ACCEPTED FOR PUBLICATION ON IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, SEPTEMBER, 2019 1A Joint Solution for Scheduling and Precoding inMultiuser MISO Downlink ChannelsAshok Bandi, Bhavani Shankar Mysore R, Symeon Chatzinotas and Björn OtterstenSnT, the University of Luxembourg, Luxembourg.Email: {ashok.bandi, bhavani.bhankar, symeon.chatzinotas, Björn.Ottersten}@uni.luMinimize the power utilized subject to scheduling and MSINRAbstract—The average performance of the MISO downlink constraints henceforth referred to as PMIN.channel, with a large number of users compared to transmitantennas of the base station, depends on the interference man- The aforementioned criteria are designed to improve theagement which necessitates the joint design of scheduling and complementary aspects of the networks. In all practical wire-precoding. Unlike the previous works which do not offer a less systems, a certain minimum received SINR is requiredtruly joint design, this paper focuses on formulating a problem for the successful transmission of information. In light of this,amenable for the joint update of scheduling and precoding. Novel to enhance the practical relevance, SINR constraints are intro-optimization formulations are investigated to reveal the hiddendifference of convex/ concave structure for three classical criteria duced in these design criteria. The WSR problem improves the(weighted sum rate, max-min signal-to-interference plus noise overall throughput of a network while satisfying the schedulingratio, and power minimization) and associated constraints are constraint and QoS requirement on the scheduled users. On theconsidered. Thereafter, we propose a convex-concave procedure contrary, the MMSINR problem improves the performance offramework based iterative algorithm where scheduling and the poorest user (in terms of SINR) among those scheduled.precoding variables are updated jointly in each iteration. Finally,we show the superiority in performance of joint solution over the Unlike WSR and MMSINR, PMIN optimizes the consumedstate-of-the-art designs through Monte-Carlo simulations. power while meeting the scheduling and SINR constraints.An elaborate discussion on each design is provided in theIndex Terms—User scheduling, Precoding, multiuser,difference-of-convex, convex-concave procedure subsequent sections.The joint design of scheduling and precoding, which wesimply refer to as joint design, is well studied during the lastI. INTRODUCTION decade (see [10] and references therein). Most of the existingWith the adoption of full frequency reuse in the next- literature on the joint design can be classified as:generation cellular networks, interference among the simul-• Non-iterative decoupled approach: In this approach,taneously served users becomes a limiting factor thwarting scheduling and precoding are treated as two decoupledthe achievement of near-optimal capacity [2]–[5]. Linear pre- problems where usually the users are scheduled accordingcoding has been largely used to achieve satisfactory interfer- to some criteria followed by precoding [8], [9], [11]–[13].ence mitigation at low complexity [6], [7]. Moreover, in a• Iterative decoupled approach: In this approach, schedul-network with a large number of users compared to the number ing and precoding are still treated as two separate prob-of BS transmit antennas, user scheduling for simultaneous lems. However, scheduling and precoding parameters aretransmission is pivotal for interference management [8], [9]. refined in each iterate to improve the objective based onOptimizing performance in such a network involves the design the feedback from the previous iterate [14]–[17].of precoding variables and user scheduling. Further, different• Joint formulation with alternate update: In this approach,perspectives to network performance motivate the need to the joint design problem is formulated as a function ofinvestigate multiple figures of merit; these include network both scheduling and precoding [18]–[20]. However, thesethroughput, user Quality of Service (QoS) power consumed formulations are not amenable for the joint update as theamong others. In this context, we address the joint design scheduling variables are coupled to precoding variablesof scheduling and precoding problem for multiuser MISO which inhibits their joint update. Hence, during the solu-downlink channels in single-cell scenario for the following net- tion stage either scheduling constraints are ignored [18]work optimization design criteria: 1) Maximize the weighted or the scheduling and precoding variables are updatedsum rate subject to user’s minimum signal-to-interference alternatively. [19].plus noise ratio (MSINR), scheduling and power constraintsreferred to in the sequel simply as WSR. 2) Maximize the The joint design is a coupled problem where the efficiencyMSINR of the scheduled users subject to scheduling and of the precoder design depends on the interference amongtotal power constraints henceforth referred to as MMSINR. 3) the users which, in turn, is a function of the scheduled users[10]. Hence, the joint update of scheduling and precoding hasThis work has been published in part at Global Conference on Signal and the potential to achieve better performance over the aforemen-Information Processing 2018 [1]This work is supported in part by Luxembourg national fund FNR project tioned approaches [8], [9], [18], [11]–[17]. The authors in [21]PROSAT. shown the user scheduling and power allocation problem to be©c 2019 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, includingreprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, orreuse of any copyrighted component of this work in other works.ACCEPTED FOR PUBLICATION ON IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, SEPTEMBER, 2019 2NP-hard. The joint design problems that are considered in this DC programming problems without SDP transformation andwork encapsulate the problem considered in [21] as a special employing a minimal number of auxiliary variables.case. Hence, the considered joint design problems are NP-hard. The aforementioned discussion reflects on the novelties ofIt is also non-convex due to the constraints on the SINR or rate the paper-based both on problem formulation and its solution.of scheduled users [14]. Hence, the optimal solution entails The contributions of the paper include:an exhaustive search over Boolean space (user scheduling)and further involves the solution of a non-convex precoding • The scheduling is handled through the power of theproblem. The exponential complexity of an exhaustive search precoding vector of the corresponding user, where non-for practical system dimensions motivates a shift towards low- zero power indicates the user being scheduled (and notcomplexity achievable solutions. In this context, we quickly scheduled otherwise). Unlike the previous works [18],review the various relevant works to place ours in perspective. [20], [23], a binary variable is used for upper boundingThe joint design problem to maximize the WSR subject to the power of the precoding vector. This renders thetotal power constraint, which is referred to as the classical formulation amenable to the joint design of schedulingWSR problem, is considered for single cell networks in [8], and precoding.[9], [11]. The channel orthogonality based scheduling followed • With the help of the aforementioned scheduling, theby zero-forcing precoding (SUS-ZF) proposed in [9] is proven joint design problem for WSR, MMSINR, and PMINto be asymptotically optimal for sum rate maximization. design criteria are formulated as mixed-integer non-linearHowever, it is easy to see that SUS-ZF is not optimal for WSR programming (MINLP) in a way that would facilitatewith non-uniform weights and QoS constraints. Similarly, the the joint updates of scheduling and precoding. Here, theclassical WSR is addressed for multicell networks in [14], nonconvexity of the problem stems from rate and SINRs[16], [17] and hierarchical networks in [18]. The joint design in the objective and constraints.problem is also considered for MMSINR in [13] and PMIN • The binary nature of the problem due to scheduling con-in [15]. However, scheduling and precoding are not jointly straints is addressed by relaxing the binary variables intoupdated in the aforementioned works. real values. This is followed by penalizing the objectiveThe coupled nature of binary variables with precoding with a novel entropy-based penalty function to promotevector appears in many other formulations [22], [23] etc. a binary solution for the scheduling variables. This stepFor example, towards maximizing the weighted sum-rate in transforms the optimization into a continuous non-convexa hierarchical network, binary variables associated with users problem.get multiplied to signal power and interference power of SINR • Unlike the classical DC formulation using SDP trans-[18]. Similarly, in [20], a binary variable is multiplied to formation [24]–[26], a novel useful reformulation ofthe rate of the users in the weighted