面向误差通道的BDMA大规模MIMO物理层安全优化beamforming
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更新于2024-08-28
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在"Robust Beamforming for Physical Layer Security in BDMA Massive MIMO"这篇研究论文中，作者 Fengchao Zhu、Feifei Gao、Hai Lin、Shi Jin、Junhui Zhao 和 Gongbin Qian探讨了在大规模多输入多输出（Massive MIMO）的宽带多址（BDMA）系统中，如何设计一种鲁棒的波束形成策略以确保物理层安全。论文针对实际应用中的一个重要挑战——信道估计误差，提出了创新的解决方案。
传统上，大规模MIMO系统通过多用户波束分割来提高频谱效率和数据传输速率。然而，当信道估计存在不确定性时，这可能导致信号被非法窃听者（eavesdropper）有效捕获，从而威胁系统的安全性。为了克服这一问题，论文提出了一种新的设计方法，目标是最大化合法用户的信号到干扰及噪声比（Signal-to-Interference-and-Noise Ratio, SINR），同时最小化基站的发射功率，并向窃听者引入人工噪声，以降低其接收质量。
作者证明，在BDMA大规模MIMO体系结构下，原非凸优化问题可以转化为一个严格的半定义编程（Semidefinite Programming, SDP）问题，这是一个更易于求解的优化形式。通过这种转化，找到了最优的秩一（rank-one）波束形成方案，确保了在信道估计误差的背景下，系统的性能最大化且安全得到保障。
值得注意的是，与单纯依赖于凸优化工具不同，文中还进一步探讨了如何直接导出最优的波束形成方向，可能采用了迭代算法或者特殊的数学技巧，以便在实际系统中实现高效的波束管理。这种方法不仅提高了系统的安全性，也兼顾了能量效率，对未来的无线通信网络设计具有重要意义。
这篇论文深入研究了物理层安全在BDMA大规模MIMO系统中的关键挑战，并提供了一种有效的解决方案，对于无线通信领域的安全性和可靠性提升具有重要的理论贡献和实践指导价值。
ZHU et al.: ROBUST BEAMFORMING FOR PHYSICAL LAYER SECURITY IN BDMA MASSIVE MIMO 777
B. Problem Formulation
Let us apply AN at BS during the downlink data trans-
mission to provide the strongest distortion to Eve, i.e., the
baseband signal from BS can be expressed as
x
b
=
K
k=1
s
k
v
k
+ wv
w
, (3)
where v
k
∼CN(0, 1) denotes the data symbol for the kth
user; v
w
∼CN(0, 1) is the AN signal for the Eve; s
k
∈ C
N×1
and w ∈ C
N×1
are information beamforming vector and AN
beamforming vector, respectively.
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
+ h
H
k
w
e
v
e
+ n
k
, (4)
where n
k
∼CN(0,σ
2
k
) represents the antenna noise of the
kth user. The downlink signal at the Eve can be expressed as
y
e
=
K
k=1
h
H
e
s
k
v
k
+ h
H
e
w
e
v
e
+ n
e
, (5)
where n
e
∼CN(0,σ
2
e
) represents the antenna noise of the
Eve. Then the secret rate for the kth user are expressed as
R
k
=log
2
⎛
⎜
⎜
⎝
1+
h
H
k
s
k
2
i=k
h
H
k
s
i
2
+h
H
k
w
e
2
+σ
2
k
⎞
⎟
⎟
⎠
−log
2
⎛
⎜
⎜
⎝
1+
h
H
e
s
k
2
i=k
h
H
e
s
i
2
+h
H
e
w
e
2
+σ
2
e
⎞
⎟
⎟
⎠
≥ log
2
(1 + γ
k
) − log
2
(1 + γ
e,k
) , (6)
where γ
k
is the receive SINR for the kth user, and γ
e,k
is
the receive SINR for Eve when Eve aims to eavesdrop the
kth user. Note that when γ
k
>γ
e,k
> 0, ∀k ∈{1,...,K},
we could always obtain non-zero secret rate for each users,
which implies that the physical layer security could be strictly
guaranteed.
