Upper-Bounding the Capacity of Relay Communications - Part I

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by  Farshad Shams

Upper-Bounding the Capacity of Relay Communications - Part I
Farshad Shams Marco Luise
Dep. of Computer Science and Engineering Dipartimento di Ingegneria dell’Informazione
IMT Institute for Advanced Studies Lucca, Italy University of Pisa, Italy
Email: [email protected] Email: [email protected]
Abstract—Thispaperfocuseson thecapacityof point-to-point nodesiandj.Thesymbolsentbytransmittersinblockindex
relay communications wherein the transmitter is assisted by an b is representedby wb. We use calligraphicletters , , to
intermediate relay. We detail the mathematical model of cutset s X Y W
indicate (finite) alphabet spaces and lower case letters x ,y
i i
3 and amplifyand forward (AF) relayingstrategy. We present the forchanneldistributions.TheconventionsX andx denote
1 upperboundcapacityofeachrelayingstrategyfrominformation A A
0 theory viewpoint and also in networks with Gaussian channels. joint vector variable and channel distribution, respectively of
2 We exemplify various outer region capacities of the addressed the indices belonging to set . C represents the comple-
A A
strategies with two different case studies. The results exhibit mentary set of . The notation C(γ ) = 1log (1+γ )
n that in low signal-to-noise ratio (SNR) environments the cutset A ij 2 2 ij
a denotes Shannon channel capacity between nodes i and j
performance is better than amplify and forward strategy.
J with SNR γ at the receiver. The symbol is used for
ij
C
1 the outer region channel capacity. To compute (conditional)
I. INTRODUCTION
1 mutual information, we need the statistical parameter:
T] inCaomopueltria-tuivseerceonmvmirounnmiceantitotnoesnhaabrleesthseinirglaen-taenntnenasnaanmdobgielnes- Σ[XA|YB],det{ΣXA|YB} (1)
I erate a virtual multiple-antenna transmitter, and consequently wherein det denotes the determinant of a matrix and
cs. e(MxpIlMoiOt s)osmyesteomf sth.eCboeonpeefiratstivoef mtraunltsimplies-siinopnutcamnulitnicprleea-oseutpthuet ΣXA|YB ={}ΣXAXA −ΣXAYB ·(ΣYBYB)−1·ΣYBXA (2a)
[
data rate, save transmission power, and extend the coverage Σ= ΣXAXA ΣXAYB (2b)
1 rangeofthe network.Asa result,itisconsideredto bea key- ΣY X ΣY Y
(cid:20) B A B B (cid:21)
v techniqueinthedevelopmentofarobustandefficientcommu-
wherein Σ denotes the covariance matrix of multivariate
9
2 nicationsystem [1]. The first idea of cooperativetransmission Gaussian distribution XA jointly with YB [2, Ch. 5]. (cid:4)
canbetracedbacktotheproposaloftherelaychannelmodel,
6 Notation 2: The power gain of the (real-valued) radio link
2 which consists of one source, one destination and one relay. between two nodes i and j at a distance d , is scaled by
ij
. A relay channel models transmissions consists of a pair of
1 2
0 transceivers communicate assisted by one intermediate node. Γ =G G λ0 d−α (3)
3 To understand how much performance improvement can be ij tx· rx· 4π · ij
(cid:18) (cid:19)
1 obtainedbyacooperativenetwork,weuseaninformationthe-
whereinG =G =1arethetransmitandreceiveantennas
v: oretical study. Such a study also results how reliable commu- gains assutmxed omrxnidirectional, the parameter λ ≅ 0.12m
0
Xi nicationsshouldtakeplaceinfuturewirelesscommunications. represents carrier wave length, and the path loss exponent is
Tothatend,westudytheinformationtheoreticaspectsaround
α = 2. Consequently, if the node i is placed close to node j
ar peer-to-peer relayed communications. We calculate the outer and far from node k then h h . (cid:4)
ij ik
region bound of cutset and AF relaying strategy. ≫
The rest of this contribution is structured as follows. In II. RELAYEDPOINT-TO-POINTCOMMUNICATION
Sec. II, we study relay assisted communication between a The relaysare able to outperforma direct-linkcommunica-
transmitter and a far destination which is out of its transmis- tion and even actualize some otherwise impossible scenario.
