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Nokia Shanghai Bell, Shanghai, China. Email: {Nuan.Song, Huan.a.Sun, Tao.a.Yang}@alcatel-sbell.com.cn. Abstract—Massive Millimeter Wave (mmWave) ...
Coordinated Hybrid Beamforming for Millimeter Wave Multi-User Massive MIMO Systems Nuan Song, Huan Sun and Tao Yang Nokia Bell Labs China Nokia Shanghai Bell, Shanghai, China Email: {Nuan.Song, Huan.a.Sun, Tao.a.Yang}@alcatel-sbell.com.cn Abstract—Massive Millimeter Wave (mmWave) Multiple Input Multiple Output (MIMO) systems utilize hybrid beamforming techniques to alleviate the implementation complexity of combining a large number of antennas. To solve the optimization problem in the hybrid design supporting multiple users and multiple data streams per user, we first propose a coordinated Radio Frequency (RF) beamforming technique based on the Generalized Low Rank Approximation of Matrices (GLRAM) approach and then develop an efficient Modified GLRAM (MGLRAM) algorithm. The proposed scheme only requires the information of the composite channel, instead of the complete physical channel matrix which is assumed to be known in the existing literature. It takes advantage of the coordination between the base station and users to achieve a maximal array gain and has no dimensionality constraint. The multiplexing gain is then exploited by applying the Block Diagonalization (BD) technique. It is shown that our proposed scheme is a practical and competing solution, which can be easily applied to both the Time Division Duplex (TDD) and Frequency Division Duplex (FDD) systems.

I. I NTRODUCTION Multiple Input Multiple Output (MIMO) communication systems benefit significantly from employing arrays with a large number of antenna elements, e.g., a hundred or more. The so called massive MIMO is one of the most promising techniques for the future wireless networks [1]. The degrees of freedom of massive MIMO (such as array gain, multiplexing gain, the capability of interference reduction, etc.), can be greatly exploited in the Millimeter Wave (mmWave) band, since very small antennas can be tightly packed into an array of a smaller size. Accordingly, well-designed transmit and receive signal processing techniques are required to achieve such advantages for MIMO systems [2], [3]. However, those traditional techniques that completely operate in the baseband are far too complicated for massive MIMO, since as many active Radio Frequency (RF) chains as the number of antennas should be implemented and then combined in the digital domain. Therefore, hybrid beamforming techniques become quite promising to handle the complexity [4], [5], [6]. The key concept of applying hybrid architectures in massive MIMO is to implement cost-effective variable phase shifters in the RF domain before sampling [7], [8], [9] and thus the reduceddimensional signal processing schemes can be carried out digitally in the baseband. Several typical references have proposed hybrid beamforming techniques in single-user MIMO [10], [11], [12], where

hybrid structures are considered for both the Base Station (BS) and the users. Some work has studied Multi-User MIMO (MU-MIMO) scenarios [13], [14], [15], [16], [17], [18]. The scenarios considered in [13], [14], [18] are restricted to the MU system but with single antenna at each user. In [15], the authors have developed a two-stage multi-user hybrid precoder algorithm with limited feedback. The considered model assumes that the BS performs the hybrid precoder while the users employ analog-only combining, which means only single data stream per user is supported. The problem will become more complicated if the transmission of multiple data streams per user and the hybrid structure implemented at users are also taken into account. Reference [16] has studied such cases and proposed an effective hybrid Block Diagonalization (BD) precoder using feedback for MU-MIMO, where both BS and users employ hybrid architectures. The scheme applies Equal Gain Transmission (EGT) method to harvest the large array gain during the RF beamforming. The EGT scheme requires the same number of RF chains at the BS as the sum of those at all users, while in practice the number of scheduled users varies and such dimensionality constraint may not hold at all times. Reference [17] only concentrates on the baseband precoding techniques but has not discussed the RF beamforming design. In this paper, we propose a coordinated hybrid beamforming scheme supporting multiple-stream transmissions for downlink MU massive MIMO systems at mmWave frequencies. Hybrid designs are considered at both the BS and the users. To solve the joint optimization problem formulated for hybrid beamforming, we split the beamforming design into two parts, i.e., the RF precoder/combiner and the digital counterparts. An efficient coordinated RF beamforming algorithm based on the Generalized Low Rank Approximation of Matrices (GLRAM) as well as its practical solution, namely Modified GLRAM (MGLRAM), are developed, which have no dimensionality constraint. From the implementation point of view, it is applicable to both Time Division Duplex (TDD) and Frequency Division Duplex (FDD) systems. To exploit the multiplexing gain, the BD precoder is applied in the digital domain. We evaluate the performance of the proposed coordinated hybrid scheme and compare it to the EGT method in [16] as well as the fully digital technique. Notation: The superscripts 𝑇 and 𝐻 represent transpose