sum-rate maximization the objective and/or SINR constraints are proposed toproblem. Please note that system models and objectives dis- manipulate the joint design as DC programming withoutcussed in [18], [20], [23] are different from each other, and SDP transformation.the emphasis is only on the occurrence of the joint design • Further, a convex-concave procedure (CCP) based itera-(coupled discrete and continuous) nature that prevails in differ- tive algorithm is proposed for WSR, MMSINR and PMINent designs. The multiplicative nature in previous formulations DC problems. A procedure is proposed to find a feasibleprecludes the joint update of scheduling and precoding. To initial point, which is sufficient for these algorithms tothe best of our knowledge, no prior work exists that update converge to a stationary point [27].the scheduling and precoding jointly for the aforementioned • Subsequently, the per iteration complexity of the CCPWSR, MMSINR and PMIN problems. Therefore, we focus based algorithms, is discussed. Further, a fast convergenceon formulating the joint design problem for WSR, MMSINR, behavior of the proposed algorithms is observed throughand PMIN that facilitates the joint scheduling and precoding extensive simulations. Finally, the efficiency of the pro-solutions. posed DC reformulations is compared to the decoupledRevisiting the WSR and MMSINR design problems for solutions using the Monte-Carlo simulations.fixed scheduled users, it is well-known that the problems The rest of the paper is organized as follows. Section IIare non-convex with difficulty to obtain a global solution. presents the system model and problem formulation of WSR,However, efficient suboptimal solutions have been proposed MMSINR, and PMIN problem. The reformulations and al-for WSR in [24] and MMSINR in [25], [26] by formulating gorithm are proposed for WSR in Section III, MMSINR inthese as difference-of-convex (DC) programming problems Section IV and PMIN in Section V respectively. Section VIwith the help of auxiliary variables and semidefinite program- presents simulation results, followed by conclusions in Sec-ming (SDP) transformations and relaxations. However, the tion VII.semidefinite relaxations for WSR and MMSINR often lead Notation: Lower or upper case letters represent scalars,to non-unity rank solutions from which the approximate rank- lower case boldface letters represent vectors, and upper case1 solutions are extracted [24]–[26]. The rank-1 approximation boldface letters represent matrices. ‖·‖ represents the Eu-results in a loss of performance. Moreover, the transformed clidean norm, |·| represen(ts) the cardinality of a set or theproblems have higher complexity than the original problems magnitude of a scalar, (·)H represents Hermitian transpose,due to auxiliary variables and SDP transformations. In this (·)T represents transpose, ab represents a choose b, and R{}work, we pose the WSR and MMSINR joint design as represents real operation, E{} represents expectation operatorACCEPTED FOR PUBLICATION ON IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, SEPTEMBER, 2019 3and ∇ represents the gradient. γi ≥ ̃i, i ∈ S̄,II. SYSTEM MODEL ︸precoding probleConsider the downlink transmission of a single cell MISO ︸ ︷︷︷︷ ︸m for selected users︸system with N users in a cell and a BS with M(≤ N) Joint schedueling and Precoding problem+ |S̄|antennas. Let h ∈ CM×1, w ∈ CM×1 and x denote where βi ∈ R , is weight and W = {w } is the matrixi i i S̄ i i=1the downlink channel, precoding vector and data of user i containing the precoding vectors of users in the set S̄. Noticerespectively. The BS is assumed to transmit independent data that to accommodate the fairness in the designs, weights orto utmost M among N users and E{|x |2} = 1,∀i. Further, let priority factors are introduced through α and β in WSR andin be the noise at user i; the noise realizations at all users are MMSINR problems respectively. Various fairness metrics areiassumed to be independent and characterized as additive white proposed in the literature, e.g. fairness in terms of rates andcomplex Gaussian with zero mean and variance σ2. Further, allocated power are considered at the physical layer. We referit is assumed that perfect channel state information of all the to [28] and references therein for details on fairness.users is available at BS and that the user channels are constant Finally, towards defining the PMIN problem, schedulingduring the transmission. Let y be the noisy received signal exactly K(≤ M) users is considered for the same reasoniof the user i and Ty , [y1, . . . yN ] , H ,H[h , . . . ,h ] , mentioned in MMSINR. With notations defined for MMSINR1 N, T TW [w , . . . ,w ], x , [x , . . . x ] , n , [n , . . . n ] . criteria, the PMIN problem is defined as:1 N 1 N 1 NThe received signal vector y of all users is given by, ∑PPMIN : min min ‖ 2wi‖ s.t. γi ≥ ̃i, i ⊆ S̄. (4)S̄⊆S ︸W 2y = HWx + n, (1) S̄ i∈S̄ ︷︷ ︸Towards defining the WSR problem mathematically, let T = ︸ PMIN probl︷em︷ for selected users ︸{1, . . . N} be the set containing indices of all users and K̄ be a Joint user scheduling and PMIN problemsubset of T with cardinality less than o∑r equal( to)M . Further, The inner optimization in (2), (3), and (4) solves the precod-let K be the collection of all the possible subsets of type K̄; ing problem for the scheduled users. The outer optimization,clearly, the cardinality of K is C , M Ni=0 i . With the on the other hand, ensures scheduling users with a maximumnotations defined, the WSR problem∑is defined as, objective value among all scheduling possibilities. Notice thatP : max max α R (2) the inner and outer optimization are coupled - the design ofWSR i i∀K̄∈K WK̄ precoder depends on the selected set of users, while the user∀i∈K̄∑≥ ∀ ∈ K̄ scheduling depends on the objectives in (2), (3) and (4) which,s.t Ri i, i , in turn, are functions of the precoder [29].‖ ‖2wi 2 ≤ PT , Towards proposing low-complexity algorithms, we begin by︸∀i∈K̄ ︷︷ ︸ addressing the user scheduling through the norm of precoding︸ ︷︷ ︸ vectors as given below{,precoding problem for selected users = 0; user not selected,where γi , ∑Joint scheduling and Precoding problem ‖wi‖2 = (5)| H |2 =6 0; user selected.hi wi , Ri , log2 (1 + γi) andσ2 + |hHw |2 The zero norm of wi ensures that all elements of wi arej=6 i∈K̄ i ji ≥ 0 are the SINR, rate and minimum rate requirement zero. Hence, the user i is not scheduled. Similarly, the non-of the user i respectively and K̄ is the set of scheduled zero norm of the precoder vector wi indicates user i beingusers. Further αi ∈ R+ denotes the weight for ith user scheduled with an assigned power of ‖2wi‖2. In the sequel,offering design flexibility, PT is the total available power, and we focus on the design of low-complexity solutions to theW = {w }|K̄| is the matrix containing the precoding vectors joint design using (5) to achieve better performance than theK̄ i i=1of users of set K̄. decoupled designs.Unlike the WSR design, scheduling of exactly K(≤ M)users is considered in MMSINR formulation. This is because III. WEIGHTED SUM RATE MAXIMIZATIONconstraining the scheduling to utmost K users always leads In (2), the weighted sum rate objective is considered toto the trivial solution of scheduling only one user and an improve the overall weighted throughput of the network. Thus,elaborate discussion is provided at the beginning of Section IV. WSR problem schedules only the users who contribute toLet S̄ be a subset o(f T) with cardinality equal to K. Let S be maximizing the objective. Given sufficient resources, the WSRthe collection of all the possible subsets of type S̄; clearly, the design schedules close to M users as the objective increasescardinality of S is NK . Letting ̃i to the MSINR requirement linearly with the number of scheduled users; on the otherof user i, ∀i, the design problem for MMSINR can then be hand, scheduling of few users with high SINRs only con-defined as, tributes logarithmically to the objective. Hence, the constraintP : max ∑max min {β γ } (3) of scheduling utmost of M users is considered as opposedMMSINR i iS̄⊆S WS̄ i⊆S̄ to the harder constraint of scheduling to exactly M users.s.t ‖ 2w ‖ ≤ P , Besides, the design is flexible to favor users by increasing thei 2 T∈S̄ corresponding weights i.e., αi to relatively larger values overiACCEPTED FOR PUBLICATION ON IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, SEPTEMBER, 2019 4the users. The minimum rate constraints preclude scheduling Similarly ηi = 1 leads to ‖wi‖22 ≤ PT which is a trivialof the users whose rates are not in the range of interest. Since upper bound compared to C4. Hence the constraint C2the scheduling of zero users in also included in the feasible along with C1 contributes only to the scheduling aspectsset, the problem (2) is always feasible. In the sequel, the of the problem.WSR problem (i.e., (2)) is transformed as a DC programming • From the objective and constraint C5, the variable ζiproblem through a sequence of novel reformulations and low- provides a lower bound for 