IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO. 5, MAY 2015
865
Performance Analysis of Cognitive Relay Networks
Over Nakagami-m Fading Channels
Xing Zhang, Senior Member, IEEE, Yan Zhang, Senior Member, IEEE,
Zhi Yan, Jia Xing, and Wenbo Wang, Member, IEEE
Abstract—In this paper, we present performance analysis for
underlay cognitive decode-and-forward relay networks with the
N th best relay selection scheme over Nakagami-m fading channels. Both the maximum tolerated interference power constraint
and the maximum transmit power limit are considered. Specifically, exact and asymptotic closed-form expressions are derived
for the outage probability of the secondary system with the N th
best relay selection scheme. The selection probability of the N th
best relay under limited feedback is discussed. In addition, we
also obtain the closed-form expression for the ergodic capacity
of the secondary system with a single relay. These expressions
facilitate in effectively evaluating the network performance in key
operation parameters and in optimizing system parameters. The
theoretical derivations are extensively validated through Monte
Carlo simulations. Both theoretical and simulation results show
that the fading severity of the secondary transmission links has
more impact on the outage performance and the capacity than
that of the interference links does. Through asymptotic analysis,
we show that the diversity order for the N th best relay selection
scheme is min(m1 , m3 ) × (M − N ) + m3 , where M denotes
the number of cognitive relays, and m1 and m3 represent the
fading severity parameters of the first-hop transmission link and
the second-hop transmission link, respectively.
Index Terms—Cognitive relay networks, N th best relay selection, Nakagami-m fading, outage probability, ergodic capacity.
I. I NTRODUCTION
R
ADIO spectrum is an important and scarce resource
which is increasingly demanded by many kinds of users.
Cognitive radio is an efficient technology to improve the spectrum resources utilization and has gained much attention in recent years [1]. There are three main cognitive radio paradigms:
underlay, overlay and interweave [2]. The underlay paradigm
allows cognitive (secondary) users to utilize the licensed specManuscript received January 4, 2014; revised May 8, 2014 and July 15, 2014;
accepted August 23, 2014. Date of publication September 30, 2014; date of
current version April 21, 2015. This work was supported in part by the National
Natural Science Foundation of China (NSFC) under Grant 61372114, by the
National 973 Program of China under Grant 2012CB316005, by the Joint Funds
of NSFC-Guangdong under Grant U1035001, and by Beijing Higher Education
Young Elite Teacher Project under Grant YETP0434.
X. Zhang, J. Xing, and W. Wang are with the Wireless Signal Processing
and Network Laboratory, Key Laboratory of Universal Wireless Communication, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing 100876, China (e-mail: zhangx@ieee.org; xjbupt@gmail.com;
wbwang@bupt.edu.cn).
Y. Zhang is with Simula Research Laboratory, Fornebu 1364, Norway
(e-mail: yanzhang@ieee.org).
Z. Yan is with the School of Electrical and Information Engineering, Hunan
University, Changsha 410082, China (e-mail: yanzhi@hnu.edu.cn).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/JSAC.2014.2361081
trum if the interference caused to primary users is below a
given interference threshold. In overlay systems, both primary
and secondary users can utilize the licensed spectrum simultaneously through sophisticated signal processing and coding.
In interweave systems, secondary users opportunistically utilize spectrum holes to communicate without interfering the
transmission of primary users. In this paper, we focus on the
underlay systems.
On the other hand, relay communication has emerged as
a powerful spatial diversity technology for effectively combating channel fading and greatly improving the transmission
performance of wireless communication systems [3]. There
are two kinds of classical relay communication protocols, i.e.,
amplify-and-forward (AF) and decode-and-forward (DF) [4]. In
multiple-relay systems, the relay-selection-based transmission
protocol can get higher spectrum efficiency compared with the
traditional relay communication protocol. Best relay selection
[5], [6] is an ideal protocol to achieve the best performance. In
practice, however, the best relay might not be available due to
some scheduling, load balancing or channel side information
(CSI) imperfect feedback conditions. In this case, the second
best relay or more generally the N th best relay might be
selected. Therefore, the study of the N th best relay selection
is of great need.
Inspired by cognitive radio and relay communication, the
cognitive relay network which combines these two techniques
is proposed. In cognitive relay networks, both the spectrum
efficiency and the transmission performance can be improved.
Recently, the research of cognitive relay networks has attracted
much attention, especially on the underlay paradigm [7]–[19].
In [7], the outage probability of a cognitive dual-hop network
with a single AF relay under the interference power constraint
was derived. The outage probability of a cognitive DF relay
network without a direct transmission link and with best relay
selection was evaluated in [8]. In [9], a rough upper bound
on outage probability for cognitive DF relay networks with
a direct transmission link and with best relay selection over
independent and identically distributed (i.i.d) Rayleigh fading
channels was obtained. However, these works can be improved
by considering the dependence among the received signal-tonoise ratios (SNRs) in the first hop. Considering this kind of
dependence, the accurate upper and lower bound on outage
probability for such systems were respectively obtained in [10]
and [11].
In [12], the exact outage probability expression for cognitive
DF relay network was derived. The authors in [13] investigated
the diversity performance of cognitive relay networks with
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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO. 5, MAY 2015
three different relaying protocols, i.e., selective AF, selective
DF, and AF with partial relay selection. In [14], the authors
studied the outage performance of a cognitive network adopting
incremental DF protocol. In [15], the outage probability, symbol error probability and ergodic capacity were derived for cognitive AF relay networks with best relay selection. References
[7]–[15] focused on Rayleigh fading channels while references
[16]–[19] studied the more general fading environment, i.e.,
Nakagami-m fading channels. In [16]–[18], the authors studied
the outage performance of cognitive relay networks with a
single relay over Nakagami-m fading channels. In [19], the
outage performance of cognitive DF relay network with best
relay selection over independent and non-identically distributed
Nakagami-m fading channels was studied.
The above mentioned references all focused on the best
relay selection scheme. Only a few studies involved the N th
best relay selection scheme. As we have mentioned, the N th
relay selection is more of practical significance. In [20], the
performance for conventional AF and DF relay networks with
the N th best relay selection over Rayleigh fading channels was
studied. The asymptotic symbol error rate for a conventional
AF relay network with the N th best relay selection over
Nakagami-m fading channels was derived in [21]. In [22],
the authors investigated the outage behavior of a conventional
dual-hop N th-best DF relay system in the presence of cochannel interference over Rayleigh fading channels. The
outage performance for cognitive relay networks with the N th
best relay selection over Rayleigh fading channels were studied
in [23]. In summary, it is observed that there have been no prior
works on the performance of cognitive relay network with
the N th best relay selection scheme over Nakagami-m fading
channels. While in practical networks, the channels will not
always be simply Rayleigh-distributed. Thus, a comprehensive
study of cognitive relay network with N th best relay selection
over the general Nakagami-m fading channel will be beneficial
for the design in practical cognitive relay systems.
