Subcarrier and Power Allocation for the Downlink of Multicarrier NOMA

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 67, NO. 12, DECEMBER 2018 11833 Subcarrier and Power Allocation for the Downlink of Multicarrier NOMA Systems Yaru Fu , Member, IEEE,LouSalaun¨ , Student Member, IEEE, Chi Wan Sung , Senior Member, IEEE, and Chung Shue Chen , Senior Member, IEEE

Abstract—This paper investigates the joint subcarrier and power cellular networks, orthogonal frequency division multiple ac- allocation problem for the downlink of a multi-carrier non- cess (OFDMA) has been widely applied, in which the whole orthogonal multiple access (MC-NOMA) system. A novel three- system bandwidth is divided into multiple orthogonal subcar- step resource allocation framework is designed to deal with the riers and each subcarrier is exclusively utilized by at most one sum rate maximization problem. In Step 1, we relax the problem user during each time slot at each base station (BS). Therefore, by assuming that each of the users can use all the subcarriers simultaneously. With this assumption, we prove the convexity of OFDMA can avoid intra-cell interference by performing user the resultant power control problem and solve it via convex pro- transmission scheduling at the BS. Moreover, it can be imple- gramming tools to get a power vector for each user. In Step 2, mented with low-complexity receivers. However, the spectral we allocate the subcarriers to users by a heuristic greedy manner efficiency is in general under-utilized due to the requirement of with the obtained power vectors in Step 1. In Step 3, the proposed channel access orthogonality. power control schemes used in Step 1 are applied once more to The mobile data traffic for cellular systems is expected to in- further improve the system performance with the obtained sub- crease by 1000 folds by 2020, seeking efficient multiple access carrier assignment of Step 2. To solve the maximization problem technologies becomes very important, as it is the key for meeting with fixed subcarrier assignments in both Step 1 and Step 3, a cen- the dramatically increasing bandwidth demand. Non-orthogonal tralized power allocation method based on projected gradient de- multiple access (NOMA) has recently received significant at- scent algorithm and two distributed power control strategies based, respectively, on pseudo-gradient algorithm and iterative waterfill- tention and has been regarded as a promising candidate for the ing algorithm are investigated. Numerical results show that our fifth generation (5G) cellular systems. In contrast to orthog- proposed three-step resource allocation algorithm could achieve onal multiple access (OMA), NOMA allows the multiplexing comparable sum rate performance to the existing near-optimal of multiple users with different power levels on the same fre- solution with much lower computational complexity and outper- quency resource block, which provides a higher system spectral forms power controlled OMA scheme. Besides, a tradeoff between efficiency [2], [3]. user fairness and sum rate performance can be achieved via ap- Since several users are multiplexed on the same subcar- plying different user power constraint strategies in the proposed rier simultaneously, successive interference cancellation (SIC) algorithm. is adopted at the receiver side to mitigate the resulting co- Index Terms—Multi-carrier non-orthogonal multiple access channel interference. Therefore, power must be allocated prop- (MC-NOMA), successive interference cancellation (SIC), game erly among the multiplexed users on each subcarrier such that theory, Nash equilibrium, gradient and pseudo-gradient descent, interfering signals can be correctly decoded and subtracted from synchronous iterative waterfilling algorithm (SIWA). the received signal of some users. Fractional transmit power control (FTPC) is a sub-optimal but commonly used power control strategy for users sum rate maximization problems, which I. INTRODUCTION allocates power according to the individual link condition of N THE third generation partnership project Long each user [2], [4]. In [5], power allocation in single-cell NOMA Term Evolution (3GPP-LTE) and LTE-Advanced (LTE-A) system is performed in a fixed manner. In [6] and [7], dis- I tributed power allocation algorithms are proposed to minimize total power consumption subject to user data rate constraints Manuscript received March 6, 2018; revised June 8, 2018 and September 4, for the downlink and uplink multi-cell NOMA systems, respec- 2018; accepted October 2, 2018. Date of publication October 11, 2018; date tively. In addition, there exists some other works that investigate of current version December 14, 2018. This work was partially supported by a the power control for NOMA in heterogeneous networks [8], grant from the Research Grants Council of the Hong Kong Special Administra- cooperative systems [9], [10] and multi-antenna scenarios [11], tive Region, China (Project No. CityU 11216416). The review of this paper was coordinated by Prof. G. Gui. This paper was presented in part at the IEEE Inter- [12]. Except power control, fairness is also an important feature national Conference on Communications, Paris, May 2017 [1]. (Corresponding of NOMA. In [13], the resource allocation fairness between author: Chung Shue Chen.) NOMA and OMA schemes is compared, based on which a hy- Y. Fu and C. W. Sung are with the Department of Electronic Engineer- brid NOMA-OMA scheme is proposed to further enhance the ing, City University of Hong Kong, Kowloon, Hong Kong SAR (e-mail:, [email protected]; [email protected]). users fairness. Recent works [14], [15] investigate NOMA with L. Salaun¨ and C. S. Chen are with the Bell Labs, Nokia Paris-Saclay, two sets of orthogonal signal waveforms, in which power imbal- 91620 Nozay, France (e-mail:, [email protected]; chung_shue. ance between different users’ signals is not needed. Moreover, [email protected]). deep learning has been applied in NOMA to estimate the chan- Color versions of one or more of the figures in this paper are available online nel state information by learning the environment automatically at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2018.2875601 via offline learning [16].

