Optimal power allocation for multi-user OFDM and distributed antenna cognitive radio with RoF

The Journal of China Universities of Posts and Telecommunications December 2010, 17(6): 41–46 www.sciencedirect.com/science/journal/10058885

http://www.jcupt.com

Optimal power allocation for multi-user OFDM and distributed antenna cognitive radio with RoF GE Wen-dong1 ( ), JI Hong1, SI Peng-bo2, LI Yi1 1. School of Information and Communication Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China 2. School of Electronics Information and Control Engineering, Beijing University of Technology, Beijing 100022, China

Abstract

In cognitive radio (CR), power allocation plays an important role in protecting primary user from disturbance of secondary user. Some existing studies about power allocation in CR utilize ‘interference temperature’ to achieve this protection, which might not be suitable for the OFDM-based CR. Thus in this paper, power allocation problem in multi-user orthogonal frequency division multiplexing (OFDM) and distributed antenna cognitive radio with radio over fiber (RoF) is firstly modeled as an optimization problem, where the limitation on secondary user is not ‘interference temperature’, but that total throughput of primary user in all the resource units (RUs) must be beyond the given threshold. Moreover, based on the theorem about maximizing the total throughput of secondary user, equal power allocation algorithm is introduced. Furthermore, as the optimization problem for power allocation is not convex, it is transformed to be a convex one with geometric programming, where the solution can be obtained using duality and Karush-Kuhn-Tucker (KKT) conditions to form the optimal power allocation algorithm. Finally, extensive simulation results illustrate the significant performance improvement of the optimal algorithm compared to the existing algorithm and equal power allocation algorithm. Keywords cognitive radio, power allocation, OFDM, distributed antenna, convex optimization, radio over fiber (RoF)

1

Introduction

CR [1], as an effective approach to solve the problem about scarcity of spectrum resource, allows secondary user to opportunistically utilize the spectrum band that is authorized to primary user under the condition that secondary user can not disturb the normal transmission of primary user. Thus power allocation takes a considerable part in protecting quality of primary link when different users share spectrum bands simultaneously. Additionally, OFDM is one of the promising solutions to provide a high performance in physical layer and has been widely adopted in standards by wireless industry [2]. Accordingly, Study in power allocation in OFDM-based CR has a significant meaning. There are many previous works about power allocation in OFDM-based CR. [3] formulates the resource allocation as a

Received date: 29-01-2010 Corresponding author: GE Wen-dong, E-mail: [email protected] DOI: 10.1016/S1005-8885(09)60523-9

multi-dimensional knapsack problem and proposes a low-complexity, greedy max-min algorithm to solve it. In Ref. [4], the authors present a wireless unlicensed system that successfully coexists with the licensed systems in the same spectrum range. And then a distributed optimization problem is formulated and solved as a dynamic selection of spectrum patterns and power allocations that are better suited to the available spectrum range without degrading the licensed system performance. In Ref. [5], Hasan et al. proposes a solution to an energy-efficient resource allocation problem which maximizes the cognitive radio link capacity taking into account the availability of the subcarriers and the limits on total interference generated to the primary users. And it consideres an energy-aware capacity expression by taking into account another factor called subcarrier availability. In Ref. [6], the authors investigate an optimal power loading algorithm for an OFDM-based cognitive radio system. And then the authors propose two suboptimal loading algorithms that are less complex.

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In this paper, the power allocation problem in multi-user OFDM and distributed antenna cognitive radio is discussed. The constraint that primary user cannot be disturbed by secondary user does not express as ‘interference temperature’ just as [5–6], but shows as the throughput of primary user should be beyond the given threshold. Meanwhile, RoF is adopted because the connection between base station and distributed antenna is fiber. The main contributions of this work are as follows: 1) The power allocation problem in multi-user OFDM and distributed antenna cognitive radio with RoF is modeled as an optimization problem in mathematics. 2) Optimization problem about power allocation is transformed into the convex problem and the optimal power allocation algorithm is deduced with theory of the convex optimization. 3) Simulation results illustrate that the optimal power allocation algorithm is superior to existing power allocation algorithm and equal power allocation algorithm. The rest of this paper is organized as follows. In Sect. 2, the system model of multi-user OFDM and distributed antenna cognitive radio is established and power allocation problem is introduced. In Sect. 3, based on the theorem about maximizing the total throughput of secondary user, equal power allocation algorithm is introduced. And then optimal power allocation algorithm is deduced. Extensive simulation results are provided in Sect. 4 to illustrate the performance comparison, and Sect. 5 concludes this study.