1 + γi.complexity sub-optimal algorithms within the framework of • The constraint C6 ensures MSINR or rate constraint ofCCP. the scheduled users. If user i is scheduled i.e., ηi = 1,from C6, ζi ≥ 1 + ̃i. Similarly, for an unscheduled userA. Joint Design Problem Formulation: WSR i, C6 becomes ζi ≥ 1. In fact for ηi = 0, constraint ismet with equality i.e., ζ = 1 due to C .Letting K̄ to be the set of scheduled users, a tractable i 2• It is easy to see that, at the optimal solution, the con-formulation of (2) using (5) is, ∑ straints C5 and C6 are met with equality.NWSR Novelty of PWSR2 : Novelty of PWSR2 lies in the formulationP1 ∥: max α[ iRi (6)∥ W,∀K̄∈K of scheduling constraint, C2. This reformulation is vital to∥ i=1 ]∥∥∥ the facilitation of the joint update of η and W as discusseds.t. C1 : ‖w∑ 1‖2 , . . . , ‖wN‖2 ≤M, in the sequel. Notice that this formulation differs from those0N in the literature ( [18], [20], [23], [30], [31], etc) where the2C : ‖w ‖ ≤ P , scheduling constraint is handled by a binary slack variable2 i 2 T which multiplies either the precoding vector or the rate ofi=1C : R ≥  , i ∈ K̄. the user, to control the user scheduling. This multiplication3 i inot only makes the constraints non-convex but also makes itRemarks: difficult to obtain the joint update of Boolean and continuous• It is clear from (5) and the definition of `0 norm, that the variables due to the coupling of variables. Moreover, theconstraint C1 imposes restrictions on the total number of constraints C5 and C6 help to reformulate the objective as aselected users to utmost M . We refer to this constraint concave function and connects the minimum rate constraints toas the user scheduling constraint throughout this section. the objective. This reformulation is crucial as it facilitates the• The constraint C2 precludes the design from using a reformulation of PWSR2 as DC programming problem withouttransmission power greater than PT . resorting to SDP transformations [24], [32]–[34].• The constraint C3 imposes a minimum rate required forthe scheduled users. B. A Novel DC reformulation: WSRA Novel MINLP formulation: The problem PWSR1 is combi- A novel rearrangement of SINR constraint C5 in PWSR thatnatorial due to the constraint C1 and2C3, and non-convex due transforms PWSR2 as a DC programming problem without SDPto the objective and constraints C1 and C3. Towards addressing transformation is,the combinatorial nature, we let ηi and ζi to be the binaryNscheduling variable and slack variable associated with user ∑WSRi respectively, and η = [η , . . . , η ]T , ζ = [ζ , . . . , ζ ]T P3 : max f (ζ,η) , αi log (ζi) (8)1 N 1 N W,ζ,ηand ̃ , 2ii − 1,∀i. With the defined notations, a tractablei=1formulation of and of PWSR then takes the form, s.t. C1,C2, C3, C4 and C6 in (7)C1 C3 1 ∑ C5 : Ii (W)∑− Gi (W, ζi) ≤ 0,∀i,NPWSR : max f (ζ,η) , α log (ζ ) (7) where∑I (W) = σ2 + |hHw |22 i i i j=6 i i j and Gi (W, ζi) =W,ζ,ηi=1 σ2N+ j=1|hHi wj |2s.t. C1 : ηi ∈ {0, 1}, ∀i, . Notice that Ii (W) is convex in W,ζi2C2 : ∑‖wi‖ ≤ PT ηi, ∀i, and for ζi > 0, Gi (W, ζi) is also jointly convex in W and ζ2 i.N Hence, (8) is a DC programming problem with combinatorialC3 : ηi ≤M, constraint C1. This is the first attempt at reformulating the∑i=1 WSR towards a tractable form without resorting to SDPN methods or use of additional slack variables thereby renderingC4 : ‖ ‖2wi 2 ≤ PT , a low-complexity solution to the problem.i=1 Beyond SDP based DC formulation: Notice that for fixedC5 : 1 + γi ≥ ζi,∀i, η, the problem PWSR3 becomes a classical WSR maximiza-C : ζ ≥ 1 + η ̃ , ∀i, tion problem subject to SINR and total power constraints6 i i i[24], [32]–[34]. The problem PWSR3 is non-convex due toRemarks: the constraint C5. Although, for fixed ζ (i.e. fixed η), the• The binary nature of ηi (i.e., C1) together with C2 constraint C5 in PWSR3 is formulated as a second-order conedetermines the scheduling of users. In other words, ηi = 0 programming (SOCP) constraint [6], a similar SOCP transfor-leads to a precoding vector containing all zero entries. mation of C5 is not known when ζ is variable. On the otherACCEPTED FOR PUBLICATION ON IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, SEPTEMBER, 2019 5hand, many previous works have exploited the DC structure approximations around the estimate of k−1(W,η, ζ)in WSR maximization problem without SINR constraint in ( ( ) ( ) ( ))[32]–[34] and with SINR constraint in [24] by transforming P̃ (η ) , λ P ηk−1i 1 i + ηi − ηk−1i ∇P ηk−1i ,it into an SDP problem. However, the SDP transformations in[24], [32]–[34], essentially increase the number of variables, G̃i(W, ζ )k−1i , −Gi(W, ζi)− thereby increasing the complexity. Moreover, SDP transforma-  [ ]k−1 N tions also introduce the non-convex rank-1 constraint on the R ∇HG {w −w }i(W, ζ )k−1 l l l=1i k−1 , (10)solutions which is difficult to handle in general; this has led to  ζi− ζi semidefinite relaxations [7] followed by extraction of feasiblerank-1 solutions. where  The problem PWSR3 is still an MINLP with a DC structure  2h hHi i wk−11in the non-convexity. This structure can be leveraged with ζk− 1the optimization tools like CCP. Now, to circumvent the  i .. combinatorial nature of PWSR3 , ηi is relaxed to a box constraint −  . k 1 k−1between 0 and 1, and penalized with P (ηi) so that the relaxed ∇Gi(W, ζi) =  2h Hihi wN  . (11)problem favours 0 or 1. The penalized reformulation of PWSR 3  ∑ζk−1 i with penalty parameter  λ1 ∈ R+ is,  Nσ2 + H k−1 2∑( ) − j=1|hi wj | N ζk−12PWSRi4 : max αi log (ζi) + λ1P (ηi) (9)W,η,ζi=1 • Optimization: The next update k+1(W,η, ζ) is obtaineds.t. C1 : 0 ≤ ηi ≤ 1, ∀i, by solving the following convex problem (which is ob-C ,C , . . . , C in (8). tained by replacing convex part of the objective and2 3 6constraints in PWSR4 with (10) and ignoring the constantWe propose a new penalty function P(η ) , η log η + terms in the objective) :i i i(1− ηi) log (1− ηi) which is a convex function in ηi ≥ 0. ∑N ( ( ))P(ηi) incurs no penalty at ηi = 0 or 1 and the penalty PWSR5 : max αi log (ζi) + λ η ∇P ηk−11 i iincreases logarithmically as ηi drifts away from ηi = 0 or 1 W,ζ,η i=1with the highest penalty at ηi = 0.5. Hence, by choosing λ1 (12)appropriately, binary nature of η is ensured. s.t C1,C2, C3, C4 and C6 in (9)( Now, notice that the objective in∑PWSR∑ 4 (a differe)nce C5 : Ii (W)− G̃i(W, ζi) ≤ 0,∀i.of concave functio)ns i.e. Nf (ζ,η) = i=1 αi log (ζi) −− Ni=1 λ1P (ηi) and constraints are convex and DC.Remarks:Hence, the problem PWSR is a DC programming problem. In • Note that the proposed JSP-WSR algorithm is based4the sequel, a CCP based algorithm is proposed [35]. on CCP framework hence a feasible initial point (FIP)is sufficient for the CCP procedure to converge to astationary point [27].C. JSP-WSR: A Joint Design Algorithm • Given the binary nature of η, at convergence, the resultingstationary point is a valid feasible solution to the originalIn this section, we propose a CCP based iterative algorithm problem PWSR. As mentioned previously, with appropri-to the DC problem in (9) which we refer to as JSP-WSR. CCP 1ate λ1 a stationary point with binary η can be obtainedis a powerful tool to find a stationary point of DC program- easily from the above iterative procedure.ming problems. Within this framework, an iterative procedure In many cases, obtaining a FIP is difficult. However, in theis performed, wherein the two steps of Convexification and next section, we propose a method which promises to obtainOptimization are executed in each iteration. In the convexifi- at least one FIP.cation step, a concave optimization problem is obtained fromPWSR4 by linearizing the objective and constraints. Hence,by definition, the modified objective and constraints lower D. Feasible Initial Point: WSRbound the actual objective and constraints of PWSR4 where CCP is an iterative algorithm and an initial feasible pointthe lower bound is tight at the previous iteration [35]. The guarantees the solutions of all iterations remain feasible.optimization step then solves the convex sub-problem globally. A trivial initial FIP is obtained by the initializing {wi =Thus, the proposed JSP-WSR algorithm iteratively executes 0}Ni=1,η = 0 and ζ = 1 where, 1 and 0 are the columnthe following two steps until convergence: vectors of length N with all ones and zeros respectively. Since• Convexification: Let k−1(W,η, ζ) be the estimates of the quality of the solution depends on the FIP, the harder task∑W,η, ζ in iteration k − 1 and Gi(W, ζ ). In itera-of finding a better FIP is considered through the followingition k, the convex part of the objective in PWSR i.e., iterative procedure.4Ni=1 λ1P (ηi), and the concave part of constraint C5 in • Step 1: Initialize η = η̂ that satisfies constraints C1 andPWSR4 for user i are replaced by their first order Taylor C3 in PWSR4 , and 0 < δ < 1.ACCEPTED FOR PUBLICATION ON IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, SEPTEMBER, 2019 6• Step 2: Solve the following optimization: Algorithm 1 JSP-WSRP : {Ŵ} : find W (13) Input: H, [1, . . . , N ] , PT ,∆, η0,W0, λ = 0, k = 1;FESWSR 1∥[ 2 Output: W,ηs.t. C̃1 : ∥‖w∥ i‖2 ≤ η̂iPT , ∀i, ]∥∥∥ while |PWSR5 (k)− PWSR5 (k − 1) |≥ ∆ do∥ H{ H } ∥ ≤ h√i wi ∀ Convexification: Convexify the problem (10)C̃2 : σ . . . hi wj j 6=i . . . , i,2 η̂ĩi Optimization: Updatek(W,η, ζ) by solving PWSR5WSRC̃ :R{hHw } ≥ 0,∀i, Update : P5 (k) , λ1, k3 i i end whileC̃4 :={hHi wi} == 0,∀i,2C̃ : ‖W‖ ≤ P . const(raints. Hence, the compu)tational complexity of PWSR55 2 Tis O 3(NM + 2N) (4N + 2) [36]. Similarly, the computa-• Step 3: If PFESWSR is feasible go to step 4 else update tional complexity of the proposed procedure in Section III-Dη = δη̂ and go to step 2. to obtain a FIP depends on the per iteration complexity of• Step 4: Let Ŵ be the solution of PFESWSR. Choose ζ̂i PFESWSR. PFESWSR is a convex problem with MN decisionsuch that 1 + η̂ĩi ≤ ζ̂i ≤ 1 + γ̂i where γ̂i is the SINR of var(iables, 2N + 1 co)nvex constraints and 2N linear con-the user i calculated using Ŵ. straints. Hence, the computational complexity of PFESWSR isRemarks: O 3(MN) (4N + 1) [36].