量化数据与随机拓扑下扩散LMS算法在传感器网络中的性能深度剖析
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本文主要探讨了"基于量化数据和随机拓扑的传感器网络扩散LMS算法性能分析"这一主题。该研究论文由Junlong Zhu等人合作完成，发表于北京邮电大学的网络与交换技术国家重点实验室、河南科技大学的信息工程学院以及北京交通大学下一代互联网互联设备国家工程实验室。作者们关注的是在分布式参数估计问题中，扩散LMS算法在量化数据和随机网络结构下的性能表现。
扩散LMS算法是一种在无线传感器网络（WSN）中广泛应用的自适应信号处理技术，其核心思想是通过局部通信和信息共享，实现全局优化。在文中，研究者们考虑了数据量化对算法精度的影响，因为实际操作中，传感器通常会受限于有限的存储和通信能力，导致数据传输存在误差。此外，他们还分析了随机拓扑在网络中的作用，因为传感器节点的位置和连接关系对信息传播和算法收敛速度有显著影响。
作者们在2016年1月30日接收了论文，并在5月10日进行了修订，最终于5月11日接受。这篇工作遵循Creative Commons Attribution License，允许无限制地使用、分发和复制，只要原作被正确引用。
论文的核心内容包括性能评估指标的选择，如收敛速度、稳定性和抗噪声能力等，这些是在随机拓扑和量化数据条件下对扩散LMS算法进行实验分析的结果。通过对各种参数设置下的仿真结果进行深入剖析，研究人员揭示了如何通过优化算法参数来改善算法在复杂网络环境中的表现。
这篇研究论文不仅提供了对量化数据和随机拓扑下扩散LMS算法性能的深入理解，也为优化此类算法设计和实际应用中的性能提升提供了有价值的参考依据。对于从事无线传感器网络、自适应信号处理或分布式计算的科研人员来说，这是一篇具有实用价值的研究文章。
International Journal of Distributed Sensor Networks 3
function (),theoptimalsolution
𝑜
satises the following
equation:
𝑁
𝑘=1
𝑢,𝑘
𝑜
=
𝑁
𝑘=1
𝑑𝑢,𝑘
,
(3)
where
𝑑𝑢,𝑘
E[d
𝑘
()u
∗
𝑘,𝑖
] and
𝑢,𝑘
E[u
∗
𝑘,𝑖
u
𝑘,𝑖
].For
convenience of presentation, we assume that
𝑢,𝑘
is positive-
denite; that is,
𝑢,𝑘
>0.Hence,
𝑢,𝑘
is invertible; then, the
optimal solution
𝑜
satises the normal equation [36]:
𝑜
=
𝑁
𝑘=1
𝑢,𝑘
−1
𝑁
𝑘=1
𝑑𝑢,𝑘
.
(4)
However, the optimal solution
𝑜
cannot be determined from
the solution of the normal equation, since {
𝑢,𝑘
,
𝑑𝑢,𝑘
} are
not known beforehand in practical. erefore, in order to
solve
𝑜
, the CTA (Combine-then-Adaptive) diusion LMS
algorithm is proposed in [21]:
𝜙
𝑘,𝑖−1
=
𝑙∈N
𝑘
𝑙𝑘
w
𝑙,𝑖−1
,
(5)
w
𝑘,𝑖
=𝜙
𝑘,𝑖−1
+
𝑘
u
∗
𝑘,𝑖
d
𝑘
(
)
−u
𝑘,𝑖
𝜙
𝑘,𝑖−1
, (6)
where
𝑘
is the local step size and w
𝑘,𝑖
is the local estimate for
the parameter
𝑜
that is evaluated by node at time index .
{
𝑙𝑘
}are nonnegative coecients, which can be treated as free
weighting parameters are chosen by the designer, and satisfy
𝑙𝑘
≥0,
𝑁
𝑙=1
𝑙𝑘
=1,
𝑙𝑘
=0
if ∉N
𝑘
.
(7)
In this section, we briey describe the diusion LMS
algorithm without quantized data and with xed topology.
However, the sources (e.g., power and bandwidth) are limited
and the topology is random in wireless sensor networks.
erefore, we consider diusion LMS algorithm with quan-
tized data and random topology in Section 3.
3. Diffusion LMS Algorithm with Quantized
Data: Random Topology
In this section, we consider the eects of quantization and
random topology in diusion LMS algorithm. erefore,
we rst establish random topology model and quantization
model and then derive diusion LMS algorithm with quan-
tized data and random topology.
3.1. Random Topology Model. In this paper, we consider an
undirected graph for simplicity; that is,
𝑙𝑘
() =
𝑘𝑙
(),
where {
𝑙𝑘
(),
𝑘𝑙
()}are coecients at time index .Tomodel
the topological randomness, we assume that the nodes and
links are random entities. We assume that, at any given time
index , the coecient (or link weight) a
𝑘𝑙
(),whichconnects
node to node , will either assume a nominal value
𝑘𝑙
=
𝑙𝑘
with probability
𝑘𝑙
=
𝑙𝑘
(since we assume
𝑘𝑙
=
𝑙𝑘
,
therefore, we can let
𝑘𝑙
=
𝑙𝑘
)orbezerowithprobability
𝑘𝑙
=1−
𝑘𝑙
;thatis,
A
𝑖
𝑘𝑙
=a
𝑘𝑙
(
)
=
𝑘𝑙
, with probability
𝑘𝑙
,
0, with probability
𝑘𝑙
=1−
𝑘𝑙
,
(8)
where A
𝑖
=[a
𝑘𝑙
()].
Since the link failures exist in the network, therefore, we
assume a normal topology
0
,includingaxednumberof
nodes and
𝑙
edges (or links). us, the
𝑙
edges give rise
to 2
𝑛
𝑙
dierent subnetworks
𝑙
,witheachsubnetwork
𝑙
with
probability
𝑙
, composed of faultless and faulty links.
Since the topology matrix A
𝑖
is random, thus we dene
themeantopologymatrixA and B,as[35]
A =E A
𝑖
=
2
𝑛
𝑙
𝑙=1
𝑙
A
𝑙
,
B =E A
𝑖
A
∗
𝑖
𝑇
=
2
𝑛
𝑙
𝑙=1
𝑙
A
𝑙
A
∗
𝑙
𝑇
,
(9)
where
𝑙
= Pr{A
𝑖
=
𝑙
}, A
𝑖
= A
𝑖
⊗
𝑀
,andA
𝑙
=
𝑙
⊗
𝑀
.
e symbols ⊗and denotetheKroneckerproductandblock
Kronecker product, respectively.
3.2. Dithered Quantization Model. In this paper, we assume
that each internode communication channel uses a uniform
quantizer with quantization step .Tomodelthecommuni-
cation channel, we introduce the quantizing function, (⋅) :
R →Q, Q ={|∈Z};thatis,
(
)
=,
−
1
2
≤<+
1
2
,
(10)
where ∈R is the channel input. erefore, () can be
written as follows:
(
)
=+
(
)
,
(11)
where ()is the quantization error, which satises
−
2
≤
(
)
<
2
.
(12)
Since the error terms are not stochastic, which fail to lead
to a reasonable solution, therefore, we introduce dither to
randomize the node states prior to quantizing the perturbed
stochastic state. As [33], the quantization error sequence,
{𝜖()}
𝑖≥0
, is dened as follows, when dither is added before
quantization:
𝜖
(
)
=
(
x
(
)
+𝛿
(
))
−
(
x
(
)
+𝛿
(
))
,
(13)
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