On the Performance of Outage Probability in Underlay Cognitive Radio with Imperfect CSI Tu Lam Thanh∗ , Vo Nguyen Quoc Bao∗ , and Beongku An† ∗

School of Electronics Engineering Posts and Telecommunications Institute of Technology, Vietnam (email: [email protected]) † School of Telecommunications Posts and Telecommunications Institute of Technology, Vietnam (e-mail: [email protected]) ‡ Dept. of Computer and Information Communications Eng. Hongik University, Republic of Korea (email: [email protected])

Abstract—In cognitive spectrum sharing systems, interference probability (IP) is one of the most important performance metrics illustrating the interference level at the primary network. In this paper, the interference probability of the primary network due to secondary underlay partial relay selection networks is investigated under the assumption of imperfect channel state information (CSI) of interference links. Numerical results show that it depends not only on the correlation efficient of interference links but also on the transmit power of secondary transmitters. It is also shown that the back-off technique is an efficient approach to guarantee the given quality of service (QoS) of the primary network. For secondary networks, we derive the outage probability over Rayleigh fading channels. Monte-Carlo simulations are performed to verify the correctness of the analysis. Index Terms—Cognitive radio, imperfect CSI, outage Probability, interference probability.

I. I NTRODUCTION Recently, cognitive radio (CR) has attracted a great attention in wireless communication network due to the capability of using spectral utilization efficiently. In CR networks, unlicensed users (secondary users - SUs) can use the same frequency band with licensed users (primary users - PUs) at the same time provided that their operations are not harmful to PUs’ quality of service (QoS). In [1], Goldsmith proposed two such approaches, which are named overlay CR paradigm and underlay CR paradigm. In the overlay scheme, SUs can only occupy the licensed bands if they are not currently used by PUs. This also means that SUs must vacate the bands as soon as PUs start using them. This method has an advantage that there is no cross-interference between the primary network and secondary network, and hence the primary network’s QoS can be guaranteed. However, the communication in the secondary network cannot be seamless because they are dependent on the presence of PUs. Different from the overlay approach, the underlay protocol permits SUs access the licensed bands at any time, provided that the interference created by the secondary operations satisfy the interference temperature at PUs. Hence, this method can be more suitable for real-time applications such as video conference, game interactive and so on. However, it still has some disadvantages such as short communication range and low QoS due to the limitation of the transmit power at the secondary transmitters.

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To enhance the performance of the underlay CR protocols, cooperative communication [2] can be used efficiently. With the help of idle relays, the communication between source and destination in cooperative protocols are more reliable than that in the traditional transmission protocol. In [2], [3], Laneman et. al. proposed various cooperative protocols such as Amplify and Forward (AF) and Decode and Forward (DF). In the AF protocol, relay nodes only amplify the source signal and then forward the amplified signal to the destination. In contrast to the AF protocol, relay nodes operating in the DF mode first decode the signal received from source, and then reencode and forward to the destination. In both protocols, the destination can receive the replicas of the source signal, which can enhance the spatial diversity gain of the communication system. By employing the advantage of cooperative communication into the underlay CR paradigm, the performance of the secondary networks can be significantly improved [4]– [6]. More particularly, in [4], the outage probability of the underlay cognitive radio with best relay selection were derived and verified. A similar work in which the authors evaluated the outage performance over Nakagami-𝑚 fading channel was studied in [5]. Recently, the optimization problem of secondary relay positions has been considered in [7]. However, all of the above-mentioned works have assumed that the perfect channel state information (CSI) of interference links, i.e., from the secondary transmitters to the primary receiver, are available at the secondary transmitters. In practice, it is hard to achieve this due to various reasons such as mobility between two terminals or errors appear when the feedback link goes down by deep fading or multi obstacles. Recently, such a effect has been studied in various network scenarios, e.g., for underlay multihop networks [8], partial relay selection [9], [10] and full relay selection [11]. Different with [8]–[11], where the maximum transmit power for secondary nodes has not been taken into account, in this paper, we derive the exact closed-form expression of outage probability for the underlay cognitive partial relay selection networks with imperfect CSI of interference links. We also study the interference probability (IP) metric, which performs level of interference at primary user caused by secondary users. Finally, various Monte-Carlo simulations are presented to verify the derivations.

