Power Minimization Techniques in Distributed Base Station Antenna Systems using Non-Orthogonal Multiple Access

Power Minimization T echniques in Distributed Base Station Antenna Systems using Non-Orthogonal Multiple Access Joumana Farah, Antoine Kilzi, Cha rbel Abdel Nour, Catherine Dou illard Abstract —This paper introduces new approaches for combining non-orthogonal multiple access (NOMA) with distributed base station (DBS) deployments. The pur pose of the study is to unloc k the true potentials of DBS syste ms in the NOMA context, since a ll previous works dealing with pow er minimization in NOMA are performed in the CBS (c entralized base station) context. This w ork targets a minimization of the total transmit p ower in each cell , under user rate and power multiplexing constraints. Different techniques are designed for the joint allocation of subcarriers , antennas and power, w ith a partic ular car e given to insuring a moderate complexity. Results show an important gain in t he tota l transmit pow er obtained b y t he DBS-NOMA combination, with respect to both DBS-OMA (orth ogonal multiple access) and CBS- NOMA deployment scen arios. 1 Index Terms —Distributed Base Station, Non Orthogonal M ultiple Access, Power Min imization, Resource allocation , Waterfilling. I. I NTRODUCTION The concept of distributed b as e stations (DBS) [1 -2] was introduced in the pa st few years in mobile communication systems to increase the cell covera ge in a cost effecti ve way, and to strengthen the network infrastructure, particularly in satur ated areas. I t consists of deploying the base station (BS) antennas in a d istributed mann er throughout the cell, ins tead of having multiple an tennas installed on a single tower at th e cell cente r. The remote units, called remote radio heads (RRH) or remote antenna units (RAU), ar e connected to a processing and control center ( PCC) throu gh coax cables or fiber optics. By reducing the average distance of each m obile user to its transmitting/receiving antenna, th e ov erall transmission power, necessary to ensure a certain quality of reception, is reduced in comparison to the centralized c onfiguration (cen tralized base stations or C BS). There fore, f rom the ecological standpo int, DBS can greatly reduce l ocal electrom agnetic radiation and C O 2 emissions of transmission syste ms. Alternatively, for the same overall transmission power as in CBS, DBS offers a h igher and more uniform capacity over each cell. Moreover, it provides a better framework for improving system robustness to fading, intra-cell and inter-cell interfe rences, shadowing, and path lo ss. It also allows the system to better adapt to the v arying user distribution. Besides, the u se of DBS will all ow the d eployment of small antennas in large scale and in discrete locations in u rban areas, e.g. on building roofs, e lectric poles, traffic and stre et lights, where they ca n be almost invisible due to their small s ize. This will significantly simplify and reduce the cost of site installation, t herefore loweri ng the capital expenditure (CAPEX ) of mobil e operators. Efficient implementation is key in squeezing the achievable potential s out of DBS systems. For this purpose, the study i n [ 3] explored the advantages of DBS and compare d the achievable ergodic capaci ty for two different trans mission scenarios: selection diversity and bl anket t ransmission. In the first one, on e o f t h e R R H s i s s e l e c t e d ( b a s e d o n a p a t h l o s s m i n i m i z a t i o n criterion) for transmitting a gi ven signal, whereas in the seco nd, J. Farah and A. Kilzi are with the Department of Electricity an d Electronics, Faculty of Engineering, Lebanese University, Roumieh, Lebanon (joumana.farah@ul.edu.lb; k ilzi.antoine@gmail.com). all an tennas in the cell participate in each transmission, thus creating a macro scopic multiple antenna system. Th e results of this study show that selection diversity achieves a better capa city in the DBS con text, compared to bla nket transmission. T he same observations are made in [4]. In [5], RRH selection is also preconized as a mean to decreas e the number of information streams that need to be assembled from or conveyed t o the involved RRHs, as well as the signaling overhe ad. Several works target the optim ization of system energy efficiency (EE) in the DBS contex t. I n [ 6 ] , t w o a n t e n n a s e l e c t i on techniques are proposed, either based on user pathloss informat ion or RRH e nergy consumpti on. Also, proportional fairness scheduling is considered for subband allocation with a utility function adapted to opti mize the EE. In [7], subcarrier assignment a nd power al location are do ne in two separate stages . In the first one, the number of subcarriers per RRH is determined, and subcarrier/RRH assignment is performed assuming initial equal p ower distribution. In th e second stage, power allocation (PA) is per formed by maximi zing the EE under the constraint s of the total tra nsmit power p er RRH, the target ed Bit Error Rate and propor tional fairness among users. Moreover, non-ort hogonal multiple access (NOMA) is currently being considered as a potential access scheme for 5G mobile communications. Se veral forms of NOMA are under evaluation [8]: Power-dom ain NOMA, sparse code multiple access (SCMA), multi-user shared access (MUSA), pattern division multiple access (PDMA) , bit division multiplexing ( B D M ) , … , t o n a m e a f e w . T h i s w o r k t a r g e t s p o w e r - d o m a i n NOMA, which applies th e multiplexing of several u sers allocated to the same subcarrier in the power domain, by taking advantage of the channel gain difference between users [9-13]. At the receiver side, user separation is done using successive interference can cellation (SIC). A pplying power multiplexing on top of the o rthogonal frequency d ivision multiplexing (OFDM) layer has pr oven t o si gnificantly increase system throughput compared to orth ogonal signaling, while also improving fairness and cell-edge user experience. In the majority of the early studies conducted on schedulin g or resource allocation for NOMA like [10, 11 ], a proportio nal fairness (PF) scheduler is used t o strike a balance between the average throughput and user f airness. Also, equal inter- subcarrier PA is assumed, while the repartition of power amo ng multiple xed users on a subcarrier is often performed using fractional transmit power allocation (FTPA) [10, 11]. Despite i ts multiple advantages, the PF scheduler is not applicable in the context o f power minimization, since it targets a tradeoff between total throughput and fairness, constrained by a fix ed total transmission power. In a former work [12], we have introduced a set of solutions for the problem of minimizing the spectrum occupancy in NOMA, u nder total BS power and user rate constraints. Also, in [13], a n efficient method is proposed to incorporate a waterfilling inter-subcarrier PA within the PF scheduler. C. Abdel Nour and C. Douillard are with Institut Mines-Telecom, C N R S UMR 6285 Lab-STICC, 29238 Brest, France, (email: charbel.abdeln our@imt- atlantique.fr; catherine. douillard@imt-atlantique.fr). However, because