sion range. Then, we extend the first scenario to a network For example, Fig. 1 depicts the simplest cooperative (relay)
whereinthedirect-linkbetweenthetransmitteranddestination communication model to realize a communication between
isavailable.We presentanupperboundonthecutsetcapacity two hidden terminals. The source s wishes to send a message
of relay communicationsin Sec. III. Sec. IV is devoted to the to the far destination d which is out of sight. We assume
well-knownrelayingstrategies:theamplifyandforward(AF). there is no fading on the wireless channels. One possible
We formally define these relaying protocols and calculate the solution is to use an intermediate node. The source s sends a
upper bound capacity for each of them. We illustrate our message to the relay r and the received noisy version of the
results in Sec. V, and then conclude in Sec. VI. original message is re-transmitted to the far destination node.
Notation 1: Upper case letters X ,Y represent the output We assume that the relay is accessible to both the source and
i i
and inputrandomvectorvariablesof node i, respectively,and destination nodes. In the example, the relay does not do any
X [b] is the bth entry of a random vector. The notation h processing (e.g. encoding, decoding) on the receive signal,
i ij
is used for the real-valued channel gain of the link between but its duty is to make possible the information exchange
Zr Zd r
s √hsr r √hrd d √hsrZr Yr Xr √hrd Zd
Fig.1. Onesource(s),onerelay(r),onefardestination (d)network. s wsEncoder Xs Decoderwˆs d
√hsd Yd
Fig.2. Onesource, onerelay,onedestination networkscenario.
froms to d. Thissimpletwo-hopmodelenlargessignificantly
the range of the network. The relaying function between the
source and destination nodes can be done in two different
maximum channel capacity of the two channels. Assigning
signaling modes: half-duplex and full-duplex. In half-duplex
p = p , the source-relay capacity is bounded by h p that
mode, the transmission from source to destination is done in s r sr r
is less than the maximum capacity of the source-relay link.
two different stages and in each stage the relay acts either as
At this pointit is easy to satisfy Eq. 4. Case (4) follows from
receiver or transmitter. In the first stage, s transmits a stream
(2).
ofinformationandroperatesasareceiveranddisidle.Inthe
Cooperative communication can be efficient also when a
secondstage,thechannelbetweensandriskeptidleanddis
direct-link between the source node s and destination d is
activetoreceivedatafromtheintermediatenode r whichacts
available.InthenetworkillustratedbyFig.2,whenthesource
only as a transmitter. On the other hand, in full-duplexmode,
node s broadcasts, a noisy version of the data comes to the
both wireless channels are simultaneously busy and the relay
relay r and another corrupted version of data approaches the
playsthe roleofa receiveranda transmitteratthesame time.
destination d. Using an intermediate node is useful when the
The function of the intermediate node is feasible using two
receiveddataat distoo weaktobe decoded.Inthissituation,
distinct antennas,one as receiverand one as transmitterusing
arelaycanhelpcommunicationbytransmittinganewversion
two orthogonalfrequencyband. In spite of the the difficulties
ofitsownreceivedsignal.Thedirect-linksignalisusedtohelp
to apply the full-duplex manner in wireless networks, in this
decoding the stronger version of original data sent by relay.
work, we concern with this model for didactic issues. Our
ThecooperativenetworkinFig.2consistsoffourfiniteran-
goal is to determine how much data we can reliably get from
domspaces: atthetransmitter, and attherelaynode,
source to destination, placing no importance on delay. Xs Yr Xr
and atthedestination.Thesourcenodewantstotransmita
In the scenarioof Fig. 1, the powerexpenditureof s and r, Yd
messagew tothedestinationthroughdirectandreliablelinks.
p and p , respectively can be set a priori or can be adjusted s
s r
Theoriginalmessagew issplitinasequenceofsub-messages
given the individual power constraints p and p and to the s
s r w1,...,wb,...,wB each uniformlyandindependentlydrawn
path gain values h and h . The objective of the power s s s
sr rd
from a set with alphabet size n and length R , represented
controlistoapproachthemaximumShannonchannelcapacity s
by = 0,1,...,2nRs 1 . The encoder at the transmitter
betweensourceanda(far)destination.Accordingtothe max- Ws { − }
is a function which maps wb to X [b] wb . We
flowmin-cuttheorem(alsoreferredtoasthecutsetbound),the Ws → Xs s s s
assume that the decoder generates a sequence of B Gaussian
maximum end-to-end channel capacity in Fig. 1 is achieved (cid:0) (cid:1)
random codewords given a constraint on average power as
when the capacity of the source-relay link is the same as that
E X2 p . We also assume E X = 0. In each block
of the r d link, i.e. C(γsr)=C(γrd). We assume that the { s} ≤ s { s}
→ withindexb,thesourcesbroadcaststheencodedsymbolsand
noises Z and Z are Gaussian random variables (0,σ2).
