Abstract
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.
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 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 [1], nonlinear least-squared estimation [2, 3], total least-squared estimation [4, 5], the discrete probability density (DPD) method [6], localization algorithm for generalized bearing [7], close-form solution for positioning based on angle of arrival measurements [8], maximum likelihood (ML) estimation, and Stansfield algorithm [9, 10].
In [1], the authors presented a closed-form solution for the emitter location based on the measurements of azimuth, associated with two-dimensional localization. Note that the scheme is used for estimating the two-dimensional coordinate of an emitter, not the three-dimensional coordinates. In addition, the algorithm presented in [1] is not iterative in the sense that it is not based on Newton-based iteration. There is also a technique called total least-squared (TLS) estimation [4, 5], which is an extension of the LS estimation.
Nonlinear least-squared estimation method [2, 3] estimates the location of an emitter by minimizing the bearing errors of the LOBs (line of bearing). On the other hand, in linear least-squared estimation [1], the location estimate of an emitter is obtained by minimizing the distances between the LOBs and the emitter location. The scheme is based on Newton-type iteration, which implies that the solution is not in closed-form.
In [6], the authors presented the discrete probability density method which is based on dividing up the AOI (area of interest) into discrete intervals. This location estimation method modifies some of the erroneous results when the input data for the fix computation are LOBs.
In [7], an unconventional localization method considering the LOBs which are not based on the Cartesian coordinates is presented. This method can estimate the location of emitter in three-dimensional space. In order to estimate the location of the emitter in three dimensions, LOB is given as azimuth and elevation. In [7], a new LOB is defined by making a slant azimuth (generalized bearing) with a given azimuth and elevation. This method reduces the computational complexity by reducing the matrix and vector dimension in the localization. The scheme is based on Gauss-Newton iteration method, which implies that the solution is not in closed-form.
In [9], the method of localization using a maximum likelihood approach is presented. Further by using a simplifying assumption that the difference between measured bearings and the bearings are small enough, Stansfield algorithm has been presented [9, 10].
In [11], the authors presented the linear LSE (least-squared error) algorithm and nonlinear LSE algorithm in case of moving emitter. By estimating the initial position, initial velocity, and constant acceleration of an emitter using least-squared estimation, the position of the emitter at a specific time can be obtained.
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 [12].
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 [1] to three-dimensional space.
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.
2. Noniterative Three-Dimensional Location Estimation
Let represent the location of the -th sensor:Let and denote the estimated azimuth and the estimated elevation estimated at . Then, let be defined asLocation estimate of the emitter is defined as :In the proposed algorithm, the location estimate is given by minimizing the sum of square of distances from to the line connecting and , for . Let denote the distance from to the line connecting and [13–16]:where, from (1) and (2), is used. The derivation of (4) is shown in Appendix A. By using the parallelogram shown in Figure 8, the minimum distance between a point and a straight line in three dimensions can be obtained.
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 is given byLet be defined as
Partial derivatives of with respect to , and are given by (10)-(12).
, , and result in a linear system of equations. Using these equations the location estimate can be given asDerivation of (13) is shown in Appendix B.
3. Numerical Results
Numerical results illustrating the validity of the proposed formulation are presented in this section. Sensor trajectory and emitter location are illustrated in Figures 1 and 2. Figure 1 shows the linear trajectory of the moving sensor. The sensor generates a number of sensor locations along the -axis. Without loss of generality, the sensor trajectory lies along the -axis between to . The distance between adjacent sensor locations is uniform. Figure 2 shows the circular movement trajectory of the sensor. The , , and coordinates of the -th sensor location are . The distance between adjacent sensor locations is uniform. The coordinate of the first sensor location is . The number of sensor locations is arbitrarily chosen to be 10, 100, and 1000. The distance between the first sensor location and the emitter is 1000(m). in Figure 1(a) is chosen to be . AOA measurement errors are assumed to be zero-mean Gaussian distributed with standard deviation varying from to . The three-dimensional emitter location is estimated using Appendix B. The number of repetitions in the Monte-Carlo simulation is . The RMSE for iterations is defined as
(a) When the location of the emitter is not biased
(b) When the location of the emitter is biased
(a) When the location of the emitter is not biased
(b) 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 1 and 2 to confirm the localization performance of the proposed algorithm when the location of the emitter is biased and when the location of the emitter is not biased based on the trajectory of the moving sensor.
Two cases when the sensor moves on a linear path are shown in Figure 1. Figure 1(a) shows sensors and an emitter geometry when the location of the emitter is not biased. Figure 1(b) shows sensors and an emitter geometry when the location of the emitter is biased. Figure 2 shows two cases when the sensor moves on a circular trajectory. Figure 2(a) shows sensors and an emitter geometry when the location of the emitter is not biased. Figure 2(b) shows sensors and an emitter geometry when the location of the emitter is biased. The average distances between the sensor location and the emitter in Figures 1(b) and 2(b) are longer than those in Figures 1(a) and 2(a).