In this paper, we investigate the performance of an underlay
cognitive DF relay network over Nakagami-m fading channels.
Our main contributions are as follows:
• The exact outage probability of the secondary system with
the N th best relay selection is derived over Nakagami-m
fading channels, which build the relationship between the
outage performance and the related system parameters. In
addition, the selection of the N th best relay in the limited
feedback scenario is discussed.
• An asymptotic analysis is carried out to get the asymptotic
outage probability of the secondary system with the N th
best relay selection. The diversity order is also obtained.
• The closed-form expression for the ergodic capacity of
the secondary system with single relay is derived over
Nakagami-m fading channels.
• The results show that the fading severity of the secondary
transmission links has more impact on the outage performance and the ergodic capacity than the fading severity of
the interference links.
The rest of this paper is organized as follows. Section II
describes the system model. Sections III and IV present the de-
Fig. 1.
System model.
tailed analysis of exact and asymptotic outage performance of
the secondary system with the N th best relay selection scheme,
respectively. The selection probability of the N th best relay
under limited feedback is discussed in Section V. In Section VI,
the exact ergodic capacity is derived and analyzed. Numerical
results are shown in Section VII. Finally, conclusions are given
in Section VIII.
coefficient and n!
Notation: Cnk represents the binomial
∞ α−1
t
e−t dt, Γ(α, x) =
represents
the
factorial
of
n.
Γ(α)
=
x α−1 0−t
∞ α−1 −t
t
e
dt
and
γ(α,
x)
=
t
e
dt
denote
the gamma
x
0
function [24, eq. (8.310.1)], the upper incomplete gamma function [24, eq. (8.350.2)] and the lower incomplete
∞gamma
−t
function [24, eq. (8.350.1)], respectively. Ei(x) = − −x e t dt
represents the exponential integral function [24, eq. (8.211.1)].
The cumulative distributed function (CDF) and the probability
density function (PDF) of random variable X are expressed as
FX (·) and fX (·), respectively.
II. S YSTEM M ODEL
We consider an underlay cognitive DF relay network, as
illustrated in Fig. 1. It involves one primary user receiver (P U )
and a secondary system. The secondary system is a dual-hop
relay communication system which consists of one secondary
source (SS), one secondary destination (SD) and M secondary relays (SRi , i = 1, . . . , M ). All nodes are equipped
with a single antenna and operate in half-duplex mode. The
interference from the primary transmitter is assumed to be
neglected as in [7]–[19]. This can be possible if the primary
transmitter is located far away from the secondary users, or the
interference is modeled as the noise term [8]. Like [13], [15],
[16], [18], etc., we assume that there is no direct link between
SS and SD due to the severe shadowing and path loss. We
employ the CSI-assisted DF relaying protocol and the N th best
relay selection scheme. A whole transmission process of the
secondary system consists of two phases. In the first phase,
the SS broadcasts messages to M relays under a transmit
power constraint which guarantees that the interference on the
primary user receiver does not exceed a threshold. In the second
phase, the N th best relay that is selected from the successful
decoding relay set based on the channel quality of the secondhop links forwards source messages to the SD. Finally, the SD
ZHANG et al.: PERFORMANCE OF COGNITIVE RELAY NETWORKS OVER NAKAGAMI-m FADING CHANNELS
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TABLE I
PARAMETERS N OTATIONS
decodes source messages. The transmitters of the secondary
system are with the maximum transmit power constraint Pmax ,
and the maximum interference power constraint of the primary
user receiver is Q. All links are assumed to be independent
Nakagami-m flat fading channels with integer values of the
fading severity parameters and unit average power. The channel
gain of SS-SRi , SS-P U , SRi -SD, and SRi -P U are denoted
as gsi , gsp , gid , and gip , whose fading severity parameters are
m1 , m2 , m3 , and m4 , respectively. The thermal noise at each
receiver is modeled as additive white Gaussian noise (AWGN)
with variance σ 2 . More details of the parameters used in this
paper are given in Table I.
III. E XACT O UTAGE P ERFORMANCE A NALYSIS
In this part, we derive the exact outage probability expression
for the previously described underlay cognitive relay network
with the N th best relay selection, which can be used to evaluate
the impact of the related parameters on the outage performance,
which include the maximum interference power constraint Q,
the maximum transmit power constraint Pmax , the fading severity parameters mi (i = 1, 2, 3, 4), the number of relays M and
the order of the selected relay N .
Considering the maximum transmit power constraint Pmax of
the secondary transmitters and the interference power constraint
Q of the primary user, the transmit power of the SS and
the ith relay SRi should be no more than min(Pmax , Q/gsp )
and min(Pmax , Q/gip ), respectively. In order to maximize
the transmission performance of the secondary system, the
secondary transmitters transmit signals with the maximum
allowable transmit power. Hence, the transmit power at SS
can be written as PS = min(Pmax , Q/gsp ), where gsp denotes
the channel coefficient of the link between SS and P U . Similarly, the transmit power at SRi is given by PRi = min(Pmax ,
Q/gip ).
In the first-hop transmission, the SS broadcasts messages to
relays. As a result, the received SNR at the ith relay SRi is
written as
min(Pmax , Q/gsp )gsi
=
γsi =
σ2
P
max gsi
,
σ2
Qgsi
σ 2 gsp ,
for Pmax < gQsp
for Pmax ≥ gQsp .
(1)
From the above equation, it is worth noting that γsi for each
i is related with gsp while Pmax > Q/gsp . Hence, the received
SNRs at relays are correlated in the first-hop transmission while
Pmax > Q/gsp , but they are independent while Pmax ≤ Q/gsp .
We denote the target transmission rate of secondary system as
R. Then the received SNR at the ith relay should meet the
following inequality if the ith relay can successfully decode
source messages.
R≤
1
log2 (1 + γsi ).
2
(2)
We define γth = 22R − 1, so the successful decoding constraint
of the ith relay can be simplified to γsi ≥ γth . We denote
the successful decoding relay set as R(s) in the first-hop
transmission.
