DSpace at VNU: On the Performance of Opportunistic Relay Selection in Cognitive Radio Networks with Primary User''s Interference and Direct Channel

Wireless Pers Commun
DOI 10.1007/s11277-016-3464-9

On the Performance of Opportunistic Relay Selection
in Cognitive Radio Networks with Primary User’s
Interference and Direct Channel
Khuong Ho-Van1 • Lien Pham-Hong2 • Son Vo-Que1
Tra Luu-Thanh1

•

Ó Springer Science+Business Media New York 2016

Abstract This paper analyzes outage performance of opportunistic relay selection in
cognitive radio networks with primary user’s interference and direct channel over independent non-identically distributed fading channels and under maximum transmit power
constraint and interference power constraint. Exact analysis is firstly proposed and then
extended to asymptotic analysis to have insights into important performance metrics such
as coding gain and diversity order. The proposed analysis can also be applied to corresponding analysis for opportunistic relay selection in cognitive radio networks with PU’s
interference and without direct channel to highlight the usefulness of direct channel in
relaying communications without any sacrifice of power and bandwidth. A multitude of
results illustrate an achievable full diversity order, a considerable system performance
deterioration due to primary user’s interference but this deterioration can be drastically
remedied by increasing the number of involved relays, and the superiority of opportunistic
relay selection with direct channel to that without it.
Keywords Opportunistic relay selection Á Cognitive radio Á Interference Á Direct channel

1 Introduction
Currently, inefficient traditional spectrum allocation by means of fixed primary users
(PUs), emergence of several new wireless applications (e.g., video calling, file transferring,
high-definition video streaming, and high-speed internet access), and a severe spectrum

& Khuong Ho-Van

1

Department of Telecommunications Engineering, HoChiMinh City University of Technology, 268
Ly Thuong Kiet Str., District 10, HoChiMinh City, Vietnam

2

Department of Computer and Communication Engineering, HoChiMinh City University of
Education Technology, 1 Vo Van Ngan Str., Thu Duc District, HoChiMinh City, Vietnam

123


K. Ho-Van et al.

under-utilization as reported in an extensive survey on frequency spectrum utilization
carried out by the Federal Communications Commission induce radio spectrum to become
more and more scarce [1]. Consequently, devising novel technologies to relieve the
pressure of the spectrum scarcity is urgent and essential. The cognitive radio technology
(e.g., [2]) is one among them, which efficiently resolves this problem by allowing secondary users (SUs) to opportunistically utilize the spectrum inherently allotted to PUs as
long as communications of PUs is not damaged.
Communications of PUs is guaranteed at an acceptable degree as long as the interference from SUs is controlled. To this end, SUs usually operates in three modes: interweave,
overlay, and underlay [3]. Due to its distinct feature of low implementation complexity, the
underlay mode has recently gained much interest, e.g., [4–19] and citations therein. This
mode properly allocates the transmit power of SUs to control the interference to PUs in
either short-term or long-term manner. According to the long-term mechanism, the
transmit power of SUs is allocated to meet the pre-determined outage probability of PUs
[4–10] while according to the short-term mechanism, the transmit power of SUs must

satisfy either interference power constraint [11–13] or both interference power constraint
and maximum transmit power constraint [14–19]. It is the limitation of transmit power of
SUs in the underlay mode that significantly shortens the transmission coverage of SUs. To
overcome this drawback, relaying communications techniques has recently been integrated
into SUs to exploit short-range point-to-point communications for low path-loss [20]. In
relaying communications, communications between the source and the destination can be
assisted by multiple relays operated in either amplify-and-forward (AF) or decode-andforward (DF) manner [21] for high performance but low bandwidth efficiency due to the
requirement of orthogonal channels for different relays in order to avoid mutual interference. Therefore, the relay selection in which a single relay among all potential candidates
is selected is preferred to optimize system resource utilization, such as power and bandwidth, in comparison with multi-relay assisted transmission while remaining the same
diversity order [18, 22].
Several relay selection schemes in cognitive radio networks1 were proposed in
[9–14, 16, 18, 19, 33–41]. To be more specific, the opportunistic relay selection was
proposed in [9–11, 18, 33–35] in which the relay with the maximum end-to-end signal-tonoise ratio (SNR) is selected; the reactive relay selection scheme, which chooses the relay
among all potential candidates (i.e., all relays are presumed to correctly recover source
information) with the largest SNR to the destination, was investigated in
[12–14, 16, 34, 36]; the Nth best-relay selection and the Lth-worst relay selection were
proposed in [19] and [37], respectively; the relay selection scheme for maximizing secrecy
capacity was studied in [38]; the relay selection scheme with the compromise between the
gain for SUs and the loss for PUs was suggested in [39]; the partial relay selection, which
simply selects the relay with the largest SNR from the source, was analyzed in [40, 41].
However, several assumptions have been imposed on these works for analysis simplicity:
(1) independent partially-identical (i.p.i) [9–11, 13, 18, 31, 33, 34, 37, 39–41] or independent identical (i.i.) fading distributions [14, 16, 19, 36]; (2) no interference from PUs;
(3) no direct channel; (4) un-correlation among received SNRs.
Among relay selection schemes, the opportunistic relay selection is theoretically proved
to be optimal (e.g., [18]), and hence, it is interesting to evaluate its information-theoretic
1

This paper focuses on the DF relays in cooperative cognitive networks and hence, relay selection schemes
with the AF relays (e.g., [23–28]) or in dual-hop cognitive networks (e.g., [29–32]) are not necessarily
surveyed.

