Optimal Power Allocation for Multiple AccessChannels in Cognitive Radio Networks

Lan Zhang†, Yan Xin†, and Ying-Chang Liang‡†Dept. of ECE, National University of Singapore, Singapore 118622

‡ Institute of Infocomm Research, 21 Heng Mui Keng Terrace, Singapore 119613Email: {zhanglan, elexy}@nus.edu.sg, ycliang@i2r.a-star.edu.sg

Abstract— Cognitive radio (CR) has been proposed as astrategy to enhance the spectrum utilization efficiency. In aCR network, the secondary users (SUs) share the same radiospectrum with the primary user (PU) under the constraint thatthe interference from the SUs to PU is below a certain threshold.In this paper, we consider the single-input multiple-output mul-tiple access channels (SIMO-MAC) for the secondary network,and study the optimal power allocation strategy for maximizingthe weighted sum rate of the SIMO-MAC under interferenceconstraints and peak transmit power constraints. Employingdecoupling techniques, we develop an iterative algorithm toobtain optimal power allocation for maximizing weighted sumrate. Simulation results are presented to verify the effectivenessof the proposed algorithm.

I. INTRODUCTION

Recently, cognitive radio (CR) has shown great potentialto dramatically enhance spectrum utilization efficiency, thusattracting much research attention [1], [2], [3], [4]. In a con-current CR, the secondary users (SUs) can simultaneous usethe spectrum licensed to the primary users (PUs) provided thatcertain constraints are satisfied. In this paper, we consider thesingle-input multiple-output multiple access channels (SIMO-MAC) in a concurrent CR network. We study the optimalpower allocation strategy for maximizing the weighted sumrate of the SIMO-MAC under interference power constraintsand peak transmit power constraints.

Most previous studies on the MAC sum rate maximiza-tion problem are under non CR settings [5], [6], [7], [8].Specifically, the work [5] develops an iterative water-fillingalgorithm, which yields the optimal solution of the multipleinput multiple output (MIMO) equally weighted sum ratemaximization problem under an individual power constraint.The MIMO-MAC equally weighted sum rate maximizationproblem subject to the sum power constraint instead of theindividual power constraint was considered in [6]. In [6], thesum power constraint was first decomposed into individualpower constraints, so that the iterative water-filling alogirithmcan be applied. In [7], a weighted sum rate maximizationproblem for the single input multiple output (SIMO) MACsubject to a sum power constraint was solved by applying acyclic coordinate ascent algorithm [8]. However, these afore-mentioned algorithms are not applicable to the CR weightedsum rate maximization problem, which is subject to notonly individual power constraints but also interference powerconstraints. More recently, under the CR setting, a weighted

......

... ......

h1

h2

hK

x1

x2

xK

g1,1

gK,1

g1,N

gK,N

y=Hx+z

PU1

PUN

(SUs)BS

Fig. 1. The system model for SIMO-MAC based cognitive radio networkswith K SUs and N PUs.

sum rate maximization problem was investigated in [9], whereonly suboptimal solutions have been derived. We in thispaper develop an efficient iterative algorithm which obtainsthe globally optimal solution of the CR weighted sum ratemaximization problem for the SIMO MAC.

The rest of the paper is organized as follows. The systemmodel and problem formulation are described in Section II.The decoupled iterative algorithm to solve the optimizationproblem and its convergence property are presented in SectionIII. Simulation results are provided in Section IV. Section Vconcludes this paper. The following notations are used in thispaper. The boldface is used to denote matrices and vectors,(·)H and (·)T denote the conjugate transpose and transposeoperation, respectively, and IM denotes an M × M identitymatrix.

II. CHANNEL MODEL AND PROBLEM FORMULATION

As depicted in Fig. 1, we consider the scenario where K SUtransmitters simultaneously communicate with their BS withNr receivers. Mathematically, the received signals at the BSof the SUs can be expressed as:

y = Hx + z, (1)

where y denotes the Nr × 1 received signal vector, x is theK × 1 transmit signal vector of SUs, H = [h1, · · · ,hK ]denotes the Nr ×K channel matrix with hi being the channelresponse from the ith SU (denoted by SUi) to the BS, and zis the additive white Gaussian noise vector at the secondaryreceiver, i.e., z ∼ CN (0, σ2I).