Lemma 1: The probability of the successful decoding relay
set R(s) is given by the expression (3), shown at the bottom of
the next page, where n = |R(s)| denotes the number of relays
m
1 −1
jwj .
in R(s), γPmax = Pmax /σ 2 , γQ = Q/σ 2 , and H =
j=0
Proof: See Appendix A.
In the second-hop transmission, the N th best relay selected
from the successful decoding relay set R(s) forwards the
source messages to the SD. N should be less than or equal
to |R(s)|. It is worth noting that the N th best relay selection
makes sense only when R(s) is not empty, since N ≥ 1.
With the N th best relay selection scheme, the received SNR
at the SD is
γrd = N th max (γid ) = N th max (γid ),
i∈R(s)
γsi ≥γth
(4)
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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO. 5, MAY 2015
where N th max(·) denotes the N th maximum item, γid =
min(Pmax ,Q/gip )gid
denotes the received SNR at the SD if the
σ2
ith relay of R(s) forwards source messages to the SD in the
second phase.
Lemma 2: The CDF of γrd is given as (5), shown at the
bottom of the page.
Proof: See Appendix B.
Finally, the SD decodes source messages. Hence, the equivalent end-to-end received SNR is γrd in the transmission procedure, and the mutual information of secondary system is given as
C=
1
log2 (1 + γrd ).
2
(6)
According to the Shannon information theory, the outage occurs when C < R. We denote the event that the N th best relay
is selected as SN . Therefore, the conditional outage probability
of the secondary system given SN and R(s) (|R(s)| ≥ N ) is
calculated as
Pr (outage|SN , R(s)) = Fγrd (γth ).
(7)
Considering all the possibilities of R(s), the outage probability of the secondary system with the N th best relay selection
scheme can be written as the following expression according to
the law of total probability.
Pr(outage|SN ) =
M
n
CM
Pr (R(s)) Fγrd (γth ).
(8)
n=N
Substituting (3) and (5) into (8), the exact outage probability for
the N th best relay selection can be obtained.
IV. A SYMPTOTIC O UTAGE P ERFORMANCE A NALYSIS
In this part, we derive the asymptotic outage probability
expression in high SNR regions to reveal the diversity performance of the secondary system with the N th best relay
selection scheme.
For a relay selection diversity communication system, the
diversity order is an important performance metric. It is defined
as d = − limγ→∞ (log Pout (γ)/ log(γ)), where γ denotes the
SNR of systems. The diversity order essentially indicates the
number of received independent fading signals at the receiver.
To derive the diversity order of the secondary system with
the N th best relay selection scheme, the asymptotic outage
probability in high SNR regions should be obtained firstly. For
the simplicity of analysis, as [13] and [25], we set γ = 1/σ 2 to
represent the SNR of the secondary system in the subsequent
discussions. Hence, the high SNR region arises while σ 2 → 0.
According to the asymptotic behavior of γ(n, x), we have
lim γ(n, x) =
x→0
n=N
In high SNR regions, the higher order terms of 1/γ can be
omitted. From (13), the n = N term (i.e., (1/γ)m1 (M −N )+m3 )
⎤n ⎡ ⎤M −n m1 γth
m2 Q
1 γth
n M
−n+i
γ
m
γ m1 , m
,
γ
m
,
1
2
γPmax
γPmax
Pmax
⎦
⎦
⎣
⎣
+
Pr (R(s)) = 1−
Γ(m1 )
Γ(m1 )
Γ(m2 )
i=0
×
Fγrd (x) =
N k−1
2
l!mm
2 Γ m2 + H,
m2 γQ +m1 γth l
γPmax
Γ(m2 )
u
Cnk−1 Ck−1
(−1)k−1−u
k=1 u=0
−
l=0
m1 γth
γQ
H ⎧ ⎨ γ m3 , γm3 x γ m4 , Pm4 Q
P
max
max
⎩
4
mm
4
Γ(m4 )
Γ(m3 )
m
3 −1
i=0
(9)
Note that the n in (9) is not necessarily an integer. Therefore,
the asymptotic outage performance analysis in this section is
applicable to the cases of arbitrary Nakagami fading parameters, including the non-integer ones.
Lemma 3: The probability of the successful decoding relay
set R(s) can be asymptotically approximated as (10), shown at
the bottom of the next page,where ∝ represents “proportional to.”
Proof: See Appendix C.
Lemma 4: The asymptotic expression for the CDF of γrd is
written as (11), shown at the bottom of the next page.
Proof: See Appendix D.
By utilizing Lemma 3 and Lemma 4, and substituting (10)
and (11) into (8), the asymptotic outage probability of the
secondary system with the N th best relay selection scheme can
be obtained as (12), shown at the bottom of the next page.
Meanwhile, we can see from (12) that
M m1 (M −n)+m3 (n−N +1)
1
. (13)
Pr(outage|SN ) ∝
γ
⎡
xn
.
n
Γ(m4 )
l
i+l
Cni CM
−n+i (−1)
w0 +...+wm1 −1 =l
γQ
m2 γQ + m1 γth l
+
m2 +H m
1 −1 j=0
1
wj !
wj 1
j!
(3)
4Q
Γ m4 , Pmmax
Γ(m4 )
⎡ ⎤⎫n−u
xm +m γ
−(m4 +i) ⎬
Γ m4 + i, γ3P 4 Q xm i xm
3
3
max
⎣
⎦
+ m4
⎭
i!
γQ
γQ
(5)
ZHANG et al.: PERFORMANCE OF COGNITIVE RELAY NETWORKS OVER NAKAGAMI-m FADING CHANNELS
is left when m1 ≤ m3 , and the n = M term (i.e.,
(1/γ)m3 (M −N +1) ) is left when m1 > m3 . To sum up,
the asymptotic outage probability of secondary system in
high SNR regions is proportional to (1/γ)m1 (M −N )+m3 when
m1 ≤ m3 , but it is proportional to (1/γ)m3 (M −N +1) when
m1 > m3 . Hence, the diversity order of the secondary system
is min(m1 , m3 ) × (M − N ) + m3 . It is indicated that the
diversity performance of the secondary system with the N th
best relay selection scheme is affected by the channel fading
severity parameters of the transmission links, as well as the
difference between the number of relays M and the order of
the selected relay N . The channel fading severity parameters
of the interference links have no impact on the diversity order.
