IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 3, NO. 2, APRIL 2014 225

A Generalized Water-Filling Algorithm withLinear Complexity and Finite Convergence Time

Suman Khakurel, Christopher Leung, and Tho Le-Ngoc

Abstract—This letter presents an algorithm with linear com-plexity and finite convergence time for solving the generalizedwater-filling (WF) problem. The WF problem is generalizedby using a weighted-sum-rate, weighted-sum-power, and peakpower constraints. The proposed algorithm solves the optimiza-tion problems with concave (power and rate) or quasi-concave(energy-efficiency) objective functions. Additionally, it can simul-taneously use maximum-power and minimum-rate constraintsand give a priority to one of the constraints in the event theygenerate an infeasible region. Through this generalization, thealgorithm can be applied to many WF-based methods proposedin the literature. Moreover, this letter shows multiple ways tofurther reduce the computational complexity and, via simulation,illustrates the effectiveness of the proposed algorithm.

Index Terms—Water-filling, optimization, energy-efficiency,rate, power adaptation, algorithm.

I. INTRODUCTION

THE well-known classical water-filling (WF) solution wasfirst proposed in [1] where the power is adapted so as

to achieve the capacity of a single-user system. The nameWF comes from the fact that this capacity-achieving poweradaptation technique has the visual interpretation of pouringwater over a surface given by the inverse of the subcarrierpower gains. In its most basic form, WF assumes full channelknowledge and a total transmit power constraint at the trans-mitter to yield an optimal solution for capacity maximization.

Most of the proposed WF methods [2] have limited them-selves to discrete bit-loading where the set of allowable bit-rate over each subcarrier is finite. Moreover, these methods arefundamentally based on the principle of iteratively increasingthe bit-rate on the most efficient subcarrier. For continuousbit-loading, monotonic property of the WF problem as afunction of the WF level can be utilized and hence, bisectioncan be applied to find the WF level. A projection approachmentioned in [3] can also be used by projecting a point onto abounded plane defined by the total transmit power constraint.However, projection only works for rate maximization and theaddition of peak power constraints transforms the problem’ssimple bounded plane into a polytope. Authors in [4] proposedpractical algorithms for a family of WF solutions with aconcave objective function where at least one of the constraintsis met at equality.

For the quasi-concave objective functions, such as energyefficiency (EE), the constraints are not necessarily met at

Manuscript received December 2, 2013. The associate editor coordinatingthe review of this letter and approving it for publication was D. Niyato.

The work presented in this paper is partly supported by the Natural Sciencesand Engineering Research Council of Canada (NSERC) grants.

The authors are with the Broadband Communications ResearchLab, ECE Department, McGill University, Montreal, Canada (e-mail:{suman.khakurel@mail., christopher.leung@mail., tho.le-ngoc@}mcgill.ca).

Digital Object Identifier 10.1109/WCL.2014.020314.130839

equality and the aforementioned algorithms may not be suit-able. A methodology requiring multiple steps resulting upto 2 calls of bisection for computing the EE WF level hasbeen discussed in [5], where the unconstrained optimizationproblem is solved first and the total power and rate constraintsare imposed next.

Since virtualization is the future of wireless communi-cation, it may be required that a scheduler need to run ageneral scheduling algorithm for multiple problems resultingin WF power allocation which will make the schedulers lesscomplex and consume less memory. This letter proposes ageneralized fast WF algorithm for the problems having eitherconcave or quasi-concave objective functions with weightson the sum-rate and the sum-power and constraints on peakpower, maximum total transmit power, and minimum rate. Theproposed algorithm has a complexity that grows linearly withthe number of subcarriers and converges in a finite number ofsteps. The weighted-sum-rate and weighted-sum-power allowthe algorithm to be applied to the band-preference WF-basedmethods. Moreover, through the use of weights, certain subcar-riers can be penalized while others are encouraged. Simulationresults indicate that the proposed algorithm is almost 3 timesfaster than the bisection algorithm and its effectiveness overbisection algorithm increases with the increase in number ofsubcarriers in the system.

The remainder of this letter is organized as follows. InSection II, the generalized WF problem is proposed andsolved. Section III develops the generalized WF algorithm.The algorithm’s performance and modifications are discussedin Section IV. Section V concludes this letter.

