Abstract
Millimeter wave (mmWave) communication has attracted increasing attention as a promising technology for 5G networks. One of the key architectural features of mmWave is the use of massive antenna arrays at both the transmitter and the receiver sides. Therefore, by employing directional beamforming (BF), both mmWave base stations (MBSs) and mmWave users (MUEs) are capable of supporting multi-beam simultaneous transmissions. However, most researches have only considered a single beam, which means that they do not make full potential of mmWave. In this context, in order to improve the performance of short-range indoor mmWave networks with multiple reflections, we investigate the challenges and potential solutions of downlink multi-user multi-beam transmission, which can be described as a high-dimensional (i.e., beamspace) multi-user multiple-input multiple-output (MU-MIMO) technique, including multi-user BF training, simultaneous users' grouping, and multi-user multibeam power allocation. Furthermore, we present the theoretical and numerical results to demonstrate that beamspace MU-MIMO compared with single beam transmission can largely improve the rate performance of mmWave systems.
Abstract
Index Terms-Millimeter wave (mmWave), beamspace MIMO, multi-user beamforming (BF) training, inter-beam interference coordination.
I. INTRODUCTION
According to Cisco forecast, global mobile data traffic will increase sevenfold between 2016 and 2021 [1] . Recent researches showed that mmWave communications, operating in 30-300 GHz bands, are promising technologies for meeting the explosive growth of mobile data demand. Compared with existing microwave systems, mmWave systems are faced with two major challenges: severe propagation loss and sensitivity to blockage. To compensate for high propagation loss, directional BF has been widely used as an essential technique to form a highly directional beam pattern with large antenna gain. Thanks to the short wavelengths of mmWave radios ranging from 10 mm to 1 mm, massive antenna arrays can be packed into the limited dimensions of mmWave transceivers. Therefore, with directional BF, it is possible to form multiple beams at both mmWave transmitter and receiver sides in mobile networks. That is, mmWave systems are in fact able to provide high-dimensional MIMO operations [2] - [4] and can realize spatial spectrum reuse at close distance [5] . However, most current work does not make full potential of mmWave. For instance, the work in [6] - [9] was focused on single beam transmission scenarios, and the work in [5] , [10] - [12] considered the scenarios where only the transmitter side was operating with multiple beams. Moreover, since mmWave radios have limited ability to diffract around obstacles (e.g., human body), the connection between each pair of transmitter and receiver is vulnerable to blockage events.
I. INTRODUCTION
In this context, aiming at increasing the achievable rate and maintaining connectivity of mmWave mobile networks, we investigated the challenges and potential solutions (including multi-beam selection, cooperative beam tracking, multi-beam power allocation, and synchronization) associated with singleuser multi-beam simultaneous transmissions (i.e., beamspace SU-MIMO) in [13] . It is worth mentioning that the proposed scheme is only applicable to the short-range scenarios with multiple NLOS paths (e.g., first or second order reflections from floor and/or ceiling in indoor scenarios). In order to further enhance the performance of mmWave systems, we extend our previous work to multi-user scenarios, namely beamspace MU-MIMO, on the basis of existing research results. To the best of our knowledge, there has been no work on this issue. Since the communication environment with multi-user is more complex than that with single user, not only need we to further expand the strategies proposed in [13] , but also will we face some new challenges for implementing beamspace MU-MIMO. For instance, due to the transmit beams selected by different MUEs may be (partially) overlapped, the interuser interference should be seriously considered in beamspace MU-MIMO. This study mainly focuses on the issues of multiuser BF training, simultaneous users' grouping, and multi-user multi-beam power allocation in beamspace MU-MIMO.
I. INTRODUCTION
The rest of the paper is organized as follows. In Section II, the network model and the basic idea of beamspace MU-MIMO are introduced. Section III first describes muti-user BF training and then proposes a multi-user grouping mechanism. In Section IV, the potential solutions of power allocation for beamspace MU-MIMO are presented and analyzed. Section V shows some numerical results to evaluate the proposed scheme. Finally, Section VI concludes the paper.
