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Outage performance of cognitive DF relay networks with nonidentical Rayleigh fading channels and maximal ratio combining
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Outage performance of cognitive DF relay networks with nonidentical Rayleigh fading channels and maximal ratio combining夽 Zhenguo Gao a,∗ , Kaichen Zhang a , Danjie Chen b , Wei Zhang a , Yibing Li c a b c
College of Automation, Harbin Engineering University, Harbin, China College of Software, Beijing University of Technology, Beijing, China College of Information and Communication Engineering, Harbin Engineering University, Harbin, China
a r t i c l e
i n f o
Article history: Received 2 May 2014 Accepted 12 August 2014 Keywords: Decode-and-forward Outage probability Symbol error rate Asymptotic expression Non-identical Rayleigh fading
a b s t r a c t In the expression of end-to-end signal-to-noise ratio (SNR) of cognitive relay networks, the channel gaina of the interference links (from secondary nodes to primary nodes in the networks) are usually shared by multiple items in the expression. These shared variables lead to the so called correlation issue, which results in the higher complexity of outage performance analysis for such networks. Till now, for cognitive relay networks with decode-and-forward (DF) relay scheme, we have not found works obtaining explicit closed-form expression of exact outage probability (OP) meanwhile treating the correlation issue completely. In this paper, for cognitive DF relay networks with maximal ratio combing (MRC) and independent non-identical Rayleigh fading channels, we obtain a closed-form expression of exact OP taking consideration of the correlation issue completely. Additionally, by shrinking or expanding the triangular integral region to suitable rectangular regions, we obtain simpler closed-form expressions of the lower and upper bounds of both OP and symbol error probability (SER). Furthermore, the asymptotic expressions of OP and SER are also obtained. The correctness of our analysis results are verified through numerical simulations. Both analysis results and simulation results show that how the correlation issue is treated can affect the coding gain whereas has no effect on the diversity gain of the network. Simulation results also indicate that, exact OP and SER are bounded from both sides by the lower bound and the upper one tightly, whereas the upper bounds of OP and SER are more tighter than the lower bounds. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction Applying multiple antennas enables the terminals to combat with channel fading effects. However, the requirements on hardware size would be prohibitive for hand-held communication devices [1,2]. To that end, cooperative communication is emerging as an alternative approach by forming a virtual antenna array to perform the function of multiple antennas [3]. By allowing secondary users (SUs) to simultaneously share the frequency band licensed to primary users (PUs) without causing harmful interferences to PUs, cognitive radio can significantly improve spectrum efficiency [4–6]. As a combination of these two attractive techniques, i.e., cooperative communication and cognitive radio, cognitive relay networks have gained much research attention in recent years. For cognitive relay networks, two typical relaying strategies are decode-and-forward (DF) and amplify-and-forward (AF) [7,8]. A DF relay decodes the received signal, re-encodes and forwards it to the destination, whereas a AF relay just amplifies the received signal directly and then forwards it to the destination node [7,8]. At the destination, the receiver employs diversity combining techniques to obtain spatial diversity from signal replicas received from SU relays and the SU source. Typical diversity combining techniques are maximal ratio combining (MRC), equal-gain combining (EGC), selection combining (SC). In cognitive relay networks, the interference power caused by an SU’s transmission on a PU receiver should not exceed the PU’s threshold. This is the so called interference power constraint, which is a main issue specific to cognitive relay networks. This interference
夽 This work has been supported by Natural Science Foundation of China (No.61073183); Natural Science Foundation of HeiLongJiang Province (No F201120); Fundamental Research Funds for the Central Universities (No. HEUCFX41311). ∗ Corresponding author. E-mail address: [email protected] (Z. Gao). http://dx.doi.org/10.1016/j.aeue.2014.08.003 1434-8411/© 2014 Elsevier GmbH. All rights reserved.
Please cite this article in press as: Gao Z, et al. Outage performance of cognitive DF relay networks with nonidentical Rayleigh fading channels and maximal ratio combining. Int J Electron Commun (AEÜ) (2014), http://dx.doi.org/10.1016/j.aeue.2014.08.003
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Fig. 1. Network model of targeted dual-hop cognitive relay networks.
power constraint of PU leads to the correlation issue in outage performance analysis of such networks, which intrinsically result from the interference channel gain of the interference link from a SU node to the PU shared by multiple signal-to-noise ratio (SNR) items in the expression of end-to-end SNR of cognitive relay networks. The correlation issue should be elaborately treated otherwise the analysis results will be incorrect. In the literature, outage performance of cognitive relay networks with DF and MRC have been investigated in many works including references [9,10]. In these works, the correlation issue was treated in different levels. For example, in [9], the SNR items in the expression of end-to-end SNR of cognitive relay networks were all considered as independent variables, thus the correlation issue is completely ignored. This treatment is obviously not appropriate, as has been verified in [10]. However, in [10], the correlation issue was only partially considered, where each set of correlating SNR items is divided into smaller subsets. Then, the correlation among SNR items in the same subset was fully considered, whereas SNR items among different subsets were treated as independent variables. It is obvious that, to obtain exact analysis results, the correlation issue should be completely considered and treated. Hence, in most recent works, such as [11,12], the correlation issue was considered and treated completely, which is also the case of our work here. In [11], closed-form expression of exact OP of cognitive DF relay networks was obtained with additional consideration of maximum transmit power limits of SUs. In the analysis in [11], the correlation issue was treated completely. However, different from our work here, all channels are assumed to be identical Rayleigh fading channels in [11]. In [12], closed-form expression of OP’s upper bound for cognitive relay networks was obtained with the assumption of non-identical Rayleigh fading channels and best relay selection. However, in [12], closed-form expression of exact OP was not obtained, and the work was targeted for networks with AF relaying scheme. In summary, for cognitive DF relay networks, although there have been some interesting works on outage performance analysis, we have not found works obtaining explicit closed-form expressions of exact outage probability (OP) and symbol error rate (SER) by treating the correlation issue completely. We do this work here. In this paper, we investigate outage performance of cognitive DF relay networks with MRC and fully non-identical Rayleigh fading channels by treating the correlation issue completely. Out work here is similar to that in [12], except that DF relay scheme is targeted here whereas AF relay scheme was focused in [12]. Closed-form expressions of exact OP and SER of targeted cognitive DF relay networks are obtained. In order to obtain less accurate but more simpler expressions, lower and upper bounds as well as asymptotic expressions of OP and SER of such networks are also obtained. The correctness of our analysis results are verified through numerical simulations. Both analysis results and simulation results show that how the correlation issue is treated can affect the coding gain whereas has no effect on the diversity gain of the network. Exact OP and SER are bounded from both sides by the lower bound and the upper one tightly, whereas the upper bound is more tighter. The remainder of this paper is organized as follows: Section 2 introduces the network model and the channel model of our targeted cognitive relay network. Then in Section 3, expressions of exact as well as upper and lower bounds of OP and SER of the network are obtained. And then in Section 4, asymptotic expressions of OP and SER are derived. The analysis results are verified through numerical simulations in Section 5. Finally a conclusion is made in Section 6.
