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Power Control for Constrained Throughput Maximization in Spectrum Shared Networks John Tadrous, Ahmed Sultan, Mohammed Nafie and Amr El-Keyi Wireless Intelligent Networks Center (WINC), Nile University, Cairo, Egypt Email: [email protected], {asultan, mnafie, aelkeyi}@nileuniversity.edu.eg

Abstract—We investigate power allocation for users in a shared spectrum network. In such a network, the primary (licensed) users communicate under a minimum guaranteed quality of service (QoS) requirements, whereas the secondary users opportunistically access the primary band. Our objective is to find a power control scheme that determines the transmit power for both primary and secondary users so that the overall network throughput is maximized while maintaining the quality of service of the primary users greater than a specified minimum limit. In the assumed model, no interference cancellation is done at the receivers resulting in a non-convex optimization problem. It has been shown previously that binary power control almost always achieves the global optimum solution when no QoS constraints are imposed. This is not necessarily the case in our scenario, however. We introduce a distributed algorithm for “ternary” power allocation to be used when individual measurements are available at each node. We show via simulations the relative efficiency of the proposed algorithm compared to previously suggested ones. If a central controller exists with available information about the system parameters, we enhance the performance of the proposed algorithm through an iterative geometric programming (GP) algorithm and prove its convergence to a better solution than ternary power allocation.

I. I NTRODUCTION Cognitive radios are spectrum agile devices that can be employed by secondary users to sense the spectrum, identify the underutilized bands, and opportunistically access those bands together with the licensed primary users such that they do not cause harmful interference to those primary users [1]. Cognitive radio technology thus allows more users to share the spectrum efficiently and enhances spectrum utilization. Cognitive radios can operate with a primary network under either the overlay or underlay spectrum sharing scheme [2]. In spectrum overlay, secondary users are allowed to access the bands that are totally free of the primary operation. Consequently, robust sensing is demanded of secondary users. In spectrum underlay, secondary users can share the same spectrum band concurrently with the primary users, so long as the interference caused on the primary users is small enough to guarantee minimum primary quality of service (QoS). The secondary users do not have to employ sophisticated sensing mechanisms. Instead, an efficient resource allocation procedure is needed so that the spectrum can be optimally utilized. In this paper we consider the problem of power allocation for primary and secondary users in a spectrum underlay system. Most of the existing work separates the power allocation for the secondary network from that of the primary

network. In [3]-[6] power allocation for secondary users under maximum tolerable interference constraint on the primary users is considered besides a minimum QoS constraint on each secondary user. In those works, two admission objectives for secondary users are discussed, namely, maximizing the number of admitted secondary links [3]-[5], and maximizing the secondary throughput [6]. In [7] distributed binary power control algorithms are introduced for cognitive links in interference limited regime where a large number of links exist and the outage probability for a primary link is limited. In this work we aim to jointly allocate power for both primary and secondary users so that the overall network throughput is maximized while protecting the QoS of the primary users by keeping their signal-to-interference-andnoise-ratio (SINR) above a minimum level. This optimization problem is non-convex, however. When no QoS constraints are considered, binary power control is shown in [8], via numerical simulations, to almost achieve the global optimum solution with negligible capacity loss. But since in our model there are QoS constraints imposed on primary users, performance of binary power allocation may be degraded. Moreover, the optimal binary power allocation is an NP-hard problem that requires exhaustive search. This has opened the area for research for suboptimal reduced complexity algorithms for binary power control [7], [8]. We consider two solutions to the aforementioned problem and make the following contributions. First, we introduce a distributed algorithm that yields a “ternary” power allocation to network nodes while considering the QoS constraints for primary users. As detailed later, the ternary power allocation produces three groups of power assignments: two operating with zero or maximum transmit power, and the third with the power level just needed to satisfy the SINR inequality constraints with equality. Algorithm execution is based on local measurements available at each node besides small amount of signaling. It is compared to the greedy algorithm of [8] and shown to outperform it. Second, we consider the case when a central node with knowledge about all system parameters exists and introduce a centralized distributed algorithm based on successive geometric programing (GP) that outperforms the ternary power allocation scheme. The convergence of the proposed algorithm is established. It is shown to result in a better performance than the optimal ternary power allocation, and it has lower computational complexity per iteration than the single condensation methods described in [11], [14] and

