User Scheduling for Massive MIMO OFDMA Systems ...

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User Scheduling for Massive MIMO OFDMA Systems with Hybrid Analog-Digital Beamforming Tadilo Endeshaw Bogale+ , Long Bao Le+ and Afshin Haghighat++ Institute National de la Recherche Scientifique (INRS)+ Interdigital++ Email: {tadilo.bogale, long.le}@emt.inrs.ca and [email protected]

Abstract— This paper proposes new user scheduling and subcarrier allocation algorithm for multiuser downlink massive multiple input multiple output (MIMO) orthogonal frequency division multiple access (OFDMA) systems with hybrid analogdigital beamforming (HB). We assume that the transmitter having N antennas is serving Ki decentralized single antenna receivers by sub-carrier i. The scheduling algorithm leverages the solutions of the digital beamforming (DB) result and is designed to maximize the total sum rate of all sub-carriers under per carrier power constraint. For this system and problem setup, the proposed algorithm is explained as follows: First, we express the HB matrix of sub-carrier i as a product of ABi , where the high dimensional matrix A ∈ C N ×Na is common to all subcarriers whereas, Bi ∈ C Na ×Ki is a low dimensional matrix which is designed for sub-carrier i, and Na is the number of RF chains satisfying N ≥ Na ≥ Ki . Second, we compute A as the first Na eigenvectors of the left singular value decomposition of the combined DB precoder matrices of sub-carriers having the highest sum rate. Finally, for fixed A, we compute Bi and its corresponding users such that the total sum rate of all sub-carriers is maximized. The performance of the proposed scheduling is studied analytically. Furthermore, the superiority of the proposed algorithm compared to that of the existing one is quantified analytically and demonstrated by computer simulations. Index Terms— Massive MIMO, Hybrid Analog-Digital beamforming, Millimeter wave, Scheduling, RF chain

I. I NTRODUCTION Beamforming is one of the potential approach to exploit the energy and spectral efficiency of a multiple input multiple output (MIMO) system. There are a bunch of digital beamforming (DB) approaches developed in the past couple of decades. However, these approaches are designed mainly for few number of antennas (around ∼ 10) [1]–[4]. Recently it has been shown that the deployment of massive number of antennas at the transmitter and/or receiver (massive MIMO) can significantly enhance the spectral and energy efficiency of wireless networks [5]. In a rich scattering environment, the performance gains of massive MIMO systems can be achieved by simple DB strategies such as zero forcing (ZF) and maximum ratio transmission [5], [6]. Most today’s wireless system operates at microwave frequencies below 6GHz. The sheer capacity requirement of the next-generation wireless network would inevitably demand us to exploit the frequency bands above 6 GHz, especially the millimeter wave (mmWave) band that spans over 30-300 GHz. Currently, the mmWave is significantly under-utilized;

when exploited it can offer a vast amount of spectrum [7]– [9]. Most importantly, as the mmWaves have extremely short wavelength, it becomes possible to pack a large number of antenna elements which consequently help realize massive MIMO systems. Systems based on DB require the same number of radio frequency (RF) chains as that of the number of base station (BS) antennas N where each RF chain requires extra circuit and power consumption. Thus, when the number of BS antennas N is very large, deploying N RF chains will be practically infeasible. For this reason, it is interesting to realize beamforming with a limited number of RF chains. One approach of achieving this goal is to deploy beamforming at both the digital and analog domains, i.e., hybrid beamforming (HB). In the digital domain, beamforming is realized using microprocessors whereas, in the analog domain beamforming is implemented by employing low cost phase shifters (PSs) (and perhaps variable gain amplifiers (VGAs)). In a typical macro BS, transceivers are designed to operate in a considerable range of bandwidths. The large bandwidth and multipath nature of wireless channels in a cellular system motivates us to consider HB for multiuser massive MIMO systems with frequency selective channels. The gain of a multiantenna system increases as the number of antennas increases. However, this gain comes at the cost of increased hardware cost and complexity, which makes antenna arrays not interesting, e.g., for low-cost mobile terminals with limited size and energy storage capability. For this reason, we consider that the antenna arrays are deployed only at the BS whereas, the mobile terminals are equipped with single antennas. We assume that perfect channel state information is available at the BS which can be achieved by the time division duplex (TDD) training method as in [6]. Recently in [10], we show that the precoding matrix of the HB design can be expressed as ABi , where A ∈ C N ×Na , the analog precoding matrix, is implemented in analog form using PSs, whereas, Bi ∈ C Na ×Ki , digital precoding matrix, is implemented using microprocessors. And when each of the real (imaginary) components of A lie in between −2 and 2 (this can be achieved by introducing simple diagonal scaling factor matrix), A can be implemented using PSs only. For this reason, we perform user scheduling and sub-carrier allocation by constraining Na ≤ N only (i.e., without incorporating a constraint on the analog precoding matrix A). Under this setting, the proposed user scheduling and subcarrier allocation algorithm is explained as follows. First, we compute A as the

