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Effect of Feedback Delay on AmplifyandForward Relay Networks With Beamforming
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R
EFERENCES
[1] L. Piazzo, “Fast algorithm for power and bit allocation in OFDM systems,”
Electron. Lett.
, vol. 35, no. 25, pp. 2173–2174, Dec. 1999.[2] W. T. Vetterling, W. H. Press, S. A. Teukolsky, and B. P. Flannery,
Numerical Recipes in C
. Cambridge, U.K.: Cambridge Univ. Press, 1992.[3] M. MohammadniaAvval, C. Snow, and L. Lampe, “Errorrate analysisfor bitloaded coded MIMOOFDM,”
IEEE Trans. Veh. Technol.
, vol. 59,no. 5, pp. 2340–2351, Jun. 2010.[4] L. Piazzo, “Fast optimal bitloading algorithm for adaptive OFDMsystems,” Univ. Rome, Rome, Italy, INFOCOM Dept., Tech. Rep. 0020403, 2003. [Online]. Available: http://infocom.uniroma1.it/~lorenz/ rep020403.ps[5] J. Campello, “Discrete bit loading for multicarrier modulation systems,”Ph.D. dissertation, Stanford Univ., Stanford, CA, 1999.
Effect of Feedback Delay on AmplifyandForwardRelay Networks With Beamforming
Himal A. Suraweera,
Member, IEEE
,Theodoros A. Tsiftsis,
Senior Member, IEEE
,George K. Karagiannidis,
Senior Member, IEEE
, andArumugam Nallanathan,
Senior Member, IEEE
Abstract
—In this paper, the decremental effect of beamforming withfeedback delay on the performance of a twohop amplifyandforward(AF) relay network over Rayleighfading channels is investigated. An antenna conﬁguration in which the source and the destination are equippedwith multiple antennas, whereas the relay is equipped with a single antenna, is assumed. We derive new expressions for the outage probabilityand the average bit error rate (BER), which are useful for a large numberof modulation schemes. To gain further insights, simple outage probabilityand average BER approximations at high signaltonoise ratio (SNR) arealso presented. It is shown that, whenever a feedback delay exists, thenetworkisnotcapableofofferingdiversitygains.Furthermore,sourceandrelay power allocation results show signiﬁcantly different behavior withfeedback delay. Numerical results supported by simulations are providedto show that feedback delay can severely degrade the performance of theconsidered AF relay system.
Index Terms
—Amplifyandforward(AF),averagebiterrorrate(BER),beamforming, feedback delay, relays.
I. I
NTRODUCTION
Wirelesscommunicationsystemscanbeneﬁtfromrelaydeploymentsince the technology promises extended signal coverage, improvedthroughputs, and spatial diversity [1], [2]. One of the relaying protocols described in the literature is amplify and forward (AF). The
Manuscript received July 20, 2010; revised December 6, 2010 andJanuary 10, 2011; accepted January 26, 2011. Date of publication February 10,2011; date of current version March 21, 2011. This paper was presented in partat the IEEE Global Communications Conference, Honolulu, HI, Nov. 30–Dec.4, 2009. The review of this paper was coordinated by Prof. C. P. Oestges.H. A. Suraweera is with the Engineering Product Development, SingaporeUniversity of Technology and Design, Singapore (email: himalsuraweera@sutd.edu.sg).T. A. Tsiftsis is with the Department of Electrical Engineering, Technological Educational Institute of Lamia, 35100 Lamia, Greece (email: tsiftsis@teilam.gr).G. K. Karagiannidis is with the Department of Electrical and ComputerEngineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece(email: geokarag@auth.gr).A. Nallanathan is with the Centre for Telecommunications Research, King’sCollege London, WC2R 2LS London, U.K. (email: arumugam.nallanathan@kcl.ac.uk).Digital Object Identiﬁer 10.1109/TVT.2011.2112786
performance of singleantenna AF relay networks has now been wellinvestigated [3]–[5].Work such as [6] has also demonstrated that signiﬁcant beneﬁtscan be gained if multiple antennas are deployed in relaying networks.A practical transmission scheme for AF relaying systems employingmultipleantennas isbeamforming[7]–[13].Theperformanceofbeamformingagainstrelayselection,consideringbothunlimitedandlimitedfeedback, has been studied in [7]. In [8], the performance of a twohoprelay network with transmit beamforming at the source and maximalratio combining (MRC) at the destination has been analyzed. Theperformance of the same system, by considering antenna correlationeffects at the source and the destination, is reported in [9]. In [10],the outage performance with beamforming, considering only limitedfeedback,hasbeenstudied.In[11],theperformanceofatwohopﬁxedgain network over Nakagami
m
fading channels has been analyzed. In[12], assuming the absence/presence of the source–destination directlink, optimal beamforming codebook designs for an AF relay systemwith limited feedback was presented. In [13], a practical scenario inwhich a multiantennaequipped source is communicating to a singleantennaequipped destination via a relay has been considered. Despitethe signiﬁcant practical interest, the authors in [13] have limited theiranalysis to a situation of
perfect