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the double condensation method introduced in [14]. Note that, although in [15] a centralized algorithm is introduced and shown to achieve the global optimum solution via monotonic optimization, its computations are rather extensive as it searches for the optimal solution over the boundary of the feasible set. The rest of this paper is organized as follows. In Section II we describe the system model. In Section III the problem is formulated. We introduce the proposed algorithms in Section IV. Simulation results are presented in Section V and the work is concluded in Section VI. II. S YSTEM M ODEL We consider a spectrum sharing model where secondary users share the same frequency band with primary users. In this model we focus on the uplink transmission from primary users toward their base station (BS), whereas the secondary users are communicating with each other in an ad-hoc fashion. The primary network is assumed to have a single BS with K primary users. We denote the transmit power of the j th primary user by ppj , j = 1, 2, · · · , K, the channel gain from pp . The SINR of the j th primary user j to the base station by g0,j link is given by pp ppj g0,j , j = 1, 2, · · · , K, N p pp s ps l=1,l=j pl g0,l + i=1 pi g0,i + N0 (1) where there are N secondary links within the coverage area of the primary BS. Each secondary link i comprises one transmitter and one corresponding receiver. The parameter psi is the transmit power of the transmitter of secondary link i, ps is the channel gain between the ith secondary transmitter g0,i and the primary BS, and N0 is the background noise power in the frequency band of operation. The channel gain between ss , a secondary transmitter i and a secondary receiver j is gj,i whereas the channel gain between a primary transmitter i and sp . Hence, we can write the SINR a secondary receiver j is gj,i of the secondary link i, xsi , as

xpj = K

ss psi gi,i , i = 1, 2, · · · , N K p sp s ss l=1,l=i pl gi,l + j=1 pj gi,j + N0 (2) For both primary and secondary networks we assume that no interference cancellation is performed and that each link can measure its own SINR locally.

xsi = N

III. P ROBLEM F ORMULATION For the given system, we consider the sum throughput. The network users are allocated powers so that the total throughput, primary and secondary, is maximized. However, in order to guarantee minimum QoS for the primary users, the power control scheme should guarantee that the SINR of each primary user does not fall below a minimum level. The QoS constraint on the primary users are xpj ≥ γjp ,

j = 1, 2, · · · , K.

(3)

This is equivalent to guaranteeing a minimum rate for each primary user. Moreover, each transmitter, secondary or primary, cannot increase its transmit power beyond a certain maximum level. The power constraints on secondary and primary users are , psi ≤ psmax i p pmax pj ≤ pj ,

i = 1, 2, · · · , N,

(4)

j = 1, 2, · · · , K.

(5)

Since the maximum achievable throughput of user i in bps/Hz is given by log2 (1 + SINRi ), the optimization problem can be written as maximize s p p ,p

subject to

N i=1

log2 (1 + xsi ) +

K j=1

log2 (1 + xpj )

xpj ≥ γjp , j = 1, 2, · · · , K

, 0 ≤ psi ≤ psmax i p pmax , 0 ≤ pj ≤ pj

(6)

i = 1, 2, · · · , N j = 1, 2, · · · , K

where ps and pp are the vectors of secondary and primary power allocations respectively. Parameters xpj and xsi are as defined in (1) and (2). We assume that this problem is feasible, i.e., there exists a power allocation vector that satisfies its constraints. This optimization problem is non-convex and has been shown to have no tractable solution [9]. In [3], a similar optimization problem is reduced to a convex problem through the approximation: log2 (1 + SINRi ) ≈ log2 (SINRi ) where the system is assumed to be in the high SINR regime. However, in our formulation there are no constraints on the SINR of the secondary network, i.e., the secondary network is opportunistic. Consequently high SINR is not guaranteed for all links. It has been shown in [8] that the optimal solution to (6) contains at least one value of maximum transmit power of a certain user. Moreover, when no minimum QoS constraints are imposed, binary power control, in which a group of users transmit with their maximum power while others are silent, is shown to achieve the global optimum solution in interference and noise limited regimes [12]. When the interference level is comparable to the noise level, binary power allocation might not be optimal. However, in [8] the authors showed, via simulations, that the loss in sum throughput is negligible when applying binary power allocation in moderate interference regime. Although binary power allocation seems a plausible solution, optimal binary power allocation is computationally complex. For L interfering users, it requires exhaustive search over 2L − 1 possible power allocation strategies and picking the one with maximum sum throughput. In the next section we propose a reduced complexity distributed algorithm for ternary power allocation that takes QoS constraints into account. Then we develop a centralized algorithm that enhances the performance of binary power allocation.