II. GENERALIZED WATER-FILLING

The WF problem can be formulated as a rate-maximizationor a power-minimization or an EE-maximization problem.Although the rate maximization is more often used in literatureand is normally associated to WF, the other problems differfrom it only by how the WF level is set. This section showsthe generalized maximization and minimization problems re-sulting in WF power allocation and their associated solution.

A. Weighted-Sum-Rate Maximization

Let p = {p1, p2 . . . , pN} be the set of allocated powersover a set of N subcarriers denoted as N = {1, 2, . . . , N}.The noise power densities σn are assumed to be normalizedby their respective channel gains. pmask

n , ωn and υn are thepeak power constraints, rate weights and power weights oneach subcarrier, respectively and Pmax is the effective totaltransmit power which is dependent on the power weights. Theweighted-sum-rate maximization problem is formulated as:

maxp

∑∀n∈N

ωn ln

(1 +

pnσn

)(1)

2162-2337/14$31.00 c© 2014 IEEE

226 IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 3, NO. 2, APRIL 2014

subject to 0 ≤ pn ≤ pmaskn , ∀n ∈ N ,

∑∀n∈N

υnpn ≤ Pmax.

The feasible region that meets every constraint in (1) formsa convex set. Thus, the solution to its Lagrange function alsosolves the maximization problem. Let λ, μ, and ν be the La-grange multipliers for the weighted-sum-power, non-negativepower values, and peak power constraints, respectively. Then,Lagrangian L of (1) can be written as:

L(p, λ, μ, ν) =∑

∀n∈Nωn ln

(1 +

pnσn

)− λ

∑∀n∈N

υnpn

−∑

∀n∈N(μnpn − νnpn).

By solving for the derivative of L and by rearranging, thefollowing solution is obtained:

pn =ωn

λυn + μn − νn− σn, ∀n ∈ N . (2)

The Lagrange multiplier λ is selected such that the weighted-sum-power inequality is tight. As for μn and νn, they areonly used when the non-negative or the peak power constraintsare violated and are zero otherwise. Eq. (2) can be furthersimplified into:

pn =

[ωna

υn− σn

]pmaskn

0

, ∀n ∈ N , (3)

where a is traditionally known as the WF level and is selectedin the same manner as λ.

B. Weighted-Sum-Power Minimization

The weighted-sum-power minimization problem can beformulated as:

maxp

−∑

∀n∈Nυnpn

subject to 0 ≤ pn ≤ pmaskn , ∀n ∈ N ,

Rmin ≤∑

∀n∈Nωn ln

(1 +

pnσn

).

(4)

Here, all variables serve the same purpose as in (1) with theaddition of the minimum rate Rmin. Alike (1), optimal solutionto this problem simplifies into (3).

C. Weighted-EE maximization problem

The weighted-EE maximization problem can be written as:

maxp

η(p) =

∑Nn=1 ωn ln

(1 + pn

σn

)

Pc +1ε

∑Nn=1 υnpn

(5)

subject to 0 ≤ pn ≤ pmaskn , ∀n,

∑∀n∈N

υnpn ≤ Pmax,

Rmin ≤∑

∀n∈Nωn ln

(1 +

pnσn

).

Here, all variables serve the same purpose as in (1) and (4).Further, Pc is the constant circuit power that correspondsto the power dissipation of the transmitter circuitry whichis independent of the transmission rate and ε is the PA

efficiency, where 0 ≤ ε ≤ 1. The objective function in (5) isquasi-concave and the Karush-Kuhn-Tucker (KKT) conditionscannot be used directly to find the optimal solution. By usingvariable transformation:

t =1

P ′c + 1T p′ , y =

p′

P ′c + 1T p′ , y ∈ R

N+ and t > 0, (6)

where p′ is a vector in RN+ with each element comprising

υnpn, 1 is an all-one vector in RN and P ′

c = εPc, we obtainthe concave maximization problem given as:

maxy∈R

N+ ,t>0

η(y/t) = t1Tρ(y/t) (7)

subject to P ′ct+ 1T y = 1, 0 ≤ yn

υnt≤ pmask

n , ∀n,1T y ≤ Pmaxt, εR

min ≤ 1Tρ(y/t), (8)

where ρn(yn/t) = εωn ln(1+1

σnυn

yn

t ) and t/ε represents theinverse of the total power consumption.

