毫米波MIMO系统：空间稀疏波束成形训练方法
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"毫米波MIMO系统的空间稀疏波束成形训练"
毫米波（Millimeter Wave，简称MMW）多输入多输出（Multiple-Input Multiple-Output，简称MIMO）系统是现代无线通信中的一个重要技术，它利用大量天线元素在发射端和接收端来实现高增益的波束成形，从而保证足够的链路预算。然而，由于功率和成本的限制，实际的MMW天线阵列中只能配备有限数量的射频（Radio Frequency，简称RF）链路，而天线元素的数量通常远大于RF链路。这种配置给传统的基于奇异值分解（Singular Value Decomposition，简称SVD）的波束形成带来了巨大的挑战，因为有效的信道估计是实现SVD波束形成的前提。
针对这一问题，本文提出了一种新的方法，即空间稀疏波束成形训练，用于解决MMW MIMO系统中的信道估计难题。论文发表于《IEEE Transactions on Wireless Communications》2017年5月第16卷第5期，由Xiantao Cheng, Niannian Lou和Shaoqian Li等人共同完成。作者们利用时分双工（Time Division Duplex，简称TDD）模式下的信道互易性，设计了两种迭代式天线训练方案，以逐步逼近理想的SVD波束形成效果。
第一种方案是基于功率迭代的方法。功率迭代是一种优化算法，通过反复应用矩阵乘法来逐渐将某一初始向量调整到矩阵的主特征向量方向，从而在没有完整信道信息的情况下逼近最佳波束形成状态。
第二种方案则采用了兰乔斯迭代（Lanczos Iteration）。兰乔斯迭代是一种用于近似计算大矩阵特征值和特征向量的有效方法，尤其适用于处理大规模稀疏矩阵。在MMW MIMO系统中，这种方法可以逐步构建出一个与原始信道矩阵相似的小型对角矩阵，进而实现近似的SVD波束形成。
这两种迭代训练策略能够降低对传统信道估计的需求，节省了系统资源，并且在一定程度上提高了波束成形的效率和性能。通过实证分析和仿真结果，文章证明了所提方法的有效性和实用性，对于未来MMW通信系统的设计具有重要的理论和实践指导价值。
总结来说，该研究论文主要探讨了如何在资源受限的MMW MIMO系统中，通过空间稀疏波束成形训练来解决信道估计难题，提出了基于功率迭代和兰乔斯迭代的两种创新性训练策略。这些方法不仅降低了对RF链路的需求，还提升了系统的整体性能，对于推进毫米波通信技术的发展具有重要意义。
3388 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 5, MAY 2017
weight r, then down-converts the obtained N
s
analog signals
into baseband data using its RF chains. Specifically, we have
y = W
H
r =
√
PW
H
HFs + W
H
n (4)
where W =[w
1
, w
2
, ···, w
N
s
]∈C
N
r
×N
s
represents the
receiver beamforming matrix.
Here it is worthy to point out that the above transmitter
and receiver beamforming matrices: F and W,relyonanalog
processing. However, in the analog domain, signals may be
only controlled by a phase shifter network [19], [24], [42],
which will greatly restricts the feasible set of beamforming
matrices. To overcome such hardware limitation, hybrid analog
and digital beamforming is suggested [19], [24], which divides
beamforming among the analog and digital domains. In other
words, we have that F = F
RF
F
BB
and W = W
RF
W
BB
,
where {F
RF
∈ C
N
t
×N
RF
, W
RF
∈ C
N
r
×N
RF
} represent the
analog beamforming matrices with the entries bearing equal
gain, {F
RF
, F
BB
}∈C
N
RF
×N
s
represent the baseband beam-
forming matrices. When incorporating F
BB
and W
BB
,the
system architecture in Fig.1 is also applicable for hybrid
beamforming.
With the signal model in (4), the achievable spectral effi-
ciency is given by [6], [9], [19]
R = log
2
I
N
s
+
P
N
s
σ
2
W
H
W
−1
W
H
HFF
H
H
H
W
(5)
It is well known that the optimal beamforming matrix pair
(F
o
, W
o
) maximizing R is determined by the SVD of the
channel matrix H [6]. Assume that the SVD of H bear the
form below
H = UV
H
= λ
1
u
1
v
H
1
+ λ
2
u
2
v
H
2
+···+λ
N
u
N
v
H
N
(6)
where U =[u
1
, ···, u
N
r
] is an N
r
× N
r
unitary matrix,
is an N
r
× N
t
diagonal matrix with non-negative singular
values {λ
i
}
N
i=1
on the diagonal, N = min(N
t
, N
r
),andV =
[v
1
, ···, v
N
t
] is an N
t
×N
t
unitary matrix. The vectors u
i
and
v
i
(i = 1, ···, N) are referred to as the left and right singular
vectors, respectively, corresponding to the singular value λ
i
.
