2012 Abstract Our contribution in this paper is the derivation of an exact closed-form outage probability formula for underlay cognitive cooperative networks operated over Rayleigh fad-i
Trang 1DOI 10.1007/s11277-012-0742-z
Exact Outage Probability of Underlay Cognitive
Cooperative Networks Over Rayleigh Fading Channels
Khuong Ho-Van
Published online: 8 July 2012
© Springer Science+Business Media, LLC 2012
Abstract Our contribution in this paper is the derivation of an exact closed-form outage probability formula for underlay cognitive cooperative networks operated over Rayleigh fad-ing channels The derivation considers the correlation among received signal-to-noise ratios, two critical constraints (interference power constraint and maximum transmit power con-straint), and non identically distributed (i.d.) channels The derived formula is corroborated
by Monte Carlo simulations and is served as an useful and effective tool to evaluate the performance behavior of underlay cognitive cooperative networks without time-consuming simulations under different operation parameters Numerical results illustrate that under-lay cognitive cooperative networks suffer the outage saturation phenomenon for a given maximum interference power level
Keywords Decode-and-forward· Cognitive radio · Underlay · Cooperative
communications· Rayleigh fading channels
1 Introduction
Cognitive radio is an emerging technology attracting a great deal of attention due to its capa-bility of improving spectrum utilization [1] In cognitive radio, unlicensed users/secondary users (SUs) are allowed to use the licensed band primarily allocated to licensed users/primary users (PUs) unless their operation does not degrade the performance of PUs in three modes: underlay, overlay, and interweave [2] In the underlay mode, SUs are allowed to use the spec-trum when the interference caused by SUs on PUs is within the range tolerated by PUs In the overlay mode, SUs simultaneously share the same spectrum with PUs while maintaining or improving the transmission of PUs In the interweave mode, SUs are only permitted to use the empty spectrum left by PUs This paper considers the underlay mode for low implementation complexity
K Ho-Van (B)
Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam
e-mail: khuong.hovan@yahoo.ca
Trang 2In order to constrain the interference not to exceed a certain level that PUs can tolerate
in the underlay mode, SUs must adaptively limit their transmit power, significantly reduc-ing their transmission range Cooperative communications in which users share their own antennas to form a virtual antenna array for achieving the potentials of the space diversity without the need of co-located antenna arrays brings many benefits such as improved perfor-mance, increased system capacity, extended coverage, etc [3] As such, it can complement and overcome the above shortage of underlay cognitive networks Among various cooperative communications schemes, decode-and-forward (DF) and amplify-and-forward (AF) have been extensively investigated [4] In DF, each relay decodes information from the source, re-encodes it, and then forwards it to the destination In AF, each relay simply amplifies the received signal and forwards it to the destination Due to its capability of regenerating noise-free relayed signals, DF is selected in this paper
The performance analysis of underlay cognitive cooperative networks has been exten-sively studied in [5 13] However, the most recently works closely related to our research are [12,13] In [12], the outage probability of underlay cognitive cooperative networks with
DF is analyzed in consideration of the correlation among received SNRs This work is highly
regarded as the first one that discovered and took such correlation into account Nevertheless, the authors just obtain a tight lower bound of the outage probability and consider only the
interference power constraint Generally, two constraints (interference power constraint and
maximum transmit power constraint) are imposed on the underlay cognitive networks An exact closed-form outage probability formula of underlay cognitive cooperative networks taking into account factors such as the correlation among received SNRs and two above constraints was reported in [13].1 Notwithstanding, [13] assumes all channels are i.d Our contribution is to generalize the derived formula in [13] by relaxing the assumption on i.d channels Our new formula is corroborated by Monte Carlo simulations and is served as an useful and effective tool to evaluate the performance behavior of underlay cognitive coop-erative networks without time-consuming simulations under different operation parameters Numerical results illustrate that underlay cognitive cooperative networks suffer the outage saturation phenomenon for a given maximum interference power level
The rest of this paper is organized as follows The next section describes the system model and analyzes the outage probability of underlay cognitive cooperative networks Simulated and analytical results are presented in Sect.3for performance evaluation Finally, the paper
is concluded in Sect.4
2 Outage Probability Analysis
The underlay cognitive cooperative system model2under consideration is depicted in Fig.1
where a secondary relay R assists the transmission of a secondary source S to a secondary destination D, and both S and R use the same spectrum as a primary user P Assume the chan-nel between the transmitter t and the receiver r experiences independent slowly varying flat
Rayleigh fading with variance 1/λ tr Hence, the channel gain g tr = |h tr|2is an exponentially
distributed random variable with the probability density function (pdf) f gtr (x) = λ tr e −λ tr x
1 The authors would like to thank one of reviewers for pointing out the useful and missed reference [ 13 ] Since it appeared after our manuscript submission, there is a coincidence in the problem statement between our work and [ 13 ] However, our result is different from and more generalized than [ 13 ].