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The rest of this paper is organized as follows. In Sect. II, we describe the cognitive system model including primary and secondary networks. In Sect. III, we will derive the exact closed-form expression of outage probability and interference probability. In Sect. IV, numerical results are presented to compare with our analyses. Finally, we conclude this paper in Sect. V. II. S YSTEM M ODEL

Data link Interference link

PU ~

fR,k

~

h1,k

where 𝐼0 (.) is the zeroth order modified Bessel function of the first kind [15]. To protect primary communications, transmit powers of secondary networks under the underlay approach are regulated not only by the maximum tolerable interference level, ℐ𝑝 , but also by the maximum transmit power, 𝒫𝑚 [16]. Mathematically, we have ⎞ ⎛ ⎟ ⎜ ℐ𝑝 𝒫𝑖 = min ⎝ 2 , 𝒫𝑚 ⎠ . ˜ 𝑓𝑖𝑗

Rk

f0

S

RN

where 𝜌 is the correlation ratio between 𝑓𝑖𝑗 and 𝑓˜𝑖𝑗 . Furthermore, 𝜖 is a circular symmetric complex Gaussian random variable with zero mean and variance 𝜆𝑖𝑗 . The joint probability density function (PDF) of ∣𝑓𝑖𝑗 ∣2 and ∣𝑓˜𝑖𝑗 ∣2 is given by [14] ) ( 𝑥+𝑦 ( ) √ exp −(1−𝜌 2)𝜆 2𝜌 𝑥𝑦 𝑖𝑗 2 (𝑥, 𝑦) = 𝑓 𝐼0 , (2) ∣𝑓𝑖𝑗 ∣2 ,𝑓˜𝑖𝑗 (1−𝜌2 ) 𝜆2𝑖𝑗 (1−𝜌2) 𝜆𝑖𝑗

The instantaneous SNR for the 𝑖 → 𝑗 link is then given by ⎞ ⎛

h2,k

⎟ ⎜ ℐ𝑝 𝛾𝑖𝑗 = min ⎝ 2 , 𝒫𝑚 ⎠ ∣ℎ𝑖𝑗 ∣2 . ˜ 𝑓𝑖𝑗

R1 h0

(3)

D

Fig. 1. Underlay partial relay selection networks.

The system model under consideration is presented in Fig. 1 including one source (s), one destination (d) and 𝒩 relay nodes, denoted as 𝑟1 , . . . , 𝑟𝑁 . The secondary system co-exists with a primary network, which represents by one PU receiver. In addition, all nodes in the entire system solely contain one antenna and operate in half duplex mode. In the first time slot (first phase), the source broadcasts its signal to both the relays and destination. Using partial relay selection [12], [13], the relay which has the highest signal-tonoise ratio (SNR) will decode the incoming signal from the source, then re-encode and forward to the destination in the second time slot. At the end of second phase, the destination will combine two information from the direct and indirect links by using maximal ratio combining (MRC). Moreover, we assume that the channel between node 𝑖 ∈ {s, r𝑘 } and 𝑗 ∈ {d, r𝑘 , p} is affected by Rayleigh fading. Let us denote ℎ𝑖𝑗 and 𝑓𝑖𝑗 as channel coefficients of the data link and interference link, respectively. As a result, the channel 2 2 gains, i.e., ∣ℎ𝑖𝑗 ∣ and ∣𝑓𝑖𝑗 ∣ , are exponential random variables (RVs) with parameter 𝜆𝑖𝑗 . For example, the data link between S and D denoted by ℎsd has the parameter 𝜆sd . To facilitate the analysis, we further assume that all the links from s → r𝑘 , r𝑘 → d and r𝑘 → p are similar, i.e., 𝜆sr𝑘 = 𝜆sr , 𝜆r𝑘 d = 𝜆rd and 𝜆r𝑘 p = 𝜆rp for all 𝑘. Let denote 𝑓˜𝑖𝑗 as the estimated version of 𝑓𝑖𝑗 , we have [14] √ (1) 𝑓˜𝑖𝑗 = 𝜌𝑓𝑖𝑗 + 1 − 𝜌2 𝜖,