first Na eigenvectors of the left singular value decomposition (SVD) of the combined DB precoder matrices of sub-carriers having the highest sum rate. Then, for fixed A, we compute Bi and its corresponding users such that the total sum rate is maximized for each sub-carrier. The performance of the proposed scheduling is studied analytically. The superiority of the proposed algorithm compared to that of the existing one is shown analytically and demonstrated by computer simulations. This paper is organized as follows. Section II discusses the considered HB system model. In Section III, a concise summary of Rayleigh fading and uniform linear array (ULA) channel models is provided. In Sections IV and V, the proposed user scheduling and sub-carrier allocation algorithm, and its performance analysis is presented. Computer simulation results are provided in Section VI. Finally, conclusions are drawn in Section VII. Notations: In this paper, upper/lower-case boldface letters denote matrices/column vectors. The X(i,j) , XT , XH and E(X) denote the (i, j)th element, transpose, conjugate transpose and expected value of X, respectively. dxe denotes the nearest integer greater than or equal to x, In is an identity matrix of size n×n, and CM ×M and (6) from the QR decomposition of B. Na , the rank constraint of (6) is violated and we will continue to the second phase. In this phase, first, we compute A from the SVD of the precoders of the first S˜ sub-carriers having the maximum sum rate, where S˜ is the minimum number of ¯ 1, B ¯ 2, · · · , B ¯ ˜ ]) ≥ Na . Then, sub-carriers ensuring rank([B S for fixed A, we re-express (3) as ˜H dˆik = h ik Bi di + nik , ∀i, k

(8)

˜ H = hH A. Finally, we calculate Bi by performing where h ik ik Phase I for the system (8). Algorithm I: User scheduling and sub-carrier allocation algorithm. Input: Users to be scheduled {1, 2, · · · , Kt }, Nf , Ki and Na . Phase I: ¯ i )old = 0 and Initialization: Set Kti = {1, 2, · · · , Kt }, f (B Ki = ∅, ∀i, where ∅ is empty set. for i = 1 : Nf do for n = 1 : Ki do 1) Set Kim = Ki ∪ {m}, ∀m ∈ Kti . ¯ im ), where fim (B ¯ d ) is the objec2) Compute fim (B im tive function of (7) with Kim users. ¯ im ), ∀m}. 3) Get m ˜ i = arg max{fi1 (B ¯i = B ¯ im ¯ new = fim ¯ ˜ i) 4) Set B ˜ i and f (Bi ) ˜ i (Bim new old ¯ ¯ 5) if f (Bi ) ≥ f (Bi ) then • Update Ki = Ki ∪ {m ˜ i }, Kti = Kti \{m ˜ i } and ¯ i )old = f (B ¯ i )new . f (B 6) else • Break. 7) end if end for end for ¯ = [B ¯ 1, B ¯ 2, · · · , B ¯ Nf ] and 8) Stack the precoders B ¯ if rank(B) ≤ Na then ¯ • Obtain A and Bi from the QR decomposition of B. else • Go to Phase II. end if Phase II: ¯ i ), ∀i in decreasing order f (B ¯ 1 ) ≥ f (B ¯ 2) ≥ 1) Sort f (B ¯ , · · · , ≥ f (BNf ). ˜ d = [B ¯ 1, B ¯ 2, · · · , B ¯ ˜ ], where S˜ is the 2) Compute B S ˜ ≥ Na . minimum number of sub-carriers with rank(B) H ˜ 3) Compute SVD(B) = UΛV , where the diagonal elements of Λ are arranged in decreasing order. 4) Set A of (6) as the first Na columns of U. 5) For fixed A, perform Phase I for the system (8). Output: The precoders of all sub-carriers Bd1 , Bd2 , · · · , BdNf and their corresponding users. From this algorithm, we can understand that a given user may or may not be scheduled to use all of the available subcarriers. We would like to mention here that Algorithm I can