978-1-5090-1328-9/16/$31.00 ©2016 IEEE

and Hermitian transpose, respectively. The operations ∣𝑿∣ and ∥𝑿∥𝐹 denote the determinant (product of eigenvalues) and Frobenius norm of a matrix 𝑿. The operator abs(⋅) takes the absolute value of a scalar. The notation 𝑿(:, 𝑎 : 𝑏) is a MATLAB notation, meaning that the columns of the matrix 𝑿 indexed from 𝑎 to 𝑏 are chosen. The phases of a complex matrix 𝑿 are extracted by the operation angle (𝑿). II. S YSTEM M ODEL The block diagram of the hybrid architecture for a downlink MU-MIMO system is shown in Figure 1. The BS is mounted by 𝑀𝑇 transmit antennas that all connect to 𝑁𝑇 RF chains, where 𝑁𝑇 < 𝑀𝑇 holds for the hybrid design. Similarly, for the 𝑘-th user (𝑘 = 1, ⋅ ⋅ ⋅ , 𝐾), the number of receive antennas is denoted by 𝑀𝑅𝑘 and the corresponding number of RF denote the total chains satisfies 𝑁𝑅𝑘 < 𝑀𝑅𝑘 . Accordingly, we∑ 𝐾 number of antennas for all users by 𝑀𝑅 = 𝑘=1 𝑀𝑅𝑘 and ∑𝐾 that of the RF chains by 𝑁𝑅 = 𝑘=1 𝑁𝑅𝑘 . The transmitted signal at the BS is given by 𝒙 = 𝑭RF 𝑭BB 𝒔 ∈ ℂ𝑟 ,

∑𝐾

(1)

where 𝑟 = 𝑘=1 𝑟𝑘 is the total number of data streams and 𝒔 = [ 𝒔𝑇1 , ⋅ ⋅ ⋅ , 𝒔𝑇𝐾 ]𝑇 is a vector of data symbols from all users. The matrices 𝑭RF ∈ ℂ𝑀𝑇 ×𝑁𝑇 and 𝑭BB = (1) (2) (𝑘) [ 𝑭BB , 𝑭BB , ⋅ ⋅ ⋅ , 𝑭BB ] ∈ ℂ𝑁𝑇 ×𝑟 represent the analog RF precoder and the digital baseband precoder, respectively. The analog precoder is implemented by variable phase shifters to alleviate hardware constraints and thus has a constant modulus [7]. The baseband precoder acts similarly as the conventional MIMO precoder at lower bands. 𝑭RF 1

𝑴𝑅 1

𝑾

(1) RF

(𝐾) BB

RF Chain𝑵𝑇

𝑯𝐾

baseband digital

𝑴𝑇

(1) BB

𝒚1

User 1

1 𝑭

𝑾

...

...

1 𝑯1

...

𝒔𝐾

1

...

RF Chain

...

(1) BB

...

𝑭

...

𝒔1

...

𝑭BB

𝑴𝑅 𝐾

𝑾

(𝐾) RF

𝑾

(𝐾) BB

Fig. 1. Block diagram of the hybrid architecture in MU massive MIMO systems.

The received signal for the user 𝑘 can be written as 𝒓𝑘 = 𝑯𝑘 𝑭RF 𝑭BB 𝒔 + 𝒏𝑘 ∈ ℂ𝑀𝑅𝑘 ,

(2)

where the channel matrix for the 𝑘-th user 𝑯𝑘 ∈ ℂ𝑀𝑅𝑘 ×𝑀𝑇 , 𝑘 = 1, ⋅ ⋅ ⋅ , 𝐾 represents physical channels (from the antenna element point of view) and the vector 𝒏𝑘 is the complex Additive White Gaussian Noise (AWGN) with variance 𝜎𝑛2 . At user 𝑘, the received signal after combining is given by (𝑘)

