Power Allocation in OFDM based NOMA Systems: A DC Programming Approach

Transcription

1 Power Allocation in OFDM based NOMA Systems: A DC Programming Approach Priyabrata Parida and Suvra Sekhar Das Indian Institute of Technology Kharagpur, India. Abstract In this work, we have considered the downlink of an Orthogonal Frequency Division Multiplexing based Non Orthogonal Multiple Access system where transmission to multiple number of users is performed on the same sub-band (time-frequency resource unit) using Superposition Coding (SC) technique. At the receiver side, the SC coded symbols are recovered with Successive Interference Cancellation (SIC). Assuming that complete channel state information is present at the base station, we propose (1) co-channel user set selection, (2) power distribution among the multiplexed users on each sub-band, and (3) power allocation across the sub-bands to maximize the weighted sum rate of the system. Since the problem is a non-convex combinatorial optimization problem, two step heuristic solution is employed. In the first step, for each of the sub-bands, a greedy user selection and iterative sub-optimal power allocation algorithm based on Difference of Convex (DC) programming is presented. In the second step, exploiting the DC structure of the modified problem, power allocation across sub-band is carried out through the same iterative power allocation algorithm. Simulation results are provided to assess and compare the performance of the proposed algorithms. I. INTRODUCTION The demand for mobile data traffic has been increasing exponentially for last two decades and is expected to be times larger in terms of traffic volume in 2020 than in 2010 [1]. To support this overwhelming demand for data traffic, researchers from both industry and academia are actively exploring the lesser investigated domains of Massive- MIMO, small cells, D2D communication, novel multiple access schemes etc. Traditionally, Orthogonal Multiple Access (OMA) schemes such as Frequency Domain Multiple Access (FDMA), Time Division Multiple Access (TDMA), Code Division Multiple Access (CDMA), and Orthogonal FDMA (OFDMA) have been used in cellular networks to serve multiple users in the system. Although OMA techniques are reasonable choices for obtaining good system level spectral efficiency with simple single user detection, they fail to achieve the capacity of the broadcast channel. However, Superposition Coding (SC) with Successive Interference Cancellation (SIC) is a well known non orthogonal signaling scheme that achieves the capacity in Gaussian broadcast channel [2], [3]. From information theoretic perspective, SC with SIC for broadcast channels has been investigated in [4], [5]. In [4], author has proposed an optimal power allocation and user decoding sequence through a greedy algorithm. The power allocation algorithm strategy is shown to be equivalent to the water-filling procedure. For a multi carrier system, superposition based multi-user access in the uplink is studied in [6], [7]. In [6], the author has proposed an optimized scheduling method and investigated the system level throughput in the uplink. System level simulation results are present in [7] for 3GPP-LTE uplink with multiple user scheduling over same time-frequency resource. For downlink of a multi carrier system (the focus of this work), SC with SIC is studied in [8] [12] where authors have proposed OFDM based NOMA scheme to enhance the spectrum efficiency of existing OFDMA based cellular networks. In [8], based on the results from [4] authors analyze the number of scheduled users over a Rayleigh fading channel on a single resource unit. In addition, numerical evaluation of long term average throughput is also carried out for different scheduling schemes in the same paper. System level performance of OFDM-NOMA system considering Fractional Transmit Power Allocation (FTPA) among multiplexed users and equal power across all sub-bands is presented in [9]. The same authors have also shown the effect of error propagation in SIC on system performance in [10]. In [13], authors have studied the performance of NOMA for a system with Fractional Frequency Reuse (FFR) to enhance the cell-edge user performance. A power assignment algorithm based on branch and bound method, which is suitable for implementation in systems employing discrete modulation and coding scheme, is presented in [11]. The enhancement of NOMA to multiantenna system is studied in [12]. Although FTPA is simple to implement, it does not distribute the power among multiplexed users in an optimum way to maximize the system utility. Moreover, enhancement in