channel state information (CSI) at thesource.In beamforming systems, the received signaltonoise ratio (SNR)maximization is achieved by providing CSI to the transmitter. Infrequencydivisionduplex systems,such knowledge is provided by thefeedbackofCSIfromthereceivertothetransmitter.Feedbackinvolvesdelay, and as a result, in practice, the available CSI at the transmitterand the actual channel may be different. The use of outdated CSI forbeamforming degrades the system performance. Although this performance degradation for pointtopoint systems is now well understood(see, e.g., [14]) and for partial relay selection [15], so far, in theexisting literature, the effect of feedback delay on the performance of AF relaying with beamforming has not been investigated.In this paper, the effect of feedback delay on the endtoend performance of a twohop AF relay network where multiple antennas at thesource are used for beamforming is investigated. We derive closedform expressions for the system’s outage probability and the averagebit error rate (BER) applicable for a range of modulation schemes. Togain valuable insights, in the highSNR regime, we also present asymptotic outage probability and average BER expressions. The impactof different antenna conﬁgurations, feedback delay, and SNR imbalance on the performance is illustrated through some analytical results.II. S
YSTEM
M
ODEL
Consider a wireless network where a source
S
equipped with
N
t
antennas communicates with a destination
D
equipped with
N
r
antennas through a single antenna relay
R
[8], [9], [11]. In this network, we
assume that
S
does not have a direct link to
D
. The communicationfrom
S
to
D
via relay
R
takes place in two time slots. In the ﬁrst timeslot,
S
beamforms its signal to
R
. The received signal at
R
can bewritten as
y
R
(
t
) =
√
P
1
w
†
t
h
sr
(
t
)
x
(
t
) +
n
1
(
t
)
(1)where
x
(
t
)
is the data symbol,
P
1
is the transmit power,
h
sr
(
t
) =[
h
1
sr
(
t
)
,...,h
N
t
sr
(
t
)]
T
is the channel vector from
S
to
R
with Rayleighfading entries, and
n
1
(
t
)
is the additive white Gaussian noise (AWGN)at
R
with onesided power spectrum density
σ
21
. The transpose andthe conjugate transpose are denoted by
(
·
)
T
and
(
·
)
†
, respectively.According to the principles of maximal ratio transmission, we choose
00189545/$26.00 © 2011 IEEE
1266 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011
w
t
= (
h
sr
(
t
−
T
d
)
/
h
sr
(
t
−
T
d
)
F
)
, where
·
F
is the Frobeniusnorm. Note that
w
t
is calculated from the
outdated channel
h
sr
(
t
−
T
d
)
, instead of the channel
h
sr
(
t
)
, with
T
d
being the feedback delay. The received scalar signal at
R
is then multiplied by a gain
G
/ and transmitted to the destination. The received signal at
D
isgiven by
y
D
(
t
) =
P
2
h
rd
(
t
)
G
P
1
w
†
t
h
sr
(
t
)
x
+
n
1
(
t
)
+
n
2
(
t
)
(2)where
P
2
is the relay transmit power,
h
rd
(
t
) = [
h
1
rd
(
t
)
,...,h
N
r
rd
(
t
)]
is the channel vector from
R
to
D
with Rayleigh fading entries, and
n
2
(
t
)
is the AWGN vector at
D
with onesided power spectral density
E
[
n
2
(
t
)
n
†
2
(
t
)] =
σ
22
I
N
r
. The expectation operator is
E
[
·
]
, and
I
N
r
denotes the identity matrix of size
N
r
. According to the principles of MRC, we then multiply the received signal
y
D
(
t
)
by a vector
w
r
=(
h
rd
(
t
)
/
h
rd
(
t
)
F
)
and write
r
D
(
t
) =
P
1
P
2
w
†
r
h
rd
(
t
)
G
w
†
t
h
sr
(
t
)
x
+
P
2
w
†
r
h
rd
(
t
)
Gn
1
(
t
) +
w
†
r
n
2
(
t
)
.
(3)Using the CSIassisted AF relay gain [5], [8]
1
G
2
= 1
P
1
w
†
t
h
sr
2
F
+
σ
21
(4)and after some manipulations, the endtoend SNR can be written as
γ
eq1
=
γ
1
γ
2
γ
1
+
γ
2
+ 1
(5)where
γ
1
=
w
†
t
h
sr
(
t
)
2
F
¯
γ
1
,
γ
2
=
h
rd
(
t
)
2
F
¯
γ
2
, and
¯
γ
i
= (
P
i
/σ
2
i
)
for
i
= 1
,
2
.III. P
ERFORMANCE
A
NALYSIS
In this section, we derive important performance metrics, e.g., theoutage probability and the average BER for the twohop network underinvestigation.
A. Outage Probability
The outage probability
P
o
is an important qualityofservice measure deﬁned as the probability that
γ
eq1
drops below an acceptableSNR threshold
γ
th
. To study the outage probability of (5), it isnecessary to obtain the cumulative distribution function (cdf)
F
Z
(
γ
th
)
of the random variable, which is deﬁned as
Z
=
γ
1
γ
2
γ
1
+
γ
2
+
c
(6)where
c
is a positive constant. After applying some algebraic manipulations along similar lines as in [8],
F
Z
(
γ
th
)
can be expressed as
F
Z
(
γ
th
)=1
−
∞
0
f
γ
1
(
x
+
γ
th
)
1
−
F
γ
2
γ
th
+
γ
2th
+
cγ
th
x
dx
(7)where
f
γ
1
(
·
)
and
F
γ
2
(
·
)
denote the probability density function (pdf)of
γ
1
and the cdf of
γ
2
, respectively.
1
In the presence of a feedback delay, the ability of the relay to perfectlytrack the current CSI is subjective to the pilot placement arrangement employedfor channel estimation at the relay. Analyzing the impact of any speciﬁc pilotplacement on the relay gain and system performance is beyond the scope of thecurrent work.
To obtain
F
Z
(
γ
th
)
, expressions for the pdf of
γ
1
and the cdf of
γ
2
are needed. To model the relationship between
h
sr
(
t
)
and
h
sr
(
t
−
T
d
)
, we employ the following widely adopted timevarying channelmodel:
h
sr
(
t
) =
ρ
d
h
sr
(
t
−
T
d
) +
1
−
ρ
d