2. Network model and channel model The network model of targeted dual-hop cognitive relay networks is shown in Fig. 1. The network model consists of a PU node p and an SU network consisting of an SU source s, an SU destination d, and N SU relays {r1 , . . ., rN }. All the nodes are equipped with mono antenna and adopt half duplex communication protocol. All channels in the network are assumed to be independent non-identical Rayleigh fading channels with instantaneous channel gain guv , which is a random variable follows exponential distribution function fguv (x) = uv e−uv x , u ∈ {s, 1, . . ., N}, v ∈ {d, 1, . . ., N}. Here for notation simplification, in the context of guv and uv , if u or v is in set {1, . . ., N}, the corresponding node is in fact node ru or rv . For easy reference, the notations used in the paper and their meanings are listed in Table 1. Please cite this article in press as: Gao Z, et al. Outage performance of cognitive DF relay networks with nonidentical Rayleigh fading channels and maximal ratio combining. Int J Electron Commun (AEÜ) (2014), http://dx.doi.org/10.1016/j.aeue.2014.08.003
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Table 1 Notations and their meanings. Notation
Meaning
p, s, d r1 , . . ., rN Q 2 guv uv
p represents the PU node, s and d represent the SU source and the SU destination, respectively List of SU relays in the network, and N is the number of SU relays Interference power constraint of node p Variance (noise power) of additive white Gaussian noise in received signal The instantaneous channel gain of link u → v, u ∈ {s, 1, . . ., N}, u ∈ {d, 1, . . ., N}. If u or v is in set {1, . . ., N}, the corresponding node is in fact node ru or rv The parameter of the exponentially distributed variable guv . fguv (x) = uv e−uv x
pu
Transmission power of SU node u, pu =
uv
SNR of the received signal from link u → v, uv =
0 Pr(e) fx (x) Fx (x) E2E th pexact out pSER
0 = Q/ 2 The probability of random event e The probability density function (PDF) of random variable x The cumulative distribution function (CDF) of random variable x The equivalent end-to-end SNR of the cognitive relay network th is the outage threshold pexact out = Pr(E2E < th ) = FE2E (th ) Symbol error rate of the cognitive relay network
D
D = {i|si ≥th } =
Q gup
, u ∈ {s, 1, . . ., N}. guv Q gup 2
gsi Q i ≥th If the size of D equals l, D can also be denoted as Dl g 2 sp
All noises are assumed to be additive white Gaussian noise (AWGN) with zero mean and variance 2 , denoted as N(0, 2 ). For any SU node, the interference power caused by the SU’s transmission on PU node p should not exceed Q, which is the interference power constraint of p. As a result, transmit power pu of SU node u is given by Eq. (1). pu =
Q , gup
u ∈ {s, 1, . . ., N}
(1)
Hence, the signal-to-noise ratio (SNR) uv of the signal received by v from link u → v can be obtained as Eq. (2). uv =
Qg uv , 2 gup
u ∈ {s, 1, . . ., N}, v ∈ {d, 1, . . ., N}
(2)
As in [9], it is pretty straightforward to obtain the cumulative distribution function (CDF) of uv as Eq. (3). Fuv (x) =
uv x , uv x + up 0
u ∈ {s, 1, . . ., N}, v ∈ {d, 1, . . ., N}
(3)
In out cognitive DF relay networks, the cooperative relay-assisted message transmission from s to d is performed in two equal-length time slots. In the first slot, s broadcasts the message with power ps = Q/gsp . The transmitted signal will be received by all the SU relays and d, and SNR of the corresponding signals satisfy Eq. (2). If the SNR of relay ri satisfies si ≥ th , ri is assumed to be able to decode the signal successfully. We denote the set of such relays as D, i.e., D = {i|si ≥th }. In D, the one with largest id is selected, which is responsible for decoding the signal and then rebroadcast the message using power pi = gQ in the second slot. All other relays will keep idle in the second ip
slot. Hence, at the end of the second slot, the SU destination will receive two replica of the message, one from the direct link s → d, and the other from the two-hop relay link s → ri → d, here ri represents the selected SU relay. The SU destination d adopts MRC to combine the signals, thus the spatial diversity is exploited. Hence, the equivalent end-to-end SNR E2E of the network can be expressed as Eq. (4). It is obvious that gsp exists in the denominator of several fractional items in Eq. (4), which leads to the correlation issue specific to cognitive relay networks. E2E = sd + max(id ) = i∈D
gsd Q g Q + max (id ) = sd 2 + gsp 2 i∈{j|sj ≥th } gsp
g Q sj ≥ i∈ j th 2 gsp max
gid Q gip 2
(4)
3. Outage performance analysis 3.1. Exact outage probability Outage occurs when the equivalent end-to-end SNR E2E of the network falls below the threshold th . Hence, outage probability pexact out can be expressed as Eq. (5). Therefore, we can obtain pexact out by obtaining the cumulative distribution function FE2E () of E2E first. pexact out = Pr(E2E < th ) = FE2E (th )
(5)
Since that the common random variable gsp in Eq. (4) leads to the statistical dependence among sd and si , i ∈ {1, . . ., N}, the CDF of some variables (such as sd and si ) conditioned on gsp should be obtained firstly. Considering the statistical independence among random Please cite this article in press as: Gao Z, et al. Outage performance of cognitive DF relay networks with nonidentical Rayleigh fading channels and maximal ratio combining. Int J Electron Commun (AEÜ) (2014), http://dx.doi.org/10.1016/j.aeue.2014.08.003
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variables guv , u ∈ {s, 1, . . ., N} and v ∈ {d, p, 1, . . ., N}, the CDF of Dl conditioned on gsp can be obtained as Eq. (6).
⎛ Pr(Dl |gsp ) =
⎝
Pr(si ≥th |gsp )⎠ ⎝
i∈Dl
⎛ =
⎞⎛
⎝
⎞⎛
e
−si th gsp 0
Pr si < th |gsp
i/ ∈D l
⎠⎝
(1 − e
⎛
⎞ ⎠=
0 si ⎝ Pr
⎞ −si th gsp 0
sp
i∈Dl
≥th |gsp
K
Using the identity of Eq. (7), where
≤ th |gsp
sp
)⎠
⎞ ⎠
(6)
K
is the abbreviation of
n1 ,...,nk
K
K K k (−1)k
k=1
k=0
k!
Pr(Dl |gsp ) = e−
Pr
i/ ∈Dl
K
n1 =1
n2 =1
...
K nk =1 n = / ··· = / nk 1 / n2 =
Eq. (6).
0 si ⎠⎝
i/ ∈D l
i∈Dl
(1 − xk ) =
⎞⎛
, we can obtain Eq. (8) from
(7)
xnt
n1 ,...,nk t=1 k
si th gsp 0
·
i∈Dl
−
N−l N (−1)k
k!
e
snt th gsp 0
t=1
(8)
n1 ...nk ∈ / Dl
k=0
Let Md = max id , the probability density function (PDF) of Md can be obtained as Eq. (9). i∈Dl
fMd (x) =
⎛
⎝
j∈Dl
i= / j∈Dl
⎞ id x ⎠ jd jp 0 2 id x + ip 0 (jd x + jp 0 )
(9)
CDF of E2E conditioned on Dl , Md and gsp can be expressed as Eq. (10). FE2E (x|Dl , Md , gsp ) = 1 − e
−sd (x−Md ) gsp 0
(10)
/ ∅ and Dl = ∅, Eq. (3) can be obtained as Eq. (11), With above results, analyzing pexact out = FE2E (th ) in two cases corresponding to Dl = where J1 and J2 are defined as Eqs. (12) and (13). pexact out = FE2E (th ) = J1 + J2
(11)
/ ∅ and Dl = ∅, respectively. These two equations correspond to the two cases of Dl = J1 =
Dl = / Ø
J2 =
th
∞
FE2E (th |Dl , Md , gsp ) · P(Dl |gsp )fMd (x)fgsp (gsp )dgsp dx
0
(12)
0
∞
FE2E (th |Dl = Ø, Md = 0, gsp ) · P(Dl = Ø|gsp )fgsp (gsp )dgsp
(13)
0
Substituting Eqs. (8)–(10) and fgsp (x) = sp e−sp x into Eq. (12), and with some formula expansion operation, J1 can be expressed in a preliminary form of integrals as Eq. (14).
J1 =
N−l N (−1)k Dl = / Ø j∈Dl k=0
k!
n1 ...nk ∈ / Dl
In Eq. (14), ϕ is defined as ϕ =
sp sd
jp jd
+
si i/ ∈Dl sp
+
si i/ ∈Dl sd
k
+
snt t=1 sp
k
snt t=1 sd
th 0
+1
+1
x+
ip id
x+
x−
(x − ϕ) x +
0
th 0 .
Next, we focus on integrating the rational function of J1 . Let = form of
·
th 0
jp jd
x+
th 0
jp jd
ip id
xl−1
i∈Dl
x+
ip id
dx
(14)
, then is a product of l + 1 items in similar
i∈Dl
. We group the l + 1 items according to the value of
ip . id
In other words, the items with equal
ip id
are grouped into the
same group. This value is called as the feature value of the group. Assume that the l + 1 items are grouped into groups with the ith group Please cite this article in press as: Gao Z, et al. Outage performance of cognitive DF relay networks with nonidentical Rayleigh fading channels and maximal ratio combining. Int J Electron Commun (AEÜ) (2014), http://dx.doi.org/10.1016/j.aeue.2014.08.003
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Fig. 2. The integral regions of exact outage probability and its bounds.
has size ai and feature value ˛i =
˛ p i ˛ d i
. Thus, in Eq. (12) can be expressed as =
i=1
(x +
˛ p ai i
) . By decomposing the rational function
˛ d i
in Eq. (14) into the so-called partial fractions using Eq.(2.102) [17], J1 can be obtained as Eq. (15).
J1 =
Dl = / Ø j∈Dl k=0
=
k!
n1 ···nk ∈ / Dl
k!
(−1) N−l
Dl = / Ø j∈Dl k=0
+1 ·
⎧ ⎪ ⎪ ⎪ ⎨
k!
jd
k sn si t + i/ ∈D l t=1 sp sp
n1 ···nk ∈ / Dl
jd
si + i/ ∈D l sp
n1 ···nk ∈ / Dl
i/ ∈D l
ai −1 ⎢ ⎢ ⎢ ⎣ ε=1
si + sp
t=1
sp
0
·
(x − ϕ)
th th +1 0
0
0
th 0
0
·
x−
i=1
xl−1
x+
A1 + x−ϕ
ai
i=1 ε=1
˛i p
ai dx
˛i d Bi,ai −ε+1 x+
ai p
ai −ε+1 dx
ai d
i=1
(15)
th
k sn 0 t t=1
−Bi,ai −ε+1
A1 ln x − ϕ +
⎪ ⎪ ⎪ ⎩
k sn t
jd
⎡
th +1 0
jp
N
k
th
jp
N−l N (−1)k Dl = / Ø j∈Dl k=0
=
jp
N−l N (−1)k
(ai − ε) x +
˛i p
sp
ai −ε
˛i d
⎤⎫ ⎪ ⎪ th ⎥⎪ ⎬ ˛i p ⎥ | 0 + Bi,1 ln x + ⎥ ˛i d ⎦⎪ ⎪ ⎪ ⎭ 0
In Eq. (15), the coefficients A1 , Bi,ai −ε+1 are given in Eq. (16).