• Notice that the updates of η̂ are always feasible. Differentη̂ in step 1 which satisfy the constraint C1 and C3 in IV. MAX MIN SINRPWSR4 may lead to different FIPs. Similarly, differentchoices of δ ∈ (0, 1) in step 1 may also lead to different In this section, we focus on the development of a low-FIPs. complexity algorithm for the MMSINR problem defined in (3).• The optimization problem in Step 2 is only a function of Dropping a user with low SINR clearly improves MSINR. ItW since η is fixed apriori and ζ can be calculated easily also reduces the interference to the other users and the powerfrom the solution of P as given in step 4. of the dropped user can be used to further improve the MSINRFESWSR• Following [6], the MSINR constraint, i.e. γ ≥ η̂ ̃ is of other users. Hence, the constraint of scheduling utmost Ki i ireformulated as a second-order cone (SOC) constraint as users leads to the global solution which has highest MSINRgiven in C̃ with the help of C̃ and C̃ . which is achieved by scheduling only one user. To avoid this,2 3 4• If P in step 2 is in-feasible for η in step 1, update scheduling exactly K users is considered for MMSINR design.FESWSRη as given in step 3 and repeat step 2. This is repeated Besides the scheduling constraint, the MSINR requirements ofuntil P in step 2 becomes feasible. the scheduled users are also considered. Without the MSINRFESWSR• If the initial iterates fail to result an non-zero based initial requirement, the design becomes superficial as the solutionfeasible point, the proposed method eventually lead to might include zero SINR or SINR values which are not usableη̂ = 0 and thus PFESWSR in step 2 becomes feasible with in practice.Ŵ = 0. Hence, the proposed methods always results an Infeasibility of MMSINR: The infeasibility of the problemFIP. By initializing η̂ close to 0, FIP can be obtained in due to the MSINR requirement is explained in [6] for fixed setfewer iterations. of users. Similarly, it may not be possible to find exactly K• The FIP obtained by this procedure may not be feasi- users while satisfying an arbitrarily chosen MSINR, power andble for the original WSR problem P in (2) unless system dimension constraints [6]; this renders the problem (3)∑ WSRP becomes feasible for {η̂ ∈ {0, 1}}N satisfying infeasible. In this work, it is assumed that problem PFESWSR MMSINRi i=1N has at least one feasible solution for the given scheduling andi=1 η̂i ≤M .• Although the FIP obtained by this method is not feasible MSINR constraints. Considering this, a low-complexity sub-for P optimal algorithm using the frame work of CCP is developedWSR, the final solution obtained by JSP-WSR withthis FIP becomes a feasible for P since the solution for the MMSINR problem in the sequel.WSRsatisfies the scheduling and SINR constraints of PWSR.Letting PWSR5 (k) be the objective value of the problem PWSR A. Joint Design Problem Formulation: MMSINR5at iteration k, the pseudo code of JSP-WSR for the joint design A tractable mathematical formulation of (3) is,problem is given in algorithm 1.PMM1 : ∥max min {β γ } (14)E. Complexity: WSR s.t. ∥∥[i iW i={1,...,N} ]∥C1 : ‖w1‖2 , . . . , ‖wN‖2 ∥∥ == K,The computational complexity of JSP-WSR depends on the 0Ncomplexities of iterative procedures proposed in Section III-C ∑ 2C : ‖w ‖ ≤ P ,and Section III-D. The proposed JSP-WSR in Section III-C 2 i 2 Ti=1is a CCP based iterative algorithm; hence, the complexity ( )C : β γ ≥ 1 ‖w ‖ ̃ ,∀iof the algorithm depends on complexity of the sub-problems 3 i i i 2 i ( )PWSR5 . The convex problem PWSR5 has (NM + 2N) decision where 1 is an indicato(r funct)ion with 1 ‖wi‖2 = 0 ifvariables and (2N + 1) convex constraints and 2N + 1 linear ‖wi‖2 = 0 otherwise 1 ‖wi‖2 = 1.ACCEPTED FOR PUBLICATION ON IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, SEPTEMBER, 2019 7∑The SINR γi is non-convex and piece-wise minimum of where I (W) = σ2 + H 2i j=6 i|hi wj | , Hi (W, ηi, t) ={γ }Ni i=1 is also non-convex. So, PMM1 maximizes a non-convex Ii (W) + β |hHw |2i i i Ii (W)objective subject to a combinatorial constraint C ; this is and Li (W, t) = . Notice that,1 t+ ηi tgenerally a NP-hard problem. Moreover obtaining a global given t > 0, Li (W, t) is jointly convex in W and t andsolution to PMM1 requires an exhaustive search over all the Hi (W, ηi, t) is also jointly convex in W, ηi and t. Hence,possible sets and solving the classical MMSINR problem for (16) is a DC constraint.each set. Towards addressing combinatorial constraint C1, followingA Novel Reformulation: In the classical MMSINR prob- the approach in Section III, the binary constraint ηi is relaxedlem, for the predefined selected users, SINRs of all users to a box constraint between 0 and 1 and ηi is penalized withis addressed with a slack variable, say s, that lower bounds J Ii (W) + β |hHw |2P i i(ηi). Letting i (W, ηi, t) = i , for theβiγi, ∀i i.e., {βiγi}Ni=1 ≥ s [37], [38]. However, this approach 1 + ηĩicannot be applied to the present joint design problem since sake of completion, with the help of variable t, (16) andthere always exist N −K unscheduled users whose SINR is penalization approach proposed in Section III, the problem{ }N PMMidentically zero. Therefore, lower bounding all β γ with 2 is reformulated as,i i i=1s, makes the problem trivial and the solution, say s∗, is always PMM : min t− λ2P (ηi) (17)zero. Letting ηi to be a binary variable associated to user i and3W,η,tS to be the set of scheduled users and adopting the epigraph s.t. C1 : 0 ≤ ηi ≤ 1,∀i,formulation, an equivalent formulation of PMM1 is, C2, C3, C4 in (15),PMM2 : max s (15) C5 : Ii (W)− Ji (W, ηi, t) ≤ 0,∀i,W,η,ss.t. ∈ { } ∀ C6 : Li (W, t)−Hi (W, ηi, t) ≤ 0, ∀i,C1 : ηi 0, 1 , i,2 C7 : t > 0,C2 : ‖∑wi‖2 ≤ ηiPT ,N where λ2 ∈ R+ is a penalty parameter of the design.C : η == K, The problem PMM3 i 3 maximizes a convex objective subject to∑i=1 convex and DC constraints. Hence PMM3 is a DC problem andN a CCP based algorithm could be solved with an FIP obtained2C4 : ‖wi‖2 ≤ PT , from Section IV-D . However, the strict equality constrainti=1 C in PMM3 3 , limits the update of the η. In order to allow theC5 : βiγi ≥ ηĩi,∀i, flexibility in choosing η, the following problem is consideredC instead,6 : βiγi ≥ ηis,∀i.  2Remarks: ∑NMM• Constraint C5 is the MSINR constraint equivalently writ- P4 : min t− λ2P (ηi) + Ω ηi −K (18)W,η,tten with the help of ηis. i=1• The variable s in C6 is active only when ηi = 1. For s.t. C1,C2, C4, C5, C6, C7 in (17),example, when user i not scheduled i.e., ηi = 0, its where Ω ∈ R+ is a penalty parameter. It is easy to see thatSINR is lower bounded by 0 which is always satisfied choosing the appropriate Ω (usually higher value) ensures theby the SINR definition. Similarly, when user i scheduled equality constraint. The problem P MM is also a DC problemi.e., ηi = 1, its SINR is lower bounded by s. Thus4and a CCP based algorithm, JSP-MMSINR, is proposed in thethe maximization of s optimizes the MSINR of only the sequel to solve it efficiently.scheduled users.B. A Novel DC reformulation: MMSINR C. JSP-MMSINR: A Joint Design AlgorithmThe problem PMM2 is a MINLP where the non-convexity isIn this section, we propose a CCP framework based iterativeMMdue to constraints C5 and C6, while the combinatorial naturealgorithm to the problem P4 , which is referred to as JSP-is due to constraint . Similar to constraint of PWSR, MMSINR, wherein the JSP-MMSINR executes the followingC1 C5 4constraint C5 of the problem PMM Convexification and Optimization steps in each iteration:2 can be formulated as a DC k−1constraint. However, the same approach cannot be applicable • Convexification: Let (W,η, t) be the estimates ofto constraint C in PMM as η and s are both variables. Wi, ηi, t in iteration k−1. In iteration k, the concave part6 2 i MMMoreover, to the best of our knowledge DC reformulation of of C5 and C6 for user i in P4 i.e., −Hi(W, ηi, t) andconstraints of type C in PMM6 2 is not known. In this section, −Ji(W, ηi, t) are replaced by its affine approximationk−1a novel procedure is proposed to transform constraints of type around (W,η, t) which is given by,C in PMM6 2 as DC constraints which involves the change of1 H̃k−1 k−1i (W,η, t) , −Hi (W,η, t)variable s by followed by rearrangement as described below,   t {w −wk−1}Nl− l l=1η H k 1 k−1 i1 + βiγi ≥ 1 + ⇒ Li (W, t)−Hi (W, ηi, t) ≤ 0, (16)− R∇ Hi (W,η, t)  ηi − ηi  ,t t− tk−1 ACCEPTED FOR PUBLICATION ON IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, SEPTEMBER, 2019 8J̃ k−1 k−1i (W,η, t) , −Ji (W,η, t)  the latter case, smaller δ leads to a FIP in few iterations and larger δ takes longer iterations to find a FIP.