(4)

It is noted from (3) that due to ∣𝑓𝑖𝑗 ∣2 ∕= ∣𝑓˜𝑖𝑗 ∣2 , the interference level received at PU can be greater than ℐ𝑝 . As a result, the interference probability, defined as the interference power received at the primary receiver exceeds the maximum allowable interference level, is non-zero [8]. In the next section, we will derive the closed-form expression of the interference probability over Rayleigh fading channels. For secondary networks, the system performance in terms of outage probability is investigated. III. P ERFORMANCE A NALYSIS A. Interference Probability at Primary Network With two hop relaying, interference probability (IP) at PU occurs because interference caused by the secondary source or the secondary relays exceeds the interference tolerance ℐ𝑝 . Based on the theorem of total probability, the interference probability at PU can be written as follows: ⎞ ⎤ ⎡ ⎛

126

⎟ ⎥ ⎢ ⎜ ℐ𝑝 2 IP = Pr ⎣min ⎝ 2 , 𝒫𝑚 ⎠ ∣𝑓sp ∣ ≥ ℐ𝑝 ⎦ ˜ 𝑓sp Ω1 ⎞ ⎤ ⎡ ⎛ ⎟ ⎥ ⎢ ⎜ ℐ𝑝 2 + Pr ⎣min ⎝ 2 , 𝒫𝑚 ⎠ ∣𝑓sp ∣ < ℐ𝑝 ⎦ ˜ 𝑓sp ⎞ ⎡ ⎛

⎤

⎟ ⎥ ⎢ ⎜ ℐ𝑝 2 × Pr ⎣min ⎝ 2 , 𝒫𝑚 ⎠ ∣𝑓r𝑘∗ p ∣ < ℐ𝑝 ⎦, ˜ 𝑓 r𝑘 ∗ p Ω2

(5)

The 2013 International Conference on Advanced Technologies for Communications (ATC'13)

where 𝑘 ∗ denotes the index of the selected relay, i.e., 𝑘 ∗ = 2 argmax𝑘=1,...,𝒩 ∣ℎsr𝑘 ∣ . To obtain IP, we need to calculate Ω1 and Ω2 . We first rewrite Ω1 under the general form as ⎛

