2011 International Conference on Advanced Technologies for Communications (ATC 2011)

Incremental Cooperative Diversity for Wireless Networks under Opportunistic Spectrum Access Vo Nguyen Quoc Bao

Nguyen Tuan Duc

Hoang Dinh Chien

Telecom. Dept. Posts and Telecom. Inst. of Tech., Vietnam Email:[email protected]

School of EE Ho Chi Minh Internaltional Uni. Email: [email protected]

Telecom. Dept. Ho Chi Minh City Uni. of Tech. Email:[email protected]

Abstract—This paper analyzes the performance of incremental cooperative networks under opportunistic spectrum access where the best relay opportunistically borrows spectrum allocated to a primary user to help the source transmission if the direct link is below a given threshold. Specially, we provide the system outage probability and bit error probability derived in closed-form expression over Rayleigh fading channels. Theoretical analysis and computer simulation are in good agreement showing that the proposed system always outperforms direct transmission and full diversity gain cannot be achieved under opportunistic spectrum access.

I. I NTRODUCTION Since the legendary work by Mitola was published twelve years ago [1] introducing cognitive radio (CR), this concept has subsequently been widely studied for the improvement of spectrum exploitation [2]. In particular, cognitive radio can solve the spectrum congestion problem due to exclusive assignment of frequency spectrum in radio communications by allowing secondary users (SUs) to intelligently identify unused spectrum bands (originally allocated to primary users) and then adaptively use them. Recent studies have shown that cooperative technique can be used in cognitive wireless systems to mitigate interference and to improve spectrum efficiency [3]–[7]. Besides, it is able to provide spatial diversity in cognitive relay networks, where secondary users (SU) can act as relaying nodes for the transmission of primary networks or secondary networks. However, as reported in [8]–[10], full diversity is not achieved when the spectrum acquisition is not always guaranteed in cognitive wireless relay networks using either the well-known repetitionbased relaying scheme [11], [12] or selection cooperation [13]–[15]. In this context, this paper proposes a multinode incremental relaying scheme operating under opportunistic spectrum access. The idea is motivated by [8] and [16] where the secondary destination will request the help from the best cognitive relay if the direct link between the secondary source and the secondary destination does not satisfy the network quality of services (QoS). The proposed scheme is useful for cognitive ad-hoc systems where CR promises to improve the network spectrum-utilization efficiency and to reduce the load for relaying nodes as compared to conventional cooperative networks.

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The main contribution of this paper is therefore to provide closed-form expression of outage probability and bit error probability for spectrum-sensing opportunistic incremental networks over independent but non-identically distributed (i.n.d.) Rayleigh channels. The advantage of the resulting formulas is to avoid the need for lengthy and time-consuming Monte Carlo simulation. Moreover, to the best of authors’ knowledge, the expressions derived in this paper are new and not reported in the literature so far. The rest of this paper is organized as follows. In section II, we introduce the model under study and describe the proposed protocol. Section IV shows the formulas allowing for evaluation of the outage probability, the average BER of the system. In Section IV, we contrast the simulations and the results yielded by theory. Finally, the paper is closed in section V. II. S YSTEM M ODEL We consider the general configuration of cognitive incremental relaying networks where the data transmission of the secondary source-destination pairs is possibly helped by N secondary relay nodes. Each secondary node is equipped with a single antenna and that all the secondary nodes are halfduplex and, thus, cannot simultaneously transmit and receive. The data transmission in the network takes place two phases: broadcast phase and incremental phase. In the broadcast phase, the secondary source transmits its information, which is received by the destination and the relays, thanks to broadcast nature of wireless channels. At the end of the first time slot, the destination checks the signal quality of the direct link between the source and the destination. If it is greater than a predetermined threshold (γth ), the destination will broadcast a feedback message indicating the successful reception at the destination to the source and all relays. Having received such the message, the source will make use of the following time slot to send a new signal while the relays will not perform any transmission, i.e. still keep idle. Otherwise, a ”failure” message is sent to request the help of the best opportunistic relay in the second time slot, named as the incremental phase. Different from standard cooperative incremental relaying networks [16], [17], each incremental phase typically consists of two essential sub-phases: 1) a spectrum sensing sub-phase, in which cognitive relaying nodes attempt to detect available spectrum holes; and 2) a data transmission sub-phase, in which the