2
e
(
t
)
(8)where
ρ
d
is the normalized correlation coefﬁcient between
h
jsr
(
t
)
and
h
jsr
(
t
−
T
d
)
, with
j
= 1
,...,N
t
. The error vector
e
(
t
)
∼
CN
(
0
,
I
N
t
)
, where CN
(
ϕ
,
Ξ
)
denotes the complex Gaussian distribution with mean
ϕ
and covariance
Ξ
, is uncorrelated with
h
sr
(
t
)
. ForJake’s fading spectrum,
ρ
d
=
J
0
(2
πf
d
T
d
)
, where
f
d
is the Dopplerfrequency, and
J
0
(
·
)
is the zerothorder Bessel function of the ﬁrstkind [16, Sec. (9.1)].The pdf of
γ
1
using [14, (15)] can be written as
2
f
γ
1
(
x
) = 1¯
γ
N
t
1
N
t
−
1
i
=0
N
t
−
1
i
(

ρ
d

2
)
N
t
−
i
−
1
(¯
γ
1
(1
−
ρ
d

2
))
i
(
N
t
−
i
−
1)!
×
x
N
t
−
i
−
1
e
−
x
¯
γ
1
(9)and the cdf of
γ
2
is given by
F
γ
2
(
x
) = 1
−
e
−
x
¯
γ
2
N
r
−
1
j
=0
x
¯
γ
2
j
j
!
.
(10)Substituting (9) and (10) into (7), we obtain
F
Z
(
γ
th
) =1
−
e
−
1¯
γ
1
+
1¯
γ
2
γ
th
¯
γ
N
t
1
×
N
t
−
1
i
=0
N
t
−
1
i
(

