Adaptive Gradient-Based Methods for Adaptive Power Allocation in OFDM-Based Cognitive Radio Networks

836 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 63, NO. 2, FEBRUARY 2014

Adaptive Gradient-Based Methods for AdaptivePower Allocation in OFDM-Based

Cognitive Radio NetworksWei-Chen Pao and Yung-Fang Chen, Member, IEEE

Abstract—A gradient-based method is designed for power al-location in orthogonal-frequency-division-multiplexing (OFDM)-based cognitive radio networks. The resource allocation problemsubject to a mutual interference constraint is considered. We uti-lize the gradient descent approach to allocate power to subcarriersin cognitive radio (CR) networks. The proposed gradient-basedpower allocation method with a well-designed step size can ap-proximate the optimal solution within a few iterations. Due tothe derived equation for power allocation in an adaptive manner,the proposed method is feasible for adaptive power allocationin time-varying channels. The analysis for the selection of thestep size is presented in this paper. For comparison purposes, agreedy power-loading method requiring numerous iterations isalso designed for this power allocation problem. The proposedgradient-based method and the greedy power-loading methodboth have a computational complexity of O(N), but the proposedgradient-based method requires far fewer iterations. As demon-strated in the simulation results, the proposed gradient-basedmethod with the adaptive step size has a fast rate to achievea near-optimal solution within an extremely small number ofiterations and has quite a low computational complexity of O(N).

Index Terms—Cognitive radio (CR), gradient descent, orthogo-nal frequency-division multiplexing (OFDM), power allocation.

I. INTRODUCTION

ACCORDING to the report of the Federal CommunicationsCommission, most of the licensed spectrum is not fully

utilized [1]. Efficient spectrum usage becomes a critical issuewhen a variety of wireless communication systems have comeinto wider use. The concept of cognitive radio (CR) basedon software-defined radio is first presented in [2] to solve theproblem of spectrum scarcity. The spectrum utilization problemcan be solved by allowing secondary users (SUs) to access thespectrum holes [3], which are bands of frequencies assignedto primary users (PUs) but not employed at a particular time.Opportunistic spectrum access would allow SUs to detect and

Manuscript received August 10, 2012; revised December 20, 2012 andMay 19, 2013; accepted June 22, 2013. Date of publication July 11, 2013;date of current version February 12, 2014. This work was supported by theNational Science Council of Taiwan under Contract NSC 101-2221-E-008-069 and Contract NSC 102-2221-E-008-005. The review of this paper wascoordinated by Dr. T. Jiang.

W.-C. Pao is with the Industrial Technology Research Institute, Hsinchu 310,Taiwan (e-mail: [email protected]).

Y.-F. Chen is with the Department of Communication Engineering, NationalCentral University, Taoyuan 320, Taiwan (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2013.2272804

access available spectrum holes. Orthogonal frequency-divisionmultiplexing (OFDM) has been suggested as a competitivetechnique in CR networks. An unlicensed SU is allowed toaccess the spectrum, which is allocated to the licensed PUwithout causing harmful interference to PUs. In [4], the locationawareness technique is presented to identify the transmissionregion where CR users will not cause harmful interference. Theissues in spectrum management and interference cancelation inCR networks are discussed in [5] and [6].

The power allocation problem for OFDM systems has beenstudied in [7]. The optimal solution can be obtained by usingthe waterfilling technique [8] for the capacity maximizationproblem with the power constraint. The waterfilling techniqueis employed for the single-user CR system without mutual inter-ference [9]. However, as SUs access the unoccupied frequencybands by the PUs, mutual interference [10], [11] would affectthe PUs. The problem with the mutual interference constraintin CR networks is different from that with the power constraint.Recently, a problem with a peak power constraint at the sec-ondary transmitter and an average interference constraint at theprimary receiver is presented in [12]. Some greedy algorithmsfor multiuser OFDM-based CR networks are presented in [13]–[17]. Based on the total power constraint, the transmit powerconstraint on the subchannel [18], [19], the total amount ofthe mutual interference [20]–[22], or the mutual interference oneach subchannel of PUs [23] are also studied. In addition, theresource allocation problem for SUs with the constraints on theinterference level of each PU is further considered in multicastnetworks [24].

Unlike the problems considered in [12]–[24], this paperfocuses on the power allocation problem with the mutual in-terference constraint in OFDM-based CR networks, which isintroduced in [25] and [26]. The mutual interference would de-pend on the transmit power, channel conditions, and the spectraldistance between the PU and the SU [10], [11]. There are twomain types of interference in CR networks. One is introducedby PUs into SUs, which leads to a signal-to-noise ratio loss inCR networks. The other one is introduced by SUs into PUs,which should be less than the interference temperature level[3] used to ensure the quality of service for PUs. Several low-complexity schemes have been presented in [25] and [26],but their performance is not near the optimal performance.This motivates us to design a new power allocation method,which can achieve a near-optimal performance and still has lowcomputational complexity. In [27]–[31], Wang et al. develop a

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PAO AND CHEN: GRADIENT-BASED METHODS FOR ADAPTIVE POWER ALLOCATION IN OFDM-BASED CR NETWORKS 837

series of related work for multiuser OFDM-based CR networksin varied problem formulations with different considerationsand constraints. In [27], Wang et al. derive a fast algorithmto tackle the optimal power allocation problem for an SU inOFDM CR networks with multiple interference constraints ofmultiple PUs. In [28], Ge and Wang try to maximize the sumrate of the nonreal-time SUs while guaranteeing the minimalrate requirements for the real-time SUs, with the total trans-mission power budget of the CR system and the interferenceconstraints of the PUs. In [29], a set of proportional rateconstraints to ensure fairness among SUs is further imposed.For more factors involved in the related algorithm design, inparticular, the work in [30] considers resource allocation inOFDM-based CR networks, under the consideration of manypractical limitations, such as imperfect spectrum sensing, lim-ited transmission power, different traffic demands of SUs, etc.Note that [27]–[31] introduce the barrier method for optimalpower allocation. The barrier-based method may be adaptedto deal with the formulated problem in this paper. However,the barrier method is one of standard convex optimizationtechniques, whose performance would approximate the opti-mal solution more accurately if some user-specific parametersare well defined. The barrier method in [27]–[31] utilizes alogarithmic barrier function. The barrier functions may decidethe accuracy of the approximation. It is not straightforwardto determine which barrier functions are suitable in differentproblems. In [27]–[31], the barrier method contains an outerloop for the barrier method and an inner loop for the Newtonmethod, including backtracking line search. Several parametersneed to be defined, such as the parameters for backtracking linesearch, the tolerances of the barrier method, the Newton step,and the parameters for the outer and inner loops. The selectionof parameters would directly determine the accuracy of theperformance and the number of iterations in the outer loop,the inner loop, and backtracking line search. Moreover, a warmstart is also required for the initial step of the barrier methodbecause a strictly feasible starting point is required to obey allthe constraints. A preparatory procedure for the barrier methodto provide and compute the existed feasible point is necessary.The barrier method may not be so straightforward to be utilizedfor different formulated problems with different constraints.