2 System model and problem formulation 2.1

System model

In the CR network that was considered in this paper, there exists one primary link (primary user) that consists of a transmitter and receiver. The primary link is assigned with the radio resource in OFDMA. Thus the channel that is authorized to the primary link includes N resource units (RU). Each RU consists of one or several subcarriers, and the number of subcarriers is same in every RU. Additionally, there are also N secondary mobile stations (secondary user) in this CR network. Each secondary user can utilize one RU for transmission under the constraint that it must maintain the normal quality of primary link. The secondary base station has M distributed antennas in this area in order to receive the date from the secondary user. The connection between antennas and secondary base station is fiber. Thus the RoF

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technology is brought in this system. Accordingly each secondary user and M antennas establish a macro-diversity SIMO system which can achieve more throughput than SISO system. That is why one RU might satisfy the quality of secondary link.

Fig. 1

System model (upper: scenario; lower: allocation of RUs)

In this paper, power allocation in uplink of the CR networks is considered. As RUs are orthogonal with each ( ps ) ( ss ) ( sp ) , hnm and hnn is other, we assumed that hn( pp ) , hnm respectively the channel gain from primary transmitter to primary receiver in the nth RU, from primary transmitter to the mth secondary receiving antenna in the nth RU, from the nth secondary user to the mth secondary receiving antenna and from the nth secondary to primary receiver, where 1≤m≤M , 1≤n≤N . Thus, the throughput of primary link in the nth RU can be expressed by ⎛ ⎞ ⎟ B ⎜ hn( pp ) 2 Pn( p ) (p) ⎟ ; 1≤n≤N Rn = lb ⎜1 + B ⎞⎟ N ⎜ ⎛ ( sp ) 2 ( s ) ⎜ Γ ⎜ hnn Pn + N 0 N ⎟ ⎟ ⎝ ⎠⎠ ⎝

(1)

where B is the total bandwidth of the channel occupied by primary user, Pn( p ) is the transmit power of primary user in the nth RU, Pn( s ) is the transmit power of the nth secondary user, N 0 is power spectrum density of AWGN and Γ is the SNR gap between practical system and theoretical limit [7], which is the function of bit error rate (BER) σ BER as follow. ln ( 5σ BER ) Γ =− 1.5

(2)

It is denoted that P = ⎡⎣ P1( s ) , P2( s ) ,..., PN(s ) ⎤⎦ . If C ( th ) is defined as the lower bound of primary user’s throughput, which is decided by QoS demand, the determination of P need to satisfy the following inequation, as secondary user cannot disturb the normal transmission of primary link.

Issue 6 N

GE Wen-dong, et al. / Optimal power allocation for multi-user OFDM and…

∑ R( )≥C ( p

n =1

th )

(3)

n

There is the fiber connecting antennas and secondary base station. As the signal is not digital processed at antenna, we must consider the properties of RoF. According to Ref. [8], an automatic gain control (AGC) unit is placed before the laser to avoid over modulation. Stronger receiving signal from the secondary user near the antenna might trigger the AGC, and weaker receiving signal from the secondary user far away from the antenna would suffer SNR loss, which lead to ‘near-far effects’. Therefore, in order to avoid this situation, the following inequations must be guaranteed.

