中继网络时变信道估计：DSTC下的跟踪与平滑技术
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"基于DSTC的中继网络的时变信道估计：跟踪，平滑和BCRB，这篇研究论文探讨了在时变平坦衰落场景下，采用分布式空间时间编码（DSTC）的中继网络中的信道估计算法。文章的主要目标是对每个中继跳的个体信道进行估计和跟踪，而不是仅仅估算复合信道。"
在无线通信领域，信道估计是确保数据传输质量和可靠性的关键环节。这篇发表在《IEEE无线通信 Transactions on Wireless Communications》2015年第14卷第9期的研究论文，由Shun Zhang, Feifei Gao, Jiandong Li和Hongyan Li等高级IEEE会员撰写，他们提出了一种新的方法来处理时变信道环境下的中继网络信道估计问题。
文章针对放大转发（AF）单向中继网络，该网络在每个中继节点采用了DSTC技术。与大多数现有工作不同，该研究聚焦于估算和追踪每个中继跳的独立信道，而不是简单的组合信道。为了降低信道参数的估计数量，作者引入了多项式基扩展模型（P-BEM），将问题转化为对每个中继跳的信道系数向量（称为内BEM-CVs）的估计。
考虑到信道的时变特性，论文采用了自回归（AR）模型来构建内BEM-CV的动态状态空间。通过这种方法，研究者利用无迹卡尔曼滤波器（UKF）以前向方式追踪内BEM-CV的动态变化。无迹卡尔曼滤波是一种有效且适应性强的滤波算法，尤其适用于非线性系统的状态估计。
此外，论文还讨论了利用无迹卡尔曼滤波器进行平滑处理，以优化信道估计的精度。平滑处理通常可以利用过去和未来的观测信息来改进当前时刻的估计，从而提高整体信道状态的准确度。
论文进一步引入了贝叶斯最小二乘（BCRB）理论，这是一种用于评估信道估计性能的理论框架。BCRB提供了最优估计理论的下界，即在给定的观测噪声条件下，任何无偏估计的方差不能低于这个界限。通过分析BCRB，作者能够评估所提出的跟踪和平滑策略的有效性，并可能指导更优的信道估计算法设计。
这篇研究论文在时变信道环境下，通过采用P-BEM、AR模型、UKF以及BCRB理论，为DSTC中继网络的信道估计提供了一套创新的解决方案，旨在提高信道跟踪和估算的效率与准确性。这对于提升无线通信系统在动态环境中的性能具有重要意义。
ZHANG et al.: TIME VARYING CHANNEL ESTIMATION FOR DSTC-BASED RELAY NETWORKS 5025
where
(l)
h
k
and
(l)
g
k
are the correlation matrices for h
k,m
and
g
k,m
, respectively, whose entries can be derived from (1-a) and
(1-b) as
(l)
h
k
i,j
= σ
2
h
J
0
(
2πf
ds
(i −j +2Nl)T
s
)
×J
0
2πf
dr,k
(i −j + 2Nl)T
s
, (8-a)
(l)
g
k
i,j
= σ
2
g
J
0
(
2πf
dd
(i − j +2Nl)T
s
)
×J
0
2πf
dr,k
(i −j + 2Nl)T
s
. (8-b)
Furthermore, the in-channel modeling mean square error
(MSE) for the m-th time segment can be expressed as
MSE
bem
=
1
2KN
K−1
k=0
E
e
g
k,m
H
e
g
k,m
+
e
h
k,m
H
e
h
k,m
=
1
2KN
K−1
k=0
tr
(I −
†
)
2
(0)
h
k
+
(0)
g
k
. (9)
Plugging (5) into (4), we can reexpress y
m
through P-BEM as
y
m
=
K−1
k=0
α(μ
k,m
)U
k,m
(λ
k,m
s
m
)+μ
k,m
t
k,m
+
K−1
k=0
αμ
k,m
U
k,m
v
m
+ n
m
. (10)
Before proceeding, let us define N × N diagonal matrices S
m
=
diag(s
m
) and T
k,m
= diag(t
k,m
) for further use.