⎞

Recalling Ω1 , (9) is rewritten as follows: Ω2 =

𝒩 ∑

∫∞ ( Ω1 (𝜆r𝑘 p , ℐ𝑝 ) Pr

𝑘=1

0

max

𝑖=1,...,𝒩 ,𝑖∕=𝑘

) ∣ℎsr𝑖 ∣ < 𝑧 𝑓∣ℎ 2

sr𝑘

2 ∣ (𝑧) 𝑑𝑧

2 ) 𝒩 𝒩 −1 ( ℓ ⎜ ℐ𝑝 ∣𝑓𝑖𝑗 ∣ ⎟ ∑ ∑ 2 𝒩 − 1 (−1) Ω1 (𝜆𝑖𝑗 , 𝑧) =Pr ⎝ 2 ≥ 𝑧, 𝒫𝑚 ∣𝑓𝑖𝑗 ∣ ≥ 𝑧 ⎠ = . (10) Ω1 (𝜆r𝑘 p , ℐ𝑝 ) ˜ ℓ ℓ+1 𝑓𝑖𝑗 𝑘=1 ℓ=0 ( ) ℐ𝑝 𝑥 𝒩∑ −1 ( ) ( ) ∫∞∫𝑧 exp − 𝑥+𝑦 √ 𝒩 −1 (−1)ℓ 2 2𝜌 𝑥𝑦 (1−𝜌 )𝜆𝑖𝑗 Making use the fact that ℓ ℓ+1 = 1/𝒩 [15, Eq. 𝑑𝑦𝑑𝑥 = ℓ=0 2 𝐼0 2) 𝜆 2 (1 − 𝜌 𝑖𝑗 (1 − 𝜌 ) (𝜆𝑖𝑗 ) (0.154.3)], we arrive at 𝑧 0 𝒫𝑚 ) ( 𝑥 ∫∞ exp − Ω2 = Ω1 (𝜆r𝑘 p , ℐ𝑝 ). (11) (1−𝜌2 )𝜆𝑖𝑗 = (6) 2 To this end, combining (7) and (11) with (5), we obtain the (1 − 𝜌2 ) (𝜆𝑖𝑗 ) 𝑧 𝒫𝑚 closed-form expression for the interference probability of the ℐ𝑝 𝑥 primary network. ( ) ( ) ∫𝑧 √ 2𝜌 𝑥𝑦 −𝑦 It is remarked that since 𝒫𝑚 → ∞, IP approaches 0.75, × exp 𝐼0 𝑑𝑦𝑑𝑥. (1 − 𝜌2 ) 𝜆𝑖𝑗 (1−𝜌2) 𝜆𝑖𝑗 which agrees with the previously reported result in [9]. Stated 0 another way, our result here is the most generalized cases and √ hence covers that in [9] as a special case. Furthermore, we By making a change of variable, i.e., 𝑡 = 𝑦, and with the can observe that unlike [9], IP in (5) depends on not only the help of [17, Eq. (10)] along with [17, Eq. (55)], we can obtain correlation efficient but also the transmission power. Ω1 as Since the back-off power control is a simple and efficient ( ) technique to guarantee the QoS of the primary networks [8], 𝑧 Ω1 (𝜆𝑖𝑗 , 𝑧) = exp − we here adopt it for the system under consideration. The 𝜆𝑖𝑗 𝒫 √ )] transmit power of secondary node 𝑖 is now [ (√ 2𝜌2 𝑧 2ℐ𝑝 ⎞ ⎛ , × 1−𝑄 2 2 (1−𝜌 ) 𝜆𝑖𝑗 𝒫𝑚 (1−𝜌 ) 𝜆𝑖𝑗 𝒫𝑚 ′ ⎟ ⎜ 𝜉ℐ𝑝 ( ) 𝒫𝑖 = min ⎝ 2 , 𝒫𝑚 ⎠ , (12) √ ( ) ) ( 2𝜌 ℐ 𝑧 1 𝑡 𝑠𝑧 ˜𝑖𝑗 𝑝 𝑓 + 𝐼0 + exp − 2 2𝑟 2𝒫 (1 − 𝜌2 ) 𝜆𝑖𝑗 𝒫𝑚 ⎛√ ⎞ √ where 𝜉 denotes the back-off coefficient, 0 < 𝜉 ≤ 1. Using 𝑡 ⎝ (𝑠 − 𝑟) 𝑧 (𝑠 + 𝑟) 𝑧 ⎠ the same approach as for (5) and recalling (7) and (11), we − 𝑄 , (7) , have the interference probability for this case as 𝑟 2𝒫𝑚 2𝒫𝑚 ′

where 𝑄 (𝑎, 𝑏) = function [15] and

∫∞ 𝑏

(

𝑥 exp − 𝑥

2

+𝑎 2

) 2

(

IP = Ω1 (𝜆sp , 𝜉ℐ𝑝 ) + [1 − Ω1 (𝜆sp , 𝜉ℐ𝑝 )]Ω1 (𝜆rp , 𝜉ℐ𝑝 ). (13) 𝐼0 (𝑎𝑥) 𝑑𝑥 is Marcum-Q

For decode-and-forward relaying, the system outage probability of the secondary networks can be given as

)

2 ℐ𝑝 𝜌2 + 1+ , 𝜆𝑖𝑗 1 − 𝜌2 (1 − 𝜌2 ) 𝑧 ( ) ℐ𝑝 2 𝜌2 − 𝑡= 1+ , 𝜆𝑖𝑗 1 − 𝜌2 (1 − 𝜌2 ) 𝑧 √ 16𝜌2 ℐ𝑝 𝑟 = 𝑠2 − 2 2 . (1 − 𝜌2 ) (𝜆𝑖𝑗 ) 𝑧

B. Outage Probability of Secondary Networks

𝑠=

OP = Pr [𝛾sd + 𝛾𝑅 ≤ 𝛾th ] ,

(8)