A1 = ϕl−1 (ϕ − th /0 )/
ϕ+
i=1
˛i p
ai
˛i d
⎧ ai ⎫(ε−1) ⎪ ⎪ ˛i p th ⎪ ⎪ l−1 ⎪ ⎪ x− x+ ⎨x ⎬ 0 ˛i d 1 Bi,ai −ε+1 = | a ˛i p j (ε − 1)! ⎪ ⎪ x=− ⎪ ⎪ ⎪ (x − ϕ) j=1 x + ˛j p ⎪ ˛ d ⎩ ⎭ ˛j d
(16)
i
Using a similar process, J2 can be obtained as Eq. (17). J2 =
N N (−1)k
k! k=0
n1 ···nk
th 0
k
snt t=1 sp
+1
th 0 th 0
k
snt t=1 sd
+
th 0
+
sp sd
(17)
Substituting Eqs. (15) and 17) into Eq. (11), the exact closed-form expression of pexact out can be obtained directly. 3.2. Lower and upper bounds of outage probability In Eq. (4), where the integral region is a triangle, sd and Md = max(id ) are tightly coupled, which results in the main complexity in i∈D
deriving the exact closed-form expression of pexact out . To overcome this complexity, we can decouple the inter-effect between the two SNR items by shrinking or expanding the integral region into regular rectangular regions. To make the bounds as tighter as possible, the lower bound region should be the maximum embedded rectangular region of the original one, whereas the upper bound region should be the minimum enclosing rectangular region. Please cite this article in press as: Gao Z, et al. Outage performance of cognitive DF relay networks with nonidentical Rayleigh fading channels and maximal ratio combining. Int J Electron Commun (AEÜ) (2014), http://dx.doi.org/10.1016/j.aeue.2014.08.003
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The integral regions of exact OP and its lower and upper bounds are shown in Fig.$2. In # $ this figure, the rectangular region of ( sd < th ) ( Md < th ) is divided into 4 parts. The integral region of exact OP equals S S S3 . It is obvious that, the region of S1 1 2 $ $ $ can be used to obtain a lower bound of pexact S2 S3 S4 can be used to obtain an upper bound of pexact out , whereas the region of S1 out . In other words, the integral region for the lower bound can be expressed as Eq. (18), whereas the integral region for the upper bound can be expressed as Eq. (19).
⎧ th ⎪ ⎨ sd < 2
(18)
th ⎪ ⎩ Md < 2
%
sd < th (19)
Md < th
By integrating over the modified regions, we could obtain simpler expressions of lower and upper bounds of pexact out . Simulation results in Section 5 show that the bounds are tight. With the new integral region, plower can be obtained as Eq. (20). out plower out
=
k!
Dl k=0
Similarly,
upper pout
upper pout
=
1 l+1 th l+1
N−l N (−1)k
0
2
n1 ···nk ∈ / Dl 1 +
si i/ ∈Dl sp
+
k
id i∈Dl ip
snt t=1 sp
th 0
·
sp sd
+
si i/ ∈Dl sd
+
1
k
snt t=1 sd
+
1 2
th 0
i∈Dl
1 id th 2 ip 0
(20)
+1
can be obtained as Eq. (21).
th l+1
N−l N (−1)k
k!
Dl k=0
n1 ···nk ∈ / Dl 1 +
0
si i/ ∈Dl sp
+
id i∈Dl ip
k
snt t=1 sp
th 0
·
sp sd
+
si i/ ∈Dl sd
+
k
1
snt t=1 sd
+1
th 0
i∈Dl
id th ip 0
(21)
+1
3.3. Symbol error rate Symbol error rate (SER) rate (also known as baud or modulation rate) is the number of symbol changes made to the transmission medium per second. According to Eq.(9) in Ref.[12] (i.e., Eq.(24) in Ref.[13]), a general expression of SER for our targeted cognitive DF relay network is given in Eq. (22) where a and b are determined by modulation scheme. The parameters for some typical modulation schemes are as follows: BPSK(a = 1, b = 1), BFSK with orthogonal signalling(a = 1, b = 0.5), M-ary PAM (a = 2(M − 1)/M, b = 3/(M2 − 1) [13]. √ ∞ e−bth a b pSER = √ F ( )dth (22) √ th E2E th 2 0 We do not find a way to obtain a closed-form expression of exact SER. However, we can obtain the lower and upper bounds of SER in a similar way as in Ref.[12]. In the expression of plower out in Eq. (20), the item
1 id th 2 ip 0
+ 1 is a product of l items in the form of
1 id th 2 ip 0
+ 1 . Just as what we did in
i∈Dl
the previous section for deriving exact OP, we group the items according to the value of groups, and the ith group has size bi and feature value
si i/ ∈Dl sp
By denoting 1 =
k
snt t=1 sp
+
and 2 =
1 ˇi d th . Thus, the item 2 ˇ p
i
si i/ ∈Dl sd
+
k
snt t=1 sd
i∈Dl
+
1 2
id . ip
1 id th 2 ip 0
Assuming that these l items are grouped into I
+ 1 can be re-written as
I 1 ˇi d th i=1
2 ˇ p 0 i
bi
+1
.
, plower can be further expressed in the form of partial out
fractions as Eq. (23). plower = out
C1 1 + 1 th 0
+
C2 sp sd
+ 2 th 0
+
bi I i=1 ε=1
fi,bi −ε+1 1 ˇi d th 2 ˇ p 0
bi −ε+1
(23)
+1
i
The coefficients C1 , C2 and fi,bi −ε+1 in Eqs. (23) are given in Eq. (24)–(26), respectively.
1 l+1 C1 =
2 sp sd
−
2
1
− 1
C2 =
1−
sp 1 sd 2
1
i∈Dl
1 l+1 2
l+1
sp sd 2
−
i∈Dl
id i∈Dl ip
1−
1 1 id 2 1 ip
l+1
1−
id i∈Dl ip
1 id sp 2 ip sd 2
(24)
(25)
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fi,bi −ε+1 =
⎧ ⎪ ⎪ ⎪ ⎨
1 l+1 2
1 (ε − 1)! ⎪
⎪ ⎪ ⎩ (1 + 1 x)
(x)l+1
sp sd
+ 2 x
id i∈Dl ip
⎫(ε−1) ⎪ ⎪ ⎪ ⎬ 1+ i 2 · bi ⎪ x=− ˇi p I ⎪ ˇ d 1 ˇj d ⎪ i 1+ x ⎭
1 ˇj d 2 ˇ p
7
ε−1
1 ˇi d x 2 ˇ p
(26)
2 ˇ p j
j=1
j
bi
, Substituting Eq. (23) into Eq. (22), and using Eq.(3.363.2) and Eq. (3.384.3) in [17], the closed form lower bound of pSER , denoted as plower SER can be obtained as Eq. (27), where erfc(·) is the complementary error function defined by Eq. (8.250.4) in [17], and Wi,j (·) is the Whittaker function defined by Eq. (9.220.4) in [17].
plower SER
⎧ ⎪ '( ) '( ) & ⎪ N−l N ⎨ a&b bsp 0 b0 a b sd 0 bsp 0 (−1)k b0 0 + C2 e sd 2 · erfc & C1 e 1 · erfc & k!
1 sd 2 ⎪ 2 1 2 sp 2 Dl k=0 n1 ···nk ∈ / Dl ⎪ ⎩ 1 2ˇi p 0
a√b fi,bi −ε+1 + √ 2 ˇ d I
bi
2
ˇ d i
bi −ε+1
i
i=1 ε=1
2(ε−bi )−3 4
2ˇ p 0
b
· exp
2(ε−bi )+1 4
bˇi p 0
W 2(ε−bi )−1
ˇi d
4
2(ε−bi )−1 4
,
2bˇi p 0 ˇi d
⎫ ⎪ ⎪ ⎬ (27)
⎪ ⎪ ⎭
i
Similarly, denoting 3 =
si i/ ∈Dl sp
+
k
snt t=1 sp
and 4 =
si i/ ∈Dl sd
+
k
snt t=1 sd
upper
+ 1 , pSER
can be obtained finally as Eq. (28) where
the coefficients C3 , C4 and Fi,bi −ε+1 are given in Eqs. (29)–(31), respectively.
upper
pSER
⎧ ⎪ '( ) '( ) & ⎪ N−l N ⎨ a&b bsp 0 b0 a b sd 0 bsp 0 (−1)k b0 0 + C4 e sd 4 · erfc & & C3 e 3 erfc k! sd 4
3 ⎪ 2 3 2 sp 4 Dl k=0 n1 ···nk ∈ / Dl ⎪ ⎩ a√b + √ 2 I
bi
i=1 ε=1
Fi,bi −ε+1 ˇ d i 2ˇ p 0
1 ˇi p 0
bi −ε+1
2
2(ε−bi )−3 4
b
ˇi d 2(ε−bi )+1 4
· exp
bˇi p 0 2ˇi d
W 2(ε−bi )−1 4
,
2(ε−bi )−1 4
bˇi p 0 ˇi d
⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭
(28)
i
C3 =
− 1
sp sd
−
1−
4
3
i∈Dl
sp sd 4
−
sp 3 sd 4
id i∈Dl ip
3
C4 =
l+1
1−
l+1
i∈Dl
(29)
id i∈Dl ip
1−
⎧ ⎪ ⎪ ⎪ ⎨
Fi,bi −ε+1
1 id
3 ip
1 = (ε − 1)! ⎪
id sp ip sd 4
xl+1
⎪ ⎪ ⎩ (1 + 3 x)
sp sd
(30)
id i∈Dl ip
+ 4 x
ˇ d j ˇ p j
ε−1 ·
1+
I j=1
ˇ d i x ˇ p
bi
i
1+
ˇ d j ˇ p j
⎫(ε−1) ⎪ ⎪ ⎪ ⎬
bi ⎪ ⎪ ⎪ x ⎭
|
x=−
ˇ p i ˇ d i
(31)
4. Asymptotic analysis 4.1. Asymptotic outage probability In the literature, as = Q2 → ∞, simplified approximated expressions of FE2E () in the form of (Gc )−Gd is usually interesting [15], where Gd and Gc are called as diversity gain and coding gain, respectively [16]. Here FE2E () is the CDF of E2E respect to . Diversity gain Gd indicates how fast the outage probability will decrease as → ∞, in other words, how effective that increasing transmission power will be on reducing outage probability. Please cite this article in press as: Gao Z, et al. Outage performance of cognitive DF relay networks with nonidentical Rayleigh fading channels and maximal ratio combining. Int J Electron Commun (AEÜ) (2014), http://dx.doi.org/10.1016/j.aeue.2014.08.003
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We then derive the asymptotic expression of FE2E () as → ∞. Applying the identity of the first order Taylor expansion 1 − e−x = x, Eqs. (6), (9) and (10) can be simplified as Eqs. (32)–(34), respectively.