𝒚𝑘 = 𝑾BB 𝑾RF 𝑯𝑘 𝑭RF 𝑭BB 𝒔𝑘 + 𝐾 ∑ (𝑘)𝐻 (𝑘)𝐻 (𝑖) 𝑾BB 𝑾RF 𝑯𝑖 𝑭RF 𝑭BB 𝒔𝑖 + 𝑾BB (𝑘)𝐻 𝑾RF (𝑘)𝐻 𝒏𝑘 , 𝑖=1,𝑖∕=𝑘

𝑯𝑘 =

(3)

(𝑘)

𝑁𝑝 𝑁𝑐 ∑ ( ) ( ) 𝑀𝑇 𝑀𝑅 𝑘 ∑ (𝑘) (𝑘) (𝑘) (𝑘) 𝛼𝑖𝑙 𝒂𝑅 𝜙𝑖𝑙 𝒂𝐻 𝜃𝑖𝑙 , (4) 𝑇 𝑁𝑐 𝑁𝑝 𝑖=1 𝑙=1

(𝑘)

where 𝛼𝑖𝑙 is complex channel gain of the 𝑖-th cluster and 𝑙-th path and follows the complex normal distribution 𝒞𝒩 (0, 1). (𝑘) (𝑘) The angles of arrival/departure 𝜙𝑖𝑙 , 𝜃𝑖𝑙 are uniformly dis(𝑘) (𝑘) (𝑘) tributed within [0, 2𝜋]. 𝒂𝑅 (𝜙𝑖𝑙 ) and 𝒂𝑇 (𝜃𝑖𝑙 ) are array (𝑘) response vectors of the BS at angle 𝜙𝑖𝑙 and of the user 𝑘 at (𝑘) angle 𝜃𝑖𝑙 , respectively. For simplicity in this paper, we apply the Uniform Linear Arrays (ULAs) and thus only azimuth angles are considered. III. C OORDINATED H YBRID B EAMFORMING D ESIGN A. Hybrid Beamforming Problem In MU-MIMO systems based on hybrid structures at both the BS and users, the joint precoding and decoding problem shown in (5) is to find the beamforming matrices (𝑘) (𝑘) 𝑭RF , 𝑭BB , 𝑾RF , 𝑾BB , ∀𝑘, in order to maximize the achievable sum rate under the constraints: 1) a transmit power constraint (𝑃𝑇 as the total transmit power) and 2) constant modulus constraints for RF beamformers that employ variable analog phase shifters. max

𝐾 ∑ 𝑘=1

Base Station

(𝑘)𝐻



𝒚𝐾

User K

RF analog

(𝑘)𝐻

(𝑘)

where 𝑾RF ∈ ℂ𝑀𝑅𝑘 ×𝑁𝑅𝑘 and 𝑾BB ∈ ℂ𝑁𝑅𝑘 ×𝑟𝑘 denote the RF combiner and the baseband decoder for the 𝑘-th user, respectively. The expression (3) indicates that the received signal at the 𝑘-th user includes the desired signal, the Multi-User Interference (MUI) as well as the noise. Hybrid beamforming takes advantage of both the analog and digital beamformers, achieving a high array gain to overcome the high propagation loss at mmWave frequencies as well as exploiting the spatial multiplexing gain. The mmWave channel is represented by the widely used clustered model [10]. For user 𝑘, the channel matrix can be expressed as

) (  (𝑘)  log2 𝑰𝑟𝑘 + 𝑹(𝑘) /𝑹𝑛𝑛  , (𝑘)𝐻

(𝑘)𝐻

(𝑘)

(𝑘)𝐻

(𝑘)𝐻

(𝑘)

𝐻 𝑹(𝑘) = 𝑾BB 𝑾RF 𝑯 ( 𝑘 𝑭RF 𝑭BB ⋅ (𝑾BB 𝑾RF 𝑯𝑘 𝑭RF 𝑭BB ) (𝑘) (𝑘)𝐻 (𝑘)𝐻 2 𝑹𝑛𝑛 = 𝑾BB 𝑾RF 𝜎𝑛 𝑰𝑀𝑅 + 𝑘 ) ∑𝐾 (𝑖) (𝑖)𝐻 𝐻 𝐻 𝑾 (𝑘) 𝑾 (𝑘) 𝑖=1,𝑖∕=𝑘 𝑯𝑖 𝑭RF 𝑭BB 𝑭BB 𝑭RF 𝑯𝑖 RF BB

s. t. ∥𝑭RF 𝑭BB ∥2𝐹 = 𝑃𝑇 ( ) (𝑘) abs (𝑭RF (𝑖, 𝑗)) = √ 1 , ∀𝑖, 𝑗, abs 𝑾RF (𝑖, 𝑗) = √𝑀1 𝑀𝑇