2 Although this paper only considers the case of a single relay, it is straightforward to extend our result to the case of multiple relay nodes with best relay selection studied in [ 12 , 13 ].
Trang 3Fig 1 System model
R
P
SP
h
RP
h
RD
h
SR
h
SD
h
Phase 1
Phase 2
for x≥ 0 Different from [13] where allλ tr’s are assumed to be the same (i.e., i.d channels),
we relax this assumption by considering non i.d channels
In the underlay cognitive networks, the transmit power P t of the transmitter t ∈ {S, R}
is imposed by two constraints [2]: interference power constraint, P t ≤ IT
g t P, and maximum
transmit power constraint, P t ≤ P m , where I T is the maximum interference power level
that PU still operates reliably, and P m is the maximum transmit power In other words,
P t ≤ minI T
gt P , P m
Consequently, the actual transmit power can be lower than the maxi-mum one (i.e., when I T
gt P ≤ P m), resulting in the coverage range of the secondary transmitter
in underlay cognitive networks less than that in other ones (e.g., interweave cognitive net-works)
The received SNR at the receiver r is given by [2]
γ tr = P t
g tr
N0 = min
I T
g t P , P m
g tr
where N0is the noise variance at the receivers
Let U be the transmission rate of secondary network According to the Shannon infor-mation theory, the outage (or unsuccessful inforinfor-mation decoding) occurs at the receiver r if
the received SNRγ r meets the inequality U ≥ 1
2log2(1 + γ r ) or γ r ≤ k with k = 2 2U−1
where 12 indicates the whole transmission process spends two phases In the first phase, S broadcasts its signal which is received and processed by R and temporarily stored by D If
R successfully recovers the source information (i.e., γ S R ≥ k), it will forward the processed signal to D in the second phase Then, D combines the signals from S and R with the
maxi-mum ratio combining (MRC) for restoring the source information In this case, the received
SNR at D is γ S D +γ R Dand an outage occurs ifγ S D +γ R D < k Otherwise (i.e., γ S R ≤ k), R keeps silent in the second phase and D bases on the only signal received from S for decoding the source information In this case, the received SNR at D is γ S D and an outage occurs if
γ S D < k According to the total probability law, the outage probability is defined as
P o = Pr {γ S D < k, γ S R < k}
T1
+ Pr {γ S D + γ R D< k, γ S R > k}
T2
It is noted from (1) that the received SNRs at R and D in the first phase, γ S R = min
IT
g S P , P m
gS R
N0 andγ S D = minIT
g S P , P m
gS D
N0 , are correlated since they are related
to g S P This correlation among the received SNRs is first discovered in [12] However, [12] only provides a tight lower bound of the outage probability and considers the interference power constraint, i.e.,γ tr = I T gtr
