the following advantages: 1) the operations of each SU are simple and ... It is proved that, the CR network with this si
3594
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 11, NOVEMBER 2011
A Simple Distributed Power Control Algorithm for Cognitive Radio Networks Yong Xiao, Guoan Bi, Senior Member, IEEE, and Dusit Niyato, Member, IEEE
AbstractβThis paper studies the power control problem for spectrum sharing based cognitive radio (CR) networks with multiple secondary source-to-destination (SD) pairs. A simple distributed algorithm is proposed for the secondary users (SUs) to iteratively adjust their transmit powers to improve the performance of the network. The proposed algorithm does not require each SU (or PU) to negotiate with other SUs (or PUs) during the communication. It is proved that the proposed algorithm can obtain a time average performance as good as that achieved when the Nash equilibrium (NE) is chosen in hindsight. More specifically, the average performance of CR networks ( ( will )) converge to an π-Nash equilibrium at a rate of ππ = π exp 1π . A sub-optimal algorithm is also introduced to further improve ( ) π the convergence rate to log ππβ² β² = π π1β² . Numerical results are π presented to show the performance of the proposed algorithms under different settings. Index TermsβCognitive radio, power allocation, subgradient methods.
I. I NTRODUCTION
C
OGNITIVE radio (CR) is one of the main technologies to solve the spectrum under-utilization problem in future generations of wireless systems. In CR networks, the unlicensed users, called secondary users (SU), can access the spectrum which is unoccupied or ineffectively used by the licensed users, called primary users (PU). In this paper, we focus on spatial spectrum sharing (SSS) based CR networks (spectrum underlay approach) [1] in which SUs and PUs can transmit signals at the same time over the same spectrum. In this system, maximizing the performance of SUs while simultaneously maintaining the interference powers of PUs under the acceptable levels, called the interference temperature limit [1], is still a challenging task. In [2], it was observed that if SUs employ the power control methods, i.e., to decrease (or increase) their transmit powers when the conditions of SU-to-PU channels are βgood" (or βbad"), the spectrum utilization efficiency and system performance can be greatly improved. However, most of the previously reported work neglects the interference among SUs or PUs and assumes each SU can have the global information, i.e., the states of other SUs and the gains of all channels in the network. This requires each SU to handle complex computation and unrealistic channel estimation, which may Manuscript received November 17, 2010; revised March 8, 2011; accepted June 17, 2011. The associate editor coordinating the review of this letter and approving it for publication was D. I. Kim. Y. Xiao and G. Bi are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore (e-mail:
[email protected],
[email protected]). D. Niyato is with the School of Computer Engineering, Nanyang Technological University, Singapore (e-mail:
[email protected]). Digital Object Identifier 10.1109/TWC.2011.090611.102049
be impossible for many practical systems. More specifically, the optimal power control methods reported in [2] [3] for the case of one secondary source-to-destination (SD) pair sharing the spectrum with one PU were derived by assuming the transmitters to know the instantaneous channel gains of both SU-to-SU and SU-to-PU channels. The results were extended to CR networks with multiple SD pairs [4] and multi-hop relaying SUs [5] based on the similar assumptions. Motivated by these limitations, a game theoretic framework is established in this paper to solve the power control problem of distributed CR networks with multiple secondary SD pairs and PUs. A simple distributed algorithm is proposed to achieve the following advantages: 1) the operations of each SU are simple and fully distributed, i.e., both SUs and PUs do not know the global information, and there is no central controller to manage the network. In addition, each PU in our setting broadcasts the same information to all SUs and each SU cannot communicate with others or obtain specific instructions from PUs, 2) the computational complexity of each SU is low and does not depend on the number of SUs, 3) the proposed algorithm can be directly applied to other resource control problems, e.g., spectrum allocation, time scheduling, etc. It is proved that, the CR network with this simple algorithm can achieve a time average performance as good as that achieved when the Nash equilibrium (NE) is chosen in hindsight. More specifically, we show that by using the proposed algorithm the average performance ( (of))each SU converges to an π-NE at a rate of ππ = π exp 1π . This is a surprising result because finding the NE for multi-user networks is generally a challenging task even when the number of users is small [6, Chapter 2], and most of the previously reported NE-approaching algorithms only focus on the instantaneous performance and require more stringent conditions than ours. To further improve the convergence rate, a sub-optimal algorithm is also proposed to (approach a neighborhood of the ) NE at a rate of logπππβ² β² = π π1β² . We discuss some potential π extensions of our work and present numerical results to verify the performance of our algorithms for different applications. The rest of this paper is organized as follows. The network model and problem formation are discussed in section II. The proposed algorithm and its convergence result are presented in Sections III. Applications and numerical results are given in Section IV and the paper is concluded in Section V. II. N ETWORK M ODEL AND G AME F ORMULATION Consider a CR network in which πΎ secondary SD pairs, labeled as π1 to π·1 , π2 to π·2 , ..., ππΎ to π·πΎ , simultaneously transmit signals in the same spectrum as that for π PUs, labeled as π1 , π2 , ..., ππ , as shown in Figure
c 2011 IEEE 1536-1276/11$25.00 β
XIAO et al.: A SIMPLE DISTRIBUTED POWER CONTROL ALGORITHM FOR COGNITIVE RADIO NETWORKS
Fig. 1.