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On the Performance of Opportunistic Relay Selection in…

performance limit (i.e., outage probability) under practical conditions such as both interference and maximum transmit power constraints, i.n.i.d. fading channels, presence of
PU’s interference and direct channel, correlation among Signal-to-Interference plus Noise
Ratios (SINRs)2. This is the objective of this paper. To the best of our knowledge, no
analysis accounts for all these practical conditions. The current work presents the following
contributions:
• Propose an exact closed-form outage probability expression for the opportunistic relay
selection in cognitive radio networks over i.n.i.d. Rayleigh fading channels and under
both maximum transmit power and interference power constraints, the presence of
PU’s interference and direct channel, and correlation among received SINRs. The
proposed expression is helpful in fast evaluating the system performance without timeconsuming simulations.
• Extend the proposed exact analysis to asymptotic analysis to obtain important
performance metrics (e.g., diversity order and coding gain), which proves that the
opportunistic relay selection achieves the full diversity order offered by all involved
relays and the source.
• Propose a corresponding analysis for the opportunistic relay selection with PU’s
interference and without direct channel for the convenience of comparing the
opportunistic relay selection with and without direct channel as well as emphasizing the
importance of the direct channel in relaying communications. It is recalled that no
matter whether the direct channel is utilized for signal combining at the destination, the
system resource utilization such as power and bandwidth is almost unchanged. As such,
it is essential to investigate how much gain the direct channel can contribute to
performance improvement of the opportunistic relay selection.
• Provide numerous results to expose useful insights into the system performance such as
diversity order, coding gain, substantial performance enhancement with respect to the

increase in the number of relays, considerable system performance degradation owing
to the PU’s interference, and superiority of cooperative relaying (i.e., relaying
communications with direct channel) to its dual-hop counterpart (i.e., relaying
communications without direct channel).
This paper is structured as follows. The next section presents the system model under
investigation. Exact and asymptotic analysis for the opportunistic relay selection with/
without direct channel is elaborately discussed in Sect. 3. Section 4 provides numerous
results to corroborate the proposed analysis and illustrates the outage behavior of the
opportunistic relay selection in key system parameters. Finally, useful conclusions close
the paper in Sect. 5.

2 System Model
Figure 1 demonstrates a system model for the opportunistic relay selection in cognitive
radio networks with PU’s interference and direct channel. In the secondary network, the
source Ss communicates the destination Sd with the assistance of the selected relay Sb in the
group of K relays, S ¼ fS1 ; S2 ; . . .; SK g. We assume that secondary transmitters operate in
the underlay mode (e.g., [5, 15, 17, 18]), and hence, the mutual interference between the
2

As will be shown in the next section, both PU’s interference and direct channel induce received SINRs to
be correlated, making the analysis more complicated but more general and practical.

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K. Ho-Van et al.
Primary transmitter

Primary receiver

Pt

Pr
Transmission
Interference

S1
Sb

Sd
Stage 2

Ss

SK
Stage 1

Secondary network

Fig. 1 System model

primary network and the secondary network is available. In other words, Ss and Sb interfere
communications between the primary transmitter Pt and the primary receiver Pr , and Pt
also causes interference to the received signals at relays and Sd . It is recalled that the
interference from the primary network to the secondary network was neglected for analysis
simplicity (e.g., [6, 8, 10, 12–19, 24, 31, 33, 40, 42–47] and citations therein). It is the
interference from PUs that makes the performance analysis complicated but general and
practical. Additionally, it is apparent that two stages of the opportunistic relay selection in
the secondary network can take place instantaneously with communications of two different primary transmitter-receiver pairs. Nevertheless, in order to have a compact figure,
Fig. 1 only illustrates one transmitter-receiver pair. However, this paper still reflects this

general case by assuming two different primary transmitter-receiver pairs throughout the
following analysis (i.e., there are two different channel coefficients from Pt to Sd for two
corresponding stages, namely htd1 and htd2 ). Towards this end, channel coefficients are
shown in Table 1.
Wireless channels are assumed to be independent, frequency-flat, and Rayleigh-distributed. Therefore, the channel coefficient, hpq , between the transmitter p and the receiver
q can be modelled as a circular symmetric complex Gaussian random variable with zero
mean and 1=kpq -variance, i.e., hpq $ CN ð0; 1=kpq Þ. In contrast to existing works in relay
selection where i.p.i.3 or i.i.4 fading distributions are assumed for simplicity of performance analysis, this paper investigates i.n.i.d. fading channels, and so, all kpq ’s, 8fp; qg are
not necessarily equal, making our work more general and practical.

3
That means that kpq ’s are partitioned into groups of equal value. For instance, ksk ’s, kkd ’s, kkr ’s, ktk ’s with
k 2 R are assumed to be equal in [9–11, 13, 18, 31, 40–43].
4

That means that kpq ’s, 8fp; qg are equal in [14, 16, 19, 33].

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On the Performance of Opportunistic Relay Selection in…
Table 1 Notations for channel
coefficients R ¼ f1; . . .; Kg

Notation

Channel coefficient between

hsd $ CN ð0; 1=ksd Þ

Ss and Sd in the stage 1

hsk $ CN ð0; 1=ksk Þ

Ss and Sk in the stage 1, k 2 R

hsr $ CN ð0; 1=ksr Þ

Ss and Pr in the stage 1

htk $ CN ð0; 1=ktk Þ

Pt and Sk in the stage 1, k 2 R

htd1 $ CN ð0; 1=ktd1 Þ

Pt and Sd in the stage 1

hkr $ CN ð0; 1=kkr Þ

Sk and Pr in the stage 2, k 2 R

hkd $ CN ð0; 1=kkd Þ

Sk and Sd in the stage 2, k 2 R

htd2 $ CN ð0; 1=ktd2 Þ

Pt and Sd in the stage 2

As demonstrated in Fig. 1, the opportunistic relay selection takes place in two stages. In
the stage 1, Ss broadcasts the signal ws with the power P s (i.e., P s ¼ E ws fjws j2 g where
E X fxg denotes the expectation operator) while Pt is concurrently transmitting the signal wt1
with the power P t1 ¼ E wt1 fjwt1 j2 g. The signals from Ss and Pt cause the mutual interference between the primary network and the secondary network. To this effect, the received
signals at Sk and Sd , correspondingly, can be modeled as
zsk ¼ hsk ws þ htk wt1 þ nsk ;

ð1Þ

zsd ¼ hsd ws þ htd1 wt1 þ nsd ;

ð2Þ

where zpq is the signal received at Sq from Sp and npq $ CN ð0; N0 Þ is the additive white
Gaussian noise (AWGN) at the receiver Sq .
It immediately follows from (1) and (2) that the SINRs at Sk and Sd in the stage 1 can be
represented, correspondingly, as
csk ¼

csd ¼

P s jhsk j2
P t1 jhtk j2 þ N0
P s jhsd j2
P t1 jhtd1 j2 þ N0

;