In this paper, we propose a gradient-descent-based methodto solve the allocation problem. Gradient-descent-based ap-proaches are used to find a local minimum of a function,which is employed in many researches [32]–[37], in which thegradient descent approach is used to solve the power allocationproblem for multicarrier systems. It is also utilized for theproblem of cross-layer optimization of multipath routing andpower allocation for ad hoc networks [34]. The gradient descentapproach is applied in multiple-input–multiple-output broad-cast channels [35]–[37] for different objectives. It is guaranteedto converge to a stationary point [38], [39].

Using the gradient-based method to solve the power al-location problem with an interference constraint in OFDM-based CR networks is first developed in this paper. Since theconsidered problem is a constrained optimization, the methodof steepest descent cannot be directly employed. The obtainedsolution has to be in the feasible region. The allocated power

must be positive. The constraint must be satisfied. Therefore,the projected gradient method [40] must be utilized. An orthog-onal projector is defined in the paper. The projection gradientalgorithm is an extension of the unconstrained steepest descentmethod, which is suitable for the optimization problem with aconvex feasible region. The Euclidean projection technique is toperform a projection of the power allocation onto the constraintset. This way, the constraint can be satisfied, and the resourcecan be fully used.

In addition, the main challenge in the gradient-based methodis to find the step size. The selection of the step size determinesthe accuracy of the approximation and the number of iterations.For a small value, the gradient-based method slowly progresses.A better performance may be achieved, but numerous iterationsare required. There is a tradeoff between the performanceand the convergence rate. Therefore, we may tend to find anadaptive method to adjust the step size in each iteration.

The analysis for the selection of the step size is well studiedand designed in this paper. The step size, which is adaptivelydetermined in iterations, is presented without a heuristic se-lection. To further improve the performance, the line searchmethod [8] can be also employed to adjust the value of thestep size, which can have the maximum capacity improvement.Simulation results demonstrate that the proposed gradient-based method with the adaptive step size well approximates theoptimal solution within three iterations on average even if theline search method is not applied. According to the analysis forthe selection of the step size proposed in this paper, the valueof the step size is adaptively and appropriately determined toachieve a good performance.

The proposed gradient-based method is based on the projec-tion gradient algorithm with the Euclidean projection techniquefor the satisfaction of the interference constraint. The proposedmethod has a low computational complexity of O(N). Theproposed method can be employed in the case of multiple PUsand multiple SUs when the interference temperature level toprotect PUs is predetermined for each CR users. Due to thederived equation for power allocation in an adaptive manner,the proposed method is feasible for adaptive power allocationin time-varying channels.

The organization of this paper is as follows: The problemformulation is given in Section II. The proposed gradient-based method is developed in Section III. Section IV showsthe analysis for the selection of the step size and the weightingfactor. The line search method for the selection of the stepsize and the weighting factor is introduced in Section V. Theproposed greedy power-loading method as the optimal solutionis presented in Section VI. Section VII gives the simulationresults for the proposed gradient-based method. Finally, theconclusion of this paper is made in Section VIII.

II. PROBLEM FORMULATION

The spectrum in the frequency domain is distributed to PUsand SUs/CR users. M frequency bands with bandwidth B, inhertz, are sensed by the CR system, as shown in Fig. 1. L bandshave been used by PUs. The unoccupied bands are assigned toCR users, which have N OFDM subcarriers. The bandwidth


Fig. 1. Spectrum distribution of PUs and SUs/CR users in the frequencydomain.

of subcarriers is Δf Hz. We assume that L PU bands areactive while one CR user is transmitting in frequency bands.Our objective is to maximize the capacity of the CR user whilekeeping the interference introduced to the PU bands below theinterference temperature level. The capacity (in bits per secondper hertz) of the CR user is defined as a convex optimizationproblem of the form [25], [26]

Minpn

−CCR =1N

N∑n=1

− log2

(1 + pn |hss

n |2 /(σ2 + J̃n))

=1N

N∑n=1

fn(pn)

subject to pn ≥ 0, for n = 1, 2, . . . , N and

N∑n=1

K̃npn = K1×NpN×1 ≤ Ith (1)

where N denotes the total number of OFDM subcarriers;hssn denotes the channel gain of subcarrier n; pn denotes the

power allocated to subcarrier n; σ2 denotes the noise poweron subcarrier n; J̃n =

∑Ll=1 J

(l)n ; J (l)

n denotes the interferenceintroduced by the lth PU to the nth subcarrier, which is relatedto channel gain hps

l between the lth PU transmitter and the CRuser’s receiver; the PU signal has been taken to be an ellipticallyfiltered white noise process with amplitude PPU [10], [11];Ith denotes the total interference constraint; K̃n =

∑Ll=1 K

(l)n ;

K(l)n pn denotes the interference introduced by the CR users to

PU bands, which is related to channel gain hspl between the CR

user’s transmitter and the lth PU receiver; K is a vector whosenth element is K̃n; the mutual interference between PUs andSUs, i.e., J (l)

n and K(l)n , are introduced in [10] and [11]; and p

is a vector whose nth element is pn.

