Ergodic capacity and outage probability optimization for secondary user in cognitive radio networks under interference outage constraint

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Ergodic capacity and outage probability optimization for secondary user in cognitive radio networks under interference outage constraint Ding Xu a,∗ , Qun Li b a Wireless Communication Key Lab of Jiangsu Province and Key Lab of Broadband Wireless Communications and Sensor Network Technology, Ministry of Education, Nanjing University of Posts and Telecommunications, Nanjing, China b School of Computer Science & Technology, Nanjing University of Posts and Telecommunications, Nanjing, China

a r t i c l e

i n f o

Article history: Received 14 October 2013 Received in revised form 9 January 2014 Accepted 24 February 2014 Keywords: Cognitive radio Ergodic capacity Outage probability Power allocation

a b s t r a c t This paper considers a cognitive radio network where a secondary user (SU) coexists with a primary user (PU). The interference outage constraint is applied to protect the primary transmission. The power allocation problem to jointly maximize the ergodic capacity and minimize the outage probability of the SU, subject to the average transmit power constraint and the interference outage constraint, is studied. Suppose that the perfect knowledge of the instantaneous channel state information (CSI) of the interference link between the SU transmitter and the PU receiver is available at the SU, the optimal power allocation strategy is then proposed. Additionally, to manage more practical situations, we further assume only the interference link channel distribution is known and derive the corresponding optimal power allocation strategy. Extensive simulation results are given to verify the effectiveness of the proposed strategies. It is shown that the proposed strategies achieve high ergodic capacity and low outage probability simultaneously, whereas optimizing the ergodic capacity (or outage probability) only leads to much higher outage probability (or lower ergodic capacity). It is also shown that the SU performance is not degraded due to partial knowledge of the interference link CSI if tight transmit power constraint is applied. © 2014 Elsevier GmbH. All rights reserved.

1. Introduction Motivated by the fact that many parts of the licensed spectrum are used inefficiently under the current traditional spectrum regulation policy, the concept of cognitive radio (CR) is proposed as a promising solution for improving the spectrum efficiency [1]. Secondary users (SUs) in CR networks can coexist with the primary users (PUs) on the same spectrum band in the spectrum sharing model provided that the interference caused by the former to the latter is below a certain threshold [2]. Power allocation for the SU is an important operation not only to ensure that the SU causes acceptable interference to the PU but also to maximize the SU utility. In this regard, many studies on the problem of power allocation for the SU have recently been appeared in the literature. In [3], under the constraint that the outage probability of the PU is not degraded by the SU, the optimal power allocation strategy was proposed to maximize the SU rate. In [4], under the peak/average transmit power and the peak/average interference power constraints, the optimal power

∗ Corresponding author. E-mail address: [email protected] (D. Xu).

allocation strategies to maximize the ergodic capacity and the outage capacity of the SU were derived and it is shown that the SU under the average power constraint achieves higher capacity gain over the peak power constraint. Under the same power constraints, [5] obtained the optimal power allocation to minimize the average bit error rate (BER) of the SU. The authors in [6] proposed the optimal power allocation strategy to maximize the SU rate under the PU rate loss constraint. In their further work [7], the optimal power allocation strategies to maximize the SU ergodic capacity and outage capacity under the PU outage constraint were derived. In [8], the optimal power allocation to maximize the SU throughput under the signal to noise ratio (SINR) constraint with PU’s limited cooperation was derived. It is shown in [8] that the SU throughput is greatly increased compared to the non-cooperative case. In [9], a family of power allocation strategies were presented and established that the family can balance the SU rate against the individual PU rate loss constraint. In [10], utilizing opportunistic subchannel access, two power allocation strategies were proposed to maximize the SU rate and the sum rate of the SU and the PU, respectively. Under both the PU and the SU SINR constraints, [11] studied the problem of power allocation to maximize the sum rate of the SU and the PU and provided the solutions for both low and high SINR scenarios. In [12], the authors proposed a distributed power allocation scheme