− R∇HJ k−1(W,η, t) {w −wk−1l l }N l=1η − ηk−1  , Notice that a FIP obtained from this process is only feasiblei i it− tk−1 to problem PMM4 but not to the problem PMMSINR since itviolates scheduling constraint and binary constraint of η.(19) However, appropriate adaptation of the penalty parameters λ2where ∇H k−1 k−1(W,η, t) and ∇J (W,η, t) are the and Ω (e.g. monotonic increment) ensures that the obtainedi ievaluated gradients of H (W,η, t) and J (W,η, t) final solution from algorithm 2 is always feasible to PMMSINR.i iat k−1MM(W,η, t) respectively. The expressions for Letting P5 (k) be the objective value of the problem PMM5∇H k−1(W,η, t) and ∇J k−1(W,η, t) can be ob- at iteration k, the pseudocode of JSP-MMSINR for the jointi itained by following (11). Similarly, the first order Taylor design problem is given in algorithm 2.series approximation of the objective in PMM4 after ignor-ing the constant terms,   Algorithm 2 JSP-MMSINR∑ ( ) ∑ 2 Input: H, [̃1, . . . , ̃ ] , P 0 0N T ,∆, η , W , λ1 = 0, k = 1;N  N  Output: W,η, tF (t,η) = t−λ η ∇P ηk−12 i i + Ω ηi −K while |PMM7 (k)− PMM7 (k − 1) |≥ ∆ doi=1 i=1 Convexification: Convexify the problem (19)Optimization: The update k is obtained by solv- Optimization: Updatek(W,η, t) by solving PMM• (W,η, t) 5ing the following convex problem: Update : PMM7 (k) , λ2, k;end whilePMM5 : max F (t,η) (20)W,η,ts.t. C1, C2, C3, C4 in (18)− E. Complexity: MM-SINRk 1C5 : Ii (W, t) + J̃ (W,η, t) ≤ 0, ∀i, Similar to JSP-WSR, the computational complexity of JSP-L H̃ k−1C6 : i (W, t) + (W,η, t) ≤ 0, ∀i. MMSINR depends on the complexities of iterative proceduresproposed in Section IV-C and Section IV-D. The proposedD. Feasible Initial Point: MM-SINR JSP-MMSINR in Section III-C is a CCP based iterativealgorithm; hence, the complexity of the algorithm dependsNotice that JSP-MMSINR is a CCP framework based al- on complexity of the sub-problems PMM5 . The problem PMMgorithm and hence a FIP is sufficient for the algorithm to 5has (NM +N(+ 1) decision variables, 2N)+ 1 convex andconverge to a stationary point [27]. Unlike WSR problem,MM 2N+1 linear constraints, hence the computational complexityobtaining a trivial FIP to the problem P4 is difficult as MM 3initializing to all zeros results in zero SINR for all the users of P5 is O (NM +N + 1) (4N + 2) . Similarly, theWand thus t = 0 where later is the violation of the constraint computational complexity of the procedure in Section IV-DC . However, one may find a FIP by the following iterative depends on the per iteration complexity of problem in step5procedure. 2 which is a convex problem with MN decision variables,2N + 1 convex constraints and 2N linear constraints. Hence,• Step 1: Initialize η = η̂ that satisfies constraints C1 andMM the computation(al complexity of )problem in step2 in Sec-C3 in P4 . 3• Step 2: For a fixed η, ignoring the constraints dependent tion IV-D is O (MN) (4N + 1) [36].on t, PMM4 can be reformulated as a convex problem by[6] or [7]. This would result similar formulation as shown V. POWER MINIMIZATIONin PFESWSR; hence omitted due to space limitation. Let In this section, we consider the joint design problem withŴ be the solution from this step. the objective of minimizing the sum power consumed at the BS• Step 3: Exit the loop if Ŵ from step 2 is feasible and subject to scheduling of K users whose MSINR requirement1t0 = else set η = δη̂ and continue to step 2. is met. As mentioned previously, scheduling utmost K usersmini{ηĩi} leads to the trivial solution of no users being scheduled whichRemarks: results in zero consumed power.• By construction, the initial η̂ from step 1 is alwaysfeasible to PMM4 . A. Joint Design Problem Formulation: PMIN• Efficient algorithms to solve the convex precoding prob-lem in step 2 is proposed in [6] and [7], and is solvable Similar to Section IV, the user scheduling is handledglobally using tools like CVX. through the norm of the precoder as shown in (5). With the• The number of iterations that are needed to obtain a FIP help of (5) and notations defined, and letting S̄ to be the setfrom above procedure depends on η̂, δ and K. Suppose, of scheduled users, a tractable formulation of PPMIN solely asif the initial η̂ ≈ 0, a FIP is obtained in one iteration with a function of precoding vectors as follows:high probability. Similarly, if the initial η̂ ≈ ∑1 the solution PPMIN 2: min ‖Wi‖ (21)from above can be infeasible in the initial iterations. For 1 S̄ 2W,i∈S̄ACCEPTED FOR PUBLIC∥ATION ON IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, SEPTEMBER, 2019 9s.t. ∥C1 :∥[ ]∥‖w1‖ ∥2 , . . . , ‖wN‖2 ∥ == K, for PMIN (JSP-PMIN) algorithm executes the following two0≥ ∈ S̄ steps iteratively until the convergence:C2 :γi ̃i, i .• Convexification: Let k−1(W,η) be the estimates ofThe problem PPMIN1 is combinatorial due to the constraints (W,η) in iteration k − 1. In iteration k, the concaveC and C and also non-convex due to {γ }N in constraint part of C in PPMIN1 2 i i=1 3 4 for user i i.e., −fi(W, ηi) isC2. Letting Υ ∈ R+ to be a constant, a mathematically replaced by its affine approximation around the estimatetractable formulation that allows us to design a low-complexity of k−1(W,η) which is given by,algorithm is  k−1f̃(W, ηi) , −f (W, η[i) −PPMIN 2 2 : min ‖W‖2 (22) ]W,η k−1 N ∈ { } ∀ R∇H k−1 {w −w }f (W, η ) l l l=1 . (25)s.t. iC1 : ηi 0, 1 , i, ηi − ηk−1i C2 : ‖ 2∑wi‖2 ≤ ηiΥ, ∀i,N • Optimization: Update k(W,η) is obtained by solving theC : η == K, following convex problem:3 i i=1 ∑ 2NC4 : γi ≥ ̃iηi, ∀i. PPMIN5 : min ‖W‖22 + µ ηi −KW,ηRemarks:• For η = 1, Υ in C provides upper bound on the power ∑i(=1N )i 2 − λ3 ηi∇P ηk−1i (26)of user i. Moreover, a lower bound on Υ would be thei=1total system power. s.t. C1 :0 ≤ ηi ≤ 1, ∀i,A DC reformulation: The problem PPMIN2 is an MINLP 2due to combinatorial constraint C1 and non-convex constraintC2 : ‖wi‖2 ≤ ηiΥ,∀i,C4. Similar to WSR and MMSINR problems, using the DC C3 :Ii (W) + f̃(W , η )k−1i i ≤ 0, ∀i.formulation of constraint C4 and penalization method for C1,PPMIN The convex problem PPMIN5 has (NM +N) decision vari-the DC formulation of the problem 2 is,∑ ables, 2N convex and 2N linear co(nstraints, hence the )com-N putational complexity of PMM is O 35 (NM +N) (4N) .PPMIN3 : min ‖ ‖2W 2 − λ3 P (ηi) (23)W,ηi=1≤ ≤ ∀ C. Feasible Initial Point: PMINs.t. C1 : 0 ηi 1, i,An initial feasible point, which suffices the convergence ofC2, C3 in (22), JSP-PMIN to a stationary point [27], for the problem PPMIN5C4 : Ii (W)− fi (W, ηi) , ∀i, is obtained by the following iterative procedure.PMINwhere λ ∈ R+ is the penalty parameter and f (W, η ) = • Step 1: Initialize η = η̂ that satisfies C1 and C3 in P .3 i i 4I | H |2 • Step 2: The precoding problem of PPMINi (W) + h w 4 for fixed η cani i .1 + ̃ η be reformulated as a convex problem by [6] or [7]. Leti iThe problem PPMIN3 is a DC problem which can be solved Ŵ be the solution from this step.using CCP. However, finding a FIP becomes difficult as for • Step 3: Exit the loop if Ŵ is feasible (see [6]) else setchosen η, PPMIN may become infeasible [6]. For the ease η = δη̂ and continue to step 2.3of finding an FIP, the constraint C in PPMIN2 is relaxed and Notice that a FIP obtained above may not be feasible to4penalized as follows:   PPMIN2 since it may violate binary and scheduling constraints.