ð3Þ

:

ð4Þ

In the stage 2, the selected relay, namely Sb , decodes the source signal and re-encodes the
decoded information before forwarding to Sd . Generally, the received signal at Sd from the
relay Sk in the stage 2 can be expressed as
zkd ¼ hkd wk þ htd2 wt2 þ nkd ;

ð5Þ

where wk is the signal transmitted by Sk with the power P k ¼ E wk fjwk j2 g and interfered by
the signal wt2 transmitted by Pt with the power P t2 ¼ E wt2 fjwt2 j2 g (i.e., two different
primary transmitters are assumed for two stages to remain the system model general).
This paper considers the opportunistic relay selection (e.g., [18, 42]), which selects the
relay Sb with the largest SINR over the relaying channel Ss À Sb À Sd . Mathematically, the
SINR generated from this relaying channel can be represented as
csbd ¼ maxðminðcsk ; ckd ÞÞ;
k2R

ð6Þ

where ckd is computed from (5) as

123


K. Ho-Van et al.

ckd ¼

P k jhkd j2
P t2 jhtd2 j2 þ N0

:

ð7Þ

It is recalled that the opportunistic relay selection can be implemented in a distributed manner
using the timer method in [22] where each relay Sk sets its timer with the value that is inversely
proportional to minðcsk ; ckd Þ and the relay with the timer that runs out first is selected.
For further performance enhancement at almost no cost of power and bandwidth, we
take advantage of the direct channel between Ss and Sd , which is different from existing
works (e.g., [18, 42]) where this channel is neglected for analysis simplicity. To combine
both signals from the relaying channel and the direct channel at Sd , the selection combining
is assumed for low complexity as compared to the maximum ratio combining [48]. To this
end, the end-to-end SINR at Sd is expressed as
ce2e ¼ maxðcsd ; csbd Þ:

ð8Þ

In the underlay mode, the secondary transmitter Sp must control its transmit power such

that the interference induced at PUs does not exceed the maximum interference power, I,
2

that PUs can tolerate, i.e., P p I =jhpr j . Additionally, each SU is designed with the
 As such, in order to meet both the interference power
maximum transmit power, P.
constraint and the maximum transmit power constraint as well as to maximize the trans

mission coverage, the transmit power of Sp should be set as P p ¼ minðI=jhpr j2 ; PÞ.
The system model under consideration is more general and practical than existing works
(e.g., [18, 42]) by accounting for both direct channel and PU’s interference, and i.n.i.d.
fading channels. These additional factors also create correlation among received SINRs,
which was neglected in the analysis of [18, 42], and it is this correlation that makes the
analysis of this paper more complicated. Specifically, the proposed system model induces
the following correlations:
 they are correlated. This
• Since csd and csk have a common term P s ¼ minðI=jhsr j2 ; PÞ,
leads to the correlation among quantities minðcsk ; ckd Þ in (6), and the correlation
between csd and csbd in (8). It is noted that both [18] and [42] assumed un-correlation
among quantities minðcsk ; ckd Þ.
• Correlation among minðcsk ; ckd Þ in (6) is also caused by the fact that ckd ’s in (7) have a
common term htd2 . This correlation is apparently present due to the PU’s interference,
which is not available in [18, 42].
These correlations can be broken down by using the conditional probability concept, which
is very useful for the analysis of the next section.

3 Performance Analysis
The derivation of the exact closed-form outage probability expression at Sd for the
opportunistic relay selection in cognitive radio networks with PU’s interference and direct
channel is firstly proposed in this section, which is then used for asymptotic analysis to
expose important performance metrics such as diversity order and coding gain. Since the
proposed analysis framework is relatively general, it is straightforwardly extended to
corresponding analysis for the opportunistic relay selection with PU’s interference and
without direct channel to illustrate the advantage of utilizing the direct channel in relaying
communications without exhaustive simulations.

123

On the Performance of Opportunistic Relay Selection in…

3.1 Exact Analysis
The outage probability is defined as the probability that ce2e is below a threshold c0 , i.e.,
PCC
c0 g where c0 ¼ 22s À 1 with s being the required transmission rate and
o ¼ Prfce2e
PrfX g is the probability of the event X. Since ce2e contains two common quantities,
x ¼ jhsr j2 and y ¼ jhtd2 j2 , which cause correlation among received SINRs as discussed in
Sect. 2, PCC
o must be evaluated in terms of conditional probabilities, i.e.,
PCC
o ¼ E x;y fPrfce2e

c0 jx; ygg
¼ E x;y fPrfmaxðcsd ; csbd Þ c0 jx; ygg
8
9
j
>
<
=
zfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl{>
¼ E x;y Prfcsd c0 jxg Prfcsbd c0 jx; yg :
>
>
:|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
;

ð9Þ

g

À
Á
Since hpq $ CN 0; 1=kpq , the probability density function (pdf) and the cumulative dis 2
tribution function (cdf) of hpq  are represented as f h 2 ðzÞ ¼ kpq eÀkpq z and F h 2 ðzÞ ¼
j pq j
j pq j
1 À eÀkpq z for z ! 0, respectively. As such, it immediately follows that
(

 )


c0 x


P s jhsd j2

g ¼ Pr

P t1 jhtd1 j2 þ N0

  9
8
<
P t1 jhtd1 j2 þ N0 c0  =
x

¼ Pr jhsd j2
 ;
:
Ps


 
99
8 8
2
==
< <
P t1 jhtd1 j þ N0 c0 
2
2
jhtd1 j ; x
¼ E jhtd1 j2 Pr jhsd j

;;
: :
Ps

 
8
9
0
1
<
=
P t1 jhtd1 j2 þ N0 c0 

jhtd1 j2 ; xA
¼ E jhtd1 j2 Fjhsd j2 @

:
;
Ps


(
)
2
ðPt1 jhtd1 j þN0 Þksd c0 
2
Ps
¼ E jhtd1 j2 1 À eÀ
jhtd1 j ; x


ð10Þ

¼ 1 À Td ;

where

)
ðPt1 jhtd1 j2 þN0 Þksd c0 
2
Ps
e
jhtd1 j ; x



(

À

Td ¼ E jhtd1 j2
¼

Z1

eÀ

ðPt1 zþN0 Þksd c0
Ps

fjhtd1 j2 ðzÞdz

0

¼

ð11Þ

Z1
e

À

ðPt1 zþN0 Þksd c0

Ps

Àktd1 z

ktd1 e

dz

0

ktd1 eÀ

¼ P

t1 ksd c0

Ps

N0 ksd c0
Ps

þ ktd1

:

123


K. Ho-Van et al.

Similarly, the j term in (9) can be written as
 '
&

j ¼ Pr maxðminðcsk ; ckd ÞÞ c0 x; y
k2R
Y
¼
Prfminðcsk ; ckd Þ c0 jx; yg
k2R

¼

Y

k2R

¼

ð12Þ

ð1 À Prfminðcsk ; ckd Þ ! c0 jx; ygÞ
1

0

YB
C
@1 À Prfcsk ! c0 jxg Prfckd ! c0 jyg A:
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

k2R
ak

bk

The ak term in (12) can be computed in the same manner as (10), i.e.,
ak ¼ Tk ;

ð13Þ

where
ktk eÀ
Tk ¼ P k c
t1 sk 0

Ps

N0 ksk c0
Ps

þ ktk

ð14Þ

:

Meanwhile, after using (7), the bk term in (12) should be rewritten as
 '
&
ðP t2 y þ N0 Þc0 

bk ¼ Pr jhkd j2 !
y :
Pk



Since P k ¼ min yIk ; P with yk ¼ jhkr j2 , the bk term is further simplified as
 '
&
ðP t2 y þ N0 Þc0 
bk ¼ 1 À Pr jhkd j2
y
Pk

& &
''
ðP t2 y þ N0 Þc0 
;
y
y
¼ 1 À E yk Pr jhkd j2
 k
Pk

&
 
'
ðP t2 y þ N0 Þc0 
¼ 1 À E yk Fjhkd j2
yk ; y

Pk
& k ðP yþN Þc 
'
À kd t2P 0 0 
k
¼ E yk e
y k ; y
 
(
)


Àkkd ðPt2 yþN0 Þc0 = min yI ;P 
k
¼ E yk e
y k ; y

¼

Z1
e

À

kkd ðPt2 yþN0 Þc0
yk
I

fjhkr j2 ðyk Þdyk þ

l

¼

eÀ

kkd ðPt2 yþN0 Þc0
P

ð16Þ

fjhkr j2 ðyk Þdyk

0

Z1
e

À

kkd ðPt2 yþN0 Þc0
yk
I

l

¼ Ek eÀDk y

123

Zl



Àkkr yk

kkr e


Bk
;
1þ
y þ Ck

dyk þ

Zl
0

eÀ

ð15Þ

kkd ðPt2 yþN0 Þc0
P

kkr eÀkkr yk dyk


On the Performance of Opportunistic Relay Selection in…

where
Ak ¼ 1 À eÀkkr l
kkr IeÀkkr l
Ak P t2 kkd c0
kkr I þ kkd c0
Ck ¼
P t2 kkd c0
P t2 kkd c0
Dk ¼
P
Bk ¼

E k ¼ Ak eÀ
I
I¼
N0
P
P¼
N0
P t2
P t2 ¼
N0
I
l¼ :
P

kkd c0
P

ð17Þ

Inserting (13) and (16) into (12) and then (10) and (12) into (9), one obtains the compact
form of PCC
o as
(
)
Y
CC
Po ¼ E x;y ð1 À Td Þ
ð1 À bk Tk Þ :
ð18Þ
k2R

Using the fact that
Y

ð1 À ak Þ ¼ 1 þ

k2R

K À1
X

ðÀ1Þu

u¼1

KÀuþ1
X KÀuþ2

X

ÁÁÁ

s1 ¼1 s2 ¼s1 þ1

K
X

Y

su ¼suÀ1 þ1 k2A

ak þ ðÀ1ÞK

Y

ak ;

ð19Þ

k2R

where A ¼ fR½s1 Š; R½s2 Š; . . .; R½su Šg,5 to expand the product in (18), one obtains
PCC
o ¼ Y ; G; þ

KÀ1
X
u¼1

ðÀ1Þu

KÀuþ1
X KÀuþ2
X

K
X

ÁÁÁ

s1 ¼1 s2 ¼s1 þ1

Y A GA þ ðÀ1ÞK Y R GR ;

ð20Þ

su ¼suÀ1 þ1

where B ¼ f;; A; Rg with ; denoting the empty set, and
(
)
Y
Y B ¼ E x ð1 À Td Þ
Tk ;

ð21Þ

k2B

(
GB ¼ E y

Y

)
bk :

ð22Þ

k2B

It is apparent that the derivation of the exact closed-form expression of PCC
o is completed
after solving Y B and GB .
5

R½jŠ denotes the jth element of the set R.

123


K. Ho-Van et al.

Theorem 1 The exact closed-form representation of Y B is given by
Y B ¼ Y^B À Y^fB;dg ;

ð23Þ

where L ¼ B or L ¼ fB; dg, and
!
k c
X
Y ktk PeÀ skP 0
Y ktk I
ksr
mk nðl; ML ; lk Þ þ As
;
Y^L ¼
P k c
P k c þ ktk P
k2L t1 sk 0
k2L
k2L t1 sk 0

ð24Þ

with As ¼ 1 À eÀksr l being defined in (17) and
lk ¼

ktk I
P t1 ksk c0
c0 X
ksk
I k2L

ML ¼ ksr þ
Y À

mk ¼

ð25Þ

lj À lk

ÁÀ1

ð26Þ
ð27Þ

;

j2Lnk

nða; b; cÞ ¼ À ebc EiðÀb½a þ cŠÞ:
P t1 ¼

ð28Þ

P t1
:
N0

ð29Þ

In (28), EiðÁÞ is the exponential integral function in [49, eq. (8.211.1)], which is a built-in
function in most computation software (e.g., Matlab).
Proof

We further decompose Y B in (21) as
(
)
(
)
Y
Y
YB ¼ Ex
Tk À E x Td
Tk ;
k2B

ð30Þ

k2B

where each term in (30) has a common form as
(
)
Y
^
Tk :
YL ¼ Ex

ð31Þ

k2L

By inserting (31) into (30), we see that (30) perfectly matches (23). Consequently, in order
to complete the proof, we should prove that (31) is represented in closed-form as (24).

