Int. J. Communications, Network and System Sciences, 2010, 3, 430-433
doi:10.4236/ijcns.2010.35056 Published Online May 2010 (http://www.SciRP.org/journal/ijcns/)
Outage Performance of Opportunistic
Amplify-and-Forward Relaying over Asymmetric Fading
Environments
Sudhan Majhi1,2,3, Youssef Nasser1,2,3, Jean François Hélard1,2,3
1European University of Brettany, Rennes, France
2Institut National des Sciences Appliquees, Rennes, France
3Institute of Electronics and Telecommunications of Rennes, Rennes, France
Received February 16, 2010; revised March 21, 2010; accepted April 22, 2010
Abstract
This letter analyzes the outage probability of opportunistic amplify-and-forward relaying over asymmetric
and independent but non-identically distributed (i.n.d) fading environments. The work investigates the sce-
narios where cooperative nodes are located at different geographical locations. As a result, the different sig-
nals are affected by different i.n.d fading channels, one may undergo Rician fading distribution and others
may undergo Rayleigh fading distribution. In this letter, a lower bound of the outage probability for various
asymmetric fading environments is derived at high SNR by applying the initial value theorem. The analytical
model is validated through Monte-Carlo simulation results.
Keywords: Outage Probability, Opportunistic Relaying, Amplify-and-Forward Relaying, Rayleigh and Rician
Fading Channels, Asymmetric Fading Channels, Independent and Non-Identically Distributed
1. Introduction
Cooperative relaying is a promising technology for fu-
ture wireless communications. It can benefit most of the
leverages of multiple input multiple output (MIMO) wit-
hout using the conventional MIMO schemes [1]. Among
the cooperative techniques, the opportunistic relaying, in
which only one relay (R) node forwards the source’s (S)
data to the destination (D) has shown its efficiency com-
pared to other techniques [2].
The outage performance of opportunistic amplify-and-
forward (AF) relaying over a symmetric fading environ-
ment is widely investigated in [1,3,4]. However, in prac-
tice, cooperative nodes are usually located in different
geographical location environments and at different dis-
tances with respect to S and D. Therefore, one link could
be either in line-of-sight (LOS) situation or in non-LOS
(NLOS) situation. For example, the fixed relay nodes
used for forwarding source’s data to a specific region
(e.g. tunnel, behind the building) often use directional
antenna, so the R-D link is usually in a LOS situation.
However, we cannot assume such a situation in all
transmission environments especially when D is in a
deep shadowing region with respect to S. The outage
performance analysis of opportunistic relaying for mixed
and i.n.d fading environments is, therefore, of practical
importance.
The asymmetric fading channel is introduced in [5].
However, the authors of this work assume additive white
Gaussian noise (AWGN) channel of the R-D link. In [6],
an approximation of the outage performance over asym-
metric fading channel, i.e., Rayleigh and Rician, is given.
However, to the best knowledge of the authors, no clo-
sed-form expression is provided. In this letter, the ana-
lytical model of the outage probability of opportunistic
AF relaying over asymmetric and i.n.d fading environm-
ents is given. Then, the lower bound of the outage prob-
ability for high SNR values is deduced and verified thro-
ugh Monte-Carlo simulations.
2. System Model and SNR Evaluation
In this framework, we consider a general 2-hop AF re-
laying network consisting of S, m relays, Ri, 12i,,...,m

and D. We assume that D performs maximum ratio com-
bining at the receiving side. The equivalent instantaneous
end-to-end signal-to-noise ratio (SNR) for opportunistic

S. MAJHI ET AL. 431
AF relaying is given as [3]:
22
2
22
1
ii
ii
ii
ii
ssr srd
srr d
ssd
di
sd ssr srd
srr d
Ph Ph
NN
Ph
SNR max
NPh Ph
NN


(1)
where 2
ab
hrepresents the channel gain of the a-b link,
a
P is the power transmitted by the node a. As mentioned
in [3], we assume that AWGN variance is 0 1
ab
N/


,
a,b where 0

is proportional to the system SNR.
For simplicity reasons, we use different notations of
the random variables of the different fading distributions.
For the Rayleigh fading, let 2
aba ab
Ph  be the instant-
taneous signal power and for the Rician fading, the in-
stantaneous signal power is denoted as ab

. The prob-
ability density function (PDF) of ab

and ab

are ex-
pressed respectively as:

1ab
ab
x/
ab
fx e





(2)



