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IEEE SIGNAL PROCESSING LETTERS, VOL. 22, NO. 11, NOVEMBER 2015
Fast Computation of Generalized Waterfilling Problems N. Kalpana, Student Member, IEEE, and Mohammed Zafar Ali Khan, Member, IEEE
Abstract—In this letter, we present a novel method of solving the Generalized Waterfilling problem (GWFP) using equivalence with a waterfilling problem (WFP) which has a closed form solution. Consequentially, a class of WFPs with only one-step called ‘1-step’ WFP is used to derive a low complexity algorithm which has the smallest worst case complexity reported. The proposed algorithm is verified by simulations. Index Terms—Fast computation, generalized waterfilling problem, iterative waterfilling algorithm.
I. INTRODUCTION
W
ATERFILLING plays an important role in resource allocation. In any (general) waterfilling problem (WFP), powers are allocated to the resources of the transmitting user in order to maximize the transmitting user’s capacity (or mutual information) while satisfying the total power budget constraint. The user’s resources can be the sub-carriers in Orthogonal Frequency Division Multiplexing (OFDM) or the normal frequency bands or the usage of the same sub-carriers in different time slots [1]. This implies that the resource’s allocated power is inversely proportional to the noise level of the resource in WFP so as to maximize capacity [2], [3]. The solution to these class of problems can be viewed as ‘pouring limited volume of water in a tank whose bottom has stair levels whose height is determined by the noise levels in each resource. The allotted power for the resource is the difference in between the constant water level and the resource’s noise level’. WFP finds applications in various fields of communication systems like Cognitive radio (CR) [4], MIMO-OFDM systems [5]–[10], DSL systems [11]–[14], Energy harvesting [15], [16], WLAN [17], etc. As the list keeps increasing, the focus of this paper is to solve a generalized class of water-filing problems. Due to the significant role played by WFP in communication systems, floating point operations (flops) taken to implement WFP need to be as small as possible; which is the major contribution of this paper. In general, WFP has been solved using the iterative waterfilling algorithms (IWFA) [6], [14], [18]. These algorithms achieve close to the exact capacity values, as the number of iterations tend to infinity [19] (and the references therein).
Manuscript received April 03, 2015; revised May 13, 2015; accepted May 23, 2015. Date of publication June 02, 2015; date of current version June 12, 2015. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Alessio Zappone. Kalpana. N and M. Z. A. Khan are with the Department of Electrical Engineering, Indian Institute of Technology, Hyderabad, TS, 502205 India (e-mail:
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2015.2440653
However, exact algorithms which give exact solutions in finite iterations, for special cases of GWFP, have been proposed [1], [5], [19], [20]. In this paper we present an exact algorithm named ‘ -Equivalence Based Waterfilling (EBWF)’, where is an integer, that has lower computational complexity among the known algorithms. The outline of the paper is as follows. Section II introduces the Generalized Waterfilling Problem (GWFP) and presents initial results on equivalence of WFP’s. Section III present the EBWF and gives a comparison of computational complexities. Example applications and simulation results are presented in Section IV. Section V concludes the paper. II. GENERALIZED WATERFILLING PROBLEM (GWFP) The generalized waterfilling problem (GWFP) can be described as follows: , correGiven a power budget, , and a sequence sponding to user resources/subchannels, maximize the user’s weighted capacity, , while the sum of the allocated powers should not exceed the power budget. Mathematically:
(1)
where ’s are the weights of the powers, ’s are the allocated powers and ’s are the weights of resources capacities. For convenience of presentation of results, the case where is called the Weighted Waterfilling Problem (WWFP); as it deals with weighted sum of individual rates. The solution to (1) is obtained by using Karush–Kuhn–Tucker (KKT) conditions and is given as
(2)
where , is the Lagrangian and indicates the ‘water level’ of the GWFP. Definition 1: (The Number of Positive Powers, ) Let be the set of indices where is positive. Then the , is the cardinality of the set, number of positive powers, . In (2), the lagrange multiplier and the number of positive powers, , are unknown and are solved using either iterative methods or exact algorithms [1], [5], [19]. Note that and are not known apriori and solving for is a non-linear problem.
