On the capacity of cognitive broadcast channels ... - Wiley Online Library

WIRELESS COMMUNICATIONS AND MOBILE COMPUTING Wirel. Commun. Mob. Comput. 2013; 13:198–203 Published online 20 January 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/wcm.1108

RESEARCH ARTICLE

On the capacity of cognitive broadcast channels with opportunistic scheduling Dong Li* Faculty of Information Technology, Macau University of Science and Technology, Avenida Wai Long, Taipa, Macau, China

ABSTRACT In this paper, we investigate the fundamental capacity limit of the cognitive broadcast channels with opportunistic scheduling, where cognitive users (CUs) can share the same spectrum with the primary user (PU) as long as the interference introduced to the PU is kept at an acceptable level. In this context, we analyze the capacity gains offered by this opportunistic spectrum sharing in Rayleigh fading environment. Specifically, we analyze the outage and effective capacity of the selected CU, and derive closed-form expressions for these capacity metrics. We also obtain closed-form expressions for the asymptotic performance as the bandwidth approaches infinity. Numerical results are provided to corroborate our theoretical analysis and quantify the effects of the system parameters. Copyright © 2011 John Wiley & Sons, Ltd. KEYWORDS cognitive radio; spectrum sharing; opportunistic scheduling; outage capacity; effective capacity *Correspondence

Dong Li, Faculty of Information Technology, Macau University of Science and Technology, Avenida Wai Long, Taipa, Macau, China. E-mail: [email protected]

1. INTRODUCTION Recently, cognitive radio has been proposed as a promising technology to relieve the problem of spectrum overcrowding caused by the increasing number of bandwidth-consuming services, and the problem of spectrum underutilization as reported by the Federal Communications Commission [1]. In this technology, the cognitive user (CU) can opportunistically access the spectrum owned by the primary user (PU) in the context of the spectrum access model [2], which, however, requires reliable information of the vacant spectrum. In the spectrum sharing model [2], the CU is allowed to operate in the same frequency band with the PU on condition that the interference introduced to the PU is kept within a given threshold. Therefore, the challenge is that the capacity performance of the CU should be improved without violation of the interference constraint on the PU in this case. Recent work on the capacity analysis in spectrum sharing systems can be found in Refs. [3--5]. Different capacity maximization problem in terms of the outage, ergodic, and minimum-rate capacity, subject to both the peak and the average interference constraint in Rayleigh fading channels were considered in Ref. [3]. Besides, Musavian et al. [4,5] extended the above results to the imperfect feedback and 198

soft sensing scenarios. It should be noted that these work focused on the case where there is only one point-to-point cognitive link. Wang et al. [6,7] analyzed and investigated the CU’s capacity by exploiting spatial diversity, and multispectrum and multi-user diversity in the spectrum sharing environment. However, CUs in Ref. [7] have to spend a lot of time and energy monitoring multiple spectra. In this paper, we consider the opportunistic spectrum sharing between CUs and the PU through multi-user diversity. In contrast to Ref. [7], CUs are assumed to share the same spectrum with the PU (which is known as spectrum underlay [2]), and different useful capacity metrics are investigated. To be specific, the contributions can be summarized as follows: (i) we derive closed-form expressions for both the outage capacity and the effective capacity, in order to provide more insight into the effect of main parameters on the system performance and (ii) we also obtain closed-form expressions for their asymptotic performance as the bandwidth approaches infinity, in order to analyze the information-theoretic capacity limit of the selected CU. The rest of this paper is organized as follows: Section 2 describes the system model. Closed-form expressions for different capacity notions with respect to outage and effective capacity are given in Section 3, followed by an extension to the asymptotic performance analysis in Copyright © 2011 John Wiley & Sons, Ltd.

D. Li

Capacity of cognitive broadcast channels with opportunistic scheduling

is selected for transmission. Then, the received SNR of the selected CU can be denoted as

CU 1 . .

PU

γmax = max γi =

.

CBS

i

CU n .

Let Z =

.