III. GRADIENT-BASED POWER ALLOCATION METHOD

This paper provides a novel gradient-based method to solvethe power allocation problem with the interference constraint.Because of the interference constraint, the gradient-basedmethod needs to project the gradient vector on the constraintvector to obtain a feasible direction. The related proof for con-vergence can be found in [41]. In the process of performing the

proposed gradient-based method, some subcarriers would beassigned with zero power. It means that these subcarriers shouldnot be considered. To maximize the capacity while satisfyingthe interference constraint, the Euclidean projection operationis performed for the power allocation onto the interferenceconstraint. In the gradient-based method, we know that thestep size should be carefully determined. The step size canbe predetermined or adaptively adjusted in iterations. Differ-ent strategies will be compared in the simulation. The pro-posed gradient-based method contains two main componentsin the framework, i.e., the gradient descent approach and theEuclidean projection.

A. Gradient Descent Approach

The gradient depends on the partial derivative of fn(pn) withrespect to pn. The gradient is given as

∇fn(pn) =∂fn(pn)

∂pn=

−1ln 2

· |hssn |2 /(σ2 + J̃n)

1 + pn |hssn |2 /(σ2 + J̃n)

.

(2)

The gradient with respect to the power vector is

∇F(p) =

[∂f1(p1)

∂p1

∂f2(p2)

∂p2. . .

∂fN (pN )

∂pN

]T(3)

where p = [p1 p2 . . . pN ]T . Each element is evaluated by usingthe current power allocated to that selected subcarrier. To satisfythe interference constraint in (1), the specific structure of theconstraint set allows us to compute the projection operatorusing the orthogonal projector [40]. The orthogonal projectionof any vector onto the null space of K involves multiplicationby the following matrix J as an orthogonal projector:

J = IN×N −KT (KKT )−1K (4)

where IN×N is the identity matrix. Power vector p is updatedby searching in direction ∇F(p) and premultiplied by orthog-onal projector matrix J, i.e.,

p̂(t+ 1) = p(t)− αJ(t)∇F (p(t))√(J(t)∇F (p(t)))T (J(t)∇F (p(t)))

(5)

where t is the iteration index, α is the positive step size, andΦ/

√ΦTΦ is a unity directional vector to cancel out the factor

in ∇F(p). Note that in case some subcarriers may be assignednegative p̂n(t+ 1) during an iteration, we do not allocate powerto these subcarriers thereafter. For those N+ subcarriers withnonnegative power assignment, the power allocation is reeval-uated by excluding these N −N+ subcarriers in the iteration.We initially allocate power for pn to each subcarrier by utilizingthe equal interference method [26], i.e.,

pn = Ith/NK̃n. (6)

After step size α is appropriately set, we can updatepower p̂n(t+ 1) by (5). The interference caused by allocatedpower p̂n(t+ 1) can be calculated according to interference


Fig. 2. Flowchart of the proposed gradient-based power allocation method.

calculation K̃np̂n. Referring to Fig. 2, if the sum of the causedinterference is less than or equal to interference constraint Ith,the next step is to evaluate the achievable capacity improve-ment, i.e., ΔC = CCR(t+ 1)− CCR(t). On the contrary, ifthe sum of the caused interference K̃np̂n exceeds interferenceconstraint Ith, the allocation power p̂n(t+ 1) for those N+

subcarriers with nonnegative power assignment should be re-allocated to satisfy the interference constraint. In the proposedmethod, the Euclidean projection technique is used.

B. Euclidean Projection

If the sum of the caused interference K̃np̂n exceeds inter-ference constraint Ith, the Euclidean projection technique isused to perform a projection of the power allocation onto theconstraint set. Therefore, the interference constraint can besatisfied. Only those N+ subcarriers with nonnegative powerassignment are selected to perform the Euclidean projectionon a polyhedron [8]. The Euclidean projection of p̂n(t+ 1) onhyperplane H = {pn|K1×N+pN+×1 = Ith} is given by

pn(t+ 1) = p̂n(t+ 1)− K̃n · |K1×N+ p̂(t+ 1)N+×1 − Ith|∑N+

n=1 |K̃n|2.

(7)

Projected power pn(t+ 1) is obtained and shown to satisfythe interference constraint. To judiciously adjust the allocatedpower to obtain an improved performance, we introduce theweighting function composed of projected power pn(t+ 1) and

the allocated power in the previous iteration pn(t) to updatepower in this paper. The weighting function is expressed as

pn(t+ 1) = (1 − δ)pn(t) + δpn(t+ 1) (8)

where δ ∈ (0, 1) is a weighting factor. Interference constraintIth is guaranteed to be satisfied for updated power pn(t+ 1).After weighting factor δ is appropriately set, the achievablecapacity improvement ΔC can be calculated. If the achiev-able capacity is improved, the proposed gradient-based methodcontinues to update the allocated power. Referring to Fig. 2,gradient (3) and orthogonal projector (4) for the next iterationis updated according to updated power pn(t+ 1) for N+

subcarriers.The procedure of the proposed gradient-based power allo-

cation method is shown in Fig. 2. Referring to Fig. 2, t is theiteration index. The capacity of the CR user in (1) is utilizedfor the stop criterion. The process of the iteration terminatesif the capacity improvement ΔC becomes negative, whichmeans the capacity starts to decrease in the iteration. Thegoal of the proposed gradient-based power allocation methodtends to find the amount of power allocated to each subcarrierdirectly. The computational complexity of the gradient-basedmethod is O(N) in each iteration, where the matrix-relatedoperations can be manipulated to achieve this complexity.

Intuitively, a simple strategy for determining the step sizeand the weighting factor is to give fixed values in advance forthe gradient-based method. While considering the performanceand the number of iterations required in the proposed method,the value of the step size and the weighting factor should beappropriately determined. The choices of the step size and theweighting factor would affect the performance. We will givethe performance comparison for the proposed method withthe different fixed step size and the different fixed weightingfactor in the simulation. However, the goal of the proposedgradient-based power allocation method is designed to obtainan improved performance in iterations. A proper strategy forthe step size and the weighting factor should be developed withthe performance improvement. Therefore, a better approach isthat the step size and the weighting factor should be adaptivelyadjusted for obtaining a better performance in each iteration.The analysis for the selection of the step size and the weightingfactor is presented in the following section. The analysis isderived based on the performance improvement under consid-eration therein.