ρ
d

2
)
N
t
−
i
−
1
(¯
γ
1
(1
−
ρ
d

2
))
i
(
N
t
−
i
−
1)!
×
N
r
−
1
j
=0
γ
th
¯
γ
2
j
j
!
I
1
(11)where
I
1
=
∞
0
(
x
+
γ
th
)
N
t
−
i
−
1
(
x
+
γ
th
+
c
)
j
x
−
j
e
−
(
x/
¯
γ
1
)
−
(
γ
2th
+
cγ
th
/x
¯
γ
2
)
dx.
Using the binomial theorem and with the help of the identity [17, eq.(3.471.9)], we obtain
I
1
=
N
t
−
i
−
1
p
=0
N
t
−
i
−
1
p
γ
N
t
−
i
−
p
−
1th
j
q
=0
jq
(
γ
th
+
c
)
j
−
q
×
2
¯
γ
1
(
γ
2th
+
cγ
th
)¯
γ
2
p
+
q
−
j
+12
K
p
+
q
−
j
+1
2
γ
2th
+
cγ
th
¯
γ
1
¯
γ
2
(12)where
K
ν
(
·
)
is the
ν
thorder modiﬁed Bessel function of the secondkind [16, Sec. (9.6)]. Now, the outage probability follows by substituting
c
= 1
into (12) and then using (11).Note that the derived outage probability (valid for arbitrary
ρ
d
) canbe further simpliﬁed in many special cases. For example, when
D
hasa single antenna and the feedback delay goes to inﬁnity, the outage
2
It is noted that, [14, eq. (15)] includes a small typo, which we have correctedin (9). The power of the term
1
/
¯
γ
1
outside the
must be
N
t
and not
N
t
−
1
.
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011 1267
probability can be obtained from (11) and (12) by substituting
N
r
= 1
,
i
=
N
t
−
1
, and

ρ
d

2
= 0
and is given by
F
γ
eq1
(
γ
th
) = 1
−
2
e
−
1¯
γ
1
+
1¯
γ
2
γ
th
γ
2th
+
γ
th
¯
γ
1
¯
γ
2
×
K
1
2
γ
2th
+
γ
th
¯
γ
1
¯
γ
2
.
(13)
B. Outage Probability at High SNR
Although (11) is exact and valid for any given SNR, it is difﬁcult togain insights such as the effect of parameters
N
t
and
ρ
d
on the outageprobability. Therefore, as
¯
γ
1
and
¯
γ
2
tend to inﬁnity with ﬁxed ratio
µ
, we can analyze the system’s asymptotic outage probability with thehelp of
Theorem 1
.
Theorem 1:
The outage probability at high SNR for
ρ
d
= 1
isgiven by
3
P
∞
o
=
1
N
t
!
γ
th
¯
γ
1
N
t
+
o
γ
th
¯
γ
1
N
t
+1
, N
t
< N
r
1
N
!
1 +
1
µ
N
γ
th
¯
γ
1
N
+
o
γ
th
¯
γ
1
N
+1
, N
t
=
N
r
=
N
1
µ
N r
N
r
!
γ
th
¯
γ
1
N
r
+
o
γ
th
¯
γ
1
N
r
+1
, N
t
> N
r
.(14)The outage probability for
ρ
d
<
1
is given by
P
∞
o
=
(1
−
ρ
d