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For multicarrier NOMA (MC-NOMA) systems, subcarrier 2) In Steps 1 and 3, the subcarrier assignment is fixed. We assignment and power control for multiplexed users are two prove that the resultant subproblem is convex, which can interacted factors for achieving high system performance. The be optimally solved in a centralized manner via gradient work in [17] demonstrates that in practical LTE cellular systems, descent algorithm. Alternatively, it can be solved more MC-NOMA has better system-level downlink performance in efficiently in a distributed manner. To do this, we model terms of user throughput than that of OFDMA networks by the problem as a multi-player concave game, and prove realistic computer simulation. In [18], a monotonic optimiza- the existence and uniqueness of its Nash equilibrium. tion method is proposed to achieve the optimal subcarrier and Two distributed algorithms, namely, pseudo-gradient de- power allocation for MC-NOMA system. However, it is as- scent and iterative waterfilling, are investigated, and their sumed that each subcarrier can accommodate at most two users, convergence properties are established. Furthermore, we which restricts the application of the designed algorithm. The analyze the asymptotic computational complexity of the authors of [19] propose a low-complexity algorithm of jointly proposed power control schemes. Except for the above- optimizing subcarrier assignment and power allocation to mini- mentioned, in Step 2, a heuristic method, which makes use mize the total transmit power of OFDM-based NOMA network. of the result obtained in Step 1, is proposed for subcarrier In addition, the energy-efficient resource allocation problems assignment. for MC-NOMA systems with perfect channel state information 3) Under our framework, different allocation algorithms can (CSI) and imperfect CSI are investigated in [20] and [21], re- be applied to solve each subproblem in the three steps. Nu- spectively. An iterative algorithm is provided in [22] to solve merical results show that the algorithms constructed under the sub-channel and power allocation problem in MC-NOMA this framework, could generally achieve comparable per- network. The designed resource allocation strategy is not so formance to LDDP with much fewer computation steps efficient as swap-matching is adopted to do the subcarrier as- and outperform standard power-controlled OMA scheme signment during each iteration, especially when the number of in terms of sum rate performance. Different tradeoffs can user is large. In [23], a greedy user selection and sub-optimal be achieved by using different algorithms in the three power allocation scheme based on difference-of-convex (DC) steps. Furthermore, a balancing between sum-rate perfor- programming is presented to maximize the weighted through- mance and user fairness can also be obtained by adopting put in a two-user NOMA system. It is observable that the scheme different user power budget strategies. has high computational complexity due to the DC programming The rest of this paper is organized as follows. In Section II, in optimizing the power allocation among different subcarriers we present the system model and formulate the sum rate maxi- and different users. In [24], various user pairing algorithms for mization problem mathematically. In Section III, the proposed multi-input and single-output (MISO) MC-NOMA system are three-step resource allocation framework is introduced. The cen- proposed. However, the performance gain of [24] is limited due tralized gradient descent algorithm and some distributed power to the use of naive power control schemes such as fixed power control methods are analyzed in Section IV and Section V, re- allocation (FPA) and FTPC among multiplexed users. Besides, spectively. In Section VI, we evaluate the performance of our a joint power and channel allocation problem for sum rate max- proposed three-step subcarrier and power allocation algorithms imization of MC-NOMA system is formulated in [25], which is and some other existing resource allocation schemes through shown NP-hard and solved by Lagrangian duality and dynamic computer simulations. Finally, the pros and cons of the designed programming (LDDP). To gauge the performance of LDDP, resource allocation schemes and the future work are presented the authors investigate the optimality bound on the global opti- in Section VII. mum and show that LDDP could achieve a close-to-upper-bound performance. Motivated by the aforementioned observations and the much II. SYSTEM MODEL AND PROBLEM FORMULATION lower latency requirement of 5G, we propose a time-efficient A. System Model joint subcarrier and power allocation algorithm for MC-NOMA, which could achieve comparable performance to LDDP. Our This section describes the system model and the problem to be main contributions are summarized as follows: solved in this paper. The downlink of a multi-user MC-NOMA 1) A resource allocation framework is designed, which con- system with one base station (BS) serving K users is consid- sists of the following three steps: ered. Denote the set of all users’ indices by K {1, 2,...,K}. a) Assume each user can use all subcarriers. By this as- The overall system bandwidth W is divided into N subcar- sumption, the problem becomes convex and can be riers. We denote the index set of these N subcarriers by solved efficiently. A centralized or some distributed N {1, 2,...,N}.For n ∈N,letWn be the bandwidth of algorithms for the resultant power control subprob- subcarrier n, where n∈N Wn = W . Assume there is no inter- lem are designed. ference among different subcarriers because of the orthogonal b) Based on the power allocation solution obtained, the frequency division. ∈K ∈N n subcarrier assignment problem is solved, taking into For k and n ,letgk be the link gain of user k on account practical system constraints. subcarrier n. We assume a frequency-flat block fading channel n ≥ c) Given the above subcarrier allocation, Step 3 for each subcarrier. Let pk 0 be the allocated transmit power solves a convex problem, i.e., the power alloca- to user k on subcarrier n.Userk is said to be multiplexed on n tion is further adjusted to optimize the final system subcarrier n if pk > 0. Additionally, we assume the BS has a to- performance. tal power budget, which is denoted by Ptotal. Moreover, we set N n ≤ The above 3-step methodology allows us to solve the prob- maximum power allocation for each user, i.e., n=1 pk p¯k , lem efficiently. This is how we handle the problem, why where p¯k > 0 and k ∈K, to ensure the effort fairness among we do 3 steps. And then, we show its near optimal perfor- users [25]–[27]. Note that the maximum power budget for each mance by comprehensive simulation results. user can be set arbitrarily for resource allocation fairness or any FU et al.: SUBCARRIER AND POWER ALLOCATION FOR THE DOWNLINK OF MULTICARRIER NOMA SYSTEMS 11835

K ≤ the problem can be formulated as follows: preference, subject to the constraint k=1 p¯k Ptotal. Fur- thermore, denote by ηn the received noise power of user k on k maximize R , (3) subcarrier n. For notation simplicity, we normalize the noise sum n n n power as η˜k ηk /gk . subject to We consider the scenario in which BS allocates subcarriers N to its attached users and multiplexes them on a given subcarrier n ≤ ∈K C1: pk p¯k ,k , using superposition coding. For n ∈N,letUn be the set of users to whom subcarrier n is assigned. Therefore, each subcarrier can n=1 n ≥ ∈K ∈N be modeled as a multi-user Gaussian broadcast channel and SIC C2: pk 0,k ,n , is applied at the receiver side when it is possible to eliminate C : |U |≤M, U ⊆K,n∈N, the intra-band interference. 3 n n n ∈U ∈N Since SIC is applied, we need to consider the decoding order C4: pk = 0,k n ,n , (4) of users on the same subcarrier. For n ∈N,letΠn be the set of all possible permutations of Un . For example, if users u, v ∈K where C1 represents the maximum power allocation for user are multiplexed on subcarrier n, i.e., Un = {u, v}, then k. Such a constraint is imposed to guarantee the effort fairness among users as mentioned in Section II-A. C2 indicates the Π = (u, v), (v,u) . nonnegativity of the allocated power for each user on each sub- n carrier. C3 restricts that the number of multiplexed users on π ∈ each subcarrier is no more than M. When M = 1, the system Let n Πn be the decoding order of the users on subcarrier reduces to orthogonal multiple access (OMA). In NOMA, we n.Letπn (i), where i ∈{1, 2,...,|Un |}, be its i-th component, consider M ≥ 2. Note that Un is a set variable to be optimized, which means that user πn (i) first decodes the signals of πn (1) − which represents the subset of users to whom subcarrier n is to πn (i 1), then subtracts these signals, and finally decodes its assigned. Besides, C4 represents the power constraint of user k intended message by treating the signals of the remaining users π on its unallocated subcarriers. on subcarrier n as noise. Note that n can be seen as a vector Note that the above maximization problem (3) is a coupled function of the normalized noise power of each multiplexed user mixed integer non-convex problem and is known NP-hard [25], n ηñ k ∈U on subcarrier , i.e., k where n [28, Section 6.2] and is which is in general difficult to solve. For this reason, a close-to- defined as follows: upper-bound solution based on LDDP has been proposed in [25]. In the following, we will design another subcarrier and power π |U | n (πn (1),πn (2),...,πn ( n )), allocation optimization method, which is more time efficient at the cost of slight degradation in sum rate performance. such that the following two criteria are satisfied: 1) The normalized noise power of multiplexed users on sub- III. THE THREE-STEP OPTIMIZATION FRAMEWORK carrier n are arranged in descending order: ηñ ≥ ηñ ≥···≥ηñ ; FOR MC-NOMA π n (1) π n (2) π n (|Un |) 2) When there is a tie, we arrange those users in ascending The sum-rate performance of a scheme in MC-NOMA system order of their indices, i.e., is principally affected by two interacted factors, namely, subcar- if ηñ =˜ηn and i

n∈Nk n ≥ ∈K ∈N B. Problem Formulation C2: pk 0,k ,n , The objective of this work is to maximize the sum of data rates n C3 : p = 0,k∈K,n/∈Nk . (6) subject to power constraints and a given maximum allowable k number of multiplexed users per subcarrier. Mathematically, We detail the three-step methodology below: 11836 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 67, NO. 12, DECEMBER 2018