IV. ANALYSIS FOR THE SELECTION OF THE STEP SIZE

AND THE WEIGHTING FACTOR

The choices of the step size and the weighting factor de-termine the behavior of the proposed gradient-based powerallocation method. For a small value, the gradient-based methodslowly progresses. A better performance may be achieved, butnumerous iterations are required. If a large value is selected, afast rate is revealed to achieve a solution, but the performancemay not be near the optimal performance. The selection of thestep size and the weighting factor becomes important. Here,


the analysis for the selection of the step size and the weightingfactor is presented.

A. Selection of the Step Size

For the analysis, power vector (5) is rewritten as

p̂(t+ 1) = p(t)− αX(t) (9)

where

X(t)=J(t)∇F (p(t))

/√(J(t)∇F (p(t)))T (J(t)∇F (p(t))) .

Power update (5) for subcarrier n is rewritten as

p̂n(t+ 1) = pn(t)− αXn(t). (10)

Regarding the selection of α, the main criterion is to make thevalue as large as possible but not to make the allocated powerof p̂n(t+ 1) become nonpositive. This way, the proposedgradient-based method can achieve a solution in a fast rate.

Step size α is selected to meet the condition, i.e., p̂n(t+ 1) ≥0, ∀n = 1 to N . We assume that one step size αn is consideredin one subcarrier. Note that there is only one value of thestep size employed in the proposed gradient-based method.Therefore, we have selected one step size α for all subcarriersamong all values of αn. For allocating nonnegative power to allsubcarriers, step size αn on a per-subcarrier basis is given as

pn(t)− αnXn(t) ≥ 0 ⇒ αn = pn(t)/Xn(t). (11)

The step size is defined to be greater than zero; the conditionof pn(t)/Xn(t) < 0 always makes the updated power greaterthan zero if Xn(t) < 0. This case is not considered for thecondition of the step size under development. Moreover, thereis no power reallocation of (9) and (10) in subcarrier n ifXn(t) = 0, such that the case is not required for considerationin setting the step size. Hence, the largest value of step size α isdefined as

α′ =Min{αn,where n ∈ S1}=Min {pn(t)/Xn(t),where n ∈ S1} (12)

where set S1 = {n : Xn(t) > 0}. The choice of the step size tomake all components nonnegative should be within the range

0 ≤ α ≤ α′ = Min {pn(t)/Xn(t),where n ∈ S1} . (13)

With this manner, the notation α in (5) is more appropriateto be modified as α(t). However, we prefer to find a range thatmay enhance the capacity in iterations. The capacity differencein consecutive iterations can be written as

CCR(t+ 1)− CCR(t)

=

N∑n=1

log2 (1+p̂n(t+1) ·Hn)−N∑

n=1

log2 (1 + pn(t) ·Hn)

= log2

(N∏

n=1

(1+p̂n(t+1)·Hn) /N∏

n=1

(1+pn(t) ·Hn)

)

(14)

where Hn = |hssn |2/(σ2 + J̃n). Each term of (14) related to

each subcarrier n can be given as

(1 + p̂n(t+ 1) ·Hn) / (1 + pn(t) ·Hn)

= (1 + (pn(t)− αXn(t)) ·Hn) / (1 + pn(t) ·Hn)

= 1 −(αXn(t)/

(pn(t) +H−1

n

)). (15)

Equation (14) can be rewritten as

CCR(t+1)−CCR(t)=log2

N∏n=1

(1−αXn(t)/

(pn(t)+H−1

n

)).

(16)

By applying (16) recursively, we have

C(t+ 1)=C(1) + log2 ΠNn=1

(1 − αXn(1)/

(pn(1) +H−1

n

))+ · · ·+ log2 Π

Nn=1

(1 − αXn(t)/

(pn(t) +H−1

n

)). (17)

Our goal is to find a step size that would improve theperformance for each iteration. We may utilize the inequalityregarding (17) for the selection of the step size to increase thecapacity for each iteration, i.e.,

N∏n=1

(1 − αXn(t)/

(pn(t) +H−1

n

))> 1. (18)

For this purpose, the individual term of (18) can be consideredin the following two cases.

Case I: All terms of 1 − αXn(t)/(pn(t) +H−1n ) are more

than 1. That is

1 − αXn(t)/(pn(t) +H−1

n

)> 1. (19)

After the manipulation, we have

αXn(t) < 0. (20)

For the feasible solution, α is known as

α < 0, if Xn(t) > 0 (21)

α > 0, if Xn(t) < 0. (22)

Equation (21) contradicts the definition of the step size, whichis a positive value. For those subcarriers belonging to the setof Xn(t) > 0 (21) to obtain (19), this case is not consideredfor the condition of the step size under development. Thestep size would be selected as any positive value accordingto (22). It is impossible to find such α for every n to satisfy1 − αXn(t)/(pn(t) +H−1

n ) > 1 because it is equivalent toincreasing each individual capacity at each subcarrier.

Case II: Some terms of 1 − αXn(t)/(pn(t) +H−1n ) are

more than 1, as discussed in Case I. Some terms of 1 −αXn(t)/(pn(t) +H−1

n ) are more than zero because of the logoperation. For a feasible solution of α, we have the inequality

1 − αXn(t)/(pn(t) +H−1

n

)> 0 ⇒ pn(t) +H−1

n > αXn(t).(23)


No power reallocation (10) is operated to subcarrier n ifXn(t) = 0. For the feasible solution, α should satisfy(

pn(t) +H−1n

)/Xn(t) > α > 0, if Xn(t) > 0 (24)

α >(pn(t) +H−1

n

)/Xn(t), if Xn(t) < 0. (25)

Generally, as observed from (10), the value of Xn(t) is takenas the indicator to increase or decrease the power assignmentat subcarrier n. (pn(t) +H−1

n )/Xn(t) < 0 (25) is always sat-isfied because step size α is preset to be greater than zero. Stepsize α in (24) should be given as

α∗ = Min{(

pn(t) +H−1n

)/Xn(t), where n ∈ S1

}. (26)

Equation (26) can be utilized in the selection of step size α.Based on Cases I and II, we know that step size α can be

selected within the range

0 ≤ α ≤ α∗=Min{(

pn(t)+H−1n

)/Xn(t), where n∈S1

}.