of the difference in probl em structure, both studies cannot be directly gener alized to the case of power minimizat ion. A few recent works tackle the power minimizat ion problem in the NOM A context. In [14], a "relax-then -adjust" procedure is used to provide a suboptim al solution to the NP- hard problem: first, the problem is relaxed from the constraint s relative to power domain multiplexing. Then, the obtained solution is iteratively adjusted using a bisection search , lead ing to a relativ ely high complex ity. In [15 ], optimal PA is first conducted assum ing fixed subcarrier assignment. Then, a deletion-based algorithm iteratively removes users from subcarriers until the constraints of the maxim um number of multiplexed users are satisfied, thus necessitating a large num ber of i terations to con verge. In [1 6], the authors propose an opti mal and a suboptimal solu tion for determining th e u ser scheduling , the SI C or der, and the PA, for th e case of a maximum of two users per subcarrier. However, the power domain multiplexing constraints are not t aken i nto c onsideration. Power minimizat io n strategies are also proposed in [17] f or multiple-inp ut multipl e- output NOMA (MIMO-NOMA), where PA and receive beamform ing design are alternated in an iterative way. Constraints on the targeted SINR (signal to interfe rence and noise ratio) are considered to guarantee successful SIC decodin g. The subcarrier allocation problem is not include d, i.e., all us ers have access to the whole spectrum. Results, provided for a moderate number of u sers (4 or 6), show an important gain of performance with respect t o or thogonal multiple access (OMA). In [18], we have introduced a set of techniques that al low th e joint allocation of subcarriers and po wer, with the aim of minimizing the total power in NOMA CBS. Particu larly, we showed that the most efficient method, from the power perspective, consists of applying user pairi ng at a subsequent stage to single-user assignmen t, i.e., aft er applying OMA signaling as the first stage, instead of jo intly a ssigning coll ocated users to subcarriers. The main objective of this work is to study the potential of applying NOMA in the DBS context. To the best of our knowledge, the pr oblem o f power m inimization in the DBS context has not been addressed in the literature: only the prob lem of EE optimization was considered in this context [6,7]. In fac t, power minimization in DBS systems is a study item worth being explored since the establishe d techniques for both OMA and NOMA in CBS do not simply extend to the generalized case of DBS. As it will be seen in this article, RRH selection, added t o subcarrier assignment, user pairin g, and PA, render the problem much more co mplex than in the CBS case. For this purpos e, we will start by redesigning our p revious CBS so lution in [ 18], by exploiting particular properties of the waterfilling procedure, so as to decrease i ts complexity, w ith out incurring any performanc e loss. Th en, we will propose several solutions for extending the study to th e DBS case. Most interestingly, we will show tha t by appropriately combining the pair ing and RRH selection steps, and using certain information-t heory properties of NOMA, two collocated use rs on a subcarrier can bot h pe rform SIC. The exploitati on of such subcarriers can allow a significant performance enhancement, and ther efore a particu lar care will b e given for t heir allocation. The paper is org anized as follows: in Section II, we start by a description of the system model, with a formulati on of the resource allo cation problem in th e con text o f DBS-NOMA. Then, in S ection III, we presen t several suboptimal solu tions f or the power minimizat ion problem , for the case of a single powering RRH per subcarri er. In Section IV, we introduce a novel approac h for allowing a mut ual SIC implem entation on certain subcarriers, and introduce sev eral allocation techn ique s for exploitin g such subcarriers. Section V prov ides a brief overview of the co mplexity of the proposed algorithms. Section VI presents a performance anal ysis of th e d ifferent allocation strategies, while Section VI I concludes the p aper. II. D ESCRIPTI ON OF THE DBS-NOMA SYSTE M AND FORMULA TION OF THE POWER MI NIMIZATION P ROBLEM This study i s conducted on a downli nk system consisting o f a total of R RRHs uniformly positioned over a cell, where K mobile users are random ly deploye d. RRHs and users are assumed to be equipped with a single antenna. Users transm it channel state information (CSI ) t o R R Hs , a n d t h e P C C c o l l e c t s all CSI from RRHs. Alternatively, in a TDD (tim e division duplexing) scenari o, the PCC can benefit from channel reciprocity to perform channel estimation by exploiting uplink transmissions. Then, it allocates subcarriers, powers and RRHs t o u s e r s i n s u c h a w a y t o g u a r a n t e e a t r a n s m i s s i o n r a t e o f R k,req [bps] for each user k . The system bandwidth B is equally divided i n t o a t o t a l o f S subcarriers. On the n th subcarrier ( 1 ≤ n ≤ S ), a max imu m of m ( n ) users { k 1 , k 2 , …, k m(n) } are chosen from the set of K users, to be collocated on n (or paired on n when m ( n ) = 2 ). Classical OMA signaling corresponds to the special case of m ( n ) = 1. The framework is schematized in figure 1. NOMA subcarriers can be served by the same RRH o r by different RRHs . Fo r in sta nce , on e can con side r ser vin g Use r 1 a nd U ser 2 on the same subcarrier by RRH 1, while User 2 and User 3 are served (paired) on another subcarri er by RRH 1 and RRH 2 respectively. Fig. 1. Distributed Base Station system using NOMA (PCC = processing and control center, R RH = remote r adio head, SC = subcarrier) Let: ,, i kn r P the power of the i th user on subcarrier n , transmitted by RRH r , ,, i kn r h the channel coefficient between user k i a nd RRH r o v er n , H the t hree-dimensi onal c hannel gain matrix with e lements h k,n,r , 1 ≤ k ≤ K, 1 ≤ n ≤ S , 1 ≤ r ≤ R , N 0 the power spectral density of additive white Gau ssian noise, including randomi zed inter-cell interference, and assumed to be constant ove r all subcarrier s. A user k i on subcarrier n can remove the inter-user interference f r o m a n y o t h e r u s e r k j , collocated on n , whose channel gain verifies ,, ji kn k n hh  [9,10] and treats the received signals from other users as noise. In the rest of the study, and withou t loss of g enerality, we wi ll c o n s i d e r a m a x i m u m n u m b e r o f c o l l o c a t e d u s e r s p e r s u b c a r r i e r of 2, i .e., m ( n ) = 1 or 2. On t he one hand, it has bee n shown [10] that the gain in performance obtain ed with the collo cation of 3 users per subca rrier, com pared to 2, is minor. On the other han d, limiting the number of multiplexed users per subcarrier reduces the SIC co mplexity in the receiv er terminals. We will d enote by first (resp. second) user the one having the higher (resp. lowe r) channel gain between the two u sers. Their theoretical throughputs ,, i kn r R , 1 ≤ i ≤ 2 , o n n a r e g i v e n b y t h e S h a n n o n capacity limit