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based on Nash bargaining solution and presented theoretic analysis of the convergence of the distributed scheme. In their further work [13], using Stackelberg game theory to model the interaction between the SU and the PU, a price-based power allocation strategy was proposed to maximize the revenues of the SU and the PU. In [14], the optimal power allocation to minimize the outage probability of the SUs was derived for interference-limited cognitive multiple access channels. In [15], a power allocation strategy to maximize the average sum rate of a CR multicast network under the service outage constraint was proposed. It is shown in [15] that the proposed strategy significantly outperforms the conventional scheme. In this paper, we study the problem of power allocation for the SU to jointly maximize the ergodic capacity and minimize the outage probability of the SU, under the average transmit power constraint and the interference outage constraint to protect the PU transmission. Ergodic capacity is the maximum achievable rate averaged over all the fading states, which is used to measure the performance of the SU on a long-term basis. Outage probability is the probability that the rate is below a predefined target rate (outage capacity), which is a performance metric of the SU on a short-term basis. Different from existing work, such as [4,7,14], which only considered ergodic capacity or outage probability as the performance indicator, this paper takes both ergodic capacity and outage probability of the SU into consideration. The optimal power allocation strategy is then derived assuming that the SU has the perfect knowledge of the instantaneous channel state information (CSI) of the interference link between the SU transmitter and the PU receiver. Besides, to manage more practical situations, we also consider the partial CSI scenario that the SU only knows the distribution of the interference link channel and derive the corresponding optimal power allocation strategy. Simulation results show that our proposed strategies can achieve high ergodic capacity and low outage probability at the same time, while the power allocation strategy optimizing the ergodic capacity (or outage probability) only leads to much higher outage probability (or lower ergodic capacity). It is also shown that, under tight transmit power constraint, the SU performance is not degraded due to partial knowledge of the interference link CSI. The remainder of the paper is organized as follows. Section 2 introduces the system model. Sections 3 and 4 derive the optimal power allocation strategies with perfect and partial CSI of the interference link, respectively. Section 5 presents the simulation results. Finally, Section 6 concludes the paper.

2. System model We consider one secondary link coexists with one primary link as in [4,5,7]. The PU and the SU share the same narrow band for transmission. The primary link consists of a PU transmitter (PT) and a PU receiver (PR), while the secondary link consists of a SU transmitter (ST) and a SU receiver (SR). Let gss , gsp and gps denote the instantaneous channel gains of the ST-SR link, the ST-PR link and the PT-SR link, respectively. All the channels involved are assumed to be independent block fading channels. The perfect knowledge of gss and gps is assumed to be available at the SU1 . As for gsp , Section 3 assumes that the SU knows gsp perfectly and Section 4 assumes that only the distribution of gsp is known at the SU. The noise power at the SR is assumed to be  2 .

1 In practice, gss can be obtained by classic channel estimation technologies, while gps can be obtained at the SU by measuring the received signal power from PT assuming that the SU knows the transmit power of PT [7].

To protect the PU transmission, the interference outage constraint is applied as Pr(ps gsp > Q ) ≤ ε,

(1)

where ps is the transmit power the ST, Q is the predefined interference power threshold and ε is the interference outage probability limit. In addition, the transmit power of the ST is also constrained as E{ps } ≤ P,

(2)

where P is the average transmit power limit. 3. Optimal power allocation with perfect CSI In this section, the problem of power allocation to jointly maximize the ergodic capacity and minimize the outage probability of the SU subject to the average transmit power constraint and the interference outage constraint is studied. The optimization problem is stated as (P1) max

E{log2 (1 + )} and min Pr (log2 (1 + ) < R)

s.t.

(1), (2),

ps ≥0

ps ≥0

(3)

where  = ps gss / ␴2 +Pp gps is the SINR of the SU, Pp is the transmit power of the PU and R is the target rate. The above multi-objective functions in P1 can be transformed to a single-objective function by applying scalarization methods [16]. To make the two objective functions in (3) comparable, we normalize the ergodic capacity as E{log2 (1 + )}/Cmax , where Cmax is the maximum ergodic capacity under the constraints in (1) and (2) which can be obtained by solving the following problem



Cmax = max

E log2 (1 + )

s.t.

(1), (2).

ps ≥0

 (4)

Therefore, the two objective functions E{log2 (1 + )}/Cmax and Pr(log 2 (1 + ) < R) are dimensionless quantities within [0, 1], which are comparable. Then P1 can be reformulated as (P2) min ps ≥0



ω Cmax



E −log2 (1 + ) + (1 − ω)Pr (log2 (1 + ) < R) (5)

s.t. (1), (2), where weight factor ω (0 ≤ ω ≤ 1) represents the relative importance of the objective function E{−log2 (1 + )}/Cmax to the objective function Pr(log 2 (1 + ) < R)2 . As for solving the problem in (4) to obtain the value of Cmax , the same power allocation strategy proposed in what follows to solve P2 can be used by setting ω = 1 and Cmax = 1 in (5). For the convenience of later analysis, the following indicator functions are introduced as



o (ps ) = and i (ps ) =



1,

log2 (1 + ) < R

0,

log2 (1 + ) ≥R,

1,

ps gsp > Q

0,

ps gsp ≤ Q.

(6)

(7)

2 Note that how to choose the value of ω depends on the preferences for the ergodic capacity and the outage probability.

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as shown in Fig. 1f and e, respectively. Whether p2s or p3s minimizes S(ps ) depends on S(p2s ) and S(p3s ) as

Thus, P2 can be rewritten as (P3) min

Cmax

ps ≥0

s.t.