Step 1: To begin with, we relax constraints C3 and C4 such p p p p Let ( 1, 2,..., K ). The convexity of problem (5) is that all users can use all N subcarriers simultane- proved below. N N ∈K ously, i.e., k = for all k . We then solve the Theorem 1: The maximization problem (5) is convex. relaxed problem (5), which is a sub-problem of (3). Proof: The basic idea is to check that the feasible power We will prove in Section IV that this optimization region is non-empty and convex, and the objective function problem is convex and can be optimally solved us- is concave. The feasible power region is non-empty because ing a centralized gradient descent algorithm or some the zero vector satisfies constraints (6). Its convexity follows distributed power control methods. The detailed anal- directly from (7), (8) and (9). ysis about the centralized and distributed power con- It remains to show that Rsum is a concave function of p.To trol strategies are given in Section IV and Section V, prove this, we rewrite (2) as follows: respectively. ⎛ ⎞ Step 2: We assign subcarriers to users based on the power N K n ∈N ⎝ pk ⎠ allocation solution obtained in Step 1. For n , Rsum = Wn log2 1 + |U | r n n n If M or more users have positive power on −1 p +˜η n=1 k=1 j=π n (k)+1 π n (j) k subcarrier n, allocate subcarrier n to the M ⎛ ⎞ |U | users who have the top-M highest individual N n pn ⎝ π n (i) ⎠ data rates on subcarrier n, with ties broken = Wn log2 1 + |U | n n n arbitrarily; n=1 i=1 j=i+ p +˜η r 1 π n (j) π n (i) If less than M (but not equal to 0) users ⎛ ⎞ |U | |U | have positive power on subcarrier n, allocate N n n pn +˜ηn ⎝ j=i π n (j) π n (i) ⎠ subcarrier n only to these users; = Wn log2 |U | r n pn +˜ηn If no one has positive power on subcarrier n, n=1 i=1 j=i+1 π n (j) π n (i) ∗ ∗ ⎛ allocate subcarrier n to user k , where k |U | n N n pn +˜ηn argmink∈K η˜k , with ties broken arbitrarily. ⎝ j=1 π n (j) π n (1) U |U |≤ = Wn log2 |U | After Step 2, n is determined and satisfies n n n n n= p +˜η M for all n ∈N. 1 j=2 π n (j) π n (1) ⎞ Step 3: Using the subcarrier assignment obtained in Step 2, |U | n n n n n p +˜η p |U | +˜η |U | we revisit the optimization problem (5) to optimize × j=2 π n (j) π n (2) ×···× π n ( n ) π n ( n ) ⎠ |U | and compute the final power allocation. n n n n p +˜η η˜π (|U |) j=3 π n (j) π n (2) n n Note that in our case and problem, when Step 3 is conducted, ⎛ the system performance cannot be further improved by adopting |U | N n pn +˜ηn another Step 2. The reason is that after Step 3 is conducted, the ⎝ j=1 π n (j) π n (1) U = Wn log2 n set of users who are allowed to use subcarrier n, denoted by n , η˜ |U | n=1 π n ( n ) has been determined for all n. It is a legitimate assignment in that ⎞ |U |≤ ∈N |U | n M where n . Moreover, this subcarrier assignment n n n n n p +˜η p |U | +˜η |U | will be different from that after Step 2 only if there are inactive × j=2 π n (j) π n (2) ×···× π n ( n ) π n ( n ) ⎠ |U | subcarriers which were originally active after Step 1. We suspect n n n pn +˜ηn p +˜η π n (|Un |) π n (|Un |−1) that such a scenario will never occur, and even if it could occur, j=2 π n (j) π n (1) ⎛ ⎞ the probability would be extremely low. That is also why we |U |− |Un | N n 1 pn +˜ηn have the three-step methodology. ⎝ j=i+1 π n (j) π n (i+1) ⎠ = Wn log2 |U | n pn +˜ηn n=1 i=1 j=i+1 π n (j) π n (i) IV. CENTRALIZED POWER ALLOCATION WITH FIXED ⎛ ⎞ |U | SUBCARRIER ASSIGNMENT N n pn +˜ηn ⎝ j=1 π n (j) π n (1) ⎠ + Wn log2 n . The power allocation problem (5) is central to the proposed η˜ |U | n=1 π n ( n ) three-step optimization scheme. We find that it is convex. The result is detailed below. (10) |U | Let n pn +˜η n n j = i + 1 π n ( j ) π n ( i + 1) For n ∈Nand i ∈Un ,letα (p) |U | and p n i n pn +˜η n k (pk )n∈Nk . (7) j = i + 1 π n ( j ) π n ( i ) |U | n pn +˜η n n p j = 1 π n ( j ) π n ( 1) Therefore, the set of all feasible powers for user k can be β ( ) ηñ , respectively. Thus, (10) can be re- π n ( |Un |) given by written as ⎛ ⎞ |U |− N n 1 P p n ≤ n ≥ ∈N k k : pk p¯k and pk 0,n k . (8) ⎝ n p n p ⎠ Rsum = Wn log2(αi ( )) + log2(β ( )) . (11) n∈N k n=1 i=1 Define P as the set of feasible powers for the system, which can ηñ −ηñ n π n ( i + 1) π n ( i ) be expressed by the Cartesian product of all users’ feasible sets It can be shown that α = 1 + |U | is a con- i n pn +˜η n and is given by j = i + 1 π n ( j ) π n ( i ) cave function of p [29] and βn is linear to p. Hence, both αn βn p αn ≥ P Pk . (9) i and are concave functions of . Moreover, i 1as ηñ ≥ ηñ . Thus, log (αn ) and log (βn ) are concave k∈K π n (i+1) π n (i) 2 i 2 FU et al.: SUBCARRIER AND POWER ALLOCATION FOR THE DOWNLINK OF MULTICARRIER NOMA SYSTEMS 11837