For Ω2 , making use again the theorem of total probability, we have [ ( ) 𝒩 ∑ ℐ𝑝 Pr min , 𝒫𝑚 ∣𝑓r𝑘 p ∣2 ≥ ℐ𝑝 Ω2 = 2 ˜ ∣𝑓 r𝑘 p ∣ 𝑘=1 ] 2 2 ∩ ∣ℎsr𝑘 ∣ > max ∣ℎsr𝑖 ∣ . (9) 𝑖=1,...,𝒩 ,𝑖∕=𝑘

(14)

where 𝛾𝑅 = min (𝛾sr𝑘∗ , 𝛾r𝑘∗ d ), and 𝛾th = 22ℛ − 1 with ℛ being the expected data rate of secondary systems. Sharing the same interference link from s → p, 𝛾sd and 𝛾𝑅 are not independent. Therefore, (14) is rewritten as ∫∞ ∫𝛾th OP = 𝐹𝛾sd ∣𝑥 (𝛾th − 𝑦) 𝑓 𝛾𝑅 ∣𝑥 (𝑦) 𝑓𝛾sp (𝑥) 𝑑𝑦𝑑𝑥. (15) 0

0

Over Rayleigh fading, 𝐹𝛾sd ∣𝑥 is of the form ⎤ ⎡ 𝛾 ⎦. ) ( 𝐹𝛾sd ( 𝛾∣ 𝑥) =1 − exp ⎣− 𝜉ℐ min 𝑥𝑝 , 𝒫𝑚 𝜆sd

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(16)

The 2013 International Conference on Advanced Technologies for Communications (ATC'13)

Assuming that all channels are independent, we can write

𝐹𝛾𝑅 (𝛾∣𝑥) = 𝐹𝛾sr𝑘∗ ∣𝑥 (𝛾)+𝐹𝛾r𝑘∗ d (𝛾)−𝐹𝛾sr𝑘∗ ∣𝑥 (𝛾)𝐹𝛾r𝑘∗ d (𝛾). (17)

With partial relay selection, the CDFs of 𝐹𝛾sr𝑘∗ ∣𝑥 and 𝐹𝛾r𝑘∗ d (𝛾) are expressed, respectively, as follows [13]: ⎡

⎞⎤𝒩

⎛

𝐹𝛾sr𝑘∗ ∣𝑥 (𝛾) =⎣1−exp ⎝−

min

(

𝛾 𝜉ℐ𝑝 𝑥 , 𝒫𝑚

)

𝜆sr𝑘

⎠⎦

(18)

and (

) 𝛾 𝛾 𝐹𝛾r𝑘 d (𝛾) =1 − exp − + 𝜆 𝒫𝜆r𝑘 d 𝛾 + 𝜉ℐ𝑝 𝜆rr𝑘 dp 𝑘 ( ) ( ) 𝜉ℐ𝑝 𝛾 × exp − exp − . (19) 𝒫 𝑚 𝜆 r𝑘 p 𝒫 𝑚 𝜆 r𝑘 d

In addition, ℬℓ with ℓ = 0, . . . , 9 are given by ( ) 𝜉𝜇 ℬ0 =1 − exp − , 𝜆sp 𝜉ℐ𝑝 𝜆sr𝑘 , ℬ1 = 𝑘𝜆sp 1 , ℬ3 =ℬ2 − 𝒫𝑚 𝜆sd 𝑘 1 + , ℬ2 = 𝒫𝑚 𝜆sr𝑘 𝒫 𝑚 𝜆 r𝑘 d 𝜇𝑘 1 + , ℬ4 = 𝜉ℐ𝑝 𝜆sr𝑘 𝒫 𝑚 𝜆 r𝑘 d 𝑘 , ℬ5 = 𝜉ℐ𝑝 𝜆sr𝑘 1 1 − , ℬ6 = 𝜉ℐ𝑝 𝜆sr 𝜉ℐ𝑝 𝜆sd 𝜉 1 𝜉 + − , ℬ7 = 𝒫𝑚 𝜆sr 𝒫𝑚 𝜆rd 𝒫𝑚 𝜆sd 𝛾 1 + , ℬ8 = 𝜉ℐ𝑝 𝜆sd 𝜆sp [ ( )] 𝜉 . ℬ9 =exp −𝛾 𝒫𝑚 𝜆sd

