A Survey of Localization in Wireless Sensor Network <s> International Journal of Distributed Sensor Networks 3 <s> A maximum likelihood (ML) acoustic source location estimation method is presented for the application in a wireless ad hoc sensor network. This method uses acoustic signal energy measurements taken at individual sensors of an ad hoc wireless sensor network to estimate the locations of multiple acoustic sources. Compared to the existing acoustic energy based source localization methods, this proposed ML method delivers more accurate results and offers the enhanced capability of multiple source localization. A multiresolution search algorithm and an expectation-maximization (EM) like iterative algorithm are proposed to expedite the computation of source locations. The Crame/spl acute/r-Rao Bound (CRB) of the ML source location estimate has been derived. The CRB is used to analyze the impacts of sensor placement to the accuracy of location estimates for single target scenario. Extensive simulations have been conducted. It is observed that the proposed ML method consistently outperforms existing acoustic energy based source localization methods. An example applying this method to track military vehicles using real world experiment data also demonstrates the performance advantage of this proposed method over a previously proposed acoustic energy source localization method. <s> BIB001 </s> A Survey of Localization in Wireless Sensor Network <s> International Journal of Distributed Sensor Networks 3 <s> This work addresses the problem of estimating the locations of multiple acoustic sources by a network of distributed energy measuring sensors. The maximum likelihood (ML) solution to this problem is related to the optimization of a non-convex function of, usually, many variables. Thus, search-based methods of high complexity are required in order to yield an accurate solution. Considerable reduction of the complexity can be achieved by means of an alternating projection (AP) algorithm that decomposes the original problem into a number of simpler, yet also non-convex, optimization steps. The particular form of the derived cost functions of each such optimization step indicates that, in some cases, an approximate form of these cost functions can be used. These approximate cost functions can be evaluated using considerably lower computational complexity. Thus, a low-complexity version of the AP algorithm is proposed. Extensive simulation results demonstrate that the proposed algorithm offers a performance close to that of the exact AP implementation, and in some cases, similar performance to that of the ML estimator. <s> BIB002 </s> A Survey of Localization in Wireless Sensor Network <s> International Journal of Distributed Sensor Networks 3 <s> Energy-based multisource localization is an important research problem in wireless sensor networks (WSNs). Existing algorithms for this problem, such as multiresolution (MR) search and exhaustive search methods, are of either high computational complexity or low estimation accuracy. In this paper, an efficient expectation-maximization (EM) algorithm for maximum-likelihood (ML) estimation is presented for energy-based multisource localization in WSNs using acoustic sensors. The basic idea of the algorithm is to decompose each sensor's energy measurement, which is a superimposition of energy signals emitted from multiple sources, into components, each of which corresponds to an individual source, and then estimate the source parameters, such as source energy and location, as well as the decay factor of the signal during propagation. An efficient sequential dominant-source (SDS) initialization scheme and an incremental parameterized search refinement scheme are introduced to speed up the algorithm and improve the estimation accuracy. Theoretic analyses on the algorithm convergence rate, the Cramer-Rao lower bound (CRLB) for localization accuracy, and the computational complexity of the algorithm are also given. The simulation results show that the proposed EM algorithm provides a good tradeoff between estimation accuracy and computational complexity. <s> BIB003
The received signal strength at ith sensor during time interval t can be written as where d ik (t) is the distance between the ith sensor and the kth source. K is the number of the sources. g i is the gain of ith sensor. ε i (t) is random variable with mean μ i and variance σ 2 i . S k (t) is the signal energy at 1 meter away for kth source. α is the attenuation exponent. We define the following matrix notations as follows: Using these notations, (2) can be represented as where H = GD. So the joint probability density function (4) can be expressed as where θ = [r 1 , r 2 , . . . , r K ; S 1 , S 2 , . . . , S K ] T . The maximum likelihood estimation is equivalent to minimizing the following function: We can obtain the maximum likelihood parameter estimation of θ by minimizing L(θ). To minimize L(θ), we should take the following operation: This condition leads to the following relation: So we get the modified cost function: And a multiresolution (MR) search and the expectation maximization (EM) method BIB001 were proposed to solve  . An efficient EM algorithm BIB003 was proposed to improve the estimation accuracy and avoid trapping into local optima through the effective sequential dominant-source initialization and incremental search schemes. An alternating projection BIB002 algorithm was proposed to decompose the multiple-source localization into a number of simpler, yet also nonconvex, optimization steps. This method could decrease the computation complexity.