贝叶斯压缩感知在多用户MIMO波束成形系统中的速率自适应反馈研究
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"多用户MIMO波束成形系统中具有贝叶斯压缩感知的速率自适应反馈"
在无线通信领域，多输入多输出（MIMO）技术被广泛视为提升链路容量和能效的关键手段。然而，充分利用MIMO优势的前提是发射端能够获取到准确的信道状态信息（CSI）。在实际系统中，CSI通常由接收端估算后通过带限信道反馈至发射端。因此，设计一种高效、低开销的反馈机制至关重要。
本文提出了一种基于贝叶斯压缩感知（BCS）的反馈机制，专门针对时变且空间和时间相关的向量自回归（VAR）无线信道。贝叶斯压缩感知是一种利用信号稀疏性的理论，它可以以较低的采样率恢复高维信号，从而减少反馈信息的传输量。在该机制中，时变的VAR无线信道的特性被考虑进来，这通常存在于移动通信环境中，信道状态随时间和空间不断变化。
文章中，作者们首先介绍了BCS的基本原理，它结合了贝叶斯统计和压缩感知的思想，能够从少量观测数据中估计出信道的统计特性。然后，他们分析了在统计学意义上的反馈率失真函数，并以封闭形式给出了表达式。这个失真函数描述了在给定反馈速率下，信道估计的精度与信息损失之间的关系。
接着，论文进一步探讨了速率自适应反馈策略的实现，这种策略可以根据信道条件和系统资源动态调整反馈速率。这样，当信道条件恶劣或者系统资源紧张时，可以降低反馈速率，节省资源；而在信道条件较好时，增加反馈速率，提高信道估计的准确性，进而优化系统性能。
此外，为了验证所提出的BCS反馈方案的有效性，文章进行了数值模拟和性能评估。结果表明，相比于传统的反馈方法，BCS反馈在保持相当的性能水平下，能显著降低所需的反馈带宽，提升了系统的整体效率。
这篇研究论文为多用户MIMO波束成形系统提供了创新的反馈策略，利用贝叶斯压缩感知理论来实现对时变VAR无线信道的高效和适应性的 CSI 反馈。这一成果对于未来5G及更高级别的无线通信系统设计具有重要的理论和实践意义，尤其是在资源有限的环境下，如何优化信道反馈以最大化系统性能是一个亟待解决的问题。
HUANG et al.: RATE-ADAPTIVE FEEDBACK WITH BCS IN MULTIUSER MIMO BEAMFORMING SYSTEMS 4841
ergodic downlink capacity, and use it to facilitate the
rate-adaptive feedback according to the transmission rate
requirement of downlink services.
The rest of this paper is organized as follows. In section II,
the feature of multiuser MIMO channel is described, and the
VAR model is used to describe the temporal correlation among
continuous CSIs. In section III, Bayesian CS model is built
to compress the dimension of the channel prediction error. In
section IV, the feedback rate-distortion function and ergodic
downlink capacity are derived, and the relationship between the
feedback rate and downlink capacity will be defined. Such a
rate-adaptive feedback can be used to support arbitrary trans-
mission rate. In section V, simulation results are given to show
the efficiency of t he proposed compressive feedback scheme in
terms of the feedback rate, distortion, ergodic downlink capac-
ity, and bit error rate (BER) performance, followed by the
conclusion of this paper in section VI.
II. VAR-B
ASED MIMO CHANNEL MODEL
In this section, we first describe the structure of MIMO
channel, which is a prerequisite knowledge for the proposed
BCS feedback scheme. Furthermore, the VAR model will be
introduced to describe the correlation between continuous CSIs
[15]. From this VAR model, we will introduce the proposed
spatiotemporal compression, in order to reduce the feedback
error and the dimension of channel vector (thus achieve a
low-dimensional codebook).