r d N w each X [b] wb experiences two different paths to approach
The power optimization problem becomes to the equation: s s
the relay and destination. The relay observes Y [b] as:
r
(cid:0) (cid:1)
h p =h p (4)
sr s rd r
Y [b]= h X [b] wb +Z (5)
r sr s s r
We study the final equation in four different cases:
p (cid:0) (cid:1)
h The receive signals at r and d are different but statisti-
i. if(h h )and(p <p )thenp =p , p = srp ;
sr≤ rd s r s s r h r cally correlated. In full-duplex mode, the relay processes the
rd
ii. if(h <h )and(p p )thenp =p , p = hsrp ; received signal in the previous block index and generates
sr rd s ≥ r s r r hrd r the information Xr[b] wsb−1 , to be sent into the relay-
iii. if(hsr≥hrd)and(ps >pr)then ps = hhrsdrps, pr =pr ; gdeensteinraatteiodnvecrhsaionnneolf. YTrh[(cid:0)eb in1fo](cid:1)ramndatiiotnisXthre[bo]utwpusbt−o1f thisearelraey-
h − (cid:0) (cid:1)
iv. if(h >h )and(p p )then p = rdp , p =p . r’s deterministic function whose input is the sequence of the
sr rd s ≤ r s hsr s r s previous received signals:
Case (1) is the situation in which the maximum capacity
ofthe source-relaylink isless thanthatrelay-destinationlink. X [b] wb−1 =fb(Y [b 1],Y [b 2],...,Y [1]); (6)
Thesourceexploitsthemaximumcapacityofthesource-relay r s r r − r − r
link, i.e. p = p . To adjust p it is enough to satisfy Eq. 4, The func(cid:0)tion f(cid:1)depends on the specific cooperative strat-
s s r
since the relay-destination channel capacity is limited to the egy as will be specified later on. Each symbol sequence
capacity of source-relay link. The same derivation applies to X [b] wb−1 is such that E X2 p , and E X = 0.
{ r s } { r} ≤ r { r}
case (3). In case (2) there is no exact relation between the The destination receives a superposition of two different
(cid:0) (cid:1)
signals: Reference [5, Proposition 2] shows that the (s;r;d) is
cutset
C
attainedbyGaussianchannels.Thefollowingtheorempresents
Y [b]= h X [b] wb + h X [b] wb−1 +Z (7)
d sd s s rd r s d the capacity of the AWGN relay channel.
It is seepn that, the (cid:0)dest(cid:1)inatipon node re(cid:0)ceives(cid:1)information Theorem 1: The AWGN cutset capacity of a point-to-point
bothaboutwbandwb−1.Itmeans,thedestinationreceivestwo relayed communication is upper bounded by:
s s
different versions of wb in two broadcast and multiple-access
s (h +h )p 1 ρ2
sctaangceesl.inTghethdeeceoffdeecrtaotfthXesd[bes−tin1a]tiownsb−d1ecofdreosmthXerm[be]sswagsbe−1wˆs. Ccutset(s;r;d)=−1s≤uρp≤1min(C sd srσw2s(cid:0) − (cid:1)!
Inparticular,thedecodingfunctionisamappingfunctionfrom
d s. The error probability(cid:0)at the(cid:1)destination’s(cid:0)decode(cid:1)r ,C hsdps+hrdpr +2ρ hsdpshrdpr (10)
Y → W σ2
is defined as: w p !)
Proof:Themutualinformationtermsarecalculatedusing
Pn =2−nRs Pr wˆ =w w was sent (8)
e · { s 6 s| s } Eqs. 9.
Xws 1 Σ[Y Y X ]
which is defined based on the assumption that the messages I(X ; Y Y X )= log d r| r =
s d r| r 2 2 Σ[Y Y X X ]
are independent and uniformly distributed over the alphabet d r r s
|
space. The rate Rs is achievable if there exists a sequence of =1log Σ[YdYr|Xr] =C (hsd+hsr)ps 1−ρ2 ;
codes n,2nRs forwhich Pen→∞ is arbitrarily close to zero. 2 2 σw2 σw2 (cid:0) (cid:1)!