We assume that L bits are used to feedback the SNRs, so
there are q = 2L quantization intervals. Given the quantized
codebook {γ̂1 , γ̂2 , · · · , γ̂q }, the quantized value of γid (i ∈
R(s)) is determined by
γ̂id = arg
The relay selection process is mainly based on the obtained
channel knowledge. In practice, due to imperfect CSI feedback,
the chosen relay may not be the best one. Limited feedback is
often used to perform relay selection [26]. In this section, we
discuss the impact of limited feedback on the relay selection
process.
The relay is selected according to the SNR of SRi -SD γid
(i ∈ R(s)). Considering channel reciprocity, we assume relay
SRi can acquire the channel fading coefficient of SRi -SD.
Therefore, γid is available at SRi . In order to select a relay,
SRi should feedback its SNR γid to the decision-making node
who performs relay selection.
Pr (R(s)) ≈
N
γ→∞ Fγrd (x) ≈
k=1
1
Γ(m1 + 1)
m3 x
γ
γ→∞
Pr(outage|SN ) ≈
m1 γth
γ
m1 M −n
m3 (n−k+1)
1
Γ(m2 )
min
γ̂∈{γ̂1 ,γ̂2 ,···,γ̂q }
|γid − γ̂|.
(14)
Then each relay transmits the index of its quantized value to
the decision-making node. We assume that γid falls into each
quantization interval with equal probability p = 1/q through
some non-uniform quantizer.
With limited feedback, the SNRs of different relays may fall
into the same quantization interval. The “best” relay selection
depends on the best quantization interval, i.e., the quantization
interval with the largest quantized value that contains at least
one relay. We denote the set of relays whose SNR falls into
the best quantization interval as R(b). R(b) is a subset of the
successful decoding relay set R(s). For a given R(s) with
n = |R(s)|, when n = 0, i.e., R(s) is empty, no relay would be
selected. When n > 0, i.e., R(s) is not empty, the probability
that there are nb relays in the best quantization interval for the
case of nb < n can be calculated as
V. S ELECTION P ROBABILITY OF THE N TH B EST R ELAY
U NDER L IMITED F EEDBACK
γ→∞
869
Pr (|R(b)| = nb |R(s)) = Cnnb pnb
q−1
(1 − ap)n−nb .
(15)
a=1
For the case of nb = n, i.e., all relays are in the same quantization interval, the probability that there are nb relays in R(b) for
m1 (M −n) m2 Q
γ m2 ,
Pmax
Pmax
m1 (M −n) 1
m2 Q
Γ m1 (M − n) + m2 ,
+
m2 Q
Pmax
1
(10)
n−k+1
1
Γ(m3 + 1)Γ(m4 )
m 3 n−k+1 m3 1
1
m4 Q
m4 Q
×
γ m4 ,
Γ m3 + m4 ,
+
Pmax
Pmax
m4 Q
Pmax
Cnk−1
(11)
m1 (M −n)+m3 (n−k+1) n k−1 M −n n−k+1
N M 1
3
γth
mm
mm
CM Cn
1
3
γ
Γ(m2 )
Γ(m1 + 1)
Γ(m3 + 1)Γ(m4 )
n=N k=1
m 3 n−k+1
m3 1
1
m4 Q
m4 Q
×
γ m4 ,
Γ m3 + m4 ,
+
Pmax
Pmax
m4 Q
Pmax
m1 (M −n) m1 (M −n) 1
1
m2 Q
m2 Q
×
γ m2 ,
Γ m1 (M − n) + m2 ,
+
(12)
Pmax
Pmax
m2 Q
Pmax
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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO. 5, MAY 2015
the given R(s) can be expressed as
= 1 − Pr{γsr > x} Pr{γrd > x}
Pr (|R(b)| = nb |R(s)) = qpn .
The “best” relay is selected from R(b) randomly, i.e., each relay
in R(b) is chosen with probability n1b . We denote the event
that the N th best relay is chosen as SN (N = 1, 2, · · · , n).
Therefore, the conditional selection probability of the N th best
relay given R(s) is expressed as
Pr (SN |R(s)) =
n
1
Pr (|R(b)| = nb |R(s)) .
nb
(17)
nb =N
According to the law of total probability, the selection probability of the N th best relay under limited feedback is given by
Pr(SN ) =
M
n
CM
Pr (R(s)) Pr (SN |R(s)) ,
where Pr(R(s)) can be calculated as (3).
Combining this with the results in Section III, we can obtain
the outage probability of the secondary system with limited
feedback as
Pout = Pr (R(s) = ∅)+
Pr(SN ) Pr(outage|SN ).
(19)
N =1
VI. E XACT E RGODIC C APACITY A NALYSIS
In this section, we derive the exact ergodic capacity of the
cognitive DF relay network over Nakagami-m fading channels.
Specifically, we study the special case where there is only one
single relay in the secondary system, i.e., M = 1 and N = 1.
Additionally, we consider symmetric channel conditions where
the channels associated with the first hop and the second hop
have the same parameters. The fading severity parameters of the
secondary transmission links are denoted as m1 = m3 = ms
while those of the interference links are denoted as m2=m4=mp .
For the case of M = 1 and N = 1, the end-to-end SNR of
the secondary system is given by
γe2e = min(γsr , γrd ).
Since there is only one relay node, the CDF of γrd is the
same as the CDF of γid in (42). Due to the symmetric channel
conditions, γsr has the same distribution as γrd . We rewrite it as
mp Q
mp Q
sx
Γ
m
γ ms , γm
,
,
γ
m
p
p
Pmax
Pmax
Pmax
+
Fγsr(x) = Fγrd(x)=
Γ(ms )
Γ(mp )
Γ(mp )
i −(mp +i)
ms −1
m
ms x
1 ms x
mp p + mp
−
Γ(mp ) i=0 i! γQ
γQ
m s x + mp γ Q
× Γ mp + i,
.
(22)
γPmax
According to the definition, the ergodic capacity of the
secondary system can be expressed as
1 ∞
C̄ =
log2 (1 + x)fγe2e (x) dx.
(23)
2 0
By using the same method as [27], we can rewrite the expression for the ergodic capacity as
∞
1
1
C̄ =
(1 − Fγe2e (x)) dx
2 ln 2 0 1 + x
∞
1
1
(1 − Fγsr (x)) (1 − Fγrd (x)) dx.