2
)
N
t
−
1
+
1
µ
γ
th
¯
γ
1
+
o
γ
th
¯
γ
1
2
, N
r
= 1
(1
−
ρ
d

2
)
N
t
−
1
γ
th
¯
γ
1
+
o
γ
th
¯
γ
1
2
, N
r
>
1
(15)where
µ
= (¯
γ
2
/
¯
γ
1
)
.
Proof:
See the Appendix.
We observe that a longer delay (low
ρ
d
) degrades the systemperformancebyincreasingtheoutageprobability.Moreimportantly,inthe presence of feedback delay, no diversity gains can be achieved. Inthis twohop system, the worst link dominates the performance at highSNR. According to (8), coefﬁcients of the noise vector
e
(
t
)
, althoughundesirable, contributes in determining the beamforming weights of the
S
−
R
channel
h
sr
(
t
)
. As such, although signals from
N
t
differentpaths are received at
R
, we do not expect an improvement in the signalquality since
e
(
t
)
is uncorrelated with
h
sr
(
t
)
.
C. Average BER
For many modulation formats used in wireless applications, theaverage BER expressed as
E
[
a Q
(
bγ
eq1
)]
is given by
P
b
=
a
√
2
π
∞
0
F
γ
eq1
t
2
b
e
−
t
22
dt
(16)where
a
,
b >
0
, and
Q
(
x
) = (1
/
√
2
π
)
∞
x
e
−
(
y
2
/
2)
dy
is theGaussian
Q
function. Moreover, our BER derivations can also be
3
Wehavedeﬁned
f
(
x
) =
o
(
g
(
x
))
,
x
→
x
0
,if
lim
x
→
x
0
(
f
(
x
)
/g
(
x
)) = 0
.
extended to square/rectangular
M
QAM since
P
b
can be written asa ﬁniteweighted sum of
a Q
(
bγ
eq1
)
terms.To the best of our knowledge, (16) does not have a closedformsolution. To overcome this, we substitute
c
= 0
into (11) and tightlylower bound the average BER as
P
b
≥
a
2
−
2
π
N
t
−
1
i
=0
N
t
−
1
i
(

ρ
d

2
)
N
t
−
i
−
1
(
b
¯
γ
1
(1
−
ρ
d

2
))
i
(
b
¯
γ
1
)
N
t
(
N
t
−
i
−
1)!
×
N
r
−
1
j
=0
(
b
¯
γ
2
)
−
j
j
!
N
t
−
i
−
1
p
=0
N
t
−
i
−
1
p
×
j
q
=0
jq
¯
γ
1
¯
γ
2
p
+
q
−
j
+12
I
2
.
(17)In (17),
I
2
=
∞
0
t
2(
N
t
−
i
+
j
)
e
−
((1
/b
¯
γ
1
)+(1
/b
¯
γ
2
)+(1
/
2))
t
2
K
p
+
q
−
j
+1
((2
t
2
/b
√
¯
γ
1
¯
γ
2
))
dt
. Applying a simple variabletransformation and using [17, eq. (6.621.3)],
I
2
has a closedformsolution given by
I
2
=
√
π
(2
α
2
)
η
2
2(
α
1
+
α
2
)
η
1
+
η
2
Γ(
η
1
+
η
2
)Γ(
η
1
−
η
2
)Γ
η
1
+
12
×
2
F
1
η
1
+
η
2
,η
2
+ 12;
η
1
+ 12;
α
1
−
α
2
α
1
+
α
2
(18)where
η
1
=
N
t
−
i
+
j
+ (1
/
2)
,
η
2
=
p
+
q
−
j
+ 1
,
α
1
=(1
/b
¯
γ
1
) + (1
/b
¯
γ
2
) + (1
/
2)
,
α
2
= (2
/b
√
¯
γ
1
¯
γ
2
)
,
Γ(
z
)
is the gammafunction, and
2
F
1
(
·
,
·
;
·
;
·
)
is the Gauss hypergeometric function[16, eq. (15.1.1)].
D. Average BER at High SNR
Asymptotic analysis on relaying networks are important since suchstudies provide valuable insights that are useful to design engineers.It is shown in [18] that, at high SNRs, the error performance canbe calculated based on the behavior of the pdf of the instantaneouschannel power gain around zero. First, consider the case of
ρ
d
= 1
.By following [18, Prop. (1)] and using (14), the average BER in thehighSNR regime can be expressed as
4
P
∞
b
=2
q
a
Ψ Γ
q
+
32
√
π
(
b
¯
γ
1
)
−
(
q
+1)
+
o
¯
γ
−
(
q
+2)1
(19)where
q
= min(
N
t
,N
r
)
−
1
, and
Ψ =
1
N
t
!
, N
t
< N
r
1
N
!
1 +
1
µ
N
, N
t
=
N
r
=
N
1
µ
N r
N
r
!
, N
t
> N
r
.(20)Equation (19) implies that the array gain
G
a
and the diversity gain
G
d
can be written as
G
a
=
b
2
q
a
Ψ Γ
q
+
32
√
π
−
1
q
+1
(21)
G
d
= min(
N
t
,N
r
)
.
(22)
4
A highSNR averagesymbolerrorrate approximation has also been presented in [8, eq. (18)]. However, our expression is much simpler and clearlyshows how network parameters affect the system’s error performance athigh SNR.
1268 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011
Fig. 1. Outage probability for various antenna conﬁgurations.
f
d
T
d
= 0
.
1
,
¯
γ
1
= 3
dB, and
¯
γ
2
= 7
dB.
From (22), we can infer that beamforming with no feedback delay can achieve the maximum possible diversity order of thissystem [8].Nowconsiderthecaseof
ρ
d
<
1
.TheaverageBERinthehighSNRregime is given by
P
∞
b
=
aψ
2
b
¯
γ
1
+
o
¯
γ
−
21
(23)where
ψ
=
(1
−
ρ
d