as follows: 11 1 2 ,, ,, ,, 2 0 log 1 / kn r kn r kn r Ph B R S NB S       , (1) 22 2 12 2 ,, ,, ,, 2 2 ,, ,, 0 log 1 / kn rkn r kn r kn r k n r Ph B R S Ph N B S       , (2) Proposition 2.1. W h e n t h e s a m e R R H i s u s e d t o p o w e r t h e signals of the two paired users on a subcarrier, only one of th e two users is ca pable of perfor ming SIC. Proof: Let 1 2 () ,, k kn r R the necessary rate at u ser k 1 to decode the signal of user k 2 : 21 1 2 11 2 ,, ,, () 2 ,, 2 ,, ,, 0 log 1 / kn rk n r k kn r kn r kn r Ph B R S Ph N B S       . Let  2 = N 0 B / S the noise power o n eac h subcarrier. k 1 can perform SIC on n if: 1 2 2 () ,, ,, k kn r kn r RR  . By writing: 1 2 2 () ,, 2 ,, log k kn r kn r B X RR S Y     , with:    11 2 1 1 2 22 2 22 ,, ,, ,, ,, ,, ,, k nr k nr k nr k nr k nr k nr XP h P h P h     and    11 1 2 2 2 22 22 2 ,, ,, ,, ,, ,, ,, k nr k nr k nr k nr k nr k nr YP h P h P h     . After some calculations, we can write:  21 2 22 2 ,, ,, ,, kn r k n r kn r XY P h h    . Therefore, X – Y ≥ 0, since 12 ,, ,, kn r k n r hh  . Knowing that l og(x) i s a monotonical ly increasing functio n of x , this proves that 1 2 2 () ,, ,, 0 k kn r kn r RR  At the same time, k 2 cannot perform SIC since: 12 2 11 1 22 2 ,, ,, () ,, 2 ,, ,, 22 ,, ,, 2 log 1 log kn r k n r k kn r kn r kn r kn rkn r Ph B RR R S Ph B Z S T              with:   12 1 1 2 1 2 22 2 2 2 ,, ,, ,, ,, ,, ,, ,, k nr k nr k n r k nr k nr k nr k nr ZT P h h P P h h     ≤ 0.  Proposition 2.1 will be of primary importance for the further development of ou r allocation techniques in the DBS-NOMA context. Let S k be the set of subcarriers allocated to a user k , such that k is eit he r th e fi rst , s eco nd o r s ole use r on a ny of i ts s ub car r iers, each of which being powered by a selected RRH. Let T k b e t h e mapping set of RRHs corresponding to user k , such that the i th element of T k correspond s to th e RRH selected fo r powering the i th subcarrier from set S k . The corresponding optimization problem can be formulated a s:   ,, * ,, ,, ,, 1 () , s . t . () ,, a r g m i n kk k n r k k k K kk k n r k n r SR P k rT i i n nS S STP P      , subject to: ,, , r e q , 1 k kn r k nS RR k K      ,, 0, 1 , , kk kn r P kK n Sr T      21 ,, ,, , , 1 k kn r k n r P Pn S k K     The third constraint is the po wer multiplexing constraint prope r to NOMA signaling. The problem i s a mixed c ombinatorial and non-convex one. Besides, compared to the case of NOMA CBS signaling, an additional dimension is added to the problem, which is the determination of the best RRH to power each allocated subcarrier to a user. In the se quel, we start by reviewing our previous soluti on in the N OMA CBS c ontext. Then, we propose several enhanc ements to this solution, so as to pave the way fo r its adaptation to the DB S context. III. R ESOURCE ALLOCATI ON TECHNIQUES FOR T HE CASE OF A SINGLE POWERING RRH PER SUBCARRIER A. The previous power mini mization technique for NOMA in CBS In [18], we showed that, in OMA signaling, the power allocation problem for a user k, over the set S k of its allocated subcarriers, can be formulated b y a recursive low-complexity waterfilling technique. The latte r provides the new waterline level for user k , after the assignm ent of a subcarrier n , as well as the power decrease  P k,n , r incurred by this assignment. Let N k = Card( S k ) and w k ( N k ) the corresponding waterline level. After addi ng a subcarrier n a to user k , the new water line level, in terms of w k ( N k ), is [18]:   /1 1/ 1 22 ,, () (1 ) / kk k a NN kk kk N kn r wN wN h     . ( 3 ) Adding n a decreases the waterline only if its channel gain verifies [18]: 2 2 ,, () a kn r kk h wN   . ( 4 ) The power decrease incurre d by adding n a is expressed as:  2 ,, 2 ,, 1( 1 ) ( ) a kn r k k k k k k kn r PN w N N w N h       . (5) Three m ain difficulties reside i n applying NOMA in the power minimization c ontext: 1. the achievable rate on ea ch s ubcarrier being dependent on th e user pairing order and on the inter-user interference term in the denominator of (2), 2. the necessity to meet K independent user rate constraints, 3. the power domain multiplexing constraints that must be respected on each subcarrier to allow proper decoding at the receivers. In [18], an efficient method was proposed for incorporati ng the waterfilling principle w ithin NOMA signaling. It is summarized in Algorithm 1. Note that, since this algorithm was designed for th e CBS case, in this pa rt of the paper, r designates the central (unique) BS antenna. The initializatio n phase Worst-Best-H [18] is useful at the beginning of the algorithm to avoid depriving cell-edge users o f their best subcarriers (essentia l in decreasing their power) in favor of cell-interior users. Af terwards, prior ity is based on the users' necessary total po wers. T he following notations are used : s ole k S is the set of su bcarr iers where user k is the sole user (i.e., m ( n )=1,  n  s ole k S ), s ole k R t h e t o t a l r a t e o f k on subcarriers in s ole k S , f irst k S (resp. sec ond k S ) the set of subcarriers where k is first (resp. second) user, collocated with a second (resp. first) one , f irst k R and sec ond k R the total rates correspon ding to f irst k S and sec ond k S , estimated using (1) and (2), S p the overall set of available subcarriers, S f the overall set of subcarriers assigned a first user without a second user, P k,tot t h e t o t a l a m o u n t o f necessary power for user k and U p the set of users whose power level can still be decreased (initially U p = {1, 2, …, K }). To estim ate 2 ,, kn r P  i n phase 4 of Algorithm 1, the po wer needed on the subcarriers 2 s ole k S is first found, constrained by 2 s ole k R . For this purpose, we calculate 22 2 2 sec ,r eq first s ole ond kk k k RR RR   . Then, a gradual waterfilling is performed on the set 2 s ole k S , s o a s t o r e a c h 2 s ole k R on this set. Following that, 2 ,, kn r P  is estimated for candidate subcarrier n . Note that gradual di chot omy-based waterf illing is performed using the procedure described in [19]. Moreover, the allocation of subcarrier n * t o k 2 may only decrease its power by a negligible amount. For this purpose, the power decrease is compared to a threshold  . The latter is chosen in s uch a way to strike a balance between power efficiency and sp ectral efficiency of the system, since unused subcarrie rs are released for use by other users or operators. Algorithm 1: NOMA-CBS Phase 1: Attribute a subcarrier to each user using the Worst-Best-H priority Phase 2: // Assigning first users to s ubcarrier using OMA signaling , *a r g m a x kt o t k kP  // identify the user with the highest priority For every n  S p verifying ( 4) Calculate ** (1 ) kk wN  using (3) Calculate *, , kn r P  using (5) End for *, , * arg min kn r n n P   Attribute n * to k *, unless  P k*,n*,r > -  and update P k*,tot If  P k*,n*,r > -  , remove k * from U p Repeat Phase 2 until no more su bcarriers can be allocated Phase 3: Search for subcar riers with negative- powers (i. e., having 22 ,, /( 1 ) kn r k k hw N   ): free subcarriers one by one and update waterlines and powers Phase 4: // Assigning second users to subcarriers using NOMA signaling, re-ini tialized by U p = {1, 2 , …, K }. 