⎛
Pr(Dl |gsp ) = ⎝
e
−si th gsp 0
⎞⎛
⎞
⎠⎝
si th gsp ⎠ 0
i/ ∈D l
i∈Dl
→∞
fMd (x) =
⎛
(32)
⎞ x jd id ⎠ ⎝ · ip 0
i= / j∈Dl
j∈Dl
(33)
jp 0
sd (th − x) gsp 0
FE2E (th Md = x, gsp ) =
(34)
Plugging Eq. (32)–(34) into Eq. (12) and (13), and using the exponential function Eq. (3.326) in [17], J1 and J2 can be approximated as Eq. (35) and (36), respectively:
⎛ ⎞⎛ ⎞ N+1 sd ⎝ id ⎠ ⎝ si ⎠ (N − l + 2) th J1 = · →∞
Dl = / Ø →∞
J2 =
sp
ip
' N
sd sp
i∈Dl
i/ ∈D l
)
(35)
0
N+1 th
· (N + 2)
si
N+1
l+1
sp
(36)
0
i=1
exact As a result, asymptotic expression of pexact out , denoted as pout,Asymp , can be approximated as Eq. (37).
pexact out,Asym =
⎛
→∞
sd
Dl
sp
⎞⎛
⎝
id
i∈Dl
ip
⎞
⎠⎝
si
i/ ∈D l
sp
N+1 ⎠ (N − l + 2) th l+1
(37)
0
upper
and pout,Asym , can be derived as Eqs. (38) and (39). Similarly, the asymptotic expressions for the bounds of OP, denoted as plower out,Asym plower out,Asym
⎛ ⎞⎛ ⎞ N+1 sd ⎝ id ⎠ ⎝ si ⎠ = (N − l + 2) th
→∞
Dl
2sp
⎛
→∞ upper pout,Asym =
sd
Dl
i∈Dl
sp
2ip
⎞⎛
⎝
id
i∈Dl
i/ ∈D l
ip
⎞
⎠⎝
si
i/ ∈D l
(38)
0
sp
sp
N+1 ⎠ (N − l + 2) th
(39)
0
Moreover, neglecting the correlation issue as in [9], the asymptotic expression of conventional analysis result of outage probability, , can be obtained as Eq. (40). denoted as pconv out,Asym
→∞
pconv out,Asym =
Dl
⎛
⎝
si
i/ ∈D l
sp
⎞⎛
⎠⎝
id
i∈Dl
ip
⎞
N+1 ⎠ sd (2) th sp l + 1
(40)
0
From Eq. (37)–(40), it can be seen that diversity gain of these 4 cases are all equal to N + 1, which is only determined by the number of disjoint paths from the source s to the destination d. On the contrary, the cases have different coding gain Gc . 4.2. Asymptotic symbol error rate Substituting FE2E () in Eq. (22) with pexact defined in Eq. (37), and using Eq.(3.371) in [17], the corresponding asymptotic expression out,Asym of SER, denoted as pexact , can be obtained as Eq. (41). Similarity, other asymptotic expressions of SER, denoted as plower and SER,Asymp SER,Asymp upper
pSER,Asymp , can be obtained as Eq. (42) and (43), respectively. In these equations, (2N + 1) ! ! is defined as (2N + 1) ! ! =1 · 3 . . . (2N + 1).
→∞
sd
pexact SER,Asymp =
Dl
plower SER,Asymp
⎝
id
i∈Dl
ip
⎞⎛
⎠⎝
si
i/ ∈D l
sp
⎞
⎠ · (N − l + 2) l+1
1 N+1 a · (2N + 1)!! 2N+2 · bN+1
0
⎛ ⎞⎛ ⎞ 1 N+1 a · (2N + 1)!! /2 /2 sd id si ⎠ ⎝ ⎠⎝ = · (N − l + 2) N+2 N+1
→∞
Dl
upper
sp
⎛
sp
→∞
sd
pSER,Asymp =
Dl
sp
⎛
i∈Dl
⎝
id
i∈Dl
ip
ip
i/ ∈Dl
⎞⎛
⎠⎝
si
i/ ∈D l
sp
0
sp
2
·b
(41)
(42)
⎞
N+1 a · (2N + 1)!! ⎠ · (N − l + 2) 1 0
2N+2 · bN+1
(43)
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9
0
10
−2
Outage Probability
10
−4
10
−6
10
−8
10
−10
10
−12
10
0
Sim. N=2 Sim. N=3 Sim. N=5 Exact OP. N=2 Exact OP. N=3 Exact OP. N=5 Asymptotic. N=2 Asymptotic. N=3 Asymptotic. N=5 5
10
15
20
25
30
Q dB Fig. 3. The effect of N on outage probability of networks. 0
Outage Probability
10
−5
10
−10
10
−15
10
0
upper bound exact OP (i.n.d) lower bound coventional OP (i.i.d) Asymptotic upper Asymptotic OP (i.n.d) Asymptotic lower Asymptotic conventional Asymptotic OP (i.i.d)
5
10
N=5
15
20
25
30
Q dB Fig. 4. The effect of how the correlation issue is treated on outage probability of networks. 0
10
−1
10
Symbol Error Rate
−2
10
−3
10
−4
10
−5
10
−6
10
−7
10
−8
10
0
exact SER upper bound lower bound Asymptotic SER Asymptotic upper bound Asymptotic lower bound
5
10
15
20
25
30
Q dB Fig. 5. Verification of simulation results and theoretical results about SER of networks with N = 2.
5. Numerical results We verify the rightness of our analysis results through four numerical experiments. In the first experiment, the effect of parameter N on outage performances is inspected. Simulations are performed for N = 2, 3, and5. 5 5 Other main parameters in the simulations are set as follows: sp = sd = 1, {si }i=1 = {2, 2.1, 2.2, 2.3, 2.4}, {id }i=1 = {1.1, 1.2, 1.3, 1.4, 1.5}, Please cite this article in press as: Gao Z, et al. Outage performance of cognitive DF relay networks with nonidentical Rayleigh fading channels and maximal ratio combining. Int J Electron Commun (AEÜ) (2014), http://dx.doi.org/10.1016/j.aeue.2014.08.003
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{ip }i=1 = {1.1, 1.2, 1.3, 1.4, 1.5}, th = 3 dB and 2 = 1 dB. For each simulation configuration, Q varies from 0 dB to 30 dB in step size 5 dB. The simulation results are shown in Fig. 3. The fine match between the simulation results and the analysis results confirms the rightness of our analysis results. The results show that increasing N can decrease outage probability significantly. Furthermore, the asymptotic lines imply that diversity gain is solely determined by N. In the second experiment, we inspect the effects of the ways how the correlation issue is treated. The simulations in this experiment correspond to 5 cases: (1) exact outage probability of networks with independent non-identical (i.n.d) channels; (2) upper bound OP of networks with i.n.d channels; (3) lower bound OP of networks with i.n.d channels; (4) conventional analysis results (where the correlation issue is completely ignored) of networks with i.n.d channels; (5) OP of networks with independent identical (i.i.d) channels. Main parameters of the first four cases are identical to those in the first experiment, except that N =5 here. In the 5th case, main parameters are set as follows: sp = ip = 2 si = sd = i,d = 1 for i ∈ {1, . . ., N}, N = 5. The asymptotic results show that these cases have identical diversity gain but different coding gains. The results in Fig. 4 show the following three facts: (1) The diversity gain is solely determined by the number of disjoint paths from the source to the destination; (2) ignoring the correlation issue leads to over-optimistic (lower) estimation of outage probability; (3) exact OP is bounded from both sides by the lower bound and the upper one tightly. The results about SER of our targeted cognitive DF relay networks are verified through a third experiment. In the simulations of the experiment, N is fixed to 2. The corresponding results are shown in Fig. 5. The results show that, the upper bound of SER is much more tighter than the lower bound. 6. Conclusion In this paper, for cognitive decode-and-forward (DF) relay networks with maximal ratio combing (MRC) and independent non-identical Rayleigh fading channels, we analyze outage performance of such networks taking consideration of the correlation issue completely. An closed-form expression of exact outage probability (OP) is obtained. Then, by shrinking or expanding the triangular integral region to suitable rectangular regions, simpler closed-form expressions of the lower and upper bounds of both OP and symbol error rate (SER) are obtained. Furthermore, the asymptotic expressions of OP and SER are also obtained. The correctness of our analysis results are verified through numerical simulations. Both analysis results and simulation results show that how the correlation issue is treated can affect the coding gain whereas has no effect on the diversity gain of the network. Simulation results indicate that, exact OP and exact SER are bounded from both sides by the lower bound and the upper one tightly, whereas the upper bounds of them are both more tighter. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [15] [16] [17]
Nosratinia A, Hunter TE. Cooperative communication in wireless networks. IEEE Commun Mag 2004;42(10):74–80. Zou Y, Yao YD, Zheng B. Opportunistic distributed space-time coding for decode-and-forward cooperation systems. IEEE Trans Signal Process 2012;60(4):1766–81. Duong TQ, Costa DBD, Elkashlan M, Bao VNQ. Cognitive amplify-and-forward relay networks over Nakagami-m fading. IEEE Trans Veh Technol 2012;61(5):1001–2. Wang J, Huang AP, Cai L, Wang W. On the queue dynamics of multi-user multi-channel cognitive radio networks. IEEE Trans Veh Technol 2013;62(3):1314–28. Marinho JM, Monteiro E. Cognitive radio: survey on communication protocols, spectrum decision issues, and future research directions. Wirel Netw 2012;18(2):147–64. Ghasemi A, Sousa ES. Fundamental limits of spectrum sharing in fading environments. IEEE Trans Wirel Commun 2007;6:649–58. Xia MH, Aïssa S. Cooperative AF relaying in spectrum-sharing systems: performance analysis under average interference power constraints and Nakagami-m fading. IEEE Trans Commun 2012;60(6):1523–33. Xia MH, Aïssa S. Cooperative AF relaying in spectrum-sharing systems: outage probability analysis under co-channel interferences and relay selection. IEEE Trans Commun 2012;60(11):3252–62. Guo Y, Kang G, Zhang N, Zhou W, Zhang P. Outage performance of relay-assisted cognitive-radio system under spectrum-sharing constraints. IEL Electron Lett 2010;46(2):182–3. Luo LP, Zhang P, Zhang GC, Qin JY. Outage performance for cognitive relay networks with underlay spectrum sharing. IEEE Commun Lett 2011;15(7):710–2. Yan Z, Zhang X, Wang WB. Exact outage performance of cognitive relay networks with maximum transmit power limits. IEEE Commun Lett 2011;5(12):1317–9. Bao VNQ, Duong TQ, Costa DBD, Alexandropoulos GC, Nallanathan A. Cognitive amplify-and-forward relaying with best relay selection in nonidentical rayleigh fading. IEEE Commun Lett 2013;17(3):475–8. McKay MR, Grant AJ, Collings IB. Performance analysis of MIMO-MRC in double-correlated Rayleigh environments. IEEE Trans Commun 2007;55(3), 497C507. Kim KJ, Duongy TQ, Tsiftsisz TA, Bao VNQ. Cognitive multihop networks in spectrum sharing environment with multiple licensed users. In: Proc. IEEE international conference on communications (ICC2013). 2013. Wang ZD, Giannakis GB. A simple and general parameterization quantifying performance in fading channels. IEEE Trans Commun 2003;51(8):1389–98. Gradshteyn IS, Ryzhik IM. Table of integrals, series, and products. 6th ed. San Diego: Academic; 2000.