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KALPANA AND KHAN: FAST COMPUTATION OF GENERALIZED WATERFILLING PROBLEMS
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A. -Equivalent WFPs In this paper we propose a novel way to obtain the solution for the number of positive powers, , using equivalence. First we show that it is sufficient to consider the WWFP. Lemma 1: For every GWFP, defined in (1), there exists a WWFP of the form
(3) Fig. 1. Equal ’sum of noise levels’ (slanted parts) for both the given WFP and in figure) (a) Given WFP (b) 1-step WFP. an -equivalent 1-step WFP s. (
where . Moreover, the WWFP has the same solution as the GWFP. Proof: Consider the GWFP given in (1) with solution given but by (2). Now construct a WWFP with same and . Substituting, this in (1) we have (3). Note that the solution is again given by (2), completing the proof. In what follows we assume a WWFP only. Rearranging (2) for a WWFP, we have (4) where . Without loss of generality, we assume that ’s are sorted in ascending order. It follows that, ; are in descending order and the first powers are positive and all the remaining powers are zero. Accordingly, (4) simplifies as
(8) where . Proof: Substituting (5) in (2) (with
), we get (9)
As
, and , we have
; from (5), for
and (10) (11)
Substituting the inequality of (10) in (9), we have
(5) Substituting (5) in the weighted power constraint given in (2), we have (6) Note that in (6) the unknowns are on the LHS and depend on the power budget, and weighted sum . Note also that the nonlinearity in (6) is due to the second term on the RHS. Most importantly, observe that the value of in (6) depends on and as such we can have two difthe weighted sum ferent GWFP’s with different ’s but have the same weighted and hence the same . Formally, sum Definition 2: ( -Equivalent Waterfilling Problems) For a given , two WFPs are said to be -Equivalent Waterfilling Problems if 1) They have the same weighted sum, , 2) The -th step is same. In other words the total ‘water’ contained in -equivalent waterfilling problems is same in the -th step. An example of ‘ -Equivalent Waterfilling Problem’ is shown in Fig. 1(a) and Fig. 1(b). Note that this allows us to equate a WWFP to a simpler WFP (for the same ) where the computation of is simpler. We will use the following proposition for calculating . Lemma 2: For every WWFP constructed as in Lemma 1, the following inequalities hold: (7)
which is same as (7). Similarly, substituting (11) in (9), we have (8). B. WFP’s with Closed Form Solution for We now present a WFP that has closed form solution for the number of non-zero powers, . Note that this WFP dose not occur in practical scenarios, but we will use it to obtain a solution for practical WWFP’s in Section III. 1) 1–step WFP: The 1–step WFP has only one step and is and , . Note that defined by . The weighted sum for this case, we have assumed that , is given by (12) In this case the number of positive powers can take only two values; (13) Note that this is an example of a WFP with closed form solution for . We can define more examples by assuming a structure on the ’s. For example assuming that the sequence follows arithmetic progression; in its simplest form, can be defined as a WFP with and where is a constant. It can be shown, for this case, that the (14)
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IEEE SIGNAL PROCESSING LETTERS, VOL. 22, NO. 11, NOVEMBER 2015
However, the 1–step WFP is the simplest WFP with closed form solution for ; to the best of our knowledge.
TABLE I COMPUTATIONAL COMPLEXITY IN NUMBER OF ARITHMETIC OPERATIONS OF VARIOUS KNOWN SOLUTIONS TO WWFP
III. SOLUTION OF WWFP BASED ON -EQUIVALENCE Based on any of the WFP’s with closed form solution (an example of which is presented in Section II-B), we can develop an algorithm for finding for any WWFP, called as -Equivalence based Waterfilling (EBWF). In what follows we give a detailed implementation of the algorithm using the 1–step WFP; as it is the simplest and requires least number of computations. Algorithm 1 -Equivalence based algorithm for obtaining Require: Inputs required are , , (in ascending order). Ensure: Output is . 1: . Denote . 2: Calculate . 3: Calculate , the second step, of the 1–step -equivalent WFP for each (denoted by ) as 4: If then . Exit the algorithm. 5: 6: else 7: , Go to 2 8: end if Note that while there is no , this initialization removes need at every . Also a scaled version of the first for addition of step, of the 1–step equivalent WFP for each (denoted by ) . is calculated in Step 2. The actual step size is given by adds and Finally observe that Algorithm 1 requires multiplies to find the value of . This algorithm lends itself to a nice geometric interpretation: gives the total area of steps; , the area of the the term ‘Noise’ steps and the difference, , is the area where the ‘water can be poured’. If the difference of the two is greater than then the number of steps is ‘enough’ to store all the water. Based on this interpretation, we can now use the following proposition to prove that the Algorithm 1 outputs . Proposition 1: The function is a monotonically increasing function of . . Proof: We need to prove Subtracting the two terms we have (15) , this completes the proof. Since We have: Theorem 1: The output of Algorithm 1 gives the optimal value of the number of positive powers, , defined in Definition 1. Proof: First observe that the conditional test in the algois equivalent to . From Proposition rithm is monotonically increasing function of . Then using 1, the geometric interpretation of the algorithm, is the smallest number of steps into which all of can be poured without ‘overflowing’. Once the value of is found, we can find the individual powers, as given in Algorithm 2.
