ADMM驱动的TDOA精确估计算法
134 浏览量
更新于2024-08-27
1
收藏 431KB PDF 举报
本文主要探讨了"基于ADMM的TDOA估计"这一主题，针对传感器网络中广泛使用的TDOA（Time Difference of Arrival）测量，提出了一种新颖的算法。TDOA是衡量两个传感器相对于共同参考源的时间差，对于定位和信号同步等应用至关重要。传统的TDOA估计通常依赖于精确的信号处理技术，然而，当数据量大或噪声干扰时，准确度可能会受到影响。
该研究者团队利用低秩特性来简化TDOA矩阵，其每个元素表示两个传感器之间的TDOA值。他们注意到，通过引入TDOA矩阵的Gram矩阵，可以提高算法的鲁棒性，因为Gram矩阵能够捕捉数据的相关结构，有助于减少噪声的影响。这种方法将TDOA估计问题转化为一个半定编程（Semidefinite Programming, SDP）问题，这是一个优化问题，其中目标函数和约束条件都与矩阵的正定性有关。
为了提升算法的可扩展性和计算效率，他们开发了一种高效的交替方向多重子梯度方法（Alternating Direction Method of Multipliers, ADMM）。ADMM是一种有效的求解大规模优化问题的算法，它通过将复杂问题分解成更易于管理的小部分，实现了并行化处理，从而显著提高了算法的执行速度。
通过仿真结果，论文作者展示了所提算法在实际场景中的性能优势，特别是在处理大规模数据集和应对噪声挑战时，表现出良好的稳健性和准确性。因此，本文不仅提供了一种新颖的TDOA估计策略，还展示了如何结合ADMM、Gram矩阵和SDP技术来提升无线传感器网络中的定位精度，对相关领域的研究和工程实践具有重要意义。关键词包括：交替方向方法（ADMM）、Gram矩阵、半定编程（SDP）和时间差分到达（TDOA）。
1406 IEEE COMMUNICATIONS LETTERS, VOL. 22, NO. 7, JULY 2018
ADMM-Based TDOA Estimation
Yanping Zhu, Bowen Deng, Aimin Jiang , Xiaofeng Liu, Yibin Tang, and Xiao Yao
Abstract—Accurate estimation of time difference of arrival
(TDOA) between sensors is a fundamental problem in various
applications of sensor networks. In this letter, a novel algorithm
is proposed to estimate TDOAs based on low-rank essence of a
TDOA matrix, whose entry (i, j) denotes TDOA value between
sensors i and j with respect to a common reference source.
To enhance the robustness of the proposed algorithm, we further
introduce the Gram matrix of the TDOA matrix, resulting in a
semidefinite programming estimation problem. An efficient alter-
nating direction method of multipliers is developed to improve the
scalability of the proposed algorithm. Simulation results validate
the performance of the proposed algorithm.
Index Terms— Alternating direction method of multipliers
(ADMM), Gram matrix, semidefinite programming (SDP), time-
difference of arrival (TDOA).
I. INTRODUCTION
T
IME difference of arrival (TDOA) measurements are
widely used in various applications of sensor networks,
e.g., source localization [1] and tracking [2]. In practice, they
can be obtained by generalized cross-correlation (GCC) [3],
but adversely affected by many factors, e.g., noise, out-
liers, and missing data. When only considering additive
Gaussian noise, TDOA estimation can be well addressed
by Gauss-Markov method [4] or standard least-squares
estimator [5]. Outliers generated by multipath and interference
have been studied from various perspectives. To identify and
remove outliers, the
p
-norm is employed in [6]. Multiple
quality metrics are also used in [7] along with the Zero-Sum
Condition (ZSC) to select reliable TDOA measurements.
Another outlier removal algorithm [8] is based on geometrical
constraints and statistical tests. Geometrical constraints are
based on minimal ZSCs, while the statistical model character-
izing the distance of TDOAs from the corresponding feasible
set is derived for multiple statistical tests to identify potential
outliers.