A. Game-Theoretic Analysis Algorithm 1: Projected Gradient Descent Algorithm. 1: Given a starting point p =(0,...,0) In this subsection, the power control problem is first formu- 2: repeat lated in a game-theoretic manner. Subsequently, the existence 3: Save the previous power vector p = p and uniqueness of this game’s Nash equilibrium are proved. We consider distributed power allocation problem, in which 4: Determine a search direction Δ ∇Rsum (p ) each user allocates power to its assigned subcarriers so as to 5: Exact line search: choose a step size α n maximize its individual sum of data rates Rk ∈N R , arg max R (p +ˆαΔ) n k k αˆ≥0 sum k ∈K. Each user treats the interference from other users after p p 6: Update ΨP ( + αΔ) SIC as additional white Gaussian noise (AWGN). From this 7: until ||p − p||2 < 2 point, the power control problem can be regarded as a K-player 8: return p p P game [33], where k in (7) and k in (8) are the power allocation strategy and the strategy set of player k, respectively. Note that P represents the strategy space of all players in the system and functions because the logarithmic operation in our scenario is is defined in (9). The strategy space is the set of all possible concave and non-decreasing. Since finite summation of concave configurations in the game. ∈K p p p p p p functions is still concave [29], we deduce that Rsum is concave. For k ,let −k ( 1, 2,..., k−1, k+1,..., K ).We define This completes the proof. According to Theorem 1, the optimization problem (5) is u (p , p ) Rn , (13) convex. Therefore, it can be optimally solved through classi- k k −k k ∈N cal convex programming methods [30]. Since (5) is a linearly n k n constrained convex problem, we choose the projected gradient as the utility function of user k, in which Rk is defined in (1). descent method [31] to solve it. The pseudocode of the pro- Additionally, let jected gradient descent algorithm for solving (5) is given in u Algorithm 1. (u1,u2,...,uK ). (14) In Algorithm 1, the operator ∇ represents the gradient with Based on the aforementioned definitions, this game is char- p RD →P respect to power vector . ΨP : denotes the euclidean acterized by the pair (P, u). A common equilibrium concept P K |N | RΩ projection on the feasible set , where D k=1 k and in game theory is the Nash equilibrium, which is defined as is the set of Ω-dimensional real vectors. is a small constant. follows: Note that the projection ΨP (p) can be obtained by solving the Definition 2: Given the link gain matrix and the normal- following problem: ized noise power of each user, a power allocation strategy p p p p ∈P ˜ (˜1, ˜2,...,˜K ) is called a Nash equilibrium if the minimize ||p − x||2 ∈K p ∈P 2 following holds for all k and any k k , subject to x ∈P, (12) p p ≥ p p uk (˜k , ˜−k ) uk ( k , ˜−k,). (15) where ||·||2 represents the l2-norm of a vector. P is the cartesian At a Nash equilibrium, given that the other players fix their product of Pk , which is given by (9). power allocation, no player would further increase the data rate Note that the detailed definitions of the projection ΨP and the unilaterally, i.e., no user has incentive to change its power strat- convergence analysis of Algorithm 1 can be found in [32] and egy at a Nash equilibrium. The absence of Nash equilibrium [31], respectively. indicates that the distributed system is inherently unstable [34]. The existence of Nash equilibrium in our MC-NOMA system is a corollary of the fundamental theorem in game theory [33], V. D ISTRIBUTED POWER ALLOCATION WITH FIXED [35]–[37]. According to the notation of this work, it can be SUBCARRIER ASSIGNMENT represented as follows: In Section IV, we solve the maximization problem (5) through Theorem 3: If for each k ∈K,i)Pk is compact and convex, p p a standard convex programming method, which is however ii) uk is continuous in , and iii) uk is concave in k for any p P u a centralized algorithm. In this section, we provide two dis- given −k , then ( , ) is called a concave game, and it has at tributed power allocation strategies by transforming the power least one Nash Equilibrium. control problem into a multi-player concave game. The dis- It is straightforward to see that all three conditions in the theo- tributed power control methods are more time efficient than rem are satisfied. Thus, there exists at least one Nash equilibrium the centralized scheme, which can balance the sum-rate perfor- in game (P, u). mance and computational cost when considering practical ap- Next, we show that the Nash equilibrium of (P, u) is unique. plications. In addition, the techniques can be extended to solve Let Φ:P→RD be the pseudo-gradient of u as defined in [33], similar problems. Specifically, one of the two distributed meth- i.e., ods is based on pseudo-gradient algorithm and the other one on p ∇ p ∇ p ∀p ∈P iterative waterfilling algorithm. The existence and uniqueness Φ( ) ( 1u1( ),..., K uK ( )), , (16) of Nash equilibrium of this game is demonstrated and pseudo- in which ∇k denotes the gradient with respect to user k’s power p gradient algorithm can be applied to reach this point. In addition, vector k , i.e., the convergence of iterative waterfilling algorithm in some case ⎛ ⎞ is also proved. Besides the above results, we analyze the asymp- ∇ p ⎝Wn · 1 ⎠ totic computational complexity of the proposed power control k uk ( )= |U | . (17) ln 2 n n n −1 p +˜η strategies at the end of this section. j=π (k) π n (j) k n n∈Nk 11838 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 67, NO. 12, DECEMBER 2018

Since the uniqueness of Nash equilibrium is related to the Therefore, the statement is true for Q = q + 1, which com- monotonicity of Φ [33], we will show that Φ is strongly mono- pletes the proof. RQ With the aforementioned lemma, we have the following tone. Let ++ be the set of Q-dimensional strictly positive real vectors. An auxiliary lemma is introduced as follows: theorem: Q Theorem 5: Φ is strongly monotone, i.e., there exists a con- Lemma 4: Let (bi ) ∈{ ··· } ∈ R be a non-increasing se- i 1 Q ++ stant c<0 such that for all p, p ∈P, quence with Q positive real values, i.e., b1 ≥ b2 ≥···≥ Q b − ≥ b > 0. Then, for all (x ) ∈{ ··· } ∈ R , Q 1 Q i i 1 Q (Φ(p) − Φ(p ))T · (p − p ) ≤ c||p − p ||2, (23) 2 Q Q Q 2 Q 2 xixj ≥ i=1 xi ≥ i=1 xi in which Φ is defined in (16). . (18) bi 2bi 2b1 Proof: For all p, p ∈P, according to (16) and (17), we have i=1 j=i Proof: The second inequality in (18) is obvious. We will (Φ(p) − Φ(p))T · (p − p) prove the first inequality by mathematical induction. K N Basis: It is evident that the statement is true for Q = 1. W 1 1 = n − (pn − p n ), (24) Inductive step: Assume that it is true for Q = q, i.e., ln 2 ˆn ˆn k k k=1 n=1 Ik Ik q q x x q x2 i j ≥ i=1 i . (19) where bi 2bi i=1 j=i |U | n Consider the case where Q = q + 1. According to (18), we Iˆn = pn +˜ηn , k π n (j) k have −1 j=π n (k) q+1 q+1 q+1 2 xixj − xi and bi 2bi i=1 j=i i=1 |U | n − ˆn n n 2 2 q 1 q+1 Ik = pπ (j) +˜ηk . xq+1 xq xq xq+1 xixj n −1 = + + + j=π n (k) bq+1 bq bq bi i=1 j=i We re-arrange (24) as follows: − 2 2 q 1 2 xq+1 xq x − + + i (Φ(p) − Φ(p))T · (p − p) 2bq+1 2bq 2bi i=1 |U | K N n − n n n − n q−1 q+1 q−1 (p p ) −1 (p p ) 2 2 2 Wn k k j=π n (k) π n (j) π n (j) xq+1 xq xq xq+1 xixj x ≥ − i = + + + ln 2 Iˆn Iˆn 2bq 2bq bq bi 2bi k=1 n=1 k k i=1 j=i i=1 |U | |U | n n n n n q−1 q+1 q−1 N n (p − p ) (p − p ) 2 2 Wn π n (i) π n (i) j=i π n (j) π n (j) (xq+1 + xq ) xixj − xi = = + . (20) ˆn ˆn 2bq bi 2bi ln 2 I I i=1 j=i i=1 n=1 i=1 π n (i) π n (i) |U | |U | ≤ N n n n n Note that the inequality in (20) holds because bq+1 bq . Wn xi xj We define an auxiliary variable x as follows: = − , (25) i ln 2 bn n=1 i=1 j=i i xi if i