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Contents lists available at ScienceDirect
International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.com/locate/aeue
Outage performance of cognitive DF relay networks with nonidentical Rayleigh fading channels and maximal ratio combining夽 Zhenguo Gao a,∗ , Kaichen Zhang a , Danjie Chen b , Wei Zhang a , Yibing Li c a b c
College of Automation, Harbin Engineering University, Harbin, China College of Software, Beijing University of Technology, Beijing, China College of Information and Communication Engineering, Harbin Engineering University, Harbin, China
a r t i c l e
i n f o
Article history: Received 2 May 2014 Accepted 12 August 2014 Keywords: Decode-and-forward Outage probability Symbol error rate Asymptotic expression Non-identical Rayleigh fading
a b s t r a c t In the expression of end-to-end signal-to-noise ratio (SNR) of cognitive relay networks, the channel gaina of the interference links (from secondary nodes to primary nodes in the networks) are usually shared by multiple items in the expression. These shared variables lead to the so called correlation issue, which results in the higher complexity of outage performance analysis for such networks. Till now, for cognitive relay networks with decode-and-forward (DF) relay scheme, we have not found works obtaining explicit closed-form expression of exact outage probability (OP) meanwhile treating the correlation issue completely. In this paper, for cognitive DF relay networks with maximal ratio combing (MRC) and independent non-identical Rayleigh fading channels, we obtain a closed-form expression of exact OP taking consideration of the correlation issue completely. Additionally, by shrinking or expanding the triangular integral region to suitable rectangular regions, we obtain simpler closed-form expressions of the lower and upper bounds of both OP and symbol error probability (SER). Furthermore, the asymptotic expressions of OP and SER are also obtained. The correctness of our analysis results are verified through numerical simulations. Both analysis results and simulation results show that how the correlation issue is treated can affect the coding gain whereas has no effect on the diversity gain of the network. Simulation results also indicate that, exact OP and SER are bounded from both sides by the lower bound and the upper one tightly, whereas the upper bounds of OP and SER are more tighter than the lower bounds. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction Applying multiple antennas enables the terminals to combat with channel fading effects. However, the requirements on hardware size would be prohibitive for hand-held communication devices [1,2]. To that end, cooperative communication is emerging as an alternative approach by forming a virtual antenna array to perform the function of multiple antennas [3]. By allowing secondary users (SUs) to simultaneously share the frequency band licensed to primary users (PUs) without causing harmful interferences to PUs, cognitive radio can significantly improve spectrum efficiency [4–6]. As a combination of these two attractive techniques, i.e., cooperative communication and cognitive radio, cognitive relay networks have gained much research attention in recent years. For cognitive relay networks, two typical relaying strategies are decode-and-forward (DF) and amplify-and-forward (AF) [7,8]. A DF relay decodes the received signal, re-encodes and forwards it to the destination, whereas a AF relay just amplifies the received signal directly and then forwards it to the destination node [7,8]. At the destination, the receiver employs diversity combining techniques to obtain spatial diversity from signal replicas received from SU relays and the SU source. Typical diversity combining techniques are maximal ratio combining (MRC), equal-gain combining (EGC), selection combining (SC). In cognitive relay networks, the interference power caused by an SU’s transmission on a PU receiver should not exceed the PU’s threshold. This is the so called interference power constraint, which is a main issue specific to cognitive relay networks. This interference
夽 This work has been supported by Natural Science Foundation of China (No.61073183); Natural Science Foundation of HeiLongJiang Province (No F201120); Fundamental Research Funds for the Central Universities (No. HEUCFX41311). ∗ Corresponding author. E-mail address: [email protected] (Z. Gao). http://dx.doi.org/10.1016/j.aeue.2014.08.003 1434-8411/© 2014 Elsevier GmbH. All rights reserved.
Please cite this article in press as: Gao Z, et al. Outage performance of cognitive DF relay networks with nonidentical Rayleigh fading channels and maximal ratio combining. Int J Electron Commun (AEÜ) (2014), http://dx.doi.org/10.1016/j.aeue.2014.08.003
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Fig. 1. Network model of targeted dual-hop cognitive relay networks.
power constraint of PU leads to the correlation issue in outage performance analysis of such networks, which intrinsically result from the interference channel gain of the interference link from a SU node to the PU shared by multiple signal-to-noise ratio (SNR) items in the expression of end-to-end SNR of cognitive relay networks. The correlation issue should be elaborately treated otherwise the analysis results will be incorrect. In the literature, outage performance of cognitive relay networks with DF and MRC have been investigated in many works including references [9,10]. In these works, the correlation issue was treated in different levels. For example, in [9], the SNR items in the expression of end-to-end SNR of cognitive relay networks were all considered as independent variables, thus the correlation issue is completely ignored. This treatment is obviously not appropriate, as has been verified in [10]. However, in [10], the correlation issue was only partially considered, where each set of correlating SNR items is divided into smaller subsets. Then, the correlation among SNR items in the same subset was fully considered, whereas SNR items among different subsets were treated as independent variables. It is obvious that, to obtain exact analysis results, the correlation issue should be completely considered and treated. Hence, in most recent works, such as [11,12], the correlation issue was considered and treated completely, which is also the case of our work here. In [11], closed-form expression of exact OP of cognitive DF relay networks was obtained with additional consideration of maximum transmit power limits of SUs. In the analysis in [11], the correlation issue was treated completely. However, different from our work here, all channels are assumed to be identical Rayleigh fading channels in [11]. In [12], closed-form expression of OP’s upper bound for cognitive relay networks was obtained with the assumption of non-identical Rayleigh fading channels and best relay selection. However, in [12], closed-form expression of exact OP was not obtained, and the work was targeted for networks with AF relaying scheme. In summary, for cognitive DF relay networks, although there have been some interesting works on outage performance analysis, we have not found works obtaining explicit closed-form expressions of exact outage probability (OP) and symbol error rate (SER) by treating the correlation issue completely. We do this work here. In this paper, we investigate outage performance of cognitive DF relay networks with MRC and fully non-identical Rayleigh fading channels by treating the correlation issue completely. Out work here is similar to that in [12], except that DF relay scheme is targeted here whereas AF relay scheme was focused in [12]. Closed-form expressions of exact OP and SER of targeted cognitive DF relay networks are obtained. In order to obtain less accurate but more simpler expressions, lower and upper bounds as well as asymptotic expressions of OP and SER of such networks are also obtained. The correctness of our analysis results are verified through numerical simulations. Both analysis results and simulation results show that how the correlation issue is treated can affect the coding gain whereas has no effect on the diversity gain of the network. Exact OP and SER are bounded from both sides by the lower bound and the upper one tightly, whereas the upper bound is more tighter. The remainder of this paper is organized as follows: Section 2 introduces the network model and the channel model of our targeted cognitive relay network. Then in Section 3, expressions of exact as well as upper and lower bounds of OP and SER of the network are obtained. And then in Section 4, asymptotic expressions of OP and SER are derived. The analysis results are verified through numerical simulations in Section 5. Finally a conclusion is made in Section 6.