Missing data of TDOA measurements are induced by
communication failure or severe non-line-of-sight (NLOS)
propagation. A recent study presented in [9] exploits algebraic
properties of a TDOA matrix to handle missing data along
Manuscript received April 2, 2018; accepted April 27, 2018. Date of
publication May 7, 2018; date of current version July 10, 2018. This
work was supported in part by the National Nature Science Foundation
of China under grants 61471157, 61401148, 61501169, 61501170, and
61772090, the Fundamental Research Funds for the Central Universities
under grant 26120182018B23014, the Key Development Program of Jiangsu
Province of China under grants BE2017071 and BE2017647, and the Projects
of International Cooperation and Exchanges of Changzhou of China under
grants CZ20170018. The associate editor coordinating the review of this
letter and approving it for publication was L. Mucchi. (Corresponding author:
Aimin Jiang.)
Y. Zhu is with the School of Information Science and Engineering,
Changzhou University, Changzhou 213164, China (e-mail: zhuyanping@
cczu.edu.cn).
B. Deng, A. Jiang, X. Liu, Y. Tang, and X. Yao are with the College
of Internet of Things Engineering, Hohai University, Changzhou Campus,
Changzhou 213022, China (e-mail: jiangam@hhuc.edu.cn).
Digital Object Identifier 10.1109/LCOMM.2018.2833546
with additive noise and outliers. For a network with n sensors,
a TDOA matrix is a skew-symmetric matrix, whose element
(i, j) represents the TDOA value between sensors i and j
with respect to a common reference source. Although this
matrix is fully determined by n(n − 1)/2 pairwise measure-
ments, further examination shows that these measurements are
redundant [10], [11] and data redundancy can be exploited
to improve the denoising performance and the accuracy of
TDOA matrix completion [9]–[11]. A noiseless TDOA matrix
has rank 2 and a singular value decomposition (SVD) with
n − 1 degrees of freedom. Using these properties, one can
readily obtain closed-form solutions to the TDOA matrix
completion problem [9]. In [11], source localization is divided
to two distinct steps: denoising TDOAs and recovering source
positions. Based on statistical assumption of additive noise
in noisy TDOAs, denoising TDOAs is achieved by the
Maximum-Likelihood estimator derived by minimal ZSCs.
Note that a specific TDOA matrix corresponds to infinite
number of sensor geometries, whose essential properties can
be described by geometrical constraints. Interested readers
can refer to [8]–[11] for the connection between these two
formulations.
In this letter, we focus on the TDOA estimation, given
noisy and incomplete measurements. In the proposed algo-
rithm, we exploit the low rank essence of the Gram matrix
of the TDOA matrix to enhance estimation accuracy. The
resulting model does not rely on any extra prior information
(e.g., the exact number of outliers, statistical assumptions of
noise and outliers, sensors’ positions). Section II describes
the estimation problem of TDOA matrix and an semidefinite
programming (SDP) problem is further formulated. Then,
an alternating direction method of multipliers (ADMM) is
developed in Section III. Numerical experiments are given in
Section IV. Finally, Section V summarizes the conclusions.
II. P
ROPOSED ESTIMATION MODEL
Suppose that one source and n sensors are located at s
0
∈
R
d
and s
i
∈ R
d
, i =1,...,n, respectively. The TDOA value
for a pair of sensors is defined as a
ij
= d
i
− d
j
where d
i
=
s
i
− s
0
2
. Collecting all the TDOA values, we can construct
the TDOA matrix A =[a
ij
] ∈ R
n×n
. In practice, TDOAs
are contaminated by errors of range measurements and NLOS
propagation, yielding noisy measurements {˜a
ij
}. Moreover,
some of TDOA values cannot be reliably attained, leading to
a noisy and incomplete TDOA matrix
˜
A.
It can be demonstrated [9] that, using noiseless measure-
ments, a TDOA matrix can be written as
A = φ(x)=x · 1
T
− 1 · x
T
(1)
where x ∈ R
n
and 1 denotes a vector of ones. TDOAs
in (1) surely satisfy the ZSCs (e.g.,
m
i=1
a
k
i
k
i+1
= a
k
1
k
m
)
for any m, which are the fundamental of geometrical-
constraints-based algorithms. Eq. (1) demonstrates an exact
1558-2558 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
下载后可阅读完整内容，剩余3页未读，立即下载
2023-02-23 上传
2023-04-18 上传
2022-10-28 上传
2021-03-23 上传
2021-03-02 上传
2021-04-14 上传
2021-09-25 上传
2023-04-18 上传
2024-05-18 上传
weixin_38681218
- 粉丝: 10
- 资源: 945
我的内容管理
展开