1
0
41
1abab ab
ab
K/K ab ab
ab
ab ab
KK
K
feI
 








(3)
where, {}
ab ab
E

 {}
ab ab
E

 and ab
K
is the
Rician factor. {.}E holds for expectation value.
The upper bound of the instantaneous SNR of (1) for
the opportunistic AF relaying is defined as:

22
2
0ii
ubs sds srs rd
i
SNRP hmaxmin P h,Ph





(4)
This instantaneous SNR value will be used in the fol-
lowing section to evaluate the outage probability.
3. Analysis of Outage Probability
In this section, we provide the lower bound of the outage
probability of opportunistic AF relaying for different
channels given in Figure 1.
3.1. Asymmetric Channel I
Theorem 1: If S-D link is Rayleigh fading channel and
S-R and R-D links are Rician fading channels, then the
lower bound of the outage probability over asymmetric
channel I is:





1
1
11
1
1
ii
srrd
ii
ii
msrr d
Im
out KK
i
sd srrd
KK
pmee











(5)
Figure 1. Different asymmetric fading channels of a coop-
erative network.
Proof: By using2
s
dssd
Ph 
, 2
ii
srs sr
Ph  and
2
ii
rds rd
Ph 
in [4], the outage probability over the
asymmetric channel I can be written as:
[]
II
out ub
pPr



 (6)
where ub
SNR is derived toI
ubsdmax


 ,




2
0
21
R/

,

12
max
max m
, ,...,

   and i




min ii
s
rrd
ξ,ξ.
The cumulative distribution function (CDF) of the
random variable i

over i.n.d is given as:








1
1
11 []1 []
21
12
21
2
iii
i
i
i
i
i
i
srr d
sr
sr
sr
rd
rd
rd
FPrPr
K
QK,
K
QK,


 




 



 









(7)
where


1
Q, is the 1st order Marcum Q-function and the
PDF of i

is obtained by differentiating above as:






1
21
2
21
2
i
ii rd
i
i
i
isr
i
i
sr
sr
sr
rd
rd
rd
K
fK, f
K
QK, f






















(8)
The CDF of the random variable max

over i.n.d fad-
ing channel can be expressed as:
 
1
max i
m
i
FF




 (9)
and the corresponding PDF of max

is obtained by dif-
ferentiating the above as:

S. MAJHI ET AL. 432
 
1 1
maxi j
m m
j
i
ji
ffF
 



(10)
Since

00
i
F

, the

1thm order derivative of
(10) at high SNR i.e., at 0


as 0

, can be
written as:
 
1
0
1
1
!0
max i
mm
m
i
f|mf
 






 (11)
The outage probability given in (6) is a CDF of
I
ub

,
which can be evaluated by using the initial value theorem
(IVT) of the Laplace Transformation (LT). The LT of the
PDF of the random variable
I
ub

can be expressed by
using Equation 15 in [3] and, then (11), as:

 
 
1
0
11
1
1
1
() 0
!00
ubsd max
sd i
m
I
mm
m
m
i
Lfff |
s
mff
s











(12)
Since

0
sd
f

and


0
i
f

are constant with respect
to the variables, the PDF of
I
ub

is obtained by apply-
ing the inverse LT (ILT) on (12) as:
 
1
()0 0
ubsd i
m
Im
i
fff



 (13)
We complete the proof by integrating (13) and substi-
tuting the vale of

0
sd
f

and

0
i
f

.
3.2. Asymmetric Channel II
Theorem 2: If S-D and S-R links are Rician fading
channels and R-D link is Rayleigh fading channel, then
the lower bound of the outage probability over asymmet-
ric channel II is:




1
1
1
11
1
i
sd sr
i
i
i
msr
sd II m
out KK
ird
sd sr
K
K
pme e












(14)
Proof: For the asymmetric channel II, we use
2
s
dssd
Ph  , 2
ii
srs sr
P
h  and 2
ii
rds rd
Ph  . The
outage probability can be expressed as:
[]
II II
out ub
pPr

 
(15)
where II
ub sdmax
g
  ,

12
max
max m
g
g ,g,...,g and

min ii
isrrd
g,

. The CDF of the random variable i
g
over i.n.d fading channel can be written as:



 1
21
12 1
i
ii rd
i
i
sr
gsr
sr
K
FQK, F



 







(16)
where


rd
i
F


is the CDF of the random variable i
rd

.
The corresponding PDF of i
g
is expressed as:






1
21
2
1
i
ii rd
i
i
rd sr
ii
sr
gsr
sr
K
fQK, f
Ff
















(17)
Similarly, by using the IVT and the ILT, the PDF of
I
I
ub

can be derived as:
 
1
00
ubsd i
m
II m
g
i
fff



 (18)
By integrating (18), we complete the proof.
3.3. Asymmetric Channel III
Theorem 3: If S-D link is Rician fading channel and S-R
and R-D link are Rayleigh fading channels, the corre-
sponding lower bound of outage probability is:



1
1
111
1sd
ii
m
sd III m
out K
isrrd
sd
K
pme




 






(19)
Proof: For the asymmetric channel III, we use
2
s
dssd
Ph  , 2
ii
srs sr
Ph  and 2
ii
rds rd
Ph  .
The outage probability can be written as:
[]
III III
out ub
pPr



 (20)
where III
ubsd max


 ,

12max m
max,,...,


and


ii
isrrd
min ,

. The corresponding PDF of the
random variable i

is given by:
 


 



11
isrrdrdsr
iiii
fFf Ff 


 
(21)
Again by using the IVT and ILT, the PDF of
I
II
ub

is
obtained as:
 
1
00
ubsd i
m
III m
i
fff




(22)
By integrating (22), we complete the proof.
Similarly, the outage probability of other possible
asymmetric channels can be derived by using the above
procedure. The upper bound of the outage probability of
the opportunistic AF relaying can be derived simply by
using the above method and Equation 8 in [7].
4. Numerical Examples
In this section, analytical and Monte-Carlo simulation
results are presented. Since the channels are i.n.d, we
set different means for different S-Ri/Ri-D links. In the

S. MAJHI ET AL. 433
Rician fading channel, the Rician factor ab
K
is uni-
formly distributed in [2,3] and the mean ab

of the
NLOS components are uniformly distributed in [0,1].
The LOS components are derived for a given ab
K
and
ab

.
It is clear from (5), (14) and (19) that the outage
probability over Rician fading channel is obtained by
substituting

11
sd
K
s
dsd sd
/Ke/

in (5) and the
outage probability over Rayleigh fading channel is ob-
tained by substituting 0
sd
K in (19). Figure 2 shows
the lower bound of the outage probability over the sym-
metric and asymmetric fading environments. Due to the
presence of LOS signal, the outage performance over
Rician fading channel outperforms all other scenarios.
Inversely, due to the absence of direct signal, the Ray-
leigh fading channel has poorer outage performance than
the other scenarios.
The opportunistic relaying provides better outage per-
formance than without cooperation. It implies that the
outage performance of opportunistic relaying depends
mainly on cooperative links (S-R and R-D links). For
this reason, asymmetric channel I provides better outage
performance than the asymmetric channel II and asym-
metric channel III due to the presence of LOS signal
inboth S-R and R-D links.
We also note that asymmetric channel II provides bett-
er outage performance than asymmetric channel III. Si-
nce S-D and R-D links undergo the same fading in both
scenarios, the LOS component existing in S-R link of
Rician, Analytical
Rayleigh, Analytical
Asymmetric I, Analytical Equation (5)
Asymmetric II, Analytical Equation (14)
Asymmetric III, Analytical Equation (19)
Simulation
Outage probability
10
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10
-7
SNR [dB]
0 2 4 6 8 10 12 14 16 18 20
Figure 2. The outage probability over asymmetric channel
I, asymmetric channel II and asymmetric channel III. Due
to the high SNR approximation for the analysis, analytical
results converge with Monte-Carlo simulation results at
medium and high SNR regime.
scenario II highly improves the outage performance. It is
clear from the above discussion that S-R is a dominating
link, therefore, it is better to localize the opportunistic
relay node in LOS environment with respect to S in order
to improve the overall outage performance. Finally, the
Monte-Carlo simulation results provided in Figure 2
shows that the analytical outage probabilities are a tight
bound at medium and high SNR regime.
5. Conclusions
In this letter, the outage performance of opportunistic AF
relaying over asymmetric and i.n.d fading environments
has been investigated. A lower bound of the outage
probability has been derived and validated through
Monte-Carlo simulation results. We show that the outage
performance is better when the relay is in LOS situation
with respect to the source rather than to the destination.
6. Acknowledgements
The authors would like to thank the European IST-FP7
WHERE project and the European Network of Excel-
lence NEWCOM++ for support of this work.
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