2. Network model and channel model The network model of targeted dual-hop cognitive relay networks is shown in Fig. 1. The network model consists of a PU node p and an SU network consisting of an SU source s, an SU destination d, and N SU relays {r1 , . . ., rN }. All the nodes are equipped with mono antenna and adopt half duplex communication protocol. All channels in the network are assumed to be independent non-identical Rayleigh fading channels with instantaneous channel gain guv , which is a random variable follows exponential distribution function fguv (x) = uv e−uv x , u ∈ {s, 1, . . ., N}, v ∈ {d, 1, . . ., N}. Here for notation simplification, in the context of guv and uv , if u or v is in set {1, . . ., N}, the corresponding node is in fact node ru or rv . For easy reference, the notations used in the paper and their meanings are listed in Table 1. Please cite this article in press as: Gao Z, et al. Outage performance of cognitive DF relay networks with nonidentical Rayleigh fading channels and maximal ratio combining. Int J Electron Commun (AEÜ) (2014), http://dx.doi.org/10.1016/j.aeue.2014.08.003
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Table 1 Notations and their meanings. Notation
Meaning
p, s, d r1 , . . ., rN Q 2 guv uv
p represents the PU node, s and d represent the SU source and the SU destination, respectively List of SU relays in the network, and N is the number of SU relays Interference power constraint of node p Variance (noise power) of additive white Gaussian noise in received signal The instantaneous channel gain of link u → v, u ∈ {s, 1, . . ., N}, u ∈ {d, 1, . . ., N}. If u or v is in set {1, . . ., N}, the corresponding node is in fact node ru or rv The parameter of the exponentially distributed variable guv . fguv (x) = uv e−uv x
pu
Transmission power of SU node u, pu =
uv
SNR of the received signal from link u → v, uv =
0 Pr(e) fx (x) Fx (x) E2E th pexact out pSER
0 = Q/ 2 The probability of random event e The probability density function (PDF) of random variable x The cumulative distribution function (CDF) of random variable x The equivalent end-to-end SNR of the cognitive relay network th is the outage threshold pexact out = Pr(E2E < th ) = FE2E (th ) Symbol error rate of the cognitive relay network
D
D = {i|si ≥th } =
Q gup
, u ∈ {s, 1, . . ., N}. guv Q gup 2
gsi Q i ≥th If the size of D equals l, D can also be denoted as Dl g 2 sp
All noises are assumed to be additive white Gaussian noise (AWGN) with zero mean and variance 2 , denoted as N(0, 2 ). For any SU node, the interference power caused by the SU’s transmission on PU node p should not exceed Q, which is the interference power constraint of p. As a result, transmit power pu of SU node u is given by Eq. (1). pu =
Q , gup
u ∈ {s, 1, . . ., N}
(1)
Hence, the signal-to-noise ratio (SNR) uv of the signal received by v from link u → v can be obtained as Eq. (2). uv =
Qg uv , 2 gup
u ∈ {s, 1, . . ., N}, v ∈ {d, 1, . . ., N}
(2)
As in [9], it is pretty straightforward to obtain the cumulative distribution function (CDF) of uv as Eq. (3). Fuv (x) =
uv x , uv x + up 0
u ∈ {s, 1, . . ., N}, v ∈ {d, 1, . . ., N}
(3)
In out cognitive DF relay networks, the cooperative relay-assisted message transmission from s to d is performed in two equal-length time slots. In the first slot, s broadcasts the message with power ps = Q/gsp . The transmitted signal will be received by all the SU relays and d, and SNR of the corresponding signals satisfy Eq. (2). If the SNR of relay ri satisfies si ≥ th , ri is assumed to be able to decode the signal successfully. We denote the set of such relays as D, i.e., D = {i|si ≥th }. In D, the one with largest id is selected, which is responsible for decoding the signal and then rebroadcast the message using power pi = gQ in the second slot. All other relays will keep idle in the second ip
slot. Hence, at the end of the second slot, the SU destination will receive two replica of the message, one from the direct link s → d, and the other from the two-hop relay link s → ri → d, here ri represents the selected SU relay. The SU destination d adopts MRC to combine the signals, thus the spatial diversity is exploited. Hence, the equivalent end-to-end SNR E2E of the network can be expressed as Eq. (4). It is obvious that gsp exists in the denominator of several fractional items in Eq. (4), which leads to the correlation issue specific to cognitive relay networks. E2E = sd + max(id ) = i∈D
gsd Q g Q + max (id ) = sd 2 + gsp 2 i∈{j|sj ≥th } gsp
g Q sj ≥ i∈ j th 2 gsp max
gid Q gip 2
(4)
3. Outage performance analysis 3.1. Exact outage probability Outage occurs when the equivalent end-to-end SNR E2E of the network falls below the threshold th . Hence, outage probability pexact out can be expressed as Eq. (5). Therefore, we can obtain pexact out by obtaining the cumulative distribution function FE2E () of E2E first. pexact out = Pr(E2E < th ) = FE2E (th )
(5)
Since that the common random variable gsp in Eq. (4) leads to the statistical dependence among sd and si , i ∈ {1, . . ., N}, the CDF of some variables (such as sd and si ) conditioned on gsp should be obtained firstly. Considering the statistical independence among random Please cite this article in press as: Gao Z, et al. Outage performance of cognitive DF relay networks with nonidentical Rayleigh fading channels and maximal ratio combining. Int J Electron Commun (AEÜ) (2014), http://dx.doi.org/10.1016/j.aeue.2014.08.003
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variables guv , u ∈ {s, 1, . . ., N} and v ∈ {d, p, 1, . . ., N}, the CDF of Dl conditioned on gsp can be obtained as Eq. (6).
⎛ Pr(Dl |gsp ) =
⎝
Pr(si ≥th |gsp )⎠ ⎝
i∈Dl
⎛ =
⎞⎛
⎝
⎞⎛
e
−si th gsp 0
Pr si < th |gsp
i/ ∈D l
⎠⎝
(1 − e
⎛
⎞ ⎠=
0 si ⎝ Pr
⎞ −si th gsp 0
sp
i∈Dl
≥th |gsp
K
Using the identity of Eq. (7), where
≤ th |gsp
sp
)⎠
⎞ ⎠
(6)
K
is the abbreviation of
n1 ,...,nk
K
K K k (−1)k
k=1
k=0
k!
Pr(Dl |gsp ) = e−
Pr
i/ ∈Dl
K
n1 =1
n2 =1
...
K nk =1 n = / ··· = / nk 1 / n2 =
Eq. (6).
0 si ⎠⎝
i/ ∈D l
i∈Dl
(1 − xk ) =
⎞⎛
, we can obtain Eq. (8) from
(7)
xnt
n1 ,...,nk t=1 k
si th gsp 0
·
i∈Dl
−
N−l N (−1)k
k!
e
snt th gsp 0
t=1
(8)
n1 ...nk ∈ / Dl
k=0
Let Md = max id , the probability density function (PDF) of Md can be obtained as Eq. (9). i∈Dl
fMd (x) =
⎛
⎝
j∈Dl
i= / j∈Dl
⎞ id x ⎠ jd jp 0 2 id x + ip 0 (jd x + jp 0 )
(9)
CDF of E2E conditioned on Dl , Md and gsp can be expressed as Eq. (10). FE2E (x|Dl , Md , gsp ) = 1 − e
−sd (x−Md ) gsp 0
(10)
/ ∅ and Dl = ∅, Eq. (3) can be obtained as Eq. (11), With above results, analyzing pexact out = FE2E (th ) in two cases corresponding to Dl = where J1 and J2 are defined as Eqs. (12) and (13). pexact out = FE2E (th ) = J1 + J2
(11)
/ ∅ and Dl = ∅, respectively. These two equations correspond to the two cases of Dl = J1 =
Dl = / Ø
J2 =
th
∞
FE2E (th |Dl , Md , gsp ) · P(Dl |gsp )fMd (x)fgsp (gsp )dgsp dx
0
(12)
0
∞
FE2E (th |Dl = Ø, Md = 0, gsp ) · P(Dl = Ø|gsp )fgsp (gsp )dgsp
(13)
0
Substituting Eqs. (8)–(10) and fgsp (x) = sp e−sp x into Eq. (12), and with some formula expansion operation, J1 can be expressed in a preliminary form of integrals as Eq. (14).
J1 =
N−l N (−1)k Dl = / Ø j∈Dl k=0
k!
n1 ...nk ∈ / Dl
In Eq. (14), ϕ is defined as ϕ =
sp sd
jp jd
+
si i/ ∈Dl sp
+
si i/ ∈Dl sd
k
+
snt t=1 sp
k
snt t=1 sd
th 0
+1
+1
x+
ip id
x+
x−
(x − ϕ) x +
0
th 0 .
Next, we focus on integrating the rational function of J1 . Let = form of
·
th 0
jp jd
x+
th 0
jp jd
ip id
xl−1
i∈Dl
x+
ip id
dx
(14)
, then is a product of l + 1 items in similar
i∈Dl
. We group the l + 1 items according to the value of
ip . id
In other words, the items with equal
ip id
are grouped into the
same group. This value is called as the feature value of the group. Assume that the l + 1 items are grouped into groups with the ith group Please cite this article in press as: Gao Z, et al. Outage performance of cognitive DF relay networks with nonidentical Rayleigh fading channels and maximal ratio combining. Int J Electron Commun (AEÜ) (2014), http://dx.doi.org/10.1016/j.aeue.2014.08.003
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5
Fig. 2. The integral regions of exact outage probability and its bounds.
has size ai and feature value ˛i =
˛ p i ˛ d i
. Thus, in Eq. (12) can be expressed as =
i=1
(x +
˛ p ai i
) . By decomposing the rational function
˛ d i
in Eq. (14) into the so-called partial fractions using Eq.(2.102) [17], J1 can be obtained as Eq. (15).
J1 =
Dl = / Ø j∈Dl k=0
=
k!
n1 ···nk ∈ / Dl
k!