system utility is possible by exploiting frequency diversity and assigning more power to the sub-bands with better channel quality. Motivated by these two ideas, for the downlink of an OFDM-NOMA system, we have proposed algorithms for (1) non orthogonal user scheduling, (2) power allocation among multiplexed users, and (3) power allocation across sub-bands to maximize overall system utility (weighted sum rate) subject to maximum power constraint. Furthermore, from a practical implementation point of view, we have considered that a maximum of two users can be multiplexed over a certain sub-band. The framed optimization problem is non-convex in nature and is solved heuristically in two steps. In the first step user selection and power allocation among the users is performed based on a greedy user assignment and iterative power allocation algorithm for each of the sub-bands. In the second step, given the information regarding the multiplexed users and the power distribution between them, the base station tries to maximize the system utility by allocating power to sub-bands depending upon the channel quality of the users /14/$ IEEE 1020

2 scheduled on them. In both steps, the objective functions for optimal power allocation is decomposed into Difference of Convex functions (DC), which is iteratively solved with successive convex approximation to arrive at an efficient sub optimal solution. Moreover, we have studied the effect of the number of users on the system performance as well. The rest of the paper is organized as follows. Section II describes the system model and focuses on problem formulation. The proposed algorithms are given in Section III. Simulation results demonstrating the effectiveness of the proposed methods are presented in Section IV. Finally, conclusion is drawn in Section V. II. SYSTEM MODEL In this work, we have considered a downlink OFDM- NOMA based cellular system with hexagonal grid layout of 19 sites. Each site is divided into 3 cells with directional antennas. M number of users are deployed uniformly in the center cell (known as desired cell) and rest of the cells contribute to the Inter Cell Interference (ICI). The total bandwidth of the system (B sys ) is equally partitioned into N number of sub-bands with bandwidth B sb. Moreover, each sub-band consists of N sc number of sub-carriers of bandwidth B sc in the frequency domain and spans over one Transmission Time Interval (T tti ) in time domain. Since the system employs multiplexing of users through superposition coding, a single sub-band can be assigned to multiple number of users. A. Signal Model Considering that k n number of users are multiplexed on n th sub-band (SB n ), a superposition coded symbol (as in (1)) will be transmitted by the base station on each of the sub-carriers in SB n. k n s n = pn,is i, (1) i=1 where s i s are complex QAM symbols drawn from a constellation with zero mean and unit variance, and p n,i is the power allocated to i th user multiplexed on SB n. If P n is the total power alloted to SB n, then k n i=1 p n,i = P n. If i th user is not scheduled on SB n then p n,i = 0. The signal that would be received by the m th user (UE m ) on SB n is given by ym n = h n m sn + wn m = p n,mh n m sm + k n i=1,i m pn,i h n m s i + w n m, (2) where h n m is the complex channel response between the desired base station and UE m on SB n. It captures the effect of path loss, shadowing, and small scale fading. wm n constitutes of both additive Gaussian white noise and ICI. Without SIC, the received Signal to Interference plus Noise Ratio (SINR) n,p re (Pre-SIC SINR) of UE m on SB n is γ p n,m h n m 2 kn i=1,i m p n,i h n m 2 + E[ wm n 2 ] = m = p n,mγn m 1 + k n i=1,i m p n,i Γ n m, (3) where Γ n m = h n m 2 /E[ w n m 2 ] represents the Channel to Interference plus Noise Ratio (CINR) of UE m on SB n. SIC thrives on the fact that if all the users having a poorer channel condition with respect to UE m are able to decode their respective transmitted symbols, then UE m would also be able to decode all the symbols transmitted to those users. Without loss of generality consider that for all the users multiplexed over SB n, the CINRs are ordered as Γ n 1 2 Γ n Γ n m 2 Γ n m Γ n k n 2. Hence, UE m would be able to successfully decode and remove the symbols of the users UE m+1, UE m+2,..., UE kn. However, the interference power from UE 1, UE 2,..., UE m 1, who are having better channel conditions compared to UE m, will be treated as noise by UE m during the decoding process. After performing SIC, the post processing SINR of UE m on SB n is given by γm n p n,m Γ n m = 1 + m 1 i=1 p. (4) n,iγ n m B. Problem