=
2 ln 2 0 1 + x
(24)
By substituting (22) into (24) and utilizing the series representation of the incomplete Gamma function [24, eq. (8.352.6)],
we can get (25), shown at the bottom of the page, where A and
B(v) are defined as
∞ i+j
x
− 2ms x
e γPmax dx
(26)
A=
1+x
0
and
(20)
∞
B(v) =
Thus the CDF of γe2e can be written as
0
Fγe2e (x) = Pr {min(γsr , γrd ) ≤ x}
xi+j
− 2ms x
e γPmax dx,
v
(1 + x)(ms x + mp γQ )
+ (mp γQ )
m Q
p
mp − Pmax
e
2
l=0
k=0
mp +i−1 mp +j−1
Γ(mp + i)Γ(mp + j)
(27)
respectively.
2
m
s −1 m
s −1
1
mi+j
1
1
mp Q
s
C̄ =
γ mp ,
A
2
i+j
2 ln 2 (Γ(mp )) i=0 j=0 i!j!
Pmax
γPmax
m+j−1
1 1
mp Q
1
mp Q
− Pmax
mp
(m
γ
)
Γ(m
+
j)e
B(mp + j − k)
+ γ mp ,
p
Q
p
Pmax γPi max
k! γPkmax
k=0
mp +i−1
1 1
mp Q
1
mp Q
− Pmax
mp
+ γ mp ,
(mp γQ ) Γ(mp + i)e
B(mp + i − l)
j
Pmax γP
l! γPl max
l=0
max
(21)
(18)
n=N
M
= 1 − (1 − Fγsr (x)) (1 − Fγrd (x)) .
(16)
1
1
B(2mp + i + j − l − k)
l!k! γPl+k
max
(25)
ZHANG et al.: PERFORMANCE OF COGNITIVE RELAY NETWORKS OVER NAKAGAMI-m FADING CHANNELS
By using [24, eq. (3.353.5)], we obtain
2ms
2ms
A = (−1)i+j−1 e γPmax Ei −
γPmax
h
i+j
γPmax
+
(h − 1)!(−1)i+j−h
.
2ms
871
(28)
h=1
To calculate the function B(v), we should consider two cases.
For the case of ms = mp γQ , by splitting the term 1/(1 +
x)(ms x + mp γQ )v , we get
∞ i+j
1
x
− γ2ms x
Pmax dx
e
B(v) =
(mp γQ − ms )v 0 1 + x
−
v
ms
a
(m
γ
p Q −ms )
a=1
Φ1
∞
0
xi+j
− γ2ms x
Pmax dx .
e
(ms x+mp γQ )v+1−a
Φ2
(29)
The integral term Φ1 is the same as A. The integral term Φ2
can be calculated with the help of the Meijer’s G-function [28].
Specifically, the term 1/(ms x + mp γQ )v+1−a in Φ2 can be
expressed as
1
(ms x + mp γQ )v+1−a
!
ms x !! a − v
1
1
1,1
G
.
=
(mp γQ )v+1−a Γ(v + 1 − a) 1,1 mp γQ ! 0
(30)
By substituting (30) into Φ2 and using [24, eq. (7.813.1)], we
have
i+j+1
γPmax
1
1
Φ2 =
(mp γQ )v+1−a Γ(v + 1 − a) 2ms
!
Pmax !! −i − j, a − v
. (31)
× G1,2
2,1
0
2mp Q !
Thus B(v) is obtained for the case of ms = mp γQ .
For the case of ms = mp γQ , B(v) can be written as
∞
xi+j
1
− 2ms x
B(v) = v
e γPmax dx.
v+1
ms 0 (1 + x)
(32)
By the same way we get Φ2 , we can obtain the closed-form
expression for (32).
In conclusion, we obtain B(v) as (33), shown at the bottom
of the page.
By substituting A and B(v) which are given by (28) and (33),
respectively, into (25), we can get the closed-form expression
for the system ergodic capacity.
Fig. 2. Outage probability versus interference power constraint to noise
ratio for different channel fading severity parameters with M = 6, N = 2,
Pmax /σ 2 = 10 dB and R = 1 bit/s/Hz.
VII. N UMERICAL R ESULTS AND D ISCUSSIONS
In this section, we present numerical results to validate our
theoretical analysis in Sections III–VI. A detailed investigation
is given on the impact of the interference power constraint,
the maximum transmit power constraint, the fading severity
parameters, the number of relays and the relay selection scheme
on the outage and diversity performance of the secondary
system. The effect of limited feedback on the relay selection
probability is also studied.
Fig. 2 illustrates the exact outage probability of the secondary
system versus the interference power constraint to noise ratio
Q/σ 2 for various channel fading severity parameters. The
number of relays M and the order of the selected relay N are set
to 6 and 2, respectively. The solid lines represent our analytical
results, and the square symbols represent the Monte Carlo
simulation results. From Fig. 2, we can see that our analytical
results match well with the simulation results, which validates
our theoretical analysis. The outage performance improves with
the increase of the fading severity parameters. It is worth noting
that the outage probability has a decline in the low Q region.
This is because when Q gets smaller, there are fewer relays
available in the successful decoding relay set, so the chance
that the N th best relay makes sense (N ≤ |R(s)|) gets smaller.
Besides, the outage performance improves with the increase of
Q in the median Q region, but will reach saturation when Q is
large enough due to the existence of Pmax .
Fig. 3 plots the outage probability of the secondary system
versus the maximum transmit power constraint to noise ratio
Pmax /σ 2 . For the similar reason of the case in Fig. 2, the outage
⎧
!
i+j+1
v
γPmax
ms
Pmax ! −i−j,a−v
1
1
⎪
⎨ (mp γQ1−ms )v A−
,
G1,2
2,1 2mp Q !
0
(mp γQ −ms )a (mp γQ )v+1−a Γ(v+1−a)
2ms
a=1
B(v) =
!
i+j+1
⎪
!
γPmax
γ
⎩ 1v 1
G1,2 Pmax ! −i−j,−v ,
ms Γ(v+1)
2ms
2,1
2ms
0
for ms = mp γQ
for ms = mp γQ
(33)
872
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO. 5, MAY 2015
Fig. 3. Outage probability versus maximum transmit power constraint to
noise ratio for different channel fading severity parameters with M = 6, N =
2, Q/σ 2 = 10 dB and R = 1 bit/s/Hz.
Fig. 5. Outage probability versus interference power constraint to noise ratio
for different orders of selected relays (N ) with M = 6, m1 = m2 = m3 =
m4 = 2, Pmax /σ 2 = 10 dB and R = 1 bit/s/Hz.