ω



E −log2 (1 + ) + (1 − ω)E{o (ps )} (8)

 p∗s =

E{i (ps )} ≤ ε,

3

(9)

p2s ,

f (p2s ) < f (p3s ) + 

p3s ,

otherwise.

(16)

and constraint (2). Under the assumption of continuous channel gain distributions, the time-sharing condition is satisfied for P3 according to [17]. It is noted that zero duality gap is achieved if time-sharing condition is satisfied [17]. Therefore, the Lagrange duality method is applied to solve P3 [18]. The Lagrangian function of P3 is written as L(ps , , ) =

  ω E −log2 (1 + ) + (1 − ω)E{o (ps )} Cmax + (E{i (ps )} − ε) + (E{ps } − P),

 (10)

where  and  are the non-negative dual variables associated with the constraints in (9) and (2), respectively. The Lagrange dual function is then expressed as D(, ) = min L(ps , , ),

(11)

ps ≥0

which can be decoupled into parallel subproblems with same structure for each fading state as min − ps ≥0

ω log2 (1 + ) + (1 − ω)o (ps ) + i (ps ) + ps . Cmax

(2R − 1)( 2 + Pp gps ) . gss

(13)

The critical value of ps , over which the value of i (ps ) changes from 0 to 1, is calculated from (7) as p2s = gQsp . Let f (ps ) = −

ω log2 (1 + ) + ps . Cmax



=

 2 + Pp gps − gss Cmax ln 2 ω

=

p1s ,

f (p1s ) +  ≤ f (p3s ) + 1 − ω

p3s ,

otherwise.

(17)

Case 4: R4 . In this case, S(p1s ) = f (p1s ) + , S(p2s ) = f (p2s ) + 1 − ω and S(p3s ) = f (p3s ) + . Since p1s ≤ p3s and f(ps ) is a monotonously decreasing function of ps for ps < p3s , we have S(p1s )≥S(p3s ). Thus, the minimum value of S(ps ) is attained at p2s or p3s , and there are two subcases for S(ps ), as shown in Fig. 1a and b, respectively. Whether p2s or p3s minimizes S(ps ) depends on S(p2s ) and S(p3s ) as

 p∗s

=

p2s ,

f (p2s ) + 1 − ω < f (p3s ) + 

p3s ,

otherwise.

(18)

Case 5: R5 . In this case, S(p1s ) = f (p1s ), S(p2s ) = f (p2s ) and S(p3s ) = f (p3s ) + 1 − ω. Since p1s ≤ p2s and f(ps ) is a monotonously increasing function of ps for ps > p3s , we have S(p1s ) ≤ S(p2s ). Thus, the minimum value of S(ps ) is attained at p1s or p3s , and there are two subcases for S(ps ), as shown in Fig. 1c and d, respectively. Whether p1s or p3s minimizes S(ps ) depends on S(p1s ) and S(p3s ) as

 p∗s =

p1s ,

f (p1s ) ≤ f (p3s ) + 1 − ω

p3s ,

otherwise.

(19)

(14)

It is easy to prove that f(ps ) is a convex function w.r.t. ps , and its minimum value is attained at p3s

p∗s

(12)

Therefore, P2 can be solved by solving (12) for each fading state with fixed  and , and iteratively updating these two variables via subgradient-based methods such as ellipsoid method [19]. The details are omitted here for brevity. In what follows, the solution to subproblem (12) is derived. Let S(ps ) = − C ω log2 (1 + ) + (1 − ω)o (ps ) + i (ps ) + ps . Note max that o (ps ) and i (ps ) in S(ps ) are in general step functions of ps . The critical value of ps , over which the value of o (ps ) changes from 1 to 0, can be calculated from (6) as p1s =

Case 3: R3 . In this case, S(p1s ) = f (p1s ) + , S(p2s ) = f (p2s ) + 1 − ω and S(p3s ) = f (p3s ) + 1 − ω. Since p2s > p3s and f(ps ) is a monotonously increasing function of ps for ps > p3s , we have S(p2s ) > S(p3s ). Thus, S(ps ) attains its minimum value at p1s or p3s , and there are two subcases for S(ps ), as shown in Fig. 1g and h, respectively. Whether p1s or p3s minimizes S(ps ) depends on S(p1s ) and S(p3s ) as