X , where X Y

CU N

y fX (yz) fY (y) dy

∞

=

yN e−yz (1 − e−yz )N−1 e−y dy

0

∞

=

yN e

−yz

0

2. SYSTEM MODEL The system under consideration consists of one primary receiver (PR), one cognitive base station (CBS) and multiple cognitive receivers (CRs), as illustrated in Figure 1. In the cognitive broadcast system, the CBS selects the CU with the best channel condition for transmission, and the transmission of the selected CU are not allowed cause harmful interference to the PU. All channels are assumed to be independent and identically Rayleigh distributed. gci and gp denote the channel power gains from the CBS to the ith CR and to the PR, respectively, which are assumed to be independent and identically exponential variables with a unit mean. Hence, the probability density functions (PDF) of the channel power gains gci and gp can be represented as fgci (gci ) = e−gci and fgp (gp ) = e−gp , respectively. Furthermore, the knowledge of gci can be sent to the CBS through the feedback channel by performing channel estimation at the CR, while the information of gp can be obtained by a band manager [8] that mediates between CUs and the PU. Note that, in spectrum sharing systems, CUs are allowed to share the same frequency band with the PU on condition that the interference introduced to the PU is within a tolerable range. Thus, the transmit power of all CUs are assumed to be the same (we use the terms the nth CU and the nth cognitive link between the CBS and the nth CR interchangeably thereafter), and can be written as (1)

where Q is the interference limit. Correspondingly, the received signal-to-noise ratio (SNR) is given by Q gc i Pgci = N0 B N0 B g p

i

0

Section 4. In Section 5, performance analysis and comparison are given. Finally, Section 6 concludes this paper.

γi =

= max gci and Y = gp in Equation (3),

∞

fZ (z) =

Figure 1. System model.

Q gp

(3)

then the PDF of the variable Z can be represented as

.

P=

gci Q max i N0 B g p

(2)

where N0 and B denotes the noise variance and received signal bandwidth, respectively. In the spectrum sharing system with opportunistic scheduling, all CUs want to transmit data to the CBS, but only the CU with the maximum SNR

=N

N−1

n=0 N−1

(−1)

n

n=0

N−1

=N

(−1)n

(−1)n

n=0

N−1 n N−1 n

N−1 n ∞

e−nyz e−y dy

y e−((n+1)z+1)y dy

0

1 ((n + 1)z + 1)2

(4)

where fX (x) = Ne−x (1−e−x )N−1 [9], and the cumulative density function (CDF) is given by

FZ (z) =

z

fv (v) dv 0

=N

N−1

(−1)

n

n=0

=N

N−1 n=0

(−1)n

N−1 n N−1 n

0

z

1 dv ((n + 1) v + 1)2

z (n + 1) z + 1

(5)

In the following section, we investigate the performance of the considered system with respect to the outage capacity and the effective capacity, respectively.

3. CAPACITY PERFORMANCE ANALYSIS 3.1. Outage capacity Outage capacity is an important performance indicator for a slow fading channel environment, where the delay requirement is small compared with the coherent time. Specifically, it is defined as the maximizing the transmission rate that can be maintained over the fading states with a given outage probability, which can be expressed as Cout = max sup {R : Pout = Pr{B ln(1 + γmax ) < R} < ς}

Wirel. Commun. Mob. Comput. 2013; 13:198–203 © 2011 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

i

Pgp=Q

(6) 199


where Pr{} denotes the probability, and R and ς represent the target transmission rate and outage probability, respectively. Note that maximizing the transmission rate for a given outage probability is equivalent to minimizing the outage probability for a given transmission rate, which can be expressed as Pout = Pr{B ln(1 + γmax ) ≤ R} = Pr

max gc

i

i

gp

≤ (e

R/B

= FZ (eR/B −1) NQ0 B =N

N−1

(−1)n

n=0

D. Li

R[m] = Tf B ln 1 +

M   −θ R[m]     lnE e m=1 . Besides, when R[m]    

lim 1 M→∞ M

= − 1θ ln E e−θTf B ln(1+γmax )

(−1)n

n=0

= − 1θ ln E (1 + γmax )−θTf B

(eR/B −1)N0 B (n+1)(eR/B −1)N0 B+Q

= − 1θ ln

1 N−1 n

n+1

(8)

Thus, in the region of Q1, the effects of the transmission rate R and signal bandwidth B on the outage probability disappear, and it only depends on the number of CUs N.