(26)

In 𝐻𝑖𝑗 , using the definition of the exponential integrals func∫∞ tion, i.e., 𝐸1 (𝑧) = exp(−𝑡) 𝑑𝑡 [15, Eq. (8.211.1)] and with 𝑡 𝑧

Substituting (18) and (19) into (17) and then making use the binomial theorem, we have

𝑓𝛾𝑅 ( 𝛾∣ 𝑥)=

𝒩 ( ) ∑ 𝒩

𝑘 ⎡

𝑘=1

(−1)

⎞⎤

⎛

× exp⎣−𝛾 ⎝ ⎡ ×⎣

𝑘+1

min

𝒜𝜉ℐ𝑝 𝛼2

(

𝑘 𝜉ℐ𝑝 𝑥 , 𝒫𝑚

⎛

2

−⎝

)

𝜆sr𝑘

(

+ 𝑘

1 ⎠⎦ 𝒫 𝑚 𝜆 r𝑘 d

the help of [15, Eq. (3.352.1)] and [15, Eq. (3.352.2)], 𝐽𝑘 with 𝑘 = 1, . . . , 4 are respectively derived as (27), (28), (29), and (30), where 𝒞𝑖 with 1 ≤ 𝑖 ≤ 𝑐 and 𝒟𝑗 with 1 ≤ 𝑗 ≤ 𝑒 are given by [ ] 1 𝑑(𝑐−𝑖) 1 𝒞𝑖 = , 𝑒 (𝑐−𝑖) (𝑐 − 𝑖)! 𝑑𝑥 (𝑥 + 𝑑) 𝑥=−𝑏 [ ] 1 𝑑(𝑑−𝑗) 1 𝒟𝑗 = . 𝑐 (𝑑−𝑗) (𝑑 − 𝑗)! 𝑑𝑥 (𝑥 + 𝑏) 𝑥=−𝑑

∫𝑏

𝜉ℐ (𝛾 + 𝜉ℐ𝑝 𝛼2 ) min 𝑥𝑝 , 𝒫𝑚 𝜆sr𝑘 )[ ]] 1 𝒜𝜉ℐ𝑝 𝛼2 + 𝒜−1− , (20) 𝒫 𝑚 𝜆 r𝑘 d 𝛾 + 𝜉ℐ𝑝 𝛼2

where 𝒜 = exp

(

𝜉𝜇 𝜆r 𝑘 p

)

1 − exp (−𝑏𝑎) 𝑎

(27)

= exp (𝑎𝑏) [𝐸1 (𝑎𝑏) − 𝐸1 (𝑎𝑐 + 𝑎𝑏)]

(28)

𝐽1 (𝑎, 𝑏) =

)

exp (−𝑥𝑎) 𝑑𝑥 = 0

∫𝑐 𝐽2 (𝑎, 𝑏, 𝑐) = 0

and 𝛼2 =

𝜆r 𝑘 d 𝜆r 𝑘 p .

Finally, from (15), (16), and (20), we can obtain the closedform expression of the outage probability of underlay partial relay selection networks as

OP = ℬ0 (𝐻11 − 𝐻12 ) + ℬ1 (𝐻21 − 𝐻22 ) ,

(21)

where 𝐻𝑖𝑗 are defined and shown on the top of the next page.

exp (−𝑥𝑎) 𝑑𝑥 𝑥+𝑏

IV. N UMERICAL R ESULTS In this section, Monte-Carlo simulations are performed to verify the correctness of the analysis. For illustration purposes, we set 𝜆sd = 1, 𝜆rp = 2 and ℛ = 1 throughout this section. In Fig. 2, we investigate the interference probability versus 𝜌 for various 𝜉. It can be observed that the simulation results are in excellent agreement with the numerical results. When 𝜉 = 1, the interference probability is always equal 0.75 regardless of

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The 2013 International Conference on Advanced Technologies for Communications (ATC'13)

𝐻11 =

𝒩 ( ) ∑ 𝒩 𝑘=1

𝐻12

𝐻21

𝑘

( = exp −

(−1)