II. SYSTEM OVERVIEW
In this study, we consider a short-range indoor mmWave network with one reference MBS and U total sparsely distributed MUEs. Let R denote the set of these MUEs. Meanwhile, both the MBS and MUEs are equipped with massive antenna arrays. Thus, with directional BF and space division technique, they are capable of supporting multiple orthogonal beams simultaneously and can realize spectrum reuse, as illustrated in Fig.1 
II. SYSTEM OVERVIEW
The multi-user multi-beam simultaneous transmission scheme investigated in this study can be described as beamspace MU-MIMO defined as Definition 1. Fig. 1 shows an example of beamspace MU-MIMO in two-dimensional (2D) perspective. Note that the analysis is also applicable to three-dimensional (3D) mode. For ease of analysis, similar to [10] , [13] , [14] , we replace the MBS with U virtual MBSs (vMBSs) located at the same position. Each vMBS serves different MUEs with different transmit beam sets. Moreover, when 1 < b u ≤ b u max , the transmission mode between MUE u and its corresponding vMBS is beamspace SU-MIMO (e.g., for MUE1 and MUE3 in Fig. 1 ) and it is beamspace SU-SISO when b u = 1 (e.g., for MUE2). In this context, beamspace MU-MIMO can be defined as a set of beamspace SU-MIMO and/or SU-SISO technologies with space division technique.
II. SYSTEM OVERVIEW
Definition 1 (Beamspace MU-MIMO): The beamspace MU-MIMO is defined as an mmWave communication mode that an MBS with multiple orthogonal beams can transmit simultaneously to a set of MUEs, where each MUE is with one or more operating beams. That is, denoting Q as the set of MUEs, b and b u as the number of transmitting and receiving beams of the MBS and MUE u (u ∈ Q), respectively, the multi-user multibeam simultaneous transmissions can be termed as N × N U beamspace MU-MIMO, where U is the number of MUEs in Q, N is the total number of transmitting and receiving (T-R) beam pairs between the MBS and the simultaneous transmitting
II. SYSTEM OVERVIEW
In order to implement beamspace MU-MIMO and, meanwhile, to achieve optimal system performance, we face many challenges as below. 1) Multi-user BF training: Since only one transmit/receive direction's link quality can be detected at a time in traditional BF training (e.g., in 802.11ad/ay), the efficiency of the optimal beam selection is generally very low. It means that the existing beam selection solutions are not entirely applicable to beamspace MU-MIMO. This study utilizes the capability of supporting multiple beams both at the MBS and MUEs to detect the quality of multiple links simultaneously, and thus to increase the efficiency of multi-beam selection for beamspace MU-MIMO. sets selected by different MUEs may be (partially) overlapped, e.g., in Fig. 1 , one NLOS link for MUE1 is in conflict with the LOS link for MUE2 over the transmit beam. And the inter-user interference will be severe in this case. To address this issue, we can switch some MUEs' initial selected T-R beam pairs with conflicting beams to a suitable candidate (if available), or assign the MUEs with the same conflicting transmit beam to different groups. MUEs in different groups will be served in time division manner.
II. SYSTEM OVERVIEW
3) Power allocation: Considering that the transmission performance of different links for different MUEs may vary widely, the appropriate power allocation strategies should be seriously considered to maximize the achievable rate of beamspace MU-MIMO.
III. BEAM/USER MANAGEMENT FOR BEAMSPACE MU-MIMO
In this section, we first give an efficient multi-user BF training mechanism for beamspace MU-MIMO. It is worth mentioning that the BF training (or beam steering) operations are designed to determine the best T-R beam pair set N u pair (u ∈ R) that best matches the LOS path and/or NLOS paths between a vMBS and its corresponding MUE, hereafter called vMBS-MUE. In this study, after the successful completion of BF training, directional BF is established. Moreover, to guarantee the network performance, we further adjust the initial selected T-R beam pair sets by analyzing inter-user interference. Table I summarizes the main notations used throughout the paper.
A. Multi-user BF Training
The multi-user BF training mechanism in this study aims at improving the downlink performance of beamspace MU-MIMO. The corresponding strategies for uplink transmission are left as our future work. For ease of illustration, we divide the region of the MBS/MUEs into a number of transmit/receive sectors (i.e., orthogonal beam directions). Meanwhile, we assume that MUEs can distinguish signals received from different beams. The proposed mechanism mainly consists of three phases and, moreover, the conceptual flow of the first two phases is illustrated in Fig. 2 . The details are as follows.
A. Multi-user BF Training
(i) Transmit Training: In this phase, all MUEs are in the quasi-omni mode and the MBS scans n tx transmit sectors' 
A. Multi-user BF Training
The set of MUEs with conflicting beam m (Cm ⊆ R) η
A. Multi-user BF Training
Threshold of SNR (or SINR) ξt
A. Multi-user BF Training
Transmitting beamwidth ξr
A. Multi-user BF Training
Receiving beamwidth
A. Multi-user BF Training
quality simultaneously with n tx directional beams. Here, different sectors are scanned by different beams which are mutually orthogonal. Assuming that the total number of transmit sectors is S MBS , we have
A. Multi-user BF Training
Hence, we only need to test SMBS ntx times to determine the best transmit beam set N u TX for MUE u (∀u ∈ R), while the traditional transmit training operating with only one beam at a time is required to test S MBS times.
A. Multi-user BF Training
(ii) Receive Training: In this phase, it reverses the scanning roles from the transmit training. That is, MUEs detect multiple receive sectors simultaneously with multiple directional beams and the MBS remains in the quasi-omni mode at this time. Similar to transmit training, MUE u (∀u ∈ R) can obtain its best receive beam set N u RX after scaning Su n u rx times, where S u is the total number of MUE u's receive sectors, n u rx is the number of simultaneous scanning beams, and
A. Multi-user BF Training
Since the number of simultaneous receive beams supported by each MUE may be different, the number of tests required for completing their respective receive training will also be different. Supposing that S u = S MUE for ∀u ∈ R, the number of tests required to complete multi-user receive training is are transmitted simultaneously. Note that MUEs make the decision independently in this phase and they may obtain several alternative (or sub-optimal) T-R beam pair sets.
B. Simultaneous MUEs' Grouping
Considering the diversity and finiteness of the number of simultaneous operating beams that can be supported by the MBS and MUEs, the MBS is generally unable to serve all MUEs in its coverage simultaneously. Moreover, the system performance of beamspace MU-MIMO is various for different combinations of simultaneous MUEs. To ensure the performance, this subsection is devoted to grouping simultaneous MUEs through the analysis of inter-user interference.
B. Simultaneous MUEs' Grouping
Since the decision of multi-beam selection for each MUE is relatively independent, the transmit beams selected by them may be (partially) overlapped. We assume that one beam can serve only one MUE at the same time. To avoid beam conflicts, we need to adjust or re-select N u pair (∀u ∈ R), e.g., by beam switching. After that, we can proceed with the selection of simultaneous MUEs as described in Algorithm 1, the main idea of which can be outlined as follows: 
IV. POWER ALLOCATION FOR BEAMSPACE MU-MIMO
Since the quality of different links may vary widely among simultaneous MUEs, we should make reasonable power allocation for beamspace MU-MIMO in order to maximize its achievable rate. The NLOS links in this study are assumed to be first order reflections, because mmWave signals are generally negligible after high-order reflections and the actual transmission paths of them are unpredictable. For tractability of the analysis, we approximate the actual antenna pattern by an ideal sectored antenna model [7] , [15] . The directivity gain can be expressed as [8] , [14] 
IV. POWER ALLOCATION FOR BEAMSPACE MU-MIMO
where ξ is the operating beamwidth and z is the average gain of side lobes, 0 ≤ z < 1. Furthermore, the path loss of mmWave can be modeled as [16] L (R) [dB] = A + 20 log 10 (f c ) + 10n log 10 (R) , (4) where f c is the carrier frequency in GHz, R is transmission distance in km, A is the attenuation value, and n is the path loss exponent. Since the T-R beam pairs for beamspace MU-MIMO are mutually orthogonal (i.e., the main lobes are nonoverlapping), we assume that the inter-beam interference is mainly caused by side lobes. Therefore, the SINR of link i To maximize the achievable rate of beamspace MU-MIMO, we first collect the variables ξ u,i t , ξ u,i r , and P i t (∀u ∈ Q, ∀i ∈ N u ) in vectors ξ t , ξ r and p, respectively, and then formulate the problem under consideration as an optimization problem (P1) given by maximize ξt,ξr,p,Q
IV. POWER ALLOCATION FOR BEAMSPACE MU-MIMO
where ξ t,min and ξ r,min are the minimum beamwidth of transmit and receive beams, respectively; p max and P max are the maximum transmission power of each transmit beam and the MBS, respectively. Note that function arguments have been discarded for notational simplicity. Considering the simplest scenario with pencil beams, i.e., z ≪ 1, we can neglect the inter-beam interference and optimize the operating beamwidth for each link individually. That is, the optimal beamwidth of transmit and receive beams are ξ u,i t * = ξ t,min and ξ u,i r * = ξ r,min , respectively. Hereafter, the optimized parameters are identified by the " * " on the upper right corner. Meanwhile, the SINR expression formulated in Eq. (5) reduces to SNR according to
IV. POWER ALLOCATION FOR BEAMSPACE MU-MIMO
As P1 is difficult to obtain its optimal solution, we investigate two low complexity and easy to implement solutions for multi-user multi-beam power allocation to suboptimally address P1 that with pencil beams.
IV. POWER ALLOCATION FOR BEAMSPACE MU-MIMO
⋆ Average Power Allocation (APA): Each link's transmission power is the same without considering the difference of link quality, i.e., P i t *
IV. POWER ALLOCATION FOR BEAMSPACE MU-MIMO
where Q * and b * u can be obtained by Algorithm 2.