The RMSEs of the estimates of , , and coordinates when the sensor moves on a linear trajectory are shown in Figure 3. The RMSEs of the estimates of , , and coordinates when the sensor moves on a linear trajectory and the location of the emitter is biased are shown in Figure 4. RMSEs in Figure 3 are smaller than those in Figure 4 since the emitter location is not biased from the linear trajectory of moving sensor. As the standard deviation of the AOA measurement error increases, the RMSE of each coordinate increases. In addition, for a given standard deviation of AOA measurement error, the RMSEs of emitter location estimate get smaller as the number of sensor locations increases. In Figure 3, the x, y, and z coordinates of the emitter are , but in Figure 4, it is set as to bias the location of the emitter. In Figure 4, the location of the emitter is biased from the trajectory of the moving sensor. Compared with Figure 3, it can be seen that the overall performance is significantly degraded. In Figure 4, the RMSE gets smaller with an increase of the number of sensor locations. In Figure 4, note also that the RMSEs for 100 sensor locations and 1000 sensor locations gets approximately equal to those for 10 sensor locations when the standard deviation of the AOA measurement error is greater than , which is due to the biased emitter location. The same is not true for in Figure 3, where the sensor location is not biased.
The RMSEs of the estimates of , , and coordinates when the sensor moves on a circular trajectory are shown in Figure 5. The RMSEs of the estimates of , , and coordinates when the sensor moves on a circular trajectory and the location of the emitter is biased are shown in Figure 6. The circular trajectory of the sensor is set as Figure 2.
In Figure 5, RMSE, for a circular trajectory shown in Figure 2(a), is illustrated with an increase of the standard deviation of AOA measurement error. For a given standard deviation of AOA measurement error, the RMSE gets smaller with the increase of the number of sensor locations. In addition, as in the case of Figure 3, the location of the emitter is not biased from the circular trajectory of the moving sensor, so that the RMSEs with respect to the standard deviation of the AOA measurement error are small overall. In Figure 5, the x, y, and z coordinates of the emitter are chosen to be . In case of Figure 6, location of the emitter is set to to bias the location of the emitter as same as the linear trajectory case. The performance of the three-dimensional location estimation with respect to the standard deviation of the AOA measurement error can be seen in Figure 6. RMSEs show significant performance degradation compared to Figure 5 because the location of the emitter is biased from the circular trajectory of the moving sensor. As shown in Figure 4, it can be seen that the RMSEs increase drastically when the number of sensor location is 100 and 1000 for large standard deviations of AOA measurement errors of and due to the location of the emitter biased from the circular trajectory of the sensor.
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 [2, 3], by minimizing the error of the azimuth using Newton iteration, the nonlinear LS localization algorithm estimates two-dimensional location of the emitter. To estimate three-dimensional location of the emitter by using the nonlinear LS method, difference of the azimuth and elevation between estimated results and measured results are needed to be considered. In [12], the nonlinear LS method based three-dimensional localization algorithm is presented. The derivation of this algorithm is shown in Appendix C.
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 , , and coordinates of the emitter are obtained analytically, the Newton iteration method are not required. Therefore, the computational complexity of the proposed algorithm is lower than that of the existing three-dimensional localization methods that use the Newton iteration method to continuously optimize the location of the emitter.
Figure 7 shows the computational complexity. The operation times of the proposed scheme in this paper and the three-dimensional nonlinear LS method as the number of snapshots increases are shown in Figure 7. From Figure 7, it can be seen that the operation time of the proposed scheme in this paper is significantly lower than that of the three-dimensional nonlinear LS method in all the sections.
4. Conclusion
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:(i)The closed-form solution is given.(ii)It is computationally efficient.(iii)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.
Appendix
A.
The derivation of (4) is shown in this appendix. Refer to Figure 8. The minimum distance between a point and a straight line in three dimensions can be obtained from a parallelogram which can be obtained by using a vector from arbitrary point on the straight line to the target and a direction vector of the straight line.
The area of the parallelogram can be defined aswhere is the cross product operator. The area of the parallelogram can be expressed by using the length of and the height of the parallelogram:
Also is the minimum distance between a point and a straight line in three dimensions. can be defined as (4):
There is a different way to derive in (8). A parametric equation of the line in Figure 8 can be written as
The squared distance between and an arbitrary point on the line in Figure 8 can be expressed as
Using , (A.5) can be written as
Perfect square expression of (A.6) is
The minimum of squared distance in (A.7) is given by
After a little manipulation, (A.8) can be expressed aswhich is the same expression as in (8).
B.
The derivation of the location estimate is shown in this appendix.
, , and result in a linear system of equations:
From (B.1), the location estimate in (13) can be obtained.
C.
The derivation of the nonlinear LS method based three-dimensional localization algorithm is shown in this appendix.
The azimuth of the -th LOB can be written as a function of and coordinates: and are the estimates of and coordinates of an emitter, respectively. By using Tayler approximation, (C.1) can be written as and are the updates of and to minimize the difference of the measured azimuth and the calculated azimuth. Equation (C.2) can be summarized as
The elevation of the -th LOB can be written as a function of , , and coordinates:, , and are estimates of , , and coordinates of an emitter, respectively. By using Tayler approximation, (C.4) can be written as, , and are the amount of updates of , , and and to minimize the difference of the measured elevation and the calculated elevation. Equation (C.5) can be summarized asFrom the left-hand side of (C.3), is defined asFrom the left-hand side of (C.6), is defined asFrom the left-hand side of (C.3), is defined asFrom the left-hand side of (C.6), is defined asBy using (C.7), (C.8), (C.9), and (C.10), (C.3) and (C.6) can be expressed into matrix expression as
Estimate of is given as
When , , and are the initial values of , , and coordinates, the estimated location of the emitter in the first iteration can be written asFrom the second iteration, the location estimates are updated:
Abbreviations
| AOA: | Angle of arrival |
| LOB: | Line of bearing |
| LS: | Least-squared |
| ML: | Maximum likelihood |
| RMSE: | Root mean squared error |
| TOA: | Time of arrival |
| TLS: | Total least-squared. |
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
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.
Acknowledgments
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07048294).