多输入多输出被动雷达的TDOA/AOA三维定位解析算法

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本文主要探讨了在多发射器-多接收器被动雷达系统中，利用时差定位（TDOA）和角度到达（AOA）测量数据来估计单个目标位置的问题。TDOA是一种通过比较不同接收器接收到信号的时间差来确定目标距离的技术，而AOA则提供了关于目标方向的信息，有助于弥补TDOA方法在估计目标高度方面的局限性。作者提出了一种闭合形式的两步定位估计算法，旨在解决多输入-多输出系统中的定位问题。首先，通过测量的AOA信息，该方法能够区分不同高度的信号，从而改善了仅依赖TDOA可能导致的垂直方向定位误差。这一步利用了角度测量的特性，因为目标在不同高度上，其信号到达各个接收器的角度是不同的。接下来，为了进一步提高精度，文中采用了一系列加权最小二乘优化方法。这些权重矩阵的引入是为了处理可能存在的噪声、不准确度量值以及传感器间的不一致性。通过合理设计权重，方法能够在复杂环境中，如广泛分离的天线配置下，优化位置估计结果，降低误差影响。在分析过程中，作者详细讨论了这种方法的理论基础，包括如何构建位置估计模型，如何选择合适的权重，以及如何处理可能的多解性和局部最优解。此外，还评估了算法的稳健性和计算效率，以及它在实际应用中的性能，比如对于目标跟踪和避障等场景的适用性。这篇研究文章提供了一种有效的策略，将TDOA和AOA结合在多输入-多输出被动雷达系统中，以实现三维目标定位，并展示了在复杂环境条件下提高定位精度的可能性。这对于无线通信、自动驾驶、无人机导航等领域具有重要意义，为未来的研究和实际工程应用奠定了坚实的基础。

IET Radar, Sonar & Navigation

Research Article

Algebraic solution for three-dimensional

TDOA/AOA localisation in multiple-input–

multiple-output passive radar

ISSN 1751-8784

Received on 31st March 2017

Revised 25th August 2017

Accepted on 3rd September 2017

E-First on 4th October 2017

doi: 10.1049/iet-rsn.2017.0117

www.ietdl.org

Ali Noroozi

, Mohammad Ali Sebt

Faculty of Electrical Engineering, K.N. Toosi University of Technology, Seyed-Khandan bridge, Shariati Avenue, Tehran, Iran

E-mail: ali_noroozi@ee.kntu.ac.ir

Abstract: The problem of estimating the location of a single target from time difference of arrival (TDOA) and angle of arrival

(AOA) measurements using multi-transmitter multi-receiver passive radar system with widely separated antennas is discussed.

A closed-form two-step target position estimator is presented and analysed. Using the measured AOAs, the method is able to

resolve the weakness of the TDOA-based methods in estimating the target height. Several weighted least-squares

minimisations are employed by the method to produce a location estimate. A weighting matrix in each step is employed to

provide a significant improvement in the performance of the algorithm. The Cramer–Rao lower bound (CRLB) for target

localisation accuracy is also developed. The proposed estimator is analytically shown to reach the CRLB for Gaussian TDOA

and AOA noises at moderate noise level. Simulation studies indicate that the proposed hybrid TDOA/AOA location scheme

performs better than any of the other algorithms, especially in the z-direction.

1 Introduction

Research on passive radars with multiple transmitters and multiple

receivers [known as multiple-input–multiple-output (MIMO)

passive radars] has received considerable attention in the past few

years [1–3]. MIMO passive radars can also play an important role

in improving the detection and localisation performances [4, 5].

This would be achieved by taking the advantage of the spatial

diversity provided by employing widely separated antenna

configuration [6, 7].

In passive radar systems, since the location of the target is

unknown, we are not able to compute the time of propagation

between transmitter–target–receiver triangles, namely the bistatic

triangle. Instead, we can measure the time difference between the

arrival of the direct and reflected signals collected from the

reference and surveillance channels, respectively. This delay is

determined by calculating the cross-ambiguity function (CAF)

between the reference signal and the reflected target echo. It can

then be converted to the bistatic range (BR), which is the sum of

transmitter–target and target–receiver ranges.

Several publications on target location estimation in the widely

distributed MIMO radars have appeared in recent years, which

indicate the importance of this issue. In general, the localisation

methods in these radars can be classified as either direct or indirect.

The former is based on processing simultaneously the observations

collected by the receivers to generate the target location estimate.

Although, the direct methods such as the maximum-likelihood

estimator (MLE) [7–9] and sparse recovery [10] are asymptotically

optimum, they, in addition to requiring high computational cost to

produce the desired results, cannot give a closed-form solution. On

the other hand, the indirect methods first measure the time delays

using the CAF. Then, by multiplying these delays by the speed of

signal propagation and performing some simple operations, the

corresponding BRs are calculated. These BRs form a set of elliptic

equations through which the target position is obtained. The best

linear unbiased estimator (BLUE) method is presented in [7, 8, 11],

which linearises the problem with a Taylor series expansion of the

non-linear equations. The major drawback of this approach is that

it requires an initial guess which is sufficiently close to the target

position. In [5], a least-squares solution is proposed in which only a

small number of the BR equations are utilised and then these

equations are changed to the range difference (RD) equations.

Using these RD equations yields the target location. Although this

approach has a closed-form solution, it does not exploit all the

possible equations. Furthermore, its performance is degraded by

converting the BR equations to the RD equations [12]. Another

solution to find the position of a moving target is presented in [13],

called the distributed approach. In this method, the measurements

are first divided into several groups based on the different

transmitters or receivers. Later, two weighted least-squares (WLSs)

estimators for each group are employed to independently generate

a target position estimate. The results from different groups are

then combined to produce the final estimate. In [14], a closed-form

two-step WLS method is proposed for finding the location of a

target. Under the conditions that the noise is small and the same in

all the measurements, this method can approximate the MLE. From

a practical viewpoint, however, they may never satisfy, especially

the second one. Recently, in [15, 16], a closed-form one-step WLS

method for target localisation in the general case of noise is

presented in two different conditions – when the measurement

noise is small, which leads to the MLE, and when it is not, which

results in the BLUE [17]. Although the method presented in [15,

16] is shown to outperform the previous works, generally it fails to

attain the Cramer–Rao lower bound (CRLB).

The major drawback of time difference of arrival (TDOA)-

based localisation with terrestrial transmitters and receivers is the

lack of accuracy in estimating the target height [16]. Although the

techniques using the TDOA measurements gathered from

terrestrial transmitters and receivers have a very good performance

in the x and y directions, they suffer from poor accuracy in the z-

direction. The main cause of this problem is the lack of the

diversity in the z-direction. To overcome this difficulty, a few

suggestions are put forward in [16], one of which is employing a

joint localisation scheme based on the TDOA and angle of arrival

(AOA) measurements. Thus, the aim of this paper is to improve the

localisation performance by utilising the measured AOAs as well

as the measured TDOAs.

This paper seeks to address the target localisation problem from

TDOA and AOA measurements in the presence of multiple

transmitters and multiple receivers. A closed-form two-step

solution is proposed, which is able to reach the CRLB under

Gaussian TDOA and AOA noise conditions. In this paper, we also

generate two weighting matrices which result in an approximate

MLE in each step under the condition that the measurement noise

is Gaussian and small. The covariance matrix of the location

estimate is derived. Analytically, we compare the covariance

IET Radar Sonar Navig., 2018, Vol. 12 Iss. 1, pp. 21-29

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