DSpace at VNU: Exact Outage Probability of Underlay Cognitive Cooperative Networks Over Rayleigh Fading Channels

Wireless Pers Commun (2013) 70:1001–1009
DOI 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 fading channels. The derivation considers the correlation among received signal-to-noise ratios,
two critical constraints (interference power constraint and maximum transmit power constraint), 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 underlay 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 capability 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 spectrum 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:

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K. Ho-Van

In 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 reducing 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 performance, 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 extensively 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 cooperative 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. 3 for performance evaluation. Finally, the paper
is concluded in Sect. 4.

2 Outage Probability Analysis
The underlay cognitive cooperative system model2 under 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 channel between the transmitter t and the receiver r experiences independent slowly varying flat
Rayleigh fading with variance 1/λtr . Hence, the channel gain gtr = |h tr |2 is an exponentially
distributed random variable with the probability density function (pdf) f gtr (x) = λtr e−λtr 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].

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Underlay Cognitive Cooperative Networks

1003

Fig. 1 System model
P

hSP
hSR
S
Phase 1

hRP
R

hSD

hRD

Phase 2

D

for x ≥ 0. Different from [13] where all λtr ’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 Pt of the transmitter t ∈ {S, R}
is imposed by two constraints [2]: interference power constraint, Pt ≤ gItTP , and maximum
transmit power constraint, Pt ≤ Pm , where IT is the maximum interference power level
that PU still operates reliably, and Pm is the maximum transmit power. In other words,
Pt ≤ min gItTP , Pm . Consequently, the actual transmit power can be lower than the maximum one (i.e., when gItTP ≤ Pm ), resulting in the coverage range of the secondary transmitter
in underlay cognitive networks less than that in other ones (e.g., interweave cognitive networks).
The received SNR at the receiver r is given by [2]
γtr = Pt

gtr
= min
N0

IT
, Pm
gt P

gtr
,
N0

(1)

where N0 is the noise variance at the receivers.
Let U be the transmission rate of secondary network. According to the Shannon information theory, the outage (or unsuccessful information decoding) occurs at the receiver r if
the received SNR γr meets the inequality U ≥ 21 log2 (1 + γr ) or γr ≤ k with k = 22U −1
where 21 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 maximum ratio combining (MRC) for restoring the source information. In this case, the received
SNR at D is γ S D +γ R D and an outage occurs if γ 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
Po = Pr {γ S D < k, γ S R < k} + Pr {γ S D + γ R D < k, γ S R > k} .
T1

(2)

T2

It is noted from (1) that the received SNRs at R and D in the first phase, γ S R =
min gISTP , Pm gNS0R and γ S D = min gISTP , Pm gNS 0D , 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., γtr = NIT0 ggttrP . 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.

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K. Ho-Van

To derive the closed forms of two terms, T1 and T2 in (2), we note that Pr {gtr < m} =
∞
λtr e−λtr x d x = 1 − e−λtr m , Pr {gtr > m} = m λtr e−λtr x d x = e−λtr m and the cumula-

m
0

tive density function of x R D = min
⎛

IT
gR P ,

−

Pm g R D can be borrowed from [2]
⎞
βλ
I
RD T
Pm

xe
Fx R D (x) = ⎝
− 1⎠ e −
x + β IT

λR D x
Pm

+ 1,

(3)

where β = λ R P /λ R D .
The derivative of Fx R D (x) results in the pdf of x R D as
⎞
⎛
βλ D I T
− RPm
βλ
I
λ

x
λR D x
R
D
T
R
D
β IT
λ R D ⎝ xe
e− Pm e− Pm −
− 1⎠ e− Pm .
f x R D (x) =
2
Pm
x + β IT
(x + β IT )

(4)

Due to γ R D = x R D /N0 , the pdf of γ R D is
f γ R D (x) = N0 f x R D (N0 x)
=

Ge−Gρ R D
(x + G)

2

e−ρ R D x +

Gρ R D e−Gρ R D −ρ R D x
e
x+G

−ρ R D e−Gρ R D − 1 e−ρ R D x ,
where ρ R D =

λ R D N0
Pm

and G =

(5)

λ R P IT
λ R D N0 .