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best relay among available ones amplifies and then forwards the received signals towards the secondary destination [14], [15]. Without any a priori knowledge of primary signals, as in [8]–[10], [18], all secondary relaying nodes perform energy detection - the simplest spectrum sensing technique - to detect currently unused bands [19], [20]. It simply treats primary signals as noise and decides on the presence or absence of the primary signal based on the energy of the observed signal. However, in practice, the relay nodes may not always be able to acquire spectrum holes resulting in the fact that such relays cannot always involve in cooperative transmission. In Rayleigh fading channels, the average probability of detection for each potential relay is reported in [20, eq. (9)]. It is assumed that every channel between the secondary nodes experiences slow, flat, Rayleigh fading. Due to Rayleigh fading, the channel powers denoted by |hSD |2 , |hSRk |2 and |hRk D |2 are independent and exponential random variables whose means are λ0 , λ1,k and λ2,k , respectively. Let us define 2 2 2 γ0 = P1 |hSD | , γ1,k = P1 |hSRk | , and γ2,k = P2 |hRk D | as the instantaneous SNRs for the links S → D, S → Rk , and Rk → D, respectively, where P1 and P2 are the corresponding average transmit signal-to-noise (SNRs) for the source and the relays. It is assumed that the receivers at the destination and relays have perfect channel state information but no transmitter channel state information is available at the source and relays. III. P ERFORMANCE A NALYSIS In this section, we focus on the derivation of a closed-form expression for the outage probability and the bit error rate of the considered system. To achieve our goal, we first develop the PDF of the instantaneous SNRs in transmission phase, which then is used for the derivation. A. Outage Probability The outage probability is an important quality of service (QoS) measure, which is defined as the probability that the end-to-end instantaneous SNR is below the predetermined threshold. Mathematically, the outage probability can be derived as follows: Pr(O)

=

Pr{γ0 ≤ γth } Pr{O|γ0 ≤ γth } + Pr{γ0 > γth } Pr{O|γ0 ≤ γth }   

=

Pr{γ0 ≤ γth } Pr{O|γ0 ≤ γth }.

=0

(1)

In contrast to conventional incremental cooperative networks, the number of relays involved in the incremental phase is not fixed, i.e. varying according to the result of the spectrum sensing sub-phase. With opportunistic relaying, only the best relay among relays successful in obtaining spectrum forwards the received data toward the destination as per the rule of amplify-and-forward (AF). Denoting R as the set of potential relays, it is obvious that its cardinality, K = |R|, is a random variable taking values from 0 to N . For each K, N  = N !(NK!−K)! possible subsets of size K. Thus, there are K

applying the law of total probability [21], the conditional outage probability, Pr{O|γ0 ≤ γth }, can be re-written as Pr{O|γ0 ≤ γth }

N K Pr(R = {Rn1 , · · · , RnK })   . (2) = K=0 n1 ,...,nK =1 ×Pr(O|R = {Rn1 , · · · , RnK }∩γ0 ≤ γth ) n1 <···
In the above, Pr(R = {Rn1 , · · · , RnK }) denotes the probability for the potential relay set R given by   Pd,i (1 − Pd,j ). (3) Pr(R = {Rn1 , · · · , RnK }) = Ri ∈R

Rj ∈R /

For example, if all secondary nodes have the same average K probability of detection, i.e., {Pd,i }i=1 = Pd , the probability of potential relay set becomes K

Pr(R = {Rn1 , · · · , RnK }) = (Pd ) (1 − Pd )N −K .

(4)

Pr(O|R = {Rn1 , · · · , RnK }) is the outage probability conditioned on R. In the incremental phase, if at least one secondary relay is successful in obtaining spectrum; the best relay is activated to retransmit the source signal. Then, the destination combines the signals received from source and the best relay using the maximum ratio combining (MRC) technique. Otherwise, the destination decodes the signal solely based on the signal of the direct link. The instantaneous SNR at the destination of the incremental phase conditioned on γ0 < γth is γ0 |γ0 ≤ γth , K=0 γΣ |γ0 ≤ γth = . (5) ∗ γ0 |γ0 ≤ γth + γK , 1 ≤ K ≤ N To calculate the PDF of γΣ |γ0 < γth , we should know the PDF ∗ . The conditional PDF of γ0 |γ0 < γth of γ0 |γ0 < γth and γK can be straightforwardly found to be as [21] γ0 e−γ/¯ , γ < γth −γth /γ ¯0 (1−e )¯ γ0 fγ0 |γ0 <γth (γ) = , (6) 0, γ ≥ γth ∗ ∗ , we start with γK = where γ¯0 = E[γ0 ] = P1 λ0 . For γK γ γ2,k [22] is the equivalent maxk=1,...,K γk where γk = γ1,k1,k +γ2,k +1 instantaneous SNR of dual AF link k. γ1,k and γ2,k are the instantaneous SNRs of the links from the secondary source to secondary relay k and secondary relay k to the secondary destination with their expected value γ¯1,k = P1 λ1,k and γ¯2,k = P2 λ2,k , respectively. To make the analysis feasible, we take an approximation approach. In particular, we adopt the well-known approximation of AF dual hop links where the equivalent instantaneous SNR of dual hop links can be approximated by the minimum of those two instantaneous SNRs at medium-to-high SNRs, namely γk ≈ min(γ1,k , γ2,k ) [23]. Making use the fact that the minimum of two exponential random variables is also exponentially distributed with parameter equal to the sum of the parameters of γ1,k and γ2,k , we have the expected value of γk given by γ¯k =