Towards this end, we firstly plug (14) into (31), and then perform some basic manipulations to simplify (31) as
Y^L ¼

Z1
ksr e
l

¼

Àksr x

Y ktk eÀ
k2L

Pt1 ksk c0
I

N0 ksk c0
x
I

x þ ktk

dx þ

Zl

Àksr x

ksr e

0

k2L

Pt1 ksk c0
P

N0 ksk c0
P

þ ktk

dx

! Z1
k c
Y ktk I
Y 1
Y ktk PeÀ skP 0
ÀML x
ksr
e
dx þ As
;
P k c
x þ lk
P k c þ ktk P
k2L t1 sk 0
k2L
k2L t1 sk 0

l

where lk and ML are defined in (25) and (26), respectively.

123

Y ktk eÀ

ð32Þ


On the Performance of Opportunistic Relay Selection in…

Q
1
Finally, using the partial fraction expansion to decompose the k2L xþl
product in (32),
k
one obtains
!
1
k c
X Z eÀML x
Y ktk PeÀ skP 0
Y ktk I
^
ksr
;
ð33Þ
YL ¼

mk
dx þ As
P k c
x þ lk
P k c þ ktk P
k2L t1 sk 0
k2L
k2L t1 sk 0
l

where mk is defined in (27).
The last integral in (33) are represented in closed-form as (28) by firstly changing the
integral variable and then using the definition of EiðÁÞ. Given (28), (33) matches (24),
completing the proof.
h
Theorem 2 GB can be represented in closed-form as
!
!
BjÀ1 jBX
jX
jÀcþ1 jBX
jÀcþ2
jBj
Y
X
U; þ
Ek
ÁÁÁ
UC þ UB ;
GB ¼

c¼1

k2B

j1 ¼1

j2 ¼j1 þ1

where C ¼ fB½j1 Š; B½j2 Š; . . .; B½jc Šg, D ¼ f;; C; Bg, and
!
X
Y
Bk ktd2
vk eCk HB EiðÀCk HB Þ;
UD ¼ À
k2D

ð34Þ

jc ¼jcÀ1 þ1

ð35Þ

k2D

with
vk ¼

Y À

Cj À Ck

j2Dnk

HB ¼ ktd2 þ

X

ÁÀ1

;
ð36Þ

Dk :

k2B

and jBj denoting the cardinality of the set B.
Proof

Inserting (16) into (22) and after some algebraic manipulations, one obtains
! ( P

)
Ày
Dk Y
Y
B
k
GB ¼

Ek E y e k2B
1þ
:
ð37Þ
y þ Ck
k2B
k2B

Q 
Bk
product, it immediately
Applying (19) once again to decompose the k2B 1 þ yþC
k
follows that (37) coincides (34) where
! ( P
Ày
Dk Y
Y
UD ¼
Bk E y e k2B
k2D

)
1
:
y þ Ck
k2D

ð38Þ

Consequently, in order to complete the proof, we firstly need to evaluate (38) and then
prove that the result of this evaluation is exactly (35). To this effect, using the partial
Q
1
fraction expansion for the k2D yþC
product in (38), one obtains
k

123


K. Ho-Van et al.

UD ¼

Y

! (

Ày

P

Bk E y e

k2D

¼

Y

!
Bk

k2D

¼

Y

!
Bk

¼

k2D

X

Bk ktd2

)

>
: y þ Ck >
;
P

Z1
vk

k2D

!

vk
k2B
y
þ
Ck
k2D
8 P 9
Ày
D
>
<e k2B k >
=

vk E y

k2D

k2D

Y

X

X

Dk

Ày

Dk

ð39Þ

e k2B
ktd2 eÀktd2 y dy
y þ Ck

0

X

Z1
vk

k2D

eÀHB y
dy;
y þ Ck

0

where vk and HB are defined in (36).
With the aid of [49, Eq. (358.4)], the last integral in (39) is solved in closed-form as
R1 eÀHB y

C k HB
EiðÀCk HB Þ. Inserting this result into (39), one can see that (39) is in a
yþCk dy ¼ Àe
0

h

perfect agreement with (35), completing the proof.

Substituting (23) and (34) into (20), one obtains the exact closed-form outage probability of the opportunistic relay selection in cognitive radio networks with PU’s interference and direct channel. It is worth emphasizing that although the opportunistic relay
selection was discussed (e.g., [18, 42]), its exact closed-form outage probability expression
in (20) has not been reported in any open literature for the general case of i.n.i.d. fading
channels, selection combining, PU’s interference, direct channel, and correlation among
received SINRs. On contrary, [18] and [42] consider i.p.i. fading channels, no direct
channel, no PU’s interference, and un-correlation among received SNRs. To the best of the
authors’ knowledge, (20) is totally novel and represented in a very convenient and compact
form for the analytical evaluation. In addition, derivations in this section are relatively
general, and thus, can be used for the corresponding analysis of the opportunistic relay
selection in cognitive radio networks with PU’s interference and without direct channel.
Indeed, the outage probability of such a system model can be represented as
PDH
o ¼ Prfcsbd

ð40Þ

c0 g:

By following the same procedure of deriving PCC
o in (9) with setting csd ¼ 0, it immediately follows that
^

PDH
o ¼ Y ; G; þ

KÀ1
X
u¼1

ðÀ1Þu

KÀuþ1
X KÀuþ2
X
s1 ¼1 s2 ¼s1 þ1

ÁÁÁ

K
X

Y^A GA þ ðÀ1ÞK Y^R GR :

ð41Þ

su ¼suÀ1 þ1

It is noted that (41) is still novel since it accounts for PU’s interference, i.n.i.d. fading
channels, and correlation among received SINRs. Therefore, the analysis in [18] and [42]
where no PU’s interference, i.p.i. fading distributions, and un-correlation among received
SNRs are assumed is only a special case of (41). Additionally, it is interesting to discover
DU

that since Y^B [ Y B , PCC
o \Po . In other words, utilizing the direct channel is always
beneficial without any significant sacrifice of system resources such as power and
bandwidth.