2.1 Derivation of T1
With γ S D and γ S R having the form in (1), T1 is written as
T1 = Pr {γ S D < k, γ S R < k}
IT
gS D
gS R
IT
= Pr min
, Pm
< k, min
, Pm
gS P

N0
gS P
N0
⎧
⎫
⎨
⎬
N0 k
N0 k
, gS R <
= Pr g S D <
⎩
min gISTP , Pm
min gISTP , Pm ⎭
⎧
⎫ ⎧
⎫
∞
⎨
⎬ ⎨
⎬
N0 k
N0 k
= Pr g S D <
Pr g S R <
f gS P (x) d x
⎩
min IT , P ⎭ ⎩
min IT , P ⎭
0

∞

=

x

⎛

⎜
⎝1 − e

λ S D N0 k
−
I
min xT ,Pm

m

⎞⎛

⎟⎜
⎠ ⎝1 − e

λ S R N0 k
−
I
min xT ,Pm

⎞

x

⎟
⎠ f gS P (x) d x

0
∞

=

1−e

−

λ S D N0 kx
IT

1−e

−

λ S R N0 kx
IT

λ S P e−λ S P x d x

IT
Pm
IT

Pm

1 − e−

+
0

123

λ S D N0 k
Pm

1 − e−

λ S R N0 k
Pm

m

λ S P e−λ S P x d x


Underlay Cognitive Cooperative Networks

1005

= 1 − e−ρ S D k − e−ρ S R k + e−(ρ S D +ρ S R )k +
+

ρ S D ke−(ρ S D k+χ )

ρS 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 ]
.
−
ρS R k + χ
(ρ S D + ρ S R ) k + χ
λ S D N0
Pm , ρ S R

where ρ S D =

=

λ S R N0
Pm , χ

λ S P IT
Pm

=

(6)

.

2.2 Derivation of T2
Rewrite T2 as
T2 = Pr {γ S D + γ R D <k, γ S R >k}

IT
gS D
gS R
IT
= Pr min
, Pm
+ γ R D , Pm
>k
gS P
N0
gS P
N0
⎫
⎧
⎛
⎞
k
∞
⎬
⎨
−
y)
N
k
N
(k
0
0
, gS R >

= ⎝ Pr g S D <
f g S P (x) d x⎠ f γ R D (y) dy
⎩
min IT , P
min IT , P ⎭
0

0

⎛
k

∞

⎝

=
0
k

(y) dy =
0

0

⎢
⎣
⎡

k

=

⎡

⎧
⎨

Pr
0
∞

⎩

m

x

(k − y) N0

gS D <

min

⎛
⎜
⎝1 − e

−

⎫
⎬

IT
x

λ S D N0 (k−y)
IT
x ,Pm

min

, Pm ⎭
⎞
λ
⎟
⎠e

−

min

⎧
⎨

Pr

⎩

x

N0 k

gS R >
min

S R N0 k
IT
x ,Pm

m

IT
x

⎢
⎢ IT
⎢ Pm
⎢
⎢
⎣
+

1−e
IT
Pm

−

λ S D N0 (k−y)x

IT

1−e

−

λ S D N0 (k−y)
Pm

e

−

e

λ S R N0 kx
IT

−

, Pm ⎭
⎤

⎞

f g S P (x) d x ⎠ f γ R D (y)dy

⎥
λ S P e−λ S P x d x ⎦ f γ R D

0
∞

⎫
⎬

⎤
λS P

λ S R N0 k
Pm

e−λ S P x d x

λ S P e−λ S P x d x

⎥
⎥
⎥
⎥ f γ R D (y) dy.
⎥
⎦

(7)

0

For simplicity, we denote

A=

ρS D k + ρS R k + χ
,
ρS D

(8)

B =

χe−(ρ S R k+χ)
+ e−ρ S R k 1 − e−χ ,
ρS R k + χ

(9)

C =

χe−(ρ S D k+ρ S R k+χ)
,
ρS D

(10)

D = e−(ρ S D +ρ S R )k 1 − e−χ .