Therefore, the last expression in (20) can be re-written as n ˆn ˆn bi = Iπ (i)Iπ (i) q−1 q+1 q−1 n n (x + x )2 x x x2 ⎛ ⎞ ⎛ ⎞ q+1 q i j i |U | |U | + − n n 2bq bi 2bi ⎝ n n ⎠ ⎝ n n ⎠ i=1 j=i i=1 = p +˜η p +˜η , ⎛ ⎞ π n (j) π n (i) π n (j) π n (i) − j=i j=i x2 q q x x x2 q 1 2 q ⎝ i j − q ⎠ − xi = + for n ∈Nand i ∈{1, 2,...,|U |}. 2bq bi bq 2bi n i=1 j=i i=1 Note that − ⎛ ⎞ ⎛ ⎞ 2 q q q 1 2 |U | |U | xq xixj x n n − − i = + bn = ⎝ pn +˜ηn ⎠ ⎝ p n +˜ηn ⎠ 2bq bi 2bi 1 π n (j) π n (1) π n (j) π n (1) i=1 j=i i=1 j=1 j=1 x2 x2 ⎛ ⎞ q q |U | 2 ≥− + by induction hypothesis n 2bq 2bq ⎝ n ⎠ Cn ≤ p¯π (j) +˜η . n π n (1) 2 = 0. (22) j=1 FU et al.: SUBCARRIER AND POWER ALLOCATION FOR THE DOWNLINK OF MULTICARRIER NOMA SYSTEMS 11839

To prove this, we first introduce the IWA and we focus on the Algorithm 2: Pseudo-Gradient Descent Algorithm. synchronous version (SIWA), in which all players adjust their 1: procedure of user k power allocation simultaneously. p 2: Given a starting point k =(0,...,0) Given fixed power allocation for other users and constant 3: repeat Iñ p p channel gains, i.e., k are fixed, the optimal power allocation 4: Save the previous power vector k = k for user k is given by the following theorem [39]: 5: Determine the search direction Δ ∇ u (p) p∗ ∈P k k k Theorem 7: k k maximizes Rk if and only if there ex- 6: Exact line search; compute step size αk ists a water level, ωk , such that arg max ≥ R (p +ˆα Δ ) ∗ αˆ k 0 k k k k p n =[ω − Iñ ]+ , for n ∈N , (27) p p k k k k 7: Update k ΨPk ( k + αk Δk ) ∀ ∈K ||p − p ||2 where 8: until k , k k 2 < 9: end procedure 0ifX ≤ 0 + = (28) X otherwise, and Based on Lemma 4, we have ∗n |U | |U | |U | p =¯p . (29) N n n n n N n k k Wn xi xj Wn ∈N − ≤− n 2 n k n (xi ) ln 2 b ln 2 · Cn n=1 i=1 j=i i n=1 i=1 With the given power vectors of the other users, we define the waterfilling function for user k as ≤ ||p − p||2 c 2, p ∗n fk ( −k ) (pk )n∈Nk, (30) where c is a constant such that − N W n ≤ c<0. This n=1 ln 2·C n ∗n where pk is defined in Theorem 7. Furthermore, we define completes the proof. P→P Note that the detailed definition of strong monotonicity in F : as the waterfilling function of the system, which is Theorem 5 can be found in [38]. given as follows: Theorem 6: The game (P, u) has unique Nash equilibrium. p p p p K F ( 1, 2,..., K ) (fk ( −k ))k= . (31) Proof: According to Theorem 5, the pseudo-gradient of this 1 game, Φ, is strongly monotone. Based on [33], [38], the game Note that SIWA is an iterative algorithm. For k ∈K and ∈N n has a unique Nash equilibrium. n k ,letpk (t) be the power of user k on subcarrier n at p(t) time t, and k be the corresponding indexed family at time t. ñ { n B. Two Distributed Power Control Algorithms According to (26), we can define Ik (t) as a function of pj (t): j ∈Un \{k}}. SIWA is then defined as follows: In this subsection, we propose two distributed power allocation schemes, which can be applied in Step 1 and Step 3 of our p(t+1) p(t+1) p(t+1) p(t) p(t) p(t) three-step methodology to solve optimization problem (5). 1 , 2 ,..., K = F 1 , 2 ,..., K , 1) The Pseudo-Gradient Descent Method: The first scheme (32) is based on pseudo-gradient descent method. It follows from p(0) ∈K with k = 0 for all k . Theorem 6 that the K-player game has a unique equilibrium Let ωk (t) be the water level of user k at time t. From (29), point. Therefore, pseudo-gradient descent method can be ap- we have plied to both Step 1 and Step 3 and it is guaranteed to converge − ñ + to the unique fixed point [33]. [ωk (t + 1) Ik (t)] =¯pk . (33) n∈N For k ∈Kand n ∈Nk , we define k |U | I˜ n Note that ωk (t + 1) can be regarded as a function of k (t) n n n n I˜ p +˜η , (26) (I˜ (t)) ∈ , and we denote it by k π n (j) k k n N k −1 j=π n (k)+1 ωk (t + 1)=gk (I˜k (t)). (34) as the normalized interference plus noise of user k on Subsequently, we will investigate the convergence of SIWA. subcarrier n. We consider two waterfilling scenarios for user k. The normal- Without loss of generality, we take user k ∈Kas an exam- ized interference at subcarrier n in the two scenarios are denoted ple and state the pseudocode of the pseudo-gradient descent by Iñ and Iñ , respectively. After performing waterfilling, we algorithm in Algorithm 2. k k denote the water levels in the two scenarios by ωk and ω , In Algorithm 2, ΨP represents the projection on P , k ∈K. k k k respectively. With this setting, we have the following lemma: The pseudo-gradient descent algorithm is distributed in the sense ∈K ñ ≥ ñ ∈N Lemma 8: For any k ,ifIk I for all n k , then that each user only needs to know the link gain between itself k and the BS and the normalized interference plus noise on each gk (I˜k ) ≥ gk (I˜ ). k subcarrier, i.e., Iñ . This can be seen at Line 5 of Algorithm 2, I˜ I˜ k Proof: Let ωk gk ( k ) and ωk gk ( k ). By contradic- in which ∇ u is given by (17). All these information can be k k tion, assume ωk <ωk . First, note that measured or estimated locally. − ñ + ≤ − ñ + ≤ − ñ + 2) Iterative Waterfilling Method: The second scheme is [ωk Ik ] [ωk Ik ] [ωk Ik ] , based on iterative waterfilling algorithm (IWA). We show that n∈Nk n∈Nk n∈Nk its convergence can be guaranteed when |Un |≤2 for all n ∈N. (35) 11840 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 67, NO. 12, DECEMBER 2018