123


On the Performance of Opportunistic Relay Selection in…

3.2 Asymptotic Analysis
The coding gain and the diversity order are important performance metrics characterizing a
relay selection scheme. The former demonstrates the performance gap between the coded
system and the uncoded one while the latter shows how fast the outage probability reduces
with respect to the SNR. To analyze them, we should investigate the outage performance in
the high SNR regime, i.e., P ! 1.
x!0

Using the
that eÀqx % 1 À qx, where q is a positive constant, and
Àl fact
Á
P s ¼ P min x ; 1 , one can approximate the third equality in (10) as

9
8

=
< P t1 jhtd1 j2 þ N0 ksd c0 
P!1

jhtd1 j2 ; x
g % E jhtd1 j2

;
:
Ps

¼

Z1 
ðP t1 z þ N0 Þksd c0
ktd1 eÀktd1 z dz
Ps

ð42Þ

0

¼

Qsd c0
À Á;
P min lx ; 1

where

Qsd ¼


P t1

þ 1 ksd :
ktd1

ð43Þ



Imitating (42) with P k ¼ P min ylk ; 1 , one can also approximate (13) and (16) as
P!1

ak % 1 À

Rsk c0
À Á;
P min lx ; 1

ð44Þ

and

&
'
kkd ðP t2 y þ N0 Þc0 
P!1
yk ; y
bk % E yk 1 À

Pk
1
0

1
Z
Zl
 t2 y þ N0 Þc0
k
ð
P
P!1
C
B kkd ðP t2 y þ N0 Þc0 yk
kd
kkr eÀkkr yk dyk þ
kkr eÀkkr yk dyk A
% 1À@
I
P
l

ð45Þ

0

Mkd ðP t2 y þ 1Þc0
;
% 1À
P

P!1

where




P t1
þ 1 ksk ;
ktk
 Àkkr l

e
þ 1 kkd :
¼
kkr l

Rsk ¼
Mkd

ð46Þ

Plugging (44) and (45) into (12), and then (42) and (12) into (9), one obtains the
approximation of PCC
o in the high SNR regime as

123


K. Ho-Van et al.

P!1
PCC
% E x;y

o

P!1

% E x;y

(
(

"
#
!!)
Qsd c0 Y
Rsk c0
Mkd ðP t2 y þ 1Þc0
À Á
À Á 1À
1À 1À
P
P min lx ;1 k2R
P min lx ;1
Qsd c0 Y
Rsk c0
M ðP y þ 1Þc0 Rsk Mkd ðP t2 y þ 1Þc20
Àl Á
Àl Á þ kd t2
À Á
À
P
P min x ;1 k2R P min x ;1

P 2 min lx ;1

!)
:
ð47Þ

By eliminating the high-order term in the above product, one can further approximate (47)
as
(
 Kþ1
) Y
Y
Y
1
y
CC P!1 c0
K
À
Á
P t2 Qsd E x;y
Gk
1þ
Mkd ;
ð48Þ
Po %
Gk
P
min lx ; 1 k2R k2R
k2R
where

!
1
Rsk
À Áþ1 :
Gk ¼
ð49Þ
P t2 Mkd min lx ; 1


Q
Utilizing (19) once again to decompose the k2R 1 þ Gyk product in (48), one can
represent (48) in closed-form as
P!1
PCC
%
o

!
 Kþ1
KÀ1 KÀuþ1
K
Y
X KÀuþ2
X
X
X
c0
K
P t2 Qsd W ; V ; þ
ÁÁÁ

WAVA þ WRVR
Mkd ;
P
u¼1 s1 ¼1 s2 ¼s1 þ1
su ¼suÀ1 þ1
k2R
ð50Þ

where B ¼ f;;A;Rg and
È

jB j

WB ¼ Ey y

É

¼

Z1

Àj B j

yjBj ktd2 eÀktd2 y dy ¼ jBj!ktd2 ;

ð51Þ

0

9

8 Q
Gk >
>
=
<
k2B
Àl Á ;
VB ¼ Ex
>
;
:min x ; 1 >

ð52Þ

with B ¼ RnB.
To complete the approximation of PCC
o in the high SNR regime, we must evaluate (52).
Towards this end, inserting (49) into (52), one obtains
(
!)
Y
1
Rsk
ÀjBj
À Á
À Á
V B ¼ P t2 E x
:
ð53Þ
1þ

min lx ; 1 k2B
Mkd min lx ; 1
Using (19) to decompose the above product, one can express (53) in closed-form as
0
1
BjÀ1 jBjÀcþ1 jBjÀcþ2
jX
jBj
X X
X
ÀjBj
V B ¼ P t2 @M; þ
ÁÁÁ
MC þ MBA:
ð54Þ
c¼1

j1 ¼1

j2 ¼j1 þ1

È
É
È
É
 B , and
 ¼ ;; C;
where C ¼ B½j1 Š; B½j2 Š; . . .; B½jc Š , D

123

jc ¼jcÀ1 þ1


On the Performance of Opportunistic Relay Selection in…

(

)
Rsk
Á
À Á
MD ¼ E x
min x ; 1 k2D Mkd min lx ; 1
&h
l iÀjDjÀ1 ' Y R
sk
¼ E x min ; 1
x
M
 kd
k2D
1
0
Z1  jDjþ1
Zl
x
C Y Rsk
B
¼@

ksr eÀksr x dx þ ksr eÀksr x dxA
l
 Mkd
0

1
Àl

Y

l

ð55Þ

k2D

0

Á1
jþ1 À 
D
jX
Àksr l
D
  þ 1 ! Y Rsk
e
A
¼ @As þ
:
Àk


 Mkd
ðksr lÞjDjþ1 k¼0 k!ðksr lÞ
k2D
It is noted that PCC
o in (50) is very compact to determine the diversity order and the coding
gain of the opportunistic relay selection in cognitive radio networks with PU’s interference
and direct channel. Indeed, according to [42], PCC
o in the high SNR regime can be represented in terms of the diversity order, JdCC , and the coding gain, JcCC , as
P!1 À CC ÁÀJdCC
PCC
. Consequently, after performing the parameter matching, we can
% Jc P
o
infer from (50) that the opportunistic relay selection achieves the full diversity order of
JdCC ¼ K þ 1 offered by all available secondary relays and the source. Furthermore, the
coding gain is offered by
JcCC

À1
"
!
#Kþ1
KÀ1 KÀuþ1
K
Y
X KÀuþ2
X
X
X

1
K
¼
Qsd P t2 W ; V ; þ
ÁÁÁ
WAVA þ WRVR
Mkd
:
c0
u¼1 s1 ¼1 s2 ¼s1 þ1
su ¼suÀ1 þ1
k2R

ð56Þ
Following the same procedure of deriving (50), one can achieve the approximation of PDH
o
in the high SNR regime as
!