also be extended straightforwardly for other design criteria and precoding method. V. P ERFORMANCE A NALYSIS In this section we provide performance analysis of the proposed user scheduling and sub-carrier allocation algorithm. By combining ZF precoding and Algorithm I, problem (6) can be solved and realized by the following three possible approaches. 1) Antenna Selection Beamforming Approach: When A = IN ×Na , the constraint function of (6) is satisfied implicitly. Thus, for such a setting, this problem can be solved independently for each sub-carrier just by employing the ZF precoding and Phase I of Algorithm I. As this approach implicitly selects the first Na antennas from N available antennas, we treat this approach as an antenna selection beamforming (ASB). As this approach is widely known in the literature [15], the ASB approach can be treated as an existing approach. 2) Proposed Hybrid Beamforming Approach: In this approach, we utilize the proposed HB architecture of Fig. 1. Here ¯ i ) and we apply the ZF precoding to design the precoders Bi (B Algorithm I to schedule the served users and sub-carriers. We denote this as the proposed HB approach. 3) Digital Beamforming Approach: The upper bound solution of problem (6) is achieved when we have N number of RF chains which is the scenario of the DB approach. In the following, we provide the performances of these three approaches for the Rayleigh fading and ULA channel models. A. Rayleigh Fading Channel In this subsection, we examine the above approaches by ˜ H , ∀k of (4) are i.i.d assuming that the channel coefficients h k Rayleigh fading. Lemma 1: Under ZF beamforming, Rayleigh fading channel ˜ H and large Kt , we can have h k RiHB ≥ RiASB when KHB = KASB , and i i

RiHB = RiDB when KHB = KDB i i KASB (RiASB ), i

KHB (RiHB ) i

KDB (RiDB ) i

where and are the served set of users (achieved sum rate) in sub-carrier i using the existing ASB, proposed HB, and DB approaches, respectively. Proof: See Appendix B of [10]. From Lemma 1, we have noticed that the proposed HB achieves the same sum rate as that of the DB one when KHB = KDB . However, in general, the set of served users i i (obtained by Algorithm I) of the HB and DB approaches may not be necessarily the same for all channel realizations. This motivates us to examine the performances of the aforementioned three approaches for the case where KASB 6= i DB KHB = 6 K for some i. For such a case, we are not able to i i quantify the relation between RiASB , RiHB and RiDB for each channel realization. Thus, we examine the performances of these three approaches by examining their achieved average rates under ZF beamforming with equal power allocation strategy as follows.

Theorem 2: Under ZF beamforming with equal power allocation, Pi = P, Ki = K and a unit variance i.i.d ˜ H , the following average rates can Rayleigh fading channel h k be achieved.   P Na −K+1 ASB (Kg )} E{R } ≤ KNf log2 1 + E{χmax K   P DB N −K+1 E{R } ≤ KNf log2 1 + E{χmax (Kg )} (9) K   P −K+1 (Ks )} E{RHB } ≤ K S˜ log2 1 + E{χN max K   ˜ log2 1 + P E{χNa −K+1 (Kg )} + K(Nf − S) max K Kt Nf t where S˜ ≥ 1, Kg = d K K e, Ks = d KNa e and the notation E{χM max (L)} denotes the expected value of the maximum of L independent Chi-square distributed random variables each with M degrees of freedom1 . Proof: See Appendix C of [10].

B. Uniform Linear Array (ULA) Channel From Theorem 2, we can observe that the proposed HB approach achieves lower average sum rates than that of the DB approach. And, this performance loss occurs due to the rank constraint of (6). For the ZF precoding of this paper, AB of (6) has the same rank as that of the combined channels of all users. Thus, the considered HB approach achieves the same performance as that of the DB if the combined channel of all of the Kt users have a maximum rank of Na . In this regard, we examine the following lemma. Lemma 2: When d˜ = λ2 and the AOD of the Kt users satisfy 1 1 sin (θkm ) ∈ n sin (θ)[− 2N , 2N ], n = 1, 2, · · · , Na , where θ is an arbitrary angle, we can achieve KHB = KDB and RiHB = RiDB , ∀i. i i Proof: See Appendix D of [10]. VI. S IMULATION This section presents simulation results. We have used Nf = 64, Lp = 8 (i.e., 8 tap channel), ρk = 1, ∀k and Kmax = 8. The signal to noise ratio (SNR) which is defined as SN R = Nf P Kmax σ 2 is controlled by varying Pi = P while keeping the noise power set to 1mW. We have used two channel models with different parameter settings, one is the Rayleigh fading channel (which may likely be valid at microwave frequency bands) and the other is the ULA channel (which arises both at microwave and mmWave frequency bands). All of the plots are generated by averaging over 1000 channel realizations and ASR denotes average sum rate. A. Rayleigh Fading Channel In this subsection, we provide simulation results for the ˜ H is taken from i.i.d scenario where the multipath channel h k Rayleigh fading channel model. 1 For the simulation, we employ simple trapezoid numerical integration approach of Matlab to compute E{χM max (L)}. As will be demonstrated in the simulation section, the bound derived in this theorem is tight.

