Adaptive robust time-of-arrival source localization algorithm based on variable step size weighted block Newton method
- Chee-Hyun Park^{1} and
- Joon-Hyuk Chang^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s13638-017-0909-0
© The Author(s) 2017
Received: 22 February 2017
Accepted: 29 June 2017
Published: 11 July 2017
Abstract
We propose a line-of-sight (LOS)/non-line-of-sight (NLOS) mixture source localization algorithms that utilize the weighted block Newton (WBN) and variable step size WBN (VSSWBN) method, in which the weighting matrix is determined in the form of the inverse of the squared error or as an exponential function with a negative exponent. The proposed WBN and VSSWBN algorithms converge in two iterations; thus, the required number of extra samples in the transient period is negligible. Also, we perform an analysis of the mean square error (MSE) of the weighted block Newton method. To verify the superiority of the proposed methods, the MSE performances are compared via extensive simulation.
Keywords
1 Introduction
The aim of the source localization system is to find a geometrical point of intersection using the measurements from each receiver, such as the time difference of arrival (TDOA), time of arrival (TOA), or received signal strength (RSS). Localizing a point source in which passive and stationary sensors are used has been a popular research issue in the areas of radar, sonar, global positioning system, video conferencing, and telecommunication. Two key issues need to be resolved in source localization. The first is that the wireless localization systems must be able to cope with rapidly varying channel conditions. To do this, adaptive filters have been important in adaptive source localization methods to stabilize localization performance under varying channel conditions [1–3]. However, these adaptive source localization algorithms were designed for the Gaussian noise situation and their localization performance is severely degraded in impulsive noise environments, such as Gaussian mixture and Student’s t distribution. Also, the convergence rate of the least mean square (LMS) algorithm adopted in [1–3] may be much slower in some adverse environments. The second key issue is the challenge of localization in environments of line-of-sight (LOS)/non-line-of-sight (NLOS) mixture. LOS/NLOS mixture scenarios occur when there are obstructions between transmitters and receivers located in indoor environments and outdoor situations such as urban areas. The localization performance of traditional approaches, which assume only LOS conditions, is severely degraded under NLOS conditions; thus, mitigation of NLOS errors has become an urgent task and has been extensively investigated in the last decade. In general, research of the LOS/NLOS mixture problem for localization takes one of two approaches: (1) the constrained least squares (LS) method using optimization, such as semidefinite relaxation and second-order cone relaxation [4–9] and (2) localization based on the “NLOS identify and discard” [10–14]. Although localization using the optimization method has relatively high accuracy, the computational load is intensive. Localization using the “NLOS identify and discard” method also has relatively high accuracy when the LOS/NLOS mixture sensors are perfectly separated from the LOS sensors. However, the complete classification of LOS sensors and LOS/NLOS mixture sensors is nearly impossible, making it evident that classification error exists and the resulting false classification incurs drastically increased localization error. Furthermore, when the number of sensors is large, the number of cases to be calculated is increased; thus, the computational burden is much high. Although our research results deal with the case in which the positions of sensors are accurately known, some recent works assume the unknown coordinates of sensors [15–18].
The motivation for this paper is as follows. Adaptive localization methods have been proposed for positioning in changing environments [1–3]. However, conventional adaptive localization algorithms, which minimize the squared error sum, are not robust to non-Gaussian noise (impulsive noise) situations, so they must be adapted to impulsive noise conditions. Our proposed algorithm minimizes the weighted squared error sum instead of the squared error sum, and the weighting matrix in the proposed algorithm counteracts the adverse effects of the LOS/NLOS mixture sensor. Namely, the weight is small when the sample is an outlier, which means an outlier-corrupted measurement is prevented from entering into the minimization of the cost function. Also, the convergence rate of the LMS method is slower than that of the Newton method and this slow convergence rate increases the number of samples in the transient period of the adaptive filter and this is clearly a waste of resources. Thus, we propose a robust weighted block Newton (WBN) and variable step size WBN (VSSWBN) algorithms that use the Newton method instead of the LMS method. As can be seen from simulation results, the proposed WBN and VSSWBN methods converge in two iterations. Therefore, it is possible to neglect the number of additional samples in the transient period of the adaptive localization algorithm.