K
KÀ1 KÀuþ1
K
Y
X KÀuþ2
X
X
X
DH P!1 c0 P t2
W ; K; þ
ÁÁÁ

W A KA þ W R KR
Mkd ;
Po %
P
u¼1 s1 ¼1 s2 ¼s1 þ1
su ¼suÀ1 þ1
k2R
ð57Þ
where B ¼ f;; A; Rg, W B is given by (51), and
0
1
BjÀ1 jBjÀcþ1 jBjÀcþ2
jX
jBj
X X
X
ÀjBj
KB ¼ P t2 @H; þ
ÁÁÁ
HC þ HBA;
c¼1

j1 ¼1

j2 ¼j1 þ1

ð58Þ

jc ¼jcÀ1 þ1

with
0

1
jDj   
Àksr l X
Y Rsk
D!
e
ðksr lÞk A
:
HD ¼ @As þ

D
 Mkd
ðk lÞj j k¼0 k!
sr

ð59Þ

k2D

From (57), one can infer the diversity order and the coding gain of the opportunistic relay
selection in cognitive radio networks with PU’s interference and without direct channel,
respectively, as

123


K. Ho-Van et al.

JdDH ¼ K;

ð60Þ

and
JcDH

1
¼
c0 P t2

"
W ; K; þ

K À1 KÀuþ1
X KÀuþ2
X
X

ÁÁÁ

u¼1 s1 ¼1 s2 ¼s1 þ1

!

K
X

W A KA þ W R KR

su ¼suÀ1 þ1

Y

#ÀK1
Mkd

:

k2R

ð61Þ
The asymptotic analysis proves that utilizing the direct channel increases the diversity
order, making the opportunistic relay selection with direct channel always better than that
without direct channel at almost no cost of bandwidth and power.

4 Illustrative Results
This section provides numerous results to demonstrate the outage performance of the
opportunistic relay selection in cognitive radio networks with PU’s interference and with/
without direct channel as well as to corroborate the proposed analytical expressions. For
illustration purpose, we only investigate a primary transmitter-receiver pair (i.e., P t1 ¼
P t2 ¼ P t and ktd1 ¼ ktd2 ¼ ktd ) and arbitrarily choose user positions as shown in Fig. 2.
The variance of the channel coefficient, hpq , between the transmitter p and the receiver q is
Àf
modeled as 1=kpq ¼ dpq
(e.g., [50]) where dpq is the distance between the transmitter p and
the receiver q while f is the path-loss exponent. To limit case-studies, we assume f ¼ 3. In
the sequel, three different relay sets fS1 g, fSk g3k¼1 , fSk g5k¼1 are illustrated for K ¼ 1; 3; 5,
correspondingly.

0.7
0.6

Pr

0.5

Pt

0.4

y

0.3
0.2
S4
0.1
0

S3
S

S
2
S1

−0.2
0

0.2

0.4

0.6

x

123

d

5

−0.1

Fig. 2 User positions

S

S

s

0.8

1


On the Performance of Opportunistic Relay Selection in…

Figure 3 demonstrates the outage performance of the opportunistic relay selection with
respect to the variation of maximum transmit power-to-noise variance ratio, P, for the
proportional factor l ¼ 0:2, the PU’s power-to-noise variance ratio P t ¼ 20 dB, and the
required transmission rate s ¼ 1 bit/s/Hz. It is seen that the exact analysis (i.e., (20) and
(41)) is in an excellent agreement with the simulation while the asymptotic analysis (i.e.,
(50) and (57)) perfectly matches the simulation at large values of P, verifying the accuracy
of the proposed expressions. Additionally, the system performance is drastically enhanced
with respect to the increase in the number of involved relays K. This is reasonable since the
larger K, the higher diversity order is achievable, and so, the smaller the outage probability.
Moreover, the increase in P also improves the system performance. This comes from the
fact that P upper bounds the transmit power of SUs and thus, the larger P, the larger the
transmit power, which ultimately reduces the corresponding outage probability. However,
due to the PU’s interference, the system is always in outage at small values of P (e.g.,
P\20 dB). For all system parameters under consideration, the opportunistic relay selection with direct channel is significantly better that without direct channel, as expected and
analyzed in Sect. 3. This encourages utilizing the direct channel in relaying
communications.
Figure 4 illustrates the impact of the outage threshold c0 (or the required transmission
rate s) on the performance of the opportunistic relay selection for P ¼ 20, P t ¼ 15 dB,
l ¼ 0:4. It is seen that the analysis excellently matches the simulation, validating the
accuracy of the derived expressions. Additionally, the outage threshold c0 ¼ 22s À 1 significantly deteriorates the outage performance of the opportunistic relay selection with/
without direct channel. This comes from the fact that given operation conditions, the more
stringent the system performance requirement (i.e., the larger outage threshold), the higher
the probability that the system is in outage. Moreover, similar to Fig. 3, increasing the
number of relays can considerably improve system performance (e.g., the outage probability is reduced more than 10 times when K increases from 1 to 3 for s ¼ 0:1 bits/s/Hz).
0

10

K=1

−2

Outage probability

10

−4

10

K=3

Sim.: no DC
Exact: no DC
Asym.: no DC
Sim.: DC
Exact: DC
Asym.: DC

−6

10

−8

10

K=5

−10

10

15

20

25

30

35

40

45

50

55

Maximum transmit power−to−noise variance ratio (dB)
Fig. 3 Outage probability versus P. ’Sim.’, ’Exact’, ’Asym.’, and ’DC’ stand for ’Simulation’, ’Exact
analysis’, ’Asymptotic analysis’, and ’Direct channel’, respectively