∑ ∑ However, the adopted penalty methods ensure the scheduling 2N N and binary constraints. Hence, the final solution obtained fromPPMIN 24 : min ‖W‖2 − Ω P (ηi) + µ ηi −K 3 is always a feasible solution to PPMIN2 .W,ηi=1 i=1 Letting PPMIN5 (k) be the objective value of the problem(24) PPMIN5 at iteration k, The pseudo code of the algorithm iss.t. C1,C2, C4 in (23) illustrated in the algorithm 3.where µ > 0 is penalty parameter. Notice that for the VI. SIMULATION RESULTSappropriate µ, equality constraint is ensured. Moreover, Theproblem PPMIN is a DC problem which solvable using CCP. A. Simulation Setup4In this section, we evaluate the performance of the proposedalgorithms for the MMSINR, WSR and PMIN problems.B. Joint Design Algorithm: PMIN The system parameters and benchmark scheduling methodIn this section, following the CCP framework proposed discussed in this paragraph are common for all the figures.in Section IV-C, the CCP based algorithm for PMIN is Entries of the channel matrix, i.e., {hij}s are drawn from theproposed. The proposed joint scheduling and precoding (JSP) complex normal distribution with zero mean and unit varianceACCEPTED FOR PUBLICATION ON IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, SEPTEMBER, 2019 10Algorithm 3 [ TABLE I: Summary of the acronyms of different benchmark algorithms basedJSP-PMIN ] on design criteria.Input: H, ¯̃1, . . . , ¯̃N ,∆, η0, W0, λ3 = 0, k = 1;Output:W,η Design crite- Scheduling Precoding schemeswhile |PPMIN5 (k)− PPMIN5 (k − 1) |≥ ∆ do ria schemesConvexification: Conve(xify the )problem (19) WSR: SUS: Semi WSR: proposedOptimization: Update Wk,ηk by solving PPMIN weighted orthogonal user precoding; WSR-Z:5Update : PPMIN (k) ,Ω, k sum rate scheduling proposed precoding5end while with a trivial FIP;RWSR: referenceand noise variances are considered to be unity i.e., σ2 = 1. precoding [24]Simulation results in all the figures are averaged over 500 MMSINR: WSUS: MMSINR: proposeddifferent channel realizations (CRs). The penalty parameter Max-Min Weighted precoding schemeλ1 is initialized to 0.5 and incremented as λ1 = 1.1λ1 until SINR SUSλ1 ≤ 10. For all the simulations of MMSINR and PMIN, PMIN: Power RUS: Random PMIN: referenceK is chosen as M . By the nature of MMSINR (PMIN) Minimization user scheduling precoding in [7]design, dropping the user with the lowest SINR (higher power)leads to a better objective. This phenomenon continues until MMSINR, ES-MMSINR, SUS-MMSINR, and WSUS-it drops N −M users and can not drop any further due to the MMSINR respectively. Similarly, RUS, ES, SUS andscheduling constraint. Since, this naturally enforces the binary WSUS based scheduling followed by the SDP basednature of η, λ2 = 0 (λ3 = 0) in MMSINR (PMIN) still power minimization proposed in [7] is used for PMINyields the binary η which is shown Section VI-D and VI-E. precoding problem and is referred to simply as RUS-Hence, λ and λ are fixed zero in all iterations. The penalty PMIN, ES-PMIN, SUS-PMIN, and WSUS-PMIN re-2 3parameters Ω and µ are initialized to 0.01 and incremented as spectively. For the ease of reference, acronyms used forΩ = 1.2Ω and µ = 1.2µ in each iteration until Ω ≤ 20 and benchmark (BM) algorithms are tabulated in Table I.µ ≤ 20. – An SDR version of DC formulation proposed in [24]also used for solving the precoding for the scheduledusers in WSR case as a reference hence is referred toB. Benchmark algorithms as RWSR. WSUS combined with RWSR is referred toTo evaluate the performance of the proposed JSP algorithms as WSUS-RWSR.- due to the lack of a comparable joint solution - the following • In step 3: If the precoding problem in step 2 is infeasiblebenchmarks (iterative decoupled solutions that execute the exit the loop else drop the user with least orthogonalityfollowing steps in sequence) are devised: and repeat step 2 for an updated set of scheduled users.• In step 1, users are scheduled according to proposed However, the precoding problems for MMSINR andweighted semi-orthogonal user scheduling (WSUS) or PMIN are assumed to be feasible.exhaustive search-based user scheduling (ES) or randomuser scheduling (RUS). The considered WSUS is an ex- C. WSR Performance Evaluationtension of the SUS algorithm proposed in [9]. In SUS, the In figure 1a, we compare the performance of JSP-WSR asusers are selected sequentially based on the orthogonality a function of N varying from 15 to 30 in steps of 5 forof their channels with those of already scheduled ones. M = 10, PT = 10dB and ̃i = 4dB, ∀i. Weights {α }Ni i=1In WSUS, orthogonality indices calculated according to are randomly drawn from the set { kN }, k = 1, . . . , N . InSUS are multiplied with their associated weights and figure 1a, RUS-WSR, SUS-WSR, WSUS-WSR and WSUS-the user with the highest weighted orthogonality index RWSR are the decoupled benchmark algorithms. The JSP-is scheduled. This process is repeated until M users are WSR initialized with a trivial solution (W0 = 0, η0 = 0)scheduled. is referred to as JSP-WSR-Z and JSP-WSR initialized with an• In step 2, the precoding problem for the scheduled users FIP obtained from Section III-D continues to be referred to asis solved by the following methods: JSP-WSR. From figure 1a, it is clear that the joint solution– It is easy to see that, retaining only the terms cor- JSP-WSR outperforms all the other decoupled benchmarks.responding to scheduled users by substituting corre- Although JSP-WSR, RUS-WSR, SUS-WSR, and WSUS-WSRsponding ηis to 1 (rest are made zero) and ignoring the have the same underlying precoding algorithm, JSP-WSRconstraint solely dependent on ηis in (9), (18) yields achieves better performance as it jointly updates schedulingthe DC formulation of the precoding problem for the and precoding. Considering weights into scheduling in WSUS-scheduled users for WSR and MMSINR respectively. WSR improves over SUS-WSR, as shown in figure 1a, but itThese problems can be solved using CCP with a FIP still outperformed by JSP-WSR. However, the gains diminishobtained from PFESWSR and PFESPMIN by substitut- as N increases as the probability of finding nearly orthogonaling corresponding ηis with 1. RUS, ES, SUS and user channels (for the considered Gaussian model) increases;WSUS combined with this proposed WSR is simply this implies that the user scheduling has minimum impactreferred to as RUS-WSR, ES-WSR, SUS-WSR, and on performance. Hence, WSUS-WSR performs close to JSP-WSUS-WSR respectively and for MMSINR as RUS- WSR for N relatively larger than M . However, the gainsACCEPTED FOR PUBLICATION ON IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, SEPTEMBER, 2019 1120 Figure 1b illustrates the complexity of algorithms as a func-tion of running time in seconds. Notice that the running time18 includes the time to calculate the FIPs and the final solutions.In the decoupled algorithms i.e., WSUS-WSR and WSUS-16 RWSR the complexity of scheduling algorithms is negligiblecompared to the latter precoding problem. Since the precodingis always performed on M users, the precoding complexity14 JSP-WSRWSUS-WSR of RUS-WSR, WSUS-WSR, and WSUS-RWSR is only aWSUS-RWSR function of M . On the contrary, joint design algorithms, JSP-12 SUS-WSRJSP-WSR-Z WSR, JSP-WSR-Z operate on N users hence the complexityRUS-WSR increases with N . However, due to the efficiency in the design10 of JSP-WSR, its complexity can be comparable to that of15 20 25 30Users Per Cell WSUS-WSR for relatively low values of N , e.g. N = 15.In table III, the number of iterations needed for JSP-WSR(a) to converge to a stationary point is illustrated as a function ofM = 10, {̃i = 4dB}Ni=1, PT = 10 dB, and N is varied from200 15 to 30 (in steps of 5). The per-iteration complexity of JSP-180 JSP-WSR WSR is known to be polynomial and Table III confirms theWSUS-RWSR number of iterations that are need for JSP-WSR to converge160 WSUS-WSRRUS-WSR is also in O (N). Figure 1-b shows an approximately linear140 SUS-WSR increase in run-time for JSP-WSR to converge, despite the120 sub-linear increase in number of iterations. This confirms the100 predominant contribution of the per-iteration complexity inPWSR5 to the total run-time. Moreover, table III confirms that80 JSP-WSR, in general, has a fast convergence rate (i.e., low-60 complexity). However, in some scenarios, JSP-WSR may not40 exhibit fast convergence (i.e., high complexity).In table II, the performance of JSP-WSR is compared with2015 20 25 30 ES-WSR and WSUS-WSR for M = 3, {̃i = 4dB}Ni=1,Users Per Cell PT = 10 dB, and N is varied from 5 to 7 in steps of 1.(b) Although the JSP-WSR is guaranteed to converge only to astationary point theoretically, the results in table II confirmsFig. 1: Comparison of different WSR optimization approaches for M =10,{̃ = 4dB}N , P = 10 dB, and N is varied from 15 to 30 (a) Achieved that these stationary points are indeed high-quality solutions.i i=1 TWSR and (b) algorithm run time On the other hand, the shortcomings of the decoupled solutionobtained by JSP-WSR even in comparison with WSUS-WSR i.e., WSUS-WSR, leads to a large performance gap from bothstill amounts up to 28% ( ). Notice that despite the ES-WSR and JSP-WSR.N = 15difference in the rate of growth, all methods benefit from The performance of the JSP-WSR is illustrated for uniformmultiuser diversity to improve SR as increases. weighted case i.e. {αi = 1}Ni=1 in figure 2a as a functionNof N . The performance gain by jointly updating schedulingNotice that JSP-WSR and JSP-WSR-Z are identical ex- and precoding in JSP-WSR over the decoupled SUS-WSR andcept the FIPs. JSP-WSR and JSP-WSR-Z are CCP based SUS-RWSR is clear from figure 2a. However, as N increasesalgorithms hence the performance differentiation depends on (N ≈ 20), SUS schedules the users with strong channel gainsFIP. Figure 1a shows that while a poor FIP like W0 = 0, and least interference; hence SUS-WSR performs close to JSP-η0 = 0 results in worse performance than decoupled solutions, WSR. Despite the efficiency of SUS in the region around N =the FIPs from Section III-D achieves better performance. 