The feasible region that meets the above constraints formsa convex set and the objective function is jointly concavein y and t. Hence, KKT conditions are both necessary andsufficient for optimality. If λ, μ and ν ∈ R are the Lagrangemultipliers, then the Lagrangian of (7) can be written as:

L = (t+ν)1Tρ(y/t)−λ(P ′ct+1T y)−μ(1T y−Pmaxt). (9)

Differentiating L with respect to yn, we get

(t+ ν)∂ρn(y/t)

∂yn− λ− μ = 0, n = 1, 2, ..., N, (10)

and∂ρn∂yn

=εωn/t

σnυn + yn/t. Substituting the value of

∂ρn∂yn

in

(10), we have

pn =ynt

=

(εωn

tυn.t+ ν

λ+ μ− σn

)pmaskn

0

, ∀n ∈ N , (11)

which is a WF power allocation procedure in frequencydomain and simplifies into the same solution as the one forproblems (1) and (4). Since EE is a strictly quasi-concavefunction in p, the optimal solution is either the global optimumwhere both the weighted-sum-rate and weighted-sum-powerinequalities are inactive, if feasible, or the boundary pointwhere either of those inequalities is tight. Moreover, it isshown in [6] that at the optimality of an unconstrained EEmaximization problem, η = λ.

III. GENERALIZED WATER-FILLING ALGORITHM

The generalized WF algorithm uses recursion to keep trackof the total transmit power and rate as it tries differentpotential WF levels. This section derives the process behindthe recursion and develops the generalized WF algorithm.

A. Calculating the Weighted-Sum-Power

The total transmit power as a function of a is given as:

P (a) =∑

n∈N :0≤pn(a)<pmaskn

υn

(ωna

υn− σn

)+

∑n∈N :pn(a)=pmask

n

υnpmaskn , (12)

KHAKUREL et al.: A GENERALIZED WATER-FILLING ALGORITHM WITH LINEAR COMPLEXITY AND FINITE CONVERGENCE TIME 227

where the first summation is for the active but not saturatedsubcarriers and the second is for the saturated subcarriers.Next, we consider two sets of WF levels of interest:

a0n =υnσn

ωn, amask

n = a0n +υnp

maskn

ωn. (13)

Here, a0n represents the WF level at which subcarrier ntransitions from inactive to active and amask

n represents the WFlevel at which subcarrier n becomes saturated by reaching itspeak power constraint. By using (13), we eliminate the upperbound on the WF level for inclusion in the first summation,and rearrange the variables of (12) to obtain:

P (a) = a

( ∑n∈N :a0

n≤a

ωn −∑

n∈N :amaskn ≤a

ωn

)

−∑

n∈N :a0n≤a

ωna0n +

∑n∈N :amask

n ≤a

ωnamaskn . (14)

With this new form, the total transmit power can be recursivelycalculated while searching for the WF level as long as it ismonotonically increasing or decreasing.

B. Calculating the Weighted-Sum-Rate

The rate as a function of the WF level is given by:

R(a) =∑

n∈N :0≤pn(a)<pmaskn

ωn ln

(ωna

υnσn

)+

∑n∈N :pn(a)=pmask

n

ωn ln

(1 +

pmaskn

σn

).

Similar to the weighted-sum-power case, by using (13), theneliminating the upper bound on the WF level from the firstsummation, and rearranging the terms, the function becomes:

R(a) = ln(a)

( ∑n∈N :a0

n≤a

ωn −∑

n∈N :amaskn ≤a

ωn

)−

∑n∈N :a0

n≤a

ωn ln(a0n) +

∑n∈N :amask

n ≤a

ωn ln(amaskn ).

Similar to transmit power, the final form can keep track of theweighted-sum-rate through recursion by performing a cumu-lative sum as long as the tried WF levels are monotonicallyincreasing or decreasing.

C. Finding the Water-Filling Level

One method to exploit the recursive forms for calculatingthe total transmit power and total rate is to try the WF levelsa0n and amask

n , ∀n ∈ N , in an ascending order. In this way,the interval between two consecutive WF levels containingthe optimal WF level can be isolated. Within this interval, thesummations are constant thus, turning the total power, rate andEE into simple functions of the WF level a. Therefore, thismethod only requires calculating a0n, amask

n and their naturallogarithms, sorting the WF levels and performing a cumulativesummation until the feasible optimum is reached. Althoughthis method can work well, sorting process is expensive withits O(N logN) complexity.