(cid:0) (cid:1)III. CUTSETUPPER BOUND 1 Var(Yd) 1 Var(Yd)
I(X X ; Y )= log = log =
In this section we derive the upperbound capacity of max- s r d 2 2 Σ[Y X X ] 2 2 σ2
d| s r w
flow min-cut or “cutset”. The cutset upper bound is used
h p +h p +2ρ h p h p
as a reference to compare the upper bound of the realistic =C sd s rd r sd s rd r .
σ2
models. M. R. Aref in [3, Th. 3.4] pioneered to establish w p !
the cutset bound in a general reliable network with multiple
relays. S. Zahedi in [4, Th. 2.2] presents another proof to Performingthe maximizationover ρ, we can easily obtainthe
upperbound.Inotherwords,underaveragepowerconstraints,
the cutset upper bound in the one relay case like Fig. 2. The
following proposition shows that a cooperativesystem can be ajointlyGaussianinputdistributionsimultaneouslymaximizes
both mutual information terms. The important result of The-
decomposed into a broadcast channel from the source node’s
viewpoint,and a multiple-accesschannelfromthe destination orem 1 is: If γsr > γrd (i.e. the relay node is in a good
point of view. position to receive signals from the transmitter rather than to
Proposition 1: [4, Th. 2.2] For any relay channel ( deliversymbolstothedestination)thenthecutsetupperbound
s
,p(y y x x ), ) the cutset capacity is uXpp×er capacity is achieved by the maximization of the multiple-
r d r s r r d
bXounded by| Y × Y access term (the second term), otherwise it is achieved by
the broadcast term.
cutset(s;r;d)= sup min I(Xs; YdYr Xr), I(XsXr; Yd) IntheAWGNcutsetupperboundcapacity,whenρ 0,the
C { | }
p(xs,xr) broadcastterm,I(X ; Y Y X ),isdecreasingfuncti≥onofρ,
s d r r
where the supremum is computed over all joint distributions |
while the multiple-access term, I(X X ; Y ), is increasing.
s r d
on complying with individual power constraints.
s r Thus, the maximum of (s;r;d) is taken at the point at
X ×X cutset
The first term is the mutual information of broadcasting C
which two terms are equal:
X toward r and d with transition probability p(y y x ).
s d r| s (h +h )p ρ2+2 h p h p ρ+(h p h p )=0
The second term is the mutual informationof multiple-access sr sd s sd s rd r rd r− sr s
(11)
of r and s at the destination node with transition probability p
p(y x x ). Thus, in general, random variables y and y We analyze the Eq. 11 in different cases. First, we assume
d s r d r
|
are statistically related to both inputs x and x through the parameter ρ 0 is fixed and it is possible for the source
s r ≥
p(y y x x ). and relay nodes to adjust the transmission powers. Then, we
d r s r
|
p(x ,x )is ajointGaussiandistributionon witha assume the source and relay transmit at the maximum power
s r s r
cross correlation coefficient of ρ = E{XsXXr}×X. In the and the network can tune the value of ρ.
E{Xs2}E{Xr2} 1) We assume Xs and Xr are statistically independent; i.e.
case of ρ =0 we have: p(x ,x )=p(x ) p(x ). ρ = 0. The capacity region of (s;r;d) achieves
s r s r cutset
p · C
At this point, We recall some useful statistics equalities its maximum with adjusting the power expenditure of the
related to Fig. 2. To review the algebraic manipulation of the source and relay nodes.In Eq. 11 with ρ =0, it is enough
following formulas, see [2, Ch. 5]. tosatisfyh p h p =0,thatisequaltothemaximum
rd r sr s
−
Var(Y )=h p +h p +2ρ h p h p +σ2 (9a) capacity problem of Fig. 1 and Eq. 4. With fixed ρ = 0
d sd s rd r sd s rd r w andanappropriatepowercontrol,the (s;r;d)takes:
ΣΣ[[YYddYYrr||XXsrX]=r](=hsdΣ+[Yhds|Xr)spXsr(cid:0)]1p=−σρw22(cid:1)+σw2 ((99bc)) Ccutset(s;r;d)=C(cid:18)(hsr+σhw2sd)ps(cid:19)=CcCuts(cid:18)ethsdpsσ+w2hrdpr(cid:19)
The relation hsrps = hrdpr means that the data rate of dsr relay (r)
the channel between the source and relay nodes is equal
d
r
to that between the relay and destination. In this case,
the s r d path has the maximum possible efficiency. source (s) dsd dest. (d)
→ →
Choosing two independent random spaces for and
s
X
guarantees minimum processing for relaying’s data
r
X Fig.3. Asinglerelaycommunication networkscenario.