Fig. 4. Outage probability versus interference power constraint to noise ratio
for different numbers of relays (M ) with N = 2, m1 = m2 = m3 = m4 =
3, Pmax /σ 2 = 10 dB and R = 1 bit/s/Hz.
Fig. 6. Outage probability versus interference power constraint to noise ratio
for different fading severity of the transmission links and the interference links
with M = 6, N = 2, Pmax /σ 2 = 10 dB and R = 1 bit/s/Hz.
probability has a decline in the low Pmax region and reaches
saturation in the high Pmax region.
Fig. 4 illustrates the impact of the number of relays M on
the outage performance of the secondary system with the N th
best relay selection (N = 2). In the median and high Q regions,
more relays can provide better outage performance. It is also
observed that the turning point between the low Q region and
the median Q region shifts left with the increase of M .
Fig. 5 gives the outage performance of the secondary system
for different relay selection schemes. The number of relays M
is set to 6. It shows that in the median and high Q regions, the
outage performance decreases with the increase of the order of
the selected relay (i.e., N ) since the performance of the second
hop is worsened. In addition, we can observe that the turning
point between the low Q region and the median Q region shifts
right with the increase of N .
The impact of the fading severity of the transmission links
and the interference links on the outage performance is illustrated in Fig. 6. It is observed that the fading severity of the
transmission links has great impact on the outage performance,
but the fading severity of the interference links has little impact
on the outage performance. This is in compliance with the
results in [29] which investigates the imperfect CSI of the
transmission links and the interference links.
Fig. 7 plots the impact of the fading severity of the firsthop links and the second-hop links on the outage performance.
From this figure, we can see that when the number of relays
M is relatively small, the fading severity of the second-hop
links has more influence on the outage performance than that
of the first-hop links. But when M is relatively large, these two
hops have nearly equal impact on the outage performance of the
secondary system.
Figs. 8 and 9 present the exact and asymptotic outage probability curves according to Sections III and IV. From these two
figures, we can observe that the asymptotic outage probability
is very close to the exact one in high SNR regions. It is
indicated that our asymptotic outage probability expression
can be used to effectively evaluate the outage performance
ZHANG et al.: PERFORMANCE OF COGNITIVE RELAY NETWORKS OVER NAKAGAMI-m FADING CHANNELS
Fig. 7. Outage probability versus interference power constraint to noise ratio
for different fading severity of first-hop links and second-hop links with N = 2,
Pmax /σ 2 = 10 dB and R = 1 bit/s/Hz.
Fig. 8. The exact and asymptotic outage probability versus system SNR (γ =
1/σ 2 ) for different fading severity of the transmission links with M = 4, N =
2 and m2 = m4 = 2, Q = 10 dB, Pmax = 10 dB and R = 1 bit/s/Hz.
of the secondary system in high SNR regions, even for noninteger fading severity parameters. According to our analysis
in Section IV, the diversity order of the secondary system
is min(m1 , m3 ) × (M − N ) + m3 , which coincides with the
slope of the curves in these figures.
In Fig. 10, the selection probability of the N th best relay
with limited feedback is illustrated. From this figure, we can
observe that the selection probability of the relay decreases
with the increase of N . This means the probability that a
better relay is selected is larger than the probability that a
worse relay is selected in the limited feedback scenario. For
the best relay, as the number of feedback bits L increases, its
selection probability gets larger. For the worse relays (N ≥ 3),
the selection probability decreases with the increase of L.
Fig. 11 depicts the ergodic capacity of the secondary system
with one single relay. It can be observed that the ergodic
capacity improves with the increase of the interference power
constraint Q and will reach saturation when Q is large enough
873
Fig. 9. The exact and asymptotic outage probability versus system SNR (γ =
1/σ 2 ) for different relay selection scheme with m1 = m2 = m3 = m4 = 2,
Q = 10 dB, Pmax = 10 dB, σ 2 = 1 and R = 1 bit/s/Hz.
Fig. 10. Selection probability of the N th best relay versus the number of feedback bits (L) with M = 10, m1 = m2 = 3, Q/σ 2 = 10 dB, Pmax /σ 2 =
10 dB and R = 1 bit/s/Hz.
Fig. 11. Ergodic capacity versus interference power constraint to noise ratio
for different fading severity of the transmission links and the interference links
with M = 1, N = 1 and Pmax /σ 2 = 10 dB.
874
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO. 5, MAY 2015
due to the maximum transmit power constraint Pmax . It can
also be seen that the fading severity of the transmission links
has much greater impact on the ergodic capacity than that of
the interference links. As the fading of the transmission links
gets severer, the capacity of the secondary system will decrease
greatly. However, as the fading of the interference links gets
severer, the capacity will increase when Q is small and will
decrease when Q is large.
VIII. C ONCLUSION AND F UTURE W ORKS
In this paper, we study the performance of an underlay
cognitive DF relay network with the N th best relay selection over Nakagami-m fading channels, considering both the
maximum transmit power limit and the interference power
constraint. The exact and asymptotic outage probability expressions for such system are derived. Through asymptotic
analysis, we obtain the diversity order of the secondary system,
which is min(m1 , m3 ) × (M − N ) + m3 , where M represents
the number of relays and m1 , m3 denote the fading severity
parameters of the first-hop transmission link and the secondhop transmission link. It is indicated that the fading severity of
channels, the number of relays and the relay selection scheme
have great impact on the outage performance of the secondary
system. The selection probability of the N th best relay is
given in the limited feedback scenario. Besides, we obtain the
exact ergodic capacity for the special case where there is one
single relay in the secondary system. The theoretical analysis
is validated by simulations. The results show that the fading
severity of the transmission links has more impact on the outage
performance as well as the ergodic capacity than that of the
interference links. In this paper the direct link for the secondary
system is not considered. We believe that considering the direct
link can be an interesting topic in our future works.
The first summand in (34) can be calculated as
n M −n
Pmax gsi
Pmax gsi
≥
γ
<
γ
Pr
I1 = Pr
th
th
σ2
σ2
Q
× Pr Pmax <
gsp
n M −n
γth σ 2
γth σ 2
Q
Fgsp
= 1−Fgsi
Fgsi
Pmax
Pmax
Pmax
⎤n ⎡
⎤M −n
⎡
1 γth
1 γth
γ m1 , m
γ m1 , m
γPmax
γPmax
⎦ ⎣
⎦
= ⎣1 −
Γ(m1 )
Γ(m1 )
×
2Q
γ m2 , Pmmax
Γ(m2 )
max
2 m2 −1 −m2 t
e
mm
2 t
dt
Γ(m2 )
⎤M −n+i
⎡ ∞
n
γ m1 , m1γγQth t
⎦
⎣
=
Cni (−1)i
Q
Γ(m1 )
P
i=0
×
max
mm2 tm2 −1 e−m2 t
dt
× 2
Γ(m2 )
According to the definition of R(s), the probability of the set
R(s) can be written as
⎡
⎤
#
#
Pr(R(s)) = Pr⎣
(γsi ≥ γth ),
(γsi < γth )⎦
= Pr⎣
i∈R(s)
σ2
(36)
Then, utilizing the following expansion for an incomplete
gamma function [24, eq. (8.352.6)]:
%
$
n−1 i
x
−x
.