Network model for CR networks with multiple SD pairs.
1. Let the channel gain between ππ and π·π be βππ ,π·π for π, π β {1, 2, ..., πΎ}, and that between ππ and ππ be βππ ,ππ for π β {1, 2, ..., πΎ}, π β {1, 2, ..., π }. In a practical system, ππ for π β {1, 2, ..., π } can only maintain a certain level of QoS if its received interference power is lower than the interference temperature limit, denoted by ππ . Let the transmit power of ππ be π€π . We can define the power constraints for SUs as follows: π― π π πβ β€ πβ , where β denotes the transpose of [π€1 , π€2 , . . . , π€πΎ ], π = [π1 , π2 , . . . , ππ ], is given by β‘ βπ1 ,π1 βπ2 ,π1 β
β
β
β’ βπ1 ,π2 βπ2 ,π2 β
β
β
β’ π―π π = β’ .. .. .. β£ . . . βπ1 ,ππ
βπ2 ,ππ
(1) a matrix, π = and π― π π β βπΓπΎ βππΎ ,π1 βππΎ ,π2 .. .
β€ β₯ β₯ β₯. β¦
(2)
β
β
β
βππΎ ,ππ
Based on the above notations, the received signal to noise ratio (SNR) of π·π can be written as ππ π
π =
β
βππ ,π·π π€π . βππ ,π·π π€π + ππ
(3)
πβ{1,2,...,πΎ} πβ=π
where if PUs keep sending signals during the SUsβ communication, ππ should contain both the transmit signals of PUs and the additive noise received by ππ . However, if PUs are absent when SUs transmit, i.e., as in the time sharing spectrum (TSS) based CR networks [7], ππ should only contain the additive noise of ππ . In this paper, the game theoretic method is used to investigate the power control problem for CR networks. In our game, players are SUs who share the spectrum with PUs. We define the revenue of the πth SD pair, denoted by ππ , to be the benefit obtained by ππ and π·π from using the PUsβ spectrum. The price paid by ππ , denoted by ππ , is defined as the price charged by PUs for using the licensed spectrum. We also define the payoff of the πth SD pair, denoted by ππ , to be the difference between the revenue and price, i.e., ππ = ππ β ππ . In this paper, we assume the revenue and payoff of ππ to be the function of ππ π
π defined in (3). Let us consider a finitely repeated game model [8] in which each SU plays the game repeatedly. By using subscript [π‘] to denote the parameters and operations in the π‘th time slot, the main objective of the πth SD pair is to maximize its
3595
β time average payoff π1 ππ‘=1 ππ (π [π‘] ) over π time slots of transmission without causing the adverse effects on other SD pairs. This is different from most of the previously reported work in repeated game where each player only cares about the payoff of the last iteration, i.e., payoff in the π th time slot. Our setting has more practical meaning for CR networks because most mobile devices can tolerate a few periods of βbad" performance if the average payoff is good. It is assumed that each player cares the performance of the future as same as that of the present and hence the time discount factor [8, Definition 6.1.2] is 1. The result with other choice of the time discount factor can be similarly obtained. The main objective is to find a balance point of the entire network, called Nash Equilibrium (NE), in which each SD pair cannot further improve its payoff by choosing a different transmit power, given the transmit powers of other SUs. The formal definition of the NE is given as follows. Definition 1. [8, Definition 3.3.4] A strategy profile π€πβ is at a Nash equilibrium (NE) if, for every player π and every strategy β ) π€π , π€πβ is at least as good as the strategy profile (π€π , π€βπ in which the player π chooses π€π while other players choose β π€βπ , i.e., for every player π, β β ππ (π€πβ , π€βπ ) β₯ ππ (π€π , π€βπ ),
(4)
where subscript βπ means all the players except player π. We also define a strategy profile to be π-Nash equilibrium (π-NE), if this strategy profile is within the distance of π to the payoff achieved by a NE, i.e., the following condition needs to be satisfied, ( ) ( ) β β β ππ π€π , π€βπ β€ π for π > 0. (5) ππ π€πβ , π€βπ Finding the NE of the multi-user network is difficult and intractable [6, Chapter 2]. In this paper, we try to develop an algorithm that, during π time slots of repeated power control games, can achieve the average performance nearly as good as the system with the NE decision being chosen in hindsight, i.e., we have π 2 ( ) 1 β min π
π[π] β π
(πβ ) 2 = 0 π ββ π πβ{1,...,π‘} π‘=1 [ ] where π
(π [π] ) = π1 (π[π] ), π2 (π [π] ), ..., ππΎ (π[π] ) .