process. Suppose the destination node consists of two
different antennas with orthogonal frequencies which are
used simultaneously for receiving data from the source
Ineachblockindexb,the sourcenodebroadcastsX [b] wb ,
and relay nodes. Thus, there is no interference between s s
and at the same time the relay just increase the amplitude of
the r d and s d links. So, it results: (cid:0) (cid:1)
→ → the analog observed signal Y [b 1] to result a normalized
r
hsdps+hrdpr transmit X [b] wb−1 as: −
Ccutset(s;r;d)=C σ2 =C(γsd+γrd). r s
Therefore, when ρ (cid:18)= 0, thwe requ(cid:19)irements to achieve Xr[b] wsb−1 (cid:0)=α[b(cid:1)].Yr[b−1] (12)
tchoenturoplp,earndboaunndadcduetrsectomcappoanceitnytaist daenstainpaptrioopnritaotesupmowtheer (cid:0) (cid:1)=α[b]. hsrXs[b−1] wsb−1 +Zr
received SNRs. Equivalently, the reliable communication (cid:16)p (cid:0) (cid:1) (cid:17)
wherein α is amplification factor and is chosen to satisfy
forms a parallel channel between s d and r d links.
the relay’s power limit. The relay node has its own power
→ →
2) If the sequences of Xs and Yr are drawn from two constraint as E X2 p , so that:
correlated code spaces with a strictly positive correlation { r}≤ r
(ρ >0), the necessary condition for the power control to p
α[b]2 r (13)
achieve the capacity Ccutset(s;r;d) is: γsr > γrd. This | | ≤ σw2 +hsrps
meansthat the data rate of r d channelis less than that
→
s r link. Hence, there mustbe a delaybetween sending Forsimplicity,we assume α[b]=α ineveryblock.As can
→
the broadcast message and the multiple-access message. be observed, if σ2 +h p p the effect of the relay is
w sr s ≫ r
3) Finally, we suppose that the source and relay nodes negligible. Combining Eq. 7 and Eq. 12 gives:
transmit at maximum power and the network is able to
tune the correlation parameter. The appropriate value of ρ Y [b]= h X [b] wb + α . h h X [b 1] wb−1
is found by resolving Eq. 11 for ρ. d sd s s | | sr rd s − s
p (cid:0) (cid:1) p+ α . hrdZr+Zd(cid:0) (14(cid:1))
| |
h p h p + h p (h p +h p h p )
ρ∗ = − sd s rd r sr s sd s sr s− rd r p
(h +h )p Thus, the maximum capacity of AF scheme turns out to be:
p psd sr s
0onthρe∗cond1i.tioIfnothnatth∆e c=on(thrsadry+∆hsr)0p,st−hehnrdthper c>ap0acaintdy (s;r;d)=C √hsd + |α|.√hsrhrd 2 .ps (15)
of≤r ≤d is higher than that of t≤he broadcast channel. CAF (cid:0) (1+|α|2hrd)σw2(cid:1) !
→
This means that the link between relay and destination
The relay node does not regenerate any new code, and
must be kept idle for receiving the broadcast message and
consequently the complexity of this scheme is low. Since the
therefore using the links is not highly efficient. Instead,
relay node amplifies whatever it receives, including noise,
the condition ∆ > 0 means the broadcast capacity is
it is mainly useful in high SNR environments. When the
higher than r d channel data rate. Using an appropriate
→ channel between the transmitter and the relay is very noisy,
memory at the relay node, all channels get busy.