IV. POWER ALLOCATION FOR BEAMSPACE MU-MIMO
⋆ Priority Power Allocation (PPA): Considering the quality of different links may vary widely, we give priority to optimize the transmission power of the links that with high link quality to address P1. Furthermore, this solution includes the following two cases.
IV. POWER ALLOCATION FOR BEAMSPACE MU-MIMO
1) Considering fairness: When the fairness of power allocation among simultaneous MUEs is taken into account, we will give priority to optimize the best link for each MUE (e.g., link ℓ for MUE u), i.e., P u,ℓ t * FP = p max . Then, we employ APA to allocate power for other links (e.g., link i), i.e., P u,i (7), we can easily obtain the maximum achievable rate with APA and with PPA, respectively. Since the length of the paper is limited, we do not give these results here. 
IV. POWER ALLOCATION FOR BEAMSPACE MU-MIMO
This section presents some numerical results on the performance of beamspace MU-MIMO. The objective of this work is two-fold: (i) to verify the effectiveness of the proposed multiuser BF training mechanism; (ii) to compare and analyze the performance of APA and PPA. To simplify simulations, we assume that ξ u,i t = ξ t , ξ u,i r = ξ r , and b u max = b MUE max , for ∀i ∈ N u , ∀u ∈ Q. Moreover, we consider a short-range indoor mmWave network with R u i = R LOS for LOS links and R u i = R NLOS for NLOS links. Table II summarizes the detailed simulation parameters. In addition, at a standard temperature of 17 • C, we let P N [dB] = −174 [dBm/Hz] + 10 log 10 (B) + N F , where N F is noise figure in dB. Fig. 3 shows that the proposed multi-beam transmit training can effectively improve the efficiency of beam selection. For example, when ξ t = 10 • , we have S MBS = 36 and 1 ≤ n tx ≤ 10 which can be known from Eq. (1). Hence, if n tx = 5, to obtain the best transmit beam sets (i.e., N u TX , ∀u ∈ Q), the MBS only needs to scan SMBS ntx = 8 times by adopting the proposed solution. However, using the traditional transmit training operating with single beam, the number of scans required is S MBS = 36. Similarly, we can verify the effectiveness of the proposed multi-beam receive training for selecting the best receive beam sets (i.e., N u RX , ∀u ∈ Q). Furthermore, the larger the values of n tx and n rx , the more superior the multi-user BF training.
IV. POWER ALLOCATION FOR BEAMSPACE MU-MIMO
In Fig. 4 , we investigate the rate performance of beamspace MU-MIMO with APA and with PPA, respectively. Here we consider the network is with three MUEs, i.e., U = 3 and, meanwhile, each of them is operating with a LOS link and two NLOS links, i.e., b = u∈Q b u = 9. Moreover, we assume that the quality of each LOS link is better than that of NLOS links. In this context, the rate performance of beamspace MU-MIMO with PPA is the same regardless of whether the fairness of power allocation among simultaneous MUEs is taken into account. Clearly, the performance of PPA is generally better than that of APA for beamspace MU-MIMO. Further, the results indicate that beamspace MU-MIMO compared with beamspace MU-SISO can largely improve the rate performance of mmWave networks. Note that U > 1 and b u = 1 (∀u ∈ Q) in beamspace MU-SISO systems. For example, when η = 10dB shown in Fig. 4(a) , we have Rate APA MU−MIMO = 120Gbps and Rate PPA MU−MIMO = 210Gbps while Rate MU−SISO = 49Gbps. To make the results more general, the inter-beam interference caused by side lobes is not ignored in our simulations, i.e., z = 0. By comparing Fig.  4 (a) with Fig. 4(b) , we can see that the greater the value of z, the greater the interference, and the more obvious the impact on the system performance.
VI. CONCLUSIONS
In this study, in order to further enhance the performance of mmWave networks with multiple reflections, we extended our previous work to multi-user scenario, namely beamspace MU-MIMO, and investigated its challenges and potential solutions for downlink transmission. First, we improved the efficiency of multi-beam selection for beamspace MU-MIMO by utilizing the capability of supporting multiple beams both at the MBS and MUEs. Second, to avoid beam conflicts, we grouped simultaneous served MUEs. Third, we analyzed two low complexity multi-user multi-beam power allocation solutions, i.e., APA and PPA. The numerical results demonstrated that they are very effective to improve the achievable rate of beamspace MU-MIMO. Furthermore, the corresponding strategies for uplink beamspace MU-MIMO will be also our future work. 

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