𝑘+1

[𝒜𝜉ℐ𝑝 𝛼2 𝐽3 (ℬ2 , 𝜉ℐ𝑝 𝛼2 , 𝛾, 2) − (𝒜 − 1) (1 − exp (−ℬ2 𝛾)) + ℬ2 𝒜𝜉ℐ𝑝 𝛼2 𝐽2 (ℬ2 , 𝜉ℐ𝑝 𝛼2 , 𝛾)] , (22)

) ( )∑ [ 𝒩 𝒩 𝛾 ℬ2 (𝒜 − 1) (1 − exp (−ℬ3 𝛾)) 𝑘+1 (−1) 𝒜𝜉ℐ𝑝 𝛼2 𝐽3 (ℬ3 , 𝜉ℐ𝑝 𝛼2 , 𝛾, 2) − 𝒫𝑚 𝜆sd ℬ3 𝑘 𝑘=1

+ℬ2 𝒜𝜉ℐ𝑝 𝛼2 𝐽2 (ℬ3 , 𝜉ℐ𝑝 𝛼2 , 𝛾)] , ( ) ( )[ 𝒩 ∑ 𝒩 𝒜𝜉ℐ𝑝 𝛼2 𝜇 ℬ4 𝑘+1 = exp − 𝐽4 (ℬ4 , 𝜉ℐ𝑝 𝛼2 , 2, ℬ2 , 1, 𝛾) − (𝒜 − 1) 𝐽2 (ℬ4 , ℬ2 , 𝛾) (−1) 𝜆sp ℬ3 ℬ3 𝑘 𝑘=1

𝐻21

ℬ4 𝒜𝜉ℐ𝑝 𝛼2 (𝒜 − 1) 𝒜𝜉ℐ𝑝 𝛼2 + 𝐽4 (ℬ4 , 𝜉ℐ𝑝 𝛼2 , 1, ℬ2 , 1, 𝛾) − 𝐽3 (ℬ4 , ℬ2 , 2, 𝛾) + 𝐽4 (ℬ4 , 𝜉ℐ𝑝 𝛼2 , 1, ℬ2 , 2, 𝛾) ℬ3 ℬ3 ℬ3 ( ) ( )[ 𝒩 ∑ 𝒩 𝒜𝜉ℐ𝑝 𝛼2 𝜇 ℬ4 𝑘+1 = exp − 𝐽4 (ℬ4 , 𝜉ℐ𝑝 𝛼2 , 2, ℬ2 , 1, 𝛾) − (𝒜 − 1) 𝐽2 (ℬ4 , ℬ2 , 𝛾) (−1) 𝜆sp ℬ3 ℬ3 𝑘 𝑘=1

ℬ4 𝒜𝜉ℐ𝑝 𝛼2 (𝒜 − 1) 𝒜𝜉ℐ𝑝 𝛼2 + 𝐽4 (ℬ4 , 𝜉ℐ𝑝 𝛼2 , 1, ℬ2 , 1, 𝛾) − 𝐽3 (ℬ4 , ℬ2 , 2, 𝛾) + 𝐽4 (ℬ4 , 𝜉ℐ𝑝 𝛼2 , 1, ℬ2 , 2, 𝛾) ℬ3 ℬ3 ℬ3

∫𝑐 𝐽3 (𝑎, 𝑏, 𝑐, 𝑑) = 0

exp (−𝑥𝑎) (𝑥 + 𝑏)

𝑑

∫∞ 𝑑𝑥 = 0

exp (−𝑥𝑎) (𝑥 + 𝑏)

𝑑

∫∞ 𝑑𝑥 − 𝑐

𝑑−1

=

exp (−𝑥𝑎) (𝑥 + 𝑏)

𝑑

(23)

] (24)

] (25)

𝑑𝑥

∑ 1 (−𝑎) 𝑑−𝑘−1 exp (𝑎𝑏) 𝐸1 (𝑎𝑏) (𝑘 − 1)! (−𝑎) + (𝑑 − 1)! (𝑑 − 1)! 𝑑−1

𝑘=1

− exp (−𝑎𝑐)