where the first inequality follows from the assumption that ωk < the convergence of SIWA in Step 3 is guaranteed based on ω and the second inequality follows from the condition that Theorem 9. k ñ ≥ ñ Ik Ik . According to (33), both sides are equal to p¯k , which implies, in particular, equality holds in the first inequality. This C. Asymptotic Computational Complexity Analysis is possible only if In this subsection, we analyze the asymptotic computational − ñ + − ñ + [ωk Ik ] =[ωk Ik ] = 0 (36) complexity of the proposed centralized and distributed power allocation algorithms, which are the key components of our ∈N for all n k . As a result, p¯k = 0, which violates our assump- designed three-step resource allocation framework. Besides, the tion in the system model. complexity analysis of both LDDP and FTPC is also discussed. |U |≤ ∈N Theorem 9: Given n 2 for all n , SIWA always 1) Complexity Analysis of Algorithm 1: Since the optimiza- converges. tion problem (5) is convex according to Theorem 1, it can be |U |≤ Proof: Since n 2, there are at most two multiplexed solved in a centralized manner by Algorithm 1 in O(log(1/)) ∈N users in subcarrier n. For each subcarrier n k ,userk may iterations [31], where is the error tolerance for algorithm ter- suffer from intra-band interference if subcarrier n is also as- mination. As shown in Line 4 of Algorithm 1, each iteration signed to another user and that user has a smaller normalized requires to compute the gradient ∇Rsum (p ) with respect to noise power than user k. We denote this subset of subcarriers by the power vector p. Each element (∇R )n of this gradient is T S S N \T ∈T sum k k , and its complement by k , i.e., k = k k .Forn k , computed by adding up the differentials of data rates Rn ,for − π n (j) we define kn as the index of the user who shares subcarrier n −1 n j<= πn (k), with respect to pk . Therefore, each iteration has with user k. 2 Since each user has a total power constraint, Iñ (t) is bounded O(NK ) time complexity. In Step 3 of our three-step method- k ology, where no more than M users are multiplexed on each from above for all k and n. Therefore, according to (33), ωk (t) ∈K subcarrier due to constraint C3, this complexity can be further is also bounded from above for all k . The convergence of 2 SIWA in Step 3 is established if refined to O(NM ). 2) Complexity Analysis of Algorithm 2: According to [33], ω(t) (ω1(t),ω2(t),...,ωK (t)) (37) Algorithm 2 is guaranteed to converge to the unique fixed point within O(log(1/)) iterations. In Line 5 of Algorithm 2, each is monotonically increasing, i.e., for any t ≥ 1, user k ∈Konly computes its pseudo-gradient ∇k uk (p ) which ω(t + 1) ω(t), (38) consists of N elements. Thus, each iteration’s time complexity is O(NK). In Step 3 of our three-step method, this complexity which we will prove in the following by induction. can be further reduced to O(NM). p(0) ñ n Basis: Since k = 0 for all k,wehaveIk (0)=˜ηk for all 3) Complexity Analysis of SIWA: By simulation results, we ñ ≥ ñ observe that SIWA converges exponentially fast. Since in Step 1 k and n. It is obvious that Ik (1) Ik (0). Lemma 8 and (34) imply ω(2) ω(1). all users can use N subcarriers simultaneously, each waterfill- Inductive step: Suppose (38) holds for t = L, i.e., ing step has O(NK) time complexity. Furthermore, the time complexity comes to be O(NM) in Step 3 as no more than M ω(L + 1) ω(L). (39) users can be multiplexed on each subcarrier. 4) Complexity Analysis of LDDP and FTPC: According to First, consider n ∈S . By the definition of S , Iñ (t)=˜ηn k k k k [25], the complexity of LDDP is O(CNMKJ2), where C rep- for all t, which implies resents the number of sub-gradient optimization iterations upon ñ ñ ∈S termination and J is the number of power levels. In LDDP, Ik (L + 1)=Ik (L), for n k . (40) the total power budget P is divided in J discrete power ∈T total Next, consider n k . According to (26), for any t,wehave steps of value Ptotal/J and power allocation is performed by n n n distributing these discrete power steps among users and subcar- I˜ (t)=p− (t)+˜η , for n ∈Tk . (41) k kn k riers. In Section VI-D, we study the trade-off between sum-rate By the definition of Tk ,user−kn experiences no intra-band performance and computational complexity of LDDP for dif- interference in subcarrier n. The waterfilling method dictates ferent values of J. We determine that for the system described that in Table II, LDDP with J = 10K is an appropriate choice as n n + a near-optimal benchmark since its sum-rate does not improve p− (t)=[ω− (t) − η˜− ] . (42) kn kn kn significantly by further increasing J. In addition, we analyze Substituting it back to (41), we obtain how the performance gain in LDDP’s optimization depends on J, N, K and M. We deduce that J = O(min{K, MN}) is a n n + n ˜ − − ∈T Ik (t)=[ω kn (t) η˜−kn ] +˜ηk , for n k , (43) good choice for achieving near-optimal sum-rate in practice. Besides, the complexity of FTPC is O(NK) by definition. which, together with the inductive hypothesis in (39), implies We summarize the above results in Table I. ñ ≥ ñ ∈T Ik (L + 1) Ik (L), for n k . (44) VI. SIMULATION RESULTS Invoking Lemma 8 with (40) and (44) and using (34), we obtain ω(L + 2) ω(L + 1), which completes the proof. In this section, we evaluate the performance of the proposed Note that M = 2 is commonly applied in practical NOMA subcarrier and power allocation strategies by extensive simula- system from implementation point of view [40]. According tions. The radius R of the cell is set to 250 meters. Within the to the subcarrier assignment in Step 2 of the proposed three- cell, there is one BS located at the center and K users uniformly step methodology, we have |Un |≤M; therefore, if M = 2, distributed inside it. The system bandwidth W is assumed to be FU et al.: SUBCARRIER AND POWER ALLOCATION FOR THE DOWNLINK OF MULTICARRIER NOMA SYSTEMS 11841

TABLE I ASYMPTOTIC COMPUTATIONAL COMPLEXITY COMPARISON OF DIFFERENT SCHEMES

TABLE II SIMULATION PARAMETERS

5 MHz and Wn = W/N for n ∈N, where N = 10. The noise power spectral density is assumed to be −174 dBm/Hz. In the radio propagation model, we follow [41] and include the distance- dependent path loss, shadow fading and small-scale fading. The distance-dependent path loss is given by 128.1 + 37.6 log10 d, in which d is the distance between the transmitter and the receiver in km. Lognormal shadowing has a standard deviation of 8 dB. For small-scale fading, each user experiences indepen- dent Rayleigh fading with variance 1. The details of simulation parameters are summarized in Table II. Wer will compare the performance of the following schemes: Double Iterative Waterﬁlling Algorithm (DIWA): we adopt the proposed three-step MC-NOMA optimization frame- r work and use SIWA in Step 1 and 3; Double Gradient Algorithm (DGA): we adopt the proposed three-step MC-NOMA optimization framework and use the projected gradient descent algorithm (Algorithm 1) in r both Step 1 and 3; Double Pseudo-Gradient Algorithm (DPGA): we apply the designed three-step MC-NOMA optimization scheme and adopt pseudo-gradient descent algorithm (Algorithm 2) in r the Step 1 and Step 3; r LDDP: the near-optimal scheme proposed in [25]; NOMA-SIWA-FTPC: we apply the proposed SIWA to NOMA with FTPC, in which SIWA and FTPC are invoked r to perform subcarrier and power allocation, respectively; OMA-FTPC: we adopt FTPC [2], [24], [42], [43] in orthogonal multiple access (OMA) system.