+

,

(15)

where (.)+ denotes max(. , 0). Define the following six regions: R1  {p1s ≤ p3s < p2s }, R2  {p1s ≤ 2 ps ≤ p3s }, R3  {p3s < p2s < p1s }, R4  {p2s < p1s ≤ p3s }, R5  {p3s < p1s ≤ p2s }, R6  {p2s ≤ p3s < p1s }. Denote p∗s as the optimal solution to (12), and we make the following discussions on p∗s in the following six cases: Case 1: R1 . In this case, S(p1s ) = f (p1s ), S(p2s ) = f (p2s ) and S(p3s ) = f (p3s ). Note that f(ps ) is a monotonously decreasing function of ps for ps < p3s and a monotonously increasing function of ps for ps > p3s . Thus, as shown in Fig. 1a, S(ps ) attains its minimum value at p3s . Therefore, p∗s = p3s . Case 2: R2 . In this case, S(p1s ) = f (p1s ), S(p2s ) = f (p2s ) and S(p3s ) = f (p3s ) + . Since p1s ≤ p2s and f(ps ) is a monotonously decreasing function of ps for ps < p3s , we have S(p1s )≥S(p2s ). Thus, S(ps ) attains its minimum value at p2s or p3s , and there are two subcases for S(ps ),

Case 6: R6 . In this case, S(p1s ) = f (p1s ) + , S(p2s ) = f (p2s ) + 1 − ω and S(p3s ) = f (p3s ) + 1 − ω + . As shown in Fig. 1i–k, respectively, S(ps ) has three possible forms depending on the relationships between p1s , p2s and p3s . Therefore, we have

p∗s =

⎧ 1 ps , f (p1s ) +  ≤ min(f (p2s ) + 1 − ω, ⎪ ⎪ ⎪ ⎪ ⎪ f (p3s ) + 1 − ω + ) ⎪ ⎪ ⎪ ⎪ ⎨ p2 , f (p2 ) + 1 − ω < min(f (p1 ) + , s s s ⎪ f (p3s ) + 1 − ω + ) ⎪ ⎪ ⎪ ⎪ ⎪ p3s , f (p3s ) + 1 − ω +  < f (p1s ) + , ⎪ ⎪ ⎪ ⎩ 3 2

(20)

f (ps ) + 1 − ω +  ≤ f (ps ) + 1 − ω

The results obtained in the above discussions can then be summarized as: Theorem 1.

The optimal solution of (12) is given by

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Fig. 1. Illustration of different forms of S(ps ) (The curves with dash-dot lines and solid lines correspond to f(ps ) and S(ps ), respectively).

⎧ 3 ps , ⎪ ⎪ ⎪ 2 ⎪ ps , ⎪ ⎪ ⎪ ⎪ ⎪ p3s , ⎪ ⎪ ⎪ ⎪ ⎪ p1s , ⎪ ⎪ ⎪ ⎪ ⎪ p3s , ⎪ ⎪ ⎪ ⎨ 2 p∗s =

ps ,

⎪ ⎪ ⎪ ⎪ ⎪ p1s , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p3s , ⎪ ⎪ ⎪ ⎪ p1s , ⎪ ⎪ ⎪ ⎪ ⎪ p2s , ⎪ ⎪ ⎩ 3 p3s ,

ps ,

R1 R2 , f (p2s ) < f (p3s ) +  R2 , f (p2s )≥f (p3s ) +  R3 , f (p1s ) +  ≤ f (p3s ) + 1 − ω R3 , f (p1s ) +  > f (p3s ) + 1 − ω R4 , f (p2s ) + 1 − ω < f (p3s ) +  R4 , f (p2s ) + 1 − ω≥f (p3s ) +  R5 , f (p1s ) ≤ f (p3s ) + 1 − ω R5 , f (p1s ) > f (p3s ) + 1 − ω R6 , f (p1s ) +  ≤ min(f (p2s ) + 1 − ω, f (p3s ) + 1 − ω + ) R6 , f (p2s ) + 1 − ω < min(f (p1s ) + , f (p3s ) + 1 − ω + ) R6 , f (p3s ) + 1 − ω +  < f (p1s ) + , f (p3s ) + 1 − ω +  ≤ f (p2s ) + 1 − ω)

(21)

Next, further elaborations on the optimal power allocation in (21) are given as follows: (1) Region R1 : In this region, since p1s ≤ p3s < p2s , the interference power limit will be satisfied and the outage of the SU will not occur even when the SU transmits with p3s to achieve maximum ergodic capacity. In practice, this may happen when the channel condition of the ST-SR link is sufficiently good or the channel gain of the ST-PR link is sufficiently small. (2) Regions R2 , R3 , R4 , R5 and R6 : In these regions, the channel conditions of the ST-SR link and the ST-PR link are not as favorable