𝑑−1 𝑑−𝑘−1 ∑ (𝑘 − 1)! (−𝑎) 𝑘=1

∫𝑓 𝐽4 (𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓 ) = 0

(𝑑 − 1)! (𝑏 + 𝑐)

Interference Probability

0.6

0.5

0.4 Analysis ξ = 0.4 Analysis ξ = 0.6 Analysis ξ = 0.8 Analysis ξ = 1 Simulation ξ = 0.4 Simulation ξ = 0.6 Simulation ξ = 0.8 Simulation ξ =1

0.1

0

0

0.2

(𝑑 ≥ 2)

(29)

ℐ𝑝 . In addition, the interference probability decreases with the decreasing of 𝜉. From this, it is worthy noting that we can trade the performance of secondary networks for guaranteeing QoS of the primary network. In Fig. 3 and Fig. 4, we show the outage probability of secondary networks where the maximum allowable interference level and the maximal transmit power are taken into account. From Fig. 3, we can see that the maximum transmit power causes a floor on the outage probability of secondary networks. It can be explained by making use the fact that the actual transmit power of secondary transmitter is determined, as in (3), by the minimum function of the maximum allowable interference level and the maximal transmit power. In Fig. 4, the secondary outage probability versus 𝒫𝑚 /𝒩0 are plotted. Similar to Fig. 3, there exists an outage probability floor since 𝒫𝑚 /𝒩0 is greater than 15 dB. We also observe that due to

0.7

0.2

𝑑−1

(−𝑎) exp (𝑎𝑏) 𝐸1 (𝑎 (𝑏 + 𝑐)) (𝑑 − 1)!

𝑐 𝑒 ∑ ∑ exp (−𝑥𝑎) 𝑑𝑥 = 𝒞 𝐽 (𝑎, 𝑏, 𝑓 )+ 𝒞 𝐽 (𝑎, 𝑏, 𝑓, 𝑖) + 𝒟 𝐽 (𝑎, 𝑑, 𝑓 )+ 𝒟𝑗 𝐽3 (𝑎, 𝑑, 𝑓, 𝑗) (30) 1 2 𝑖 3 1 2 𝑐 𝑒 (𝑥+𝑏) (𝑥+𝑑) 𝑖=2 𝑗=2

0.8

0.3

𝑘

−

0.4

0.6

0.8

1

Ip /N0

Fig. 2. IP versus 𝜌 with ℐ𝑝 = 5 dB, 𝒫𝑚 = 15 dB, 𝜆sr = 4, 𝜆sp = 1 and 𝜆rd = 3.

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The 2013 International Conference on Advanced Technologies for Communications (ATC'13)

form expression, while the outage probability is obtained for secondary networks. Numerical results confirm that the interference probability is a function of the maximal tolerable and the maximum transmit power.

0

10

ACKNOWLEDGMENT

−1

Outage Probability

10

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.04-2012.20.

−2

10

R EFERENCES Analysis N = 1 Analysis N = 2 Analysis N = 4 Simulation N = 1 Simulation N = 2 Simulation N = 4

−3

10

−4

10

0

5

10

15 Ip /N0

20

25

30

Fig. 3. OP versus ℐ𝑝 /𝒩0 with 𝜆sr = 10, 𝜆sp = 4, 𝜆rd = 10, 𝜌 = 0.8 and 𝒫𝑚 = 15 dB.

0

10

Analysis N = 1 Analysis N = 2 Analysis N = 4 Simulation N = 1 Simulation N = 2 Simulation N = 4

Outage Probability

−1

10

−2

10

−3

10

0

5

10

15 Pm /N0

20

25

30

Fig. 4. OP versus 𝒫𝑚 /𝒩0 with 𝜆sr = 10, 𝜆sp = 4, 𝜆rd = 10, 𝜌 = 0.8 and ℐ𝑝 = 15 dB.

partial relay selection, the outage probability is improved when increasing the number of the secondary relays. V. C ONCLUSIONS In this paper, the imperfect CSI effects of interference links on performance of cognitive networks have been investigated over Rayleigh fading channels. For the primary network, the interference probability is derived in a closed

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