{

}

( ss ) 2 ( s ) max h1(mss ) 2 P1( s ) , h2( ssm) 2 P2( s ) ,..., hNm PN ≤φm ; 1≤m≤M

(4)

where φm is upper threshold to trigger the AGC in the mth secondary receiving antenna. 2.2

Problem formulation

When all the antennas receive the signal from each secondary user, maximal ratio combining (MRC) is adopted to combine these signals. So the throughput of the nth secondary user can be shown as ⎛ ⎞ ( ss ) 2 ( s ) ⎜ ⎟ M B h P nm n ⎟ ; 1≤n≤N Rn(s ) = lb ⎜1 + ∑ (5) B ⎞⎟ N ⎜ m =1 ⎛ ( ps ) 2 ( p ) h P N + Γ 0 ⎜ nm n ⎟ ⎜ N ⎠ ⎟⎠ ⎝ ⎝

Thus, the power allocation problem in this paper is to find out a group of secondary users’ transmit powers in order to maximize the total throughput of secondary users under the constraint mentioned above, which could be stated as N ⎫ max G = ∑ Rn( s ) ⎪ P n =1 ⎪ ⎪ ⎧ N ( p) ( th ) ⎪⎪ ⎪∑ Rn ≥C (6) ⎪ n =1 ⎬ ( ss ) 2 ( s ) ( ss ) 2 ( s ) ⎪ s.t. ⎨ max h1m P1 ,..., hNm PN ≤φm ; 1≤m≤M ⎪ ⎪ ⎪ ⎪ ⎪0 ≺ P ≺ Pmax ; 1≤n≤N ⎪ s s ( ) ( ) ⎪ R ≥R ; 1≤n≤N min ⎩ n ⎭⎪

{

throughput of all secondary users arrives at its maximum value when Eq. (3) is equation. Moreover, equal power allocation is introduced. Furthermore, optimal power allocation is deduced using theory of convex optimization. 3.1

(s) the upper bound of secondary users’ transmit power and Rmin

is the minimum throughput to satisfy the QoS demands of secondary users.

3 Power allocation algorithm In this section, the Theorem 1 is demonstrated that the total

Theorem 1

Theorem 1 G arrives at its maximum value when Eq. (3) is equation. Proof The Proof is provided in Appendix A.

3.2

Equal power allocation

In equal power allocation algorithm, all the secondary users transmit data with equal power. With the conclusion of Theorem 1, P can be deduced from ⎛ ⎞ ⎟ N B ⎜ hn( pp ) 2 Pn( p ) ⎟ = C ( th ) lb ⎜1 + (7) ∑ B N ⎛ ⎞ ⎜ ⎟ sp s ( ) ( ) 2 n =1 + h P N Γ 0 ⎜ nn equal ⎟ ⎜ N ⎠ ⎟⎠ ⎝ ⎝ (s) . It is difficult to obtain the where P1(s ) = P2( s ) = ... = PN( s ) = Pequal

close form solution of Eq. (7). However, the method of numerical analysis can be applied to solve this equation, such as Bisection method, Newton’s method and Tchebycheff’s iteration method. After solving the Eq. (7) and the solution is assumed to be (s) ′ Pequal , the last three constraints in problem Eq. (6) also need to be satisfied. Thus the final transmit power of secondary user can be represented by N (s) Rmin ⎧ −1 2B ⎪ ( ss ) 2 ⎪M (s) h , Pequal = max ⎨ nm ⎪∑ B⎞ ⎛ ( ps ) 2 ( p ) m =1 ⎪ Γ ⎜ hnm Pn + N 0 N ⎟ ⎝ ⎠ ⎩ ⎫ ⎪ ⎧⎪ ( s ) φm ⎫ φ ⎪⎪ min ⎨ Pequal′ , (ss ) 2 ,..., ( ssm) 2 , Pmax ⎬⎬ h1m hNm ⎪ ⎩⎪ ⎭⎪ ⎪ ⎭

}

where G is the total throughput of secondary users, Pmax is

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3.3

(8)

Optimal power allocation

In this subsection, the optimal solution of problem Eq. (6) will be deduced. As the problem Eq. (6) is not the convex optimization, we first transform it to be a convex one, using geometric programming. When primary user has higher SINR [9] and the background noise is trivial, comparing with interference from secondary user to primary user, the first