as those in R1 . Consequently, whether the SU is in an outage or the interference power is in an outage depends on the relative channel conditions of the ST-SR link to the ST-PR link. Take region R2 as an example. In region R2 , since p1s ≤ p2s ≤ p3s , we have i (p1s ) = 0, i (p3s ) = 1, o (p2s ) = 0 and o (p3s ) = 0. Thus, if the SU transmits with p1s or p2s , the interference power limit and the target rate of the SU will be satisfied. Clearly, with p1s ≤ p2s , the SU will have a higher rate with transmit power p2s over p1s . Therefore, transmit power p2s is better than p1s . On the other hand, with p2s ≤ p3s , the SU will achieve a higher rate with transmit power p3s over p2s . However, the interference power limit is not satisfied if the SU transmits with p3s . Therefore, the SU has to balance the SU rate increase and the cost to the interference outage constraint before allocating its transmit power to be p2s or p3s . We may regard  as the penalty that the SU has to pay when an interference outage event happens. Then, if the SU transmits with power p2s , it will cause no additional penalty and the total penalty for the SU will be f (p2s ), while when the SU transmits with power p3s , it will cause a penalty  and the total penalty for the SU will be f (p3s ) + . Clearly, if f (p2s ) < f (p3s ) + , the SU should transmit with power p2s , otherwise, the SU should transmit with power p3s . Note that the elaborations on regions R3 , R4 , R5 and R6 are similar to those for the region R2 and are thus omitted here for brevity. The complexity of the proposed power allocation strategy with perfect CSI is analyzed briefly as follows. For fixed  and , O(1) operations are required for obtaining the optimal power allocation in (21). To obtain optimal * and * , the ellipsoid method converges in O(4/ı2 ) of iterations to achieve ı-optimality for the dual function D(, ), i.e., D(, ) − D(* , * ) < ı [18]. Consequently, the total complexity is in order of O(1/ı2 ).

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4. Optimal power allocation with partial CSI In this section, the problem of power allocation is studied assuming that only the distribution of gsp is known at the SU. Without loss of generality, the channel of the ST-PR link is assumed to follow Nakagami-m fading which includes Rayleigh fading as a special case with fading parameter m = 1. Then, the channel gain gsp is Gamma distributed with probability density function (PDF) given by [20] mm xm−1 −mx/˝ , e ˝m (m)

fgsp (x) =

x≥0

(22)

˝ = E{gsp }, (·) is the Gamma function defined as, (x) = where ∞ x−1 −t t

0

e

dt [21], and cumulative distribution function (CDF) [20]



Fgsp (x) = 1 −

m, (mx/˝) (m)

x≥0

,

(23)

where (·, ·) is the incomplete Gamma function defined as, ∞ (˛, x) = x e−t t ˛−1 dt [21]. Using (23), the interference outage constraint in (1) can be rewritten as ε≥

Pr(ps gsp > Q )

=

1 − Fgsp (



Q ) ps

=

(24)

m, (mQ/˝ps ) (m)

,

Based on (24), the transmit power the SU is constrained as ps ≤

mQ , ˝ −1 (m, ε (m))

(25)

where −1 (· , ·) denotes the inverse incomplete Gamma function which can be efficiently solved by MATLAB function gammaincinv. Let P = mQ/˝ −1 (m, ε (m)). The optimization problem to jointly maximize the ergodic capacity and minimize the outage probability of the SU subject to the average transmit power constraint and the interference outage constraint is formulated as (P4) max

0≤ps ≤P

s.t.



E log2 (1 + )



and min Pr (log2 (1 + ) < R) ps ≥0

(26)

min



ω

0≤ps ≤P

Cmax 

s.t.

(2),



E −log2 (1 + ) + (1 − ω)Pr (log2 (1 + ) < R) (27)

where



Cmax  = max

E log2 (1 + )

s.t.

(2)

0≤ps ≤P

ω

0≤ps ≤P

Cmax

s.t.

(2).

 p∗s

=





E −log2 (1 + ) + (1 − ω)E{o (ps )}

min f (ps ) + (1 − ω)o (ps ),

p3s ,

f (p3s ) + 1 − ω < f (p1s )

p1s ,

f (p3s ) + 1 − ω≥f (p1s )

(31)

Define the following four regions: R1   {P ≤ p3s }, R2   {p3s < P ≤ p1s }, R3   {p1s ≤ p3s < P}, R4   {p3s < p1s ≤ P}. The results obtained in the above cases lead to the following theorem: Theorem 2.

p∗s =

The optimal solution of (30) is given by

⎧ P, R1  ⎪ ⎪ ⎪ ⎪ ⎪ p3 , R2  ⎪ ⎨ s p3 ,

R3 

p1s ,

R4 , f (p3s ) + 1 − ω≥f (p1s )

s ⎪ ⎪ ⎪ ⎪ p3s , R4 , f (p3s ) + 1 − ω < f (p1s ) ⎪ ⎪ ⎩

(32)

(28)

(29)

Similar to the techniques used in Section 3, P6 can be decoupled into parallel subproblems with same structure for each fading state as 0≤ps ≤P