Instead of sorting the WF levels, we use a selection al-gorithm, which mimics the quicksort algorithm, to isolate the

interval. We partition the WF levels into two sets using a pivot.Every WF level not exceeding the pivot is put into the lowset, and the other WF levels into the high set. Then, the totalpower and the rate are computed as a function of the pivot.Note that only the values of the smaller WF levels are neededin the calculation. If the optimal WF level is smaller, thenthe process is repeated with the low set, else the process isrepeated with the high set. In the latter case, with the largerWF level, the total power and the rate from the previousstep can be reused. This is because the previous calculationincludes the information of all WF levels in the low set. Thisallows the total power and the rate to be tracked recursivelyand the interval to be determined in O(N) time.

The following are the variables used by the generalized WFalgorithm: ωall

i = {−ωi if 0 < i ≤ N and ω(i−N) if N <i ≤ 2N}, aalli = {a0i if 0 < i ≤ N and amask

(i−N)if N < i ≤2N}, lalli = {ln(a0i ) if 0 < i ≤ N and ln(amask

(i−N)) if N <i ≤ 2N}. The algorithm for finding the WF level is defined

Algorithm 1 Finding the WF level

1: I ← {1, 2, . . . , 2N}2: k ← 0, W (0) ← 0, P (0) ← 0, R(0) ← 03: while I �= ∅ do4: k ← k + 1, I low ← ∅, Ihigh ← ∅5: Choose pivot(k) from set I6: W (k) ←W (k−1), P (k) ← P (k−1), R(k) ← R(k−1)

7: for all i ∈ I do8: if aalli ≤ aall

pivot(k) then9: W (k) ←W (k) − ωall

i , P (k) ← P (k) + ωalli aalli

10: R(k) ← R(k) + ωalli lalli , I low ← I low ∪ i

11: else12: Ihigh ← Ihigh ∪ i13: end if14: end for15: if Pmax < P (k) + aall

pivot(k)W(k) then

16: I ← I low\pivot(k)17: W (k) ←W (k−1), P (k) ← P (k−1), R(k) ← R(k−1)

18: else if R(k) + lallpivot(k)W

(k) ≤ Rmin then19: I ← Ihigh

20: else if1

aallpivot(k)

>R(k) + lall

pivot(k)W(k)

P ′c + P (k) + aall

pivot(k)W (k)then

21: I ← Ihigh22: else23: I ← I low\pivot(k)24: W (k) ←W (k−1), P (k) ← P (k−1), R(k) ← R(k−1)

25: end if26: end while27: aPmax ← (Pmax − P (k))/W (k)

28: aRmin ← exp((Rmin −R(k))/W (k)

)29: Solve aEE(R

(k)+ln(aEE)W(k)) = P ′

c+P (k)+aEEW(k)

30: a← min{aPmax , aRmin , aEE}31: for n = 1 to N do32: pn ←

[ωnaυn− σn

]pmaskn

033: end for

in Algorithm 1. The variable k in Algorithm 1 is only therefor notational convenience and is not essential. In line 5,an acceptable pivot selection strategy is to randomly choose

228 IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 3, NO. 2, APRIL 2014

2 4 6 8 102

4

6

8

10

12

Number of subcarriers

Tim

e in

s

Proposed algorithmBisection algorithm

Fig. 1. Time taken by the proposed and bisection algorithm versus numberof subcarriers.

an index from I. The optimal pivot selection strategy isbeyond the scope of this letter. Line 6 keeps track of thepreviously calculated summations in the weighted-sum-powerand weighted-sum-rate functions. Lines 9 to 10 accumulate theinformation from the WF levels in the low set. Lines 15 to 25compare the results to the constraints. Lines 15 to 17 and lines18 to 19 are for the power and rate constraints, respectively.We know that dR

dP |P=P (a) = 1a and dη

dP =d(εR/P+P ′

c)dP =

εP ′

c+PdRdP − εR

(P ′c+P )2 . Thus, dη

dP |P=P (a) > 0 if 1a > R

P ′c+P .