lim
(6)
III. M AIN R ESULTS Algorithm 1 is illustrated below. 1) Initialization: Let the transmit signal of ππ be π₯π . Consider the transmission of the πth SD pair. During the first two time slots, i.e., π‘ β {1, 2}, a) ππ first sends signals π₯π[1] and π₯π[2] with powers π€π[1] and π€π[2] , respectively, b) After receiving the signal sent by ππ , π·π , observed its received SNR, feedbacks the revenue functions ππ[1] and ππ[2] in the first and second time slots, respectively. Each PU ππ also sends the prices ππ[1] and ππ[2] to ππ during the first two time slots. 2) Iteration: For π‘ = 3 : π , a) In βπΎtime slot π‘, ππ monitors its interference power π=1 βππ ,ππ π€π[π‘] . If the interference powers of all
3596
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 11, NOVEMBER 2011
PUs satisfy (1), ππ only sends price ππ[π‘] to ππ , and then ππ updates its transmit power as follows, )+ ( (7) π€π[π‘] = π€π[π‘β1] β πΏπ[π‘β1] ππ[π‘] , where (β
)+ = max{0, β
}, ππ[π‘] is a subgradient [9] of ππ[π‘] which is a function that satisfies the relation ( ) ( ) β₯ ππ[π‘β1] π[π‘β1] ππ[π‘] π[π‘] ( ) +ππ[π‘] π[π‘] β π π[π‘β1] , (8) and πΏπ[π‘β1] is the step size of the π‘th iteration, defined π’π as πΏπ[π‘β1] = π‘β1 where π’π is the step size coefficient which is a constant controlling the iteration speed, b) However, if the interference powers of some PUs exceed the interference temperature limit, each of these PUs should transmit more information to help SUs to adjust the transmit powers during the subsequent iterations. More specifically, β assuming ππ observes πΎ high interference power, i.e., π=1 βππ ,ππ π€π[π‘] > ππ , the operations of ππ , except for sending the price to SUs, are described as follows, i) ππ broadcasts a high-interference-message to inform all SUs that its interference power exceeds the limit. ππ can decode this message and exploit it to obtain βππ ,ππ , ii) Because we assume each PU monitors its interference power in each iteration, ππ can calculate the interference power exceeding value πΌπ[π‘] =
πΎ β
βππ ,ππ π€π[π‘] β ππ ,
(9)
π=1
and the change of interference power in time slot π‘ π½π[π‘] =
πΎ β
βππ ,ππ π€π[π‘] β
π=1
πΎ β
βππ ,ππ π€π[π‘β1] , (10)
π=1
iii) By assuming the channel gains to be known by the receivers (i.e., assume a training code is involved in the transmit signals of SUs), ππ can exploit its received signal to calculate a constant βπΎ Ξ¦π[π‘] = π=1 β2ππ ,ππ . Each PU ππ broadcasts the constants πΌπ[π‘] , π½π[π‘] and Ξ¦π[π‘] to all SUs, iv) ππ , knowing βππ ,ππ (obtained in Step b-i)), updates its transmit power by π€π[π‘]
=
[
π€π[π‘β1] β πΏπ[π‘β1] ππ[π‘] (11) ( )]+ β1 ββππ ,ππ Ξ¦π[π‘β1] πΌπ[π‘β1] β π½π[π‘β1] .
If more than one PU detects higher-than-tolerate interference power, they should sequentially repeat the operations from Steps b-i) to b-iv) until the transmit powers of all SUs satisfy (1). 3) Termination: The above process continues until the obtained payoff is close to the optimal value within an acceptable range, i.e., β₯ππ[π‘] (π [π‘] ) β ππ[π‘] (πβ )β₯22 β€ π, or the number of time slots reaches π , i.e., π‘ = π . The main idea of Algorithm 1 is that if the transmit powers of SUs satisfy (1), each SU maximizes its payoff by using
the subgradient method, as shown in Step 2-a). If any PUs detect the higher-than-tolerable interference levels, they will broadcast several constants, i.e., ππ broadcasts Ξ¦π , πΌπ and π½π defined in Step 2) to all SUs, and SUs use these constants to project their transmit powers into the convex hull defined in (1). Note that simply projecting the transmit powers of all SUs to (1) requires each PU to know the information of all other PUs, which does not meet the requirements that PUs cannot communicate with each other. Hence, in Algorithm 1, if more than one PU detect higher-than-tolerable interference power, each of them informs SUs to project their powers to the corresponding linear equation in the convex hull defined in (1) one by one. This setting will not result in much adverse effects on the convergence performance of Algorithm 1 because we always assume π β« πΎ. Also if the transmit powers of SUs increase gradually, only a few PUs will first detect high-thantolerable interference powers in each time slot. In other words, the number of iterations used on the projection operation is negligible as the total number of iterations becomes large. Note that our algorithm is different from that proposed in [10] in which each SU needs to communicate with its nearby SUs to make decisions based on the βconsensus" among them. In our algorithm, each PU (or SU) cannot communicate with its nearby PUs (or SUs), and hence can be directly applied into many practical systems, i.e., the cellular based mobile network. Theorem 1. If the following assumptions A1) The iteration step sizes are bounded, i.e., β₯ππ[π‘] β₯22 β€ ππ+ , where ππ+ is a constant, A2) The difference of power values between two iterations is bounded, i.e., β₯π€π[π‘] β π€π[π ] β₯22 β€ π€π+ for π‘ β= π and π‘, π β {1, ..., π }, where π€π+ is a constant, A3) The payoff for the πth SD pair is a continuous function of π€π , are satisfied, then Algorithm 1 has the following properties P1) It converges to a stationary point if π is large enough, P2) If all SD pairs use Algorithm 1, the time average payoff of ππ converges to an π-NE at a rate of ( ( )) π exp 1π 1 , i.e., 2 ππ 1 β ) ( π