We next consider the weighted sum rate maximizationproblem where the transmit power of the SUs is subject tothe individual power constraints as well as the interference

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power constraint due to the secondary usage requirement.Mathematically, the problem can be formulated as follows:

Problem 1 (Weighted Rate Maximization):

maxK∑

i=1

wiri, (2)

subject to: pi ≤ P̄i, i = 1, · · · ,K, (3)K∑

k=1

gkpk ≤ Pt, (4)

where wi is the weight of SUi, and ri is the rate of SUi.We assume that pi is the power allocated to SUi, whichis subject to a peak power P̄i. Let gi be the power gainbetween SUi to the PU1, and Pt be the interference constraintfor PU. The interference power received by PU from allSUs is characterized by gT p where g = [g1, · · · , gK ]T andp := [p1, · · · , pK ]T .

Moreover, it is worth mentioning that any boundary pointof the capacity regions of the MAC can be expressed as aweighted rate sum for a certain choice of weights [10]. Thus,by varying the weights of SUs in Problem 1, the entire capacityregion of the CR SIMO-MAC can be obtained.

III. DUAL DECOMPOSITION ALGORITHM

It is well known that the capacity of the MAC is achievedby the successive interference cancelation (SIC) scheme, andthe achievable rate of SUi can be written as:

ri = log|INr

+∑i

j=1 hπjhH

πjpπj

||INr

+∑i−1

j=1 hπjhH

πjpπj

| , (5)

where π = π1, · · · , πK , a permutation of 1, . . . , K, representsa decoding order (πK is decoded first and π1 is decoded last).According to [10], the optimal weighted sum rate of (2) canbe achieved by a decoding order π which sorts the weights innonincreasing order as:

wπ1 ≥ wπ2 ≥ · · · ≥ wπK. (6)

Without loss of generality, we assume πk = k throughoutthe paper. Therefore, the objective function of (2) can betransformed as:

max f(p) =K∑

k=1

∆k log |INr+

k∑j=1

hjhHj pj |, (7)

where ∆k = wk − wk+1, and wK+1 = 0. From Eq. (7), itis easy to observe that Problem 1 is a convex optimizationproblem. It can be solved through standard convex opti-mization packages directly, such as interior-point algorithm.However, the standard interior-point method does not exploitthe special structure of the problem, and the problem may stillbe computationally intensive if the Nr and K are large. In thefollowing, we develop a new method based on the Karush-Kuhn-Tucker (KKT) conditions to solve this problem.

1We first consider a single PU case. The power gain from SUi to the jthPU (PUj ) is denoted by gi,j . The case with multiple PUs is considered inSection III-A.

Problem 1 consists of two types of constraints, the indi-vidual transmit power constraint and the interference powerconstraint, which can be viewed as a weighted sum powerconstraint. The major difficulty in solving Problem 1 is thatall variables pi are coupled in the interference constraint. Onemethod that overcomes the difficulty of the coupled constraintis to use the dual decomposition algorithm [6], which candecompose the sum power constraint into individual powerconstraints. Thus, by applying the dual decomposition method,Problem 1 can be further transformed as follows:

maxp,q

f(p), (8)

subject to pi ≤ P̄i, gipi ≤ qi,K∑

k=1

qk ≤ Pt, i = 1, · · · ,K, (9)

where qi denotes the interference from the SUi to the PU,and is termed as interference allocation. Moreover, we defineq := [q1, · · · , qK ]T . To further simplify the computation, wecan incorporate the constraint

∑Ki=1 qi ≤ Pt into the objective

function by forming the Lagrangian function with respect tothis constraint, and transform the problem into the followingform:

Problem 2 (Equivalent Problem):

minλ≥0

maxp,q

f(p) − λ(K∑

k=1

qk − Pt), (10)

subject to pi ≤ P̄i, (11)

gipi ≤ qi, i = 1, · · · ,K. (12)Problem 2 is a minimax problem, with the optimizationvariables λ,p, and q. In the following lemma, we show thatProblem 1 and Problem 2 have the same optimal solution.

Lemma 1: The optimal solutions of Problem 1 and Problem2 are the same.

Proof: The Lagrangian function of Problem 1 withrespect to the coupled constraint

∑Ki=1 gipi ≤ Pt is

L(p, q, λ) = f(p) − λ(K∑

k=1

qk − Pt). (13)

Moreover, the dual objective function is

g(λ) = maxp,q

L(p, q, λ), (14)

where the constraints of the maximization are pi ≤ P̄i andgipi ≤ qi. Thus, the dual problem of Problem 1 can be writtenas follows:

minλ

g(λ) s.t. λ ≥ 0. (15)

Due to the convexity of Problem 1, the dual problem (15) andProblem 1 achieve the same optimal solution. Since the dualproblem (15) is equivalent to Problem 2, the proof followsimmediately.