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constraint in problem Eq. (6) can be represented by N B ⎛ hn( pp ) 2 Pn( p ) ⎞ ( th ) lb ⎜⎜ (9) ∑ ( sp ) 2 ( s ) ⎟ ⎟≥C h P Γ n =1 N ⎝ nn n ⎠ X = [ x1 , x2 ,...xN ], ( −∞ < xn < +∞,1≤n≤N ) is defined to subject to Pn( s ) = 2 xn , (1≤n≤N ) . Inequation (9) can be transform to N ⎛ hn( pp ) 2 Pn( p ) ⎞ N ( th ) x ≤ lb ⎜⎜ ∑ ∑ n ( sp ) 2 ⎟ ⎟− C n =1 n =1 ⎝ Γ hnn ⎠ B N

(10)

According to Theorem 1, we can rewrite problem Eq. (6) ( ss ) 2 xn M ⎫ ⎛ ⎞ hnm 2 N B ⎜1 + ∑ ⎪ ⎟ min G′ = −∑ lb ⎜ m =1 ⎛ ( ps ) 2 ( p ) B ⎞⎟ ⎪ X h P N Γ + n =1 N 0 nm n ⎜ ⎟ ⎜ ⎪ N ⎠ ⎟⎠ ⎝ ⎝ ⎪ N ⎧N ⎪ ⎛ hn( pp ) 2 Pn( p ) ⎞ N ( th ) − C ⎪∑ xn = ∑ lb ⎜⎜ ⎪ ( sp ) 2 ⎟ ⎟ n =1 ⎪ n =1 ⎪ ⎝ Γ hnn ⎠ B ⎪ ⎪ ⎪ x − lb ⎛ φm ⎞≤0; 1≤m≤M , 1≤n≤N ⎪ ⎜ ⎟ n ⎬ (11) ⎜ h(ss ) ⎟ ⎪ nm ⎠ ⎝ ⎪ ⎪⎪ ⎪ s.t. ⎨ xn − lb ( Pmax )≤0; 1≤n≤N ⎪ ⎪ N (s) ⎪ Rmin ⎛ ⎞ ⎪ −1 2B ⎪ ⎜ ⎟ ⎪ ⎪ ( ss ) 2 ⎟ ⎪− x + lb ⎜ M h nm ⎜∑ ⎟≤0; 1≤n≤N ⎪ ⎪ n ⎪ ⎜ m =1 Γ ⎛ h( ps ) 2 P ( p ) + N B ⎞ ⎟ ⎪ 0 ⎜ nm n ⎟⎟ ⎜ ⎪ N⎠⎠ ⎝ ⎝ ⎩⎪ ⎭ Since log-sum-exp functions are convex [10]. This optimization problem is a convex one. Now the Lagrange dual method will be applied to solve this problem. Firstly, the Lagrange dual function is defined as ⎛ N ⎜ g ( λ , u, x , η ) = inf L ( X , λ , u, x, η ) = inf ⎜ −∑ lb (1 + H ( n ) 2 xn ) + ⎜ n =1 ⎝

⎛

N

⎞

N

M

⎛ ⎜ ⎝

⎛ φm ⎞ ⎞ N ( ss ) ⎟ ⎟ ⎟⎟ + ∑ ξ n xn − ⎝ hnm ⎠ ⎠ n =1

λ ⎜ ∑ xn − Q ⎟ + ∑∑υmn ⎜ xn − lb ⎜⎜ ⎝ n =1

⎠

n =1 m =1

(

(s) ⎛ ⎛ NB Rmin ⎞⎞⎞ −1⎟ ⎟ ⎟ 2 ⎜ ⎜ lb ( Pmax ) ) − ∑η n ⎜ xn − lb ⎜ ⎟⎟⎟ n =1 ⎜ H (n) ⎟ ⎟ ⎟ ⎜ ⎝ ⎠⎠⎠ ⎝

N

where

(12)