(37)
γ(n, x) = Γ(n) 1 − e
i!
i=0
⎤
#
Pmax gsi
Q⎦
≥ γth ,
< γth , Pmax<
σ2
gsp
i
∈R(s)
I2 can be transformed into
I2 =
where the first summand denotes that all relays can successfully
decode in the set R(s), and the second summand denotes that
the other relays decode failed.
Cni (−1)i
∞
Q
Pmax
⎤M −n+i
⎡ γ m1 , m1γγQth t
⎦
⎣
Γ(m1 )
mm2 tm2 −1 e−m2 t
dt
× 2
Γ(m2 )
I2
(34)
n
i=0
I1
⎤
# # Qgsi
Qg
Q
si
⎦,
+ Pr⎣
≥ γth ,
< γth , Pmax ≥
σ 2 gsp
σ 2 gsp
gsp
i∈R(s)
i
∈R(s)
⎡
i
∈R(s)
i∈R(s)
# Pmax gsi
(35)
In the second summand, it is found that all parts are correlated with the variable gsp . Hence, the second summand can be
written as
n M −n
∞ Qgsi
Qgsi
≥ γth
< γth
I2 =
fgsp (t)dt
Pr
Pr
Q
σ2 t
σ2 t
Pmax
⎤n ⎡ ⎤M −n
⎡
∞
γ m1 , m1γγQth t
γ m1 , m1γγQth t
⎦ ⎣
⎦
⎣1 −
=
Q
Γ(m
)
Γ(m
)
1
1
P
A PPENDIX A
P ROOF OF L EMMA 1
⎡
.
=
n
Cni (−1)i
i=0
×
⎡
∞
Q
Pmax
⎢
⎣1−e
j ⎤M −n+i
m γ t
− 1γ th
Q
2 m2 −1 −m2 t
e
mm
2 t
dt
Γ(m2 )
m1 γth t
m
1 −1
γQ
j=0
j!
⎥
⎦
ZHANG et al.: PERFORMANCE OF COGNITIVE RELAY NETWORKS OVER NAKAGAMI-m FADING CHANNELS
=
n M
−n+i
i=0
l
i+l
Cni CM
−n+i (−1)
l=0
×e
−
⎡
m1 γth l+m2 γQ
γQ
t
⎣
m
1 −1
j=0
1
j!
2
mm
2
Γ(m2 )
m1 γth t
γQ
∞
tm2 −1
=
Q
Pmax
j
⎤l
⎦ dt
=
(38)
N
k=1
N
875
Cnk−1 [Pr(γid ≤ x)]n−k+1 [Pr(γid < x)]k−1
Cnk−1 [Fγid (x)]n−k+1 [1−Fγid (x)]k−1 ,
(41)
k=1
where Fγid (x) is expressed as
According
m
1 −1
⎡
⎣
1
j!
j=0
m
1 −1
1
j!
j=0
to
the multinomial theorem, the
j l
m1 γth t
in (38) can be expanded as
γQ
m1 γth t
γQ
j
⎤l
⎦
m
1 −1
= l!
w0 +w1 +···+wm1 −1 =l j=0
j wj 1 m1 γth t
1
wj ! j!
γQ
⎧
⎨
= l!
term
w0 +w1 +···+wm1 −1 =l
m1 γth t
⎩ γQ
H m
1 −1
j=0
⎫
wj⎬
1
1
,
⎭
wj ! j!
(39)
where H =
m
1 −1
jwj . So I2 can be rewritten as
j=0
I2 =
n M
−n+i
i=0
l=0
×
×
=
l
i+l
Cni CM
−n+i (−1)
w0 +w1 +...+wm1 −1 =l
m1 γth
γQ
∞
t
H m
1 −1 j=0
m2 −1+H −
e
1
wj !
wj 1
j!
m1 γth l+m2 γQ
γQ
Q
Pmax
n M
−n+i
i=0
l=0
2
mm
2 l!
Γ(m2 )
t
dt
l
i+l
Cni CM
−n+i (−1)
w0 +w1 +...+wm1 −1 =l
2
mm
2 l!
Γ(m2 )
H m2 +H
m1 γth
γQ
γQ
m2 γQ + m1 γth l
m
wj 1 −1
1
1
m2 γQ + m1 γth l
× Γ m2 +H,
×
.
γPmax
wj ! j!
j=0
×
min(Pmax , Q/gip )gid
Fγid (x) = Pr(γid ≤ x) = Pr
≤
x
σ2
Pmax gid
Qgid
Q
Q
≤ x, Pmax<
≤ x, Pmax ≥
= Pr
+Pr 2
σ2
g
σ gip
gip
ip
Pmax gid
Q
= Pr
≤ x Pr Pmax <
σ2
gip
J1
∞
Qgid
≤
x
fgip (t)dt .
+
Pr
(42)
Q
σ2 t
Pmax
J2
Utilizing the CDF of gip and gid , J1 can be easily calculated as
m4 Q
3x
γ m3 , γm
,
γ
m
4 Pmax
Pmax
.
(43)
J1 =
Γ(m3 )
Γ(m4 )
For J2 , the integral can be calculated as following by using
the expansion expression (37) as follows:
∞ γ m3 , m3 xt
4 m4 −1 −m4 t
γQ
mm
e
4 t
dt
J2 =
Q
Γ(m
)
Γ(m
)
3
4
Pmax
i ∞
m3 −1 4
m x
mm
− γ 3 t 1 m3 x
m4 −1 −m4 t
4
Q
=
1−e
dt
t
e
t
Γ(m4 ) P Q
i! γQ
i=0
max
i
4Q
3 −1
Γ m4 , Pmmax
4 m
mm
1 xm3
4
−
=
Γ(m4 )
Γ(m4 ) i=0 i! γQ
−(m4 +i) xm3
xm3 + m4 γQ
×
+ m4
Γ m4 + i,
.