The organization of this paper is as follows. Section 2 explains the LOS/NLOS mixture source localization problem to be solved in this paper. In Section 3, the details of the existing localization methods are addressed. The proposed adaptive localization methods using the weighting matrix and variable step size are addressed in Section 4. The MSE analysis of the proposed WBN algorithm is performed in Section 5. The estimation performances of the proposed methods are evaluated via simulation results in Section 6, comparing them with those of the existing algorithms. Finally, the conclusion is presented in Section 7.
2 Problem formulation
where \({n}_{{i}} \sim (1-\epsilon){N}(0,\sigma _{1}^{2})+ \epsilon {N}(\mu _{2},\sigma _{2}^{2}), \sigma _{1}^{2}\ll \sigma _{2}^{2},\;\;{i}=1,2,{\ldots },{M},\;\;\;\) with M denoting the number of sensors [7, 19–21]. Also, r _{ i } is the measured distance between the source and the ith sensor and d _{ i } is the range (distance) model between the source and the ith sensor. The measurement noise n _{ i } is modeled as a Gaussian mixture distribution, where the LOS noise is distributed according to \({N}\left (0,\sigma _{1}^{2}\right)\) with a probability (1−ε) and the NLOS noise distributed by \({N}\left (\mu _{2},\sigma _{2}^{2}\right)\) with a probability of ε. It is assumed that while the statistics of the inlier can be obtained, the mean and variance of the outlier distribution are unknown. In practice, \(\sigma _{1}^{2}\) can be estimated by observing the energy bins in an absence of the transmitted signal and indeed the sample variance is usually adopted. Here, ε (0≤ε≤1) is the contamination ratio (i.e., fraction of contamination) which is a small number (typically smaller than 0.1) [7, 19–21]. Also, [x y]^{ T } is the true source position and [x _{ i } y _{ i }]^{ T } is the position of the ith sensor. Note that, throughout this paper, a lowercase boldface letter denotes a vector, an uppercase boldface letter indicates a matrix and the superscript T signifies the vector/matrix transpose. The purpose of this paper is to determine the source position that minimizes the MSE of the position estimate.
3 Review of the existing TOA localization methods
In this section, we briefly discuss the block LMS algorithm, M, and LMedS estimators in terms of the formulation of the source localization.
3.1 Block LMS source localization [1]
where e ^{(k)}=b _{ k }−A x ^{(k)}, superscript is the iteration number, and μ is a positive step size (see [1] for details). Note that the input signal in the conventional LMS algorithm was substituted with A ^{ T } in the block LMS method.
3.2 LMedS estimator
- (1)
Calculate the m subsets of three measurements.
- (2)
For each subset S, compute a location by trilateration P _{ S }.
- (3)For each solution P _{ S }, the residues R _{ S } are obtained as$$\begin{array}{@{}rcl@{}} {R}_{S}=\left[({r}_{1}-\hat{d}_{1})^{2},({r}_{2}-\hat{d}_{2})^{2},\cdots,({r}_{M}-\hat{d}_{M})^{2}\right] \end{array} $$(5)
where r _{ i } is the same with (1), \(\hat {d}_{i}=\sqrt {\left ({x}_{ps}-{x}_{i}\right)^{2}+ \left ({y}_{ps}-{y}_{i}\right)^{2}}\) and the median of the residuals R _{ S } is computed.
- (4)
The solution P _{ S }, which gives the minimum median, is determined as the source location.
3.3 M-estimator
and sign(·) is the sign function defined as \(\text {sign}(\nu)=\left \{\begin {array}{lll}1, & \text {if }\nu >0 \\0, & \text {if }\nu =0 \\ -1, & \text {if }\nu < 0. \end {array} \right.\)
4 Proposed adaptive robust localization methods
In this paper, the LOS/NLOS mixture state is divided into the LOS and LOS/NLOS states. The LOS state denotes the case where the contamination ratio is zero (ε=0) and the LOS/NLOS state is the condition in which 0<ε≤1. The adaptive robust localization algorithms in this paper can be represented as follows.