(27)

The range of the step size is not guaranteed to enhance thecapacity in every iteration. However, the performance may beenhanced if the step size is selected within the range. Simula-tion results will show the outperformance of the gradient-basedmethod with the adaptive step size. Finally, α′ (12) and α∗ (26)are combined to form the range of the step size for the selec-tion, i.e.,

0 ≤ α(t) ≤ α′ < α∗ (28)

where α′ = Minn∈S1{pn(t)/Xn(t)} in (12) and α∗ =

Minn∈S1{(pn(t) +H−1

n )/Xn(t)} in (26). If the proposedgradient-based method is performed with the version of theevaluation α(t) in (28) for each iteration, the evaluation, whichincludes Min{•} and summation operations, requires anadditional complexity of O(N) in each iteration. The totalcomplexity is still O(N). On the other hand, a fixed α could beemployed in all iterations, but the value may be set accordingto the experiences of experimental results.

B. Selection of the Weighting Factor

Referring to Fig. 2, for the design of the proposed gradient-based method, the allocated power multiplied by the inter-

ference factor should be equal to interference constraint (1).Otherwise, the Euclidean projection operation is performed. Byputting (7) and (10) into (8) for the analysis, updated power(8) is rewritten as (29), shown at the bottom of the page,where |K1×N+ p̂(t+ 1)N+×1 − Ith| means the difference inthe interference constraint. Note that α and Xn(t) have beendetermined and calculated in the process of the selection of thestep size. Therefore, updated power (8) according to (29) is

pn(t+ 1) = pn(t)− δYn(t) (30)

where Yn(t) is a term for subcarrier n, being calculated withoutunknown values of the parameters here. The analysis for theweighting factor is made in the following, which is similar tothat of the step size in the previous section. The goal of theselection of δ is to make it as large as possible but not to makeallocated power pn(t+ 1) become nonpositive. Weighting fac-tor δn for each subcarrier is given as

pn(t)− δnYn(t) ≥ 0 ⇒ δn = pn(t)/Yn(t). (31)

The largest value of weighting factor δ is defined as

δ′ =Min{δn, where n ∈ S2}=Min {pn(t)/Yn(t), where n ∈ S2} (32)

where set S2 = {n : Yn(t) > 0}. On the other hand, one wouldfind a range that may enhance the capacity in consecutiveiterations based on applying the capacity difference recursively.The weighting factor is as

δ∗ = Min{(

pn(t) +H−1n

)/Yn(t), where n ∈ S2

}. (33)

δ′ in (32) and δ∗ in (33) are combined to form the selectionrange of the weighting factor

0 ≤ δ(t) ≤ δ′ < δ∗ (34)

where δ′ = Minn∈S2{pn(t)/Yn(t)} in (32), and δ∗ =

Minn∈S2{(pn(t) +H−1

n )/Yn(t)} in (33). Note that weightingfactor δ is defined to be between 0 and 1. If the value is outof the range, the weighting factor is adjusted by multiplyinga factor. The evaluation requires an additional complexity ofO(N) in each iteration. Similarly, a fixed δ can be employedand set according to the experiences of experimental results.

pn(t+ 1) = (1 − δ)pn(t) + δpn(t+ 1)

= (1 − δ)pn(t) + δ

(p̂n(t+ 1)− K̃n · |K1×N+ p̂(t+ 1)N+×1 − Ith|∑N+

n=1 |K̃n|2

)

=(1 − δ)pn(t) + δ

(pn(t)− αXn(t)−

K̃n · |K1×N+ p̂(t+ 1)N+×1 − Ith|∑N+

n=1 |K̃n|2

)

= pn(t)− δ

(αXn(t) +

K̃n · |K1×N+ p̂(t+ 1)N+×1 − Ith|∑N+

n=1 |K̃n|2

)(29)


Simulation results will reveal the effect of the gradient-based method with the adaptive values of the step size andthe weighting factor. The proposed gradient-based powerallocation method with the adaptive step size and the adaptiveweighting factor achieves near-optimal solution with anextremely small number of iterations.

In this paper, the line search method [8] can be additionallyemployed for the proposed scheme in both the fixed and theadaptive modes to determine the step size and the weightingfactor by truly calculating and comparing the achievable capac-ity. The line search method can be viewed as a performance im-provement approach and is discussed in the following section.

V. LINE SEARCH METHOD FOR THE SELECTION OF THE

STEP SIZE AND THE WEIGHTING FACTOR

Generally, the suitable step size and the suitable weightingfactor may not easily be determined. In addition to employingfixed values of the step size and the weighting factor in theproposed method, we also use the line search method [8] todynamically adjust the step size and the weighting factor forthe proposed method. The performance of the method withthe fixed values of the step size and the weighting factor willbe compared with that of the method of the step size and theweighting factor determined with the help of the proposed linesearch method in the simulation.

The concept of the line search method is to calculate thecost of the optimization problem by varying variable σ. Its goaltends to find variable σ, which has the maximized value of afunction, i.e.,

σ = argmaxσ≥0

f(σ). (35)

Based on this concept, the proposed line search method forthe step size and the weighting factor is developed as follows.Step size α and weighting factor δ are defined as

α = σmA and δ = σm

B (36)

where σA and σB are two parameters that are needed to be setin advance; m in the power is a positive value. Two parameters,i.e., σA and σB , can be fixed or adaptively determined initerations, as proposed in Section IV. In addition to the fixedσA and σB values, the values determined by (28) and (34) canalso be parameters σA and σB , respectively, if the line searchmethod is applied to the proposed gradient-based method withthe adaptive step size and the adaptive weighting factor. Theproposed line search method is presented as follows.

Step 1: Initialization: σA and σB are two fixed positive param-eters in the beginning.