Formulation Using (4), the total number of bits that can be transmitted to UE m on SB n during a time slot can be written as R n m( p n) = B sb T ttilog 2 (1 + γ n m( p n)), (5) where p n R M contains the power assigned to each of the multiplexed user on SB n. The objective of scheduling and resource allocation unit is to (i) select the best set of co-channel users for each sub-band, (ii) distribute power among the multiplexed users on each sub-band, and (iii) allocate power across the subbands subject to total power constraint (P max ) so that total system utility (weighted sum rate) can be maximized. The optimization problem to be solved by the scheduling unit is given as max p n 0,α i,n {0,1} subject to M α i,n W i Ri n ( pn) i=1 p n 1 = P max. In (6), W i is the weight associated with UE i. In our case, we have considered the weight as the reciprocal of past average data rate of the user. In that sense, the utility function that we have considered is equivalent to proportional fair utility function [14]. Furthermore, α i,n is a binary variable defined as follows { 1 if UE i is assigned to SB n α i,n = (7) 0 Otherwise C. Blueprint to Obtain Efficient Sub-Optimal Solution Problem (6) is a combinatorial optimization problem and for simple OFDMA based systems (where only one user per subband is allowed), this problem is NP-Hard in nature. From a practical implementation point of view and guided by the results presented in [8], [9], in this work we have considered that maximum two users can be multiplexed over a particular sub-band. This consideration is also important from the point of view of keeping the receiver complexity comparatively less (6) 1021

3 and to restrict the error propagation due to SIC to a single level only. Hence, (6) can be written in the following form max p n 0 subject to 2 W n,i Ri n ( pn) i=1 (8) p n 1 = P max In (8), without loss of generality, W n,1 represents the weight of the user who performs SIC on SB n and W n,2 represents the user who does not perform SIC. Furthermore, R n 1 ( p n ) and R n 2 ( p n ) are the rates of the SIC and non-sic users respectively on SB n. Due to the complexity of the above problem, it is very difficult to obtain the global optimum solution within allowable time limit. In this work we have attempted to solve the problem in two steps and find an efficient sub-optimal solution. In the first step, for each of the sub-bands, 1) Find the two users to be multiplexed in the power domain (Discussed in Section III-A). 2) Find the power ratio (β) based on which multiplexed users will be assigned fractions of powers available to the total resource band (Discussed in Section III-B). We define the parameter β as the ratio of the power allocated to the user who will perform SIC to the total power available for that sub-band. In the second step, Power allocation is performed across the sub-bands such that total weighted sum rate of the system gets maximized (Discussed in Section III-C). III. PROPOSED ALGORITHMS The performance of NOMA is principally guided by the selection of user set over a particular sub-band and allocation of the power to the multiplexed users on the sub-band. A. A Brief Description of User Multiplexing Algorithm 1) User Selection Through Exhaustive Search [9]: Selection of the best set of users to be multiplexed is a combinatorial optimization problem and is also coupled with the power allocation algorithm among the multiplexed users. For a given power distribution algorithm, one approach to select the best set is to search over all possible combinations of users and select the one which maximizes the weighted sum rate on the sub-band as mentioned in [9]. Complexity: The search space depends on the number of users to be multiplexed over a certain sub-band. If out of M users, m number of users are to be multiplexed, then the scheduler needs to search over ( M m ) + ( M m 1) ( M 1 ) combinations. The overall complexity of the algorithm would be O(2 M ) [15]. 