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γ ¯1,k γ ¯2,k γ ¯1,k +¯ γ2,k [21]. According to [15], [17], the ∗ γK = maxk=1,...,K min(γ1,k , γ2,k ) is given by

∗ (γ) = fγK

K

(−1)

n1 ,...,nk =1 n1 <···
k=1



K

k−1

joint PDF of

B. Bit Error Probability With incremental relaying, we can express the probability of a bit error in Rayleigh fading channels as

1 − χγ e k, χk

(7)

Pr(E)

=

Pr (γ0 ≥ γth ) Pr(E|γ0 ≥ γth ) + Pr (γ0 < γth ) Pr(E|γ0 < γth ).

(11)

−1

k where χk = ¯n−1 . Under the assumption of inde- The error probability Pr(E|γ0 ≥ γth ) is given as l=1 γ l pendence of γ0 and γk and without getting into the details of +∞  the derivation, the PDF of γK |γ0 < γth with 1 ≤ K ≤ N is Pe (E|γ)fγ0 |γ0 ≥γth (γ)dγ, (12) Pr(E|γ0 ≥ γth ) = derived as 0 ⎧ K K   ⎪ k−1 uL (¯ γ0 ,χk ) ⎪ where Pe (E |γ0 ) and fγ0 |γ0 ≥γth (γ) represent the conditional ⎪ , γ ≤ γth ⎪ (−1) −γth /γ ¯0 ⎪ ⎨k=1 n1 ,...,nk =11−e error probability for M -ary square quadrature amplitude (Mn1 <··· γth tively. When Gray code is used for bit-to-symbol mappings, ⎪ −γth /γ ¯0 ⎪ n1 ,...,nk =11−e ⎩i=1 Pe (E |γ0 ) is given by [24] n1 <···
where uL (¯ γ0 , χk ) and uG (¯ γ0 , χk ) are defined, respectively, as follows: ⎧  γ¯  γ 1 − γ¯0 0 ⎪ e + ⎪ γ ¯ −χ γ ¯ 0 0 k ⎨   γ¯0 = χk γ χk 1 − χk uL (¯ γ0 , χ k ) = , χk −¯ γ0 χk e ⎪ γ ⎪ ⎩ γ − γ¯0 e γ¯0 = χk γ ¯2 0

uG (¯ γ0 , χ k ) =

 1 1 

⎧ −γ − ⎪ ⎨ 1−e th γ¯0 χk ⎪ ⎩

1−¯ γ0 /χk

γ

1 − χk χk e

,

γ

γth − γ¯0 e γ ¯02

γ¯0 = χk

.

Pe (E |γ0 ) =

j=1

Pr(O|R = {Rn1 , · · · , RnK } ∩ γ0 ≤ γth ) =

Pr(E|γ0 ≥ γth ) =

=

1 1−e−γth /γ¯0



K 

(−1)

K 

k−1

n1 ,...,nk =1 0 n1<···
k=1

γ0 , χ k ) = Introducing UL (¯

γth

γth

γ0 , χk )dγ L (¯

γ0 , χk )dγ L (¯

. (9)

γth

γth

γth

= e γ¯0

√ ϕjh erfc ( ςh γ)

j=1 h=0 √ log2 M υj ∞

= e γ¯0



j=1 h=0γ th √ M υj



log2

γth

e γ¯0 − γ¯γ e 0 dγ γ¯0

γ 1 √ ϕjh erfc ( ςh γ) e− γ¯0 dγ γ¯0

  I1 ϕjh , ςh , γ¯0 , γth .