20, SUS-RWSR performs poor due to the inefficiency of theThis shows the efficiency of the FIP mechanism detailed in RWSR precoding scheme.Section III-D. In particular, W0 = 0, η0 = 0 is a bad choice Figure 2b illustrates the convergence behavior of the JSP-since it is the solution that achieves lowest WSR i.e., zero and WSR and the convergence of η to binary values as a functionhence the solutions of JSP-WSR-Z are generally the stationary of iterations. The SR obtained in each iteration is shown bypoints around the lowest objective. the red curve while the penalized SR is shown by the blueDespite having the same WSUS scheduling algorithm and curve. As the FIP of JSP-WSR contains a non-binary η, thethe same FIP for precoding, WSUS-WSR outperforms WSUS- solutions obtained in the initial iterations include the non-RWSR due to the difference in precoding algorithms as shown binary η; hence, the difference between SR (red curve) and SRin figure 1a. Although classical WSR can be formulated as a plus penalty (blue curve). However, as the penalty factor (λ1)DC problem using proposed reformulations and also by the increases over the iterations, JSP-WSR favors the solutionsapproach in [24], due to the efficiency of proposed reformu- with binary ηis. As a result, the penalty approaches zero overlations, WSUS-WSR achieves the better objective which is the iterations i.e., P (ηi) ≈ 0,∀i. This behavior is clear fromconfirmed by figure 1a. iteration 8 onwards. Moreover, the convergence behavior ofRun time in Seconds WSR in bps/HzACCEPTED FOR PUBLICATION ON IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, SEPTEMBER, 2019 125.625JSP-MMSINR24 WSUS-MMSINRSUS-MMSINR4.623M=10, N=152221 3.6JSP-WSRSUS-PWSR20 SUS-RWSR19 2.612 16 20 1 2 3 4N SINR levels(a) (a)22 721.7 JSP-MMSINRM=10, N=20 WSUS-MMSINR21 6 SUS-MMSINRRUS-MMSINR20 519 20.9 48 13 1718 3 (log  )i 2 i17  (log  +  P( )) 2i 2 i 1 i16 11 5 9 13 17 1 2 3 4Iteration SINR levels(b) (b)Fig. 2: Comparison of various WSR optimization methods for uniform 65weighted case with M = 10, {̃i = 4dB}Ni=1, PT = 10dB (a) Nvarying from 12 to 20. (b) convergence of the JSP-WSR (with penalty) andconvergence of η to binary for N = 20. 55 JSP-MMSINRRUS-MMSINRM=10, N=20TABLE II: Comparison of various WSR solutions for M =3, {̃ = SUS-MMSINRi 454dB}Ni=1, PT = 10 dB, and N varying from 5 to 7 .WSUS-MMSINRUsers in weighted sum rate in bps/Hz 35cell (N )ES-WSR JSP-WSR WSUS- 25WSR15N = 5 5.83 5.67 5.32 1 2 3 4N = 6 6.51 5.97 5.59 SINR levelsN = 7 6.64 6.33 6.02 (c)the JSP-WSR to a stationary point of PWSR is shown by Fig. 3: Comparison of different MMSINR optimization approaches for P5 T =the convergence of the blue curve which depicts its objective 10dB, {̃i = 0dB}Ni=1 and SINR levels are varied from 1 to 4 (a) N = 15(b) N = 20 (C) algorithm run time.value.average weighted MSINR) as a function of SINR levels.D. MMSINR Performance Evaluation For SINR level 1, 2, 3 and 4, the weight βi associatedFigure 3 illustrates the weighted MSINR of the scheduled with user i is randomly drawn from the sets {1}, {0.5, 1},users (averaged over 500 different CRs and referred to as {0.333, 0.6666, 0.9999} and {0.25, 0.5, 0.75, 1} respectively.TABLE III: Convergence rate of JSP-WSR for M = 10, {̃ = 4dB}N , For example, for SINR levels 2, βi is randomly selected fromi i=1PT = 10 dB as a function of N . {0.5, 1}. Hence the MMSINR requirement of each user is̃i/0.5 or ̃i (also ̃i = 1). Notice that a higher value of βiNumber of users in a Average number increases the likeliness of user i being scheduled.cell of iterations to The performance of JSP-MMSINR is compared with SUS-converge MMSINR and WSUS-MMSINR for M = 10, {̃i =N = 15 161 (0 dB) }Ni=1, PT = 10dB and N = 15 in figure 3a andN = 20 20.3 N = 20 in figure 3b. It is clear from figure 3a and 3bN = 25 22.5 , that the joint solution JSP-MMSINR improves the per-N = 30 24.8 formance over the decoupled design RUS-MMSINR, SUS-Objective SR in bps/HzRun time in seconds Average weighted MSINR in dB Average weighted MSINR in dBACCEPTED FOR PUBLICATION ON IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, SEPTEMBER, 2019 13TABLE IV: Performance of various MMSINR solutions for M =3, SINR6.2level 4, PT = 10dB and N varying from 5 to 7 JSP-MMSINR6 SUS-MMSINRUsers in Average MSINR in dB with 5.8cell (N )5.6ES- JSP- WSUS-5.4MMSINR MMSINR MMSINRN = 5 5.23 5.15 4.63 5.2N = 6 5.95 5.79 5.03 5N = 7 6.71 6.56 5.99 4.812 14 16 18 20NMMSINR, and WSUS-MMSINR. Despite identical underlyingprecoding scheme in JSP-MMSINR, RUS-MMSINR, SUS- (a)MMSINR, and WSUS-MMSINR, the systematic approach ofjoint scheduling and precoder update considering the weights 0.570.55helps JSP-MMSINR to achieve better performance. The naive t +  ( -M)2i iuser scheduling based method i.e., RUS-MMSINR clearly tperforms poorer than other benchmark methods. Although 0.45WSUS-MMSINR achieves better performance over SUS-MMSINR by considering the weights into scheduling, it still0.35performs worse than JSP-MMSINR showing the inefficiencyof decoupled design. The gains obtained by JSP-MMSINRcompared to best performing decoupled method i.e., WSUS- 0.25MMSINR amounts up to 10% (figure 3b, SINR level 4). 1 5 9 13 17 20IterationIn figure 3c, the run time of the algorithms is illustrated (b)as a function of SINR levels for M = 10, N = 20 andPT = 10dB. Figure 3c shows that the gains of JSP-MMSINR Fig. 4: Comparison of MMSINR for uniform weighted case, M = 10, βi =are achieved at the expense of high computational complexity 1, ̃i = 0 dB, ∀i, PT = 10 dB (a) N varies from 12 to 20. (b) Convergence ofthe JSP-MMSINR (with penalty) and convergence of η to binary for N = 20.as illustrated in figure 3c. Moreover, the complexity of JSP-MMSINR increases as SINR levels increase. The increase in WSUS-MMSINR and ES-WSR for M = 3, SINR level 4,SINR levels enforces the inclusion of users with higher SINR PT = 10 dB. The relatively similar performance of JSP-requirement since the users with lower SINR requirement may MMSINR and ES-MMSINR confirms the efficiency of JSP-not be sufficient to schedule exactly M users. Hence, JSP- MMSINR and high-quality nature of stationary points that theMMSINR takes relatively longer time to converge compared JSP-MMSINR converges to.lower SINR levels.The performance of JSP-MMSINR is illustrated for uniform E. PMIN Performance Evaluationweighted case i.e., {βi = 1}Ni=1 in figure 4 for M = 10 and The total power consumed by the scheduled users (for eachPT = 10dB. In figure 4a, the average MSINR is illustrated as a channel realization) is averaged over 500 channel realizationsfunction of N varying from 12 to 18 in steps of 2. The superior (CRs) which is referred to as average total power per CR.performance of JSP-MMSINR over SUS-MMSINR is clear In figure 5, the average total power per CR is depicted as afrom 4a. However, the gains diminish as N increases as the function of SINR levels for M = 10, N = 15 in figure 5a andSUS based solution becomes efficient as mentioned previously. N = 20 in figure 5b. The SINR level 1, 2, 3 and 4 (chosen∑In figure 4b, the convergence behavior of the algorithm and differently than MMSINR design) on the x-axis indicate thatprogression towards achieving exact scheduling constraint i.e., ̃i is randomly chosen from the sets {1}, {1, 2}, {1, 2, 3} andNi=1 ηi == M are illustrated as a function of the iteration {1, 2, 3, 4} for user i respectively. For example, for the SINRnumber. While the blue curve depicts the inverse of MSINR level 2, ̃i for user i is randomly chosen from the set {1, 2}.(i.e., t in PMM4 ) achieved over the iteration, the red curve Figure 5a and 5b clear shows that the joint solutiondepicts the penalized objective where the penalty aims to JSP-PMIN outperforms RUS-PMIN, SUS-PMIN and WSUS-satisfy the constraint of scheduling exactly M users. As FIPs PMIN. Although the precoding problem for the scheduledviolate the exact scheduling constraint, the penalized objective users by RUS, SUS, and WSUS is solved globally using(red curve) is far from the objective (blue curve). However, [7], the inefficient scheduling leads to poorer performanceincreasing the penalty parameter Ω over the iterations until compared to JSP-PMIN. On the contrary, the system design inΩ ≤ 20 ensures the scheduling constraint. This behavior JSP-PMIN helps to gain up to 25% ( SINR level 4, figure 5b)is observed from iteration 8 in figure 4b as the difference in comparison with WSUS-PMIN. In figure 5c, the run timebetween penalized objective and objective is approximately of algorithms is illustrated in seconds as a function of SINRzero. Moreover, the binary nature of η is also achieved over levels. As shown in figure 5c, the performance gains of JSP-the iterations due to nature of MMSINR for fixed λ2 = 0 in MMSINR incur higher computational complexity.figure 3 and 4. In table V, the performance of JSP-PMIN and WSUS-PMINTable IV compares the performance of JSP-MMSINR, is compared with ES-PMIN for M = 5, ̃i ∈ {0, 1, 2, 3},∀i forObjective Average MSINR in dBACCEPTED FOR PUBLICATION ON IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, SEPTEMBER, 2019 1412 TABLE V: Peformance comparison of various PMIN solutions for M =3,SINR level 4 and N is varied from 5 to 7.11.5SUS-PMINWSUS-PMINJSP-PMIN Users in Average power consumed in dB with10.5cell (N )9.5 