2) Heuristic Algorithm for User Selection: In order to reduce the complexity of the exhaustive search, the user selection can be performed in a greedy manner for each of the sub-bands. The first user to be multiplexed over the sub-band would be the one which has the maximum weighted sum rate (similar to Proportional Fair Scheduling [14]). The next user is selected out of M 1 remaining users subject to condition that it improves the weighted sum rate over that sub-band. The algorithm is presented in Algorithm 1. Complexity: Let us consider that it takes an average of T 1 iteration for the DC algorithm, presented in Section III-B2, to converge. So over all complexity of the user selection algorithm for a system consisting of M number of users and N number of sub-bands would be O(MN) + T 1O(MN) [15]. Algorithm 1: Greedy User Selection Algorithm Data: Set of Users in the System (U M ), Set of Γ i s, Set of User Weights (W i s), Average User Data Rates ( R i (t)s). Result: Desired Suboptimal User Set K sopt. 1 for each sub-band do 2 K sopt = 3 W SR sopt = 4 K sopt argmax i UM Ri n ( pn) R i (t) 5 Store the Single User Weighted Sum Rate (W SR 1 ) R n i ( pn) R i (t) U M U M argmax i UM 6 for each i in U M do 7 Find Power Ratio Using (9) or Algorithm 2 8 Find the Weighted Sum Rate (W SR 2,i ) using the objective function of Problem (10) 9 W SR sopt W SR 2,i 10 W SR 2 = max(w SR sopt ) 11 if W SR 1 W SR 2 then 12 K sopt argmax i UM W SR sopt B. Power Allocation between Multiplexed Users As mentioned in the previous section, the user selection and power allocation processes are coupled with each other. In literature, such as [9] and [10], the use of Fractional Transmit Power Allocation (FTPA), prevalent for power control in LTE uplink, is used for OFDM-NOMA downlink systems. In this section, we provide a brief description of FTPA as well as, the proposed power allocation scheme based on DC programming approach. 1) Fractional Transmit Power Allocation: In FTPA, the transmit power to the multiplexed user UE m is dynamically allocated as per the channel gains of all the multiplexed users on that sub-band. The FTPA scheme is given as follows p n P nγ n m ( σ) m = j U Opt,n Γ, (9) n ( σ) j where σ is the decay factor. From the observation of (9), it is clear that σ = 0 corresponds to equal power transmission among all the users, and as σ increases more power is alloted to the channel with poorer channel condition. However, FTPA power allocation scheme does not take into consideration the weights of the multiplexed users while assigning power to them. Hence, from weighted sum rate maximization point of view, FTPA is sub-optimal. In the following subsection we address the power allocation among multiplexed users by solving the weighted sum rate maximization problem on each sub-band. 2) DC Programming Based Power Allocation: As already mentioned in Section II, in this work we have considered only two users to be multiplexed over a certain sub-band. Let us consider a dummy scenario where two users UE n 1 and UE n

4 are to be multiplexed over SB n with Γ 1 and Γ 2 as their CINRs and W 1 and W 2 as their respective weights. Without loss of generality we consider that Γ 1 Γ 2. Hence, UE1 n performs SIC to remove the interfering symbol power of UE2 n, whereas UE2 n decodes its data treating UE1 n symbol power as noise. As the objective is to maximize the weighted sum rate over the sub-band for a given power ratio (β), the problem can be stated as ( ) max W1log 2 (1 + βpnγ1) + W2log (1 β)pnγ β [0,1] 1 + βp nγ 2 ( ) 1 + PnΓ 2 min W 1log 2 (1 + βp nγ 1) W 2log 2 β [0,1] 1 + βp nγ 2 min q(β) = min f(β) g(β), β [0,1] β [0,1] (10) where ( f(β) is ) W 1log 2 (1 + βp nγ 1) and g(β) is W 1+PnΓ 2log βP nγ 2. It can be shown that both f(β) and g(β) are convex with respect to the β. Problem (10) is not convex in general. However, since the objective function is the difference of two convex functions, it can be efficiently solved using numerical methods to get local and at times global optimum solution [16], [17]. Using successive convex approximation approach efficient suboptimal solution can be found by iteratively solving a sequence of convex sub-problems. The convex sub-problems are obtained by linearizing the non-convex part of the objective function. Let Q(x) be an objective function which can be expressed as the difference of two convex functions F(x) and G(x) i.e. Q(x) = F(x) G(x). Then the convex sub problems are obtained by replacing G(x) with its first order approximation at poing x (k) i.e. G(x) = G(x (k) ) + G(x (k) ) T (x x (k) ). The generic algorithm to solve optimization problems involving difference of convex functions is given in Algorithm 2. In (11), Q (k) (x) Algorithm 2: Iterative, suboptimal solution for DC Problems Data: Initial Point (x (0) ), Max Iterations (K max), Tolerance (ɛ), Objective function Q(x), Convex functions F(x) and G(x). Result: Desired Suboptimal Solution x subopt. 