(11)

Case 1:λ R D = λ S D 3
(λ

−λ

)N

y

RD
SD 0
3 When inserting (5) into (7), there appear some integrals in the form of k e−
Pm
dy. Therefore,
0
in order to compute this integral, two cases (λ R D = λ S D and λ R D = λ S D ) must be considered.

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K. Ho-Van

Inserting f γ R D (x) in (5) into (7), we have
1
e−ρ R D k
−
G
k+G

T2 = BGe−Gρ R D
+

Dλ R D e−Gρ R D − 1
(λ R D − λ S D )

− B e−Gρ R D − 1

1 − e−ρ R D k

1 − e−(ρ R D −ρ S D )k

1
e−(ρ R D −ρ S D )k
−
G
k+G
⎞
RD
C Gρ R D e
C Ge−Gρ R D
+
+ DGρ R D e−Gρ R D
2
A+G
(A+G)
⎠ I2 (k, ρ R D − ρ S D , G)
−⎝
−Gρ
− DGe−Gρ R D + C GeA+G R D (ρ R D − ρ S D )

− DGe−Gρ R D +

⎛
−Gρ

+

C Ge−Gρ R D
(A+G)2

−Cρ R D

C Ge−Gρ R D
A+G

C Gρ R D e−Gρ R D
A+G
e−Gρ R D − 1

+

I1 (k, ρ R D − ρ S D , A) ,

(12)

where
k

I2 (k, ρ, G) =
0
k

I1 (k, ρ, A) =
0

e−ρy
dy = eρG (Ei (−ρk − ρG) − Ei (−ρG)) ,
y+G

(13)

e−ρy
dy = e−ρ A (Ei (ρ (A − k)) − Ei (ρ A)) .
y−A

(14)

Here the exponential integral function Ei(x) is defined in [14] as Ei (x) = − ∫∞
−x
is a built-in function in most computation softwares (e.g., Matlab).
Case 2: λ R D = λ S D
Again inserting f γ R D (x) in (5) into (7) and considering λ R D = λ S D , we have
1
e−ρ R D k
−
G
k+G

T2 = BGe−Gρ R D
−
+

+ DGe−Gρ R D +

C Ge−Gρ R D
A+G

e−t
t

dt which

1
1
−
k+G
G

k+G
C Ge−Gρ R D
C Gρ R D e−Gρ R D
+
+ DGρ R D e−Gρ R D ln
A+G
G
(A + G)2
C Ge−Gρ R D
(A + G)

2

+

−B e−Gρ R D − 1

C Gρ R D e−Gρ R D
− Cρ R D e−Gρ R D − 1
A+G
1 − e−ρ R D k + Dρ R D e−Gρ R D − 1 k,

ln

A−k
A

(15)

where ln(x) is the natural logarithm of x.

3 Illustrative Results
For illustration purpose, we randomly select λ 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 = 1 bps/Hz.
Figure 2 investigate the effect of IT on the outage performance. We fix Pm at 25 dB. It is shown
that analytical and simulated results4 are perfectly matched for both cases, confirming the accuracy
of the derived formula in (2). Additionally, the outage performance is improved with respect to the
4 108 channel realizations are generated to obtain simulated results.