2
)
N
t
−
1
+
1
µ
, N
r
= 1
(1
−
ρ
d

2
)
N
t
−
1
, N
r
>
1
.(24)Equation (23) implies that the
G
a
and
G
d
of the system withfeedback delay can be written as
G
a
= (2
b/aψ
)
and
G
d
= 1
,respectively.IV. N
UMERICAL AND
S
IMULATION
R
ESULTS
In this section, we conﬁrm the derived analytical results throughcomparison with Monte Carlo simulations and discuss the impact of feedback delay on system’s performance. We assume Jake’s fadingspectrum, and in the examples,
ρ
d
= 0
.
903
,
0
.
642
,
and
0
.
472
correspond to
f
d
T
d
= 0
.
1
,
0
.
2
,
and
0
.
3
, respectively.Fig. 1 shows the outage probability for various antenna conﬁgurations. We see that the analytical results obtained using (11) exactlymatch the simulations. As expected, the outage probability is signiﬁcantly improved as the number of antenna increases. Thus, deploymentof multiple antennas improve the performance of this network.Fig. 2 shows the outage probability for various feedback delays.We see that a large feedback delay can signiﬁcantly degrade systemperformance. When the feedback delay is large, the system’s outageperformance is mostly determined by low SNR thresholds. The outagecurves shift to high SNR thresholds when the feedback delay tendsto zero, indicating that the system performance improves for smallfeedback delays.Fig. 3 shows the outage probability using source and relay powerallocations
β
using our analytical results in (11). In particular, weset
¯
γ
1
+ ¯
γ
2
= ¯
γ
and have
¯
γ
1
=
β
¯
γ
and
¯
γ
2
= (1
−
β
)¯
γ
. As expected,with no feedback delay
(
ρ
d
= 1)
, equal power allocation
(
β
= 0
.
5)
is
Fig. 2. Outage probability for different
ρ
d
’s.
N
t
=
N
r
= 3
,
¯
γ
1
= 3
dB, and
¯
γ
2
= 7
dB.Fig. 3. Outage probability for different
β
’s with
N
t
=
N
r
= 2
,
γ
th
=
−
5
dB, and
¯
γ
= 20
dB.
optimal, i.e., yields the lowest outage probability. This observation isexpected since, for the considered antenna conﬁguration (equal number of antennas at the source and the destination), the network is symmetrical with respect to
S
−
R
and
R
−
D
link SNRs. However, interestingly, when there is a feedback delay,
β
= 0
.
5
yields a suboptimalresult. The optimal
β
is close to 0.9. The optimal
β
slightly changesfor different
ρ
d
’s. In the presence of feedback delays, the
S
−
R
and
R
−
D
linksarenotsymmetrical.The
S
−
R
linkisweaker,comparedwith the
R
−
D
link, and becomes the bottleneck for the network’sperformance. Therefore, to improve the performance, more powermust be allocated to the
S
−
R
link. For large feedback delays
(
ρ
d
=0
.
2
−
0
.
6)
, the outage probability is barely affected by the variationsof
β
. For example, when
ρ
d
= 0
.
2
, the outage probabilities at
β
=0
.
1
and
0
.
9
are approximately
3
×
10
−
2
and
4
×
10
−
3
, respectively.However, when the feedback delay is decreased
(
ρ
d
= 0
.
8
→
1)
, thesituationissigniﬁcantlyimproved.Whenalargefeedbackdelayexists,the
S
−
R
link is severely weakened, and although allocating morepower to the
S
−
R
link improves the system’s outage probability, itdoessomarginally.Fig.3alsoshowsthat,forthegiven
β
’sintherangeof 0.1–0.9, when a large feedback delay exists
(
ρ
d
= 0
.
2
−
0
.
6)
, only