BDMA大规模MIMO系统下的鲁棒下行波束forming设计
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本文主要探讨了在大规模多输入多输出（Massive MIMO）的波分多址（BDMA）系统中，设计一种鲁棒的下行信道束形（downlink beamforming）策略，以应对信道状态信息（CSI）不精确的情况。论文发表于2018年4月的《IEEE Transactions on Communications》期刊，作者是Fengchao Zhu、Feifei Gao（IEEE资深会员）、Shi Jin（IEEE会员）、HaiLin（IEEE资深会员）和Minli Yao。
在大规模MIMO系统中，BDMA是一种多用户同时服务的技术，通过将信号空间分成多个独立的子空间，每个用户占用一个子空间进行通信。然而，由于实际操作中 CSI 的不确定性，如测量误差和无线环境变化，对下行链路的精确控制变得至关重要。因此，研究的目标是在考虑到这些不确定性因素时，设计出能够降低基站（BS）功率消耗的同时满足信号与干扰及噪声比（Signal-to-Interference-plus-Noise Ratio, SINR）约束的鲁棒策略。
作者采用最坏情况下的确定性模型来建立问题，即在给定一组SINR阈值的情况下，寻求最小化BS的功率消耗。为了将初始的非凸优化问题转化为一个可解的优化问题，论文利用了S-Procedure方法，这是一种数学工具，它允许将非线性约束转化为线性等价形式，从而使问题变得更容易处理。接着，作者引入了半定义松弛（Semidefinite Relaxation, SDR）技术，将原问题转化为一个凸的半定义编程问题，这使得求解变得更加高效。
进一步，作者证明了SDR解的最优性，关键在于展示了在BDMA体系下，由于用户间的正交性，优化后的束形向量具有秩一特性。这意味着可以通过解析方法得到最优的束形方向和功率分配，这是论文的核心贡献之一。这种优化方法不仅保证了系统的性能，还能有效地管理干扰，提升整个系统的效率和鲁棒性。
本文的研究为BDMA大规模MIMO系统设计了一种有效的鲁棒下行信道束形方案，通过理论分析和算法优化，实现了在不确定的CSI条件下，既保证了服务质量，又降低了系统复杂度。这一成果对于提升5G和未来无线网络的容量和可靠性具有重要意义。
1498 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 66, NO. 4, APRIL 2018
h
k
for the kth user lies within a ball of radius
k
around the
estimated channel vector
˜
h
k
, i.e.,
h
k
∈ U
k
=
˜
h
k
+ δ
k
|δ
k
≤
k
, (3)
where δ
k
∈ C
N×1
is the channel estimation error whose norm
is assumed to be bounded by
k
.
For a general training scheme, there is no obvious relation-
ship among
˜
h
k
. Hence, in this paper, we adopt the recently
popular BDMA scheme [14], [15] such that pilots of different
users are transmitted through orthogonal spatial directions.
1
In this case, the estimated channels
˜
h
i
and
˜
h
j
for different
users must be orthogonal,
2
i.e.,
˜
h
H
i
˜
h
j
= 0, ∀i = j, (4)
which is the key point of our proposed design. Please refer
to [14] and [15] for detailed discussion of BDMA.
B. Problem Formulation
The downlink signal at the kth user can be expressed as
y
k
= h
H
k
s
k
v
k
+
i=k,i∈K
h
H
k
s
i
v
i
+ n
k
, (5)
where v
k
∼ CN (0, 1) denotes the data symbol for the
kth user; s
k
is the corresponding downlink beamforming vector
to be designed; n
k
∼ CN (0,σ
2
k
) represents the receive noise.
To meet the requirement of the energy efficiency of
5G communications, our target here is to minimize the transmit
power of BS, at the same time providing intended users with
satisfied SINR. The worst-case robust transmit beamforming
design can then be formulated as [30], [31]
P1 : min
{s
k
}
K
k=1
s
k
2
(6)
s.t.
h
H
k
s
k
2
i=k
h
H
k
s
i
2
+ σ
2
k
≥ γ
k
, ∀h
k
∈ U
k
, (7)
k = 1, 2,...,K, (8)
where γ
k
> 0 is the desired SINR for the kth user. In small
MIMO systems,
3
P1 is NP hard, and it is difficult to obtain
the optimal beamforming vectors {s
k
, k = 1,...,K }. Thus,
different approximation algorithms [30], [31] are designed to
extract suboptimal solutions. Nevertheless, we will next show
that the optimal solutions could be derived in closed-form
thanks to (4) under BDMA massive MIMO scheme.
