Error Source Analysis of Target Localization Based on Weighted Linear Least Squares in Wireless Acoustic Sensor Network

Each sensor node in wireless sensor network (WSN) uses only a single microphone instead of microphone arrays as its sensing device, which can guarantee the most important characteristics of WSN, such as small size, low power consumption and convenient deployment. In this paper, weighted linear least squares method (WLLS) was proposed to realize target localization. By dividing the localization equations and introducing new parameters, the localization problem is linearized and computational complexity is reduced. Then the most influencing factors on localization accuracy were analyzed, including the source position, background noise, sensing range, attenuation coefficient and node location error, which will provide an effective instruction on WASN design and nodes selection. Introduction Wireless sensor network (WSN) is composed of multiple sensor nodes, which realizing the communication and information sharing by self-organized network. WSN has great application potentials in many areas, such as target tracking, environmental monitoring. Various kinds of sensors may exist at a single node in a sensor network, Wireless Acoustic Sensor Network (WASN) is the type that only with one acoustic sensor at the sensor node. Wireless Microphone Array Sensor Network (WMASN), is the type with a microphone array at each node [1,2]. Each node independently measures the noise of the source at first, and then sensing data of multiple nodes is fused to achieve target localization through self-organizing network. Because of the special application requirements of the wireless acoustic sensor network, RSS target location algorithm with low hardware requirement and the small calculation amount has a huge potential for application. The target localization algorithms using RSS include nearest point method, centroid method, weighted centroid method, least squares method (LS), maximum likelihood estimation (ML) and so on. Nearest Point Method, centroid method and weighted centroid method have low positioning accuracy and require a higher accuracy of the sensor node's own position. The advantage of ML algorithm is the ability of multi-target sound source localization, but its value function is highly nonlinear, so it is difficult to get the global optimal solution because of many local optimum solutions. The ML algorithm was treated as semidefinite programming problem in Ouyang and Meng’s research [3,4], then gets the global convergence ability. Expectation Maximization (EM) algorithm is used to simplify the ML optimization process. When the estimated parameters follow a normal distribution, the ML algorithm is consistent with the least-squares method [5]. The LS or WLS algorithm is a set of overdetermined nonlinear equations about the target coordinates, and the amount of calculation is large. A sub-optimal method that simplifies the nonlinear formulation to linearity is often used, resulting in received signal strength of at least 4 nodes are needed to locate the in-plane object. Tarrio linearize the equation system by removing the square parameters through substrate the equations [6]. In So’s research, the problems linearized by using Best Linear Unbiased Estimator (BLUE) method [7] and computation burden is smaller than Tarrio’s method. Yuan and Wang proposed the Approximate Least Squares Method to reduce the computation burden to obtain an approximate 324 solution to the target location[8,9]. This paper divides the positioning equations and introduces new parameters to transform the weighted least squares (WLS) into the weighted linear least squares (WLLS). Acoustic Source Localization Algorithm Bases on WLLS Assuming there are m sensor nodes and one gateway node in WASN, and the gateway node has sufficient energy and strong computational capability which satisfy the requirement of data collection and source location estimation. Assuming the coordinate of source is (x,y), the sensor node is (xi,yi), where subscript i is the order of sensor node. Considering totally n nodes are able to detect the target, energy received by each node is represented as 1 2 3 [ , , , , ] T n I I I I  I , here i I is the energy received by the ith node. Acoustic attenuation equation ( ) ( ) ( ) i i i i S t I t g t d     and let 1 i g  then we have localization equation of node i: 2 2 2 ( ) ( ) ( ) ( ) i i i S t x x y y I t          (1) Where S(t) represents the sound energy value at 1m from the sound source, gi is the gain of acoustic sensor, ( ) i t  is White Gaussian Noise with average value of 0, and variance of 2  . Dividing the positioning equations of the two nodes, removing the unknown S(t), and introducing another variable R to linearize the nonlinear equation. 2/ 2/ 2/ 2/ 2/ 2/ 2 2 2 2/ 2 2/ 2 2/ 2 2/ 2 2 2 2 ( ) j j i i j j i i i j j j j j i i i i I x I x x I y I y y I I x y x I y I x I y I                                 (2) Let 2 2 R x y   , 2/ i i E I   and 2/ j j E I   , with the matrix form:  Aθ b (3) Where b is a (n-1)×1 type matrix. A is (n-1)×3 type matrix. 2 2 2 2 1 1 1 2 2 2 2 2 2 2 2 2 2 2 1 1 ( ) ( ) ( ) ( )