𝑅𝑘

, ∀𝑖, 𝑗, 𝑘. (5)

It is quite challenging to solve the above sum rate maximization problem and no direct solution exists. Therefore, it is more efficient to decouple the beamforming design into two stages, i.e., to find RF beamforming matrices and then to design the baseband beamforming, separately [15], [16], [17]. The main purpose of the RF analog precoder/combiner technique is to exploit the large array gain in massive MIMO systems, while the digital baseband precoder/decoder processing tries to suppress the interference and exploits the spatial multiplexing gain.

B. Coordinated RF Beamforming In the MU scenario, the effective channel for the 𝑘-th user after the RF beamforming and combining is given by ˇ 𝑘 = 𝑾 (𝑘)𝐻 𝑯𝑘 𝑭RF . If 𝑾 (𝑘) and 𝑭RF are unitary matrices, 𝑯 RF RF the original mmWave channel 𝑯𝑘 , which is generally lowrank, is equivalent in distribution to the effective channel ˇ 𝑘 [19]. In the discussed hybrid architecture, 𝑾 (𝑘) and 𝑯 RF 𝑭RF perform the low-rank transformation, the transformed ˇ 𝑘 should keep the rank characteristic (or effective) channel 𝑯 of 𝑯𝑘 as much as possible, which can be addressed in the problem of approximating a collection of matrices with a low rank, i.e., 2  (𝑘) 𝐻  𝑯𝑘 − 𝑾RF 𝑩𝑘 𝑭RF 

min

(𝑘)

𝐹

(6)

= 𝑰𝑁𝑅𝑘 , ∀𝑘,

(𝑘) 𝑾RF ,∀𝑘

s. t.

𝑭RF , 𝐻 𝑭RF 𝑭RF

=

𝐾  2 ∑  (𝑘)𝐻  𝑾RF 𝑯𝑘 𝑭RF 

𝑘=1

(𝑘)𝐻 (𝑘) 𝑰𝑁𝑇 , 𝑾RF 𝑾RF

(7)

𝐹

= 𝑰𝑁𝑅𝑘 , ∀𝑘.

For the user 𝑘, the effective array gain can be interpreted by the gain of the effective channel, which is proportional to the receive Signal-to-Noise-Ratio (SNR) at the RF chain’s level [19]. Therefore, the optimization problem (7) aims at maximizing the effective array gain by the joint design of (𝑘) 𝑭RF and 𝑾RF , ∀𝑘. The existing solution to design the joint RF beamformers in [16] applies an EGT method to harvest large array gains by enlarging the sum of the squares of the diagonal entries in the entire effective channel. The 𝑭RF is obtained by extracting the phases of channel ] [ the conjugate transpose of the composite (1)𝐻 (𝐾)𝐻 based on the matrix 𝑾RF 𝑯1 , ⋅ ⋅ ⋅ , 𝑾RF 𝑯𝐾 (𝑘)

(8)

𝑷

𝐻 s. t. 𝑭RF 𝑭RF = 𝑰𝑁𝑇

and

⎞ 𝐾 ∑ ⎜ (𝑘)𝐻 (𝑘) ⎟ 𝐻 trace ⎝ 𝑾RF 𝑯𝑘 𝑭RF 𝑭RF 𝑯𝑘𝐻 𝑾RF ⎠ ,

  ⎛

max (𝑘)

𝑾RF ,∀𝑘

𝑘=1

𝑸𝑘

(𝑘)

s. t. 𝑾RF 𝑾RF = 𝑰𝑁𝑅𝑘 , ∀𝑘.

where 𝑩𝑘 ∈ ℂ𝑁𝑅𝑘 ×𝑁𝑇 . It has been shown in [20] that if the (𝑘) problem (6) has optimal solutions for 𝑾RF , 𝑭RF , and 𝑩𝑘 , it can be equivalent to the following reformulated optimization problem max