λ , υ = [υ mn ]M × N , (1≤m≤M , 1≤n≤N ) , ξ = [ξ1 , ξ 2 ,

..., ξ N ] , η = [η1 ,η2 ,...,η N ] are Lagrange multipliers, and

( ss ) 2 ⎛M ⎞ hnm ⎜ ⎟ ∑ (13) H ( n ) = ⎜ m =1 ⎛ ( ps ) 2 ( p ) B ⎞⎟ ⎜ Γ ⎜ hnm Pn + N 0 N ⎟ ⎟ ⎝ ⎠ ⎝ ⎠ It is supposed that g ( λ , u, x,η ) arrives at its maximum

value d ∗ when λ = λ ∗ , υ = υ ∗ , ξ = ξ ∗ and η = η ∗ and G′ ( X ) reaches its minimum value when X = X ∗ . As

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problem Eq. (11) is convex and there exists an X subject to ⎫ ⎛ φ ⎞ xn − lb ⎜⎜ (mss ) ⎟⎟ < 0; 1≤m≤M , 1≤n≤N ⎪ ⎪ ⎝ hnm ⎠ ⎪ xn − lb ( Pmax ) < 0; 1≤n≤N ⎪ (14) ⎬ N (s) ⎪ ⎛ B Rmin ⎞ −1⎟ 2 ⎪ − xn + lb ⎜⎜ ⎪ ⎟ < 0; 1≤n≤N H n ( ) ⎜ ⎟ ⎪⎭ ⎝ ⎠ According to the Slater condition, strong duality holds between the primary problem and dual problem. Namely d ∗ = p∗ (15) Thus we can obtain KKT optimality conditions ⎛ ⎛ φ ⎞⎞ ∗ υmn ⎜ xn∗ − lb ⎜⎜ (mss ) ⎟⎟ ⎟ = 0; 1≤m≤M ,1≤n≤N ⎜ ⎟ ⎝ hnm ⎠ ⎠ ⎝

⎫ ⎪ ⎪ ⎪ ∗ ∗ ξ n ( xn − lb ( Pmax ) ) = 0; 1≤n≤N ⎪ ⎪ (s) ⎪ ⎛ ⎛ NB Rmin ⎞⎞ ⎬ 2 −1 ⎟ ⎟ ⎜ ∗⎜ ∗ η n ⎜ xn − lb ⎜ ⎪ ⎟ ⎟ = 0; 1≤n≤N ⎪ ⎜ H (n) ⎟ ⎟ ⎜ ⎝ ⎠⎠ ⎝ ⎪ x∗ ⎪ N M N N 1 H ( n ) 2 n ln 2 ∗ ∗ ∗ ∗ ⎪ − ∗ + λ + ∑∑υ mn + ∑ ξ n − ∑η n = 0 ⎪⎭ ln 2 1 + H ( n ) 2 xn n =1 m =1 n =1 n =1 When (s) ⎛ NB Rmin ⎞ 2 −1⎟ ∗ ⎜ lb ⎜ ⎟ < xn < ⎜ H ( n) ⎟ ⎝ ⎠ ⎧⎪ ⎛ φ1 ⎞ ⎫⎪ ⎛φ ⎞ min ⎨lb ⎜⎜ ( ss ) ⎟⎟ ,...,lb ⎜⎜ (Mss ) ⎟⎟ ,lb ( Pmax ) ⎬ ; 1≤n≤N ⎝ hnM ⎠ ⎩⎪ ⎝ hn1 ⎠ ⎭⎪

(16)

(17)

From the first, second and third equations of KKT condition, we can obtain ∗ = 0; 1≤m≤M ,1≤n≤N ⎫ υmn ⎪ ∗ (18) ξ n = 0; 1≤n≤N ⎬ ⎪ ∗ η n = 0; 1≤n≤N ⎭ Then from the last equation of KKT condition we can obtain ⎛ ⎞ λ∗ ⎟ ; 1≤n≤N xn = lb ⎜ (19) ⎜ ( H ( n ) − λ∗H ( n)) ⎟ ⎝ ⎠ Using the first constraint of problem Eq. (11), we can deduce N