increasing the amplification factor α increases the noise at
the destination. The relay should thus not always transmit
For strictly negative ρ, from the formula of (s;r;d)
Ccutset with maximum power. Reference [6, p. 46] demonstrates that
in Theorem 1, it is derived that reducing coefficient ρ toward
under the condition: α γ , the (s;r;d) outperforms
1 yields decreasing either broadcast and multiple-access | | ≤ sr CAF
− thecapacityofthemaximalratiocombining(MRC)technique
capacities.Thesolutiontocompensatetheaffectofanegative
that is:
ρ is to raise significantlythe upperboundlimits of the source
and relay power consumption. γ γ
(s;r;d)=C γ + sr rd (16)
MRC sd
C γ +γ
IV. AMPLIFY AND FORWARD TECHNIQUE (cid:18) sr rd(cid:19)
IntheAFtechnique,thetransmitmessagews isasequence Comparing (15) and (16) we find that the MRC technique
of B sub-messages ws1,...,wsb,...,wsB which are indepen- performs better than AF under the following condition:
dently and uniformly drawn from the message set =
s
c{o0d,1ed,..t.o,2XnRs[sb]−w1s}b. Euancdherstuhbe-mcoensssatrgaeinwt sbthaist sEe{pXars2a}teWl≤y epns-. γsγrsr+γsγdsd <|α|2.hrd (17)
(cid:0) (cid:1)
3 1.5x 10−4 1
cutset
AF
2.5 cutset r ρ
nt
MRC e
Hz] 2 Hz] 1 fici
/s/ relay off /s/ oef
b 1.5 b MRC 0.5c
[ [ n
e e o
Rat 1 r Rat0.5 AF orrelati
0.5 C
relay off
−00.5 0 0.5 1 1.5 −−0110000 00 110000 220000 330000 440000 550000 6600000
dsr[m] dsr[m]
Fig. 4. Rates for one relay with ps = pr = 100mW, σw2 = 1µW, Fig. 5. Rates for one relay with ps = pr = 100mW, σw2 = 1µW,
dsd=1m,anddr =0.1m. dsd=500m,anddr =10m.
V. CASE STUDY derived that the AF technique is not quite useful in a low
SNR network, whereas the MRC technique performs much
In this section, we exemplify the various outer region
betterthanAF.Thisisbecausethereceivedsignalattherelay
boundspresentedsofarinthissection.Weconsiderapoint-to-
is very noisy, and also the scaling factor is higher than that
pointrelayedcommunicationwith Gaussianchannelswherein
in the previous scenario. In this situation (almost) no signal
the transmitter s, relay r, and sink d are located as sketched
perfectly approaches to the destination. Our experiments in a
in Fig. 3. We assume a vertical distance of d between the
r
given scenario with different parameters result that reducing
relay and the s d direct-link. The path condition values
→ theamplificationfactordoesnoteffecttheAFdatarate.Ascan
h =h =h =1 are scaled with respect to Notation 2.
sr rd sd
beseen,inlowSNR,thecodingtechniqueshowssignificantly
First,weexperimentahighSNRenvironmentandsupposethe
higher rates than in the direct-link.
source and destination are located at a distance of d =1m,
sd
andtherelayislocatedataverticaldistanceofdr =0.1mand VI. SUMMARY
it is horizontally moving from d = 0.5m to d =1.5m.
sr − sr The data rate of a relayed communication is a function
Fig. 4 plots various data rates for p = p = 100mW, and
s r of both relaying strategy and positions of the nodes. The
σ2 = 1µW. The curve labeled AF shows the outer region
w cutset upper bound capacity is decomposed into two terms:
of AF strategy with the largest possible scaling factor α in
broadcast capacity from transmitter’s viewpoint and multiple
Eq. 13. The curve labeled ρ plots a particular value of the
channel access at the destination. When the relay is close to
correlation coefficient we tried for this example.
the transmitter, the upper bound capacity of relaying com-
Inthestudiedcasestudy,theAFandMRCtechniquesshow
munication is equal to that multiple-access channel. On the
a very good performance. This is due to the fact of the short
other hand, when the relay is close to the destination, the
distances result in a high SNR. As the relay moves toward
upper bound capacity of relaying communication is equal to
the destination (d 1), the achieved signal at the relay
sr → that broadcast channel. In a low SNR environment, cutset
becomes weaker and this significantly decreases the AF data
technique achieves a significantly high data rate, whereas AF
rate. Generally, the AF and MRC techniques would be useful
shows a good performance in high SNR regime.
when the relay is located so as to be able to perfectly receive
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d =10m.We set the same α,p ,p , and σ2 as the previous
r s r w [6] K.J.R.Liu,A.K.Sadek,W.Su,andA.Kwasinski,CooperativeCom-
simulation. Fig. 5 plots various data rates. munications andNetworking. New York,NY:Cambridge University
We draw a different experimental function for correlation Press,2009.
value which is the curve labeled ρ. From Fig. 5, it is clearly