π[π‘] β π
(πβ ) β€ π. (12) ππ π‘=1
2
Proof: See Appendix A. Note that the convergence performance of Algorithm 1 is closely related to the combination of two important parameters: the initial value π€π[1] of transmit powers and the iteration step size π’π . If π€π[1] and π’π satisfies β£π€π[1] β π€πβ β£ β π’π , Algorithm 1 could approach to the neighborhood of a NE within a few iterations. However, if π€π[1] and π’π are improperly chosen, the convergence rate of Algorithm 1 will be very slow. This slow convergence rate is due to the fact that we require π€π to reach π€πβ exactly without assuming any information about the systems or channel gains to be known by SUs. The slow 1 In this paper, we follows Bachmann-Landau notations: π = π(π) if (π) < +β. lim ππ(π)
πββ
XIAO et al.: A SIMPLE DISTRIBUTED POWER CONTROL ALGORITHM FOR COGNITIVE RADIO NETWORKS
π€π+ πΏ
and π€π[π‘] is calculated by using either (7) (if the power constraint in (1) is satisfied) or (11) (if (1) is not satisfied) in Algorithm 1. The NE is a balance point where each player achieves locally optimal solution and hence has no incentive to deviate from this point. In other words, we can always find a neighborhood, denoted as π = [π1 , π2 , ..., ππΎ ], for the NE that all the elements in this neighborhood cannot achieve β )β₯ higher performance than that of the NE, i.e., ππ (π€πβ , π€βπ β ππ (π Β± Ξπ) where Ξπ = [Ξπ€1 , Ξπ€2 , ..., Ξπ€πΎ , ] and 0 < Ξπ€π < ππ for π β {1, 2, ..., πΎ}. It is easy to observe that if π [π‘] β π[π‘+1] > π, Algorithm 2 may not converge to a NE. Let us summarize our observations and present the main result for Algorithm 2 as follows. πΏΛ π+
Theorem 2. If π€π β π²π , ππ β« π2 π β π β {1, 2, ..., πΎ} and Assumptions A1)-A3) 1 ]are satisfied, Algorithm 2 [ Λ + Λin +Theorem + πΏΛπΎ ππΎ πΏ1 π1 πΏ2 π2 neighborhood of a NE, approaches a 2 , 2 , ..., 2 ) ( ) 2 ( πβ² 1 πβ πΏΛπ ππ+ πΏΛπ ππ+ β β βπ
π Β± 2 β€ i.e., π β² π
π€π[π‘] , π€βπ[π‘] Β± 2 π π‘=1 2 (1) πΏΛπ ππ+ ππβ² β² β² π , at a rate of log π β² = π πβ² , where π = β£π β 2 β£ and π π€+ πΏΛπ = π . πΏ
Proof: See Appendix B. IV. A PPLICATIONS AND N UMERICAL R ESULTS In this section, the proposed algorithm is applied to the network utility maximization problem to evaluate its performance. We assume that each SU has an objective to maximize its achievable rate and define the utility function of ππ as ππ (π€π ) = πΎπ log (1 + ππ π
π ) where ππ π
π is defined in (3) and πΎπ is a constant. Since Algorithm 2 can be regarded as a special case of Algorithm 1, we only present the numerical results for Algorithm 1. Consider a special case where SUs are far from each other β π€ π π . Assume PUs do not charge any and ππ π
π β ππ ,π· ππ prices for SUs if the transmit powers of SUs satisfy (1). It
0.6
t=1
Average Revenue: (1/T) Ξ£ T r[t]
convergence performance of Algorithm 1 can be improved by assuming more information to be observed by SUs. For example, if we assume that ππ knows the payoff functions, or both the revenue and pricing functions, the optimization problem can be converted into the Lagrangian dual problem [11] which will be discussed in the next section. Another way to improve the convergence rate of Algorithm 1 is to decrease the required accuracy of the results. In practical digital systems, mobile devices cannot adjust their parameters with an infinite accuracy, i.e., the value of π€π can only be chosen from a finite discrete set. In the rest of this section, we consider a simple case in which the transmit power of each SU is linearly quantized into πΏ levels and π€π can only be chosen from this finite set, i.e., π€π β π²π where π€ + 2π€ + π²π = {0, πΏπ , πΏπ , ..., π€π+ }. Let us describe Algorithm 2 as follows. Algorithm 2: Each SU adjusts its transmit power by using the exact same procedures as Algorithm 1. The only difference is that, in the π‘th iteration, the transmit power of ππ is π€π[π‘] = Λ Λ ππΏΛπ if ππΏΛπ β πΏ2π < π€π[π‘] < ππΏΛπ + πΏ2π where π β {1, 2, ..., πΏ}, πΏΛπ =
3597
0.5
0.4 u=2 u=4 u=6 Dual Method Optimal Value
0.3
0.2
0.1
0
0
20
40
60
80 Iteration
100
120
140
Fig. 2. Comparison of different algorithms, where π = 1, πΎ = 10, ππ = 2, for π β {1, 2, ..., πΎ}, π β {1, 2, ..., π }.