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IV. P ROPOSED A LGORITHMS Different reduced complexity algorithms have been proposed to get efficient binary power allocation with performance sufficiently close to the exhaustive search solution. In [7] a distributed algorithm is introduced where each secondary user makes a decision according to its own SINR whether to transmit with maximum power or remain silent. This algorithm, however, is derived for the interference limited regime where the number of secondary users is extremely large. In [8], two different suboptimal techniques are introduced. The first is clustering in which the set of users is divided into smaller disjoint sets and optimal binary power allocation is done for every set separately. The second is a greedy approach, with O(L2 ) complexity, in which the user that maximizes the sum throughput at a certain iteration transmits with maximum power. The iterations continue until all users are active or have reached a state at which activating one more user degrades the sum throughput. The greedy approach is shown to outperform the clustering method. A. Distributed Ternary Power Allocation Since QoS constraints are imposed on primary users, we define a ternary power allocation strategy as a strategy in which the users are clustered into three disjoint sets. The first set, P, contains users, primary or secondary, sending with their maximum power, the second set, Q, contains primary users with QoS constraints sending with transmit powers that is sufficient to make each of them operate at the minimum QoS requirement, and the third set, S, is of secondary users, with no QoS constraints, sending with zero transmit powers. The power allocation for users in Q depends on users in P. 1) Handling QoS Constraints: In order to allocate power for primary users in Q at a certain ternary power allocation strategy, a distributed algorithm called Distributed Constrained Power Control (DCPC) [16] can be used. Note that |P| + |Q| + |S| = N + K = L. Hence, DCPC algorithm can be applied to the primary users in Q as follows. Each user i ∈ Q synchronously or asynchronously with other users updates its transmit power iteratively according to γip p pmax p , i ∈ Q, (7) , pi (T ) p pi (T + 1) = min pi xi (T ) where ppi (T ) is the transmit power for primary user i at iteration T and xpi (T ) is its measured SINR at the same iteration. DCPC converges to a unique power vector that contains values of minimum transmit powers for users in Q that make each of them supported exactly at γip , or some maximum power allocation indicating that the current ternary power allocation strategy is infeasible. The optimal ternary power allocation is obtained by testing all possible P sets and the corresponding Q and S then picking the ternary power allocation strategy that yields maximum sum throughput with feasible power allocation, i.e., with QoS for primary users is guaranteed.

2) Distributed Algorithm: In this algorithm we consider testing different sequences of users that lead to always increasing sum throughput when transmitting with their maximum power and at the same time maintain the feasibility of the network. To do that, we denote the throughput of secondary user i by Ris and the throughput of primary user j by Rjp . The change in throughput between any two successive power allocation strategies s1 , s2 for secondary user i is ΔRis = Ris (s2 ) − Ris (s1 ) and for primary user j is ΔRjp = Rjp (s2 )−Rjp (s1 ). This change in throughput can be calculated locally as a direct function of measured SINR. The overall change in sum throughput is then obtained as ΔRsum =

N i=1

ΔRis

+

K j=1

ΔRjp

(8)

Moreover, we assume that all users are ordered circularly in a set L, i.e., the order of user i + L is i for L = |L| = K + N . Also we use the statement “some users join Q and S accordingly” to mean that all primary users other than those in P join Q and secondary users other than those in P join S. The distributed algorithm is described as follows. Algorithm 1 Distributed Binary Power Allocation 1: Initialize: P = {}, primary users in Q and secondary users in S. 2: DCPC(Q). 3: for root = 1 to L do 4: P = {root}, and other users join Q and S accordingly. 5: DCPC(Q) 6: if Q is infeasible then 7: P = P − {root} where root joins Q or S accordingly. 8: Go to next iteration. 9: end if 10: for j = root + 1 to L + root − 1 do 11: P = P + {j} and other users remain in their respective sets Q and S. 12: DCPC(Q). 13: if the strategy is infeasible or ΔRsum < 0 then 14: P = P − {j} where j joins Q or S accordingly. 15: Go to previous power allocation strategy. 16: end if 17: end for 18: sroot =current power allocation strategy. 19: root sends Rsum (sroot ) to all users. 20: end for 21: Users operate according to s∗ where s∗ = arg maxsroot Rsum (sroot ). For the root user to keep track of Rsum we assume that each primary user initially broadcasts its own minimum QoS requirements. K Consequently, before the algorithm execution min = j=1 log2 (1 + γjp ) is calculated at each user. Then Rsum after each DCPC, in turns, user i, i = 1, · · · , L, calculates the cumulative change in sum throughput as ΔR(i)sum =