ES-PMIN JSP- WSUS-PMIN PMIN8.5 M=10,N=15 N = 5 5.4477 5.99 6.2448N = 6 5.116 5.6164 5.87717.5N = 7 3.95 4.5365 5.12706.51 2 3 4 In figure 6b, the convergence behavior of the JSP-PMINSINR levels algorithm (red curve) and the progression towards ensuring(a) the exact scheduling constraint is depicted as a function ofiterations for N = 15. The FIP may include the solutions that10 violate exact scheduling constraint due to which the penalizedRUS-PMINSUS-PMIN objective and objective differs by a large factor in the initial8 WSUS-PMIN iterations. However, the increment in the penalty parameter µJSP-PMINensures the exact scheduling constraint over the iterations. This6 is confirmed by figure 6b, as the difference between penalizedobjective and objective, becomes approximately zero. For the4 reasons at the beginning of this section, λ3 = 0 still achievesM=10, N=20 the binary nature of η over iterations.22.4SUS-PMIN:  = 0 dBi0 2.2 JSP-PMIN:  = 0 dB1 1.5 2 2.5 3 3.5 4 iSINR levels2(b)1.845 1.6JSP-PMIN 1.440RSUS-PMINSUS-PMIN 1.2WSUS-PMIN35115 20 25 3030 N(a)2520 702 260 ||W||  +  (   - M)i i15 21 1.5 2 2.5 3 3.5 4 ||W||50SINR levels(c) 405Fig. 5: Comparison of different PMIN optimization approaches for M = 10, 30PT = 10 dB and SINR levels varying from 1 to 4 (a) N = 15 (b) N = 20 4(c) algorithm run time. 203different N . Despite the theoretical guarantees of convergence 102JSP-PMIN only to a stationary point, JSP-PMIN performs 01 6 6 1111 1166 20close to ES-PMIN as can be observed in table V. This justifies Iterationthe efficiency of JSP-PMIN approach. (b)The performance JSP-PMIN for uniform weighted case (i.e.,Fig. 6: Comparison of different PMIN optimization approaches for M = 10,all users with same MSINR requirement) is illustrated in PT = 10dB and {i = 0dB}Ni=1 (a) N varying from 10 to 30 in steps offigure 6 for M = 10 and {̃ = 1}Ni i=1. In figure 6a, the 5 (b) convergence of the JSP-PMIN (with penalty) and convergence of η toaverage total power per CR in dB is depicted as a function binary for N = 15.of N varying from 15 to 30 in steps of 5. The superiorperformance of JSP-PMIN over SUS-PMIN is clear fromfigure 6a. However, the gains diminish as increases as VII. CONCLUSIONSNthe SUS based scheduling becomes efficient (kindly refer to In this paper, the joint scheduling and precoding problemsimilar discussion on WSR results). was considered for multiuser MISO downlink channels forRun time in seconds Average power in dB Average power in dBObjective Average power in dBACCEPTED FOR PUBLICATION ON IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, SEPTEMBER, 2019 15three performance optimization criteria: weighted sum rate [16] M. Li, I. B. Collings, S. V. Hanly, C. Liu, and P. Whiting, “Multicellmaximization, maximization of minimum SINR and power Coordinated Scheduling With Multiuser Zero-Forcing Beamforming,”minimization. Unlike the existing works, the design is for- IEEE Trans. Wireless Commun., vol. 15, no. 2, pp. 827–842, Feb 2016.[17] M. Kountouris, D. Gesbert, and T. 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Tassiulas, “Convex 7283–7298, Dec 2016.approximation techniques for joint multiuser downlink beamforming andadmission control,” IEEE Trans. Wireless Commun., vol. 7, no. 7, pp.2682–2693, July 2008.ACCEPTED FOR PUBLICATION ON IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, SEPTEMBER, 2019 16A shok BANDI (S18) was born in Kunkalagunta, Björn Ottersten (S’87-M’89-SM’99-F’04) wasAndhra Pradesh, India, in 1988. He received the born in Stockholm, Sweden, in 1961. He receivedM.Tech. degree in electronics and communication the M.S. degree in electrical engineering and ap-engineering from the National Institute of Technol- plied physics from Linkping University, Linköping,ogy (NIT), Tiruchirappalli, India, in 2012. He has Sweden, in 1986, and the Ph.D. degree in electri-worked on Physical layer design and development cal engineering from Stanford University, Stanford,for WLAN 802.11a/n/ac at Imgaination Technolo- CA, USA, in 1990. He has held research posi-gies, Hyderabad, India from 2012 to 2015 and tions with the Department of Electrical Engineering,at National Instruments, Bangalore, India (2015- Linköping University, the Information Systems Lab-2016). He was worked as a project associate in oratory, Stanford University, the Katholieke Univer-the department of ECE from 2016 to May 2017. siteit Leuven, Leuven, Belgium, and the UniversityHe is currently pursuing the Ph.D. degree in electrical engineering with of Luxembourg, Luxembourg. From 1996 to 1997, he was the Directorthe University of Luxembourg. He joined the Interdisciplinary Centre for of Research with ArrayComm, Inc., a start-up in San Jose, CA, USA,Security, Reliability, and Trust, University of Luxembourg, Luxembourg, in based on his patented technology. In 1991, he was appointed a Professor ofJune 2017. He is working on sparse signal Recovery and joint update of integer signal processing with the Royal Institute of Technology (KTH), Stockholm,and non-linear variables in MINLP problems that apprear in for Wireless Sweden. From 1992 to 2004, he was the Head of the Department for Signals,communications within the Project PROSAT(on-board PROcessing techniques Sensors, and Systems, KTH, and from 2004 to 2008, he was the Dean offor high throughput SATellites), funded under FNR CORE-PPP Framework. the School of Electrical Engineering, KTH. He is currently the Director forthe Interdisciplinary Centre for Security, Reliability and Trust, University ofLuxembourg. As Digital Champion of Luxembourg, he acts as an Adviser tothe European Commission.He was a recipient of the IEEE Signal Processing Society TechnicalAchievement Award in 2011, EURASIP Meritorious Service Award 2010, andthe European Research Council advanced research grant twice, in 2009-2013M . R. Bhavani Shankar (M11SM15) received Mas- and in 2017-2022. He has co-authored journal papers that received the IEEEters and Ph. D in Electrical Communication Engi- Signal Processing Society Best Paper Award in 1993, 2001, 2006, and 2013,neering from Indian Institute of Science, Bangalore and seven other IEEE conference papers best paper awards. He has served asin 2000 and 2007 respectively. He was a Post Doc Editor-in-Chief of EURASIP Signal Processing Journal, Associate Editor forat the ACCESS Linnaeus Centre, Signal Processing the IEEE TRANSACTIONS ON SIGNAL PROCESSING, and the EditorialLab, Royal Institute of Technology (KTH), Sweden Board of the IEEE Signal Processing Magazine. He is currently a memberfrom 2007 to September 2009. He joined SnT in Oc- of the editorial boards of EURASIP Signal Processing Journal, EURASIPtober 2009 as a Research Associate and is currently Journal of Advances Signal Processing and Foundations and Trends of Signala Research Scientist at SnT. He was with Beceem Processing. He is a fellow of IEEE and EURASIP.Communications, Bangalore from 2006 to 2007 asa Staff Design Engineer working on Physical Layeralgorithms for WiMAX compliant chipsets. He was a visiting student atthe Communication Theory Group, ETH Zurich, headed by Prof. HelmutBlcskei during 2004. Prior to joining Ph. D, he worked on Audio Codingalgorithms in Sasken Communications, Bangalore as a Design Engineer from2000 to 2001. His research interests include Design and Optimization ofMIMO Communication Systems, Automotive Radar and Array Processing,polynomial signal processing, Satellite communication systems, ResourceAllocation, Game Theory and Fast Algorithms for Structured Matrices. Heis currently on the Executive Committee of the IEEE Benelux joint chapteron communications and vehicular technology, member of the EURASIPTechnical Area Committee on Theoretical and Methodological Trends inSignal Processing and serves as handling editor for Elsevier Signal Processing.He was a co-recipient of the 2014 Distinguished Contributions to SatelliteCommunications Award, from the Satellite and Space Communications Tech-nical Committee of the IEEE Communications Society.Symeon Chatzinotas (S’06-M’09-SM’13) is cur-rently Assistant Professor and Deputy Head of theSIGCOM Research Group at SnT University ofLuxembourg. In the past, he has been a VisitingProfessor at the University of Parma, Italy and hewas involved in numerous Research and Develop-ment projects for the National Center for ScientificResearch Demokritos, the Center of Research andTechnology Hellas and the Center of Communica-tion Systems Research, University of Surrey. Hereceived the M.Eng. degree in telecommunicationsfrom the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 2003,and the M.Sc. and Ph.D. degrees in electronic engineering from the Universityof Surrey, Surrey, U.K., in 2006 and 2009, respectively. He was a co-recipientof the 2014 IEEE Distinguished Contributions to Satellite CommunicationsAward, the CROWNCOM 2015 Best Paper Award and the 2018 EURASICJWCN Best Paper Award. He has authored more than 350 technical papers inrefereed international journals, conferences and scientific books. He has beenthe PI or Co-PI of more than 30 projects funded by FNR, ESA and H2020.