Beamspace MIMO-NOMA for Millimeter-Wave Communications Using Lens Antenna Arrays

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1 Beamsace MIMO-NOMA for Millimeter-Wave Communications Using Lens Antenna Arrays Bichai Wang 1, Linglong Dai 1,XiqiGao, and Lajos anzo 3 1 Tsinghua National Laboratory for Information Science and Technology (TNList), Tsinghua University, Beijing , China National Mobile Communications Research Laboratory, Southeast University, Nanjing 10096, China 3 Electronics and Comuter Science, University of Southamton, Southamton SO17 1BJ, U.K. wbc15@mails.tsinghua.edu.cn, daill@tsinghua.edu.cn, xqgao@seu.edu.cn, lh@ecs.soton.ac.u Abstract The recent concet of beamsace multile-inut multile-outut (MIMO) is caable of significantly reducing the number of radio-frequency (RF) chains required by millimeterwave (mmwave) massive MIMO systems. owever, the fundamental limit of the existing beamsace MIMO is that, the number of suorted users cannot be higher than the number of RF chains using the same time-frequency resources. To brea this limit, beamsace MIMO is integrated with non-orthogonal multile access (NOMA) in the roosed MIMO-NOMA system in this aer, where the number of suorted users can be higher than the number of RF chains. To reduce the interbeam interference, a transmit recoding (TPC) scheme based on the rincile of zero-forcing (ZF) is designed. Furthermore, a dynamic ower allocation scheme is roosed for imizing the achievable sum rate. Moreover, a low-comlexity iterative otimization algorithm is conceived for dynamic ower allocation. Simulation results show that the roosed beamsace MIMO- NOMA achieves a higher sectrum and energy efficiency than the existing beamsace MIMO for mmwave communications. I. INTRODUCTION Millimeter-wave (mmwave) massive multile-inut multile-outut (MIMO) is able to significantly imrove the data rate in future 5G communication systems [1]. owever, it is a big challenge to realize mmwave massive MIMO in ractice due to high transceiver comlexity and energy consumtion caused by a large number of radio-frequency (RF) chains []. The recent concet of beamsace MIMO can significantly reduce the number of RF chains by using a lens antenna array. Nevertheless, a fundamental limit of beamsace MIMO is that, each RF chain can only suort a single user using the same time-frequency resources, hence the imum number of users that can be suorted cannot exceed the number of RF chains [3] [5]. In this aer, we brea this fundamental limit by roosing a sectrum- and energy-efficient mmwave transmission scheme that integrates beamsace MIMO with non-orthogonal multile access (NOMA), i.e., beamsace MIMO-NOMA 1. Secifically, NOMA erforms suerosition coding at the transmitter and successive interference cancellation (SIC) at the receiver to realize non-orthogonal multilexing [6] [8]. By 1 Simulation codes are rovided to reroduce the results resented in this aer: htt://oa.ee.tsinghua.edu.cn/dailinglong/ublications/ublications.html. integrating NOMA into beamsace MIMO, where the intrabeam suerosition coding is erformed, more than one user can be simultaneously suorted by a single beam, which is more efficient than beamsace MIMO using a single beam to serve a single user. Thus, the number of suorted users can be higher than the number of beams. To reduce the interbeam interference, the equivalent channel vector of each beam is determined to realize the transmit recoding (TPC) based on the rincile of zero-forcing (ZF). Furthermore, a dynamic ower allocation scheme is roosed for jointly otimizing the ower assigned to different users by imizing the achievable sum rate, and an iterative otimization algorithm is develoed for ower allocation. Our simulation results show that the roosed beamsace MIMO-NOMA achieves a higher sectrum and energy efficiency than conventional beamsace MIMO for mmwave communications. The rest