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Closed-form expression of three-dimensional emitter location estimation using azimuth and elevation measurements at multiple locations is presented in this paper. The three-dimensional location estimate is obtained from three-dimensional sensor locations and the azimuth and elevation measurements at each sensor location. Since the formulation is not iterative, it is not computationally intensive and does not need initial location estimate. Numerical results are presented to show the validity of the proposed scheme.

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 the use of these parameters to estimate location. The localization parameters can be either AOA (angle of arrival) or TOA (time of arrival).

In this paper, we consider AOA-based localization. The AOA-based localization algorithm can be classified as follows: linear least-squared (LS) estimation [

In [

Nonlinear least-squared estimation method [

In [

In [

In [

In [

In two-dimensional algorithm, the emitter is assumed to lie in the plane defined by the trajectory of the sensors. On the other hand, in three-dimensional algorithm, the emitter is not necessarily lie in the plane defined by the trajectory of the sensors.

As far as the authors know, there has been no study on closed-form expression of three- dimensional localization via AOA measurements of azimuth and elevation. In this paper, we derive an explicit closed-form expression for three-dimensional localization. Newton-based iterative approach for localizing three-dimensional coordinates of an emitter will be submitted as a separate manuscript [

The method proposed in this paper can estimate the location of the signal source by using results of multiple LOB measurements from moving antenna array. The method proposed in this paper is different from the conventional location estimation algorithm of estimating the location of the signal source through the Newton iteration method. The proposed method can estimate the location of signal source on a three-dimensional space without iteration. The scheme is an extension of the Brown algorithm [

In this paper an additive noise associated with noisy LOB measurement is assumed to be zero-mean Gaussian distributed. We are concerned with estimating the location of a single stationary target by using the received signals at the moving sensor. We assume that the locations of the moving sensor are available. The validity of the scheme is illustrated using the numerical results. We assume that the position of the sensor is available without uncertainty. That is, there is no error in the estimation of sensor position. The computational cost of the proposed scheme will be compared with that of the Newton-based iterative approach.

Let

An explicit expression in terms of the sensor location and the emitter location in Cartesian coordinates is

After some algebraic manipulations, it can be shown that

Partial derivatives of

Numerical results illustrating the validity of the proposed formulation are presented in this section. Sensor trajectory and emitter location are illustrated in Figures

The linear trajectory of the moving sensor.

When the location of the emitter is not biased

When the location of the emitter is biased

The circular trajectory of the moving sensor.

When the location of the emitter is not biased

When the location of the emitter is biased

Comparing with the case when the position of the emitter is not biased, when the location of the emitter is biased, even a small LOB error has a greater influence on the location estimation.

In this paper, the location of the emitter is set as shown in Figures

Two cases when the sensor moves on a linear path are shown in Figure

The RMSEs of the estimates of

RMSE of X, Y, and Z coordinates when the sensor moves on a linear trajectory.

RMSE of X, Y, and Z coordinates when the sensor moves on a linear trajectory (the emitter is in a biased location).

The RMSEs of the estimates of

RMSE of X, Y, and Z coordinates when the sensor moves on a circular trajectory.

RMSE of X, Y, and Z coordinates when the sensor moves on a circular trajectory (the emitter is in a biased location).

In Figure

The proposed scheme and the nonlinear iterative LS localization algorithm are compared to show the superiority of the proposed scheme on the computational complexity.

In [

In the passive system, the existing localization methods which estimate the location of the emitter on the three-dimensional space using only the LOB require the initial values and perform the location estimation of the emitter by optimizing the initial values using Newton iteration method. Typically these localization methods have two problems.

The first problem is that convergence of Newton iteration method heavily depends on the initial value. Even if many iterations are performed with appropriate convergence conditions, the coordinates of the emitter cannot be estimated correctly if there is a problem with the initial value.

The second problem is the setting of the convergence condition when optimizing the Newton iteration method. If the convergence condition is strictly given, accurate location estimation can be expected, but many iterations are required. This causes a lot of computation.

In case of the proposed algorithm in this paper, since the

Figure

Operation times of proposed scheme and three-dimensional nonlinear LS method.

Parallelogram.

The estimation of an emitter location for three-dimensional localization using three-dimensional AOA measurements at known locations has been addressed in this paper, and a noniterative formulation has been derived and verified. The formulation presented in this paper is not based on Newton-type iteration which implies that the proposed scheme has the following advantages:

The closed-form solution is given.

It is computationally efficient.

It does not require an initial guess and is not subject to a local minimum problem.

In our future study, an evaluation of the performance analysis of the proposed scheme will be conducted to quantitatively analyze the performance of the proposed location estimation algorithm.

The derivation of (

The area of the parallelogram

Also

There is a different way to derive

The squared distance between

Using

Perfect square expression of (

The minimum of squared distance in (

After a little manipulation, (

The derivation of the location estimate

From (

The derivation of the nonlinear LS method based three-dimensional localization algorithm is shown in this appendix.

The azimuth of the

The elevation of the

Estimate of

When

Angle of arrival

Line of bearing

Least-squared

Maximum likelihood

Root mean squared error

Time of arrival

Total least-squared.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Ji Woong Paik made a Matlab simulation and wrote the initial draft. Joon-Ho Lee originally derived the mathematical formulation of the proposed scheme and corrected the manuscript. In addition, Joon-Ho Lee checked the numerical results.

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07048294).