2, arg max kt o t k kP  // identify the user with the highe st priority For every n  S f s.t. 21 ,, ,, kn r k n r hh  // k 1 is the first user on n Calculate 2 ,, kn r P using FTPA: 21 2 1 22 ,, ,, ,, ,, / kn r k n rkn r k n r PP h h     , (6) Calculate 2 f irst k R , 2 sec ond k R and 2 s ole k R Perform gradual waterfilling on 2 s ole k S constrained by 2 s ole k R // Calculate the new total power of k 2 22 2 2 2 sec 22 2 (2) ,, ,, ,, ,, , sole first ond kk k kn r kn r kn r kn r kt o t nS nS nS PP P P P        2 22 (2 ) ( 1 ) ,, ,, kn r kt o t kt o t PP P   // 2 (1) , kt o t P is the previous total power of k 2 End for 2 ,, *a r g m i n kn r n nP   If 2 ,* , knr P  < -  Assign k 2 as second user on n * 1 ,* , kn r P and 2 ,* , kn r P are fixed in the following iterations Else free zero-power subcarriers of k 2 one by one, update its waterline and power levels; then, remove k 2 from U p Repeat Phase 4 until no more s ubcarriers can be allocated secon d users B. Runtime enhancement of th e NOMA CBS solution and adaptation to th e DBS c ontext First, we start by revisiting the waterfilling principles summarized in section III.A, in order to introduce several procedures for reducing the complexity of Algorithm 1, prior to its adaptation to the DBS context (Algorithm 2). In this sectio n, we consider t he case where the first and second users on a subcarrier are powered by the same RRH, thus the name of the method NOM A-DBS-SRRH . Proposition 3.1. In the OMA phase of NOMA-CBS (Cf. phase 2 of A lgorithm 1) , the subcarrier n a that ensures the lowest power decrease to user k corresponds to the one with the h ighest channel gain among the available subcarriers, provided that it verifies (4). Proof: Using (3), we ca n rewrite the pow er variation as:   1/ 1 2 ,, 22 2 ,, ,, () 1( ) / k k aa N N kk kn r k k k k kn r kn r wN PN N w N hh            .(7) By taking the derivative of ,, kn r P  with respect to ,, a kn r h , we get:      1/ 1 2 2 ,, /1 23 1 ,, ,, 1 ,, 2 2( ) k kk a a k a N kn r NN kk kn r kn r N kn r P wN h h h           . Therefore, we can verify that:    1/ 1 22 ,, /1 22 ,, , , ,, 0( ) k kk aa a N kn r NN kk kn r kn r kn r P wN hh h            , (8) which directly leads to (4). We deduce that ,, kn r P  is a monotonically d ecreasing function of ,, a kn r h , which concludes the proof of Proposition 3. 1 .  Proposition 3.1 als o m eans that the su bcarriers se quentially assigned by Algorithm 1 to a user k are in decreasing order of channel gain (i.e., the fi rst subcarrier assigned t o k has the highest channel gain in S k ). These results will allow us to significantly reduce the complexity of Phase 2 in Algorithm 1. Furthermore, in case 2 2 ,, () a kn r kk h wN   , one can verify u sing (7) that ,, 0 kn r P  . Hence, any subcarrier verifying (4) not only reduces the waterline, as stat ed in [18], but also guarantees a decrease of the necessa ry power, i.e., ,, 0 kn r P  . Proposition 3.2. After assigning a subcarrier n a ve rif yi ng (4 ) to a user k , all subcarriers of k , including n a , have a pos itive power level (i.e., 2 2 ,, (1 ) a kk kn r wN h   ). Proof: the power allocated to subcarrier n a is: 2 ,, 2 ,, (1 ) a a kn r k k kn r Pw N h    . It can be written in terms of w k ( N k ) as:  /1 2 /1 2 ,, ,, 1/ 1 2 ,, 2 () kk kk a a k a NN NN kk kn r kn r N kn r wN h P h                . This allows us to verify that ,, 0 a kn r P  as long as (4) is verified, i.e., the allocated p ower to the added subcarrier n a i s e ns ur ed to be positive. Besides, the power allocated to the subcarriers n  S k of u ser k , after each waterline update, is higher for subcarriers with higher channel gains. Therefore, the most recently added subcarrier n a i s t h e o n e w i t h t h e l o w est channel gain and the lo we st all oca ted p ow er am ong st the sub ca rri ers in S k , due to the consequence of Proposition 3.1 . Since this power is positive, the powers of all subcarriers in S k are positive. This concludes the proof of Propositio n 3.2 and allows us to completely rule out phase 3 in Alg orithm 1.  Next, we turn our attention to the pairing phase, i.e., the assignment of second users to subcarriers using NOMA. From the runtime perspective, the mos t constraining step in this pha se of Algorithm 1 is the dichotom y-based waterfilling calculation. The latter aims at compensating for the additional rate brought to u ser k by the newly added subcarrier n (as second user) by removing this additional rate from the one that should be achieved on the sole subcarriers of k (i.e., subcarriers in set s ole k S ). Therefore, we propose to replace this dichotomy-based waterline estimation by an effi cient iterative wa terline update as follows: Recall that the total rate s ole k R to be distributed on the set s ole k S of user k can be expressed as: 2 ,, ,, 2 2 log , 1 sole k sole kn r kn r k nS B Ph R S         wher e  2 ,, 2 ,, sole kn r k k kn r Pw N h   , with   Card , s ole sole kk NS  and   s ole kk wN is the waterline on the sol e subcarriers of k . Therefore, s ole k R can be rewritten as:   2 ,, 2 2 log sole k sole sole kk k n r k nS wN h B R S         , l eadi ng to:  1/ 2 2 ,, 2 s ole k sole k sole k N RS B sole kk kn r nS wN h         . If a user k is assigned as a second user to a subcarrier n s , he will gain a rate on n s , calculated using (6) and (2). This rate corresponds to the rate decrease ,, 0 s kn r R  that should be compensated for on the sole subcarriers of k , s o a s to ens ur e t he global rate constraint R k , req . Let ' ,, s sole sole kk k n r RR R   the new rate to be distribute d on s ole k S . The correspond in g new waterline can be expressed as:  ' 1/ 2 2 ,, '2 s ole k sole k sole k N RS B sole kk kn r nS wN h         . Therefore, it ca n be shown that:   ,, '2 kn r s sole k RS NB s ole sole kk kk wN w N   . (9) The rate variation ,, s kn r R  lowers the water le vel and may cause some subcarriers to have negative powers. Such subcarriers must be removed from s ole k S . In the case wher e the change in wa terline does not provoke any subcarrier removal, t he resulting change i n the total power of user k is:       ,, ,, ' s s sole sole sole k n r k kk kk k n r PN w N w N P    . (10) When the waterline decrease i n d u c e s t h e r e m o v a l o f s o m e subcarriers, those subcarriers are the ones with the weakest channel gains in s ole k S . Let us sort the elements of s ole k S in decreasing order of magnitude, that is, () ,1, 1 , 2 , 2 ,, ( ) ) ( s ole sole kk kr k r kN r N hh h    , wh ere b y r ( i ) we denote the RRH powering the i th subcarrier of user k (those subcarriers are assumed to be num bered from 1 to s ole k N ). If the last subcarrier is first rem ove d, the resulting waterline is:    /1 1 /1 22 ,, 1( / ) sole sole sole kk k k NN N sole sole kk k k k N r wN w N h     . Since ,, 0 sole k kN r P  , i.e., 22 ,, () sole k sole kk kN r hw N   , we get:    1 s ole sole kk kk wN wN  . This means that r emoving a subcarrier with a negative power always decreases the waterline. Therefore, any other subcarrier n s u c h t h a t () ,, ,( ,) s ole sole kk kr kr NN nn hh  with