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ΔR(i−1)sum + ΔRiu , where ΔR(1)sum = ΔR1u and u ∈ {s, p}, and sends it to the next user i + 1. After completing one cycle, i.e., when i = L the overall value of ΔRsum is available at user L. By sending ΔRsum to root it can calculate the current value of Rsum . The idea of this algorithm is to start with a certain user, root ∈ L, and set its transmit power to pumax root , u ∈ {s, p} while other users are transmitting with their minimum transmit powers that satisfy the constraints of (6). Then for all remaining users, sequentially, each user is tested if it transmits with maximum power and an increase in sum throughput happens while the current power strategy is feasible then this user will remain in P. Otherwise, the system goes back to the previous power allocation strategy, then the next user is tested. The algorithm continues by trying this method over all different roots and picking the strategy that maximizes the sum throughput. The complexity of this algorithm in unconstrained scenario is O(L2 ) as the greedy algorithm of [8]. But in case of constrained scenario, complexity grows because of the distributed execution of DCPC algorithm. However, this increase can be reduced using the fact that if any set P results in an infeasible strategy, then P cannot be a subset of a larger set P and the strategy is feasible. The proof of this fact is omitted for space limitation. Making use of this fact, different sets will not be tested and the complexity will be reduced.

where x0 is the solution to the approximate problem in the previous iteration and f0 (x) is the original objective function. The above three conditions are sufficient to guarantee that the solution of each approximate problem increases the objective function, and after convergence of the series of the approximate problems, the Karush-Kuhn-Tucker conditions of the original problem will be satisfied [11], [13]. Approximations to objective functions similar to that of (6) have been introduced based on successive GP. However, the single condensation method introduced in [11] and used in [8] requires about (L + 1)L variables to be updated each iteration. And the double condensation method proposed in [14] requires very complex computations to update everyone of L variables. Moreover, it is not proved to satisfy the above three conditions. Hence, no proof on convergence is given. Because of the monotonicity of the log function, the original objective function of (6) can be written as

B. Centralized Enhancement

Then the approximate optimization problem would be

The main idea of binary power control is that one of the corner points of the rate region almost always achieves the global optimal solution. The number of those corner points is 2L − 1 and they are obtained through binary power allocation in the unconstrained case. However, under SINR constraints there may exist some other corner points that are not achieved by the ternary power allocation described in Subsection IV-A. Instead, they are obtained by power allocation strategies that contain some users in P, some users in Q, some users in S and some other users, primary or secondary, in a set I such that their transmit power is smaller than the maximum power but results in at most one primary user j in P supported exactly at its γjp . Note that, |P| + |Q| + |S| + |I| = L. Characterizing those corner points is computationally complex as it depends on the system parameters, like noise variance and channel gains, and the number of those corner points may be very large. So, in order to enhance the performance of the proposed algorithm, we suggest a centralized technique based on GP. Since the objective function of (6) is non-convex, the idea of the centralized technique is to start with the sum throughout obtained from Algorithm 1 as an initial point to successive GP approximations to (6). Finally, we get a local optimum solution that has larger throughput than the initial one. To attain that, the approximate objective function f˜0 (x) should satisfy the following conditions: 1) f0 (x) ≥ f˜0 (x) for all x, 2) f0 (x0 ) = f˜0 (x0 ), 3) ∇f0 (x0 ) = ∇f˜0 (x0 ),

f0 (x) =

N

K

(1 + xsi )

i=1

j=1

(1 + xpj ).