(−1) N−l
Dl = / Ø j∈Dl k=0
+1 ·
⎧ ⎪ ⎪ ⎪ ⎨
k!
jd
k sn si t + i/ ∈D l t=1 sp sp
n1 ···nk ∈ / Dl
jd
si + i/ ∈D l sp
n1 ···nk ∈ / Dl
i/ ∈D l
ai −1 ⎢ ⎢ ⎢ ⎣ ε=1
si + sp
t=1
sp
0
·
(x − ϕ)
th th +1 0
0
0
th 0
0
·
x−
i=1
xl−1
x+
A1 + x−ϕ
ai
i=1 ε=1
˛i p
ai dx
˛i d Bi,ai −ε+1 x+
ai p
ai −ε+1 dx
ai d
i=1
(15)
th
k sn 0 t t=1
−Bi,ai −ε+1
A1 ln x − ϕ +
⎪ ⎪ ⎪ ⎩
k sn t
jd
⎡
th +1 0
jp
N
k
th
jp
N−l N (−1)k Dl = / Ø j∈Dl k=0
=
jp
N−l N (−1)k
(ai − ε) x +
˛i p
sp
ai −ε
˛i d
⎤⎫ ⎪ ⎪ th ⎥⎪ ⎬ ˛i p ⎥ | 0 + Bi,1 ln x + ⎥ ˛i d ⎦⎪ ⎪ ⎪ ⎭ 0
In Eq. (15), the coefficients A1 , Bi,ai −ε+1 are given in Eq. (16).
A1 = ϕl−1 (ϕ − th /0 )/
ϕ+
i=1
˛i p
ai
˛i d
⎧ ai ⎫(ε−1) ⎪ ⎪ ˛i p th ⎪ ⎪ l−1 ⎪ ⎪ x− x+ ⎨x ⎬ 0 ˛i d 1 Bi,ai −ε+1 = | a ˛i p j (ε − 1)! ⎪ ⎪ x=− ⎪ ⎪ ⎪ (x − ϕ) j=1 x + ˛j p ⎪ ˛ d ⎩ ⎭ ˛j d
(16)
i
Using a similar process, J2 can be obtained as Eq. (17). J2 =
N N (−1)k
k! k=0
n1 ···nk
th 0
k
snt t=1 sp
+1
th 0 th 0
k
snt t=1 sd
+
th 0
+
sp sd
(17)
Substituting Eqs. (15) and 17) into Eq. (11), the exact closed-form expression of pexact out can be obtained directly. 3.2. Lower and upper bounds of outage probability In Eq. (4), where the integral region is a triangle, sd and Md = max(id ) are tightly coupled, which results in the main complexity in i∈D
deriving the exact closed-form expression of pexact out . To overcome this complexity, we can decouple the inter-effect between the two SNR items by shrinking or expanding the integral region into regular rectangular regions. To make the bounds as tighter as possible, the lower bound region should be the maximum embedded rectangular region of the original one, whereas the upper bound region should be the minimum enclosing rectangular region. Please cite this article in press as: Gao Z, et al. Outage performance of cognitive DF relay networks with nonidentical Rayleigh fading channels and maximal ratio combining. Int J Electron Commun (AEÜ) (2014), http://dx.doi.org/10.1016/j.aeue.2014.08.003
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The integral regions of exact OP and its lower and upper bounds are shown in Fig.$2. In # $ this figure, the rectangular region of ( sd < th ) ( Md < th ) is divided into 4 parts. The integral region of exact OP equals S S S3 . It is obvious that, the region of S1 1 2 $ $ $ can be used to obtain a lower bound of pexact S2 S3 S4 can be used to obtain an upper bound of pexact out , whereas the region of S1 out . In other words, the integral region for the lower bound can be expressed as Eq. (18), whereas the integral region for the upper bound can be expressed as Eq. (19).
⎧ th ⎪ ⎨ sd < 2
(18)
th ⎪ ⎩ Md < 2
%
sd < th (19)
Md < th
By integrating over the modified regions, we could obtain simpler expressions of lower and upper bounds of pexact out . Simulation results in Section 5 show that the bounds are tight. With the new integral region, plower can be obtained as Eq. (20). out plower out
=
k!
Dl k=0
Similarly,
upper pout
upper pout
=
1 l+1 th l+1
N−l N (−1)k
0
2
n1 ···nk ∈ / Dl 1 +
si i/ ∈Dl sp
+
k
id i∈Dl ip
snt t=1 sp
th 0
·
sp sd
+
si i/ ∈Dl sd
+
1
k
snt t=1 sd
+
1 2
th 0
i∈Dl
1 id th 2 ip 0
(20)
+1
can be obtained as Eq. (21).
th l+1
N−l N (−1)k
k!
Dl k=0
n1 ···nk ∈ / Dl 1 +
0
si i/ ∈Dl sp
+
id i∈Dl ip
k
snt t=1 sp
th 0
·
sp sd
+
si i/ ∈Dl sd
+
k
1
snt t=1 sd
+1
th 0
i∈Dl
id th ip 0
(21)
+1
3.3. Symbol error rate Symbol error rate (SER) rate (also known as baud or modulation rate) is the number of symbol changes made to the transmission medium per second. According to Eq.(9) in Ref.[12] (i.e., Eq.(24) in Ref.[13]), a general expression of SER for our targeted cognitive DF relay network is given in Eq. (22) where a and b are determined by modulation scheme. The parameters for some typical modulation schemes are as follows: BPSK(a = 1, b = 1), BFSK with orthogonal signalling(a = 1, b = 0.5), M-ary PAM (a = 2(M − 1)/M, b = 3/(M2 − 1) [13]. √ ∞ e−bth a b pSER = √ F ( )dth (22) √ th E2E th 2 0 We do not find a way to obtain a closed-form expression of exact SER. However, we can obtain the lower and upper bounds of SER in a similar way as in Ref.[12]. In the expression of plower out in Eq. (20), the item
1 id th 2 ip 0
+ 1 is a product of l items in the form of
1 id th 2 ip 0
+ 1 . Just as what we did in
i∈Dl
the previous section for deriving exact OP, we group the items according to the value of groups, and the ith group has size bi and feature value
si i/ ∈Dl sp
By denoting 1 =
k
snt t=1 sp
+
and 2 =
1 ˇi d th . Thus, the item 2 ˇ p
i
si i/ ∈Dl sd
+
k
snt t=1 sd
i∈Dl
+
1 2
id . ip
1 id th 2 ip 0
Assuming that these l items are grouped into I
+ 1 can be re-written as
I 1 ˇi d th i=1
2 ˇ p 0 i
bi
+1
.
, plower can be further expressed in the form of partial out
fractions as Eq. (23). plower = out
C1 1 + 1 th 0
+
C2 sp sd
+ 2 th 0
+
bi I i=1 ε=1
fi,bi −ε+1 1 ˇi d th 2 ˇ p 0
bi −ε+1
(23)
+1
i
The coefficients C1 , C2 and fi,bi −ε+1 in Eqs. (23) are given in Eq. (24)–(26), respectively.
1 l+1 C1 =
2 sp sd
−
2
1
− 1
C2 =
1−
sp 1 sd 2
1
i∈Dl
1 l+1 2
l+1
sp sd 2
−
i∈Dl
id i∈Dl ip
1−
1 1 id 2 1 ip
l+1
1−
id i∈Dl ip
1 id sp 2 ip sd 2
(24)
(25)
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fi,bi −ε+1 =
⎧ ⎪ ⎪ ⎪ ⎨
1 l+1 2
1 (ε − 1)! ⎪
⎪ ⎪ ⎩ (1 + 1 x)
(x)l+1
sp sd
+ 2 x
id i∈Dl ip
⎫(ε−1) ⎪ ⎪ ⎪ ⎬ 1+ i 2 · bi ⎪ x=− ˇi p I ⎪ ˇ d 1 ˇj d ⎪ i 1+ x ⎭
1 ˇj d 2 ˇ p
7
ε−1
1 ˇi d x 2 ˇ p
(26)
2 ˇ p j
j=1
j
bi
, Substituting Eq. (23) into Eq. (22), and using Eq.(3.363.2) and Eq. (3.384.3) in [17], the closed form lower bound of pSER , denoted as plower SER can be obtained as Eq. (27), where erfc(·) is the complementary error function defined by Eq. (8.250.4) in [17], and Wi,j (·) is the Whittaker function defined by Eq. (9.220.4) in [17].
plower SER
⎧ ⎪ '( ) '( ) & ⎪ N−l N ⎨ a&b bsp 0 b0 a b sd 0 bsp 0 (−1)k b0 0 + C2 e sd 2 · erfc & C1 e 1 · erfc & k!
1 sd 2 ⎪ 2 1 2 sp 2 Dl k=0 n1 ···nk ∈ / Dl ⎪ ⎩ 1 2ˇi p 0
a√b fi,bi −ε+1 + √ 2 ˇ d I
bi
2
ˇ d i
bi −ε+1
i
i=1 ε=1
2(ε−bi )−3 4
2ˇ p 0
b
· exp
2(ε−bi )+1 4
bˇi p 0
W 2(ε−bi )−1
ˇi d
4
2(ε−bi )−1 4
,
2bˇi p 0 ˇi d
⎫ ⎪ ⎪ ⎬ (27)
⎪ ⎪ ⎭
i
Similarly, denoting 3 =
si i/ ∈Dl sp
+
k
snt t=1 sp
and 4 =
si i/ ∈Dl sd
+
k
snt t=1 sd
upper
+ 1 , pSER
can be obtained finally as Eq. (28) where
the coefficients C3 , C4 and Fi,bi −ε+1 are given in Eqs. (29)–(31), respectively.