an initially negative power before the removal of the last subcarrier will get an eve n more negative power after this removal. This leads us to the conclusion that the negative-power subcarriers can all be removed at once, rather than one by one as usually d one in waterfilling algorithms [20]. The corresponding power variation is:     , 2 1 ,, 2 0 ,j , ( ) s kk s ole sole sole sole kL k k k k k k L kn r j kN r N j PN L w N L N w N P h           , (11) where L is the number of n egativ e-power subcarriers and  sole kk wN L  the water level after the removal of L subcarriers:    1 1 22 ,, 0 ( ) / sole k sole sole kk k N L sole sole NL NL kk k k k N j r j wN L w N h        (12) After this removal some new negative-power subcarriers may arise that should be removed as well. However, statistical estimations on Mont e-Carlo simulati ons of our algorithms have shown that negative-power sub ca rriers are very rare (less than one allocatio n case out of a thous and). Algorithm 2: NOMA-DBS-SRRH Phase 1: Worst-Best-H subcarrier and RRH allocation Phase 2: // single-user assignment , *a r g m a x kt o t k kP  // identify the user w ith the highest priori ty *, , ( , ), s.t . and ( 4 ) (* , * ) a r g m a x p kn r nr n S nr h   // using proposition 3.1 Calculate ** (1 ) kk wN  using (3) and *, *, * kn r P  using (5) If  P k*,n*,r* < -  attribute n * to k *, and u pdate P k*,tot Else remove k * from U p Repeat Phase 2 until no more subcarriers can be allocated Phase 3: // NO MA pairing 2, arg max kt o t k kP  For every n  S f s.t. 21 ,, ,, kn r k n r hh  Calculate 2 ,, kn r P using (6) // r is the RRH powering user k 1 on n Calculate 2 f irst k R , 2 sec ond k R and 2 s ole k R Calculate 2 ,, kn r P  using (9) a nd (10) End for 2 ,, *a r g m i n kn r n nP  If 2 ,* , knr P  < -  Assign k 2 on n * Fix 1 ,* , * kn r P and 2 ,* ,* kn r P and update 22 ,, , s ole kn r k Pn S  Else free zero-power subca rriers of k 2 using (1 1) and (12); then, remove k 2 from U p Repeat Phase 3 until no more su bcarriers can be allocated secon d users C. Enhancemen t of the NOMA DBS solution th rough local power optimizatio n The power decrease incurred by a candidate subcarrier in t he third phase of NOMA-DBS-SRRH is greatly influenced by the amount of power 2 ,, kn r P allocated to user k 2 o n n u s i n g F T P A . Indeed, the addition of a new subcarrier translates in a rise o f the user power on the one hand, and in a power decrease due to the subsequent w aterline reduction on his sole subcarriers (through (9)), on the other. Therefore, w e propose to optimize the valu e of 2 ,, kn r P in such a way th at the c onsequent user power redu ction is minimized: 2 ,, 2 ,, Mi n kn r kn r P P  subject to: 21 ,, ,, kn r k n r PP  on n By injecting (9) into (10), and expressing 2 ,, kn r R  using (2), the corresponding Lagr angian can be written as:    2 2 2 22 2 2 12 1 22 1 2 ,, 22 ,, ,, ,, ,, ,, ,, ,, ,1 1 sole k N kn r sole sole kk k kn r k n r kn r kn r kn r kn r kn r Ph LP N w N Ph PP P                    where  is the Lagrange multiplie r. The correspond ing Karush-Ku hn-Tucker (KKT) conditions are :   2 22 2 22 12 1 2 21 1 1 2 2 ,, ,, ,, 22 22 ,, ,, , ,, ,, ,, 11 0 0 sole k sole N kk k n r kn rkn r kn r k n r kn k n r kn r k n r wN h Ph Ph P h PP                      W e c a n v e r i f y t h a t t h e s e c o n d d e r i v a t i v e o f t h e L a g r a n g i a n i s always positive, thus the correspond ing solution constitutes a unique m inimum. For  = 0, this optim um is:  2 2 22 2 12 2 12 2 2 22 1 ,, ,, ,, * ,, 22 2 ,, ,, ,, 1 sole k sole k N sole N kk k n r kn r k n r kn r kn r k n r k n r wN h Ph P Ph h                 (13) For  ≠ 0, we get 21 ,, ,, kn r k n r PP  . However, in t he latter case, a certain gap must be set between th e power levels of the two p aired u sers, in suc h a way to guarante e successful SIC decoding at the first user level. Indeed, a SINR level should be guaranteed to a llow efficient SIC, as shown in [17]. Therefore, we will take: 21 ,, ,, (1 ) kn r k n r PP    , ( 1 4 ) with µ a positive safety power margin that depends on pr actical SIC impl ementation. In other terms, if the obtained 2 * ,, kn r P verifies the power constraint inequality, it is retained as the optimal so lution; otherwise, it is taken as in ( 14). This metho d, referred to as "NOMA-DBS-SRRH-L PO", operates similarly to Algorithm 2, except that (6) in Phase 3 is replaced by either ( 13) or (14). D. NOMA DBS solution with optimal powe r allocation In this method, we propose to jointly optimize the inter- subcarrier and intra-subcarrier PA by applying the Relax-then- adjust procedure in [14] based on successive variabl e substitution. This technique is applied in our work subsequentl y to NOMA-DBS-SRRH-LP O, as shown in Algorithm 3. Algorithm 3: NOMA-DBS-SRRH-OPA Phase 1: Apply NOMA-DBS-SRRH -LPO to determine first and second user assignments to subcarriers, as well as a provisional power allocation. Phase 2: Apply optim al PA using t he procedure in [1 4]. IV. R ESOURCE ALLOCATI ON TECHNIQUES IN DB S FOR THE CASE OF MUTUAL SIC A. Theoretical founda tion In this section, we consider the case where the users k 1 and k 2 , collocated on subcarrier n , are powered by two different RRHs, respectively r 1 and r 2 . Proposition 4.1. Users k 1 and k 2 can both per form SIC, if: 12 2 2 ,, ,, kn r k n r hh  ( 1 5 ) 21 1 1 ,, ,, kn r k n r hh  ( 1 6 ) The corresponding p ower multiplexing constraint is: 11 2 2 2 1 11 12 2 2 22 ,, ,, ,, 22 ,, ,, ,, kn r k n r k n r kn r kn r k n r hP h P hh  . ( 1 7 ) Proof: User k 1 can perform SIC on n i f 1 22 22 () ,, ,, k kn r kn r RR  with the following power multiplexing condition : 11 11 2 2 1 2 22 ,, ,, ,, ,, kn r kn r k n r kn r Ph P h  . (18) Note that if r 1 = r 2 = r , (18) reverts to the previous case of a single RRH per subcarrier, i.e. , 12 ,, ,, kn r k n r PP  . Similarly, k 2 can perform SIC on n i f 2 11 11 () ,, ,, k kn r kn r RR  , with 22 22 1 1 2 1 22 ,, ,, ,, ,, k nr k nr k nr k nr Ph P h  ( 1 9 ) We can th en identify the conditions that g uarantee a mutual SIC , that is, both users performing SIC. In such co nditions, the reachable rates by users k 1 and k 2 become: 11 11 11 2 ,, ,, ,, 2 2 log 1 kn r kn r kn r B Ph R S        ( 2 0 ) 22 22 22 2 ,, ,, ,, 2 2 log 1 kn r kn r kn r B Ph R S        ( 2 1 ) Following the same reasoning as in the proof of Proposi tion 2.1 , we can write: 22 12 1 22 22 22 11 11 2 ,, ,, () ,, 2 ,, ,, 22 ,, ,, 2 log 1 log kn r k n r k kn r kn r kn r kn r kn r Ph B RR R S Ph B X S Y              where   22 1 2 22 1 1 22 1 1 22 22 2 2 2 ,, ,, ,, ,, ,, ,, ,, k nr k nr k nr k nr k nr k nr k nr XY P h h P P h h     Similarly: 2 11 11 () ,, 2 ,, log k kn r kn r B Z RR S T     , with   11 2 1 11 11 2 2 11 2 2 22 2 2 2 ,, ,, ,, ,, ,, ,, ,, kn r k n r kn r kn r k n r kn r k n r ZT P h h P P h h     In practical transmission situations, the second term in the expression of X - Y is much smaller th an the first, and the same goes for Z - T . Therefore, the second terms can be neglected, leading to the conditions (15 ) and (16) for mutual SIC.  