Problem 2 consists of two layers of optimization. The innerlayer involves the maximization of L(p, q, λ) over p and q

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for fixed λ as given in (14), whereas the outer one involvesthe minimization g(λ) in (15) over λ.

We start with considering the inner layer. For the fixedvariable λ, Eq. (14) can be considered as a convex maximiza-tion problem over p and q. We can rewrite the inner layeroptimization as follows:

maxp,q

f(p) − λ(K∑

k=1

qk − Pt), (16)

subject to pi ≤ P̄i, (17)

gipi ≤ qi, i = 1, · · · ,K, (18)

where λ is a fixed constant. The two-constraint problem can besolved via an iterative algorithm similar to the iterative waterfilling [5]. In each iteration step, we sequentially computethe optimal power allocation pi and interference allocationqi assuming that pj and qj , where j �= i, are fixed. In theiterative process, the objective value will increase continuouslyuntil the global optimal power allocation is achieved due tothe convexity of the problem. In the following, we first presentthe power allocation update rule in each iteration step.

In each iteration step, we update pi according to the solutionof the following problem

maxpi,qi

f(p) − λ(K∑

k=1

qk − Pt), s.t. pi ≤ P̄i, gipi ≤ qi. (19)

In order to obtain the optimal solution of (19), we firstdecouple this two-constraint optimization problem (19) intotwo single-constraint sub-problems as follows:

Subproblem 1:

maxpi,qi

f(p) − λ(K∑

k=1

qk − Pt), s.t. pi ≤ P̄i, (20)

Subproblem 2:

maxpi,qi

f(p) − λ(K∑

k=1

qk − Pt), s.t. gipi ≤ qi. (21)

In each iteration step, we assume that p(1)i and p

(2)i are the

optimal solutions for Subproblems 1 and 2, respectively. Sincepi is a scalar, only one constraint between (21) and (20) is aneffective constraint. Thus, in each iteration step, we update pi

to the minimum value between p(1)i and p

(2)i .

For Subproblem 1, the optimal solution is p(1)i = P̄i,

obviously. However, for Subproblem 2, the optimal solutioncan not be obtained directly. The Lagrangian function ofSubproblem 2 is given by

L(p, q, ν) = f(p) − λ(K∑

k=1

qk − Pt) − ν(gipi − qi). (22)

Thus, the KKT conditions are obtained by applying ∂L/∂pi =0 and ∂L/∂qi = 0, which are

∂f(p)∂pi

− νgi = 0, (23)

ν = λ, (24)

ν(gipi − qi) = 0, i = 1, · · · ,K. (25)

For each i and j ≥ i, we have:

∂f(p)∂pi

=K∑

j=i

∆jhHj (Qi,j(p) + hih

Hi pi)−1hj , (26)

where Qi,j(p) = INr+

∑jk=1,k �=i hkhH

k pk. Applying thematrix inversion lemma,

(A + cd∗)−1 = A−1 − A−1cd∗A−1

1 + c∗A−1d, (27)

where A is a nonsingular matrix and c and d are vectors withappropriate dimensions, we can rewrite Eq. (26) as:

∂f(p)∂pi

=K∑

j=i

∆j

hHj Qi,j(p)−1hj

1 + pihHj Qi,j(p)−1hj

. (28)

Substituting (24) and (28) into (23), we have

K∑j=i

∆j

hHj Qi,j(p)−1hj

1 + pihHj Qi,j(p)−1hj

= λgi. (29)

Since λ is a constant same for all the SUs, it can be viewedas a generalized water level in the water filling principle. Ifthe SUj’s power is fixed, where j �= i, it is easy to observedthat the value Qi,j(p) is constant. Due to the monotonicallydecreasing property of the left hand side of (29) in pi, thereis a unique p∗i satisfying the (29). A simple bisection canefficiently solve for pi. Moreover, Eq. (29) belongs to a wellknown family of non linear equations called secular equationsfor which there are highly efficient root finding algorithms[11]. After obtaining the optimal solution of p

(1)i and p

(2)i , it is

easy to observe that p∗i = min(p(1)i , p

(2)i ) is the optimal power

solution for the problem (19), while pj is constant, wherej �= i. Consequently, we have the corresponding interferenceallocation q∗i = gip

∗i .