N

λ∗ =

2Q ∏ H ( n ) n =1

N ⎛ ⎞ ⎜ 1 + N 2Q ∏ H ( n ) ⎟ ⎜ ⎟ n =1 ⎝ ⎠

where

(20)

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GE Wen-dong, et al. / Optimal power allocation for multi-user OFDM and…

N ⎛ h( pp ) 2 P ( p ) ⎞ N Q = ∑ lb ⎜⎜ n ( spn) 2 ⎟⎟ − C ( th ) n =1 ⎝ Γ hnn ⎠ B

(21)

So the optimal solution of problem Eq. (6) under the condition Eq. (17) is N

N

(s)

Pn = 2 = xn

2Q ∏ H ( n ) n =1

H ( n)

be 50 kbit/s. In Fig. 2, the performance of optimal power allocation algorithm is compared with existing power allocation algorithm based on ‘interference temperature’. The constraint of the existing algorithm can be represented as N

; 1≤n≤N

(22)

When condition Eq. (17) is false, the solution of problem Eq. (6) could be represented as ⎧ ⎪ ⎧⎪ φ ⎫⎪ φ ⎪⎧ ⎛ φ ⎞ ⎪min ⎨ ( ss1 ) ,..., (Mss ) , Pmax ⎬ ; if xn∗≥ min ⎨lb ⎜ ( ss1 ) ⎟ ,..., ⎜ ⎟ hnM ⎪ ⎪⎩ hn1 ⎪⎭ ⎪⎩ ⎝ hn1 ⎠ ⎪ ⎫⎪ ⎪ ⎛φ ⎞ Pn(s ) = ⎨ lb ⎜⎜ (Mss ) ⎟⎟ ,lb ( Pmax ) ⎬ (23) ⎝ hnM ⎠ ⎪ ⎭⎪ ⎪ N (s) (s) ⎛ NB Rmin ⎞ ⎪ 2 B Rmin − 1 −1⎟ 2 ⎜ ∗ ⎪ ; if xn≤lb ⎜ ⎟ , 1≤n≤N ⎪ H ( n) ⎜ H (n) ⎟ ⎝ ⎠ ⎩

However, we must verify whether P = ⎡⎣ P1( s ) , P2( s ) ,..., PN( s ) ⎤⎦

45

∑ h( n =1

sp ) 2 ( s ) nn n

P ≤I th , where I th is the interference threshold.

The details of this algorithm can be found in Refs. [3,6]. In these simulations, M, I th and BER are assumed to be 8, 27 dBm and 10−3 respectively. From this picture, the total throughput of secondary user in optimal power allocation algorithm is higher than that in existing power allocation algorithm in different C ( th ) s. The reason is that existing power allocation algorithm limit the transmit power of secondary users with ‘interference temperature’ in every subcarriers, while optimal power allocation algorithm only enable secondary users to guarantee the total throughput of primary user. Thus those secondary users with higher channel gain could enhance their transmit power to some degree.

will satisfy the first constraint of problem Eq. (6). If it is not satisfied, the elements of P that are equal to

(2

N (s) R B min

(2

N (s) R B min

)

H ( n ) will be fixed and other elements will be −1 deduced again with the theorem of convex optimization mentioned above until this constraint is satisfied. If the facts that all the transmission powers of SUs are equal to

)

H ( n ) cannot satisfy this constraint, the feasible −1 region of the problem Eq. (11) is null set. Thus there is no solution.

4

Performance analysis

In this section, extensive simulation results are firstly shown to compare the performance of existing power allocation algorithm, equal power allocation algorithm and optimal power allocation algorithm. The parameters in the simulations are chosen based on the parameters adopted commonly [2,6,11] as follows. All the channels involved are assumed to be Rayleigh fading. The bandwidth of the channel occupied by primary user B is assumed to be 100 kHz, the quantity of RU or secondary user N is supposed to be 10, The transmit power Pn( p ) of primary

Fig. 2 Comparison of power allocation based on ‘interference temperature’ and optimal power allocation