In the special case where r 1 = r 2 = r , only one of the two conditions (15) and (16) can be verified at a time, i.e., only one o f t h e t w o u s e r s c a n p e r f o r m S I C . O t h e r w i s e , w h e n ( 1 5 ) a n d ( 1 6 ) are both veri fied, we also ha ve: 11 2 1 12 2 2 22 ,, ,, 22 ,, ,, kn r k n r kn r k n r hh hh  . Therefore, in light o f (18) and (19), one can conclude that whe n ( 1 5 ) a n d ( 1 6 ) a r e s i m u l t a n e o u s l y v e r i f i e d , a P A s c h e m e c a n b e found to allow a mutual S IC by ensur ing (17). B. Optimal solution for th e unconstrained case In the case where the power multiplexing const raints (18) and (19) are discard ed, one can verify, using ( 20) and (21) and sim ple Lagrangian optimization, that the PA in the pairing phase re ver ts to the user-specific waterfilling solution, similarly to the ph ase 2 in NOMA-DBS-SRRH or NOMA -DBS-SRRH-LPO. The only difference resides in that only candidate su bcarriers verify ing (15) and ( 16) are cons idered for pairing. This technique, used as a lower-bound benchmark on the to tal power, will be referred to as "NOMA-DBS-MutSIC- UC". C. Optimal formulation for the c onstrained case The fully constrained case can b e cast as the solution of the following optimization problem: ,, ,, {} 111 max kn r KS R kn r P knr P       , subject to: 2 ,, ,, 2, r e q 2 log 1 , 1 k kn r kn r k nS Ph Rk K           22 2 1 11 22 2 ,, ,, 2 ,, ,, , kn r kn r mSIC kn r kn r Ph nS P h   22 1 1 11 12 2 ,, ,, 2 ,, ,, , kn r k n r mSIC kn r kn r Ph nS P h     ,, 0, , , kn r Pk n r  S mSIC is the set of subcarriers wher e mutual SIC is performed. The corresponding Lagrangian with multipliers  k and β i,n is:  21 2 2 11 22 22 1 1 11 12 2 ,, ,, 12 , , 1 , 2 ,, 111 ,, 2 ,, ,, 2, 2 ,, ,, 2 ,, ,, ,2 2 1 ,, , log 1 mSIC mSIC k KS R kn r k n r kn r n kn r knr n S kn r kn r k n r n kn r nS kn r kn r kn r kk r e q kn S hP LP P P h Ph P h Ph R                                     K  Writing th e KKT conditions (not pr esented here for the sak e of concision) leads to a system of N e non-linear equations with N e variables, where N e = 3 Ca r d ( S mSIC )+ K + S (taking into account the S -Card( S mSIC ) power variables on non-p aired subcarriers). Using the fact that β 1,n and β 2, n cannot be simulta neously non-zero on n , ther e are Card ( ) 3 mSIC S combinations to solve, with 2Card( S mSIC )+ K + S variab les. This leads to a p rohibitive complexity; therefore, next we elaborate different resource allocation strategies in which we account for the power multiplexi ng constraints at every subcarrier assignm ent iterati on. D. Suboptimal sol ution for the constrained cas e using direct power adjustment (DPA) In this section, a simpler y et efficient power adjustment based technique is proposed, where the adjustm ent is carried out afte r every subcarrier allocati on. Focusi ng on the pairin g step, and following a similar reasoning to the one in III.C, we can w rite the power variati on, due to the pairing of user k 2 on n with mutual SIC, as:  2 22 22 22 2 2 2 22 1 2 ,, ,, ,, ,, 2 11 sole k N kn r kn r sole sole kn r k k k kn r Ph PN w N P                (22) Using simple mathematical derivation, we can verify that:     2 2 22 22 22 22 22 22 2 22 1 2 ,, * ,, ,, 12 ,, ,, 22 1 ,, 0 / sole k sole k sole k N sole N kk kn r kn r kn r kn r kn r N kn r wN P PP h P h           In fact, 22 * ,, kn r P is the p ower corresponding t o the waterfilling applied to the candidate subcarrier n and the sole subcarriers of user k 2 . Also, we can verify that the second derivative of 22 ,, kn r P  wi t h r es p e c t to 22 ,, kn r P is always po sitive. Therefore, we deduce that for any v alue of 22 ,, kn r P greater (resp. lower) than 22 * ,, kn r P , 22 ,, kn r P  is strictly increasing (r esp. decreasing) with 22 ,, kn r P . Consequentl y, the further 22 ,, kn r P is apart from 22 * ,, kn r P , the greater is 22 ,, kn r P  . Hence, the best choice of 22 * ,, kn r P , when it does not verify (17), should be at the limits of the inequalit y (17 ) . This leads us to the subopti mal Algorithm 4. Algorithm 4: NOMA-DBS-MutSIC-DPA Phase 1: Worst-Best-H subcarrier and RRH allocation followed by OMA single-user assi gnment Phase 2: // NO MA pairing 2, arg max kt o t k kP      2 , s.t. ( 15), ( 16) & ( 4) are verified c Sn r  For every candidate couple ( n,r 2 )  S c Calculate 22 ,, kn r P and 22 ,, kn r P  using (3) and (5) If 22 ,, kn r P verifies (17), set 22 22 * ,, ,, kn r kn r PP  If 22 1 1 11 12 2 ,, ,, 2 ,, ,, kn r k n r kn r kn r Ph P h  , set 11 22 1 1 12 2 ,, * ,, ,, 2 ,, (1 ) kn r kn r k n r kn r h PP h   a n d estimate 22 ,, kn r P  using (9) an d (10) If 22 2 1 11 22 2 ,, ,, 2 ,, ,, kn r kn r kn r kn r Ph P h  , set 21 22 1 1 22 2 ,, * ,, ,, 2 ,, (1 ) kn r kn r k n r kn r h PP h   and estimate 22 ,, kn r P  using (9) an d (10) End for 22 2 2, , (, ) (* , * ) a r g m i n kn r nr nr P   Continue similarly to NOMA-DBS-SRRH E. Suboptimal solutions for the c onstrained case usin g sequential optimization for power adjustment In order to further optimize the NOMA-DBS-Mu tSIC-DPA technique, we propose to replace the adjustment and power estimation steps by a sequential power optimization. Instead o f optimizing the choice of 22 ,, kn r P over t he candidate couple ( n , r 2 ), we look fo r a wider optimization in which powers of both fir st and second users on the considered subcarrier are adjusted, in a way that their global power variation is optimal:    1 1 22 1 1 22 ,, ,, 11 2 2 * ,, ,, ,, ,, , ,a r g m a x kn r k n r kn r k n r kn r k n r PP PP P P P        subject to: 22 1 1 22 2 1 11 2 2 11 11 12 2 2 22 ,, ,, ,, ,, ,, ,, 22 ,, ,, ,, ,, ,, 0 , 0 k n rk n r k n rk n r kn r k n r kn r kn r kn r k n r Ph Ph PP PP hh    22 ,, kn r P  is expressed as in (22), while 11 ,, kn r P  is given by:   ,, 1 1 11 1 1 11 11 1 ,, , ,, ,, 12 1 kn r sole k RS NB sole I kn r k I k kn r kn r PN W P P             where 11 ,, I kn r P is the initial power allocated o n n to k 1 and W I,k 1 the initial waterline of k 1 (before pairing w ith user k 2 a n d p o w e r adjustment). Also, the rate variation of user k 1 on n can be written as: 11 11 1 11 11 22 ,, ,, ,, 2 22 ,, ,, log kn r kn r kn r I kn r kn r Ph B R S Ph          . The Lagrangian of this problem is:   11 2 2 11 2 2 11 2 1 1 1 22 22 1 1 12 2 2 ,, ,, 1 2 ,, ,, 22 ,, ,, 1 ,, ,, 2 ,, ,, 22 ,, ,, ,, , kn r k n r kn r k n r kn r k n r k nr k nr k nr k nr kn r k n r LP P P P hh PP P P hh                      The solution of this problem must verify the following conditions:   11 2 2 ,, ,, 1 2 ,, , 0 kn r k n r LP P     1111 1 2 2 2 22 1 12 1 22 11 2 2 22 1, , , , , , , , 22 2, , , , , , , , 12 ,, ,, 0 0 ,0 ,0 kn r kn r kn r k n r k nr k nr k nr k nr kn r k n r Ph h P PP h h PP        Four cases are identified: 1. λ 1 = 0, λ 2 = 0 2. λ 1 ≠ 0, λ 2 = 0  2 2 11 11 1 2 22 ,, ,, ,, ,, / k nr k nr k nr k nr PP h h  3. λ 1 = 0, λ 2 ≠ 0  22 1 1 2 1 22 22 ,, ,, ,, ,, / kn r k n r kn r kn r PP h h  4. λ 1 ≠ 0, λ 2 ≠ 0 Case 1 corresponds to the uncon strained waterfilling solution applied separately to the two concerned users. Case 4 is genera lly impossible, since the two bou ndaries of the inequality (17) would be equal. Considering Case 2, by replacing 22 ,, kn r P in terms of 11 ,, kn r P in the Lagrangian and by taking the derivative with respect to 11 ,, kn r P , we