Step 2: Step size α and weighting factor δ are obtained by vary-ing m in a range. m ∈ {1, 2, . . . ,mmax} (e.g., mmax = 4).Correspondingly, power vector p̂(t+ 1) (5) and updatedpower pn(t+ 1) (8) can be calculated when step size αand weighting factor δ are set according to definition (36).The achievable capacity (1) with the allocated power, i.e.,(5) and (8), can be calculated.

Step 3: After that, find a value m in (36) to determine step sizeα, weighting factor δ, and power (5) and (8), which resultsin the maximum capacity, respectively.

In Step 2, the updated power for some subcarriers could benonpositive. According to the design of the proposed gradient-based method, those subcarriers with nonpositive power are notconsidered while evaluating the capacity. If the sum of the inter-ference caused by those N+ subcarriers with allocated positivepower exceeds interference constraint Ith, the scaling operationis performed to satisfy the interference constraint. The scaledpower is obtained according to the following equation:

p̂n+(t+1)=Ith · K̃n+ p̂n+(t+1)

/K̃n+

N+∑n+=1

K̃n+ p̂n+(t+1).

(37)

The scaling operation is only applied at finding the step size inthe line search method when calculating the achievable capacityin Step 2.

The computation complexity of the gradient-based methodwith the line search method is O(Nmmax) in each iteration.If mmax is a preset fixed constant, the complexity can beexpressed as O(N). Simulation results reveal that the proposedgradient-based method with the line search may have a goodperformance, but the number of iterations is increased. If thecomplexity is an issue under consideration, the solution of theproposed gradient-based method without the line search hasbeen be pretty close to the optimal solution when the step sizeand the weighting factor are adaptively determined by (28)and (34).

VI. MODIFIED GREEDY POWER-LOADING METHOD

Recently, the barrier method in obtaining optimal powerallocation for the resource allocation problem in CR networkshas been investigated in [27]–[31]. The barrier method is oneof standard convex optimization techniques whose performancewould approximate the optimal solution more accurately ifuser-specific parameters are well defined. User-specific param-eters may include those for the barrier method, the Newtonmethod, and backtracking line search. The tolerance of thebarrier method and the Newton method is also required. Thereis a tradeoff between the accuracy of performance and thenumber of iterations. Moreover, it may need a strictly feasiblestarting point to obey all the constraints in the initial stepof the barrier method. The computational complexity may beincreased because of the computation of the inverse of theHessian matrix if fast algorithms are not feasible for differentformulated problems. The barrier functions may also determinethe accuracy of the approximation. In addition to the Newtonmethod, the greedy-based method can be also utilized to obtainthe optimal solution [42]–[44] as the role of performance com-parison. The difference for the problem under considerationis that an interference constraint is imposed. To make theiterative procedure suitable for the optimization problem underconsideration in this paper, the greedy power-loading method ismodified and proposed for the problem in the following.


Assume that each subcarrier has assigned power pn associ-ated with its capacity gn, where there is a constraint of In, i.e.,

pn = In/K̃n (38)

gn(pn) = − fn(pn)=log2

(1+pn |hss

n |2/(σ2+J̃n)

)(39)

where In denotes the interference that subcarrier n produces.

Step 1: The initial capacity for subcarrier n: gn = 0, ∀n = 1to N and the initial amount of the produced interferencefor each subcarrier: In = 0 ∀n = 1 to N . Interferenceconstraint Ith is divided into a small amount

ΔIth = Ith/D (40)

where D is the number of the small amount.Step 2: Consider the amount of ΔIth for “loading” at a time

I ′n = In +ΔIth, ∀n = 1 to N. (41)

The corresponding allocated power and the capacity arecalculated by (38) and (39). If the amount of ΔIth causedby the nth subcarrier has the maximum capacity improve-ment, it is expressed as

Δn = gn (p′n)− gn(pn). (42)

The capacity improvement Δn for each subcarrier is thedifference between previous capacity gn(pn) and currentcapacity gn(p

′n) by loading one additional amount of ΔIth

in subcarrier n. After the nth subcarrier with the largestcapacity improvement Δn is selected, some relevant pa-rameters are updated for all subcarriers. The amount ofthe interference constraint is reduced to Ith = Ith −ΔIth.The amount of interference caused by the nth subcarrieris increased to In = In +ΔIth. The allocated power andthe capacity for the nth subcarrier is calculated by (38)and (39).

Step 3: Repeat Step 2 until the sum of all the amount ofthe interference introduced by all subcarriers is equal tointerference constraint Ith. The total achievable capacity(1) is obtained.

Its performance can be much close to that of the optimumsolution as the number of D is exceedingly large. The solutionof this method is taken as an optimal solution for comparisonin the simulation by using a pretty large value D. The compu-tational complexity is O(N) in each iteration, where the logoperation in (1) is required.

The most recent suboptimal schemes are developed in [26].The equal interference scheme, which is the best suboptimalscheme in [26], allocates power to subcarriers to make allsubcarriers have the same interference constraint. It has alow complexity of O(1) but cannot provide much competitiveperformance because it assumes that all subcarriers should beallocated with power. Its performance is not close to the op-timal performance. The performance of the equal interferencescheme is compared in the simulation.

Fig. 3. Capacity difference compared with the optimal solution for thegradient-based methods with a heuristic selection of the step size, the weightingfactor, and the parameters.

Fig. 4. Behavior in capacity with the gradient-based methods by varying thenumber of iterations. Interference threshold Ith = 0.005W .

VII. SIMULATION RESULTS

Assume that there are six bands with a bandwidth of180 kHz. Two bands unoccupied by PUs are assigned to CRusers. Each band contains 12 subcarriers with a bandwidth of15 kHz, as shown in Fig. 1. The value of Ts, σ2, and PPU are83. 33 μs, 0.01 W, and 10 mW in the simulations. D is 10 000for the proposed greedy loading-based method. Channel gainshssn , hsp

l , and hpsl are assumed to be Rayleigh distribution with

an average channel power of 1. The simulation results shown inFigs. 3–9 are the average values from 1000 trial runs.

The first simulation is conducted for the proposed gradient-based method with a fixed step size α in (5) and a fixedweighting factor δ in (8). The step sizes α are 0.2, 0.5, and0.7. The weighting factors δ are also set to be equal to 0.2, 0.5,and 0.7. Note that the step size and the weighting factor in thissimulation are set according to the experiences of experimentalresults. It may need a lot of experiments to find the suitable


Fig. 5. Capacity difference compared with the optimal solution for thegradient-based methods with an adaptive selection of the step size and theweighting factor.