is observed that in this case the payoff function ππ (π) of ππ has the following features: 1) it is continuous in π, 2) it is concave in π€π , 3) it has a continuous first derivative with respect to π€π , and 4) there exists π = {π1 , π2 , ..., ππΎ } > 0 such βπΎ that π=1 ππ ππ (π) is diagonally strictly concave. Following the same methods as in [12], we can claim that for this CR network the NE is unique and corresponds to the global utility maximization point [11]. In other words, if all SUs use Algorithm 1 to adjust their transmit powers, the average payoff of the network will converge to the global utility maximization π ( ) β βπΎ 1 π[π‘] π [π‘] for π[π‘] = πΎ point, defined as max π1 π=1 ππ[π‘] π
π‘=1
under the power constraint defined in (1). To demonstrate the convergence performance of Algorithm 1, the results of the Lagrangian dual methods discussed in [11] are also provided. In this method, ππ iteratively chooses the optimal Lagrangian coefficients to maximize its payoff (see [11] for the detailed description of the dual method). In Figure 2, we compare the convergence rates of Algorithm 1 and the dual method. It is observed that the dual method generally converges to the optimal value at a faster rate than Algorithm 1. However, each SU in the dual method needs to know the payoff functions of all SUs and has to search for the optimal power to minimize the objective function in each iteration, which incurs much more computational complexity and communication overhead than those of our proposed algorithm. In addition, Figure 2 shows that if the step size is chosen optimally, the convergence rate of Algorithm 1 could outperform that of the dual methods. Let us consider a more general case as follows. Define the revenue of ππ as ππ = πΌπ log (1 + ππ π
π ) where ππ π
π is given in (3) and πΌπ is a constant. The pricing function charged π β by all PUs from ππ is given by ππ = π½π βππ ,ππ π€π where π=1
π½π is a constant. Note that this problem is not convex and is generally impossible to efficiently find the global optimal solution [13]. In this case, Algorithm 1 may not converge to the global utility maximization point, but still can converge to a NE. The convergence results of Algorithm 1 with different parameters in this case are presented in Figures 3 and 4. As
3598
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 11, NOVEMBER 2011
0.14
1
0.12
0.9
w decided by the PUs (w = h
0.1
Average Payoff: (1/T) Ξ£T Ο[t]
u=1.5 u=2 u=2.5 u=3
0.08 0.06
i[1] S ,P i
\q )
k
k
0.8
t=1
Average Payoff: (1/T) Ξ£T Ο
t=1 [t]
i[1]
Power Control with Algorithm 1 (400 iterations) Power Control with Algorithm 1 (300 iterations) Power Control with Algorithm 1 (200 iterations) Power Control with Algorithm 1 (100 iterations)
0.04 0.02 0
0.7 0.6 0.5 0.4 0.3
β0.02 β0.04
50
100 150 Number of Iterations
200
250
0.2
1
1.5
2
2.5
3 3.5 Distance dS ,D i
Fig. 3. Convergence rate of Algorithm 1 with different step sizes, where π€π[1] is the least squares solution of π― π π π β = π β , πΌπ = 10, π½π = 0.1, π = 5, πΎ = 10, ππ = 5, for π β {1, 2, ..., πΎ}, π β {1, 2, ..., π }.
4
4.5
5
i
Fig. 5. Payoffs of power control achieved by Algorithm 1 and fixed transmit power (π€π[1] = βππ ,ππ βππ ) under different πππ ,π·π , where πΌπ = 10, π½π = 0.1, π = 5, πΎ = 10, ππ = 2, π’π = β₯π 1 β₯2 , for π β {1, 2, ..., πΎ}, π β {1, 2, ..., π }.