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1) Verification of Theoretical Rates: In this simulation, we examine the tightness of the upper bound rates given in (9) under equal power allocation policy. To this end, we take N = 64, Na = 16, Ki = Kmax and Kt = 8. Fig. 2 shows the rates achieved by simulation and theory. As can be seen from this figure, the bound derived in (9) is very tight. Furthermore, as expected the rate achieved by the proposed HB approach is higher than that of the existing ASB approach, and superior performance is achieved by the DB approach. 2) Effect of Power Allocation and Number of Users (Ki ): As can be observed from Section V, the theoretical average sum rate expressions of (9) is derived by assuming that Ki is fixed a priori. And Fig. 2 is plotted for fixed Ki = Kmax . However, when we employ Algorithm I, the number of served users per sub-carrier is updated adaptively. Hence the number of served users per sub-carrier may vary from one channel realization to another. Furthermore, from fundamentals of MIMO communications, ZF precoding with water filling power allocation achieves better performance than that of the equal power allocation. This simulation demonstrates the joint benefits of the ZF precoding with water filling power allocation and Algorithm I (i.e., choosing Ki adaptively). To this end, we set Ki ≤ Kmax , Kt = 16 and N = 64. Fig. 3 shows the performances of the existing ASB, proposed HB, and DB approaches for these parameter settings. As we can see from this figure, for all approaches, performing power allocation with adaptive Ki is advantageous which is expected2 . In the subsequent simulations, we employ ZF precoding with water filling power allocation and Algorithm I (i.e., the number of served users of sub-carrier i Ki ≤ Kmax is chosen adaptively). 3) Comparison of Proposed HB and Existing ASB Approaches: In this simulation, we examine and compare the performances of the proposed HB and existing ASB approaches for different parameter settings. Fig. 4 shows the average sum rate achieved by these approaches for different SNR and Kt . From this figure, we can observe that increasing 2 Note that the complexity of water filling power allocation is almost the same as that of the equal power allocation.

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Fig. 3. Comparison of ASRs achieved by ZF precoding with equal power and Ki = Kmax versus ZF precoding with water filling power allocation and adaptive Ki .

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Fig. 2. Comparison of theoretical and simulated ASR of the existing ASB, proposed HB, and DB approaches under ZF precoding and equal power allocation.

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Kt increases the average sum rate of both approaches (for all SNR values) slightly up to some Kt . This is expected because limKt ≥Kto E{χL max (Kt )} ≈ c, ∃Kto for fixed L. Next we evaluate the effect of the number of RF chains on the performances of these approaches when Kt = 32 which is shown in Fig. 5. From this figure, one can observe that increasing Na increases the average sum rate. B. Uniform Linear Array Channel This subsection provides simulation results for the ULA channel model. To this end, we set Ls = 8, Ki ≤ Kmax , Kt = 32 and N = 64. Under such settings, we examine the sum rates of the aforementioned three approaches when θkm , ∀m, k are selected as in the condition stated by Lemma 2 (Fig. 6). As we can see from this figure, the proposed HB approach achieves the same performance as that of the DB and inferior performance is achieved by the existing ASB approach which is in line with the result of Lemma 2.

of the DB result and is designed to maximize the total sum rate of all sub-carriers under per carrier power constraint. The performance of the proposed scheduling is examined analytically. The superiority of the proposed algorithm compared to that of the existing one is shown analytically. Extensive computer simulations are performed to validate theoretical results, and study the effects of different parameters such as Na , N and Kt . Numerical results also demonstrate that the proposed HB achieves significantly better performance than those of the existing HBs in both flat fading and frequency selective channels.

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Fig. 6. The ASR of ASB, proposed HB, and DB for ULA channels with AODs are as in Lemma 2 with θ = π2 .

From Figs. 4 - 6, one can notice that the proposed HB approach and scheduling algorithm achieves better performance than that of the existing ASB approach and scheduling algorithm. VII. C ONCLUSIONS This paper proposes a new user scheduling and sub-carrier allocation algorithm for multiuser downlink massive MIMO OFDMA systems employing hybrid analog-digital beamforming. We consider that the transmitter having N antennas is serving Ki decentralized single antenna receivers in subcarrier i. The scheduling algorithm leverages the solutions

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