of this aer is organized as follows. The system model of beamsace MIMO is introduced in Section II, and Section III discusses the roosed beamsace MIMO- NOMA. Simulation results are rovided in Section IV. Finally, conclusions are drawn in Section V. II. SYSTEM MODEL OF BEAMSPACE MIMO We consider a single-cell downlin mmwave communication system, where the base station (BS) is equied with a lens antenna array having N antennas as well as N RF RF chains, and K single-antenna users are simultaneously served by the BS [4] [5]. By emloying a lens antenna array in beamsace MIMO, the satial channel can be transformed to the beamsace channel [4]. Secifically, the mathematical function of the lens antenna array is to realize the satial discrete Fourier transformation with the aid of the N N transform matrix U [5], which contains the array s steering vectors of N directions covering the entire sace as follows: U = [ a ( θ1 ) ) )],, a ( θ,, a ( θn (1) where θ ( ) n = 1 N n N+1 for n = 1,,,N are the redefined satial directions /17/$ IEEE

2 Then, the received signal vector ȳ in the downlin can be reresented as ȳ = U WPs + v = WPs + v, () where s = [s 1,s,,s K ] T is the K-element transmitted signal vector for all K users with a normalized ower of E ( ss ) = I K, P =diag{} includes the transmitted ower of all K users with = [ 1,,, ] K satisfying K P (the imum transmitted ower at the BS), =1 W =[w 1, w,, w K ] is the (N K)-element TPC matrix with w =1for =1,,,K, and v is the noise vector obeying the comlex Gaussian distribution CN ( ) 0,σ I K. =[h 1, h,, h K ] of size (N K) is the channel matrix, where h of size (N 1) denotes the satial channel vector between the BS and the th user. In this aer, we consider the widely used Saleh-Valenzuela channel model for mmwave communications [3] [5], hence h can be reresented as ( ) L ( ) h = β (0) a θ (0) + β (l) a θ (l), (3) where β (0) a(θ(0) ) is the line-of-sight (LoS) comonent of the th user with β (0) being the comlex-valued gain and a(θ (0) ) being the array s steering vector. Furthermore, β (l) a(θ(l) ) for 1 l L is the lth non-line-of-sight (NLoS) comonent of the th user, where L is the total number of NLoS comonents. Finally, the beamsace channel matrix is defined as =U=[Uh 1, Uh,, Uh K ]= [ h1, h,, h ] K, (4) l=1 where h = Uh is the beamsace channel vector between the BS and the th user, which is actually the Fourier transformation of the satial channel vector h in (3). It has been shown for mmwave communications that the number of dominant elements of each beamsace channel vector h is much smaller than N, since the number of dominant scatters is limited [5]. In this case, the beamsace channel matrix has a sarse nature [4], which can be exloited for reducing the number of RF chains without obvious erformance loss by beam selection. Secifically, according to the sarse beamsace channel matrix, only a small number of beams can be selected for simultaneously serving K users. Then, the received signal vector in () can be rewritten as ȳ = r W r Ps + v, (5) where r = (i, :) i Γ of size ( Γ K) is the dimensionreduced beamsace channel matrix including the selected beams, and Γ is the index set of selected beams. Furthermore, W r of size ( Γ K) is the dimension-reduced TPC matrix. Since the row dimension of W r is much smaller than N, the number of required RF chains can be significantly reduced, and we have N RF = Γ [5]. owever, a single beam can only suort a single user in existing beamsace MIMO systems [3] [5], otherwise signal for different users cannot be searated by linear oerations, which is a fundamental limit of beamsace MIMO. Note that linear oerations have been considered in almost all of massive MIMO systems due to high comlexity caused by non-linear oerations. To brea this limit, in the next section we roose a new transmission scheme called beamsace MIMO-NOMA that integrates NOMA with