Journal of Sensors

Journal of Sensors / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 9346142 | https://doi.org/10.1155/2020/9346142

Byung-Kwon Son, Do-Jin An, Joon-Ho Lee, "Performance Analysis of AOA-Based Localization Using the LS Approach: Explicit Expression of Mean-Squared Error", Journal of Sensors, vol. 2020, Article ID 9346142, 22 pages, 2020. https://doi.org/10.1155/2020/9346142

Performance Analysis of AOA-Based Localization Using the LS Approach: Explicit Expression of Mean-Squared Error

Academic Editor: Vincenzo Stornelli
Received11 Dec 2019
Accepted14 May 2020
Published18 Aug 2020

Abstract

In this paper, a passive localization of the emitter using noisy angle-of-arrival (AOA) measurements, called Brown DWLS (Distance Weighted Least Squares) algorithm, is considered. The accuracy of AOA-based localization is quantified by the mean-squared error. Various estimates of the AOA-localization algorithm have been derived (Doğançay and Hmam, 2008). Explicit expression of the location estimate of the previous study is used to get an analytic expression of the mean-squared 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). AOA-based localization schemes have been considered [19]. The AOA-based localization algorithm can be classified as follows: linear least-squared (LS) estimation [10], nonlinear least-squared estimation [11, 12], minimum mean-squared error (MMSE) estimation with Kalman filter [13], and maximum likelihood (ML) estimation [14].

In [10], least-squared (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 least-squared 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 AOA-based localization have been considered in [14], where performances in terms of the bias and the covariance matrix for these AOA-based localization algorithms have been presented analytically. In the proposed scheme, least-squared (LS) algorithm is employed to get location estimate using noisy AOA measurements. Explicit expressions of localization error and the mean-squared error (MSE) are derived. There is also a technique called total least-squared (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 LS-based linear bearing-based 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 LS-based localization algorithm. In this paper, two approximations named -approximation and -approximation have been employed to derive the mean-squared 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. LS-Based 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 least-squared (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 th-order Taylor series expansion, respectively: where and are defined from

If the first-order -approximation, corresponding to , is applied, (11) and (12) can be written as

The explicit expressions of and based on the first-order Taylor series are derived in [16]

The explicit expressions of , , , and are derived in Appendices A and B, where denotes that the second- and third-order 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 first-order Taylor series, the second-order Taylor series, the third-order Taylor series, and the associated error bounds for each entry of .


ConstantFirst-order termSecond-order termThird-order termFourth-order term



Constant
First-order term
Second-order term
Third-order term
Fourth-order term
Error bound

First-order Taylor series
Second-order Taylor series
Third-order Taylor series

Note that the upper bound of absolute value of the error between the original function value and the th-order Taylor series is given by the th-order term of the Taylor series expansion. For example, the difference between the original function value and the first-order Taylor series is given by the second-order term of the Taylor series expansion, which is described in the second row of Table 2. The corresponding error bounds for the second-order Taylor series and the third-order Taylor series are given in the third row and the fourth row of Table 2, respectively.

Similarly, Tables 36 tabulate the same quantities for , , , and , respectively.


Constant
First-order-term
Second-order-term
Third-order term
Fourth-order term
Error bound

First-order Taylor series
Second-order Taylor series
Third-order Taylor series


Constant
First-order term
Second-order term
Third-order term
Fourth-order term
Error bound

First-order Taylor series
Second-order Taylor series
Third-order Taylor series


Constant
First-order term
Second-order term
Third-order term
Fourth-order term
Error bound

First-order Taylor series
Second-order Taylor series
Third-order Taylor series


Constant
First-order term
Second-order term
Third-order term
Fourth-order term
Error bound

First-order Taylor series
Second-order Taylor series