(14)

h=0

In the above, I1 is derived as √  ∞ γ a erfc bγ 1c e− c dγ I1 (a, b, c, γth ) = γth  γ   bc √  γth  . (15) − th c =a e erfc bγth − 1+bc erfc (1 + bc) c

and after perform-

0

ing integration, we have ⎧    γ − γ¯th γ ¯0 ⎪ 0 1 − e ⎪ γ ¯0 −χk ⎪ ⎨    γ − χth k k U(¯ γ0 , χk ) = 1 − e + χkχ−¯ γ 0 ⎪  γth  ⎪ ⎪ ⎩ e− γ¯0 1 − 1 + γγ¯th 0

∞ log υj 2 M

j=1

fγΣ (γ)dγ

0

(13)

h=0

√ M/(2M − where υj = (1−2−j ) M −1, ςh = (2h + 1)2 3log √ 2 j h.2j−1 / M j−1 (−1) − 2), and ϕh  = (2  √ √ √ j−1 h.2 / M + 1/2 )/( M log2 M ). Furthermore, we define . and erfc(.) as the floor and complementary error function, respectively. Substituting (13) into (12) and after some manipulation, (12) can be rewritten as

The advantage of the form in (8) is that it lends itself nicely to obtaining simple closed-form expressions not only for the outage probability but also for the average bit error rate. Having the PDF of γΣ |γ0 < γth in hands, we can now obtain the closed-form expression of Pr(O|R = {Rn1 , · · · , RnK } ∩ γ0 ≤ γth ) as follows: γth

√ ϕjh erfc ( ςh γ),

√

γ¯0 = χk

,

√

υj

M

log2

Following the approach leading to (16), we have

, γ¯0 = χk

Pr{E|γ0 ≤ γth }

N K Pr(R = {Rn1 , · · · , RnK })   . (16) . (10) = K=0 n1 ,...,nK =1 ×Pr(E|R = {Rn1 , · · · , RnK }∩γ0 ≤ γth ) n1 <···
, γ¯0 = χk

After combining (9) and (10) with (1), we finally yield the end-to-end outage probability.

One can easily recognize that the probability Pr(R = {Rn1 , · · · , RnK }) has the same form of (4). In the sequel, we provide the closed-form expression for the Pr(E|R =

123

{Rn1 , · · · , RnK } ∩ γ0 ≤ γth ), namely

0

10

Analysis Simulation

Pr(E|R = {Rn1 , · · · , RnK } ∩ γ0 ≤ γth ) √

γth

√

υj K M  

log2

j=1

K 

(−1)k−1

n1 ,...,nk =1 n1 <···
h=0 k=1

=

−1

10

⎤

Bit Error Probability

υj 2 M  √  ∞ log ϕjh erfc ςh γ fγΣ |γ0 <γth (γ)dγ = j=1 h=0 0 ⎡ γth √  √ ϕjh erfc ςh γ fγΣ |γ0 <γth (γ)dγ υj ⎢ log 2 M ⎢0 ∞ = √  ⎣  j j=1 h=0 + ϕh erfc ςh γ fγΣ |γ0 <γth (γ)dγ

⎥ ⎥ , (17) ⎦

Λ(ϕjk ,ςk ,¯ γ0 ,χk )

−2

10

−3

10

P ={0, 0.3, 0.6, 0.9, 0.95, 0.99, 1} d

Λ(ϕjk , ςk , γ¯0 , χk )

−4

is derived as shown in (18) at the top where of the next page. Furthermore, I2 (.) and I3 (.) can be solved in closed-form as [25]

10

0

2

4

6

8

10 E /N b

12

14

16

18

20

o

Fig. 2. Bit error probability versus average SNRs, M = 4, N = 2, η = 3, d = 0.5, and γth = 3.

γth √  γ a erfc bγ 1c e− c dγ I2 (a, b, c, γth ) =    bc 0√  γth  , γth erf (1+bc) = a 1−e− c erfc bγth − 1+bc c 0

0

10

√  γ I3 (a, b, c, γth ) = a erfc bγ cγ2 e− c dγ 0 ⎡  √   bγth − (bc+1)γth ⎤ γth  c 1−e− c 1+ γcth erfc bγth + π(bc+1) 2e ⎢ ⎥ # "  =a ⎣ ⎦ (bc+3/2)2 bc (bc+1)γth − erf c (bc+1)3 γth

−1

Outage Probability

10

where erf(.) denotes the error function.

P = 0.7

−2

d

10

−3

10

−4

10

IV. N UMERICAL R ESULTS AND D ISCUSSION 0

10

−6

10

Pd = 1

N=1 N=2 N=3 N=4

−5

10

0

5

10

15 E /N b

20

25

30

o

−1

Outage Probability

10

Fig. 3. Effect of number of relays on the system outage probability, Pd = 0.7, η = 3, d = 0.5, and γth = 3.

