,4
S−DF, perfect CSI and ideal feedback link, M=2
10−1 10−1
FD and CE Error
ρe=0.9
10−2 10−2
out10−3 SER10−3 ρe=0.95
P A
ρe=1
10−4 10−4
ideal feedback Link M=4 M=3
ideal feedback link and perfect CSI
10−5 10−5
S−DF, with FD, simulation.
S−DF, with FD, analytical
10−60 5 10 15 20 25 30 10−60 5 10 15 20 25 30
SNR(dB) SNR(dB)
Fig. 3. Outage Probability for M = 2,3,4 and perfect CSI in presence of Fig.6.ASERforM =2presenceofdelayinthefeedbacklinkandchannel
delayinthefeedback link. Inthisfigureρ =1andρ =0.9. estimation error.Inthisfigureρ =0.9.
e f f
100 1.6
FD and CE error diversity order with FD and perfect CSI
ρe=0.9 1.4
10−1
1.2
ρe=0.95
10−2 der 1
Pout ρe=1 sity Or0.8
10−3 ver
Di0.6
ideal feedback link with perfect CSI
0.4
10−4
S−DF, with FD and CE error, simulation, M=2 0.2
S−DF, with FD and CE error, analytical, M=2
10−5 0
0 5 10 15 20 25 30 0 100 200 300 400 500
SNR(dB) SNR(dB)
Fig.4.Outageprobability forM =2presenceofdelayinthefeedback link Fig.7.EffectivediversityorderversusSNR.ρ =1andM =4,Thefigure
andchannel estimation error.Inthisfigureρf =0.9. isplotted fordifferent values ofρf andisobteained from(46).
10 IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS,SUBMITTED:SEPT.2010.
1.6 DERIVING THEPROBABILITY DENSITY FUNCTION OFγm⋆d
Let γ denote the normalized received SNR at the desti-
1.4 m⋆d
nation terminal over the S R D link. Then, the PDF
M=4 → i⋆ →
of γ is written as
1.2 m⋆d
M=3
der 1 M=2 fγm⋆d(x)=Pr(all relaysareoff)fγm⋆d (allrelaysareoff)(x)
Diversity Or00..68 +|D(XDs)(|s)≥P1r(relays in D(s) areon)fγm⋆d(cid:12)(relay(cid:12)(cid:12)sin D(s)areon)(x). (47)
(cid:12)
The ith relay decodes its received signal erroneously with
0.4 probabilityB which is given in (23). Since all S R links
i i
→
are statistically, we have
0.2
M
00 50 100 150 200 250 300 350 400 450 500 Pr(allrelays areoff)= Bi. (48)
SNR(dB) i=1
Y
On the other hand, if all the relaysare off, then no communi-
Fig. 8.Effective diversity order versus SNR. ρ = 1, ρ = 0.9 and M =
2,3,4.Thefigureisobtained from(46). e f cation would occur between source and destination terminal.
The received SNR at the destination terminal would be zero.
3
Therefore, the conditional PDF can be written as [29]
2.5 ρf=ρe=1 f (x)=δ(x). (49)
γ (allrelaysareoff)
m⋆d
The probability of(cid:12)decoding set, given that there is at least
2 (cid:12)
one relay in (s), is given by
der D
Or
ersity 1.5 Pr(relays in (s) areon)= (1 Bi) Bi . (50)
v D  − 
Di 1 i∈/YD(s) i∈YD(s)
 
Inserting (48), (49) and (50) in (47) yields (22).
ρf=0.9
0.5
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