1 Set k = 0 2 Repeat Step (3) to (5) until Q(x (k+1) ) Q(x (k) ) ɛ or k > K max 3 Convex Approximation of Q(x) at x (k) : 4 Solve: 5 k k + 1. Q (k) (x) = F(x) G(x (k) ) G(x (k) ) T (x x (k) ) (11) x (k) = argmin Q (k) (x) (12) x χ is convex w.r.t. x. In (12) the minimization is performed over the set of feasible solutions (χ). If the constraint set (χ) is compact and continuous, by Cauchy s Theorem, the sequence {Q (k) (x)} always converges. Hence, the iterative process terminates at no solution improvement with some tolerance limit. Furthermore, if F(x) and G(x) are continuously differentiable on the constraint set, Algorithm 2 always returns a stationary point of the objective function Q (k) (x) [18]. However, the attainment of a stationary point does not guarantee global optimal solution. With repsect to Problem (10), the variable x of Algorithm 2 is the scalar quantity β and (12) reduces to an one dimensional search problem which can be efficiently solved using any line search method such as bisection method. C. Power Allocation Across Sub-bands In this section, we discuss two different methods to allocate power across sub-bands given user allocation and power ratios on the sub-bands. 1) DC Programming Approach: Since the scheduler has the information regarding the users to be scheduled on each of the sub-bands and the power ratio to distribute power between the scheduled users, the optimization Problem (8) can be reframed as follows ( ) 1 + PnΓ max W n,1 log 2 (1 + β np nγ n,1 ) + W n,2 log n,2 2 P n β np nγ n,2 subject to P n = P max (13) where β n is the power ratio on SB n and P n = p n 1 is the power allocated to SB n. Problem (13) is non convex w.r.t. P n. Moreover, Problem (13) can also be written as subject to min Y( P ) Z( P ) P n 0 P 1 = P max, (14) where Z( P ) is N Wn,2log 2 (1 + βnpnγn,2), Y( P ) is N [ Wn,1log 2 (1 + βnpnγn,1) Wn,2log 2 (1 + PnΓn,2)], and P R N contains the power allocated to each of the sub-bands. It is easy to show that 2 Y( P ) and 2 Z( P ) are positive semi-definite matrices. We can exploit the difference of convex function structure of Problem (14) to get the iterative sub-optimal solution using Algorithm 2. Note that in case of Problem (14), the variable x of Algorithm 2 is the power vector P and (12) can be solved by interior point method or sequential quadratic programming. Once the power on each of the sub-bands gets decided, the scheduling takes place with pre-calculated power ratio and user allocation decision. 2) Equal Power Across Sub-bands: Instead of allocating optimized power values across the sub-bands, equal power can be allocated to each of the sub-bands which would result in reduction in both complexity and signaling overhead. IV. SIMULATION RESULTS In this section, we study the performance of proposed algorithms through extensive Monte Carlo simulation. 1023

5 Parameters TABLE I SYSTEM PARAMETERS Values Network Layout Hexagonal grid with 19 sites, 3 sectors per cell Distance Dependent Path loss log 10 (d km ) enb Transmit Power System Bandwidth Subcarrier Spacing Number of Sub-bands TTI Duration Antenna Configuration Rx Noise Figure Shadow Fading Throughput Calculation A. Simulation Parameters 41 dbm 5 MHz 15 KHz 10 (30 Sub-carriers / SB) 1 ms (12 OFDM Symbols) SISO 7 db Log normal distribution standard deviation 8 db Based on Shannon s Formula (max 6 b/s/hz) We have considered a 19 cell hexagonal cellular model with unity frequency reuse. The inter-site distance is considered to be 500 m. Users with mean velocity of 3 Km/hr are uniformly distributed within the cell. The minimum distance of the users from the base station is kept to be 35 meters. Power delay profile of Sub-Urban Macro Non Line of Sight (SuMa-NLoS) [19] is considered to generate the small scale variation in the channel. ICI is generated assuming the neighbouring cells are operating at full load condition. Moreover, there is no coordination among neighbouring cells to avoid ICI. Rest of the system parameters is presented in Table I. In order to average SINR across the sub-carriers within a sub-band capacity based averaging 1 method is used. We have considered a full buffer best effort traffic model for simulation purpose. B. Numerical Results Legends used for different power allocation schemes is given in Table II. Algorithm FTPA- Equal TABLE II LEGENDS User Power Assignment within a sub-band FTPA Power allocation across the sub-band Equal Power Allocation FTPA-DC FTPA Iterative DC Algo. DC-Equal Iterative DC Algo. Equal Power Allocation DC-DC Iterative DC Algo. Iterative DC Algo. On Power Allocation Schemes: Fig. 1 presents the distribution of downlink user throughput for following two cases: (1) power allocation within a sub-band and across the subbands through DC algorithm, (2) power allocation within a sub-band through FTPA and across sub-bands through DC algorithm. In both the cases, greedy user selection scheme is 1 B sb log(1 + γ sb ) = B sc Nsc i=1 log(1 + γi sc ) where γ sb is the effective (average) SINR over the sub-band and γ i sc is the SINR over ith sub-carrier within the sub-band. Prob (User Throughput Abscisa) Percentage of User Pairing DC DC FTPA DC OFDMA Downlink User Throughput (b / s) x 10 6 Fig. 1. Distribution of Downlink User Throughput FTPA Equal FTPA DC DC Equal DC DC Num of Users Fig. 2. Percentage of User Pairing with respect to Number of Users employed with 10 users in the system. It is observed that DC- DC power allocation scheme is better than FTPA-DC in terms of user throughput. Further, Mean Spectral Efficiency (MSE) for all the four cases is presented in the first column of Table IV. It is observed that, in all the four cases user throughput is better than OFDMA scheme. The user throughput distribution for OFDMA scheme is obtained through proportional fair scheduling with an averaging window of 100 TTIs. The percentage of improvement of MSE w.r.t. OFDMA is given within brackets in the first column of Table IV. Moreover, for power allocation between users within a sub-band, iterative DC algorithm gives better performance in comparison to FTPA scheme. The reason behind this increase in system throughput can be attributed to the fact that using DC algorithm the percentage of time two users are multiplexed increases as shown in Fig. 2. It also reveals that with increase in the number of users in the system percentage of user pairing also increases. However, the values saturate beyond a particular number of users in the system. As shown in Fig. 3, irrespective of the number of users in the system, DC-DC gives the maximum benefit followed by DC-Equal, FTPA-DC, and FTPA-Equal. Hence, distributing the power across sub-bands has clear benefits in terms of mean system throughput at the cost of increased system complexity and control channel overhead. On User Selection Schemes: Table III presents the mean 1024

6 Mean System Throughput (Mbps) FTPA Equal FTPA DC DC Equal DC DC Num of Users Fig. 3. Mean System Throughput as a Function of Number of Users system throughput for both the user selection schemes when 10 number users are present in the system. It is clear that the performance of the proposed greedy user selection method is as good as the exhaustive search method for all the four cases of power allocation. TABLE III COMPARISON BETWEEN USER SELECTION SCHEMES Algorithm Exhaustive Greedy FTPA-Equal Mbps Mbps FTPA-DC Mbps Mbps DC-Equal Mbps Mbps DC-DC Mbps Mbps On Convergence of Power Allocation Algorithms: The iterative DC programming based algorithm (Algorithm 2) converges quite fast. In both the cases, power allocation among multiplexed users and power allocation across the sub-bands, the convergence occurs within 5-10 iterations for a tolerance limit (ɛ) of It is also observed that the number of steps required to converge is independent of the starting point. On Multi User Diversity: From Fig. 3 and Table IV it is clear that increasing the number of users in the system improves the mean system throughput for all the schemes, which signifies the multi user diversity gain. TABLE IV MULTI USER DIVERSITY GAIN MSE Algorithm 10 Users 15 Users 20 Users 25 Users FTPA-Equal (9.44 %) FTPA-DC (10.44 %) DC-Equal (12.44 %) DC-DC (15.83 %) V. CONCLUSION In this paper we have addressed the issue of non orthogonal user selection, power allocation among the selected users and power allocation across the sub-bands for an OFDM based NOMA system. From practical implementation point of view, we have considered that maximum two users can be multiplexed over a sub-band. Given a power distribution algorithm among the multiplexed users, we have proposed a greedy user selection algorithm for sub-bands. It is seen that greedy user selection performs as good as exhaustive user selection scheme. The proposed power distribution among users within a sub-band based on iterative DC algorithm performs better than existing FTPA scheme. Moreover, it is shown that further improvement in the system capacity is possible by allocating power across sub-bands using the proposed iterative DC programming algorithm. Through extensive system level simulation, it is shown that the proposed power allocation algorithm along with greedy user selection algorithm gives more than 15% benefit in terms of mean spectral efficiency. REFERENCES [1] Qualcomm, 3GPP RAN Rel-12 and beyond, Qualcomm, Tech. Rep., [2] D. Tse and P. Viswanath, Fundamentals of Wireless Communications, [3] T. M. Cover and J. A. Thomas, Elements of Information Theory. Wiley- Interscience, [4] D. Tse, Optimal power allocation over parallel gaussian broadcast channels, Online: dntse/papers/broadcast2.pdf, [5] L. Li and A. 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