−2

10

−3

10

P ={0, 0.3, 0.6, 0.9, 0.95, 0.99, 1} d

−4

10

Analysis Simulation 0

2

4

6

8

10 E /N b

12

14

16

18

20

o

Fig. 1. Outage probability versus average SNRs, N = 2, η = 3, d = 0.5, and γth = 3.

In this section, we conduct numerical evaluations for the proposed scheme showing the advantages as well as demonstrating the validity and usefulness of our analytical expressions. We assume that all secondary nodes are located in a straight line and all relay nodes are between the source and the destination, which are placed in the both ends. We normalized the distance between the source and the destination to one and

let d denote the distance between the source to all relays. As a results, the average channel powers are λ0 = 1, λ1,k = d−η and λ2,k = (1 − d)−η for all k. In addition, the transmission power at the secondary source and the relays is set to P1 = P2 . Fig. 1 depicts the outage performance of the proposed system for some Pd assumptions while in Fig. 2 the same comparison in terms of average BEP is presented. As can be clearly seen from both figures, analytical and simulated curves match excellently in the medium-to-high SNR regime. As expected, the system performance increases as Pd increases, yet in a non-linear manner. We note that the results in (1) and (11) generalize the outage probability and the bit error rate for direct transmission and conventional incremental cooperative networks. In particular, for the specific case of Pd = 0 (i.e., direct transmission), it is easy to see that (1) and (11) reduce to the classical result given in [26]. Similarly, for the particular case of Pd = 1 (i.e., conventional incremental cooperative

124

⎧ ⎨

−γ

( 1 − 1 )

th γ ¯ χ γ ¯0 I (ϕj , ς , γ¯ , γ )+ χk I (ϕj , ς , χ , γ )+ 1−e 1−¯γ0 /χ0k k Λ(ϕjk , ςk , γ¯0 , χk ) = γ¯0−χk 2 k k 0 th χk−¯γ0 2 k k k th ⎩ I3 (ϕjk , ςk , γ¯0 , γth )+¯ γth /¯ γ0 I1 (ϕjk , ςk , γ¯0 , γth )

N=1 N=2 N=3 N=4 −2

Bit Error Probability

10

Pd = 0.7 −3

10

N increasing −4

10

Pd = 1 −5

0

5

10

15 E /N b

20

25

30

o

Fig. 4. Effect of number of relays on the system bit error probability, M = 4, Pd = 0.7, η = 3, d = 0.5, and γth = 3.

, γ¯0 = χk

(18)

V. C ONCLUSION We conducted a study of incremental cooperative networks under opportunistic spectrum access. In particular, we derived the system outage probability, which is valid not only for i.i.d. but also for i.n.d. Rayleigh fading channels. Together with the outage probability, a closed-form expression for bit error probability was also derived for square M -QAM. Numerical results demonstrate that the proposed schemes outperforms direct transmission and the probability of spectrum detection has a significant effect on the system performance. As further work, it may be important to analyze optimal power allocation between the source and the relays as well as the effect of the imperfect feedback channel on the system performance.

0

10

Analysis Simulation M = {4, 16, 64, 256} −1

Bit Error Probability

, γ¯0 = χk

incremental relaying, we still get some advantage compared to that of direct transmission. For example, at the target BER at 10−3 , the performance of the proposed system is about 5 dB superior to that of direct transmission. Beside, one may observe in Fig. 4 that unlike the conventional incremental relaying networks where all BER curves converge to a lower bound at high SNRs regardless the increase of N , for the proposed system there is a certain gap between BER curves. The effect of modulation level on the bit error is shown in Fig. 5. The main result extracted from Fig. 5 is that lower modulation levels result in better error performance and vice versa. For instance, if the modulation level is increased from 4 to 16, the error probability reduction amounts are about 5 dB. In addition, Fig. 5 manifests the validity of the analyzed approach used in this paper.

−1

10

10

I1 (ϕjk , ςk , χk , γth )

10

−2

10

ACKNOWLEDGMENT −3

10

0

5

10 Eb/No

15

This research was supported by the Vietnam’s National Foundation for Science and Technology Development (NAFOSTED) (No. 102.99-2010.10).

20

Fig. 5. Effects of modulation levels on the system bit error probability, N = 2, η = 3, d = 0.5, and γth = 3.

R EFERENCES

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