123


K. Ho-Van et al.
0

Outage probability

10

Sim.: no DC & K = 1
Exact: no DC & K = 1
Sim.: DC & K = 1
Exact: DC & K = 1
Sim.: no DC & K = 3
Exact: no DC & K = 3
Sim.: DC & K = 3
Exact: DC & K = 3
Sim.: no DC & K = 5
Exact: no DC & K = 5
Sim.: DC & K = 5
Exact: DC & K = 5

−1

10

−2

10

−3

10

0.2

0.4

0.6

0.8

1

1.2

1.4

Required transmission rate (bits/s/Hz)
Fig. 4 Outage probability versus the required transmission rate. ’Sim.’, ’Exact’, and ’DC’ stand for
’Simulation’, ’Exact analysis’, and ’Direct channel’, respectively

0

10

K=1

−1

Outage probability

10

K=3
−2

10

Sim.: no DC
Exact: no DC
Sim.: DC
Exact: DC

−3

10

K=5

−4

10

0

0.2

0.4

0.6

0.8

1

Proportional factor µ
Fig. 5 Outage probability versus the proportional factor l. ’Sim.’, ’Exact’, and ’DC’ stand for ’Simulation’,
’Exact analysis’, and ’Direct channel’, respectively

Furthermore, for all operation parameters under investigation, cooperative relaying always
outperforms dual-hop relaying.
Figure 5 illustrates the effect of the proportional factor l on the outage performance of
the opportunistic relay selection with/without direct channel for P ¼ 20, P t ¼ 15 dB,

123


On the Performance of Opportunistic Relay Selection in…
0

10

K=1
−1

Outage probability

10

K=3

−2

10

Sim.: no DC
Exact: no DC
Sim.: DC
Exact: DC

−3

10

K=5

−4

10

0

5

10

15

20

25

30

P (dB)
t

Fig. 6 Outage probability versus P t

s ¼ 0:2 bits/s/Hz. It is observed that the simulation perfectly agrees the analysis, again
corroborating the validity of the proposed expressions. Moreover, the increase in l considerably enhances the outage performance. This is reasonable since the increase in l ¼
I =P is equivalent to the increase in I and hence, inducing PUs more tolerable with the
interference from SUs. As a result, SUs can operate with high transmit powers, eventually
mitigating their outage probability. Furthermore, the performance of the opportunistic
relay selection is dramatically improved with the higher number of involved relays. In
addition, cooperative relaying is always superior to dual-hop relaying for all operation
parameters under consideration.
Figure 6 illustrates the effect of the PU’s interference on the outage behavior of the
opportunistic relay selection for P ¼ 20 dB, l ¼ 0:2, and s ¼ 0:5 bits/s/Hz. Results
illustrate the excellent agreement between the analysis and the simulation, again confirming the validity of the proposed expressions. In addition, the increase in the PU’s
transmit power, which is equivalent to the increase in the PU’s interference, drastically
degrades the performance of SUs. Fortunately, the impact of the PU’s interference can be
considerably remedied by increasing the number of relays (e.g., the outage probability is
reduced more than 10 times when K increases from 1 to 3 at P t ¼ 0 dB for both cases of
with and without direct channel). Moreover, utilizing the direct channel can significantly
improve the performance of the opportunistic relay selection for any considered system
parameters.

5 Conclusions
This paper firstly proposes an exact outage analysis framework for the opportunistic relay
selection in cognitive radio networks with PU’s interference and with/without direct
channel under a general scenario: i.n.i.d. Rayleigh fading channels, the mutual interference

123


K. Ho-Van et al.

between the secondary network and the primary network, maximum transmit power
constraint and interference power constraint, and correlation among received SINRs. Then
the proposed framework is extended to asymptotic analysis to have insights into important
performance metrics such as coding gain and diversity order. Simulations corroborate the
validity of the proposed analysis. Also, numerous results illustrate that (1) the relay
selection is essential in cognitive radio networks not only because of performance
improvement with respect to the increase in the number of relays but also because of low
total transmit power and small transmission bandwidth; (2) the PU’s interference significantly degrades the performance of the opportunistic relay selection; (3) cooperative
relaying achieves higher diversity order and better performance than dual-hop relaying
under any system parameters.
Acknowledgments This research is funded by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) Under Grant Number 102.04-2014.42

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Khuong Ho-Van received the B.E. (with the first-rank honor) and the
M.S. degrees in Electronics and Telecommunications Engineering
from HoChiMinh 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. During 2007–2011, he joined
McGill University, Canada as a postdoctoral fellow. Currently, he is an
Associate professor at HoChiMinh City University of Technology. His

major research interests are modulation and coding techniques,
diversity technique, digital signal processing, and cognitive radio.

Lien Pham-Hong received the B.E. degree in Telecommunications
Engineering from HaNoi City University of Technology, Vietnam, in
1957 and the Ph.D. degree in Electrical Engineering from University of
Slovakia, Slovakia, in 1993. Currently, she is an Associate professor at
Ho Chi Minh City University of Technical Education. Her major
research interests are modulation and coding techniques, diversity
technique, digital communication systems, wireless systems, and
cognitive radio.

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On the Performance of Opportunistic Relay Selection in…
Son Vo-Que was born in Quang Ngai, Vietnam. He received the B.E.
degree in Electrical and Electronics Engineering from Ho Chi Minh
City University of Technology, Vietnam in 2003. Two years later, he
got the M.E. degree in Telecommunications Engineering from the
same university. He received the Ph.D. degree in Electrical Engineering from University of Bremen, Germany in 2011. Since 2004, Dr.
Son has been a faculty member of Department of Electronic Engineering at Ho Chi Minh City University of Technology. His research
interests lie in Mobile and Wireless Communications, Wireless Sensor
Networks, Internet of Things, autonomous/M2M communications. He
is an author of several journals and tens of conference papers.

Tra Luu-Thanh received the B.E. degree in Electrical Engineering
from Ho Chi Minh City University of Technology, Vietnam in 2001,
and the Ph.D. degree in Telecommunication Engineering from Telecom ParisTech, France in 2006. His research interests are telecommunications engineering, computer network, computer & network
security, and embedded system.

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