(1) P ≤ p1s . In this subcase, the objective function f(ps ) + (1 − ω)o (ps ) is equal to f(ps ) + 1 − ω for 0 ≤ ps ≤ P , which is a convex function with its minimum value attained at ps = p3s , as shown in Fig. 2c. Thus, we have p∗s = p3s . (2) P > p1s , p1s ≤ p3s . In this subcase, o (ps ) = 1, if 0 ≤ ps ≤ p1s ; otherwise, o (ps ) = 0. Thus, the objective function f(ps ) + (1 − ω)o (ps ) is equal to f(ps ) + 1 − ω for 0 ≤ ps ≤ p1s , and is equal to f(ps ) for p1s ≤ ps ≤ p3s , as shown in Fig. 2d. Since f(ps ) is a monotonously decreasing function of ps for p1s ≤ ps ≤ p3s , we have p∗s = p3s . (3) P > p1s , p1s > p3s . In this subcase, o (ps ) = 1, if 0 ≤ ps ≤ p1s ; otherwise, o (ps ) = 0. Thus, the objective function f(ps ) + (1 − ω)o (ps ) is equal to f(ps ) + 1 − ω for 0 ≤ ps ≤ p1s , and is equal to f(ps ) for p1s ≤ ps ≤ P. Depending on the values of the objective function at p1s and p3s , the objective function has two forms as shown in Fig. 2e and f, respectively. Thus, we have

Next, we give detailed elaborations on the optimal power allocation in (32) as follows:



It is noted that the problem in (28) can be solved by following traditional water-filling power allocation with peak power constraint P [22]. Using indicator function o (ps ) in (6), P5 can be rewritten as (P6) min

where function f(ps ) is defined in (14) with its minimum value attained at p3s in (15). We make the following discussions on the optimal solution to (30), p∗s , in the following two cases: Case 1: P ≤ p3s . In this case, depending on the relationship between P and p1s , where p1s is the critical value for the function o (ps ) given in (13), the objective function f(ps ) + (1 − ω)o (ps ) has two possible forms, as shown in Fig. 2a and b, respectively. Since f(ps ) is a monotonously increasing function of ps for ps > p3s , regardless of the relationship between p1s and P , the minimum value of the objective function, f(ps ) + (1 − ω)o (ps ), is attained at p∗s = P. Case 2: P > p3s . In this case, the objective function f(ps ) + (1 − ω)o (ps ) has four possible forms, as shown in Fig. 2c, d, e, f, respectively. Consequently, the following subcases are considered:

(2).

Follow the similar procedures in Section 3, P4 can be reformulated as (P5)

5

(30)

(1) Region R1 : In this region, the SU will achieve its maximum rate if it transmits with its maximum allowable transmit power P . There will be no outage of the SU if P≥p1s ; otherwise the SU will be in outage. In both cases, transmitting with maximum allowable transmit power P is optimum in order to increase the SU rate. In practice, this may happen when the channel condition of the ST-PR link is very good and the performance of the SU is constrained by the interference outage constraint. (2) Region R2 : In this region, the SU is always in outage. Thus, the SU can allocate the transmit power solely to maximize its achievable rate. In practice, this happens when the channel condition of the ST-SR link is not good enough to maintain its target rate. (3) Region R3 : In this region, the maximum allowable transmit P is large enough that the SU can allocate the transmit power

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Fig. 2. Illustration of different forms of function f(ps ) + (1 − ω)o (ps ) (The curves with dash-dot lines and solid lines correspond to f(ps ) and f(ps ) + (1 − ω)o (ps ), respectively).

without considering the interference to the PU. The SU will achieve its maximum achievable rate and not be in outage when it transmits with the optimal power p3s . This happens when the channel condition of the ST-PR link is very poor and the channel condition of the ST-SR link is very good. (4) Region R4 : In this region, similar to region R3 , the maximum allowable transmit P is large enough that the SU can allocate the transmit power without considering the interference to the PU. However, different from region R3 , if the SU transmits with the optimal power p3s , it will cause an outage to the SU; if the SU transmits with p1s , it will not do so. Note that p3s is the optimum transmit power achieving maximum rate which takes power consumption into consideration. Thus, the SU has to balance the two optimization objectives, i.e., ergodic capacity and outage probability. The objective function is f (p3s ) + 1 − ω at p3s and is f (p1s ) at p1s . Obviously, when f (p3s ) + 1 − ω is smaller than f (p1s ), the SU should transmit with power p3s ; otherwise, the SU should transmit with power p1s .