Without loss of generality, hereafter we assume that N = N
t
and λ
1
>λ
2
> ··· >λ
N
. When considering N
s
≤ N data
streams, the optimal transmitter and receiver beamforming
matrices are F
o
=[v
1
, ···, v
N
s
] and W
o
=[u
1
, ···, u
N
s
],
respectively. The conventional way of obtaining F
o
and W
o
is to first estimate the channel matrix H and then conduct
the SVD of H. However, the involved channel estimation is
formidably challenging for MIMO MMW systems with large
arrays but a limited number of RF chains. Motivated by this,
in next sections we will develop two efficient beamforming
training schemes to realize the SVD beamforming, which
circumvent the burdensome channel estimation.
III. P
OWER ITERATION BASED BEAMFORMING TRAINING
A. Power Iteration (PWI) Principle
Note that u
H
i
u
j
= δ
i, j
, v
H
i
v
j
= δ
i, j
, i, j = 1, ···, N,
where δ
i, j
denotes the Kronecker delta. We can easily obtain
from (6) that
H
t
= H
H
H = λ
2
1
v
1
v
H
1
+···+λ
2
N
v
N
v
H
N
(7)
which in fact is the eigenvalue decomposition (EVD) of H
t
.
Specifically, we have H
t
v
i
= λ
2
i
v
i
.Thismeansthat{λ
2
i
}
N
i=1
and {v
i
}
N
i=1
, respectively, are the eigenvalues and eigenvectors
of H
t
.
Let us first consider single-stream (SS) beamforming, i.e.
N
s
= 1. In this case, the optimal transmitter and receiver
beamforming matrices reduce to v
1
and u
1
, respectively. Given
an initial vector v
0
∈ C
N×1
, power iteration (PWI) [29] tries
to obtain v
1
by exploiting that
˜v = lim
n→∞
H
n
t
v
0
1
= lim
n→∞
a
1
λ
2n
1
v
1
+ a
2
λ
2n
2
v
2
+···+a
N
λ
2n
N
v
N
2
= lim
n→∞
a
1
λ
2n
1
v
1
+
a
2
λ
2n
2
a
1
λ
2n
1
v
2
+···+
a
m
λ
2n
N
a
1
λ
2n
1
v
N
3
= a
1
λ
2n
1
v
1
(8)
where in the 1
st
step we assume that v
0
=
N
i=1
a
i
v
i
and
use (7), in the 3
rd
step we use the fact λ
1
>λ
2
> ···>λ
N
.
Obviously, ˜v/˜v=v
1
.SinceHv
1
= λ
1
u
1
, we can easily
obtain u
1
= Hv
1
/Hv
1
.
For multi-stream (MS) beamforming, we need to obtain
F
o
=[v
1
, ···, v
N
s
], which can be achieved by using N
s
-stage
PWI. During the first stage, the same process as in (8) is used
to obtain v
1
. During the I-th (2 ≤ I ≤ N
s
) stage, we try
to obtain v
I
with the knowledge of {v
i
}
I−1
i=1
. Specifically, we
choose an initial vector v
0
∈ C
N×1
, and project it onto the
null space of {v
i
}
I−1
i=1
, thereby obtaining
¯v
(I)
0
= (I
m
− P
(I)
)v
0
= v
0
−
I−1
i=1
(v
H
i
v
0
)v
i
=
N
i=I
b
i
v
i
(9)
where P
(I)
=[v
1
, ···, v
I−1
][v
1
, ···, v
I−1
]
H
is the orthogo-
nal projection matrix associated with {v
i
}
I−1
i=1
, and we assume
that v
0
=
N
i=1
b
i
v
i
. Similar to (8), we have that ¯v =
lim
n→∞
H
n
t
¯v
(I)
0
= b
I
λ
2n
I
v
I
and thus ¯v/¯v=v
I
. For the receiver
beamforming matrix W
o
=[u
1
, ···, u
N
s
], the vector u
i
can
be obtained as u
i
= Hv
i
/Hv
i
(i = 1, ···, N
s
).
B. PWI Based Beamforming Training
Let us focus on the SS beamforming for the moment. In the
beginning, the transmitter adopts an N
t
× 1 beam vector
f
0
=[1, 0, ···, 0]
T
. Note that f
0
is omnidirectional, since
at present we do not know the position of the receiver with
respect to the transmitter. If we use directional beam vector, the
receiver may encounter extremely low SNR if it does not lie in
the beam direction. For f
0
, we only activate the first antenna
element while keeping the other antenna elements inactive.
Considering that per-antenna transmit power is limited, we can
repeatedly transmit f
0
with n
p
times, and average the received
signals at the receiver, which can improve the received SNR by
10 log
10
n
p
dB. Simulations (to be given in Section V) show
that this doing can lead to satisfactory performance.
With F = f
0
,theN
r
× 1 signal in (3) changes to
r =
√
PHf
0
+ n
r
=
P
L
L
l=1
{α
l
a
H
t
(φ
l
)f}a
r
(θ
l
) + n
r
(10)
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