can verify that 11 * ,, kn r P is the solution of the following nonlinear equation: 1 11 1 1 11 1 1 1 11 11 11 11 1 2 2 11 2 2 11 2 2 11 2 12 1 1 22 2 2 1 ,, , , , , , , , 22 22 2 ,, ,, ,, ,, ,, 1 22 2 2 ,, ,, ,, ,, ,, , 22 2 ,, 1 1 sole k sole k N I k kn r kn r kn r kn r II k n rk n r k n rk n r k n r N kn r k n r kn r k n r kn r Ik kn r Wh P h h Ph Ph h hh P h h W h                   1 0   (23) N o t e t h a t i n p r a c t i c e , w e a l s o t ake into consideration the safe ty power margin µ in the calculation of 11 ,, kn r P . Similar calculations are performed for Case 3. The so lution that yields the lowest  P is retained. Also, if none of the cases provides positive power solutions, the curren t candidate couple ( n , r 2 ) is discarded. This method of optimal power a djustment (OPAd) will be referred to as "NOMA-DBS-Mu tSIC-OPAd". In order to decrease the complexity of "NOMA-DBS- MutSIC-OPAd ", inherent to the resolution of nonlinear equations, we consider a semi-o ptim al variant of this technique , called "NOMA-DBS-MutSIC-SO P Ad": at the stage where candidate couples ( n , r 2 ) are c onsidered for potential assignment to user k 2 , DPA is u sed for power adjustment, in order to determine the best candidate in a cost-effective way. Then, the preceding OPAd solution is applied to allocate power levels to users k 1 and k 2 on the retained candidate. F. Combination of the allocation of mutu al and single SIC subcarriers in DBS The case of two different power ing RRHs per subcarrier with only one user perform ing SIC is studied based on information theory developments similar to t he ones perf ormed in section IV.A. For insta nce, if (15) is verified and (16) is not, k 1 p er f o r ms SIC on subcarrier n , while k 2 does not. The corresponding power multiplexing conditions become: 11 11 2 2 1 2 22 ,, ,, ,, ,, kn r kn r k n r kn r Ph P h  and 22 22 1 1 2 1 22 ,, ,, ,, ,, k nr k nr k nr k nr Ph P h  . In order to furt her exploit the space diversity inherent to DBS systems, w e propose to first apply NOMA-DBS-M utSIC- SOPAd in order to identify and allocate subcarriers allowing mutual SIC. Then, in a subsequent phase, the remaining set of solely assigned subca rriers is further exa mined for potential allocation of a second user, using either the same or a differe nt RRH from that o f the first assigned user, but such that only th e latter performs SIC. LPO is use d for power allocation in this second phase. This m ethod will be referred to as "NOMA-DBS- Mut&SingSIC". V. C OMPLEXITY A NALYSIS In this section, w e analyze th e complexity of the different allocation techniques proposed in this study. The complexity of OMA-CBS, NOM A-CBS and OMA- DBS is studied by considering an impleme ntation that includes the runtime enhancement proce dures introduced i n section III.B. Starting with OMA-CBS, we consi d er t hat the c han ne l mat ri x is reordered such as for each user the subcarriers are sorted b y the decreasing order of channel gain. This step, that accelerat es the subsequent subcarrier allocation stages, has a complexity o f O( KS log( S )). Following the Worst-Be st-H phase, each iteration complexity is mainly dominated by the search of the most power consuming user with a cost O( K ) . T h i s l e a d s t o a t o t a l o f O( KS log( S )+( S-K ) K ). Each all ocation step in t he pairing phase of NOMA-CBS consists of the iden tif ication of the most power consu ming user , followed by a search over the subcarrier space, and a power update over the set of t he user' s sole subcarriers, with an ave rage number of S / K subcarriers. Therefore, the total complexity of NOMA-CBS is O( KS log( S )+( S-K ) K+S ( K + S + S / K )). In OMA-DBS, we consider an i nitial sorting of each user subcarrier gains, separately for each RRH, with a cost of O( KSR log( S )). Then, an allocation cycle will consist of user identification, followed by the search of the RRH providing the subcarrier with the highest cha nnel gain. This corresponds to a complexity of O ( K+R ). Therefore, the total complexity is: O( KSR log( S) +( S-K )( K+R )). Consequently, the total complexity of NOMA-DBS-SRRH and NOMA-DBS-SRRH-LPO is O( KSR log( S )+( S-K )( K + R )+ S ( K + S + S / K )). The most constraining part in NOMA-DBS-SRRH-OP A is the resolution of a s et of N OPA non-linear equations with N OPA unknowns, in the phase 2 of Algorithm 3. N OPA = Card( S mux )+ K + S , where S mux is the set of NOMA m ultiplexed subcarriers. Therefore, the co mplexity of this algorithm is f comp ( N OPA ), where f comp is a function that could either be exponential or p olynomial in ter ms of N OPA , depending on the resolution meth od. Concerning NOMA-DBS-M utSIC-UC, by followi ng the same reasoning as for OMA-DBS, and accounting for the search of an eventual collocated user for at most S sub ca rri ers , w e ge t a tot al of O( KSR log( S )+( S-K )( K+R )+ S ( K+R )). As for NOMA-DBS-MutSI C-DPA , the total complexity is O( KSR log( S )+( S-K )( K+R )+ S ( K+S ( R- 1)+ S / K )), where the S ( R- 1) term stems from the fact t hat the search over the subca rrier space in the pairing phase is conducted over all combinations o f subcarriers and RRHs, except for th e RRH of th e first user on t he candidate subcarrier . I n N O M A - D B S - M u t S I C - O P A d , l e t C b e t h e c o m p l e x i t y o f solving the nonlinear equation ( 23). The total complexity is therefore O( KSR log( S )+( S-K )( K + R )+ S ( K + S ( R -1) C + S / K )). Given that NOMA-DBS-MutSIC-SOP Ad solves (23) only once per allocation step, its complexity is O( KSR log( S )+( S- K )( K + R )+ S ( K + S ( R -1)+ S / K + C )). Consequent ly, the complexity of NOMA-DBS-Mut&SingSIC is O( KSR log( S )+( S- K )( K + R ))+ S ( K + S ( R -1)+ S / K + C )+ SR ( K + S + S/K ). The additional term corresponds to the Single SIC phase which is similar to th e pairing phase in NOMA-CB S exce pt that the search space is enlarged by a factor R . Table 1 summarizes the approximate complexity of the different technique s. TABLE. 1. Approximate complexity of the different allocation techniques. VI. P RACTICAL R ESULTS The per formance of the different allocation tec hniques are assessed through intensive simulations in the LTE/LTE- Advanced context [21]. The cell model is a hexagonal one with a radius R d of 500 m. For the DBS system, we consider a number R o f RRHs of 4 or 7. In each case, one antenna is located at the cell center, while the others ar e equally distanced and p ositio ned o n a c ir c l e of r ad i u s 2 R d /3 centered at the cell center. The system bandwidth B is 10 MHz. The transmission me dium is a frequency-sele ctive Rayleigh fading chan nel with an rms of 500 ns. We c onsider distance-dependent path loss with a decay facto r of 3.76 and lognormal shadowing with an 8 dB variance. The noise power spectral density N 0 is 4.10 -18 mW/Hz. In this study, we assume perfect knowledge of the user channel gains by the PCC. The α decay f actor in FTPA is taken equal to 0 .5, while the power thre shold  is 0.01 Watt as in [18]. The safety power margin µ is s e t t o 0 . 