A. MIMO Channel
In this paper, we consider a general MIMO wireless commu-
nication system with M transmit antennas at the base-station as
well as K users (each with a single antenna). Since the users are
in different places, their channel vectors are uncorrelated with
each other. The channel vector of user i , h
i
, can be modeled
as [13]
h
T
i
= αh
T
iid
R
1
2
T
X
(1)
where α is a scalar that reflects the channel gain; h
iid
repre-
sents a M × 1 vector consisting of independent and identically
distributed (i.i.d.) complex Gaussian random components with
zero-mean and unit variance, and (·)
T
represents transpose
operation; R
T
X
is a M × M correlation matrix at the transmit-
ter side (i.e., the base-station). We also assume that uniformly
separated linear antenna arrays are installed at the base-station,
and the spatial correlations in h
i
are solely determined by the
Jakes model [8]. Every element R
ij
in R
T
X
represents the cor-
relation coefficient between the i
th
and the j
th
antennas at the
base-station, and can be formulated as
R
ij
= J
0
(
2πd
ij
λ
) (2)
where d
ij
is the distance between the i
th
and the j
th
antennas
at the base-station, λ is the wavelength of the central carrier,
and J
0
(·) represents the zer o
th
-order Bessel function of the
first kind. From Eqn. (1), one can see that each component
of h
i
is the combination of vector h
iid
with different weights.
Thus there are internal correlations in the channel vector h
i
.
We assume that a perfect channel vector h
i
can be estimated
at the user i. This paper assumes single antenna in each user.
However, it can be easily extended to multi-antenna users by
using methods such as [16].
The CDI feedback information is the index of the quantized
channel vector, which is a codevector in the codebook. As in
other related works (especially [7]), we assume that the forward
error correction (FEC) can be adopted to mitigate the transmis-
sion errors. Hence, the successful collection of CDI feedbacks
can be guaranteed. After successfully collecting and selecting
(i.e., multiuser selection) CSI feedbacks from multiple users
(i.e., H = [h
1
, h
2
,...,h
M
]
T
), the base-station performs pre-
coding (i.e., ZFDPC or ZFBF), and obtains spatial multiplexing
gain. The signal received by a user i can be represented as
r
i
= h
T
i
x + n
i
, i = 1, 2,...,M (3)
where x is the transmitted symbol-vector that contains infor-
mation symbols of the selected users with an average power
constraint E{
x
2
}=P, and r
i
is the signal received by user i;
and n
i
represents additive white Gaussian noise (AWGN) with
variance σ
2
. Please note that x (i.e., M × 1 vector) is a precoded
result over original information symbol s (i.e., M × 1 vector),
through ZFDPC or ZFBF with the knowledge of CSI feedback
information H.
B. VAR Model
From the above description, we can see that it is critical for
a base-station to get to know the downlink CSIs, in order to
obtain the multiplexing gain and multiuser gain. For a specific
user i , it only knows its own CSI h
i
, and needs to feedback
such a channel vector to the base-station. In the following dis-
cussions, we omit index i of h
i
, and denote the channel vector
at time instant t as h
t
. As we mentioned earlier, it is very useful
to compress h
t
in order to obtain a lower dimension representa-
tion (thus obtaining low-dimension codebook). As we know, for
a time-varying channel (especially slowly varying one), there
are correlations between continuous CSIs [17]. Recently, much
research work on channel tracking has been done [8]–[11], [18],
and LP (i.e., VAR model) has been proved as t he best model to
describe the correlations among the CSI time sequences [9].
In a fast fading channel, the VAR model may not track the
CSI very well, and the prediction error will have large variance
in statistics. However, the proposed feedback scheme can still
benefit from the BCS model, which can compress the dimen-
sionality of the feedback CSI and thus a lower dimensional
codebook can be used in VQC.
AR( p) model (with order p) is commonly used to describe
the temporally correlated fading channel in single input single
output system (SISO) [3], [18]. Here, we adopt its extension
version (i.e., VAR( p)) to describe the vector channel, that is
h
t
= φ
1
h
t−1
+ φ
2
h
t−2
+···+φ
p
h
t− p
+ μ
t
(4)
where μ
t
is a M × 1 vector with complex zero-mean, covari-
ance A, multivariate normal distribution, and μ
t
represents
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