×

0

∞ 0

N−1

(−1)n

n=0

1+

Q N0 B

−θTf B

Q z N0 B

fZ (z) dz

N−1 n

−θTf B z

1 ((n+1) z+1)2

dz

where Tf represents the frame duration, and the time index m is omitted for the sake of simplicity. In Equation (10), the integration in the last equality can be expressed as

3.2. Effective capacity =

   M  −θ R[m]    −(−θ) 1   Ceff = =− lim lnE e m=1  M→∞ Mθ θ    

1+

(10)

0

For many delay-insensitive applications, it is desirable to maximize channel capacity based on ergodic capacity criterion since it will not lead to a significant delay. However, due to the unpredictable nature of wireless channels, it is appropriate to use the statistical delay to guarantee the quality of service (QoS). In this section, we consider using effective capacity which is related to the statistical delay and the dual of effective bandwidth [10], to evaluate the system performance with real-time applications. More specifically, it is defined as the maximum constant arrival-rate that can be supported by the channel while satisfying the delay QoS requirement [11], which is given by

∞

= − 1θ ln N

thus leading to a closed-form expression for the outage probability of the considered system. If the interference limit is low, i.e., Q1, the outage probability in (7) can be simplified as

, m= 1, 2, · · · , M

gp [m]

is the instantaneous transmission rate and (−θ) =   

(7)

Pout = N

i

Ceff = − 1θ ln(E{e−θR })

max gci [m]

is uncorrelated, the effective capacity can be further expressed as

−1) NQ0 B

N−1 n

N−1

Q N0 B

∞

(1 + az)−c

1 dz (bz + 1)2

 1−c+ a a −1−c (c−1)cπ csc(cπ)+c F 1,1,2−c; b 2 1 a b b  a = b, c = 1 (a−b)(c−1) 

b−a+a ln a b a = b, c (a−b)2 1 a=b a(c+1)

=1

(11) where a = NQ0 B , b = n + 1 and c = θTf B. 2 F1 [α, β, χ, δ] denotes the Gauss’s hypergeometric function.

4. ASYMPTOTIC PERFORMANCE ANALYSIS In the study so far, we have considered the system with finite bandwidth. In this section, we asymptotically analyze the outage capacity and the effective capacity of the considered system as the signal bandwidth B approaches infinity, in order to investigate the effects of main system parameters on the system capacity.

(9) where θ = − lim

x→∞

ln(Pr{q(∞)>x}) x

denotes the delay QoS

exponent, and q(m) is the transmit buffer length at time m. Note that θ → 0 indicates there is no delay constraint while θ → ∞ refers to strict delay constraint. 200

4.1. Asymptotic outage capacity In parallel with the former analysis, we also focus on the outage probability. Taking the limit of (7) as B goes to infinity,


D. Li


we can obtain the asymptotic outage probability as

= Pr

B→∞

= Pr = Pr

max gci

Q N0

gp i

i

Q

N−1

(−1)n

N−1 n

n=0

where d =

RN0 , Q

N=1

≤R

gp

N=5

RN0 Q

RN0

i

≤

gp

max gci

Q N0 B

≤R

i

max gc

= FZ =N

B ln 1 +

lim

Outage Probability

aym Pout

0

10

d bd+1

(12)

−1

10

N = 10

−2

10 −10

Analysis Simulation −5

and the first equality follows from the fact

that lim δ ln 1 + δ→∞

∂ δ

= ∂. Note that when Q1, this is

the outage probability given in (8), which further verifies the correctness of our analysis as a double check.

0 Q (dB)

5

10

Figure 2. Outage probability versus the interference constraint Q for different number of CUs N, where R = 2 nats/s.

4.2. Asymptotic effective capacity

representing the Euler-Mascheroni constant [13], while the second equality follows from the fact that ln(1 + x) ≈ x as x1.

The asymptotic effective capacity can be obtained by taking the limit of (10) as B goes to infinity

5. NUMERICAL RESULTS

asy Ceff (θ)

= lim B→∞

− 1θ

= − 1θ ln

 

= − 1θ ln

ln

0

N

=

ln

1+

B→∞

e

−

N−1

θTf Qz N0

(−1)n

n=0

N

lim

0

∞

− 1θ

0

∞

= − 1θ ln

∞

N−1

(−1)n

−θTf B

Q N0 B z

fZ (z) dz

NQz0 B

Q



− θTNf Q z 0

1+ N Bz 0



fZ (z) dz

fZ (z) dz

N−1 ∞ n

0

N−1 1 g

e

−

θTf Q z N0

b

eg/b

+∞

−t

b

n

1 (bz+1)2

g

Ei − b

dz

+1

(13)

n=0

e dt is the exponenwhere g = θTNf0Q and Ei(x) = x t tial integral function. Besides, the third equality follows from the fact that lim (1 + 1δ )δ = e. In addition, for the spe-