4.1 WBN method
where H ^{(k)} is the Hessian matrix, g ^{(k)} is the gradient vector of the cost function in the kth iteration, \( {Q}^{({k})}=\text {diag}\left [\!\frac {1}{({b}_{{1},{k}}- {a}_{1}^{T} {x}^{({k})})^{2}}\cdots \frac {1}{({b}_{{M},{k}}- {a}_{M}^{T} {x}^{({k})})^{2}}\right ]\) or \(\text {diag}\left [e^{-\frac {({b}_{{1},{k}}- {a}_{1}^{T} {x}^{({k})})^{2}}{2\zeta }}\cdots \right.\left. e^{-\frac {({b}_{{M},{k}}- {a}_{M}^{T} {x}^{({k})})^{2}}{2\zeta }}\right ]\), ζ is a tuning parameter to be determined through offline work, e ^{(k)}=b _{ k }−A x ^{(k)}, b _{ k }=[b _{1,k }⋯b _{ M,k }]^{ T }, and b _{ i,k } denotes the sample of the ith sensor at the kth iteration. We use the weighting matrix Q ^{(k)} to alleviate the adverse effect of the LOS/NLOS mixture components. Note that the weights are small for samples contaminated by outliers because the squared residuals are large and they are large for samples in the LOS state because the squared residuals are small. The advantage of using the weighting matrix (Q ^{(k)}) is that it alleviates the adverse effects of outliers. Meanwhile, the disadvantages are that the estimation performance may be inferior to the LOS-based algorithm in the LOS situation and the weight can diverge to infinity when the squared residual is close to zero. In this case, the small positive value can be added to the squared residual for the stability of the algorithm.
4.2 WCBN method
4.3 VSSWBN algorithm
where J ^{(k)}=e ^{(k)} ^{ T } Q ^{(k)} e ^{(k)}, e ^{(k)}=b _{ k }−A x ^{(k)}, and μ _{max},ρ are generally selected through experiments to provide the maximum convergence rate preserving steady state misadjustment error small. The value of μ _{min} is determined as the level of steady-state misadjustment and the required tracking capabilities. The length of the transient period of the adaptive source localization method is desired to be short because the extra samples are consumed until the algorithm converges compared to the existing robust localization algorithm. Therefore, the step size is determined as large as possible in the proposed method to aid fast convergence. In our simulation results, the VSSWBN algorithm showed a superior MSE performance and a similar convergence rate compared to the WBN method for large step sizes (both algorithms converged in two iterations). However, the MSE performances of the VSSWBN and WBN algorithms were similar for small or medium step sizes.
5 Performance analysis
5.1 MSE performance analysis
and Q ^{(l)} was treated as the constant matrix for the ease of derivation. The WBN algorithm converges when 0<μ<1 from (18). In the derivation of the last term of (19), it is assumed that \( {Q}^{({k}-1)}\simeq {R}_{{k}-1}^{-1}\) (Q is the instantaneous form of R ^{−1}) and the summation is approximately equal to μ ^{2}{(A ^{ T } Q ^{(k−1)} A)^{−1} A ^{ T } Q ^{(k−1)} R _{ k−1} Q ^{(k−1)} A(A ^{ T } Q ^{(k−1)} A)^{−1}} because 0<1−μ<1. Note that the inverse of the squared error is relatively large in the LOS sensor and is small when the outliers exist. Then, although σ _{2} is much larger than σ _{1}, the MSE is nearly constant with respect to the standard deviation of NLOS noise and bias because the effect of large error standard deviation of LOS/NLOS mixture sensor in R _{ l } is attenuated by the weighting matrix Q ^{(l)} (see (19) and (20)).
5.2 Computational complexity analysis
Comparison of the computational complexity
Algorithm | Computational complexity |
---|---|
Block Newton | O(M ^{2} N) |
WBN | O(2M ^{2} N) |
VSSWBN | O(2M ^{2} N+M ^{2}) |
WCBN | O(M ^{2}) |
6 Simulation results
7 Conclusions
The WBN algorithm was developed by modifying the block LMS algorithm to make it robust to outliers and the proposed method employed a weighting matrix. Furthermore, the VSSWBN method was proposed to improve the MSE performances of the WBN algorithm. We also analyzed the MSE of the WBN algorithm. In the simulation results, the MSE averages of the proposed methods were smaller than that of the other adaptive localization methods and robust positioning algorithms.
Declarations
Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT, and Future Planning (No.2014R1A2A1A10049735).
Authors’ contributions
In this research paper, the authors proposed a robust localization algorithm. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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