0 1 . O M A - C B S a n d O M A - D B S s c e n a r i o s a r e also shown for comparison, where only phases 1 and 2 of Algorithm 2 are app lied, using either R =1 (for OMA-CBS) or R ≠1 (for OMA-DBS). Figure 2 represents the total tr ansmit power in the cell in t er ms of the req uested rate, for the case of 15 use rs, 64 subcarriers and 4 RRHs. It shows th at the DBS configuration greatly outperforms CBS: a large leap in power with a factor around 16 is achieved with both OMA and NOMA signaling. At a target rate of 12 Mbps, the required total power using OMA-DBS, NOMA-DBS-SRRH, NOMA-DBS-SRRH-LPO and NOMA- DBS-SRRH-O PA is respectively 39.79, 32.34, 30.04 and 29.41 W. This shows a clear advantage of NOMA over OMA in the DBS context. Besides, applying LPO allows a power reduction o f 7 . 7 % o v e r FT P A , w i t h a s i m i l a r com putational load, while the margin over optimal PA is of only 2% at 12 Mbps. Fig. 2. Total po wer in terms of R k,req , for K = 15, S = 64 and R = 4. We now turn o ur attention to the evaluation of mutual SIC and single SIC configurations. Fi gure 3 shows that all three constrained conf igurations b ased on pure mutual SIC (NOMA- DBS-MutSIC-DP A, NOMA-DBS-M utSIC-SOPAd and NOMA-DBS-M utSIC-OPAd) largely outperform NOMA-DBS- SRRH-LPO. Their gain towards the latter is respectively 10.91, 16.18 and 22.19 W, at a requested rate o f 13 Mbps. The significant gain of optimal power adjustment towards its suboptimal counterpart comes at the cost of a significant complexity increase, as shown in Section V. The most power- efficient mutual SIC implementation is obviously NOMA-DBS- MutSIC-UC, since it is designed to s o l v e a r e l a x e d v e r s i o n o f t he power minimization problem , by dro pping all power multiplexing constraints. Therefore, it essentially serves as a benchmark for assessing the other methods, because power multiplexing conditions are essen tial for allowing correct sign al decoding at the receiver si de. Fig. 3. Total po wer in terms of R k,req , for K = 15, S = 64 and R = 4. Technique Complexity OMA-CBS O( KS log( S )) N OMA-CBS O( S 2 + KS log( S )) OMA-DBS O( KSR log( S )) N OMA-DBS-SRRH O( S 2 + KSR log( S )) N OMA-DBS-SRRH-LPO O( S 2 + KSR log( S )) N OMA-DBS-SRRH-OPA f comp ( N OPA ) N OMA-DBS-MutSIC-UC O( KSR log( S )) N OMA-DBS-MutSIC-DPA O( S 2 R) N OMA-DBS-MutSIC-OPAd O( S 2 RC ) N OMA-DBS-MutSIC-SOPAd O( S 2 R + SC ) N OMA-DBS-Mut&Si ngSIC O( S 2 R + SC ) The best global strategy remai ns the combination of mutual and single SIC subcar riers, sin ce it allows a power reduction o f 10.9 and 32.73 W, respectiv ely at 12 and 13 Mbps, towards NOMA-DBS-M utSIC-SOPAd. Figure 4 shows the influence of increasing the number of RRHs on the system performance. As expected, increasing the number of spread antennas greatly reduces the overall power, either with single SIC or combined mutual and single SIC configurations. A significant leap in power reduction is observ ed when R is increased from 4 to 5, followed by a more moderate one when going from 5 to 7 antennas, and the same behavior is expected for larger values of R . However, considerations of practical order would suggest limiting the number of deployed RRHs in the cell to a certain e xtent, mainly because of the inherent over head of CSI si gnaling exchange, not to mention geographical deploym ent constraints. Fig. 4. Total power in terms of R k,req , for K = 15, S = 64 and R = 4, 5 or 7. In figure 5, we show the perfo rmance for a varying number of users, for the case of 4 RRHs and 128 subcarriers. Fig. 5. Total power in terms of K , for R k,req = 5 M b p s , S = 1 2 8 a n d R = 4. Results prove that the allocation strategies based on mutual SIC, or combined mutual and single SIC, scale much better to crowded areas, com pared to si ngle SIC solutions . The power reduction of NOMA-DBS-Mut&SingSIC towards NOMA- DBS-SRRH-LPO is 62.4% and 71.7% for 36 and 40 user s respectively. I n T a b l e 2 , w e s h o w t h e s t a t i s t i c s o f t h e n u m b e r o f n o n - multiplexed subcarriers, the number of subcarriers where a mutual SIC is performed, the number of subcarrie rs where a Single SIC is performed while t he multiplexed users are powered by the same RRH, and th e number of subcarrier s where a Single SIC is performed while powering t he paired users from different RRHs. On average, NOMA-DBS-SRRH-LPO uses Single SIC NOMA on 25% (resp. 32%) of the s ubcarrier s fo r R k,req = 9 M b p s ( r e s p . 1 2 M b p s ) , w h i l e t h e r e s t o f t h e s u b c a r r i e r s i s m o s t l y dedicated to a single user (a small proportion is not allocated a t all, depending on th e power threshold  ). On the othe r hand, the proportions are respectively 7% and 9% with NOMA-DBS- MutSIC-SOPAd. Therefore, in the light of the results of figures 3 and 5, NOMA-DB S-MutSIC-SOPAd not only outperform s NOMA-DBS-SRRH-L PO from the power perspective, but it also presents the advantage of yielding a reduced complexity a t the User Equipment ( UE) level, by requiring a smaller amount of SIC procedures at the recei ver side. This shows the efficien cy of the mutual SIC strategy, combined with appropriate powe r adjustment, over classical s ingle SIC configurations. TABLE. 2 . Statistics of user multiplexing, for K = 15, S = 64 and R = 4. It can be noted that in NOMA-DBS-M ut&SingSIC, 31% (resp. 44%) of the subcarrie rs are powe red from different antenna s, using either mutual or single SI C . This shows the im portance of exploiting the additional spatial diversity, combined with NOMA, inhe rent to DBS systems. VII. C ONCLUSION In this paper, various resource allocation techniques were presented for minimizing the to tal downlink transmit power in DBS systems for 5G and beyond networks. We first proposed several enhancements to a previously developed method in the CBS context, prior to extending it to the DBS context. Furthermore, we unveiled the hidden potentials o f DBS for NOMA systems and developed new techniques to make the most out of these advantages, while extracting their best characteristics and tradeoffs. Pa rticularly, this study has ena bled the design o f NOMA with S IC dec oding at both paired UE sides. Simulation results have shown th e superiority of the proposed methods with respect to Single SIC configurations. They also promoted mutual SIC with suboptimal power adjustment to the Resource allocation technique Non Mux SC SC MutSIC SC SingSIC SRRH SC SingSIC DRRH  R k,req = 9 Mbps N OMA-DBS-SRRH-LPO 48.156 - 15.844 - N OMA-DBS-MutSIC-SOPAd 59.399 4.601 - - N OMA-DBS-Mut&SingSIC 39. 332 4.601 4.984 15.083 R k,req = 12 Mbps N OMA-DBS-SRRH-LPO 43.203 - 20.797 - N OMA-DBS-MutSIC-SOPAd 58.199 5.801 - - N OMA-DBS-Mut&SingSIC 28. 64 5.801 7.489 22.07 best tra deoff between transmit power and com plexity at both the PCC and the UE levels. Several aspects of this work can be further explored, since m any a dditional challenges need to be addressed to enhance the NOMA-DBS-specific resource allocation schemes. For instance, the study can be enriched by the use of MIMO antenna systems , in a distributed context. Furthermore, practical considera tions can be incorporated in th e study, such as imperfect antenna synchron ization and limited CSI ex change. VIII. 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