π[π‘] 2
β0.14 β0.16
Average Payoff: (1/T) Ξ£T Ο
t=1 [t]
β0.18
wi[1] = 0.75*hik\q
β0.2
wi[1] = hik\q wi[1] = 1.5* hik\q
β0.22
wi[1] = 2* hik\q
β0.24 β0.26 β0.28 β0.3 β0.32 β0.34
0
100
200 300 Number of Iterations
400
500
Fig. 4. Convergence rate of Algorithm 1 with different initial values, where π€π[1] = βππ ,ππ βππ means π€π[1] is the least squares solution of π― π π πβ = π, πΌπ = 10, π½π = 0.1, π = 5, πΎ = 10, ππ = 5, π’π = β₯π 1 β₯2 , for π β {1, 2, ..., πΎ}, π β {1, 2, ..., π }.
π[π‘] 2
observed in Section III, the convergence rate of Algorithm 1 depends on the initial transmit powers of SUs and the iteration step size. It is observed that if π€π‘[1] has a distance of π’π from a NE, only a few number of iterations are required to approach the performance of this NE. However, if the initial transmit powers are not properly chosen, the adjustment of Algorithm 1 on the transmit powers of SUs may require a large number of iterations. Similarly, an optimal value of π’π should be close to the distance between π€π[1] and π€πβ . However, if no information about initial transmit power and the optimal power are known, on one hand, a large step size may lead to high fluctuations during the first few iterations and, on the other hand, a small step size will cause slow convergence rate. The above observation shows the importance of choosing proper combinations of initial powers and step sizes. In a practical system, SUs can first send the training code to let PUs to decide proper initial values of SUsβ transmit powers. Also, the step size of each iteration, chosen by each SD pair, can
be adjusted if more information can be observed or more feedback messages can be received during the communication, i.e., the dual methods. Finding the distributed algorithms to allow iterative improvement of the step size in Algorithm 1 will be our future work. Let us consider the numerical results of CR networks with Algorithm 1. Denote the distance between two users π and π as ππ,π for π, π β [π1 , π2 , ..., ππΎ ] βͺ [π·1 , π·2 , ..., π·πΎ ] βͺ [π1 , π2 , ..., ππ ] and consider the following channel models, π Λ ππ /ππ Λ Λ βππ = β ππ ,π·π and βππ = βππ /πππ ,ππ , where βππ and Λ βππ are average channel fading coefficients unrelated to the distance of the transmission and π is the channel attenuation exponent. Assume that SUs are located on a regular planar network, i.e., πππ ,ππ = πππ ,ππ , πππ ,π·π = πππ ,π·π , β 2 πππ ,ππ + π2ππ ,π·π and πππ ,ππ+1 = ππ·π ,π·π+1 , for πππ ,π·π = π, π β {1, 2, ..., πΎ} and π, π β {1, 2, ..., π }. In Figure 5, the average payoff of the network obtained by using Algorithm 1 with different πππ ,π·π is shown and compared with that of the fixed transmit power method. It is observed that the proposed algorithm greatly improves the performance of CR networks. More importantly, even with a small number of iterations, Algorithm 1 can achieve a significant payoff improvement.
V. C ONCLUSION In this paper, the power control problem for SSS based CR networks with multiple secondary SD pairs and PUs has been investigated. A simple distributed algorithm has been proposed to iteratively improve the performance of CR networks. It has been proved that the proposed converges to an π)) ( ( algorithm NE at a speed of ππ = π exp 1π . A sub-optimal algorithm has been also proposed to converge ( ) to a neighborhood of a NE at a rate of logπππβ² β² = π π1β² . Numerical results have π been presented to show the convergence rates under different settings.