Therefore, total transmit power must be increased as long asthis condition is satisfied (see line 20). Algorithm 1 in itscurrent form gives the priority to the total transmit powerconstraint and hence, is checked first. Finally, lines 27 to30 calculate the optimal WF level. If the solution lies at theboundary points where transmit power and rate constraints areactive then the optimal WF levels are given by lines 27 and28, respectively. In case of EE maximization problem, wherethe solution is globally optimal none of the constraints maybe active and the Lagrange multiplier, μ = ν = 0 (becauseof the complementary slackness condition). Hence, WF level,

a = εωn/υnλ and η = λ =εωn

υna=

εωnR(a)

υn(P ′c + P (a))

. Thus,

aR(a) = P ′c + P (a). Substituting R(a) = R(k) + ln(a)W (k)

and P (a) = P (k) + aW (k), a can be computed by solvingthe equation mentioned above (see line 29). Finally, line 30calculates the WF level according to the priority strategy.

Algorithm 1 in its current form is capable of solvingproblems (1), (4) and (5). For the rate maximization problem,Rmin should be set to 0 and for the power minimizationproblem, Pmax should be set to∞. If rate or EE maximizationis desired with a priority on the rate constraint, lines 15 to 17need to be switched with lines 18 to 19. Additionally, lines23 to 24 need to be replaced with line 21 and the minimumin line 30 needs to be replaced by a maximum.

IV. PERFORMANCE AND MODIFICATIONS

The expected complexity of Algorithm 1 is of O(N) givenrandomly chosen pivots. Moreover, each element of ωall

i andlalli are expected to be accessed once, and aalli twice. Hence,

with every constraint being enforced, the number of arrayaccesses is 8N . With the sorting method, just the sorting ofthe 2N WF levels, aalli , requires 2N log 2N element accesses

and another expected 3N for the cumulative sum. The bisec-tion method requires many iterations to obtain an acceptableinterval of error. At each iteration, 4N element accesses andN logarithm operations are required.

Next, we perform simulations to analyze the performanceof the proposed algorithm. The simulation scenario considersa system bandwidth of 2 MHz and subcarrier bandwith of180 kHz. Hence, the number of subcarriers (N ) is 10. The PAefficiency (ε) is assumed to be 1, circuit power (Pc) is 36 W,spectral mask pmask

n is 5 W for each subcarrier and maximumtransmit power of BS (Pmax) is 40 W ≈ 46 dBm unlessindicated otherwise. Also, noise spectral density is −174dBm/Hz. We simulate 10, 000 independent trials in order toobtain the average value of parameters. The CPU processorused is a core duo processor with a speed of 1.83GHz andmemory of 1GB RAM, the software used is MATLAB 8.2and the precision level of bisection algorithm is 10−6. Wecompare the effectiveness of the proposed algorithm with themulti-level WF bisection algorithm [5] as can be seen in Fig. 1.We can observe that the proposed algorithm is almost 3 timesfaster than the bisection algorithm and the effectiveness of thealgorithm increases with the increase in number of subcarriers.

V. CONCLUSION

The weighted-sum-rate maximization, weighted-sum-powerminimization and weighted-EE maximization problems wereintroduced and were shown to have the same WF solutionwith the exception lying in the WF level. Then, the weighted-sum-rate and weighted-sum-power as a recursive function ofthe WF level were developed. Using these recursive functionsand a partition process, the generalized WF algorithm withlinear complexity and finite convergence time was proposed.Simulation results has shown that the proposed algorithmis almost 3 times faster than the bisection algorithm andits effectiveness over bisection algorithm increases with theincrease in number of subcarriers in the system.

REFERENCES

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[3] G. Scutari, D. P. Palomar, and S. Barbarossa, “Asynchronous iterativewaterfilling for Gaussian frequency-selective interference channels: aunified framework,” in Proc. 2007 Information Theory and ApplicationsWorkshop, pp. 349-358.

[4] D. Palomar and J. Fonollosa, “Practical algorithms for a family ofwaterfilling solutions,” IEEE Trans. Signal Process., vol. 53, no. 2, pp.686-695, Feb. 2005.

[5] R. Prabhu and B. Daneshrad, “An energy-efficient water-filling algorithmfor OFDM systems,” in Proc. 2010 IEEE Int. Conf. Commun.

[6] C. Isheden and G. P. Fettweis, “Energy-efficient link adaptation onparallel channels,” in Proc. 2011 European Signal Process. Conf.