γQ
γPmax
(44)
Substituting (43) and (44) into (42), we can get Fγid (x).
Then from (41), we can obtain Fγrd (x) as (5).
(40)
A PPENDIX C
P ROOF OF L EMMA 3
Substituting (35) and (40) into (34), we can obtain (3).
A PPENDIX B
P ROOF OF L EMMA 2
According to the expression of γrd (4), the CDF of γrd can
be written as
Fγrd (x) = Pr N th max γid ≤ x
i∈R(s)
According to (9), I1 can be written as the following expression in high SNR regions:
m1 ⎤n⎡ m1 ⎤M −n ⎡
m1 γth
m1 γth
m2 Q
γ
m
,
2
γ→∞
Pmax γ
Pmax
⎦ ⎣ Pmax γ
⎦
I1 ≈ ⎣1−
Γ(m1 +1)
Γ(m1 +1)
Γ(m2 )
≈
1
Γ(m1 + 1)
m1 M −n γ m , m2 Q
2
Pmax
m1 γth
.
Pmax γ
Γ(m2 )
(45)
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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO. 5, MAY 2015
For I2 , it can be approximated written as
m 1 ⎤ n ⎡ m1 ⎤M −n
⎡
m1 γth t
m1 γth t
∞
γ→∞
Qγ
Qγ
⎦ ⎣
⎦
⎣1 −
I2 ≈
Q
Γ(m
+
1)
Γ(m
1
1 + 1)
P
max
mm2 tm2 −1 e−m2 t
dt
× 2
Γ(m2 )
m1 ⎤M −n m1(M−n)
⎡
≈⎣
m1 γth
Qγ
Γ(m1 +1)
⎦
1
m2
Γ(m2 )
m2 Q
Γ m1 (M −n)+m2 ,
.
Pmax
(46)
Substituting (45) and (46) into (34), we can obtain (10).
A PPENDIX D
P ROOF OF L EMMA 4
According to (9), J1 can be written as the following expression in high SNR regions:
m 3 γ m , m 4 Q
4 Pmax
γ→∞
m3 x
1
.
(47)
J1 ≈
Γ(m3 + 1) Pmax γ
Γ(m4 )
For J2 , it can be written as
m3 m4 m4 −1 −m4 t
∞
γ→∞
1
m4 t
e
m3 xt
dt
J2 ≈
Q
Γ(m3 + 1)
Qγ
Γ(m4 )
Pmax
⎞
⎛ m 3 Γ m + m , m 4 Q
3
4 Pmax
m3 x
⎠.
⎝
=
(48)
m4 Qγ
Γ(m3 + 1)Γ(m4 )
Substituting (47) and (48) into (42), we can obtain the
asymptotic expression for the CDF of γid as
m 3
m 3
γ→∞ m3 x
1
1
Fγid (x) ≈
γ
Γ(m3 + 1)Γ(m4 )
Pmax
m 3 1
m4 Q
m4 Q
× γ m4 ,
Γ m3 +m4 ,
+
. (49)
Pmax
m4 Q
Pmax
From (49), we can easily obtain that
γ→∞
1 − Fγid (x) ≈ 1.
(50)
Substituting (49) and (50) into (41), we can obtain the
asymptotic expression for the CDF of γrd as (11).
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Feb. 2005.
[2] A. Goldsmith, S. Jafar, I. Maric, and S. Srinivasa, “Breaking spectrum
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ZHANG et al.: PERFORMANCE OF COGNITIVE RELAY NETWORKS OVER NAKAGAMI-m FADING CHANNELS
Xing Zhang (M’10–SM’14) received the Ph.D. degree from Beijing University of Posts and Telecommunications (BUPT), Beijing, China, in 2007. Since
July 2007, he has been with the School of Information and Communications Engineering, BUPT,
where he is currently an Associate Professor. He is
the author/coauthor of two technical books and more
than 50 papers in top journals and international conferences and filed more than 30 patents (12 granted).
His research interests are mainly in wireless communications and networks, green communications
and energy-efficient design, cognitive radio and cooperative communications,
traffic modeling, and network optimization.
Prof. Zhang has served on the editorial boards of several international journals, including KSII Transactions on Internet and Information Systems and the
International Journal of Distributed Sensor Networks, and as a TPC Cochair/
TPC member for a number of major international conferences, including MobiQuitous 2012, IEEE ICC/GLOBECOM/WCNC, CROWNCOM, Chinacom,
etc. He received the Best Paper Award in the 9th International Conference
on Communications and Networking in China (Chinacom 2014) and the 17th
International Symposium on Wireless Personal Multimedia Communications
(WPMC 2014).
Yan Zhang (SM’10) received the Ph.D. degree from
Nanyang Technological University, Singapore. Since
August 2006, he has been with Simula Research
Laboratory, Fornebu, Norway, where he is currently
the Head of the Department of Networks. His recent
research interests include wireless networks,
machine-to-machine communications, and smart
grid communications. He is a Regional Editor or an
Associate Editor on the editorial board and a Guest
Editor of a number of international journals.
877
Zhi Yan received the B.Sc. degree in mechanical engineering and automation and the Ph.D. degree in communication and information system from
Beijing University of Posts and Telecommunications (BUPT), Beijing, China, in 2007 and 2012,
respectively. From August 2012 to March 2014, he
was a Researcher with the Network Technology Research Center, China Unicom Research Institute. He
is currently an Assistant Professor with the School
of Electrical and Information Engineering, Hunan
University, Changsha, China. His current research
interests are in the cognitive radio, cooperative communication, and cellular
network traffic analysis and modeling.
Jia Xing received the B.S. degree in communication engineering in 2012 from Beijing University
of Posts and Telecommunications, Beijing, China,
in 2012 where she is currently working toward
the M.S. degree in the Key Laboratory of Universal Wireless Communications, School of Information and Communication Engineering. Her research
interests include cognitive radio and cooperative
communication.
Wenbo Wang received the B.S., M.S., and Ph.D. degrees from Beijing University of Posts and Telecommunications (BUPT), Beijing, China, in 1986, 1989,
and 1992, respectively. He is currently a Professor
with and the Executive Vice Dean of the Graduate
School, BUPT. He is also the Deputy Director of the
Key Laboratory of Universal Wireless Communication, Ministry of Education. He has published more
than 200 journal and international conference papers
and 6 books. His current research interests include
radio transmission technology, wireless network theory, and software radio technology.