(9)

We suggest the following approximate objective function: f˜0 (x) = c

N

(xsi )

i=1

maximize s p s p p ,p ,x ,x

c

N

s

(xsi )λi

K

λsi

K j=1

p

(xpj )λj .

(10)

p

(xpj )λj

i=1 j=1 p p −1 subject to, γi xi ≤ 1, N p K p pp ps xj ( k=1,k=j pk g0,k + l=1 psl g0,l + N0 ) ≤ 1, p pp (pj g0,j ) N K sp ss xsi ( j=1,j=i psj gi,j + k=1 ppk gi,k + N0 ) ≤ 1, s ss (pi gi,i ) ppj (ppmax )−1 ≤ 1, j s smax −1 pi (pi ) ≤ 1,

∀i ∀j (11) ∀i ∀j ∀i,

where xs = [xs1 , · · · , xsN ] and xp = [xp1 , · · · , xpK ]. Note that, this problem is written in the form of a geometric program, where the objective function is a monomial and the constraints are posynonmials [10]. Moreover, if the original objective function (9) is used instead of the approximate one in (11) the new problem is equivalent to that of (6). Next we determine the values of c and λ’s in order to satisfy the given three conditions per iteration. Let x = [x1 , x2 · · · , xL ] where xj = xpj , j = 1, 2, · · · , K, xj = xsi , j = K + i, i = 1, 2, · · · , N , Λ = [λ1 , λ2 , · · · , λL ] where λj = λpj , j = 1, 2, · · · , K, λj = λsi , j = K + i and i = 1, 2, · · · , N . From the third condition we have: x0i , i = 1, 2, · · · , L (12) λi = 1 + x0i

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From the second condition we have: c

L

x0i λi =

i=1

L

(1 + x0i )

i=1

L (1 + x0i ) c = i=1 L λi i=1 (x0i )

(13)

By obtaining c and Λ the approximate objective function f˜(x) is determined for the next iteration. What remains is to check the satisfaction of the first condition L

(1 + xi ) ≥ c

i=1

L i=1

xλi i .

Equivalently, by substituting for c and λi from (13) and (12) it is required to check that L

(1 + xi ) ≥

i=1

L (1 + x0i ) i=1

x0i ( 1+x 0i

x0i

)

x

0i ) ( 1+x

.xi

0i

.

(14)

Theorem 1: Fix i, for all xi ≥ 0 the following inequality is satisfied: 1 + xi ≥

(1 + x0i ) x0i ( 1+x 0i

x0i

)

x

0i ) ( 1+x

.xi

0i

,

xi ≥ 0

(15)

Proof: By combining both sides of (15), and taking the log we define: g(xi ) = log

1 + x0i x0i xi + log 1 + xi 1 + x0i x0i

(16)

Now we prove that g(xi ) ≤ 0 by showing that the maximum of g(xi ) is zero. First, the stationary point is obtained by: ∂g(xi ) |xi =x∗i = 0 ∂xi x0i −1 = 0 ⇒ x∗i = x0i + ∗ ∗ 1 + xi xi (1 + x0i ) By substituting with x∗i in (16): g(x∗i = x0i ) = 0. The second derivative of g(xi ) is: −1/x0i ∂ 2 g(x∗i ) = 2 (1 + x0i )2 ∂x∗i which is non-positive for x0i ≥ 0. Therefore, g(xi ) ≤ 0 for xi ≥ 0, and consequently the inequality of (15) is satisfied. Since the inequality of (15) is satisfied ∀i = 1, · · · , L, inequality (14) is also satisfied, and the first condition on the approximate function is maintained. Thus, using the approximate function defined in (10) only L + 1 parameters need to be updated per iteration, namely λs and c, and computing each value of them is simpler than the cases of single or double condensation. The algorithm then is summarized as

Algorithm 2 Successive GP 1: Initialize Λ and c from (12) and (13), respectively, based on the solution to Algorithm 1. 2: while Target accuracy is not reached do 3: Solve (11). 4: Update Λ and c from (12) and (13) respectively. 5: end while