In this section, we present and evaluate the capacity performance of the cognitive broadcast channel with opportunistic scheduling. The asymptotic performance of the outage and effective capacity is also provided for comparison. In this simulation, Tf B = 1 and N0 B = 1 are assumed. Here, B refers to the finite bandwidth. Furthermore, N = 1 denotes no user scheduling. We start by comparing the analytical and simulation results of the considered system with respect to the outage probability and the effective capacity, respectively. The analytical results are generated by combining (7) and (11), while the simulation results are generated based on the Monte-Carlo method. Figures 2 and 3 show that the outage probability and the effective capacity versus the interference constraint Q for different number of CUs N under the tar4

δ→∞

cial case of small interference limit Q1, the asymptotic performance of the effective capacity can be simplified as asy

Ceff (θ) = − 1θ ln N

1 g

×b

b

×b

b

n=0

1+

= − 1θ ln N

1 g

N−1

g b

(−1)n

1+

g b

(−1)n

N−1 n

−ϒ + ln gb + 1

N−1 n=0

Effective Capacity (nats/s/Hz)

N−1 n

−ϒ−1 +

g b

+1

3.5

Analysis Simulation

N = 10

3 N=5 2.5 2 N=1 1.5 1 0.5 0 −10

(14)

where the first equality follows from the fact that when x1, ex ≈ 1 + x and Ei(x) ≈ −ϒ−lnx [12] with ϒ = 0.577216

−5

0 Q (dB)

5

10

Figure 3. Effective capacity versus the interference constraint Q for different number of CUs N, where Â = 0.1.


201


interference constraint Q for different delay QoS exponents θ with N = 10. It is seen that the effective capacities and their corresponding asymptotic ones are monotonically decreasing function of θ increases, and it can be also deduced that both of them will be zero as θ approaches infinity. In addition, it is also observed that the gain in the asymptotic performance is bigger than that in the effective capacity by increasing Q, even for small Q. This is because that the former one increases as a cube function of Q as illustrated in Equation (14).

0

10

−1

Outage Probability

10

Asypmtotic Performance

−2

10

−3

10

−4

R = 2 nats/s R = 3 nats/s R = 4 nats/s

10 −10

−5

0 Q (dB)

5

10

Figure 4. Outage probability and corresponding asymptotic performance versus the interference constraint Q for different target transmission rates R, where N = 10. 25

Effective Capacity (nats/s/Hz)

20

D. Li

θ = 0.1 θ = 0.5 θ=1

15

6. CONCLUSIONS In this paper, we have analyzed and investigated the capacity performance of cognitive broadcast channels with opportunistic scheduling, subject to the interference constraint on the PU. Specifically, we have studied the capacity gains provided by this opportunistic spectrum sharing system, and derived closed-form expressions for the outage and effective capacity in Rayleigh fading channels with finite and infinite bandwidth, respectively. Numerical results and comparisons confirm the effectiveness of our theoretical analysis.

Asypmtotic Performance 10

REFERENCES 5

0 −10

−5

0 Q (dB)

5

10

Fig. 5. Effective capacity and corresponding asymptotic performance versus the interference constraint Q for different delay QoS exponents Â, where N = 10.

get transmission rate and the delay QoS exponent θ = 0.1, respectively. From both Figures 2 and 3, we can safely arrive at the conclusion that all the analytical results are accurate since they match precisely with the simulation results. Next, we investigate the performance comparison of both the outage probability and the effective capacity and their corresponding asymptotic performance, respectively. Figure 4 shows the outage probability and its asymptotic performance when N = 10. As shown in Figure 4, when the interference constraint Q is small, the outage probabilities for different R are almost the same. This is because the effect of R on the outage capacity performance vanishes if Q1, which is in accordance with the analytical result in Equation (8). However, when Q is sufficiently high, the selected CU can benefit significantly from decreasing R, which means that R becomes the dominant factor in this case. Besides, it can be observed that the gain in the outage probability is bigger than that in the asymptotic performance by decreasing R for higher value of Q. Figure 5 shows that the effective capacity and its asymptotic performance versus the 202

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AUTHOR’S BIOGRAPHY Dong Li received the B.E. degree in Communication Engineering from Yunnan University, Kunming, China, in 2004, and the M.E. and Ph.D. degrees in Electronics and Communication Engineering from Sun Yat-Sen University, Guangzhou, China, in 2006 and 2010, respectively. From 2006 to 2007, he was with the Department of Electronics and Information Engineering, Jilin University, Zhuhai, China. Since 2010, he has been with the Faculty of Information Technology, Macau University of Science and Technology, Macau, China, where he is now an Assistant Professor. His research interests include Cooperative and Cognitive Communications, Multiple Input and Multiple Output (MIMO) and Orthogonal Frequency Division Multiplexing (OFDM).


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