In summary, the problem (16) can be solved by the iterativealgorithm. In each iteration step, pi is cyclicly updated bycomputing the decoupled Subproblems 1 and 2, while keepingpj with j �= i as constant. The iterative process continues untila fixed point achieved. The convergence of the proposed algo-rithm is guaranteed by the monotonically increasing propertyof each step.

We next turn our attention to the outer layer of Problem 2,i.e., seek to find the optimal λ for (15). Since the Lagrangianfunction g(λ) is convex over λ, the optimal λ can be obtainedthrough a one-dimensional search. However, because g(λ) isnot necessarily differentiable, the gradient algorithm cannot beapplied. Alternatively, the subgradient method can be used tofind the optimal solution. In each iteration step, λ is updatedaccording to the subgradient direction.

Lemma 2: The subgradient of g(λ) is Pt −∑K

i=1 qi, whereλ ≥ 0, and qi is the corresponding optimal interferenceallocation for a fixed λ in (14).

Proof: Let s be the subgradient of g(λ̃). For a givenλ̃ ≥ 0, the subgradient s of g(λ̃) satisfies

g(λ′) ≥ g(λ̃) + s(λ′ − λ̃), (30)

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where λ′ is any feasible value. Since

g(λ′) = maxp,q

(f(p) − λ′(

K∑i=1

qi − Pt))

= f(p′) − λ′( K∑

i=1

q′i − Pt

)

≥ f(p̃) − λ′( K∑

i=1

q̃i − Pt

)(31)

= f(p̃)−λ̃( K∑

i=1

q̃i − Pt

)+λ̃

( K∑i=1

q̃i − Pt

)

− λ′( K∑

i=1

q̃i − Pt

)

= g(λ̃) +(Pt −

K∑i=1

q̃i

)(λ′ − λ̃),

where s := Pt−∑K

i=1 q̃i is the subgradient of g(λ̃), p′ and q′iare the optimal variables in (14) for λ = λ′, and p̃ and q̃i arethe optimal variables in (14) for λ = λ̃. The inequality (31) isbecause p′ and q′i are the optimal variables in (14) for λ = λ′.

Lemma 2 indicates that the value of λ should increase, if∑Ki=1 qi > Pt, and vice versa.In the following, we outline the iterative algorithm to solve

Problem 2.Algorithm 1:

1) Initialization: λmin, λmax, p.2) repeat

a) λ = (λmin + λmax)/2, m = 1,b) repeat,

i) for i = 1, · · · ,K,compute p

(2)i according to Eq. (29)

pi(m) = min(P̄i, p(2)i ), qi(m) = gi · pi(m),

ii) m = m + 1,c) until it convergesd) if

∑Kk=1 qk > Pt, then λmin = λ, elseif∑K

k=1 qk < Pt, then λmax = λ,3) until |λmin − λmax| ≤ ε,

where m denotes the mth iteration step, and ε > 0 is aconstant. The following proposition shows the convergenceproperty of Algorithm 1.

Proposition 1: Algorithm 1 converges to an optimal solu-tion of Problem 2.

Proof: Algorithm 1 includes the inner and outer loops.The inner loop is to compute pi qi for i = 1, · · · ,K. In eachiteration step to update pi, we need to fix the other covariancematrices pj , where j �= i. Since the objective function (14) isnondecreasing with each iteration, the algorithm converges toa fixed point. Moreover, due to the convexity of (14), the fixedpoint is the optimal solution. The outer loop is to compute theLagrangian coefficient λ. According to the Lagrangian dualfunction’s convexity [12], there is a unique optimal λ which

achieves the optimal value of (15). Hence, the one dimensionalline bisection search converges to the optimal λ. The prooffollows.

Remark 1: If weights for all SUs are equal, then the objec-tive function (7) can be expressed in the form of determinant,and correspondingly the problem reduces the determinantmaximization problem, which can be solved by using theinterior point method MAXDET [13]. In Section IV, wecompare the MAXDET with our algorithm.