Fig. 6. Behavior in capacity with the gradient-based methods by varying thenumber of iterations. Interference threshold Ith = 0.005W .

Fig. 7. Capacity comparison by varying the interference temperature levelconstraints among various methods.

Fig. 8. Behavior of the step size with the gradient-based method for a giveninterference threshold Ith = 0.005W .

Fig. 9. Behavior of the gradient with the gradient-based method for a giveninterference threshold Ith = 0.005W .

values in the simulation. The values may be various with differ-ent simulation parameters. Fig. 3 shows the capacity differencecompared with the greedy method, which is taken as an optimalsolution for comparison in the simulation. Performances of theproposed methods with a fixed step size and a fixed weightingfactor are not near the optimal performance. From the result,the proposed method with a smaller step size and a smallerweighting factor (α = 0.2, δ = 0.2) has a better performancecompared with the methods with a larger step size and alarger weighting factor, i.e., (α = 0.5, δ = 0.5) and (α = 0.7,δ = 0.7).

The illustration of the behavior of the proposed methods witha fixed step size and a fixed weighting factor is presented inFig. 4 in terms of the number of iterations when interferenceconstraint Ith is 0.005 W. The optimal solution is also com-pared, which does not vary with the increase in the number ofiterations. For convenience of comparison, a horizontal curveis plotted. The proposed methods with a fixed step size and a


TABLE IPERFORMANCE COMPARISON

fixed weighting factor may achieve a local maximum in a fastrate. As revealed in Table I, the average number of iterationscan be less than 5 when proper values of the step size and theweighting factor are employed. However, their performancesare not near the optimal performance. In general, the gradient-based method with a fixed step size and a fixed weightingfactor can only provide fair performances. The performancemay be close to the optimal performance if quite small valuesof the step size and the weighting factor are given. However,numerous iterations are needed. The values of the step sizeand the weighting factor may be difficult to determine whileconsidering the performance and the number of required itera-tions. Owing to this drawback, the proposed line search methodis employed to adjust the values dynamically in the followingsimulation.

The performance of the method with the fixed parameters,i.e., σA and σB , is also shown in Figs. 3 and 4 by applyingthe line search method. Parameter σA in (36) is 0.2, 0.5, and0.7, respectively. Parameter σB is also set to be 0.2, 0.5, and0.7. Compared with the performance of the method without theline search method shown in Fig. 3, the performance of thatwith the line search method is greatly improved and close to theoptimal performance. With resorting to the line search method,the method with small parameters (σA = 0.2, σB = 0.2) hasthe best performance. However, the method with the line searchmethod shown in Fig. 4 converges pretty slowly. Its averagednumber of iterations is from 500 to 1000 when small valuesof the parameters (σA = 0.2, σB = 0.2) are employed. Theaveraged numbers of iterations for the methods with the linesearch method are compared in Table I. The method with largerparameters (σA = 0.7, σB = 0.7) achieves a solution in a fastrate but has a degraded performance.

From the result, it indicates that the line search method forthe selection of the step size and the weighting factor improvesthe performance but increases the number of iterations toachieve a solution. The parameters, i.e., σA and σB , wouldbe set according to the experiences of experimental results.The next simulation shows the effect of the proposed methodwith the adaptive step size and the adaptive weighting factorproposed in Section IV.

Fig. 5 shows the performance of the gradient-based methodwith the proposed adaptive method in setting the step size andthe weighting factor in the iteration. The upper limits of the stepsize in (28) and the weighting factor in (34) are set for theirvalues in the proposed method, i.e.,

α(t) =Min{(

pn(t) +H−1n

)/Xn(t)

}(43)

δ(t) =Min{(

pn(t) +H−1n

)/Yn(t)

}. (44)

Their values (43) and (44) are adaptively adjusted in it-erations. The performance of the method with the adaptivestep size and the adaptive weighting factor is near the optimalperformance in Fig. 5. Fig. 6 is the illustration of the behaviorof the method with the adaptive step size and the adaptiveweighting factor for a given interference threshold, i.e., Ith =0.005W . The optimal solution is also compared, but it doesnot vary with the increase in the number of iterations. Forconvenience of comparison, a horizontal curve is plotted. Itachieves a solution within three iterations on average, as shownin Table I. Owing to the adaptive selection of the step size (43)and the weighting factor (44), the method with the adaptive stepsize and the adaptive weighting factor outperforms the methodsthat have heuristic selection of the fixed step size and the fixedweighting factor.

Fig. 7 shows the capacity performance by varying the in-terference introduced to the PU bands. The equal interfer-ence scheme and the nulling mechanism [26] are compared.Compared with the equal interference scheme, the capacityimprovement for the method with the adaptive selection of thestep size and the weighting factor can be observed. It achievesmore than 98.8% of the optimal solution in a fast rate. Theproposed method with the adaptive step size and the adaptiveweighting factor achieves a solution quickly and performs well.

In Fig. 7, the line search method is also applied for themethod with adaptive parameters σA and σB . The parametersare adaptively determined by taking (43) and (44) as parametersσA and σB , respectively. This method also well approximatesthe optimal solution, and its performance is even closer to theoptimal performance while the averaged number of iterations isless than 200. Table I shows the capacity performance relativeto the optimal solution and the averaged number of iterations toachieve a solution. Obviously, even if the line search method isnot utilized to improve the performance, the proposed gradient-based method with the adaptive selection of the step size andthe weighting factor still provides a near-optimal performancewithin three iterations on average and has a pretty low compu-tational complexity of O(N).

