International Journal of Antennas and Propagation

Volume 2016, Article ID 3293418, 11 pages

http://dx.doi.org/10.1155/2016/3293418

## Geolocation of a Known Altitude Target Using TDOA and GROA in the Presence of Receiver Location Uncertainty

^{1}Zhengzhou Institute of Information Science and Technology, Zhengzhou, Henan 450002, China^{2}National Key Laboratory of Science and Technology on Blind Signal Processing, Chengdu, Sichuan 610041, China^{3}School of Internet of Things (IoT) Engineering, Jiangnan University, Wuxi, Jiangsu 214122, China

Received 2 June 2016; Accepted 17 August 2016

Academic Editor: Sotirios K. Goudos

Copyright © 2016 Bing Deng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper considers the problem of geolocating a target on the Earth surface using the target signal time difference of arrival (TDOA) and gain ratio of arrival (GROA) measurements when the receiver positions are subject to random errors. The geolocation Cramer-Rao lower bound (CRLB) is derived and the performance improvement due to the use of target altitude information is quantified. An algebraic geolocation solution is developed and its approximate efficiency under small Gaussian noise is established analytically. Its sensitivity to the target altitude error is also studied. Simulations justify the validity of the theoretical developments and illustrate the good performance of the proposed geolocation method.

#### 1. Introduction

Passive target localization is a classical problem which has gained considerable attention in different application contexts, such as radar, sonar, navigation, tracking, and wireless communications [1–3]. Localization techniques have been extensively investigated for positioning parameters including the angle of arrival (AOA) [4], time difference of arrival (TDOA), and frequency difference of arrival (FDOA) of the target signal captured at spatially distributed receivers [2–4].

More recently, the use of the received signal strength (RSS) has been considered for target localization via, for example, microphone arrays [5]. Under the free-space propagation condition, the received signal energy is inversely proportional to the distance squared between the target and the receiver [5, 6]. This leads to the development of several received signal strength indicator- (RSSI-) based localization methods (see [5–10] and the references therein). But they require that the target transmit power is known, which renders them unsuitable for the passive localization of uncooperative targets. On the other hand, noting that the signal energies received at different receivers would be different, the utilization of the gain difference of arrival (GROA) measurement has been recognized to be useful for passive localization [11–13]. It needs the reciprocal of the received signal amplitude with respect to a reference receiver only to locate a target. The requirement for knowing the target signal transmit power is thus eliminated.

In the literature, several techniques have been proposed for target localization using GROA. Specifically, Cui et al. [14] considered using the signal TDOA and interaural level difference (ILD) obtained at two microphones for 2D sound source localization. Ho and Sun [11] utilized TDOA and GROA measurements jointly in 3D localization. They assumed the use of more than four sensors and proposed a closed-form two-step solution, which will be referred to as the two-step weighted least-squares (TSWLS) technique. The contribution of the GROA measurements to the improvement of target localization accuracy was studied. Different from the study in [11], Hao et al. [12, 13] considered the practical scenario where the known sensor positions have errors. They proposed in [12] a new closed-form algorithm that estimates both the unknown source and the sensor positions from TDOA and GROA measurements [12]. In [13], two bias mitigation methods, called BiasSub and BiasRed, were developed to reduce the estimation bias of the original TSWLS method [11].

In this work, we consider the passive geolocation of a target on the Earth surface using TDOA and GROA measurements in the presence of receiver position errors. The target altitude information can come from, for example, an altimeter [15, 16] or simply the prior information that the target is on the ground. The study begins with mathematically formulating the geolocation problem and deriving the geolocation Cramer-Rao lower bound (CRLB). The contribution of the target altitude information to improving the geolocation accuracy is investigated. The target geolocation problem is then cast into an equality-constrained optimization problem, where the equality constraint comes from the target altitude information and the cost function takes into account the presence of receiver position errors. An improved constrained weighted least-squares (ICWLS) solution is derived by following a similar approach as in [15]. Its approximate efficiency is established analytically. The sensitivity of the geolocation accuracy to the error in the target altitude information is quantitatively analyzed. Simulations corroborate the theoretical developments and show better performance of the proposed geolocation technique over a benchmark method.

The rest of this paper is organized as follows. Section 2 formulates the geolocation problem in consideration. Section 3 derives the geolocation CRLB. Section 4 presents the proposed ICWLS geolocation technique and the performance analysis with respect to the target position CRLB. Section 5 investigates the impact of the target altitude uncertainty on the geolocation accuracy. Section 6 gives the simulation results and Section 7 concludes the paper.

#### 2. Problem Formulation

Consider the geolocation scenario shown in Figure 1. The target is located on the surface of Earth modeled as an oblate spheroid. The unknown target position vector in the geocentric coordinate system is denoted by , the elements of which are related to the geodetic coordinates of the target viaHere, and denote the geodetic latitude and longitude of the target. , where km is the equatorial radius of the spheroid Earth and is the eccentricity. is the target altitude. This work assumes that is known to the geolocation algorithm. Under this assumption, eliminating and in (1) yields an equality constraint on the target geocentric position , which is