Diffusion Kalman Filter Algorithm for Adaptive
Network with Quantized Information Exchange
Shujie Yang, Changqiao Xu, Jianfeng Guan
State Key Laboratory of Networking and Switching Technology,
Beijing University of Posts and Telecommunications, Beijing 100876, China
zongshi@bupt.edu.cn, cqxu@bupt.edu.com, guanjian863@163.com
Abstract—We study the distributed Kalman filter in sensor
networks where multiple sensors collaborate to achieve a common
objective. Diffusion Kalman filtering algorithms have been a
popular topic in linear dynamic system estimation problems, In
there algorithms, nodes cooperate with their direct neighbors and
diffuse the information across the entire network through a se-
quence of Kalman iterations and data-aggregation. In this article,
we propose the diffusion Kalman filtering with quantized global
(DKFQFI)algorithm because of the limited sources in the wireless
environment, where nodes exchange their quantized states with
neighbors to reduce the consumption of resources. To prove the
convergence of the DKFQFI algorithm, we derive the theoretical
expressions of the mean and mean-square performance. From
the expressions, we show that the mean performance and mean-
square performance of the proposed algorithm are unbiased and
stable. Therefore, the feasibility of the algorithm is verified.
Moreover, the proposed algorithm achieves an outperform by
simulation.
Index Terms—Kalman filter, distributed algorithm, quantized
information, MSD.
I. INTRODUCTION
In recent years, many novel concepts attract more and more
attention in the wireless network[1,2], in which the concept of
adaptive networks was put forwarded and got more and more
attentions [3,4,5]. In adaptive networks, agents with processing
and learning abilities are directly linked together and cooperate
with their neighbors to solve several real-time problems, such
as distributed estimation, optimization and inference problems.
Through cooperation, nodes will have better performance
than working independently. For example, if one adaptive
agent cooperates with another independent adaptive agent,
it can improve the performance of both agents. In [3], it
provided an overview of diffusion strategies for adaptation
over networks. In [4], it showed recent advances related to
adaptation, learning, and optimization over networks. It also
discussed various distributed strategies to interact locally in
response to learn and adapt continually with time. Based on
this, a large number of strategies for adaptive networks were
proposed in which the diffusion strategies performed better.
In earlier works [6]-[11], it has been shown how diffusion
strategies obtained a better performance for adaptive networks.
In these literatures, diffusion algorithms such as recursive least
squares algorithm ( RLS) and least mean square algorithm
(LMS) were proposed, in which nodes can adjust their perfor-
mance based on streaming data and network conditions by the
continuous diffusion of information across the network. All of
these literatures have proved advantages of the diffusion strate-
gies for adaptive networks. However, because of limitations of
these filters, they are failure to obtain accurate measurement
for moving targets. And they are all belongs to wiener filtering
as well. As we all known, the winner filtering requires that
the signal and noise must be a smooth process, which greatly
restricts its application. Considering the instability of noise
and a decentralized control problem, Kalman filtering appears
to be a better solution.
Like diffusion LMS and RLS, diffusion strategies can also
be applied to the solution of distributed state-space filtering
problems. A diffusion Kalman filtering (DKF) has been in-
troduced in [12,13]. We can see the diffusion Kalman filter
algorithm, which is comprised of the incremental update step
and the diffusion update s tep, has an excellent performance in
tracing a moving target.
All of the literatures above assumed the information ex-
change between nodes was continuous. However, in wireless
networks, quantization is usually required before data is ex-
changed [14,15], since the limited sources, such as bandwidth
and power, will prevent the exchange of high-precision data.
These issues motivated the study of adaptive networks in this
condition. In [16], the innovation was quantized by either an
iterative binary quantizer or a single-shot batch quantizer, and
a recursive state estimator was introduced. In [17], Kalman
filters based on both quantized observations and quantized
innovations were proposed, in which the tradeoff between
energy consumption and estimation accuracy was studied. In
[18], a quantized gossip-based interactive Kalman filtering
algorithm was implemented, where the weak convergence was
proved by studying the estimation error variance sequence at
a randomly selected sensor.
Here, we utilize the advantages stated above in adaptive net-
works, and propose a Diffusion Kalman Filter with Quantized
Global Information exchange algorithm (DKFQGI), which is
a fully distributed Kalman filtering solution to communicate
information with their neighbors. The main contributions of
this paper are as follows:
∙ We start to give a diffusion Kalman filter algorithm with
only global information exchange for simplicity, which
lays the foundation of this literature. We do this for two
reasons. Firstly, the global information exchange plays a
prime role compared with the local information exchange.
Secondly, ignoring local information exchange can save
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