1
In particular, the columns of discrete Fourier transform (DFT) matrix with
non-overlap indices are assigned to different users as the training directions.
Such a scheme is not optimal in terms of training design but could be
implemented with much higher efficiency. Moreover, as long as N is large,
the performance loss of the channel estimation accuracy is small compared
to the optimal one.
2
Note that the BDMA scheme can also be applied in millimeter wave
communication systems [15], [37], where the estimated channels can be
presented as orthogonal vectors with BDMA channel estimation algorithms.
3
For contrast, we use small MIMO system to represent the conventional
MIMO system where the number of transmit antennas is small.
III. THE OPTIMAL ROBUST BEAMFORMING
A. Problem Re-formulation
The main difficulty for solving P1 lies in the constraints (7).
Usually, the S-Procedure [38] is adopted to convert the con-
straints (7) to easier cases [30], [31]. Substituting (3) into (7),
we can equivalently rewrite (7) as
⎧
⎪
⎪
⎨
⎪
⎪
⎩
˜
h
k
+ δ
k
H
⎛
⎝
1
γ
k
s
k
s
H
k
−
i=k
s
i
s
H
i
⎞
⎠
˜
h
k
+ δ
k
− σ
2
k
≥ 0,
−δ
H
k
Iδ
k
+
2
k
≥ 0, k = 1, 2,...,K.
(9)
Lemma 1 (S-Procedure [38]): For any b
1
∈ C
n×1
, b
2
∈
C
n×1
, c
1
∈ R, c
2
∈ R and for any complex Hermitian matrices
A
1
∈ C
n×n
, A
2
∈ C
n×n
,define f
1
(x) and f
2
(x) as
f
1
(x) = x
H
A
1
x + b
H
1
x + x
H
b
1
+ c
1
,
f
2
(x) = x
H
A
2
x + b
H
2
x + x
H
b
2
+ c
2
, (10)
respectively. The condition f
1
(x) ≥ 0 ⇒ f
2
(x) ≥ 0 holds
true if and only if there exists a nonnegative μ, such that
A
2
b
2
b
H
2
c
2
− μ
A
1
b
1
b
H
1
c
1
0.
According to Lemma 1, we know that the constraints in (9)
hold true if and only if there exists μ
k
≥ 0, k = 1, 2,...,K
such that
X
k
+ μ
k
IX
k
˜
h
k
˜
h
H
k
X
H
k
˜
h
H
k
X
k
˜
h
k
− σ
2
k
− μ
k
2
k
0, (11)
where for simplicity, we define X
k
as
X
k
=
1
γ
k
s
k
s
H
k
−
i=k
s
i
s
H
i
. (12)
It is then clear that P1 can be equivalently expressed as
P1 − EQV :
min
{μ
k
},{s
k
},{S
k
}
K
k=1
Tr(S
k
) (13)
s.t.
X
k
+ μ
k
IX
k
˜
h
k
˜
h
H
k
X
H
k
˜
h
H
k
X
k
˜
h
k
− σ
2
k
− μ
k
2
k
0, (14)
X
k
=
1
γ
k
S
k
−
i=k
S
i
, (15)
μ
k
≥ 0, S
k
= s
k
s
H
k
, k = 1, 2,...,K. (16)
Note that the nonlinear constraint S
k
= s
k
s
H
k
in (16) is
equivalent to: S
k
0 and Rank(S
k
) = 1.
However, P1 − EQV is still NP hard due to the fact
that the rank constraint (16) is non-convex. Using the SDR
technique [35] (i.e., dropping the constraints Rank(S
k
) = 1,
k = 1, 2,...,K), we can obtain the following relaxed convex
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