upper
pSER
⎧ ⎪ '( ) '( ) & ⎪ N−l N ⎨ a&b bsp 0 b0 a b sd 0 bsp 0 (−1)k b0 0 + C4 e sd 4 · erfc & & C3 e 3 erfc k! sd 4
3 ⎪ 2 3 2 sp 4 Dl k=0 n1 ···nk ∈ / Dl ⎪ ⎩ a√b + √ 2 I
bi
i=1 ε=1
Fi,bi −ε+1 ˇ d i 2ˇ p 0
1 ˇi p 0
bi −ε+1
2
2(ε−bi )−3 4
b
ˇi d 2(ε−bi )+1 4
· exp
bˇi p 0 2ˇi d
W 2(ε−bi )−1 4
,
2(ε−bi )−1 4
bˇi p 0 ˇi d
⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭
(28)
i
C3 =
− 1
sp sd
−
1−
4
3
i∈Dl
sp sd 4
−
sp 3 sd 4
id i∈Dl ip
3
C4 =
l+1
1−
l+1
i∈Dl
(29)
id i∈Dl ip
1−
⎧ ⎪ ⎪ ⎪ ⎨
Fi,bi −ε+1
1 id
3 ip
1 = (ε − 1)! ⎪
id sp ip sd 4
xl+1
⎪ ⎪ ⎩ (1 + 3 x)
sp sd
(30)
id i∈Dl ip
+ 4 x
ˇ d j ˇ p j
ε−1 ·
1+
I j=1
ˇ d i x ˇ p
bi
i
1+
ˇ d j ˇ p j
⎫(ε−1) ⎪ ⎪ ⎪ ⎬
bi ⎪ ⎪ ⎪ x ⎭
|
x=−
ˇ p i ˇ d i
(31)
4. Asymptotic analysis 4.1. Asymptotic outage probability In the literature, as = Q2 → ∞, simplified approximated expressions of FE2E () in the form of (Gc )−Gd is usually interesting [15], where Gd and Gc are called as diversity gain and coding gain, respectively [16]. Here FE2E () is the CDF of E2E respect to . Diversity gain Gd indicates how fast the outage probability will decrease as → ∞, in other words, how effective that increasing transmission power will be on reducing outage probability. Please cite this article in press as: Gao Z, et al. Outage performance of cognitive DF relay networks with nonidentical Rayleigh fading channels and maximal ratio combining. Int J Electron Commun (AEÜ) (2014), http://dx.doi.org/10.1016/j.aeue.2014.08.003
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We then derive the asymptotic expression of FE2E () as → ∞. Applying the identity of the first order Taylor expansion 1 − e−x = x, Eqs. (6), (9) and (10) can be simplified as Eqs. (32)–(34), respectively.
⎛
Pr(Dl |gsp ) = ⎝
e
−si th gsp 0
⎞⎛
⎞
⎠⎝
si th gsp ⎠ 0
i/ ∈D l
i∈Dl
→∞
fMd (x) =
⎛
(32)
⎞ x jd id ⎠ ⎝ · ip 0
i= / j∈Dl
j∈Dl
(33)
jp 0
sd (th − x) gsp 0
FE2E (th Md = x, gsp ) =
(34)
Plugging Eq. (32)–(34) into Eq. (12) and (13), and using the exponential function Eq. (3.326) in [17], J1 and J2 can be approximated as Eq. (35) and (36), respectively:
⎛ ⎞⎛ ⎞ N+1 sd ⎝ id ⎠ ⎝ si ⎠ (N − l + 2) th J1 = · →∞
Dl = / Ø →∞
J2 =
sp
ip
' N
sd sp
i∈Dl
i/ ∈D l
)
(35)
0
N+1 th
· (N + 2)
si
N+1
l+1
sp
(36)
0
i=1
exact As a result, asymptotic expression of pexact out , denoted as pout,Asymp , can be approximated as Eq. (37).
pexact out,Asym =
⎛
→∞
sd
Dl
sp
⎞⎛
⎝
id
i∈Dl
ip
⎞
⎠⎝
si
i/ ∈D l
sp
N+1 ⎠ (N − l + 2) th l+1
(37)
0
upper
and pout,Asym , can be derived as Eqs. (38) and (39). Similarly, the asymptotic expressions for the bounds of OP, denoted as plower out,Asym plower out,Asym
⎛ ⎞⎛ ⎞ N+1 sd ⎝ id ⎠ ⎝ si ⎠ = (N − l + 2) th
→∞
Dl
2sp
⎛
→∞ upper pout,Asym =
sd
Dl
i∈Dl
sp
2ip
⎞⎛
⎝
id
i∈Dl
i/ ∈D l
ip
⎞
⎠⎝
si
i/ ∈D l
(38)
0
sp
sp
N+1 ⎠ (N − l + 2) th
(39)
0
Moreover, neglecting the correlation issue as in [9], the asymptotic expression of conventional analysis result of outage probability, , can be obtained as Eq. (40). denoted as pconv out,Asym
→∞
pconv out,Asym =
Dl
⎛
⎝
si
i/ ∈D l
sp
⎞⎛
⎠⎝
id
i∈Dl
ip
⎞
N+1 ⎠ sd (2) th sp l + 1
(40)
0
From Eq. (37)–(40), it can be seen that diversity gain of these 4 cases are all equal to N + 1, which is only determined by the number of disjoint paths from the source s to the destination d. On the contrary, the cases have different coding gain Gc . 4.2. Asymptotic symbol error rate Substituting FE2E () in Eq. (22) with pexact defined in Eq. (37), and using Eq.(3.371) in [17], the corresponding asymptotic expression out,Asym of SER, denoted as pexact , can be obtained as Eq. (41). Similarity, other asymptotic expressions of SER, denoted as plower and SER,Asymp SER,Asymp upper
pSER,Asymp , can be obtained as Eq. (42) and (43), respectively. In these equations, (2N + 1) ! ! is defined as (2N + 1) ! ! =1 · 3 . . . (2N + 1).
→∞
sd
pexact SER,Asymp =
Dl
plower SER,Asymp
⎝
id
i∈Dl
ip
⎞⎛
⎠⎝
si
i/ ∈D l
sp
⎞
⎠ · (N − l + 2) l+1
1 N+1 a · (2N + 1)!! 2N+2 · bN+1
0
⎛ ⎞⎛ ⎞ 1 N+1 a · (2N + 1)!! /2 /2 sd id si ⎠ ⎝ ⎠⎝ = · (N − l + 2) N+2 N+1
→∞
Dl
upper
sp
⎛
sp
→∞
sd
pSER,Asymp =
Dl
sp
⎛
i∈Dl
⎝
id
i∈Dl
ip
ip
i/ ∈Dl
⎞⎛
⎠⎝
si
i/ ∈D l
sp
0
sp
2
·b
(41)
(42)
⎞
N+1 a · (2N + 1)!! ⎠ · (N − l + 2) 1 0
2N+2 · bN+1
(43)
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9
0
10
−2
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5. Numerical results We verify the rightness of our analysis results through four numerical experiments. In the first experiment, the effect of parameter N on outage performances is inspected. Simulations are performed for N = 2, 3, and5. 5 5 Other main parameters in the simulations are set as follows: sp = sd = 1, {si }i=1 = {2, 2.1, 2.2, 2.3, 2.4}, {id }i=1 = {1.1, 1.2, 1.3, 1.4, 1.5}, Please cite this article in press as: Gao Z, et al. Outage performance of cognitive DF relay networks with nonidentical Rayleigh fading channels and maximal ratio combining. Int J Electron Commun (AEÜ) (2014), http://dx.doi.org/10.1016/j.aeue.2014.08.003
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{ip }i=1 = {1.1, 1.2, 1.3, 1.4, 1.5}, th = 3 dB and 2 = 1 dB. For each simulation configuration, Q varies from 0 dB to 30 dB in step size 5 dB. The simulation results are shown in Fig. 3. The fine match between the simulation results and the analysis results confirms the rightness of our analysis results. The results show that increasing N can decrease outage probability significantly. Furthermore, the asymptotic lines imply that diversity gain is solely determined by N. In the second experiment, we inspect the effects of the ways how the correlation issue is treated. The simulations in this experiment correspond to 5 cases: (1) exact outage probability of networks with independent non-identical (i.n.d) channels; (2) upper bound OP of networks with i.n.d channels; (3) lower bound OP of networks with i.n.d channels; (4) conventional analysis results (where the correlation issue is completely ignored) of networks with i.n.d channels; (5) OP of networks with independent identical (i.i.d) channels. Main parameters of the first four cases are identical to those in the first experiment, except that N =5 here. In the 5th case, main parameters are set as follows: sp = ip = 2 si = sd = i,d = 1 for i ∈ {1, . . ., N}, N = 5. The asymptotic results show that these cases have identical diversity gain but different coding gains. The results in Fig. 4 show the following three facts: (1) The diversity gain is solely determined by the number of disjoint paths from the source to the destination; (2) ignoring the correlation issue leads to over-optimistic (lower) estimation of outage probability; (3) exact OP is bounded from both sides by the lower bound and the upper one tightly. The results about SER of our targeted cognitive DF relay networks are verified through a third experiment. In the simulations of the experiment, N is fixed to 2. The corresponding results are shown in Fig. 5. The results show that, the upper bound of SER is much more tighter than the lower bound. 6. Conclusion In this paper, for cognitive decode-and-forward (DF) relay networks with maximal ratio combing (MRC) and independent non-identical Rayleigh fading channels, we analyze outage performance of such networks taking consideration of the correlation issue completely. An closed-form expression of exact outage probability (OP) is obtained. Then, by shrinking or expanding the triangular integral region to suitable rectangular regions, simpler closed-form expressions of the lower and upper bounds of both OP and symbol error rate (SER) are obtained. Furthermore, the asymptotic expressions of OP and SER are also obtained. The correctness of our analysis results are verified through numerical simulations. Both analysis results and simulation results show that how the correlation issue is treated can affect the coding gain whereas has no effect on the diversity gain of the network. Simulation results indicate that, exact OP and exact SER are bounded from both sides by the lower bound and the upper one tightly, whereas the upper bounds of them are both more tighter. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [15] [16] [17]
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Please cite this article in press as: Gao Z, et al. Outage performance of cognitive DF relay networks with nonidentical Rayleigh fading channels and maximal ratio combining. Int J Electron Commun (AEÜ) (2014), http://dx.doi.org/10.1016/j.aeue.2014.08.003