Research Article  Open Access
ByungKwon Son, DoJin An, JoonHo Lee, "Performance Analysis of AOABased Localization Using the LS Approach: Explicit Expression of MeanSquared Error", Journal of Sensors, vol. 2020, Article ID 9346142, 22 pages, 2020. https://doi.org/10.1155/2020/9346142
Performance Analysis of AOABased Localization Using the LS Approach: Explicit Expression of MeanSquared Error
Abstract
In this paper, a passive localization of the emitter using noisy angleofarrival (AOA) measurements, called Brown DWLS (Distance Weighted Least Squares) algorithm, is considered. The accuracy of AOAbased localization is quantified by the meansquared error. Various estimates of the AOAlocalization algorithm have been derived (DoÄŸanÃ§ay and Hmam, 2008). Explicit expression of the location estimate of the previous study is used to get an analytic expression of the meansquared error (MSE) of one of the various estimates. To validate the derived expression, we compare the MSE from the Monte Carlo simulation with the analytically derived MSE.
1. Introduction
There has been a great deal of research on the determination of emitter location. Localization consists of two parts: measuring localization parameters between nodes and with these parameters to estimate location. The localization parameters can be AOA and TOA (time of arrival). AOAbased localization schemes have been considered [1â€“9]. The AOAbased localization algorithm can be classified as follows: linear leastsquared (LS) estimation [10], nonlinear leastsquared estimation [11, 12], minimum meansquared error (MMSE) estimation with Kalman filter [13], and maximum likelihood (ML) estimation [14].
In [10], leastsquared (LS) algorithm for emitter localization is proposed. In the algorithm, the distances between the given bearing lines to emitter location estimate are calculated as a function of emitter location estimate. Cost function is defined from the sum of squares of the distances, and the location estimate is obtained from the location minimizing the cost function. In [11, 12], nonlinear leastsquared algorithm is used for the emitter localization. In [13], since the bearing angle measurements are noisy, the measurements are combined using a nonlinear least squares filter or an extended Kalman filter to obtain the optimal filtered position estimate. The maximum likelihood (ML) and the Stansfield algorithm for AOAbased localization have been considered in [14], where performances in terms of the bias and the covariance matrix for these AOAbased localization algorithms have been presented analytically. In the proposed scheme, leastsquared (LS) algorithm is employed to get location estimate using noisy AOA measurements. Explicit expressions of localization error and the meansquared error (MSE) are derived. There is also a technique called total leastsquared (TLS) estimation [15], which is an extension of the LS estimation.
In this paper, we are concerned with estimating the location of the single stationary target by using the received signals at the moving sensor. We assume that the locations of the moving sensor are available. It is assumed that these is no uncertainty in sensor location. Also, it is assumed that the speed of the moving sensor is constant, which implies, given the trajectory of the moving sensor, every location of the moving sensor at the instants, when the LOB measurement is given, can be specified by the LOB measurement interval.
Note that, in the LSbased linear bearingbased localization algorithm, the speed of the moving sensor is not necessarily constant. The assumption of constant moving speed is just for convenience in implementation of the algorithm in the numerical results. Gaussian random variable representing LOB measurement error at each sensor location is adopted. Means of these Gaussian random variables are set to be identically zero. Generally, it can be regarded as a thermal noise component generated inside the receiver. By exploiting this observation, we try to get an analytic expression of the location estimate and the analytic expression of the MSE of the location estimate.
AOA estimation errors as well as sensor location errors are responsible for the errors in the location estimate. In [16], it is assumed that sensor locations are available without uncertainty and that the AOA estimation errors can be modelled as Gaussian random variables. Explicit expression of the location estimate in these assumptions has been derived in [16]. In this paper, using the results presented in [16], an explicit expression of the MSE of the location estimate is derived. In [17], an iterative solution based on ML approach has been presented and its performance has been illustrated.
In this paper, we present more explicit expression of the MSE. Our expression is explicit and intuitive in that the estimation error and the MSE are expressed in terms of the AOA estimation error. Therefore, the contribution of this paper is to present an analytic expression of the MSE of the LSbased localization algorithm. In this paper, two approximations named approximation and approximation have been employed to derive the meansquared error (MSE) of location estimate.
Taylor series expansion has been used in approximation. approximation has been adopted to get polynomial approximations of various sinusoids. Many studies have been conducted on error analysis due to various nonlinear approximations [18, 19].
2. LSBased Location Estimate [10]
Let and denote the th sensor location and the AOA measurement at . Given and for , where is the number of sensor coordinates, we are to estimate , which denotes the true emitter location. x estimation of the emitter location can be written as [10] where
The linear leastsquared (LS) estimation algorithm for estimating the emitter location is briefly described. Since is not invertible, the following normal equation is obtained from (1) for the LS estimate:
The location estimate for the noiseless AOA measurements can be written as
Noiseless AOA is denoted by and noisy AOA is denoted , where denotes an error in AOA measurement. Under these conditions, denotes an estimate of emitter location. The location estimate can be obtained from the least squares solution of where and are defined as
The normal equation of (7) is given by
The location estimate is given by
3. Approximation of and
From (6) and (7), it is easy to show that the entries of and can be expressed as where and denote the entry at the th row and the th column of and , respectively. Note that is a matrix and that is a matrix.
Let and denote the approximations of and based on the thorder Taylor series expansion, respectively: where and are defined from
If the firstorder approximation, corresponding to , is applied, (11) and (12) can be written as
The explicit expressions of and based on the firstorder Taylor series are derived in [16]
The explicit expressions of , , , and are derived in Appendices A and B, where denotes that the second and thirdorder Taylor expansion has been adopted.
4. Error Bound for the Approximation
In this section, we describe the error bound due to various orders of approximation for each of the entries of and . In Table 1, all the terms of Taylor series expansion for each entries of and are tabulated. Table 2 tabulates the firstorder Taylor series, the secondorder Taylor series, the thirdorder Taylor series, and the associated error bounds for each entry of .


Note that the upper bound of absolute value of the error between the original function value and the thorder Taylor series is given by the thorder term of the Taylor series expansion. For example, the difference between the original function value and the firstorder Taylor series is given by the secondorder term of the Taylor series expansion, which is described in the second row of Table 2. The corresponding error bounds for the secondorder Taylor series and the thirdorder Taylor series are given in the third row and the fourth row of Table 2, respectively.
Similarly, Tables 3â€“6 tabulate the same quantities for , , , and , respectively.