Remark: There is one specific case for the precoding matri-
ces U
k,m
, i.e., U
k,m
is diagonal. For diagonal U
k,m
, it can be
obtained
(μ
k,m
) U
k,m
(λ
k,m
s
m
)
= U
k,m
diag
⎛
⎜
⎝
(λ
k,m
⊗ μ
k,m
)
c
k,m
⎞
⎟
⎠
s
m
, (11)
where c
k,m
is one (2Q −1) × 1 vector, and
is the N ×(2Q −
1) P-BEM matrix with the similar structure with . With (11),
y
m
can be rewritten as
y
m
=
K−1
k=0
U
k,m
S
m
c
k,m
+ T
k,m
μ
k,m
+
K−1
k=0
αdiag(μ
k,m
)U
k,m
v
m
+ n
m
. (12)
With U
k,m
being diagonal, the original signal model in (4) can
be reexpressed as
y
m
= α
K−1
k=0
U
k,m
S
m
(h
k,m
g
k,m
) + α
K−1
k=0
T
k,m
g
k,m
+
˜
n
m
.
(13)
Comparing (12) and (13), we have the following observations
when the precoding matrices U
k,m
are diagonal: the co-channel
vectors h
k,m
g
k,m
for the link from S to R
k
and then to D can
be approximated by P-BEM with 2Q −1 compositive BEM-
CV c
k,m
. If we consider valid data transmission process with no
superimposed training from R
k
, only the composite BEM-CV
c
k,m
are needed for ML data detection, which is similar to the
case under time-invariant scenario [14]. Notice that the above
results can be applied for CE-BEM but not for DKL-BEM.
Nonetheless, in order to gain better diversity performance,
the precoding matrices U
k,m
should be optimized according to
different criteria, i.e., data detection MSE, bit-error-rate (BER),
throughput and others, and the resultant U
k,m
’s are usually
not diagonal. Unfortunately, for non-diagonal matrices U
k,m
,
we cannot obtain results similar with that for diagonal pre-
coding matrices in (12), (13). From (10), it can be found
that the equivalent transmission channel from S to R
k
and
then to D can only be characterized by one N × N matrix
diag(μ
k,m
)U
k,m
diag(λ
k,m
) but not the equivalent composite
BEM-CV for non-diagonal U
k,m
. Thus if D implements
ML data detection with non-diagonal U
k,m
at R
k
,the
estimation of in-BEM-CVs λ
k,m
, μ
k,m
should be ob-
tained to construct the estimated transmission matrices, i.e.,
diag( ˆμ
k,m
)U
k,m
diag(
ˆ
λ
k,m
),where
ˆ
λ
k,m
( ˆμ
k,m
) represents
the estimation of λ
k,m
(μ
k,m
). The above phenomenon is quite
different from the case under time-invariant fading scenario
[14] and that with diagonal precoding matrices for time-varying
fading scenario.
For generality, we will adopt (10) as the observation signal
model, and focus on the estimation of in-BEM-CVs λ
k,m
, μ
k,m
,
k = 0,...,K − 1, m = 0,...,M − 1. Moreover, the general
model (10) can be written as the compacted matrix form as
y
m
= α[λ
m
](I ⊗)μ
m
+ ϒ
m
(I ⊗ )μ
m
+ α[μ
m
]
˜
v
m
+ n
m
= α[μ
m
]
m
(I ⊗)λ
m
+ ϒ
m
(I ⊗)μ
m
+ α[μ
m
]
˜
v
m
+ n
m
˜w
m
, (14)
where
[λ
m
]=
diag(U
0,m
S
m
λ
0,m
),...,diag(U
K−1,m
S
m
λ
K−1,m
)
,
[μ
m
]=
diag(μ
0,m
),...,diag(μ
K−1,m
)
,
m
= blkdiag(U
0,m
S
m
,...,U
K−1,m
S
m
),
ϒ
m
=[T
0,m
,...,T
K−1,m
],
˜
v
m
=
(U
0,m
v
0,m
)
T
,...,(U
K−1,m
v
K−1,m
)
T
T
, (15)
and
˜
w
m
is the equivalent noise at D corresponding to
˜
n
m
in (4).
Note that y
m
is a nonlinear function with respect to μ
m
, λ
m
due
to the first term on right hand side (RHS) of (14).
III. I
N-BEM-CV ESTIMATION
A. MAP In-BEM-CV Estimator
Define the KQ × 1 in-BEM-CVs λ
m
=[λ
T
0,m
,...,λ
T
K−1,m
]
T
,
μ
m
=[μ
T
0,m
,...,μ
T
K−1,m
]
T
,2KQ × 1 vector τ
m
=[λ
T
m
, μ
T
m
]
T
,
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