XIAO et al.: A SIMPLE DISTRIBUTED POWER CONTROL ALGORITHM FOR COGNITIVE RADIO NETWORKS
{ π[π‘+1] =
π[π‘] β πΏ[π‘] π [π‘] ,
( )β1 ( ) π― π π π[π‘] β π― π π πΉ [π‘] π [π‘] β π , π[π‘] β πΏ[π‘] π [π‘] β π― β π π π― π π π― β π π
A PPENDIX A P ROOF OF T HEOREM 1
1 π
π‘=1
=
(π)
β€
(π)
β₯π[π‘+1] β πβ β₯22 =
1 π
π β
β₯π[π‘] β π β β πΉ [π‘] π [π‘] β₯22
π‘=1
π ( ) 1 β[ β₯π[π‘] β πβ β₯22 β 2πΉ [π‘] π [π‘] π[π‘] β π β π π‘=1 ] +πΉ 2[π‘] β₯π [π‘] β₯22 π ) ( ( ) 1 β[ β₯π[π‘] β π β β₯22 β 2πΉ [π‘] π
π [π‘] β π
(πβ ) π π‘=1 ] +πΉ 2[π‘] β₯π [π‘] β₯22 , (14)
where (π) and (π) are obtained by using (7) and (8) in Algorithm 1, respectively. Equation (14) can be rewritten recursively as follows: π 1 β π [π‘+1] β πβ 2 β€ π[1] β π β 2 2 2 π π‘=1
β
π π‘ ( ( ) ) 2 ββ πΉ [π] π
π[π] β π
(π β ) π π‘=1 π=1
π π‘ 1 ββ 2 2 + πΉ [π] π [π] . π π‘=1 2
if π― π π π[π‘] β€ π π ,
(13)
otherwise,
as follows:
From Definition 1, it is observed that for a NE, a neighborhood can always be found in which all the elements in this neighborhood have lower payoff than the NE. By assuming all payoff functions of SUs to be continuous, the neighborhood of the NE can be regarded as a quasiconcave hull, i.e., for π = [π1 , π2 , ..., ππΎ ] neighborhood of a NE, we have ππ (πβ Β± Ξπ) < ππ (π) where π β [πβ β Ξπ, πβ + Ξπ] and Ξπ€π β€ ππ . Since Algorithm 1 uses a decreasing step size for iteration, it will finally fall into a small neighborhood of the NE. Therefore, if Algorithm 1 is proved to converge to a stationary point in which all elements in a neighborhood of that point has lower payoffs, we can claim that this point is the NE and Algorithm 1 converges. Defining π [π‘] = [π1[π‘] , π2[π‘] , ..., ππΎ[π‘] ]β , we can re-write (11) in Algorithm 1 to the vector form in (13) at the top of next page. In this figure, π― π π is a sub-matrix of π― π π which only contains the channel gains connected with one PU observing high interference noises in its rows, and πΉ [π‘] = diag[πΏ1[π‘] , πΏ2[π‘] , ..., πΏπΎ[π‘] ]. Let us first consider the case that the transmit powers of the SUs always satisfy (1). Denoting a NE achieving power β β ] , control schemes for all the SUs as π β = [π€1β , π€2β , ..., π€πΎ we have π β
3599
(15)
π=1
βπ Note that we have π1 π‘=1 β₯π[π‘+1] β πβ β₯22 β₯ 0, and the second term in the right-hand-side of (15) can be expressed
π π‘ ( ( ) ) 2 ββ πΉ [π] π
π[π] β π
(πβ ) (16) π π‘=1 π=1 ( π‘ ) π ] [ ( ) 2 β β β₯ πΉ [π] min π
π[π] β π
(πβ ) π π‘=1 πβ[1,π‘] π=1 ( ) π π π‘ (π) [ ( ) ] 2 ββ 1β β₯ πΉ [π] min π
π[π] β π
(π β ) π π‘=1 π π‘=1 πβ[1,π‘] π=1
where (π) comes from that fact that βπ βπ‘ 1 1 π‘=1 π=1 π . π Substituting (16) into (15), we have π [ ( ) ] 1 β min π
π [π‘] β π
(πβ ) π π‘=1 πβ[1,π‘]
β€
π[1] β π β 2 + 2 2 π
1 π
π β
π β π‘ β
π‘=1 π=1 π‘ β
π‘=1 π=1
βπ‘
1 π=1 π
2 πΉ 2[π] π [π] 2
.
β₯
(17)
πΉ [π]
Substituting the step size function πΉ [π] = ππ for π = [π’ equation and using the result β] βinto1 the above β1π‘, π’2 ,1..., π’πΎ π < = , and the properties of Harmonic π=1 π2 β π=1 π2 6 π‘ 1 β1 1 β2 1 number: β 12 π‘ + π(π‘β4 ) β π=1 π = log π‘ + 2 π‘ 1 β1 1 β2 Λ the following results can be obtained: log π‘+ 2 π‘ β 12 π‘ + π
, π ] [ ( ) 1 β min π
π[π] β π
(πβ ) π π‘=1 πβ[1,π‘] + βπ π (π) π + + ππ π‘=1 6 β€ 2 βπ β1 + π
) Λ π‘=1 (log π‘ + 0.5π‘ π π
Β¨ β 2 βπ log π +0.5π β1 +π
Λ log π‘ + + 2π
Λ π‘=1 π π
(18)
where (π) is obtained from Assumptions A1) and A2) in Theorem 1, and β π
Β¨ is a constant defined as π
Β¨ = π+ + π +π 2 and hence π 6 . Because π‘=1 log π‘ = Ξ(π log π ) βπ log π +0.5π β1 1 β limπ ββ π π‘=1 log π‘ β β and limπ ββ π 0, it can be claimed that Algorithm 1 converges to a stationary point. Assume that for π β₯ ππ and ππ is a very large number, we try to achieve the accuracy of π [ ( ) ] β 1 min π
π[π] β π
(πβ ) = π. From (18) and conπ π‘=1 πβ[1,π‘]
cavity of logarithm function, the following results can be 2 Again, we follow Bachmann-Landau notations: π = Ξ(π) if π = π(π) and π = π(π)
3600
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 11, NOVEMBER 2011
obtained, π
Β¨ π
ππ log ππ + 0.5ππβ1 1 β log π‘ + + 2π
Λ β₯ lim ππ ββ ππ ππ π‘=1 ( ) ( ) ππ ππ + 1 1 β = lim log π‘ = log π ββ ππ π‘=1 2 ( ( )) 1 β ππ = π exp . π
Therefore, we can claim that by using Algorithm 1, the average payoff )) networks converges to π-NE at a speed ( of ( SU of ππ = π exp 1π . Consider the case that at least one PU observes the higherthan-tolerable interference power. By multiplying (13) with π― π π , it can be easily shown that π― π π πβ [π‘+1] always converges to π β . Substituting π― π π πβ [π‘+1] = π β into (13), (13) Λ [π‘+1] π [π‘+1] , can be rewritten as π[π‘+2] = π [π‘+1] β πΉ [π‘+1] π ( )β1 Λ [π‘+1] = π° + π― β π π π― π π π― β π π π― π π and π° is where π the identity matrix. If we denote π [π‘] = πΛ [π‘] π [π‘] and assume β₯Λ π [π‘] π [π‘] β₯22 β€ π + , the same method as the previous case can be used to obtain the similar convergence rate.