N0gt P In [13], the authors investigate both constraints making the received SNRs of the form in (1) and derive the exact closed-form outage probability expression Nevertheless, the assumption on i.d channels is made there In this paper, we relax this assumption for more generalized
Trang 4To derive the closed forms of two terms, T1and T2in (2), we note that Pr{g tr < m} =
m
0 λ tr e −λ tr x d x = 1 − e −λ tr m, Pr{g tr > m} = m∞λ tr e −λ tr x d x = e −λ tr mand the
cumula-tive density function of x R D = min I T
gR P , P m
g R Dcan be borrowed from [2]
F x R D (x) =
⎛
⎝ xe−
βλRD IT Pm
x + β I T − 1
⎞
whereβ = λ R P /λ R D
The derivative of F x R D (x) results in the pdf of x R Das
f xR D (x) = β I T
(x + β I T )2e−βλRD IT Pm e−λRD x Pm −λ R D
P m
⎛
⎝ xe−
βλRD IT Pm
x + β I T − 1
⎞
⎠ e−λRD x Pm (4) Due toγ R D = x R D /N0, the pdf ofγ R Dis
f γ R D (x) = N0f xR D (N0x )
= Ge −Gρ R D
(x + G)2e −ρ R D x+ G ρ R D e −Gρ R D
x + G e −ρ R D x
−ρ R D
whereρ R D=λ R D N0
P m and G= λ R P IT
λ R D N0
2.1 Derivation of T1
Withγ S Dandγ S Rhaving the form in (1), T1is written as
T1= Pr {γ S D < k, γ S R < k}
= Pr
min
I T
g S P , P m
g S D
N0 < k, min
I T
g S P , P m
g S R
N0 < k
= Pr
⎧
⎨
⎩g S D < N0k
min
I T
gS P , P m
, g S R < N0k
min
I T
gS P , P m
⎫
⎬
⎭
=
∞
0
Pr
⎧
⎨
⎩g S D < N0k
min
IT
x , P m
⎫
⎬
⎭Pr
⎧
⎨
⎩g S R < N0k
min
IT
x , P m
⎫
⎬
⎭ f gS P (x) dx
=
∞
0
⎛
⎜
⎝1 − e
− λSD N0k min
IT
x ,Pm
⎞
⎟
⎛
⎜
⎝1 − e
− λSR N0k min
IT
x ,Pm
⎞
⎟
⎠ f gS P (x) dx
=
∞
IT
Pm
1− e−λSD N0kx IT
1− e−λSR N0kx IT
λ S P e −λ S Px d x
+
IT
Pm
0
1− e−λSD N0k Pm
1− e−λSR N0k Pm
λ S P e −λ S Px d x
Trang 5= 1 − e −ρ S D k − e −ρ S Rk + e −(ρ S D +ρ S R )k+ρ S D ke −(ρ S D k +χ)
ρ S D k + χ
+ρ S R ke −(ρ S R k +χ)
(ρ S D + ρ S R ) ke −[(ρ S D +ρ S R )k+χ]
whereρ S D = λ S D N0
P m , ρ S R =λ S RN0
Pm , χ = λ S P IT
Pm
2.2 Derivation of T2
Rewrite T2as
T2= Pr {γ S D + γ R D <k, γ S R >k}
= Pr
min
N0 + γ R D <k, min
N0 >k
=
k
0
⎛
⎝
∞
0
Pr
⎧
⎨
⎩g S D < (k − y) N0
min
I T
x , P m
, g S R > N0k
min
I T
x , P m
⎫
⎬
⎭ f g S P (x) dx
⎞
⎠ f γ R D (y) dy
=
k
0
⎛
⎝
∞
0
Pr
⎧
⎨
⎩g S D < (k − y) N0
min
I T
x , P m
⎫
⎬
⎭Pr
⎧
⎨
⎩g S R > N0k
min
I T
x , P m
⎫
⎬
⎭ f g S P (x) dx
⎞
⎠ f γ R D (y)dy
(y) dy =
k
0
⎡
⎢∞
0
⎛
⎜
⎝1 − e
−λSD N0(k−y) min
IT x ,Pm
⎞
⎟
⎠ e
− λSR N0k min
IT x ,Pm
λ S P e −λ S P x d x
⎤
⎥
⎦ f γ R D
=
k
0
⎡
⎢
⎢
⎢
∞
IT
Pm
1− e−λSD N0(k−y)x IT
e−λSR N0kx IT λ S P e −λ S P x d x
+
IT
Pm
0
1− e−λSD N0(k−y) Pm
e−λSR N0k Pm λ S P e −λ S P x d x
⎤
⎥
⎥
⎥f γ R D (y) dy. (7)
For simplicity, we denote
S R k + χ + e −ρ S R k 1− e −χ
!
3 When inserting ( 5 ) into ( 7 ), there appear some integrals in the form of 0k e−( λRD−λSD )N0 y
in order to compute this integral, two cases (λR D = λ S DandλR D = λ S D) must be considered.
Trang 6Inserting f γ R D (x) in (5) into (7), we have
T2= BGe −Gρ R D
1
− Be −Gρ R D− 1 1− e −ρ R D k +D λ R D e −Gρ R D− 1
!