Fig. 3 compares the performance between equal power allocation algorithm and optimal power allocation algorithm in different lower bound of primary user’s throughput. Parameters of these simulations are similar to Fig. 2. From this picture, the total throughput of secondary user in two algorithms all decrease with the increment of C ( th ) , which should be ascribed to the fact that the increment of C ( th )

assumed to be −117 dBm . The upper bound of secondary user’s transmit power Pmax is 1 W. The minimum throughput

constrain the transmit power of secondary user to lead to lower throughput. Moreover, optimal power allocation algorithm is superior to equal power allocation algorithm. That is because the secondary user has a higher transmit power in some RUs where the interference from primary user to secondary user is low. Furthermore, when C ( th ) is very

(s) is set to to satisfy the QoS demands of secondary users Rmin

high, performance of equal power allocation algorithm is

user is assumed to be 0.5 W. Background noise N 0 is

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approximately equivalent to optimal power allocation one. That is because the extremely higher C ( th ) reduce the transmit power of secondary users dramatically, which abate improvement of optimal power allocation algorithm.

Fig. 3 Comparison of equal power allocation and optimal power allocation

In Fig. 4 the effect of quantity of antennas M on the performance of optimal power allocation algorithm is illustrated. Here, BER is set to be 10−3 , and M varies from 1 to 8. In this figure, the total throughput of secondary user increase with the increment of M, which is ascribed to the fact that more antennas can enhance the SINR of secondary user in the base station after MRC. Additionally, when M is beyond 4, the improvement of secondary user’s throughput is not apparent. Thus it is unnecessary to increase the quantity of antennas when the expenditure of establishing the networks is limited. Meanwhile, the performance of this algorithm is similar in different M when C ( th ) is higher. Accordingly, in this situation the number of secondary links to cope with could abate in order to reduce the burden of calculation at Base station.

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Conclusions

In this paper, the power allocation problem in multi-user OFDM and distributed antenna cognitive radio with radio over fiber was discussed, where the constraint that primary user can not be disturbed by secondary user was achieved by the rules that throughput of primary user should be beyond the given threshold. Moreover, the theorem that total throughput of secondary user arrives at its maximum value when throughput of primary user is equal to the threshold was justified. Furthermore, equal power allocation algorithm and optimal power allocation algorithm were deduced with the theory of convex optimization problem. Finally, extensive simulation results show that the performance of the optimal power allocation algorithm is superior to existing power allocation algorithm and equal power allocation algorithm. Acknowledgements This work was supported by the National Natural Science Foundation of China (60832009), the Beijing National Science Foundation (4102044), the Fundamental Research Funds for the Central Universities (BUPT2009RC0119), the New Generation of Broadband Wireless Mobile Communication Networks of National Major

Projects

for

Science

and

Technology

Development

(2009ZX03003-003-01).

Appendix A Proof of Theorem 1 Reduction to absurdity is adopted to justify this theorem. It is assumed that P ′ = ⎢⎡ P1( s )′ , P2( s )′ ,..., PN( s )′ ⎥⎤ can ⎣ ⎦ Proof

enable G to arrive at its maximum value when ⎛ ⎞ hn( pp ) 2 Pn( p ) N B ⎜1 + ⎟ ( th ) (A.1) lb ⎜ B ⎞⎟−C > 0 ⎛ ( sp ) 2 ( s )′ ∑ h P N Γ + N n =1 0 ⎜ nn n ⎟⎟ ⎜ N ⎠⎠ ⎝ ⎝ As the left side of Eq. (A.1) is monotone decreasing function with P1( s )′ , thus ∃ P ′′ = ⎡⎢ P1( s )′′ , P2( s )′ ,..., PN( s )′ ⎤⎥ can ⎣ ⎦

force Inequation (A.1) to turn to the equation and P1(s )′′ > P1( s )′ . While G is monotone increasing function with P1( s ) . So it is obvious that G ( P ′′ ) > G ( P ′ )

(A.2)

Thus G ( P ′ ) is not the maximum value of G, which is in Fig. 4 Comparison of G in optimal power allocation with different M

contradiction to assumption. Accordingly, when Inequation (3) is equation, G arrives at its maximum value.

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