beamsace MIMO for mmwave communications. III. PROPOSED BEAMSPACE MIMO-NOMA In this section, the basic rincile of the roosed beamsace MIMO-NOMA will be introduced at first. Then, the corresonding TPC scheme and the dynamic ower allocation scheme will be discussed, searately. A. The basic rincile of beamsace MIMO-NOMA In the roosed beamsace MIMO-NOMA system, beam selection algorithms, such as the imum magnitude (MM) selection and signal-to-interference-lus-noise ratio (SINR) imization based selection [5] for the existing beamsace MIMO, can be used for selecting one beam for each user, where each RF chain corresonds to a single beam. Note that different users may select the same beam, which are termed as interfering users in this aer. In contrast to the existing beamsace MIMO systems, where user scheduling is erformed to select only one user out of these interfering users [5], they can be simultaneously served within the same beam in the roosed beamsace MIMO-NOMA system. Thus, although the number of selected beams is equal to N RF, the number of users K can be higher than N RF, i.e., K N RF. Let S n for n = 1,,,N RF denote the set of users served by the nth beam with S i S j = Φ for i = j and NRF S n = K. The beamsace channel vector with n=1 N RF elements between the BS and the mth user in the nth beam after beam selection is denoted by h m,n, and w n of size (N RF 1) denotes the uniform TPC vector invoed for users in the nth beam. Without loss of generality, we assume that h 1,n w n h,n w n h S w n,n n for n =1,,,N RF. Then, in the nth beam using NOMA with SIC, the ith (i >m) user s signal is detectable at the mth user, rovided that it is detectable at itself [6]. Therefore, the mth user can detect the ith user s signals for 1 m<i S n, and then remodulate and remove the detected signals from its received signals, in a successive manner [6]. Then, the remaining signal received at the mth user in the nth beam can be written as m 1 ˆy m,n =h m,nw n m,n s m,n + h m,nw n i,n s i,n + h m,n w j i,j s i,j + v m,n, j =n (6)

3 where s m,n and m,n are the transmitted signal and the ower transmitted to the mth user in the nth beam, while v m,n is the noise obeying the comlex Gaussian distribution CN ( 0,σ ). Then, according to (6), the SINR at the mth user in the nth beam can be reresented as h m,n w n γ m,n = m,n, (7) ξ m,n where we have ξ m,n = m 1 h m,nw n i,n + S h m,nw j j i,j + σ. j =n (8) As a result, the achievable rate of the mth user in the nth beam is R m,n =log (1 + γ m,n ). (9) Finally, the achievable sum rate of the roosed beamsace MIMO-NOMA system is R sum = R m,n, (10) which can be imroved by carefully designing the recoding {w n } NRF n=1 and ower allocation { m,n} Sn,NRF m=1,n=1, which will be introduced in the next two subsections, searately. B. Precoding Design To reduce the inter-beam interference, TPC should be carefully designed. In existing beamsace MIMO systems, where only a single user can be served in each beam (i.e., K N RF ), the low-comlexity linear ZF TPC can be utilized for removing the inter-beam interference [4] [5], which can be simly realized by alying the seudo-inverse of the beamsace channel matrix to all users. owever, in the roosed beamsace MIMO-NOMA system, the number of users is higher than the number of beams, i.e., K N RF, which means that the seudo-inverse of the beamsace channel matrix of size (N RF K) does not exist. As a result, the conventional ZF recoding cannot be directly used. To address this roblem, an equivalent channel can be derived for each beam for generating the TPC vector. Secifically, since the LoS comonent is dominant in the mmwave channel and the beamsace channel matrix has a sarse structure reresenting the directions of different users [5], the beamsace channel vectors of different users in the same beam are highly correlated. ence, one of the beamsace channel vectors of users