⎟ ⎜ 𝐾 ⎟ ⎜ 𝐻 ∑ (𝑘) (𝑘)𝐻 𝐻 ⎟ 𝑭 𝑯 𝑾 𝑾 𝑯 𝑭 max trace ⎜ 𝑘 RF ⎟ RF 𝑘 RF RF ⎜ 𝑭RF ⎠ ⎝ 𝑘=1

 

(𝑘)𝐻

𝐾  ∑

𝑭RF , 𝑾RF ,∀𝑘, 𝑩𝑘 𝑘=1 (𝑘)𝐻 (𝑘) 𝐻 s. t. 𝑭RF 𝑭RF = 𝑰𝑁𝑇 , 𝑾RF 𝑾RF

design. According to linear algebra, (7) can be equivalently written as ⎞ ⎛

feedback of the RF combiners from all users 𝑾RF , ∀𝑘. It is obtained by selecting vectors from the DFT basis set. However, the scheme in [16] assumes that the original downlink channel matrix 𝑯 is known and the EGT scheme has a dimensionality constraint 𝑁𝑇 = 𝑁𝑅 . In the following, we propose a coordinated RF beamforming algorithm, which is an efficient and practical solution to the maximization problem of the effective array gain in (7). 1) GLRAM based Solution: In general the maximization in (7) does not have a closed-form solution. Reference [20] derives an iterative algorithm, namely GLRAM, to solve a similar mathematical problem for image compression and retrieval. Therefore, inspired by [20], we reformulate the problem in (7) and apply the GLRAM in the RF beamforming

(9) In our case, the concept of GLRAM is to iteratively update (𝑘) 𝑭RF and 𝑾RF so that the effective array gain is maximized. The GLRAM-based RF beamforming algorithm is summarized as follows. ]𝑇 [ (𝑘) , ∀𝑘 = a) Initialize: 𝑾RF = 𝑰𝑁𝑅𝑘 , 0𝑁𝑅𝑘 ,𝑀𝑅𝑘 −𝑁𝑅𝑘 1 ⋅ ⋅ ⋅ 𝐾, the iteration index 𝑖, and the threshold 𝜖 ∈ ℝ (an arbitrary small number). b) Set 𝑖 = 𝑖 + 1 and compute 𝑷 (𝑖) as 𝑷 (𝑖) =

𝐾 ∑ 𝑘=1

(𝑘)

(𝑘)𝐻

𝑯𝑘𝐻 𝑾RF (𝑖)𝑾RF (𝑖)𝑯𝑘

(10)

and solve the optimization problem in (8) via Singular Value Decomposition (SVD) of 𝑷 (𝑖) and obtain 𝑭RF (𝑖) = 𝑼𝑃 (:, 1 : 𝑁𝑇 ), which are the 𝑁𝑇 eigenvectors of 𝑷 (𝑖) corresponding to 𝑁𝑇 largest eigenvalues. c) For each user 𝑘, ∀𝑘, compute 𝑸𝑘 (𝑖) as 𝐻 𝑸𝑘 (𝑖) = 𝑯𝑘 𝑭RF (𝑖)𝑭RF (𝑖)𝑯𝑘𝐻

(11)

and solve the optimization problem in (9) via SVD of (𝑘) 𝑸𝑘 (𝑖) and obtain 𝑾RF (𝑖) = 𝑼𝑄𝑘 (:, 1 : 𝑁𝑅𝑘 ), which is the 𝑁𝑅𝑘 eigenvectors of 𝑸𝑘 (𝑖) corresponding to 𝑁𝑅𝑘 largest eigenvalues.  ∑  ) (∑   𝐾 ˇ  ˇ , if d) Compute abs  𝐾 𝑘=1 𝑯𝑘 (𝑖) −  𝑘=1 𝑯𝑘 (𝑖 − 1) 𝐹 𝐹 it is greater than 𝜖, go to Step b); otherwise, the convergence is achieved. During the iterations of the GLRAM-based scheme, the RF precoder design in Step b) is carried out at the BS and the RF combiners are computed in Step c) at individual users. Since both the BS and users should exchange their information to solve the corresponding maximization problems, we consider the proposed GLRAM-based algorithm as the coordinated RF beamforming. It will be shown in Section IV that the proposed beamforming converges very fast. 2) Modified GLRAM based Solution: From the implementation point of view, it is always not efficient to have several iterations since a frequent exchange of information requires a large overhead. Furthermore, in the case where the RF beamformer is implemented by analog phase shifters, the constant modulus constraint should be considered. Therefore,