The complexity of the proposed power allocation strategy with partial CSI is in order of O(1/ı2 ), which can be analyzed by following

a similar approach as we did for the case of perfect CSI. The details are omitted here for brevity. 5. Simulation results In this section, simulation results are given to evaluate the performance of the proposed power allocation strategies. We assume all the channels involved are Nakagami-m fading with parameter m = 2 and channel gains are Gamma distributed with unit mean for gss and mean of 0.5 for gps and gsp . The noise is assumed to be 1, i.e.,  2 = 1. The transmit power of the PU is Pp = 10 dB. The predefined interference power threshold is Q = 10 dB. The target rate of the SU is R = 1 bit/s/Hz. Figs. 3 and 4 plot the ergodic capacity and the outage probability of the SU against the average transmit power constraint P with different values of ω, respectively. It is noted that, ω = 0 corresponds to the case of minimizing the outage probability only and ω = 1 corresponds to the case of maximizing the ergodic capacity only. In the figures, both the cases of perfect CSI and partial CSI are plotted. From Fig. 3, it is seen that the curves almost overlap when P is small. This indicates that P limits the ergodic capacity of the SU. In this case, increasing the values of ω can not improve the ergodic

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Fig. 3. Ergodic capacity of the SU against the average transmit power constraint P (ε = 0.2).

capacity, and obtaining perfect CSI has no capacity gain compared to the case of partial CSI. This observation is very meaningful which allows the SU to use partial CSI instead of perfect CSI when P is small which causes almost zero capacity loss. For the case of partial CSI, as P increases, the ergodic capacities increase and eventually saturate beyond certain points as P increases further. This indicates that the transmit power constraint in (25) becomes the dominant constraint, in other words, P ≤ P. For the case of perfect CSI, the ergodic capacities increase as P increases and get saturated when P is sufficiently large for zero ω. This is due to the fact that, for zero ω, the SU aims to minimize the outage probability without considering the ergodic capacity, and for sufficiently large P, the SU will allocate just enough transmit power to avoid outage events which leaves remaining transmit power budget unused. On the other hand, for nonzero ω, the SU aims to not only minimize the outage probability but also maximize the ergodic capacity, and for sufficiently large P, besides avoiding outage events, the SU will allocate as much transmit power as possible to maximize the ergodic capacity which leads to much higher ergodic capacity. In addition, it is seen that, when P is large, the ergodic capacity with perfect CSI is higher than

Fig. 4. Outage probability of the SU against the average transmit power constraint P (ε = 0.2).

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Fig. 5. Ergodic capacity of the SU against the weight factor ω (ε = 0.2).

that with partial CSI, especially for nonzero ω. This indicates that, having perfect CSI can lead to ergodic capacity improvement only when P is large. From Fig. 4, it is seen that the curves with perfect CSI and partial CSI overlap when P is small. This indicates that P limits the outage probability of the SU and obtaining perfect CSI cannot decrease it. It is also seen that, for the case of partial CSI, as P increases, the outage probabilities with different values of ω decrease and eventually saturate to the same level when P is sufficiently large. This indicates that the transmit power constant in (25) becomes the dominant constraint and varying values of ω cannot change the outage probability. For the case of perfect CSI, the outage probability decreases as P increases and becomes zero or gets saturated when P is sufficiently large. It is interesting to observe that, the outage probability with ω = 0.5 does not decrease monotonically with the increase of P and there is a region of P where the outage probability increases as P increases. The possible explanation is that with the further increase of P, the increase of ergodic capacity may lead to lower value of the optimization objective function in (5), which may result in allocating more resource to increase the ergodic capacity and the outage probability may increase accordingly. In addition, it is seen that, when P is large, the outage probability with perfect CSI is much lower than that with partial CSI. This indicates that, having perfect CSI can decrease outage probability significantly when P is large. Figs. 5 and 6 plot the ergodic capacity and the outage probability of the SU against the weight factor ω with different values of P, respectively. It is seen that, when P is large, perfect CSI results in much higher ergodic capacity and lower outage probability compared to partial CSI. It is also seen that when P is large and ω changes from zero to nonzero, the ergodic capacity increases significantly while the outage probability is almost unchanged. This indicates that optimizing the ergodic capacity and outage probability of the SU together can greatly improve the ergodic capacity while maintaining the outage probability almost unchanged. It is also seen that, for nonzero ω, as ω increases, the ergodic capacity increases very slowly, and the outage probability increases (except when P is large, the outage probability with partial CSI is unchanged, which is consistent with the observation from Fig. 4). From the above observations, we suggest to choose small value of ω to not only achieve high ergodic capacity but also low outage probability. Figs. 7 and 8 plot the ergodic capacity and the outage probability of the SU against the interference outage constraint ε with

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different values of P, respectively. It is seen that, for small P, the ergodic capacities or the outage probabilities with perfect CSI and partial CSI are the same. It is also seen that, the ergodic capacities and the outage probabilities are unchanged as ε increases in small P region. In this case, P limits the performance of the SU and increasing ε or having perfect CSI cannot improve it. On the other hand, for large P, it is seen that, perfect CSI results in much higher ergodic capacity and lower outage probability compared to partial CSI, and with the increase of ε, the ergodic capacity increases and the outage probability decreases.

6. Conclusions

Fig. 6. Outage probability of the SU against the weight factor ω (ε = 0.2).