Our target is to design simultaneous information beamform-
ing vectors {s
k
} and AN beamforming vector w
e
to provide
legal users and the Eve with different SINRs, meanwhile
minimizing the transmit power of BS. Taking the channel
estimation errors into consideration, the robust transmit beam-
forming design can be formulated as
2
P1 :min
{s
k
},w
e
w
e
2
+
K
k=1
s
k
2
(7)
s.t.
h
H
e
s
k
2
i=k
h
H
e
s
i
2
+h
H
e
w
e
2
+σ
2
e
≤ γ
e,k
, h
e
∈U
e
,
(8)
h
H
k
s
k
2
i=k
h
H
k
s
i
2
+h
H
k
w
e
2
+σ
2
k
≥ γ
k
, ∀h
k
∈U
k
,
(9)
k =1, 2,...,K, (10)
2
Note that the similar problem formulation in non-robust communication
systems could be found in [36]–[38].
where γ
e,k
> 0 is the maximum allowable SINR for Eve,
and γ
k
> 0 is the desired SINR for the kth user. Note
that the optimal solutions of P1 are very hard to obtain in
conventional MIMO systems.
3
Nevertheless, we will next
show that P1 could be globally solved utilizing property (1)
of BDMA massive MIMO systems.
III. O
PTIMAL ROBUST BEAMFORMING
A. Problem Reformulation
The main difficulty of solving P1 lies in the constraints (8)
and (9). Define X
e,k
and X
s,k
as
X
e,k
=
i=k
s
i
s
H
i
+ w
e
w
H
e
−
1
γ
e,k
s
k
s
H
k
, (11)
X
s,k
=
1
γ
k
s
k
s
H
k
−
i=k
s
i
s
H
i
− w
e
w
H
e
, (12)
respectively. Substituting (2) into (8), we can equivalently
rewrite (8) as
˜
h
e
+ δ
e
H
X
e,k
˜
h
e
+ δ
e
+ σ
2
e
≥ 0,
−δ
H
e
Iδ
e
+
2
e
≥ 0,k=1, 2,...,K.
(13)
Substituting (2) into (9), we can equivalently rewrite (9) as
˜
h
k
+ δ
k
H
X
s,k
˜
h
k
+ δ
k
− σ
2
k
≥ 0,
−δ
H
k
Iδ
k
+
2
k
≥ 0,k=1, 2,...,K.
(14)
According to the Lemma of S-Procedure [39], we know
that the constraints in (13) hold true if and only if there exists
μ
e,k
≥ 0,k=1, 2,...,K such that
X
e,k
+ μ
e,k
IX
e,k
˜
h
e
˜
h
H
e
X
H
e,k
˜
h
H
e
X
e,k
˜
h
e
+ σ
2
e
− μ
e,k
2
e
0. (15)
Meanwhile, the constraints in (14) hold true if and only if
there exists μ
s,k
≥ 0,k=1, 2,...,K such that
X
s,k
+ μ
s,k
IX
s,k
˜
h
k
˜
h
H
k
X
H
s,k
˜
h
H
k
X
s,k
˜
h
k
− σ
2
k
− μ
s,k
2
k
0. (16)
It is then clear that P1 can be equivalently re-expressed as
P1−EQV :
min
{μ
e,k
},{μ
s,k
},w
e
,{s
k
}
Tr(W
e
)+
K
k=1
Tr(S
k
) (17)
s.t.
X
e,k
+μ
e,k
IX
e,k
˜
h
e
˜
h
H
e
X
H
e,k
˜
h
H
e
X
e,k
˜
h
e
+σ
2
e
−μ
e,k
2
e
0,
(18)
X
s,k
+μ
s,k
IX
s,k
˜
h
k
˜
h
H
k
X
H
s,k
˜
h
H
k
X
s,k
˜
h
k
−σ
2
k
−μ
s,k
2
k
0,
(19)
X
e,k
=
i=k
S
i
+ W
e
−
1
γ
e,k
S
k
, (20)
X
s,k
=
1
γ
k
S
k
−
i=k
S
i
− W
e
, (21)
μ
e,k
≥ 0,μ
s,k
≥ 0, (22)
W
e
= w
e
w
H
e
, S
k
= s
k
s
H
k
,k=1, 2,...,K,
(23)
3
This non-convex problem cannot be solved efficiently for large values
of N and is thus not suited for real-time signal processing applications in
conventional communication systems [34].
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