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Underlay Cognitive Cooperative Networks

1007

0

10

Case 1: Analysis
Case 1: Simulation
Case 2: Analysis
Case 2: Simulation
−1

Outage probability

10

−2

10

−3

10

−4

10

0

5

10

15

20

IT (dB)
Fig. 2 Outage probability versus I T (Pm = 25 dB)

10

0

Outage probability

Case 1: Analysis
Case 1: Simulation
Case 2: Analysis
Case 2: Simulation

10

−1

10

−2

10

−3

0

5

10

15

20

25

30

Pm (dB)
Fig. 3 Outage probability versus Pm (I T = 15 dB)

increase in IT . This is obvious since IT imposes a constraint on the transmit power and the higher
is IT , the higher can the transmit power be, eventually enhancing communication reliability.
Figure 3 compares simulated and analytical results when Pm varies from 0 to 30 dB while IT is
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 Pm . This saturation phenomenon comes from the fact that the
transmit power is limited by the minimum of the maximum interference power level, IT , and the

maximum transmit power, Pm . As such, when Pm exceeds a certain value (e.g., around 15 dB in
Fig. 3), the transmit power is completely controlled by IT , resulting in the same outage probability
for any increase in Pm .

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K. Ho-Van

4 Conclusions
In this paper, the exact closed-form outage probability expression for underlay cognitive cooperative 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 maximum interference power level.

References
1. Yucek, T., & Arslan, H. (2009). A survey of spectrum sensing algorithms for cognitive radio
applications. IEEE Communications Surveys & Tutorials, 11, 116–130.
2. Lee, J., Wang, H., Andrews, J. G., & Hong, D. (2011). Outage probability of cognitive relay networks
with interference constraints. IEEE Transactions on Wireless Communications, 10, 390–395.
3. Nosratinia, A., Hunter, T. E., & Hedayat, A. (2004). Cooperative communication in wireless networks. IEEE Communications Magazine, 42, 74–80.
4. Laneman, J. N., Tse, D. N. C., & Wornell, G. W. (2004). Cooperative diversity in wireless networks:
Efficient protocols and outage behavior. IEEE Transactions on Information Theory, 50, 3062–3080.
5. Caijun, Z., Ratnarajah, T., & Kai-Kit, W. (2011). Outage analysis of decode-and-forward cognitive
dual-hop systems with the interference constraint in Nakagami-m fading channels. IEEE Transactions
on Vehicular Technology, 60, 2875–2879.
6. Lee, K., & Yener, A. (2006). Outage performance of cognitive wireless relay networks. In Proceeding

of IEEE GLOBECOM, Nov. 2006.
7. Zhang, Q., Jia, J., & Zhang, J. (2009). Cooperative relay to improve diversity in cognitive radio
networks. IEEE Communications Magazine, 47(2), 111–117.
8. Zhang, W., & Letaief, K. B. (2009). Cooperative communications for cognitive radio networks. Proceedings of IEEE, 97(5), 878–893.
9. Guo, Y., Kang, G., Zhang, N., Zhou, W., & Zhang, P. (2010). Outage performance of relay-assisted
cognitive-radio system under spectrum-sharing constraints. Electronics Letters, 46(2), 182–183.
10. Sagong, S., Lee, J., & Hong, D. (2011). Capacity of reactive DF scheme in cognitive relay networks. IEEE
Transactions on Wireless Communications, 10(10), 3133–3138.
11. Choi, M., Park, J., & Choi, S. (2011). Low complexity multiple relay selection scheme for cognitive
relay networks. In Proceeding of IEEE VTC, Fall (pp. 1–5).
12. Liping, L., Ping, Z., Guangchi, Z., & Jiayin, Q. (2011). Outage performance for cognitive relay
networks with underlay spectrum sharing. IEEE Communications Letters, 15, 710–712.
13. Yan, Z., Zhang, X., & Wang, W. (2011). Exact outage performance of cognitive relay networks with
maximum transmit power limits. IEEE Communications Letters, 15(12), 1317–1319.
14. Gradshteyn, I. S., & Ryzhik, I. M. (2007). Table of integrals, series, and products, 7th Ed. New
York: Academic Press.

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Author 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.

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