A. The Convergence Time To begin with, we investigate the convergence time of the proposed power control algorithms. Fig. 1 shows the convergence of SIWA, DGA and DPGA in both Step 1 and Step 3 of Fig. 1. Convergence of the proposed power control algorithms. 11842 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 67, NO. 12, DECEMBER 2018

Fig. 2. Sum of data rates vs. different number of users, M = 2. Fig. 4. Performance loss vs. different number of users, M = 2.

Fig. 3. Sum of data rates vs. different number of users, M = 3. Fig. 5. Performance loss vs. different number of users, M = 3. our three-step resource allocation methodology. The number of users K and the number of multiplexed users M is set to be 10 Fig. 2 and Fig. 3, we choose to compare its sum-rate performance and 2, respectively. applied in the three-step algorithm (i.e., DGA according to our Fig. 1(a) and Fig. 1(b) show the convergence performance of aforementioned definition) to that of LDDP and OMA-FTPC SIWAin Step 1 and Step 3, respectively. We choose several users with different number of users. The number of multiplexed users on an arbitrary chosen instance and use their water levels during M of Fig. 2 and Fig. 3 is set to 2 and 3, respectively. This setting iterations to demonstrate. The x-axis represents the number of is based on an implementation point of view, i.e., reducing the iterations, while the y-axis denotes the water level of each user. receiver complexity and error propagation due to SIC [40]. We can observe that SIWA in both Step 1 and Step 3 takes In Fig. 2, it can be seen that the sum of data rates of each only a few iterations to converge. As expected, the water level scheme increases with the number of users K. For any given K, of each user in Step 3 is monotonically increasing. Fig. 1(c) LDDP has the best performance among the three schemes. Nev- illustrates the convergence of DGA (projected gradient descent ertheless, DGA achieves comparable sum of data rates to that of algorithm) and DPGA (pseudo-gradient descent algorithm) in LDDP,see for example, when K = 20 and M = 2, the proposed Step 1 and Step 3, respectively. The x-axis indicates the number DGA has a performance loss of only 3.14% when compared with of iterations, while the y-axis represents the sum of user data LDDP. In addition, OMA-FTPC has the worst system perfor- rates. It can be seen that these gradient descent based power mance among all. Similar conclusions can be drawn from Fig. 3. control algorithms converge rapidly in both Steps 1 and 3. It is worth mentioning that the proposed scheme becomes more efficient with the increasing of M. For example, when K = 20 B. Sum of Data Rates and M = 3, the performance loss of DGA when compared to In this subsection, we evaluate the sum of data rates perfor- LDDP is reduced to 1.71%. mance of the aforementioned subcarrier and power allocation In Fig. 4 and Fig. 5, we compare the sum of data rates per- schemes. Note that each data point is obtained by averaging over formance of the three-step scheme under DGA, DPGA, DIWA, 2,000 random instances. and NOMA-SIWA-FTPC. The x-axis indicates the number of The purpose of Fig. 2 and Fig. 3 is to compare the perfor- users, while the y-axis represents the sum-rate performance loss mance between NOMA system given by LDDP, our three-step compared to that of LDDP, which is defined as follows: method and the performance in OMA system given by OMA- FTPC. Since projected gradient descent algorithm (Algorithm 1) Rsum (LLDP ) − Rsum , (45) achieves the optimal solution to power control problem (5), in Rsum (LLDP ) FU et al.: SUBCARRIER AND POWER ALLOCATION FOR THE DOWNLINK OF MULTICARRIER NOMA SYSTEMS 11843

Fig. 6. Sum of data rates vs. different number of multiplexed users.

Fig. 7. Number of operations vs. different number of users, M = 2. where Rsum (LLDP ) and Rsum refer to the solutions of LDDP and the three-step algorithm, respectively. Fig. 4 shows the performance loss versus different number of users K of the four proposed schemes, where M = 2. It can be seen that, for fixed K, DGA has the smallest performance loss among these four schemes. Meanwhile, DIWA has a similar performance to that of DPGA. In addition, for any given K, NOMA-SIWA-FTPC has the largest performance loss among all. Fig. 5 represents the performance loss of our proposed resource allocation strategies when compared with LDDP where the number of multiplexed users is assumed to be 3. The similar conclusions can be achieved to the scenario where 2 users are multiplexed on each subcarrier (Fig. 4), which are omitted here. Fig. 6 depicts the sum of data rates of LDDP and our proposed resource allocation algorithms versus the number of multiplexed users M, in which the number of users K is assumed to be 6. It can be seen that with the increasing of M, the performance gap between LDDP and our proposed schemes decreases. For example, when M = 4, the proposed DGA and SIWA have almost Fig. 8. Number of operations vs. different number of users, M = 3. the same performance to that of LDDP, i.e., a performance loss of approximately 0.0016% and 0.0074%, respectively, while 0.1%, 0.2%, and 0.27% of that required by LDDP, respectively. DPGA stagnates at 0.036% and NOMA-SIWA-FTPC has a per- Besides, when K = 20 and M = 3, the needed operations of formance loss of 0.37%. DIWA, DPGA and DGA is around 0.01%, 0.07%, and 0.1% of that needed by LDDP, respectively. C. Number of Operations Fig. 9 compares the number of operations required under different number of multiplexed users M with fixed number of Fig. 7 to Fig. 9 show the number of operations required by users K = 6. We can see that for any given M, the number of the different schemes. For each scheme, we count the number of operations of LDDP is the highest of all. In addition, with the additions, multiplications, and comparisons used, which reflects increasing of M, the computational complexity of LDDP has the time complexity, as an estimation. the fastest growth rate among all the schemes. In Fig. 7 and Fig. 8, the number of multiplexed users M is fixed to 2 and 3, respectively. We see that except OMA-FTPC, D. Impact of the Power Levels J on LDDP’s Performance the number of operations for each scheme increases with the increasing number of users, K. For any given K, DGA has We show in this subsection how the sum-rate and complex- the highest computational complexity among the proposed al- ity of LDDP depends on the number of power levels J and gorithms (DGA, DPGA, DIWA, NOMA-SIWA-FTPC). OMA- the system’s parameters. We present in Fig. 10 and Fig. 11 the FTPC requires the fewest operations of all. In addition, the sum-rate and number of operations of LDDP under different proposed DIWA uses approximately the same number of opera- number of power levels J = 10, 20, 50, 200 and 10K, and com- tions as NOMA-SIWA-FTPC and slightly more operations than pare them to the performance of DGA. As expected from [25], OMA-FTPC but much less operations than DPGA and DGA. when J increases, the sum-rate of LDDP increases until reach- In any case, all of the proposed schemes are much more time ing the optimal, while the computational complexity increases in efficient than LDDP especially when the number of users is O(CNMKJ2). The number of power levels considered here is large. For example, when K = 20 and M = 2, the number of always greater than 10, otherwise LDDP would not have enough operations required by DIWA, DPGA and DGA is less than discrete power steps to serve all N = 10 subcarriers. 11844 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 67, NO. 12, DECEMBER 2018