(λ R D − λ S D )
1− e −(ρ R D −ρ S D )k
−
1
−
⎛
⎝
C G ρ R D e −GρRD
A +G +C Ge −GρRD
(A+G)2 + DGρ R D e −Gρ R D
−DGe −Gρ R D+C Ge −GρRD
A +G
(ρ R D − ρ S D )
⎞
⎠ I2(k, ρ R D − ρ S D , G)
+
"
C Ge −GρRD
(A+G)2 +C G ρ R D e −GρRD
A +G
−Cρ R D e −Gρ R D− 1!
#
I1(k, ρ R D − ρ S D , A) , (12)
where
I2(k, ρ, G) =
k
0
y + G d y = e ρG (Ei (−ρk − ρG) − Ei (−ρG)) , (13)
I1(k, ρ, A) =
k
0
y − A d y = e −ρ A (Ei (ρ (A − k)) − Ei (ρ A)) (14) Here the exponential integral function Ei(x) is defined in [14] as Ei (x) = − ∫∞−x e −t
is a built-in function in most computation softwares (e.g., Matlab)
Again inserting f γ R D (x) in (5) into (7) and consideringλ R D = λ S D, we have
T2 = BGe −Gρ R D
1
+
1
1
G
−
(A + G)2 + DGρ R D e −Gρ R D
ln
G
+
(A + G)2 +C G ρ R D e −Gρ R D
A
−Be −Gρ R D− 1 1− e −ρ R D k
+ Dρ R D
where ln(x) is the natural logarithm of x.
3 Illustrative Results
For illustration purpose, we randomly selectλ S P = 0.6366, λ S R = 0.2530, λ R P = 0.0316, λ R D=
0.0894 Two cases are considered: Case 1 (λ S D = 1 = λ R D) and Case 2 (λ S D = λ R D) We
assume the noise variance is normalized such that N0 = 0 dB and the required transmission rate
U= 1bps/Hz
Figure2investigate the effect of I T on the outage performance We fix P mat 25 dB It is shown that analytical and simulated results4are perfectly matched for both cases, confirming the accuracy
of the derived formula in (2) Additionally, the outage performance is improved with respect to the
4 108channel realizations are generated to obtain simulated results.
Trang 70 5 10 15 20
10−4
10−3
10−2
10−1
100
IT (dB)
Case 1: Analysis Case 1: Simulation Case 2: Analysis Case 2: Simulation
Fig 2 Outage probability versus I T(Pm = 25 dB)
10−3
10−2
10−1
100
Pm (dB)
Case 1: Analysis Case 1: Simulation Case 2: Analysis Case 2: Simulation
Fig 3 Outage probability versus P m (IT = 15 dB)
increase in I T This is obvious since I Timposes a constraint on the transmit power and the higher
is I T, the higher can the transmit power be, eventually enhancing communication reliability Figure3compares simulated and analytical results when P m varies from 0 to 30 dB while I Tis fixed at 15 dB It is seen that both analytical and simulated results are in the good agreement, again validating the proposed formula Additionally, the results show that underlay cognitive cooperative
networks are quickly stable at high P m This saturation phenomenon comes from the fact that the
transmit power is limited by the minimum of the maximum interference power level, I T, and the
maximum transmit power, P m As such, when P m exceeds a certain value (e.g., around 15 dB in Fig.3), the transmit power is completely controlled by I T, resulting in the same outage probability
for any increase in P m
Trang 84 Conclusions
In this paper, the exact closed-form outage probability expression for underlay cognitive coop-erative networks under the general conditions such as the correlation among the received SNRs, two constraints (interference power constraint and maximum transmit power constraint), and non i.d channels is derived and validated by simulated results Numerical results show that underlay cognitive cooperative networks suffer the outage saturation phenomenon for a certain maximum interference power level, and their performance is better with respect to the increase in the maxi-mum interference power level
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Trang 9Author Biography
Khuong Ho-Van received the B.E (with the first-rank honor) and the
M.S degrees in Electronics and Telecommunications Engineering from
Ho Chi Minh City University of Technology, Vietnam, in 2001 and
2003, respectively, and the Ph.D degree in Electrical Engineering from University of Ulsan, Korea in 2006 From April 2001 to September
2004, he was a lecturer at Telecommunications Department, Ho Chi Minh City University of Technology During 2007–2011, he joined McGill University, Canada as a postdoctoral fellow Currently, he is an assistant professor at Ho Chi Minh City University of Technology His major research interests are modulation and coding techniques, MIMO system, digital signal processing, cooperative communications.