A PPENDIX B P ROOF OF T HEOREM 2 Following the same steps as in Appendix A, let us consider the case in which each SU can only choose from πΏ quantized power levels, i.e., π€π β π²π . In this case, if the initial point is not appropriately chosen, Algorithm 2 will be unstable for the first few iterations, i.e., fluctuating between different neighborhood of NEs. However, when the number of iterations is large, the transmit powers of SUs will approach to the neighborhood of a NE. Therefore, for a large number of iterations, Algorithm 2 can be regarded as Algorithm 1 with + a constant step size πΉ [π] = πΉΛ where πΉΛ = ππΏ is a constant. Let us substitute this step size in (17), the following results can be obtained,
π [ ( ) ] 1 β min π π[π] β π (πβ ) π π‘=1 πβ[1,π‘]
β€
βπ‘ 2 π 1 β π+ + (πΉΛ π + /π‘) π=1 π Λ βπ‘ π π π‘=1 (2πΉ/π‘) π=1
2 π 1 β π+ + πΉΛ π + (π‘ + 1) = . π π‘=1 2πΉΛ (π‘ + 1)
(19)
By using the same method as in Appendix A, we can claim that if the step size is fixed, the number of iterations for ( ) Λ + πβ² Algorithm 2 is given by logππ β² = π π1β² where πβ² = π β πΉπ2 π + and πΉΛ = π . πΏ
R EFERENCES [1] S. Haykin, βCognitive radio: brain-empowered wireless communications,β IEEE J. Sel. Areas Commun., vol. 23, pp. 201β220, 2005. [2] G. Amir and S. S. Elvino, βFundamental limits of spectrum-sharing in fading environments,β IEEE Trans. Wireless Commun., vol. 6, no. 2, pp. 649β658, 2007. [3] M. Gastpar, βOn capacity under receive and spatial spectrum-sharing constraints,β IEEE Trans. Inf. Theory, vol. 53, no. 2, pp. 471β487, 2007. [4] Z. Lan, Y.-C. Liang, and X. Yan, βJoint beamforming and power allocation for multiple access channels in cognitive radio networks,β IEEE J. Sel. Areas Commun., vol. 26, pp. 38β51, 2008. [5] Y. Xiao, G. Bi, and D. Niyato, βGame theoretic analysis for spectrum sharing with multi-hop relaying,β IEEE Trans. Wireless Commun., vol. 10, no. 5, pp. 1527β1537, May 2011. [6] N. Nisan, T. Roughgarden, E. Tardos, and V. V. Vazirani, Algorithmic Game Theory. Cambridge University Press, 2007. [7] Q. Zhao and B. Sadler, βA survey of dynamic spectrum access,β IEEE Signal Process. Mag., vol. 24, no. 3, pp. 79β89, 2007. [8] Y. Shoham and K. Leyton-Brown, Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations. Cambridge University Press, 2008. [9] S. P. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. [10] I. Lobel and A. Ozdaglar, βDistributed subgradient methods for convex optimization over random networks,β IEEE Trans. Automatic Control, vol. 56, no. 6, pp. 1291β1306, June 2011. [11] Y. Wei and R. Lui, βDual methods for nonconvex spectrum optimization of multicarrier systems,β IEEE Trans. Commun., vol. 54, pp. 1310β1322, 2006. [12] D. Mosk-Aoyama, βConvergence to and quality of equilibria in distributed systems,β Ph.D. dissertation, 2008. [13] G. Scutari and D. P. Palomar, βMIMO cognitive radio: a game theoretical approach," IEEE Trans. Signal Process., vol. 58, no. 2, pp. 761β780, 2010.