The next simulation for the method with the adaptive param-eters and the line search method is conducted for the behaviorof the step size. Fig. 8 shows the scaled value of the stepsize by averaging from 500 trial runs with a given interferencethreshold, i.e., Ith = 0.005W . Based on the observation in theoptimal solution, only some subcarriers are selected to be useddue to the interference constraint. It means that some subcar-riers should not be selected to be considered in iterations forthe proposed gradient-based method. The issue in determining


Fig. 10. (a) Capacity gap between the optimal solution and the proposed gradient-based method for spectrum distribution Scenario I. (b) Capacity gap betweenthe optimal solution and the proposed gradient-based method for spectrum distribution Scenario II. (c) Capacity gap between the optimal solution and the proposedgradient-based method for spectrum distribution Scenario III. (d) Capacity gap between the optimal solution and the proposed gradient-based method for spectrumdistribution Scenario IV.

the value of the step size related to the number of selectedsubcarriers with allocating power is discussed in the following.

In view of the proposed range of the step size (28) derivedin Section IV, if the step size is chosen between α′ and α∗,it means that some subcarriers are allocated with nonposi-tive power. In the procedure of the proposed method, thosesubcarriers are not considered thereafter. The number of theselected subcarrier with allocated power is changed. Referringto Fig. 8, a larger value would be determined in the beginning sothat subcarriers with positive power are selected. If subcarriersselected to be allocated with power are not changed after afew iterations, the value of the step size should be decreasedto less than α′ (28). The scheme adjusts the amount of powerallocated to these subcarriers such that it would achieve theoptimal performance. As revealed in Fig. 8, the scaled valueof the step size decreases as the number of iterations increases.Based on the given discussion and the behavior shown in Fig. 8,the step size should be adaptively set, instead of a fixed value.The analysis for the selection of the step size (28) is developedto determine its value adaptively. The performance is enhanced

after iterations. According to [45], the norm of the gradient (3)is getting smaller if the current solution is sufficiently closeto a minimum true point. The corresponding behavior of thegradient varying with iterations is shown in Fig. 9. The gradientvanishes at a minimum point. The proposed method actuallyapproaches the optimal solution after iterations. Referring toFig. 7, the method with adaptive step size α and adaptiveweighting factor δ provides a good performance in an extremelysmall number of iterations and just has a computational com-plexity of O(N).

We also set up the COST 207 propagation models [46]to evaluate the proposed gradient-based method along withusing the modified Jakes’ model [47]. Typical cases for rural,hilly terrain, typical urban, and bad urban areas are tested andcompared with Rayleigh distribution, as adopted in Figs. 3–9.Simulation results are shown in Fig. 10(a)–(d) for four spectrumdistribution scenarios. Note that the bands assigned to the CRuser are fixed in the previous simulations as one specific casewith random generated channel gains. In general, availablespectrum holes/bands for the CR user are not fixed because the


preference on the frequency domains of PU varies in time. Toconfirm the superiority of the proposed gradient-based method,we consider a more general situation in the next simulation.After assigning frequency bands to the PU, available bandsfor the CR user are assumed to be varied in time. The patternof available bands for the CR user is randomly generated inthe simulation. Fig. 10(a) is Scenario I with six bands, wheretwo bands are assigned to CR users, and each band contains12 subcarriers. Fig. 10(b) is Scenario II with 15 bands, wherefive bands are assigned to CR users, and each band contains12 subcarriers. Fig. 10(c) is Scenario III with six bands, wheretwo bands are assigned to CR users, and each band contains24 subcarriers. Fig. 10(d) is Scenario IV with 15 bands, wherefive bands are assigned to CR users, and each band contains 24subcarriers. The capacity gap denotes the difference betweenthe performances of the optimal solution and the proposedgradient-based method with the adaptive selection of the stepsize and the weighting factor. It reveals that the capacity gapis about 0.01 bits/s/Hz for different COST 207 typical repre-sentative cases and spectrum distribution scenarios. It meansthat the proposed gradient-based method can approximate theoptimal solution accurately. Therefore, the proposed gradient-based method would be applicable for all scenarios, and the gapwill be small if the developed selection of the step size is used,which is verified in the simulations. A similar performancetrend can be observed for other different settings.

VIII. CONCLUSION

The power allocation problem with the mutual interferenceconstraint in OFDM-based CR networks is resolved by us-ing the proposed gradient-based method with the Euclideanprojection technique and the method of the adaptive selectionfor the step size and the weighting factor, which has beenproposed. The effect of the selection of the step size and theweighting factor has been analyzed in this paper. The valuesof the step size and the weighting factor should be adaptivelyset in iterations to achieve a near-optimal solution in a fastrate. From the result, the performance of the proposed gradient-based method with the adaptive step size and the adaptiveweighting factor is pretty near the optimal performance, whichis obtained by the proposed modified greedy loading method.With the help of the adaptive selection scheme for the step sizeand the weighting factor, the proposed gradient-based methodwith a low complexity of O(N) achieves a good performancein a quite small number of iterations. The proposed line-search-based method can be additionally applied to improve theperformance even closer to that of the optimal solution.

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Wei-Chen Pao received the B.S. degree in electri-cal engineering from Yuan Ze University, Taoyuan,Taiwan, in 2004; the M.S. degree in electronic engi-neering from National Yunlin University of Scienceand Technology, Yunlin, Taiwan, in 2006; and thePh.D. degree in communication engineering fromNational Central University, Taoyuan, in 2011.

He is currently with the Industrial TechnologyResearch Institute, Hsinchu, Taiwan. His current re-search interest includes resource management algo-rithm designs for wireless communication systems.

Yung-Fang Chen (S’95–M’98) received the B.S. de-gree in computer science and information engineer-ing from National Taiwan University, Taipei, Taiwan,in 1990; the M.S. degree in electrical engineeringfrom the University of Maryland, College Park, MD,USA, in 1994; and the Ph.D. degree in electricalengineering from Purdue University, West Lafayette,IN, USA, in 1998.

From 1998 to 2000, he was with Lucent Technolo-gies, Whippany, NJ, USA, where he worked withthe CDMA Radio Technology Performance Group.

Since 2000, he has been with the faculty of the Department of Communi-cation Engineering, National Central University, Taoyuan, Taiwan, where heis currently a Professor. His research interests include resource managementalgorithm designs for communication systems and signal processing algorithmdesigns for wireless communication systems.

Documents

Adaptive Gradient-Based Methods for Adaptive Power Allocation in OFDM-Based Cognitive Radio Networks