Fig. 9. Number of operations vs. different number of multiplexed users. Fig. 11. Number of operations for different values of J, M = 3.

respectively 1.04% and 1.74% for J = 50 and J = 10K = 200, while the increase in complexity are approximately 100 and 1000 folds. Moreover, DGA outperforms both LDDP with J = 10 and J = 20 in terms of higher sum-rate and lower complexity. For K = 20, the performance loss of LDDP with J = 10 and J = 20 compared to DGA are respectively 3.20% and 1.54%. In summary, DGA has close to the best LDDP’s sum- rate, while requiring less computational complexity. The latter statement can be explained by comparing the asymptotic complexity of LDDP iteration O(NMKJ2) and DGA iteration O(NK2). As discussed previously, we can assume J = O(min{K, MN}).IfK ≤ MN (which is the case considered in our simulations), LDDP iteration’s complexity can be written as O(NMK3). In this case, DGA requires MK times less operations than LDDP, which explains the results shown in Fig. 10. Sum of data rates for different values of J, M = 3. Figs. 7, 8, 9 and Fig. 11. Otherwise if K>MN, then LDDP iteration’s complexity becomes O(N 3M 3K). This complexity is in practice higher than DGA’s iteration complexity. Indeed, let’s We observe in Fig. 10 that LDDP’s sum-rate for J = 200 consider a 3GPP-LTE use case [44] with a bandwidth of 10 MHz and J = 10K are very close, i.e., they have less than 0.01% divided in N = 50 resource blocks, and let M = 2, DGA re- of difference for any number of users. Hence, LDDP’s sum- mains advantageous in terms of computational complexity as rate does not improve signiﬁcantly by further increasing J over long as K

Fig. 12. Fairness index vs. different number of users, M = 2.

Fig. 14. Fairness index over a scheduling frame, M = 2.

Fig. 13. Sum of data rates vs. different number of users, M = 2. as follows: ( gn )−1 n∈N k ∈K p¯k = n −1 Ptotal,k . (46) j∈K( n∈N gj ) Fig. 15. Sum of data rates over a scheduling frame, M = 2. From Fig. 12 we see that, given the number of users, DGA under proportional power constraints has higher fairness index than deﬁne user k’s average rate prior to time slot t as follows: that of DGA with equal power constraint. For example, when 1 1 K = 20, DGA with proportional power constraints improves R¯k (t)= 1 − R¯k (t − 1)+ Rk (t). (47) Jain’s fairness index by 60% when compared to that of equal T T power constraint scheme. At each time t, LDDP is applied to optimize the weighted sum- Fig. 13 is applied to illustrate the sum rate performance of rate k wk (t)Rk (t), where the weight wk (t) of each user k DGA under different power constraint methods where two users is set inversely proportional to its average rate, i.e., wk (t)= are multiplexed on each subcarrier. For each value of K, DGA 1/R¯k (t). Besides, at each time t, the individual power budgets under equal power budget has better sum rate performance of DGA proportional power constraints are deﬁned similarly to than DGA with proportional power constraints since the lat- (46) with additional weight factors quoted below: ter scheme assigns more power to users with weak channel w (t)( gn )−1 conditions, resulting in sum rate performance degradation, e.g., k n∈N k ∈K p¯k (t)= n −1 Ptotal,k . (48) when K = 20, DGA with proportional power budget has 7.3% j∈K wk (t)( n∈N gj ) sum rate loss when compared to that of equal power budget. Fig. 14 and Fig. 15 present respectively Jain’s fairness index and sum-rate of the aforementioned schemes over a scheduling F. Fairness Performance Over a Scheduling Frame frame, in which DGA equal power constraints is also consid- In this subsection, we compare the fairness performance of ered. Fig. 14 shows that LDDP and DGA proportional power DGA with proportional power constraint to LDDP with weight- constraints outperforms DGA equal power constraints in terms ing factor proposed in [25]. In the same way as in [25], we con- of fairness. This result is consistent with the fact that the latter sider a scheduling frame of T = 20 times slots. The channel state scheme do not take into account previous data rates. information is collected at the beginning of the frame. Let Rk (t) In Fig. 14, DGA proportional power constraints has greater be user k’s achievable data rate at time slot t ∈{1,...,T}.We fairness than LDDP for K<10, and inversely for K ≥ 10. For 11846 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 67, NO. 12, DECEMBER 2018

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Chung Shue Chen (S’02–M’05–SM’16) received the B.Eng., M.Phil., and Ph.D. degrees in information engineering from The Chinese University of Hong Kong (CUHK), Hong Kong, in 1999, 2001, and 2005, respectively. He is a Research Scientist (MTS) with Nokia Bell Labs, Nozay, France. Prior to joining Bell Labs, he worked at INRIA, in the research group on Network Theory and Communications (TREC, Yaru Fu received the B.S. (Hons.) degree in telecom- INRIA-ENS). He was an Assistant Professor with munication engineering from Northeast Electric CUHK. He was an ERCIM Alain Bensoussan Fel- Power University, Jilin, China, in 2011, M.S. (Hons.) low with Norwegian University of Science and Tech- degree in communication and information system nology (NTNU), Norway, and the National Center for Mathematics and Com- from Nanjing University of Posts and Telecommuni- puter Science (CWI), The Netherlands. His research interests include wireless cations, Nanjing, China, in 2014 and Ph.D. degree in communications and networking, optimization and algorithms, 5G, IoT, and electronic engineering from City University of Hong intelligent systems. He was a TPC in international conferences including IEEE Kong, Hong Kong SAR, in 2018. In September 2018, ICC, Globecom, WCNC, PIMRC, VTC, CCNC, and WiOpt. He is an Editor of she started her journey as a Postdoc Research Fellow Transactions on Emerging Telecommunications Technologies and an Associate with Singapore University of Technology and De- Editor for IEEE ACCESS. He also holds a position of Permanent Research Mem- sign, Singapore, under the supervision of Prof. Tony ber at the Laboratory of Information, Networking and Communication Sciences Quek. Her research interests include cross-layer optimization for wireless com- (LINCS) in Paris. He was the recipient of Sir Edward Youde Memorial Fellow- munication, power control, and resource management for NOMA systems. ship and ERCIM Fellowship.