we modify the GLRAM method and propose a sub-optimal MGLRAM-based RF beamforming algorithm. Instead of initializing the RF combiners by Step a) in Section III-B1, (𝑘) similarly to [16], we aim to find 𝑾RF for the initialization by solving  2  (𝑘)𝐻  max ∀𝑘 = 1, ⋅ ⋅ ⋅ , 𝐾, (12) 𝑾RF 𝑯𝑘  , 𝐹

(𝑘)

𝒘RF [𝑖]∈𝒲,∀𝑖 (𝑘)

(𝑘)

where 𝒘RF [𝑖] is the 𝑖-th column of 𝑾RF and 𝒲 denotes the DFT codebook of size 𝑀𝑅𝑘 . Then we apply only one iteration (𝑘) to solve 𝑭RF and 𝑾RF successively. The MGLRAM-based RF beamforming algorithm is summarized as follows. (𝑘) i) Initialize: 𝑾RF (0) by solving (12). ii) Compute 𝑷 by (10) and obtain 𝑭RF by taking the phases of eigenvectors 𝑭RF = √

1 𝑒𝑗⋅angle(𝑼𝑃 (:,1:𝑁𝑇 )) . 𝑀𝑇

(13)

iii) For 𝑘 = 1, ⋅ ⋅ ⋅ , 𝐾, compute 𝑸𝑘 by (11) and obtain (𝑘) 𝑾RF according to (𝑘) 𝑾RF = √

1 𝑒𝑗⋅angle(𝑼𝑄𝑘 (:,1:𝑁𝑅𝑘 )) . 𝑀𝑅 𝑘

(14)

Implementation: The above three-step MGLRAM-based coordinated RF beamforming scheme can be easily carried out in practical systems. Steps i) can be implemented by beam sweeping with much fewer efforts, as the number of antennas at each user is much smaller than that of the BS. During Step ii) only the information of 𝑷 or the composite 𝐻 (0)𝑯𝑘 needs to be estimated either at the BS channel 𝑾RF or at users with feedback to the BS, depending on whether TDD or FDD is applied. Furthermore, to avoid the complicated calculations of SVD in Steps ii) and iii), a low-complexity and practical solution is to use the power method for obtaining the eigenvectors [21]. C. Digital Baseband Beamforming (𝑘)

After the RF beamforming matrices 𝑭RF and 𝑾RF , ∀𝑘 are obtained, a digital BD precoding scheme operating in a reduced dimension in the baseband is applied to maximize the achievable sum rate under a transmit power constraint. In our hybrid case, the BD technique is designed based on the ˇ 𝑘 , ∀𝑘. The concept of BD is that effective channel matrix 𝑯 by the decomposition of a multi-user broadcast channel into multiple parallel independent single-user MIMO channels, the precoding matrix 𝑭BB is found to lie in the null space of the other users channel matrices to eliminate all MUI. ˜ 𝑘 ∈ ℂ(𝑁𝑅 −𝑁𝑅𝑘 )×𝑁𝑇 as Thus we define 𝑯 ] [ ˜𝑘 = 𝑯 ˇ 𝑇, ⋅⋅⋅ , 𝑯 ˇ𝑇 , 𝑯 ˇ 𝑇 , ⋅⋅⋅ , 𝑯 ˇ𝑇 𝑇 . 𝑯 1 𝐾 𝑘−1 𝑘+1 (15) This definition requires 𝑁𝑇 ≥ 𝑁𝑅 so that there is null space ˜ 𝑘 to ensure zero MUI. We compute the SVD of 𝑯 ˜ 𝑘 who for 𝑯 ˜ has a rank 𝐿𝑘 and obtain [ ]𝐻 ˜ 𝑘 𝑽˜ (1) 𝑽˜ (0) ˜𝑘 = 𝑼 ˜𝑘Σ 𝑯 . (16) 𝑘 𝑘