V. N UMERICAL R ESULTS The simulation setup we use is close to that used in [4]. We assume that the secondary transmitters and the primary users are located in a square area of size Dp × Dp with BS of the primary network located at the center. The receiving node of each secondary link is placed randomly in a Ds × Ds square with its transmitting node at the center. The channel gain between any transmitter j and receiver i is modeled as gi,j = K0 × 10βi,j /10 × d−4 i,j , where di,j is the corresponding distance, βi,j is a random Gaussian variable with zero mean and a standard deviation of 6 dB to account for shadowing effects, and K0 = 103 is a factor that includes some system parameters such as antenna gain and carrier frequency. The background noise power is N0 = 10−10 Watt. The maximum transmit power on each secondary transmitter is psmax = 0.1 Watt, and on each primary transmitter is ppmax = 1 Watt. For simulations that require solving (11) we use the CVX, a package for specifying and solving convex programs [17]. A. Comparison with Greedy Algorithm In this subsection we compare the proposed distributed algorithm to the greedy algorithm of [8]. Since the greedy algorithm was established under no SINR constraints, in this comparison we consider no such constraints. The dimension Dp = 200m, whereas Ds = 100m. The performance of both algorithms is compared to the optimal binary power allocation for different number of links as follows. The number of primary users is fixed at K = 2 and the number of secondary users is varied from N = 2 to N = 12. For each number of users, L = N + K, normalized sum throughput is obtained for the distributed and greedy algorithms by dividing their values by the sum throughput obtained via exhaustive search over binary power allocation strategy. The result is an average over 103 simulation runs where in each simulation the locations of the L users are generated randomly. Fig. 1 depicts the result of the comparison. It is obvious that the performance of both algorithms degrades as L increases, however, the distributed algorithm has better performance than the greedy algorithm and its decay rate against L is smaller. Although both algorithms have almost the same performance for small number of users, the performance of the greedy algorithm degrades to 93.1% at L = 14 whereas the distributed algorithm achieves 98.4% of the optimal binary power allocation performance. B. Successive GP The proposed successive GP algorithm is compared to both the optimal ternary power allocation and the proposed

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Normalized throughput to that of exhaustive search binary power allocation

1.02 Proposed algorithm Greedy algorithm 1

0.98

0.96

0.94

0.92

0.9

4

6

8 10 Number of users

12

14

Fig. 1. Normalized sum throughput vs. number of users L = N + K where K = 2 is fixed and N varies from 2 to 12

Successive GP Distributed algorithm

Normarlized sum throughput to exhaustive search ternary power allocation

1.1

this problem is non-convex, however, in unconstrained scenarios binary power allocation has been shown to almost achieve the global optimum solution. However, the optimal binary power allocation requires exhaustive search over all possible binary power allocation strategies, which grows exponentially with the number of users. In this paper, we introduced the definition of ternary power control to consider the satisfaction of the QoS constraints, and developed a less complex distributed suboptimal algorithm that we show, via simulations, its relative efficiency over previous schemes. Moreover, we introduced a centralized enhancement to the performance of the distributed algorithm using GP. We established an algorithm based on successive solutions to approximate GP problems that yields better performance than the optimal ternary power allocation. We proved the convergence of this algorithm and showed that it requires simpler computations between iterations than previous techniques, namely, single and double condensation. R EFERENCES

1.05

1

0.95

0.9

7

8

9 10 11 12 13 Minimum QoS constraint per primary user (dB)

14

15

Fig. 2. Normalized sum throughput vs. QoS constraints on each primary user for K = 2 and N = 3.

distributed algorithm. In this simulation setup Dp = 800m and Ds = 1000m. The normalized sum throughput is obtained for the proposed successive GP and proposed distributed algorithms, averaged over 103 simulation runs and plotted against the minimum SINR requirement per primary user in a network consisting of K = 2 and N = 3 links. Primary users are assumed to have equal QoS requirements at each simulation. For this type of simulation we assume that both primary and secondary users use spreading gain factor of 80. Fig. 2 shows the result of this comparison. The average performance of the proposed GP algorithm is better than the optimal ternary power allocation described in Subsection IV-A although its initial point is given from the suboptimal proposed distributed algorithm. However, the loss of the distributed algorithm can be tolerated in completely distributed systems as it is not much lower than the performance of the iterative GP algorithm. VI. C ONCLUSION We have considered power allocation for spectrum shared cognitive radio network. Our objective is to allocate power jointly to primary and secondary users so that the overall sum throughput is maximized while primary users have minimum QoS constraints in terms of SINR. The objective function of

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