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 [1–9]. 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 .
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 3–6 tabulate the same quantities for , , , and , respectively.
5. Approximation of
Substituting (11) and (12) in (7) results in
-approximations of is denoted by . In Appendix C, based on the perturbation of the solution of linear system, it is shown that can be written as
6. Analytic Expression of the MSE of the Location Estimate
In this section, we derive explicit closed-form expressions of the mean-squared error (MSE) of the location estimates in (18). Let and denote the -coordinate and -coordinate of , respectively:
Similarly, and denote the -coordinate and -coordinate of , respectively:
Euclidean distance between and is given by
Similarly, the distance between and is written as
Substituting (18) in (22) yields
From (23), the MSE of the location estimate is
Let and be defined as
In terms of in (25) and in (26), the MSE in (24) can be rewritten as where denote and the th entry of and denote the entry at the th row and theth column of , respectively.
, , and are explicitly expressed as where the mean values in (28), (29), and (30) are derived in Appendices D, E, and F. Note that the first-order Taylor expansion, the second-order Taylor expansion, and third-order Taylor series have been employed in Appendices D, E, and F, respectively.
The summary of the localization algorithm is tabulated in Table 7.
7. Results and Discussion
Trajectory of sensor locations is given in Figure 1.
(a) Linear trajectory
(b) Circular trajectory
Simulation parameters are as follows:
Speed of sensor: 0.280 km/sec
Sampling interval: 2 sec
Number of sensor locations: 100
Number of Monte Carlo simulation:
The empirical MSE of , and is defined as where the lower script denotes the estimate associated with the th repetition out of repetitions.
Similarly, the empirical MSEs of , , , and are given by
in (31) is defined in (9). and in (32) and (33) are given by (17) and (18) with . Similarly, and in (34) and (35) are given by (17) and (18) with . and in (36) and (37) are given by (17) and (18) with .
The results with can be calculated using (9) and (18), respectively. In Figures 2(a)–2(c), we illustrate how the location estimates are distributed with , , and for the linear trajectory. It is clear that is more accurate than and that is more accurate than . Similarly, is closer to than and is closer to than . Similar observations can be made in Figure 3, where the results for the circular trajectory have been illustrated.
(a)
(b)
(c)
(a)
(b)
(c)
Linear trajectory in Figure 1(a) is considered to get actual error and error bound in Figure 4. In Figure 4(a), the actual errors for and their error bounds associated with the first-order Taylor series for 50 repetitions are illustrated. Note that the -axis represents each independent trial. It is clearly shown that the errors for all the cases are actually smaller than the error bounds. Figure 4(b) represents corresponding results for , respectively.
(a)
(b)
The results validating the derived MSEs are illustrated in Figures 5 and 6. The results in Figures 5 and 6 correspond to the linear trajectory in Figure 1(a) and the circular trajectory in Figure 1(b), respectively.
(a)
(b)
(c)
(a)
(b)
(c)
Standard deviations of AOA estimation error are set to 10 logarithmically equally spaced values between 0.01 and 10 degrees. More specifically, the standard deviations are approximately given by 0.01, 0.0215, 0.0464, 0.1, 0.2154, 0.4642, 1.0, 2.1544, 4.6416, and 10.
The MSEs with respect to the standard derivation of the LOB error are illustrated in Figures 5(a) and 5(b) for the linear trajectory. The results in Figure 5(a) with , , , and are obtained from (31), (32), (33) and (27), respectively.
is not equal to , since -approximation is used to get from . Similarly, is not equal to , since -approximation is adopted to get from . Although, they are not exactly equal, it is clear from Figure 5 that is approximately equal to and that is approximately equal to .
It should be noted that and show excellent agreements, which validates (18) with . Therefore, we can use analytically derived expression to see how .
The same observations can be made in Figure 5(b). The results in Figure 5(b) with , , , and are obtained from (31), (34), (35), and (27), respectively. Note that the results in Figure 5(b) are for the second-order -approximation and that the results in Figure 5(c) are for the third-order -approximation. The results in Figure 5(b) with , , , and are obtained from (31), (36), (37), and (27), respectively.
To reduce the error due to -approximation, the second-order Taylor series can be used, which is illustrated in Figure 5(b). In Figure 5(a), difference between and is much greater than that between and for large standard deviation of AOA estimation. On the other hand, difference between and is much smaller than that between and for large standard deviation of AOA estimation in Figure 5(c). Actually, and in Figure 5(c) is approximately equal for all standard deviation of AOA estimation error.
In Figure 6(a), it is clear that the difference between and gets larger with the increase of standard deviation of AOA estimation, which implies that the error due to the first-order -approximation becomes greater with the increase of standard deviation of AOA. and are approximately equal for standard deviation of , which implies that error due to -approximation is very small for standard deviation of . The difference between and quantifies how much error occurs due to -approximation. The difference between and is negligible for standard deviation of , which means that -approximation results in approximately no error for standard deviation of . Comparing , , and , the first-order -approximation induces much larger error than -approximation.
The error due to -approximation can be reduced by adopting the second-order Taylor series in the -approximation, which is illustrated in Figure 6(b). Compared with the results in Figure 6(b), the difference between and for the second-order -approximation is smaller than that between and for the first-order -approximation.
The error due to -approximation can be made smaller by adopting the second-order -approximation, which is illustrated in Figure 6(c). In Figure 6(c), it is shown that and show agreement for all standard deviation of AOA estimation, which is quite different from and in Figure 6(a).
8. Conclusion
Performance analysis of AOA-based localization is considered in this paper. Monte Carlo-based performance analysis is computationally very intensive, especially for a large number of repetitions. Closed-form expression of the mean-squared error (MSE) of location estimate has been derived, and the validity is shown in the numerical results. The usefulness of the derivation lies in the fact that the MSE can be analytically obtained in a closed-form without computationally intensive Monte Carlo simulation. Since error due to -approximation and -approximation is highly dependent on the noise variance, the scheme is more useful for high SNR. To reduce the error due to -approximation and -approximation, higher order Taylor series can be adopted to reduce error at the expense of more computation in calculating the analytic MSE for taking higher order terms.
Appendix
A. Second-Order Approximation of and
The second-order Taylor expansion-based approximation of the entries of and is presented.
B. Third-Order Approximation of and
The third-order Taylor expansion-based approximation of the entries of and is presented.
C. The Perturbation of the Solution of Linear System, -Approximation of
Let denote an unperturbed linear system with and .
Consider a perturbed linear system, where and . If is nonsingular, then it is clear that is differentiable in a neighborhood of zero.
Let the first derivative of be denoted by . Differentiation of (C.2) yields
Substituting in (C.3) yields
Comparing (C.1) and (C.2), it is clear that in (C.4) is equal to in (C.1):
Using (C.5) in (C.4) results in
Solving for yields
First-order approximation of based on the Taylor expansion can be written as
Comparing (C.2) and (16), and in (C.2) correspond to and in (16), respectively. Therefore, the approximation of and , is obtained from (C.3) by substituting and .
D. Mean of Entries of and Based on the First-Order Taylor Series
Explicit expression of many expectation terms in (28), (29) and (30) is derived. The expectation of can be written as
Explicit expression of the expectation of is derived:
Since the other entries can be derived similarly, we list the final expressions for the other entries:
E. Mean of Entries of and Based on the Second-Order Taylor Series
F. Mean of Entries of and Based on the Third-Order Taylor Series
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 they have no conflicts of interest.
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).