This paper proposes optimal power allocation strategies for the SU to jointly maximize the ergodic capacity and minimize the outage probability of the SU, subject to the average transmit power constraint and the interference outage constraint. Two scenarios are considered. In one scenario, the perfect knowledge of the CSI of the instantaneous interference link between the SU transmitter and the PU receiver is assumed to be available at the SU. In another scenario, a more practical assumption that only the interference link channel distribution is known at the SU is adopted. It is shown that, compared to the power allocation strategy that optimizes one objective (ergodic capacity or outage probability), the proposed strategies achieve high ergodic capacity and low outage probability simultaneously. It is also shown that partial knowledge of the interference link channel will not degrade the SU performance under tight transmit power constraint.

References

Fig. 7. Ergodic capacity of the SU against the interference outage constraint ε (ω = 0.5).

Fig. 8. Outage probability of the SU against the interference outage constraint ε (ω = 0.5).

[1] Mitola J, Maguire GQ. Cognitive radio: making software radios more personal. IEEE Pers Commun 1999;4:13–8. [2] Haykin S. Cognitive radio: brain-empowered wireless communications. IEEE J Sel Areas Commun 2005;2:201–20. [3] Chen Y, Yu G, Zhang Z, Chen HH, Qiu P. On cognitive radio networks with opportunistic power control strategies in fading channels. IEEE Trans Wireless Commun 2008;7:2752–61. [4] Kang X, Liang YC, Nallanathan A, Garg HK, Zhang R. Optimal power allocation for fading channels in cognitive radio networks: ergodic capacity and outage capacity. IEEE Trans Wireless Commun 2009;2:940–50. [5] Xu D, Feng ZY, Zhang P. Minimum average BER power allocation for fading channels in cognitive radio networks. In: Proceedings of IEEE wireless communications and networking conference. 2011. p. 78–83. [6] Kang X, Garg HK, Liang YC, Zhang R. Optimal power allocation for OFDM-based cognitive radio with new primary transmission protection criteria. IEEE Trans Wireless Commun 2010;6:2066–75. [7] Kang X, Zhang R, Liang YC, Garg HK. Optimal power allocation strategies for fading cognitive radio channels with primary user outage constraint. IEEE J Sel Areas Commun 2011;2:374–83. [8] Xu D, Feng ZY, Li YZ, Zhang P. Optimal power control of cognitive radio under SINR constraint with primary user’s cooperation. IEICE Trans Commun 2011;9:2685–9. [9] Chowdhury M, Singla A, Chaturvedi AK. A family of power allocation schemes achieving high secondary user rates in spectrum sharing OFDM cognitive radio. In: Proceedings of IEEE global communications conference. 2012. p. 1144–9. [10] Xu D, Feng ZY, Zhang P. Power allocation schemes for downlink cognitive radio networks with opportunistic sub-channel access. KSII Trans Internet Inf Syst 2012;7:1777–91. [11] Singh S, Teal TD, Dmochowski PA, Coulson AJ. Interference management in cognitive radio systems-a convex optimisation approach. In: Proceedings of IEEE international conference on communications. 2012. p. 1884–9. [12] Wang Z, Jiang L, He C. A distributed power control algorithm in cognitive radio networks based on Nash bargaining solution. In: Proceedings of IEEE international conference on communications. 2012. p. 1672–6. [13] Wang Z, Jiang L, He C. A novel price-based power control algorithm in cognitive radio networks. IEEE Commun Lett 2013;1:43–6. [14] Li D, Liang YC. Power allocation for interference-limited cognitive multiple access channels. IEEE Commun Lett 2013;3:291–4. [15] Xu D, Feng ZY, Wang Y, Zhang P. Joint power and rate allocation for spectrum sharing cognitive radio multicast networks under service outage constraint. AEU Int J Electron Commun 2013;7:585–7. [16] Marler RT, Arora JS. Survey of multi-objective optimization methods for engineering. Structural and multidisciplinary optimization 2004;6:369–95.

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G Model AEUE-51169; No. of Pages 9

ARTICLE IN PRESS D. Xu, Q. Li / Int. J. Electron. Commun. (AEÜ) xxx (2014) xxx–xxx

[17] Yu W, Lui R. Dual methods for nonconvex spectrum optimization of multicarrier systems. IEEE Trans Commun 2006;7:1310–22. [18] Boyd S, Vandenberghe L. Convex Optimization. Cambridge, UK: Cambridge University Press; 2004. [19] Bland R, Goldfarb D, Todd M. The ellipsoid method: a survey. Oper Res 1981;6:1039–91.

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[20] Simon MK, Alouini MS. Digital communication over fading channels. New York, US: John Wiley & Sons Press; 2005. [21] Gradshteyn IS, Ryzhik IM. Table of integrals, series, and products. New York, US: Academic Press; 2007. [22] Goldsmith AJ, Varaiya PP. Capacity of fading channels with channel side information. IEEE Trans Inf Theory 1997;6:1986–92.

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