˜ 𝑘 corresponds to the last right singular The null space of 𝑯 ˜ (0) vectors 𝑽˜𝑘 ∈ ℂ𝑁𝑇 ×(𝑁𝑇 −𝐿𝑘 ) . The effective channel of ˇ 𝑘 𝑽˜ (0) ∈ user 𝑘 after eliminating the MUI is identified as 𝑯 𝑘 ˜ ℂ𝑁𝑅𝑘 ×(𝑁𝑇 −𝐿𝑘 ) . This effective channel is equivalent to a con˜ 𝑘 transmit ventional single-user MIMO channel with 𝑁𝑇 − 𝐿 antennas and 𝑁𝑅𝑘 receive antennas. Define the SVD [ ]𝐻 ˜ ˇ 𝑘 𝑽˜ (0) = 𝑼𝑘 Σ𝑘 𝑽 (1) 𝑽 (0) ∈ ℂ𝑁𝑅𝑘 ×(𝑁𝑇 −𝐿𝑘 ) . 𝑯 𝑘 𝑘 𝑘 (17) and let the rank of the 𝑘-th user’s effective channel matrix (0) (1) be 𝑟𝑘 . The product 𝑽˜𝑘 𝑽𝑘 produces an orthogonal basis of dimension 𝑟𝑘 and represent the transmission vectors that maximize the information rate of user 𝑘 subject to the zeros MUI constraint. Thus the basedband precoding matrix 𝑭BB for all users can be defined as ] 1 [ 𝑭BB = 𝑽˜1(0) 𝑽1(1) , 𝑽˜2(0) 𝑽2(1) , ⋅ ⋅ ⋅ , 𝑽˜𝐾(0) 𝑽𝐾(1) Λ 2 , (18) where Λ ∈ ℂ𝑟×𝑟 is a diagonal matrix whose elements 𝜆𝑘 are the power loading coefficients which can be found by water-filling on the singular values Σ𝑘 from all user collected together, assuming a total power constraint. The receive beam(𝑘) forming vectors of user 𝑘 are chosen as 𝑾BB = 𝑼𝑘 . IV. S IMULATION R ESULTS In this section we evaluate the proposed algorithm for a MUMIMO system supporting multiple-stream transmission per user and compare it to the “Hybrid EGT-BD” scheme proposed in [16] as well as the fully digital BD precoding technique [2]. It is assumed that the BS has 𝑀𝑇 = 256 antennas and 𝑁𝑇 = 32 RF chains. There are total 𝐾 = 8 users, where each user is mounted with 𝑀𝑅𝑘 = 8 antennas connecting to 𝑁𝑅𝑘 RF chains. For the mmWave channel model, the number of clusters 𝑁𝑐 = 8 and the number of paths 𝑁𝑝 = 10 are chosen. We consider ULAs at both BS and users, where the inter-element distance equals to the half wavelength. A. Convergence Performance Figure 2 shows the effective array gain (defined as the target function to be maximized in (7)) as a function of the number of iterations for two variants of the proposed RF beamforming algorithm. The scheme “GLRAM+I” refers to the approach with an initialization of Step a) in GLRAM and “GLRAM+DFT” is the one with a DFT initialization by solving (12). It can be observed that both the iterative GLRAM variants converge quickly in 3 or 4 iterations and the difference in effective array gain between two consecutive iterations is very small. Compared to the converged case, the performance of the sub-optimal MGLRAM-based method only shows a degradation of less than 1 dB. Please note that even though the “GLRAM+DFT” outperforms “GLRAM+I”, in implementation the DFT initialization includes a round of beam sweeping and thus the “true” number of iterations should be shifted one to the right. This results in almost the same performance for both. But from the implementation point of view, the “GLRAM+I” with two iterations requires twice

Convergence analysis 21 20.8

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20.6

Figure 4 shows the throughput of the system versus SNR when the number of data streams varies. We can observe that our proposed scheme “Hybrid MGLRAM-BD” outperforms the one proposed in [16] and approaches the fully digital BD scheme. 250 Hybrid EGT−BD Hybrid MGLRAM−BD Fully digital BD 200

𝑁𝑅𝑘 = 𝑟𝑘 = 4

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Fig. 4. Throughput performance of the proposed scheme for transmitting different data streams.