Algorithm 2 Algorithm for obtaining Require: Inputs required are , , (defined in (in ascending order). Algorithm 3), Ensure: Output is . 1: Calculate the water level using (6) as 2: Calculate the allotted powers using (5) as
. (16)
This requires additions/subtractions and multiplications/divisions. Resulting in an overall computational com. Since is not known apriori, the worst plexity of . Table I gives the number case complexity is given by of flops required for PWFA, IWFA, algorithm of [5], algorithm of [1], Geometric Waterfilling algorithm (GWF) of [19] and the proposed equivalent WFP algorithm to implement WWFP as reported in the references given. Note that bisection can be used iterations [5]. Obfor both PWFA, IWFA requiring serve that the proposed equivalence based waterfilling (EBWF) reduces the complexity by atleast half as compared to the best known WWFP, the GWF. IV. PARTICULARIZATION TO EXAMPLES OF WFPS AND SIMULATION RESULTS In this section we show that the EBWF algorithm developed in Section III can be used to solve a wide family of practical waterfilling problems and verify the results through simulations. 1) Maximization of mutual information, subject to sum power constraint: This problem is of particular interest as it results in the capacity achieving solution [21]–[25]. Substituting in (1), particularizes the GWFP to the CWFP. The EBWF particularizes to this case with reduction in complexity due to absence of multipilcation by . by 2) Weighted Sum Rate Maximization, subject to sum power in (1), particularizes the constraint: Substituting GWFP to this case. Weighted sum rate maximization typically appears in MAC scheduling wherein the weights, , correspond to the priorities of the users or the length of the queues of the users [26], [27]. Another scenario arises in Gaussian MIMO broadcast channel where the system stability region is achieved by maximizing a weighted sum rate that depend on the queue buffer sizes, under random packet arrival and transmission queues [26], [28]. The weights, , in this case, are chosen for stability and by varying the weights, different points on the boundary of achieavble region can be obtained [27]. 3) Sum Rate Maximization, subject to interfernce constraint: in (1), particularizes the GWFP to this Substituting case. This scenario arises in cognitive radio where the sum rate is maximized subject to interference temperature constraints [29], [30]. The ’s in this case correspond to the -th subcarrier gain of the secondary user to the primary. The EBWF algorithm is applicable to this case using the Lemma 1.
KALPANA AND KHAN: FAST COMPUTATION OF GENERALIZED WATERFILLING PROBLEMS
TABLE II COMPUTATIONAL COMPLEXITIES OF EXISTING AND THE PROPOSED SOLUTION FOR WWFP,
4) Weighted Sum Rate Maximization, subject to interfernce constraint: This is the most general case and can be thought of as a generalization of previous case, where in the MAC schedules for secondary users in cognitive radio network [31]. The weights, , correspond to the priorities of the users and ’s in this case correspond to the -th subcarrier gain of the secondary user to the primary. The EBWF algorithm is applicable to this case using the Lemma 1. A. Simulation Results Simulations are done in MATLAB R2010b software. We aswhere is AWGN variance and is channel sume gain, as in a wireless communication channel. , and are assumed to be exponentially distributed with variance 1. Setting , simulations show that same capacities are obtained for the user’s resources from all the algorithms with computational complexities as given in Table II. For sake of comparison, IWFA is implementted using bisection. V. CONCLUSION In this paper, we have used ‘equivalence’ in WWFP to obtain the optimal number of non-zero powers. The resulting algorithm reduces the computational complexity by more than half. Simulation results show that the proposed solutions requires significantly less number of flops. Application of the ‘equivalence’ principle to more complicated waterfilling problems like ‘cave waterfilling’ and ‘multiple waterlevel’ problems [5], [32], [33] needs to be studied as it is not straight forward. REFERENCES [1] E. Altman, K. Avrachenkov, and A. Garnaev, “Closed form solutions for water-filling problems in optimization and game frameworks,” Telecommun. Syst., vol. 47, no. 1–2, pp. 153–164, 2011. [2] D. Tse and P. Vishwananth, Fundamentals of Wireless Communication. Cambridge, U.K.: Cambridge Univ. Press, May 2005. [3] P. Wang, M. Zhao, L. Xiao, S. Zhou, and J. Wang, “Power allocation in ofdm-based cognitive radio systems,” in Proc. GLOBECOM, 2007, pp. 1–5. [4] G. Bansal, M. J. Hossain, and V. K. Bhargava, “Optimal and suboptimal power allocation schemes for OFDM-based Cognitive Radio Systems,” IEEE Trans. Wireless Commun., vol. 7, no. 11, pp. 4710–4718, Nov. 2008. [5] D. P. Palomar and J. R. Fonollosa, “Practical algorithms for a family of waterfilling solutions,” IEEE Trans. Signal Process., vol. 53, no. 2, pp. 686–695, Feb. 2005. [6] A. Liu, Y. Liu, V. K. N. Lau, H. Xiang, and W. Luo, “Polite waterfilling for Weighted sum-rate maximization in MIMO B-MAC networks under Multiple Linear Constraints,” , 2011 [Online]. Available: http://ecee.colorado.edu/liue/publications/index.html [7] B.-C. Liang, R. Zhang, and J. M. Cioffi, “Sub-channel growing and statistical water-filling for mimo-ofdm systems,” in Proc. Signals, Systems and Computers, Nov. 2003, pp. 997–1001. [8] W. Yu, “Multiuser water-filling in the presence of crosstalk,” in Proc. Information Theory and Applications Workshop, Feb. 2007, pp. 414–420. [9] G. Mnz, S. Pfletschinger, and J. Speidel, “An efficient waterfilling algorithm for multiple access ofdm,” in Proc. IEEE Global Telecommunications Conf., 2002, pp. 681–685.
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