A. The multiple PU case

The algorithm can also be extended to the case with multiplePUs. In that case, the equivalent problem is changed asfollows:

minλ≥0

maxp,q

f(p) −N∑

j=1

λj(K∑

i=1

qi,j − P(j)t ), (32)

subject to pi ≤ P̄i, (33)

gi,jpi ≤ qi,j , i = 1, · · · ,K, j = 1, · · · , N, (34)

where P(j)t is the threshold of PUj , λj is the Lagrangian

coefficient for the interference constraint of PUj , and qi,j

denotes the interference from the SUi to PUj . Since thereare N Lagrangian coefficients, the optimal values of the La-grangian coefficients can not be obtained through the bisectionsearching algorithm. However, the problem can be solvedthrough the subgradient algorithm.Algorithm 2:

1) Initialization: λj(0), j = 1, · · · , N , n = 1,2) repeat

a) Initialization: pi, i = 1, · · · ,K, m = 1,b) repeat,

i) for i = 1, · · · ,KFor decoupled subproblem with a single PUj

interference constraint j = 1, · · · , N , find theoptimal solution p∗i,j according to Eq. (29),pi(m) = min(p∗i,1, · · · , p∗i,N , P̄i), qi(m) = gi ·pi(m),

ii) m = m + 1,

c) Until it converges.d) Update λj(n) through a subgradient algorithm

λj(n + 1) = λj(n) + t(∑K

k=1 qk − Pt),e) n = n + 1,

3) Stop when |λj(∑K

k=1 qk −P(j)t )| < ε for j = 1, · · · , N

are satisfied simultaneously,

where n denotes the nth iteration step of the outer loop, and tdenotes the step size of the subgradient algorithm. It has beenshown in [14] that the subgradient algorithm converges to theoptimal solution.

IV. SIMULATION RESULTS

In this section, we provide several simulation examples toillustrate the effectiveness of the proposed algorithms. Forsimplicity, we assume in our simulation examples that all SUsare located at the same distance, l1, to the BS, and the same

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0 5 10 15 20 25 304

6

8

10

12

14

16

18

Individual Power (dB)

Sum

−ra

te (

bps/

Hz)

Algorithm 1Maxdet algorithm

Fig. 2. Comparison of the achievable sum-rate obtained by Algorithm1 and the MAXDET algorithm with K = 3, and Nr = 3.

distance l(n)2 , to PUn (For the single PU case we omit the

superscript and simply use l2). Suppose that the same pathloss model can be used to describe the transmissions from theSUs to the BS and to the PUs, and the path loss exponentis 4. The elements of matrix H are assumed to be circularlysymmetric complex Gaussian (CSCG) RVs with mean zeroand variance one, and the power gain factor from the SUi tothe PUn can be modelled as gi,n = (l1/l

(n)2 )4|αi,n|2, where

αi,n is also modelled as CSCG with mean zero and varianceone. The noise covariance matrix at BS is assumed to be theidentity matrix, the individual power and interference powerare defined in dB relative to the noise power, and Pt is chosento be 0 dB.

Example 1: In the simulation network, we have K = 3, andNr = 3. The individual power constraint ranges from 2 dB to30 dB. We compare the result obtained by Algorithm 1 and theMAXDET algorithm. In Fig. 2, as can be seen from the figure,the two results obtained by the two algorithms are coincidedwith each other. This is because both algorithms can reach theglobally optimal solution of the optimization problem.

Example 2: In this simulation, we show that the influenceof the PU interference constraint. In the network, we haveK = Nr = 3, w1 = 2, and w2 = w3 = 1. In Fig. 3, weplot the achievable weighted sum rate versus the individualpower constraint in the case with one PU and two PUs. It canbe observed from the figure that the two curves coincide witheach other in the low individual power regime. This is becausein this regime the individual power constraint dominates theachievable rate. While in the high individual power regime,the achievable weighted sum rate of the two PU case is lessthan that of the single PU case. This is because the extra PUinterference constraint reduces the freedom of the transmitters.

V. CONCLUSIONS

We proposed an efficient iterative algorithm to solve theweighted sum rate maximization problem for the CR SIMO-

0 5 10 15 20 25 303

4

5

6

7

8

9

10

11

12

Individual Power (dB)

Wei

ghte

d su

m−

rate

(bp

s/H

z)

two PUs existone PU exists

Fig. 3. Comparison of the achievable weighted sum-rate of the casewith one PU and two PUs.

MAC under individual power constraints and an interferenceconstraint. We showed that the proposed algorithm convergesto the globally optimal solution. Furthermore, we demon-strated that the proposed algorithm can be extended to amultiple PU case.

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