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Throughput (bps/Hz)

𝐻 of the estimation and the feedback of 𝑾RF 𝑯𝑘 as well as twice of the SVD computation, which has a higher complexity than that of the “GLRAM+DFT”. Therefore, the proposed MGLRAM scheme applies DFT initialization and one iteration to update the RF beamformers. To evaluate the performance degradation due to the constant modulus constraint, we plot the throughput versus SNR at different iterations in Figure 3, when DFT initialization is applied. It can be observed that the performance of the sub-optimal solution by taking only phases of eigenvectors for the GLRAM method is slightly worse than that of the original one. Additionally, the number of iterations has a small influence on the throughput performance.

On the one hand, as the total number of transmitted data streams is 𝑟 = 16 or 32 in the discussed cases, which is much smaller than the total number of received antennas at users, i.e., 𝑀𝑅 = 64, the fully digital BD with degrees of freedom of 64 that applies 16 or 32 data streams cannot fully exploit the multiplexing gain. On the other hand, the hybrid beamforming at users takes advantage of analog combining and accordingly achieves an additional array gain in terms of SNR. Thus, we compare these algorithms when users employ fully digital combiners. Figure 5 shows the performance in such a scenario, where 𝑟𝑘 = 𝑁𝑅𝑘 = 𝑀𝑅𝑘 = 4 is considered for all users so that the degrees of freedom in terms of the multiplexing gain (i.e., 𝑟 = 𝑀𝑅 = 32) can be fully exploited. We also implement the iterative coordinated algorithm without the constant modulus constraint “Hybrid GLRAM-BD”. The number of iterations for the “Hybrid GLRAM-BD” is set to 4, which is enough for the convergence. It can be observed that the best performance is achieved by the fully digital BD techniques as well as the iterative “Hybrid GLRAM-BD”. The “Hybrid MGLRAM-BD” still outperforms the one in [16] and a small performance degradation as compared to the best case. V. C ONCLUSIONS

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Fig. 3. Throughput performance with the number of iterations for two variants with DFT initialization with 𝑁𝑅𝑘 = 𝑟𝑘 = 4.

This paper proposes a coordinated hybrid beamforming scheme supporting multiple data streams for downlink MU massive MIMO systems at mmWave frequencies. The hybrid beamforming is designed by splitting the beamforming problem into the design of the RF precoder/combiner and

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Fig. 5. Throughput performance of the proposed scheme when fully digital combining at users is applied .

the baseband digital precoder/decoder. We develop an efficient coordinated RF beamforming algorithm based on the iterative GLRAM approach, which takes advantage of the coordination between the BS and users. It has no dimensionality constraint and converges only within 3 or 4 iterations. To facilitate the implementation, we further derive a three-step MGLRAM based coordinated RF beamforming solution, which requires the beam sweeping and one iteration to determine the RF beamformers. The proposed algorithm only needs to estimate the composite channel from the antenna elements at the BS to the output RF chains, instead of the complete original channel. The estimation of the composite channel can be carried out either at the BS or at users, depending on whether TDD or FDD is applied. For the digital baseband beamforming, BD is applied to exploit the multiplexing gain. Simulation results show that our proposed coordinated hybrid beamforming solution outperforms the EGT scheme proposed in [16] and approaches the fully digital BD technique. ACKNOWLEDGMENT This work was supported by the National Science and Technology Major Projects under Grants 2015ZX03002002. R EFERENCES [1] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta, O. Edfors, and F. Tufvesson, “Scaling up MIMO: Opportunities and challenges with very large arrays,” IEEE Signal Processing Magazine, vol. 30, no. 1, pp. 40–60, 2013. [2] Q. H. Spencer and A. L. Swindlehurstand M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” IEEE Transactions on Signal Processing, vol. 52, no. 2, pp. 461– 471, 2004. [3] Q. H. Spencer, C. B. Peel, A. L. Swindlehurst, and M. Haardt, “An introduction to the multi-user MIMO downlink,” IEEE Communications Magazine, vol. 42, no. 10, pp. 60–67, 2004. [4] W. Roh, J-Y. Seol, J. Park, B. Lee, J. Lee, Y. Kim, J. Cho, K. Cheun, and F. Aryanfar, “Millimeter-wave beamforming as an enabling technology for 5G cellular communications: theoretical feasibility and prototype results,” IEEE Communications Magazine, vol. 52, no. 2, pp. 106–113, 2014.

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