multilexed in the nth beam can be regarded as the equivalent channel vector of the nth beam, and this user will not suffer from substantial inter-beam interference. More articularly, leaning in mind that the first user in each beam should erform SIC to decode all the other users signals in this beam, we use the beamsace channel vector of the first user in each beam as the equivalent channel vector. Secifically, the equivalent channel matrix of size (N RF N RF ) for all N RF beams can be written as =[h 1,1, h 1,,, h 1,NRF ]. (11) Then, the TPC matrix of size (N RF N RF ) can be generated by W =[ w 1, w,, w NRF ]=( ) = ( ) 1. (1) After normalizing the TPC vectors, the recoding vector for the nth beam (n =1,,,N RF ) can be written as w n = w n. (13) w n C. Dynamic Power Allocation After obtaining the TPC vectors in (13), the ower allocation should be otimized. Similar to the existing MIMO- NOMA systems [7] [8], both the inter-beam interference and intra-beam interference should be reduced to imrove the achievable sum rate. owever, in contrast to the existing MIMO-NOMA systems, where fixed inter-beam ower allocation and only two users er beam are usually considered, multile users (e.g., 1,, or 3 users) are allowed in each beam in the roosed beamsace MIMO-NOMA system. Accordingly, the dynamic ower allocation is roosed for imizing the achievable sum rate by solving the joint ower otimization roblem, which includes both the intra-beam ower otimization and the inter-beam ower otimization. Secifically, the ower allocation roblem can be formulated as { m,n} R m,n, s.t. C 1 : m,n 0, C : n, m, m,n P, C 3 : R m,n R min, n, m, (14) where R m,n is the achievable rate of the mth user in the nth beam, as defined in (9). Furthermore, the constraint C 1 indicates that the ower allocated to each user must be ositive, C is the transmit ower constraint with P being the imum total ower transmitted by the BS, and C 3 is the data rate constraint for each user with R min being the minimum data rate of each user. By substituting (7)-(9) into the constraint C 3 in (14), we have h m,n w n m,n η m 1 h m,n w n η h m,n w j i,j ω, j =n i,n (15) where η = Rmin 1 and ω = ησ. In this way, the non-linear constraint C 3 has been transformed into a linear constraint. Then, the otimization roblem (14) can be rewritten as { m,n} log (1+γ m,n ), s.t. C 1, C, (15). (16)

4 We can see from (16) that all constraints are linear inequality constraints, while the objective function is non-convex. Therefore, this otimization roblem is NP-hard, and it remains an oen challenge to obtain the closed-form solution of the otimal ower allocation roblem (16). To solve this difficult non-convex roblem (16), we roose an iterative otimization algorithm for carrying out the ower allocation. Secifically, according to the extension of the Sherman-Morrison-Woodbury formula [9], we have (1 + γ m,n ) 1 =1 h m,nw n m,n ( ) h m,nw n 1, m,n + ξ m,n (17) where n =1,,,N RF and m =1,,, S n. We can find that the exression (17) has the same form as the MMSE. Secifically, if MMSE detection is used for detecting s m,n from ˆy m,n in (6), it can be formulated as where c o m,n =argmin c m,n e m,n, (18) { e m,n =E s m,n c m,nˆy m,n } (19) is the mean square error (MSE), c m,n is the channel equalization coefficient, and c o m,n is the otimal value of c m,n required for minimizing the MSE. Substituting (6) into (19), we have e m,n = 1 cm,n m,n h m,nw n + cm,n σ + c m,n h m,nw n + c m,n j =n m 1 i,n h m,n w j i,j. (0) Then, by solving (18) based on (0), the otimal equalization coefficient c o m,n can be calculated by c o m,n = ( m,n h ) ( ) m,nw n m,n h m,n w n 1. +ξ m,n (1) Substituting (1) into (0), we obtain the MMSE as e o m,n =1 ( h m,n w n h ) m,n m,n w n 1, m,n + ξ m,n () which is equal to (1 + γ m,n ) 1 in (17). Then, the achievable rate of the mth user in the nth beam can be written as R m,n =log (1 + γ m,n )=( log e m,n ). c m,n (3) To remove the log function in (3), we introduce the following roosition [9]. Proosition 1: Letf (a) = ab ln +log a + 1 ln and a be a ositive real number. Then we have f (a) = log b, a>0 where the otimal value of a is a o = 1 b. Using Proosition 1, (3) can be rewritten as ( R m,n = a m,ne m,n +log c m,n a m,n>0 ln a m,n + 1 ). (4) ln As a result, the objective function of the otimization roblem (16) has been transformed into a quadratic rogramming function, and (16) can be reformulated as { m,n} ( am,ne m,n c m,n a ln +log a m,n + 1 ln ), m,n>0 s.t. C 1, C, (15). (5) To solve the reformulated otimization roblem (5), we roose to iteratively otimize {c m,n }, {a m,n }, and { m,n }. Secifically, given the otimal ower allocation solution { m,n (t 1) {c (t) } in the (t 1)th iteration, the otimal solution of m,n} in the tth iteration can be obtained according to (1). The corresonding MMSE {em,n} o(t) can be obtained according to (), and then the otimal solution of {a (t) m,n} in the tth iteration can be obtained by a (t) m,n = 1. em,n o(t) After obtaining the otimal {c (t) m,n} and {a (t) m,n} in the tth iteration, the otimal { (t) m,n} in the tth iteration can be obtained by solving the following roblem: RF S min { (t) m,n} a (t) m,ne (t) m,n, s.t. C 1, C, (15). (6) Then, the convex otimization roblem (6) can be solved according to the Karush-Kuhn-Tucer (KKT) [8] conditions. IV. SIMULATION RESULTS In this section, we rovide simulation results for verifying the erformance of the roosed beamsace MIMO-NOMA. Secifically, we consider a tyical downlin mmwave massive MIMO system, where the BS is equied with N = 56 antennas and communicates with K = 3 users. One LoS comonent and L =NLoS comonents are assumed for all users channels. We consider the channel arameters of user as follows: 1) β (0) CN(0, 1), β (l) CN ( 0, 10 1) for 1 l L; )θ (0) and θ (l) for 1 l L obey the uniform distribution within [ 1, ] 1. The signal-to-noise ratio (SNR) is defined as E b σ in this aer. In the simulations, we consider the following four tyical mmwave massive MIMO schemes for comarison: 1) Fully digital MIMO, where each antenna is connected to one RF chain, i.e., N RF = N; ) Beamsace MIMO [5], where each beam only suorts a single user with N RF = K; 3) MIMO- OMA [8] with K N RF, where OMA is erformed for the interfering users, and users in the same beam are allocated with orthogonal frequency resources; 4) Proosed beamsace MIMO-NOMA with K N RF, which integrates NOMA into beamsace MIMO, and different users share the same timefrequency resources. Particularly, ZF recoding is considered in the fully digital MIMO and beamsace MIMO. In this aer, the sectrum efficiency is defined as the achievable sum rate (10), and the energy-efficiency ε is defined as the ratio between the achievable sum rate R sum and the total ower consumtion [10], i.e., ε = R sum P + N RF P RF + N RF P SW + P BB (bs/z/w), (7)

5 Sectrum efficiency Energy efficiency Fully digital MIMO Proosed beamsace MIMO NOMA MIMO OMA [8] Beamsace MIMO [5] SNR (db) Fig. 1. Sectrum efficiency against SNR. Fully digital MIMO Proosed beamsace MIMO NOMA MIMO OMA [8] Beamsace MIMO [5] SNR (db) Fig.. Energy efficiency against SNR. where P is the imum transmitted ower, P RF is the ower consumed by each RF chain, P SW is the ower consumtion of the beam switch, and P BB is the baseband ower consumtion. Secifically, we adot the tyical values of P = 3 mw, P RF = 300 mw, P SW =5mW, and P BB = 00 mw [10]. Fig. 1 shows the sectrum efficiency against SNR, where the number of iterations required for solving the ower allocation otimization roblem is set to 10 (We have verified through simulations that the sectrum efficiency tends to a constant after 10 iterations). We find that the roosed beamsace MIMO-NOMA achieves a higher sectrum efficiency than beamsace MIMO [5] as well as MIMO-OMA [8]. Quantitatively, the roosed beamsace MIMO-NOMA has about 3 db SNR gain over beamsace MIMO, which benefits from the emloyment of NOMA to serve multile users in each beam. Additionally, the roosed beamsace MIMO-NOMA also outerforms MIMO-OMA, since NOMA can achieve a higher sectrum efficiency than that of OMA [6]. Fig. shows the energy efficiency against SNR. We find that the roosed beamsace MIMO-NOMA achieves a higher energy efficiency than the other three schemes. Particularly, the roosed beamsace MIMO-NOMA has about 4 db SNR gain over beamsace MIMO, which benefits from the emloyment of NOMA to serve multile users in each beam. Additionally, the roosed beamsace MIMO-NOMA achieves a higher energy efficiency than the fully digital MIMO, where the number of RF chains is equal to the number of BS antennas. By contrast, the number of RF chains is lower than the number of antennas in the roosed beamsace MIMO-NOMA. V. CONCLUSIONS In this aer, we have roosed a new mmwave transmission scheme called beamsace MIMO-NOMA to integrate beamsace MIMO and NOMA in order to brea the fundamental limit of beamsace MIMO, where more than one user can be served in each beam using the same time-frequency resources. To mitigate the inter-beam interference, the equivalent channel vector was determined for each beam for ZF-based TPC. Furthermore, we roosed to jointly otimize the ower allocation of all users by imizing the achievable sum rate, and an iterative otimization algorithm has been develoed for ower allocation. Our simulation results have shown that the roosed beamsace MIMO-NOMA achieves better erformance than beamsace MIMO in terms of sectrum and energy efficiency. ACKNOWLEDGEMENTS This wor was suorted by the National Natural Science Foundation of China (Grant Nos and ), and the Royal Academy of Engineering under the UK China Industry Academia Partnershi Programme Scheme (Grant No. UK-CIAPP 49). REFERENCES [1] S. Mumtaz, J. Rodriquez, and L. Dai, MmWave Massive MIMO: A Paradigm for 5G, Academic Press, Elsevier, 016. [] R. W. eath, N. Gonzlez-Prelcic, S. Rangan, W. Roh, and A. M. Sayeed, An overview of signal rocessing techniques for millimeter wave MIMO systems, IEEE J. Sel. Toics Signal Process., vol. 10, no. 3, , Ar [3] J. Brady, N. Behdad, and A. Sayeed, Beamsace MIMO for millimeterwave communications: System architecture, modeling, analysis, and measurements, IEEE Trans. Antennas Proag., vol. 61, no. 7, , Jul [4] Y. Zeng and R. Zhang, Millimeter wave MIMO with lens antenna array: A new ath division multilexing aradigm, IEEE Trans. Commun.,vol. 64, no. 4, , Ar [5] X. Gao, L. Dai, Z. Chen, Z. Wang, and Z. Zhang, Near-otimal beam selection for beamsace mmwave massive MIMO systems, IEEE Commun. Lett., vol. 0, no. 5, , May 016. [6] L. Dai, B. Wang, Y. Yuan, S. an, C.-L. I, and Z. Wang, Nonorthogonal multile access for 5G: Solutions, challenges, oortunities, and future research trends, IEEE Commun. Mag., vol. 53, no. 9, , Se [7] Z. Ding, F. Adachi, and. V. Poor, The alication of MIMO to nonorthogonal multile access, IEEE Trans. Wireless Commun., vol. 15, no. 1, , Jan [8] M. S. Ali,. Tabassum, and E. ossain, Dynamic user clustering and ower allocation in non-orthogonal multile access (NOMA) systems, IEEE Access, vol. 4, , Aug [9] Q. Zhang, Q. Li, and J. Qin, Robust beamforming for non-orthogonal multile access systems in MISO channels, to aear in IEEE Trans. Veh. Technol., 017. [10] X. Gao, L. Dai, S. an, C.-L. I, and R. W. eath, Energy-efficient hybrid analog and digital recoding for mmwave MIMO systems with large antenna arrays, IEEE J. Sel. Areas Commun., vol. 34, no. 4, , Ar. 016.

User Selection and Power Allocation for MmWave-NOMA Networks

User Selection and Power Allocation for MmWave-NOMA Networks Jingjing Cui, Yuanwei Liu, Zhiguo Ding, Pingzhi Fan and Arumugam Nallanathan Southwest Jiaotong University, Chengdu, P. R. China Queen Mary