Robust Beamforming for BDMA Massive MIMO
Fengchao Zhu
1,2
,FeifeiGao
2
, Hai Lin
3
,andShiJin
4
1
High-Tech Institute of Xi’an, Shaanxi, China
2
Tsinghua National Laboratory for Information Science and Technology
3
Department of Electrical and Information Systems, Osaka Prefecture University
4
National Communications Research Laboratory, Southeast University
fengchao
zhu@126.com, feifeigao@ieee.org, hai.lin@ieee.org, jinshi@seu.edu.cn
Abstract—In this paper, we design robust downlink beamform-
ing against the imperfect channel state information (CSI) for
beam division multiple access (BDMA) massive multiple-input
multiple output (MIMO) systems. Different from conventional
approach, the optimality of semi-definite relaxation (SDR) is
strictly proved by showing the rank-one property of the optimal
beamforming with the orthogonal BDMA massive channels,
where globally optimal robust beamforming solutions are derived.
More importantly, we make one step further by deriving the
optimal beamforming directions and optimal beamforming power
allocation of the SDR in closed-form, which greatly reduces the
optimization complexity and makes the proposed design practical
for a real word massive MIMO system. Simulation results are
provided to demonstrate the efficiency of the proposed algorithm.
I. INTRODUCTION
Massive multiple-input multiple-output (MIMO) systems [1]
are currently hot topics of 5G wireless communications that
demand for high spectral utilization to support the quality of
service (QoS) for a wide variety of multimedia applications.
Most of the potential advantages of massive MIMO system rely
heavily on the perfect channel state information (CSI) at the
base station (BS) [2]. A lot of prominent researches have been
implemented to reduce the computational complexity and make
channel estimation feasible in massive MIMO systems. Based
on the assumption that the angular spread (AS) of the incident
signals at BS from each user is narrow, the authors in [3]–
[5] proposed an angle domain spatial basis expansion model
(SBEM) that could represent the corresponding channels with
only a few parameters. A technique called beam division
multiple access (BDMA) was presented in [6], [7], where
it is shown that the channel after discrete Fourier transform
(DFT) is approximately sparse, and then assign different users
with orthogonal beams for transmission. It is worth noting
that the basic idea of the above mentioned channel estimation
algorithms [3]–[5] is to approximate massive MIMO channels
by various orthogonal bases, by which means the effect param-
eters can be reduced. However, perfect channel estimations are
usually unavailable for massive MIMO systems due to channel
estimation errors.
To handle to imperfect CSI, robust beamforming has drawn
considerable interest [8], [9]. In general, there are two major
types of robust design: the stochastic robust beamforming and
This work is supported in part by the the National Natural Science Founda-
tion of China under Grant 61422109, Grant 61531011 and Grant 61601474,
in part by the National Science Foundation (NSFC) for Distinguished Young
Scholars of China with Grant 61625106, and in part by JSPS KAKENHI Grant
Number JP17K06435
the worst-case robust beamforming. For the stochastic robust
beamforming [10], [11], the design is usually formulated as
minimizing the transmission power while ensuring that all
users’ signal-to-interference-and-noise ratio (SINR) require-
ments are satisfied with high probabilities, where the CSI
errors are modeled as random variables with known statistical
properties. In contrast, the worst-case robust beamforming
[12], [13] is usually designed to satisfy the SINR require-
ments for all possible channel realizations in the uncertainty
regions, where the CSI errors belong to some known bounded
uncertainty sets. However, it has been shown in [14] that the
downlink multiuser robust beamforming for multi-input single-
output (MISO) channel needs to solve NP-hard optimizations
[15], where just suboptimal solutions could be found through
different rank-one approximations. Moreover, existing robust
beamforming algorithms [14] need to solve a sequence of
semi-definite programming (SDP) problems, which could be
highly unavailable in massive MIMO systems due to its
forbiddenly high computational complexity.
In this paper, we consider robust beamforming for multiuser
massive MIMO systems following BDMA scheme [6], [7],
where the estimated channels are lied orthogonally to each
other and the channel estimation errors are bounded by ball
constants. We demonstrate that global optimal robust beam-
forming could be derived in closed-form, which will greatly
reduce the computational complexity for massive MIMO sys-
tems. Compared to directly applying the conventional SDP
method [14] onto massive MIMO, the proposed scheme yield
the optimal solution with much less computational complexity
and is applicable for real world massive MIMO system.
II. S
YSTEM MODEL AND PROBLEM FORMULATION
A. System Model
Let us consider a multiuser massive MIMO system shown in
Fig. 1, where BS is equipped with 𝑁 ≫ 1 antennas in the form
of uniform linear array (ULA). There are 𝐾 single-antenna
users with index set 𝒦 = {1,...,𝐾} randomly distributed in
the coverage area. The 𝑘th user is located at 𝐷
𝑘
meters away
from BS and is surrounded by a ring of 𝑃 ≫ 1 local scatterers
with the radius 𝑅
𝑘
[5]. The channel from the 𝑘th user to BS
is composed of 𝑃 rays and can be expressed as [5]:
𝒉
𝑘
=
1
√
𝑃
𝑃
𝑝=1
𝛼
𝑘,𝑝
𝒂(𝜃
𝑘,𝑝
), (1)
where 𝛼
𝑘,𝑝
∼𝒞𝒩(0,𝜁
𝑘,𝑝
) represents the complex gain of the
𝑝th ray. Moreover, 𝒂(𝜃
𝑘,𝑝
) ∈ ℂ
𝑁×1
is the steering vector and
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